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Investments
ZVI BODIE Boston University
ALEX KANE University of California, San Diego
ALAN J. MARCUS Boston College
T E N T H E D I T I O N
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INVESTMENTS, TENTH EDITION
Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2014 by McGraw-Hill
Education. All rights reserved. Printed in the United States of America. Previous editions © 2011, 2009, and
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Library of Congress Cataloging-in-Publication Data
Bodie, Zvi.
Investments / Zvi Bodie, Boston University, Alex Kane, University of California,
San Diego, Alan J. Marcus, Boston College.—10th Edition.
pages cm.—(The McGraw-Hill/Irwin series in finance, insurance and real estate)
Includes index.
ISBN-13: 978-0-07-786167-4 (alk. paper)
ISBN-10: 0-07-786167-1 (alk. paper)
1. Investments. 2. Portfolio management. I. Kane, Alex. II. Marcus, Alan J. III. Title.
HG4521.B564 2014
332.6—dc23
2013016066
The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does
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v
ZVI BODIE Boston University
Zvi Bodie is the Norman
and Adele Barron Professor
of Management at Boston
University. He holds a PhD
from the Massachusetts
Institute of Technology and
has served on the finance fac-
ulty at the Harvard Business
School and MIT’s Sloan
School of Management.
Professor Bodie has published
widely on pension finance
and investment strategy in
leading professional jour-
nals. In cooperation with the
Research Foundation of the
CFA Institute, he has recently
produced a series of Webcasts
and a monograph entitled The Future of Life Cycle Saving and Investing.
ALEX KANE University of California, San Diego
Alex Kane is professor of
finance and economics at
the Graduate School of
International Relations
and Pacific Studies at the
University of California,
San Diego. He has been visit-
ing professor at the Faculty
of Economics, University of
Tokyo; Graduate School of
Business, Harvard; Kennedy
School of Government,
Harvard; and research associ-
ate, National Bureau of
Economic Research. An
author of many articles in
finance and management
journals, Professor Kane’s
research is mainly in corporate
finance, portfolio management,
and capital markets, most
recently in the measurement
of market volatility and
pricing of options.
ALAN J. MARCUS Boston College
Alan Marcus is the Mario J.
Gabelli Professor of Finance
in the Carroll School of
Management at Boston
College. He received his PhD
in economics from MIT.
Professor Marcus has been
a visiting professor at the
Athens Laboratory of
Business Administration and
at MIT’s Sloan School of
Management and has served
as a research associate at the
National Bureau of Economic
Research. Professor Marcus
has published widely in the
fields of capital markets and
portfolio management. His
consulting work has ranged
from new-product develop-
ment to provision of expert
testimony in utility rate
proceedings. He also spent
2 years at the Federal Home
Loan Mortgage Corporation
(Freddie Mac), where he
developed models of mortgage
pricing and credit risk. He cur-
rently serves on the Research
Foundation Advisory Board
of the CFA Institute.
About the Authors
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vi
Preface xvi
PART I
Introduction 1 1
The Investment Environment 1
2 Asset Classes and Financial
Instruments 28
3 How Securities Are Traded 59
4 Mutual Funds and Other Investment
Companies 92
PART II
Portfolio Theory and Practice 117
5 Risk, Return, and the Historical
Record 117
6 Capital Allocation to Risky Assets 168
7 Optimal Risky Portfolios 205
8 Index Models 256
Brief Contents
PART III
Equilibrium in Capital Markets 291
9 The Capital Asset Pricing Model 291
10 Arbitrage Pricing Theory and Multifactor
Models of Risk and Return 324
11 The Efficient Market Hypothesis 349
12 Behavioral Finance and Technical
Analysis 388
13 Empirical Evidence on Security Returns 414
PART IV
Fixed-Income Securities 445
14 Bond Prices and Yields 445
15 The Term Structure of Interest Rates 487
16 Managing Bond Portfolios 515
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Brief Contents
vii
PART VII
Applied Portfolio Management 835
24 Portfolio Performance Evaluation 835
25 International Diversification 882
26 Hedge Funds 926
27 The Theory of Active Portfolio
Management 951
28 Investment Policy and the Framework of the
CFA Institute 977
REFERENCES TO CFA PROBLEMS 1015
GLOSSARY G-1
NAME INDEX I-1
SUBJECT INDEX I-4
PART V
Security Analysis 557 17
Macroeconomic and Industry Analysis 557
18 Equity Valuation Models 591
19 Financial Statement Analysis 635
PART VI
Options, Futures, and Other Derivatives 678
20 Options Markets: Introduction 678
21 Option Valuation 722
22 Futures Markets 770
23 Futures, Swaps, and Risk Management 799
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viii
Reverses / Federal Funds / Brokers’ Calls / The LIBOR Market / Yields on Money Market Instruments
2.2 The Bond Market 34
Treasury Notes and Bonds / Inflation-Protected Treasury Bonds / Federal Agency Debt / International Bonds / Municipal Bonds / Corporate Bonds / Mortgages and Mortgage-Backed Securities
2.3 Equity Securities 41
Common Stock as Ownership Shares / Characteristics of Common Stock / Stock Market Listings / Preferred Stock / Depository Receipts
2.4 Stock and Bond Market Indexes 44
Stock Market Indexes / Dow Jones Averages / Standard & Poor’s Indexes / Other U.S. Market-Value Indexes / Equally Weighted Indexes / Foreign and International Stock Market Indexes / Bond Market Indicators
2.5 Derivative Markets 51
Options / Futures Contracts
End of Chapter Material 54–58
CHAPTER 3
How Securities Are Traded 59
3.1 How Firms Issue Securities 59
Privately Held Firms / Publicly Traded Companies / Shelf Registration / Initial Public Offerings
3.2 How Securities Are Traded 63
Types of Markets
Direct Search Markets / Brokered Markets / Dealer Markets / Auction Markets
Types of Orders
Market Orders / Price-Contingent Orders
Trading Mechanisms
Dealer Markets / Electronic Communication Networks (ECNs) / Specialist Markets
Preface xvi
PART I
Introduction 1
CHAPTER 1
The Investment Environment 1
1.1 Real Assets versus Financial Assets 2
1.2 Financial Assets 3
1.3 Financial Markets and the Economy 5
The Informational Role of Financial Markets / Consumption Timing / Allocation of Risk / Separation of Ownership and Management / Corporate Governance and Corporate Ethics
1.4 The Investment Process 8
1.5 Markets Are Competitive 9
The Risk–Return Trade-Off / Efficient Markets
1.6 The Players 11
Financial Intermediaries / Investment Bankers / Venture Capital and Private Equity
1.7 The Financial Crisis of 2008 15
Antecedents of the Crisis / Changes in Housing Finance / Mortgage Derivatives / Credit Default Swaps / The Rise of Systemic Risk / The Shoe Drops / The Dodd-Frank Reform Act
1.8 Outline of the Text 23
End of Chapter Material 24–27
CHAPTER 2
Asset Classes and Financial Instruments 28
2.1 The Money Market 29
Treasury Bills / Certificates of Deposit / Commercial Paper / Bankers’ Acceptances / Eurodollars / Repos and
Contents
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5.1 Determinants of the Level of Interest Rates 118
Real and Nominal Rates of Interest / The Equilibrium Real Rate of Interest / The Equilibrium Nominal Rate of Interest / Taxes and the Real Rate of Interest
5.2 Comparing Rates of Return for Different Holding Periods 122
Annual Percentage Rates / Continuous Compounding
5.3 Bills and Inflation, 1926–2012 125
5.4 Risk and Risk Premiums 127
Holding-Period Returns / Expected Return and Standard Deviation / Excess Returns and Risk Premiums
5.5 Time Series Analysis of Past Rates of Return 130
Time Series versus Scenario Analysis / Expected Returns and the Arithmetic Average / The Geometric (Time- Weighted) Average Return / Variance and Standard Deviation / Mean and Standard Deviation Estimates from Higher-Frequency Observations / The Reward-to- Volatility (Sharpe) Ratio
5.6 The Normal Distribution 135
5.7 Deviations from Normality and Risk Measures 137
Value at Risk / Expected Shortfall / Lower Partial Standard Deviation and the Sortino Ratio / Relative Frequency of Large, Negative 3-Sigma Returns
5.8 Historic Returns on Risky Portfolios 141
Portfolio Returns / A Global View of the Historical Record
5.9 Long-Term Investments 152
Normal and Lognormal Returns / Simulation of Long- Term Future Rates of Return / The Risk-Free Rate Revisited / Where Is Research on Rates of Return Headed? / Forecasts for the Long Haul
End of Chapter Material 161–167
CHAPTER 6
Capital Allocation to Risky Assets 168
6.1 Risk and Risk Aversion 168
Risk, Speculation, and Gambling / Risk Aversion and Utility Values / Estimating Risk Aversion
6.2 Capital Allocation across Risky and Risk-Free Portfolios 175
6.3 The Risk-Free Asset 177
6.4 Portfolios of One Risky Asset and a Risk-Free Asset 178
6.5 Risk Tolerance and Asset Allocation 182
Nonnormal Returns
6.6 Passive Strategies: The Capital Market Line 187
End of Chapter Material 190–199
Appendix A: Risk Aversion, Expected Utility, and the St. Petersburg Paradox 199
3.3 The Rise of Electronic Trading 68
3.4 U.S. Markets 69
NASDAQ / The New York Stock Exchange / ECNs
3.5 New Trading Strategies 71
Algorithmic Trading / High-Frequency Trading / Dark Pools / Bond Trading
3.6 Globalization of Stock Markets 74
3.7 Trading Costs 76
3.8 Buying on Margin 76
3.9 Short Sales 80
3.10 Regulation of Securities Markets 83
Self-Regulation / The Sarbanes-Oxley Act / Insider Trading
End of Chapter Material 87–91
CHAPTER 4
Mutual Funds and Other Investment Companies 92
4.1 Investment Companies 92
4.2 Types of Investment Companies 93
Unit Investment Trusts / Managed Investment Companies / Other Investment Organizations
Commingled Funds / Real Estate Investment Trusts (REITs) / Hedge Funds
4.3 Mutual Funds 96
Investment Policies
Money Market Funds / Equity Funds / Sector Funds / Bond Funds / International Funds / Balanced Funds / Asset Allocation and Flexible Funds / Index Funds
How Funds Are Sold
4.4 Costs of Investing in Mutual Funds 99
Fee Structure
Operating Expenses / Front-End Load / Back-End Load / 12b-1 Charges
Fees and Mutual Fund Returns
4.5 Taxation of Mutual Fund Income 103
4.6 Exchange-Traded Funds 103
4.7 Mutual Fund Investment Performance: A First Look 107
4.8 Information on Mutual Funds 110
End of Chapter Material 112–116
PART II
Portfolio Theory and Practice 117
CHAPTER 5
Risk, Return, and the Historical Record 117
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Appendix B: Utility Functions and Equilibrium Prices of Insurance Contracts 203
Appendix C: The Kelly Criterion 203
CHAPTER 7
Optimal Risky Portfolios 205
7.1 Diversification and Portfolio Risk 206
7.2 Portfolios of Two Risky Assets 208
7.3 Asset Allocation with Stocks, Bonds, and Bills 215
Asset Allocation with Two Risky Asset Classes
7.4 The Markowitz Portfolio Optimization Model 220
Security Selection / Capital Allocation and the Separation Property / The Power of Diversification / Asset Allocation and Security Selection / Optimal Portfolios and Nonnormal Returns
7.5 Risk Pooling, Risk Sharing, and the Risk of Long- Term Investments 230
Risk Pooling and the Insurance Principle / Risk Sharing / Investment for the Long Run
End of Chapter Material 234–244
Appendix A: A Spreadsheet Model for Efficient Diversification 244
Appendix B: Review of Portfolio Statistics 249
CHAPTER 8
Index Models 256
8.1 A Single-Factor Security Market 257
The Input List of the Markowitz Model / Normality of Returns and Systematic Risk
8.2 The Single-Index Model 259
The Regression Equation of the Single-Index Model / The Expected Return–Beta Relationship / Risk and Covariance in the Single-Index Model / The Set of Estimates Needed for the Single-Index Model / The Index Model and Diversification
8.3 Estimating the Single-Index Model 264
The Security Characteristic Line for Hewlett-Packard / The Explanatory Power of the SCL for HP / Analysis of Variance / The Estimate of Alpha / The Estimate of Beta / Firm-Specific Risk / Correlation and Covariance Matrix
8.4 Portfolio Construction and the Single-Index Model 271
Alpha and Security Analysis / The Index Portfolio as an Investment Asset / The Single-Index-Model Input List / The Optimal Risky Portfolio in the Single-Index Model / The Information Ratio / Summary of Optimization Procedure / An Example
Risk Premium Forecasts / The Optimal Risky Portfolio
8.5 Practical Aspects of Portfolio Management with the Index Model 278
Is the Index Model Inferior to the Full-Covariance Model? / The Industry Version of the Index Model / Predicting Betas / Index Models and Tracking Portfolios
End of Chapter Material 284–290
PART III
Equilibrium in Capital Markets 291
CHAPTER 9
The Capital Asset Pricing Model 291
9.1 The Capital Asset Pricing Model 291
Why Do All Investors Hold the Market Portfolio? / The Passive Strategy Is Efficient / The Risk Premium of the Market Portfolio / Expected Returns on Individual Securities / The Security Market Line / The CAPM and the Single-Index Market
9.2 Assumptions and Extensions of the CAPM 302
Assumptions of the CAPM / Challenges and Extensions to the CAPM / The Zero-Beta Model / Labor Income and Nontraded Assets / A Multiperiod Model and Hedge Portfolios / A Consumption-Based CAPM / Liquidity and the CAPM
9.3 The CAPM and the Academic World 313
9.4 The CAPM and the Investment Industry 315
End of Chapter Material 316–323
CHAPTER 10
Arbitrage Pricing Theory and Multifactor Models of Risk
and Return 324
10.1 Multifactor Models: An Overview 325
Factor Models of Security Returns
10.2 Arbitrage Pricing Theory 327
Arbitrage, Risk Arbitrage, and Equilibrium / Well- Diversified Portfolios / Diversification and Residual Risk in Practice / Executing Arbitrage / The No-Arbitrage Equation of the APT
10.3 The APT, the CAPM, and the Index Model 334
The APT and the CAPM / The APT and Portfolio Optimization in a Single-Index Market
10.4 A Multifactor APT 338
10.5 The Fama-French (FF) Three-Factor Model 340
End of Chapter Material 342–348
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CHAPTER 11
The Efficient Market Hypothesis 349
11.1 Random Walks and the Efficient Market Hypothesis 350
Competition as the Source of Efficiency / Versions of the Efficient Market Hypothesis
11.2 Implications of the EMH 354
Technical Analysis / Fundamental Analysis / Active versus Passive Portfolio Management / The Role of Portfolio Management in an Efficient Market / Resource Allocation
11.3 Event Studies 359
11.4 Are Markets Efficient? 362
The Issues
The Magnitude Issue / The Selection Bias Issue / The Lucky Event Issue
Weak-Form Tests: Patterns in Stock Returns
Returns over Short Horizons / Returns over Long Horizons
Predictors of Broad Market Returns / Semistrong Tests: Market Anomalies
The Small-Firm-in-January Effect / The Neglected- Firm Effect and Liquidity Effects / Book-to-Market Ratios / Post–Earnings-Announcement Price Drift
Strong-Form Tests: Inside Information / Interpreting the Anomalies
Risk Premiums or Inefficiencies? / Anomalies or Data Mining? / Anomalies over Time
Bubbles and Market Efficiency
11.5 Mutual Fund and Analyst Performance 375
Stock Market Analysts / Mutual Fund Managers / So, Are Markets Efficient?
End of Chapter Material 380–387
CHAPTER 12
Behavioral Finance and Technical Analysis 388
12.1 The Behavioral Critique 389
Information Processing
Forecasting Errors / Overconfidence / Conservatism / Sample Size Neglect and Representativeness
Behavioral Biases
Framing / Mental Accounting / Regret Avoidance
Affect
Prospect Theory
Limits to Arbitrage
Fundamental Risk / Implementation Costs / Model Risk
Limits to Arbitrage and the Law of One Price
“Siamese Twin” Companies / Equity Carve-Outs / Closed-End Funds
Bubbles and Behavioral Economics / Evaluating the Behavioral Critique
12.2 Technical Analysis and Behavioral Finance 400
Trends and Corrections
Momentum and Moving Averages / Relative Strength / Breadth
Sentiment Indicators
Trin Statistic / Confidence Index / Put/Call Ratio
A Warning
End of Chapter Material 407–413
CHAPTER 13
Empirical Evidence on Security Returns 414
13.1 The Index Model and the Single-Factor APT 415
The Expected Return–Beta Relationship
Setting Up the Sample Data / Estimating the SCL / Estimating the SML
Tests of the CAPM / The Market Index / Measurement Error in Beta
13.2 Tests of the Multifactor CAPM and APT 421
Labor Income / Private (Nontraded) Business / Early Versions of the Multifactor CAPM and APT / A Macro Factor Model
13.3 Fama-French-Type Factor Models 426
Size and B/M as Risk Factors / Behavioral Explanations / Momentum: A Fourth Factor
13.4 Liquidity and Asset Pricing 433
13.5 Consumption-Based Asset Pricing and the Equity Premium Puzzle 435
Consumption Growth and Market Rates of Return / Expected versus Realized Returns / Survivorship Bias / Extensions to the CAPM May Resolve the Equity Premium Puzzle / Liquidity and the Equity Premium Puzzle / Behavioral Explanations of the Equity Premium Puzzle /
End of Chapter Material 442–444
PART IV
Fixed-Income Securities 445
CHAPTER 14
Bond Prices and Yields 445
14.1 Bond Characteristics 446
Treasury Bonds and Notes
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Accrued Interest and Quoted Bond Prices
Corporate Bonds
Call Provisions on Corporate Bonds / Convertible Bonds / Puttable Bonds / Floating-Rate Bonds
Preferred Stock / Other Domestic Issuers / International Bonds / Innovation in the Bond Market
Inverse Floaters / Asset-Backed Bonds / Catastrophe Bonds / Indexed Bonds
14.2 Bond Pricing 452
Bond Pricing between Coupon Dates
14.3 Bond Yields 458
Yield to Maturity / Yield to Call / Realized Compound Return versus Yield to Maturity
14.4 Bond Prices over Time 463
Yield to Maturity versus Holding-Period Return / Zero- Coupon Bonds and Treasury Strips / After-Tax Returns
14.5 Default Risk and Bond Pricing 468
Junk Bonds / Determinants of Bond Safety / Bond Indentures
Sinking Funds / Subordination of Further Debt / Dividend Restrictions / Collateral
Yield to Maturity and Default Risk / Credit Default Swaps / Credit Risk and Collateralized Debt Obligations
End of Chapter Material 479–486
CHAPTER 15
The Term Structure of Interest Rates 487
15.1 The Yield Curve 487
Bond Pricing
15.2 The Yield Curve and Future Interest Rates 490
The Yield Curve under Certainty / Holding-Period Returns / Forward Rates
15.3 Interest Rate Uncertainty and Forward Rates 495
15.4 Theories of the Term Structure 497
The Expectations Hypothesis / Liquidity Preference
15.5 Interpreting the Term Structure 501
15.6 Forward Rates as Forward Contracts 504
End of Chapter Material 506–514
CHAPTER 16
Managing Bond Portfolios 515
16.1 Interest Rate Risk 516
Interest Rate Sensitivity / Duration / What Determines Duration?
Rule 1 for Duration / Rule 2 for Duration / Rule 3 for Duration / Rule 4 for Duration / Rule 5 for Duration
16.2 Convexity 525
Why Do Investors Like Convexity? / Duration and Convexity of Callable Bonds / Duration and Convexity of Mortgage-Backed Securities
16.3 Passive Bond Management 533
Bond-Index Funds / Immunization / Cash Flow Matching and Dedication / Other Problems with Conventional Immunization
16.4 Active Bond Management 543
Sources of Potential Profit / Horizon Analysis
End of Chapter Material 545–556
PART V
Security Analysis 557
CHAPTER 17
Macroeconomic and Industry Analysis 557
17.1 The Global Economy 558
17.2 The Domestic Macroeconomy 560
17.3 Demand and Supply Shocks 562
17.4 Federal Government Policy 563
Fiscal Policy / Monetary Policy / Supply-Side Policies
17.5 Business Cycles 566
The Business Cycle / Economic Indicators / Other Indicators
17.6 Industry Analysis 571
Defining an Industry / Sensitivity to the Business Cycle /
Sector Rotation / Industry Life Cycles
Start-Up Stage / Consolidation Stage / Maturity Stage / Relative Decline
Industry Structure and Performance
Threat of Entry / Rivalry between Existing Competi- tors / Pressure from Substitute Products / Bargaining Power of Buyers / Bargaining Power of Suppliers
End of Chapter Material 582–590
CHAPTER 18
Equity Valuation Models 591
18.1 Valuation by Comparables 591
Limitations of Book Value
18.2 Intrinsic Value versus Market Price 593
18.3 Dividend Discount Models 595
The Constant-Growth DDM / Convergence of Price to Intrinsic Value / Stock Prices and Investment Opportunities / Life Cycles and Multistage Growth Models / Multistage Growth Models
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18.4 Price–Earnings Ratio 609
The Price–Earnings Ratio and Growth Opportunities / P/E Ratios and Stock Risk / Pitfalls in P/E Analysis / Combining P/E Analysis and the DDM / Other Comparative Valuation Ratios
Price-to-Book Ratio / Price-to-Cash-Flow Ratio / Price-to-Sales Ratio
18.5 Free Cash Flow Valuation Approaches 617
Comparing the Valuation Models / The Problem with DCF Models
18.6 The Aggregate Stock Market 622
End of Chapter Material 623–634
CHAPTER 19
Financial Statement Analysis 635
19.1 The Major Financial Statements 635
The Income Statement / The Balance Sheet / The Statement of Cash Flows
19.2 Measuring Firm Performance 640
19.3 Profitability Measures 641
Return on Assets, ROA / Return on Capital, ROC / Return on Equity, ROE / Financial Leverage and ROE / Economic Value Added
19.4 Ratio Analysis 645
Decomposition of ROE / Turnover and Other Asset Utilization Ratios / Liquidity Ratios / Market Price Ratios: Growth versus Value / Choosing a Benchmark
19.5 An Illustration of Financial Statement Analysis 655
19.6 Comparability Problems 658
Inventory Valuation / Depreciation / Inflation and Interest Expense / Fair Value Accounting / Quality of Earnings and Accounting Practices / International Accounting Conventions
19.7 Value Investing: The Graham Technique 665
End of Chapter Material 665–677
PART VI
Options, Futures, and Other Derivatives 678
CHAPTER 20
Options Markets: Introduction 678
20.1 The Option Contract 679
Options Trading / American and European Options / Adjustments in Option Contract Terms / The Options Clearing Corporation / Other Listed Options
Index Options / Futures Options / Foreign Currency Options / Interest Rate Options
20.2 Values of Options at Expiration 685
Call Options / Put Options / Option versus Stock Investments
20.3 Option Strategies 689
Protective Put / Covered Calls / Straddle / Spreads / Collars
20.4 The Put-Call Parity Relationship 698
20.5 Option-Like Securities 701
Callable Bonds / Convertible Securities / Warrants / Collateralized Loans / Levered Equity and Risky Debt
20.6 Financial Engineering 707
20.7 Exotic Options 709
Asian Options / Barrier Options / Lookback Options / Currency-Translated Options / Digital Options
End of Chapter Material 710–721
CHAPTER 21
Option Valuation 722
21.1 Option Valuation: Introduction 722
Intrinsic and Time Values / Determinants of Option Values
21.2 Restrictions on Option Values 725
Restrictions on the Value of a Call Option / Early Exercise and Dividends / Early Exercise of American Puts
21.3 Binomial Option Pricing 729
Two-State Option Pricing / Generalizing the Two-State Approach / Making the Valuation Model Practical
21.4 Black-Scholes Option Valuation 737
The Black-Scholes Formula / Dividends and Call Option Valuation / Put Option Valuation / Dividends and Put Option Valuation
21.5 Using the Black-Scholes Formula 746
Hedge Ratios and the Black-Scholes Formula / Portfolio Insurance / Option Pricing and the Crisis of 2008–2009 / Option Pricing and Portfolio Theory / Hedging Bets on Mispriced Options
21.6 Empirical Evidence on Option Pricing 758
End of Chapter Material 759–769
CHAPTER 22
Futures Markets 770
22.1 The Futures Contract 771
The Basics of Futures Contracts / Existing Contracts
22.2 Trading Mechanics 775
The Clearinghouse and Open Interest / The Margin Account and Marking to Market / Cash versus Actual Delivery / Regulations / Taxation
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Contents
xiv
22.3 Futures Markets Strategies 781
Hedging and Speculation / Basis Risk and Hedging
22.4 Futures Prices 785
The Spot-Futures Parity Theorem / Spreads / Forward versus Futures Pricing
22.5 Futures Prices versus Expected Spot Prices 791
Expectations Hypothesis / Normal Backwardation / Contango / Modern Portfolio Theory
End of Chapter Material 793–798
CHAPTER 23
Futures, Swaps, and Risk Management 799
23.1 Foreign Exchange Futures 799
The Markets / Interest Rate Parity / Direct versus Indirect Quotes / Using Futures to Manage Exchange Rate Risk
23.2 Stock-Index Futures 806
The Contracts / Creating Synthetic Stock Positions: An Asset Allocation Tool / Index Arbitrage / Using Index Futures to Hedge Market Risk
23.3 Interest Rate Futures 813
Hedging Interest Rate Risk
23.4 Swaps 815
Swaps and Balance Sheet Restructuring / The Swap Dealer / Other Interest Rate Contracts / Swap Pricing / Credit Risk in the Swap Market / Credit Default Swaps
23.5 Commodity Futures Pricing 822
Pricing with Storage Costs / Discounted Cash Flow Analysis for Commodity Futures
End of Chapter Material 825–834
PART VII
Applied Portfolio Management 835
CHAPTER 24
Portfolio Performance Evaluation 835
24.1 The Conventional Theory of Performance Evaluation 835
Average Rates of Return / Time-Weighted Returns versus Dollar-Weighted Returns / Dollar-Weighted Return and Investment Performance / Adjusting Returns for Risk / The M2 Measure of Performance / Sharpe’s Ratio Is the Criterion for Overall Portfolios / Appropriate Performance Measures in Two Scenarios
Jane’s Portfolio Represents Her Entire Risky Invest- ment Fund / Jane’s Choice Portfolio Is One of Many Portfolios Combined into a Large Investment Fund
The Role of Alpha in Performance Measures / Actual Performance Measurement: An Example / Performance Manipulation and the Morningstar Risk-Adjusted Rating / Realized Returns versus Expected Returns
24.2 Performance Measurement for Hedge Funds 851
24.3 Performance Measurement with Changing Portfolio Composition 854
24.4 Market Timing 855
The Potential Value of Market Timing / Valuing Market Timing as a Call Option / The Value of Imperfect Forecasting
24.5 Style Analysis 861
Style Analysis and Multifactor Benchmarks / Style Analysis in Excel
24.6 Performance Attribution Procedures 864
Asset Allocation Decisions / Sector and Security Selection Decisions / Summing Up Component Contributions
End of Chapter Material 870–881
CHAPTER 25
International Diversification 882
25.1 Global Markets for Equities 883
Developed Countries / Emerging Markets / Market Capitalization and GDP / Home-Country Bias
25.2 Risk Factors in International Investing 887
Exchange Rate Risk / Political Risk
25.3 International Investing: Risk, Return, and Benefits from Diversification 895
Risk and Return: Summary Statistics / Are Investments in Emerging Markets Riskier? / Are Average Returns Higher in Emerging Markets? / Is Exchange Rate Risk Important in International Portfolios? / Benefits from International Diversification / Misleading Representation of Diversification Benefits / Realistic Benefits from International Diversification / Are Benefits from International Diversification Preserved in Bear Markets?
25.4 Assessing the Potential of International Diversification 911
25.5 International Investing and Performance Attribution 916
Constructing a Benchmark Portfolio of Foreign Assets / Performance Attribution
End of Chapter Material 920–925
CHAPTER 26
Hedge Funds 926
26.1 Hedge Funds versus Mutual Funds 927
26.2 Hedge Fund Strategies 928
Directional and Nondirectional Strategies / Statistical Arbitrage
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Contents
xv
CHAPTER 28
Investment Policy and the Framework of the CFA Institute 977
28.1 The Investment Management Process 978
Objectives / Individual Investors / Personal Trusts / Mutual Funds / Pension Funds / Endowment Funds / Life Insurance Companies / Non–Life Insurance Companies / Banks
28.2 Constraints 983
Liquidity / Investment Horizon / Regulations / Tax Considerations / Unique Needs
28.3 Policy Statements 985
Sample Policy Statements for Individual Investors
28.4 Asset Allocation 992
Taxes and Asset Allocation
28.5 Managing Portfolios of Individual Investors 994
Human Capital and Insurance / Investment in Residence / Saving for Retirement and the Assumption of Risk / Retirement Planning Models / Manage Your Own Portfolio or Rely on Others? / Tax Sheltering
The Tax-Deferral Option / Tax-Deferred Retirement Plans / Deferred Annuities / Variable and Universal Life Insurance
28.6 Pension Funds 1000
Defined Contribution Plans / Defined Benefit Plans / Pension Investment Strategies
Investing in Equities / Wrong Reasons to Invest in Equities
28.7 Investments for the Long Run 1003
Target Investing and the Term Structure of Bonds / Making Simple Investment Choices / Inflation Risk and Long-Term Investors
End of Chapter Material 1004–1014
REFERENCES TO CFA PROBLEMS 1015
GLOSSARY G-1
NAME INDEX I-1
SUBJECT INDEX I-4
26.3 Portable Alpha 931
An Example of a Pure Play
26.4 Style Analysis for Hedge Funds 933
26.5 Performance Measurement for Hedge Funds 935
Liquidity and Hedge Fund Performance / Hedge Fund Performance and Survivorship Bias / Hedge Fund Performance and Changing Factor Loadings / Tail Events and Hedge Fund Performance
26.6 Fee Structure in Hedge Funds 943
End of Chapter Material 946–950
CHAPTER 27
The Theory of Active Portfolio Management 951
27.1 Optimal Portfolios and Alpha Values 951
Forecasts of Alpha Values and Extreme Portfolio Weights / Restriction of Benchmark Risk
27.2 The Treynor-Black Model and Forecast Precision 958
Adjusting Forecasts for the Precision of Alpha / Distribution of Alpha Values / Organizational Structure and Performance
27.3 The Black-Litterman Model 962
Black-Litterman Asset Allocation Decision / Step 1: The Covariance Matrix from Historical Data / Step 2: Determination of a Baseline Forecast / Step 3: Integrating the Manager’s Private Views / Step 4: Revised (Posterior) Expectations / Step 5: Portfolio Optimization
27.4 Treynor-Black versus Black-Litterman: Complements, Not Substitutes 968
The BL Model as Icing on the TB Cake / Why Not Replace the Entire TB Cake with the BL Icing?
27.5 The Value of Active Management 970
A Model for the Estimation of Potential Fees / Results from the Distribution of Actual Information Ratios / Results from Distribution of Actual Forecasts / Results with Reasonable Forecasting Records
27.6 Concluding Remarks on Active Management 972
End of Chapter Material 973–974
Appendix A: Forecasts and Realizations of Alpha 974
Appendix B: The General Black-Litterman Model 975
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xvi
become an investment professional, or simply a sophisti-
cated individual investor, you will find these skills essen-
tial, especially in today’s rapidly evolving environment.
Our primary goal is to present material of practical
value, but all three of us are active researchers in finan-
cial economics and find virtually all of the material in this
book to be of great intellectual interest. Fortunately, we
think, there is no contradiction in the field of investments
between the pursuit of truth and the pursuit of money.
Quite the opposite. The capital asset pricing model, the
arbitrage pricing model, the efficient markets hypothesis,
the option-pricing model, and the other centerpieces of
modern financial research are as much intellectually satis-
fying subjects of scientific inquiry as they are of immense
practical importance for the sophisticated investor.
In our effort to link theory to practice, we also have
attempted to make our approach consistent with that of the
CFA Institute. In addition to fostering research in finance,
the CFA Institute administers an education and certifi-
cation program to candidates seeking designation as a
Chartered Financial Analyst (CFA). The CFA curriculum
represents the consensus of a committee of distinguished
scholars and practitioners regarding the core of knowledge
required by the investment professional.
Many features of this text make it consistent with and
relevant to the CFA curriculum. Questions from past CFA
exams appear at the end of nearly every chapter, and, for
students who will be taking the exam, those same ques-
tions and the exam from which they’ve been taken are
listed at the end of the book. Chapter 3 includes excerpts
from the “Code of Ethics and Standards of Professional
Conduct” of the CFA Institute. Chapter 28, which dis-
cusses investors and the investment process, presents the
W e’ve just ended three decades of rapid and pro-
found change in the investments industry as
well as a financial crisis of historic magnitude.
The vast expansion of financial markets during this period
was due in part to innovations in securitization and credit
enhancement that gave birth to new trading strategies.
These strategies were in turn made feasible by develop-
ments in communication and information technology, as
well as by advances in the theory of investments.
Yet the financial crisis also was rooted in the cracks
of these developments. Many of the innovations in secu-
rity design facilitated high leverage and an exaggerated
notion of the efficacy of risk transfer strategies. This
engendered complacency about risk that was coupled
with relaxation of regulation as well as reduced trans-
parency, masking the precarious condition of many big
players in the system. Of necessity, our text has evolved
along with financial markets and their influence on
world events.
Investments, Tenth Edition, is intended primarily as a textbook for courses in investment analysis. Our guiding
principle has been to present the material in a framework
that is organized by a central core of consistent fundamen-
tal principles. We attempt to strip away unnecessary math-
ematical and technical detail, and we have concentrated
on providing the intuition that may guide students and
practitioners as they confront new ideas and challenges in
their professional lives.
This text will introduce you to major issues currently
of concern to all investors. It can give you the skills to
conduct a sophisticated assessment of watershed current
issues and debates covered by the popular media as well
as more-specialized finance journals. Whether you plan to
Preface
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Preface
xvii
CFA Institute’s framework for systematically relating
investor objectives and constraints to ultimate investment
policy. End-of-chapter problems also include questions
from test-prep leader Kaplan Schweser.
In the Tenth Edition, we have continued our systematic
collection of Excel spreadsheets that give tools to explore
concepts more deeply than was previously possible. These
spreadsheets, available on the Web site for this text ( www. mhhe.com/bkm ), provide a taste of the sophisticated ana- lytic tools available to professional investors.
UNDERLYING PHILOSOPHY
In the Tenth Edition, we address many of the changes in
the investment environment, including the unprecedented
events surrounding the financial crisis.
At the same time, many basic principles remain impor- tant. We believe that attention to these few important
principles can simplify the study of otherwise difficult
material and that fundamental principles should orga-
nize and motivate all study. These principles are crucial
to understanding the securities traded in financial markets
and in understanding new securities that will be intro-
duced in the future, as well as their effects on global mar-
kets. For this reason, we have made this book thematic,
meaning we never offer rules of thumb without reference
to the central tenets of the modern approach to finance.
The common theme unifying this book is that security markets are nearly efficient, meaning most securities are usually priced appropriately given their risk and return
attributes. Free lunches are rarely found in markets as
competitive as the financial market. This simple observa-
tion is, nevertheless, remarkably powerful in its implica-
tions for the design of investment strategies; as a result,
our discussions of strategy are always guided by the
implications of the efficient markets hypothesis. While
the degree of market efficiency is, and always will be, a
matter of debate (in fact we devote a full chapter to the
behavioral challenge to the efficient market hypothesis),
we hope our discussions throughout the book convey a
good dose of healthy criticism concerning much conven-
tional wisdom.
Distinctive Themes Investments is organized around several important themes:
1. The central theme is the near-informational-efficiency of well-developed security markets, such as those in the
United States, and the general awareness that competi-
tive markets do not offer “free lunches” to participants.
A second theme is the risk–return trade-off. This too
is a no-free-lunch notion, holding that in competi-
tive security markets, higher expected returns come
only at a price: the need to bear greater investment
risk. However, this notion leaves several questions
unanswered. How should one measure the risk of
an asset? What should be the quantitative trade-
off between risk (properly measured) and expected
return? The approach we present to these issues is
known as modern portfolio theory, which is another organizing principle of this book. Modern portfolio
theory focuses on the techniques and implications of
efficient diversification, and we devote considerable attention to the effect of diversification on portfolio
risk as well as the implications of efficient diversi-
fication for the proper measurement of risk and the
risk–return relationship.
2. This text places greater emphasis on asset allocation than most of its competitors. We prefer this empha-
sis for two important reasons. First, it corresponds to
the procedure that most individuals actually follow.
Typically, you start with all of your money in a bank
account, only then considering how much to invest in
something riskier that might offer a higher expected
return. The logical step at this point is to consider
risky asset classes, such as stocks, bonds, or real
estate. This is an asset allocation decision. Second,
in most cases, the asset allocation choice is far more
important in determining overall investment perfor-
mance than is the set of security selection decisions.
Asset allocation is the primary determinant of the
risk–return profile of the investment portfolio, and so
it deserves primary attention in a study of investment
policy.
3. This text offers a much broader and deeper treat- ment of futures, options, and other derivative secu-
rity markets than most investments texts. These
markets have become both crucial and integral to
the financial universe. Your only choice is to become
conversant in these markets—whether you are to be
a finance professional or simply a sophisticated indi-
vidual investor.
NEW IN THE TENTH EDITION
The following is a guide to changes in the Tenth Edition.
This is not an exhaustive road map, but instead is meant to
provide an overview of substantial additions and changes
to coverage from the last edition of the text.
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Preface
xviii
concerning the use of financial ratios as tools to evaluate
firm performance.
Chapter 21 Option Valuation We have added substantial new sections on risk-neutral
valuation methods and their implementation in the bino-
mial option-pricing model, as well as the implications
of the option pricing model for tail risk and financial
instability.
Chapter 24 Portfolio Performance Evaluation New sections on the vulnerability of standard perfor-
mance measures to manipulation, manipulation-free mea-
sures, and the Morningstar Risk-Adjusted Return have
been added.
ORGANIZATION AND CONTENT
The text is composed of seven sections that are fairly inde-
pendent and may be studied in a variety of sequences.
Because there is enough material in the book for a two-
semester course, clearly a one-semester course will require
the instructor to decide which parts to include.
Part One is introductory and contains important insti- tutional material focusing on the financial environment.
We discuss the major players in the financial markets,
provide an overview of the types of securities traded in
those markets, and explain how and where securities are
traded. We also discuss in depth mutual funds and other
investment companies, which have become an increas-
ingly important means of investing for individual inves-
tors. Perhaps most important, we address how financial
markets can influence all aspects of the global economy,
as in 2008.
The material presented in Part One should make it
possible for instructors to assign term projects early in
the course. These projects might require the student to
analyze in detail a particular group of securities. Many
instructors like to involve their students in some sort of
investment game, and the material in these chapters will
facilitate this process.
Parts Two and Three contain the core of modern portfolio theory. Chapter 5 is a general discussion of risk
and return, making the general point that historical returns
on broad asset classes are consistent with a risk–return
trade-off, and examining the distribution of stock returns.
We focus more closely in Chapter 6 on how to describe
investors’ risk preferences and how they bear on asset
allocation. In the next two chapters, we turn to portfolio
optimization (Chapter 7) and its implementation using
index models (Chapter 8).
Chapter 1 The Investment Environment This chapter contains updated coverage of the consequences
of the financial crisis as well as the Dodd-Frank act.
Chapter 2 Asset Classes and Financial Instruments We devote additional attention to money markets, includ-
ing recent controversies concerning the regulation of
money market mutual funds as well as the LIBOR scandal.
Chapter 3 How Securities Are Traded We have extensively rewritten this chapter and included
new sections that detail the rise of electronic markets,
algorithmic and high-speed trading, and changes in mar-
ket structure.
Chapter 5 Risk, Return, and the Historical Record This chapter has been updated with considerable attention
paid to evidence on tail risk and extreme stock returns.
Chapter 9 The Capital Asset Pricing Model We have streamlined the explanation of the simple CAPM
and updated and integrated the sections dealing with
extensions of the CAPM, tying together extra-market
hedging demands and factor risk premia.
Chapter 10 Arbitrage Pricing Theory The chapter contains new material on the practical feasi-
bility of creating well-diversified portfolios and the impli-
cations for asset pricing.
Chapter 11 The Efficient Market Hypothesis We have added new material documenting the behavior of
market anomalies over time, suggesting how market inef-
ficiencies seem to be corrected.
Chapter 13 Empirical Evidence on Security Returns Increased attention is given to tests of multifactor models
of risk and return and the implications of these tests for
the importance of extra-market hedging demands.
Chapter 14 Bond Prices and Yields This chapter includes new material on sovereign credit
default swaps.
Chapter 18 Equity Valuation Models This chapter includes a new section on the practical prob-
lems entailed in using DCF security valuation models and
the response of value investors to these problems.
Chapter 19 Financial Statement Analysis We have added a new introduction to the discussion of
ratio analysis, providing greater structure and rationale
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Preface
xix
After our treatment of modern portfolio theory in Part
Two, we investigate in Part Three the implications of that
theory for the equilibrium structure of expected rates of
return on risky assets. Chapter 9 treats the capital asset
pricing model and Chapter 10 covers multifactor descrip-
tions of risk and the arbitrage pricing theory. Chapter 11
covers the efficient market hypothesis, including its ratio-
nale as well as evidence that supports the hypothesis and
challenges it. Chapter 12 is devoted to the behavioral
critique of market rationality. Finally, we conclude Part
Three with Chapter 13 on empirical evidence on security
pricing. This chapter contains evidence concerning the
risk–return relationship, as well as liquidity effects on
asset pricing.
Part Four is the first of three parts on security valu- ation. This part treats fixed-income securities—bond
pricing (Chapter 14), term structure relationships (Chap-
ter 15), and interest-rate risk management (Chapter 16).
Parts Five and Six deal with equity securities and derivative securities. For a course emphasizing security
analysis and excluding portfolio theory, one may pro-
ceed directly from Part One to Part Four with no loss in
continuity.
Finally, Part Seven considers several topics important for portfolio managers, including performance evalua-
tion, international diversification, active management, and
practical issues in the process of portfolio management.
This part also contains a chapter on hedge funds.
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xx
A Guided Tour
This book contains several features designed to make it easy for students to understand, absorb, and apply the concepts and techniques presented.
1 1 AN INVESTMENT IS the current commitment of money or other resources in the expecta- tion of reaping future benefits. For example, an individual might purchase shares of stock anticipating that the future proceeds from the shares will justify both the time that her money is tied up as well as the risk of the investment. The time you will spend studying this text (not to mention its cost) also is an investment. You are forgoing either current leisure or the income you could be earning at a job in the expectation that your future career will be suf- ficiently enhanced to justify this commitment of time and effort. While these two invest- ments differ in many ways, they share one key attribute that is central to all investments: You
Broadly speaking, this chapter addresses three topics that will provide a useful perspec- tive for the material that is to come later. First, before delving into the topic of “investments,” we consider the role of financial assets in the economy. We discuss the relationship between securities and the “real” assets that actually produce goods and services for consumers, and we consider why financial assets are important to the functioning of a developed economy.
Given this background, we then take a first look at the types of decisions that con- front investors as they assemble a portfolio of assets. These investment decisions are made in an environment where higher returns usually can be obtained only at the price of
The Investment Environment
CHAPTER ONE
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NUMBERED EXAMPLES NUMBERED AND TITLED examples are integrated throughout chapters. Using the worked-out solutions to these examples as models, students can learn how to solve specific problems step-by-step as well as gain insight into general principles by seeing how they are applied to answer concrete questions.
Here are fees for different classes of the Dreyfus High Yield Fund in 2012. Notice the trade-off between the front-end loads versus 12b-1 charges in the choice between Class A and Class C shares. Class I shares are sold only to institutional investors and carry lower fees.
Example 4.2 Fees for Various Classes
Class A Class C Class I
Front-end load 0–4.5%a 0 0 Back-end load 0 0–1%b 0%b
12b-1 feesc .25% 1.0% 0% Expense ratio .70% .70% .70%
a Depending on size of investment. b Depending on years until holdings are sold. c Including service fee.
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CHAPTER OPENING VIGNETTES SERVE TO OUTLINE the upcoming material in the chapter and provide students with a road map of what they will learn.
claim and limited liability features. Residual claim means that stockholders are the last in line of all those who have a
claim on the assets and income of the corporation. In a liquidation of the firm’s assets the
shareholders have a claim to what is left after all other claimants such as the tax authorities,
employees, suppliers, bondholders, and other creditors have been paid. For a firm not in
liquidation, shareholders have claim to the part of operating income left over after inter-
est and taxes have been paid. Management can either pay
this residual as cash dividends to shareholders or reinvest
it in the business to increase the value of the shares. Limited liability means that the most shareholders
can lose in the event of failure of the corporation is their
original investment. Unlike owners of unincorporated
businesses, whose creditors can lay claim to the personal
assets of the owner (house, car, furniture), corporate
shareholders may at worst have worthless stock. They are
not personally liable for the firm’s obligations.
Stock Market Listings
a. If you buy 100 shares of IBM stock, to what are you entitled?
b. What is the most money you can make on this investment over the next year?
c. If you pay $180 per share, what is the most money you could lose over the year?
CONCEPT CHECK 2.3
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CONCEPT CHECKS A UNIQUE FEATURE of this book! These self-test questions and problems found in the body of the text enable the students to determine whether they’ve understood the preceding material. Detailed solutions are provided at the end of each chapter.
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xxi
Investors Sour on Pro Stock Pickers
Investors are jumping out of mutual funds managed by professional stock pickers and shifting massive amounts of money into lower-cost funds that echo the broader market.
Through November 2012, investors pulled $119.3 billion from so-called actively managed U.S. stock funds accord- ing to the latest data from research firm Morningstar Inc. At the same time, they poured $30.4 billion into U.S. stock exchange-traded funds.
The move reflects the fact that many money manag- ers of stock funds, which charge fees but also dangle the prospect of higher returns, have underperformed the benchmark stock indexes. As a result, more investors are choosing simply to invest in funds tracking the indexes, which carry lower fees and are perceived as having less risk.
The mission of stock pickers in a managed mutual fund is to outperform the overall market by actively trading individual stocks or bonds, with fund managers receiving higher fees for their effort. In an ETF (or indexed mutual fund), managers balance the share makeup of the fund so it accurately reflects the performance of its underlying index, charging lower fees.
Morningstar says that when investors have put money in stock funds, they have chosen low-cost index funds and ETFs. Some index ETFs cost less than 0.1% of assets a year, while many actively managed stock funds charge 1% a year or more.
While the trend has put increasing pressure lately on stock pickers, it is shifting the fortunes of some of the big- gest players in the $14 trillion mutual-fund industry.
Fidelity Investments and American Funds, among the largest in the category, saw redemptions or weak investor interest compared with competitors, according to an anal- ysis of mutual-fund flows done for The Wall Street Journal by research firm Strategic Insight, a unit of New York-based Asset International.
At the other end of the spectrum, Vanguard, the world’s largest provider of index mutual funds, pulled in a net $141 billion last year through December, according to the company.
Many investors say they are looking for a way to invest cheaply, with less risk.
Source: Adapted from Kirsten Grind, “Investors Sour on Pro Stock Pickers” The Wall Street Journal, January 3, 2013.
W O
R D
S FR O
M T
H E ST
R E E T
or a mutual fund company that operates a market index fund. Vanguard, for example, oper-
ates the Index 500 Portfolio that mimics the S&P 500 index fund. It purchases shares of the
firms constituting the S&P 500 in proportion to the market values of the outstanding equity
of each firm, and therefore essentially replicates the S&P 500 index. The fund thus dupli-
cates the performance of this market index. It has one of the lowest operating expenses
(as a percentage of assets) of all mutual stock funds precisely because it requires minimal
managerial effort.
A second reason to pursue a passive strategy is the free-rider benefit. If there are many
active, knowledgeable investors who quickly bid up prices of undervalued assets and force
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eXcel APPLICATIONS: Two–Security Model
T he accompanying spreadsheet can be used to mea- sure the return and risk of a portfolio of two risky
assets. The model calculates the return and risk for vary- ing weights of each security along with the optimal risky and minimum-variance portfolio. Graphs are automatically generated for various model inputs. The model allows you to specify a target rate of return and solves for optimal combinations using the risk-free asset and the optimal risky portfolio. The spreadsheet is constructed with the
two-security return data from Table 7.1 . This spreadsheet is available at www.mhhe.com/bkm. Excel Question
1. Suppose your target expected rate of return is 11%.
a. What is the lowest-volatility portfolio that provides that expected return?
b. What is the standard deviation of that portfolio? c. What is the composition of that portfolio?
0
Standard Deviation (%)
0 5
5
11
10 15 20 25 3530
Expected
Return (%)A B C D E F
1
2 Expected Standard Correlation
3 Return Deviation Coefficient Covariance
4 Security 1
5 Security 2
6 T-Bill
7
8 Weight Weight Expected Standard Reward to
9 Security 1 Security 2 Return Deviation Volatility
10
11
12
13
14
Asset Allocation Analysis: Risk and Return
0.08 0.12 0.3 0.0072
0.13 0.2
0.05 0
1 0 0.08000 0.12000 0.25000
0.9 0.1 0.08500 0.11559 0.30281
0.8 0.2 0.09000 0.11454 0.34922
0.7 0.3 0.09500 0.11696 0.38474
0.6 0.4 0.10000 0.407710.12264
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C D E F
Period Implicitly Assumed Probability = 1/5
Squared Deviation
Gross HPR = 1 + HPR
Wealth Index*
2001 .2 −0.1189 0.0196 0.0586 0.0707 0.0077
0.1774
0.0008
0.1983
0.8811 0.8811 0.6864 0.8833 0.9794 1.0275
Check: 1.0054^5=
0.7790 1.2869 1.1088 1.0491
0.0054 1.0275
−0.2210 0.2869 0.1088 0.0491 0.0210 0.0210
HPR (decimal)
.2
.2
.2
.2
2002 2003 2004 2005
Arithmetic average Expected HPR SUMPRODUCT(B5:B9, C5:C9) =
SUMPRODUCT(B5:B9, D5:D9)^.5 = STDEV(C5:C9) = Geometric average return
*The value of $1 invested at the beginning of the sample period (1/1/2001). GEOMEAN(E5:E9) − 1 =
Standard deviation
AVERAGE(C5:C9) =
Spreadsheet 5.2
Time series of HPR for the S&P 500
e X c e l Please visit us at www.mhhe.com/bkm
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EXCEL APPLICATIONS THE TENTH EDITION features Excel Spreadsheet Applications with new Excel questions. A sample spreadsheet is presented in the text with an interactive version available on the book’s Web site at www.mhhe.com/bkm .
EXCEL EXHIBITS SELECTED EXHIBITS ARE set as Excel spreadsheets and are denoted by an icon. They are also available on the book’s Web site at www.mhhe.com/bkm .
WORDS FROM THE STREET BOXES SHORT ARTICLES FROM business periodicals, such as The Wall Street Journal, are included in boxes throughout the text. The articles are chosen for real-world relevance and clarity of presentation.
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End-of-Chapter Features
PROBLEM SETS WE STRONGLY BELIEVE that practice in solving problems is critical to understanding investments, so a good variety of problems is provided. For ease of assignment we separated the questions by level of difficulty Basic, Intermediate, and Challenge.
EXAM PREP QUESTIONS PRACTICE QUESTIONS for the CFA ® exams provided by Kaplan Schweser, A Global Leader in CFA ® Education, are available in selected chapters for additional test practice. Look for the Kaplan Schweser logo. Learn more at www.schweser.com .
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C H A P T E R 5 Risk, Return, and the Historical Record 163
Basic
PROBLEM SETS 1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation
imply that this increase will result in a fall in the real rate of interest? Explain.
2. You’ve just stumbled on a new dataset that enables you to compute historical rates of return on
U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these
data to help estimate the expected rate of return on U.S. stocks over the coming year?
3. You are considering two alternative 2-year investments: You can invest in a risky asset with a
positive risk premium and returns in each of the 2 years that will be identically distributed and
uncorrelated, or you can invest in the risky asset for only 1 year and then invest the proceeds in
a risk-free asset. Which of the following statements about the first investment alternative (com-
pared with the second) are true?
a. Its 2-year risk premium is the same as the second alternative. b. The standard deviation of its 2-year return is the same. c. Its annualized standard deviation is lower. d. Its Sharpe ratio is higher. e. It is relatively more attractive to investors who have lower degrees of risk aversion.
4. You have $5,000 to invest for the next year and are considering three alternatives:
a. A money market fund with an average maturity of 30 days offering a current yield of 6% per year.
b. A 1-year savings deposit at a bank offering an interest rate of 7.5%. c. A 20-year U.S. Treasury bond offering a yield to maturity of 9% per year.
What role does your forecast of future interest rates play in your decisions?
5. Use Figure 5.1 in the text to analyze the effect of the following on the level of real interest rates:
a. Businesses become more pessimistic about future demand for their products and decide to reduce their capital spending.
b. Households are induced to save more because of increased uncertainty about their future Social Security benefits.
c. The Federal Reserve Board undertakes open-market purchases of U.S. Treasury securities in order to increase the supply of money.
Intermediate
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5. Characterize each company in the previous problem as underpriced, overpriced, or properly
priced.
6. What is the expected rate of return for a stock that has a beta of 1.0 if the expected return on the
market is 15%?
a. 15%. b. More than 15%. c. Cannot be determined without the risk-free rate.
7. Kaskin, Inc., stock has a beta of 1.2 and Quinn, Inc., stock has a beta of .6. Which of the follow-
ing statements is most accurate?
a. The expected rate of return will be higher for the stock of Kaskin, Inc., than that of Quinn, Inc. b. The stock of Kaskin, Inc., has more total risk than Quinn, Inc. c. The stock of Quinn, Inc., has more systematic risk than that of Kaskin, Inc.
8. You are a consultant to a large manufacturing corporation that is considering a project with the
following net after-tax cash flows (in millions of dollars):
Years from Now After-Tax Cash Flow
Intermediate
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SUMMARY AT THE END of each chapter, a detailed summary outlines the most important concepts presented. A listing of related Web sites for each chapter can also be found on the book’s Web site at www. mhhe.com/bkm . These sites make it easy for students to research topics further and retrieve financial data and information.
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1. Unit investment trusts, closed-end management companies, and open-end management compa- nies are all classified and regulated as investment companies. Unit investment trusts are essen-
tially unmanaged in the sense that the portfolio, once established, is fixed. Managed investment
companies, in contrast, may change the composition of the portfolio as deemed fit by the portfo-
lio manager. Closed-end funds are traded like other securities; they do not redeem shares for their
investors. Open-end funds will redeem shares for net asset value at the request of the investor.
2. Net asset value equals the market value of assets held by a fund minus the liabilities of the fund divided by the shares outstanding.
3. Mutual funds free the individual from many of the administrative burdens of owning individual securities and offer professional management of the portfolio. They also offer advantages that are
available only to large-scale investors, such as discounted trading costs. On the other hand, funds
are assessed management fees and incur other expenses, which reduce the investor’s rate of return.
Funds also eliminate some of the individual’s control over the timing of capital gains realizations.
4. Mutual funds are often categorized by investment policy. Major policy groups include money market funds; equity funds, which are further grouped according to emphasis on income versus
growth; fixed-income funds; balanced and income funds; asset allocation funds; index funds; and
specialized sector funds.
5. Costs of investing in mutual funds include front-end loads, which are sales charges; back-end loads, which are redemption fees or, more formally, contingent-deferred sales charges; fund oper-
ating expenses; and 12b-1 charges, which are recurring fees used to pay for the expenses of mar-
keting the fund to the public.
6. Income earned on mutual fund portfolios is not taxed at the level of the fund. Instead, as long as the fund meets certain requirements for pass-through status, the income is treated as being earned
by the investors in the fund.
SUMMARY
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xxiii
E-INVESTMENTS BOXES THESE EXERCISES PROVIDE students with simple activities to enhance their experi- ence using the Internet. Easy-to-follow instructions and questions are presented so students can utilize what they have learned in class and apply it to today’s Web-driven world.
EXCEL PROBLEMS SELECTED CHAPTERS CONTAIN prob- lems, denoted by an icon, specifically linked to Excel templates that are available on the book’s Web site at www.mhhe.com/bkm .
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49.25 500 54.75 300
49.00 200 58.25 100
48.50 600
a. If a market buy order for 100 shares comes in, at what price will it be filled? b. At what price would the next market buy order be filled? c. If you were a security dealer, would you want to increase or decrease your inventory of this stock?
9. You are bullish on Telecom stock. The current market price is $50 per share, and you have
$5,000 of your own to invest. You borrow an additional $5,000 from your broker at an interest
rate of 8% per year and invest $10,000 in the stock.
a. What will be your rate of return if the price of Telecom stock goes up by 10% during the next year? The stock currently pays no dividends.
b. How far does the price of Telecom stock have to fall for you to get a margin call if the main- tenance margin is 30%? Assume the price fall happens immediately.
10. You are bearish on Telecom and decide to sell short 100 shares at the current market price of
$50 per share.
a. How much in cash or securities must you put into your brokerage account if the broker’s initial margin requirement is 50% of the value of the short position?
b. How high can the price of the stock go before you get a margin call if the maintenance mar- gin is 30% of the value of the short position?
e X c e l Please visit us at www.mhhe.com/bkm
e X c e l Please visit us at www.mhhe.com/bkm
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E-INVESTMENTS EXERCISES The Federal Reserve Bank of St. Louis has information available on interest rates and eco- nomic conditions. A publication called Monetary Trends contains graphs and tables with information about current conditions in the capital markets. Go to the Web site www. stls.frb.org and click on Economic Research on the menu at the top of the page. Find the most recent issue of Monetary Trends in the Recent Data Publications section and answer these questions.
1. What is the professionals’ consensus forecast for inflation for the next 2 years? (Use the Federal Reserve Bank of Philadelphia line on the graph to answer this.)
2. What do consumers expect to happen to inflation over the next 2 years? (Use the University of Michigan line on the graph to answer this.)
3. Have real interest rates increased, decreased, or remained the same over the last 2 years?
4. What has happened to short-term nominal interest rates over the last 2 years? What about long-term nominal interest rates?
5. How do recent U.S. inflation and long-term interest rates compare with those of the other countries listed?
6. What are the most recently available levels of 3-month and 10-year yields on Treasury securities?
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a. McCracken was correct and Stiles was wrong. b. Both were correct. c. Stiles was correct and McCracken was wrong.
17. A ssume a universe of n (large) securities for which the largest residual variance is not larger than nsM
2 . Construct as many different weighting schemes as you can that generate well- diversified
portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT
security market line:
a. For a single-factor market. b. For a multifactor market.
19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor.
a. Explain why. b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit
of the merged company. Would you expect the stock market behavior of the merged firm to
differ from that of a portfolio of the two previously independent firms? How does the merger
affect market capitalization? What is the prediction of the Fama-French model for the risk
premium on the combined firm? Do we see here a flaw in the FF model?
Challenge
1. J effrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing
CAPM and APT, the consultant made the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio. b. Neither the CAPM nor APT assumes normally distributed security returns. c. The CAPM assumes that one specific factor explains security returns but APT does not.
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CFA PROBLEMS WE PROVIDE SEVERAL questions from past CFA examinations in applicable chapters. These questions represent the kinds of questions that professionals in the field believe are relevant to the “real world.” Located at the back of the book is a list- ing of each CFA question and the level and year of the CFA exam it was included in for easy reference when studying for the exam.
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xxiv
and feedback is provided and EZ Test’s grade book is
designed to export to your grade book.
• PowerPoint Presentation These presentation slides, also prepared by Anna Kovalenko, contain figures and
tables from the text, key points, and summaries in a
visually stimulating collection of slides that you can
customize to fit your lecture.
• Solutions Manual Updated by Marc-Anthony Isaacs, this Manual provides detailed solutions to the end-of-
chapter problem sets. This supplement is also available
for purchase by your students or can be packaged with
your text at a discount.
F O R T H E S T U D E N T
• Excel Templates are available for selected spread- sheets featured within the text, as well as those fea-
tured among the Excel Applications boxes. Selected
end-of-chapter problems have also been designated
as Excel problems, for which the available tem-
plate allows students to solve the problem and gain
experience using spreadsheets. Each template can
also be found on the book’s Web site www.mhhe. com/bkm .
• Related Web Sites A list of suggested Web sites is provided for each chapter. To keep Web addresses up-
to-date, the suggested sites as well as their links are
provided online. Each chapter summary contains a ref-
erence to its related sites.
• Online Quizzes These multiple-choice questions are provided as an additional testing and reinforcement
tool for students. Each quiz is organized by chapter to
test the specific concepts presented in that particular
chapter. Immediate scoring of the quiz occurs upon
submission and the correct answers are provided.
F O R T H E I N S T R U C T O R
Online Learning Center www.mhhe.com/bkm Find a wealth of information online! At this book’s Web
site instructors have access to teaching supports such as
electronic files of the ancillary materials. Students have
access to study materials created specifically for this text
and much more. All Excel spreadsheets, denoted by an
icon in the text are located at this site. Links to the addi-
tional support material are also included.
• Instructor’s Manual Prepared by Anna Kovalenko, Virginia Tech University, the Manual has been revised
and improved for this edition. Each chapter includes a
Chapter Overview, Learning Objectives, and Presenta-
tion of Material.
• Test Bank Prepared by John Farlin, Ohio Dominican University, the Test Bank has been revised to improve
the quality of questions. Each question is ranked by
level of difficulty, which allows greater flexibility in
creating a test and also provides a rationale for the
solution.
• Computerized Test Bank A comprehensive bank of test questions is provided within a computerized test
bank powered by McGraw-Hill’s flexible electronic
testing program EZ Test Online ( www.eztestonline. com ). You can select questions from multiple McGraw- Hill test banks or author your own, and then print the
test for paper distribution or give it online. This user-
friendly program allows you to sort questions by for-
mat, edit existing questions or add new ones, and
scramble questions for multiple versions of the same
test. You can export your tests for use in WebCT, Black-
board, PageOut, and Apple’s iQuiz. Sharing tests with
colleagues, adjuncts, and TAs is easy! Instant scoring
Supplements
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Supplements
xxv
improve student engagement in and out of class. You can
select and use any asset that enhances your lecture. The
Connect Finance Instructor Library includes all of the instructor supplements for this text.
Student Study Center The Connect Finance Stu- dent Study Center is the place for students to access addi-
tional resources. The Student Study Center:
• Offers students quick access to lectures, practice mate-
rials, eBooks, and more.
• Provides instant practice material and study questions,
easily accessible on the go.
Diagnostic and Adaptive Learning of Concepts: LearnSmart Students want to make the best use of their study time. The LearnSmart adaptive self-study tech-
nology within Connect Finance provides students with a seamless combination of practice, assessment, and reme-
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• Adapts automatically to each student, so students spend
less time on the topics they understand and practice
more those they have yet to master.
• Provides continual reinforcement and remediation but
gives only as much guidance as students need.
• Integrates diagnostics as part of the learning experience.
• Enables you to assess which concepts students have
efficiently learned on their own, thus freeing class time
for more applications and discussion.
Student Progress Tracking Connect Finance keeps instructors informed about how each student, section, and
class is performing, allowing for more productive use of
lecture and office hours. The progress-tracking function
enables you to:
• View scored work immediately and track individual or
group performance with assignment and grade reports.
• Access an instant view of student or class performance
relative to learning objectives.
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MCGRAW-HILL’S CONNECT FINANCE
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McGraw-Hill’s Connect Finance is an online assignment and assessment solution that connects students with the
tools and resources they’ll need to achieve success.
McGraw-Hill’s Connect Finance helps prepare students for their future by enabling faster learning, more efficient
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McGraw-Hill’s Connect Finance Features Connect Finance offers a number of powerful tools and features to make managing assignments easier, so faculty
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and efficient. Connect Finance offers you the features described below.
Simple Assignment Management With Connect Finance, creating assignments is easier than ever, so you can spend more time teaching and less time managing.
The assignment management function enables you to:
• Create and deliver assignments easily with selectable
end-of-chapter questions and test bank items.
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Smart Grading When it comes to studying, time is pre- cious. Connect Finance helps students learn more efficiently by providing feedback and practice material when they need
it, where they need it. When it comes to teaching, your time
also is precious. The grading function enables you to:
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dents immediate feedback on their work and side-by-
side comparisons with correct answers.
• Access and review each response; manually change
grades or leave comments for students to review.
• Reinforce classroom concepts with practice tests and
instant quizzes.
Instructor Library The Connect Finance Instruc- tor Library is your repository for additional resources to
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Supplements
xxvi
connecting selected questions in the text and the test bank
to the six general knowledge and skill guidelines in the
AACSB standards.
The statements contained in Investments Tenth Edition are provided only as a guide for the users of this textbook.
The AACSB leaves content coverage and assessment
within the purview of individual schools, the mission
of the school, and the faculty. While Investments Tenth Edition and the teaching package make no claim of any
specific AACSB qualification or evaluation, within this
edition we have labeled selected questions according to
the six general knowledge and skills areas.
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tance and value of AACSB accreditation, Investments Tenth Edition recognizes the curricula guidelines detailed
in the AACSB standards for business accreditation by
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xxvii
Mahmoud Haddad
Wayne State University
Greg Hallman
University of Texas at Austin
Robert G. Hansen
Dartmouth College
Joel Hasbrouck
New York University
Andrea Heuson
University of Miami
Eric Higgins
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Shalom J. Hochman
University of Houston
Stephen Huffman
University of Wisconsin at Oshkosh
Eric Hughson
University of Colorado
Delroy Hunter
University of South Florida
A. James Ifflander
A. James Ifflander and Associates
Robert Jennings
Indiana University
George Jiang
University of Arizona
Richard D. Johnson
Colorado State University
Susan D. Jordan
University of Kentucky
G. Andrew Karolyi
Ohio State University
Ajay Khorana
Georgia Institute of Technology
Anna Kovalenko
Virginia Tech University
Josef Lakonishok
University of Illinois at Champaign/Urbana
Anna Craig
Emory University
Elton Daal
University of New Orleans
David C. Distad
University of California at Berkeley
Craig Dunbar
University of Western Ontario
David Durr
Murray State University
Bjorn Eaker
Duke University
John Earl
University of Richmond
Michael C. Ehrhardt
University of Tennessee at Knoxville
Venkat Eleswarapu
Southern Methodist University
David Ellis
Babson College
Andrew Ellul
Indiana University
John Fay
Santa Clara University
Greg Filbeck
University of Toledo
James Forjan
York College of Pennsylvania
David Gallagher
University of Technology, Sydney
Jeremy Goh
Washington University
Richard Grayson
Loyola College
John M. Griffin
Arizona State University
Weiyu Guo
University of Nebraska at Omaha
Throughout the development of this text,
experienced instructors have provided crit-
ical feedback and suggestions for improve-
ment. These individuals deserve a special
thanks for their valuable insights and con-
tributions. The following instructors played
a vital role in the development of this and
previous editions of Investments:
J. Amanda Adkisson
Texas A&M University
Sandro Andrade
University of Miami at Coral Gables
Tor-Erik Bakke
University of Wisconsin
Richard J. Bauer Jr.
St. Mary’s University
Scott Besley
University of Florida
John Binder
University of Illinois at Chicago
Paul Bolster
Northwestern University
Phillip Braun
University of Chicago
Leo Chan
Delaware State University
Charles Chang
Cornell University
Kee Chaung
SUNY Buffalo
Ludwig Chincarini
Pomona College
Stephen Ciccone
University of New Hampshire
James Cotter
Wake Forest University
L. Michael Couvillion
Plymouth State University
Acknowledgments
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xxviii
Acknowledgments
Charles A. Trzincka
SUNY Buffalo
Yiuman Tse
Binghamton SUNY
Joe Ueng
University of St. Thomas
Gopala Vasuderan
Suffolk University
Joseph Vu
DePaul University
Qinghai Wang
Georgia Institute of Technology
Richard Warr
North Carolina State University
Simon Wheatley
University of Chicago
Marilyn K. Wiley
Florida Atlantic University
James Williams
California State University at Northridge
Michael Williams
University of Denver
Tony R. Wingler
University of North Carolina at Greensboro
Guojun Wu
University of Michigan
Hsiu-Kwang Wu
University of Alabama
Geungu Yu
Jackson State University
Thomas J. Zwirlein
University of Colorado at Colorado Springs
Edward Zychowicz
Hofstra University
For granting us permission to include
many of their examination questions in the
text, we are grateful to the CFA Institute.
Much credit is due to the development
and production team at McGraw-Hill/
Irwin: our special thanks go to Noelle
Bathurst, Development Editor; Chuck
Synovec, Executive Brand Manager; Bruce
Gin, Content Project Manager; Melissa
Caughlin, Senior Marketing Manager;
Jennifer Jelinski, Marketing Specialist;
Michael McCormick, Senior Buyer; and
Debra Kubiak, Designer.
Finally, we thank Judy, Hava, and
Sheryl, who contribute to the book with
their support and understanding.
Zvi Bodie Alex Kane
Alan J. Marcus
Andrew Prevost
Ohio University
Herbert Quigley
University of the District of Columbia
Murli Rajan
University of Scranton
Speima Rao
University of Southwestern Louisiana
Rathin Rathinasamy
Ball State University
William Reese
Tulane University
Craig Rennie
University of Arkansas
Maurico Rodriquez
Texas Christian University
Leonard Rosenthal
Bentley College
Anthony Sanders
Ohio State University
Gary Sanger
Louisiana State University
Don Seeley
University of Arizona
John Settle
Portland State University
Edward C. Sims
Western Illinois University
Robert Skena
Carnegie Mellon University
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University of North Carolina at Chapel Hill
Keith V. Smith
Purdue University
Patricia B. Smith
University of New Hampshire
Ahmad Sohrabian
California State Polytechnic University– Pomova
Eileen St. Pierre
University of Northern Colorado
Laura T. Starks
University of Texas
Mick Swartz
University of Southern California
Manuel Tarrazo
University of San Francisco
Steve Thorley
Brigham Young University
Ashish Tiwari
University of Iowa
Jack Treynor
Treynor Capital Management
Malek Lashgari
University of Hartford
Dennis Lasser
Binghamton SUNY
Hongbok Lee
Western Illinois University
Bruce Lehmann
University of California at San Diego
Jack Li
Northeastern University
Larry Lockwood
Texas Christian University
Christopher K. Ma
Texas Tech University
Anil K. Makhija
University of Pittsburgh
Davinder Malhotra
Philadelphia University
Steven Mann
University of South Carolina
Deryl W. Martin
Tennessee Technical University
Jean Masson
University of Ottawa
Ronald May
St. John’s University
William McDonald
University of Notre Dame
Rick Meyer
University of South Florida
Bruce Mizrach
Rutgers University at New Brunswick
Mbodja Mougoue
Wayne State University
Kyung-Chun (Andrew) Mun
Truman State University
Carol Osler
Brandeis University
Gurupdesh Pandner
DePaul University
Don B. Panton
University of Texas at Arlington
Dimitris Papanikolaou
Northwestern University
Dilip Patro
Rutgers University
Robert Pavlik
Southwest Texas State
Marianne Plunkert
University of Colorado at Denver
Jeffrey Pontiff
Boston College
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1 1 PA
R T I
AN INVESTMENT IS the current commitment of money or other resources in the expecta- tion of reaping future benefits. For example, an individual might purchase shares of stock anticipating that the future proceeds from the shares will justify both the time that her money is tied up as well as the risk of the investment. The time you will spend studying this text (not to mention its cost) also is an investment. You are forgoing either current leisure or the income you could be earning at a job in the expectation that your future career will be suf- ficiently enhanced to justify this commitment of time and effort. While these two invest- ments differ in many ways, they share one key attribute that is central to all investments: You sacrifice something of value now, expecting to benefit from that sacrifice later.
This text can help you become an informed practitioner of investments. We will focus on investments in securities such as stocks, bonds, or options and futures contracts, but much of what we discuss will be useful in the analysis of any type of investment. The text will pro- vide you with background in the organization of various securities markets; will survey the valuation and risk-management principles useful in particular markets, such as those for bonds or stocks; and will introduce you to the principles of portfolio construction.
Broadly speaking, this chapter addresses three topics that will provide a useful perspec- tive for the material that is to come later. First, before delving into the topic of “investments,” we consider the role of financial assets in the economy. We discuss the relationship between securities and the “real” assets that actually produce goods and services for consumers, and we consider why financial assets are important to the functioning of a developed economy.
Given this background, we then take a first look at the types of decisions that con- front investors as they assemble a portfolio of assets. These investment decisions are made in an environment where higher returns usually can be obtained only at the price of greater risk and in which it is rare to find assets that are so mispriced as to be obvi- ous bargains. These themes—the risk–return trade-off and the efficient pricing of financial assets—are central to the investment process, so it is worth pausing for a brief discussion of their implications as we begin the text. These implications will be fleshed out in much greater detail in later chapters.
We provide an overview of the organiza- tion of security markets as well as the vari- ous players that participate in those markets. Together, these introductions should give you a feel for who the major participants are in
The Investment
Environment
CHAPTER ONE
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2
(concluded)
the securities markets as well as the setting in which they act. Finally, we discuss the financial crisis that began playing out in 2007 and peaked in 2008. The crisis dramatically illustrated the
connections between the financial system and the “real” side of the economy. We look at the origins of the crisis and the lessons that may be drawn about systemic risk. We close the chapter with an overview of the remainder of the text.
1 You might wonder why real assets held by households in Table 1.1 amount to $23,774 billion, while total real
assets in the domestic economy ( Table 1.2 ) are far larger, at $48,616 billion. A big part of the difference reflects
the fact that real assets held by firms, for example, property, plant, and equipment, are included as financial assets of the household sector, specifically through the value of corporate equity and other stock market investments.
Also, Table 1.2 includes assets of noncorporate businesses. Finally, there are some differences in valuation meth-
ods. For example, equity and stock investments in Table 1.1 are measured by market value, whereas plant and
equipment in Table 1.2 are valued at replacement cost.
The material wealth of a society is ultimately determined by the productive capacity of its
economy, that is, the goods and services its members can create. This capacity is a function
of the real assets of the economy: the land, buildings, machines, and knowledge that can be used to produce goods and services.
In contrast to real assets are financial assets such as stocks and bonds. Such securi- ties are no more than sheets of paper or, more likely, computer entries, and they do not
contribute directly to the productive capacity of the economy. Instead, these assets are the
means by which individuals in well-developed economies hold their claims on real assets.
Financial assets are claims to the income generated by real assets (or claims on income
from the government). If we cannot own our own auto plant (a real asset), we can still buy
shares in Ford or Toyota (financial assets) and thereby share in the income derived from the
production of automobiles.
While real assets generate net income to the economy, financial assets simply define the
allocation of income or wealth among investors. Individuals can choose between consum-
ing their wealth today or investing for the future. If they choose to invest, they may place
their wealth in financial assets by purchasing various securities. When investors buy these
securities from companies, the firms use the money so raised to pay for real assets, such as
plant, equipment, technology, or inventory. So investors’ returns on securities ultimately
come from the income produced by the real assets that were financed by the issuance of
those securities.
The distinction between real and financial assets is apparent when we compare the bal-
ance sheet of U.S. households, shown in Table 1.1 , with the composition of national wealth
in the United States, shown in Table 1.2 . Household wealth includes financial assets such as
bank accounts, corporate stock, or bonds. However, these
securities, which are financial assets of households, are
liabilities of the issuers of the securities. For example, a bond that you treat as an asset because it gives you a
claim on interest income and repayment of principal from
Toyota is a liability of Toyota, which is obligated to make
these payments to you. Your asset is Toyota’s liability.
Therefore, when we aggregate over all balance sheets,
these claims cancel out, leaving only real assets as the net
wealth of the economy. National wealth consists of struc-
tures, equipment, inventories of goods, and land. 1
1.1 Real Assets versus Financial Assets
Are the following assets real or financial?
a. Patents
b. Lease obligations
c. Customer goodwill
d. A college education
e. A $5 bill
CONCEPT CHECK 1.1
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C H A P T E R 1 The Investment Environment 3
We will focus almost exclusively on financial assets. But you shouldn’t lose sight of the
fact that the successes or failures of the financial assets we choose to purchase ultimately
depend on the performance of the underlying real assets.
Assets $ Billion % Total Liabilities and Net Worth $ Billion % Total
Real assets Liabilities Real estate $18,608 24.4% Mortgages $ 9,907 13.0% Consumer durables 4,821 6.3 Consumer credit 2,495 3.3 Other 345 0.5 Bank and other loans 195 0.3
Total real assets $23,774 31.2% Security credit 268 0.4 Other 568 0.7
Total liabilities $13,433 17.6% Financial assets Deposits $ 8,688 11.4% Life insurance reserves 1,203 1.6 Pension reserves 13,950 18.3 Corporate equity 9,288 12.2 Equity in noncorp. business 7,443 9.8 Mutual fund shares 5,191 6.8 Debt securities 5,120 6.7 Other 1,641 2.2
Total financial assets 52,524 68.8 Net worth 62,866 82.4
Total 76,298 100.0% $76,298 100.0%
Table 1.1
Balance sheet of U.S. households Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.
Assets $ Billion
Commercial real estate $12,781 Residential real estate 23,460 Equipment and software 5,261 Inventories 2,293 Consumer durables 4,821
Total $48,616
Table 1.2
Domestic net worth
Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.
1.2 Financial Assets
It is common to distinguish among three broad types of financial assets: fixed income,
equity, and derivatives. Fixed-income or debt securities promise either a fixed stream of income or a stream of income determined by a specified formula. For example, a corporate
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4 P A R T I Introduction
bond typically would promise that the bondholder will receive a fixed amount of interest
each year. Other so-called floating-rate bonds promise payments that depend on current
interest rates. For example, a bond may pay an interest rate that is fixed at 2 percentage
points above the rate paid on U.S. Treasury bills. Unless the borrower is declared bankrupt,
the payments on these securities are either fixed or determined by formula. For this reason,
the investment performance of debt securities typically is least closely tied to the financial
condition of the issuer.
Nevertheless, fixed-income securities come in a tremendous variety of maturities and
payment provisions. At one extreme, the money market refers to debt securities that are short term, highly marketable, and generally of very low risk. Examples of money market
securities are U.S. Treasury bills or bank certificates of deposit (CDs). In contrast, the
fixed-income capital market includes long-term securities such as Treasury bonds, as well as bonds issued by federal agencies, state and local municipalities, and corporations. These
bonds range from very safe in terms of default risk (for example, Treasury securities) to
relatively risky (for example, high-yield or “junk” bonds). They also are designed with
extremely diverse provisions regarding payments provided to the investor and protection
against the bankruptcy of the issuer. We will take a first look at these securities in Chapter 2
and undertake a more detailed analysis of the debt market in Part Four.
Unlike debt securities, common stock, or equity, in a firm represents an ownership share in the corporation. Equityholders are not promised any particular payment. They
receive any dividends the firm may pay and have prorated ownership in the real assets of
the firm. If the firm is successful, the value of equity will increase; if not, it will decrease.
The performance of equity investments, therefore, is tied directly to the success of the firm
and its real assets. For this reason, equity investments tend to be riskier than investments in
debt securities. Equity markets and equity valuation are the topics of Part Five.
Finally, derivative securities such as options and futures contracts provide payoffs that are determined by the prices of other assets such as bond or stock prices. For example, a call option on a share of Intel stock might turn out to be worthless if Intel’s share price
remains below a threshold or “exercise” price such as $20 a share, but it can be quite valu-
able if the stock price rises above that level. 2 Derivative securities are so named because
their values derive from the prices of other assets. For example, the value of the call option
will depend on the price of Intel stock. Other important derivative securities are futures and
swap contracts. We will treat these in Part Six.
Derivatives have become an integral part of the investment environment. One use of
derivatives, perhaps the primary use, is to hedge risks or transfer them to other parties.
This is done successfully every day, and the use of these securities for risk management is
so commonplace that the multitrillion-dollar market in derivative assets is routinely taken
for granted. Derivatives also can be used to take highly speculative positions, however.
Every so often, one of these positions blows up, resulting in well-publicized losses of
hundreds of millions of dollars. While these losses attract considerable attention, they are
in fact the exception to the more common use of such securities as risk management tools.
Derivatives will continue to play an important role in portfolio construction and the finan-
cial system. We will return to this topic later in the text.
Investors and corporations regularly encounter other financial markets as well. Firms
engaged in international trade regularly transfer money back and forth between dollars and
2 A call option is the right to buy a share of stock at a given exercise price on or before the option’s expiration
date. If the market price of Intel remains below $20 a share, the right to buy for $20 will turn out to be valueless.
If the share price rises above $20 before the option expires, however, the option can be exercised to obtain the
share for only $20.
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C H A P T E R 1 The Investment Environment 5
other currencies. Well more than a trillion dollars of currency is traded each day in the mar-
ket for foreign exchange, primarily through a network of the largest international banks.
Investors also might invest directly in some real assets. For example, dozens of commod-
ities are traded on exchanges such as the New York Mercantile Exchange or the Chicago
Board of Trade. You can buy or sell corn, wheat, natural gas, gold, silver, and so on.
Commodity and derivative markets allow firms to adjust their exposure to various busi-
ness risks. For example, a construction firm may lock in the price of copper by buying
copper futures contracts, thus eliminating the risk of a sudden jump in the price of its raw
materials. Wherever there is uncertainty, investors may be interested in trading, either to
speculate or to lay off their risks, and a market may arise to meet that demand.
1.3 Financial Markets and the Economy
We stated earlier that real assets determine the wealth of an economy, while financial assets
merely represent claims on real assets. Nevertheless, financial assets and the markets in
which they trade play several crucial roles in developed economies. Financial assets allow
us to make the most of the economy’s real assets.
The Informational Role of Financial Markets Stock prices reflect investors’ collective assessment of a firm’s current performance and
future prospects. When the market is more optimistic about the firm, its share price will
rise. That higher price makes it easier for the firm to raise capital and therefore encour-
ages investment. In this manner, stock prices play a major role in the allocation of capi-
tal in market economies, directing capital to the firms and applications with the greatest
perceived potential.
Do capital markets actually channel resources to the most efficient use? At times, they
appear to fail miserably. Companies or whole industries can be “hot” for a period of time
(think about the dot-com bubble that peaked in 2000), attract a large flow of investor capi-
tal, and then fail after only a few years. The process seems highly wasteful.
But we need to be careful about our standard of efficiency. No one knows with certainty
which ventures will succeed and which will fail. It is therefore unreasonable to expect that
markets will never make mistakes. The stock market encourages allocation of capital to
those firms that appear at the time to have the best prospects. Many smart, well-trained, and well-paid professionals analyze the prospects of firms whose shares trade on the stock
market. Stock prices reflect their collective judgment.
You may well be skeptical about resource allocation through markets. But if you are,
then take a moment to think about the alternatives. Would a central planner make fewer
mistakes? Would you prefer that Congress make these decisions? To paraphrase Winston
Churchill’s comment about democracy, markets may be the worst way to allocate capital
except for all the others that have been tried.
Consumption Timing Some individuals in an economy are earning more than they currently wish to spend.
Others, for example, retirees, spend more than they currently earn. How can you shift your
purchasing power from high-earnings periods to low-earnings periods of life? One way is
to “store” your wealth in financial assets. In high-earnings periods, you can invest your
savings in financial assets such as stocks and bonds. In low-earnings periods, you can sell
these assets to provide funds for your consumption needs. By so doing, you can “shift”
your consumption over the course of your lifetime, thereby allocating your consumption to
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6 P A R T I Introduction
periods that provide the greatest satisfaction. Thus, financial markets allow individuals to
separate decisions concerning current consumption from constraints that otherwise would
be imposed by current earnings.
Allocation of Risk Virtually all real assets involve some risk. When Ford builds its auto plants, for example, it
cannot know for sure what cash flows those plants will generate. Financial markets and the
diverse financial instruments traded in those markets allow investors with the greatest taste
for risk to bear that risk, while other, less risk-tolerant individuals can, to a greater extent,
stay on the sidelines. For example, if Ford raises the funds to build its auto plant by selling
both stocks and bonds to the public, the more optimistic or risk-tolerant investors can buy
shares of its stock, while the more conservative ones can buy its bonds. Because the bonds
promise to provide a fixed payment, the stockholders bear most of the business risk but
reap potentially higher rewards. Thus, capital markets allow the risk that is inherent to all
investments to be borne by the investors most willing to bear that risk.
This allocation of risk also benefits the firms that need to raise capital to finance their
investments. When investors are able to select security types with the risk-return character-
istics that best suit their preferences, each security can be sold for the best possible price.
This facilitates the process of building the economy’s stock of real assets.
Separation of Ownership and Management Many businesses are owned and managed by the same individual. This simple organiza-
tion is well suited to small businesses and, in fact, was the most common form of business
organization before the Industrial Revolution. Today, however, with global markets and
large-scale production, the size and capital requirements of firms have skyrocketed. For
example, in 2012 General Electric listed on its balance sheet about $70 billion of property,
plant, and equipment, and total assets of $685 billion. Corporations of such size simply
cannot exist as owner-operated firms. GE actually has more than half a million stockhold-
ers with an ownership stake in the firm proportional to their holdings of shares.
Such a large group of individuals obviously cannot actively participate in the day-to-
day management of the firm. Instead, they elect a board of directors that in turn hires and
supervises the management of the firm. This structure means that the owners and manag-
ers of the firm are different parties. This gives the firm a stability that the owner-managed
firm cannot achieve. For example, if some stockholders decide they no longer wish to hold
shares in the firm, they can sell their shares to other investors, with no impact on the man-
agement of the firm. Thus, financial assets and the ability to buy and sell those assets in the
financial markets allow for easy separation of ownership and management.
How can all of the disparate owners of the firm, ranging from large pension funds hold-
ing hundreds of thousands of shares to small investors who may hold only a single share,
agree on the objectives of the firm? Again, the financial markets provide some guidance.
All may agree that the firm’s management should pursue strategies that enhance the value
of their shares. Such policies will make all shareholders wealthier and allow them all to
better pursue their personal goals, whatever those goals might be.
Do managers really attempt to maximize firm value? It is easy to see how they might
be tempted to engage in activities not in the best interest of shareholders. For example,
they might engage in empire building or avoid risky projects to protect their own jobs or
overconsume luxuries such as corporate jets, reasoning that the cost of such perquisites is
largely borne by the shareholders. These potential conflicts of interest are called agency problems because managers, who are hired as agents of the shareholders, may pursue their own interests instead.
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C H A P T E R 1 The Investment Environment 7
Several mechanisms have evolved to mitigate potential agency problems. First, com-
pensation plans tie the income of managers to the success of the firm. A major part of the
total compensation of top executives is often in the form of stock or stock options, which
means that the managers will not do well unless the stock price increases, benefiting share-
holders. (Of course, we’ve learned more recently that overuse of options can create its own
agency problem. Options can create an incentive for managers to manipulate information
to prop up a stock price temporarily, giving them a chance to cash out before the price
returns to a level reflective of the firm’s true prospects. More on this shortly.) Second,
while boards of directors have sometimes been portrayed as defenders of top management,
they can, and in recent years, increasingly have, forced out management teams that are
underperforming. The average tenure of CEOs fell from 8.1 years in 2006 to 6.6 years in
2011, and the percentage of incoming CEOs who also serve as chairman of the board of
directors fell from 48% in 2002 to less than 12% in 2009. 3 Third, outsiders such as security
analysts and large institutional investors such as mutual funds or pension funds monitor the
firm closely and make the life of poor performers at the least uncomfortable. Such large
investors today hold about half of the stock in publicly listed firms in the U.S.
Finally, bad performers are subject to the threat of takeover. If the board of directors is lax
in monitoring management, unhappy shareholders in principle can elect a different board.
They can do this by launching a proxy contest in which they seek to obtain enough proxies (i.e., rights to vote the shares of other shareholders) to take control of the firm and vote in
another board. However, this threat is usually minimal. Shareholders who attempt such a
fight have to use their own funds, while management can defend itself using corporate cof-
fers. Most proxy fights fail. The real takeover threat is from other firms. If one firm observes
another underperforming, it can acquire the underperforming business and replace manage-
ment with its own team. The stock price should rise to reflect the prospects of improved
performance, which provides incentive for firms to engage in such takeover activity.
3 “Corporate Bosses Are Much Less Powerful than They Used To Be,” The Economist, January 21, 2012.
In February 2008, Microsoft offered to buy Yahoo! by paying its current shareholders $31 for each of their shares, a considerable premium to its closing price of $19.18 on the day before the offer. Yahoo’s management rejected that offer and a better one at $33 a share; Yahoo’s CEO Jerry Yang held out for $37 per share, a price that Yahoo! had not reached in more than 2 years. Billionaire investor Carl Icahn was outraged, arguing that management was protecting its own position at the expense of shareholder value. Icahn notified Yahoo! that he had been asked to “lead a proxy fight to attempt to remove the current board and to establish a new board which would attempt to negoti- ate a successful merger with Microsoft.” To that end, he had purchased approximately 59 million shares of Yahoo! and formed a 10-person slate to stand for election against the current board. Despite this challenge, Yahoo’s management held firm in its refusal of Microsoft’s offer, and with the support of the board, Yang managed to fend off both Microsoft and Icahn. In July, Icahn agreed to end the proxy fight in return for three seats on the board to be held by his allies. But the 11-person board was still dominated by current Yahoo management. Yahoo’s share price, which had risen to $29 a share during the Microsoft negotiations, fell back to around $21 a share. Given the difficulty that a well-known billionaire faced in defeating a determined and entrenched management, it is no wonder that proxy contests are rare. Historically, about three of four proxy fights go down to defeat.
Example 1.1 Carl Icahn’s Proxy Fight with Yahoo!
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8 P A R T I Introduction
Corporate Governance and Corporate Ethics We’ve argued that securities markets can play an important role in facilitating the deploy-
ment of capital resources to their most productive uses. But market signals will help to
allocate capital efficiently only if investors are acting on accurate information. We say that
markets need to be transparent for investors to make informed decisions. If firms can mis- lead the public about their prospects, then much can go wrong.
Despite the many mechanisms to align incentives of shareholders and managers, the
three years from 2000 through 2002 were filled with a seemingly unending series of scan-
dals that collectively signaled a crisis in corporate governance and ethics. For example,
the telecom firm WorldCom overstated its profits by at least $3.8 billion by improperly
classifying expenses as investments. When the true picture emerged, it resulted in the
largest bankruptcy in U.S. history, at least until Lehman Brothers smashed that record in
2008. The next-largest U.S. bankruptcy was Enron, which used its now-notorious “special-
purpose entities” to move debt off its own books and similarly present a misleading picture
of its financial status. Unfortunately, these firms had plenty of company. Other firms such
as Rite Aid, HealthSouth, Global Crossing, and Qwest Communications also manipulated
and misstated their accounts to the tune of billions of dollars. And the scandals were hardly
limited to the United States. Parmalat, the Italian dairy firm, claimed to have a $4.8 billion
bank account that turned out not to exist. These episodes suggest that agency and incentive
problems are far from solved.
Other scandals of that period included systematically misleading and overly optimistic
research reports put out by stock market analysts. (Their favorable analysis was traded for
the promise of future investment banking business, and analysts were commonly compen-
sated not for their accuracy or insight, but for their role in garnering investment banking
business for their firms.) Additionally, initial public offerings were allocated to corporate
executives as a quid pro quo for personal favors or the promise to direct future business
back to the manager of the IPO.
What about the auditors who were supposed to be the watchdogs of the firms? Here
too, incentives were skewed. Recent changes in business practice had made the consulting
businesses of these firms more lucrative than the auditing function. For example, Enron’s
(now-defunct) auditor Arthur Andersen earned more money consulting for Enron than by
auditing it; given Arthur Andersen’s incentive to protect its consulting profits, we should
not be surprised that it, and other auditors, were overly lenient in their auditing work.
In 2002, in response to the spate of ethics scandals, Congress passed the Sarbanes-Oxley
Act to tighten the rules of corporate governance. For example, the act requires corporations
to have more independent directors, that is, more directors who are not themselves manag-
ers (or affiliated with managers). The act also requires each CFO to personally vouch for
the corporation’s accounting statements, created an oversight board to oversee the audit-
ing of public companies, and prohibits auditors from providing various other services to
clients.
1.4 The Investment Process
An investor’s portfolio is simply his collection of investment assets. Once the portfolio is established, it is updated or “rebalanced” by selling existing securities and using the
proceeds to buy new securities, by investing additional funds to increase the overall size of
the portfolio, or by selling securities to decrease the size of the portfolio.
Investment assets can be categorized into broad asset classes, such as stocks, bonds, real
estate, commodities, and so on. Investors make two types of decisions in constructing their
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C H A P T E R 1 The Investment Environment 9
portfolios. The asset allocation decision is the choice among these broad asset classes, while the security selection decision is the choice of which particular securities to hold within each asset class.
Asset allocation also includes the decision of how much of one’s portfolio to place
in safe assets such as bank accounts or money market securities versus in risky assets.
Unfortunately, many observers, even those providing financial advice, appear to incor-
rectly equate saving with safe investing. 4 “Saving” means that you do not spend all of your
current income, and therefore can add to your portfolio. You may choose to invest your
savings in safe assets, risky assets, or a combination of both.
“Top-down” portfolio construction starts with asset allocation. For example, an individ-
ual who currently holds all of his money in a bank account would first decide what propor-
tion of the overall portfolio ought to be moved into stocks, bonds, and so on. In this way,
the broad features of the portfolio are established. For example, while the average annual
return on the common stock of large firms since 1926 has been better than 11% per year,
the average return on U.S. Treasury bills has been less than 4%. On the other hand, stocks
are far riskier, with annual returns (as measured by the Standard & Poor’s 500 index) that
have ranged as low as –46% and as high as 55%. In contrast, T-bills are effectively risk-
free: You know what interest rate you will earn when you buy them. Therefore, the decision
to allocate your investments to the stock market or to the money market where Treasury
bills are traded will have great ramifications for both the risk and the return of your portfo-
lio. A top-down investor first makes this and other crucial asset allocation decisions before
turning to the decision of the particular securities to be held in each asset class.
Security analysis involves the valuation of particular securities that might be included in the portfolio. For example, an investor might ask whether Merck or Pfizer is more attrac-
tively priced. Both bonds and stocks must be evaluated for investment attractiveness, but
valuation is far more difficult for stocks because a stock’s performance usually is far more
sensitive to the condition of the issuing firm.
In contrast to top-down portfolio management is the “bottom-up” strategy. In this pro-
cess, the portfolio is constructed from the securities that seem attractively priced without
as much concern for the resultant asset allocation. Such a technique can result in unin-
tended bets on one or another sector of the economy. For example, it might turn out that
the portfolio ends up with a very heavy representation of firms in one industry, from one
part of the country, or with exposure to one source of uncertainty. However, a bottom-up
strategy does focus the portfolio on the assets that seem to offer the most attractive invest-
ment opportunities.
4 For example, here is a brief excerpt from the Web site of the Securities and Exchange Commission. “Your
‘savings’ are usually put into the safest places or products . . . When you ‘invest,’ you have a greater chance of
losing your money than when you ‘save.’” This statement is incorrect: Your investment portfolio can be invested
in either safe or risky assets, and your savings in any period is simply the difference between your income and
consumption.
1.5 Markets Are Competitive
Financial markets are highly competitive. Thousands of intelligent and well-backed ana-
lysts constantly scour securities markets searching for the best buys. This competition
means that we should expect to find few, if any, “free lunches,” securities that are so under-
priced that they represent obvious bargains. This no-free-lunch proposition has several
implications. Let’s examine two.
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10 P A R T I Introduction
The Risk–Return Trade-Off Investors invest for anticipated future returns, but those returns rarely can be predicted
precisely. There will almost always be risk associated with investments. Actual or real-
ized returns will almost always deviate from the expected return anticipated at the start of
the investment period. For example, in 1931 (the worst calendar year for the market since
1926), the S&P 500 index fell by 46%. In 1933 (the best year), the index gained 55%. You
can be sure that investors did not anticipate such extreme performance at the start of either
of these years.
Naturally, if all else could be held equal, investors would prefer investments with the
highest expected return. 5 However, the no-free-lunch rule tells us that all else cannot be
held equal. If you want higher expected returns, you will have to pay a price in terms of
accepting higher investment risk. If higher expected return can be achieved without bear-
ing extra risk, there will be a rush to buy the high-return assets, with the result that their
prices will be driven up. Individuals considering investing in the asset at the now-higher
price will find the investment less attractive: If you buy at a higher price, your expected
rate of return (that is, profit per dollar invested) is lower. The asset will be considered
attractive and its price will continue to rise until its expected return is no more than com-
mensurate with risk. At this point, investors can anticipate a “fair” return relative to the
asset’s risk, but no more. Similarly, if returns were independent of risk, there would be
a rush to sell high-risk assets. Their prices would fall (and their expected future rates of
return rise) until they eventually were attractive enough to be included again in investor
portfolios. We conclude that there should be a risk–return trade-off in the securities markets, with higher-risk assets priced to offer higher expected returns than lower-risk
assets.
Of course, this discussion leaves several important questions unanswered. How should
one measure the risk of an asset? What should be the quantitative trade-off between risk
(properly measured) and expected return? One would think that risk would have some-
thing to do with the volatility of an asset’s returns, but this guess turns out to be only
partly correct. When we mix assets into diversified portfolios, we need to consider the
interplay among assets and the effect of diversification on the risk of the entire portfolio.
Diversification means that many assets are held in the portfolio so that the exposure to any particular asset is limited. The effect of diversification on portfolio risk, the implica-
tions for the proper measurement of risk, and the risk–return relationship are the topics of
Part Two. These topics are the subject of what has come to be known as modern portfolio theory. The development of this theory brought two of its pioneers, Harry Markowitz and William Sharpe, Nobel Prizes.
Efficient Markets Another implication of the no-free-lunch proposition is that we should rarely expect to find
bargains in the security markets. We will spend all of Chapter 11 examining the theory and
evidence concerning the hypothesis that financial markets process all available infor-
mation about securities quickly and efficiently, that is, that the security price usually
reflects all the information available to investors concerning its value. According to this
hypothesis, as new information about a security becomes available, its price quickly
5 The “expected” return is not the return investors believe they necessarily will earn, or even their most likely
return. It is instead the result of averaging across all possible outcomes, recognizing that some outcomes are more
likely than others. It is the average rate of return across possible economic scenarios.
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C H A P T E R 1 The Investment Environment 11
adjusts so that at any time, the security price equals the market consensus estimate of the
value of the security. If this were so, there would be neither underpriced nor overpriced
securities.
One interesting implication of this “efficient market hypothesis” concerns the choice
between active and passive investment-management strategies. Passive management calls for holding highly diversified portfolios without spending effort or other resources attempt-
ing to improve investment performance through security analysis. Active management is the attempt to improve performance either by identifying mispriced securities or by timing
the performance of broad asset classes—for example, increasing one’s commitment to
stocks when one is bullish on the stock market. If markets are efficient and prices reflect
all relevant information, perhaps it is better to follow passive strategies instead of spending
resources in a futile attempt to outguess your competitors in the financial markets.
If the efficient market hypothesis were taken to the extreme, there would be no point in
active security analysis; only fools would commit resources to actively analyze securities.
Without ongoing security analysis, however, prices eventually would depart from “correct”
values, creating new incentives for experts to move in. Therefore, even in environments
as competitive as the financial markets, we may observe only near -efficiency, and profit opportunities may exist for especially diligent and creative investors. In Chapter 12, we
examine such challenges to the efficient market hypothesis, and this motivates our discus-
sion of active portfolio management in Part Seven. More important, our discussions of
security analysis and portfolio construction generally must account for the likelihood of
nearly efficient markets.
1.6 The Players
From a bird’s-eye view, there would appear to be three major players in the financial
markets:
1. Firms are net demanders of capital. They raise capital now to pay for investments
in plant and equipment. The income generated by those real assets provides the
returns to investors who purchase the securities issued by the firm.
2. Households typically are net suppliers of capital. They purchase the securities
issued by firms that need to raise funds.
3. Governments can be borrowers or lenders, depending on the relationship between
tax revenue and government expenditures. Since World War II, the U.S. government
typically has run budget deficits, meaning that its tax receipts have been less than its
expenditures. The government, therefore, has had to borrow funds to cover its budget
deficit. Issuance of Treasury bills, notes, and bonds is the major way that the govern-
ment borrows funds from the public. In contrast, in the latter part of the 1990s, the
government enjoyed a budget surplus and was able to retire some outstanding debt.
Corporations and governments do not sell all or even most of their securities directly
to individuals. For example, about half of all stock is held by large financial institutions
such as pension funds, mutual funds, insurance companies, and banks. These financial
institutions stand between the security issuer (the firm) and the ultimate owner of the
security (the individual investor). For this reason, they are called financial intermediaries. Similarly, corporations do not market their own securities to the public. Instead, they hire
agents, called investment bankers, to represent them to the investing public. Let’s examine
the roles of these intermediaries.
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12 P A R T I Introduction
Financial Intermediaries Households want desirable investments for their savings, yet the small (financial) size
of most households makes direct investment difficult. A small investor seeking to lend
money to businesses that need to finance investments doesn’t advertise in the local news-
paper to find a willing and desirable borrower. Moreover, an individual lender would not
be able to diversify across borrowers to reduce risk. Finally, an individual lender is not
equipped to assess and monitor the credit risk of borrowers.
For these reasons, financial intermediaries have evolved to bring the suppliers of capital (investors) together with the demanders of capital (primarily corporations and the
federal government). These financial intermediaries include banks, investment companies,
insurance companies, and credit unions. Financial intermediaries issue their own securities
to raise funds to purchase the securities of other corporations.
For example, a bank raises funds by borrowing (taking deposits) and lending that money
to other borrowers. The spread between the interest rates paid to depositors and the rates
charged to borrowers is the source of the bank’s profit. In this way, lenders and borrowers
do not need to contact each other directly. Instead, each goes to the bank, which acts as an
intermediary between the two. The problem of matching lenders with borrowers is solved
when each comes independently to the common intermediary.
Financial intermediaries are distinguished from other businesses in that both their
assets and their liabilities are overwhelmingly financial. Table 1.3 presents the aggregated
balance sheet of commercial banks, one of the largest sectors of financial intermediaries.
Notice that the balance sheet includes only very small amounts of real assets. Compare
Assets $ Billion % Total Liabilities and Net Worth $ Billion % Total
Real assets Liabilities Equipment and premises $ 121.3 0.9% Deposits $10,260.3 73.7% Other real estate 44.8 0.3 Debt and other borrowed funds 743.5 5.3
Total real assets $ 166.1 1.2% Federal funds and repurchase agreements
478.8 3.4
Other 855.8 6.1
Total liabilities $12,338.4 88.6% Financial assets Cash $ 1,335.9 9.6% Investment securities 2,930.6 21.0 Loans and leases 7,227.7 51.9 Other financial assets 1,161.5 8.3
Total financial assets $12,655.7 90.9% Other assets Intangible assets $ 371.4 2.7% Other 732.8 5.3
Total other assets $ 1,104.2 7.9% Net worth $ 1,587.6 11.4%
Total $13,926.0 100.0% $13,926.0 100.0%
Table 1.3
Balance sheet of FDIC-insured commercial banks and savings institutions Note: Column sums may differ from total because of rounding error. Source: Federal Deposit Insurance Corporation, www.fdic.gov , July 2012.
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C H A P T E R 1 The Investment Environment 13
Table 1.3 to the aggregated balance sheet of the nonfinancial corporate sector in Table 1.4
for which real assets are about half of all assets. The contrast arises because intermediaries
simply move funds from one sector to another. In fact, the primary social function of such
intermediaries is to channel household savings to the business sector.
Other examples of financial intermediaries are investment companies, insurance com-
panies, and credit unions. All these firms offer similar advantages in their intermediary
role. First, by pooling the resources of many small investors, they are able to lend con-
siderable sums to large borrowers. Second, by lending to many borrowers, intermediaries
achieve significant diversification, so they can accept loans that individually might be too
risky. Third, intermediaries build expertise through the volume of business they do and can
use economies of scale and scope to assess and monitor risk.
Investment companies , which pool and manage the money of many investors, also arise out of economies of scale. Here, the problem is that most household portfolios are not
large enough to be spread across a wide variety of securities. In terms of brokerage fees
and research costs, purchasing one or two shares of many different firms is very expensive.
Mutual funds have the advantage of large-scale trading and portfolio management, while
participating investors are assigned a prorated share of the total funds according to the size
of their investment. This system gives small investors advantages they are willing to pay
for via a management fee to the mutual fund operator.
Investment companies also can design portfolios specifically for large investors with partic-
ular goals. In contrast, mutual funds are sold in the retail market, and their investment philoso-
phies are differentiated mainly by strategies that are likely to attract a large number of clients.
Like mutual funds, hedge funds also pool and invest the money of many clients. But they are open only to institutional investors such as pension funds, endowment funds, or
wealthy individuals. They are more likely to pursue complex and higher-risk strategies.
They typically keep a portion of trading profits as part of their fees, whereas mutual funds
charge a fixed percentage of assets under management.
Assets $ Billion % Total Liabilities and Net Worth $ Billion % Total
Real assets Liabilities Equipment and software $ 4,259 13.9% Bonds and mortgages $ 5,935 19.4% Real estate 9,051 29.5 Bank loans 612 2.0 Inventories 2,010 6.6 Other loans 1,105 3.6
Total real assets $15,320 50.0% Trade debt 1,969 6.4 Other 4,267 13.9
Financial assets Total liabilities $13,887 45.3% Deposits and cash $ 967 3.2% Marketable securities 769 2.5 Trade and consumer credit 2,555 8.3 Direct investment abroad 4,055 13.2 Other 6,983 22.8 Total financial assets $15,329 50.0% Net worth $16,762 54.7% Total $30,649 100.0% $30,649 100.0%
Table 1.4
Balance sheet of U.S. nonfinancial corporations Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.
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14
Economies of scale also explain the proliferation of analytic services available to inves-
tors. Newsletters, databases, and brokerage house research services all engage in research
to be sold to a large client base. This setup arises naturally. Investors clearly want infor-
mation, but with small portfolios to manage, they do not find it economical to personally
gather all of it. Hence, a profit opportunity emerges: A firm can perform this service for
many clients and charge for it.
Investment Bankers Just as economies of scale and specialization create profit opportunities for financial
intermediaries, so do these economies create niches for firms that perform specialized
services for businesses. Firms raise much of their capital by selling securities such as
stocks and bonds to the public. Because these firms do not do so frequently, however,
investment bankers that specialize in such activities can offer their services at a cost below that of maintaining an in-house security issuance division. In this role, they are
called underwriters. Investment bankers advise the issuing corporation on the prices it can charge for the
securities issued, appropriate interest rates, and so forth. Ultimately, the investment bank-
ing firm handles the marketing of the security in the primary market , where new issues
Separating Commercial Banking from Investment Banking
Until 1999, the Glass-Steagall Act had prohibited banks in the United States from both accepting deposits and underwriting securities. In other words, it forced a sepa- ration of the investment and commercial banking indus- tries. But when Glass-Steagall was repealed, many large commercial banks began to transform themselves into “universal banks” that could offer a full range of com- mercial and investment banking services. In some cases, commercial banks started their own investment banking divisions from scratch, but more frequently they expanded through merger. For example, Chase Manhattan acquired J.P. Morgan to form JPMorgan Chase. Similarly, Citigroup acquired Salomon Smith Barney to offer wealth manage- ment, brokerage, investment banking, and asset man- agement services to its clients. Most of Europe had never forced the separation of commercial and investment bank- ing, so their giant banks such as Credit Suisse, Deutsche Bank, HSBC, and UBS had long been universal banks. Until 2008, however, the stand-alone investment banking sector in the U.S. remained large and apparently vibrant, includ- ing such storied names as Goldman Sachs, Morgan-Stanley, Merrill Lynch, and Lehman Brothers.
But the industry was shaken to its core in 2008, when several investment banks were beset by enormous losses on their holdings of mortgage-backed securities. In March, on the verge of insolvency, Bear Stearns was merged into JPMorgan Chase. On September 14, 2008, Merrill Lynch, also suffering steep mortgage-related losses, negotiated an agreement to be acquired by Bank of America. The next day, Lehman Brothers entered into the largest bankruptcy in U.S. history, having failed to find an acquirer able and willing to rescue it from its steep losses. The next week, the
only two remaining major independent investment banks, Goldman Sachs and Morgan Stanley, decided to convert from investment banks to traditional bank holding com- panies. In doing so, they became subject to the supervision of national bank regulators such as the Federal Reserve and the far tighter rules for capital adequacy that govern commercial banks. The firms decided that the greater sta- bility they would enjoy as commercial banks, particularly the ability to fund their operations through bank deposits and access to emergency borrowing from the Fed, justified the conversion. These mergers and conversions marked the effective end of the independent investment banking industry—but not of investment banking. Those services now will be supplied by the large universal banks.
Today, the debate about the separation between com- mercial and investment banking that seemed to have ended with the repeal of Glass-Steagall has come back to life. The Dodd-Frank Wall Street Reform and Consumer Protection Act places new restrictions on bank activities. For example, the Volcker Rule, named after former chair- man of the Federal Reserve Paul Volcker, prohibits banks from “proprietary trading,” that is, trading securities for their own accounts, and restricts their investments in hedge funds or private equity funds. The rule is meant to limit the risk that banks can take on. While the Volcker Rule is far less restrictive than Glass-Steagall had been, they both are motivated by the belief that banks enjoying Federal guarantees should be subject to limits on the sorts of activities in which they can engage. Proprietary trading is a core activity for investment banks, so limitations on this activity for commercial banks would reintroduce a separa- tion between their business models.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 1 The Investment Environment 15
of securities are offered to the public. Later, investors can trade previously issued securities
among themselves in the so-called secondary market . For most of the last century, investment banks and commercial banks in the U.S. were
separated by law. While those regulations were effectively eliminated in 1999, the industry
known as “Wall Street” was until 2008 still comprised of large, independent investment
banks such as Goldman Sachs, Merrill Lynch, and Lehman Brothers. But that stand-alone
model came to an abrupt end in September 2008, when all the remaining major U.S. invest-
ment banks were absorbed into commercial banks, declared bankruptcy, or reorganized as
commercial banks. The nearby box presents a brief introduction to these events.
Venture Capital and Private Equity While large firms can raise funds directly from the stock and bond markets with help from
their investment bankers, smaller and younger firms that have not yet issued securities
to the public do not have that option. Start-up companies rely instead on bank loans and
investors who are willing to invest in them in return for an ownership stake in the firm.
The equity investment in these young companies is called venture capital (VC) . Sources of venture capital are dedicated venture capital funds, wealthy individuals known as angel investors, and institutions such as pension funds.
Most venture capital funds are set up as limited partnerships. A management company
starts with its own money and raises additional capital from limited partners such as pen-
sion funds. That capital may then be invested in a variety of start-up companies. The man-
agement company usually sits on the start-up company’s board of directors, helps recruit
senior managers, and provides business advice. It charges a fee to the VC fund for oversee-
ing the investments. After some period of time, for example, 10 years, the fund is liqui-
dated and proceeds are distributed to the investors.
Venture capital investors commonly take an active role in the management of a start-up
firm. Other active investors may engage in similar hands-on management but focus instead
on firms that are in distress or firms that may be bought up, “improved,” and sold for a
profit. Collectively, these investments in firms that do not trade on public stock exchanges
are known as private equity investments.
1.7 The Financial Crisis of 2008
This chapter has laid out the broad outlines of the financial system, as well as some of the
links between the financial side of the economy and the “real” side in which goods and
services are produced. The financial crisis of 2008 illustrated in a painful way the intimate
ties between these two sectors. We present in this section a capsule summary of the crisis,
attempting to draw some lessons about the role of the financial system as well as the causes
and consequences of what has become known as systemic risk. Some of these issues are complicated; we consider them briefly here but will return to them in greater detail later in
the text once we have more context for analysis.
Antecedents of the Crisis In early 2007, most observers thought it inconceivable that within two years, the world
financial system would be facing its worst crisis since the Great Depression. At the time,
the economy seemed to be marching from strength to strength. The last significant macro-
economic threat had been from the implosion of the high-tech bubble in 2000–2002. But
the Federal Reserve responded to an emerging recession by aggressively reducing interest
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16 P A R T I Introduction
rates. Figure 1.1 shows that Treasury bill rates dropped drastically between 2001 and 2004,
and the LIBOR rate, which is the interest rate at which major money-center banks lend to
each other, fell in tandem. 6 These actions appeared to have been successful, and the reces-
sion was short-lived and mild.
By mid-decade the economy was apparently healthy once again. Although the stock
market had declined substantially between 2001 and 2002, Figure 1.2 shows that it
reversed direction just as dramatically beginning in 2003, fully recovering all of its post-
tech-meltdown losses within a few years. Of equal importance, the banking sector seemed
healthy. The spread between the LIBOR rate (at which banks borrow from each other)
and the Treasury-bill rate (at which the U.S. government borrows), a common measure of
credit risk in the banking sector (often referred to as the TED spread 7 ), was only around .25% in early 2007 (see the bottom curve in Figure 1.1 ), suggesting that fears of default or
“counterparty” risk in the banking sector were extremely low.
Indeed, the apparent success of monetary policy in this recession, as well as in the last
30 years more generally, had engendered a new term, the “Great Moderation,” to describe
the fact that recent business cycles—and recessions in particular—seemed so mild com-
pared to past experience. Some observers wondered whether we had entered a golden age
for macroeconomic policy in which the business cycle had been tamed.
The combination of dramatically reduced interest rates and an apparently stable econ-
omy fed a historic boom in the housing market. Figure 1.3 shows that U.S. housing prices
Figure 1.1 Short-term LIBOR and Treasury-bill rates and the TED spread
P e rc
e n
t
Ja n
-0 0
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0 0
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-1 2
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0
1
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3
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5
6
7
8
3-month LIBOR 3-month T-Bill TED spread
6 LIBOR stands for London Interbank Offer Rate. It is a rate charged on dollar-denominated loans in an interbank
lending market outside of the U.S. (largely centered in London). The rate is typically quoted for 3-month loans.
The LIBOR rate is closely related to the Federal Funds rate in the U.S. The Fed Funds rate is the rate charged on
loans between U.S. banks, usually on an overnight basis. 7 TED stands for Treasury–Eurodollar spread. The Eurodollar rate in this spread is in fact LIBOR.
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C H A P T E R 1 The Investment Environment 17
began rising noticeably in the late 1990s and accelerated dramatically after 2001 as inter-
est rates plummeted. In the 10 years beginning 1997, average prices in the U.S. approxi-
mately tripled.
But the newfound confidence in the power of macroeconomic policy to reduce risk, the
impressive recovery of the economy from the high-tech implosion, and particularly the
housing price boom following the aggressive reduction in interest rates may have sown
the seeds for the debacle that played out in 2008. On the one hand, the Fed’s policy of
reducing interest rates had resulted in low yields on a wide variety of investments, and
investors were hungry for higher-yielding alternatives. On the other hand, low volatility
and growing complacency about risk encouraged greater tolerance for risk in the search
Figure 1.2 Cumulative returns on the S&P 500 index
1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 0
5
10
15
C u
m u
la ti
ve V
a lu
e o
f a $
1 I n
ve st
m e n
t
20
25
30
35
40
Figure 1.3 The Case-Shiller index of U.S. housing prices
0 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013
50
100
150
200
In d
e x
(J a n
u a ry
2 0 0 0 =
1 0 0 )
250
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18 P A R T I Introduction
for these higher-yielding investments. Nowhere was this more evident than in the explod-
ing market for securitized mortgages. The U.S. housing and mortgage finance markets
were at the center of a gathering storm.
Changes in Housing Finance Prior to 1970, most mortgage loans would come from a local lender such as a neigh-
borhood savings bank or credit union. A homeowner would borrow funds for a home
purchase and repay the loan over a long period, commonly 30 years. A typical thrift insti-
tution would have as its major asset a portfolio of these long-term home loans, while its
major liability would be the accounts of its depositors. This landscape began to change
when Fannie Mae (FNMA, or Federal National Mortgage Association) and Freddie Mac
(FHLMC, or Federal Home Loan Mortgage Corporation) began buying mortgage loans
from originators and bundling them into large pools that could be traded like any other
financial asset. These pools, which were essentially claims on the underlying mortgages,
were soon dubbed mortgage-backed securities, and the process was called securitization . Fannie and Freddie quickly became the behemoths of the mortgage market, between them
buying around half of all mortgages originated by the private sector.
Figure 1.4 illustrates how cash flows passed from the original borrower to the ulti-
mate investor in a mortgage-backed security. The loan originator, for example, the savings
and loan, might make a $100,000 home loan to a homeowner. The homeowner would
repay principal and interest (P&I) on the loan over 30 years. But then the originator would
sell the mortgage to Freddie Mac or Fannie Mae and recover the cost of the loan. The
originator could continue to service the loan (collect monthly payments from the home-
owner) for a small servicing fee, but the loan payments net of that fee would be passed
along to the agency. In turn, Freddie or Fannie would pool the loans into mortgage-backed
securities and sell the securities to investors such as pension funds or mutual funds. The
agency (Fannie or Freddie) typically would guarantee the credit or default risk of the loans
included in each pool, for which it would retain a guarantee fee before passing along the
rest of the cash flow to the ultimate investor. Because the mortgage cash flows were passed
along from the homeowner to the lender to Fannie or Freddie to the investor, the mortgage-
backed securities were also called pass-throughs. Until the last decade, the vast majority of securitized mortgages were held or guaran-
teed by Freddie Mac or Fannie Mae. These were low-risk conforming mortgages, meaning that eligible loans for agency securitization couldn’t be too big, and homeowners had to
meet underwriting criteria establishing their ability to repay the loan. For example, the
ratio of loan amount to house value could be no more than 80%. But securitization gave
rise to a new market niche for mortgage lenders: the “originate to distribute” (versus origi-
nate to hold) business model.
Figure 1.4 Cash flows in a mortgage pass-through security
$100 K
P&I P&I – servicing – guarantee fee
$100 K $100 K
OriginatorHomeowner
P&I – servicing fee
InvestorAgency
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C H A P T E R 1 The Investment Environment 19
Whereas conforming loans were pooled almost entirely through Freddie Mac and
Fannie Mae, once the securitization model took hold, it created an opening for a new
product: securitization by private firms of nonconforming “subprime” loans with higher default risk. One important difference between the government agency pass-throughs
and these so-called private-label pass-throughs was that the investor in the private-label
pool would bear the risk that homeowners might default on their loans. Thus, originating
mortgage brokers had little incentive to perform due diligence on the loan as long as the loans could be sold to an investor. These investors, of course, had no direct contact with the borrowers, and they could not perform detailed underwriting concerning loan quality.
Instead, they relied on borrowers’ credit scores, which steadily came to replace conven-
tional underwriting.
A strong trend toward low-documentation and then no-documentation loans, entailing
little verification of a borrower’s ability to carry a loan, soon emerged. Other subprime
underwriting standards quickly deteriorated. For example, allowed leverage on home
loans (as measured by the loan-to-value ratio) rose dramatically. By 2006, the majority
of subprime borrowers purchased houses by borrowing the entire purchase price! When housing prices began falling, these loans were quickly “underwater,” meaning that the
house was worth less than the loan balance, and many homeowners decided to walk away
from their loans.
Adjustable-rate mortgages (ARMs) also grew in popularity. These loans offered bor-
rowers low initial or “teaser” interest rates, but these rates eventually would reset to current
market interest yields, for example, the Treasury bill rate plus 3%. Many of these borrow-
ers “maxed out” their borrowing capacity at the teaser rate, yet, as soon as the loan rate was
reset, their monthly payments would soar, especially if market interest rates had increased.
Despite these obvious risks, the ongoing increase in housing prices over the last
decade seemed to lull many investors into complacency, with a widespread belief that
continually rising home prices would bail out poorly performing loans. But starting in
2004, the ability of refinancing to save a loan began to diminish. First, higher interest
rates put payment pressure on homeowners who had taken out adjustable-rate mortgages.
Second, as Figure 1.3 shows, housing prices peaked by 2006, so homeowners’ ability to
refinance a loan using built-up equity in the house declined. Housing default rates began
to surge in 2007, as did losses on mortgage-backed securities. The crisis was ready to
shift into high gear.
Mortgage Derivatives One might ask: Who was willing to buy all of these risky subprime mortgages? Secu-
ritization, restructuring, and credit enhancement provide a big part of the answer. New
risk-shifting tools enabled investment banks to carve out AAA-rated securities from
original-issue “junk” loans. Collateralized debt obligations, or CDOs, were among the
most important and eventually damaging of these innovations.
CDOs were designed to concentrate the credit (i.e., default) risk of a bundle of loans on
one class of investors, leaving the other investors in the pool relatively protected from that
risk. The idea was to prioritize claims on loan repayments by dividing the pool into senior
versus junior slices, called tranches. The senior tranches had first claim on repayments from the entire pool. Junior tranches would be paid only after the senior ones had received
their cut. 8 For example, if a pool were divided into two tranches, with 70% of the pool
allocated to the senior tranche and 30% allocated to the junior one, the senior investors
8 CDOs and related securities are sometimes called structured products. “Structured” means that original cash
flows are sliced up and reapportioned across tranches according to some stipulated rule.
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20 P A R T I Introduction
would be repaid in full as long as 70% or more of the loans in the pool performed, that is,
as long as the default rate on the pool remained below 30%. Even with pools composed of
risky subprime loans, default rates above 30% seemed extremely unlikely, and thus senior
tranches were frequently granted the highest (i.e., AAA) rating by the major credit rating
agencies, Moody’s, Standard & Poor’s, and Fitch. Large amounts of AAA-rated securities
were thus carved out of pools of low-rated mortgages. (We will describe CDOs in more
detail in Chapter 14.)
Of course, we know now that these ratings were wrong. The senior-subordinated
structure of CDOs provided far less protection to senior tranches than investors antici-
pated. When housing prices across the entire country began to fall in unison, defaults in
all regions increased, and the hoped-for benefits from spreading the risks geographically
never materialized.
Why had the rating agencies so dramatically underestimated credit risk in these sub-
prime securities? First, default probabilities had been estimated using historical data from
an unrepresentative period characterized by a housing boom and an uncommonly prosper-
ous and recession-free macroeconomy. Moreover, the ratings analysts had extrapolated
historical default experience to a new sort of borrower pool—one without down payments,
with exploding-payment loans, and with low- or no-documentation loans (often called liar loans ). Past default experience was largely irrelevant given these profound changes in the market. Moreover, the power of cross-regional diversification to minimize risk engendered
excessive optimism.
Finally, agency problems became apparent. The ratings agencies were paid to provide
ratings by the issuers of the securities—not the purchasers. They faced pressure from the
issuers, who could shop around for the most favorable treatment, to provide generous ratings.
When Freddie Mac and Fannie Mae pooled mortgages into securities, they guaranteed the underlying mortgage loans against homeowner defaults. In contrast, there were no guarantees on the mortgages pooled into subprime mortgage-backed securities, so investors would bear credit risk. Was either of these arrangements necessarily a better way to manage and allocate default risk?
CONCEPT CHECK 1.2
Credit Default Swaps In parallel to the CDO market, the market in credit default swaps also exploded in this period. A credit default swap, or CDS, is in essence an insurance contract against the
default of one or more borrowers. (We will describe these in more detail in Chapter 14.)
The purchaser of the swap pays an annual premium (like an insurance premium) for protec-
tion from credit risk. Credit default swaps became an alternative method of credit enhance-
ment, seemingly allowing investors to buy subprime loans and insure their safety. But in
practice, some swap issuers ramped up their exposure to credit risk to unsupportable lev-
els, without sufficient capital to back those obligations. For example, the large insurance
company AIG alone sold more than $400 billion of CDS contracts on subprime mortgages.
The Rise of Systemic Risk By 2007, the financial system displayed several troubling features. Many large banks and
related financial institutions had adopted an apparently profitable financing scheme: bor-
rowing short term at low interest rates to finance holdings in higher-yielding, long-term
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C H A P T E R 1 The Investment Environment 21
illiquid assets, 9 and treating the interest rate differential between their assets and liabilities
as economic profit. But this business model was precarious: By relying primarily on short-
term loans for their funding, these firms needed to constantly refinance their positions (i.e.,
borrow additional funds as the loans matured), or else face the necessity of quickly selling
off their less-liquid asset portfolios, which would be difficult in times of financial stress.
Moreover, these institutions were highly leveraged and had little capital as a buffer against
losses. Large investment banks on Wall Street in particular had sharply increased leverage,
which added to an underappreciated vulnerability to refunding requirements—especially
if the value of their asset portfolios came into question. Even small portfolio losses could
drive their net worth negative, at which point no one would be willing to renew outstanding
loans or extend new ones.
Another source of fragility was widespread investor reliance on “credit enhancement”
through products like CDOs. Many of the assets underlying these pools were illiquid, hard
to value, and highly dependent on forecasts of future performance of other loans. In a
widespread downturn, with rating downgrades, these assets would prove difficult to sell.
The steady displacement of formal exchange trading by informal “over-the-counter”
markets created other problems. In formal exchanges such as futures or options markets,
participants must put up collateral called margin to guarantee their ability to make good on their obligations. Prices are computed each day, and gains or losses are continually added
to or subtracted from each trader’s margin account. If a margin account runs low after a
series of losses, the investor can be required to either contribute more collateral or to close
out the position before actual insolvency ensues. Positions, and thus exposures to losses,
are transparent to other traders. In contrast, the over-the-counter markets where CDS con-
tracts trade are effectively private contracts between two parties with less public disclosure
of positions, less standardization of products (which makes the fair value of a contract hard
to discover), and consequently less opportunity to recognize either cumulative gains or
losses over time or the resultant credit exposure of each trading partner.
This new financial model was brimming with systemic risk , a potential breakdown of the financial system when problems in one market spill over and disrupt others. When
lenders such as banks have limited capital and are afraid of further losses, they may ratio-
nally choose to hoard their capital instead of lending it to customers such as small firms,
thereby exacerbating funding problems for their customary borrowers.
The Shoe Drops By fall 2007, housing price declines were widespread ( Figure 1.3 ), mortgage delinquen-
cies increased, and the stock market entered its own free fall ( Figure 1.2 ). Many investment
banks, which had large investments in mortgages, also began to totter.
The crisis peaked in September 2008. On September 7, the giant federal mortgage agen-
cies Fannie Mae and Freddie Mac, both of which had taken large positions in subprime
mortgage–backed securities, were put into conservatorship. (We will have more to say
on their travails in Chapter 2.) The failure of these two mainstays of the U.S. housing and
mortgage finance industries threw financial markets into a panic. By the second week of
September, it was clear that both Lehman Brothers and Merrill Lynch were on the verge
of bankruptcy. On September 14, Merrill Lynch was sold to Bank of America, again with
the benefit of government brokering and protection against losses. The next day, Lehman
Brothers, which was denied equivalent treatment, filed for bankruptcy protection. Two
9 Liquidity refers to the speed and the ease with which investors can realize the cash value of an investment. Illiq-
uid assets, for example, real estate, can be hard to sell quickly, and a quick sale may require a substantial discount
from the price at which the asset could be sold in an unrushed situation.
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22 P A R T I Introduction
days later, on September 17, the government reluctantly lent $85 billion to AIG, reasoning
that its failure would have been highly destabilizing to the banking industry, which was
holding massive amounts of its credit guarantees (i.e., CDS contacts). The next day, the
Treasury unveiled its first proposal to spend $700 billion to purchase “toxic” mortgage-
backed securities.
A particularly devastating fallout of the Lehman bankruptcy was on the “money mar-
ket” for short-term lending. Lehman had borrowed considerable funds by issuing very
short-term debt, called commercial paper. Among the major customers in commercial
paper were money market mutual funds, which invest in short-term, high-quality debt of
commercial borrowers. When Lehman faltered, the Reserve Primary Money Market Fund,
which was holding large amounts of (AAA-rated!) Lehman commercial paper, suffered
investment losses that drove the value of its assets below $1 per share. 10
Fears spread that
other funds were similarly exposed, and money market fund customers across the country
rushed to withdraw their funds. The funds in turn rushed out of commercial paper into safer
and more liquid Treasury bills, essentially shutting down short-term financing markets.
The freezing up of credit markets was the end of any dwindling possibility that the finan-
cial crisis could be contained to Wall Street. Larger companies that had relied on the com-
mercial paper market were now unable to raise short-term funds. Banks similarly found it
difficult to raise funds. (Look back to Figure 1.1 , where you will see that the TED spread,
a measure of bank insolvency fears, skyrocketed in 2008.) With banks unwilling or unable
to extend credit to their customers, thousands of small businesses that relied on bank lines
of credit also became unable to finance their normal business operations. Capital-starved
companies were forced to scale back their own operations precipitously. The unemploy-
ment rate rose rapidly, and the economy was in its worst recession in decades. The turmoil
in the financial markets had spilled over into the real economy, and Main Street had joined
Wall Street in a bout of protracted misery.
The Dodd-Frank Reform Act The crisis engendered many calls for reform of Wall Street. These eventually led to the
passage in 2010 of the Dodd-Frank Wall Street Reform and Consumer Protection Act,
which proposes several mechanisms to mitigate systemic risk.
The act calls for stricter rules for bank capital, liquidity, and risk management practices,
especially as banks become larger and their potential failure would be more threatening to
other institutions. With more capital supporting banks, the potential for one insolvency to
trigger another could be contained. In addition, when banks have more capital, they have
less incentive to ramp up risk, as potential losses will come at their own expense and not
the FDIC’s.
Dodd-Frank also mandates increased transparency, especially in derivatives markets.
For example, one suggestion is to standardize CDS contracts so they can trade in central-
ized exchanges where prices can be determined in a deep market and gains or losses can be
settled on a daily basis. Margin requirements, enforced daily, would prevent CDS partici-
pants from taking on greater positions than they can handle, and exchange trading would
facilitate analysis of the exposure of firms to losses in these markets.
The act also attempts to limit the risky activities in which banks can engage. The so-
called Volcker Rule, named after former chairman of the Federal Reserve Paul Volcker,
10
Money market funds typically bear very little investment risk and can maintain their asset values at $1 per share.
Investors view them as near substitutes for checking accounts. Until this episode, no other retail fund had “broken
the buck.”
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C H A P T E R 1 The Investment Environment 23
prohibits banks from trading for their own accounts and limits total investments in hedge
funds or private equity funds.
The law also addresses shortcomings of the regulatory system that became apparent in
2008. The U.S. has several financial regulators with overlapping responsibility, and some
institutions were accused of “regulator shopping,” seeking to be supervised by the most
lenient regulator. Dodd-Frank seeks to unify and clarify lines of regulatory authority and
responsibility in one or a smaller number of government agencies.
The act addresses incentive issues. Among these are proposals to force employee com-
pensation to reflect longer-term performance. The act requires public companies to set
“claw-back provisions” to take back executive compensation if it was based on inaccu-
rate financial statements. The motivation is to discourage excessive risk-taking by large
financial institutions in which big bets can be wagered with the attitude that a successful
outcome will result in a big bonus while a bad outcome will be borne by the company, or
worse, the taxpayer.
The incentives of the bond rating agencies are also a sore point. Few are happy with a
system that has the ratings agencies paid by the firms they rate. The act creates an Office
of Credit Ratings within the Securities and Exchange Commission to oversee the credit
rating agencies.
It is still too early to know which, if any, of these reforms will stick. The implementa-
tion of Dodd-Frank is still subject to considerable interpretation by regulators, and the act
is still under attack by some members of Congress. But the crisis surely has made clear the
essential role of the financial system in the functioning of the real economy.
1.8 Outline of the Text
The text has seven parts, which are fairly independent and may be studied in a variety of
sequences. Part One is an introduction to financial markets, instruments, and trading of
securities. This part also describes the mutual fund industry.
Parts Two and Three contain the core of what has come to be known as “modern port-
folio theory.” We start in Part Two with a general discussion of risk and return and the les-
sons of capital market history. We then focus more closely on how to describe investors’
risk preferences and progress to asset allocation, efficient diversification, and portfolio
optimization.
In Part Three, we investigate the implications of portfolio theory for the equilibrium
relationship between risk and return. We introduce the capital asset pricing model, its
implementation using index models, and more advanced models of risk and return. This
part also treats the efficient market hypothesis as well as behavioral critiques of theories
based on investor rationality and closes with a chapter on empirical evidence concerning
security returns.
Parts Four through Six cover security analysis and valuation. Part Four is devoted to
debt markets and Part Five to equity markets. Part Six covers derivative assets, such as
options and futures contracts.
Part Seven is an introduction to active investment management. It shows how different
investors’ objectives and constraints can lead to a variety of investment policies. This part
discusses the role of active management in nearly efficient markets and considers how one
should evaluate the performance of managers who pursue active strategies. It also shows
how the principles of portfolio construction can be extended to the international setting and
examines the hedge fund industry.
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24 P A R T I Introduction
1. Real assets create wealth. Financial assets represent claims to parts or all of that wealth. Financial assets determine how the ownership of real assets is distributed among investors.
2. Financial assets can be categorized as fixed income, equity, or derivative instruments. Top- down portfolio construction techniques start with the asset allocation decision—the alloca-
tion of funds across broad asset classes—and then progress to more specific security-selection
decisions.
3. Competition in financial markets leads to a risk–return trade-off, in which securities that offer higher expected rates of return also impose greater risks on investors. The presence of risk, how-
ever, implies that actual returns can differ considerably from expected returns at the beginning of
the investment period. Competition among security analysts also promotes financial markets that
are nearly informationally efficient, meaning that prices reflect all available information concern-
ing the value of the security. Passive investment strategies may make sense in nearly efficient
markets.
4. Financial intermediaries pool investor funds and invest them. Their services are in demand because small investors cannot efficiently gather information, diversify, and monitor portfolios.
The financial intermediary sells its own securities to the small investors. The intermediary invests
the funds thus raised, uses the proceeds to pay back the small investors, and profits from the
difference (the spread).
5. Investment banking brings efficiency to corporate fund-raising. Investment bankers develop expertise in pricing new issues and in marketing them to investors. By the end of 2008, all the
major stand-alone U.S. investment banks had been absorbed into commercial banks or had reor-
ganized themselves into bank holding companies. In Europe, where universal banking had never
been prohibited, large banks had long maintained both commercial and investment banking
divisions.
6. The financial crisis of 2008 showed the importance of systemic risk. Systemic risk can be limited by transparency that allows traders and investors to assess the risk of their counterparties, capital
requirements to prevent trading participants from being brought down by potential losses, fre-
quent settlement of gains or losses to prevent losses from accumulating beyond an institution’s
ability to bear them, incentives to discourage excessive risk taking, and accurate and unbiased
analysis by those charged with evaluating security risk.
SUMMARY
investment
real assets
financial assets
fixed-income (debt) securities
equity
derivative securities
agency problem
asset allocation
security selection
security analysis
risk–return trade-off
passive management
active management
financial intermediaries
investment companies
investment bankers
primary market
secondary market
venture capital
private equity
securitization
systemic risk
KEY TERMS
Related Web sites for this chapter are available at www. mhhe.com/bkm
1. Financial engineering has been disparaged as nothing more than paper shuffling. Critics argue
that resources used for rearranging wealth (that is, bundling and unbundling financial assets) might be better spent on creating wealth (that is, creating real assets). Evaluate this criticism. Are any benefits realized by creating an array of derivative securities from various primary
securities?
2. Why would you expect securitization to take place only in highly developed capital markets?
3. What is the relationship between securitization and the role of financial intermediaries in the
economy? What happens to financial intermediaries as securitization progresses?
PROBLEM SETS
Basic
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C H A P T E R 1 The Investment Environment 25
4. Although we stated that real assets constitute the true productive capacity of an economy, it is
hard to conceive of a modern economy without well-developed financial markets and security
types. How would the productive capacity of the U.S. economy be affected if there were no
markets in which to trade financial assets?
5. Firms raise capital from investors by issuing shares in the primary markets. Does this imply that
corporate financial managers can ignore trading of previously issued shares in the secondary
market?
6. Suppose housing prices across the world double.
a. Is society any richer for the change? b. Are homeowners wealthier? c. Can you reconcile your answers to ( a ) and ( b )? Is anyone worse off as a result of the change?
7. Lanni Products is a start-up computer software development firm. It currently owns computer
equipment worth $30,000 and has cash on hand of $20,000 contributed by Lanni’s owners. For
each of the following transactions, identify the real and/or financial assets that trade hands. Are
any financial assets created or destroyed in the transaction?
a. Lanni takes out a bank loan. It receives $50,000 in cash and signs a note promising to pay back the loan over 3 years.
b. Lanni uses the cash from the bank plus $20,000 of its own funds to finance the development of new financial planning software.
c. Lanni sells the software product to Microsoft, which will market it to the public under the Microsoft name. Lanni accepts payment in the form of 1,500 shares of Microsoft stock.
d. Lanni sells the shares of stock for $80 per share and uses part of the proceeds to pay off the bank loan.
8. Reconsider Lanni Products from the previous problem.
a. Prepare its balance sheet just after it gets the bank loan. What is the ratio of real assets to total assets?
b. Prepare the balance sheet after Lanni spends the $70,000 to develop its software product. What is
the ratio of real assets to total assets?
c. Prepare the balance sheet after Lanni accepts the payment of shares from Microsoft. What is the
ratio of real assets to total assets?
9. Examine the balance sheet of commercial banks
in Table 1.3 . What is the ratio of real assets to
total assets? What is that ratio for nonfinancial
firms ( Table 1.4 )? Why should this difference be
expected?
10. Consider Figure 1.5 , which describes an issue of
American gold certificates.
a. Is this issue a primary or secondary market transaction?
b. Are the certificates primitive or derivative assets? c. What market niche is filled by this offering?
11. Discuss the advantages and disadvantages of the
following forms of managerial compensation in
terms of mitigating agency problems, that is, poten-
tial conflicts of interest between managers and
shareholders.
a. A fixed salary. b. Stock in the firm that must be held for five years. c. A salary linked to the firm’s profits.
Intermediate
Figure 1.5 A gold-backed security
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26 P A R T I Introduction
12. We noted that oversight by large institutional investors or creditors is one mechanism to reduce
agency problems. Why don’t individual investors in the firm have the same incentive to keep an
eye on management?
13. Give an example of three financial intermediaries and explain how they act as a bridge between
small investors and large capital markets or corporations.
14. The average rate of return on investments in large stocks has outpaced that on investments in
Treasury bills by about 7% since 1926. Why, then, does anyone invest in Treasury bills?
15. What are some advantages and disadvantages of top-down versus bottom-up investing styles?
16. You see an advertisement for a book that claims to show how you can make $1 million with no
risk and with no money down. Will you buy the book?
17. Why do financial assets show up as a component of household wealth, but not of national
wealth? Why do financial assets still matter for the material well-being of an economy?
18. Wall Street firms have traditionally compensated their traders with a share of the trading profits
that they generated. How might this practice have affected traders’ willingness to assume risk?
What is the agency problem this practice engendered?
19. What reforms to the financial system might reduce its exposure to systemic risk?
E-INVESTMENTS EXERCISES
1. Log on to finance.yahoo.com and enter the ticker symbol RRD in the Get Quotes box to find information about R.R. Donnelley & Sons.
a. Click on Profi le. What is Donnelly’s main line of business?
b. Now go to Key Statistics. How many shares of the company’s stock are outstand- ing? What is the total market value of the fi rm? What were its profi ts in the most recent fi scal year?
c. Look up Major Holders of the company’s stock. What fraction of total shares is held by insiders?
d. Now go to Analyst Opinion. What is the average price target (i.e., the predicted stock price of the Donnelly shares) of the analysts covering this fi rm? How does that compare to the price at which the stock is currently trading?
e. Look at the company’s Balance Sheet. What were its total assets at the end of the most recent fi scal year?
2. a. Go to the Securities and Exchange Commission Web site, www.sec.gov . What is the mission of the SEC? What information and advice does the SEC offer to begin- ning investors?
b. Go to the NASD Web site, www.fi nra.org . What is its mission? What information and advice does it offer to beginners?
c. Go to the IOSCO Web site, www.iosco.org . What is its mission? What information and advice does it offer to beginners?
1. a. Real
b. Financial
c. Real
d. Real
e. Financial
SOLUTIONS TO CONCEPT CHECKS
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C H A P T E R 1 The Investment Environment 27
2. The central issue is the incentive to monitor the quality of loans when originated as well as
over time. Freddie and Fannie clearly had incentive to monitor the quality of conforming loans
that they had guaranteed, and their ongoing relationships with mortgage originators gave them
opportunities to evaluate track records over extended periods of time. In the subprime mortgage
market, the ultimate investors in the securities (or the CDOs backed by those securities), who
were bearing the credit risk, should not have been willing to invest in loans with a disproportionate
likelihood of default. If they properly understood their exposure to default risk, then the
(correspondingly low) prices they would have been willing to pay for these securities would have
imposed discipline on the mortgage originators and servicers. The fact that they were willing to
hold such large positions in these risky securities suggests that they did not appreciate the extent
of their exposure. Maybe they were led astray by overly optimistic projections for housing prices
or by biased assessments from the credit-reporting agencies. In principle, either arrangement
for default risk could have provided the appropriate discipline on the mortgage originators; in
practice, however, the informational advantages of Freddie and Fannie probably made them the
better “recipients” of default risk. The lesson is that information and transparency are some of the
preconditions for well-functioning markets.
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2 YOU LEARNED IN Chapter 1 that the pro- cess of building an investment portfolio usu- ally begins by deciding how much money to allocate to broad classes of assets, such as safe money market securities or bank accounts, longer term bonds, stocks, or even asset classes like real estate or precious metals. This process is called asset allocation. Within each class the investor then selects specific assets from a more detailed menu. This is called security selection.
Each broad asset class contains many spe- cific security types, and the many variations on a theme can be overwhelming. Our goal in this chapter is to introduce you to the important features of broad classes of securi- ties. Toward this end, we organize our tour of financial instruments according to asset class.
Financial markets are traditionally seg- mented into money markets and capital
markets . Money market instruments include short-term, marketable, liquid, low-risk debt securities. Money market instruments some- times are called cash equivalents, or just cash for short. Capital markets, in contrast, include longer term and riskier securities. Securities in the capital market are much more diverse than those found within the money market. For this reason, we will subdivide the capital market into four segments: longer term bond markets, equity markets, and the derivative markets for options and futures.
We first describe money market instru- ments. We then move on to debt and equity securities. We explain the structure of various stock market indexes in this chapter because market benchmark portfolios play an impor- tant role in portfolio construction and evalua- tion. Finally, we survey the derivative security markets for options and futures contracts.
CHAPTER TWO
Asset Classes and
Financial Instruments 2 PA
R T I
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C H A P T E R 2 Asset Classes and Financial Instruments 29
The money market is a subsector of the fixed-income market. It consists of very short-
term debt securities that usually are highly marketable. Table 2.1 lists outstanding volume
in 2012 for some of the major instruments in this market. Many of these securities trade
in large denominations, and so are out of the reach of individual investors. Money mar-
ket funds, however, are easily accessible to small investors. These mutual funds pool the
resources of many investors and purchase a wide variety of money market securities on
their behalf.
Treasury Bills U.S. Treasury bills (T-bills, or just bills, for short) are the most marketable of all money market instruments. T-bills represent the simplest form of borrowing: The government
raises money by selling bills to the public. Investors buy the bills at a discount from the
stated maturity value. At the bill’s maturity, the holder receives from the government a
payment equal to the face value of the bill. The difference between the purchase price and
ultimate maturity value constitutes the investor’s earnings.
T-bills are issued with initial maturities of 4, 13, 26, or 52 weeks. Individuals can pur-
chase T-bills directly, at auction, or on the secondary market from a government securi-
ties dealer. T-bills are highly liquid; that is, they are easily converted to cash and sold
at low transaction cost and with not much price risk. Unlike most other money market
instruments, which sell in minimum denominations of $100,000, T-bills sell in minimum
denominations of only $100, although $10,000 denominations are far more common. The
income earned on T-bills is exempt from all state and local taxes, another characteristic
distinguishing them from other money market instruments.
Figure 2.1 is a partial listing of T-bill rates. Rather than providing prices of each bill,
the financial press reports yields based on those prices. You will see yields corresponding
to both bid and ask prices. The ask price is the price you would have to pay to buy a T-bill from a securities dealer. The bid price is the slightly lower price you would receive if you wanted to sell a bill to a dealer. The bid–ask spread is the difference in these prices, which is the dealer’s source of profit. (Notice in Figure 2.1 that the bid yield is higher than the ask yield. This is because prices and yields are inversely related.)
2.1 The Money Market
*Small denominations are less than $100,000. Sources: Economic Report of the President, U.S. Government Printing Office, 2012; Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, September 2012.
$ Billion
Repurchase agreements $1,141 Small-denomination time deposits and savings deposits*
7,202
Large-denomination time deposits* 1,603 Treasury bills 1,478 Commercial paper 1,445 Money market mutual funds 2,645
Table 2.1
Major components of the money market
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30 P A R T I Introduction
The first two yields in Figure 2.1 are reported using the
bank-discount method. This means that the bill’s discount from its maturity or face value is “annualized” based on
a 360-day year, and then reported as a percentage of face
value. For example, for the highlighted bill maturing on
December 20, 2012, days to maturity are 156 and the yield
under the column labeled “ASKED” is given as .125%. This
means that a dealer was willing to sell the bill at a discount
from face value of .125% 3 (156/360) 5 .0542%. So a bill
with $10,000 face value could be purchased for $10,000 3
(1 2 .000542) 5 $9,994.58. Similarly, on the basis of the
bid yield of .130%, a dealer would be willing to purchase the
bill for $10,000 3 (1 2 .00130 3 156/360) 5 $9,994.367.
The bank discount method for computing yields has a
long tradition, but it is flawed for at least two reasons. First,
it assumes that the year has only 360 days. Second, it com-
putes the yield as a fraction of par value rather than of the
price the investor paid to acquire the bill. 1 An investor who buys the bill for the ask price
and holds it until maturity will see her investment grow over 156 days by a multiple of
$10,000/$9,994.58 5 1.000542, for a gain of .0542%. Annualizing this gain using a 365-day
year results in a yield of .0542% 3 365/156 5 .127%, which is the value reported in the last
column under “ASK YLD.” This last value is called the Treasury-bill’s bond-equivalent yield.
Certificates of Deposit A certificate of deposit, or CD, is a time deposit with a bank. Time deposits may not be withdrawn on demand. The bank pays interest and principal to the depositor only at the
end of the fixed term of the CD. CDs issued in denominations greater than $100,000
are usually negotiable, however; that is, they can be sold to another investor if the owner
needs to cash in the certificate before its maturity date. Short-term CDs are highly market-
able, although the market significantly thins out for maturities of 3 months or more. CDs
are treated as bank deposits by the Federal Deposit Insurance Corporation, so they are cur-
rently insured for up to $250,000 in the event of a bank insolvency.
Commercial Paper Large, well-known companies often issue their own short-term unsecured debt notes rather
than borrow directly from banks. These notes are called commercial paper. Very often, commercial paper is backed by a bank line of credit, which gives the borrower access to
cash that can be used (if needed) to pay off the paper at maturity.
Commercial paper maturities range up to 270 days; longer maturities would require
registration with the Securities and Exchange Commission and so are almost never issued.
Most often, commercial paper is issued with maturities of less than 1 or 2 months. Usually,
it is issued in multiples of $100,000. Therefore, small investors can invest in commercial
paper only indirectly, via money market mutual funds.
Commercial paper is considered to be a fairly safe asset, because a firm’s condition
presumably can be monitored and predicted over a term as short as 1 month.
1 Both of these “errors” were dictated by computational simplicity in precomputer days. It is easier to compute
percentage discounts from a round number such as par value rather than purchase price. It is also easier to annual-
ize using a 360-day year, because 360 is an even multiple of so many numbers.
Sep 06 12 51 0.080 0.075 0.000 0.076 Oct 04 12 79 0.085 0.080 0.000 0.081 Nov 01 12 107 0.110 0.100 0.005 0.101 Nov 29 12 135 0.110 0.105 0.000 0.106 Dec 20 12 156 0.130 0.125 0.005 0.127 Apr 04 13 261 0.160 0.150 0.005 0.152
BIDMATURITY DAYS TO
MAT ASKED ASK YLDCHG
Treasury Bills
Figure 2.1 Treasury bill yields Source: Compiled from data obtained from The Wall Street Journal Online, July 17, 2012.
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C H A P T E R 2 Asset Classes and Financial Instruments 31
While most commercial paper is issued by nonfinancial firms, in recent years there was
a sharp increase in asset-backed commercial paper issued by financial firms such as banks. This was short-term commercial paper typically used to raise funds for the institution to
invest in other assets, most notoriously, subprime mortgages. These assets were in turn
used as collateral for the commercial paper—hence the label “asset backed.” This practice
led to many difficulties starting in the summer of 2007 when the subprime mortgagors
began defaulting. The banks found themselves unable to issue new commercial paper to
refinance their positions as the old paper matured.
Bankers’ Acceptances A banker’s acceptance starts as an order to a bank by a bank’s customer to pay a sum of money at a future date, typically within 6 months. At this stage, it is similar to a postdated
check. When the bank endorses the order for payment as “accepted,” it assumes respon-
sibility for ultimate payment to the holder of the acceptance. At this point, the acceptance
may be traded in secondary markets like any other claim on the bank. Bankers’ acceptances
are considered very safe assets because traders can substitute the bank’s credit standing for
their own. They are used widely in foreign trade where the creditworthiness of one trader
is unknown to the trading partner. Acceptances sell at a discount from the face value of the
payment order, just as T-bills sell at a discount from par value.
Eurodollars Eurodollars are dollar-denominated deposits at foreign banks or foreign branches of American banks. By locating outside the United States, these banks escape regulation by
the Federal Reserve. Despite the tag “Euro,” these accounts need not be in European banks,
although that is where the practice of accepting dollar-denominated deposits outside the
United States began.
Most Eurodollar deposits are for large sums, and most are time deposits of less than
6 months’ maturity. A variation on the Eurodollar time deposit is the Eurodollar certifi-
cate of deposit, which resembles a domestic bank CD except that it is the liability of a
non-U.S. branch of a bank, typically a London branch. The advantage of Eurodollar CDs
over Eurodollar time deposits is that the holder can sell the asset to realize its cash value
before maturity. Eurodollar CDs are considered less liquid and riskier than domestic CDs,
however, and thus offer higher yields. Firms also issue Eurodollar bonds, which are dollar-
denominated bonds outside the U.S., although bonds are not a money market investment
because of their long maturities.
Repos and Reverses Dealers in government securities use repurchase agreements, also called “repos” or “RPs,” as a form of short-term, usually overnight, borrowing. The dealer sells govern-
ment securities to an investor on an overnight basis, with an agreement to buy back those
securities the next day at a slightly higher price. The increase in the price is the overnight
interest. The dealer thus takes out a 1-day loan from the investor, and the securities serve
as collateral.
A term repo is essentially an identical transaction, except that the term of the implicit loan can be 30 days or more. Repos are considered very safe in terms of credit risk because
the loans are backed by the government securities. A reverse repo is the mirror image of a repo. Here, the dealer finds an investor holding government securities and buys them,
agreeing to sell them back at a specified higher price on a future date.
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32 P A R T I Introduction
Federal Funds Just as most of us maintain deposits at banks, banks maintain deposits of their own at
a Federal Reserve bank. Each member bank of the Federal Reserve System, or “the
Fed,” is required to maintain a minimum balance in a reserve account with the Fed.
The required balance depends on the total deposits of the bank’s customers. Funds in
the bank’s reserve account are called federal funds, or fed funds. At any time, some banks have more funds than required at the Fed. Other banks, primarily big banks in
New York and other financial centers, tend to have a shortage of federal funds. In the
federal funds market, banks with excess funds lend to those with a shortage. These
loans, which are usually overnight transactions, are arranged at a rate of interest called
the federal funds rate. Although the fed funds market arose primarily as a way for banks to transfer balances
to meet reserve requirements, today the market has evolved to the point that many large
banks use federal funds in a straightforward way as one component of their total sources of
funding. Therefore, the fed funds rate is simply the rate of interest on very short-term loans
among financial institutions. While most investors cannot participate in this market, the fed
funds rate commands great interest as a key barometer of monetary policy.
Brokers’ Calls Individuals who buy stocks on margin borrow part of the funds to pay for the stocks from
their broker. The broker in turn may borrow the funds from a bank, agreeing to repay the
bank immediately (on call) if the bank requests it. The rate paid on such loans is usually
about 1% higher than the rate on short-term T-bills.
The LIBOR Market The London Interbank Offered Rate (LIBOR) is the rate at which large banks in London are willing to lend money among themselves. This rate, which is quoted on dollar-
denominated loans, has become the premier short-term interest rate quoted in the European
money market, and it serves as a reference rate for a wide range of transactions. For exam-
ple, a corporation might borrow at a floating rate equal to LIBOR plus 2%.
LIBOR interest rates may be tied to currencies other than the U.S. dollar. For example,
LIBOR rates are widely quoted for transactions denominated in British pounds, yen, euros,
and so on. There is also a similar rate called EURIBOR (European Interbank Offered Rate)
at which banks in the euro zone are willing to lend euros among themselves.
LIBOR is a key reference rate in the money market, and many trillions of dollars of
loans and derivative assets are tied to it. Therefore, the 2012 scandal involving the fixing
of LIBOR deeply shook these markets. The nearby box discusses those events.
Yields on Money Market Instruments Although most money market securities are of low risk, they are not risk-free. The secu-
rities of the money market promise yields greater than those on default-free T-bills, at
least in part because of greater relative riskiness. In addition, many investors require more
liquidity; thus they will accept lower yields on securities such as T-bills that can be quickly
and cheaply sold for cash. Figure 2.2 shows that bank CDs, for example, consistently have
paid a premium over T-bills. Moreover, that premium increased with economic crises such
as the energy price shocks associated with the two OPEC disturbances, the failure of Penn
Square bank, the stock market crash in 1987, the collapse of Long Term Capital Manage-
ment in 1998, and the credit crisis beginning with the breakdown of the market in subprime
mortgages beginning in 2007. If you look back to Figure 1.1 in Chapter 1, you’ll see that
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33
The LIBOR Scandals
LIBOR was designed initially as a survey of interbank lend- ing rates but soon became a key determinant of short- term interest rates with far-reaching significance. Around $350 trillion of derivative contracts have payoffs tied to it, and perhaps another $400 trillion of loans and bonds with floating interest rates linked to LIBOR are currently outstanding. LIBOR is quoted for loans in several curren- cies, e.g., the dollar, yen, euro, and British pound, and for maturities ranging from a day to a year, although 3 months is the most common.
However, LIBOR is not a rate at which actual transac- tions occur; instead, it is just a survey of “estimated” bor- rowing rates, and this has made it vulnerable to tampering. Several large banks are asked to report the rate at which they believe they can borrow in the interbank market. Out- liers are trimmed from the sample of responses, and LIBOR is calculated as the average of the mid-range estimates.
Over time, several problems surfaced. First, it appeared that banks understated the rates at which they claimed they could borrow in an effort to make themselves look financially stronger. Other surveys that asked for estimates of the rates at which other banks could borrow resulted in higher values. Moreover, LIBOR did not seem to reflect current market conditions. A majority of LIBOR submissions were unchanged from day to day even when other interest rates fluctuated, and LIBOR spreads showed surprisingly
low correlation with other measures of credit risk such as spreads on credit default swaps. Even worse, once the market came under scrutiny, it emerged that participating banks were colluding to manipulate their LIBOR submis- sions to enhance profits on their derivatives trades. Trad- ers used e-mails and instant messages to tell each other whether they wanted to see higher or lower submissions. Members of this informal cartel essentially set up a “favor bank” to help each other move the survey average up or down depending on their trading positions.
To date, around $2.5 billion of fines have been paid: Royal Bank of Scotland paid $612 million, Barclays $450 million, and UBS $1,500 million. Other banks remain under investigation. But government fines may be only the tip of the iceberg. Private lawsuits are sure to come, as anyone trading a LIBOR derivative against these banks or anyone who participated in a loan with an interest rate tied to LIBOR can claim to have been harmed.
Several reforms have been suggested. The British Bankers Association, which until recently ran the LIBOR survey, yielded responsibility for LIBOR to British regulators. Other proposals are to increase the number of submissions to make collusion more difficult and to eliminate LIBOR in less active currencies and maturities where collusion is easier. More substantive proposals would replace the survey rate with one based on actual, verifiable, transactions—i.e., real loans among banks.
W O
R D
S FR O
M T
H E ST
R E E T
0.0
0.5
1.0
1.5
2.5
3.5
4.5
2.0
3.0
P e rc
e n
ta g
e P
o in
ts 4.0
5.0 OPEC I
OPEC lI Penn Square
Market Crash
LTCM
Credit Crisis
1 9 7 0
1 9 7 2
1 9 7 4
1 9 7 6
1 9 7 8
1 9 8 0
1 9 8 2
1 9 8 4
1 9 8 6
1 9 8 8
1 9 9 0
1 9 9 2
1 9 9 4
1 9 9 6
1 9 9 8
2 0 0 0
2 0 0 2
2 0 0 4
2 0 0 6
2 0 0 8
2 0 1 0
2 0 1 2
2 0 1 4
Figure 2.2 The spread between 3-month CD and Treasury bill rates
the TED spread, the difference between the LIBOR rate and Treasury bills, also peaked
during periods of financial stress.
Money market funds are mutual funds that invest in money market instruments and
have become major sources of funding to that sector. The nearby box discusses the fallout
of the credit crisis of 2008 on those funds.
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34
The bond market is composed of longer term borrowing or debt instruments than those
that trade in the money market. This market includes Treasury notes and bonds, corporate
bonds, municipal bonds, mortgage securities, and federal agency debt.
These instruments are sometimes said to comprise the fixed-income capital market, because most of them promise either a fixed stream of income or a stream of income that
is determined according to a specific formula. In practice, these formulas can result in a
flow of income that is far from fixed. Therefore, the term fixed income is probably not fully appropriate. It is simpler and more straightforward to call these securities either debt
instruments or bonds.
Treasury Notes and Bonds The U.S. government borrows funds in large part by selling Treasury notes and Treasury bonds . T-notes are issued with maturities ranging up to 10 years, while bonds are issued with maturities ranging from 10 to 30 years. Both notes and bonds may be
2.2 The Bond Market
Money Market Funds and the Credit Crisis of 2008
Money market funds are mutual funds that invest in the short-term debt instruments that comprise the money mar- ket. In 2013, these funds had investments totaling about $2.6 trillion. They are required to hold only short-maturity debt of the highest quality: The average maturity of their holdings must be maintained at less than 3 months. Their biggest investments tend to be in commercial paper, but they also hold sizable fractions of their portfolios in cer- tificates of deposit, repurchase agreements, and Treasury securities. Because of this very conservative investment profile, money market funds typically experience extremely low price risk. Investors for their part usually acquire check- writing privileges with their funds and often use them as a close substitute for a bank account. This is feasible because the funds almost always maintain share value at $1.00 and pass along all investment earnings to their investors as interest.
Until 2008, only one fund had “broken the buck,” that is, suffered losses large enough to force value per share below $1. But when Lehman Brothers filed for bankruptcy protection on September 15, 2008, several funds that had invested heavily in its commercial paper suffered large losses. The next day, the Reserve Primary Fund, the oldest money market fund, broke the buck when its value per share fell to only $.97.
The realization that money market funds were at risk in the credit crisis led to a wave of investor redemptions similar to a run on a bank. Only three days after the Lehman bank- ruptcy, Putman’s Prime Money Market Fund announced that it was liquidating due to heavy redemptions. Fear- ing further outflows, the U.S. Treasury announced that it would make federal insurance available to money market funds willing to pay an insurance fee. This program would
thus be similar to FDIC bank insurance. With the federal insurance in place, the outflows were quelled.
However, the turmoil in Wall Street’s money market funds had already spilled over into “Main Street.” Fearing further investor redemptions, money market funds had become afraid to commit funds even over short periods, and their demand for commercial paper had effectively dried up. Firms throughout the economy had come to depend on those markets as a major source of short- term finance to fund expenditures ranging from salaries to inventories. Further breakdown in the money markets would have had an immediate crippling effect on the broad economy. To end the panic and stabilize the money mar- kets, the federal government decided to guarantee invest- ments in money market funds. The guarantee did in fact calm investors and end the run, but it put the government on the hook for a potential liability of up to $3 trillion—the assets held in money market funds at the time.
To prevent another occurrence of this crisis, the SEC later proposed that money market funds no longer be allowed to “round off” value per share to $1, but instead be forced to recognize daily changes in value. Alterna- tively, funds wishing to maintain share value at $1 would be required to set aside reserves against potential invest- ment losses. But the mutual fund industry lobbied vehe- mently against these reforms, arguing that their customers demanded stable share prices and that the proposed capi- tal requirements would be so costly that the industry would no longer be viable. In the face of this opposition, the SEC commissioners voted in 2012 against the reforms, but they were given new life when the Financial Stability Oversight Council weighed in to support them. It is still too early to predict the final resolution of the debate.
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T H
E S
T R
E E T
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C H A P T E R 2 Asset Classes and Financial Instruments 35
Listing of Treasury Issues MATURITY COUPON BID ASKED CHG
ASK YLD
Nov 15 13 Nov 15 15 Feb 15 18 Feb 15 20 Aug 15 25 May 15 30 Feb 15 36 May 15 42
4.250 4.500 3.500 8.500 6.875 6.250 4.500 3.000
105.3281 113.5078 115.0703 154.3906 158.6797 161.1094 138.0469 108.2969
105.3438 113.5391 115.1172 154.4375 158.7578 161.1875 138.1250 108.3594
–0.0078 –0.0859 –0.1406 –0.2734 –0.6641 –0.8906 –0.9375 –0.9297
0.212 0.398 0.729 1.107 1.809 2.113 2.378 2.596
Figure 2.3 Listing of Treasury bonds and notes Source: Compiled from data obtained from the Wall Street Journal Online, July 17, 2012.
What were the bid price, ask price, and yield to maturity of the 4.5% February 2036 Treasury bond displayed in Figure 2.3 ? What was its ask price the previous day?
CONCEPT CHECK 2.1
issued in increments of $100 but far more commonly trade
in denominations of $1,000. Both notes and bonds make
semiannual interest payments called coupon payments, a name derived from precomputer days, when investors
would literally clip coupons attached to the bond and pres-
ent a coupon to receive the interest payment.
Figure 2.3 is a listing of Treasury issues. Notice the high-
lighted note that matures in November 2015. Its bid price
is 113.5078. (This is the decimal version of 11365/128 . The
minimum tick size, or price increment in the Wall Street Journal listing, is generally 1/128 of a point.) Although bonds are typically traded in denominations of $1,000 par value,
prices are quoted as a percentage of par. Thus, the bid price
should be interpreted as 113.5078% of par, or $1,135.078
for the $1,000 par value bond. Similarly, the ask price at
which the bond could be sold to a dealer is 113.5391% of
par, or $1,135.391. The 2 .0859 change means that the clos-
ing price on this day fell by .0859% of par value (equivalently, by 11/128 of a point) from the
previous day’s close. Finally, the yield to maturity based on the ask price is .398%.
The yield to maturity reported in the financial pages is calculated by determining the semiannual yield and then doubling it, rather than compounding it for two half-year
periods. This use of a simple interest technique to annualize means that the yield is
quoted on an annual percentage rate (APR) basis rather than as an effective annual yield.
The APR method in this context is also called the bond equivalent yield. We discuss the yield to maturity in more detail in Part Four.
Inflation-Protected Treasury Bonds The best place to start building an investment portfolio is at the least risky end of the
spectrum. Around the world, governments of many countries, including the United States,
have issued bonds that are linked to an index of the cost of living in order to provide their
citizens with an effective way to hedge inflation risk.
In the United States inflation-protected Treasury bonds are called TIPS (Treasury
Inflation-Protected Securities). The principal amount on these bonds is adjusted in propor-
tion to increases in the Consumer Price Index. Therefore, they provide a constant stream
of income in real (inflation-adjusted) dollars. Yields on TIPS bonds should be interpreted
as real or inflation-adjusted interest rates. We return to TIPS bonds in more detail in
Chapter 14.
Federal Agency Debt Some government agencies issue their own securities to finance their activities. These
agencies usually are formed to channel credit to a particular sector of the economy that
Congress believes might not receive adequate credit through normal private sources.
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36 P A R T I Introduction
The major mortgage-related agencies are the Federal Home Loan Bank (FHLB), the
Federal National Mortgage Association (FNMA, or Fannie Mae), the Government National
Mortgage Association (GNMA, or Ginnie Mae), and the Federal Home Loan Mortgage
Corporation (FHLMC, or Freddie Mac). The FHLB borrows money by issuing securities
and lends this money to savings and loan institutions to be lent in turn to individuals bor-
rowing for home mortgages.
Although the debt of federal agencies was never explicitly insured by the federal govern-
ment, it had long been assumed that the government would assist an agency nearing default.
Those beliefs were validated when Fannie Mae and Freddie Mac encountered severe finan-
cial distress in September 2008. With both firms on the brink of insolvency, the government
stepped in and put them both into conservatorship, assigned the Federal Housing Finance
Agency to run the firms, but did in fact agree to make good on the firm’s bonds.
International Bonds Many firms borrow abroad and many investors buy bonds from foreign issuers. In addition
to national capital markets, there is a thriving international capital market, largely centered
in London.
A Eurobond is a bond denominated in a currency other than that of the country in which it is issued. For example, a dollar-denominated bond sold in Britain would be called
a Eurodollar bond. Similarly, investors might speak of Euroyen bonds, yen-denominated
bonds sold outside Japan. Because the European currency is called the euro, the term
Eurobond may be confusing. It is best to think of them simply as international bonds.
In contrast to bonds that are issued in foreign currencies, many firms issue bonds in for-
eign countries but in the currency of the investor. For example, a Yankee bond is a dollar-
denominated bond sold in the United States by a non-U.S. issuer. Similarly, Samurai bonds
are yen-denominated bonds sold in Japan by non-Japanese issuers.
Municipal Bonds Municipal bonds are issued by state and local governments. They are similar to Treasury and corporate bonds except that their interest income is exempt from federal income taxa-
tion. The interest income also is usually exempt from state and local taxation in the issuing
state. Capital gains taxes, however, must be paid on “munis” when the bonds mature or if
they are sold for more than the investor’s purchase price.
General obligation bonds are backed by the “full faith and credit” (i.e., the taxing power) of the issuer, while revenue bonds are issued to finance particular projects and are backed either by the revenues from that project or by the particular municipal agency oper-
ating the project. Typical issuers of revenue bonds are airports, hospitals, and turnpike or
port authorities. Obviously, revenue bonds are riskier in terms of default than general obli-
gation bonds. Figure 2.4 plots outstanding amounts of both types of municipal securities.
An industrial development bond is a revenue bond that is issued to finance commercial enterprises, such as the construction of a factory that can be operated by a private firm. In
effect, these private-purpose bonds give the firm access to the municipality’s ability to bor-
row at tax-exempt rates, and the federal government limits the amount of these bonds that
may be issued. 2
Like Treasury bonds, municipal bonds vary widely in maturity. A good deal of the debt
issued is in the form of short-term tax anticipation notes, which raise funds to pay for
2 A warning, however. Although interest on industrial development bonds usually is exempt from federal tax, it
can be subject to the alternative minimum tax if the bonds are used to finance projects of for-profit companies.
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C H A P T E R 2 Asset Classes and Financial Instruments 37
expenses before actual collection of taxes. Other municipal debt is long term and used to
fund large capital investments. Maturities range up to 30 years.
The key feature of municipal bonds is their tax-exempt status. Because investors pay
neither federal nor state taxes on the interest proceeds, they are willing to accept lower
yields on these securities.
An investor choosing between taxable and tax-exempt bonds must compare after-tax
returns on each bond. An exact comparison requires a computation of after-tax rates of
return that explicitly accounts for taxes on income and realized capital gains. In practice,
there is a simpler rule of thumb. If we let t denote the investor’s combined federal plus local marginal tax bracket and r denote the total before-tax rate of return available on tax- able bonds, then r (1 2 t ) is the after-tax rate available on those securities. 3 If this value exceeds the rate on municipal bonds, r m , the investor does better holding the taxable bonds. Otherwise, the tax-exempt municipals provide higher after-tax returns.
One way to compare bonds is to determine the interest rate on taxable bonds that would be
necessary to provide an after-tax return equal to that of municipals. To derive this value, we
set after-tax yields equal, and solve for the equivalent taxable yield of the tax-exempt bond. This is the rate a taxable bond must offer to match the after-tax yield on the tax-free municipal.
r (1 2 t) 5 rm (2.1)
or
r 5 rm /(12 t) (2.2)
Thus the equivalent taxable yield is simply the tax-free rate divided by 1 2 t. Table 2.2 presents equivalent taxable yields for several municipal yields and tax rates.
3 An approximation to the combined federal plus local tax rate is just the sum of the two rates. For example,
if your federal tax rate is 28% and your state rate is 5%, your combined tax rate would be approximately
33%. A more precise approach would recognize that state taxes are deductible at the federal level. You owe
federal taxes only on income net of state taxes. Therefore, for every dollar of income, your after-tax proceeds
would be (1 2 t federal ) 3 (1 2 t state ). In our example, your after-tax proceeds on each dollar earned would be (1 2 .28) 3 (1 2 .05) 5 .684, which implies a combined tax rate of 1 2 .684 5 .316, or 31.6%.
Figure 2.4 Tax-exempt debt outstanding Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.
1 9 7 9
1 9 8 0
1 9 8 1
1 9 8 2
1 9 8 4
1 9 8 6
1 9 8 8
1 9 9 0
1 9 9 2
1 9 9 4
1 9 9 6
1 9 9 8
1 9 8 3
1 9 8 5
1 9 8 7
1 9 8 9
1 9 9 1
1 9 9 3
1 9 9 5
1 9 9 7
1 9 9 9
2 0 0 1
2 0 0 0
2 0 0 2
2 0 0 4
2 0 0 6
2 0 0 8
2 0 1 0
2 0 1 2
2 0 0 3
2 0 0 5
2 0 0 7
2 0 0 9
2 0 1 1
Industrial Revenue Bonds
General Obligation
0
500
1,000
1,500$ B
il li o
n
2,000
2,500
3,000
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38 P A R T I Introduction
This table frequently appears in the marketing literature for tax-exempt mutual bond
funds because it demonstrates to high-tax-bracket investors that municipal bonds offer
highly attractive equivalent taxable yields. Each entry is calculated from Equation 2.2. If
the equivalent taxable yield exceeds the actual yields offered on taxable bonds, the investor
is better off after taxes holding municipal bonds. Notice that the equivalent taxable interest
rate increases with the investor’s tax bracket; the higher the bracket, the more valuable the
tax-exempt feature of municipals. Thus high-tax-bracket investors tend to hold municipals.
We also can use Equation 2.1 or 2.2 to find the tax bracket at which investors are indif-
ferent between taxable and tax-exempt bonds. The cutoff tax bracket is given by solving
Equation 2.2 for the tax bracket at which after-tax yields are equal. Doing so, we find that
t 5 1 2 rm r
(2.3)
Thus the yield ratio r m / r is a key determinant of the attractiveness of municipal bonds. The higher the yield ratio, the lower the cutoff tax bracket, and the more individuals will prefer
to hold municipal debt. Figure 2.5 plots the ratio of 20-year municipal debt yields to the
Table 2.2
Equivalent taxable yields corresponding to various tax-exempt yields
Tax-Exempt Yield
Marginal Tax Rate 1% 2% 3% 4% 5%
20% 1.25% 2.50% 3.75% 5.00% 6.25% 30 1.43 2.86 4.29 5.71 7.14
40 1.67 3.33 5.00 6.67 8.33
50 2.00 4.00 6.00 8.00 10.00
Figure 2.5 Ratio of yields on municipal debt to corporate Baa-rated debt Source: Authors’ calculations, using data from www.federalreserve.gov/releases/h15/data.htm.
0.5
1 9 5 3
1 9 5 6
1 9 5 9
1 9 6 2
1 9 6 5
1 9 6 8
1 9 7 1
1 9 7 4
1 9 7 7
1 9 8 0
1 9 8 3
1 9 8 6
1 9 8 9
1 9 9 2
1 9 9 5
1 9 9 8
2 0 0 1
2 0 0 4
2 0 0 7
2 0 1 0
2 0 1 3
0.6
0.7
R a
ti o
0.8
0.9
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C H A P T E R 2 Asset Classes and Financial Instruments 39
yield on Baa-rated corporate debt. The default risk of these corporate and municipal bonds
may be comparable, but certainly will fluctuate over time. For example, the sharp run-up
in the ratio in 2011 probably reflects increased concern at the time about the precarious
financial condition of several states and municipalities.
Figure 2.5 shows that in recent years, the ratio of tax-exempt to taxable yields has fluc- tuated around .70. What does this imply about the cutoff tax bracket above which tax-exempt bonds provide higher after-tax yields? Equation 2.3 shows that an inves- tor whose tax bracket (federal plus local) exceeds 1 2 .70 5 .30, or 30%, will derive a greater after-tax yield from municipals. Note, however, that it is difficult to control precisely for differences in the risks of these bonds, so the cutoff tax bracket must be taken as approximate.
Example 2.1 Taxable versus Tax-Exempt Yields
Suppose your tax bracket is 30%. Would you prefer to earn a 6% taxable return or a 4% tax-free return? What is the equivalent taxable yield of the 4% tax-free yield?
CONCEPT CHECK 2.2
Corporate Bonds Corporate bonds are the means by which private firms borrow money directly from the
public. These bonds are similar in structure to Treasury issues—they typically pay semi-
annual coupons over their lives and return the face value to the bondholder at maturity.
They differ most importantly from Treasury bonds in degree of risk. Default risk is a real
consideration in the purchase of corporate bonds, and Chapter 14 discusses this issue in
considerable detail. For now, we distinguish only among secured bonds, which have spe- cific collateral backing them in the event of firm bankruptcy; unsecured bonds, called
debentures, which have no collateral; and subordinated debentures, which have a lower priority claim to the firm’s assets in the event of bankruptcy.
Corporate bonds sometimes come with options attached. Callable bonds give the firm the option to repurchase the bond from the holder at a stipulated call price. Convertible bonds give the bondholder the option to convert each bond into a stipulated number of shares of stock. These options are treated in more detail in Chapter 14.
Mortgages and Mortgage-Backed Securities Because of the explosion in mortgage-backed securities, almost anyone can invest in a
portfolio of mortgage loans, and these securities have become a major component of the
fixed-income market. As described in Chapter 1, a mortgage-backed security is either an ownership claim in a pool of mortgages or an obligation that is secured by such a
pool. Most pass-throughs have traditionally been comprised of conforming mortgages, which means that the loans must satisfy certain underwriting guidelines (standards for
the credit-worthiness of the borrower) before they may be purchased by Fannie Mae
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40 P A R T I Introduction
or Freddie Mac. In the years leading up to the financial crisis, however, a large amount
of subprime mortgages, that is, riskier loans made to financially weaker borrowers, were bundled and sold by “private-label” issuers. Figure 2.6 illustrates the explosive
growth of both agency and private-label mortgage-backed securities, at least until
the crisis.
In an effort to make housing more affordable to low-income households, Fannie and
Freddie had been encouraged to buy subprime mortgage securities. As we saw in Chapter 1,
these loans turned out to be disastrous, with trillion-dollar losses spread among banks,
hedge funds and other investors, and Freddie and Fannie, which lost billions of dollars on
the subprime mortgage pools they had purchased. You can see from Figure 2.6 that start-
ing in 2007, the market in private-label mortgage pass-throughs began to shrink rapidly.
Agency pass-throughs shrank even more precipitously following an agreement for Freddie
and Fannie to wind down purchases of mortgages for new pass-throughs. At the same time,
existing pass-throughs shrank as healthy loans were paid off and delinquent loans were
removed from outstanding pools.
Despite these troubles, few believe that securitization itself will cease, although prac-
tices in this market are highly likely to become far more conservative than in previous
years, particularly with respect to the credit standards that must be met by the ultimate
borrower. Indeed, securitization has become an increasingly common staple of many
credit markets. For example, car loans, student loans, home equity loans, credit card
loans, and even debt of private firms now are commonly bundled into pass-through secu-
rities that can be traded in the capital market. Figure 2.7 documents the rapid growth of
nonmortgage asset-backed securities. The market expanded more than five-fold in the
decade ending 2007. After the financial crisis, it contracted considerably as the perceived
risks of credit card and home equity loans skyrocketed, but the asset-backed market is
still substantial.
Figure 2.6 Mortgage-backed securities outstanding Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.
0
1 9 7 9
1 9 8 2
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1 9 9 1
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2 0 0 0
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2 0 1 2
1,000
2,000
3,000
4,000
5,000
$ B
il li o
n s
6,000
7,000
8,000
Federal Agencies Private Issuers
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C H A P T E R 2 Asset Classes and Financial Instruments 41
Figure 2.7 Asset-backed securities outstanding Source: The Securities & Industry and Financial Markets Association, www.sifma.org .
0
1 9 9 6
1 9 9 7
1 9 9 8
1 9 9 9
2 0 0 0
2 0 0 1
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200
400
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800
1,000
1,200
1,400
1,600
1,800
2,000
Automobile
Credit Card
Equipment
Home Equity
Manufactured Housing
Student Loans
Common Stock as Ownership Shares Common stocks, also known as equity securities or equities, represent ownership shares in a corporation. Each share of common stock entitles its owner to one vote on any matters
of corporate governance that are put to a vote at the corporation’s annual meeting and to a
share in the financial benefits of ownership. 4
The corporation is controlled by a board of directors elected by the shareholders. The
board, which meets only a few times each year, selects managers who actually run the
corporation on a day-to-day basis. Managers have the authority to make most business
decisions without the board’s specific approval. The board’s mandate is to oversee the
management to ensure that it acts in the best interests of shareholders.
The members of the board are elected at the annual meeting. Shareholders who do
not attend the annual meeting can vote by proxy, empowering another party to vote in their name. Management usually solicits the proxies of shareholders and normally gets a
vast majority of these proxy votes. Thus, management usually has considerable discretion
2.3 Equity Securities
4 A corporation sometimes issues two classes of common stock, one bearing the right to vote, the other not.
Because of its restricted rights, the nonvoting stock might sell for a lower price.
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42 P A R T I Introduction
to run the firm as it sees fit—without daily oversight from the equityholders who actually
own the firm.
We noted in Chapter 1 that such separation of ownership and control can give rise to
“agency problems,” in which managers pursue goals not in the best interests of sharehold-
ers. However, there are several mechanisms that alleviate these agency problems. Among
these are compensation schemes that link the success of the manager to that of the firm;
oversight by the board of directors as well as outsiders such as security analysts, creditors,
or large institutional investors; the threat of a proxy contest in which unhappy shareholders
attempt to replace the current management team; or the threat of a takeover by another firm.
The common stock of most large corporations can be bought or sold freely on one
or more stock exchanges. A corporation whose stock is not publicly traded is said to be
closely held. In most closely held corporations, the owners of the firm also take an active
role in its management. Therefore, takeovers are generally not an issue.
Characteristics of Common Stock The two most important characteristics of common stock as an investment are its residual claim and limited liability features.
Residual claim means that stockholders are the last in line of all those who have a
claim on the assets and income of the corporation. In a liquidation of the firm’s assets the
shareholders have a claim to what is left after all other claimants such as the tax authorities,
employees, suppliers, bondholders, and other creditors have been paid. For a firm not in
liquidation, shareholders have claim to the part of operating income left over after inter-
est and taxes have been paid. Management can either pay
this residual as cash dividends to shareholders or reinvest
it in the business to increase the value of the shares. Limited liability means that the most shareholders
can lose in the event of failure of the corporation is their
original investment. Unlike owners of unincorporated
businesses, whose creditors can lay claim to the personal
assets of the owner (house, car, furniture), corporate
shareholders may at worst have worthless stock. They are
not personally liable for the firm’s obligations.
Stock Market Listings Figure 2.8 presents key trading data for a small sample of stocks traded on the New York
Stock Exchange. The NYSE is one of several markets in which investors may buy or sell
shares of stock. We will examine these markets in detail in Chapter 3.
To interpret Figure 2.8 , consider the highlighted listing for General Electric. The table
provides the ticker symbol (GE), the closing price of the stock ($19.72), and its change
( 1 $.13) from the previous trading day. About 45.3 million shares of GE traded on this
day. The listing also provides the highest and lowest price at which GE has traded in the
last 52 weeks. The .68 value in the Dividend column means that the last quarterly divi-
dend payment was $.17 per share, which is consistent with annual dividend payments of
$.17 3 4 5 $.68. This corresponds to an annual dividend yield (i.e., annual dividend per
dollar paid for the stock) of .68/19.72 5 .0345, or 3.45%.
The dividend yield is only part of the return on a stock investment. It ignores prospective
capital gains (i.e., price increases) or losses. Low-dividend firms presumably offer greater prospects for capital gains, or investors would not be willing to hold these stocks in their port-
folios. If you scan Figure 2.8 , you will see that dividend yields vary widely across companies.
a. If you buy 100 shares of IBM stock, to what are you entitled?
b. What is the most money you can make on this investment over the next year?
c. If you pay $180 per share, what is the most money you could lose over the year?
CONCEPT CHECK 2.3
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C H A P T E R 2 Asset Classes and Financial Instruments 43
The P/E ratio, or price–earnings ratio, is the ratio of the current stock price to last year’s earnings per share. The P/E ratio tells us how much stock purchasers must pay per
dollar of earnings that the firm generates. For GE, the ratio of price to earnings is 16.01.
The P/E ratio also varies widely across firms. Where the dividend yield and P/E ratio are
not reported in Figure 2.8 , the firms have zero dividends, or zero or negative earnings. We
shall have much to say about P/E ratios in Chapter 18. Finally, we see that GE’s stock price
has increased by 10.11% since the beginning of the year.
Preferred Stock Preferred stock has features similar to both equity and debt. Like a bond, it promises to pay to its holder a fixed amount of income each year. In this sense preferred stock
is similar to an infinite-maturity bond, that is, a perpetuity. It also resembles a bond in
that it does not convey voting power regarding the management of the firm. Preferred
stock is an equity investment, however. The firm retains discretion to make the dividend
payments to the preferred stockholders; it has no contractual obligation to pay those
dividends. Instead, preferred dividends are usually cumulative; that is, unpaid dividends cumulate and must be paid in full before any dividends may be paid to holders of com-
mon stock. In contrast, the firm does have a contractual obligation to make the interest
payments on the debt. Failure to make these payments sets off corporate bankruptcy
proceedings.
Preferred stock also differs from bonds in terms of its tax treatment for the firm.
Because preferred stock payments are treated as dividends rather than interest, they are
not tax-deductible expenses for the firm. This disadvantage is somewhat offset by the fact
that corporations may exclude 70% of dividends received from domestic corporations in
the computation of their taxable income. Preferred stocks therefore make desirable fixed-
income investments for some corporations.
Even though preferred stock ranks after bonds in terms of the priority of its claims
to the assets of the firm in the event of corporate bankruptcy, preferred stock often
sells at lower yields than do corporate bonds. Presumably, this reflects the value of the
dividend exclusion, because the higher risk of preferred would tend to result in higher
yields than those offered by bonds. Individual investors, who cannot use the 70% tax
Game Stop CI A GME 16.74 0.17 1,584,470 26.66 16.36 0.60 3.58 6.80 230.63 Gannett GCI 14.65 20.04 4,277,615 16.26 8.28 0.80 5.46 8.07 9.57 Gap GPS 28.47 0.50 5,952,210 29.23 15.08 0.50 1.76 16.88 53.48 Gardner Denver GDI 48.30 0.08 797,258 92.93 45.54 0.20 0.41 9.01 237.32 Gartner IT 45.71 20.14 580,056 46.69 31.98 .... .... 30.98 31.46 GasLog GLOG 10.24 0.08 126,666 13.34 8.76 .... .... 217.49 GATX GMT 38.76 0.27 226,825 45.50 28.90 1.20 3.10 15.19 211.22 Gaylord Entertainment GET 37.78 0.44 273,954 40.37 17.39 .... .... 105.03 56.50 Gazit-Globe GZT 9.30 20.11 2,152 11.07 8.41 0.43 4.61 .... 21.59 GenCorp GY 6.69 20.01 321,785 7.27 3.74 .... .... 93.96 25.75 Genco Shipping&Trading GNK 2.79 0.04 355,441 10.14 2.72 .... .... ....dd 258.73 Generac Holdings GNRC 22.21 20.14 169,519 30.61 15.41 .... .... 4.36 220.76 General Cable BGC 26.80 0.47 322,159 45.20 20.21 .... .... 19.43 7.16 General Dynamics GD 64.69 0.65 1,359,120 74.54 53.95 2.04 3.15 9.31 22.59 General Electric GE 19.72 0.13 45,307,178 21.00 14.02 0.68 3.45 16.01 10.11
NAME YTD% CHGP/EYIELDDIV52 WK LOW52 WK HIGHVOLUMENET CHGCLOSESYMBOL
....
Figure 2.8 Listing of stocks traded on the New York Stock Exchange Source: Compiled from data from The Wall Street Journal Online, July 18, 2012.
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44 P A R T I Introduction
exclusion, generally will find preferred stock yields unattractive relative to those on
other available assets.
Preferred stock is issued in variations similar to those of corporate bonds. It may be
callable by the issuing firm, in which case it is said to be redeemable. It also may be con- vertible into common stock at some specified conversion ratio. Adjustable-rate preferred
stock is another variation that, like adjustable-rate bonds, ties the dividend to current mar-
ket interest rates.
Depository Receipts American Depository Receipts, or ADRs, are certificates traded in U.S. markets that repre-
sent ownership in shares of a foreign company. Each ADR may correspond to ownership of
a fraction of a foreign share, one share, or several shares of the foreign corporation. ADRs
were created to make it easier for foreign firms to satisfy U.S. security registration require-
ments. They are the most common way for U.S. investors to invest in and trade the shares
of foreign corporations.
Stock Market Indexes The daily performance of the Dow Jones Industrial Average is a staple portion of the
evening news report. Although the Dow is the best-known measure of the performance of
the stock market, it is only one of several indicators. Other more broadly based indexes
are computed and published daily. In addition, several indexes of bond market perfor-
mance are widely available.
The ever-increasing role of international trade and investments has made indexes of
foreign financial markets part of the general news as well. Thus foreign stock exchange
indexes such as the Nikkei Average of Tokyo and the Financial Times index of London are
fast becoming household names.
Dow Jones Averages The Dow Jones Industrial Average (DJIA) of 30 large, “blue-chip” corporations has been
computed since 1896. Its long history probably accounts for its preeminence in the public
mind. (The average covered only 20 stocks until 1928.)
Originally, the DJIA was calculated as the average price of the stocks included in the
index. Thus, one would add up the prices of the 30 stocks in the index and divide by 30.
The percentage change in the DJIA would then be the percentage change in the average
price of the 30 shares.
This procedure means that the percentage change in the DJIA measures the return
(excluding dividends) on a portfolio that invests one share in each of the 30 stocks in the
index. The value of such a portfolio (holding one share of each stock in the index) is the
sum of the 30 prices. Because the percentage change in the average of the 30 prices is the same as the percentage change in the sum of the 30 prices, the index and the portfolio have the same percentage change each day.
Because the Dow corresponds to a portfolio that holds one share of each component
stock, the investment in each company in that portfolio is proportional to the company’s
share price. Therefore, the Dow is called a price-weighted average.
2.4 Stock and Bond Market Indexes
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C H A P T E R 2 Asset Classes and Financial Instruments 45
Consider the data in Table 2.3 for a hypothetical two-stock version of the Dow Jones Average. Let’s compare the changes in the value of the portfolio holding one share of each firm and the price-weighted index. Stock ABC starts at $25 a share and increases to $30. Stock XYZ starts at $100, but falls to $90.
Portfolio: Initial value 5 $25 1 $100 5 $125 Final value 5 $30 1 $90 5 $120 Percentage change in portfolio value 5 2 5/125 5 2 .04 5 2 4% Index: Initial index value 5 (25 1 100)/2 5 62.5 Final index value 5 (30 1 90)/2 5 60 Percentage change in index 5 2 2.5/62.5 5 2 .04 5 2 4%
The portfolio and the index have identical 4% declines in value. Notice that price-weighted averages give higher-priced shares more weight
in determining performance of the index. For example, although ABC increased by 20%, while XYZ fell by only 10%, the index dropped in value. This is because the 20% increase in ABC represented a smaller price gain ($5 per share) than the 10% decrease in XYZ ($10 per share). The “Dow portfolio” has four times as much invested in XYZ as in ABC because XYZ’s price is four times that of ABC. Therefore, XYZ dominates the average. We conclude that a high-price stock can dominate a price-weighted average.
Example 2.2 Price-Weighted Average
You might wonder why the DJIA is now (in early 2013) at a level of about 14,000 if it is
supposed to be the average price of the 30 stocks in the index. The DJIA no longer equals
the average price of the 30 stocks because the averaging procedure is adjusted whenever a
stock splits or pays a stock dividend of more than 10%, or when one company in the group
of 30 industrial firms is replaced by another. When these events occur, the divisor used to
compute the “average price” is adjusted so as to leave the index unaffected by the event.
Suppose XYZ were to split two for one so that its share price fell to $50. We would not want the average to fall, as that would incorrectly indicate a fall in the general level of market prices. Following a split, the divisor must be reduced to a value that leaves the average unaffected. Table 2.4 illustrates this point. The initial share price of XYZ, which
Example 2.3 Splits and Price-Weighted Averages
Stock Initial Price
Final Price
Shares (million)
Initial Value of Outstanding
Stock ($ million)
Final Value of Outstanding
Stock ($ million)
ABC $25 $30 20 $500 $600 XYZ 100 90 1 100 90
Total $600 $690
Table 2.3
Data to construct stock price indexes
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46 P A R T I Introduction
In the same way that the divisor is updated for stock splits, if one firm is dropped from
the average and another firm with a different price is added, the divisor has to be updated
to leave the average unchanged by the substitution. By 2013, the divisor for the Dow Jones
Industrial Average had fallen to a value of about .1302.
Because the Dow Jones averages are based on small numbers of firms, care must be
taken to ensure that they are representative of the broad market. As a result, the composi-
tion of the average is changed every so often to reflect changes in the economy. Table 2.5
presents the composition of the Dow industrials in 1928 as well as its composition as of
mid-2013. The table presents striking evidence of the changes in the U.S. economy in the
last 85 years. Many of the “bluest of the blue chip” companies in 1928 no longer exist, and
the industries that were the backbone of the economy in 1928 have given way to some that
could not have been imagined at the time.
was $100 in Table 2.3 , falls to $50 if the stock splits at the beginning of the period. Notice that the number of shares outstanding doubles, leaving the market value of the total shares unaffected.
We find the new divisor as follows. The index value before the stock split 5 125/2 5 62.5. We must find a new divisor, d, that leaves the index unchanged after XYZ splits and its price falls to $50. Therefore, we solve for d in the following equation:
Price of ABC 1 Price of XYZ
d 5
25 1 50 d
5 62.5
which implies that the divisor must fall from its original value of 2.0 to a new value of 1.20. Because the split changes the price of stock XYZ, it also changes the relative weights
of the two stocks in the price-weighted average. Therefore, the return of the index is affected by the split.
At period-end, ABC will sell for $30, while XYZ will sell for $45, representing the same negative 10% return it was assumed to earn in Table 2.3 . The new value of the price-weighted average is (30 1 45)/1.20 5 62.5, the same as its value at the start of the year; therefore, the rate of return is zero, rather than the 2 4% return that we calcu- lated in the absence of a split.
The split reduces the relative weight of XYZ because its initial price is lower; because XYZ is the poorer performing stock, the performance of the average is higher. This example illustrates that the implicit weighting scheme of a price- weighted average is somewhat arbitrary, being determined by the prices rather than by the outstanding market values (price per share times number of shares) of the shares in the average.
Table 2.4
Data to construct stock price indexes after a stock split
Stock Initial Price
Final Price
Shares (million)
Initial Value of Outstanding
Stock ($ million)
Final Value of Outstanding
Stock ($ million)
ABC $25 $30 20 $500 $600 XYZ 50 45 2 100 90 Total $600 $690
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C H A P T E R 2 Asset Classes and Financial Instruments 47
Suppose the price of XYZ in Table 2.3 increases to $110, while ABC falls to $20. Find the percentage change in the price-weighted average of these two stocks. Compare that to the percentage return of a portfolio holding one share in each company.
CONCEPT CHECK 2.4
Table 2.5
Companies included in the Dow Jones Industrial Average: 1928 and 2013
Dow Industrials in 1928 Current Dow Companies Ticker Symbol Industry
Year Added to Index
Wright Aeronautical 3M MMM Diversified industrials 1976
Allied Chemical Alcoa AA Aluminum 1959
North American American Express AXP Consumer finance 1982
Victor Talking Machine AT&T T Telecommunications 1999
International Nickel Bank of America BAC Banking 2008
International Harvester Boeing BA Aerospace & defense 1987
Westinghouse Caterpillar CAT Construction 1991
Texas Gulf Sulphur Chevron CVX Oil and gas 2008
General Electric Cisco Systems CSCO Computer equipment 2009
American Tobacco Coca-Cola KO Beverages 1987
Texas Corp DuPont DD Chemicals 1935
Standard Oil (NJ) ExxonMobil XOM Oil & gas 1928
Sears Roebuck General Electric GE Diversified industrials 1907
General Motors Hewlett-Packard HPQ Computers 1997
Chrysler Home Depot HD Home improvement retailers 1999
Atlantic Refining Intel INTC Semiconductors 1999
Paramount Publix IBM IBM Computer services 1979
Bethlehem Steel Johnson & Johnson JNJ Pharmaceuticals 1997
General Railway Signal JPMorgan Chase JPM Banking 1991
Mack Trucks McDonald’s MCD Restaurants 1985
Union Carbide Merck MRK Pharmaceuticals 1979
American Smelting Microsoft MSFT Software 1999
American Can Pfizer PFE Pharmaceuticals 2004
Postum Inc Procter & Gamble PG Household products 1932
Nash Motors Travelers TRV Insurance 2009
American Sugar UnitedHealth Group UNH Health insurance 2012
Goodrich United Technologies UTX Aerospace 1939
Radio Corp Verizon VZ Telecommunications 2004
Woolworth Wal-Mart WMT Retailers 1997
U.S. Steel Walt Disney DIS Broadcasting & entertainment 1991
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48 P A R T I Introduction
Standard & Poor’s Indexes The Standard & Poor’s Composite 500 (S&P 500) stock index represents an improvement
over the Dow Jones Averages in two ways. First, it is a more broadly based index of 500
firms. Second, it is a market-value-weighted index . In the case of the firms XYZ and ABC in Example 2.2, the S&P 500 would give ABC five times the weight given to XYZ
because the market value of its outstanding equity is five times larger, $500 million versus
$100 million.
The S&P 500 is computed by calculating the total market value of the 500 firms in the
index and the total market value of those firms on the previous day of trading. The percent-
age increase in the total market value from one day to the next represents the increase in
the index. The rate of return of the index equals the rate of return that would be earned by
an investor holding a portfolio of all 500 firms in the index in proportion to their market
values, except that the index does not reflect cash dividends paid by those firms.
Actually, most indexes today use a modified version of market-value weights. Rather
than weighting by total market value, they weight by the market value of free float, that is, by the value of shares that are freely tradable among investors. For example, this procedure
does not count shares held by founding families or governments. These shares are effec-
tively not available for investors to purchase. The distinction is more important in Japan
and Europe, where a higher fraction of shares are held in such nontraded portfolios.
To illustrate how value-weighted indexes are computed, look again at Table 2.3 . The final value of all outstanding stock in our two-stock universe is $690 million. The initial value was $600 million. Therefore, if the initial level of a market-value-weighted index of stocks ABC and XYZ were set equal to an arbitrarily chosen starting value such as 100, the index value at year-end would be 100 3 (690/600) 5 115. The increase in the index reflects the 15% return earned on a portfolio consisting of those two stocks held in proportion to outstanding market values.
Unlike the price-weighted index, the value-weighted index gives more weight to ABC. Whereas the price-weighted index fell because it was dominated by higher-price XYZ, the value-weighted index rises because it gives more weight to ABC, the stock with the higher total market value.
Note also from Tables 2.3 and 2.4 that market-value-weighted indexes are unaffected by stock splits. The total market value of the outstanding XYZ stock decreases from $100 million to $90 million regardless of the stock split, thereby rendering the split irrelevant to the performance of the index.
Example 2.4 Value-Weighted Indexes
Reconsider companies XYZ and ABC from Concept Check 2.4. Calculate the percentage change in the market-value-weighted index. Compare that to the rate of return of a portfolio that holds $500 of ABC stock for every $100 of XYZ stock (i.e., an index portfolio).
CONCEPT CHECK 2.5
A nice feature of both market-value-weighted and price-weighted indexes is that
they reflect the returns to straightforward portfolio strategies. If one were to buy shares
in each component firm in the index in proportion to its outstanding market value, the
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C H A P T E R 2 Asset Classes and Financial Instruments 49
value-weighted index would perfectly track capital gains on the underlying portfolio.
Similarly, a price-weighted index tracks the returns on a portfolio comprised of an equal
number of shares of each firm.
Investors today can easily buy market indexes for their portfolios. One way is to pur-
chase shares in mutual funds that hold shares in proportion to their representation in the
S&P 500 or another index. These index funds yield a return equal to that of the index and so provide a low-cost passive investment strategy for equity investors. Another approach
is to purchase an exchange-traded fund, or ETF, which is a portfolio of shares that can be bought or sold as a unit, just as one can buy or sell a single share of stock. Available ETFs
range from portfolios that track extremely broad global market indexes all the way to nar-
row industry indexes. We discuss both mutual funds and ETFs in detail in Chapter 4.
Standard & Poor’s also publishes a 400-stock Industrial Index, a 20-stock Transportation
Index, a 40-stock Utility Index, and a 40-stock Financial Index.
Other U.S. Market-Value Indexes The New York Stock Exchange publishes a market-value-weighted composite index of
all NYSE-listed stocks, in addition to subindexes for industrial, utility, transportation,
and financial stocks. These indexes are even more broadly based than the S&P 500. The
National Association of Securities Dealers publishes an index of more than 3,000 firms
traded on the NASDAQ market.
The ultimate U.S. equity index so far computed is the Wilshire 5000 index of the market
value of essentially all actively traded stocks in the U.S. Despite its name, the index actu-
ally includes more than 5,000 stocks. The performance of many of these indexes appears
daily in The Wall Street Journal.
Equally Weighted Indexes Market performance is sometimes measured by an equally weighted average of the returns
of each stock in an index. Such an averaging technique, by placing equal weight on each
return, corresponds to an implicit portfolio strategy that invests equal dollar values in each
stock. This is in contrast to both price weighting (which requires equal numbers of shares
of each stock) and market-value weighting (which requires investments in proportion to
outstanding value).
Unlike price- or market-value-weighted indexes, equally weighted indexes do not corre-
spond to buy-and-hold portfolio strategies. Suppose that you start with equal dollar invest-
ments in the two stocks of Table 2.3 , ABC and XYZ. Because ABC increases in value
by 20% over the year while XYZ decreases by 10%, your portfolio no longer is equally
weighted. It is now more heavily invested in ABC. To reset the portfolio to equal weights,
you would need to rebalance: Sell off some ABC stock and/or purchase more XYZ stock.
Such rebalancing would be necessary to align the return on your portfolio with that on the
equally weighted index.
Foreign and International Stock Market Indexes Development in financial markets worldwide includes the construction of indexes for these
markets. Among these are the Nikkei (Japan), FTSE (U.K.; pronounced “footsie”), DAX
(Germany), Hang Seng (Hong Kong), and TSX (Canada).
A leader in the construction of international indexes has been MSCI (Morgan Stanley
Capital International), which computes over 50 country indexes and several regional
indexes. Table 2.6 presents many of the indexes computed by MSCI.
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50 P A R T I Introduction
Bond Market Indicators Just as stock market indexes provide guidance concerning the performance of the overall
stock market, several bond market indicators measure the performance of various catego-
ries of bonds. The three most well-known groups of indexes are those of Merrill Lynch,
Barclays (formerly, the Lehman Brothers index), and Salomon Smith Barney (now part
of Citigroup). Figure 2.9 shows the components of the U.S. fixed-income market in 2012.
The major problem with bond market indexes is that true rates of return on many
bonds are difficult to compute because the infrequency with which the bonds trade
makes reliable up-to-date prices difficult to obtain. In practice, some prices must
be estimated from bond-valuation models. These “matrix” prices may differ from true
market values.
Table 2.6
Sample of MSCI stock indexes Source: MSCI, www.msci.com. Used with permission.
Regional Indexes Countries
Developed Markets Emerging Markets Developed Markets Emerging Markets
EAFE (Europe, Australia, Far East) Emerging Markets (EM) Australia Brazil
EASEA (EAFE excluding Japan) EM Asia Austria Chile
Europe EM Far East Belgium China
European Monetary Union (EMU) EM Latin America Canada Colombia
Far East EM Eastern Europe Denmark Czech Republic
Kokusai (World excluding Japan) EM Europe Finland Egypt
Nordic countries EM Europe & Middle East France Hungary
North America Germany India
Pacific Greece Indonesia
World Hong Kong Korea
G7 countries Ireland Malaysia
World excluding U.S. Israel Mexico
Italy Morocco
Japan Peru
Netherlands Poland
New Zealand Russia
Norway South Africa
Portugal Taiwan
Singapore Thailand
Spain Turkey
Sweden
Switzerland
U.K.
U.S.
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C H A P T E R 2 Asset Classes and Financial Instruments 51
$6,202.0
Treasury Debt
Federal Agency and Gov’t Sponsored Enterprise
Corporate Bonds
Tax-Exempt*
Mortgage-Backed Securities
Other Asset-Backed Securities $5,192.5
$3,428.0
$10,827.5
$2,953.1 $1,049.3
Figure 2.9 The U.S. fixed-income market (values in $ billions) Source: Flow of Funds Accounts of the United States: Flows & Outstandings, Board of Governors of the Federal Reserve System, June 2012.
One of the most significant developments in financial markets in recent years has been
the growth of futures, options, and related derivatives markets. These instruments provide
payoffs that depend on the values of other assets such as commodity prices, bond and stock
prices, or market index values. For this reason these instruments sometimes are called
derivative assets. Their values derive from the values of other assets.
Options A call option gives its holder the right to purchase an asset for a specified price, called the exercise or strike price , on or before a specified expiration date. For example, a July call option on IBM stock with an exercise price of $180 entitles its owner to purchase IBM
stock for a price of $180 at any time up to and including the expiration date in July. Each
option contract is for the purchase of 100 shares. However, quotations are made on a per-
share basis. The holder of the call need not exercise the option; it will be profitable to exer-
cise only if the market value of the asset that may be purchased exceeds the exercise price.
When the market price exceeds the exercise price, the option holder may “call away”
the asset for the exercise price and reap a payoff equal to the difference between the stock
price and the exercise price. Otherwise, the option will be left unexercised. If not exercised
before the expiration date of the contract, the option simply expires and no longer has
value. Calls therefore provide greater profits when stock prices increase and thus represent
bullish investment vehicles.
In contrast, a put option gives its holder the right to sell an asset for a specified exercise price on or before a specified expiration date. A July put on IBM with an exercise price
of $180 thus entitles its owner to sell IBM stock to the put writer at a price of $180 at any
time before expiration in July, even if the market price of IBM is lower than $180. Whereas
profits on call options increase when the asset increases in value, profits on put options
2.5 Derivative Markets
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52 P A R T I Introduction
increase when the asset value falls. The put is exercised only if its holder can deliver an
asset worth less than the exercise price in return for the exercise price.
Figure 2.10 is an excerpt of the options quotations for IBM from the online edition of
The Wall Street Journal. The price of IBM shares on this date was $183.65. The first two columns give the expiration month and exercise (or strike) price for each option. We have
included listings for call and put options with exercise prices of $180 and $185 per share
and with expiration dates in July, August, and October 2012 and January 2013.
The next columns provide the closing prices, trading volume, and open interest (out-
standing contracts) of each option. For example, 1,998 contracts traded on the July 2012
expiration call with an exercise price of $180. The last trade was at $5.50, meaning that
an option to purchase one share of IBM at an exercise price of $180 sold for $5.50. Each
option contract (on 100 shares) therefore costs $550. Notice that the prices of call options decrease as the exercise price increases. For exam-
ple, the July expiration call with exercise price $185 costs only $2.80. This makes sense,
because the right to purchase a share at a higher price is less valuable. Conversely, put
prices increase with the exercise price. The right to sell IBM at a price of $180 in July costs
$2.11, while the right to sell at $185 costs $4.20.
Option prices also increase with time until expiration. Clearly, one would rather have
the right to buy IBM for $180 at any time until October rather than at any time until July.
Not surprisingly, this shows up in a higher price for the October expiration options. For
example, the call with exercise price $180 expiring in October sells for $9.70 compared to
only $5.50 for the July call.
Open Interest
8123
3621
4984
3196
7370
3367
2692
10731
Volume
847
245
76
2725
634
783
243
Last
2.11 3080
3.70
6.85
10.25
4.20
6.26
9.10
12.01
Open Interest
1998
2105
424
2372
3897
2656
969
3156
Volume
620
406
184
2231
656
843
135
Last
5.50
6.85
9.70
2.80
4.10
6.99
9.75
Strike
180.00
180.00
180.00
180.00
185.00
185.00
185.00
185.00
Expiration
Jul
Aug
Oct
Jan
Jul
Aug
Oct
Jan
Underlying stock price: 183.65
PRICES AT CLOSE, July 17, 2012
PutCall
12.58 52
IBM (IBM)
Figure 2.10 Stock options on IBM Source: Compiled from data downloaded from The Wall Street Journal Online, July 17, 2012.
What would be the profit or loss per share to an investor who bought the July 2012 expiration IBM call option with exercise price $180 if the stock price at the expira- tion date is $187? What about a purchaser of the put option with the same exercise price and expiration?
CONCEPT CHECK 2.6
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C H A P T E R 2 Asset Classes and Financial Instruments 53
Futures Contracts A futures contract calls for delivery of an asset (or in some cases, its cash value) at a spec- ified delivery or maturity date for an agreed-upon price, called the futures price, to be paid
at contract maturity. The long position is held by the trader who commits to purchasing the asset on the delivery date. The trader who takes the short position commits to delivering the asset at contract maturity.
Figure 2.11 illustrates the listing of the corn futures contract on the Chicago Board of
Trade for July 17, 2012. Each contract calls for delivery of 5,000 bushels of corn. Each row
details prices for contracts expiring on various dates. The first row is for the nearest term
or “front” contract, with maturity in September 2012. The most recent price was $7.95 per
bushel. (The numbers after the apostrophe denote eighths of a cent.) That price is up $.155
from yesterday’s close. The next columns show the contract’s opening price as well as the
high and low price during the trading day. Volume is the number of contracts trading that
day; open interest is the number of outstanding contracts.
The trader holding the long position profits from price increases. Suppose that at con-
tract maturity, corn is selling for $7.97 per bushel. The long position trader who entered
the contract at the futures price of $7.95 on July 17 would pay the previously agreed-upon
$7.95 for each bushel of corn, which at contract maturity would be worth $7.97.
Because each contract calls for delivery of 5,000 bushels, the profit to the long posi-
tion would equal 5,000 3 ($7.97 2 $7.95) 5 $1,000. Conversely, the short position must
deliver 5,000 bushels for the previously agreed-upon futures price. The short position’s
loss equals the long position’s profit.
The right to purchase the asset at an agreed-upon price, as opposed to the obligation,
distinguishes call options from long positions in futures contracts. A futures contract
obliges the long position to purchase the asset at the futures price; the call option, in con- trast, conveys the right to purchase the asset at the exercise price. The purchase will be made only if it yields a profit.
Clearly, a holder of a call has a better position than the holder of a long position on a
futures contract with a futures price equal to the option’s exercise price. This advantage, of
course, comes only at a price. Call options must be purchased; futures contracts are entered
into without cost. The purchase price of an option is called the premium. It represents the com- pensation the purchaser of the call must pay for the ability to exercise the option only when it
is profitable to do so. Similarly, the difference between a put option and a short futures posi-
tion is the right, as opposed to the obligation, to sell an asset at an agreed-upon price.
Figure 2.11 Corn futures prices in the Chicago Board of Trade, July 17, 2012 Source: The Wall Street Journal Online, July 17, 2012.
Open Int
369243
499807
135778
21882
57618
9120
54205
Volume
179014
24738
8119
12310
1833
4510
Low
763’4 83008
755’6
757’2
755’0
749’0
660’0
625’0
High
797’0
785’6
784’4
780’4
774’0
673’0
637’0
Open
780’0
772’4
772’4
763’2
670’0
633’0
Chg
15’4
12’2
11’4
10’6
–1’6
–1’0
Last
795’0
783’4
783’2
779’6
773’4
669’4
634’0
Month
Sep 2012
Dec 2012
Mar 2013
May 2013
Jul 2013
Sep 2013
Dec 2013
11’2 769’2
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money market
capital markets
ask price
bid price
bid–ask spread
certificate of deposit
commercial paper
banker’s acceptance
Eurodollars
repurchase agreements
federal funds
London Interbank Offered
Rate (LIBOR)
Treasury notes
Treasury bonds
yield to maturity
municipal bonds
equivalent taxable yield
equities
residual claim
limited liability
capital gains
price–earnings ratio
preferred stock
price-weighted average
market-value-weighted index
index funds
derivative assets
call option
exercise (strike) price
put option
futures contract
KEY TERMS
54 P A R T I Introduction
1. Money market securities are very short-term debt obligations. They are usually highly market- able and have relatively low credit risk. Their low maturities and low credit risk ensure minimal
capital gains or losses. These securities trade in large denominations, but they may be purchased
indirectly through money market funds.
2. Much of U.S. government borrowing is in the form of Treasury bonds and notes. These are coupon-paying bonds usually issued at or near par value. Treasury notes and bonds are similar in
design to coupon-paying corporate bonds.
3. Municipal bonds are distinguished largely by their tax-exempt status. Interest payments (but not capital gains) on these securities are exempt from federal income taxes. The equivalent taxable
yield offered by a municipal bond equals r m /(1 2 t ), where r m is the municipal yield and t is the investor’s tax bracket.
4. Mortgage pass-through securities are pools of mortgages sold in one package. Owners of pass- throughs receive the principal and interest payments made by the borrowers. The originator that
issued the mortgage merely services it, simply “passing through” the payments to the purchas-
ers of the mortgage. A federal agency may guarantee the payments of interest and principal on
mortgages pooled into its pass-through securities, but these guarantees are absent in private-label
pass-throughs.
5. Common stock is an ownership share in a corporation. Each share entitles its owner to one vote on matters of corporate governance and to a prorated share of the dividends paid to shareholders.
Stock, or equity, owners are the residual claimants on the income earned by the firm.
6. Preferred stock usually pays fixed dividends for the life of the firm; it is a perpetuity. A firm’s failure to pay the dividend due on preferred stock, however, does not precipitate corporate bank-
ruptcy. Instead, unpaid dividends simply cumulate. Newer varieties of preferred stock include
convertible and adjustable-rate issues.
7. Many stock market indexes measure the performance of the overall market. The Dow Jones averages, the oldest and best-known indicators, are price-weighted indexes. Today, many broad-
based, market-value-weighted indexes are computed daily. These include the Standard & Poor’s
500 stock index, the NYSE index, the NASDAQ index, the Wilshire 5000 index, and indexes of
many non-U.S. stock markets.
8. A call option is a right to purchase an asset at a stipulated exercise price on or before an expiration date. A put option is the right to sell an asset at some exercise price. Calls increase in value while
puts decrease in value as the price of the underlying asset increases.
9. A futures contract is an obligation to buy or sell an asset at a stipulated futures price on a maturity date. The long position, which commits to purchasing, gains if the asset value increases while the
short position, which commits to delivering, loses.
SUMMARY
Related Web sites for this chapter are available at www. mhhe.com/bkm
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C H A P T E R 2 Asset Classes and Financial Instruments 55
KEY EQUATIONS Equivalent taxable yield: rmuni
1 2 tax rate , where rmuni is the rate on tax-free municipal debt
Cutoff tax rate (for indifference to taxable versus tax-free bonds): 1 2 rmuni
rtaxable
PROBLEM SETS 1. In what ways is preferred stock like long-term debt? In what ways is it like equity? 2. Why are money market securities sometimes referred to as “cash equivalents”?
3. Which of the following correctly describes a repurchase agreement?
a. The sale of a security with a commitment to repurchase the same security at a specified future date and a designated price.
b. The sale of a security with a commitment to repurchase the same security at a future date left unspecified, at a designated price.
c. The purchase of a security with a commitment to purchase more of the same security at a specified future date.
4. What would you expect to happen to the spread between yields on commercial paper and Trea-
sury bills if the economy were to enter a steep recession?
5. What are the key differences between common stock, preferred stock, and corporate bonds?
6. Why are high-tax-bracket investors more inclined to invest in municipal bonds than low-bracket
investors?
7. Turn back to Figure 2.3 and look at the Treasury bond maturing in May 2030.
a. How much would you have to pay to purchase one of these bonds? b. What is its coupon rate? c. What is the yield to maturity of the bond?
8. Suppose investors can earn a return of 2% per 6 months on a Treasury note with 6 months
remaining until maturity. What price would you expect a 6-month maturity Treasury bill to sell
for?
9. Find the after-tax return to a corporation that buys a share of preferred stock at $40, sells it at
year-end at $40, and receives a $4 year-end dividend. The firm is in the 30% tax bracket.
10. Turn to Figure 2.8 and look at the listing for General Dynamics.
a. How many shares could you buy for $5,000? b. What would be your annual dividend income from those shares? c. What must be General Dynamics earnings per share? d. What was the firm’s closing price on the day before the listing?
11. Consider the three stocks in the following table. P t represents price at time t, and Q t represents shares outstanding at time t. Stock C splits two for one in the last period.
P0 Q0 P1 Q1 P2 Q2
A 90 100 95 100 95 100 B 50 200 45 200 45 200 C 100 200 110 200 55 400
a. Calculate the rate of return on a price-weighted index of the three stocks for the first period ( t 5 0 to t 5 1).
b. What must happen to the divisor for the price-weighted index in year 2? c. Calculate the rate of return for the second period ( t 5 1 to t 5 2).
Basic
Intermediate
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56 P A R T I Introduction
12. Using the data in the previous problem, calculate the first-period rates of return on the following
indexes of the three stocks:
a. A market-value-weighted index. b. An equally weighted index.
13. An investor is in a 30% tax bracket. If corporate bonds offer 9% yields, what must municipals
offer for the investor to prefer them to corporate bonds?
14. Find the equivalent taxable yield of a short-term municipal bond currently offering yields of 4%
for tax brackets of zero, 10%, 20%, and 30%.
15. What problems would confront a mutual fund trying to create an index fund tied to an equally
weighted index of a broad stock market?
16. Which security should sell at a greater price?
a. A 10-year Treasury bond with a 9% coupon rate versus a 10-year T-bond with a 10% coupon. b. A 3-month expiration call option with an exercise price of $40 versus a 3-month call on the
same stock with an exercise price of $35.
c. A put option on a stock selling at $50, or a put option on another stock selling at $60 (all other relevant features of the stocks and options may be assumed to be identical).
17. Look at the futures listings for the corn contract in Figure 2.11 .
a. Suppose you buy one contract for March delivery. If the contract closes in March at a level of 787.25, what will your profit be?
b. How many March maturity contracts are outstanding?
18. Turn back to Figure 2.10 and look at the IBM options. Suppose you buy a January 2013 expira-
tion call option with exercise price $180.
a. Suppose the stock price in January is $193. Will you exercise your call? What is the profit on your position?
b. What if you had bought the January call with exercise price $185? c. What if you had bought a January put with exercise price $185?
19. Why do call options with exercise prices greater than the price of the underlying stock sell for
positive prices?
20. Both a call and a put currently are traded on stock XYZ; both have strike prices of $50 and
expirations of 6 months. What will be the profit to an investor who buys the call for $4 in the
following scenarios for stock prices in 6 months? What will be the profit in each scenario to an
investor who buys the put for $6?
a. $40 b. $45 c. $50 d. $55 e. $60
21. Explain the difference between a put option and a short position in a futures contract.
22. Explain the difference between a call option and a long position in a futures contract.
Challenge
1. A firm’s preferred stock often sells at yields below its bonds because
a. Preferred stock generally carries a higher agency rating. b. Owners of preferred stock have a prior claim on the firm’s earnings. c. Owners of preferred stock have a prior claim on a firm’s assets in the event of liquidation. d. Corporations owning stock may exclude from income taxes most of the dividend income they
receive.
2. A municipal bond carries a coupon of 6¾% and is trading at par. What is the equivalent taxable
yield to a taxpayer in a combined federal plus state 34% tax bracket?
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C H A P T E R 2 Asset Classes and Financial Instruments 57
3. Which is the most risky transaction to undertake in the stock index option markets if the stock market is expected to increase substantially after the transaction is completed?
a. Write a call option. b. Write a put option. c . Buy a call option. d. Buy a put option.
4. Short-term municipal bonds currently offer yields of 4%, while comparable taxable bonds pay
5%. Which gives you the higher after-tax yield if your tax bracket is:
a. Zero b. 10% c. 20% d. 30%
5. The coupon rate on a tax-exempt bond is 5.6%, and the rate on a taxable bond is 8%. Both bonds
sell at par. At what tax bracket (marginal tax rate) would an investor be indifferent between the
two bonds?
E-INVESTMENTS EXERCISES Barclays maintains a Web site at www.barcap.com/inflation/index.shtml with informa- tion about inflation around the world and tools to help issuers and investors understand the inflation-linked asset class. Inflation-linked bonds were issued by a number of coun- tries after 1945, including Israel, Argentina, Brazil, and Iceland. However, the modern mar- ket is generally deemed to have been born in 1981, when the first index-linked gilts were issued in the U.K. The other large markets adopted somewhat different calculations to those used by the U.K., mostly copying the more straightforward model first employed by Canada in 1991. In chronological order, the markets are the U.K. (1981), Australia (1985), Canada (1991), Sweden (1994), the United States (1997), France (1998), Italy (2003), and Japan (2004).
SOLUTIONS TO CONCEPT CHECKS 1. The bid price of the bond is 138.0469% of par, or $1,380.469, and the ask price is 138.125% of
par, or $1,381.25. This ask price corresponds to a yield of 2.378%. The ask price fell .9375 from
its level yesterday, so the ask price then must have been 139.0625, or $1,390.625.
2. A 6% taxable return is equivalent to an after-tax return of 6(1 2 .30) 5 4.2%. Therefore, you
would be better off in the taxable bond. The equivalent taxable yield of the tax-free bond is
4/(1 2 .30) 5 5.71%. So a taxable bond would have to pay a 5.71% yield to provide the same
after-tax return as a tax-free bond offering a 4% yield.
3. a. You are entitled to a prorated share of IBM’s dividend payments and to vote in any of IBM’s stockholder meetings.
b. Your potential gain is unlimited because IBM’s stock price has no upper bound. c. Your outlay was $180 3 100 5 $18,000. Because of limited liability, this is the most you can
lose.
4. The price-weighted index increases from 62.5 [i.e., (100 1 25)/2] to 65 [i.e., (110 1 20)/2], a
gain of 4%. An investment of one share in each company requires an outlay of $125 that would
increase in value to $130, for a return of 4% (i.e., 5/125), which equals the return to the price-
weighted index.
5. The market-value-weighted index return is calculated by computing the increase in the
value of the stock portfolio. The portfolio of the two stocks starts with an initial value
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58 P A R T I Introduction
of $100 million 1 $500 million 5 $600 million and falls in value to $110 million 1
$400 million 5 $510 million, a loss of 90/600 5 .15, or 15%. The index portfolio return is
a weighted average of the returns on each stock with weights of 1⁄6 on XYZ and 5⁄6 on ABC (weights proportional to relative investments). Because the return on XYZ is 10%, while
that on ABC is 2 20%, the index portfolio return is 1⁄6 3 10% 1 5⁄6 3 ( 2 20%) 5 2 15%, equal to the return on the market-value-weighted index.
6. The payoff to the call option is $7 per share at expiration. The option cost is $5.50 per share. The
dollar profit is therefore $1.50. The put option expires worthless. Therefore, the investor’s loss is
the cost of the put, or $2.11.
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3 THIS CHAPTER WILL provide you with a broad introduction to the many venues and procedures available for trading securities in the United States and international markets. We will see that trading mechanisms range from direct negotiation among market partic- ipants to fully automated computer crossing of trade orders.
The first time a security trades is when it is issued to the public. Therefore, we begin with a look at how securities are first marketed to the public by investment bankers, the midwives of securities. We turn next to a broad survey of how already-issued securities may be traded among investors, focusing on the differences
between dealer markets, electronic markets, and specialist markets. With this background, we consider specific trading arenas such as the New York Stock Exchange, NASDAQ, and sev- eral all-electronic markets. We compare the mechanics of trade execution and the impact of cross-market integration of trading.
We then turn to the essentials of some specific types of transactions, such as buying on margin and short-selling stocks. We close the chapter with a look at some important aspects of the regulations governing security trading, including insider trading laws and the role of security markets as self-regulating organizations.
How Securities
Are Traded
CHAPTER THREE
3.1 How Firms Issue Securities
Firms regularly need to raise new capital to help pay for their many investment projects.
Broadly speaking, they can raise funds either by borrowing money or by selling shares in
the firm. Investment bankers are generally hired to manage the sale of these securities in
what is called a primary market for newly issued securities. Once these securities are issued, however, investors might well wish to trade them among themselves. For example,
you may decide to raise cash by selling some of your shares in Apple to another investor.
This transaction would have no impact on the total outstanding number of Apple shares.
Trades in existing securities take place in the secondary market. Shares of publicly listed firms trade continually on well-known markets such as the
New York Stock Exchange or the NASDAQ Stock Market. There, any investor can choose
to buy shares for his or her portfolio. These companies are also called publicly traded, publicly owned, or just public companies. Other firms, however, are private corporations,
3 PA
R T I
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60 P A R T I Introduction
whose shares are held by small numbers of managers and investors. While ownership
stakes in the firm are still determined in proportion to share ownership, those shares do not
trade in public exchanges. While many private firms are relatively young companies that
have not yet chosen to make their shares generally available to the public, others may be
more established firms that are still largely owned by the company’s founders or families,
and others may simply have decided that private organization is preferable.
Privately Held Firms A privately held company is owned by a relatively small number of shareholders. Privately
held firms have fewer obligations to release financial statements and other information to
the public. This saves money and frees the firm from disclosing information that might be
helpful to its competitors. Some firms also believe that eliminating requirements for quar-
terly earnings announcements gives them more flexibility to pursue long-term goals free
of shareholder pressure.
At the moment, however, privately held firms may have only up to 499 shareholders.
This limits their ability to raise large amounts of capital from a wide base of investors.
Thus, almost all of the largest companies in the U.S. are public corporations.
When private firms wish to raise funds, they sell shares directly to a small number of
institutional or wealthy investors in a private placement. Rule 144A of the SEC allows them to make these placements without preparing the extensive and costly registration
statements required of a public company. While this is attractive, shares in privately held
firms do not trade in secondary markets such as a stock exchange, and this greatly reduces
their liquidity and presumably reduces the prices that investors will pay for them. Liquidity has many specific meanings, but generally speaking, it refers to the ability to buy or sell
an asset at a fair price on short notice. Investors demand price concessions to buy illiquid
securities.
As firms increasingly chafe against the informational requirements of going public,
federal regulators have come under pressure to loosen the constraints entailed by private
ownership, and they are currently reconsidering some of the restrictions on private com-
panies. They may raise beyond 499 the number of shareholders that private firms can have
before they are required to disclose financial information, and they may make it easier to
publicize share offerings.
Trading in private corporations also has evolved in recent years. To get around the
499-investor restriction, middlemen have formed partnerships to buy shares in private
companies; the partnership counts as only one investor, even though many individuals may
participate in it.
Very recently, some firms have set up computer networks to enable holders of private-
company stock to trade among themselves. However, unlike the public stock markets reg-
ulated by the SEC, these networks require little disclosure of financial information and
provide correspondingly little oversight of the operations of the market. For example, in
the run-up to its 2012 IPO, Facebook enjoyed huge valuations in these markets, but skep-
tics worried that investors in these markets could not obtain a clear view of the firm, the
interest among other investors in the firm, or the process by which trades in the firm’s
shares were executed.
Publicly Traded Companies When a private firm decides that it wishes to raise capital from a wide range of inves-
tors, it may decide to go public. This means that it will sell its securities to the general public and allow those investors to freely trade those shares in established securities
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C H A P T E R 3 How Securities Are Traded 61
markets. The first issue of shares to the general public is called the firm’s initial public offering, or IPO. Later, the firm may go back to the public and issue additional shares. A seasoned equity offering is the sale of additional shares in firms that already are pub- licly traded. For example, a sale by Apple of new shares of stock would be considered a
seasoned new issue.
Public offerings of both stocks and bonds typically are marketed by investment bankers
who in this role are called underwriters. More than one investment banker usually markets the securities. A lead firm forms an underwriting syndicate of other investment bankers to
share the responsibility for the stock issue.
Investment bankers advise the firm regarding the terms on which it should attempt to
sell the securities. A preliminary registration statement must be filed with the Securities
and Exchange Commission (SEC), describing the issue and the prospects of the
company. When the statement is in final form and accepted by the SEC, it is called the
prospectus. At this point, the price at which the securities will be offered to the public is announced.
In a typical underwriting arrangement, the investment bankers purchase the securities
from the issuing company and then resell them to the public. The issuing firm sells the
securities to the underwriting syndicate for the public offering price less a spread that
serves as compensation to the underwriters. This procedure is called a firm commitment. In addition to the spread, the investment banker also may receive shares of common stock or
other securities of the firm. Figure 3.1 depicts the relationships among the firm issuing the
security, the lead underwriter, the underwriting syndicate, and the public.
Shelf Registration An important innovation in the issuing of securities
was introduced in 1982 when the SEC approved Rule
415, which allows firms to register securities and grad-
ually sell them to the public for 2 years following the
initial registration. Because the securities are already
registered, they can be sold on short notice, with little
additional paperwork. Moreover, they can be sold
in small amounts without incur-
ring substantial flotation costs.
The securities are “on the shelf,”
ready to be issued, which has
given rise to the term shelf registration.
Initial Public Offerings Investment bankers manage the
issuance of new securities to
the public. Once the SEC has
commented on the registration
statement and a preliminary
prospectus has been distrib-
uted to interested investors, the
invest ment bankers organize
road shows in which they travel around the country to publicize
Figure 3.1 Relationships among a firm issuing securities, the under- writers, and the public
Issuing Firm
Lead Underwriter
Investment Banker A
Private Investors
Investment Banker D
Investment Banker B
Investment Banker C
Underwriting Syndicate
Why does it make sense for shelf registration to be limited in time?
CONCEPT CHECK 3.1
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62 P A R T I Introduction
the imminent offering. These road shows serve two purposes. First, they generate interest
among potential investors and provide information about the offering. Second, they pro-
vide information to the issuing firm and its underwriters about the price at which they will
be able to market the securities. Large investors communicate their interest in purchasing
shares of the IPO to the underwriters; these indications of interest are called a book and the process of polling potential investors is called bookbuilding. The book provides valu- able information to the issuing firm because institutional investors often will have useful
insights about the market demand for the security as well as the prospects of the firm and
its competitors. Investment bankers frequently revise both their initial estimates of the
offering price of a security and the number of shares offered based on feedback from the
investing community.
Why do investors truthfully reveal their interest in an offering to the investment
banker? Might they be better off expressing little interest, in the hope that this will
drive down the offering price? Truth is the better policy in this case because truth
telling is rewarded. Shares of IPOs are allocated across investors in part based on the
strength of each investor’s expressed interest in the offering. If a firm wishes to get a
large allocation when it is optimistic about the security, it needs to reveal its optimism.
In turn, the underwriter needs to offer the security at a bargain price to these investors
to induce them to participate in bookbuilding and share their information. Thus, IPOs
commonly are underpriced compared to the price at which they could be marketed.
Such underpricing is reflected in price jumps that occur on the date when the shares
are first traded in public security markets. The November 2011 IPO of Groupon was
a typical example of underpricing. The company issued about 35 million shares to the
public at a price of $20. The stock price closed that day at $26.11, a bit more than 30%
above the offering price.
While the explicit costs of an IPO tend to be around 7% of the funds raised, such under-
pricing should be viewed as another cost of the issue. For example, if Groupon had sold its
shares for the $26.11 that investors obviously were willing to pay for them, its IPO would
have raised 30% more money than it actually did. The money “left on the table” in this case
far exceeded the explicit cost of the stock issue. Nevertheless, underpricing seems to be
a universal phenomenon. Figure 3.2 presents average first-day returns on IPOs of stocks
across the world. The results consistently indicate that IPOs are marketed to investors at
attractive prices.
Pricing of IPOs is not trivial and not all IPOs turn out to be underpriced. Some do
poorly after issue. Facebook’s 2012 IPO was a notable disappointment. Within a week of
its IPO, Facebook’s share price was 15% below the $38 offer price, and five months later,
its shares were selling at about half the offer price.
Interestingly, despite their typically attractive first-day returns, IPOs have been poor
long-term investments. Ritter calculates the returns to a hypothetical investor who bought
equal amounts of each U.S. IPO between 1980 and 2009 at the close of trading on the
first day the stock was listed and held each position for three years. That portfolio would
have underperformed the broad U.S. stock market on average by 19.8% for three-year
holding periods and underperformed “style-matched” portfolios of firms with comparable
size and ratio of book value to market value by 7.3%. 1 Other IPOs cannot even be fully
sold to the market. Underwriters left with unmarketable securities are forced to sell them
at a loss on the secondary market. Therefore, the investment banker bears price risk for an
underwritten issue.
1 Professor Jay Ritter’s Web site contains a wealth of information and data about IPOs: http://bear.warrington.
ufl.edu/ritter/ipodata.htm.
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C H A P T E R 3 How Securities Are Traded 63
0%
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50%
75%
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A ve
ra g
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Figure 3.2 Average first-day returns on IPOs from around the world Source: Provided by Professor J. Ritter of the University of Florida, 2008. This is an updated version of the information contained in T. Loughran, J. Ritter, and K. Rydqvist, “Initial Public Offerings,” Pacific-Basin Finance Journal 2 (1994), pp. 165–199. Copyright 1994 with permission from Elsevier Science.
3.2 How Securities Are Traded
Financial markets develop to meet the needs of particular traders. Consider what would
happen if organized markets did not exist. Any household wishing to invest in some type
of financial asset would have to find others wishing to sell. Soon, venues where interested
traders could meet would become popular. Eventually, financial markets would emerge
from these meeting places. Thus, a pub in old London called Lloyd’s launched the mari-
time insurance industry. A Manhattan curb on Wall Street became synonymous with the
financial world.
Types of Markets We can differentiate four types of markets: direct search markets, brokered markets, dealer
markets, and auction markets.
Direct Search Markets A direct search market is the least organized market. Buyers and sellers must seek each other out directly. An example of a transaction in such a market
is the sale of a used refrigerator where the seller advertises for buyers in a local newspaper
or on Craigslist. Such markets are characterized by sporadic participation and low-priced
and nonstandard goods. Firms would find it difficult to profit by specializing in such an
environment.
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64 P A R T I Introduction
Brokered Markets The next level of organization is a brokered market. In markets where trading in a good is active, brokers find it profitable to offer search services to buyers
and sellers. A good example is the real estate market, where economies of scale in searches
for available homes and for prospective buyers make it worthwhile for participants to pay
brokers to help them conduct the searches. Brokers in particular markets develop special-
ized knowledge on valuing assets traded in that market.
Notice that the primary market, where new issues of securities are offered to the public, is an example of a brokered market. In the primary market, investment bankers who market
a firm’s securities to the public act as brokers; they seek investors to purchase securities
directly from the issuing corporation.
Dealer Markets When trading activity in a particular type of asset increases, dealer markets arise. Dealers specialize in various assets, purchase these assets for their own accounts, and later sell them for a profit from their inventory. The spreads between dealers’
buy (or “bid”) prices and sell (or “ask”) prices are a source of profit. Dealer markets save
traders on search costs because market participants can easily look up the prices at which they
can buy from or sell to dealers. A fair amount of market activity is required before dealing in a
market is an attractive source of income. Most bonds trade in over-the-counter dealer markets.
Auction Markets The most integrated market is an auction market, in which all traders converge at one place (either physically or “electronically”) to buy or sell an
asset. The New York Stock Exchange (NYSE) is an example of an auction market. An
advantage of auction markets over dealer markets is that one need not search across
dealers to find the best price for a good. If all participants converge, they can arrive at
mutually agreeable prices and save the bid–ask spread.
Notice that both over-the-counter dealer markets and stock exchanges are secondary
markets. They are organized for investors to trade existing securities among themselves.
Types of Orders Before comparing alternative trading practices and com-
peting security markets, it is helpful to begin with an
overview of the types of trades an investor might wish to
have executed in these markets. Broadly speaking, there
are two types of orders: market orders and orders contin-
gent on price.
Market Orders Market orders are buy or sell orders that are to be executed immediately at current market
prices. For example, our investor might call her broker and
ask for the market price of FedEx. The broker might report back that the best bid price is $90 and the best ask price is $90.05, meaning that the investor would need to pay $90.05 to pur- chase a share, and could receive $90 a share if she wished to sell some of her own holdings
of FedEx. The bid–ask spread in this case is $.05. So an order to buy 100 shares “at market” would result in purchase at $90.05, and an order to “sell at market” would be executed at $90.
This simple scenario is subject to a few potential complications. First, the posted price
quotes actually represent commitments to trade up to a specified number of shares. If the
market order is for more than this number of shares, the order may be filled at multiple prices.
For example, if the ask price is good for orders up to 1,000 shares, and the investor wishes
to purchase 1,500 shares, it may be necessary to pay a slightly higher price for the last 500
shares. Figure 3.3 shows the average depth of the markets for shares of stock (i.e., the total
Many assets trade in more than one type of market. What types of markets do the following trade in?
a. Used cars
b. Paintings
c. Rare coins
CONCEPT CHECK 3.2
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C H A P T E R 3 How Securities Are Traded 65
number of shares offered for trading at the
best bid and ask prices). Notice that depth
is considerably higher for the large stocks in
the S&P 500 than for the smaller stocks that
constitute the Russell 2000 index. Depth is
considered another component of liquidity.
Second, another trader may beat our investor
to the quote, meaning that her order would
then be executed at a worse price. Finally,
the best price quote may change before her
order arrives, again causing execution at a
price different from the one at the moment
of the order.
Price-Contingent Orders Investors also may place orders specifying prices
at which they are willing to buy or sell a
security. A limit buy order may instruct the
broker to buy some number of shares if and
when FedEx may be obtained at or below a stipulated price. Conversely, a limit sell
instructs the broker to sell if and when the
stock price rises above a specified limit. A collection of limit orders waiting to be exe- cuted is called a limit order book.
Figure 3.4 is a portion of the limit order
book for shares in FedEx taken from the
Bid
FDX FedEx Corporation
FDX Go>>
ARCA
ARCA
ARCA
ARCA
ARCA
90.04
90.03
90.02
90.00
89.99
14:05:22
14:05:25
14:05:25
14:05:18
14:05:21
NYSE Arca.
ID Price Size Time
Ask
ARCA
ARCA
ARCA
ARCA
ARCA
90.05
90.06
90.07
90.09
90.10
400
104
303
303
404
14:05:21
14:05:21
14:05:21
14:05:18
ARCA 90.01 14:05:17 ARCA 90.08 303 14:05:18
ARCA 89.98 14:05:14 ARCA 90.11 404 14:04:00 ARCA 89.97
100
302
204
302
403
1604
1003 1103 14:05:20 ARCA 90.12 802 14:05:23
14:04:55
ID Price Size Time
Figure 3.4 The limit order book for FedEx on the NYSE Arca market Source: New York Stock Exchange Euronext, www.nyse.com , June 22, 2012.
Figure 3.3 Average market depth for large (S&P 500) and small (Russell 2000) firms
Source: James J. Angel, Lawrence E. Harris, and Chester Spatt, ”Equity Trading in the 21st Century,” Quarterly Journal of Finance 1 (2011), pp. 1–53; Knight Capital Group.
6,000
5,000
4,000
3,000
2,000
1,000
1 1 /2
0 0 3
0 8 /2
0 0 4
0 5 /2
0 0 5
0 2 /2
0 0 6
1 1 /2
0 0 6
0 8 /2
0 0 7
1 1 /2
0 0 9
0 8 /2
0 1 0
0 5 /2
0 1 1
0 5 /2
0 0 8
0 2 /2
0 0 9
A ve
ra g
e D
e p
th (
Su m
o f
Sh a re
s O
ff e re
d a
t B
e st
B id
a n
d A
sk P
ri ce
s)
–
S&P 500 firms
Russell 2000 firms
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66 P A R T I Introduction
What type of trading order might you give to your broker in each of the following circumstances?
a. You want to buy shares of FedEx, to diversify your portfolio. You believe the share price is approxi- mately at the “fair” value, and you want the trade done quickly and cheaply.
b. You want to buy shares of FedEx, but believe that the current stock price is too high given the firm’s prospects. If the shares could be obtained at a price 5% lower than the current value, you would like to purchase shares for your portfolio.
c. You plan to purchase a condominium sometime in the next month or so and will sell your shares of Intel to provide the funds for your down payment. While you believe that the Intel share price is going to rise over the next few weeks, if you are wrong and the share price drops suddenly, you will not be able to afford the purchase. Therefore, you want to hold on to the shares for as long as pos- sible, but still protect yourself against the risk of a big loss.
CONCEPT CHECK 3.3
Trading Mechanisms An investor who wishes to buy or sell shares will place an order with a brokerage firm.
The broker charges a commission for arranging the trade on the client’s behalf. Brokers
have several avenues by which they can execute that trade, that is, find a buyer or seller and
arrange for the shares to be exchanged.
Broadly speaking, there are three trading systems employed in the United States: over-
the-counter dealer markets, electronic communication networks, and specialist markets.
The best-known markets such as NASDAQ or the New York Stock Exchange actually use
a variety of trading procedures, so before you delve into specific markets, it is useful to
understand the basic operation of each type of trading system.
Buy
Sell
Price above the Limit
Price below the Limit
Limit-Buy Order
Limit-Sell Order
Stop-Buy Order
Stop-Loss Order
A ct
io n
Condition
Figure 3.5 Price-contingent orders
NYSE Arca exchange (one of several electronic exchanges; more on
these shortly). Notice that the best orders are at the top of the list:
the offers to buy at the highest price and to sell at the lowest price.
The buy and sell orders at the top of the list—$90.04 and $90.05—
are called the inside quotes; they are the highest buy and lowest sell orders. For FedEx, the inside spread at this time was only 1 cent.
Note, however, that order sizes at the inside quotes can be fairly
small. Therefore, investors interested in larger trades face an effective spread greater than the nominal one because they cannot execute their
entire trades at the inside price quotes.
Stop orders are similar to limit orders in that the trade is not to be executed unless the stock hits a price limit. For stop-loss orders, the stock is to be sold if its price falls below a stipulated level. As the name suggests, the order lets the stock be sold to stop further
losses from accumulating. Similarly, stop-buy orders specify that a stock should be bought when its price rises above a limit. These trades often accom-
pany short sales (sales of securities you don’t own but have borrowed from your broker) and are used to limit potential losses from the short position. Short sales are discussed
in greater detail later in this chapter. Figure 3.5 organizes these types of trades in a
convenient matrix.
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C H A P T E R 3 How Securities Are Traded 67
Dealer Markets Roughly 35,000 securities trade on the over-the-counter or OTC market. Thousands of brokers register with the SEC as security dealers. Dealers quote prices at which they are willing to buy or sell securities. A broker then executes a trade by
contacting a dealer listing an attractive quote.
Before 1971, all OTC quotations were recorded manually and published daily on so-
called pink sheets. In 1971, the National Association of Securities Dealers introduced its
Automatic Quotations System, or NASDAQ, to link brokers and dealers in a computer
network where price quotes could be displayed and revised. Dealers could use the network
to display the bid price at which they were willing to purchase a security and the ask price
at which they were willing to sell. The difference in these prices, the bid–ask spread, was
the source of the dealer’s profit. Brokers representing clients could examine quotes over
the computer network, contact the dealer with the best quote, and execute a trade.
As originally organized, NASDAQ was more of a price-quotation system than a trading
system. While brokers could survey bid and ask prices across the network of dealers in the
search for the best trading opportunity, actual trades required direct negotiation (often over
the phone) between the investor’s broker and the dealer in the security. However, as we will
see, NASDAQ is no longer a mere price quotation system. While dealers still post bid and
ask prices over the network, what is now called the NASDAQ Stock Market allows for electronic execution of trades, and the vast majority of trades are executed electronically.
Electronic Communication Networks (ECNs) Electronic communication networks allow participants to post market and limit orders over computer networks. The limit-order book is available to all participants. An example of such an order book from
NYSE Arca, one of the leading ECNs, appears in Figure 3.4 . Orders that can be “crossed,”
that is, matched against another order, are done automatically without requiring the inter-
vention of a broker. For example, an order to buy a share at a price of $50 or lower will be
immediately executed if there is an outstanding ask price of $50. Therefore, ECNs are true
trading systems, not merely price-quotation systems.
ECNs offer several attractions. Direct crossing of trades without using a broker-dealer
system eliminates the bid–ask spread that otherwise would be incurred. Instead, trades are
automatically crossed at a modest cost, typically less than a penny per share. ECNs are
attractive as well because of the speed with which a trade can be executed. Finally, these
systems offer investors considerable anonymity in their trades.
Specialist Markets Specialist systems have been largely replaced by electronic commu- nication networks, but as recently as a decade ago, they were still a dominant form of market
organization for trading in stocks. In these systems, exchanges such as the NYSE assign respon-
sibility for managing the trading in each security to a specialist. Brokers wishing to buy or sell shares for their clients direct the trade to the specialist’s post on the floor of the exchange. While
each security is assigned to only one specialist, each specialist firm makes a market in many
securities. The specialist maintains the limit order book of all outstanding unexecuted limit
orders. When orders can be executed at market prices, the specialist executes, or “crosses,” the
trade. The highest outstanding bid price and the lowest outstanding ask price “win” the trade.
Specialists are also mandated to maintain a “fair and orderly” market when the book of
limit buy and sell orders is so thin that the spread between the highest bid price and lowest
ask price becomes too wide. In this case, the specialist firm would be expected to offer to buy
and sell shares from its own inventory at a narrower bid-ask spread. In this role, the specialist
serves as a dealer in the stock and provides liquidity to other traders. In this context, liquidity
providers are those who stand willing to buy securities from or sell securities to other traders.
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68 P A R T I Introduction
3.3 The Rise of Electronic Trading
When first established, NASDAQ was primarily an over-the-counter dealer market and
the NYSE was a specialist market. But today both are primarily electronic markets. These
changes were driven by an interaction of new technologies and new regulations. New regu-
lations allowed brokers to compete for business, broke the hold that dealers once had on
information about best-available bid and ask prices, forced integration of markets, and
allowed securities to trade in ever-smaller price increments (called tick sizes ). Technology made it possible for traders to rapidly compare prices across markets and direct their trades
to the markets with the best prices. The resulting competition drove down the cost of trade
execution to a tiny fraction of its value just a few decades ago.
In 1975, fixed commissions on the NYSE were eliminated, which freed brokers to com-
pete for business by lowering their fees. In that year also, Congress amended the Securities
Exchange Act to create the National Market System to at least partially centralize trading
across exchanges and enhance competition among different market makers. The idea was
to implement centralized reporting of transactions as well as a centralized price quotation
system to give traders a broader view of trading opportunities across markets.
The aftermath of a 1994 scandal at NASDAQ turned out to be a major impetus in the
further evolution and integration of markets. NASDAQ dealers were found to be colluding
to maintain wide bid-ask spreads. For example, if a stock was listed at $30 bid—$301/2
ask, a retail client who wished to buy shares from a dealer would pay $301/2 while a client
who wished to sell shares would receive only $30. The dealer would pocket the 1/2-point
spread as profit. Other traders may have been willing to step in with better prices (e.g., they
may have been willing to buy shares for $301/8 or sell them for $303/8), but those better
quotes were not made available to the public, enabling dealers to profit from artificially
wide spreads at the public’s expense. When these practices came to light, an antitrust law-
suit was brought against NASDAQ.
In response to the scandal, the SEC instituted new order-handling rules. Published
dealer quotes now had to reflect limit orders of customers, allowing them to effectively
compete with dealers to capture trades. As part of the antitrust settlement, NASDAQ agreed
to integrate quotes from ECNs into its public display, enabling the electronic exchanges
to also compete for trades. Shortly after this settlement, the SEC adopted Regulation ATS
(Alternative Trading Systems), giving ECNs the right to register as stock exchanges. Not
surprisingly, they captured an ever-larger market share, and in the wake of this new compe-
tition, bid–ask spreads narrowed.
Even more dramatic narrowing of trading costs came in 1997, when the SEC allowed
the minimum tick size to fall from one-eighth of a dollar to one-sixteenth. Not long after,
in 2001, “decimalization” allowed the tick size to fall to 1 cent. Bid–ask spreads again fell
dramatically. Figure 3.6 shows estimates of the “effective spread” (the cost of a transac-
tion) during three distinct time periods defined by the minimum tick size. Notice how
dramatically effective spread falls along with the minimum tick size.
Technology was also changing trading practices. The first ECN, Instinet, was estab-
lished in 1969. By the 1990s, exchanges around the world were rapidly adopting fully
electronic trading systems. Europe led the way in this evolution, but eventually American
exchanges followed suit. The National Association of Securities Dealers (NASD) spun
off the NASDAQ Stock Market as a separate entity in 2000, which quickly evolved into
a centralized limit-order matching system—effectively a large ECN. The NYSE acquired
the electronic Archipelago Exchange in 2006 and renamed it NYSE Arca.
In 2005, the SEC adopted Regulation NMS (for National Market System), which was
fully implemented in 2007. The goal was to link exchanges electronically, thereby creating,
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C H A P T E R 3 How Securities Are Traded 69
in effect, one integrated electronic market. The regulation required exchanges to honor
quotes of other exchanges when they could be executed automatically. An exchange that
could not handle a quote electronically would be labeled a “slow market” under Reg NMS
and could be ignored by other market participants. The NYSE, which was still devoted to
the specialist system, was particularly at risk of being passed over, and in response to this
pressure, it moved aggressively toward automated execution of trades. Electronic trading
networks and the integration of markets in the wake of Reg NMS made it much easier for
exchanges around the world to compete; the NYSE lost its effective monopoly in the trad-
ing of its own listed stocks, and by the end of the decade, its share in the trading of NYSE-
listed stocks fell from about 75% to 25%.
While specialists still exist, trading today is overwhelmingly electronic, at least for
stocks. Bonds are still traded in more traditional dealer markets. In the U.S., the share of
electronic trading in equities rose from about 16% in 2000 to over 80% by the end of the
decade. In the rest of the world, the dominance of electronic trading is even greater.
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Figure 3.6 The effective spread (measured in dollars per share) fell dramatically as the minimum tick size fell (value-weighted average of NYSE-listed shares)
Source: Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam, “Liquidity and Market Efficiency,” Journal of Financial Economics 87 (2008), 249–268. Copyright © February 2008, with permission from Elsevier.
3.4 U.S. Markets The NYSE and the NASDAQ Stock Market remain the two largest U.S. stock markets. But
electronic communication networks have steadily increased their market share. Figure 3.7
shows the comparative trading volume of NYSE-listed shares on the NYSE and NASDAQ
as well as on the major ECNs, namely, BATS, NYSE Arca, and Direct Edge. The “Other”
category, which recently has risen above 30%, includes so-called dark pools, which we
will discuss shortly.
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70 P A R T I Introduction
NASDAQ The NASDAQ Stock Market lists around 3,000 firms. It has steadily introduced ever-more
sophisticated trading platforms, which today handle the great majority of its trades. The
current version, called the NASDAQ Market Center, consolidates NASDAQ’s previous
electronic markets into one integrated system. NASDAQ merged in 2008 with OMX, a
Swedish-Finnish company that controls seven Nordic and Baltic stock exchanges to form
NASDAQ OMX Group. In addition to maintaining the NASDAQ Stock Market, it also
maintains several stock markets in Europe as well as an options and futures exchange in
the U.S.
NASDAQ has three levels of subscribers. The highest, level 3 subscribers, are registered
market makers. These are firms that make a market in securities, maintain inventories of
securities, and post bid and ask prices at which they are willing to buy or sell shares. Level
3 subscribers can enter and change bid–ask quotes continually and have the fastest execu-
tion of trades. They profit from the spread between bid and ask prices.
Level 2 subscribers receive all bid and ask quotes but cannot enter their own quotes.
They can see which market makers are offering the best prices. These subscribers tend to
be brokerage firms that execute trades for clients but do not actively deal in stocks for their
own account.
Level 1 subscribers receive only inside quotes (i.e., the best bid and ask prices), but do
not see how many shares are being offered. These subscribers tend to be investors who are
not actively buying or selling but want information on current prices.
Figure 3.7 Market share of trading in NYSE-listed shares Source: James J. Angel, Lawrence E. Harris, and Chester Spatt, “Equity Trading in the 21 st Century,” Quarterly Journal of Finance 1 (2011), 1–53.
90%
Feb-02 Feb-03 Feb-04 Feb-05 Feb-06 Feb-07 Feb-08 Feb-09 Feb-10 Feb-11
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NASDAQ OMX BATS
OtherDirect Edge
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C H A P T E R 3 How Securities Are Traded 71
The New York Stock Exchange The NYSE is the largest U.S. stock exchange as measured by the value of the stocks listed on the exchange. Daily trading volume on the NYSE is about a billion shares. In 2006, the
NYSE merged with the Archipelago Exchange to form a publicly held company called
the NYSE Group, and then in 2007, it merged with the European exchange Euronext to
form NYSE Euronext. The firm acquired the American Stock Exchange in 2008, which
has since been renamed NYSE Amex and focuses on small firms. NYSE Arca is the firm’s
electronic communications network, and this is where the bulk of exchange-traded funds
trade. In 2012, NYSE Euronext agreed to be purchased by InternationalExchange (ICE),
whose main business to date has been energy-futures trading. ICE plans to retain the NYSE
Euronext name as well as the fabled trading floor on Wall Street.
The NYSE was long committed to its specialist trading system, which relied heavily
on human participation in trade execution. It began its transition to electronic trading for
smaller trades in 1976 with the introduction of its DOT (Designated Order Turnaround),
and later SuperDOT systems, which could route orders directly to the specialist. In 2000,
the exchange launched Direct 1 , which could automatically cross smaller trades (up to
1,099 shares) without human intervention, and in 2004, it began eliminating the size restric-
tions on Direct 1 trades. The change of emphasis dramatically accelerated in 2006 with the
introduction of the NYSE Hybrid Market, which allowed brokers to send orders either for
immediate electronic execution or to the specialist, who could seek price improvement
from another trader. The Hybrid system allowed the NYSE to qualify as a fast market for
the purposes of Regulation NMS, but still offer the advantages of human intervention for
more complicated trades. In contrast, NYSE’s Arca marketplace is fully electronic.
ECNs Over time, more fully automated markets have gained market share at the expense of less
automated ones, in particular, the NYSE. Some of the biggest ECNs today are Direct Edge,
BATS, and NYSE Arca. Brokers that have an affiliation with an ECN have computer access
and can enter orders in the limit order book. As orders are received, the system determines
whether there is a matching order, and if so, the trade is immediately crossed.
Originally, ECNs were open only to other traders using the same system. But following
the implementation of Reg NMS, ECNs began listing limit orders on other networks. Traders
could use their computer systems to sift through the limit order books of many ECNs and
instantaneously route orders to the market with the best prices. Those cross-market links have
become the impetus for one of the more popular strategies of so-called high-frequency trad-
ers, which seek to profit from even small, transitory discrepancies in prices across markets.
Speed is obviously of the essence here, and ECNs compete in terms of the speed they can
offer. Latency refers to the time it takes to accept, process, and deliver a trading order. BATS, for example, advertises latency times of around 200 microseconds, i.e., .0002 second.
3.5 New Trading Strategies
The marriage of electronic trading mechanisms with computer technology has had far-
ranging impacts on trading strategies and tools. Algorithmic trading delegates trading decisions to computer programs. High frequency trading is a special class of algorith- mic trading in which computer programs initiate orders in tiny fractions of a second,
far faster than any human could process the information driving the trade. Much of the
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72 P A R T I Introduction
market liquidity that once was provided by brokers making a market in a security has been
displaced by these high-frequency traders. But when high-frequency traders abandon the
market, as in the so-called flash crash of 2010, liquidity can likewise evaporate in a flash.
Dark pools are trading venues that preserve anonymity, but also affect market liquidity. We will address these emerging issues later in this section.
Algorithmic Trading Algorithmic trading is the use of computer programs to make trading decisions. Well more than half of all equity volume in the U.S. is believed to be initiated by computer
algorithms. Many of these trades exploit very small discrepancies in security prices and
entail numerous and rapid cross-market price comparisons that are well suited to com-
puter analysis. These strategies would not have been feasible before decimalization of the
minimum tick size.
Some algorithmic trades attempt to exploit very short-term trends (as short as a few
seconds) as new information about a firm becomes reflected in its stock price. Others use
versions of pairs trading in which normal price relations between pairs (or larger groups) of stocks seem temporarily disrupted and offer small profit opportunities as they move
back into alignment. Still others attempt to exploit discrepancies between stock prices and
prices of stock-index futures contracts.
Some algorithmic trading involves activities akin to traditional market making. The
traders seek to profit from the bid–ask spread by buying a stock at the bid price and rap-
idly selling it at the ask price before the price can change. While this mimics the role of a
market maker who provides liquidity to other traders in the stock, these algorithmic traders
are not registered market makers and so do not have an affirmative obligation to maintain
both bid and ask quotes. If they abandon a market during a period of turbulence, the shock
to market liquidity can be disruptive. This seems to have been a problem during the flash
crash of May 6, 2010, when the stock market encountered extreme volatility, with the Dow
Jones average falling by 1,000 points before recovering around 600 points in intraday trad-
ing. The nearby box discusses this amazing and troubling episode.
High-Frequency Trading It is easy to see that many algorithmic trading strategies require extremely rapid trade
initiation and execution. High-frequency trading is a subset of algorithmic trading that relies on computer programs to make extremely rapid decisions. High-frequency traders
compete for trades that offer very small profits. But if those opportunities are numerous
enough, they can accumulate to big money.
We pointed out that one high-frequency strategy entails a sort of market making,
attempting to profit from the bid–ask spread. Another relies on cross-market arbitrage,
in which even tiny price discrepancies across markets allow the firm to buy a security at
one price and simultaneously sell it at a slightly higher price. The competitive advantage
in these strategies lies with the firms that are quickest to identify and execute these profit
opportunities. There is a tremendous premium on being the first to “hit” a bid or ask price.
Trade execution times for high-frequency traders are now measured in milliseconds,
even microseconds. This has induced trading firms to “co-locate” their trading centers next
to the computer systems of the electronic exchanges. When execution or latency periods
are less than a millisecond, the extra time it takes for a trade order to travel from a remote
location to a New York exchange would be enough to make it nearly impossible to win
the trade.
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C H A P T E R 3 How Securities Are Traded 73
To understand why co-location has become a key issue, consider this calculation.
Even light can travel only 186 miles in 1 millisecond, so an order originating in Chicago
transmitted at the speed of light would take almost 5 milliseconds to reach New York.
But ECNs today claim latency periods considerably less than 1 millisecond, so an
order from Chicago could not possibly compete with one launched from a co-located
facility.
In some ways, co-location is a new version of an old phenomenon. Think about why,
even before the advent of the telephone, so many brokerage firms originally located their
headquarters in New York: They were “co-locating” with the NYSE so that their brokers
could bring trades to the exchange quickly and efficiently. Today, trades are transmitted
electronically, but competition among traders for fast execution means that the need to be
near the market (now embodied in computer servers) remains.
Dark Pools Many large traders seek anonymity. They fear that if others see them executing a large buy
or sell program, their intentions will become public and prices will move against them.
Very large trades (called blocks, usually defined as a trade of more than 10,000 shares) traditionally were brought to “block houses,” brokerage firms specializing in matching
block buyers and sellers. Part of the expertise of block brokers is in identifying trad-
ers who might be interested in a large purchase or sale if given an offer. These brokers
discreetly arrange large trades out of the public eye, and so avoid moving prices against
their clients.
Block trading today has been displaced to a great extent by dark pools, trading systems in which participants can buy or sell large blocks of securities without showing their hand.
Not only are buyers and sellers in dark pools hidden from the public, but even trades may
not be reported, or if they are reported, they may be lumped with other trades to obscure
information about particular participants.
Dark pools are somewhat controversial because they contribute to the fragmentation
of markets. When many orders are removed from the consolidated limit order book, there
are fewer orders left to absorb fluctuations in demand for the security, and the public price
may no longer be “fair” in the sense that it reflects all the potentially available information
about security demand.
Another approach to dealing with large trades is to split them into many small trades,
each of which can be executed on electronic markets, attempting to hide the fact that the
total number of shares ultimately to be bought or sold is large. This trend has led to rapid
decline in average trade size, which today is less than 300 shares.
Bond Trading In 2006, the NYSE obtained regulatory approval to expand its bond-trading system to
include the debt issues of any NYSE-listed firm. Until then, each bond needed to be regis-
tered before listing, and such a requirement was too onerous to justify listing most bonds.
In conjunction with these new listings, the NYSE has expanded its electronic bond-trading
platform, which is now called NYSE Bonds and is the largest centralized bond market of
any U.S. exchange.
Nevertheless, the vast majority of bond trading occurs in the OTC market among bond
dealers, even for bonds that are actually listed on the NYSE. This market is a network of
bond dealers such as Merrill Lynch (now part of Bank of America), Salomon Smith Barney
(a division of Citigroup), and Goldman Sachs that is linked by a computer quotation
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74
system. However, because these dealers do not carry extensive inventories of the wide
range of bonds that have been issued to the public, they cannot necessarily offer to sell
bonds from their inventory to clients or even buy bonds for their own inventory. They
may instead work to locate an investor who wishes to take the opposite side of a trade. In
practice, however, the corporate bond market often is quite “thin,” in that there may be few
investors interested in trading a specific bond at any particular time. As a result, the bond
market is subject to a type of liquidity risk, for it can be difficult to sell one’s holdings
quickly if the need arises.
3.6 Globalization of Stock Markets
Figure 3.8 shows that NYSE-Euronext is by far the largest equity market as measured by the
total market value of listed firms. All major stock markets today are effectively electronic.
Securities markets have come under increasing pressure in recent years to make inter-
national alliances or mergers. Much of this pressure is due to the impact of electronic
trading. To a growing extent, traders view stock markets as computer networks that link
them to other traders, and there are increasingly fewer limits on the securities around
the world that they can trade. Against this background, it becomes more important for
exchanges to provide the cheapest and most efficient mechanism by which trades can be
executed and cleared. This argues for global alliances that can facilitate the nuts and bolts
of cross-border trading and can benefit from economies of scale. Exchanges feel that they
The Flash Crash of May 2010
At 2:42 New York time on May 6, 2010, the Dow Jones Industrial Average was already down about 300 points for the day. The market was demonstrating concerns about the European debt crisis, and nerves were already on edge. Then, in the next 5 minutes, the Dow dropped an additional 600 points. And only 20 minutes after that, it had recovered most of those 600 points. Besides the stag- gering intraday volatility of the broad market, trading in individual shares and ETFs was even more disrupted. The iShares Russell 1000 Value fund temporarily fell from $59 a share to 8 cents. Shares in the large consulting company Accenture, which had just sold for $38, traded at 1 cent only a minute or two later. At the other extreme, share prices of Apple and Hewlett-Packard momentarily increased to over $100,000. These markets were clearly broken.
The causes of the flash crash are still debated. An SEC report issued after the trade points to a $4 billion sale of market index futures contracts by a mutual fund. As market prices began to tumble, many algorithmic trad- ing programs withdrew from the markets, and those that remained became net sellers, further pushing down equity prices. As more and more of these algorithmic traders shut down, liquidity in these markets evaporated: Buyers for many stocks simply disappeared.
W O
R D
S FR
O M
T H
E S
T R
E E T
Finally, trading was halted for a short period. When it resumed, buyers decided to take advantage of many severely depressed stock prices, and the market rebounded almost as quickly as it had crashed. Given the intra- day turbulence and the clearly distorted prices at which some trades had been executed, the NYSE and NASDAQ decided to cancel all trades that were executed more than 60% away from a “reference price” close to the opening price of the day. Almost 70% of those canceled trades involved ETFs.
The SEC has since approved experimentation with new circuit breakers to halt trading for 5 minutes in large stocks that rise or fall by more than 10% in a 5-minute period. The idea is to prevent trading algorithms from moving share prices quickly before human traders have a chance to determine whether those prices are moving in response to fundamental information.
The flash crash highlighted the fragility of markets in the face of huge variation in trading volume created by algorithmic traders. The potential for these high- frequency traders to withdraw from markets in periods of turbulence remains a concern, and many observers are not convinced that we are protected from future flash crashes.
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C H A P T E R 3 How Securities Are Traded 75
eventually need to offer 24-hour global markets and platforms that allow trading of differ-
ent security types, for example, both stocks and derivatives. Finally, companies want to be
able to go beyond national borders when they wish to raise capital.
These pressures have resulted in a broad trend toward market consolidation. In the
last decade, most of the mergers were “local,” that is, involving exchanges operating on
the same continent. In the U.S., the NYSE merged with the Archipelago ECN in 2006,
and in 2008 acquired the American Stock Exchange. NASDAQ acquired Instinet (which
operated another major ECN, INET) in 2005 and the Boston Stock Exchange in 2007.
In the derivatives market, the Chicago Mercantile Exchange acquired the Chicago Board
of Trade in 2007 and the New York Mercantile Exchange in 2008, thus moving almost all
futures trading in the U.S. onto one exchange. In Europe, Euronext was formed by the
merger of the Paris, Brussels, Lisbon, and Amsterdam exchanges and shortly thereafter
purchased Liffe, the derivatives exchange based in London. The LSE merged in 2007 with
Borsa Italiana, which operates the Milan exchange.
There has also been a wave of intercontinental consolidation. The NYSE Group and
Euronext merged in 2007. Germany’s Deutsche Börse and the NYSE Euronext agreed
to merge in late 2011. The merged firm would be able to support trading in virtually
every type of investment. However, in early 2012, the proposed merger ran aground when
European Union antitrust regulators recommended that the combination be blocked. Still,
the attempt at the merger indicates the thrust of market pressures, and other combinations
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Source: World Federation of Exchanges, 2012.
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76 P A R T I Introduction
continue to develop. The NYSE and the Tokyo stock exchange have announced their
intention to link their networks to give customers of each access to both markets. In 2007,
the NASDAQ Stock Market merged with OMX, which operates seven Nordic and Baltic
stock exchanges, to form NASDAQ OMX Group. In 2008, Eurex took over International
Securities Exchange (ISE), to form a major options exchange.
3.7 Trading Costs
Part of the cost of trading a security is obvious and explicit. Your broker must be paid a
commission. Individuals may choose from two kinds of brokers: full-service or discount
brokers. Full-service brokers who provide a variety of services often are referred to as
account executives or financial consultants.
Besides carrying out the basic services of executing orders, holding securities for safe-
keeping, extending margin loans, and facilitating short sales, brokers routinely provide
information and advice relating to investment alternatives.
Full-service brokers usually depend on a research staff that prepares analyses and
forecasts of general economic as well as industry and company conditions and often makes
specific buy or sell recommendations. Some customers take the ultimate leap of faith
and allow a full-service broker to make buy and sell decisions for them by establishing
a discretionary account. In this account, the broker can buy and sell prespecified securi- ties whenever deemed fit. (The broker cannot withdraw any funds, though.) This action
requires an unusual degree of trust on the part of the customer, for an unscrupulous broker
can “churn” an account, that is, trade securities excessively with the sole purpose of gen-
erating commissions.
Discount brokers, on the other hand, provide “no-frills” services. They buy and sell
securities, hold them for safekeeping, offer margin loans, facilitate short sales, and that
is all. The only information they provide about the securities they handle is price quota-
tions. Discount brokerage services have become increasingly available in recent years.
Many banks, thrift institutions, and mutual fund management companies now offer such
services to the investing public as part of a general trend toward the creation of one-stop
“financial supermarkets.” Stock trading fees have fallen steadily over the last decade, and
discount brokerage firms such as Schwab, E*Trade, or TD Ameritrade now offer commis-
sions below $10.
In addition to the explicit part of trading costs—the broker’s commission—there is an
implicit part—the dealer’s bid–ask spread. Sometimes the broker is also a dealer in the
security being traded and charges no commission but instead collects the fee entirely in
the form of the bid–ask spread. Another implicit cost of trading that some observers would
distinguish is the price concession an investor may be forced to make for trading in quanti-
ties greater than those associated with the posted bid or ask price.
3.8 Buying on Margin
When purchasing securities, investors have easy access to a source of debt financing called
broker’s call loans. The act of taking advantage of broker’s call loans is called buying on margin.
Purchasing stocks on margin means the investor borrows part of the purchase price of
the stock from a broker. The margin in the account is the portion of the purchase price
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C H A P T E R 3 How Securities Are Traded 77
The percentage margin is defined as the ratio of the net worth, or the “equity value,” of the account to the market value of the securities. To demonstrate, suppose an investor initially pays $6,000 toward the purchase of $10,000 worth of stock (100 shares at $100 per share), borrowing the remaining $4,000 from a broker. The initial balance sheet looks like this:
Assets Liabilities and Owners’ Equity
Value of stock $10,000 Loan from broker $4,000 Equity $6,000
The initial percentage margin is
Margin 5 Equity in account
Value of stock 5
$6,000 $10,000
5 .60, or 60%
If the price declines to $70 per share, the account balance becomes:
Assets Liabilities and Owners’ Equity
Value of stock $7,000 Loan from broker $4,000 Equity $3,000
The assets in the account fall by the full decrease in the stock value, as does the equity. The percentage margin is now
Margin 5 Equity in account
Value of stock 5
$3,000 $7,000
5 .43, or 43%
Example 3.1 Margin
contributed by the investor; the remainder is borrowed from the broker. The brokers in
turn borrow money from banks at the call money rate to finance these purchases; they then
charge their clients that rate (defined in Chapter 2), plus a service charge for the loan. All
securities purchased on margin must be maintained with the brokerage firm in street name,
for the securities are collateral for the loan.
The Board of Governors of the Federal Reserve System limits the extent to which stock
purchases can be financed using margin loans. The current initial margin requirement is
50%, meaning that at least 50% of the purchase price must be paid for in cash, with the
rest borrowed.
If the stock value in Example 3.1 were to fall below $4,000, owners’ equity would
become negative, meaning the value of the stock is no longer sufficient collateral to cover
the loan from the broker. To guard against this possibility, the broker sets a maintenance margin. If the percentage margin falls below the maintenance level, the broker will issue a margin call, which requires the investor to add new cash or securities to the margin account. If the investor does not act, the broker may sell securities from the account to pay
off enough of the loan to restore the percentage margin to an acceptable level.
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eXcel APPLICATIONS: Buying on Margin
T he Online Learning Center ( www.mhhe.com/bkm ) con- tains the Excel spreadsheet model below, which makes
it easy to analyze the impacts of different margin levels and the volatility of stock prices. It also allows you to com- pare return on investment for a margin trade with a trade using no borrowed funds.
Excel Questions
1. Suppose you buy 100 shares share of stock initially selling for $50, borrowing 25% of the necessary funds from your
broker, i.e., the initial margin on your purchase is 25%. You pay an interest rate of 8% on margin loans.
a. How much of your own money do you invest? How much do you borrow from your broker?
b. What will be your rate of return for the following stock prices at the end of a 1-year holding period? (1) $40, (2) $50, (3) $60.
2. Repeat Question 1 assuming your initial margin was 50%. How does margin affect the risk and return of your position?
78
A B C D E F G H
Initial Equity Investment Amount Borrowed Initial Stock Price Shares Purchased Ending Stock Price Cash Dividends During Hold Per. Initial Margin Percentage Maintenance Margin Percentage
Rate on Margin Loan Holding Period in Months
Return on Investment Capital Gain on Stock Dividends Interest on Margin Loan Net Income Initial Investment
Return on Investment
Ending St Price
$20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00
−41.60% −121.60% −101.60%
−81.60% −61.60% −41.60% −21.60%
−1.60% 18.40% 38.40% 58.40% 78.40% 98.40%
118.40%
Ending St Price
$20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00
LEGEND: Enter data
Value calculated
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
−18.80% −58.80% −48.80% −38.80% −28.80% −18.80%
−8.80% 1.20%
11.20% 21.20% 31.20% 41.20% 51.20% 61.20%
Suppose the maintenance margin is 30%. How far could the stock price fall before the investor would get a margin call?
Let P be the price of the stock. The value of the investor’s 100 shares is then 100 P, and the equity in the account is 100 P 2 $4,000. The percentage margin is (100 P 2 $4,000)/100 P. The price at which the percentage margin equals the mainte- nance margin of .3 is found by solving the equation
100P 2 4,000 100 P
5 .3
which implies that P 5 $57.14. If the price of the stock were to fall below $57.14 per share, the investor would get a margin call.
Example 3.2 Maintenance Margin
Suppose the maintenance margin in Example 3.2 is 40%. How far can the stock price fall before the investor gets a margin call?
CONCEPT CHECK 3.4
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C H A P T E R 3 How Securities Are Traded 79
Suppose that in this margin example, the investor borrows only $5,000 at the same interest rate of 9% per year. What will the rate of return be if the price of FinCorp goes up by 30%? If it goes down by 30%? If it remains unchanged?
CONCEPT CHECK 3.5
Why do investors buy securities on margin? They do so when they wish to invest an
amount greater than their own money allows. Thus, they can achieve greater upside poten-
tial, but they also expose themselves to greater downside risk.
To see how, let’s suppose an investor is bullish on FinCorp stock, which is selling for
$100 per share. An investor with $10,000 to invest expects FinCorp to go up in price by
30% during the next year. Ignoring any dividends, the expected rate of return would be
30% if the investor invested $10,000 to buy 100 shares.
But now assume the investor borrows another $10,000 from the broker and invests
it in FinCorp, too. The total investment in FinCorp would be $20,000 (for 200 shares).
Assuming an interest rate on the margin loan of 9% per year, what will the investor’s rate
of return be now (again ignoring dividends) if FinCorp stock goes up 30% by year’s end?
The 200 shares will be worth $26,000. Paying off $10,900 of principal and interest on
the margin loan leaves $15,100 (i.e., $26,000 2 $10,900). The rate of return in this case
will be
$15,000 2 $10,000
$10,000 5 51%
The investor has parlayed a 30% rise in the stock’s price into a 51% rate of return on the
$10,000 investment.
Doing so, however, magnifies the downside risk. Suppose that, instead of going up
by 30%, the price of FinCorp stock goes down by 30% to $70 per share. In that case, the
200 shares will be worth $14,000, and the investor is left with $3,100 after paying off the
$10,900 of principal and interest on the loan. The result is a disastrous return of
$3,100 2 $10,000
$10,000 5 269%
Table 3.1 summarizes the possible results of these hypothetical transactions. If there is
no change in FinCorp’s stock price, the investor loses 9%, the cost of the loan.
Change in Stock Price
End-of-Year Value of Shares
Repayment of Principal and Interest*
Investor’s Rate of Return
30% increase $26,000 $10,900 51% No change 20,000 10,900 29
30% decrease 14,000 10,900 269
* Assuming the investor buys $20,000 worth of stock, borrowing $10,000 of the purchase price at an interest rate of 9% per year.
Table 3.1
Illustration of buying stock on margin
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eXcel APPLICATIONS: Short Sale
T he Online Learning Center ( www.mhhe.com/bkm ) con- tains this Excel spreadsheet model, built using the text
example for Dot Bomb. The model allows you to analyze the effects of returns, margin calls, and different levels of initial and maintenance margins. The model also includes a sensitivity analysis for ending stock price and return on investment.
Excel Questions
1. Suppose you sell short 100 shares of stock initially selling for $100 a share. Your initial margin requirement is 50% of the
value of the stock sold. You receive no interest on the funds placed in your margin account.
a. How much do you need to contribute to your margin account?
b. What will be your rate of return for the following stock prices at the end of a 1-year holding period? Assume the stock pays no dividends. (1) $90, (2) $100, (3) $110.
2. Repeat Question 1 (b) but now assume that the stock pays dividends of $2 per share at year-end. What is the relation- ship between the total rate of return on the stock and the return to your short position?
A B C D E
Initial Investment Initial Stock Price Number of Shares Sold Short Ending Stock Price Cash Dividends Per Share Initial Margin Percentage Maintenance Margin Percentage
Return on Short Sale Capital Gain on Stock Dividends Paid Net Income Initial Investment Return on Investment
Margin Positions Margin Based on Ending Price
Price for Margin Call
$50,000.00 $100.00
1,000 $70.00 $0.00
50.00% 30.00%
$30,000.00 $0.00
$30,000.00 $50,000.00
60.00%
114.29%
$115.38
Ending St Price
$170.00 160.00 150.00 140.00 130.00 120.00 110.00 100.00 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00
60.00% −140.00% −120.00% −100.00%
−80.00% −60.00% −40.00% −20.00%
0.00% 20.00% 40.00% 60.00% 80.00%
100.00% 120.00% 140.00% 160.00% 180.00%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
LEGEND: Enter data
Value calculated
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3.9 Short Sales
2 Naked short-selling is a variant on conventional short-selling. In a naked short, a trader sells shares that have not
yet been borrowed, assuming that the shares can be acquired in time to meet any delivery deadline. While naked
short-selling is prohibited, enforcement has been spotty, as many firms have engaged in it based on their “reason-
able belief ” that they will be able to acquire the stock by the time delivery is required. Now the SEC is requiring
that short-sellers have made firm arrangements for delivery before engaging in the sale.
Normally, an investor would first buy a stock and later sell it. With a short sale, the order
is reversed. First, you sell and then you buy the shares. In both cases, you begin and end
with no shares.
A short sale allows investors to profit from a decline in a security’s price. An investor borrows a share of stock from a broker and sells it. Later, the short-seller must purchase
a share of the same stock in order to replace the share that was borrowed. This is called
covering the short position. Table 3.2 compares stock purchases to short sales. 2 The short-seller anticipates the stock price will fall, so that the share can be purchased
later at a lower price than it initially sold for; if so, the short-seller will reap a profit.
Short-sellers must not only replace the shares but also pay the lender of the security any
dividends paid during the short sale.
In practice, the shares loaned out for a short sale are typically provided by the short-
seller’s brokerage firm, which holds a wide variety of securities of its other investors in
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C H A P T E R 3 How Securities Are Traded 81
To illustrate the mechanics of short-selling, suppose you are bearish (pessimistic) on Dot Bomb stock, and its market price is $100 per share. You tell your broker to sell short 1,000 shares. The broker borrows 1,000 shares either from another customer’s account or from another broker.
The $100,000 cash proceeds from the short sale are credited to your account. Suppose the broker has a 50% margin requirement on short sales. This means you must have other cash or securities in your account worth at least $50,000 that can serve as margin on the short sale. Let’s say that you have $50,000 in Treasury bills. Your account with the broker after the short sale will then be:
Assets Liabilities and Owners’ Equity
Cash $100,000 Short position in Dot Bomb stock (1,000 shares owed)
$100,000
T-bills 50,000 Equity 50,000
Example 3.3 Short Sales
Purchase of Stock
Time Action Cash Flow*
0 Buy share 2 Initial price
1 Receive dividend, sell share Ending price 1 Dividend
Profit 5 (Ending price 1 Dividend) 2 Initial price
Short Sale of Stock
Time Action Cash Flow*
0 Borrow share; sell it 1 Initial price 1 Repay dividend and buy share to
replace the share originally borrowed 2 (Ending price 1 Dividend)
Profit 5 Initial price 2 (Ending price 1 Dividend)
*A negative cash flow implies a cash outflow.
Table 3.2
Cash flows from purchasing versus short-selling shares of stock
street name (i.e., the broker holds the shares registered in its own name on behalf of the
client). The owner of the shares need not know that the shares have been lent to the short-
seller. If the owner wishes to sell the shares, the brokerage firm will simply borrow shares
from another investor. Therefore, the short sale may have an indefinite term. However, if
the brokerage firm cannot locate new shares to replace the ones sold, the short-seller will
need to repay the loan immediately by purchasing shares in the market and turning them
over to the brokerage house to close out the loan.
Finally, exchange rules require that proceeds from a short sale must be kept on account
with the broker. The short-seller cannot invest these funds to generate income, although
large or institutional investors typically will receive some income from the proceeds of a
short sale being held with the broker. Short-sellers also are required to post margin (cash or
collateral) with the broker to cover losses should the stock price rise during the short sale.
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82 P A R T I Introduction
a. Construct the balance sheet if Dot Bomb in Example 3.4 goes up to $110.
b. If the short position maintenance margin in the Dot Bomb example is 40%, how far can the stock price rise before the investor gets a margin call?
CONCEPT CHECK 3.6
Like investors who purchase stock on margin, a short-seller must be concerned about
margin calls. If the stock price rises, the margin in the account will fall; if margin falls to
the maintenance level, the short-seller will receive a margin call.
Your initial percentage margin is the ratio of the equity in the account, $50,000, to the current value of the shares you have borrowed and eventually must return, $100,000:
Percentage margin 5 Equity
Value of stock owed 5
$50,000 $100,000
5 .50
Suppose you are right and Dot Bomb falls to $70 per share. You can now close out your position at a profit. To cover the short sale, you buy 1,000 shares to replace the ones you borrowed. Because the shares now sell for $70, the purchase costs only $70,000. 3 Because your account was credited for $100,000 when the shares were bor- rowed and sold, your profit is $30,000: The profit equals the decline in the share price times the number of shares sold short.
3 Notice that when buying on margin, you borrow a given amount of dollars from your broker, so the amount of
the loan is independent of the share price. In contrast, when short-selling you borrow a given number of shares, which must be returned. Therefore, when the price of the shares changes, the value of the loan also changes.
Suppose the broker has a maintenance margin of 30% on short sales. This means the equity in your account must be at least 30% of the value of your short position at all times. How much can the price of Dot Bomb stock rise before you get a margin call?
Let P be the price of Dot Bomb stock. Then the value of the shares you must pay back is 1,000 P and the equity in your account is $150,000 2 1,000 P. Your short position margin ratio is equity/value of stock 5 (150,000 2 1,000 P )/1,000 P. The critical value of P is thus
Equity
Value of shares owed 5
15,000 2 1,000P 1,000P
5 .3
which implies that P 5 $115.38 per share. If Dot Bomb stock should rise above $115.38 per share, you will get a margin call, and you will either have to put up additional cash or cover your short position by buying shares to replace the ones borrowed.
Example 3.4 Margin Calls on Short Positions
You can see now why stop-buy orders often accompany short sales. Imagine that you
short-sell Dot Bomb when it is selling at $100 per share. If the share price falls, you will
profit from the short sale. On the other hand, if the share price rises, let’s say to $130, you
will lose $30 per share. But suppose that when you initiate the short sale, you also enter a
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C H A P T E R 3 How Securities Are Traded 83
stop-buy order at $120. The stop-buy will be executed if the share price surpasses $120,
thereby limiting your losses to $20 per share. (If the stock price drops, the stop-buy will
never be executed.) The stop-buy order thus provides protection to the short-seller if the
share price moves up.
Short-selling periodically comes under attack, particularly during times of financial
stress when share prices fall. The last few years have been no exception to this rule. For
example, following the 2008 financial crisis, the SEC voted to restrict short sales in stocks
that decline by at least 10% on a given day. Those stocks may now be shorted on that day
and the next only at a price greater than the highest bid price across national stock markets.
The nearby box examines the controversy surrounding short sales in greater detail.
3.10 Regulation of Securities Markets
Trading in securities markets in the United States is regulated by a myriad of laws. The major
governing legislation includes the Securities Act of 1933 and the Securities Exchange Act of
1934. The 1933 act requires full disclosure of relevant information relating to the issue of new
securities. This is the act that requires registration of new securities and issuance of a prospec-
tus that details the financial prospects of the firm. SEC approval of a prospectus or financial
report is not an endorsement of the security as a good investment. The SEC cares only that the
relevant facts are disclosed; investors must make their own evaluation of the security’s value.
The 1934 act established the Securities and Exchange Commission to administer the
provisions of the 1933 act. It also extended the disclosure principle of the 1933 act by
requiring periodic disclosure of relevant financial information by firms with already-issued
securities on secondary exchanges.
The 1934 act also empowers the SEC to register and regulate securities exchanges,
OTC trading, brokers, and dealers. While the SEC is the administrative agency responsible
for broad oversight of the securities markets, it shares responsibility with other regula-
tory agencies. The Commodity Futures Trading Commission (CFTC) regulates trading in
futures markets, while the Federal Reserve has broad responsibility for the health of the
U.S. financial system. In this role, the Fed sets margin requirements on stocks and stock
options and regulates bank lending to security market participants.
The Securities Investor Protection Act of 1970 established the Securities Investor
Protection Corporation (SIPC) to protect investors from losses if their brokerage firms
fail. Just as the Federal Deposit Insurance Corporation provides depositors with federal
protection against bank failure, the SIPC ensures that investors will receive securities held
for their account in street name by a failed brokerage firm up to a limit of $500,000 per
customer. The SIPC is financed by levying an “insurance premium” on its participating, or
member, brokerage firms.
In addition to federal regulations, security trading is subject to state laws, known gener-
ally as blue sky laws because they are intended to give investors a clearer view of invest- ment prospects. Varying state laws were somewhat unified when many states adopted
portions of the Uniform Securities Act, which was enacted in 1956.
The 2008 financial crisis also led to regulatory changes, some of which we detailed
in Chapter 1. The Financial Stability Oversight Council (FSOC) was established by the
Dodd-Frank Wall Street Reform and Consumer Protection Act to monitor the stability of
the U.S. financial system. It is largely concerned with risks arising from potential failures
of large, interconnected banks, but its voting members are the chairpersons of the main
U.S. regulatory agencies, and therefore the FSOC serves a broader role to connect and
coordinate key financial regulators.
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84
Self-Regulation In addition to government regulation, the securities market exercises considerable self-
regulation. The most important overseer in this regard is the Financial Industry Regula-
tory Authority (FINRA), which is the largest nongovernmental regulator of all securities
firms in the United States. FINRA was formed in 2007 through the consolidation of the
National Association of Securities Dealers (NASD) with the self-regulatory arm of the
New York Stock Exchange. It describes its broad mission as the fostering of investor
protection and market integrity. It examines securities firms, writes and enforces rules
concerning trading practices, and administers a dispute-resolution forum for investors
and registered firms.
In addition to being governed by exchange regulation, there is also self-regulation
among the community of investment professionals. For example, the CFA Institute has
developed standards of professional conduct that govern the behavior of members with the
Chartered Financial Analysts designation, commonly referred to as CFAs. The nearby box
presents a brief outline of those principles.
The Sarbanes-Oxley Act The scandals of 2000–2002 centered largely on three broad practices: allocations of shares
in initial public offerings, tainted securities research and recommendations put out to the
Short-selling has long been viewed with suspicion, if not outright hostility. England banned short sales for a good part of the 18th century. Napoleon called short-sellers enemies of the state. In the U.S., short-selling was widely viewed as contributing to the market crash of 1929, and in 2008, short-sellers were blamed for the collapse of the investment banks Bear Stearns and Lehman Brothers. With share prices of other financial firms tumbling in September 2008, the SEC instituted a temporary ban on short-selling of nearly 1,000 of those firms. Similarly, the Financial Ser- vices Authority, the financial regulator in the U.K., pro- hibited short sales on about 30 financial companies, and Australia banned shorting altogether.
The rationale for these bans is that short sales put downward pressure on share prices that in some cases may be unwarranted: Rumors abound of investors who first put on a short sale and then spread negative rumors about the firm to drive down its price. More often, however, short- ing is a legitimate bet that a share price is too high and is due to fall. Nevertheless, during the market stresses of late 2008, the widespread feeling was that even if short positions were legitimate, regulators should do what they could to prop up the affected institutions.
Hostility to short-selling may well stem from confusion between bad news and the bearer of that news. Short- selling allows investors whose analysis indicates a firm is overpriced to take action on that belief—and to profit if they are correct. Rather than causing the stock price
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Short-Selling Comes Under Fire—Again
to fall, shorts may be anticipating a decline in the stock price. Their sales simply force the market to reflect the deteriorating prospects of troubled firms sooner than it might have otherwise. In other words, short-selling is part of the process by which the full range of information and opinion—pessimistic as well as optimistic—is brought to bear on stock prices.
For example, short-sellers took large (negative) posi- tions in firms such as WorldCom, Enron, and Tyco even before these firms were exposed by regulators. In fact, one might argue that these emerging short positions helped regulators identify the previously undetected scandals. And in the end, Lehman and Bear Stearns were brought down by their very real losses on their mortgage-related investments—not by unfounded rumors.
Academic research supports the conjecture that short sales contribute to efficient “price discovery.” For example, the greater the demand for shorting a stock, the lower its future returns tend to be; moreover, firms that attack short-sellers with threats of legal action or bad publicity tend to have especially poor future returns. 1 Short-sale bans may in the end be nothing more than an understand- able, but nevertheless misguided, impulse to “shoot the messenger.”
1See, for example, C. Jones and O. A. Lamont, “Short Sale Constraints and Stock Returns,” Journal of Financial Economics, November 2002, pp. 207–39, or O. A. Lamont, “Go Down Fighting: Short Sellers vs. Firms,” Yale ICF Working Paper No. 04–20, July 2004.
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85
public, and, probably most important, misleading financial statements and accounting
practices. The Sarbanes-Oxley Act was passed by Congress in 2002 in response to these
problems. Among the key reforms are:
• Creation of the Public Company Accounting Oversight Board to oversee the
auditing of public companies.
• Rules requiring independent financial experts to serve on audit committees of a
firm’s board of directors.
• CEOs and CFOs must now personally certify that their firms’ financial reports
“fairly represent, in all material respects, the operations and financial condition
of the company,” and are subject to personal penalties if those reports turn out to
be misleading. Following the letter of the rules may still be necessary, but it is no
longer sufficient accounting practice.
Excerpts from CFA Institute Standards of Professional Conduct
I. Professionalism
• Knowledge of law. Members must understand, have knowledge of, and comply with all applicable laws, rules, and regulations including the Code of Ethics and Standards of Professional Conduct.
• Independence and objectivity. Members shall main- tain independence and objectivity in their profes- sional activities.
• Misrepresentation. Members must not knowingly misrepresent investment analysis, recommendations, or other professional activities.
II. Integrity of Capital Markets
• Non-public information. Members must not exploit material non-public information.
• Market manipulation. Members shall not attempt to distort prices or trading volume with the intent to mislead market participants.
III. Duties to Clients
• Loyalty, prudence, and care. Members must place their clients’ interests before their own and act with reasonable care on their behalf.
• Fair dealing. Members shall deal fairly and objec- tively with clients when making investment recom- mendations or taking actions.
• Suitability. Members shall make a reasonable inquiry into a client’s financial situation, investment expe- rience, and investment objectives prior to making appropriate investment recommendations.
• Performance presentation. Members shall attempt to ensure that investment performance is presented fairly, accurately, and completely.
• Confidentiality. Members must keep information about clients confidential unless the client permits disclosure.
IV. Duties to Employers
• Loyalty. Members must act for the benefit of their employer.
• Compensation. Members must not accept compen- sation from sources that would create a conflict of interest with their employer’s interests without written consent from all involved parties.
• Supervisors. Members must make reasonable efforts to detect and prevent violation of applica- ble laws and regulations by anyone subject to their supervision.
V. Investment Analysis and Recommendations
• Diligence. Members must exercise diligence and have reasonable basis for investment analysis, rec- ommendations, or actions.
• Communication. Members must distinguish fact from opinion in their presentation of analysis and disclose general principles of investment processes used in analysis.
VI. Conflicts of Interest
• Disclosure of conflicts. Members must disclose all matters that reasonably could be expected to impair their objectivity or interfere with their other duties.
• Priority of transactions. Transactions for clients and employers must have priority over transactions for the benefit of a member.
VII. Responsibilities as Member of CFA Institute
• Conduct. Members must not engage in conduct that compromises the reputation or integrity of the CFA Institute or CFA designation.
Source: “Code of Ethics and Standards of Professional Conduct,” reproduced with permission from the Standards of Practice Handbook, 10th Ed. 2010, CFA Institute, Copyright 2010. www.cfainstitute.org/ centre/codes/ethics
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86 P A R T I Introduction
4 Jeffrey E. Jaffee, “Special Information and Insider Trading,” Journal of Business 47 (July 1974).
• Auditors may no longer provide several other services to their clients. This is
intended to prevent potential profits on consulting work from influencing the
quality of their audit.
• The Board of Directors must be composed of independent directors and hold
regular meetings of directors in which company management is not present (and
therefore cannot impede or influence the discussion).
More recently, there has been a fair amount of pushback on Sarbanes-Oxley. Many
observers believe that the compliance costs associated with the law are too onerous,
especially for smaller firms, and that heavy-handed regulatory oversight is giving for-
eign locales an undue advantage over the United States when firms decide where to
list their securities. Moreover, the efficacy of single-country regulation is being tested
in the face of increasing globalization and the ease with which funds can move across
national borders.
Insider Trading Regulations also prohibit insider trading. It is illegal for anyone to transact in securities
to profit from inside information, that is, private information held by officers, directors, or major stockholders that has not yet been divulged to the public. But the definition of
insiders can be ambiguous. While it is obvious that the chief financial officer of a firm is
an insider, it is less clear whether the firm’s biggest supplier can be considered an insider.
Yet a supplier may deduce the firm’s near-term prospects from significant changes in
orders. This gives the supplier a unique form of private information, yet the supplier
is not technically an insider. These ambiguities plague security analysts, whose job is
to uncover as much information as possible concerning the firm’s expected prospects.
The dividing line between legal private information and illegal inside information can be
fuzzy.
The SEC requires officers, directors, and major stockholders to report all transactions
in their firm’s stock. A compendium of insider trades is published monthly in the SEC’s
Official Summary of Securities Transactions and Holdings. The idea is to inform the public of any implicit vote of confidence or no confidence made by insiders.
Insiders do exploit their knowledge. Three forms of evidence support this conclusion. First, there have been well-publicized convictions of principals in insider trading schemes.
Second, there is considerable evidence of “leakage” of useful information to some trad-
ers before any public announcement of that information. For example, share prices of firms
announcing dividend increases (which the market interprets as good news concerning the
firm’s prospects) commonly increase in value a few days before the public announcement of the increase. Clearly, some investors are acting on the good news before it is released
to the public. Share prices still rise substantially on the day of the public release of good
news, however, indicating that insiders, or their associates, have not fully bid up the price
of the stock to the level commensurate with the news.
A third form of evidence on insider trading has to do with returns earned on trades by
insiders. Researchers have examined the SEC’s summary of insider trading to measure
the performance of insiders. In one of the best known of these studies, Jaffee 4 examined
the abnormal return of stocks over the months following purchases or sales by insid-
ers. For months in which insider purchasers of a stock exceeded insider sellers of the
stock by three or more, the stock had an abnormal return in the following 8 months of
about 5%. Moreover, when insider sellers exceeded insider buyers, the stock tended to
perform poorly.
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C H A P T E R 3 How Securities Are Traded 87
1. Firms issue securities to raise the capital necessary to finance their investments. Investment bankers market these securities to the public on the primary market. Investment bankers generally
act as underwriters who purchase the securities from the firm and resell them to the public at a
markup. Before the securities may be sold to the public, the firm must publish an SEC-accepted
prospectus that provides information on the firm’s prospects.
2. Already-issued securities are traded on the secondary market, that is, on organized stock markets; on the over-the-counter market; and occasionally for very large trades, through direct negotiation.
Only license holders of exchanges may trade on the exchange. Brokerage firms holding licenses
to trade on the exchange sell their services to individuals, charging commissions for executing
trades on their behalf.
3. Trading may take place in dealer markets, via electronic communication networks, or in specialist markets. In dealer markets, security dealers post bid and ask prices at which they are willing to
trade. Brokers for individuals execute trades at the best available prices. In electronic markets, the
existing book of limit orders provides the terms at which trades can be executed. Mutually agree-
able offers to buy or sell securities are automatically crossed by the computer system operating the
market. In specialist markets, the specialist acts to maintain an orderly market with price continuity.
Specialists maintain a limit-order book, but also sell from or buy for their own inventories of stock.
4. NASDAQ was traditionally a dealer market in which a network of dealers negotiated directly over sales of securities. The NYSE was traditionally a specialist market. In recent years, however, both
exchanges have dramatically increased their commitment to electronic and automated trading.
Trading activity today is overwhelmingly electronic.
5. Buying on margin means borrowing money from a broker to buy more securities than can be purchased with one’s own money alone. By buying securities on a margin, an investor magnifies
both the upside potential and the downside risk. If the equity in a margin account falls below the
required maintenance level, the investor will get a margin call from the broker.
6. Short-selling is the practice of selling securities that the seller does not own. The short-seller bor- rows the securities sold through a broker and may be required to cover the short position at any time
on demand. The cash proceeds of a short sale are kept in escrow by the broker, and the broker usu-
ally requires that the short-seller deposit additional cash or securities to serve as margin (collateral).
7. Securities trading is regulated by the Securities and Exchange Commission, by other government agencies, and through self-regulation of the exchanges. Many of the important regulations have
to do with full disclosure of relevant information concerning the securities in question. Insider
trading rules also prohibit traders from attempting to profit from inside information.
SUMMARY
Related Web sites for this chapter are available at www. mhhe.com/bkm
KEY TERMS primary market secondary market
private placement
initial public offerings (IPOs)
underwriters
prospectus
dealer markets
auction market
bid price
ask price
bid–ask spread
limit order
stop orders
over-the-counter (OTC)
market
NASDAQ Stock Market
electronic communication
networks (ECNs)
specialist
stock exchanges
latency
algorithmic trading
high-frequency trading
blocks
dark pools
margin
short sale
inside information
PROBLEM SETS 1. What are the differences between a stop-loss order, a limit sell order, and a market order? 2. Why have average trade sizes declined in recent years?
3. How do margin trades magnify both the upside potential and the downside risk of an investment
position?
Basic
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88 P A R T I Introduction
4. A market order has:
a. Price uncertainty but not execution uncertainty. b. Both price uncertainty and execution uncertainty. c. Execution uncertainty but not price uncertainty.
5. Where would an illiquid security in a developing country most likely trade?
a. Broker markets. b. Electronic crossing networks. c. Electronic limit-order markets.
6. Dée Trader opens a brokerage account and purchases 300 shares of Internet Dreams at $40 per share.
She borrows $4,000 from her broker to help pay for the purchase. The interest rate on the loan is 8%.
a. What is the margin in Dée’s account when she first purchases the stock? b. If the share price falls to $30 per share by the end of the year, what is the remaining margin in
her account? If the maintenance margin requirement is 30%, will she receive a margin call?
c. What is the rate of return on her investment?
7. Old Economy Traders opened an account to short sell 1,000 shares of Internet Dreams from
the previous problem. The initial margin requirement was 50%. (The margin account pays no
interest.) A year later, the price of Internet Dreams has risen from $40 to $50, and the stock has
paid a dividend of $2 per share.
a. What is the remaining margin in the account? b. If the maintenance margin requirement is 30%, will Old Economy receive a margin call? c. What is the rate of return on the investment?
8. Consider the following limit-order book for a share of stock. The last trade in the stock occurred
at a price of $50.
Limit Buy Orders Limit Sell Orders
Price Shares Price Shares
$49.75 500 $50.25 100
49.50 800 51.50 100
49.25 500 54.75 300
49.00 200 58.25 100
48.50 600
a. If a market buy order for 100 shares comes in, at what price will it be filled? b. At what price would the next market buy order be filled? c. If you were a security dealer, would you want to increase or decrease your inventory of this stock?
9. You are bullish on Telecom stock. The current market price is $50 per share, and you have
$5,000 of your own to invest. You borrow an additional $5,000 from your broker at an interest
rate of 8% per year and invest $10,000 in the stock.
a. What will be your rate of return if the price of Telecom stock goes up by 10% during the next year? The stock currently pays no dividends.
b. How far does the price of Telecom stock have to fall for you to get a margin call if the main- tenance margin is 30%? Assume the price fall happens immediately.
10. You are bearish on Telecom and decide to sell short 100 shares at the current market price of
$50 per share.
a. How much in cash or securities must you put into your brokerage account if the broker’s initial margin requirement is 50% of the value of the short position?
b. How high can the price of the stock go before you get a margin call if the maintenance mar- gin is 30% of the value of the short position?
Intermediate
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C H A P T E R 3 How Securities Are Traded 89
11. Suppose that Intel currently is selling at $20 per share. You buy 1,000 shares using $15,000 of
your own money, borrowing the remainder of the purchase price from your broker. The rate on
the margin loan is 8%.
a. What is the percentage increase in the net worth of your brokerage account if the price of Intel immediately changes to: (i) $22; (ii) $20; (iii) $18? What is the relationship between your percentage return and the percentage change in the price of Intel?
b. If the maintenance margin is 25%, how low can Intel’s price fall before you get a margin call? c. How would your answer to ( b ) change if you had financed the initial purchase with only
$10,000 of your own money?
d. What is the rate of return on your margined position (assuming again that you invest $15,000 of your own money) if Intel is selling after 1 year at: (i) $22; (ii) $20; (iii) $18? What is the relationship between your percentage return and the percentage change in the price of Intel?
Assume that Intel pays no dividends.
e. Continue to assume that a year has passed. How low can Intel’s price fall before you get a margin call?
12. Suppose that you sell short 1,000 shares of Intel, currently selling for $20 per share, and give
your broker $15,000 to establish your margin account.
a. If you earn no interest on the funds in your margin account, what will be your rate of return after 1 year if Intel stock is selling at: (i) $22; (ii) $20; (iii) $18? Assume that Intel pays no
dividends.
b. If the maintenance margin is 25%, how high can Intel’s price rise before you get a margin call? c. Redo parts ( a ) and ( b ), but now assume that Intel also has paid a year-end dividend of $1
per share. The prices in part ( a ) should be interpreted as ex-dividend, that is, prices after the dividend has been paid.
13. Here is some price information on Marriott:
Bid Ask
Marriott 39.95 40.05
You have placed a stop-loss order to sell at $40. What are you telling your broker? Given market
prices, will your order be executed?
14. Here is some price information on FinCorp stock. Suppose that FinCorp trades in a dealer market.
Bid Ask
55.25 55.50
a. Suppose you have submitted an order to your broker to buy at market. At what price will your trade be executed?
b. Suppose you have submitted an order to sell at market. At what price will your trade be executed?
c. Suppose you have submitted a limit order to sell at $55.62. What will happen? d. Suppose you have submitted a limit order to buy at $55.37. What will happen?
15. You’ve borrowed $20,000 on margin to buy shares in Disney, which is now selling at $40 per
share. Your account starts at the initial margin requirement of 50%. The maintenance margin is
35%. Two days later, the stock price falls to $35 per share.
a. Will you receive a margin call? b. How low can the price of Disney shares fall before you receive a margin call?
16. On January 1, you sold short one round lot (that is, 100 shares) of Four Sisters stock at $21 per
share. On March 1, a dividend of $2 per share was paid. On April 1, you covered the short sale
by buying the stock at a price of $15 per share. You paid 50 cents per share in commissions for
each transaction. What is the value of your account on April 1?
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90 P A R T I Introduction
1. F B N Inc. has just sold 100,000 shares in an initial public offering. The underwriter’s explicit fees
were $70,000. The offering price for the shares was $50, but immediately upon issue, the share
price jumped to $53.
a. What is your best guess as to the total cost to FBN of the equity issue? b. Is the entire cost of the underwriting a source of profit to the underwriters?
2. If you place a stop-loss order to sell 100 shares of stock at $55 when the current price is $62, how
much will you receive for each share if the price drops to $50?
a. $50. b. $55. c. $54.87. d. Cannot tell from the information given.
3. Specialists on the New York Stock Exchange do all of the following except:
a. Act as dealers for their own accounts. b. Execute limit orders. c. Help provide liquidity to the marketplace. d. Act as odd-lot dealers.
SOLUTIONS TO CONCEPT CHECKS
E-INVESTMENTS EXERCISES When you are choosing which brokerage firm(s) to use to execute your trades, you should consider several factors. Also, a wide range of services claim to objectively recommend bro- kerage firms. However, many are actually sponsored by the brokerage firms themselves. Go to the Web site www.consumersearch.com/online-brokers/reviews and read the information provided under “Our Sources.” Then follow the link for the Barron’s rat- ings. Here you can read the Barron’s annual broker survey and download the “How the Brokers Stack Up” report, which contains a list of fees. Suppose that you have $3,000 to invest and want to put it in a non-IRA account.
1. Are all of the brokerage firms suitable if you want to open a cash account? Are they all suitable if you want a margin account?
2. Choose two of the firms listed. Assume that you want to buy 200 shares of LLY stock using a market order. If the order is filled at $42 per share, how much will the commis- sion be for the two firms if you place an online order?
3. Are there any maintenance fees associated with the account at either brokerage firm?
4. Now assume that you have a margin account and the balance is $3,000. Calculate the interest rate you would pay if you borrowed money to buy stock.
1. Limited-time shelf registration was introduced because its cost savings outweighed the
disadvantage of slightly less up-to-date disclosures. Allowing unlimited shelf registration would
circumvent “blue sky” laws that ensure proper disclosure as the financial circumstances of the
firm change over time.
2. a. Used cars trade in dealer markets (used-car lots or auto dealerships) and in direct search markets when individuals advertise in local newspapers or on the Web.
b. Paintings trade in broker markets when clients commission brokers to buy or sell art for them, in dealer markets at art galleries, and in auction markets.
c. Rare coins trade mostly in dealer markets in coin shops, but they also trade in auctions and in direct search markets when individuals advertise they want to buy or sell coins.
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C H A P T E R 3 How Securities Are Traded 91
3. a. You should give your broker a market order. It will be executed immediately and is the cheapest type of order in terms of brokerage fees.
b. You should give your broker a limit-buy order, which will be executed only if the shares can be obtained at a price about 5% below the current price.
c. You should give your broker a stop-loss order, which will be executed if the share price starts falling. The limit or stop price should be close to the current price to avoid the possibility of
large losses.
4. Solving
100P 2 $4,000
100P 5 .4
yields P 5 $66.67 per share.
5. The investor will purchase 150 shares, with a rate of return as follows:
Year-End Change in Price
Year-End Value of Shares
Repayment of Principal and Interest
Investor’s Rate of Return
30% $19,500 $5,450 40.5%
No change 15,000 5,450 24.5
230% 10,500 5,450 249.5
6. a. Once Dot Bomb stock goes up to $110, your balance sheet will be:
Assets Liabilities and Owner’s Equity
Cash $100,000 Short position in Dot Bomb $110,000
T-bills 50,000 Equity 40,000
b. Solving
$150,000 2 1,000P
1,000P 5 .4
yields P 5 $107.14 per share.
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44 PA
R T I
THE PREVIOUS CHAPTER introduced you to the mechanics of trading securities and the structure of the markets in which securities trade. Commonly, however, individual inves- tors do not trade securities directly for their own accounts. Instead, they direct their funds to investment companies that purchase secu- rities on their behalf. The most important of these financial intermediaries are open- end investment companies, more commonly known as mutual funds, to which we devote most of this chapter. We also touch briefly on other types of investment companies such as unit investment trusts, hedge funds, and closed-end funds. We begin the chap- ter by describing and comparing the various
types of investment companies available to investors. We then examine the functions of mutual funds, their investment styles and pol- icies, and the costs of investing in these funds. Next we take a first look at the investment performance of these funds. We consider the impact of expenses and turnover on net per- formance and examine the extent to which performance is consistent from one period to the next. In other words, will the mutual funds that were the best past performers be the best future performers? Finally, we dis- cuss sources of information on mutual funds, and we consider in detail the information provided in the most comprehensive guide, Morningstar’s Mutual Fund Sourcebook.
CHAPTER FOUR
Mutual Funds and Other
Investment Companies
Investment companies are financial intermediaries that collect funds from individual investors and invest those funds in a potentially wide range of securities or other assets.
Pooling of assets is the key idea behind investment companies. Each investor has a claim to
the portfolio established by the investment company in proportion to the amount invested.
These companies thus provide a mechanism for small investors to “team up” to obtain the
benefits of large-scale investing.
Investment companies perform several important functions for their investors:
1. Record keeping and administration. Investment companies issue periodic status reports, keeping track of capital gains distributions, dividends, investments, and
redemptions, and they may reinvest dividend and interest income for shareholders.
4.1 Investment Companies
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C H A P T E R 4 Mutual Funds and Other Investment Companies 93
2. Diversification and divisibility. By pooling their money, investment companies enable investors to hold fractional shares of many different securities. They can act
as large investors even if any individual shareholder cannot.
3. Professional management. Investment companies can support full-time staffs of security analysts and portfolio managers who attempt to achieve superior invest-
ment results for their investors.
4. Lower transaction costs. Because they trade large blocks of securities, investment companies can achieve substantial savings on brokerage fees and commissions.
While all investment companies pool assets of individual investors, they also need to
divide claims to those assets among those investors. Investors buy shares in investment
companies, and ownership is proportional to the number of shares purchased. The value
of each share is called the net asset value, or NAV. Net asset value equals assets minus liabilities expressed on a per-share basis:
Net asset value 5 Market value of assets minus liabilities
Shares outstanding
Consider a mutual fund that manages a portfolio of securities worth $120 million. Suppose the fund owes $4 million to its investment advisers and owes another $1 million for rent, wages due, and miscellaneous expenses. The fund has 5 million shares outstanding.
Net asset value 5 $120 million 2 $5 million
5 million shares 5 $23 per share
Example 4.1 Net Asset Value
Consider these data from the March 2012 balance sheet of Vanguard’s Growth and Income Fund. What was the net asset value of the fund? Assets: $2,877.06 million Liabilities: $ 14.73 million Shares: 95.50 million
CONCEPT CHECK 4.1
In the United States, investment companies are classified by the Investment Company Act
of 1940 as either unit investment trusts or managed investment companies. The portfolios of
unit investment trusts are essentially fixed and thus are called “unmanaged.” In contrast, man-
aged companies are so named because securities in their investment portfolios continually are
bought and sold: The portfolios are managed. Managed companies are further classified as
either closed-end or open-end. Open-end companies are what we commonly call mutual funds.
Unit Investment Trusts Unit investment trusts are pools of money invested in a portfolio that is fixed for the life of the fund. To form a unit investment trust, a sponsor, typically a brokerage firm, buys a
4.2 Types of Investment Companies
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94 P A R T I Introduction
portfolio of securities that are deposited into a trust. It then sells shares, or “units,” in the
trust, called redeemable trust certificates. All income and payments of principal from the portfolio are paid out by the fund’s trustees (a bank or trust company) to the shareholders.
There is little active management of a unit investment trust because once established,
the portfolio composition is fixed; hence these trusts are referred to as unmanaged. Trusts tend to invest in relatively uniform types of assets; for example, one trust may
invest in municipal bonds, another in corporate bonds. The uniformity of the portfolio is
consistent with the lack of active management. The trusts provide investors a vehicle to
purchase a pool of one particular type of asset that can be included in an overall portfolio
as desired.
Sponsors of unit investment trusts earn their profit by selling shares in the trust at a pre-
mium to the cost of acquiring the underlying assets. For example, a trust that has purchased
$5 million of assets may sell 5,000 shares to the public at a price of $1,030 per share,
which (assuming the trust has no liabilities) represents a 3% premium over the net asset
value of the securities held by the trust. The 3% premium is the trustee’s fee for establish-
ing the trust.
Investors who wish to liquidate their holdings of a unit investment trust may sell the
shares back to the trustee for net asset value. The trustees can either sell enough securi-
ties from the asset portfolio to obtain the cash necessary to pay the investor, or they may
instead sell the shares to a new investor (again at a slight premium to net asset value). Unit
investment trusts have steadily lost market share to mutual funds in recent years. Assets in
such trusts declined from $105 billion in 1990 to only $60 billion in 2012.
Managed Investment Companies There are two types of managed companies: closed-end and open-end. In both cases, the
fund’s board of directors, which is elected by shareholders, hires a management company
to manage the portfolio for an annual fee that typically ranges from .2% to 1.5% of assets.
In many cases the management company is the firm that organized the fund. For example,
Fidelity Management and Research Corporation sponsors many Fidelity mutual funds and
is responsible for managing the portfolios. It assesses a management fee on each Fidelity
fund. In other cases, a mutual fund will hire an outside portfolio manager. For example,
Vanguard has hired Wellington Management as the investment adviser for its Wellington
Fund. Most management companies have contracts to manage several funds.
Open-end funds stand ready to redeem or issue shares at their net asset value (although both purchases and redemptions may involve sales charges). When investors in open-end
funds wish to “cash out” their shares, they sell them back
to the fund at NAV. In contrast, closed-end funds do not redeem or issue shares. Investors in closed-end funds
who wish to cash out must sell their shares to other inves-
tors. Shares of closed-end funds are traded on organized
exchanges and can be purchased through brokers just like
other common stock; their prices, therefore, can differ
from NAV. In early 2013, about $265 billion of assets were
held in closed-end funds.
Figure 4.1 is a listing of closed-end funds. The first col-
umn gives the name and ticker symbol of the fund. The
next two columns give the fund’s most recent net asset
value and closing share price. The premium or discount
in the next column is the percentage difference between
price and NAV: (Price – NAV)/NAV. Notice that there are
FUND
Gabelli Equity Trust (GAB) General Amer Investors (GAM)
J Hancock Tx-Adv Div Inc (HTD) Liberty All-Star Equity (USA) Liberty All-Star Growth (ASG)
Guggenheim Enh Eq Strat (GGE)
Nuveen Tx-Adv TR Strat (JTA)
Gabelli Div & Inc Tr (GDV)
NAV
5.22 31.73 19.13 20.15 5.04 4.20
11.01
17.44
MKT PRICE
27.17 17.47 18.69 4.54 3.90
10.21
15.99 5.31
PREM/ DISC %
1.72 214.37 28.68 27.25 29.92 27.14 27.27
28.31
52-WEEK MARKET RETURN %
21.89 22.49
5.34 19.89
26.59 28.96
0.73
1.59
Figure 4.1 Closed-end mutual funds Source: Data compiled from The Wall Street Journal Online, July 24, 2012.
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C H A P T E R 4 Mutual Funds and Other Investment Companies 95
more funds selling at discounts to NAV (indicated by negative differences) than premiums.
Finally, the 52-week return based on the percentage change in share price plus dividend
income is presented in the last column.
The common divergence of price from net asset value, often by wide margins, is a
puzzle that has yet to be fully explained. To see why this is a puzzle, consider a closed-
end fund that is selling at a discount from net asset value. If the fund were to sell all the
assets in the portfolio, it would realize proceeds equal to net asset value. The difference
between the market price of the fund and the fund’s NAV would represent the per-share
increase in the wealth of the fund’s investors. Moreover, fund premiums or discounts tend
to dissipate over time, so funds selling at a discount receive a boost to their rate of return
as the discount shrinks. Pontiff estimates that a fund selling at a 20% discount would have
an expected 12-month return more than 6% greater than funds selling at net asset value. 1
Interestingly, while many closed-end funds sell at a discount from net asset value, the
prices of these funds when originally issued are often above NAV. This is a further puzzle,
as it is hard to explain why investors would purchase these newly issued funds at a pre-
mium to NAV when the shares tend to fall to a discount shortly after issue.
In contrast to closed-end funds, the price of open-end funds cannot fall below NAV,
because these funds stand ready to redeem shares at NAV. The offering price will exceed
NAV, however, if the fund carries a load. A load is, in effect, a sales charge. Load funds are sold by securities brokers and directly by mutual fund groups.
Unlike closed-end funds, open-end mutual funds do not trade on organized exchanges.
Instead, investors simply buy shares from and liquidate through the investment company at
net asset value. Thus the number of outstanding shares of these funds changes daily.
Other Investment Organizations Some intermediaries are not formally organized or regulated as investment companies, but
nevertheless serve similar functions. Three of the more important are commingled funds,
real estate investment trusts, and hedge funds.
Commingled Funds Commingled funds are partnerships of investors that pool funds. The management firm that organizes the partnership, for example, a bank or insur-
ance company, manages the funds for a fee. Typical partners in a commingled fund might
be trust or retirement accounts with portfolios much larger than those of most individual
investors, but still too small to warrant managing on a separate basis.
Commingled funds are similar in form to open-end mutual funds. Instead of shares,
though, the fund offers units, which are bought and sold at net asset value. A bank or insur- ance company may offer an array of different commingled funds, for example, a money
market fund, a bond fund, and a common stock fund.
Real Estate Investment Trusts (REITs) A REIT is similar to a closed-end fund. REITs invest in real estate or loans secured by real estate. Besides issuing shares, they raise
capital by borrowing from banks and issuing bonds or mortgages. Most of them are highly
leveraged, with a typical debt ratio of 70%.
There are two principal kinds of REITs. Equity trusts invest in real estate directly, whereas mortgage trusts invest primarily in mortgage and construction loans. REITs gen- erally are established by banks, insurance companies, or mortgage companies, which then
serve as investment managers to earn a fee.
1 Jeffrey Pontiff, “Costly Arbitrage: Evidence from Closed-End Funds,” Quarterly Journal of Economics 111
(November 1996), pp. 1135–51.
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96 P A R T I Introduction
Hedge Funds Like mutual funds, hedge funds are vehicles that allow private inves- tors to pool assets to be invested by a fund manager. Unlike mutual funds, however, hedge
funds are commonly structured as private partnerships and thus subject to only minimal
SEC regulation. They typically are open only to wealthy or institutional investors. Many
require investors to agree to initial “lock-ups,” that is, periods as long as several years in
which investments cannot be withdrawn. Lock-ups allow hedge funds to invest in illiq-
uid assets without worrying about meeting demands for redemption of funds. Moreover,
because hedge funds are only lightly regulated, their managers can pursue investment strat-
egies involving, for example, heavy use of derivatives, short sales, and leverage; such strat-
egies typically are not open to mutual fund managers.
Hedge funds by design are empowered to invest in a wide range of investments, with
various funds focusing on derivatives, distressed firms, currency speculation, convertible
bonds, emerging markets, merger arbitrage, and so on. Other funds may jump from one
asset class to another as perceived investment opportunities shift.
Hedge funds enjoyed great growth in the last several years, with assets under manage-
ment ballooning from about $50 billion in 1990 to just about $2 trillion in 2012. We devote
all of Chapter 26 to these funds.
Mutual funds are the common name for open-end investment companies. This is the
dominant investment company today, accounting for more than 90% of investment com-
pany assets. Assets under management in the U.S. mutual fund industry were approxi-
mately $13.5 trillion in early 2013, and approximately another $13 trillion was held in
non-U.S. funds.
Investment Policies Each mutual fund has a specified investment policy, which is described in the fund’s pro-
spectus. For example, money market mutual funds hold the short-term, low-risk instru-
ments of the money market (see Chapter 2 for a review of these securities), while bond
funds hold fixed-income securities. Some funds have even more narrowly defined man-
dates. For example, some bond funds will hold primarily Treasury bonds, others primarily
mortgage-backed securities.
Management companies manage a family, or “complex,” of mutual funds. They orga-
nize an entire collection of funds and then collect a management fee for operating them.
By managing a collection of funds under one umbrella, these companies make it easy
for investors to allocate assets across market sectors and to switch assets across funds
while still benefiting from centralized record keeping. Some of the most well-known man-
agement companies are Fidelity, Vanguard, Barclays, and T. Rowe Price. Each offers an
array of open-end mutual funds with different investment policies. In 2013, there were
nearly 8,000 mutual funds in the U.S., which were offered by a bit more than 700 fund
complexes.
Funds are commonly classified by investment policy into one of the following groups.
Money Market Funds These funds invest in money market securities such as com- mercial paper, repurchase agreements, or certificates of deposit. The average maturity of
4.3 Mutual Funds
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C H A P T E R 4 Mutual Funds and Other Investment Companies 97
these assets tends to be a bit more than 1 month. Money market funds usually offer check-
writing features, and net asset value is fixed at $1 per share, 2 so that there are no tax impli-
cations such as capital gains or losses associated with redemption of shares.
Equity Funds Equity funds invest primarily in stock, although they may, at the portfo- lio manager’s discretion, also hold fixed-income or other types of securities. Equity funds
commonly will hold between 4% and 5% of total assets in money market securities to
provide the liquidity necessary to meet potential redemption of shares.
Stock funds are traditionally classified by their emphasis on capital appreciation versus
current income. Thus, income funds tend to hold shares of firms with consistently high dividend yields. Growth funds are willing to forgo current income, focusing instead on pros- pects for capital gains. While the classification of these funds is couched in terms of income
versus capital gains, in practice, the more relevant distinction concerns the level of risk these
funds assume. Growth stocks, and therefore growth funds, are typically riskier and respond
more dramatically to changes in economic conditions than do income funds.
Sector Funds Some equity funds, called sector funds, concentrate on a particular industry. For example, Fidelity markets dozens of “select funds,” each of which invests in
a specific industry such as biotechnology, utilities, energy, or telecommunications. Other
funds specialize in securities of particular countries.
Bond Funds As the name suggests, these funds specialize in the fixed-income sec- tor. Within that sector, however, there is considerable room for further specialization. For
example, various funds will concentrate on corporate bonds, Treasury bonds, mortgage-
backed securities, or municipal (tax-free) bonds. Indeed, some municipal bond funds invest
only in bonds of a particular state (or even city!) to satisfy the investment desires of resi-
dents of that state who wish to avoid local as well as federal taxes on interest income. Many
funds also specialize by maturity, ranging from short-term to intermediate to long-term, or
by the credit risk of the issuer, ranging from very safe to high-yield, or “junk,” bonds.
International Funds Many funds have international focus. Global funds invest in securities worldwide, including the United States. In contrast, international funds invest in securities of firms located outside the United States. Regional funds concentrate on a partic- ular part of the world, and emerging market funds invest in companies of developing nations.
Balanced Funds Some funds are designed to be candidates for an individual’s entire investment portfolio. These balanced funds hold both equities and fixed-income securi-
ties in relatively stable proportions. Life-cycle funds are balanced funds in which the asset mix can range from aggressive (primarily marketed to younger investors) to conservative
(directed at older investors). Static allocation life-cycle funds maintain a stable mix across
stocks and bonds, while targeted-maturity funds gradually become more conservative as the investor ages.
Many balanced funds are in fact funds of funds. These are mutual funds that primarily invest in shares of other mutual funds. Balanced funds of funds invest in equity and bond
funds in proportions suited to their investment goals.
2 The box in Chapter 2 noted that money market funds are able to maintain NAV at $1.00 because they invest in
short-maturity debt of the highest quality with minimal price risk. In only the rarest circumstances have any funds
incurred losses large enough to drive NAV below $1.00. In September 2008, however, Reserve Primary Fund, the
nation’s oldest money market fund, “broke the buck” when it suffered losses on its holding of Lehman Brothers
commercial paper, and its NAV fell to $.97.
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98 P A R T I Introduction
Asset Allocation and Flexible Funds These funds are similar to balanced funds in that they hold both stocks and bonds. However, asset allocation funds may dramatically
vary the proportions allocated to each market in accord with the portfolio manager’s fore-
cast of the relative performance of each sector. Hence these funds are engaged in market
timing and are not designed to be low-risk investment vehicles.
Index Funds An index fund tries to match the performance of a broad market index. The fund buys shares in securities included in a particular index in proportion to each
security’s representation in that index. For example, the Vanguard 500 Index Fund is a
mutual fund that replicates the composition of the Standard & Poor’s 500 stock price
index. Because the S&P 500 is a value-weighted index, the fund buys shares in each
S&P 500 company in proportion to the market value of that company’s outstanding
equity. Investment in an index fund is a low-cost way for small investors to pursue a pas-
sive investment strategy—that is, to invest without engaging in security analysis. About
15% of equity funds in 2012 were indexed. Of course, index funds can be tied to noneq-
uity indexes as well. For example, Vanguard offers a bond index fund and a real estate
index fund.
Table 4.1 breaks down the number of mutual funds by investment orientation.
Sometimes a fund name describes its investment policy. For example, Vanguard’s GNMA
fund invests in mortgage-backed securities, the Municipal Intermediate fund invests in
intermediate-term municipal bonds, and the High-Yield Corporate bond fund invests
Table 4.1
U.S. mutual funds by investment classification
Note: Column sums subject to rounding error. Source: Investment Company Institute, 2012 Mutual Fund Fact Book.
Assets ($ billion)
% of Total Assets
Number of Funds
Equity funds Capital appreciation focus $ 2,356 20.3% 2,686 World/international 1,359 11.7 1,285 Total return 1,490 12.8 610 Total equity funds $ 5,205 44.8% 4,581
Bond funds Corporate $ 452 3.9% 252 High yield 212 1.8 179 World 259 2.2 205 Government 261 2.2 246 Strategic income 1,204 10.4 484 Single-state municipal 159 1.4 347 National municipal 338 2.9 216 Total bond funds $ 2,885 24.8% 1,929
Hybrid (bond/stock) funds $ 839 7.2% 495 Money market funds Taxable $ 2,400 20.7% 431 Tax-exempt 292 2.5 201 Total money market funds $ 2,692 23.2% 632 Total $11,621 100.0% 7,637
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C H A P T E R 4 Mutual Funds and Other Investment Companies 99
in large part in speculative grade, or “junk,” bonds with high yields. However, names
of common stock funds often reflect little or nothing about their investment policies.
Examples are Vanguard’s Windsor and Wellington funds.
How Funds Are Sold Mutual funds are generally marketed to the public either directly by the fund underwriter
or indirectly through brokers acting on behalf of the underwriter. Direct-marketed funds
are sold through the mail, various offices of the fund, over the phone, or, more so, over the
Internet. Investors contact the fund directly to purchase shares.
About half of fund sales today are distributed through a sales force. Brokers or financial
advisers receive a commission for selling shares to investors. (Ultimately, the commission
is paid by the investor. More on this shortly.)
Investors who rely on their broker’s advice to select their mutual funds should be aware
that brokers may have a conflict of interest with regard to fund selection. This can arise
from a practice called revenue sharing, in which fund companies pay the brokerage firm for preferential treatment when making investment recommendations.
Many funds also are sold through “financial supermarkets” that sell shares in funds
of many complexes. Instead of charging customers a sales commission, the broker splits
management fees with the mutual fund company. Another advantage is unified record
keeping for all funds purchased from the supermarket, even if the funds are offered by
different complexes. On the other hand, many contend that these supermarkets result in
higher expense ratios because mutual funds pass along the costs of participating in these
programs in the form of higher management fees.
Fee Structure An individual investor choosing a mutual fund should consider not only the fund’s stated
investment policy and past performance but also its management fees and other expenses.
Comparative data on virtually all important aspects of mutual funds are available in
Morningstar’s Mutual Fund Sourcebook, which can be found in many academic and public libraries. You should be aware of four general classes of fees.
Operating Expenses Operating expenses are the costs incurred by the mutual fund in operating the portfolio, including administrative expenses and advisory fees paid to the
investment manager. These expenses, usually expressed as a percentage of total assets under
management, may range from 0.2% to 2%. Shareholders do not receive an explicit bill for
these operating expenses; however, the expenses periodically are deducted from the assets
of the fund. Shareholders pay for these expenses through the reduced value of the portfolio.
The simple average of the expense ratio of equity funds in the U.S. was 1.43% in 2011.
But larger funds tend to have lower expense ratios, so the average expense ratio weighted
by assets under management is considerably smaller, 0.79%. Not surprisingly, the average
expense ratio of actively managed funds is considerably higher than that of indexed funds,
.93% versus .14% (weighted by assets under management).
In addition to operating expenses, many funds assess fees to pay for marketing and
distribution costs. These charges are used primarily to pay the brokers or financial advis-
ers who sell the funds to the public. Investors can avoid these expenses by buying shares
directly from the fund sponsor, but many investors are willing to incur these distribution
fees in return for the advice they may receive from their broker.
4.4 Costs of Investing in Mutual Funds
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100 P A R T I Introduction
Front-End Load A front-end load is a commission or sales charge paid when you purchase the shares. These charges, which are used primarily to pay the brokers who sell
the funds, may not exceed 8.5%, but in practice they are rarely higher than 6%. Low-load funds have loads that range up to 3% of invested funds. No-load funds have no front-end sales charges. Loads effectively reduce the amount of money invested. For example, each
$1,000 paid for a fund with a 6% load results in a sales charge of $60 and fund investment
of only $940. You need cumulative returns of 6.4% of your net investment (60/940 5 .064)
just to break even.
Back-End Load A back-end load is a redemption, or “exit,” fee incurred when you sell your shares. Typically, funds that impose back-end loads start them at 5% or 6% and
reduce them by 1 percentage point for every year the funds are left invested. Thus an exit
fee that starts at 6% would fall to 4% by the start of your third year. These charges are
known more formally as “contingent deferred sales charges.”
12b-1 Charges The Securities and Exchange Commission allows the managers of so-called 12b-1 funds to use fund assets to pay for distribution costs such as advertis-
ing, promotional literature including annual reports and prospectuses, and, most important,
commissions paid to brokers who sell the fund to investors. These 12b-1 fees are named after the SEC rule that permits use of these plans. Funds may use 12b-1 charges instead of,
or in addition to, front-end loads to generate the fees with which to pay brokers. As with
operating expenses, investors are not explicitly billed for 12b-1 charges. Instead, the fees
are deducted from the assets of the fund. Therefore, 12b-1 fees (if any) must be added to
operating expenses to obtain the true annual expense ratio of the fund. The SEC requires
that all funds include in the prospectus a consolidated expense table that summarizes all
relevant fees. The 12b-1 fees are limited to 1% of a fund’s average net assets per year. 3
Many funds offer “classes” that represent ownership in the same portfolio of securities,
but with different combinations of fees. For example, Class A shares might have front-end
loads while Class B shares rely on 12b-1 fees.
3 The maximum 12b-1 charge for the sale of the fund is .75%. However, an additional service fee of .25% of the
fund’s assets also is allowed for personal service and/or maintenance of shareholder accounts.
Here are fees for different classes of the Dreyfus High Yield Fund in 2012. Notice the trade-off between the front-end loads versus 12b-1 charges in the choice between Class A and Class C shares. Class I shares are sold only to institutional investors and carry lower fees.
Example 4.2 Fees for Various Classes
Class A Class C Class I
Front-end load 0–4.5%a 0 0 Back-end load 0 0–1%b 0%b
12b-1 feesc .25% 1.0% 0% Expense ratio .70% .70% .70%
a Depending on size of investment. b Depending on years until holdings are sold. c Including service fee.
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C H A P T E R 4 Mutual Funds and Other Investment Companies 101
Each investor must choose the best combination of fees. Obviously, pure no-load no-fee
funds distributed directly by the mutual fund group are the cheapest alternative, and these
will often make most sense for knowledgeable investors. However, as we have noted, many
investors are willing to pay for financial advice, and the commissions paid to advisers
who sell these funds are the most common form of payment. Alternatively, investors may
choose to hire a fee-only financial manager who charges directly for services instead of
collecting commissions. These advisers can help investors select portfolios of low- or no-
load funds (as well as provide other financial advice). Independent financial planners have
become increasingly important distribution channels for funds in recent years.
If you do buy a fund through a broker, the choice between paying a load and paying
12b-1 fees will depend primarily on your expected time horizon. Loads are paid only once
for each purchase, whereas 12b-1 fees are paid annually. Thus, if you plan to hold your
fund for a long time, a one-time load may be preferable to recurring 12b-1 charges.
Fees and Mutual Fund Returns The rate of return on an investment in a mutual fund is measured as the increase or decrease
in net asset value plus income distributions such as dividends or distributions of capital
gains expressed as a fraction of net asset value at the beginning of the investment period.
If we denote the net asset value at the start and end of the period as NAV 0 and NAV 1 ,
respectively, then
Rate of return 5 NAV1 2 NAV0 1 Income and capital gain distributions
NAV0
For example, if a fund has an initial NAV of $20 at the start of the month, makes income
distributions of $.15 and capital gain distributions of $.05, and ends the month with NAV
of $20.10, the monthly rate of return is computed as
Rate of return 5 $20.10 2 $20.00 1 $.15 1 $.05
$20.00 5 .015, or 1.5%
Notice that this measure of the rate of return ignores any commissions such as front-end
loads paid to purchase the fund.
On the other hand, the rate of return is affected by the fund’s expenses and 12b-1 fees.
This is because such charges are periodically deducted from the portfolio, which reduces
net asset value. Thus the investor’s rate of return equals the gross return on the underlying
portfolio minus the total expense ratio.
To see how expenses can affect rate of return, consider a fund with $100 million in assets at the start of the year and with 10 million shares outstanding. The fund invests in a portfolio of stocks that provides no income but increases in value by 10%. The expense ratio, including 12b-1 fees, is 1%. What is the rate of return for an investor in the fund?
The initial NAV equals $100 million/10 million shares 5 $10 per share. In the absence of expenses, fund assets would grow to $110 million and NAV would grow to $11 per share, for a 10% rate of return. However, the expense ratio of the fund is 1%. Therefore, $1 million will be deducted from the fund to pay these fees, leaving the portfolio worth only $109 million, and NAV equal to $10.90. The rate of return on the fund is only 9%, which equals the gross return on the underlying portfolio minus the total expense ratio.
Example 4.3 Fees and Net Returns
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102 P A R T I Introduction
Fees can have a big effect on performance. Table 4.2 considers an investor who starts
with $10,000 and can choose among three funds that all earn an annual 12% return on invest-
ment before fees but have different fee structures. The table shows the cumulative amount
in each fund after several investment horizons. Fund A has total operating expenses of .5%,
no load, and no 12b-1 charges. This might represent a low-cost producer like Vanguard.
Fund B has no load but has 1% in management expenses and .5% in 12b-1 fees. This level
of charges is fairly typical of actively managed equity funds. Finally, Fund C has 1% in man-
agement expenses, has no 12b-1 charges, but assesses an 8% front-end load on purchases.
Note the substantial return advantage of low-cost Fund A. Moreover, that differential is
greater for longer investment horizons.
Table 4.2
Impact of costs on investment performance
Cumulative Proceeds (All Dividends Reinvested)
Fund A Fund B Fund C
Initial investment* $10,000 $10,000 $ 9,200 5 years 17,234 16,474 15,502
10 years 29,699 27,141 26,123
15 years 51,183 44,713 44,018
20 years 88,206 73,662 74,173
* After front-end load, if any. Notes: 1. Fund A is no-load with .5% expense ratio. 2. Fund B is no-load with 1.5% expense ratio. 3. Fund C has an 8% load on purchases and a 1% expense ratio. 4. Gross return on all funds is 12% per year before expenses.
The Equity Fund sells Class A shares with a front-end load of 4% and Class B shares with 12b-1 fees of .5% annually as well as back-end load fees that start at 5% and fall by 1% for each full year the investor holds the portfolio (until the fifth year). Assume the rate of return on the fund portfolio net of operating expenses is 10% annually. What will be the value of a $10,000 investment in Class A and Class B shares if the shares are sold after ( a ) 1 year, ( b ) 4 years, ( c ) 10 years? Which fee structure provides higher net pro- ceeds at the end of each investment horizon?
CONCEPT CHECK 4.2
Although expenses can have a big impact on net investment performance, it is
sometimes difficult for the investor in a mutual fund to measure true expenses accurately.
This is because of the practice of paying for some expenses in soft dollars. A portfo- lio manager earns soft-dollar credits with a brokerage firm by directing the fund’s trades
to that broker. On the basis of those credits, the broker will pay for some of the mutual
fund’s expenses, such as databases, computer hardware, or stock-quotation systems. The
soft-dollar arrangement means that the stockbroker effectively returns part of the trading
commission to the fund. Purchases made with soft dollars are not included in the fund’s
expenses, so funds with extensive soft-dollar arrangements may report artificially low
expense ratios to the public. However, the fund may have paid its broker needlessly high
commissions to obtain its soft-dollar “rebate.” The impact of the higher trading commis-
sion shows up in net investment performance rather than the reported expense ratio.
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C H A P T E R 4 Mutual Funds and Other Investment Companies 103
Investment returns of mutual funds are granted “pass-through status” under the U.S. tax
code, meaning that taxes are paid only by the investor in the mutual fund, not by the fund
itself. The income is treated as passed through to the investor as long as the fund meets sev-
eral requirements, most notably that virtually all income is distributed to shareholders. A
fund’s short-term capital gains, long-term capital gains, and dividends are passed through
to investors as though the investor earned the income directly.
The pass-through of investment income has one important disadvantage for individual
investors. If you manage your own portfolio, you decide when to realize capital gains and
losses on any security; therefore, you can time those realizations to efficiently manage
your tax liabilities. When you invest through a mutual fund, however, the timing of the
sale of securities from the portfolio is out of your control, which reduces your ability to
engage in tax management. 4
A fund with a high portfolio turnover rate can be particularly “tax inefficient.” Turnover is the ratio of the trading activity of a portfolio to the assets of the portfolio. It measures the
fraction of the portfolio that is “replaced” each year. For example, a $100 million portfolio
with $50 million in sales of some securities and purchases of other securities would have
a turnover rate of 50%. High turnover means that capital gains or losses are being realized
constantly, and therefore that the investor cannot time the realizations to manage his or her
overall tax obligation.
Turnover rates in equity funds in the last decade have typically been around 60% when
weighted by assets under management. By contrast, a low-turnover fund such as an index
fund may have turnover as low as 2%, which is both tax-efficient and economical with
respect to trading costs.
4.5 Taxation of Mutual Fund Income
4 An interesting problem that an investor needs to be aware of derives from the fact that capital gains and divi-
dends on mutual funds are typically paid out to shareholders once or twice a year. This means that an investor
who has just purchased shares in a mutual fund can receive a capital gain distribution (and be taxed on that dis-
tribution) on transactions that occurred long before he or she purchased shares in the fund. This is particularly a
concern late in the year when such distributions typically are made.
An investor’s portfolio currently is worth $1 million. During the year, the investor sells 1,000 shares of FedEx at a price of $80 per share and 4,000 shares of Cisco at a price of $20 per share. The proceeds are used to buy 800 shares of IBM at $200 per share.
a. What was the portfolio turnover rate?
b. If the shares in FedEx originally were purchased for $70 each and those in Cisco were purchased for $17.50, and the investor’s tax rate on capital gains income is 20%, how much extra will the investor owe on this year’s taxes as a result of these transactions?
CONCEPT CHECK 4.3
Exchange-traded funds (ETFs), first introduced in 1993, are offshoots of mutual funds that allow investors to trade index portfolios just as they do shares of stock. The first ETF
was the “spider,” a nickname for SPDR, or Standard & Poor’s Depository Receipt, which
is a unit investment trust holding a portfolio matching the S&P 500 Index. Unlike mutual
4.6 Exchange-Traded Funds
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104 P A R T I Introduction
funds, which can be bought or sold only at the end of the day when NAV is calculated,
investors can trade spiders throughout the day, just like any other share of stock. Spiders
gave rise to many similar products such as “diamonds” (based on the Dow Jones Indus-
trial Average, ticker DIA), “cubes” (based on the NASDAQ 100 index, ticker QQQ), and
“WEBS” (World Equity Benchmark Shares, which are shares in portfolios of foreign stock
market indexes). By 2012, about $1 trillion was invested in more than 1,100 U.S. ETFs.
Table 4.3 , panel A presents some of the major sponsors of ETFs, and panel B gives a small
flavor of the types of funds offered.
Figure 4.2 shows the rapid growth in the ETF market since 1998. Until 2008, most
ETFs were required to track specified indexes, and ETFs tracking broad indexes still domi-
nate the industry. However, there are dozens of industry-sector ETFs, and as Figure 4.2
makes clear, commodity, bond, and international ETFs have grown especially dramatically
in recent years. While only $1 billion was invested in commodity ETFs in 2004, by 2011
this value had grown to $109 billion. Gold and silver ETFs dominate this sector, account-
ing for about three-quarters of commodity-based funds. Indeed, ETFs have become the
main way for investors to speculate in precious metals. Figure 4.3 shows that by 2011
ETFs had captured a significant portion of the assets under management in the investment
company universe.
Table 4.3
ETF sponsors and products
A. ETF Sponsors
Sponsor Product Name
BlackRock Global Investors iShares Merrill Lynch HOLDRS (Holding Company Depository Receipts: “Holders”) StateStreet/Merrill Lynch Select Sector SPDRs (S&P Depository Receipts: “Spiders”) Vanguard Vanguard ETF
B. Sample of ETF Products
Name Ticker Index Tracked
Broad U.S. indexes Spiders SPY S&P 500 Diamonds DIA Dow Jones Industrials Cubes QQQ NASDAQ 100 iShares Russell 2000 IWM Russell 2000 Total Stock Market (Vanguard) VTI Wilshire 5000
Industry indexes Energy Select Spider XLE S&P 500 energy companies iShares Energy Sector IYE Dow Jones energy companies Financial Sector Spider XLF S&P 500 financial companies iShares Financial Sector IYF Dow Jones financial companies
International indexes WEBS United Kingdom EWU MSCI U.K. Index WEBS France EWQ MSCI France Index WEBS Japan EWJ MSCI Japan Index
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C H A P T E R 4 Mutual Funds and Other Investment Companies 105
Barclays Global Investors was long the market leader in the ETF market, using the
product name iShares. Since Barclays’s 2009 merger with Blackrock, iShares has operated
under the Blackrock name. The firm sponsors ETFs for several dozen equity index funds,
including many broad U.S. equity indexes, broad international and single-country funds,
and U.S. and global industry sector funds. Blackrock also offers several bond ETFs and
0
100
200
300
400
500
600
700
800
900
1000
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
ET F
A ss
e ts
( $
m ill
io n
) Bond Commodities Global/Int’l Equity U.S. Equity: Sector U.S. Equity: Broad Index
Figure 4.2 Growth of U.S. ETFs over time Source: Investment Company Institute, 2012 Investment Company Fact Book.
Figure 4.3 Investment company assets under management, 2011 ($ billion) Source: Investment Company Institute, 2012 Investment Company Fact Book.
$11,600
$1,047
$239 $60
Mutual Funds
Exchange-Traded Funds
Closed-End Funds
Unit Investment Trusts
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106 P A R T I Introduction
a few commodity funds such as ones for gold and silver. For more information on these
funds, go to www.iShares.com. More recently, a variety of new ETF products have been devised. Among these are lev-
eraged ETFs, with daily returns that are a targeted multiple of the returns on an index, and inverse ETFs, which move in the opposite direction to an index. In addition, there is now
a small number of actively managed ETF funds that, like actively managed mutual funds,
attempt to outperform market indexes. But these account for only about 3% of assets under
management in the ETF industry.
Other even more exotic variations are so-called synthetic ETFs such as exchange-
traded notes (ETNs) or exchange-traded vehicles (ETVs). These are nominally debt
securities, but with payoffs linked to the performance of an index. Often that index mea-
sures the performance of an illiquid and thinly traded asset class, so the ETF gives the
investor the opportunity to add that asset class to his or her portfolio. However, rather
than invest in those assets directly, the ETF achieves this exposure by entering a “total
return swap” with an investment bank in which the bank agrees to pay the ETF the return
on the index in exchange for a relatively fixed fee. These have become controversial, as
the ETF is then exposed to risk that in a period of financial stress the investment bank
will be unable to fulfill its obligation, leaving investors without the returns they were
promised.
ETFs offer several advantages over conventional mutual funds. First, as we just noted,
a mutual fund’s net asset value is quoted—and therefore, investors can buy or sell their
shares in the fund—only once a day. In contrast, ETFs trade continuously. Moreover,
like other shares, but unlike mutual funds, ETFs can be sold short or purchased on
margin.
ETFs also offer a potential tax advantage over mutual funds. When large numbers
of mutual fund investors redeem their shares, the fund must sell securities to meet the
redemptions. This can trigger capital gains taxes, which are passed through to and must be
paid by the remaining shareholders. In contrast, when small investors wish to redeem their
position in an ETF, they simply sell their shares to other traders, with no need for the fund
to sell any of the underlying portfolio. Large investors can exchange their ETF shares for
shares in the underlying portfolio; this form of redemption also avoids a tax event.
ETFs are often cheaper than mutual funds. Investors who buy ETFs do so through bro-
kers rather than buying directly from the fund. Therefore, the fund saves the cost of mar-
keting itself directly to small investors. This reduction in expenses may translate into lower
management fees.
There are some disadvantages to ETFs, however. First, while mutual funds can be
bought at no expense from no-load funds, ETFs must be purchased from brokers for a
fee. In addition, because ETFs trade as securities, their prices can depart from NAV, at
least for short periods, and these price discrepancies can easily swamp the cost advan-
tage that ETFs otherwise offer. While those discrepancies typically are quite small, they
can spike unpredictably when markets are stressed. Chapter 2 briefly discussed the so-
called flash crash of May 6, 2010, when the Dow Jones Industrial Average fell by 583
points in seven minutes, leaving it down nearly 1,000 points for the day. Remarkably, the index recovered more than 600 points in the next 10 minutes. In the wake of this incred-
ible volatility, the stock exchanges canceled many trades that had gone off at what were
viewed as distorted prices. Around one-fifth of all ETFs changed hands on that day at
prices less than one-half of their closing price, and ETFs accounted for about two-thirds
of all canceled trades.
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C H A P T E R 4 Mutual Funds and Other Investment Companies 107
We noted earlier that one of the benefits of mutual funds for the individual investor is the
ability to delegate management of the portfolio to investment professionals. The investor
retains control over the broad features of the overall portfolio through the asset allocation
decision: Each individual chooses the percentages of the portfolio to invest in bond funds
versus equity funds versus money market funds, and so forth, but can leave the specific
security selection decisions within each investment class to the managers of each fund.
Shareholders hope that these portfolio managers can achieve better investment perfor-
mance than they could obtain on their own.
What is the investment record of the mutual fund industry? This seemingly straightfor-
ward question is deceptively difficult to answer because we need a standard against which
to evaluate performance. For example, we clearly would not want to compare the invest-
ment performance of an equity fund to the rate of return available in the money market.
The vast differences in the risk of these two markets dictate that year-by-year as well as
average performance will differ considerably. We would expect to find that equity funds
outperform money market funds (on average) as compensation to investors for the extra
risk incurred in equity markets. How then can we determine whether mutual fund portfolio
managers are performing up to par given the level of risk they incur? In other words, what is the proper benchmark against which investment performance ought to be evaluated?
Measuring portfolio risk properly and using such measures to choose an appropriate
benchmark is far from straightforward. We devote all of Parts Two and Three of the text
to issues surrounding the proper measurement of portfolio risk and the trade-off between
risk and return. In this chapter, therefore, we will satisfy ourselves with a first look at the
question of fund performance by using only very simple performance benchmarks and
ignoring the more subtle issues of risk differences across funds. However, we will return
to this topic in Chapter 11, where we take a closer look at mutual fund performance after
adjusting for differences in the exposure of portfolios to various sources of risk.
Here we use as a benchmark for the performance of equity fund managers the rate of
return on the Wilshire 5000 index. Recall from Chapter 2 that this is a value-weighted
index of essentially all actively traded U.S. stocks. The performance of the Wilshire
5000 is a useful benchmark with which to evaluate professional managers because it
corresponds to a simple passive investment strategy: Buy all the shares in the index in
proportion to their outstanding market value. Moreover, this is a feasible strategy for
even small investors, because the Vanguard Group offers an index fund (its Total Stock
Market Portfolio) designed to replicate the performance of the Wilshire 5000 index.
Using the Wilshire 5000 index as a benchmark, we may pose the problem of evaluat-
ing the performance of mutual fund portfolio managers this way: How does the typical
performance of actively managed equity mutual funds compare to the performance of a
passively managed portfolio that simply replicates the composition of a broad index of
the stock market?
4.7 Mutual Fund Investment Performance: A First Look
At least two problems were exposed in this episode. First, when markets are not work-
ing properly, it can be hard to measure the net asset value of the ETF portfolio, especially
for ETFs that track less liquid assets. And, reinforcing this problem, some ETFs may be
supported by only a very small number of dealers. If they drop out of the market during a
period of turmoil, prices may swing wildly.
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108 P A R T I Introduction
Casual comparisons of the performance of the Wilshire 5000 index versus that of
professionally managed mutual funds reveal disappointing results for active managers.
Figure 4.4 shows that the average return on diversified equity funds was below the return on
the Wilshire index in 25 of the 41 years from 1971 to 2011. The average annual return
on the index was 11.75%, which was 1% greater than that of the average mutual fund. 5
This result may seem surprising. After all, it would not seem unreasonable to expect
that professional money managers should be able to outperform a very simple rule such as
“hold an indexed portfolio.” As it turns out, however, there may be good reasons to expect
such a result. We explore them in detail in Chapter 11, where we discuss the efficient mar-
ket hypothesis.
Of course, one might argue that there are good managers and bad managers, and that
good managers can, in fact, consistently outperform the index. To test this notion, we
examine whether managers with good performance in one year are likely to repeat that per-
formance in a following year. Is superior performance in any particular year due to luck,
and therefore random, or due to skill, and therefore consistent from year to year?
To answer this question, we can examine the performance of a large sample of equity
mutual fund portfolios, divide the funds into two groups based on total investment return,
and ask: “Do funds with investment returns in the top half of the sample in one period con-
tinue to perform well in a subsequent period?”
5 Of course, actual funds incur trading costs while indexes do not, so a fair comparison between the returns on
actively managed funds versus those on a passive index should first reduce the return on the Wilshire 5000 by an
estimate of such costs. Vanguard’s Total Stock Market Index portfolio, which tracks the Wilshire 5000, charges
an expense ratio of less than .10%, and, because it engages in little trading, incurs low trading costs. Therefore,
it would be reasonable to reduce the returns on the index by about .15%. This reduction would not erase the dif-
ference in average performance.
Figure 4.4 Rates of return on actively managed equity funds versus Wilshire 5000 index, 1971–2011 Source: For Wilshire returns, see www.wilshire.com. Used with permission of Wilshire Associates. For active fund returns, see http:// www.fiscalisadvisory.com/assets/pdfs/spiva_report_year_end_2011.pdf .
240
230
220
210
0
10
20
30
40
R at
e o
f R
e tu
rn (
% )
Active Funds Wilshire 5000
1 9
7 1
1 9
7 5
1 9
7 9
1 9
8 3
1 9
8 7
1 9
9 1
1 9
9 5
1 9
9 9
2 0
0 3
2 0
0 7
2 0
1 1
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C H A P T E R 4 Mutual Funds and Other Investment Companies 109
Table 4.4 presents such an analysis from a study by Malkiel. 6 The table shows the frac-
tion of “winners” (i.e., top-half performers) in each year that turn out to be winners or los-
ers in the following year. If performance were purely random from one period to the next,
there would be entries of 50% in each cell of the table, as top- or bottom-half performers
would be equally likely to perform in either the top or bottom half of the sample in the
following period. On the other hand, if performance were due entirely to skill, with no
randomness, we would expect to see entries of 100% on the diagonals and entries of 0% on
the off-diagonals: Top-half performers would all remain in the top half while bottom-half
performers similarly would all remain in the bottom half. In fact, the table shows that 65.1% of
initial top-half performers fall in the top half of the sample in the following period, while 64.5%
of initial bottom-half performers fall in the bottom half in the following period. This evidence
is consistent with the notion that at least part of a fund’s performance is a function of skill as
opposed to luck, so that relative performance tends to persist from one period to the next. 7
On the other hand, this relationship does not seem stable across different sample
periods. While initial-year performance predicts subsequent-year performance in the 1970s
(panel A), the pattern of persistence in performance virtually disappears in the 1980s (panel B).
To summarize, the evidence that performance is consistent from one period to the next is
suggestive, but it is inconclusive.
Other studies suggest that there is little performance persistence among professional
managers, and if anything, bad performance is more likely to persist than good perfor-
mance. 8 This makes some sense: It is easy to identify fund characteristics that will result in
consistently poor investment performance, notably high expense ratios, and high turnover
ratios with associated trading costs. It is far harder to identify the secrets of successful
stock picking. (If it were easy, we would all be rich!) Thus the consistency we do observe
in fund performance may be due in large part to the poor performers. This suggests that
the real value of past performance data is to avoid truly poor funds, even if identifying the
future top performers is still a daunting task.
Table 4.4
Consistency of investment results
Successive Period Performance
Initial Period Performance Top Half Bottom Half
A. Malkiel study, 1970s Top half 65.1% 34.9%
Bottom half 35.5 64.5
B. Malkiel study, 1980s
Top half 51.7 48.3
Bottom half 47.5 52.5
Source: Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. Used with permission of John Wiley and Sons, via Copyright Clearance Center.
6 Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June
1995), pp. 549–72. 7 Another possibility is that performance consistency is due to variation in fee structure across funds. We return to
this possibility in Chapter 11. 8 See for example, Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52
(1997), 57–82. Carhart’s study also addresses survivorship bias, the tendency for better-performing funds to stay
in business and thus remain in the sample. We return to his study in Chapter 11.
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110 P A R T I Introduction
Suppose you observe the investment performance of 400 portfolio managers and rank them by invest- ment returns during the year. Twenty percent of all managers are truly skilled, and therefore always fall in the top half, but the others fall in the top half purely because of good luck. What fraction of this year’s top-half managers would you expect to be top-half performers next year?
CONCEPT CHECK 4.4
The first place to find information on a mutual fund is in its prospectus. The Securities and
Exchange Commission requires that the prospectus describe the fund’s investment objec-
tives and policies in a concise “Statement of Investment Objectives” as well as in lengthy
discussions of investment policies and risks. The fund’s investment adviser and its port-
folio manager are also described. The prospectus also presents the costs associated with
purchasing shares in the fund in a fee table. Sales charges such as front-end and back-end
loads as well as annual operating expenses such as management fees and 12b-1 fees are
detailed in the fee table.
Funds provide information about themselves in two other sources. The Statement of
Additional Information (SAI), also known as Part B of the prospectus, includes a list of the
securities in the portfolio at the end of the fiscal year, audited financial statements, a list
of the directors and officers of the fund—as well as their personal investments in the fund,
and data on brokerage commissions paid by the fund. However, unlike the fund prospec-
tus, investors do not receive the SAI unless they specifically request it; one industry joke
is that SAI stands for “something always ignored.” The fund’s annual report also includes
portfolio composition and financial statements, as well as a discussion of the factors that
influenced fund performance over the last reporting period.
With thousands of funds to choose from, it can be difficult to find and select the
fund that is best suited for a particular need. Several publications now offer “encyclope-
dias” of mutual fund information to help in the search process. One prominent source is
Morningstar’s Mutual Fund Sourcebook. Morningstar’s Web site, www.morningstar.com, is another excellent source of information, as is Yahoo!’s site, finance.yahoo.com/funds. The Investment Company Institute ( www.ici.org ), the national association of mutual funds, closed-end funds, and unit investment trusts, publishes an annual Directory of Mutual Funds that includes information on fees as well as phone numbers to contact funds. To illustrate the range of information available about funds, we consider Morningstar’s
report on Wells Fargo’s Advantage Growth Fund, reproduced in Figure 4.5 .
The table on the left labeled “Performance” first shows the fund’s quarterly returns
in the last few years, and just below that, returns over longer periods. You can compare
returns to both its standard index (the S&P 500) and its category index (the Russell 1000)
in the rows labeled 1 /2 Index, as well as its percentile rank within its comparison group,
or category. Continuing down the left column we see data on fees and expenses, as well
as several measures of the fund’s risk and return characteristics. (We will discuss all of
these measures in Part 2 of the text.) The fund has provided good returns compared to risk,
earning it the coveted Morningstar 5-star rating. Of course, we are all accustomed to the
disclaimer that “past performance is not a reliable measure of future results,” and this is
true as well of Morningstar’s star ratings. Consistent with this disclaimer, past results have
little predictive power for future performance, as we saw in Table 4.4 .
4.8 Information on Mutual Funds
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C H A P T E R 4 Mutual Funds and Other Investment Companies 111
Figure 4.5 Morningstar report Source: Morningstar Mutual Funds, © 2012 Morningstar, Inc. All rights reserved. Used with permission.
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112 P A R T I Introduction
More data on the performance of the fund are provided in the graph near the top of the
figure. The line graph compares the growth of $10,000 invested in the fund versus its two
comparison indexes over the last 10 years. Below the graph are boxes for each year that
depict the relative performance of the fund for that year. The shaded area on the box shows
the quartile in which the fund’s performance falls relative to other funds with the same
objective. If the shaded band is at the top of the box, the firm was a top quartile performer
in that period, and so on. The table below the bar charts presents historical data on the year-
by-year performance of the fund.
Below the table, the “Portfolio Analysis” table shows the asset allocation of the fund, and
then Morningstar’s well-known style box. In this box, Morningstar evaluates style along
two dimensions: One dimension is the size of the firms held in the portfolio as measured
by the market value of outstanding equity; the other dimension is a value/growth measure.
Morningstar defines value stocks as those with low ratios of market price per share to vari- ous measures of value. It puts stocks on a growth-value continuum based on the ratios of
stock price to the firm’s earnings, book value, sales, cash flow, and dividends. Value stocks
are those with a low price relative to these measures of value. In contrast, growth stocks have high ratios, suggesting that investors in these firms must believe that the firm will
experience rapid growth to justify the prices at which the stocks sell. The shaded box shows
that the fund tends to hold larger firms (top row) and growth stocks (right column).
Finally, the tables in the right column provide information on the current composition
of the portfolio. You can find the fund’s 15 “Top Holdings” there as well as the weighting
of the portfolio across various sectors of the economy.
1. Unit investment trusts, closed-end management companies, and open-end management compa- nies are all classified and regulated as investment companies. Unit investment trusts are essen-
tially unmanaged in the sense that the portfolio, once established, is fixed. Managed investment
companies, in contrast, may change the composition of the portfolio as deemed fit by the portfo-
lio manager. Closed-end funds are traded like other securities; they do not redeem shares for their
investors. Open-end funds will redeem shares for net asset value at the request of the investor.
2. Net asset value equals the market value of assets held by a fund minus the liabilities of the fund divided by the shares outstanding.
3. Mutual funds free the individual from many of the administrative burdens of owning individual securities and offer professional management of the portfolio. They also offer advantages that are
available only to large-scale investors, such as discounted trading costs. On the other hand, funds
are assessed management fees and incur other expenses, which reduce the investor’s rate of return.
Funds also eliminate some of the individual’s control over the timing of capital gains realizations.
4. Mutual funds are often categorized by investment policy. Major policy groups include money market funds; equity funds, which are further grouped according to emphasis on income versus
growth; fixed-income funds; balanced and income funds; asset allocation funds; index funds; and
specialized sector funds.
5. Costs of investing in mutual funds include front-end loads, which are sales charges; back-end loads, which are redemption fees or, more formally, contingent-deferred sales charges; fund oper-
ating expenses; and 12b-1 charges, which are recurring fees used to pay for the expenses of mar-
keting the fund to the public.
6. Income earned on mutual fund portfolios is not taxed at the level of the fund. Instead, as long as the fund meets certain requirements for pass-through status, the income is treated as being earned
by the investors in the fund.
SUMMARY
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C H A P T E R 4 Mutual Funds and Other Investment Companies 113
7. The average rate of return of the average equity mutual fund in the last four decades has been below that of a passive index fund holding a portfolio to replicate a broad-based index like
the S&P 500 or Wilshire 5000. Some of the reasons for this disappointing record are the costs
incurred by actively managed funds, such as the expense of conducting the research to guide
stock-picking activities, and trading costs due to higher portfolio turnover. The record on the
consistency of fund performance is mixed. In some sample periods, the better-performing funds
continue to perform well in the following periods; in other sample periods they do not.
Related Web sites for this chapter are available at www. mhhe.com/bkm
investment company
net asset value (NAV)
unit investment trust
open-end fund
closed-end fund
load
hedge fund
funds of funds
12b-1 fees
soft dollars
turnover
exchange-traded funds
KEY TERMS
PROBLEM SETS 1. Would you expect a typical open-end fixed-income mutual fund to have higher or lower operat- ing expenses than a fixed-income unit investment trust? Why?
2. What are some comparative advantages of investing in the following:
a. Unit investment trusts. b. Open-end mutual funds. c. Individual stocks and bonds that you choose for yourself.
3. Open-end equity mutual funds find it necessary to keep a significant percentage of total invest-
ments, typically around 5% of the portfolio, in very liquid money market assets. Closed-end
funds do not have to maintain such a position in “cash equivalent” securities. What difference
between open-end and closed-end funds might account for their differing policies?
4. Balanced funds, life-cycle funds, and asset allocation funds all invest in both the stock and bond
markets. What are the differences among these types of funds?
5. Why can closed-end funds sell at prices that differ from net asset value while open-end funds
do not?
6. What are the advantages and disadvantages of exchange-traded funds versus mutual funds?
7. An open-end fund has a net asset value of $10.70 per share. It is sold with a front-end load of
6%. What is the offering price?
8. If the offering price of an open-end fund is $12.30 per share and the fund is sold with a front-
end load of 5%, what is its net asset value?
9. The composition of the Fingroup Fund portfolio is as follows:
Stock Shares Price
A 200,000 $35 B 300,000 40 C 400,000 20 D 600,000 25
The fund has not borrowed any funds, but its accrued management fee with the portfolio man-
ager currently totals $30,000. There are 4 million shares outstanding. What is the net asset value
of the fund?
10. Reconsider the Fingroup Fund in the previous problem. If during the year the portfolio
manager sells all of the holdings of stock D and replaces it with 200,000 shares of stock E
at $50 per share and 200,000 shares of stock F at $25 per share, what is the portfolio turn-
over rate?
Intermediate
Basic
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114 P A R T I Introduction
11. The Closed Fund is a closed-end investment company with a portfolio currently worth $200
million. It has liabilities of $3 million and 5 million shares outstanding.
a. What is the NAV of the fund? b. If the fund sells for $36 per share, what is its premium or discount as a percent of net asset
value?
12. Corporate Fund started the year with a net asset value of $12.50. By year-end, its NAV equaled
$12.10. The fund paid year-end distributions of income and capital gains of $1.50. What was the
(pretax) rate of return to an investor in the fund?
13. A closed-end fund starts the year with a net asset value of $12.00. By year-end, NAV equals
$12.10. At the beginning of the year, the fund was selling at a 2% premium to NAV. By the end
of the year, the fund is selling at a 7% discount to NAV. The fund paid year-end distributions of
income and capital gains of $1.50.
a. What is the rate of return to an investor in the fund during the year? b. What would have been the rate of return to an investor who held the same securities as the
fund manager during the year?
14. a. Impressive Fund had excellent investment performance last year, with portfolio returns that placed it in the top 10% of all funds with the same investment policy. Do you expect it to be
a top performer next year? Why or why not?
b. Suppose instead that the fund was among the poorest performers in its comparison group. Would you be more or less likely to believe its relative performance will persist into the fol-
lowing year? Why?
15. Consider a mutual fund with $200 million in assets at the start of the year and with 10 million
shares outstanding. The fund invests in a portfolio of stocks that provides dividend income
at the end of the year of $2 million. The stocks included in the fund’s portfolio increase in
price by 8%, but no securities are sold, and there are no capital gains distributions. The fund
charges 12b-1 fees of 1%, which are deducted from portfolio assets at year-end. What is
net asset value at the start and end of the year? What is the rate of return for an investor in
the fund?
16. The New Fund had average daily assets of $2.2 billion last year. The fund sold $400 million
worth of stock and purchased $500 million during the year. What was its turnover ratio?
17. If New Fund’s expense ratio (see the previous problem) was 1.1% and the management fee was
.7%, what were the total fees paid to the fund’s investment managers during the year? What
were other administrative expenses?
18. You purchased 1,000 shares of the New Fund at a price of $20 per share at the beginning of
the year. You paid a front-end load of 4%. The securities in which the fund invests increase in
value by 12% during the year. The fund’s expense ratio is 1.2%. What is your rate of return on
the fund if you sell your shares at the end of the year?
19. Loaded-Up Fund charges a 12b-1 fee of 1.0% and maintains an expense ratio of .75%. Econ-
omy Fund charges a front-end load of 2% but has no 12b-1 fee and an expense ratio of .25%.
Assume the rate of return on both funds’ portfolios (before any fees) is 6% per year. How much
will an investment in each fund grow to after:
a. 1 year. b. 3 years. c. 10 years.
20. City Street Fund has a portfolio of $450 million and liabilities of $10 million.
a. If 44 million shares are outstanding, what is net asset value? b. If a large investor redeems 1 million shares, what happens to the portfolio value, to shares
outstanding, and to NAV?
21. The Investments Fund sells Class A shares with a front-end load of 6% and Class B shares with
12b-1 fees of .5% annually as well as back-end load fees that start at 5% and fall by 1% for each
full year the investor holds the portfolio (until the fifth year). Assume the portfolio rate of return
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C H A P T E R 4 Mutual Funds and Other Investment Companies 115
net of operating expenses is 10% annually. If you plan to sell the fund after 4 years, are Class A
or Class B shares the better choice for you? What if you plan to sell after 15 years?
22. You are considering an investment in a mutual fund with a 4% load and expense ratio of .5%.
You can invest instead in a bank CD paying 6% interest.
a. If you plan to invest for 2 years, what annual rate of return must the fund portfolio earn for you to be better off in the fund than in the CD? Assume annual compounding of returns.
b. How does your answer change if you plan to invest for 6 years? Why does your answer change?
c. Now suppose that instead of a front-end load the fund assesses a 12b-1 fee of .75% per year. What annual rate of return must the fund portfolio earn for you to be better off in the fund
than in the CD? Does your answer in this case depend on your time horizon?
23. Suppose that every time a fund manager trades stock, transaction costs such as commissions
and bid–ask spreads amount to .4% of the value of the trade. If the portfolio turnover rate is
50%, by how much is the total return of the portfolio reduced by trading costs?
24. You expect a tax-free municipal bond portfolio to provide a rate of return of 4%. Management
fees of the fund are .6%. What fraction of portfolio income is given up to fees? If the manage-
ment fees for an equity fund also are .6%, but you expect a portfolio return of 12%, what frac-
tion of portfolio income is given up to fees? Why might management fees be a bigger factor in
your investment decision for bond funds than for stock funds? Can your conclusion help explain
why unmanaged unit investment trusts tend to focus on the fixed-income market?
25. Suppose you observe the investment performance of 350 portfolio managers for 5 years and
rank them by investment returns during each year. After 5 years, you find that 11 of the funds
have investment returns that place the fund in the top half of the sample in each and every year
of your sample. Such consistency of performance indicates to you that these must be the funds
whose managers are in fact skilled, and you invest your money in these funds. Is your conclu-
sion warranted?
Challenge
E-INVESTMENTS EXERCISES Go to www.morningstar.com. In the Morningstar Tools section, click on the link for the Mutual Fund Screener. Set the criteria you desire, then click on the Show Results tab. If you get no funds that meet all of your criteria, choose the criterion that is least important to you and relax that constraint. Continue the process until you have several funds to compare.
1. Examine all of the views available in the drop-down box menu ( Snapshot, Performance, Portfolio, and Nuts and Bolts ) to answer the following questions:
a. Which fund has the best expense ratio?
b. Which funds have the lowest Morningstar Risk rating?
c. Which fund has the best 3-year return? Which has the best 10-year return?
d. Which fund has the lowest turnover ratio? Which has the highest?
e. Which fund has the longest manager tenure? Which has the shortest?
f. Do you need to eliminate any of the funds from consideration due to a minimum initial investment that is higher than you are capable of making?
2. Based on what you know about the funds, which one do you think would be the best one for your investment?
3. Select up to five funds that are of the most interest to you. Click on the button that says Score These Results. Customize the criteria listed by indicating their importance to you. Examine the score results. Does the fund with the highest score match the choice you made in part 2?
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116 P A R T I Introduction
1. NAV 5 $2,877.06 2 $14.73
95.50 5 $29.97
2. The net investment in the Class A shares after the 4% commission is $9,600. If the fund earns a
10% return, the investment will grow after n years to $9,600 3 (1.10) n . The Class B shares have no front-end load. However, the net return to the investor after 12b-1 fees will be only 9.5%.
In addition, there is a back-end load that reduces the sales proceeds by a percentage equal to
(5 – years until sale) until the fifth year, when the back-end load expires.
Class A Shares Class B Shares
Horizon $9,600 3 (1.10)n $10,000 3 (1.095)n 3 (1 2 percentage exit fee)
1 year $10,560 $10,000 3 (1.095) 3 (1 2 .04) 5 $10,512 4 years $14,055 $10,000 3 (1.095)4 3 (1 2 .01) 5 $14,233 10 years $24,900 $10,000 3 (1.095)10 5 $24,782
For a very short horizon such as 1 year, the Class A shares are the better choice. The front-end and
back-end loads are equal, but the Class A shares don’t have to pay the 12b-1 fees. For moderate
horizons such as 4 years, the Class B shares dominate because the front-end load of the Class A
shares is more costly than the 12b-1 fees and the now-smaller exit fee. For long horizons of
10 years or more, Class A again dominates. In this case, the one-time front-end load is less
expensive than the continuing 12b-1 fees.
3. a. Turnover 5 $160,000 in trades per $1 million of portfolio value 5 16%. b. Realized capital gains are $10 3 1,000 5 $10,000 on FedEx and $2.50 3 4,000 5 $10,000 on
Cisco. The tax owed on the capital gains is therefore .20 3 $20,000 5 $4,000.
4. Twenty percent of the managers are skilled, which accounts for .2 3 400 5 80 of those managers
who appear in the top half. There are 120 slots left in the top half, and 320 other managers, so
the probability of an unskilled manager “lucking into” the top half in any year is 120/320, or
.375. Therefore, of the 120 lucky managers in the first year, we would expect .375 3 120 5 45
to repeat as top-half performers next year. Thus, we should expect a total of 80 1 45 5 125, or
62.5%, of the better initial performers to repeat their top-half performance.
SOLUTIONS TO CONCEPT CHECKS
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5 CASUAL OBSERVATION AND formal research both suggest that investment risk is as impor- tant to investors as expected return. While we have theories about the relationship between risk and expected return that would prevail in rational capital markets, there is no theory about the levels of risk we should find in the marketplace. We can at best estimate the level of risk likely to confront investors from his- torical experience.
This situation is to be expected because prices of investment assets fluctuate in response to news about the fortunes of corporations, as well as to macroeconomic developments. There is no theory about the frequency and impor- tance of such events; hence we cannot deter- mine a “natural” level of risk.
Compounding this difficulty is the fact that neither expected returns nor risk are directly observable. We observe only realized rates of return. Hence, to make forecasts about future expected returns and risk, we must learn how to “forecast” their past values, that is, the expected returns and risk that investors actu- ally anticipated, from historical data. (There is an old saying that forecasting the future
is even more difficult than forecasting the past.) Moreover, in learning from a historical record we face what has become known as the “black swan” problem. 1 No matter how long a historical record, there is never a guar- antee that it exhibits the worst (and best) that nature can throw at us in the future. This problem is particularly daunting when con- sidering the risk of long-run investments. In this chapter, we present the essential tools for estimating expected returns and risk from the historical record and consider implications for future investments.
We begin with interest rates and invest- ments in safe assets and examine the history of risk-free investments in the U.S over the last 86 years. Moving to risky assets, we begin with scenario analysis of risky investments and the data inputs necessary to conduct it. With this in mind, we develop statistical tools needed to make inferences from historical time series of portfolio returns. We present a global view of the history of stock and bond returns worldwide. We end with implications of the historical record for future investments and risk measures commonly used in the industry.
Risk, Return, and the
Historical Record
CHAPTER FIVE 5 PA
R T II
1 Black swans are a metaphor for highly improbable—but highly impactful—events. Until the discovery of Australia, Europeans,
having observed only white swans, believed that a black swan was outside the realm of reasonable possibility or, in statistical jargon,
an extreme “outlier” relative to their “sample” of observations.
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118 P A R T I I Portfolio Theory and Practice
5.1 Determinants of the Level of Interest Rates
Interest rates and forecasts of their future values are among the most important inputs
into an investment decision. For example, suppose you have $10,000 in a savings
account. The bank pays you a variable interest rate tied to some short-term reference rate
such as the 30-day Treasury bill rate. You have the option of moving some or all of your
money into a longer-term certificate of deposit that offers a fixed rate over the term of
the deposit.
Your decision depends critically on your outlook for interest rates. If you think rates
will fall, you will want to lock in the current higher rates by investing in a relatively long-
term CD. If you expect rates to rise, you will want to postpone committing any funds to
long-term CDs.
Forecasting interest rates is one of the most notoriously difficult parts of applied macro-
economics. Nonetheless, we do have a good understanding of the fundamental factors that
determine the level of interest rates:
1. The supply of funds from savers, primarily households.
2. The demand for funds from businesses to be used to finance investments in plant,
equipment, and inventories (real assets or capital formation).
3. The government’s net demand for funds as modified by actions of the Federal
Reserve Bank.
Before we elaborate on these forces and resultant interest rates, we need to distinguish
real from nominal interest rates.
Real and Nominal Rates of Interest An interest rate is a promised rate of return denominated in some unit of account (dollars,
yen, euros, or even purchasing power units) over some time period (a month, a year,
20 years, or longer). Thus, when we say the interest rate is 5%, we must specify both the
unit of account and the time period.
Assuming there is no default risk, we can refer to the promised rate of interest as a
risk-free rate for that particular unit of account and time period. But if an interest rate is
risk-free for one unit of account and time period, it will not be risk-free for other units or
periods. For example, interest rates that are absolutely safe in dollar terms will be risky
when evaluated in terms of purchasing power because of inflation uncertainty.
To illustrate, consider a 1-year dollar (nominal) risk-free interest rate. Suppose
exactly 1 year ago you deposited $1,000 in a 1-year time deposit guaranteeing a rate
of interest of 10%. You are about to collect $1,100 in cash. What is the real return on
your investment? That depends on what money can buy these days, relative to what you
could buy a year ago. The consumer price index (CPI) measures purchasing power by averaging the prices of goods and services in the consumption basket of an average
urban family of four.
Suppose the rate of inflation (the percent change in the CPI, denoted by i ) for the last year amounted to i 5 6%. This tells you that the purchasing power of money is reduced by 6% a year. The value of each dollar depreciates by 6% a year in terms of the goods it can buy.
Therefore, part of your interest earnings are offset by the reduction in the purchasing
power of the dollars you will receive at the end of the year. With a 10% interest rate, after
you net out the 6% reduction in the purchasing power of money, you are left with a net
increase in purchasing power of about 4%. Thus we need to distinguish between a nominal interest rate —the growth rate of your money—and a real interest rate —the growth rate
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C H A P T E R 5 Risk, Return, and the Historical Record 119
of your purchasing power. If we call rn the nominal rate, rr the real rate, and i the inflation rate, then we conclude
rr < rn 2 i (5.1)
In words, the real rate of interest is the nominal rate reduced by the loss of purchasing
power resulting from inflation.
In fact, the exact relationship between the real and nominal interest rate is given by
1 1 rr 5 1 1 rn
1 1 i (5.2)
This is because the growth factor of your purchasing power, 1 1 rr, equals the growth factor of your money, 1 1 rn, divided by the growth factor of prices, 1 1 i. The exact rela- tionship can be rearranged to
rr 5 rn 2 i
1 1 i (5.3)
which shows that the approximation rule (Equation 5.1) overstates the real rate by the
factor 1 1 i.
If the nominal interest rate on a 1-year CD is 8%, and you expect inflation to be 5% over the coming year, then using the approximation formula, you expect the real rate of interest to be rr 5 8% 2 5% 5 3%. Using the exact formula, the real rate
is rr 5 .08 2 .05 1 1 .05
5 .0286, or 2.86%. Therefore, the approximation rule overstates the
expected real rate by .14% (14 basis points). The approximation rule is more exact for small inflation rates and is perfectly exact for continuously compounded rates. We discuss further details in the next section.
Example 5.1 Approximating the Real Rate
2 You can find the real rate for a desired maturity from inflation-indexed bonds issued by the U.S. Treasury, called
TIPS. (See Chapter 14 for a fuller discussion.) The difference between TIPS yields and yields on comparable
nominal Treasury bonds provides an estimate of the market’s expectation of future inflation.
Notice that conventional certificates of deposit offer a guaranteed nominal rate of interest. Thus you can only infer the expected real rate on these investments by adjusting
for your expectation of the rate of inflation.
It is always possible to calculate the real rate after the fact. The inflation rate is pub-
lished by the Bureau of Labor Statistics (BLS). The future real rate, however, is unknown,
and one has to rely on expectations. In other words, because future inflation is risky, the
real rate of return is risky even when the nominal rate is risk-free. 2
The Equilibrium Real Rate of Interest Three basic factors—supply, demand, and government actions—determine the real inter- est rate. The nominal interest rate is the real rate plus the expected rate of inflation. And
thus a fourth factor affecting the nominal interest rate is expected inflation.
Although there are many different interest rates economywide (as many as there are types
of debt securities), these rates tend to move together, so economists frequently talk as if
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120 P A R T I I Portfolio Theory and Practice
Real Interest Rate
Equilibrium Real Rate
of Interest
Equilibrium Funds Lent
Funds
E
E'
Demand
Supply
Figure 5.1 Determination of the equilibrium real rate of interest
3 Experts disagree significantly on the extent to which household saving increases in response to an increase in
the real interest rate.
there were a single represen-
tative rate. We can use this
abstraction to gain some
insights into the real rate of
interest if we consider the
supply and demand curves
for funds.
Figure 5.1 shows a
downward-sloping demand
curve and an upward-
sloping supply curve. On
the horizontal axis, we mea-
sure the quantity of funds,
and on the vertical axis,
we measure the real rate of
interest. The supply curve slopes
up from left to right because
the higher the real interest
rate, the greater the supply
of household savings. The assumption is that at higher real interest rates households will
choose to postpone some current consumption and set aside or invest more of their dispos-
able income for future use. 3
The demand curve slopes down from left to right because the lower the real interest
rate, the more businesses will want to invest in physical capital. Assuming that businesses
rank projects by the expected real return on invested capital, firms will undertake more
projects the lower the real interest rate on the funds needed to finance those projects.
Equilibrium is at the point of intersection of the supply and demand curves, point E in Figure 5.1 .
The government and the central bank (the Federal Reserve) can shift these supply
and demand curves either to the right or to the left through fiscal and monetary policies.
For example, consider an increase in the government’s budget deficit. This increases the
government’s borrowing demand and shifts the demand curve to the right, which causes
the equilibrium real interest rate to rise to point E 9 . That is, a forecast that indicates higher than previously expected government borrowing increases expected future interest rates. The
Fed can offset such a rise through an expansionary monetary policy, which will shift the
supply curve to the right.
Thus, although the fundamental determinants of the real interest rate are the propensity
of households to save and the expected profitability of investment in physical capital, the
real rate can be affected as well by government fiscal and monetary policies.
The Equilibrium Nominal Rate of Interest We’ve seen that the nominal rate of return on an asset is approximately equal to the real
rate plus inflation. Because investors should be concerned with real returns—the increase
in purchasing power—we would expect higher nominal interest rates when inflation is
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C H A P T E R 5 Risk, Return, and the Historical Record 121
higher. This higher nominal rate is necessary to maintain the expected real return offered
by an investment.
Irving Fisher (1930) argued that the nominal rate ought to increase one-for-one with
expected inflation, E(i ). The so-called Fisher equation is
rn 5 rr 1 E(i) (5.4)
The equation implies that when real rates are reasonably stable, changes in nominal rates
ought to predict changes in inflation rates. This claim has been debated and empirically
investigated with mixed results. Although the data do not strongly support the Fisher
equation, nominal interest rates seem to predict inflation as well as alternative methods,
in part because we are unable to forecast inflation well with any method. It is difficult to
determine the empirical validity of the Fisher hypothesis because real rates also change
unpredictably over time. Nominal interest rates can be viewed as the sum of the required
real rate on nominally risk-free assets, plus a “noisy” forecast of inflation.
In Chapter 15 we discuss the relationship between short- and long-term interest rates.
Longer rates incorporate forecasts for long-term inflation. For this reason alone, inter-
est rates on bonds of different maturity may diverge. In addition, we will see that prices
of longer-term bonds are more volatile than those of short-term bonds. This implies that
expected returns on longer-term bonds may include a risk premium, so that the expected
real rate offered by bonds of varying maturity also may vary.
a. Suppose the real interest rate is 3% per year and the expected inflation rate is 8%. According to the Fisher hypothesis, what is the nominal interest rate?
b. Suppose the expected inflation rate rises to 10%, but the real rate is unchanged. What happens to the nominal interest rate?
CONCEPT CHECK 5.1
Taxes and the Real Rate of Interest Tax liabilities are based on nominal income and the tax rate determined by the investor’s tax bracket. Congress recognized the resultant “bracket creep” (when nominal income
grows due to inflation and pushes taxpayers into higher brackets) and mandated index-
linked tax brackets in the Tax Reform Act of 1986.
Index-linked tax brackets do not provide relief from the effect of inflation on the taxa-
tion of savings, however. Given a tax rate ( t ) and a nominal interest rate, rn, the after-tax interest rate is rn (1 2 t ). The real after-tax rate is approximately the after-tax nominal rate minus the inflation rate:
rn(1 2 t) 2 i 5 (rr 1 i)(1 2 t) 2 i 5 rr (1 2 t) 2 it (5.5)
Thus the after-tax real rate falls as inflation rises. Investors suffer an inflation penalty
equal to the tax rate times the inflation rate. If, for example, you are in a 30% tax bracket
and your investments yield 12%, while inflation runs at the rate of 8%, then your before-
tax real rate is approximately 4%, and you should, in an inflation-protected tax system, net after taxes a real return of 4%(1 2 .3) 5 2.8%. But the tax code does not recognize that
the first 8% of your return is just compensation for inflation—not real income—and hence
your after-tax return is reduced by 8% 3 .3 5 2.4%, so that your after-tax real interest rate,
at .4%, is almost wiped out.
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122 P A R T I I Portfolio Theory and Practice
5.2 Comparing Rates of Return for Different Holding Periods
Consider an investor who seeks a safe investment, say, in U.S. Treasury securities. 4 We
observe zero-coupon Treasury securities with several different maturities. Zero-coupon
bonds, discussed more fully in Chapter 14, are sold at a discount from par value and pro-
vide their entire return from the difference between the purchase price and the ultimate
repayment of par value. 5 Given the price, P ( T ), of a Treasury bond with $100 par value and
maturity of T years, we calculate the total risk-free return available for a horizon of T years as the percentage increase in the value of the investment.
rf (T ) 5 100
P(T ) 2 1 (5.6)
For T 5 1, Equation 5.6 provides the risk-free rate for an investment horizon of 1 year.
Suppose prices of zero-coupon Treasuries with $100 face value and various maturities are as follows. We find the total return of each security by using Equation 5.6:
Horizon, T Price, P ( T ) [100/ P ( T )] 2 1 Risk-Free Return
for Given Horizon
Half-year $97.36 100/97.36 2 1 5 .0271 r f (.5) 5 2.71% 1 year $95.52 100/95.52 2 1 5 .0469 r f (1) 5 4.69% 25 years $23.30 100/23.30 2 1 5 3.2918 r f (25) 5 329.18%
Example 5.2 Annualized Rates of Return
Not surprisingly, longer horizons in Example 5.2 provide greater total returns. How
should we compare returns on investments with differing horizons? This requires that we
express each total return as a rate of return for a common period. We typically express all investment returns as an effective annual rate (EAR), defined as the percentage increase in funds invested over a 1-year horizon.
For a 1-year investment, the EAR equals the total return, r f (1), and the gross return, (1 1 EAR), is the terminal value of a $1 investment. For investments that last less
than 1 year, we compound the per-period return for a full year. For the 6-month bill in
Example 5.2, we compound 2.71% half-year returns over two semiannual periods to obtain
a terminal value of 1 1 EAR 5 (1.0271) 2 5 1.0549, implying that EAR 5 5.49%.
For investments longer than a year, the convention is to express the EAR as the annual
rate that would compound to the same value as the actual investment. For example, the
4 Yields on Treasury bills and bonds of various maturities are widely available on the Web, for example at Yahoo!
Finance, MSN Money, or directly from the Federal Reserve.
5 The U.S. Treasury issues T-bills, which are pure discount (or zero-coupon) securities with maturities of up to
1 year. However, financial institutions create zero-coupon Treasury bonds called Treasury strips with maturities
up to 30 years by buying coupon-paying T-bonds, “stripping” off the coupon payments, and selling claims to the
coupon payments and final payment of face value separately. See Chapter 14 for further details.
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C H A P T E R 5 Risk, Return, and the Historical Record 123
investment in the 25-year bond in Example 5.2 grows by its maturity by a factor of 4.2918
(i.e., 1 1 3.2918), so its EAR is found by solving
(1 1 EAR)25 5 4.2918
1 1 EAR 5 4.29181/25 5 1.0600
In general, we can relate EAR to the total return, r f ( T ), over a holding period of length T by using the following equation:
1 1 EAR 5 31 1 rf (T )41/T (5.7) We illustrate with an example.
For the 6-month Treasury in Example 5.2, T 5 ½, and 1/ T 5 2. Therefore,
1 1 EAR 5 (1.0271)2 5 1.0549 and EAR 5 5.49%
For the 25-year Treasury in Example 5.2, T 5 25. Therefore,
1 1 EAR 5 4.29181/25 5 1.060 and EAR 5 6.0%
Example 5.3 Effective Annual Rate versus Total Return
In Table 5.1 we use Equation 5.8 to find the APR corresponding to an EAR of 5.8% with various compounding periods. Conversely, we find values of EAR implied by an APR of 5.8%.
Example 5.4 EAR versus APR
Annual Percentage Rates Annualized rates on short-term investments (by convention, T , 1 year) often are reported using simple rather than compound interest. These are called annual percentage rates, or APRs. For example, the APR corresponding to a monthly rate such as that charged on a credit card is reported as 12 times the monthly rate. More generally, if there are n compounding periods in a year, and the per-period rate is r f ( T ), then the APR 5 n 3 r f ( T ). Conversely, you can find the per-period rate from the APR as r f ( T ) 5 T 3 APR.
Using this procedure, the APR of the 6-month bond in Example 5.2 with a 6-month
rate of 2.71% is 2 3 2.71 5 5.42%. To generalize, note that for short-term investments
of length T, there are n 5 1/ T compounding periods in a year. Therefore, the relationship among the compounding period, the EAR, and the APR is
1 1 EAR 5 31 1 rf (T )4n 5 31 1 rf (T)41/ T 5 31 1 T 3 APR41/ T (5.8) Equivalently,
APR 5 (1 1 EAR)T 2 1
T
Continuous Compounding It is evident from Table 5.1 (and Equation 5.8) that the difference between APR and EAR
grows with the frequency of compounding. This raises the question: How far will these
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124 P A R T I I Portfolio Theory and Practice
two rates diverge as the compounding frequency continues to grow? Put differently, what
is the limit of [1 1 T 3 APR] 1/ T , as T gets ever smaller? As T approaches zero, we effec- tively approach continuous compounding (CC), and the relation of EAR to the annual percentage rate, denoted by r cc for the continuously compounded case, is given by the exponential function
1 1 EAR 5 exp(rcc) 5 e rcc (5.9)
where e is approximately 2.71828. To find r cc from the effective annual rate, we solve Equation 5.9 for r cc as follows:
ln(1 1 EAR) 5 rcc
where ln(•) is the natural logarithm function, the inverse of exp(•). Both the exponen-
tial and logarithmic functions are available in Excel, and are called EXP(•) and LN(•),
respectively.
The continuously compounded annual percentage rate, r cc , that provides an EAR of 5.8% is 5.638% (see Table 5.1 ). This is virtually the same as the APR for daily compounding. But for less frequent compounding, for example, semiannually, the APR necessary to provide the same EAR is noticeably higher, 5.718%. With less frequent compounding, a higher APR is necessary to provide an equivalent effective return.
Example 5.5 Continuously Compounded Rates
While continuous compounding may at first seem to be a mathematical nuisance, work-
ing with such rates can actually simplify calculations of expected return and risk. For
example, given a continuously compounded rate, the total return for any period T, r cc ( T ), is simply exp( T 3 r cc ).
6 In other words, the total return scales up in direct proportion to the
time period, T. This is far simpler than working with the exponents that arise using discrete
6 This follows from Equation 5.9. If 1 1 EAR 5 ercc, then (1 1 EAR)T 5 erccT.
Compounding Period
EAR 5 [1 1 rf (T )] 1/T2 1 5 .058 APR 5 rf (T)*(1/T) 5 .058
T rf (T) APR 5 [(1 1 EAR)^T 2 1]/T rf (T) EAR 5 (1 1 APR*T)^(1/T) 2 1
1 year 1.0000 .0580 .05800 .0580 .05800 6 months 0.5000 .0286 .05718 .0290 .05884 1 quarter 0.2500 .0142 .05678 .0145 .05927 1 month 0.0833 .0047 .05651 .0048 .05957 1 week 0.0192 .0011 .05641 .0011 .05968 1 day 0.0027 .0002 .05638 .0002 .05971 Continuous rcc 5 ln(1 1 EAR) 5 .05638 EAR 5 exp(rcc ) 2 1 5 .05971
Table 5.1
Annual percentage rates (APR) and effective annual rates (EAR). In the first set of columns, we hold the equivalent annual rate (EAR) fixed at 5.8% and find APR for each holding period. In the second set of columns, we hold APR fixed at 5.8% and solve for EAR.
e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 5 Risk, Return, and the Historical Record 125
period compounding. As another example, look again at Equation 5.1. There, the relation-
ship between the real rate rr, the nominal rate rn, and the inflation rate i, rr < rn 2 i, was only an approximation, as demonstrated by Equation 5.3. But if we express all rates as
continuously compounded, then Equation 5.1 is exact, 7 that is, rr cc 5 rn cc 2 i cc .
7 1 1 r (real) 5
1 1 r (nominal)
1 1 inflation
1 ln 31 1 r (real) 4 5 lna1 1 r (nominal) 1 1 inflation
b 5 ln 31 1 r (nominal) 4 2 ln(1 1 inflation) 1 rcc(real) 5 rcc(nominal) 2 icc
A bank offers two alternative interest schedules for a savings account of $100,000 locked in for 3 years: ( a ) a monthly rate of 1%; ( b ) an annually, continuously compounded rate ( r cc ) of 12%. Which alternative should you choose?
CONCEPT CHECK 5.2
Financial time series often begin in July 1926, the starting date of a widely used accu-
rate return database from the Center for Research in Security Prices at the University of
Chicago.
Table 5.2 summarizes the history of short-term interest rates in the U.S., the inflation
rate, and the resultant real rate. You can find the entire post-1926 history of the monthly
rates of these series on the text’s Web site, www.mhhe.com/bkm (link to the student mate- rial for Chapter 5). The real rate is computed from the monthly T-bill rate and the percent
change in the CPI.
The first set of columns of Table 5.2 lists average annual rates for the various series.
The average interest rate over the more recent half of our history (1969–2012), 5.35%, was
noticeably higher than in the earlier half, 1.79%. The reason is inflation, the main driver
of T-bill rates, which also had a noticeably higher average value in the recent half of the
sample, 4.36%, than in the earlier period, 1.74%. Nevertheless, nominal interest rates over
the recent period were still high enough to leave a higher average real rate, 0.95%, com-
pared with a paltry 10 basis points (.10%) for the earlier half.
5.3 Bills and Inflation, 1926–2012
Table 5.2
Statistics for T-bill rates, inflation rates, and real rates, 1926–2012
Annualized Average Rates Annualized Standard Deviations *
T-Bills Inflation Real T-Bill T-Bills Inflation Real T-Bill
All months 3.55 3.04 0.52 2.95 4.06 3.95 First half 1.79 1.74 0.10 1.56 4.66 4.98 Recent half 5.35 4.36 0.95 3.02 2.82 2.44
*Annualized standard deviations include the effect of serial correlations at lags 1–11 months. Sources: Annual rates of return from rolling over 1-month T-bills: Kenneth French and Morningstar, Inc.; annual inflation rates: Bureau of Labor Statistics.
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126 P A R T I I Portfolio Theory and Practice
An important lesson from this history is that even a moderate rate of inflation can erase
most of the nominal gains provided by these low-risk investments. In both halves of the
sample, the real return was less than one-fifth the nominal return.
We can demonstrate the cumulative gain from a particular investment over a given
period using a wealth index. Assuming $1 was invested at the beginning of a period, we compound the investment value, year by year, by 1 plus the gross annual rate of return.
The wealth index at the end of the last year shows the total increase in wealth per dollar
invested over the full investment period. Figure 5.2 shows that $1 invested in T-bills at the
beginning of 1970 would have grown to $9.20 by September 2012, a seemingly impres-
sive gain for a low-risk investment. However, the wealth index in real (inflation-adjusted)
dollars shows a meager ending value of only $1.20. Similarly, a dollar invested in T-bills
for the entire 1926–2012 period is shown (in the inset) to reach $20.25, but with a real
value of only $1.55. The standard deviation (SD) panel in Table 5.2 shows a lower SD of inflation in the
recent half, 2.82%, than in the earlier period, 4.66%. This contributed to a lower standard
deviation of the real rate in the recent half of the sample, 2.44%, compared with the earlier
half, 4.98%. Notice that the SD of the nominal rate actually was higher in the recent period
(3.02%) than in the earlier one (1.56%), indicating that the lower variation in realized real
returns must be attributed to T-bill rates more closely tracking inflation in that period.
Indeed, Figure 5.3 documents a moderation in the rate of inflation as well as a consider-
ably tighter fit between inflation and nominal rates. As we emphasized earlier, investors presumably focus on the real returns they can
earn on their investments. For them to realize an acceptable real rate, they must earn a
higher nominal rate when inflation is expected to be higher. Therefore, the nominal T-bill
rate observed at the beginning of a period should reflect anticipations of inflation over
that period. When the expected real rate is stable and realized inflation matches initial
0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
1
2
3
4
W e a lt
h I n
d e x
5
6
7
8
9
10
0
1926 1936 1946 1956 1966 1976 1986 1996 2006 2016
2 4 6 8
10 12 14 16 18 20 Nominal Bills
Real Bills
Figure 5.2 Nominal and real wealth indexes for investments in Treasury bills, 1970–2012 (inset figure is for 1926–2012)
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C H A P T E R 5 Risk, Return, and the Historical Record 127
expectations, the correlation between inflation and nominal T-bill rates will be close to
perfect (1.0), while the correlation between inflation and the realized real rate will be close
to zero. At the other extreme, if investors either ignored or were very poor at predict-
ing inflation, its correlation with nominal T-bill rates would be zero and the correlation
between inflation and real rates would be perfectly negative ( 2 1.0), since the real rate
would then fall one-for-one with any increase in inflation.
The following table compares correlations of inflation with nominal and real T-bill rates
in early and recent subperiods. The results suggest improving accuracy in anticipating
inflation. The correlation of inflation with nominal T-bills increased from virtually zero to
0.48, while correlation with real T-bills increased from 2 0.98 to 2 0.67. 8
1926–1968 1969–2012
Correlation with nominal T-bill rate 2 0.03 0.48 Correlation with real T-bill rate 2 0.98 2 0.67
215
210
25
0
P e rc
e n
ta g
e P
o in
ts
5
10
15
20
T-Bills Inflation
1 9 2 7
1 9 3 2
1 9 3 7
1 9 4 2
1 9 4 7
1 9 5 2
1 9 5 7
1 9 6 2
1 9 6 7
1 9 7 2
1 9 7 7
1 9 8 2
1 9 8 7
1 9 9 2
1 9 9 7
2 0 0 2
2 0 0 7
2 0 1 2
Figure 5.3 Interest and inflation rates, 1926–2012
8 Examination of Figure 5.3 suggests the correlation in very recent years has fallen off due to the active interven-
tion of the Federal Reserve following the financial crisis.
Holding-Period Returns You are considering investing in a stock-index fund. The fund currently sells for $100 per share.
With an investment horizon of 1 year, the realized rate of return on your investment will depend
on ( a ) the price per share at year’s end and ( b ) the cash dividends you will collect over the year.
5.4 Risk and Risk Premiums
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128 P A R T I I Portfolio Theory and Practice
Suppose the price per share at year’s end is $110 and cash dividends over the year
amount to $4. The realized return, called the holding-period return, HPR (in this case, the holding period is 1 year), is defined as
HPR 5 Ending price of a share 2 Beginning price 1 Cash dividend
Beginning price (5.10)
Here we have
HPR 5 $110 2 $100 1 $4
$100 5 .14, or 14%
This definition of the HPR treats the dividend as paid at the end of the holding period.
When dividends are received earlier, the HPR ignores reinvestment income between the
receipt of the payment and the end of the holding period. The percent return from dividends
is called the dividend yield, and so dividend yield plus the rate of capital gains equals HPR.
Expected Return and Standard Deviation There is considerable uncertainty about the price of a share plus dividend income 1 year
from now, however, so you cannot be sure about your eventual HPR. We can quantify our
beliefs about the state of the market and the stock-index fund in terms of four possible
scenarios with probabilities as presented in columns A through E of Spreadsheet 5.1 .
How can we evaluate this probability distribution? To start, we characterize probability
distributions of rates of return by their expected or mean return, E ( r ), and standard devia- tion, s . The expected rate of return is a probability-weighted average of the rates of return
in each scenario. Calling p ( s ) the probability of each scenario and r ( s ) the HPR in each scenario, where scenarios are labeled or “indexed” by s, we write the expected return as
E(r) 5 a s
p(s)r (s) (5.11)
Applying this formula to the data in Spreadsheet 5.1 , the expected rate of return on the
index fund is
E(r) 5 (.25 3 .31) 1 (.45 3 .14) 1 3.25 3 (2.0675) 4 1 3.05 3 (2.52) 4 5 .0976
A B C D E F HG I
Purchase Price 5 $100
State of the Market Probability Excellent Good Poor
0.05
0.25 0.45 0.25
Expected Value (mean)
Standard Deviation of HPR Variance of HPR
Crash
Year-End Price
46.00
126.50 110.00 89.75
2.00
4.50 4.00 3.50
20.5600
0.2700 0.1000
20.1075 0.3815
0.0451 0.0018 0.0273
20.5200
0.3100 0.1400
20.0675
Cash Dividends
T-bill Rate 5 0.04
HPR
20.6176
0.2124 0.0424
20.1651
Squared Deviations
from Mean Deviations
from Mean
0.3815
0.0451 0.0018 0.0273
0.0380 0.1949
0.0976
Excess Returns
Squared Deviations
from Mean
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
SUMPRODUCT(B8:B11, E8:E11) 5 SUMPRODUCT(B8:B11, G8:G11) 5
SQRT(G13) 5 0.0576Risk Premium
Standard Deviation of Excess Return 0.1949SQRT(SUMPRODUCT(B8:B11, I8:I11)) 5 SUMPRODUCT(B8:B11, H8:H11) 5
Spreadsheet 5.1
Scenario analysis of holding period return of the stock-index fund
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C H A P T E R 5 Risk, Return, and the Historical Record 129
Spreadsheet 5.1 shows that this sum can be evaluated easily in Excel, using the
SUMPRODUCT function, which first calculates the products of a series of number
pairs, and then sums the products. Here, the number pair is the probability of each
scenario and the rate of return.
The standard deviation of the rate of return ( s ) is a measure of risk. It is defined as the
square root of the variance, which in turn is the expected value of the squared deviations
from the expected return. The higher the volatility in outcomes, the higher will be the aver-
age value of these squared deviations. Therefore, variance and standard deviation provide
one measure of the uncertainty of outcomes. Symbolically,
s2 5 a s
p(s)3r(s) 2 E(r)42 (5.12) Therefore, in our example
s2 5 .25(.31 2 .0976)2 1 .45(.14 2 .0976)2 1 .25(2.0675 2 .0976)2
1 .05(2.52 2 .0976)2 5 .0380
This value is calculated in cell G13 of Spreadsheet 5.1 using the SUMPRODUCT func-
tion. The standard deviation is calculated in cell G14 as
s 5 ".0380 5 .1949 5 19.49% Clearly, what would trouble potential investors in the index fund is the downside risk
of a crash or poor market, not the upside potential of a good or excellent market. The stan-
dard deviation of the rate of return does not distinguish between good or bad surprises; it
treats both simply as deviations from the mean. As long as the probability distribution is
more or less symmetric about the mean, s is a reasonable measure of risk. In the special
case where we can assume that the probability distribution is normal—represented by the
well-known bell-shaped curve— E ( r ) and s completely characterize the distribution.
Excess Returns and Risk Premiums How much, if anything, should you invest in our index fund? First, you must ask how
much of an expected reward is offered for the risk involved in investing money in stocks.
We measure the reward as the difference between the expected HPR on the index stock fund and the risk-free rate, that is, the rate you can earn by leaving money in risk-free assets such as T-bills, money market funds, or the bank. We call this difference the risk premium on common stocks. The risk-free rate in our example is 4% per year, and the expected index fund return is 9.76%, so the risk premium on stocks is 5.76% per year.
The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate is called the excess return. Therefore, the risk premium is the expected value of the excess return, and the standard deviation of the excess return is a
measure of its risk. (See Spreadsheet 5.1 for these calculations.)
The degree to which investors are willing to commit funds to stocks depends on risk aversion. Investors are risk averse in the sense that, if the risk premium were zero, they would not invest any money in stocks. In theory, there must always be a positive risk
premium on stocks in order to induce risk-averse investors to hold the existing supply of
stocks instead of placing all their money in risk-free assets.
Although the scenario analysis illustrates the concepts behind the quantification of risk
and return, you may still wonder how to get a more realistic estimate of E ( r ) and s for common stocks and other types of securities. Here, history has insights to offer. Analysis
of the historical record of portfolio returns makes use of a variety of concepts and statisti-
cal tools, and so we first turn to a preparatory discussion.
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130 P A R T I I Portfolio Theory and Practice
You invest $27,000 in a corporate bond selling for $900 per $1,000 par value. Over the coming year, the bond will pay interest of $75 per $1,000 of par value. The price of the bond at year’s end will depend on the level of interest rates that will prevail at that time. You construct the following scenario analysis:
Interest Rates Probability Year-End Bond Price
High .2 $850 Unchanged .5 915 Low .3 985
Your alternative investment is a T-bill that yields a sure rate of return of 5%. Calculate the HPR for each scenario, the expected rate of return, and the risk premium on your investment. What is the expected end- of-year dollar value of your investment?
CONCEPT CHECK 5.3
Time Series versus Scenario Analysis In a forward-looking scenario analysis we determine a set of relevant scenarios and asso-
ciated investment rates of return, assign probabilities to each, and conclude by comput-
ing the risk premium (reward) and standard deviation (risk) of the proposed investment.
In contrast, asset return histories come in the form of time series of realized returns that do
not explicitly provide investors’ original assessments of the probabilities of those returns;
we observe only dates and associated HPRs. We must infer from this limited data the prob-
ability distributions from which these returns might have been drawn or, at least, expected
return and standard deviation.
Expected Returns and the Arithmetic Average When we use historical data, we treat each observation as an equally likely “scenario.”
So if there are n observations, we substitute equal probabilities of 1/ n for each p ( s ) in Equation 5.11. The expected return, E ( r ), is then estimated by the arithmetic average of the sample rates of return:
E(r) 5 a n
s 5 1 p(s)r (s) 5
1
n a n
s 5 1 r (s)
(5.13)
5 Arithmetic average of rates of return
5.5 Time Series Analysis of Past Rates of Return
To illustrate, Spreadsheet 5.2 presents a (very short) time series of annual holding-period returns for the S&P 500 index over the period 2001–2005. We treat each HPR of the n 5 5 observations in the time series as an equally likely annual outcome during the sam- ple years and assign it an equal probability of 1/5, or .2. Column B in Spreadsheet 5.2 therefore uses .2 as probabilities, and Column C shows the annual HPRs. Applying Equation 5.13 (using Excel’s SUMPRODUCT function) to the time series in Spread- sheet 5.2 demonstrates that adding up the products of probability times HPR amounts to taking the arithmetic average of the HPRs (compare cells C10 and C11).
Example 5.6 Arithmetic Average and Expected Return
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C H A P T E R 5 Risk, Return, and the Historical Record 131
Example 5.6 illustrates the logic for the wide use of the arithmetic average in invest-
ments. If the time series of historical returns fairly represents the true underlying prob-
ability distribution, then the arithmetic average return from a historical period provides a
forecast of the investment’s expected future HPR.
The Geometric (Time-Weighted) Average Return The arithmetic average provides an unbiased estimate of the expected future return. But what does the time series tell us about the actual performance of a portfolio over the past sample period? Let’s continue Example 5.6 using its very short sample period just to illus-
trate. We will present results for meaningful periods later in the chapter.
Column F in Spreadsheet 5.2 shows the wealth index from investing $1 in an S&P 500
index fund at the beginning of 2001. The value of the wealth index at the end of 2005,
$1.0275, is the terminal value of the $1 investment, which implies a 5 - year holding-period return (HPR) of 2.75%.
An intuitive measure of performance over the sample period is the (fixed) annual HPR
that would compound over the period to the same terminal value as obtained from the
sequence of actual returns in the time series. Denote this rate by g, so that
Terminal value 5 (1 1 r1) 3 (1 1 r2) 3 ? ? ? 3 (1 1 r5) 5 1.0275
(1 1 g)n 5 Terminal value 5 1.0275 (cell F9 in Spreadsheet 5.2) (5.14)
g 5 Terminal value1/n 2 1 5 1.0751/5 2 1 5 .0054 5 .54% (cell E14)
where 1 1 g is the geometric average of the gross returns (1 1 r ) from the time series (which can be computed with Excel’s GEOMEAN function) and g is the annual HPR that would replicate the final value of our investment.
Practitioners call g the time-weighted (as opposed to dollar-weighted) average return, to emphasize that each past return receives an equal weight in the process of averaging.
This distinction is important because investment managers often experience significant
changes in funds under management as investors purchase or redeem shares. Rates of
return obtained during periods when the fund is large produce larger dollar profits than
rates obtained when the fund is small. We discuss this distinction in the chapter on perfor-
mance evaluation.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C D E F
Period Implicitly Assumed Probability = 1/5
Squared Deviation
Gross HPR = 1 + HPR
Wealth Index*
2001 .2 −0.1189 0.0196 0.0586 0.0707 0.0077
0.1774
0.0008
0.1983
0.8811 0.8811 0.6864 0.8833 0.9794 1.0275
Check: 1.0054^5=
0.7790 1.2869 1.1088 1.0491
0.0054 1.0275
−0.2210 0.2869 0.1088 0.0491 0.0210 0.0210
HPR (decimal)
.2
.2
.2
.2
2002 2003 2004 2005
Arithmetic average Expected HPR SUMPRODUCT(B5:B9, C5:C9) =
SUMPRODUCT(B5:B9, D5:D9)^.5 = STDEV(C5:C9) = Geometric average return
*The value of $1 invested at the beginning of the sample period (1/1/2001). GEOMEAN(E5:E9) − 1 =
Standard deviation
AVERAGE(C5:C9) =
Spreadsheet 5.2
Time series of HPR for the S&P 500
e X c e l Please visit us at www.mhhe.com/bkm
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132 P A R T I I Portfolio Theory and Practice
The larger the swings in rates of return, the greater the discrepancy between the arith-
metic and geometric averages, that is, between the compound rate earned over the sample
period and the average of the annual returns. If returns come from a normal distribution,
the expected difference is exactly half the variance of the distribution, that is,
E3Geometric average4 5 E3Arithmetic average4 2 ½ s2 (5.15) (A warning: To use Equation 5.15, you must express returns as decimals, not percentages.)
When returns are approximately normal, Equation 5.15 will be a good approximation. 9
Variance and Standard Deviation When thinking about risk, we are interested in the likelihood of deviations from the
expected return. In practice, we usually cannot directly observe expectations, so we esti- mate the variance by averaging squared deviations from our estimate of the expected
The geometric average in Example 5.6 (.54%) is substantially less than the arithmetic average (2.10%). This discrepancy sometimes is a source of confusion. It arises from the asymmetric effect of positive and negative rates of returns on the terminal value of the portfolio.
Observe the returns in years 2002 ( 2 .2210) and 2003 (.2869). The arithmetic aver- age return over the 2 years is ( 2 .2210 1 .2869)/2 5 .03295 (3.295%). However, if you had invested $100 at the beginning of 2002, you would have only $77.90 at the end of the year. In order to simply break even, you would then have needed to earn $21.10 in 2003, which would amount to a whopping return of 27.09% (21.10/77.90). Why is such a high rate necessary to break even, rather than the 22.10% you lost in 2002? The value of your investment in 2003 was much smaller than $100; the lower base means that it takes a greater subsequent percentage gain to just break even. Even a rate as high as the 28.69% realized in 2003 yields a portfolio value in 2003 of $77.90 3 1.2869 5 $100.25, barely greater than $100. This implies a 2-year annually compounded rate (the geometric average) of only .12%, significantly less than the arith- metic average of 3.295%.
Example 5.7 Geometric versus Arithmetic Average
You invest $1 million at the beginning of 2018 in an S&P 500 stock-index fund. Given the rate of return for 2018, 2 40%, what rate of return in 2019 will be necessary for your portfolio to recover to its original value?
CONCEPT CHECK 5.4
9 We are told to measure historical performance over a particular sample period by the geometric average but to
estimate expected future performance from that same sample as the arithmetic average. The question naturally arises: If the same sample were to recur in the future, performance would be measured by the geometric aver- age, so isn’t that the best estimator of expected return? Surprisingly, the answer is no. Future results will always
contain both positive and negative surprises compared to prior expectations. When compounded, a run of positive
surprises has greater impact on final wealth than a run of equal-sized negative ones. Because of this asymmetry,
the sample geometric average is actually a downward-biased estimate of the future average return drawn from
the same distribution. The bias turns out to be half the variance, and thus using the arithmetic average corrects
for this bias.
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C H A P T E R 5 Risk, Return, and the Historical Record 133
return, the arithmetic average, r . Adapting Equation 5.12 for historic data, we again use equal probabilities for each observation, and use the sample average in place of the unob-
servable E ( r ).
Variance 5 Expected value of squared deviations
s2 5 a p(s)3r (s) 2 E(r)42 Using historical data with n observations, we could estimate variance as
ŝ2 5 1
n a n
s 5 1
3r (s) 2 r 42 (5.16) where ŝ replaces s to denote that it is an estimate.
Take another look at Spreadsheet 5.2 . Column D shows the square deviations from the arithmetic average, and cell D12 gives the standard deviation as .1774, which is the square root of the sum of products of the (equal) probabilities times the squared deviations.
Example 5.8 Variance and Standard Deviation
The variance estimate from Equation 5.16 is biased downward, however. The reason
is that we have taken deviations from the sample arithmetic average, r , instead of the unknown, true expected value, E ( r ), and so have introduced an estimation error. Its effect on the estimated variance is sometimes called a degrees of freedom bias. We can elimi- nate the bias by multiplying the arithmetic average of squared deviations by the factor
n /( n 2 1). The variance and standard deviation then become
ŝ2 5 a n n 2 1
b 3 1 n a
n
s 5 1
3r (s) 2 r 42 5 1 n 2 1 a
n
s 5 1
3r (s) 2 r 42 ŝ 5 Å 1n 2 1 a
n
s 5 1
3r (s) 2 r 42 (5.17) Cell D13 shows the unbiased estimate of standard deviation, .1983, which is higher than
the .1774 value obtained in cell D12. For large samples, n /( n 2 1) is close to 1, and the adjustment for degrees of freedom becomes trivially small.
Mean and Standard Deviation Estimates from Higher-Frequency Observations Do more frequent observations lead to more accurate estimates? The answer to this ques-
tion is surprising: Observation frequency has no impact on the accuracy of mean estimates.
It is the duration of a sample time series (as opposed to the number of observations) that improves accuracy.
The total 10-year return divided by 10 is as accurate an estimate of the expected annual
return as 12 times the average of 120 monthly returns. The average monthly return must
be consistent with the average 10-year return, so the extra intra-year observations yield
no additional information about average return. However, a longer sample, for example, a
100-year return, will provide a more accurate estimate of the mean return than a 10-year
return, provided the probability distribution of returns remains unchanged over the 100 years.
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134 P A R T I I Portfolio Theory and Practice
This suggests a rule: Use the longest sample that you still believe comes from the same
return distribution. Unfortunately, in practice, old data may be less informative. Are return
data from the 19th century relevant to estimating expected returns in the 21st century?
Quite possibly not, implying that we face severe limits to the accuracy of our estimates of
mean returns.
In contrast to the mean, the accuracy of estimates of the standard deviation and
higher moments (all computed using deviations from the average ) can be made more precise by increasing the number of observations. Thus, we can improve accuracy
of estimates of SD and higher moments of the distribution by using more frequent
observations.
Estimates of standard deviation begin with the variance. When monthly returns are
uncorrelated from one month to another, monthly variances simply add up. Thus, when the
variance is the same every month, we annualize by: 10
sA 2 5 12sM
2 . In general, the T -month variance is T times the 1-month variance. Consequently, standard deviation grows at the
rate of "T, that is: sA 5 "12sM. While the mean and variance grow in direct proportion to time, SD grows at the rate of square root of time.
The Reward-to-Volatility (Sharpe) Ratio Finally, it is worth noting that investors presumably are interested in the expected
excess return they can earn by replacing T-bills with a risky portfolio, as well as the risk they would thereby incur. While the T-bill rate is not constant over the entire
period, we still know with certainty what nominal rate we will earn if we purchase a
bill and hold it to maturity. Other investments typically entail accepting some risk in
return for the prospect of earning more than the safe T-bill rate. Investors price risky
assets so that the risk premium will be commensurate with the risk of that expected
excess return, and hence it’s best to measure risk by the standard deviation of excess, not total, returns.
The importance of the trade-off between reward (the risk premium) and risk (as mea-
sured by standard deviation or SD) suggests that we measure the attraction of a portfolio
by the ratio of risk premium to SD of excess returns.
Sharpe ratio 5 Risk premium
SD of excess return (5.18)
This reward-to-volatility measure (first proposed by William Sharpe and hence called the
Sharpe ratio ) is widely used to evaluate the performance of investment managers. Notice that the Sharpe ratio divides the risk premium (which rises in direct proportion
to time) by the standard deviation (which rises in direct proportion to square root of unit of
time). Therefore, the Sharpe ratio will be higher when annualized from higher frequency
returns. For example, to annualize the Sharpe ratio (SR) from monthly rates, we multiply
the numerator by 12 and the denominator by "12. Hence the annualized Sharpe ratio is SRA 5 SRM "12 . In general, the Sharpe ratio of a long-term investment over T years will increase by a factor of "T when T -period rates replace annual rates. 10
When returns are uncorrelated, we do not have to worry about covariances among them. Therefore, the variance
of the sum of 12 monthly returns (i.e., the variance of the annual return) is the sum of the 12 monthly variances.
If returns are correlated across months, annualizing is more involved and requires adjusting for the structure of
serial correlation.
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C H A P T E R 5 Risk, Return, and the Historical Record 135
5.6 The Normal Distribution
The bell-shaped normal distribution appears naturally in many applications. For exam- ple, heights and weights of newborns are well described by the normal distribution. In fact,
many variables that are the end result of multiple random influences will exhibit a normal
distribution, for example, the error of a machine that aims to fill containers with exactly
1 gallon of liquid. By the same logic, if return expectations implicit in asset prices are
rational, actual rates of return should be normally distributed around these expectations.
To see why the normal curve is “normal,” consider a newspaper stand that turns a profit
of $100 on a good day and breaks even on a bad day, with equal probabilities of .5. Thus,
the mean daily profit is $50 dollars. We can build a tree that compiles all the possible
outcomes at the end of any period. Here is an event tree showing outcomes after 2 days:
Two good days, profit = $200
Two bad days, profit = 0
One good and one bad day, profit = $100
Notice that 2 days can produce three different outcomes and, in general, n days can produce n 1 1 possible outcomes. The most likely 2-day outcome is “one good and
Take another look at Spreadsheet 5.1 . The scenario analysis for the proposed investment in the stock-index fund resulted in a risk premium of 5.76%, and standard deviation of excess returns of 19.49%. This implies a Sharpe ratio of .30, a value in line with the historical performance of stock-index funds. We will see that while the Sharpe ratio is an adequate measure of the risk–return trade-off for diversified portfolios (the subject of this chapter), it is inadequate when applied to individual assets such as shares of stock.
Example 5.9 Sharpe Ratio
Using the annual returns for years 2003–2005 in Spreadsheet 5.2 ,
a. Compute the arithmetic average return.
b. Compute the geometric average return.
c. Compute the standard deviation of returns.
d. Compute the Sharpe ratio assuming the risk-free rate was 6% per year.
CONCEPT CHECK 5.5
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136 P A R T I I Portfolio Theory and Practice
one bad day,” which can happen in two ways (first a good day, or first a bad day). The
probability of this outcome is .5. Less likely are the two extreme outcomes (both good
days or both bad days) with probability .25 each.
What is the distribution of profits at the end of 200 days? There are 201 possible out-
comes and, again, the midrange outcomes are the most likely because there are more
sequences that lead to them. While only one sequence can result in 200 consecutive bad
days, an enormous number of sequences result in 100 good days and 100 bad days. The
probability distribution will eventually take on the familiar bell shape. 11
Figure 5.4 shows a graph of the normal curve with mean of 10% and standard deviation
of 20%. A smaller SD means that possible outcomes cluster more tightly around the mean,
while a higher SD implies more diffuse distributions. The likelihood of realizing any
particular outcome when sampling from a normal distribution is fully determined by the
number of standard deviations that separate that outcome from the mean. Put differently,
the normal distribution is completely characterized by two parameters, the mean and SD. Investment management is far more tractable when rates of return can be well approxi-
mated by the normal distribution. First, the normal distribution is symmetric, that is, the
probability of any positive deviation above the mean is equal to that of a negative devia-
tion of the same magnitude. Absent symmetry, measuring risk as the standard deviation
of returns is inadequate. Second, the normal distribution belongs to a special family of
distributions characterized as “stable,” because of the following property: When assets
with normally distributed returns are mixed to construct a portfolio, the portfolio return
also is normally distributed. Third, scenario analysis is greatly simplified when only two
parameters (mean and SD) need to be estimated to obtain the probabilities of future sce-
narios. Fourth, when constructing portfolios of securities, we must account for the statis-
tical dependence of returns across securities. Generally, such dependence is a complex,
23s 22s 21s 0 11s 12s 13s
250 230 210 30 50 70
68.26%
95.44% 99.74%
10
Figure 5.4 The normal distribution with mean 10% and standard deviation 20%.
11 Early descriptions of the normal distribution in the 18th century were based on the outcomes of a “binomial
tree” like that of the newspaper stand. This representation is used in practice to price many option contracts, as we
will see in Chapter 21. For a demonstration of how the binomial distribution quickly approximates the normal, go
to www.jcu.edu/math/isep/Quincunx/Quincunx.html .
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C H A P T E R 5 Risk, Return, and the Historical Record 137
multilayered relationship. But when securities are normally distributed, the statistical rela-
tionships between returns can be summarized with a straightforward correlation coeffi-
cient. Thus we need to estimate only one parameter to summarize the dependence of any
two securities.
How closely must actual return distributions fit the normal curve to justify its use in
investment management? Clearly, the normal curve cannot be a perfect description of real-
ity. For example, actual returns cannot be less than 2 100%, which the normal distribution
would not rule out. But this does not mean that the normal curve cannot still be useful.
A similar issue arises in many other contexts. For example, birth weight is typically evalu-
ated in comparison to a normal curve of newborn weights, although no baby is born with a
negative weight. The normal distribution still is useful here because the SD of the weight
is small relative to its mean, and the likelihood of a negative weight would be too trivial
to matter. 12
In a similar spirit, we must identify criteria to determine the adequacy of the
normality assumption for rates of return.
Suppose the monthly rate of return on the S&P 500 is approximately normally distrib- uted with a mean of 1% and standard deviation of 6%. What is the probability that the return on the index in any month will be negative? We can use Excel’s built-in func- tions to quickly answer this question. The probability of observing an outcome less than some cutoff according to the normal distribution function is given as NORMDIST(cutoff, mean, standard deviation, TRUE). In this case, we want to know the probability of an outcome below zero, when the mean is 1% and the standard deviation is 6%, so we compute NORMDIST(0, 1, 6, TRUE) 5 .4338. We could also use Excel’s built-in standard normal function, NORMSDIST, which uses a mean of 0 and a standard deviation of 1, and ask for the probability of an outcome 1/6 of a standard deviation below the mean: NORMSDIST( 2 1/6) 5 .4338.
Example 5.10 Normal Distribution Function in Excel
What is the probability that the return on the index in Example 5.10 will be below 2 15%?
CONCEPT CHECK 5.6
12
In fact, the standard deviation is 511 grams while the mean is 3,958 grams. A negative weight would therefore
be 7.74 standard deviations below the mean, and according to the normal distribution would have probability of
only 4.97 3 10 2 15
. The issue of negative birth weight clearly isn’t a practical concern.
As we noted earlier (but you can’t repeat it too often!), normality of excess returns hugely
simplifies portfolio selection. Normality assures us that standard deviation is a complete
measure of risk and hence the Sharpe ratio is a complete measure of portfolio performance.
Unfortunately, deviations from normality of asset returns are quite significant and difficult
to ignore.
Deviations from normality may be discerned by calculating the higher moments of
return distributions. The n th central moment of a distribution of excess returns, R, is esti- mated as the average value of (R 2 R)n. The first moment ( n 5 1) is necessarily zero
5.7 Deviations from Normality and Risk Measures
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138 P A R T I I Portfolio Theory and Practice
(the average deviation from the sample average
must be zero). The second moment ( n 5 2) is the estimate of the variance of returns, ŝ2.
13
A measure of asymmetry called skew uses the ratio of the average cubed deviations from the average, called the third moment, to the
cubed standard deviation to measure asymme-
try or “skewness” of a distribution.
Skew 5 Average B(R 2 R)3 ŝ3
R (5.19) Cubing deviations maintains their sign (the
cube of a negative number is negative). When
a distribution is “skewed to the right,” as is the
dark curve in Figure 5.5A , the extreme posi-
tive values, when cubed, dominate the third
moment, resulting in a positive skew. When a
distribution is “skewed to the left,” the cubed
extreme negative values dominate, and the
skew will be negative.
When the distribution is positively skewed (skewed
to the right), the standard deviation overestimates
risk, because extreme positive surprises (which do not
concern investors) nevertheless increase the estimate
of volatility. Conversely, and more important, when
the distribution is negatively skewed, the SD will
underestimate risk.
Another potentially important deviation from nor-
mality, kurtosis, concerns the likelihood of extreme
values on either side of the mean at the expense
of a smaller likelihood of moderate deviations.
Graphically speaking, when the tails of a distribu-
tion are “fat,” there is more probability mass in the
tails of the distribution than predicted by the normal
distribution, at the expense of “slender shoulders,”
that is, less probability mass near the center of the
distribution. Figure 5.5B superimposes a “fat-tailed”
distribution on a normal with the same mean and SD.
Although symmetry is still preserved, the SD will
underestimate the likelihood of extreme events: large
losses as well as large gains.
13 For distributions that are symmetric about the average, as is the case for the normal distribution, all odd
moments ( n 5 1, 3, 5, . . .) have expectations of zero. For the normal distribution, the expectations of all higher even moments ( n 5 4, 6, . . .) are functions only of the standard deviation, s . For example, the expected fourth moment ( n 5 4) is 3 s 4 , and for n 5 6, it is 15 s 6 . Thus, for normally distributed returns the standard devia- tion, s , provides a complete measure of risk, and portfolio performance may be measured by the Sharpe ratio,
R/s. For other distributions, however, asymmetry may be measured by higher nonzero odd moments. Higher even moments (in excess of those consistent with the normal distribution), combined with large, negative odd
moments, indicate higher probabilities of extreme negative outcomes.
Figure 5.5A Normal and skewed distributions (mean 5 6%, SD 5 17%)
.00
Rate of Return
.20 .40 .60−.60
P ro
b a b
il it
y
−.40 −.20
.030
.025
.020
.015
.010
.005
.000
Skew = −.75 Skew = .75
Negatively Skewed Normal
Positively Skewed
Figure 5.5B Normal and fat-tailed distributions (mean 5 .1, SD 5 .2)
−.4 −.2−.6 .2 .4 .60 Rate of Return
Kurtosis =.35
P ro
b a b
il it
y D
e n
si ty
.8
Normal Fat-Tailed
.00
.10
.20
.30
.50
.40
.60
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C H A P T E R 5 Risk, Return, and the Historical Record 139
Kurtosis measures the degree of fat tails. We use deviations from the average raised to the fourth power, scaled by the fourth power of the SD,
Kurtosis 5 Average B(R 2 R)4 ŝ4
R 2 3 (5.20) We subtract 3 in Equation 5.20, because the ratio for a
normal distribution is 3. Thus, the kurtosis of a normal
distribution is defined as zero, and any kurtosis above
zero is a sign of fatter tails. The kurtosis of the distribu-
tion in Figure 5.5B , which has visible fat tails, is .35.
In addition to a shift of observations between the
shoulders and the tails, kurtosis can be affected by a shift
from the shoulders to the center of the distribution (which decreases kurtosis), or vice
versa. This element of kurtosis is called peakedness, as it affects the height of the peak of the distribution at its center. It is not shown in Figure 5.5B , but we often encounter peaked-
ness in histograms of actual distributions.
Notice that both skew and kurtosis are pure numbers. They do not change when annual-
ized from higher frequency observations. Higher frequency of extreme negative returns may result from negative skew and/or
kurtosis (fat tails). Therefore, we would like a risk measure that indicates vulnerability
to extreme negative returns. We discuss four such measures that are most frequently used
in practice: value at risk, expected shortfall, lower partial standard deviation, and the
frequency of extreme (3-sigma) returns.
Value at Risk The value at risk (denoted VaR to distinguish it from Var, the abbreviation for vari- ance) is the loss corresponding to a very low percentile of the entire return distribution,
for example, the 5th or 1st percentile return. VaR is actually written into regulation
of banks and closely watched by risk managers. It is another name for quantile of a distribution. The quantile, q, of a distribution is the value below which lies q % of the possible values. Thus the median is q 5 50th quantile. Practitioners commonly estimate the 5% VaR, meaning that 95% of returns will exceed the VaR, and 5% will be worse.
Therefore, the 5% VaR may be viewed as the best rate of return out of the 5% worst- case future scenarios.
When portfolio returns are normally distributed, the VaR is determined by the mean and
SD of the distribution. Recalling that 2 1.65 is the 5th percentile of the standard normal
distribution (with mean 5 0 and SD 5 1), the VaR for a normal distribution is
VaR(.05, normal) 5 Mean 2 1.65 SD
To obtain a sample estimate of VaR, we sort the observations from high to low. The VaR
is the return at the 5th percentile of the sample distribution. Almost always, 5% of the
number of observations will not be an integer, and so we must interpolate. Suppose a
sample comprises 84 annual returns so that 5% of the number of observations is 4.2. We
must interpolate between the fourth and fifth observation from the bottom. Suppose the
bottom five returns are
225.03% 2 25.69% 2 33.49% 2 41.03% 2 45.64%
The VaR is therefore between 2 25.03% and 2 25.69% and would be calculated as
VaR 5 225.69 1 0.2(25.69 2 25.03) 5 225.56%
Estimate the skew and kurtosis of the five rates in Spreadsheet 5.2 .
CONCEPT CHECK 5.7
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140 P A R T I I Portfolio Theory and Practice
Expected Shortfall When we assess tail risk by looking at the 5% worst-case scenarios, the VaR is the most
optimistic measure of risk as it takes the highest return (smallest loss) of all these cases. A
more realistic view of downside exposure would focus instead on the expected loss given that we find ourselves in one of the worst-case scenarios. This value, unfortunately, has
two names: either expected shortfall (ES) or conditional tail expectation (CTE); the latter emphasizes that this expectation is conditioned on being in the left tail of the distri-
bution. ES is the more commonly used terminology.
Extending the previous VaR example, we assume equal probabilities for all values.
Hence, we need to average across the bottom 5% of the observations. We sum the bottom
4 returns plus 0.2 of the fifth from the bottom, and divide by 4.2 to find ES 5 2 35.94%,
significantly less than the 2 25.56% VaR. 14
Lower Partial Standard Deviation and the Sortino Ratio The use of standard deviation as a measure of risk when the return distribution is non-
normal presents two problems: (1) the asymmetry of the distribution suggests we should
look at negative outcomes separately; and (2) because an alternative to a risky portfolio is
a risk-free investment, we should look at deviations of returns from the risk-free rate rather
than from the sample average, that is, at negative excess returns. A risk measure that addresses these issues is the lower partial standard deviation
(LPSD) of excess returns, which is computed like the usual standard deviation, but using only “bad” returns. Specifically, it uses only negative deviations from the risk-free rate
(rather than negative deviations from the sample average), squares those deviations to obtain
an analog to variance, and then takes the square root to obtain a “left-tail standard devia-
tion.” The LPSD is therefore the square root of the average squared deviation, conditional on a negative excess return. Notice that this measure ignores the frequency of negative excess
returns, that is, portfolios with the same average squared negative excess returns will yield
the same LPSD regardless of the relative frequency of negative excess returns.
Practitioners who replace standard deviation with this LPSD typically also replace the
Sharpe ratio (the ratio of average excess return to standard deviation) with the ratio of aver-
age excess returns to LPSD. This variant on the Sharpe ratio is called the Sortino ratio.
Relative Frequency of Large, Negative 3-Sigma Returns Here we concentrate on the relative frequency of large, negative returns compared with those
frequencies in a normal distribution with the same mean and standard deviation. Extreme
returns are often called jumps, as the stock price makes a big sudden movement. We compare the fraction of observations with returns 3 or more standard deviations below the mean to
the relative frequency of negative 3-sigma returns in the corresponding normal distribution.
14
A formula for the ES in the case of normally distributed returns is given in Jonathan Treussard, “The Non-
monotonicity of Value-at-Risk and the Validity of Risk Measures over Different Horizons,” IFCAI Journal of Financial Risk Management, March 2007. The formula is
ES 5 1
.05 exp(m)N 32s 2 F (.95) 4 2 1
where m is the mean of the continuously compounded returns, s is the SD, N (• ) is the cumulative standard nor- mal, and F is its inverse. In the sample above, m and s were estimated as 5.47% and 19.54%. Assuming normal- ity, we would have ES 5 2 30.57%, suggesting that this distribution has a larger left tail than the normal. It should
be noted, however, that estimates of VaR and ES from historical samples, while unbiased, are subject to large
estimation errors because they are estimated from a small number of extreme returns.
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Some Funds Stop Grading on the Curve
In 2008 a typical investment portfolio of 60% stocks and 40% bonds lost roughly a fifth of its value. Standard portfolio- construction tools assume that will happen only once every 111 years. Though mathematicians and many investors have long known market behavior isn’t a pretty picture, standard portfolio construction assumes returns fall along a tidy, bell- curve-shaped distribution. With that approach, a 2008-type decline would fall near the skinny left tail, indicating its rarity.
Recent history would suggest such meltdowns aren’t so rare. In a little more than two decades, investors have been buffeted by the 1987 market crash, the implosion of hedge fund Long-Term Capital Management, the bursting of the tech-stock bubble, and other crises.
Many of Wall Street’s new tools assume market returns fall along a “fat-tailed” distribution, where, say, the nearly 40% stock-market decline in 2008 would be more common than previously thought. These new assumptions present a far different picture of risk. Consider the 60% stock, 40% bond portfolio that fell about 20%. Under the fat-tailed distribution, that should occur once every 40 years, not once every 111 years as assumed under a bell-curve-type distribution. (The last year as bad as 2008 was 1931.)
One potential pitfall: Number-crunchers have a smaller supply of historical observations to construct models focused on rare events. “Data are intrinsically sparse,” says
Lisa Goldberg, executive director of analytic initiatives at MSCI Barra.
Many of the new tools also limit the role of conven- tional risk measures. Standard deviation, proposed as a risk measure by Nobel Prize–winning economist Harry Markowitz in the 1950s, can be used to gauge how much an investment’s returns vary over time. But it is equally affected by upside and downside moves, whereas many investors fear losses much more than they value gains. And it doesn’t fully gauge risk in a fat-tailed world.
A newer measure that has gained prominence in recent decades ignores potential gains and looks at downside risk. That measure, called “value at risk,” might tell you that you have a 5% chance of losing 3% or more in a single day, but it doesn’t home in on the worst downside scenarios.
To focus on extreme risk, many firms have begun using a measure called “expected shortfall” or “conditional value at risk,” which is the expected portfolio loss when value at risk has been breached. Conditional value at risk helps estimate the magnitudes of expected loss on the very bad days. Firms such as J.P. Morgan and MSCI Barra are employing the measure.
Source: Eleanor Laise, The Wall Street Journal, September 8, 2009, p. C1. Reprinted with permission. © 2009 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
W O
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M T
H E ST
R E E T
141
This measure can be quite informative about downside risk, but in practice is most use-
ful for large, high-frequency samples. Observe from Figure 5.4 that the relative frequency
of negative 3-sigma jumps in a standard normal distribution is only 0.13%, that is, 1.3
observations per 1,000. Thus in a small sample, it is hard to obtain a representative out-
come, one that reflects true statistical expectations of extreme events.
In the analysis of the history of some popular investment vehicles in the next section
we will show why practitioners need this plethora of statistics and performance measures
to analyze risky investments. The nearby box discusses the growing popularity of these
measures, and particularly the new focus on fat tails and extreme events.
We can now apply the analytical tools worked out in previous sections to six interesting
risky portfolios.
Our base portfolio is the broadest possible U.S. equity portfolio, including all stocks
listed on the NYSE, AMEX, and NASDAQ. We shall denote it as “All U.S.” Logic sug-
gests that an unmanaged (passive) portfolio should invest more in larger firms, and hence
the natural benchmark is a value-weighted portfolio. Firm capitalization, or market cap, is highly skewed to the right, with many small but a few gigantic firms. Because it is value
weighted, the All U.S. portfolio is dominated by the large-firm corporate sector.
The data include monthly excess returns on all U.S. stocks from July 1926 to September
2012, a sample period spanning just over 86 years. We break this period into three subpe-
riods, different in length and economic circumstances. We would like to see how portfolio
performance varies across the subperiods.
5.8 Historic Returns on Risky Portfolios
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142 P A R T I I Portfolio Theory and Practice
The early subperiod stretches over the second quarter of the 20th century, July 1926 to
December 1949, and its 282 months span the Great Depression, WWII, and its immediate
aftermath.
The second subperiod is the second half of the 20th century (January 1950 to December
1999) and its 600 months cover a relatively stable period, albeit with three wars (Korea,
Vietnam, and the first Gulf War) and eight relatively mild recessions and recoveries. It ends
with the tech bubble of the late 1990s.
The third subperiod covers the first 153 months of the 21st century, a difficult period.
It includes two deep recessions of different types: one following the bursting of the tech
bubble in 2001, and the other following the bursting of the housing-price bubble beginning
in 2007; each episode slashed stock values by about 40%. The prolonged Iraq war and the
Afghan conflict put further strain on the U.S. economy in these years.
We also present four portfolios to compare with the benchmark All U.S portfolio. These
comparison groups are motivated by empirical evidence that two variables (other than risk)
have been associated with stock returns: firm size (measured by market capitalization) and
the ratio of firm book value to market value of equity.
Average realized returns have generally been higher for stocks of small rather than large
capitalization firms, other things equal. “Other things” in this context means risk as best as
we can measure it. Hence, two of the four portfolios include stocks of firms in the top half
of the distribution of market capitalization, while the other two portfolios include firms
from the bottom half.
The accounting value of a firm reported on its balance sheet reflects the historical cost
of its past investments in assets, often dubbed assets in place and therefore is a backward- looking measure of value. Book value of equity equals total firm value minus the par value
of outstanding debt. In contrast, the market value of equity reflects the present value of future cash flows from existing lines of business, expected growth in those businesses, as well as cash flows from projects yet to be started, often dubbed growth opportunities. A good portion of the difference between the book value and market value of equity will
depend on the relative proportions of assets in place versus growth opportunities. A low
ratio of book-to-market value (B/M) is typical of firms whose market value derives mostly
from growth prospects. A high B/M ratio is typical of “value” firms, whose market values
derive mostly from assets in place. Realized average returns, other things equal, histori-
cally have been higher for value firms than for growth firms.
Eugene Fama and Kenneth French extensively documented the firm size and B/M regu-
larities, and these patterns have since been corroborated in stock exchanges around the
world. 15
The Fama-French database includes returns on portfolios of U.S. stocks sorted
by size (Big; Small) and by B/M ratios (High; Medium; Low) and rebalances these six
portfolios every midyear. 16
We drop the medium B/M portfolios, identify high B/M firms as “Value firms” and low
B/M firms as “Growth firms,” and thus obtain four comparison portfolios as Big/Value;
Big/Growth; Small/Value; Small/Growth. While it is common to use value weighting, we
will use equally weighted versions of these portfolios in order to give greater emphasis to
smaller firms and thus sharpen the contrast with the large-cap heavy All U.S. benchmark.
Table 5.3 shows the number of firms, average firm size, and average B/M ratio for each
portfolio at the outset and end of each subperiod. The number of stocks in the portfolios
steadily increases, except during the 21st century, when bad times killed off a large num-
ber of small firms. The exit of so many small firms from the sample generally increased
the average capitalization of the remaining firms, despite their loss of market value. The
15
This literature began in earnest with their 1992 publication: “The Cross Section of Expected Stock Returns,”
Journal of Finance 47, 427–465.
16
The database is available at: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
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C H A P T E R 5 Risk, Return, and the Historical Record 143
average market cap of all firms grew from $57 million to $4.47 billion, an annual growth
rate of 5.2% over the 86 years, slower than average growth of nominal GDP (6.7%), but
2 percentage points higher than average inflation (3.2%).
Figure 5.6 presents six histograms of the 1,035 total monthly returns on T-bills and excess
returns on the five risky portfolios. 17
Bear in mind that even small differences in average
monthly returns will have great impact on final wealth when compounded over long periods.
The histogram in panel A shows T-bills rates in the range of 2 .05% to 1.5%, 18
while panel
B shows excess monthly returns on the All U.S. stock portfolio lying in the range between
2 20% and 1 20%; annualized, this is equivalent to a range of 2 93% to 891%! The vertical
axis shows the fraction of returns in each bin. (The dark columns in the histograms are based
on the historical sample while the light columns describe a normal distribution.) The bins
are 2.5 basis points (.025%) wide for T-bill returns, and 50 basis points wide for the All U.S.
portfolio. The extreme bins at the far right and left of each histogram are actually the sums
of the frequencies of all returns beyond the reported range (less than 2 20% or greater than
20%). Panels C and D show histograms of excess returns on the two “Big” or large-cap port-
folios (one for Big/Value stocks and the other for Big/Growth stocks) while panels E and F
are excess return histograms on the two “Small” portfolios (Small/Value and Small/Growth).
A first look at the actual excess returns on stocks verifies that the tails are uniformly fat-
ter than would be observed in a normal distribution, implying higher incidence of extreme
results. Given the potential outsized impact of extreme returns, it is customary to fit a
All U.S. a Big/Value Big/Growth Small/Value Small/Growth
July 1926 Number of firms 427 37 85 90 43 Average capitalization ($ mil) 57 39 108 5 8 Average B/M ratio b 1.02 2.36 0.45 3.6 0.81 January 1950 Number of firms 899 73 196 197 75 Average capitalization ($ mil) 69 77 186 7 11 Average B/M ratio b 1.18 2.60 0.50 2.95 0.67 January 2000 Number of firms 5,495 150 576 1,709 1,158 Average capitalization ($ mil) 2,545 3,542 18,246 106 299 Average B/M ratio b 0.52 1.38 0.14 1.70 0.22 September 2012 Number of firms 3,383 153 408 1,065 672 Average capitalization ($ mil) 4,470 13,325 18,070 297 582 Average B/M ratio b 0.68 1.32 0.25 1.33 0.26
Table 5.3
Number of firms, average capitalization, and average B/M ratios of portfolios Notes: a Value weighted, hence dominated by big stocks b B/M ratio are sampled in midyears Source: Professor Kenneth French’s Web site, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html .
17 These are excess returns of the continuously compounded rates, which are the ones most appropriate to assume
are normally distributed. We discuss this issue in the next section. 18
T-bills made their debut during the 1940s. For earlier dates, commercial paper is used as the closest approxima-
tion to short-term risk-free rates. In a few instances they were issued slightly above par and thus yielded slightly
negative rates.
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144 P A R T I I Portfolio Theory and Practice
Figure 5.6 Frequency distribution of annual rates of return, 1926–2012 Source: Prepared from data in Table 5.3.
0.00
0.02
0.04
0.06
0.08
0.10
0.00
0.01
0.02
0.03
0.04
0.05
0.06 Estimated Normal Mean = 0.71 SD = 3.89 Negative Jumps Average = 215.44 SD = 17.18 Positive Jumps Average = 15.23 SD = 16.62
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0
0.01
0.02
0.03
0.04
0.05
0.06
225 220 215 29.5 24.5 0.5 5.5 10.5 15.5 20.5
225 220 215 29.5 24.5 0.5 5.5 10.5 15.5 20.5
225 220 215 29.5 24.5 0.5 5.5 10.5 15.5 20.5
225 220 215 29.5 24.5 0.5 5.5 10.5 15.5 20.5
225 220 215 29.5 24.5 0.5 5.5 10.5 15.5 20.5 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0
0.01
0.02
0.03
0.04
0.05
0.06
20.05 0.475 0.975
Average = 0.29 SD = 0.25
A: T-Bill Rates
C: Big/Value
E: Small/Value F: Small/Growth
D: Big/Growth
B: All U.S.
1.475
Actual Average = 0.48 SD = 5.44
Actual Average = 0.73 SD = 7.54
Actual Average = 0.73 SD = 6.02
Actual Average = 1.81 SD = 8.26
Actual Average = 0.35 SD = 8.23
Estimated Normal Mean = 0.40 SD = 4.18 Negative Jumps Average = 217.04 SD = 19.31 Positive Jumps Average = 14.24 SD = 14.64
Estimated Normal Mean = 0.47 Stdev = 4.67 Negative Jumps Average = 216.93 SD = 18.08 Positive Jumps Average = 15.62 SD = 17.52
Estimated Normal Mean = 0.92 SD = 4.39 Negative Jumps Average = 219.41 SD = 22.98 Positive Jumps Average = 16.24 SD = 18.32
Estimated Normal Mean = 0.85 Stdev = 4.15 Negative Jumps Average = 217.39 SD = 19.79 Positive Jumps Average = 16.39 SD = 19.03
corresponding normal distribution using only the moderate range of excess returns, and
estimate the distribution of extreme rates separately. Accordingly, the light-colored col-
umns in panels B through F show the expected frequency from a normal distribution with
mean and SD matched to those of the actual returns in the range of 6 10%. The boxes
to the left of the histograms show the average and SD of the actual distributions, while
the boxes on the right show the statistics for subgroups of returns: the midrange (returns
within 6 10% of the mean), the negative jumps (extreme returns less than 2 10%), and the
positive jumps (returns greater than 10%). The mean and SD of the jump components are
calculated using differences from the full-sample average. The SD of the jumps indicates
the contribution of the positive and negative jumps to the variance of the full distribution.
The histograms give us a first, quite vivid look at the risk involved in owning common
stocks. This risk is dominated by the frequency and size of negative jumps. We need more for-
mal analysis to determine whether these deviations from normality are economically decisive.
Table 5.4 presents a wide range of statistics for the five stock portfolios for the entire
86-year period, as well as the three subperiods. With 1,035 monthly observations, aver-
age excess returns are all statistically significantly above zero, verifying a positive risk
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1 4 5
Statistic All U.S. a Big/Value b Big/Growth c Small/Value d Small/Growth e
All 1,035 Months: July 1926–September 2012 Average excess return 7.52 12.34 10.98 26.28 8.38 Standard deviation 20.46 29.25 20.79 41.41 32.80 Checks on normality Lower partial SD (LPSD) 21.67 26.78 20.88 31.57 30.36 Skew f 2 0.54 0.34 2 0.58 1.19 0.17 Kurtosis f 6.58 11.40 5.25 13.31 6.19 VaR 5% actual f 2 8.01 2 10.08 2 7.92 2 8.30 2 11.64 normal f 2 8.13 2 11.01 2 8.76 2 11.12 2 12.35 ES (CTE) 5% actual f 2 12.40 2 16.35 2 13.05 2 14.85 2 17.39 normal f 2 10.17 2 13.74 2 11.01 2 14.08 2 15.27 Negative 3-sigma (obs/1000), actual 7.7 4.8 9.7 2.9 3.9
normal 0.9 0.9 0.8 0.6 1.0
1-month SD conditional on 10% loss, actual 17.18 19.79 19.31 22.98 18.08 normal 12.82 14.95 13.47 16.85 15.16 Performance Sharpe ratio (annualized) 0.37 0.42 0.53 0.63 0.26 Sortino ratio (annualized) 0.35 0.46 0.53 0.83 0.28 The 21st Century So Far: January 2000–September 2012 (153 months) Average excess return 1.82 8.80 14.51 17.89 4.83 Standard deviation 20.08 24.08 20.93 28.93 29.49 Checks on normality Lower partial SD (LPSD) 23.02 26.02 19.46 27.40 28.05 Skew 2 0.74 2 0.59 2 0.34 2 0.32 0.10 Kutosis 1.21 2.60 1.43 1.62 1.72 VaR 5%, actual 2 8.86 2 10.68 2 8.11 2 9.44 2 13.77 normal 2 7.85 2 8.97 2 8.45 2 8.77 2 13.40 ES 5%, actual 2 11.09 2 14.02 2 11.74 2 12.36 2 17.46 normal 2 9.73 2 11.22 2 10.69 2 11.14 2 16.47 Performance Sharpe ratio (annualized) 0.09 0.37 0.69 0.62 0.16 Sortino ratio (annualized) 0.08 0.34 0.75 0.65 0.17
Table 5.4
Annualized statistics from the history of monthly excess returns on common stocks, July 1926–September 2012 continued
b o d 6 1 6 7 1 _ ch
0 5 _ 1 1 7 -1
6 7 .in
d d 1
4 5
b o d 6 1 6 7 1 _ ch
0 5 _ 1 1 7 -1
6 7 .in
d d 1
4 5
6 /1
8 /1
3 8
:0 3 P
M 6 /1
8 /1
3 8
:0 3 P
M
F inal P
D F
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1 4 6
Statistic All U.S.a Big/Valueb Big/Growthc Small/Valued Small/Growthe
The 20th Century, Second Half: January 1950–December 1999 (600 months) Average excess return 8.44 11.50 9.83 17.05 7.20 Standard deviation 14.99 17.21 16.51 21.41 25.60 Checks on normality Lower partial SD (LPSD) 15.87 16.39 16.69 20.14 26.40 Skew 2 0.81 2 0.15 2 0.70 2 0.22 2 0.77 Kutosis 3.50 2.28 3.76 5.09 4.23 VaR 5%, actual 2 6.02 2 6.67 2 6.94 2 6.86 2 9.51 normal 2 6.08 2 6.48 2 7.13 2 7.33 2 9.85 ES 5%, actual 2 9.06 2 8.98 210.07 210.36 214.30 normal 2 7.70 2 8.24 2 9.01 2 9.37 212.25 Performance Sharpe ratio (annualized) 0.56 0.67 0.60 0.80 0.28 Sortino ratio (annualized) 0.53 0.70 0.59 0.85 0.27 The 20th Century, Second Quarter: July 1926–December 1949 (282 months) Average excess return 8.64 16.02 11.49 50.48 12.81 Standard deviation 28.72 46.59 27.61 63.74 45.08 Checks on normality Lower partial SD (LPSD) 29.92 40.28 28.43 44.04 37.54 Skew 2 0.30 0.40 2 0.50 0.96 0.61 Kutosis 4.60 4.88 4.41 6.25 5.36 VaR 5%, actual 212.55 217.54 211.68 216.73 215.70 normal 211.39 217.46 211.60 216.34 215.88 ES 5%, actual 217.36 224.16 218.22 222.61 221.22 normal 214.14 221.41 214.43 220.59 219.52 Performance Sharpe ratio (annualized) 0.30 0.34 0.42 0.79 0.28 Sortino ratio (annualized) 0.29 0.40 0.40 1.15 0.34
Table 5.4—concluded
Annualized statistics from the history of monthly excess returns on common stocks, July 1926–September 2012
Notes: a Stocks trading on NYSE, AMEX, and NASDAQ, value weighted b Firms in the top 1/2 by market capitalization of equity and top 1/3 by ratio of book equity/market equity (B/M), equally weighted c Firms in the top 1/2 by capitalization and bottom 1/3 by B/M ratio, equally weighted d Firms in the bottom 1/2 by capitalization and top 1/3 by B/M ratio, equally weighted e Firms in the bottom 1/2 by capitalization and bottom 1/3 by B/M ratio, equally weighted f Calculated from monthly, continuously compounded rates Source: Author’s calculations, using data from Professor Kenneth French’s Web site, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
b o d 6 1 6 7 1 _ ch
0 5 _ 1 1 7 -1
6 7 .in
d d 1
4 6
b o d 6 1 6 7 1 _ ch
0 5 _ 1 1 7 -1
6 7 .in
d d 1
4 6
6 /1
8 /1
3 8
:0 3 P
M 6 /1
8 /1
3 8
:0 3 P
M
F inal P
D F
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C H A P T E R 5 Risk, Return, and the Historical Record 147
All U.S. Big/
Value Big/
Growth Small/ Value
Small/ Growth
Average of Four Comparison Portfolios
All years 20.46 29.25 20.79 41.41 32.80 31.06 21st century 20.08 24.08 20.93 28.93 29.49 25.86 20th cent. 2 nd half 14.99 17.21 16.51 21.41 25.60 20.18 20th cent. 2 nd quarter 28.72 46.59 27.61 63.74 45.08 45.76
Table 5.4B
Standard deviations over time
All U.S. Big/
Value Big/
Growth Small/ Value
Small/ Growth
Average of Four Comparison Portfolios
All years 0.37 0.42 0.53 0.63 0.26 0.46 21st century 0.09 0.37 0.69 0.62 0.16 0.46 20th cent. 2 nd half 0.56 0.67 0.60 0.80 0.28 0.59 20th cent. 2 nd quarter 0.30 0.34 0.42 0.79 0.28 0.46
Table 5.4C
Sharpe ratios over time
premium. Compared to subperiod averages, the 21st century so far has been particularly
hard on very large firms, as we see from the value-weighted All U.S. portfolio. Not sur-
prisingly, the second half of the 20th century, politically and economically the most stable
subperiod, offered the highest average returns, particularly for the equally-weighted port-
folios. Table 5.4A , which reports a subset of Table 5.4 , shows these average returns.
As we would expect, the second quarter of the 20th century, dominated by the Great
Depression and legendary for upheaval in stock values, exhibits the highest standard devia-
tions ( Table 5.4B ).
All portfolios attained their highest Sharpe ratios over the second half of the 20th
century ( Table 5.4C ). The 21st century has witnessed the lowest performance from the large
cap-weighted All U.S. portfolio and a middling performance from the equally weighted
portfolios. More surprising is the fact that average returns were not particularly low over
the second quarter of the 20th century, despite the deep setbacks of the Depression period.
All U.S. Big/
Value Big/
Growth Small/ Value
Small/ Growth
Average of Four Comparison Portfolios
All years 7.52 12.34 10.98 26.28 8.38 14.49 21st century 1.82 8.80 14.51 17.89 4.83 11.51 20th cent. 2 nd half 8.64 16.02 11.49 50.48 12.81 22.70 20th cent. 2 nd quarter 8.44 11.50 9.83 17.05 7.20 11.40
Table 5.4A
Average excess returns over time
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148 P A R T I I Portfolio Theory and Practice
However, given the considerable imprecision of these estimates (standard errors of around
0.20 or 20%), we cannot be sure Sharpe ratios are all that different either across subperiods
or across portfolios.
Portfolio Returns The major objective is to compare the five equity portfolios. We started with the prem-
ise that the value-weighted All U.S. portfolio is a natural choice for passive investors.
We chose the other four portfolios because empirical evidence suggests that size (big vs.
small) and B/M ratios (value vs. growth) are important drivers of performance.
The average returns in Table 5.4A show that the Small/Value portfolio did in fact offer
a higher average return in all periods, and the differences from the averages of the other
portfolios are all statistically significant. 19
In addition, the average of the returns on the
equally weighted comparison portfolios (the right-most column in Table 5.4A ) was signif-
icantly higher than that of the All U.S. portfolio. But before deeming these performances
superior or inferior, it must be shown that the differences in their average returns cannot
be explained by differences in risk. Here, we must question the use of standard deviation
as a measure of risk for any particular asset or portfolio. Standard deviation measures
overall volatility and hence is a legitimate risk measure only for portfolios that are consid-
ered appropriate for an investor’s entire wealth-at-risk, that is, for broad capital allocation.
Assets or portfolios that are considered to be added to the rest of an investor’s entire- wealth portfolio must be judged on the basis of incremental risk. This distinction requires risk measures other than standard deviation, and we will return to this issue in great detail
in later chapters.
Table 5.4B shows the large standard deviation involved in these broad-based stock
investments. Annual SD ranges from 15% to as much as 63%. Even using the smallest SD
suggests that losing 15% of portfolio value in one year would not be so unusual. Apparently,
size is correlated with volatility, as suggested by the higher SD of the two small portfolios,
and the lowest volatility of the large-cap All U.S. portfolio. While it appears that value
portfolios generally are more volatile than growth portfolios, the difference is not suffi-
cient to make us confident of this assertion.
Regardless of how we resolve the question of performance of these portfolios, we
must determine whether SD is an adequate measure of risk in the first place, in view of
deviations from normality. Table 5.4 shows that negative skew is present in some of the
portfolios some of the time, and positive kurtosis is present in all portfolios all the time. This implies that we must carefully evaluate the effect of these deviations on value-at-risk
(VaR), expected shortfall (ES), and negative 3-sigma frequencies. Finally, since Figure 5.6
separates the distributions of monthly excess returns to those within a range of 6 10% and
those outside that range, we can quantify the implication of extreme returns.
We start with the difference between VaR from the actual distribution of returns and the
equivalent normal distribution (with the same mean and variance). Recall that the 5% VaR
is the loss corresponding to the 5th percentile of the rate of return distribution. It is one
measure of the risk of extreme outcomes, often called tail risk because it focuses on out- comes in the far left tail of the distribution. We compare historical tail risk to that predicted
by the normal distribution by comparing actual VaR to the VaR of the equivalent normal
distribution. The excess VaR is the VaR of the historical distribution versus the VaR of the
corresponding normal, where negative numbers indicate greater losses.
Table 5.4D shows that for the overall period, VaR indicates no greater tail risk than is
characteristic of the equivalent normal. The worst excess VaR compared to the normal
19 The t-statistic of the difference in average return is: Average difference/SD(Difference).
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C H A P T E R 5 Risk, Return, and the Historical Record 149
All U.S. Big/
Value Big/
Growth Small/ Value
Small/ Growth
Number of negative jumps 7.7 4.8 9.7 2.9 3.9 Expected jumps for equivalent normal 0.9 0.9 0.8 0.6 1.0 Difference 5 excess jumps 6.8 4.0 8.9 2.3 2.8 Average years between excess jumps 12.24 21.06 9.42 36.23 29.37 Expected excess returns between extra jumps (in units of SD) 16.90 32.91 17.16 104.23 28.98
Table 5.4F
Incidence of negative 3-sigma returns
All U.S.
Big/ Value
Big/ Growth
Small/ Value
Small/ Growth
Average of Four Comparison Portfolios
All years 0.12 0.93 0.85 2.81 0.72 1.33 21st century 2 1.02 2 1.71 0.34 2 0.67 2 0.37 2 0.60 20th cent. 2 nd quarter 0.06 2 0.19 0.19 0.47 0.34 0.20 20th cent. 2 nd half 2 1.16 2 0.09 2 0.08 2 0.39 0.18 2 0.09
Table 5.4D
VaR (excess over normal distribution) expressed as a fraction of monthly standard deviation
All U.S. Big/
Value Big/
Growth Small/ Value
Small/ Growth
Average of Four Comparison Portfolios
All years 2 0.41 2 0.33 2 0.34 2 0.08 2 0.25 2 0.25 21st century 2 0.28 2 0.47 2 0.17 2 0.19 2 0.11 2 0.24 20th cent. 2 nd quarter 2 0.18 2 0.06 2 0.13 2 0.07 2 0.18 2 0.11 20th cent. 2 nd half 2 0.77 2 0.59 2 0.77 2 0.37 2 0.26 2 0.50
Table 5.4E
Expected shortfall (excess over normal distribution) expressed as a fraction of monthly standard deviation
( 2 1.71% for the Big/Value portfolio in the 21st century) is less than a third of the monthly
SD of this portfolio, 6.01%. Hence, VaR figures indicate that the normal is a decent
approximation to the actual return distribution.
However, other measures indicate that tail risk may be somewhat greater than in the
normal distribution. The expected shortfall (ES) figures in Table 5.4 are more negative
for the actual than for the equivalent normal excess returns (consistent with the fat tails
indicated by the positive kurtosis). To assess the economic significance of the differ-
ences from normal, we present them in Table 5.4E as fractions of the monthly SDs of
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150 P A R T I I Portfolio Theory and Practice
the various portfolios. The negative signs tell us that while the most negative 5% of the
actual observations are always worse than the equivalent normal, the differences are not
substantial: The magnitudes are never larger than 0.77 of the portfolio SD. Measured over
the entire period, the excess shortfall does not exceed 0.41 of the monthly standard devia-
tion. Here, again, we don’t see evidence that seriously undermines the adequacy of the
normality assumption.
Table 5.4F shows the actual number of negative monthly returns or “jumps” of magni-
tude greater than 3 standard deviations, compared with the expected number correspond-
ing to the equivalent normal distributions. The actual numbers range from 2.9 to 9.7 per
1,000 months, compared with only 0.6 to 1.0 for equivalent normal distributions. What are
we to make of this? Negative 3-sigma returns are very bad surprises indeed. To help inter-
pret these differences, we compute the expected length of time (number of years) between
“extra jumps,” i.e., jumps beyond the expected number based on the normal distribution.
We also calculate the expected total return over this period, also in units of standard devia-
tion of the actual distribution.
Table 5.4F shows the results of these calculations. We see that one excess jump is
observed every 9 to 36 years, and that over such periods, the portfolios are expected to
yield excess returns of 16 to 104 standard deviations compared with the loss of 3 SD or
more due to these jumps. Thus, jump risk does not appear large enough to affect the risk
and return of long-term stock returns.
Finally, we interpret the size of the jumps outside the range of 6 10% that appear so ominous in Figure 5.6 . To quantify this risk, we ask: “When we look at all excess returns
below 2 10% in our history of 1,035 months, what is the SD of all these (extremely bad)
returns?” And a follow-up question: “What would be the tail SD of a normal return with
the same mean and overall SD as our sample, conditional on return falling below 2 10%?”
Table 5.4G answers these two questions. It is evident that the actual history suggests a
larger SD than a normal distribution would imply, consistent with Figure 5.6 . The differ-
ence can be as large as 43% of the SD of the equivalent normal in the extreme negative
range. Of all the statistics we have examined so far, this is the most damning for a straight-
forward approximation of actual distributions by the normal.
We can conclude from all this that a simple normal distribution is generally not a bad
approximation of portfolio returns, despite the fact that in some circumstances it may
understate investment risk. However, we can make up for this pitfall by more careful esti-
mation of the SD of extreme returns. Nevertheless, we should be cautious about application
of theories and inferences that require normality of returns. In general, one should verify
that standard deviations assumed for assets or portfolios adequately represent tail risk.
In the next chapters we will return to these portfolios and ask whether the All U.S.
portfolio is the most efficient in terms of its risk-return trade-off. We will also con-
sider adjustments in view of the performance of the size-B/M portfolios as well as other
All U.S. Big/Value Big/Growth Small/Value Small/Growth
From 1,035-month history 17.18 19.79 19.31 22.98 18.08 From an equivalent normal 12.82 14.95 13.47 16.85 15.16 % Difference 33.99 32.35 43.35 36.39 19.23
Table 5.4G
Standard deviation conditional on excess return less than 210%
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C H A P T E R 5 Risk, Return, and the Historical Record 151
Figure 5.7 Nominal and real equity returns around the world, 1900–2000 Source: Elroy Dimson, Paul Marsh, and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton: Princeton University Press, 2002), p. 50. Reprinted by permission of the Princeton University Press.
A n
n u
a li ze
d P
e rc
e n
ta g
e R
e tu
rn
15
Real
Nominal
Bel
2.5
8.2
2.7
12.0
3.6
9.7
3.6
10.0
3.8
12.1
4.5
12.5
4.6
8.9
4.8
9.5
5.0
7.6
5.8
10.1
5.8
9.0
6.4
9.7
6.7
10.1
6.8
12.0
7.5
11.9
7.6
11.6
Ita Ger Spa Fra Jap Den Ire Swi U.K. Neth Can U.S. SAf Aus Swe
12
9
6
3
0
empirically driven wrinkles. It is comforting that the assumption of approximately nor-
mal distributions of asset returns, which makes this investigation tractable, is also rea-
sonably accurate.
A Global View of the Historical Record As financial markets around the world grow and become more transparent, U.S. investors
look to improve diversification by investing internationally. Foreign investors that tradi-
tionally used U.S. financial markets as a safe haven to supplement home-country invest-
ments also seek international diversification to reduce risk. The question arises as to how
historical U.S. experience compares with that of stock markets around the world.
Figure 5.7 shows a century-long history (1900–2000) of average nominal and real
returns in stock markets of 16 developed countries. We find the United States in fourth
place in terms of average real returns, behind Sweden, Australia, and South Africa.
Figure 5.8 shows the standard deviations of real stock and bond returns for these same
countries. We find the United States tied with four other countries for third place in terms
of lowest standard deviation of real stock returns. So the United States has done well, but
not abnormally so, compared with these countries.
One interesting feature of these figures is that the countries with the worst results,
measured by the ratio of average real returns to standard deviation, are Italy, Belgium,
Germany, and Japan—the countries most devastated by World War II. The top-performing
countries are Australia, Canada, and the United States, the countries least devastated by
the wars of the 20th century. Another, perhaps more telling feature is the insignificant
difference between the real returns in the different countries. The difference between the
highest average real rate (Sweden, at 7.6%) and the average return across the 16 countries
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152 P A R T I I Portfolio Theory and Practice
(5.1%) is 2.5%. Similarly, the difference between the average and the lowest country return
(Belgium, at 2.5%) is 2.6%. Using the average standard deviation of 23%, the t -statistic for a difference of 2.6% with 100 observations is
t-Statistic 5 Difference in mean
Standard deviation /"n 5 2.6
23 /"100 5 1.3 which is far below conventional levels of statistical significance. We conclude that the U.S.
experience cannot be dismissed as an outlier. Hence, using the U.S. stock market as a yard-
stick for return characteristics may be reasonable.
These days, practitioners and scholars are debating whether the historical U.S. average
risk-premium of large stocks over T-bills of 7.52% ( Table 5.4 ) is a reasonable forecast
for the long term. This debate centers around two questions: First, do economic factors
that prevailed over that historic period (1926–2012) adequately represent those that may
prevail over the forecasting horizon? Second, is the arithmetic average from the available
history a good yardstick for long-term forecasts?
Figure 5.8 Standard deviations of real equity and bond returns around the world, 1900–2000 Source: Elroy Dimson, Paul Marsh, and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton: Princeton University Press, 2002), p. 61. Reprinted by permission of the Princeton University Press.
St a n
d a rd
D e vi
a ti
o n
o f
A n
n u
a l R
e a l R
e tu
rn (
% )
35
Equities
Bonds
Can
17
11
18
13
20
15
20
12
20
10
20
8
21
9
22
12
22
13
23
12
23
11
23
13
23
14
29
14
30
21
32
16
Aus U.K. U.S. Swi Neth Spa Ire Bel SAf Swe Fra Ita Jap GerDen
30
25
20
15
10
5
0
Consider an investor saving $1 today toward retirement in 25 years, or 300 months. Invest-
ing the dollar in a risky stock portfolio (reinvesting dividends until retirement) with an
expected rate of return of 1% per month, this retirement “fund” is expected to grow almost
20-fold to a terminal value of (1 1 .01) 300
5 $19.79 (providing total growth of 1,879%).
5.9 Long-Term Investments *
*The material in this and the next subsection addresses important and ongoing debates about risk and return, but it
is more challenging. It may be skipped in shorter courses without impairing the ability to understand later chapters.
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C H A P T E R 5 Risk, Return, and the Historical Record 153
Compare this impressive result to a 25-year investment in a safe Treasury bond with a
monthly return of .5% that grows by retirement to only 1.005 300
5 $4.46. We see that
a monthly risk premium of just .5% produces a retirement fund that is more than four times
that of the risk-free alternative. Such is the power of compounding. Why, then, would any-
one invest in Treasuries? Obviously, this is an issue of trading excess return for risk. What
is the nature of this return-to-risk trade-off? The risk of an investment that compounds at
fluctuating rates over the long run is important, but is widely misunderstood.
We can construct the probability distribution of the stock-fund terminal value from
a binomial tree just as we did earlier for the newspaper stand, except that instead of
adding monthly profits, the portfolio value compounds monthly by a rate drawn from a given distribution. For example, suppose we can approximate the portfolio monthly
distribution as follows: Each month the rate of return is either 5.54% or 2 3.54%,
with equal probabilities of .5. This configuration generates an expected return of 1%
per month. The portfolio volatility is measured as the monthly standard deviation:
".5 3 (5.54 2 1)2 1 .5 3 (23.54 2 1)2 5 4.54%. After 2 months, the event tree looks like this:
Portfolio value = $1 × 1.0554 × 1.0554 = $1.1139
Portfolio value = $1 × 1.0554 × .9646 = $1.0180
Portfolio value = $1 × .9646 × .9646 = $.9305
“Growing” the tree for 300 months will result in 301 different possible outcomes. The
probability of each outcome can be obtained from Excel’s BINOMDIST function. From
the 301 possible outcomes and associated probabilities we compute the mean ($19.79) and
the standard deviation ($18.09) of the terminal value. Can we use this standard deviation as a
measure of risk to be weighed against the risk premium of 19.79 2 4.29 5 15.5 (1,550%)?
Recalling the effect of asymmetry on the validity of standard deviation as a measure of
risk, we must first view the shape of the probability distribution at the end of the tree.
Figure 5.9 plots the probability of possible outcomes against the terminal value. The
asymmetry of the distribution is striking. The highly positive skewness suggests the stan-
dard deviation of terminal value will not be useful in this case. Indeed, the binomial dis-
tribution, when period returns compound, converges to a lognormal, rather than a normal, distribution. The lognormal describes the distribution of a variable whose logarithm is normally distributed.
Normal and Lognormal Returns As we mentioned earlier, one of the important properties of the normal distribution is its
stability, in the sense that adding up normally distributed returns results in a return that
also is normally distributed. But this property does not extend to multiplying normally distributed returns; yet this is what we need to do to find returns over longer horizons. For
example, even if two returns, r 1 and r 2 , are normal, the two-period return will compound
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154 P A R T I I Portfolio Theory and Practice
Figure 5.9 Probability of investment outcomes after 25 years with a lognormal distribution (approximated from a binomial tree)
0
.005
.01
.015
.02
.025
.03
.035
.04
.045
.05
0 20 40 60 80 100 120 140 160 180 200
P ro
b a b
il it
y o
f O
u tc
o m
e
Investment Outcome (truncated at $200)
Tail = $0.00002
Tail = $10,595,634
to (1 1 r 1 )(1 1 r 2 ) 2 1, which is not normally distributed. Perhaps the
normal distribution does not qualify
as the simplifying distribution we
purported it to be. But the lognor-
mal distribution does! What is this
distribution?
Technically, a random variable X is lognormal if its logarithm, ln( X ), is normally distributed. It turns
out that if stock prices are “instan-
taneously” normal (i.e., returns
over the shortest time intervals are
normally distributed) then their
longer-term compounded returns and
the future stock price will be log-
normal. 20
Conversely, if stock prices
are distributed lognormally, then
the continuously compounded rate
of return will be normally distrib-
uted. Thus, if we work with continuously compounded (CC) returns rather than effec-
tive per period rates of return, we can preserve the simplification provided by the
normal distribution, since those CC returns will be normal regardless of the invest-
ment horizon.
Recall that the continuously compounded rate is r CC 5 ln(1 1 r ), so if we observe effective rates of return, we can use this formula to compute the CC rate. With r CC nor- mally distributed, we can do all our analyses and calculations using the normally distrib-
uted CC rates. If needed, we can always recover the effective rate, r , from the CC rate from: r 5 erCC 2 1.
Let’s see what the rules are when a stock price is lognormally distributed. Suppose the
log of the stock price is normally distributed with an expected annual growth rate of g and a SD of s . When a normal rate compounds by random shocks from instant to instant,
the fluctuations do not produce symmetric effects on price. A positive uptick raises the
base, so the next tick is expected to be larger than the previous one. The reverse is true
after a downtick; the base is smaller and the next tick is expected to be smaller. As a
result, a sequence of positive shocks will have a larger upward effect than the downward
effect of a sequence of negative shocks. Thus, an upward drift is created just by volatil-
ity, even if g is zero. How big is this extra drift? It depends on the amplitude of the ticks; in fact, it amounts to half their variance. Therefore m, the expected continuously com- pounded expected rate of return, is larger than g. The rule for the expected CC annual rate becomes,
E(rCC) 5 m 5 g 1 ½ s 2 (5.21)
20 We see a similar phenomenon in the binomial tree example depicted in Figure 5.9 . Even with many bad
returns, stock prices cannot become negative, so the distribution is bounded at zero. But many good returns can
increase stock prices without limit, so the compound return after many periods has a long right tail, but a left tail
bounded by a worst-case cumulative return of 2 100%. This gives rise to the asymmetric skewed shape that is
characteristic of the log-normal distribution.
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C H A P T E R 5 Risk, Return, and the Historical Record 155
With a normally distributed CC rate, we expect that some initial wealth of $ W 0 will com- pound over one year to W0e
g 1½ s 2
5 Wem, and hence the expected effective rate of return is
E(r) 5 eg 1½ s 2
2 1 5 em 2 1 (5.22)
If an annual CC rate applies to an investment over any period T, either longer or shorter than one year, the investment will grow by the proportion r (T ) 5 erCC T 2 1. The expected cumulative return, r CC T , is proportional to T , that is, E(rCCT ) 5 mT 5 gT 1 ½ s
2T and expected final wealth is
E(WT) 5 W0 e mT 5 W0e
(g 1½ s 2 )T (5.23)
The variance of the cumulative return is also proportional to the time horizon:
Var( r CC T ) 5 T Var( r CC ), 21
but standard deviation rises only in proportion to the square root
of time: s(rCCT ) 5 "T Var (rCC) 5 s"T. This appears to offer a mitigation of investment risk in the long run: Because
the expected return increases with horizon at a faster rate than the standard deviation, the
expected return of a long-term risky investment becomes ever larger relative to its standard
deviation. Perhaps shortfall risk declines as investment horizon increases. We look at this
possibility in Example 5.11.
21
The variance of the effective annual rate when returns are lognormally distributed is: Var( r ) 5 e 2 m ( es 2
2 1).
22
In some versions of Excel, the function is NORM.S.DIST(z, TRUE).
An expected effective monthly rate of return of 1% is equivalent to a CC rate of ln(1.01) 5 0.00995 (0.995% per month). The risk-free rate is assumed to be 0.5% per month, equivalent to a CC rate of ln(1.005) 5 0.4988%. The effective SD of 4.54% implies (see footnote 21 ) a monthly SD of the CC rate of 4.4928%. Hence the monthly CC risk premium is 0.995 2 0.4988 5 0.4963%, with a SD of 4.4928%, and a Sharpe ratio of .4963/4.4928 5 0.11. In other words, returns would have to be .11 standard deviations below the mean before the stock portfolio underperformed T-bills. Using the normal distribution, we see that the probability of a rate of return shortfall relative to the risk-free rate is 45.6%. (You can confirm this by entering 2 .11 in Excel’s NORMSDIST function. 22 ) This is the probability of investor “regret,” that after the fact, the investor would have been better off in T-bills than investing in the stock portfolio.
For a 300-month horizon, however, the expected value of the cumulative excess return is .4963% 3 300 5 148.9% and the standard deviation is 4.4928"300 5 77.82, implying a whopping Sharpe ratio of 1.91. Enter 2 1.91 in Excel’s NORMSDIST function, and you will see that the probability of shortfall over a 300-month horizon is only .029.
Example 5.11 Shortfall Risk in the Short Run and the Long Run
A warning: The probability of a shortfall is an incomplete measure of investment risk. Such probability does not take into account the size of potential losses, which for some of the possible outcomes (however unlikely) amount to complete ruin. The worst-case
scenarios for the 25-year investment are far worse than for the 1-month investment. We demonstrate the buildup of risk over the long run graphically in Figures 5.10 and 5.11 .
A better way to quantify the risk of a long-term investment would be the market price
of insuring it against a shortfall. An insurance premium must take into account both the
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156 P A R T I I Portfolio Theory and Practice
Figure 5.10 Annually compounded, 25-year HPRs from bootstrapped history and a normal distribution (50,000 observations)
SD Min 25-year HPR
Avg. 1-year HPR Avg. 25-year HPR
Worst-case terminal loss (%) Max Probability (loss) Probability (shortfall) Skew Kurtosis
*Bootstrapped history
Actual*
12.13 10.24 10.29 4.27
211.53
95 28.88 .0095 .1044
2.0854 .0040
Normal
12.15
4.32 26.67
82 29.32 .0064 .0603 .1135
2.0121
SD Min 25-year HPR
Avg. 1-year HPR Avg. 25-year HPR
Worst-case terminal loss (%) Max Probability (loss) Probability (shortfall) Skew Kurtosis
*Bootstrapped history
Actual*
17.97 12.28 12.21 7.28 217
99 47.03 .0415 .1178 .1362 .0678
Normal
17.95
7.41 214.9
98 48.34 .0428 .1232 .2000 .0598
0 1 2 3 4 5 6 7 8 9
10
215 210 25 0 5 10 15 20 25 30 35 40 45
Fr e q
u e n
cy
Rate of Return
B: Small Stocks
0 1 2 3 4 5 6 7 8 9
10
215 210 25 0 5 10 15 20 25 30 35 40 45
Fr e q
u e n
cy
A: Large Stocks
History Normal
History Normal
probability of possible losses and the magnitude of these losses. We show in later chap-
ters how the fair market price of portfolio insurance can be estimated from option-pricing
models.
Despite the low probability that a portfolio insurance policy would have to pay up (only
2.9% for the 25-year policy), the magnitude and timing 23
of possible losses would make
such long-term insurance surprisingly costly. For example, standard option-pricing models
suggest that the value of insurance against shortfall risk over a 10-year horizon would cost
nearly 20% of the initial value of the portfolio. And contrary to any intuition that a longer
horizon reduces shortfall risk, the value of portfolio insurance increases dramatically with
the maturity of the contract. For example, a 25-year policy would be about 50% more
costly, or about 30% of the initial portfolio value.
23 By “timing,” we mean that a decline in stock prices is associated with a bad economy when extra income would
be most important to an investor. The fact that the insurance policy would pay off in these scenarios contributes
to its market value.
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C H A P T E R 5 Risk, Return, and the Historical Record 157
Simulation of Long-Term Future Rates of Return The frequency distributions in Figure 5.6 provide only rough descriptions of the nature of
the return distributions and are even harder to interpret for long-term investments. One way
to learn from history about the distribution of long-term future returns is to simulate these
future returns from that history. The method to accomplish this task is called bootstrapping. Bootstrapping is a procedure that avoids any assumptions about the return distribution,
except that all rates of return in the sample history are equally likely. For example, we
could simulate 25 years of possible future returns by sampling (with replacement) 25 ran-
domly selected annual returns from our available history. We compound those 25 returns
to obtain one possible 25-year holding-period return. This procedure is repeated thousands
of times to generate a probability distribution of long-term total returns that is anchored in
the historical frequency distribution.
The cardinal decision when embarking on a bootstrapping exercise is the choice of
how far into the past we should go to draw observations for “future” return sequences. We
Figure 5.11 Annually compounded, 25-year HPRs from bootstrapped history (50,000 observations)
A1: 25 Years in Large Stocks
B2: 10 Years in Small Stocks and 15 Years in T-bills
A2: 10 Years in Large Stocks and 15 Years in T-bills
B1: 25 Years in Small Stocks
0
2
4
6
8
10
12
14
21521025 0 5 10 15 20 25 30 35 40 45
Avg. 25-year HPR
Terminal loss (%)
Probability (loss) Avg. 1-year HPR 12.13
10.24 4.27
211.53 95 28.88 .0095
SD Min
Max
Fr e q
u e n
cy
0
2
4
6
8
10
12
14
21521025 0 5 10 15 20 25 30 35 40 45
Avg. 25-year HPR
Terminal loss (%)
Probability (loss) Avg. 1-year HPR 17.97
12.28 7.28
217.00 99 47.03 .0415
SD Min
Max
Fr e q
u e n
cy
0
2
4
6
8
10
12
14
21521025 0 5 10 15 20 25 30 35 40 45
Avg. 25-year HPR
Terminal loss (%)
Probability (loss) Avg. 1-year HPR 9.40
7.01 4.44
210.77 94 26.63 .0552
SD Min
Max
Fr e q
u e n
cy
0
2
4
6
8
10
12
14
21521025 0 5 10 15 20 25 30 35 40 45
Avg. 25-year HPR
Terminal loss (%)
Probability (loss) Avg. 1-year HPR 7.10
6.27 2.65
25.38 75 16.02 .0127
SD Min
Max
Fr e q
u e n
cy
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158 P A R T I I Portfolio Theory and Practice
will use our entire historical sample so that we are more likely to include low-probability
events of extreme value.
At this point, it is well to bring up again Nassim Taleb’s metaphor of the black swan. 24
Taleb uses the black swan, once unknown to Europeans, as an example of events that may
occur without any historical precedent. The black swan is a symbol of tail risk—highly
unlikely but extreme and important outcomes that are all but impossible to predict from
experience. The implication for bootstrapping is that limiting possible future returns to the
range of past returns, or extreme returns to their historical frequency, may easily underes-
timate actual exposure to tail risk. Notice that when simulating from a normal distribution,
we do allow for unbounded bad outcomes, although without allowing for fat tails, we may
greatly underestimate their probabilities. However, using any particular probability dis- tribution predetermines the shape of future events based on measurements from the past.
The dilemma of how to describe uncertainty largely comes down to how investors
should respond to the possibility of low-probability disasters. Those who argue that an
investment is less risky in the long run implicitly downplay extreme events. The high price
of portfolio insurance provides proof positive that a majority of investors certainly do not
ignore them. As far as the present exercise is concerned, we show that even a simulation
based on generally benign past U.S. history will produce cases of investor ruin.
An important objective of this exercise is to assess the potential effect of deviations
from normality on the probability distribution of a long-term investment in U.S. stocks. For
this purpose, we bootstrap 50,000 25-year simulated “histories” for large and small stocks,
and produce for each history the average annual return. We contrast these samples to simi-
lar samples drawn from normal distributions that (due to compounding) result in lognor-
mally distributed long-term total returns. Results are shown in Figure 5.10 . Panel A shows
frequency distributions of large U.S. stocks, constructed by sampling both from actual
returns and from the normal distribution. Panel B shows the same frequency distributions
for small U.S. stocks. The boxes inside Figure 5.10 show the statistics of the distributions.
We first review the results for large stocks in panel A. We see that the difference in
frequency distributions between the simulated history and the normal curve is small but
distinct. Despite the very small differences between the averages of 1-year and 25-year
annual returns, as well as between the standard deviations, the small differences in
skewness and kurtosis combine to produce significant differences in the probabilities of
shortfalls and losses, as well as in the potential terminal loss. For small stocks, shown in
panel B, the smaller differences in skewness and kurtosis lead to almost identical figures
for the probability and magnitude of losses.
We should also consider the risk of long-term investments. The probability of ruin is
miniscule, and indeed, as the following table indicates, the probability of any loss is less than 1% for large stocks and 5% for small stocks. This is in line with our calculations in
Example 5.11 showing that shortfall probabilities fall as the investment horizon extends.
But look at the top line of the table, showing the potential size of your loss in the (admit- tedly unlikely) worst-case scenarios. Risk depends on both the probability and the size of
the potential loss, and here that worst-case scenario is very bad indeed.
Statistic
Actual Normal Distribution
Large Stocks Small Stocks Large Stocks Small Stocks
Maximum terminal loss (%) 95 99 82 98 Probability of loss .0095 .0415 .0064 .0428 Probability of shortfall .1044 .1178 .0603 .1232
24
The Black Swan: The Impact of the Highly Improbable, New York, Random House, 2010.
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C H A P T E R 5 Risk, Return, and the Historical Record 159
What about risk for investors with
other long-term horizons? Figure 5.11
compares 25-year to 10-year invest-
ments in large and small stocks. For
an appropriate comparison we must
supplement the 10-year investment
with a 15-year investment in T-bills.
To accomplish this comparison, we
bootstrap 15-year samples from the
history of T-bill rates and augment
each sample with 10 annual rates
drawn from the history of the risky
investment. Panels A1 and A2 in
Figure 5.11 show the comparison for
large stocks. The frequency distribu-
tions reveal a substantial difference in
the risks of the terminal portfolio. This
difference is clearly manifested in the
portfolio performance statistics. The
same picture arises in panels B1 and
B2 for small stocks. Notice that even
a 10-year investment in small stocks
could lead to a terminal loss of 94%.
Figure 5.12 shows trajectories of
wealth indexes for some of the pos-
sible outcomes of a 25-year invest-
ment in large stocks, compared with the wealth index of the average outcome of a T-bill
portfolio. The outcomes of the stock portfolio in Figure 5.12 range from the worst, through
the bottom 1% and 5% of terminal value, and up to the mean and median terminal values.
The bottom 5% still results in a significant shortfall relative to the T-bill portfolio. In sum,
the analysis clearly rejects the notion that stocks become less risky in the long run.
Yet many practitioners hold on to the view that investment risk is less pertinent to long-
term investors. A typical demonstration shown in the nearby box relies on the fact that the
standard deviation (or range of likely outcomes) of annualized returns is lower for longer- term horizons. But the demonstration is silent on the range of total returns.
The Risk-Free Rate Revisited At the outset of this chapter we put forward a simple view of the real and nominal risk-
free rate, where we were not very explicit about investment horizon. But as a general
rule, the maturity of the risk-free rate should match the investment horizon. Investors
with long maturities will view the rate on long-term safe bonds as providing their bench-
mark risk-free rate. Interest rates generally vary with maturity and, surely, inflation is
more difficult to predict over longer horizons. Thus inflation risk becomes more potent
with maturity.
It is important to realize that the risk premium on risky assets is a real quantity. The expected rate on a risky asset equals the risk-free rate plus a risk premium. That risk pre-
mium is incremental to the risk-free rate and makes for the same incremental addition, whether we state the risk-free rate in real or nominal terms.
An investor views the real rate for each maturity as the benchmark for investments of that maturity, and hence a real risky rate should be displayed as a real risk-free rate plus
Figure 5.12 Wealth indexes of selected outcomes of large stock portfolios and the average T-bill portfolio. Inset: Focus on worst, 1%, and 5% outcomes versus bills.
0 0 5 10 15 20 25
2
4
6
8
10
12
14
16
18
20 Worst 1% 5%
Average Median T-Bills
Years
W e a lt
h I n
d e x
0 0 5 10 15 20 25
0.5 1
1.5 2
2.5 3
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160
a risk premium. Even default-free nominal rates on long-maturity Treasury bonds may
embody a risk premium due to uncertainty about future inflation and interest rates.
Enter TIPS, the Treasury bond that promises investors an inflation-indexed real rate for
a desired maturity. Now we can think of the expected real rate on a risky investment of a
given maturity as the rate on the same-maturity TIPS bond plus a risk premium.
The existence of both nominal Treasuries and TIPS also has informational value. The
difference in the expected rates on these bonds is called the forward rate of inflation, which includes both the expected rate and the appropriate risk premium.
Why then do we see excess returns usually stated relative to one-month T-bill rates?
This is because most discussions refer to short-term investments. To seriously consider a
long-term investment, however, we must account for the relevant real, risk-free rate.
Where Is Research on Rates of Return Headed? In order to learn more about the distribution of returns, particularly the behavior of rela-
tively rare extreme events, we need a lot more data. The speed with which we accumulate
even daily rates of return will not get us there; by the time we have a large enough sample,
the distributions may well have changed. But it may be that help is on the way.
The highest frequency we can obtain for rates of return comes from trade-by-trade data.
Statistical methods recently developed by astrophysicists can glean from these observa-
tions the essential components of the return distributions. The return process can usefully
be described as a sum of an instantaneous normal that develops into a lognormal compound
Time vs. Risk
MANY BEGINNING INVESTORS eye the stock market with a bit of suspicion. They view equity investing as an anx- ious game of Russian roulette: The longer they stay in, the greater their chance of experiencing more losses. In fact, history shows that the opposite is true. The easiest way to reduce the risk of investing in equities—and improve the gain—is to increase the time you hang on to your portfolio.
See for yourself. The demonstration below uses histori- cal data from 1950 through 2005 to compare investment returns over different lengths of time for small-cap stocks, large caps, long-term bonds, and T-bills.
Source: CRSP, Federal Reserve
Time vs. Risk
125% 100% 75% 50% 25% 0% −25% −50% −75%
BondsT–billsLarge cap
Small cap
1–year periods Best
Average
Worst
zoom in
adjust for inflation
Range of annualized returns over 1–year periods,
The graph starts out showing results for investments held over one-year periods. There’s no doubt about it: Over such short intervals, small-cap stocks are definitely the risk- iest bet.
But what about investing for more than a year? If you move the slider at the bottom right of the graph, you can see the range of returns for longer time periods. Even investing for two years instead of one cuts your risk sig- nificantly. As the length of time increases, the volatility of equities decreases sharply—so much so that you may need to click the “zoom in” button to get a closer view. Over 10-year periods, government bonds look safer than large- cap equities on the downside. Click the “adjust for infla- tion” box, however, and you’ll see that bond “safety” can be illusory. Inflation has an uncanny ability to erode the value of securities that don’t grow fast enough.
Now move the slider all the way to the right to see the results of investing for 20-year intervals. Adjusting for inflation, the best 20-year gain a portfolio of long- term Treasury bonds could muster is much lower than that achieved by small- and large-cap stocks. And contrary to popular belief, over their worst 20-year period, long- term bonds actually lost money when adjusted for infla- tion. Meanwhile, small-cap investors still had gains over a 20-year-period, even when stocks were at their worst.
Source: Abridged from ”Time vs. Risk,” SmartMoney.com, July 31, 2010. © 2013 Dow Jones & Company, Inc. Used with permission via icopyright.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 5 Risk, Return, and the Historical Record 161
return plus jumps that generate deviation from normality. The jump process itself can be
decomposed to an aggregation of small jumps plus large jumps that dominate the tails of
the distributions. 25
We expect that before long practitioners will be able to purchase output of such research
and obtain accurate risk parameters of a large array of investments. These will add rel-
evance to the insights and investment practices described in future chapters.
Forecasts for the Long Haul We use arithmetic averages to forecast future rates of return because they are unbiased
estimates of expected rates over equivalent holding periods. But the arithmetic average of
short-term returns can be misleading when used to forecast long-term cumulative returns.
This is because sampling errors in the estimate of expected return will have asymmetric
impact when compounded over long periods. Positive sampling variation will compound
to greater upward errors than negative variation.
Jacquier, Kane, and Marcus show that an unbiased forecast of total return over long
horizons requires compounding at a weighted average of the arithmetic and geometric
historical averages. 26
The proper weight applied to the geometric average equals the ratio
of the length of the forecast horizon to the length of the estimation period. For example, if
we wish to forecast the cumulative return for a 25-year horizon from an 80-year history, an
unbiased estimate would be to compound at a rate of
Geometric average 3 25
80 1 Arithmetic average 3
(80 2 25)
80
This correction would take about .6% off the historical arithmetic average risk premium on
large stocks and about 2% off the arithmetic average premium of small stocks. A forecast
for the next 80 years would require compounding at only the geometric average, and for
longer horizons at an even lower number. The forecast horizons that are relevant for cur-
rent investors would depend on their life expectancies.
25
For an introduction to this approach, see Yacine Aït-Sahalia and Jean Jacod, “Analyzing the Spectrum of Asset
Returns: Jump and Volatility Components in High Frequency Data,” Journal of Economic Literature 50 (2012), pp. 1007–50.
26
Eric Jacquier, Alex Kane, and Alan J. Marcus, “Geometric or Arithmetic Means: A Reconsideration,” Financial Analysts Journal, November/December 2003.
1. The economy’s equilibrium level of real interest rates depends on the willingness of households to save, as reflected in the supply curve of funds, and on the expected profitability of business
investment in plant, equipment, and inventories, as reflected in the demand curve for funds. It
depends also on government fiscal and monetary policy.
2. The nominal rate of interest is the equilibrium real rate plus the expected rate of inflation. In gen- eral, we can directly observe only nominal interest rates; from them, we must infer expected real
rates, using inflation forecasts.
3. The equilibrium expected rate of return on any security is the sum of the equilibrium real rate of interest, the expected rate of inflation, and a security-specific risk premium.
4. Investors face a trade-off between risk and expected return. Historical data confirm our intui- tion that assets with low degrees of risk provide lower returns on average than do those of
higher risk.
SUMMARY
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162 P A R T I I Portfolio Theory and Practice
5. Assets with guaranteed nominal interest rates are risky in real terms because the future inflation rate is uncertain.
6. Historical rates of return over the 20th century in developed capital markets suggest the U.S. history of stock returns is not an outlier compared to other countries.
7. Investments in risky portfolios do not become safer in the long run. On the contrary, the longer a risky investment is held, the greater the risk. The basis of the argument that stocks are safe in
the long run is the fact that the probability of a shortfall becomes smaller. However, probability
of shortfall is a poor measure of the safety of an investment. It ignores the magnitude of possible
losses.
8. Historical returns on stocks exhibit more frequent large negative deviations from the mean than would be predicted from a normal distribution. The lower partial standard deviation
(LPSD), the skew, and kurtosis of the actual distribution quantify the deviation from nor-
mality. The LPSD, instead of the standard deviation, is sometimes used by practitioners as a
measure of risk.
9. Widely used measures of tail risk are value at risk (VaR) and expected shortfall or, equivalently, conditional tail expectations. VaR measures the loss that will be exceeded with a specified prob-
ability such as 5%. The VaR does not add new information when returns are normally distrib-
uted. When negative deviations from the average are larger and more frequent than the normal
distribution, the 5% VaR will be more than 1.65 standard deviations below the average return.
Expected shortfall (ES) measures the expected rate of return conditional on the portfolio falling
below a certain value. Thus, 1% ES is the expected return of all possible outcomes in the bottom
1% of the distribution.
Related Web sites for this chapter are available at www. mhhe.com/bkm
nominal interest rate
real interest rate
effective annual rate (EAR)
annual percentage rate (APR)
dividend yield
risk-free rate
risk premium
excess return
risk aversion
normal distribution
event tree
skew
kurtosis
value at risk (VaR)
expected shortfall (ES)
conditional tail expectation
(CTE)
lower partial standard
deviation (LPSD)
Sortino ratio
lognormal distribution
KEY TERMS
Arithmetic average of n returns: ( r 1 1 r 2 1 ? ? ? 1 r n )/ n
Geometric average of n returns: [(1 1 r 1 ) (1 1 r 2 ) ? ? ? (1 1 r n )] 1/ n
2 1
Continuously compounded rate of return, r cc : ln(1 1 Effective annual rate)
Expected return: S [prob(Scenario) 3 Return in scenario]
Variance: S [prob(Scenario) 3 (Deviation from mean in scenario) 2 ]
Standard deviation: "Variance Sharpe ratio:
Portfolio risk premium
Standard deviation of excess return 5
E(rP) 2 rf
sP
Real rate of return: 1 1 Nominal return
1 1 Inflation rate 2 1
Real rate of return (continuous compounding): r nominal 2 Inflation rate
KEY EQUATIONS
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C H A P T E R 5 Risk, Return, and the Historical Record 163
Basic
PROBLEM SETS 1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation
imply that this increase will result in a fall in the real rate of interest? Explain.
2. You’ve just stumbled on a new dataset that enables you to compute historical rates of return on
U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these
data to help estimate the expected rate of return on U.S. stocks over the coming year?
3. You are considering two alternative 2-year investments: You can invest in a risky asset with a
positive risk premium and returns in each of the 2 years that will be identically distributed and
uncorrelated, or you can invest in the risky asset for only 1 year and then invest the proceeds in
a risk-free asset. Which of the following statements about the first investment alternative (com-
pared with the second) are true?
a. Its 2-year risk premium is the same as the second alternative. b. The standard deviation of its 2-year return is the same. c. Its annualized standard deviation is lower. d. Its Sharpe ratio is higher. e. It is relatively more attractive to investors who have lower degrees of risk aversion.
4. You have $5,000 to invest for the next year and are considering three alternatives:
a. A money market fund with an average maturity of 30 days offering a current yield of 6% per year.
b. A 1-year savings deposit at a bank offering an interest rate of 7.5%. c. A 20-year U.S. Treasury bond offering a yield to maturity of 9% per year.
What role does your forecast of future interest rates play in your decisions?
5. Use Figure 5.1 in the text to analyze the effect of the following on the level of real interest rates:
a. Businesses become more pessimistic about future demand for their products and decide to reduce their capital spending.
b. Households are induced to save more because of increased uncertainty about their future Social Security benefits.
c. The Federal Reserve Board undertakes open-market purchases of U.S. Treasury securities in order to increase the supply of money.
6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD
offering an interest rate of 5% and a 1-year “Inflation-Plus” CD offering 1.5% per year plus the
rate of inflation.
a. Which is the safer investment? b. Which offers the higher expected return? c. If you expect the rate of inflation to be 3% over the next year, which is the better investment?
Why?
d. If we observe a risk-free nominal interest rate of 5% per year and a risk-free real rate of 1.5% on inflation-indexed bonds, can we infer that the market’s expected rate of inflation is 3.5%
per year?
7. Suppose your expectations regarding the stock price are as follows:
State of the Market Probability Ending Price HPR (including dividends)
Boom .35 $140 44.5% Normal growth .30 110 14.0 Recession .35 80 2 16.5
Use Equations 5.11 and 5.12 to compute the mean and standard deviation of the HPR on stocks.
Intermediate
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164 P A R T I I Portfolio Theory and Practice
8. Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an
8% coupon if it is currently selling at par and the probability distribution of its yield to maturity
a year from now is as follows:
State of the Economy Probability YTM
Boom .20 11.0% Normal growth .50 8.0 Recession .30 7.0
For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every
6 months.
9. What is the standard deviation of a random variable q with the following probability distribution:
Value of q Probability
0 .25 1 .25
2 .50
10. The continuously compounded annual return on a stock is normally distributed with a mean
of 20% and standard deviation of 30%. With 95.44% confidence, we should expect its
actual return in any particular year to be between which pair of values? Hint: Look again at Figure 5.4 .
a. 2 40.0% and 80.0% b. 2 30.0% and 80.0% c. 2 20.6% and 60.6% d. 2 10.4% and 50.4%
11. Using historical risk premiums over the 7/1926–9/2012 period as your guide, what would be
your estimate of the expected annual HPR on the Big/Value portfolio if the current risk-free
interest rate is 3%?
12. Visit Professor Kenneth French’s data library website: http://mba.tuck.dartmouth.edu/ pages/faculty/ken.french/data_library.html and download the monthly returns of “6 port- folios formed on size and book-to-market (2 3 3).” Choose the value-weighted series for the
period from 1/1928–12/2012 (1,020 months). Split the sample in half and compute the aver-
age, SD, skew, and kurtosis for each of the six portfolios for the two halves. Do the six split-
halves statistics suggest to you that returns come from the same distribution over the entire
period?
13. During a period of severe inflation, a bond offered a nominal HPR of 80% per year. The infla-
tion rate was 70% per year.
a. What was the real HPR on the bond over the year? b. Compare this real HPR to the approximation rr < rn 2 i.
14. Suppose that the inflation rate is expected to be 3% in the near future. Using the historical data
provided in this chapter, what would be your predictions for:
a. The T-bill rate? b. The expected rate of return on the Big/Value portfolio? c. The risk premium on the stock market?
15. An economy is making a rapid recovery from steep recession, and businesses foresee a need
for large amounts of capital investment. Why would this development affect real interest
rates?
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C H A P T E R 5 Risk, Return, and the Historical Record 165
16. You are faced with the probability distribution of the HPR on the stock market index fund given
in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund
with exercise price of $110 and time to expiration of 1 year is $12.
a. What is the probability distribution of the HPR on the put option? b. What is the probability distribution of the HPR on a portfolio consisting of one share of
the index fund and a put option?
c. In what sense does buying the put option constitute a purchase of insurance in this case?
17. Take as given the conditions described in the previous problem, and suppose the risk-free inter-
est rate is 6% per year. You are contemplating investing $107.55 in a 1-year CD and simultane-
ously buying a call option on the stock market index fund with an exercise price of $110 and
expiration of 1 year. What is the probability distribution of your dollar return at the end of the
year?
18. Consider these long-term investment data:
• The price of a 10-year $100 par zero coupon inflation-indexed bond is $84.49.
• A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the
(effective) quarterly rate of 10%.
a. Compute the annual rate on the real bond. b. Compute the CC annual risk premium on the real-estate investment. c. Use the appropriate formula and Excel Solver or Goal Seek to find the SD of the
CC annual excess return on the real-estate investment.
d. What is the probability of loss or shortfall after 10 years?
Challenge
1. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities
versus risk-free T-bills (U.S. Treasury bills) based on the following table?
Action Probability Expected Return
Invest in equities .6 $50,000 .4 2$30,000
Invest in risk-free T-bill 1.0 $ 5,000
2. Based on the scenarios below, what is the expected return for a portfolio with the following return
profile?
Market Condition
Bear Normal Bull
Probability .2 .3 .5 Rate of return 2 25% 10% 24%
Use the following scenario analysis for Stocks X and Y to answer CFA Problems 3 through 6 (round to the nearest percent).
Bear Market Normal Market Bull Market
Probability 0.2 0.5 0.3 Stock X 2 20% 18% 50%
Stock Y 2 15% 20% 10%
3. What are the expected rates of return for Stocks X and Y?
4. What are the standard deviations of returns on Stocks X and Y?
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166 P A R T I I Portfolio Theory and Practice
5. Assume that of your $10,000 portfolio, you invest $9,000 in Stock X and $1,000 in Stock Y.
What is the expected return on your portfolio?
6. Probabilities for three states of the economy and probabilities for the returns on a particular stock
in each state are shown in the table below.
State of Economy Probability of
Economic State Stock Performance
Probability of Stock Performance in Given Economic State
Good .3 Good .6 Neutral .3 Poor .1
Neutral .5 Good .4 Neutral .3 Poor .3
Poor .2 Good .2 Neutral .3 Poor .5
What is the probability that the economy will be neutral and the stock will experience poor performance?
7. An analyst estimates that a stock has the following probabilities of return depending on the state
of the economy:
State of Economy Probability Return
Good .1 15% Normal .6 13 Poor .3 7
What is the expected return of the stock?
E-INVESTMENTS EXERCISES The Federal Reserve Bank of St. Louis has information available on interest rates and eco- nomic conditions. A publication called Monetary Trends contains graphs and tables with information about current conditions in the capital markets. Go to the Web site www. stls.frb.org and click on Economic Research on the menu at the top of the page. Find the most recent issue of Monetary Trends in the Recent Data Publications section and answer these questions.
1. What is the professionals’ consensus forecast for inflation for the next 2 years? (Use the Federal Reserve Bank of Philadelphia line on the graph to answer this.)
2. What do consumers expect to happen to inflation over the next 2 years? (Use the University of Michigan line on the graph to answer this.)
3. Have real interest rates increased, decreased, or remained the same over the last 2 years?
4. What has happened to short-term nominal interest rates over the last 2 years? What about long-term nominal interest rates?
5. How do recent U.S. inflation and long-term interest rates compare with those of the other countries listed?
6. What are the most recently available levels of 3-month and 10-year yields on Treasury securities?
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V is
it u
s a t
w w
w .m
h h
e .c
o m
/b k m
C H A P T E R 5 Risk, Return, and the Historical Record 167
SOLUTIONS TO CONCEPT CHECKS 1. a. 1 1 rn 5 (1 1 rr)(1 1 i ) 5 (1.03)(1.08) 5 1.1124
rn 5 11.24%
b. 1 1 rn 5 (1.03)(1.10) 5 1.133 rn 5 13.3%
2. a. EAR 5 (1 1 .01) 12 2 1 5 .1268 5 12.68%
b. EAR 5 e .12 2 1 5 .1275 5 12.75%
Choose the continuously compounded rate for its higher EAR.
3. Number of bonds bought is 27,000/900 5 30
Interest Rates Probability Year-end Bond Price HPR End-of-Year Value
High .2 $850 (75 1 850)/900 2 1 5 .0278 (75 1 850)30 5 $27,750 Unchanged .5 915 .1000 $29,700
Low .3 985 .1778 $31,800
Expected rate of return .1089
Expected end-of-year dollar value $29,940
Risk premium .0589
4. (1 1 Required rate)(1 2 .40) 5 1
Required rate 5 .667, or 66.7%
5. a. Arithmetic return 5 (1/3)(.2869) 1 (1/3)(.1088) 1 (1/3)(0.0491) 5 .1483 5 14.83%
b. Geometric average 5 3"1.2869 3 1.1088 3 1.0491 2 1 5 .1439 5 14.39%
c. Standard deviation 5 12.37%
d. Sharpe ratio 5 (14.83 2 6.0)/12.37 5 .71
6. The probability of a more extreme bad month, with return below 2 15%, is much lower:
NORMDIST( 2 15,1,6,TRUE) 5 .00383. Alternatively, we can note that 2 15% is 16/6
standard deviations below the mean return, and use the standard normal function to compute
NORMSDIST( 2 16/6) 5 .00383.
7. If the probabilities in Spreadsheet 5.2 represented the true return distribution, we would use
Equations 5.19 and 5.20 to obtain: Skew 5 0.0931; Kurtosis 5 2 1.2081. However, in this case,
the data in the table represent a (short) historical sample, and correction for degrees-of-freedom
bias is required (in a similar manner to our calculations for standard deviation). You can use
Excel functions to obtain: SKEW(C5:C9) 5 0.1387; KURT(C5:C9) 5 2 0.2832.
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66 THE PROCESS OF constructing an overall portfolio requires you to: (1) select the com- position of the risky portfolio and (2) decide how much to invest in it, directing the remaining investment budget to a risk-free investment. The second step is called capital allocation to risky assets.
Clearly, to decide on your capital allocation you need to know the risky portfolio and evalu- ate its properties. Can the construction of that risky portfolio be delegated to an expert? An affirmative answer is necessary for a viable investments management industry. A negative answer would require every investor to learn and implement portfolio management for himself.
To understand the existence of a portfolio management industry in the face of personal
investor preferences, we need insight into the nature of risk aversion. We characterize a per- sonal utility function that provides a score for the attractiveness of candidate overall portfo- lios on the basis of expected return and risk. By choosing the portfolio with the highest score, investors maximize their satisfaction with their choice of investments; that is, they achieve the optimal allocation of capital to risky assets.
The utility model also reveals the appropri- ate objective function for the construction of an optimal risky portfolio and thus explains how an industry can serve investors with highly diverse preferences without the need to know each of them personally.
CHAPTER SIX
Capital Allocation
to Risky Assets
PA R
T I I
6.1 Risk and Risk Aversion
In Chapter 5 we introduced the concepts of the holding-period return (HPR) and the excess
rate of return over the risk-free rate. We also discussed estimation of the risk premium (the expected excess return) and the standard deviation of the excess return, which we use as the measure of portfolio risk. We demonstrated these concepts with a scenario analysis
of a specific risky portfolio (Spreadsheet 5.1). To emphasize that bearing risk typically
must be accompanied by a reward in the form of a risk premium, we first differentiate
between speculation and gambling.
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C H A P T E R 6 Capital Allocation to Risky Assets 169
Risk, Speculation, and Gambling One definition of speculation is “the assumption of considerable investment risk to obtain commensurate gain.” However, this definition is useless without specifying what is meant
by “considerable risk” and “commensurate gain.”
By “considerable risk” we mean that the risk is sufficient to affect the decision. An indi-
vidual might reject an investment that has a positive risk premium because the potential
gain is insufficient to make up for the risk involved. By “commensurate gain” we mean a
positive risk premium, that is, an expected profit greater than the risk-free alternative.
To gamble is “to bet or wager on an uncertain outcome.” The central difference between
gambling and speculation is the lack of “commensurate gain.” Economically speaking,
a gamble is the assumption of risk for enjoyment of the risk itself, whereas speculation
is undertaken in spite of the risk involved because one perceives a favorable risk–return trade-off. To turn a gamble into a speculative venture requires an adequate risk premium to
compensate risk-averse investors for the risks they bear. Hence, risk aversion and specula- tion are consistent. Notice that a risky investment with a risk premium of zero, sometimes called a fair game , amounts to a gamble. A risk-averse investor will reject it.
In some cases a gamble may appear to be speculation. Suppose two investors disagree sharply about the future exchange rate of the U.S. dollar against the British pound. They
may choose to bet on the outcome: Paul will pay Mary $100 if the value of £1 exceeds
$1.60 one year from now, whereas Mary will pay Paul if the pound is worth less than
$1.60. There are only two relevant outcomes: (1) the pound will exceed $1.60, or (2) it
will fall below $1.60. If both Paul and Mary agree on the probabilities of the two possible
outcomes, and if neither party anticipates a loss, it must be that they assign p 5 .5 to each outcome. In that case the expected profit to both is zero and each has entered one side of a
gambling prospect.
What is more likely, however, is that Paul and Mary assign different probabilities to
the outcome. Mary assigns it p . .5, whereas Paul’s assessment is p , .5. They perceive, subjectively, two different prospects. Economists call this case of differing beliefs “het-
erogeneous expectations.” In such cases investors on each side of a financial position see
themselves as speculating rather than gambling.
Both Paul and Mary should be asking, Why is the other willing to invest in the side of
a risky prospect that I believe offers a negative expected profit? The ideal way to resolve
heterogeneous beliefs is for Paul and Mary to “merge their information,” that is, for each
party to verify that he or she possesses all relevant information and processes the informa-
tion properly. Of course, the acquisition of information and the extensive communication
that is required to eliminate all heterogeneity in expectations is costly, and thus up to a point
heterogeneous expectations cannot be taken as irrational. If, however, Paul and Mary enter
such contracts frequently, they would recognize the information problem in one of two ways:
Either they will realize that they are creating gambles when each wins half of the bets, or the
consistent loser will admit that he or she has been betting on the basis of inferior forecasts.
Assume that dollar-denominated T-bills in the United States and pound-denominated bills in the United Kingdom offer equal yields to maturity. Both are short-term assets, and both are free of default risk. Neither offers investors a risk premium. However, a U.S. investor who holds U.K. bills is subject to exchange rate risk, because the pounds earned on the U.K. bills eventually will be exchanged for dollars at the future exchange rate. Is the U.S. investor engaging in speculation or gambling?
CONCEPT CHECK 6.1
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170 P A R T I I Portfolio Theory and Practice
Risk Aversion and Utility Values The history of rates of return on various asset classes, as well as elaborate empirical studies,
leave no doubt that risky assets command a risk premium in the marketplace. This implies
that most investors are risk averse.
Investors who are risk averse reject investment portfolios that are fair games or worse. Risk-averse investors consider only risk-free or speculative prospects with positive risk
premiums. Loosely speaking, a risk-averse investor “penalizes” the expected rate of return
of a risky portfolio by a certain percentage (or penalizes the expected profit by a dollar
amount) to account for the risk involved. The greater the risk, the larger the penalty. We
believe that most investors would accept this view from simple introspection, but we dis-
cuss the question more fully in Appendixes A through C of this chapter.
To illustrate the issues we confront when choosing among portfolios with varying
degrees of risk, suppose the risk-free rate is 5% and that an investor considers three
alternative risky portfolios as shown in Table 6.1 . The risk premiums and degrees of risk
(standard deviation, SD) represent the properties of low-risk bonds ( L ), high-risk bonds ( M ), and large stocks ( H ). Accordingly, these portfolios offer progressively higher risk premiums to compensate for greater risk. How might investors choose among them?
Intuitively, a portfolio is more attractive when its expected return is higher and its risk
is lower. But when risk increases along with return, the most attractive portfolio is not
obvious. How can investors quantify the rate at which they are willing to trade off return
against risk?
We will assume that each investor can assign a welfare, or utility , score to competing portfolios on the basis of the expected return and risk of those portfolios. Higher utility
values are assigned to portfolios with more attractive risk–return profiles. Portfolios
receive higher utility scores for higher expected returns and lower scores for higher
volatility. Many particular “scoring” systems are legitimate. One reasonable function that
has been employed by both financial theorists and the CFA Institute assigns a portfolio
with expected return E ( r ) and variance of returns s 2 the following utility score:
U 5 E(r) 2 ½A s 2 (6.1)
where U is the utility value and A is an index of the investor’s risk aversion. The factor of ½ is just a scaling convention. To use Equation 6.1, rates of return must be expressed as
decimals rather than percentages. Notice that the portfolio in question here is the all-wealth
investment. Hence, assuming normality, standard deviation is the appropriate measure
of risk.
Equation 6.1 is consistent with the notion that utility is enhanced by high expected
returns and diminished by high risk. Notice that risk-free portfolios receive a utility
score equal to their (known) rate of return, because they receive no penalty for risk.
The extent to which the variance of risky portfolios lowers utility depends on A, the
Portfolio Risk Premium Expected Return Risk (SD)
L (low risk) 2% 7% 5% M (medium risk) 4 9 10 H (high risk) 8 13 20
Table 6.1
Available risky portfolios (Risk-free rate 5 5%)
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C H A P T E R 6 Capital Allocation to Risky Assets 171
investor’s degree of risk aversion. More risk-averse investors (who have the larger values
of A ) penalize risky investments more severely. Investors choosing among competing investment portfolios will select the one providing the highest utility level. The box on
page 174 discusses some techniques that financial advisers use to gauge the risk aversion
of their clients.
We can interpret the utility score of risky portfolios as a certainty equivalent rate of return. The certainty equivalent rate is the rate that a risk-free investment would need to
offer to provide the same utility score as the risky portfolio. In other words, it is the rate
that, if earned with certainty, would provide a utility score equivalent to that of the port-
folio in question. The certainty equivalent rate of return is a natural way to compare the
utility values of competing portfolios.
A portfolio can be desirable only if its certainty equivalent return exceeds that of the
risk-free alternative. A sufficiently risk-averse investor may assign any risky portfolio,
even one with a positive risk premium, a certainty equivalent rate of return that is below
the risk-free rate, which will cause the investor to reject the risky portfolio. At the same
time, a less risk-averse investor may assign the same portfolio a certainty equivalent rate
that exceeds the risk-free rate and thus will prefer the portfolio to the risk-free alternative.
If the risk premium is zero or negative to begin with, any downward adjustment to utility
only makes the portfolio look worse. Its certainty equivalent rate will be below that of the
risk-free alternative for all risk-averse investors.
Investor Risk Aversion ( A )
Utility Score of Portfolio L [ E ( r ) 5 .07; s 5 .05]
Utility Score of Portfolio M [ E ( r ) 5 .09; s 5 .10]
Utility Score of Portfolio H [ E ( r ) 5 .13; s 5 .20]
2.0 .07 2 ½ 3 2 3 .05 2 5 .0675 .09 2 ½ 3 2 3 .1 2 5 .0800 .13 2 ½ 3 2 3 .2 2 5 .09
3.5 .07 2 ½ 3 3.5 3 .05 2 5 .0656 .09 2 ½ 3 3.5 3 .1 2 5 .0725 .13 2 ½ 3 3.5 3 .2 2 5 .06
5.0 .07 2 ½ 3 5 3 .05 2 5 .0638 .09 2 ½ 3 5 3 .1 2 5 .0650 .13 2 ½ 3 5 3 .2 2 5 .03
Table 6.2
Utility scores of alternative portfolios for investors with varying degrees of risk aversion
Consider three investors with different degrees of risk aversion: A 1 5 2, A 2 5 3.5, and A 3 5 5, all of whom are evaluating the three portfolios in Table 6.1 . Because the risk-free rate is assumed to be 5%, Equation 6.1 implies that all three investors would assign a utility score of .05 to the risk-free alternative. Table 6.2 presents the utility scores that would be assigned by each investor to each portfolio. The portfolio with the highest util- ity score for each investor appears in bold. Notice that the high-risk portfolio, H, would be chosen only by the investor with the lowest degree of risk aversion, A 1 5 2, while the low-risk portfolio, L, would be passed over even by the most risk-averse of our three investors. All three portfolios beat the risk-free alternative for the investors with levels of risk aversion given in the table.
Example 6.1 Evaluating Investments by Using Utility Scores
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172 P A R T I I Portfolio Theory and Practice
A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills offer a safe rate of return of 7%. Would an investor with risk-aversion parameter A 5 4 prefer to invest in T-bills or the risky portfolio? What if A 5 2?
CONCEPT CHECK 6.2
In contrast to risk-averse investors, risk-neutral investors (with A 5 0) judge risky prospects solely by their expected rates of return. The level of risk is irrelevant to the risk-
neutral investor, meaning that there is no penalty for risk. For this investor a portfolio’s
certainty equivalent rate is simply its expected rate of return.
A risk lover (for whom A , 0) is happy to engage in fair games and gambles; this investor adjusts the expected return upward to take into account the “fun” of confronting the prospect’s risk. Risk lovers will always take a fair game because their upward adjust-
ment of utility for risk gives the fair game a certainty equivalent that exceeds the alterna-
tive of the risk-free investment.
We can depict the individual’s trade-off between risk and return by plotting the char-
acteristics of portfolios that would be equally attractive on a graph with axes measuring
the expected value and standard deviation of portfolio returns. Figure 6.1 plots the char-
acteristics of one portfolio denoted P. Portfolio P, which has expected return E ( r P ) and standard deviation s P , is preferred
by risk-averse investors to any portfolio in quadrant IV because its expected return is
equal to or greater than any portfolio in that quadrant and its standard deviation is equal
to or smaller than any portfolio in that quadrant. Conversely, any portfolio in quadrant I
dominates portfolio P because its expected return is equal to or greater than P ’s and its standard deviation is equal to or smaller than P ’s.
P
E(rP)
E(r)
σP σ
I II
III IV
Northwest (preferred direction)
Figure 6.1 The trade-off between risk and return of a potential investment portfolio, P
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C H A P T E R 6 Capital Allocation to Risky Assets 173
This is the mean-standard devia-
tion, or equivalently, mean- variance (M-V) criterion . It can be stated as follows: portfolio A dominates B if
E(rA) $ E(rB)
and
sA # sB
and at least one inequality is strict
(to rule out indifference).
In the expected return–standard
deviation plane in Figure 6.1 , the
preferred direction is northwest,
because in this direction we simul-
taneously increase the expected
return and decrease the variance of the rate of return. Any portfolio that
lies northwest of P is superior to it. What can be said about portfolios
in quadrants II and III? Their desir-
ability, compared with P, depends on the exact nature of the investor’s risk
aversion. Suppose an investor iden-
tifies all portfolios that are equally attractive as portfolio P. Starting at P, an increase in stan- dard deviation lowers utility; it must be compensated for by an increase in expected return.
Thus point Q in Figure 6.2 is equally desirable to this investor as P. Investors will be equally attracted to portfolios with high risk and high expected returns compared with other port-
folios with lower risk but lower expected returns. These
equally preferred portfolios will lie in the mean–standard
deviation plane on a curve called the indifference curve , which connects all portfolio points with the same utility
value ( Figure 6.2 ).
To determine some of the points that appear on the
indifference curve, examine the utility values of sev-
eral possible portfolios for an investor with A 5 4, presented in Table 6.3 . Note that each portfolio offers
Figure 6.2 The indifference curve
P E(rP)
E(r)
σP
σ
Q
Indifference Curve
a. How will the indifference curve of a less risk- averse investor compare to the indifference curve drawn in Figure 6.2 ?
b. Draw both indifference curves passing through point P.
CONCEPT CHECK 6.3
Expected Return, E ( r ) Standard Deviation, s Utility 5 E ( r ) 2 ½ A s 2
.10 .200 .10 2 .5 3 4 3 .04 5 .02 .15 .255 .15 2 .5 3 4 3 .065 5 .02 .20 .300 .20 2 .5 3 4 3 .09 5 .02 .25 .339 .25 2 .5 3 4 3 .115 5 .02
Table 6.3
Utility values of possible portfolios for investor with risk aversion, A 5 4
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Time for Investing’s Four-Letter Word
What four-letter word should pop into mind when the stock market takes a harrowing nose dive?
No, not those. R-I-S-K. Risk is the potential for realizing low returns or even
losing money, possibly preventing you from meeting important objectives, like sending your kids to the college of their choice or having the retirement lifestyle you crave.
But many financial advisers and other experts say that when times are good, some investors don’t take the idea of risk as seriously as they should, and overexpose themselves to stocks. So before the market goes down and stays down, be sure that you understand your tolerance for risk and that your portfolio is designed to match it.
Assessing your risk tolerance, however, can be tricky. You must consider not only how much risk you can afford to take but also how much risk you can stand to take.
Determining how much risk you can stand—your tem- peramental tolerance for risk—is more difficult. It isn’t easy to quantify.
To that end, many financial advisers, brokerage firms and mutual-fund companies have created risk quizzes to help people determine whether they are conservative, moderate or aggressive investors. Some firms that offer such quizzes include Merrill Lynch, T. Rowe Price Associates Inc., Baltimore, Zurich Group Inc.’s Scudder Kemper Invest- ments Inc., New York, and Vanguard Group in Malvern, Pa.
Typically, risk questionnaires include seven to 10 ques- tions about a person’s investing experience, financial secu- rity and tendency to make risky or conservative choices.
The benefit of the questionnaires is that they are an objective resource people can use to get at least a rough idea of their risk tolerance. “It’s impossible for someone to assess their risk tolerance alone,” says Mr. Bernstein. “I may say I don’t like risk, yet will take more risk than the average person.”
Many experts warn, however, that the questionnaires should be used simply as a first step to assessing risk toler- ance. “They are not precise,” says Ron Meier, a certified public accountant.
The second step, many experts agree, is to ask yourself some difficult questions, such as: How much you can stand to lose over the long term?
“Most people can stand to lose a heck of a lot tempo- rarily,” says Mr. Schatsky, a financial adviser in New York. The real acid test, he says, is how much of your portfolio’s value you can stand to lose over months or years.
As it turns out, most people rank as middle-of-the-road risk-takers, say several advisers. “Only about 10% to 15% of my clients are aggressive,” says Mr. Roge.
WHAT’S YOUR RISK TOLERANCE? Circle the letter that corresponds to your answer
1. Just 60 days after you put money into an investment, its price falls 20%. Assuming none of the fundamen- tals have changed, what would you do?
a. Sell to avoid further worry and try something else b. Do nothing and wait for the investment to come
back c. Buy more. It was a good investment before; now
it’s a cheap investment, too
2. Now look at the previous question another way. Your investment fell 20%, but it’s part of a portfolio being used to meet investment goals with three different time horizons.
2A. What would you do if the goal were five years away?
a. Sell b. Do nothing c. Buy more
W O
R D
S FR
O M
T H
E S
T R
E E T
174
identical utility, because the portfolios with higher expected return also have higher risk
(standard deviation).
Estimating Risk Aversion How can we estimate the levels of risk aversion of individual investors? A number of meth-
ods may be used. The questionnaire in the nearby box is of the simplest variety and, indeed,
can distinguish only between high (conservative), medium (moderate), or low (aggressive)
levels of the coefficient of risk aversion. More complex questionnaires, allowing subjects
to pinpoint specific levels of risk aversion coefficients, ask would-be investors to choose
from various sets of hypothetical lotteries. Access to investment accounts of active investors would provide observations of how
portfolio composition changes over time. Coupling this information with estimates of the
risk–return combinations of these positions would in principle allow us to calculate inves-
tors’ implied risk aversion coefficients.
Finally, researchers track behavior of groups of individuals to obtain average degrees
of risk aversion. These studies range from observed purchase of insurance policies and
durables warranties to labor supply and aggregate consumption behavior.
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175
2B. What would you do if the goal were 15 years away?
a. Sell b. Do nothing c. Buy more
2C. What would you do if the goal were 30 years away?
a. Sell b. Do nothing c. Buy more
3. The price of your retirement investment jumps 25% a month after you buy it. Again, the fundamentals haven’t changed. After you finish gloating, what do you do?
a. Sell it and lock in your gains b. Stay put and hope for more gain c. Buy more; it could go higher
4. You’re investing for retirement, which is 15 years away. Which would you rather do?
a. Invest in a money-market fund or guaranteed invest- ment contract, giving up the possibility of major gains, but virtually assuring the safety of your principal
b. Invest in a 50-50 mix of bond funds and stock funds, in hopes of getting some growth, but also giving yourself some protection in the form of steady income
c. Invest in aggressive growth mutual funds whose value will probably fluctuate significantly during the year, but have the potential for impressive gains over five or 10 years
5. You just won a big prize! But which one? It’s up to you.
a. $2,000 in cash b. A 50% chance to win $5,000 c. A 20% chance to win $15,000
6. A good investment opportunity just came along. But you have to borrow money to get in. Would you take out a loan?
a. Definitely not b. Perhaps c. Yes
7. Your company is selling stock to its employees. In three years, management plans to take the company public. Until then, you won’t be able to sell your shares and you will get no dividends. But your investment could multiply as much as 10 times when the company goes public. How much money would you invest?
a. None b. Two months’ salary c. Four months’ salary
SCORING YOUR RISK TOLERANCE To score the quiz, add up the number of answers you gave in each category a–c, then multiply as shown to find your score
( a ) answers ______ 3 1 5 ______ points
( b ) answers ______ 3 2 5 ______ points
( c ) answers ______ 3 3 5 ______ points
YOUR SCORE ______ points
If you scored . . . You may be a:
9–14 points Conservative investor
5–21 points Moderate investor
22–27 points Aggressive investor
Source: Reprinted with permission from The Wall Street Journal. © 1998 by Dow Jones & Company. All Rights Reserved Worldwide.
W O
R D
S FR O
M T
H E ST
R E E T
6.2 Capital Allocation across Risky and Risk-Free Portfolios
History shows us that long-term bonds have been riskier investments than Treasury bills
and that stocks have been riskier still. On the other hand, the riskier investments have
offered higher average returns. Investors, of course, do not make all-or-nothing choices
from these investment classes. They can and do construct their portfolios using securities
from all asset classes. Some of the portfolio may be in risk-free Treasury bills, some in
high-risk stocks.
The most straightforward way to control the risk of the portfolio is through the fraction
of the portfolio invested in Treasury bills and other safe money market securities versus
risky assets. A capital allocation decision implies an asset allocation choice among broad
investment classes, rather than among the specific securities within each asset class. Most
investment professionals consider asset allocation the most important part of portfolio
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176 P A R T I I Portfolio Theory and Practice
construction. Consider this statement by John Bogle, made when he was chairman of the
Vanguard Group of Investment Companies:
The most fundamental decision of investing is the allocation of your assets: How much
should you own in stock? How much should you own in bonds? How much should you
own in cash reserves? . . . That decision [has been shown to account] for an astonish-
ing 94% of the differences in total returns achieved by institutionally managed pension
funds. . . . There is no reason to believe that the same relationship does not also hold true
for individual investors. 1
Therefore, we start our discussion of the risk–return trade-off available to investors by
examining the most basic asset allocation choice: the choice of how much of the portfolio
to place in risk-free money market securities versus other risky asset classes.
We denote the investor’s portfolio of risky assets as P and the risk-free asset as F. We assume for the sake of illustration that the risky component of the investor’s overall portfo-
lio comprises two mutual funds, one invested in stocks and the other invested in long-term
bonds. For now, we take the composition of the risky portfolio as given and focus only on
the allocation between it and risk-free securities. In the next chapter, we turn to asset allo-
cation and security selection across risky assets.
When we shift wealth from the risky portfolio to the risk-free asset, we do not change
the relative proportions of the various risky assets within the risky portfolio. Rather, we
reduce the relative weight of the risky portfolio as a whole in favor of risk-free assets.
For example, assume that the total market value of an initial portfolio is $300,000, of
which $90,000 is invested in the Ready Asset money market fund, a risk-free asset for
practical purposes. The remaining $210,000 is invested in risky securities—$113,400 in
equities ( E ) and $96,600 in long-term bonds ( B ). The equities and bond holdings comprise “the” risky portfolio, 54% in E and 46% in B:
E: wE 5 113,400
210,000 5 .54
B: wB 5 96,600
210,000 5 .46
The weight of the risky portfolio, P, in the complete portfolio , including risk-free and risky investments, is denoted by y:
y 5 210,000
300,000 5 .7 (risky assets)
1 2 y 5 90,000
300,000 5 .3 (risk-free assets)
The weights of each asset class in the complete portfolio are as follows:
E: $113,400
$300,000 5 .378
B: $96,600
$300,000 5 .322
Risky portfolio 5 E 1 B 5 .700
The risky portfolio makes up 70% of the complete portfolio.
1 John C. Bogle, Bogle on Mutual Funds (Burr Ridge, IL: Irwin Professional Publishing, 1994), p. 235.
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C H A P T E R 6 Capital Allocation to Risky Assets 177
Rather than thinking of our risky holdings as E and B separately, we may view our hold- ings as if they were in a single fund that holds equities and bonds in fixed proportions. In
this sense we may treat the risky fund as a single risky asset, that asset being a particular
bundle of securities. As we shift in and out of safe assets, we simply alter our holdings of
that bundle of securities commensurately.
With this simplification, we turn to the desirability of reducing risk by changing the
risky/risk-free asset mix, that is, reducing risk by decreasing the proportion y. As long as we do not alter the weights of each security within the risky portfolio, the probability dis-
tribution of the rate of return on the risky portfolio remains unchanged by the asset reallo-
cation. What will change is the probability distribution of the rate of return on the complete portfolio that consists of the risky asset and the risk-free asset.
Suppose that the owner of this portfolio wishes to decrease risk by reducing the alloca- tion to the risky portfolio from y 5 .7 to y 5 .56. The risky portfolio would then total only .56 3 $300,000 5 $168,000, requiring the sale of $42,000 of the original $210,000 of risky holdings, with the proceeds used to purchase more shares in Ready Asset (the money market fund). Total holdings in the risk-free asset will increase to $300,000 3 (1 2 .56) 5 $132,000, the original holdings plus the new contribution to the money market fund:
$90,000 1 $42,000 5 $132,000
The key point, however, is that we leave the proportions of each asset in the risky port- folio unchanged. Because the weights of E and B in the risky portfolio are .54 and .46, respectively, we sell .54 3 $42,000 5 $22,680 of E and .46 3 $42,000 5 $19,320 of B. After the sale, the proportions of each asset in the risky portfolio are in fact unchanged:
E: wE 5 113,400 2 22,680 210,000 2 42,000
5 .54
B: wB 5 96,600 2 19,320 210,000 2 42,000
5 .46
Example 6.2 The Risky Portfolio
What will be the dollar value of your position in equities ( E ), and its proportion in your overall portfolio, if you decide to hold 50% of your investment budget in Ready Asset?
CONCEPT CHECK 6.4
6.3 The Risk-Free Asset
By virtue of its power to tax and control the money supply, only the government can issue
default-free bonds. Even the default-free guarantee by itself is not sufficient to make the
bonds risk-free in real terms. The only risk-free asset in real terms would be a perfectly
price-indexed bond. Moreover, a default-free perfectly indexed bond offers a guaranteed
real rate to an investor only if the maturity of the bond is identical to the investor’s desired
holding period. Even indexed bonds are subject to interest rate risk, because real interest
rates change unpredictably through time. When future real rates are uncertain, so is the
future price of indexed bonds.
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178 P A R T I I Portfolio Theory and Practice
Nevertheless, it is
com mon practice to
view Treasury bills as
“the” risk-free asset . Their short-term nature
makes their values
insensitive to inter-
est rate fluctuations.
Indeed, an investor can
lock in a short-term
nominal return by buy-
ing a bill and holding it
to maturity. Moreover,
inflation uncertainty
over the course of a few
weeks, or even months,
is negligible compared
with the uncertainty of
stock market returns.
In practice, most
investors use a broad
range of money market
instruments as a risk-
free asset. All the money market instruments are virtually free of interest rate risk because
of their short maturities and are fairly safe in terms of default or credit risk.
Most money market funds hold, for the most part, three types of securities—Treasury
bills, bank certificates of deposit (CDs), and commercial paper (CP)—differing slightly
in their default risk. The yields to maturity on CDs and CP for an identical maturity, for
example, are always somewhat higher than those of T-bills. The recent history of this yield
spread for 90-day CDs is shown in Figure 6.3 .
Money market funds have changed their relative holdings of these securities over time
but, by and large, T-bills make up only about 15% of their portfolios. 2 Nevertheless, the risk
of such blue-chip short-term investments as CDs and CP is minuscule compared with that of
most other assets such as long-term corporate bonds, common stocks, or real estate. Hence
we treat money market funds as the most easily accessible risk-free asset for most investors.
Figure 6.3 Spread between 3-month CD and T-bill rates
1 9 7 0
1 9 7 2
1 9 7 4
1 9 7 6
1 9 7 8
1 9 8 0
1 9 8 2
1 9 8 4
1 9 8 6
1 9 8 8
1 9 9 0
1 9 9 2
1 9 9 4
1 9 9 6
1 9 9 8
2 0 0 0
2 0 0 2
2 0 0 4
2 0 0 6
2 0 0 8
2 0 1 0
2 0 1 2
2 0 1 4
OPEC I
OPEC ll
Penn Square
Market Crash
LTCM
Credit Crisis
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
P e rc
e n
ta g
e P
o in
ts
2 See http://www.icifactbook.org/ , Section 4 of Data Tables.
6.4 Portfolios of One Risky Asset and a Risk-Free Asset
In this section we examine the feasible risk–return combinations available to investors
when the choice of the risky portfolio has already been made. This is the “technical” part
of capital allocation. In the next section we address the “personal” part of the problem—
the individual’s choice of the best risk–return combination from the feasible set.
Suppose the investor has already decided on the composition of the risky portfolio, P. Now the concern is with capital allocation, that is, the proportion of the investment budget,
y, to be allocated to P. The remaining proportion, 1 2 y, is to be invested in the risk-free asset, F.
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C H A P T E R 6 Capital Allocation to Risky Assets 179
Denote the risky rate of return of P by r P , its expected rate of return by E ( r P ), and its standard deviation by s P . The rate of return on the risk-free asset is denoted as r f . In the numerical example we assume that E ( r P ) 5 15%, s P 5 22%, and the risk-free rate is r f 5 7%. Thus the risk premium on the risky asset is E ( r P ) 2 r f 5 8%.
With a proportion, y, in the risky portfolio, and 1 2 y in the risk-free asset, the rate of return on the complete portfolio, denoted C, is r C where
rC 5 yrP 1 (1 2 y)rf (6.2)
Taking the expectation of this portfolio’s rate of return,
E(rC) 5 yE(rP) 1 (1 2 y)rf 5 rf 1 y3E(rP) 2 rf4 5 7 1 y(15 2 7) (6.3)
This result is easily interpreted. The base rate of return for any portfolio is the risk-free
rate. In addition, the portfolio is expected to earn a proportion, y, of the risk premium of the risky portfolio, E ( r P ) 2 r f . Investors are assumed risk averse and unwilling to take a risky position without a positive risk premium.
With a proportion y in a risky asset, the standard deviation of the complete portfolio is the standard deviation of the risky asset multiplied by the weight, y, of the risky asset in that portfolio.
3 Because the standard deviation of the risky portfolio is s P 5 22%,
sC 5 ysP 5 22y (6.4)
which makes sense because the standard deviation of the portfolio is proportional to both
the standard deviation of the risky asset and the proportion invested in it. In sum, the
expected return of the complete portfolio is E ( r C ) 5 r f 1 y [ E ( r P ) 2 r f ] 5 7 1 8 y and the standard deviation is s C 5 22 y.
The next step is to plot the portfolio characteristics (with various choices for y ) in the expected return–standard deviation plane in Figure 6.4 . The risk-free asset, F, appears on the vertical axis because its standard
deviation is zero. The risky asset, P, is plotted with a standard deviation of 22%,
and expected return of 15%. If an investor
chooses to invest solely in the risky asset,
then y 5 1.0, and the complete portfolio is P. If the chosen position is y 5 0, then 1 2 y 5 1.0, and the complete portfolio is the risk-free portfolio F.
What about the more interesting mid-
range portfolios where y lies between 0 and 1? These portfolios will graph on the
straight line connecting points F and P. The slope of that line is [ E ( r P ) 2 r f ]/ s P (rise/run), in this case, 8/22.
The conclusion is straightforward.
Increasing the fraction of the overall port-
folio invested in the risky asset increases
expected return at a rate of 8%, according
to Equation 6.3. It also increases portfolio
3 This is an application of a basic rule from statistics: If you multiply a random variable by a constant, the standard
deviation is multiplied by the same constant. In our application, the random variable is the rate of return on the
risky asset, and the constant is the fraction of that asset in the complete portfolio. We will elaborate on the rules
for portfolio return and risk in the following chapter.
Figure 6.4 The investment opportunity set with a risky asset and a risk-free asset in the expected return–standard deviation plane
E(r)
σP = 22%
E(rP) = 15%
σ
rƒ = 7% F
P
E(rP) − rƒ = 8%
CAL = Capital Allocation Line
S = 8/22
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180 P A R T I I Portfolio Theory and Practice
standard deviation at the rate of 22%, according to Equation 6.4. The extra return per extra risk
is thus 8/22 5 .36.
To derive the exact equation for the straight line between F and P, we rearrange Equation 6.4 to find that y 5 s C / s P , and we substitute for y in Equation 6.3 to describe the expected return–standard deviation trade-off:
E(rC) 5 rf 1 y3E(rP) 2 rf4 5 rf 1
sC
sP 3E(rP) 2 rf4 5 7 1 8
22 sC
(6.5)
Thus the expected return of the complete portfolio as a function of its standard devia-
tion is a straight line, with intercept r f and slope
S 5 E(rP) 2 rf
sP 5
8
22 (6.6)
Figure 6.4 graphs the investment opportunity set, which is the set of feasible expected return and standard deviation pairs of all portfolios resulting from different values of y. The graph is a straight line originating at r f and going through the point labeled P.
This straight line is called the capital allocation line (CAL). It depicts all the risk–return combinations available to investors. The slope of the CAL, denoted S, equals the increase in the expected return of the complete portfolio per unit of additional standard deviation—
in other words, incremental return per incremental risk. For this reason, the slope is called
the reward-to-volatility ratio . It also is called the Sharpe ratio (see Chapter 5). A portfolio equally divided between the risky asset and the risk-free asset, that is, where
y 5 .5, will have an expected rate of return of E ( r C ) 5 7 1 .5 3 8 5 11%, implying a risk premium of 4%, and a standard deviation of s C 5 .5 3 22 5 11%. It will plot on the line
FP midway between F and P. The reward-to-volatility ratio is S 5 4/11 5 .36, precisely the same as that of portfolio P.
What about points on the CAL to the right of portfolio P ? If investors can borrow at the (risk-free) rate of r f 5 7%, they can construct portfolios that may be plotted on the CAL to the right of P.
Can the reward-to-volatility (Sharpe) ratio, S 5 [ E ( r C ) 2 r f ]/ s C , of any combination of the risky asset and the risk-free asset be different from the ratio for the risky asset taken alone, [ E ( r P ) 2 r f ]/ s P , which in this case is .36?
CONCEPT CHECK 6.5
Suppose the investment budget is $300,000 and our investor borrows an additional $120,000, investing the total available funds in the risky asset. This is a levered position in the risky asset, financed in part by borrowing. In that case
y 5 420,000 300,000
5 1.4
and 1 2 y 5 1 2 1.4 5 2 .4, reflecting a short (borrowing) position in the risk-free asset. Rather than lending at a 7% interest rate, the investor borrows at 7%. The distribution of the portfolio rate of return still exhibits the same reward-to-volatility ratio:
E(rC) 5 7% 1 (1.4 3 8%) 5 18.2%
Example 6.3 Leverage
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C H A P T E R 6 Capital Allocation to Risky Assets 181
sC 5 1.4 3 22% 5 30.8%
S 5 E(rC ) 2 rf
sC 5
18.2 2 7 30.8
5 .36
As one might expect, the levered portfolio has a higher standard deviation than does an unlevered position in the risky asset.
Clearly, nongovernment investors can-
not borrow at the risk-free rate. The risk
of a borrower’s default induces lend-
ers to demand higher interest rates on
loans. Therefore, the nongovernment
investor’s borrowing cost will exceed
the lending rate of r f 5 7%. Suppose the borrowing rate is r f
B 5 9%. Then
in the borrowing range, the reward-to-
volatility ratio, the slope of the CAL, will
be 3E(rP) 2 rfB4/sP 5 6/22 5 .27. The CAL will therefore be “kinked” at point
P, as shown in Figure 6.5 . To the left of P the investor is lending at 7%, and the slope of the CAL is .36. To the right of
P, where y . 1, the investor is borrowing at 9% to finance extra investments in the
risky asset, and the slope is .27.
In practice, borrowing to invest in the
risky portfolio is easy and straightfor-
ward if you have a margin account with
a broker. All you have to do is tell your broker that you want to buy “on margin.” Margin
purchases may not exceed 50% of the purchase value. Therefore, if your net worth in
the account is $300,000, the broker is allowed to lend you up to $300,000 to purchase
additional stock. 4 You would then have $600,000 on the asset side of your account and
$300,000 on the liability side, resulting in y 5 2.0.
Figure 6.5 The opportunity set with differential borrowing and lending rates
E(r)
E(rP) = 15%
σ
rƒ = 7%
P
σP = 22%
CAL
S(y ≤ 1) = .36 rƒ
B = 9%
S(y > 1) = .27
4 Margin purchases require the investor to maintain the securities in a margin account with the broker. If the value
of the securities falls below a “maintenance margin,” a “margin call” is sent out, requiring a deposit to bring the
net worth of the account up to the appropriate level. If the margin call is not met, regulations mandate that some or
all of the securities be sold by the broker and the proceeds used to reestablish the required margin. See Chapter 3,
Section 3.6, for further discussion.
Suppose that there is an upward shift in the expected rate of return on the risky asset, from 15% to 17%. If all other parameters remain unchanged, what will be the slope of the CAL for y ≤ 1 and y . 1?
CONCEPT CHECK 6.6
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182 P A R T I I Portfolio Theory and Practice
6.5 Risk Tolerance and Asset Allocation
We have shown how to develop the CAL, the graph of all feasible risk–return combina-
tions available for capital allocation. The investor confronting the CAL now must choose
one optimal portfolio, C, from the set of feasible choices. This choice entails a trade-off between risk and return. Individual differences in risk aversion lead to different capital
allocation choices even when facing an identical opportunity set (that is, a risk-free rate
and a reward-to-volatility ratio). In particular, more risk-averse investors will choose to
hold less of the risky asset and more of the risk-free asset.
The expected return on the complete portfolio is given by Equation 6.3:
E ( r C ) 5 r f 1 y [ E ( r P ) 2 r f ]. Its variance is, from Equation 6.4, sC 2 5 y2sP
2 . Investors
choose the allocation to the risky asset, y, that maximizes their utility function as given by Equation 6.1: U 5 E ( r ) 2 ½ A s 2 . As the allocation to the risky asset increases (higher y ), expected return increases, but so does volatility, so utility can increase or decrease. Table 6.4 shows utility levels corresponding to different values of y. Initially, utility increases as y increases, but eventually it declines.
Figure 6.6 is a plot of the utility function from Table 6.4 . The graph shows that utility
is highest at y 5 .41. When y is less than .41, investors are willing to assume more risk to increase expected return. But at higher levels of y, risk is higher, and additional allocations to the risky asset are undesirable—beyond this point, further increases in risk dominate the
increase in expected return and reduce utility.
To solve the utility maximization problem more generally, we write the problem as follows:
Max y
U 5 E(rC) 2 ½ AsC 2 5 rf 1 y3E(rP) 2 rf4 2 ½ Ay2sP2
Students of calculus will recognize that the maximization problem is solved by setting
the derivative of this expression to zero. Doing so and solving for y yields the optimal posi- tion for risk-averse investors in the risky asset, y *, as follows: 5
y* 5 E(rP) 2 rf
AsP 2
(6.7)
(1) y
(2) E ( r C )
(3) s C
(4) U 5 E(r) 2 ½ A s
2
0 .070 0 .0700 0.1 .078 .022 .0770 0.2 .086 .044 .0821 0.3 .094 .066 .0853 0.4 .102 .088 .0865 0.5 .110 .110 .0858 0.6 .118 .132 .0832 0.7 .126 .154 .0786 0.8 .134 .176 .0720 0.9 .142 .198 .0636 1.0 .150 .220 .0532
Table 6.4
Utility levels for various positions in risky assets ( y ) for an investor with risk aversion A 5 4
5 The derivative with respect to y equals E(rP) 2 rf 2 yAsP
2 . Setting this expression equal to zero and solving for
y yields Equation 6.7.
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C H A P T E R 6 Capital Allocation to Risky Assets 183
This solution shows that the optimal position in the risky asset is inversely proportional to the level of risk aversion and the level of risk (as measured by the variance) and directly
proportional to the risk premium offered by the risky asset.
Figure 6.6 Utility as a function of allocation to the risky asset, y
U ti
li ty
0 0
0.2 0.4 0.6 Allocation to Risky Asset, y
0.8 1 1.2
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
A graphical way of presenting this decision problem is to use indifference curve
analysis. To illustrate how to build an indifference curve, consider an investor with
risk aversion A 5 4 who currently holds all her wealth in a risk-free portfolio yielding
Using our numerical example [ r f 5 7%, E ( r P ) 5 15%, and s P 5 22%], and expressing all returns as decimals, the optimal solution for an investor with a coefficient of risk aver- sion A 5 4 is
y* 5 .15 2 .07
4 3 .222 5 .41
In other words, this particular investor will invest 41% of the investment budget in the risky asset and 59% in the risk-free asset. As we saw in Figure 6.6 , this is the value of y at which utility is maximized.
With 41% invested in the risky portfolio, the expected return and standard deviation of the complete portfolio are
E(rC) 5 7 1 3.41 3 (15 2 7) 4 5 10.28% sC 5 .41 3 22 5 9.02%
The risk premium of the complete portfolio is E ( r C ) 2 r f 5 3.28%, which is obtained by taking on a portfolio with a standard deviation of 9.02%. Notice that 3.28/9.02 = .36, which is the reward-to-volatility (Sharpe) ratio of any complete portfolio given the parameters of this example.
Example 6.4 Capital Allocation
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184 P A R T I I Portfolio Theory and Practice
r f 5 5%. Because the variance of such a portfolio is zero, Equation 6.1 tells us that its utility value is U 5 .05. Now we find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio, say, with s 5 1%. We use Equation 6.1 to find how much E ( r ) must increase to compensate for the higher value of s :
U 5 E(r) 2 ½ 3 A 3 s2
.05 5 E(r) 2 ½ 3 4 3 .012
This implies that the necessary expected return increases to
Required E(r) 5 .05 1 ½ 3 A 3 s2
5 .05 1 ½ 3 4 3 .012 5 .0502 (6.8)
We can repeat this calculation for other levels of s , each time finding the value of E ( r ) necessary to maintain U 5 .05. This process will yield all combinations of expected return and volatility with utility level of .05; plotting these combinations gives us the indifference
curve.
We can readily generate an investor’s indifference curves using a spreadsheet. Table 6.5
contains risk–return combinations with utility values of .05 and .09 for two investors, one
with A 5 2 and the other with A 5 4. The plot of these indifference curves appears in Figure 6.7 . Notice that the intercepts of the indifference curves are at .05 and .09, exactly
the level of utility corresponding to the two curves.
Any investor would prefer a portfolio on the higher indifference curve with a higher cer-
tainty equivalent (utility). Portfolios on higher indifference curves offer a higher expected
return for any given level of risk. For example, both indifference curves for A 5 2 have the same shape, but for any level of volatility, a portfolio on the curve with utility of .09
offers an expected return 4% greater than the corresponding portfolio on the lower curve,
for which U 5 .05. Figure 6.7 demonstrates that more risk-averse investors have steeper indifference
curves than less risk-averse investors. Steeper curves mean that investors require a greater
increase in expected return to compensate for an increase in portfolio risk.
A 5 2 A 5 4
s U 5 .05 U 5 .09 U 5 .05 U 5 .09
0 .0500 .0900 .050 .090 .05 .0525 .0925 .055 .095 .10 .0600 .1000 .070 .110 .15 .0725 .1125 .095 .135 .20 .0900 .1300 .130 .170 .25 .1125 .1525 .175 .215 .30 .1400 .1800 .230 .270 .35 .1725 .2125 .295 .335 .40 .2100 .2500 .370 .410 .45 .2525 .2925 .455 .495 .50 .3000 .3400 .550 .590
Table 6.5
Spreadsheet calculations of indifference curves (Entries in columns 2–4 are expected returns necessary to provide specified utility value.)
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C H A P T E R 6 Capital Allocation to Risky Assets 185
Higher indifference curves cor-
respond to higher levels of utility.
The investor thus attempts to find
the complete portfolio on the high-
est possible indifference curve.
When we superimpose plots of
indifference curves on the invest-
ment opportunity set represented
by the capital allocation line as
in Figure 6.8 , we can identify the
highest possible indifference curve that still touches the CAL. That
indifference curve is tangent to the
CAL, and the tangency point corre-
sponds to the standard deviation and
expected return of the optimal com-
plete portfolio.
To illustrate, Table 6.6 provides
calculations for four indifference
curves (with utility levels of .07,
.078, .08653, and .094) for an inves-
tor with A 5 4. Columns (2)–(5) use Equation 6.8 to calculate the
expected return that must be paired
Figure 6.7 Indifference curves for U 5 .05 and U 5 .09 with A 5 2 and A 5 4
E(r)
0
U = .09
A = 4
A = 4
A = 2
A = 2
U = .05
.10 .20 .30 .40 .50 σ
.60
.40
.20
σc = .0902 σP = .22
σ
E(r)
E(rP) = .15
E(rc) = .1028
rf = .07
C
P
CAL
0
U = .094 U = .08653 U = .078 U = .07
Figure 6.8 Finding the optimal complete portfolio by using indifference curves
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186 P A R T I I Portfolio Theory and Practice
with the standard deviation in column (1) to provide the utility value corresponding to each
curve. Column (6) uses Equation 6.5 to calculate E ( r C ) on the CAL for the standard devia- tion s C in column (1):
E(rC) 5 rf 1 3E(rP) 2 rf4 sC sP
5 7 1 315 2 74 sC 22
Figure 6.8 graphs the four indifference curves and the CAL. The graph reveals that the
indifference curve with U 5 .08653 is tangent to the CAL; the tangency point corresponds to the complete portfolio that maximizes utility. The tangency point occurs at s C 5 9.02%
and E ( r C ) 5 10.28%, the risk–return parameters of the optimal complete portfolio with y * 5 0.41. These values match our algebraic solution using Equation 6.7.
We conclude that the choice for y *, the fraction of overall investment funds to place in the risky portfolio, is determined by risk aversion (the slope of indifference curves) and the
Sharpe ratio (the slope of the opportunity set).
In sum, capital allocation determines the complete portfolio, which constitutes the
investor’s entire wealth. Portfolio P represents all-wealth-at-risk. Hence, when returns are normally distributed, standard deviation is the appropriate measure of risk. In future chap-
ters we will consider augmenting P with “good” additions, meaning assets that improve the feasible risk-return trade-off. The risk of these potential additions will have to be mea-
sured by their incremental effect on the standard deviation of P.
Nonnormal Returns In the foregoing analysis we assumed normality of returns by taking the standard devia-
tion as the appropriate measure of risk. But as we discussed in Chapter 5, departures from
normality could result in extreme losses with far greater likelihood than would be plausible
under a normal distribution. These exposures, which are typically measured by value at
risk (VaR) or expected shortfall (ES), also would be important to investors.
Therefore, an appropriate extension of our analysis would be to present investors with
forecasts of VaR and ES. Taking the capital allocation from the normal-based analysis
as a benchmark, investors facing fat-tailed distributions might consider reducing their
allocation to the risky portfolio in favor of an increase in the allocation to the risk-free
vehicle.
s U 5 .07 U 5 .078 U 5 .08653 U 5 .094 CAL
0 .0700 .0780 .0865 .0940 .0700 .02 .0708 .0788 .0873 .0948 .0773 .04 .0732 .0812 .0897 .0972 .0845 .06 .0772 .0852 .0937 .1012 .0918 .08 .0828 .0908 .0993 .1068 .0991 .0902 .0863 .0943 .1028 .1103 .1028 .10 .0900 .0980 .1065 .1140 .1064 .12 .0988 .1068 .1153 .1228 .1136 .14 .1092 .1172 .1257 .1332 .1209 .18 .1348 .1428 .1513 .1588 .1355 .22 .1668 .1748 .1833 .1908 .1500 .26 .2052 .2132 .2217 .2292 .1645 .30 .2500 .2580 .2665 .2740 .1791
Table 6.6
Expected returns on four indifference curves and the CAL. Investor’s risk aver- sion is A 5 4.
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C H A P T E R 6 Capital Allocation to Risky Assets 187
There are signs of advances in dealing with extreme values (in addition to new tech-
niques to handle transaction data mentioned in Chapter 5). Back in the early 20th century,
Frank Knight, one of the great economists of the time, distinguished risk from uncertainty, the difference being that risk is a known problem in which probabilities can be ascertained
while uncertainty is characterized by ignorance even about probabilities (reminiscent of
the black swan problem). Hence, Knight argued, we must use different methods to handle
uncertainty and risk.
Probabilities of moderate outcomes in finance can be readily assessed from experience
because of the high relative frequency of such observations. Extreme negative values are
blissfully rare, but for that very reason, accurately assessing their probabilities is virtually
impossible. However, the Bayesian statistics that took center stage in decision making in
later periods rejected Knight’s approach on the argument that even if probabilities are hard
to estimate objectively, investors nevertheless have a notion, albeit subjective, of what they
may be and must use those beliefs to make economic decisions. In the Bayesian frame-
work, these so-called priors must be used even if they apply to unprecedented events that
characterize uncertainty. Accordingly, in this school of thought, the distinction between
risk and uncertainty is deemed irrelevant.
Economists today are coming around to Knight’s position. Advanced utility functions
attempt to distinguish risk from uncertainty and give these uncertain outcomes a larger role
in the choice of portfolios. These approaches have yet to enter everyday practice, but as
they are developed, practical measures are certain to follow.
6.6 Passive Strategies: The Capital Market Line
a. If an investor’s coefficient of risk aversion is A 5 3, how does the optimal asset mix change? What are the new values of E ( r C ) and s C ?
b. Suppose that the borrowing rate, r f B 5 9% is greater than the lending rate, r f 5 7%. Show graphically
how the optimal portfolio choice of some investors will be affected by the higher borrowing rate. Which investors will not be affected by the borrowing rate?
CONCEPT CHECK 6.7
The CAL is derived with the risk-free and “the” risky portfolio, P. Determination of the assets to include in P may result from a passive or an active strategy. A passive strategy describes a portfolio decision that avoids any direct or indirect security analysis. 6 At first blush, a passive strategy would appear to be naive. As will become apparent, however,
forces of supply and demand in large capital markets may make such a strategy the reason-
able choice for many investors.
In Chapter 5, we presented a compilation of the history of rates of return on differ-
ent portfolios. The data are available at Professor Kenneth French’s Web site, mba.tuck .dartmouth.edu/pages/faculty/ken.french/data_library.html . We can use these data to examine various passive strategies.
A natural candidate for a passively held risky asset would be a well-diversified portfolio
of common stocks such as “All U.S.” described in Chapter 5. Because a passive strategy
6 By “indirect security analysis” we mean the delegation of that responsibility to an intermediary such as a
professional money manager.
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188 P A R T I I Portfolio Theory and Practice
requires that we devote no resources to acquiring information on any individual stock or
group of stocks, we must follow a “neutral” diversification strategy. One way is to select
a diversified portfolio of stocks that mirrors the value of the corporate sector of the U.S.
economy. This results in a portfolio in which, for example, the proportion invested in
Microsoft stock will be the ratio of Microsoft’s total market value to the market value of
all listed stocks.
The most popular value-weighted index of U.S. stocks is the Standard & Poor’s
Composite Index of 500 large capitalization U.S. corporations (the S&P 500). Table 6.7
summarizes the performance of the S&P 500 portfolio over the 87-year period 1926–2012,
as well as for four subperiods. Table 6.7 shows the average return for the portfolio, the return
on rolling over 1-month T-bills for the same period, as well as the resultant average excess
return and its standard deviation. The Sharpe ratio was .40 for the overall period, 1926–2012.
In other words, stock market investors enjoyed a .40% average excess return over the T-bill
rate for every 1% of standard deviation. The large standard deviation of the excess return
(20.48%) is one reason we observe a wide range of average excess returns and Sharpe ratios
across subperiods (varying from .21 to .74). Using the statistical distribution of the difference
between the Sharpe ratios of two portfolios, we can estimate the probability of observing a
deviation of the Sharpe ratio for a particular subperiod from that of the overall period, assum-
ing the latter is the true value. The last column of Table 6.7 shows that the probabilities of
finding such widely different Sharpe ratios over the subperiods are actually quite substantial.
We call the capital allocation line provided by 1-month T-bills and a broad index of
common stocks the capital market line (CML). A passive strategy generates an invest- ment opportunity set that is represented by the CML.
How reasonable is it for an investor to pursue a passive strategy? We cannot answer
such a question without comparing the strategy to the costs and benefits accruing to an
active portfolio strategy. Some thoughts are relevant at this point, however.
First, the alternative active strategy is not free. Whether you choose to invest the time
and cost to acquire the information needed to generate an optimal active portfolio of risky
assets, or whether you delegate the task to a professional who will charge a fee, constitution
of an active portfolio is more expensive than a passive one. The passive portfolio requires
negligible cost to purchase T-bills and management fees to either an exchange-traded fund
Average Annual Returns S&P 500 Portfolio
Period S&P 500 Portfolio
1-Month T-Bills
Risk Premium
Standard Deviation
Sharpe Ratio (Reward-to- Volatility) Probability *
1926–2012 11.67 3.58 8.10 20.48 0.40 — 1989–2012 11.10 3.52 7.59 18.22 0.42 0.94 1968–1988 10.91 7.48 3.44 16.71 0.21 0.50 1947–1967 15.35 2.28 13.08 17.66 0.74 0.24 1926–1946 9.40 1.04 8.36 27.95 0.30 0.71
Table 6.7
Average annual return on large stocks and 1-month T-bills; standard deviation and Sharpe ratio of large stocks over time
*The probability that the estimate of the Sharpe ratio over 1926–2012 equals the true value and that we observe the reported, or an even more different Sharpe ratio for the subperiod.
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Investors Sour on Pro Stock Pickers
Investors are jumping out of mutual funds managed by professional stock pickers and shifting massive amounts of money into lower-cost funds that echo the broader market.
Through November 2012, investors pulled $119.3 billion from so-called actively managed U.S. stock funds accord- ing to the latest data from research firm Morningstar Inc. At the same time, they poured $30.4 billion into U.S. stock exchange-traded funds.
The move reflects the fact that many money manag- ers of stock funds, which charge fees but also dangle the prospect of higher returns, have underperformed the benchmark stock indexes. As a result, more investors are choosing simply to invest in funds tracking the indexes, which carry lower fees and are perceived as having less risk.
The mission of stock pickers in a managed mutual fund is to outperform the overall market by actively trading individual stocks or bonds, with fund managers receiving higher fees for their effort. In an ETF (or indexed mutual fund), managers balance the share makeup of the fund so it accurately reflects the performance of its underlying index, charging lower fees.
Morningstar says that when investors have put money in stock funds, they have chosen low-cost index funds and ETFs. Some index ETFs cost less than 0.1% of assets a year, while many actively managed stock funds charge 1% a year or more.
While the trend has put increasing pressure lately on stock pickers, it is shifting the fortunes of some of the big- gest players in the $14 trillion mutual-fund industry.
Fidelity Investments and American Funds, among the largest in the category, saw redemptions or weak investor interest compared with competitors, according to an anal- ysis of mutual-fund flows done for The Wall Street Journal by research firm Strategic Insight, a unit of New York-based Asset International.
At the other end of the spectrum, Vanguard, the world’s largest provider of index mutual funds, pulled in a net $141 billion last year through December, according to the company.
Many investors say they are looking for a way to invest cheaply, with less risk.
Source: Adapted from Kirsten Grind, The Wall Street Journal, January 3, 2013. Reprinted with permission. © 2013 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
W O
R D
S FR O
M T
H E ST
R E E T
189
or a mutual fund company that operates a market index fund. Vanguard, for example, oper-
ates the Index 500 Portfolio that mimics the S&P 500 index fund. It purchases shares of the
firms constituting the S&P 500 in proportion to the market values of the outstanding equity
of each firm, and therefore essentially replicates the S&P 500 index. The fund thus dupli-
cates the performance of this market index. It has one of the lowest operating expenses
(as a percentage of assets) of all mutual stock funds precisely because it requires minimal
managerial effort.
A second reason to pursue a passive strategy is the free-rider benefit. If there are many
active, knowledgeable investors who quickly bid up prices of undervalued assets and force
down prices of overvalued assets (by selling), we have to conclude that at any time most
assets will be fairly priced. Therefore, a well-diversified portfolio of common stock will
be a reasonably fair buy, and the passive strategy may not be inferior to that of the average
active investor. (We will elaborate on this argument and provide a more comprehensive
analysis of the relative success of passive strategies in later chapters.) The nearby box
points out that passive index funds have actually outperformed most actively managed
funds in the past decades and that investors are responding to the lower costs and better
performance of index funds by directing their investments into these products.
To summarize, a passive strategy involves investment in two passive portfolios: virtu-
ally risk-free short-term T-bills (or, alternatively, a money market fund) and a fund of com-
mon stocks that mimics a broad market index. The capital allocation line representing such
a strategy is called the capital market line. Historically, based on 1926 to 2012 data, the passive risky portfolio offered an average risk premium of 8.1% and a standard deviation
of 20.48%, resulting in a reward-to-volatility ratio of .40.
Passive investors allocate their investment budgets among instruments according to
their degree of risk aversion. We can use our analysis to deduce a typical investor’s risk-
aversion parameter. From Table 1.1 in Chapter 1, we estimate that approximately 65.6%
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190 P A R T I I Portfolio Theory and Practice
of net worth is invested in a broad array of risky assets. 7 We assume this portfolio has
the same reward–risk characteristics that the S&P 500 has exhibited since 1926, as docu-
mented in Table 6.7. Substituting these values in Equation 6.7, we obtain
y* 5 E(rM) 2 rf
AsM 2
5 .081
A 3 .20482 5 .656
which implies a coefficient of risk aversion of
A 5 .081
.656 3 .20482 5 2.94
Of course, this calculation is highly speculative. We have assumed that the average
investor holds the naive view that historical average rates of return and standard devia-
tions are the best estimates of expected rates of return and risk, looking to the future.
To the extent that the average investor takes advantage of contemporary information
in addition to simple historical data, our estimate of A 5 2.94 would be an unjustified inference. Nevertheless, a broad range of studies, taking into account the full range of
available assets, places the degree of risk aversion for the representative investor in the
range of 2.0 to 4.0. 8
7 We include in the risky portfolio real assets, half of pension reserves, corporate and noncorporate equity, and
half of mutual fund shares. This portfolio sums to $50.05 trillion, which is 65.6% of household net worth. (See
Table 1.1.)
8 See, for example, I. Friend and M. Blume, “The Demand for Risky Assets,” American Economic Review 64
(1974); or S. J. Grossman and R. J. Shiller, “The Determinants of the Variability of Stock Market Prices,”
American Economic Review 71 (1981).
Suppose that expectations about the S&P 500 index and the T-bill rate are the same as they were in 2012, but you find that a greater proportion is invested in T-bills today than in 2012. What can you conclude about the change in risk tolerance over the years since 2012?
CONCEPT CHECK 6.8
SUMMARY 1. Speculation is the undertaking of a risky investment for its risk premium. The risk premium has to be large enough to compensate a risk-averse investor for the risk of the investment.
2. A fair game is a risky prospect that has a zero risk premium. It will not be undertaken by a risk- averse investor.
3. Investors’ preferences toward the expected return and volatility of a portfolio may be expressed by a utility function that is higher for higher expected returns and lower for higher portfolio
variances. More risk-averse investors will apply greater penalties for risk. We can describe these
preferences graphically using indifference curves.
4. The desirability of a risky portfolio to a risk-averse investor may be summarized by the cer- tainty equivalent value of the portfolio. The certainty equivalent rate of return is a value that, if
it is received with certainty, would yield the same utility as the risky portfolio.
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C H A P T E R 6 Capital Allocation to Risky Assets 191
5. Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Other methods involve diversification of the risky portfolio and hedging. We take up these
methods in later chapters.
6. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of other assets such as
long-term bonds and common stocks, so for the purpose of our analysis we consider T-bills as
the risk-free asset. Money market funds hold, in addition to T-bills, short-term relatively safe
obligations such as CP and CDs. These entail some default risk, but again, the additional risk is
small relative to most other risky assets. For convenience, we often refer to money market funds
as risk-free assets.
7. An investor’s risky portfolio (the risky asset) can be characterized by its reward-to- volatility ratio, S 5 [ E ( r P ) 2 r f ]/ s P . This ratio is also the slope of the CAL, the line that, when graphed, goes from the risk-free asset through the risky asset. All combinations of the risky
asset and the risk-free asset lie on this line. Other things equal, an investor would prefer a
steeper-sloping CAL, because that means higher expected return for any level of risk. If the
borrowing rate is greater than the lending rate, the CAL will be “kinked” at the point of the
risky asset.
8. The investor’s degree of risk aversion is characterized by the slope of his or her indifference curve. Indifference curves show, at any level of expected return and risk, the required risk pre-
mium for taking on one additional percentage point of standard deviation. More risk-averse
investors have steeper indifference curves; that is, they require a greater risk premium for taking
on more risk.
9. The optimal position , y *, in the risky asset, is proportional to the risk premium and inversely proportional to the variance and degree of risk aversion:
y* 5 E(rP) 2 rf
AsP 2
Graphically, this portfolio represents the point at which the indifference curve is tangent to the CAL.
10. A passive investment strategy disregards security analysis, targeting instead the risk-free asset and a broad portfolio of risky assets such as the S&P 500 stock portfolio. If in 2012 inves-
tors took the mean historical return and standard deviation of the S&P 500 as proxies for its
expected return and standard deviation, then the values of outstanding assets would imply a
degree of risk aversion of about A 5 2.94 for the average investor. This is in line with other stud- ies, which estimate typical risk aversion in the range of 2.0 through 4.0.
Related Web sites for this chapter are available at www. mhhe.com/bkm
risk premium
fair game
risk averse
utility
certainty equivalent rate
risk neutral
risk lover
mean-variance (M-V) criterion
indifference curve
complete portfolio
risk-free asset
capital allocation line
reward-to-volatility ratio
passive strategy
capital market line
KEY TERMS
KEY EQUATIONS Utility score: U = E(r) – ½ As2
Optimal allocation to risky portfolio: y* 5 E(rP) 2 rf
AsP 2
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192 P A R T I I Portfolio Theory and Practice
1. Which of the following choices best completes the following statement? Explain. An investor
with a higher degree of risk aversion, compared to one with a lower degree, will prefer invest-
ment portfolios
a. with higher risk premiums. b. that are riskier (with higher standard deviations). c. with lower Sharpe ratios. d. with higher Sharpe ratios. e. None of the above is true.
2. Which of the following statements are true? Explain.
a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio. b. The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios. c. With a fixed risk-free rate, doubling the expected return and standard deviation of the risky
portfolio will double the Sharpe ratio.
d. Holding constant the risk premium of the risky portfolio, a higher risk-free rate will increase the Sharpe ratio of investments with a positive allocation to the risky asset.
3. What do you think would happen to the expected return on stocks if investors perceived higher
volatility in the equity market? Relate your answer to Equation 6.7.
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either
$70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in
T-bills pays 6% per year.
a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
b. Suppose that the portfolio can be purchased for the amount you found in ( a ). What will be the expected rate of return on the portfolio?
c. Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay?
d. Comparing your answers to ( a ) and ( c ), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio
will sell?
5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of
18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for
which the risky portfolio is still preferred to bills?
6. Draw the indifference curve in the expected return–standard deviation plane corresponding
to a utility level of .05 for an investor with a risk aversion coefficient of 3. ( Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return
providing a utility level of .05. Then plot the expected return–standard deviation points so
derived.)
7. Now draw the indifference curve corresponding to a utility level of .05 for an investor
with risk aversion coefficient A 5 4. Comparing your answer to Problem 6, what do you conclude?
8. Draw an indifference curve for a risk-neutral investor providing utility level .05.
9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover.
For Problems 10 through 12: Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors’ expec- tations for future performance and that the current T-bill rate is 5%.
Basic
Intermediate
PROBLEM SETS
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C H A P T E R 6 Capital Allocation to Risky Assets 193
10. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500
index with weights as follows:
W bills W index
0 1.0 0.2 0.8 0.4 0.6 0.6 0.4 0.8 0.2 1.0 0
11. Calculate the utility levels of each portfolio of Problem 10 for an investor with A 5 2. What do you conclude?
12. Repeat Problem 11 for an investor with A 5 3. What do you conclude?
Use these inputs for Problems 13 through 19: You manage a risky portfolio with expected rate of return of 18% and standard deviation of 28%. The T-bill rate is 8%.
13. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market
fund. What is the expected value and standard deviation of the rate of return on his portfolio?
14. Suppose that your risky portfolio includes the following investments in the given proportions:
Stock A 25% Stock B 32% Stock C 43%
What are the investment proportions of your client’s overall portfolio, including the position in
T-bills?
15. What is the reward-to-volatility ratio ( S ) of your risky portfolio? Your client’s?
16. Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the
slope of the CAL? Show the position of your client on your fund’s CAL.
17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%.
a. What is the proportion y ? b. What are your client’s investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client’s portfolio?
18. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s
standard deviation will not exceed 18%.
a. What is the investment proportion, y ? b. What is the expected rate of return on the complete portfolio?
19. Your client’s degree of risk aversion is A 5 3.5.
a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return on your client’s opti-
mized portfolio?
20. Look at the data in Table 6.7 on the average risk premium of the S&P 500 over T-bills, and the
standard deviation of that risk premium. Suppose that the S&P 500 is your risky portfolio.
a. If your risk-aversion coefficient is A 5 4 and you believe that the entire 1926–2012 period is representative of future expected performance, what fraction of your portfolio should be
allocated to T-bills and what fraction to equity?
b. What if you believe that the 1968–1988 period is representative? c. What do you conclude upon comparing your answers to (a) and (b)?
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194 P A R T I I Portfolio Theory and Practice
21. Consider the following information about a risky portfolio that you manage, and a risk-free
asset: E ( r P ) 5 11%, s P 5 15%, r f 5 5%.
a. Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What
proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset?
b. What will be the standard deviation of the rate of return on her portfolio? c. Another client wants the highest return possible subject to the constraint that you limit his
standard deviation to be no more than 12%. Which client is more risk averse?
22. Investment Management Inc. (IMI) uses the capital market line to make asset allocation recom-
mendations. IMI derives the following forecasts:
• Expected return on the market portfolio: 12%.
• Standard deviation on the market portfolio: 20%.
• Risk-free rate: 5%.
Samuel Johnson seeks IMI’s advice for a portfolio asset allocation. Johnson informs IMI that
he wants the standard deviation of the portfolio to equal half of the standard deviation for the
market portfolio. Using the capital market line, what expected return can IMI provide subject to
Johnson’s risk constraint?
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the S&P 500 index has an expected return of 13% and standard devia- tion of 25%, that r f 5 5%, and that your fund has the parameters given in Problem 21.
23. Draw a diagram of your client’s CML, accounting for the higher borrowing rate. Superimpose
on it two sets of indifference curves, one for a client who will choose to borrow, and one who
will invest in both the index fund and a money market fund.
24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for
which y 5 1?
25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund.
26. What is the largest percentage fee that a client who currently is lending ( y , 1) will be willing to pay to invest in your fund? What about a client who is borrowing ( y . 1)?
For Challenge Problems 27, 28, and 29: You estimate that a passive portfolio, that is, one invested in a risky portfolio that mimics the S&P 500 stock index, yields an expected rate of return of 13% with a standard deviation of 25%. You manage an active port- folio with expected return 18% and standard deviation 28%. The risk-free rate is 8%.
27. Draw the CML and your funds’ CAL on an expected return–standard deviation diagram.
a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of your fund over the passive fund.
28. Your client ponders whether to switch the 70% that is invested in your fund to the passive
portfolio.
a. Explain to your client the disadvantage of the switch. b. Show him the maximum fee you could charge (as a percentage of the investment in your
fund, deducted at the end of the year) that would leave him at least as well off investing in
your fund as in the passive one. ( Hint: The fee will lower the slope of his CAL by reducing the expected return net of the fee.)
29. Consider again the client in Problem 19 with A 5 3.5.
a. If he chose to invest in the passive portfolio, what proportion, y, would he select? b. Is the fee (percentage of the investment in your fund, deducted at the end of the year) that
you can charge to make the client indifferent between your fund and the passive strategy
affected by his capital allocation decision (i.e., his choice of y )?
Challenge
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C H A P T E R 6 Capital Allocation to Risky Assets 195
Use the following data in answering CFA Problems 1–3:
Utility Formula Data
Investment Expected Return, E(r) Standard Deviation, s
1 .12 .30 2 .15 .50 3 .21 .16 4 .24 .21
U 5 E ( r ) 2 ½ A s 2 , where A 5 4
1. On the basis of the utility formula above, which investment would you select if you were risk
averse with A 5 4?
2. On the basis of the utility formula above, which investment would you select if you were risk neutral?
3. The variable ( A ) in the utility formula represents the:
a. investor’s return requirement. b. investor’s aversion to risk. c. certainty equivalent rate of the portfolio. d. preference for one unit of return per four units of risk.
Use the following graph to answer CFA Problems 4 and 5.
Risk, �
Expected Return, E(r)
H
G
F
E
Capital Allocation Line (CAL)
1
1
2
2
3
3
4
4
0
4. Which indifference curve represents the greatest level of utility that can be achieved by the
investor?
5. Which point designates the optimal portfolio of risky assets?
6. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities
versus risk-free T-bills on the basis of the following table?
Action Probability Expected Return
Invest in equities .6 $50,000 .4 2$30,000
Invest in risk-free T-bills 1.0 $ 5,000
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196 P A R T I I Portfolio Theory and Practice
7. The change from a straight to a kinked capital allocation line is a result of the:
a. Reward-to-volatility ratio increasing. b. Borrowing rate exceeding the lending rate. c. Investor’s risk tolerance decreasing. d. Increase in the portfolio proportion of the risk-free asset.
8. You manage an equity fund with an expected risk premium of 10% and an expected standard
deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her
portfolio in your equity fund and $40,000 in a T-bill money market fund. What is the expected
return and standard deviation of return on your client’s portfolio?
9. What is the reward-to-volatility ratio for the equity fund in CFA Problem 8?
E-INVESTMENTS EXERCISES There is a difference between an investor’s willingness to take risk and his or her ability to take risk. Take the quizzes offered at the Web sites below and compare the results. If they are significantly different, which one would you use to determine an investment strategy?
http://mutualfunds.about.com/library/personalitytests/blrisktolerance.htm
http://mutualfunds.about.com/library/personalitytests/blriskcapacity.htm
SOLUTIONS TO CONCEPT CHECKS 1. The investor is taking on exchange rate risk by investing in a pound-denominated asset. If the
exchange rate moves in the investor’s favor, the investor will benefit and will earn more from
the U.K. bill than the U.S. bill. For example, if both the U.S. and U.K. interest rates are 5%, and
the current exchange rate is $2 per pound, a $2 investment today can buy 1 pound, which can be
invested in England at a certain rate of 5%, for a year-end value of 1.05 pounds. If the year-end
exchange rate is $2.10 per pound, the 1.05 pounds can be exchanged for 1.05 3 $2.10 5 $2.205
for a rate of return in dollars of 1 1 r 5 $2.205/$2 5 1.1025, or r 5 10.25%, more than is available from U.S. bills. Therefore, if the investor expects favorable exchange rate movements,
the U.K. bill is a speculative investment. Otherwise, it is a gamble.
2. For the A 5 4 investor the utility of the risky portfolio is
U 5 .02 2 (½ 3 4 3 .32) 5 .02
while the utility of bills is
U 5 .07 2 (½ 3 4 3 0) 5 .07
The investor will prefer bills to the risky portfolio. (Of course, a mixture of bills and the portfolio
might be even better, but that is not a choice here.)
Even for the A 5 2 investor, the utility of the risky portfolio is
U 5 .20 2 (½ 3 2 3 .32) 5 .11
while the utility of bills is again .07. The less risk-averse investor prefers the risky portfolio.
3. The less risk-averse investor has a shallower indifference curve. An increase in risk requires less
increase in expected return to restore utility to the original level.
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P E(rP)
E(r)
σP σ
Less Risk Averse
More Risk Averse
C H A P T E R 6 Capital Allocation to Risky Assets 197
4. Holding 50% of your invested capital in Ready Assets means that your investment proportion in
the risky portfolio is reduced from 70% to 50%.
Your risky portfolio is constructed to invest 54% in E and 46% in B. Thus the proportion of E in your overall portfolio is .5 3 54% 5 27%, and the dollar value of your position in E is $300,000 3 .27 5 $81,000.
5. In the expected return–standard deviation plane all portfolios that are constructed from the
same risky and risk-free funds (with various proportions) lie on a line from the risk-free rate
through the risky fund. The slope of the CAL (capital allocation line) is the same everywhere;
hence the reward-to-volatility ratio is the same for all of these portfolios. Formally, if you invest
a proportion, y, in a risky fund with expected return E ( r P ) and standard deviation s P , and the remainder, 1 2 y, in a risk-free asset with a sure rate r f , then the portfolio’s expected return and standard deviation are
E(rC) 5 rf 1 y3E(rP) 2 rf4 sC 5 ysP
and therefore the reward-to-volatility ratio of this portfolio is
SC 5 E(rC) 2 rf
sC 5
y3E(rP) 2 rf4 ysP
5 E(rP) 2 rf
sP
which is independent of the proportion y.
6. The lending and borrowing rates are unchanged at rf 5 7%, rf B 5 9%. The standard deviation of
the risky portfolio is still 22%, but its expected rate of return shifts from 15% to 17%.
The slope of the two-part CAL is
E(rP) 2 rf
sP for the lending range
E(rP) 2 r f
B
sP for the borrowing range
Thus in both cases the slope increases: from 8/22 to 10/22 for the lending range, and from 6/22 to
8/22 for the borrowing range.
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198 P A R T I I Portfolio Theory and Practice
7. a. The parameters are r f 5 .07, E ( r P ) 5 .15, s P 5 .22. An investor with a degree of risk aversion A will choose a proportion y in the risky portfolio of
y 5 E(rP) 2 rf
AsP 2
With the assumed parameters and with A 5 3 we find that
y 5 .15 2 .07
3 3 .0484 5 .55
When the degree of risk aversion decreases from the original value of 4 to the new value of 3,
investment in the risky portfolio increases from 41% to 55%. Accordingly, both the expected
return and standard deviation of the optimal portfolio increase:
E(rC) 5 .07 1 (.55 3 .08) 5 .114 (before: .1028)
sC 5 .55 3 .22 5 .121 (before: .0902)
b. All investors whose degree of risk aversion is such that they would hold the risky portfolio in a proportion equal to 100% or less ( y ≤ 1.00) are lending rather than borrowing, and so are unaffected by the borrowing rate. The least risk-averse of these investors hold 100% in the
risky portfolio ( y 5 1). We can solve for the degree of risk aversion of these “cut off” investors from the parameters of the investment opportunities:
y 5 1 5 E(rP) 2 rf
AsP 2
5 .08
.0484 A
which implies
A 5 .08
.0484 5 1.65
Any investor who is more risk tolerant (that is, A , 1.65) would borrow if the borrowing rate were 7%. For borrowers,
y 5 E(rP) 2 rf
B
AsP 2
Suppose, for example, an investor has an A of 1.1. When rf 5 r f B 5 7%, this investor chooses
to invest in the risky portfolio:
y 5 .08
1.1 3 .0484 5 1.50
which means that the investor will borrow an amount equal to 50% of her own investment
capital. Raise the borrowing rate, in this case to rf B 5 9%, and the investor will invest less in
the risky asset. In that case:
y 5 .06
1.1 3 .0484 5 1.13
and “only” 13% of her investment capital will be borrowed. Graphically, the line from r f to the risky portfolio shows the CAL for lenders. The dashed part would be relevant if the borrowing rate equaled the lending rate. When the borrowing rate exceeds the lending rate, the CAL is
kinked at the point corresponding to the risky portfolio.
The following figure shows indifference curves of two investors. The steeper indifference
curve portrays the more risk-averse investor, who chooses portfolio C 0 , which involves lending. This investor’s choice is unaffected by the borrowing rate. The more risk-tolerant
investor is portrayed by the shallower-sloped indifference curves. If the lending rate equaled
the borrowing rate, this investor would choose portfolio C 1 on the dashed part of the CAL. When the borrowing rate goes up, this investor chooses portfolio C 2 (in the borrowing range of the kinked CAL), which involves less borrowing than before. This investor is hurt by the
increase in the borrowing rate.
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C H A P T E R 6 Capital Allocation to Risky Assets 199
E(r)
E(rP)
rf
C0
C2
C1
rf B
sP
s
8. If all the investment parameters remain unchanged, the only reason for an investor to decrease the
investment proportion in the risky asset is an increase in the degree of risk aversion. If you think
that this is unlikely, then you have to reconsider your faith in your assumptions. Perhaps the S&P
500 is not a good proxy for the optimal risky portfolio. Perhaps investors expect a higher real rate
on T-bills.
APPENDIX A: Risk Aversion, Expected Utility, and the St. Petersburg Paradox
We digress in this appendix to examine the rationale behind our contention that investors
are risk averse. Recognition of risk aversion as central in investment decisions goes back at
least to 1738. Daniel Bernoulli, one of a famous Swiss family of distinguished mathemati-
cians, spent the years 1725 through 1733 in St. Petersburg, where he analyzed the follow-
ing coin-toss game. To enter the game one pays an entry fee. Thereafter, a coin is tossed
until the first head appears. The number of tails, denoted by n, that appears until the first head is tossed is used to compute the payoff, $ R, to the participant, as
R(n) 5 2n
The probability of no tails before the first head ( n 5 0) is 1/2 and the corresponding payoff is 2
0 5 $1. The probability of one tail and then heads ( n 5 1) is 1/2 3 1/2 with payoff 2 1 5 $2,
the probability of two tails and then heads ( n 5 2) is 1/2 3 1/2 3 1/2, and so forth. The following table illustrates the probabilities and payoffs for various outcomes:
Tails Probability Payoff 5 $ R(n) Probability 3 Payoff
0 1/2 $1 $1/2 1 1/4 $2 $1/2 2 1/8 $4 $1/2 3 1/16 $8 $1/2
… … … …
n (1/2)n 1 1 $2n $1/2
The expected payoff is therefore
E(R) 5 a `
n 5 0 Pr(n)R(n) 5 1@2 1
1@2 1 c5 `
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200 P A R T I I Portfolio Theory and Practice
The evaluation of this game is called the “St. Petersburg Paradox.” Although the expected
payoff is infinite, participants obviously will be willing to purchase tickets to play the
game only at a finite, and possibly quite modest, entry fee.
Bernoulli resolved the paradox by noting that investors do not assign the same value
per dollar to all payoffs. Specifically, the greater their wealth, the less their “apprecia-
tion” for each extra dollar. We can make this insight mathematically precise by assigning a
welfare or utility value to any level of investor wealth. Our utility function should increase
as wealth is higher, but each extra dollar of wealth should increase utility by progres-
sively smaller amounts. 9 (Modern economists would say that investors exhibit “decreasing
marginal utility” from an additional payoff dollar.) One particular function that assigns a
subjective value to the investor from a payoff of $ R, which has a smaller value per dollar the greater the payoff, is the function ln( R ) where ln is the natural logarithm function. If this function measures utility values of wealth, the subjective utility value of the game is
indeed finite, equal to .693. 10
The certain wealth level necessary to yield this utility value
is $2.00, because ln(2.00) 5 .693. Hence the certainty equivalent value of the risky
payoff is $2.00, which is the maximum amount that this investor will pay to play the game.
Von Neumann and Morgenstern adapted this approach to investment theory in a com-
plete axiomatic system in 1946. Avoiding unnecessary technical detail, we restrict our-
selves here to an intuitive exposition of the rationale for risk aversion.
Imagine two individuals who are identical twins, except that one of them is less fortunate
than the other. Peter has only $1,000 to his name while Paul has a net worth of $200,000.
How many hours of work would each twin be willing to offer to earn one extra dollar? It is
likely that Peter (the poor twin) has more essential uses for the extra money than does Paul.
Therefore, Peter will offer more hours. In other words, Peter derives a greater personal wel-
fare or assigns a greater “utility” value to the 1,001st dollar than Paul does to the 200,001st.
Figure 6A.1 depicts graphically the relationship between the wealth and the utility value of
wealth that is consistent with this notion of decreasing marginal utility.
Individuals have different rates of decrease in their marginal utility of wealth. What
is constant is the principle that the per-dollar increment to utility decreases with wealth. Functions that exhibit the property of decreasing per-unit value as the number of units
grows are called concave. A simple example is the log function, familiar from high school
mathematics. Of course, a log function will not fit all investors, but it is consistent with the
risk aversion that we assume for all investors.
Now consider the following simple prospect:
$100,000
$150,000
$50,000
p 5 ½
1 2 p 5 ½
This is a fair game in that the expected profit is zero. Suppose, however, that the curve
in Figure 6A.1 represents the investor’s utility value of wealth, assuming a log utility func-
tion. Figure 6A.2 shows this curve with numerical values marked.
Figure 6A.2 shows that the loss in utility from losing $50,000 exceeds the gain from win-
ning $50,000. Consider the gain first. With probability p 5 .5, wealth goes from $100,000
9 This utility is similar in spirit to the one that assigns a satisfaction level to portfolios with given risk and return
attributes. However, the utility function here refers not to investors’ satisfaction with alternative portfolio choices
but only to the subjective welfare they derive from different levels of wealth.
10
If we substitute the “utility” value, ln( R ), for the dollar payoff, R, to obtain an expected utility value of the game (rather than expected dollar value), we have, calling V ( R ) the expected utility,
V(R) 5 a `
n 5 0 Pr(n) ln 3R(n) 4 5 a`
n 5 0
(1/2)n 1 1 ln(2n) 5 .693
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C H A P T E R 6 Capital Allocation to Risky Assets 201
to $150,000. Using the log utility function, utility goes from ln(100,000) 5 11.51 to
ln(150,000) 5 11.92, the distance G on the graph. This gain is G 5 11.92 2 11.51 5 .41. In expected utility terms, then, the gain is pG 5 .5 3 .41 5 .21.
Now consider the possibility of coming up on the short end of the prospect. In that case,
wealth goes from $100,000 to $50,000. The loss in utility, the distance L on the graph, is L 5 ln(100,000) 2 ln(50,000) 5 11.51 2 10.82 5 .69. Thus the loss in expected utility
Figure 6A.2 Fair games and expected utility
Y
G
L
W
U(W) = ln(W)
U(150,000) = 11.92
U(100,000) = 11.51
E[U(W)] = 11.37
U(50,000) = 10.82
E(W) = 100,000W1 = 50,000 WCE W2 = 150,000
Figure 6A.1 Utility of wealth with a log utility function
U (W)
W
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202 P A R T I I Portfolio Theory and Practice
terms is (1 2 p ) L 5 .5 3 .69 5 .35, which exceeds the gain in expected utility from the possibility of winning the game.
We compute the expected utility from the risky prospect:
E3U(W)4 5 pU(W1) 1 (1 2 p)U(W2) 5 ½ ln(50,000) 1 ½ ln(150,000) 5 11.37
If the prospect is rejected, the utility value of the (sure) $100,000 is ln(100,000) 5 11.51, greater
than that of the fair game (11.37). Hence the risk-averse investor will reject the fair game.
Using a specific investor utility function (such as the log utility function) allows us to com-
pute the certainty equivalent value of the risky prospect to a given investor. This is the amount
that, if received with certainty, she would consider equally attractive as the risky prospect.
If log utility describes the investor’s preferences toward wealth outcomes, then
Figure 6A.2 can also tell us what is, for her, the dollar value of the prospect. We ask, What
sure level of wealth has a utility value of 11.37 (which equals the expected utility from the
prospect)? A horizontal line drawn at the level 11.37 intersects the utility curve at the level
of wealth W CE . This means that
ln(WCE) 5 11.37 which implies that
WCE 5 e 11.37 5 $86,681.87
W CE is therefore the certainty equivalent of the prospect. The distance Y in Figure 6A.2 is the penalty, or the downward adjustment, to the expected profit that is attributable to the
risk of the prospect.
Y 5 E(W) 2 WCE 5 $100,000 2 $86,681.87 5 $13,318.13
This investor views $86,681.87 for certain as being equal in utility value as $100,000 at
risk. Therefore, she would be indifferent between the two.
Suppose the utility function is U(W ) 5 "W a. What is the utility level at wealth levels $50,000 and $150,000?
b. What is expected utility if p still equals .5?
c. What is the certainty equivalent of the risky prospect?
d. Does this utility function also display risk aversion?
e. Does this utility function display more or less risk aversion than the log utility function?
CONCEPT CHECK 6A.1
PROBLEMS: APPENDIX A
1. Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a
risk-free asset paying an annual interest rate of 6%. There is a probability of .001 that your house
will burn to the ground and its value will be reduced to zero. With a log utility of end-of-year
wealth, how much would you be willing to pay for insurance (at the beginning of the year)?
(Assume that if the house does not burn down, its end-of-year value still will be $200,000.)
2. If the cost of insuring your house is $1 per $1,000 of value, what will be the certainty equivalent
of your end-of-year wealth if you insure your house at:
a. ½ its value. b. Its full value. c. 1½ times its value.
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C H A P T E R 6 Capital Allocation to Risky Assets 203
APPENDIX B: Utility Functions and Equilibrium Prices of Insurance Contracts
APPENDIX B: Utility Functions and Equilibrium Prices of Insurance Contracts
APPENDIX B: Utility Functions and Equilibrium Prices of Insurance Contracts
The utility function of an individual investor allows us to measure the subjective value the
individual would place on a dollar at various levels of wealth. Essentially, a dollar in bad times
(when wealth is low) is more valuable than a dollar in good times (when wealth is high).
Suppose that all investors hold the risky S&P 500 portfolio. Then, if the portfolio value
falls in a worse-than-expected economy, all investors will, albeit to different degrees, expe-
rience a “low-wealth” scenario. Therefore, the equilibrium value of a dollar in the low-
wealth economy would be higher than the value of a dollar when the portfolio performs
better than expected. This observation helps explain the apparently high cost of portfolio
insurance that we encountered when considering long-term investments in the previous
chapter. It also helps explain why an investment in a stock portfolio (and hence in individ-
ual stocks) has a risk premium that appears to be so high and results in probability of short-
fall that is so low. Despite the low probability of shortfall risk, stocks still do not dominate
the lower-return risk-free bond, because if an investment shortfall should transpire, it will
coincide with states in which the value of dollar returns is high.
Does revealed behavior of investors demonstrate risk aversion? Looking at prices
and past rates of return in financial markets, we can answer with a resounding yes. With
remarkable consistency, riskier bonds are sold at lower prices than are safer ones with
otherwise similar characteristics. Riskier stocks also have provided higher average rates
of return over long periods of time than less risky assets such as T-bills. For example, over
the 1926 to 2012 period, the average rate of return on the S&P 500 portfolio exceeded the
T-bill return by around 8% per year.
It is abundantly clear from financial data that the average, or representative, investor
exhibits substantial risk aversion. For readers who recognize that financial assets are priced
to compensate for risk by providing a risk premium and at the same time feel the urge for
some gambling, we have a constructive recommendation: Direct your gambling impulse to
investment in financial markets. As Von Neumann once said, “The stock market is a casino
with the odds in your favor.” A small risk-seeking investment may provide all the excite-
ment you want with a positive expected return to boot!
APPENDIX C: The Kelly Criterion
To take a step upwards from the gamble of the St. Petersburg Paradox, consider a
sequence of identical one-period investment prospects, each with two possible payoffs
(with rates of return expressed as decimals): a positive excess return, b, with prob- ability p, and a negative excess return, 2a (a . 0), with probability q 5 1 2 p. J.L. Kelly
11 considered this a basic form of a capital allocation problem and determined the
optimal investment in such a sequence of bets for an investor with a log utility function
(described in Appendix A).
Investing a fraction y in the prospect and the remainder in the risk-free asset provides a total rate of return of 1 1 r 1 by with probability p, or 1 1 r – ay with probability q. Because Kelly employs a log utility function, the expected utility of the prospect, per dol-
lar of initial wealth, is:
E3U(y)4 5 p ln(1 1 r 1 yb) 1 q ln(1 1 r 2 ay) (6.C.1) 11
J.L. Kelly Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal 35 (1956), 917–56.
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204 P A R T I I Portfolio Theory and Practice
The investment that maximizes the expected utility has become known as the Kelly crite-
rion (or Kelly formula). The criterion states that the fraction of total wealth invested in the
risky prospect is independent of wealth and is given by:
y 5 (1 1 r)ap a
2 q
b b (6.C.2)
This will be the investor’s asset allocation in each period.
The Kelly formula calls for investing more in the prospect when p and b are large and less when q and a are large. Risk aversion stands out since, when the gains and losses are equal, i.e., when a 5 b, y 5 (1 1 r)(p 2 q)/a, the larger the win/loss spread (correspond- ing to larger values of a and b), the smaller the fraction invested. A higher interest rate also increases risk taking (an income effect).
Kelly’s rule is based on the log utility function. One can show that investors who have
such a utility function will, in each period, attempt to maximize the geometric mean of the
portfolio return. So the Kelly formula also is a rule to maximize geometric mean, and it
has several interesting properties: (1) It never risks ruin, since the fraction of wealth in the
risky asset in Equation 6C.2 never exceeds 1/a. (2) The probability that it will outperform any other strategy goes to 1 as the investment horizon goes to infinity. (3) It is myopic,
meaning the optimal strategy is the same regardless of the investment horizon. (4) If you
have a specified wealth goal (e.g., $1 million), the strategy has the shortest expected time
to that goal. Considerable literature has been devoted to the Kelly criterion. 12
12 See, for example, L.C. MacLean, E.O. Thorp, W.T. Ziemba, Eds., The Kelly Capital Growth Criterion: Theory
and Practice (World Scientific Handbook in Financial Economic Series), Singapore: World Scientific Publishing Co., 2010.
SOLUTION TO CONCEPT CHECK
A.1. a. U(W) 5 "W U(50,000) 5 "50,000 5 223.61 U(150,000) 5 387.30
b. E ( U ) 5 (.5 3 223.61) + (.5 3 387.30) 5 305.45 c. We must find W CE that has utility level 305.45. Therefore
"WCE 5 305.45 WCE 5 305.45
2 5 $93,301
d. Yes. The certainty equivalent of the risky venture is less than the expected outcome of $100,000.
e. The certainty equivalent of the risky venture to this investor is greater than it was for the log utility investor considered in the text. Hence this utility function displays less risk
aversion.
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7 THE INVESTMENT DECISION can be viewed as a top-down process: (i) Capital allocation between the risky portfolio and risk-free assets, (ii) asset allocation in the risky portfo- lio across broad asset classes (e.g., U.S. stocks, international stocks, and long-term bonds), and (iii) security selection of individual assets within each asset class.
Capital allocation, as we saw in Chapter 6, determines the investor’s exposure to risk. The optimal capital allocation is determined by risk aversion as well as expectations for the risk–return trade-off of the optimal risky port- folio. In principle, asset allocation and security selection are technically identical; both aim at identifying that optimal risky portfolio, namely, the combination of risky assets that provides the best risk–return trade-off. In practice, however, asset allocation and secu- rity selection are typically separated into two steps, in which the broad outlines of the port- folio are established first (asset allocation), while details concerning specific securities are filled in later (security selection). After we show how the optimal risky portfolio may be constructed, we will consider the costs and benefits of pursuing this two-step approach.
We first motivate the discussion by illustrat- ing the potential gains from simple diversifi- cation into many assets. We then proceed to
examine the process of efficient diversification from the ground up, starting with an invest- ment menu of only two risky assets, then adding the risk-free asset, and finally, incor- porating the entire universe of available risky securities. We learn how diversification can reduce risk without affecting expected returns. This accomplished, we re-examine the hierar- chy of capital allocation, asset allocation, and security selection. Finally, we offer insight into the power of diversification by drawing an analogy between it and the workings of the insurance industry.
The portfolios we discuss in this and the fol- lowing chapters are of a short-term-horizon— even if the overall investment horizon is long, portfolio composition can be rebalanced or updated almost continuously. For these short horizons, the assumption of normality is suf- ficiently accurate to describe holding-period returns, and we will be concerned only with portfolio means and variances.
In Appendix A, we demonstrate how con- struction of the optimal risky portfolio can easily be accomplished with Excel. Appendix B provides a review of portfolio statistics with emphasis on the intuition behind covariance and correlation measures. Even if you have had a good quantitative methods course, it may well be worth skimming.
Optimal Risky Portfolios
CHAPTER SEVEN 7 PA
R T II
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206 P A R T I I Portfolio Theory and Practice
7.1 Diversification and Portfolio Risk
Suppose your portfolio is composed of only one stock, say, Dell Inc. What would be the
sources of risk to this “portfolio”? You might think of two broad sources of uncertainty.
First, there is the risk that comes from conditions in the general economy, such as the
business cycle, inflation, interest rates, and exchange rates. None of these macroeconomic
factors can be predicted with certainty, and all affect the rate of return on Dell stock. In
addition to these macroeconomic factors there are firm-specific influences, such as Dell’s
success in research and development, and personnel changes. These factors affect Dell
without noticeably affecting other firms in the economy.
Now consider a naive diversification strategy, in which you include additional securi- ties in your portfolio. For example, place half your funds in ExxonMobil and half in Dell.
What should happen to portfolio risk? To the extent that the firm-specific influences on the
two stocks differ, diversification should reduce portfolio risk. For example, when oil prices
fall, hurting ExxonMobil, computer prices might rise, helping Dell. The two effects are
offsetting and stabilize portfolio return.
But why end diversification at only two stocks? If we diversify into many more
securities, we continue to spread out our exposure to firm-specific factors, and portfo-
lio volatility should continue to fall. Ultimately, however, even with a large number of
stocks we cannot avoid risk altogether, because virtually all securities are affected by
the common macroeconomic factors. For example, if all stocks are affected by the busi-
ness cycle, we cannot avoid exposure to business cycle risk no matter how many stocks
we hold.
When all risk is firm-specific, as in Figure 7.1 , panel A, diversification can reduce risk
to arbitrarily low levels. The reason is that with all risk sources independent, the exposure
to any particular source of risk is reduced to a negligible level. The reduction of risk to
very low levels in the case of independent risk sources is sometimes called the insurance principle, because of the notion that an insurance company depends on the risk reduction achieved through diversification when it writes many policies insuring against many inde-
pendent sources of risk, each policy being a small part of the company’s overall portfolio.
(See Section 7.5 for a discussion of the insurance principle.)
When common sources of risk affect all firms, however, even extensive diversifica- tion cannot eliminate risk. In Figure 7.1 , panel B, portfolio standard deviation falls as the
number of securities increases, but it cannot be reduced to zero. The risk that remains even
after extensive diversification is called market risk, risk that is attributable to marketwide risk sources. Such risk is also called systematic risk, or nondiversifiable risk . In contrast, the risk that can be eliminated by diversification is called unique risk , firm-specific risk , nonsystematic risk , or diversifiable risk .
This analysis is borne out by empirical studies. Figure 7.2 shows the effect of portfolio
diversification, using data on NYSE stocks. 1 The figure shows the average standard devia-
tion of equally weighted portfolios constructed by selecting stocks at random as a function
of the number of stocks in the portfolio. On average, portfolio risk does fall with diversifi-
cation, but the power of diversification to reduce risk is limited by systematic or common
sources of risk.
1 Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative
Analysis 22 (September 1987).
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C H A P T E R 7 Optimal Risky Portfolios 207
A ve
ra g
e P
o rt
fo li o
S ta
n d
a rd
D e vi
a ti
o n
( %
)
50
0 2 4 6 8 10 12 14 16 18 20 100 200 300 400 500 600 700 800 900 1,000 0
100
75
50 40
R is
k C
o m
p a re
d t
o a
O n
e -S
to ck
P o
rt fo
li o
( %
)
Number of Stocks in Portfolio
40 45
35 30 25 20 15 10 5 0
Figure 7.2 Portfolio diversification. The average standard deviation of returns of portfolios composed of only one stock was 49.2%. The average portfolio risk fell rapidly as the number of stocks included in the portfolio increased. In the limit, portfolio risk could be reduced to only 19.2%.
Source: From Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22 (September 1987). Reprinted by permission.
A
n
B
n
Unique Risk
Market Risk
σ σ
Figure 7.1 Portfolio risk as a function of the number of stocks in the portfolio Panel A: All risk is firm specific. Panel B: Some risk is systematic, or marketwide.
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208 P A R T I I Portfolio Theory and Practice
In the last section we considered naive diversification using equally weighted portfolios
of several securities. It is time now to study efficient diversification, whereby we construct risky portfolios to provide the lowest possible risk for any given level of expected return.
Portfolios of two risky assets are relatively easy to analyze, and they illustrate the prin-
ciples and considerations that apply to portfolios of many assets. It makes sense to think
about a two-asset portfolio as an asset allocation decision, and so we consider two mutual
funds, a bond portfolio specializing in long-term debt securities, denoted D, and a stock fund that specializes in equity securities, E. Table 7.1 lists the parameters describing the rate-of-return distribution of these funds.
A proportion denoted by w D is invested in the bond fund, and the remainder, 1 2 w D , denoted w E , is invested in the stock fund. The rate of return on this portfolio, r p , will be
2
rp 5 wD rD 1 wE rE (7.1)
where r D is the rate of return on the debt fund and r E is the rate of return on the equity fund. The expected return on the portfolio is a weighted average of expected returns on the
component securities with portfolio proportions as weights:
E (rp) 5 wDE (rD) 1 wE E (rE ) (7.2)
The variance of the two-asset portfolio is
sp 2 5 wD
2 sD 2 1 wE
2 sE 2 1 2wDwE Cov(rD, rE ) (7.3)
Our first observation is that the variance of the portfolio, unlike the expected return, is
not a weighted average of the individual asset variances. To understand the formula for the portfolio variance more clearly, recall that the covariance of a variable with itself is the
variance of that variable; that is
Cov(rD, rD) 5 a scenarios
Pr(scenario)3rD 2 E(rD)4 3rD 2 E(rD)4 5 a
scenarios
Pr (scenario)3rD 2 E(rD)42 (7.4) 5 sD
2
Therefore, another way to write the variance of the portfolio is
sp 2 5 wDwD Cov(rD , rD) 1 wE wE Cov(rE, rE) 1 2wDwE Cov(rD , rE) (7.5)
7.2 Portfolios of Two Risky Assets
Debt Equity
Expected return, E ( r ) 8% 13% Standard deviation, s 12% 20% Covariance, Cov( r D , r E ) 72 Correlation coefficient, r DE .30
Table 7.1
Descriptive statistics for two mutual funds
2 See Appendix B of this chapter for a review of portfolio statistics.
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C H A P T E R 7 Optimal Risky Portfolios 209
In words, the variance of the portfolio is a weighted sum of covariances, and each weight
is the product of the portfolio proportions of the pair of assets in the covariance term.
Table 7.2 shows how portfolio variance can be calculated from a spreadsheet. Panel A
of the table shows the bordered covariance matrix of the returns of the two mutual funds. The bordered matrix is the covariance matrix with the portfolio weights for each fund
placed on the borders, that is, along the first row and column. To find portfolio variance,
multiply each element in the covariance matrix by the pair of portfolio weights in its row
and column borders. Add up the resultant terms, and you have the formula for portfolio
variance given in Equation 7.5.
We perform these calculations in panel B, which is the border-multiplied covariance matrix: Each covariance has been multiplied by the weights from the row and the column
in the borders. The bottom line of panel B confirms that the sum of all the terms in this
matrix (which we obtain by adding up the column sums) is indeed the portfolio variance
in Equation 7.5.
This procedure works because the covariance matrix is symmetric around the diagonal,
that is, Cov( r D , r E ) 5 Cov( r E , r D ). Thus each covariance term appears twice. This technique for computing the variance from the border-multiplied covariance matrix
is general; it applies to any number of assets and is easily implemented on a spreadsheet.
Concept Check 1 asks you to try the rule for a three-asset portfolio. Use this problem to
verify that you are comfortable with this concept.
A. Bordered Covariance Matrix
Portfolio Weights wD wE
wD Cov(rD, rD) Cov(rD, rE) wE Cov(rE, rD) Cov(rE, rE)
B. Border-Multiplied Covariance Matrix
Portfolio Weights wD wE
wD wDwDCov(rD, rD) wDwECov(rD, rE) wE wEwDCov(rE, rD) wEwECov(rE, rE)
wD 1 wE 5 1 wDwDCov(rD, rD) 1 wEwDCov(rE, rD) wDwECov(rD, rE) 1 wEwECov(rE, rE)
Portfolio variance wDwDCov(rD, rD) 1 wEwDCov(rE, rD) 1 wDwECov(rD, rE) 1 wEwECov(rE, rE)
Table 7.2
Computation of portfolio variance from the covariance matrix
a. First confirm for yourself that our simple rule for computing the variance of a two-asset portfolio from the bordered covariance matrix is consistent with Equation 7.3.
b. Now consider a portfolio of three funds, X, Y, Z, with weights w X , w Y , and w Z . Show that the portfolio variance is
wX 2sX
2 1 wY 2 sY
2 1 wZ 2sZ
2 1 2wXwY Cov(rX , rY)
1 2wX wZ Cov(rX , rZ) 1 2wY wZ Cov(rY, rZ)
CONCEPT CHECK 7.1
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210 P A R T I I Portfolio Theory and Practice
Equation 7.3 reveals that variance is reduced if the covariance term is negative. It is
important to recognize that even if the covariance term is positive, the portfolio standard
deviation still is less than the weighted average of the individual security standard devia- tions, unless the two securities are perfectly positively correlated.
To see this, notice that the covariance can be computed from the correlation coefficient,
r DE , as
Cov(rD, rE) 5 rDE sDsE (7.6)
Therefore,
sp 2 5 wD
2 sD 2 1 wE
2 sE 2 1 2wDwE sDsErDE (7.7)
Other things equal, portfolio variance is higher when r DE is higher. In the case of perfect
positive correlation, r DE 5 1, the right-hand side of Equation 7.7 is a perfect square and
simplifies to
sp 2 5 (wDsD 1 wEsE)
2 (7.8)
or
sp 5 wDsD 1 wEsE (7.9)
Therefore, the standard deviation of the portfolio with perfect positive correlation is just
the weighted average of the component standard deviations. In all other cases, the cor-
relation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations.
A hedge asset has negative correlation with the other assets in the portfolio. Equation 7.7 shows that such assets will be particularly effective in reducing total risk. Moreover,
Equation 7.2 shows that expected return is unaffected by correlation between returns.
Therefore, other things equal, we will always prefer to add to our portfolios assets with low
or, even better, negative correlation with our existing position.
Because the portfolio’s expected return is the weighted average of its component
expected returns, whereas its standard deviation is less than the weighted average of the
component standard deviations, portfolios of less than perfectly correlated assets always offer some degree of diversification benefit. The lower the correlation between the assets, the greater the gain in efficiency.
How low can portfolio standard deviation be? The lowest possible value of the correla-
tion coefficient is 2 1, representing perfect negative correlation. In this case, Equation 7.7
simplifies to
sp 2 5 (wDsD 2 wE sE)
2 (7.10)
and the portfolio standard deviation is
sp 5 Absolute value (wDsD 2 wE sE) (7.11)
When r 5 2 1, a perfectly hedged position can be obtained by choosing the portfolio pro-
portions to solve
wDsD 2 wE sE 5 0
The solution to this equation is
wD 5 sE
sD 1 sE
wE 5 sD
sD 1 sE 5 1 2 wD
(7.12)
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C H A P T E R 7 Optimal Risky Portfolios 211
These weights drive the standard deviation of the portfolio to zero.
We can experiment with different portfolio proportions to observe the effect on portfo-
lio expected return and variance. Suppose we change the proportion invested in bonds. The
effect on expected return is tabulated in Table 7.3 and plotted in Figure 7.3 . When the pro-
portion invested in debt varies from zero to 1 (so that the proportion in equity varies from
1 to zero), the portfolio expected return goes from 13% (the stock fund’s expected return)
to 8% (the expected return on bonds).
What happens when w D . 1 and w E , 0? In this case portfolio strategy would be to sell the equity fund short and invest the proceeds of the short sale in the debt fund. This will
decrease the expected return of the portfolio. For example, when w D 5 2 and w E 5 2 1, expected portfolio return falls to 2 3 8 1 ( 2 1) 3 13 5 3%. At this point the value of the
bond fund in the portfolio is twice the net worth of the account. This extreme position is
financed in part by short-selling stocks equal in value to the portfolio’s net worth.
Table 7.3
Expected return and standard deviation with various correla- tion coefficients
Portfolio Standard Deviation for Given Correlation
wD wE E(rP) r 5 21 r 5 0 r 5 .30 r 5 1
0.00 1.00 13.00 20.00 20.00 20.00 20.00 0.10 0.90 12.50 16.80 18.04 18.40 19.20 0.20 0.80 12.00 13.60 16.18 16.88 18.40 0.30 0.70 11.50 10.40 14.46 15.47 17.60 0.40 0.60 11.00 7.20 12.92 14.20 16.80 0.50 0.50 10.50 4.00 11.66 13.11 16.00 0.60 0.40 10.00 0.80 10.76 12.26 15.20 0.70 0.30 9.50 2.40 10.32 11.70 14.40 0.80 0.20 9.00 5.60 10.40 11.45 13.60 0.90 0.10 8.50 8.80 10.98 11.56 12.80 1.00 0.00 8.00 12.00 12.00 12.00 12.00
Minimum Variance Portfolio
wD 0.6250 0.7353 0.8200 — wE 0.3750 0.2647 0.1800 — E(rP) 9.8750 9.3235 8.9000 — sP 0.0000 10.2899 11.4473 —
Let us apply this analysis to the data of the bond and stock funds as presented in Table 7.1 . Using these data, the formulas for the expected return, variance, and standard deviation of the portfolio as a function of the portfolio weights are
E (rp ) 5 8wD 1 13wE sp
2 5 122wD 2 1 202wE
2 1 2 3 12 3 20 3 .3 3 wDwE 5 144wD
2 1 400wE 2 1 144wDwE
sp 5 "sp2
Example 7.1 Portfolio Risk and Return
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212 P A R T I I Portfolio Theory and Practice
Expected Return
13%
8%
Equity Fund
Debt Fund
w (stocks)
w (bonds) = 1 − w (stocks) − 0.5 0 1.0 2.0
1.5 1.0 0 −1.0
Figure 7.3 Portfolio expected return as a function of investment proportions
ρ = .30
−.50 .500 1.501.0 Weight in Stock Fund
Portfolio Standard Deviation (%)
ρ = −1
ρ = 0
ρ = 1
35
30
25
20
15
10
5
0
Figure 7.4 Portfolio standard deviation as a function of investment proportions
The reverse happens when w D , 0 and w E . 1. This strategy calls for selling the bond fund short and using the proceeds to finance
additional purchases of the equity fund.
Of course, varying investment proportions
also has an effect on portfolio standard devia-
tion. Table 7.3 presents portfolio standard
deviations for different portfolio weights cal-
culated from Equation 7.7 using the assumed
value of the correlation coefficient, .30, as well
as other values of r . Figure 7.4 shows the rela-
tionship between standard deviation and port-
folio weights. Look first at the solid curve for
r DE 5 .30. The graph shows that as the port-
folio weight in the equity fund increases from
zero to 1, portfolio standard deviation first falls
with the initial diversification from bonds into
stocks, but then rises again as the portfolio
becomes heavily concentrated in stocks, and
again is undiversified. This pattern will gener-
ally hold as long as the correlation coefficient
between the funds is not too high. 3 For a pair
of assets with a large positive correlation of
3 As long as r , s D / s E , volatility will initially fall when we start with all bonds and begin to move into stocks.
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C H A P T E R 7 Optimal Risky Portfolios 213
returns, the portfolio standard deviation will increase monotonically from the low-risk
asset to the high-risk asset. Even in this case, however, there is a positive (if small) benefit
from diversification.
What is the minimum level to which portfolio standard deviation can be held? For the
parameter values stipulated in Table 7.1 , the portfolio weights that solve this minimization
problem turn out to be 4
wMin(D) 5 .82
wMin(E) 5 1 2 .82 5 .18
This minimum-variance portfolio has a standard deviation of
sMin 5 3(.822 3 122) 1 (.182 3 202) 1 (2 3 .82 3 .18 3 72)41/2 5 11.45% as indicated in the last line of Table 7.3 for the column r 5 .30.
The solid colored line in Figure 7.4 plots the portfolio standard deviation when r 5 .30
as a function of the investment proportions. It passes through the two undiversified port-
folios of w D 5 1 and w E 5 1. Note that the minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets. This illustrates the effect of diversification.
The other three lines in Figure 7.4 show how portfolio risk varies for other values of the
correlation coefficient, holding the variances of each asset constant. These lines plot the
values in the other three columns of Table 7.3 .
The solid dark straight line connecting the undiversified portfolios of all bonds or all
stocks, w D 5 1 or w E 5 1, shows portfolio standard deviation with perfect positive cor- relation, r 5 1. In this case there is no advantage from diversification, and the portfo-
lio standard deviation is the simple weighted average of the component asset standard
deviations.
The dashed colored curve depicts portfolio risk for the case of uncorrelated assets,
r 5 0. With lower correlation between the two assets, diversification is more effective and
portfolio risk is lower (at least when both assets are held in positive amounts). The mini-
mum portfolio standard deviation when r 5 0 is 10.29% (see Table 7.3 ), again lower than the standard deviation of either asset.
Finally, the triangular broken line illustrates the perfect hedge potential when the two
assets are perfectly negatively correlated ( r 5 2 1). In this case the solution for the mini-
mum-variance portfolio is, by Equation 7.12,
wMin(D; r 5 21) 5 sE
sD 1 sE 5
20
12 1 20 5 .625
wMin(E; r 5 21) 5 1 2 .625 5 .375
and the portfolio variance (and standard deviation) is zero.
We can combine Figures 7.3 and 7.4 to demonstrate the relationship between portfolio
risk (standard deviation) and expected return—given the parameters of the available assets.
4 This solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from
Equation 7.3, substitute 1 2 w D for w E , differentiate the result with respect to w D , set the derivative equal to zero, and solve for w D to obtain
wMin(D) 5 sE
2 2 Cov(rD, rE)
sD 2 1 sE
2 2 2 Cov(rD, rE)
Alternatively, with a spreadsheet program such as Excel, you can obtain an accurate solution by using the Solver
to minimize the variance. See Appendix A for an example of a portfolio optimization spreadsheet.
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214 P A R T I I Portfolio Theory and Practice
14
13
12
11
10
9
8
7
6
5
Standard Deviation (%) 0 2 4 6 8 10 12 14 16 18 20
Expected Return (%)
D
E
ρ = −1 ρ = 0
ρ = .30 ρ = 1
Figure 7.5 Portfolio expected return as a function of standard deviation
This is done in Figure 7.5 . For any pair of invest-
ment proportions, w D , w E , we read the expected return from Figure 7.3 and the standard deviation
from Figure 7.4 . The resulting pairs of expected
return and standard deviation are tabulated in
Table 7.3 and plotted in Figure 7.5 .
The solid colored curve in Figure 7.5 shows the
portfolio opportunity set for r 5 .30. We call it the portfolio opportunity set because it shows all
combinations of portfolio expected return and stan-
dard deviation that can be constructed from the two
available assets. The other lines show the portfolio
opportunity set for other values of the correlation
coefficient. The solid black line connecting the two
funds shows that there is no benefit from diversifi-
cation when the correlation between the two is per-
fectly positive ( r 5 1). The opportunity set is not
“pushed” to the northwest. The dashed colored line
demonstrates the greater benefit from diversification
when the correlation coefficient is lower than .30.
Finally, for r 5 2 1, the portfolio opportunity
set is linear, but now it offers a perfect hedging
opportunity and the maximum advantage from
diversification.
To summarize, although the expected return of
any portfolio is simply the weighted average of the asset expected returns, this is not true
of the standard deviation. Potential benefits from diversification arise when correlation is
less than perfectly positive. The lower the correlation, the greater the potential benefit from
diversification. In the extreme case of perfect negative correlation, we have a perfect hedg-
ing opportunity and can construct a zero-variance portfolio.
Suppose now an investor wishes to select the optimal portfolio from the opportunity set.
The best portfolio will depend on risk aversion. Portfolios to the northeast in Figure 7.5
provide higher rates of return but impose greater risk.
The best trade-off among these choices is a matter of per-
sonal preference. Investors with greater risk aversion will
prefer portfolios to the southwest, with lower expected
return but lower risk. 5
5 Given a level of risk aversion, one can determine the portfolio that provides the highest level of utility. Recall
from Chapter 6 that we were able to describe the utility provided by a portfolio as a function of its expected return,
E ( r p ), and its variance, sp 2, according to the relationship U 5 E(rp) 2 0.5Asp
2. The portfolio mean and variance are
determined by the portfolio weights in the two funds, w E and w D , according to Equations 7.2 and 7.3. Using those equations and some calculus, we find the optimal investment proportions in the two funds. A warning: To use the fol-
lowing equation (or any equation involving the risk aversion parameter, A ), you must express returns in decimal form.
wD 5 E(rD) 2 E(rE) 1 A(sE
2 2 sDsE rDE)
A(sD 2 1 sE
2 2 2sDsE rDE)
wE 5 1 2 wD
Here, too, Excel’s Solver or similar software can be used to maximize utility subject to the constraints of
Equations 7.2 and 7.3, plus the portfolio constraint that w D 1 w E 5 1 (i.e., that portfolio weights sum to 1).
Compute and draw the portfolio opportunity set for the debt and equity funds when the correla- tion coefficient between them is r 5 .25.
CONCEPT CHECK 7.2
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C H A P T E R 7 Optimal Risky Portfolios 215
When optimizing capital allocation, we want to work with the capital allocation line (CAL)
offering the highest slope or Sharpe ratio. The steeper the CAL, the greater is the expected
return corresponding to any level of volatility. Now we proceed to asset allocation: con-
structing the risky portfolio of major asset classes, here a bond and a stock fund, with the
highest possible Sharpe ratio.
The asset allocation decision requires that we consider T-bills or another safe asset
along with the risky asset classes. The reason is that the Sharpe ratio we seek to maximize
is defined as the risk premium in excess of the risk-free rate, divided by the standard devia- tion. We use T-bill rates as the risk-free rate in evaluating the Sharpe ratios of all possible
portfolios. The portfolio that maximizes the Sharpe ratio is the solution to the asset alloca-
tion problem. Using only stocks, bonds, and bills is actually not so restrictive, as it includes
all three major asset classes. As the nearby box emphasizes, most investment professionals
recognize that “the really critical decision is how to divvy up your money among stocks,
bonds, and super-safe investments such as Treasury bills.”
Asset Allocation with Two Risky Asset Classes What if our risky assets are still confined to the bond and stock funds, but we can also
invest in risk-free T-bills yielding 5%? We start with a graphical solution. Figure 7.6 shows
the opportunity set based on the properties of the bond and stock funds, using the data
from Table 7.1 and assuming that r 5 .3.
Two possible capital allocation lines (CALs) are drawn from the risk-free rate ( r f 5 5%) to two feasible portfolios. The first possible CAL is drawn through the minimum-variance
portfolio A, which is invested 82% in bonds and 18% in stocks ( Table 7.3 , bottom panel, last column). Portfolio A ’s expected return is 8.90%, and its standard deviation is 11.45%. With a T-bill rate of 5%, its Sharpe ratio, which is the slope of the CAL, is
SA 5 E(rA) 2 rf
sA 5
8.9 2 5
11.45 5 .34
Now consider the CAL that uses portfolio B instead of A. Portfolio B invests 70% in bonds and 30% in stocks. Its expected return is 9.5%
(a risk premium of 4.5%), and its standard devi-
ation is 11.70%. Thus the Sharpe ratio on the
CAL supported by portfolio B is
SB 5 9.5 2 5
11.7 5 .38
which is higher than the Sharpe ratio of the
CAL using the minimum-variance portfolio and
T-bills. Hence, portfolio B dominates A. But why stop at portfolio B? We can con-
tinue to ratchet the CAL upward until it ulti-
mately reaches the point of tangency with the
investment opportunity set. This must yield the
CAL with the highest feasible Sharpe ratio.
Therefore, the tangency portfolio, labeled P in
7.3 Asset Allocation with Stocks, Bonds, and Bills
13
12
11
10
9
8
7
6
5
Standard Deviation (%)
0 5 10 15 20 25
Expected Return (%)
D
E
A
B
CAL(A)
CAL(B)
Figure 7.6 The opportunity set of the debt and equity funds and two feasible CALs
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Recipe for Successful Investing: First, Mix Assets Well
First things first. If you want dazzling investment results, don’t start
your day foraging for hot stocks and stellar mutual funds. Instead, say investment advisers, the really critical decision is how to divvy up your money among stocks, bonds, and supersafe investments such as Treasury bills.
In Wall Street lingo, this mix of investments is called your asset allocation. “The asset-allocation choice is the first and most important decision,” says William Droms, a finance professor at Georgetown University. “How much you have in [the stock market] really drives your results.”
“You cannot get [stock market] returns from a bond portfolio, no matter how good your security selection is or how good the bond managers you use,” says William John Mikus, a managing director of Financial Design, a Los Angeles investment adviser.
For proof, Mr. Mikus cites studies such as the 1991 anal- ysis done by Gary Brinson, Brian Singer and Gilbert Bee- bower. That study, which looked at the 10-year results for 82 large pension plans, found that a plan’s asset-allocation policy explained 91.5% of the return earned.
DESIGNING A PORTFOLIO Because your asset mix is so important, some mutual fund companies now offer free services to help investors design their portfolios.
Gerald Perritt, editor of the Mutual Fund Letter, a Chicago newsletter, says you should vary your mix of assets depending on how long you plan to invest. The further
away your investment horizon, the more you should have in stocks. The closer you get, the more you should lean toward bonds and money-market instruments, such as Treasury bills. Bonds and money-market instruments may generate lower returns than stocks. But for those who need money in the near future, conservative investments make more sense, because there’s less chance of suffering a devastating short-term loss.
SUMMARIZING YOUR ASSETS “One of the most important things people can do is sum- marize all their assets on one piece of paper and figure out their asset allocation,” says Mr. Pond.
Once you’ve settled on a mix of stocks and bonds, you should seek to maintain the target percentages, says Mr. Pond. To do that, he advises figuring out your asset allocation once every six months. Because of a stock- market plunge, you could find that stocks are now a far smaller part of your portfolio than you envisaged. At such a time, you should put more into stocks and lighten up on bonds.
When devising portfolios, some investment advisers consider gold and real estate in addition to the usual trio of stocks, bonds and money-market instruments. Gold and real estate give “you a hedge against hyperinflation,” says Mr. Droms.
Source: Jonathan Clements, “Recipe for Successful Investing: First, Mix Assets Well,” The Wall Street Journal, October 6, 1993. Reprinted by permission of The Wall Street Journal, © 1993 Dow Jones & Company, Inc. All rights reserved worldwide.
W O
R D
S FR
O M
T H
E S
T R
E E T
216
Figure 7.7 , is the optimal risky portfolio to mix with T-bills. We can read the expected
return and standard deviation of portfolio P from the graph in Figure 7.7: E ( r P ) 5 11% and s P 5 14.2%.
In practice, when we try to construct optimal risky portfolios from more than two risky
assets, we need to rely on a spreadsheet (which we present in Appendix A) or another
computer program. To start, however, we will demonstrate the solution of the portfolio
construction problem with only two risky assets and a risk-free asset. In this simpler case,
we can find an explicit formula for the weights of each asset in the optimal portfolio, mak-
ing it easier to illustrate general issues.
The objective is to find the weights w D and w E that result in the highest slope of the CAL. Thus our objective function is the Sharpe ratio:
Sp 5 E(rp) 2 rf
sp
For the portfolio with two risky assets, the expected return and standard deviation of
portfolio p are
E(rp) 5 wD E (rD) 1 wE E (rE)
5 8wD 1 13wE sp 5 3wD2 sD2 1 wE2 sE2 1 2wDwE Cov(rD, rE)41/2 5 3144wD2 1 400wE2 1 (2 3 72wDwE )41/2
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C H A P T E R 7 Optimal Risky Portfolios 217
When we maximize the objective function, S p , we have to satisfy the constraint that the portfolio weights sum to 1.0, that is, w D 1 w E 5 1. Therefore, we solve an optimization problem formally written as
Max wi
Sp 5 E(rp) 2 rf
sp
subject to S w i 5 1. This is a maximization problem that can be solved using standard tools of calculus.
In the case of two risky assets, the solution for the weights of the optimal risky portfolio, P, is given by Equation 7.13. Notice that the solution employs excess returns (denoted R ) rather than total returns (denoted r ). 6
wD 5 E(RD)sE
2 2 E(RE) Cov(RD, RE)
E(RD)sE 2 1 E(RE)sD
2 2 3E(RD) 1 E(RE) 4 Cov(RD, RE) (7.13) wE 5 1 2 wD
6 The solution procedure for two risky assets is as follows. Substitute for E ( r P ) from Equation 7.2 and for s P from
Equation 7.7. Substitute 1 2 w D for w E . Differentiate the resulting expression for S p with respect to w D , set the derivative equal to zero, and solve for w D .
Standard Deviation (%) 0 5 10 15 20 25 30
Expected Return (%)
D
E
P
rf = 5%
CAL(P)
Opportunity Set of Risky Assets
2
0
4
6
8
10
12
14
16
18
Figure 7.7 The opportunity set of the debt and equity funds with the optimal CAL and the optimal risky portfolio
Using our data, the solution for the optimal risky portfolio is
wD 5 (8 2 5)400 2 (13 2 5)72
(8 2 5)400 1 (13 2 5)144 2 (8 2 5 1 13 2 5)72 5 .40
wE 5 1 2 .40 5 .60
Example 7.2 Optimal Risky Portfolio
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218 P A R T I I Portfolio Theory and Practice
In Chapter 6 we found the optimal complete portfolio given an optimal risky portfolio and the CAL generated by a combination of this portfolio and T-bills. Now that we have
constructed the optimal risky portfolio, P, we can use the individual investor’s degree of risk aversion, A, to calculate the optimal proportion of the complete portfolio to invest in the risky component.
The expected return and standard deviation of this optimal risky portfolio are
E (rP ) 5 (.4 3 8) 1 (.6 3 13) 5 11%
sP 5 [(.4 2 3 144) 1 (.62 3 400) 1 (2 3 .4 3 .6 3 72)]1/2 5 14.2%
This asset allocation produces an optimal risky portfolio whose CAL has a slope of
SP 5 11 2 5
14.2 5 .42
which is the Sharpe ratio of portfolio P. Notice that this slope exceeds the slope of any of the other feasible portfolios that we have considered, as it must if it is to be the slope of the best feasible CAL.
7 Notice that we express returns as decimals in Equation 7.14. This is necessary when using the risk aversion
parameter, A, to solve for capital allocation.
Now that asset allocation is decided, we can find each investor’s optimal capital allo- cation. An investor with a coefficient of risk aversion A 5 4 would take a position in portfolio P of 7
y 5 E(rP) 2 rf
AsP 2
5 .11 2 .05
4 3 .1422 5 .7439 (7.14)
Thus the investor will invest 74.39% of his or her wealth in portfolio P and 25.61% in T-bills. Portfolio P consists of 40% in bonds, so the fraction of wealth in bonds will be yw D 5 .4 3 .7439 5 .2976, or 29.76%. Similarly, the investment in stocks will be yw E 5 .6 3 .7439 5 .4463, or 44.63%. The graphical solution of this asset allocation problem is presented in Figures 7.8 and 7.9.
Example 7.3 The Optimal Complete Portfolio
Once we have reached this point, generalizing to the case of many risky assets is
straightforward. Before we move on, let us briefly summarize the steps we followed to
arrive at the complete portfolio.
1. Specify the return characteristics of all securities (expected returns, variances,
covariances).
2. Establish the risky portfolio (asset allocation):
a. Calculate the optimal risky portfolio, P (Equation 7.13).
b. Calculate the properties of portfolio P using the weights determined in step ( a ) and Equations 7.2 and 7.3.
3. Allocate funds between the risky portfolio and the risk-free asset (capital allocation):
a. Calculate the fraction of the complete portfolio allocated to portfolio P (the risky portfolio) and to T-bills (the risk-free asset) (Equation 7.14).
b. Calculate the share of the complete portfolio invested in each asset and in T-bills.
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C H A P T E R 7 Optimal Risky Portfolios 219
Recall that our two risky assets, the bond and stock mutual funds, are already diversified
portfolios. The diversification within each of these portfolios must be credited for a good deal of the risk reduction compared to undiversified single securities. For example, the standard
deviation of the rate of return on an average stock is about 50% (see Figure 7.2 ). In contrast,
the standard deviation of our stock-index fund is only 20%, about equal to the historical
standard deviation of the S&P 500 portfolio. This is evidence of the importance of diver-
sification within the asset class. Optimizing the asset allocation between bonds and stocks
contributed incrementally to the improvement in the Sharpe ratio of the complete portfolio.
The CAL using the optimal combination of stocks and bonds (see Figure 7.8 ) shows that one
can achieve an expected return of 13% (matching that of the stock portfolio) with a standard
deviation of 18%, which is less than the 20% standard deviation of the stock portfolio.
Portfolio P 74.39%
Stocks 44.63%
Bonds 29.76%
T-bills 25.61%
Figure 7.9 The proportions of the optimal complete portfolio
Standard Deviation (%)
0 5 10 15 20 25 30
Expected Return (%)
D
C
E
P
CAL(P)
Opportunity Set of Risky Assets
Optimal Risky Portfolio
Indifference Curve
Optimal Complete Portfolio2
0
4
6
8
10
12
14
16
18
rf = 5%
Figure 7.8 Determination of the optimal complete portfolio
The universe of available securities includes two risky stock funds, A and B, and T-bills. The data for the universe are as follows:
Expected Return Standard Deviation
A 10% 20% B 30 60 T-bills 5 0
The correlation coefficient between funds A and B is 2 .2.
a. Draw the opportunity set of funds A and B.
b. Find the optimal risky portfolio, P, and its expected return and standard deviation.
c. Find the slope of the CAL supported by T-bills and portfolio P.
d. How much will an investor with A 5 5 invest in funds A and B and in T-bills?
CONCEPT CHECK 7.3
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220 P A R T I I Portfolio Theory and Practice
8 When short sales are prohibited, single securities may lie on the frontier. For example, the security with the high-
est expected return must lie on the frontier, as that security represents the only way that one can obtain a return that high, and so it must also be the minimum-variance way to obtain that return. When short sales are feasible,
however, portfolios can be constructed that offer the same expected return and lower variance. These portfolios
typically will have short positions in low-expected-return securities.
Security Selection We can generalize the portfolio con-
struction problem to the case of many
risky securities and a risk-free asset.
As in the two risky assets example,
the problem has three parts. First,
we identify the risk–return combina-
tions available from the set of risky
assets. Next, we identify the optimal
portfolio of risky assets by finding
the portfolio weights that result in
the steepest CAL. Finally, we choose
an appropriate complete portfolio
by mixing the risk-free asset with
the optimal risky portfolio. Before
describing the process in detail, let
us first present an overview.
The first step is to determine the
risk–return opportunities available to
the investor. These are summarized
by the minimum-variance frontier of risky assets. This frontier is a graph
of the lowest possible variance that can be attained for a given portfolio expected return.
Given the input data for expected returns, variances, and covariances, we can calculate the
minimum-variance portfolio for any targeted expected return. The plot of these expected
return–standard deviation pairs is presented in Figure 7.10 .
Notice that all the individual assets lie to the right inside the frontier, at least when we
allow short sales in the construction of risky portfolios. 8 This tells us that risky portfolios
comprising only a single asset are inefficient. Diversifying investments leads to portfolios
with higher expected returns and lower standard deviations.
All the portfolios that lie on the minimum-variance frontier from the global minimum-
variance portfolio and upward provide the best risk–return combinations and thus are
candidates for the optimal portfolio. The part of the frontier that lies above the global
minimum-variance portfolio, therefore, is called the efficient frontier of risky assets. For any portfolio on the lower portion of the minimum-variance frontier, there is a portfolio
with the same standard deviation and a greater expected return positioned directly above it.
Hence the bottom part of the minimum-variance frontier is inefficient.
The second part of the optimization plan involves the risk-free asset. As before, we
search for the capital allocation line with the highest Sharpe ratio (that is, the steepest
slope) as shown in Figure 7.11 .
The CAL that is supported by the optimal portfolio, P, is tangent to the efficient fron- tier. This CAL dominates all alternative feasible lines (the broken lines that are drawn
through the frontier). Portfolio P, therefore, is the optimal risky portfolio.
7.4 The Markowitz Portfolio Optimization Model
Figure 7.10 The minimum-variance frontier of risky assets
E(r)
Minimum-Variance Frontier
Individual Assets
Global Minimum- Variance Portfolio
Efficient Frontier
σ
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E(r)
CAL(P) Efficient Frontier
P
rf
σ
Figure 7.11 The efficient frontier of risky assets with the optimal CAL
eXcel APPLICATIONS: Two–Security Model
T he accompanying spreadsheet can be used to mea- sure the return and risk of a portfolio of two risky
assets. The model calculates the return and risk for vary- ing weights of each security along with the optimal risky and minimum-variance portfolio. Graphs are automatically generated for various model inputs. The model allows you to specify a target rate of return and solves for optimal combinations using the risk-free asset and the optimal risky portfolio. The spreadsheet is constructed with the
two-security return data from Table 7.1 . This spreadsheet is available at www.mhhe.com/bkm. Excel Question
1. Suppose your target expected rate of return is 11%.
a. What is the lowest-volatility portfolio that provides that expected return?
b. What is the standard deviation of that portfolio? c. What is the composition of that portfolio?
0
Standard Deviation (%)
0 5
5
11
10 15 20 25 3530
Expected
Return (%)A B C D E F
1
2 Expected Standard Correlation
3 Return Deviation Coefficient Covariance
4 Security 1
5 Security 2
6 T-Bill
7
8 Weight Weight Expected Standard Reward to
9 Security 1 Security 2 Return Deviation Volatility
10
11
12
13
14
Asset Allocation Analysis: Risk and Return
0.08 0.12 0.3 0.0072
0.13 0.2
0.05 0
1 0 0.08000 0.12000 0.25000
0.9 0.1 0.08500 0.11559 0.30281
0.8 0.2 0.09000 0.11454 0.34922
0.7 0.3 0.09500 0.11696 0.38474
0.6 0.4 0.10000 0.407710.12264
221
Finally, in the last part of the problem the individual investor chooses the appropriate
mix between the optimal risky portfolio P and T-bills, exactly as in Figure 7.8 . Now let us consider each part of the portfolio construction problem in more detail. In
the first part of the problem, risk–return analysis, the portfolio manager needs as inputs
a set of estimates for the expected returns of each security and a set of estimates for the
covariance matrix. (In Part Five on security analysis we will examine the security valuation
techniques and methods of financial
analysis that analysts use. For now,
we will assume that analysts already
have spent the time and resources to
prepare the inputs.)
The portfolio manager is now
armed with the n estimates of E ( r i ) and the n 3 n estimates of the covari- ance matrix in which the n diagonal elements are estimates of the vari-
ances, si 2, and the n 2 2 n 5 n ( n 2 1)
off-diagonal elements are the esti-
mates of the covariances between
each pair of asset returns. (You can
verify this from Table 7.2 for the case
n 5 2.) We know that each covariance appears twice in this table, so actually
we have n ( n 2 1)/2 different covari- ance estimates. If our portfolio man-
agement unit covers 50 securities,
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222 P A R T I I Portfolio Theory and Practice
our security analysts need to deliver 50 estimates of expected returns, 50 estimates
of variances, and 50 3 49/2 5 1,225 different estimates of covariances. This is a daunting
task! (We show later how the number of required estimates can be reduced substantially.)
Once these estimates are compiled, the expected return and variance of any risky port-
folio with weights in each security, w i , can be calculated from the bordered covariance matrix or, equivalently, from the following extensions of Equations 7.2 and 7.3:
E(rp) 5 a n
i51 wiE(ri) (7.15)
sp 2 5 a
n
i51 a
n
j51 wiwj Cov(ri, rj) (7.16)
An extended worked example showing how to do this using a spreadsheet is presented in
Appendix A of this chapter.
We mentioned earlier that the idea of diversification is age-old. The phrase “don’t put all
your eggs in one basket” existed long before modern finance theory. It was not until 1952,
however, that Harry Markowitz published a formal model of portfolio selection embodying
diversification principles, thereby paving the way for his 1990 Nobel Prize in Economics. 9
His model is precisely step one of portfolio management: the identification of the efficient
set of portfolios, or the efficient frontier of risky assets. The principal idea behind the frontier set of risky portfolios is that, for any risk level,
we are interested only in that portfolio with the highest expected return. Alternatively, the
frontier is the set of portfolios that minimizes the variance for any target expected return.
Indeed, the two methods of computing the efficient set of risky portfolios are equiva-
lent. To see this, consider the graphical representation of these procedures. Figure 7.12
shows the minimum-variance frontier.
9 Harry Markowitz, “Portfolio Selection,” Journal of Finance, March 1952.
E(r)
E(r3)
E(r2)
E(r1)
σA σB σC
Efficient Frontier of Risky Assets
Global Minimum- Variance Portfolio
σ
Figure 7.12 The efficient portfolio set
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C H A P T E R 7 Optimal Risky Portfolios 223
The points marked by squares are the result of a variance-minimization program. We
first draw the constraints, that is, horizontal lines at the level of required expected returns.
We then look for the portfolio with the lowest standard deviation that plots on each hori-
zontal line—we look for the portfolio that will plot farthest to the left (smallest standard
deviation) on that line. When we repeat this for many levels of required expected returns,
the shape of the minimum-variance frontier emerges. We then discard the bottom (dashed)
half of the frontier, because it is inefficient.
In the alternative approach, we draw a vertical line that represents the standard devia-
tion constraint. We then consider all portfolios that plot on this line (have the same stan-
dard deviation) and choose the one with the highest expected return, that is, the portfolio
that plots highest on this vertical line. Repeating this procedure for many vertical lines
(levels of standard deviation) gives us the points marked by circles that trace the upper por-
tion of the minimum-variance frontier, the efficient frontier.
When this step is completed, we have a list of efficient portfolios, because the solution
to the optimization program includes the portfolio proportions, w i , the expected return, E ( r p ), and the standard deviation, s p .
Let us restate what our portfolio manager has done so far. The estimates generated by
the security analysts were transformed into a set of expected rates of return and a covari-
ance matrix. This group of estimates we shall call the input list. This input list is then fed into the optimization program.
Before we proceed to the second step of choosing the optimal risky portfolio from the
frontier set, let us consider a practical point. Some clients may be subject to additional
constraints. For example, many institutions are prohibited from taking short positions in
any asset. For these clients the portfolio manager will add to the optimization program
constraints that rule out negative (short) positions in the search for efficient portfolios.
In this special case it is possible that single assets may be, in and of themselves, efficient
risky portfolios. For example, the asset with the highest expected return will be a frontier
portfolio because, without the opportunity of short sales, the only way to obtain that rate of
return is to hold the asset as one’s entire risky portfolio.
Short-sale restrictions are by no means the only such constraints. For example, some
clients may want to ensure a minimal level of expected dividend yield from the optimal
portfolio. In this case the input list will be expanded to include a set of expected dividend
yields d 1 , . . . , d n and the optimization program will include an additional constraint that ensures that the expected dividend yield of the portfolio will equal or exceed the desired
level, d. Portfolio managers can tailor the efficient set to conform to any desire of the client. Of
course, any constraint carries a price tag in the sense that an efficient frontier constructed
subject to extra constraints will offer a Sharpe ratio inferior to that of a less constrained
one. The client should be made aware of this cost and should carefully consider constraints
that are not mandated by law.
Another type of constraint is aimed at ruling out investments in industries or countries
considered ethically or politically undesirable. This is referred to as socially responsible investing, which entails a cost in the form of a lower Sharpe ratio on the resultant con- strained, optimal portfolio. This cost can be justifiably viewed as a contribution to the
underlying cause.
Capital Allocation and the Separation Property Now that we have the efficient frontier, we proceed to step two and introduce the risk-
free asset. Figure 7.13 shows the efficient frontier plus three CALs representing various-
portfolios from the efficient set. As before, we ratchet up the CAL by selecting different
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eXcel APPLICATIONS: Optimal Portfolios
A spreadsheet model featuring optimal risky portfolios is available on the Online Learning Center at www
.mhhe.com/bkm. It contains a template that is similar to the template developed in this section. The model can be used to find optimal mixes of securities for targeted levels of returns for both restricted and unrestricted portfolios. Graphs of the efficient frontier are generated for each set of inputs. The example available at our Web site applies the model to portfolios constructed from equity indexes (called WEBS securities) of several countries.
Excel Questions
1. Find the optimal risky portfolio formed from the eight coun- try index portfolios using the data provided in this box. What is the mean and variance of that portfolio’s rate of return?
2. Does the optimal risky portfolio entail a short position in any index? If it does, redo Question 1 but now impose a con- straint barring short positions. Explain why this constrained portfolio offers a less attractive risk-return trade-off than the unconstrained portfolio in Question 1.
A B C D E F
1
2
Standard3
Deviation4
5
6
7
8
9
10
11
12
Mean
ReturnWEBS
EWD
EWH
EWI
EWJ
EWL
EWP
EWW
S&P 500
Country
Sweden
Hong Kong
Italy
Japan
Switzerland
Spain
Mexico
15.5393
6.3852
26.5999
1.4133
18.0745
18.6347
16.2243
17.2306
26.4868
41.1475
26.0514
26.0709
21.6916
25.0779
38.7686
17.1944
Efficient Frontier for World Equity Benchmark Securities (WEBS)
224
portfolios until we reach portfolio P, which is the tangency point of a line from F to the efficient frontier. Portfolio P maximizes the Sharpe ratio, the slope of the CAL from F to portfolios on the efficient frontier. At this point our portfolio manager is done. Portfolio P is
the optimal risky portfolio for the manager’s clients.
There is yet another way to find the best risky
portfolio, achievable by introducing the risk-free
(T-bill) rate from the outset. In this approach, we
ask the spreadsheet program to maximize the Sharpe
ratio of portfolio P. The reason this is worth mention- ing is that we can skip the charting of the efficient
frontier altogether and proceed directly to find the
portfolio that produces the steepest CAL. The pro-
gram maximizes the Sharpe ratio with no constraint
on expected return or variance at all (using just the
feasibility constraint that portfolio weights sum to
1.0). Examination of Figure 7.13 shows that the solu-
tion strategy is to find the portfolio producing the
highest slope of the CAL (Sharpe ratio) regardless of
expected return or SD. Expected return and standard
deviation are easily computed from the optimal port-
folio weights applied to the input list in Equations
7.15 and 7.16.
While this last approach does not immediately
produce the entire minimum-variance frontier, this
E(r)
(Global Minimum-Variance Portfolio)
Efficient Frontier of Risky Assets
CAL(P)
CAL(A)
CAL(G)
A
G
P
F
σ
Figure 7.13 Capital allocation lines with various portfolios from the efficient set
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C H A P T E R 7 Optimal Risky Portfolios 225
shortcoming can be rectified by directly identifying two portfolios on the frontier. The first
is the already familiar Global Minimum Variance portfolio, identified in Figure 7.12 as
G. Portfolio G is achieved by minimizing variance without any constraint on the expected return; check this in Figure 7.13 . The expected return on portfolio G is higher than the risk- free rate (its risk premium will be positive).
Another portfolio that will be of great interest to us later is the portfolio on the ineffi-
cient portion of the minimum-variance frontier with zero covariance (or correlation) with
the optimal risky portfolio. We will call this portfolio Z. Once we identify portfolio P, we can find portfolio Z by solving in Excel for the portfolio that minimizes standard deviation subject to having zero covariance with P. In later chapters we will return to this portfolio.
An important property of frontier portfolios is that any portfolio formed by combining
two portfolios from the minimum-variance frontier will also be on that frontier, with loca-
tion along the frontier depending on the weights of that mix. Therefore, portfolio P plus either G or Z can be used to easily trace out the entire efficient frontier.
This is a good time to ponder our results and their implementation. The most striking
conclusion of all this analysis is that a portfolio manager will offer the same risky portfo-
lio, P, to all clients regardless of their degree of risk aversion. 10 The degree of risk aversion of the client comes into play only in capital allocation, the selection of the desired point
along the CAL. Thus the only difference between clients’ choices is that the more risk- averse client will invest more in the risk-free asset and less in the optimal risky portfolio
than will a less risk-averse client. However, both will use portfolio P as their optimal risky investment vehicle.
This result is called a separation property; it tells us that the portfolio choice problem may be separated into two independent tasks.
11 The first task, determination of the optimal
risky portfolio, is purely technical. Given the manager’s input list, the best risky portfolio
is the same for all clients, regardless of risk aversion. However, the second task, capital
allocation, depends on personal preference. Here the client is the decision maker.
The crucial point is that the optimal portfolio P that the manager offers is the same for all clients. Put another way, investors with varying degrees of risk aversion would be satis-
fied with a universe of only two mutual funds: a money market fund for risk-free invest-
ments and a mutual fund that holds the optimal risky portfolio, P, on the tangency point of the CAL and the efficient frontier. This result makes professional management more
efficient and hence less costly. One management firm can serve any number of clients with
relatively small incremental administrative costs.
In practice, however, different managers will estimate different input lists, thus deriv-
ing different efficient frontiers, and offer different “optimal” portfolios to their clients.
The source of the disparity lies in the security analysis. It is worth mentioning here that the
universal rule of GIGO (garbage in–garbage out) also applies to security analysis. If
the quality of the security analysis is poor, a passive portfolio such as a market index fund
will result in a higher Sharpe ratio than an active portfolio that uses low-quality security
analysis to tilt portfolio weights toward seemingly favorable (mispriced) securities.
One particular input list that would lead to a worthless estimate of the efficient frontier
is based on recent security average returns. If sample average returns over recent years are
10
Clients who impose special restrictions (constraints) on the manager, such as dividend yield, will obtain another
optimal portfolio. Any constraint that is added to an optimization problem leads, in general, to a different and
inferior optimum compared to an unconstrained program.
11
The separation property was first noted by Nobel laureate James Tobin, “Liquidity Preference as Behavior
toward Risk,” Review of Economic Statistics 25 (February 1958), pp. 65–86.
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226 P A R T I I Portfolio Theory and Practice
used as proxies for the future return on the security, the noise in those estimates will make
the resultant efficient frontier virtually useless for portfolio construction.
Consider a stock with an annual standard deviation of 50%. Even if one were to use
a 10-year average to estimate its expected return (and 10 years is almost ancient his-
tory in the life of a corporation), the standard deviation of that estimate would still be
50/"10 5 15.8%. The chances that this average represents expected returns for the com- ing year are negligible.
12 In Chapter 25, we demonstrate that efficient frontiers constructed
from past data may be wildly optimistic in terms of the apparent opportunities they offer to improve Sharpe ratios.
As we have seen, optimal risky portfolios for different clients also may vary because
of portfolio constraints such as dividend-yield requirements, tax considerations, or other
client preferences. Nevertheless, this analysis suggests that a limited number of portfolios
may be sufficient to serve the demands of a wide range of investors. This is the theoretical
basis of the mutual fund industry.
The (computerized) optimization technique is the easiest part of the portfolio construc-
tion problem. The real arena of competition among portfolio managers is in sophisticated
security analysis. This analysis, as well as its proper interpretation, is part of the art of
portfolio construction. 13
12
Moreover, you cannot avoid this problem by observing the rate of return on the stock more frequently. In
Chapter 5 we pointed out that the accuracy of the sample average as an estimate of expected return depends on
the length of the sample period, and is not improved by sampling more frequently within a given sample period.
13
You can find a nice discussion of some practical issues in implementing efficient diversification in a white
paper prepared by Wealthcare Capital Management at this address: www.financeware.com/ruminations/ WP_EfficiencyDeficiency.pdf. A copy of the report is also available at the Online Learning Center for this text, www.mhhe.com/bkm.
Suppose that two portfolio managers who work for competing investment management houses each employ a group of security analysts to prepare the input list for the Markowitz algorithm. When all is com- pleted, it turns out that the efficient frontier obtained by portfolio manager A dominates that of manager B. By dominate, we mean that A ’s optimal risky portfolio lies northwest of B ’s. Hence, given a choice, inves- tors will all prefer the risky portfolio that lies on the CAL of A.
a. What should be made of this outcome?
b. Should it be attributed to better security analysis by A ’s analysts?
c. Could it be that A ’s computer program is superior?
d. If you were advising clients (and had an advance glimpse at the efficient frontiers of various manag- ers), would you tell them to periodically switch their money to the manager with the most northwest- erly portfolio?
CONCEPT CHECK 7.4
The Power of Diversification Section 7.1 introduced the concept of diversification and the limits to the benefits of diver-
sification resulting from systematic risk. Given the tools we have developed, we can recon-
sider this intuition more rigorously and at the same time sharpen our insight regarding the
power of diversification.
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C H A P T E R 7 Optimal Risky Portfolios 227
Recall from Equation 7.16, restated here, that the general formula for the variance of a
portfolio is
sp 2 5 a
n
i51 a
n
j51 wiwj Cov(ri , rj ) (7.16)
Consider now the naive diversification strategy in which an equally weighted portfolio is constructed, meaning that w i 5 1/ n for each security. In this case Equation 7.16 may be rewritten as follows, where we break out the terms for which i 5 j into a separate sum, noting that Cov(ri, ri) 5 si
2:
sp 2 5
1
n a n
i51
1
n si
2 1 a n
j51 j2i
a n
i51
1
n2 Cov(ri, rj) (7.17)
Note that there are n variance terms and n ( n 2 1) covariance terms in Equation 7.17. If we define the average variance and average covariance of the securities as
s 2 5 1
n a n
i51 si
2 (7.18)
Cov 5 1
n(n 2 1) a n
j51 j2i
a n
i51 Cov(ri, rj) (7.19)
we can express portfolio variance as
sp 2 5
1
n s 2 1
n 2 1 n
Cov (7.20)
Now examine the effect of diversification. When the average covariance among security
returns is zero, as it is when all risk is firm-specific, portfolio variance can be driven to zero.
We see this from Equation 7.20. The second term on the right-hand side will be zero in this
scenario, while the first term approaches zero as n becomes larger. Hence when security returns are uncorrelated, the power of diversification to reduce portfolio risk is unlimited.
However, the more important case is the one in which economywide risk factors impart
positive correlation among stock returns. In this case, as the portfolio becomes more highly
diversified ( n increases), portfolio variance remains positive. Although firm-specific risk, represented by the first term in Equation 7.20, is still diversified away, the second term
simply approaches Cov as n becomes greater. [Note that ( n 2 1)/ n 5 1 2 1/ n, which approaches 1 for large n. ] Thus the irreducible risk of a diversified portfolio depends on the covariance of the returns of the component securities, which in turn is a function of the
importance of systematic factors in the economy.
To see further the fundamental relationship between systematic risk and security corre-
lations, suppose for simplicity that all securities have a common standard deviation, s , and
all security pairs have a common correlation coefficient, r . Then the covariance between
all pairs of securities is r s 2 , and Equation 7.20 becomes
sp 2 5
1
n s2 1
n 2 1 n
rs2 (7.21)
The effect of correlation is now explicit. When r 5 0, we again obtain the insurance
principle, where portfolio variance approaches zero as n becomes greater. For r . 0, how- ever, portfolio variance remains positive. In fact, for r 5 1, portfolio variance equals s
2
regardless of n, demonstrating that diversification is of no benefit: In the case of perfect
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228 P A R T I I Portfolio Theory and Practice
correlation, all risk is systematic. More generally, as n becomes greater, Equation 7.21 shows that systematic risk becomes r s
2 .
Table 7.4 presents portfolio standard deviation as we include ever-greater numbers of
securities in the portfolio for two cases, r 5 0 and r 5 .40. The table takes s to be 50%. As
one would expect, portfolio risk is greater when r 5 .40. More surprising, perhaps, is that
portfolio risk diminishes far less rapidly as n increases in the positive correlation case. The correlation among security returns limits the power of diversification.
Note that for a 100-security portfolio, the standard deviation is 5% in the uncorrelated
case—still significant compared to the potential of zero standard deviation. For r 5 .40,
the standard deviation is high, 31.86%, yet it is very close to undiversifiable systematic
risk in the infinite-sized security universe, "rs2 5 ".4 3 502 5 31.62%. At this point, further diversification is of little value.
Perhaps the most important insight from the exercise is this: When we hold diversified
portfolios, the contribution to portfolio risk of a particular security will depend on the
covariance of that security’s return with those of other securities, and not on the security’s variance. As we shall see in Chapter 9, this implies that fair risk premiums also should
depend on covariances rather than total variability of returns.
Table 7.4
Risk reduction of equally weighted portfolios in correlated and uncorrelated universes
r 5 0 r 5 .40
Universe Size n
Portfolio Weights
w 5 1/n (%)
Standard Deviation
(%) Reduction
in s Standard
Deviation (%) Reduction in s
1 100 50.00 14.64 50.00 8.17 2 50 35.36 41.83 5 20 22.36 1.95 36.06 0.70 6 16.67 20.41 35.36
10 10 15.81 0.73 33.91 0.20 11 9.09 15.08 33.71 20 5 11.18 0.27 32.79 0.06 21 4.76 10.91 32.73
100 1 5.00 0.02 31.86 0.00 101 0.99 4.98 31.86
Suppose that the universe of available risky securities consists of a large number of stocks, identically dis- tributed with E ( r ) 5 15%, s 5 60%, and a common correlation coefficient of r 5 .5.
a. What are the expected return and standard deviation of an equally weighted risky portfolio of 25 stocks?
b. What is the smallest number of stocks necessary to generate an efficient portfolio with a standard deviation equal to or smaller than 43%?
c. What is the systematic risk in this security universe?
d. If T-bills are available and yield 10%, what is the slope of the CML? (Because of the symmetry assumed for all securities in the investment universe, the market index in this economy will be an equally weighted portfolio of all stocks.)
CONCEPT CHECK 7.5
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C H A P T E R 7 Optimal Risky Portfolios 229
Asset Allocation and Security Selection As we have seen, the theories of security selection and asset allocation are identical. Both
activities call for the construction of an efficient frontier, and the choice of a particular
portfolio from along that frontier. The determination of the optimal combination of secu-
rities proceeds in the same manner as the analysis of the optimal combination of asset
classes. Why, then, do we (and the investment community) distinguish between asset allo-
cation and security selection?
Three factors are at work. First, as a result of greater need and ability to save (for col-
lege educations, recreation, longer life in retirement, health care needs, etc.), the demand
for sophisticated investment management has increased enormously. Second, the widen-
ing spectrum of financial markets and financial instruments has put sophisticated invest-
ment beyond the capacity of many amateur investors. Finally, there are strong economies
of scale in investment analysis. The end result is that the size of a competitive invest-
ment company has grown with the industry, and efficiency in organization has become an
important issue.
A large investment company is likely to invest both in domestic and international mar-
kets and in a broad set of asset classes, each of which requires specialized expertise. Hence
the management of each asset-class portfolio needs to be decentralized, and it becomes
impossible to simultaneously optimize the entire organization’s risky portfolio in one
stage, although this would be prescribed as optimal on theoretical grounds. In future chap- ters we will see how optimization of decentralized portfolios can be mindful as well of the
entire portfolio of which they are a part.
The practice is therefore to optimize the security selection of each asset-class portfo-
lio independently. At the same time, top management continually updates the asset allo-
cation of the organization, adjusting the investment budget allotted to each asset-class
portfolio.
Optimal Portfolios and Nonnormal Returns The portfolio optimization techniques we have used so far assume normal distributions
of returns in that standard deviation is taken to be a fully adequate measure of risk.
However, potential nonnormality of returns requires us to pay attention as well to risk
measures that focus on worst-case losses such as value at risk (VaR) or expected short-
fall (ES).
In Chapter 6 we suggested that capital allocation to the risky portfolio should be recon-
sidered in the face of fat-tailed distributions that can result in extreme values of VaR and
ES. Specifically, forecasts of greater than normal VaR and ES should encourage more
moderate capital allocations to the risky portfolio. Accounting for the effect of diversifica-
tion on VaR and ES would be useful as well. Unfortunately, the impact of diversification
on tail risk cannot be easily anticipated.
A practical way to estimate values of VaR and ES in the presence of fat tails is boot-
strapping (described in Section 5.9). We start with a historical sample of returns of each
asset in our prospective portfolio. We compute the portfolio return corresponding to a draw
of one return from each asset’s history. We thus calculate as many of these random port-
folio returns as we wish. Fifty thousand returns produced in this way can provide a good
estimate of VaR and ES values. The forecasted values for VaR and ES of the mean-variance
optimal portfolio can then be compared to other candidate portfolios. If these other portfo-
lios yield sufficiently better VaR and ES values, we may prefer one of those to the mean-
variance efficient portfolio.
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230 P A R T I I Portfolio Theory and Practice
7.5 Risk Pooling, Risk Sharing, and the Risk of Long-Term Investments *
Diversification means that we spread our investment budget across a variety of assets and
thus limit overall risk. Sometimes it is argued that spreading investments across time, so
that average performance reflects returns in several investment periods, offers an analo-
gous benefit dubbed “time diversification.” A common belief is that time diversification
can make long-term investing safer.
Is this extension of diversification to investments over time valid? The question of how
risk increases when the horizon of a risky investment lengthens is analogous to risk pool-
ing, the process by which an insurance company aggregates a large portfolio (or pool) of
uncorrelated risks. However, the application of risk pooling to investment risk is widely
misunderstood, as is the application of “the insurance principle” to long-term investments.
In this section, we try to clarify these issues and explore the appropriate extension of the
insurance principle to investment risk.
Risk Pooling and the Insurance Principle Risk pooling means merging uncorrelated, risky projects as a means to reduce risk. Applied to the insurance business, risk pooling entails selling many uncorrelated insurance
policies. This application of risk pooling has come to be known as the insurance principle.
Conventional wisdom holds that risk pooling reduces risk, and that such pooling is the
driving force behind risk management for the insurance industry.
But even brief reflection should convince you that risk pooling cannot be the entire
story. How can adding bets that are independent of your other bets reduce your total expo- sure to risk? This would be little different from a gambler in Las Vegas arguing that a few
more trips to the roulette table will reduce his total risk by diversifying his overall “port-
folio” of wagers. You would immediately realize that the gambler now has more money at
stake, and his overall potential swing in wealth is clearly wider: While his average gain or
loss per bet may become more predictable as he repeatedly returns to the table, his total proceeds become less so. As we will see, the insurance principle is sometimes similarly
misapplied to long-term investments by incorrectly extending what it implies about aver- age returns to predictions about total returns.
Imagine a rich investor, Warren, who holds a $1 billion portfolio, P. The fraction of the portfolio invested in a risky asset, A, is y, leaving the fraction 1 2 y invested in the risk- free rate. Asset A ’s risk premium is R, and its standard deviation is s . From Equations 6.3 and 6.4, the risk premium of the complete portfolio P is R P 5 yR, its standard deviation is s P 5 y s , and the Sharpe ratio is S P 5 R / s . Now Warren identifies another risky asset, B, with the same risk premium and standard deviation as A. Warren estimates that the correla- tion (and therefore covariance) between the two investments is zero, and he is intrigued at
the potential this offers for risk reduction through diversification.
Given the benefits that Warren anticipates from diversification, he decides to take a
position in asset B equal in size to his existing position in asset A. He therefore transfers another fraction, y, of wealth from the risk-free asset to asset B. This leaves his total port- folio allocated as follows: The fraction y is still invested in asset A, an additional invest- ment of y is invested in B, and 1 2 2 y is in the risk-free asset. Notice that this strategy is
*The material in this section is more challenging. It may be skipped without impairing the ability to understand
later chapters.
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C H A P T E R 7 Optimal Risky Portfolios 231
analogous to pure risk pooling; Warren has taken on additional risky (albeit uncorrelated)
bets, and his risky portfolio is larger than it was previously. We will denote Warren’s new
portfolio as Z. We can compute the risk premium of portfolio Z from Equation 7.2, its variance from
Equation 7.3, and thus its Sharpe ratio. Remember that capital R denotes the risk premium of each asset and the risk premium of the risk-free asset is zero. When calculating portfolio
variance, we use the fact that covariance is zero. Thus, for Portfolio Z:
RZ 5 yR 1 yR 1 (1 2 2y)0 5 2yR (double RP)
sZ 2 5 y 2s2 1 y2s2 1 0 5 2y2s 2 (double the variance of P)
sZ 5 "sZ2 5 ys"2 ("2 5 1.41 times the standard deviation of P) SZ 5 RZ/sZ 5 2yR/( ys"2) 5 "2R/s ("2 5 1.41 times Sharpe ratio of P)
The good news from these results is that the Sharpe ratio of Z is higher than that of P by the factor "2. Its excess rate of return is double that of P, yet its standard deviation is only "2 times larger. The bad news is that by increasing the scale of the risky investment, the standard deviation of the portfolio also increases by "2.
We might now imagine that instead of two uncorrelated assets, Warren has access to
many. Repeating our analysis, we would find that with n assets the Sharpe ratio under strategy Z increases (relative to its original value) by a factor of "n to "n 3 R/s. But the total risk of the pooling strategy Z will increase by the same multiple, to s"n.
This analysis illustrates both the opportunities and limitations of pure risk pooling:
Pooling increases the scale of the risky investment (from y to 2 y ) by adding an additional position in another, uncorrelated asset. This addition of another risky bet also increases the
size of the risky budget. So risk pooling by itself does not reduce risk, despite the fact that it benefits from the lack of correlation across policies.
The insurance principle tells us only that risk increases less than proportionally to the
number of policies insured when the policies are uncorrelated; hence profitability—in this
application, the Sharpe ratio—increases. But this effect does not actually reduce risk.
This might limit the potential economies of scale of an ever-growing portfolio such
as that of a large insurer. You can interpret each “asset” in our analysis as one insurance
policy. Each policy written requires the insurance company to set aside additional capital
to cover potential losses. The insurance company invests its capital until it needs to pay out
on claims. Selling more policies entails increasing the total position in risky investments
and therefore the capital that must be allocated to those policies. As the company invests
in more uncorrelated assets (insurance policies), the Sharpe ratio continuously increases
(which is good), but since more funds are invested in risky policies, the overall risk of the
portfolio rises (which is bad). As the number of policies grows, the risk of the pool will
certainly grow—despite “diversification” across policies. Eventually, that growing risk
will overwhelm the company’s available capital.
Insurance analysts often think in terms of probability of loss. Their mathematically cor-
rect interpretation of the insurance principle is that the probability of loss declines with risk
pooling. This interpretation relates to the fact that the Sharpe ratio (profitability) increases
with risk pooling. But to equate the declining probability of loss to reduction in total risk
is erroneous; the latter is measured by overall standard deviation, which increases with
risk pooling. (Again, think about the gambler in Las Vegas. As he returns over and over
again to the roulette table, the probability that he will lose becomes ever more certain, but
the magnitude of potential dollar gains or losses becomes ever greater.) Thus risk pooling
allows neither investors nor insurance companies to shed risk. However, the increase in
risk can be overcome when risk pooling is augmented by risk sharing, as discussed in the next subsection.
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232 P A R T I I Portfolio Theory and Practice
Risk Sharing Now think about a variation on the risk pooling portfolio Z. Imagine that Warren has iden- tified several attractive insurance policies and wishes to invest in all of them. For simplic-
ity, we will look at the case of two policies, so the pool will have the same properties as
portfolio Z. We saw that if Warren invested in this two-policy pool, his total risk would be sZ 5 ys"2. But if this is more risk than he is willing to bear, what might he do?
His solution is risk sharing, the act of selling shares in an attractive risky portfolio to limit risk and yet maintain the Sharpe ratio (profitability) of the resultant position. Suppose
that every time a new risky asset is added to the portfolio, Warren sells off a portion of
his investment in the pool to maintain the total funds invested in risky assets unchanged.
For example, when a second asset is added, he sells half of his position to other investors.
While the total investment budget directed into risky assets is therefore unchanged, it is
equally divided between assets A and B, with weights in each of y /2. In following this strat- egy, the risk-free component of his complete portfolio remains fixed with weight 1 2 y. We will call this strategy V.
If you compare the risk-pooling strategy Z with the risk-pooling-plus-risk-sharing strat- egy V, you will notice that they both entail an investment in the pool of two assets; the only difference between them is that the risk-sharing strategy sells off half the combined pool to
maintain a risky portfolio of fixed size. While the weight of the total risky pool in strategy Z is 2 y, in the risk-sharing strategy, the risky weight is only one-half that level. Therefore, we can find the properties of the risk-sharing portfolio by substituting y for 2 y in each formula or, equivalently, substituting y /2 for y in the following table.
oiloftroP:gnilooPksiR Z oiloftroP:gnirahSksiR V
RZ 2 R RV 2(y/2)R yR
Z 2 5
5
2y 2 2 V 2 5
5
2 y /2 2 2 5 5
y 2 2 /2
Z 5 Z 2 5 y 2 V 5 V
2 5 y / 2 S ( )Z 5 RZ/ Z 5 2yR/ y 2 5 2R/ SV 5 RV/ V 5 2R/
y
We observe that portfolio V matches the attractive Sharpe ratio of portfolio Z, but with lower volatility. Thus risk sharing combined with risk pooling is the key to the insurance industry. True diversification means spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio.
To control his total risk, Warren had to sell off a fraction of the pool of assets. This
implies that a portion of those assets must now be held by someone else. For example,
if the assets are insurance policies, other investors must be sharing the risk, perhaps by
buying shares in the insurance company. Alternatively, insurance companies commonly
“reinsure” their risk by selling off portions of the policies to other investors or insurance
companies, thus explicitly sharing the risk.
We can easily generalize Warren’s example to the case of more than two assets.
Suppose the risky pool has n assets. Then the volatility of the risk-sharing portfolio will be sV 5 ys/"n, and its Sharpe ratio will be "nR /s. Clearly, both of these improve as n increases. Think back to our gambler at the roulette wheel one last time. He was wrong to
argue that diversification means that 100 bets are less risky than 1 bet. His intuition would
be correct, however, if he shared those 100 bets with 100 of his friends. A 1/100 share of
100 bets is in fact less risky than one bet. Fixing the amount of his total money at risk as
that money is spread across more independent bets is the way for him to reduce risk. 14
14
For the Las Vegas gambler, risk sharing makes the gambles ever more certain to produce a negative rate of return, highlighting the illness that characterizes compulsive gambling.
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C H A P T E R 7 Optimal Risky Portfolios 233
With risk sharing, one can set up an insurance company of any size, amassing a large
portfolio of policies and limiting total risk by selling shares among many investors. As the
Sharpe ratio steadily increases with the number of policies written, while the risk to each
diversified shareholder falls, the size of ever-more-profitable insurance companies appears
unlimited. In reality, however, two problems put a damper on this process. First, burdens
related to problems of managing very large firms will sooner or later eat into the increased
gross margins. More important, the issue of “too big to fail” may emerge. The possibility of
error in assessing the risk of each policy or misestimating the correlations across losses on
the pooled policies (or worse yet, intentional underestimation of risk) can cause an insur-
ance company to fail. As we saw in Chapter 1, too big to fail means that such failure can
lead to related failures among the firm’s trading partners. This is similar to what happened
in the financial crisis of 2008. The jury is still out on the role of lack of scruples in this affair.
It is hoped that future regulation will put real limits on exaggerated optimism concerning
the power of diversification to limit risk, despite the appealing mitigation of risk sharing.
Investment for the Long Run Now we turn to the implications of risk pooling and risk sharing for long-term investing.
Think of extending an investment horizon for another period (which adds the uncertainty
of that period’s risky return) as analogous to adding another risky asset or insurance policy
to a pool of assets.
Examining the impact of an extension of the investment horizon requires us to clarify
what the alternative is. Suppose you consider an investment in a risky portfolio over the
next 2 years, which we’ll call the “long-term investment.” How should you compare this
decision to a “short-run investment”? We must compare these two strategies over the same
period, that is, 2 years. The short-term investment therefore must be interpreted as invest-
ing in the risky portfolio over 1 year and in the risk-free asset over the other.
Once we agree on this comparison, and assuming the risky return on the first year is
uncorrelated with that of the second, it becomes clear that the “long-term” strategy is
analogous to portfolio Z. This is because holding on to the risky investment in the sec- ond year (rather than withdrawing to the risk-free rate) piles up more risk, just as selling
another insurance policy does. Put differently, the long-term investment may be considered
analogous to risk pooling. While extending a risky investment to the long run improves the
Sharpe ratio (as does risk pooling), it also increases risk. Thus “time diversification” is not
really diversification.
The more accurate analogy to risk sharing for a long-term horizon is to spread the risky
investment budget across each of the investment periods. Compare the following three
strategies applied to the whole investment budget over a 2-year horizon:
1. Invest the whole budget at risk for one period, and then withdraw the entire
proceeds, placing them in a risk-free asset in the other period. Because you
are invested in the risky asset for only 1 year, the risk premium over the whole
investment period is R, the 2-year SD is s , and the 2-year Sharpe ratio is S 5 R / s .
2. Invest the whole budget in the risky asset for both periods. The 2-year risk premium
is 2 R (assuming continuously compounded rates), the 2-year variance is 2 s 2 , the
2-year SD is s"2, and the 2-year Sharpe ratio is S 5 R"2/s. This is analogous to risk pooling, taking two “bets” on the risky portfolio instead of one (as in Strategy 1).
3. Invest half the investment budget in the risky position in each of two periods, plac-
ing the remainder of funds in the risk-free asset. The 2-year risk premium is R,
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234 P A R T I I Portfolio Theory and Practice
the 2-year variance is 2 3 (½ s ) 2 5 s
2 /2, the SD is s/"2, and the Sharpe ratio is
S 5 R"2/s. This is analogous to risk sharing, taking a fractional position in each year’s investment return.
Strategy 3 is less risky than either alternative. Its expected total return equals Strategy 1’s,
yet its risk is lower and therefore its Sharpe ratio is higher. It achieves the same Sharpe
ratio as Strategy 2 but with standard deviation reduced by a factor of 2. In summary, its
Sharpe ratio is at least as good as either alternative and, more to the point, its total risk is
less than either.
We conclude that risk does not fade in the long run. An investor who can invest in an
attractive portfolio for only one period, and chooses to invest a given budget in that period,
would find it preferable to put money at risk in that portfolio in as many periods as allowed
but will decrease the risky budget in each period. Simple risk pooling, or in this case, time diversification, does not reduce risk.
1. The expected return of a portfolio is the weighted average of the component security expected returns with the investment proportions as weights.
2. The variance of a portfolio is the weighted sum of the elements of the covariance matrix with the product of the investment proportions as weights. Thus the variance of each asset is weighted by
the square of its investment proportion. The covariance of each pair of assets appears twice in the
covariance matrix; thus the portfolio variance includes twice each covariance weighted by the
product of the investment proportions in each of the two assets.
3. Even if the covariances are positive, the portfolio standard deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively
correlated. Thus portfolio diversification is of value as long as assets are less than perfectly
correlated.
4. The greater an asset’s covariance with the other assets in the portfolio, the more it contributes to portfolio variance. An asset that is perfectly negatively correlated with a portfolio can serve as a
perfect hedge. The perfect hedge asset can reduce the portfolio variance to zero.
5. The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfolio on the efficient
frontier.
6. A portfolio manager identifies the efficient frontier by first establishing estimates for asset expected returns and the covariance matrix. This input list is then fed into an optimization pro-
gram that reports as outputs the investment proportions, expected returns, and standard deviations
of the portfolios on the efficient frontier.
7. In general, portfolio managers will arrive at different efficient portfolios because of differences in methods and quality of security analysis. Managers compete on the quality of their security
analysis relative to their management fees.
8. If a risk-free asset is available and input lists are identical, all investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors
with identical input lists will hold an identical risky portfolio, differing only in how much each
allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the
separation principle of portfolio construction.
9. Diversification is based on the allocation of a fixed portfolio across several assets, limiting the exposure to any one source of risk. Adding additional risky assets to a portfolio, thereby
increasing the total amounts invested, does not reduce dollar risk, even if it makes the rate of
return more predictable. This is because that uncertainty is applied to a larger investment base.
SUMMARY
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C H A P T E R 7 Optimal Risky Portfolios 235
Nor does investing over longer horizons reduce risk. Increasing the investment horizon is analo-
gous to investing in more assets. It increases total risk. Analogously, the key to the insurance
industry is risk sharing—the spreading of risk across many investors, each of whom takes on
only a small exposure to any given source of risk. Risk pooling—the assumption of ever-more
sources of risk—may increase rate of return predictability, but not the p redictability of total
dollar returns.
Related Web sites for this chapter are available at www. mhhe.com/bkm
diversification
insurance principle
market risk
systematic risk
nondiversifiable risk
unique risk
firm-specific risk
nonsystematic risk
diversifiable risk
minimum-variance portfolio
portfolio opportunity set
Sharpe ratio
optimal risky portfolio
minimum-variance frontier
efficient frontier of risky
assets
input list
separation property
risk pooling
risk sharing
KEY TERMS
PROBLEM SETS 1. Which of the following factors reflect pure market risk for a given corporation? a. Increased short-term interest rates. b. Fire in the corporate warehouse. c. Increased insurance costs. d. Death of the CEO. e. Increased labor costs.
2. When adding real estate to an asset allocation program that currently includes only stocks, bonds,
and cash, which of the properties of real estate returns affect portfolio risk ? Explain.
a. Standard deviation. b. Expected return. c. Correlation with returns of the other asset classes.
3. Which of the following statements about the minimum variance portfolio of all risky securities
are valid? (Assume short sales are allowed.) Explain.
a. Its variance must be lower than those of all other securities or portfolios. b. Its expected return can be lower than the risk-free rate. c. It may be the optimal risky portfolio.
d. It must include all individual securities.
Basic
KEY EQUATIONS The expected rate of return on a portfolio: E ( r p ) 5 w D E ( r D ) 1 w E E ( r E ) The variance of the return on a portfolio: sp
2 5 (wDsD) 2 1 (wE sE)
2 1 2(wDsD)(wE sE)rD E
The Sharpe ratio of a portfolio: Sp 5 E(rp) 2 rf
sp
Sharpe ratio maximizing portfolio weights with two risky assets ( D and E ) and a risk-free asset:
wD 5 3E(rD) 2 rf 4sE2 2 3E(rE) 2 rf 4sDsE rD E3E(rD) 2 rf 4sE2 1 3E(rE) 2 rf 4sD2 2 3E(rD) 2 rf 1 E(rE) 2 rf 4sDsErDE
wE 5 1 2 wD
Optimal capital allocation to the risky asset, y: E(rp) 2 rf
Asp 2
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236 P A R T I I Portfolio Theory and Practice
The following data apply to Problems 4 through 10: A pension fund manager is con- sidering three mutual funds. The first is a stock fund, the second is a long-term govern- ment and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:
Expected Return Standard Deviation
Stock fund (S) 20% 30% Bond fund (B) 12 15
The correlation between the fund returns is .10.
4. What are the investment proportions in the minimum-variance portfolio of the two risky funds,
and what is the expected value and standard deviation of its rate of return?
5. Tabulate and draw the investment opportunity set of the two risky funds. Use investment pro-
portions for the stock fund of zero to 100% in increments of 20%.
6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the
expected return and standard deviation of the optimal portfolio?
7. Solve numerically for the proportions of each asset and for the expected return and standard
deviation of the optimal risky portfolio.
8. What is the Sharpe ratio of the best feasible CAL?
9. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the
best feasible CAL.
a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds?
10. If you were to use only the two risky funds, and still require an expected return of 14%, what
would be the investment proportions of your portfolio? Compare its standard deviation to that of
the optimized portfolio in Problem 9. What do you conclude?
11. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an
expected return of 10% with a standard deviation of 30%.
a. In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
b. Given the data above, reanswer ( a ) with the additional assumption that the correlation coef- ficient between gold and stocks equals 1. Draw a graph illustrating why one would or would
not hold gold in one’s portfolio. Could this set of assumptions for expected returns, standard
deviations, and correlation represent an equilibrium for the security market?
12. Suppose that there are many stocks in the security market and that the characteristics of stocks
A and B are given as follows:
Stock Expected Return Standard Deviation
A 10% 5% B 15 10
Correlation 5 21
Suppose that it is possible to borrow at the risk-free rate, r f . What must be the value of the risk- free rate? ( Hint: Think about constructing a risk-free portfolio from stocks A and B. )
13. Assume that expected returns and standard deviations for all securities (including the risk-free
rate for borrowing and lending) are known. In this case all investors will have the same optimal
risky portfolio. (True or false?)
14. The standard deviation of the portfolio is always equal to the weighted average of the standard
deviations of the assets in the portfolio. (True or false?)
15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3
chance of halving your investment in a year. What is the standard deviation of the rate of return
on this investment?
Intermediate
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C H A P T E R 7 Optimal Risky Portfolios 237
16. Suppose that you have $1 million and the following two opportunities from which to construct
a portfolio:
a. Risk-free asset earning 12% per year. b. Risky asset with expected return of 30% per year and standard deviation of 40%.
If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return?
The following data are for Problems 17 through 19: The correlation coefficients between pairs of stocks are as follows: Corr( A, B ) 5 .85; Corr( A, C ) 5 .60; Corr( A, D ) 5 .45. Each stock has an expected return of 8% and a standard deviation of 20%.
17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice):
a. B. c. D. b. C. d. Need more data.
18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant investors? Explain.
19. Suppose that in addition to investing in one more stock you can invest in T-bills as well. Would
you change your answers to Problems 17 and 18 if the T-bill rate is 8%?
The following table of compound annual returns by decade applies to Challenge Problems 20 and 21.
1920s* 1930s 1940s 1950s 1960s 1970s 1980s 1990s
Small-company stocks 23.72% 7.28% 20.63% 19.01% 13.72% 8.75% 12.46% 13.84%
Large-company stocks 18.36 21.25 9.11 19.41 7.84 5.90 17.60 18.20
Long-term government 3.98 4.60 3.59 0.25 1.14 6.63 11.50 8.60 Intermediate-term government
3.77 3.91 1.70 1.11 3.41 6.11 12.01 7.74
Treasury bills 3.56 0.30 0.37 1.87 3.89 6.29 9.00 5.02 Inflation 21.00 22.04 5.36 2.22 2.52 7.36 5.10 2.93
*Based on the period 1926–1929.
20. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns
for each asset class and for inflation. Also find the correlation between the returns of various
asset classes. What do the data indicate?
21. Convert the asset returns by decade presented in the table into real rates. Repeat the analysis of
Challenge Problem 20 for the real rates of return.
The following information applies to Problems 22 through 26: Greta, an elderly inves- tor, has a degree of risk aversion of A 5 3 when applied to return on wealth over a 3-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of 3-year strategies. (All rates are annual, continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a SD of 20%. The hedge fund risk premium is estimated at 10% with a SD of 35%. The return on each of these portfolios in any year is uncorrelated with its return or the return of any other portfo- lio in any other year. The hedge fund management claims the correlation coefficient between the annual returns on the S&P 500 and the hedge fund in the same year is zero, but Greta believes this is far from certain.
22. Compute the estimated 3-year risk premiums, SDs, and Sharpe ratios for the two portfolios.
23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what
would be the optimal asset allocation? What should be Greta’s capital allocation?
24. If the correlation coefficient between annual portfolio returns is 0.3, what is the annual covariance?
25. With correlation of 0.3, what is the covariance between the 3-year returns?
26. Repeat Problem 15 using an annual correlation of 0.3. (If you cannot calculate the 3-year covar-
iance in Problem 17, assume it is 0.05.)
Challenge
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238 P A R T I I Portfolio Theory and Practice
The following data apply to CFA Problems 1 through 3: Hennessy & Associates manages a $30 million equity portfolio for the multimanager Wilstead Pension Fund. Jason Jones, financial vice
president of Wilstead, noted that Hennessy had rather consistently achieved the best record among
the Wilstead’s six equity managers. Performance of the Hennessy portfolio had been clearly superior
to that of the S&P 500 in 4 of the past 5 years. In the one less-favorable year, the shortfall was trivial.
Hennessy is a “bottom-up” manager. The firm largely avoids any attempt to “time the market.” It
also focuses on selection of individual stocks, rather than the weighting of favored industries.
There is no apparent conformity of style among Wilstead’s six equity managers. The five manag-
ers, other than Hennessy, manage portfolios aggregating $250 million made up of more than 150
individual issues.
Jones is convinced that Hennessy is able to apply superior skill to stock selection, but the favor-
able returns are limited by the high degree of diversification in the portfolio. Over the years, the
portfolio generally held 40–50 stocks, with about 2%–3% of total funds committed to each issue.
The reason Hennessy seemed to do well most years was that the firm was able to identify each year
10 or 12 issues that registered particularly large gains.
On the basis of this overview, Jones outlined the following plan to the Wilstead pension committee:
Let’s tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the
commitments to the stocks that it really favors, and eliminate the remainder. Except for this
one new restriction, Hennessy should be free to manage the portfolio exactly as before.
All the members of the pension committee generally supported Jones’s proposal because all
agreed that Hennessy had seemed to demonstrate superior skill in selecting stocks. Yet the proposal
was a considerable departure from previous practice, and several committee members raised ques-
tions. Respond to each of the following questions.
1. a. Will the limitation to 20 stocks likely increase or decrease the risk of the portfolio? Explain. b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without signifi-
cantly affecting risk? Explain.
2. One committee member was particularly enthusiastic concerning Jones’s proposal. He suggested
that Hennessy’s performance might benefit further from reduction in the number of issues to 10.
If the reduction to 20 could be expected to be advantageous, explain why reduction to 10 might
be less likely to be advantageous. (Assume that Wilstead will evaluate the Hennessy portfolio
independently of the other portfolios in the fund.)
3. Another committee member suggested that, rather than evaluate each managed portfolio indepen-
dently of other portfolios, it might be better to consider the effects of a change in the Hennessy
portfolio on the total fund. Explain how this broader point of view could affect the committee
decision to limit the holdings in the Hennessy portfolio to either 10 or 20 issues.
4. Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?
Portfolio Expected Return (%) Standard Deviation (%)
a. W 15 36 b. X 12 15 c. Z 5 7 d. Y 9 21
5. Which statement about portfolio diversification is correct?
a. Proper diversification can reduce or eliminate systematic risk. b. Diversification reduces the portfolio’s expected return because it reduces a portfolio’s total risk. c. As more securities are added to a portfolio, total risk typically would be expected to fall at a
decreasing rate.
d. The risk-reducing benefits of diversification do not occur meaningfully until at least 30 indi- vidual securities are included in the portfolio.
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C H A P T E R 7 Optimal Risky Portfolios 239
6. The measure of risk for a security held in a diversified portfolio is:
a. Specific risk. b. Standard deviation of returns. c. Reinvestment risk.
d. Covariance.
7. Portfolio theory as described by Markowitz is most concerned with:
a. The elimination of systematic risk. b. The effect of diversification on portfolio risk. c. The identification of unsystematic risk.
d. Active portfolio management to enhance return.
8. Assume that a risk-averse investor owning stock in Miller Corporation decides to add the stock
of either Mac or Green Corporation to her portfolio. All three stocks offer the same expected
return and total variability. The covariance of return between Miller and Mac is 2 .05 and
between Miller and Green is 1 .05. Portfolio risk is expected to:
a. Decline more when the investor buys Mac. b. Decline more when the investor buys Green. c. Increase when either Mac or Green is bought. d. Decline or increase, depending on other factors.
9. Stocks A, B, and C have the same expected return and standard deviation. The following table shows the correlations between the returns on these stocks.
Stock A Stock B Stock B
Stock A 11.0 Stock B 10.9 11.0 Stock C 10.1 20.4 11.0
Given these correlations, the portfolio constructed from these stocks having the lowest risk is a
portfolio:
a. Equally invested in stocks A and B. b. Equally invested in stocks A and C. c. Equally invested in stocks B and C. d. Totally invested in stock C.
10. Statistics for three stocks, A, B, and C, are shown in the following tables.
Standard Deviations of Returns
Stock: A B C
Standard deviation (%): 40 20 40
Correlations of Returns
Stock A B C
A 1.00 0.90 0.50 B 1.00 0.10 C 1.00
Only on the basis of the information provided in the tables, and given a choice between a port- folio made up of equal amounts of stocks A and B or a portfolio made up of equal amounts of stocks B and C, which portfolio would you recommend? Justify your choice.
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240 P A R T I I Portfolio Theory and Practice
11. George Stephenson’s current portfolio of $2 million is invested as follows:
Summary of Stephenson’s Current Portfolio
Value Percent of
Total Expected Annual
Return Annual Standard
Deviation
Short-term bonds $ 200,000 10% 4.6% 1.6% Domestic large-cap equities 600,000 30 12.4 19.5 Domestic small-cap equities 1,200,000 60 16.0 29.9 Total portfolio $2,000,000 100% 13.8 23.1
Stephenson soon expects to receive an additional $2 million and plans to invest the entire
amount in an index fund that best complements the current portfolio. Stephanie Coppa, CFA,
is evaluating the four index funds shown in the following table for their ability to produce a
portfolio that will meet two criteria relative to the current portfolio: (1) maintain or enhance
expected return and (2) maintain or reduce volatility.
Each fund is invested in an asset class that is not substantially represented in the current
portfolio.
Index Fund Characteristics
Index Fund Expected Annual Return Expected Annual
Standard Deviation Correlation of Returns with Current Portfolio
Fund A 15% 25% 10.80 Fund B 11 22 10.60 Fund C 16 25 10.90 Fund D 14 22 10.65
State which fund Coppa should recommend to Stephenson. Justify your choice by describing
how your chosen fund best meets both of Stephenson’s criteria. No calculations are required.
12. Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Com-
pany common stock worth $100,000. Her financial adviser provided her with the following
forecast information:
Risk and Return Characteristics
Expected Monthly Returns
Standard Deviation of Monthly Returns
Original Portfolio 0.67% 2.37% ABC Company 1.25 2.95
The correlation coefficient of ABC stock returns with the original portfolio returns is .40.
a. The inheritance changes Grace’s overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the:
i. Expected return of her new portfolio which includes the ABC stock.
ii. Covariance of ABC stock returns with the original portfolio returns.
iii. Standard deviation of her new portfolio, which includes the ABC stock.
b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the gov-
ernment securities, calculate the
i. Expected return of her new portfolio, which includes the government securities.
ii. Covariance of the government security returns with the original portfolio returns.
iii. Standard deviation of her new portfolio, which includes the government securities.
c. Determine whether the systematic risk of her new portfolio, which includes the government securities, will be higher or lower than that of her original portfolio.
d. On the basis of conversations with her husband, Grace is considering selling the $100,000 of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ stock
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C H A P T E R 7 Optimal Risky Portfolios 241
has the same expected return and standard deviation as ABC stock. Her husband comments,
“It doesn’t matter whether you keep all of the ABC stock or replace it with $100,000 of XYZ
stock.” State whether her husband’s comment is correct or incorrect. Justify your response.
e. In a recent discussion with her financial adviser, Grace commented, “If I just don’t lose money in my portfolio, I will be satisfied.” She went on to say, “I am more afraid of losing
money than I am concerned about achieving high returns.”
i. Describe one weakness of using standard deviation of returns as a risk measure for Grace. ii. Identify an alternate risk measure that is more appropriate under the circumstances.
13. Dudley Trudy, CFA, recently met with one of his clients. Trudy typically invests in a master list of
30 equities drawn from several industries. As the meeting concluded, the client made the follow-
ing statement: “I trust your stock-picking ability and believe that you should invest my funds in
your five best ideas. Why invest in 30 companies when you obviously have stronger opinions on a
few of them?” Trudy plans to respond to his client within the context of modern portfolio theory.
a. Contrast the concepts of systematic risk and firm-specific risk, and give an example of each type of risk.
b. Critique the client’s suggestion. Discuss how both systematic and firm-specific risk change as the number of securities in a portfolio is increased.
E-INVESTMENTS EXERCISES Go to the www.investopedia.com/articles/basics/03/050203.asp Web site to learn more about diversification, the factors that influence investors’ risk preferences, and the types of investments that fit into each of the risk categories. Then check out www. investopedia.com/articles/pf/05/061505.asp for asset allocation guidelines for various types of portfolios from conservative to very aggressive. What do you conclude about your own risk preferences and the best portfolio type for you? What would you expect to happen to your attitude toward risk as you get older? How might your portfolio compo- sition change?
SOLUTIONS TO CONCEPT CHECKS 1. a. The first term will be wD 3 wD 3 sD
2 , because this is the element in the top corner of
the matrix (sD 2 ) times the term on the column border ( w D ) times the term on the row
border ( w D ). Applying this rule to each term of the covariance matrix results in the sum wD
2 sD 2 1 wDwE Cov(rE, rD) 1 wE wD Cov(rD, rE) 1 wE
2 sE 2 , which is the same as Equation 7.3,
because Cov( r E , r D ) 5 Cov( r D , r E ). b. The bordered covariance matrix is
wX wY wZ
wX s2x Cov(rX, rY) Cov(rX, rZ)
wY Cov(rY, rX) s2Y Cov(rY, rZ)
wZ Cov(rZ, rX) Cov(rZ, rY) s2Z
There are nine terms in the covariance matrix. Portfolio variance is calculated from these nine
terms:
sP 2 5 wX
2 sX 2 1 wY
2 sY 2 1 wZ
2 sZ 2
1 wXwY Cov(rX , rY) 1 wY wX Cov(rY , rX) 1 wX wZ Cov(rX , rZ) 1 wZ wX Cov(rZ , rX) 1 wY wZ Cov (rY , rZ) 1 wZ wY Cov(rZ , rY)
5 wX 2 sX
2 1 wY 2 sY
2 1 wZ 2 sZ
2
1 2wX wY Cov(rX , rY) 1 2wX wZ Cov(rX , rZ) 1 2wY wZ Cov(rY , rZ)
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242 P A R T I I Portfolio Theory and Practice
2. The parameters of the opportunity set are E ( r D ) 5 8%, E ( r E ) 5 13%, s D 5 12%, s E 5 20%, and r ( D, E ) 5 .25. From the standard deviations and the correlation coefficient we generate the covariance matrix:
Fund D E
D 144 60 E 60 400
The global minimum-variance portfolio is constructed so that
wD 5 sE
2 2 Cov(rD, rE)
sD 2 1 sE
2 2 2 Cov(rD, rE)
5 400 2 60
(144 1 400) 2 (2 3 60) 5 .8019
wE 5 1 2 wD 5 .1981
Its expected return and standard deviation are
E(rP) 5 (.8019 3 8) 1 (.1981 3 13) 5 8.99%
sP 5 3wD2 sD2 1 wE2 sE2 1 2 wD wE Cov (rD , rE) 41/2 5 3(.80192 3 144) 1 (.19812 3 400) 1 (2 3 .8019 3 .1981 3 60) 41/2 5 11.29%
For the other points we simply increase w D from .10 to .90 in increments of .10; accordingly, w E ranges from .90 to .10 in the same increments. We substitute these portfolio proportions in the formulas for expected return and standard deviation. Note that when w E 5 1.0, the portfolio parameters equal those of the stock fund; when w D 5 1, the portfolio parameters equal those of the debt fund.
We then generate the following table:
wE wD E(r) s
0.0 1.0 8.0 12.00 0.1 0.9 8.5 11.46 0.2 0.8 9.0 11.29 0.3 0.7 9.5 11.48 0.4 0.6 10.0 12.03 0.5 0.5 10.5 12.88 0.6 0.4 11.0 13.99 0.7 0.3 11.5 15.30 0.8 0.2 12.0 16.76 0.9 0.1 12.5 18.34 1.0 0.0 13.0 20.00 0.1981 0.8019 8.99 11.29 minimum variance portfolio
You can now draw your graph.
3. a. The computations of the opportunity set of the stock and risky bond funds are like those of Question 2 and will not be shown here. You should perform these computations, however, in
order to give a graphical solution to part a. Note that the covariance between the funds is
Cov(rA, rB) 5 r(A, B) 3 sA 3 sB 5 2.2 3 20 3 60 5 2240
b. The proportions in the optimal risky portfolio are given by
wA 5 (10 2 5)602 2 (30 2 5)(2240)
(10 2 5)602 1 (30 2 5)202 2 30(2240)
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C H A P T E R 7 Optimal Risky Portfolios 243
5 .6818
wB 5 1 2 wA 5 .3182
The expected return and standard deviation of the optimal risky portfolio are
E(rP) 5 (.6818 3 10) 1 (.3182 3 30) 5 16.36%
sP 5 {(.6818 2 3 20
2 ) 1 (.3182
2 3 60
2 ) 1 [2 3 .6818 3 .3182(2240)]}
1/2
5 21.13%
Note that portfolio P is not the global minimum-variance portfolio. The proportions of the latter are given by
wA 5 602 2 (2240)
602 1 202 2 2(2240) 5 .8571
wB 5 1 2 wA 5 .1429
With these proportions, the standard deviation of the minimum-variance portfolio is
s(min) 5 (.85712 3 202) 1 (.14292 3 602) 1 32 3 .8571 3 .1429 3 (2240) 41/2 5 17.57%
which is less than that of the optimal risky portfolio.
c. The CAL is the line from the risk-free rate through the optimal risky portfolio. This line represents all efficient portfolios that combine T-bills with the optimal risky portfolio. The
slope of the CAL is
S 5 E(rP) 2 rf
sP 5
16.36 2 5
21.13 5 .5376
d. Given a degree of risk aversion, A, an investor will choose a proportion, y, in the optimal risky portfolio of (remember to express returns as decimals when using A ):
y 5 E(rP) 2 rf
AsP 2
5 .1636 2 .05
5 3 .21132 5 .5089
This means that the optimal risky portfolio, with the given data, is attractive enough for an
investor with A 5 5 to invest 50.89% of his or her wealth in it. Because stock A makes up 68.18% of the risky portfolio and stock B makes up 31.82%, the investment proportions for this investor are
Stock A: .5089 3 68.18 5 34.70%
Stock B: .5089 3 31.82 5 16.19%
Total 50.89%
4. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on
various securities and estimates of risk, that is, the covariance matrix. The forecasts themselves
do not control outcomes. Thus preferring managers with rosier forecasts (northwesterly frontiers)
is tantamount to rewarding the bearers of good news and punishing the bearers of bad news.
What we should do is reward bearers of accurate news. Thus if you get a glimpse of the frontiers (forecasts) of portfolio managers on a regular basis, what you want to do is develop the track
record of their forecasting accuracy and steer your advisees toward the more accurate forecaster.
Their portfolio choices will, in the long run, outperform the field.
5. The parameters are E ( r ) 5 15, s 5 60, and the correlation between any pair of stocks is r 5 .5.
a. The portfolio expected return is invariant to the size of the portfolio because all stocks have identical expected returns. The standard deviation of a portfolio with n 5 25 stocks is
sP 5 3s2/n 1 r 3 s2(n 2 1)/n41/2 5 3602/25 1 .5 3 602 3 24/2541/2 5 43.27%
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APPENDIX A: A Spreadsheet Model for Efficient Diversification
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244 P A R T I I Portfolio Theory and Practice
b. Because the stocks are identical, efficient portfolios are equally weighted. To obtain a standard deviation of 43%, we need to solve for n:
432 5 602
n 1 .5 3
602(n 2 1)
n 1,849n 5 3,600 1 1,800n 2 1,800
n 5 1,800
49 5 36.73
Thus we need 37 stocks and will come in with volatility slightly under the target.
c. As n gets very large, the variance of an efficient (equally weighted) portfolio diminishes, leaving only the variance that comes from the covariances among stocks, that is
sP 5 "r 3 s2 5 ".5 3 602 5 42.43% Note that with 25 stocks we came within .84% of the systematic risk, that is, the nonsystematic
risk of a portfolio of 25 stocks is only .84%. With 37 stocks the standard deviation is 43%, of
which nonsystematic risk is .57%.
d. If the risk-free is 10%, then the risk premium on any size portfolio is 15 2 10 5 5%. The standard deviation of a well-diversified portfolio is (practically) 42.43%; hence the slope of
the CAL is
S 5 5/42.43 5 .1178
Several software packages can be used to generate the efficient frontier. We will dem-
onstrate the method using Microsoft Excel. Excel is far from the best program for this
purpose and is limited in the number of assets it can handle, but working through a simple
portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in
more sophisticated “black-box” programs. You will find that even in Excel, the computa-
tion of the efficient frontier is fairly easy.
We apply the Markowitz portfolio optimization program to a practical problem of inter-
national diversification. We take the perspective of a portfolio manager serving U.S. clients,
who wishes to construct for the next year an optimal risky portfolio of large stocks in the U.S
and six developed capital markets (Japan, Germany, U.K., France, Canada, and Australia).
First we describe the input list: forecasts of risk premiums and the covariance matrix. Next,
we describe Excel’s Solver, and finally we show the solution to the manager’s problem.
The Covariance Matrix To capture recent risk parameters the manager compiles an array of 60 recent monthly
(annualized) rates of return, as well as the monthly T-bill rates for the same period.
The standard deviations of excess returns are shown in Spreadsheet 7A.1 (column C).
They range from 14.93% (U.K. large stocks) to 22.7% (Germany). For perspective on how
these parameters can change over time, standard deviations for the period 1991–2000 are
also shown (column B). In addition, we present the correlation coefficient between large
stocks in the six foreign markets with U.S. large stocks for the same two periods. Here we
see that correlations are higher in the more recent period, consistent with the process of
globalization.
The covariance matrix shown in Spreadsheet 7A.2 was estimated from the array of 60
returns of the seven countries using the COVARIANCE function from the dialog box of
Data Analysis in Excel’s Tools menu. Due to a quirk in the Excel software, the covariance
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C H A P T E R 7 Optimal Risky Portfolios 245
Spreadsheets 7A.1, 7A.2, 7A.3 Spreadsheet model for international diversifi cation
A B C D E F G H
1 2
7A.1 Country Index Statistics and Forecasts of Excess Returns Correlation with the
U.S. Average Excess Return Forecast
3
4 Country
−0.0148 0.0094
0.0247
0.0209
0.1225
0.0398
0.1009
0.1108
0.0536
0.0837
0.0473
0.0468
−0.0177 0.0727
1
0.83
0.83
0.85
0.81
0.43
0.79
1
0.64
0.54
0.53
0.52
0.41
0.72
Standard Deviation 1991–2000
0.1295
0.1466
0.1741
0.1538
0.1808
0.2432
0.1687
0.1495
0.1493
0.2008
0.2270
0.1617
0.1878
0.1727
2001–2005 1991–2000 2001–2005
US
UK
France
Germany
Australia
Japan
Canada
5 6 7
8
9
10
11 12
1991–2000 2001–2005 2006
0.0600
0.0530
0.0700
0.0800
0.0580
0.0450
0.0590
A B C D E F G H I
7A.2 The Bordered Covariance Matrix
Portfolio
Weights 1.0000 0.0000 0.0000 0.0000
1.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
1.0000
0.0600
0.1495
0.4013
Cell A18 - A24 A18 is set arbitrarily to 1 while A19 to A24 are set to 0
Formula in cell C16 =A18 ... Formula in cell I16 =A24
Formula in cell A25 =SUM(A18:A24)
Formula in cell C25 =C16*SUMPRODUCT($A$18:$A$24,C18:C24)
Formula in cell D25-I25 Copied from C25 (note the absolute addresses)
Formula in cell A26 =SUMPRODUCT($A$18:$A$24,H6:H12)
Formula in cell A27 =SUM(C25:I25)^0.5
Formula in cell A28 =A26/A27
0.0224
0.0184
0.0250
0.0288
0.0195
0.0121
0.0205
0.0224
0.0184
0.0223
0.0275
0.0299
0.0204
0.0124
0.0206
0.0000
0.0288
0.0299
0.0438
0.0515
0.0301
0.0183
0.0305
0.0000
0.0121
0.0124
0.0177
0.0183
0.0147
0.0353
0.0158
0.0000
US UK 0.0000
0.0250
0.0275
0.0403
0.0438
0.0259
0.0177
0.0273
0.0000
France Germany 0.0000
0.0195
0.0204
0.0259
0.0301
0.0261
0.0147
0.0234
0.0000
Australia Japan US
UK
France
Germany
Australia
Japan
Canada
0.0000
0.0205
0.0206
0.0273
0.0305
0.0234
0.0158
0.0298
0.0000
Canada
13
14
15
16
17 18 19 20 21 22 23 24 25 26 27 28 29
30
31
32
33
34
35
36
37
38
A B C D E F G H I J K L
US
Mean SD 0.1
Min Var Optimum 0.0383
0.1132
0.3386
0.6112
0.8778
−0.2140 −0.5097 0.0695
0.2055
−0.0402 0.0465
0.0400
0.1135
0.3525
0.6195
0.8083
−0.2029 −0.4610 0.0748
0.1987
−0.0374 0.0466
0.0450
0.1168
0.3853
0.6446
0.5992
−0.1693 −0.3144 0.0907
0.1781
−0.0288 0.0480
0.0500
0.1238
0.4037
0.6696
0.3900
−0.1357 −0.1679 0.1067
0.1575
−0.0203 0.0509
0.0550
0.1340
0.4104
0.6947
0.1809
−0.1021 −0.0213 0.1226
0.1369
−0.0118 0.0550
0.0564
0.1374
0.4107
0.7018
0.1214
−0.0926 0.0205
0.1271
0.1311
−0.0093 0.0564
0.0575
0.1401
0.4106
0.7073
0.0758
−0.0852 0.0524
0.1306
0.1266
−0.0075 0.0575
0.0600
0.1466
0.4092
0.7198
−0.0283 −0.0685 0.1253
0.1385
0.1164
−0.0032 0.0602
0.0700
0.1771
0.3953
0.7699
−0.4465 −0.0014 0.4185
0.1704
0.0752
0.0139
0.0727
0.0800
0.2119
0.3774
0.8201
−0.8648 0.0658
0.7117
0.2023
0.0341
0.0309
0.08710.0411
Slope
UK
France
Germany
Australia
Japan
Canada
CAL*
*Risk premium on CAL = SD � slope of optimal risky portfolio
44 45 46 47 48 49 50 51 52 53 54 55
43
Cell to store constraint on risk premium
7A.3 The Efficient Frontier39
40
41
42
0.0400
Mean
SD
Slope
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246 P A R T I I Portfolio Theory and Practice
matrix is not corrected for degrees-of-freedom bias; hence, each of the elements in the
matrix was multiplied by 60/59 to eliminate downward bias.
Expected Returns While estimation of the risk parameters (the covariance matrix) from excess returns is
a simple technical matter, estimating the risk premium (the expected excess return) is a
daunting task. As we discussed in Chapter 5, estimating expected returns using histori-
cal data is unreliable. Consider, for example, the negative average excess returns on U.S.
large stocks over the period 2001–2005 (cell G6) and, more generally, the big differences
in average returns between the 1991–2000 and 2001–2005 periods, as demonstrated in
columns F and G.
In this example, we simply present the manager’s forecasts of future returns as shown in
column H. In Chapter 8 we will establish a framework that makes the forecasting process
more explicit.
The Bordered Covariance Matrix and Portfolio Variance The covariance matrix in Spreadsheet 7A.2 is bordered by the portfolio weights, as
explained in Section 7.2 and Table 7.2 . The values in cells A18–A24, to the left of the cova-
riance matrix, will be selected by the optimization program. For now, we arbitrarily input
1.0 for the U.S. and zero for the others. Cells A16–I16, above the covariance matrix, must
be set equal to the column of weights on the left, so that they will change in tandem as the
column weights are changed by Excel’s Solver. Cell A25 sums the column weights and is
used to force the optimization program to set the sum of portfolio weights to 1.0.
Cells C25–I25, below the covariance matrix, are used to compute the portfolio variance
for any set of weights that appears in the borders. Each cell accumulates the contribution
to portfolio variance from the column above it. It uses the function SUMPRODUCT to
accomplish this task. For example, row 33 shows the formula used to derive the value that
appears in cell C25.
Finally, the short column A26–A28 below the bordered covariance matrix presents port-
folio statistics computed from the bordered covariance matrix. First is the portfolio risk
premium in cell A26, with formula shown in row 35, which multiplies the column of port-
folio weights by the column of forecasts (H6–H12) from Spreadsheet 7A.1 . Next is the
portfolio standard deviation in cell A27. The variance is given by the sum of cells C25–I25
below the bordered covariance matrix. Cell A27 takes the square root of this sum to pro-
duce the standard deviation. The last statistic is the portfolio Sharpe ratio, cell A28, which
is the slope of the CAL (capital allocation line) that runs through the portfolio constructed
using the column weights (the value in cell A28 equals cell A26 divided by cell A27). The
optimal risky portfolio is the one that maximizes the Sharpe ratio.
Using the Excel Solver Excel’s Solver is a user-friendly, but quite powerful, optimizer. It has three parts: (1) an
objective function, (2) decision variables, and (3) constraints. Figure 7A.1 shows three
pictures of the Solver. For the current discussion we refer to picture A.
The top panel of the Solver lets you choose a target cell for the “objective function,”
that is, the variable you are trying to optimize. In picture A, the target cell is A27, the port-
folio standard deviation. Below the target cell, you can choose whether your objective is to
maximize, minimize, or set your objective function equal to a value that you specify. Here
we choose to minimize the portfolio standard deviation.
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C H A P T E R 7 Optimal Risky Portfolios 247
The next panel contains the decision variables. These are cells that the Solver can
change in order to optimize the objective function in the target cell. Here, we input cells
A18–A24, the portfolio weights that we select to minimize portfolio volatility.
The bottom panel of the Solver can include any number of constraints. One constraint
that must always appear in portfolio optimization is the “feasibility constraint,” namely,
that portfolio weights sum to 1.0. When we bring up the constraint dialogue box, we spec-
ify that cell A25 (the sum of weights) be set equal to 1.0.
Finding the Minimum Variance Portfolio It is helpful to begin by identifying the global minimum variance portfolio ( G ). This pro- vides the starting point of the efficient part of the frontier. Once we input the target cell, the
decision variable cells, and the feasibility constraint, as in picture A, we can select “solve”
and the Solver returns portfolio G. We copy the portfolio statistics and weights to our output Spreadsheet 7A.3 . Column C in Spreadsheet 7A.3 shows that the lowest standard deviation
(SD) that can be achieved with our input list is 11.32%. Notice that the SD of portfolio G is considerably lower than even the lowest SD of the individual indexes. From the risk premium
of portfolio G (3.83%) we begin building the efficient frontier with ever-larger risk premiums.
Charting the Efficient Frontier of Risky Portfolios We determine the desired risk premiums (points on the efficient frontier) that we wish to
use to construct the graph of the efficient frontier. It is good practice to choose more points
in the neighborhood of portfolio G because the frontier has the greatest curvature in that region. It is sufficient to choose for the highest point the highest risk premium from the
Figure 7A.1 Solver dialog box
B
C
A
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248 P A R T I I Portfolio Theory and Practice
input list (here, 8% for Germany). You can produce the entire efficient frontier in minutes
following this procedure.
1. Input to the Solver a constraint that says: Cell A26 (the portfolio risk premium)
must equal the value in cell E41. The Solver at this point is shown in picture B of
Figure 7A.1 . Cell E41 will be used to change the required risk premium and thus
generate different points along the frontier.
2. For each additional point on the frontier, you input a different desired risk premium
into cell E41, and ask the Solver to solve again.
3. Every time the Solver gives you a solution to the request in (2), copy the results into
Spreadsheet 7A.3 , which tabulates the collection of points along the efficient fron-
tier. For the next step, change cell E41 and repeat from step 2.
Finding the Optimal Risky Portfolio on the Efficient Frontier Now that we have an efficient frontier, we look for the portfolio with the highest Sharpe
ratio. This is the efficient frontier portfolio that is tangent to the CAL. To find it, we just need
to make two changes to the Solver. First, change the target cell from cell A27 to cell A28,
the Sharpe ratio of the portfolio, and request that the value in this cell be maximized. Next,
eliminate the constraint on the risk premium that may be left over from the last time you
used the Solver. At this point the Solver looks like picture C in Figure 7A.1 .
The Solver now yields the optimal risky portfolio. Copy the statistics for the optimal
portfolio and its weights to your Spreadsheet 7A.3 . In order to get a clean graph, place
the column of the optimal portfolio in Spreadsheet 7A.3 so that the risk premiums of all
portfolios in the spreadsheet are steadily increasing from the risk premium of portfolio G (3.83%) all the way up to 8%.
The efficient frontier is graphed using the data in cells C45–I45 (the horizontal or
x -axis is portfolio standard deviation) and C44–I44 (the vertical or y -axis is portfolio risk premium). The resulting graph appears in Figure 7A.2 .
The Optimal CAL It is instructive to superimpose on the graph of the efficient frontier in Figure 7A.2 the CAL
that identifies the optimal risky portfolio. This CAL has a slope equal to the Sharpe ratio of
Figure 7A.2 Efficient frontier and CAL for country stock indexes
.0900
.0800
.0700
.0600
.0500
.0400
.0300 .1000 .1200 .1400 .1600 .1800 .2000 .2200 .2400
Standard Deviation
R is
k P
re m
iu m
Risk Premium SD Slope
Efficient Pf No short
No short Min. Var pf
0.0564 0.0575 0.0383 0.0540
0.1374 0.1401 0.1132 0.1350
0.4107 0.4104 0.3386 0.3960
Eff Frontier — No Short sales
Efficient Frontier Capital Allocation Line
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C H A P T E R 7 Optimal Risky Portfolios 249
APPENDIX B: Review of Portfolio Statistics
We base this review of scenario analysis on a two-asset portfolio. We denote the assets
D and E (which you may think of as debt and equity), but the risk and return parameters we use in this appendix are not necessarily consistent with those used in Section 7.2.
Expected Returns We use “expected value” and “mean” interchangeably. For an analysis with n scenarios, where the rate of return in scenario i is r ( i ) with probability p ( i ), the expected return is
E(r) 5 a n
i51 p(i)r (i) (7B.1)
If you were to increase the rate of return assumed for each scenario by some amount D,
then the mean return will increase by D. If you multiply the rate in each scenario by a
factor w, the new mean will be multiplied by that factor:
a n
i51 p(i) 3 3r (i) 1 D4 5 an
i51 p(i) 3 r (i) 1 Da
n
i51 p(i) 5 E(r) 1 D (7B.2)
a n
i51 p(i) 3 3wr(i)4 5 wan
i51 p(i) 3 r(i) 5 wE(r)
the optimal risky portfolio. Therefore, we add at the bottom of Spreadsheet 7A.3 a row with
entries obtained by multiplying the SD of each column’s portfolio by the Sharpe ratio of the
optimal risky portfolio from cell H46. This results in the risk premium for each portfolio along
the CAL efficient frontier. We now add a series to the graph with the standard deviations in
B45–I45 as the x -axis and cells B54–I54 as the y -axis. You can see this CAL in Figure 7A.2 .
The Optimal Risky Portfolio and the Short-Sales Constraint With the input list used by the portfolio manager, the optimal risky portfolio calls for signifi-
cant short positions in the stocks of France and Canada (see column H of Spreadsheet 7A.3 ).
In many cases the portfolio manager is prohibited from taking short positions. If so, we need
to amend the program to preclude short sales.
To accomplish this task, we repeat the exercise, but with one change. We add to the
Solver the following constraint: Each element in the column of portfolio weights, A18–
A24, must be greater than or equal to zero. You should try to produce the short-sale con-
strained efficient frontier in your own spreadsheet. The graph of the constrained frontier is
also shown in Figure 7A.2 .
Column C of Spreadsheet 7B.1 shows scenario rates of return for debt, D. In column D we add 3% to each scenario return and in column E we multiply each rate by .4.
The spreadsheet shows how we compute the expected return for columns C, D, and
E. It is evident that the mean increases by 3% (from .08 to .11) in column D and is
multiplied by .4 (from .08 to 0.032) in column E.
Example 7B.1 Expected Rates of Return
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250 P A R T I I Portfolio Theory and Practice
Now let’s construct a portfolio that invests a fraction of the investment budget, w ( D ), in bonds and the fraction w ( E ) in stocks. The portfolio’s rate of return in each scenario and its expected return are given by
rP(i) 5 wDrD(i) 1 wErE(i) (7B.3)
E(rP) 5 a p(i)3wDrD(i) 1 wErE(i)4 5 a p(i)wDrD(i) 1 a p(i)wErE(i) 5 wDE(rD) 1 wEE(rE)
The rate of return on the portfolio in each scenario is the weighted average of the com-
ponent rates. The weights are the fractions invested in these assets, that is, the portfolio
weights. The expected return on the portfolio is the weighted average of the asset means.
Spreadsheet 7B.2 lays out rates of return for both stocks and bonds. Using assumed
weights of .4 for debt and .6 for equity, the portfolio return in each scenario appears
in column L. Cell L8 shows the portfolio expected return as .1040, obtained using
the SUMPRODUCT function, which multiplies each scenario return (column L) by
the scenario probability (column I) and sums the results.
Example 7B.2 Portfolio Rate of Return
Spreadsheet 7B.1 Scenario analysis for bonds
A B C D E GF
1
2
3
4
5
6
7
8
9
10
11
12
Scenario rates of return
rD(i) rD(i) + 0.03 0.4*rD(i)
Mean
Cell C8
-0.10
0.00
0.10
0.32
0.080
=SUMPRODUCT($B$4:$B$7,C4:C7)
-0.07
0.03
0.13
0.35
0.110
-0.040
0.000
0.040
0.128
0.032
0.14
0.36
0.30
0.20
Probability
1
2
3
4
Scenario
e X c e l Please visit us at www.mhhe.com/bkm
Spreadsheet 7B.2 Scenario analysis for bonds and stocks
H I J K L
1
2
3
4
5
6
7
8
9
10
11
12
Scenario rates of return Portfolio return 0.4*rD(i)+0.6*rE(i)rD(i) rE(i)
0.14
0.36
0.30
0.20
-0.10
0.00
0.10
0.32
0.08Mean
Cell L4 =0.4*J4+0.6*K4
Cell L8 =SUMPRODUCT($I$4:$I$7,L4:L7)
-0.35
0.20
0.45
-0.19
0.12
-0.2500
0.1200
0.3100
0.0140
0.1040
1
2
3
4
ProbabilityScenario
e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 7 Optimal Risky Portfolios 251
Variance and Standard Deviation The variance and standard deviation of the rate of return on an asset from a scenario analy-
sis are given by 15
s2(r) 5 a n
i51 p(i)3r(i) 2 E(r)42 (7B.4)
s(r) 5 "s2(r) Notice that the unit of variance is percent squared. In contrast, standard deviation, the
square root of variance, has the same dimension as the original returns, and therefore is
easier to interpret as a measure of return variability.
When you add a fixed incremental return, D, to each scenario return, you increase the
mean return by that same increment. Therefore, the deviation of the realized return in each
scenario from the mean return is unaffected, and both variance and SD are unchanged. In
contrast, when you multiply the return in each scenario by a factor w, the variance is mul- tiplied by the square of that factor (and the SD is multiplied by w ):
Var(wr) 5 a n
i51 p(i) 3 3wr (i) 2 E(wr)42 5 w2 an
i51 p(i)3r(i) 2 E(r)42 5 w2s2
SD(wr) 5 "w2s2 5 ws(r) (7B.5) Excel does not have a direct function to compute variance and standard deviation for
a scenario analysis. Its STDEV and VAR functions are designed for time series. We need
to calculate the probability-weighted squared deviations directly. To avoid having to
first compute columns of squared deviations from the mean, however, we can simplify
our problem by expressing the variance as a difference between two easily computable
terms:
s 2 (r) 5 E[r 2 E(r)]2 5 E{r2 1 [E(r)]2 2 2rE(r)}
5 E(r2) 1 3E(r)42 2 2E(r)E(r) (7B.6) 5 E(r2) 2 3E(r)42 5 an
i51 p(i)r(i)2 2 B a
n
i51 p(i)r(i)R2
15
Variance (here, of an asset rate of return) is not the only possible choice to quantify variability. An alternative
would be to use the absolute deviation from the mean instead of the squared deviation. Thus, the mean absolute deviation (MAD) is sometimes used as a measure of variability. The variance is the preferred measure for several
reasons. First, working with absolute deviations is mathematically more difficult. Second, squaring deviations
gives more weight to larger deviations. In investments, giving more weight to large deviations (hence, losses) is
compatible with risk aversion. Third, when returns are normally distributed, the variance is one of the two param-
eters that fully characterize the distribution.
You can compute the first expression, E ( r 2 ), in Equation 7B.6 using Excel’s SUM- PRODUCT function. For example, in Spreadsheet 7B.3 , E ( r 2 ) is first calculated in cell C21 by using SUMPRODUCT to multiply the scenario probability times the
asset return times the asset return again. Then [ E ( r )] 2 is subtracted (notice the sub- traction of C20
2 in cell C21), to arrive at variance.
Example 7B.3 Calculating the Variance of a Risky Asset in Excel
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252 P A R T I I Portfolio Theory and Practice
The variance of a portfolio return is not as simple to compute as the mean. The portfolio variance is not the weighted average of the asset variances. The deviation of the portfolio rate of return in any scenario from its mean return is
rP 2 E(rP) 5 wDrD(i) 1 wErE(i) 2 3wDE(rD) 1 wEE(rE)4 5 wD 3rD(i) 2 E(rD)4 1 wE 3rE(i) 2 E(rE)4 (7B.7)
5 wDd(i) 1 wEe(i)
where the lowercase variables denote deviations from the mean:
d(i ) = rD(i) 2 E(rD)
e(i ) 5 rE(i ) 2 E(rE )
We express the variance of the portfolio return in terms of these deviations from the mean
in Equation 7B.7:
sP 2 5 a
n
i51 p(i)3rP 2 E(rP)42 5 an
i51 p(i)3wD d(i) 1 wEe(i)42
5 a n
i51 p(i)3wD2 d(i)2 1 wE2 e(i)2 1 2wDwE d(i)e(i)4
5 wD 2 a
n
i51 p(i)d(i)2 1 wE
2 a
n
i51 p(i)e(i)2 1 2wDwE a
n
i51 p(i)d(i)e(i)
5 wD 2 sD
2 1 wE 2 sE
2 1 2wDwE a n
i51 p(i)d(i)e(i)
(7B.8)
The last line in Equation 7B.8 tells us that the variance of a portfolio is the weighted sum
of portfolio variances (notice that the weights are the squares of the portfolio weights),
plus an additional term that, as we will soon see, makes all the difference.
Notice also that d ( i ) 3 e ( i ) is the product of the deviations of the scenario returns of the two assets from their respective means. The probability-weighted average of this product is
its expected value, which is called covariance and is denoted Cov( r D , r E ). The covariance between the two assets can have a big impact on the variance of a portfolio.
Covariance The covariance between two variables equals
Cov(rD, rE) 5 E(d 3 e) 5 E5 3rD 2 E(rD)4 3rE 2 E(rE)46 (7B.9) 5 E(rDrE) 2 E(rD)E(rE)
Spreadsheet 7B.3 Scenario analysis for bonds
e X c e l Please visit us at www.mhhe.com/bkm
A B C D E F G
13
14
15
16
17
18
19
20
21
22
23
24
Scenario rates of return
rD(i) rD(i) + 0.03 0.4*rD(i)
0.14
0.36
0.30
0.20
Mean
Cell C21
Variance
SD
-0.10
0.00
0.10
0.32
0.0800
0.0185
0.1359
=SUMPRODUCT($B$16:$B$19,C16:C19,C16:C19)-C20^2 Cell C22 =C21^0.5
-0.07
0.03
0.13
0.35
0.1100
0.0185
0.1359
-0.040
0.000
0.040
0.128
0.0240
0.0034
0.0584
1
2
3
4
ProbabilityScenario
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C H A P T E R 7 Optimal Risky Portfolios 253
The covariance is an elegant way to quantify the covariation of two variables. This is easi-
est seen through a numerical example.
Imagine a three-scenario analysis of stocks and bonds as given in Spreadsheet 7B.4 . In
scenario 1, bonds go down (negative deviation) while stocks go up (positive deviation). In
scenario 3, bonds are up, but stocks are down. When the rates move in opposite directions, as
in this case, the product of the deviations is negative; conversely, if the rates moved in the same
direction, the sign of the product would be positive. The magnitude of the product shows the
extent of the opposite or common movement in that scenario. The probability-weighted aver-
age of these products therefore summarizes the average tendency for the variables to co-vary across scenarios. In the last line of the spreadsheet, we see that the covariance is 2 80 (cell H6).
Suppose our scenario analysis had envisioned stocks generally moving in the same direction
as bonds. To be concrete, let’s switch the forecast rates on stocks in the first and third scenarios,
that is, let the stock return be 2 10% in the first scenario and 30% in the third. In this case, the
absolute value of both products of these scenarios remains the same, but the signs are positive,
and thus the covariance is positive, at 1 80, reflecting the tendency for both asset returns to vary
in tandem. If the levels of the scenario returns change, the intensity of the covariation also may
change, as reflected by the magnitude of the product of deviations. The change in the magni-
tude of the covariance quantifies the change in both direction and intensity of the covariation.
If there is no comovement at all, because positive and negative products are equally
likely, the covariance is zero. Also, if one of the assets is risk-free, its covariance with any
risky asset is zero, because its deviations from its mean are identically zero.
The computation of covariance using Excel can be made easy by using the last line in
Equation 7B.9. The first term, E ( r D 3 r E ), can be computed in one stroke using Excel’s SUMPRODUCT function. Specifically, in Spreadsheet 7B.4 , SUMPRODUCT(A3:A5,
B3:B5, C3:C5) multiplies the probability times the return on debt times the return on
equity in each scenario and then sums those three products.
Notice that adding D to each rate would not change the covariance because deviations
from the mean would remain unchanged. But if you multiply either of the variables by a fixed factor, the covariance will increase by that factor. Multiplying both variables results
in a covariance multiplied by the products of the factors because
Cov(wDrD , wE rE) 5 E5 3wDrD 2 wD E(rD)4 3wE rE 2 wE E(rE)46 (7B.10) 5 wDwE Cov(rD, rE)
The covariance in Equation 7B.10 is actually the term that we add (twice) in the last line of
the equation for portfolio variance, Equation 7B.8. So we find that portfolio variance is the
weighted sum (not average) of the individual asset variances, plus twice their covariance weighted by the two portfolio weights ( w D 3 w E ).
Like variance, the dimension (unit) of covariance is percent squared. But here we can-
not get to a more easily interpreted dimension by taking the square root, because the aver-
age product of deviations can be negative, as it was in Spreadsheet 7B.4 . The solution in
this case is to scale the covariance by the standard deviations of the two variables, produc-
ing the correlation coefficient.
Spreadsheet 7B.4 Three-scenario analysis for stocks and bonds
e X c e l Please visit us at www.mhhe.com/bkm
A
Rates of Return Deviation from Mean
B C D E F G H
1
2
3 0.25
0.50
0.25
Mean:
-2
6
14
6
30
10
-10
10
-8
0
8
0
20
0
-20
0
-160
0
-160
-80
4
5
6
Probability Bonds Stocks Bonds Stocks
Product of
Deviations
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V is
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s a t
w w
w .m
h h
e .c
o m
/b k m
254 P A R T I I Portfolio Theory and Practice
Correlation Coefficient Dividing the covariance by the product of the standard deviations of the variables will
generate a pure number called correlation. We define correlation as follows:
Corr (rD, rE) 5 Cov(rD, rE)
sDsE (7B.11)
The correlation coefficient must fall within the range [ 2 1, 1]. This can be explained as
follows. What two variables should have the highest degree comovement? Logic says a
variable with itself, so let’s check it out.
Cov(rD, rD) 5 E5 3rD 2 E(rD)4 3 3rD 2 E(rD)46 5 E 3rD 2 E(rD)42 5 sD2 (7B.12)
Corr(rD, rD) 5 Cov(rD, rD)
sD sD 5
sD 2
sD 2 5 1
Similarly, the lowest (most negative) value of the correlation coefficient is 2 1. (Check this
for yourself by finding the correlation of a variable with its own negative.)
An important property of the correlation coefficient is that it is unaffected by both
addition and multiplication. Suppose we start with a return on debt, r D , multiply it by a constant, w D , and then add a fixed amount D. The correlation with equity is unaffected:
Corr(D 1 wDrD, rE ) 5 Cov(D 1 wDrD, rE)
"Var(D 1 wDrD) 3 sE (7B.13) 5
wD Cov(rD, rE)
"wD2 sD2 3 sE 5 wD Cov(rD, rE)
wDsD 3 sE
5 Corr(rD, rE)
Because the correlation coefficient gives more intuition about the relationship between
rates of return, we sometimes express the covariance in terms of the correlation coefficient.
Rearranging Equation 7B.11, we can write covariance as
Cov(rD, rE) 5 sD sE Corr (rD, rE) (7B.14)
Spreadsheet 7B.5 shows the covariance and correlation between stocks and bonds
using the same scenario analysis as in the other examples in this appendix. Covari-
ance is calculated using Equation 7B.9. The SUMPRODUCT function used in cell
J22 gives us E ( r D 3 r E ), from which we subtract E ( r D ) 3 E ( r E ) (i.e., we subtract J20 3 K20). Then we calculate correlation in cell J23 by dividing covariance by the
product of the asset standard deviations.
Example 7B.4 Calculating Covariance and Correlation
Portfolio Variance We have seen in Equation 7B.8, with the help of Equation 7B.10, that the variance of a
two-asset portfolio is the sum of the individual variances multiplied by the square of the
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C H A P T E R 7 Optimal Risky Portfolios 255
portfolio weights, plus twice the covariance between the rates, multiplied by the product
of the portfolio weights:
sP 2 5 wD
2 sD 2 1 wE
2 sE 2 1 2wDwE Cov(rD, rE) (7B.15)
5 wD 2 sD
2 1 wE 2 sE
2 1 2wDwE sD sE Corr (rD, rE)
H I J K L M
13
14
15
16
17
18
19
20
21
22
23
24
25
Scenario rates of return
rD(i) rE(i)
0.14
0.36
0.30
0.20
-0.10 0.00
0.10
0.32
0.08
0.1359
-0.0034 -0.0847
Mean
SD
Covariance
Correlation
Cell J22
Cell J23
=SUMPRODUCT(I16:I19,J16:J19,K16:K19)-J20*K20 =J22/(J21*K21)
-0.35 0.20
0.45
-0.19 0.12
0.2918
1
2
3
4
ProbabilityScenario
Spreadsheet 7B.5 Scenario analysis for bonds and stocks
e X c e l Please visit us at www.mhhe.com/bkm
We calculate portfolio variance in Spreadsheet 7B.6 . Notice there that we calcu-
late the portfolio standard deviation in two ways: once from the scenario portfolio
returns (cell E35) and again (in cell E36) using the first line of Equation 7B.15. The
two approaches yield the same result. You should try to repeat the second calculation
using the correlation coefficient from the second line in Equation 7B.15 instead of
covariance in the formula for portfolio variance.
Example 7B.5 Calculating Portfolio Variance
A B C D E F G 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Cell E35 =SUMPRODUCT(B30:B33,E30:E33,E30:E33)-E34^2)^0.5 Cell E36 =(0.4*C35)^2+(0.6*D35)^2+2*0.4*0.6*C36)^0.5
Scenario rates of return Portfolio return rD(i) rE(i) 0.4*rD(i)+0.6rE(i)
0.14
0.36
0.30
0.20
Mean
SD
Covariance
Correlation
-0.10 0.00
0.10
0.32
0.08
0.1359
-0.0034 -0.0847
-0.35 0.20
0.45
-0.19 0.12
0.2918
SD:
-0.25 0.12
0.31
0.014
0.1040
0.1788
0.1788
1
2
3
4
ProbabilityScenario
Spreadsheet 7B.6 Scenario analysis for bonds and stocks
e X c e l Please visit us at www.mhhe.com/bkm
Suppose that one of the assets, say, E, is replaced with a money market instrument, that is, a risk-free asset. The variance of E is then zero, as is the covariance with D. In that case, as seen from Equation 7B.15, the portfolio standard deviation is just w D s D . In other words, when we mix a risky portfolio with the risk-free asset, portfolio standard deviation
equals the risky asset’s standard deviation times the weight invested in that asset. This
result was used extensively in Chapter 6.
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8 THE MARKOWITZ PROCEDURE introduced in the preceding chapter suffers from two drawbacks. First, the model requires a huge number of estimates to fill the covariance matrix. Second, the model does not provide any guideline to the forecasting of the secu- rity risk premiums that are essential to con- struct the efficient frontier of risky assets. Because past returns are unreliable guides to expected future returns, this drawback can be telling.
In this chapter we introduce index mod- els that simplify estimation of the covari- ance matrix and greatly enhance the analysis of security risk premiums. By allowing us to explicitly decompose risk into systematic and firm-specific components, these models also shed considerable light on both the power and the limits of diversification. Further, they allow us to measure these components of risk for particular securities and portfolios.
We begin the chapter by describing a single-factor security market and show how it can justify a single-index model of security returns. Once its properties are analyzed, we proceed to an extensive example of estima- tion of the single-index model. We review the
statistical properties of these estimates and show how they relate to the practical issues facing portfolio managers.
Despite the simplification they offer, index models remain true to the concepts of the efficient frontier and portfolio opti- mization. Empirically, index models are as valid as the assumption of normality of the rates of return on available securities. To the extent that short-term returns are well approximated by normal distributions, index models can be used to select optimal port- folios nearly as accurately as the Markowitz algorithm. Finally, we examine optimal risky portfolios constructed using the index model. While the principles are the same as those employed in the previous chapter, the properties of the portfolio are easier to derive and interpret in this context. We illustrate how to use the index model by con- structing an optimal risky portfolio using a small sample of firms. This portfolio is com- pared to the corresponding portfolio con- structed from the Markowitz model. We conclude with a discussion of several practi- cal issues that arise when implementing the index model.
CHAPTER EIGHT
Index Models 8 PA
R T I I
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C H A P T E R 8 Index Models 257
The Input List of the Markowitz Model The success of a portfolio selection rule depends on the quality of the input list, that is,
the estimates of expected security returns and the covariance matrix. In the long run, effi-
cient portfolios will beat portfolios with less reliable input lists and consequently inferior
reward-to-risk trade-offs.
Suppose your security analysts can thoroughly analyze 50 stocks. This means that your
input list will include the following:
n 5 50 estimates of expected returns
n 5 50 estimates of variances
(n2 2 n)/2 5 1,225 estimates of covariances 1,325 total estimates
This is a formidable task, particularly in light of the fact that a 50-security portfolio
is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n 5 3,000, roughly the number of NYSE stocks, we need more than 4.5 million estimates.
Another difficulty in applying the Markowitz model to portfolio optimization is that
errors in the assessment or estimation of correlation coefficients can lead to nonsensical
results. This can happen because some sets of correlation coefficients are mutually incon-
sistent, as the following example demonstrates: 1
Asset Standard
Deviation (%)
Correlation Matrix
A B C
A 20 1.00 0.90 0.90 B 20 0.90 1.00 0.00 C 20 0.90 0.00 1.00
Suppose that you construct a portfolio with weights 2 1.00; 1.00; 1.00, for assets A; B; C,
respectively, and calculate the portfolio variance. You will find that the portfolio variance
appears to be negative ( 2 200). This of course is not possible because portfolio
variances cannot be negative: We conclude that the inputs in the estimated correlation matrix
must be mutually inconsistent. Of course, true correlation coefficients are always consistent. 2 But we do not know these true correlations and can only estimate them with some impreci-
sion. Unfortunately, it is difficult to determine at a quick glance whether a correlation matrix
is inconsistent, providing another motivation to seek a model that is easier to implement.
Introducing a model that simplifies the way we describe the sources of security risk
allows us to use a smaller, consistent set of estimates of risk parameters and risk pre-
miums. The simplification emerges because positive covariances among security returns
arise from common economic forces that affect the fortunes of most firms. Some examples
of common economic factors are business cycles, interest rates, and the cost of natural
resources. The unexpected changes in these variables cause, simultaneously, unexpected
8.1 A Single-Factor Security Market
1 We are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management,
Northwestern University, for this example.
2 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is “positive
definite.”
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258 P A R T I I Portfolio Theory and Practice
changes in the rates of return on the entire stock market. By decomposing uncertainty into
these systemwide versus firm-specific sources, we vastly simplify the problem of estimat-
ing covariance and correlation.
Normality of Returns and Systematic Risk We can always decompose the rate of return on any security, i, into the sum of its expected plus unanticipated components:
ri 5 E(ri ) 1 ei (8.1)
where the unexpected return, e i , has a mean of zero and a standard deviation of s i that measures the uncertainty about the security return.
When security returns can be well approximated by normal distributions that are cor-
related across securities, we say that they are joint normally distributed. This assumption alone implies that, at any time, security returns are driven by one or more common vari-
ables. When more than one variable drives normally distributed security returns, these
returns are said to have a multivariate normal distribution. We begin with the simpler case where only one variable drives the joint normally distributed returns, resulting in a
single-factor security market. Extension to the multivariate case is straightforward and is
discussed in later chapters.
Suppose the common factor, m, that drives innovations in security returns is some macroeconomic variable that affects all firms. Then we can decompose the sources of
uncertainty into uncertainty about the economy as a whole, which is captured by m, and uncertainty about the firm in particular, which is captured by e i . In this case, we amend Equation 8.1 to accommodate two sources of variation in return:
ri 5 E(ri) 1 m 1 ei (8.2)
The macroeconomic factor, m, measures unanticipated macro surprises. As such, it has a mean of zero (over time, surprises will average out to zero), with standard deviation of
s m . In contrast, e i measures only the firm-specific surprise. Notice that m has no subscript because the same common factor affects all securities. Most important is the fact that m and e i are uncorrelated, that is, because e i is firm-specific, it is independent of shocks to the common factor that affect the entire economy. The variance of r i thus arises from two uncorrelated sources, systematic and firm specific. Therefore,
si 2 5 sm
2 1 s2(ei ) (8.3)
The common factor, m, generates correlation across securities, because all securities will respond to the same macroeconomic news, while the firm-specific surprises, captured
by e i , are assumed to be uncorrelated across firms. Because m is also uncorrelated with any of the firm-specific surprises, the covariance between any two securities i and j is
Cov(ri, rj) 5 Cov(m 1 ei, m 1 ej ) 5 sm 2 (8.4)
Finally, we recognize that some securities will be more sensitive than others to macro-
economic shocks. For example, auto firms might respond more dramatically to changes in
general economic conditions than pharmaceutical firms. We can capture this refinement by
assigning each firm a sensitivity coefficient to macro conditions. Therefore, if we denote
the sensitivity coefficient for firm i by the Greek letter beta, b i , we modify Equation 8.2 to obtain the single-factor model:
ri 5 E(ri) 1 bim 1 ei (8.5)
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C H A P T E R 8 Index Models 259
3 Practitioners often use a “modified” index model that is similar to Equation 8.8 but that uses total rather than
excess returns. This practice is most common when daily data are used. In this case the rate of return on bills is on
the order of only about .01% per day, so total and excess returns are almost indistinguishable.
Equation 8.5 tells us the systematic risk of security i is determined by its beta coefficient. “Cyclical” firms have greater sensitivity to the market and therefore higher systematic risk.
The systematic risk of security i is bi 2sm
2 , and its total risk is
si 2 5 bi
2sm 2 1 s2(ei ) (8.6)
The covariance between any pair of securities also is determined by their betas:
Cov(ri, rj) 5 Cov(bim 1 ei, bjm 1 ej) 5 bibjsm 2 (8.7)
In terms of systematic risk and market exposure, this equation tells us that firms are close
substitutes. Equivalent beta securities give equivalent market exposures.
Up to this point we have used only statistical implications from the joint normality of
security returns. Normality of security returns alone guarantees that portfolio returns are
also normal (from the “stability” of the normal distribution discussed in Chapter 5) and that
there is a linear relationship between security returns and the common factor. This greatly
simplifies portfolio analysis. Statistical analysis, however, does not identify the common
factor, nor does it specify how that factor might operate over a longer investment period.
However, it seems plausible (and can be empirically verified) that the variance of the com-
mon factor usually changes relatively slowly through time, as do the variances of indi-
vidual securities and the covariances among them. We seek a variable that can proxy for
this common factor. To be useful, this variable must be observable, so we can estimate its
volatility as well as the sensitivity of individual securities returns to variation in its value.
8.2 The Single-Index Model
A reasonable approach to making the single-factor model operational is to assert that the
rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the
common macroeconomic factor. This approach leads to an equation similar to the single-
factor model, which is called a single-index model because it uses the market index to proxy for the common factor.
The Regression Equation of the Single-Index Model Because rates of return on market indexes such as the S&P 500 can be observed, we have
a considerable amount of past data with which to estimate systematic risk. We denote the
market index by M, with excess return of R M 5 r M 2 r f , and standard deviation of s M . Because the index model is linear, we can estimate the sensitivity (or beta) coefficient of a
security on the index using a single-variable linear regression. We regress the excess return
of a security, R i 5 r i 2 r f , on the excess return of the index, R M . To estimate the regression, we collect a historical sample of paired observations, R i ( t ) and R M ( t ), where t denotes the date of each pair of observations (e.g., the excess returns on the stock and the index in a
particular month). 3 The regression equation is
Ri(t) 5 ai 1 bi RM(t) 1 ei(t) (8.8)
The intercept of this equation (denoted by the Greek letter alpha, or a ) is the security’s
expected excess return when the market excess return is zero. The slope coefficient, b i , is
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260 P A R T I I Portfolio Theory and Practice
the security beta. Beta is the security’s sensitivity to the index: It is the amount by which the
security return tends to increase or decrease for every 1% increase or decrease in the return
on the index. e i is the zero-mean, firm-specific surprise in the security return in time t, also called the residual.
The Expected Return–Beta Relationship Because E ( e i ) 5 0, if we take the expected value of E ( R i ) in Equation 8.8, we obtain the expected return–beta relationship of the single-index model:
E(Ri) 5 ai 1 biE(RM) (8.9)
The second term in Equation 8.9 tells us that part of a security’s risk premium is due to the
risk premium of the index. The market risk premium is multiplied by the relative sensitiv-
ity, or beta, of the individual security. We call this the systematic risk premium because it derives from the risk premium that characterizes the entire market, which proxies for the
condition of the full economy or economic system.
The remainder of the risk premium is given by the first term in the equation, a . Alpha is
a nonmarket premium. For example, a may be large if you think a security is underpriced and therefore offers an attractive expected return. Later on, we will see that when secu-
rity prices are in equilibrium, such attractive opportunities ought to be competed away, in
which case a will be driven to zero. But for now, let’s assume that each security analyst
comes up with his or her own estimates of alpha. If managers believe that they can do a
superior job of security analysis, then they will be confident in their ability to find stocks
with nonzero values of alpha.
We will see shortly that the index model decomposition of an individual security’s risk
premium to market and nonmarket components greatly clarifies and simplifies the opera-
tion of macroeconomic and security analysis within an investment company.
Risk and Covariance in the Single-Index Model Remember that one of the problems with the Markowitz model is the overwhelming
number of parameter estimates required to implement it. Now we will see that the index
model simplification vastly reduces the number of parameters that must be estimated.
Equation 8.8 yields the systematic and firm-specific components of the overall risk of each
security, and the covariance between any pair of securities. Both variances and covariances
are determined by the security betas and the properties of the market index:
Total risk 5 Systematic risk 1 Firm-specific risk
si 2 5 bi
2sM 2 1 s2(ei)
Covariance 5 Product of betas 3 Market-index risk
Cov(ri , rj) 5 bibjsM 2 (8.10)
Correlation 5 Product of correlations with the market index
Corr(ri , rj ) 5 bibjsM
2
sisj 5
bisM 2 bjsM
2
sisMsjsM 5 Corr (ri , rM) 3 Corr (rj, rM)
Equations 8.9 and 8.10 imply that the set of parameter estimates needed for the single-
index model consists of only a , b , and s ( e ) for the individual securities, plus the risk pre- mium and variance of the market index.
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C H A P T E R 8 Index Models 261
The Set of Estimates Needed for the Single-Index Model We summarize the results for the single-index model in the table below.
The data below describe a three-stock financial market that satisfies the single-index model.
Stock Capitalization Beta Mean Excess
Return Standard Deviation
A $3,000 1.0 10% 40% B $1,940 0.2 2 30 C $1,360 1.7 17 50
The standard deviation of the market-index portfolio is 25%.
a. What is the mean excess return of the index portfolio?
b. What is the covariance between stock A and stock B ?
c. What is the covariance between stock B and the index?
d. Break down the variance of stock B into its systematic and firm-specific components.
CONCEPT CHECK 8.1
Symbol
1. The stock’s expected return if the market is neutral, that is, if the market’s excess return, r M 2 r f , is zero a i
2. The component of return due to movements in the overall market; b i is the security’s responsiveness to market movements b i ( r M 2 r f )
3. The unexpected component of return due to unexpected events that are relevant only to this security (firm specific) e i
4. The variance attributable to the uncertainty of the common macroeconomic factor bi
2sM 2
5. The variance attributable to firm-specific uncertainty s 2 ( e i )
These calculations show that if we have:
• n estimates of the extra-market expected excess returns, a i
• n estimates of the sensitivity coefficients, b i
• n estimates of the firm-specific variances, s 2 ( e i )
• 1 estimate for the market risk premium, E ( R M )
• 1 estimate for the variance of the (common) macroeconomic factor, sM 2
then these (3 n 1 2) estimates will enable us to prepare the entire input list for this single- index-security universe. Thus for a 50-security portfolio we will need 152 estimates rather
than 1,325; for the entire New York Stock Exchange, about 3,000 securities, we will need
9,002 estimates rather than approximately 4.5 million!
It is easy to see why the index model is such a useful abstraction. For large universes of
securities, the number of estimates required for the Markowitz procedure using the index
model is only a small fraction of what otherwise would be needed.
Another advantage is less obvious but equally important. The index model abstraction
is crucial for specialization of effort in security analysis. If a covariance term had to be
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262 P A R T I I Portfolio Theory and Practice
calculated directly for each security pair, then security analysts could not specialize by
industry. For example, if one group were to specialize in the computer industry and another
in the auto industry, who would have the common background to estimate the covariance
between IBM and GM? Neither group would have the deep understanding of other indus- tries necessary to make an informed judgment of co-movements among industries. In con-
trast, the index model suggests a simple way to compute covariances. Covariances among
securities are due to the influence of the single common factor, represented by the market
index return, and can be easily estimated using the regression Equation 8.8.
The simplification derived from the index model assumption is, however, not without
cost. The “cost” of the model lies in the restrictions it places on the structure of asset
return uncertainty. The classification of uncertainty into a simple dichotomy—macro ver-
sus micro risk—oversimplifies sources of real-world uncertainty and misses some impor-
tant sources of dependence in stock returns. For example, this dichotomy rules out industry
events, events that may affect many firms within an industry without substantially affect-
ing the broad macroeconomy.
This last point is potentially important. Imagine that the single-index model is perfectly
accurate, except that the residuals of two stocks, say, British Petroleum (BP) and Royal
Dutch Shell, are correlated. The index model will ignore this correlation (it will assume it
is zero), while the Markowitz algorithm (which accounts for the full covariance between
every pair of stocks) will automatically take the residual correlation into account when
minimizing portfolio variance. If the universe of securities from which we must construct
the optimal portfolio is small, the two models will yield substantively different optimal
portfolios. The portfolio of the Markowitz algorithm will place a smaller weight on both
BP and Shell (because their mutual covariance reduces their diversification value), result-
ing in a portfolio with lower variance. Conversely, when correlation among residuals
is negative, the index model will ignore the potential
diversification value of these securities. The resulting
“optimal” portfolio will place too little weight on these
securities, resulting in an unnecessarily high variance. The optimal portfolio derived from the single-index
model therefore can be significantly inferior to that of
the full-covariance (Markowitz) model when stocks with
correlated residuals have large alpha values and account
for a large fraction of the portfolio. If many pairs of the
covered stocks exhibit residual correlation, it is possible
that a multi-index model, which includes additional fac- tors to capture those extra sources of cross-security cor-
relation, would be better suited for portfolio analysis and
construction. We will demonstrate the effect of correlated
residuals in the spreadsheet example in this chapter, and
discuss multi-index models in later chapters.
The Index Model and Diversification The index model, first suggested by Sharpe,
4 also offers insight into portfolio diversifica-
tion. Suppose that we choose an equally weighted portfolio of n securities. The excess rate of return on each security is given by
Ri 5 ai 1 biRM 1 ei
Suppose that the index model for the excess returns of stocks A and B is estimated with the following results:
RA 5 1.0% 1 .9RM 1 eA RB 5 22.0% 1 1.1RM 1 eB sM 5 20%
s(eA) 5 30%
s(eB) 5 10%
Find the standard deviation of each stock and the covariance between them.
CONCEPT CHECK 8.2
4 William F. Sharpe, “A Simplified Model of Portfolio Analysis,” Management Science, January 1963.
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C H A P T E R 8 Index Models 263
Similarly, we can write the excess return on the portfolio of stocks as
RP 5 aP 1 bPRM 1 eP (8.11)
We now show that, as the number of stocks included in this portfolio increases, the part of
the portfolio risk attributable to nonmarket factors becomes ever smaller. This part of the
risk is diversified away. In contrast, market risk remains, regardless of the number of firms
combined into the portfolio.
To understand these results, note that the excess rate of return on this equally weighted
portfolio, for which each portfolio weight w i 5 1/ n, is
RP 5 a n
i51 wi Ri 5
1
n a n
i51 Ri 5
1
n a n
i51
(ai 1 bi RM 1 ei )
5 1
n a n
i51 ai 1 a1n a
n
i51 bibRM 1 1n a
n
i51 ei (8.12)
Comparing Equations 8.11 and 8.12, we see that the portfolio has a sensitivity to the
market given by
bP 5 1
n a n
i51 bi (8.13)
which is the average of the individual b i s. It has a nonmarket return component of
aP 5 1
n a n
i51 ai (8.14)
which is the average of the individual alphas, plus the zero mean variable
eP 5 1
n a n
i51 ei (8.15)
which is the average of the firm-specific components. Hence the portfolio’s variance is
sP 2 5 bP
2 sM 2 1 s2(eP) (8.16)
The systematic risk component of the portfolio variance, which we defined as the compo-
nent that depends on marketwide movements, is bP 2 sM
2 and depends on the sensitivity coef-
ficients of the individual securities. This part of the risk depends on portfolio beta and sM 2 and
will persist regardless of the extent of portfolio diversification. No matter how many stocks
are held, their common exposure to the market will be reflected in portfolio systematic risk. 5
In contrast, the nonsystematic component of the portfolio variance is s 2 ( e P ) and is
attributable to firm-specific components, e i . Because these e i s are independent, and all have zero expected value, the law of averages can be applied to conclude that as more and
more stocks are added to the portfolio, the firm-specific components tend to cancel out,
resulting in ever-smaller nonmarket risk. Such risk is thus termed diversifiable. To see this more rigorously, examine the formula for the variance of the equally weighted “portfolio”
of firm-specific components. Because the e i s are uncorrelated,
s2(eP) 5 a n
i51 a1nb
2
s2(ei) 5 1
n s – 2(e) (8.17)
where s – 2(e) is the average of the firm-specific variances. Because this average is indepen-
dent of n, when n gets large, s 2 ( e P ) becomes negligible.
5 One can construct a portfolio with zero systematic risk by mixing negative b and positive b assets. The point of
our discussion is that the vast majority of securities have a positive b , implying that well-diversified portfolios
with small holdings in large numbers of assets will indeed have positive systematic risk.
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264 P A R T I I Portfolio Theory and Practice
To summarize, as diversification increases, the total variance of a portfolio approaches
the systematic variance, defined as the variance of the market factor multiplied by the
square of the portfolio sensitivity coefficient, bP 2 . This is shown in Figure 8.1 .
Figure 8.1 shows that as more and more securities are combined into a portfolio, the
portfolio variance decreases because of the diversification of firm-specific risk. However,
the power of diversification is limited. Even for very large n, part of the risk remains because of the exposure of virtually all assets to the com-
mon, or market, factor. Therefore, this systematic risk is
said to be nondiversifiable. This analysis is borne out by empirical evidence. We
saw the effect of portfolio diversification on portfolio
standard deviations in Figure 7.2. These empirical results
are similar to the theoretical graph presented here in
Figure 8.1 .
Figure 8.1 The variance of an equally weighted portfolio with risk coefficient b P in the single-factor economy
σ2P
n
σ2(eP) 5 σ 2(e)/ n
Diversifiable Risk
Systematic Risk
β2Pσ 2 M
Reconsider the two stocks in Concept Check 2. Suppose we form an equally weighted portfolio of A and B. What will be the nonsystematic stan- dard deviation of that portfolio?
CONCEPT CHECK 8.3
8.3 Estimating the Single-Index Model
Armed with the theoretical underpinnings of the single-index model, we now provide an
extended example that begins with estimation of the regression equation (8.8) and contin-
ues through to the estimation of the full covariance matrix of security returns.
To keep the presentation manageable, we focus on only six large U.S. corporations:
Hewlett-Packard and Dell from the information technology (IT) sector of the S&P 500,
Target and Walmart from the retailing sector, and British Petroleum and Royal Dutch Shell
from the energy sector.
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C H A P T E R 8 Index Models 265
We work with monthly observations of rates of return for the six stocks, the S&P 500
portfolio, and T-bills over a 5-year period (60 observations). As a first step, the excess
returns on the seven risky assets are computed. We start with a detailed look at the prepara-
tion of the input list for Hewlett-Packard (HP), and then proceed to display the entire input
list. Later in the chapter, we will show how these estimates can be used to construct the
optimal risky portfolio.
The Security Characteristic Line for Hewlett-Packard The index model regression Equation 8.8 restated for Hewlett-Packard (HP) is
RHP(t) 5 aHP 1 bHPRS&P500(t) 1 eHP(t)
The equation describes the (linear) dependence of HP’s excess return on changes in the
state of the economy as represented by the excess returns of the S&P 500 index portfolio.
The regression estimates describe a straight line with intercept a HP and slope b HP , which
we call the security characteristic line (SCL) for HP. Figure 8.2 shows a graph of the excess returns on HP and the S&P 500 portfolio over
the 60-month period. The graph shows that HP returns generally follow those of the index,
but with much larger swings. Indeed, the annualized standard deviation of the excess return
on the S&P 500 portfolio over the period was 13.58%, while that of HP was 38.17%. The
swings in HP’s excess returns suggest a greater-than-average sensitivity to the index, that
is, a beta greater than 1.0.
The relationship between the returns of HP and the S&P 500 is made clearer by the
scatter diagram in Figure 8.3 , where the regression line is drawn through the scatter. The vertical distance of each point from the regression line is the value of HP’s residual, e HP ( t ), corresponding to that particular month. The rates in Figure 8.2 are not annualized, and
the scatter diagram shows monthly swings of over 6 30% for HP, but returns in the range
of 2 11% to 8.5% for the S&P 500. The
regression analysis output obtained by
using Excel is shown in Table 8.1 .
The Explanatory Power of the SCL for HP Considering the top panel of Table 8.1
first, we see that the correlation of HP
with the S&P 500 is quite high (.7238),
telling us that HP tracks changes in the
returns of the S&P 500 fairly closely. The
R -square (.5239) tells us that variation in the S&P 500 excess returns explains about
52% of the variation in the HP series.
The adjusted R -square (which is slightly smaller) corrects for an upward bias in
R -square that arises because we use the fitted values of two parameters,
6 the slope
6 In general, the adjusted R -square (RA
2 ) is derived from the unadjusted by RA 2 5 1 2 (1 2 R2)
n 2 1
n 2 k 2 1 , where k
is the number of independent variables (here, k 5 1). An additional degree of freedom is lost to the estimate of the intercept.
Figure 8.2 Excess returns on HP and S&P 500
−.4
−.3
−.2
−.1
.0
.1
60100 20 30 40 50
.2
.3
.4
Observation Month
E xc
e ss
R e tu
rn s
(% )
S&P 500 HP
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266 P A R T I I Portfolio Theory and Practice
(beta) and intercept (alpha), rather than
their true, but unobservable, values. With
60 observations, this bias is small. The
standard error of the regression is the
standard deviation of the residual, which
we discuss in more detail shortly. This is
a measure of the slippage in the average
relationship between the stock and the
index due to the impact of firm-specific
factors, and is based on in-sample data. A more severe test is to look at returns
from periods after the one covered by the
regression sample and test the power of
the independent variable (the S&P 500)
to predict the dependent variable (the
return on HP). Correlation between
regression forecasts and realizations
of out-of-sample data is almost always considerably lower than in-sample
correlation.
Analysis of Variance The next panel of Table 8.1 shows the
analysis of variance (ANOVA) for the
SCL. The sum of squares (SS) of the regression (.3752) is the portion of the variance of
the dependent variable (HP’s return) that is explained by the independent variable (the S&P
500 return); it is equal to bHP 2 sS&P 500
2 . The MS column for the residual (.0059) shows the vari-
ance of the unexplained portion of HP’s return, that is, the portion of return that is independent of the market index. The square root of this value is the standard error (SE) of the regression
(.0767) reported in the first panel. If you divide the total SS of the regression (.7162) by 59,
A
Figure 8.3 Scatter diagram of HP, the S&P 500, and the security characteristic line (SCL) for HP
−.4
−.3
−.2
−.1
0
.1
.2
.3
.4
−.15 −.10 −.05 0 .05 .10
Excess Returns, S&P 500
E xc
e ss
R e tu
rn , H
P
Regression Statistics
Multiple R .7238 R -square .5239 Adjusted R -square .5157 Standard error .0767 Observations 60
ANOVA
df SS MS
Regression 1 .3752 .3752 Residual 58 .3410 .0059 Total 59 .7162
Coefficients Standard Error t-Stat p-Value
Intercept 0.0086 .0099 0.8719 .3868 S&P 500 2.0348 .2547 7.9888 .0000
Table 8.1
Excel output: Regression statistics for the SCL of Hewlett-Packard
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C H A P T E R 8 Index Models 267
7 When monthly data are annualized, average return and variance are multiplied by 12. However, because variance
is multiplied by 12, standard deviation is multiplied by "12. 8 R-Square 5
bHP 2 sS&P500
2
bHP 2 sS&P500
2 1 s2(eHP) 5
.3752
.7162 5 .5239
Equivalently, R -square equals 1 minus the fraction of variance that is not explained by market returns, i.e., 1 minus the ratio of firm-specific risk to total risk. For HP, this is
1 2 s2(eHP)
bHP 2 sS&P500
2 1 s2(eHP) 5 1 2
.3410
.7162 5 .5239
9 We can relate the standard error of the alpha estimate to the standard error of the residuals as follows:
SE(aHP) 5 s(eHP)Å1n 1 (AvgS&P500)2
Var(S&P500) 3 (n 2 1)
10
The t -statistic is based on the assumption that returns are normally distributed. In general, if we standardize the estimate of a normally distributed variable by computing its difference from a hypothesized value and dividing
by the standard error of the estimate (to express the difference as a number of standard errors), the resulting vari-
able will have a t -distribution. With a large number of observations, the bell-shaped t -distribution approaches the normal distribution.
you will obtain the estimate of the variance of the dependent variable (HP), .012 per month,
equivalent to a monthly standard deviation of 11%. When it is annualized, 7 we obtain an annu-
alized standard deviation of 38.17%, as reported earlier. Notice that the R -square (the ratio of explained to total variance) equals the explained (regression) SS divided by the total SS.
8
The Estimate of Alpha We move to the bottom panel. The intercept (.0086 5 .86% per month) is the estimate of
HP’s alpha for the sample period. Although this is an economically large value (10.32% on
an annual basis), it is statistically insignificant. This can be seen from the three statistics
next to the estimated coefficient. The first is the standard error of the estimate (0.0099). 9
This is a measure of the imprecision of the estimate. If the standard error is large, the range
of likely estimation error is correspondingly large.
The t -statistic reported in the bottom panel is the ratio of the regression parameter to its standard error. This statistic equals the number of standard errors by which our estimate
exceeds zero, and therefore can be used to assess the likelihood that the true but unob-
served value might actually equal zero rather than the estimate derived from the data. 10
The intuition is that if the true value were zero, we would be unlikely to observe estimated
values far away (i.e., many standard errors) from zero. So large t -statistics imply low prob- abilities that the true value is zero.
In the case of alpha, we are interested in the average value of HP’s return net of the
impact of market movements. Suppose we define the nonmarket component of HP’s return
as its actual return minus the return attributable to market movements during any period.
Call this HP’s firm-specific return, which we abbreviate as R fs .
Rfirm-specific 5 Rfs 5 RHP 2 bHPRS&P500
If R fs were normally distributed with a mean of zero, the ratio of its estimate to its standard error would have a t -distribution. From a table of the t -distribution (or using Excel’s TINV function) we can find the probability that the true alpha is actually zero
or even lower given the positive estimate of its value and the standard error of the esti-
mate. This is called the level of significance or, as in Table 8.1 , the probability or p-value. The conventional cutoff for statistical significance is a probability of less than 5%, which
requires a t -statistic of about 2.0. The regression output shows the t -statistic for HP’s alpha
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268 P A R T I I Portfolio Theory and Practice
to be .8719, indicating that the estimate is not significantly different from zero. That is,
we cannot reject the hypothesis that the true value of alpha equals zero with an accept-
able level of confidence. The p -value for the alpha estimate (.3868) indicates that if the true alpha were zero, the probability of obtaining an estimate as high as .0086 (given the
large standard error of .0099) would be .3868, which is not so unlikely. We conclude that
the sample average of R fs is too low to reject the hypothesis that the true value of alpha is zero. But even if the alpha value were both economically and statistically significant within
the sample, we still would not use that alpha as a forecast for a future period. Overwhelming empirical evidence shows that 5-year alpha values do not persist over time, that is, there
seems to be virtually no correlation between estimates from one sample period to the next.
In other words, while the alpha estimated from the regression tells us the average return
on the security when the market was flat during that estimation period, it does not forecast what the firm’s performance will be in future periods. This is why security analysis is so
hard. The past does not readily foretell the future. We elaborate on this issue in Chapter 11
on market efficiency.
The Estimate of Beta The regression output in Table 8.1 shows the beta estimate for HP to be 2.0348, more than
twice that of the S&P 500. Such high market sensitivity is not unusual for technology
stocks. The standard error (SE) of the estimate is .2547. 11
The value of beta and its SE produce a large t -statistic (7.9888), and a p -value of prac- tically zero. We can confidently reject the hypothesis that HP’s true beta is zero. A more
interesting t -statistic might test a null hypothesis that HP’s beta is greater than the market- wide average beta of 1. This t -statistic would measure how many standard errors separate the estimated beta from a hypothesized value of 1. Here too, the difference is easily large
enough to achieve statistical significance:
Estimated value 2 Hypothesized value
Standard error 5
2.03 2 1
.2547 5 4.00
However, we should bear in mind that even here, precision is not what we might like it to
be. For example, if we wanted to construct a confidence interval that includes the true but
unobserved value of beta with 95% probability, we would take the estimated value as the
center of the interval and then add and subtract about two standard errors. This produces a
range between 1.43 and 2.53, which is quite wide.
Firm-Specific Risk The monthly standard deviation of HP’s residual is 7.67%, or 26.6% annually. This is quite
large, on top of HP’s already high systematic risk. The standard deviation of systematic
risk is b 3 s (S&P 500) 5 2.03 3 13.58 5 27.57%. Notice that HP’s firm-specific risk is
as large as its systematic risk, a common result for individual stocks.
Correlation and Covariance Matrix Figure 8.4 graphs the excess returns of the pairs of securities from each of the three sectors
with the S&P 500 index on the same scale. We see that the IT sector is the most variable,
followed by the retail sector, and then the energy sector, which has the lowest volatility.
Panel 1 in Spreadsheet 8.1 shows the estimates of the risk parameters of the S&P 500
portfolio and the six analyzed securities. You can see from the high residual standard
11
SE(b) 5 s(eHP)
sHP"n 2 1
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C H A P T E R 8 Index Models 269
deviations (column E) how important diversification is. These securities have tremendous
firm-specific risk. Portfolios concentrated in these (or other) securities would have unnec-
essarily high volatility and inferior Sharpe ratios.
Panel 2 shows the correlation matrix of the residuals from the regressions of excess
returns on the S&P 500. The shaded cells show correlations of same-sector stocks, which
are as high as .7 for the two oil stocks (BP and Shell). This is in contrast to the assumption
of the index model that all residuals are uncorrelated. Of course, these correlations are, to
Figure 8.4 Excess returns on portfolio assets
−.3
−.2
−.1
.0
.1
.2
.3
0 10 20 30 40 50 60 Month
M o
n th
ly R
a te
s (%
)
S&P 500 HP DELL
−.3
−.2
−.1
.0
.1
.2
.3
0 10 20 30 40 50 60 Month
M o
n th
ly R
a te
s (%
)
S&P 500 WMT TARGET
−.3
−.2
−.1
.0
.1
.2
.3
0 10 20 30 40 50 60 Month
M o
n th
ly R
a te
s (%
)
S&P 500 BP SHELL
A
B
C
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270 P A R T I I Portfolio Theory and Practice
Spreadsheet 8.1
Implementing the index model
A B C D E F 1 2
3 4 5 6
0.1358
0.3817
0.2901
0.1935
0.2611
0.1822
0.1988
HP
1.00
7 8 9 10
0.1720 0.1981 0.66
11
0.0634 0.1722 0.35
12
0.0914
0.2762
0.1358
0.1672
0.0841
0.0234
0.0234
0.0086
0.0086
0.0124
0.0150
0.1371
0.5505
1.0000
1.0922
20.0100
0.0639
20.0050
0.0322
0.0075
0.0835
0.0025
0.0429
0.012
0.0400
0.0124
0.0375
0.0184
0.0227
0.0227
0.0114
0.0114
0.1780
0.2656
0
0.2392
0.1757
0.46
0.72
0.58
0.43
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
50
51
52
53
54
55
56
57
58 59
G H I J
60 61
SD of Excess Return
SD of Residual
SD of Systematic ComponentBeta
Correlation with the S&P 500
2.03
1.23
0.62
1.27
0.47
0.67
1.00
1.00
0.0475
0.0475
0.0175
0.0175
0.0253
0.0253
0.0375
0.0462
0.0462
0.0232
0.2126
0.3863
0.1492
0.0705
0.0404
0.1691
0.17180.8282
2.0348
0.1371 0.0639
0.1358
0.0878
0.2497
0.0222
0.1911
0.3472
0.1205
0.0392
0.4045
0.7349
0.5400
0.0297
0.0789
0.1433
0.0205
0.0317
20.1748
1.2315
0.0322 0.0835
0.6199 1.2672
0.0400
0.4670
0.0429
0.6736
0.0648
0.1422
0.46
1.0158
20.3176
0.1009
0.0572
20.1619
20.2941
0.0865
0.0309
0.0232
0.0288
0.0288
0.0106
0.0106
0.0153
0.01530.0842
0.0141
0.0145
0.0145
0.0053
0.0053
0.0077
0.0077
0.0682
0.0109
0.0109
0.0157
0.0157
0.0332
0.0058
0.0058
0.0395
0.0374
0.0141
2.03Beta
Beta
Risk premium
S&P 500
0.0600
0
S&P 500
2.03
1.23
0.62
1.27
0.47
0.67
0.67
1S&P 500
S&P 500
HP DELL WMT
TARGET BP
SHELL
DELL WMT
TARGET BP
SHELL
HP
DELL WMT
TARGET BP
SHELL
HP
S&P 500 DELL WMT TARGET BP SHELL Overall PfHPActive Pf A
DELL WMT TARGET BP
HP DELL WMT TARGET BP
1
0.08
20.34
20.10
20.20
20.06
1
0.17
0.12
20.28
20.19
1
1
1
0.50
0.70
0.62 0.47
20.19
20.24
20.13
20.22
0.1457
SHELL
HP DELL WMT TARGET BP SHELL
1.23 1.27
Off-diagonal cells equal to covariance
multiplies beta from row and column by index variance
formula in cell C26
formula in cell C27
5 B4 ^ 2
Alpha
Risk premium
s2(e)
s2(e A )
a A
a/s2(e)
w0(i)
[w0(i)]2
w0A w*(Risky portf)
SD
Sharpe ratio
1
0.06
0.44 0.35
Cells on the diagonal (shadowed) equal to variance
5 C$25*$B27*$B$4 ^ 2
Panel 5: Computation of the Optimal Risky Portfolio
Panel 4: Macro Forecast and Forecasts of Alpha Values
Panel 3: The Index Model Covariance Matrix
Panel 2: Correlation of Residuals
Panel 1: Risk Parameters of the Investable Universe (annualized)
e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 8 Index Models 271
a great extent, high by design, because we selected pairs of firms from the same industry.
Cross-industry correlations are typically far smaller, and the empirical estimates of corre-
lations of residuals for industry indexes (rather than individual stocks in the same industry)
would be far more in accord with the model. In fact, a few of the stocks in this sample actu-
ally seem to have negatively correlated residuals. Of course, correlation also is subject to
statistical sampling error, and this may be a fluke.
Panel 3 produces covariances derived from Equation 8.10 of the single-index model.
Variances of the S&P 500 index and the individual covered stocks appear on the diagonal.
The variance estimates for the individual stocks equal bi 2sM
2 1 s2(ei). The off-diagonal terms are covariance values and equal bibjsM
2 .
8.4 Portfolio Construction and the Single-Index Model
In this section, we look at the implications of the index model for portfolio construction. 12
We will see that the model offers several advantages, not only in terms of parameter
estimation, but also for the analytic simplification and organizational decentralization that
it makes possible.
Alpha and Security Analysis Perhaps the most important advantage of the single-index model is the framework it pro-
vides for macroeconomic and security analysis in the preparation of the input list that is so
critical to the efficiency of the optimal portfolio. The Markowitz model requires estimates
of risk premiums for each security. The estimate of expected return depends on both mac-
roeconomic and individual-firm forecasts. But if many different analysts perform security
analysis for a large organization such as a mutual fund company, a likely result is incon-
sistency in the macroeconomic forecasts that partly underlie expectations of returns across
securities. Moreover, the underlying assumptions for market-index risk and return often
are not explicit in the analysis of individual securities.
The single-index model creates a framework that separates these two quite different sources
of return variation and makes it easier to ensure consistency across analysts. We can lay down
a hierarchy of the preparation of the input list using the framework of the single-index model.
1. Macroeconomic analysis is used to estimate the risk premium and risk of the
market index.
2. Statistical analysis is used to estimate the beta coefficients of all securities and their
residual variances, s 2 ( e i ).
3. The portfolio manager uses the estimates for the market-index risk premium and
the beta coefficient of a security to establish the expected return of that security
absent any contribution from security analysis. The market-driven expected return is conditional on information common to all securities, not on information gleaned
from security analysis of particular firms. This market-driven expected return can
be used as a benchmark.
4. Security-specific expected return forecasts (specifically, security alphas) are derived
from various security-valuation models (such as those discussed in Part Five). Thus,
the alpha value distills the incremental risk premium attributable to private informa- tion developed from security analysis.
12
The use of the index model to construct optimal risky portfolios was originally developed in Jack Treynor and
Fischer Black, “How to Use Security Analysis to Improve Portfolio Selection,” Journal of Business, January 1973.
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272 P A R T I I Portfolio Theory and Practice
In the context of Equation 8.9, the risk premium on a security not subject to security
analysis would be b i E ( R M ). In other words, the risk premium would derive solely from the security’s tendency to follow the market index. Any expected return beyond this bench-
mark risk premium (the security alpha) would be due to some nonmarket factor that would
be uncovered through security analysis.
The end result of security analysis is the list of alpha values. Statistical methods of esti-
mating beta coefficients are widely known and standardized; hence, we would not expect
this portion of the input list to differ greatly across portfolio managers. In contrast, macro
and security analysis are far less of an exact science and therefore provide an arena for dis-
tinguished performance. Using the index model to disentangle the premiums due to market
and nonmarket factors, a portfolio manager can be confident that macro analysts compiling
estimates of the market-index risk premium and security analysts compiling alpha values
are using consistent estimates for the overall market.
In the context of portfolio construction, alpha is more than just one of the components
of expected return. It is the key variable that tells us whether a security is a good or a
bad buy. Consider an individual stock for which we have a beta estimate from statistical
considerations and an alpha value from security analysis. We easily can find many other
securities with identical betas and therefore identical systematic components of their risk
premiums. Therefore, what really makes a security attractive or unattractive to a portfolio
manager is its alpha value. In fact, we’ve suggested that a security with a positive alpha is
providing a premium over and above the premium it derives from its tendency to track the
market index. This security is a bargain and therefore should be overweighted in the over-
all portfolio compared to the passive alternative of using the market-index portfolio as the
risky vehicle. Conversely, a negative-alpha security is overpriced and, other things equal,
its portfolio weight should be reduced. In more extreme cases, the desired portfolio weight
might even be negative, that is, a short position (if permitted) would be desirable.
The Index Portfolio as an Investment Asset The process of charting the efficient frontier using the single-index model can be pursued
much like the procedure we used in Chapter 7, where we used the Markowitz model to find
the optimal risky portfolio. Here, however, we can benefit from the simplification the index
model offers for deriving the input list. Moreover, portfolio optimization highlights another
advantage of the single-index model, namely, a simple and intuitively revealing representation
of the optimal risky portfolio. Before we get into the mechanics of optimization in this setting,
however, we start by considering the role of the index portfolio in the optimal portfolio.
Suppose the prospectus of an investment company limits the universe of investable
assets to only stocks included in the S&P 500 portfolio. In this case, the S&P 500 index
captures the impact of the economy on the large stocks the firm may include in its portfolio.
Suppose that the resources of the company allow coverage of only a relatively small subset
of this so-called investable universe. If these analyzed firms are the only ones allowed in the portfolio, the portfolio manager may well be worried about limited diversification.
A simple way to avoid inadequate diversification is to include the S&P 500 portfolio as
one of the assets of the portfolio. Examination of Equations 8.8 and 8.9 reveals that if we
treat the S&P 500 portfolio as the market index, it will have a beta of 1.0 (its sensitivity to
itself), no firm-specific risk, and an alpha of zero—there is no nonmarket component in its
expected return. Equation 8.10 shows that the covariance of any security, i, with the index is bisM
2 . To distinguish the S&P 500 from the n securities covered by the firm, we will designate it the ( n 1 1)th asset. We can think of the S&P 500 as a passive portfolio that the manager would select in the absence of security analysis. It gives broad market exposure
without the need for expensive security analysis. However, if the manager is willing to
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C H A P T E R 8 Index Models 273
engage in such research, she may devise an active portfolio that can be mixed with the index to provide an even better risk–return trade-off.
The Single-Index-Model Input List If the portfolio manager plans to compile a portfolio from a list of n actively researched firms plus a passive market-index portfolio, the input list will include the following estimates:
1. Risk premium on the S&P 500 portfolio.
2. Standard deviation of the S&P 500 portfolio.
3. n sets of estimates of (a) beta coefficients, (b) stock residual variances, and (c) alpha values. (The alpha values, together with the risk premium of the S&P
500 and the beta of each security, determine the expected return on each security.)
The Optimal Risky Portfolio in the Single-Index Model The single-index model allows us to solve for the optimal risky portfolio directly and to
gain insight into the nature of the solution. First we confirm that we easily can set up the
optimization process to chart the efficient frontier in this framework along the lines of the
Markowitz model.
With the estimates of the beta and alpha coefficients, plus the risk premium of the
index portfolio, we can generate the n 1 1 expected returns using Equation 8.9. With the estimates of the beta coefficients and residual variances, together with the variance of
the index portfolio, we can construct the covariance matrix using Equation 8.10. Given
a column of risk premiums and the covariance matrix, we can conduct the optimization
program described in Chapter 7.
We can take the description of how diversification works in the single-index framework
of Section 8.2 a step further. We showed earlier that the alpha, beta, and residual variance
of an equally weighted portfolio are the simple averages of those parameters across
component securities. This result is not limited to equally weighted portfolios. It applies
to any portfolio, where we need only replace “simple average” with “weighted average,”
using the portfolio weights. Specifically,
aP 5 a n11
i51 wiai and for the index, an11 5 aM 5 0
bP 5 a n11
i51 wibi and for the index, bn11 5 bM 5 1 (8.18)
s2(eP) 5 a n11
i51 wi
2s2(ei) and for the index, s 2(en11) 5 s
2(eM) 5 0
The objective is to maximize the Sharpe ratio of the portfolio by using portfolio weights,
w 1 , . . . , w n 1 1 . With this set of weights, the expected return, standard deviation, and Sharpe ratio of the portfolio are
E(RP) 5 aP 1 E(RM)bP 5 a n11
i51 wiai 1 E(RM) a
n11
i51 wibi
sP 5 3bP2 sM2 1 s2(eP)41/2 5 BsM2 ¢ an11 i51
wibi≤ 2
1 a n11
i51 wi
2s2(ei)R 1/2
(8.19)
SP 5 E(RP)
sP
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274 P A R T I I Portfolio Theory and Practice
13
The definition of correlation implies that r(RA, RM) 5 Cov(RA, RM)
sAsM 5 bA
sM
sA . Therefore, given the ratio of SD,
a higher beta implies higher correlation and smaller benefit from diversification than when b 5 1 in Equation 8.20.
This requires the modification of Equation 8.21.
At this point, as in the Markowitz procedure, we could use Excel’s optimization program
to maximize the Sharpe ratio subject to the adding-up constraint that the portfolio weights
sum to 1. However, this is not necessary because when returns follow the index model,
the optimal portfolio can be derived explicitly, and the solution for the optimal portfolio
provides insight into the efficient use of security analysis in portfolio construction. It is
instructive to outline the logical thread of the solution. We will not show every algebraic
step, but will instead present the major results and interpretation of the procedure.
Before delving into the results, let us first explain the basic trade-off the model reveals.
If we were interested only in diversification, we would just hold the market index. Security
analysis gives us the chance to uncover securities with a nonzero alpha and to take a differ-
ential position in those securities. The cost of that differential position is a departure from
efficient diversification, in other words, the assumption of unnecessary firm-specific risk.
The model shows us that the optimal risky portfolio trades off the search for alpha against
the departure from efficient diversification.
The optimal risky portfolio turns out to be a combination of two component portfolios:
(1) an active portfolio, denoted by A, comprised of the n analyzed securities (we call this the active portfolio because it follows from active security analysis), and (2) the market- index portfolio, the ( n 1 1)th asset we include to aid in diversification, which we call the passive portfolio and denote by M.
Assume first that the active portfolio has a beta of 1. In that case, the optimal weight
in the active portfolio would be proportional to the ratio a A / s 2 ( e A ). This ratio balances the
contribution of the active portfolio (its alpha) against its contribution to the portfolio vari-
ance (via residual variance). The analogous ratio for the index portfolio is E(RM)/sM 2 , and
hence the initial position in the active portfolio (i.e., if its beta were 1) is
wA 0 5
aA
sA 2
E(RM)
sM 2
(8.20)
Next, we amend this position to account for the actual beta of the active portfolio. For
any level of sA 2, the correlation between the active and passive portfolios is greater when
the beta of the active portfolio is higher. This implies less diversification benefit from the
passive portfolio and a lower position in it. Correspondingly, the position in the active
portfolio increases. The precise modification for the position in the active portfolio is: 13
wA * 5
wA 0
1 1 (1 2 bA)wA 0 (8.21)
Notice that when bA 5 1, wA * 5 wA
0 .
The Information Ratio Equations 8.20 and 8.21 yield the optimal position in the active portfolio once we know its
alpha, beta, and residual variance. With wA * in the active portfolio and 1 2 wA
* invested in
the index portfolio, we can compute the expected return, standard deviation, and Sharpe
ratio of the optimal risky portfolio. The Sharpe ratio of an optimally constructed risky port-
folio will exceed that of the index portfolio (the passive strategy). The exact relationship is
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C H A P T E R 8 Index Models 275
SP 2 5 SM
2 1 B aA s(eA)
R2 (8.22) Equation 8.22 shows us that the contribution of the active portfolio (when held in its opti-
mal weight, wA * ) to the Sharpe ratio of the overall risky portfolio is determined by the ratio
of its alpha to its residual standard deviation. This important ratio is called the information ratio. It measures the extra return we can obtain from security analysis compared to the firm-specific risk we incur when we over- or underweight securities relative to the passive
market index. Equation 8.22 therefore implies that to maximize the overall Sharpe ratio,
we must maximize the information ratio of the active portfolio.
It turns out that the information ratio of the active portfolio will be maximized if we
invest in each security in proportion to its ratio of a i / s 2 ( e i ). Scaling this ratio so that the
total position in the active portfolio adds up to wA * , the weight in each security is
wi * 5 wA
*
ai
s2(ei)
a n
i51
ai
s2(ei)
(8.23)
With this set of weights, the contribution of each security to the information ratio of the
active portfolio is the square of its own information ratio, that is,
B aA s(eA)
R2 5 a n
i51 B ai
s(ei) R2 (8.24)
The model thus reveals the central role of the information ratio in efficiently taking
advantage of security analysis. The positive contribution of a security to the portfolio is
made by its addition to the nonmarket risk premium (its alpha). Its negative impact is to
increase the portfolio variance through its firm-specific risk (residual variance).
In contrast to alpha, the market (systematic) component of the risk premium, b i E ( R M ), is offset by the security’s nondiversifiable (market) risk, bi
2sM 2 , and both are driven by the
same beta. This trade-off is not unique to any security, as any security with the same beta
makes the same balanced contribution to both risk and return. Put differently, the beta of a
security is neither vice nor virtue. It is a property that simultaneously affects the risk and risk premium of a security. Hence we are concerned only with the aggregate beta of the
active portfolio, rather than the beta of each individual security.
We see from Equation 8.23 that if a security’s alpha is negative, the security will assume
a short position in the optimal risky portfolio. If short positions are prohibited, a negative-
alpha security would simply be taken out of the optimization program and assigned a port-
folio weight of zero. As the number of securities with nonzero alpha values (or the number
with positive alphas if short positions are prohibited) increases, the active portfolio will
itself be better diversified and its weight in the overall risky portfolio will increase at the
expense of the passive index portfolio.
Finally, we note that the index portfolio is an efficient portfolio only if all alpha values
are zero. This makes intuitive sense. Unless security analysis reveals that a security has a
nonzero alpha, including it in the active portfolio would make the portfolio less attractive.
In addition to the security’s systematic risk, which is compensated for by the market risk
premium (through beta), the security would add its firm-specific risk to portfolio variance.
With a zero alpha, however, the latter is not compensated by an addition to the nonmarket
risk premium. Hence, if all securities have zero alphas, the optimal weight in the active
portfolio will be zero, and the weight in the index portfolio will be 1. However, when
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276 P A R T I I Portfolio Theory and Practice
security analysis uncovers securities with nonmarket risk premiums (nonzero alphas), the
index portfolio is no longer efficient.
Summary of Optimization Procedure Once security analysis is complete, the optimal risky portfolio is formed from the index-
model estimates of security and market index parameters using these steps:
1. Compute the initial position of each security in the active portfolio as
wi 0 5 ai/s
2(ei).
2. Scale those initial positions to force portfolio weights to sum to 1 by dividing by
their sum, that is, wi 5 wi
0
a n
i51 wi
0
.
3. Compute the alpha of the active portfolio: aA 5 a n
i51 wiai.
4. Compute the residual variance of the active portfolio: s2(eA) 5 a n
i51 wi
2s2(ei).
5. Compute the initial position in the active portfolio: w e
E RA
A
A
M
M
0 2 ( )
( ) .
2
6. Compute the beta of the active portfolio: bA 5 a n
i51 wibi.
7. Adjust the initial position in the active portfolio: wA * 5
wA 0
1 1 (1 2 bA)wA 0 .
8. Note: the optimal risky portfolio now has weights: wM * 5 1 2 wA
* ; wi * 5 wA
* wi.
9. Calculate the risk premium of the optimal risky portfolio from the risk
premium of the index portfolio and the alpha of the active portfolio:
E(RP) 5 (wM * 1 wA
* bA)E(RM) 1 wA * aA. Notice that the beta of the risky portfolio
is wM * 1 wA
* bA because the beta of the index portfolio is 1.
10. Compute the variance of the optimal risky portfolio from the variance
of the index portfolio and the residual variance of the active portfolio:
sP 2 5 (wM
* 1 wA * bA)
2sM 2 1 3wA* s(eA)42.
An Example We can illustrate the implementation of the index model by constructing an optimal port-
folio from the S&P 500 index and the six stocks for which we analyzed risk parameters in
Section 8.3.
This example entails only six analyzed stocks, but by virtue of selecting three pairs of firms from the same industry with relatively high residual correlations, we put the index
model to a severe test. This is because the model ignores the correlation between residuals
when producing estimates for the covariance matrix. Therefore, comparison of results from
the index model with the full-blown covariance (Markowitz) model should be instructive.
Risk Premium Forecasts Panel 4 of Spreadsheet 8.1 contains estimates of alpha and the risk premium for each stock. These alphas would be the most important production of
the investment company in a real-life procedure. Statistics plays a small role here; in this
arena, macro/security analysis is king. In this example, we simply use illustrative values to
demonstrate the portfolio construction process and possible results. You may wonder why we
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C H A P T E R 8 Index Models 277
have chosen such small, forecast alpha values. The reason is that even when security analysis
uncovers a large apparent mispricing, that is, large alpha values, these forecasts must be
substantially trimmed to account for the fact that such forecasts are subject to large estima-
tion error. We discuss the important procedure of adjusting actual forecasts in Chapter 27.
The Optimal Risky Portfolio Panel 5 of Spreadsheet 8.1 displays calculations for the optimal risky portfolio. They follow the summary procedure of Section 8.4 (you should
try to replicate these calculations in your own spreadsheet). In this example we allow short
sales. Notice that the weight of each security in the active portfolio (see row 52) has the
same sign as the alpha value. Allowing short sales, the positions in the active portfolio are
quite large (e.g., the position in BP is .7349); this is an aggressive portfolio. As a result,
the alpha of the active portfolio (2.22%) is larger than that of any of the individual alpha
forecasts. However, this aggressive stance also results in a large residual variance (.0404,
which corresponds to a residual standard deviation of 20%). Therefore, the position in the
active portfolio is scaled down (see Equation 8.20) and ends up quite modest (.1718; cell
C57), reinforcing the notion that diversification considerations are paramount in the opti-
mal risky portfolio.
The optimal risky portfolio has a risk premium of 6.48%, standard deviation of 14.22%,
and a Sharpe ratio of .46 (cells J58–J61). By comparison, the Sharpe ratio of the index
portfolio is .06/.1358 5 .44 (cell B61), which is quite close to that of the optimal risky
portfolio. The small improvement is a result of the modest alpha forecasts that we used.
In Chapter 11 on market efficiency and Chapter 24 on performance evaluation we demon-
strate that such results are common in the mutual fund industry. Of course, a few portfolio
managers can and do produce portfolios with better performance.
The interesting question here is the extent to which the index model produces results
that are inferior to that of the full-covariance (Markowitz) model. Figure 8.5 shows the
efficient frontiers from the two models with the example data. We find that the difference is
in fact small. Table 8.2 compares the compositions and expected performance of the global
minimum variance ( G ) and the optimal risky portfolios derived from the two models. The standard deviations of efficient portfolios produced from the Markowitz model and
the index model are calculated from the covariance matrixes used in each model. As dis-
cussed earlier, we cannot be sure that the covariance estimates from the full covariance model are more accurate than those from
the more restrictive single-index model.
However, by assuming the full covariance
model to be more accurate, we get an idea
of how far off the two models can be.
Figure 8.5 shows that for conservative
portfolios (closer to the minimum-variance
portfolio G), the index model underesti- mates the volatility and hence overestimates
performance. The reverse happens with
portfolios that are riskier than the index,
which also include the region near the opti-
mal portfolio. Despite these differences,
what stands out from this comparison is
that the outputs of the two models are in
fact extremely similar, with the index model
perhaps calling for a more conservative
position. This is where we would like to be
with a model relying on approximations.
Figure 8.5 Efficient frontiers with the index model and full-covariance matrix
.08
.07
.06
.05
.04
.03 .10 .11 .12 .13 .14 .15 .16 .17
Standard Deviation
R is
k P
re m
iu m
Efficient Frontier (Full Covariance)
Efficient Frontier (Index Model)
S&P500
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278 P A R T I I Portfolio Theory and Practice
8.5 Practical Aspects of Portfolio Management with the Index Model
Table 8.2
Comparison of portfolios from the single-index and full- covariance models
Global Minimum
Variance Portfolio Optimal Portfolio
Full-Covariance Model
Index Model
Full-Covariance Model
Index Model
Mean .0371 .0354 .0677 .0649 SD .1089 .1052 .1471 .1423 Sharpe ratio .3409 .3370 .4605 .4558
Portfolio Weights
S&P 500 .88 .83 .75 .83 HP 2 .11 2 .17 .10 .07 DELL 2 .01 2 .05 2 .04 2 .06 WMT .23 .14 2 .03 2 .05 TARGET 2 .18 2 .08 .10 .06 BP .22 .20 .25 .13 SHELL 2 .02 .12 2 .12 .03
The tone of our discussions in this chapter indicates that the index model may be
preferred for the practice of portfolio management. Switching from the Markowitz to an
index model is an important decision and hence the first question is whether the index
model really is inferior to the Markowitz full-covariance model.
Is the Index Model Inferior to the Full-Covariance Model? This question is partly related to a more general question of the value of parsimonious
models. As an analogy, consider the question of adding additional explanatory variables
in a regression equation. We know that adding explanatory variables will in most cases
increase R -square, and in no case will R -square fall. But this does not necessarily imply a better regression equation.
14 A better criterion is contribution to the predictive power of the
regression. The appropriate question is whether inclusion of a variable that contributes to in-
sample explanatory power is likely to contribute to out-of-sample forecast precision. Add-
ing variables, even ones that may appear significant, sometimes can be hazardous to forecast
precision. Put differently, a parsimonious model that is stingy about inclusion of indepen-
dent variables is often superior. Predicting the value of the dependent variable depends on
two factors, the precision of the coefficient estimates and the precision of the forecasts of
the independent variables. When we add variables, we introduce errors on both counts.
This problem applies as well to replacing the single-index with the full-blown
Markowitz model, or even a multi-index model of security returns. To add another index,
we need both a forecast of the risk premium of the additional index portfolio and estimates
of security betas with respect to that additional factor. The Markowitz model allows far
14 In fact, the adjusted R -square may fall if the additional variable does not contribute enough explanatory power
to compensate for the extra degree of freedom it uses.
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C H A P T E R 8 Index Models 279
more flexibility in our modeling of asset covariance structure compared to the single-index
model. But that advantage may be illusory if we can’t estimate those covariances with a
sufficient degree of accuracy. Using the full-covariance matrix invokes estimation risk of
thousands of terms. Even if the full Markowitz model would be better in principle, it is very possible that the cumulative effect of so many estimation errors will result in a portfolio
that is actually inferior to that derived from the single-index model.
Against the potential superiority of the full-covariance model, we have the clear practi-
cal advantage of the single-index framework. Its aid in decentralizing macro and security
analysis is another decisive advantage.
The Industry Version of the Index Model Not surprisingly, the index model has attracted the attention of practitioners. To the extent
that it is approximately valid, it provides a convenient benchmark for security analysis.
A portfolio manager who has neither special information about a security nor insight
that is unavailable to the general public will take the security’s alpha value as zero, and,
according to Equation 8.9, will forecast a risk premium for the security equal to b i R M . If we restate this forecast in terms of total returns, one would expect
E(ri) 5 rf 1 bi 3E(rM) 2 rf4 (8.25) A portfolio manager who has a forecast for the market index, E ( r M ), and observes the
risk-free T-bill rate, r f , can use the model to determine the benchmark expected return for any stock. The beta coefficient, the market risk, sM
2 , and the firm-specific risk, s 2 ( e ), can
be estimated from historical SCLs, that is, from regressions of security excess returns on
market index excess returns.
There are several proprietary sources for such regression results, sometimes called “beta
books.” The Web sites for this chapter at the Online Learning Center ( www.mhhe.com/bkm ) also provide security betas. Table 8.3 is a sample of a typical page from a beta book. Beta
books typically use the S&P 500 as the proxy for the market portfolio. They commonly
employ the 60 most recent monthly observations to calculate regression parameters, and use
total returns, rather than excess returns (deviations from T-bill rates) in the regressions. In
this way they estimate a variant of our index model, which is
r 5 a 1 brM 1 e * (8.26)
instead of
r 2 rf 5 a 1 b(rM 2 rf) 1 e (8.27)
To see the effect of this departure, we can rewrite Equation 8.27 as
r 5 rf 1 a 1 brM 2 brf 1 e 5 a 1 rf (1 2 b) 1 brM 1 e (8.28)
Comparing Equations 8.26 and 8.28, you can see that if r f is constant over the sample period, both equations have the same independent variable, r M , and residual, e. Therefore, the slope coefficient will be the same in the two regressions.
15
However, the intercept that beta books call ALPHA, as in Table 8.3 , is really an estimate
of a 1 r f (1 2 b ). The apparent justification for this procedure is that, on a monthly basis, r f (1 2 b ) is small and is likely to be swamped by the volatility of actual stock returns. But it is worth noting that for b ≠ 1, the regression intercept in Equation 8.26 will not equal the index model a as it does when excess returns are used as in Equation 8.27.
15
Actually, r f does vary over time and so should not be grouped casually with the constant term in the regression. However, variations in r f are tiny compared with the swings in the market return. The actual volatility in the T-bill rate has only a small impact on the estimated value of b .
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280 P A R T I I Portfolio Theory and Practice
Always remember as well that these alpha estimates are ex post (after the fact) mea-
sures. They do not mean that anyone could have forecast these alpha values ex ante (before
the fact). In fact, the name of the game in security analysis is to forecast alpha values
ahead of time. A well-constructed portfolio that includes long positions in future positive-
alpha stocks and short positions in future negative-alpha stocks will outperform the market
index. The key term here is “well constructed,” meaning that the portfolio has to balance
concentration on high-alpha stocks with the need for risk-reducing diversification as dis-
cussed earlier in the chapter.
Much of the other output in Table 8.3 is similar to the Excel output ( Table 8.1 ) that we
discussed when estimating the index model for Hewlett-Packard. The R -square statistic is the ratio of systematic variance to total variance, the fraction of total volatility attribut-
able to market movements. For most firms, R -square is substantially below .5, indicating that stocks have far more firm-specific than systematic risk. This highlights the practical
importance of diversification.
Ticker Symbol Security Name BETA ALPHA RSQ
Residual Std Dev
Std Error Beta
Standard Error Alpha
Adjusted Beta
AMZN Amazon.com 2.25 0.006 0.238 0.1208 0.5254 0.0156 1.84 F Ford 1.64 2 0.012 0.183 0.1041 0.4525 0.0135 1.43 NEM Newmont Mining Corp. 0.44 0.002 0.023 0.0853 0.3709 0.0110 0.62 INTC Intel Corporation 1.60 2 0.010 0.369 0.0627 0.2728 0.0081 1.40 MSFT Microsoft Corporation 0.87 0.001 0.172 0.0569 0.2477 0.0074 0.91 DELL Dell Inc. 1.36 2 0.014 0.241 0.0723 0.3143 0.0094 1.24 BA Boeing Co. 1.42 0.004 0.402 0.0517 0.2250 0.0067 1.28 MCD McDonald’s Corp. 0.92 0.016 0.312 0.0409 0.1777 0.0053 0.95 PFE Pfizer Inc. 0.65 2 0.006 0.131 0.0504 0.2191 0.0065 0.77 DD DuPont 0.97 2 0.002 0.311 0.0434 0.1887 0.0056 0.98 DIS Walt Disney Co. 0.91 0.005 0.278 0.0440 0.1913 0.0057 0.94 XOM ExxonMobil Corp. 0.87 0.011 0.216 0.0497 0.2159 0.0064 0.91 IBM IBM Corp. 0.88 0.004 0.248 0.0459 0.1997 0.0059 0.92 WMT Walmart 0.06 0.002 0.002 0.0446 0.1941 0.0058 0.38 HNZ HJ Heinz Co. 0.43 0.009 0.110 0.0368 0.1599 0.0048 0.62 LTD Limited Brands Inc. 1.30 0.001 0.216 0.0741 0.3223 0.0096 1.20 ED Consolidated Edison Inc. 0.15 0.004 0.101 0.0347 0.1509 0.0045 0.43 GE General Electric Co. 0.65 2 0.002 0.173 0.0425 0.1850 0.0055 0.77 MEAN 0.97 0.001 0.207 0.0589 0.2563 0.0076 0.98 STD DEVIATION 0.56 0.008 0.109 0.0239 0.1039 0.0031 0.37
Table 8.3
Market sensitivity statistics: Regressions of total stock returns on total S&P 500 returns over 60 months, 2004–2008
Source: Compiled from CRSP (University of Chicago) database.
What was Intel’s index-model a per month during the period covered by the Table 8.3 regression if during this period the average monthly rate of return on T-bills was .2%?
CONCEPT CHECK 8.4
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C H A P T E R 8 Index Models 281
The Resid Std Dev column is the standard deviation of the monthly regression residuals, also sometimes called the standard error of the regression. The standard errors of the alpha
and beta estimates allow us to evaluate the precision of the estimates. Notice that the stan-
dard errors of alpha tend to be far greater multiples of the estimated value of alpha than is
the case for beta estimates.
Intel’s Resid Std Dev is 6.27% per month and its R 2 is .369. This tells us that sIntel
2 (e) 5 6.272 5 39.31 and, because R 2 5 1 2 s 2 ( e )/ s 2 , we can solve for Intel’s total standard deviation by rearranging as follows:
sIntel 5 BsIntel 2 (e)
1 2 R2 R1/2 5 a39.31
.631 b1/2 5 7.89% per month
This is Intel’s monthly standard deviation for the sample period. Therefore, the annualized
standard deviation for that period was 7.89 "12 5 27.33% . The last column is called Adjusted Beta. The motivation for adjusting beta estimates
is that, on average, the beta coefficients of stocks seem to move toward 1 over time. One
explanation for this phenomenon is intuitive. A business enterprise usually is established
to produce a specific product or service, and a new firm may be more unconventional than
an older one in many ways, from technology to management style. As it grows, however, a
firm often diversifies, first expanding to similar products and later to more diverse opera-
tions. As the firm becomes more conventional, it starts to resemble the rest of the economy
even more. Thus its beta coefficient will tend to change in the direction of 1.
Another explanation for this phenomenon is statistical. We know that the average beta
over all securities is 1. Thus, before estimating the beta of a security, our best forecast
would be that it is 1. When we estimate this beta coefficient over a particular sample
period, we sustain some unknown sampling error of the estimated beta. The greater the dif-
ference between our beta estimate and 1, the greater is the chance that we incurred a large
estimation error and that beta in a subsequent sample period will be closer to 1.
The sample estimate of the beta coefficient is the best guess for that sample period.
Given that beta has a tendency to evolve toward 1, however, a forecast of the future beta
coefficient should adjust the sample estimate in that direction.
Table 8.3 adjusts beta estimates in a simple way. 16
It takes the sample estimate of beta
and averages it with 1, using weights of two-thirds and one-third:
Adjusted beta 5 2⁄3 sample beta 1 1⁄3 (1) (8.29)
16
A more sophisticated method is described in Oldrich A. Vasicek, “A Note on Using Cross-Sectional Information
in Bayesian Estimation of Security Betas,” Journal of Finance 28 (1973), pp. 1233–39.
For the 60 months used in Table 8.3 , Intel’s beta was estimated at 1.60. Therefore, its adjusted beta is 2⁄3 3 1.60 1 1⁄3 5 1.40, taking it a third of the way toward 1.
In the absence of special information concerning Intel, if our forecast for the market index is 10% and T-bills pay 4%, we learn from the beta book that the forecast for the rate of return on Intel stock is
E(rIntel) 5 rf 1 Adjusted beta 3 3E(rM) 2 rf 4 5 4 1 1.40(10 2 4) 5 12.40%
The sample period regression alpha is 2 1.0%. Because Intel’s beta is greater than 1, we know that this means that the estimate of the index model a is somewhat larger.
Example 8.1 Adjusted Beta
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282 P A R T I I Portfolio Theory and Practice
As in Equation 8.28, we have to subtract (1 2 b ) r f from the regression alpha to obtain the index model a . In any event, the standard error of the alpha estimate is .81%. The estimate of alpha is far less than twice its standard error. Consequently, we cannot reject the hypothesis that the true alpha is zero.
Predicting Betas Adjusted betas are a simple way to recognize that betas estimated from past data may not
be the best estimates of future betas: Betas seem to drift toward 1 over time. This suggests
that we might want a forecasting model for beta.
One simple approach would be to collect data on beta in different periods and then esti-
mate a regression equation:
Current beta 5 a 1 b (Past beta) (8.30)
Given estimates of a and b, we would then forecast future betas using the rule
Forecast beta 5 a 1 b (Current beta) (8.31)
There is no reason, however, to limit ourselves to such simple forecasting rules. Why not
also investigate the predictive power of other financial variables in forecasting beta? For
example, if we believe that firm size and debt ratios are two determinants of beta, we might
specify an expanded version of Equation 8.30 and estimate
Current beta 5 a 1 b1(Past beta) 1 b2(Firm size) 1 b3(Debt ratio)
Now we would use estimates of a and b 1 through b 3 to forecast future betas. Such an approach was followed by Rosenberg and Guy,
17 who found the following
variables to help predict betas:
1. Variance of earnings.
2. Variance of cash flow.
3. Growth in earnings per share.
4. Market capitalization (firm size).
5. Dividend yield.
6. Debt-to-asset ratio.
Rosenberg and Guy also found that even after controlling for a firm’s financial charac-
teristics, industry group helps to predict beta. For example, they found that the beta values
of gold mining companies are on average .827 lower than would be predicted based on
financial characteristics alone. This should not be surprising; the 2 .827 “adjustment factor”
for the gold industry reflects the fact that gold values are inversely related to market returns.
Table 8.4 presents beta estimates and adjustment factors for a subset of firms in the
Rosenberg and Guy study.
17
Barr Rosenberg and J. Guy, “Prediction of Beta from Investment Fundamentals, Parts 1 and 2,” Financial Analysts Journal, May–June and July–August 1976.
Compare the first five and last four industries in Table 8.4 . What characteristic seems to determine whether the adjustment factor is positive or negative?
CONCEPT CHECK 8.5
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283
Index Models and Tracking Portfolios Suppose a portfolio manager believes she has identified an underpriced portfolio. Her
security analysis team estimates the index model equation for this portfolio (using the S&P
500 index) in excess return form and obtains the following estimates:
RP 5 .04 1 1.4RS&P500 1 eP (8.32)
Alpha Betting
For those who believe in efficient markets, the recent explosion in the number of exchange-traded funds rep- resents a triumph. ETFs are quoted securities that track a particular index, for a fee that is normally just a fraction of a percentage point. They enable investors to assemble a low-cost portfolio covering a wide range of assets from international equities, through government and corporate bonds, to commodities.
But as fast as the assets of ETFs and index-tracking mutual funds are growing, another section of the indus- try seems to be flourishing even faster. Watson Wyatt, a firm of actuaries, estimates that “alternative asset invest- ment” (ranging from hedge funds through private equity to property) grew by around 20% in 2005, to $1.26 trillion. Investors who take this route pay much higher fees in the hope of better performance. One of the fastest-growing assets, funds of hedge funds, charge some of the highest fees of all.
Why are people paying up? In part, because investors have learned to distinguish between the market return, dubbed beta, and managers’ outperformance, known as alpha. “Why wouldn’t you buy beta and alpha sepa- rately?” asks Arno Kitts of Henderson Global Investors, a fund-management firm. “Beta is a commodity and alpha is about skill.”
Clients have become convinced that no one firm can produce good performance in every asset class. That has
led to a “core and satellite” model, in which part of the portfolio is invested in index trackers with the rest in the hands of specialists. But this creates its own problems. Rela- tions with a single balanced manager are simple. It is much harder to research and monitor the performance of spe- cialists. That has encouraged the middlemen—managers of managers (in the traditional institutional business) and funds-of-funds (in the hedge-fund world), which are usu- ally even more expensive.
That their fees endure might suggest investors can identify outperforming fund managers in advance. How- ever, studies suggest this is extremely hard. And even where you can spot talent, much of the extra perfor- mance may be siphoned off into higher fees. “A dispro- portionate amount of the benefits of alpha go to the manager, not the client,” says Alan Brown at Schroders, an asset manager.
In any event, investors will probably keep pursuing alpha, even though the cheaper alternatives of ETFs and tracking funds are available. Craig Baker of Watson Wyatt, says that, although above-market returns may not be avail- able to all, clients who can identify them have a “first mover” advantage. As long as that belief exists, managers can charge high fees.
Source: The Economist, September 14, 2006. Copyright © The Economist Newspaper Limited, London.
W O
R D
S FR O
M T
H E ST
R E E T
Industry Beta Adjustment Factor
Agriculture 0.99 2 .140 Drugs and medicine 1.14 2 .099 Telephone 0.75 2 .288 Energy utilities 0.60 2 .237 Gold 0.36 2 .827 Construction 1.27 .062 Air transport 1.80 .348 Trucking 1.31 .098 Consumer durables 1.44 .132
Table 8.4
Industry betas and adjustment factors
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284 P A R T I I Portfolio Theory and Practice
Therefore, P has an alpha value of 4% and a beta of 1.4. The manager is confident in the quality of her security analysis but is wary about the performance of the broad market in
the near term. If she buys the portfolio, and the market as a whole turns down, she still
could lose money on her investment (which has a large positive beta) even if her team is
correct that the portfolio is underpriced on a relative basis. She would like a position that
takes advantage of her team’s analysis but is independent of the performance of the overall
market.
To this end, a tracking portfolio ( T ) can be constructed. A tracking portfolio for port- folio P is a portfolio designed to match the systematic component of P ’s return. The idea is for the portfolio to “track” the market-sensitive component of P ’s return. This means the tracking portfolio must have the same beta on the index portfolio as P and as little nonsys- tematic risk as possible. This procedure is also called beta capture.
A tracking portfolio for P will have a levered position in the S&P 500 to achieve a beta of 1.4. Therefore, T includes positions of 1.4 in the S&P 500 and 2 .4 in T-bills. Because T is constructed from the index and bills, it has an alpha value of zero.
Now consider buying portfolio P but at the same time offsetting systematic risk by assuming a short position in the tracking portfolio. The short position in T cancels out the systematic exposure of the long position in P: the overall combined position is thus market neutral. Therefore, even if the market does poorly, the combined position should not be affected. But the alpha on portfolio P will remain intact. The combined portfolio, C, pro- vides an excess return per dollar of
RC 5 RP 2 RT 5 (.04 1 1.4RS&P500 1 eP) 2 1.4RS&P500 5 .04 1 eP (8.33)
While this portfolio is still risky (due to the residual risk, e P ), the systematic risk has been eliminated, and if P is reasonably well-diversified, the remaining nonsystematic risk will be small. Thus the objective is achieved: The manager can take advantage of the 4% alpha
without inadvertently taking on market exposure. The process of separating the search for
alpha from the choice of market exposure is called alpha transport. This “long-short strategy” is characteristic of the activity of many hedge funds. Hedge
fund managers identify an underpriced security and then try to attain a “pure play” on the
perceived underpricing. They hedge out all extraneous risk, focusing the bet only on the
perceived “alpha” (see the box on p. 283). Tracking funds are the vehicle used to hedge
the exposures to which they do not want exposure. Hedge fund managers use index regres- sions such as those discussed here, as well as more-sophisticated variations, to create the
tracking portfolios at the heart of their hedging strategies.
SUMMARY 1. A single-factor model of the economy classifies sources of uncertainty as systematic (macro- economic) factors or firm-specific (microeconomic) factors. The index model assumes that the
macro factor can be represented by a broad index of stock returns.
2. The single-index model drastically reduces the necessary inputs in the Markowitz portfolio selec- tion procedure. It also aids in specialization of labor in security analysis.
3. According to the index model specification, the systematic risk of a portfolio or asset equals b2sM 2
and the covariance between two assets equals bibjsM 2 .
4. The index model is estimated by applying regression analysis to excess rates of return. The slope of the regression curve is the beta of an asset, whereas the intercept is the asset’s alpha during the
sample period. The regression line is also called the security characteristic line.
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C H A P T E R 8 Index Models 285
5. Optimal active portfolios constructed from the index model include analyzed securities in propor- tion to their information ratios. The full risky portfolio is a mixture of the active portfolio and the
passive market-index portfolio. The index portfolio is used to enhance the diversification of the
overall risky position.
6. Practitioners routinely estimate the index model using total rather than excess rates of return. This makes their estimate of alpha equal to a 1 r f (1 2 b ).
7. Betas show a tendency to evolve toward 1 over time. Beta forecasting rules attempt to predict this drift. Moreover, other financial variables can be used to help forecast betas.
Related Web sites for this chapter are available at www. mhhe.com/bkm
KEY TERMS single-factor model single-index model
regression equation
residuals
security characteristic line
scatter diagram
information ratio
tracking portfolio
KEY EQUATIONS Single-index model (in excess returns): Ri(t) 5 ai 1 biRM(t) 1 ei(t) Security risk in index model:
Total risk 5 Systematic risk 1 firm-specific risk
s2 5 b2sM 2 1 s2(e)
Covariance 5 Cov(ri , rj) 5 Product of betas 3 Market-index risk 5 bibjsM 2
Active portfolio management in the index model
Sharpe ratio of optimal risky portfolio: SP 2 5 SM
2 1 B aA s(eA)
R2
Asset weight in active portfolio: wi * 5 wA
*
ai
s2(ei)
a n
i51
ai
s2(ei)
Information ratio of active portfolio: B aA s(eA)
R2 5 a n
i51 B ai
s(ei) R2
PROBLEM SETS 1. What are the advantages of the index model compared to the Markowitz procedure for obtaining an efficiently diversified portfolio? What are its disadvantages?
2. What is the basic trade-off when departing from pure indexing in favor of an actively managed
portfolio?
3. H ow does the magnitude of firm-specific risk affect the extent to which an active investor will be
willing to depart from an indexed portfolio?
4. Why do we call alpha a “nonmarket” return premium? Why are high-alpha stocks desirable
investments for active portfolio managers? With all other parameters held fixed, what would
happen to a portfolio’s Sharpe ratio as the alpha of its component securities increased?
Basic
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286 P A R T I I Portfolio Theory and Practice
5. A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient
portfolio using only these 60 securities.
a. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?
b. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?
6. The following are estimates for two stocks.
Stock Expected Return Beta Firm-Specific Standard Deviation
A 13% 0.8 30% B 18 1.2 40
The market index has a standard deviation of 22% and the risk-free rate is 8%.
a. What are the standard deviations of stocks A and B ? b. Suppose that we were to construct a portfolio with proportions:
Stock A: .30 Stock B: .45 T-bills: .25
Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of
the portfolio.
7. Consider the following two regression lines for stocks A and B in the following figure.
Intermediate
rA − rf rB − rf
rM − rf rM − rf
a. Which stock has higher firm-specific risk? b. Which stock has greater systematic (market) risk? c. Which stock has higher R 2 ? d. Which stock has higher alpha? e. Which stock has higher correlation with the market?
8. Consider the two (excess return) index model regression results for A and B:
R A 5 1% 1 1.2 R M
R -square 5 .576
Residual standard deviation 5 10.3%
R B 5 2 2% 1 .8 R M
R -square 5 .436
Residual standard deviation 5 9.1%
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C H A P T E R 8 Index Models 287
a. Which stock has more firm-specific risk? b. Which has greater market risk? c. For which stock does market movement explain a greater fraction of return variability? d. If r f were constant at 6% and the regression had been run using total rather than excess
returns, what would have been the regression intercept for stock A ?
Use the following data for Problems 9 through 14. Suppose that the index model for stocks A and B is estimated from excess returns with the following results:
RA 5 3% 1 .7RM 1 eA RB 5 22% 1 1.2RM 1 eB sM 5 20%; R-squareA 5 .20; R-squareB 5 .12
9. What is the standard deviation of each stock?
10. Break down the variance of each stock to the systematic and firm-specific components.
11. What are the covariance and correlation coefficient between the two stocks?
12. What is the covariance between each stock and the market index?
13. For portfolio P with investment proportions of .60 in A and .40 in B, rework Problems 9, 10, and 12.
14. Rework Problem 13 for portfolio Q with investment proportions of .50 in P, .30 in the market index, and .20 in T-bills.
15. A stock recently has been estimated to have a beta of 1.24:
a. What will a beta book compute as the “adjusted beta” of this stock? b. Suppose that you estimate the following regression describing the evolution of beta over time:
bt 5 .3 1 .7bt21
What would be your predicted beta for next year?
16. Based on current dividend yields and expected growth rates, the expected rates of return on stocks
A and B are 11% and 14%, respectively. The beta of stock A is .8, while that of stock B is 1.5. The T-bill rate is currently 6%, while the expected rate of return on the S&P 500 index is 12%. The
standard deviation of stock A is 10% annually, while that of stock B is 11%. If you currently hold a passive index portfolio, would you choose to add either of these stocks to your holdings?
17. A portfolio manager summarizes the input from the macro and micro forecasters in the follow-
ing table:
Micro Forecasts
Asset Expected Return (%) Beta Residual Standard Deviation (%)
Stock A 20 1.3 58 Stock B 18 1.8 71 Stock C 17 0.7 60 Stock D 12 1.0 55
Macro Forecasts
Asset Expected Return (%) Standard Deviation (%)
T-bills 8 0 Passive equity portfolio 16 23
a. Calculate expected excess returns, alpha values, and residual variances for these stocks. b. Construct the optimal risky portfolio. c. What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the
active portfolio?
d. What should be the exact makeup of the complete portfolio for an investor with a coefficient of risk aversion of 2.8?
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288 P A R T I I Portfolio Theory and Practice
18. Recalculate Problem 17 for a portfolio manager who is not allowed to short sell securities.
a. What is the cost of the restriction in terms of Sharpe’s measure? b. What is the utility loss to the investor ( A 5 2.8) given his new complete portfolio?
19. Suppose that on the basis of the analyst’s past record, you estimate that the relationship between
forecast and actual alpha is:
Actual abnormal return 5 .3 3 Forecast of alpha
Use the alphas from Problem 17. How much is expected performance affected by recognizing
the imprecision of alpha forecasts?
20. S uppose that the alpha forecasts in row 44 of Spreadsheet 8.1 are doubled. All the other data
remain the same. Recalculate the optimal risky portfolio. Before you do any calculations, how-
ever, use the Summary of Optimization Procedure to estimate a back-of-the-envelope calcula-
tion of the information ratio and Sharpe ratio of the newly optimized portfolio. Then recalculate
the entire spreadsheet example and verify your back-of-the-envelope calculation.
Challenge
1. When the annualized monthly percentage rates of return for a stock market index were regressed
against the returns for ABC and XYZ stocks over a 5-year period ending in 2013, using an ordi-
nary least squares regression, the following results were obtained:
Statistic ABC XYZ
Alpha 2 3.20% 7.3%
Beta 0.60 0.97 R 2 0.35 0.17 Residual standard deviation 13.02% 21.45%
Explain what these regression results tell the analyst about risk–return relationships for each
stock over the sample period. Comment on their implications for future risk–return relationships,
assuming both stocks were included in a diversified common stock portfolio, especially in view
of the following additional data obtained from two brokerage houses, which are based on 2 years
of weekly data ending in December 2013.
Brokerage House Beta of ABC Beta of XYZ
A .62 1.45 B .71 1.25
2. Assume the correlation coefficient between Baker Fund and the S&P 500 Stock Index is .70.
What percentage of Baker Fund’s total risk is specific (i.e., nonsystematic)?
3. The correlation between the Charlottesville International Fund and the EAFE Market Index is
1.0. The expected return on the EAFE Index is 11%, the expected return on Charlottesville Inter-
national Fund is 9%, and the risk-free return in EAFE countries is 3%. Based on this analysis,
what is the implied beta of Charlottesville International?
4. The concept of beta is most closely associated with:
a. Correlation coefficients. b. Mean-variance analysis. c. Nonsystematic risk. d. Systematic risk.
5. Beta and standard deviation differ as risk measures in that beta measures:
a. Only unsystematic risk, while standard deviation measures total risk. b. Only systematic risk, while standard deviation measures total risk. c. Both systematic and unsystematic risk, while standard deviation measures only unsystematic risk. d. Both systematic and unsystematic risk, while standard deviation measures only systematic risk.
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C H A P T E R 8 Index Models 289
SOLUTIONS TO CONCEPT CHECKS
E-INVESTMENTS EXERCISES Go to http://finance.yahoo.com and click on Stocks link under the Investing tab. Look for the Stock Screener link under Research Tools. The Java Yahoo! Finance Screener lets you create your own screens. In the Click to Add Criteria box, find Trading and Volume on the menu and choose Beta. In the Conditions box, choose , 5 and in the Values box, enter 1. Hit the Enter key and then request the top 200 matches in the Return Top_Matches box. Click on the Run Screen button.
Select the View Table tab and sort the results to show the lowest betas at the top of the list by clicking on the Beta column header. Which firms have the lowest betas? In which industries do they operate?
Select the View Histogram tab and when the histogram appears, look at the bottom of the screen to see the Show Histogram for box. Use the menu that comes up when you click on the down arrow to select beta. What pattern(s), if any, do you see in the distributions of betas for firms that have betas less than 1?
1. a. Total market capitalization is 3,000 1 1,940 1 1,360 5 6,300. Therefore, the mean excess return of the index portfolio is
3,000
6,300 3 10 1
1,940
6,300 3 2 1
1,360
6,300 3 17 5 9.05% 5 .0905
b. The covariance between stocks A and B equals
Cov(RA, RB) 5 bAbB sM 2 5 1 3 .2 3 .252 5 .0125
c. The covariance between stock B and the index portfolio equals
Cov(RB, RM) 5 bBsM 2 5 .2 3 .252 5 .0125
d. The total variance of B equals
sB 2 5 Var (bBRM 1 eB) 5 bB
2 sM 2 1 s2(eB)
Systematic risk equals bB 2 sM
2 5 .22 3 .252 5 .0025.
Thus the firm-specific variance of B equals
s2(eB) 5 sB 2 2 bB
2 sM 2 5 .302 2 .22 3 .252 5 .0875
2. The variance of each stock is b2sM 2 1 s2(e).
For stock A, we obtain
sA 2 5 .92(20)2 1 302 5 1,224
sA 5 35%
For stock B,
sB 2 5 1.12(20)2 1 102 5 584
sB 5 24%
The covariance is
bAbB sM 2 5 .9 3 1.1 3 202 5 396
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290 P A R T I I Portfolio Theory and Practice
3. s 2 (eP) 5 (
1⁄2) 2 [s
2 (eA) 1 s
2 (eB)]
5 1⁄4 (.302 1 .102)
5 .0250
Therefore s ( e P ) 5 .158 5 15.8%
4. The regression ALPHA is related to the index-model a by
ALPHA 5 aindex model 1 (1 2 b)rf
For Intel, ALPHA 5 2 1.0%, b 5 1.60, and we are told that r f was .2%. Thus
aindex model 5 21.0% 2 (1 2 1.60).2% 5 2.88%
Intel’s return was somewhat disappointing. It underperformed its “benchmark” return by an
average of .88% per month.
5. The industries with positive adjustment factors are most sensitive to the economy. Their betas
would be expected to be higher because the business risk of the firms is higher. In contrast, the
industries with negative adjustment factors are in business fields with a lower sensitivity to the
economy. Therefore, for any given financial profile, their betas are lower.
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9 THE CAPITAL ASSET pricing model, almost always referred to as the CAPM, is a center- piece of modern financial economics. The model gives us a precise prediction of the rela- tionship that we should observe between the risk of an asset and its expected return. This relationship serves two vital functions. First, it provides a benchmark rate of return for evaluating possible investments. For exam- ple, if we are analyzing securities, we might be interested in whether the expected return we forecast for a stock is more or less than its
“fair” return given its risk. Second, the model helps us to make an educated guess as to the expected return on assets that have not yet been traded in the marketplace. For example, how do we price an initial public offering of stock? How will a major new investment proj- ect affect the return investors require on a company’s stock? Although the CAPM does not fully withstand empirical tests, it is widely used because of the insight it offers and because its accuracy is deemed acceptable for important applications.
The Capital Asset
Pricing Model
CHAPTER NINE
9.1 The Capital Asset Pricing Model
The capital asset pricing model is a set of predictions concerning equilibrium expected
returns on risky assets. Harry Markowitz laid down the foundation of modern portfolio
management in 1952. The CAPM was published 12 years later in articles by William
Sharpe, 1 John Lintner,
2 and Jan Mossin.
3 The time for this gestation indicates that the
leap from Markowitz’s portfolio selection model to the CAPM is not trivial.
Shooting straight to the heart of the CAPM, suppose all investors optimized their port-
folios á la Markowitz. That is, each investor uses an input list (expected returns and covari-
ance matrix) to draw an efficient frontier employing all available risky assets and identifies
an efficient risky portfolio, P, by drawing the tangent CAL (capital allocation line) to the frontier as in Figure 9.1 , panel A (which is just a reproduction of Figure 7.11). As a result,
each investor holds securities in the investable universe with weights arrived at by the
Markowitz optimization process.
1 William Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium,” Journal of Finance, September 1964.
2 John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and
Capital Budgets,” Review of Economics and Statistics, February 1965. 3 Jan Mossin, “Equilibrium in a Capital Asset Market,” Econometrica, October 1966.
9 PA
R T
III
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292 P A R T I I I Equilibrium in Capital Markets
The CAPM asks what would
happen if all investors shared an
identical investable universe and
used the same input list to draw
their efficient frontiers. Obviously,
their efficient frontiers would be
identical. Facing the same risk-free
rate, they would then draw an iden-
tical tangent CAL and naturally
all would arrive at the same risky
portfolio, P. All investors there- fore would choose the same set of
weights for each risky asset. What
must be these weights?
A key insight of the CAPM is
this: Because the market portfolio
is the aggregation of all of these
identical risky portfolios, it too will
have the same weights. Therefore,
if all investors choose the same
risky portfolio, it must be the mar- ket portfolio, that is, the value- weighted portfolio of all assets in
the investable universe. Therefore,
the capital allocation line based
on each investor’s optimal risky
portfolio will in fact also be the
capital market line, as depicted in Figure 9.1 , panel B. This impli-
cation will allow us to say much
about the risk–return trade-off.
Why Do All Investors Hold the Market Portfolio? What is the market portfolio?
When we sum over, or aggregate,
the portfolios of all individual
investors, lending and borrowing
will cancel out (because each lender has a corresponding borrower), and the value of the
aggregate risky portfolio will equal the entire wealth of the economy. This is the market portfolio, M. The proportion of each stock in this portfolio equals the market value of the stock (price per share times number of shares outstanding) divided by the sum of the mar-
ket value of all stocks. 4 This implies that if the weight of GE stock, for example, in each
common risky portfolio is 1%, then GE also will constitute 1% of the market portfolio. The
same principle applies to the proportion of any stock in each investor’s risky portfolio. As
a result, the optimal risky portfolio of all investors is simply a share of the market portfolio
in Figure 9.1 .
E(r)
CAL(P) Efficient Frontier
A: The Efficient Frontier of Risky Assets with the Optimal CAL
B: The Efficient Frontier and the Capital Market Line
P
rf
σ
E(rP)
σP
E(r)
CML Efficient Frontier
M
rf
σ
E(rM)
σM
Figure 9.1 Capital allocation line and the capital market line
4 We use the term “stock” for convenience; the market portfolio properly includes all assets in the economy.
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C H A P T E R 9 The Capital Asset Pricing Model 293
Now suppose that the optimal portfolio of our investors does not include the stock of
some company, such as Delta Airlines. When all investors avoid Delta stock, the demand
is zero, and Delta’s price takes a free fall. As Delta stock gets progressively cheaper, it
becomes ever more attractive and other stocks look relatively less attractive. Ultimately,
Delta reaches a price where it is attractive enough to include in the optimal stock portfolio.
Such a price adjustment process guarantees that all stocks will be included in the opti-
mal portfolio. It shows that all assets have to be included in the market portfolio. The only issue is the price at which investors will be willing to include a stock in their optimal risky
portfolio.
The Passive Strategy Is Efficient In Chapter 6 we defined the CML as the CAL that is constructed from a money market
account (or T-bills) and the market portfolio. Perhaps now you can fully appreciate why
the CML is an interesting CAL. In the simple world of the CAPM, M is the optimal tan- gency portfolio on the efficient frontier.
In this scenario, the market portfolio held by all investors is based on the common input
list, thereby incorporating all relevant information about the universe of securities. This
means that investors can skip the trouble of doing security analysis and obtain an efficient
portfolio simply by holding the market portfolio. (Of course, if everyone were to follow
this strategy, no one would perform security analysis and this result would no longer hold.
We discuss this issue in greater depth in Chapter 11 on market efficiency.)
Thus the passive strategy of investing in a market-index portfolio is efficient. For this
reason, we sometimes call this result a mutual fund theorem . The mutual fund theorem is another incarnation of the separation property discussed in Chapter 7. If all investors
would freely choose to hold a common risky portfolio identical to the market portfolio,
they would not object if all stocks in the market were replaced with shares of a single
mutual fund holding that market portfolio.
In reality, different investment managers do create risky portfolios that differ from the
market index. We attribute this in part to the use of different input lists in the formation of
their optimal risky portfolios. Nevertheless, the practical
significance of the mutual fund theorem is that a passive
investor may view the market index as a reasonable first
approximation to an efficient risky portfolio.
The nearby box contains a parable illustrating the argu-
ment for indexing. If the passive strategy is efficient, then
attempts to beat it simply generate trading and research
costs with no offsetting benefit, and ultimately inferior
results.
The Risk Premium of the Market Portfolio In Chapter 6 we discussed how individual investors go about deciding capital allocation.
If all investors choose to invest in portfolio M and the risk-free asset, what can we deduce about the equilibrium risk premium of portfolio M ?
Recall that each individual investor chooses a proportion y, allocated to the optimal portfolio M, such that
y 5 E ( rM ) 2 rf
AsM 2
(9.1)
where E ( r M ) 2 r f 5 E ( R M ) is the risk premium (expected excess return) on the market portfolio.
If there are only a few investors who perform security analysis, and all others hold the market portfolio, M, would the CML still be the efficient CAL for investors who do not engage in security analysis? Why or why not?
CONCEPT CHECK 9.1
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294
In the simplified CAPM economy, risk-free investments involve borrowing and lend-
ing among investors. Any borrowing position must be offset by the lending position of
the creditor. This means that net borrowing and lending across all investors must be zero,
and therefore, substituting the representative investor’s risk aversion, A, for A, the average position in the risky portfolio is 100%, or y 5 1. Setting y 5 1 in Equation 9.1 and rear- ranging, we find that the risk premium on the market portfolio is related to its variance by
the average degree of risk aversion:
E ( RM ) 5 AsM 2 (9.2)
The Parable of the Money Managers
Some years ago, in a land called Indicia, revolution led to the overthrow of a socialist regime and the restoration of a system of private property. Former government enterprises were reformed as corporations, which then issued stocks and bonds. These securities were given to a central agency, which offered them for sale to individuals, pension funds, and the like (all armed with newly printed money).
Almost immediately a group of money managers came forth to assist these investors. Recalling the words of a venerated elder, uttered before the previous revolution (“Invest in Corporate Indicia”), they invited clients to give them money, with which they would buy a cross-section of all the newly issued securities. Investors considered this a reasonable idea, and soon everyone held a piece of Corporate Indicia.
Before long the money managers became bored because there was little for them to do. Soon they fell into the habit of gathering at a beachfront casino where they passed the time playing roulette, craps, and similar games, for low stakes, with their own money.
After a while, the owner of the casino suggested a new idea. He would furnish an impressive set of rooms which would be designated the Money Managers’ Club. There the members could place bets with one another about the fortunes of various corporations, industries, the level of the Gross Domestic Product, foreign trade, etc. To make the betting more exciting, the casino owner suggested that the managers use their clients’ money for this purpose.
The offer was immediately accepted, and soon the money managers were betting eagerly with one another. At the end of each week, some found that they had won money for their clients, while others found that they had lost. But the losses always exceeded the gains, for a certain
amount was deducted from each bet to cover the costs of the elegant surroundings in which the gambling took place.
Before long a group of professors from Indicia U. sug- gested that investors were not well served by the activi- ties being conducted at the Money Managers’ Club. “Why pay people to gamble with your money? Why not just hold your own piece of Corporate Indicia?” they said.
This argument seemed sensible to some of the inves- tors, and they raised the issue with their money managers. A few capitulated, announcing that they would henceforth stay away from the casino and use their clients’ money only to buy proportionate shares of all the stocks and bonds issued by corporations.
The converts, who became known as managers of Indicia funds, were initially shunned by those who contin- ued to frequent the Money Managers’ Club, but in time, grudging acceptance replaced outright hostility. The wave of puritan reform some had predicted failed to material- ize, and gambling remained legal. Many managers contin- ued to make their daily pilgrimage to the casino. But they exercised more restraint than before, placed smaller bets, and generally behaved in a manner consonant with their responsibilities. Even the members of the Lawyers’ Club found it difficult to object to the small amount of gam- bling that still went on.
And everyone but the casino owner lived happily ever after.
Source: William F. Sharpe, “The Parable of the Money Managers,” The Financial Analysts’ Journal 32 (July/August 1976), p. 4. Copyright 1976, CFA Institute. Reproduced from The Financial Analysts’ Journal with permission from the CFA Institute. All rights reserved.
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Data from the last eight decades for the S&P 500 index yield the following statistics: average excess return, 7.9%; standard deviation, 23.2%.
a. To the extent that these averages approximated investor expectations for the period, what must have been the average coefficient of risk aversion?
b. If the coefficient of risk aversion were actually 3.5, what risk premium would have been consistent with the market’s historical standard deviation?
CONCEPT CHECK 9.2
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C H A P T E R 9 The Capital Asset Pricing Model 295
Expected Returns on Individual Securities The CAPM is built on the insight that the appropriate risk premium on an asset will be
determined by its contribution to the risk of investors’ overall portfolios. Portfolio risk is
what matters to investors and is what governs the risk premiums they demand.
Remember that in the CAPM, all investors use the same input list, that is, the same
estimates of expected returns, variances, and covariances. To calculate the variance of
the market portfolio, we use the bordered covariance matrix with the market portfolio
weights, as discussed in Chapter 7. We highlight GE in this depiction of the n stocks in the market portfolio so that we can measure the contribution of GE to the risk of the market
portfolio.
Recall that we calculate the variance of the portfolio by summing over all the elements
of the covariance matrix, first multiplying each element by the portfolio weights from the
row and the column. The contribution of one stock to portfolio variance therefore can be
expressed as the sum of all the covariance terms in the column corresponding to the stock,
where each covariance is first multiplied by both the stock’s weight from its row and the
weight from its column. 5
5 An alternative approach would be to measure GE’s contribution to market variance as the sum of the elements in the
row and the column corresponding to GE. In this case, GE’s contribution would be twice the sum in Equation 9.3. The approach that we take in the text allocates contributions to portfolio risk among securities in a convenient man-
ner in that the sum of the contributions of each stock equals the total portfolio variance, whereas the alternative mea-
sure of contribution would sum to twice the portfolio variance. This results from a type of double-counting, because
adding both the rows and the columns for each stock would result in each entry in the matrix being added twice.
Thus, the contribution of GE’s stock to the variance of the market portfolio is
wGE 3w1Cov ( R1, RGE ) 1 w2Cov ( R2, RGE ) 1 . . . 1 wGECov ( RGE, RGE ) 1 . . . (9.3)
1 wnCov ( Rn, RGE )4 Notice that every term in the square brackets can be slightly rearranged as follows:
w i Cov ( R i , R GE ) 5 Cov ( w i R i , R GE ). Moreover, because covariance is additive, the sum of the terms in the square brackets is
a n
i51 wiCov ( Ri, RGE ) 5 a
n
i51 Cov (wiRi, RGE ) 5 Cov aan
i51 wiRi, RGEb (9.4)
Portfolio Weights w 1 w 2 . . . w GE . . . w n
w 1 Cov( R 1 , R 1 ) Cov( R 1 , R 2 ) . . . Cov( R 1 , R GE ) . . . Cov( R 1 , R n ) w 2 Cov( R 2 , R 1 ) Cov( R 2 , R 2 ) . . . Cov( R 2 , R GE ) . . . Cov( R 2 , R n )
.
.
. . . .
.
.
.
. . .
. . .
w GE Cov( R GE , R 1 ) Cov( R GE , R 2 ) . . . Cov( R GE , R GE ) . . . Cov( R GE , R n )
. . .
.
.
. . . .
. . .
. . .
w n Cov( R n , R 1 ) Cov( R n , R 2 ) . . . Cov( R n , R GE ) . . . Cov( R n , R n )
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296 P A R T I I I Equilibrium in Capital Markets
But because a n
i51 wiRi 5 RM, Equation 9.4 implies that
a n
i51 wiCov (Ri, RGE) 5 Cov (RM, RGE)
and therefore, GE’s contribution to the variance of the market portfolio (Equation 9.3) may
be more simply stated as w GE Cov( R M , R GE ). This should not surprise us. For example, if the covariance between GE and the rest
of the market is negative, then GE makes a “negative contribution” to portfolio risk: By
providing excess returns that move inversely with the rest of the market, GE stabilizes the
return on the overall portfolio. If the covariance is positive, GE makes a positive contribu-
tion to overall portfolio risk because its returns reinforce swings in the rest of the portfolio. 6
We also observe that the contribution of GE to the risk premium of the market portfolio
is w GE E ( R GE ). Therefore, the reward-to-risk ratio for investments in GE can be expressed as
GE’s contribution to risk premium
GE’s contribution to variance 5
wGEE(RGE)
wGECov(RGE, RM) 5
E(RGE)
Cov(RGE, RM)
The market portfolio is the tangency (efficient mean-variance) portfolio. The reward-to-
risk ratio for investment in the market portfolio is
Market risk premium
Market variance 5
E( RM )
sM 2
(9.5)
The ratio in Equation 9.5 is often called the market price of risk because it quantifies the extra return that investors demand to bear portfolio risk. Notice that for components of the efficient portfolio, such as shares of GE, we measure risk as the contribution to portfolio variance (which depends on its covariance with the market). In contrast, for the efficient portfolio itself, variance is the appropriate measure of risk.
7
A basic principle of equilibrium is that all investments should offer the same reward-
to-risk ratio. If the ratio were better for one investment than another, investors would rear-
range their portfolios, tilting toward the alternative with the better trade-off and shying
away from the other. Such activity would impart pressure on security prices until the ratios
were equalized. Therefore we conclude that the reward-to-risk ratios of GE and the market
portfolio should be equal:
E( RGE )
Cov( RGE, RM ) 5
E( RM )
sM 2
(9.6)
To determine the fair risk premium of GE stock, we rearrange Equation 9.6 slightly to obtain
E( RGE ) 5 Cov(RGE, RM )
sM 2
E( RM ) (9.7)
6 A positive contribution to variance doesn’t imply that diversification isn’t beneficial. Excluding GE from the
portfolio would require that its weight be assigned to the remaining stocks, and that reallocation would increase
variance even more. Variance is reduced by including more stocks and reducing the weight of all (i.e., diversify-
ing), despite the fact that each positive-covariance security makes some contribution to variance.
7 Unfortunately the market portfolio’s Sharpe ratio
E(rM) 2 rf
sM
sometimes is referred to as the market price of risk, but it is not. The unit of risk is variance, and the price of risk
relates risk premium to variance (or to covariance for incremental risk).
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C H A P T E R 9 The Capital Asset Pricing Model 297
The ratio Cov(RGE, RM)/sM 2 measures the contribution of GE stock to the variance of the
market portfolio as a fraction of the total variance of the market portfolio. The ratio is
called beta and is denoted by b . Using this measure, we can restate Equation 9.7 as
E(rGE ) 5 rf 1 bGE 3E(rM) 2 rf4 (9.8) This expected return–beta ( or mean-beta) relationship is the most familiar expression of the CAPM to practitioners.
If the expected return–beta relationship holds for any individual asset, it must hold for
any combination of assets. Suppose that some portfolio P has weight w k for stock k, where k takes on values 1, . . . , n. Writing out the CAPM Equation 9.8 for each stock, and multi- plying each equation by the weight of the stock in the portfolio, we obtain these equations,
one for each stock:
w1E( r1) 5 w1rf 1 w1b1 3E( rM ) 2 rf4 1w2E( r2 ) 5 w2rf 1 w2b2 3E( rM ) 2 rf4
1 c5 c
1wnE( rn ) 5 wnrf 1 wnbn 3E( rM ) 2 rf4 E( rP ) 5 rf 1 bP 3E( rM ) 2 rf4
Summing each column shows that the CAPM holds for the overall portfolio because
E(rP) 5 g k
wkE(rk ) is the expected return on the portfolio, and bP 5 g k
wk bk is the portfolio
beta. Incidentally, this result has to be true for the market portfolio itself,
E( rM ) 5 rf 1 bM3E( rM ) 2 rf4 Indeed, this is a tautology because b M 5 1, as we can verify by noting that
bM 5 Cov( RM, RM )
sM 2
5 sM
2
sM 2
This also establishes 1 as the weighted-average value of beta across all assets. If the market
beta is 1, and the market is a portfolio of all assets in the economy, the weighted-average
beta of all assets must be 1. Hence betas greater than 1 are considered aggressive in that
investment in high-beta stocks entails above-average sensitivity to market swings. Betas
below 1 can be described as defensive.
A word of caution: We often hear that well-managed firms will provide high rates of
return. We agree this is true if one measures the firm’s return on its investments in plant and equipment. The CAPM, however, predicts returns on investments in the securities of the firm.
Let’s say that everyone knows a firm is well run. Its stock price will therefore be bid
up, and consequently returns to stockholders who buy at those high prices will not be
excessive. Security prices, in other words, already reflect public information about a firm’s
prospects; therefore only the risk of the company (as measured by beta in the context of
the CAPM) should affect expected returns. In a well-functioning market, investors receive
high expected returns only if they are willing to bear risk.
Investors do not directly observe or determine expected returns on securities. Rather,
they observe security prices and bid those prices up or down. Expected rates of return are
determined by the prices investors must pay compared to the cash flows those investments
might garner.
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298 P A R T I I I Equilibrium in Capital Markets
The Security Market Line We can view the expected return–beta relationship as a reward–risk equation. The beta of
a security is the appropriate measure of its risk because beta is proportional to the risk the
security contributes to the optimal risky portfolio.
Risk-averse investors measure the risk of the optimal risky portfolio by its variance.
Hence, we would expect the risk premium on individual assets to depend on the contribu- tion of the asset to the risk of the portfolio. The beta of a stock measures its contribution to the variance of the market portfolio and therefore the required risk premium is a function
of beta. The CAPM confirms this intuition, stating further that the security’s risk premium
is directly proportional to both the beta and the risk premium of the market portfolio; that
is, the risk premium equals b [ E ( rM ) 2 rf ]. The expected return–beta relationship can be portrayed graphically as the security
market line (SML) in Figure 9.2 . Because the market’s beta is 1, the slope is the risk pre- mium of the market portfolio. At the point on the horizontal axis where b 5 1, we can read
off the vertical axis the expected return on the market portfolio.
It is useful to compare the security market line to the capital market line. The CML
graphs the risk premiums of efficient portfolios (i.e., portfolios composed of the market and the risk-free asset) as a function of portfolio standard deviation. This is appropriate
because standard deviation is a valid measure of risk for efficiently diversified portfolios
that are candidates for an investor’s overall portfolio.
The SML, in contrast, graphs individual asset risk premiums as a function of asset risk. The relevant
measure of risk for individual assets held as parts of
well-diversified portfolios is not the asset’s standard
deviation or variance; it is, instead, the contribution of
the asset to the portfolio variance, which we measure
by the asset’s beta. The SML is valid for both efficient
portfolios and individual assets.
The security market line provides a benchmark
for the evaluation of investment performance. Given
the risk of an investment, as measured by its beta, the
SML provides the required rate of return necessary to
compensate investors for risk as well as the time value
of money.
Because the security market line is the graphic rep-
resentation of the expected return–beta relationship,
“fairly priced” assets plot exactly on the SML; that
is, their expected returns are commensurate with their
risk. All securities must lie on the SML in market
equilibrium. We see here how the CAPM may be of
use in the money-management industry. Suppose that
the SML relation is used as a benchmark to assess the Figure 9.2 The security market line
E(r)
E(rM)
E(rM) 2 rf 5 Slope of SML
SML
rf
βM 51.0
1
β
Suppose that the risk premium on the market portfolio is estimated at 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested 25% in Toyota and 75% in Ford, if they have betas of 1.10 and 1.25, respectively?
CONCEPT CHECK 9.3
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C H A P T E R 9 The Capital Asset Pricing Model 299
fair expected return on a risky asset. Then security
analysis is performed to calculate the return actually
expected. (Notice that we depart here from the simple
CAPM world in that some investors now apply their
own unique analysis to derive an “input list” that may
differ from their competitors’.) If a stock is perceived
to be a good buy, or underpriced, it will provide an
expected return in excess of the fair return stipulated
by the SML. Underpriced stocks therefore plot above
the SML: Given their betas, their expected returns
are greater than dictated by the CAPM. Overpriced
stocks plot below the SML.
The difference between the fair and actually
expected rates of return on a stock is called the stock’s
alpha , denoted by a . For example, if the market return is expected to be 14%, a stock has a beta of 1.2, and the
T-bill rate is 6%, the SML would predict an expected
return on the stock of 6 1 1.2(14 2 6) 5 15.6%. If
one believed the stock would provide an expected
return of 17%, the implied alpha would be 1.4% (see
Figure 9.3 ).
One might say that security analysis (which we
treat in Part Five) is about uncovering securities with
nonzero alphas. This analysis suggests that the starting point of portfolio management can
be a passive market-index portfolio. The portfolio manager will then increase the weights
of securities with positive alphas and decrease the weights of securities with negative
alphas. We showed one strategy for adjusting the portfolio weights in such a manner in
Chapter 8.
The CAPM is also useful in capital budgeting decisions. For a firm considering a new
project, the CAPM can provide the required rate of return that the project needs to yield, based on its beta, to be acceptable to investors. Managers can use the CAPM to obtain this
cutoff internal rate of return (IRR), or “hurdle rate” for the project.
The nearby box describes how the CAPM can be used in capital budgeting. It also
discusses some empirical anomalies concerning the model, which we address in detail in
Chapters 11–13.
Figure 9.3 The SML and a positive-alpha stock
E(r) (%)
SML
6
14
15.6
1.21.0
17 Stock
M
α
β
Yet another use of the CAPM is in utility rate-making cases. 8 In this case the issue is the rate of return that a regulated utility should be allowed to earn on its investment in plant and equipment. Suppose that the equityholders have invested $100 million in the firm and that the beta of the equity is .6. If the T-bill rate is 6% and the market risk premium is 8%, then the fair profits to the firm would be assessed as 6 1 .6 3 8 5 10.8% of the $100 million investment, or $10.8 million. The firm would be allowed to set prices at a level expected to generate these profits.
Example 9.1 Using the CAPM
8 This application is becoming less common, as many states are in the process of deregulating their public utilities
and allowing a far greater degree of free market pricing. Nevertheless, a considerable amount of rate setting still
takes place.
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300
Tales From the Far Side
Financial markets’ evaluation of risk determines the way firms invest. What if the markets are wrong? Investors are rarely praised for their good sense. But for the past two decades a growing number of firms have based their decisions on a model which assumes that peo- ple are perfectly rational. If they are irrational, are busi- nesses making the wrong choices?
The model, known as the “capital-asset pricing model,” or CAPM, has come to dominate modern finance. Almost any manager who wants to defend a project—be it a brand, a factory or a corporate merger—must justify his decision partly based on the CAPM. The reason is that the model tells a firm how to calculate the return that its inves- tors demand. If shareholders are to benefit, the returns from any project must clear this “hurdle rate.”
Although the CAPM is complicated, it can be reduced to five simple ideas:
1. Investors can eliminate some risks—such as the risk that workers will strike, or that a firm’s boss will quit—by diversifying across many regions and sectors.
2. Some risks, such as that of a global recession, cannot be eliminated through diversification. So even a basket of all of the stocks in a stock market will still be risky.
3. People must be rewarded for investing in such a risky basket by earning returns above those that they can get on safer assets, such as Treasury bills.
4. The rewards on a specific investment depend only on the extent to which it affects the market basket’s risk.
5. Conveniently, that contribution to the market basket’s risk can be captured by a single measure—dubbed “beta”—which expresses the relationship between the investment’s risk and the market’s.
Beta is what makes the CAPM so powerful. Although an investment may face many risks, diversified investors should care only about those that are related to the market
basket. Beta not only tells managers how to measure those risks, but it also allows them to translate them directly into a hurdle rate. If the future profits from a project will not exceed that rate, it is not worth shareholders’ money.
The diagram shows how the CAPM works. Safe invest- ments, such as Treasury bills, have a beta of zero. Riskier investments should earn a premium over the risk-free rate which increases with beta. Those whose risks roughly match the market’s have a beta of one, by definition, and should earn the market return.
So suppose that a firm is considering two projects, A and B. Project A has a beta of ½: when the market rises or falls by 10%, its returns tend to rise or fall by 5%. So its risk premium is only half that of the market. Project B ’s
W O
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Return
Market Return
Risk-Free Return
A
B
1 2 b
rB
rA
Beta Power
1 2
Stock XYZ has an expected return of 12% and risk of b 5 1. Stock ABC has expected return of 13% and b 5 1.5. The market’s expected return is 11%, and r f 5 5%.
a. According to the CAPM, which stock is a better buy?
b. What is the alpha of each stock? Plot the SML and each stock’s risk–return point on one graph. Show the alphas graphically.
The risk-free rate is 8% and the expected return on the market portfolio is 16%. A firm considers a project that is expected to have a beta of 1.3.
a. What is the required rate of return on the project?
b. If the expected IRR of the project is 19%, should it be accepted?
CONCEPT CHECK 9.4 and 9.5
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301
risk premium is twice that of the market, so it must earn a higher return to justify the expenditure.
NEVER KNOWINGLY UNDERPRICED But there is one small problem with the CAPM: Financial economists have found that beta is not much use for explaining rates of return on firms’ shares. Worse, there appears to be another measure which explains these returns quite well.
That measure is the ratio of a firm’s book value (the value of its assets at the time they entered the balance sheet) to its market value. Several studies have found that, on average, companies that have high book-to-market ratios tend to earn excess returns over long periods, even after adjusting for the risks that are associated with beta.
The discovery of this book-to-market effect has sparked a fierce debate among financial economists. All of them agree that some risks ought to carry greater rewards. But they are now deeply divided over how risk should be mea- sured. Some argue that since investors are rational, the book-to-market effect must be capturing an extra risk fac- tor. They conclude, therefore, that managers should incor- porate the book-to-market effect into their hurdle rates. They have labeled this alternative hurdle rate the “new estimator of expected return,” or NEER.
Other financial economists, however, dispute this approach. Since there is no obvious extra risk associated with a high book-to-market ratio, they say, investors must be mistaken. Put simply, they are underpricing high book- to-market stocks, causing them to earn abnormally high returns. If managers of such firms try to exceed those inflated hurdle rates, they will forgo many profitable investments. With economists now at odds, what is a con- scientious manager to do?
Jeremy Stein, an economist at the Massachusetts Insti- tute of Technology’s business school, offers a paradoxical
answer. * If investors are rational, then beta cannot be the only measure of risk, so managers should stop using it. Conversely, if investors are irrational, then beta is still the right measure in many cases. Mr. Stein argues that if beta captures an asset’s fundamental risk—that is, its contribu- tion to the market basket’s risk—then it will often make sense for managers to pay attention to it, even if investors are somehow failing to.
Often, but not always. At the heart of Mr. Stein’s argu- ment lies a crucial distinction—that between ( a ) boosting a firm’s long-term value and ( b ) trying to raise its share price. If investors are rational, these are the same thing: any deci- sion that raises long-term value will instantly increase the share price as well. But if investors are making predictable mistakes, a manager must choose.
For instance, if he wants to increase today’s share price—perhaps because he wants to sell his shares, or to fend off a takeover attempt—he must usually stick with the NEER approach, accommodating investors’ mispercep- tions. But if he is interested in long-term value, he should usually continue to use beta. Showing a flair for market- ing, Mr. Stein labels this far-sighted alternative to NEER the “fundamental asset risk”—or FAR—approach.
Mr. Stein’s conclusions will no doubt irritate many com- pany bosses, who are fond of denouncing their investors’ myopia. They have resented the way in which CAPM—with its assumption of investor infallibility—has come to play an important role in boardroom decision making. But it now appears that if they are right, and their investors are wrong, then those same far-sighted managers ought to be the CAPM’s biggest fans.
*Jeremy Stein, “Rational Capital Budgeting in an Irrational World,” The Journal of Business, October 1996. Source: “Tales from the FAR Side,” The Economist Group, Inc. November 16, 1996, p. 8. © The Economist Newspaper Limited, London.
The CAPM and the Single-Index Market The key implications of the CAPM can be summarized by these two statements:
1. The market portfolio is efficient.
2. The risk premium on a risky asset is proportional to its beta.
While these two statements often are thought of as complementary, they are actually sub-
stitutes because one can be derived from the other (one is true if and only if the other is
as well). We have focused on one direction, proceeding from the efficiency of the market
portfolio to the mean-beta equation. We now proceed from the mean return–beta relation-
ship to the efficiency of the market portfolio using the index-model market structure we
described in Chapter 8.
Deriving the CAPM is even more intuitive when starting from a single-index market.
Rather than beginning with investors who all apply the Markowitz algorithm to identical
input lists, suppose instead that they all face a market where excess stock returns, R i , are
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302 P A R T I I I Equilibrium in Capital Markets
normally distributed and driven by one systematic factor. The effect of the macro factor is
assumed captured by the return on a broad, value-weighted stock-index portfolio, M. The excess return on any stock is described by Equation 8.11 and restated here.
Ri 5 ai 1 biRM 1 ei (9.9)
Each firm-specific, zero-mean residual, e i , is uncorrelated across stocks and uncorrelated with the market factor, R M . Residuals represent diversifiable, nonsystematic, or unique risk. The total risk of a stock is then just the sum of the variance of the systematic com-
ponent, b i R M , and the variance of e i . In sum, the risk premium (mean excess return) and variance are:
E( Ri ) 5 ai 1 biE( RM )
si 2 5 bi
2sM 2 1 s2(ei)
(9.10)
The return on a portfolio, Q, constructed from N stocks (ordered by k 5 1, . . . , N ) with a set of weights, w k , must satisfy Equation 9.11, which states that the portfolio alpha, beta, and residual will be the weighted average of the respective parameters of the component
securities.
RQ 5 a N
k51 wkak 1 a
N
k51 wkbkRM 1 a
N
k51 wkek 5 aQ 1 bQRM 1 eQ (9.11)
Investors have two considerations when forming their portfolios: First, they can diversify
nonsystematic risk. Since the residuals are uncorrelated, residual risk, s 2 (eQ) 5 g
N k 51wk
2s2(e k ),
becomes ever smaller as diversification reduces portfolio weights. Second, by choosing stocks
with positive alpha, or taking short positions in negative-alpha stocks, the risk premium on Q can be increased.
9
As a result of these considerations, investors will relentlessly pursue positive alpha stocks,
and shun (or short) negative-alpha stocks. Consequently, prices of positive alpha stocks will
rise and prices of negative alpha stocks will fall. This will continue until all alpha values
are driven to zero. At this point, investors will be content to minimize risk by completely
eliminating unique risk, that is, by holding the broadest possible, market portfolio. When all
stocks have zero alphas, the market portfolio is the optimal risky portfolio. 10
9 The systematic part of the portfolio is of no relevance in this endeavor, since, if desired, the beta of Q can be
increased by leverage (borrow and invest in M ), or decreased by including in Q a short position in M. The pro- ceeds from the short position in M can be invested in the risk-free asset, thus leaving the alpha and nonsystematic risk unchanged.
10
Recall from Chapter 8 that the weight of a stock in an active portfolio will be zero if its alpha is zero (see
Equation 8.20); hence if all alphas are zero, the passive market portfolio will be the optimal risky portfolio.
9.2 Assumptions and Extensions of the CAPM
Now that we understand the basic insights of the CAPM, we can more explicitly iden-
tify the set of simplifying assumptions on which it relies. A model consists of (i) a set of
assumptions, (ii) logical/mathematical development of the model through manipulation
of those assumptions, and (iii) a set of predictions. Assuming the logical/mathematical
manipulations are free of errors, we can test a model in two ways, normative and positive. Normative tests examine the assumptions of the model, while positive tests examine the
predictions.
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C H A P T E R 9 The Capital Asset Pricing Model 303
If a model’s assumptions are valid, and the development is error-free, then the predic-
tions of the model must be true. In this case, testing the assumptions is synonymous with
testing the model. But few, if any, models can pass the normative test. In most cases, as with
the CAPM, the assumptions are admittedly invalid—we recognize that we have simplified
reality, and therefore to this extent are relying on “untrue” assumptions. The motivation for
invoking unrealistic assumptions is clear; we simply cannot solve a model that is perfectly
consistent with the full complexity of real-life markets. As we’ve noted, the need to use sim-
plifying assumptions is not peculiar to economics—it characterizes all of science.
Assumptions are chosen first and foremost to render the model solvable. But we prefer
assumptions to which the model is “robust.” A model is robust with respect to an assump-
tion if its predictions are not highly sensitive to violation of the assumption. If we use
only assumptions to which the model is robust, the model’s predictions will be reason-
ably accurate despite its shortcomings. The upshot of all this is that tests of models are
almost always positive—we judge a model on the success of its empirical predictions.
This standard brings statistics into any science and requires us to take a stand on what are
acceptable levels of significance and power. 11
Because the nonrealism of the assumptions
precludes a normative test, the positive test is really a test of the robustness of the model
to its assumptions.
Assumptions of the CAPM Table 9.1 enumerates the list of assumptions underlying the CAPM. In our discussion so
far, we have cited explicitly only these three assumptions:
1.a. Investors are rational, mean-variance optimizers.
1.c. Investors use identical input lists, referred to as homogeneous expectations . 2.a. All assets are publicly traded (short positions are allowed) and investors can
borrow or lend at a common risk-free rate.
The first assumption is far-reaching. Its “visible” part is that investors are not concerned
with higher moments (skew and kurtosis) that may “fatten” the left tail of the return distri-
bution. We can ascertain the validity of this assumption from statistical tests of the normal-
ity of return distributions as we did in Chapter 5.
Less visible is that, by assuming that only the mean and variance of wealth matter to
investors, Assumption 1(a) rules out concern with the correlation of asset returns with
either inflation or prices of important consumption items such as housing or energy. The
extra demand for assets that can be used to hedge these “extra market” risks would increase
their prices and reduce their risk premiums relative to the prediction of the CAPM.
11
To illustrate the meanings of significance and power, consider a test of the efficacy of a new drug. The agency
testing the drug may make two possible errors. The drug may be useless (or even harmful), but the agency may
conclude that it is useful. This is called a “Type I” error. The significance level of a test is the probability of a Type I error. Typical practice is to fix the level of significance at some low level, for example, 5%. In the case of drug
testing, for example, the first goal is to avoid introducing ineffective or harmful treatments. The other possible
error is that the drug is actually useful, but the testing procedure concludes it is not. This mistake, called “Type
II” error, would lead us to discard a useful treatment. The power of the test is the probability of avoiding Type II error (i.e., one minus the probability of making such an error), that is, the probability of accepting the drug if it
is indeed useful. We want tests that, at a given level of significance, have the most power, so we will admit effec-
tive drugs with high probability. In social sciences in particular, available tests often have low power, in which
case they are susceptible to Type II error and will reject a correct model (a “useful drug”) with high frequency.
“The drug is useful” is analogous in the CAPM to alphas being zero. When the test data reject the hypothesis that
observed alphas are zero at the desired level of significance, the CAPM fails. However, if the test has low power,
the probability that we accept the model when not true is too high.
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304 P A R T I I I Equilibrium in Capital Markets
Similar extra-market risk factors would arise in a multiperiod model, which requires
the addition of Assumption 1(b), limiting investors to a common single-period horizon.
Consider a possible decline in future interest rates. Investors would be unhappy about this
event to the extent that it would reduce the expected income their investments could throw
off in the future. Assets whose returns are negatively correlated with interest rates (e.g.,
long-term bonds) would hedge this risk and thus command higher prices and lower risk
premiums. Because of such hedging demands, correlation with any parameter describing
future investment opportunities can result in violations of the CAPM mean-beta equation
(and therefore with the efficiency of the market portfolio). A single-period investor horizon
eliminates these possibilities.
Interestingly, Assumption 1(c) (investors optimize with the same input list), appears
ominously restrictive, but it actually is not all that problematic. With the addition of
Assumption 2(b) (all information is public), investors generally will be close to agreement.
Moreover, trades of investors who derive different input lists will offset and prices will
reflect consensus expectations. We will later allow for the likelihood that some investors
expend resources to obtain private information and exploit prices that don’t reflect the
insights derived from this information. But regardless of their success, it is reasonable to
assert that, absent private information, investors should assume alpha values are zero.
The assumption that all assets are tradable (2a) is essential for identical input lists.
It allows us to ignore federal and state assets and liabilities. More importantly, privately
held but nontraded assets such as human capital and private business can create large dif-
ferences in investor portfolios. Consider owners of a family business. Prudence dictates
that they avoid assets that are highly correlated with their businesses. Similarly, investors
should avoid stock returns that are positively correlated with their personal income; for
example, Boeing employees should avoid investing in the airline and related businesses.
Differential demands arising from this consideration can lead to violation of the mean-beta
equation and derail the mean-variance efficiency of the index portfolio.
Restrictions on borrowing (or significantly higher rates on borrowed funds), which vio-
lates Assumption 2(a), also can create problems for the CAPM, because borrowers and lend-
ers will arrive at different tangency portfolios and thus different optimal risky portfolios.
Taxes create conditions in which two investors can realize different after-tax returns
from the same stock. Such distortions could, in principle, lead to different after-tax opti-
mal risky portfolios to different investors; hence Assumption 2(c) (no taxes). Despite an
extension to the CAPM that incorporates personal taxes on dividends and capital gains, 12
Table 9.1
The assumptions of the CAPM
1. Individual behavior a. Investors are rational, mean-variance optimizers. b. Their planning horizon is a single period. c. Investors have homogeneous expectations (identical input lists).
2. Market structure a. All assets are publicly held and trade on public exchanges, short positions are allowed,
and investors can borrow or lend at a common risk-free rate. b. All information is publicly available. c. No taxes. d. No transaction costs.
12 Michael J. Brennan, “Taxes, Market Valuation, and Corporate Finance Policy,” National Tax Journal,
December 1973.
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C H A P T E R 9 The Capital Asset Pricing Model 305
there is no decisive evidence that taxes are a major factor in stock returns. A plausible
explanation for this negative finding relies on “clientele” and supply effects. If high tax-
bracket investors shy away from high-yield (dividend-paying) stocks and thus force down
their prices, tax-exempt investors will view the stocks as a bargain and take up the slack
in demand. On the other end, if corporations see that high dividend yields reduce stock
prices, they simply will substitute stock repurchases for dividends, reinforcing the clientele
effect in neutralizing tax effects.
Finally, transaction costs inhibit trades and thus the reaction to changes in information;
hence Assumption 2(d) (no transaction costs). While in reality trading costs have fallen,
remaining differentials in trading costs may still play an important role in stock returns.
Challenges and Extensions to the CAPM Which assumptions are most worrisome? We start with the fact that short positions are not
as easy to take as long ones for three reasons:
1. The liability of investors who hold a short position in an asset is potentially unlim-
ited, since the price may rise without limit. Hence a large short position requires
large collateral, and proceeds cannot be used to invest in other risky assets.
2. There is a limited supply of shares of any stock to be borrowed by would-be short
sellers. It often happens that investors simply cannot find shares to borrow in order
to short.
3. Many investment companies are prohibited from short sales. The U.S. and other
countries further restrict short sales by regulation.
Why are short sales important? Notice that Assumption 1(a) begins with “investors are
rational . . .” When investors exhibit “irrational exuberance” (excessive optimism) about
an asset and, as a result, prices rise above intrinsic values, rational investors will take
short positions, thus holding down the price. But with effective restrictions, short sales
can fail to prevent prices rising to unsustainable levels that are precursors to a correction
or even a crash. This really defines a “bubble.”
Three unrealistic assumptions, 2(a) (all assets trade) and 2(d) (there are no transac-
tion costs), combined with 1(b) (single-period horizon), generate the major challenges to
the model. These challenges have motivated a set of extensions that are, even today, still
“under construction” in one way or another. For this reason, none of the extensions has
decisively superseded the simple CAPM in the industry. It is an impressive phenomenon,
that despite failing many empirical tests, the compelling logic of the CAPM keeps it at the
center of the investments industry. However, for better insight to the CAPM, it is useful to
understand the extensions of the model.
The Zero-Beta Model Efficient frontier portfolios have a number of interesting characteristics, independently
derived by Merton and Roll. 13
Two of these are
1. Any portfolio that is a combination of two frontier portfolios is itself on the efficient
frontier.
2. Every portfolio on the efficient frontier, except for the global minimum-variance
portfolio, has a “companion” portfolio on the bottom (inefficient) half of the frontier
13
Robert C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis, 1972. Richard Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and Potential Testability of the Theory,” Journal of Financial Economics 4 (1977).
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306 P A R T I I I Equilibrium in Capital Markets
with which it is uncorrelated. Because it is uncorrelated, the companion portfolio
is referred to as the zero-beta portfolio of the efficient portfolio. If we choose the market portfolio M and its zero-beta companion portfolio Z, then we obtain a CAPM-like equation
E( ri ) 2 E( rZ ) 5 3E( RM ) 2 E( RZ )4 Cov( ri, rM ) sM
2 5 bi3E( rM ) 2 E( rZ )4 (9.12)
Equation 9.12 resembles the SML of the CAPM, except that the risk-free rate is
replaced with the expected return on the zero-beta companion of the market-index
portfolio.
Fischer Black used these properties to show that Equation 9.12 is the CAPM equation
that results when investors face restrictions on borrowing. 14
In this case, at least some
investors will choose portfolios on the high risk-premium portion of the efficient frontier.
Put differently, investors who would otherwise wish to borrow and leverage their portfo-
lios but who find it impossible or costly will instead tilt their portfolios toward high-beta
stocks and away from low-beta ones. As a result, prices of high beta stocks will rise, and
their risk premiums will fall. The SML will be flatter than in the simple CAPM. You see
from Equation 9.12 that the risk premium on the market portfolio is smaller (because the
expected return on the zero-beta portfolio is greater than the risk-free rate) and therefore
the reward to bearing beta risk is smaller.
Labor Income and Nontraded Assets Two important asset classes that are not traded are human capital and privately held busi- nesses. The discounted value of future labor income exceeds the total market value of
traded assets. The market value of privately held corporations and businesses is of the same
order of magnitude. Human capital and private enterprises are different types of assets with
possibly different implications for equilibrium returns on traded securities.
Privately held businesses may be the lesser of the two sources of departures from the
CAPM. Suppose that privately held businesses have risk characteristics similar to those
of traded assets. In this case, individuals can partially offset the diversification problems
posed by their nontraded entrepreneurial assets by reducing their portfolio demand for
securities of similar, traded assets. Thus, the CAPM expected return–beta equation may
not be greatly disrupted by the presence of entrepreneurial income.
To the extent that risk characteristics of private enterprises differ from those of traded
securities, a portfolio of traded assets that best hedges the risk of typical private business
would enjoy excess demand from the population of private business owners. The price of
assets in this portfolio will be bid up relative to the CAPM considerations, and the expected
returns on these securities will be lower in relation to their systematic risk. Conversely,
securities highly correlated with such risk will have high equilibrium risk premiums and
may appear to exhibit positive alphas relative to the conventional SML. In fact, Heaton and
Lucas show that adding proprietary income to a standard asset-pricing model improves its
predictive performance. 15
The size of labor income and its special nature is of greater concern for the validity of
the CAPM. The possible effect of labor income on equilibrium returns can be appreciated
14
Fischer Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, July 1972.
15
John Heaton and Deborah Lucas, “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial
Risk,” Journal of Finance 55 (June 2000). This paper offers evidence of the effect of entrepreneurial risk on both portfolio choice and the risk–return relationship.
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C H A P T E R 9 The Capital Asset Pricing Model 307
from its important effect on personal portfolio choice. Despite the fact that an individual
can borrow against labor income (via a home mortgage) and reduce some of the uncertainty
about future labor income via life insurance, human capital is less “portable” across time
and may be more difficult to hedge using traded securities than nontraded business. This
may induce pressure on security prices and result in departures from the CAPM expected
return–beta equation. Thus, the demand for stocks of labor-intensive firms with high wage
expenses may be good hedges for uncertain labor income, and these stocks may require a
lower expected return than predicted by the CAPM.
Mayers 16
derives the equilibrium expected return–beta equation for an economy in
which individuals are endowed with labor income of varying size relative to their nonlabor
capital. The resultant SML equation is
E( Ri ) 5 E( RM )
Cov( Ri, RM ) 1 PH PM
Cov( Ri, RH )
sM 2 1
PH PM
Cov( RM, RH ) (9.13)
where
P H 5 value of aggregate human capital
P M 5 market value of traded assets (market portfolio)
R H 5 excess rate of return on aggregate human capital
The CAPM measure of systematic risk, beta, is replaced in the extended model by an
adjusted beta that also accounts for covariance with the portfolio of aggregate human capi-
tal. Notice that the ratio of human capital to market value of all traded assets, P H / P M , may well be greater than 1, and hence the effect of the covariance of a security with labor
income, Cov( R i , R H ), relative to the average, Cov( R M , R H ), is likely to be economically significant. When Cov( R i , R H ) is positive, the adjusted beta is greater when the CAPM beta is smaller than 1, and vice versa. Because we expect Cov( R i , R H ) to be positive for the aver- age security, the risk premium in this model will be greater, on average, than predicted by
the CAPM for securities with beta less than 1, and smaller for securities with beta greater
than 1. The model thus predicts a security market line that is less steep than that of the stan-
dard CAPM. This may help explain the average negative alpha of high-beta securities and
positive alpha of low-beta securities that lead to the statistical failure of the CAPM equa-
tion. In Chapter 13 on empirical evidence we present additional results along these lines.
A Multiperiod Model and Hedge Portfolios Robert C. Merton revolutionized financial economics by using continuous-time models to
extend models of asset pricing. 17
While his (Nobel Prize–winning) contributions to option-
pricing theory and financial engineering (along with those of Fischer Black and Myron
Scholes) may have had greater impact on the investment industry, his solo contribution to
portfolio theory was equally important for our understanding of the risk–return relationship.
In his basic model, Merton relaxes the “single-period” myopic assumptions about inves-
tors. He envisions individuals who optimize a lifetime consumption/investment plan, and
who continually adapt consumption/investment decisions to current wealth and planned
retirement age. When uncertainty about portfolio returns is the only source of risk and
16
David Mayers, “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty,” in Studies in the Theory of Capital Markets, ed. M. C. Jensen (New York: Praeger, 1972).
17
Merton’s classic works are collected in Continuous-Time Finance (Oxford, U.K.: Basil Blackwell, 1992).
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308 P A R T I I I Equilibrium in Capital Markets
investment opportunities remain unchanged through time, that is, there is no change in the
risk-free rate or the probability distribution of the return on the market portfolio or indi-
vidual securities, Merton’s so-called intertemporal capital asset pricing model (ICAPM)
predicts the same expected return–beta relationship as the single-period equation. 18
But the situation changes when we include additional sources of risk. These extra risks
are of two general kinds. One concerns changes in the parameters describing investment
opportunities, such as future risk-free rates, expected returns, or the risk of the market
portfolio. Suppose that the real interest rate may change over time. If it falls in some future
period, one’s level of wealth will now support a lower stream of real consumption. Future
spending plans, for example, for retirement spending, may be put in jeopardy. To the extent
that returns on some securities are correlated with changes in the risk-free rate, a portfolio
can be formed to hedge such risk, and investors will bid up the price (and bid down the
expected return) of those hedge assets. Investors will sacrifice some expected return if they
can find assets whose returns will be higher when other parameters (in this case, the real
risk-free rate) change adversely.
The other additional source of risk concerns the prices of the consumption goods that
can be purchased with any amount of wealth. Consider inflation risk. In addition to the
expected level and volatility of nominal wealth, investors must be concerned about the cost
of living—what those dollars can buy. Therefore, inflation risk is an important extramarket
source of risk, and investors may be willing to sacrifice some expected return to purchase
securities whose returns will be higher when the cost of living changes adversely. If so,
hedging demands for securities that help to protect against inflation risk would affect port-
folio choice and thus expected return. One can push this conclusion even further, arguing
that empirically significant hedging demands may arise for important subsectors of con-
sumer expenditures; for example, investors may bid up share prices of energy companies
that will hedge energy price uncertainty. These sorts of effects may characterize any assets
that hedge important extramarket sources of risk.
More generally, suppose we can identify K sources of extramarket risk and find K asso- ciated hedge portfolios. Then, Merton’s ICAPM expected return–beta equation would gen-
eralize the SML to a multi-index version:
E( Ri ) 5 biME( RM ) 1 a K
k51 bikE( Rk ) (9.14)
where b iM is the familiar security beta on the market-index portfolio, and b ik is the beta on
the k th hedge portfolio. Other multifactor models using additional factors that do not arise from extramarket
sources of risk have been developed and lead to SMLs of a form identical to that of the
ICAPM. These models also may be considered extensions of the CAPM in the broad sense.
We examine these models in the next chapter.
A Consumption-Based CAPM The logic of the CAPM together with the hedging demands noted in the previous subsec-
tion suggest that it might be useful to center the model directly on consumption. Such
models were first proposed by Mark Rubinstein, Robert Lucas, and Douglas Breeden. 19
18
Eugene F. Fama also made this point in “Multiperiod Consumption-Investment Decisions,” American Economic Review 60 (1970).
19
Mark Rubinstein, “The Valuation of Uncertain Income Streams and the Pricing of Options,” Bell Journal of Economics and Management Science 7 (1976), pp. 407–25; Robert Lucas, “Asset Prices in an Exchange Econ- omy,” Econometrica 46 (1978), pp. 1429–45; Douglas Breeden, “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities,” Journal of Financial Economics 7 (1979), pp. 265–96.
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C H A P T E R 9 The Capital Asset Pricing Model 309
In a lifetime consumption plan, the investor must in each period balance the allocation
of current wealth between today’s consumption and the savings and investment that will
support future consumption. When optimized, the utility value from an additional dollar
of consumption today must be equal to the utility value of the expected future consump-
tion that can be financed by that additional dollar of wealth. 20
Future wealth will grow
from labor income, as well as returns on that dollar when invested in the optimal complete
portfolio.
Suppose risky assets are available and you wish to increase expected consumption
growth by allocating some of your savings to a risky portfolio. How would we measure the
risk of these assets? As a general rule, investors will value additional income more highly
during difficult economic times (when resources are scarce) than in affluent times (when
consumption is already abundant). An asset will therefore be viewed as riskier in terms
of consumption if it has positive covariance with consumption growth—in other words,
if its payoff is higher when consumption is already high and lower when consumption is
relatively restricted. Therefore, equilibrium risk premiums will be greater for assets that
exhibit higher covariance with consumption growth. Developing this insight, we can write
the risk premium on an asset as a function of its “consumption risk” as follows:
E( Ri ) 5 biCRPC (9.15)
where portfolio C may be interpreted as a consumption-tracking portfolio (also called a consumption-mimicking portfolio ), that is, the portfolio with the highest correlation with consumption growth; b iC is the slope coefficient in the regression of asset i ’s excess returns, R i , on those of the consumption-tracking portfolio; and, finally, RP C is the risk premium associated with consumption uncertainty, which is measured by the expected excess return
on the consumption-tracking portfolio:
RPC 5 E( RC ) 5 E( rC ) 2 rf (9.16)
Notice how similar this conclusion is to the conventional CAPM. The consumption-
tracking portfolio in the CCAPM plays the role of the market portfolio in the conven-
tional CAPM. This is in accord with its focus on the risk of consumption opportunities rather than the risk and return of the dollar value of the portfolio. The excess return on the consumption-tracking portfolio plays the role of the excess return on the market portfolio,
M. Both approaches result in linear, single-factor models that differ mainly in the identity of the factor they use.
In contrast to the CAPM, the beta of the market portfolio on the market factor of the
CCAPM is not necessarily 1. It is perfectly plausible and empirically evident that this
beta is substantially greater than 1. This means that in the linear relationship between the
market-index risk premium and that of the consumption portfolio,
E( RM ) 5 aM 1 bMCE( RC ) 1 eM (9.17)
where a M and á M allow for empirical deviation from the exact model in Equation 9.15, and
b MC is not necessarily equal to 1.
Because the CCAPM is so similar to the CAPM, one might wonder about its usefulness.
Indeed, just as the CAPM is empirically flawed because not all assets are traded, so is the
20
Wealth at each point in time equals the market value of assets in the balance sheet plus the present value of
future labor income. These models of consumption and investment decisions are often made tractable by assum-
ing investors exhibit constant relative risk aversion, or CRRA. CRRA implies that an individual invests a constant
proportion of wealth in the optimal risky portfolio regardless of the level of wealth. You might recall that our
prescription for optimal capital allocation in Chapter 6 also called for an optimal investment proportion in the
risky portfolio regardless of the level of wealth. The utility function we employed there also exhibited CRRA.
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310 P A R T I I I Equilibrium in Capital Markets
CCAPM. The attractiveness of this model is in that it compactly incorporates consump-
tion hedging and possible changes in investment opportunities, that is, in the parameters
of the return distributions in a single-factor framework. There is a price to pay for this
compactness, however. Consumption growth figures are published infrequently (monthly
at the most) compared with financial assets, and are measured with significant error.
Nevertheless, recent empirical research 21
indicates that this model is more successful in
explaining realized returns than the CAPM, which is a reason why students of investments
should be familiar with it. We return to this issue, as well as empirical evidence concerning
the CCAPM, in Chapter 13.
Liquidity and the CAPM Despite Assumption 2(d) saying that securities can be traded costlessly, the CAPM has
little to say about trading activity. In the equilibrium of the CAPM, all investors share
all available information and demand identical portfolios of risky assets. The awkward
implication of this result is that there is no reason for trade. If all investors hold identical
portfolios of risky assets, then when new (unexpected) information arrives, prices will
change commensurately, but each investor will continue to hold a piece of the market
portfolio, which requires no exchange of assets. How do we square this implication with
the observation that on a typical day, trading volume amounts to several billion shares?
One obvious answer is heterogeneous expectations, that is, beliefs not shared by the entire
market. Diverse beliefs will give rise to trading as investors attempt to profit by rearranging
portfolios in accordance with their now-heterogeneous demands. In reality, trading (and
trading costs) will be of great importance to investors.
The liquidity of an asset is the ease and speed with which it can be sold at fair market value. Part of liquidity is the cost of engaging in a transaction, particularly the bid–ask
spread. Another part is price impact—the adverse movement in price one would encoun-
ter when attempting to execute a larger trade. Yet another component is immediacy—the
ability to sell the asset quickly without reverting to fire-sale prices. Conversely, illiquidity can be measured in part by the discount from fair market value a seller must accept if the
asset is to be sold quickly. A perfectly liquid asset is one that would entail no illiquidity
discount.
Liquidity (or the lack of it) has long been recognized as an important characteristic
that affects asset values. In legal cases, courts have routinely applied very steep discounts
to the values of businesses that cannot be publicly traded. But liquidity has not always
been appreciated as an important factor in security markets, presumably due to the rela-
tively small trading cost per transaction compared with the large costs of trading assets
such as real estate. The breakthrough came in the work of Amihud and Mendelson 22
and
today, liquidity is increasingly viewed as an important determinant of prices and expected
returns. We supply only a brief synopsis of this important topic here and provide empirical
evidence in Chapter 13.
One important component of trading cost is the bid–ask spread. For example, in elec-
tronic markets, the limit-order book contains the “inside spread,” that is, the difference
between the highest price at which some investor will purchase any shares and the lowest
21
Ravi Jagannathan and Yong Wang, “Lazy Investors, Discretionary Consumption, and the Cross-Section of
Stock Returns,” Journal of Finance 62 (August 2007), pp. 1633–61.
22
Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Economics 17 (1986). A summary of the ensuing large body of literature on liquidity can be found in Yakov Amihud, Haim
Mendelson, and Lasse Heje Pedersen, Market Liquidity: Asset Pricing Risk and Crises, Cambridge University Press, New York: 2013.
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C H A P T E R 9 The Capital Asset Pricing Model 311
price at which another investor is willing to sell. The effective bid–ask spread will also
depend on the size of the desired transaction. Larger purchases will require a trader to
move deeper into the limit-order book and accept less-attractive prices. While inside
spreads on electronic markets often appear extremely low, effective spreads can be much
larger, because most limit orders are good for only small numbers of shares.
There is greater emphasis today on the component of the spread due to asymmetric
information. Asymmetric information is the potential for one trader to have private informa-
tion about the value of the security that is not known to the trading partner. To see why such
an asymmetry can affect the market, think about the problems facing someone buying a
used car. The seller knows more about the car than the buyer, so the buyer naturally won-
ders if the seller is trying to get rid of the car because it is a “lemon.” At the least, buyers
worried about overpaying will shave the prices they are willing to pay for a car of uncer-
tain quality. In extreme cases of asymmetric information, trading may cease altogether. 23
Similarly, traders who post offers to buy or sell at limit prices need to be worried about
being picked off by better-informed traders who hit their limit prices only when they are
out of line with the intrinsic value of the firm.
Broadly speaking, we may envision investors trading securities for two reasons. Some
trades are driven by “noninformational” motives, for example, selling assets to raise cash
for a big purchase, or even just for portfolio rebalancing. These sorts of trades, which are
not motivated by private information that bears on the value of the traded security, are
called noise trades. Security dealers will earn a profit from the bid–ask spread when trans- acting with noise traders (also called liquidity traders because their trades may derive from needs for liquidity, i.e., cash).
Other transactions are initiated by traders who believe they have come across informa-
tion that a security is mispriced. But if that information gives them an advantage, it must
be disadvantageous to the other party in the transaction. In this manner, information trad-
ers impose a cost on both dealers and other investors who post limit orders. Although on
average dealers make money from the bid–ask spread when transacting with liquidity trad-
ers, they will absorb losses from information traders. Similarly, any trader posting a limit
order is at risk from information traders. The response is to increase limit-ask prices and
decrease limit-bid orders—in other words, the spread must widen. The greater the relative
importance of information traders, the greater the required spread to compensate for the
potential losses from trading with them. In the end, therefore, liquidity traders absorb most
of the cost of the information trades because the bid–ask spread that they must pay on their
“innocent” trades widens when informational asymmetry is more severe.
The discount in a security price that results from illiquidity can be surprisingly large, far
larger than the bid–ask spread. Consider a security with a bid–ask spread of 1%. Suppose
it will change hands once a year for the next 3 years and then will be held forever by
the third buyer. For the last trade, the investor will pay for the security 99.5% or .995
of its fair price; the price is reduced by half the spread that will be incurred when the
stock is sold. The second buyer, knowing the security will be sold a year later for .995
of fair value, and having to absorb half the spread upon purchase, will be willing to pay
.995 2 .005/1.05 5 .9902 (i.e., 99.02% of fair value), if the spread from fair value is dis-
counted at a rate of 5%. Finally, the current buyer, knowing the loss next year, when the
stock will be sold for .9902 of fair value (a discount of .0098), will pay for the security only
.995 2 .0098/1.05 5 .9857. Thus the discount has ballooned from .5% to 1.43%. In other
23
The problem of informational asymmetry in markets was introduced by the 2001 Nobel laureate George A.
Akerlof and has since become known as the lemons problem. A good introduction to Akerlof’s contributions can be found in George A. Akerlof, An Economic Theorist’s Book of Tales (Cambridge, U.K.: Cambridge University Press, 1984).
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312 P A R T I I I Equilibrium in Capital Markets
words, the present values of all three future trading costs (spreads) are discounted into the
current price. 24
To extend this logic, if the security will be traded once a year forever, its
current illiquidity cost will equal immediate cost plus the present value of a perpetuity of
.5%. At an annual discount rate of 5%, this sum equals .005 1 .005/.05 5 .105, or 10.5%!
Obviously, liquidity is of potentially large value and should not be ignored in deriving the
equilibrium value of securities.
As trading costs are higher, the illiquidity discount will be greater. Of course, if some-
one can buy a share at a lower price, the expected rate of return will be higher. Therefore,
we should expect to see less-liquid securities offer higher average rates of return. But this
illiquidity premium need not rise in direct proportion to trading cost. If an asset is less
liquid, it will be shunned by frequent traders and held instead by longer term traders who
are less affected by high trading costs. Hence in equilibrium, investors with long holding
periods will, on average, hold more of the illiquid securities, while short-horizon inves-
tors will prefer liquid securities. This “clientele effect” mitigates the effect of the bid–ask
spread for illiquid securities. The end result is that the liquidity premium should increase
with trading costs (measured by the bid–ask spread) at a decreasing rate. Figure 9.4 con-
firms this prediction.
So far, we have shown that the expected level of liquidity can affect prices, and there-
fore expected rates of return. What about unanticipated changes in liquidity? In some circumstances, liquidity can unexpectedly dry up. For example, in the financial crisis of
2008, as many investors attempted to reduce leverage and cash out their positions, finding
buyers for some assets became difficult. Many mortgage-backed securities stopped trading
Figure 9.4 The relationship between illiquidity and average returns Source: Derived from Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Economics 17 (1986), pp. 223–49. Copyright 1986, with permission from Elsevier.
1.2
0 0.5 1 1.5 2 2.5 3 3.5
Bid–Ask Spread (%)
Average Monthly Return (% per month)
0
0.2
0.4
0.6
0.8
1
24 We will see another instance of such capitalization of trading costs in Chapter 13, where one explanation for
large discounts on closed-end funds is the substantial present value of a stream of apparently small per-period expenses.
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C H A P T E R 9 The Capital Asset Pricing Model 313
altogether. Liquidity had evaporated. Nor was this an unheard-of phenomenon. The market
crash of 1987, as well as the failure of Long-Term Capital Management in 1998, also saw
large declines in liquidity across broad segments of the market.
In fact, several studies have investigated variation in a number of measures of liquidity
for large samples of stocks and found that when liquidity in one stock decreases, it tends to
decrease in other stocks at the same time; thus liquidity across stocks shows significant cor-
relation. 25
In other words, variation in liquidity has an important systematic component. Not
surprisingly, investors demand compensation for exposure to liquidity risk. The extra expected return for bearing liquidity risk modifies the CAPM expected return–beta relationship.
Following up on this insight, Amihud demonstrates that firms with greater liquidity
uncertainty have higher average returns. 26
Later studies focus on exposure to marketwide liquidity risk, as measured by a “liquidity beta.” Analogously to a traditional market beta,
the liquidity beta measures the sensitivity of a firm’s returns to changes in market liquidity
(whereas the traditional beta measures return sensitivity to the market return). Firms that
provide better returns when market liquidity falls offer some protection against liquidity
risk, and thus should be priced higher and offer lower expected returns. In fact, we will
see in Chapter 13 that firms with high liquidity betas have offered higher average returns,
just as theory predicts. 27
Moreover, the liquidity premium that emerges from these studies
appears to be of roughly the same order of magnitude as the market risk premium, suggest-
ing that liquidity should be a first-order consideration when thinking about security pricing.
25
See, for example, Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam, “Commonality in Liquidity,”
Journal of Financial Economics 56 (2000), pp. 3–28, or J. Hasbrouck and D. H. Seppi, “Common Factors in Prices, Order Flows and Liquidity,” Journal of Financial Economics 59 (2001), pp. 383–411.
26
Yakov Amihud, “Illiquidity and Stock Returns: Cross-Section and Time-Series Effects,” Journal of Financial Markets 9 (2002), pp. 31–56.
27
See L. Pástor and R. F. Stambaugh, “Liquidity Risk and Expected Stock Returns,” Journal of Political Economy 111 (2003), pp. 642–685, or V. V. Acharya and L. H. Pedersen, “Asset Pricing with Liquidity Risk,” Journal of Financial Economics 77 (2005), pp. 375–410.
9.3 The CAPM and the Academic World
The thorn in the side of academic researchers is Assumption 1(a) (all assets trade) that
leads to the result that the efficient portfolio must include all risky assets in the economy.
In reality, we cannot even observe all the assets that do trade, let alone properly account
for those that do not. The theoretical market portfolio, which is central to the CAPM, is
impossible to pin down in practice.
Since the theoretical CAPM market portfolio cannot be observed, tests of the CAPM
must be directed at the mean-beta relationship as applied to all observed assets with respect
to an observed, but perhaps inefficient, stock index portfolio. These tests face surprisingly
difficult hurdles.
The objective is to test the SML equation, E ( R i ) 5 b i R M . We do so with a regression of excess returns of a sample of stocks ( i 5 1, . . . , N ) over a given period, t, against the betas of each stock:
Ri,t 5 l0 1 l1bi 1 l2sei 2 1 hi,t (9.18)
The CAPM predicts that (1) l 0 5 0, that is, the average alpha in the sample will be zero;
(2) l 1 5 R M , that is, the slope of the SML equals the market-index risk premium; and (3) l 2 5 0, that is, unique risk, sei
2 , doesn’t earn a risk premium. h i is the zero-mean
residual of this regression.
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314 P A R T I I I Equilibrium in Capital Markets
Where, you may ask, do we obtain the beta coefficients and residual variances for the
N stocks in the regression? We have to estimate this pair for each stock from a time series of stock returns. And therein lies the snag: We estimate these parameters with large errors.
Moreover, these errors may be correlated: First, beta may be correlated with the residual
variance of each stock (as well as errors in these estimates), and second, the error terms in
the regression may be correlated across stocks. These measurement errors can result in a
downward bias in the slope of the SML ( l 1 ), and an upward bias in the average alpha ( l 0 ).
We can’t even predict the sign of the bias in ( l 2 ).
An example of this hazard was pointed out in an early paper by Miller and Scholes, 28
who demonstrated how econometric problems could lead one to reject the CAPM even if
it were perfectly valid. They considered a checklist of difficulties encountered in testing
the model and showed how these problems potentially could bias conclusions. To prove
the point, they simulated rates of return that were constructed to satisfy the predictions of the CAPM and used these rates to test the model with standard statistical techniques of
the day. The result of these tests was a rejection of the model that looks surprisingly similar
to what we find in tests of returns from actual data—this despite the fact that the data were
constructed to satisfy the CAPM. Miller and Scholes thus demonstrated that econometric
technique alone could be responsible for the rejection of the model in actual tests.
Moreover, both coefficients, alpha and beta, as well as residual variance, are likely time
varying. There is nothing in the CAPM that precludes such time variation, but standard
regression techniques rule it out and thus may lead to false rejection of the model. There
are now well-known techniques to account for time-varying parameters. In fact, Robert
Engle won the Nobel Prize for his pioneering work on econometric techniques to deal
with time-varying volatility, and a good portion of the applications of these new tech-
niques have been in finance. 29
Moreover, betas may vary not purely randomly over time,
but in response to changing economic conditions. A “conditional” CAPM allows risk and
return to change with a set of “conditioning variables.” 30
As importantly, Campbell and
Vuolteenaho 31
find that the beta of a security can be decomposed into two components,
one that measures sensitivity to changes in corporate profitability and another that mea-
sures sensitivity to changes in the market’s discount rates. These are found to be quite
different in many cases. Improved econometric techniques such as those proposed in this
short survey may help resolve part of the empirical failure of the simple CAPM.
A strand of research that has not yet yielded fruit is the search for portfolios that hedge the
price risk of specific consumption items, as in Merton’s Equation 9.14. But the jury is still
out on the empirical content of this equation with respect to future investment opportunities.
As mentioned in Chapter 5, Fama and French documented the explanatory power of
size and book-to-market ratios (B/M). They interpret portfolios formed to align with these
characteristics as hedging portfolios in the context of Equation 9.14. Following their lead,
other papers have now suggested a number of other extra-market risk factors (discussed in
the next chapter). But we don’t really know what uncertainties in future investment oppor-
tunities are hedged by these factors, leading many to be skeptical of empirically driven
identification of extra-market hedging portfolios.
28
Merton H. Miller and Myron Scholes, “Rates of Return in Relations to Risk: A Re-examination of Some Recent
Findings,” in Studies in the Theory of Capital Markets, Michael C. Jensen, ed. (New York: Praeger, 1972).
29
Engle’s work gave rise to the widespread use of so-called ARCH models. ARCH stands for autoregressive
conditional heteroskedasticity, which is a fancy way of saying that volatility changes over time, and that recent
levels of volatility can be used to form optimal estimates of future volatility.
30
There is now a large literature on conditional models of security market equilibrium. Much of it derives from
Ravi Jagannathan and Zhenyu Wang, “The Conditional CAPM and the Cross-Section of Expected Returns,”
Journal of Finance 51 (March 1996), pp. 3–53.
31
John Campbell and Tuomo Vuolteenaho, “Bad Beta, Good Beta,” American Economic Review 94 (December 2004), pp. 1249–75.
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C H A P T E R 9 The Capital Asset Pricing Model 315
The bottom line is that in the academic world the single-index CAPM is considered
passé. We don’t yet know, however, what shape the successful extension to replace it will
take. Stay tuned for future editions of this text.
9.4 The CAPM and the Investment Industry
While academics have been riding multiple-index models in search of a CAPM that best
explains returns, the industry has steadfastly stayed with the single-index CAPM.
This interesting phenomenon can be explained by a “test of the non-testable.” Presumably,
the CAPM tenet that the market portfolio is efficient cannot be tested because the true mar-
ket portfolio cannot be observed in the first place. But as time has passed, it has become
ever more evident that consistently beating a (not very broad) index portfolio such as the
S&P 500 appears to be beyond the power of most investors.
Indirect evidence on the efficiency of the market portfolio can be found in a study by
Burton Malkiel, 32
who estimates alpha values for a large sample of equity mutual funds.
The results, which appear in Figure 9.5 , show that the distribution of alphas is roughly bell
shaped, with a mean that is slightly negative but statistically indistinguishable from zero.
On average, it does not appear that mutual funds outperform the market index (the S&P
500) on a risk-adjusted basis. 33
32
Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72.
33
Notice that the study included all mutual funds with at least 10 years of continuous data. This suggests the
average alpha from this sample would be upward biased because funds that failed after less than 10 years were
ignored and omitted from the left tail of the distribution. This survivorship bias makes the finding that the average fund underperformed the index even more telling. We discuss survivorship bias further in Chapter 11.
Figure 9.5 Estimates of individual mutual fund alphas, 1972–1991. This is a plot of the frequency distri- bution of estimated alphas for all-equity mutual funds with 10-year continuous records.
Source: Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. Used with permission of John Wiley and Sons, via Copyright Clearance Center.
Alpha (%)
Fr e q
u e n
cy
36
32
28
24
16
20
12
8
4
0 −3 −2 −1 0 1 2
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316 P A R T I I I Equilibrium in Capital Markets
This result is quite meaningful. While we might expect realized alpha values of indi-
vidual securities to center around zero, professionally managed mutual funds might be
expected to demonstrate average positive alphas. Funds with superior performance (and
we do expect this set to be nonempty) should tilt the sample average to a positive value.
The small impact of superior funds on this distribution suggests the difficulty in beating
the passive strategy that the CAPM deems to be optimal.
From the standpoint of the industry, an index portfolio that can be beaten by only a
small fraction of professional managers over a 10-year period may well be taken as ex-ante
efficient for all practical purposes, that is, to be used as: (1) a diversification vehicle to mix
with an active portfolio from security analysis (discussed in Chapter 8); (2) a benchmark
for performance evaluation and compensation (discussed in Chapter 24); (3) a means to
adjudicate law suits about fair compensation to various risky enterprises; and (4) a means
to determine proper prices in regulated industries, allowing shareholders to earn a fair rate
of return on their investments, but no more.
SUMMARY 1. The CAPM assumes that investors are single-period planners who agree on a common input list from security analysis and seek mean-variance optimal portfolios.
2. The CAPM assumes that security markets are ideal in the sense that:
a. They are large, and investors are price-takers. b. There are no taxes or transaction costs. c. All risky assets are publicly traded. d. Investors can borrow and lend any amount at a fixed risk-free rate.
3. With these assumptions, all investors hold identical risky portfolios. The CAPM holds that in equilibrium the market portfolio is the unique mean-variance efficient tangency portfolio. Thus a
passive strategy is efficient.
4. The CAPM market portfolio is a value-weighted portfolio. Each security is held in a proportion equal to its market value divided by the total market value of all securities.
5. If the market portfolio is efficient and the average investor neither borrows nor lends, then the risk premium on the market portfolio is proportional to its variance, sM
2 , and to the average coefficient
of risk aversion across investors, A:
E( rM ) 2 rf 5 AsM 2
6. The CAPM implies that the risk premium on any individual asset or portfolio is the product of the risk premium on the market portfolio and the beta coefficient:
E( ri ) 2 rf 5 bi3E( rM ) 2 rf4 where the beta coefficient is the covariance of the asset with the market portfolio as a fraction of
the variance of the market portfolio:
bi 5 Cov( ri, rM )
sM 2
7. When risk-free investments are restricted but all other CAPM assumptions hold, then the simple version of the CAPM is replaced by its zero-beta version. Accordingly, the risk-free rate in the
expected return–beta relationship is replaced by the zero-beta portfolio’s expected rate of return:
E( ri ) 5 E(rZ) 1 bi[E(rM) 2 E(rZ)]
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C H A P T E R 9 The Capital Asset Pricing Model 317
Market risk premium: E( RM ) 5 AsM 2
Beta: bi 5 Cov( Ri, RM )
sM 2
Security market line: E ( ri ) 5 rf 1 bi 3E ( rM ) 2 rf 4 Zero-beta SML: E ( ri ) 5 E ( rZ ) 1 bi 3E ( rM ) 2 E ( rZ )4 Multifactor SML (in excess returns): E( Ri ) 5 biM E( RM ) 1 a
K
k51 E( Rk )
8. The simple version of the CAPM assumes that investors have a single-period time hori- zon. When investors are assumed to be concerned with lifetime consumption and bequest
plans, but investors’ tastes and security return distributions are stable over time, the market
portfolio remains efficient and the simple version of the expected return–beta relationship
holds. But if those distributions change unpredictably, or if investors seek to hedge nonmar-
ket sources of risk to their consumption, the simple CAPM will give way to a multifactor
version in which the security’s exposure to these nonmarket sources of risk command risk
premiums.
9. The consumption-based capital asset pricing model (CCAPM) is a single-factor model in which the market portfolio excess return is replaced by that of a consumption-tracking port-
folio. By appealing directly to consumption, the model naturally incorporates consumption-
hedging considerations and changing investment opportunities within a single-factor
framework.
10. The security market line of the CAPM must be modified to account for labor income and other significant nontraded assets.
11. Liquidity costs and liquidity risk can be incorporated into the CAPM relationship. Investors demand compensation for expected costs of illiquidity as well as the risk surrounding those
costs.
Related Web sites for this chapter are available at www. mhhe.com/bkm
market portfolio
mutual fund theorem
market price of risk
beta
expected return–beta (or mean-beta) relationship
security market line (SML)
alpha
homogeneous expectations
zero-beta portfolio
liquidity
illiquidity
KEY TERMS
KEY EQUATIONS
PROBLEM SETS
Basic
1. What must be the beta of a portfolio with E ( r P ) 5 18%, if r f 5 6% and E ( r M ) 5 14%?
2. The market price of a security is $50. Its expected rate of return is 14%. The risk-free rate is 6%
and the market risk premium is 8.5%. What will be the market price of the security if its correla-
tion coefficient with the market portfolio doubles (and all other variables remain unchanged)?
Assume that the stock is expected to pay a constant dividend in perpetuity.
3. Are the following true or false? Explain.
a. Stocks with a beta of zero offer an expected rate of return of zero. b. The CAPM implies that investors require a higher return to hold highly volatile securities. c. You can construct a portfolio with beta of .75 by investing .75 of the investment budget in
T-bills and the remainder in the market portfolio.
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318 P A R T I I I Equilibrium in Capital Markets
4. Here are data on two companies. The T-bill rate is 4% and the market risk premium is 6%.
Company $1 Discount Store Everything $5
Forecasted return 12% 11% Standard deviation of returns 8% 10% Beta 1.5 1.0
What would be the fair return for each company, according to the capital asset pricing model
(CAPM)?
5. Characterize each company in the previous problem as underpriced, overpriced, or properly
priced.
6. What is the expected rate of return for a stock that has a beta of 1.0 if the expected return on the
market is 15%?
a. 15%. b. More than 15%. c. Cannot be determined without the risk-free rate.
7. Kaskin, Inc., stock has a beta of 1.2 and Quinn, Inc., stock has a beta of .6. Which of the follow-
ing statements is most accurate?
a. The expected rate of return will be higher for the stock of Kaskin, Inc., than that of Quinn, Inc. b. The stock of Kaskin, Inc., has more total risk than Quinn, Inc. c. The stock of Quinn, Inc., has more systematic risk than that of Kaskin, Inc.
8. You are a consultant to a large manufacturing corporation that is considering a project with the
following net after-tax cash flows (in millions of dollars):
Years from Now After-Tax Cash Flow
0 240 1–10 15
The project’s beta is 1.8. Assuming that r f 5 8% and E ( r M ) 5 16%, what is the net present value of the project? What is the highest possible beta estimate for the project before its NPV
becomes negative?
9. Consider the following table, which gives a security analyst’s expected return on two stocks for
two particular market returns:
Market Return Aggressive Stock Defensive Stock
5% 22% 6%
25 38 12
a. What are the betas of the two stocks? b. What is the expected rate of return on each stock if the market return is equally likely to be
5% or 25%?
c. If the T-bill rate is 6% and the market return is equally likely to be 5% or 25%, draw the SML for this economy.
d. Plot the two securities on the SML graph. What are the alphas of each? e. What hurdle rate should be used by the management of the aggressive firm for a project with
the risk characteristics of the defensive firm’s stock?
For Problems 10 to 16: If the simple CAPM is valid, which of the following situations are possible? Explain. Consider each situation independently.
10. Portfolio Expected Return Beta
A 20 1.4 B 25 1.2
Intermediate
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C H A P T E R 9 The Capital Asset Pricing Model 319
11. Portfolio Expected Return
Standard Deviation
A 30 35 B 40 25
12. Portfolio Expected Return
Standard Deviation
Risk-free 10 0 Market 18 24
A 16 12
13. Portfolio Expected Return
Standard Deviation
Risk-free 10 0 Market 18 24
A 20 22
14. Portfolio Expected Return Beta
Risk-free 10 0 Market 18 1.0
A 16 1.5
15. Portfolio Expected Return Beta
Risk-free 10 0 Market 18 1.0
A 16 0.9
16. Portfolio Expected Return
Standard Deviation
Risk-free 10 0 Market 18 24
A 16 22
For Problems 17 to 19 assume that the risk-free rate of interest is 6% and the expected rate of return on the market is 16%.
17. A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year.
Its beta is 1.2. What do investors expect the stock to sell for at the end of the year?
18. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I
think the beta of the firm is .5, when in fact the beta is really 1, how much more will I offer for the firm than it is truly worth?
19. A stock has an expected rate of return of 4%. What is its beta?
20. Two investment advisers are comparing performance. One averaged a 19% rate of return and
the other a 16% rate of return. However, the beta of the first investor was 1.5, whereas that of the
second was 1.
a. Can you tell which investor was a better selector of individual stocks (aside from the issue of general movements in the market)?
b. If the T-bill rate were 6% and the market return during the period were 14%, which investor would be the superior stock selector?
c. What if the T-bill rate were 3% and the market return were 15%?
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320 P A R T I I I Equilibrium in Capital Markets
1. a. John Wilson is a portfolio manager at Austin & Associates. For all of his clients, Wilson manages portfolios that lie on the Markowitz efficient frontier. Wilson asks Mary Regan,
CFA, a managing director at Austin, to review the portfolios of two of his clients, the Eagle
Manufacturing Company and the Rainbow Life Insurance Co. The expected returns of the two
portfolios are substantially different. Regan determines that the Rainbow portfolio is virtually
identical to the market portfolio and concludes that the Rainbow portfolio must be superior
to the Eagle portfolio. Do you agree or disagree with Regan’s conclusion that the Rainbow
portfolio is superior to the Eagle portfolio? Justify your response with reference to the capital
market line.
b. Wilson remarks that the Rainbow portfolio has a higher expected return because it has greater nonsystematic risk than Eagle’s portfolio. Define nonsystematic risk and explain why you
agree or disagree with Wilson’s remark.
2. Wilson is now evaluating the expected performance of two common stocks, Furhman Labs Inc.
and Garten Testing Inc. He has gathered the following information:
• The risk-free rate is 5%.
• The expected return on the market portfolio is 11.5%.
• The beta of Furhman stock is 1.5.
• The beta of Garten stock is .8.
Based on his own analysis, Wilson’s forecasts of the returns on the two stocks are 13.25% for
Furhman stock and 11.25% for Garten stock. Calculate the required rate of return for Furhman
Labs stock and for Garten Testing stock. Indicate whether each stock is undervalued, fairly val-
ued, or overvalued.
3. The security market line depicts:
a. A security’s expected return as a function of its systematic risk. b. The market portfolio as the optimal portfolio of risky securities. c. The relationship between a security’s return and the return on an index. d. The complete portfolio as a combination of the market portfolio and the risk-free asset.
21. Suppose the rate of return on short-term government securities (perceived to be risk-free) is
about 5%. Suppose also that the expected rate of return required by the market for a portfolio
with a beta of 1 is 12%. According to the capital asset pricing model:
a. What is the expected rate of return on the market portfolio? b. What would be the expected rate of return on a stock with b 5 0? c. Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 divi-
dends next year and you expect it to sell then for $41. The stock risk has been evaluated at
b 5 2 .5. Is the stock overpriced or underpriced?
22. Suppose that borrowing is restricted so that the zero-beta version of the CAPM holds. The
expected return on the market portfolio is 17%, and on the zero-beta portfolio it is 8%. What is
the expected return on a portfolio with a beta of .6?
23. a. A mutual fund with beta of .8 has an expected rate of return of 14%. If r f 5 5%, and you expect the rate of return on the market portfolio to be 15%, should you invest in this fund?
What is the fund’s alpha?
b. What passive portfolio comprised of a market-index portfolio and a money market account would have the same beta as the fund? Show that the difference between the expected rate of
return on this passive portfolio and that of the fund equals the alpha from part ( a ).
24. Outline how you would incorporate the following into the CCAPM:
a. Liquidity. b. Nontraded assets. (Do you have to worry about labor income?)
Challenge
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C H A P T E R 9 The Capital Asset Pricing Model 321
4. Within the context of the capital asset pricing model (CAPM), assume:
• Expected return on the market 5 15%.
• Risk-free rate 5 8%.
• Expected rate of return on XYZ security 5 17%.
• Beta of XYZ security 5 1.25.
Which one of the following is correct?
a. XYZ is overpriced. b. XYZ is fairly priced. c. XYZ’s alpha is 2 .25%. d. XYZ’s alpha is .25%.
5. What is the expected return of a zero-beta security?
a. Market rate of return. b. Zero rate of return. c. Negative rate of return. d. Risk-free rate of return.
6. Capital asset pricing theory asserts that portfolio returns are best explained by:
a. Economic factors. b. Specific risk. c. Systematic risk. d. Diversification.
7. According to CAPM, the expected rate of return of a portfolio with a beta of 1.0 and an alpha of 0 is:
a. Between r M and r f . b. The risk-free rate, r f . c. b ( r M 2 r f ). d. The expected return on the market, r M .
The following table shows risk and return measures for two portfolios.
Portfolio Average Annual Rate of Return Standard Deviation Beta
R 11% 10% 0.5 S&P 500 14% 12% 1.0
8. When plotting portfolio R on the preceding table relative to the SML, portfolio R lies:
a. On the SML. b. Below the SML. c. Above the SML. d. Insufficient data given.
9. When plotting portfolio R relative to the capital market line, portfolio R lies:
a. On the CML. b. Below the CML. c. Above the CML. d. Insufficient data given.
10. Briefly explain whether investors should expect a higher return from holding portfolio A versus portfolio B under capital asset pricing theory (CAPM). Assume that both portfolios are well diversified.
Portfolio A Portfolio B
Systematic risk (beta) 1.0 1.0 Specific risk for each individual security High Low
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322 P A R T I I I Equilibrium in Capital Markets
11. Joan McKay is a portfolio manager for a bank trust department. McKay meets with two clients,
Kevin Murray and Lisa York, to review their investment objectives. Each client expresses an
interest in changing his or her individual investment objectives. Both clients currently hold
well-diversified portfolios of risky assets.
a. Murray wants to increase the expected return of his portfolio. State what action McKay should take to achieve Murray’s objective. Justify your response in the context of
the CML.
b. York wants to reduce the risk exposure of her portfolio but does not want to engage in bor- rowing or lending activities to do so. State what action McKay should take to achieve York’s
objective. Justify your response in the context of the SML.
12. Karen Kay, a portfolio manager at Collins Asset Management, is using the capital asset pricing
model for making recommendations to her clients. Her research department has developed the
information shown in the following exhibit.
Forecast Returns, Standard Deviations, and Betas
Forecast Return Standard Deviation Beta
Stock X 14.0% 36% 0.8
Stock Y 17.0 25 1.5
Market index 14.0 15 1.0
Risk-free rate 5.0
a. Calculate expected return and alpha for each stock. b. Identify and justify which stock would be more appropriate for an investor who wants to
i. add this stock to a well-diversified equity portfolio. ii. hold this stock as a single-stock portfolio.
E-INVESTMENTS EXERCISES Fidelity provides data on the risk and return of its funds at www.fidelity.com . Click on the Research link, then choose Mutual Funds from the submenu. In the Fund Evaluator section, search over all open no-load funds. On the next screen, click on Risk/ Volatility Measures and indicate that you want to screen for funds with betas less than or equal to .50. Click Search Funds to see the results. Select five funds from the resulting list and click Compare. Rank the five funds according to their betas and then according to their stan- dard deviations. Do both lists rank the funds in the same order? How would you explain any difference in the rankings? Repeat the exercise to compare five funds that have betas greater than or equal to 1.50. Why might the degree of agreement when ranking funds by beta versus standard deviation differ when using high versus low beta funds?
SOLUTIONS TO CONCEPT CHECKS 1. We can characterize the entire population by two representative investors. One is the
“uninformed” investor, who does not engage in security analysis and holds the market portfolio,
whereas the other optimizes using the Markowitz algorithm with input from security analysis.
The uninformed investor does not know what input the informed investor uses to make portfolio
purchases. The uninformed investor knows, however, that if the other investor is informed, the
market portfolio proportions will be optimal. Therefore, to depart from these proportions would
constitute an uninformed bet, which will, on average, reduce the efficiency of diversification with
no compensating improvement in expected returns.
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C H A P T E R 9 The Capital Asset Pricing Model 323
2. a. Substituting the historical mean and standard deviation in Equation 9.2 yields a coefficient of risk aversion of
A 5 E(rM) 2 rf
sM 2
5 .079
.2322 5 1.47
b. This relationship also tells us that for the historical standard deviation and a coefficient of risk aversion of 3.5 the risk premium would be
E(rM) 2 rf 5 AsM 2 5 3.5 3 .2322 5 .188 5 18.8%
3. For these investment proportions, w Ford , w Toyota , the portfolio b is
bP 5 wFordbFord 1 wToyotabToyota 5 (.75 3 1.25) 1 (.25 3 1.10) 5 1.2125
As the market risk premium, E ( r M ) 2 r f , is 8%, the portfolio risk premium will be
E(rP) 2 rf 5 bP 3E(rM) 2 rf 4 5 1.2125 3 8 5 9.7%
4. The alpha of a stock is its expected return in excess of that required by the CAPM.
a 5 E(r) 2 5rf 1 b 3E(rM) 2 rf 46 aX Y Z 5 12 2 35 1 1.0(11 2 5) 4 5 1% aA B C 5 13 2 35 1 1.5(11 2 5) 4 5 21%
ABC plots below the SML, while XYZ plots above.
E(r), Percent
b
.5 1 1.5
14
αXYZ � 0 XYZ
Market
SML
0
αABC � 0
E(rM) � 11
rf � 5
ABC12
5. The project-specific required return is determined by the project beta coupled with the market
risk premium and the risk-free rate. The CAPM tells us that an acceptable expected rate of return
for the project is
rf 1 b 3E( rM ) 2 rf4 5 8 1 1.3(16 2 8) 5 18.4% which becomes the project’s hurdle rate. If the IRR of the project is 19%, then it is desirable. Any
project with an IRR equal to or less than 18.4% should be rejected.
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THE EXPLOITATION OF security mispricing in such a way that risk-free profits can be earned is called arbitrage. It involves the simultane- ous purchase and sale of equivalent securi- ties in order to profit from discrepancies in their prices. Perhaps the most basic principle of capital market theory is that equilibrium market prices are rational in that they rule out arbitrage opportunities. If actual secu- rity prices allow for arbitrage, the result will be strong pressure to restore equilibrium. Therefore, security markets ought to satisfy a “no-arbitrage condition.” In this chapter, we show how such no-arbitrage conditions together with the factor model introduced in Chapter 8 allow us to generalize the security market line of the CAPM to gain richer insight into the risk–return relationship.
We begin by showing how the decompo- sition of risk into market versus firm-specific influences that we introduced in earlier chapters can be extended to deal with the multifaceted nature of systematic risk. Mul- tifactor models of security returns can be used to measure and manage exposure to each of many economywide factors such as business-cycle risk, interest or inflation rate
risk, energy price risk, and so on. These models also lead us to a multifactor version of the security market line in which risk premiums derive from exposure to multiple risk sources, each with their own risk premium.
We show how factor models combined with a no-arbitrage condition lead to a simple relationship between expected return and risk. This approach to the risk–return trade- off is called arbitrage pricing theory, or APT. In a single-factor market where there are no extra-market risk factors, the APT leads to a mean return–beta equation identical to that of the CAPM. In a multifactor market with one or more extra-market risk factors, the APT delivers a mean-beta equation similar to Merton’s intertemporal extension of the CAPM (his ICAPM). We ask next what factors are likely to be the most important sources of risk. These will be the factors generating substantial hedging demands that brought us to the multifactor CAPM introduced in Chapter 9. Both the APT and the CAPM there- fore can lead to multiple-risk versions of the security market line, thereby enriching the insights we can derive about the risk–return relationship.
CHAPTER TEN 10 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 11 PA
R T I II
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 325
The index model introduced in Chapter 8 gave us a way of decomposing stock variability
into market or systematic risk, due largely to macroeconomic events, versus firm-specific
or idiosyncratic effects that can be diversified in large portfolios. In the single-index model,
the return on a broad market-index portfolio summarized the impact of the macro factor.
In Chapter 9 we introduced the possibility that asset-risk premiums may also depend on
correlations with extra-market risk factors, such as inflation, or changes in the parameters
describing future investment opportunities: interest rates, volatility, market-risk premiums,
and betas. For example, returns on an asset whose return increases when inflation increases
can be used to hedge uncertainty in the future inflation rate. Its risk premium may fall as a
result of investors’ extra demand for this asset.
Risk premiums of individual securities should reflect their sensitivities to changes in
extra-market risk factors just as their betas on the market index determine their risk premi-
ums in the simple CAPM. When securities can be used to hedge these factors, the resulting
hedging demands will make the SML multifactor, with each risk source that can be hedged
adding an additional factor to the SML. Risk factors can be represented either by returns
on these hedge portfolios (just as the index portfolio represents the market factor), or more
directly by changes in the risk factors themselves, for example, changes in interest rates
or inflation.
Factor Models of Security Returns We begin with a familiar single-factor model like the one introduced in Chapter 8. Uncer-
tainty in asset returns has two sources: a common or macroeconomic factor and firm-
specific events. The common factor is constructed to have zero expected value, because
we use it to measure new information concerning the macroeconomy, which, by definition, has zero expected value.
If we call F the deviation of the common factor from its expected value, b i the sensi- tivity of firm i to that factor, and e i the firm-specific disturbance, the factor model states that the actual excess return on firm i will equal its initially expected value plus a (zero expected value) random amount attributable to unanticipated economywide events, plus
another (zero expected value) random amount attributable to firm-specific events.
Formally, the single-factor model of excess returns is described by Equation 10.1:
Ri 5 E(Ri) 1 biF 1 ei (10.1)
where E ( R i ) is the expected excess return on stock i. Notice that if the macro factor has a value of 0 in any particular period (i.e., no macro surprises), the excess return on the secu-
rity will equal its previously expected value, E ( R i ), plus the effect of firm-specific events only. The nonsystematic components of returns, the e i s, are assumed to be uncorrelated across stocks and with the factor F.
10.1 Multifactor Models: An Overview
To make the factor model more concrete, consider an example. Suppose that the macro factor, F, is taken to be news about the state of the business cycle, measured by the unexpected percentage change in gross domestic product (GDP), and that the consen- sus is that GDP will increase by 4% this year. Suppose also that a stock’s b value is 1.2.
Example 10.1 Factor Models
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326 P A R T I I I Equilibrium in Capital Markets
The factor model’s decomposition of returns into systematic and firm-specific
components is compelling, but confining systematic risk to a single factor is not. Indeed,
when we motivated systematic risk as the source of risk premiums in Chapter 9, we noted
that extra market sources of risk may arise from a number of sources such as uncertainty
about interest rates, inflation, and so on. The market return reflects macro factors as well as
the average sensitivity of firms to those factors.
It stands to reason that a more explicit representation of systematic risk, allowing for
different stocks to exhibit different sensitivities to its various components, would constitute
a useful refinement of the single-factor model. It is easy to see that models that allow for
several factors— multifactor models —can provide better descriptions of security returns. Apart from their use in building models of equilibrium security pricing, multifactor
models are useful in risk management applications. These models give us a simple way
to measure investor exposure to various macroeconomic risks and construct portfolios to
hedge those risks.
Let’s start with a two-factor model. Suppose the two most important macroeconomic
sources of risk are uncertainties surrounding the state of the business cycle, news of which
we will again measure by unanticipated growth in GDP and changes in interest rates. We
will denote by IR any unexpected change in interest rates. The return on any stock will
respond both to sources of macro risk and to its own firm-specific influences. We can write
a two-factor model describing the excess return on stock i in some time period as follows:
Ri 5 E (Ri ) 1 biGDPGDP 1 biIRIR 1 ei (10.2)
The two macro factors on the right-hand side of the equation comprise the system-
atic factors in the economy. As in the single-factor model, both of these macro factors
have zero expectation: They represent changes in these variables that have not already been
anticipated. The coefficients of each factor in Equation 10.2 measure the sensitivity of share
returns to that factor. For this reason the coefficients are sometimes called factor loadings or, equivalently, factor betas. An increase in interest rates is bad news for most firms, so we would expect interest rate betas generally to be negative. As before, e i reflects firm- specific influences.
To illustrate the advantages of multifactor models, consider two firms, one a regulated
electric-power utility in a mostly residential area, the other an airline. Because residential
demand for electricity is not very sensitive to the business cycle, the utility has a low beta
If GDP increases by only 3%, then the value of F would be 2 1%, representing a 1% disappointment in actual growth versus expected growth. Given the stock’s beta value, this disappointment would translate into a return on the stock that is 1.2% lower than previously expected. This macro surprise, together with the firm-specific disturbance, e i , determines the total departure of the stock’s return from its originally expected value.
Suppose you currently expect the stock in Example 10.1 to earn a 10% rate of return. Then some macroeconomic news suggests that GDP growth will come in at 5% instead of 4%. How will you revise your estimate of the stock’s expected rate of return?
CONCEPT CHECK 10.1
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 327
on GDP. But the utility’s stock price may have a relatively high sensitivity to interest rates.
Because the cash flow generated by the utility is relatively stable, its present value behaves
much like that of a bond, varying inversely with interest rates. Conversely, the performance
of the airline is very sensitive to economic activity but is less sensitive to interest rates. It
will have a high GDP beta and a lower interest rate beta. Suppose that on a particular day, a
news item suggests that the economy will expand. GDP is expected to increase, but so are
interest rates. Is the “macro news” on this day good or bad? For the utility, this is bad news:
Its dominant sensitivity is to rates. But for the airline, which responds more to GDP, this
is good news. Clearly a one-factor or single-index model cannot capture such differential
responses to varying sources of macroeconomic uncertainty.
1 Stephen A. Ross, “Return, Risk and Arbitrage,” in I. Friend and J. Bicksler, eds., Risk and Return in Finance
(Cambridge, MA: Ballinger, 1976).
Factor betas can provide a framework for a hedging strategy. The idea for an investor
who wishes to hedge a source of risk is to establish an opposite factor exposure to offset
that particular source of risk. Often, futures contracts can be used to hedge particular factor
exposures. We explore this application in Chapter 22.
As it stands, however, the multifactor model is no more than a description of the fac- tors that affect security returns. There is no “theory” in the equation. The obvious question
left unanswered by a factor model like Equation 10.2 is where E ( R ) comes from, in other words, what determines a security’s expected excess rate of return. This is where we need a
theoretical model of equilibrium security returns. We therefore now turn to arbitrage pric-
ing theory to help determine the expected value, E ( R ), in Equations 10.1 and 10.2.
10.2 Arbitrage Pricing Theory
Stephen Ross developed the arbitrage pricing theory (APT) in 1976. 1 Like the CAPM, the APT predicts a security market line linking expected returns to risk, but the path it takes
to the SML is quite different. Ross’s APT relies on three key propositions: (1) security
returns can be described by a factor model; (2) there are sufficient securities to diversify
away idiosyncratic risk; and (3) well-functioning security markets do not allow for the per-
sistence of arbitrage opportunities. We begin with a simple version of Ross’s model, which
assumes that only one systematic factor affects security returns.
Suppose we estimate the two-factor model in Equation 10.2 for Northeast Airlines and find the following result:
R 5 .133 1 1.2(GDP) 2 .3(IR) 1 e
This tells us that, based on currently available information, the expected excess rate of return for Northeast is 13.3%, but that for every percentage point increase in GDP beyond current expectations, the return on Northeast shares increases on average by 1.2%, while for every unanticipated percentage point that interest rates increases, Northeast’s shares fall on average by .3%.
Example 10.2 Risk Assessment Using Multifactor Models
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328 P A R T I I I Equilibrium in Capital Markets
Arbitrage, Risk Arbitrage, and Equilibrium An arbitrage opportunity arises when an investor can earn riskless profits without making a net investment. A trivial example of an arbitrage opportunity would arise if shares of a
stock sold for different prices on two different exchanges. For example, suppose IBM sold
for $195 on the NYSE but only $193 on NASDAQ. Then you could buy the shares on
NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per
share without tying up any of your own capital. The Law of One Price states that if two assets are equivalent in all economically relevant respects, then they should have the same
market price. The Law of One Price is enforced by arbitrageurs: If they observe a violation
of the law, they will engage in arbitrage activity —simultaneously buying the asset where it is cheap and selling where it is expensive. In the process, they will bid up the price where
it is low and force it down where it is high until the arbitrage opportunity is eliminated.
The idea that market prices will move to rule out arbitrage opportunities is perhaps the
most fundamental concept in capital market theory. Violation of this restriction would indi-
cate the grossest form of market irrationality.
The critical property of a risk-free arbitrage portfolio is that any investor, regardless of
risk aversion or wealth, will want to take an infinite position in it. Because those large posi-
tions will quickly force prices up or down until the opportunity vanishes, security prices
should satisfy a “no-arbitrage condition,” that is, a condition that rules out the existence of
arbitrage opportunities.
There is an important difference between arbitrage and risk–return dominance argu-
ments in support of equilibrium price relationships. A dominance argument holds that
when an equilibrium price relationship is violated, many investors will make limited port-
folio changes, depending on their degree of risk aversion. Aggregation of these limited
portfolio changes is required to create a large volume of buying and selling, which in turn
restores equilibrium prices. By contrast, when arbitrage opportunities exist, each inves-
tor wants to take as large a position as possible; hence it will not take many investors to
bring about the price pressures necessary to restore equilibrium. Therefore, implications
for prices derived from no-arbitrage arguments are stronger than implications derived from
a risk–return dominance argument.
The CAPM is an example of a dominance argument, implying that all investors hold
mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their
portfolios toward the underpriced and away from the overpriced securities. Pressure on
equilibrium prices results from many investors shifting their portfolios, each by a relatively
small dollar amount. The assumption that a large number of investors are mean-variance
optimizers is critical. In contrast, the implication of a no-arbitrage condition is that a few
investors who identify an arbitrage opportunity will mobilize large dollar amounts and
quickly restore equilibrium.
Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict definition. Arbitrageur often refers to a professional searching for mispriced securities in
specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free)
arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it from pure arbitrage.
To leap ahead, in Part Four we will discuss “derivative” securities such as futures and
options, whose market values are determined by prices of other securities. For example,
the value of a call option on a stock is determined by the price of the stock. For such secu-
rities, strict arbitrage is a practical possibility, and the condition of no-arbitrage leads to
exact pricing. In the case of stocks and other “primitive” securities whose values are not
determined strictly by another bundle of assets, no-arbitrage conditions must be obtained
by appealing to diversification arguments.
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 329
Well-Diversified Portfolios Consider the risk of a portfolio of stocks in a single-factor market. We first show that if a
portfolio is well diversified, its firm-specific or nonfactor risk becomes negligible, so that
only factor (or systematic) risk remains. The excess return on an n -stock portfolio with weights w i , S w i 5 1, is
RP 5 E(RP) 1 bPF 1 eP (10.3)
where
bP 5 g wibi; E(RP) 5 gwiE(Ri)
are the weighted averages of the b i and risk premiums of the n securities. The portfolio nonsystematic component (which is uncorrelated with F ) is e P 5 S w i e i , which similarly is a weighted average of the e i of the n securities.
We can divide the variance of this portfolio into systematic and nonsystematic sources:
sP 2 5 bP
2 sF 2 1 s2(eP)
where sF 2 is the variance of the factor F and s 2 ( e P ) is the nonsystematic risk of the portfo-
lio, which is given by
s2(eP) 5 Variance(gwi ei) 5 gwi 2 s2(ei)
Note that in deriving the nonsystematic variance of the portfolio, we depend on the fact
that the firm-specific e i s are uncorrelated and hence that the variance of the “portfolio” of nonsystematic e i s is the weighted sum of the individual nonsystematic variances with the square of the investment proportions as weights.
If the portfolio were equally weighted, w i 5 1/ n, then the nonsystematic variance would be
s2(eP) 5 Variance (gwiei ) 5 a a1nb2s2(ei ) 5 1n a s2(ei )n 5 1n s2(ei ) where the last term is the average value of nonsystematic variance across securities. In
words, the nonsystematic variance of the portfolio equals the average nonsystematic vari-
ance divided by n. Therefore, when n is large, nonsystematic variance approaches zero. This is the effect of diversification.
This property is true of portfolios other than the equally weighted one. Any portfolio for which each w i becomes consistently smaller as n gets large (more precisely, for which each wi
2 approaches zero as n increases) will satisfy the condition that the portfolio nonsystem- atic risk will approach zero. This property motivates us to define a well-diversified portfolio as one with each weight, w i , small enough that for practical purposes the nonsystematic variance, s
2 ( e P ), is negligible.
a. A portfolio is invested in a very large number of shares ( n is large). However, one-half of the portfolio is invested in stock 1, and the rest of the portfolio is equally divided among the other n 2 1 shares. Is this portfolio well diversified?
b. Another portfolio also is invested in the same n shares, where n is very large. Instead of equally weighting with portfolio weights of 1/ n in each stock, the weights in half the securities are 1.5/ n while the weights in the other shares are .5/ n. Is this portfolio well diversified?
CONCEPT CHECK 10.2
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330 P A R T I I I Equilibrium in Capital Markets
Because the expected value of e P for any well-diversified portfolio is zero, and its variance also is effectively zero, we can conclude that any realized value of e P will be virtually zero. Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio,
for all practical purposes
RP 5 E(RP) 1 bPF
The solid line in Figure 10.1, panel A plots the excess return of a well-diversified port-
folio A with E ( R A ) 5 10% and b A 5 1 for various realizations of the systematic factor. The expected return of portfolio A is 10%; this is where the solid line crosses the vertical axis. At this point the systematic factor is zero, implying no macro surprises. If the macro factor
is positive, the portfolio’s return exceeds its expected value; if it is negative, the portfolio’s
return falls short of its mean. The excess return on the portfolio is therefore
E(RA) 1 bAF 5 10% 1 1.0 3 F
Compare panel A in Figure 10.1 with panel B, which is a similar graph for a single stock ( S ) with b s 5 1. The undiversified stock is subject to nonsystematic risk, which is seen in
a scatter of points around the line. The well-diversified portfolio’s return, in contrast, is
determined completely by the systematic factor.
In a single-factor world, all pairs of well-diversified portfolios are perfectly correlated:
Their risk is fully determined by the same systematic factor. Consider a second well-
diversified portfolio, Portfolio Q, with R Q 5 E ( R Q ) 1 b Q F. We can compute the standard deviations of P and Q, as well as the covariance and correlation between them:
sP 5 bPsF; sQ 5 bQsF
Cov(RP, RQ) 5 Cov(bP F, bQF) 5 bPbQsF 2
rPQ 5 Cov(RP, RQ)
sPsQ 5 1
Figure 10.1 Excess returns as a function of the systematic factor. Panel A, Well-diversified portfolio A. Panel B, Single stock ( S ).
0 F
A
10
Excess Return (%)
0 F
S
10
Excess Return (%)
A B
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 331
Residual SD of Each Stock 5 50% Residual SD of Each Stock 5 100%
N SD( eP ) N SD( eP ) Equal weights: w i 5 1/ N
4 25.00 4 50.00 60 6.45 60 12.91
200 3.54 200 7.07 1,000 1.58 1,000 3.16
10,000 0.50 10,000 1.00 Sets of four relative weights: w 1 5 0.65, w 2 5 0.2, w 3 5 0.1, w 4 5 0.05
4 36.23 4 72.46 60 9.35 60 18.71
200 5.12 200 10.25 1,000 2.29 1,000 4.58
10,000 0.72 10,000 1.45
Table 10.1
Residual variance with even and uneven portfolio weights
Perfect correlation means that in a plot of expected return versus standard deviation (such
as Figure 7.5), any two well-diversified portfolios lie on a straight line. We will see later
that this common line is the CML.
Diversification and Residual Risk in Practice What is the effect of diversification on portfolio residual SD in practice, where portfo- lio size is not unlimited? In reality, we may find (annualized) residual SDs as high as
50% for large stocks and even 100% for small stocks. To illustrate the impact of diversi-
fication, we examine portfolios of two configurations. One portfolio is equally weighted;
this achieves the highest benefits of diversification with equal-SD stocks. For compari-
son, we form the other portfolio using far-from-equal weights. We select stocks in groups
of four, with relative weights in each group of 70%, 15%, 10%, and 5%. The highest
weight is 14 times greater than the lowest, which will severely reduce potential benefits of
diversification. However, extended diversification in which we add to the portfolio more
and more groups of four stocks with the same relative weights will overcome this prob-
lem because the highest portfolio weight still falls with additional diversification. In an
equally weighted 1,000-stock portfolio, each weight is 0.1%; in the unequally weighted
portfolio, with 1,000/4 5 250 groups of four stocks, the highest and lowest weights are
70%/250 5 0.28% and 5%/250 5 0.02%, respectively.
What is a large portfolio? Many widely held ETFs each include hundreds of stocks, and
some funds such as the Wilshire 5000 hold thousands. These portfolios are accessible to
the public since the annual expense ratios of investment companies that offer such funds
are of the order of only 10 basis points. Thus a portfolio of 1,000 stocks is not unheard of,
but a portfolio of 10,000 stocks is.
Table 10.1 shows portfolio residual SD as a function of the number of stocks. Equally
weighted, 1,000-stock portfolios achieve small but not negligible standard deviations of
1.58% when residual risk is 50% and 3.16% when residual risk is 100%. The SDs for the
unbalanced portfolios are about double these values. For 10,000-stock portfolios, the SDs
are negligible, verifying that diversification can eliminate risk even in very unbalanced
portfolios, at least in principle, if the investment universe is large enough.
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332 P A R T I I I Equilibrium in Capital Markets
Executing Arbitrage Imagine a single-factor market where the well-diversified portfolio, M, represents the market factor, F, of Equation 10.1. The excess return on any security is given by R i 5 a i 1 b i R M 1 e i , and that of a well-diversified (therefore zero residual) portfolio, P, is
RP 5 aP 1 bPRM (10.4)
E(RP) 5 aP 1 bPE(RM) (10.5)
Now suppose that security analysis reveals that portfolio P has a positive alpha. 2 We also estimate the risk premium of the index portfolio, M, from macro analysis.
Since neither M nor portfolio P have residual risk, the only risk to the returns of the two portfolios is systematic, derived from their betas on the common factor (the beta of the
index is 1.0). Therefore, you can eliminate the risk of P altogether: Construct a zero-beta portfolio, called Z, from P and M by appropriately selecting weights w P and w M 5 1 2 w P on each portfolio:
bZ 5 wPbP 1 (1 2 wP)bM 5 0
bM 5 1
wP 5 1
1 2 bP ; wM 5 1 2 wP 5
2bP
1 2 bP
(10.6)
Therefore, portfolio Z is riskless, and its alpha is
aZ 5 wPaP 1 (1 2 wP)aM 5 wPaP (10.7)
The risk premium on Z must be zero because the risk of Z is zero. If its risk premium were not zero, you could earn arbitrage profits. Here is how:
Since the beta of Z is zero, Equation 10.5 implies that its risk premium is just its alpha. Using Equation 10.7, its alpha is w P a P , so
E(RZ) 5 wPaP 5 1
1 2 bP aP (10.8)
You now form a zero-net-investment arbitrage portfolio: If b P , 1 and the risk premium
of Z is positive (implying that Z returns more than the risk-free rate), borrow and invest the proceeds in Z . For every borrowed dollar invested in Z, you get a net return (i.e., net of
paying the interest on your loan) of 1
1 2 bP aP. This is a money machine, which you would
work as hard as you can. 3 Similarly if b P . 1, Equation 10.8 tells us that the risk pre-
mium is negative; therefore, sell Z short and invest the proceeds at the risk-free rate. Once again, a money machine has been created. Neither situation can persist, as the large volume
of trades from arbitrageurs pursuing these strategies will push prices until the arbitrage
opportunity disappears (i.e., until the risk premium of portfolio Z equals zero).
2 If the portfolio alpha is negative, we can still pursue the following strategy. We would simply switch to a short
position in P, which would have a positive alpha of the same absolute value as P ’s, and a beta that is the negative of P ’s.
3 The function in Equation 10.8 becomes unstable at b P 5 1. For values of b P near 1, it becomes infinitely large
with the sign of a P . This isn’t an economic absurdity, since in that case, the sizes of your long position in P and short position in M will be almost identical, and the arbitrage profit you earn per dollar invested will be nearly infinite.
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 333
Figure 10.2 Returns as a function of the systematic factor: an arbitrage opportunity
0 F (Realization of Macro Factor)
A B
10 8
Return (%)
The No-Arbitrage Equation of the APT We’ve seen that arbitrage activity will quickly pin the risk premium of any zero-beta well-
diversified portfolio to zero. 4 Setting the expression in Equation 10.8 to zero implies that
the alpha of any well-diversified portfolio must also be zero. From Equation 10.5, this means that for any well-diversified P,
E(RP) 5 bPE(RM) (10.9)
In other words, the risk premium (expected excess return) on portfolio P is the product of its beta and the market-index risk premium. Equation 10.9 thus establishes that the SML
of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage”
requirement of the APT.
Another demonstration that the APT results in the same SML as the CAPM is more
graphical in nature. First we show why all well-diversified portfolios with the same beta must
have the same expected return. Figure 10.2 plots the returns on two such portfolios, A and B, both with betas of 1, but with differing expected returns: E ( r A ) 5 10% and E ( r B ) 5 8%. Could portfolios A and B coexist with the return pattern depicted? Clearly not: No matter what the systematic factor turns out to be, portfolio A outperforms portfolio B, leading to an arbitrage opportunity.
If you sell short $1 million of B and buy $1 million of A, a zero-net-investment strategy, you would have a riskless payoff of $20,000, as follows:
(.10 1 1.0 3 F) 3 $1 million
2(.08 1 1.0 3 F) 3 $1 million
.02 3 $1 million 5 $20,000
from long position in A
from short position in B
net proceeds
Your profit is risk-free because the factor risk cancels out across the long and short
positions. Moreover, the strategy requires zero-net-investment. You should pursue it on an
4 As an exercise, show that when a P , 0 you reverse the position of P in Z, and the arbitrage portfolio will still
earn a riskless excess return.
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334 P A R T I I I Equilibrium in Capital Markets
rf = 4
β
6
.5 1
10
7
F
C
D
A
Risk Premium
(With Respect to Macro Factor)
Expected Return (%)
Figure 10.3 An arbitrage opportunity
infinitely large scale until the return discrepancy between the two portfolios disappears.
Well-diversified portfolios with equal betas must have equal expected returns in market
equilibrium, or arbitrage opportunities exist.
What about portfolios with different betas? Their risk premiums must be propor-
tional to beta. To see why, consider Figure 10.3 . Suppose that the risk-free rate is 4%
and that a well-diversified portfolio, C, with a beta of .5, has an expected return of 6%. Portfolio C plots below the line from the risk-free asset to portfolio A. Consider, there- fore, a new portfolio, D, composed of half of portfolio A and half of the risk-free asset. Portfolio D ’s beta will be (.5 3 0 1 .5 3 1.0) 5 .5, and its expected return will be (.5 3 4 1 .5 3 10) 5 7%. Now portfolio D has an equal beta but a greater expected return than portfolio C. From our analysis in the previous paragraph we know that this consti- tutes an arbitrage opportunity. We conclude that, to preclude arbitrage opportunities, the
expected return on all well-diversified portfolios must lie on the straight line from the
risk-free asset in Figure 10.3 .
Notice in Figure 10.3 that risk premiums are indeed proportional to portfolio betas. The
risk premium is depicted by the vertical arrow, which measures the distance between the
risk-free rate and the expected return on the portfolio. As in the simple CAPM, the risk
premium is zero for b 5 0 and rises in direct proportion to b .
10.3 The APT, the CAPM, and the Index Model
Equation 10.9 raises three questions:
1. Does the APT also apply to less-than-well-diversified portfolios?
2. Is the APT as a model of risk and return superior or inferior to the CAPM? Do we
need both models?
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 335
3. Suppose a security analyst identifies a positive-alpha portfolio with some remaining
residual risk. Don’t we already have a prescription for this case from the Treynor-
Black (T-B) procedure applied to the index model (Chapter 8)? Is this framework
preferred to the APT?
The APT and the CAPM The APT is built on the foundation of well-diversified portfolios. However, we’ve seen,
for example in Table 10.1 , that even large portfolios may have non-negligible residual risk.
Some indexed portfolios may have hundreds or thousands of stocks, but active portfolios
generally cannot, as there is a limit to how many stocks can be actively analyzed in search
of alpha. How does the APT stand up to these limitations?
Suppose we order all portfolios in the universe by residual risk. Think of Level 0 portfo-
lios as having zero residual risk; in other words, they are the theoretically well-diversified
portfolios of the APT. Level 1 portfolios have very small residual risk, say up to 0.5%.
Level 2 portfolios have yet greater residual SD, say up to 1%, and so on.
If the SML described by Equation 10.9 applies to all well-diversified Level 0 portfolios,
it must at least approximate the risk premiums of Level 1 portfolios. Even more important,
while Level 1 risk premiums may deviate slightly from Equation 10.9, such deviations
should be unbiased, with alphas equally likely to be positive or negative. Deviations should
be uncorrelated with beta or residual SD and should average to zero.
We can apply the same logic to portfolios of slightly higher Level 2 residual risk. Since
all Level 1 portfolios are still well approximated by Equation 10.9, so must be risk pre-
miums of Level 2 portfolios, albeit with slightly less accuracy. Here too, we may take
comfort in the lack of bias and zero average deviations from the risk premiums pre-
dicted by Equation 10.9. But still, the precision of predictions of risk premiums from
Equation 10.9 consistently deteriorates with increasing residual risk. (One might ask why
we don’t transform Level 2 portfolios into Level 1 or even Level 0 portfolios by further
diversifying, but as we’ve pointed out, this may not be feasible in practice for assets with
considerable residual risk when active portfolio size or the size of the investment universe
is limited.) If residual risk is sufficiently high and the impediments to complete diversifica-
tion are too onerous, we cannot have full confidence in the APT and the arbitrage activities
that underpin it.
Despite this shortcoming, the APT is valuable. First, recall that the CAPM requires that
almost all investors be mean-variance optimizers. We may well suspect that they are not.
The APT frees us of this assumption. It is sufficient that a small number of sophisticated
arbitrageurs scour the market for arbitrage opportunities. This alone produces an SML,
Equation 10.9, that is a good and unbiased approximation for all assets but those with
significant residual risk.
Perhaps even more important is the fact that the APT is anchored by observable port-
folios such as the market index. The CAPM is not even testable because it relies on an
unobserved, all-inclusive portfolio. The reason that the APT is not fully superior to the
CAPM is that at the level of individual assets and high residual risk, pure arbitrage may
be insufficient to enforce Equation 10.9. Therefore, we need to turn to the CAPM as a
complementary theoretical construct behind equilibrium risk premiums.
It should be noted, however, that when we replace the unobserved market portfolio of
the CAPM with an observed, broad index portfolio that may not be efficient, we no longer
can be sure that the CAPM predicts risk premiums of all assets with no bias. Neither model
therefore is free of limitations. Comparing the APT arbitrage strategy to maximization of
the Sharpe ratio in the context of an index model may well be the more useful framework
for analysis.
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336 P A R T I I I Equilibrium in Capital Markets
The APT and Portfolio Optimization in a Single-Index Market The APT is couched in a single-factor market
5 and applies with perfect accuracy to well-
diversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk pre- mium of a well-diversified portfolio deviates from Equation 10.9. The trades executed by
these arbitrageurs are the enforcers of the accuracy of this equation.
In effect, the APT shows how to take advantage of security mispricing when diversifica-
tion opportunities are abundant. When you lock in and scale up an arbitrage opportunity
you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio,
but only if the arbitrage portfolio is truly risk-free! However, if the arbitrage position is not
perfectly well diversified, an increase in its scale (borrowing cash, or borrowing shares to
go short) will increase the risk of the arbitrage position, potentially without bound. The
APT ignores this complication.
Now consider an investor who confronts the same single factor market, and whose
security analysis reveals an underpriced asset (or portfolio), that is, one whose risk
premium implies a positive alpha. This investor can follow the advice weaved throughout
Chapters 6–8 to construct an optimal risky portfolio. The optimization process will
consider both the potential profit from a position in the mispriced asset, as well as the risk
of the overall portfolio and efficient diversification. As we saw in Chapter 8, the Treynor-
Black (T-B) procedure can be summarized as follows. 6
1. Estimate the risk premium and standard deviation of the benchmark (index) portfo-
lio, RP M and s M .
2. Place all the assets that are mispriced into an active portfolio. Call the alpha of the
active portfolio a A , its systematic-risk coefficient b A , and its residual risk s ( e A ). Your optimal risky portfolio will allocate to the active portfolio a weight, wA
* :
wA *
wA 0
1 1 wA 0 (1 2 bA)
w e
E RA
A
A
M
M
0 2 ( )
( ) ;
2
The allocation to the passive portfolio is then, wM * 5 1 2 wA
* . With this alloca-
tion, the increase in the Sharpe ratio of the optimal portfolio, S P , over that of the passive portfolio, S M , depends on the size of the information ratio of the active portfolio, IR A 5 a A / s ( e A ). The optimized portfolio can attain a Sharpe ratio of
SP 5 "SM2 1 IRA2 . 3. To maximize the Sharpe ratio of the risky portfolio, you maximize the IR of the
active portfolio. This is achieved by allocating to each asset in the active portfolio
a portfolio weight proportional to: w Ai 5 a i / s 2 ( e i ). When this is done, the square
of the information ratio of the active portfolio will be the sum of the squared indi-
vidual information ratios: IRA 2 5 a IRi
2.
Now see what happens in the T-B model when the residual risk of the active portfolio
is zero. This is essentially the assumption of the APT, that a well-diversified portfolio
(with zero residual risk) can be formed. When the residual risk of the active portfolio goes
to zero, the position in it goes to infinity. This is precisely the same implication as the
APT: When portfolios are well-diversified, you will scale up an arbitrage position without
5 The APT is easily extended to a multifactor market as we show later.
6 The tediousness of some of the expressions involved in the T-B method should not deter anyone. The calcula-
tions are pretty straightforward, especially in a spreadsheet. The estimation of the risk parameters also is a rela-
tively straightforward statistical task. The real difficulty is to uncover security alphas and the macro-factor risk
premium, RP M .
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 337
Table 10.2 summarizes a rudimentary experiment that compares the prescriptions and predictions of the APT and T-B model in the presence of realistic values of residual risk. We use relatively small alpha values (1 and 3%), three levels of residual risk consistent with values in Table 10.1 (2, 3, and 4%), and two levels of beta (0.5 and 2) to span the likely range of reasonable parameters.
The first set of columns in Table 10.2 , titled Active Portfolio, show the parameter values in each example. The second set of columns, titled Zero-Net-Investment, Arbitrage (Zero-Beta), shows the weight in the active portfolio and resultant information ratio of the active portfolio. This would be the Sharpe ratio if the arbitrage position (the positive- alpha, zero-beta portfolio) made up the entire risky portfolio (as would be prescribed
Example 10.3 Exploiting Alpha
bound. Similarly, when the residual risk of an asset in the active T-B portfolio is zero, it
will displace all other assets from that portfolio, and thus the residual risk of the active
portfolio will be zero and elicit the same extreme portfolio response.
When residual risks are nonzero, the T-B procedure produces the optimal risky port-
folio, which is a compromise between seeking alpha and shunning potentially diversifi-
able risk. The APT ignores residual risk altogether, assuming it has been diversified away.
Obviously, we have no use for the APT in this context. When residual risk can be made
small through diversification, the T-B model prescribes very aggressive (large) positions
in mispriced securities that exert great pressure on equilibrium risk premiums to eliminate
nonzero alpha values. The T-B model does what the APT is meant to do but with more
flexibility in terms of accommodating the practical limits to diversification. In this sense,
Treynor and Black anticipated the development of the APT.
Index Risk Premium 5 7 Index SD 5 20 Index Sharpe Ratio 5 0.35
Active Portfolio
Zero-Net-Investment, Arbitrage (Zero-Beta)
Portfolio Treynor-Black Procedure
Alpha (%) Residual SD Beta w in Active Info Ratio w (beta 5 0) w (beta) Sharpe Ratio
Incremental Sharpe Ratio
1 4 0.5 2 0.25 3.57 1.28 0.43 0.18 1 4 2 1 0.25 3.57 1.00 0.43 0.18 1 3 0.5 2 0.33 6.35 1.52 0.48 0.15 1 3 2 1 0.33 6.35 1.00 0.48 0.15 1 2 0.5 2 0.50 14.29 1.75 0.61 0.11 1 2 2 1 0.50 14.29 1.00 0.61 0.11 3 4 0.5 2 0.75 10.71 1.69 0.83 0.08 3 4 2 1 0.75 10.71 1.00 0.83 0.08 3 3 0.5 2 1.00 19.05 1.81 1.06 0.06 3 3 2 1 1.00 19.05 1.00 1.06 0.06 3 2 0.5 2 1.50 42.86 1.91 1.54 0.04 3 2 2 1 1.50 42.86 1.00 1.54 0.04
Table 10.2
Performance of APT vs. Index Model when diversification of residual SD is incomplete
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338 P A R T I I I Equilibrium in Capital Markets
10.4 A Multifactor APT
We have assumed so far that only one systematic factor affects stock returns. This simpli-
fying assumption is in fact too simplistic. We’ve noted that it is easy to think of several
factors driven by the business cycle that might affect stock returns: interest rate fluctua-
tions, inflation rates, and so on. Presumably, exposure to any of these factors will affect a
stock’s risk and hence its expected return. We can derive a multifactor version of the APT
to accommodate these multiple sources of risk.
Suppose that we generalize the single-factor model expressed in Equation 10.1 to a
two-factor model:
Ri 5 E(Ri) 1 bi1F1 1 bi2F2 1 ei (10.10)
In Example 10.2, factor 1 was the departure of GDP growth from expectations, and factor 2
was the unanticipated change in interest rates. Each factor has zero expected value because
each measures the surprise in the systematic variable rather than the level of the variable. Similarly, the firm-specific component of unexpected return, e i , also has zero expected value. Extending such a two-factor model to any number of factors is straightforward.
We can now generalize the simple APT to a more general multifactor version. But first
we must introduce the concept of a factor portfolio, which is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of zero on any other factor.
We can think of a factor portfolio as a tracking portfolio. That is, the returns on such a port- folio track the evolution of particular sources of macroeconomic risk but are uncorrelated
with other sources of risk. It is possible to form such factor portfolios because we have a
large number of securities to choose from, and a relatively small number of factors. Factor
portfolios will serve as the benchmark portfolios for a multifactor security market line.
The multidimensional SML predicts that exposure to each risk factor contributes to the
security’s total risk premium by an amount equal to the factor beta times the risk premium
of the factor portfolio tracking that source of risk. We illustrate with an example.
by the APT). The last set of columns shows the T-B position in the active portfolio that maximizes the Sharpe ratio of the overall risky portfolio. The final column shows the increment to the Sharpe ratio of the T-B portfolio relative to the APT portfolio.
Keep in mind that even when the two models call for a similar weight in the active portfolio (compare w in Active for the APT model to w(beta) for the T-B model), they nevertheless prescribe a different overall risky portfolio. The APT assumes zero invest- ment beyond what is necessary to hedge out the market risk of the active portfolio. In contrast, the T-B procedure chooses a mix of active and index portfolios to maximize the Sharpe ratio. With identical investment in the active portfolio, the T-B portfolio can still include additional investment in the index portfolio.
To obtain the Sharpe ratio of the risky portfolio, we need the Sharpe ratio of the index portfolio. As an estimate, we use the average return and standard deviation of the broad market index (NYSE 1 AMEX 1 NASDAQ) over the period 1926–2012. The top row (over the column titles) of Table 10.2 shows an annual Sharpe ratio of 0.35. The rows of the table are ordered by the information ratio of the active portfolio.
Table 10.2 shows that the T-B procedure noticeably improves the Sharpe ratio beyond the information ratio of the APT (for which the IR is also the Sharpe ratio). However, as the information ratio of the active portfolio increases, the difference in the T-B and APT active portfolio positions declines, as does the difference between their Sharpe ratios. Put differently, the higher the information ratio, the closer we are to a risk-free arbitrage opportunity, and the closer are the prescriptions of the APT and T-B models.
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 339
To generalize the argument in Example 10.4, note that the factor exposures of any port-
folio, P, are given by its betas, b P 1 and b P 2 . A competing portfolio, Q, can be formed by investing in factor portfolios with the following weights: b P 1 in the first factor portfolio,
b P 2 in the second factor portfolio, and 1 2 b P 1 2 b P 2 in T-bills. By construction, portfolio
Q will have betas equal to those of portfolio P and expected return of
E (rQ) 5 bP1E (r1) 1 bP2E (r2) 1 (1 2 bP1 2 bP2)rf
5 rf 1 bP1 3E(r1) 2 rf4 1 bP2 3E(r2) 2 rf4 (10.11) Using the numbers in Example 10.4:
E(rQ) 5 4 1 .5 3 (10 2 4) 1 .75 3 (12 2 4) 5 13%
Suppose that the two factor portfolios, portfolios 1 and 2, have expected returns E ( r 1 ) 5 10% and E ( r 2 ) 5 12%. Suppose further that the risk-free rate is 4%. The risk premium on the first factor portfolio is 10% 2 4% 5 6%, whereas that on the second factor portfolio is 12% 2 4% 5 8%.
Now consider a well-diversified portfolio, portfolio A, with beta on the first factor, b A 1 5 .5, and beta on the second factor, b A 2 5 .75. The multifactor APT states that the overall risk premium on this portfolio must equal the sum of the risk premiums required as compensation for each source of systematic risk. The risk premium attributable to risk factor 1 should be the portfolio’s exposure to factor 1, b A 1 , multiplied by the risk pre- mium earned on the first factor portfolio, E ( r 1 ) 2 r f . Therefore, the portion of portfolio A ’s risk premium that is compensation for its exposure to the first factor is b A 1 [ E ( r 1 ) 2 r f ] 5 .5(10% 2 4%) 5 3%, whereas the risk premium attributable to risk factor 2 is b A 2 [ E ( r 2 ) 2 r f ] 5 .75(12% 2 4%) 5 6%. The total risk premium on the portfolio should be 3% 1 6% 5 9% and the total return on the portfolio should be 4% 1 9% 5 13%.
Example 10.4 Multifactor SML
Suppose that the expected return on portfolio A from Example 10.4 were 12% rather than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from the factor portfolios with the same betas as portfolio A. This requires weights of .5 on the first factor portfolio, .75 on the second factor portfolio, and 2 .25 on the risk-free asset. This portfolio has exactly the same factor betas as portfolio A: It has a beta of .5 on the first factor because of its .5 weight on the first factor portfolio, and a beta of .75 on the second factor. (The weight of 2 .25 on risk-free T-bills does not affect the sensitiv- ity to either factor.)
Now invest $1 in portfolio Q and sell (short) $1 in portfolio A. Your net investment is zero, but your expected dollar profit is positive and equal to
$1 3 E(rQ) 2 $1 3 E (rA) 5 $1 3 .13 2 $1 3 .12 5 $.01
Moreover, your net position is riskless. Your exposure to each risk factor cancels out because you are long $1 in portfolio Q and short $1 in portfolio A, and both of these well-diversified portfolios have exactly the same factor betas. Thus, if portfolio A ’s expected return differs from that of portfolio Q ’s, you can earn positive risk-free profits on a zero-net-investment position. This is an arbitrage opportunity.
Example 10.5 Mispricing and Arbitrage
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340 P A R T I I I Equilibrium in Capital Markets
Because portfolio Q in Example 10.5 has precisely the same exposures as portfolio A to the two sources of risk, their expected returns also ought to be equal. So portfolio A also ought to have an expected return of 13%. If it does not, then there will be an arbitrage opportunity.
7
We conclude that any well-diversified portfolio with betas b P 1 and b P 2 must
have the return given in Equation 10.11 if arbitrage opportunities are to be precluded.
Equation 10.11 simply generalizes the one-factor SML.
Finally, the extension of the multifactor SML of Equation 10.11 to individual assets
is precisely the same as for the one-factor APT. Equation 10.11 cannot be satisfied by
every well-diversified portfolio unless it is satisfied approximately by individual securities.
Equation 10.11 thus represents the multifactor SML for an economy with multiple sources
of risk.
We pointed out earlier that one application of the CAPM is to provide “fair” rates
of return for regulated utilities. The multifactor APT can be used to the same ends. The
nearby box summarizes a study in which the
APT was applied to find the cost of capital
for regulated electric companies. Notice that
empirical estimates for interest rate and infla-
tion risk premiums in the box are negative, as
we argued was reasonable in our discussion
of Example 10.2.
Using the factor portfolios of Example 10.4, find the equilib- rium rate of return on a portfolio with b 1 5 .2 and b 2 5 1.4.
CONCEPT CHECK 10.3
7 The risk premium on portfolio A is 9% (more than the historical risk premium of the S&P 500) despite the fact
that its betas, which are both below 1, might seem defensive. This highlights another distinction between multi- factor and single-factor models. Whereas a beta greater than 1 in a single-factor market is aggressive, we cannot
say in advance what would be aggressive or defensive in a multifactor economy where risk premiums depend on
the sum of the contributions of several factors.
8 Eugene F. Fama and Kenneth R. French, “Multifactor Explanations of Asset Pricing Anomalies,” Journal of
Finance 51 (1996), pp. 55–84.
10.5 The Fama-French (FF) Three-Factor Model
The currently dominant approach to specifying factors as candidates for relevant sources
of systematic risk uses firm characteristics that seem on empirical grounds to proxy for
exposure to systematic risk. The factors chosen are variables that on past evidence seem to
predict average returns well and therefore may be capturing risk premiums. One example
of this approach is the Fama and French three-factor model and its variants, which have
come to dominate empirical research and industry applications: 8
Rit 5 ai 1 biMRMt 1 biSMBSMBt 1 biHMLHMLt 1 eit (10.12)
where
SMB 5 Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the
return on a portfolio of large stocks.
HML 5 High Minus Low, i.e., the return of a portfolio of stocks with a high
book-to-market ratio in excess of the return on a portfolio of stocks with a
low book-to-market ratio.
Note that in this model the market index does play a role and is expected to capture sys-
tematic risk originating from macroeconomic factors.
These two firm-characteristic variables are chosen because of long-standing observa-
tions that corporate capitalization (firm size) and book-to-market ratio predict deviations
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341
Using the APT to Find Cost of Capital
Elton, Gruber, and Mei * use the APT to derive the cost of capital for electric utilities. They assume that the relevant risk factors are unanticipated developments in the term structure of interest rates, the level of interest rates, infla- tion rates, the business cycle (measured by GDP), foreign exchange rates, and a summary measure they devise to measure other macro factors.
Their first step is to estimate the risk premium associ- ated with exposure to each risk source. They accomplish this in a two-step strategy (which we will describe in con- siderable detail in Chapter 13):
1. Estimate “factor loadings” (i.e., betas) of a large sample of firms. Regress returns of 100 randomly selected stocks against the systematic risk factors. They use a time-series regression for each stock (e.g., 60 months of data), therefore estimating 100 regressions, one for each stock.
2. Estimate the reward earned per unit of exposure to each risk factor. For each month, regress the return of each stock against the five betas estimated. The coefficient on each beta is the extra average return earned as beta increases, i.e., it is an estimate of the risk premium for that risk factor from that month’s data. These estimates are of course subject to sam- pling error. Therefore, average the risk premium esti- mates across the 12 months in each year. The average response of return to risk is less subject to sampling error.
The risk premiums are in the middle column of the table at the top of the next column.
Notice that some risk premiums are negative. The interpretation of this result is that risk premium should be positive for risk factors you don’t want exposure to, but negative for factors you do want exposure to. For example, you should desire securities that have higher returns when inflation increases and be willing to accept lower expected returns on such securities; this shows up as a negative risk premium.
Factor Factor Risk Premium
Factor Betas for Niagara Mohawk
Term structure .425 1.0615 Interest rates 2 .051 2 2.4167 Exchange rates 2 .049 1.3235 Business cycle .041 .1292 Infl ation 2 .069 2 .5220 Other macro factors .530 .3046
Therefore, the expected return on any security should be related to its factor betas as follows:
rf 1 .425 bterm struc 2 .051 bint rate 2.049 bex rate 1 .041 bbus cycle 2 .069 binflation 1 .530 bother
Finally, to obtain the cost of capital for a particular firm, the authors estimate the firm’s betas against each source of risk, multiply each factor beta by the “cost of factor risk” from the table above, sum over all risk sources to obtain the total risk premium, and add the risk-free rate.
For example, the beta estimates for Niagara Mohawk appear in the last column of the table above. Therefore, its cost of capital is
Cost of capital 5 rf 1 .425 3 1.0615 2 .051(22.4167) 2.049(1.3235) 1 .041(.1292) 2.069(2.5220) 1 .530(.3046)
5 rf 1 .72
In other words, the monthly cost of capital for Niagara Mohawk is .72% above the monthly risk-free rate. Its annu- alized risk premium is therefore .72% 3 12 5 8.64%.
*Edwin J. Elton, Martin J. Gruber, and Jianping Mei, “Cost of Capital Using Arbitrage Pricing Theory: A Case Study of Nine New York Utilities,” Financial Markets, Institutions, and Instruments 3 (August 1994), pp. 46–68.
W O
R D
S FR O
M T
H E ST
R E E T
of average stock returns from levels consistent with the CAPM. Fama and French jus-
tify this model on empirical grounds: While SMB and HML are not themselves obvious
candidates for relevant risk factors, the argument is that these variables may proxy for
yet-unknown more-fundamental variables. For example, Fama and French point out that
firms with high ratios of book-to-market value are more likely to be in financial distress
and that small stocks may be more sensitive to changes in business conditions. Thus, these
variables may capture sensitivity to risk factors in the macroeconomy. More evidence on
the Fama-French model appears in Chapter 13.
The problem with empirical approaches such as the Fama-French model, which use prox-
ies for extramarket sources of risk, is that none of the factors in the proposed models can
be clearly identified as hedging a significant source of uncertainty. Black 9 points out that
9 Fischer Black, “Beta and Return,” Journal of Portfolio Management 20 (1993), pp. 8–18.
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342 P A R T I I I Equilibrium in Capital Markets
when researchers scan and rescan the database of security returns in search of explanatory
factors (an activity often called data-snooping), they may eventually uncover past “patterns”
that are due purely to chance. Black observes that return premiums to factors such as firm
size have proven to be inconsistent since first discovered. However, Fama and French have
shown that size and book-to-market ratios have predicted average returns in various time
periods and in markets all over the world, thus mitigating potential effects of data-snooping.
The firm-characteristic basis of the Fama-French factors raises the question of whether
they reflect a multi-index ICAPM based on extra-market hedging demands or just repre-
sent yet-unexplained anomalies, where firm characteristics are correlated with alpha val-
ues. This is an important distinction for the debate over the proper interpretation of the
model, because the validity of FF-style models may signify either a deviation from rational
equilibrium (as there is no rational reason to prefer one or another of these firm character-
istics per se), or that firm characteristics identified as empirically associated with average
returns are correlated with other (yet unknown) risk factors.
The issue is still unresolved and is discussed in Chapter 13.
1. Multifactor models seek to improve the explanatory power of single-factor models by explicitly accounting for the various systematic components of security risk. These models use indicators
intended to capture a wide range of macroeconomic risk factors.
2. Once we allow for multiple risk factors, we conclude that the security market line also ought to be multidimensional, with exposure to each risk factor contributing to the total risk premium of the
security.
3. A (risk-free) arbitrage opportunity arises when two or more security prices enable investors to construct a zero-net-investment portfolio that will yield a sure profit. The presence of arbitrage
opportunities will generate a large volume of trades that puts pressure on security prices. This
pressure will continue until prices reach levels that preclude such arbitrage.
4. When securities are priced so that there are no risk-free arbitrage opportunities, we say that they satisfy the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are
important because we expect them to hold in real-world markets.
5. Portfolios are called “well-diversified” if they include a large number of securities and the invest- ment proportion in each is sufficiently small. The proportion of a security in a well-diversified
portfolio is small enough so that for all practical purposes a reasonable change in that security’s
rate of return will have a negligible effect on the portfolio’s rate of return.
6. In a single-factor security market, all well-diversified portfolios have to satisfy the expected return–beta relationship of the CAPM to satisfy the no-arbitrage condition. If all well-diversified
portfolios satisfy the expected return–beta relationship, then individual securities also must sat-
isfy this relationship, at least approximately.
7. The APT does not require the restrictive assumptions of the CAPM and its (unobservable) market portfolio. The price of this generality is that the APT does not guarantee this relationship for all
securities at all times.
8. A multifactor APT generalizes the single-factor model to accommodate several sources of sys- tematic risk. The multidimensional security market line predicts that exposure to each risk factor
contributes to the security’s total risk premium by an amount equal to the factor beta times the
risk premium of the factor portfolio that tracks that source of risk.
9. A multifactor extension of the single-factor CAPM, the ICAPM, is a model of the risk–return trade-off that predicts the same multidimensional security market line as the APT. The ICAPM
suggests that priced risk factors will be those sources of risk that lead to significant hedging
demand by a substantial fraction of investors.
SUMMARY
Related Web sites for this chapter are available at www. mhhe.com/bkm
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 343
single-factor model
multifactor model
factor loading
factor beta
arbitrage pricing theory
arbitrage
Law of One Price
risk arbitrage
well-diversified portfolio
factor portfolio KEY TERMS
Single factor model: R i 5 E ( R i ) 1 b 1 F 1 e i
Multifactor model (here, 2 factors, F 1 and F 2 ): R i 5 E ( R i ) 1 b 1 F 1 1 b 2 F 2 1 e i
Single-index model: R i 5 a i 1 b i R M 1 e i
Multifactor SML (here, 2 factors, labeled 1 and 2)
E(ri) 5 rf 1 b13E(r1) 2 rf4 1 b23E(r2) 2 rf4 5 rf 1 b1E(R1) 1 b2E(R2)
where the risk premiums on the two factor portfolios are E ( R 1 ) and E ( R 2 )
KEY EQUATIONS
1. Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial
production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a
beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12%. If industrial
production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised
estimate of the expected rate of return on the stock?
2. T he APT itself does not provide guidance concerning the factors that one might expect to deter-
mine risk premiums. How should researchers decide which factors to investigate? Why, for example,
is industrial production a reasonable factor to test for a risk premium?
3. If the APT is to be a useful theory, the number of systematic factors in the economy must be
small. Why?
4. S uppose that there are two independent economic factors, F 1 and F 2 . The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 45%. The
following are well-diversified portfolios:
Portfolio Beta on F 1 Beta on F 2 Expected Return
A 1.5 2.0 31% B 2.2 2 0.2 27%
What is the expected return–beta relationship in this economy?
5. Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio E ( r ) Beta
A 12% 1.2 F 6% 0.0
Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
6. Assume that both portfolios A and B are well diversified, that E ( r A ) 5 12%, and E ( r B ) 5 9%. If the economy has only one factor, and b A 5 1.2, whereas b B 5 .8, what must be the risk-
free rate?
Basic
Intermediate
PROBLEM SETS
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344 P A R T I I I Equilibrium in Capital Markets
7. Assume that stock market returns have the market index as a common factor, and that all stocks
in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard
deviation of 30%.
Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 1 2%, and
the other half an alpha of 2 2%. Suppose the analyst buys $1 million of an equally weighted
portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of
the negative alpha stocks.
a. What is the expected profit (in dollars) and standard deviation of the analyst’s profit? b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100
stocks?
8. Assume that security returns are generated by the single-index model,
Ri 5 ai 1 biRM 1 ei
where R i is the excess return for security i and R M is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the following data:
Security b i E ( R i ) s ( e i )
A 0.8 10% 25% B 1.0 12 10 C 1.2 14 20
a. If s M 5 20%, calculate the variance of returns of securities A, B, and C. b. Now assume that there are an infinite number of assets with return characteristics identical to
those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolio’s excess returns? What about portfolios
composed only of type B or C stocks? c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically.
9. The SML relationship states that the expected risk premium on a security in a one-factor model
must be directly proportional to the security’s beta. Suppose that this were not the case. For exam-
ple, suppose that expected return rises more than proportionately with beta as in the figure below.
B
C
A
E(r)
β
a. How could you construct an arbitrage portfolio? ( Hint: Consider combinations of portfolios A and B, and compare the resultant portfolio to C. )
b. Some researchers have examined the relationship between average returns on diversified port- folios and the b and b
2 of those portfolios. What should they have discovered about the effect
of b 2 on portfolio return?
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 345
10. Consider the following multifactor (APT) model of security returns for a particular stock.
Factor Factor Beta Factor Risk Premium
Inflation 1.2 6% Industrial production 0.5 8 Oil prices 0.3 3
a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the mar- ket views the stock as fairly priced.
b. Suppose that the market expected the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2. Calculate the revised expecta-
tions for the rate of return on the stock once the “surprises” become known.
Factor Expected Rate of Change Actual Rate of Change
Inflation 5% 4% Industrial production 3 6 Oil prices 2 0
11. Suppose that the market can be described by the following three sources of systematic risk with
associated risk premiums.
Factor Risk Premium
Industrial production ( I ) 6% Interest rates ( R ) 2 Consumer confidence ( C ) 4
The return on a particular stock is generated according to the following equation:
r 5 15% 1 1.0 I 1 .5R 1 .75C 1 e
Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6%. Is the stock
over- or underpriced? Explain.
12. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh
insights concerning the firm’s cost of capital. She decides that this would be a good opportunity
to try out the new material on the APT that she learned last semester. She decides that three
promising factors would be (i) the return on a broad-based index such as the S&P 500; (ii) the
level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and
(iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta
of Pork Products against each of these factors by using a multiple regression and to estimate the
risk premium associated with each exposure factor. Comment on Jennifer’s choice of factors.
Which are most promising with respect to the likely impact on her firm’s cost of capital? Can
you suggest improvements to her specification?
Use the following information to Answer Problems 13–16: Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The
only model that Orb’s senior management has promoted in the past is the capital asset pricing model
(CAPM). Now Orb’s management has asked one of its analysts, Kevin McCracken, CFA, to investi-
gate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitiv-
ity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk
premium for real GDP is 8% while the factor risk premium for inflation is 2%. He estimates for
Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively.
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346 P A R T I I I Equilibrium in Capital Markets
Using his APT results, he computes the expected return of the fund. For comparison purposes, he
then uses fundamental analysis to also compute the expected return of Orb’s High Growth Fund.
McCracken finds that the two estimates of the Orb High Growth Fund’s expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of
Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the
expected return is 8.5% above the risk-free rate. McCracken then applies the APT model to the Large
Cap Fund. He finds that the sensitivities to real GDP and inflation are .75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to compose a portfolio that has a unit
sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT
estimates for the High Growth Fund and the Large Cap Fund. He then computes the sensitivities for
a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken
will use his APT results for these three funds to accomplish the task of creating a portfolio with a
unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles
says such a GDP Fund would be good for clients who are retirees who live off the steady income of
their investments. McCracken says that the fund would be a good choice if upcoming supply side
macroeconomic policies of the government are successful.
13. According to the APT, if the risk-free rate is 4%, what should be McCracken’s estimate of the
expected return of Orb’s High Growth Fund?
14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information
Kwon provides, is an arbitrage opportunity available?
15. The GDP Fund composed from the other three funds would have a weight in Utility Fund equal to ( a ) 2 2.2; ( b ) 2 3.2; or ( c ) .3.
16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund
would be appropriate:
a. McCracken was correct and Stiles was wrong. b. Both were correct. c. Stiles was correct and McCracken was wrong.
17. A ssume a universe of n (large) securities for which the largest residual variance is not larger than nsM
2 . Construct as many different weighting schemes as you can that generate well-
diversified portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT
security market line:
a. For a single-factor market. b. For a multifactor market.
19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor.
a. Explain why. b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit
of the merged company. Would you expect the stock market behavior of the merged firm to
differ from that of a portfolio of the two previously independent firms? How does the merger
affect market capitalization? What is the prediction of the Fama-French model for the risk
premium on the combined firm? Do we see here a flaw in the FF model?
Challenge
1. J effrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced
securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing
CAPM and APT, the consultant made the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio. b. Neither the CAPM nor APT assumes normally distributed security returns. c. The CAPM assumes that one specific factor explains security returns but APT does not.
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C H A P T E R 1 0 Arbitrage Pricing Theory and Multifactor Models of Risk and Return 347
State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each incor-
rect argument, why the argument is incorrect.
2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%.
Portfolio Expected Return Beta
X 16% 1.00 Y 12 0.25
In this situation you would conclude that portfolios X and Y:
a. Are in equilibrium. b. Offer an arbitrage opportunity. c. Are both underpriced. d. Are both fairly priced.
3. A zero-investment portfolio with a positive alpha could arise if:
a. The expected return of the portfolio equals zero. b. The capital market line is tangent to the opportunity set. c. The Law of One Price remains unviolated. d. A risk-free arbitrage opportunity exists.
4. According to the theory of arbitrage:
a. High-beta stocks are consistently overpriced. b. Low-beta stocks are consistently overpriced. c. Positive alpha investment opportunities will quickly disappear. d. Rational investors will pursue arbitrage consistent with their risk tolerance.
5. The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing
model (CAPM) because the APT:
a. Places more emphasis on market risk. b. Minimizes the importance of diversification. c. Recognizes multiple unsystematic risk factors. d. Recognizes multiple systematic risk factors.
6. An investor takes as large a position as possible when an equilibrium price relationship is vio-
lated. This is an example of:
a. A dominance argument. b. The mean-variance efficient frontier. c. Arbitrage activity. d. The capital asset pricing model.
7. The feature of the general version of the arbitrage pricing theory (APT) that offers the greatest
potential advantage over the simple CAPM is the:
a. Identification of anticipated changes in production, inflation, and term structure of interest rates as key factors explaining the risk–return relationship.
b. Superior measurement of the risk-free rate of return over historical time periods. c. Variability of coefficients of sensitivity to the APT factors for a given asset over time. d. Use of several factors instead of a single market index to explain the risk–return relationship.
8. In contrast to the capital asset pricing model, arbitrage pricing theory:
a. Requires that markets be in equilibrium. b. Uses risk premiums based on micro variables. c. Specifies the number and identifies specific factors that determine expected returns. d. Does not require the restrictive assumptions concerning the market portfolio.
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348 P A R T I I I Equilibrium in Capital Markets
SOLUTIONS TO CONCEPT CHECKS
E-INVESTMENTS EXERCISES One of the factors in the APT model specified by Chen, Roll, and Ross is the percent change in unanticipated inflation. Who gains and who loses when inflation changes? Go to http://hussmanfunds.com/rsi/infsurprises.htm to see a graph of the Inflation Surprise Index and Economists’ Inflation Forecasts.
1. The GDP beta is 1.2 and GDP growth is 1% better than previously expected. So you will increase
your forecast for the stock return by 1.2 3 1% 5 1.2%. The revised forecast is for an 11.2%
return.
2. a. This portfolio is not well diversified. The weight on the first security does not decline as n increases. Regardless of how much diversification there is in the rest of the portfolio, you will
not shed the firm-specific risk of this security.
b. This portfolio is well diversified. Even though some stocks have three times the weight of other stocks (1.5/ n versus .5/ n ), the weight on all stocks approaches zero as n increases. The impact of any individual stock’s firm-specific risk will approach zero as n becomes ever larger.
3. The equilibrium return is E ( r ) 5 r f 1 b P 1 [ E ( r 1 ) 2 r f ] 1 b P 2[ E ( r 2 ) 2 r f ]. Using the data in Example 10.4:
E(r) 5 4 1 .2 3 (10 2 4) 1 1.4 3 (12 2 4) 5 16.4%
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111 PA
R T
III
ONE OF THE early applications of comput- ers in economics in the 1950s was to analyze economic time series. Business cycle theo- rists felt that tracing the evolution of several economic variables over time would clarify and predict the progress of the economy through boom and bust periods. A natural candidate for analysis was the behavior of stock market prices over time. Assuming that stock prices reflect the prospects of the firm, recurrent patterns of peaks and troughs in economic performance ought to show up in those prices.
Maurice Kendall examined this proposi- tion in 1953. 1 He found to his great surprise that he could identify no predictable pat- terns in stock prices. Prices seemed to evolve randomly. They were as likely to go up as they were to go down on any particular day, regardless of past performance. The data pro- vided no way to predict price movements.
At first blush, Kendall’s results were dis- turbing to some financial economists. They seemed to imply that the stock market is dominated by erratic market psychology, or “animal spirits”—that it follows no logical rules. In short, the results appeared to con- firm the irrationality of the market. On fur- ther reflection, however, economists came to reverse their interpretation of Kendall’s study.
It soon became apparent that random price movements indicated a well-functioning or efficient market, not an irrational one. In this chapter we explore the reasoning behind what may seem a surprising conclu- sion. We show how competition among ana- lysts leads naturally to market efficiency, and we examine the implications of the efficient market hypothesis for investment policy. We also consider empirical evidence that sup- ports and contradicts the notion of market efficiency.
The Efficient Market
Hypothesis
CHAPTER ELEVEN
1 Maurice Kendall, “The Analysis of Economic Time Series, Part I: Prices,” Journal of the Royal Statistical Society 96 (1953).
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350 P A R T I I I Equilibrium in Capital Markets
11.1 Random Walks and the Efficient Market Hypothesis
Suppose Kendall had discovered that stock price changes are predictable. What a gold mine
this would have been. If they could use Kendall’s equations to predict stock prices, inves-
tors would reap unending profits simply by purchasing stocks that the computer model
implied were about to increase in price and by selling those stocks about to fall in price.
A moment’s reflection should be enough to convince yourself that this situation could
not persist for long. For example, suppose that the model predicts with great confidence
that XYZ stock price, currently at $100 per share, will rise dramatically in 3 days to $110.
What would all investors with access to the model’s prediction do today? Obviously, they
would place a great wave of immediate buy orders to cash in on the prospective increase in
stock price. No one holding XYZ, however, would be willing to sell. The net effect would
be an immediate jump in the stock price to $110. The forecast of a future price increase will lead instead to an immediate price increase. In other words, the stock price will imme-
diately reflect the “good news” implicit in the model’s forecast.
This simple example illustrates why Kendall’s attempt to find recurrent patterns in
stock price movements was likely to fail. A forecast about favorable future performance leads instead to favorable current performance, as market participants all try to get in on the action before the price jump.
More generally, one might say that any information that could be used to predict stock
performance should already be reflected in stock prices. As soon as there is any informa-
tion indicating that a stock is underpriced and therefore offers a profit opportunity, inves-
tors flock to buy the stock and immediately bid up its price to a fair level, where only
ordinary rates of return can be expected. These “ordinary rates” are simply rates of return
commensurate with the risk of the stock.
However, if prices are bid immediately to fair levels, given all available information,
it must be that they increase or decrease only in response to new information. New infor-
mation, by definition, must be unpredictable; if it could be predicted, then the prediction
would be part of today’s information. Thus stock prices that change in response to new
(that is, previously unpredicted) information also must move unpredictably.
This is the essence of the argument that stock prices should follow a random walk, that is, that price changes should be random and unpredictable.
2 Far from a proof of market
irrationality, randomly evolving stock prices would be the necessary consequence of intel-
ligent investors competing to discover relevant information on which to buy or sell stocks
before the rest of the market becomes aware of that information.
Don’t confuse randomness in price changes with irrationality in the level of prices. If prices are determined rationally, then only new information will cause them to change.
Therefore, a random walk would be the natural result of prices that always reflect all current
knowledge. Indeed, if stock price movements were predictable, that would be damning evi-
dence of stock market inefficiency, because the ability to predict prices would indicate that
2 Actually, we are being a little loose with terminology here. Strictly speaking, we should characterize stock
prices as following a submartingale, meaning that the expected change in the price can be positive, presumably as
compensation for the time value of money and systematic risk. Moreover, the expected return may change over
time as risk factors change. A random walk is more restrictive in that it constrains successive stock returns to be
independent and identically distributed. Nevertheless, the term “random walk” is commonly used in the looser sense that price changes are essentially unpredictable. We will follow this convention.
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C H A P T E R 1 1 The Efficient Market Hypothesis 351
all available information was not already
reflected in stock prices. Therefore, the
notion that stocks already reflect all avail-
able information is referred to as the
efficient market hypothesis (EMH). 3 Figure 11.1 illustrates the response
of stock prices to new information in an
efficient market. The graph plots the price
response of a sample of firms that were
targets of takeover attempts. In most take-
overs, the acquiring firm pays a substan-
tial premium over current market prices.
Therefore, announcement of a takeover
attempt should cause the stock price to
jump. The figure shows that stock prices
jump dramatically on the day the news
becomes public. However, there is no
further drift in prices after the announce- ment date, suggesting that prices reflect
the new information, including the likely
magnitude of the takeover premium, by
the end of the trading day.
Even more dramatic evidence of
rapid response to new information may
be found in intraday prices. For exam-
ple, Patell and Wolfson show that most
of the stock price response to corporate
dividend or earnings announcements occurs within 10 minutes of the announcement. 4
A nice illustration of such rapid adjustment is provided in a study by Busse and Green,
who track minute-by-minute stock prices of firms that are featured on CNBC’s “Morning”
or “Midday Call” segments. 5 Minute 0 in Figure 11.2 is the time at which the stock is
mentioned on the midday show. The top line is the average price movement of stocks
that receive positive reports, while the bottom line reports returns on stocks with negative
reports. Notice that the top line levels off, indicating that the market has fully digested the
news within 5 minutes of the report. The bottom line levels off within about 12 minutes.
Competition as the Source of Efficiency Why should we expect stock prices to reflect “all available information”? After all, if you
are willing to spend time and money on gathering information, it might seem reasonable
that you could turn up something that has been overlooked by the rest of the investment
community. When information is costly to uncover and analyze, one would expect invest-
ment analysis calling for such expenditures to result in an increased expected return.
3 Market efficiency should not be confused with the idea of efficient portfolios introduced in Chapter 7. An infor-
mationally efficient market is one in which information is rapidly disseminated and reflected in prices. An efficient portfolio is one with the highest expected return for a given level of risk.
4 J. M. Patell and M. A. Wolfson, “The Intraday Speed of Adjustment of Stock Prices to Earnings and Dividend
Announcements,” Journal of Financial Economics 13 (June 1984), pp. 223–52. 5 J. A. Busse and T. C. Green, “Market Efficiency in Real Time,” Journal of Financial Economics 65 (2002),
pp. 415–37. You can find an intraday movie version of this figure at www.bus.emory.edu/cgreen/docs/cnbc/ cnbc.html.
36
32
28
24
20
16
12
8
4
0
−4
−8
−12
−16 −135 −120 −105 −90 −75 −60 −45 −30 −15 0 15 30
Days Relative to Announcement Date C
u m
u la
ti ve
A b
n o
rm a l R
e tu
rn (
% )
Figure 11.1 Cumulative abnormal returns before takeover attempts: target companies
Source: Arthur Keown and John Pinkerton, “Merger Announcements and Insider Trading Activity,” Journal of Finance 36 (September 1981). Used with permission of John Wiley and Sons, via Copyright Clearance Center. Updates courtesy of Jinghua Yan.
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352 P A R T I I I Equilibrium in Capital Markets
This point has been stressed by
Grossman and Stiglitz. 6 They argued
that investors will have an incen-
tive to spend time and resources to
analyze and uncover new informa-
tion only if such activity is likely to
generate higher investment returns.
Thus, in market equilibrium, effi-
cient information-gathering activ-
ity should be fruitful. Moreover,
it would not be surprising to find
that the degree of efficiency differs
across various markets. For example,
emerging markets that are less inten-
sively analyzed than U.S. markets
or in which accounting disclosure
requirements are less rigorous may
be less efficient than U.S. markets.
Small stocks that receive relatively
little coverage by Wall Street ana-
lysts may be less efficiently priced
than large ones. Still, while we
would not go so far as to say that you
absolutely cannot come up with new
information, it makes sense to con-
sider and respect your competition.
6 Sanford J. Grossman and Joseph E. Stiglitz, “On the Impossibility of Informationally Efficient Markets,” American
Economic Review 70 (June 1980).
−15 −10 −5 0 5 10 15 Minutes Relative to Mention
0.75
0.50
0.25
0.00
−0.25
−0.75
−0.50
−1.00
−1.25
−1.50
C u
m u
la ti
ve R
e tu
rn (
% )
Midday-Positive Midday-Negative
Figure 11.2 Stock price reaction to CNBC reports. The figure shows the reaction of stock prices to on-air stock reports during the “Midday Call” segment on CNBC. The chart plots cumulative returns beginning 15 minutes before the stock report.
Source: Reprinted from J. A. Busse and T. C. Green, “Market Efficiency in Real Time,” Journal of Financial Economics 65 (2002), p. 422. Copyright 2002, with permission from Elsevier.
Consider an investment management fund currently managing a $5 billion portfolio. Suppose that the fund manager can devise a research program that could increase the portfolio rate of return by one-tenth of 1% per year, a seemingly modest amount. This program would increase the dollar return to the portfolio by $5 billion 3 .001, or $5 million. Therefore, the fund would be willing to spend up to $5 million per year on research to increase stock returns by a mere tenth of 1% per year. With such large rewards for such small increases in investment performance, it should not be surpris- ing that professional portfolio managers are willing to spend large sums on industry analysts, computer support, and research effort, and therefore that price changes are, generally speaking, difficult to predict.
With so many well-backed analysts willing to spend considerable resources on research, easy pickings in the market are rare. Moreover, the incremental rates of return on research activity may be so small that only managers of the largest portfolios will find them worth pursuing.
Example 11.1 Rewards for Incremental Performance
Although it may not literally be true that “all” relevant information will be uncovered, it
is virtually certain that there are many investigators hot on the trail of most leads that seem
likely to improve investment performance. Competition among these many well-backed,
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Matchmakers for the Information Age
The most precious commodity on Wall Street is informa- tion, and informed players can charge handsomely for providing it. An industry of so-called expert network providers has emerged for selling access to experts with unique insights about a wide variety of firms and indus- tries to investors who need that information to make deci- sions. These firms have been dubbed matchmakers for the information age. Experts can range from doctors who help predict the release of blockbuster drugs to meteorologists who forecast weather that can affect commodity prices to business executives who can provide specialized insight about companies and industries.
But some of those experts have peddled prohibited inside information. In 2011, Winifred Jiau, a consultant for Primary Global Research, was convicted of selling informa- tion about Nvidia and Marvell Technologies to the hedge fund SAC Capital Advisors. Several other employees of Primary Global also were charged with insider trading.
Expert firms are supposed to provide only public infor- mation, along with the expert’s insights and perspective. But the temptation to hire experts with inside informa- tion and charge handsomely for access to them is obvious.
The SEC has raised concerns about the boundary between legitimate and illegal services, and several hedge funds in 2011 shut down after raids searched for evidence of such illicit activity.
In the wake of increased scrutiny, compliance efforts of both buyers and sellers of expert information have mushroomed. The largest network firm is Gerson Lehrman Group with a stable of 300,000 experts. It now maintains records down to the minute of which of its experts talks to whom and the topics they have discussed. 7 These records could be turned over to authorities in the event of an insider trading investigation. For their part, some hedge funds have simply ceased working with expert-network firms or have promulgated clearer rules for when their employees may talk with consultants.
Even with these safeguards, however, there remains room for trouble. For example, an investor may meet an expert through a legitimate network and then the two may establish a consulting relationship on their own. This legal matchmaking becomes the precursor to the illegal selling of insider tips. Where there is a will to cheat, there usually will be a way.
W O
R D
S FR O
M T
H E ST
R E E T
353
highly paid, aggressive analysts ensures that, as a general rule, stock prices ought to reflect
available information regarding their proper levels.
Information is often said to be the most precious commodity on Wall Street, and the
competition for it is intense. Sometimes the quest for a competitive advantage can tip over
into a search for illegal inside information. In 2011, Raj Rajaratnam, the head of the Galleon
Group hedge fund which once managed $6.5 billion, was convicted on insider trading
charges for soliciting tips from a network of corporate insiders and traders. Rajaratnam’s
was only one of several major insider trading cases working their way through the courts in
2011. While Galleon’s practices were egregious, drawing a clear line separating legitimate
and prohibited sources of information often can be difficult. For example, a large industry
of expert network firms has emerged in the last decade to connect (for a fee) investors to industry experts who can provide unique perspective on a company. As the nearby box dis-
cusses, this sort of arrangement can easily cross the line into insider trading.
Versions of the Efficient Market Hypothesis It is common to distinguish among three versions of the EMH: the weak, semistrong, and
strong forms of the hypothesis. These versions differ by their notions of what is meant by
the term “all available information.”
The weak-form hypothesis asserts that stock prices already reflect all information that can be derived by examining market trading data such as the history of past prices, trad-
ing volume, or short interest. This version of the hypothesis implies that trend analysis is
fruitless. Past stock price data are publicly available and virtually costless to obtain. The
weak-form hypothesis holds that if such data ever conveyed reliable signals about future
performance, all investors already would have learned to exploit the signals. Ultimately,
7 “Expert Networks Are the Matchmakers for the Information Age,” The Economist, June 16, 2011.
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354 P A R T I I I Equilibrium in Capital Markets
the signals lose their value as they become widely known because a buy signal, for instance,
would result in an immediate price increase.
The semistrong-form hypothesis states that all publicly available information regard- ing the prospects of a firm must be reflected already in the stock price. Such information
includes, in addition to past prices, fundamental data on the firm’s product line, quality
of management, balance sheet composition, patents held, earning forecasts, and account-
ing practices. Again, if investors have access to such information from publicly available
sources, one would expect it to be reflected in stock prices.
Finally, the strong-form version of the efficient market hypothesis states that stock prices reflect all information relevant to the firm, even including information available only to
company insiders. This version of the hypothesis is quite extreme. Few would argue with the
proposition that corporate officers have access to pertinent information long enough before
public release to enable them to profit from trading on that information. Indeed, much of
the activity of the Securities and Exchange Commission is directed toward preventing insid-
ers from profiting by exploiting their privileged situation. Rule 10b-5 of the Security Exchange
Act of 1934 sets limits on trading by corporate officers, directors, and substantial owners,
requiring them to report trades to the SEC. These insiders, their relatives, and any associates
who trade on information supplied by insiders are considered in violation of the law.
Defining insider trading is not always easy, however. After all, stock analysts are in
the business of uncovering information not already widely known to market participants.
As we saw in Chapter 3 as well as in the nearby box, the distinction between private and
inside information is sometimes murky.
Notice one thing that all versions of the EMH have in common: They all assert that
prices should reflect available information. We do not expect traders to be superhuman or market prices to always be right. We will always wish for more information about a com-
pany’s prospects than will be available. Sometimes market prices will turn out in retrospect
to have been outrageously high, at other times absurdly low. The EMH asserts only that
at the given time, using current information, we cannot be sure if today’s prices will ulti-
mately prove themselves to have been too high or too low. If markets are rational, however,
we can expect them to be correct on average.
a. Suppose you observed that high-level managers make superior returns on investments in their company’s stock. Would this be a violation of weak-form market efficiency? Would it be a violation of strong-form market efficiency?
b. If the weak form of the efficient market hypothesis is valid, must the strong form also hold? Conversely, does strong-form efficiency imply weak-form efficiency?
CONCEPT CHECK 11.1
Technical Analysis Technical analysis is essentially the search for recurrent and predictable patterns in stock prices. Although technicians recognize the value of information regarding future economic
prospects of the firm, they believe that such information is not necessary for a successful
trading strategy. This is because whatever the fundamental reason for a change in stock
price, if the stock price responds slowly enough, the analyst will be able to identify a trend
11.2 Implications of the EMH
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C H A P T E R 1 1 The Efficient Market Hypothesis 355
that can be exploited during the adjustment period. The key to successful technical analy-
sis is a sluggish response of stock prices to fundamental supply-and-demand factors. This
prerequisite, of course, is diametrically opposed to the notion of an efficient market.
Technical analysts are sometimes called chartists because they study records or charts of past stock prices, hoping to find patterns they can exploit to make a profit. As an example of
technical analysis, consider the relative strength approach. The chartist compares stock per- formance over a recent period to performance of the market or other stocks in the same indus-
try. A simple version of relative strength takes the ratio of the stock price to a market indicator
such as the S&P 500 index. If the ratio increases over time, the stock is said to exhibit relative
strength because its price performance is better than that of the broad market. Such strength
presumably may continue for a long enough period of time to offer profit opportunities.
One of the most commonly heard components of technical analysis is the notion of
resistance levels or support levels. These values are said to be price levels above which it is difficult for stock prices to rise, or below which it is unlikely for them to fall, and they
are believed to be levels determined by market psychology.
Consider stock XYZ, which traded for several months at a price of $72 and then declined to $65. If the stock eventually begins to increase in price, $72 is considered a resistance level (according to this theory) because investors who bought originally at $72 will be eager to sell their shares as soon as they can break even on their investment. Therefore, at prices near $72 a wave of selling pressure would exist. Such activity imparts a type of “memory” to the market that allows past price history to influence current stock prospects.
Example 11.2 Resistance Levels
The efficient market hypothesis implies that technical analysis is without merit. The past
history of prices and trading volume is publicly available at minimal cost. Therefore, any
information that was ever available from analyzing past prices has already been reflected
in stock prices. As investors compete to exploit their common knowledge of a stock’s price
history, they necessarily drive stock prices to levels where expected rates of return are
exactly commensurate with risk. At those levels one cannot expect abnormal returns.
As an example of how this process works, consider what would happen if the market
believed that a level of $72 truly was a resistance level for stock XYZ in Example 11.2.
No one would be willing to purchase the stock at a price of $71.50, because it would have
almost no room to increase in price, but ample room to fall. However, if no one would
buy it at $71.50, then $71.50 would become a resistance level. But then, using a similar
analysis, no one would buy it at $71, or $70, and so on.
The notion of a resistance level is a logical conundrum.
Its simple resolution is the recognition that if the stock is
ever to sell at $71.50, investors must believe that the price can as easily increase as fall. The fact that investors are
willing to purchase (or even hold) the stock at $71.50 is
evidence of their belief that they can earn a fair expected
rate of return at that price. An interesting question is whether a technical rule that seems to work will continue
to work in the future once it becomes widely recognized. A clever analyst may occasion-
ally uncover a profitable trading rule, but the real test of efficient markets is whether the
rule itself becomes reflected in stock prices once its value is discovered. Once a useful
If everyone in the market believes in resistance levels, why do these beliefs not become self- fulfilling prophecies?
CONCEPT CHECK 11.2
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356 P A R T I I I Equilibrium in Capital Markets
technical rule (or price pattern) is discovered, it ought to be invalidated when the mass of
traders attempts to exploit it. In this sense, price patterns ought to be self-destructing. Thus the market dynamic is one of a continual search for profitable trading rules,
followed by destruction by overuse of those rules found to be successful, followed by more
searching for yet-undiscovered rules.
Fundamental Analysis Fundamental analysis uses earnings and dividend prospects of the firm, expectations of future interest rates, and risk evaluation of the firm to determine proper stock prices.
Ultimately, it represents an attempt to determine the present discounted value of all the
payments a stockholder will receive from each share of stock. If that value exceeds the
stock price, the fundamental analyst would recommend purchasing the stock.
Fundamental analysts usually start with a study of past earnings and an examination
of company balance sheets. They supplement this analysis with further detailed economic
analysis, ordinarily including an evaluation of the quality of the firm’s management, the
firm’s standing within its industry, and the prospects for the industry as a whole. The hope
is to attain insight into future performance of the firm that is not yet recognized by the rest
of the market. Chapters 17 through 19 provide a detailed discussion of the types of analy-
ses that underlie fundamental analysis.
Once again, the efficient market hypothesis predicts that most fundamental analysis also is doomed to failure. If the analyst relies on publicly available earnings and industry infor-
mation, his or her evaluation of the firm’s prospects is not likely to be significantly more
accurate than those of rival analysts. Many well-informed, well-financed firms conduct
such market research, and in the face of such competition it will be difficult to uncover data
not also available to other analysts. Only analysts with a unique insight will be rewarded.
Fundamental analysis is much more difficult than merely identifying well-run firms with
good prospects. Discovery of good firms does an investor no good in and of itself if the rest of
the market also knows those firms are good. If the knowledge is already public, the investor
will be forced to pay a high price for those firms and will not realize a superior rate of return.
The trick is not to identify firms that are good, but to find firms that are better than everyone else’s estimate. Similarly, poorly run firms can be great bargains if they are not
quite as bad as their stock prices suggest.
This is why fundamental analysis is difficult. It is not enough to do a good analysis of
a firm; you can make money only if your analysis is better than that of your competitors
because the market price will already reflect all commonly recognized information.
Active versus Passive Portfolio Management By now it is apparent that casual efforts to pick stocks are not likely to pay off. Competi-
tion among investors ensures that any easily implemented stock evaluation technique will
be used widely enough so that any insights derived will be reflected in stock prices. Only
serious analysis and uncommon techniques are likely to generate the differential insight necessary to yield trading profits.
Moreover, these techniques are economically feasible only for managers of large
portfolios. If you have only $100,000 to invest, even a 1% per year improvement in
performance generates only $1,000 per year, hardly enough to justify herculean efforts.
The billion-dollar manager, however, reaps extra income of $10 million annually from the
same 1% increment.
If small investors are not in a favored position to conduct active portfolio management,
what are their choices? The small investor probably is better off investing in mutual funds.
By pooling resources in this way, small investors can gain from economies of scale.
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C H A P T E R 1 1 The Efficient Market Hypothesis 357
More difficult decisions remain, though. Can investors be sure that even large mutual
funds have the ability or resources to uncover mispriced stocks? Furthermore, will any
mispricing be sufficiently large to repay the costs entailed in active portfolio management?
Proponents of the efficient market hypothesis believe that active management is largely
wasted effort and unlikely to justify the expenses incurred. Therefore, they advocate a
passive investment strategy that makes no attempt to outsmart the market. A passive strategy aims only at establishing a well-diversified portfolio of securities without attempt-
ing to find under- or overvalued stocks. Passive management is usually characterized by a
buy-and-hold strategy. Because the efficient market theory indicates that stock prices are
at fair levels, given all available information, it makes no sense to buy and sell securities
frequently, which generates large trading costs without increasing expected performance.
One common strategy for passive management is to create an index fund, which is a fund designed to replicate the performance of a broad-based index of stocks. For example,
Vanguard’s 500 Index Fund holds stocks in direct proportion to their weight in the Standard
& Poor’s 500 stock price index. The performance of the 500 Index Fund therefore repli-
cates the performance of the S&P 500. Investors in this fund obtain broad diversification
with relatively low management fees. The fees can be kept to a minimum because Vanguard
does not need to pay analysts to assess stock prospects and does not incur transaction costs
from high portfolio turnover. Indeed, while the typical annual charge for an actively man-
aged equity fund is around 1% of assets, the expense ratio of the 500 Index Fund is only
.17%. Vanguard’s 500 Index Fund is among the largest equity mutual funds with over $100
billion of assets in 2012, and about 15% of assets in equity funds are indexed.
Indexing need not be limited to the S&P 500, however. For example, some of the funds
offered by the Vanguard Group track the broader-based CRSP 8 index of the total U.S.
equity market, the Barclays U.S. Aggregate Bond Index, the CRSP index of small-
capitalization U.S. companies, and the Financial Times indexes of the European and Pacific Basin equity markets. Several other mutual fund complexes offer indexed portfolios, but
Vanguard dominates the retail market for indexed products.
Exchange-traded funds, or ETFs, are a close (and often lower-expense) alternative to
indexed mutual funds. As noted in Chapter 4, these are shares in diversified portfolios that
can be bought or sold just like shares of individual stock.
ETFs matching several broad stock market indexes such
as the S&P 500 or CRSP indexes and dozens of interna-
tional and industry stock indexes are available to inves-
tors who want to hold a diversified sector of a market
without attempting active security selection.
The Role of Portfolio Management in an Efficient Market If the market is efficient, why not pick stocks by throwing darts at The Wall Street Journal instead of trying rationally to choose a stock portfolio? This is a tempting conclusion to
draw from the notion that security prices are fairly set, but it is far too facile. There is a role
for rational portfolio management, even in perfectly efficient markets.
You have learned that a basic principle in portfolio selection is diversification. Even
if all stocks are priced fairly, each still poses firm-specific risk that can be eliminated
through diversification. Therefore, rational security selection, even in an efficient market,
calls for the selection of a well-diversified portfolio providing the systematic risk level
that the investor wants.
Rational investment policy also requires that tax considerations be reflected in secu-
rity choice. High-tax-bracket investors generally will not want the same securities that low
8 CRSP is the Center for Research in Security Prices at the University of Chicago.
What would happen to market efficiency if all investors attempted to follow a passive strategy?
CONCEPT CHECK 11.3
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358 P A R T I I I Equilibrium in Capital Markets
bracket investors find favorable. At an obvious level, high-bracket investors find it advanta-
geous to buy tax-exempt municipal bonds despite their relatively low pretax yields, whereas
those same bonds are unattractive to low-tax-bracket or tax-exempt investors. At a more sub-
tle level, high-bracket investors might want to tilt their portfolios in the direction of capital
gains as opposed to interest income, because capital gains are taxed less heavily and because
the option to defer the realization of capital gains income is more valuable the higher the
current tax bracket. Hence these investors may prefer stocks that yield low dividends yet
offer greater expected capital gains income. They also will be more attracted to investment
opportunities for which returns are sensitive to tax benefits, such as real estate ventures.
A third argument for rational portfolio management relates to the particular risk profile
of the investor. For example, a Toyota executive whose annual bonus depends on Toyota’s
profits generally should not invest additional amounts in auto stocks. To the extent that
his or her compensation already depends on Toyota’s well-being, the executive is already
overinvested in Toyota and should not exacerbate the lack of diversification. This lesson
was learned with considerable pain in September 2008 by Lehman Brothers employees
who were famously invested in their own firm when the company failed. Roughly 30% of
the shares in the firm were owned by its 24,000 employees, and their losses on those shares
totaled around $10 billion.
Investors of varying ages also might warrant different portfolio policies with regard
to risk bearing. For example, older investors who are essentially living off savings might
choose to avoid long-term bonds whose market values fluctuate dramatically with changes
in interest rates (discussed in Part Four). Because these investors are living off accumulated
savings, they require conservation of principal. In contrast, younger investors might be
more inclined toward long-term inflation-indexed bonds. The steady flow of real income
over long periods of time that is locked in with these bonds can be more important than
preservation of principal to those with long life expectancies.
In conclusion, there is a role for portfolio management even in an efficient market.
Investors’ optimal positions will vary according to factors such as age, tax bracket, risk
aversion, and employment. The role of the portfolio manager in an efficient market is to
tailor the portfolio to these needs, rather than to beat the market.
Resource Allocation We’ve focused so far on the investment implications of the efficient market hypothesis.
Deviations from efficiency may offer profit opportunities to better-informed traders at the
expense of less-informed ones.
However, deviations from informational efficiency would also result in a large cost that
will be borne by all citizens, namely, inefficient resource allocation. Recall that in a capital-
ist economy, investments in real assets such as plant, equipment, and know-how are guided in large part by the prices of financial assets. For example, if the value of telecommunica-
tion capacity reflected in stock market prices exceeds the cost of installing such capacity,
managers might justifiably conclude that telecom investments seem to have positive net
present value. In this manner, capital market prices guide allocation of real resources.
If markets were inefficient and securities commonly mispriced, then resources would
be systematically misallocated. Corporations with overpriced securities would be able to
obtain capital too cheaply, and corporations with undervalued securities might forgo invest-
ment opportunities because the cost of raising capital would be too high. Therefore, ineffi-
cient capital markets would diminish one of the most potent benefits of a market economy.
As an example of what can go wrong, consider the dot-com bubble of the late 1990s,
which sent a strong but, as it turned out, wildly overoptimistic signal about prospects for
Internet and telecommunication firms and ultimately led to substantial overinvestment in
those industries.
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C H A P T E R 1 1 The Efficient Market Hypothesis 359
Before writing off markets as a means to guide resource allocation, however, one has
to be reasonable about what can be expected from market forecasts. In particular, you
shouldn’t confuse an efficient market, where all available information is reflected in prices,
with a perfect-foresight market. As we said earlier, “all available information” is still far
from complete information, and generally rational market forecasts will sometimes be
wrong; sometimes, in fact, they will be very wrong.
The notion of informationally efficient markets leads to a powerful research methodology.
If security prices reflect all currently available information, then price changes must reflect
new information. Therefore, it seems that one should be able to measure the importance of
an event of interest by examining price changes during the period in which the event occurs.
An event study describes a technique of empirical financial research that enables an observer to assess the impact of a particular event on a firm’s stock price. A stock market
analyst might want to study the impact of dividend changes on stock prices, for example. An
event study would quantify the relationship between dividend changes and stock returns.
Analyzing the impact of any particular event is more difficult than it might at first appear.
On any day, stock prices respond to a wide range of economic news such as updated fore-
casts for GDP, inflation rates, interest rates, or corporate profitability. Isolating the part of a
stock price movement that is attributable to a specific event is not a trivial exercise.
The general approach starts with a proxy for what the stock’s return would have been
in the absence of the event. The abnormal return due to the event is estimated as the dif- ference between the stock’s actual return and this benchmark. Several methodologies for
estimating the benchmark return are used in practice. For example, a very simple approach
measures the stock’s abnormal return as its return minus that of a broad market index. An
obvious refinement is to compare the stock’s return to those of other stocks matched accord-
ing to criteria such as firm size, beta, recent performance, or ratio of price to book value per
share. Another approach estimates normal returns using an asset pricing model such as the
CAPM or one of its multifactor generalizations such as the Fama-French three-factor model.
Many researchers have used a “market model” to estimate abnormal returns. This
approach is based on the index models we introduced in Chapter 9. Recall that a single-
index model holds that stock returns are determined by a market factor and a firm-specific
factor. The stock return, r t , during a given period t, would be expressed mathematically as
rt 5 a 1 brMt 1 et (11.1)
where r Mt is the market’s rate of return during the period and e t is the part of a security’s return resulting from firm-specific events. The parameter b measures sensitivity to the market return, and a is the average rate of return the stock would realize in a period with a zero market return.
9 Equation 11.1 therefore provides a decomposition of r t into market
and firm-specific factors. The firm-specific or abnormal return may be interpreted as the
unexpected return that results from the event.
Determination of the abnormal return in a given period requires an estimate of e t . Therefore, we rewrite Equation 11.1:
et 5 rt 2 (a 2 brMt) (11.2)
11.3 Event Studies
9 We know from Chapter 9 that the CAPM implies that the intercept a in Equation 11.1 should equal r f (1 2 b ).
Nevertheless, it is customary to estimate the intercept in this equation empirically rather than imposing the CAPM
value. One justification for this practice is that empirically fitted security market lines seem flatter than predicted
by the CAPM (see Chapter 13), which would make the intercept implied by the CAPM too small.
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360 P A R T I I I Equilibrium in Capital Markets
Equation 11.2 has a simple interpretation: The residual, e t , that is, the component presum- ably due to the event in question, is the stock’s return over and above what one would predict
based on broad market movements in that period, given the stock’s sensitivity to the market.
The market model is a highly flexible tool, because it can be generalized to include
richer models of benchmark returns, for example, by including industry as well as broad
market returns on the right-hand side of Equation 11.1, or returns on indexes constructed
to match characteristics such as firm size. However, one must be careful that regression
parameters in Equation 11.1 (the intercept a and slope b ) are estimated properly. In par- ticular, they must be estimated using data sufficiently separated in time from the event in
question that they are not affected by event-period abnormal stock performance. In part
because of this vulnerability of the market model, returns on characteristic-matched port-
folios have become more widely used benchmarks in recent years.
Suppose that the analyst has estimated that a 5 .05% and b 5 .8. On a day that the market goes up by 1%, you would predict from Equation 11.1 that the stock should rise by an expected value of .05% 1 .8 3 1% 5 .85%. If the stock actually rises by 2%, the analyst would infer that firm-specific news that day caused an additional stock return of 2% 2 .85% 5 1.15%. This is the abnormal return for the day.
Example 11.3 Abnormal Returns
We measure the impact of an event by estimating the abnormal return on a stock (or
group of stocks) at the moment the information about the event becomes known to the
market. For example, in a study of the impact of merger attempts on the stock prices of tar-
get firms, the announcement date is the date on which the public is informed that a merger
is to be attempted. The abnormal returns of each firm surrounding the announcement date
are computed, and the statistical significance and magnitude of the typical abnormal return
are assessed to determine the impact of the newly released information.
One concern that complicates event studies arises from leakage of information. Leakage occurs when information regarding a relevant event is released to a small group of inves-
tors before official public release. In this case the stock price might start to increase (in the
case of a “good news” announcement) days or weeks before the official announcement
date. Any abnormal return on the announcement date is then a poor indicator of the total
impact of the information release. A better indicator would be the cumulative abnormal return, which is simply the sum of all abnormal returns over the time period of interest. The cumulative abnormal return thus captures the total firm-specific stock movement for
an entire period when the market might be responding to new information.
Figure 11.1 (earlier in the chapter) presents the results from a fairly typical event study.
The authors of this study were interested in leakage of information before merger announce-
ments and constructed a sample of 194 firms that were targets of takeover attempts. In
most takeovers, stockholders of the acquired firms sell their shares to the acquirer at sub-
stantial premiums over market value. Announcement of a takeover attempt is good news
for shareholders of the target firm and therefore should cause stock prices to jump.
Figure 11.1 confirms the good-news nature of the announcements. On the announce-
ment day, called day 0, the average cumulative abnormal return (CAR) for the sample of
takeover candidates increases substantially, indicating a large and positive abnormal return
on the announcement date. Notice that immediately after the announcement date the CAR no longer increases or decreases significantly. This is in accord with the efficient mar-
ket hypothesis. Once the new information became public, the stock prices jumped almost
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C H A P T E R 1 1 The Efficient Market Hypothesis 361
immediately in response to the good news. With prices once again fairly set, reflecting the
effect of the new information, further abnormal returns on any particular day are equally
likely to be positive or negative. In fact, for a sample of many firms, the average abnormal
return should be extremely close to zero, and thus the CAR will show neither upward nor
downward drift. This is precisely the pattern shown in Figure 11.1 .
The pattern of returns for the days preceding the public announcement date yields some
interesting evidence about efficient markets and information leakage. If insider trading
rules were perfectly obeyed and perfectly enforced, stock prices should show no abnormal
returns on days before the public release of relevant news, because no special firm-specific
information would be available to the market before public announcement. Instead, we
should observe a clean jump in the stock price only on the announcement day. In fact,
Figure 11.1 shows that the prices of the takeover targets clearly start an upward drift
30 days before the public announcement. It appears that information is leaking to some
market participants who then purchase the stocks before the public announcement. Such
evidence of leakage appears almost universally in event studies, suggesting at least some
abuse of insider trading rules.
Actually, the SEC also can take some comfort from patterns such as that in Figure 11.1 .
If insider trading rules were widely and flagrantly violated, we would expect to see abnor-
mal returns earlier than they appear in these results. For example, in the case of mergers,
the CAR would turn positive as soon as acquiring firms decided on their takeover targets,
because insiders would start trading immediately. By the time of the public announce-
ment, the insiders would have bid up the stock prices of target firms to levels reflecting the
merger attempt, and the abnormal returns on the actual public announcement date would
be close to zero. The dramatic increase in the CAR that we see on the announcement
date indicates that a good deal of these announcements are indeed news to the market
and that stock prices did not already reflect complete knowledge about the takeovers. It
would appear, therefore, that SEC enforcement does have a substantial effect on restricting
insider trading, even if some amount of it still persists.
Event study methodology has become a widely accepted tool to measure the economic
impact of a wide range of events. For example, the SEC regularly uses event studies to
measure illicit gains captured by traders who may have violated insider trading or other
securities laws. 10
Event studies are also used in fraud cases, where the courts must assess
damages caused by a fraudulent activity.
10
For a review of SEC applications of this technique, see Mark Mitchell and Jeffry Netter, “The Role of Financial
Economics in Securities Fraud Cases: Applications at the Securities and Exchange Commission,” The Business Lawyer 49 (February 1994), pp. 545–90.
Suppose the stock of a company with market value of $100 million falls by 4% on the day that news of an accounting scandal surfaces. The rest of the market, however, generally did well that day. The market indexes were up sharply, and on the basis of the usual relationship between the stock and the market, one would have expected a 2% gain on the stock. We would conclude that the impact of the scandal was a 6% drop in value, the difference between the 2% gain that we would have expected and the 4% drop actually observed. One might then infer that the damages sustained from the scan- dal were $6 million, because the value of the firm (after adjusting for general market movements) fell by 6% of $100 million when investors became aware of the news and reassessed the value of the stock.
Example 11.4 Using Abnormal Returns to Infer Damages
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362 P A R T I I I Equilibrium in Capital Markets
Suppose that we see negative abnormal returns (declining CARs) after an announce- ment date. Is this a violation of efficient markets?
CONCEPT CHECK 11.4
The Issues Not surprisingly, the efficient market hypothesis does not exactly arouse enthusiasm in the
community of professional portfolio managers. It implies that a great deal of the activity
of portfolio managers—the search for undervalued securities—is at best wasted effort,
and quite probably harmful to clients because it costs money and leads to imperfectly
diversified portfolios. Consequently, the EMH has never been widely accepted on Wall
Street, and debate continues today on the degree to which security analysis can improve
investment performance. Before discussing empirical tests of the hypothesis, we want to
note three factors that together imply that the debate probably never will be settled: the
magnitude issue, the selection bias issue, and the lucky event issue.
The Magnitude Issue We noted that an investment manager overseeing a $5 billion portfolio who can improve performance by only .1% per year will increase invest-
ment earnings by .001 3 $5 billion 5 $5 million annually. This manager clearly would
be worth her salary! Yet can we, as observers, statistically measure her contribution?
Probably not: A .1% contribution would be swamped by the yearly volatility of the
market. Remember, the annual standard deviation of the well-diversified S&P 500 index
has been around 20%. Against these fluctuations, a small increase in performance would
be hard to detect.
All might agree that stock prices are very close to fair values and that only managers of
large portfolios can earn enough trading profits to make the exploitation of minor mispric-
ing worth the effort. According to this view, the actions of intelligent investment managers
are the driving force behind the constant evolution of market prices to fair levels. Rather
than ask the qualitative question, Are markets efficient? we ought instead to ask a more
quantitative question: How efficient are markets?
The Selection Bias Issue Suppose that you discover an investment scheme that could really make money. You have two choices: either publish your technique in The Wall Street Journal to win fleeting fame, or keep your technique secret and use it to earn millions of dollars. Most investors would choose the latter option, which presents us with
a conundrum. Only investors who find that an investment scheme cannot generate abnor-
mal returns will be willing to report their findings to the whole world. Hence opponents
of the efficient markets’ view of the world always can use evidence that various techniques
do not provide investment rewards as proof that the techniques that do work simply are
not being reported to the public. This is a problem in selection bias; the outcomes we are able to observe have been preselected in favor of failed attempts. Therefore, we cannot
fairly evaluate the true ability of portfolio managers to generate winning stock market
strategies.
11.4 Are Markets Efficient?
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363
How to Guarantee a Successful Market Newsletter
Suppose you want to make your fortune publishing a mar- ket newsletter. You need first to convince potential sub- scribers that you have talent worth paying for. But what if you have no talent? The solution is simple: Start eight newsletters.
In year 1, let four of your newsletters predict an up- market and four a down-market. In year 2, let half of the originally optimistic group of newsletters continue to pre- dict an up-market and the other half a down-market. Do the same for the originally pessimistic group. Continue in this manner to obtain the pattern of predictions in the table that follows (U 5 prediction of an up-market, D 5 prediction of a down-market).
After 3 years, no matter what has happened to the market, one of the newsletters would have had a perfect prediction record. This is because after 3 years there are 2 3 5 8 outcomes for the market, and we have covered all eight possibilities with the eight newsletters. Now, we simply slough off the seven unsuccessful newsletters, and market the eighth newsletter based on its perfect track record. If we want to establish a newsletter with a per- fect track record over a 4-year period, we need 2 4 5 16
newsletters. A 5-year period requires 32 newsletters, and so on.
After the fact, the one newsletter that was always right will attract attention for your uncanny foresight and investors will rush to pay large fees for its advice. Your fortune is made, and you have never even researched the market!
WARNING: This scheme is illegal! The point, however, is that with hundreds of market newsletters, you can find one that has stumbled onto an apparently remarkable string of successful predictions without any real degree of skill. After the fact, someone’s prediction history can seem to imply great forecasting skill. This person is the one we will read about in The Wall Street Journal; the others will be forgotten.
Newsletter Predictions
Year 1 2 3 4 5 6 7 8
1 U U U U D D D D
2 U U D D U U D D
3 U D U D U D U D
W O
R D
S FR O
M T
H E ST
R E E T
The Lucky Event Issue In virtually any month it seems we read an article about some investor or investment company with a fantastic investment performance over the
recent past. Surely the superior records of such investors disprove the efficient market
hypothesis.
Yet this conclusion is far from obvious. As an analogy to the investment game, consider
a contest to flip the most number of heads out of 50 trials using a fair coin. The expected
outcome for any person is, of course, 50% heads and 50% tails. If 10,000 people, however,
compete in this contest, it would not be surprising if at least one or two contestants flipped
more than 75% heads. In fact, elementary statistics tells us that the expected number of
contestants flipping 75% or more heads would be two. It would be silly, though, to crown
these people the “head-flipping champions of the world.” Obviously, they are simply the
contestants who happened to get lucky on the day of the event. (See the nearby box.)
The analogy to efficient markets is clear. Under the hypothesis that any stock is fairly
priced given all available information, any bet on a stock is simply a coin toss. There is
equal likelihood of winning or losing the bet. However, if many investors using a variety
of schemes make fair bets, statistically speaking, some of those investors will be lucky and win a great majority of the bets. For every big winner, there may be many big losers, but we
never hear of these managers. The winners, though, turn up in The Wall Street Journal as the latest stock market gurus; then they can make a fortune publishing market newsletters.
Our point is that after the fact there will have been at least one successful investment
scheme. A doubter will call the results luck; the successful investor will call it skill. The
proper test would be to see whether the successful investors can repeat their performance
in another period, yet this approach is rarely taken.
With these caveats in mind, we turn now to some of the empirical tests of the efficient
market hypothesis.
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364 P A R T I I I Equilibrium in Capital Markets
Weak-Form Tests: Patterns in Stock Returns Returns over Short Horizons Early tests of efficient markets were tests of the weak form. Could speculators find trends in past prices that would enable them to earn
abnormal profits? This is essentially a test of the efficacy of technical analysis.
One way of discerning trends in stock prices is by measuring the serial correlation of stock market returns. Serial correlation refers to the tendency for stock returns to be
related to past returns. Positive serial correlation means that positive returns tend to fol-
low positive returns (a momentum type of property). Negative serial correlation means
that positive returns tend to be followed by negative returns (a reversal or “correction”
property). Both Conrad and Kaul 11
and Lo and MacKinlay 12
examine weekly returns of
NYSE stocks and find positive serial correlation over short horizons. However, the cor-
relation coefficients of weekly returns tend to be fairly small, at least for large stocks for
which price data are the most reliably up-to-date. Thus, while these studies demonstrate
weak price trends over short periods, 13
the evidence does not clearly suggest the existence
of trading opportunities.
While broad market indexes demonstrate only weak serial correlation, there appears to
be stronger momentum in performance across market sectors exhibiting the best and worst
recent returns. In an investigation of intermediate-horizon stock price behavior (using 3- to
12-month holding periods), Jegadeesh and Titman 14
found a momentum effect in which good or bad recent performance of particular stocks continues over time. They conclude
that while the performance of individual stocks is highly unpredictable, portfolios of the best-performing stocks in the recent past appear to outperform other stocks with enough
reliability to offer profit opportunities. Thus, it appears that there is evidence of short- to
intermediate-horizon price momentum in both the aggregate market and cross-sectionally
(i.e., across particular stocks).
Returns over Long Horizons Although studies of short- to intermediate-horizon returns have detected momentum in stock market prices, tests of long-horizon returns (i.e.,
returns over multiyear periods) have found suggestions of pronounced negative long-term
Legg Mason’s Value Trust, managed by Bill Miller, outperformed the S&P 500 in each of the 15 years end- ing in 2005. Is Miller’s performance sufficient to dissuade you from a belief in efficient markets? If not, would any performance record be sufficient to dissuade you? Now consider that in the next 3 years, the fund dramatically underperformed the S&P 500; by the end of 2008, its cumulative 18-year performance was barely different from the index. Does this affect your opinion?
CONCEPT CHECK 11.5
11
Jennifer Conrad and Gautam Kaul, “Time-Variation in Expected Returns,” Journal of Business 61 (October 1988), pp. 409–25.
12
Andrew W. Lo and A. Craig MacKinlay, “Stock Market Prices Do Not Follow Random Walks: Evidence from
a Simple Specification Test,” Review of Financial Studies 1 (1988), pp. 41–66.
13
On the other hand, there is evidence that share prices of individual securities (as opposed to broad market
indexes) are more prone to reversals than continuations at very short horizons. See, for example, B. Lehmann,
“Fads, Martingales and Market Efficiency,” Quarterly Journal of Economics 105 (February 1990), pp. 1–28; and N. Jegadeesh, “Evidence of Predictable Behavior of Security Returns,” Journal of Finance 45 (September 1990), pp. 881–98. However, as Lehmann notes, this is probably best interpreted as due to liquidity problems after big
movements in stock prices as market makers adjust their positions in the stock.
14
Narasimhan Jegadeesh and Sheridan Titman, “Returns to Buying Winners and Selling Losers: Implications for
Stock Market Efficiency,” Journal of Finance 48 (March 1993), pp. 65–91.
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C H A P T E R 1 1 The Efficient Market Hypothesis 365
15
Eugene F. Fama and Kenneth R. French, “Permanent and Temporary Components of Stock Prices,” Journal of Political Economy 96 (April 1988), pp. 24–73; James Poterba and Lawrence Summers, “Mean Reversion in Stock Prices: Evidence and Implications,” Journal of Financial Economics 22 (October 1988), pp. 27–59.
16
Werner F. M. DeBondt and Richard Thaler, “Does the Stock Market Overreact?” Journal of Finance 40 (1985), pp. 793–805.
17
Navin Chopra, Josef Lakonishok, and Jay R. Ritter, “Measuring Abnormal Performance: Do Stocks Overre-
act?” Journal of Financial Economics 31 (1992), pp. 235–68.
18
Eugene F. Fama and Kenneth R. French, “Dividend Yields and Expected Stock Returns,” Journal of Financial Economics 22 (October 1988), pp. 3–25.
serial correlation in the performance of the aggregate market. 15
The latter result has given
rise to a “fads hypothesis,” which asserts that the stock market might overreact to relevant
news. Such overreaction leads to positive serial correlation (momentum) over short time
horizons. Subsequent correction of the overreaction leads to poor performance following
good performance and vice versa. The corrections mean that a run of positive returns even-
tually will tend to be followed by negative returns, leading to negative serial correlation
over longer horizons. These episodes of apparent overshooting followed by correction give
the stock market the appearance of fluctuating around its fair value.
These long-horizon results are dramatic but still not conclusive. An alternative interpre-
tation of these results holds that they indicate only that the market risk premium varies over
time. For example, when the risk premium and the required return on the market rises, stock
prices will fall. When the market then rises (on average) at this higher rate of return, the data
convey the impression of a stock price recovery. The apparent overshooting and correction
are in fact no more than a rational response of market prices to changes in discount rates.
In addition to studies suggestive of overreaction in overall stock market returns over
long horizons, many other studies suggest that over long horizons, extreme performance in
particular securities also tends to reverse itself: The stocks that have performed best in the
recent past seem to underperform the rest of the market in following periods, while the worst
past performers tend to offer above-average future performance. DeBondt and Thaler 16
and
Chopra, Lakonishok, and Ritter 17
find strong tendencies for poorly performing stocks in one
period to experience sizable reversals over the subsequent period, while the best-performing
stocks in a given period tend to follow with poor performance in the following period.
For example, the DeBondt and Thaler study found that if one were to rank the perfor-
mance of stocks over a 5-year period and then group stocks into portfolios based on invest-
ment performance, the base-period “loser” portfolio (defined as the 35 stocks with the
worst investment performance) outperformed the “winner” portfolio (the top 35 stocks) by
an average of 25% (cumulative return) in the following 3-year period. This reversal effect, in which losers rebound and winners fade back, suggests that the stock market overreacts
to relevant news. After the overreaction is recognized, extreme investment performance is
reversed. This phenomenon would imply that a contrarian investment strategy—investing in recent losers and avoiding recent winners—should be profitable. Moreover, these returns
seem pronounced enough to be exploited profitably.
Thus it appears that there may be short-run momentum but long-run reversal patterns in
price behavior both for the market as a whole and across sectors of the market. One inter-
pretation of this pattern is that short-run overreaction (which causes momentum in prices)
may lead to long-term reversals (when the market recognizes its past error).
Predictors of Broad Market Returns Several studies have documented the ability of easily observed variables to predict mar-
ket returns. For example, Fama and French 18
showed that the return on the aggregate
stock market tends to be higher when the dividend/price ratio, the dividend yield, is high.
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366 P A R T I I I Equilibrium in Capital Markets
Campbell and Shiller 19
found that the earnings yield can predict market returns. Keim and
Stambaugh 20
showed that bond market data such as the spread between yields on high- and
low-grade corporate bonds also help predict broad market returns.
Again, the interpretation of these results is difficult. On the one hand, they may imply that
abnormal stock returns can be predicted, in violation of the efficient market hypothesis. More
probably, however, these variables are proxying for variation in the market risk premium. For
example, given a level of dividends or earnings, stock prices will be lower and dividend and
earnings yields will be higher when the risk premium (and therefore the expected market return)
is higher. Thus a high dividend or earnings yield will be associated with higher market returns.
This does not indicate a violation of market efficiency. The predictability of market returns is
due to predictability in the risk premium, not in risk-adjusted abnormal returns.
Fama and French 21
showed that the yield spread between high- and low-grade bonds
has greater predictive power for returns on low-grade bonds than for returns on high-grade
bonds, and greater predictive power for stock returns than for bond returns, suggesting that
the predictability in returns is in fact a risk premium rather than evidence of market inef-
ficiency. Similarly, the fact that the dividend yield on stocks helps to predict bond market
returns suggests that the yield captures a risk premium common to both markets rather
than mispricing in the equity market.
Semistrong Tests: Market Anomalies Fundamental analysis uses a much wider range of information to create portfolios than
does technical analysis. Investigations of the efficacy of fundamental analysis ask whether
publicly available information beyond the trading history of a security can be used to
improve investment performance, and therefore they are tests of semistrong-form market
efficiency. Surprisingly, several easily accessible statistics, for example, a stock’s price–
earnings ratio or its market capitalization, seem to predict abnormal risk-adjusted returns.
Findings such as these, which we will review in the following pages, are difficult to rec-
oncile with the efficient market hypothesis, and therefore are often referred to as efficient
market anomalies. A difficulty in interpreting these tests is that we usually need to adjust for portfolio risk
before evaluating the success of an investment strategy. Some tests, for example, have used
the CAPM to adjust for risk. However, we know that even if beta is a relevant descriptor
of stock risk, the empirically measured quantitative trade-off between risk as measured by
beta and expected return differs from the predictions of the CAPM. (We review this evi-
dence in Chapter 13.) If we use the CAPM to adjust portfolio returns for risk, inappropriate
adjustments may lead to the conclusion that various portfolio strategies can generate supe-
rior returns, when in fact it simply is the risk adjustment procedure that has failed.
Another way to put this is to note that tests of risk-adjusted returns are joint tests of the efficient market hypothesis and the risk adjustment procedure. If it appears that a portfolio strategy can generate superior returns, we must then choose between rejecting the EMH
and rejecting the risk adjustment technique. Usually, the risk adjustment technique is based
on more-questionable assumptions than is the EMH; by opting to reject the procedure, we
are left with no conclusion about market efficiency.
19
John Y. Campbell and Robert Shiller, “Stock Prices, Earnings and Expected Dividends,” Journal of Finance 43 (July 1988), pp. 661–76.
20
Donald B. Keim and Robert F. Stambaugh, “Predicting Returns in the Stock and Bond Markets,” Journal of Financial Economics 17 (1986), pp. 357–90.
21
Eugene F. Fama and Kenneth R. French, “Business Conditions and Expected Returns on Stocks and Bonds,”
Journal of Financial Economics 25 (November 1989), pp. 3–22.
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C H A P T E R 1 1 The Efficient Market Hypothesis 367
An example of this issue is the discovery by Basu 22
that portfolios of low price–earnings
(P/E) ratio stocks have provided higher returns than high P/E portfolios. The P/E effect holds up even if returns are adjusted for portfolio beta. Is this a confirmation that the market system-
atically misprices stocks according to P/E ratio? This would be an extremely surprising and, to
us, disturbing conclusion, because analysis of P/E ratios is such a simple procedure. Although
it may be possible to earn superior returns by using hard work and much insight, it hardly
seems plausible that such a simplistic technique is enough to generate abnormal returns.
Another interpretation of these results is that returns are not properly adjusted for risk.
If two firms have the same expected earnings, the riskier stock will sell at a lower price and
lower P/E ratio. Because of its higher risk, the low P/E stock also will have higher expected
returns. Therefore, unless the CAPM beta fully adjusts for risk, P/E will act as a useful
additional descriptor of risk, and will be associated with abnormal returns if the CAPM is
used to establish benchmark performance.
The Small-Firm-in-January Effect The so-called size or small-firm effect, originally documented by Banz,
23 is illustrated in Figure 11.3 . It shows the historical per-
formance of portfolios formed by dividing the NYSE stocks into 10 portfolios each year
according to firm size (i.e., the total value of outstanding equity). Average annual returns
between 1926 and 2011 are consistently higher on the small-firm portfolios. The differ-
ence in average annual return between portfolio 10 (with the largest firms) and portfolio 1
(with the smallest firms) is 8.52%. Of course, the smaller-firm portfolios tend to be riskier.
22 Sanjoy Basu, “The Investment Performance of Common Stocks in Relation to Their Price-Earnings Ratios: A
Test of the Efficient Market Hypothesis,” Journal of Finance 32 (June 1977), pp. 663–82; and “The Relationship between Earnings Yield, Market Value, and Return for NYSE Common Stocks: Further Evidence,” Journal of Financial Economics 12 (June 1983). 23
Rolf Banz, “The Relationship between Return and Market Value of Common Stocks,” Journal of Financial Economics 9 (March 1981).
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
A n
n u
a l R
e tu
rn (
% )
Size Decile: 1 = Smallest, 10 = Largest
19.38
16.64 16.35 15.70 15.01 14.91 14.40 13.37 12.71
10.86
Figure 11.3 Average annual return for 10 size-based portfolios, 1926–2011 Source: Authors’ calculations, using data obtained from Professor Ken French’s data library at http:// mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
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368 P A R T I I I Equilibrium in Capital Markets
But even when returns are adjusted for risk using the CAPM, there is still a consistent pre-
mium for the smaller-sized portfolios.
Imagine earning a premium of this size on a billion-dollar portfolio. Yet it is remarkable
that following a simple (even simplistic) rule such as “invest in low-capitalization stocks”
should enable an investor to earn excess returns. After all, any investor can measure firm
size at little cost. One would not expect such minimal effort to yield such large rewards.
Later studies (Keim, 24
Reinganum, 25
and Blume and Stambaugh 26
) showed that the
small-firm effect is concentrated in January, in fact, in the first 2 weeks of January. The
size effect is largely a “small-firm-in-January” effect.
The Neglected-Firm Effect and Liquidity Effects Arbel and Strebel 27 gave another interpretation of the small-firm-in-January effect. Because small firms tend to be
neglected by large institutional traders, information about smaller firms is less available.
This information deficiency makes smaller firms riskier investments that command higher
returns. “Brand-name” firms, after all, are subject to considerable monitoring from insti-
tutional investors, which promises high-quality information, and presumably investors do
not purchase “generic” stocks without the prospect of greater returns.
As evidence for the neglected-firm effect, Arbel 28 divided firms into highly researched, moderately researched, and neglected groups based on the number of institutions hold-
ing the stock. The January effect was in fact largest for the neglected firms. An article
by Merton 29
shows that neglected firms might be expected to earn higher equilibrium
returns as compensation for the risk associated with limited information. In this sense the
neglected-firm premium is not strictly a market inefficiency, but is a type of risk premium.
Work by Amihud and Mendelson 30
on the effect of liquidity on stock returns might be
related to both the small-firm and neglected-firm effects. As we noted in Chapter 9, inves-
tors will demand a rate-of-return premium to invest in less-liquid stocks that entail higher
trading costs. In accord with this hypothesis, Amihud and Mendelson showed that these
stocks have a strong tendency to exhibit abnormally high risk-adjusted rates of return.
Because small and less-analyzed stocks as a rule are less liquid, the liquidity effect might
be a partial explanation of their abnormal returns. However, this theory does not explain
why the abnormal returns of small firms should be concentrated in January. In any case,
exploiting these effects can be more difficult than it would appear. The high trading costs
on small stocks can easily wipe out any apparent abnormal profit opportunity.
Book-to-Market Ratios Fama and French 31 showed that a powerful predic- tor of returns across securities is the ratio of the book value of the firm’s equity to the
24
Donald B. Keim, “Size Related Anomalies and Stock Return Seasonality: Further Empirical Evidence,” Journal of Financial Economics 12 (June 1983).
25
Marc R. Reinganum, “The Anomalous Stock Market Behavior of Small Firms in January: Empirical Tests for
Tax-Loss Effects,” Journal of Financial Economics 12 (June 1983).
26
Marshall E. Blume and Robert F. Stambaugh, “Biases in Computed Returns: An Application to the Size Effect,”
Journal of Financial Economics, 1983.
27
Avner Arbel and Paul J. Strebel, “Pay Attention to Neglected Firms,” Journal of Portfolio Management, Winter 1983.
28
Avner Arbel, “Generic Stocks: An Old Product in a New Package,” Journal of Portfolio Management, Summer 1985.
29
Robert C. Merton, “A Simple Model of Capital Market Equilibrium with Incomplete Information,” Journal of Finance 42 (1987), pp. 483–510.
30
Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Econom- ics 17 (December 1986), pp. 223–50; and “Liquidity, Asset Prices, and Financial Policy,” Financial Analysts Journal 47 (November/December 1991), pp. 56–66.
31
Eugene F. Fama and Kenneth R. French, “The Cross Section of Expected Stock Returns,” Journal of Finance 47 (1992), pp. 427–65.
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C H A P T E R 1 1 The Efficient Market Hypothesis 369
market value of equity. Fama
and French stratified firms into
10 groups according to book-
to-market ratios and examined
the average monthly rate of
return of each of the 10 groups.
Figure 11.4 is an updated ver-
sion of their results. The decile
with the highest book-to-market
ratio had an average annual
return of 16.87%, while the
lowest-ratio decile averaged
only 10.92%. The dramatic
dependence of returns on book-
to-market ratio is independent
of beta, suggesting either that
high book-to-market ratio firms
are relatively underpriced, or
that the book-to-market ratio
is serving as a proxy for a risk
factor that affects equilibrium
expected returns.
In fact, Fama and French
found that after controlling for
the size and book-to-market effects, beta seemed to have no power to explain average security returns.
32 This finding is an important challenge to the notion of rational markets,
because it seems to imply that a factor that should affect returns—systematic risk—seems
not to matter, while a factor that should not matter—the book-to-market ratio—seems
capable of predicting future returns. We will return to the interpretation of this anomaly.
Post–Earnings-Announcement Price Drift A fundamental principle of effi- cient markets is that any new information ought to be reflected in stock prices very rapidly.
When good news is made public, for example, the stock price should jump immediately.
A puzzling anomaly, therefore, is the apparently sluggish response of stock prices to firms’
earnings announcements, as uncovered by Ball and Brown. 33
Their results were later con-
firmed and extended in many other papers. 34
32
However, a study by S. P. Kothari, Jay Shanken, and Richard G. Sloan, “Another Look at the Cross-Section of
Expected Stock Returns,” Journal of Finance 50 (March 1995), pp. 185–224, finds that when betas are estimated using annual rather than monthly returns, securities with high beta values do in fact have higher average returns.
Moreover, the authors find a book-to-market effect that is attenuated compared to the results in Fama and French
and furthermore is inconsistent across different samples of securities. They conclude that the empirical case for
the importance of the book-to-market ratio may be somewhat weaker than the Fama and French study would
suggest.
33
R. Ball and P. Brown, “An Empirical Evaluation of Accounting Income Numbers,” Journal of Accounting Research 9 (1968), pp. 159–78.
34
There is a voluminous literature on this phenomenon, often referred to as post–earnings-announcement price
drift. For papers that focus on why such drift may be observed, see V. Bernard and J. Thomas, “Evidence That
Stock Prices Do Not Fully Reflect the Implications of Current Earnings for Future Earnings,” Journal of Account- ing and Economics 13 (1990), pp. 305–40, or R. H. Battalio and R. Mendenhall, “Earnings Expectation, Inves- tor Trade Size, and Anomalous Returns Around Earnings Announcements,” Journal of Financial Economics 77 (2005), pp. 289–319.
18
16
14
12
10
8
6
4
2
0
Book-to-Market Decile: 1 = Lowest, 10 = Highest 1 2 3 4 5 6 7 8 9 10
A n
n u
a l M
o n
th ly
R e tu
rn (
% )
10.92 11.66 11.64 11.59
12.98 13.25 13.32
15.31 15.87
16.87
Figure 11.4 Average return as a function of book-to-market ratio, 1926–2011
Source: Authors’ calculations, using data obtained from Professor Ken French’s data library at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
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370 P A R T I I I Equilibrium in Capital Markets
The “news content” of an earn-
ings announcement can be evaluated
by comparing the announcement of
actual earnings to the value previ-
ously expected by market partici-
pants. The difference is the “earnings
surprise.” (Market expectations of
earnings can be roughly measured
by averaging the published earnings
forecasts of Wall Street analysts or
by applying trend analysis to past
earnings.) Rendleman, Jones, and
Latané 35
provide an influential study
of sluggish price response to earn-
ings announcements. They calculate
earnings surprises for a large sample
of firms, rank the magnitude of the
surprise, divide firms into 10 deciles
based on the size of the surprise, and
calculate abnormal returns for each
decile. Figure 11.5 plots cumulative
abnormal returns by decile.
Their results are dramatic. The
correlation between ranking by earn-
ings surprise and abnormal returns
across deciles is as predicted. There
is a large abnormal return (a jump
in cumulative abnormal return) on
the earnings announcement day
(time 0). The abnormal return is pos-
itive for positive-surprise firms and
negative for negative-surprise firms.
The more remarkable, and interesting, result of the study concerns stock price move-
ment after the announcement date. The cumulative abnormal returns of positive-surprise stocks continue to rise—in other words, exhibit momentum—even after the earnings
information becomes public, while the negative-surprise firms continue to suffer negative
abnormal returns. The market appears to adjust to the earnings information only gradually,
resulting in a sustained period of abnormal returns.
Evidently, one could have earned abnormal profits simply by waiting for earnings
announcements and purchasing a stock portfolio of positive-earnings-surprise companies.
These are precisely the types of predictable continuing trends that ought to be impossible
in an efficient market.
Strong-Form Tests: Inside Information It would not be surprising if insiders were able to make superior profits trading in their firm’s
stock. In other words, we do not expect markets to be strong-form efficient; we regulate and
35
Richard J. Rendleman Jr., Charles P. Jones, and Henry A. Latané, “Empirical Anomalies Based on Unexpected
Earnings and the Importance of Risk Adjustments,” Journal of Financial Economics 10 (November 1982), pp. 269–87.
10
8
6
4
2
0
−2
−4
−6
−8
−10
−20 −10 0 10 20 30 40 50 60 70 80 90
C u
m u
la ti
ve A
ve ra
g e E
xc e ss
R e tu
rn (
% )
Days from Earnings Announcement
10 9
8
7
6
5
4
3
2
1
Earnings Surprise Decile
Figure 11.5 Cumulative abnormal returns in response to earnings announcements
Source: Reprinted from R. J. Rendleman Jr., C. P. Jones, and H. A. Latané, “Empirical Anomalies Based on Unexpected Earnings and the Importance of Risk Adjustments,” Journal of Financial Economics 10 (1982), pp. 269–287. Copyright 1982, with permission from Elsevier.
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C H A P T E R 1 1 The Efficient Market Hypothesis 371
limit trades based on inside information. The ability of insiders to trade profitably in their
own stock has been documented in studies by Jaffe, 36
Seyhun, 37
Givoly and Palmon, 38
and
others. Jaffe’s was one of the earlier studies that documented the tendency for stock prices
to rise after insiders intensively bought shares and to fall after intensive insider sales.
Can other investors benefit by following insiders’ trades? The Securities and Exchange
Commission requires all insiders to register their trading activity and it publishes these
trades in an Official Summary of Security Transactions and Holdings. Since 2002, insiders must report large trades to the SEC within 2 business days. Once the Official Summary is published, the trades become public information. At that point, if markets are efficient,
fully and immediately processing the information released in the Official Summary of trad- ing, an investor should no longer be able to profit from following the pattern of those
trades. Several Internet sites contain information on insider trading. See the Web sites at
our Online Learning Center ( www.mhhe.com/bkm ) for some suggestions. The study by Seyhun, which carefully tracked the public release dates of the Official
Summary, found that following insider transactions would be to no avail. Although there is some tendency for stock prices to increase even after the Official Summary reports insider buying, the abnormal returns are not of sufficient magnitude to overcome transaction costs.
Interpreting the Anomalies How should we interpret the ever-growing anomalies literature? Does it imply that markets
are grossly inefficient, allowing for simplistic trading rules to offer large profit opportuni-
ties? Or are there other, more-subtle interpretations?
Risk Premiums or Inefficiencies? The price-earnings, small-firm, market-to- book, momentum, and long-term reversal effects are currently among the most puzzling
phenomena in empirical finance. There are several interpretations of these effects. First note
that to some extent, some of these phenomena may be related. The feature that small firms,
low-market-to-book firms, and recent “losers” seem to have in common is a stock price that
has fallen considerably in recent months or years. Indeed, a firm can become a small firm or
a low-market-to-book firm by suffering a sharp drop in price. These groups therefore may
contain a relatively high proportion of distressed firms that have suffered recent difficulties.
Fama and French 39
argue that these effects can be explained as manifestations of risk
premiums. Using their three-factor model, introduced in the previous chapter, they show
that stocks with higher betas (also known as factor loadings) on size or market-to-book
factors have higher average returns; they interpret these returns as evidence of a risk pre-
mium associated with the factor. This model does a much better job than the one-factor
CAPM in explaining security returns. While size or book-to-market ratios per se are obvi-
ously not risk factors, they perhaps might act as proxies for more fundamental deter-
minants of risk. Fama and French argue that these patterns of returns may therefore be
consistent with an efficient market in which expected returns are consistent with risk. In
this regard, it is worth noting that returns to “style portfolios,” for example, the return
on portfolios constructed based on the ratio of book-to-market value (specifically, the
36
Jeffrey F. Jaffe, “Special Information and Insider Trading,” Journal of Business 47 (July 1974).
37
H. Nejat Seyhun, “Insiders’ Profits, Costs of Trading and Market Efficiency,” Journal of Financial Economics 16 (1986).
38
Dan Givoly and Dan Palmon, “Insider Trading and Exploitation of Inside Information: Some Empirical Evi-
dence,” Journal of Business 58 (1985).
39
Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics 33 (1993), pp. 3–56.
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372 P A R T I I I Equilibrium in Capital Markets
Fama-French high minus low book-to-market portfolio) or firm size (the return on the
small-minus big-firm portfolio) do indeed seem to predict business cycles in many coun-
tries. Figure 11.6 shows that returns on these portfolios tend to have positive returns in
years prior to rapid growth in gross domestic product. We examine the Fama-French paper
in more detail in Chapter 13.
The opposite interpretation is offered by Lakonishok, Shleifer, and Vishny, 40
who argue
that these phenomena are evidence of inefficient markets, more specifically, of systematic
errors in the forecasts of stock analysts. They believe that analysts extrapolate past perfor-
mance too far into the future, and therefore overprice firms with recent good performance
and underprice firms with recent poor performance. Ultimately, when market participants
recognize their errors, prices reverse. This explanation is consistent with the reversal effect
and also, to a degree, is consistent with the small-firm and book-to-market effects because
firms with sharp price drops may tend to be small or have high book-to-market ratios.
If Lakonishok, Shleifer, and Vishny are correct, we ought to find that analysts system-
atically err when forecasting returns of recent “winner” versus “loser” firms. A study by
La Porta 41
is consistent with this pattern. He finds that equity of firms for which analysts
predict low growth rates of earnings actually perform better than those with high expected
35
30
25
20
15
10
5
0
−5
−10
−15
−20
P a st
Y e a r
R e tu
rn (
% )
A u
st ra
li a
C a n
a d
a
Fr a n
ce
G e rm
a n
y
It a ly
Ja p
a n
N e th
e rl
a n
d s
Sw it
ze rl
a n
d
U .K
.
U .S
.
HML SMB
Figure 11.6 Return to style portfolios as predictors of GDP growth. Average difference in the return on the style portfolio in years before good GDP growth versus in years with bad GDP growth. Positive value means the style portfolio does better in years prior to good macroeconomic performance. HML 5 high minus low portfolio, sorted on ratio of book-to-market value. SMB 5 small minus big portfolio, sorted on firm size.
Source: Reprinted from J. Liew and M. Vassalou, “Can Book-to-Market, Size, and Momentum Be Risk Factors That Predict Economic Growth?” Journal of Financial Economics 57 (2000), pp. 221–45. Copyright 2000, with permission from Elsevier.
40 Josef Lakonishok, Andrei Shleifer, and Robert W. Vishny, “Contrarian Investment, Extrapolation, and Risk,”
Journal of Finance 50 (1995), pp. 541–78. 41
Raphael La Porta, “Expectations and the Cross Section of Stock Returns,” Journal of Finance 51 (December 1996), pp. 1715–42.
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C H A P T E R 1 1 The Efficient Market Hypothesis 373
earnings growth. Analysts seem overly pessimistic about firms with low growth prospects
and overly optimistic about firms with high growth prospects. When these too-extreme
expectations are “corrected,” the low-expected-growth firms outperform high-expected-
growth firms.
Anomalies or Data Mining? We have covered many of the so-called anomalies cited in the literature, but our list could go on and on. Some wonder whether these anoma-
lies are really unexplained puzzles in financial markets, or whether they instead are an
artifact of data mining. After all, if one reruns the computer database of past returns over
and over and examines stock returns along enough dimensions, simple chance will cause
some criteria to appear to predict returns. In this regard, it is noteworthy that some anomalies have not shown much staying
power after being reported in the academic literature. For example, after the small-firm
effect was published in the early 1980s, it promptly disappeared for much of the rest of
the decade.
Still, even acknowledging the potential for data mining, a common thread seems to
run through many of the anomalies we have considered, lending support to the notion that
there is a real puzzle to explain. Value stocks—defined by low P/E ratio, high book-to-
market ratio, or depressed prices relative to historic levels—seem to have provided higher
average returns than “glamour” or growth stocks.
One way to address the problem of data mining is to find a dataset that has not already
been researched and see whether the relationship in question shows up in the new data.
Such studies have revealed size, momentum, and book-to-market effects in security mar-
kets around the world. While these phenomena may be a manifestation of a systematic risk
premium, the precise nature of that risk is not fully understood.
Anomalies over Time We pointed out earlier that while no market can be perfectly efficient, in well-functioning markets, anomalies ought to be self-destructing. As market
participants learn of profitable trading strategies, their attempts to exploit them should
move prices to levels at which abnormal profits are no longer available. Chordia, Sub-
ramanyam, and Tong 42
look for this dynamic in the pattern of many of the anomalies
discussed in this chapter. They focus on abnormal returns associated with several char-
acteristics including size, book-to-market ratio, momentum, and turnover (which may
be inversely related to the neglected firm effect). They break their sample at 1993 and
show that the abnormal returns associated with these characteristics in the pre-1993 period
largely disappear in the post-1993 period (with the notable exception of the book-to-
market effect). Their interpretation is that the market has become more efficient as knowl-
edge about these anomalies percolated through the investment community. Interestingly,
they find that that the attenuation of alphas is greatest in the most liquid stocks, where
trading activity is least costly.
McLean and Pontiff 43
provide further insight into this phenomenon. They identify more
than 80 characteristics recognized in the academic literature as associated with abnormal
returns. Rather than using a common break point for all characteristics, they carefully track
both the publication date of each finding as well as the date the paper was first posted to the
42
T. Chordia, A. Subrahmanyam, and Q. Tong, “Trends in the Cross-Section of Expected Stock Returns” (May 2,
2012). Available at SSRN: http://ssrn.com/abstract=2029057 or http://dx.doi.org/10.2139/ssrn.2029057.
43
David R. McLean and Jeffrey E. Pontiff, “Does Academic Research Destroy Stock Return Predictability?”
(October 3, 2012). AFFI/EUROFIDAI, Paris, December 2012 Finance Meetings Paper. Available at SSRN:
http://ssrn.com/abstract=2156623 or http://dx.doi.org/10.2139/ssrn.2156623.
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374 P A R T I I I Equilibrium in Capital Markets
Social Science Research Network. This allows them to break the sample for each finding
at dates corresponding to when that particular finding became public. They conclude that
the post-publication decay in abnormal return is about 35% (e.g., a 5% abnormal return
pre-publication falls on average to 3.25% after publication). 44
They show that trading vol-
ume and variance in stocks identified with anomalies increases, as does short interest in
“overpriced” stocks. These patterns are consistent with informed participants attempting to
exploit newly recognized mispricing. Moreover, the decay in alpha is most pronounced for
stocks that are larger, more liquid, and with low idiosyncratic risk. These are precisely the
stocks for which trading activity in pursuit of reliable abnormal returns is most feasible.
Thus, while abnormal returns do not fully disappear, these results are consistent with a
market groping its way toward greater efficiency over time.
Bubbles and Market Efficiency Every so often, asset prices seem (at least in retrospect) to lose their grounding in reality.
For example, in the tulip mania in 17th-century Holland, tulip prices peaked at several
times the annual income of a skilled worker. This episode has become the symbol of a
speculative “bubble” in which prices appear to depart from any semblance of intrinsic
value. Bubbles seem to arise when a rapid run-up in prices creates a widespread expecta-
tion that they will continue to rise. As more and more investors try to get in on the action,
they push prices even further. Inevitably, however, the run-up stalls and the bubble ends in
a crash.
Less than a century after tulip mania, the South Sea Bubble in England became almost
as famous. In this episode, the share price of the South Sea Company rose from £128 in
January 1720 to £550 in May, and peaked at around £1,000 in August—just before the
bubble burst and the share price collapsed to £150 in September, leading to widespread
bankruptcies among those who had borrowed to buy shares on credit. In fact, the company
was a major lender of money to investors willing to buy (and thus bid up) its shares. This
sequence may sound familiar to anyone who lived through the dot-com boom and bust of
1995–2002 45
or, more recently, the financial turmoil of 2008, with origins widely attrib-
uted to a collapsing housing price bubble.
It is hard to defend the position that security prices in these instances represented ratio-
nal, unbiased assessments of intrinsic value. And in fact, some economists, most notably
Hyman Minsky, have suggested that bubbles arise naturally. During periods of stability
and rising prices, investors extrapolate that stability into the future and become more will-
ing to take on risk. Risk premiums shrink, leading to further increases in asset prices, and
expectations become even more optimistic in a self-fulfilling cycle. But in the end, pricing
and risk taking become excessive and the bubble bursts. Ironically, the initial period of
stability fosters behavior that ultimately results in instability.
But beware of jumping to the conclusion that asset prices may generally be thought of
as arbitrary and obvious trading opportunities abundant. First, most bubbles become “obvi-
ous” only in retrospect. At the time, the price run-up often seems to have a defensible ratio-
nale. In the dot-com boom, for example, many contemporary observers rationalized stock
44
About a third of that decay occurs between the final date of the sample and the publication date, which the
authors note may reflect the portion of apparent abnormal returns that actually are due to data mining. The
remaining decay would then be attributable to the actions of sophisticated investors whose trades move anoma-
lous prices back toward intrinsic value.
45
The dot-com boom gave rise to the term irrational exuberance. In this vein, consider that one company going public in the investment boom of 1720 described itself simply as “a company for carrying out an undertaking of
great advantage, but nobody to know what it is.”
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C H A P T E R 1 1 The Efficient Market Hypothesis 375
price gains as justified by the prospect of a new and more profitable economy, driven by
technological advances. Even the irrationality of the tulip mania may have been overblown
in its later retelling. 46
In addition, security valuation is intrinsically difficult. Given the
considerable imprecision of estimates of intrinsic value, large bets on perceived mispricing
may entail hubris.
Moreover, even if you suspect that prices are in fact “wrong,” taking advantage of them
can be difficult. We explore these issues in more detail in the following chapter, but for
now, we simply point out some impediments to making aggressive bets against an asset,
among them, the costs of short selling overpriced securities as well as potential problems
obtaining the securities to sell short, and the possibility that even if you are ultimately
correct, the market may disagree and prices still can move dramatically against you in the
short term, thus wiping out your portfolio.
46
For interesting discussions of this possibility, see Peter Garber, Famous First Bubbles: The Fundamentals of Early Manias (Cambridge: MIT Press, 2000), and Anne Goldgar, Tulipmania: Money, Honor, and Knowledge in the Dutch Golden Age (Chicago: University of Chicago Press, 2007).
47
This problem may be less severe in the future; one recent reform intended to mitigate the conflict of interest in
having brokerage firms that sell stocks also provide investment advice is to separate analyst coverage from the
other activities of the firm.
48
B. Barber, R. Lehavy, M. McNichols, and B. Trueman, “Can Investors Profit from the Prophets? Security Ana-
lyst Recommendations and Stock Returns,” Journal of Finance 56 (April 2001), pp. 531–63.
49
K. L. Womack, “Do Brokerage Analysts’ Recommendations Have Investment Value?” Journal of Finance 51 (March 1996), pp. 137–67.
We have documented some of the apparent chinks in the armor of efficient market propo-
nents. For investors, the issue of market efficiency boils down to whether skilled investors
can make consistent abnormal trading profits. The best test is to look at the performance
of market professionals to see if they can generate performance superior to that of a pas-
sive index fund that buys and holds the market. We will look at two facets of professional
performance: that of stock market analysts who recommend investment positions and that
of mutual fund managers who actually manage portfolios.
Stock Market Analysts Stock market analysts historically have worked for brokerage firms, which presents an
immediate problem in interpreting the value of their advice: Analysts have tended to be
overwhelmingly positive in their assessment of the prospects of firms. 47
For example, on
a scale of 1 (strong buy) to 5 (strong sell), the average recommendation for 5,628 cov-
ered firms in 1996 was 2.04. 48
As a result, we cannot take positive recommendations
(e.g., to buy) at face value. Instead, we must look at either the relative enthusiasm of ana-
lyst recommendations compared to those for other firms, or at the change in consensus
recommendations.
Womack 49
focuses on changes in analysts’ recommendations and finds that positive
changes are associated with increased stock prices of about 5%, and negative changes
result in average price decreases of 11%. One might wonder whether these price changes
reflect the market’s recognition of analysts’ superior information or insight about firms
or, instead, simply result from new buy or sell pressure brought on by the recommenda-
tions themselves. Womack argues that price impact seems to be permanent, and therefore
11.5 Mutual Fund and Analyst Performance
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376 P A R T I I I Equilibrium in Capital Markets
consistent with the hypothesis that analysts do in fact reveal new information. Jegadeesh,
Kim, Krische, and Lee 50
also find that changes in consensus recommendations are associ-
ated with price changes, but that the level of consensus recommendations is an inconsistent predictor of future stock performance.
Barber, Lehavy, McNichols, and Trueman 51
focus on the level of consensus recom-
mendations and show that firms with the most-favorable recommendations outperform
those with the least-favorable recommendations. While their results seem impressive, the
authors note that portfolio strategies based on analyst consensus recommendations would
result in extremely heavy trading activity with associated costs that probably would wipe
out the potential profits from the strategy.
In sum, the literature suggests some value is added by analysts, but ambiguity remains.
Are superior returns following analyst upgrades due to revelation of new information or
due to changes in investor demand in response to the changed outlook? Also, are these
results exploitable by investors who necessarily incur trading costs?
Mutual Fund Managers As we pointed out in Chapter 4, casual evidence does not support the claim that profes-
sionally managed portfolios can consistently beat the market. Figure 4.2 in that chapter
demonstrated that between 1972 and 2011 the returns of a passive portfolio indexed to the
Wilshire 5000 typically would have been better than those of the average equity fund. On
the other hand, there was some (admittedly inconsistent) evidence of persistence in per-
formance, meaning that the better managers in one period tended to be better managers in
following periods. Such a pattern would suggest that the better managers can with some
consistency outperform their competitors, and it would be inconsistent with the notion that
market prices already reflect all relevant information.
The analyses cited in Chapter 4 were based on total returns; they did not properly adjust
returns for exposure to systematic risk factors. In this section we revisit the question of
mutual fund performance, paying more attention to the benchmark against which perfor-
mance ought to be evaluated.
As a first pass, we might examine the risk-adjusted returns (i.e., the alpha, or return in
excess of required return based on beta and the market-index return in each period) of a
large sample of mutual funds. But the market index may not be an adequate benchmark
against which to evaluate mutual fund returns. Because mutual funds tend to maintain con-
siderable holdings in equity of small firms, whereas the capitalization-weighted index is
dominated by large firms, mutual funds as a whole will tend to outperform the index when
small firms outperform large ones and underperform when small firms fare worse. Thus a
better benchmark for the performance of funds would be an index that separately incorpo-
rates the stock market performance of smaller firms.
The importance of the benchmark can be illustrated by examining the returns on small
stocks in various subperiods. 52
In the 20-year period between 1945 and 1964, for example,
a small-stock index underperformed the S&P 500 by about 4% per year (i.e., the alpha
of the small-stock index after adjusting for systematic risk was 24%). In the following
20-year period between 1965 and 1984, small stocks outperformed the S&P index by 10%.
50
N. Jegadeesh, J. Kim, S. D. Krische, and C. M. Lee, “Analyzing the Analysts: When Do Recommendations Add
Value?” Journal of Finance 59 (June 2004), pp. 1083–124.
51
Barber et al., op. cit.
52
This illustration and the statistics cited are based on E. J. Elton, M. J. Gruber, S. Das, and M. Hlavka, “Effi-
ciency with Costly Information: A Reinterpretation of Evidence from Managed Portfolios,” Review of Financial Studies 6 (1993), pp. 1–22, which is discussed shortly.
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C H A P T E R 1 1 The Efficient Market Hypothesis 377
Thus if one were to examine mutual fund returns in the earlier period, they would tend
to look poor, not necessarily because fund managers were poor stock pickers, but simply
because mutual funds as a group tended to hold more small stocks than were represented
in the S&P 500. In the later period, funds would look better on a risk-adjusted basis rela-
tive to the S&P 500 because small stocks performed better. The “style choice,” that is, the
exposure to small stocks (which is an asset allocation decision) would dominate the evalu-
ation of performance even though it has little to do with managers’ stock-picking ability. 53
The conventional performance benchmark today is a four-factor model, which employs
the three Fama-French factors (the return on the market index, and returns to portfolios
based on size and book-to-market ratio) augmented by a momentum factor (a portfolio
constructed based on prior-year stock return). Alphas constructed using an expanded index
model using these four factors control for a wide range of mutual fund style choices that
may affect average returns, for example, an inclination to growth versus value or small-
versus large-capitalization stocks. Figure 11.7 shows a frequency distribution of four-
factor alphas for U.S. domestic equity funds. 54
The results show that the distribution of
alpha is roughly bell shaped, with a slightly negative mean. On average, it does not appear
that these funds outperform their style-adjusted benchmarks.
Consistent with Figure 11.7 , Fama and French 55
use the four-factor model to assess the
performance of equity mutual funds and show that, while they may exhibit positive alphas
53 Remember that the asset allocation decision is usually in the hands of the individual investor. Investors allocate
their investment portfolios to funds in asset classes they desire to hold, and they can reasonably expect only that
mutual fund portfolio managers will choose stocks advantageously within those asset classes. 54
We are grateful to Professor Richard Evans for these data. 55
Eugene F. Fama, and Kenneth R. French. “Luck versus Skill in the Cross-Section of Mutual Fund Returns.”
Journal of Finance 65 (2010), pp. 1915–47.
Alpha (% per Month)
Fr e q
u e n
cy
20%
25%
15%
10%
5%
0%
−2.0 −1.67 −1.33 −1.0 −0.67 −0.33 0 0.33 0.67 1.0 1.33 1.67
Figure 11.7 Mutual fund alphas computed using a four-factor model of expected return, 1993–2007. (The best and worst 2.5% of observations are excluded from this distribution.)
Source: Professor Richard Evans, University of Virginia, Darden School of Business.
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378 P A R T I I I Equilibrium in Capital Markets
before fees, after the fees charged to their customers, alphas were negative. Likewise, Wermers,
56 who uses both style portfolios as well as the characteristics of the stocks held
by mutual funds to control for performance, also finds positive gross alphas but negative
net alphas after controlling for fees and risk.
Carhart 57
reexamines the issue of consistency in mutual fund performance and finds
that, after controlling for these factors, there is only minor persistence in relative perfor-
mance across managers. Moreover, much of that persistence seems due to expenses and
transactions costs rather than gross investment returns.
However, Bollen and Busse 58
do find evidence of performance persistence, at least over
short horizons. They rank mutual fund performance using the four-factor model over a
base quarter, assign funds into one of ten deciles according to base-period alpha, and then
look at performance in the following quarter. Figure 11.8 illustrates their results. The solid
line is the average alpha of funds within each of the deciles in the base period (expressed
on a quarterly basis). The steepness of that curve reflects the considerable dispersion in
performance in the ranking period. The dashed line is the average performance of the funds
in each decile in the following quarter. The shallowness of this curve indicates that most
of the original performance differential disappears. Nevertheless, the plot is still clearly
downward sloping, so it appears that, at least over a short horizon such as one quarter,
there is some performance consistency. However, that persistence is probably too small a
fraction of the original performance differential to justify performance chasing by mutual
fund customers.
Q u
a rt
e rl
y R
e tu
rn (
% )
−2
−4
−6
0
6
4
2
Performance Decile in Ranking Quarter
Ranking Quarter Post-Ranking Quarter
1 2 3 4 5 6 7 8 9 10
Figure 11.8 Risk-adjusted performance in ranking quarter and following quarter
56 R. R. Wermers, “Mutual Fund Performance: An Empirical Decomposition into Stock-Picking Talent, Style,
Transaction Costs, and Expenses.” Journal of Finance 55 (2000), pp. 1655–1703. 57
Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52 (1997), pp. 57–82. 58
Nicolas P. B. Bollen and Jeffrey A. Busse, “Short-Term Persistence in Mutual Fund Performance,” Review of Financial Studies 19 (2004), pp. 569–97.
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C H A P T E R 1 1 The Efficient Market Hypothesis 379
This pattern is actually consistent with the prediction of an influential paper by Berk
and Green. 59
They argue that skilled mutual fund managers with abnormal performance
will attract new funds until the additional costs and complexity of managing those extra
funds drive alphas down to zero. Thus, skill will show up not in superior returns, but rather
in the amount of funds under management. Therefore, even if managers are skilled, alphas
will be short-lived, as they seem to be in Figure 11.8 .
Del Guercio and Reuter 60
offer a finer interpretation of mutual fund performance and
the Berk and Green hypothesis. They split mutual fund investors into those who buy funds
directly for themselves versus those who purchase funds through brokers, reasoning that
the direct-sold segment may be more financially literate while the broker-sold segment is
less comfortable making financial decisions without professional advice. Consistent with
this hypothesis, they show that direct-sold investors direct their assets to funds with posi-
tive alphas (consistent with the Berk-Green model), but broker-sold investors generally do
not. This provides a greater incentive for direct-sold funds to invest relatively more in alpha-
generating inputs such as talented portfolio managers or analysts. Moreover, they show that
the after-fee performance of direct-sold funds is as good as that of index funds (again, consis-
tent with Berk-Green), while the performance of broker-sold funds is considerably worse. It
thus appears that the average underperformance of actively managed mutual funds is driven
largely by broker-sold funds and that this underperformance may be interpreted as an implicit
cost that less informed investors pay for the advice they get from their brokers.
In contrast to the extensive studies of equity fund managers, there have been few studies
of the performance of bond fund managers. Blake, Elton, and Gruber 61
examined the per-
formance of fixed-income mutual funds. They found that, on average, bond funds under-
perform passive fixed-income indexes by an amount roughly equal to expenses, and that
there is no evidence that past performance can predict future performance. More recently,
Chen, Ferson, and Peters (2010) find that on average, bond mutual funds outperform pas-
sive bond indexes in terms of gross returns but underperform once the fees they charge
their investors are subtracted, a result similar to those others have found for equity funds.
Thus the evidence on the risk-adjusted performance of professional managers is mixed
at best. We conclude that the performance of professional managers is broadly consistent
with market efficiency. The amounts by which professional managers as a group beat or
are beaten by the market fall within the margin of statistical uncertainty. In any event, it is
quite clear that performance superior to passive strategies is far from routine. Studies show
either that most managers cannot outperform passive strategies or that if there is a margin
of superiority, it is small.
On the other hand, a small number of investment superstars—Peter Lynch (formerly of
Fidelity’s Magellan Fund), Warren Buffett (of Berkshire Hathaway), John Templeton (of
Templeton Funds), and Mario Gabelli (of GAMCO) among them—have compiled career
records that show a consistency of superior performance hard to reconcile with abso-
lutely efficient markets. In a careful statistical analysis of mutual fund “stars,” Kosowski,
Timmerman, Wermers, and White 62
conclude that the stock-picking ability of a minority of
59
J. B. Berk and R. C. Green, “Mutual Fund Flows and Performance in Rational Markets,” Journal of Political Economy 112 (2004), pp. 1269–95.
60
Diane Del Guercio and Jonathan Reuter, “Mutual Fund Performance and the Incentive to Generate Alpha,”
Journal of Finance, forthcoming, 2013.
61
Christopher R. Blake, Edwin J. Elton, and Martin J. Gruber, “The Performance of Bond Mutual Funds,” Jour- nal of Business 66 (July 1993), pp. 371–404.
62
R. Kosowski, A. Timmerman, R. Wermers, and H. White. “Can Mutual Fund ‘Stars’ Really Pick Stocks? New
Evidence from a Bootstrap Analysis,” Journal of Finance 61 (December 2006), pp. 2551–95.
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380 P A R T I I I Equilibrium in Capital Markets
managers is sufficient to cover their costs, and that their superior performance tends to per-
sist over time. However, Nobel Prize–winner Paul Samuelson 63
reviewed this investment
hall of fame and pointed out that the records of the vast majority of professional money
managers offer convincing evidence that there are no easy strategies to guarantee success
in the securities markets.
So, Are Markets Efficient? There is a telling joke about two economists walking down the street. They spot a $20 bill
on the sidewalk. One stoops to pick it up, but the other one says, “Don’t bother; if the bill
were real someone would have picked it up already.”
The lesson is clear. An overly doctrinaire belief in efficient markets can paralyze the
investor and make it appear that no research effort can be justified. This extreme view is
probably unwarranted. There are enough anomalies in the empirical evidence to justify the
search for underpriced securities that clearly goes on.
The bulk of the evidence, however, suggests that any supposedly superior investment
strategy should be taken with many grains of salt. The market is competitive enough that only differentially superior information or insight will earn money; the easy pickings have
been picked. In the end it is likely that the margin of superiority that any professional man-
ager can add is so slight that the statistician will not easily be able to detect it.
We conclude that markets are generally very efficient, but that rewards to the especially
diligent, intelligent, or creative may in fact be waiting.
63
Paul Samuelson, “The Judgment of Economic Science on Rational Portfolio Management,” Journal of Portfolio Management 16 (Fall 1989), pp. 4–12.
1. Statistical research has shown that to a close approximation stock prices seem to follow a random walk with no discernible predictable patterns that investors can exploit. Such findings are now
taken to be evidence of market efficiency, that is, evidence that market prices reflect all cur-
rently available information. Only new information will move stock prices, and this information
is equally likely to be good news or bad news.
2. Market participants distinguish among three forms of the efficient market hypothesis. The weak form asserts that all information to be derived from past trading data already is reflected in stock
prices. The semistrong form claims that all publicly available information is already reflected.
The strong form, which generally is acknowledged to be extreme, asserts that all information,
including insider information, is reflected in prices.
3. Technical analysis focuses on stock price patterns and on proxies for buy or sell pressure in the market. Fundamental analysis focuses on the determinants of the underlying value of the firm,
such as current profitability and growth prospects. Because both types of analysis are based on
public information, neither should generate excess profits if markets are operating efficiently.
4. Proponents of the efficient market hypothesis often advocate passive as opposed to active invest- ment strategies. The policy of passive investors is to buy and hold a broad-based market index.
They expend resources neither on market research nor on frequent purchase and sale of stocks.
Passive strategies may be tailored to meet individual investor requirements.
5. Event studies are used to evaluate the economic impact of events of interest, using abnormal stock returns. Such studies usually show that there is some leakage of inside information to some
market participants before the public announcement date. Therefore, insiders do seem to be able
to exploit their access to information to at least a limited extent.
SUMMARY
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C H A P T E R 1 1 The Efficient Market Hypothesis 381
6. Empirical studies of technical analysis do not generally support the hypothesis that such analysis can generate superior trading profits. One notable exception to this conclusion is the apparent
success of momentum-based strategies over intermediate-term horizons.
7. Several anomalies regarding fundamental analysis have been uncovered. These include the P/E effect, the small-firm-in-January effect, the neglected-firm effect, post–earnings-announcement
price drift, and the book-to-market effect. Whether these anomalies represent market inefficiency
or poorly understood risk premiums is still a matter of debate.
8. By and large, the performance record of professionally managed funds lends little credence to claims that most professionals can consistently beat the market.
Related Web sites for this chapter are available at www. mhhe.com/bkm
random walk
efficient market hypothesis
weak-form EMH
semistrong-form EMH
strong-form EMH
technical analysis
resistance levels
support levels
fundamental analysis
passive investment strategy
index fund
event study
abnormal return
cumulative abnormal return
momentum effect
reversal effect
anomalies
P/E effect
small-firm effect
neglected-firm effect
book-to-market effect
KEY TERMS
KEY EQUATIONS Abnormal return 5 Actual return 2 Expected return given the return on a market index 5 rt 2 (a 1 brMt)
PROBLEM SETS 1. If markets are efficient, what should be the correlation coefficient between stock returns for two nonoverlapping time periods?
2. A successful firm like Microsoft has consistently generated large profits for years. Is this a viola-
tion of the EMH?
3. “If all securities are fairly priced, all must offer equal expected rates of return.” Comment.
4. Steady Growth Industries has never missed a dividend payment in its 94-year history. Does this
make it more attractive to you as a possible purchase for your stock portfolio?
5. At a cocktail party, your co-worker tells you that he has beaten the market for each of the last
3 years. Suppose you believe him. Does this shake your belief in efficient markets?
6. “Highly variable stock prices suggest that the market does not know how to price stocks.”
Comment.
7. Why are the following “effects” considered efficient market anomalies? Are there rational expla-
nations for any of these effects?
a. P/E effect. b. Book-to-market effect. c. Momentum effect. d. Small-firm effect.
8. If prices are as likely to increase as decrease, why do investors earn positive returns from the
market on average?
9. Which of the following most appears to contradict the proposition that the stock market is weakly efficient? Explain.
a. Over 25% of mutual funds outperform the market on average. b. Insiders earn abnormal trading profits. c. Every January, the stock market earns abnormal returns.
Basic
Intermediate
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382 P A R T I I I Equilibrium in Capital Markets
10. Which of the following sources of market inefficiency would be most easily exploited?
a. A stock price drops suddenly due to a large sale by an institution. b. A stock is overpriced because traders are restricted from short sales. c. Stocks are overvalued because investors are exuberant over increased productivity in the
economy.
11. Suppose that, after conducting an analysis of past stock prices, you come up with the following
observations. Which would appear to contradict the weak form of the efficient market hypoth- esis? Explain.
a. The average rate of return is significantly greater than zero. b. The correlation between the return during a given week and the return during the following
week is zero.
c. One could have made superior returns by buying stock after a 10% rise in price and selling after a 10% fall.
d. One could have made higher-than-average capital gains by holding stocks with low dividend yields.
12. Which of the following statements are true if the efficient market hypothesis holds?
a. It implies that future events can be forecast with perfect accuracy. b. It implies that prices reflect all available information. c. It implies that security prices change for no discernible reason. d. It implies that prices do not fluctuate.
13. Respond to each of the following comments.
a. If stock prices follow a random walk, then capital markets are little different from a casino. b. A good part of a company’s future prospects are predictable. Given this fact, stock prices
can’t possibly follow a random walk.
c. If markets are efficient, you might as well select your portfolio by throwing darts at the stock listings in The Wall Street Journal.
14. Which of the following would be a viable way to earn abnormally high trading profits if markets
are semistrong-form efficient?
a. Buy shares in companies with low P/E ratios. b. Buy shares in companies with recent above-average price changes. c. Buy shares in companies with recent below-average price changes. d. Buy shares in companies for which you have advance knowledge of an improvement in the
management team.
15. Suppose you find that prices of stocks before large dividend increases show on average consis-
tently positive abnormal returns. Is this a violation of the EMH?
16. “If the business cycle is predictable, and a stock has a positive beta, the stock’s returns also must
be predictable.” Respond.
17. Which of the following phenomena would be either consistent with or a violation of the effi-
cient market hypothesis? Explain briefly.
a. Nearly half of all professionally managed mutual funds are able to outperform the S&P 500 in a typical year.
b. Money managers that outperform the market (on a risk-adjusted basis) in one year are likely to outperform in the following year.
c. Stock prices tend to be predictably more volatile in January than in other months. d. Stock prices of companies that announce increased earnings in January tend to outperform
the market in February.
e. Stocks that perform well in one week perform poorly in the following week.
18. An index model regression applied to past monthly returns in Ford’s stock price produces the
following estimates, which are believed to be stable over time:
rF 5 .10% 1 1.1rM
If the market index subsequently rises by 8% and Ford’s stock price rises by 7%, what is the
abnormal change in Ford’s stock price?
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C H A P T E R 1 1 The Efficient Market Hypothesis 383
19. The monthly rate of return on T-bills is 1%. The market went up this month by 1.5%. In addi-
tion, AmbChaser, Inc., which has an equity beta of 2, surprisingly just won a lawsuit that awards
it $1 million immediately.
a. If the original value of AmbChaser equity were $100 million, what would you guess was the rate of return of its stock this month?
b. What is your answer to ( a ) if the market had expected AmbChaser to win $2 million?
20. In a recent closely contested lawsuit, Apex sued Bpex for patent infringement. The jury came
back today with its decision. The rate of return on Apex was r A 5 3.1%. The rate of return on Bpex was only r B 5 2.5%. The market today responded to very encouraging news about the unemployment rate, and r M 5 3%. The historical relationship between returns on these stocks and the market portfolio has been estimated from index model regressions as:
Apex: rA 5 .2% 1 1.4rM Bpex: rB 5 2.1% 1 .6rM
On the basis of these data, which company do you think won the lawsuit?
21. Investors expect the market rate of return in the coming year to be 12%. The T-bill rate is 4%. Changing Fortunes Industries’ stock has a beta of .5. The market value of its outstanding equity
is $100 million.
a. What is your best guess currently as to the expected rate of return on Changing Fortunes’ stock? You believe that the stock is fairly priced.
b. If the market return in the coming year actually turns out to be 10%, what is your best guess as to the rate of return that will be earned on Changing Fortunes’ stock?
c. Suppose now that Changing Fortunes wins a major lawsuit during the year. The settlement is $5 million. Changing Fortunes’ stock return during the year turns out to be 10%. What is your
best guess as to the settlement the market previously expected Changing Fortunes to receive from the lawsuit? (Continue to assume that the market return in the year turned out to be 10%.)
The magnitude of the settlement is the only unexpected firm-specific event during the year.
22. Dollar-cost averaging means that you buy equal dollar amounts of a stock every period, for
example, $500 per month. The strategy is based on the idea that when the stock price is low,
your fixed monthly purchase will buy more shares, and when the price is high, fewer shares.
Averaging over time, you will end up buying more shares when the stock is cheaper and fewer
when it is relatively expensive. Therefore, by design, you will exhibit good market timing. Eval-
uate this strategy.
23. We know that the market should respond positively to good news and that good-news events
such as the coming end of a recession can be predicted with at least some accuracy. Why, then,
can we not predict that the market will go up as the economy recovers?
24. You know that firm XYZ is very poorly run. On a scale of 1 (worst) to 10 (best), you would give
it a score of 3. The market consensus evaluation is that the management score is only 2. Should
you buy or sell the stock?
25. Suppose that during a certain week the Fed announces a new monetary growth policy, Congress
surprisingly passes legislation restricting imports of foreign automobiles, and Ford comes out
with a new car model that it believes will increase profits substantially. How might you go about
measuring the market’s assessment of Ford’s new model?
26. Good News, Inc., just announced an increase in its annual earnings, yet its stock price fell. Is
there a rational explanation for this phenomenon?
27. Shares of small firms with thinly traded stocks tend to show positive CAPM alphas. Is this a
violation of the efficient market hypothesis?
28. Examine the accompanying figure, 64
which presents cumulative abnormal returns both before
and after dates on which insiders buy or sell shares in their firms. How do you interpret this
figure? What are we to make of the pattern of CARs before and after the event date?
Challenge
64
Reprinted from Nejat H. Seyhun, “Insiders, Profits, Costs of Trading and Market Efficiency,” Journal of Financial Economics 16 (1986). Copyright 1986, with permission from Elsevier.
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384 P A R T I I I Equilibrium in Capital Markets
Event Day Relative to Insider Trading Day
C u
m u
la ti
ve D
a il y
A ve
ra g
e P
re d
ic ti
o n
E rr
o rs
( %
)
2200 2100 0 100 200 300
Sales
Purchases
22
21
0
1
2
3
4
29. Suppose that as the economy moves through a business cycle, risk premiums also change. For
example, in a recession when people are concerned about their jobs, risk tolerance might be
lower and risk premiums might be higher. In a booming economy, tolerance for risk might be
higher and premiums lower.
a. Would a predictably shifting risk premium such as described here be a violation of the effi- cient market hypothesis?
b. How might a cycle of increasing and decreasing risk premiums create an appearance that stock prices “overreact,” first falling excessively and then seeming to recover?
1. The semistrong form of the efficient market hypothesis asserts that stock prices:
a. Fully reflect all historical price information. b. Fully reflect all publicly available information. c. Fully reflect all relevant information, including insider information. d. May be predictable.
2. Assume that a company announces an unexpectedly large cash dividend to its shareholders. In an
efficient market without information leakage, one might expect:
a. An abnormal price change at the announcement. b. An abnormal price increase before the announcement. c. An abnormal price decrease after the announcement. d. No abnormal price change before or after the announcement.
3. Which one of the following would provide evidence against the semistrong form of the efficient market theory?
a. About 50% of pension funds outperform the market in any year. b. All investors have learned to exploit signals about future performance. c. Trend analysis is worthless in determining stock prices. d. Low P/E stocks tend to have positive abnormal returns over the long run.
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C H A P T E R 1 1 The Efficient Market Hypothesis 385
4. According to the efficient market hypothesis:
a. High-beta stocks are consistently overpriced. b. Low-beta stocks are consistently overpriced. c. Positive alphas on stocks will quickly disappear. d. Negative alpha stocks consistently yield low returns for arbitrageurs.
5. A “random walk” occurs when:
a. Stock price changes are random but predictable. b. Stock prices respond slowly to both new and old information. c. Future price changes are uncorrelated with past price changes. d. Past information is useful in predicting future prices.
6. Two basic assumptions of technical analysis are that security prices adjust:
a. Gradually to new information, and study of the economic environment provides an indica- tion of future market movements.
b. Rapidly to new information, and study of the economic environment provides an indication of future market movements.
c. Rapidly to new information, and market prices are determined by the interaction between supply and demand.
d. Gradually to new information, and prices are determined by the interaction between supply and demand.
7. When technical analysts say a stock has good “relative strength,” they mean:
a. The ratio of the price of the stock to a market or industry index has trended upward. b. The recent trading volume in the stock has exceeded the normal trading volume. c. The total return on the stock has exceeded the total return on T-bills. d. The stock has performed well recently compared to its past performance.
8. Your investment client asks for information concerning the benefits of active portfolio manage-
ment. She is particularly interested in the question of whether active managers can be expected
to consistently exploit inefficiencies in the capital markets to produce above-average returns
without assuming higher risk.
The semistrong form of the efficient market hypothesis asserts that all publicly available
information is rapidly and correctly reflected in securities prices. This implies that investors
cannot expect to derive above-average profits from purchases made after information has
become public because security prices already reflect the information’s full effects.
a. Identify and explain two examples of empirical evidence that tend to support the EMH implication stated above.
b. Identify and explain two examples of empirical evidence that tend to refute the EMH impli- cation stated above.
c. Discuss reasons why an investor might choose not to index even if the markets were, in fact, semistrong-form efficient.
9. a. Briefly explain the concept of the efficient market hypothesis (EMH) and each of its three forms—weak, semistrong, and strong—and briefly discuss the degree to which existing
empirical evidence supports each of the three forms of the EMH.
b. Briefly discuss the implications of the efficient market hypothesis for investment policy as it applies to:
i. Technical analysis in the form of charting.
ii. Fundamental analysis.
c. Briefly explain the roles or responsibilities of portfolio managers in an efficient market environment.
10. Growth and value can be defined in several ways. “Growth” usually conveys the idea of a
portfolio emphasizing or including only issues believed to possess above-average future rates
of per-share earnings growth. Low current yield, high price-to-book ratios, and high price-to-
earnings ratios are typical characteristics of such portfolios. “Value” usually conveys the idea of
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386 P A R T I I I Equilibrium in Capital Markets
portfolios emphasizing or including only issues currently showing low price-to-book ratios, low
price-to-earnings ratios, above-average levels of dividend yield, and market prices believed to
be below the issues’ intrinsic values.
a. Identify and provide reasons why, over an extended period of time, value-stock investing might outperform growth-stock investing.
b. Explain why the outcome suggested in ( a ) should not be possible in a market widely regarded as being highly efficient.
E-INVESTMENTS EXERCISES 1. Use data from finance.yahoo.com to answer the following questions:
a. Collect the following data for 25 fi rms of your choosing.
i. Book-to-market ratio. ii. Price–earnings ratio. iii. Market capitalization (size). iv. Price–cash flow ratio (i.e., market capitalization/operating cash flow). v. Another criterion that interests you.
You can fi nd this information by choosing a company and then clicking on Key Sta- tistics. Rank the fi rms based on each of the criteria separately, and divide the fi rms into fi ve groups based on their ranking for each criterion. Calculate the average rate of return for each group of fi rms.
Do you confi rm or reject any of the anomalies cited in this chapter? Can you uncover a new anomaly? Note: For your test to be valid, you must form your port- folios based on criteria observed at the beginning of the period. Why?
b. Use the price history from the Historical Prices tab to calculate the beta of each of the fi rms in part ( a ). Use this beta, the T-bill rate, and the return on the S&P 500 to calculate the risk-adjusted abnormal return of each stock group. Does any anomaly uncovered in the previous question persist after controlling for risk?
c. Now form stock groups that use two criteria simultaneously. For example, form a portfolio of stocks that are both in the lowest quintile of price–earnings ratio and in the highest quintile of book-to-market ratio. Does selecting stocks based on more than one characteristic improve your ability to devise portfolios with abnor- mal returns? Repeat the analysis by forming groups that meet three criteria simul- taneously. Does this yield any further improvement in abnormal returns?
2. Several Web sites list information on earnings surprises. Much of the information sup- plied is from Zacks.com. Each day the largest positive and negative surprises are listed. Go to www.zacks.com/research/earnings/today_eps.php and identify the top pos- itive and the top negative earnings surprises for the day. The table will list the time and date of the announcement. Do you notice any difference between the times of day positive announcements tend to be made versus negative announcements?
Identify the tickers for the top three positive surprises. Once you have identified the top surprises, go to finance.yahoo.com . Enter the ticker symbols and obtain quotes for these securities. Examine the 5-day charts for each of the companies. Is the infor- mation incorporated into price quickly? Is there any evidence of prior knowledge or anticipation of the disclosure in advance of the trading?
Choose one of the stocks listed and click on its symbol to follow the link for more information. Click on the link for Interactive Chart that appears under the graph. You can move the cursor over various parts of the graph to investigate what happened to the price and trading volume of the stock on each trading day. Do you notice any patterns?
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C H A P T E R 1 1 The Efficient Market Hypothesis 387
SOLUTIONS TO CONCEPT CHECKS 1. a. A high-level manager might well have private information about the firm. Her ability to
trade profitably on that information is not surprising. This ability does not violate weak-form
efficiency: The abnormal profits are not derived from an analysis of past price and trading
data. If they were, this would indicate that there is valuable information that can be gleaned
from such analysis. But this ability does violate strong-form efficiency. Apparently, there is
some private information that is not already reflected in stock prices.
b. The information sets that pertain to the weak, semistrong, and strong form of the EMH can be described by the following illustration:
Strong- form set
Semistrong- form set
Weak- form set
The weak-form information set includes only the history of prices and volumes. The semistrong-
form set includes the weak form set plus all publicly available information. In turn, the strong-form set includes the semistrong set plus insiders’ information. It is illegal to act on this incremental information (insiders’ private information). The direction of valid implication is
Strong-form EMH 1 Semistrong-form EMH 1 Weak-form EMH The reverse direction implication is not valid. For example, stock prices may reflect all past price
data (weak-form efficiency) but may not reflect relevant fundamental data (semistrong-form
inefficiency).
2. The point made in the preceding discussion is that the very fact that we observe stock prices near
so-called resistance levels belies the assumption that the price can be a resistance level. If a stock
is observed to sell at any price, then investors must believe that a fair rate of return can be earned if the stock is purchased at that price. It is logically impossible for a stock to have a resistance
level and offer a fair rate of return at prices just below the resistance level. If we accept that prices are appropriate, we must reject any presumption concerning resistance levels.
3. If everyone follows a passive strategy, sooner or later prices will fail to reflect new information. At this point there are profit opportunities for active investors who uncover mispriced securities.
As they buy and sell these assets, prices again will be driven to fair levels.
4. Predictably declining CARs do violate the EMH. If one can predict such a phenomenon, a profit
opportunity emerges: Sell (or short sell) the affected stocks on an event date just before their
prices are predicted to fall.
5. The answer depends on your prior beliefs about market efficiency. Miller’s record through 2005 was
incredibly strong. On the other hand, with so many funds in existence, it is less surprising that some fund would appear to be consistently superior after the fact. Exceptional past performance of a small
number of managers is possible by chance even in an efficient market. A better test is provided in
“continuation studies.” Are better performers in one period more likely to repeat that performance in
later periods? Miller’s record after 2005 fails the continuation or consistency criterion.
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12 1 THE EFFICIENT MARKET hypothesis makes two important predictions. First, it implies that security prices properly reflect whatever information is available to investors. A second implication follows immediately: Active trad- ers will find it difficult to outperform passive strategies such as holding market indexes. To do so would require differential insight; this in a highly competitive market is very hard to come by.
Unfortunately, it is hard to devise measures of the “true” or intrinsic value of a security, and correspondingly difficult to test directly whether prices match those values. Therefore, most tests of market efficiency have focused on the performance of active trading strate- gies. These tests have been of two kinds. The anomalies literature has examined strate- gies that apparently would have provided superior risk-adjusted returns (e.g., investing in stocks with momentum or in value rather than glamour stocks). Other tests have looked at the results of actual investments by asking whether professional managers have been able to beat the market.
Neither class of tests has proven fully con- clusive. The anomalies literature suggests that several strategies would have provided superior returns. But there are questions as to whether some of these apparent anomalies
reflect risk premiums not captured by simple models of risk and return, or even if they merely reflect data mining. Moreover, the apparent inability of the typical money man- ager to turn these anomalies into superior returns on actual portfolios casts additional doubt on their “reality.”
A relatively new school of thought, behav- ioral finance, argues that the sprawling litera- ture on trading strategies has missed a larger and more important point by overlooking the first implication of efficient markets—the correctness of security prices. This may be the more important implication, because market economies rely on prices to allocate resources efficiently. The behavioral school argues that even if security prices are wrong, to exploit them still can be difficult and, therefore, the failure to uncover obviously successful trad- ing rules or traders cannot be taken as proof of market efficiency.
Whereas conventional theories presume that investors are rational, behavioral finance starts with the assumption that they are not. We will examine some of the information- processing and behavioral irrationalities uncovered by psychologists in other contexts and show how these tendencies applied to financial markets might result in some of the anomalies discussed in the previous chapter.
CHAPTER TWELVE
Behavioral Finance
and Technical Analysis
PA R
T I II
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(concluded)
389
We then consider the limitations of strategies designed to take advantage of behaviorally induced mispricing. If the limits to such arbi- trage activity are severe, mispricing can survive even if some rational investors attempt to
exploit it. We turn next to technical analysis and show how behavioral models give some sup- port to techniques that clearly would be use- less in efficient markets. We close the chapter with a brief survey of some of these technical strategies.
12.1 The Behavioral Critique
The premise of behavioral finance is that conventional financial theory ignores how real people make decisions and that people make a difference.
1 A growing number of econ-
omists have come to interpret the anomalies literature as consistent with several “irra-
tionalities” that seem to characterize individuals making complicated decisions. These
irrationalities fall into two broad categories: first, that investors do not always process
information correctly and therefore infer incorrect probability distributions about future
rates of return; and second, that even given a probability distribution of returns, they often
make inconsistent or systematically suboptimal decisions.
Of course, the existence of irrational investors would not by itself be sufficient to render
capital markets inefficient. If such irrationalities did affect prices, then sharp-eyed arbitra-
geurs taking advantage of profit opportunities might be expected to push prices back to
their proper values. Thus, the second leg of the behavioral critique is that in practice the
actions of such arbitrageurs are limited and therefore insufficient to force prices to match
intrinsic value.
This leg of the argument is important. Virtually everyone agrees that if prices are right
(i.e., price 5 intrinsic value), then there are no easy profit opportunities. But the reverse is
not necessarily true. If behaviorists are correct about limits to arbitrage activity, then the
absence of profit opportunities does not necessarily imply that markets are efficient. We’ve
noted that most tests of the efficient market hypothesis have focused on the existence of
profit opportunities, often as reflected in the performance of money managers. But their
failure to systematically outperform passive investment strategies need not imply that mar-
kets are in fact efficient.
We will start our summary of the behavioral critique with the first leg of the argument,
surveying a sample of the informational processing errors uncovered by psychologists in
other areas. We next examine a few of the behavioral irrationalities that seem to character-
ize decision makers. Finally, we look at limits to arbitrage activity, and conclude with a
tentative assessment of the import of the behavioral debate.
Information Processing Errors in information processing can lead investors to misestimate the true probabilities
of possible events or associated rates of return. Several such biases have been uncovered.
Here are four of the more important ones.
1 The discussion in this section is largely based on Nicholas Barberis and Richard Thaler, “A Survey of Behavioral
Finance,” in the Handbook of the Economics of Finance, eds. G. M. Constantinides, M. Harris, and R. Stulz (Amsterdam: Elsevier, 2003).
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390 P A R T I I I Equilibrium in Capital Markets
Forecasting Errors A series of experiments by Kahneman and Tversky 2 indicate that people give too much weight to recent experience compared to prior beliefs when mak-
ing forecasts (sometimes dubbed a memory bias ) and tend to make forecasts that are too extreme given the uncertainty inherent in their information. DeBondt and Thaler
3 argue
that the P/E effect can be explained by earnings expectations that are too extreme. In this
view, when forecasts of a firm’s future earnings are high, perhaps due to favorable recent
performance, they tend to be too high relative to the objective prospects of the firm. This results in a high initial P/E (due to the excessive optimism built into the stock price) and
poor subsequent performance when investors recognize their error. Thus, high P/E firms
tend to be poor investments.
Overconfidence People tend to overestimate the precision of their beliefs or fore- casts, and they tend to overestimate their abilities. In one famous survey, 90% of drivers
in Sweden ranked themselves as better-than-average drivers. Such overconfidence may be
responsible for the prevalence of active versus passive investment management—itself an
anomaly to adherents of the efficient market hypothesis. Despite the growing popularity
of indexing, only about 15% of the equity in the mutual fund industry is held in indexed
accounts. The dominance of active management in the face of the typical underperfor-
mance of such strategies (consider the generally disappointing performance of actively
managed mutual funds reviewed in Chapter 4 as well as in the previous chapter) is consis-
tent with a tendency to overestimate ability.
An interesting example of overconfidence in financial markets is provided by Barber
and Odean, 4 who compare trading activity and average returns in brokerage accounts of
men and women. They find that men (in particular, single men) trade far more actively than
women, consistent with the generally greater overconfidence among men well documented
in the psychology literature. They also find that trading activity is highly predictive of
poor investment performance. The top 20% of accounts ranked by portfolio turnover had
average returns 7 percentage points lower than the 20% of the accounts with the lowest
turnover rates. As they conclude, “trading [and by implication, overconfidence] is hazard-
ous to your wealth.”
Overconfidence appears to be a widespread phenomenon, also showing up in many
corporate finance contexts. For example, overconfident CEOs are more likely to over-
pay for target firms when making corporate acquisitions. 5 Just as overconfidence can
degrade portfolio investments, it also can lead such firms to make poor investments in
real assets.
Conservatism A conservatism bias means that investors are too slow (too conserva- tive) in updating their beliefs in response to new evidence. This means that they might
initially underreact to news about a firm, so that prices will fully reflect new information
only gradually. Such a bias would give rise to momentum in stock market returns.
2 D. Kahneman and A. Tversky, “On the Psychology of Prediction,” Psychology Review 80 (1973), pp. 237–51,
and “Subjective Probability: A Judgment of Representativeness,” Cognitive Psychology 3 (1972), pp. 430–54.
3 W.F.M. De Bondt and R. H. Thaler, “Do Security Analysts Overreact?” American Economic Review 80 (1990),
pp. 52–57.
4 Brad Barber and Terrance Odean, “Boys Will Be Boys: Gender, Overconfidence, and Common Stock
Investment,” Quarterly Journal of Economics 16 (2001), pp. 262–92, and “Trading Is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors,” Journal of Finance 55 (2000), pp. 773–806.
5 U. Malmendier and G. Tate, “Who Makes Acquisitions? CEO Overconfidence and the Market’s Reaction,”
Journal of Financial Economics 89 (July 2008), pp. 20–43.
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 391
Sample Size Neglect and Representativeness The notion of representative- ness bias holds that people commonly do not take into account the size of a sample, acting as if a small sample is just as representative of a population as a large one. They may there-
fore infer a pattern too quickly based on a small sample and extrapolate apparent trends
too far into the future. It is easy to see how such a pattern would be consistent with over-
reaction and correction anomalies. A short-lived run of good earnings reports or high stock
returns would lead such investors to revise their assessments of likely future performance,
and thus generate buying pressure that exaggerates the price run-up. Eventually, the gap
between price and intrinsic value becomes glaring and the market corrects its initial error.
Interestingly, stocks with the best recent performance suffer reversals precisely in the few
days surrounding earnings announcements, suggesting that the correction occurs just as
investors learn that their initial beliefs were too extreme. 6
6 N. Chopra, J. Lakonishok, and J. Ritter, “Measuring Abnormal Performance: Do Stocks Overreact?” Journal of
Financial Economics 31 (1992), pp. 235–68.
We saw in the previous chapter that stocks seem to exhibit a pattern of short- to middle-term momentum, along with long-term reversals. How might this pattern arise from an interplay between the conservatism and representativeness biases?
CONCEPT CHECK 12.1
Behavioral Biases Even if information processing were perfect, many studies conclude that individuals would
tend to make less-than-fully-rational decisions using that information. These behavioral
biases largely affect how investors frame questions of risk versus return, and therefore
make risk–return trade-offs.
Framing Decisions seem to be affected by how choices are framed . For example, an individual may reject a bet when it is posed in terms of the risk surrounding possible gains
but may accept that same bet when described in terms of the risk surrounding potential
losses. In other words, individuals may act risk averse in terms of gains but risk seeking in
terms of losses. But in many cases, the choice of how to frame a risky venture—as involv-
ing gains or losses—can be arbitrary.
Consider a coin toss with a payoff of $50 for tails. Now consider a gift of $50 that is bundled with a bet that imposes a loss of $50 if that coin toss comes up heads. In both cases, you end up with zero for heads and $50 for tails. But the former description frames the coin toss as posing a risky gain while the latter frames the coin toss in terms of risky losses. The difference in framing can lead to different attitudes toward the bet.
Example 12.1 Framing
Mental Accounting Mental accounting is a specific form of framing in which people segregate certain decisions. For example, an investor may take a lot of risk with one
investment account but establish a very conservative position with another account that is
dedicated to her child’s education. Rationally, it might be better to view both accounts as
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392 P A R T I I I Equilibrium in Capital Markets
part of the investor’s overall portfolio with the risk–return profiles of each integrated into
a unified framework. Nevertheless, Statman 7 points out that a central distinction between
conventional and behavioral finance theory is that the behavioral approach views inves-
tors as building their portfolios in “distinct mental account layers in a pyramid of assets,”
where each layer may be tied to particular goals and elicit different levels of risk aversion.
In another paper, Statman 8 argues that mental accounting is consistent with some inves-
tors’ irrational preference for stocks with high cash dividends (they feel free to spend divi-
dend income, but would not “dip into capital” by selling a few shares of another stock
with the same total rate of return) and with a tendency to ride losing stock positions for
too long (because “behavioral investors” are reluctant to realize losses). In fact, investors
are more likely to sell stocks with gains than those with losses, precisely contrary to a tax-
minimization strategy. 9
Mental accounting effects also can help explain momentum in stock prices. The house money effect refers to gamblers’ greater willingness to accept new bets if they currently are ahead. They think of (i.e., frame) the bet as being made with their “winnings account,”
that is, with the casino’s and not with their own money, and thus are more willing to accept
risk. Analogously, after a stock market run-up, individuals may view investments as largely
funded out of a “capital gains account,” become more tolerant of risk, discount future cash
flows at a lower rate, and thus further push up prices.
Regret Avoidance Psychologists have found that individuals who make decisions that turn out badly have more regret (blame themselves more) when that decision was
more unconventional. For example, buying a blue-chip portfolio that turns down is not
as painful as experiencing the same losses on an unknown start-up firm. Any losses on
the blue-chip stocks can be more easily attributed to bad luck rather than bad decision
making and cause less regret. De Bondt and Thaler 10
argue that such regret avoidance is consistent with both the size and book-to-market effect. Higher book-to-market firms
tend to have depressed stock prices. These firms are “out of favor” and more likely to be
in a financially precarious position. Similarly, smaller, less well known firms are also less
conventional investments. Such firms require more “courage” on the part of the inves-
tor, which increases the required rate of return. Mental
accounting can add to this effect. If investors focus on
the gains or losses of individual stocks, rather than on
broad portfolios, they can become more risk averse con-
cerning stocks with recent poor performance, discount
their cash flows at a higher rate, and thereby create a
value-stock risk premium.
Affect Conventional models of portfolio choice focus on asset risk and return. But behavioral
finance focuses as well on affect, which is a feeling of “good” or “bad” that consumers may attach to a potential purchase or investors to a stock. For example, firms with reputations
7 Meir Statman, 2008. Handbook of Finance, vol. II, Ch 9, “What is Behavioral Finance?” ed. Frank J. Fabozzi,
(Hoboken, NJ: John Wiley & Sons, Inc.): 79–84.
8 Meir Statman, “Behavioral Finance,” Contemporary Finance Digest 1 (Winter 1997), pp. 5–22.
9 H. Shefrin and M. Statman, “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and
Evidence,” Journal of Finance 40 (July 1985), pp. 777–90; and T. Odean, “Are Investors Reluctant to Realize Their Losses?” Journal of Finance 53 (1998), pp. 1775–98.
10
W.F.M. De Bondt and R. H. Thaler, “Further Evidence on Investor Overreaction and Stock Market Seasonality,”
Journal of Finance 42 (1987), pp. 557–81.
How might the P/E effect (discussed in the previ- ous chapter) also be explained as a consequence of regret avoidance?
CONCEPT CHECK 12.2
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 393
for socially responsible policies or
attractive working conditions or those
producing popular products may gen-
erate higher affect in public percep-
tion. If investors favor stocks with
good affect, that might drive up prices
and drive down average rates of return.
Statman, Fisher, and Anginer 11
looked
for evidence that affect influences
security pricing. They found that
stocks ranked high in Fortune’s survey of most admired companies (i.e., with
high affect) tended to have lower
average risk-adjusted returns than the
least admired firms, suggesting that
their prices have been bid up relative
to their underlying profitability, and
therefore, that their expected future
returns are lower.
Prospect Theory Prospect theory modifies the analytic description of ratio-
nal risk-averse investors found in stan-
dard financial theory. 12
Figure 12.1 ,
panel A, illustrates the conventional
description of a risk-averse investor.
Higher wealth provides higher sat-
isfaction, or “utility,” but at a dimin-
ishing rate (the curve flattens as the
individual becomes wealthier). This
gives rise to risk aversion: A gain of
$1,000 increases utility by less than
a loss of $1,000 reduces it; therefore,
investors will reject risky prospects
that don’t offer a risk premium.
Figure 12.1 , panel B, shows a com-
peting description of preferences char-
acterized by “loss aversion.” Utility
depends not on the level of wealth, as in panel A, but on changes in wealth from current levels. Moreover, to the
left of zero (zero denotes no change
from current wealth), the curve is convex rather than concave. This has several impli-
cations. Whereas many conventional utility functions imply that investors may become
11 Meir Statman, Kenneth L. Fisher, and Deniz Angier, “Affect in a Behavioral Asset-Pricing Model,” Financial
Analysts Journal 64 (2008), 20–29. 12
Prospect theory originated with a highly influential paper about decision making under uncertainty by
D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decision under Risk,” Econometrica 47 (1979), pp. 263–91.
Figure 12.1 Prospect theory. Panel A: A conventional utility function is defined in terms of wealth and is concave, resulting in risk aversion. Panel B: Under loss aversion, the utility function is defined in terms of losses relative to current wealth. It is also convex to the left of the origin, giving rise to risk-seeking behavior in terms of losses.
A: Conventional Utility Function
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 5 10 15 20 25 30
Wealth U
ti li ty
B: Utility Function under Prospect Theory
−3
−2
−1
1
2
3
−20 −15 −10 −5 5 10 15 20
Change in Wealth
U ti
li ty
0 0
4
25
−4
−25
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394 P A R T I I I Equilibrium in Capital Markets
less risk averse as wealth increases, the function in panel B always re-centers on current
wealth, thereby ruling out such decreases in risk aversion and possibly helping to explain
high average historical equity risk premiums. Moreover, the convex curvature to the left of
the origin in panel B will induce investors to be risk seeking rather than risk averse when it
comes to losses. Consistent with loss aversion, traders in the T-bond futures contract have
been observed to assume significantly greater risk in afternoon sessions following morning
sessions in which they have lost money. 13
These are only a sample of many behavioral biases uncovered in the literature. Many
have implications for investor behavior. The nearby box offers some good examples.
Limits to Arbitrage Behavioral biases would not matter for stock pricing if rational arbitrageurs could fully
exploit the mistakes of behavioral investors. Trades of profit-seeking investors would cor-
rect any misalignment of prices. However, behavioral advocates argue that in practice,
several factors limit the ability to profit from mispricing. 14
Fundamental Risk Suppose that a share of IBM is underpriced. Buying it may pre- sent a profit opportunity, but it is hardly risk-free, because the presumed market under-
pricing can get worse. While price eventually should converge to intrinsic value, this may
not happen until after the trader’s investment horizon. For example, the investor may be
a mutual fund manager who may lose clients (not to mention a job!) if short-term per-
formance is poor, or a trader who may run through her capital if the market turns against
her, even temporarily. A comment often attributed to the famous economist John Maynard
Keynes is that “markets can remain irrational longer than you can remain solvent.” The
fundamental risk incurred in exploiting apparent profit opportunities presumably will limit the activity of traders.
13
J. D. Coval and T. Shumway, “Do Behavioral Biases Affect Prices?” Journal of Finance 60 (February 2005), pp. 1–34.
14
Some of the more influential references on limits to arbitrage are J. B. DeLong, A. Schleifer, L. Summers,
and R. Waldmann, “Noise Trader Risk in Financial Markets,” Journal of Political Economy 98 (August 1990), pp. 704–38; and A. Schleifer and R. Vishny, “The Limits of Arbitrage,” Journal of Finance 52 (March 1997), pp. 35–55.
In mid-2012, the NASDAQ index fluctuated at a level around 2,900. From that perspec- tive, the value the index had reached in 2000, around 5,000, seemed obviously crazy. Surely some investors living through the Internet “bubble” of the late 1990s must have identified the index as grossly overvalued, suggesting a good selling opportunity. But this hardly would have been a riskless arbitrage opportunity. Consider that NASDAQ may also have been overvalued in 1999 when it first crossed above 3,500 (20% higher than its value in 2012). An investor in 1999 who believed (as it turns out, quite correctly) that NASDAQ was overvalued at 3,500 and decided to sell it short would have suffered enormous losses as the index increased by another 1,500 points before finally peaking at 5,000. While the investor might have derived considerable satisfaction at eventually being proven right about the overpricing, by entering a year before the market “cor- rected,” he might also have gone broke.
Example 12.2 Fundamental Risk
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395
Implementation Costs Exploiting overpricing can be particularly difficult. Short- selling a security entails costs; short-sellers may have to return the borrowed security on
little notice, rendering the horizon of the short sale uncertain; other investors such as many
pension or mutual fund managers face strict limits on their discretion to short securities.
This can limit the ability of arbitrage activity to force prices to fair value.
Model Risk One always has to worry that an apparent profit opportunity is more apparent than real. Perhaps you are using a faulty model to value the security, and the
price actually is right. Mispricing may make a position a good bet, but it is still a risky one,
which limits the extent to which it will be pursued.
Limits to Arbitrage and the Law of One Price While one can debate the implications of much of the anomalies literature, surely the Law
of One Price (positing that effectively identical assets should have identical prices) should
be satisfied in rational markets. Yet there are several instances where the law seems to have
been violated. These instances are good case studies of the limits to arbitrage.
Why It’s So Tough to Fix Your Portfolio
If your portfolio is out of whack, you could ask an invest- ment adviser for help. But you might have better luck with your therapist.
It’s a common dilemma: You know you have the wrong mix of investments, but you cannot bring yourself to fix the mess. Why is it so difficult to change? At issue are three mental mistakes.
CHASING WINNERS Looking to lighten up on bonds and get back into stocks? Sure, you know stocks are a long-term investment and, sure, you know they are best bought when cheap.
Yet it’s a lot easier to pull the trigger and buy stocks if the market has lately been scoring gains. “People are influ- enced by what has happened most recently, and then they extrapolate from that,” says Meir Statman, a finance pro- fessor at Santa Clara University in California. “But often, they end up being optimistic and pessimistic at just the wrong time.”
Consider some results from the UBS Index of Investor Optimism, a monthly poll conducted by UBS and the Gal- lup Organization. Each month, the poll asks investors what gain they expect from their portfolio during the next 12 months. Result? You guessed it: The answers rise and fall with the stock market.
For instance, during the bruising bear market, investors grew increasingly pessimistic, and at the market bottom they were looking for median portfolio gains of just 5%. But true to form, last year’s rally brightened investors’ spir- its and by January they were expecting 10% returns.
GETTING EVEN This year’s choppy stock market hasn’t scared off just bond investors. It has also made it difficult for stock investors to rejigger their portfolios.
Blame it on the old “get even, then get out” syndrome. With stocks treading water, many investors are reluctant to sell, because they are a long way from recovering their bear-market losses. To be sure, investors who bought near the peak are underwater, whether they sell or not. But sell- ing losers is still agonizing, because it means admitting you made a mistake.
“If you’re rational and you have a loss, you sell, take the tax loss and move on,” Prof. Statman says. “But if you’re a normal person, selling at a loss tears your heart out.”
MUSTERING COURAGE Whether you need to buy stocks or buy bonds, it takes con- fidence to act. And right now, investors just aren’t confi- dent. “There’s this status-quo bias,” says John Nofsinger, a finance professor at Washington State University in Pull- man, Washington. “We’re afraid to do anything, because we’re afraid we’ll regret it.”
Once again, it’s driven by recent market action. When markets are flying high, folks attribute their portfolio’s gains to their own brilliance. That gives them the confi- dence to trade more and to take greater risks. Overreacting to short-term market results is, of course, a great way to lose a truckload of money. But with any luck, if you are aware of this pitfall, maybe you will avoid it.
Or maybe [this is] too optimistic. “You can tell some- body that investors have all these behavioral biases,” says Terrance Odean, a finance professor at the University of California at Berkeley. “So what happens? The investor thinks, ‘Oh, that sounds like my husband. I don’t think many investors say, ‘Oh, that sounds like me.’”
Source: Jonathan Clements, The Wall Street Journal Online, June 23, 2004. © 2004 Dow Jones & Company, Inc. All rights reserved worldwide.
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396 P A R T I I I Equilibrium in Capital Markets
“Siamese Twin” Companies 15 In 1907, Royal Dutch Petroleum and Shell Trans- port merged their operations into one firm. The two original companies, which continued
to trade separately, agreed to split all profits from the joint company on a 60/40 basis.
Shareholders of Royal Dutch receive 60% of the cash flow, and those of Shell receive 40%.
One would therefore expect that Royal Dutch should sell for exactly 60/40 5 1.5 times the
price of Shell. But this is not the case. Figure 12.2 shows that the relative value of the two
firms has departed considerably from this “parity” ratio for extended periods of time.
Doesn’t this mispricing give rise to an arbitrage opportunity? If Royal Dutch sells for
more than 1.5 times Shell, why not buy relatively underpriced Shell and short sell over-
priced Royal? This seems like a reasonable strategy, but if you had followed it in February
1993 when Royal sold for about 10% more than its parity value, Figure 12.2 shows that
you would have lost a lot of money as the premium widened to about 17% before finally
reversing after 1999. As in Example 12.2, this opportunity posed fundamental risk.
Equity Carve-Outs Several equity carve-outs also have violated the Law of One Price.
16 To illustrate, consider the case of 3Com, which in 1999 decided to spin off its Palm
division. It first sold 5% of its stake in Palm in an IPO, announcing that it would distribute
the remaining 95% of its Palm shares to 3Com shareholders 6 months later in a spinoff.
Each 3Com shareholder would receive 1.5 shares of Palm in the spinoff.
15
This discussion is based on K. A. Froot and E. M. Dabora, “How Are Stock Prices Affected by the Location of
Trade?” Journal of Financial Economics 53 (1999), pp. 189–216. 16
O. A. Lamont and R. H. Thaler, “Can the Market Add and Subtract? Mispricing in Tech Carve-outs,” Journal of Political Economy 111 (2003), pp. 227–68.
Figure 12.2 Pricing of Royal Dutch relative to Shell ( deviation from parity ) Source: O. A. Lamont and R. H. Thaler, “Anomalies: The Law of One Price in Financial Markets,” Journal of Economic Perspectives 17 (Fall 2003), pp. 191–202. Figure 1, p. 196. Used with permission of American Economic Association.
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 397
Once Palm shares began trading, but prior to the spinoff, the share price of 3Com should
have been at least 1.5 times that of Palm. After all, each share of 3Com entitled its owner to 1.5 shares of Palm plus an ownership stake in a profitable company. Instead, Palm shares at the IPO actually sold for more than the 3Com shares. The stub value of 3Com (i.e., the value of each 3Com share net of the value of the claim to Palm represented by that share)
could be computed as the price of 3Com minus 1.5 times the price of Palm. This calcula-
tion, however, implies that 3Com’s stub value was negative, despite the fact that it was a
profitable company with cash assets alone of about $10 per share.
Again, an arbitrage strategy seems obvious. Why not buy 3Com and sell Palm? The
limit to arbitrage in this case was the inability of investors to sell Palm short. Virtually
all available shares in Palm were already borrowed and sold short, and the negative stub
values persisted for more than 2 months.
Closed-End Funds We noted in Chapter 4 that closed-end funds often sell for sub- stantial discounts or premiums from net asset value. This is “nearly” a violation of the
Law of One Price, because one would expect the value of the fund to equal the value of
the shares it holds. We say nearly because, in practice, there are a few wedges between the
value of the closed-end fund and its underlying assets. One is expenses. The fund incurs
expenses that ultimately are paid for by investors, and these will reduce share price. On the
other hand, if managers can invest fund assets to generate positive risk-adjusted returns,
share price might exceed net asset value.
Lee, Shleifer, and Thaler 17
argue that the patterns of discounts and premiums on
closed-end funds are driven by changes in investor sentiment. They note that discounts on
various funds move together and are correlated with the return on small stocks, suggest-
ing that all are affected by common variation in sentiment. One might consider buying
funds selling at a discount from net asset value and selling those trading at a premium, but
discounts and premiums can widen, subjecting this strategy to fundamental risk. Pontiff 18
demonstrates that deviations of price from net asset value in closed-end funds tend to be
higher in funds that are more difficult to arbitrage, for example, those with more idiosyn-
cratic volatility.
Closed-end fund discounts are a good example of apparent anomalies that also may
have rational explanations. Ross demonstrates that they can be reconciled with rational
investors even if expenses or fund abnormal returns are modest. 19
He shows that if a fund
has a dividend yield of d , an alpha (risk-adjusted abnormal return) of a , and expense ratio
of á, then using the constant-growth dividend discount model (see Chapter 18), the pre-
mium of the fund over its net asset value will be
Price 2 NAV
NAV 5
a 2 á
d 1 á 2 a
If the fund manager’s performance more than compensates for expenses (i.e., if a > á),
the fund will sell at a premium to NAV; otherwise it will sell at a discount. For example,
suppose a 5 .015, the expense ratio is á 5 .0125, and the dividend yield is a 5 .02. Then
the premium will be .14, or 14%. But if the market turns sour on the manager and revises
its estimate of a downward to .005, that premium quickly turns into a discount of 27%.
17
C. M. Lee, A. Shleifer, and R. H. Thaler, “Investor Sentiment and the Closed-End Fund Puzzle,” Journal of Finance 46 (March 1991), pp. 75–109.
18
Jeffrey Pontiff, “Costly Arbitrage: Evidence from Closed-End Funds,” Quarterly Journal of Economics 111 (November 1996), pp. 1135–51.
19
S. A. Ross, “Neoclassical Finance, Alternative Finance and the Closed End Fund Puzzle,” European Financial Management 8 (2002), pp. 129–37, http://ssrn.com/abstract 5 313444 .
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398 P A R T I I I Equilibrium in Capital Markets
This analysis might explain why the public is willing to purchase closed-end funds at a
premium; if investors do not expect a to exceed á, they won’t purchase shares in the fund.
But the fact that most premiums eventually turn into discounts indicates how difficult it is
for management to fulfill these expectations. 20
Fundamental risk may be limited by a “deadline” that forces a convergence between price and intrinsic value. What do you think would happen to a closed-end fund’s discount if the fund announced that it plans to liquidate in 6 months, at which time it will distribute NAV to its shareholders?
CONCEPT CHECK 12.3
Bubbles and Behavioral Economics In Example 12.2 above, we pointed out that the stock market run-up of the late 1990s,
and even more spectacularly, the run-up of the technology-heavy NASDAQ market, seems
in retrospect to have been an obvious bubble. In a 6-year period beginning in 1995, the
NASDAQ index increased by a factor of more than 6. Former Fed Chairman Alan Green-
span famously characterized the dot-com boom as an example of “irrational exuberance,”
and his assessment turned out to be correct: by October 2002, the index fell to less than
one-fourth the peak value it had reached only 2½ years earlier. This episode seems to be
a case in point for advocates of the behavioral school, exemplifying a market moved by
irrational investor sentiment. Moreover, in accord with behavioral patterns, as the dot-com
boom developed, it seemed to feed on itself, with investors increasingly confident of their
investment prowess (overconfidence bias) and apparently willing to extrapolate short-term
patterns into the distant future (representativeness bias).
Only 5 years later, another bubble, this time in housing prices, was under way. As in
the dot-com bubble, prospects of further price increases fueled speculative demand by
purchasers. Shortly thereafter, of course, housing prices stalled and then fell. The bursting
bubble set off the worst financial crisis in 75 years.
Bubbles are a lot easier to identify as such once they are over. While they are going on,
it is not as clear that prices are irrationally exuberant and, indeed, many financial commen-
tators during the dot-com bubble justified the boom as consistent with glowing forecasts
for the “new economy.” A simple example shows how hard it can be to tie down the fair
value of stock investments. 21
In 2000, near the peak of the dot-com boom, the dividends paid by the firms included in the S&P 500 totaled $154.6 million. If the discount rate for the index was 9.2% and the expected dividend growth rate was 8%, the value of these shares according to the con- stant-growth dividend discount model (see Chapter 18 for more on this model) would be
Value 5 Dividend
Discount rate 2 Growth rate 5
$154.6 .092 2 .08
5 $12,883 million
Example 12.3 A Stock Market Bubble?
20
We might ask why this logic of discounts and premiums does not apply to open-end mutual funds because they
incur similar expense ratios. Because investors in these funds can redeem shares for NAV, the shares cannot sell
at a discount to NAV. Expenses in open-end funds reduce returns in each period rather than being capitalized into
price and inducing a discount.
21
The following example is taken from R. A. Brealey, S. C. Myers, and F. Allen, Principles of Corporate Finance, 8th ed. (New York: McGraw-Hill Irwin, 2006).
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 399
22
P. R. Rau, O. Dimitrov, and M. Cooper, “A Rose.com by Any Other Name,” Journal of Finance 56 (2001), pp. 2371–88.
23
E. F. Fama, “Market Efficiency, Long-Term Returns, and Behavioral Finance,” Journal of Financial Economics 49 (September 1998), pp. 283–306.
Still, other evidence seems to tag the dot-com boom as at least partially irrational.
Consider, for example, the results of a study documenting that firms adding “.com” to the
end of their names during this period enjoyed a meaningful stock price increase. 22
That
doesn’t sound like rational valuation.
Evaluating the Behavioral Critique As investors, we are concerned with the existence of profit opportunities. The behavioral
explanations of efficient market anomalies do not give guidance as to how to exploit any
irrationality. For investors, the question is still whether there is money to be made from
mispricing, and the behavioral literature is largely silent on this point.
However, as we emphasized above, one of the important implications of the efficient
market hypothesis is that security prices serve as reliable guides to the allocation of real
assets. If prices are distorted, then capital markets will give misleading signals (and incen-
tives) as to where the economy may best allocate resources. In this crucial dimension, the
behavioral critique of the efficient market hypothesis is certainly important irrespective of
any implication for investment strategies.
There is considerable debate among financial economists concerning the strength of
the behavioral critique. Many believe that the behavioral approach is too unstructured,
in effect allowing virtually any anomaly to be explained by some combination of irra-
tionalities chosen from a laundry list of behavioral biases. While it is easy to “reverse
engineer” a behavioral explanation for any particular anomaly, these critics would like
to see a consistent or unified behavioral theory that can explain a range of behavioral anomalies.
More fundamentally, others are not convinced that the anomalies literature as a whole is
a convincing indictment of the efficient market hypothesis. Fama 23
notes that the anoma-
lies are inconsistent in terms of their support for one type of irrationality versus another.
For example, some papers document long-term corrections (consistent with overreac-
tion), while others document long-term continuations of abnormal returns (consistent with
underreaction). Moreover, the statistical significance of many of these results is hard to
assess. Even small errors in choosing a benchmark against which to compare returns can
cumulate to large apparent abnormalities in long-term returns.
This was quite close to the actual total value of those firms at the time. But the estimate is highly sensitive to the input values, and even a small reassessment of their prospects would result in a big revision of price. Suppose the expected dividend growth rate fell to 7.4%. This would reduce the value of the index to
Value 5 Dividend
Discount rate 2 Growth rate 5
$154.6 .092 2 .074
5 $8,589 million
which was about the value to which the S&P 500 firms had fallen by October 2002. In light of this example, the run-up and crash of the 1990s seems easier to reconcile with rational behavior.
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400 P A R T I I I Equilibrium in Capital Markets
The behavioral critique of full rationality in investor decision making is well taken, but
the extent to which limited rationality affects asset pricing remains controversial. Whether
or not investor irrationality affects asset prices, however, behavioral finance already makes
important points about portfolio management. Investors who are aware of the potential
pitfalls in information processing and decision making that seem to characterize their peers
should be better able to avoid such errors. Ironically, the insights of behavioral finance
may lead to some of the same policy conclusions embraced by efficient market advocates.
For example, an easy way to avoid some of the behavioral minefields is to pursue passive,
largely indexed, portfolio strategies. It seems that only rare individuals can consistently
beat passive strategies; this conclusion may hold true whether your fellow investors are
behavioral or rational.
12.2 Technical Analysis and Behavioral Finance
Technical analysis attempts to exploit recurring and predictable patterns in stock prices to
generate superior investment performance. Technicians do not deny the value of funda-
mental information, but believe that prices only gradually close in on intrinsic value. As
fundamentals shift, astute traders can exploit the adjustment to a new equilibrium.
For example, one of the best-documented behavioral tendencies is the disposition effect , which refers to the tendency of investors to hold on to losing investments. Behavioral
investors seem reluctant to realize losses. This disposition effect can lead to momentum in
stock prices even if fundamental values follow a random walk. 24
The fact that the demand
of “disposition investors” for a company’s shares depends on the price history of those
shares means that prices could close in on fundamental values only over time, consistent
with the central motivation of technical analysis.
Behavioral biases may also be consistent with technical analysts’ use of volume data.
An important behavioral trait noted above is overconfidence, a systematic tendency to
overestimate one’s abilities. As traders become overconfident, they may trade more, induc-
ing an association between trading volume and market returns. 25
Technical analysis thus
uses volume data as well as price history to direct trading strategy.
Finally, technicians believe that market fundamentals can be perturbed by irrational or
behavioral factors, sometimes labeled sentiment variables. More or less random price fluc-
tuations will accompany any underlying price trend, creating opportunities to exploit cor-
rections as these fluctuations dissipate. The nearby box explores the link between technical
analysis and behavioral finance.
Trends and Corrections Much of technical analysis seeks to uncover trends in market prices. This is in effect a
search for momentum. Momentum can be absolute, in which case one searches for upward
price trends, or relative, in which case the analyst looks to invest in one sector over another
(or even take on a long-short position in the two sectors). Relative strength statistics are
designed to uncover these potential opportunities.
24
Mark Grinblatt and Bing Han, “Prospect Theory, Mental Accounting, and Momentum,” Journal of Financial Economics 78 (November 2005), pp. 311–39.
25
S. Gervais and T. Odean, “Learning to Be Overconfident,” Review of Financial Studies 14 (2001), pp. 1–27.
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401
Momentum and Moving Averages While we all would like to buy shares in firms whose prices are trending upward, this begs the question of how to identify the
underlying direction of prices, if in fact such trends actually exist. A popular tool used for
this purpose is the moving average.
The moving average of a stock price is the average price over a given interval, where
that interval is updated as time passes. For example, a 50-day moving average traces the
average price over the previous 50 days. The average is recomputed each day by dropping
the oldest observation and adding the newest. Figure 12.3 is a moving average chart for
Intel. Notice that the moving average (the colored curve) is a “smoothed” version of the
original data series (the jagged dark curve).
After a period in which prices have been falling, the moving average will be above the
current price (because the moving average continues to average in the older and higher
prices until they leave the sample period). In contrast, when prices have been rising, the
moving average will be below the current price.
Prices breaking through the moving average from below, as at point A in Figure 12.3 , is
taken as a bullish signal, because it signifies a shift from a falling trend (with prices below
the moving average) to a rising trend (with prices above the moving average). Conversely,
when prices drop below the moving average, as at point B, analysts might conclude that
market momentum has become negative.
Technical Failure
Practical traders, who believe themselves to be quite exempt from any intellectual influences, are usually slaves of some defunct mathematician. That is what Keynes might have said had he considered the faith placed by some investors in the work of Leonardo of Pisa, a 12th and 13th century number-cruncher.
Better known as Fibonacci, Leonardo produced the sequence formed by adding consecutive components of a series—1, 1, 2, 3, 5, 8 and so on. Numbers in this series crop up frequently in nature and the relationship between components tends towards 1.618, a figure known as the golden ratio in architecture and design.
If it works for plants (and appears in “The Da Vinci Code”), why shouldn’t it work for financial markets? Some traders believe that markets will change trend when they reach, say, 61.8% of the previous high, or are 61.8% above their low.
Believers in Fibonacci numbers are part of a school known as technical analysis, or chartism, which believes the future movement of asset prices can be divined from past data. But there is bad news for the numerologists. A new study * by Professor Roy Batchelor and Richard Ramyar of the Cass Business School, finds no evidence that Fibonacci numbers work in American stockmarkets.
This research may well fall on stony ground. Experience suggests that chartists defend their territory with an almost religious zeal. But their arguments are often anecdotal: “If technical analysis doesn’t work, how come so-and-so is a multi-millionaire?” This “survivorship bias” ignores the many traders whose losses from using charts drive them
out of the market. Furthermore, the recommendations of technical analysts can be so hedged about with qualifica- tions that they can validate almost any market outcome.
If the efficient market theory is correct, technical anal- ysis should not work at all; the prevailing market price should reflect all information, including past price move- ments. However, academic fashion has moved in favor of behavioral finance, which suggests that investors may not be completely rational and that their psychological biases could cause prices to deviate from their “correct” level. Technical analysts also make the perfectly fair argument that those who analyze markets on the basis of fundamen- tals (such as economic statistics or corporate profits) are no more successful.
All that talk of long waves is distinctly mystical and seems to take the deterministic view of history that human activity is subject to some pre-ordained pattern. Chartists fall prey to their own behavioral flaw, finding “confir- mation” of patterns everywhere, as if they were reading clouds in their coffee futures.
Besides, technical analysis tends to increase trading activity, creating extra costs. Hedge funds may be able to rise above these costs; small investors will not. As illusion- ists often proclaim, don’t try this at home.
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*“No Magic in the Dow—Debunking Fibonacci’s Code,” working paper, Cass Business School, September 2006.
Source: The Economist, September 21, 2006. © The Economist Newspaper Limited, London.
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402 P A R T I I I Equilibrium in Capital Markets
Consider the following price data. Each observation represents the closing level of the Dow Jones Industrial Average (DJIA) on the last trading day of the week. The 5-week moving average for each week is the average of the DJIA over the previous 5 weeks. For example, the first entry, for week 5, is the average of the index value between weeks 1 and 5: 13,290, 13,380, 13,399, 13,379, and 13,450. The next entry is the average of the index values between weeks 2 and 6, and so on.
Figure 12.4 plots the level of the index and the 5-week moving average. Notice that while the index itself moves up and down rather abruptly, the moving average is a rela- tively smooth series, because the impact of each week’s price movement is averaged with that of the previous weeks. Week 16 is a bearish point according to the moving average rule. The price series crosses from above the moving average to below it, signify- ing the beginning of a downward trend in stock prices.
Example 12.4 Moving Averages
Week DJIA
5-Week Moving Average Week DJIA
5-Week Moving Average
1 13,290 11 13,590 13,555 2 13,380 12 13,652 13,586 3 13,399 13 13,625 13,598 4 13,379 14 13,657 13,624 5 13,450 13,380 15 13,699 13,645 6 13,513 13,424 16 13,647 13,656 7 13,500 13,448 17 13,610 13,648 8 13,565 13,481 18 13,595 13,642 9 13,524 13,510 19 13,499 13,610
10 13,597 13,540 20 13,466 13,563
Figure 12.3 Share price and 50-day moving average for Intel Source: Yahoo! Finance, August 27, 2012 (finance.yahoo.com).
30 29 28 27 26 25 24 23 22 21
20
19 Nov11Sep11 Jan12 Mar12 May12 Jul12
INTC 50–day MA
A
B
as of 24-Aug-2012Intel Corporation
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 403
Other techniques also are used to uncover potential momentum in stock prices. Two
of the more famous ones are Elliott wave theory and Kondratieff waves. Both posit the
existence of long-term trends in stock market prices that may be disturbed by shorter-term
trends as well as daily fluctuations of little importance. Elliott wave theory superimposes
long-term and short-term wave cycles in an attempt to describe the complicated pattern of
actual price movements. Once the longer-term waves are identified, investors presumably
can buy when the long-term direction of the market is positive. While there is consider-
able noise in the actual evolution of stock prices, by properly interpreting the wave cycles,
one can, according to the theory, predict broad movements. Similarly, Kondratieff waves
are named after a Russian economist who asserted that the macroeconomy (and therefore
the stock market) moves in broad waves lasting between 48 and 60 years. Kondratieff’s
assertion is hard to evaluate empirically, however, because cycles that last about 50 years
provide only two independent data points per century, which is hardly enough data to test
the predictive power of the theory.
Relative Strength Relative strength measures the extent to which a security has outperformed or underperformed either the market as a whole or its particular industry.
Relative strength is computed by calculating the ratio of the price of the security to a price
index for the industry. For example, the relative strength of Toyota versus the auto industry
would be measured by movements in the ratio of the price of Toyota divided by the level
of an auto industry index. A rising ratio implies Toyota has been outperforming the rest of
the industry. If relative strength can be assumed to persist over time, then this would be a
signal to buy Toyota.
Similarly, the strength of an industry relative to the whole market can be computed by
tracking the ratio of the industry price index to the market price index.
Breadth The breadth of the market is a measure of the extent to which movement in a market index is reflected widely in the price movements of all the stocks in the market.
The most common measure of breadth is the spread between the number of stocks that
advance and decline in price. If advances outnumber declines by a wide margin, then
Figure 12.4 Moving averages
13,700
13,600
13,500
13,400
13,300
13,200
13,100
13,000
Week
D o
w J
o n
e s
In d
u st
ri a l A
ve ra
g e
1 3 5 7 9 11 13 15 17 19
DJIA Moving Average
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404 P A R T I I I Equilibrium in Capital Markets
the market is viewed as being stronger
because the rally is widespread. These
numbers are reported in The Wall Street Journal (see Figure 12.5 ).
Some analysts cumulate breadth data
each day as in Table 12.1 . The cumulative
breadth for each day is obtained by adding
that day’s net advances (or declines) to the
previous day’s total. The direction of the
cumulated series is then used to discern
broad market trends. Analysts might use a
moving average of cumulative breadth to
gauge broad trends.
Sentiment Indicators Behavioral finance devotes considerable
attention to market sentiment, which may be interpreted as the general level of opti-
mism among investors. Technical analysts
have devised several measures of senti-
ment; we review a few of them.
Trin Statistic Market volume is sometimes used to measure the strength of a market rise or fall. Increased investor participation in a market advance or retreat is viewed as
a measure of the significance of the movement. Technicians consider market advances
to be a more favorable omen of continued price increases when they are associated with
increased trading volume. Similarly, market reversals are considered more bearish when
associated with higher volume. The trin statistic is defined as
Trin 5 Volume declining/Number declining
Volume advancing/Number advancing
Therefore, trin is the ratio of average volume in declining issues to average volume in
advancing issues. Ratios above 1.0 are considered bearish because the falling stocks would
then have higher average volume than the advancing stocks, indicating net selling pressure.
The Wall Street Journal Online provides the data necessary to compute trin in its Markets Diary section. Using the data in Figure 12.5 , trin for the NYSE on this day was:
Trin 5 $1,058,313/1,553
$852,581/1,455 5 1.16
Figure 12.5 Market Diary Source: The Wall Street Journal Online, August 27, 2012.
Issues Advancing Declining Unchanged Total
Issues at New 52 Week High New 52 Week Low
Share Volume Total Advancing Declining Unchanged
NYSE 1,455 1,553
105 3,113
104 17
1,944,911,334 852,581,038
1,058,312,638 34,017,658
Amex 200 202
22 424
3 7
48,215,758 18,651,981 27,715,823
1,847,954
NASDAQ 1,176 1,274
112 2,562
56 31
1,112,153,552 521,407,205 579,005,241
11,741,106
3:31 p.m. EDT 08/27/12
Trading Diary: Volume, Advancers, Decliners Markets Diary
Day Advances Declines Net Advances Cumulative Breadth
1 1,302 1,248 54 54 2 1,417 1,140 277 331 3 1,203 1,272 2 69 262 4 1,012 1,622 2 610 2 348 5 1,133 1,504 2 371 2 719
Note: The sum of advances plus declines varies across days because some stock prices are unchanged.
Table 12.1
Breadth
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 405
Remember, however, that for every buyer, there must be a seller of stock. Rising volume in
a rising market should not necessarily indicate a larger imbalance of buyers versus sellers.
For example, a trin statistic above 1.0, which is considered bearish, could equally well be
interpreted as indicating that there is more buying activity in declining issues.
Confidence Index Barron’s computes a confidence index using data from the bond market. The presumption is that actions of bond traders reveal trends that will emerge soon
in the stock market.
The confidence index is the ratio of the average yield on 10 top-rated corporate bonds divided by the average yield on 10 intermediate-grade corporate bonds. The ratio
will always be below 100% because higher-rated bonds will offer lower promised yields
to maturity. When bond traders are optimistic about the economy, however, they might
require smaller default premiums on lower-rated debt. Hence, the yield spread will narrow,
and the confidence index will approach 100%. Therefore, higher values of the confidence
index are bullish signals.
Yields on lower-rated debt will rise after fears of recession have spread through the economy. This will reduce the confidence index. Should the stock market now be expected to fall or will it already have fallen?
CONCEPT CHECK 12.4
Put/Call Ratio Call options give investors the right to buy a stock at a fixed “exercise” price and therefore are a way of betting on stock price increases. Put options give the right
to sell a stock at a fixed price and therefore are a way of betting on stock price decreases. 26
The ratio of outstanding put options to outstanding call options is called the put/call ratio . Typically, the put/call ratio hovers around 65%. Because put options do well in falling
markets while call options do well in rising markets, deviations of the ratio from historical
norms are considered to be a signal of market sentiment and therefore predictive of market
movements.
Interestingly, however, a change in the ratio can be given a bullish or a bearish inter-
pretation. Many technicians see an increase in the ratio as bearish, as it indicates growing
interest in put options as a hedge against market declines. Thus, a rising ratio is taken as
a sign of broad investor pessimism and a coming market decline. Contrarian investors,
however, believe that a good time to buy is when the rest of the market is bearish because
stock prices are then unduly depressed. Therefore, they would take an increase in the put/
call ratio as a signal of a buy opportunity.
A Warning The search for patterns in stock market prices is nearly irresistible, and the ability of the
human eye to discern apparent patterns is remarkable. Unfortunately, it is possible to
perceive patterns that really don’t exist. Consider Figure 12.6 , which presents simulated
and actual values of the Dow Jones Industrial Average during 1956 taken from a famous
study by Harry Roberts. 27
In Figure 12.6, panel B, the market appears to present a classic
26
Puts and calls were defined in Chapter 2, Section 2.5. They are discussed more fully in Chapter 20.
27
H. Roberts, “Stock Market ‘Patterns’ and Financial Analysis: Methodological Suggestions,” Journal of Finance 14 (March 1959), pp. 11–25.
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406 P A R T I I I Equilibrium in Capital Markets
head-and-shoulders pattern where the middle hump (the head) is flanked by two shoulders.
When the price index “pierces the right shoulder”—a technical trigger point—it is believed
to be heading lower, and it is time to sell your stocks. Figure 12.6, panel A also looks like
a “typical” stock market pattern.
Can you tell which of the two graphs is constructed from the real value of the Dow
and which from the simulated data? Figure 12.6, panel A is based on the real data. The
graph in panel B was generated using “returns” created by a random-number generator.
These returns by construction were patternless, but the simulated price path that is plotted appears to follow a pattern much like that of panel A.
Figure 12.7 shows the weekly price changes behind the two panels in Figure 12.6 . Here the randomness in both series—the stock price as well as the simulated sequence—is
obvious.
A problem related to the tendency to perceive patterns where they don’t exist is data
mining. After the fact, you can always find patterns and trading rules that would have
generated enormous profits. If you test enough rules, some will have worked in the past.
Unfortunately, picking a theory that would have worked after the fact carries no guarantee
of future success.
In evaluating trading rules, you should always ask whether the rule would have seemed
reasonable before you looked at the data. If not, you might be buying into the one arbitrary rule among many that happened to have worked in the recent past. The hard but crucial
question is whether there is reason to believe that what worked in the past should continue
to work in the future.
Figure 12.6 Actual and simulated levels for stock market prices of 52 weeks Source: Harry Roberts, “Stock Market ‘Patterns’ and Financial Analysis: Methodological Suggestions,” Journal of Finance 14 (March 1959), pp. 1–10. Used with permission of John Wiley and Sons, via Copyright Clearance Center.
Friday closing levels, December 30, 1955–December 28, 1956, Dow Jones Industrial Average
A
5 10 15 20 25 30 35 40 45 50
525 520 515 510 505 500 495 490 485 480 475 470 465 460
Level
Week
5 10 15 20 25 30 35 40 45 50
485 480 475 470 465 460 455 450 445 440 435 430 425 420
B
Level
Week
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 407
1. Behavioral finance focuses on systematic irrationalities that characterize investor decision making. These “behavioral shortcomings” may be consistent with several efficient market
anomalies.
2. Among the information processing errors uncovered in the psychology literature are memory bias, overconfidence, conservatism, and representativeness. Behavioral tendencies include fram-
ing, mental accounting, regret avoidance, and loss aversion.
3. Limits to arbitrage activity impede the ability of rational investors to exploit pricing errors induced by behavioral investors. For example, fundamental risk means that even if a security is
mispriced, it still can be risky to attempt to exploit the mispricing. This limits the actions of arbi-
trageurs who take positions in mispriced securities. Other limits to arbitrage are implementation
costs, model risk, and costs to short-selling. Occasional failures of the Law of One Price suggest
that limits to arbitrage are sometimes severe.
4. The various limits to arbitrage mean that even if prices do not equal intrinsic value, it still may be difficult to exploit the mispricing. As a result, the failure of traders to beat the market may not be
proof that markets are in fact efficient, with prices equal to intrinsic value.
5. Technical analysis is the search for recurring and predictable patterns in stock prices. It is based on the premise that prices only gradually close in on intrinsic value. As fundamentals shift, astute
traders can exploit the adjustment to a new equilibrium.
6. Technical analysis also uses volume data and sentiment indicators. These are broadly consis- tent with several behavioral models of investor activity. Moving averages, relative strength, and
breadth are used in other trend-based strategies.
7. Some sentiment indicators are the trin statistic, the confidence index, and the put/call ratio.
Related Web sites for this chapter are available at www. mhhe.com/bkm
Figure 12.7 Actual and simulated changes in weekly stock prices for 52 weeks Source: Harry Roberts, “Stock Market ‘Patterns’ and Financial Analysis: Methodological Suggestions,” Journal of Finance 14 (March 1959), pp. 1–10. Used with permission of John Wiley and Sons, via Copyright Clearance Center.
Changes from Friday to Friday (closing) January 6, 1956–December 28, 1956, Dow Jones Industrial Average
25 20 15 10
5 0 −5 −10 −15 −20 −25 −30
1 5 10 15 20 3025 35 40 45 50
Week
A
Change 30 25 20 15 10 5 0 −5 −10 −15 −20 −25
1 5 10 15 20 3025 35 40 45 50
Week
B
Change
SUMMARY
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408 P A R T I I I Equilibrium in Capital Markets
1. Explain how some of the behavioral biases discussed in the chapter might contribute to the suc-
cess of technical trading rules.
2. Why would an advocate of the efficient market hypothesis believe that even if many inves-
tors exhibit the behavioral biases discussed in the chapter, security prices might still be set
efficiently?
3. What sorts of factors might limit the ability of rational investors to take advantage of any “pricing
errors” that result from the actions of “behavioral investors”?
4. Even if behavioral biases do not affect equilibrium asset prices, why might it still be important for
investors to be aware of them?
5. Some advocates of behavioral finance agree with efficient market advocates that indexing is the
optimal investment strategy for most investors. But their reasons for this conclusion differ greatly.
Compare and contrast the rationale for indexing according to both of these schools of thought.
6. Jill Davis tells her broker that she does not want to sell her stocks that are below the price she
paid for them. She believes that if she just holds on to them a little longer they will recover, at
which time she will sell them. What behavioral characteristic does Davis have as the basis for her
decision making?
a. Loss aversion. b. Conservatism. c. Representativeness.
7. After Polly Shrum sells a stock, she avoids following it in the media. She is afraid that it may
subsequently increase in price. What behavioral characteristic does Shrum have as the basis for
her decision making?
a. Fear of regret. b. Representativeness. c. Mental accounting.
8. All of the following actions are consistent with feelings of regret except:
a. Selling losers quickly. b. Hiring a full-service broker. c. Holding on to losers too long.
9. Match each example to one of the following behavioral characteristics.
behavioral finance
conservatism
representativeness bias
framing
mental accounting
regret avoidance
prospect theory
fundamental risk
disposition effect
relative strength
breadth
trin statistic
confidence index
put/call ratio
KEY TERMS
PROBLEM SETS
Basic
Example Characteristic
a. Investors are slow to update their beliefs when given new evidence.
i. Disposition effect.
b. Investors are reluctant to bear losses caused by their unconventional decisions.
ii. Representativeness bias.
c. Investors exhibit less risk tolerance in their retirement accounts versus their other stock accounts.
iii. Regret avoidance.
d. Investors are reluctant to sell stocks with “paper” losses. iv. Conservatism bias.
e. Investors disregard sample size when forming views about the future from the past.
v. Mental accounting.
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 409
10. What do we mean by fundamental risk, and why may such risk allow behavioral biases to
persist for long periods of time?
11. What is meant by data mining, and why must technical analysts be careful not to engage in it?
12. Even if prices follow a random walk, they still may not be informationally efficient. Explain
why this may be true, and why it matters for the efficient allocation of capital in our economy.
13. Use the data from The Wall Street Journal in Figure 12.5 to verify the trin ratio for the NYSE. Is the trin ratio bullish or bearish?
14. Calculate breadth for the NYSE using the data in Figure 12.5 . Is the signal bullish or bearish?
15. Collect data on the DJIA for a period covering a few months. Try to identify primary trends.
Can you tell whether the market currently is in an upward or downward trend?
16. Suppose Baa-rated bonds currently yield 6%, while Aa-rated bonds yield 5%. Suppose that due
to an increase in the expected inflation rate, the yields on both bonds increase by 1%. What
would happen to the confidence index? Would this be interpreted as bullish or bearish by a
technical analyst? Does this make sense to you?
17. Table 12A presents price data for Computers, Inc., and a computer industry index. Does Com-
puters, Inc., show relative strength over this period?
18. Use again the data in Table 12A to compute a 5-day moving average for Computers, Inc. Can
you identify any buy or sell signals?
19. Yesterday, the Dow Jones industrials gained 54 points. However, 1,704 issues declined in price
while 1,367 advanced. Why might a technical analyst be concerned even though the market
index rose on this day?
Intermediate
Table 12A
Computers, Inc., stock price history
Trading Day
Computers, Inc.
Industry Index
1 19.63 50.0
2 20 50.1
3 20.50 50.5
4 22 50.4
5 21.13 51.0
6 22 50.7
7 21.88 50.5
8 22.50 51.1
9 23.13 51.5
10 23.88 51.7
11 24.50 51.4
12 23.25 51.7
13 22.13 52.2
14 22 52.0
15 20.63 53.1
16 20.25 53.5
17 19.75 53.9
18 18.75 53.6
19 17.50 52.9
20 19 53.4
Trading Day
Computers, Inc.
Industry Index
21 19.63 54.1
22 21.50 54.0
23 22 53.9
24 23.13 53.7
25 24 54.8
26 25.25 54.5
27 26.25 54.6
28 27 54.1
29 27.50 54.2
30 28 54.8
31 28.50 54.2
32 28 54.8
33 27.50 54.9
34 29 55.2
35 29.25 55.7
36 29.50 56.1
37 30 56.7
38 28.50 56.7
39 27.75 56.5
40 28 56.1
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410 P A R T I I I Equilibrium in Capital Markets
20. Table 12B contains data on market advances and declines. Calculate cumulative breadth and
decide whether this technical signal is bullish or bearish.
21. If the trading volume in advancing shares on day 1 in the previous problem was 330 million
shares, while the volume in declining issues was 240 million shares, what was the trin statistic
for that day? Was trin bullish or bearish?
22. Given the following data, is the confidence index rising or falling? What might explain the
pattern of yield changes?
This Year Last Year
Yield on top-rated corporate bonds 8% 8.5%
Yield on intermediate-grade corporate bonds
10.5 10
Table 12B
Market advances and declines
Day Advances Declines
1 906 704
2 653 986
3 721 789
4 503 968
5 497 1,095
Day Advances Declines
6 970 702
7 1,002 609
8 903 722
9 850 748
10 766 766
23. Go to www.mhhe.com/bkm and link to the material for Chapter 12, where you will find 5 years of weekly returns for the S&P 500.
a. Set up a spreadsheet to calculate the 26-week moving average of the index. Set the value of the index at the beginning of the sample period equal to 100. The index value in each week is then
updated by multiplying the previous week’s level by (1 1 rate of return over previous week).
b. Identify every instance in which the index crosses through its moving average from below. In how many of the weeks following a cross-through does the index increase? Decrease?
c. Identify every instance in which the index crosses through its moving average from above. In how many of the weeks following a cross-through does the index increase? Decrease?
d. How well does the moving average rule perform in identifying buy or sell opportunities?
24. Go to www.mhhe.com/bkm and link to the material for Chapter 12, where you will find 5 years of weekly returns for the S&P 500 and Fidelity’s Select Banking Fund (ticker FSRBX).
a. Set up a spreadsheet to calculate the relative strength of the banking sector compared to the broad market. Hint: As in the previous problem, set the initial value of the sector index and
the S&P 500 index equal to 100, and use each week’s rate of return to update the level of
each index.
b. Identify every instance in which the relative strength ratio increases by at least 5% from its value 5 weeks earlier. In how many of the weeks following a substantial increase in relative
strength does the banking sector outperform the S&P 500? In how many of those weeks does
the banking sector underperform the S&P 500?
c. Identify every instance in which the relative strength ratio decreases by at least 5% from its value 5 weeks earlier. In how many of the weeks following a substantial decrease in relative
strength does the banking sector underperform the S&P 500? In how many of those weeks
does the banking sector outperform the S&P 500?
d. How well does the relative strength rule perform in identifying buy or sell opportunities?
e X c e l Please visit us at www.mhhe.com/bkm
e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 411
25. One seeming violation of the Law of One Price is the pervasive discrepancy of closed-end fund
prices from their net asset values. Would you expect to observe greater discrepancies on diversi-
fied or less-diversified funds? Why?
Challenge
1. Don Sampson begins a meeting with his financial adviser by outlining his investment phi-
losophy as shown below:
Statement Number Statement
1 Investments should offer strong return potential but with very limited risk. I prefer to be conservative and to minimize losses, even if I miss out on substantial growth opportunities.
2 All nongovernmental investments should be in industry-leading and financially strong companies.
3 Income needs should be met entirely through interest income and cash dividends. All equity securities held should pay cash dividends.
4 Investment decisions should be based primarily on consensus forecasts of general economic conditions and company-specific growth.
5 If an investment falls below the purchase price, that security should be retained until it returns to its original cost. Conversely, I prefer to take quick profits on successful investments.
6 I will direct the purchase of investments, including derivative securities, periodically. These aggressive investments result from personal research and may not prove consistent with my investment policy. I have not kept records on the performance of similar past investments, but I have had some “big winners.”
Select the statement from the table above that best illustrates each of the following behavioral finance
concepts. Justify your selection.
a. Mental accounting. b. Overconfidence (illusion of control). c. Reference dependence (framing).
2. Monty Frost’s tax-deferred retirement account is invested entirely in equity securities.
Because the international portion of his portfolio has performed poorly in the past, he
has reduced his international equity exposure to 2%. Frost’s investment adviser has rec-
ommended an increased international equity exposure. Frost responds with the following
comments:
a. On the basis of past poor performance, I want to sell all my remaining international equity securities once their market prices rise to equal their original cost.
b. Most diversified international portfolios have had disappointing results over the past 5 years. During that time, however, the market in Country XYZ has outperformed all other markets,
even our own. If I do increase my international equity exposure, I would prefer that the
entire exposure consist of securities from Country XYZ.
c. International investments are inherently more risky. Therefore, I prefer to purchase any international equity securities in my “speculative” account, my best chance at becoming
rich. I do not want them in my retirement account, which has to protect me from poverty in
my old age.
Frost’s adviser is familiar with behavioral finance concepts but prefers a traditional or standard
finance approach (modern portfolio theory) to investments.
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412 P A R T I I I Equilibrium in Capital Markets
Indicate the behavioral finance concept that Frost most directly exhibits in each of his three
comments. Explain how each of Frost’s comments can be countered by using an argument from
standard finance.
3. Louise and Christopher Maclin live in London, United Kingdom, and currently rent an apartment
in the metropolitan area. During an initial discussion of the Maclins’ financial plans, Christopher
Maclin makes the following statements to the Maclins’ financial adviser, Grant Webb:
a. “I have used the Internet extensively to research the outlook for the housing market over the next 5 years, and I believe now is the best time to buy a house.”
b. “I do not want to sell any bond in my portfolio for a lower price than I paid for the bond.” c. “I will not sell any of my company stock because I know my company and I believe it has
excellent prospects for the future.”
For each statement ( a )–( c ) identify the behavioral finance concept most directly exhibited. Explain how each behavioral finance concept is affecting Maclin’s investment decision making.
4. During an interview with her investment adviser, a retired investor made the following two
statements:
a. “I have been very pleased with the returns I’ve earned on Petrie stock over the past 2 years and I am certain that it will be a superior performer in the future.”
b. “I am pleased with the returns from the Petrie stock because I have specific uses for that money. For that reason, I certainly want my retirement fund to continue owning the Petrie stock.”
Identify which principle of behavioral finance is most consistent with each of the investor’s two
statements.
5. Claire Pierce comments on her life circumstances and investment outlook:
I must support my parents who live overseas on Pogo Island. The Pogo Island economy has grown rapidly over the past 2 years with minimal inflation, and consensus forecasts call for a continuation of these favorable trends for the foreseeable future. Economic growth has resulted from the export of a natural resource used in an exciting new technology application.
I want to invest 10% of my portfolio in Pogo Island government bonds. I plan to purchase long-term bonds because my parents are likely to live more than 10 years. Experts uniformly do not foresee a resurgence of inflation on Pogo Island, so I am certain that the total returns produced by the bonds will cover my parents’ spending needs for many years to come. There should be no exchange rate risk because the bonds are denominated in local currency. I want to buy the Pogo Island bonds, but am not willing to distort my portfolio’s long-term asset allocation to do so. The overall mix of stocks, bonds, and other investments should not change. Therefore, I am considering selling one of my U.S. bond funds to raise cash to buy the Pogo Island bonds. One possibility is my High Yield Bond Fund, which has declined 5% in value year to date. I am not excited about this fund’s prospects; in fact I think it is likely to decline more, but there is a small probability that it could recover very quickly. So I have decided instead to sell my Core Bond Fund that has appreciated 5% this year. I expect this investment to continue to deliver attractive returns, but there is a small chance this year’s gains might disappear quickly.
Once that shift is accomplished, my investments will be in great shape. The sole exception is my Small Company Fund, which has performed poorly. I plan to sell this investment as soon as the price increases to my original cost.
Identify three behavioral finance concepts illustrated in Pierce’s comments and describe each of the
three concepts. Discuss how an investor practicing standard or traditional finance would challenge
each of the three concepts.
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C H A P T E R 1 2 Behavioral Finance and Technical Analysis 413
1. Conservatism implies that investors will at first respond too slowly to new information, leading to
trends in prices. Representativeness can lead them to extrapolate trends too far into the future and
overshoot intrinsic value. Eventually, when the pricing error is corrected, we observe a reversal.
2. Out-of-favor stocks will exhibit low prices relative to various proxies for intrinsic value such as
earnings. Because of regret avoidance, these stocks will need to offer a more attractive rate of
return to induce investors to hold them. Thus, low P/E stocks might on average offer higher rates
of return.
3. At liquidation, price will equal NAV. This puts a limit on fundamental risk. Investors need only
carry the position for a few months to profit from the elimination of the discount. Moreover, as
the liquidation date approaches, the discount should dissipate. This greatly limits the risk that
the discount can move against the investor. At the announcement of impending liquidation, the
discount should immediately disappear, or at least shrink considerably.
4. By the time the news of the recession affects bond yields, it also ought to affect stock prices. The
market should fall before the confidence index signals that the time is ripe to sell.
E-INVESTMENTS EXERCISES
1. Log on to finance.yahoo.com to find the monthly dividend-adjusted closing prices for the most recent 4 years for Abercrombie & Fitch (ANF). Also collect the closing level of the S&P 500 Index over the same period.
a. Calculate the 4-month moving average of both the stock and the S&P 500 over time. For each series, use Excel to plot the moving average against the actual level of the stock price or index. Examine the instances where the moving average and price series cross. Is the stock more or less likely to increase when the price crosses through the moving average? Does it matter whether the price crosses the mov- ing average from above or below? How reliable would an investment rule based on moving averages be? Perform your analysis for both the stock price and the S&P 500.
b. Calculate and plot the relative strength of the stock compared to the S&P 500 over the sample period. Find all instances in which relative strength of the stock increases by more than 10 percentage points (e.g., an increase in the relative strength index from .93 to 1.03) and all those instances in which relative strength of the stock decreases by more than 10 percentage points. Is the stock more or less likely to outperform the S&P in the following 2 months when relative strength has increased or to underperform when relative strength has decreased? In other words, does relative strength continue? How reliable would an investment rule based on relative strength be?
2. The Yahoo! Finance charting function allows you to specify comparisons between companies by choosing the Technical Analysis tab. Short interest ratios are found in the Key Statistics table. Prepare charts of moving averages and obtain short interest ratios for GE and SWY. Prepare a 1-year chart of the 50- and 200-day average price of GE, SWY, and the S&P 500 Index.
a. Which, if either, of the companies is priced above its 50- and 200-day averages?
b. Would you consider their charts as bullish or bearish? Why?
c. What are the short interest ratios for the two companies?
SOLUTIONS TO CONCEPT CHECKS
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13 1 PA
R T I II
IN THIS CHAPTER, we turn to the vast liter- ature testing models of risk and return. The very existence of such a vast literature sug- gests a serious problem is involved— testing these models is not trivial. Indeed, an impor- tant part of the work here is to understand the challenges in doing so.
All models of capital asset pricing have two parts. First, they derive the optimal portfo- lio of an individual investor, conditional on a utility function (describing how an investor trades off risk against expected return) and an input list that includes estimates of port- folio expected returns and risk. Second, they derive predictions about expected returns on capital assets in equilibrium, when investors complete the trades necessary to arrive at their personal optimal portfolios.
Obviously, the flow of new information alone will change input lists and thus desired portfolios. Here is where the efficient mar- ket hypothesis (EMH) kicks in. If asset prices reflect all available information, then changes of asset prices resulting from new informa- tion will have zero means, that is, prices will follow random walks. 1 The response to new
information will introduce noise around the predictions of the model, but by itself this should not cause any difficulty that cannot be overcome with appropriate statistical meth- ods and lots of data. But when the EMH is off, even temporarily, by economically signifi- cant margins, changes in prices and expected returns will not change randomly and model predictions can be affected. This is why a test of an asset pricing model is of necessity a joint test of the EMH.
The single-factor CAPM has one key impli- cation that can be expressed in either of two ways: The market portfolio is mean-variance efficient, and (equivalently) the risk premium (expected excess return) on each individual asset is proportional to its beta, E ( R i ) 5 b i E ( R M ). The first statement is, in practice, untestable because we do not observe the market port- folio. However, if a broad index is sufficiently well diversified, even if not mean-variance efficient, it may nevertheless support the mean-beta relationship (the SML) using the arguments of the APT.
Testing the ex-ante mean-variance effi- ciency of a particular market index can never
CHAPTER THIRTEEN
Empirical Evidence
on Security Returns
1 Actually, prices will show an upward drift since expected rates of return are positive. But over short time horizons, this drift is
trivial compared to volatility. For example, at a daily horizon, the expected rate of return is around 5 basis points (corresponding to
an annual return of 12%). The daily standard deviation of stock prices is an order of magnitude higher, typically exceeding 2% for
individual stocks.
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(concluded)
415
13.1 The Index Model and the Single-Factor APT
be a conclusive test of the CAPM. In any sample, there always is an ex-post efficient portfolio that will never be identical to the index. How do we measure “distance from efficiency,” and what would constitute a rejection of the model? Given these difficulties, the mean-beta equa- tion has been the test arena of most research. However, most of these tests are better inter- preted as tests of the APT (rather than the CAPM) since we know from the outset that the index may not be mean-variance efficient but may nevertheless be well-diversified.
We begin with tests of the single-factor security market line, the theater where the basic meth- odologies have been developed, and then pro- ceed to multifactor models with emphasis on the empirically motivated Fama-French three-factor model. We show how this research may be inter- preted as tests of Merton’s multifactor ICAPM. We end this part of the chapter with a section that brings liquidity into the empirical framework. We devote a section to the theoretically appealing consumption CAPM in order to present the equity premium puzzle, and end with an assessment of where research into asset pricing is headed.
The Expected Return–Beta Relationship Recall that if the expected return–beta relationship holds with respect to an observable ex
ante efficient index, M, the expected rate of return on any security i is
E(ri) 5 rf 1 bi 3E(rM) 2 rf4 (13.1) where b i is defined as Cov(ri, rM)/sM
2 .
This is the most commonly tested implication of the CAPM. Early simple tests
followed three basic steps: establishing sample data, estimating the SCL (security charac-
teristic line), and estimating the SML (security market line).
Setting Up the Sample Data Determine a sample period of, for example, 60 monthly holding periods (5 years). For each of the 60 holding periods, collect the rates of return on
100 stocks, a market portfolio proxy (e.g., the S&P 500), and 1-month (risk-free) T-bills. Your
data thus consist of
r it 5 6,000 returns on the 100 stocks over the 60-month sample period; i 5 1, . . . , 100, and t 5 1, . . . , 60.
r Mt 5 60 observations of the returns on the S&P 500 index over the sample period (one each month).
r ft 5 60 observations of the risk-free rate (one each month).
This constitutes a table of 102 3 60 5 6,120 rates of return.
Estimating the SCL View Equation 13.1 as a security characteristic line (SCL), as in Chapter 8. For each stock, i, you estimate the beta coefficient as the slope of a first-pass regression equation. (The terminology first-pass regression is due to the fact that the esti- mated coefficients will be used as input into a second-pass regression. )
rit 2 rft 5 ai 1 bi(rMt 2 rft) 1 eit
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416 P A R T I I I Equilibrium in Capital Markets
You will use the following statistics in later analysis:
ri 2 rf 5 Sample averages (over the 60 observations) of the excess return on each of the 100 stocks.
b i 5 Sample estimates of the beta coefficients of each of the 100 stocks.
rM 2 rf 5 Sample average of the excess return of the market index.
s 2 ( e i ) 5 Estimates of the variance of the residuals for each of the 100 stocks.
The sample average excess returns on each stock and the market portfolio are taken as
estimates of expected excess returns, and the values of b i are estimates of the true beta coefficients for the 100 stocks during the sample period. s
2 ( e i ) estimates the nonsystem-
atic risk of each of the 100 stocks. It is understood that all these statistics include estima-
tion errors.
a. How many regression estimates of the SCL do we have from the sample?
b. How many observations are there in each of the regressions?
c. According to the CAPM, what should be the intercept in each of these regressions?
CONCEPT CHECK 13.1
Estimating the SML Now view Equation 13.1 as a security market line (SML) with 100 observations for the stocks in your sample. You can estimate g 0 and g 1 in the following
second-pass regression equation with the estimates b i from the first pass as the independent variable:
ri 2 rf 5 g0 1 g1bi i 5 1, c, 100 (13.2)
Compare Equations 13.1 and 13.2; you should conclude that if the CAPM is valid, then g 0
and g 1 should satisfy
g0 5 0 and g15 rM 2 rf
In fact, however, you can go a step further and argue that the key property of the expected
return–beta relationship described by the SML is that the expected excess return on securi-
ties is determined only by the systematic risk (as measured by beta) and should be inde- pendent of the nonsystematic risk, as measured by the variance of the residuals, s
2 ( e i ),
which also were estimated from the first-pass regression. These estimates can be added as
a variable in Equation 13.2 of an expanded SML that now looks like this:
ri 2 rf 5 g0 1 g1bi 1 g2s 2(ei) (13.3)
This second-pass regression equation is estimated with the hypotheses
g0 5 0; g1 5 rM 2 rf ; g2 5 0
The hypothesis that g 2 5 0 is consistent with the notion that nonsystematic risk should
not be “priced,” that is, that there is no risk premium earned for bearing nonsystem-
atic risk. More generally, according to the CAPM, the risk premium depends only on
beta. Therefore, any additional right-hand-side variable in Equation 13.3 beyond beta should have a coefficient that is insignificantly different from zero in the second-pass
regression.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 417
Tests of the CAPM Early tests of the CAPM performed by John Lintner,
2 and later replicated by Merton Miller
and Myron Scholes, 3 used annual data on 631 NYSE stocks for 10 years, 1954 to 1963, and
produced the following estimates (with returns expressed as decimals rather than percentages):
Coefficient: g 0 5 .127 g 1 5 .042 g 2 5 .310
Standard error: .006 .006 .026
Sample average: rM 2 rf 5 .165
These results are inconsistent with the CAPM. First, the estimated SML is “too flat”;
that is, the g 1 coefficient is too small. The slope should equal rM 2 rf 5 .165 (16.5% per year), but it is estimated at only .042. The difference, .122, is about 20 times the standard
error of the estimate, .006, which means that the measured slope of the SML is less than it
should be by a statistically significant margin. At the same time, the intercept of the esti-
mated SML, g 0 , which is hypothesized to be zero, in fact equals .127, which is more than
20 times its standard error of .006.
2 John Lintner, “Security Prices, Risk and Maximal Gains from Diversification,” Journal of Finance 20 (December 1965).
3 Merton H. Miller and Myron Scholes, “Rate of Return in Relation to Risk: A Reexamination of Some Recent
Findings,” in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger, 1972). 4 Richard Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and Potential Testability of the
Theory,” Journal of Financial Economics 4 (1977).
a. What is the implication of the empirical SML being “too flat”?
b. Do high- or low-beta stocks tend to outperform the predictions of the CAPM?
c. What is the implication of the estimate of g 2 ?
CONCEPT CHECK 13.2
The two-stage procedure employed by these researchers (i.e., first estimate security
betas using a time-series regression and then use those betas to test the SML relationship
between risk and average return) seems straightforward, and the rejection of the CAPM
using this approach is disappointing. However, it turns out that there are several difficulties
with this approach. First and foremost, stock returns are extremely volatile, which lessens
the precision of any tests of average return. For example, the average standard deviation of
annual returns of the stocks in the S&P 500 is about 40%; the average standard deviation
of annual returns of the stocks included in these tests is probably even higher.
In addition, there are fundamental concerns about the validity of the tests. First, the market
index used in the tests is surely not the “market portfolio” of the CAPM. Second, in light of
asset volatility, the security betas from the first-stage regressions are necessarily estimated
with substantial sampling error and therefore cannot readily be used as inputs to the second-
stage regression. Finally, investors cannot borrow at the risk-free rate, as assumed by the
simple version of the CAPM. Let us investigate the implications of these problems in turn.
The Market Index In what has come to be known as Roll’s critique, Richard Roll 4 pointed out that:
1. There is a single testable hypothesis associated with the CAPM: The market portfo-
lio is mean-variance efficient.
2. All the other implications of the model, the best-known being the linear relation
between expected return and beta, follow from the market portfolio’s efficiency
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418 P A R T I I I Equilibrium in Capital Markets
and therefore are not independently testable. There is an “if and only if” relation
between the expected return–beta relationship and the efficiency of the market
portfolio.
3. In any sample of observations of individual returns there will be an infinite number
of ex post (i.e., after the fact) mean-variance efficient portfolios using the sample-
period returns and covariances (as opposed to the ex ante expected returns and covariances). Sample betas of individual assets estimated against each such ex-post
efficient portfolio will be exactly linearly related to the sample average returns of
these assets. In other words, if betas are calculated against such portfolios, they will
satisfy the SML relation exactly whether or not the true market portfolio is mean-
variance efficient in an ex ante sense.
4. The CAPM is not testable unless we know the exact composition of the true market
portfolio and use it in the tests. This implies that the theory is not testable unless all individual assets are included in the sample.
5. Using a proxy such as the S&P 500 for the market portfolio is subject to two dif-
ficulties. First, the proxy itself might be mean-variance efficient even when the true
market portfolio is not. Conversely, the proxy may turn out to be inefficient, but
obviously this alone implies nothing about the true market portfolio’s efficiency.
Furthermore, most reasonable market proxies will be very highly correlated with
each other and with the true market portfolio whether or not they are mean-variance
efficient. Such a high degree of correlation will make it seem that the exact compo-
sition of the market portfolio is unimportant, whereas the use of different proxies
can lead to quite different conclusions. This problem is referred to as benchmark error , because it refers to the use of an incorrect benchmark (market proxy) portfo- lio in the tests of the theory.
Roll and Ross 5 and Kandel and Stambaugh
6 expanded Roll’s critique. Essentially, they
argued that tests that reject a positive relationship between average return and beta point
to inefficiency of the market proxy used in those tests, rather than refuting the theoretical
expected return–beta relationship. They demonstrate that even if the CAPM is true, highly
diversified portfolios, such as the value- or equally weighted portfolios of all stocks in the
sample, may fail to produce a significant average return–beta relationship.
Kandel and Stambaugh considered the properties of the usual two-pass test of the
CAPM in an environment in which borrowing is restricted but the zero-beta version of
the CAPM holds. In this case, you will recall that the expected return–beta relationship
describes the expected returns on a stock, a portfolio E on the efficient frontier, and that portfolio’s zero-beta companion, Z (see Equation 9.12):
E(ri) 2 E(rZ) 5 bi 3E(rE) 2 E(rZ)4 (13.4) where b i denotes the beta of security i on efficient portfolio E.
We cannot construct or observe the efficient portfolio E (because we do not know expected returns and covariances of all assets), and so we cannot estimate Equation 13.4
directly. Kandel and Stambaugh asked what would happen if we followed the common
procedure of using a market proxy portfolio M in place of E, and used as well the more
5 Richard Roll and Stephen A. Ross, “On the Cross-Sectional Relation between Expected Return and Betas,”
Journal of Finance 50 (1995), pp. 185–224. 6 Schmuel Kandel and Robert F. Stambaugh, “Portfolio Inefficiency and the Cross-Section of Expected Returns,”
Journal of Finance 50 (1995), pp. 185–224; “A Mean-Variance Framework for Tests of Asset Pricing Models,” Review of Financial Studies 2 (1989), pp. 125–56; “On Correlations and Inferences about Mean-Variance Efficiency,” Journal of Financial Economics 18 (1987), pp. 61–90.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 419
efficient generalized least squares regression procedure in estimating the second-pass
regression for the zero-beta version of the CAPM, that is,
ri 2 rZ 5 g0 1 g1 3 (Estimated bi)
They showed that the estimated values of g 0 and g 1 will be biased by a term proportional
to the relative efficiency of the market proxy. If the market index used in the regression is
fully efficient, the test will be well specified. But the second-pass regression will provide a
poor test of the CAPM if the proxy for the market portfolio is not efficient. Thus, we still
cannot test the model in a meaningful way without a reasonably efficient market proxy.
Unfortunately, it is impossible to determine how efficient our market index is, so we can-
not tell how good our tests are.
Given the impossibility of testing the CAPM directly, we can retreat to testing the APT,
which produces the same mean-beta equation (the security market line). 7 This model
depends only on the index portfolio being well diversified. Choosing a broad market index
allows us to test the SML as applied to the chosen index.
Measurement Error in Beta It is well known in statistics that if the right-hand-side variable of a regression equation
is measured with error (in our case, beta is measured with error and is the right-hand-side
variable in the second-pass regression), then the slope coefficient of the regression equation
will be biased downward and the intercept biased upward. This is consistent with the findings
cited above; g 0 was higher than predicted by the CAPM and g 1 was lower than predicted.
Indeed, a well-controlled simulation test by Miller and Scholes 8 confirms these argu-
ments. In this test a random-number generator simulated rates of return with covariances
similar to observed ones. The average returns were made to agree exactly with the CAPM.
Miller and Scholes then used these randomly generated rates of return in the tests we
have described as if they were observed from a sample of stock returns. The results of this
“simulated” test were virtually identical to those reached using real data, despite the fact
that the simulated returns were constructed to obey the SML, that is, the true g coefficients were g0 5 0, g1 5 rM 2 rf , and g 2 5 0.
9
This postmortem of the early test gets us back to square one. We can explain away
the disappointing test results, but we have no positive results to support the CAPM-APT
implications.
The next wave of tests was designed to overcome the measurement error problem that
led to biased estimates of the SML. The innovation in these tests, pioneered by Black,
Jensen, and Scholes, 10
was to use portfolios rather than individual securities. Combining
securities into portfolios diversifies away most of the firm-specific part of returns, thereby
7 Although the APT strictly applies only to well-diversified portfolios, the discussion in Chapter 9 shows that
optimization in a single-index market as prescribed by Treynor and Black will generate strong pressure on single
securities to satisfy the mean-beta equation as well. 8 Miller and Scholes, “Rate of Return in Relation to Risk.”
9 In statistical tests, there are two possible errors: Type I and Type II. A Type I error means that you reject a null
hypothesis (for example, a hypothesis that beta does not affect expected returns) when it is actually true. This
is sometimes called a false positive, in which you incorrectly decide that a relationship exists when it actually does not. The probability of this error is called the significance level of the test statistic. Thresholds for rejection of the null hypothesis are usually chosen to limit the probability of Type I error to below 5%. Type II error is
a false negative, in which a relationship actually does exist, but you fail to detect it. The power of a test equals (1 2 probability of Type II). Miller and Scholes’s experiment showed that early tests of the CAPM had low
power. 10
Fischer Black, Michael C. Jensen, and Myron Scholes, “The Capital Asset Pricing Model: Some Empirical
Tests,” in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger, 1972).
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420 P A R T I I I Equilibrium in Capital Markets
enhancing the precision of the estimates of beta and the expected rate of return of the
portfolio of securities. This mitigates the statistical problems that arise from measurement
error in the beta estimates.
Testing the model with diversified portfolios rather than individual securities completes
our retreat to the APT. Additionally, combining stocks into portfolios reduces the number
of observations left for the second-pass regression. Suppose we group the 100 stocks into
five portfolios of 20 stocks each. If the residuals of the 20 stocks in each portfolio are prac-
tically uncorrelated, the variance of the portfolio residual will be about one-twentieth the
residual variance of the average stock. Thus the portfolio beta in the first-pass regression
will be estimated with far better accuracy. However, with portfolios of 20 stocks each, we
are left with only five observations for the second-pass regression.
To get the best of this trade-off, we need to construct portfolios with the largest pos-
sible dispersion of beta coefficients. Other things equal, a regression yields more accu-
rate estimates the more widely spaced the observations of the independent variables.
We therefore will attempt to maximize the range of the independent variable of the
second-pass regression, the portfolio betas. Rather than allocate 20 stocks to each portfolio
randomly, we first rank stocks by betas. Portfolio 1 is formed from the 20 highest-beta
stocks and portfolio 5 the 20 lowest-beta stocks. A set of portfolios with small nonsys-
tematic components, e P , and widely spaced betas will yield reasonably powerful tests of the SML.
Fama and MacBeth (FM) 11
used this methodology to verify that the observed relation-
ship between average excess returns and beta is indeed linear and that nonsystematic risk
does not explain average excess returns. Using 20 portfolios constructed according to the
Black, Jensen, and Scholes methodology, FM expanded the estimation of the SML equa-
tion to include the square of the beta coefficient (to test for linearity of the relationship
between returns and betas) and the estimated standard deviation of the residual (to test for
the explanatory power of nonsystematic risk). For a sequence of many subperiods, they
estimated for each subperiod the equation
ri 5 g0 1 g1bi 1 g2bi 2 1 g3s(ei) (13.5)
The term g 2 measures potential nonlinearity of return, and g 3 measures the explanatory
power of nonsystematic risk, s ( e i ). According to the CAPM, both g 2 and g 3 should have coefficients of zero in the second-pass regression.
FM estimated Equation 13.5 for every month of the period January 1935 through June
1968. The results are summarized in Table 13.1 , which shows average coefficients and
t -statistics for the overall period as well as for three subperiods. FM observed that the coefficients on residual standard deviation (nonsystem-
atic risk), denoted by g 3 , fluctuated greatly from month
to month, and its t -statistics were insignificant despite large average values. Thus, the overall test results were
reasonably favorable to the security market line of the
CAPM (or perhaps more accurately of the APT that FM
actually tested). But time has not been favorable to the
CAPM since.
Recent replications of the FM test show that results
deteriorate in later periods (since 1968). Worse, even for
the FM period, 1935–1968, when the equally weighted
11 Eugene Fama and James MacBeth, “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political
Economy 81 (March 1973).
a. According to the CAPM and the data in Table 13.1 , what are the predicted values of g 0 , g 1 , g 2 , and g 3 in the Fama-MacBeth regressions for the period 1946–1955?
b. What would you conclude if you performed the Fama and MacBeth tests and found that the coefficients on b 2 and s ( e ) were positive?
CONCEPT CHECK 13.3
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C H A P T E R 1 3 Empirical Evidence on Security Returns 421
NYSE-stock portfolio they used as the market index is replaced with the more appropriate
value-weighted index, results turn against the model. In particular, the slope of the SML
clearly is too flat.
Table 13.1
Summary of Fama and MacBeth (1973) study (all rates in basis points per month)
Period 1935/6–1968 1935–1945 1946–1955 1956/6–1968
Av. r f 13 2 9 26 Av. g 0 2 r f 8 10 8 5 Av. t ( g 0 2 r f ) 0.20 0.11 0.20 0.10
Av. r M 2 r f 130 195 103 95 Av. g 1 114 118 209 34 Av. t ( g 1 ) 1.85 0.94 2.39 0.34
Av. g 2 2 26 2 9 2 76 0 Av. t ( g 2 ) 2 0.86 2 0.14 2 2.16 0
Av. g 3 516 817 2 378 960 Av. t ( g 3 ) 1.11 0.94 2 0.67 1.11
Av. R -SQR 0.31 0.31 0.32 0.29
13.2 Tests of the Multifactor CAPM and APT
Three types of factors are likely candidates to augment the market risk factor in a multi-
factor SML: (1) Factors that hedge consumption against uncertainty in prices of impor-
tant consumption categories (e.g., housing or energy) or general inflation; (2) factors that
hedge future investment opportunities (e.g., interest rates or the market risk premium); and
(3) factors that hedge assets missing from the market index (e.g., labor income or private
business).
As we learned from Merton’s ICAPM (Chapter 9), these extra-market sources of risk
will command a risk premium if there is significant demand to hedge them. We begin with
the third source because there is little doubt that nontraded assets in the personal portfolios
of investors affect demand for traded risky assets. Hence, a factor representing these assets,
that is, one correlated with their returns, should affect risk premiums.
Labor Income The major factors in the omitted asset category are labor income and private business.
Taking on labor income first, Mayers 12
viewed each individual as being endowed with
labor income but able to trade only securities and an index portfolio. His model creates
a wedge between betas measured against the traded, index portfolio and betas measured
against the true market portfolio, which includes aggregate labor income. The result of his
model is an SML that is flatter than the simple CAPM’s. Most of this income is positively
correlated with the market index, and it has substantial value compared to the market
value of the securities in the market index. Its absence from the index pushes the slope
12 David Mayers, “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty,” in Studies in the
Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger, 1972), pp. 223–48.
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422 P A R T I I I Equilibrium in Capital Markets
of the observed SML (return vs. beta measured against the index) below the return of the
index portfolio. 13
If the value of labor income is not perfectly correlated with the market-index portfolio,
then the possibility of negative returns to labor will represent a source of risk not fully cap-
tured by the index. But suppose investors can trade a portfolio that is correlated with the
return on aggregate human capital. Then their hedging demands against the risk to the value
of their human capital might meaningfully influence security prices and risk premia. If so,
human capital risk (or some empirical proxy for it) can serve as an additional factor in a
multifactor SML. Stocks with a positive beta on the value of labor exaggerate exposure to
this risk factor; therefore, they will command lower prices, or equivalently, provide a larger-
than-CAPM risk premium. Thus, by adding this factor, the SML becomes multidimensional. Jagannathan and Wang
14 used the rate of change in aggregate labor income as a proxy
for changes in the value of human capital. In addition to the standard security betas esti-
mated using the value-weighted stock market index, which we denote b vw
, they also esti-
mated the betas of assets with respect to labor income growth, which we denote b labor
.
Finally, they considered the possibility that business cycles affect asset betas, an issue that
has been examined in a number of other studies. 15
These may be viewed as conditional betas, as their values are conditional on the state of the economy. Jagannathan and Wang
used the spread between the yields on low- and high-grade corporate bonds as a proxy for
the state of the business cycle and estimate asset betas relative to this business cycle vari-
able; we denote this beta as b prem
. With the estimates of these three betas for several stock
portfolios, Jagannathan and Wang estimated a second-pass regression which includes firm
size (market value of equity, denoted ME):
E(Ri) 5 c0 1 csize log(ME) 1 cvwb vw 1 cpremb
prem 1 claborb labor (13.6)
Jagannathan and Wang test their model with 100 portfolios that are designed to spread
securities on the basis of size and beta. Stocks are sorted into 10 size portfolios, and the stocks
within each size decile are further sorted by beta into 10 subportfolios, resulting in 100 port-
folios in total. Table 13.2 shows a subset of the various versions of the second-pass estimates.
The first two rows in the table show the coefficients and t -statistics of a test of the CAPM along the lines of the Fama and MacBeth tests introduced in the previous section. The result
is a sound rejection of the model, as the coefficient on beta is negative, albeit not significant.
The next two rows show that the model is not helped by the addition of the size factor.
The dramatic increase in R -square (from 1.35% to 57%) shows that size explains variations in average returns quite well while beta does not. Substituting the default premium and
labor income for size (panel B) results in a similar increase in explanatory power ( R -square of 55%), but the CAPM expected return–beta relationship is not redeemed. The default
premium is significant, while labor income is borderline significant. When we add size as
13 Asset betas on the index portfolio are likely positively correlated with their betas on the omitted asset (for
example, aggregate labor income). Therefore, the coefficient on asset beta in the SML regression (of returns on
index beta) will be downward biased, resulting in a slope smaller than average RM. In Equation 9.13 the observed
beta of most assets will be larger than the true beta whenever biM . biH sH
2
sM 2
.
14 Ravi Jagannathan and Zhenyu Wang, “The Conditional CAPM and the Cross-Section of Expected Returns,”
Journal of Finance 51 (March 1996), pp. 3–54. 15
For example, Campbell Harvey, “Time-Varying Conditional Covariances in Tests of Asset Pricing Models,”
Journal of Financial Economics 24 (October 1989), pp. 289–317; Wayne Ferson and Campbell Harvey, “The Variation of Economic Risk Premiums,” Journal of Political Economy 99 (April 1991), pp. 385–415; and Wayne Ferson and Robert Korajczyk, “Do Arbitrage Pricing Models Explain the Predictability of Stock Returns?”
Journal of Business 68 (July 1995), pp. 309–49.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 423
well, in the last two rows, we find it is no longer significant and only marginally increases
explanatory power.
Despite the clear rejection of the CAPM, we do learn two important facts from Table 13.2 .
First, conventional first-pass estimates of security betas are greatly deficient. They clearly
do not fully capture the cyclicality of stock returns and thus do not accurately measure the
systematic risk of stocks. This actually can be interpreted as good news for the CAPM in
that it may be possible to replace the simple beta with better estimates of systematic risk
and transfer the explanatory power of instrumental variables such as size and the default
premium to the index rate of return. Second, and more relevant to the work of Jagannathan
and Wang, is the conclusion that human capital will be important in any version of the
CAPM that better explains the systematic risk of securities.
Private (Nontraded) Business Whereas Jagannathan and Wang focus on labor income, Heaton and Lucas
16 estimate the
importance of proprietary business. We expect that private-business owners will reduce
demand for traded securities that are positively correlated with their specific entrepreneurial
income. If this effect is sufficiently important, aggregate demand for traded securities will
be determined in part by the covariance with aggregate noncorporate business income. The
risk premium on securities with high covariance with noncorporate business income should
be commensurately higher.
Table 13.2
Evaluation of various CAPM specifications
This table gives the estimates for the cross-sectional regression model
E(Rit) 5 c0 1 csizelog(MEi) 1 cvwbi vw 1 cprembi
prem 1 claborbi
labor
with either a subset or all of the variables. Here, Rit is the return on portfolio i (i 5 1, 2, . . . , 100) in month t (July 1963–December 1990), Rt
vw is the return on the value-weighted index of stocks, Rt21 prem is the yield spread between low- and high-grade corporate
bonds, and Rt labor is the growth rate in per capita labor income. The bi
vw is the slope coefficient in the OLS regression of Rit on a constant and Rt
vw. The other betas are estimated in a similar way. The portfolio size, log(MEi), is calculated as the equally weighted average of the logarithm of the market value (in millions of dollars) of the stocks in portfolio i. The regression models are estimated by using the Fama-MacBeth procedure. The “corrected t-values” take sampling errors in the estimated betas into account. All R2s are reported as percentages.
Coefficient c0 cvw cprem clabor csize R 2
A. The Static CAPM without Human Capital Estimate 1.24 20.10 1.35 t-value 5.16 20.28
Estimate 2.08 20.32 20.11 57.56 t-value 5.77 20.94 22.30
B. The Conditional CAPM with Human Capital Estimate 1.24 20.40 0.34 0.22 55.21 t-value 4.10 20.88 1.73 2.31
Estimate 1.70 20.40 0.20 0.10 20.07 64.73 t-value 4.14 21.06 2.72 2.09 21.30
16 John Heaton and Debora Lucas, “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk,”
Journal of Finance 55, no. 3 (June 2000), pp. 1163–98.
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424 P A R T I I I Equilibrium in Capital Markets
Consistent with theory, Heaton and Lucas find that households with higher invest-
ments in private business do in fact reduce the fraction of total wealth invested in equity.
Table 13.3 presents excerpts from their regression analysis, in which allocation of the
overall portfolio to stocks is the dependent variable. The share of private business in total
wealth (labeled “relative business”) receives negative and statistically significant coeffi-
cients in these regressions. Notice also the negative and significant coefficient on risk atti-
tude based on a self-reported degree of risk aversion.
Finally, Heaton and Lucas extend Jagannathan and Wang’s equation to include the rate
of change in proprietary-business wealth. They find that this variable also is significant and
improves the explanatory power of the regression. Here, too, the market rate of return does
not help explain the rate of return on individual securities and, hence, this implication of
the CAPM still must be rejected.
Early Versions of the Multifactor CAPM and APT The multifactor CAPM and APT are elegant theories of how exposure to systematic risk
factors should influence expected returns, but they provide little guidance concerning
which factors (sources of risk) ought to result in risk premiums. A test of this hypothesis
would require three stages:
Share of Stock in Assets
Stock Relative to Liquid Assets
Stock Relative to Financial Assets
Stock Relative to Total Assets
Intercept 0.71 0.53 0.24 (14.8) (21.28) (10.54)
Total income 3 10210 21.80 2.416 21.72 (20.435) (20.19) (20.85)
Net worth 3 10210 2.75 5.04 7.37 (0.895) (3.156) (5.02)
Relative business 20.14 20.50 20.32 (24.34) (229.31) (220.62)
Age of respondent 27.94 3 1024 26.99 3 1025 2.44 3 1023
(21.26) (20.21) (24.23) Risk attitude 20.05 20.02 20.02
(24.74) (23.82) (24.23) Relative mortgage 0.05 0.43 0.30
(1.31) (20.90) (16.19) Relative pension 0.07 20.41 20.31
(1.10) (211.67) (29.60) Relative real estate 20.04 20.44 20.31
(21.41) (227.00) (220.37) Adjusted R-square 0.03 0.48 0.40
Note: t-statistics in parentheses. Source: John Heaton and Debora Lucas, “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk,” Journal of Finance 55, no. 3 (June 2000), pp. 1163–98. Reprinted by permission of the publisher, Blackwell Publishing, Inc.
Table 13.3
Determinants of stockholdings
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C H A P T E R 1 3 Empirical Evidence on Security Returns 425
1. Specification of risk factors.
2. Identification of portfolios that hedge these fundamental risk factors.
3. Test of the explanatory power and risk premiums of the hedge portfolios.
A Macro Factor Model Chen, Roll, and Ross
17 identify several possible variables that might proxy for systematic
factors:
IP 5 Growth rate in industrial production.
EI 5 Changes in expected inflation measured by changes in short-term (T-bill)
interest rates.
UI 5 Unexpected inflation defined as the difference between actual and expected
inflation.
CG 5 Unexpected changes in risk premiums measured by the difference between the
returns on corporate Baa-rated bonds and long-term government bonds.
GB 5 Unexpected changes in the term premium measured by the difference between
the returns on long- and short-term government bonds.
With the identification of these potential economic factors, Chen, Roll, and Ross skipped
the procedure of identifying factor portfolios (the portfolios that have the highest correla-
tion with the factors). Instead, by using the factors themselves, they implicitly assumed
that factor portfolios exist that can proxy for the factors. They use these factors in a test
similar to that of Fama and MacBeth.
A critical part of the methodology is the grouping of stocks into portfolios. Recall that in
the single-factor tests, portfolios were constructed to span a wide range of betas to enhance
the power of the test. In a multifactor framework the efficient criterion for grouping is less
obvious. Chen, Roll, and Ross chose to group the sample stocks into 20 portfolios by size
(market value of outstanding equity), a variable that is known to be associated with average
stock returns.
They first used 5 years of monthly data to estimate the factor betas of the 20 portfolios
in 20 first-pass regressions.
r 5 a 1 bMrM 1 bIPIP 1 bEIEI 1 bUIUI 1 bCGCG 1 bGBGB 1 e (13.7a)
where M stands for the stock market index. Chen, Roll, and Ross used as the market index both the value-weighted NYSE index (VWNY) and the equally weighted NYSE index
(EWNY).
Using the 20 sets of first-pass estimates of factor betas as the independent variables,
they now estimated the second-pass regression (with 20 observations):
r 5 g0 1 gMbM 1 gIPbIP 1 gEIbEI 1 gUIbUI 1 gCGbCG 1 gGBbGB 1 e (13.7b)
where the gammas become estimates of the risk premiums on the factors.
Chen, Roll, and Ross ran this second-pass regression for every month of their sample
period, reestimating the first-pass factor betas once every 12 months. The estimated risk pre-
miums (the values for the parameters, g) were averaged over all the second-pass regressions.
Note in Table 13.4 that the two market indexes EWNY and VWNY are not statistically
significant (their t -statistics of 1.218 and 2 .633 are less than 2). Note also that the VWNY
17 Nai-Fu Chen, Richard Roll, and Stephen Ross, “Economic Forces and the Stock Market,” Journal of Business
59 (1986).
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426 P A R T I I I Equilibrium in Capital Markets
factor has the “wrong” sign in that it seems to imply a negative market-risk premium.
Industrial production (IP), the risk premium on corporate bonds (CG), and unanticipated
inflation (UI) are the factors that appear to have significant explanatory power.
A EWNY IP EI UI CG GB Constant
5.021 14.009 20.128 20.848 0.130 25.017 6.409 (1.218) (3.774) (21.666) (22.541) (2.855) (21.576) (1.848)
B VWNY IP EI UI CG GB Constant 22.403 11.756 20.123 20.795 8.274 25.905 10.713 (20.633) (3.054) (21.600) (22.376) (2.972) (21.879) (2.755)
VWNY 5 Return on the value-weighted NYSE index; EWNY 5 Return on the equally weighted NYSE index; IP 5 Monthly growth rate in industrial production; EI 5 Change in expected inflation; UI 5 Unanticipated inflation; CG 5 Unanticipated change in the risk premium (Baa and under return 2 Long-term government bond return); GB 5 Unanticipated change in the term structure (long- term government bond return 2 Treasury-bill rate); Note that t-statistics are in parentheses.
Source: Modified from Nai-Fu Chen, Richard Roll, and Stephen Ross, “Economic Forces and the Stock Market,” Journal of Business 59 (1986). Reprinted by permission of the publisher, The University of Chicago Press.
Table 13.4
Economic variables and pricing (percent per month 3 10), multivariate approach
13.3 Fama-French-Type Factor Models
The multifactor models that currently occupy center stage are the three-factor models
introduced by Fama and French (FF) and its close relatives. 18
The systematic factors in
the FF model are firm size and book-to-market ratio (B/M) as well as the market index.
These additional factors are empirically motivated by the observations, documented in
Chapter 11, that historical-average returns on stocks of small firms and on stocks with
high ratios of book equity to market equity (B/M) are higher than predicted by the secu-
rity market line of the CAPM.
However, Fama and French did more than document the empirical role of size and B/M
in explaining rates of return. They also introduced a general method to generate factor port-
folios and applied their method to these firm characteristics. Exploring this innovation is a
useful way to understand the empirical building blocks of a multifactor asset pricing model.
Suppose you find, as Fama and French did, that stock market capitalization (or “market
cap”) seems to predict alpha values in a CAPM equation. On average, the smaller the mar-
ket cap, the greater the alpha of a stock. This finding would add size to the list of anomalies
that refute the CAPM.
But suppose you believe that size varies with sensitivity to changes in future investment
opportunities. Then, what appears as alpha in a single factor CAPM is really an extra-market
source of risk in a multifactor CAPM. If this sounds far-fetched, here’s a story: When
investors anticipate a market downturn, they adjust their portfolios to minimize their expo-
sure to losses. Suppose that small stocks generally are harder hit in down markets, akin to a
larger beta in bad times. Then investors will avoid such stocks in favor of the less-sensitive
stocks of larger firms. This would explain a risk premium to small size beyond the beta on
contemporaneous market returns. An “alpha” for size may be instead an ICAPM risk pre-
mium for assets with greater sensitivity to deterioration in future investment opportunities.
18 Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal
of Financial Economics 33 (1993), pp. 3–56.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 427
The FF innovation is a method to quantify the size risk premium. Recall that the dis-
tribution of corporate size is asymmetric: a few big and many small corporations. Since
the NYSE is the exchange where bigger stocks trade, Fama and French first determine
the median size of NYSE stocks. They use this median to classify all traded U.S. stocks
(NYSE 1 AMEX 1 NASDAQ) as big or small and create one portfolio from big stocks
and another from small stocks. Finally, each of these portfolios is value-weighted for effi-
cient diversification.
As in the APT, Fama and French construct a zero-net-investment size-factor portfolio
by going long the small- and going short the big-stock portfolio. The return of this portfo-
lio, called SMB (small minus big), is simply the return on the small-stock portfolio minus
the return on the big-stock portfolio. If size is priced, then this portfolio will exhibit a risk
premium. Because the SMB is practically well diversified (on the order of 4,000 stocks),
it joins the market-index portfolio in a two-factor APT model with size as the extra-market
source of risk. In the two-factor SML, the risk premium on any asset should be determined
by its loadings (betas) on the two factor portfolios. This is a testable hypothesis.
Fama and French use this approach to form both size and book-to-market ratio (B/M)
factors. To create these two extra-market risk factors, they double-sort stocks by both size
and B/M. They break the U.S. stock population into three groups based on B/M ratio: the
bottom 30% (low), the middle 40% (medium), and the top 30% (high). 19
Now six portfo-
lios are created based on the intersections of the size and B/M sorts: Small/Low; Small/
Medium; Small/High; Big/Low; Big/Medium; Big/High. Each of these six portfolios is
value weighted.
The returns on the Big and Small portfolio are:
RS 5 1@3(RS /L 1 RS/M 1 RS /H); RB 5
1@3(RB /L 1 RB/M 1 RB/H)
Similarly, the returns on the high and low (Value and Growth 20
) portfolios are:
RH 5 1@2(RSH 1 RBH); RL 5
1@2(RSL 1 RBL)
The returns of the zero-net-investment factors SMB (Small minus Big, i.e., Long Small
and Short Big), and HML (High minus Low, i.e., Long High B/M and Short Low B/M) are
created from these portfolios:
RSMB 5 RS 2 RB; RHML 5 RH 2 RL
We measure the sensitivity of individual stocks to the factors by estimating the factor betas
from first-pass regressions of stock excess returns on the excess return of the market index
as well as on R SMB and R HML . These factor betas should, as a group, predict the total risk premium. Therefore, the Fama-French three-factor asset-pricing model is
21
E(ri) 2 rf 5 ai 1 bi3E(rM) 2 rf4 1 siE3SMB4 1 hiE3HML4 (13.8) 19
Fama and French could have experimented with optimal break points for the three B/M groups, but such an
approach might quickly give way to data mining. 20
High B/M stocks are called value assets because, for the large part, their market values derive from assets already in place. Low B/M are called growth stocks because their market values derive from expected growth in future cash flows. One needs to assume high growth to justify the prices at which the assets trade. At the same
time, however, a firm that falls into hard times will see its market price fall and its B/M ratio rise. So some of
the so-called value firms may actually be distressed firms. This subgroup of the value-firm portfolio may well
account for the value premium of the B/M factor. 21
We subtract the risk-free rate from the return on the market portfolio, but not from the SMB and HML returns
because the SMB and HML factors are zero-net-investment portfolios. Hence their entire return already is a pre- mium. There is no opportunity cost from giving up the risk-free investment to switch into these portfolios.
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428 P A R T I I I Equilibrium in Capital Markets
The coefficients b i , s i , and h i are the betas (also called loadings in this con- text) of the stock on the three factors.
If these are the only risk factors, excess
returns on all assets should be fully
explained by risk premiums due to
these factor loadings. In other words,
if these factors fully explain asset
returns, the intercept of the equation
should be zero.
Goyal 22
surveys asset pricing
tests. He applies Equation 13.8 to the
returns of 25 portfolios of all U.S.
stocks sorted by size and B/M ratio.
Figure 13.1 shows the average actual
return of each portfolio over the period
1946–2010 against returns predicted
by the CAPM (panel A) and by the FF
three-factor model. In this test, the FF
model provides a clear improvement
over the CAPM.
Notice in panel A that the predicted
returns are almost the same for all
portfolios. This is indeed a weakness
of tests with portfolios that are sorted
on size and B/M, but not on beta. As a
result, all portfolios have betas near 1.0.
Adding a sort on beta to a 5 3 5 sort
on size and B/M will raise the number
of portfolios from 25 to 125. This is
unwieldy. But advances in economet-
rics and computing power will allow
these types of tests to advance.
Size and B/M as Risk Factors Liew and Vassalou
23 show that returns
on style portfolios (HML or SMB)
seem to predict GDP growth, and
thus may in fact capture some aspects
of business cycle risk. Each bar in
Figure 13.2 is the average difference
in the return on the HML or SMB portfolio in years before good GDP growth versus in
years with poor GDP growth. Positive values mean the portfolio does better in years prior
to good macroeconomic performance. The predominance of positive values leads them to
conclude that the returns on the HML and SMB portfolios are positively related to future
Figure 13.1 CAPM versus the Fama and French model. The figure plots the average actual returns versus returns predicted by CAPM and the FF model for 25 size and book-to-market double-sorted portfolios.
Source: Amit Goyal, “Empirical Cross Sectional Asset Pricing: A Survey,” Financial Markets and Portfolio Management 26 (2012), pp. 3–38.
A: CAPM Fit
A ct
u a l R
e tu
rn (
% p
e r
m o
n th
)
0.0 0.2
Predicted Return from CAPM (% per month)
0.4 0.6 0.8 1.0 1.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
B: Fama and French (1993) Fit
A ct
u a l R
e tu
rn (
% p
e r
m o
n th
)
0.0 0.2
Predicted Return from Fama and French Model (% per month)
0.4 0.6 0.8 1.0 1.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
22
Amit Goyal, “Empirical Cross Sectional Asset Pricing: A Survey,” Financial Markets and Portfolio Manage- ment 26 (2012), pp. 3–38.
23
J. Liew and M. Vassalou, “Can Book-to-Market, Size and Momentum Be Risk Factors That Predict Economic
Growth?” Journal of Financial Economics 57 (2000), pp. 221–45.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 429
growth in the macroeconomy, and so may be proxies for business cycle risk. Thus, at least
part of the size and value premiums may reflect rational rewards for greater risk exposure.
Petkova and Zhang 24
also try to tie the average return premium on value (high B/M)
portfolios to risk premiums. Their approach uses a conditional CAPM. In the conventional
CAPM, we treat both the market risk premium and firm betas as given parameters. In con-
trast, as we noted earlier in the chapter, the conditional CAPM allows both of these terms
to vary over time, and possibly to co-vary. If a stock’s beta is higher when the market risk
premium is high, this positive association leads to a “synergy” in its risk premium, which
is the product of its incremental beta and market risk premium.
What might lead to such an association between beta and the market risk premium?
Zhang 25
focuses on irreversible investments. He notes that firms classified as value firms
(with high book-to-market ratios) on average will have greater amounts of tangible capital.
Investment irreversibility puts such firms more at risk for economic downturns because
in a severe recession, they will suffer from excess capacity from assets already in place.
In contrast, growth firms are better able to deal with a downturn by deferring investment
plans. The greater exposure of high book-to-market firms to recessions will result in higher
down-market betas. Moreover, some evidence suggests that the market risk premium also
is higher in down markets, when investors are feeling more economic pressure and anxiety.
24 Ralitsa Petkova and Lu Zhang, “Is Value Riskier than Growth?” Journal of Financial Economics 78 (2005),
pp. 187–202. 25
Lu Zhang, “The Value Premium,” Journal of Finance 60 (2005), pp. 67–103.
Figure 13.2 Difference in return to factor portfolios in year prior to above-average versus below-average GDP growth. Both SMB and HML portfolio returns tend to be higher in years preceding better GDP growth.
Source: J. Liew and M. Vassalou, “Can Book-to-Market, Size and Momentum Be Risk Factors That Predict Economic Growth?” Journal of Financial Economics 57 (2000), pp. 221–45. © 2000 with permission from Elsevier.
−20
−15
−10
−5
0
5
10
15
20
25
30
35
A u
st ra
li a
C a n
a d
a
Fr a n
ce
G e rm
a n
y
It a ly
Ja p
a n
N e
th e rl
a n
d s
Sw it
ze rl
a n
d
U .K
.
U .S
.P a st
Y e a r
R e tu
rn (
% )
HML SMB
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430 P A R T I I I Equilibrium in Capital Markets
The combination of these two factors might impart a positive correlation between the beta
of high B/M firms and the market risk premium.
To quantify these notions, Petkova and Zhang attempt to fit both beta and the market
risk premium to a set of “state variables,” that is, variables that summarize the state of the
economy. These are:
DIV 5 Market dividend yield.
DEFLT 5 Default spread on corporate bonds (Baa – Aaa rates).
TERM 5 Term structure spread (10-year–1-year Treasury rates).
TB 5 1-month T-bill rate.
They estimate a first-pass regression, but first substitute these state variables for beta as
follows:
rHML 5 a 1 brMt 1 ei
5 a 1 3b0 1 b1DIVt 1 b2DEFLTt 1 b3TERMt 1 b4TBt4rMt 1 ei ('''''''''')''''''''''*
5 bt d a time-varying beta
The strategy is to estimate parameters b 0 through b 4 and then fit beta using the values of the four state variables at each date. In this way, they can estimate beta in each period.
Similarly, one can directly estimate the determinants of a time-varying market risk pre-
mium, using the same set of state variables:
rMkt, t 2 rft 5 c0 1 c1DIVt 1 c2DEFLTt 1 c3TERMt 1 c4TBt 1 et
The fitted value from this regression is the estimate of the market risk premium.
Finally, Petkova and Zhang examine the relationship between beta and the market risk
premium. They define the state of economy by the size of the premium. A peak is defined
as the periods with the 10% lowest risk premiums; a trough has the 10% highest risk pre-
miums. The results, presented in Figure 13.3 , support the notion of a countercyclical value
Figure 13.3 HML beta in different economic states. The beta of the HML portfo- lio is higher when the market risk premium is higher.
Source: Ralitsa Petkova and Lu Zhang, “Is Value Riskier than Growth?” Journal of Financial Economics 78 (2005), pp. 187–202. © 2005 with permission from Elsevier.
.40
.05
−.33
−.15
−.40 −.30 −.20 −.10 .00 .10 .20 .30 .40 .50
B e ta
o f
H M
L P o
rt fo
li o
Peak Expansion
Value Beta < Growth Beta
TroughRecession
Value Beta > Growth Beta
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C H A P T E R 1 3 Empirical Evidence on Security Returns 431
beta: The beta of the HML portfolio is negative in good economies, meaning that the beta
of value stocks (high book-to-market) is less than that of growth stocks (low B/M), but the
reverse is true in recessions. While the covariance between the HML beta and the market
risk premium is not sufficient to explain by itself the average return premium on value port-
folios, it does suggest that at least part of the explanation may be a rational risk premium.
Behavioral Explanations On the other side of the debate, several authors make the case that the value premium is a
manifestation of market irrationality. The essence of the argument is that analysts tend to
extrapolate recent performance too far out into the future, and thus tend to overestimate
the value of firms with good recent performance. When the market realizes its mistake, the
prices of these firms fall. Thus, on average, “glamour firms,” which are characterized by
recent good performance, high prices, and lower book-to-market ratios, tend to underper-
form “value firms” because their high prices reflect excessive optimism relative to those
lower book-to-market firms.
Figure 13.4 , from a study by Chan, Karceski, and Lakonishok, 26
makes the case for
overreaction. Firms are sorted into deciles based on income growth in the past 5 years. By
construction, the growth rates uniformly increase from the first through the tenth decile.
The book-to-market ratio for each decile at the end of the 5-year period (the dashed line) tracks recent growth very well. B/M falls steadily with growth over the past 5 years. This
is evidence that past growth is extrapolated and then impounded in price. High past growth leads to higher prices and lower B/M ratios.
26 L.K.C. Chan, J. Karceski, and J. Lakonishok, “The Level and Persistence of Growth Rates,” Journal of Finance
58 (April 2003), pp. 643–84.
Figure 13.4 The book-to-market ratio reflects past growth, but not future growth prospects. B/M tends to fall with income growth experienced at the end of a 5-year period, but actually increases slightly with future income growth rates.
Source: L.K.C. Chan, J. Karceski, and J. Lakonishok, “The Level and Persistence of Growth Rates,” Journal of Finance 58 (April 2003), pp. 643–84. Used with permission of John Wiley and Sons, via Copyright Clear- ance Center.
B o
o k /M
a rk
e t
R a ti
o
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1 2 3 4 5 6 7 8 9 10
Growth Rate Decile
Median growth rate Beginning B/M Ending B/M
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432 P A R T I I I Equilibrium in Capital Markets
But B/M at the beginning of a 5-year period shows little or even a positive association with subsequent growth (the solid colored line), implying that market capitalization today
is inversely related to growth prospects. In other words, the firms with lower B/M (glamour firms) experience no better or even worse average future income growth than other firms.
The implication is that the market ignores evidence that past growth cannot be extrapolated
far into the future. Book-to-market may reflect past growth better than future growth, con-
sistent with extrapolation error.
More direct evidence supporting extrapolation error is provided by La Porta,
Lakonishok, Shleifer, and Vishny, 27
who examine stock price performance when actual
earnings are released to the public. Firms are classified as growth versus value stocks, and
the stock price performance at earnings announcements for 4 years following the classifi-
cation date is then examined. Figure 13.5 demonstrates that growth stocks underperform
value stocks surrounding these announcements. We conclude that when news of actual
earnings is released to the public, the market is relatively disappointed in stocks it has been
pricing as growth firms.
Momentum: A Fourth Factor Since the seminal Fama-French three-factor model was introduced, a fourth factor has
come to be added to the standard controls for stock return behavior. This is a momentum
factor. As we first saw in Chapter 11, Jegadeesh and Titman uncovered a tendency for
good or bad performance of stocks to persist over several months, a sort of momentum
property. 28
Carhart added this momentum effect to the three-factor model as a tool to
Figure 13.5 Value minus glamour returns surrounding earnings announcements, 1971–1992. Announcement effects are measured for each of 4 years following classification as a value versus a growth firm.
Source: R. La Porta, J. Lakonishok, A. Shleifer, and R.W. Vishny, “Good News for Value Stocks,” Journal of Finance 52 (1997), pp. 859–874. Used with permission of John Wiley and Sons, via Copyright Clearance Center.
3.22
2.26
1.18
1.60
2.79
0.0
1.0
2.0
3.0
4.0
1 2 3 4 5
Postformation Year
D if
fe re
n ce
i n
R e tu
rn s
(% )
27 R. La Porta, J. Lakonishok, A. Shleifer, and R.W. Vishny, “Good News for Value Stocks,” Journal of Finance
52 (1997), pp. 859–874. 28
Narasimhan Jegadeesh and Sheridan Titman, “Returns to Buying Winners and Selling Losers: Implications for
Stock Market Efficiency,” Journal of Finance 48 (March 1993), pp. 65–91.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 433
evaluate mutual fund performance. 29
The factor is constructed in the same way and is
denoted by WML (winners minus losers). Versions of this factor take winners/losers based
on 1–12 months of past returns. Carhart found that much of what appeared to be the alpha
of many mutual funds could in fact be explained as due to their loadings or sensitivities to
market momentum. The original Fama-French model augmented with a momentum fac-
tor has become a common four-factor model used to evaluate abnormal performance of a
stock portfolio.
Of course, this additional factor presents further conundrums of interpretation. To char-
acterize the original Fama-French factors as reflecting obvious sources of risk is already
a bit of a challenge. A momentum factor seems even harder to position as reflecting a
risk–return trade-off.
29
Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52 (March 1997), pp. 57–82.
30
L. Pástor and R. F. Stambaugh, “Liquidity Risk and Expected Stock Returns,” Journal of Political Economy 111 (2003), pp. 642–85.
In Chapter 9 we saw that an important extension of the CAPM incorporates consider-
ations of asset liquidity. Unfortunately, measuring liquidity is far from trivial. The effect of
liquidity on an asset’s expected return is composed of two factors:
1. Transaction costs that are dominated by the bid–ask spread that dealers set to com-
pensate for losses incurred when trading with informed traders.
2. Liquidity risk resulting from covariance between changes in asset liquidity cost with both changes in market-index liquidity cost and with market-index rates of return.
Neither of these factors are directly observable and their effect on equilibrium rates of
return is difficult to estimate.
Liquidity embodies several characteristics such as trading costs, ease of sale, necessary
price concessions to effect a quick transaction, market depth, and price predictability. As
such, it is difficult to measure with any single statistic. Popular measures of liquidity, or,
more precisely, illiquidity, focus on the price impact dimension: What price concession
might a seller have to offer in order to accomplish a large sale of an asset or, conversely,
what premium must a buyer offer to make a large purchase?
One measure of illiquidity is employed by Pástor and Stambaugh, who look for evi-
dence of price reversals, especially following large trades. 30
Their idea is that if stock price
movements tend to be partially reversed on the following day, then we can conclude that
part of the original price change was not due to perceived changes in intrinsic value (these
price changes would not tend to be reversed), but was instead a symptom of price impact
associated with the original trade. Reversals suggest that part of the original price change
was a concession on the part of trade initiators who needed to offer higher purchase prices
or accept lower selling prices to complete their trades in a timely manner. Pástor and
Stambaugh use regression analysis to show that reversals do in fact tend to be larger when
associated with higher trading volume—exactly the pattern that one would expect if part
of the price move is a liquidity phenomenon. They run a first-stage regression of returns
on lagged returns and trading volume. The coefficient on the latter term measures the ten-
dency of high-volume trades to be accompanied by larger reversals.
13.4 Liquidity and Asset Pricing
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434 P A R T I I I Equilibrium in Capital Markets
Another measure of illiquidity, proposed by Amihud, also focuses on the association
between large trades and price movements. 31
His measure is:
ILLIQ 5 Monthly average of daily c Absolute value(Stock return) Dollar volume
d This measure of illiquidity is based on the price impact per dollar of transactions in the
stock and can be used to estimate both liquidity cost and liquidity risk.
Finally, Sadka uses trade-by-trade data to devise a third measure of liquidity. 32
He
begins with the observation that part of price impact, a major component of illiquidity cost,
is due to asymmetric information. (Turn back to our discussion of liquidity in Chapter 9
for a review of asymmetric information and the bid–ask spread.) He then uses regression
analysis to break out the component of price impact that is due to information issues. The
liquidity of firms can wax or wane as the prevalence of informationally motivated trades
varies, giving rise to liquidity risk.
Any of these liquidity measures can be averaged over stocks to devise measures of mar-
ketwide illiquidity. Given market illiquidity, we can then measure the “liquidity beta” of
any individual stock (the sensitivity of returns to changes in market liquidity) and estimate
the impact of liquidity risk on expected return. If stocks with high liquidity betas have
higher average returns, we conclude that liquidity is a “priced factor,” meaning that expo-
sure to it offers higher expected return as compensation for the risk.
Pástor and Stambaugh conclude that liquidity risk is in fact a priced factor, and that
the risk premium associated with it is quantitatively significant. They sort portfolios into
deciles based on liquidity beta and then compute the average alphas of the stocks in each
decile using two models that ignore liquidity: the CAPM and the Fama-French three-factor model. Figure 13.6 shows that the alpha computed under either model rises substantially
across liquidity-beta deciles, clear evidence that when controlling for other factors, aver-
age return rises along with liquidity risk. Not surprisingly, the relationship between liquid-
ity risk and alpha across deciles is more regular for the Fama-French model, as it controls
for a wider range of other influences on average return.
Pástor and Stambaugh also test the impact of the liquidity beta on alpha computed from
a four-factor model (that also controls for momentum) and obtain similar results . In fact,
they suggest that liquidity risk factor may account for a good part of the apparent profit-
ability of the momentum strategy.
Acharya and Pedersen use Amihud’s measure to test for price effects associated with
the average level of illiquidity as well as a liquidity risk premium. 33 They demonstrate that expected stock returns depend on the average level of illiquidity. ( Figure 9.4 in Chapter 9
shows a similar result.) But Acharya and Pedersen demonstrate that stock returns depend
on several liquidity betas as well: the sensitivity of individual stock illiquidity to market
illiquidity; the sensitivity of stock returns to market illiquidity; and the sensitivity of stock
illiquidity to market return. They conclude that adding these liquidity effects to the conven-
tional CAPM increases our ability to explain expected asset returns.
31
Yakov Amihud, “Illiquidity and Stock Returns: Cross-Section and Time-Series Effects,” Journal of Financial Markets 5 (2002), pp. 31–56.
32
Ronnie Sadka, “Momentum and Post-earnings Announcement Drift Anomalies: The Role of Liquidity Risk,”
Journal of Financial Economics 80 (2006), pp. 309–49.
33
V. V. Acharya and L. H. Pedersen, “Asset Pricing with Liquidity Risk,” Journal of Financial Economics 77 (2005), pp. 375–410.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 435
34
Jarnish Mehra and Edward Prescott, “The Equity Premium: A Puzzle,” Journal of Monetary Economics, March 1985.
Figure 13.6 Alphas of value-weighted portfolios sorted on liquidity betas Source: L. Pástor and R. F. Stambaugh, “Liquidity Risk and Expected Stock Returns,” Journal of Political Economy 111 (2003), pp. 642–85, Table 4. Copyright © 2003, The University of Chicago Press.
−2
−4
−6
−8
4
2
Lowest 2 3 4
Illiquidity Decile
5 6 7 8 9 Highest
0
A lp
h a (
% p
e r
ye a r)
CAPM alpha FF alpha
13.5 Consumption-Based Asset Pricing and the Equity Premium Puzzle
In a classic article, Mehra and Prescott observed that historical excess returns on risky
assets in the U.S. are too large to be consistent with economic theory and reasonable levels
of risk aversion. 34
This observation has come to be known as the “equity premium puzzle.”
The debate about the equity premium puzzle suggests that forecasts of the market risk
premium should be lower than historical averages. The question of whether past returns
provide a guideline to future returns is sufficiently important to justify stretching the scope
of our discussions of equilibrium in capital markets.
Consumption Growth and Market Rates of Return The ICAPM is derived from a lifetime consumption/investment plan of a representative
consumer/investor. Each individual’s plan is set to maximize a utility function of lifetime
consumption, and consumption/investment in each period is based on age and current
wealth, as well as the risk-free rate and the market portfolio’s risk and risk premium.
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436 P A R T I I I Equilibrium in Capital Markets
The consumption model implies that what matters to investors is not their wealth per
se, but their lifetime flow of consumption. There can be slippage between wealth and con-
sumption due to variation in factors such as the risk-free rate, the market portfolio risk
premium, or prices of major consumption items. Therefore, a better measure of consumer
well-being than wealth is the consumption flow that such wealth can support.
Given this framework, the generalization of the basic CAPM is that instead of measur-
ing security risk based on the covariance of returns with the market return (a measure that
focuses only on wealth), we are better off using the covariance of returns with aggregate
consumption. Hence, we would expect the risk premium of the market index to be related
to that covariance as follows:
E(rM) 2 rf 5 ACov (rM, rC ) (13.10)
where A depends on the average coefficient of risk aversion and r C is the rate of return on a consumption-tracking portfolio constructed to have the highest possible correlation with
growth in aggregate consumption. 35
The first wave of attempts to estimate consumption-based asset pricing models used
consumption data directly rather than returns on consumption-tracking portfolios. By and
large, these tests found the CCAPM no better than the conventional CAPM in explaining
risk premiums. The equity premium puzzle refers to the fact that using reasonable estimates of A, the covariance of consumption growth with the market-index return, Cov( r M , r C ), is far too low to justify observed historical-average excess returns on the market-index port-
folio, shown on the left-hand side of Equation 13.10. 36
Thus, the risk premium puzzle says
in effect that historical excess returns are too high and/or our inferences about risk aversion
are too low.
Recent research improves the quality of estimation in several ways. First, rather than
using consumption growth directly, it uses consumption-tracking portfolios. The available
(infrequent) data on aggregate consumption is used only to construct the consumption-
tracking portfolio. The frequent and accurate data on the return on these portfolios may then
be used to test the asset pricing model. (On the other hand, any inaccuracy in the construc-
tion of the consumption-mimicking portfolios will muddy the relationship between asset
returns and consumption risk.) For example, a study by Jagannathan and Wang focuses
on year-over-year fourth-quarter consumption and employs a consumption-tracking
portfolio. 37
Table 13.5 , excerpted from their study, shows that the Fama-French factors are
in fact associated with consumption betas as well as excess returns. The top panel contains
familiar results: Moving across each row, we see that higher book-to-market ratios are asso-
ciated with higher average returns. Similarly, moving down each column, we see that larger
size generally implies lower average returns. The novel results are in the lower panel: A
high book-to-market ratio is associated with higher consumption beta, and larger firm size
is associated with lower consumption beta. The suggestion is that the explanatory power of
the Fama-French factors for average returns may in fact reflect differences in consumption
35
This equation is analogous to the equation for the risk premium in the conventional CAPM, i.e., that E ( r M ) 2 r f 5 A Cov( r M , r M ) 5 A Var( r M ). In the multifactor version of the ICAPM, however, the market is no longer mean- variance efficient, so the risk premium of the market index will not be proportional to its variance. The APT also
implies a linear relationship between risk premium and covariance with relevant factors, but it is silent about the
slope of the relationship because it avoids assumptions about utility.
36
Notice that the conventional CAPM does not pose such problems. In the CAPM, E ( r M ) 2 r f 5 A Var( r M ). A risk premium of .085 (8.5%) and a standard deviation of .20 (20%, or variance of .04) imply a coefficient of risk aver-
sion of .085/.04 5 2.125, which is quite plausible.
37
Ravi Jagannathan and Yong Wang, “Lazy Investors, Discretionary Consumption, and the Cross-Section of
Stock Returns,” Journal of Finance 62 (August 2006), pp. 1623–61.
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C H A P T E R 1 3 Empirical Evidence on Security Returns 437
risk of those portfolios. Figure 13.7 shows
that the average returns of the 25 Fama-
French portfolios are strongly associated
with their consumption betas. Other tests
reported by Jagannathan and Wang show
that the CCAPM explains returns even
better than the Fama-French three-factor
model, which in turn is superior to the
single-factor CAPM.
Moreover, the standard CCAPM fo-
cuses on a representative consumer/investor,
thereby ignoring information about hetero-
geneous investors with different levels of
wealth and consumption habits. To improve
the model’s power to explain returns, some
newer studies allow for several classes of
investors with differences in wealth and
consumption behavior. For example, the
covariance between market returns and
consumption is far higher when we focus
on the consumption risk of households that
actually hold financial securities. 38
This
observation mitigates the equity risk pre-
mium puzzle.
Book-to-Market
Size Low Medium High
Average annual excess returns* (%) Small 6.19 12.24 17.19 Medium 6.93 10.43 13.94 Big 7.08 8.52 9.5
Consumption beta* Small 3.46 4.26 5.94 Medium 2.88 4.35 5.71 Big 3.39 2.83 4.41
Table 13.5
Annual excess returns and consumption betas
*Average annual excess returns on the 25 Fama-French portfolios from 1954 to 2003. Consumption betas estimated by the time series regression
Ri,t 5 ai 1 bi,cgct 1 ei,t ,
where R i, t is the excess return over the risk-free rate, and g ct is annual consumption growth calculated using fourth-quarter consumption data.
Source: Ravi Jagannathan and Yong Wang, “Lazy Investors, Discretionary Consumption, and the Cross-Section of Stock Returns,” Journal of Finance 62 (August 2006), pp. 1623–61.
38 C. J. Malloy, T. Moskowitz, and A. Vissing-Jørgensen, “Long-Run Stockholder Consumption Risk and Asset
Returns,” Journal of Finance 64 (December 2009), pp. 2427–80.
Figure 13.7 Cross section of stock returns: Fama-French 25 portfolios, 1954–2003
Annual excess returns and consumption betas. This figure plots the average annual excess returns on the 25 Fama-French portfolios and their consumption betas. Each two-digit number represents one portfolio. The first digit refers to the size quintile (1 5 smallest, 5 5 largest), and the second digit refers to the book-to-market quintile (1 5 lowest, 5 5 highest).
E xc
e ss
R e tu
rn s
(% )
0 1 2
Consumption Betas
3 4 5 6 7 8
10
8 5241 31 21 11
51 42
53 22
54 55
3233 43
13 44
34 23
24
14 25
15
35
1245
6
4
2
0
20
18
16
14
12
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438 P A R T I I I Equilibrium in Capital Markets
Expected versus Realized Returns Fama and French offer another interpretation of the equity premium puzzle.
39 Using stock
index returns from 1872 to 1999, they report the average risk-free rate, average stock
market return (represented by the S&P 500 index), and resultant risk premium for the over-
all period and subperiods:
Period Risk-Free Rate S&P 500 Return Equity Premium
1872–1999 4.87 10.97 6.10 1872–1949 4.05 8.67 4.62 1950–1999 6.15 14.56 8.41
The big increase in the average excess return on equity after 1949 suggests that the equity
premium puzzle is largely a creature of modern times.
Fama and French suspect that estimating the risk premium from average realized
returns may be the problem. They use the constant-growth dividend-discount model (see
an introductory finance text or Chapter 18) to estimate expected returns and find that for
the period 1872–1949, the dividend discount model (DDM) yields similar estimates of
the expected risk premium as the average realized excess return. But for the period 1950– 1999, the DDM yields a much smaller risk premium, which suggests that the high average
excess return in this period may have exceeded the returns investors actually expected to
earn at the time.
In the constant-growth DDM, the expected capital gains rate on the stock will equal the
growth rate of dividends. As a result, the expected total return on the firm’s stock will be
the sum of dividend yield (dividend/price) plus the expected dividend growth rate, g:
E(r) 5 D1 P0 1 g (13.11)
where D 1 is end-of-year dividends and P 0 is the current price of the stock. Fama and French treat the S&P 500 as representative of the average firm, and use Equation 13.11 to produce
estimates of E ( r ). For any sample period, t 5 1, . . . , T, Fama and French estimate expected return from the
sum of the dividend yield ( D t / P t 2 1 ) plus the dividend growth rate ( g t 5 D t / D t 2 1 2 1). In contrast, the realized return is the dividend yield plus the rate of capital gains ( P t / P t 2 1 2 1). Because the dividend yield is common to both estimates, the difference between the
expected and realized return equals the difference between the dividend growth and capital
gains rates. While dividend growth and capital gains were similar in the earlier period, cap-
ital gains significantly exceeded the dividend growth rate in modern times. Hence, Fama
and French conclude that the equity premium puzzle may be due at least in part to unan-
ticipated capital gains in the latter period.
Fama and French argue that dividend growth rates produce more reliable estimates of
the capital gains investors actually expected to earn than the average of their realized capi-
tal gains. They point to three reasons:
1. Average realized returns over 1950–1999 exceeded the internal rate of return on
corporate investments. If those average returns were representative of expectations,
we would have to conclude that firms were willingly engaging in negative-NPV
investments.
39
Eugene Fama and Kenneth French, “The Equity Premium,” Journal of Finance 57, no. 2 (2002).
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C H A P T E R 1 3 Empirical Evidence on Security Returns 439
2. The statistical precision of estimates from the DDM are far higher than those using
average historical returns. The standard error of the estimates of the risk premium
from realized returns greatly exceed the standard error from the dividend discount
model (see the following table).
3. The reward-to-volatility (Sharpe) ratio derived from the DDM is far more stable
than that derived from realized returns. If risk aversion remains the same over time,
we would expect the Sharpe ratio to be stable.
The evidence for the second and third points is shown in the following table, where
estimates from the dividend discount model (DDM) and from realized returns (Realized)
are shown side by side.
Mean Return Standard Error t-Statistic Sharpe Ratio
Period DDM Realized DDM Realized DDM Realized DDM Realized
1872–1999 4.03 6.10 1.14 1.65 3.52 3.70 0.22 0.34 1872–1949 4.35 4.62 1.76 2.20 2.47 2.10 0.23 0.24 1950–1999 3.54 8.41 1.03 2.45 3.42 3.43 0.21 0.51
Fama and French’s study provides a simple explanation for the equity premium puzzle,
namely, that observed rates of return in the recent half-century were unexpectedly high.
It also implies that forecasts of future excess returns will be lower than past averages.
(Coincidentally, their study was published in 1999, and so far appears prophetic in light of
low subsequent average returns since then.)
Work by Goetzmann and Ibbotson lends support to Fama and French’s argument. 40
Goetzmann and Ibbotson combine research that extends data on rates of return on stocks
and long-term corporate bonds back to 1792. Summary statistics for these values between
1792 and 1925 are as follows:
Arithmetic Average Geometric Average Standard Deviation
NYSE total return 7.93% 6.99% 14.64% U.S. bond yields 4.17% 4.16% 4.17%
These statistics suggest a risk premium that is much lower than the historical aver-
age for 1926–2009 (much less 1950–1999), which is the period that produces the equity
premium puzzle. 41
Thus, the period for which Fama and French claim realized rates were
unexpected is actually relatively short in historical perspective.
Survivorship Bias The equity premium puzzle emerged from long-term averages of U.S. stock returns. There
are reasons to suspect that these estimates of the risk premium are subject to survivorship
bias, as the United States has arguably been the most successful capitalist system in the
world, an outcome that probably would not have been anticipated several decades ago.
40
William N. Goetzmann and Roger G. Ibbotson, “History and the Equity Risk Premium,” working paper, Yale
University, October 18, 2005.
41
The short-term risk-free rate is a lot more difficult to assess because short-term bonds in this period were quite
risky and average rates exceeded the yields on long-term corporate bonds.
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440 P A R T I I I Equilibrium in Capital Markets
Jurion and Goetzmann assembled a
database of capital appreciation indexes
for the stock markets of 39 countries over
the period 1921–1996. 42
Figure 13.8 shows
that U.S. equities had the highest real return
of all countries, at 4.3% annually, versus a
median of .8% for other countries. More-
over, unlike the United States, many other
countries have had equity markets that
actually closed, either permanently or for
extended periods of time.
The implication of these results is that
using average U.S. data may impart a
form of survivorship bias to our estimate
of expected returns, because unlike many
other countries, the United States has never
been a victim of such extreme problems.
Estimating risk premiums from the expe-
rience of the most successful country and
ignoring the evidence from stock markets
that did not survive for the full sample period
will impart an upward bias in estimates of
expected returns. The high realized equity
premium obtained for the United States may
not be indicative of required returns.
As an analogy, think of the effect of survivorship bias in the mutual fund industry.
We know that some companies regularly close down their worst-performing mutual funds.
If performance studies include only mutual funds for which returns are available during an
entire sample period, the average returns of the funds that make it into the sample will be
reflective of the performance of long-term survivors only. With the failed funds excluded
from the sample, the average measured performance of mutual fund managers will be bet-
ter than one could reasonably expect from the full sample of managers. Think back to the
box in Chapter 11, “How to Guarantee a Successful Market Newsletter.” If one starts many
newsletters with a range of forecasts, and continues only the newsletters that turned out to
have successful advice, then it will appear from the sample of survivors that the average newsletter had forecasting skill.
Extensions to the CAPM May Resolve the Equity Premium Puzzle Constantinides argues that the standard CAPM can be extended to account for observed
excess returns by relaxing some of its assumptions, in particular, by recognizing that con-
sumers face uninsurable and idiosyncratic income shocks, for example, the loss of employ-
ment. 43
The prospect of such events is higher in economic downturns and this observation
takes us a long way toward understanding the means and variances of asset returns as well
as their variation along the business cycle.
In addition, life-cycle considerations are important and often overlooked. Borrowing
constraints become important when placed in the context of the life cycle. The imaginary
42 Philippe Jurion and William N. Goetzmann, “Global Stock Markets in the Twentieth Century,” Journal
of Finance 54, no. 3 (June 1999). 43
George M. Constantinides, “Understanding the Equity Risk Premium Puzzle,” in Handbooks in Finance: Hand- book of the Equity Risk Premium, ed. Rajnish Mehra (Amsterdam: Elsevier, 2008), pp. 331–59.
Figure 13.8 Real returns on global stock markets. The figure displays average real returns for 39 markets over the period 1921 to 1996. Markets are sorted by years of existence. The graph shows that markets with long histo- ries typically have higher returns. An asterisk indicates that the market suffered a long-term break.
P e rc
e n
t p
e r
A n
n u
m
0 20 40 Years of Existence since Inception
Uruguay Hungary
Czechoslovakia Israel
Brazil
Pakistan
Egypt
Poland
Greece Argentina*
Philippines
South Africa Venezuela
India
Colombia Peru*
Spain*
Portugal* Japan* New Zealand Belgium
Italy France NetherlandsAustria*Ireland
Finland Mexico Denmark
Germany* U.K. Canada
Switzerland Sweden
Chile* Norway
Australia
60 80 100
U.S.
−6 −5 −4 −3 −2 −1
6 5 4 3 2 1 0
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C H A P T E R 1 3 Empirical Evidence on Security Returns 441
“representative consumer” who holds all stock and bond market wealth does not face
borrowing constraints. Young consumers, however, do face meaningful borrowing con-
straints. Constantinides traces their impact on the equity premium, the demand for bonds,
and on the limited participation of many consumers in the capital markets. Finally, he
shows that adding habit formation to the conventional utility function helps explain higher
risk premiums than those that would be justified by the covariance of stock returns with
aggregate consumption growth. He argues that integrating the notions of habit formation,
incomplete markets, the life cycle, borrowing constraints, and other sources of limited
stock market participation is a promising vantage point from which to study the prices
of assets and their returns, both theoretically and empirically within the class of rational
asset-pricing models.
Liquidity and the Equity Premium Puzzle We’ve seen that liquidity risk is potentially important in explaining the cross section of
stock returns. The illiquidity premium may be on the same order of magnitude as the mar-
ket risk premium. Therefore, the common practice of treating the average excess return on
a market index as an estimate of a risk premium per se is almost certainly too simplistic.
Part of that average excess return is almost certainly compensation for liquidity risk rather than just the (systematic) volatility of returns. If this is recognized, the equity premium puzzle may be less of a puzzle than it first appears.
Behavioral Explanations of the Equity Premium Puzzle Barberis and Huang explain the puzzle as an outcome of irrational investor behavior.
44
The key elements of their approach are loss aversion and narrow framing, two well-known
features of decision making under risk in experimental settings. Narrow framing is the idea
that investors evaluate every risk they face in isolation. Thus, investors will ignore low
correlation of the risk of a stock portfolio with other components of wealth, and therefore
require a higher risk premium than rational models would predict. Combined with loss
aversion, investor behavior will generate large risk premiums despite the fact that
traditionally measured risk aversion is low. (See Chapter 12 for more discussion of such
behavioral biases.)
Models that incorporate these effects can generate a large equilibrium equity risk pre-
mium and a low and stable risk-free rate, even when consumption growth is smooth and
only weakly correlated with the stock market. Moreover, they can do so for parameter
values that correspond to plausible predictions about attitudes to independent monetary
gambles. The analysis for the equity premium also has implications for a closely related
portfolio puzzle, the stock market participation puzzle. They suggest some possible direc-
tions for future research.
The approach of Barberis and Huang, when accounting for heterogeneity of prefer-
ences, can explain why a segment of the population that one would expect to participate
in the stock market still avoids it. Narrow framing also explains the disconnect between
consumption growth and market rates of return. The assessment of stock market return in
isolation ignores the limited impact on consumption via smoothing and other hedges. Loss
aversion that exaggerates disutility of losses relative to a reference point magnifies this
effect. The development of empirical literature on the tenets of these theories may deter-
mine the validity and implications of the equity premium puzzle.
44
Nicholas Barberis and Ming Huang, “The Loss Aversion/Narrow Framing Approach to the Equity Premium
Puzzle,” in Handbooks in Finance: Handbook of the Equity Risk Premium ed. Rajnish Mehra (Amsterdam: Elsevier, 2008), pp. 199–229.
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442 P A R T I I I Equilibrium in Capital Markets
1. Although the single-factor expected return–beta relationship has not been confirmed by scientific standards, its use is already commonplace in economic life.
2. Early tests of the single-factor CAPM rejected the SML, finding that nonsystematic risk was related to average security returns.
3. Later tests controlling for the measurement error in beta found that nonsystematic risk does not explain portfolio returns but also that the estimated SML is too flat compared with what the
CAPM would predict.
4. Roll’s critique implied that the usual CAPM test is a test only of the mean-variance efficiency of a prespecified market proxy and therefore that tests of the linearity of the expected return–beta relationship do not bear on the validity of the model.
5. Tests of the mean-variance efficiency of professionally managed portfolios against the bench- mark of a prespecified market index conform with Roll’s critique in that they provide evidence
on the efficiency of the market index. Empirical evidence suggests that most professionally man-
aged portfolios are outperformed by market indexes, which corroborates the efficiency of those
indexes and hence the CAPM.
6. Tests of the single-index model that account for human capital and cyclical variations in asset betas are far more consistent with the single-index CAPM and APT. These tests suggest that extra-market
macroeconomic variables are not necessary to explain expected returns. Moreover, anomalies such
as effects of size and book-to-market ratios disappear once these variables are accounted for.
7. The dominant multifactor models today are variants of the Fama-French model, incorporating market, size, value, momentum, and sometimes liquidity factors. Debate continues on whether
returns associated with these extra-market factors reflect rational risk premia or behaviorally
induced mispricing.
8. The equity premium puzzle originates from the observation that equity returns exceeded the risk- free rate to an extent that is inconsistent with reasonable levels of risk aversion—at least when
average rates of return are taken to represent expectations. Fama and French show that the puzzle
emerges primarily from excess returns over the last 50 years. Alternative estimates of expected
returns using the dividend growth model instead of average returns suggest that excess returns on
stocks were high because of unexpected large capital gains. The study suggests that future excess
returns will be lower than realized in recent decades.
9. Early research on consumption-based capital asset pricing models was disappointing, but more recent work is far more encouraging. In some studies, consumption betas explain average port-
folio returns as well as the Fama-French three-factor model. These results support Fama and
French’s conjecture that their factors proxy for more fundamental sources of risk.
Related Web sites for this chapter are available at www. mhhe.com/bkm
SUMMARY
first-pass regression second-pass regression benchmark error KEY TERMS
KEY EQUATIONS First-pass regression equation: r it 2 r ft 5 a i 1 b i ( r Mt 2 r ft ) 1 e it Second-pass regression equation: rt 2 rf 5 g0 1 g1bi
Fama-French 3-factor model: E ( r i ) 2 r f 5 a i 1 b i [ E ( r M ) 2 r f ] 1 s i E [SMB] 1 h i E [HML]
PROBLEM SETS 1. Suppose you find, as research indicates, that in the cross-section regression of the CCAPM, the coefficients of factor loadings on the Fama-French model are significant predictors of average
return factors (in addition to consumption beta). How would you explain this phenomenon?
2. Search the Internet for a recent graph of market volatility. What does this history suggest about
the history of consumption growth? Basic
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C H A P T E R 1 3 Empirical Evidence on Security Returns 443
The following annual excess rates of return were obtained for nine individual stocks and a market index:
Year Market Index
Stock Excess Returns (%)
A B C D E F G H I
1 29.65 33.88 225.20 36.48 42.89 239.89 39.67 74.57 40.22 90.19
2 211.91 249.87 24.70 225.11 254.39 44.92 254.33 279.76 271.58 226.64
3 14.73 65.14 225.04 18.91 239.86 23.91 25.69 26.73 14.49 18.14
4 27.68 14.46 238.64 223.31 20.72 23.21 92.39 23.82 13.74 0.09
5 5.18 15.67 61.93 63.95 232.82 44.26 242.96 101.67 24.24 8.98
6 25.97 232.17 44.94 219.56 69.42 90.43 76.72 1.72 77.22 72.38
7 10.64 231.55 274.65 50.18 74.52 15.38 21.95 243.95 213.40 28.95
8 1.02 223.79 47.02 242.28 28.61 217.64 28.83 98.01 28.12 39.41
9 18.82 24.59 28.69 20.54 2.32 42.36 18.93 22.45 37.65 94.67
10 23.92 28.03 48.61 23.65 26.26 23.65 23.31 15.36 80.59 52.51
11 241.61 78.22 285.02 20.79 268.70 285.71 245.64 2.27 272.47 280.26
12 26.64 4.75 42.95 248.60 26.27 13.24 234.34 254.47 21.50 224.46
e X c e l Please visit us at www.mhhe.com/bkm
3. Perform the first-pass regressions and tabulate the summary statistics.
4. Specify the hypotheses for a test of the second-pass regression for the SML.
5. Perform the second-pass SML regression by regressing the average excess return of each port-
folio on its beta.
6. Summarize your test results and compare them to the results reported in the text.
7. Group the nine stocks into three portfolios, maximizing the dispersion of the betas of the three
resultant portfolios. Repeat the test and explain any changes in the results.
8. Explain Roll’s critique as it applies to the tests performed in Problems 3 to 7.
9. Plot the capital market line (CML), the nine stocks, and the three portfolios on a graph of aver-
age returns versus standard deviation. Compare the mean-variance efficiency of the three port-
folios and the market index. Does the comparison support the CAPM?
Suppose that, in addition to the market factor that has been considered in Problems 3 to 9, a second factor is considered. The values of this factor for years 1 to 12 were as follows:
Year % Change in Factor Value Year
% Change in Factor Value
1 29.84 7 23.52
2 6.46 8 8.43
3 16.12 9 8.23
4 216.51 10 7.06
5 17.82 11 215.74
6 213.31 12 2.03
10. Perform the first-pass regressions as did Chen, Roll, and Ross and tabulate the relevant sum-
mary statistics. ( Hint: Use a multiple regression as in a standard spreadsheet package. Estimate the betas of the 12 stocks on the two factors.)
11. Specify the hypothesis for a test of a second-pass regression for the two-factor SML.
12. Do the data suggest a two-factor economy?
13. Can you identify a factor portfolio for the second factor?
14. Suppose you own your own business, which now makes up about half your net worth. On the basis of
what you have learned in this chapter, how would you structure your portfolio of financial assets?
Intermediate
Challenge
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444 P A R T I I I Equilibrium in Capital Markets
1. Identify and briefly discuss three criticisms of beta as used in the capital asset pricing model.
2. Richard Roll, in an article on using the capital asset pricing model (CAPM) to evaluate portfolio
performance, indicated that it may not be possible to evaluate portfolio management ability if
there is an error in the benchmark used.
a. In evaluating portfolio performance, describe the general procedure, with emphasis on the benchmark employed.
b. Explain what Roll meant by the benchmark error and identify the specific problem with this benchmark.
c. Draw a graph that shows how a portfolio that has been judged as superior relative to a “mea- sured” security market line (SML) can be inferior relative to the “true” SML.
d. Assume that you are informed that a given portfolio manager has been evaluated as superior when compared to the Dow Jones Industrial Average, the S&P 500, and the NYSE Composite
Index. Explain whether this consensus would make you feel more comfortable regarding the
portfolio manager’s true ability.
e. Although conceding the possible problem with benchmark errors as set forth by Roll, some contend this does not mean the CAPM is incorrect, but only that there is a measurement
problem when implementing the theory. Others contend that because of benchmark errors the
whole technique should be scrapped. Take and defend one of these positions.
3. Bart Campbell, CFA, is a portfolio manager who has recently met with a prospective client, Jane
Black. After conducting a survey market line (SML) performance analysis using the Dow Jones
Industrial Average as her market proxy, Black claims that her portfolio has experienced supe-
rior performance. Campbell uses the capital asset pricing model as an investment performance
measure and finds that Black’s portfolio plots below the SML. Campbell concludes that Black’s
apparent superior performance is a function of an incorrectly specified market proxy, not superior
investment management. Justify Campbell’s conclusion by addressing the likely effects of an
incorrectly specified market proxy on both beta and the slope of the SML.
SOLUTIONS TO CONCEPT CHECKS 1. The SCL is estimated for each stock; hence we need to estimate 100 equations. Our sample
consists of 60 monthly rates of return for each of the 100 stocks and for the market index. Thus
each regression is estimated with 60 observations. Equation 13.1 in the text shows that when
stated in excess return form, the SCL should pass through the origin, that is, have a zero intercept.
2. When the SML has a positive intercept and its slope is less than the mean excess return on the market
portfolio, it is flatter than predicted by the CAPM. Low-beta stocks therefore have yielded returns
that, on average, were higher than they should have been on the basis of their beta. Conversely, high-
beta stocks were found to have yielded, on average, lower returns than they should have on the basis
of their betas. The positive coefficient on g 2 implies that stocks with higher values of firm-specific
risk had on average higher returns. This pattern, of course, violates the predictions of the CAPM.
3. a. According to Equation 13.5, g 0 is the average return earned on a stock with zero beta and zero firm-specific risk. According to the CAPM, this should be the risk-free rate, which for the
1946–1955 period was 9 basis points, or .09% per month (see Table 13.1 ). According to the
CAPM, g 1 should equal the average market risk premium, which for the 1946–1955 period
was 103 basis points, or 1.03% per month. Finally, the CAPM predicts that g 3 , the coefficient
on firm-specific risk, should be zero.
b. A positive coefficient on beta-squared would indicate that the relationship between risk and return is nonlinear. High-beta securities would provide expected returns more than proportional
to risk. A positive coefficient on s ( e ) would indicate that firm-specific risk affects expected return, a direct contradiction of the CAPM and APT.
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14 IN THE PREVIOUS chapters on risk and return relationships, we treated securities at a high level of abstraction. We assumed implicitly that a prior, detailed analysis of each security already had been performed, and that its risk and return features had been assessed.
We turn now to specific analyses of partic- ular security markets. We examine valuation principles, determinants of risk and return, and portfolio strategies commonly used within and across the various markets.
We begin by analyzing debt securities . A debt security is a claim on a specified periodic stream of income. Debt securities are often called fixed-income securities because they promise either a fixed stream of income or one that is determined according to a speci- fied formula. These securities have the advan- tage of being relatively easy to understand because the payment formulas are specified in advance. Uncertainty about their cash flows is minimal as long as the issuer of the secu- rity is sufficiently creditworthy. That makes these securities a convenient starting point
for our analysis of the universe of potential investment vehicles.
The bond is the basic debt security, and this chapter starts with an overview of the universe of bond markets, including Treasury, corporate, and international bonds. We turn next to bond pricing, showing how bond prices are set in accordance with market interest rates and why bond prices change with those rates. Given this background, we can compare the myriad measures of bond returns such as yield to maturity, yield to call, holding-period return, and realized com- pound rate of return. We show how bond prices evolve over time, discuss certain tax rules that apply to debt securities, and show how to calculate after-tax returns. Finally, we consider the impact of default or credit risk on bond pricing and look at the determinants of credit risk and the default premium built into bond yields. Credit risk is central to both collateralized debt obligations and credit default swaps, so we examine these instru- ments as well.
Bond Prices and Yields
CHAPTER FOURTEEN 4 PA
R T IV
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446 P A R T I V Fixed-Income Securities
14.1 Bond Characteristics
A bond is a security that is issued in connection with a borrowing arrangement. The bor- rower issues (i.e., sells) a bond to the lender for some amount of cash; the bond is the
“IOU” of the borrower. The arrangement obligates the issuer to make specified payments
to the bondholder on specified dates. A typical coupon bond obligates the issuer to make
semiannual payments of interest to the bondholder for the life of the bond. These are called
coupon payments because in precomputer days, most bonds had coupons that investors would clip off and present to claim the interest payment. When the bond matures, the issuer
repays the debt by paying the bond’s par value (equivalently, its face value ). The coupon rate of the bond determines the interest payment: The annual payment is the coupon rate times the bond’s par value. The coupon rate, maturity date, and par value of the bond are
part of the bond indenture , which is the contract between the issuer and the bondholder. To illustrate, a bond with par value of $1,000 and coupon rate of 8% might be sold to a
buyer for $1,000. The bondholder is then entitled to a payment of 8% of $1,000, or $80 per
year, for the stated life of the bond, say, 30 years. The $80 payment typically comes in two
semiannual installments of $40 each. At the end of the 30-year life of the bond, the issuer
also pays the $1,000 par value to the bondholder.
Bonds usually are issued with coupon rates set just high enough to induce investors to
pay par value to buy the bond. Sometimes, however, zero-coupon bonds are issued that make no coupon payments. In this case, investors receive par value at the maturity date but
receive no interest payments until then: The bond has a coupon rate of zero. These bonds
are issued at prices considerably below par value, and the investor’s return comes solely
from the difference between issue price and
the payment of par value at maturity. We will
return to these bonds later.
Treasury Bonds and Notes Figure 14.1 is an excerpt from the listing of
Treasury issues. Treasury notes are issued
with original maturities ranging between 1
and 10 years, while Treasury bonds are issued
with maturities ranging from 10 to 30 years.
Both bonds and notes may be purchased
directly from the Treasury in denominations
of only $100, but denominations of $1,000
are far more common. Both make semiannual
coupon payments.
The highlighted bond in Figure 14.1
matures on July 31, 2018. Its coupon rate is
2.25%. Par value typically is $1,000; thus the
bond pays interest of $22.50 per year in two
semiannual payments of $11.25. Payments
are made in January and July of each year.
Although bonds usually are sold in denomi-
nations of $1,000, the bid and ask prices
are quoted as a percentage of par value. 1
U.S. Treasury Quotes
MATURITY
Jun 15 13
Jan 15 15
Jun 30 16
Jul 31 18
Nov 15 18
Feb 15 21
Feb 15 26
May 15 30
Feb 15 36
May 15 42
COUPON
1.125
0.250
1.500
2.250
9.000
7.875
6.000
6.250
4.500
3.000
BID
100.8203
100.0391
104.1563
108.4922
150.3750
154.2344
149.1484
160.2190
138.4063
108.8047
ASKED
100.8281
100.0469
104.1875
108.5391
150.4219
154.3281
149.2266
161.0000
138.4844
108.8672
CHANGE
0.0078
0.0547
0.1016
0.0938
0.1719
0.1875
0.1484
0.0781
0.1641
0.1328
ASKED YIELD (%)
0.177
0.231
0.421
0.790
0.770
1.172
1.869
2.117
2.362
2.572
Figure 14.1 Prices and yields of U.S. Treasury bonds Source: The Wall Street Journal Online, July 31, 2012. Reprinted by permission of Dow Jones & Company, Inc. © 2012 Dow Jones & Company. All Rights Reserved Worldwide.
1 Recall that the bid price is the price at which you can sell the bond to a dealer. The ask price, which is slightly
higher, is the price at which you can buy the bond from a dealer.
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C H A P T E R 1 4 Bond Prices and Yields 447
Therefore, the ask price is 108.5391% of par, or $1,085.391. The minimum price
increment, or tick size, in The Wall Street Journal listing is 1/128, so this bond may also be viewed as selling for 108 69⁄128 percent of par value.
2
The last column, labeled “Ask yield,” is the yield to maturity on the bond based on the
ask price. The yield to maturity is a measure of the average rate of return to an investor
who purchases the bond for the ask price and holds it until its maturity date. We will have
much to say about yield to maturity below.
Accrued Interest and Quoted Bond Prices The bond prices that you see quoted in the financial pages are not actually the prices that investors pay for the bond.
This is because the quoted price does not include the interest that accrues between coupon
payment dates.
If a bond is purchased between coupon payments, the buyer must pay the seller for
accrued interest, the prorated share of the upcoming semiannual coupon. For example, if
30 days have passed since the last coupon payment, and there are 182 days in the semian-
nual coupon period, the seller is entitled to a payment of accrued interest of 30/182 of the
semiannual coupon. The sale, or invoice, price of the bond would equal the stated price (sometimes called the flat price) plus the accrued interest.
In general, the formula for the amount of accrued interest between two dates is
Accrued interest 5 Annual coupon payment
2 3
Days since last coupon payment
Days separating coupon payments
Suppose that the coupon rate is 8%. Then the annual coupon is $80 and the semiannual coupon payment is $40. Because 30 days have passed since the last coupon payment, the accrued interest on the bond is $40 3 (30/182) 5 $6.59. If the quoted price of the bond is $990, then the invoice price will be $990 1 $6.59 5 $996.59.
Example 14.1 Accrued Interest
2 Bonds traded on formal exchanges are subject to minimum tick sizes set by the exchange. For example, the
minimum price increment on the 2-year Treasury bond futures contract (traded on the Chicago Board of Trade) is
1/128, although longer-term T-bonds have larger tick sizes. Private traders can negotiate their own tick size. For
example, one can find price quotes on Bloomberg screens with tick sizes as low as 1/256.
3 In contrast to bonds, stocks do not trade at flat prices with adjustments for “accrued dividends.” Whoever owns
the stock when it goes “ex-dividend” receives the entire dividend payment, and the stock price reflects the value
of the upcoming dividend. The price therefore typically falls by about the amount of the dividend on the “ex-day.”
There is no need to differentiate between reported and invoice prices for stocks.
The practice of quoting bond prices net of accrued interest explains why the price of
a maturing bond is listed at $1,000 rather than $1,000 plus one coupon payment. A pur-
chaser of an 8% coupon bond 1 day before the bond’s maturity would receive $1,040 (par
value plus semiannual interest) on the following day and so should be willing to pay a total
price of $1,040 for the bond. The bond price is quoted net of accrued interest in the finan-
cial pages and thus appears as $1,000. 3
Corporate Bonds Like the government, corporations borrow money by issuing bonds. Figure 14.2 is a
sample of listings for a few actively traded corporate bonds. Although some bonds trade
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448 P A R T I V Fixed-Income Securities
electronically on the NYSE Bonds platform, most bonds are traded over-the-counter in
a network of bond dealers linked by a computer quotation system. In practice, the bond
market can be quite “thin,” with few investors interested in trading a particular issue at any
particular time.
The bond listings in Figure 14.2 include the coupon, maturity, price, and yield to matu-
rity of each bond. The “rating” column is the estimation of bond safety given by the three
major bond-rating agencies—Moody’s, Standard & Poor’s, and Fitch. Bonds with grada-
tions of A ratings are safer than those with B ratings or below. As a general rule, safer
bonds with higher ratings promise lower yields to maturity than other bonds with similar
maturities. We will return to this topic toward the end of the chapter.
Call Provisions on Corporate Bonds Some corporate bonds are issued with call provisions allowing the issuer to repurchase the bond at a specified call price before the maturity date. For example, if a company issues a bond with a high coupon rate when market
interest rates are high, and interest rates later fall, the firm might like to retire the high-coupon
debt and issue new bonds at a lower coupon rate to reduce interest payments. This is called
refunding. Callable bonds typically come with a period of call protection, an initial time dur- ing which the bonds are not callable. Such bonds are referred to as deferred callable bonds.
The option to call the bond is valuable to the firm,
allowing it to buy back the bonds and refinance at lower
interest rates when market rates fall. Of course, the firm’s
benefit is the bondholder’s burden. Holders of called
bonds must forfeit their bonds for the call price, thereby
giving up the attractive coupon rate on their original
investment. To compensate investors for this risk, call-
able bonds are issued with higher coupons and promised
yields to maturity than noncallable bonds.
Convertible Bonds Convertible bonds give bondholders an option to exchange each bond for a specified number of shares of common stock of the firm. The conversion ratio is the number of shares for which each bond may be exchanged. Suppose a convert- ible bond is issued at par value of $1,000 and is convertible into 40 shares of a firm’s
Suppose that Verizon issues two bonds with identical coupon rates and maturity dates. One bond is callable, however, whereas the other is not. Which bond will sell at a higher price?
CONCEPT CHECK 14.1
YIELD %
0.5662
4.5187
2.2590
0.4663
1.1635
1.1569
ISSUER NAME
WACHOVIA CORP GLOBAL MTN
GOLDMAN SACHS GROUP INC
ANHEUSER BUSCH INBEV WORLDWIDE
JPMORGAN CHASE & CO
HOUSEHOLD FIN CORP
ANHEUSER BUSCH INBEV WORLDWIDE
SYMBOL
WFC.PO
GD.AEH
BUD3876843
JPM.KPG
HBC.IGQ
BUD3876840
COUPON
5.500%
5.750%
2.500%
4.750%
4.750%
1.375%
MATURITY
May 2013
Jan 2022
Jul 2022
May 2013
Jul 2013
Jul 2017
MOODY'S/S&P/ FITCH
A2 /A+ /AA–
A3 /A– /A
A3 /A /A
A2 /A /A+
Baa1 /A /AA–
A3 /A /A
HIGH
103.7060
111.8040
104.1200
103.1790
105.3220
101.3000
LOW
103.4240
100.3290
101.8410
102.6250
103.0760
100.9150
LAST
103.6590
109.4040
102.1360
103.1790
103.3780
101.0460
1.3090
0.2034
0.4110
0.2720
0.2280
–0.0820
CHANGE
Figure 14.2 Listing of corporate bonds Source: FINRA (Financial Industry Regulatory Authority), August 1, 2012.
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C H A P T E R 1 4 Bond Prices and Yields 449
stock. The current stock price is $20 per share, so the option to convert is not profitable
now. Should the stock price later rise to $30, however, each bond may be converted prof-
itably into $1,200 worth of stock. The market conversion value is the current value of the shares for which the bonds may be exchanged. At the $20 stock price, for example,
the bond’s conversion value is $800. The conversion premium is the excess of the bond value over its conversion value. If the bond were selling currently for $950, its premium
would be $150.
Convertible bondholders benefit from price appreciation of the company’s stock.
Again, this benefit comes at a price: Convertible bonds offer lower coupon rates and stated
or promised yields to maturity than do nonconvertible bonds. However, the actual return
on the convertible bond may exceed the stated yield to maturity if the option to convert
becomes profitable.
We discuss convertible and callable bonds further in Chapter 20.
Puttable Bonds While the callable bond gives the issuer the option to extend or retire the bond at the call date, the extendable or put bond gives this option to the bondholder. If the bond’s coupon rate exceeds current market yields, for instance, the bondholder will
choose to extend the bond’s life. If the bond’s coupon rate is too low, it will be optimal not
to extend; the bondholder instead reclaims principal, which can be invested at current yields.
Floating-Rate Bonds Floating-rate bonds make interest payments that are tied to some measure of current market rates. For example, the rate might be adjusted annually
to the current T-bill rate plus 2%. If the 1-year T-bill rate at the adjustment date is 4%, the
bond’s coupon rate over the next year would then be 6%. This arrangement means that the
bond always pays approximately current market rates.
The major risk involved in floaters has to do with changes in the firm’s financial
strength. The yield spread is fixed over the life of the security, which may be many years. If
the financial health of the firm deteriorates, then investors will demand a greater yield pre-
mium than is offered by the security. In this case, the price of the bond will fall. Although
the coupon rate on floaters adjusts to changes in the general level of market interest rates,
it does not adjust to changes in the financial condition of the firm.
Preferred Stock Although preferred stock strictly speaking is considered to be equity, it often is included
in the fixed-income universe. This is because, like bonds, preferred stock promises to pay
a specified stream of dividends. However, unlike bonds, the failure to pay the promised
dividend does not result in corporate bankruptcy. Instead, the dividends owed simply
cumulate, and the common stockholders may not receive any dividends until the preferred
stockholders have been paid in full. In the event of bankruptcy, preferred stockholders’
claims to the firm’s assets have lower priority than those of bondholders but higher priority
than those of common stockholders.
Preferred stock commonly pays a fixed dividend. Therefore, it is in effect a perpetuity,
providing a level cash flow indefinitely. In contrast, floating-rate preferred stock is much
like floating-rate bonds. The dividend rate is linked to a measure of current market interest
rates and is adjusted at regular intervals.
Unlike interest payments on bonds, dividends on preferred stock are not considered
tax-deductible expenses to the firm. This reduces their attractiveness as a source of capital
to issuing firms. On the other hand, there is an offsetting tax advantage to preferred stock.
When one corporation buys the preferred stock of another corporation, it pays taxes on
only 30% of the dividends received. For example, if the firm’s tax bracket is 35%, and
it receives $10,000 in preferred-dividend payments, it will pay taxes on only $3,000 of
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450 P A R T I V Fixed-Income Securities
that income: Total taxes owed on the income will be .35 3 $3,000 5 $1,050. The firm’s
effective tax rate on preferred dividends is therefore only .30 3 35% 5 10.5%. Given this
tax rule, it is not surprising that most preferred stock is held by corporations.
Preferred stock rarely gives its holders full voting privileges in the firm. However, if
the preferred dividend is skipped, the preferred stockholders may then be provided some
voting power.
Other Domestic Issuers There are, of course, several issuers of bonds in addition to the Treasury and private corpo-
rations. For example, state and local governments issue municipal bonds. The outstanding
feature of these is that interest payments are tax-free. We examined municipal bonds, the
value of the tax exemption, and the equivalent taxable yield of these bonds in Chapter 2.
Government agencies such as the Federal Home Loan Bank Board, the Farm Credit
agencies, and the mortgage pass-through agencies Ginnie Mae, Fannie Mae, and Freddie
Mac also issue considerable amounts of bonds. These too were reviewed in Chapter 2.
International Bonds International bonds are commonly divided into two categories, foreign bonds and Eurobonds. Foreign bonds are issued by a borrower from a country other than the one in which the bond is sold. The bond is denominated in the currency of the country in which
it is marketed. For example, if a German firm sells a dollar-denominated bond in the
United States, the bond is considered a foreign bond. These bonds are given colorful names
based on the countries in which they are marketed. For example, foreign bonds sold in the
United States are called Yankee bonds. Like other bonds sold in the United States, they are registered with the Securities and Exchange Commission. Yen-denominated bonds sold
in Japan by non-Japanese issuers are called Samurai bonds. British pound-denominated foreign bonds sold in the United Kingdom are called bulldog bonds.
In contrast to foreign bonds, Eurobonds are denominated in one currency, usually that
of the issuer, but sold in other national markets. For example, the Eurodollar market refers
to dollar-denominated bonds sold outside the United States (not just in Europe), although
London is the largest market for Eurodollar bonds. Because the Eurodollar market falls
outside U.S. jurisdiction, these bonds are not regulated by U.S. federal agencies. Similarly,
Euroyen bonds are yen-denominated bonds selling outside Japan, Eurosterling bonds are
pound-denominated Eurobonds selling outside the United Kingdom, and so on.
Innovation in the Bond Market Issuers constantly develop innovative bonds with unusual features; these issues illustrate
that bond design can be extremely flexible. Here are examples of some novel bonds. They
should give you a sense of the potential variety in security design.
Inverse Floaters These are similar to the floating-rate bonds we described earlier, except that the coupon rate on these bonds falls when the general level of interest rates rises. Investors in these bonds suffer doubly when rates rise. Not only does the present
value of each dollar of cash flow from the bond fall as the discount rate rises, but the level
of those cash flows falls as well. Of course, investors in these bonds benefit doubly when
rates fall.
Asset-Backed Bonds Miramax once issued bonds with coupon rates tied to the finan- cial performance of Pulp Fiction and other films. Domino’s Pizza has issued bonds with
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C H A P T E R 1 4 Bond Prices and Yields 451
payments backed by revenues from its pizza franchises. These are examples of asset-backed
securities. The income from a specified group of assets is used to service the debt. More
conventional asset-backed securities are mortgage-backed securities or securities backed by
auto or credit card loans, as we discussed in Chapter 2.
Catastrophe Bonds Oriental Land Company, which manages Tokyo Disneyland, issued a bond in 1999 with a final payment that depended on whether there had been an
earthquake near the park. More recently, FIFA (the Fédération Internationale de Foot-
ball Association) issued catastrophe bonds with payments that would be halted if terror-
ism forced the cancellation of the 2006 World Cup. These bonds are a way to transfer
“catastrophe risk” from the firm to the capital markets. Investors in these bonds receive
compensation for taking on the risk in the form of higher coupon rates. But in the event
of a catastrophe, the bondholders will give up all or part of their investments. “Disaster”
can be defined by total insured losses or by criteria such as wind speed in a hurricane or
Richter level in an earthquake. Issuance of catastrophe bonds has grown in recent years
as insurers have sought ways to spread their risks across a wider spectrum of the capital
market.
Indexed Bonds Indexed bonds make payments that are tied to a general price index or the price of a particular commodity. For example, Mexico has issued bonds with pay-
ments that depend on the price of oil. Some bonds are indexed to the general price level.
The United States Treasury started issuing such inflation-indexed bonds in January 1997.
They are called Treasury Inflation Protected Securities (TIPS). By tying the par value of
the bond to the general level of prices, coupon payments as well as the final repayment of
par value on these bonds increase in direct proportion to the Consumer Price Index. There-
fore, the interest rate on these bonds is a risk-free real rate.
To illustrate how TIPS work, consider a newly issued bond with a 3-year maturity, par
value of $1,000, and a coupon rate of 4%. For simplicity, we will assume the bond makes
annual coupon payments. Assume that inflation turns out to be 2%, 3%, and 1% in the
next 3 years. Table 14.1 shows how the bond cash flows will be calculated. The first pay-
ment comes at the end of the first year, at t 5 1. Because inflation over the year was 2%, the par value of the bond increases from $1,000 to $1,020; because the coupon rate is 4%,
the coupon payment is 4% of this amount, or $40.80. Notice that par value increases by the
inflation rate, and because the coupon payments are 4% of par, they too increase in propor-
tion to the general price level. Therefore, the cash flows paid by the bond are fixed in real terms. When the bond matures, the investor receives a final coupon payment of $42.44 plus
the (price-level-indexed) repayment of principal, $1,061.11. 4
Table 14.1
Principal and interest payments for a Treasury Inflation Protected Security
Time Inflation in Year
Just Ended Par Value Coupon Payment 1
Principal Repayment 5 Total Payment
0 $1,000.00 1 2% 1,020.00 $40.80 $ 0 $ 40.80 2 3 1,050.60 42.02 0 42.02 3 1 1,061.11 42.44 1,061.11 1,103.55
4 By the way, total nominal income (i.e., coupon plus that year’s increase in principal) is treated as taxable income
in each year.
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452 P A R T I V Fixed-Income Securities
The nominal rate of return on the bond in the first year is
Nominal return 5 Interest 1 Price appreciation
Initial price 5
40.80 1 20
1,000 5 6.08%
The real rate of return is precisely the 4% real yield on the bond:
Real return 5 1 1 Nominal return
1 1 Inflation 2 1 5
1.0608
1.02 2 1 5 .04, or 4%
One can show in a similar manner (see Problem 18 in the end-of-chapter problems) that
the rate of return in each of the 3 years is 4% as long as the real yield on the bond remains
constant. If real yields do change, then there will be capital gains or losses on the bond. In
mid-2013, the real yield on long-term TIPS bonds was less than 0.5%.
14.2 Bond Pricing
Because a bond’s coupon and principal repayments all occur months or years in the future,
the price an investor would be willing to pay for a claim to those payments depends on the
value of dollars to be received in the future compared to dollars in hand today. This “pres-
ent value” calculation depends in turn on market interest rates. As we saw in Chapter 5,
the nominal risk-free interest rate equals the sum of (1) a real risk-free rate of return and (2)
a premium above the real rate to compensate for expected inflation. In addition, because
most bonds are not riskless, the discount rate will embody an additional premium that
reflects bond-specific characteristics such as default risk, liquidity, tax attributes, call risk,
and so on.
We simplify for now by assuming there is one interest rate that is appropriate for
discounting cash flows of any maturity, but we can relax this assumption easily. In prac-
tice, there may be different discount rates for cash flows accruing in different periods. For
the time being, however, we ignore this refinement.
To value a security, we discount its expected cash flows by the appropriate discount
rate. The cash flows from a bond consist of coupon payments until the maturity date plus
the final payment of par value. Therefore,
Bond value 5 Present value of coupons 1 Present value of par value
If we call the maturity date T and call the interest rate r, the bond value can be written as
Bond value 5 a T
t51
Coupon
(1 1 r)t 1
Par value
(1 1 r)T (14.1)
The summation sign in Equation 14.1 directs us to add the present value of each coupon
payment; each coupon is discounted based on the time until it will be paid. The first term
on the right-hand side of Equation 14.1 is the present value of an annuity. The second term
is the present value of a single amount, the final payment of the bond’s par value.
You may recall from an introductory finance class that the present value of a $1 annuity
that lasts for T periods when the interest rate equals r is 1
r B1 2 1
(1 1 r)T R. We call this
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C H A P T E R 1 4 Bond Prices and Yields 453
expression the T -period annuity factor for an interest rate of r. 5 Similarly, we call 1
(1 1 r)T
the PV factor, that is, the present value of a single payment of $1 to be received in T peri- ods. Therefore, we can write the price of the bond as
Price 5 Coupon 3 1
r B1 2 1
(1 1 r)T R 1 Par value 3 1
(1 1 r)T
5 Coupon 3 Annuity factor(r, T) 1 Par value 3 PV factor(r, T) (14.2)
We discussed earlier an 8% coupon, 30-year maturity bond with par value of $1,000 paying 60 semiannual coupon payments of $40 each. Suppose that the interest rate is 8% annually, or r 5 4% per 6-month period. Then the value of the bond can be written as
Price 5 a 60
t51
$40 (1.04)t
1 $1,000 (1.04)60
5 $40 3 Annuity factor(4%, 60) 1 $1,000 3 PV factor(4%, 60)
(14.3)
It is easy to confirm that the present value of the bond’s 60 semiannual coupon pay- ments of $40 each is $904.94 and that the $1,000 final payment of par value has a present value of $95.06, for a total bond value of $1,000. You can calculate the value directly from Equation 14.2, perform these calculations on any financial calculator (see Example 14.3 below), use a spreadsheet program (see the Excel Applications box), or use a set of present value tables.
In this example, the coupon rate equals the market interest rate, and the bond price equals par value. If the interest rate were not equal to the bond’s coupon rate, the bond would not sell at par value. For example, if the interest rate were to rise to 10% (5% per 6 months), the bond’s price would fall by $189.29 to $810.71, as follows:
$40 3 Annuity factor(5%, 60) 1 $1,000 3 PV factor(5%, 60) 5 $757.17 1 $53.54 5 $810.71
At a higher interest rate, the present value of the payments to be received by the bondholder is lower. Therefore, bond prices fall as market interest rates rise. This illus- trates a crucial general rule in bond valuation. 6
Example 14.2 Bond Pricing
5 Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting T periods can be
viewed as equivalent to a perpetuity whose first payment comes at the end of the current period less another per- petuity whose first payment comes at the end of the ( T 1 1) st period. The immediate perpetuity net of the delayed perpetuity provides exactly T payments. We know that the value of a $1 per period perpetuity is $1/ r. Therefore,
the present value of the delayed perpetuity is $1/ r discounted for T additional periods, or 1
r 3
1
(1 1 r)T .
The present value of the annuity is the present value of the first perpetuity minus the present value of the delayed
perpetuity, or 1
r B1 2 1
(1 1 r)T R.
6 Here is a trap to avoid. You should not confuse the bond’s coupon rate, which determines the interest paid to the
bondholder, with the market interest rate. Once a bond is issued, its coupon rate is fixed. When the market interest rate increases, investors discount any fixed payments at a higher discount rate, which implies that present values
and bond prices fall.
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454 P A R T I V Fixed-Income Securities
Bond prices are tedious to calculate without a spreadsheet or a financial calculator,
but they are easy to calculate with either. Financial calculators designed with present and
future value formulas already programmed can greatly simplify calculations of the sort we
just encountered in Example 14.2. The basic financial calculator uses five keys that cor-
respond to the inputs for time-value-of-money problems such as bond pricing:
1. n is the number of time periods. In the case of a bond, n equals the number of periods until the bond matures. If the bond makes semiannual payments, n is the number of half-year periods or, equivalently, the number of semiannual coupon
payments. For example, if the bond has 10 years until maturity, you would enter
20 for n, since each payment period is one-half year.
2. i is the interest rate per period, expressed as a percentage (not as a decimal). For example, if the interest rate is 6%, you would enter 6, not .06.
3. PV is the present value. Many calculators require that PV be entered as a negative number, in recognition of the fact that purchase of the bond is a cash outflow, while the receipt of coupon payments and face value are cash inflows.
4. FV is the future value or face value of the bond. In general, FV is interpreted as a one-time future payment of a cash flow, which, for bonds, is the face (i.e., par)
value.
5. PMT is the amount of any recurring payment. For coupon bonds, PMT is the cou- pon payment; for zero-coupon bonds, PMT will be zero.
Given any four of these inputs, the calculator will solve for the fifth. We can illustrate with
the bond in Example 14.2.
To find the bond’s price when the annual market interest rate is 8%, you would enter these inputs (in any order):
n 60 The bond has a maturity of 30 years, so it makes 60 semian- nual payments.
i 4 The semiannual market interest rate is 4%. FV 1,000 The bond will provide a one-time cash flow of $1,000 when
it matures. PMT 40 Each semiannual coupon payment is $40.
On most calculators, you now punch the “compute” key (labeled COMP or CPT ) and then enter PV to obtain the bond price, that is the present value today of the bond’s cash flows. If you do this, you should find a value of 2 1,000. The negative sign signifies that while the investor receives cash flows from the bond, the price paid to buy the bond is a cash out flow, or a negative cash flow.
If you want to find the value of the bond when the interest rate is 10% (the second part of Example 14.2), just enter 5% for the semiannual interest rate (type “5” and then “ i ”), and when you compute PV, you will find that it is 2 810.71.
Example 14.3 Bond Pricing on a Financial Calculator
Figure 14.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates,
including 8%, at which the bond sells at par, and 10%, at which it sells for $810.71. The
negative slope illustrates the inverse relationship between prices and yields. The shape
of the curve in Figure 14.3 implies that an increase in the interest rate results in a price
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C H A P T E R 1 4 Bond Prices and Yields 455
decline that is smaller than the price gain resulting from a decrease of equal magnitude
in the interest rate. This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in
the interest rate result in progressively smaller reductions in the bond price. 7 Therefore, the
price curve becomes flatter at higher interest rates. We return to convexity in Chapter 16.
0 5 8 10 15 20 Interest Rate (%)
B o
n d
P ri
ce (
$ )
0
500
810.71 1,000
1,500
2,000
2,500
3,000
3,500
4,000
Figure 14.3 The inverse relationship between bond prices and yields. Price of an 8% coupon bond with 30-year maturity making semiannual payments
7 The progressively smaller impact of interest increases results largely from the fact that at higher rates the bond
is worth less. Therefore, an additional increase in rates operates on a smaller initial base, resulting in a smaller
price decline.
Calculate the price of the 30-year, 8% coupon bond for a market interest rate of 3% per half-year. Compare the capital gains for the interest rate decline to the losses incurred when the rate increases to 5%.
CONCEPT CHECK 14.2
Corporate bonds typically are issued at
par value. This means that the underwriters
of the bond issue (the firms that market the
bonds to the public for the issuing corpora-
tion) must choose a coupon rate that very
closely approximates market yields. In a
primary issue, the underwriters attempt to
sell the newly issued bonds directly to their
customers. If the coupon rate is inadequate,
investors will not pay par value for the
bonds.
After the bonds are issued, bondholders
may buy or sell bonds in secondary mar-
kets. In these markets, bond prices fluctuate
inversely with the market interest rate.
The inverse relationship between price
and yield is a central feature of fixed-
income securities. Interest rate fluctuations
represent the main source of risk in the
fixed-income market, and we devote con-
siderable attention in Chapter 16 to assess-
ing the sensitivity of bond prices to market yields. For now, however, we simply highlight
one key factor that determines that sensitivity, namely, the maturity of the bond.
As a general rule, keeping all other factors the same, the longer the maturity of the
bond, the greater the sensitivity of price to fluctuations in the interest rate. For example,
consider Table 14.2 , which presents the price of an 8% coupon bond at different market
yields and times to maturity. For any departure of the interest rate from 8% (the rate
at which the bond sells at par value), the change in the bond price is greater for longer
times to maturity.
This makes sense. If you buy the bond at par with an 8% coupon rate, and market
rates subsequently rise, then you suffer a loss: You have tied up your money earning 8%
when alternative investments offer higher returns. This is reflected in a capital loss on
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456 P A R T I V Fixed-Income Securities
the bond—a fall in its market price. The longer the period for which your money is tied
up, the greater the loss, and correspondingly the greater the drop in the bond price. In
Table 14.2 , the row for 1-year maturity bonds shows little price sensitivity—that is, with
only 1 year’s earnings at stake, changes in interest rates are not too threatening. But for
30-year maturity bonds, interest rate swings have a large impact on bond prices. The
force of discounting is greatest for the longest-term bonds.
This is why short-term Treasury securities such as T-bills are considered to be the safest.
They are free not only of default risk but also largely of price risk attributable to interest
rate volatility.
Bond Pricing between Coupon Dates Equation 14.2 for bond prices assumes that the next coupon payment is in precisely one
payment period, either a year for an annual payment bond or 6 months for a semiannual
payment bond. But you probably want to be able to price bonds all 365 days of the year,
not just on the one or two dates each year that it makes a coupon payment!
In principle, the fact that the bond is between coupon dates does not affect the pric-
ing problem. The procedure is always the same: Compute the present value of each
remaining payment and sum up. But if you are between coupon dates, there will be
fractional periods remaining until each payment, and this does complicate the arithmetic
computations.
Fortunately, bond pricing functions are included in most spreadsheet programs such
as Excel. The spreadsheet allows you to enter today’s date as well as the maturity date
of the bond, and so can provide prices for bonds at any date. The nearby box shows
you how.
As we pointed out earlier, bond prices are typically quoted net of accrued interest. These
prices, which appear in the financial press, are called flat prices. The actual invoice price that a buyer pays for the bond includes accrued interest. Thus,
Invoice price 5 Flat price 1 Accrued interest
When a bond pays its coupon, flat price equals invoice price, because at that moment
accrued interest reverts to zero. However, this will be the exceptional case, not the rule.
Excel pricing functions provide the flat price of the bond. To find the invoice price, we
need to add accrued interest. Fortunately, Excel also provides functions that count the days
since the last coupon payment date and thus can be used to compute accrued interest. The
nearby box also illustrates how to use these functions. The box provides examples using
bonds that have just paid a coupon and so have zero accrued interest, as well as a bond that
is between coupon dates.
Table 14.2
Bond prices at different interest rates (8% cou- pon bond, coupons paid semiannually)
Bond Price at Given Market Interest Rate
Time to Maturity 2% 4% 6% 8% 10%
1 year 1,059.11 1,038.83 1,019.13 1,000.00 981.41
10 years 1,541.37 1,327.03 1,148.77 1,000.00 875.35 20 years 1,985.04 1,547.11 1,231.15 1,000.00 828.41 30 years 2,348.65 1,695.22 1,276.76 1,000.00 810.71
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457
E xcel and most other spreadsheet programs provide built-in functions to compute bond prices and yields. They typically ask you to input both the date you buy the bond (called the settlement date ) and the maturity date of the bond. The Excel function for bond price is
5 PRICE(settlement date, maturity date, annual coupon rate, yield to maturity, redemption value as percent of par value, number of coupon payments per year)
For the 2.25% coupon July 2018 maturity bond high- lighted in Figure 14.1 , we would enter the values in Spreadsheet 14.1 . (Notice that in spreadsheets, we must enter interest rates as decimals, not percentages). Alter- natively, we could simply enter the following function in Excel:
5 PRICE(DATE(2012,7,31), DATE(2018,7,31), .0225, .0079, 100, 2)
The DATE function in Excel, which we use for both the settlement and maturity date, uses the format DATE(year,month,day). The first date is July 31, 2012, when the bond is purchased, and the second is July 31, 2018, when it matures. Most bonds pay coupons either on the 15th or the last business day of the month.
Notice that the coupon rate and yield to maturity are expressed as decimals, not percentages. In most cases, redemption value is 100 (i.e., 100% of par value), and the resulting price similarly is expressed as a percent of par value. Occasionally, however, you may encounter bonds that pay off at a premium or discount to par value. One example would be callable bonds, discussed shortly.
The value of the bond returned by the pricing func- tion is 108.5392 (cell B12), which nearly matches the price reported in Table 14.1 . (The yield to maturity is
reported to only three decimal places, which results in a little rounding error.) This bond has just paid a coupon. In other words, the settlement date is precisely at the begin- ning of the coupon period, so no adjustment for accrued interest is necessary.
To illustrate the procedure for bonds between coupon payments, consider the 6.25% coupon May 2030 bond, also appearing in Figure 14.1 . Using the entries in column D of the spreadsheet, we find in cell D12 that the (flat) price of the bond is 161.002, which matches the price given in the figure except for a few cents’ rounding error.
What about the bond’s invoice price? Rows 13 through 16 make the necessary adjustments. The function described in cell C13 counts the days since the last coupon. This day count is based on the bond’s settlement date, maturity date, coupon period (1 5 annual; 2 5 semian- nual), and day count convention (choice 1 uses actual days). The function described in cell C14 counts the total days in each coupon payment period. Therefore, the entries for accrued interest in row 15 are the semiannual coupon multiplied by the fraction of a coupon period that has elapsed since the last payment. Finally, the invoice prices in row 16 are the sum of flat price plus accrued interest.
As a final example, suppose you wish to find the price of the bond in Example 14.2. It is a 30-year maturity bond with a coupon rate of 8% (paid semiannually). The market interest rate given in the latter part of the example is 10%. However, you are not given a specific settlement or matu- rity date. You can still use the PRICE function to value the bond. Simply choose an arbitrary settlement date (January 1, 2000, is convenient) and let the maturity date be 30 years hence. The appropriate inputs appear in column F of the spreadsheet, with the resulting price, 81.0707% of face value, appearing in cell F16.
e X c e l Please visit us at www.mhhe.com/bkm
A B C D E F G
=B12 + B15
Formula in column B 2.25% coupon bond,
maturing July 31, 2018 6.25% coupon bond, maturing May 2030
8% coupon bond, 30-year maturity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Settlement date Maturity date Annual coupon rate Yield to maturity Redemption value (% of face value) Coupon payments per year
Flat price (% of par)
Invoice price Accrued interest Days in coupon period Days since last coupon
= DATE (2012, 7, 31) = DATE (2018, 7, 31)
=PRICE(B4,B5,B6,B7,B8,B9) =COUPDAYBS(B4,B5,2,1) =COUPDAYS(B4,B5,2,1) =(B13/B14)*B6*100/2
7/31/2012 7/31/2018
0.0225 0.0079
100 2 2 2
0 108.5392
108.5392
184
7/31/2012 5/15/2030
0.0625 0.02117
100
161.0020
162.3097
184 1.308
77
1/1/2000 1/1/2030
0.08 0.1
100
81.0707
81.0707
182 0
00
Spreadsheet 14.1
Bond Pricing in Excel
eXcel APPLICATIONS: Bond Pricing
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458 P A R T I V Fixed-Income Securities
Suppose an 8% coupon, 30-year bond is selling at $1,276.76. What average rate of return would be earned by an investor purchasing the bond at this price? We find the interest rate at which the present value of the remaining 60 semiannual payments equals the bond price. This is the rate consistent with the observed price of the bond. There- fore, we solve for r in the following equation:
$1,276.76 5 a 60
t51
$40 (1 1 r)t
1 $1,000
(1 1 r)60
or, equivalently,
1,276.76 5 40 3 Annuity factor(r, 60) 1 1,000 3 PV factor(r, 60)
These equations have only one unknown variable, the interest rate, r. You can use a financial calculator or spreadsheet to confirm that the solution is r 5 .03, or 3%, per half-year. 8 This is the bond’s yield to maturity.
The financial press reports yields on an annualized basis, and annualizes the bond’s semiannual yield using simple interest techniques, resulting in an annual percentage rate, or APR. Yields annualized using simple interest are also called “bond equivalent yields.” Therefore, the semiannual yield would be doubled and reported in the newspa- per as a bond equivalent yield of 6%. The effective annual yield of the bond, however, accounts for compound interest. If one earns 3% interest every 6 months, then after 1 year, each dollar invested grows with interest to $1 3 (1.03) 2 5 $1.0609, and the effective annual interest rate on the bond is 6.09%.
Example 14.4 Yield to Maturity
8 On your financial calculator, you would enter the following inputs: n 5 60 periods; PV 5 2 1276.76; FV 5 1000;
PMT 5 40; then you would compute the interest rate (COMP i or CPT i ). Notice that we enter the present value, or PV, of the bond as minus $1,276.76. Again, this is because most calculators treat the initial purchase price of the bond as a cash outflow. Spreadsheet 14.2 shows how to find yield to maturity using Excel. Without a
financial calculator or spreadsheet, you still could solve the equation, but you would need to use a trial-and-error
approach.
14.3 Bond Yields
Most bonds do not sell for par value. But ultimately, barring default, they will mature to
par value. Therefore, we would like a measure of rate of return that accounts for both cur-
rent income and the price increase or decrease over the bond’s life. The yield to maturity is
the standard measure of the total rate of return. However, it is far from perfect, and we will
explore several variations of this measure.
Yield to Maturity In practice, an investor considering the purchase of a bond is not quoted a promised rate
of return. Instead, the investor must use the bond price, maturity date, and coupon pay-
ments to infer the return offered by the bond over its life. The yield to maturity (YTM) is defined as the interest rate that makes the present value of a bond’s payments equal to its
price. This interest rate is often interpreted as a measure of the average rate of return that
will be earned on a bond if it is bought now and held until maturity. To calculate the yield
to maturity, we solve the bond price equation for the interest rate given the bond’s price.
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C H A P T E R 1 4 Bond Prices and Yields 459
Excel also provides a function for yield to maturity that is especially useful in-between
coupon dates. It is
5 YIELD(settlement date, maturity date, annual coupon rate, bond price, redemption
value as percent of par value, number of coupon payments per year)
The bond price used in the function should be the reported flat price, without accrued inter-
est. For example, to find the yield to maturity of the bond in Example 14.4, we would use
column B of Spreadsheet 14.2 . If the coupons were paid only annually, we would change
the entry for payments per year to 1 (see cell D8), and the yield would fall slightly to
5.99%.
The bond’s yield to maturity is the internal rate of return on an investment in the bond.
The yield to maturity can be interpreted as the compound rate of return over the life of the
bond under the assumption that all bond coupons can be reinvested at that yield. 9 Yield to
maturity is widely accepted as a proxy for average return.
Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price. For example, for the 8%, 30-year bond cur-
rently selling at $1,276.76, the current yield would be $80/$1,276.76 5 .0627, or 6.27%,
per year. In contrast, recall that the effective annual yield to maturity is 6.09%. For this
bond, which is selling at a premium over par value ($1,276 rather than $1,000), the coupon
rate (8%) exceeds the current yield (6.27%), which exceeds the yield to maturity (6.09%).
The coupon rate exceeds current yield because the coupon rate divides the coupon pay-
ments by par value ($1,000) rather than by the bond price ($1,276). In turn, the current
yield exceeds yield to maturity because the yield to maturity accounts for the built-in capi-
tal loss on the bond; the bond bought today for $1,276 will eventually fall in value to
$1,000 at maturity.
Example 14.4 illustrates a general rule: For premium bonds (bonds selling above par value), coupon rate is greater than current yield, which in turn is greater than yield
Spreadsheet 14.2
Finding yield to maturity in Excel
A B C D E
The formula entered here is: =YIELD(B3,B4,B5,B6,B7,B8)
Annual couponsSemiannual coupons1 2 3 4 5 6 7 8 9
10
12 11
Settlement date Maturity date Annual coupon rate Bond price (flat) Redemption value (% of face value) Coupon payments per year
Yield to maturity (decimal)
1/1/2000 1/1/2030
0.08 127.676
100 2
0.0600
1/1/2000 1/1/2030
0.08 127.676
100 1
0.0599
e X c e l Please visit us at www.mhhe.com/bkm
9 If the reinvestment rate does not equal the bond’s yield to maturity, the compound rate of return will differ from
YTM. This is demonstrated below in Examples 14.6 and 14.7.
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460 P A R T I V Fixed-Income Securities
to maturity. For discount bonds (bonds selling below par value), these relationships are reversed (see Concept Check 3).
It is common to hear people talking loosely about the yield on a bond. In these cases,
they almost always are referring to the yield to maturity.
Yield to Call Yield to maturity is calculated on the assumption that the bond will be held until maturity.
What if the bond is callable, however, and may be retired prior to the maturity date? How
should we measure average rate of return for bonds subject to a call provision?
Figure 14.4 illustrates the risk of call to the bondholder. The top line is the value of a
“straight” (i.e., noncallable) bond with par value $1,000, an 8% coupon rate, and a 30-year
time to maturity as a function of the market interest rate. If interest rates fall, the bond
price, which equals the present value of the promised payments, can rise substantially.
Now consider a bond that has the same coupon rate and maturity date but is callable
at 110% of par value, or $1,100. When interest rates fall, the present value of the bond’s
scheduled payments rises, but the call provision allows the issuer to repurchase the bond at the call price. If the call price is less than the present value of the scheduled payments, the
issuer may call the bond back from the bondholder.
The bottom line in Figure 14.4 is the value of the callable bond. At high interest rates,
the risk of call is negligible because the present value of scheduled payments is less than
the call price; therefore the values of the
straight and callable bonds converge. At
lower rates, however, the values of the bonds
begin to diverge, with the difference reflect-
ing the value of the firm’s option to reclaim
the callable bond at the call price. At very
low rates, the present value of scheduled pay-
ments exceeds the call price, so the bond is
called. Its value at this point is simply the call
price, $1,100.
This analysis suggests that bond market
analysts might be more interested in a bond’s
yield to call rather than yield to maturity,
especially if the bond is likely to be called.
The yield to call is calculated just like the
yield to maturity except that the time until
call replaces time until maturity, and the call
price replaces the par value. This computa-
tion is sometimes called “yield to first call,”
as it assumes the issuer will call the bond as
soon as it may do so.
What will be the relationship among coupon rate, current yield, and yield to maturity for bonds selling at discounts from par? Illustrate using the 8% (semiannual payment) coupon bond, assuming it is selling at a yield to maturity of 10%.
CONCEPT CHECK 14.3
2,000
5 6 7 8 9 10 11 12 133 4
Interest Rate (%)
P ri
ce s
($ )
Callable Bond
Straight Bond
0
200
400
600
800
1,000 1,100 1,200 1,400
1,600
1,800
Figure 14.4 Bond prices: Callable and straight debt. Coupon 5 8%; maturity 5 30 years; semiannual payments.
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C H A P T E R 1 4 Bond Prices and Yields 461
Suppose the 8% coupon, 30-year maturity bond sells for $1,150 and is callable in 10 years at a call price of $1,100. Its yield to maturity and yield to call would be calcu- lated using the following inputs:
Yield to Call Yield to Maturity
Coupon payment $40 $40 Number of semiannual periods 20 periods 60 periods Final payment $1,100 $1,000 Price $1,150 $1,150
Yield to call is then 6.64%. [To confirm this on a calculator, input n 5 20; PV 5 ( 2 )1150; FV 5 1100; PMT 5 40; compute i as 3.32%, or 6.64% bond equivalent yield.] Yield to maturity is 6.82%. [To confirm, input n 5 60; PV 5 ( 2 )1150; FV 5 1000; PMT 5 40; compute i as 3.41% or 6.82% bond equivalent yield.] In Excel, you can cal- culate yield to call as 5 YIELD(DATE(2000,01,01), DATE(2010,01,01), .08, 115, 110, 2). Notice that redemption value is input as 110, i.e., 110% of par value.
Example 14.5 Yield to Call
a. The yield to maturity on two 10-year maturity bonds currently is 7%. Each bond has a call price of $1,100. One bond has a coupon rate of 6%, the other 8%. Assume for simplicity that bonds are called as soon as the present value of their remaining payments exceeds their call price. What will be the capital gain on each bond if the market interest rate suddenly falls to 6%?
b. A 20-year maturity 9% coupon bond paying coupons semiannually is callable in 5 years at a call price of $1,050. The bond currently sells at a yield to maturity of 8%. What is the yield to call?
CONCEPT CHECK 14.4
We have noted that most callable bonds are issued with an initial period of call protec-
tion. In addition, an implicit form of call protection operates for bonds selling at deep dis-
counts from their call prices. Even if interest rates fall a bit, deep-discount bonds still will
sell below the call price and thus will not be subject to a call.
Premium bonds that might be selling near their call prices, however, are especially apt
to be called if rates fall further. If interest rates fall, a callable premium bond is likely to
provide a lower return than could be earned on a discount bond whose potential price
appreciation is not limited by the likelihood of a call. Investors in premium bonds therefore
may be more interested in the bond’s yield to call than its yield to maturity because it may
appear to them that the bond will be retired at the call date.
Realized Compound Return versus Yield to Maturity We have noted that yield to maturity will equal the rate of return realized over the life of
the bond if all coupons are reinvested at an interest rate equal to the bond’s yield to matu-
rity. Consider, for example, a 2-year bond selling at par value paying a 10% coupon once a
year. The yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest
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462 P A R T I V Fixed-Income Securities
rate of 10%, the $1,000 investment in the bond
will grow after 2 years to $1,210, as illustrated
in Figure 14.5 , panel A. The coupon paid in the
first year is reinvested and grows with interest to
a second-year value of $110, which together with
the second coupon payment and payment of par
value in the second year results in a total value of
$1,210.
To summarize, the initial value of the invest-
ment is V 0 5 $1,000. The final value in 2 years is V 2 5 $1,210. The compound rate of return, there- fore, is calculated as follows:
V0(1 1 r) 2 5 V2
$1,000(1 1 r)2 5 $1,210
r 5 .10 5 10%
With a reinvestment rate equal to the 10% yield to
maturity, the realized compound return equals yield to maturity.
But what if the reinvestment rate is not 10%?
If the coupon can be invested at more than 10%,
funds will grow to more than $1,210, and the real-
ized compound return will exceed 10%. If the
reinvestment rate is less than 10%, so will be the
realized compound return. Consider the following
example.
A: Reinvestment Rate 5 10%
B: Reinvestment Rate 5 8%
Future Value:
Time:
$1,100
$100Cash Flow:
0
$1,100 = $1,100
100 × 1.10 = $ 110
$1,210
Future Value:
Time:
$1,100
$100Cash Flow:
0
$1,100 = $1,100
100 × 1.08 = $ 108
$1,208
21
21
Figure 14.5 Growth of invested funds
If the interest rate earned on the first coupon is less than 10%, the final value of the investment will be less than $1,210, and the realized compound return will be less than 10%. To illustrate, suppose the interest rate at which the coupon can be invested is only 8%. The following calculations are illustrated in Figure 14.5 , panel B.
Future value of first coupon payment with interest earnings 5 $100 3 1.08 5 $ 108
1 Cash payment in second year (final coupon plus par value) $1,100
5 Total value of investment with reinvested coupons $1,208
The realized compound return is the compound rate of growth of invested funds, assum- ing that all coupon payments are reinvested. The investor purchased the bond for par at $1,000, and this investment grew to $1,208.
V0(1 1 r) 2 5 V2
$1,000(1 1 r)2 5 $1,208
r 5 .0991 5 9.91%
Example 14.6 Realized Compound Return
Example 14.6 highlights the problem with conventional yield to maturity when rein-
vestment rates can change over time. Conventional yield to maturity will not equal realized
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C H A P T E R 1 4 Bond Prices and Yields 463
compound return. However, in an economy with future interest rate uncertainty, the rates at
which interim coupons will be reinvested are not yet known. Therefore, although realized
compound return can be computed after the investment period ends, it cannot be computed in advance without a forecast of future reinvestment rates. This reduces much of the attrac-
tion of the realized return measure.
Forecasting the realized compound yield over various holding periods or investment
horizons is called horizon analysis . The forecast of total return depends on your forecasts of both the price of the bond when you sell it at the end of your horizon and the rate at which you are able to reinvest coupon income. The sales price depends in turn on the yield
to maturity at the horizon date. With a longer investment horizon, however, reinvested cou-
pons will be a larger component of your final proceeds.
Suppose you buy a 30-year, 7.5% (annual payment) coupon bond for $980 (when its yield to maturity is 7.67%) and plan to hold it for 20 years. Your forecast is that the bond’s yield to maturity will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%. At the end of your investment horizon, the bond will have 10 years remaining until expiration, so the forecast sales price (using a yield to maturity of 8%) will be $966.45. The 20 coupon payments will grow with compound interest to $2,758.92. (This is the future value of a 20-year $75 annuity with an interest rate of 6%.)
On the basis of these forecasts, your $980 investment will grow in 20 years to $966.45 1 $2,758.92 5 $3,725.37. This corresponds to an annualized compound return of 6.90%:
V0(1 1 r) 20 5 V20
$980(1 1 r)20 5 $3,725.37
r 5 .0690 5 6.90%
Example 14.7 Horizon Analysis
Examples 14.6 and 14.7 demonstrate that as interest rates change, bond investors are
actually subject to two sources of offsetting risk. On the one hand, when rates rise, bond
prices fall, which reduces the value of the portfolio. On the other hand, reinvested coupon
income will compound more rapidly at those higher rates. This reinvestment rate risk will offset the impact of price risk. In Chapter 16, we will explore this trade-off in more
detail and will discover that by carefully tailoring their bond portfolios, investors can pre-
cisely balance these two effects for any given investment horizon.
14.4 Bond Prices over Time
As we noted earlier, a bond will sell at par value when its coupon rate equals the market
interest rate. In these circumstances, the investor receives fair compensation for the time
value of money in the form of the recurring coupon payments. No further capital gain is
necessary to provide fair compensation.
When the coupon rate is lower than the market interest rate, the coupon payments alone
will not provide investors as high a return as they could earn elsewhere in the market. To
receive a competitive return on such an investment, investors also need some price appre-
ciation on their bonds. The bonds, therefore, must sell below par value to provide a “built-
in” capital gain on the investment.
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464 P A R T I V Fixed-Income Securities
When bond prices are set according to the present
value formula, any discount from par value provides an
anticipated capital gain that will augment a below-market
coupon rate by just enough to provide a fair total rate of
return. Conversely, if the coupon rate exceeds the market
interest rate, the interest income by itself is greater than
that available elsewhere in the market. Investors will bid
up the price of these bonds above their par values. As the
bonds approach maturity, they will fall in value because
fewer of these above-market coupon payments remain. The resulting capital losses offset
the large coupon payments so that the bondholder again receives only a competitive rate
of return.
Problem 14 at the end of the chapter asks you to work through the case of the high-
coupon bond. Figure 14.6 traces out the price paths of high- and low-coupon bonds (net of
accrued interest) as time to maturity approaches, at least for the case in which the market
interest rate is constant. The low-coupon bond enjoys capital gains, whereas the high-
coupon bond suffers capital losses. 11
We use these examples to show that each bond offers investors the same total rate of
return. Although the capital gains versus income components differ, the price of each
bond is set to provide competitive rates, as we should expect in well-functioning capital
markets. Security returns all should be comparable on an after-tax risk-adjusted basis. If
they are not, investors will try to sell low-return securities, thereby driving down their
prices until the total return at the now-lower price is competitive with other securities.
To illustrate built-in capital gains or losses, suppose a bond was issued several years ago when the interest rate was 7%. The bond’s annual coupon rate was thus set at 7%. (We will suppose for simplicity that the bond pays its coupon annually.) Now, with 3 years left in the bond’s life, the interest rate is 8% per year. The bond’s market price is the present value of the remaining annual coupons plus payment of par value. That present value is 10
$70 3 Annuity factor(8%, 3) 1 $1,000 3 PV factor(8%, 3) 5 $974.23
which is less than par value. In another year, after the next coupon is paid and remaining maturity falls to two
years, the bond would sell at
$70 3 Annuity factor(8%, 2) 1 $1,000 3 PV factor(8%, 2) 5 $982.17
thereby yielding a capital gain over the year of $7.94. If an investor had purchased the bond at $974.23, the total return over the year would equal the coupon payment plus capital gain, or $70 1 $7.94 5 $77.94. This represents a rate of return of $77.94/$974.23, or 8%, exactly the rate of return currently available elsewhere in the market.
Example 14.8 Fair Holding-Period Return
At what price will the bond in Example 14.8 sell in yet another year, when only 1 year remains until maturity? What is the rate of return to an investor who purchases the bond when its price is $982.17 and sells it 1 year hence?
CONCEPT CHECK 14.5
10
Using a calculator, enter n 5 3, i 5 8, PMT 5 70, FV 5 1000, and compute PV.
11
If interest rates are volatile, the price path will be “jumpy,” vibrating around the price path in Figure 14.6 and
reflecting capital gains or losses as interest rates fluctuate. Ultimately, however, the price must reach par value at
the maturity date, so the price of the premium bond will fall over time while that of the discount bond will rise.
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C H A P T E R 1 4 Bond Prices and Yields 465
Prices should continue to adjust until
all securities are fairly priced in that
expected returns are comparable, given
appropriate risk and tax adjustments.
We see evidence of this price
adjustment in Figure 14.1 . Compare
the highlighted bond with the one
just below it. The July 2018 bond
has a coupon rate of 2.25%, while
the November 2018 bond has a
much higher coupon rate, 9%. But
the higher coupon rate on that bond
does not mean that it offers a higher
return; instead, it sells at a much
higher price. The yields to maturity on
the two bonds are nearly identical, a
shade below .8%. This makes sense,
since investors should care about their
total return, including both coupon
income as well as price change. In the
end, prices of similar-maturity bonds
adjust until yields are pretty much
equalized.
Of course, the yields across bonds in Figure 14.1 are not all equal. Clearly, longer term
bonds at this time offered higher promised yields, a common pattern, and one that reflects
the relative risks of the bonds. We will explore the relation between yield and time to matu-
rity in the next chapter.
Yield to Maturity versus Holding-Period Return In Example 14.8, the holding-period return and the yield to maturity were equal. The bond
yield started and ended the year at 8%, and the bond’s holding-period return also equaled
8%. This turns out to be a general result. When the yield to maturity is unchanged over the
period, the rate of return on the bond will equal that yield. As we noted, this should not be
surprising: The bond must offer a rate of return competitive with those available on other
securities.
However, when yields fluctuate, so will a bond’s rate of return. Unanticipated changes
in market rates will result in unanticipated changes in bond returns and, after the fact, a
bond’s holding-period return can be better or worse than the yield at which it initially sells.
An increase in the bond’s yield acts to reduce its price, which reduces the holding period
return. In this event, the holding period return is likely to be less than the initial yield to
maturity. 12
Conversely, a decline in yield will result in a holding-period return greater than
the initial yield.
0 0 5 10 15 20 25 30
P ri
ce (
% o
f P a r
V a lu
e )
Time (years)
140
160
120
100
80
60
40
20
Coupon = 12% Coupon = 4%
Figure 14.6 Price path of two 30-year maturity bonds, each selling at a yield to maturity of 8%. Bond price approaches par value as maturity date approaches.
12 We have to be a bit careful here. When yields increase, coupon income can be reinvested at higher rates, which
offsets the impact of the initial price decline. If your holding period is sufficiently long, the positive impact of the
higher reinvestment rate can more than offset the initial price decline. But common performance evaluation peri-
ods for portfolio managers are no more than 1 year, and over these shorter horizons the price impact will almost
always dominate the impact of the reinvestment rate. We discuss the trade-off between price risk and reinvestment
rate risk more fully in Chapter 16.
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466 P A R T I V Fixed-Income Securities
Here is another way to think about the difference between yield to maturity and
holding-period return. Yield to maturity depends only on the bond’s coupon, current price, and par value at maturity. All of these values are observable today, so yield to
maturity can be easily calculated. Yield to maturity can be interpreted as a measure of
the average rate of return if the investment in the bond is held until the bond matures. In contrast, holding-period return is the rate of return over a particular investment period
and depends on the market price of the bond at the end of that holding period; of course
this price is not known today. Because bond prices over the holding period will respond to unanticipated changes in interest rates, holding-period return can at most be forecast.
Zero-Coupon Bonds and Treasury Strips Original-issue discount bonds are less common than coupon bonds issued at par. These are bonds that are issued intentionally with low coupon rates that cause the bond to sell at a dis-
count from par value. The most common example of this type of bond is the zero-coupon bond, which carries no coupons and provides all its return in the form of price appreciation. Zeros provide only one cash flow to their owners, on the maturity date of the bond.
U.S. Treasury bills are examples of short-term zero-coupon instruments. If the bill has
face value of $10,000, the Treasury issues or sells it for some amount less than $10,000,
agreeing to repay $10,000 at maturity. All of the investor’s return comes in the form of
price appreciation.
Longer-term zero-coupon bonds are commonly created from coupon-bearing notes and
bonds. A bond dealer who purchases a Treasury coupon bond may ask the Treasury to break
down the cash flows to be paid by the bond into a series of independent securities, where
each security is a claim to one of the payments of the original bond. For example, a 10-year
coupon bond would be “stripped” of its 20 semiannual coupons, and each coupon payment
would be treated as a stand-alone zero-coupon bond. The maturities of these bonds would
thus range from 6 months to 10 years. The final payment of principal would be treated as
another stand-alone zero-coupon security. Each of the payments is now treated as an inde-
pendent security and is assigned its own CUSIP number (by the Committee on Uniform
Securities Identification Procedures), the security identifier that allows for electronic trad-
ing over the Fedwire system, a network that connects all Federal Reserve banks and their
Show that if yield to maturity increases, then holding-period return is less than initial yield. For example, suppose in Example 14.9 that by the end of the first year, the bond’s yield to maturity is 8.5%. Find the 1-year holding-period return and compare it to the bond’s initial 8% yield to maturity.
CONCEPT CHECK 14.6
Consider a 30-year bond paying an annual coupon of $80 and selling at par value of $1,000. The bond’s initial yield to maturity is 8%. If the yield remains at 8% over the year, the bond price will remain at par, so the holding-period return also will be 8%. But if the yield falls below 8%, the bond price will increase. Suppose the yield falls and the price increases to $1,050. Then the holding-period return is greater than 8%:
Holding-period return 5 $80 1 ($1,050 2 $1,000)
$1,000 5 .13, or 13%
Example 14.9 Yield to Maturity versus Holding-Period Return
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C H A P T E R 1 4 Bond Prices and Yields 467
branches. The payments are still considered obli-
gations of the U.S. Treasury. The Treasury pro-
gram under which coupon stripping is performed
is called STRIPS (Separate Trading of Registered
Interest and Principal of Securities), and these
zero-coupon securities are called Treasury strips. What should happen to prices of zeros as time
passes? On their maturity dates, zeros must sell
for par value. Before maturity, however, they
should sell at discounts from par, because of the
time value of money. As time passes, price should
approach par value. In fact, if the interest rate is
constant, a zero’s price will increase at exactly
the rate of interest.
To illustrate, consider a zero with 30 years
until maturity, and suppose the market interest
rate is 10% per year. The price of the bond today
is $1,000/(1.10) 30
5 $57.31. Next year, with only
29 years until maturity, if the yield is still 10%,
the price will be $1,000/(1.10) 29
5 $63.04, a 10%
increase over its previous-year value. Because the
par value of the bond is now discounted for one less year, its price has increased by the
1-year discount factor.
Figure 14.7 presents the price path of a 30-year zero-coupon bond for an annual market
interest rate of 10%. The bond prices rise exponentially, not linearly, until its maturity.
After-Tax Returns The tax authorities recognize that the “built-in” price appreciation on original-issue dis-
count (OID) bonds such as zero-coupon bonds represents an implicit interest payment to
the holder of the security. The IRS, therefore, calculates a price appreciation schedule to
impute taxable interest income for the built-in appreciation during a tax year, even if the
asset is not sold or does not mature until a future year. Any additional gains or losses that
arise from changes in market interest rates are treated as capital gains or losses if the OID
bond is sold during the tax year.
Maturity Date
Today
Time (years)
P ri
ce (
$ )
1,000 900 800 700 600 500 400 300 200 100
0
30 6 9
1 2
1 5
1 8
2 1
2 4
2 7
3 0
Figure 14.7 The price of a 30-year zero-coupon bond over time at a yield to maturity of 10%. Price equals 1,000/(1.10) T , where T is time until maturity.
If the interest rate originally is 10%, the 30-year zero would be issued at a price of $1,000/(1.10) 30 5 $57.31. The following year, the IRS calculates what the bond price would be if the yield were still 10%. This is $1,000/(1.10) 29 5 $63.04. Therefore, the IRS imputes interest income of $63.04 2 $57.31 5 $5.73. This amount is subject to tax. Notice that the imputed interest income is based on a “constant yield method” that ignores any changes in market interest rates.
If interest rates actually fall, let’s say to 9.9%, the bond price will be $1,000/(1.099) 29 5 $64.72. If the bond is sold, then the difference between $64.72 and $63.04 is treated as capital gains income and taxed at the capital gains tax rate. If the bond is not sold, then the price difference is an unrealized capital gain and does not result in taxes in that year. In either case, the investor must pay taxes on the $5.73 of imputed interest at the rate on ordinary income.
Example 14.10 Taxation of Original-Issue Discount Bonds
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468 P A R T I V Fixed-Income Securities
The procedure illustrated in Example 14.10 applies as well to the taxation of other
original-issue discount bonds, even if they are not zero-coupon bonds. Consider, as an
example, a 30-year maturity bond that is issued with a coupon rate of 4% and a yield to
maturity of 8%. For simplicity, we will assume that the bond pays coupons once annually.
Because of the low coupon rate, the bond will be issued at a price far below par value,
specifically at $549.69. If the bond’s yield to maturity is still 8%, then its price in
1 year will rise to $553.66. (Confirm this for yourself.) This would provide a pretax
holding-period return (HPR) of exactly 8%:
HPR 5 $40 1 ($553.66 2 $549.69)
$549.69 5 .08
The increase in the bond price based on a constant yield, however, is treated as interest
income, so the investor is required to pay taxes on the explicit coupon income, $40, as well
as the imputed interest income of $553.66 2 $549.69 5 $3.97. If the bond’s yield actually
changes during the year, the difference between the bond’s price and the constant-yield
value of $553.66 would be treated as capital gains income if the bond is sold.
Suppose that the yield to maturity of the 4% coupon, 30-year maturity bond falls to 7% by the end of the first year and that the investor sells the bond after the first year. If the investor’s federal plus state tax rate on interest income is 38% and the combined tax rate on capital gains is 20%, what is the investor’s after- tax rate of return?
CONCEPT CHECK 14.7
14.5 Default Risk and Bond Pricing
Although bonds generally promise a fixed flow of income, that income stream is not risk- less unless the investor can be sure the issuer will not default on the obligation. While U.S.
government bonds may be treated as free of default risk, this is not true of corporate bonds.
Therefore, the actual payments on these bonds are uncertain, for they depend to some
degree on the ultimate financial status of the firm.
Bond default risk, usually called credit risk , is measured by Moody’s Investor Services, Standard & Poor’s Corporation, and Fitch Investors Service, all of which provide finan-
cial information on firms as well as quality ratings of large corporate and municipal bond
issues. International sovereign bonds, which also entail default risk, especially in emerg-
ing markets, also are commonly rated for default risk. Each rating firm assigns letter
grades to the bonds of corporations and municipalities to reflect their assessment of the
safety of the bond issue. The top rating is AAA or Aaa, a designation awarded to only
about a dozen firms. Moody’s modifies each rating class with a 1, 2, or 3 suffix (e.g.,
Aaa1, Aaa2, Aaa3) to provide a finer gradation of ratings. The other agencies use a 1 or 2
modification.
Those rated BBB or above (S&P, Fitch) or Baa and above (Moody’s) are considered
investment-grade bonds , whereas lower-rated bonds are classified as speculative- grade or junk bonds . Defaults on low-grade issues are not uncommon. For example,
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C H A P T E R 1 4 Bond Prices and Yields 469
almost half of the bonds rated CCC by Standard & Poor’s at issue have defaulted within
10 years. Highly rated bonds rarely default, but even these bonds are not free of credit risk.
For example, in 2001 WorldCom sold $11.8 billion of bonds with an investment-grade
rating. Only a year later, the firm filed for bankruptcy and its bondholders lost more than
80% of their investment. Certain regulated institutional investors such as insurance compa-
nies have not always been allowed to invest in speculative-grade bonds.
Figure 14.8 provides the definitions of each bond rating classification.
Bond Ratings
Very High Quality High Quality Speculative Very Poor
Standard & Poor’s AAA AA A BBB BB B CCC D
Moody’s Aaa Aa A Baa Ba B Caa C
At times both Moody’s and Standard & Poor’s have used adjustments to these ratings: S&P uses plus and minus signs: A+ is the strongest A rating and A− the weakest. Moody’s uses a 1, 2, or 3 designation, with 1 indicating the strongest.
Moody’s S&P
Aaa AAA Debt rated Aaa and AAA has the highest rating. Capacity to pay interest and principal is extremely strong.
Aa AA Debt rated Aa and AA has a very strong capacity to pay interest and repay principal. Together with the highest rating, this group comprises the high- grade bond class.
A A Debt rated A has a strong capacity to pay interest and repay principal, although it is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than debt in higher-rated categories.
Baa BBB Debt rated Baa and BBB is regarded as having an adequate capacity to pay interest and repay principal. Whereas it normally exhibits adequate protection parameters, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity to pay interest and repay principal for debt in this category than in higher-rated categories. These bonds are medium-grade obligations.
Ba BB Debt rated in these categories is regarded, on balance, as predominantly B B speculative with respect to capacity to pay interest and repay principal in Caa CCC accordance with the terms of the obligation. BB and Ba indicate the lowest Ca CC degree of speculation, and CC and Ca the highest degree of speculation.
Although such debt will likely have some quality and protective characteristics, these are outweighed by large uncertainties or major risk exposures to adverse conditions. Some issues may be in default.
C C This rating is reserved for income bonds on which no interest is being paid.
D D Debt rated D is in default, and payment of interest and/or repayment of principal is in arrears.
Figure 14.8 Definitions of each bond rating class Source: Stephen A. Ross and Randolph W. Westerfield, Corporate Finance, Copyright 1988 (St. Louis: Times Mirror/ Mosby College Publishing, reproduced with permission from the McGraw-Hill Companies, Inc.). Data from various edi- tions of Standard & Poor’s Bond Guide and Moody’s Bond Guide.
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470 P A R T I V Fixed-Income Securities
Junk Bonds Junk bonds, also known as high-yield bonds, are nothing more than speculative-grade (low-rated or unrated) bonds. Before 1977, almost all junk bonds were “fallen angels,” that
is, bonds issued by firms that originally had investment-grade ratings but that had since
been downgraded. In 1977, however, firms began to issue “original-issue junk.”
Much of the credit for this innovation is given to Drexel Burnham Lambert, and espe-
cially its trader Michael Milken. Drexel had long enjoyed a niche as a junk bond trader and
had established a network of potential investors in junk bonds. Firms not able to muster an
investment-grade rating were happy to have Drexel (and other investment bankers) market
their bonds directly to the public, as this opened up a new source of financing. Junk issues
were a lower-cost financing alternative than borrowing from banks.
High-yield bonds gained considerable notoriety in the 1980s when they were used as
financing vehicles in leveraged buyouts and hostile takeover attempts. Shortly thereaf-
ter, however, the junk bond market suffered. The legal difficulties of Drexel and Michael
Milken in connection with Wall Street’s insider trading scandals of the late 1980s tainted
the junk bond market.
At the height of Drexel’s difficulties, the high-yield bond market nearly dried up. Since
then, the market has rebounded dramatically. However, it is worth noting that the average
credit quality of newly issued high-yield debt issued today is higher than the average qual-
ity in the boom years of the 1980s. Of course, junk bonds are more vulnerable to economic
distress than investment-grade bonds. During the financial crisis of 2008–2009, prices on
these bonds fell dramatically, and their yields to maturity rose equally dramatically. The
spread between yields on B-rated bonds and Treasuries widened from around 3% in early
2007 to an astonishing 19% by the beginning of 2009.
Determinants of Bond Safety Bond rating agencies base their quality ratings largely on an analysis of the level and trend
of some of the issuer’s financial ratios. The key ratios used to evaluate safety are
1. Coverage ratios —Ratios of company earnings to fixed costs. For example, the times-interest-earned ratio is the ratio of earnings before interest payments and taxes to interest obligations. The fixed-charge coverage ratio includes lease pay- ments and sinking fund payments with interest obligations to arrive at the ratio of
earnings to all fixed cash obligations (sinking funds are described below). Low or
falling coverage ratios signal possible cash flow difficulties.
2. Leverage ratio, debt-to-equity ratio —A too-high leverage ratio indicates excessive indebtedness, signaling the possibility the firm will be unable to earn enough to
satisfy the obligations on its bonds.
3. Liquidity ratios —The two most common liquidity ratios are the current ratio (current assets/current liabilities) and the quick ratio (current assets excluding inventories/current liabilities). These ratios measure the firm’s ability to pay bills
coming due with its most liquid assets.
4. Profitability ratios —Measures of rates of return on assets or equity. Profitabil- ity ratios are indicators of a firm’s overall financial health. The return on assets (earnings before interest and taxes divided by total assets) or return on equity (net income/equity) are the most popular of these measures. Firms with higher returns
on assets or equity should be better able to raise money in security markets because
they offer prospects for better returns on the firm’s investments.
5. Cash flow-to-debt ratio —This is the ratio of total cash flow to outstanding debt.
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C H A P T E R 1 4 Bond Prices and Yields 471
Standard & Poor’s periodically computes median values of selected ratios for firms in
several rating classes, which we present in Table 14.3 . Of course, ratios must be evalu-
ated in the context of industry standards, and analysts differ in the weights they place on
particular ratios. Nevertheless, Table 14.3 demonstrates the tendency of ratios to improve
along with the firm’s rating class. And default rates vary dramatically with bond rating.
Historically, only about 1% of industrial bonds originally rated AA or better at issuance
had defaulted after 15 years. That ratio is around 7.5% for BBB-rated bonds, and 40% for
B-rated bonds. Credit risk clearly varies dramatically across rating classes.
Many studies have tested whether financial ratios can in fact be used to predict default
risk. One of the best-known series of tests was conducted by Edward Altman, who used
discriminant analysis to predict bankruptcy. With this technique a firm is assigned a score
based on its financial characteristics. If its score exceeds a cut-off value, the firm is deemed
creditworthy. A score below the cut-off value indicates significant bankruptcy risk in the
near future.
To illustrate the technique, suppose that we were to collect data on the return on
equity (ROE) and coverage ratios of a sample of firms, and then keep records of any
corporate bankruptcies. In Figure 14.9 we plot the
ROE and coverage ratios for each firm, using X for firms that eventually went bankrupt and O for those that remained solvent. Clearly, the X and O firms show different patterns of data, with the sol-
vent firms typically showing higher values for the
two ratios.
The discriminant analysis determines the equa-
tion of the line that best separates the X and O observations. Suppose that the equation of the line
is .75 5 .9 3 ROE 1 .4 3 Coverage. Then, based
on its own financial ratios, each firm is assigned
a “ Z -score” equal to .9 3 ROE 1 .4 3 Coverage. If its Z -score exceeds .75, the firm plots above the line and is considered a safe bet; Z -scores below .75 foretell financial difficulty.
Table 14.3
Financial ratios by rating class, long-term debt
Note: EBITDA is earnings before interest, taxes, depreciation, and amortization Source: Corporate Rating Criteria, Standard & Poor’s, 2006.
3-year medians
AAA AA A BBB BB B CCC
EBIT interest coverage multiple 23.8 19.5 8.0 4.7 2.5 1.2 0.4 EBITDA interest coverage multiple 25.5 24.6 10.2 6.5 3.5 1.9 0.9 Funds from operations/total debt (%) 203.3 79.9 48.0 35.9 22.4 11.5 5.0 Free operating cash flow/total debt (%) 127.6 44.5 25.0 17.3 8.3 2.8 (2.1) Total debt/EBITDA multiple 0.4 0.9 1.6 2.2 3.5 5.3 7.9 Return on capital (%) 27.6 27.0 17.5 13.4 11.3 8.7 3.2 Total debt/total debt 1 equity (%) 12.4 28.3 37.5 42.5 53.7 75.9 113.5
R O
E
Coverage Ratio
Figure 14.9 Discriminant analysis
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472 P A R T I V Fixed-Income Securities
Altman found the following equation to best separate failing and nonfailing firms:
Z 5 3.1 EBIT
Total assets 1 1.0
Sales
Assets 1 .42
Shareholders’ equity
Total liabilities
1 .85 Retained earnings
Total assets 1 .72
Working capital
Total assets
where EBIT 5 earnings before interest and taxes. 13
Z -scores below 1.23 indicate vulner- ability to bankruptcy, scores between 1.23 and 2.90 are a gray area, and scores above 2.90
are considered safe.
13
Altman’s original work was published in Edward I. Altman, “Financial Ratios, Discriminant Analysis, and the
Prediction of Corporate Bankruptcy,” Journal of Finance 23 (September 1968). This equation is from his updated study, Corporate Financial Distress and Bankruptcy, 2nd ed. (New York: Wiley, 1993), p. 29. Altman’s analysis is updated and extended in W. H. Beaver, M. F. McNichols, and J-W. Rhie, “Have Financial Statements become
Less Informative? Evidence from the Ability of Financial Ratios to Predict Bankruptcy,” Review of Accounting Studies 10 (2005), pp. 93–122.
14
Although it is less common, the sinking fund provision also may call for periodic payments to a trustee, with the
payments invested so that the accumulated sum can be used for retirement of the entire issue at maturity.
Suppose we add a new variable equal to current liabilities/current assets to Altman’s equation. Would you expect this variable to receive a positive or negative coefficient?
CONCEPT CHECK 14.8
Bond Indentures A bond is issued with an indenture, which is the contract between the issuer and the bond- holder. Part of the indenture is a set of restrictions that protect the rights of the bondholders.
Such restrictions include provisions relating to collateral, sinking funds, dividend policy,
and further borrowing. The issuing firm agrees to these protective covenants in order to market its bonds to investors concerned about the safety of the bond issue.
Sinking Funds Bonds call for the payment of par value at the end of the bond’s life. This payment constitutes a large cash commitment for the issuer. To help ensure the com-
mitment does not create a cash flow crisis, the firm agrees to establish a sinking fund to spread the payment burden over several years. The fund may operate in one of two ways:
1. The firm may repurchase a fraction of the outstanding bonds in the open market
each year.
2. The firm may purchase a fraction of the outstanding bonds at a special call price
associated with the sinking fund provision. The firm has an option to purchase the
bonds at either the market price or the sinking fund price, whichever is lower. To
allocate the burden of the sinking fund call fairly among bondholders, the bonds
chosen for the call are selected at random based on serial number. 14
The sinking fund call differs from a conventional bond call in two important ways.
First, the firm can repurchase only a limited fraction of the bond issue at the sinking fund
call price. At best, some indentures allow firms to use a doubling option, which allows repurchase of double the required number of bonds at the sinking fund call price. Second,
while callable bonds generally have call prices above par value, the sinking fund call price
usually is set at the bond’s par value.
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C H A P T E R 1 4 Bond Prices and Yields 473
Although sinking funds ostensibly protect bondholders by making principal repayment
more likely, they can hurt the investor. The firm will choose to buy back discount bonds
(selling below par) at market price, while exercising its option to buy back premium bonds
(selling above par) at par. Therefore, if interest rates fall and bond prices rise, firms will
benefit from the sinking fund provision that enables them to repurchase their bonds at
below-market prices. In these circumstances, the firm’s gain is the bondholder’s loss.
One bond issue that does not require a sinking fund is a serial bond issue, in which the firm sells bonds with staggered maturity dates. As bonds mature sequentially, the principal
repayment burden for the firm is spread over time, just as it is with a sinking fund. One
advantage of serial bonds over sinking fund issues is that there is no uncertainty introduced
by the possibility that a particular bond will be called for the sinking fund. The disadvan-
tage of serial bonds, however, is that bonds of different maturity dates are not interchange-
able, which reduces the liquidity of the issue.
Subordination of Further Debt One of the factors determining bond safety is total outstanding debt of the issuer. If you bought a bond today, you would be understand-
ably distressed to see the firm tripling its outstanding debt tomorrow. Your bond would be
riskier than it appeared when you bought it. To prevent
firms from harming bondholders in this manner, subordi- nation clauses restrict the amount of additional borrow- ing. Additional debt might be required to be subordinated
in priority to existing debt; that is, in the event of bank-
ruptcy, subordinated or junior debtholders will not be paid unless and until the prior senior debt is fully paid off.
Dividend Restrictions Covenants also limit the dividends firms may pay. These limitations protect the
bondholders because they force the firm to retain assets
rather than paying them out to stockholders. A typical
restriction disallows payments of dividends if cumulative
dividends paid since the firm’s inception exceed cumula-
tive retained earnings plus proceeds from sales of stock.
Collateral Some bonds are issued with specific col- lateral behind them. Collateral is a particular asset that the bondholders receive if the firm defaults on the bond. If the
collateral is property, the bond is called a mortgage bond. If the collateral takes the form of other securities held by the
firm, the bond is a collateral trust bond. In the case of equip- ment, the bond is known as an equipment obligation bond. This last form of collateral is used most commonly by firms
such as railroads, where the equipment is fairly standard and
can be easily sold to another firm should the firm default.
Collateralized bonds generally are considered safer
than general debenture bonds, which are unsecured, meaning they do not provide for specific collateral. Credit
risk of unsecured bonds depends on the general earning
power of the firm. If the firm defaults, debenture owners
become general creditors of the firm. Because they are
safer, collateralized bonds generally offer lower yields
than general debentures.
Figure 14.10 shows the terms of a bond issued by
Mobil as described in Moody’s Industrial Manual. The
Figure 14.10 Callable bond issued by Mobil Source: Mergent’s Industrial Manual, Mergent’s Investor Services, 1994. Reprinted with permission. All rights reserved.
& Mobil Corp. debenture 8s, due 2032: Rating — Aa2
2003..........105.007 2006..........104.256 2009..........103.505 2012..........102.754 2015..........102.003 2018..........101.252 2021..........100.501
2005..........104.506 2008..........103.755 2011..........103.004 2014..........102.253 2017..........101.502 2020..........100.751
2004..........104.756 2007..........104.005 2010..........103.254 2013..........102.503 2016..........101.752 2019..........101.001 2022..........100.250
and thereafter at 100 plus accrued interest. SECURITY----Not secured. Ranks equally with all other unsecured and unsubordinated indebtedness of Co. Co. nor any Affiliate will not incur any indebtedness; provided that Co. will not create as security for any indebtedness for borrowed money, any mortgage, pledge, security interest or lien on any stock or indebtedness is directly owned by Co. without effectively providing that the debt securities shall be secured equally and ratably with such indebtedness. so long as such indebtedness shall be so secured. INDENTURE M O D I F I C A T I O N - - - - I n d e n t u r e may be modified, except as provided with, consent of 66 2/3% of debs. outstg. RIGHTS ON DEFAULT----Trustee, or 25% of debs. outstg., may declare principal due and paya- ble (30 days' grace for payment of interest).
LISTED----On New York Stock Exchange. PURPOSE----Proceeds used for general corporate purposes. OFFERED----($250,000,000) at 99.51 plus accrued interest (proceeds to Co., 99.11) on Aug. 5, 1992 thru Merrill Lynch & Co., Donaldson, Lufkin & Jenerette Securities Corp., PaineWebber Inc., Pru- dential Securities Inc., Smith Barney, Harris Upham & Co. Inc. and associates.
AUTH----$250,000,000. OUTSTG----Dec. 31, 1993, $250,000,000. DATED----Oct. 30, 1991. INTEREST----F&A 12. TRUSTEE----Chemical Bank. DENOMINATION----Fully registered, $1,000 and integral multiplies thereof. Transferable and exchangeable without service charge. CALLABLE----As a whole or in part, at any time, on or after Aug. 12, 2002, at the option of Co. on at least 30 but not more than the 60 days' notice to each Aug. 11 as follows:
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474 P A R T I V Fixed-Income Securities
bond is registered and listed on the NYSE. It was issued in 1991 but was not callable until
2002. Although the call price started at 105.007% of par value, it declines gradually until
reaching par after 2020. Most of the terms of the bond are typical and illustrate many of the
indenture provisions we have mentioned. However, in recent years there has been a marked
trend away from the use of call provisions.
Yield to Maturity and Default Risk Because corporate bonds are subject to default risk, we must distinguish between the
bond’s promised yield to maturity and its expected yield. The promised or stated yield will
be realized only if the firm meets the obligations of the bond issue. Therefore, the stated
yield is the maximum possible yield to maturity of the bond. The expected yield to maturity must take into account the possibility of a default.
For example, at the height of the financial crisis in October 2008, as Ford Motor
Company struggled, its bonds due in 2028 were rated CCC and were selling at about 33%
of par value, resulting in a yield to maturity of about 20%. Investors did not really believe
the expected rate of return on these bonds was 20%. They recognized that there was a
decent chance that bondholders would not receive all the payments promised in the bond
contract and that the yield based on expected cash flows was far less than the yield based on promised cash flows. As it turned out, of course, Ford weathered the storm, and inves- tors who purchased its bonds made a very nice profit: The bonds were selling in mid-2012
for about 110% of par value, more than triple their value in 2008.
Suppose a firm issued a 9% coupon bond 20 years ago. The bond now has 10 years left until its maturity date, but the firm is having financial difficulties. Investors believe that the firm will be able to make good on the remaining interest payments, but at the maturity date, the firm will be forced into bankruptcy, and bondholders will receive only 70% of par value. The bond is selling at $750.
Yield to maturity (YTM) would then be calculated using the following inputs:
Expected YTM Stated YTM
Coupon payment $45 $45 Number of semiannual periods 20 periods 20 periods Final payment $700 $1,000 Price $750 $750
The yield to maturity based on promised payments is 13.7%. Based on the expected payment of $700 at maturity, however, the yield to maturity would be only 11.6%. The stated yield to maturity is greater than the yield investors actually expect to receive.
Example 14.11 Expected vs. Promised Yield to Maturity
Example 14.11 suggests that when a bond becomes more subject to default risk,
its price will fall, and therefore its promised yield to maturity will rise. Similarly, the
default premium, the spread between the stated yield to maturity and that on otherwise-
comparable Treasury bonds, will rise. However, its expected yield to maturity, which
ultimately is tied to the systematic risk of the bond, will be far less affected. Let’s
continue Example 14.11.
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C H A P T E R 1 4 Bond Prices and Yields 475
Suppose that the condition of the firm in Example 14.11 deteriorates further, and inves- tors now believe that the bond will pay off only 55% of face value at maturity. Investors now demand an expected yield to maturity of 12% (i.e., 6% semiannually), which is .4% higher than in Example 14.11. But the price of the bond will fall from $750 to $688 [ n 5 20; i 5 6; FV 5 550; PMT 5 $45]. At this price, the stated yield to maturity based on promised cash flows is 15.2%. While the expected yield to maturity has increased by .4%, the drop in price has caused the promised yield to maturity to rise by 1.5%.
Example 14.12 Default Risk and the Default Premium
What is the expected yield to maturity in Example 14.12 if the firm is in even worse condition? Investors expect a final payment of only $500, and the bond price has fallen to $650.
CONCEPT CHECK 14.9 To compensate for the possibility of default, corporate
bonds must offer a default premium. The default pre- mium is the difference between the promised yield on a
corporate bond and the yield of an otherwise-identical
government bond that is riskless in terms of default. If the
firm remains solvent and actually pays the investor all of
the promised cash flows, the investor will realize a higher
yield to maturity than would be realized from the govern-
ment bond. If, however, the firm goes bankrupt, the corporate bond is likely to provide a
lower return than the government bond. The corporate bond has the potential for both bet-
ter and worse performance than the default-free Treasury bond. In other words, it is riskier.
The pattern of default premiums offered on risky bonds is sometimes called the risk structure of interest rates. The greater the default risk, the higher the default premium. Figure 14.11 shows spreads between yields to maturity of bonds of different risk classes.
You can see here clear evidence of credit-risk premiums on promised yields. Note, for
example, the incredible run-up of
credit spreads during the financial
crisis of 2008–2009.
Credit Default Swaps A credit default swap (CDS) is in effect an insurance policy
on the default risk of a bond or
loan. To illustrate, the annual pre-
mium in July 2012 on a 5-year
German government CDS was
about 0.75%, meaning that the
CDS buyer would pay the seller
an annual premium of $.75 for
each $100 of bond principal. The
seller collects these annual pay-
ments for the term of the contract
but must compensate the buyer for
loss of bond value in the event of a
default. 15
–4
0
1 9 7 0
1 9 7 3
1 9 7 6
1 9 7 9
1 9 8 2
1 9 8 5
1 9 8 8
1 9 9 1
1 9 9 4
1 9 9 7
2 0 0 0
2 0 0 3
2 0 0 6
2 0 0 9
2 0 1 2
4
8
12
16
20
Y ie
ld S
p re
a d
( %
)
High Yield Baa-Rated Aaa-Rated
Figure 14.11 Yield spreads between corporate and 10-year Treasury bonds
Source: Federal Reserve Bank of St. Louis.
15 Actually, credit default swaps may pay off even short of an actual default. The contract specifies the particular
“credit events” that will trigger a payment. For example, restructuring (rewriting the terms of a firm’s outstanding
debt as an alternative to formal bankruptcy proceedings) may be defined as a triggering credit event.
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476 P A R T I V Fixed-Income Securities
As originally envisioned, credit default swaps were designed to allow lenders to buy pro-
tection against default risk. The natural buyers of CDSs would then be large bondholders
or banks that wished to enhance the creditworthiness of their outstanding loans. Even if the
borrower had a shaky credit standing, the “insured” debt would be as safe as the issuer of the
CDS. An investor holding a bond with a BB rating could, in principle, raise the effective
quality of the debt to AAA by buying a CDS on the issuer.
This insight suggests how CDS contracts should be priced. If a BB-rated corporate
bond bundled with insurance via a CDS is effectively equivalent to a AAA-rated bond,
then the premium on the swap ought to approximate the yield spread between AAA-rated
and BB-rated bonds. 16
The risk structure of interest rates and CDS prices ought to be
tightly aligned.
Figure 14.12 , panel A, shows the premiums on 5-year CDSs on German government
debt between 2008 and 2012. Even as the strongest economy in the eurozone, German
CDS prices nevertheless reflect financial strain, first in the deep recession of 2009 and
then again in 2011 as the prospects of defaults (and German-led bailouts) of Greece and
B: Premiums on Spanish Sovereign Debt CDS Contracts
250
500
750
900
0
Ja n
0 8
M a y
0 8
P re
m iu
m (
b p
/y e a r)
Se p
0 8
Ja n
0 9
M a y
0 9
Se p
0 9
Ja n
1 0
M a y
1 0
Se p
1 0
Ja n
1 1
M a y
1 1
Se p
1 1
Ja n
1 2
M a y
1 2
50
100
150
200
0
Ja n
0 8
M a y
0 8
P re
m iu
m (
b p
/y e a r)
Se p
0 8
Ja n
0 9
A: Premiums on German Sovereign Debt CDS Contracts
M a y
0 9
Se p
0 9
Ja n
1 0
M a y
1 0
Se p
1 0
Ja n
1 1
M a y
1 1
Se p
1 1
Ja n
1 2
M a y
1 2
Figure 14.12 Pricing of 5-year credit default swaps Source: Bloomberg, August 1, 2012, http://www.bloomberg.com/quote/CDBR1U5:IND/chart .
16
We say approximately because there are some differences between highly rated bonds and bonds synthetically
enhanced with credit default swaps. For example, the term of the swap may not match the maturity of the bond.
Tax treatment of coupon payments versus swap payments may differ, as may the liquidity of the bonds. Finally,
some CDSs may entail one-time up-front payments as well as annual premiums.
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477
other eurozone countries worsened. As its perceived credit risk increased, so did the cost
of insuring its debt.
Panel B of Figure 14.12 shows the premiums on 5-year CDS contracts on Spanish
government debt. Spain’s economy was far shakier than Germany’s, and its CDS prices
reflected this fact. By the summer of 2012, its 5-year CDS premiums were around 600
basis points, about eight times the price to insure German debt.
While CDSs were conceived as a form of bond insurance, it wasn’t long before inves-
tors realized that they could be used to speculate on the financial health of particular issu-
ers. As Figure 14.12 makes clear, someone in early 2011 wishing to bet against Spain
might have purchased CDS contracts on its sovereign debt and would have profited as
CDS prices spiked over the next 18 months. The nearby box discusses the role of credit
default swaps in the financial crisis of 2008–2009.
Credit Risk and Collateralized Debt Obligations Collateralized debt obligations, or CDO s, emerged in the last decade as a major mecha- nism to reallocate credit risk in the fixed-income markets. To create a CDO, a financial
institution, commonly a bank, first would establish a legally distinct entity to buy and later
resell a portfolio of bonds or other loans. A common vehicle for this purpose was the so-
called Structured Investment Vehicle (SIV). 17
An SIV raises funds, often by issuing short-
term commercial paper, and uses the proceeds to buy corporate bonds or other forms of debt
Credit Default Swaps, Systemic Risk, and the Financial
Crisis of 2008–2009
The credit crisis of 2008–2009, when lending among banks and other financial institutions effectively seized up, was in large measure a crisis of transparency. The biggest problem was a widespread lack of confidence in the financial stand- ing of counterparties to a trade. If one institution could not be confident that another would remain solvent, it would understandably be reluctant to offer it a loan. When doubt about the credit exposure of customers and trading part- ners spiked to levels not seen since the Great Depression, the market for loans dried up.
Credit default swaps were particularly cited for foster- ing doubts about counterparty reliability. By August 2008, $63 trillion of such swaps were reportedly outstanding. (By comparison, U.S. gross domestic product in 2008 was about $14 trillion.) As the subprime mortgage market col- lapsed and the economy entered a deep recession, the potential obligations on these contracts ballooned to levels previously considered unimaginable and the ability of CDS sellers to honor their commitments appeared in doubt. For example, the huge insurance firm AIG alone had sold more than $400 billion of CDS contracts on sub- prime mortgages and other loans and was days from insolvency. But AIG’s insolvency could have triggered the insolvency of other firms that had relied on its promise of protection against loan defaults. These in turn might have triggered further defaults. In the end, the government
felt compelled to rescue AIG to prevent a chain reaction of insolvencies.
Counterparty risk and lax reporting requirements made it effectively impossible to tease out firms’ exposures to credit risk. One problem was that CDS positions do not have to be accounted for on balance sheets. And the possibility of one default setting off a sequence of further defaults means that lenders may be exposed to the default of an institution with which they do not even directly trade. Such knock-on effects create systemic risk, in which the entire financial system can freeze up. With the ripple effects of bad debt extending in ever-widening circles, lending to anyone can seem imprudent.
In the aftermath of the credit crisis, the Dodd-Frank Act called for new regulation and reforms. One proposal is for a central clearinghouse for credit derivatives such as CDS contracts. Such a system would foster transparency of posi- tions, would allow the clearinghouse to replace traders’ off- setting long and short positions with a single net position, and would require daily recognition of gains or losses on positions through a margin or collateral account. If losses were to mount, positions would have to be unwound before growing to unsustainable levels. Allowing traders to accurately assess counterparty risk, and limiting such risk through margin accounts and the extra back-up of the clear- inghouse, would go a long way in limiting systemic risk.
W O
R D
S FR O
M T
H E ST
R E E T
17
The legal separation of the bank from the SIV allows the ownership of the loans to be conducted off the bank’s
balance sheet, and thus avoids capital requirements the bank would otherwise encounter.
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478 P A R T I V Fixed-Income Securities
Bank Structured investment vehicle, SIV
Senior tranche
Senior-Subordinated Tranche Structure
Typical Terms
70–90% of notional principal, coupon similar to Aa-Aaa rated bonds
5–15% of principal, investment-grade rating
5–15% of principal, higher- quality junk rating <2%, unrated, coupon rate with 20% credit spread
Mezzanine 1
Mezzanine 2
Equity/first loss/ residual tranche
Figure 14.13 Collateralized debt obligations
such as mortgage loans or credit card debt. These loans are first pooled together and then
split into a series of classes known as tranches. ( Tranche is the French word for “slice.”) Each tranche is given a different level of seniority in terms of its claims on the underlying
loan pool, and each can be sold as a stand-alone security. As the loans in the underlying pool
make their interest payments, the proceeds are distributed to pay interest to each tranche in
order of seniority. This priority structure implies that each tranche has a different exposure
to credit risk.
Figure 14.13 illustrates a typical setup. The senior tranche is on top. Its investors may
account for perhaps 80% of the principal of the entire pool. But it has first claim on all the debt service. Using our numbers, even if 20% of the debt pool defaults, the senior tranche
can be paid in full. Once the highest seniority tranche is paid off, the next-lower class (e.g.,
the mezzanine 1 tranche in Figure 14.13 ) receives the proceeds from the pool of loans until
its claims also are satisfied. Using junior tranches to insulate senior tranches from credit
risk in this manner, one can create Aaa-rated bonds even from a junk-bond portfolio.
Of course, shielding senior tranches from default risk means that the risk is concen-
trated on the lower tranches. The bottom tranche—called alternatively the equity, first-loss,
or residual tranche—has last call on payments from the pool of loans, or, put differently, is
at the head of the line in terms of absorbing default or delinquency risk.
Not surprisingly, investors in tranches with the greatest exposure to credit risk demand
the highest coupon rates. Therefore, while the lower mezzanine and equity tranches bear
the most risk, they will provide the highest returns if credit experience turns out favorably.
Mortgage-backed CDOs were an investment disaster in 2007–2009. These were CDOs
formed by pooling subprime mortgage loans made to individuals whose credit standing did
not allow them to qualify for conventional mortgages. When home prices stalled in 2007
and interest rates on these typically adjustable-rate loans reset to market levels, mortgage
delinquencies and home foreclosures soared, and investors in these securities lost billions
of dollars. Even investors in highly rated tranches experienced large losses.
Not surprisingly, the rating agencies that had certified these tranches as investment-
grade came under considerable fire. Questions were raised concerning conflicts of inter-
est: Because the rating agencies are paid by bond issuers, the agencies were accused of
responding to pressure to ease their standards.
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C H A P T E R 1 4 Bond Prices and Yields 479
Related Web sites for this chapter are available at www. mhhe.com/bkm
1. Fixed-income securities are distinguished by their promise to pay a fixed or specified stream of income to their holders. The coupon bond is a typical fixed-income security.
2. Treasury notes and bonds have original maturities greater than 1 year. They are issued at or near par value, with their prices quoted net of accrued interest.
3. Callable bonds should offer higher promised yields to maturity to compensate investors for the fact that they will not realize full capital gains should the interest rate fall and the bonds be called away
from them at the stipulated call price. Bonds often are issued with a period of call protection. In
addition, discount bonds selling significantly below their call price offer implicit call protection.
4. Put bonds give the bondholder rather than the issuer the option to terminate or extend the life of the bond.
5. Convertible bonds may be exchanged, at the bondholder’s discretion, for a specified number of shares of stock. Convertible bondholders “pay” for this option by accepting a lower coupon rate
on the security.
6. Floating-rate bonds pay a coupon rate at a fixed premium over a reference short-term interest rate. Risk is limited because the rate is tied to current market conditions.
7. The yield to maturity is the single interest rate that equates the present value of a security’s cash flows to its price. Bond prices and yields are inversely related. For premium bonds, the coupon
rate is greater than the current yield, which is greater than the yield to maturity. The order of
these inequalities is reversed for discount bonds.
8. The yield to maturity is often interpreted as an estimate of the average rate of return to an inves- tor who purchases a bond and holds it until maturity. This interpretation is subject to error,
however. Related measures are yield to call, realized compound yield, and expected (versus
promised) yield to maturity.
9. Prices of zero-coupon bonds rise exponentially over time, providing a rate of appreciation equal to the interest rate. The IRS treats this built-in price appreciation as imputed taxable interest
income to the investor.
10. When bonds are subject to potential default, the stated yield to maturity is the maximum pos- sible yield to maturity that can be realized by the bondholder. In the event of default, however,
that promised yield will not be realized. To compensate bond investors for default risk, bonds
must offer default premiums, that is, promised yields in excess of those offered by default-free
government securities. If the firm remains healthy, its bonds will provide higher returns than
government bonds. Otherwise the returns may be lower.
11. Bond safety is often measured using financial ratio analysis. Bond indentures are another safe- guard to protect the claims of bondholders. Common indentures specify sinking fund require-
ments, collateralization of the loan, dividend restrictions, and subordination of future debt.
12. Credit default swaps provide insurance against the default of a bond or loan. The swap buyer pays an annual premium to the swap seller, but collects a payment equal to lost value if the loan
later goes into default.
13. Collateralized debt obligations are used to reallocate the credit risk of a pool of loans. The pool is sliced into tranches, with each tranche assigned a different level of seniority in terms of its
claims on the cash flows from the underlying loans. High seniority tranches are usually quite
safe, with credit risk concentrated on the lower level tranches. Each tranche can be sold as a
stand-alone security.
SUMMARY
debt securities
bond
par value
face value
coupon rate
bond indenture
zero-coupon bonds
callable bonds
convertible bonds
put bond
floating-rate bonds
yield to maturity
KEY TERMS
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480 P A R T I V Fixed-Income Securities
current yield
premium bonds
discount bonds
realized compound return
horizon analysis
reinvestment rate risk
credit risk
investment-grade bonds
speculative-grade or junk
bonds
sinking fund
subordination clauses
collateral
debenture
default premium
credit default swap (CDS)
collateralized debt obligations
(CDOs)
Price of a coupon bond:
Price 5 Coupon 3 1
r B1 2 1
(1 1 r)T R 1 Par value 3 1
(1 1 r)T
5 Coupon 3 Annuity factor(r, T) 1 Par value 3 PV factor(r, T)
KEY EQUATIONS
1. Define the following types of bonds:
a. Catastrophe bond. b. Eurobond. c. Zero-coupon bond. d. Samurai bond. e. Junk bond. f. Convertible bond. g. Serial bond. h. Equipment obligation bond. i. Original issue discount bond. j. Indexed bond. k. Callable bond. l. Puttable bond.
2. T wo bonds have identical times to maturity and coupon rates. One is callable at 105, the other at
110. Which should have the higher yield to maturity? Why?
3. The stated yield to maturity and realized compound yield to maturity of a (default-free) zero-
coupon bond will always be equal. Why?
4. W hy do bond prices go down when interest rates go up? Don’t lenders like high interest rates?
5. A bond with an annual coupon rate of 4.8% sells for $970. What is the bond’s current yield?
6. Which security has a higher effective annual interest rate?
a. A 3-month T-bill selling at $97,645 with par value $100,000. b. A coupon bond selling at par and paying a 10% coupon semiannually.
7. Treasury bonds paying an 8% coupon rate with semiannual payments currently sell at par value. What coupon rate would they have to pay in order to sell at par if they paid their coupons annu- ally? (Hint: What is the effective annual yield on the bond?)
8. Consider a bond with a 10% coupon and with yield to maturity 5 8%. If the bond’s yield to
maturity remains constant, then in 1 year, will the bond price be higher, lower, or unchanged?
Why?
9. Consider an 8% coupon bond selling for $953.10 with 3 years until maturity making annual coupon payments. The interest rates in the next 3 years will be, with certainty, r 1 5 8%, r 2 5 10%, and r 3 5 12%. Calculate the yield to maturity and realized compound yield of the bond.
Basic
Intermediate
PROBLEM SETS
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C H A P T E R 1 4 Bond Prices and Yields 481
10. Assume you have a 1-year investment horizon and are trying to choose among three bonds. All
have the same degree of default risk and mature in 10 years. The first is a zero-coupon bond that
pays $1,000 at maturity. The second has an 8% coupon rate and pays the $80 coupon once per
year. The third has a 10% coupon rate and pays the $100 coupon once per year.
a. If all three bonds are now priced to yield 8% to maturity, what are their prices? b. If you expect their yields to maturity to be 8% at the beginning of next year, what will their
prices be then? What is your before-tax holding-period return on each bond? If your tax
bracket is 30% on ordinary income and 20% on capital gains income, what will your after-
tax rate of return be on each?
c. Recalculate your answer to ( b ) under the assumption that you expect the yields to maturity on each bond to be 7% at the beginning of next year.
11. A 20-year maturity bond with par value of $1,000 makes semiannual coupon payments at a
coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond
if the bond price is:
a. $950. b. $1,000. c. $1,050.
12. Repeat Problem 11 using the same data, but assuming that the bond makes its coupon payments
annually. Why are the yields you compute lower in this case?
13. Fill in the table below for the following zero-coupon bonds, all of which have par values of $1,000.
Price Maturity (years) Bond-Equivalent Yield to Maturity
$400 20 — $500 20 — $500 10 — — 10 10% — 10 8% $400 — 8%
14. Consider a bond paying a coupon rate of 10% per year semiannually when the market interest
rate is only 4% per half-year. The bond has 3 years until maturity.
a. Find the bond’s price today and 6 months from now after the next coupon is paid. b. What is the total (6-month) rate of return on the bond?
15. A bond with a coupon rate of 7% makes semiannual coupon payments on January 15 and
July 15 of each year. The Wall Street Journal reports the ask price for the bond on January 30 at 100.125. What is the invoice price of the bond? The coupon period has 182 days.
16. A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond selling above or
below par value? Explain.
17. Is the coupon rate of the bond in Problem 16 more or less than 9%?
18. Return to Table 14.1 and calculate both the real and nominal rates of return on the TIPS bond in
the second and third years.
19. A newly issued 20-year maturity, zero-coupon bond is issued with a yield to maturity of 8%
and face value $1,000. Find the imputed interest income in the first, second, and last year of the
bond’s life.
20. A newly issued 10-year maturity, 4% coupon bond making annual coupon payments is sold to the public at a price of $800. What will be an investor’s taxable income from the bond over the
coming year? The bond will not be sold at the end of the year. The bond is treated as an original-
issue discount bond.
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482 P A R T I V Fixed-Income Securities
21. A 30-year maturity, 8% coupon bond paying coupons semiannually is callable in 5 years
at a call price of $1,100. The bond currently sells at a yield to maturity of 7% (3.5% per
half-year).
a. What is the yield to call? b. What is the yield to call if the call price is only $1,050? c. What is the yield to call if the call price is $1,100, but the bond can be called in 2 years
instead of 5 years?
22. A 10-year bond of a firm in severe financial distress has a coupon rate of 14% and sells for
$900. The firm is currently renegotiating the debt, and it appears that the lenders will allow the
firm to reduce coupon payments on the bond to one-half the originally contracted amount.
The firm can handle these lower payments. What is the stated and expected yield to maturity
of the bonds? The bond makes its coupon payments annually.
23. A 2-year bond with par value $1,000 making annual coupon payments of $100
is priced at $1,000. What is the yield to maturity of the bond? What will be the realized com-
pound yield to maturity if the 1-year interest rate next year turns out to be ( a ) 8%, ( b ) 10%, ( c ) 12%?
24. Suppose that today’s date is April 15. A bond with a 10% coupon paid semiannually every
January 15 and July 15 is listed in The Wall Street Journal as selling at an ask price of 101.25. If you buy the bond from a dealer today, what price will you pay for it?
25. Assume that two firms issue bonds with the following characteristics. Both bonds are issued
at par.
ABC Bonds XYZ Bonds
Issue size $1.2 billion $150 million Maturity 10 years * 20 years Coupon 9% 10% Collateral First mortgage General debenture Callable Not callable In 10 years Call price None 110 Sinking fund None Starting in 5 years
*Bond is extendible at the discretion of the bondholder for an additional 10 years.
Ignoring credit quality, identify four features of these issues that might account for the lower
coupon on the ABC debt. Explain.
26. An investor believes that a bond may temporarily increase in credit risk. Which of the following
would be the most liquid method of exploiting this?
a. The purchase of a credit default swap. b. The sale of a credit default swap. c. The short sale of the bond.
27. Which of the following most accurately describes the behavior of credit default swaps?
a. When credit risk increases, swap premiums increase. b. When credit and interest rate risk increases, swap premiums increase. c. When credit risk increases, swap premiums increase, but when interest rate risk increases,
swap premiums decrease.
28. What would be the likely effect on the yield to maturity of a bond resulting from:
a. An increase in the issuing firm’s times-interest-earned ratio. b. An increase in the issuing firm’s debt-to-equity ratio. c. An increase in the issuing firm’s quick ratio.
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C H A P T E R 1 4 Bond Prices and Yields 483
29. A large corporation issued both fixed- and floating-rate notes 5 years ago, with terms given in
the following table:
9% Coupon Notes Floating-Rate Note
Issue size $250 million $280 million Original maturity 20 years 10 years Current price (% of par) 93 98 Current coupon 9% 8% Coupon adjusts Fixed coupon Every year Coupon reset rule — 1-year T-bill rate 1 2% Callable 10 years after issue 10 years after issue Call price 106 102.50 Sinking fund None None Yield to maturity 9.9% — Price range since issued $85–$112 $97–$102
a. Why is the price range greater for the 9% coupon bond than the floating-rate note? b. What factors could explain why the floating-rate note is not always sold at par value? c. Why is the call price for the floating-rate note not of great importance to investors? d. Is the probability of a call for the fixed-rate note high or low? e. If the firm were to issue a fixed-rate note with a 15-year maturity, what coupon rate would it
need to offer to issue the bond at par value?
f. Why is an entry for yield to maturity for the floating-rate note not appropriate?
30. Masters Corp. issues two bonds with 20-year maturities. Both bonds are callable at $1,050. The
first bond is issued at a deep discount with a coupon rate of 4% and a price of $580 to yield
8.4%. The second bond is issued at par value with a coupon rate of 8¾%.
a. What is the yield to maturity of the par bond? Why is it higher than the yield of the discount bond?
b. If you expect rates to fall substantially in the next 2 years, which bond would you prefer to hold? c. In what sense does the discount bond offer “implicit call protection”?
31. A newly issued bond pays its coupons once annually. Its coupon rate is 5%, its maturity is
20 years, and its yield to maturity is 8%.
a. Find the holding-period return for a 1-year investment period if the bond is selling at a yield to maturity of 7% by the end of the year.
b. If you sell the bond after 1 year, what taxes will you owe if the tax rate on interest income is 40% and the tax rate on capital gains income is 30%? The bond is subject to original-issue
discount tax treatment.
c. What is the after-tax holding-period return on the bond? d. Find the realized compound yield before taxes for a 2-year holding period, assuming that
(1) you sell the bond after 2 years, (2) the bond yield is 7% at the end of the second year, and
(3) the coupon can be reinvested for 1 year at a 3% interest rate.
e. Use the tax rates in ( b ) above to compute the after-tax 2-year realized compound yield. Remember to take account of OID tax rules.
Challenge
1. L eaf Products may issue a 10-year maturity fixed-income security, which might include a sinking
fund provision and either refunding or call protection.
a. Describe a sinking fund provision. b. Explain the impact of a sinking fund provision on:
i. The expected average life of the proposed security.
ii. Total principal and interest payments over the life of the proposed security.
c. From the investor’s point of view, explain the rationale for demanding a sinking fund provision.
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484 P A R T I V Fixed-Income Securities
2. Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in 5 years, and have
a 7% annual coupon rate paid semiannually.
a. Calculate the:
i. Current yield.
ii. Yield to maturity (to the nearest whole percent, i.e., 3%, 4%, 5%, etc.).
iii. Realized compound yield for an investor with a 3-year holding period and a reinvestment
rate of 6% over the period. At the end of 3 years the 7% coupon bonds with 2 years remain-
ing will sell to yield 7%.
b. Cite one major shortcoming for each of the following fixed-income yield measures:
i. Current yield.
ii. Yield to maturity.
iii. Realized compound yield.
3. On May 30, 2012, Janice Kerr is considering one of the newly issued 10-year AAA corporate
bonds shown in the following exhibit.
Description Coupon Price Callable Call Price
Sentinal, due May 30, 2022 6.00% 100 Noncallable NA Colina, due May 30, 2022 6.20% 100 Currently callable 102
a. Suppose that market interest rates decline by 100 basis points (i.e., 1%). Contrast the effect of this decline on the price of each bond.
b. Should Kerr prefer the Colina over the Sentinal bond when rates are expected to rise or to fall? c. What would be the effect, if any, of an increase in the volatility of interest rates on the prices
of each bond?
4. A convertible bond has the following features:
Coupon 5.25% Maturity June 15, 2030 Market price of bond $77.50 Market price of underlying common stock $28.00 Annual dividend $1.20 Conversion ratio 20.83 shares
Calculate the conversion premium for this bond.
5. a. Explain the impact on the offering yield of adding a call feature to a proposed bond issue. b. Explain the impact on the bond’s expected life of adding a call feature to a proposed bond
issue.
c. Describe one advantage and one disadvantage of including callable bonds in a portfolio.
6. a. An investment in a coupon bond will provide the investor with a return equal to the bond’s yield to maturity at the time of purchase if:
i. The bond is not called for redemption at a price that exceeds its par value.
ii. All sinking fund payments are made in a prompt and timely fashion over the life of the issue.
iii. The reinvestment rate is the same as the bond’s yield to maturity and the bond is held until
maturity.
iv. All of the above.
b. A bond with a call feature:
i. Is attractive because the immediate receipt of principal plus premium produces a high return.
ii. Is more apt to be called when interest rates are high because the interest savings will be
greater.
iii. Will usually have a higher yield to maturity than a similar noncallable bond.
iv. None of the above.
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C H A P T E R 1 4 Bond Prices and Yields 485
c. In which one of the following cases is the bond selling at a discount?
i. Coupon rate is greater than current yield, which is greater than yield to maturity.
ii. Coupon rate, current yield, and yield to maturity are all the same.
iii. Coupon rate is less than current yield, which is less than yield to maturity.
iv. Coupon rate is less than current yield, which is greater than yield to maturity.
d. Consider a 5-year bond with a 10% coupon that has a present yield to maturity of 8%. If inter- est rates remain constant, 1 year from now the price of this bond will be:
i. Higher.
ii. Lower.
iii. The same.
iv. Par.
E-INVESTMENTS EXERCISES
1. Go to the Web site of Standard & Poor’s at www.standardandpoors.com . Look for Rating Services (Find a Rating). Find the ratings on bonds of at least 10 companies. Try to choose a sample with a wide range of ratings. Then go to a Web site such as money.msn.com or finance.yahoo.com and obtain, for each firm, as many of the financial ratios tabulated in Table 14.3 as you can find. Which ratios seem to best explain credit ratings?
2. At www.bondsonline.com review the Industrial Spreads for various ratings (click the links on the left-side menus to follow the links to Today’s Markets, Corporate Bond Spreads ). These are spreads above U.S. Treasuries of comparable maturities. What fac- tors tend to explain the yield differences? How might these yield spreads differ during an economic boom versus a recession?
SOLUTIONS TO CONCEPT CHECKS 1. The callable bond will sell at the lower price. Investors will not be willing to pay as much if they
know that the firm retains a valuable option to reclaim the bond for the call price if interest rates
fall.
2. At a semiannual interest rate of 3%, the bond is worth $40 3 Annuity factor (3%, 60) 1 $1,000 3
PV factor(3%, 60) 5 $1,276.76, which results in a capital gain of $276.76. This exceeds the
capital loss of $189.29 (i.e., $1,000 2 $810.71) when the semiannual interest rate increased to 5%.
3. Yield to maturity exceeds current yield, which exceeds coupon rate. Take as an example the 8%
coupon bond with a yield to maturity of 10% per year (5% per half-year). Its price is $810.71, and
therefore its current yield is 80/810.71 5 .0987, or 9.87%, which is higher than the coupon rate
but lower than the yield to maturity.
4. a. The bond with the 6% coupon rate currently sells for 30 3 Annuity factor(3.5%, 20) 1 1,000 3 PV factor(3.5%, 20) 5 $928.94. If the interest rate immediately drops to 6% (3% per
half-year), the bond price will rise to $1,000, for a capital gain of $71.06, or 7.65%. The 8%
coupon bond currently sells for $1,071.06. If the interest rate falls to 6%, the present value of
the scheduled payments increases to $1,148.77. However, the bond will be called at $1,100, for a capital gain of only $28.94, or 2.70%.
b. The current price of the bond can be derived from its yield to maturity. Using your calculator, set: n 5 40 (semiannual periods); payment 5 $45 per period; future value 5 $1,000; interest rate 5 4% per semiannual period. Calculate present value as $1,098.96. Now we can calculate
yield to call. The time to call is 5 years, or 10 semiannual periods. The price at which the
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V is
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s a t
w w
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486 P A R T I V Fixed-Income Securities
bond will be called is $1,050. To find yield to call, we set: n 5 10 (semiannual periods); payment 5 $45 per period; future value 5 $1,050; present value 5 $1,098.96. Calculate yield
to call as 3.72%.
5. Price 5 $70 3 Annuity factor(8%, 1) 1 $1,000 3 PV factor(8%, 1) 5 $990.74
Rate of return to investor 5 $70 1 ($990.74 2 $982.17)
$982.17 5 .080 5 8%
6. By year-end, remaining maturity is 29 years. If the yield to maturity were still 8%, the bond
would still sell at par and the holding-period return would be 8%. At a higher yield, price
and return will be lower. Suppose, for example, that the yield to maturity rises to 8.5%. With
annual payments of $80 and a face value of $1,000, the price of the bond will be $946.70
[ n 5 29; i 5 8.5%; PMT 5 $80; FV 5 $1,000]. The bond initially sold at $1,000 when issued at the start of the year. The holding-period return is
HPR 5 80 1 (946.70 2 1,000)
1,000 5 .0267 5 2.67%
which is less than the initial yield to maturity of 8%.
7. At the lower yield, the bond price will be $631.67 [ n 5 29, i 5 7%, FV 5 $1,000, PMT 5 $40]. Therefore, total after-tax income is
Coupon $40 3 (1 2 .38) 5 $24.80 Imputed interest ($553.66 2 $549.69) 3 (1 2 .38) 5 2.46 Capital gains ($631.67 2 $553.66) 3 (1 2 .20) 5 62.41 Total income after taxes $89.67
Rate of return 5 89.67/549.69 5 .163 5 16.3%.
8. It should receive a negative coefficient. A high ratio of liabilities to assets is a bad omen for a
firm, and that should lower its credit rating.
9. The coupon payment is $45. There are 20 semiannual periods. The final payment is assumed
to be $500. The present value of expected cash flows is $650. The expected yield to maturity is
6.317% semiannual or annualized, 12.63%, bond equivalent yield.
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155 PA
R T IV
IN CHAPTER 14 we assumed for the sake of simplicity that the same constant inter- est rate is used to discount cash flows of any maturity. In the real world this is rarely the case. We have seen, for example, that in 2012 short-term Treasury bonds and notes carried yields to maturity less than 1% while the longest-term bonds offered yields of about 2.5%. At the time that these bond prices were quoted, anyway, the longer-term securities had higher yields. This, in fact, is a typical pattern, but as we shall see below, the relationship between time to maturity and yield to maturity can vary dramatically from one period to another. In this chapter we explore the pattern of interest rates for different-term assets. We attempt to identify the factors that account for that pattern and
determine what information may be derived from an analysis of the so-called term struc- ture of interest rates , the structure of interest rates for discounting cash flows of different maturities.
We demonstrate how the prices of Treasury bonds may be derived from prices and yields of stripped zero-coupon Treasury securities. We also examine the extent to which the term structure reveals market-consensus forecasts of future interest rates and how the presence of interest rate risk may affect those infer- ences. Finally, we show how traders can use the term structure to compute forward rates that represent interest rates on “forward,” or deferred, loans, and consider the relationship between forward rates and future interest rates.
The Term Structure
of Interest Rates
CHAPTER FIFTEEN
15.1 The Yield Curve
Figure 14.1 demonstrated that bonds of different maturities typically sell at different yields
to maturity. When these bond prices and yields were compiled, long-term bonds sold at
higher yields than short-term bonds. Practitioners commonly summarize the relation-
ship between yield and maturity graphically in a yield curve , which is a plot of yield to maturity as a function of time to maturity. The yield curve is one of the key concerns of
fixed-income investors. It is central to bond valuation and, as well, allows investors to
gauge their expectations for future interest rates against those of the market. Such a com-
parison is often the starting point in the formulation of a fixed-income portfolio strategy.
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488 P A R T I V Fixed-Income Securities
In 2012, the yield curve was rising, with long-term bonds offering yields higher than
those of short-term bonds. But the relationship between yield and maturity can vary
widely. Figure 15.1 illustrates yield curves of several different shapes. Panel A is the
almost-flat curve of early 2006. Panel B is a more typical upward-sloping curve from
2012. Panel C is a downward-sloping or “inverted” curve, and panel D is hump-shaped,
first rising and then falling.
Bond Pricing If yields on different-maturity bonds are not all equal, how should we value coupon bonds
that make payments at many different times? For example, suppose that yields on zero-
coupon Treasury bonds of different maturities are as given in Table 15.1 . The table tells us
that zero-coupon bonds with 1-year maturity sell at a yield to maturity of y 1 5 5%, 2-year zeros sell at yields of y 2 5 6%, and 3-year zeros sell at yields of y 3 5 7%. Which of these rates should we use to discount bond cash flows? The answer: all of them. The trick is to
consider each bond cash flow—either coupon or principal payment—as at least potentially
sold off separately as a stand-alone zero-coupon bond.
Recall the Treasury STRIPS program we introduced in the last chapter (Section 14.4).
Stripped Treasuries are zero-coupon bonds created by selling each coupon or principal
payment from a whole Treasury bond as a separate cash flow. For example, a 1-year
Figure 15.1 Treasury yield curves Source: Various editions of The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1989, 2000, 2006, and 2012 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
Treasury Yield Curve Yields as of 4:30 P.M. Eastern time
Percent 6.0
0.0
2.0
1.0
3.0
4.0
5.0
Year MaturitiesMonths B. (December 2012) Rising Yield Curve
Treasury Yield Curve Yields as of 4:30 P.M. Eastern time
Treasury Yield Curve Yields as of 4:30 P.M. Eastern time
Treasury Yield Curve Yields as of 4:30 P.M. Eastern time
Percent 6.0
1 3 6 2 5 10 1.0
2.0
3.0
4.0
5.0
30
MaturitiesYearMonths D. (October 4, 1989)
Hump-Shaped Yield Curve C. (September 11, 2000)
Inverted Yield Curve A. (January 2006)
Flat Yield Curve
6.50
3 6 1 2 3 5 3010
Percent
Months Year Maturities
5.25
5.50
5.75
6.00
6.25
8.60
3 6 1 2 3 45 710 301 3 6 1 2 56 710 30
Percent
MaturitiesYearMonths
7.40
7.60
7.80
8.00
8.20
8.40
Table 15.1
Prices and yields to maturity on zero-coupon bonds ($1,000 face value)
Maturity (years) Yield to Maturity (%) Price
1 5% $952.38 5 $1,000/1.05 2 6 $890.00 5 $1,000/1.062
3 7 $816.30 5 $1,000/1.073
4 8 $735.03 5 $1,000/1.084
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C H A P T E R 1 5 The Term Structure of Interest Rates 489
maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero
(by selling the first coupon payment as a stand-alone security) and a 12-month zero
(corresponding to payment of final coupon and principal). Treasury stripping suggests
exactly how to value a coupon bond. If each cash flow can be (and in practice often is) sold
off as a separate security, then the value of the whole bond should be the same as the value
of its cash flows bought piece by piece in the STRIPS market.
What if it weren’t? Then there would be easy profits to be made. For example, if
investment bankers ever noticed a bond selling for less than the amount at which the
sum of its parts could be sold, they would buy the bond, strip it into stand-alone zero-
coupon securities, sell off the stripped cash flows, and profit by the price difference. If
the bond were selling for more than the sum of the values of its individual cash flows, they would run the process in reverse: buy the individual zero-coupon securities in the
STRIPS market, reconstitute (i.e., reassemble) the cash flows into a coupon bond, and sell the whole bond for more than the cost of the pieces. Both bond stripping and bond reconstitution offer opportunities for arbitrage —the exploitation of mispricing among two or more securities to clear a riskless economic profit. Any violation of the Law of
One Price, that identical cash flow bundles must sell for identical prices, gives rise to
arbitrage opportunities.
Now, we know how to value each stripped cash flow. We simply look up its appropriate
discount rate in The Wall Street Journal. Because each coupon payment matures at a different time, we discount by using the yield appropriate to its particular maturity—this
is the yield on a Treasury strip maturing at the time of that cash flow. We can illustrate
with an example.
1 Remember that the yield to maturity of a coupon bond is the single interest rate at which the present value
of cash flows equals market price. To calculate the bond’s yield to maturity on your calculator or spreadsheet,
set n 5 3; price 5 2 1,082.17; future value 5 1,000; payment 5 100. Then compute the interest rate.
Calculate the yield to maturity of the coupon bond in Example 15.1, and you may be
surprised. Its yield to maturity is 6.88%; so while its maturity matches that of the 3-year
zero in Table 15.1 , its yield is a bit lower. 1 This reflects the fact that the 3-year coupon
bond may usefully be thought of as a portfolio of three implicit zero-coupon bonds, one corresponding to each cash flow. The yield on the coupon bond is then an amalgam of the
yields on each of the three components of the “portfolio.” Think about what this means: If
their coupon rates differ, bonds of the same maturity generally will not have the same yield
to maturity.
What then do we mean by “the” yield curve? In fact, in practice, traders refer to
several yield curves. The pure yield curve refers to the curve for stripped, or zero-coupon,
Suppose the yields on stripped Treasuries are as given in Table 15.1 , and we wish to value a 10% coupon bond with a maturity of 3 years. For simplicity, assume the bond makes its payments annually. Then the first cash flow, the $100 coupon paid at the end of the first year, is discounted at 5%; the second cash flow, the $100 coupon at the end of the second year, is discounted at 6%; and the final cash flow consisting of the final coupon plus par value, or $1,100, is discounted at 7%. The value of the coupon bond is therefore
100 1.05
1 100
1.062 1
1,100
1.073 5 95.238 1 89.000 1 897.928 5 $1,082.17
Example 15.1 Valuing Coupon Bonds
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490 P A R T I V Fixed-Income Securities
Treasuries. In contrast, the on-the-run yield curve refers to the plot of yield as a function of maturity for recently issued coupon bonds selling at or near par value. As we’ve just seen,
there may be significant differences in these two curves. The yield curves published in the
financial press, for example, in Figure 15.1 , are typically on-the-run curves. On-the-run
Treasuries have the greatest liquidity, so traders have keen interest in their yield curve.
15.2 The Yield Curve and Future Interest Rates
We’ve told you what the yield curve is, but we haven’t yet had much to say about where
it comes from. For example, why is the curve sometimes upward-sloping and other times
downward-sloping? How do expectations for the evolution of interest rates affect the shape
of today’s yield curve?
These questions do not have simple answers, so we will begin with an admittedly ideal-
ized framework, and then extend the discussion to more realistic settings. To start, consider
a world with no uncertainty, specifically, one in which all investors already know the path
of future interest rates.
The Yield Curve under Certainty If interest rates are certain, what should we make of the fact that the yield on the 2-year
zero coupon bond in Table 15.1 is greater than that on the 1-year zero? It can’t be that one
bond is expected to provide a higher rate of return than the other. This would not be pos-
sible in a certain world—with no risk, all bonds (in fact, all securities!) must offer identical
returns, or investors will bid up the price of the high-return bond until its rate of return is
no longer superior to that of other bonds.
Instead, the upward-sloping yield curve is evidence that short-term rates are going to
be higher next year than they are now. To see why, consider two 2-year bond strategies.
The first strategy entails buying the 2-year zero offering a 2-year yield to maturity of
y 2 5 6%, and holding it until maturity. The zero with face value $1,000 is purchased today for $1,000/1.06
2 5 $890 and matures in 2 years to $1,000. The total 2-year growth factor
for the investment is therefore $1,000/$890 5 1.06 2 5 1.1236.
Now consider an alternative 2-year strategy. Invest the same $890 in a 1-year zero-
coupon bond with a yield to maturity of 5%. When that bond matures, reinvest the pro-
ceeds in another 1-year bond. Figure 15.2 illustrates these two strategies. The interest rate
that 1-year bonds will offer next year is denoted as r 2 . Remember, both strategies must provide equal returns—neither entails any risk.
Therefore, the proceeds after 2 years to either strategy must be equal:
Buy and hold 2-year zero 5 Roll over 1-year bonds
$890 3 1.062 5 $890 3 1.05 3 (1 1 r2)
We find next year’s interest rate by solving 1 1 r 2 5 1.06 2 /1.05 5 1.0701, or r 2 5 7.01%. So
while the 1-year bond offers a lower yield to maturity than the 2-year bond (5% versus 6%),
Calculate the price and yield to maturity of a 3-year bond with a coupon rate of 4% making annual coupon payments. Does its yield match that of either the 3-year zero or the 10% coupon bond considered in Example 15.1? Why is the yield spread between the 4% bond and the zero smaller than the yield spread between the 10% bond and the zero?
CONCEPT CHECK 15.1
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C H A P T E R 1 5 The Term Structure of Interest Rates 491
we see that it has a compensating advantage: It allows you to roll over your funds into
another short-term bond next year when rates will be higher. Next year’s interest rate is
higher than today’s by just enough to make rolling over 1-year bonds equally attractive as
investing in the 2-year bond.
To distinguish between yields on long-term bonds versus short-term rates that will be
available in the future, practitioners use the following terminology. They call the yield to
maturity on zero-coupon bonds the spot rate , meaning the rate that prevails today for a time period corresponding to the zero’s maturity. In contrast, the short rate for a given time interval (e.g., 1 year) refers to the interest rate for that interval available at different points in
time. In our example, the short rate today is 5%, and the short rate next year will be 7.01%.
Not surprisingly, the 2-year spot rate is an average of today’s short rate and next year’s
short rate. But because of compounding, that average is a geometric one. 2 We see this by
again equating the total return on the two competing 2-year strategies:
(1 1 y2 ) 2 5 (1 1 r1) 3 (1 1 r2 ) (15.1)
1 1 y2 5 3(1 1 r1) 3 (1 1 r2 )41/2 Equation 15.1 begins to tell us why the yield curve might take on different shapes at
different times. When next year’s short rate, r 2 , is greater than this year’s short rate, r 1 , the average of the two rates is higher than today’s rate, so y 2 . r 1 and the yield curve slopes upward. If next year’s short rate were less than r 1 , the yield curve would slope downward.
2 In an arithmetic average, we add n numbers and divide by n. In a geometric average, we multiply n numbers and
take the n th root.
Figure 15.2 Two 2-year investment programs
0 1 2 Time Line
Alternative 1: Buy and hold 2-year zero
Alternative 2: Buy a 1-year zero, and reinvest proceeds in another 1-year zero
$890
$890 $934.50(1 + r2)
$890 ×1.062 = $1000
2-Year Investment
1-Year Investment 1-Year Investment
$890 × 1.05 = $934.50
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492 P A R T I V Fixed-Income Securities
Thus, at least in part, the yield curve reflects the market’s assessments of coming interest rates.
The following example uses a similar analysis to find the short rate that will prevail in year 3.
Use Table 15.1 to find the short rate that will prevail in the fourth year. Confirm that the discount factor on the 4-year zero is a geometric average of 1 1 the short rates in the next 4 years.
CONCEPT CHECK 15.2
Figure 15.3 summarizes the results of our analysis and emphasizes the difference
between short rates and spot rates. The top line presents the short rates for each year. The
lower lines present spot rates—or, equivalently, yields to maturity on zero-coupon bonds
for different holding periods—extending from the present to each relevant maturity date.
Holding-Period Returns We’ve argued that the multiyear cumulative returns on all of our competing bonds ought
to be equal. What about holding-period returns over shorter periods such as a year? You
might think that bonds selling at higher yields to maturity will offer higher 1-year returns,
but this is not the case. In fact, once you stop to think about it, it’s clear that this cannot be true. In a world of certainty, all bonds must offer identical returns, or investors will flock
to the higher-return securities, bidding up their prices, and reducing their returns. We can
illustrate by using the bonds in Table 15.1 .
The 1-year bond in Table 15.1 can be bought today for $1,000/1.05 5 $952.38 and will mature to its par value in 1 year. It pays no coupons, so total investment income is just its price appreciation, and its rate of return is ($1,000 2 $952.38)/$952.38 5 .05.
Example 15.3 Holding-Period Returns on Zero-Coupon Bonds
Now we compare two 3-year strategies. One is to buy a 3-year zero, with a yield to maturity from Table 15.1 of 7%, and hold it until maturity. The other is to buy a 2-year zero yielding 6%, and roll the proceeds into a 1-year bond in year 3, at the short rate r 3 . The growth factor for the invested funds under each policy will be:
Buy and hold 3-year zero 5 Buy 2-year zero; roll proceeds into 1-year bond
(1 1 y3) 3 5 (1 1 y2)
2 3 (1 1 r3)
1.073 5 1.062 3 (1 1 r3)
which implies that r 3 5 1.07 3 /1.06 2 2 1 5 .09025 5 9.025%. Again, notice that the yield
on the 3-year bond reflects a geometric average of the discount factors for the next 3 years:
1 1 y3 5 [(1 1 r1) 3 (1 1 r2) 3 (1 1 r3)] 1/3
1.07 5 [1.05 3 1.0701 3 1.09025]1/3
We conclude that the yield or spot rate on a long-term bond reflects the path of short rates anticipated by the market over the life of the bond.
Example 15.2 Finding a Future Short Rate
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C H A P T E R 1 5 The Term Structure of Interest Rates 493
Forward Rates The following equation generalizes our approach to inferring a future short rate from the
yield curve of zero-coupon bonds. It equates the total return on two n -year investment strategies: buying and holding an n -year zero-coupon bond versus buying an ( n 2 1)-year zero and rolling over the proceeds into a 1-year bond.
(1 1 yn) n 5 (1 1 yn21)
n21 3 (1 1 rn) (15.2)
where n denotes the period in question, and y n is the yield to maturity of a zero-coupon bond with an n -period maturity. Given the observed yield curve, we can solve Equation 15.2 for the short rate in the last period:
(1 1 rn) 5 (1 1 yn)
n
(1 1 yn21) n21
(15.3)
The 2-year bond can be bought for $1,000/1.06 2 5 $890.00. Next year, the bond will have a remaining maturity of 1 year and the 1-year interest rate will be 7.01%. Therefore, its price next year will be $1,000/1.0701 5 $934.49, and its 1-year holding-period rate of return will be ($934.49 2 $890.00)/$890.00 5 .05, for an identical 5% rate of return.
Show that the rate of return on the 3-year zero in Table 15.1 also will be 5%. Hint: Next year, the bond will have a maturity of 2 years. Use the short rates derived in Figure 15.3 to compute the 2-year spot rate that will prevail a year from now.
CONCEPT CHECK 15.3
Year
Short Rate in Each Year
Current Spot Rates (Yields to Maturity) for Various Maturities
1-Year Investment
2-Year Investment
3-Year Investment
4-Year Investment
1 2 3 4
r2 = 7.01%r1 = 5% r3 = 9.025% r4 = 11.06%
y1 = 5%
y2 = 6%
y3 = 7%
y4 = 8%
Figure 15.3 Short rates versus spot rates
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494 P A R T I V Fixed-Income Securities
Equation 15.3 has a simple interpretation. The numerator on the right-hand side is the
total growth factor of an investment in an n -year zero held until maturity. Similarly, the denominator is the growth factor of an investment in an ( n 2 1)-year zero. Because the former investment lasts for one more year than the latter, the difference in these growth
factors must be the rate of return available in year n when the ( n 2 1)-year zero can be rolled over into a 1-year investment.
Of course, when future interest rates are uncertain, as they are in reality, there is no
meaning to inferring “the” future short rate. No one knows today what the future interest
rate will be. At best, we can speculate as to its expected value and associated uncertainty.
Nevertheless, it still is common to use Equation 15.3 to investigate the implications of the
yield curve for future interest rates. Recognizing that future interest rates are uncertain,
we call the interest rate that we infer in this matter the forward interest rate rather than the future short rate, because it need not be the interest rate that actually will prevail at the future date.
If the forward rate for period n is denoted f n , we then define f n by the equation
(1 1 fn) 5 (1 1 yn)
n
(1 1 yn21) n21
(15.4)
Equivalently, we may rewrite Equation 15.4 as
(1 1 yn) n 5 (1 1 yn21)
n21(1 1 fn) (15.5)
In this formulation, the forward rate is defined as the “break-even” interest rate that equates the return on an n -period zero-coupon bond to that of an ( n 2 1)-period zero-coupon bond rolled over into a 1-year bond in year n. The actual total returns on the two n -year strate- gies will be equal if the short interest rate in year n turns out to equal f n .
Suppose a bond trader uses the data presented in Table 15.1 . The forward rate for year 4 would be computed as
1 1 f4 5 (1 1 y4)
4
(1 1 y3) 3 5
1.084
1.073 5 1.1106
Therefore, the forward rate is f 4 5 .1106, or 11.06%.
Example 15.4 Forward Rates
We emphasize again that the interest rate that actually will prevail in the future need
not equal the forward rate, which is calculated from today’s data. Indeed, it is not even
necessarily the case that the forward rate
equals the expected value of the future
short interest rate. This is an issue that we
address in the next section. For now, how-
ever, we note that forward rates equal future
short rates in the special case of interest rate certainty.
You’ve been exposed to many “rates” in the last few pages. Explain the differences between spot rates, short rates, and forward rates.
CONCEPT CHECK 15.4
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eXcel APPLICATIONS: Spot and Forward Yields
T he spreadsheet below (available at www.mhhe.com/bkm ) can be used to estimate prices and yields of coupon bonds and to calculate the forward rates for both single-year and multiyear periods. Spot yields are derived for the yield curve of bonds that are selling at their par value, also referred to as the current coupon or “on-the-run” bond yield curve.
The spot rates for each maturity date are used to calcu- late the present value of each period’s cash flow. The sum of these cash flows is the price of the bond. Given its price, the bond’s yield to maturity can then be computed. If you were to err and use the yield to maturity of the on-the-run
bond to discount each of the bond’s coupon payments, you could find a significantly different price. That difference is calculated in the worksheet.
Excel Questions
1. Change the spot rate in the spreadsheet to 8% for all maturities. The forward rates will all be 8%. Why is this not surprising?
2. The spot rates in column B decrease for longer maturities, and the forward rates decrease even more rapidly with maturity. What happens to the pattern of forward rates if you input spot rates that increase with maturity? Why?
Forward Rate Calculations
Spot Rate 1-yr for. Period
1 8.0000% 7.9792% 7.6770% 2 7.9896% 7.3757% 6.9205% 3 7.7846% 6.4673% 6.2695% 4 7.4537% 6.0720% 6.3065% 5 7.1760% 6.5414% 6.2920% 6 7.0699% 6.0432% 6.4299% 7 6.9227% 6.8181% 5.8904% 8 6.9096% 4.9707% 5.3993%
2-yr for. 3-yr for. 4-yr for. 5-yr for. 6-yr for.
5.2209% 5.1149% 5.1988%
A C D E F G HB 56 57 58 59 60 61 62 63 64 65 66 67
7.2723% 6.6369% 6.3600% 6.2186% 6.4671% 5.9413% 5.8701% 5.2521%
6.9709% 6.6131% 6.2807% 6.3682% 6.0910% 5.9134% 5.6414%
6.8849% 6.4988% 6.3880% 6.0872% 6.0387% 5.7217% 5.5384%
6.7441% 6.5520% 6.1505% 6.0442% 5.8579% 5.6224% 5.3969%
495
15.3 Interest Rate Uncertainty and Forward Rates
Let us turn now to the more difficult analysis of the term structure when future interest
rates are uncertain. We have argued so far that, in a certain world, different investment
strategies with common terminal dates must provide equal rates of return. For example,
two consecutive 1-year investments in zeros would need to offer the same total return as an
equal-sized investment in a 2-year zero. Therefore, under certainty,
(1 1 r1)(1 1 r2) 5 (1 1 y2) 2 (15.6)
What can we say when r 2 is not known today? For example, suppose that today’s rate is r 1 5 5% and that the expected short rate for
the following year is E ( r 2 ) 5 6%. If investors cared only about the expected value of the interest rate, then the yield to maturity on a 2-year zero would be determined by using
the expected short rate in Equation 15.6:
(1 1 y2) 2 5 (1 1 r1) 3 31 1 E(r2)4 5 1.05 3 1.06
The price of a 2-year zero would be $1,000/(1 1 y 2 ) 2 5 $1,000/(1.05 3 1.06) 5 $898.47.
But now consider a short-term investor who wishes to invest only for 1 year. She can
purchase the 1-year zero for $1,000/1.05 5 $952.38, and lock in a riskless 5% return
because she knows that at the end of the year, the bond will be worth its maturity value of
$1,000. She also can purchase the 2-year zero. Its expected rate of return also is 5%: Next year, the bond will have 1 year to maturity, and we expect that the 1-year interest rate will
be 6%, implying a price of $943.40 and a holding-period return of 5%.
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496 P A R T I V Fixed-Income Securities
But the rate of return on the 2-year bond is risky. If next year’s interest rate turns out to
be above expectations, that is, greater than 6%, the bond price will be below $943.40; con-
versely if r 2 turns out to be less than 6%, the bond price will exceed $943.40. Why should this short-term investor buy the risky 2-year bond when its expected return is 5%, no better than that of the risk-free 1-year bond? Clearly, she would not hold the 2-year bond unless it offered a higher expected rate of return. This requires that the 2-year bond sell at a price
lower than the $898.47 value we derived when we ignored risk.
The result in Example 15.5—that the forward rate exceeds the expected short rate—
should not surprise us. We defined the forward rate as the interest rate that would need to
prevail in the second year to make the long- and short-term investments equally attractive,
ignoring risk. When we account for risk, it is clear that short-term investors will shy away from the long-term bond unless it offers an expected return greater than that of the 1-year
bond. Another way of putting this is to say that investors will require a risk premium to hold
the longer-term bond. The risk-averse investor would be willing to hold the long-term bond
only if the expected value of the short rate is less than the break-even value, f 2 , because the lower the expectation of r 2 , the greater the anticipated return on the long-term bond.
Therefore, if most individuals are short-term investors, bonds must have prices that
make f 2 greater than E ( r 2 ). The forward rate will embody a premium compared with the expected future short-interest rate. This liquidity premium compensates short-term inves- tors for the uncertainty about the price at which they will be able to sell their long-term
bonds at the end of the year. 3
Suppose that the required liquidity premium for the short-term investor is 1%. What must E ( r 2 ) be if f 2 is 7%?
CONCEPT CHECK 15.5
Perhaps surprisingly, we also can imagine scenarios in which long-term bonds can be
perceived by investors to be safer than short-term bonds. To see how, we now consider a “long-term” investor, who wishes to invest for a full 2-year period. Suppose that the investor
3 Liquidity refers to the ability to sell an asset easily at a predictable price. Because long-term bonds have greater
price risk, they are considered less liquid in this context and thus must offer a premium.
Suppose that most investors have short-term horizons and therefore are willing to hold the 2-year bond only if its price falls to $881.83. At this price, the expected holding- period return on the 2-year bond is 7% (because 943.40/881.83 5 1.07). The risk pre- mium of the 2-year bond, therefore, is 2%; it offers an expected rate of return of 7% versus the 5% risk-free return on the 1-year bond. At this risk premium, investors are willing to bear the price risk associated with interest rate uncertainty.
When bond prices reflect a risk premium, however, the forward rate, f 2 , no longer equals the expected short rate, E ( r 2 ). Although we have assumed that E ( r 2 ) 5 6%, it is easy to confirm that f 2 5 8%. The yield to maturity on the 2-year zeros selling at $881.83 is 6.49%, and
1 1 f2 5 (1 1 y2)
2
1 1 y1 5
1.06492
1.05 5 1.08
Example 15.5 Bond Prices and Forward Rates with Interest Rate Risk
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C H A P T E R 1 5 The Term Structure of Interest Rates 497
can purchase a $1,000 par value 2-year zero-coupon bond for $890 and lock in a guaranteed
yield to maturity of y 2 5 6%. Alternatively, the investor can roll over two 1-year invest- ments. In this case an investment of $890 would grow in 2 years to 890 3 1.05 3 (1 1 r 2 ), which is an uncertain amount today because r 2 is not yet known. The break-even year-2 interest rate is, once again, the forward rate, 7.01%, because the forward rate is defined as
the rate that equates the terminal value of the two investment strategies.
The expected value of the payoff of the rollover strategy is 890 3 1.05 3 [1 1 E ( r 2 )]. If E ( r 2 ) equals the forward rate, f 2 , then the expected value of the payoff from the rollover strategy will equal the known payoff from the 2-year-maturity bond strategy.
Is this a reasonable presumption? Once again, it is only if the investor does not care
about the uncertainty surrounding the final value of the rollover strategy. Whenever that
risk is important, however, the long-term investor will not be willing to engage in the
rollover strategy unless its expected return exceeds that of the 2-year bond. In this case the
investor would require that
(1.05)31 1 E(r2)4 . (1.06)2 5 (1.05)(1 1 f2) which implies that E ( r 2 ) exceeds f 2 . The investor would require that the expected value of next year’s short rate exceed the forward rate.
Therefore, if all investors were long-term investors, no one would be willing to hold
short-term bonds unless those bonds offered a reward for bearing interest rate risk. In this
situation bond prices would be set at levels such that rolling over short bonds resulted in
greater expected return than holding long bonds. This would cause the forward rate to be
less than the expected future spot rate.
For example, suppose that in fact E ( r 2 ) 5 8%. The liquidity premium therefore is nega- tive: f 2 2 E ( r 2 ) 5 7.01% 2 8% 5 2 .99%. This is exactly opposite from the conclusion that we drew in the first case of the short-term investor. Clearly, whether forward rates will equal
expected future short rates depends on investors’ readiness to bear interest rate risk, as well
as their willingness to hold bonds that do not correspond to their investment horizons.
15.4 Theories of the Term Structure
The Expectations Hypothesis The simplest theory of the term structure is the expectations hypothesis . A common ver- sion of this hypothesis states that the forward rate equals the market consensus expecta-
tion of the future short interest rate; that is, f 2 5 E ( r 2 ), and liquidity premiums are zero. If f 2 5 E ( r 2 ), we may relate yields on long-term bonds to expectations of future interest rates. In addition, we can use the forward rates derived from the yield curve to infer market
expectations of future short rates. For example, with (1 1 y 2 ) 2 5 (1 1 r 1 ) 3 (1 1 f 2 ) from
Equation 15.5, if the expectations hypothesis is correct we may also write that (1 1 y 2 ) 2 5
(1 1 r 1 ) 3 [1 1 E ( r 2 )]. The yield to maturity would thus be determined solely by current and expected future one-period interest rates. An upward-sloping yield curve would be
clear evidence that investors anticipate increases in interest rates.
If the expectations hypothesis is valid, what can we conclude about the premiums necessary to induce investors to hold bonds of different maturities from their investment horizons?
CONCEPT CHECK 15.6
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498
By the way, nothing limits us to nominal bonds when using the expectations hypothesis.
The nearby box points out that we can apply the theory to the term structure of real
interest rates as well, and thereby learn something about market expectations of coming
inflation rates.
Liquidity Preference We noted earlier that short-term investors will be unwilling to hold long-term bonds unless
the forward rate exceeds the expected short interest rate, f 2 . E ( r 2 ), whereas long-term investors will be unwilling to hold short bonds unless
E ( r 2 ) . f 2 . In other words, both groups of investors require a premium to hold bonds with maturities dif-
ferent from their investment horizons. Advocates of the
liquidity preference theory of the term structure believe that short-term investors dominate the market so that the
forward rate will generally exceed the expected short
rate. The excess of f 2 over E ( r 2 ), the liquidity premium, is predicted to be positive.
To illustrate the differing implications of these theories
for the term structure of interest rates, suppose the short interest rate is expected to be
constant indefinitely. Suppose that r 1 5 5% and that E ( r 2 ) 5 5%, E ( r 3 ) 5 5%, and so on. Under the expectations hypothesis the 2-year yield to maturity could be derived from the
following:
(1 1 y2) 2 5 (1 1 r1)31 1 E(r2) 4 5 (1.05)(1.05)
so that y 2 equals 5%. Similarly, yields on bonds of all maturities would equal 5%.
The liquidity premium hypothesis also holds that issuers of bonds prefer to issue long-term bonds to lock in borrowing costs. How would this preference contribute to a positive liquidity premium?
CONCEPT CHECK 15.7
The Expectations Hypothesis and Forward Inflation Rates
Forward rates derived from conventional bonds are nomi- nal interest rates. But using price-level-indexed bonds such as TIPS, we can also calculate forward real interest rates. Recall that the difference between the real rate and the nominal rate is approximately the expected inflation rate. Therefore, comparing real and nominal forward rates might give us a glimpse of the market’s expectation of future inflation rates. The real versus nominal spread is a sort of forward inflation rate.
As part of its monetary policy, the Federal Reserve Board periodically reduces its target federal funds rate in an attempt to stimulate the economy. The following page capture from a Bloomberg screen shows the minute- by-minute spread between the 5-year forward nominal interest rate and forward real rate on one day the Fed announced such a policy change. The spread immediately widened at the announcement, signifying that the market expected the more expansionary monetary policy to even- tually result in a higher inflation rate. The increase in the inflation rate implied by the graph is fairly mild, about
.05%, from about 2.53% to 2.58%, but the impact of the announcement is very clear, and the speed of adjustment to the announcement was impressive.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 1 5 The Term Structure of Interest Rates 499
In contrast, under the liquidity preference theory, f 2 would exceed E ( r 2 ). To illustrate, suppose the liquidity premium is 1%, so f 2 is 6%. Then, for 2-year bonds:
(1 1 y2) 2 5 (1 1 r1)(1 1 f2)
5 1.05 3 1.06 5 1.113
implying that 1 1 y 2 5 1.055. Similarly, if f 3 also equals 6%, then the yield on 3-year bonds would be determined by
(1 1 y3) 3 5 (1 1 r1)(1 1 f2)(1 1 f3)
5 1.05 3 1.06 3 1.06 5 1.17978
Figure 15.4 Yield curves. Panel A, Constant expected short rate. Liquidity premium of 1%. Result is a rising yield curve. Panel B, Declining expected short rates. Increasing liquidity premiums. Result is a rising yield curve despite falling expected interest rates. (continued on next page)
4.0 0 1 2 3 4 5
Maturity 6 7 8 9 10
4.5
5.0
5.5
In te
re st
R a te
( %
)
6.0
6.5
7.0 YTM E(r) Forward Rate
Constant Liquidity Premium
A
4.0 0 1 2 3 4 5
Maturity 6 7 8 9 10
4.5
5.0
5.5
In te
re st
R a te
( %
)
6.0
6.5
7.0 YTM E(r) Forward Rate
Liquidity Premium Increases with Maturity
B
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500 P A R T I V Fixed-Income Securities
implying that 1 1 y 3 5 1.0567. The plot of the yield curve in this situation would be given as in Figure 15.4 , panel A. Such an upward-sloping yield curve is commonly observed in
practice.
If interest rates are expected to change over time, then the liquidity premium may be
overlaid on the path of expected spot rates to determine the forward interest rate. Then
the yield to maturity for each date will be an average of the single-period forward rates.
Several such possibilities for increasing and declining interest rates appear in Figure 15.4 ,
panels B to D.
Figure 15.4 (Concluded) Panel C, Declining expected short rates. Constant liquidity premiums. Result is a hump-shaped yield curve. Panel D, Increasing expected short rates. Increasing liquidity premiums. Result is a sharply rising yield curve.
4.0 0 1 2 3 4 5
Maturity 6 7 8 9 10
4.5
5.0
5.5
In te
re st
R a te
( %
)
6.0
6.5
7.0 YTM E(r) Forward Rate
Constant Liquidity Premium
C
4.0 0 1 2 3 4 5
Maturity 6 7 8 9 10
4.5
5.0
5.5
In te
re st
R a te
( %
)
6.0
6.5
7.0 YTM E(r) Forward Rate
Liquidity Premium Increases with Maturity
D
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C H A P T E R 1 5 The Term Structure of Interest Rates 501
15.5 Interpreting the Term Structure
If the yield curve reflects expectations of future short rates, then it offers a potentially
powerful tool for fixed-income investors. If we can use the term structure to infer the
expectations of other investors in the economy, we can use those expectations as bench-
marks for our own analysis. For example, if we are relatively more optimistic than other
investors that interest rates will fall, we will be more willing to extend our portfolios into
longer-term bonds. Therefore, in this section, we will take a careful look at what informa-
tion can be gleaned from a careful analysis of the term structure. Unfortunately, while the
yield curve does reflect expectations of future interest rates, it also reflects other factors
such as liquidity premiums. Moreover, forecasts of interest rate changes may have different
investment implications depending on whether those changes are driven by changes in
the expected inflation rate or the real rate, and this adds another layer of complexity to
the proper interpretation of the term structure.
We have seen that under certainty, 1 plus the yield to maturity on a zero-coupon bond
is simply the geometric average of 1 plus the future short rates that will prevail over the
life of the bond. This is the meaning of Equation 15.1, which we give in general form
here:
1 1 yn 5 3(1 1 r1)(1 1 r2)c(1 1 rn)41/n When future rates are uncertain, we modify Equation 15.1 by replacing future short rates
with forward rates:
1 1 yn 5 3(1 1 r1)(1 1 f2)(1 1 f3)c(1 1 fn)41/n (15.7) Thus there is a direct relationship between yields on various maturity bonds and forward
interest rates.
First, we ask what factors can account for a rising yield curve. Mathematically, if the
yield curve is rising, f n 1 1 must exceed y n . In words, the yield curve is upward-sloping at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity. This rule follows from the notion of the yield to maturity as an aver-
age (albeit a geometric average) of forward rates.
If the yield curve is to rise as one moves to longer maturities, it must be the case that
extension to a longer maturity results in the inclusion of a “new” forward rate that is higher
than the average of the previously observed rates. This is analogous to the observation
that if a new student’s test score is to increase the class average, that student’s score must
exceed the class’s average without her score. To increase the yield to maturity, an above-
average forward rate must be added to the other rates used in the averaging computation.
If the yield to maturity on 3-year zero-coupon bonds is 7%, then the yield on 4-year bonds will satisfy the following equation:
(1 1 y4) 4 5 (1.07)3(1 1 f4)
If f 4 5 .07, then y 4 also will equal .07. (Confirm this!) If f 4 is greater than 7%, y 4 will exceed 7%, and the yield curve will slope upward. For example, if f 4 5 .08, then (1 1 y 4 )
4 5 (1.07) 3 (1.08) 5 1.3230, and y 4 5 .0725.
Example 15.6 Forward Rates and the Slopes of the Yield Curve
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502 P A R T I V Fixed-Income Securities
Given that an upward-sloping yield curve is always associated with a forward rate higher
than the spot, or current, yield to maturity, we ask next what can account for that higher for-
ward rate. Unfortunately, there always are two possible answers to this question. Recall that
the forward rate can be related to the expected future short rate according to this equation:
fn 5 E(rn) 1 Liquidity premium (15.8)
where the liquidity premium might be necessary to induce investors to hold bonds of
maturities that do not correspond to their preferred investment horizons.
By the way, the liquidity premium need not be positive, although that is the position
generally taken by advocates of the liquidity premium hypothesis. We showed previously that
if most investors have long-term horizons, the liquidity premium in principle could be negative.
In any case, Equation 15.8 shows that there are two reasons that the forward rate could be
high. Either investors expect rising interest rates, meaning that E ( r n ) is high, or they require a large premium for holding longer-term bonds. Although it is tempting to infer from a rising
yield curve that investors believe that interest rates will eventually increase, this is not a valid
inference. Indeed, panel A in Figure 15.4 provides a simple counter-example to this line of
reasoning. There, the short rate is expected to stay at 5% forever. Yet there is a constant 1%
liquidity premium so that all forward rates are 6%. The result is that the yield curve continu-
ally rises, starting at a level of 5% for 1-year bonds, but eventually approaching 6% for long-
term bonds as more and more forward rates at 6% are averaged into the yields to maturity.
Therefore, although it is true that expectations of increases in future interest rates can
result in a rising yield curve, the converse is not true: A rising yield curve does not in and
of itself imply expectations of higher future interest rates. The effects of possible liquidity
premiums confound any simple attempt to extract expectations from the term structure.
But estimating the market’s expectations is crucial because only by comparing your own
expectations to those reflected in market prices can you determine whether you are rela-
tively bullish or bearish on interest rates.
One very rough approach to deriving expected future spot rates is to assume that liquidity
premiums are constant. An estimate of that premium can be subtracted from the forward rate
to obtain the market’s expected interest rate. For example, again making use of the example
plotted in panel A of Figure 15.4 , the researcher would estimate from historical data that a
typical liquidity premium in this economy is 1%. After calculating the forward rate from the
yield curve to be 6%, the expectation of the future spot rate would be determined to be 5%.
This approach has little to recommend it for two reasons. First, it is next to impossible
to obtain precise estimates of a liquidity premium. The general approach to doing so would
be to compare forward rates and eventually realized future short rates and to calculate
the average difference between the two. However, the deviations between the two val-
ues can be quite large and unpredictable because of unanticipated economic events that
affect the realized short rate. The data are too noisy to calculate a reliable estimate of the
expected premium. Second, there is no reason to believe that the liquidity premium should
be constant. Figure 15.5 shows the rate of return variability of prices of long-term Treasury
bonds since 1971. Interest rate risk fluctuated dramatically during the period. So we should
expect risk premiums on various maturity bonds to fluctuate, and empirical evidence
suggests that liquidity premiums do in fact fluctuate over time.
Look back at Table 15.1 . Show that y 4 will exceed y 3 if and only if the forward interest rate for period 4 is greater than 7%, which is the yield to maturity on the 3-year bond, y 3 .
CONCEPT CHECK 15.8
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C H A P T E R 1 5 The Term Structure of Interest Rates 503
Still, very steep yield
curves are interpreted by
many market profession-
als as warning signs of
impending rate increases.
In fact, the yield curve is
a good predictor of the
business cycle as a whole,
because long-term rates
tend to rise in anticipation
of an expansion in eco-
nomic activity.
The usually observed
upward slope of the yield
curve, especially for short
maturities, is the empiri-
cal basis for the liquid-
ity premium doctrine that
long-term bonds offer a
positive liquidity premium.
Because the yield curve
normally has an upward
slope due to risk premiums,
a downward-sloping yield curve is taken as a strong indication that yields are more likely
than not to fall. The prediction of declining interest rates is in turn often interpreted as a
signal of a coming recession. Short-term rates exceeded long-term ones in each of the seven
recessions since 1970. For this reason, it is not surprising that the slope of the yield curve is
one of the key components of the index of leading economic indicators.
Figure 15.6 presents a history of yields on 90-day Treasury bills and 10-year Treasury
bonds. Yields on the longer-term bonds generally exceed those on the bills, mean- ing that the yield curve generally slopes upward. Moreover, the exceptions to this rule
do seem to precede episodes of falling short rates, which, if anticipated, would induce a
downward-sloping yield curve. For example, the figure shows that 1980–1981 were years
in which 90-day yields exceeded long-term yields. These years preceded both a drastic
drop in the general level of rates and a steep recession.
Why might interest rates fall? There are two factors to consider: the real rate and the
inflation premium. Recall that the nominal interest rate is composed of the real rate plus a
factor to compensate for the effect of inflation:
1 1 Nominal rate 5 (1 1 Real rate)(1 1 Inflation rate)
or, approximately,
Nominal rate < Real rate 1 Inflation rate
Therefore, an expected change in interest rates can be due to changes in either expected
real rates or expected inflation rates. Usually, it is important to distinguish between these
two possibilities because the economic environments associated with them may vary sub-
stantially. High real rates may indicate a rapidly expanding economy, high government
budget deficits, and tight monetary policy. Although high inflation rates can arise out of
a rapidly expanding economy, inflation also may be caused by rapid expansion of the
money supply or supply-side shocks to the economy such as interruptions in oil supplies.
6.0
5.0
4.0
3.0
2.0
1.0
0.0
1 9 7 1
1 9 7 3
1 9 7 5
1 9 7 7
1 9 7 9
1 9 8 1
1 9 8 3
1 9 8 5
1 9 8 7
1 9 8 9
1 9 9 1
1 9 9 3
1 9 9 5
1 9 9 7
1 9 9 9
2 0 0 1
2 0 0 3
2 0 0 5
2 0 0 7
2 0 0 9
2 0 1 3
2 0 1 1
St a n
d a rd
D e vi
a ti
o n
o f
M o
n th
ly R
e tu
rn s
(% )
Figure 15.5 Price volatility of long-term Treasury bonds
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504 P A R T I V Fixed-Income Securities
These factors have very different implications for investments. Even if we conclude from
an analysis of the yield curve that rates will fall, we need to analyze the macroeconomic
factors that might cause such a decline.
16
12
8
4
0
−4
In te
re st
R a te
( %
)
1 9 7 2
1 9 7 4
1 9 7 6
1 9 7 8
1 9 8 0
1 9 8 2
1 9 8 4
1 9 8 6
1 9 8 8
1 9 9 0
1 9 9 2
1 9 9 4
1 9 9 6
1 9 9 8
2 0 0 0
2 0 0 2
2 0 0 4
2 0 0 6
10-Year Treasury 90-Day T-Bills Difference
2 0 0 8
2 0 1 0
1 9 7 0
2 0 1 4
2 0 1 2
Figure 15.6 Term spread: Yields on 10-year versus 90-day Treasury securities
15.6 Forward Rates as Forward Contracts
We have seen that forward rates may be derived from the yield curve, using Equation 15.5.
In general, forward rates will not equal the eventually realized short rate, or even today’s
expectation of what that short rate will be. But there is still an important sense in which
the forward rate is a market interest rate. Suppose that you wanted to arrange now to make a loan at some future date. You would agree today on the interest rate that will be charged,
but the loan would not commence until some time in the future. How would the interest
rate on such a “forward loan” be determined? Perhaps not surprisingly, it would be the
forward rate of interest for the period of the loan. Let’s use an example to see how this
might work.
Suppose the price of 1-year maturity zero-coupon bonds with face value $1,000 is $952.38 and the price of 2-year zeros with $1,000 face value is $890. The yield to maturity on the 1-year bond is therefore 5%, while that on the 2-year bond is 6%. The forward rate for the second year is thus
f2 5 (1 1 y2)
2
(1 1 y1) 2 1 5
1.062
1.05 2 1 5 .0701, or 7.01%
Example 15.7 Forward Interest Rate Contract
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C H A P T E R 1 5 The Term Structure of Interest Rates 505
Now consider the strategy laid out in the following table. In the first column we present data for this example, and in the last column we generalize. We denote by B 0 ( T ) today’s price of a zero-coupon bond with face value $1,000 maturing at time T.
Initial Cash Flow In General
Buy a 1-year zero-coupon bond 2952.38 2B0(1) Sell 1.0701 2-year zeros 1890 3 1.0701 5 952.38 1B0(2) 3 (1 1 f2)
0 0
The initial cash flow (at time 0) is zero. You pay $952.38, or in general B 0 (1), for a zero maturing in 1 year, and you receive $890, or in general B 0 (2), for each zero you sell matur- ing in 2 years. By selling 1.0701 of these bonds, you set your initial cash flow to zero. 4
At time 1, the 1-year bond matures and you receive $1,000. At time 2, the 2-year maturity zero-coupon bonds that you sold mature, and you have to pay 1.0701 3 $1,000 5 $1,070.10. Your cash flow stream is shown in Figure 15.7 , panel A. Notice that you have created a “synthetic” forward loan: You effectively will borrow $1,000 a year from now, and repay $1,070.10 a year later. The rate on this forward loan is there- fore 7.01%, precisely equal to the forward rate for the second year.
4 Of course, in reality one cannot sell a fraction of a bond, but you can think of this part of the transaction as
follows. If you sold one of these bonds, you would effectively be borrowing $890 for a 2-year period. Selling
1.0701 of these bonds simply means that you are borrowing $890 3 1.0701 5 $952.38.
In general, to construct the synthetic forward loan, you sell (1 1 f 2 ) 2-year zeros for every 1-year zero that you buy. This makes your initial cash flow zero because the prices of
the 1- and 2-year zeros differ by the factor (1 1 f 2 ); notice that
B0(1) 5 $1,000
(1 1 y1) while B0(2) 5
$1,000
(1 1 y2) 2 5
$1,000
(1 1 y1)(1 1 f2)
−$1,070.10
$1,000
20 1
−$1,000(1 + f2)
$1,000
20 1 r1 r2
B: For a General Forward Rate. The short rates in the two periods are r1 (which is observable today) and r2 (which is not). The rate that can be locked in for a one-period-ahead loan is f2.
A: Forward Rate = 7.01%
Figure 15.7 Engineering a synthetic forward loan
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V is
it u
s a t
w w
w .m
h h
e .c
o m
/b k m
506 P A R T I V Fixed-Income Securities
Therefore, when you sell (1 1 f 2 ) 2-year zeros you generate just enough cash to buy one 1-year zero. Both zeros mature to a face value of $1,000, so the difference between the
cash inflow at time 1 and the cash outflow at time 2 is the same factor, 1 1 f 2 , as illustrated in Figure 15.7 , panel B. As a result, f 2 is the rate on the forward loan.
Obviously, you can construct a synthetic forward loan for periods beyond the second
year, and you can construct such loans for multiple periods. Challenge Problems 18 and 19
at the end of the chapter lead you through some of these variants.
Suppose that the price of 3-year zero-coupon bonds is $816.30. What is the forward rate for the third year? How would you construct a synthetic 1-year forward loan that commences at t 5 2 and matures at t 5 3?
CONCEPT CHECK 15.9
1. The term structure of interest rates refers to the interest rates for various terms to maturity embod- ied in the prices of default-free zero-coupon bonds.
2. In a world of certainty all investments must provide equal total returns for any investment period. Short-term holding-period returns on all bonds would be equal in a risk-free economy,
and all equal to the rate available on short-term bonds. Similarly, total returns from rolling over
short-term bonds over longer periods would equal the total return available from long-maturity
bonds.
3. The forward rate of interest is the break-even future interest rate that would equate the total return from a rollover strategy to that of a longer-term zero-coupon bond. It is defined by the
equation
(1 1 yn) n(1 1 fn11) 5 (1 1 yn11)
n11
where n is a given number of periods from today. This equation can be used to show that yields to maturity and forward rates are related by the equation
(1 1 yn) n 5 (1 1 r1)(1 1 f2)(1 1 f3)c(1 1 fn)
4. A common version of the expectations hypothesis holds that forward interest rates are unbiased estimates of expected future interest rates. However, there are good reasons to believe that for-
ward rates differ from expected short rates because of a risk premium known as a liquidity pre- mium. A positive liquidity premium can cause the yield curve to slope upward even if no increase in short rates is anticipated.
5. The existence of liquidity premiums makes it extremely difficult to infer expected future inter- est rates from the yield curve. Such an inference would be made easier if we could assume the
liquidity premium remained reasonably stable over time. However, both empirical and theoretical
considerations cast doubt on the constancy of that premium.
6. Forward rates are market interest rates in the important sense that commitments to forward (i.e., deferred) borrowing or lending arrangements can be made at these rates.
SUMMARY
Related Web sites for this chapter are available at www. mhhe.com/bkm
term structure of interest rates
yield curve
bond stripping
bond reconstitution
pure yield curve
on-the-run yield curve
spot rate
short rate
forward interest rate
liquidity premium
expectations hypothesis
liquidity preference theory
KEY TERMS
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C H A P T E R 1 5 The Term Structure of Interest Rates 507
1. What is the relationship between forward rates and the market’s expectation of future short rates?
Explain in the context of both the expectations and liquidity preference theories of the term struc-
ture of interest rates.
2. U nder the expectations hypothesis, if the yield curve is upward-sloping, the market must expect
an increase in short-term interest rates. True/false/uncertain? Why?
3. Under the liquidity preference theory, if inflation is expected to be falling over the next few years,
long-term interest rates will be higher than short-term rates. True/false/uncertain? Why?
4. If the liquidity preference hypothesis is true, what shape should the term structure curve have in a
period where interest rates are expected to be constant?
a. Upward sloping. b. Downward sloping. c. Flat.
5. W hich of the following is true according to the pure expectations theory? Forward rates:
a. Exclusively represent expected future short rates. b. Are biased estimates of market expectations. c. Always overestimate future short rates.
6. A ssuming the pure expectations theory is correct, an upward-sloping yield curve implies:
a. Interest rates are expected to increase in the future. b. Longer-term bonds are riskier than short-term bonds. c. Interest rates are expected to decline in the future.
7. T he following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields
to maturity of each bond and the implied sequence of forward rates.
Maturity (Years) Price of Bond
1 $943.40
2 898.47
3 847.62
4 792.16
8. Assuming that the expectations hypothesis is valid, compute the expected price path of the 4-year
bond in the previous problem as time passes. What is the rate of return of the bond in each year?
Show that the expected return equals the forward rate for each year.
9. Consider the following $1,000 par value zero-coupon bonds:
Bond Years to Maturity YTM(%)
A 1 5%
B 2 6
C 3 6.5
D 4 7
According to the expectations hypothesis, what is the expected 1-year interest rate 3 years from
now?
Basic
Intermediate
PROBLEM SETS
Forward rate of interest: 1 1 fn 5 (1 1 yn)
n
(1 1 yn21) n21
Yield to maturity given sequence of forward rates: 1 1 y n 5 [(1 1 r 1 ) (1 1 f 2 ) (1 1 f 3 ) ? ? ? (1 1 f n )] 1/ n
Liquidity premium 5 Forward rate 2 Expected short rate
KEY EQUATIONS
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508 P A R T I V Fixed-Income Securities
10. The term structure for zero-coupon bonds is currently:
Maturity (Years) YTM (%)
1 4%
2 5
3 6
Next year at this time, you expect it to be:
Maturity (Years) YTM (%)
1 5%
2 6
3 7
a. What do you expect the rate of return to be over the coming year on a 3-year zero-coupon bond? b. Under the expectations theory, what yields to maturity does the market expect to observe on
1- and 2-year zeros at the end of the year? Is the market’s expectation of the return on the
3-year bond greater or less than yours?
11. The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2-year zeros is
8%. The Treasury plans to issue a 2-year maturity coupon bond, paying coupons once per year with a coupon rate of 9%. The face value of the bond is $100.
a. At what price will the bond sell? b. What will the yield to maturity on the bond be? c. If the expectations theory of the yield curve is correct, what is the market expectation of the
price that the bond will sell for next year?
d. Recalculate your answer to ( c ) if you believe in the liquidity preference theory and you believe that the liquidity premium is 1%.
12. Below is a list of prices for zero-coupon bonds of various maturities.
Maturity (Years) Price of $1,000 Par Bond
(Zero-Coupon)
1 $943.40
2 873.52
3 816.37
a. An 8.5% coupon $1,000 par bond pays an annual coupon and will mature in 3 years. What should the yield to maturity on the bond be?
b. If at the end of the first year the yield curve flattens out at 8%, what will be the 1-year holding-period return on the coupon bond?
13. Prices of zero-coupon bonds reveal the following pattern of forward rates:
Year Forward Rate
1 5%
2 7
3 8
In addition to the zero-coupon bond, investors also may purchase a 3-year bond making annual
payments of $60 with par value $1,000.
a. What is the price of the coupon bond? b. What is the yield to maturity of the coupon bond? c. Under the expectations hypothesis, what is the expected realized compound yield of the
coupon bond?
d. If you forecast that the yield curve in 1 year will be flat at 7%, what is your forecast for the expected rate of return on the coupon bond for the 1-year holding period?
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C H A P T E R 1 5 The Term Structure of Interest Rates 509
14. You observe the following term structure:
Effective Annual YTM
1-year zero-coupon bond 6.1%
2-year zero-coupon bond 6.2
3-year zero-coupon bond 6.3
4-year zero-coupon bond 6.4
a. If you believe that the term structure next year will be the same as today’s, will the 1-year or the 4-year zeros provide a greater expected 1-year return?
b. What if you believe in the expectations hypothesis?
15. The yield to maturity (YTM) on 1-year zero-coupon bonds is 5% and the YTM on 2-year zeros
is 6%. The yield to maturity on 2-year-maturity coupon bonds with coupon rates of 12% (paid
annually) is 5.8%. What arbitrage opportunity is available for an investment banking firm?
What is the profit on the activity?
16. Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34,
while a 2-year zero sells at $84.99. You are considering the purchase of a 2-year-maturity bond
making annual coupon payments. The face value of the bond is $100, and the coupon rate is 12% per year.
a. What is the yield to maturity of the 2-year zero? The 2-year coupon bond? b. What is the forward rate for the second year? c. If the expectations hypothesis is accepted, what are (1) the expected price of the coupon bond
at the end of the first year and (2) the expected holding-period return on the coupon
bond over the first year?
d. Will the expected rate of return be higher or lower if you accept the liquidity preference hypothesis?
17. T he current yield curve for default-free zero-coupon bonds is as follows:
Maturity (Years) YTM (%)
1 10%
2 11
3 12
a. What are the implied 1-year forward rates? b. Assume that the pure expectations hypothesis of the term structure is correct. If market
expectations are accurate, what will be the pure yield curve (that is, the yields to maturity on
1- and 2-year zero coupon bonds) next year?
c. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over the next year? What if you purchase a 3-year zero-coupon bond? ( Hint: Compute the current and expected future prices.) Ignore taxes.
d. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid annually? If you purchased it at that price, what would your total expected rate of return be
over the next year (coupon plus price change)? Ignore taxes.
18. Suppose that the prices of zero-coupon bonds with various maturities are given in the following
table. The face value of each bond is $1,000.
Maturity (Years) Price
1 $925.93
2 853.39
3 782.92
4 715.00
5 650.00
Challenge
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510 P A R T I V Fixed-Income Securities
1. B riefly explain why bonds of different maturities have different yields in terms of the expecta-
tions and liquidity preference hypotheses. Briefly describe the implications of each hypothesis
when the yield curve is (1) upward-sloping and (2) downward-sloping.
2. Which one of the following statements about the term structure of interest rates is true?
a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates exceed current short-term rates.
b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short- term rate.
c. The liquidity premium theory indicates that, all else being equal, longer maturities will have lower yields.
d. The liquidity preference theory contends that lenders prefer to buy securities at the short end of the yield curve.
3. The following table shows yields to maturity of zero-coupon Treasury securities.
Term to Maturity (Years) Yield to Maturity (%)
1 3.50%
2 4.50
3 5.00
4 5.50
5 6.00
10 6.60
a. Calculate the forward 1-year rate of interest for year 3. b. Describe the conditions under which the calculated forward rate would be an unbiased esti-
mate of the 1-year spot rate of interest for that year.
c. Assume that a few months earlier, the forward 1-year rate of interest for that year had been significantly higher than it is now. What factors could account for the decline in the forward rate?
4. The 6-month Treasury bill spot rate is 4%, and the 1-year Treasury bill spot rate is 5%. What is
the implied 6-month forward rate for 6 months from now?
5. The tables below show, respectively, the characteristics of two annual-pay bonds from the same
issuer with the same priority in the event of default, and spot interest rates. Neither bond’s price
is consistent with the spot rates. Using the information in these tables, recommend either bond A or bond B for purchase.
a. Calculate the forward rate of interest for each year. b. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on
that loan equals the forward rate.
c. Repeat ( b ) for a 1-year forward loan beginning in year 4.
19. Continue to use the data in the preceding problem. Suppose that you want to construct a 2-year maturity forward loan commencing in 3 years.
a. Suppose that you buy today one 3-year maturity zero-coupon bond. How many 5-year matu- rity zeros would you have to sell to make your initial cash flow equal to zero?
b. What are the cash flows on this strategy in each year? c. What is the effective 2-year interest rate on the effective 3-year-ahead forward loan? d. Confirm that the effective 2-year interest rate equals (1 1 f 4 ) 3 (1 1 f 5 ) 2 1. You therefore
can interpret the 2-year loan rate as a 2-year forward rate for the last 2 years. Alternatively,
show that the effective 2-year forward rate equals
(1 1 y5)
5
(1 1 y3) 3 2 1
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C H A P T E R 1 5 The Term Structure of Interest Rates 511
Bond Characteristics
Bond A Bond B
Coupons Annual Annual
Maturity 3 years 3 years
Coupon rate 10% 6%
Yield to maturity 10.65% 10.75%
Price 98.40 88.34
Spot Interest Rates
Term (Years) Spot Rates (Zero-Coupon)
1 5%
2 8
3 11
6. Sandra Kapple is a fixed-income portfolio manager who works with large institutional clients.
Kapple is meeting with Maria VanHusen, consultant to the Star Hospital Pension Plan, to discuss
management of the fund’s approximately $100 million Treasury bond portfolio. The current U.S.
Treasury yield curve is given in the following table. VanHusen states, “Given the large differential
between 2- and 10-year yields, the portfolio would be expected to experience a higher return over
a 10-year horizon by buying 10-year Treasuries, rather than buying 2-year Treasuries and rein-
vesting the proceeds into 2-year T-bonds at each maturity date.”
Maturity Yield Maturity Yield
1 year 2.00% 6 years 4.15%
2 2.90 7 4.30
3 3.50 8 4.45
4 3.80 9 4.60
5 4.00 10 4.70
a. Indicate whether VanHusen’s conclusion is correct, based on the pure expectations hypothesis. b. VanHusen discusses with Kapple alternative theories of the term structure of interest rates and
gives her the following information about the U.S. Treasury market:
Maturity (years) 2 3 4 5 6 7 8 9 10
Liquidity premium (%) .55 .55 .65 .75 .90 1.10 1.20 1.50 1.60
Use this additional information and the liquidity preference theory to determine what the slope of
the yield curve implies about the direction of future expected short-term interest rates.
7. A portfolio manager at Superior Trust Company is structuring a fixed-income portfolio to meet
the objectives of a client. The portfolio manager compares coupon U.S. Treasuries with zero-
coupon stripped U.S. Treasuries and observes a significant yield advantage for the stripped bonds:
Term Coupon
U.S. Treasuries Zero-Coupon Stripped
U.S. Treasuries
3 years 5.50% 5.80%
7 6.75 7.25
10 7.25 7.60
30 7.75 8.20
Briefly discuss why zero-coupon stripped U.S. Treasuries could yield more than coupon U.S.
Treasuries with the same final maturity.
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512 P A R T I V Fixed-Income Securities
8. The shape of the U.S. Treasury yield curve appears to reflect two expected Federal Reserve
reductions in the federal funds rate. The current short-term interest rate is 5%. The first reduc-
tion of approximately 50 basis points (bp) is expected 6 months from now and the second
reduction of approximately 50 bp is expected 1 year from now. The current U.S. Treasury term
premiums are 10 bp per year for each of the next 3 years (out through the 3-year benchmark).
However, the market also believes that the Federal Reserve reductions will be reversed in a
single 100-bp increase in the federal funds rate 2½ years from now. You expect liquidity premi-
ums to remain 10 bp per year for each of the next 3 years (out through the 3-year benchmark).
Describe or draw the shape of the Treasury yield curve out through the 3-year bench-
mark. Which term structure theory supports the shape of the U.S. Treasury yield curve you’ve
described?
9. U.S. Treasuries represent a significant holding in many pension portfolios. You decide to ana-
lyze the yield curve for U.S. Treasury notes.
a. Using the data in the table below, calculate the 5-year spot and forward rates assuming annual compounding. Show your calculations.
U.S. Treasury Note Yield Curve Data
Years to Maturity Par Coupon
Yield to Maturity Calculated Spot Rates
Calculated Forward Rates
1 5.00 5.00 5.00
2 5.20 5.21 5.42
3 6.00 6.05 7.75
4 7.00 7.16 10.56
5 7.00 ? ?
b. Define and describe each of the following three concepts:
i. Short rate
ii. Spot rate
iii. Forward rate
Explain how these concepts are related. c. You are considering the purchase of a zero-coupon U.S. Treasury note with 4 years to matu-
rity. On the basis of the above yield-curve analysis, calculate both the expected yield to
maturity and the price for the security. Show your calculations.
10. The spot rates of interest for five U.S. Treasury securities are shown in the following exhibit.
Assume all securities pay interest annually.
Spot Rates of Interest
Term to Maturity Spot Rate of Interest
1 year 13.00%
2 12.00
3 11.00
4 10.00
5 9.00
a. Compute the 2-year implied forward rate for a deferred loan beginning in 3 years. b. Compute the price of a 5-year annual-pay Treasury security with a coupon rate of 9% by
using the information in the exhibit.
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C H A P T E R 1 5 The Term Structure of Interest Rates 513
SOLUTIONS TO CONCEPT CHECKS
E-INVESTMENTS EXERCISES Go to www.smartmoney.com . Access the Living Yield Curve (look for the Economy and Bonds tab), a moving picture of the yield curve. Is the yield curve usually upward- or downward-sloping? What about today’s yield curve? How much does the slope of the curve vary? Which varies more: short-term or long-term rates? Can you explain why this might be the case?
1. The price of the 3-year bond paying a $40 coupon is
40
1.05 1
40
1.062 1
1040
1.073 5 38.095 1 35.600 1 848.950 5 $922.65
At this price, the yield to maturity is 6.945% [ n 5 3; PV 5 ( 2 )922.65; FV 5 1,000; PMT 5 40]. This bond’s yield to maturity is closer to that of the 3-year zero-coupon bond than is the yield to
maturity of the 10% coupon bond in Example 15.1. This makes sense: This bond’s coupon rate is
lower than that of the bond in Example 15.1. A greater fraction of its value is tied up in the final
payment in the third year, and so it is not surprising that its yield is closer to that of a pure 3-year
zero-coupon security.
2. We compare two investment strategies in a manner similar to Example 15.2:
Buy and hold 4-year zero 5 Buy 3-year zero; roll proceeds into 1-year bond
(1 1 y4) 4 5 (1 1 y3)
3 3 (1 1 r4)
1.084 5 1.073 3 (1 1 r4)
which implies that r 4 5 1.08 4 /1.07
3 2 1 5 .11056 5 11.056%. Now we confirm that the yield on
the 4-year zero is a geometric average of the discount factors for the next 3 years:
1 1 y4 5 3(1 1 r1) 3 (1 1 r2 ) 3 (1 1 r3) 3 (1 1 r4) 41/4 1.08 5 31.05 3 1.0701 3 1.09025 3 1.1105641/4
3. The 3-year bond can be bought today for $1,000/1.07 3 5 $816.30. Next year, it will have a
remaining maturity of 2 years. The short rate in year 2 will be 7.01% and the short rate in year 3
will be 9.025%. Therefore, the bond’s yield to maturity next year will be related to these short
rates according to
(1 1 y2) 2 5 1.0701 3 1.09025 5 1.1667
and its price next year will be $1,000/(1 1 y 2 ) 2 5 $1,000/1.1667 5 $857.12. The 1-year holding-
period rate of return is therefore ($857.12 2 $816.30)/$816.30 5 .05, or 5%.
4. The n -period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods. The short rate for period n is the one-period interest rate that will prevail in period n. Finally, the forward rate for period n is the short rate that would satisfy a “break-even condition” equating the total returns on two n -period investment strategies. The first strategy is an investment in an n -period zero-coupon bond; the second is an investment in an n 2 1 period zero-coupon bond “rolled over” into an investment in a one-period zero. Spot rates and forward rates are observable
today, but because interest rates evolve with uncertainty, future short rates are not. In the special case in which there is no uncertainty in future interest rates, the forward rate calculated from the yield curve would equal the short rate that will prevail in that period.
5. 7% 2 1% 5 6%.
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514 P A R T I V Fixed-Income Securities
6. The risk premium will be zero.
7. If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest
costs on long bonds over short bonds. This willingness combines with investors’ demands for
higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.
8. In general, from Equation 15.5, (1 1 y n ) n 5 (1 1 y n 2 1 )
n 2 1 3 (1 1 f n ). In this case,
(1 1 y 4 ) 4 5 (1.07)
3 3 (1 1 f 4 ). If f 4 5 .07, then (1 1 y 4 )
4 5 (1.07)
4 and y 4 5 .07. If f 4 is greater
than .07, then y 4 also will be greater, and conversely if f 4 is less than .07, then y 4 will be as well.
9. The 3-year yield to maturity is a 1,000 816.30
b1/3 2 1 5 .07 5 7.0% The forward rate for the third year is therefore
f3 5 (1 1 y3)
3
(1 1 y2) 2 2 1 5
1.073
1.062 2 1 5 .0903 5 9.03%
(Alternatively, note that the ratio of the price of the 2-year zero to the price of the 3-year zero is
1 1 f 3 5 1.0903.) To construct the synthetic loan, buy one 2-year maturity zero, and sell 1.0903 3-year maturity zeros. Your initial cash flow is zero, your cash flow at time 2 is 1 $1,000, and
your cash flow at time 3 is 2 $1,090.30, which corresponds to the cash flows on a 1-year forward
loan commencing at time 2 with an interest rate of 9.03%.
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166 PA
R T IV
IN THIS CHAPTER we turn to various strate- gies that bond portfolio managers can pur- sue, making a distinction between passive and active strategies. A passive investment strategy takes market prices of securities as set fairly. Rather than attempting to beat the market by exploiting superior infor- mation or insight, passive managers act to maintain an appropriate risk–return balance given market opportunities. One special case of passive management is an immuni- zation strategy that attempts to insulate or immunize the portfolio from interest rate risk. In contrast, an active investment strat- egy attempts to achieve returns greater than those commensurate with the risk borne. In the context of bond management this style of management can take two forms. Active managers use either interest rate forecasts to predict movements in the entire bond market or some form of intramarket analy- sis to identify particular sectors of the mar- ket or particular bonds that are relatively mispriced.
Because interest rate risk is crucial to for- mulating both active and passive strategies, we begin our discussion with an analysis of the sensitivity of bond prices to interest rate fluctuations. This sensitivity is measured by the duration of the bond, and we devote con- siderable attention to what determines bond duration. We discuss several passive invest- ment strategies, and show how duration- matching techniques can be used to immunize the holding-period return of a portfolio from interest rate risk. After examining the broad range of applications of the duration mea- sure, we consider refinements in the way that interest rate sensitivity is measured, focusing on the concept of bond convexity. Duration is important in formulating active investment strategies as well, and we conclude the chap- ter with a discussion of active fixed-income strategies. These include policies based on interest rate forecasting as well as intramar- ket analysis that seeks to identify relatively attractive sectors or securities within the fixed-income market.
Managing Bond Portfolios
CHAPTER SIXTEEN
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516 P A R T I V Fixed-Income Securities
We have seen already that bond prices and yields are inversely related, and we know that
interest rates can fluctuate substantially. As interest rates rise and fall, bondholders expe-
rience capital losses and gains. These gains or losses make fixed-income investments
risky, even if the coupon and principal payments are guaranteed, as in the case of Treasury
obligations.
Why do bond prices respond to interest rate fluctuations? Remember that in a
competitive market all securities must offer investors fair expected rates of return.
If a bond is issued with an 8% coupon when competitive yields are 8%, then it will sell at
par value. If the market rate rises to 9%, however, who would purchase an 8% coupon bond
at par value? The bond price must fall until its expected return increases to the competi-
tive level of 9%. Conversely, if the market rate falls to 7%, the 8% coupon on the bond is
attractive compared to yields on alternative investments. In response, investors eager for
that return would bid up the bond price until the total rate of return for someone purchasing
at that higher price is no better than the market rate.
Interest Rate Sensitivity The sensitivity of bond prices to changes in market interest rates is obviously of great
concern to investors. To gain some insight into the determinants of interest rate risk, turn
to Figure 16.1 , which presents the percentage change in price corresponding to changes
in yield to maturity for four bonds that differ according to coupon rate, initial yield to
maturity, and time to maturity. All four bonds illustrate that bond prices decrease when
16.1 Interest Rate Risk
P e rc
e n
ta g
e C
h a n
g e i n
B o
n d
P ri
ce
Change in Yield to Maturity (%)
D C B A
A
B
C
D
5−5 −4 −3 −2 −1 0 1 2 3 4
200
150
100
50
0
−50
Initial Bond Coupon Maturity YTM
A 12% 5 years 10% B 12% 30 years 10% C 3% 30 years 10% D 3% 30 years 6%
Figure 16.1 Change in bond price as a function of change in yield to maturity
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C H A P T E R 1 6 Managing Bond Portfolios 517
yields rise, and that the price curve is convex, meaning that decreases in yields have bigger
impacts on price than increases in yields of equal magnitude. We summarize these obser-
vations in the following two propositions:
1. Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise.
2. An increase in a bond’s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude.
Now compare the interest rate sensitivity of bonds A and B, which are identical except for maturity. Figure 16.1 shows that bond B, which has a longer maturity than bond A, exhibits greater sensitivity to interest rate changes. This illustrates another general property:
3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds.
This is not surprising. If rates increase, for example, the bond is less valuable as its cash
flows are discounted at a now-higher rate. The impact of the higher discount rate will be
greater as that rate is applied to more-distant cash flows.
Notice that while bond B has six times the maturity of bond A, it has less than six times the interest rate sensitivity. Although interest rate sensitivity seems to increase with matu-
rity, it does so less than proportionally as bond maturity increases. Therefore, our fourth
property is that:
4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. In other words, interest rate risk is less than proportional to bond maturity.
Bonds B and C, which are alike in all respects except for coupon rate, illustrate another point. The lower-coupon bond exhibits greater sensitivity to changes in interest rates. This
turns out to be a general property of bond prices:
5. Interest rate risk is inversely related to the bond’s coupon rate. Prices of low-coupon bonds are more sensitive to changes in interest rates than prices of high-coupon bonds.
Finally, bonds C and D are identical except for the yield to maturity at which the bonds currently sell. Yet bond C, with a higher yield to maturity, is less sensitive to changes in yields. This illustrates our final property:
6. The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling.
The first five of these general properties were described by Malkiel 1 and are sometimes
known as Malkiel’s bond-pricing relationships. The last property was demonstrated by
Homer and Liebowitz. 2
Maturity is a major determinant of interest rate risk. However, maturity alone is not suffi-
cient to measure interest rate sensitivity. For example, bonds B and C in Figure 16.1 have the same maturity, but the higher-coupon bond has less price sensitivity to interest rate changes.
Obviously, we need to know more than a bond’s maturity to quantify its interest rate risk.
1 Burton G. Malkiel, “Expectations, Bond Prices, and the Term Structure of Interest Rates,” Quarterly Journal
of Economics 76 (May 1962), pp. 197–218.
2 Sidney Homer and Martin L. Liebowitz, Inside the Yield Book: New Tools for Bond Market Strategy (Englewood
Cliffs, NJ: Prentice Hall, 1972).
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518 P A R T I V Fixed-Income Securities
To see why bond characteristics such as coupon rate or yield to maturity affect interest
rate sensitivity, let’s start with a simple numerical example. Table 16.1 gives bond prices
for 8% semiannual coupon bonds at different yields to maturity and times to maturity, T. [The interest rates are expressed as annual percentage rates (APRs), meaning that the true
6-month yield is doubled to obtain the stated annual yield.] The shortest-term bond falls in
value by less than 1% when the interest rate increases from 8% to 9%. The 10-year bond
falls by 6.5%, and the 20-year bond by over 9%.
Now look at a similar computation using a zero-coupon bond rather than the 8% coupon
bond. The results are shown in Table 16.2 . Notice that for each maturity, the price of the
zero-coupon bond falls by a greater proportional amount than the price of the 8% coupon
bond. Because we know that long-term bonds are more sensitive to interest rate movements
than are short-term bonds, this observation suggests that in some sense a zero-coupon bond
must represent a longer-term bond than an equal-time-to-maturity coupon bond.
In fact, this insight about the effective maturity of a bond is a useful one that we can
make mathematically precise. To start, note that the times to maturity of the two bonds in
this example are not perfect measures of the long- or short-term nature of the bonds. The
20-year 8% bond makes many coupon payments, most of which come years before the
bond’s maturity date. Each of these payments may be considered to have its own “maturity.”
In the previous chapter, we pointed out that it can be useful to view a coupon bond as a
“portfolio” of coupon payments. The effective maturity of the bond is therefore some sort
of average of the maturities of all the cash flows. The zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is, therefore, a well-defined concept.
Higher-coupon-rate bonds have a higher fraction of value tied to coupons rather than
final payment of par value, and so the “portfolio of coupons” is more heavily weighted
toward the earlier, short-maturity payments, which gives it lower “effective maturity.” This
explains Malkiel’s fifth rule, that price sensitivity falls with coupon rate.
Similar logic explains our sixth rule, that price sensitivity falls with yield to maturity. A
higher yield reduces the present value of all of the bond’s payments, but more so for more-
distant payments. Therefore, at a higher yield, a higher fraction of the bond’s value is due
to its earlier payments, which have lower effective maturity and interest rate sensitivity.
The overall sensitivity of the bond price to changes in yields is thus lower.
Table 16.1
Prices of 8% coupon bond (coupons paid semiannually)
Yield to Maturity (APR) T 5 1 Year T 5 10 Years T 5 20 Years
8% 1,000.00 1,000.00 1,000.00 9% 990.64 934.96 907.99 Fall in price (%)* 0.94% 6.50% 9.20%
*Equals value of bond at a 9% yield to maturity divided by value of bond at (the original) 8% yield, minus 1.
Yield to Maturity (APR) T 5 1 Year T 5 10 Years T 5 20 Years
8% 924.56 456.39 208.29 9% 915.73 414.64 171.93 Fall in price (%)* 0.96% 9.15% 17.46%
*Equals value of bond at a 9% yield to maturity divided by value of bond at (the original) 8% yield, minus 1.
Table 16.2
Prices of zero-coupon bond (semiannual compounding)
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C H A P T E R 1 6 Managing Bond Portfolios 519
Duration To deal with the ambiguity of the “maturity” of a bond making many payments, we need a
measure of the average maturity of the bond’s promised cash flows. We would like also to
use such an effective maturity measure as a guide to the sensitivity of a bond to interest rate
changes, because we have noted that price sensitivity tends to increase with time to maturity.
Frederick Macaulay 3 termed the effective maturity concept the duration of the bond.
Macaulay’s duration equals the weighted average of the times to each coupon or princi- pal payment. The weight associated with each payment time clearly should be related to
the “importance” of that payment to the value of the bond. In fact, the weight applied
to each payment time is the proportion of the total value of the bond accounted for by that
payment, that is, the present value of the payment divided by the bond price.
We define the weight, w t , associated with the cash flow made at time t (denoted CF t ) as:
wt 5 CFt /(1 1 y)
t
Bond price
where y is the bond’s yield to maturity. The numerator on the right-hand side of this equa- tion is the present value of the cash flow occurring at time t while the denominator is the value of all the bond’s payments. These weights sum to 1.0 because the sum of the cash
flows discounted at the yield to maturity equals the bond price.
Using these values to calculate the weighted average of the times until the receipt of
each of the bond’s payments, we obtain Macaulay’s duration formula:
D 5 a T
t51 t 3 wt (16.1)
3 Frederick Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and
Stock Prices in the United States since 1856 (New York: National Bureau of Economic Research, 1938).
Spreadsheet 16.1 Calculating the duration of two bonds
Column sums subject to rounding error.
A C D E F G Time until PV of CF Column (C) Payment (Discount rate times
Period (Years) Cash Flow 5% per period) Weight* Column (F) A. 8% coupon bond 1 0.5 38.095 0.0395
2 1.0 36.281 0.0376 0.0376 3 1.5 34.554 0.0358 0.0537 4 2.0 855.611 0.8871 1.7741
Sum: 964.540 1.0000 1.8852
B. Zero-coupon 1 0.5 0.000 0.0000 0.0000 2 1.0 0.000 0.0000 0.0000 3 1.5 0.000 0.0000 0.0000 4 2.0 822.702 1.0000 2.0000
Sum: 822.702 1.0000 2.0000
Semiannual int rate:
0.0197
*Weight = Present value of each payment (column E) divided by the bond price.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
B
40 40 40
1040
0 0 0
1000
0.05
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520 P A R T I V Fixed-Income Securities
As an example of the application of Equation 16.1, we derive in Spreadsheet 16.1 the
durations of an 8% coupon and zero-coupon bond, each with 2 years to maturity. We
assume that the yield to maturity on each bond is 10%, or 5% per half-year. The present
value of each payment is discounted at 5% per period for the number of (semiannual) peri-
ods shown in column B. The weight associated with each payment time (column F) is the
present value of the payment for that period (column E) divided by the bond price (the sum
of the present values in column E).
The numbers in column G are the products of time to payment and payment weight.
Each of these products corresponds to one of the terms in Equation 16.1. According to that
equation, we can calculate the duration of each bond by adding the numbers in column G.
The duration of the zero-coupon bond is exactly equal to its time to maturity, 2 years.
This makes sense, because with only one payment, the average time until payment must
be the bond’s maturity. In contrast, the 2-year coupon bond has a shorter duration of
1.8852 years.
Spreadsheet 16.2 shows the spreadsheet formulas used to produce the entries in
Spreadsheet 16.1 . The inputs in the spreadsheet—specifying the cash flows the bond will
pay—are given in columns B–D. In column E we calculate the present value of each cash flow
using the assumed yield to maturity, in column F we calculate the weights for Equation 16.1,
and in column G we compute the product of time to payment and payment weight. Each of
these terms corresponds to one of the values that is summed in Equation 16.1. The sums com-
puted in cells G8 and G14 are therefore the durations of each bond. Using the spreadsheet,
you can easily answer several “what if ” questions such as the one in Concept Check 1.
Suppose the interest rate decreases to 9% as an annual percentage rate. What will happen to the prices and durations of the two bonds in Spreadsheet 16.1 ?
CONCEPT CHECK 16.1
Spreadsheet 16.2
Spreadsheet formulas for calculating duration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time until PV of CF Column (C) Payment (Discount rate times
Period (Years) Cash Flow 5% per period) Weight Column (F) A. 8% coupon bond 1
2 3 4
Sum:
B. Zero-coupon 1 2 3 4
0.5 1
1.5 2
0.5 1
1.5 2
Sum:
=D4/(1+$B$16)^B4 =D5/(1+$B$16)^B5 =D6/(1+$B$16)^B6 =D7/(1+$B$16)^B7 =SUM(E4:E7)
=D10/(1+$B$16)^B10 =D11/(1+$B$16)^B11 =D12/(1+$B$16)^B12 =D13/(1+$B$16)^B13 =SUM(E10:E13)
=E4/E$8 =E5/E$8 =E6/E$8 =E7/E$8 =SUM(F4:F7)
=E10/E$14 =E11/E$14 =E12/E$14 =E13/E$14 =SUM(F10:F13)
Semiannual int rate:
=F5*C5 =F6*C6 =F7*C7 =SUM(G4:G7)
=F10*C10 =F11*C11 =F12*C12 =F13*C13 =SUM(G10:G13)
=F4*C4
A C D E F GB
40 40 40
1040
0 0 0
1000
0.05
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C H A P T E R 1 6 Managing Bond Portfolios 521
Duration is a key concept in fixed-income portfolio management for at least three rea-
sons. First, as we have noted, it is a simple summary statistic of the effective average matu-
rity of the portfolio. Second, it turns out to be an essential tool in immunizing portfolios
from interest rate risk. We explore this application in Section 16.3. Third, duration is a
measure of the interest rate sensitivity of a portfolio, which we explore here.
We have seen that a bond’s price sensitivity to interest rate changes generally increases
with maturity. Duration enables us to quantify this relationship. Specifically, it can be
shown that when interest rates change, the proportional change in a bond’s price can
be related to the change in its yield to maturity, y, according to the rule
DP
P 5 2D 3 BD(1 1 y)
1 1 y R (16.2)
The proportional price change equals the proportional change in 1 plus the bond’s yield
times the bond’s duration.
Practitioners commonly use Equation 16.2 in a slightly different form. They define
modified duration as D * 5 D /(1 1 y ), note that D(1 1 y ) 5 D y, and rewrite Equation 16.2 as
DP
P 5 2D*Dy (16.3)
The percentage change in bond price is just the product of modified duration and the
change in the bond’s yield to maturity. Because the percentage change in the bond price
is proportional to modified duration, modified duration is a natural measure of the bond’s
exposure to changes in interest rates. Actually, as we will see below, Equation 16.2, or
equivalently 16.3, is only approximately valid for large changes in the bond’s yield. The
approximation becomes exact as one considers smaller, or localized, changes in yields. 4
Consider the 2-year maturity, 8% coupon bond in Spreadsheet 16.1 making semiannual coupon payments and selling at a price of $964.540, for a yield to maturity of 10%. The duration of this bond is 1.8852 years. For comparison, we will also consider a zero-coupon bond with maturity and duration of 1.8852 years. As we found in Spreadsheet 16.1 , because the coupon bond makes payments semiannually, it is best to treat one “period” as a half-year. So the duration of each bond is 1.8852 3 2 5 3.7704 (semiannual) periods, with a per period interest rate of 5%. The modified duration of each bond is therefore 3.7704/1.05 5 3.591 periods.
Suppose the semiannual interest rate increases from 5% to 5.01%. According to Equation 16.3, the bond prices should fall by
DP/P 5 2D*Dy 5 23.591 3 .01% 5 2.03591%
Example 16.1 Duration and Interest Rate Risk
4 Students of calculus will recognize that modified duration is proportional to the derivative of the bond’s price
with respect to changes in the bond’s yield. For small changes in yield, Equation 16.3 can be restated as
D* 5 2 1
P dP
dy
As such, it gives a measure of the slope of the bond price curve only in the neighborhood of the current price.
In fact, Equation 16.3 can be derived by differentiating the following bond pricing equation with respect to y:
P 5 a T
t51
CFt
(1 1 y)t
where CF t is the cash flow paid to the bondholder at date t; CF t represents either a coupon payment before maturity or final coupon plus par value at the maturity date.
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522 P A R T I V Fixed-Income Securities
What Determines Duration? Malkiel’s bond price relations, which we laid out in the previous section, characterize the
determinants of interest rate sensitivity. Duration allows us to quantify that sensitivity. For
example, if we wish to speculate on interest rates, duration tells us how strong a bet we are
making. Conversely, if we wish to remain “neutral” on rates, and simply match the interest
rate sensitivity of a chosen bond-market index, duration allows us to measure that sensitiv-
ity and mimic it in our own portfolio. For these reasons, it is crucial to understand the deter-
minants of duration. Therefore, in this section, we present several “rules” that summarize
most of its important properties. These rules are also illustrated in Figure 16.2 , where dura-
tions of bonds of various coupon rates, yields to maturity, and times to maturity are plotted.
We have already established:
Rule 1 for Duration The duration of a zero-coupon bond equals its time to maturity.
We have also seen that a coupon bond has a lower duration than a zero with equal matu-
rity because coupons early in the bond’s life lower the bond’s weighted average time until
payments. This illustrates another general property:
Rule 2 for Duration Holding maturity constant, a bond’s duration is lower when the coupon rate is higher.
This property corresponds to Malkiel’s fifth relationship and is attributable to the impact
of early coupon payments on the weighted-average maturity of a bond’s payments. The
higher these coupons, the higher the weights on the early payments and the lower is the
weighted average maturity of the payments. In other words, a higher fraction of the total
value of the bond is tied up in the (earlier) coupon payments whose values are relatively
insensitive to yields rather than the (later and more yield-sensitive) repayment of par value.
Compare the plots in Figure 16.2 of the durations of the 3% coupon and 15% coupon
Now compute the price change of each bond directly. The coupon bond, which initially sells at $964.540, falls to $964.1942 when its yield increases to 5.01%, which is a per- centage decline of .0359%. The zero-coupon bond initially sells for $1,000/1.05 3.7704 5 831.9704. At the higher yield, it sells for $1,000/1.0501 3.7704 5 831.6717. This price also falls by .0359%.
We conclude that bonds with equal durations do in fact have equal interest rate sensitivity and that (at least for small changes in yields) the percentage price change is the modified duration times the change in yield. 5
5 Notice another implication of Example 16.1: We see from the example that when the bond makes payments
semiannually, it is convenient to treat each payment period as a half-year. This implies that when we calculate
modified duration, we divided Macaulay’s duration by (1 1 Semiannual yield to maturity). It is more common
to write this divisor as (1 1 Bond equivalent yield/2). In general, if a bond were to make n payments a year, modified duration would be related to Macaulay’s duration by D * 5 D /(1 1 y BEY / n ).
a. In Concept Check 1, you calculated the price and duration of a 2-year maturity, 8% coupon bond mak- ing semiannual coupon payments when the market interest rate is 9%. Now suppose the interest rate increases to 9.05%. Calculate the new value of the bond and the percentage change in the bond’s price.
b. Calculate the percentage change in the bond’s price predicted by the duration formula in Equation 16.2 or 16.3. Compare this value to your answer for (a).
CONCEPT CHECK 16.2
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C H A P T E R 1 6 Managing Bond Portfolios 523
bonds, each with identical yields of 15%. The plot of the duration of the 15% coupon bond
lies below the corresponding plot for the 3% coupon bond.
Rule 3 for Duration Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Duration always increases with maturity for bonds
selling at par or at a premium to par.
This property of duration corresponds to Malkiel’s third relationship, and it is fairly
intuitive. What is surprising is that duration need not always increase with time to
maturity. It turns out that for some deep-discount bonds (such as the 3% coupon bond in
Figure 16.2 ), duration may eventually fall with increases in maturity. However, for virtu-
ally all traded bonds it is safe to assume that duration increases with maturity.
Notice in Figure 16.2 that for the zero-coupon bond, maturity and duration are equal.
However, for coupon bonds, duration increases by less than a year with a year’s increase in
maturity. The slope of the duration graph is less than 1.0.
Although long-maturity bonds generally will be high-duration bonds, duration is a
better measure of the long-term nature of the bond because it also accounts for coupon
payments. Time to maturity is an adequate statistic only when the bond pays no coupons;
then, maturity and duration are equal.
Notice also in Figure 16.2 that the two 15% coupon bonds have different durations
when they sell at different yields to maturity. The lower-yield bond has longer duration.
This makes sense, because at lower yields the more distant payments made by the bond
have relatively greater present values and account for a greater share of the bond’s total
value. Thus in the weighted-average calculation of duration the distant payments receive
greater weights, which results in a higher duration measure. This establishes rule 4:
Rule 4 for Duration Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower.
Figure 16.2 Bond duration versus bond maturity
D u
ra ti
o n
( ye
a rs
) Zero-Coupon Bond
15% Coupon YTM = 15%
3% Coupon YTM = 15%
15% Coupon YTM = 6%
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Maturity
30
25
20
15
10
5
0
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524 P A R T I V Fixed-Income Securities
As we noted above, the intuition for this property is that while a higher yield reduces the
present value of all of the bond’s payments, it reduces the value of more-distant payments
by a greater proportional amount. Therefore, at higher yields a higher fraction of the total
value of the bond lies in its earlier payments, thereby reducing effective maturity. Rule 4,
which is the sixth bond-pricing relationship above, applies to coupon bonds. For zeros, of
course, duration equals time to maturity, regardless of the yield to maturity.
Finally, we present a formula for the duration of a perpetuity. This rule is derived from
and consistent with the formula for duration given in Equation 16.1 but may be easier to
use for infinitely lived bonds.
Rule 5 for Duration The duration of a level perpetuity is
Duration of perpetuity 5 1 1 y
y (16.4)
For example, at a 10% yield, the duration of a perpetuity that pays $100 once a year forever
is 1.10/.10 5 11 years, but at an 8% yield it is 1.08/.08 5 13.5 years.
Show that the duration of the perpetuity increases as the interest rate decreases in accordance with rule 4.
CONCEPT CHECK 16.3
Equation 16.4 makes it obvious that maturity and duration can differ substantially. The
maturity of the perpetuity is infinite, whereas the duration of the instrument at a 10% yield
is only 11 years. The present-value-weighted cash flows early on in the life of the perpetu-
ity dominate the computation of duration.
Notice from Figure 16.2 that as their maturities become ever longer, the durations of the
two coupon bonds with yields of 15% both converge to the duration of the perpetuity with
the same yield, 7.67 years.
The equations for the durations of coupon bonds are somewhat tedious and spreadsheets
like Spreadsheet 16.1 are cumbersome to modify for different maturities and coupon rates.
Moreover, they assume that the bond is at the beginning of a coupon payment period. Fortu-
nately, spreadsheet programs such as Excel come with generalizations of these equations that
can accommodate bonds between coupon payment dates. Spreadsheet 16.3 illustrates how to
Spreadsheet 16.3
Using Excel functions to compute duration
1
2
3
4
5
6
7
8
9
10
A B C
Inputs
Settlement date
Formula in column B
Modified duration
Macaulay duration
Outputs
Coupons per year Yield to maturity
Coupon rate Maturity date
=MDURATION(B2,B3,B4,B5,B6) =DURATION(B2,B3,B4,B5,B6)
2
0.10
0.08
=DATE(2002,1,1) =DATE(2000,1,1)1/1/2000
1.7955
1.8852
2
0.10
1/1/2002
0.08
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C H A P T E R 1 6 Managing Bond Portfolios 525
use Excel to compute duration. The spreadsheet uses many of the same conventions as the
bond-pricing spreadsheets described in Chapter 14.
The settlement date (i.e., today’s date) and maturity date are entered in cells B2 and B3
using Excel’s date function, DATE(year, month, day). The coupon and maturity rates are
entered as decimals in cells B4 and B5, and the payment periods per year are entered in cell
B6. Macaulay and modified duration appear in cells B9 and B10. The spreadsheet confirms
that the duration of the bond we looked at in Spreadsheet 16.1 is indeed 1.8852 years. For
this 2-year maturity bond, we don’t have a specific settlement date. We arbitrarily set the
settlement date to January 1, 2000, and use a maturity date precisely 2 years later.
Coupon Rates (per Year)
Years to Maturity 6% 8% 10% 12%
1 0.985 0.981 0.976 0.972 5 4.361 4.218 4.095 3.990 10 7.454 7.067 6.772 6.541 20 10.922 10.292 9.870 9.568
Infinite (perpetuity) 13.000 13.000 13.000 13.000
Table 16.3
Bond durations (yield to maturity 5 8% APR; semiannual coupons)
6 Notice that because the bonds pay their coupons semiannually, we calculate modified duration using the semian-
nual yield to maturity, 4%, in the denominator.
Use Spreadsheet 16.3 to test some of the rules for duration presented a few pages ago. What happens to duration when you change the coupon rate of the bond? The yield to maturity? The maturity? What happens to duration if the bond pays its coupons annually rather than semiannually? Why intuitively is duration shorter with semiannual coupons?
CONCEPT CHECK 16.4
Durations can vary widely among traded bonds. Table 16.3 presents durations com-
puted from Spreadsheet 16.3 for several bonds all paying semiannual coupons and yielding
4% per half-year. Notice that duration decreases as coupon rates increase, and increases
with time to maturity. According to Table 16.3 and Equation 16.2, if the interest rate were
to increase from 8% to 8.1%, the 6% coupon 20-year bond would fall in value by about
10.922 3 .1%/1.04 5 1.05%, whereas the 10% coupon 1-year bond would fall by only
.976 3 .1%/1.04 5 .094%. 6 Notice also from Table 16.3 that duration is independent of
coupon rate only for perpetuities.
As a measure of interest rate sensitivity, duration clearly is a key tool in fixed-income
portfolio management. Yet the duration rule for the impact of interest rates on bond prices
is only an approximation. Equation 16.2, or its equivalent, 16.3, which we repeat here,
states that the percentage change in the value of a bond approximately equals the product
of modified duration times the change in the bond’s yield:
DP
P 5 2D*Dy
16.2 Convexity
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526 P A R T I V Fixed-Income Securities
This equation asserts that
the percentage price change
is directly proportional to the
change in the bond’s yield.
If this were exactly so, how- ever, a graph of the percent-
age change in bond price as a
function of the change in its
yield would plot as a straight
line, with slope equal to
2 D *. Yet Figure 16.1 makes it clear that the relationship
between bond prices and
yields is not linear. The dura- tion rule is a good approxi-
mation for small changes
in bond yield, but it is less
accurate for larger changes.
Figure 16.3 illustrates this
point. Like Figure 16.1 , the
figure presents the percent-
age change in bond price in
response to a change in the
bond’s yield to maturity. The
curved line is the percentage price change for a 30-year maturity, 8% annual payment cou-
pon bond, selling at an initial yield to maturity of 8%. The straight line is the percentage
price change predicted by the duration rule. The slope of the straight line is the modified
duration of the bond at its initial yield to maturity. The modified duration of the bond at this
yield is 11.26 years, so the straight line is a plot of 2 D *D y 5 2 11.26 3 D y. Notice that the two plots are tangent at the initial yield. Thus for small changes in the bond’s yield to
maturity, the duration rule is quite accurate. However, for larger changes in yield, there is
progressively more “daylight” between the two plots, demonstrating that the duration rule
becomes progressively less accurate.
Notice from Figure 16.3 that the duration approximation (the straight line) always
understates the value of the bond; it underestimates the increase in bond price when the
yield falls, and it overestimates the decline in price when the yield rises. This is due to the
curvature of the true price-yield relationship. Curves with shapes such as that of the price-
yield relationship are said to be convex, and the curvature of the price-yield curve is called the convexity of the bond.
We can quantify convexity as the rate of change of the slope of the price-yield curve,
expressed as a fraction of the bond price. 7 As a practical rule, you can view bonds with
higher convexity as exhibiting higher curvature in the price-yield relationship. The
7 We pointed out in footnote 4 that Equation 16.3 for modified duration can be written as dP / P 5 2 D * dy. Thus
D * 5 1/ P 3 dP / dy is the slope of the price-yield curve expressed as a fraction of the bond price. Similarly, the convexity of a bond equals the second derivative (the rate of change of the slope) of the price-yield curve divided by bond price: Convexity 5 1/ P 3 d 2 P / dy 2 . The formula for the convexity of a bond with a maturity of T years making annual coupon payments is
Convexity 5 1
P 3 (1 1 y) 2 a T
t51 B CFt
(1 1 y) t (t 2 1 t)R
where CF t is the cash flow paid to the bondholder at date t; CF t represents either a coupon payment before maturity or final coupon plus par value at the maturity date.
P e rc
e n
ta g
e C
h a n
g e i n
B o
n d
P ri
ce
Change in Yield to Maturity (%)
Actual Price Change
100
80
60
40
20
0
−20
−40
−60
−5 −4 −3 −2 −1 0 1 2 3 4 5
Duration Approximation
Figure 16.3 Bond price convexity: 30-year maturity, 8% coupon bond; initial yield to maturity 5 8%
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C H A P T E R 1 6 Managing Bond Portfolios 527
convexity of noncallable bonds such as that in Figure 16.3 is positive: The slope increases
(i.e., becomes less negative) at higher yields.
Convexity allows us to improve the duration approximation for bond price changes.
Accounting for convexity, Equation 16.3 can be modified as follows: 8
DP
P 5 2D*Dy 1 1⁄2 3 Convexity 3 (Dy)2 (16.5)
The first term on the right-hand side is the same as the duration rule, Equation 16.3. The
second term is the modification for convexity. Notice that for a bond with positive convex-
ity, the second term is positive, regardless of whether the yield rises or falls. This insight
corresponds to the fact noted just above that the duration rule always underestimates the
new value of a bond following a change in its yield. The more accurate Equation 16.5,
which accounts for convexity, always predicts a higher bond price than Equation 16.2. Of
course, if the change in yield is small, the convexity term, which is multiplied by (D y ) 2 in Equation 16.5, will be extremely small and will add little to the approximation. In this case,
the linear approximation given by the duration rule will be sufficiently accurate. Thus con-
vexity is more important as a practical matter when potential interest rate changes are large.
8 To use the convexity rule, you must express interest rates as decimals rather than percentages.
The bond in Figure 16.3 has a 30-year maturity, an 8% coupon, and sells at an initial yield to maturity of 8%. Because the coupon rate equals yield to maturity, the bond sells at par value, or $1,000. The modified duration of the bond at its initial yield is 11.26 years, and its convexity is 212.4, which can be verified using the formula in footnote 7. (You can find a spreadsheet to calculate the convexity of a 30-year bond at the Online Learning Center at www.mhhe.com/bkm. ) If the bond’s yield increases from 8% to 10%, the bond price will fall to $811.46, a decline of 18.85%. The duration rule, Equation 16.2, would predict a price decline of
DP P
5 2D*Dy 5 211.26 3 .02 5 2.2252, or 222.52%
which is considerably more than the bond price actually falls. The duration-with-convexity rule, Equation 16.4, is far more accurate:
DP P
5 2D*Dy 1 ½ 3 Convexity 3 (Dy)2
5 211.26 3 .02 1 ½ 3 212.4 3 (.02)2 5 2.1827, or 218.27%
which is far closer to the exact change in bond price. (Notice that when we use Equation 16.5, we must express interest rates as decimals rather than percentages. The change in rates from 8% to 10% is represented as D y 5 .02.)
If the change in yield were smaller, say, .1%, convexity would matter less. The price of the bond actually would fall to $988.85, a decline of 1.115%. Without accounting for convexity, we would predict a price decline of
DP P
5 2D*Dy 5 211.26 3 .001 5 2.01126, or 21.126%
Accounting for convexity, we get almost the precisely correct answer:
DP P
5 211.26 3 .001 1 ½ 3 212.4 3 (.001)2 5 2.01115, or 21.115%
Nevertheless, the duration rule is quite accurate in this case, even without accounting for convexity.
Example 16.2 Convexity
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528 P A R T I V Fixed-Income Securities
Why Do Investors Like Convexity? Convexity is generally consid-
ered a desirable trait. Bonds
with greater curvature gain
more in price when yields
fall than they lose when
yields rise. For example, in
Figure 16.4 bonds A and B have the same duration at the
initial yield. The plots of their
proportional price changes
as a function of interest rate
changes are tangent, mean-
ing that their sensitivities to
changes in yields at that point
are equal. However, bond A is more convex than bond B. It enjoys greater price increases
and smaller price decreases
when interest rates fluctuate
by larger amounts. If inter-
est rates are volatile, this is an attractive asymmetry that increases the expected return on
the bond, because bond A will benefit more from rate decreases and suffer less from rate increases. Of course, if convexity is desirable, it will not be available for free: Investors
will have to pay higher prices and accept lower yields to maturity on bonds with greater
convexity.
Duration and Convexity of Callable Bonds Look at Figure 16.5 , which depicts the price-yield curve for a callable bond. When inter-
est rates are high, the curve is convex, as it would be for a straight bond. For example, at
an interest rate of 10%, the price-yield curve lies above its tangency line. But as rates fall,
there is a ceiling on the possible price: The bond cannot be worth more than its call price.
So as rates fall, we sometimes say that the bond is subject to price compression—its value
is “compressed” to the call price. In this region, for example at an interest rate of 5%,
the price-yield curve lies below its tangency line, and the curve is said to have negative convexity. 9
Notice that in the region of negative convexity, the price-yield curve exhibits an
unattractive asymmetry. Interest rate increases result in a larger price decline than the price gain corresponding to an interest rate decrease of equal magnitude. The asymmetry arises
from the fact that the bond issuer has retained an option to call back the bond. If rates
rise, the bondholder loses, as would be the case for a straight bond. But if rates fall, rather
than reaping a large capital gain, the investor may have the bond called back from her.
The bondholder is thus in a “heads I lose, tails I don’t win” position. Of course, she was
compensated for this unattractive situation when she purchased the bond. Callable bonds
sell at lower initial prices (higher initial yields) than otherwise comparable straight bonds.
9 If you’ve taken a calculus course, you will recognize that the curve is concave in this region. However, rather
than saying that these bonds exhibit concavity, bond traders prefer the terminology “negative convexity.”
P e rc
e n
ta g
e C
h a n
g e i n
B o
n d
P ri
ce
Change in Yield to Maturity (%)
Bond A
Bond B
100
80
60
40
20
0
−20
−40
−60
−5 −4 −3 −2 −1 0 1 2 3 4 5
Figure 16.4 Convexity of two bonds
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C H A P T E R 1 6 Managing Bond Portfolios 529
The effect of negative convexity is highlighted in Equation 16.5. When convexity is
negative, the second term on the right-hand side is necessarily negative, meaning that bond
price performance will be worse than would be predicted by the duration approximation.
However, callable bonds or, more generally, bonds with “embedded options,” are diffi-
cult to analyze in terms of Macaulay’s duration. This is because in the presence of such
options, the future cash flows provided by the bonds are no longer known. If the bond may
be called, for example, its cash flow stream may be terminated and its principal repaid
earlier than was initially anticipated. Because cash flows are random, we can hardly take a
weighted average of times until each future cash flow, as would be necessary to compute
Macaulay’s duration.
The convention on Wall Street is to compute the effective duration of bonds with embedded options. Effective duration cannot be computed with a simple formula such as
16.1 that requires known cash flows. Instead, more complex bond valuation approaches
that account for the embedded options are used, and effective duration is defined as the proportional change in the bond price per unit change in market interest rates:
Effective duration 5 2 DP/P
Dr (16.6)
This equation seems merely like a slight manipulation of the modified duration formula 16.3. However, there are important differences. First, note that we do not compute effec-
tive duration relative to a change in the bond’s own yield to maturity. (The denominator is
D r, not D y . ) This is because for bonds with embedded options, which may be called early, the yield to maturity is often not a relevant statistic. Instead, we calculate price change
relative to a shift in the level of the term structure of interest rates. Second, the effective
duration formula relies on a pricing methodology that accounts for embedded options.
Figure 16.5 Price-yield curve for a callable bond
B o
n d
P ri
ce
Region of Negative Convexity (Price-yield curve is below its tangency line.)
Region of Positive Convexity
0 5% 10% Interest Rate
Call Price
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530 P A R T I V Fixed-Income Securities
This means that the effective duration will be a function of variables that would not matter
to conventional duration, for example, the volatility of interest rates. In contrast, modified
or Macaulay duration can be computed directly from the promised bond cash flows and
yield to maturity.
Suppose that a callable bond with a call price of $1,050 is selling today for $980. If the yield curve shifts up by .5%, the bond price will fall to $930. If it shifts down by .5%, the bond price will rise to $1,010. To compute effective duration, we compute:
Dr 5 Assumed increase in rates 2 Assumed decrease in rates 5 .5% 2 (2.5%) 5 1% 5 .01
DP 5 Price at .5% increase in rates 2 Price at .5% decrease in rates 5 $930 2 $1,010 5 2$80
Then the effective duration of the bond is
Effective duration 5 2 DP/P Dr
5 2 2$80/$980
.01 5 8.16 years
In other words, the bond price changes by 8.16% for a 1 percentage point swing in rates around current values.
Example 16.3 Effective Duration
What are the differences between Macaulay duration, modified duration, and effective duration?
CONCEPT CHECK 16.5
Duration and Convexity of Mortgage-Backed Securities In practice, the biggest market for which call provisions are important is the market for
mortgage-backed securities. In recent years, firms have been less apt to issue bonds with
call provisions, and the number of outstanding callable corporate bonds has steadily
declined. In contrast, the mortgage-backed market grew rapidly over the last two decades,
at least until the financial crisis. Even in 2012, however, Fannie Mae and Freddie Mac
together issued more than $1 trillion of new mortgage-backed securities.
As described in Chapter 1, lenders that originate mortgage loans commonly sell those
loans to federal agencies such as the Federal National Mortgage Association (FNMA,
or Fannie Mae) or the Federal Home Loan Mortgage Corporation (FHLMC, or Freddie
Mac). The original borrowers (the homeowners) continue to make their monthly payments
to their lenders, but the lenders pass these payments along to the agency that has purchased
the loan. In turn, the agencies may combine many mortgages into a pool called a mortgage-
backed security, and then sell that security in the fixed-income market. These securities are
called pass-throughs because the cash flows from the borrowers are first passed through to the agency (Fannie Mae or Freddie Mac) and then passed through again to the ultimate
purchaser of the mortgage-backed security.
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C H A P T E R 1 6 Managing Bond Portfolios 531
As an example, suppose that ten 30-year mortgages, each with principal value of
$100,000, are grouped together into a million-dollar pool. If the mortgage rate is 8%, then
the monthly payment on each loan would be $733.76. (The interest component of the first
payment is .08 3 1/12 3 $100,000 5 $666.67; the remaining $67.09 is “amortization,”
or scheduled repayment of principal. In later periods, with a lower principal balance, less
of the monthly payments goes to interest and more to amortization.) The owner of the
mortgage-backed security would receive $7,337.60, the total payment from the 10 mort-
gages in the pool. 10
But now recall that the homeowner has the right to prepay the loan at any time. For
example, if mortgage rates go down, the homeowner may very well decide to take a new
loan at a lower rate, using the proceeds to pay off the original loan. The right to prepay
the loan is, of course, precisely analogous to the right to refund a callable bond. The call
price for the mortgage is simply the remaining principal balance on the loan. Therefore,
the mortgage-backed security is best viewed as a portfolio of callable amortizing loans. Mortgage-backs are subject to the same negative convexity as other callable bonds.
When rates fall and homeowners prepay their mortgages, the repayment of principal is
passed through to the investors. Rather than enjoying capital gains on their investment,
they simply receive the outstanding principal balance on the loan. Therefore, the value of
the mortgage-backed security as a function of interest rates, presented in Figure 16.6 , looks
much like the plot for a callable bond.
10 Actually, the financial institution that continues to service the loan and the pass-through agency that guarantees
the loan each retain a portion of the monthly payment as a charge for their services. Thus, the monthly payment
received by the investor is a bit less than the amount paid by the borrower.
Figure 16.6 Price-yield curve for a mortgage-backed security
B o
n d
P ri
ce
0
Interest Rate
Principal Balance
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532 P A R T I V Fixed-Income Securities
There are some differences between the mortgage-backs and callable corporate bonds,
however. For example, you will commonly find mortgage-backs selling for more than
their principal balance. This is because homeowners do not refinance their loans as soon
as interest rates drop. Some homeowners do not want to incur the costs or hassles of refi-
nancing unless the benefit is great enough, others may decide not to refinance if they are
planning to move shortly, and others may simply be unsophisticated in making the refi-
nancing decision. Therefore, while the mortgage-backed security exhibits negative con-
vexity at low rates, its implicit call price (the principal balance on the loan) is not a firm
ceiling on its value.
Simple mortgage-backs have also given rise to a rich set of mortgage-backed deriva-
tives. For example, a CMO (collateralized mortgage obligation) further redirects the cash
flow stream of the mortgage-backed security to several classes of derivative securities
called “tranches.” These tranches may be designed to allocate interest rate risk to investors
most willing to bear that risk. 11
The following table is an example of a very simple CMO structure. The underlying
mortgage pool is divided into three tranches, each with a different effective maturity and
therefore interest rate risk exposure. Suppose the original pool has $10 million of 15-year-
maturity mortgages, each with an interest rate of 10.5%, and is subdivided into three
tranches as follows:
Tranche A 5 $4 million principal “Short-pay” tranche
Tranche B 5 $3 million principal “Intermediate-pay” tranche
Tranche C 5 $3 million principal “Long-pay” tranche
Suppose further that in each year, 8% of outstanding loans in the pool prepay. Then total
cash flows in each year to the whole mortgage pool are given in panel A of Figure 16.7 .
Total payments shrink by 8% each year, as that percentage of the loans in the original pool
is paid off. The light portions of each bar represent interest payments, while the dark
portions are principal payments, including both loan amortization and prepayments.
In each period, each tranche receives the interest owed it based on the promised coupon
rate and outstanding principal balance. But initially, all principal payments, both prepay- ments and amortization, go to tranche A ( Figure 16.7 , panel B). Notice from panels C and
D that tranches B and C receive only interest payments until tranche A is retired. Once
tranche A is fully paid off, all principal payments go to tranche B. Finally, when B is
retired, all principal payments go to C. This makes tranche A a “short-pay” class, with the
lowest effective duration, while tranche C is the longest-pay tranche. This is therefore a
relatively simple way to allocate interest rate risk among tranches.
Many variations on the theme are possible and employed in practice. Different tranches
may receive different coupon rates. Some tranches may be given preferential treatment in
terms of uncertainty over mortgage prepayment speeds. Complex formulas may be used to
dictate the cash flows allocated to each tranche. In essence, the mortgage pool is treated as
a source of cash flows that can be reallocated to different investors in accordance with the
tastes of different investors.
11 In Chapter 14, we examined how collateralized debt obligations or CDOs used tranche structures to reallocate
credit risk among different classes. Credit risk in agency-sponsored mortgage-backed securities is not really an issue because the mortgage payments are guaranteed by the agency, and for now, the federal government; in the
CMO market, tranche structure is used to allocate interest rate risk rather than credit risk across classes.
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C H A P T E R 1 6 Managing Bond Portfolios 533
Figure 16.7 Panel A: cash flows to whole mortgage pool; Panels B–D: cash flows to three tranches.
25,000
20,000
15,000
10,000
5,000
0 1 3 5 7 9 11 13 15
Principal Interest
A: Whole Mortgage
20,000
25,000
15,000
10,000
5,000
0 1 3 5 7 9 11 13 15
Principal Interest
B: Tranche A
Principal Interest
1 3 5 7 9 11 13 15
15,000
20,000
25,000
10,000
5,000
0
C: Tranche B
15,000
20,000
25,000
10,000
5,000
0
Principal Interest
1 3 5 7 9 11 13 15
D: Tranche C
Year
Passive managers take bond prices as fairly set and seek to control only the risk of their
fixed-income portfolio. Two broad classes of passive management are pursued in the fixed-
income market. The first is an indexing strategy that attempts to replicate the performance
of a given bond index. The second broad class of passive strategies is known as immuniza-
tion techniques; they are used widely by financial institutions such as insurance companies
and pension funds, and are designed to shield the overall financial status of the institution
from exposure to interest rate fluctuations.
16.3 Passive Bond Management
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534 P A R T I V Fixed-Income Securities
Although indexing and immunization strategies are alike in that they accept market
prices as correctly set, they are very different in terms of risk exposure. A bond-index port-
folio will have the same risk–reward profile as the bond market index to which it is tied.
In contrast, immunization strategies seek to establish a virtually zero-risk profile, in which
interest rate movements have no impact on the value of the firm. We discuss both types of
strategies in this section.
Bond-Index Funds In principle, bond market indexing is similar to stock market indexing. The idea is to create
a portfolio that mirrors the composition of an index that measures the broad market. In the
U.S. equity market, for example, the S&P 500 is the most commonly used index for stock-
index funds, and these funds simply buy shares of each firm in the S&P 500 in proportion
to the market value of outstanding equity. A similar strategy is used for bond-index funds,
but as we shall see shortly, several modifications are required because of difficulties unique
to the bond market and its indexes.
Three major indexes of the broad bond market are the Barclays Capital U.S. (formerly
Lehman) Aggregate Bond Index, the Salomon Broad Investment Grade (BIG) Index
(now run by Citigroup), and the Merrill Lynch Domestic Master index. All are market-
value-weighted indexes of total returns. All three include government, corporate, mortgage-
backed, and Yankee bonds in their universes. (Yankee bonds are dollar-denominated,
SEC-registered bonds of foreign issuers sold in the United States.)
The first problem that arises in the formation of an indexed bond portfolio arises from
the fact that these indexes include thousands of securities, making it quite difficult to pur-
chase each security in the index in proportion to its market value. Moreover, many bonds
are very thinly traded, meaning that identifying their owners and purchasing the securities
at a fair market price can be difficult.
Bond-index funds also face more difficult rebalancing problems than do stock-index
funds. Bonds are continually dropped from the index as their maturities fall below 1 year.
Moreover, as new bonds are issued, they are added to the index. Therefore, in contrast to
equity indexes, the securities used to compute bond indexes constantly change. As they
do, the manager must update or rebalance the portfolio to ensure a close match between
the composition of the portfolio and the bonds included in the index. The fact that bonds
generate considerable interest income that must be reinvested further complicates the job
of the index fund manager.
In practice, it is infeasible to precisely replicate the broad bond indexes. Instead, a
stratified sampling or cellular approach is often pursued. Figure 16.8 illustrates the idea behind the cellular approach. First, the bond market is stratified into several subclasses.
Figure 16.8 shows a simple two-way breakdown by maturity and issuer; in practice, how-
ever, criteria such as the bond’s coupon rate or the credit risk of the issuer also would be
used to form cells. Bonds falling within each cell are then considered reasonably homo-
geneous. Next, the percentages of the entire universe (i.e., the bonds included in the index
that is to be matched) falling within each cell are computed and reported, as we have done
for a few cells in Figure 16.8 . Finally, the portfolio manager establishes a bond portfolio
with representation for each cell that matches the representation of that cell in the bond
universe. In this way, the characteristics of the portfolio in terms of maturity, coupon rate,
credit risk, industrial representation, and so on, will match the characteristics of the index,
and the performance of the portfolio likewise should match the index.
Retail investors can buy mutual funds or exchange-traded funds that track the broad
bond market. For example, Vanguard’s Total Bond Market Index Fund and Barclays
Aggregate Bond Fund iShare (ticker AGG) both track the Barclays Aggregate index.
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C H A P T E R 1 6 Managing Bond Portfolios 535
Immunization In contrast to indexing strategies, many institutions try to insulate their portfolios from
interest rate risk altogether. Generally, there are two ways of viewing this risk. Some insti-
tutions, such as banks, are concerned with protecting current net worth or net market value
against interest rate fluctuations. Other investors, such as pension funds, may face an
obligation to make payments after a given number of years. These investors are more
concerned with protecting the future values of their portfolios.
What is common to all investors, however, is interest rate risk. The net worth of the
firm or the ability to meet future obligations fluctuates with interest rates. Immunization techniques refer to strategies used by such investors to shield their overall financial status
from interest rate risk.
Many banks and thrift institutions have a natural mismatch between asset and liability
maturity structures. Bank liabilities are primarily the deposits owed to customers, most
of which are very short-term in nature and, consequently, of low duration. Bank assets by
contrast are composed largely of outstanding commercial and consumer loans or mort-
gages. These assets are of longer duration, and their values are correspondingly more
sensitive to interest rate fluctuations. When interest rates increase unexpectedly, banks
can suffer serious decreases in net worth—their assets fall in value by more than their
liabilities.
Similarly, a pension fund may have a mismatch between the interest rate sensitivity of
the assets held in the fund and the present value of its liabilities—the promise to make pay-
ments to retirees. The nearby box illustrates the dangers that pension funds face when they
neglect to consider the interest rate exposure of both assets and liabilities. For example, in some recent years pension funds lost ground despite the fact that they enjoyed excellent
investment returns. As interest rates fell, the value of their liabilities grew even faster than
the value of their assets. The lesson is that funds should match the interest rate exposure
of assets and liabilities so that the value of assets will track the value of liabilities whether
rates rise or fall. In other words, the financial manager might want to immunize the fund against interest rate volatility.
Figure 16.8 Stratification of bonds into cells
Treasury Sector
Term to Maturity
Agency Mortgage- backed
Industrial Finance Utility Yankee
12.1%
4.1%
9.2%
0.1%
3.4%
<1 year
1−3 years
3−5 years
5−7 years
7−10 years
10−15 years
15−30 years
30+ years
5.4%
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536
Pension funds are not alone in this concern. Any institution with a future fixed obliga-
tion might consider immunization a reasonable risk management policy. Insurance compa-
nies, for example, also pursue immunization strategies. In fact, the notion of immunization
was introduced by F. M. Redington, 12
an actuary for a life insurance company. The idea is
that duration-matched assets and liabilities let the asset portfolio meet the firm’s obliga-
tions despite interest rate movements.
Consider, for example, an insurance company that issues a guaranteed investment
contract, or GIC, for $10,000. (Essentially, GICs are zero-coupon bonds issued by the
insurance company to its customers. They are popular products for individuals’ retirement-
savings accounts.) If the GIC has a 5-year maturity and a guaranteed interest rate of 8%,
the insurance company promises to pay $10,000 3 (1.08) 5 5 $14,693.28 in 5 years.
Suppose that the insurance company chooses to fund its obligation with $10,000 of 8%
annual coupon bonds, selling at par value, with 6 years to maturity. As long as the market interest rate stays at 8%, the company has fully funded the obligation, as the present value
of the obligation exactly equals the value of the bonds.
Table 16.4 , panel A shows that if interest rates remain at 8%, the accumulated funds from
the bond will grow to exactly the $14,693.28 obligation. Over the 5-year period, year-end
coupon income of $800 is reinvested at the prevailing 8% market interest rate. At the
end of the period, the bonds can be sold for $10,000; they still will sell at par value because
the coupon rate still equals the market interest rate. Total income after 5 years from rein-
vested coupons and the sale of the bond is precisely $14,693.28.
If interest rates change, however, two offsetting influences will affect the ability of the
fund to grow to the targeted value of $14,693.28. If interest rates rise, the fund will suffer
a capital loss, impairing its ability to satisfy the obligation. The bonds will be worth less in
5 years than if interest rates had remained at 8%. However, at a higher interest rate, rein-
vested coupons will grow at a faster rate, offsetting the capital loss. In other words, fixed-
income investors face two offsetting types of interest rate risk: price risk and reinvestment rate risk. Increases in interest rates cause capital losses but at the same time increase the rate at which reinvested income will grow. If the portfolio duration is chosen appropriately,
12
F. M. Redington, “Review of the Principle of Life-Office Valuations,” Journal of the Institute of Actuaries 78 (1952).
Pension Funds Lost Ground Despite
Broad Market Gains
With the S&P 500 providing a 16% rate of return, 2012 was a good year for the stock market, and this performance helped boost the balance sheets of U.S. pension funds. Yet despite the increase in the value of their assets, the total estimated pension deficit of 400 large U.S. companies rose by nearly $80 billion, and many of these firms entered 2013 needing to shore up their pension funds with billions of dollars of additional cash. Ford Motor Co. alone predicted that it would contribute $5 billion to its fund. *
How could this happen? Blame the decline in interest rates during the year that were in part the force behind the stock market gains. As rates fell, the present value of pension obligations to retirees rose even faster than the value of the assets backing those promises. It turns out that the value of pension liabilities is more sensitive to interest rate changes than the value of the typical assets
held in those funds. So even though falling rates tend to pump up asset returns, they pump up liabilities even more so. In other words, the duration of fund investments tends to be shorter than the duration of its obligations. This duration mismatch makes funds vulnerable to interest rate declines.
Why don’t funds better match asset and liability dura- tions? One reason is that fund managers are often evalu- ated based on their performance relative to standard bond market indexes. Those indexes tend to have far shorter durations than pension fund liabilities. So to some extent, managers may be keeping their eyes on the wrong ball, one with the wrong interest rate sensitivity.
*These estimates appear in Mike Ramsey and Vipal Monga, “Low Rates Force Companies to Pour Cash into Pensions,” The Wall Street Journal, February 3, 2013. W
O R
D S
FR O
M T
H E S
T R
E E T
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C H A P T E R 1 6 Managing Bond Portfolios 537
these two effects will cancel out exactly. When the portfolio duration is set equal to the
investor’s horizon date, the accumulated value of the investment fund at the horizon date
will be unaffected by interest rate fluctuations. For a horizon equal to the portfolio’s dura- tion, price risk and reinvestment risk are precisely offsetting.
In our example, the duration of the 6-year maturity bonds used to fund the GIC is
5 years. Because the fully funded plan has equal duration for its assets and liabilities,
the insurance company should be immunized against interest rate fluctuations. To confirm
this, let’s check that the bond can generate enough income to pay off the obligation in
5 years regardless of interest rate movements.
In Table 16.4 , panels B and C illustrate two possible interest rate scenarios: Rates either
fall to 7%, or increase to 9%. In both cases, the annual coupon payments are reinvested at
the new interest rate, which is assumed to change before the first coupon payment, and the
bond is sold in year 5 to help satisfy the obligation of the GIC.
Table 16.4 , panel B shows that if interest rates fall to 7%, the total funds will accumulate
to $14,694.05, providing a small surplus of $.77. If rates increase to 9% as in Table 16.4 ,
panel C, the fund accumulates to $14,696.02, providing a small surplus of $2.74.
Payment Number
Years Remaining until Obligation
Accumulated Value of Invested Payment
A. Rates remain at 8% 1 4 800 3 (1.08)4 5 1,088.39 2 3 800 3 (1.08)3 5 1,007.77 3 2 800 3 (1.08)2 5 933.12 4 1 800 3 (1.08)1 5 864.00 5 0 800 3 (1.08)0 5 800.00
Sale of bond 0 10,800/1.08 5 10,000.00 14,693.28
B. Rates fall to 7% 1 4 800 3 (1.07)4 5 1,048.64 2 3 800 3 (1.07)3 5 980.03 3 2 800 3 (1.07)2 5 915.92 4 1 800 3 (1.07)1 5 856.00 5 0 800 3 (1.07)0 5 800.00
Sale of bond 0 10,800/1.07 5 10,093.46 14,694.05
C. Rates increase to 9% 1 4 800 3 (1.09)4 5 1,129.27 2 3 800 3 (1.09)3 5 1,036.02 3 2 800 3 (1.09)2 5 950.48 4 1 800 3 (1.09)1 5 872.00 5 0 800 3 (1.09)0 5 800.00
Sale of bond 0 10,800/1.09 5 9,908.26 14,696.02
Table 16.4
Terminal value of a bond portfolio after 5 years (all proceeds reinvested) Note: The sale price of the bond portfolio equals the portfolio’s final payment ($10,800) divided by 1 1 r, because the time to maturity of the bonds will be 1 year at the time of sale.
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538 P A R T I V Fixed-Income Securities
Several points are worth
highlighting. First, duration
matching balances the differ-
ence between the accumulated
value of the coupon payments
(reinvestment rate risk) and the
sale value of the bond (price
risk). That is, when interest
rates fall, the coupons grow
less than in the base case, but
the higher value of the bond
offsets this. When interest rates
rise, the value of the bond falls,
but the coupons more than
make up for this loss because
they are reinvested at the higher
rate. Figure 16.9 illustrates this
case. The solid curve traces the
accumulated value of the bonds
if interest rates remain at 8%.
The dashed curve shows that
value if interest rates happen
to increase. The initial impact
is a capital loss, but this loss
eventually is offset by the now-
faster growth rate of reinvested
funds. At the 5-year horizon date, equal to the bond’s duration, the two effects just
cancel, leaving the company able to satisfy its obligation with the accumulated proceeds
from the bond.
We can also analyze immunization in terms of present as opposed to future values.
Panel A in Table 16.5 shows the initial balance sheet for the insurance company’s GIC.
Both assets and the obligation have market values of $10,000, so the plan is just fully
funded. Panels B and C in the table show that whether the interest rate increases or
decreases, the value of the bonds funding the GIC and the present value of the company’s
Accumulated Value of Invested Funds
0 t* D t
Funds
Figure 16.9 Growth of invested funds. The solid colored curve rep- resents the growth of portfolio value at the original interest rate. If interest rates increase at time t *, the portfolio value initially falls but increases thereafter at the faster rate represented by the broken curve. At time D (duration) the curves cross.
Table 16.5
Market value balance sheet
Assets Liabilities
A. Interest rate 5 8% Bonds $10,000 Obligation $10,000
B. Interest rate 5 7% Bonds $10,476.65 Obligation $10,476.11
C. Interest rate 5 9% Bonds $9,551.41 Obligation $9,549.62
Notes: Value of bonds 5 800 3 Annuity factor( r, 6) 1 10,000 3 PV factor ( r, 6)
Value of obligation 5 14,693.28
(1 1 r)5 5 14,693.28 3 PV factor (r, 5)
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539
obligation change by virtually identical amounts. Regardless of the interest rate change,
the plan remains fully funded, with the surplus in panels B and C in Table 16.5 just about
zero. The duration-matching strategy has ensured that both assets and liabilities react
equally to interest rate fluctuations.
Figure 16.10 is a graph of the present values of the bond and the single-payment obliga-
tion as a function of the interest rate. At the current rate of 8%, the values are equal, and the
obligation is fully funded by the bond. Moreover, the two present value curves are tangent
at y 5 8%. As interest rates change, the change in value of both the asset and the obliga- tion is equal, so the obligation remains fully funded. For greater changes in the interest
rate, however, the present value curves diverge. This reflects the fact that the fund actually
shows a small surplus in Table 16.4 at market interest rates other than 8%.
If the obligation was immunized, why is there any surplus in the fund? The answer is convexity. Figure 16.10 shows that the coupon bond has greater convexity than the obliga-
tion it funds. Hence, when rates move substantially, the bond value exceeds the present
value of the obligation by a noticeable amount.
This example highlights the importance of rebalancing immunized portfolios. As inter- est rates and asset durations change, a manager must rebalance the portfolio to realign its
duration with the duration of the obligation. Moreover, even if interest rates do not change,
asset durations will change solely because of the passage of time. Recall from Figure 16.2 that duration generally decreases less rapidly than does maturity. Thus, even if an obliga-
tion is immunized at the outset, as time passes the durations of the asset and liability will
fall at different rates. Without portfolio rebalancing, durations will become unmatched.
eXcel APPLICATIONS: Holding-Period Immunization
T he Online Learning Center ( www.mhhe.com/bkm ) contains a spreadsheet that is useful in understanding
the concept of holding-period immunization. The spread- sheet calculates duration and holding-period returns on bonds of any maturity. The spreadsheet shows how price risk and reinvestment risk offset if a bond is sold at its duration.
Excel Questions
1. When rates increase by 100 basis points (bp), what is the change in the future sales price of the bond? The value of reinvested coupons?
2. What if rates increase by 200 bp?
3. What is the relation between price risk and reinvestment rate risk as we consider larger changes in bond yields?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
If YTM increases 200 basis points: If YTM increases 200 basis points:
A B DC E F G H
Yield to maturity
Internal rate of return Accumulated value Sale of bond Future value of coupons Yield to maturity
Market price Duration Holding period Par value Years to maturity Coupon rate
11.580%
11.5981% $1,924.693 $1,006.954
$917.739 13.580%
$1,111.929 5.000251
5.0 $1,000.00
7.0 14.000%
1,006.954 $917.739
2.00%
$1,111.929 5.000251
Internal rate of return
Yield to maturity Future value of coupons Sale of bond Accumulated value
12.580%
11.5845% $1,923.522 $1,023.817
$899.705
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540 P A R T I V Fixed-Income Securities
Figure 16.10 Immunization. The coupon bond fully funds the obligation at an inter- est rate of 8%. Moreover, the present value curves are tangent at 8%, so the obliga- tion will remain fully funded even if rates change by a small amount.
Values ($)
6,000
Coupon Bond
Single Payment Obligation
Interest Rate (%)
0 5 8 10 15 20
14,000
12,000
10,000
8,000
Obviously, immunization is a passive strategy only in the sense that it does not involve
attempts to identify undervalued securities. Immunization managers still actively update
and monitor their positions.
An insurance company must make a payment of $19,487 in 7 years. The market interest rate is 10%, so the present value of the obligation is $10,000. The company’s portfolio manager wishes to fund the obligation using 3-year zero-coupon bonds and perpetuities paying annual coupons. (We focus on zeros and perpetuities to keep the algebra simple.) How can the manager immunize the obligation?
Immunization requires that the duration of the portfolio of assets equal the duration of the liability. We can proceed in four steps:
1. Calculate the duration of the liability. In this case, the liability duration is simple to compute. It is a single-payment obligation with duration of 7 years.
2. Calculate the duration of the asset portfolio. The portfolio duration is the weighted average of duration of each component asset, with weights proportional to the funds placed in each asset. The duration of the zero- coupon bond is simply its maturity, 3 years. The duration of the perpetuity is 1.10/.10 5 11 years. Therefore, if the fraction of the portfolio invested in
Example 16.4 Constructing an Immunized Portfolio
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C H A P T E R 1 6 Managing Bond Portfolios 541
the zero is called w and the fraction invested in the perpetuity is (1 2 w ), the portfolio duration will be
Asset duration 5 w 3 3 years 1 (1 2 w ) 3 11 years
3. Find the asset mix that sets the duration of assets equal to the 7-year duration of liabilities. This requires us to solve for w in the following equation:
w 3 3 years 1 (1 2 w ) 3 11 years 5 7 years
This implies that w 5 ½. The manager should invest half the portfolio in the zero and half in the perpetuity. This will result in an asset duration of 7 years.
4. Fully fund the obligation. Because the obligation has a present value of $10,000, and the fund will be invested equally in the zero and the perpetuity, the manager must purchase $5,000 of the zero-coupon bond and $5,000 of the perpetuity. (Note that the face value of the zero will be $5,000 3 (1.10) 3 5 $6,655.)
Even if a position is immunized, however, the portfolio manager still cannot rest. This
is because of the need for rebalancing in response to changes in interest rates. Moreover,
even if rates do not change, the passage of time also will affect duration and require rebal-
ancing. Let us continue Example 16.4 and see how the portfolio manager can maintain an
immunized position.
Suppose that 1 year has passed, and the interest rate remains at 10%. The portfo- lio manager of Example 16.4 needs to reexamine her position. Is the position still fully funded? Is it still immunized? If not, what actions are required?
First, examine funding. The present value of the obligation will have grown to $11,000, as it is 1 year closer to maturity. The manager’s funds also have grown to $11,000: The zero-coupon bonds have increased in value from $5,000 to $5,500 with the passage of time, while the perpetuity has paid its annual $500 coupon and remains worth $5,000. Therefore, the obligation is still fully funded.
The portfolio weights must be changed, however. The zero-coupon bond now will have a duration of 2 years, while the perpetuity duration remains at 11 years. The obli- gation is now due in 6 years. The weights must now satisfy the equation
w 3 2 1 (1 2 w ) 3 11 5 6
which implies that w 5 5/9 . To rebalance the portfolio and maintain the duration match, the manager now must invest a total of $11,000 3 5/9 5 $6,111.11 in the zero-coupon bond. This requires that the entire $500 coupon payment be invested in the zero, with an additional $111.11 of the perpetuity sold and invested in the zero-coupon bond.
Example 16.5 Rebalancing
Of course, rebalancing of the portfolio entails transaction costs as assets are bought
or sold, so one cannot rebalance continuously. In practice, an appropriate compromise
must be established between the desire for perfect immunization, which requires continual
rebalancing, and the need to control trading costs, which dictates less frequent rebalancing.
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542 P A R T I V Fixed-Income Securities
Cash Flow Matching and Dedication The problems associated with immunization seem to have a simple solution. Why not sim-
ply buy a zero-coupon bond with face value equal to the projected cash outlay? If we follow
the principle of cash flow matching we automatically immunize the portfolio from interest rate risk because the cash flow from the bond and the obligation exactly offset each other.
Cash flow matching on a multiperiod basis is referred to as a dedication strategy. In this case, the manager selects either zero-coupon or coupon bonds with total cash flows
in each period that match a series of obligations. The advantage of dedication is that it
is a once-and-for-all approach to eliminating interest rate risk. Once the cash flows are
matched, there is no need for rebalancing. The dedicated portfolio provides the cash neces-
sary to pay the firm’s liabilities regardless of the eventual path of interest rates.
Cash flow matching is not more widely pursued probably because of the constraints that
it imposes on bond selection. Immunization or dedication strategies are appealing to firms
that do not wish to bet on general movements in interest rates, but these firms may want to
immunize using bonds that they perceive are undervalued. Cash flow matching, however,
places so many more constraints on the bond selection process that it can be impossible
to pursue a dedication strategy using only “underpriced” bonds. Firms looking for under-
priced bonds give up exact and easy dedication for the possibility of achieving superior
returns from the bond portfolio.
Sometimes, cash flow matching is simply not possible. To cash flow match for a pension
fund that is obligated to pay out a perpetual flow of income to current and future retirees,
the pension fund would need to purchase fixed-income securities with maturities ranging
up to hundreds of years. Such securities do not exist, making exact dedication infeasible.
Look again at Example 16.5. What would be the immunizing weights in the second year if the interest rate had fallen to 8%?
CONCEPT CHECK 16.6
How would an increase in trading costs affect the attractiveness of dedication versus immunization?
CONCEPT CHECK 16.7
Other Problems with Conventional Immunization If you look back at the definition of duration in Equation 16.1, you note that it uses the
bond’s yield to maturity to calculate the weight applied to the time until each coupon
payment. Given this definition and limitations on the proper use of yield to maturity, it is
perhaps not surprising that this notion of duration is strictly valid only for a flat yield curve
for which all payments are discounted at a common interest rate.
If the yield curve is not flat, then the definition of duration must be modified and
CF t /(1 1 y ) t replaced with the present value of CF t , where the present value of each cash
flow is calculated by discounting with the appropriate spot interest rate from the zero-
coupon yield curve corresponding to the date of the particular cash flow, instead of by discounting with the bond’s yield to maturity. Moreover, even with this modification,
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C H A P T E R 1 6 Managing Bond Portfolios 543
duration matching will immunize portfolios only for parallel shifts in the yield curve.
Clearly, this sort of restriction is unrealistic. As a result, much work has been devoted to
generalizing the notion of duration. Multifactor duration models have been developed to
allow for tilts and other distortions in the shape of the yield curve, in addition to shifts in its
level. However, the added complexity of such models does not appear to pay off in terms
of substantially greater effectiveness. 13
Finally, immunization can be an inappropriate goal in an inflationary environment.
Immunization is essentially a nominal notion and makes sense only for nominal liabilities.
It makes no sense to immunize a projected obligation that will grow with the price level
using nominal assets such as bonds. For example, if your child will attend college in
15 years and if the annual cost of tuition is expected to be $50,000 at that time, immunizing
your portfolio at a locked-in terminal value of $50,000 is not necessarily a risk-reducing
strategy. The tuition obligation will vary with the realized inflation rate, whereas the asset
portfolio’s final value will not. In the end, the tuition obligation will not necessarily be
matched by the value of the portfolio.
13
G. O. Bierwag, G. C. Kaufman, and A. Toevs, eds., Innovations in Bond Portfolio Management: Duration Analysis and Immunization (Greenwich, CT: JAI Press, 1983).
Sources of Potential Profit Broadly speaking, there are two sources of potential value in active bond management. The
first is interest rate forecasting, which tries to anticipate movements across the entire spec-
trum of the fixed-income market. If interest rate declines are anticipated, managers will
increase portfolio duration (and vice versa). The second source of potential profit is iden-
tification of relative mispricing within the fixed-income market. An analyst, for example,
might believe that the default premium on one particular bond is unnecessarily large and
therefore that the bond is underpriced.
These techniques will generate abnormal returns only if the analyst’s information or
insight is superior to that of the market. You cannot profit from knowledge that rates are
about to fall if prices already reflect this information. You know this from our discussion
of market efficiency. Valuable information is differential information. In this context it is
worth noting that interest rate forecasters have a notoriously poor track record. If you con-
sider this record, you will approach attempts to time the bond market with caution.
Homer and Liebowitz (see footnote 2) coined a popular taxonomy of active bond port-
folio strategies. They characterize portfolio rebalancing activities as one of four types of
bond swaps. In the first two swaps the investor typically believes that the yield relation- ship between bonds or sectors is only temporarily out of alignment. When the aberration
is eliminated, gains can be realized on the underpriced bond. The period of realignment is
called the workout period.
1. The substitution swap is an exchange of one bond for a nearly identical substitute. The substituted bonds should be of essentially equal coupon, maturity, quality, call
features, sinking fund provisions, and so on. This swap would be motivated by a
belief that the market has temporarily mispriced the two bonds, and that the dis-
crepancy between the prices of the bonds represents a profit opportunity.
An example of a substitution swap would be a sale of a 20-year maturity,
6% coupon Toyota bond that is priced to provide a yield to maturity of 6.05%,
16.4 Active Bond Management
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544 P A R T I V Fixed-Income Securities
coupled with a purchase of a 6% coupon Honda bond with the same time to
maturity that yields 6.15%. If the bonds have about the same credit rating, there
is no apparent reason for the Honda bonds to provide a higher yield. Therefore,
the higher yield actually available in the market makes the Honda bond seem
relatively attractive. Of course, the equality of credit risk is an important condi-
tion. If the Honda bond is in fact riskier, then its higher yield does not represent
a bargain.
2. The intermarket spread swap is pursued when an investor believes that the yield spread between two sectors of the bond market is temporarily out of line. For
example, if the current spread between corporate and government bonds is consid-
ered too wide and is expected to narrow, the investor will shift from government
bonds into corporate bonds. If the yield spread does in fact narrow, corporates will
outperform governments. For example, if the yield spread between 10-year Trea-
sury bonds and 10-year Baa-rated corporate bonds is now 3%, and the historical
spread has been only 2%, an investor might consider selling holdings of Treasury
bonds and replacing them with corporates. If the yield spread eventually narrows,
the Baa-rated corporate bonds will outperform the Treasuries.
Of course, the investor must consider carefully whether there is a good reason
that the yield spread seems out of alignment. For example, the default premium
on corporate bonds might have increased because the market is expecting a severe
recession. In this case, the wider spread would not represent attractive pricing of
corporates relative to Treasuries, but would simply be an adjustment for a perceived
increase in credit risk.
3. The rate anticipation swap is pegged to interest rate forecasting. In this case if investors believe that rates will fall, they will swap into bonds of longer duration.
Conversely, when rates are expected to rise, they will swap into shorter dura-
tion bonds. For example, the investor might sell a 5-year maturity Treasury bond,
replacing it with a 25-year maturity Treasury bond. The new bond has the same
lack of credit risk as the old one, but has longer duration.
4. The pure yield pickup swap is pursued not in response to perceived mispricing, but as a means of increasing return by holding higher-yield bonds. When the yield
curve is upward-sloping, the yield pickup swap entails moving into longer-term
bonds. This must be viewed as an attempt to earn an expected term premium in
higher-yield bonds. The investor is willing to bear the interest rate risk that this
strategy entails. The investor who swaps the shorter-term bond for the longer
one will earn a higher rate of return as long as the yield curve does not shift up
during the holding period. Of course if it does, the longer-duration bond will suffer
a greater capital loss.
We can add a fifth swap, called a tax swap, to this list. This simply refers to a swap to exploit some tax advantage. For example, an investor may swap from one bond that has
decreased in price to another if realization of capital losses is advantageous for tax purposes.
Horizon Analysis One form of interest rate forecasting, which we encountered in Chapter 14, is called horizon analysis. The analyst using this approach selects a particular holding period and predicts the yield curve at the end of that period. Given a bond’s time to maturity at the end of the hold-
ing period, its yield can be read from the predicted yield curve and its end-of-period price
calculated. Then the analyst adds the coupon income and prospective capital gain of the
bond to obtain the total return on the bond over the holding period.
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C H A P T E R 1 6 Managing Bond Portfolios 545
A 20-year maturity bond with a 10% coupon rate (paid annually) currently sells at a yield to maturity of 9%. A portfolio manager with a 2-year horizon needs to forecast the total return on the bond over the coming 2 years. In 2 years, the bond will have an 18-year maturity. The analyst forecasts that 2 years from now, 18-year bonds will sell at yields to maturity of 8%, and that coupon payments can be reinvested in short-term securities over the coming 2 years at a rate of 7%.
To calculate the 2-year return on the bond, the analyst would perform the following calculations:
1. Current price 5 $100 3 Annuity factor (9%, 20 years)
1 $1,000 3 PV factor (9%, 20 years)
5 $1,091.29
2. Forecast price 5 $100 3 Annuity factor (8%, 18 years)
1 $1,000 3 PV factor (8%, 18 years)
5 $1,187.44
3. The future value of reinvested coupons will be ($100 3 1.07) 1 $100 5 $207
4. The 2-year return is $207 1 ($1,187.44 2 $1,091.29)
$1,091.29 5 0.278, or 27.8%
The annualized rate of return over the 2-year period would then be (1.278) 1/2 2 1 5 0.13, or 13%.
Example 16.6 Horizon Analysis
What will be the rate of return in Example 16.6 if the manager forecasts that in 2 years the yield on 18-year bonds will be 10%, and that the reinvestment rate for coupons will be 8%?
CONCEPT CHECK 16.8
1. Even default-free bonds such as Treasury issues are subject to interest rate risk. Longer-term bonds generally are more sensitive to interest rate shifts than are short-term bonds. A measure of the aver-
age life of a bond is Macaulay’s duration, defined as the weighted average of the times until each
payment made by the security, with weights proportional to the present value of the payment.
2. Duration is a direct measure of the sensitivity of a bond’s price to a change in its yield. The pro- portional change in a bond’s price equals the negative of duration multiplied by the proportional
change in 1 1 y.
3. Convexity refers to the curvature of a bond’s price-yield relationship. Accounting for convexity can substantially improve on the accuracy of the duration approximation for the response of bond
prices to changes in yields.
4. Immunization strategies are characteristic of passive fixed-income portfolio management. Such strategies attempt to render the individual or firm immune from movements in interest rates. This
may take the form of immunizing net worth or, instead, immunizing the future accumulated value
of a fixed-income portfolio.
SUMMARY
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546 P A R T I V Fixed-Income Securities
5. Immunization of a fully funded plan is accomplished by matching the durations of assets and liabilities. To maintain an immunized position as time passes and interest rates change, the
portfolio must be periodically rebalanced. Classic immunization also depends on parallel shifts
in a flat yield curve. Given that this assumption is unrealistic, immunization generally will be
less than complete. To mitigate the problem, multifactor duration models can be used to allow for
variation in the shape of the yield curve.
6. A more direct form of immunization is dedication, or cash flow matching. If a portfolio is per- fectly matched in cash flow with projected liabilities, rebalancing will be unnecessary.
7. Active bond management consists of interest rate forecasting techniques and intermarket spread analysis. One popular taxonomy classifies active strategies as substitution swaps, intermarket
spread swaps, rate anticipation swaps, and pure yield pickup swaps.
8. Horizon analysis is a type of interest rate forecasting. In this procedure the analyst forecasts the position of the yield curve at the end of some holding period, and from that yield curve predicts
corresponding bond prices. Bonds then can be ranked according to expected total returns (coupon
plus capital gain) over the holding period.
Related Web sites for this chapter are available at www. mhhe.com/bkm
Macaulay’s duration
modified duration
convexity
effective duration
immunization
rebalancing
cash flow matching
dedication strategy
substitution swap
intermarket spread swap
rate anticipation swap
pure yield pickup swap
tax swap
horizon analysis
KEY TERMS
KEY EQUATIONS Macaulay’s duration: D 5 a
T
t51 t 3 wt
Modified duration and bond price risk: DP
P 5 2D 3 BD(1 1 y)
1 1 y R
Duration of perpetuity 5 1 1 y
y
Bond price risk including convexity: DP
P 5 2D*Dy 1 1/2 3 Convexity 3 (Dy)2
Effective duration 5 2 DP/P
Dr
PROBLEM SETS 1. Prices of long-term bonds are more volatile than prices of short-term bonds. However, yields to maturity of short-term bonds fluctuate more than yields of long-term bonds. How do you recon-
cile these two empirical observations?
2. How can a perpetuity, which has an infinite maturity, have a duration as short as 10 or 20 years?
3. A 9-year bond has a yield of 10% and a duration of 7.194 years. If the market yield changes by 50
basis points, what is the percentage change in the bond’s price?
4. Find the duration of a 6% coupon bond making annual coupon payments if it has 3 years until maturity and has a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?
5. Find the duration of the bond in Problem 4 if the coupons are paid semiannually.
Basic Intermediate
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C H A P T E R 1 6 Managing Bond Portfolios 547
6. The historical yield spread between AAA bonds and Treasury bonds widened dramatically during
the financial crisis in 2008. If you believed that the spread would soon return to more typical his-
torical levels, what should you have done? This would be an example of what sort of bond swap?
7. You predict that interest rates are about to fall. Which bond will give you the highest capital
gain?
a. Low coupon, long maturity. b. High coupon, short maturity. c. High coupon, long maturity. d. Zero coupon, long maturity.
8. Rank the durations or effective durations of the following pairs of bonds:
a. Bond A is a 6% coupon bond, with a 20-year time to maturity selling at par value. Bond B is a 6% coupon bond, with a 20-year maturity time selling below par value.
b. Bond A is a 20-year noncallable coupon bond with a coupon rate of 6%, selling at par. Bond B is a 20-year callable bond with a coupon rate of 7%, also selling at par.
9. An insurance company must make payments to a customer of $10 million in 1 year and $4 million
in 5 years. The yield curve is flat at 10%.
a. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
b. What must be the face value and market value of that zero-coupon bond?
10. Long-term Treasury bonds currently are selling at yields to maturity of nearly 6%. You expect
interest rates to fall. The rest of the market thinks that they will remain unchanged over the
coming year. In each question, choose the bond that will provide the higher holding-period
return over the next year if you are correct. Briefly explain your answer.
a. i. A Baa-rated bond with coupon rate 6% and time to maturity 20 years. ii. An Aaa-rated bond with coupon rate of 6% and time to maturity 20 years.
b. i. An A-rated bond with coupon rate 3% and maturity 20 years, callable at 105. ii. An A-rated bond with coupon rate 6% and maturity 20 years, callable at 105.
c. i. A 4% coupon noncallable T-bond with maturity 20 years and YTM 5 6%. ii. A 7% coupon noncallable T-bond with maturity 20 years and YTM 5 6%.
11. Currently, the term structure is as follows: 1-year zero-coupon bonds yield 7%, 2-year bonds
yield 8%, 3-year bonds and longer-maturity bonds all yield 9%. You are choosing between 1-,
2-, and 3-year maturity bonds all paying annual coupons of 8%. Which bond should you buy if
you strongly believe that at year-end the yield curve will be flat at 9%?
12. You will be paying $10,000 a year in tuition expenses at the end of the next 2 years. Bonds
currently yield 8%.
a. What is the present value and duration of your obligation? b. What maturity zero-coupon bond would immunize your obligation? c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now
suppose that rates immediately increase to 9%. What happens to your net position, that is,
to the difference between the value of the bond and that of your tuition obligation? What if
rates fall to 7%?
13. Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely,
the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a
pension fund with obligations to make perpetual payments of $2 million per year to beneficiar-
ies. The yield to maturity on all bonds is 16%.
a. If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is 4 years and the duration of 20-year maturity bonds with coupon rates of 6% (paid annually) is
11 years, how much of each of these coupon bonds (in market value) will you want to hold
to both fully fund and immunize your obligation?
b. What will be the par value of your holdings in the 20-year coupon bond?
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14. You are managing a portfolio of $1 million. Your target duration is 10 years, and you can choose
from two bonds: a zero-coupon bond with maturity of 5 years, and a perpetuity, each currently
yielding 5%.
a. How much of each bond will you hold in your portfolio? b. How will these fractions change next year if target duration is now 9 years?
15. My pension plan will pay me $10,000 once a year for a 10-year period. The first payment will
come in exactly 5 years. The pension fund wants to immunize its position.
a. What is the duration of its obligation to me? The current interest rate is 10% per year. b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position,
how much money ought to be placed in each bond? What will be the face value of the hold- ings in each zero?
16. A 30-year maturity bond making annual coupon payments with a coupon rate of 12% has dura-
tion of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Use a financial calculator or spreadsheet to find the price of the bond if its yield to maturity falls
to 7% or rises to 9%. What prices for the bond at these new yields would be predicted by the
duration rule and the duration-with-convexity rule? What is the percentage error for each rule?
What do you conclude about the accuracy of the two rules?
17. Frank Meyers, CFA, is a fixed-income portfolio manager for a large pension fund. A member
of the Investment Committee, Fred Spice, is very interested in learning about the management of
fixed-income portfolios. Spice has approached Meyers with several questions. Specifically,
Spice would like to know how fixed-income managers position portfolios to capitalize on their
expectations of future interest rates.
Meyers decides to illustrate fixed-income trading strategies to Spice using a fixed-rate bond
and note. Both bonds have semiannual coupon periods. Unless otherwise stated all interest rate
changes are parallel. The characteristics of these securities are shown in the following table. He
also considers a 9-year floating-rate bond (floater) that pays a floating rate semiannually and is
currently yielding 5%.
Characteristics of Fixed-Rate Bond and Fixed-Rate Note
Fixed-Rate Bond Fixed-Rate Note
Price 107.18 100.00
Yield to maturity 5.00% 5.00%
Period to maturity 18 8
Modified duration 6.9848 3.5851
Spice asks Meyers about how a fixed-income manager would position his portfolio to capitalize
on expectations of increasing interest rates. Which of the following would be the most appropri-
ate strategy?
a. Shorten his portfolio duration. b. Buy fixed-rate bonds. c. Lengthen his portfolio duration.
18. Spice asks Meyers (see previous problem) to quantify price changes from changes in inter-
est rates. To illustrate, Meyers computes the value change for the fixed-rate note in the table.
Specifically, he assumes an increase in the level of interest rate of 100 basis points. Using the
information in the table, what is the predicted change in the price of the fixed-rate note?
19. A 30-year maturity bond has a 7% coupon rate, paid annually. It sells today for $867.42. A
20-year maturity bond has 6.5% coupon rate, also paid annually. It sells today for $879.50.
A bond market analyst forecasts that in 5 years, 25-year maturity bonds will sell at yields to
maturity of 8% and 15-year maturity bonds will sell at yields of 7.5%. Because the yield curve
is upward sloping, the analyst believes that coupons will be invested in short-term securities at
a rate of 6%. Which bond offers the higher expected rate of return over the 5-year period?
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C H A P T E R 1 6 Managing Bond Portfolios 549
1. a. Explain the impact on the offering yield of adding a call feature to a proposed bond issue. b. Explain the impact on both effective bond duration and convexity of adding a call feature to
a proposed bond issue.
2. a. A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the
predicted change in price using the duration concept?
20. a. Use a spreadsheet to calculate the durations of the two bonds in Spreadsheet 16.1 if the annual interest rate increases to 12%. Why does the duration of the coupon bond fall while
that of the zero remains unchanged? ( Hint: Examine what happens to the weights computed in column F.)
b. Use the same spreadsheet to calculate the duration of the coupon bond if the coupon were 12% instead of 8% and the semiannual interest rate is again 5%. Explain why duration is
lower than in Spreadsheet 16.1 . (Again, start by looking at column F.)
21. a. Footnote 7 presents the formula for the convexity of a bond. Build a spreadsheet to calculate the convexity of a 5-year, 8% coupon bond making annual payments at the initial yield to
maturity of 10%.
b. What is the convexity of a 5-year zero-coupon bond?
22. A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual
yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% cou-
pon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly
identical duration—11.79 years—but considerably higher convexity of 231.2.
a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual per- centage capital loss on each bond? What percentage capital loss would be predicted by the
duration-with-convexity rule?
b. Repeat part ( a ), but this time assume the yield to maturity decreases to 7%. c. Compare the performance of the two bonds in the two scenarios, one involving an increase
in rates, the other a decrease. Based on the comparative investment performance, explain the
attraction of convexity.
d. In view of your answer to ( c ), do you think it would be possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity
if the yields on both bonds always increased or decreased by equal amounts, as in this
example? Would anyone be willing to buy the bond with lower convexity under these
circumstances?
23. A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon pay-
ments coming once annually). The bond sells at par value.
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years).
c. What price would be predicted by the duration rule (Equation 16.3)? What is the percentage error of that rule?
d. What price would be predicted by the duration-with-convexity rule (Equation 16.5)? What is the percentage error of that rule?
24. a. Use a spreadsheet to answer this question and assume the yield curve is flat at a level of 4%. Calculate the convexity of a “bullet” fixed-income portfolio, that is, a portfolio with a single
cash flow. Suppose a single $1,000 cash flow is paid in year 5.
b. Now calculate the convexity of a “barbell” fixed-income portfolio, that is, a portfolio with equal cash flows over time. Suppose the security makes $100 cash flows in each of years
1–9, so that its duration is close to the bullet in part a. c. Do barbells or bullets have greater convexity?
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550 P A R T I V Fixed-Income Securities
b. A 6% coupon bond with semiannual coupons has a convexity (in years) of 120, sells for 80% of par, and is priced at a yield to maturity of 8%. If the YTM increases to 9.5%, what is the
predicted contribution to the percentage change in price due to convexity?
c. A bond with annual coupon payments has a coupon rate of 8%, yield to maturity of 10%, and Macaulay duration of 9 years. What is the bond’s modified duration?
d. When interest rates decline, the duration of a 30-year bond selling at a premium:
i. Increases.
ii. Decreases.
iii. Remains the same.
iv. Increases at first, then declines.
e. If a bond manager swaps a bond for one that is identical in terms of coupon rate, maturity, and credit quality but offers a higher yield to maturity, the swap is:
i. A substitution swap.
ii. An interest rate anticipation swap.
iii. A tax swap.
iv. An intermarket spread swap.
f. Which bond has the longest duration?
i. 8-year maturity, 6% coupon.
ii. 8-year maturity, 11% coupon.
iii. 15-year maturity, 6% coupon.
iv. 15-year maturity, 11% coupon.
3. A newly issued bond has the following characteristics:
Coupon Yield to Maturity Maturity Macaulay Duration
8% 8% 15 years 10 years
a. Calculate modified duration using the information above. b. Explain why modified duration is a better measure than maturity when calculating the bond’s
sensitivity to changes in interest rates.
c. Identify the direction of change in modified duration if:
i. The coupon of the bond were 4%, not 8%.
ii. The maturity of the bond were 7 years, not 15 years.
d. Define convexity and explain how modified duration and convexity are used to approximate the bond’s percentage change in price, given a change in interest rates.
4. Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in 5 years, and have
a 7% annual coupon rate paid semiannually.
a. Calculate each of the following yields:
i. Current yield.
ii. Yield to maturity (to the nearest whole percent, i.e., 3%, 4%, 5%, etc.).
iii. Horizon yield (also called total compound return) for an investor with a 3-year holding
period and a reinvestment rate of 6% over the period. At the end of 3 years the 7% coupon
bonds with 2 years remaining will sell to yield 7%.
b. Cite a major shortcoming for each of the following fixed-income yield measures:
i. Current yield.
ii. Yield to maturity.
iii. Horizon yield (also called total compound return).
5. Sandra Kapple presents Maria VanHusen with a description, given in the following table, of the
bond portfolio held by the Star Hospital Pension Plan. All securities in the bond portfolio are
noncallable U.S. Treasury securities.
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C H A P T E R 1 6 Managing Bond Portfolios 551
Price If Yields Change
Par Value (U.S. $)
Treasury Security
Market Value
(U.S. $) Current
Price
Up 100 Basis
Points
Down 100 Basis
Points Effective Duration
48,000,000 2.375% due 2011 48,667,680 101.391 99.245 103.595 2.15
50,000,000 4.75% due 2036 50,000,000 100.000 86.372 116.887 —
98,000,000 Total Bond Portfolio 98,667,680 — — — —
a. Calculate the effective duration of each of the following:
i. The 4.75% Treasury security due 2036.
ii. The total bond portfolio.
b. VanHusen remarks to Kapple, “If you changed the maturity structure of the bond portfolio to result in a portfolio duration of 5.25 years, the price sensitivity of the portfolio would be iden-
tical to that of a single, noncallable Treasury security that also has a duration of 5.25 years.”
In what circumstance would VanHusen’s remark be correct?
6. One common goal among fixed-income portfolio managers is to earn high incremental returns on
corporate bonds versus government bonds of comparable durations. The approach of some corporate-
bond portfolio managers is to find and purchase those corporate bonds having the largest initial spreads
over comparable-duration government bonds. John Ames, HFS’s fixed-income manager, believes that
a more rigorous approach is required if incremental returns are to be maximized.
The following table presents data relating to one set of corporate/government spread relation-
ships present in the market at a given date:
Bond Rating Initial Spread over
Governments Expected
Horizon Spread Initial Duration Expected Duration 1 Year from Now
Aaa 31 b.p. 31 b.p. 4 years 3.1 years
Aa 40 b.p. 50 b.p. 4 years 3.1 years
Note: 1 b.p. means 1 basis point, or .01%.
a. Recommend purchase of either Aaa or Aa bonds for a 1-year investment horizon given a goal of maximizing incremental returns.
b. Ames chooses not to rely solely on initial spread relationships. His analytical framework con- siders a full range of other key variables likely to impact realized incremental returns, including
call provisions and potential changes in interest rates. Describe variables, in addition to those
identified above, that Ames should include in his analysis and explain how each of these could
cause realized incremental returns to differ from those indicated by initial spread relationships.
7. Patrick Wall is considering the purchase of one of the two bonds described in the following table.
Wall realizes his decision will depend primarily on effective duration, and he believes that interest
rates will decline by 50 basis points at all maturities over the next 6 months.
Characteristic CIC PTR
Market price 101.75 101.75
Maturity date June 1, 2022 June 1, 2022
Call date Noncallable June 1, 2017
Annual coupon 5.25% 6.35%
Interest payment Semiannual Semiannual
Effective duration 7.35 5.40
Yield to maturity 5.02% 6.10%
Credit rating A A
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552 P A R T I V Fixed-Income Securities
a. Calculate the percentage price change forecasted by effective duration for both the CIC and PTR bonds if interest rates decline by 50 basis points over the next 6 months.
b. Calculate the 6-month horizon return (in percent) for each bond, if the actual CIC bond price equals 105.55 and the actual PTR bond price equals 104.15 at the end of 6 months.
c. Wall is surprised by the fact that although interest rates fell by 50 basis points, the actual price change for the CIC bond was greater than the price change forecasted by effective duration,
whereas the actual price change for the PTR bond was less than the price change forecasted by
effective duration. Explain why the actual price change would be greater for the CIC bond and
the actual price change would be less for the PTR bond.
8. You are the manager for the bond portfolio of a pension fund. The policies of the fund allow for
the use of active strategies in managing the bond portfolio.
It appears that the economic cycle is beginning to mature, inflation is expected to acceler-
ate, and in an effort to contain the economic expansion, central bank policy is moving toward
constraint. For each of the situations below, state which one of the two bonds you would prefer.
Briefly justify your answer in each case.
a. Government of Canada (Canadian pay) 4% due in 2017 and priced at 98.75 to yield 4.50% to maturity.
or
Government of Canada (Canadian pay) 4% due in 2027 and priced at 91.75 to yield 5.19% to
maturity.
b. Texas Power and Light Co. 51/2 due in 2022, rated AAA, and priced at 90 to yield 7.02% to maturity.
or
Arizona Public Service Co. 5.45 due in 2022, rated A–, and priced at 85 to yield 8.05%
to maturity.
c. Commonwealth Edison 23/4 due in 2021, rated Baa, and priced at 81 to yield 7.2% to maturity.
or
Commonwealth Edison 93/8 due in 2021, rated Baa, and priced at 114.40 to yield 7.2%
to maturity.
d. Shell Oil Co. 61/2 sinking fund debentures due in 2027, rated AAA (sinking fund begins September 2013 at par), and priced at 89 to yield 7.1% to maturity.
or
Warner-Lambert 67/8 sinking fund debentures due in 2027, rated AAA (sinking fund begins
April 2020 at par), and priced at 89 to yield 7.1% to maturity.
e. Bank of Montreal (Canadian pay) 5% certificates of deposit due in 2015, rated AAA, and priced at 100 to yield 5% to maturity.
or
Bank of Montreal (Canadian pay) floating-rate note due in 2017, rated AAA. Coupon cur-
rently set at 3.7% and priced at 100 (coupon adjusted semiannually to .5% above the 3-month
Government of Canada Treasury bill rate).
9. A member of a firm’s investment committee is very interested in learning about the management
of fixed-income portfolios. He would like to know how fixed-income managers position portfo-
lios to capitalize on their expectations concerning three factors which influence interest rates:
a. Changes in the level of interest rates. b. Changes in yield spreads across/between sectors. c. Changes in yield spreads as to a particular instrument.
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C H A P T E R 1 6 Managing Bond Portfolios 553
Formulate and describe a fixed-income portfolio management strategy for each of these factors
that could be used to exploit a portfolio manager’s expectations about that factor. ( Note: Three strategies are required, one for each of the listed factors.)
10. Carol Harrod is the investment officer for a $100 million U.S. pension fund. The fixed-
income portion of the portfolio is actively managed, and a substantial portion of the
fund’s large capitalization U.S. equity portfolio is indexed and managed by Webb Street
Advisors.
Harrod has been impressed with the investment results of Webb Street’s equity index strat-
egy and is considering asking Webb Street to index a portion of the actively managed fixed-
income portfolio.
a. Describe advantages and disadvantages of bond indexing relative to active bond management.
b. Webb Street manages indexed bond portfolios. Discuss how an indexed bond portfolio is constructed under stratified sampling (cellular) methods.
c. Describe the main source of tracking error for the cellular method.
11. Janet Meer is a fixed-income portfolio manager. Noting that the current shape of the yield curve
is flat, she considers the purchase of a newly issued, 7% coupon, 10-year maturity, option-free
corporate bond priced at par. The bond has the following features:
Change in Yields
Up 10 Basis Points Down 10 Basis Points
Price 99.29 100.71
Convexity measure 35.00
Convexity adjustment 0.0035
a. Calculate the modified duration of the bond. b. Meer is also considering the purchase of a newly issued, 7.25% coupon, 12-year maturity
option-free corporate bond. She wants to evaluate this second bond’s price sensitivity to an
instantaneous, downward parallel shift in the yield curve of 200 basis points. Based on the
following data, what will be its price change in this yield-curve scenario?
Original issue price Par value, to yield 7.25%
Modified duration (at original price) 7.90
Convexity measure 41.55
Convexity adjustment (yield change of 200 basis points) 1.66
c. Meer asks her assistant to analyze several callable bonds, given the expected downward parallel shift in the yield curve. Meer’s assistant argues that if interest rates fall enough,
convexity for a callable bond will become negative. Is the assistant’s argument correct?
12. Noah Kramer, a fixed-income portfolio manager based in the country of Sevista, is consider-
ing the purchase of a Sevista government bond. Kramer decides to evaluate two strategies for
implementing his investment in Sevista bonds. Table 16A gives the details of the two strategies,
and Table 16B contains the assumptions that apply to both strategies.
Before choosing one of the two bond-investment strategies, Kramer wants to analyze
how the market value of the bonds will change if an instantaneous interest rate shift occurs
immediately after his investment. The details of the interest rate shift are shown in Table 16C .
Calculate, for the instantaneous interest rate shift shown in Table 16C , the percent change in the
market value of the bonds that will occur under each strategy.
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554 P A R T I V Fixed-Income Securities
13. As part of your analysis of debt issued by Monticello Corporation, you are asked to evaluate
two of its bond issues, shown in the following table.
Bond A (Callable)
Bond B (Noncallable)
Maturity 2020 2020
Coupon 11.50% 7.25%
Current price 125.75 100.00
Yield to maturity 7.70% 7.25%
Modified duration to maturity 6.20 6.80
Call date 2014 —
Call price 105 —
Yield to call 5.10% —
Modified duration to call 3.10 —
a. Using the duration and yield information in the table above, compare the price and yield behavior of the two bonds under each of the following two scenarios:
i. Strong economic recovery with rising inflation expectations.
ii. Economic recession with reduced inflation expectations.
b. Using the information in the table, calculate the projected price change for bond B if its yield to maturity falls by 75 basis points.
c. Describe the shortcoming of analyzing bond A strictly to call or to maturity.
Market value of bonds $10 million
Bond maturities 5 and 25 years or 15 years
Bond coupon rates 0.00% (zero coupon)
Target modified duration 15 years
Table 16B
Investment strategy assumptions
Maturity Interest Rate Change
5 year Down 75 basis points (bps)
15 year Up 25 bps
25 year Up 50 bps
Table 16C
Instantaneous interest rate shift immediately after investment
E-INVESTMENTS EXERCISES Go to www.investinginbonds.com/story.asp?id 5 207. Choose the link for the general- purpose bond calculator. The calculator provides yield to maturity, modified duration, and bond convexity as the bond’s price changes. Experiment by trying different inputs. What happens to duration and convexity as coupon increases? As maturity increases? As price increases (holding coupon fixed)?
Table 16A
Investment strategies (amounts are market value invested)
Strategy
5-Year Maturity (Modified
Duration 5 4.83)
15-Year Maturity (Modified
Duration 5 14.35)
25-Year Maturity (Modified
Duration 5 23.81)
I $5 million 0 $5 million
II 0 $10 million 0
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C H A P T E R 1 6 Managing Bond Portfolios 555
SOLUTIONS TO CONCEPT CHECKS 1. Use Spreadsheet 16.1 with a semiannual discount rate of 4.5%.
Period
Time until Payment (Years) Cash Flow
PV of CF (Discount rate 5 4.5% per
period) Weight Weight 3 Time
A. 8% coupon bond 1 0.5 40 38.278 0.0390 0.0195
2 1.0 40 36.629 0.0373 0.0373
3 1.5 40 35.052 0.0357 0.0535
4 2.0 1,040 872.104 0.8880 1.7761
Sum: 982.062 1.0000 1.8864
B. Zero-coupon 1 0.5 0 0.000 0.0000 0.0000
2 1.0 0 0.000 0.0000 0.0000
3 1.5 0 0.000 0.0000 0.0000
4 2.0 1,000 838.561 1.0000 2.0000
Sum: 838.561 1.0000 2.0000
The duration of the 8% coupon bond increases to 1.8864 years. Price increases to $982.062.
The duration of the zero-coupon bond is unchanged at 2 years, although its price also increases
(to $838.561) when the interest rate falls.
2. a. If the interest rate increases from 9% to 9.05%, the bond price falls from $982.062 to $981.177. The percentage change in price is 2 0.0901%.
b. Using the initial semiannual rate of 4.5%, duration is 1.8864 years (see Concept Check 1), so the duration formula would predict a price change of
2 1.8864
1.045 3 .0005 5 2.000903 5 2.0903%
which is almost the same answer that we obtained from direct computation in part ( a ).
3. The duration of a level perpetuity is (1 1 y )/ y or 1 1 1/ y, which clearly falls as y increases. Tabulating duration as a function of y we get
y D
.01 101 years
.02 51
.05 21
.10 11
.20 6
.25 5
.40 3.5
4. In accord with the duration rules presented in the chapter, you should find that duration is shorter
when the coupon rate or yield to maturity is higher. Duration increases with maturity for most
bonds. Duration is shorter when coupons are paid semiannually rather than annually because on
average, payments come earlier. Instead of waiting until year-end to receive the annual coupon,
investors receive half the coupon midyear.
5. Macaulay’s duration is defined as the weighted average of the time until receipt of each bond
payment. Modified duration is defined as Macaulay’s duration divided by 1 1 y (where y is yield per payment period, e.g., a semiannual yield if the bond pays semiannual coupons). One
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556 P A R T I V Fixed-Income Securities
can demonstrate that for a straight bond, modified duration approximately equals the percentage
change in bond price per change in yield. Effective duration captures this last property of modified
duration. It is defined as percentage change in bond price per change in market interest rates. Effective duration for a bond with embedded options requires a valuation method that allows
for such options in computing price changes. Effective duration cannot be related to a weighted
average of times until payments, because those payments are themselves uncertain.
6. The perpetuity’s duration now would be 1.08/.08 5 13.5. We need to solve the following equation
for w:
w 3 2 1 (1 2 w) 3 13.5 5 6
Therefore w 5 .6522.
7. Dedication would be more attractive. Cash flow matching eliminates the need for rebalancing and
thus saves transaction costs.
8. Current price 5 $1,091.29
Forecasted price 5 $100 3 Annuity factor (10%, 18 years) 1 $1,000 3 PV factor (10%, 18 years)
5 $1,000
The future value of reinvested coupons will be ($100 3 1.08) 1 $100 5 $208
The 2-year return is $208 1 ($1,000 2 $1,091.29)
$1,091.29 5 0.107, or 10.7%
The annualized rate of return over the 2-year period would then be (1.107) 1/2
2 1 5 .052, or 5.2%.
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17 7 PA
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THE INTRINSIC VALUE OF A stock depends on the dividend and earnings that can be expected from the firm. This is the heart of fundamental analysis —that is, the analysis of the determinants of value such as earnings prospects. Ultimately, the business success of the firm determines the dividends it can pay to shareholders and the price it will command in the stock market. Because the prospects of the firm are tied to those of the broader economy, however, fundamental analysis must consider the business environment in which the firm operates. For some firms, macroeconomic and industry circumstances might have a greater influence on profits than the firm’s relative performance within its industry. In other words, investors need to keep the big economic picture in mind.
Therefore, in analyzing a firm’s prospects it often makes sense to start with the broad economic environment, examining the state of the aggregate economy and even the
international economy. From there, one con- siders the implications of the outside envi- ronment on the industry in which the firm operates. Finally, the firm’s position within the industry is examined.
This chapter treats the broad-based aspects of fundamental analysis—macroeconomic and industry analysis. The two chapters fol- lowing cover firm-specific analysis. We begin with a discussion of international factors relevant to firm performance, and move on to an overview of the significance of the key variables usually used to summarize the state of the macroeconomy. We then discuss gov- ernment macroeconomic policy. We conclude the analysis of the macroenvironment with a discussion of business cycles. Finally, we move to industry analysis, treating issues concern- ing the sensitivity of the firm to the business cycle, the typical life cycle of an industry, and strategic issues that affect industry performance.
Macroeconomic and
Industry Analysis
CHAPTER SEVENTEEN
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558 P A R T V Security Analysis
17.1 The Global Economy
Stock Market Return, 2012 (%)
In Local Currency In U.S. Dollars
Forecasted growth in GDP, 2013 (%)
Brazil 10.2 0.8 3.5 Britain 8.2 13.3 1.0 Canada 4.9 8.6 2.0 China 3.1 4.2 8.5 France 18.2 20.5 0.0 Germany 31.9 34.5 0.7 Greece 38.3 41.1 2 5.0 Hong Kong 26.5 26.7 2.6 India 27.6 26.7 6.5 Italy 12.0 14.2 2 0.9 Japan 18.0 4.2 0.8 Mexico 19.5 30.9 3.7 Russia 3.7 10.5 3.7 Singapore 21.0 28.5 2.9 South Korea 11.2 20.5 3.4 Spain 2 0.6 1.4 2 1.7 Thailand 37.3 42.7 4.2 U.S. 9.8 9.8 1.0
Source: The Economist, January 5, 2013.
Table 17.1
Economic performance
A top-down analysis of a firm’s prospects must start with the global economy. The interna-
tional economy might affect a firm’s export prospects, the price competition it faces from
competitors, or the profits it makes on investments abroad. Table 17.1 shows the impor-
tance of the global or broad regional macroeconomy to firms’ prospects. Much of Europe
at this time was still mired in the euro zone crisis and was expected to show anemic or even
negative growth. In contrast, other broad regions, such as Asia, were expected to exhibit
very healthy growth rates. The so-called BRICS countries (Brazil, Russia, India, China,
and South Africa), often grouped together because of their rapid recent development, were
by and large expected to continue that performance .
In addition to the variation in regional macroeconomic conditions, economic perfor-
mance varies considerably across countries even within regions. In Europe, the Greek
economy was expected to continue its painful contraction, with a forecast decline in GDP
of 5.0% in 2013, while Germany was expected to show positive, albeit modest, growth.
Perhaps surprisingly, the best stock market returns did not align with the best macro-
economic expectations. This reflects the impact of near market efficiency. China’s growth
was the highest in the Table 17.1 sample, but at the start of the year its growth already was
expected to be high. With actual growth actually a bit lower than prior expectations, stock
market performance was lackluster. In contrast, German growth was tepid, but its stock
market shined as its risks from the euro zone difficulties seemed to subside.
These data illustrate that the national economic environment can be a crucial determi-
nant of industry performance. It is far harder for businesses to succeed in a contracting
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559
economy than in an expanding one. This observation highlights the role of a big-picture
macroeconomic analysis as a fundamental part of the investment process.
In addition, the global environment presents political risks of considerable magnitude.
The euro crisis offers a compelling illustration of the interplay between politics and eco-
nomics. The prospects of a bailout for Greece, as well as support for struggling but much
larger economies such as Spain, have been in large part political issues but with enormous
consequences for the world economy. Similarly, the 2012 debate over the so-called fiscal
cliff in the United States was the stage for pitched political battles with huge economic
consequences. The ongoing political battle over government budget deficits and how to
address them is of tremendous import for the economy. At this level of analysis, it is clear
that politics and economics are intimately entwined.
Other political issues that are less sensational but still extremely important to economic
growth and investment returns include issues of protectionism and trade policy, the free
flow of capital, and the status of a nation’s workforce.
One obvious factor that affects the international competitiveness of a country’s indus-
tries is the exchange rate between that country’s currency and other currencies. The
exchange rate is the rate at which domestic currency can be converted into foreign cur- rency. For example, in early 2013, it took about 88 Japanese yen to purchase 1 U.S. dollar.
We would say that the exchange rate is ¥88 per dollar or, equivalently, $.0144 per yen.
As exchange rates fluctuate, the dollar value of goods priced in foreign currency simi-
larly fluctuates. For example, in 1980, the dollar–yen exchange rate was about $.0045 per
yen. Because the exchange rate in 2013 was $.0114 per yen, a U.S. citizen would need
2.5 times as many dollars in 2013 to buy a product selling for ¥10,000 as would have
been required in 1980. If the Japanese producer were to maintain a fixed yen price for its
product, the price expressed in U.S. dollars would more than double. This would make
Japanese products more expensive to U.S. consumers, however, and result in lost sales.
Obviously, appreciation of the yen creates a problem for Japanese producers that must
compete with U.S. producers.
The nearby box discusses Honda’s response to the dramatic increase in the value of
the yen. It is moving a good part of its manufacturing operations to North America to take
advantage of the reduced cost of production (as measured in yen) in the U.S. and Mexico.
Honda Revs Up Outside Japan
Honda Motor Co. plans to shift a major chunk of its manufacturing to North America over the next two years, bulking up production capacity in the region by as much as 40% to combat a strengthening yen that has made Japanese cars too expensive to export around the world.
The drive to bulk up in North America is led by the yen’s strength against the U.S. dollar, a change that is causing Honda and other Japanese auto makers to lose money on many of the vehicles they now export from Japan. A stron- ger yen erodes the value of dollar-denominated profit and makes exports less price competitive.
Honda, which produced 1.29 million vehicles in North America in 2010, plans to build a new plant in Celaya, Mexico, and expand all seven of its existing assembly plants, aiming to build just short of 2 million cars and trucks a year, Tetsuo Iwamura, president of American Honda, the company’s North American arm, said in an interview with The Wall Street Journal.
The strategic shift is “directly linked to the yen,” Mr. Iwamura said. “It is virtually impossible to make money [on exporting vehicles from Japan] in the short and medium term.”
Honda’s shift is indicative of the broad impact the yen is having on Japanese auto makers facing a currency that has strengthened by nearly 40% in the last four years. The yen was trading at 77.89 to the dollar Tuesday and as recently as 2007 was at 120 to the dollar.
The dramatic strengthening of the yen makes it par- ticularly hard to make money on small cars because profit margins are already thin. To help reduce the number of Fits that Honda exports from Japan, the company recently began shipping Fit cars from China to Canadian dealers as a stopgap measure.
Source: Mike Ramsey and Neal Boudette, “Honda Revs Up Plants in America,” The Wall Street Journal, Dec. 21, 2011. Reprinted with permission. © 2011 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
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560 P A R T V Security Analysis
17.2 The Domestic Macroeconomy
The macroeconomy is the environment in which all firms operate. The importance of the
macroeconomy in determining investment performance is illustrated in Figure 17.2 , which
compares the level of the S&P 500 stock price index to forecasts of earnings per share of
the S&P 500 companies. The graph shows that stock prices tend to rise along with earnings.
While the exact ratio of stock price to earnings varies with factors such as interest rates, risk,
inflation rates, and other variables, the graph does illustrate that as a general rule the ratio
has tended to be in the range of 12 to 25. Given “normal” price–earnings ratios, we would
expect the S&P 500 index to fall within these boundaries. While the earnings- multiplier
rule clearly is not perfect—note the dramatic increase in the price–earnings multiple during
Figure 17.1 Change in real exchange rate: U.S. dollar versus major currencies, 2001–2011
Source: Computed from data in the Economic Report of the President, 2012.
−30.7%
−14.9%
−30.9%
−16.0%
−35% −30% −25% −20% −15% −10% −5% 0%
Canada
Japan
Euro
U.K.
Moreover, by moving some production to North America, Honda diversifies its exposure
to future exchange rate fluctuations.
Figure 17.1 shows the change in the purchasing power of the U.S. dollar relative to
the purchasing power of the currencies of several major industrial countries from 2001
through 2011. The ratio of purchasing powers is called the “real,” or inflation-adjusted,
exchange rate. The change in the real exchange rate measures how much more or less
expensive foreign goods have become to U.S. citizens, accounting for both exchange rate
fluctuations and inflation differentials across countries. A positive value in Figure 17.1
means that the dollar has gained purchasing power relative to another currency; a negative
number indicates a depreciating dollar. The figure demonstrates that in the last decade, the
U.S. dollar has depreciated in real terms relative to each currency in Figure 17.1 . Goods
priced in foreign currencies have become more expensive to U.S. consumers; conversely,
goods priced in U.S. dollars have become more affordable to consumers abroad.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 561
the dot-com boom of the late 1990s—it also seems clear that the level of the broad market
and aggregate earnings do trend together. Thus the first step in forecasting the performance
of the broad market is to assess the status of the economy as a whole.
The ability to forecast the macroeconomy can translate into spectacular investment per-
formance. But it is not enough to forecast the macroeconomy well. You must forecast it
better than your competitors to earn abnormal profits. In this section, we will review some of the key economic statistics used to describe the state of the macroeconomy.
Gross Domestic Product Gross domestic product , or GDP , is the measure of the econo- my’s total production of goods and services. Rapidly growing GDP indicates an expanding
economy with ample opportunity for a firm to increase sales. Another popular measure of
the economy’s output is industrial production. This statistic provides a measure of eco- nomic activity more narrowly focused on the manufacturing side of the economy.
Employment The unemployment rate is the percentage of the total labor force (i.e., those who are either working or actively seeking employment) yet to find work. The unem-
ployment rate measures the extent to which the economy is operating at full capacity. The
unemployment rate is a factor related to workers only, but further insight into the strength
of the economy can be gleaned from the unemployment rate for other factors of produc-
tion. Analysts also look at the factory capacity utilization rate, which is the ratio of actual output from factories to potential output.
Inflation The rate at which the general level of prices rise is called inflation . High rates of inflation often are associated with “overheated” economies, that is, economies where
the demand for goods and services is outstripping productive capacity, which leads to
upward pressure on prices. Most governments walk a fine line in their economic policies.
Figure 17.2 S&P 500 Index versus earnings per share Source: Authors’ calculations using data from The Economic Report of the President.
0
500
1,000
1,500
2,000
2,500
1 9 5 5
1 9 5 9
1 9 6 3
1 9 6 7
1 9 7 1
1 9 7 5
1 9 7 9
1 9 8 3
1 9 8 7
1 9 9 1
1 9 9 5
1 9 9 9
2 0 0 3
2 0 0 7
2 0 1 1
12 x EPS
18 x EPS
25 x EPS
S&P 500
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562 P A R T V Security Analysis
They hope to stimulate their economies enough to maintain nearly full employment, but
not so much as to bring on inflationary pressures. The perceived trade-off between infla-
tion and unemployment is at the heart of many macroeconomic policy disputes. There is
considerable room for disagreement as to the relative costs of these policies as well as the
economy’s relative vulnerability to these pressures at any particular time.
Interest Rates High interest rates reduce the present value of future cash flows, thereby reducing the attractiveness of investment opportunities. For this reason, real interest
rates are key determinants of business investment expenditures. Demand for housing
and high-priced consumer durables such as automobiles, which are commonly financed,
also is highly sensitive to interest rates because interest rates affect interest payments. (In
Chapter 5, Section 5.1, we examined the determinants of interest rates.)
Budget Deficit The budget deficit of the federal government is the difference between government spending and revenues. Any budgetary shortfall must be offset by government
borrowing. Large amounts of government borrowing can force up interest rates by increas-
ing the total demand for credit in the economy. Economists generally believe excessive
government borrowing will “crowd out” private borrowing and investing by forcing up
interest rates and choking off business investment.
Sentiment Consumers’ and producers’ optimism or pessimism concerning the economy is an important determinant of economic performance. If consumers have confidence in
their future income levels, for example, they will be more willing to spend on big-ticket
items. Similarly, businesses will increase production and inventory levels if they anticipate
higher demand for their products. In this way, beliefs influence how much consumption
and investment will be pursued and affect the aggregate demand for goods and services.
Consider an economy where the dominant industry is automobile production for domestic consumption as well as export. Now suppose the auto market is hurt by an increase in the length of time people use their cars before replacing them. Describe the probable effects of this change on ( a ) GDP, ( b ) unemploy- ment, ( c ) the government budget deficit, and ( d ) interest rates.
CONCEPT CHECK 17.1
17.3 Demand and Supply Shocks
A useful way to organize your analysis of the factors that might influence the macro-
economy is to classify any impact as a supply or demand shock. A demand shock is an event that affects the demand for goods and services in the economy. Examples of positive
demand shocks are reductions in tax rates, increases in the money supply, increases in
government spending, or increases in foreign export demand. A supply shock is an event that influences production capacity and costs. Examples of supply shocks are changes in
the price of imported oil; freezes, floods, or droughts that might destroy large quantities of
agricultural crops; changes in the educational level of an economy’s workforce; or changes
in the wage rates at which the labor force is willing to work.
Demand shocks are usually characterized by aggregate output moving in the same direc-
tion as interest rates and inflation. For example, a big increase in government spending will
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 563
tend to stimulate the economy and increase GDP. It also might increase interest rates by
increasing the demand for borrowed funds by the government as well as by businesses that
might desire to borrow to finance new ventures. Finally, it could increase the inflation rate
if the demand for goods and services is raised to a level at or beyond the total productive
capacity of the economy.
Supply shocks are usually characterized by aggregate output moving in the opposite
direction of inflation and interest rates. For example, a big increase in the price of imported
oil will be inflationary because costs of production will rise, which eventually will lead
to increases in prices of finished goods. The increase in inflation rates over the near term
can lead to higher nominal interest rates. Against this background, aggregate output will
be falling. With raw materials more expensive, the productive capacity of the economy
is reduced, as is the ability of individuals to purchase goods at now-higher prices. GDP,
therefore, tends to fall.
How can we relate this framework to investment analysis? You want to identify the
industries that will be most helped or hurt in any macroeconomic scenario you envision.
For example, if you forecast a tightening of the money supply, you might want to avoid
industries such as automobile producers that might be hurt by the likely increase in interest
rates. We caution you again that these forecasts are no easy task. Macroeconomic predic-
tions are notoriously unreliable. And again, you must be aware that in all likelihood your
forecast will be made using only publicly available information. Any investment advantage
you have will be a result only of better analysis—not better information.
17.4 Federal Government Policy
As the previous section would suggest, the government has two broad classes of macro-
economic tools—those that affect the demand for goods and services and those that affect
the supply. For much of postwar history, demand-side policy was of primary interest. The
focus was on government spending, tax levels, and monetary policy. Since the 1980s, how-
ever, increasing attention has been focused on supply-side economics. Broadly interpreted,
supply-side concerns have to do with enhancing the productive capacity of the economy,
rather than increasing the demand for the goods and services the economy can produce.
In practice, supply-side economists have focused on the appropriateness of the incentives
to work, innovate, and take risks that result from our system of taxation. However, issues
such as national policies on education, infrastructure (such as communication and trans-
portation systems), and research and development also are properly regarded as part of
supply-side macroeconomic policy.
Fiscal Policy Fiscal policy refers to the government’s spending and tax actions and is part of “demand- side management.” Fiscal policy is probably the most direct way either to stimulate or
to slow the economy. Decreases in government spending directly deflate the demand for
goods and services. Similarly, increases in tax rates immediately siphon income from con-
sumers and result in fairly rapid decreases in consumption.
Ironically, although fiscal policy has the most immediate impact on the economy, the
formulation and implementation of such policy is usually painfully slow and involved.
This is because fiscal policy requires enormous amounts of compromise between the exec-
utive and legislative branches. Tax and spending policy must be initiated and voted on by
Congress, which requires considerable political negotiations, and any legislation passed
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564
must be signed by the president, requiring more negotiation. Thus, although the impact of
fiscal policy is relatively immediate, its formulation is so cumbersome that fiscal policy
cannot in practice be used to fine-tune the economy.
Moreover, much of government spending, such as that for Medicare or Social Security,
is nondiscretionary, meaning that it is determined by formula rather than policy and cannot
be changed in response to economic conditions. This places even more rigidity into the
formulation of fiscal policy.
A common way to summarize the net impact of government fiscal policy is to look at the
government’s budget deficit or surplus, which is simply the difference between revenues and
expenditures. A large deficit means the government is spending considerably more than it is
taking in by way of taxes. The net effect is to increase the demand for goods (via spending)
by more than it reduces the demand for goods (via taxes), thereby stimulating the economy.
The debate over proper fiscal policy has become increasingly bitter in recent years. The
nearby box summarizes the fiscal cliff impasse of 2012.
Monetary Policy Monetary policy refers to the manipulation of the money supply to affect the macro- economy and is the other main leg of demand-side policy. Monetary policy works largely
through its impact on interest rates. Increases in the money supply lower short-term interest
rates, ultimately encouraging investment and consumption demand. Over longer periods,
however, most economists believe a higher money supply leads only to a higher price level
and does not have a permanent effect on economic activity. Thus the monetary authorities
face a difficult balancing act. Expansionary monetary policy probably will lower interest
rates and thereby stimulate investment and some consumption demand in the short run, but
these circumstances ultimately will lead only to higher prices. The stimulation/inflation
trade-off is implicit in all debate over proper monetary policy.
Fiscal policy is cumbersome to implement but has a fairly direct impact on the econ-
omy, whereas monetary policy is easily formulated and implemented but has a less imme-
diate impact. Monetary policy is determined by the Board of Governors of the Federal
Fiscal Policy and the Fiscal Cliff
The so-called fiscal cliff crisis at the end of 2012 came close to being an unintended experiment in extreme fiscal pol- icy. The seed of the crisis was planted in August 2011 when Congress, until then unable to agree on an expansion of the federal debt ceiling, passed an interim compromise. The debt ceiling would be allowed to rise, but with the pro- vision that Congress would, by the end of 2012, pass legis- lation reducing the federal budget deficit by $1.2 trillion over 10 years. A bipartisan “super-committee” would be formed to propose a package of tax increases and spend- ing reductions. If no agreement were reached, 2013 tax rates would automatically revert to the higher levels pre- vailing during the Clinton presidency, and across-the-board annual spending cuts or “sequestrations” of about $110 billion would be split equally between domestic and defense programs.
This compromise was intended to create a sword of Damocles to hang over Congress’s head, ensuring that it would be forced to arrive at a legislative compromise. If not, then the resulting contractionary fiscal policy, entailing
both steep spending cuts and dramatic tax increases, was widely regarded as sure to lead to another recession. The Congressional Budget Office estimated that the threatened tax increases and spending reductions would act to reduce GDP growth in 2013 from around 1.7% to 2 0.5% and increase the unemployment rate by more than a percentage point. This was the fiscal cliff Congress risked jumping from.
In the end, Congress narrowly avoided jumping by enacting yet another stop-gap compromise. It delayed sequestration by two months, allowing budgetary negotia- tions to continue, and it paid for the cost of that delay by raising some tax rates, including those on capital gains and household income above $450,000.
But the fiscal fight is far from over. The tax increases agreed to at the end of 2012 were far from enough to meet the original target of a $1.2 trillion reduction in the 10-year deficit. Spending cut discussions are still con- tentious. And the government will bump up against the new debt ceiling in early 2013. The fiscal debate continues unabated.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 565
Reserve System. Board members are appointed by the president for 14-year terms and
are reasonably insulated from political pressure. The board is small enough, and often
sufficiently dominated by its chairperson, that policy can be formulated and modulated
relatively easily.
Implementation of monetary policy also is quite direct. The most widely used tool is
the open market operation, in which the Fed buys or sells bonds for its own account. When
the Fed buys securities, it simply “writes a check,” thereby increasing the money supply.
(Unlike us, the Fed can pay for the securities without drawing down funds at a bank account.)
Conversely, when the Fed sells a security, the money paid for it leaves the money supply.
Open market operations occur daily, allowing the Fed to fine-tune its monetary policy.
Other tools at the Fed’s disposal are the discount rate, which is the interest rate it charges
banks on short-term loans, and the reserve requirement, which is the fraction of deposits
that banks must hold as cash on hand or as deposits with the Fed. Reductions in the dis-
count rate signal a more expansionary monetary policy. Lowering reserve requirements
allows banks to make more loans with each dollar of deposits and stimulates the economy
by increasing the effective money supply.
While the discount rate is under the direct control of the Fed, it is changed relatively
infrequently. The federal funds rate is by far the better guide to Federal Reserve policy. The federal funds rate is the interest rate at which banks make short-term, usually overnight,
loans to each other. These loans occur because some banks need to borrow funds to meet
reserve requirements, while other banks have excess funds. Unlike the discount rate, the
fed funds rate is a market rate, meaning that it is determined by supply and demand rather
than being set administratively. Nevertheless, the Federal Reserve Board targets the fed
funds rate, expanding or contracting the money supply through open market operations
as it nudges the fed funds rate to its targeted value. This is the benchmark short-term U.S.
interest rate, and as such it has considerable influence on other interest rates in the United
States and the rest of the world.
Monetary policy affects the economy in a more roundabout way than fiscal policy.
Whereas fiscal policy directly stimulates or dampens the economy, monetary policy works
largely through its impact on interest rates. Increases in the money supply lower interest
rates, which stimulates investment demand. As the quantity of money in the economy
increases, investors will find that their portfolios of assets include too much money. They
will rebalance their portfolios by buying securities such
as bonds, forcing bond prices up and interest rates down.
In the longer run, individuals may increase their holdings
of stocks as well and ultimately buy real assets, which
stimulates consumption demand directly. The ultimate
effect of monetary policy on investment and consump-
tion demand, however, is less immediate than that of fiscal
policy.
Supply-Side Policies Fiscal policy and monetary policy are demand-oriented tools that affect the economy by
stimulating the total demand for goods and services. The implicit belief is that the econ-
omy will not by itself arrive at a full employment equilibrium and that macroeconomic
policy can push the economy toward this goal. In contrast, supply-side policies treat the
issue of the productive capacity of the economy. The goal is to create an environment in
which workers and owners of capital have the maximum incentive and ability to produce
and develop goods.
Suppose the government wants to stimulate the economy without increasing interest rates. What combination of fiscal and monetary policy might accomplish this goal?
CONCEPT CHECK 17.2
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566 P A R T V Security Analysis
Supply-side economists also pay considerable attention to tax policy. Whereas
demand-siders look at the effect of taxes on consumption demand, supply-siders focus
on incentives and marginal tax rates. They argue that
lowering tax rates will elicit more investment and
improve incentives to work, thereby enhancing eco-
nomic growth. Some go so far as to claim that reduc-
tions in tax rates can lead to increases in tax revenues
because the lower tax rates will cause the economy and
the revenue tax base to grow by more than the tax rate
is reduced.
17.5 Business Cycles
We’ve looked at the tools the government uses to fine-tune the economy, attempting to
maintain low unemployment and low inflation. Despite these efforts, economies repeat-
edly seem to pass through good and bad times. One determinant of the broad asset alloca-
tion decision of many analysts is a forecast of whether the macroeconomy is improving or
deteriorating. A forecast that differs from the market consensus can have a major impact
on investment strategy.
The Business Cycle The economy recurrently experiences periods of expansion and contraction, although the
length and depth of those cycles can be irregular. This recurring pattern of recession and
recovery is called the business cycle . Figure 17.3 presents graphs of several measures of production and output. The production series all show clear variation around a gener-
ally rising trend. The bottom graph of capacity utilization also evidences a clear cyclical
(although irregular) pattern.
The transition points across cycles are called peaks and troughs, indicated by the left
and right edges of the shaded regions in Figure 17.3 . A peak is the transition from the end of an expansion to the start of a contraction. A trough occurs at the bottom of a recession just as the economy enters a recovery. The shaded areas in Figure 17.3 therefore all repre-
sent periods of recession.
As the economy passes through different stages of the business cycle, the relative perfor-
mance of different industry groups might be expected to vary. For example, at a trough,
just before the economy begins to recover from a recession, one would expect that cyclical industries , those with above-average sensitivity to the state of the economy, would tend to outperform other industries. Examples of cyclical industries are producers of durable goods
such as automobiles. Because purchases of these goods can be deferred during a recession,
sales are particularly sensitive to macroeconomic conditions. Other cyclical industries are pro-
ducers of capital goods, that is, goods used by other firms to produce their own products. When
demand is slack, few companies will be expanding and purchasing capital goods. Therefore,
the capital goods industry bears the brunt of a slowdown but does well in an expansion.
In contrast to cyclical firms, defensive industries have little sensitivity to the business cycle. These are industries that produce goods for which sales and profits are least sensitive
to the state of the economy. Defensive industries include food producers and processors,
pharmaceutical firms, and public utilities. These industries will outperform others when
the economy enters a recession.
The cyclical/defensive classification corresponds well to the notion of systematic or
market risk introduced in our discussion of portfolio theory. When perceptions about the
health of the economy become more optimistic, for example, the prices of most stocks will
Large tax cuts in 2001 were followed by relatively rapid growth in GDP. How would demand-side and supply-side economists differ in their inter- pretations of this phenomenon?
CONCEPT CHECK 17.3
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 567
Figure 17.3 Cyclical indicators Source: The Conference Board, Business Cycle Indicators, December 2008. Used with permission.
16,000 4 /5
8
4 /6
0 2 /6
1
1 2 /6
9 1 1 /7
0
1 1 /7
3 3 /7
5
1 /8
0 7 /8
0 7 /8
1 1 1 /8
2
7 /9
0 3 /9
1
3 /0
1 1 1 /0
1
1 2 /0
7
8,000
4,000
2,000 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08
06 08
06 08
06 08
58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04
58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04
58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04
160
120
80
40
120 100
80
60
40
20
100
90
80
70
60
55. Gross Domestic Product-Ann. Rate, Bil. 2000$ Q [C,C,C]
73. Industrial Production Index, Durable Manufacturers [C,C,C] 74. Industrial Production Index, Nondurable Manufacturers [C,L,L]
75. Industrial Production Index, Consumer Goods [C,L,C]
82. Capacity Utilization Rate, Manufacturing Sector-Percent [L,C,L]
increase as forecasts of profitability rise. Because the cyclical firms are most sensitive to
such developments, their stock prices will rise the most. Thus firms in cyclical industries
will tend to have high-beta stocks. In general, then, stocks of cyclical firms will show
the best results when economic news is positive but the worst results when that news is
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568 P A R T V Security Analysis
bad. Conversely, defensive firms will have low betas and performance that is relatively
unaffected by overall market conditions.
If your assessments of the state of the business cycle were reliably more accurate than
those of other investors, you would simply choose cyclical industries when you are rela-
tively more optimistic about the economy and defensive firms when you are relatively
more pessimistic. Unfortunately, it is not so easy to determine when the economy is
passing through a peak or a trough. It if were, choosing between cyclical and defensive
industries would be easy. As we know from our discussion of efficient markets, however,
attractive investment choices will rarely be obvious. It usually is not apparent that a reces-
sion or expansion has started or ended until several months after the fact. With hindsight,
the transitions from expansion to recession and back might be apparent, but it is often quite
difficult to say whether the economy is heating up or slowing down at any moment.
Economic Indicators Given the cyclical nature of the business cycle, it is not surprising that to some extent
the cycle can be predicted. A set of cyclical indicators computed by the Conference
Board helps forecast, measure, and interpret short-term fluctuations in economic activity.
Leading economic indicators are those economic series that tend to rise or fall in advance of the rest of the economy. Coincident and lagging indicators, as their names suggest, move in tandem with or somewhat after the broad economy.
Ten series are grouped into a widely followed composite index of leading economic
indicators. Similarly, four coincident and seven lagging indicators form separate indexes.
The composition of these indexes appears in Table 17.2 .
Source: The Conference Board, Business Cycle Indicators, November 2012.
A. Leading indicators
1. Average weekly hours of production workers (manufacturing) 2. Initial claims for unemployment insurance 3. Manufacturers’ new orders (consumer goods and materials industries) 4. Fraction of companies reporting slower deliveries 5. New orders for nondefense capital goods 6. New private housing units authorized by local building permits 7. Yield curve slope: 10-year Treasury minus federal funds rate 8. Stock prices, 500 common stocks 9. Money supply (M2) growth rate
10. Index of consumer expectations
B. Coincident indicators
1. Employees on nonagricultural payrolls 2. Personal income less transfer payments 3. Industrial production 4. Manufacturing and trade sales
C. Lagging indicators
1. Average duration of unemployment 2. Ratio of trade inventories to sales 3. Change in index of labor cost per unit of output 4. Average prime rate charged by banks 5. Commercial and industrial loans outstanding 6. Ratio of consumer installment credit outstanding to personal income 7. Change in consumer price index for services
Table 17.2
Indexes of economic indicators
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 569
Figure 17.4 graphs these three series. The dates at the top of the chart correspond to
turning points between expansions and contractions. While the index of leading indica-
tors consistently turns before the rest of the economy, its lead time is somewhat erratic.
Moreover, the lead time for peaks is consistently longer than that for troughs.
The stock market price index is a leading indicator. This is as it should be, as stock
prices are forward-looking predictors of future profitability. Unfortunately, this makes the
series of leading indicators much less useful for investment policy—by the time the series
predicts an upturn, the market has already made its move. Although the business cycle may
be somewhat predictable, the stock market may not be. This is just one more manifestation
of the efficient markets hypothesis.
The money supply is another leading indicator. This makes sense in light of our earlier
discussion concerning the lags surrounding the effects of monetary policy on the economy.
Figure 17.4 Indexes of leading, coincident, and lagging indicators Note: Shaded areas represent recessions.
Source: The Conference Board, Business Cycle Indicators, December 2008. Used with permission.
120
100
80
60
40
58 646260 66 7068 7472 807876 8482 908886 969492 0098 08060402
920. U.S. Composite Index of 4 Coincident Indicators 120
80
40 0
0
−2 +1
+3 +15
+13 +3 +2
+3 +6
+6
+16
+11
+10 +6
+1 +1
+3
+1
0
−3
0 0
58 646260 66 7068 7472 807876 8482 908886 969492 0098 08060402
930. U.S. Composite Index of 7 Lagging Indicators
+17
−3
58 646260 66 7068 7472 807876 8482 908886 969492 0098 08060402
1 2 /6
9 1 1 /7
0
1 1 /7
3 3 /7
5
1 /8
0 7 /8
0 7 /8
1 1 1 /8
2
7 /9
0 3 /9
1
3 /0
1 1 1 /0
1
1 2 /0
7
910. U.S. Composite Index of 10 Leading Indicators 120
80
40
4 /5
8
4 /6
0 2 /6
1
−9 −11
−8 −9 −15
−11 −7
−2
−10 −2−7
−2 −8
−18
0
0
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570 P A R T V Security Analysis
An expansionary monetary policy can be observed fairly quickly, but it might not affect the
economy for several months. Therefore, today’s monetary policy might well predict future
economic activity.
Other leading indicators focus directly on decisions made today that will affect production
in the near future. For example, manufacturers’ new orders for goods, contracts and orders
for plant and equipment, and housing starts all signal a coming expansion in the economy.
A wide range of economic indicators is released to the public on a regular “economic
calendar.” Table 17.3 is an “economic calendar,” listing the public announcement dates and
sources for about 20 statistics of interest. These announcements are reported in the finan-
cial press, for example, The Wall Street Journal, as they are released. They also are avail- able at many sites on the Web, for example, at the Yahoo! Finance Web site. Figure 17.5
Statistic Release Date * Source Web Site
Auto and truck sales 2nd of month Commerce Department commerce.gov Business inventories 15th of month Commerce Department commerce.gov Construction spending 1st business day of month Commerce Department commerce.gov Consumer confidence Last Tuesday of month Conference Board conference-board.org Consumer credit 5th business day of month Federal Reserve Board federalreserve.gov Consumer price index (CPI) 13th of month Bureau of Labor Statistics bls.gov Durable goods orders 26th of month Commerce Department commerce.gov Employment cost index End of first month of quarter Bureau of Labor Statistics bls.gov Employment record (unemployment, average workweek, nonfarm payrolls)
1st Friday of month Bureau of Labor Statistics bls.gov
Existing home sales 25th of month National Association of Realtors
realtor.org
Factory orders 1st business day of month Commerce Department commerce.gov Gross domestic product 3rd–4 th week of month Commerce Department commerce.gov Housing starts 16th of month Commerce Department commerce.gov Industrial production 15th of month Federal Reserve Board federalreserve.gov Initial claims for jobless benefits
Thursdays Department of Labor dol.gov
International trade balance 20th of month Commerce Department commerce.gov Index of leading economic indicators
Beginning of month Conference Board conference-board.org
Money supply Thursdays Federal Reserve Board federalreserve.gov New home sales Last business day of month Commerce Department commerce.gov Producer price index 11th of month Bureau of Labor Statistics bls.gov Productivity and costs 2nd month in quarter
(approx. 7th day of month) Bureau of Labor Statistics bls.gov
Retail sales 13th of month Commerce Department commerce.gov Survey of purchasing managers
1st business day of month Institute for Supply Management
ism.ws
Table 17.3
Economic calendar *Many of these release dates are approximate.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 571
is a brief excerpt from the Economic Calendar page at Yahoo! The page gives a list of the
announcements released the week of January 3, 2013. Notice that recent forecasts of each
variable are provided along with the actual value of each statistic. This is useful, because
in an efficient market, security prices already will reflect market expectations. The new information in the announcement will determine the market response.
Other Indicators You can find lots of important information about the state of the economy from sources
other than the official components of the economic calendar or the components of business
cycle indicators. Table 17.4, which is derived from some suggestions in Inc. magazine, 1 contains a few.
1 Gene Sperling, “The Insider’s Guide to Economic Forecasting,” Inc., August 2003, p. 96.
Figure 17.5 Economic calendar at Yahoo!, week of January 3, 2013 Source: Yahoo! Finance, finance.yahoo.com, January 3, 2013.
Jan 4
Jan 4
Jan 4
Jan 4
PriorActualForStatisticDate
Unemployment rate
Hourly earnings
8:30 AM
8:30 AM
8:30 AM
8:30 AM
10:00 AM
10:00 AM
Dec
Dec
Dec
Dec
Nov
Dec
165K
7.7%
0.2%
34.5
0.5%
53.5
145K
7.7%
0.2%
34.5
0.5%
53.5
Jan 4
Jan 4
Factory orders
ISM services
Average workweek
Time (ET)
Briefing Forecast
Market Expects
Revised From
Nonfarm private payrolls 168K
34.5
7.8%
0.3%
0.0%
56.1
171K
34.4
7.8%
0.3%
0.8%
54.7
147K
7.7%
0.2%
–
–
–
Economic Calendar
Week of December 31, 2012
17.6 Industry Analysis
Industry analysis is important for the same reason that macroeconomic analysis is. Just as
it is difficult for an industry to perform well when the macroeconomy is ailing, it is unusual
for a firm in a troubled industry to perform well. Similarly, just as we have seen that eco-
nomic performance can vary widely across countries, performance also can vary widely
across industries. Figure 17.6 illustrates the dispersion of industry performance. It shows
return on equity based on 2012 profitability for several major industry groups. ROE ranged
from 6.7% for money center banks to 29.6% in the restaurant industry.
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572 P A R T V Security Analysis
CEO polls www.businessroundtable.org
The business roundtable surveys CEOs about planned capital spending, a good measure of their optimism about the economy.
Temp jobs (search for “Temporary Help Services”) www.bls.gov
A useful leading indicator. Businesses often hire temporary work- ers as the economy first picks up, until it is clear that an upturn is going to be sustained. This series is available at the Bureau of Labor Statistics Web site.
Walmart sales www.walmartstores.com
Walmart sales are a good indicator of the retail sector. It publishes its same-store sales weekly.
Commercial and industrial loans www.federalreserve.gov
These loans are used by small and medium-sized firms. Informa- tion is published weekly by the Federal Reserve.
Semiconductors www.semi.org
The book-to-bill ratio (i.e., new sales versus actual shipments) indicates whether demand in the technology sector is increasing (ratio . 1) or falling. This ratio is published by Semiconductor Equipment and Materials International.
Commercial structures http://bea.doc.gov
Investment in structures is an indicator of businesses’ forecasts of demand for their products in the near future. This series is com- piled by the Bureau of Economic Analysis as part of its GDP series.
Table 17.4
Useful economic indicators
Figure 17.6 Return on equity by industry, 2012 Source: Yahoo! Finance, finance.yahoo.com, September 12, 2012.
0 5 10 15 20 25 30
Restaurants
Rate of Return (%)
Application Software
Chemical Products
Health Care Plans
Aerospace/Defense
Pharmaceuticals
Home Improvement
Computer Systems
Telecom Services
Integrated Oil & Gas
Business Software
Asset Management
Food Products
Biotech
Trucking
Auto Manufacturers
Electric Utilities
Heavy Construction
Money Center Banks
29.6 24.9
19.6
17.2
17.1
15.8
15.6
14.1
14.1
11.4
11.4
10.3
9.0
8.7
8.5
7.9
7.1
7.0
6.7
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 573
Figure 17.7 Industry stock price performance, 2012
0 10 20 30 40 50 60
Restaurants
Application Software
Chemical Products
Health Care Plans
Aerospace/Defense
Pharmaceuticals
Home Improvement
Computer Systems
Telecom Services
Integrated Oil & Gas
Business Software
Asset Management
Food Products
Biotech
Trucking
Auto Manufacturers
Electric Utilities
Heavy Construction
Money Center Banks
7.0
5.5
18.1
30.0
21.4
13.5
14.3
57.3 19.8
18.6
2.6
15.7
28.3
15.4
36.6
11.1
26.2
38.0
22.4
Rate of Return (%)
Given the wide variation in profitability, it is not surprising that industry groups
exhibit considerable dispersion in their stock market performance. Figure 17.7 presents
the stock market performance of the same industries included in Figure 17.6 . The spread
in performance is remarkable, ranging from a 57.3% gain in the home improvement
industry to only 2.6% in the oil and gas industry. This range of performance was very
much available to virtually all investors in 2012. Recall that iShares are exchange-traded
funds (see Chapter 4) that trade like stocks and thus allow even small investors to take
a position in each traded industry. Alternatively, one can invest in mutual funds with an
industry focus. For example, Fidelity offers over 40 sector funds, each with a particular
industry focus.
Defining an Industry Although we know what we mean by an “industry,” deciding where to draw the line
between one industry and another can be difficult in practice. Consider, for example, one
of the industries depicted in Figure 17.6 , application software firms. Industry ROE in 2012
was 24.9%. But there is substantial variation within this group by focus, and one might
well be justified in further dividing these firms into distinct subindustries. Their differ-
ences may result in considerable dispersion in financial performance. Figure 17.8 shows
ROE for a sample of the firms included in this industry, confirming that 2012 performance
did indeed vary widely: from 7.9% for Nuance to 30.3% for Intuit.
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574 P A R T V Security Analysis
A useful way to define industry groups in practice is given by the North American
Industry Classification System, or NAICS codes . 2 These are codes assigned to group firms for statistical analysis. The first two digits of the NAICS codes denote very broad industry
classifications. For example, Table 17.5 shows that the codes for all construction firms start
with 23. The next digits define the industry grouping more narrowly. For example, codes
starting with 236 denote building construction, 2361 denotes residential construction, and 236115 denotes single-family construction. The first five digits of the NAICS codes are common across all three NAFTA countries. The sixth digit is country-specific and allows
for a finer partition of industries. Firms with the same four-digit NAICS codes are
commonly taken to be in the same industry.
NAICS industry classifications are not perfect. For example, both JCPenney and Neiman
Marcus might be classified as “Department Stores.” Yet the former is a high-volume
“value” store, whereas the latter is a high-margin elite retailer. Are they really in the same
industry? Still, these classifications are a tremendous aid in conducting industry analysis
because they provide a means of focusing on very broad or fairly narrowly defined groups
of firms.
Several other industry classifications are provided by other analysts; for example,
Standard & Poor’s reports on the performance of about 100 industry groups. S&P com-
putes stock price indexes for each group, which is useful in assessing past investment
Figure 17.8 ROE of major software development firms Source: Yahoo! Finance, finance.yahoo.com, December 15, 2012.
0 5 10 15 20 25 30 35
Nuance
Adobe
CA (Computer Assoc.)
SAP
Microsoft
Oracle
Intuit
ROE (%)
7.9
13.4
17.0
23.7
24.5
24.7
30.3
2 These codes are used for firms operating inside the NAFTA (North American Free Trade Agreement) region,
which includes the U.S., Mexico, and Canada. NAICS codes replaced the Standard Industry Classification or SIC
codes previously used in the U.S.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 575
performance. The Value Line Investment Survey reports on the conditions and prospects of about 1,700 firms, grouped into about 90 industries. Value Line’s analysts prepare fore-
casts of the performance of industry groups as well as of each firm.
Sensitivity to the Business Cycle Once the analyst forecasts the state of the macroeconomy, it is necessary to determine the
implication of that forecast for specific industries. Not all industries are equally sensitive
to the business cycle.
For example, Figure 17.9 plots changes in retail sales (year over year) in two indus-
tries: jewelry and grocery stores. Clearly, sales of jewelry, which is a luxury good, fluctu-
ate more widely than those of groceries. Jewelry sales jumped in 1999 at the height of
NAICS Code NAICS Title
23 Construction 236 Construction of Buildings 2361 Residential Building Construction 23611 Residential Building Construction 236115 New Single-Family Housing Construction 236116 New Multifamily Housing Construction 236118 Residential Remodelers 2362 Nonresidential Building Construction 23621 Industrial Building Construction 23622 Commercial and Institutional Building Construction
Table 17.5
Examples of NAICS industry codes
Figure 17.9 Industry cyclicality
−15%
−10%
−5%
0%
5%
10%
15%
A n
n u
a l Sa
le s
G ro
w th
( %
)
1 9 9 4
1 9 9 5
1 9 9 6
1 9 9 7
1 9 9 8
1 9 9 9
2 0 0 0
2 0 0 1
2 0 0 2
2 0 0 3
2 0 0 4
2 0 0 5
2 0 0 6
2 0 0 7
2 0 0 8
2 0 0 9
2 0
1 0
Grocery Jewelry
1 9 9 3
2 0
1 1
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576 P A R T V Security Analysis
the dot-com boom but fell steeply in the recessions of 2001 and 2008–2009. In contrast,
sales growth in the grocery industry is relatively stable, with no years in which sales
meaningfully decline. These patterns reflect the fact that jewelry is a discretionary good,
whereas most grocery products are staples for which demand will not fall significantly
even in hard times.
Three factors will determine the sensitivity of a firm’s earnings to the business cycle.
First is the sensitivity of sales. Necessities will show little sensitivity to business condi-
tions. Examples of industries in this group are food, drugs, and medical services. Other
industries with low sensitivity are those for which income is not a crucial determinant of
demand. Tobacco products are an example of this type of industry. Another industry in
this group is movies, because consumers tend to substitute movies for more expensive
sources of entertainment when income levels are low. In contrast, firms in industries such
as machine tools, steel, autos, and transportation are highly sensitive to the state of the
economy.
The second factor determining business cycle sensitivity is operating leverage, which
refers to the division between fixed and variable costs. (Fixed costs are those the firm
incurs regardless of its production levels. Variable costs are those that rise or fall as the
firm produces more or less product.) Firms with greater amounts of variable as opposed to
fixed costs will be less sensitive to business conditions. This is because in economic down-
turns, these firms can reduce costs as output falls in response to falling sales. Profits for
firms with high fixed costs will swing more widely with sales because costs do not move to
offset revenue variability. Firms with high fixed costs are said to have high operating lever-
age, because small swings in business conditions can have large impacts on profitability.
Consider two firms operating in the same industry with identical revenues in all phases of the business cycle: recession, normal, and expansion. Firm A has short-term leases on most of its equipment and can reduce its lease expenditures when production slackens. It has fixed costs of $5 million and variable costs of $1 per unit of output. Firm B has long-term leases on most of its equipment and must make lease payments regardless of economic conditions. Its fixed costs are higher, $8 million, but its variable costs are only $.50 per unit. Table 17.6 shows that Firm A will do better in recessions than Firm B, but not as well in expansions. A’s costs move in conjunction with its revenues to help perfor- mance in downturns and impede performance in upturns.
Example 17.1 Operating Leverage
Recession Normal Expansion
A B A B A B
Sales (million units) 5 5 6 6 7 7 Price per unit $ 2 $ 2 $ 2 $ 2 $ 2 $ 2 Revenue ($ million) 10 10 12 12 14 14 Fixed costs ($ million) 5 8 5 8 5 8 Variable costs ($ million) 5 2.5 6 3 7 3.5 Total costs ($ million) $10 $10.5 $11 $11 $12 $11.5 Profits $ 0 $ (0.5) $ 1 $ 1 $ 2 $ 2.5
Table 17.6
Operating leverage of firms A and B throughout business cycle
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 577
We can quantify operating leverage by measuring how sensitive profits are to changes
in sales. The degree of operating leverage , or DOL, is defined as
DOL 5 Percentage change in profits
Percentage change in sales
DOL greater than 1 indicates some operating leverage. For example, if DOL 5 2, then for
every 1% change in sales, profits will change by 2% in the same direction, either up or down.
We have seen that the degree of operating leverage increases with a firm’s exposure to
fixed costs. In fact, one can show that DOL depends on fixed costs in the following manner: 3
DOL 5 1 1 Fixed costs
Profits
What will be profits in the three scenarios for Firm C with fixed costs of $2 million and variable costs of $1.50 per unit? What are your conclusions regarding operating leverage and business risk?
CONCEPT CHECK 17.4
Return to the two firms illustrated in Table 17.6 and compare profits and sales in the normal scenario for the economy with those in a recession. Profits of Firm A fall by 100% (from $1 million to zero) when sales fall by 16.7% (from $6 million to $5 million):
DOL(Firm A) 5 Percentage change in profits
Percentage change in sales 5 2100% 216.7%
5 6
We can confirm the relationship between DOL and fixed costs as follows:
DOL(Firm A) 5 1 1 Fixed costs
Profits 5 1 1
$5 million $1 million
5 6
Firm B has higher fixed costs, and its operating leverage is higher. Again, compare data for a normal scenario to a recession. Profits for Firm B fall by 150%, from $1 million to 2 $.5 million. Operating leverage for Firm B is therefore
DOL(Firm B) 5 Percentage change in profits
Percentage change in sales 5 2150% 216.7%
5 9
which reflects its higher level of fixed costs:
DOL(Firm B) 5 1 1 Fixed costs
Profits 5 1 1
$8 million $1 million
5 9
Example 17.2 Degree of Operating Leverage
3 Operating leverage and DOL are treated in more detail in most corporate finance texts.
The third factor influencing business cycle sensitivity is financial leverage, which is
the use of borrowing. Interest payments on debt must be paid regardless of sales. They are
fixed costs that also increase the sensitivity of profits to business conditions. (We will have
more to say about financial leverage in Chapter 19.)
Investors should not always prefer industries with lower sensitivity to the business cycle.
Firms in sensitive industries will have high-beta stocks and are riskier. But while they swing
lower in downturns, they also swing higher in upturns. As always, the issue you need to address
is whether the expected return on the investment is fair compensation for the risks borne.
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578 P A R T V Security Analysis
Sector Rotation One way that many analysts think about the relationship between industry analysis and
the business cycle is the notion of sector rotation . The idea is to shift the portfolio more heavily into industry or sector groups that are expected to outperform based on one’s
assessment of the state of the business cycle.
Figure 17.10 is a stylized depiction of the business cycle. Near the peak of the business
cycle, the economy might be overheated with high inflation and interest rates, and price
pressures on basic commodities. This might be a good time to invest in firms engaged in
natural resource extraction and processing such as minerals or petroleum.
Following a peak, when the economy
enters a contraction or recession, one
would expect defensive industries that
are less sensitive to economic conditions,
for example, pharmaceuticals, food, and
other necessities, to be the best perform-
ers. At the height of the contraction,
financial firms will be hurt by shrinking
loan volume and higher default rates.
Toward the end of the recession, however,
contractions induce lower inflation and
interest rates, which favor financial firms.
At the trough of a recession, the
economy is poised for recovery and sub-
sequent expansion. Firms might thus be
spending on purchases of new equipment
to meet anticipated increases in demand.
This, then, would be a good time to invest
in capital goods industries, such as equip-
ment, transportation, or construction.
Finally, in an expansion, the economy
is growing rapidly. Cyclical industries such as
consumer durables and luxury items will be most
profitable in this stage of the cycle. Banks might
also do well in expansions, since loan volume will be
high and default exposure low when the economy is
growing rapidly.
Figure 17.11 illustrates sector rotation. When
investors are relatively pessimistic about the econ-
omy, they will shift into noncyclical industries such
as consumer staples or health care. When antici-
pating an expansion, they will prefer more cyclical
industries such as materials and technology.
Let us emphasize again that sector rotation, like
any other form of market timing, will be successful
only if one anticipates the next stage of the business
cycle better than other investors. The business cycle
depicted in Figure 17.10 is highly stylized. In real
life, it is never as clear how long each phase of the
cycle will last, nor how extreme it will be. These
forecasts are where analysts need to earn their keep.
Economic Activity
Time
Peak
Trough
Expansion
Contraction
Peak
Figure 17.10 A stylized depiction of the business cycle
Figure 17.11 Sector rotation Source: Sam Stovall, BusinessWeek Online, “A Cyclical Take on Performance.” Reprinted with special permission from the July 8, 2004, issue of BusinessWeek. © 2004 McGraw-Hill Companies, Inc.
Energy
Health Care
Consumer Staples
Utilities
FinancialsTechnology
Consumer
Discretionary
Materials
Industrials
Ex pa
ns io
n
Typical Sector Rotation Through an Average
Economic Cycle
Co nt
ra ct
io n
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 579
Industry Life Cycles Examine the biotechnology industry and you will find many firms with high rates of invest-
ment, high rates of return on investment, and low dividend payout rates. Do the same for
the public utility industry and you will find lower rates of return, lower investment rates,
and higher dividend payout rates. Why should this be?
The biotech industry is still new. Recently, available technologies have created opportu-
nities for highly profitable investment of resources. New products are protected by patents,
and profit margins are high. With such lucrative investment opportunities, firms find it
advantageous to put all profits back into the firm. The companies grow rapidly on average.
Eventually, however, growth must slow. The high profit rates will induce new firms to
enter the industry. Increasing competition will hold down prices and profit margins. New
technologies become proven and more predictable, risk levels fall, and entry becomes even
easier. As internal investment opportunities become less attractive, a lower fraction of prof-
its is reinvested in the firm. Cash dividends increase.
Ultimately, in a mature industry, we observe “cash cows,” firms with stable dividends
and cash flows and little risk. Growth rates might be similar to that of the overall economy.
Industries in early states of their life cycles offer high-risk/high-potential-return invest-
ments. Mature industries offer lower-risk, lower-return profiles.
This analysis suggests that a typical industry life cycle might be described by four stages: a start-up stage, characterized by extremely rapid growth; a consolidation stage,
characterized by growth that is less rapid but
still faster than that of the general economy; a
maturity stage, characterized by growth no faster
than the general economy; and a stage of rela-
tive decline, in which the industry grows less
rapidly than the rest of the economy, or actually
shrinks. This industry life cycle is illustrated in
Figure 17.12 . Let us turn to an elaboration of
each of these stages.
Start-Up Stage The early stages of an industry are often characterized by a new tech-
nology or product such as desktop personal com-
puters in the 1980s, cell phones in the 1990s,
or the new generation of smart phones intro-
duced more recently. At this stage, it is difficult
to predict which firms will emerge as industry
leaders. Some firms will turn out to be wildly
In which phase of the business cycle would you expect the following industries to enjoy their best performance?
a. Newspapers
b. Machine tools
c. Beverages
d. Timber
CONCEPT CHECK 17.5
Figure 17.12 The industry life cycle
Sales
Start-up Consol- idation
Maturity Relative Decline
Rapid and Increasing Growth
Stable Growth
Slowing Growth
Minimal or Negative Growth
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580 P A R T V Security Analysis
successful, and others will fail altogether. Therefore, there is considerable risk in selecting
one particular firm within the industry. For example, in the smart phone industry, there is
still a battle among competing technologies, such as Google’s Android phones and Apple’s
iPhone, and it is difficult to predict ultimate market shares.
At the industry level, however, it is clear that sales and earnings will grow at an extremely
rapid rate because the new product has not yet saturated its market. For example, in 2010,
relatively few households had smart phones. The potential market for the product therefore
was huge. In contrast to this situation, consider the market for a mature product like refrig-
erators. Almost all households in the United States already have refrigerators, so the mar-
ket for this good is primarily composed of households replacing old ones. Obviously, the
growth rate in this market in the next decade will be far lower than that for smart phones.
Consolidation Stage After a product becomes established, industry leaders begin to emerge. The survivors from the start-up stage are more stable, and market share is easier
to predict. Therefore, the performance of the surviving firms will more closely track the
performance of the overall industry. The industry still grows faster than the rest of the
economy as the product penetrates the marketplace and becomes more commonly used.
Maturity Stage At this point, the product has reached its full potential for use by consumers. Further growth might merely track growth in the general economy. The prod-
uct has become far more standardized, and producers are forced to compete to a greater
extent on the basis of price. This leads to narrower profit margins and further pressure on
profits. Firms at this stage sometimes are characterized as cash cows, having reasonably
stable cash flow but offering little opportunity for profitable expansion. The cash flow is
best “milked from” rather than reinvested in the company.
We pointed to desktop PCs as a start-up industry in the 1980s. By the mid-1990s it
was a mature industry, with high market penetration, considerable price competition, low
profit margins, and slowing sales. By the 1990s, desktops were progressively giving way
to laptops, which were in their own start-up stage. Within a dozen years, laptops had in
turn entered a maturity stage, with standardization, considerable market penetration, and
dramatic price competition. Today, tablet computers are in a start-up stage.
Relative Decline In this stage, the industry might grow at less than the rate of the overall economy, or it might even shrink. This could be due to obsolescence of the product,
competition from new low-cost suppliers, or competition from new products; consider for
example, the steady displacement of desktops, first by laptops and now by tablets.
At which stage in the life cycle are investments in an industry most attractive?
Conventional wisdom is that investors should seek firms in high-growth industries. This
recipe for success is simplistic, however. If the security prices already reflect the likeli-
hood for high growth, then it is too late to make money from that knowledge. Moreover,
high growth and fat profits encourage competition from other producers. The exploitation
of profit opportunities brings about new sources of supply that eventually reduce prices,
profits, investment returns, and finally growth. This is the dynamic behind the progression
from one stage of the industry life cycle to another. The famous portfolio manager Peter
Lynch makes this point in One Up on Wall Street:
Many people prefer to invest in a high-growth industry, where there’s a lot of sound and
fury. Not me. I prefer to invest in a low-growth industry. . . . In a low-growth industry,
especially one that’s boring and upsets people [such as funeral homes or the oil-drum
retrieval business], there’s no problem with competition. You don’t have to protect your
flanks from potential rivals . . . and this gives you the leeway to continue to grow. 4
4 Peter Lynch with John Rothchild, One Up on Wall Street (New York: Penguin, p. 131).
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 581
In fact, Lynch uses an industry classification system in a very similar spirit to the life-
cycle approach we have described. He places firms in the following six groups:
Slow Growers Large and aging companies that will grow only slightly faster than the broad economy. These firms have matured from their earlier fast-growth phase. They usu-
ally have steady cash flow and pay a generous dividend, indicating that the firm is generat-
ing more cash than can be profitably reinvested in the firm.
Stalwarts Large, well-known firms like Coca-Cola, Hershey’s, or Colgate-Palmolive. They grow faster than the slow growers, but are not in the very rapid growth start-up stage.
They also tend to be in noncyclical industries that are relatively unaffected by recessions.
Fast Growers Small and aggressive new firms with annual growth rates in the neigh- borhood of 20% to 25%. Company growth can be due to broad industry growth or to an
increase in market share in a more mature industry.
Cyclicals These are firms with sales and profits that regularly expand and contract along with the business cycle. Examples are auto companies, steel companies, or the construc-
tion industry.
Turnarounds These are firms that are in bankruptcy or soon might be. If they can recover from what might appear to be imminent disaster, they can offer tremendous investment
returns. A good example of this type of firm would be Chrysler in 1982, when it required a
government guarantee on its debt to avoid bankruptcy. The stock price rose 15-fold in the
next 5 years.
Asset Plays These are firms that have valuable assets not currently reflected in the stock price. For example, a company may own or be located on valuable real estate that is worth
as much as or more than the company’s business enterprises. Sometimes the hidden asset
can be tax-loss carryforwards. Other times the assets may be intangible. For example,
a cable company might have a valuable list of cable subscribers. These assets do not
immediately generate cash flow, and so may be more easily overlooked by other analysts
attempting to value the firm.
Industry Structure and Performance The maturation of an industry involves regular changes in the firm’s competitive environ-
ment. As a final topic, we examine the relationship among industry structure, competitive
strategy, and profitability. Michael Porter has highlighted these five determinants of com-
petition: threat of entry from new competitors, rivalry between existing competitors, price
pressure from substitute products, bargaining power of buyers, and bargaining power of
suppliers. 5
Threat of Entry New entrants to an industry put pressure on price and profits. Even if a firm has not yet entered an industry, the potential for it to do so places pressure on
prices, because high prices and profit margins will encourage entry by new competitors.
Therefore, barriers to entry can be a key determinant of industry profitability. Barriers can
take many forms. For example, existing firms may already have secure distribution chan-
nels for their products based on long-standing relationships with customers or suppliers
5 Michael Porter, Competitive Advantage: Creating and Sustaining Superior Performance (New York: Free Press,
1985).
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582 P A R T V Security Analysis
that would be costly for a new entrant to duplicate. Brand loyalty also makes it difficult
for new entrants to penetrate a market and gives firms more pricing discretion. Propri-
etary knowledge or patent protection also may give firms advantages in serving a market.
Finally, an existing firm’s experience in a market may give it cost advantages due to the
learning that takes place over time.
Rivalry between Existing Competitors When there are several competitors in an industry, there will generally be more price competition and lower profit margins as
competitors seek to expand their share of the market. Slow industry growth contributes to
this competition, because expansion must come at the expense of a rival’s market share.
High fixed costs also create pressure to reduce prices, because fixed costs put greater pres-
sure on firms to operate near full capacity. Industries producing relatively homogeneous
goods are also subject to considerable price pressure, because firms cannot compete on the
basis of product differentiation.
Pressure from Substitute Products Substitute products means that the industry faces competition from firms in related industries. For example, sugar producers compete
with corn syrup producers. Wool producers compete with synthetic fiber producers. The
availability of substitutes limits the prices that can be charged to customers.
Bargaining Power of Buyers If a buyer purchases a large fraction of an indus- try’s output, it will have considerable bargaining power and can demand price concessions.
For example, auto producers can put pressure on suppliers of auto parts. This reduces the
profitability of the auto parts industry.
Bargaining Power of Suppliers If a supplier of a key input has monopolistic control over the product, it can demand higher prices for the good and squeeze profits
out of the industry. One special case of this issue pertains to organized labor as a supplier
of a key input to the production process. Labor unions engage in collective bargaining to
increase the wages paid to workers. When the labor market is highly unionized, a signifi-
cant share of the potential profits in the industry can be captured by the workforce.
The key factor determining the bargaining power of suppliers is the availability of sub-
stitute products. If substitutes are available, the supplier has little clout and cannot extract
higher prices.
1. Macroeconomic policy aims to maintain the economy near full employment without aggravat- ing inflationary pressures. The proper trade-off between these two goals is a source of ongoing
debate.
2. The traditional tools of macropolicy are government spending and tax collection, which consti- tute fiscal policy, and manipulation of the money supply via monetary policy. Expansionary fiscal
policy can stimulate the economy and increase GDP but tends to increase interest rates. Expan-
sionary monetary policy works by lowering interest rates.
3. The business cycle is the economy’s recurring pattern of expansions and recessions. Leading eco- nomic indicators can be used to anticipate the evolution of the business cycle because their values
tend to change before those of other key economic variables.
4. Industries differ in their sensitivity to the business cycle. More sensitive industries tend to be those producing high-priced durable goods for which the consumer has considerable discretion as
to the timing of purchase. Examples are jewelry, automobiles, or consumer durables. Other sensi-
tive industries are those that produce capital equipment for other firms. Operating leverage and
financial leverage increase sensitivity to the business cycle.
Related Web sites for this chapter are available at www. mhhe.com/bkm
SUMMARY
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 583
1. What monetary and fiscal policies might be prescribed for an economy in a deep recession?
2. If you believe the U.S. dollar will depreciate more dramatically than do other investors, what
will be your stance on investments in U.S. auto producers?
3. Choose an industry and identify the factors that will determine its performance in the next
3 years. What is your forecast for performance in that time period?
4. What are the differences between bottom-up and top-down approaches to security valuation?
What are the advantages of a top-down approach?
5. What characteristics will give firms greater sensitivity to business cycles?
6. Unlike other investors, you believe the Fed is going to loosen monetary policy. What would be
your recommendations about investments in the following industries?
a. Gold mining b. Construction
7. According to supply-side economists, what will be the long-run impact on prices of a reduction
in income tax rates?
8. Which of the following is consistent with a steeply upwardly sloping yield curve?
a. Monetary policy is expansive and fiscal policy is expansive. b. Monetary policy is expansive while fiscal policy is restrictive. c. Monetary policy is restrictive and fiscal policy is restrictive.
9. Which of the following is not a governmental structural policy that supply-side economists believe would promote long-term growth in an economy?
a. A redistributive tax system. b. A promotion of competition. c. Minimal government interference in the economy.
10. Consider two firms producing smart phones. One uses a highly automated robotics process,
whereas the other uses workers on an assembly line and pays overtime when there is heavy
production demand.
a. Which firm will have higher profits in a recession? In a boom? b. Which firm’s stock will have a higher beta?
11. Here are four industries and four forecasts for the macroeconomy. Match the industry to the
scenario in which it is likely to be the best performer.
Basic
Intermediate
fundamental analysis
exchange rate
gross domestic product (GDP)
unemployment rate
inflation
budget deficit
demand shock
supply shock
fiscal policy
monetary policy
business cycle
peak
trough
cyclical industries
defensive industries
leading economic indicators
NAICS codes
degree of operating leverage
sector rotation
industry life cycle
KEY TERMS
PROBLEM SETS
Industry Economic Forecast
a. Housing construction b. Health care c. Gold mining d. Steel production
(i) Deep recession: falling inflation, interest rates, and GDP (ii) Superheated economy: rapidly rising GDP, increasing inflation
and interest rates (iii) Healthy expansion: rising GDP, mild inflation, low
unemployment (iv) Stagflation: falling GDP, high inflation
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584 P A R T V Security Analysis
12. In which stage of the industry life cycle would you place the following industries?
( Note: There is considerable room for disagreement concerning the “correct” answers to this question.)
a. Oil well equipment. b. Computer hardware. c. Computer software. d. Genetic engineering. e. Railroads.
13. For each pair of firms, choose the one that you think would be more sensitive to the business
cycle.
a. General Autos or General Pharmaceuticals. b. Friendly Airlines or Happy Cinemas.
14. Why do you think the index of consumer expectations is a useful leading indicator of the macro-
economy? (See Table 17.2 .)
15. Why do you think the change in the index of labor cost per unit of output is a useful lagging
indicator of the macroeconomy? (See Table 17.2 .)
16. General Weedkillers dominates the chemical weed control market with its patented product
Weed-ex. The patent is about to expire, however. What are your forecasts for changes in the
industry? Specifically, what will happen to industry prices, sales, the profit prospects of General
Weedkillers, and the profit prospects of its competitors? What stage of the industry life cycle do
you think is relevant for the analysis of this market?
17. Your business plan for your proposed start-up firm envisions first-year revenues of $120,000,
fixed costs of $30,000, and variable costs equal to one-third of revenue.
a. What are expected profits based on these expectations? b. What is the degree of operating leverage based on the estimate of fixed costs and expected
profits?
c. If sales are 10% below expectation, what will be the decrease in profits? d. Show that the percentage decrease in profits equals DOL times the 10% drop in sales. e. Based on the DOL, what is the largest percentage shortfall in sales relative to original expec-
tations that the firm can sustain before profits turn negative? What are break-even sales at
this point?
f. Confirm that your answer to ( e ) is correct by calculating profits at the break-even level of sales.
Use the following case in answering Problems 18–21: Institutional Advisors for All Inc., or IAAI, is a consulting firm that primarily advises all types of institutions such as foundations, endowments,
pension plans, and insurance companies. IAAI also provides advice to a select group of individual
investors with large portfolios. One of the claims the firm makes in its advertising is that IAAI
devotes considerable resources to forecasting and determining long-term trends; then it uses com-
monly accepted investment models to determine how these trends should affect the performance of
various investments. The members of the research department of IAAI recently reached some con-
clusions concerning some important macroeconomic trends. For instance, they have seen an upward
trend in job creation and consumer confidence and predict that this should continue for the next few
years. Other domestic leading indicators that the research department at IAAI wishes to consider are
industrial production, average weekly hours in manufacturing, S&P 500 stock prices, M2 money
supply, and the index of consumer expectations.
In light of the predictions for job creation and consumer confidence, the investment advisers at
IAAI want to make recommendations for their clients. They use established theories that relate job
creation and consumer confidence to inflation and interest rates and then incorporate the forecast
movements in inflation and interest rates into established models for explaining asset prices. Their
primary concern is to forecast how the trends in job creation and consumer confidence should affect
bond prices and how those trends should affect stock prices.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 585
The members of the research department at IAAI also note that stocks have been trending up
in the past year, and this information is factored into the forecasts of the overall economy that they
deliver. The researchers consider an upward-trending stock market a positive economic indicator in
itself; however, they disagree as to the reason this should be the case.
18. The researchers at IAAI have forecast positive trends for both job creation and consumer confi-
dence. Which, if either, of these trends should have a positive effect on stock prices?
19. Stock prices are useful as a leading indicator. To explain this phenomenon, which of the follow-
ing is most accurate? Stock prices:
a. Predict future interest rates and reflect the trends in other indicators. b. Do not predict future interest rates, nor are they correlated with other leading indicators; the
usefulness of stock prices as a leading indicator is a mystery.
c. Reflect the trends in other leading indicators only and do not have predictive power of their own.
20. Which of the domestic series that the IAAI research department listed for use as leading indica-
tors is least appropriate?
a. Industrial production. b. Manufacturing average weekly hours. c. M2 money supply.
21. IAAI uses primarily historical data in its calculations and forecasts. Which of the following
regarding the actions of IAAI is most accurate?
a. Credit risk premiums may be useful to IAAI because they are based on actual market expectations.
b. IAAI should use a moving average of recent stock returns when times are bad because it will result in a high expected equity risk premium.
c. Long time spans should be used so that regime changes can be factored into the forecasts.
Use the following case in answering Problems 22–25: Mary Smith, a Level II CFA candidate, was recently hired for an analyst position at the Bank of Ireland. Her first assignment is to examine the
competitive strategies employed by various French wineries.
Smith’s report identifies four wineries that are the major players in the French wine industry. Key
characteristics of each are cited in Table 17A . In the body of Smith’s report, she includes a discus-
sion of the competitive structure of the French wine industry. She notes that over the past 5 years, the
French wine industry has not responded to changing consumer tastes. Profit margins have declined
steadily and the number of firms representing the industry has decreased from 10 to 4. It appears that
participants in the French wine industry must consolidate in order to survive.
Smith’s report notes that French consumers have strong bargaining power over the industry. She
supports this conclusion with five key points, which she labels “Bargaining Power of Buyers”:
• Many consumers are drinking more beer than wine with meals and at social occasions.
• Increasing sales over the Internet have allowed consumers to better research the wines, read
opinions from other customers, and identify which producers have the best prices.
• The French wine industry is consolidating and consists of only 4 wineries today compared
to 10 wineries 5 years ago.
South Winery North Winery East Winery West Winery
Founding date 1750 1903 1812 1947
Generic competitive strategy ? Cost leadership Cost leadership Cost leadership
Major customer market (more than 80% concentration)
France France England U.S.
Production site France France France France
Table 17A
Characteristics of Four Major French Wineries
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586 P A R T V Security Analysis
• More than 65% of the business for the French wine industry consists of purchases from restau-
rants. Restaurants typically make purchases in bulk, buying four to fi ve cases of wine at a time.
• Land where the soil is fertile enough to grow grapes necessary for the wine production
process is scarce in France.
After completing the first draft of her report, Smith takes it to her boss, RonVanDriesen, to review.
VanDriesen tells her that he is a wine connoisseur himself, and often makes purchases from the
South Winery. Smith tells VanDriesen, “In my report I have classified the South Winery as a stuck-
in-the-middle firm. It tries to be a cost leader by selling its wine at a price that is slightly below the
other firms, but it also tries to differentiate itself from its competitors by producing wine in bottles
with curved necks, which increases its cost structure. The end result is that the South Winery’s profit
margin gets squeezed from both sides.” VanDriesen replies, “I have met members of the management
team from the South Winery at a couple of the wine conventions I have attended. I believe that the
South Winery could succeed at following both a cost leadership and a differentiation strategy if its
operations were separated into distinct operating units, with each unit pursuing a different competi-
tive strategy.” Smith makes a note to do more research on generic competitive strategies to verify
VanDriesen’s assertions before publishing the final draft of her report.
22. If the French home currency were to greatly appreciate in value compared to the English
currency, what is the likely impact on the competitive position of the East Winery?
a. Make the firm less competitive in the English market. b. No impact, since the major market for East Winery is England, not France. c. Make the firm more competitive in the English market.
23. Which of Smith’s points effectively support the conclusion that consumers have strong bargain-
ing power over the industry?
24. Smith notes in her report that the West Winery might differentiate its wine product on attributes
that buyers perceive to be important. Which of the following attributes would be the most likely area of focus for the West Winery to create a differentiated product?
a. The method of delivery for the product. b. The price of the product. c. A focus on customers aged 30 to 45.
25. Smith knows that a firm’s generic strategy should be the centerpiece of a firm’s strategic plan.
On the basis of a compilation of research and documents, Smith makes three observations about
the North Winery and its strategic planning process:
i. North Winery’s price and cost forecasts account for future changes in the structure of the
French wine industry.
ii. North Winery places each of its business units into one of three categories: build, hold, or
harvest.
iii. North Winery uses market share as the key measure of its competitive position.
Which of these observation(s) least support the conclusion that the North Winery’s strategic planning process is guided and informed by its generic competitive strategy?
1. Briefly discuss what actions the U.S. Federal Reserve would likely take in pursuing an expan- sionary monetary policy using each of the following three monetary tools:
a. Reserve requirements. b. Open market operations. c. Discount rate.
2. An unanticipated expansionary monetary policy has been implemented. Indicate the impact of
this policy on each of the following four variables:
a. Inflation rate. b. Real output and employment. c. Real interest rate. d. Nominal interest rate.
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 587
3. Universal Auto is a large multinational corporation headquartered in the United States. For seg-
ment reporting purposes, the company is engaged in two businesses: production of motor vehicles
and information processing services.
The motor vehicle business is by far the larger of Universal’s two segments. It consists mainly
of domestic U.S. passenger car production, but it also includes small truck manufacturing opera-
tions in the United States and passenger car production in other countries. This segment of Uni-
versal has had weak operating results for the past several years, including a large loss in 2013.
Although the company does not reveal the operating results of its domestic passenger car seg-
ments, that part of Universal’s business is generally believed to be primarily responsible for the
weak performance of its motor vehicle segment.
Idata, the information processing services segment of Universal, was started by Universal
about 15 years ago. This business has shown strong, steady growth that has been entirely internal;
no acquisitions have been made.
An excerpt from a research report on Universal prepared by Paul Adams, a CFA candidate,
states: “Based on our assumption that Universal will be able to increase prices significantly on
U.S. passenger cars in 2014, we project a multibillion dollar profit improvement.”
a. Discuss the concept of an industrial life cycle by describing each of its four phases. b. Identify where each of Universal’s two primary businesses—passenger cars and information
processing—is in such a cycle.
c. Discuss how product pricing should differ between Universal’s two businesses, based on the location of each in the industrial life cycle.
4. Adams’s research report (see the preceding problem) continued as follows: “With a business
recovery already under way, the expected profit surge should lead to a much higher price for
Universal Auto stock. We strongly recommend purchase.”
a. Discuss the business cycle approach to investment timing. (Your answer should describe actions to be taken on both stocks and bonds at different points over a typical business cycle.)
b. Assuming Adams’s assertion is correct (that a business recovery is already under way), evalu- ate the timeliness of his recommendation to purchase Universal Auto, a cyclical stock, based
on the business cycle approach to investment timing.
5. Janet Ludlow is preparing a report on U.S.-based manufacturers in the electric toothbrush indus-
try and has gathered the information shown in Tables 17B and 17C . Ludlow’s report concludes
that the electric toothbrush industry is in the maturity (i.e., late) phase of its industry life cycle.
a. Select and justify three factors from Table 17B that support Ludlow’s conclusion. b. Select and justify three factors from Table 17C that refute Ludlow’s conclusion.
2007 2008 2009 2010 2011 2012
Return on equity
Electric toothbrush industry index 12.5% 12.0% 15.4% 19.6% 21.6% 21.6%
Market index 10.2 12.4 14.6 19.9 20.4 21.2
Average P/E
Electric toothbrush industry index 28.5 3 23.2 3 19.6 3 18.7 3 18.5 3 16.2 3
Market index 10.2 12.4 14.6 19.9 18.1 19.1
Dividend payout ratio
Electric toothbrush industry index 8.8% 8.0% 12.1% 12.1% 14.3% 17.1%
Market index 39.2 40.1 38.6 43.7 41.8 39.1
Average dividend yield
Electric toothbrush industry index 0.3% 0.3% 0.6% 0.7% 0.8% 1.0%
Market index 3.8 3.2 2.6 2.2 2.3 2.1
Table 17B
Ratios for electric toothbrush industry index and broad stock market index
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588 P A R T V Security Analysis
6. As a securities analyst you have been asked to review a valuation of a closely held business,
Wigwam Autoparts Heaven, Inc. (WAH), prepared by the Red Rocks Group (RRG). You are
to give an opinion on the valuation and to support your opinion by analyzing each part of the
valuation. WAH’s sole business is automotive parts retailing. The RRG valuation includes a sec-
tion called “Analysis of the Retail Autoparts Industry,” based completely on the data in Table 17D
and the following additional information:
• WAH and its principal competitors each operated more than 150 stores at year-end 2012.
• The average number of stores operated per company engaged in the retail autoparts industry is 5.3.
• The major customer base for autoparts sold in retail stores consists of young owners of old
vehicles. These owners do their own automotive maintenance out of economic necessity.
a. One of RRG’s conclusions is that the retail autoparts industry as a whole is in the maturity stage of the industry life cycle. Discuss three relevant items of data from Table 17D that sup-
port this conclusion.
b. Another RRG conclusion is that WAH and its principal competitors are in the consolidation stage of their life cycle.
i. Cite three relevant items of data from Table 17D that support this conclusion.
ii. Explain how WAH and its principal competitors can be in a consolidation stage while
their industry as a whole is in the maturity stage.
7. a. If the exchange rate value of the British pound goes from U.S.$1.75 to U.S.$1.55, then the pound has:
i. Appreciated and the British will find U.S. goods cheaper.
ii. Appreciated and the British will find U.S. goods more expensive.
iii. Depreciated and the British will find U.S. goods more expensive.
iv. Depreciated and the British will find U.S. goods cheaper.
b. Changes in which of the following are likely to affect interest rates?
i. Inflation expectations.
ii. Size of the federal deficit.
iii. Money supply.
c. According to the supply-side view of fiscal policy, if the impact on total tax revenues is the same, does it make any difference whether the government cuts taxes by either reducing mar-
ginal tax rates or increasing the personal exemption allowance?
i. No, both methods of cutting taxes will exert the same impact on aggregate supply.
ii. No, people in both cases will increase their saving, expecting higher future taxes, and
thereby offset the stimulus effect of lower current taxes.
• Industry Sales Growth —Industry sales have grown at 15–20% per year in recent years and are expected to grow at 10–15% per year over the next 3 years.
• Non-U.S. Markets —Some U.S. manufacturers are attempting to enter fast-growing non-U.S. markets, which remain largely unexploited.
• Mail Order Sales —Some manufacturers have created a new niche in the industry by selling electric toothbrushes directly to customers through mail order. Sales for this industry segment are growing at 40% per year.
• U.S. Market Penetration —The current penetration rate in the United States is 60% of households and will be difficult to increase.
• Price Competition —Manufacturers compete fiercely on the basis of price, and price wars within the industry are common.
• Niche Markets —Some manufacturers are able to develop new, unexploited niche markets in the United States based on company reputation, quality, and service.
• Industry Consolidation —Several manufacturers have recently merged, and it is expected that consolidation in the industry will increase.
• New Entrants —New manufacturers continue to enter the market.
Table 17C
Characteristics of the electric toothbrush manufacturing industry
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C H A P T E R 1 7 Macroeconomic and Industry Analysis 589
iii. Yes, the lower marginal tax rates alone will increase the incentive to earn marginal income
and thereby stimulate aggregate supply.
iv. Yes, interest rates will increase if marginal tax rates are lowered, whereas they will tend to
decrease if the personal exemption allowance is raised.
Table 17D
Selected retail autoparts industry data
2013 2012 2011 2010 2009 2008 2007 2006 2005 2004
Population 18–29 years old (percentage change) 2 1.8% 2 2.0% 2 2.1% 2 1.4% 2 0.8% 2 0.9% 2 1.1% 2 0.9% 2 0.7% 2 0.3%
Number of households with income more than $35,000 (percentage change) 6.0% 4.0% 8.0% 4.5% 2.7% 3.1% 1.6% 3.6% 4.2% 2.2%
Number of households with income less than $35,000 (percentage change) 3.0% 2 1.0% 4.9% 2.3% 2 1.4% 2.5% 1.4% 2 1.3% 0.6% 0.1%
Number of cars 5–15 years old (percentage change) 0.9% 2 1.3% 2 6.0% 1.9% 3.3% 2.4% 2 2.3% 2 2.2% 2 8.0% 1.6%
Automotive aftermarket industry retail sales (percentage change) 5.7% 1.9% 3.1% 3.7% 4.3% 2.6% 1.3% 0.2% 3.7% 2.4%
Consumer expenditures on automotive parts and accessories (percentage change) 2.4% 1.8% 2.1% 6.5% 3.6% 9.2% 1.3% 6.2% 6.7% 6.5%
Sales growth of retail autoparts companies with 100 or more stores 17.0% 16.0% 16.5% 14.0% 15.5% 16.8% 12.0% 15.7% 19.0% 16.0%
Market share of retail autoparts companies with 100 or more stores 19.0% 18.5% 18.3% 18.1% 17.0% 17.2% 17.0% 16.9% 15.0% 14.0%
Average operating margin of retail autoparts companies with 100 or more stores 12.0% 11.8% 11.2% 11.5% 10.6% 10.6% 10.0% 10.4% 9.8% 9.0%
Average operating margin of all retail autoparts companies 5.5% 5.7% 5.6% 5.8% 6.0% 6.5% 7.0% 7.2% 7.1% 7.2%
E-INVESTMENTS EXERCISES
1. Is the U.S. economy in a recession or not? Check the “official” opinion at the National Bureau of Economic Research (NBER) at www.nber.org/data . Link to the Official Business Cycle Dates. How does the NBER select the beginning or end of a recession (follow the available link for a discussion of this topic)? What period in U.S. economic history was the longest expansion? Contraction? Look at the Announcement Dates section toward the bottom of the page. How much of a time lag is there between when a peak or a trough occurs and when it is announced? What implication does this have for investors?
2. Use data from finance.yahoo.com to answer the following questions.
a. Go to the Investing tab and click on Industries. Find the price/book ratios for Medical Instruments & Supplies and for Electric Utilities. Do the differences make sense in light of their different stages in the industry life cycle?
b. Now look at each industry’s Price/Earnings ratio and Dividend Yield. Again, do the differences make sense in light of their different stages in the industry life cycle?
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590 P A R T V Security Analysis
SOLUTIONS TO CONCEPT CHECKS 1. The downturn in the auto industry will reduce the demand for the product of this economy. The
economy will, at least in the short term, enter a recession. This would suggest that:
a. GDP will fall. b. The unemployment rate will rise. c. The government deficit will increase. Income tax receipts will fall, and government
expenditures on social welfare programs probably will increase.
d. Interest rates should fall. The contraction in the economy will reduce the demand for credit. Moreover, the lower inflation rate will reduce nominal interest rates.
2. Expansionary fiscal policy coupled with expansionary monetary policy will stimulate the
economy, with the loose monetary policy keeping down interest rates.
3. A traditional demand-side interpretation of the tax cuts is that the resulting increase in after-tax
income increased consumption demand and stimulated the economy. A supply-side interpretation
is that the reduction in marginal tax rates made it more attractive for businesses to invest and for
individuals to work, thereby increasing economic output.
4. Firm C has the lowest fixed cost and highest variable costs. It should be least sensitive to the
business cycle. In fact, it is. Its profits are highest of the three firms in recessions but lowest in
expansions.
Recession Normal Expansion
Revenue $10 $12 $14
Fixed cost 2 2 2
Variable cost 7.5 9 10.5
Profits $ 0.5 $ 1 $ 1.5
5. a. Newspapers will do best in an expansion when advertising volume is increasing. b. Machine tools are a good investment at the trough of a recession, just as the economy is about
to enter an expansion and firms may need to increase capacity.
c. Beverages are defensive investments, with demand that is relatively insensitive to the business cycle. Therefore, they are relatively attractive investments if a recession is forecast.
d. Timber is a good investment at a peak period, when natural resource prices are high and the economy is operating at full capacity.
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188 PA
R T
V
AS OUR DISCUSSION of market efficiency indicated, finding undervalued securities is hardly easy. At the same time, there are enough chinks in the armor of the efficient market hypothesis that the search for such securities should not be dismissed out of hand. Moreover, it is the ongoing search for mispriced securities that maintains a nearly efficient market. Even minor mispricing would allow a stock market analyst to earn his salary.
This chapter describes the valuation mod- els that stock market analysts use to uncover mispriced securities. The models presented are those used by fundamental analysts, those analysts who use information concern- ing the current and prospective profitability of a company to assess its fair market value. We start with a discussion of alternative mea- sures of the value of a company. From there,
we progress to quantitative tools called divi- dend discount models, which security ana- lysts commonly use to measure the value of a firm as an ongoing concern. Next we turn to price–earnings, or P/E, ratios, explaining why they are of such interest to analysts but also highlighting some of their shortcomings. We explain how P/E ratios are tied to dividend valuation models and, more generally, to the growth prospects of the firm.
We close the chapter with a discussion and extended example of free cash flow models used by analysts to value firms based on fore- casts of the cash flows that will be generated from the firms’ business endeavors. Finally, we apply the several valuation tools covered in the chapter to a real firm and find some disparity in their conclusions—a conundrum that will confront any security analyst—and consider reasons for these discrepancies.
Equity Valuation Models
CHAPTER EIGHTEEN
18.1 Valuation by Comparables
The purpose of fundamental analysis is to identify stocks that are mispriced relative to some
measure of “true” value that can be derived from observable financial data. There are many con-
venient sources of relevant data. For U.S. companies, the Securities and Exchange Commission
provides information at its EDGAR Web site, www.sec.gov/edgar.shtml . The SEC requires all public companies (except foreign companies and companies with less than $10 million in assets
and 500 shareholders) to file registration statements, periodic reports, and other forms electroni-
cally through EDGAR. Anyone can access and download this information. Many Web sites such
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592 P A R T V Security Analysis
as finance.yahoo.com , money.msn.com , or finance.google.com also provide analysis and data derived from the EDGAR reports.
Table 18.1 shows some financial highlights for Microsoft as well as some comparable
data for other firms in the software applications industry. The price per share of Microsoft’s
common stock is $30.63, and the total market value or capitalization of those shares (called
market cap for short) is $258 billion. Under the heading “Valuation,” Table 18.1 shows the ratio of Microsoft’s stock price to five benchmarks. Its share price is 15.4 times its (per share)
earnings in the most recent 12 months, 3.9 times its recent book value, 3.5 times its sales, and
10.9 times its cash flow. The last valuation ratio, PEG, is the P/E ratio divided by the growth
rate of earnings. We would expect more rapidly growing firms to sell at higher multiples of
current earnings (more on this below), so PEG normalizes the P/E ratio by the growth rate. These valuation ratios are commonly used to assess the valuation of one firm compared
to others in the same industry, and we will consider all of them. The column to the right
gives comparable ratios for other firms in the software applications industry. For example,
an analyst might note that Microsoft’s price/earnings ratio and price/CF ratio are both
below the industry average. Microsoft’s ratio of market value to book value, the net worth of the company as reported on the balance sheet, is also below industry norms, 3.9 versus
10.5. These ratios might indicate that its stock is underpriced. However, Microsoft is a
more mature firm than many in the industry, and perhaps this discrepancy reflects a lower
expected future growth rate. In fact, its PEG ratio is nearly identical to the industry aver-
age. Clearly, rigorous valuation models will be necessary to sort through these sometimes
conflicting signals of value.
Limitations of Book Value Shareholders in a firm are sometimes called “residual claimants,” which means that the
value of their stake is what is left over when the liabilities of the firm are subtracted from
Table 18.1
Financial highlights for Microsoft Corporation, September 12, 2012
Price per share $ 30.63 Common shares outstanding (billion) 8.38 Market capitalization ($ billion) $258 Latest 12 Months Sales ($ billion) $ 73.72 EBITDA ($ billion) $ 30.71 Net income ($ billion) $ 16.98 Earnings per share $ 2.00 Valuation Microsoft Industry Avg Price/Earnings 15.4 17.5 Price/Book 3.9 10.5 Price/Sales 3.5 Price/Cash flow 10.9 20.5 PEG 1.1 1.2 Profitability ROE (%) 27.5 24.9 ROA (%) 15.0 Operating profit margin (%) 37.9 Net profit margin (%) 23.0 23.2
Source: Compiled from data available at finance.yahoo.com , September 12, 2012.
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C H A P T E R 1 8 Equity Valuation Models 593
its assets. Shareholders’ equity is this net worth. However, the values of both assets and
liabilities recognized in financial statements are based on historical—not current—values.
For example, the book value of an asset equals the original cost of acquisition less some adjustment for depreciation, even if the market price of that asset has changed over time.
Moreover, depreciation allowances are used to allocate the original cost of the asset over
several years, but do not reflect loss of actual value.
Whereas book values are based on original cost, market values measure current values of assets and liabilities. The market value of the shareholders’ equity investment equals the
difference between the current values of all assets and liabilities. We’ve emphasized that
current values generally will not match historical ones. Equally or even more important,
many assets, such as the value of a good brand name or specialized expertise developed
over many years, may not even be included on the financial statements. Market prices
therefore reflect the value of the firm as a going concern. It would be unusual if the market
price of a stock were exactly equal to its book value.
Can book value represent a “floor” for the stock’s price, below which level the market
price can never fall? Although Microsoft’s book value per share in 2012 was less than
its market price, other evidence disproves this notion. While it is not common, there are
always some firms selling at a market price below book value. In 2012, for example, such
troubled firms included Sprint/Nextel, Bank of America, Mitsubishi, and AOL.
A better measure of a floor for the stock price is the firm’s liquidation value per share. This represents the amount of money that could be realized by breaking up the firm, selling
its assets, repaying its debt, and distributing the remainder to the shareholders. If the mar-
ket price of equity drops below liquidation value, the firm becomes attractive as a takeover
target. A corporate raider would find it profitable to buy enough shares to gain control and
then actually to liquidate.
Another measure of firm value is the replacement cost of assets less liabilities. Some analysts believe the market value of the firm cannot remain for long too far above its
replacement cost because if it did, competitors would enter the market. The resulting com-
petitive pressure would drive down the market value of all firms until they fell to replace-
ment cost.
This idea is popular among economists, and the ratio of market price to replacement cost
is known as Tobin’s q, after the Nobel Prize–winning economist James Tobin. In the long run, according to this view, the ratio of market price to replacement cost will tend toward
1, but the evidence is that this ratio can differ significantly from 1 for very long periods.
Although focusing on the balance sheet can give some useful information about a firm’s
liquidation value or its replacement cost, the analyst must usually turn to expected future
cash flows for a better estimate of the firm’s value as a going concern. We therefore turn
to the quantitative models that analysts use to value common stock based on forecasts of
future earnings and dividends.
18.2 Intrinsic Value versus Market Price
The most popular model for assessing the value of a firm as a going concern starts from
the observation that an investor in stock expects a return consisting of cash dividends and
capital gains or losses. We begin by assuming a 1-year holding period and supposing that
ABC stock has an expected dividend per share, E ( D 1 ), of $4; the current price of a share, P 0 , is $48; and the expected price at the end of a year, E ( P 1 ), is $52. For now, don’t worry about how you derive your forecast of next year’s price. At this point we ask only whether
the stock seems attractively priced today given your forecast of next year’s price.
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594 P A R T V Security Analysis
The expected holding-period return is E ( D 1 ) plus the expected price appreciation, E ( P 1 ) 2 P 0 , all divided by the current price, P 0 :
Expected HPR 5 E(r) 5 E(D1) 1 3E(P1) 2 P0 4
P0
5 4 1 (52 2 48)
48 5 .167, or 16.7%
Thus, the stock’s expected holding-period return is the sum of the expected dividend
yield, E ( D 1 )/ P 0 , and the expected rate of price appreciation, the capital gains yield, [ E ( P 1 ) 2 P 0 ]/ P 0 .
But what is the required rate of return for ABC stock? The CAPM states that when
stock market prices are at equilibrium levels, the rate of return that investors can expect
to earn on a security is r f 1 b [ E ( r M ) 2 r f ]. Thus, the CAPM may be viewed as providing an estimate of the rate of return an investor can reasonably expect to earn on a security
given its risk as measured by beta. This is the return that investors will require of any other
investment with equivalent risk. We will denote this required rate of return as k. If a stock is priced “correctly,” it will offer investors a “fair” return, that is, its expected return will equal its required return. Of course, the goal of a security analyst is to find stocks that are mispriced. For example, an underpriced stock will provide an expected return greater than
the required return.
Suppose that r f 5 6%, E ( r M ) 2 r f 5 5%, and the beta of ABC is 1.2. Then the value of k is
k 5 6% 1 1.2 3 5% 5 12%
The expected holding-period return, 16.7%, therefore exceeds the required rate of return
based on ABC’s risk by a margin of 4.7%. Naturally, the investor will want to include more
of ABC stock in the portfolio than a passive strategy would indicate.
Another way to see this is to compare the intrinsic value of a share of stock to its market
price. The intrinsic value, denoted V 0 , is defined as the present value of all cash payments to the investor in the stock, including dividends as well as the proceeds from the ultimate
sale of the stock, discounted at the appropriate risk-adjusted interest rate, k. If the intrinsic value, or the investor’s own estimate of what the stock is really worth, exceeds the market
price, the stock is considered undervalued and a good investment. For ABC, using a 1-year
investment horizon and a forecast that the stock can be sold at the end of the year at price
P 1 5 $52, the intrinsic value is
V0 5 E(D1) 1 E(P1)
1 1 k 5
$4 1 $52
1.12 5 $50
Equivalently, at a price of $50, the investor would derive a 12% rate of return—just
equal to the required rate of return—on an investment in the stock. However, at the current
price of $48, the stock is underpriced compared to intrinsic value. At this price, it provides
better than a fair rate of return relative to its risk. Using the terminology of the CAPM, it is
a positive-alpha stock, and investors will want to buy more of it than they would following
a passive strategy.
If the intrinsic value turns out to be lower than the current market price, investors should
buy less of it than under the passive strategy. It might even pay to go short on ABC stock,
as we discussed in Chapter 3.
In market equilibrium, the current market price will reflect the intrinsic value estimates
of all market participants. This means the individual investor whose V 0 estimate differs
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C H A P T E R 1 8 Equity Valuation Models 595
from the market price, P 0 , in effect must disagree with some or all of the market consensus estimates of E ( D 1 ), E ( P 1 ), or k. A common term for the market consensus value of the required rate of return, k, is the market capitalization rate, which we use often through- out this chapter.
You expect the price of IBX stock to be $59.77 per share a year from now. Its current market price is $50, and you expect it to pay a dividend 1 year from now of $2.15 per share.
a. What are the stock’s expected dividend yield, rate of price appreciation, and holding-period return?
b. If the stock has a beta of 1.15, the risk-free rate is 6% per year, and the expected rate of return on the market portfolio is 14% per year, what is the required rate of return on IBX stock?
c. What is the intrinsic value of IBX stock, and how does it compare to the current market price?
CONCEPT CHECK 18.1
18.3 Dividend Discount Models
Consider an investor who buys a share of Steady State Electronics stock, planning to hold
it for 1 year. The intrinsic value of the share is the present value of the dividend to be
received at the end of the first year, D 1 , and the expected sales price, P 1 . We will henceforth use the simpler notation P 1 instead of E ( P 1 ) to avoid clutter. Keep in mind, though, that future prices and dividends are unknown, and we are dealing with expected values, not
certain values. We’ve already established
V0 5 D1 1 P1
1 1 k (18.1)
Although this year’s dividends are fairly predictable given a company’s history, you
might ask how we can estimate P 1 , the year-end price. According to Equation 18.1, V 1 (the year-end intrinsic value) will be
V1 5 D2 1 P2
1 1 k
If we assume the stock will be selling for its intrinsic value next year, then V 1 5 P 1 , and we can substitute this value for P 1 into Equation 18.1 to find
V0 5 D1
1 1 k 1
D2 1 P2 (1 1 k)2
This equation may be interpreted as the present value of dividends plus sales price
for a 2-year holding period. Of course, now we need to come up with a forecast of P 2 . Continuing in the same way, we can replace P 2 by ( D 3 1 P 3 )/(1 1 k ), which relates P 0 to the value of dividends plus the expected sales price for a 3-year holding period.
More generally, for a holding period of H years, we can write the stock value as the present value of dividends over the H years, plus the ultimate sale price, P H :
V0 5 D1
1 1 k 1
D2 (1 1 k)2
1 c1 DH 1 PH (1 1 k)H
(18.2)
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596 P A R T V Security Analysis
Note the similarity between this formula and the bond valuation formula developed
in Chapter 14. Each relates price to the present value of a stream of payments (coupons
in the case of bonds, dividends in the case of stocks) and a final payment (the face value
of the bond, or the sales price of the stock). The key differences in the case of stocks are
the uncertainty of dividends, the lack of a fixed maturity date, and the unknown sales
price at the horizon date. Indeed, one can continue to substitute for price indefinitely, to
conclude
V0 5 D1
1 1 k 1
D2 (1 1 k)2
1 D3
(1 1 k)3 1 c (18.3)
Equation 18.3 states that the stock price should equal the present value of all expected
future dividends into perpetuity. This formula is called the dividend discount model (DDM) of stock prices.
It is tempting, but incorrect, to conclude from Equation 18.3 that the DDM focuses
exclusively on dividends and ignores capital gains as a motive for investing in stock. Indeed,
we assume explicitly in Equation 18.1 that capital gains (as reflected in the expected sales
price, P 1 ) are part of the stock’s value. Our point is that the price at which you can sell a stock in the future depends on dividend forecasts at that time.
The reason only dividends appear in Equation 18.3 is not that investors ignore capital
gains. It is instead that those capital gains will reflect dividend forecasts at the time the
stock is sold. That is why in Equation 18.2 we can write the stock price as the present value
of dividends plus sales price for any horizon date. P H is the present value at time H of all dividends expected to be paid after the horizon date. That value is then discounted back
to today, time 0. The DDM asserts that stock prices are determined ultimately by the cash
flows accruing to stockholders, and those are dividends. 1
The Constant-Growth DDM Equation 18.3 as it stands is still not very useful in valuing a stock because it requires
dividend forecasts for every year into the indefinite future. To make the DDM practical,
we need to introduce some simplifying assumptions. A useful and common first pass at the
problem is to assume that dividends are trending upward at a stable growth rate that we
will call g. For example, if g 5 .05, and the most recently paid dividend was D 0 5 3.81, expected future dividends are
D1 5 D0(1 1 g) 5 3.81 3 1.05 5 4.00
D2 5 D0(1 1 g) 2 5 3.81 3 (1.05)2 5 4.20
D3 5 D0(1 1 g) 3 5 3.81 3 (1.05)3 5 4.41
and so on. Using these dividend forecasts in Equation 18.3, we solve for intrinsic
value as
V0 5 D0(1 1 g)
1 1 k 1
D0(1 1 g) 2
(1 1 k)2 1
D0(1 1 g) 3
(1 1 k)3 1 c
1 If investors never expected a dividend to be paid, then this model implies that the stock would have no value. To
reconcile the DDM with the fact that non-dividend-paying stocks do have a market value, one must assume that
investors expect that some day it may pay out some cash, even if only a liquidating dividend.
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C H A P T E R 1 8 Equity Valuation Models 597
This equation can be simplified to 2
V0 5 D0(1 1 g)
k 2 g 5
D1 k 2 g
(18.4)
Note in Equation 18.4 that we divide D 1 (not D 0 ) by k 2 g to calculate intrinsic value. If the market capitalization rate for Steady State is 12%, we can use Equation 18.4 to show
that the intrinsic value of a share of Steady State stock is
$3.81(1 1 .05)
.12 2 .05 5
$4.00
.12 2 .05 5 $57.14
Equation 18.4 is called the constant-growth DDM, or the Gordon model, after Myron J. Gordon, who popularized the model. It should remind you of the formula for the pres-
ent value of a perpetuity. If dividends were expected not to grow, then the dividend stream
would be a simple perpetuity, and the valuation formula would be 3 V 0 5 D 1 / k. Equation
18.4 is a generalization of the perpetuity formula to cover the case of a growing perpetuity. As g increases (for a given value of D 1 ), the stock price also rises.
2 We prove that the intrinsic value, V 0 , of a stream of cash dividends growing at a constant rate g is equal to
D1 k 2 g
as follows. By definition,
V0 5 D1
1 1 k 1
D1(1 1 g)
(1 1 k)2 1
D1(1 1 g) 2
(1 1 k)3 1 c (a)
Multiplying through by (1 1 k )/(1 1 g ), we obtain
(1 1 k)
(1 1 g) V0 5
D1 (1 1 g)
1 D1
(1 1 k) 1
D1(1 1 g)
(1 1 k)2 1 c (b)
Subtracting equation (a) from equation (b), we find that
1 1 k
1 1 g V0 2 V0 5
D1 (1 1 g)
which implies
(k 2 g)V0 (1 1 g)
5 D1
(1 1 g)
V0 5 D1
k 2 g
3 Recall from introductory finance that the present value of a $1 per year perpetuity is 1/ k. For example, if
k 5 10%, the value of the perpetuity is $1/.10 5 $10. Notice that if g 5 0 in Equation 18.4, the constant-growth DDM formula is the same as the perpetuity formula.
Preferred stock that pays a fixed dividend can be valued using the constant-growth divi- dend discount model. The constant-growth rate of dividends is simply zero. For example, to value a preferred stock paying a fixed dividend of $2 per share when the discount rate is 8%, we compute
V0 5 $2
.08 2 0 5 $25
Example 18.1 Preferred Stock and the DDM
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598 P A R T V Security Analysis
The constant-growth DDM is valid only when g is less than k. If dividends were expected to grow forever at a rate faster than k, the value of the stock would be infinite. If an analyst derives an estimate of g greater than k, that growth rate must be unsustainable in the long run. The appropriate valuation model to use in this case is a multistage DDM such
as those discussed below.
The constant-growth DDM is so widely used by stock market analysts that it is worth
exploring some of its implications and limitations. The constant-growth rate DDM implies
that a stock’s value will be greater:
1. The larger its expected dividend per share.
2. The lower the market capitalization rate, k.
3. The higher the expected growth rate of dividends.
Another implication of the constant-growth model is that the stock price is expected to
grow at the same rate as dividends. To see this, suppose Steady State stock is selling at its
intrinsic value of $57.14, so that V 0 5 P 0 . Then
P0 5 D1
k 2 g
Note that price is proportional to dividends. Therefore, next year, when the dividends
paid to Steady State stockholders are expected to be higher by g 5 5%, price also should increase by 5%. To confirm this, note
D2 5 $4(1.05) 5 $4.20
P1 5 D2
k 2 g 5
$4.20
.12 2 .05 5 $60.00
which is 5% higher than the current price of $57.14. To generalize,
P1 5 D2
k 2 g 5
D1(1 1 g)
k 2 g 5
D1 k 2 g
(1 1 g) 5 P0(1 1 g)
Therefore, the DDM implies that in the case of constant growth of dividends, the
rate of price appreciation in any year will equal that constant-growth rate, g. For a stock
High Flyer Industries has just paid its annual dividend of $3 per share. The dividend is expected to grow at a constant rate of 8% indefinitely. The beta of High Flyer stock is 1.0, the risk-free rate is 6%, and the market risk premium is 8%. What is the intrinsic value of the stock? What would be your estimate of intrinsic value if you believed that the stock was riskier, with a beta of 1.25?
Because a $3 dividend has just been paid and the growth rate of dividends is 8%, the forecast for the year-end dividend is $3 3 1.08 5 $3.24. The market capitalization rate (using the CAPM) is 6% 1 1.0 3 8% 5 14%. Therefore, the value of the stock is
V0 5 D1
k 2 g 5
$3.24 .14 2 .08
5 $54
If the stock is perceived to be riskier, its value must be lower. At the higher beta, the market capitalization rate is 6% 1 1.25 3 8% 5 16%, and the stock is worth only
$3.24
.16 2 .08 5 $40.50
Example 18.2 The Constant-Growth DDM
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C H A P T E R 1 8 Equity Valuation Models 599
whose market price equals its intrinsic value ( V 0 5 P 0 ), the expected holding-period return will be
E(r) 5 Dividend yield 1 Capital gains yield
5 D1 P0 1
P1 2 P0 P0
5 D1 P0 1 g (18.5)
This formula offers a means to infer the market capitalization rate of a stock, for if the
stock is selling at its intrinsic value, then E ( r ) 5 k, implying that k 5 D 1 / P 0 1 g. By observing the dividend yield, D 1 / P 0 , and estimating the growth rate of dividends, we can compute k. This equation is also known as the discounted cash flow (DCF) formula.
This approach is often used in rate hearings for regulated public utilities. The regulatory
agency responsible for approving utility pricing decisions is mandated to allow the firms to
charge just enough to cover costs plus a “fair” profit, that is, one that allows a competitive
return on the investment the firm has made in its productive capacity. In turn, that return is
taken to be the expected return investors require on the stock of the firm. The D 1 / P 0 1 g formula provides a means to infer that required return.
Suppose that Steady State Electronics wins a major contract for its new computer chip. The very profitable contract will enable it to increase the growth rate of dividends from 5% to 6% without reducing the current dividend from the projected value of $4.00 per share. What will happen to the stock price? What will happen to future expected rates of return on the stock?
The stock price ought to increase in response to the good news about the contract, and indeed it does. The stock price jumps from its original value of $57.14 to a post- announcement price of
D1
k 2 g 5
$4.00 .12 2 .06
5 $66.67
Investors who are holding the stock when the good news about the contract is announced will receive a substantial windfall.
On the other hand, at the new price the expected rate of return on the stock is 12%, just as it was before the new contract was announced.
E(r) 5 D1 P0 1 g 5
$4.00 $66.67
1 .06 5 .12, or 12%
This result makes sense. Once the news about the contract is reflected in the stock price, the expected rate of return will be consistent with the risk of the stock. Because the risk of the stock has not changed, neither should the expected rate of return.
Example 18.3 The Constant-Growth Model
a. IBX’s stock dividend at the end of this year is expected to be $2.15, and it is expected to grow at 11.2% per year forever. If the required rate of return on IBX stock is 15.2% per year, what is its intrinsic value?
b. If IBX’s current market price is equal to this intrinsic value, what is next year’s expected price?
c. If an investor were to buy IBX stock now and sell it after receiving the $2.15 dividend a year from now, what is the expected capital gain (i.e., price appreciation) in percentage terms? What are the dividend yield and the holding-period return?
CONCEPT CHECK 18.2
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600 P A R T V Security Analysis
Convergence of Price to Intrinsic Value Now suppose that the current market price of ABC stock is only $48 per share and,
therefore, that the stock is undervalued by $2 per share. In this case the expected rate of
price appreciation depends on an additional assumption about whether the discrepancy
between the intrinsic value and the market price will disappear, and if so, when.
One fairly common assumption is that the discrepancy will never disappear and that the
market price will trend upward at rate g forever. This implies that the discrepancy between intrinsic value and market price also will grow at the same rate. In our example:
Now Next Year
V 0 5 $50 V 1 5 $50 3 1.04 5 $52 P 0 5 $48 P 1 5 $48 3 1.04 5 $49.92 V 0 2 P 0 5 $2 V 1 2 P 1 5 $2 3 1.04 5 $2.08
Under this assumption the expected HPR will exceed the required rate, because the
dividend yield is higher than it would be if P 0 were equal to V 0 . In our example the divi- dend yield would be 8.33% instead of 8%, so that the expected HPR would be 12.33%
rather than 12%:
E(r) 5 D1 P0 1 g 5
$4
$48 1 .04 5 .0833 1 .04 5 .1233
An investor who identifies this undervalued stock can get an expected dividend that
exceeds the required yield by 33 basis points. This excess return is earned each year, and the market price never catches up to intrinsic value.
An alternative assumption is that the gap between market price and intrinsic value will
disappear by the end of the year. In that case we would have P 1 5 V 1 5 $52, and
E(r) 5 D1 P0 1
P1 2 P0 P0
5 4
48 1
52 2 48
48 5 .0833 1 .0833 5 .1667
The assumption of complete catch-up to intrinsic value produces a much larger 1-year
HPR. In future years, however, the stock is expected to generate only fair rates of return.
Many stock analysts assume that a stock’s price will approach its intrinsic value gradu-
ally over time—for example, over a 5-year period. This puts their expected 1-year HPR
somewhere between the bounds of 12.33% and 16.67%.
Stock Prices and Investment Opportunities Consider two companies, Cash Cow, Inc., and Growth Prospects, each with expected earn-
ings in the coming year of $5 per share. Both companies could in principle pay out all
of these earnings as dividends, maintaining a perpetual dividend flow of $5 per share. If
the market capitalization rate were k 5 12.5%, both companies would then be valued at D 1 / k 5 $5/.125 5 $40 per share. Neither firm would grow in value, because with all earn- ings paid out as dividends, and no earnings reinvested in the firm, both companies’ capital
stock and earnings capacity would remain unchanged over time; earnings 4 and dividends
would not grow.
4 Actually, we are referring here to earnings net of the funds necessary to maintain the productivity of the firm’s
capital, that is, earnings net of “economic depreciation.” In other words, the earnings figure should be interpreted
as the maximum amount of money the firm could pay out each year in perpetuity without depleting its productive
capacity. For this reason, the net earnings number may be quite different from the accounting earnings figure that
the firm reports in its financial statements. We explore this further in the next chapter.
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C H A P T E R 1 8 Equity Valuation Models 601
Now suppose one of the firms, Growth Prospects,
engages in projects that generate a return on invest-
ment of 15%, which is greater than the required rate
of return, k 5 12.5%. It would be foolish for such a company to pay out all of its earnings as dividends.
If Growth Prospects retains or plows back some of
its earnings into its profitable projects, it can earn a
15% rate of return for its shareholders, whereas if
it pays out all earnings as dividends, it forgoes the
projects, leaving shareholders to invest the dividends
in other opportunities at a fair market rate of only
12.5%. Suppose, therefore, that Growth Prospects
chooses a lower dividend payout ratio (the fraction of earnings paid out as dividends), reducing payout
from 100% to 40%, maintaining a plowback ratio (the fraction of earnings reinvested in the firm) at
60%. The plowback ratio is also referred to as the
earnings retention ratio. The dividend of the company, therefore, will be $2 (40% of $5 earnings) instead of $5.
Will share price fall? No—it will rise! Although dividends initially fall under the earn-
ings reinvestment policy, subsequent growth in the assets of the firm because of rein-
vested profits will generate growth in future dividends, which will be reflected in today’s
share price.
Figure 18.1 illustrates the dividend streams generated by Growth Prospects under two
dividend policies. A low-reinvestment-rate plan allows the firm to pay higher initial divi-
dends, but results in a lower dividend growth rate. Eventually, a high-reinvestment-rate
plan will provide higher dividends. If the dividend growth generated by the reinvested
earnings is high enough, the stock will be worth more under the high-reinvestment
strategy.
How much growth will be generated? Suppose Growth Prospects starts with plant and
equipment of $100 million and is all equity financed. With a return on investment or equity
(ROE) of 15%, total earnings are ROE 3 $100 million 5 .15 3 $100 million 5 $15 million.
There are 3 million shares of stock outstanding, so earnings per share are $5, as posited
above. If 60% of the $15 million in this year’s earnings is reinvested, then the value of the
firm’s assets will increase by .60 3 $15 million 5 $9 million, or by 9%. The percentage
increase in assets is the rate at which income was generated (ROE) times the plowback
ratio (the fraction of earnings reinvested in the firm), which we will denote as b. Now endowed with 9% more assets, the company earns 9% more income, and pays out
9% higher dividends. The growth rate of the dividends, therefore, is 5
g 5 ROE 3 b 5 .15 3 .60 5 .09
If the stock price equals its intrinsic value, it should sell at
P0 5 D1
k 2 g 5
$2
.125 2 .09 5 $57.14
D iv
id e n
d s
p e r
Sh a re
( $ )
Year
12
0 302010
Low Reinvestment
High Reinvestment
0
2
4
6
8
10
Figure 18.1 Dividend growth for two earnings reinvestment policies
5 We can derive this relationship more generally by noting that with a fixed ROE, earnings (which equal
ROE 3 book value) will grow at the same rate as the book value of the firm. Abstracting from issuance of new
shares of stock, the growth rate of book value equals reinvested earnings/book value. Therefore,
g 5 Reinvested earnings
Book value 5
Reinvested earnings
Total earnings 3
Total earnings
Book value 5 b 3 ROE
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602 P A R T V Security Analysis
When Growth Prospects pursued a no-growth policy and paid out all earnings as
dividends, the stock price was only $40. Therefore, you can think of $40 as the value per
share of the assets the company already has in place.
When Growth Prospects decided to reduce current dividends and reinvest some of its
earnings in new investments, its stock price increased. The increase in the stock price
reflects the fact that the planned investments provide an expected rate of return greater than
the required rate. In other words, the investment opportunities have positive net present
value. The value of the firm rises by the NPV of these investment opportunities. This net
present value is also called the present value of growth opportunities, or PVGO. Therefore, we can think of the value of the firm as the sum of the value of assets already
in place, or the no-growth value of the firm, plus the net present value of the future invest-
ments the firm will make, which is the PVGO. For Growth Prospects, PVGO 5 $17.14 per
share:
Price 5 No-growth value per share 1 PVGO
P0 5 E1 k 1 PVGO
57.14 5 40 1 17.14
(18.6)
We know that in reality, dividend cuts almost always are accompanied by steep drops
in stock prices. Does this contradict our analysis? Not necessarily: Dividend cuts are usu-
ally taken as bad news about the future prospects of the firm, and it is the new information about the firm—not the reduced dividend yield per se—that is responsible for the stock
price decline.
For example, when J.P. Morgan cut its quarterly dividend from 38 cents to 5 cents a
share in 2009, its stock price actually increased by about 5%. The company was able to
convince investors that the cut would conserve cash and prepare the firm to weather a
severe recession. When investors were convinced that the dividend cut made sense, the
stock price actually increased. Similarly, when BP announced in the wake of the massive
2010 Gulf oil spill that it would suspend dividends for the rest of the year, its stock price
did not budge. The cut already had been widely anticipated, so it was not new information.
These examples show that stock price declines in response to dividend cuts are really a
response to the information conveyed by the cut.
It is important to recognize that growth per se is not what investors desire. Growth
enhances company value only if it is achieved by investment in projects with attractive
profit opportunities (i.e., with ROE . k ). To see why, let’s now consider Growth Prospects’s unfortunate sister company, Cash Cow, Inc. Cash Cow’s ROE is only 12.5%, just equal to
the required rate of return, k. The net present value of its investment opportunities is zero. We’ve seen that following a zero-growth strategy with b 5 0 and g 5 0, the value of Cash Cow will be E 1 / k 5 $5/.125 5 $40 per share. Now suppose Cash Cow chooses a plowback ratio of b 5 .60, the same as Growth Prospects’s plowback. Then g would increase to
g 5 ROE 3 b 5 .125 3 .60 5 .075
but the stock price is still $40:
P0 5 D1
k 2 g 5
$2
.125 2 .075 5 $40
which is no different from the no-growth strategy.
In the case of Cash Cow, the dividend reduction used to free funds for reinvest-
ment in the firm generates only enough growth to maintain the stock price at the cur-
rent level. This is as it should be: If the firm’s projects yield only what investors can
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C H A P T E R 1 8 Equity Valuation Models 603
earn on their own, shareholders cannot be made better off by a high-reinvestment-rate
policy. This demonstrates that “growth” is not the same as growth opportunities. To jus-
tify reinvestment, the firm must engage in projects with better prospective returns than
those shareholders can find elsewhere. Notice also that the PVGO of Cash Cow is zero:
PVGO 5 P 0 2 E 1 / k 5 40 2 40 5 0. With ROE 5 k, there is no advantage to plowing funds back into the firm; this shows up as PVGO of zero. In fact, this is why firms with
considerable cash flow but limited investment prospects are called “cash cows.” The cash
these firms generate is best taken out of, or “milked from,” the firm.
a. Calculate the price of a firm with a plowback ratio of .60 if its ROE is 20%. Current earnings, E 1 , will be $5 per share, and k 5 12.5%.
b. What if ROE is 10%, which is less than the market capitalization rate? Compare the firm’s price in this instance to that of a firm with the same ROE and E 1 , but a plowback ratio of b 5 0.
CONCEPT CHECK 18.3
Life Cycles and Multistage Growth Models As useful as the constant-growth DDM formula is, you need to remember that it is based
on a simplifying assumption, namely, that the dividend growth rate will be constant for-
ever. In fact, firms typically pass through life cycles with very different dividend profiles in
Takeover Target is run by entrenched management that insists on reinvesting 60% of its earnings in projects that provide an ROE of 10%, despite the fact that the firm’s capi- talization rate is k 5 15%. The firm’s year-end dividend will be $2 per share, paid out of earnings of $5 per share. At what price will the stock sell? What is the present value of growth opportunities? Why would such a firm be a takeover target for another firm?
Given current management’s investment policy, the dividend growth rate will be
g 5 ROE 3 b 5 10% 3 .60 5 6%
and the stock price should be
P0 5 $2
.15 2 .06 5 $22.22
The present value of growth opportunities is
PVGO 5 Price per share 2 No-growth value per share
5 $22.22 2 E1/k 5 $22.22 2 $5/.15 5 2$11.11
PVGO is negative. This is because the net present value of the firm’s projects is negative: The rate of return on those assets is less than the opportunity cost of capital.
Such a firm would be subject to takeover, because another firm could buy the firm for the market price of $22.22 per share and increase the value of the firm by changing its investment policy. For example, if the new management simply paid out all earnings as dividends, the value of the firm would increase to its no-growth value, E 1 / k 5 $5/.15 5 $33.33.
Example 18.4 Growth Opportunities
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604 P A R T V Security Analysis
different phases. In early years, there are ample opportunities for profitable reinvestment in
the company. Payout ratios are low, and growth is correspondingly rapid. In later years, the
firm matures, production capacity is sufficient to meet market demand, competitors enter
the market, and attractive opportunities for reinvestment may become harder to find. In this
mature phase, the firm may choose to increase the dividend payout ratio, rather than retain
earnings. The dividend level increases, but thereafter it grows at a slower rate because the
company has fewer growth opportunities.
Table 18.2 illustrates this pattern. It gives Value Line’s forecasts of return on capital,
dividend payout ratio, and 3-year projected growth rate in earnings per share for a sample
of the firms included in the computer software industry versus those of East Coast electric
utilities. (We compare return on capital rather than return on equity because the latter is
affected by leverage, which tends to be far greater in the electric utility industry than in the
software industry. Return on capital measures operating income per dollar of total long-
term financing, regardless of whether the source of the capital supplied is debt or equity.
We will return to this issue in the next chapter.)
By and large, the software firms have attractive investment opportunities. The median
return on capital of these firms is forecast to be 16.5%, and the firms have responded with
high plowback ratios. Most of these firms pay no dividends at all. The high return on capi-
tal and high plowback result in rapid growth. The median projected growth rate of earnings
per share in this group is 13.2%.
Table 18.2
Financial ratios in two industries
Ticker Return on Capital (%)
Payout Ratio (%)
Growth Rate 2014–2016
Computer Software Adobe Systems ADBE 12.0 % 0.0 % 13.2% Cognizant CTSH 18.5 0.0 20.5 Compuware CPWR 13.5 0.0 16.6 Intuit INTU 20.0 22.0 10.9 Microsoft MSFT 31.5 34.0 11.7 Oracle ORCL 20.5 12.0 7.0 Red Hat RHT 13.0 0.0 18.2 Parametric Tech PMTC 15.0 0.0 16.0 SAP SAP 16.5 28.0 9.1 Median 16.5 % 0.0% 13.2% Electric Utilities (East Coast) Central Hudson G&E CHG 6.0 % 66.0 % 2.0% Consolidated Edison ED 6.5 58.0 2.9 Duke Energy DUK 5.5 66.0 4.0 Northeast Utilities NU 6.0 53.0 7.7 Pennsylvania Power PPL 7.0 58.0 7.7 Public Service Enterprise PEG 7.5 53.0 6.3 South Carolina E & G SCG 6.0 57.0 3.8 Southern Company SO 7.0 69.0 5.1 Tampa Electric TE 7.5 59.0 8.3 United Illuminating UIL 6.0 71.0 2.1 Median 6.3 % 58.5 % 4.5%
Source: Value Line Investment Survey, July and August, 2012. Reprinted with permission of Value Line Investment Survey. © 2012 Value Line Publishing, Inc. All rights reserved.
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C H A P T E R 1 8 Equity Valuation Models 605
In contrast, the electric utilities are more representative of mature firms. Their median
return on capital is lower, 6.3%; dividend payout is higher, 58.5%; and median growth is
lower, 4.5%. We conclude that the higher payouts of the electric utilities reflect their more
limited opportunities to reinvest earnings at attractive rates of return.
To value companies with temporarily high growth, analysts use a multistage version of
the dividend discount model. Dividends in the early high-growth period are forecast and
their combined present value is calculated. Then, once the firm is projected to settle down
to a steady-growth phase, the constant-growth DDM is applied to value the remaining
stream of dividends.
We can illustrate this with a real-life example. Figure 18.2 is a Value Line Investment
Survey report on Honda Motor Co. Some of the relevant information for 2012 is highlighted.
Honda’s beta appears at the circled A, its recent stock price at the B, the per-share
dividend payments at the C, the ROE (referred to as return on shareholder equity ) at the D, and the dividend payout ratio (referred to as all dividends to net profits ) at the E. 6 The rows ending at C, D, and E are historical time series. The boldfaced, italicized entries
under 2013 are estimates for that year. Similarly, the entries in the far right column (labeled
15–17) are forecasts for some time between 2015–2017, which we will take to be 2016.
Value Line projects fairly rapid growth in the near term, with dividends rising from $.78
in 2013 to $1.00 in 2016. This growth rate cannot be sustained indefinitely. We can obtain
dividend inputs for this initial period by using the explicit forecasts for 2013 and 2016 and
linear interpolation for the years between:
2013 $.78 2015 $ .92
2014 $.85 2016 $1.00
Now let us assume the dividend growth rate levels off in 2016. What is a good guess for
that steady-state growth rate? Value Line forecasts a dividend payout ratio of .25 and an
ROE of 10%, implying long-term growth will be
g 5 ROE 3 b 5 10.0% 3 (1 2 .25) 5 7.5%
Our estimate of Honda’s intrinsic value using an investment horizon of 2016 is therefore
obtained from Equation 18.2, which we restate here:
V2012 5 D2013 1 1 k
1 D2014
(1 1 k)2 1
D2015 (1 1 k)3
1 D2016 1 P2016
(1 1 k)4
5 .78
1 1 k 1
.85
(1 1 k)2 1
.92
(1 1 k)3 1
1.00 1 P2016 (1 1 k)4
Here, P 2016 represents the forecast price at which we can sell our shares at the end of 2016, when dividends are assumed to enter their constant-growth phase. That price, according to
the constant-growth DDM, should be
P2016 5 D2017 k 2 g
5 D2016(1 1 g)
k 2 g 5
1.00 3 1.075
k 2 .075
The only variable remaining to be determined to calculate intrinsic value is the market
capitalization rate, k.
6 Because Honda is a Japanese firm, Americans would hold its shares via ADRs, or American Depository
Receipts. ADRs are not shares of the firm, but are claims to shares of the underlying foreign stock that are then traded in U.S. security markets. Value Line notes that each Honda ADR is a claim on one common share, but in
other cases, each ADR may represent a claim to multiple shares or even fractional shares.
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606 P A R T V Security Analysis
B
A
C
D
E
Figure 18.2 Value Line Investment Survey report on Honda Motor Co. Source: Jason A. Smith, Value Line Investment Survey, May 25, 2012. Reprinted with permission of Value Line Investment Survey © 2012 Value Line Publishing, Inc. All rights reserved.
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C H A P T E R 1 8 Equity Valuation Models 607
7 When valuing long-term assets such as stocks, it is common to treat the long-term Treasury bond, rather than
short-term T-bills, as the risk-free asset.
One way to obtain k is from the CAPM. Observe from the Value Line report that Honda’s beta is .95. The risk-free rate on long-term Treasury bonds in 2012 was about
2.0%. 7 Suppose that the market risk premium were forecast at 8%, roughly in line with its
historical average. This would imply that the forecast for the market return was
Risk-free rate 1 Market risk premium 5 2% 1 8% 5 10%
Therefore, we can solve for the market capitalization rate as
k 5 rf 1 b3E(rM) 2 rf 4 5 2% 1 .95(10% 2 2%) 5 9.6% Our forecast for the stock price in 2016 is thus
P2016 5 $1.00 3 1.075
.096 2 .075 5 $51.19
and today’s estimate of intrinsic value is
V2012 5 .78
1.096 1
.85
(1.096)2 1
.92
(1.096)3 1
1.00 1 51.19
(1.096)4 5 $38.29
We know from the Value Line report that Honda’s actual price was $32.88 (at the cir-
cled B). Our intrinsic value analysis indicates that the stock was underpriced. Should we
increase our holdings?
Perhaps. But before betting the farm, stop to consider how firm our estimate is. We’ve
had to guess at dividends in the near future, the ultimate growth rate of those dividends,
and the appropriate discount rate. Moreover, we’ve assumed Honda will follow a relatively
simple two-stage growth process. In practice, the growth of dividends can follow more
complicated patterns. Even small errors in these approximations could upset a conclusion.
For example, suppose that we have overestimated Honda’s growth prospects and that
the actual ROE in the post-2016 period will be 9% rather than 10%. Using the lower return
on equity in the dividend discount model would result in an intrinsic value in 2012 of only
$28.77, which is less than the stock price. Our conclusion regarding intrinsic value versus price is reversed.
The exercise also highlights the importance of performing sensitivity analysis when
you attempt to value stocks. Your estimates of stock values are no better than your assump-
tions. Sensitivity analysis will highlight the inputs that need to be most carefully exam-
ined. For example, even modest changes in the estimated ROE for the post-2016 period
can result in big changes in intrinsic value. Similarly, small changes in the assumed capi-
talization rate would change intrinsic value substantially. On the other hand, reasonable
changes in the dividends forecast between 2013 and 2016 would have a small impact on
intrinsic value.
Confirm that the intrinsic value of Honda using ROE 5 9% is $28.77. ( Hint: First calculate the stock price in 2016. Then calculate the present value of all interim dividends plus the present value of the 2016 sales price.)
CONCEPT CHECK 18.4
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608 P A R T V Security Analysis
Multistage Growth Models The two-stage growth model that we just considered for Honda is a good start toward
realism, but clearly we could do even better if our valuation model allowed for more
flexible patterns of growth. Multistage growth models allow dividends per share to grow
at several different rates as the firm matures. Many analysts use three-stage growth mod-
els. They may assume an initial period of high dividend growth (or instead make year-
by-year forecasts of dividends for the short term), a final period of sustainable growth,
and a transition period between, during which dividend growth rates taper off from
the initial rapid rate to the ultimate sustainable rate. These models are conceptually no
harder to work with than a two-stage model, but they require many more calculations
and can be tedious to do by hand. It is easy, however, to build an Excel spreadsheet for
such a model.
Spreadsheet 18.1 is an example of such a model. Column B contains the inputs we have
used so far for Honda. Column E contains dividend forecasts. In cells E2 through E5 we
present the Value Line estimates for the next 4 years. Dividend growth in this period is
about 8.6% annually. Rather than assume a sudden transition to constant dividend growth
starting in 2016, we assume instead that the dividend growth rate in 2016 will be 8.6% and
that it will decline steadily through 2027, finally reaching the constant terminal growth rate
of 7.5% (see column F). Each dividend in the transition period is the previous year’s divi-
dend times that year’s growth rate. Terminal value once the firm enters a constant-growth
stage (cell G17) is computed from the constant-growth DDM. Finally, investor cash flow
in each period (column H) equals dividends in each year plus the terminal value in 2027.
The present value of these cash flows is computed in cell H19 as $40.29, about 5% higher
than the value we found from the two-stage model. We obtain a greater intrinsic value in
this case because we assume that dividend growth only gradually declines to its steady-
state value.
Spreadsheet 18.1
A three-stage growth model for Honda Motor Co.
e X c e l Please visit us at www.mhhe.com/bkm
A B C D E F G H I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Inputs beta mkt_prem rf k_equity plowback roe term_gwth
Value line
annual dividends forecasts of
Transitional period with slowing dividend growth
Beginning of constant growth period
0.95 0.08 0.02
0.0960 0.75 0.1
0.075
Year 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027
Dividend Div growth Term value Investor CF 0.78 0.85 0.92 1.00 1.09 1.18 1.28 1.38 1.50 1.62 1.75 1.88 2.03 2.18 2.35 2.52
0.0863 0.0852 0.0841 0.0829 0.0818 0.0807 0.0795 0.0784 0.0773 0.0761 0.0750 0.0750 129.18
0.78 0.85 0.92 1.00 1.09 1.18 1.28 1.38 1.50 1.62 1.75 1.88 2.03 2.18 2.35
131.71
E17*(1+F17)/(B5 − F17) 40.29
NPV(B5,H2:H17)
= PV of CF
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C H A P T E R 1 8 Equity Valuation Models 609
18.4 Price–Earnings Ratio
The Price–Earnings Ratio and Growth Opportunities Much of the real-world discussion of stock market valuation concentrates on the firm’s
price–earnings multiple, the ratio of price per share to earnings per share, commonly called the P/E ratio. Our discussion of growth opportunities shows why stock market ana-
lysts focus on the P/E ratio. Both companies considered, Cash Cow and Growth Prospects,
had earnings per share (EPS) of $5, but Growth Prospects reinvested 60% of earnings in
prospects with an ROE of 15%, whereas Cash Cow paid out all earnings as dividends.
Cash Cow had a price of $40, giving it a P/E multiple of 40/5 5 8.0, whereas Growth Pros-
pects sold for $57.14, giving it a multiple of 57.14/5 5 11.4. This observation suggests the
P/E ratio might serve as a useful indicator of expectations of growth opportunities.
We can see how growth opportunities are reflected in P/E ratios by rearranging
Equation 18.6 to
P0 E1 5
1
k ¢1 1 PVGO
E/k ≤ (18.7)
When PVGO 5 0, Equation 18.7 shows that P 0 5 E 1 / k. The stock is valued like a non- growing perpetuity of E 1 , and the P/E ratio is just 1/ k. However, as PVGO becomes an increasingly dominant contributor to price, the P/E ratio can rise dramatically.
The ratio of PVGO to E / k has a straightforward interpretation. It is the ratio of the component of firm value due to growth opportunities to the component of value due to
assets already in place (i.e., the no-growth value of the firm, E / k ). When future growth opportunities dominate the estimate of total value, the firm will command a high price
relative to current earnings. Thus a high P/E multiple indicates that a firm enjoys ample
growth opportunities.
P/E multiples do vary with growth prospects. Between 1996 and 2012, for example,
FedEx’s P/E ratio averaged about 17.4 while Consolidated Edison’s (an electric utility)
average P/E was only 15.7. These numbers do not necessarily imply that FedEx was over-
priced compared to Con Ed. If investors believed FedEx would grow faster than Con Ed,
the higher price per dollar of earnings would be justified. That is, an investor might well
pay a higher price per dollar of current earnings if he or she expects that earnings stream to grow more rapidly. In fact, FedEx’s growth rate has been consistent with its higher P/E
multiple. Over this period, its earnings per share grew at 10.2% per year while Con Ed’s
earnings growth rate was only 1.6%. Figure 18.4 on page 615 shows the EPS history of the
two companies.
We conclude that the P/E ratio reflects the market’s optimism concerning a firm’s
growth prospects. Analysts must decide whether they are more or less optimistic than the
belief implied by the market multiple. If they are more optimistic, they will recommend
buying the stock.
There is a way to make these insights more precise. Look again at the constant-growth
DDM formula, P 0 5 D 1 /( k 2 g ). Now recall that dividends equal the earnings that are not reinvested in the firm: D 1 5 E 1 (1 2 b ). Recall also that g 5 ROE 3 b. Hence, substituting for D 1 and g, we find that
P0 5 E1(1 2 b)
k 2 ROE 3 b
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610 P A R T V Security Analysis
implying the P/E ratio is
P0 E1 5
1 2 b
k 2 ROE 3 b (18.8)
It is easy to verify that the P/E ratio increases with ROE. This makes sense, because high-
ROE projects give the firm good opportunities for growth. 8 We also can verify that the P/E
ratio increases for higher plowback, b, as long as ROE exceeds k. This too makes sense. When a firm has good investment opportunities, the market will reward it with a higher
P/E multiple if it exploits those opportunities more aggressively by plowing back more
earnings into those opportunities.
Remember, however, that growth is not desirable for its own sake. Examine Table 18.3
where we use Equation 18.8 to compute both growth rates and P/E ratios for different com-
binations of ROE and b. Although growth always increases with the plowback rate (move across the rows in panel A), the P/E ratio does not (move across the rows in panel B). In
the top row of panel B, the P/E falls as the plowback rate increases. In the middle row, it is
unaffected by plowback. In the third row, it increases.
This pattern has a simple interpretation. When the expected ROE is less than the
required return, k, investors prefer that the firm pay out earnings as dividends rather than reinvest earnings in the firm at an inadequate rate of return. That is, for ROE lower than
k, the value of the firm falls as plowback increases. Conversely, when ROE exceeds k, the firm offers attractive investment opportunities, so the value of the firm is enhanced as those
opportunities are more fully exploited by increasing the plowback rate.
Finally, where ROE just equals k, the firm offers “break-even” investment opportuni- ties with a fair rate of return. In this case, investors are indifferent between reinvestment
of earnings in the firm or elsewhere at the market capitalization rate, because the rate of
return in either case is 12%. Therefore, the stock price is unaffected by the plowback rate.
We conclude that the higher the plowback rate, the higher the growth rate, but a higher
plowback rate does not necessarily mean a higher P/E ratio. Higher plowback increases
P/E only if investments undertaken by the firm offer an expected rate of return greater than
the market capitalization rate. Otherwise, increasing plowback hurts investors because
more money is sunk into projects with inadequate rates of return.
8 Note that Equation 18.8 is a simple rearrangement of the DDM formula, with ROE 3 b 5 g. Because that
formula requires that g , k, Equation 18.8 is valid only when ROE 3 b , k.
Table 18.3
Effect of ROE and plowback on growth and the P/E ratio
Plowback Rate ( b )
0 .25 .50 .75
ROE A. Growth rate, g 10% 0 2.5% 5.0% 7.5% 12 0 3.0 6.0 9.0 14 0 3.5 7.0 10.5
B. P/E ratio 10% 8.33 7.89 7.14 5.56 12 8.33 8.33 8.33 8.33 14 8.33 8.82 10.00 16.67
Assumption: k 5 12% per year.
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C H A P T E R 1 8 Equity Valuation Models 611
Notwithstanding these fine points, P/E ratios frequently are taken as proxies for the
expected growth in dividends or earnings. In fact, a common Wall Street rule of thumb is
that the growth rate ought to be roughly equal to the P/E ratio. In other words, the ratio of
P/E to g, often called the PEG ratio, should be about 1.0. Peter Lynch, the famous portfolio manager, puts it this way in his book One Up on Wall Street:
The P/E ratio of any company that’s fairly priced will equal its growth rate. I’m talking
here about growth rate of earnings here. . . . If the P/E ratio of Coca Cola is 15, you’d
expect the company to be growing at about 15% per year, etc. But if the P/E ratio is less
than the growth rate, you may have found yourself a bargain.
The importance of growth opportunities is most evident in the valuation of start-up
firms. For example, in the dot-com boom of the late 1990s, many companies that had yet to
turn a profit were valued by the market at billions of dollars. The perceived value of these
companies was exclusively as growth opportunities. For example, the online auction firm eBay had 1998 profits of $2.4 million, far less than the $45 million profit earned by the
traditional auctioneer Sotheby’s; yet eBay’s market value was more than 10 times greater:
Let’s try Lynch’s rule of thumb. Assume that
r f 5 8% (roughly the value when Peter Lynch was writing)
r M 2 r f 5 8% (about the historical average market risk premium)
b 5 .4 (a typical value for the plowback ratio in the United States)
Therefore, r M 5 r f 1 market risk premium 5 8% 1 8% 5 16%, and k 5 16% for an average ( b 5 1) company. If we also accept as reasonable that ROE 5 16% (the same value as the expected return on the stock), we conclude that
g 5 ROE 3 b 5 16% 3 .4 5 6.4%
and
P E 5
1 2 .4 .16 2 .064
5 6.26
Thus, the P/E ratio and g are about equal using these assumptions, consistent with the rule of thumb.
However, note that this rule of thumb, like almost all others, will not work in all circumstances. For example, the yield on long-term Treasury bonds today is more like 2%, so a comparable forecast of r M today would be
rf 1 Market risk premium 5 2% 1 8% 5 10%
If we continue to focus on a firm with b 5 1, and if ROE still is about the same as k, then
g 5 10% 3 .4 5 4.0%
while
P E 5
1 2 .4 .10 2 .04
5 10
The P/E ratio and g now diverge and the PEG ratio is now 2.5. Nevertheless, lower- than-average PEG ratios are still widely seen as signaling potential underpricing.
Example 18.5 P/E Ratio versus Growth Rate
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612 P A R T V Security Analysis
$22 billion versus $1.9 billion. (As it turns out, the market was quite right to value eBay
so much more aggressively than Sotheby’s. Its net income in 2011 was $1.8 billion, more
than 10 times that of Sotheby’s.
Of course, when company valuation is determined primarily by growth opportunities,
those values can be very sensitive to reassessments of such prospects. When the market
became more skeptical of the business prospects of most Internet retailers at the close of
the 1990s, that is, as it revised the estimates of growth opportunities downward, their stock
prices plummeted.
As perceptions of future prospects wax and wane, share price can swing wildly. Growth
prospects are intrinsically difficult to tie down; ultimately, however, those prospects drive
the value of the most dynamic firms in the economy.
The nearby box contains a simple valuation analysis. As Facebook headed toward its
highly anticipated IPO in 2012, there was widespread speculation about the price at which
it would eventually trade in the stock market. Notice that the discussion in the article
focused on two key questions. First, what was a reasonable projection of the growth rate
of Facebook’s profits? Second, what multiple of earnings was appropriate to translate an
earnings forecast into a price forecast? These are precisely the questions addressed by our
stock valuation models.
ABC stock has an expected ROE of 12% per year, expected earnings per share of $2, and expected divi- dends of $1.50 per share. Its market capitalization rate is 10% per year.
a. What are its expected growth rate, its price, and its P/E ratio?
b. If the plowback ratio were .4, what would be the expected dividend per share, the growth rate, price, and the P/E ratio?
CONCEPT CHECK 18.5
P/E Ratios and Stock Risk One important implication of any stock-valuation model is that (holding all else equal)
riskier stocks will have lower P/E multiples. We can see this quite easily in the context of
the constant-growth model by examining the formula for the P/E ratio (Equation 18.8):
P
E 5
1 2 b
k 2 g
Riskier firms will have higher required rates of return, that is, higher values of k. There- fore, the P/E multiple will be lower. This is true even outside the context of the constant-
growth model. For any expected earnings and dividend stream, the present value of those cash flows will be lower when the stream is perceived to be riskier. Hence the stock price
and the ratio of price to earnings will be lower.
Of course, you can find many small, risky, start-up companies with very high P/E mul-
tiples. This does not contradict our claim that P/E multiples should fall with risk; instead it
is evidence of the market’s expectations of high growth rates for those companies. This is
why we said that high-risk firms will have lower P/E ratios holding all else equal. Given a growth projection, the P/E multiple will be lower when risk is perceived to be higher.
Pitfalls in P/E Analysis No description of P/E analysis is complete without mentioning some of its pitfalls. First,
consider that the denominator in the P/E ratio is accounting earnings, which are influenced
by somewhat arbitrary accounting rules such as the use of historical cost in depreciation
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613
and inventory valuation. In times of high inflation, historic cost depreciation and inventory
costs will tend to underrepresent true economic values, because the replacement cost of
both goods and capital equipment will rise with the general level of prices. As Figure 18.3
demonstrates, P/E ratios generally have been inversely related to the inflation rate. In part,
this reflects the market’s assessment that earnings in high inflation periods are of “lower
quality,” artificially distorted by inflation, and warranting lower P/E ratios.
Earnings management is the practice of using flexibility in accounting rules to improve the apparent profitability of the firm. We will have much to say on this topic in the
next chapter on interpreting financial statements. A version of earnings management that
became common in the 1990s was the reporting of “pro forma earnings” measures.
Facebook’s $100 Billion Question
WHAT IS FACEBOOK WORTH? As investors dug into the company’s freshly released financials Wednesday, analysts and investors began circulating a range of values—from as little as $50 billion to as much as $125 billion—for the social-networking Web site.
It will be months before the market sets a final price, but already the valuation question has become a tug of war over two essential questions: Just how fast can the company continue to grow? And can it extract value from advertising in the way it plans?
Facebook’s revenue grew 88% in 2011, and net income grew 65%. Facebook’s growth has already decelerated from 154% from 2009 to 2010 to the 88% it experienced last year.
Francis Gaskins, president of IPOdesktop.com, which analyzes IPOs for investors, says he doesn’t believe Face- book is worth more than $50 billion—50 times its reported profits for 2011 of $1 billion, or more than triple the mar- ket’s average price-to-earnings ratio. Google Inc.’s profits are 10 times that of Facebook, but its stock-market value is $190 billion, he notes.
A $100 billion valuation “would have us believe that Facebook is worth 53% of Google, even though Google’s sales and profits are 10 times that of Facebook,” he said.
Martin Pyykkonen, an analyst at Denver banking bou- tique Wedge Partners, is more bullish, saying the value could top $100 billion. He says Facebook could trade at 15 to 18 times next year’s expected earnings before interest, taxes, and certain noncash charges, a cash-flow measure known as EBITDA. By comparison, he says, mature compa- nies trade at 8 to 10 times EBITDA. Microsoft Corp. trades at 7 times, and Google about 10 times.
While that math only justifies an $81 billion valuation, he says Facebook may be able to unlock faster growth in ad spending and reach $5.5 billion in EBITDA, which could justify a higher multiple of 20 times, implying a $110 billion valuation.
Source: Randall Smith, “Facebook’s $100 Billion Question,” The Wall Street Journal, February 3, 2012. Reprinted by permission of The Wall Street Journal, Copyright © 2012 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
W O
R D
S FR O
M T
H E ST
R E E T
Figure 18.3 P/E ratios of the S&P 500 Index and inflation
0
10
20
30
40
50
60
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
P/E Ratio
Inflation Rate
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614 P A R T V Security Analysis
Pro forma earnings are calculated ignoring certain expenses, for example, restructur-
ing charges, stock-option expenses, or write-downs of assets from continuing operations.
Firms argue that ignoring these expenses gives a clearer picture of the underlying profit-
ability of the firm. Comparisons with earlier periods probably would make more sense if
those costs were excluded.
But when there is too much leeway for choosing what to exclude, it becomes hard for
investors or analysts to interpret the numbers or to compare them across firms. The lack of
standards gives firms considerable leeway to manage earnings.
Even GAAP allows firms considerable discretion to manage earnings. For example,
in the late 1990s, Kellogg took restructuring charges, which are supposed to be one-time
events, nine quarters in a row. Were these really one-time events, or were they more appro-
priately treated as ordinary expenses? Given the available leeway in managing earnings,
the justified P/E multiple becomes difficult to gauge.
Another confounding factor in the use of P/E ratios is related to the business cycle. We
were careful in deriving the DDM to define earnings as being net of economic deprecia- tion, that is, the maximum flow of income that the firm could pay out without depleting
its productive capacity. But reported earnings are computed in accordance with generally
accepted accounting principles and need not correspond to economic earnings. Beyond
this, however, notions of a normal or justified P/E ratio, as in Equation 18.7 or 18.8, assume
implicitly that earnings rise at a constant rate, or, put another way, on a smooth trend line.
In contrast, reported earnings can fluctuate dramatically around a trend line over the course
of the business cycle.
Another way to make this point is to note that the “normal” P/E ratio predicted by
Equation 18.8 is the ratio of today’s price to the trend value of future earnings, E 1 . The P/E ratio reported in the financial pages of the newspaper, by contrast, is the ratio of price
to the most recent past accounting earnings. Current accounting earnings can differ con- siderably from future economic earnings. Because ownership of stock conveys the right to
future as well as current earnings, the ratio of price to most recent earnings can vary sub-
stantially over the business cycle, as accounting earnings and the trend value of economic
earnings diverge by greater and lesser amounts.
As an example, Figure 18.4 graphs the earnings per share of FedEx and Con Ed since
1996. Note that FedEx’s EPS is far more variable. Because the market values the entire
stream of future dividends generated by the company, when earnings are temporar-
ily depressed, the P/E ratio should tend to be high—that is, the denominator of the ratio
responds more sensitively to the business cycle than the numerator. This pattern is borne
out well.
Figure 18.5 graphs the P/E ratios of the two firms. FedEx has greater earnings volatility
and more variability in its P/E ratio. Its clearly higher average growth rate shows up in its
generally higher P/E ratio. The only period in which Con Ed’s ratio exceeded FedEx’s was
in 2012, a year when FedEx’s earnings rose at a far faster rate than its underlying trend.
The market seems to have decided that this earnings performance was not likely to be sus-
tainable, and FedEx’s price rose less dramatically than its annual earnings. Consequently,
its P/E ratio declined.
This example shows why analysts must be careful in using P/E ratios. There is no way
to say P/E ratio is overly high or low without referring to the company’s long-run growth
prospects, as well as to current earnings per share relative to the long-run trend line.
Nevertheless, Figures 18.4 and 18.5 demonstrate a clear relation between P/E ratios and
growth. Despite considerable short-run fluctuations, FedEx’s EPS clearly trended upward
over the period. Con Ed’s earnings were essentially flat. FedEx’s growth prospects are
reflected in its consistently higher P/E multiple.
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C H A P T E R 1 8 Equity Valuation Models 615
This analysis suggests
that P/E ratios should vary
across industries, and in
fact they do. Figure 18.6
shows P/E ratios in 2012
for a sample of industries.
Notice that the industries
with the highest multiples—
such as business software
or biotech—have attractive
investment opportunities and
relatively high growth rates,
whereas the industries
with the lowest ratios—for
example, aerospace or com-
puter manufacturers—are
in more mature or less prof-
itable industries with lim-
ited growth opportunities.
The relationship between
P/E and growth is not per-
fect, which is not surpris-
ing in light of the pitfalls
discussed in this section,
but as a general rule, the
P/E multiple does appear to
track growth opportunities.
Combining P/E Analysis and the DDM Some analysts use P/E
ratios in conjunction with
earnings forecasts to esti-
mate the price of a stock
at an investor’s horizon
date. The Honda analysis in
Figure 18.2 shows that
Value Line forecast a P/E ratio for 2016 of 14. EPS for 2016 was forecast at $4, implying
a price in 2016 of 14 3 $4 5 $56. Given an estimate of $56 for the 2016 sales price, we
would compute intrinsic value in 2012 as
V2012 5 .78
1.096 1
.85
(1.096)2 1
.92
(1.096)3 1
1.00 1 56
(1.096)4 5 $41.62
Other Comparative Valuation Ratios The price–earnings ratio is an example of a comparative valuation ratio. Such ratios are
used to assess the valuation of one firm versus another based on a fundamental indicator
such as earnings. For example, an analyst might compare the P/E ratios of two firms in the
0
1
2
3
4
5
6
1996 1998 2000 2002 2004 2006 2008 2010 2012 E a rn
in g
s p
e r
Sh a re
( 1 9 9 2
= 1
.0 )
Con Ed
FedEx
Figure 18.4 Earnings growth for two companies
Figure 18.5 Price–earnings ratios
5
10
15
20
25
1996 1998 2000 2002 2004 2006 2008 2010 2012
P /E
R a ti
o
P/E FedEx
P/E Con Ed
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616 P A R T V Security Analysis
same industry to test whether the market is valuing one firm “more aggressively” than the
other. Other such comparative ratios are commonly used:
Price-to-Book Ratio This is the ratio of price per share divided by book value per share. As we noted earlier in this chapter, some analysts view book value as a useful mea-
sure of value and therefore treat the ratio of price to book value as an indicator of how
aggressively the market values the firm.
Price-to-Cash-Flow Ratio Earnings as reported on the income statement can be affected by the company’s choice of accounting practices, and thus are commonly viewed
as subject to some imprecision and even manipulation. In contrast, cash flow—which
tracks cash actually flowing into or out of the firm—is less affected by accounting deci-
sions. As a result, some analysts prefer to use the ratio of price to cash flow per share
rather than price to earnings per share. Some analysts use operating cash flow when cal-
culating this ratio; others prefer “free cash flow,” that is, operating cash flow net of new
investment.
Price-to-Sales Ratio Many start-up firms have no earnings. As a result, the price– earnings ratio for these firms is meaningless. The price-to-sales ratio (the ratio of stock
price to the annual sales per share) has recently become a popular valuation benchmark for
these firms. Of course, price-to-sales ratios can vary markedly across industries, because
profit margins vary widely.
Figure 18.6 P/E ratios for different industries Source: Yahoo! Finance, finance.yahoo.com, September 12, 2012.
0 10 20 30 40 50 60
Aerospace/Defense Money Center Banks
Health Care Plans Computer Systems
Telecom Services Iron Steel
Industrial Metals Electric Utilities
Home Improvement Integrated Oil & Gas
Pharmaceuticals Chemical Products Asset Management
Application Software Food Products
Restaurants Auto Manufacturers
Trucking Heavy Construction
Business Software Biotech
P/E Ratio
8.5 11.3 11.8
13.1 14.7 14.8 14.9 15.6 16.5 16.8 17.2 17.4 17.5 17.5
21.1 21.4
25.3 28.0
32.4 34.7 57.8
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C H A P T E R 1 8 Equity Valuation Models 617
Be Creative Sometimes a standard valuation ratio will simply not be available, and you will have to devise your own. In the 1990s, some analysts valued retail Internet firms based
on the number of hits their Web sites received. As it turns out, they valued these firms
using too-generous “price-to-hits” ratios. Nevertheless, in a new investment environment,
these analysts used the information available to them to devise the best valuation tools they
could.
Figure 18.7 presents the behavior of several valuation measures. While the levels of
these ratios differ considerably, for the most part, they track each other fairly closely, with
upturns and downturns at the same times.
Figure 18.7 Market valuation statistics
Price Sales
Price Cash Flow
Price Earnings
40
35
30
25
20
15
10
5
0 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
18.5 Free Cash Flow Valuation Approaches
An alternative approach to the dividend discount model values the firm using free cash
flow, that is, cash flow available to the firm or its equityholders net of capital expendi-
tures. This approach is particularly useful for firms that pay no dividends, for which the
dividend discount model would be difficult to implement. But free cash flow models
may be applied to any firm and can provide useful insights about firm value beyond
the DDM.
One approach is to discount the free cash flow for the firm (FCFF) at the weighted- average cost of capital to obtain the value of the firm, and subtract the then-existing value
of debt to find the value of equity. Another is to focus from the start on the free cash flow to
equityholders (FCFE), discounting those directly at the cost of equity to obtain the market value of equity.
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618 P A R T V Security Analysis
The free cash flow to the firm is the after-tax cash flow generated by the firm’s
operations, net of investments in capital, and net working capital. It includes cash flows
available to both debt- and equityholders. 9 It equals:
FCFF 5 EBIT (1 2 tc) 1 Depreciation 2 Capital expenditures 2 Increase in NWC (18.9)
where
EBIT 5 earnings before interest and taxes
t c 5 the corporate tax rate
NWC 5 net working capital
Alternatively, we can focus on cash flow available to equityholders. This will differ
from free cash flow to the firm by after-tax interest expenditures, as well as by cash flow
associated with net issuance or repurchase of debt (i.e., principal repayments minus pro-
ceeds from issuance of new debt).
FCFE 5 FCFF 2 Interest expense 3 (1 2 tc) 1 Increases in net debt (18.10)
A free cash flow to the firm valuation model discounts year-by-year cash flows plus
some estimate of terminal value, V T . In Equation 18.11, we use the constant-growth model to estimate terminal value and discount at the weighted-average cost of capital.
Firm value 5 a T
t51
FCFFt
(1 1 WACC)t 1
VT (1 1 WACC)T
, where VT 5 FCFFT11
WACC 2 g (18.11)
To find equity value, we subtract the existing market value of debt from the derived value
of the firm.
Alternatively, we can discount free cash flows to equity (FCFE) at the cost of equity, k E .
Intrinsic value of equity 5 a T
t51
FCFEt
(1 1 kE) t 1
VT (1 1 kE)
T, where VT 5 FCFET11
kE 2 g (18.12)
As in the dividend discount model, free cash flow models use a terminal value to avoid
adding the present values of an infinite sum of cash flows. That terminal value may simply
be the present value of a constant-growth perpetuity (as in the formulas above) or it may
be based on a multiple of EBIT, book value, earnings, or free cash flow. As a general rule,
estimates of intrinsic value depend critically on terminal value.
Spreadsheet 18.2 presents a free cash flow valuation of Honda using the data supplied
by Value Line in Figure 18.2 . We start with the free cash flow to the firm approach given in
Equation 18.9. Panel A of the spreadsheet lays out values supplied by Value Line. Entries
for middle years are interpolated from beginning and final values. Panel B calculates free
cash flow. The sum of after-tax profits in row 11 (from Value Line) plus after-tax interest
payments in row 12 [i.e., interest expense 3 (1 2 t c )] equals EBIT(1 2 t c ). In row 13 we subtract the change in net working capital, in row 14 we add back depreciation, and in row
15 we subtract capital expenditures. The result in row 17 is the free cash flow to the firm,
FCFF, for each year between 2013 and 2016.
To find the present value of these cash flows, we will discount at WACC, which is
calculated in panel C. WACC is the weighted average of the after-tax cost of debt and the
9 This is firm cash flow assuming all-equity financing. Any tax advantage to debt financing is recognized by using
an after-tax cost of debt in the computation of weighted-average cost of capital. This issue is discussed in any
introductory corporate finance text.
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C H A P T E R 1 8 Equity Valuation Models 619
cost of equity in each year. When computing WACC, we must account for the change in
leverage forecast by Value Line. To compute the cost of equity, we will use the CAPM as
in our earlier (dividend discount model) valuation exercise, but accounting for the fact that
equity beta will decline each year as the firm reduces leverage. 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
2012 2013 2014 2015 2016 A. Input data P/E 14.35
2.65 30000 1800 3.10
36825
5700.0 702.0
14.25 14.17 14.08 14.00 Cap spending/shr 2.70 2.82 2.93 3.05 LT Debt 28500 27333 26167 25000 Shares 1800 1798 1797 1795 EPS 3.75 3.83 3.92 4.00 Working Capital 37750 41173 44597 48020
B. Cash flow calculations Profits (after tax) 6850.0 6970.0 7090.0 7210.0 Interest (after tax) 666.9 639.6 612.3 585.0 = (1-tax_rate) × r_debt × LT Debt Chg Working Cap 925.0 3423.3 3423.3 3423.3 Depreciation 7250.0 7166.67 7083 7000.0 Cap Spending 4860.0
7000.0 5064.9 5269.8 5474.8
Terminal value FCFF 8981.9 6288.0 6092.5 5896.9 104098.2 FCFE 6815.0 4481.8 4313.5 4145.3 83203.2
C. Discount rate calculations Current beta 0.95 from Value Line
from Value Line
assumes fixed debt ratio after 2016
Unlevered beta 0.767 current beta/[1 + (1-tax)*debt/equity] terminal growth 0.02 tax_rate 0.35 r_debt 0.036 YTM in 2012 on A+ rated LT debt risk-free rate 0.02 market risk prem 0.08 MV equity 81795
0.27 0.950 0.096 0.077
100940 Row 3 × Row 11 Debt/Value 0.25 0.23 0.22 0.20 linear trend from initial to final value Levered beta 0.934 0.919 0.904 0.891 unlevered beta × [1 + (1-tax)*debt/equity] k_equity 0.095 0.094 0.092 0.091 0.091 from CAPM and levered beta WACC 0.077 0.077 0.077 0.078 0.078 (1-t)*r_debt*D/V + k_equity*(1-D/V) PV factor for FCFF 1.000 0.929 0.862 0.800 0.742 0.742 Discount each year at WACC PV factor for FCFE 1.000 0.913 0.835 0.765 0.701 0.701 Discount each year at k_equity
D. Present values Intrinsic val Equity val Intrin/share PV(FCFF) 8341 5421 4875 4378 77283 100298 70298 39.05 PV(FCFE) 6225 3744 3299 2905 58307 74479 74479 41.38
IA C D E F G H J LK MB
Spreadsheet 18.2
Free cash flow valuation of Honda Motor Co. *2012 P/E ratio is from Yahoo! Finance. Other inputs are from Value Line.
e X c e l Please visit us at www.mhhe.com/bkm
10
Call b L the firm’s equity beta at the initial level of leverage as provided by Value Line. Equity betas reflect both
business risk and financial risk. When a firm changes its capital structure (debt/equity mix), it changes financial risk,
and therefore equity beta changes. How should we recognize the change in financial risk? As you may remember
from an introductory corporate finance class, you must first unleverage beta. This leaves us with business risk. We
use the following formula to find unleveraged beta, b U (where D/E is the firm’s current debt-equity ratio):
bU 5 bL
1 1 (D/E)(1 2 tc)
Then, we re-leverage beta in any particular year using the forecast capital structure for that year (which reintro-
duces the financial risk associated with that year’s capital structure):
bL 5 bU31 1 (D/E)(1 2 tc) 4
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620 P A R T V Security Analysis
To find Honda’s cost of debt, we note that its long-term bonds were rated A 1 in 2012
and that yields to maturity on this quality debt at the time were about 3.6%. Honda’s debt-
to-value ratio (assuming its debt is selling near par value) is computed in row 29. In 2012,
the ratio was .27. Based on Value Line forecasts, it will fall to .20 by 2016. We interpolate
the debt-to-value ratio for the intermediate years. WACC is computed in row 32. WACC
increases slightly over time as the debt-to-value ratio declines between 2012 and 2016.
The present value factor for cash flows accruing in each year is the previous year’s factor
divided by (1 1 WACC) for that year. The present value of each cash flow (row 37) is the
free cash flow times the cumulative discount factor.
The terminal value of the firm (cell H17) is computed from the constant-growth model
as FCFF 2016 3 (1 1 g )/(WACC 2016 2 g ), where g (cell B23) is the assumed value for the steady growth rate. We assume in the spreadsheet that g 5 .02, roughly in line with the long-run growth rate of the broad economy.
11 Terminal value is also discounted back to
2012 (cell H37), and the intrinsic value of the firm is thus found as the sum of discounted
free cash flows between 2013 and 2016 plus the discounted terminal value. Finally, the
value of debt in 2012 is subtracted from firm value to arrive at the intrinsic value of equity
in 2012 (cell K37), and value per share is calculated in cell L37 as equity value divided by
number of shares in 2012.
The free cash flow to equity approach yields a similar intrinsic value for the stock. 12
FCFE (row 18) is obtained from FCFF by subtracting after-tax interest expense and net
debt repurchases. The cash flows are then discounted at the equity rate. Like WACC, the
cost of equity changes each period as leverage changes. The present value factor for equity
cash flows is presented in row 34. Equity value is reported in cell J38, which is put on a per
share basis in cell L38.
Spreadsheet 18.2 is available at the Online Learning Center for this text, www.mhhe. com/bkm .
Comparing the Valuation Models In principle, the free cash flow approach is fully consistent with the dividend discount
model and should provide the same estimate of intrinsic value if one can extrapolate to
a period in which the firm begins to pay dividends growing at a constant rate. This was
demonstrated in two famous papers by Modigliani and Miller. 13
However, in practice, you
will find that values from these models may differ, sometimes substantially. This is due
to the fact that in practice, analysts are always forced to make simplifying assumptions.
For example, how long will it take the firm to enter a constant-growth stage? How should
depreciation best be treated? What is the best estimate of ROE? Answers to questions
like these can have a big impact on value, and it is not always easy to maintain consistent
assumptions across the models.
11
In the long run a firm can’t grow forever at a rate higher than the aggregate economy. So by the time we assert
that growth is in a stable stage, it seems reasonable that the growth rate should not be significantly greater than
that of the overall economy (although it can be less if the firm is in a declining industry).
12
Over the 2013–2016 period, Value Line predicts that Honda will retire a considerable fraction of its outstand-
ing debt. The implied debt repurchases are a use of cash and reduce the cash flow available to equity. Such
repurchases cannot be sustained indefinitely, however, for debt outstanding would soon be run down to zero.
Therefore, in our estimate of the terminal value of equity, we compute the final cash flow assuming that starting
in 2016 Honda will begin issuing enough debt to maintain its debt-to-value ratio. This approach is consistent with the assumption of constant growth and constant discount rates after 2016.
13
Franco Modigliani and M. Miller, “The Cost of Capital, Corporation Finance, and the Theory of Investment,”
American Economic Review, June 1958, and “Dividend Policy, Growth, and the Valuation of Shares,” Journal of Business, October 1961.
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C H A P T E R 1 8 Equity Valuation Models 621
We have now valued Honda using several approaches, with estimates of intrinsic value
as follows:
Model Intrinsic Value
Two-stage dividend discount model $38.81 DDM with earnings multiple terminal value 41.62 Three-stage DDM 40.29 Free cash flow to the firm 39.05 Free cash flow to equity 41.38 Market price (from Value Line) 32.88
What should we make of these differences? All the estimates are somewhat higher than
Honda’s actual stock price, perhaps indicating that they use an unrealistically high value
for the ultimate constant growth rate. For example, in the long run, it seems unlikely that
Honda will be able to grow as rapidly as Value Line’s forecast for 2016 growth, 7.5%.
The two-stage dividend discount model is the most conservative of the estimates, largely
because it assumes that Honda’s dividend growth rate will fall to its terminal value after
only 3 years. In contrast, the three-stage DDM allows growth to taper off over a longer
period. Given the consistency with which all these estimates exceed market price, perhaps
the stock is indeed underpriced compared to its intrinsic value.
On balance, however, this valuation exercise suggests that finding bargains is not as easy as
it seems. While these models are easy to apply, establishing proper inputs is more of a
challenge. This should not be surprising. In even a moderately efficient market, finding
profit opportunities will be more involved than analyzing Value Line data for a few hours.
These models are extremely useful to analysts, however, because they provide ballpark
estimates of intrinsic value. More than that, they force rigorous thought about underlying
assumptions and highlight the variables with the greatest impact on value and the greatest
payoff to further analysis.
The Problem with DCF Models Our estimates of Honda’s intrinsic value are all based on discounted cash flow (DCF) mod-
els, in which we calculate the present value of forecasted cash flows and a terminal sales
price at some future date. It is clear from the calculations for Honda that most of the action
in these models is in the terminal value and that this value can be highly sensitive to even
small changes in some input values (see, for example, Concept Check 18.4). Therefore,
you must recognize that DCF valuation estimates are almost always going to be imprecise.
Growth opportunities and future growth rates are especially hard to pin down.
For this reason, many value investors employ a hierarchy of valuation. They view the most
reliable components of value as the items on the balance sheet that allow for the most precise
estimates of market value. Real estate, plant, and equipment would fall in this category.
A somewhat less reliable component of value is the economic profit on assets already in
place. For example, a company like Intel earns a far higher ROE on its investments in chip-
making facilities than its cost of capital. The present value of these “economic profits,” or
economic value added, 14
is a major component of Intel’s market value. This component
of value is less certain than its balance sheet assets, however, because there is always a
concern that new competitors will enter the market, force down prices and profit margins,
and reduce the return on Intel’s investments. Thus, one needs to carefully assess the barri-
ers to entry that protect Intel’s pricing and profit margins. We noted some of these barriers
in the last chapter, where we discussed the role of industry analysis, market structure, and
competitive position (see Section 17.7).
14
We discuss economic value added in greater detail in the next chapter.
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622 P A R T V Security Analysis
Finally, the least reliable components of value are growth opportunities, the purported
ability of firms like Intel to invest in positive-NPV ventures that contribute to high mar-
ket valuations today. Value investors don’t deny that such opportunities exist, but they are
skeptical that precise values can be attached to them and, therefore, tend to be less willing
to make investment decisions that depend on the value of those opportunities.
18.6 The Aggregate Stock Market
The most popular approach to valuing the overall stock market is the earnings multiplier
approach applied at the aggregate level. The first step is to forecast corporate profits for the
coming period. Then we derive an estimate of the earnings multiplier, the aggregate P/E
ratio, based on a forecast of long-term interest rates. The product of the two forecasts is the
estimate of the end-of-period level of the market.
The forecast of the P/E ratio of the market is sometimes derived from a graph similar to
that in Figure 18.8 , which plots the earnings yield (earnings per share divided by price per share, the reciprocal of the P/E ratio) of the S&P 500 and the yield to maturity on 10-year
Treasury bonds. The two series clearly move in tandem over time and suggest that this
relationship and the current yield on 10-year Treasury bonds could help in forecasting the
earnings yield on the S&P 500. Given that earnings yield, a forecast of earnings could be
used to predict the level of the S&P in some future period. Let’s consider a simple example
of this procedure.
Figure 18.8 Earnings yield of S&P 500 versus 10-year Treasury-bond yield
0
2
4
6
8
10
12
14
16
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Y ie
ld (
% )
Treasury Yield
Earnings Yield
In mid-2012, the forecast for 12-month earnings per share for the S&P 500 portfolio was about $110. The 10-year Treasury bond yield was about 2.9%. As a first approach, we might posit that the spread between the earnings yield and the Treasury yield, which was around 3.9%, will remain at that level by the end of the year. Given a Trea- sury yield of 2.9%, this would imply an earnings yield for the S&P of 6.8%, and a
Example 18.6 Forecasting the Aggregate Stock Market
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C H A P T E R 1 8 Equity Valuation Models 623
Some analysts use an aggregate version of the dividend discount model rather than an
earnings multiplier approach. All of these models, however, rely heavily on forecasts of
such macroeconomic variables as GDP, interest rates, and the rate of inflation, which are
difficult to predict accurately.
Because stock prices reflect expectations of future dividends, which are tied to the eco-
nomic fortunes of firms, it is not surprising that the performance of a broad-based stock
index like the S&P 500 is taken as a leading economic indicator, that is, a predictor of the
performance of the aggregate economy. Stock prices are viewed as embodying consen-
sus forecasts of economic activity and are assumed to move up or down in anticipation
of movements in the economy. The government’s index of leading economic indicators,
which is taken to predict the progress of the business cycle, is made up in part of recent
stock market performance. However, the predictive value of the market is far from perfect.
A well-known joke, often attributed to Paul Samuelson, is that the market has forecast
eight of the last five recessions.
P/E ratio of 1/.068 5 14.7. Our forecast for the level of the S&P index would then be 14.7 3 110 5 1,617. Given a current value for the S&P 500 of 1,457, this would imply a 1-year capital gain on the index of 160/1,457 5 11.0%.
Of course, there is uncertainty regarding all three inputs into this analysis: the actual earnings on the S&P 500 stocks, the level of Treasury yields at year-end, and the spread between the Treasury yield and the earnings yield. One would wish to perform sensitivity or scenario analysis to examine the impact of changes in all of these variables. To illus- trate, consider Table 18.4 , which shows a simple scenario analysis treating possible effects of variation in the Treasury bond yield. The scenario analysis shows that forecast level of the stock market varies inversely and with dramatic sensitivity to interest rate changes.
Table 18.4
S&P 500 index fore- casts under various scenarios
Pessimistic Scenario
Most Likely Scenario
Optimistic Scenario
Treasury bond yield 3.4% 2.9% 2.4% Earnings yield 7.3% 6.8% 6.3% Resulting P/E ratio 13.7 14.7 15.9 EPS forecast 110 110 110 Forecast for S&P 500 1,507 1,617 1,749
Forecast for the earnings yield on the S&P 500 equals Treasury bond yield plus 3.9%. The P/E ratio is the reciprocal of the forecast earnings yield.
1. One approach to firm valuation is to focus on the firm’s book value, either as it appears on the balance sheet or as adjusted to reflect current replacement cost of assets or liquidation value.
Another approach is to focus on the present value of expected future dividends.
2. The dividend discount model holds that the price of a share of stock should equal the present value of all future dividends per share, discounted at an interest rate commensurate with the risk
of the stock.
3. Dividend discount models give estimates of the intrinsic value of a stock. If price does not equal intrinsic value, the rate of return will differ from the equilibrium return based on the stock’s risk.
The actual return will depend on the rate at which the stock price is predicted to revert to its
intrinsic value.
SUMMARY
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624 P A R T V Security Analysis
4. The constant-growth version of the DDM asserts that if dividends are expected to grow at a constant rate forever, the intrinsic value of the stock is determined by the formula
V0 5 D1
k 2 g
This version of the DDM is simplistic in its assumption of a constant value of g. There are more-sophisticated multistage versions of the model for more-complex environments. When the
constant-growth assumption is reasonably satisfied and the stock is selling for its intrinsic value,
the formula can be inverted to infer the market capitalization rate for the stock:
k 5 D1 P0 1 g
5. The constant-growth dividend discount model is best suited for firms that are expected to exhibit stable growth rates over the foreseeable future. In reality, however, firms progress through life
cycles. In early years, attractive investment opportunities are ample and the firm responds with
high plowback ratios and rapid dividend growth. Eventually, however, growth rates level off to
more sustainable values. Three-stage growth models are well suited to such a pattern. These
models allow for an initial period of rapid growth, a final period of steady dividend growth, and
a middle, or transition, period in which the dividend growth rate declines from its initial high
rate to the lower sustainable rate.
6. Stock market analysts devote considerable attention to a company’s price-to-earnings ratio. The P/E ratio is a useful measure of the market’s assessment of the firm’s growth opportunities.
Firms with no growth opportunities should have a P/E ratio that is just the reciprocal of the capi-
talization rate, k. As growth opportunities become a progressively more important component of the total value of the firm, the P/E ratio will increase.
7. The expected growth rate of earnings is related both to the firm’s expected profitability and to its dividend policy. The relationship can be expressed as
g 5 (ROE on new investment) 3 (1 2 Dividend payout ratio)
8. You can relate any DDM to a simple capitalized earnings model by comparing the expected ROE on future investments to the market capitalization rate, k. If the two rates are equal, then the stock’s intrinsic value reduces to expected earnings per share (EPS) divided by k.
9. Many analysts form their estimates of a stock’s value by multiplying their forecast of next year’s EPS by a predicted P/E multiple. Some analysts mix the P/E approach with the dividend
discount model. They use an earnings multiplier to forecast the terminal value of shares at a
future date, and add the present value of that terminal value with the present value of all interim
dividend payments.
10. The free cash flow approach is the one used most often in corporate finance. The analyst first estimates the value of the entire firm as the present value of expected future free cash flows to
the entire firm and then subtracts the value of all claims other than equity. Alternatively, the free
cash flows to equity can be discounted at a discount rate appropriate to the risk of the stock.
11. The models presented in this chapter can be used to explain and forecast the behavior of the aggregate stock market. The key macroeconomic variables that determine the level of stock
prices in the aggregate are interest rates and corporate profits.
Related Web sites for this chapter are available at www. mhhe.com/bkm
book value
liquidation value
replacement cost
Tobin’s q intrinsic value of a share
market capitalization rate
dividend discount model (DDM)
constant-growth DDM
dividend payout ratio
plowback ratio
earnings retention ratio
present value of growth
opportunities (PVGO)
price–earnings multiple
earnings management
KEY TERMS
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C H A P T E R 1 8 Equity Valuation Models 625
Intrinsic value: V0 5 D1
1 1 k 1
D2 (1 1 k)2
1 c1 DH 1 PH (1 1 k)H
Constant growth DDM: V0 5 D1
k 2 g
Growth opportunities: Price 5 E1 k 1 PVGO
Determinant of P/E ratio: P0 E1 5
1
k ¢1 1 PVGO
E1/k ≤
Free cash flow to the firm:
FCFF 5 EBIT(1 2 tc) 1 Depreciation 2 Capital expenditures 2 Increases in NWC
Free cash flow to equity: FCFE 5 FCFF 2 Interest expense 3 (1 2 tc) 1 Increases in net debt
KEY EQUATIONS
1. In what circumstances would you choose to use a dividend discount model rather than a free
cash flow model to value a firm?
2. In what circumstances is it most important to use multistage dividend discount models rather
than constant-growth models?
3. I f a security is underpriced (i.e., intrinsic value . price), then what is the relationship between
its market capitalization rate and its expected rate of return?
4. Deployment Specialists pays a current (annual) dividend of $1.00 and is expected to grow at
20% for 2 years and then at 4% thereafter. If the required return for Deployment Specialists is
8.5%, what is the intrinsic value of Deployment Specialists stock?
5. Jand, Inc., currently pays a dividend of $1.22, which is expected to grow indefinitely at 5%.
If the current value of Jand’s shares based on the constant-growth dividend discount model is
$32.03, what is the required rate of return?
6. A firm pays a current dividend of $1.00 which is expected to grow at a rate of 5% indefinitely. If
current value of the firm’s shares is $35.00, what is the required return applicable to the invest-
ment based on the constant-growth dividend discount model (DDM)?
7. Tri-coat Paints has a current market value of $41 per share with earnings of $3.64. What is the
present value of its growth opportunities (PVGO) if the required return is 9%?
8. a. Computer stocks currently provide an expected rate of return of 16%. MBI, a large com- puter company, will pay a year-end dividend of $2 per share. If the stock is selling at $50 per
share, what must be the market’s expectation of the growth rate of MBI dividends?
b. I f dividend growth forecasts for MBI are revised downward to 5% per year, what will hap- pen to the price of MBI stock? What (qualitatively) will happen to the company’s price–
earnings ratio?
9. a. MF Corp. has an ROE of 16% and a plowback ratio of 50%. If the coming year’s earnings are expected to be $2 per share, at what price will the stock sell? The market capitalization
rate is 12%.
b. What price do you expect MF shares to sell for in 3 years?
10. The market consensus is that Analog Electronic Corporation has an ROE 5 9%, has a beta of
1.25, and plans to maintain indefinitely its traditional plowback ratio of 2/3. This year’s earn-
ings were $3 per share. The annual dividend was just paid. The consensus estimate of the com-
ing year’s market return is 14%, and T-bills currently offer a 6% return.
a. Find the price at which Analog stock should sell. b. Calculate the P/E ratio.
Basic
Intermediate
PROBLEM SETS
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626 P A R T V Security Analysis
c. Calculate the present value of growth opportunities. d. Suppose your research convinces you Analog will announce momentarily that it will immediately
reduce its plowback ratio to 1/3. Find the intrinsic value of the stock. The market is still unaware
of this decision. Explain why V 0 no longer equals P 0 and why V 0 is greater or less than P 0 .
11. The FI Corporation’s dividends per share are expected to grow indefinitely by 5% per year.
a. If this year’s year-end dividend is $8 and the market capitalization rate is 10% per year, what must the current stock price be according to the DDM?
b. If the expected earnings per share are $12, what is the implied value of the ROE on future investment opportunities?
c. How much is the market paying per share for growth opportunities (i.e., for an ROE on future investments that exceeds the market capitalization rate)?
12. The stock of Nogro Corporation is currently selling for $10 per share. Earnings per share in the
coming year are expected to be $2. The company has a policy of paying out 50% of its earn-
ings each year in dividends. The rest is retained and invested in projects that earn a 20% rate of
return per year. This situation is expected to continue indefinitely.
a. Assuming the current market price of the stock reflects its intrinsic value as computed using the constant-growth DDM, what rate of return do Nogro’s investors require?
b. By how much does its value exceed what it would be if all earnings were paid as dividends and nothing were reinvested?
c. If Nogro were to cut its dividend payout ratio to 25%, what would happen to its stock price? What if Nogro eliminated the dividend?
13. The risk-free rate of return is 8%, the expected rate of return on the market portfolio is 15%,
and the stock of Xyrong Corporation has a beta coefficient of 1.2. Xyrong pays out 40% of its
earnings in dividends, and the latest earnings announced were $10 per share. Dividends were
just paid and are expected to be paid annually. You expect that Xyrong will earn an ROE of 20%
per year on all reinvested earnings forever.
a. What is the intrinsic value of a share of Xyrong stock? b. If the market price of a share is currently $100, and you expect the market price to be equal
to the intrinsic value 1 year from now, what is your expected 1-year holding-period return on
Xyrong stock?
14. The Digital Electronic Quotation System (DEQS) Corporation pays no cash dividends currently
and is not expected to for the next 5 years. Its latest EPS was $10, all of which was reinvested
in the company. The firm’s expected ROE for the next 5 years is 20% per year, and during this
time it is expected to continue to reinvest all of its earnings. Starting in year 6, the firm’s ROE
on new investments is expected to fall to 15%, and the company is expected to start paying out
40% of its earnings in cash dividends, which it will continue to do forever after. DEQS’s market
capitalization rate is 15% per year.
a. What is your estimate of DEQS’s intrinsic value per share? b. Assuming its current market price is equal to its intrinsic value, what do you expect to hap-
pen to its price over the next year? The year after?
c. What effect would it have on your estimate of DEQS’s intrinsic value if you expected DEQS to pay out only 20% of earnings starting in year 6?
15. Recalculate the intrinsic value of Honda in each of the following scenarios by using the three-
stage growth model of Spreadsheet 18.1 (available at www.mhhe.com/bkm ; link to Chapter 18 material). Treat each scenario independently.
a. ROE in the constant-growth period will be 9%. b. Honda’s actual beta is 1.0. c. The market risk premium is 8.5%.
16. Recalculate the intrinsic value of Honda shares using the free cash flow model of Spreadsheet
18.2 (available at www.mhhe.com/bkm ; link to Chapter 18 material) under each of the follow- ing assumptions. Treat each scenario independently.
e X c e l Please visit us at www.mhhe.com/bkm
e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 1 8 Equity Valuation Models 627
a. Honda’s P/E ratio starting in 2016 will be 15. b. Honda’s unlevered beta is 0.7. c. The market risk premium is 9%.
17. The Duo Growth Company just paid a dividend of $1 per share. The dividend is expected to
grow at a rate of 25% per year for the next 3 years and then to level off to 5% per year forever.
You think the appropriate market capitalization rate is 20% per year.
a. What is your estimate of the intrinsic value of a share of the stock? b. If the market price of a share is equal to this intrinsic value, what is the expected dividend
yield?
c. What do you expect its price to be 1 year from now? Is the implied capital gain consistent with your estimate of the dividend yield and the market capitalization rate?
18. The Generic Genetic (GG) Corporation pays no cash dividends currently and is not
expected to for the next 4 years. Its latest EPS was $5, all of which was reinvested in the
company. The firm’s expected ROE for the next 4 years is 20% per year, during which time
it is expected to continue to reinvest all of its earnings. Starting in year 5, the firm’s ROE on
new investments is expected to fall to 15% per year. GG’s market capitalization rate is 15%
per year.
a. What is your estimate of GG’s intrinsic value per share? b. Assuming its current market price is equal to its intrinsic value, what do you expect to
happen to its price over the next year?
19. The MoMi Corporation’s cash flow from operations before interest and taxes was $2 million
in the year just ended, and it expects that this will grow by 5% per year forever. To make this
happen, the firm will have to invest an amount equal to 20% of pretax cash flow each year. The
tax rate is 35%. Depreciation was $200,000 in the year just ended and is expected to grow at the
same rate as the operating cash flow. The appropriate market capitalization rate for the unlever-
aged cash flow is 12% per year, and the firm currently has debt of $4 million outstanding. Use
the free cash flow approach to value the firm’s equity.
20. C hiptech, Inc., is an established computer chip firm with several profitable existing products as
well as some promising new products in development. The company earned $1 a share last year,
and just paid out a dividend of $.50 per share. Investors believe the company plans to maintain
its dividend payout ratio at 50%. ROE equals 20%. Everyone in the market expects this situa-
tion to persist indefinitely.
a. What is the market price of Chiptech stock? The required return for the computer chip indus- try is 15%, and the company has just gone ex-dividend (i.e., the next dividend will be paid a
year from now, at t 5 1). b. Suppose you discover that Chiptech’s competitor has developed a new chip that will elimi-
nate Chiptech’s current technological advantage in this market. This new product, which
will be ready to come to the market in 2 years, will force Chiptech to reduce the prices of
its chips to remain competitive. This will decrease ROE to 15%, and, because of falling
demand for its product, Chiptech will decrease the plowback ratio to .40. The plowback ratio
will be decreased at the end of the second year, at t 5 2: The annual year-end dividend for the second year (paid at t 5 2) will be 60% of that year’s earnings. What is your estimate of Chiptech’s intrinsic value per share? ( Hint: Carefully prepare a table of Chiptech’s earnings and dividends for each of the next 3 years. Pay close attention to the change in the payout
ratio in t 5 2.) c. No one else in the market perceives the threat to Chiptech’s market. In fact, you are confi-
dent that no one else will become aware of the change in Chiptech’s competitive status until
the competitor firm publicly announces its discovery near the end of year 2. What will be the
rate of return on Chiptech stock in the coming year (i.e., between t 5 0 and t 5 1)? In the second year (between t 5 1 and t 5 2)? The third year (between t 5 2 and t 5 3)? ( Hint: Pay attention to when the market catches on to the new situation. A table of dividends and market prices over time might help.)
Challenge
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1. At Litchfield Chemical Corp. (LCC), a director of the company said that the use of dividend
discount models by investors is “proof ” that the higher the dividend, the higher the stock price.
a. Using a constant-growth dividend discount model as a basis of reference, evaluate the direc- tor’s statement.
b. Explain how an increase in dividend payout would affect each of the following (holding all other factors constant):
i. Sustainable growth rate.
ii. Growth in book value.
2. Helen Morgan, CFA, has been asked to use the DDM to determine the value of Sundanci, Inc.
Morgan anticipates that Sundanci’s earnings and dividends will grow at 32% for 2 years and 13%
thereafter. Calculate the current value of a share of Sundanci stock by using a two-stage dividend
discount model and the data from Tables 18A and 18B .
3. Abbey Naylor, CFA, has been directed to determine the value of Sundanci’s stock using the Free
Cash Flow to Equity (FCFE) model. Naylor believes that Sundanci’s FCFE will grow at 27%
for 2 years and 13% thereafter. Capital expenditures, depreciation, and working capital are all
expected to increase proportionately with FCFE.
a. Calculate the amount of FCFE per share for the year 2011, using the data from Table 18A . b. Calculate the current value of a share of Sundanci stock based on the two-stage FCFE
model.
c. i. Describe one limitation of the two-stage DDM model that is addressed by using the two- stage FCFE model.
ii. Describe one limitation of the two-stage DDM model that is not addressed by using the two-stage FCFE model.
Income Statement 2010 2011
Revenue $ 474 $ 598
Depreciation 20 23
Other operating costs 368 460
Income before taxes 86 115
Taxes 26 35
Net income 60 80
Dividends 18 24
Earnings per share $0.714 $0.952
Dividend per share $0.214 $0.286
Common shares outstanding (millions) 84.0 84.0
Balance Sheet 2010 2011
Current assets $ 201 $ 326
Net property, plant and equipment 474 489
Total assets 675 815
Current liabilities 57 141
Long-term debt 0 0
Total liabilities 57 141
Shareholders’ equity 618 674
Total liabilities and equity 675 815
Capital expenditures 34 38
Table 18A
Sundanci actual 2010 and 2011 financial statements for fiscal years ending May 31 ($ million, except per- share data)
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C H A P T E R 1 8 Equity Valuation Models 629
4. Christie Johnson, CFA, has been assigned to analyze Sundanci using the constant dividend growth
price/earnings (P/E) ratio model. Johnson assumes that Sundanci’s earnings and dividends will
grow at a constant rate of 13%.
a. Calculate the P/E ratio based on information in Tables 18A and 18B and on Johnson’s assump- tions for Sundanci.
b. Identify, within the context of the constant dividend growth model, how each of the following factors would affect the P/E ratio.
• Risk (beta) of Sundanci.
• Estimated growth rate of earnings and dividends.
• Market risk premium.
5. Dynamic Communication is a U.S. industrial company with several electronics divisions. The
company has just released its 2013 annual report. Tables 18C and 18D present a summary of
Dynamic’s financial statements for the years 2012 and 2013. Selected data from the financial
statements for the years 2009 to 2011 are presented in Table 18E .
a. A group of Dynamic shareholders has expressed concern about the zero growth rate of divi- dends in the last 4 years and has asked for information about the growth of the company.
Calculate Dynamic’s sustainable growth rates in 2010 and 2013. Your calculations should use
beginning-of-year balance sheet data.
b. Determine how the change in Dynamic’s sustainable growth rate (2013 compared to 2010) was affected by changes in its retention ratio and its financial leverage. ( Note: Your calcula- tions should use beginning-of-year balance sheet data.)
Table 18C
Dynamic Communica- tion balance sheets
$ Million
2013 2012
Cash and equivalents $ 149 $ 83
Accounts receivable 295 265
Inventory 275 285
Total current assets $ 719 $ 633
Gross fixed assets 9,350 8,900
Accumulated depreciation (6,160) (5,677)
Net fixed assets $3,190 $3,223
Total assets $3,909 $3,856
Accounts payable $ 228 $ 220
Notes payable 0 0
Accrued taxes and expenses 0 0
Total current liabilities $ 228 $ 220
Long-term debt $1,650 $1,800
Common stock 50 50
Additional paid-in capital 0 0
Retained earnings 1,981 1,786
Total shareholders’ equity $2,031 $1,836
Total liabilities and shareholders’ equity $3,909 $3,856
Required rate of return on equity 14%
Growth rate of industry 13%
Industry P/E ratio 26
Table 18B
Selected financial information
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630 P A R T V Security Analysis
6. Mike Brandreth, an analyst who specializes in the electronics industry, is preparing a research
report on Dynamic Communication. A colleague suggests to Brandreth that he may be able to
determine Dynamic’s implied dividend growth rate from Dynamic’s current common stock price,
using the Gordon growth model. Brandreth believes that the appropriate required rate of return
for Dynamic’s equity is 8%.
a. Assume that the firm’s current stock price of $58.49 equals intrinsic value. What sustained rate of dividend growth as of December 2013 is implied by this value? Use the constant-
growth dividend discount model (i.e., the Gordon growth model).
b. The management of Dynamic has indicated to Brandreth and other analysts that the com- pany’s current dividend policy will be continued. Is the use of the Gordon growth model to
value Dynamic’s common stock appropriate or inappropriate? Justify your response based on
the assumptions of the Gordon growth model.
7. Peninsular Research is initiating coverage of a mature manufacturing industry. John Jones, CFA,
head of the research department, gathered the following fundamental industry and market data to
help in his analysis:
Forecast industry earnings retention rate 40% Forecast industry return on equity 25% Industry beta 1.2 Government bond yield 6% Equity risk premium 5%
2013 2012
Total revenues $3,425 $3,300
Operating costs and expenses 2,379 2,319
Earnings before interest, taxes, depreciation and amortization (EBITDA)
$1,046 $ 981
Depreciation and amortization 483 454
Operating income (EBIT) $ 563 $ 527
Interest expense 104 107
Income before taxes $ 459 $ 420
Taxes (40%) 184 168
Net income $ 275 $ 252
Dividends $ 80 $ 80
Change in retained earnings $ 195 $ 172
Earnings per share $ 2.75 $ 2.52
Dividends per share $ 0.80 $ 0.80
Number of shares outstanding (millions) 100 100
Table 18D
Dynamic Communica- tion statements of income (U.S. $ millions except for share data)
2011 2010 2009
Total revenues $3,175 $3,075 $3,000 Operating income (EBIT) 495 448 433 Interest expense 104 101 99 Net income $ 235 $ 208 $ 200 Dividends per share $ 0.80 $ 0.80 $ 0.80 Total assets $3,625 $3,414 $3,230 Long-term debt $1,750 $1,700 $1,650 Total shareholders’ equity $1,664 $1,509 $1,380
Number of shares outstanding (millions) 100 100 100
Table 18E
Dynamic Communica- tion selected data from financial state- ments (U.S. $ millions except for share data)
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C H A P T E R 1 8 Equity Valuation Models 631
a. Compute the price-to-earnings ( P 0 / E 1 ) ratio for the industry based on this fundamental data. b. Jones wants to analyze how fundamental P/E ratios might differ among countries. He gath-
ered the following economic and market data:
Fundamental Factors Country A Country B
Forecast growth in real GDP 5% 2% Government bond yield 10% 6% Equity risk premium 5% 4%
Determine whether each of these fundamental factors would cause P/E ratios to be generally higher for Country A or higher for Country B.
8. Janet Ludlow’s firm requires all its analysts to use a two-stage dividend discount model (DDM)
and the capital asset pricing model (CAPM) to value stocks. Using the CAPM and DDM, Ludlow
has valued QuickBrush Company at $63 per share. She now must value SmileWhite Corporation.
a. Calculate the required rate of return for SmileWhite by using the information in the follow- ing table:
QuickBrush SmileWhite
Beta 1.35 1.15 Market price $45.00 $30.00 Intrinsic value $63.00 ? Notes: Risk-free rate 4.50% Expected market return 14.50%
b. Ludlow estimates the following EPS and dividend growth rates for SmileWhite:
First 3 years 12% per year
Years thereafter 9% per year
Estimate the intrinsic value of SmileWhite by using the table above, and the two-stage DDM. Dividends per share in the most recent year were $1.72.
c. Recommend QuickBrush or SmileWhite stock for purchase by comparing each company’s intrinsic value with its current market price.
d. Describe one strength of the two-stage DDM in comparison with the constant-growth DDM. Describe one weakness inherent in all DDMs.
9. Rio National Corp. is a U.S.-based company and the largest competitor in its industry. Tables
18F – 18I present financial statements and related information for the company. Table 18J pre-
sents relevant industry and market data.
The portfolio manager of a large mutual fund comments to one of the fund’s analysts, Katrina
Shaar: “We have been considering the purchase of Rio National Corp. equity shares, so I would
like you to analyze the value of the company. To begin, based on Rio National’s past perfor-
mance, you can assume that the company will grow at the same rate as the industry.”
a. Calculate the value of a share of Rio National equity on December 31, 2013, using the Gordon constant-growth model and the capital asset pricing model.
b. Calculate the sustainable growth rate of Rio National on December 31, 2013. Use 2013 beginning-of-year balance sheet values.
10. While valuing the equity of Rio National Corp. (from the previous problem), Katrina Shaar
is considering the use of either cash flow from operations (CFO) or free cash flow to equity
(FCFE) in her valuation process.
a. State two adjustments that Shaar should make to cash flow from operations to obtain free cash flow to equity.
b. Shaar decides to calculate Rio National’s FCFE for the year 2013, starting with net income. Determine for each of the five supplemental notes given in Table 18H whether an adjustment
should be made to net income to calculate Rio National’s free cash flow to equity for the
year 2013, and the dollar amount of any adjustment.
c. Calculate Rio National’s free cash flow to equity for the year 2013.
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632 P A R T V Security Analysis
2013 2012
Cash $ 13.00 $ 5.87 Accounts receivable 30.00 27.00 Inventory 209.06 189.06
Current assets $252.06 $221.93 Gross fixed assets 474.47 409.47 Accumulated depreciation (154.17 ) (90.00 )
Net fixed assets 320.30 319.47
Total assets $572.36 $541.40 Accounts payable $ 25.05 $ 26.05 Notes payable 0.00 0.00 Current portion of long-term debt 0.00 0.00
Current liabilities $ 25.05 $ 26.05 Long-term debt 240.00 245.00
Total liabilities $265.05 $271.05 Common stock 160.00 150.00 Retained earnings 147.31 120.35
Total shareholders’ equity $307.31 $270.35
Total liabilities and shareholders’ equity $572.36 $541.40
Table 18F
Rio National Corp. summary year-end balance sheets (U.S. $ millions)
Revenue $300.80 Total operating expenses (173.74 )
Operating profit 127.06 Gain on sale 4.00
Earnings before interest, taxes, depreciation & amortization (EBITDA)
131.06
Depreciation and amortization (71.17 )
Earnings before interest & taxes (EBIT) 59.89 Interest (16.80) Income tax expense (12.93 )
Net income $ 30.16
Table 18G
Rio National Corp. summary income statement for the year ended December 31, 2013 (U.S. $ millions)
Note 1: Rio National had $75 million in capital expenditures during the year. Note 2: A piece of equipment that was originally purchased for $10 million was sold for
$7 million at year-end, when it had a net book value of $3 million. Equipment sales are unusual for Rio National.
Note 3: The decrease in long-term debt represents an unscheduled principal repayment; there was no new borrowing during the year.
Note 4: On January 1, 2013, the company received cash from issuing 400,000 shares of common equity at a price of $25.00 per share.
Note 5: A new appraisal during the year increased the estimated market value of land held for investment by $2 million, which was not recognized in 2013 income.
Table 18H
Rio National Corp. supplemental notes for 2013
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C H A P T E R 1 8 Equity Valuation Models 633
Dividends paid (U.S. $ millions) $3.20
Weighted-average shares outstanding during 2013 16,000,000 Dividend per share $0.20 Earnings per share $1.89 Beta 1.80
Note: The dividend payout ratio is expected to be constant.
Table 18I
Rio National Corp. common equity data for 2013
Risk-free rate of return 4.00% Expected rate of return on market index 9.00% Median industry price/earnings (P/E) ratio 19.90 Expected industry earnings growth rate 12.00%
Table 18J
Industry and market data December 31, 2013
11. Shaar (from the previous problem) has revised slightly her estimated earnings growth rate for
Rio National and, using normalized (underlying) EPS, which is adjusted for temporary impacts
on earnings, now wants to compare the current value of Rio National’s equity to that of the
industry, on a growth-adjusted basis. Selected information about Rio National and the industry
is given in Table 18K .
Compared to the industry, is Rio National’s equity overvalued or undervalued on a P/E-to-
growth (PEG) basis, using normalized (underlying) earnings per share? Assume that the risk of
Rio National is similar to the risk of the industry.
Rio National
Estimated earnings growth rate 11.00%
Current share price $25.00 Normalized (underlying) EPS for 2011 $1.71 Weighted-average shares outstanding during 2011 16,000,000
Industry
Estimated earnings growth rate 12.00% Median price/earnings (P/E) ratio 19.90
Table 18K
Rio National Corp. vs. industry
E-INVESTMENTS EXERCISES Go to the MoneyCentral Investor page at moneycentral.msn.com/investor/home.asp . Use the Research Wizard function under Guided Research to obtain fundamentals, price history, price target, catalysts, and comparison for Walmart (WMT). For comparison, use Target (TGT), BJ’s Wholesale Club (BJ), and the industry.
1. What has been the 1-year sales and income growth for Walmart?
2. What has been the company’s 5-year profit margin? How does that compare with the other two firms’ profit margins and the industry’s profit margin?
3. What have been the percentage price changes for the last 3, 6, and 12 months? How do they compare with the other firms’ price changes and the industry’s price changes?
4. What are the estimated high and low prices for Walmart for the coming year based on its current P/E multiple?
5. Compare the price performance of Walmart with that of Target and BJ’s. Which of the companies appears to be the most expensive in terms of current earnings? Which of the companies is the least expensive in terms of current earnings?
6. What are the firms’ Stock Scouter Ratings? How are these ratings interpreted?
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634 P A R T V Security Analysis
SOLUTIONS TO CONCEPT CHECKS 1. a. Dividend yield 5 $2.15/$50 5 4.3%. Capital gains yield 5 (59.77 2 50)/50 5 19.54%. Total return 5 4.3% 1 19.54% 5 23.84%.
b. k 5 6% 1 1.15(14% 2 6%) 5 15.2%.
c. V 0 5 ($2.15 1 $59.77)/1.152 5 $53.75, which exceeds the market price. This would indicate a “buy” opportunity.
2. a. D 1 /( k 2 g ) 5 $2.15/(.152 2 .112) 5 $53.75.
b. P 1 5 P 0 (1 1 g ) 5 $53.75(1.112) 5 $59.77.
c. The expected capital gain equals $59.77 2 $53.75 5 $6.02, for a percentage gain of 11.2%. The dividend yield is D 1 / P 0 5 2.15/53.75 5 4%, for a holding-period return of 4% 1 11.2% 5 15.2%.
3. a. g 5 ROE 3 b 5 20% 3 .60 5 12%.
D1 5 .4 3 E1 5 .4 3 $5 5 $2. P0 5 $2/(.125 2 .12) 5 $400.
b. When the firm invests in projects with ROE less than k, its stock price falls. If b 5 .60, then g 5 10% 3 .60 5 6% and P 0 5 $2/(.125 2 .06) 5 $30.77. In contrast, if b 5 0, then P 0 5 $5/.125 5 $40.
4. V2012 5 .78
(1.096) 1
.85
(1.096)2 1
.92
(1.096)3 1
1.00 1 P2016 (1.096)4
Now compute the sales price in 2016 using the constant-growth dividend discount model. The
growth rate will be g 5 ROE 3 b 5 9% 3 .75 5 6.75%.
P2016 5 1.00 3 (1 1 g)
k 2 g 5
$1.0675
.096 2 .0675 5 $37.46
Therefore, V 2012 5 $28.77.
5. a. ROE 5 12%. b 5 $.50/$2.00 5 .25. g 5 ROE 3 b 5 12% 3 .25 5 3%. P0 5 D1/(k 2 g) 5 $1.50/(.10 2 .03) 5 $21.43. P0/E1 5 $21.43/$2.00 5 10.71.
b. If b 5 .4, then .4 3 $2 5 $.80 would be reinvested and the remainder of earnings, or $1.20, would be paid as dividends.
g 5 12% 3 .4 5 4.8%. P0 5 D1/(k 2 g) 5 $1.20/(.10 2 .048) 5 $23.08. P0/E1 5 $23.08/$2.00 5 11.54.
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199 PA
R T
V
IN THE PREVIOUS chapter, we explored equity valuation techniques. These techniques take the firm’s dividends and earnings prospects as inputs. Although the valuation analyst is interested in economic earnings streams, only financial accounting data are readily available. What can we learn from a company’s account- ing data that can help us estimate the intrinsic value of its common stock? In this chapter, we show how investors can use financial data as inputs into stock valuation analysis. We start by reviewing the basic sources of such data— the income statement, the balance sheet, and the statement of cash flows. We note the dif- ference between economic and accounting earnings. Although economic earnings are
more important for issues of valuation, they can at best be estimated, so, in practice, ana- lysts always begin their evaluation of the firm using accounting data. We show how analysts use financial ratios to explore the sources of a firm’s profitability and evaluate the “quality” of its earnings in a systematic fashion. We also examine the impact of debt policy on various financial ratios.
Finally, we conclude with a discussion of the challenges you will encounter when using financial statement analysis as a tool in uncov- ering mispriced securities. Some of these issues arise from differences in firms’ account- ing procedures. Others are due to inflation- induced distortions in accounting numbers.
Financial Statement
Analysis
CHAPTER NINETEEN
The Income Statement The income statement is a summary of the profitability of the firm over a period of time, such as a year. It presents revenues generated during the operating period, the expenses
incurred during that same period, and the company’s net earnings or profits, which are
simply the difference between revenues and expenses.
It is useful to distinguish four broad classes of expenses: cost of goods sold, which is
the direct cost attributable to producing the product sold by the firm; general and adminis-
trative expenses, which correspond to overhead expenses, salaries, advertising, and other
costs of operating the firm that are not directly attributable to production; interest expense
on the firm’s debt; and taxes on earnings owed to federal and local governments.
19.1 The Major Financial Statements
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636 P A R T V Security Analysis
Table 19.1 presents an income statement for Home Depot (HD). At the top are the
company’s revenues from operations. Next come operating expenses, the costs incurred
in the course of generating those revenues, including a depreciation allowance. The differ-
ence between operating revenues and operating costs is called operating income. Income or expenses from other, primarily nonrecurring, sources are then added or subtracted to
obtain earnings before interest and taxes (EBIT), which is what the firm would have earned
if not for obligations to its creditors and the tax authorities. EBIT is a measure of the profit-
ability of the firm’s operations, ignoring any interest burden attributable to debt financing.
The income statement then goes on to subtract net interest expense from EBIT to arrive at
taxable income. Finally, the income tax due the government is subtracted to arrive at net
income, the “bottom line” of the income statement.
Analysts also commonly prepare a common-size income statement, in which all items on the income statement are expressed as a fraction of total revenue. This makes it easier
to compare firms of different sizes. The right-hand column of Table 19.1 is Home Depot’s
common-size income statement.
In the previous chapter, we saw that stock valuation models require a measure of
economic earnings —the sustainable cash flow that can be paid out to stockholders without impairing the productive capacity of the firm. In contrast, accounting earnings are affected by several conventions regarding the valuation of assets such as inventories (e.g., LIFO
versus FIFO treatment), and by the way some expenditures such as capital investments are
recognized over time (as depreciation expenses). We discuss problems with some of these
accounting conventions in greater detail later in the chapter. In addition to these accounting
issues, as the firm makes its way through the business cycle, its earnings will rise above or
fall below the trend line that might more accurately reflect sustainable economic earnings.
This introduces an added complication in interpreting net income figures. One might won-
der how closely accounting earnings approximate economic earnings and, correspondingly,
how useful accounting data might be to investors attempting to value the firm.
$ Million
Percent of Revenue
Operating revenues Net sales 70,395 100.0% Operating expenses Cost of goods sold 46,133 65.5% Selling, general & administrative expenses 14,346 20.4% Other 1,560 2.2% Depreciation 1,682 2.4% Earnings before interest and income taxes 6,674 9.5% Interest expense 606 0.9%
Taxable income 6,068 8.6% Taxes 2,185 3.1%
Net income 3,883 5.5% Allocation of net income Dividends 1,632 2.3% Addition to retained earnings 2,251 3.2%
Note: Sums subject to rounding error. Source: Home Depot Annual Report, year ending January 2012.
Table 19.1
Consolidated statement of income for Home Depot
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C H A P T E R 1 9 Financial Statement Analysis 637
In fact, the net income figure on the firm’s income statement does convey considerable
information concerning a firm’s prospects. We see this in the fact that stock prices tend
to increase when firms announce earnings greater than market analysts or investors had
anticipated.
The Balance Sheet While the income statement provides a measure of profitability over a period of time, the
balance sheet provides a “snapshot” of the financial condition of the firm at a particular moment. The balance sheet is a list of the firm’s assets and liabilities at that moment. The
difference in assets and liabilities is the net worth of the firm, also called shareholders’ or stockholders’ equity. Like income statements, balance sheets are reasonably standardized in presentation. Table 19.2 is HD’s balance sheet.
The first section of the balance sheet gives a listing of the assets of the firm. Current
assets are presented first. These are cash and other items such as accounts receivable or
inventories that will be converted into cash within 1 year. Next comes a listing of long-term
or “fixed” assets. Tangible fixed assets are items such as buildings, equipment, or vehicles. HD also has several intangible assets such as a respected brand name and expertise. But
accountants generally are reluctant to include these assets on the balance sheet, as they are
so hard to value. However, when one firm purchases another for a premium over its book
value, that difference, called “goodwill,” is listed on the balance sheet as an intangible fixed asset. HD lists goodwill at $1,120 million. 1 The sum of current and fixed assets is total assets, the last line of the assets section of the balance sheet.
The liability and shareholders’ equity (also called stockholders’ equity) section is
arranged similarly. First come short-term, or “current,” liabilities such as accounts pay-
able, accrued taxes, and debts that are due within 1 year. Following this is long-term debt
and other liabilities due in more than 1 year. The difference between total assets and total
liabilities is stockholders’ equity. This is the net worth, or book value, of the firm. Stock-
holders’ equity is divided into par value of stock, additional paid-in capital, and retained
earnings, although this division is usually unimportant. Briefly, par value plus additional
paid-in capital represent the proceeds realized from the sale of stock to the public, whereas
retained earnings represent the buildup of equity from profits plowed back into the firm.
Even if the firm issues no new equity, book value typically will increase each year due to
reinvested earnings.
The entries in the left columns of the balance sheet in Table 19.2 present the dollar
value of each asset. Just as they compute common-size income statements, however, ana-
lysts also find it convenient to use common-size balance sheets when comparing firms of different sizes. Each item is expressed as a percentage of total assets. These entries appear
in the right columns of Table 19.2 .
The Statement of Cash Flows The income statement and balance sheets are based on accrual methods of accounting,
which means that revenues and expenses are recognized at the time of a sale even if no
cash has yet been exchanged. In contrast, the statement of cash flows tracks the cash
1 Firms are required to test their goodwill assets for “impairment” each year. If the value of the acquired firm is
clearly less than its purchase price, that amount must be charged off as an expense. For example, in 2012 Hewlett-
Packard wrote off $8.8 billion on its earlier purchase of the software company Autonomy Corp. amid charges that
Autonomy had overstated its profitability prior to the purchase. AOL Time Warner set a record when it recognized
an impairment of $99 billion in 2002 following the January 2001 merger of Time Warner with AOL.
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6 3 8
P A
R T
V
S e c u rity
A n a ly
sis
Assets $ Million Percent of
Total Assets Liabilities and Shareholders’ Equity $ Million Percent of
Total Assets
Current assets Current liabilities Cash and marketable securities 1,987 4.9% Debt due for repayment 30 0.1% Receivables 1,245 3.1% Accounts payable 8,199 20.2% Inventories 10,325 25.5% Other current liabilities 1,147 2.8% Other current assets 963 2.4% Total current liabilities 9,376 23.1%
Total current assets 14,520 35.8%
Fixed Assets Long-term debt 10,758 26.6% Tangible fixed assets Other long-term liabilities 2,486 6.1% Property, plant, and equipment 24,448 60.3% Other long-term assets 430 1.1% Total liabilities 22,620 55.8%
Total tangible fixed assets 24,878 61.4% Shareholders’ equity
Intangible fixed assets Common stock and other paid-in capital 652 1.6%
Goodwill 1,120 2.8% Retained earnings 17,246 42.6%
Total fixed assets 25,998 64.2% Total shareholders’ equity 17,898 44.2%
Total assets 40,518 100.0% Total liabilities and shareholders’ equity 40,518 100.0%
Table 19.2
Consolidated balance sheet for Home Depot
Note: Column sums subject to rounding error. Source: Home Depot Annual Report, year ending January 2012.
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C H A P T E R 1 9 Financial Statement Analysis 639
implications of transactions. For example, if goods are sold now, with payment due in
60 days, the income statement will treat the revenue as generated when the sale occurs, and
the balance sheet will be immediately augmented by accounts receivable, but the statement
of cash flows will not show an increase in available cash until the bill is paid.
Table 19.3 is the statement of cash flows for HD. The first entry listed under cash
provided by operations is net income. The next entries modify that figure for components
of income that have been recognized but for which cash has not yet changed hands. For
example, HD’s accounts receivable increased by $170 million. This portion of sales was
claimed on the income statement, but the cash had not yet been collected. Increases in
accounts receivable are in effect an investment in working capital, and therefore reduce
the cash flows realized from operations. Similarly, increases in accounts payable mean
that expenses have been recognized, but cash has not yet left the firm. Any payment delay
increases the company’s net cash flows in this period.
Another major difference between the income statement and the statement of cash flows
involves depreciation, which is a major addition to income in the adjustment section of the
statement of cash flows in Table 19.3 . The income statement attempts to “smooth” large
capital expenditures over time. The depreciation expense on the income statement does this
by recognizing such expenditures over a period of many years rather than at the specific
time of purchase. In contrast, the statement of cash flows recognizes the cash implication
of a capital expenditure when it occurs. Therefore, it adds back the depreciation “expense”
Source: Home Depot Annual Report, year ending January 2012.
$ Million
Cash provided by operations Net income $ 3,883 Adjustments to net income Depreciation 1,682 Changes in working capital Decrease (increase) in receivables (170) Decrease (increase) in inventories 256 Increase (decrease) in other current liabilities 405
Changes due to other operating activities 595 Total adjustments $ 2,768 Cash provided by operations 6,651
Cash flows from investments Gross investment in tangible fixed assets $(1,221)
Investments in other assets 92 Cash provided by (used for) investments $(1,129)
Cash Provided by (Used for) Financing Activities Additions to (reductions in) long-term debt $ 966 Net issues (repurchases of) shares (3,164) Dividends (1,632) Other (218)
Cash provided by (used for) financing activities $(4,048)
Net increase in cash $ 1,474
Table 19.3
Statement of cash flows for Home Depot
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640 P A R T V Security Analysis
that was used to compute net income; instead, it acknowledges a capital expenditure when
it is paid. It does so by reporting cash flows separately for operations, investing, and
financing activities. This way, any large cash flows, such as those for big investments, can
be recognized without affecting the measure of cash provided by operations.
The second section of the statement of cash flows is the accounting of cash flows from
investing activities. For example, HD used $1,221 million of cash investing in tangible
fixed assets. These entries are investments in the assets necessary for the firm to maintain
or enhance its productive capacity.
Finally, the last section of the statement lists the cash flows realized from financing
activities. Issuance of securities will contribute positive cash flows, while repurchase or
redemption of outstanding securities uses up cash. For example, HD expended $3,164 million
to repurchase shares of its stock, which was a major use of cash. Its dividend payments,
$1,632 million, also used cash. In total, HD’s financing activities absorbed $4,048 million.
To summarize, HD’s operations generated a cash flow of $6,651 million. Some of that
cash flow, $1,129 million, went to pay for new investments. Another part, $4,048 million,
went to pay dividends and retire outstanding securities. HD’s cash holdings therefore
increased by 6,651 2 1,129 2 4,048 5 $1,474 million. This is reported on the last line
of Table 19.3 .
The statement of cash flows provides important evidence on the well-being of a firm. If
a company cannot pay its dividends and maintain the productivity of its capital stock out
of cash flow from operations, for example, and it must resort to borrowing to meet these
demands, this is a serious warning that the firm cannot maintain the dividend payout at
its current level in the long run. The statement of cash flows will reveal this developing
problem when it shows that cash flow from operations is inadequate and that borrowing is
being used to maintain dividend payments at unsustainable levels.
In Chapter 1, we noted that a natural goal of the firm is to maximize value, but that various
agency problems, or conflicts of interest, may impede that goal. How can we measure how
well the firm is actually performing? Financial analysts have come up with a mind-numbing
list of financial ratios that measure many aspects of firm performance. Before getting lost
in the trees, however, let’s first pause to consider what sorts of ratios may be related to the
ultimate objective of added value.
Two broad activities are the responsibility of a firm’s financial managers: investment
decisions and financing decisions. Investment, or capital budgeting, decisions pertain to
the firm’s use of capital: the business activities in which it is engaged. Here, the questions we will wish to answer pertain to the profitability of those projects. How should profit-
ability be measured? How does the acceptable level of profitability depend on risk and
the opportunity cost of the funds used to pay for the firm’s many projects? In contrast,
financial decisions pertain to the firm’s sources of capital. Is there a sufficient supply of financing to meet projected needs for growth? Does the financing plan rely too heavily on
borrowed funds? Is there sufficient liquidity to deal with unexpected cash needs?
These questions suggest that we organize the ratios we choose to construct along the
lines given in Figure 19.1 . The figure shows that when evaluating the firm’s investment
activities, we will ask two questions: How efficiently does the firm deploy its assets, and
how profitable are its sales? In turn, aspects of both efficiency and profitability can be mea-
sured with several ratios: Efficiency is typically assessed using several turnover ratios, while
19.2 Measuring Firm Performance
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C H A P T E R 1 9 Financial Statement Analysis 641
the profitability of sales is commonly measured with various profit margins. Similarly, when
evaluating financing decisions, we look at both leverage and liquidity, and we will see that
aspects of each of these two concepts also can be measured with an array of statistics.
The next section shows how to calculate and interpret some of these key financial ratios
and shows how many of them are related.
What is the profitability of investments in real assets? Are financing decisions prudent? Return on assets•
• • •
• • • •
• •
Return on equity Return on capital Economic value added
Are assets used efficiently? Turnover ratios
Profitability of sales? Profit margins
Is leverage excessive? Debt ratios Coverage ratios
Is there sufficient liquidity? Current, quick, cash ratios Net working capital
Figure 19.1 Important financial questions and some ratios that help answer them
Big firms naturally earn greater profits than smaller ones. Therefore, most profitability
measures focus on earnings per dollar employed. The most common measures are return
on assets, return on capital, and return on equity.
Return on Assets, ROA Return on assets equals EBIT as a fraction of the firm’s total assets. 2
ROA 5 EBIT
Total assets
The numerator of this ratio may be viewed as total operating income of the firm. There-
fore, ROA tells us the income earned per dollar deployed in the firm.
Return on Capital, ROC Return on capital expresses EBIT as a fraction of long-term capital, shareholders’ equity
plus long-term debt. It tells us the income earned per dollar of long-term capital invested
in the firm.
ROC 5 EBIT
Long-term capital
19.3 Profitability Measures
2 ROA sometimes is computed using EBIT 3 (1 2 Tax rate) in the numerator. Sometimes it is computed using
after-tax operating income, i.e., Net income 1 Interest 3 (1 2 Tax rate). Sometimes, it even is calculated using
just net income in the numerator, although this definition ignores altogether the income the firm has generated for
debt investors. Unfortunately, definitions of many key financial ratios are not fully standardized.
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642 P A R T V Security Analysis
Return on Equity, ROE Whereas ROA and ROC measure profitability relative to funds raised by both debt and
equity financing, return on equity focuses only on the profitability of equity invest- ments. It equals net income realized by shareholders per dollar they have invested in
the firm.
ROE 5 Net income
Shareholders’ equity
We noted in Chapter 18 that ROE is one of the two basic factors in determining a firm’s
growth rate of earnings. Sometimes it is reasonable to assume that future ROE will approx-
imate its past value, but a high ROE in the past does not necessarily imply a firm’s future
ROE will be high. A declining ROE, on the other hand, is evidence that the firm’s new
investments have offered a lower ROE than its past investments. The vital point for a secu-
rity analyst is not to accept historical values as indicators of future values. Data from the
recent past may provide information regarding future performance, but the analyst should
always keep an eye on the future. Expectations of future dividends and earnings determine the intrinsic value of the company’s stock.
Not surprisingly, ROA and ROE are linked, but as we will see next, their relationship is
affected by the firm’s financial policies.
Financial Leverage and ROE An analyst interpreting the past behavior of a firm’s ROE or forecasting its future value
must pay careful attention to the firm’s debt-equity mix and to the interest rate on its debt.
An example will show why. Suppose Nodett is a firm that is all-equity financed and has
total assets of $100 million. Assume it pays corporate taxes at the rate of 40% of taxable
earnings.
Table 19.4 shows the behavior of sales, earnings before interest and taxes, and net
profits under three scenarios representing phases of the business cycle. It also shows the
behavior of two of the most commonly used profitability measures: operating ROA, which
equals EBIT/assets, and ROE, which equals net profits/equity.
Somdett is an otherwise identical firm to Nodett, but $40 million of its $100 million
of assets are financed with debt bearing an interest rate of 8%. It pays annual interest
expenses of $3.2 million. Table 19.5 shows how Somdett’s ROE differs from Nodett’s.
Note that annual sales, EBIT, and therefore ROA for both firms are the same in each
of the three scenarios; that is, business risk for the two companies is identical. But their
financial risk differs. Although Nodett and Somdett have the same ROA in each scenario,
Somdett’s ROE exceeds that of Nodett in normal and good years and is lower in bad
years.
Scenario Sales
($ millions) EBIT
($ millions) ROA
(% per year) Net Profit
($ millions) ROE
(% per year)
Bad year 80 5 5 3 3 Normal year 100 10 10 6 6 Good year 120 15 15 9 9
Table 19.4
Nodett’s profitability over the business cycle
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C H A P T E R 1 9 Financial Statement Analysis 643
We can summarize the exact relationship among ROE, ROA, and leverage in the
following equation: 3
ROE 5 (1 2 Tax rate) BROA 1 (ROA 2 Interest rate) Debt Equity
R (19.1) The relationship has the following implications. If there is no debt or if the firm’s ROA
equals the interest rate on its debt, its ROE will simply equal (1 2 Tax rate) 3 ROA. If
its ROA exceeds the interest rate, then its ROE will exceed (1 2 Tax rate) 3 ROA by an
amount that will be greater the higher the debt-to-equity ratio.
This result makes sense: If ROA exceeds the borrowing rate, the firm earns more on its
money than it pays out to creditors. The surplus earnings are available to the firm’s owners, the
equityholders, which increases ROE. If, on the other hand, ROA is less than the interest rate
paid on debt, then ROE will decline by an amount that depends on the debt-to-equity ratio.
3 The derivation of Equation 19.1 is as follows:
ROE 5 Net profit
Equity 5
EBIT 2 Interest 2 Taxes
Equity 5
(1 2 Tax rate)(EBIT 2 Interest)
Equity
5 (1 2 Tax rate) B (ROA 3 Assets) 2 (Interest rate 3 Debt) Equity
R 5 (1 2 Tax rate) BROA 3 Equity 1 Debt
Equity 2 Interest rate 3
Debt
Equity R
5 (1 2 Tax rate) BROA 1 (ROA 2 Interest rate) Debt Equity
R
To illustrate the application of Equation 19.1, we can use the numerical example in Table 19.5 . In a normal year, Nodett has an ROE of 6%, which is .6 (i.e., 1 minus the tax rate) times its ROA of 10%. However, Somdett, which borrows at an interest rate of 8% and maintains a debt-to-equity ratio of 2/3, has an ROE of 6.8%. The calculation using Equation 19.1 is
ROE 5 .6[10% 1 (10% 2 8%)2 ⁄3]
5 .6[10% 1 4 ⁄3%] 5 6.8%
The important point is that increased debt will make a positive contribution to a firm’s ROE only if the firm’s ROA exceeds the interest rate on the debt.
Example 19.1 Leverage and ROE
Nodett Somdett
Scenario EBIT
($ millions) Net Profits ($ millions)
ROE (%)
Net Profit* ($ millions)
ROE† (%)
Bad year 5 3 3 1.08 1.8 Normal year 10 6 6 4.08 6.8 Good year 15 9 9 7.08 11.8
*Somdett’s after-tax profits are given by .6(EBIT 2 $3.2 million). † ROE 5 net profit/equity. Somdett’s equity is only $60 million.
Table 19.5
Impact of financial leverage on ROE
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644 P A R T V Security Analysis
Notice that financial leverage increases the risk of the equityholder returns. Table 19.5
shows that ROE of Somdett is worse than that of Nodett in bad years. Conversely, in good
years, Somdett outperforms Nodett because the excess of ROA over ROE provides addi-
tional funds for equityholders. The presence of debt makes Somdett’s ROE more sensitive
to the business cycle than Nodett’s. Even though the two companies have equal business
risk (reflected in their identical EBITs in all three scenarios), Somdett’s stockholders bear
greater financial risk than Nodett’s because all of the firm’s business risk is absorbed by a
smaller base of equity investors.
Even if financial leverage increases the expected ROE of Somdett relative to Nodett (as
it seems to in Table 19.5 ), this does not imply that Somdett’s share price will be higher.
Financial leverage increases the risk of the firm’s equity as surely as it raises the expected
ROE, and the higher discount rate will offset the higher expected earnings.
Mordett is a company with the same assets as Nodett and Somdett but a debt-to-equity ratio of 1.0 and an interest rate of 9%. What would its net profit and ROE be in a bad year, a normal year, and a good year?
CONCEPT CHECK 19.1
Economic Value Added While profitability measures such as ROA, ROC, or ROE are commonly used to measure
performance, profitability is really not enough. A firm should be viewed as successful
only if the return on its projects is better than the rate investors could expect to earn for
themselves (on a risk-adjusted basis) in the capital market. Plowing back funds into the
firm increases stock price only if the firm earns a higher rate of return on the reinvested funds than the opportunity cost of capital, that is, the market capitalization rate. To account
for this opportunity cost, we might measure the success of the firm using the difference between the return on capital, ROC, and the opportunity cost of capital, k. Economic value added is the spread between ROC and k multiplied by the capital invested in the firm. It therefore measures the dollar value of the firm’s return in excess of its opportunity cost.
Another term for EVA (the term coined by Stern Stewart, a consulting firm that has pro-
moted its use) is residual income.
In 2012, Intel had a weighted-average cost of capital of 7.8% (based on its cost of debt, its capital structure, its equity beta, and estimates derived from the CAPM for the cost of equity). Its return on capital was 13.9%, fully 6.1% greater than the oppor- tunity cost of capital on its investments in plant, equipment, and know-how. In other words, each dollar invested by Intel earned about 6.1 cents more than the return that investors could have anticipated by investing in equivalent-risk stocks. Intel earned this superior rate of return on a capital base of $56.34 billion. Its economic value added, that is, its return in excess of opportunity cost, was therefore (.139 2 .078) 3 $56.34 5 $3.44 billion.
Example 19.2 Economic Value Added
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C H A P T E R 1 9 Financial Statement Analysis 645
Table 19.6 shows EVA for a small sample of firms. The EVA leader in this sample
was Microsoft. Notice that ExxonMobil’s EVA was greater than Intel’s, despite a smaller
margin between its ROC and the cost of capital. This is because ExxonMobil applied its
margin to a much larger capital base. At the other extreme, AT&T earned less than its
opportunity cost of capital, which resulted in a negative EVA.
Notice that even the EVA “losers” in Table 19.6 reported positive accounting profits.
For example, by conventional standards, AT&T was solidly profitable in 2012, with an
ROC of 3.9%. But its cost of capital was higher, at 4.9%. By this standard, it did not
cover its opportunity cost of capital, and returned a negative EVA in 2012. EVA treats
the opportunity cost of capital as a real cost that, like other costs, should be deducted
from revenues to arrive at a more meaningful “bottom line.” A firm that is earning prof-
its but is not covering its opportunity cost might be able to redeploy its capital to bet-
ter uses. Therefore, a growing number of firms now calculate EVA and tie managers’
compensation to it.
Source: Authors’ calculations using data from finance.yahoo.com. Actual EVA estimates reported by Stern Stewart differ from the values in Table 19.6 because of adjustments to the accounting data involving issues such as treatment of research and development expenses, taxes, advertising expenses, and depreciation. The estimates in Table 19.6 are designed to show the logic behind EVA but must be taken as imprecise.
Table 19.6
Economic value added, 2012
Ticker EVA
($ billion) Capital
($ billion) ROC (%) Cost of
Capital (%)
Microsoft MSFT 4.76 81.2 14.2 8.4 ExxonMobil XOM 3.63 179.06 9.3 7.3 Intel INTC 3.44 56.34 13.9 7.8 GlaxoSmithKline GSK 2.13 38.10 11.0 5.4 Google GOOG 1.36 75.95 10.5 8.7 Home Depot HD 1.07 28.57 11.2 7.4 Hewlett Packard HPQ 2 0.58 50.88 4.9 6.0 AT&T T 2 1.59 164.38 3.9 4.9
Decomposition of ROE To understand the factors affecting a firm’s ROE, particularly its trend over time and its
performance relative to competitors, analysts often “decompose” ROE into the product of
a series of ratios. Each component ratio is in itself meaningful, and the process serves to
focus the analyst’s attention on the separate factors influencing performance. This kind of
decomposition of ROE is often called the DuPont system. One useful decomposition of ROE is
ROE 5 Net profit
Equity 5
Net profits
Pretax profits 3
Pretax profits
EBIT 3
EBIT
Sales 3
Sales
Assets 3
Assets
Equity
(1) 3 (2) 3 (3) 3 (4) 3 (5)
(19.2)
19.4 Ratio Analysis
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646 P A R T V Security Analysis
Table 19.7 shows all these ratios for Nodett and Somdett Corporations under the three dif-
ferent economic scenarios. Let us first focus on factors 3 and 4. Notice that their product,
EBIT/Assets, gives us the firm’s ROA.
Factor 3 is known as the firm’s operating profit margin or return on sales (ROS), which equals operating profit per dollar of sales. In a normal year, profit margin is .10, or
10%; in a bad year, it is .0625, or 6.25%; and in a good year, .125, or 12.5%.
Factor 4, the ratio of sales to total assets, is known as total asset turnover (ATO). It indicates the efficiency of the firm’s use of assets in the sense that it measures the annual
sales generated by each dollar of assets. In a normal year, ATO for both firms is 1.0 per
year, meaning that sales of $1 per year were generated per dollar of assets. In a bad year,
this ratio declines to .8 per year, and in a good year, it rises to 1.2 per year.
Comparing Nodett and Somdett, we see that factors 3 and 4 do not depend on a firm’s
financial leverage. The firms’ ratios are equal to each other in all three scenarios. Similarly,
factor 1, the ratio of net income after taxes to pretax profit, is the same for both firms.
We call this the tax-burden ratio. Its value reflects both the government’s tax code and the policies pursued by the firm in trying to minimize its tax burden. In our example it does not
change over the business cycle, remaining a constant .6.
Although factors 1, 3, and 4 are not affected by a firm’s capital structure, factors 2 and
5 are. Factor 2 is the ratio of pretax profits to EBIT. The firm’s pretax profits will be great-
est when there are no interest payments to be made to debtholders. In fact, another way to
express this ratio is
Pretax profits
EBIT 5
EBIT 2 Interest expense
EBIT
We will call this factor the interest-burden ratio. It takes on its highest possible value, 1, for Nodett, which has no financial leverage. The higher the degree of financial leverage,
the lower the interest burden ratio. Nodett’s ratio does not vary over the business cycle.
It is fixed at 1.0, reflecting the total absence of interest payments. For Somdett, how-
ever, because interest expense is fixed in a dollar amount while EBIT varies, the interest
burden ratio varies from a low of .36 in a bad year to a high of .787 in a good year.
ROE
(1)
Net Profit/ Pretax Profit
(2)
Pretax Profit/EBIT
(3)
EBIT/Sales (Margin)
(4)
Sales/Assets (Turnover)
(5)
Assets/ Equity
(6)
Compound Leverage
Factor (2) 3 (5)
Bad year Nodett .030 .6 1.000 .0625 0.800 1.000 1.000 Somdett .018 .6 0.360 .0625 0.800 1.667 0.600 Normal year Nodett .060 .6 1.000 .1000 1.000 1.000 1.000 Somdett .068 .6 0.680 .1000 1.000 1.667 1.134 Good year Nodett .090 .6 1.000 .1250 1.200 1.000 1.000 Somdett .118 .6 0.787 .1250 1.200 1.667 1.311
Table 19.7
Ratio decomposition analysis for Nodett and Somdett
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C H A P T E R 1 9 Financial Statement Analysis 647
A closely related statistic to the interest burden ratio is the interest coverage ratio, or times interest earned. The ratio is defined as
Interest coverage 5 EBIT/Interest expense
A high coverage ratio indicates that the likelihood of bankruptcy is low because annual
earnings are significantly greater than annual interest obligations. It is widely used by both
lenders and borrowers in determining the firm’s debt capacity and is a major determinant
of the firm’s bond rating.
Factor 5, the ratio of assets to equity, is a measure of the firm’s degree of financial lever-
age. It is called the leverage ratio and is equal to 1 plus the total debt-to-equity ratio. 4 In our numerical example in Table 19.7 , Nodett has a leverage ratio of 1, while Somdett’s is 1.667.
From our discussion in Section 19.2, we know that financial leverage helps boost ROE
only if ROA is greater than the interest rate on the firm’s debt. How is this fact reflected in
the ratios of Table 19.7 ?
The answer is that to measure the full impact of leverage in this framework, the analyst
must take the product of the interest burden and leverage ratios (i.e., factors 2 and 5, shown
in Table 19.7 as column 6). For Nodett, factor 6, which we call the compound leverage factor, remains a constant 1.0 under all three scenarios. But for Somdett, we see that the compound
leverage factor is greater than 1 in normal years (1.134) and in good years (1.311), indicating
the positive contribution of financial leverage to ROE. It is less than 1 in bad years, reflecting
the fact that when ROA falls below the interest rate, ROE falls with increased use of debt.
We can summarize all of these relationships as follows. From Equation 19.2,
ROE 5 Tax burden 3 Interest burden 3 Margin 3 Turnover 3 Leverage
Because
ROA 5 Margin 3 Turnover (19.3)
and
Compound leverage factor 5 Interest burden 3 Leverage
we can decompose ROE equivalently as follows:
ROE 5 Tax burden 3 ROA 3 Compound leverage factor (19.4)
Equation 19.3 shows that ROA is the product of margin and turnover. High values of one of these ratios are often accompanied by low values of the other. For example, Walmart
has low profit margins but high turnover, while Tiffany has high margins but low turnover.
Firms would love to have high values for both margin and turnover, but this generally will
not be possible: Retailers with high markups will sacrifice sales volume, and conversely,
those with low turnover need high margins just to remain viable. Therefore, comparing
these ratios in isolation usually is meaningful only in evaluating firms following similar
strategies in the same industry. Cross-industry comparison can be misleading.
Figure 19.2 shows evidence of the turnover-profit margin trade-off. Industries with high
turnover such as groceries or retail apparel tend to have low profit margins, while indus-
tries with high margins such as utilities tend to have low turnover. The two curved lines
in the figure trace out turnover-margin combinations that result in an ROA of either 3%
or 6%. You can see that most industries lie inside of this range, so ROA across industries
demonstrates far less variation than either turnover or margin taken in isolation.
4 Assets
Equity 5
Equity 1 Debt
Equity 5 1 1
Debt
Equity
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648 P A R T V Security Analysis
Consider two firms with the same ROA of 10% per year. The first is a discount super- market chain, the second is a gas and electric utility.
As Table 19.8 shows, the supermarket chain has a “low” profit margin of 2% and achieves a 10% ROA by “turning over” its assets five times per year. The capital-intensive utility, on the other hand, has a “low” asset turnover ratio of only .5 times per year and achieves its 10% ROA through its higher, 20%, profit margin. The point here is that a “low” margin or asset turnover ratio need not indicate a troubled firm. Each ratio must be interpreted in light of industry norms.
Example 19.3 Margin versus Turnover
0.5 1.0 1.5
Asset Turnover
2.0 2.5 3.00.0 0
2
4
P ro
fi t
M a rg
in (
% )
6
8
10
12
14
ROA = 6%
Utilities
Oil & Gas Extraction
Amusements
Hotels
Petroleum Personal Services
Food Products
Transportation Equipment
Restaurants Industrial Equipment Wholesalers–
Nondurables
Wholesalers– Durables
Retailing–Apparel
Retailing–General Merchandise
Grocery Stores
Paper Lumber
Metals
Communications
Agricultural Production
Printing and Publishing
Airlines
Health Services
ROA = 3%
Figure 19.2 Median ROA, profit margin, and asset turnover for 23 industries, 1990–2004 Source: “Figure D: ROAs of Sample Firms (1977–1986)” from Thomas I. Selling and Clyde P. Stickney, ”The Effects of Business Environ- ments and Strategy on a Firm’s Rate of Return on Assets.” Copyright 1989. CFA Institute, Reproduced and republished from Financial Analysis Journal, January–February 1989, pp. 43–52, with permission from the CFA Institute. All rights reserved. Updates courtesy of Professors James Wahlen, Stephen Baginski, and Mark Bradshaw.
Do a ratio decomposition analysis for the Mordett Corporation of Concept Check 1, preparing a table similar to Table 19.7 .
CONCEPT CHECK 19.2
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C H A P T E R 1 9 Financial Statement Analysis 649
Turnover and Other Asset Utilization Ratios It is often helpful in understanding a firm’s ratio of sales to assets to compute comparable
efficiency-of-utilization, or turnover, ratios for subcategories of assets. For example, we
can think about turnover relative to fixed rather than total assets:
Fixed-asset turnover 5 Sales
Fixed assets
This ratio measures sales per dollar of the firm’s money tied up in fixed assets.
To illustrate how you can compute this and other ratios from a firm’s financial state-
ments, consider Growth Industries, Inc. (GI). GI’s historical income statement and opening
and closing balance sheets for the years 2010–2013 appear in Table 19.9 .
Margin 3 ATO 5 ROA
Supermarket chain 2% 5.0 10% Utility 20% 0.5 10%
Table 19.8
Differences between profit margin and asset turnover across industries
2010 2011 2012 2013
Income statements Sales revenue $100,000 $120,000 $144,000 Cost of goods sold (including depreciation) 55,000 66,000 79,200 Depreciation 15,000 18,000 21,600 Selling and administrative expenses 15,000 18,000 21,600 Operating income 30,000 36,000 43,200 Interest expense 10,500 19,095 34,391 Taxable income 19,500 16,905 8,809 Income tax (40% rate) 7,800 6,762 3,524 Net income $ 11,700 $ 10,143 $ 5,285
Balance sheets (end of year) Cash and marketable securities $ 50,000 $ 60,000 $ 72,000 $ 86,400 Accounts receivable 25,000 30,000 36,000 43,200 Inventories 75,000 90,000 108,000 129,600 Net plant and equipment 150,000 180,000 216,000 259,200
Total assets $300,000 $360,000 $432,000 $518,400
Accounts payable $ 30,000 $ 36,000 $ 43,200 $ 51,840 Short-term debt 45,000 87,300 141,957 214,432 Long-term debt (8% bonds maturing in 2025) 75,000 75,000 75,000 75,000
Total liabilities $150,000 $198,300 $260,157 $341,272
Shareholders’ equity (1 million shares outstanding) $150,000 $161,700 $171,843 $177,128
Other data Market price per common share at year-end $ 93.60 $ 61.00 $ 21.00
Table 19.9
Growth Industries financial statements ($ thousand)
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650 P A R T V Security Analysis
GI’s total asset turnover in 2013 was .303, which was below the industry average
of .4. To understand better why GI underperformed, we can compute asset utilization ratios
separately for fixed assets, inventories, and accounts receivable.
GI’s sales in 2013 were $144 million. Its only fixed assets were plant and equip-
ment, which were $216 million at the beginning of the year and $259.2 million at year’s
end. Average fixed assets for the year were, therefore, $237.6 million [($216 million
1 $259.2 million)/2]. GI’s fixed-asset turnover for 2013 therefore was $144 million per
year/$237.6 million 5 .606 per year. In other words, for every dollar of fixed assets, there
were $.606 in sales.
Comparable figures for the fixed-asset turnover ratio for 2011 and 2012 and the 2013
industry average are
2011 2012 2013 2013 Industry Average
.606 .606 .606 .700
GI’s fixed asset turnover has been stable over time and below the industry average.
Notice that when a financial ratio includes one item from the income statement, which
covers a period of time, and another from a balance sheet, which is a “snapshot” at a
particular time, common practice is to take the average of the beginning and end-of-year
balance sheet figures. Thus in computing the fixed-asset turnover ratio we divided sales
(from the income statement) by average fixed assets (from the balance sheet).
Another widely followed turnover ratio is the inventory turnover ratio, which is the ratio of cost of goods sold per dollar of average inventory. (We use the cost of goods sold
instead of sales revenue in the numerator to maintain consistency with inventory, which is
valued at cost.) This ratio measures the speed with which inventory is turned over.
In 2011, GI’s cost of goods sold (excluding depreciation) was $40 million, and its
average inventory was $82.5 million [($75 million 1 $90 million)/2]. Its inventory turn-
over was .485 per year ($40 million/$82.5 million). In 2012 and 2013, inventory turnover
remained the same, which was below the industry average of .5 per year. In other words,
GI was burdened with a higher level of inventories per dollar of sales than its competitors.
This higher investment in working capital in turn resulted in a higher level of assets per
dollar of sales or profits, and a lower ROA than its competitors.
Another aspect of efficiency surrounds management of accounts receivable, which is
often measured by days sales in receivables, that is, the average level of accounts receiv- able expressed as a multiple of daily sales. It is computed as average accounts receivable/
sales 3 365 and may be interpreted as the number of days’ worth of sales tied up in
accounts receivable. You can also think of it as the average lag between the date of sale and
the date payment is received, and is therefore also called the average collection period. For GI in 2013 the average collection period was 100.4 days:
($36 million 1 $43.2 million)/2
$144 million 3 365 5 100.4 days
The industry average was only 60 days. This statistic tells us that GI’s average receivables
per dollar of sales exceeds that of its competitors. Again, this implies a higher required
investment in working capital, and ultimately a lower ROA.
In summary, these ratios show us that GI’s poor total asset turnover relative to the indus-
try is in part caused by lower-than-average fixed-asset turnover and inventory turnover
and higher-than-average days receivables. This suggests GI may be having problems with
excess plant capacity along with poor inventory and receivables management practices.
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C H A P T E R 1 9 Financial Statement Analysis 651
Liquidity Ratios Leverage is one measure of the safety of a firm’s debt. Debt ratios compare the firm’s
indebtedness to broad measures of its assets, and coverage ratios compare various mea-
sures of earning power against the cash flow needed to satisfy debt obligations. But lever-
age is not the only determinant of financial prudence. You also want to know that a firm
can lay its hands on cash either to pay its scheduled obligations or to meet unforeseen
obligations. Liquidity is the ability to convert assets into cash at short notice. Liquidity is commonly measured using the current ratio, quick ratio, and cash ratio.
1. Current ratio: Current assets/current liabilities. This ratio measures the ability of the firm to pay off its current liabilities by liquidating its current assets (i.e., turning
them into cash). It indicates the firm’s ability to avoid insolvency in the short run.
GI’s current ratio in 2011, for example, was (60 1 30 1 90)/(36 1 87.3) 5 1.46.
In other years, it was
2011 2012 2013 2013 Industry Average
1.46 1.17 .97 2.0
This represents an unfavorable time trend and poor standing relative to the indus-
try. This troublesome pattern is not surprising given the working capital burden
resulting from GI’s subpar performance with respect to receivables and inventory
management.
2. Quick ratio: (Cash 1 marketable securities 1 receivables)/current liabilities. This ratio is also called the acid test ratio. It has the same denominator as the current ratio, but its numerator includes only cash, cash equivalents, and receivables. The
quick ratio is a better measure of liquidity than the current ratio for firms whose
inventory is not readily convertible into cash. GI’s quick ratio shows the same dis-
turbing trends as its current ratio:
2011 2012 2013 2013 Industry Average
.73 .58 .49 1.0
3. Cash ratio. A company’s receivables are less liquid than its holdings of cash and marketable securities. Therefore, in addition to the quick ratio, analysts also com-
pute a firm’s cash ratio, defined as
Cash ratio 5 Cash 1 Marketable securities
Current liabilities
GI’s cash ratios are
2011 2012 2013 2013 Industry Average
.487 .389 .324 .70
GI’s liquidity ratios have fallen dramatically over this 3-year period, and by 2013, its
liquidity measures are far below industry averages. The decline in the liquidity ratios com-
bined with the decline in coverage ratio (you can confirm that times interest earned has
also fallen over this period) suggests that its credit rating has been declining as well, and,
no doubt, GI is considered a relatively poor credit risk in 2013.
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652 P A R T V Security Analysis
Market Price Ratios: Growth versus Value The market–book-value ratio (P/B) equals the market price of a share of the firm’s common stock divided by its book value, that is, shareholders’ equity per share. Some analysts consider the stock of a firm with a low market–book value to be a “safer” investment, seeing the book
value as a “floor” supporting the market price. These analysts presumably view book value
as the level below which market price will not fall because the firm always has the option
to liquidate, or sell, its assets for their book values. However, this view is questionable. In
fact, some firms do sell below book value. For example, in mid-2012, shares in both Bank
of America and Citigroup sold for less than 50% of book value. Nevertheless, a low market–
book-value ratio is seen by some as providing a “margin of safety,” and some analysts will
screen out or reject high-P/B firms in their stock selection process.
In fact, a better interpretation of the market-price-to-book ratio is as a measure of
growth opportunities. Recall from the previous chapter that we may view the two com-
ponents of firm value as assets in place and growth opportunities. As the next example
illustrates, firms with greater growth opportunities will tend to exhibit higher multiples of
market-price-to-book value.
Consider two firms, both with book value per share of $10, both with a market capital- ization rate of 15%, and both with plowback ratios of .60.
Bright Prospects has an ROE of 20%, which is well in excess of the market capitaliza- tion rate; this ROE implies that the firm is endowed with ample growth opportunities. With ROE 5 .20, Bright Prospects will earn $2 per share this year. With its plowback ratio of .60, it pays out a dividend of D 1 5 (1 2 .6) 3 $2 5 $.80, has a growth rate of g 5 b 3 ROE 5 .60 3 .20 5 .12, and a stock price of D 1 /( k 2 g ) 5 $.80/(.15 2 .12) 5 $26.67. Its price–book ratio is 26.67/10 5 2.667.
In contrast, Past Glory has an ROE of only 15%, just equal to the market capital- ization rate. It therefore will earn $1.50 per share this year and will pay a dividend of D 1 5 .4 3 $1.50 5 $.60. Its growth rate is g 5 b 3 ROE 5 .60 3 .15 5 .09, and its stock price is D 1 /( k 2 g ) 5 $.60/(.15 2 .09) 5 $10. Its price–book ratio is $10/$10 5 1.0. Not surprisingly, a firm that earns just the required rate of return on its investments will sell for book value, and no more.
We conclude that the market-price-to-book-value ratio is determined in large part by growth prospects.
Example 19.4 Price to Book and Growth Options
Another measure used to place firms along a growth versus value spectrum is the price– earnings (P/E) ratio. In fact, we saw in the last chapter that the ratio of the present value of growth options to the value of assets in place largely determines the P/E multiple. While
low-P/E stocks allow you to pay less per dollar of current earnings, the high-P/E stock may still be a better bargain if its earnings are expected to grow quickly enough.
5
Many analysts nevertheless believe that low-P/E stocks are more attractive than high-
P/E stocks. And in fact, low-P/E stocks have generally been positive-alpha investments
using the CAPM as a return benchmark. But an efficient market adherent would discount
this track record, arguing that such a simplistic rule could not really generate abnormal
returns, and that the CAPM may not be a good benchmark for returns in this case.
5 Remember, though, P/E ratios reported in the financial pages are based on past earnings, while price is deter-
mined by the firm’s prospects of future earnings. Therefore, reported P/E ratios may reflect variation in current earnings around a trend line.
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C H A P T E R 1 9 Financial Statement Analysis 653
In any event, the important points to remember are that ownership of the stock conveys
the right to future as well as current earnings and, therefore, that a high P/E ratio may
best be interpreted as a signal that the market views the firm as enjoying attractive growth
opportunities.
Before leaving the P/B and P/E ratios, it is worth pointing out an important relationship
between them:
ROE 5 Earnings
Book value 5
Market price
Book value 4
Market price
Earnings (19.5)
5 P/B ratio 4 P/E ratio
By rearranging the terms, we find that a firm’s P/E ratio equals its price-to-book ratio
divided by ROE:
P
E 5
P/B
ROE
Thus a company with a high P/B ratio nevertheless can have a relatively low P/E if its ROE
is high enough.
Wall Street often distinguishes between “good firms” and “good investments.” A good firm
may be highly profitable, with a correspondingly high ROE. But if its stock price is bid up to a
level commensurate with this ROE, its P/B ratio will also be high, and the stock price may be
a relatively large multiple of earnings, thus reducing its attractiveness as an investment. The
high ROE of the firm does not by itself imply that the stock is a good investment. Conversely, troubled firms with low ROEs can be good investments if their prices are low enough.
Table 19.10 summarizes the ratios reviewed in this section.
What were GI’s ROE, P/E, and P/B ratios in the year 2013? How do they compare to the industry average ratios, which were: ROE 5 8.64%; P/E 5 8; P/B 5 .69? How does GI’s earnings yield in 2013 compare to the industry average?
CONCEPT CHECK 19.3
Choosing a Benchmark We have discussed how to calculate
the principal financial ratios. To evalu-
ate the performance of a given firm,
however, you need a benchmark to
which you can compare its ratios.
One obvious benchmark is the ratio
for the same company in earlier years.
For example, Figure 19.3 shows HD’s
return on assets, profit margin, and
asset turnover ratio for the last few
years. You can see there that a good
part of the decline in ROA between
2005 and 2009 was due to HD’s fall-
ing profit margin. In 2008, margin
improved but turnover fell, resulting
in a further drop in ROA.
0
5
10
15
20
25
30
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
R a ti
o
ROA
Turnover × 10
Profit Margin
Figure 19.3 DuPont decomposition for Home Depot
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654 P A R T V Security Analysis
Table 19.10
Summary of key financial ratios
Leverage
Interest burden EBIT 2 Interest expense
EBIT
Interest coverage (Times interest earned)
EBIT Interest expense
Leverage Assets Equity
5 1 1 Debt Equity
Compound leverage factor Interest burden 3 Leverage
Asset utilization
Total asset turnover Sales
Average total assets
Fixed asset turnover Sales
Average fixed assets
Inventory turnover Cost of goods sold
Average inventories
Days sales in receivables Average accounts receivable
Annual sales 3 365
Liquidity
Current ratio Current assets
Current liabilities
Quick ratio Cash 1 Marketable securities 1 Receivables
Current liabilities
Cash ratio Cash 1 Marketable securities
Current liabilities
Profitability
Return on assets EBIT
Average total assets
Return on equity Net income
Average stockholders’ equity
Return on sales (Profit margin) EBIT Sales
Market price
Market-to-book Price per share
Book value per share
Price–earnings ratio Price per share
Earnings per share
Earnings yield Earnings per share
Price per share
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C H A P T E R 1 9 Financial Statement Analysis 655
Table 19.11
Financial ratios for major industry groups
Source: U.S. Department of Commerce , Quarterly Financial Report for Manufacturing, Mining and Trade Corporations, third quarter 2012. Available at http://www2.census.gov/econ/qfr/current/qfr_pub.pdf.
LT Debt Assets
Interest Coverage
Current Ratio
Quick Ratio
Asset Turnover
Profit Margin (%)
Return on Assets (%)
Return on Equity (%)
Payout Ratio
All manufacturing 0.20 4.80 1.35 0.92 0.80 7.91 6.36 14.76 0.37 Food products 0.28 4.64 1.32 0.76 1.14 7.25 8.23 14.00 0.32 Clothing 0.18 7.41 2.26 1.25 1.25 9.57 11.95 23.44 0.28 Printing/ publishing
0.40 3.50 1.51 1.20 1.36 8.08 10.97 35.35 0.27
Chemicals 0.26 3.33 1.05 0.74 0.47 11.65 5.43 13.95 0.46 Drugs 0.26 3.03 0.96 0.70 0.33 15.33 5.02 14.33 0.42 Machinery 0.18 5.92 1.38 0.87 0.80 9.49 7.58 17.54 0.24 Electrical 0.11 4.76 1.07 0.67 0.52 7.56 3.91 11.03 0.56 Motor vehicles 0.13 4.31 1.39 1.07 1.11 3.92 4.36 13.18 0.23 Computer and electronic
0.16 5.66 1.57 1.28 0.50 9.25 4.67 9.41 0.60
It is also helpful to compare financial ratios to those of other firms in the same industry.
Financial ratios for industries are published by the U.S. Department of Commerce (see
Table 19.11 ), Dun & Bradstreet ( Industry Norms and Key Business Ratios ), and the Risk Management Association, or RMA ( Annual Statement Studies ). A broad range of financial ratios is also easily accessible on the Web.
Table 19.11 presents ratios for a sample of major industry groups to give you a feel for
some of the differences across industries. You should note that while some ratios such as
asset turnover or total debt ratio tend to be relatively stable over time, others such as return
on assets or equity will be more sensitive to current business conditions.
In her 2015 annual report to the shareholders of Growth Industries, Inc., the president
wrote: “2015 was another successful year for Growth Industries. As in 2014, sales, assets,
and operating income all continued to grow at a rate of 20%.”
Is she right?
We can evaluate her statement by conducting a full-scale ratio analysis of Growth
Industries. Our purpose is to assess GI’s performance in the recent past, to evaluate its
future prospects, and to determine whether its market price reflects its intrinsic value.
Table 19.12 shows the key financial ratios we can compute from GI’s financial state-
ments. The president is certainly right about the growth rate in sales, assets, and operating
income. Inspection of GI’s key financial ratios, however, contradicts her first sentence: 2015
was not another successful year for GI—it appears to have been another miserable one.
19.5 An Illustration of Financial Statement Analysis
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656 P A R T V Security Analysis
ROE has been declining steadily from 7.51% in 2013 to 3.03% in 2015. A comparison
of GI’s 2015 ROE to the 2015 industry average of 8.64% makes the deteriorating time
trend appear especially alarming. The low and falling market-to-book-value ratio and the
falling price–earnings ratio indicate investors are less and less optimistic about the firm’s
future profitability.
The fact that ROA has not been declining, however, tells us that the source of the
declining time trend in GI’s ROE must be related to financial leverage. And we see that as
GI’s leverage ratio climbed from 2.117 in 2013 to 2.723 in 2015, its interest-burden ratio
( column 2) worsened from .650 to .204—with the net result that the compound leverage
factor fell from 1.376 to .556.
The rapid increase in short-term debt from year to year and the concurrent increase in
interest expense (see Table 19.9 ) make it clear that to finance its 20% growth rate in sales,
GI has incurred sizable amounts of short-term debt at high interest rates. The firm is pay-
ing rates of interest greater than the ROA it is earning on the investment financed with the
new borrowing. As the firm has expanded, its situation has become ever more precarious.
In 2015, for example, the average interest rate on GI’s short-term debt was 20% ver-
sus an ROA of 9.09%. (You can calculate the interest rate on GI’s short-term debt using
the data in Table 19.9 as follows. The balance sheet shows us that the coupon rate on its
long-term debt was 8%, and its par value was $75 million. Therefore the interest paid on
the long-term debt was .08 3 $75 million 5 $6 million. Total interest paid in 2015 was
$34,391,000, so the interest paid on the short-term debt must have been $34,391,000 2
$6,000,000 5 $28,391,000. This is 20% of GI’s short-term debt at the start of the year.)
GI’s problems become clear when we examine its statement of cash flows in Table 19.13 .
The statement is derived from the income statement and balance sheet data in Table 19.9 .
GI’s cash flow from operations is falling steadily, from $12,700,000 in 2013 to $6,725,000
in 2015. The firm’s investment in plant and equipment, by contrast, has increased greatly.
Net plant and equipment (i.e., net of depreciation) rose from $150,000,000 in 2012 to
$259,200,000 in 2015 (see Table 19.9 ). This near doubling of capital assets makes the
decrease in cash flow from operations all the more troubling.
The source of the difficulty is GI’s enormous amount of short-term borrowing. In a
sense, the company is being run as a pyramid scheme. It borrows more and more each
year to maintain its 20% growth rate in assets and income. However, the new assets are not
(1) (2) (3) (4) (5) (6) (7)
Year ROE
Net Profit/ Pretax Profit
Pretax Profit/ EBIT
EBIT/ Sales
(Margin)
Sales/ Assets
(Turnover) Assets/ Equity
Compound Leverage
Factor (2) 3 (5)
ROA (3) 3 (4) P/E P/B
2013 7.51% .6 .650 30% .303 2.117 1.376 9.09% 8 .58 2014 6.08 .6 .470 30 .303 2.375 1.116 9.09 6 .35 2015 3.03 .6 .204 30 .303 2.723 0.556 9.09 4 .12 Industry average 8.64 .6 .800 30 .400 1.500 1.200 12.00 8 .69
Table 19.12
Key financial ratios of Growth Industries, Inc.
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C H A P T E R 1 9 Financial Statement Analysis 657
generating enough cash flow to support the extra interest burden of the debt, as the falling
cash flow from operations indicates. Eventually, when the firm loses its ability to borrow
further, its growth will be at an end.
At this point GI stock might be an attractive investment. Its market price is only 12% of
its book value, and with a P/E ratio of 4, its earnings yield is 25% per year. GI is a likely
candidate for a takeover by another firm that might replace GI’s management and build
shareholder value through a radical change in policy.
2013 2014 2015
Cash flow from operating activities Net income $ 11,700 $ 10,143 $ 5,285 1 Depreciation 15,000 18,000 21,600
1 Decrease (increase) in accounts receivable (5,000) (6,000) (7,200)
1 Decrease (increase) in inventories (15,000) (18,000) (21,600)
1 Increase in accounts payable 6,000 7,200 8,640
Cash provided by operations $ 12,700 $ 11,343 $ 6,725
Cash flow from investing activities Investment in plant and equipment* $(45,000) $(54,000) $(64,800)
Cash flow from financing activities Dividends paid† $ 0 $ 0 $ 0 Short-term debt issued 42,300 54,657 72,475
Change in cash and marketable securities‡ $ 10,000 $ 12,000 $ 14,400
Table 19.13
Growth Industries statement of cash flows ($ thousand)
*Gross investment equals increase in net plant and equipment plus depreciation. † We can conclude that no dividends are paid because stockholders’ equity increases each year by the full amount of net income, imply- ing a plowback ratio of 1.0. ‡ Equals cash flow from operations plus cash flow from investment activities plus cash flow from financing activities. Note that this equals the yearly change in cash and marketable securities on the balance sheet.
You have the following information for IBX Corporation for the years 2013 and 2015 (all figures are in $ million):
2013 2015
Net income $ 253.7 $ 239.0 Pretax income 411.9 375.6 EBIT 517.6 403.1 Average assets 4,857.9 3,459.7 Sales 6,679.3 4,537.0 Shareholders’ equity 2,233.3 2,347.3
What is the trend in IBX’s ROE; how can you account for it in terms of tax burden, margin, turnover, and financial leverage?
CONCEPT CHECK 19.4
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658 P A R T V Security Analysis
Financial statement analysis gives us a good amount of ammunition for evaluating a com-
pany’s performance and future prospects. But comparing financial results of different com-
panies is not so simple. There is more than one acceptable way to represent various items
of revenue and expense according to generally accepted accounting principles (GAAP).
This means two firms may have exactly the same economic income yet very different
accounting incomes.
Furthermore, interpreting a single firm’s performance over time is complicated when
inflation distorts the dollar measuring rod. Comparability problems are especially acute
in this case because the impact of inflation on reported results often depends on the par-
ticular method the firm adopts to account for inventories and depreciation. The security
analyst must adjust the earnings and the financial ratio figures to a uniform standard before
attempting to compare financial results across firms and over time.
Comparability problems can arise out of the flexibility of GAAP guidelines in account-
ing for inventories and depreciation and in adjusting for the effects of inflation. Other
important potential sources of noncomparability include the capitalization of leases and
other expenses, the treatment of pension costs, and allowances for reserves.
Inventory Valuation There are two commonly used ways to value inventories: LIFO (last-in first-out) and FIFO (first-in first-out). We can explain the difference using a numerical example.
Suppose Generic Products, Inc. (GPI), has a constant inventory of 1 million units of
generic goods. The inventory turns over once per year, meaning the ratio of cost of goods
sold to inventory is 1.
The LIFO system calls for valuing the million units used up during the year at the cur-
rent cost of production, so that the last goods produced are considered the first ones to be
sold. They are valued at today’s cost.
The FIFO system assumes that the units used up or sold are the ones that were added to
inventory first, and goods sold should be valued at original cost.
If the price of generic goods has been constant, at the level of $1, say, the book value
of inventory and the cost of goods sold would be the same, $1 million under both systems.
But suppose the price of generic goods rises by 10 cents per unit during the year as a result
of general inflation.
LIFO accounting would result in a cost of goods sold of $1.1 million, whereas the end-
of-year balance sheet value of the 1 million units in inventory remains $1 million. The
balance sheet value of inventories is given as the cost of the goods still in inventory. Under
LIFO the last goods produced are assumed to be sold at the current cost of $1.10; the
goods remaining are the previously produced goods, at a cost of only $1. You can see that
although LIFO accounting accurately measures the cost of goods sold today, it understates
the current value of the remaining inventory in an inflationary environment.
In contrast, under FIFO accounting, the cost of goods sold would be $1 million, and the
end-of-year balance sheet value of the inventory would be $1.1 million. The result is that
the LIFO firm has both a lower reported profit and a lower balance sheet value of invento-
ries than the FIFO firm.
LIFO is preferred over FIFO in computing economic earnings (i.e., real sustainable
cash flow) because it uses up-to-date prices to evaluate the cost of goods sold. However,
19.6 Comparability Problems
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C H A P T E R 1 9 Financial Statement Analysis 659
LIFO accounting induces balance sheet distortions when it values investment in invento-
ries at original cost. This practice results in an upward bias in ROE because the investment
base on which return is earned is undervalued.
Depreciation Another source of problems is the measurement of depreciation, which is a key factor in
computing true earnings. The accounting and economic measures of depreciation can dif-
fer markedly. According to the economic definition, depreciation is the amount of a firm’s operating cash flow that must be reinvested in the firm to sustain its real productive capacity
at the current level.
The accounting measurement is quite different. Accounting depreciation is the amount of the original acquisition cost of an asset that is allocated to each accounting period over
an arbitrarily specified life of the asset. This is the figure reported in financial statements.
Assume, for example, that a firm buys machines with a useful economic life of
20 years at $100,000 apiece. In its financial statements, however, the firm can depreci-
ate the machines over 10 years using the straight-line method, for $10,000 per year
in depreciation. Thus after 10 years a machine will be fully depreciated on the books,
even though it remains a productive asset that will not need replacement for another
10 years.
In computing accounting earnings, this firm will overestimate depreciation in the first
10 years of the machine’s economic life and underestimate it in the last 10 years. This will
cause reported earnings to be understated compared with economic earnings in the first
10 years and overstated in the last 10 years.
Depreciation comparability problems add one more wrinkle. A firm can use different
depreciation methods for tax purposes than for other reporting purposes. Most firms use
accelerated depreciation methods for tax purposes and straight-line depreciation in pub-
lished financial statements. There also are differences across firms in their estimates of the
depreciable life of plant, equipment, and other depreciable assets.
Another complication arises from inflation. Because conventional depreciation is based
on historical costs rather than on the current replacement cost of assets, measured depreci-
ation in periods of inflation is understated relative to replacement cost, and real economic income (sustainable cash flow) is correspondingly overstated.
For example, suppose Generic Products, Inc., has a machine with a 3-year useful
life that originally cost $3 million. Annual straight-line depreciation is $1 million,
regardless of what happens to the replacement cost of the machine. Suppose inflation
in the first year turns out to be 10%. Then the true annual depreciation expense is
$1.1 million in current terms, whereas conventionally measured depreciation remains
fixed at $1 million per year. Accounting income overstates real economic income by $.1 million.
Inflation and Interest Expense Although inflation can cause distortions in the measurement of a firm’s inventory and
depreciation costs, it has perhaps an even greater effect on calculation of real interest expense. Nominal interest rates include an inflation premium that compensates the lender
for inflation-induced erosion in the real value of principal. From the perspective of both
lender and borrower, therefore, part of what is conventionally measured as interest expense
should be treated more properly as repayment of principal.
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660 P A R T V Security Analysis
Mismeasurement of real interest means inflation deflates the computation of real
income. The effects of inflation on the reported values of inventories and depreciation that
we have discussed work in the opposite direction.
Suppose Generic Products has debt outstanding with a face value of $10 million at an interest rate of 10% per year. Interest expense as conventionally measured is $1 million per year. However, suppose inflation during the year is 6%, so that the real interest rate is 4%. Then $.6 million of what appears as interest expense on the income state- ment is really an inflation premium, or compensation for the anticipated reduction in the real value of the $10 million principal; only $.4 million is real interest expense. The $.6 million reduction in the purchasing power of the outstanding principal may be thought of as repayment of principal, rather than as an interest expense. Real income of the firm is, therefore, understated by $.6 million.
Example 19.5 Inflation and Real Income
In a period of rapid inflation, companies ABC and XYZ have the same reported earnings. ABC uses LIFO inventory accounting, has relatively fewer depreciable assets, and has more debt than XYZ. XYZ uses FIFO inventory accounting. Which company has the higher real income, and why?
CONCEPT CHECK 19.5
Fair Value Accounting Many major assets and liabilities are not traded in financial markets and do not have eas-
ily observable values. For example, we cannot simply look up the values of employee
stock options, health care benefits for retired employees, or buildings and other real estate.
While the true financial status of a firm may depend critically on these values, which can
swing widely over time, common practice has been to simply value them at historic cost.
Proponents of fair value accounting, also known as mark-to-market accounting, argue that financial statements would give a truer picture of the firm if they better reflected the
current market values of all assets and liabilities.
The Financial Accounting Standards Board’s Statement No. 157 on fair value account-
ing places assets in one of three “buckets.” Level 1 assets are traded in active markets and
therefore should be valued at their market price. Level 2 assets are not actively traded,
but their values still may be estimated using observable market data on similar assets.
They can be “marked to a matrix” of comparable securities. Level 3 assets are hardest
to value. Here it is difficult even to identify other assets that are similar enough to serve
as benchmarks for their market values; one has to resort to pricing models to estimate
their intrinsic values. Rather than mark to market, these values are often called “mark
to model,” although they are also disparagingly known as mark-to-make believe, as the
estimates are so prone to manipulation by creative use of model inputs. Since 2012, firms
have been required to disclose more about the methods and assumptions used in their
valuation models and to describe the sensitivity of their valuation estimates to changes in
their methodology.
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C H A P T E R 1 9 Financial Statement Analysis 661
6 D. Bergstresser, M. Desai, and J. Rauh, “Earnings Manipulation, Pension Assumptions, and Managerial Invest-
ment Decisions,” Quarterly Journal of Economics 121 (2006), pp. 157–95.
Mark-to-Market Accounting: Cure or Disease?
As banks and other institutions holding mortgage-backed securities revalued their portfolios throughout 2008, their net worth fell along with the value of those securities. The losses on these securities were painful enough, but their knock-on effects only increased the banks’ woes. For example, banks are required to maintain adequate levels of capital relative to assets. If capital reserves decline, a bank may be forced to shrink until its remaining capital is once again adequate compared to its asset base. But such shrinkage may require the bank to cut back on its lending, which restricts its customers’ access to credit. It may also require asset sales, and if many banks attempt to shrink their portfolios at once, waves of forced sales may put fur- ther pressure on prices, resulting in additional write-downs and reductions to capital in a self-feeding cycle. Critics of mark-to-market accounting therefore contend that it exac- erbated the problems of an already reeling economy.
Advocates, however, argue that the critics confuse the message with the messenger. Mark-to-market accounting makes transparent losses that have already been incurred, but it does not cause those losses. Critics retort that when markets are faltering, market prices may be unreliable. If trading activity has largely broken down, and assets can be sold only at fire-sale prices, then those prices may no longer be indicative of fundamental value. Markets can- not be efficient if they are not even functioning. In the turmoil surrounding the defaulted mortgages weighing down bank portfolios, one of the early proposals of then– Treasury secretary Henry Paulson was for the government
to buy bad assets at “hold to maturity” prices based on estimates of intrinsic value in a normally functioning mar- ket. In that spirit, FASB approved new guidelines in 2009 allowing valuation based on an estimate of the price that would prevail in an orderly market rather than the one that could be received in a forced liquidation.
Waiving write-down requirements may best be viewed as thinly veiled regulatory forbearance. Regulators know that losses have been incurred and that capital has been impaired. But by allowing firms to carry assets on their books at model rather than market prices, the unpleas- ant implications of that fact for capital adequacy may be politely ignored for a time. Even so, if the goal is to avoid forced sales in a distressed market, transparency may nev- ertheless be the best policy. Better to acknowledge losses and explicitly modify capital regulations to help institu- tions recover their footing in a difficult economy than to deal with losses by ignoring them. After all, why bother preparing financial statements if they are allowed to obscure the true condition of the firm?
Before abandoning fair value accounting, it would be prudent to consider the alternative. Traditional historic- cost accounting, which would allow firms to carry assets on the books at their original purchase price, has even less to recommend it. It would leave investors without an accu- rate sense of the condition of shaky institutions, and by the same token lessen the pressure on those firms to get their houses in order. Dealing with losses must surely require acknowledging them.
W O
R D
S FR O
M T
H E ST
R E E T
Critics of fair value accounting argue that it relies too heavily on estimates. Such esti-
mates potentially introduce considerable noise in firms’ accounts and can induce great
profit volatility as fluctuations in asset valuations are recognized. Even worse, subjective
valuations may offer management a tempting tool to manipulate earnings or the appar-
ent financial condition of the firm at opportune times. As just one example, Bergstresser,
Desai, and Rauh 6 find that firms make more aggressive assumptions about returns on
defined benefit pension plans (which lowers the computed present value of pension obliga-
tions) during periods in which executives are actively exercising their stock options.
A contentious debate over the application of fair value accounting to troubled financial
institutions erupted in 2008 when even values of financial securities such as subprime
mortgage pools and derivative contracts backed by these pools came into question as trad-
ing in these instruments dried up. Without well-functioning markets, estimating (much
less observing) market values was, at best, a precarious exercise. In 2012, for example,
employees at Credit Suisse were convicted of intentionally overstating the value of thinly
traded mortgage bonds during the financial crisis to improve the apparent profitability of
their trading activities.
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662 P A R T V Security Analysis
Many observers feel that mark-to-market accounting exacerbated the financial melt-
down by forcing banks to excessively write down asset values; others feel that a failure to
mark would have been tantamount to willfully ignoring reality and abdicating the respon-
sibility to redress problems at nearly or already insolvent banks. The nearby box discusses
the debate.
Quality of Earnings and Accounting Practices Many firms will make accounting choices that present their financial statements in the
best possible light. The different choices that firms can make give rise to the comparabil-
ity problems we have discussed. As a result, earnings statements for different companies
may be more or less rosy presentations of true economic earnings—sustainable cash flow
that can be paid to shareholders without impairing the firm’s productive capacity. Analysts
commonly evaluate the quality of earnings reported by a firm. This concept refers to the realism and conservatism of the earnings number, in other words, the extent to which we
might expect the reported level of earnings to be sustained.
Examples of the types of factors that influence quality of earnings are:
• Allowance for bad debt. Most firms sell goods using trade credit and must make an allowance for bad debt. An unrealistically low allowance reduces the quality of
reported earnings.
• Nonrecurring items. Some items that affect earnings should not be expected to recur regularly. These include asset sales, effects of accounting changes, effects of
exchange rate movements, or unusual investment income. For example, in years
with large stock market returns, some firms enjoy large capital gains on securities
held. These contribute to that year’s earnings, but should not be expected to repeat
regularly. They would be considered a “low-quality” component of earnings.
Similarly, investment gains in corporate pension plans generated large but one-off
contributions to reported earnings.
• Earnings smoothing. In 2003, Freddie Mac was the subject of an accounting scandal, when it emerged that it had improperly reclassified mortgages held in
its portfolio in an attempt to reduce its current earnings. Why would it take such actions? Because later, if earnings turned down, Freddie could “release” earnings
by reversing these transactions, and thereby create the appearance of steady
earnings growth. Indeed, almost until its sudden collapse in 2008, Freddie Mac’s
nickname on Wall Street was “Steady Freddie.” Similarly, in the four quarters
ending in October 2012, the four largest U.S. banks released $18.2 billion in
reserves, which accounted for nearly one-quarter of their pretax income. 7 Such
“earnings” are clearly not sustainable over the long-term and therefore must be
considered low quality.
• Revenue recognition. Under GAAP accounting, a firm is allowed to recognize a sale before it is paid. This is why firms have accounts receivable. But sometimes
it can be hard to know when to recognize sales. For example, suppose a
computer firm signs a contract to provide products and services over a 5-year
period. Should the projected revenue be booked immediately or spread out over
5 years? A more extreme version of this problem is called “channel stuffing,”
7 Michael Rapoport, “Bank Profit Spigot to Draw Scrutiny,” The Wall Street Journal, October 11, 2012.
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C H A P T E R 1 9 Financial Statement Analysis 663
in which firms “sell” large quantities of goods to customers, but give them the
right to later either refuse delivery or return the product. The revenue from the
“sale” is booked now, but the likely returns are not recognized until they occur
(in a future accounting period). Hewlett-Packard argued in 2012 that it was led
to overpay for its acquisition of Autonomy Corp. when Autonomy artificially
enhanced its financial performance using channel stuffing. For example,
Autonomy apparently sold software valued at over £4 billion to Tikit Group; it
booked the entire deal as revenue but would not be paid until Tikit actually sold
software to its clients. 8 Thus, several years’ worth of only tentative future sales
was recognized in 2010.
If you see accounts receivable increasing far faster than sales, or becoming a
larger percentage of total assets, beware of these practices. Given the wide latitude
firms have in how they recognize revenue, many analysts choose instead to concen-
trate on cash flow, which is far harder for a company to manipulate.
• Off-balance-sheet assets and liabilities. Suppose that one firm guarantees the outstanding debt of another firm, perhaps a firm in which it has an ownership
stake. That obligation ought to be disclosed as a contingent liability, because it may require payments down the road. But these obligations may not be reported
as part of the firm’s outstanding debt. Similarly, leasing may be used to manage
off-balance-sheet assets and liabilities. Airlines, for example, may show no
aircraft on their balance sheets but have long-term leases that are virtually
equivalent to debt-financed ownership. However, if the leases are treated as
operating rather than capital leases, they may appear only as footnotes to the
financial statements.
International Accounting Conventions The examples cited above illustrate some of the problems that analysts can encounter when
attempting to interpret financial data. Even greater problems arise in the interpretation of
the financial statements of foreign firms. This is because these firms do not follow GAAP
guidelines. Accounting practices in various countries differ to greater or lesser extents
from U.S. standards. Here are some of the major issues that you should be aware of when
using the financial statements of foreign firms:
• Reserving practices. Many countries allow firms considerably more discretion in setting aside reserves for future contingencies than is typical in the United
States. Because additions to reserves result in a charge against income,
reported earnings are far more subject to managerial discretion than in the
United States.
• Depreciation. In the United States, firms typically maintain separate sets of accounts for tax and reporting purposes. For example, accelerated depreciation
is typically used for tax purposes, whereas straight-line depreciation is used for
reporting purposes. In contrast, most other countries do not allow dual sets of
accounts, and most firms in foreign countries use accelerated depreciation to
minimize taxes despite the fact that it results in lower reported earnings. This
8 Ben Worthen, Paul Sonne, and Justin Scheck, “Long Before H-P Deal, Autonomy’s Red Flags,” The Wall Street
Journal, November 26, 2012.
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664 P A R T V Security Analysis
Australia
0 8070605040302010
Reported P/E
24.1 9.1
12.6 11.1
26.5 17.1
78.145.1 12.4
10.7 10.0 9.5United Kingdom
Switzerland
Japan
Germany
France Adjusted P/E
Figure 19.4 Adjusted versus reported price–earnings ratios Source: “Figure J: Adjusted versus Reported Price/Earnings Ratio” from Lawrence S. Speidell and Vinod Bavishi, “GAAP Arbitrage: Valuation Opportunities in International Accounting Standards,” Financial Analysts Journal, November–December 1992, pp. 58–66. Copyright 1992, CFA Institute. Reproduced from the Financial Analysts Journal with permission from the CFA Institute.
makes reported earnings of foreign firms lower than they would be if the firms were
allowed to use the U.S. practice.
• Intangibles. Treatment of intangibles such as goodwill can vary widely. Are they amortized or expensed? If amortized, over what period? Such issues can have a
large impact on reported profits.
The effect of different accounting practices can be substantial. Figure 19.4 compares
P/E ratios in different countries as reported and restated on a common basis. While P/E
multiples have changed considerably since this study was published, these results illustrate
how different accounting rules can have a big impact on these ratios.
Some of the differences between U.S. and European accounting standards arise from
different philosophies regarding regulating accounting practice. GAAP accounting in the
U.S. is rules-based, with detailed, explicit, and lengthy rules governing almost any circum-
stance that can be anticipated. In contrast, the international financial reporting standards (IFRS) used in the European Union are principles-based, setting out general approaches
for the preparation of financial statements. While EU rules are more flexible, firms must be
prepared to demonstrate that their accounting choices are consistent with IFRS principles.
IFRS seem on their way to becoming global standards, even outside of the European
Union. By 2008, over 100 countries had adopted them, and they are making inroads even
in the United States. In November 2007, the SEC began allowing foreign firms to issue
securities in the U.S. if their financial statements are prepared using IFRS. In 2008, the
SEC went even further when it proposed allowing large U.S. multinational firms to report
earnings using IFRS rather than GAAP starting in 2010. A final integration of U.S. rules
with IFRS has been long expected but repeatedly delayed. However, even without formal
adoption of IFRS, the widespread belief is that the U.S. will continue to change GAAP
over time to more closely conform to IFRS rules. The goal is to make cross-border financial
statements more consistent and comparable, thereby improving the quality of information
available to investors.
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C H A P T E R 1 9 Financial Statement Analysis 665
9 As cited by John Train in Money Masters (New York: Harper & Row, 1987).
1. The primary focus of the security analyst should be the firm’s real economic earnings rather than its reported earnings. Accounting earnings as reported in financial statements can be a biased esti-
mate of real economic earnings, although empirical studies reveal that reported earnings convey
considerable information concerning a firm’s prospects.
2. A firm’s ROE is a key determinant of the growth rate of its earnings. ROE is affected profoundly by the firm’s degree of financial leverage. An increase in a firm’s debt-to-equity ratio will raise its ROE
and hence its growth rate only if the interest rate on the debt is less than the firm’s return on assets.
SUMMARY
No presentation of fundamental security analysis would be complete without a discussion
of the ideas of Benjamin Graham, the greatest of the investment “gurus.” Until the evolu-
tion of modern portfolio theory in the latter half of the 20th century, Graham was the single
most important thinker, writer, and teacher in the field of investment analysis. His influ-
ence on investment professionals remains very strong.
Graham’s magnum opus is Security Analysis, written with Columbia Professor David Dodd in 1934. Its message is similar to the ideas presented in this chapter. Graham
believed careful analysis of a firm’s financial statements could turn up bargain stocks. Over
the years, he developed many different rules for determining the most important financial
ratios and the critical values for judging a stock to be undervalued. Through many editions,
his book has been so influential and successful that widespread adoption of Graham’s tech-
niques has led to the elimination of the very bargains they are designed to identify.
In a 1976 seminar Graham said: 9
I am no longer an advocate of elaborate techniques of security analysis in order to find
superior value opportunities. This was a rewarding activity, say, forty years ago, when
our textbook “Graham and Dodd” was first published; but the situation has changed a
good deal since then. In the old days any well-trained security analyst could do a good
professional job of selecting undervalued issues through detailed studies; but in the light
of the enormous amount of research now being carried on, I doubt whether in most cases
such extensive efforts will generate sufficiently superior selections to justify their cost.
To that very limited extent I’m on the side of the “efficient market” school of thought
now generally accepted by the professors.
Nonetheless, in that same seminar, Graham suggested a simplified approach to identify-
ing bargain stocks:
My first, more limited, technique confines itself to the purchase of common stocks at
less than their working-capital value, or net current-asset value, giving no weight to the
plant and other fixed assets, and deducting all liabilities in full from the current assets.
We used this approach extensively in managing investment funds, and over a 30-odd-year
period we must have earned an average of some 20 percent per year from this source.
I consider it a foolproof method of systematic investment—once again, not on the basis
of individual results but in terms of the expectable group income.
There are two convenient sources of information for those interested in trying out the
Graham technique: Both Standard & Poor’s Outlook and The Value Line Investment Survey carry lists of stocks selling below net working capital value.
19.7 Value Investing: The Graham Technique
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666 P A R T V Security Analysis
3. It is often helpful to the analyst to decompose a firm’s ROE ratio into the product of several accounting ratios and to analyze their separate behavior over time and across companies within
an industry. A useful breakdown is
ROE 5 Net profits
Pretax profits 3
Pretax profits
EBIT 3
EBIT
Sales 3
Sales
Assets 3
Assets
Equity
4. Other accounting ratios that have a bearing on a firm’s profitability and/or risk are fixed-asset turnover, inventory turnover, days sales in receivables, and the current, quick, and interest
coverage ratios.
5. Two ratios that make use of the market price of the firm’s common stock in addition to its finan- cial statements are the ratios of market to book value and price to earnings. Analysts sometimes
take low values for these ratios as a margin of safety or a sign that the stock is a bargain.
6. Good firms are not necessarily good investments. Stock market prices of successful firms may be bid up to levels that reflect that success. If so, the price of these firms relative to their earn-
ings prospects may not constitute a bargain.
7. A major problem in the use of data obtained from a firm’s financial statements is comparabil- ity. Firms have a great deal of latitude in how they choose to compute various items of revenue
and expense. It is, therefore, necessary for the security analyst to adjust accounting earnings
and financial ratios to a uniform standard before attempting to compare financial results across
firms.
8. Comparability problems can be acute in a period of inflation. Inflation can create distortions in accounting for inventories, depreciation, and interest expense.
9. Fair value or mark-to-market accounting requires that most assets be valued at current market value rather than historical cost. This policy has proved to be controversial because ascertaining
true market value in many instances is difficult, and critics contend that financial statements
are therefore unduly volatile. Advocates argue that financial statements should reflect the best
estimate of current asset values.
10. International financial reporting standards have become progressively accepted throughout the world, including the United States. They differ from traditional U.S. GAAP procedures in that
they are principles-based rather than rules-based.
Related Web sites for this chapter are available at www. mhhe.com/bkm
income statement
economic earnings
accounting earnings
balance sheet
statement of cash flows
return on assets (ROA)
return on equity (ROE)
economic value added
residual income
DuPont system
profit margin
return on sales
total asset turnover
interest coverage ratio
times interest earned
leverage ratio
inventory turnover ratio
average collection period
liquidity
current ratio
quick ratio
acid test ratio
cash ratio
market–book-value ratio
price–earnings (P/E) ratio
LIFO
FIFO
fair value accounting
mark-to-market accounting
quality of earnings
international financial
reporting standards
KEY TERMS
KEY EQUATIONS ROE and leverage: ROE 5(1 2Tax rate)aROA 1(ROA 2Interest rate) Debt
Equity b
DuPont formula: ROE 5 Net profit
Pretax profit 3
Pretax profit
EBIT 3
EBIT
Sales 3
Sales
Assets 3
Assets
Equity
Another DuPont formula: ROA 5 Margin 3 Turnover
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C H A P T E R 1 9 Financial Statement Analysis 667
PROBLEM SETS 1. What is the major difference in approach of international financial reporting standards and U.S. GAAP accounting? What are the advantages and disadvantages of each?
2. If markets are truly efficient, does it matter whether firms engage in earnings management? On
the other hand, if firms manage earnings, what does that say about management’s view on effi-
cient markets?
3. What financial ratios would a credit rating agency such as Moody’s or Standard & Poor’s be most
interested in? Which ratios would be of most interest to a stock market analyst deciding whether
to buy a stock for a diversified portfolio?
4. The Crusty Pie Co., which specializes in apple turnovers, has a return on sales higher than the
industry average, yet its ROA is the same as the industry average. How can you explain this?
5. The ABC Corporation has a profit margin on sales below the industry average, yet its ROA is
above the industry average. What does this imply about its asset turnover?
6. Firm A and firm B have the same ROA, yet firm A ’s ROE is higher. How can you explain this?
7. Use the DuPont system and the following data to find return on equity.
Leverage ratio (assets/equity) 2.2
Total asset turnover 2.0
Net profit margin 5.5%
Dividend payout ratio 31.8%
8. Recently, Galaxy Corporation lowered its allowance for doubtful accounts by reducing bad debt
expense from 2% of sales to 1% of sales. Ignoring taxes, what are the immediate effects on ( a ) operating income and ( b ) operating cash flow?
Use the following case in answering Problems 9–11: Hatfield Industries is a large manufacturing conglomerate based in the United States with annual sales in excess of $300 million. Hatfield is cur-
rently under investigation by the Securities and Exchange Commission (SEC) for accounting irregu-
larities and possible legal violations in the presentation of the company’s financial statements. A due
diligence team from the SEC has been sent to Hatfield’s corporate headquarters in Philadelphia for a
complete audit in order to further assess the situation.
Several unique circumstances at Hatfield are discovered by the SEC due diligence team during
the course of the investigation:
• Management has been involved in ongoing negotiations with the local labor union, of
which approximately 40% of its full-time labor force are members. Labor offi cials are
seeking increased wages and pension benefi ts, which Hatfi eld’s management states is
not possible at this time due to decreased profi tability and a tight cash fl ow situation.
Labor offi cials have accused Hatfi eld’s management of manipulating the company’s
fi nancial statements to justify not granting any concessions during the course of
negotiations.
• All new equipment obtained over the past several years has been established on
Hatfi eld’s books as operating leases, although past acquisitions of similar equipment
were nearly always classifi ed as capital leases. Financial statements of industry peers
indicate that capital leases for this type of equipment are the norm. The SEC wants
Hatfi eld’s management to provide justifi cation for this apparent deviation from “normal”
accounting practices.
• Inventory on Hatfield’s books has been steadily increasing for the past few years in
comparison to sales growth. Management credits improved operating efficiencies
in its production methods that have contributed to boosts in overall production. The
SEC is seeking evidence that Hatfield somehow may have manipulated its inventory
accounts.
The SEC due diligence team is not necessarily searching for evidence of fraud but of possible
manipulation of accounting standards for the purpose of misleading shareholders and other interested
Basic
Intermediate
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668 P A R T V Security Analysis
parties. Initial review of Hatfield’s financial statements indicates that at a minimum, certain practices
have resulted in low-quality earnings.
9. Labor officials believe that the management of Hatfield is attempting to understate its net
income to avoid making any concessions in the labor negotiations. Which of the following
actions by management will most likely result in low-quality earnings?
a. Lengthening the life of a depreciable asset in order to lower the depreciation expense. b. Lowering the discount rate used in the valuation of the company’s pension obligations. c. The recognition of revenue at the time of delivery rather than when payment is received.
10. Hatfield has begun recording all new equipment leases on its books as operating leases, a change
from its consistent past use of capital leases, in which the present value of lease payments is
classified as a debt obligation. What is the most likely motivation behind Hatfield’s change in accounting methodology? Hatfield is attempting to:
a. Improve its leverage ratios and reduce its perceived leverage. b. Reduce its cost of goods sold and increase its profitability. c. Increase its operating margins relative to industry peers.
11. The SEC due diligence team is searching for the reason behind Hatfield’s inventory build-up
relative to its sales growth. One way to identify a deliberate manipulation of financial results by
Hatfield is to search for:
a. A decline in inventory turnover. b. Receivables that are growing faster than sales. c. A delay in the recognition of expenses.
12. A firm has an ROE of 3%, a debt-to-equity ratio of .5, a tax rate of 35%, and pays an interest
rate of 6% on its debt. What is its operating ROA?
13. A firm has a tax burden ratio of .75, a leverage ratio of 1.25, an interest burden of .6, and a
return on sales of 10%. The firm generates $2.40 in sales per dollar of assets. What is the firm’s
ROE?
14. Use the following cash flow data for Rocket Transport to find Rocket’s
a. Net cash provided by or used in investing activities. b. Net cash provided by or used in financing activities. c. Net increase or decrease in cash for the year.
Cash dividend $ 80,000
Purchase of bus $ 33,000
Interest paid on debt $ 25,000
Sales of old equipment $ 72,000
Repurchase of stock $ 55,000
Cash payments to suppliers $ 95,000
Cash collections from customers $300,000
15. Here are data on two firms.
Equity ($ million)
Debt ($ million)
ROC (%)
Cost of Capital (%)
Acme 100 50 17% 9% Apex 450 150 15% 10%
a. Which firm has the higher economic value added? b. Which has the higher economic value added per dollar of invested capital?
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C H A P T E R 1 9 Financial Statement Analysis 669
1. The information in the following exhibit comes from the notes to the financial statements of
QuickBrush Company and SmileWhite Corporation:
QuickBrush SmileWhite
Goodwill The company amortizes goodwill over 20 years.
The company amortizes goodwill over 5 years.
Property, plant, and equipment
The company uses a straight-line depreciation method over the economic lives of the assets, which range from 5 to 20 years for buildings.
The company uses an accelerated depreciation method over the economic lives of the assets, which range from 5 to 20 years for buildings.
Accounts receivable
The company uses a bad debt allowance of 2% of accounts receivable.
The company uses a bad debt allowance of 5% of accounts receivable.
Determine which company has the higher quality of earnings by discussing each of the three notes.
2. Scott Kelly is reviewing MasterToy’s financial statements in order to estimate its sustainable
growth rate. Consider the information presented in the following exhibit.
MasterToy Inc.: Actual 2013 and estimated 2014 financial statements for fiscal year ending December 31 ($ million, except per-share data)
2013 2014 Change (%)
Income Statement
Revenue $4,750 $5,140 7.6%
Cost of goods sold 2,400 2,540
Selling, general, and administrative 1,400 1,550
Depreciation 180 210
Goodwill amortization 10 10
Operating income $ 760 $ 830 8.4
Interest expense 20 25
Income before taxes $ 740 $ 805
Income taxes 265 295
Net income $ 475 $ 510
Earnings per share $ 1.79 $ 1.96 8.6
Average shares outstanding (millions) 265 260
Balance Sheet
Cash $ 400 $ 400
Accounts receivable 680 700
Inventories 570 600
Net property, plant, and equipment 800 870
Intangibles 500 530
Total assets $2,950 $3,100
Current liabilities 550 600
Long-term debt 300 300
Total liabilities $ 850 $ 900
Stockholders’ equity 2,100 2,200
Total liabilities and equity $2,950 $3,100
Book value per share $ 7.92 $ 8.46
Annual dividend per share $ 0.55 $ 0.60
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670 P A R T V Security Analysis
a. Identify and calculate the components of the DuPont formula. b. Calculate the ROE for 2014 using the components of the DuPont formula. c. Calculate the sustainable growth rate for 2014 from the firm’s ROE and plowback ratios.
3. This problem should be solved using the following data:
Cash payments for interest $(12)
Retirement of common stock (32)
Cash payments to merchandise suppliers (85)
Purchase of land (8)
Sale of equipment 30
Payments of dividends (37)
Cash payment for salaries (35)
Cash collection from customers 260 Purchase of equipment (40)
a. What are cash flows from operating activities? b. Using the data above, calculate cash flows from investing activities. c. Using the data above, calculate cash flows from financing activities.
4. Janet Ludlow is a recently hired analyst. After describing the electric toothbrush industry, her
first report focuses on two companies, QuickBrush Company and SmileWhite Corporation, and
concludes:
QuickBrush is a more profitable company than SmileWhite, as indicated by the 40% sales growth and substantially higher margins it has produced over the last few years. SmileWhite’s sales and earnings are growing at a 10% rate and pro- duce much lower margins. We do not think SmileWhite is capable of growing faster than its recent growth rate of 10% whereas QuickBrush can sustain a 30% long-term growth rate.
a. Criticize Ludlow’s analysis and conclusion that QuickBrush is more profitable, as defined by return on equity (ROE), than SmileWhite and that it has a higher sustainable growth rate. Use
only the information provided in Tables 19A and 19B . Support your criticism by calculating
and analyzing:
• The fi ve components that determine ROE.
• The two ratios that determine sustainable growth: ROE and plowback.
b. Explain how QuickBrush has produced an average annual earnings per share (EPS) growth rate of 40% over the last 2 years with an ROE that has been declining. Use only the informa-
tion provided in Table 19A .
Use the following in answering CFA Problems 5–8: Eastover Company (EO) is a large, diversified forest products company. Approximately 75% of its sales are from paper and forest products, with
the remainder from financial services and real estate. The company owns 5.6 million acres of timber-
land, which is carried at very low historical cost on the balance sheet.
Peggy Mulroney, CFA, is an analyst at the investment counseling firm of Centurion Investments.
She is assigned the task of assessing the outlook for Eastover, which is being considered for pur-
chase, and comparing it to another forest products company in Centurion’s portfolios, Southampton
Corporation (SHC). SHC is a major producer of lumber products in the United States. Building
products, primarily lumber and plywood, account for 89% of SHC’s sales, with pulp accounting for
the remainder. SHC owns 1.4 million acres of timberland, which is also carried at historical cost on
the balance sheet. In SHC’s case, however, that cost is not as far below current market as Eastover’s.
Mulroney began her examination of Eastover and Southampton by looking at the five compo-
nents of return on equity (ROE) for each company. For her analysis, Mulroney elected to define
equity as total shareholders’ equity, including preferred stock. She also elected to use year-end data
rather than averages for the balance sheet items.
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C H A P T E R 1 9 Financial Statement Analysis 671
5. a. On the basis of the data shown in Tables 19C and 19D , calculate each of the five ROE compo- nents for Eastover and Southampton in 2013. Using the five components, calculate ROE for
both companies in 2013.
b. Referring to the components calculated in part ( a ), explain the difference in ROE for Eastover and Southampton in 2013.
c. Using 2013 data, calculate the sustainable growth rate for both Eastover and Southampton. Discuss the appropriateness of using these calculations as a basis for estimating future growth.
Income Statement December
2011 December
2012 December
2013
Revenue $3,480 $5,400 $7,760
Cost of goods sold 2,700 4,270 6,050
Selling, general, and admin. expense 500 690 1,000
Depreciation and amortization 30 40 50
Operating income (EBIT) $ 250 $ 400 $ 660
Interest expense 0 0 0
Income before taxes $ 250 $ 400 $ 660
Income taxes 60 110 215
Income after taxes $ 190 $ 290 $ 445
Diluted EPS $ 0.60 $ 0.84 $ 1.18
Average shares outstanding (000) 317 346 376
Financial Statistics December
2011 December
2012 December
2013 3-Year
Average
COGS as % of sales 77.59% 79.07% 77.96% 78.24%
General & admin. as % of sales 14.37 12.78 12.89 13.16
Operating margin 7.18 7.41 8.51
Pretax income/EBIT 100.00 100.00 100.00
Tax rate 24.00 27.50 32.58
Balance Sheet December
2011 December
2012 December
2013
Cash and cash equivalents $ 460 $ 50 $ 480
Accounts receivable 540 720 950
Inventories 300 430 590
Net property, plant, and equipment 760 1,830 3,450
Total assets $2,060 $3,030 $5,470
Current liabilities $ 860 $1,110 $1,750
Total liabilities $ 860 $1,110 $1,750
Stockholders’ equity 1,200 1,920 3,720
Total liabilities and equity $2,060 $3,030 $5,470
Market price per share $21.00 $30.00 $45.00
Book value per share $ 3.79 $ 5.55 $ 9.89
Annual dividend per share $ 0.00 $ 0.00 $ 0.00
Table 19A
QuickBrush Company financial statements: yearly data ($000 except per-share data)
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672 P A R T V Security Analysis
6. a. Mulroney recalled from her CFA studies that the constant-growth discounted dividend model was one way to arrive at a valuation for a company’s common stock. She collected current
dividend and stock price data for Eastover and Southampton, shown in Table 19E . Using 11%
as the required rate of return (i.e., discount rate) and a projected growth rate of 8%, com-
pute a constant-growth DDM value for Eastover’s stock and compare the computed value for
Eastover to its stock price indicated in Table 19F .
b. Mulroney’s supervisor commented that a two-stage DDM may be more appropriate for com- panies such as Eastover and Southampton. Mulroney believes that Eastover and Southampton
Income Statement December
2011 December
2012 December
2013
Revenue $104,000 $110,400 $119,200
Cost of goods sold 72,800 75,100 79,300
Selling, general, and admin. expense 20,300 22,800 23,900
Depreciation and amortization 4,200 5,600 8,300
Operating income $ 6,700 $ 6,900 $ 7,700
Interest expense 600 350 350
Income before taxes $ 6,100 $ 6,550 $ 7,350
Income taxes 2,100 2,200 2,500
Income after taxes $ 4,000 $ 4,350 $ 4,850
Diluted EPS $ 2.16 $ 2.35 $ 2.62
Average shares outstanding (000) 1,850 1,850 1,850
Financial Statistics December
2011 December
2012 December
2013 3-Year
Average
COGS as % of sales 70.00% 68.00% 66.53% 68.10%
General & admin. as % of sales 19.52 20.64 20.05 20.08
Operating margin 6.44 6.25 6.46
Pretax income/EBIT 91.04 94.93 95.45
Tax rate 34.43 33.59 34.01
Balance Sheet December
2011 December
2012 December
2013
Cash and cash equivalents $ 7,900 $ 3,300 $ 1,700
Accounts receivable 7,500 8,000 9,000
Inventories 6,300 6,300 5,900
Net property, plant, and equipment 12,000 14,500 17,000
Total assets $ 33,700 $ 32,100 $ 33,600
Current liabilities $ 6,200 $ 7,800 $ 6,600
Long-term debt 9,000 4,300 4,300
Total liabilities $ 15,200 $ 12,100 $ 10,900
Stockholders’ equity 18,500 20,000 22,700
Total liabilities and equity $ 33,700 $ 32,100 $ 33,600
Market price per share $ 23.00 $ 26.00 $ 30.00
Book value per share $ 10.00 $ 10.81 $ 12.27
Annual dividend per share $ 1.42 $ 1.53 $ 1.72
Table 19B
SmileWhite Corporation financial statements: yearly data ($000 except per-share data)
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C H A P T E R 1 9 Financial Statement Analysis 673
could grow more rapidly over the next 3 years and then settle in at a lower but sustainable rate
of growth beyond 2017. Her estimates are indicated in Table 19G . Using 11% as the required
rate of return, compute the two-stage DDM value of Eastover’s stock and compare that value
to its stock price indicated in Table 19F .
c. Discuss advantages and disadvantages of using a constant-growth DDM. Briefly discuss how the two-stage DDM improves upon the constant-growth DDM.
7. In addition to the discounted dividend model approach, Mulroney decided to look at the price–
earnings ratio and price–book ratio, relative to the S&P 500, for both Eastover and Southampton.
Mulroney elected to perform this analysis using 2010–2014 and current data.
a. Using the data in Tables 19E and 19F , compute both the current and the 5-year (2010–2014) average relative price–earnings ratios and relative price–book ratios for Eastover and
Southampton (i.e., ratios relative to those for the S&P 500). Discuss each company’s current
relative price–earnings ratio compared to its 5-year average relative price–earnings ratio and
each company’s current relative price–book ratio as compared to its 5-year average relative
price–book ratio.
b. Briefly discuss one disadvantage for each of the relative price–earnings and relative price– book approaches to valuation.
2009 2010 2011 2012 2013
Income Statement Summary
Sales $5,652 $6,990 $7,863 $8,281 $7,406
Earnings before interest and taxes (EBIT) $ 568 $ 901 $1,037 $ 708 $ 795
Interest expense (net) (147) (188) (186) (194) (195)
Income before taxes $ 421 $ 713 $ 851 $ 514 $ 600
Income taxes (144) (266) (286) (173) (206)
Tax rate 34% 37% 33% 34% 34%
Net income $ 277 $ 447 $ 565 $ 341 $ 394
Preferred dividends (28) (17) (17) (17) (0)
Net income to common $ 249 $ 430 $ 548 $ 324 $ 394
Common shares outstanding (millions) 196 204 204 205 201
Balance Sheet Summary
Current assets $1,235 $1,491 $1,702 $1,585 $1,367
Timberland assets 649 625 621 612 615
Property, plant, and equipment 4,370 4,571 5,056 5,430 5,854
Other assets 360 555 473 472 429
Total assets $6,614 $7,242 $7,852 $8,099 $8,265
Current liabilities $1,226 $1,186 $1,206 $1,606 $1,816
Long-term debt 1,120 1,340 1,585 1,346 1,585
Deferred taxes 1,000 1,000 1,016 1,000 1,000
Equity–preferred 364 350 350 400 0
Equity–common 2,904 3,366 3,695 3,747 3,864
Total liabilities and equity $6,614 $7,242 $7,852 $8,099 $8,265
Table 19C
Eastover Company ($ million, except shares outstanding)
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674 P A R T V Security Analysis
8. Mulroney previously calculated a valuation for Southampton for both the constant-growth and
two-stage DDM as shown below:
Constant-Growth Approach Two-Stage Approach
$29 $35.50
Using only the information provided and your answers to CFA Problems 5–7, select the stock
(EO or SHC) that Mulroney should recommend as the better value, and justify your selection.
9. In reviewing the financial statements of the Graceland Rock Company, you note that net income
increased while cash flow from operations decreased from 2013 to 2014.
a. Explain how net income could increase for Graceland Rock Company while cash flow from operations decreased. Give some illustrative examples.
b. Explain why cash flow from operations may be a good indicator of a firm’s “quality of earnings.”
10. A firm has net sales of $3,000, cash expenses (including taxes) of $1,400, and depreciation of
$500. If accounts receivable increase over the period by $400, what would be cash flow from
operations?
11. A company’s current ratio is 2.0. Suppose the company uses cash to retire notes payable due
within 1 year. What would be the effect on the current ratio and asset turnover ratio?
12. Jones Group has been generating stable after-tax return on equity (ROE) despite declining oper-
ating income. Explain how it might be able to maintain its stable after-tax ROE.
2009 2010 2011 2012 2013
Income Statement Summary
Sales $1,306 $1,654 $1,799 $2,010 $1,793
Earnings before interest and taxes (EBIT) $ 120 $ 230 $ 221 $ 304 $ 145
Interest expense (net) (13) (36) (7) (12) (8)
Income before taxes $ 107 $ 194 $ 214 $ 292 $ 137
Income taxes (44) (75) (79) (99) (46)
Tax rate 41% 39% 37% 34% 34%
Net income $ 63 $ 119 $ 135 $ 193 $ 91
Common shares outstanding (millions) 38 38 38 38 38
Balance Sheet Summary
Current assets $ 487 $ 504 $ 536 $ 654 $ 509
Timberland assets 512 513 508 513 518
Property, plant, and equipment 648 681 718 827 1,037
Other assets 141 151 34 38 40
Total assets $1,788 $1,849 $1,796 $2,032 $2,104
Current liabilities $ 185 $ 176 $ 162 $ 180 $ 195
Long-term debt 536 493 370 530 589
Deferred taxes 123 136 127 146 153
Equity 944 1,044 1,137 1,176 1,167
Total liabilities and equity $1,788 $1,849 $1,796 $2,032 $2,104
Table 19D
Southampton Corporation ($ million, except shares outstanding)
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C H A P T E R 1 9 Financial Statement Analysis 675
2009 2010 2011 2012 2013 2014 5-Year Average
(2010–2014)
Eastover Company
Earnings per share $ 1.27 $ 2.12 $ 2.68 $ 1.56 $ 1.87 $ 0.90
Dividends per share 0.87 0.90 1.15 1.20 1.20 1.20
Book value per share 14.82 16.54 18.14 18.55 19.21 17.21
Stock price
High 28 40 30 33 28 30
Low 20 20 23 25 18 20
Close 25 26 25 28 22 27
Average P/E 18.9 14.2 9.9 18.6 12.3 27.8
Average P/B 1.6 1.8 1.5 1.6 1.2 1.5
Southampton Corporation
Earnings per share $ 1.66 $ 3.13 $ 3.55 $ 5.08 $ 2.46 $ 1.75
Dividends per share 0.77 0.79 0.89 0.98 1.04 1.08
Book value per share 24.84 27.47 29.92 30.95 31.54 32.21
Stock price
High 34 40 38 43 45 46
Low 21 22 26 28 20 26
Close 31 27 28 39 27 44
Average P/E 16.6 9.9 9.0 7.0 13.2 20.6
Average P/B 1.1 1.1 1.1 1.2 1.0 1.1
S&P 500
Average P/E 15.8 16.0 11.1 13.9 15.6 19.2 15.2
Average P/B 1.8 2.1 1.9 2.2 2.1 2.3 2.1
Table 19E
Valuation of Eastover Company and Southampton Corporation compared to S&P 500
Current Share Price
Current Dividends Per Share
2015 EPS Estimate
Current Book Value Per Share
Eastover $ 28 $ 1.20 $ 1.60 $ 17.32
Southampton 48 1.08 3.00 32.21
S&P 500 1660 48.00 82.16 639.32
Table 19F
Current information
Next 3 Years (2015, 2016, 2017) Growth Beyond 2017
Eastover 12% 8%
Southampton 13% 7%
Table 19G
Projected growth rates as of year-end 2014
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676 P A R T V Security Analysis
13. The DuPont formula defines the net return on shareholders’ equity as a function of the follow-
ing components:
• Operating margin.
• Asset turnover.
• Interest burden.
• Financial leverage.
• Income tax rate.
Using only the data in Table 19H :
a. Calculate each of the five components listed above for 2010 and 2014, and calculate the return on equity (ROE) for 2010 and 2014, using all of the five components.
b. Briefly discuss the impact of the changes in asset turnover and financial leverage on the change in ROE from 2010 to 2014.
2010 2014
Income Statement Data
Revenues $542 $979
Operating income 38 76
Depreciation and amortization 3 9
Interest expense 3 0
Pretax income 32 67
Income taxes 13 37
Net income after tax 19 30
Balance Sheet Data
Fixed assets $ 41 $ 70
Total assets 245 291
Working capital 123 157
Total debt 16 0
Total shareholders’ equity 159 220
Table 19H
Income statements and balance sheets
E-INVESTMENTS EXERCISES This chapter introduced the idea of economic value added (EVA) as a means to measure firm performance. A related measure is market value added (MVA), which is the difference between the market value of a firm and its book value. You can find the firms with the best such measures at www.evadimensions.com. You will see there that EVA leaders do not necessarily have the highest return on capital. Why not? Are the EVA leaders also the MVA leaders? Why not?
SOLUTIONS TO CONCEPT CHECKS 1. A debt-to-equity ratio of 1 implies that Mordett will have $50 million of debt and $50 million of
equity. Interest expense will be .09 3 $50 million, or $4.5 million per year. Mordett’s net profits
and ROE over the business cycle will therefore be
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C H A P T E R 1 9 Financial Statement Analysis 677
Nodett Mordett
Scenario EBIT Net Profits ROE Net Profits* ROE†
Bad year $ 5 million $3 million 3% $0.3 million .6%
Normal year 10 6 6 3.3 6.6
Good year 15 9 9 6.3 12.6
*Mordett’s after-tax profits are given by .6 (EBIT 2 $4.5 million). †Mordett’s equity is only $50 million.
2.
3. GI’s ROE in 2015 was 3.03%, computed as follows:
ROE 5 $5,285
.5($171,843 1$177,128) 5.0303, or 3.03%
Its P/E ratio was 4 5 $21/$5.285 and its P/B ratio was .12 5 $21/$177. Its earnings yield was
25% compared with an industry average of 12.5%.
Note that in our calculations P/E does not equal (P/B)/ROE because (following common prac-
tice) we have computed ROE with average shareholders’ equity in the denominator and P/B with end -of-year shareholders’ equity in the denominator.
4.
ROE increased despite a decline in operating margin and a decline in the tax burden ratio because of
increased leverage and turnover. Note that ROA declined from 11.65% in 2013 to 10.65% in 2015.
5. LIFO accounting results in lower reported earnings than does FIFO. Fewer assets to depreciate
result in lower reported earnings because there is less bias associated with the use of historic cost.
More debt results in lower reported earnings because the inflation premium in the interest rate is
treated as part of interest expense and not as repayment of principal. If ABC has the same reported
earnings as XYZ despite these three sources of downward bias, its real earnings must be greater.
Ratio Decomposition Analysis for Mordett Corporation
(1) (2) (3) (4) (5) (6)
ROE Net Profit/
Pretax Profit Pretax
Profit/EBIT EBIT/Sales (Margin)
Sales/Assets (turnover)
Assets/ Equity
Combined Leverage Factor (2) 3 (5)
Bad year
Nodett .030 .6 1.000 .0625 0.800 1.000 1.000
Somdett .018 .6 0.360 .0625 0.800 1.667 0.600
Mordett .006 .6 0.100 .0625 0.800 2.000 0.200
Normal year
Nodett .060 .6 1.000 .100 1.000 1.000 1.000
Somdett .068 .6 0.680 .100 1.000 1.667 1.134
Mordett .066 .6 0.550 .100 1.000 2.000 1.100
Good year
Nodett .090 .6 1.000 .125 1.200 1.000 1.000
Somdett .118 .6 0.787 .125 1.200 1.667 1.311
Mordett .126 .6 0.700 .125 1.200 2.000 1.400
IBX Ratio Analysis
(1) (2) (3) (4) (5) (6) (7)
Year ROE
Net Profit/ Pretax Profit
Pretax Profit/ EBIT
EBIT/Sales (Margin)
Sales/ Assets
(turnover) Assets/ Equity
Combined Leverage
Factor (2) 3 (5)
ROA (3) 3 (4)
2015 11.4% .616 .796 7.75% 1.375 2.175 1.731 10.65%
2013 10.2 .636 .932 8.88 1.311 1.474 1.374 11.65
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20 DERIVATIVE SECURITIES, OR more simply derivatives, play a large and increasingly important role in financial markets. These are securities whose prices are determined by, or “derive from,” the prices of other securities.
Options and futures contracts are both derivative securities. Their payoffs depend on the value of other securities. Swaps, which we will discuss in Chapter 23, also are derivatives. Because the value of derivatives depends on the value of other securities, they can be powerful tools for both hedging and specula- tion. We will investigate these applications in the next four chapters, starting in this chapter with options.
Trading of standardized options contracts on a national exchange started in 1973 when the Chicago Board Options Exchange (CBOE) began listing call options. These contracts were almost immediately a great success,
crowding out the previously existing over- the-counter options market.
Option contracts are traded now on several exchanges. They are written on common stock, stock indexes, foreign exchange, agricultural commodities, precious metals, and interest rate futures. In addition, the over-the-counter market has enjoyed a tremendous resurgence in recent years as trading in custom-tailored options has exploded. Popular and potent tools in modifying portfolio characteristics, options have become essential tools a port- folio manager must understand.
This chapter is an introduction to options markets. It explains how puts and calls work and examines their investment characteris- tics. Popular option strategies are considered next. Finally, we examine a range of securities with embedded options such as callable or convertible bonds, and we take a quick look at some so-called exotic options.
CHAPTER TWENTY 2 Options Markets: Introduction PA
R T V
I
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C H A P T E R 2 0 Options Markets: Introduction 679
A call option gives its holder the right to purchase an asset for a specified price, called the exercise , or strike, price , on or before some specified expiration date. For example, a February call option on IBM stock with exercise price $195 entitles its owner to
purchase IBM stock for a price of $195 at any time up to and including the expiration
date in February. The holder of the call is not required to exercise the option. She will
choose to exercise only if the market value of the underlying asset exceeds the exercise
price. In that case, the option holder may “call away” the asset for the exercise price.
Otherwise, the option may be left unexercised. If it is not exercised before the expira-
tion date of the contract, a call option simply expires and becomes valueless. Therefore,
if the stock price is greater than the exercise price on the expiration date, the value of
the call option equals the difference between the stock price and the exercise price; but
if the stock price is less than the exercise price at expiration, the call will be worthless.
The net profit on the call is the value of the option minus the price originally paid to purchase it.
The purchase price of the option is called the premium . It represents the compensation the purchaser of the call must pay for the right to exercise the option only when exercise
is desirable.
Sellers of call options, who are said to write calls, receive premium income now as pay- ment against the possibility they will be required at some later date to deliver the asset in
return for an exercise price less than the market value of the asset. If the option is left to
expire worthless, the writer of the call clears a profit equal to the premium income derived
from the initial sale of the option. But if the call is exercised, the profit to the option writer
is the premium income minus the difference between the value of the stock that must be delivered and the exercise price that is paid for those shares. If that difference is larger than
the initial premium, the writer will incur a loss.
20.1 The Option Contract
Consider the February 2013 expiration call option on a share of IBM with an exercise price of $195 selling on January 18, 2013, for $3.65. Exchange-traded options expire on the third Friday of the expiration month, which for this option was February 15. Until the expiration date, the call holder may buy shares of IBM for $195. On January 18, IBM sells for $194.47. Because the stock price is only $194.47, it clearly would not make sense at the moment to exercise the option to buy at $195. Indeed, if IBM remains below $195 by the expiration date, the call will be left to expire worthless. On the other hand, if IBM is selling above $195 at expiration, the call holder will find it optimal to exercise. For example, if IBM sells for $197 on February 15, the option will be exercised, as it will give its holder the right to pay $195 for a stock worth $197. The value of each option on the expiration date would then be
Value at expiration 5 Stock price 2 Exercise price 5 $197 2 $195 5 $2
Despite the $2 payoff at expiration, the call holder still realizes a loss of $1.65 on the investment because the initial purchase price was $3.65:
Profit 5 Final value 2 Original investment 5 $2.00 2 $3.65 5 2$1.65
Example 20.1 Profits and Losses on a Call Option
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680 P A R T V I Options, Futures, and Other Derivatives
A put option gives its holder the right to sell an asset for a specified exercise or strike price on or before some expiration date. A February expiration put on IBM with an exer-
cise price of $195 entitles its owner to sell IBM stock to the put writer at a price of $195 at
any time before expiration in February even if the market price of IBM is less than $195.
Whereas profits on call options increase when the asset price rises, profits on put options
increase when the asset price falls. A put will be exercised only if the exercise price is greater than the price of the underlying asset, that is, only if its holder can deliver for the
exercise price an asset with market value less than that amount. (One doesn’t need to own
the shares of IBM to exercise the IBM put option. Upon exercise, the investor’s broker
purchases the necessary shares of IBM at the market price and immediately delivers, or
“puts them,” to an option writer for the exercise price. The owner of the put profits by the
difference between the exercise price and market price.)
Now consider the February 2013 expiration put option on IBM with an exercise price of $195, selling on January 18 for $5.00. It entitled its owner to sell a share of IBM for $195 at any time until February 15. If the holder of the put buys a share of IBM and immediately exercises the right to sell it at $195, net proceeds will be $195 2 $194.47 5 $.53. Obviously, an investor who pays $5 for the put has no inten- tion of exercising it immediately. If, on the other hand, IBM were selling for $188 at expiration, the put would turn out to be a profitable investment. Its value at expiration would be
Value at expiration 5 Exercise price 2 Stock price 5 $195 2 $188 5 $7
and the investor’s profit would be $7 2 $5 5 $2. This is a holding period return of $2/$5 5 .40, or 40%—over only 28 days! Obviously, put option sellers on January 18 (who are on the other side of the transaction) did not consider this outcome very likely.
Example 20.2 Profits and Losses on a Put Option
An option is described as in the money when its exercise would produce a positive cash flow. Therefore, a call option is in the money when the asset price is greater than
the exercise price, and a put option is in the money when the asset price is less than the
exercise price. Conversely, a call is out of the money when the asset price is less than the exercise price; no one would exercise the right to purchase for the strike price an asset
worth less than that amount. A put option is out of the money when the exercise price
is less than the asset price. Options are at the money when the exercise price and asset price are equal.
Nevertheless, exercise of the call is optimal at expiration if the stock price exceeds the exercise price because the exercise proceeds will offset at least part of the purchase price. The call buyer will clear a profit if IBM is selling above $198.65 at the expiration date. At that stock price, the proceeds from exercise will just cover the original cost of the call.
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C H A P T E R 2 0 Options Markets: Introduction 681
Options Trading Some options trade on over-the-counter markets. The OTC
market offers the advantage that the terms of the option
contract—the exercise price, expiration date, and number
of shares committed—can be tailored to the needs of the
traders. The costs of establishing an OTC option contract,
however, are higher than for exchange-traded options.
Options contracts traded on exchanges are standard-
ized by allowable expiration dates and exercise prices for
each listed option. Each stock option contract provides
for the right to buy or sell 100 shares of stock (except
when stock splits occur after the contract is listed and
the contract is adjusted for the terms of the split).
Standardization of the terms of listed option contracts
means all market participants trade in a limited and uni-
form set of securities. This increases the depth of trad-
ing in any particular option, which lowers trading costs
and results in a more competitive market. Exchanges,
therefore, offer two important benefits: ease of trading,
which flows from a central marketplace where buy-
ers and sellers or their representatives congregate; and
a liquid secondary market where buyers and sellers of
options can transact quickly and cheaply.
Until recently, most options trading in the United
States took place on the Chicago Board Options Exchange. However, by 2003 the Inter-
national Securities Exchange, an electronic exchange based in New York, displaced the
CBOE as the largest options market. Options trading in Europe is uniformly transacted in
electronic exchanges.
Figure 20.1 is a selection of listed stock option quotations for IBM. The last recorded
price on the New York Stock Exchange for IBM shares was $194.47 per share. 1 The exer-
cise (or strike) prices bracket the stock price. While exercise prices generally are set at
five-point intervals, larger intervals sometimes are set for stocks selling above $100, and
intervals of $2.50 may be used for stocks selling at low prices. If the stock price moves out-
side the range of exercise prices of the existing set of options, new options with appropriate
exercise prices may be offered. Therefore, at any time, both in-the-money and out-of-the-
money options will be listed, as in this example.
Figure 20.1 shows both call and put options listed for each expiration date and exercise
price. The three sets of columns for each option report closing price, trading volume in
contracts, and open interest (number of outstanding contracts). When we compare prices
of call options with the same expiration date but different exercise prices in Figure 20.1 ,
we see that the value of a call is lower when the exercise price is higher. This makes sense,
because the right to purchase a share at a lower exercise price is more valuable than the
right to purchase at a higher price. Thus the February expiration IBM call option with
strike price $195 sells for $3.65 whereas the $200 exercise price February call sells for
1 Occasionally, this price may not match the closing price listed for the stock on the stock market page. This is
because some NYSE stocks also trade on exchanges that close after the NYSE, and the stock pages may reflect
the more recent closing price. The options exchanges, however, close with the NYSE, so the closing NYSE stock
price is appropriate for comparison with the closing option price.
PRICES AT CLOSE JANUARY 18, 2013
Expiration
Jan Feb Apr Jul Jan Feb Apr Jul Jan Feb Apr Jul Jan Feb Apr Jul
185 185 185 185 190 190 190 190 195 195 195 195 200 200 200 200
9.15 10.60 12.00 14.35
4.40 6.75 8.85
10.95 0.01 3.65 5.90 8.45 1.10 1.61 3.70 6.10
307 299
41 37
815 402 107
15 2451 1337 1785
13 1248 1053 629
80
2431 2
706 134
5697 2808 1866
645 11718 11902
2928 5773 2966 5530 3236 1257
0.76 1.82 3.60 6.55 1.75 3.00 5.20 8.54 0.70 5.00 7.30
10.85 5.55 8.09
10.05 ...
302 710 104
37 507
3553 527
6 4090
860 934
22 637 546 375
...
2488 3645 2047 1354 2496
10377 2177 1142 8862 3156 1141 3419 6199
967 1903 1105
Call Put
I B M (IBM) Underlying Stock Price: 194.47
Strike Last Volume Open
Interest Last Volume Open
Interest
Figure 20.1 Stock options on IBM closing prices as of January 18, 2013
Source: The Wall Street Journal Online, January 18, 2013.
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682 P A R T V I Options, Futures, and Other Derivatives
only $1.61. Conversely, put options are worth more when the exercise price is higher: You would rather have the right to sell shares for $200 than for $195 and this is reflected in
the prices of the puts. The February expiration put option with strike price $200 sells for
$8.09, whereas the $195 exercise price February put sells for only $5.
If an option does not trade on a given day, three dots will appear in the volume and price
columns. Because trading is infrequent, it is not unusual to find option prices that appear
out of line with other prices. You might see, for example, two calls with different exercise
prices that seem to sell for the same price. This discrepancy arises because the last trades
for these options may have occurred at different times during the day. At any moment, the
call with the lower exercise price must be worth more than an otherwise-identical call with
a higher exercise price.
Expirations of most exchange-traded options tend to be fairly short, ranging up to only
several months. For larger firms and several stock indexes, however, longer-term options
are traded with expirations ranging up to several years. These options are called LEAPS
(for L ong- T erm E quity A ntici P ation S ecurities).
a. What will be the proceeds and net profits to an investor who purchases the February expiration IBM calls with exercise price $195 if the stock price at expiration is $205? What if the stock price at expira- tion is $185?
b. Now answer part ( a ) for an investor who purchases a February expiration IBM put option with exercise price $195.
CONCEPT CHECK 20.1
American and European Options An American option allows its holder to exercise the right to purchase (if a call) or sell (if a put) the underlying asset on or before the expiration date. European options allow for exercise of the option only on the expiration date. American options, because they allow
more leeway than their European counterparts, generally will be more valuable. Virtually
all traded options in the United States are American style. Foreign currency options and
stock index options are notable exceptions to this rule, however.
Adjustments in Option Contract Terms Because options convey the right to buy or sell shares at a stated price, stock splits would
radically alter their value if the terms of the options contract were not adjusted to account
for the stock split. For example, reconsider the IBM call options in Figure 20.1 . If IBM
were to announce a 2-for-1 split, its share price would fall from about $195 to about
$97.50. A call option with exercise price $195 would be just about worthless, with virtu-
ally no possibility that the stock would sell at more than $195 before the options expired.
To account for a stock split, the exercise price is reduced by a factor of the split, and the
number of options held is increased by that factor. For example, each original call option
with exercise price of $195 would be altered after a 2-for-1 split to two new options, with
each new option carrying an exercise price of $97.50. A similar adjustment is made for stock
dividends of more than 10%; the number of shares covered by each option is increased in
proportion to the stock dividend, and the exercise price is reduced by that proportion.
In contrast to stock dividends, cash dividends do not affect the terms of an option con-
tract. Because payment of a cash dividend reduces the selling price of the stock without
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C H A P T E R 2 0 Options Markets: Introduction 683
inducing offsetting adjustments in the option contract, the value of the option is affected by
dividend policy. Other things being equal, call option values are lower for high-dividend
payout policies, because such policies slow the rate of increase of stock prices; conversely,
put values are higher for high-dividend payouts. (Of course, the option values do not nec-
essarily rise or fall on the dividend payment or ex-dividend dates. Dividend payments are
anticipated, so the effect of the payment already is built into the original option price.)
Suppose that IBM’s stock price at the exercise date is $200, and the exercise price of the call is $195. What is the payoff on one option contract? After a 2-for-1 split, the stock price is $100, the exercise price is $97.50, and the option holder now can purchase 200 shares. Show that the split leaves the payoff from the option unaffected.
CONCEPT CHECK 20.2
The Options Clearing Corporation The Options Clearing Corporation (OCC), the clearinghouse for options trading, is jointly
owned by the exchanges on which stock options are traded. Buyers and sellers of options
who agree on a price will strike a deal. At this point, the OCC steps in. The OCC places
itself between the two traders, becoming the effective buyer of the option from the writer
and the effective writer of the option to the buyer. All individuals, therefore, deal only with
the OCC, which effectively guarantees contract performance.
When an option holder exercises an option, the OCC arranges for a member firm with
clients who have written that option to make good on the option obligation. The mem-
ber firm selects from its clients who have written that option to fulfill the contract. The
selected client must deliver 100 shares of stock at a price equal to the exercise price for
each call option contract written or must purchase 100 shares at the exercise price for each
put option contract written.
Because the OCC guarantees contract performance, it requires option writers to post
margin to guarantee that they can fulfill their contract obligations. The margin required is
determined in part by the amount by which the option is in the money, because that value
is an indicator of the potential obligation of the option writer. When the required margin
exceeds the posted margin, the writer will receive a margin call. In contrast, the holder of
the option need not post margin because the holder will exercise the option only if it is
profitable to do so. After purchase of the option, no further money is at risk.
Margin requirements are determined in part by the other securities held in the investor’s
portfolio. For example, a call option writer owning the stock against which the option is
written can satisfy the margin requirement simply by allowing a broker to hold that stock
in the brokerage account. The stock is then guaranteed to be available for delivery should
the call option be exercised. If the underlying security is not owned, however, the margin
requirement is determined by the value of the underlying security as well as by the amount
by which the option is in or out of the money. Out-of-the-money options require less mar-
gin from the writer, for expected payouts are lower.
Other Listed Options Options on assets other than stocks are also widely traded. These include options on mar-
ket indexes and industry indexes, on foreign currency, and even on the futures prices of
agricultural products, gold, silver, fixed-income securities, and stock indexes. We will dis-
cuss these in turn.
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684 P A R T V I Options, Futures, and Other Derivatives
Index Options An index option is a call or put based on a stock market index such as the S&P 500 or the NASDAQ 100. Index options are traded on several broad-based
indexes as well as on several industry-specific indexes and even commodity price indexes.
We discussed many of these indexes in Chapter 2.
The construction of the indexes can vary across contracts or exchanges. For example,
the S&P 100 index is a value-weighted average of the 100 stocks in the Standard & Poor’s
100 stock group. The weights are proportional to the market value of outstanding equity
for each stock. The Dow Jones Industrial Index, by contrast, is a price-weighted average
of 30 stocks.
Option contracts on many foreign stock indexes also trade. For example, options on
the (Japanese) Nikkei Stock Index trade on the Singapore as well as the Chicago Mercan-
tile Exchange. Options on European indexes such as the Financial Times Share Exchange
(FTSE 100) trade on the NYSE-Euronext Exchange. The Chicago Board Options Exchange
also lists options on industry indexes such as the oil or high-tech industries.
In contrast to stock options, index options do not require that the call writer actually
“deliver the index” upon exercise or that the put writer “purchase the index.” Instead, a
cash settlement procedure is used. The payoff that would accrue upon exercise of the
option is calculated, and the option writer simply pays that amount to the option holder.
The payoff is equal to the difference between the exercise price of the option and the
value of the index. For example, if the S&P index is at 1400 when a call option on the
index with exercise price 1390 is exercised, the holder of the call receives a cash pay-
ment of the difference, 1400 2 1390, times the contract multiplier of $100, or $1,000 per
contract.
Options on the major indexes, that is, the S&P 100 (often called the OEX after its
ticker symbol), the S&P 500 (the SPX), the NASDAQ 100 (the NDX), and the Dow Jones
Industrials (the DJX), are the most actively traded contracts on the CBOE. Together, these
contracts dominate CBOE volume.
Futures Options Futures options give their holders the right to buy or sell a speci- fied futures contract, using as a futures price the exercise price of the option. Although the
delivery process is slightly complicated, the terms of futures options contracts are designed
in effect to allow the option to be written on the futures price itself. The option holder
receives upon exercise a net payoff equal to the difference between the current futures
price on the specified asset and the exercise price of the option. Thus if the futures price is,
say, $37, and the call has an exercise price of $35, the holder who exercises the call option
on the futures gets a payoff of $2.
Foreign Currency Options A currency option offers the right to buy or sell a quantity of foreign currency for a specified amount of domestic currency. Currency
option contracts call for purchase or sale of the currency in exchange for a specified num-
ber of U.S. dollars. Contracts are quoted in cents or fractions of a cent per unit of foreign
currency.
There is an important difference between currency options and currency futures options. The former provide payoffs that depend on the difference between the exercise price and
the exchange rate at expiration. The latter are foreign exchange futures options that provide
payoffs that depend on the difference between the exercise price and the exchange rate
futures price at expiration. Because exchange rates and exchange rate futures prices gener- ally are not equal, the options and futures-options contracts will have different values, even
with identical expiration dates and exercise prices. Trading volume in currency futures
options dominates trading in currency options.
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C H A P T E R 2 0 Options Markets: Introduction 685
Interest Rate Options Options are traded on Treasury notes and bonds, Treasury bills, and government bonds of other major economies such as the U.K. or Japan. Options
on several interest rates also trade. Among these are contracts on Treasury bond, Treasury
note, federal funds, LIBOR, Euribor, 2 and Eurodollar futures.
2 The Euribor market is similar to the LIBOR market (see Chapter 2), but the interest rate charged in the Euribor
market is the interbank rate for euro-denominated deposits.
Call Options Recall that a call option gives the right to purchase a security at the exercise price. Suppose
you hold a call option on FinCorp stock with an exercise price of $100, and FinCorp is now
selling at $110. You can exercise your option to purchase the stock at $100 and simultane-
ously sell the shares at the market price of $110, clearing $10 per share. Yet if the shares
sell below $100, you can sit on the option and do nothing, realizing no further gain or loss.
The value of the call option at expiration equals
Payoff to call holder 5 bST 2 X if ST . X 0 if ST # X
where S T is the value of the stock at expiration and X is the exercise price. This formula emphasizes the option property because the payoff cannot be negative. The option is exer-
cised only if S T exceeds X. If S T is less than X, the option expires with zero value. The loss to the option holder in this case equals the price originally paid for the option. More gen-
erally, the profit to the option holder is the option payoff at expiration minus the original purchase price.
The value at expiration of the call with exercise price $100 is given by the schedule:
Stock price: $90 $100 $110 $120 $130 Option value: 0 0 10 20 30
For stock prices at or below $100, the option is worthless. Above $100, the option is worth
the excess of the stock price over $100. The option’s value increases by $1 for each dollar
increase in the stock price. This relationship can be depicted graphically as in Figure 20.2 .
The solid line in Figure 20.2 is the value of the call at expiration. The net profit to the holder of the call equals the gross payoff less the initial investment in the call. Suppose
the call cost $14. Then the profit to the call holder would be given by the dashed (bottom)
line of Figure 20.2 . At option expiration, the investor suffers a loss of $14 if the stock price
is less than or equal to $100.
Profits do not become positive until the stock price at expiration exceeds $114. The break-
even point is $114, because at that price the payoff to the call, S T 2 X 5 $114 2 $100 5 $14, equals the initial cost of the call.
Conversely, the writer of the call incurs losses if the stock price is high. In that scenario,
the writer will receive a call and will be obligated to deliver a stock worth S T for only X dollars:
Payoff to call writer 5 b2(ST 2 X) if ST . X 0 if ST # X
20.2 Values of Options at Expiration
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686 P A R T V I Options, Futures, and Other Derivatives
The call writer, who is exposed to losses
if the stock price increases, is willing to bear
this risk in return for the option premium.
Figure 20.3 depicts the payoff and profit
diagrams for the call writer. These are the
mirror images of the corresponding diagrams
for call holders. The break-even point for the
option writer also is $114. The (negative) pay-
off at that point just offsets the premium origi-
nally received when the option was written.
Put Options A put option is the right to sell an asset at
the exercise price. In this case, the holder
will not exercise the option unless the asset
is worth less than the exercise price. For example, if FinCorp shares were to fall to
$90, a put option with exercise price $100
could be exercised to clear $10 for its holder. The holder would purchase a share for $90
and simultaneously deliver it to the put option writer for the exercise price of $100.
The value of a put option at expiration is
Payoff to put holder 5 b0 if ST $ X X 2 ST if ST , X
The solid line in Figure 20.4 illustrates the payoff at expiration to the holder of a put option
on FinCorp stock with an exercise price of $100. If the stock price at expiration is above
$100, the put has no value, as the right to sell the shares at $100 would not be exercised.
Below a price of $100, the put value at expiration increases by $1 for each dollar the stock
price falls. The dashed line in Figure 20.4 is a graph of the put option owner’s profit at
expiration, net of the initial cost of the put.
Writing puts naked (i.e., writing a put without an offsetting short position in the
stock for hedging purposes) exposes the
writer to losses if the market falls. Writ-
ing naked, deep-out-of-the-money puts
was once considered an attractive way to
generate income, as it was believed that
as long as the market did not fall sharply
before the option expiration, the option
premium could be collected without the
put holder ever exercising the option
against the writer. Because only sharp
drops in the market could result in losses
to the put writer, the strategy was not
viewed as overly risky. However, in the
wake of the market crash of October 1987,
such put writers suffered huge losses. Par-
ticipants now perceive much greater risk to
this strategy.
$30
ST 80 90 100 110 120
Cost of Option Profit
Payoff = Value at Expiration
−$14 −$10
0
$10
$20
Figure 20.2 Payoff and profit to call option at expiration
$14
0 $100 $114
Profit
Payoff
ST
Figure 20.3 Payoff and profit to call writers at expiration
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C H A P T E R 2 0 Options Markets: Introduction 687
Option versus Stock Investments Purchasing call options is a bullish strat-
egy; that is, the calls provide profits when
stock prices increase. Purchasing puts, in
contrast, is a bearish strategy. Symmetri-
cally, writing calls is bearish, whereas
writing puts is bullish. Because option val-
ues depend on the price of the underlying
stock, purchase of options may be viewed
as a substitute for direct purchase or sale of
a stock. Why might an option strategy be
preferable to direct stock transactions?
For example, why would you purchase a
call option rather than buy shares of stock
directly? Maybe you have some informa-
tion that leads you to believe the stock will
increase in value from its current level,
which in our examples we will take to be
$100. You know your analysis could be incorrect, however, and that shares also could
fall in price. Suppose a 6-month maturity call option with exercise price $100 currently
sells for $10, and the interest rate for the period is 3%. Consider these three strategies for
investing a sum of money, say, $10,000. For simplicity, suppose the firm will not pay any
dividends until after the 6-month period.
Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100.
Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling
for $10. (This would require 10 contracts, each for 100 shares.)
Strategy C: Purchase 100 call options for $1,000. Invest your remaining $9,000 in
6-month T-bills, to earn 3% interest. The bills will grow in value from $9,000 to
$9,000 3 1.03 5 $9,270.
Let us trace the possible values of these three portfolios when the options expire in
6 months as a function of the stock price at that time:
Payoff = Value of Put at Expiration Profit
$100
$100 0
Price of Put ST
Figure 20.4 Payoff and profit to put option at expiration
Consider these four option strategies: (i) buy a call; (ii) write a call; (iii) buy a put; (iv) write a put.
a. For each strategy, plot both the payoff and profit diagrams as a function of the final stock price. b. Why might one characterize both buying calls and writing puts as “bullish” strategies? What is the
difference between them? c. Why might one characterize both buying puts and writing calls as “bearish” strategies? What is the
difference between them?
CONCEPT CHECK 20.3
Stock Price
Portfolio $95 $100 $105 $110 $115 $120
Portfolio A: All stock $9,500 $10,000 $10,500 $11,000 $11,500 $12,000 Portfolio B: All options 0 0 5,000 10,000 15,000 20,000 Portfolio C: Call plus bills 9,270 9,270 9,770 10,270 10,770 11,270
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688 P A R T V I Options, Futures, and Other Derivatives
Portfolio A will be worth 100 times the share price. Portfolio B is worthless unless shares sell for more than the exercise price of the call. Once that point is reached, the portfolio is
worth 1,000 times the excess of the stock price over the exercise price. Finally, portfolio C is worth $9,270 from the investment in T-bills plus any profits from the 100 call options.
Remember that each of these portfolios involves the same $10,000 initial investment. The
rates of return on these three portfolios are as follows:
Stock Price
Portfolio $95 $100 $105 $110 $115 $120
Portfolio A: All stock 25.0% 0.0% 5.0% 10.0% 15.0% 20.0% Portfolio B: All options 2100.0 2100.0 250.0 0.0 50.0 100.0 Portfolio C: Call plus bills 27.3 27.3 22.3 2.7 7.7 12.7
These rates of return are graphed in Figure 20.5 .
Comparing the returns of portfolios B and C to those of the simple investment in stock represented by portfolio A, we see that options offer two interesting features. First, an option offers leverage. Compare the returns of portfolios B and A. Unless the stock increases from its initial value of $100, the value of portfolio B falls pre- cipitously to zero—a rate of return of negative 100%. Conversely, modest increases
in the rate of return on the stock result in disproportionate increases in the option rate
of return. For example, a 4.3% increase in the stock price from $115 to $120 would
increase the rate of return on the call from 50% to 100%. In this sense, calls are a
levered investment on the stock. Their values respond more than proportionately to
changes in the stock value.
Figure 20.5 vividly illustrates this
point. The slope of the all-option portfo-
lio is far steeper than that of the all-stock
portfolio, reflecting its greater propor-
tional sensitivity to the value of the
underlying security. The leverage factor
is the reason investors (illegally) exploit-
ing inside information commonly choose
options as their investment vehicle.
The potential insurance value of
options is the second interesting feature,
as portfolio C shows. The T-bill- plus- option portfolio cannot be worth less
than $9,270 after 6 months, as the option
can always be left to expire worthless.
The worst possible rate of return on port-
folio C is 2 7.3%, compared to a (theo- retically) worst possible rate of return on
the stock of 2 100% if the company were
to go bankrupt. Of course, this insurance
comes at a price: When the share price
increases, portfolio C, the option-plus- bills portfolio, does not perform as well
as portfolio A, the all-stock portfolio.
R a te
o f
R e tu
rn (
% )
100
ST
−120
−100
−80
−60
−40
−20
0
20
40
60
80
8 6
8 8
9 0
9 2
9 4
9 6
9 8
1 0 0
1 0 2
1 0 4
1 0 6
1 0 8
1 1 0
1 1 2
1 1 4
1 1 6
1 1 8
1 2 0
A: All Stocks
C: Call Plus Bills
B: All Options
8 4
Figure 20.5 Rate of return to three strategies
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C H A P T E R 2 0 Options Markets: Introduction 689
This simple example makes an important point. Although options can be used by
speculators as effectively leveraged stock positions, as in portfolio B, they also can be used by investors who desire to tailor their risk exposures in creative ways, as in port-
folio C. For example, the call-plus-bills strategy of portfolio C provides a rate of return profile quite unlike that of the stock alone. The absolute limitation on downside risk
is a novel and attractive feature of this strategy. We next discuss several option strate-
gies that provide other novel risk profiles that might be attractive to hedgers and other
investors.
An unlimited variety of payoff patterns can be achieved by combining puts and calls with
various exercise prices. We explain in this section the motivation and structure of some of
the more popular ones.
Protective Put Imagine you would like to invest in a stock, but you are unwilling to bear potential losses
beyond some given level. Investing in the stock alone seems risky to you because in prin-
ciple you could lose all the money you invest. You might consider instead investing in
stock and purchasing a put option on the stock. Table 20.1 shows the total value of your
portfolio at option expiration: Whatever happens to the stock price, you are guaranteed a
payoff at least equal to the put option’s exercise price because the put gives you the right to
sell your shares for that price.
20.3 Option Strategies
Figure 20.6 illustrates the payoff and profit to this protective put strategy. The solid line in Figure 20.6, panel C is the total payoff. The dashed line is displaced
downward by the cost of establishing the position, S 0 1 P. Notice that potential losses are limited.
It is instructive to compare the profit on the protective put strategy with that of the
stock investment. For simplicity, consider an at-the-money protective put, so that X 5 S 0 .
Suppose the strike price is X 5 $100 and the stock is selling at $97 at option expiration. Then the value of your total portfolio is $100. The stock is worth $97 and the value of the expiring put option is
X 2 ST 5 $100 2 $97 5 $3
Another way to look at it is that you are holding the stock and a put contract giving you the right to sell the stock for $100. The right to sell locks in a minimum portfolio value of $100. On the other hand, if the stock price is above $100, say, $104, then the right to sell a share at $100 is worthless. You allow the put to expire unexercised, ending up with a share of stock worth S T 5 $104.
Example 20.3 Protective Put
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690 P A R T V I Options, Futures, and Other Derivatives
Table 20.1
Value of a protective put portfolio at option expiration
ST " X ST + X
Stock ST ST 1 Put X 2 ST 0 5 TOTAL X ST
ST
ST
ST
Payoff of Stock
Payoff of Option
Payoff of Protective Put
Payoff
Profit
X
X
X
X
X − (S0 + P)
A: Stock
+ B: Put
= C: Protective Put
Figure 20.6 Value of a protective put position at option expiration
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C H A P T E R 2 0 Options Markets: Introduction 691
Figure 20.7 compares the profits for the two
strategies. The profit on the stock is zero
if the stock price remains unchanged and
ST 5 S 0 . It rises or falls by $1 for every dollar swing in the ultimate stock price. The profit
on the protective put is negative and equal
to the cost of the put if S T is below S 0 . The profit on the protective put increases one for
one with increases in the stock price once S T exceeds S 0 .
Figure 20.7 makes it clear that the pro-
tective put offers some insurance against
stock price declines in that it limits losses.
Therefore, protective put strategies provide
a form of portfolio insurance. The cost of the protection is that, in the case of stock
price increases, your profit is reduced by
the cost of the put, which turned out to be
unneeded.
This example also shows that despite
the common perception that derivatives
mean risk, derivative securities can be used
effectively for risk management. In fact, such risk management is becoming accepted as part of the fiduciary responsibility of
financial managers. Indeed, in one often-cited court case, Brane v. Roth, a company’s board of directors was successfully sued for failing to use derivatives to hedge the
price risk of grain held in storage. Such hedging might have been accomplished using
protective puts.
The claim that derivatives are best viewed as risk management tools may seem surpris-
ing in light of the credit crisis of the last few years. The crisis was immediately precipi-
tated when the highly risky positions that many financial institutions had established in
credit derivatives blew up 2007–2008, resulting in large losses and government bailouts.
Still, the same characteristics that make derivatives potent tools to increase risk also make
them highly effective in managing risk, at least when used properly. Derivatives have aptly
been compared to power tools: very useful in skilled hands, but also very dangerous when
not handled with care. The nearby box makes the case for derivatives as central to risk
management.
Covered Calls A covered call position is the purchase of a share of stock with a simultaneous sale of a call option on that stock. The call is “covered” because the potential obligation
to deliver the stock can be satisfied using the stock held in the portfolio. Writing an
option without an offsetting stock position is called by contrast naked option writ- ing. The value of a covered call position at expiration, presented in Table 20.2 , equals the stock value minus the value of the call. The call value is subtracted because the covered call position involves writing a call to another investor who may exercise it at
your expense.
Profits
Profit on Stock
ST
Profit on Protective Put Portfolio
−P
−S0
S0 = X
Figure 20.7 Protective put versus stock investment (at-the-money option)
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The Case for Derivatives
They’ve been dubbed financial weapons of mass destruc- tion, attacked for causing the financial turmoil sweeping the nation and identified as the kryptonite that brought down the global economy. Yet few Main Streeters really know what derivatives are—namely, financial contracts between a buyer and a seller that derive value from an underlying asset, such as a mortgage or a stock. There seems to be near consensus that derivatives were a source of undue risk.
And then there’s Robert Shiller. The Yale economist believes just the opposite is true. A champion of financial innovation and an expert in management of risk, Shiller contends that derivatives, far from being a problem, are actually the solution. Derivatives, Shiller says, are merely a risk-management tool the same way insurance is. “You pay a premium and if an event happens, you get a pay- ment.” That tool can be used well or, as happened recently, used badly. Shiller warns that banishing the tool gets us nowhere.
For all the trillions in derivative trading, there were very few traders. Almost all the subprime mortgages that were bundled and turned into derivatives were sold by a hand- ful of Wall Street institutions, working with a small num- ber of large institutional buyers. It was a huge but illiquid and opaque market.
Meanwhile, the system was built on the myriad deci- sions of individual homeowners and lenders around the world. None of them, however, could hedge their bets the way large institutions can. Those buying a condo in Miami had no way to protect themselves if the market went down.
Derivatives, according to Shiller, could be used by home- owners—and, by extension, lenders—to insure themselves against falling prices. In Shiller’s scenario, you would be able to go to your broker and buy a new type of financial instrument, perhaps a derivative that is inversely related to a regional home-price index. If the value of houses in your
area declined, the financial instrument would increase in value, offsetting the loss. Lenders could do the same thing, which would help them hedge against foreclosures. The idea is to make the housing market more liquid. More buyers and sellers mean that markets stay liquid and func- tional even under pressure.
Some critics dismiss Shiller’s basic premise that more derivatives would make the housing market more liquid and more stable. They point out that futures contracts haven’t made equity markets or commodity markets immune from massive moves up and down. They add that a ballooning world of home-based derivatives wouldn’t lead to homeowners’ insurance: it would lead to a new playground for speculators.
In essence, Shiller is laying the intellectual ground- work for the next financial revolution. We are now suffer- ing through the first major crisis of the Information Age economy. Shiller’s answers may be counterintuitive, but no more so than those of doctors and scientists who centu- ries ago recognized that the cure for infectious diseases was not flight or quarantine, but purposely infecting more people through vaccinations. “We’ve had a major glitch in derivatives and securitization,” says Shiller. “The Titanic sank almost a century ago, but we didn’t stop sailing across the Atlantic.”
Of course, people did think twice about getting on a ship, at least for a while. But if we listen only to our fears, we lose the very dynamism that has propelled us this far. That is the nub of Shiller’s call for more derivatives and more innovation. Shiller’s appeal is a tough sell at a time when derivatives have produced so much havoc. But he reminds us that the tools that got us here are not to blame; they can be used badly and they can be used well. And trying to stem the ineffable tide of human creativity is a fool’s errand.
Source: Zachary Karabell, “The Case for Derivatives,” Newsweek, February 2, 2009.
W O
R D
S FR
O M
T H
E S
T R
E E T
692
The solid line in Figure 20.8, panel C is the payoff. You see that the total position is
worth S T when the stock price at time T is below X and rises to a maximum of X when S T exceeds X. In essence, the sale of the call options means the call writer has sold the claim to any stock value above X in return for the initial premium (the call price). Therefore, at expiration, the position is worth at most X. The dashed line of Figure 20.8, panel C is the net profit to the covered call.
Writing covered call options has been a popular investment strategy among institutional
investors. Consider the managers of a fund invested largely in stocks. They might find it
appealing to write calls on some or all of the stock in order to boost income by the premi-
ums collected. Although they thereby forfeit potential capital gains should the stock price
rise above the exercise price, if they view X as the price at which they plan to sell the stock anyway, then the call may be viewed as a kind of “sell discipline.” The written call guaran-
tees the stock sale will occur as planned.
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C H A P T E R 2 0 Options Markets: Introduction 693
Table 20.2
Value of a covered call position at option expiration
ST " X ST + X
Payoff of stock ST ST 1 Payoff of written call 20 2(ST 2 X ) 5 TOTAL ST X
ST
Payoff of Stock
Payoff of Written Call
Payoff of Covered Call
Payoff
Profit
X
A: Stock
+ B: Write Call
= C: Covered Call
ST
ST
X
X
X
− (S0 − C)
Figure 20.8 Value of a covered call position at expiration
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694
Straddle A long straddle is established by buying both a call and a put on a stock, each with the same exercise price, X, and the same expiration date, T. Straddles are useful strategies for investors who believe a stock will move a lot in price but are uncertain about the direc-
tion of the move. For example, suppose you believe an important court case that will
make or break a company is about to be settled, and the market is not yet aware of the
situation. The stock will either double in value if the case is settled favorably or will drop
by half if the settlement goes against the company. The straddle position will do well
regardless of the outcome because its value rises when the stock price makes extreme
upward or downward moves from X. The worst-case scenario for a straddle is no movement in the stock price. If S T equals
X, both the call and the put expire worthless, and the investor’s outlay for the purchase of both options is lost. Straddle positions, therefore, are bets on volatility. An investor who
Assume a pension fund holds 1,000 shares of stock, with a current price of $100 per share. Suppose the portfolio manager intends to sell all 1,000 shares if the share price hits $110, and a call expiring in 60 days with an exercise price of $110 currently sells for $5. By writing 10 call contracts (for 100 shares each) the fund can pick up $5,000 in extra income. The fund would lose its share of profits from any movement of the stock price above $110 per share, but given that it would have sold its shares at $110, it would not have realized those profits anyway.
Example 20.4 Covered Call
eXcel APPLICATIONS: Spreads and Straddles
U sing spreadsheets to analyze combinations of options is very helpful. Once the basic models are built, it is easy
to extend the analysis to different bundles of options. The Excel model “Spreads and Straddles” shown below can be used to evaluate the profitability of different strategies. You can find a link to this spreadsheet at www.mhhe.com/bkm.
Excel Question
1. Use the data in this spreadsheet to plot the profit on a bullish spread (see Figure 20.10) with X1 5 120 and X2 5 130.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Stock Prices
Buying Options: Call Options Strike
Put Options Strike
Straddle
Spreads and Straddles
Payoff
Payoff
Payoff
Profit
Profit
Profit
Return %
Return %
Return %
Ending Ending X 120 StraddleX 110 Straddle
A B C D E F G H I J K L
Beginning Market Price Ending Market Price
116.5 130
120 110
130 120 110
140
170 160 150
180
210 200 190
100 90 80 70 60 50
130 120 110
140
170 160 150
180
210 200 190
100 90 80 70 60 50
130 120 110
140
130 140
130 120 110
140
22.80
0.00 0.00 0.00
10.00
20.00 10.00 0.00
10.00
16.80 13.60 10.30
12.60 17.20 23.60 30.50
35.40 34.00 37.20 40.80
−12.60 −17.20 −23.60 −20.50
−15.40 −24.00
−15.40 24.60
24.60 34.60 44.60 54.60 64.60
14.60
14.60
4.60
4.60
−5.40 −15.40 −25.40 −35.40 −25.40 −15.40
−5.40
−24.00 36.00
16.00 26.00 36.00 46.00 56.00
26.00
6.00
16.00
−4.00
6.00 −4.00
−14.00 −24.00 −34.00 −24.00 −14.00
−37.20 −30.80
−100.00% −100.00% −100.00%
−67.21%
−43.50% −70.59%
−100.00% −75.49%
Price
Price
Price
Stock Price Profit
Stock Price Profit
20.00 10.00 0.00 0.00
−2.80 −6.80
−13.60 −10.30
−12.28% −40.48%
−100.00% −100.00%
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C H A P T E R 2 0 Options Markets: Introduction 695
establishes a straddle must view the stock as more volatile than the market does. Con-
versely, investors who write straddles—selling both a call and a put—must believe the stock is less volatile. They accept the option premiums now, hoping the stock price will not
change much before option expiration.
The payoff to a straddle is presented in Table 20.3 . The solid line of Figure 20.9,
panel C illustrates this payoff. Notice the portfolio payoff is always positive, except
at the one point where the portfolio has zero value, S T 5 X. You might wonder why all investors don’t pursue such a seemingly “no-lose” strategy. The reason is that the
straddle requires that both the put and call be purchased. The value of the portfolio at
expiration, while never negative, still must exceed the initial cash outlay for a straddle
investor to clear a profit.
The dashed line of Figure 20.9, panel C is the profit diagram. The profit line lies below
the payoff line by the cost of purchasing the straddle, P 1 C. It is clear from the dia- gram that the straddle generates a loss unless the stock price deviates substantially from X. The stock price must depart from X by the total amount expended to purchase the call and the put for the straddle
to clear a profit.
Strips and straps are variations of straddles. A strip is two puts and one call on a security with the same exercise
price and expiration date. A strap is two calls and one put.
Spreads A spread is a combination of two or more call options (or two or more puts) on the same stock with differing exercise prices or times to maturity. Some options are bought, whereas
others are sold, or written. A money spread involves the purchase of one option and the simultaneous sale of another with a different exercise price. A time spread refers to the sale and purchase of options with differing expiration dates.
Consider a money spread in which one call option is bought at an exercise price X 1 , whereas another call with identical expiration date, but higher exercise price, X 2 , is writ- ten. The payoff will be the difference in the value of the call held and the value of the call
written, as in Table 20.4 .
There are now three instead of two outcomes to distinguish: the lowest-price region
where S T is below both exercise prices, a middle region where S T is between the two exer- cise prices, and a high-price region where S T exceeds both exercise prices. Figure 20.10 illustrates the payoff and profit to this strategy, which is called a bullish spread because the payoff either increases or is unaffected by stock price increases. Holders of bullish spreads
benefit from stock price increases.
One motivation for a bullish spread might be that the investor thinks one option is over-
priced relative to another. For example, an investor who believes an X 5 $100 call is cheap compared to an X 5 $110 call might establish the spread, even without a strong desire to take a bullish position in the stock.
Collars A collar is an options strategy that brackets the value of a portfolio between two bounds. Sup- pose that an investor currently is holding a large position in FinCorp stock, which is currently
selling at $100 per share. A lower bound of $90 can be placed on the value of the portfolio
by buying a protective put with exercise price $90. This protection, however, requires that the
investor pay the put premium. To raise the money to pay for the put, the investor might write a
call option, say, with exercise price $110. The call might sell for roughly the same price as the
Graph the profit and payoff diagrams for strips and straps.
CONCEPT CHECK 20.4
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696 P A R T V I Options, Futures, and Other Derivatives
ST
Payoff of Call
Payoff of Put
Payoff of Straddle
Payoff
Profit
X
X
A: Call
+ B: Put
= C : Straddle
ST
ST
Profit
Profit
Payoff
P + C
Payoff
X
X X − P
− C
−P
0
0
X
X − P − C
− (P + C)
0
Table 20.3
Value of a straddle position at option expiration
ST * X ST # X
Payoff of call 0 ST 2 X 1 Payoff of put X 2 ST 0 5 TOTAL X 2 ST ST 2 X
Figure 20.9 Value of a straddle at expiration
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C H A P T E R 2 0 Options Markets: Introduction 697
ST
Payoff
Payoff and Profit
Payoff
X1
X2 − X1
C2 − C1
Profit
ST
ST
Payoff Profit
Payoff
Profit
Payoff
X2
A: Call Held (Strike price 5 X1)
B: Call Written (Strike price 5 X2)
− C1 X1 X2
X1
C2 0
0
C: Bullish Spread
0
X2
Figure 20.10 Value of a bullish spread position at expiration
ST " X1 X1 * ST " X2 ST # X2
Payoff of purchased call, exercise price 5 X1 0 ST 2 X1 ST 2 X1 1 Payoff of written call, exercise price 5 X2 20 20 2(ST 2 X2) 5 TOTAL 0 ST 2 X1 X2 2 X1
Table 20.4
Value of a bullish spread position at expiration
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698 P A R T V I Options, Futures, and Other Derivatives
put, meaning that the net outlay for the two options positions is approximately zero. Writing
the call limits the portfolio’s upside potential. Even if the stock price moves above $110, the
investor will do no better than $110, because at a higher price the stock will be called away.
Thus the investor obtains the downside protection represented by the exercise price of the put
by selling her claim to any upside potential beyond the exercise price of the call.
A collar would be appropriate for an investor who has a target wealth goal in mind but is unwilling to risk losses beyond a certain level. If you are contemplating buying a house for $220,000, for example, you might set this figure as your goal. Your current wealth may be $200,000, and you are unwilling to risk losing more than $20,000. A collar established by (1) purchasing 2,000 shares of stock currently selling at $100 per share, (2) purchasing 2,000 put options (20 options contracts) with exercise price $90, and (3) writing 2,000 calls with exercise price $110 would give you a good chance to realize the $20,000 capital gain without risking a loss of more than $20,000.
Example 20.5 Collars
Graph the payoff diagram for the collar described in Example 20.5.
CONCEPT CHECK 20.5
We saw in the previous section that a protective put portfolio, comprising a stock position
and a put option on that position, provides a payoff with a guaranteed minimum value,
but with unlimited upside potential. This is not the only way to achieve such protection,
however. A call-plus-bills portfolio also can provide limited downside risk with unlimited
upside potential.
Consider the strategy of buying a call option and, in addition, buying Treasury bills with
face value equal to the exercise price of the call, and with maturity date equal to the expira-
tion date of the option. For example, if the exercise price of the call option is $100, then
each option contract (which is written on 100 shares) would require payment of $10,000
upon exercise. Therefore, you would purchase a T-bill with a maturity value of $10,000.
More generally, for each option that you hold with exercise price X, you would purchase a risk-free zero-coupon bond with face value X.
Examine the value of this position at time T, when the options expire and the zero- coupon bond matures:
20.4 The Put-Call Parity Relationship
ST " X ST + X
Value of call option 0 ST 2 X Value of zero-coupon bond X X
TOTAL X ST
If the stock price is below the exercise price, the call is worthless, but the bond matures to
its face value, X. It therefore provides a floor value to the portfolio. If the stock price
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C H A P T E R 2 0 Options Markets: Introduction 699
exceeds X, then the payoff to the call, S T 2 X, is added to the face value of the bond to provide a total payoff of S T . The payoff to this portfolio is precisely identical to the payoff of the protective put that we derived in Table 20.1 .
If two portfolios always provide equal values, then they must cost the same amount to
establish. Therefore, the call-plus-bond portfolio must cost the same as the stock-plus-put
portfolio. Each call costs C. The riskless zero-coupon bond costs X /(1 1 r f ) T . Therefore,
the call-plus-bond portfolio costs C 1 X /(1 1 r f ) T . The stock costs S 0 to purchase now (at
time zero), while the put costs P. Therefore, we conclude that
C 1 X
(1 1 rf ) T 5 S0 1 P (20.1)
Equation 20.1 is called the put-call parity theorem because it represents the proper relationship between put and call prices. If the parity relation is ever violated, an arbi-
trage opportunity arises. For example, suppose you collect these data for a certain
stock:
Stock price $110 Call price (1-year expiration, X 5 $105) $ 17 Put price (1-year expiration, X 5 $105) $ 5 Risk-free interest rate 5% per year
We can use these data in Equation 20.1 to see if parity is violated:
C 1 X
(1 1 rf) T 5
?
S0 1 P
17 1 105
1.05 5
?
110 1 5
117 2 115
This result, a violation of parity—117 does not equal 115—indicates mispricing. To exploit
the mispricing, you buy the relatively cheap portfolio (the stock-plus-put position repre-
sented on the right-hand side of the equation) and sell the relatively expensive portfolio
(the call-plus-bond position corresponding to the left-hand side). Therefore, if you buy the stock, buy the put, write the call, and borrow $100 for 1 year (because borrowing money is the opposite of buying a bond), you should earn arbitrage profits.
Let’s examine the payoff to this strategy. In 1 year, the stock will be worth S T . The $100 borrowed will be paid back with interest, resulting in a cash outflow of $105. The written
call will result in a cash outflow of S T 2 $105 if S T exceeds $105. The purchased put pays off $105 2 S T if the stock price is below $105.
Table 20.5 summarizes the outcome. The immediate cash inflow is $2. In 1 year, the
various positions provide exactly offsetting cash flows: The $2 inflow is realized with-
out any offsetting outflows. This is an arbitrage opportunity that investors will pursue on
a large scale until buying and selling pressure restores the parity condition expressed in
Equation 20.1.
Equation 20.1 actually applies only to options on stocks that pay no dividends before
the expiration date of the option. The extension of the parity condition for European call
options on dividend-paying stocks is, however, straightforward. Problem 12 at the end of
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700 P A R T V I Options, Futures, and Other Derivatives
the chapter leads you through the demonstration. The more general formulation of the
put-call parity condition is
P 5 C 2 S0 1 PV(X) 1 PV(dividends) (20.2)
where PV(dividends) is the present value of the dividends that will be paid by the stock
during the life of the option. If the stock does not pay dividends, Equation 20.2 becomes
identical to Equation 20.1.
Notice that this generalization would apply as well to European options on assets other
than stocks. Instead of using dividend income in Equation 20.2, we would let any income
paid out by the underlying asset play the role of the stock dividends. For example, European
put and call options on bonds would satisfy the same parity relationship, except that the
bond’s coupon income would replace the stock’s dividend payments in the parity formula.
Even this generalization, however, applies only to European options, as the cash flow
streams from the two portfolios represented by the two sides of Equation 20.2 will match
only if each position is held until expiration. If a call and a put may be optimally exercised
at different times before their common expiration date, then the equality of payoffs cannot
be assured, or even expected, and the portfolios will have different values.
Table 20.5
Arbitrage strategy
Immediate Cash Flow
Cash Flow in 1 Year
Position ST * 105 ST # 105
Buy stock 2110 ST ST Borrow $105/1.05 5 $100 1100 2105 2105 Sell call 117 0 2(ST 2 105) Buy put 25 105 2 ST 0 TOTAL 2 0 0
Let’s see how well parity works using the data in Figure 20.1 on the IBM options. The February expiration call with exercise price $195 and time to expiration of 28 days cost $3.65 while the corresponding put option cost $5. IBM was selling for $194.47, and the annualized short-term interest rate on this date was 0.1%. IBM was expected to pay a dividend of $.85 with an ex-dividend date of February 8, 18 days hence. According to parity, we should find that
P 5 C 1 PV(X) 2 S0 1 PV(Dividends)
5.00 5 3.65 1 195
(1.001)28/365 2194.47 1
.85
(1.001)18/365
5.00 5 3.65 1 194.985 2 194.47 1 .85
5.00 5 5.015
So parity is violated by about $.015 per share. Is this a big enough difference to exploit? Almost certainly not. You have to weigh the potential profit against the trading costs of the call, put, and stock. More important, given the fact that options trade relatively infre- quently, this deviation from parity might not be “real,” but may instead be attributable to “stale” (i.e., out-of-date) price quotes at which you cannot actually trade.
Example 20.6 Put-Call Parity
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C H A P T E R 2 0 Options Markets: Introduction 701
20.5 Option-Like Securities
Suppose you never traded an option directly. Why do you need to appreciate the properties
of options in formulating an investment plan? Many financial instruments and agreements
have features that convey implicit or explicit options to one or more parties. To value and
use these securities correctly, you must understand their embedded option attributes.
Callable Bonds You know from Chapter 14 that many corporate bonds are issued with call provisions enti-
tling the issuer to buy bonds back from bondholders at some time in the future at a specified
call price. The bond issuer holds a call option with exercise price equal to the price at which
the bond can be repurchased. A callable bond arrangement therefore is essentially a sale of a
straight bond (a bond with no option features such as callability or convertibility) to the inves- tor and the concurrent issuance of a call option by the investor to the bond-issuing firm.
There must be some compensation for the firm’s implicit call option. If the callable
bond were issued with the same coupon rate as a straight bond, it would sell at a lower
price than the straight bond: the price difference would equal the value of the call. To sell
callable bonds at par, firms must issue them with coupon rates higher than the coupons
on straight debt. The higher coupons are the investor’s compensation for the call option
retained by the issuer.
Figure 20.11 illustrates this optionlike property. The horizontal axis is the value of a
straight bond with otherwise identical terms to the callable bond. The dashed 45-degree
line represents the value of straight debt. The solid line is the value of the callable bond,
and the dotted line is the value of the call option retained by the firm. A callable bond’s
potential for capital gains is limited by the firm’s option to repurchase at the call price.
How is a callable bond similar to a covered call strategy on a straight bond?
CONCEPT CHECK 20.6
The option inherent in callable bonds actually is more complex than an ordinary call
option, because usually it may be exercised only after some initial period of call protec-
tion. The price at which the bond is callable may change over time also. Unlike exchange-
listed options, these features are defined in the initial bond covenant and will depend on
the needs of the issuing firm and its perception of the market’s tastes.
Suppose the period of call protection is extended. How will the coupon rate the company needs to offer on its bonds change to enable the issuer to sell the bonds at par value?
CONCEPT CHECK 20.7
Convertible Securities Convertible bonds and convertible preferred stock convey options to the holder of the
security rather than to the issuing firm. A convertible security typically gives its holder the
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702 P A R T V I Options, Futures, and Other Derivatives
right to exchange each bond or share
of preferred stock for a fixed number
of shares of common stock, regard-
less of the market prices of the secu-
rities at the time.
For example, a bond with a con- version ratio of 10 allows its holder to convert one bond of par value $1,000
into 10 shares of common stock.
Alternatively, we say the conversion price in this case is $100: To receive 10 shares of stock, the investor sacri-
fices bonds with face value $1,000 or,
put another way, $100 of face value
per share. If the present value of the
bond’s scheduled payments is less
than 10 times the value of one share
of stock, it may pay to convert; that is,
the conversion option is in the money.
A bond worth $950 with a conversion
ratio of 10 could be converted profit-
ably if the stock were selling above
$95, as the value of the 10 shares
received for each bond surrendered would exceed $950. Most convertible bonds are issued
“deep out of the money.” That is, the issuer sets the conversion ratio so that conversion will
not be profitable unless there is a substantial increase in stock prices and/or decrease in bond
prices from the time of issue.
A bond’s conversion value equals the value it would have if you converted it into stock immediately. Clearly, a bond must sell for at least its conversion value. If it did not, you could
purchase the bond, convert it, and clear an immediate profit. This condition could never persist,
for all investors would pursue such a strategy and ultimately would bid up the price of the bond.
The straight bond value, or “bond floor,” is the value the bond would have if it were not
convertible into stock. The bond must sell for more than its straight bond value because
a convertible bond has more value; it is in fact a straight bond plus a valuable call option.
Therefore, the convertible bond has two lower bounds on its market price: the conversion
value and the straight bond value.
Value of Straight Debt
Value of Callable Bond
Call Price
Value of Firm’s Call Option
Value of Straight Debt
Figure 20.11 Values of callable bonds compared with straight bonds
Should a convertible bond issued at par value have a higher or lower coupon rate than a nonconvertible bond issued at par?
CONCEPT CHECK 20.8
Figure 20.12 illustrates the optionlike properties of the convertible bond. Figure 20.12,
panel A shows the value of the straight debt as a function of the stock price of the issu-
ing firm. For healthy firms, the straight debt value is almost independent of the value of
the stock because default risk is small. However, if the firm is close to bankruptcy (stock
prices are low), default risk increases, and the straight bond value falls. Panel B shows the
conversion value of the bond. Panel C compares the value of the convertible bond to these
two lower bounds.
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C H A P T E R 2 0 Options Markets: Introduction 703
When stock prices are low, the straight bond value is the effective lower bound, and the
conversion option is nearly irrelevant. The convertible will trade like straight debt. When
stock prices are high, the bond’s price is determined by its conversion value. With conver-
sion all but guaranteed, the bond is essentially equity in disguise.
We can illustrate with two examples:
Straight Debt Value
Stock Price
A
Conversion Value
Stock Price
B
Convertible Bond Value
Straight Debt
Conversion Value Convertible Bond Value
Stock Price
C
Figure 20.12 Value of a convertible bond as a function of stock price. Panel A, Straight debt value, or bond floor. Panel B, Conversion value of the bond. Panel C, Total value of convertible bond.
Bond A Bond B
Annual coupon $80 $80 Maturity date 10 years 10 years Quality rating Baa Baa Conversion ratio 20 25 Stock price $30 $50 Conversion value $600 $1,250 Market yield on 10-year Baa-rated bonds 8.5% 8.5% Value as straight debt $967 $967 Actual bond price $972 $1,255 Reported yield to maturity 8.42% 4.76%
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704 P A R T V I Options, Futures, and Other Derivatives
Bond A has a conversion value of only $600. Its value as straight debt, in contrast, is $967. This is the present value of the coupon and principal payments at a market rate for
straight debt of 8.5%. The bond’s price is $972, so the premium over straight bond value is
only $5, reflecting the low probability of conversion. Its reported yield to maturity based
on scheduled coupon payments and the market price of $972 is 8.42%, close to that of
straight debt.
The conversion option on bond B is in the money. Conversion value is $1,250, and the bond’s price, $1,255, reflects its value as equity (plus $5 for the protection the bond offers
against stock price declines). The bond’s reported yield is 4.76%, far below the compa-
rable yield on straight debt. The big yield sacrifice is attributable to the far greater value of
the conversion option.
In theory, we could value convertible bonds by treating them as straight debt plus call
options. In practice, however, this approach is often impractical for several reasons:
1. The conversion price frequently increases over time, which means the exercise
price of the option changes.
2. Stocks may pay several dividends over the life of the bond, further complicating the
option-valuation analysis.
3. Most convertibles also are callable at the discretion of the firm. In essence, both
the investor and the issuer hold options on each other. If the issuer exercises its
call option to repurchase the bond, the bondholders typically have a month during
which they still can convert. When issuers use a call option, knowing bondholders
will choose to convert, the issuer is said to have forced a conversion. These condi- tions together mean the actual maturity of the bond is indeterminate.
Warrants Warrants are essentially call options issued by a firm. One important difference between calls and warrants is that exercise of a warrant requires the firm to issue a new share of
stock—the total number of shares outstanding increases. Exercise of a call option requires
only that the writer of the call deliver an already-issued share of stock to discharge the
obligation. In that case, the number of shares outstanding remains fixed. Also unlike call
options, warrants result in a cash flow to the firm when the warrant holder pays the exer-
cise price. These differences mean that warrant values will differ somewhat from the val-
ues of call options with identical terms.
Like convertible debt, warrant terms may be tailored to meet the needs of the firm. Also
like convertible debt, warrants generally are protected against stock splits and dividends
in that the exercise price and the number of warrants held are adjusted to offset the effects
of the split.
Warrants are often issued in conjunction with another security. Bonds, for example,
may be packaged together with a warrant “sweetener,” frequently a warrant that may be
sold separately. This is called a detachable warrant. Issue of warrants and convertible securities creates the potential for an increase in out-
standing shares of stock if exercise occurs. Exercise obviously would affect financial sta-
tistics that are computed on a per-share basis, so annual reports must provide earnings per
share figures under the assumption that all convertible securities and warrants are exer-
cised. These figures are called fully diluted earnings per share. 3
3 We should note that the exercise of a convertible bond need not reduce EPS. Diluted EPS will be less than undi-
luted EPS only if interest saved (per share) on the convertible bonds is less than the prior EPS.
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C H A P T E R 2 0 Options Markets: Introduction 705
The executive and employee stock options that became so popular in the 1990s
actually were warrants. Some of these grants were huge, with payoffs to top executives in
excess of $100 million. Yet firms almost uniformly chose not to acknowledge these grants
as expenses on their income statements until new reporting rules that took effect in 2006
required such recognition.
Collateralized Loans Many loan arrangements require that the borrower put up collateral to guarantee the loan
will be paid back. In the event of default, the lender takes possession of the collateral. A
nonrecourse loan gives the lender no recourse beyond the right to the collateral. That is, the lender may not sue the borrower for further payment if the collateral turns out not to be
valuable enough to repay the loan.
This arrangement gives an implicit call option to the borrower. Assume the borrower is
obligated to pay back L dollars at the maturity of the loan. The collateral will be worth S T dollars at maturity. (Its value today is S 0 .) The borrower has the option to wait until loan maturity and repay the loan only if the collateral is worth more than the L dollars necessary to satisfy the loan. If the collateral is worth less than L, the borrower can default on the loan, discharging the obligation by forfeiting the collateral, which is worth only S T .
4
Another way of describing such a loan is to view the borrower as turning over the col-
lateral to the lender but retaining the right to reclaim it by paying off the loan. The transfer
of the collateral with the right to reclaim it is equivalent to a payment of S 0 dollars, less a simultaneous recovery of a sum that resembles a call option with exercise price L. In effect, the borrower turns over collateral but keeps an option to “repurchase” it for L dollars at the maturity of the loan if L turns out to be less than S T . This is a call option.
A third way to look at a collateralized loan is to assume that the borrower will repay the
L dollars with certainty but also retain the option to sell the collateral to the lender for L dollars, even if S T is less than L. In this case, the sale of the collateral would generate the cash necessary to satisfy the loan. The ability to “sell” the collateral for a price of L dollars represents a put option, which guarantees the borrower can raise enough money to satisfy
the loan simply by turning over the collateral.
It is perhaps surprising to realize that we can describe the same loan as involving either
a put option or a call option, as the payoffs to calls and puts are so different. Yet the equiva-
lence of the two approaches is nothing more than a reflection of the put-call parity rela-
tionship. In our call-option description of the loan, the value of the borrower’s liability is
S 0 2 C: The borrower turns over the asset, which is a transfer of S 0 dollars, but retains a call worth C dollars. In the put-option description, the borrower is obligated to pay L dollars but retains the put, which is worth P: The present value of this net obligation is L /(1 1 r f )
T 2 P. Because these alternative descriptions are equivalent ways of viewing the
same loan, the value of the obligations must be equal:
S0 2 C 5 L
(1 1 rf) T 2 P (20.3)
Treating L as the exercise price of the option, Equation 20.3 is simply the put-call parity relationship.
4 In reality, of course, defaulting on a loan is not so simple. There are losses of reputation involved as well as
considerations of ethical behavior. This is a description of a pure nonrecourse loan where both parties agree from
the outset that only the collateral backs the loan and that default is not to be taken as a sign of bad faith if the col-
lateral is insufficient to repay the loan.
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706 P A R T V I Options, Futures, and Other Derivatives
Figure 20.13 illustrates this fact. Figure 20.13, panel A is the value of the payment to
be received by the lender, which equals the minimum of ST or L. Panel B shows that this amount can be expressed as ST minus the payoff of the call implicitly written by the lender and held by the borrower. Panel C shows it also can be viewed as a receipt of L dollars minus the proceeds of a put option.
Figure 20.13 Collateralized loan. Panel A, Payoff to collateralized loan. Panel B, Lender can be viewed as collecting the collateral from the borrower, but issuing an option to the borrower to call back the collateral for the face value of the loan. Panel C, Lender can be viewed as collecting a risk-free loan from the bor- rower, but issuing a put to the borrower to sell the collateral for the face value of the loan.
L
L ST
L dollars minus the payoff to the implicit put option
Payoff to a put with exercise price L
Payoff
L
L ST
ST dollars minus the payoff to the implicit call option
Payoff to call with exercise price L
Payoff
A
B
C
Payoff
L
L ST
Payoff to lender
When ST exceeds L, the loan is repaid and the collateral is reclaimed. Otherwise, the collateral is forfeited and the total loan repayment is worth only ST
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C H A P T E R 2 0 Options Markets: Introduction 707
Levered Equity and Risky Debt Investors holding stock in incorporated firms are protected by limited liability, which
means that if the firm cannot pay its debts, the firm’s creditors may attach only the firm’s
assets, not sue the corporation’s equityholders for further payment. In effect, any time the
corporation borrows money, the maximum possible collateral for the loan is the total of
the firm’s assets. If the firm declares bankruptcy, we can interpret this as an admission that
the assets of the firm are insufficient to satisfy the claims against it. The corporation may
discharge its obligations by transferring ownership of the firm’s assets to the creditors.
Just as is true for nonrecourse collateralized loans, the required payment to the credi-
tors represents the exercise price of the implicit option, while the value of the firm is the
underlying asset. The equityholders have a put option to transfer their ownership claims on
the firm to the creditors in return for the face value of the firm’s debt.
Alternatively, we may view the equityholders as retaining a call option. They have,
in effect, already transferred their ownership claim to the firm to the creditors but have
retained the right to reacquire that claim by paying off the loan. Hence the equityholders
have the option to “buy back” the firm for a specified price: They have a call option.
The significance of this observation is that analysts can value corporate bonds using
option-pricing techniques. The default premium required of risky debt in principle can be
estimated by using option-valuation models. We consider some of these models in the next
chapter.
One of the attractions of options is the ability they provide to create investment positions
with payoffs that depend in a variety of ways on the values of other securities. We have
seen evidence of this capability in the various options strategies examined in Section 20.4.
Options also can be used to
custom-design new securities
or portfolios with desired pat-
terns of exposure to the price
of an underlying security. In
this sense, options (and futures
contracts, to be discussed in
Chapters 22 and 23) provide
the ability to engage in finan- cial engineering, the creation of portfolios with specified
payoff patterns.
A simple example of a prod-
uct engineered with options
is the index-linked certifi-
cate of deposit. Index-linked
CDs enable retail investors to
take small positions in index
options. Unlike conventional
CDs, which pay a fixed rate of
interest, these CDs pay deposi-
tors a specified fraction of the
20.6 Financial Engineering
Rate of Return on Index-Linked CD
Slope = .7
rM = Market Rate of Return
Figure 20.14 Return on index-linked CD
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708 P A R T V I Options, Futures, and Other Derivatives
rate of return on a market index such as the S&P 500, while guaranteeing a minimum rate
of return should the market fall. For example, the index-linked CD may offer 70% of any
market increase, but protect its holder from any market decrease by guaranteeing at least
no loss.
The index-linked CD is clearly a type of call option. If the market rises, the depositor
profits according to the participation rate or multiplier, in this case 70%; if the market falls, the investor is insured against loss. Just as clearly, the bank offering these CDs is in
effect writing call options and can hedge its position by buying index calls in the options
market. Figure 20.14 shows the nature of the bank’s obligation to its depositors.
How might the bank set the appropriate multiplier? To answer this, note various fea-
tures of the option:
1. The price the depositor is paying for the options is the forgone interest on the
conventional CD that could be purchased. Because interest is received at the end
of the period, the present value of the interest payment on each dollar invested is
r f /(1 1 r f ). Therefore, the depositor trades a sure payment with present value per dollar invested of r f /(1 1 r f ) for a return that depends on the market’s performance. Conversely, the bank can fund its obligation using the interest that it would have
paid on a conventional CD.
2. The option we have described is an at-the-money option, meaning that the exercise
price equals the current value of the stock index. The option goes into the money as
soon as the market index increases from its level at the inception of the contract.
3. We can analyze the option on a per-dollar-invested basis. For example, the option
costs the depositor r f /(1 1 r f ) dollars per dollar placed in the index-linked CD. The market price of the option per dollar invested is C / S 0 : The at-the-money option costs C dollars and is written on one unit of the market index, currently at S 0 .
Now it is easy to determine the multiplier that the bank can offer on the CDs. It receives
from its depositors a “payment” of r f /(1 1 r f ) per dollar invested. It costs the bank C / S 0 to purchase the call option on a $1 investment in the market index. Therefore, if r f /(1 1 r f ) is, for example, 70% of C / S 0 , the bank can purchase at most .7 call option on the $1 investment and the multiplier will be .7. More generally, the break-even multiplier on an index-linked
CD is r f /(1 1 r f ) divided by C / S 0 .
Suppose that r f 5 6% per year, and that 6-month maturity at-the-money calls on the market index currently cost $50. The index is at 1,000. Then the option costs 50/1,000 5 $.05 per dollar of market value. The CD rate is 3% per 6 months, meaning that r f /(1 1 r f ) 5 .03/1.03 5 .0291. Therefore, the multiplier would be .0291/.05 5 .5825.
Example 20.7 Indexed-Linked CDs
The index-linked CD has several variants. Investors can purchase similar CDs that guar-
antee a positive minimum return if they are willing to settle for a smaller multiplier. In this
case, the option is “purchased” by the depositor for ( r f 2 r min )/(1 1 r f ) dollars per dol- lar invested, where r min is the guaranteed minimum return. Because the purchase price is lower, fewer options can be purchased, which results in a lower multiplier. Another variant
of the “bullish” CD we have described is the bear CD, which pays depositors a fraction of any fall in the market index. For example, a bear CD might offer a rate of return of .6 times any percentage decline in the S&P 500.
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C H A P T E R 2 0 Options Markets: Introduction 709
Continue to assume that r f 5 3% per half-year, that at-the-money calls sell for $50, and that the market index is at 1,000. What would be the multiplier for 6-month bullish equity-linked CDs offering a guaran- teed minimum return of .5% over the term of the CD?
CONCEPT CHECK 20.9
Options markets have been tremendously successful. Investors clearly value the portfolio
strategies made possible by trading options; this is reflected in the heavy trading volume
in these markets. Success breeds imitation, and in recent years we have witnessed con-
siderable innovation in the range of option instruments available to investors. Part of this
innovation has occurred in the market for customized options, which now trade in active
over-the-counter markets. Many of these options have terms that would have been highly
unusual even a few years ago; they are therefore called exotic options. In this section we survey a few of the more interesting variants of these new instruments.
Asian Options You already have been introduced to American- and European-style options. Asian-style
options are options with payoffs that depend on the average price of the underlying asset during at least some portion of the life of the option. For example, an Asian call option may
have a payoff equal to the average stock price over the last 3 months minus the strike price
if that value is positive, and zero otherwise. These options may be of interest, for example,
to firms that wish to hedge a profit stream that depends on the average price of a commod-
ity over some period of time.
Barrier Options Barrier options have payoffs that depend not only on some asset price at option expiration, but
also on whether the underlying asset price has crossed through some “barrier.” For example,
a down-and-out option is one type of barrier option that automatically expires worthless if
and when the stock price falls below some barrier price. Similarly, down-and-in options will
not provide a payoff unless the stock price does fall below some barrier at least once during the life of the option. These options also are referred to as knock-out and knock-in options.
Lookback Options Lookback options have payoffs that depend in part on the minimum or maximum price
of the underlying asset during the life of the option. For example, a lookback call option
might provide a payoff equal to the maximum stock price during the life of the option minus the exercise price, instead of the final stock price minus the exercise price. Such an option provides (for a price, of course) a form of perfect market timing, providing the call
holder with a payoff equal to the one that would accrue if the asset were purchased for X dollars and later sold at what turns out to be its high price.
Currency-Translated Options Currency-translated options have either asset or exercise prices denominated in a foreign
currency. For example, the quanto allows an investor to fix in advance the exchange rate at
20.7 Exotic Options
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1. A call option is the right to buy an asset at an agreed-upon exercise price. A put option is the right to sell an asset at a given exercise price.
2. American-style options allow exercise on or before the expiration date. European options allow exercise only on the expiration date. Most traded options are American in nature.
3. Options are traded on stocks, stock indexes, foreign currencies, fixed-income securities, and sev- eral futures contracts.
4. Options can be used either to lever up an investor’s exposure to an asset price or to provide insur- ance against volatility of asset prices. Popular option strategies include covered calls, protective
puts, straddles, spreads, and collars.
5. The put-call parity theorem relates the prices of put and call options. If the relationship is vio- lated, arbitrage opportunities will result. Specifically, the relationship that must be satisfied is
P 5 C 2 S0 1 PV(X) 1 PV(dividends)
where X is the exercise price of both the call and the put options, PV( X ) is the present value of a claim to X dollars to be paid at the expiration date of the options, and PV(dividends) is the present value of dividends to be paid before option expiration.
6. Many commonly traded securities embody option characteristics. Examples of these securities are callable bonds, convertible bonds, and warrants. Other arrangements such as collateralized loans
and limited-liability borrowing can be analyzed as conveying implicit options to one or more parties.
7. Trading in so-called exotic options now takes place in an active over-the-counter market.
SUMMARY
Related Web sites for this chapter are available at www. mhhe.com/bkm
call option
exercise or strike price
premium
put option
in the money
out of the money
at the money
American option
European option
protective put
covered call
straddle
spread
collar
put-call parity theorem
warrant
KEY TERMS
Payoff to call investor 5 bST 2 X if ST . X 0 if ST # X
Payoff to put investor 5 b0 if ST . X X 2 ST if ST # X
Put-call parity: P 5 C 2 S0 1 PV(X ) 1 PV(Div)
KEY EQUATIONS
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710 P A R T V I Options, Futures, and Other Derivatives
which an investment in a foreign currency can be converted back into dollars. The right to
translate a fixed amount of foreign currency into dollars at a given exchange rate is a sim-
ple foreign exchange option. Quantos are more interesting because the amount of currency
that will be translated into dollars depends on the investment performance of the foreign
security. Therefore, a quanto in effect provides a random number of options.
Digital Options Digital options, also called binary or “bet” options, have fixed payoffs that depend on
whether a condition is satisfied by the price of the underlying asset. For example, a digital
call option might pay off a fixed amount of $100 if the stock price at maturity exceeds the
exercise price.
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C H A P T E R 2 0 Options Markets: Introduction 711
1. We said that options can be used either to scale up or reduce overall portfolio risk. What are some
examples of risk-increasing and risk-reducing options strategies? Explain each.
2. What are the trade-offs facing an investor who is considering buying a put option on an existing
portfolio?
3. What are the trade-offs facing an investor who is considering writing a call option on an existing
portfolio?
4. Why do you think the most actively traded options tend to be the ones that are near the money?
5. Turn back to Figure 20.1 , which lists prices of various IBM options. Use the data in the figure to
calculate the payoff and the profits for investments in each of the following February expiration
options, assuming that the stock price on the expiration date is $195.
a. Call option, X 5 $190. b. Put option, X 5 $190. c. Call option, X 5 $195. d. Put option, X 5 $195. e. Call option, X 5 $200. f. Put option, X 5 $200.
6. Suppose you think FedEx stock is going to appreciate substantially in value in the next 6 months.
Say the stock’s current price, S 0 , is $100, and the call option expiring in 6 months has an exercise price, X, of $100 and is selling at a price, C, of $10. With $10,000 to invest, you are considering three alternatives.
a. Invest all $10,000 in the stock, buying 100 shares. b. Invest all $10,000 in 1,000 options (10 contracts). c. Buy 100 options (one contract) for $1,000, and invest the remaining $9,000 in a money market
fund paying 4% in interest over 6 months (8% per year).
What is your rate of return for each alternative for the following four stock prices 6 months
from now? Summarize your results in the table and diagram below.
Basic
Intermediate
PROBLEM SETS
ST0
Rate of Return
Price of Stock 6 Months from Now
$80 $100 $110 $120
a. All stocks (100 shares)
b. All options (1,000 shares)
c. Bills 1 100 options
7. The common stock of the P.U.T.T. Corporation has been trading in a narrow price range for the
past month, and you are convinced it is going to break far out of that range in the next 3 months.
You do not know whether it will go up or down, however. The current price of the stock is $100
per share, and the price of a 3-month call option at an exercise price of $100 is $10.
a. If the risk-free interest rate is 10% per year, what must be the price of a 3-month put option on P.U.T.T. stock at an exercise price of $100? (The stock pays no dividends.)
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712 P A R T V I Options, Futures, and Other Derivatives
b. What would be a simple options strategy to exploit your conviction about the stock price’s future movements? How far would it have to move in either direction for you to make a
profit on your initial investment?
8. The common stock of the C.A.L.L. Corporation has been trading in a narrow range around $50
per share for months, and you believe it is going to stay in that range for the next 3 months. The
price of a 3-month put option with an exercise price of $50 is $4.
a. If the risk-free interest rate is 10% per year, what must be the price of a 3-month call option on C.A.L.L. stock at an exercise price of $50 if it is at the money? (The stock pays no
dividends.)
b. What would be a simple options strategy using a put and a call to exploit your conviction about the stock price’s future movement? What is the most money you can make on this
position? How far can the stock price move in either direction before you lose money?
c. How can you create a position involving a put, a call, and riskless lending that would have the same payoff structure as the stock at expiration? What is the net cost of establishing that
position now?
9. You are a portfolio manager who uses options positions to customize the risk profile of your
clients. In each case, what strategy is best given your client’s objective?
a. • Performance to date: Up 16%. • Client objective: Earn at least 15%.
• Your scenario: Good chance of large gains or large losses between now and end of year.
i. Long straddle.
ii. Long bullish spread.
iii. Short straddle.
b. • Performance to date: Up 16%. • Client objective: Earn at least 15%.
• Your scenario: Good chance of large losses between now and end of year.
i. Long put options.
ii. Short call options.
iii. Long call options.
10. An investor purchases a stock for $38 and a put for $.50 with a strike price of $35. The inves-
tor sells a call for $.50 with a strike price of $40. What is the maximum profit and loss for this
position? Draw the profit and loss diagram for this strategy as a function of the stock price at
expiration.
11. Imagine that you are holding 5,000 shares of stock, currently selling at $40 per share. You are
ready to sell the shares but would prefer to put off the sale until next year for tax reasons. If
you continue to hold the shares until January, however, you face the risk that the stock will drop
in value before year-end. You decide to use a collar to limit downside risk without laying out
a good deal of additional funds. January call options with a strike of $35 are selling at $2, and
January puts with a strike price of $45 are selling at $3. What will be the value of your portfolio
in January (net of the proceeds from the options) if the stock price ends up at: ( a ) $30, ( b ) $40, or ( c ) $50? Compare these proceeds to what you would realize if you simply continued to hold the shares.
12. In this problem, we derive the put-call parity relationship for European options on stocks that
pay dividends before option expiration. For simplicity, assume that the stock makes one divi-
dend payment of $ D per share at the expiration date of the option.
a. What is the value of a stock-plus-put position on the expiration date of the option? b. Now consider a portfolio comprising a call option and a zero-coupon bond with the same
maturity date as the option and with face value ( X 1 D ). What is the value of this portfolio on the option expiration date? You should find that its value equals that of the stock-plus-put
portfolio regardless of the stock price.
c. What is the cost of establishing the two portfolios in parts ( a ) and ( b )? Equate the costs of these portfolios, and you will derive the put-call parity relationship, Equation 20.2.
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C H A P T E R 2 0 Options Markets: Introduction 713
13. a. A butterfly spread is the purchase of one call at exercise price X 1 , the sale of two calls at exercise price X 2 , and the purchase of one call at exercise price X 3 . X 1 is less than X 2 , and X 2 is less than X 3 by equal amounts, and all calls have the same expiration date. Graph the payoff diagram to this strategy.
b. A vertical combination is the purchase of a call with exercise price X 2 and a put with exercise price X 1 , with X 2 greater than X 1 . Graph the payoff to this strategy.
14. A bearish spread is the purchase of a call with exercise price X 2 and the sale of a call with exercise price X 1 , with X 2 greater than X 1 . Graph the payoff to this strategy and compare it to Figure 20.10 .
15. Joseph Jones, a manager at Computer Science, Inc. (CSI), received 10,000 shares of company
stock as part of his compensation package. The stock currently sells at $40 a share. Joseph
would like to defer selling the stock until the next tax year. In January, however, he will need to
sell all his holdings to provide for a down payment on his new house. Joseph is worried about
the price risk involved in keeping his shares. At current prices, he would receive $400,000 for
the stock. If the value of his stock holdings falls below $350,000, his ability to come up with the
necessary down payment would be jeopardized. On the other hand, if the stock value rises to
$450,000, he would be able to maintain a small cash reserve even after making the down pay-
ment. Joseph considers three investment strategies:
a. Strategy A is to write January call options on the CSI shares with strike price $45. These calls are currently selling for $3 each.
b. Strategy B is to buy January put options on CSI with strike price $35. These options also sell for $3 each.
c. Strategy C is to establish a zero-cost collar by writing the January calls and buying the January puts.
Evaluate each of these strategies with respect to Joseph’s investment goals. What are the advan-
tages and disadvantages of each? Which would you recommend?
16. Use the spreadsheet from the Excel Application boxes on spreads and straddles (available at
www.mhhe.com/bkm ; link to Chapter 20 material) to answer these questions.
a. Plot the payoff and profit diagrams to a straddle position with an exercise (strike) price of $130. Assume the options are priced as they are in the Excel Application.
b. Plot the payoff and profit diagrams to a bullish spread position with exercise (strike) prices of $120 and $130. Assume the options are priced as they are in the Excel Application.
17. Some agricultural price support systems have guaranteed farmers a minimum price for their
output. Describe the program provisions as an option. What is the asset? The exercise price?
18. In what ways is owning a corporate bond similar to writing a put option? A call option?
19. An executive compensation scheme might provide a manager a bonus of $1,000 for every dollar
by which the company’s stock price exceeds some cutoff level. In what way is this arrangement
equivalent to issuing the manager call options on the firm’s stock?
20. Consider the following options portfolio. You write a January expiration call option on IBM
with exercise price $195. You write a January IBM put option with exercise price $190.
a. Graph the payoff of this portfolio at option expiration as a function of IBM’s stock price at that time.
b. What will be the profit/loss on this position if IBM is selling at $198 on the option expiration date? What if IBM is selling at $205? Use The Wall Street Journal listing from Figure 20.1 to answer this question.
c. At what two stock prices will you just break even on your investment? d. What kind of “bet” is this investor making; that is, what must this investor believe about
IBM’s stock price to justify this position?
21. Consider the following portfolio. You write a put option with exercise price 90 and buy a put
option on the same stock with the same expiration date with exercise price 95.
a. Plot the value of the portfolio at the expiration date of the options. b. On the same graph, plot the profit of the portfolio. Which option must cost more?
e X c e l Please visit us at www.mhhe.com/bkm
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714 P A R T V I Options, Futures, and Other Derivatives
22. A FinCorp put option with strike price 60 trading on the Acme options exchange sells for $2. To
your amazement, a FinCorp put with the same maturity selling on the Apex options exchange
but with strike price 62 also sells for $2. If you plan to hold the options positions to expiration,
devise a zero-net-investment arbitrage strategy to exploit the pricing anomaly. Draw the profit
diagram at expiration for your position.
23. Assume a stock has a value of $100. The stock is expected to pay a dividend of $2 per share at
year-end. An at-the-money European-style put option with one-year maturity sells for $7. If the
annual interest rate is 5%, what must be the price of a 1-year at-the-money European call option
on the stock?
24. You buy a share of stock, write a 1-year call option with X 5 $10, and buy a 1-year put option with X 5 $10. Your net outlay to establish the entire portfolio is $9.50. What is the risk-free interest rate? The stock pays no dividends.
25. You write a put option with X 5 100 and buy a put with X 5 110. The puts are on the same stock and have the same expiration date.
a. Draw the payoff graph for this strategy. b. Draw the profit graph for this strategy. c. If the underlying stock has positive beta, does this portfolio have positive or negative beta?
26. Joe Finance has just purchased a stock index fund, currently selling at $1,200 per share.
To protect against losses, Joe also purchased an at-the-money European put option on the
fund for $60, with exercise price $1,200, and 3-month time to expiration. Sally Calm, Joe’s
financial adviser, points out that Joe is spending a lot of money on the put. She notes that
3-month puts with strike prices of $1,170 cost only $45, and suggests that Joe use the
cheaper put.
a. Analyze Joe’s and Sally’s strategies by drawing the profit diagrams for the stock-plus-put positions for various values of the stock fund in 3 months.
b. When does Sally’s strategy do better? When does it do worse? c. Which strategy entails greater systematic risk?
27. You write a call option with X 5 50 and buy a call with X 5 60. The options are on the same stock and have the same expiration date. One of the calls sells for $3; the other sells for $9.
a. Draw the payoff graph for this strategy at the option expiration date. b. Draw the profit graph for this strategy. c. What is the break-even point for this strategy? Is the investor bullish or bearish on the stock?
28. Devise a portfolio using only call options and shares of stock with the following value (payoff)
at the option expiration date. If the stock price is currently 53, what kind of bet is the investor
making?
50
50 60 110
Payoff
ST
29. You are attempting to formulate an investment strategy. O n the one hand, you think there is
great upward potential in the stock market and would like to participate in the upward move
if it materializes. However, you are not able to afford substantial stock market losses and so
cannot run the risk of a stock market collapse, which you think is also a possibility. Your
investment adviser suggests a protective put position: Buy both shares in a market index
stock fund and put options on those shares with 3-month expiration and exercise price of
$1,170. The stock index fund is currently selling for $1,350. However, your uncle suggests
Challenge
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C H A P T E R 2 0 Options Markets: Introduction 715
you instead buy a 3-month call option on the index fund with exercise price $1,260 and buy
3-month T-bills with face value $1,260.
a. On the same graph, draw the payoffs to each of these strategies as a function of the stock fund value in 3 months. ( Hint: Think of the options as being on one “share” of the stock index fund, with the current price of each share of the fund equal to $1,350.)
b. Which portfolio must require a greater initial outlay to establish? ( Hint: Does either portfo- lio provide a final payout that is always at least as great as the payoff of the other portfolio?)
c. Suppose the market prices of the securities are as follows:
Stock fund $1,350
T-bill (face value $1,260) $1,215 Call (exercise price $1,260) $ 180 Put (exercise price $1,170) $ 9
Make a table of the profits realized for each portfolio for the following values of the stock
price in 3 months: S T 5 $1,000, $1,260, $1,350, $1,440. Graph the profits to each portfolio as a function of S T on a single graph.
d. Which strategy is riskier? Which should have a higher beta? e. Explain why the data for the securities given in part ( c ) do not violate the put-call parity
relationship.
30. FedEx is selling for $100 a share. A FedEx call option with one month until expiration and an
exercise price of $105 sells for $2 while a put with the same strike and expiration sells for $6.94.
What is the market price of a zero-coupon bond with face value $105 and 1 month maturity?
What is the risk-free interest rate expressed as an effective annual yield?
31. Demonstrate that an at-the-money call option on a given stock must cost more than an at-the-
money put option on that stock with the same expiration. The stock will pay no dividends until
after the expiration date. ( Hint: Use put-call parity.)
1. Donna Donie, CFA, has a client who believes the common stock price of TRT Materials (cur-
rently $58 per share) could move substantially in either direction in reaction to an expected court
decision involving the company. The client currently owns no TRT shares, but asks Donie for
advice about implementing a strangle strategy to capitalize on the possible stock price movement.
A strangle is a portfolio of a put and a call with a higher exercise price but the same expiration
date. Donie gathers the TRT option-pricing data:
Characteristic Call Option Put Option
Price $ 5 $ 4
Strike price $60 $55
Time to expiration 90 days from now 90 days from now
a. Recommend whether Donie should choose a long strangle strategy or a short strangle strategy to achieve the client’s objective.
b. Calculate, at expiration for the appropriate strangle strategy in part ( a ), the:
i. Maximum possible loss per share.
ii. Maximum possible gain per share.
iii. Break-even stock price(s).
2. Martin Bowman is preparing a report distinguishing traditional debt securities from structured
note securities. Discuss how the following structured note securities differ from a traditional debt
security with respect to coupon and principal payments:
a. Equity index-linked notes. b. Commodity-linked bear bond.
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716 P A R T V I Options, Futures, and Other Derivatives
3. Suresh Singh, CFA, is analyzing a convertible bond. The characteristics of the bond and the
underlying common stock are given in the following exhibit:
Convertible Bond Characteristics
Par value $1,000 Annual coupon rate (annual pay) 6.5% Conversion ratio 22 Market price 105% of par value Straight value 99% of par value
Underlying Stock Characteristics
Current market price $40 per share Annual cash dividend $1.20 per share
Compute the bond’s:
a. Conversion value. b. Market conversion price.
4. Rich McDonald, CFA, is evaluating his investment alternatives in Ytel Incorporated by analyzing
a Ytel convertible bond and Ytel common equity. Characteristics of the two securities are given in
the following exhibit:
Characteristics Convertible Bond Common Equity
Par value $1,000 — Coupon (annual payment) 4% — Current market price $980 $35 per share Straight bond value $925 — Conversion ratio 25 — Conversion option At any time — Dividend — $0 Expected market price in 1 year $1,125 $45 per share
a. Calculate, based on the exhibit, the:
i. Current market conversion price for the Ytel convertible bond.
ii. Expected 1-year rate of return for the Ytel convertible bond.
iii. Expected 1-year rate of return for the Ytel common equity.
One year has passed and Ytel’s common equity price has increased to $51 per share. Also, over
the year, the interest rate on Ytel’s nonconvertible bonds of the same maturity increased, while
credit spreads remained unchanged.
b. Name the two components of the convertible bond’s value. Indicate whether the value of each component should decrease, stay the same, or increase in response to the:
i. Increase in Ytel’s common equity price.
ii. Increase in interest rates.
5. a. Consider a bullish spread option strategy using a call option with a $25 exercise price priced at $4 and a call option with a $40 exercise price priced at $2.50. If the price of the stock
increases to $50 at expiration and each option is exercised on the expiration date, the net profit
per share at expiration (ignoring transaction costs) is:
i. $8.50
ii. $13.50
iii. $16.50
iv. $23.50
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C H A P T E R 2 0 Options Markets: Introduction 717
b. A put on XYZ stock with a strike price of $40 is priced at $2.00 per share, while a call with a strike price of $40 is priced at $3.50. What is the maximum per-share loss to
the writer of the uncovered put and the maximum per-share gain to the writer of the
uncovered call?
Maximum Loss to Put Writer
Maximum Gain to Call Writer
i. $38.00 $ 3.50
ii. $38.00 $36.50
iii. $40.00 $ 3.50
iv. $40.00 $40.00
SOLUTIONS TO CONCEPT CHECKS
E-INVESTMENTS EXERCISES
1. Go to www.nasdaq.com and select IBM in the quote section. Once you have the infor- mation quote, request the information on options. You will be able to access the prices for the calls and puts that are closest to the money. For example, if the price of IBM is $196.72, you will use the options with the $195 exercise price. Use near-term options. For example, in February, you would select April and July expirations.
a. What are the prices for the put and call with the nearest expiration date?
b. What would be the cost of a straddle using these options?
c. At expiration, what would be the break-even stock prices for the straddle?
d. What would be the percentage increase or decrease in the stock price required to break even?
e. What are the prices for the put and call with a later expiration date?
f. What would be the cost of a straddle using the later expiration date? At expiration, what would be the break-even stock prices for the straddle?
g. What would be the percentage increase or decrease in the stock price required to break even?
1. a. Denote the stock price at call option expiration by S T , and the exercise price by X. Value at expiration 5 S T 2 X 5 S T 2 $195 if this value is positive; otherwise the call expires worthless.
Profit 5 Final value 2 Price of call option 5 Proceeds 2 $3.65.
ST 5 $185 ST 5 $205
Proceeds $0 $10
Profits 2$3.65 $ 6.35
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718 P A R T V I Options, Futures, and Other Derivatives
S Payoff
Profit
Buy Call
S Payoff
Profit
Write Put
S
Payoff
Profit
Write Call
S Payoff
Profit
Buy Put
b. Value at expiration 5 X 2 S T 5 $195 2 S T if this value is positive; otherwise the put expires worthless.
Profit 5 Final value 2 Price of put option 5 Proceeds 2 $5.00.
ST 5 $185 ST 5 $205
Proceeds $10 $0
Profits 1$ 5.00 2$5.00
2. Before the split, the final payoff would have been 100 3 ($200 2 $195) 5 $500. After the split,
the payoff is 200 3 ($100 2 $97.50) 5 $500. The payoff is unaffected.
3. a.
b. The payoffs and profits to both buying calls and writing puts generally are higher when the stock price is higher. In this sense, both positions are bullish. Both involve potentially taking
delivery of the stock. However, the call holder will choose to take delivery when the stock price is high, while the put writer is obligated to take delivery when the stock price is low.
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C H A P T E R 2 0 Options Markets: Introduction 719
c. The payoffs and profits to both writing calls and buying puts generally are higher when the stock price is lower. In this sense, both positions are bearish. Both involve potentially making
delivery of the stock. However, the put holder will choose to make delivery when the stock price is low, while the call writer is obligated to make delivery when the stock price is high.
4.
Slope = −2
Slope = 1
Payoff and Profit
Payoff
Profit
− 2P − C
2X − 2P − C
X
2X
ST
Payoff to a Strip
ST " X ST + X
2 Puts 2(X 2 ST) 0
1 Call 0 ST 2 X
Payoff to a Strap
ST " X ST + X
1 Put X 2 ST 0
2 Calls 0 2(ST 2 X)
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720 P A R T V I Options, Futures, and Other Derivatives
5. The payoff table on a per-share basis is as follows:
Slope = −1
Slope = 2
Payoff and Profit
Payoff
Profit
− P − 2C
X − P − 2C
X
ST X
ST " 90 90 " ST " 110 ST + 110
Buy put (X 5 90) 90 2 ST 0 0
Share ST ST ST Write call (X 5 110) 0 0 2(ST 2 110)
TOTAL 90 ST 110
The graph of the payoff is as follows. If you multiply the per-share values by 2,000, you will see that
the collar provides a minimum payoff of $180,000 (representing a maximum loss of $20,000) and a
maximum payoff of $220,000 (which is the cost of the house).
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C H A P T E R 2 0 Options Markets: Introduction 721
6. The covered call strategy would consist of a straight bond with a call written on the bond. The
value of the strategy at option expiration as a function of the value of the straight bond is given by
the solid colored payoff line in the following figure, which is virtually identical to Figure 20.11 .
$90
$110
$90 $110
Collar
Payoff
ST
7. The call option is worth less as call protection is expanded. Therefore, the coupon rate need not
be as high.
8. Lower. Investors will accept a lower coupon rate in return for the conversion option.
9. The depositor’s implicit cost per dollar invested is now only ($.03 2 $.005)/1.03 5 $.02427 per
6-month period. Calls cost 50/1,000 5 $.05 per dollar invested in the index. The multiplier falls
to .02427/.05 5 .4854.
Value of Straight Bond
Call Written
Value of Straight Bond
Payoff of Covered Call
X
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21 IN THE PREVIOUS chapter we examined option markets and strategies. We noted that many securities contain embedded options that affect both their values and their risk– return characteristics. In this chapter, we turn our attention to option-valuation issues. To understand most option-valuation models requires considerable mathematical and sta- tistical background. Still, many of the ideas and insights of these models can be demon- strated in simple examples, and we will con- centrate on these.
We start with a discussion of the factors that ought to affect option prices. After this discussion, we present several bounds within which option prices must lie. Next we turn to quantitative models, starting with a simple “two-state” option-valuation model, and then showing how this approach can be
generalized into a useful and accurate pric- ing tool. We then move on to one particular valuation formula, the famous Black-Scholes model, one of the most significant break- throughs in finance theory in several decades. Finally, we look at some of the more impor- tant applications of option-pricing theory in portfolio management and control.
Option-pricing models allow us to “back out” market estimates of stock-price volatil- ity, and we will examine these measures of implied volatility. Next we turn to some of the more important applications of option-pricing theory in risk management. Finally, we take a brief look at some of the empirical evidence on option pricing, and the implications of that evidence concerning the limitations of the Black-Scholes model.
CHAPTER TWENTY-ONE
Option Valuation
21.1 Option Valuation: Introduction
Intrinsic and Time Values Consider a call option that is out of the money at the moment, with the stock price below
the exercise price. This does not mean the option is valueless. Even though immediate
exercise today would be unprofitable, the call retains a positive value because there is
always a chance the stock price will increase sufficiently by the expiration date to allow
for profitable exercise. If not, the worst that can happen is that the option will expire with
zero value.
2 PA
R T V
I
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C H A P T E R 2 1 Option Valuation 723
The value S 0 2 X is sometimes called the intrinsic value of in-the-money call options because it gives the payoff that could be obtained by immediate exercise. Intrinsic value is
set equal to zero for out-of-the-money or at-the-money options. The difference between the
actual call price and the intrinsic value is commonly called the time value of the option. “Time value” is unfortunate terminology because it may confuse the option’s time value
with the time value of money. Time value in the options context refers simply to the dif-
ference between the option’s price and the value the option would have if it were expiring
immediately. It is the part of the option’s value that may be attributed to the fact that it still
has positive time to expiration.
Most of an option’s time value typically is a type of “volatility value.” Because the
option holder can choose not to exercise, the payoff cannot be worse than zero. Even if a
call option is out of the money now, it still will sell for a positive price because it offers the
potential for a profit if the stock price increases, while imposing no risk of additional loss
should the stock price fall. The volatility value lies in the value of the right not to exercise the call if that action would be unprofitable. The option to exercise, as opposed to the obli-
gation to exercise, provides insurance against poor stock price performance.
As the stock price increases substantially, it becomes likely that the call option will
be exercised by expiration. Ultimately, with exercise all but assured, the volatility value
becomes minimal. As the stock price gets ever larger, the option value approaches the
“adjusted” intrinsic value, the stock price minus the present value of the exercise price,
S 0 2 PV( X ). Why should this be? If you are virtually certain the option will be exercised and the
stock purchased for X dollars, it is as though you own the stock already. The stock cer- tificate, with a value today of S 0 , might as well be sitting in your safe-deposit box now, as it will be there in only a few months. You just haven’t paid for it yet. The present
value of your obligation is the present value of X, so the net value of the call option is S 0 2 PV( X ).
1
Figure 21.1 illustrates the call option valuation function. The value curve shows that
when the stock price is very low, the option is nearly worthless, because there is almost
no chance that it will be exercised. When the stock price is very high, the option value
approaches adjusted intrinsic value. In the midrange case, where the option is approxi-
mately at the money, the option curve diverges from the straight lines corresponding to
adjusted intrinsic value. This is because although exercise today would have a negligible
(or negative) payoff, the volatility value of the option is quite high in this region.
The call always increases in value with the stock price. The slope is greatest, however,
when the option is deep in the money. In this case, exercise is all but assured, and the
option increases in price one-for-one with the stock price.
Determinants of Option Values We can identify at least six factors that should affect the value of a call option: the stock
price, the exercise price, the volatility of the stock price, the time to expiration, the inter-
est rate, and the dividend rate of the stock. The call option should increase in value with
the stock price and decrease in value with the exercise price because the payoff to a call,
1 This discussion presumes that the stock pays no dividends until after option expiration. If the stock does pay
dividends before expiration, then there is a reason you would care about getting the stock now rather than at expiration—getting it now entitles you to the interim dividend payments. In this case, the adjusted intrinsic value
of the option must subtract the value of the dividends the stock will pay out before the call is exercised. Adjusted
intrinsic value would more generally be defined as S 0 2 PV( X ) 2 PV( D ), where D is the dividend to be paid before option expiration.
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724 P A R T V I Options, Futures, and Other Derivatives
if exercised, equals S T 2 X. The magnitude of the expected payoff increases with the difference S 0 2 X.
Call option values also increase with the volatility of the underlying stock price. To see
why, consider circumstances where possible stock prices at expiration may range from $10
to $50 compared to a situation where they range only from $20 to $40. In both cases, the
expected, or average, stock price will be $30. Suppose the exercise price on a call option is
also $30. What are the option payoffs?
High-Volatility Scenario
Stock price $10 $20 $30 $40 $50 Option payoff 0 0 0 10 20
Low-Volatility Scenario
Stock price $20 $25 $30 $35 $40 Option payoff 0 0 0 5 10
If each outcome is equally likely, with probability .2, the expected payoff to the option
under high-volatility conditions will be $6, but under low-volatility conditions the expected
payoff is half as much, only $3.
Despite the fact that the average stock price in each scenario is $30, the average option
payoff is greater in the high-volatility scenario. The source of this extra value is the limited
loss an option holder can suffer, or the volatility value of the call. No matter how far below
Option Value
Out of the Money
In the Money
S0
Time Value
PV(X) X
Value of Call Option S0 – PV(X)
Value of Option if Now at Expiration = S – X = Intrinsic Value
Figure 21.1 Call option value before expiration
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C H A P T E R 2 1 Option Valuation 725
$30 the stock price drops, the option holder will get zero. Obviously, extremely poor stock
price performance is no worse for the call option holder than moderately poor performance.
In the case of good stock performance, however, the call will expire in the money, and
it will be more profitable the higher the stock price. Thus extremely good stock outcomes
can improve the option payoff without limit, but extremely poor outcomes cannot worsen
the payoff below zero. This asymmetry means that volatility in the underlying stock price
increases the expected payoff to the option, thereby enhancing its value. 2
Similarly, longer time to expiration increases the value of a call option. For more distant
expiration dates, there is more time for unpredictable future events to affect prices, and the
range of likely stock prices increases. This has an effect similar to that of increased volatil-
ity. Moreover, as time to expiration lengthens, the present value of the exercise price falls,
thereby benefiting the call option holder and increasing the option value. As a corollary
to this issue, call option values are higher when interest rates rise (holding the stock price
constant) because higher interest rates also reduce the present value of the exercise price. Finally, the dividend payout policy of the firm affects
option values. A high-dividend payout policy puts a drag
on the rate of growth of the stock price. For any expected
total rate of return on the stock, a higher dividend yield
must imply a lower expected rate of capital gain. This drag
on stock price appreciation decreases the potential pay-
off from the call option, thereby lowering the call value.
Table 21.1 summarizes these relationships.
If This Variable Increases . . . The Value of a Call Option
Stock price, S Increases Exercise price, X Decreases Volatility, s Increases Time to expiration, T Increases Interest rate, r f Increases Dividend payouts Decreases
Table 21.1
Determinants of call option values
Prepare a table like Table 21.1 for the deter- minants of put option values. How should American put values respond to increases in S, X, s , T, r f , and dividend payouts?
CONCEPT CHECK 21.1
2 You should be careful interpreting the relationship between volatility and option value. Neither the focus of this
analysis on total (as opposed to systematic) volatility nor the conclusion that options buyers seem to like volatil-
ity contradicts modern portfolio theory. In conventional discounted cash flow analysis, we find the discount rate
appropriate for a given distribution of future cash flows. Greater risk implies a higher discount rate and lower present value. Here, however, the cash flow from the option depends on the volatility of the stock. The option value increases not because traders like risk but because the expected cash flow to the option holder increases
along with the volatility of the underlying asset.
21.2 Restrictions on Option Values
Several quantitative models of option pricing have been devised, and we will examine
some of them later in this chapter. All models, however, rely on simplifying assump-
tions. You might wonder which properties of option values are truly general and which
depend on the particular simplifications. To start with, we will consider some of the more
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726 P A R T V I Options, Futures, and Other Derivatives
important general properties of option prices. Some of these properties have important
implications for the effect of stock dividends on option values and the possible profitability
of early exercise of an American option.
Restrictions on the Value of a Call Option The most obvious restriction on the value of a call option is that its value cannot be nega-
tive. Because the option need not be exercised, it cannot impose any liability on its holder;
moreover, as long as there is any possibility that at some point the option can be exercised
profitably, it will command a positive price. Its payoff is zero at worst, and possibly posi-
tive, so it has some positive value.
We can place another lower bound on the value of a call option. Suppose that the stock
will pay a dividend of D dollars just before the expiration date of the option, denoted by T (where today is time 0). Now compare two portfolios, one consisting of a call option on one share of stock and the other a leveraged equity position consisting of that share and
borrowing of ( X 1 D )/(1 1 r f ) T dollars. The loan repayment is X 1 D dollars, due on the
expiration date of the option. For example, for a half-year maturity option with exercise
price $70, dividends to be paid of $5, and effective annual interest of 10%, you would pur-
chase one share of stock and borrow $75/(1.10) 1/2
5 $71.51. In 6 months, when the loan
matures, the payment due is $75.
At that time, the payoff to the leveraged equity position would be
In General Our Numbers
Stock value S T 1 D S T 1 5 2 Payback of loan 2 ( X 1 D ) 2 75 TOTAL S T 2 X S T 2 70
where S T denotes the stock price at the option expiration date. Notice that the payoff to the stock is the ex-dividend stock value plus dividends received. Whether the total payoff to
the stock-plus-borrowing position is positive or negative depends on whether S T exceeds X. The net cash outlay required to establish this leveraged equity position is S 0 2 $71.51, or, more generally, S 0 2 ( X 1 D )/(1 1 r f )
T , that is, the current price of the stock, S 0 , less the
initial cash inflow from the borrowing position.
The payoff to the call option will be S T 2 X if the option expires in the money and zero otherwise. Thus the option payoff is equal to the leveraged equity payoff when that payoff
is positive and is greater when the leveraged equity position has a negative payoff. Because
the option payoff is always greater than or equal to that of the leveraged equity position,
the option price must exceed the cost of establishing that position.
Therefore, the value of the call must be greater than S 0 2 ( X 1 D )/(1 1 r f ) T , or, more
generally,
C $ S0 2 PV(X) 2 PV(D)
where PV( X ) denotes the present value of the exercise price and PV( D ) is the present value of the dividends the stock will pay at the option’s expiration. More generally, we can inter-
pret PV( D ) as the present value of any and all dividends to be paid prior to the option expi- ration date. Because we know already that the value of a call option must be nonnegative,
we may conclude that C is greater than the maximum of either 0 or S 0 2 PV( X ) 2 PV( D ). We also can place an upper bound on the possible value of the call; this bound is simply
the stock price. No one would pay more than S 0 dollars for the right to purchase a stock currently worth S 0 dollars. Thus C # S 0 .
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C H A P T E R 2 1 Option Valuation 727
Figure 21.2 demonstrates graphically
the range of prices that is ruled out by these
upper and lower bounds for the value of a call
option. Any option value outside the shaded
area is not possible according to the restric-
tions we have derived. Before expiration, the
call option value normally will be within the allowable range, touching neither the upper
nor lower bound, as in Figure 21.3 .
Early Exercise and Dividends A call option holder who wants to close out
that position has two choices: exercise the
call or sell it. If the holder exercises at time t, the call will provide a payoff of S t 2 X, assuming, of course, that the option is in the
money. We have just seen that the option can
be sold for at least S t 2 PV( X ) 2 PV ( D ). Therefore, for an option on a non-dividend-
paying stock, C is greater than S t 2 PV ( X ). Because the present value of X is less than X itself, it follows that
C $ St 2 PV(X) . St 2 X
The implication here is that the pro-
ceeds from a sale of the option (at price C ) must exceed the proceeds from an exercise
( S t 2 X ). It is economically more attractive to sell the call, which keeps it alive, than to
exercise and thereby end the option. In other
words, calls on non-dividend-paying stocks
are “worth more alive than dead.”
If it never pays to exercise a call option
before expiration, the right to exercise
early actually must be valueless. We con-
clude that the values of otherwise identi-
cal American and European call options
on stocks paying no dividends are equal.
This simplifies matters, because any valu-
ation formula that applies to the European
call, for which only one exercise date
need be considered, also must apply to an
American call.
As most stocks do pay dividends, you
may wonder whether this result is just a theo-
retical curiosity. It is not: Reconsider our
argument and you will see that all that we really require is that the stock pay no dividends
until the option expires. This condition will be true for many real-world options.
Call Value
Lower Bound 5 Adjusted Intrinsic Value 5 S0 2 PV(X) 2 PV(D)
Up pe
r B ou
nd 5
S 0 Al
lo w
ab le
R an
ge
PV(X) 1 PV(D) S0
Figure 21.2 Range of possible call option values
Value of Call for Given Stock Price
C
S0 PV(X) 1 PV(D)
Figure 21.3 Call option value as a function of the current stock price
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728 P A R T V I Options, Futures, and Other Derivatives
Early Exercise of American Puts For American put options, the optimality of early exercise is most definitely a possibility. To see why, consider a simple example. Suppose that you purchase a put option on a stock. Soon
the firm goes bankrupt, and the stock price falls to zero. Of course you want to exercise now,
because the stock price can fall no lower. Immediate exercise gives you immediate receipt of
the exercise price, which can be invested to start generating income. Delay in exercise means a
time-value-of-money cost. The right to exercise a put option before expiration must have value.
Now suppose instead that the firm is only nearly bankrupt, with the stock selling at just a few cents. Immediate exercise may still be optimal. After all, the stock price can fall by
only a very small amount, meaning that the proceeds from future exercise cannot be more
than a few cents greater than the proceeds from immediate exercise. Against this possibil-
ity of a tiny increase in proceeds must be weighed the time-value-of-money cost of defer-
ring exercise. Clearly, there is some stock price below which early exercise is optimal.
This argument also proves that the American put must be worth more than its European
counterpart. The American put allows you to exercise anytime before expiration. Because
the right to exercise early may be useful in some circumstances, it will command a
premium in the capital market. The American put therefore will sell for a higher price than
a European put with otherwise identical terms.
Figure 21.4 , panel A illustrates the value of an American put option as a function of the
current stock price, S 0 . Once the stock price drops below a critical value, denoted S * in the figure, exercise becomes optimal. At that point the option-pricing curve is tangent to the
straight line depicting the intrinsic value of the option. If and when the stock price reaches
S *, the put option is exercised and its payoff equals its intrinsic value. In contrast, the value of the European put, which is graphed in Figure 21.4 , panel B, is
not asymptotic to the intrinsic value line. Because early exercise is prohibited, the maxi-
mum value of the European put is PV( X ), which occurs at the point S 0 5 0. Obviously, for a long enough horizon, PV( X ) can be made arbitrarily small.
A: American Put Value B: European Put Value
X X
S* X XPV(X) S0 S0
PV(X) Put Value
Time Value
Figure 21.4 Put option values as a function of the current stock price
In light of this discussion, explain why the put-call parity relationship is valid only for European options on non-dividend-paying stocks. If the stock pays no dividends, what inequality for American options would correspond to the parity theorem?
CONCEPT CHECK 21.2
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C H A P T E R 2 1 Option Valuation 729
21.3 Binomial Option Pricing
Two-State Option Pricing A complete understanding of commonly used option-valuation formulas is difficult with-
out a substantial mathematics background. Nevertheless, we can develop valuable insight
into option valuation by considering a simple special case. Assume that a stock price can
take only two possible values at option expiration: The stock will either increase to a given
higher price or decrease to a given lower price. Although this may seem an extreme simpli-
fication, it allows us to come closer to understanding more complicated and realistic mod-
els. Moreover, we can extend this approach to describe far more reasonable specifications
of stock price behavior. In fact, several major financial firms employ variants of this simple
model to value options and securities with optionlike features.
Suppose the stock now sells at S 0 5 $100, and the price will either increase by a factor of u 5 1.20 to $120 ( u stands for “up”) or fall by a factor of d 5 .9 to $90 ( d stands for “down”) by year-end. A call option on the stock might specify an exercise price of $110
and a time to expiration of 1 year. The interest rate is 10%. At year-end, the payoff to the
holder of the call option will be either zero, if the stock falls, or $10, if the stock price goes
to $120.
These possibilities are illustrated by the following value “trees”:
120 10
100 C
90 0
Stock price Call option value
Compare the payoff of the call to that of a portfolio consisting of one share of the
stock and borrowing of $81.82 at the interest rate of 10%. The payoff of this portfolio also
depends on the stock price at year-end:
Value of stock at year-end $90 $120 2 Repayment of loan with interest 2 90 2 90 TOTAL $ 0 $ 30
We know the cash outlay to establish the portfolio is $18.18: $100 for the stock, less the
$81.82 proceeds from borrowing. Therefore the portfolio’s value tree is
30
18.18
0
The payoff of this portfolio is exactly three times that of the call option for either value
of the stock price. In other words, three call options will exactly replicate the payoff to the
portfolio; it follows that three call options should have the same price as the cost of estab-
lishing the portfolio. Hence the three calls should sell for the same price as this replicating portfolio. Therefore,
3C 5 $18.18
or each call should sell at C 5 $6.06. Thus, given the stock price, exercise price, interest rate, and volatility of the stock price (as represented by the spread between the up or down
movements), we can derive the fair value for the call option.
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730 P A R T V I Options, Futures, and Other Derivatives
This valuation approach relies heavily on the notion of replication. With only two pos-
sible end-of-year values of the stock, the payoffs to the levered stock portfolio replicate the
payoffs to three call options and, therefore, command the same market price. Replication is
behind most option-pricing formulas. For more complex price distributions for stocks, the
replication technique is correspondingly more complex, but the principles remain the same.
One way to view the role of replication is to note that, using the numbers assumed for
this example, a portfolio made up of one share of stock and three call options written is
perfectly hedged. Its year-end value is independent of the ultimate stock price:
Stock value $90 $120 2 Obligations from 3 calls written 2 0 2 30 Net payoff $90 $ 90
The investor has formed a riskless portfolio, with a payout of $90. Its value must be the
present value of $90, or $90/1.10 5 $81.82. The value of the portfolio, which equals $100
from the stock held long, minus 3 C from the three calls written, should equal $81.82. Hence $100 2 3 C 5 $81.82, or C 5 $6.06.
The ability to create a perfect hedge is the key to this argument. The hedge locks in the
end-of-year payout, which therefore can be discounted using the risk-free interest rate. To find the value of the option in terms of the value of the stock, we do not need to know either
the option’s or the stock’s beta or expected rate of return. The perfect hedging, or replica-
tion, approach enables us to express the value of the option in terms of the current value
of the stock without this information. With a hedged position, the final stock price does
not affect the investor’s payoff, so the stock’s risk and return parameters have no bearing.
The hedge ratio of this example is one share of stock to three calls, or one-third. This
ratio has an easy interpretation in this context: It is the ratio of the range of the values of the
option to those of the stock across the two possible outcomes. The stock, which originally
sells for S 0 5 100, will be worth either d 3 $100 5 $90 or u 3 $100 5 $120, for a range of $30. If the stock price increases, the call will be worth C u 5 $10, whereas if the stock price decreases, the call will be worth C d 5 0, for a range of $10. The ratio of ranges, 10/30, is one-third, which is the hedge ratio we have established.
The hedge ratio equals the ratio of ranges because the option and stock are perfectly
correlated in this two-state example. Because they are perfectly correlated, a perfect hedge
requires that they be held in a fraction determined only by relative volatility.
We can generalize the hedge ratio for other two-state option problems as
H 5 Cu 2 Cd
uS0 2 dS0
where C u or C d refers to the call option’s value when the stock goes up or down, respec- tively, and uS 0 and dS 0 are the stock prices in the two states. The hedge ratio, H, is the ratio of the swings in the possible end-of-period values of the option and the stock. If the
investor writes one option and holds H shares of stock, the value of the portfolio will be unaffected by the stock price. In this case, option pricing is easy: Simply set the value of
the hedged portfolio equal to the present value of the known payoff.
Using our example, the option-pricing technique would proceed as follows:
1. Given the possible end-of-year stock prices, uS 0 5 120 and dS 0 5 90, and the exercise price of 110, calculate that C u 5 10 and C d 5 0. The stock price range is 30, while the option price range is 10.
2. Find that the hedge ratio of 10/30 5 1⁄3.
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C H A P T E R 2 1 Option Valuation 731
3. Find that a portfolio made up of 1⁄3 share with one written option would have an
end-of-year value of $30 with certainty.
4. Show that the present value of $30 with a 1-year interest rate of 10% is $27.27.
5. Set the value of the hedged position to the present value of the certain payoff:
1@3 S0 2 C0 5 $27.27
$33.33 2 C0 5 $27.27
6. Solve for the call’s value, C 0 5 $6.06.
What if the option is overpriced, perhaps selling for $6.50? Then you can make arbi-
trage profits. Here is how:
Cash Flow in 1 Year for Each Possible Stock Price
Initial Cash Flow S1 5 90 S1 5 120
1. Write 3 options $ 19.50 $ 0 $230 2. Purchase 1 share 2100 90 120 3. Borrow $80.50 at 10% interest
Repay in 1 year 80.50 288.55 288.55
TOTAL $ 0 $ 1.45 $ 1.45
Although the net initial investment is zero, the payoff in 1 year is positive and riskless.
If the option were underpriced, one would simply reverse this arbitrage strategy: Buy the
option, and sell the stock short to eliminate price risk. Note, by the way, that the present
value of the profit to the arbitrage strategy above exactly equals three times the amount by
which the option is overpriced. The present value of the risk-free profit of $1.45 at a 10%
interest rate is $1.318. With three options written in the strategy above, this translates to
a profit of $.44 per option, exactly the amount by which the option was overpriced: $6.50
versus the “fair value” of $6.06.
Suppose the call option had been underpriced, selling at $5.50. Formulate the arbitrage strategy to exploit the mispricing, and show that it provides a riskless cash flow in 1 year of $.6167 per option purchased. Compare the present value of this cash flow to the option mispricing.
CONCEPT CHECK 21.3
Generalizing the Two-State Approach Although the two-state stock price model seems simplistic, we can generalize it to incor-
porate more realistic assumptions. To start, suppose we were to break up the year into
two 6-month segments, and then assert that over each half-year segment the stock price
could take on two values. In this example, we will say it can increase 10% (i.e., u 5 1.10) or decrease 5% (i.e., d 5 .95). A stock initially selling at 100 could follow these possible paths over the course of the year:
121 110 100 104.50 95
90.25
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732 P A R T V I Options, Futures, and Other Derivatives
The midrange value of 104.50 can be attained by two paths: an increase of 10% followed
by a decrease of 5%, or a decrease of 5% followed by a 10% increase.
There are now three possible end-of-year values for the stock and three for the option:
Cuu Cu C Cud 5 Cdu Cd Cdd
Using methods similar to those we followed above, we could value C u from knowl- edge of C uu and C ud , then value C d from knowledge of C du and C dd , and finally value C from knowledge of C u and C d . And there is no reason to stop at 6-month intervals. We could next break the year into four 3-month units, or twelve 1-month units, or 365 1-day
units, each of which would be posited to have a two-state process. Although the calcula-
tions become quite numerous and correspondingly tedious, they are easy to program into
a computer, and such computer programs are used widely by participants in the options
market.
Suppose that the risk-free interest rate is 5% per 6-month period and we wish to value a call option with exercise price $110 on the stock described in the two-period price tree just above. We start by finding the value of C u . From this point, the call can rise to an expiration-date value of C uu 5 $11 (because at this point the stock price is u 3 u 3 S 0 5 $121) or fall to a final value of C ud 5 0 (because at this point, the stock price is u 3 d 3 S 0 5 $104.50, which is less than the $110 exercise price). Therefore the hedge ratio at this point is
H 5 Cuu 2 Cud
uuS0 2 udS0 5
$11 2 0 $121 2 104.50
5 2 3
Thus, the following portfolio will be worth $209 at option expiration regardless of the ultimate stock price:
udS 5 $104.50 uuS 0 5 $121
Buy 2 shares at price uS 0 5 $110 $209 $242 Write 3 calls at price C u 0 2 33 TOTAL $209 $209
The portfolio must have a current market value equal to the present value of $209:
2 3 110 2 3Cu 5 $209/1.05 5 $199.047
Solve to find that C u 5 $6.984. Next we find the value of C d . It is easy to see that this value must be zero. If we reach
this point (corresponding to a stock price of $95), the stock price at option expiration will be either $104.50 or $90.25; in either case, the option will expire out of the money. (More formally, we could note that with C ud 5 C dd 5 0, the hedge ratio is zero, and a portfolio of zero shares will replicate the payoff of the call!)
Finally, we solve for C using the values of C u and C d . Concept Check 4 leads you through the calculations that show the option value to be $4.434.
Example 21.1 Binomial Option Pricing
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C H A P T E R 2 1 Option Valuation 733
Making the Valuation Model Practical As we break the year into progressively finer subintervals, the range of possible year-end stock
prices expands. For example, when we increase the number of subperiods to three, the number
of possible stock prices increases to four, as demonstrated in the following stock price tree:
u3S0 u2S0 uS0 u2dS0 S0 udS0 dS0 ud2S0 d2S0
d3S0
Thus, by allowing for an ever-greater number of subperiods, we can overcome one of the
apparent limitations of the valuation model: that the number of possible end-of-period
stock prices is small.
Notice that extreme events such as u 3 S 0 or d 3 S 0 are relatively rare, as they require either
three consecutive increases or decreases in the three subintervals. More moderate, or mid-
range, results such as u 2 dS 0 can be arrived at by more than one path—any combination of two price increases and one decrease will result in stock price u 2 dS 0 . There are three of these paths: uud, udu, duu. In contrast, only one path, uuu, results in a stock price of u 3 S 0 . Thus midrange values are more likely. As we make the model more realistic and break
up the option maturity into more and more subperiods, the probability distribution for the
final stock price begins to resemble the familiar bell-shaped curve with highly unlikely
extreme outcomes and far more likely midrange outcomes. The probability of each out-
come is given by the binomial probability distribution, and this multiperiod approach to
option pricing is therefore called the binomial model. But we still need to answer an important practical question. Before the binomial
model can be used to value actual options, we need a way to choose reasonable val-
ues for u and d. The spread between up and down movements in the price of the stock reflects the volatility of its rate of return, so the choice for u and d should depend on that volatility. Call s your estimate of the standard deviation of the stock’s continuously
compounded annualized rate of return, and Δ t the length of each subperiod. To make the standard deviation of the stock in the binomial model match your estimate of s , it
turns out that you can set u 5 exp(s"Dt) and d 5 exp(2s"Dt). 3 You can see that the
Show that the initial value of the call option in Example 21.1 is $4.434.
a. Confirm that the spread in option values is C u 2 C d 5 $6.984.
b. Confirm that the spread in stock values is uS 0 2 dS 0 5 $15.
c. Confirm that the hedge ratio is .4656 shares purchased for each call written.
d. Demonstrate that the value in one period of a portfolio comprised of .4656 shares and one call written is riskless.
e. Calculate the present value of this payoff.
f. Solve for the option value.
CONCEPT CHECK 21.4
3 Notice that d 5 1/ u. This is the most common, but not the only, way to calibrate the model to empirical
volatility. For alternative methods, see Robert L. McDonald, Derivatives Markets, 3rd ed., Pearson/Addison- Wesley, Boston: 2013, Ch. 10.
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734 P A R T V I Options, Futures, and Other Derivatives
proportional difference between u and d increases with both annualized volatility as well as the duration of the subperiod. This makes sense, as both higher s and longer holding
periods make future stock prices more uncertain. The following example illustrates how
to use this calibration.
4 Using this probability, the continuously compounded expected rate of return on the stock is .10. In general,
the formula relating the probability of an upward movement with the annual expected rate of return, r, is
p 5 exp(rDt) 2 d
u 2 d .
Suppose you are using a 3-period model to value a 1-year option on a stock with volatility (i.e., annualized standard deviation) of s 5 .30. With a time to expiration of T 5 1 year, and three subperiods, you would calculate Dt 5 T/n 5 1/3, u 5 exp(s"Dt) 5 exp (.30"1/3) 5 1.189 and d 5 exp(2s"Dt) 5 exp(2.30"1/3) 5 .841. Given the probability of an up movement, you could then work out the probability of any final stock price. For example, suppose the probability that the stock price increases is .554 and the probability that it decreases is .446. 4 Then the probability of stock prices at the end of the year would be as follows:
We plot this probability distribution in Figure 21.5 , panel A. Notice that the two middle end-of-period stock prices are, in fact, more likely than either extreme.
Example 21.2 Calibrating u and d to Stock Volatility
Event Possible Paths Probability Final Stock Price
3 down movements
ddd .446 3 5 0.089 59.48 5 100 3 .841 3
2 down and 1 up
ddu, dud, udd
3 3 .446 2 3 .554 5 0.330 84.10 5 100 3 1.189 3 .841 2
1 down and 2 up
uud, udu, duu
3 3 .446 3 .554 2 5 0.411 118.89 5 100 3 1.189 2 3 .841
3 up movements
uuu .554 3 5 0.170 168.09 5 100 3 1.189 3
Now we can extend Example 21.2 by breaking up the option maturity into ever-shorter
subintervals. As we do, the stock price distribution becomes increasingly plausible, as we
demonstrate in Example 21.3.
In Example 21.2, we broke up the year into three subperiods. Let’s now look at the cases of six and 20 subperiods.
Subperiods, n D t 5 T/n u 5 exp(s"Dt) d 5 exp(2s"Dt)
3 .333 exp(.173) 5 1.189 exp( 2 .173) 5 .841 6 .167 exp(.123) 5 1.130 exp( 2 .095) 5 .885 20 .015 exp(.067) 5 1.069 exp( 2 .067) 5 .935
Example 21.3 Increasing the Number of Subperiods
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C H A P T E R 2 1 Option Valuation 735
Figure 21.5 Probability distributions for final stock price. Possible outcomes and associated probabilities. In each panel, the stock’s annualized, continuously compounded expected rate of return is 10% and its standard deviation is 30%. Panel A. Three subintervals. In each subinterval, the stock can increase by 18.9% or fall by 15.9%. Panel B. Six subintervals. In each subinterval, the stock can increase by 13.0% or fall by 11.5%. Panel C. Twenty subintervals. In each subinterval, the stock can increase by 6.9% or fall by 6.5%.
.00
.05
.10
.15
.20
.25
.30
.35
.40
25 50 75 100 125 150 175 200 225 250
Final Stock Price
.45
P ro
b a b
il it
y
A
.00
.05
.10
.15
.20
.25
.30
.35
25 50 75 100 125 150 175 200 225 250 Final Stock Price
P ro
b a b
il it
y
B
.00
.05
.10
.15
.20
25 50 75 100 125 150 175 200 225 250
Final Stock Price
C
P ro
b a b
il it
y
We plot the resulting probability distributions in panels B and C of Figure 21.5 . 5 Notice that the right tail of the distribution in panel C is noticeably longer than
the left tail. In fact, as the number of intervals increases, the distribution progressively approaches the skewed log-normal (rather than the symmetric normal) distribution. Even if the stock price were to decline in each subinterval, it can never drop below zero. But there is no corresponding upper bound on its potential performance. This asymmetry gives rise to the skewness of the distribution.
Eventually, as we divide the option maturity into an ever-greater number of subintervals,
each node of the event tree would correspond to an infinitesimally small time interval. The
possible stock price movement within that time interval would be correspondingly small.
As those many intervals passed, the end-of-period stock price would more and more closely
5 We adjust the probabilities of up versus down movements using the formula in footnote 4 to make the distribu-
tions in Figure 21.5 comparable. In each panel, p is chosen so that the stock’s expected annualized, continuously compounded rate of return is 10%.
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736
resemble a lognormal distribution. 6 Thus the apparent oversimplification of the two-state
model can be overcome by progressively subdividing any period into many subperiods.
At any node, one still could set up a portfolio that would be perfectly hedged over the
next tiny time interval. Then, at the end of that interval, on reaching the next node, a new
hedge ratio could be computed and the portfolio composition could be revised to remain
6 Actually, more complex considerations enter here. The limit of this process is lognormal only if we assume
also that stock prices move continuously, by which we mean that over small time intervals only small price
movements can occur. This rules out rare events such as sudden, extreme price moves in response to dramatic
information (like a takeover attempt). For a treatment of this type of “jump process,” see John C. Cox and Stephen
A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics 3 (January–March 1976), pp. 145–66, or Robert C. Merton, “Option Pricing When Underlying Stock Returns Are
Discontinuous,” Journal of Financial Economics 3 (January–March 1976), pp. 125–44.
A Risk-Neutral Shortcut
We pointed out earlier in the chapter that the binomial model valuation approach is arbitrage-based. We can value the option by replicating it with shares of stock plus bor- rowing. The ability to replicate the option means that its price relative to the stock and the interest rate must be based only on the technology of replication and not on risk preferences. It cannot depend on risk aversion or the capi- tal asset pricing model or any other model of equilibrium risk-return relationships.
This insight—that the pricing model must be inde- pendent of risk aversion—leads to a very useful shortcut to valuing options. Imagine a risk-neutral economy, that is, an economy in which all investors are risk-neutral. This hypothetical economy must value options the same as our real one because risk aversion cannot affect the valuation formula.
In a risk-neutral economy, investors would not demand risk premiums and would therefore value all assets by discounting expected payoffs at the risk-free rate of interest. Therefore, a security such as a call option would be valued by discounting its expected cash flow
at the risk-free rate: “E”(CF)
C 5 11rf
. We put the expectation
operator E in quotation marks to signify that this is not the true expectation, but the expectation that would prevail in the hypothetical risk-neutral economy. To be consistent, we must calculate this expected cash flow using the rate of return the stock would have in the risk- neutral economy, not using its true rate of return. But if we successfully maintain consistency, the value derived for the hypothetical economy should match the one in our own.
How do we compute the expected cash flow from the option in the risk-neutral economy? Because there are no risk premiums, the stock’s expected rate of return must equal the risk-free rate. Call p the probability that the stock price increases. Then p must be chosen to equate the expected rate of increase of the stock price to the risk-free rate (we ignore dividends here):
“E”(S1) 5 p(uS) 1 (1 2 p)dS 5 (1 1 rf)S
This implies that p 5 1 1 rf 2 d
u 2 d . We call p a risk-neutral
probability to distinguish it from the true, or “objective,” probability. To illustrate, in our two-state example at the beginning of Section 21.2, we had u 5 1.2, d 5 .9, and
r f 5 .10. Given these values, p 5 1 1 .10 2 .9
1.2 2 .9 5
2 3
.
Now let’s see what happens if we use the discounted cash flow formula to value the option in the risk-neutral economy. We continue to use the two-state example from Section 21.2. We find the present value of the option pay- off using the risk-neutral probability and discount at the risk-free interest rate:
“E”(CF) C 5
11rf
p Cu1(12p) Cd 5
11rf
2/3310 1 1/330 5
1.10 6.065
This answer exactly matches the value we found using our no-arbitrage approach!
We repeat: This is not truly an expected discounted value.
• The numerator is not the true expected cash flow from the option because we use the risk-neutral probability, p, rather than the true probability.
• The denominator is not the proper discount rate for option cash flows because we do not account for the risk.
• In a sense, these two “errors” cancel out. But this is not just luck: We are assured to get the correct result because the no-arbitrage approach implies that risk preferences cannot affect the option value. Therefore, the value computed for the risk-neutral economy must equal the value that we obtain in our economy.
When we move to the more realistic multiperiod model, the calculations are more cumbersome, but the idea is the same. Footnote 4 shows how to relate p to any expected rate of return and volatility estimate. Simply set the expected rate of return on the stock equal to the risk- free rate, use the resulting probability to work out the expected payoff from the option, discount at the risk-free rate, and you will find the option value. These calculations are actually fairly easy to program in Excel.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 2 1 Option Valuation 737
hedged over the coming small interval. By continuously revising the hedge position, the
portfolio would remain hedged and would earn a riskless rate of return over each inter-
val. This is called dynamic hedging, the continued updating of the hedge ratio as time passes. As the dynamic hedge becomes ever finer, the resulting option-valuation proce-
dure becomes more precise. The nearby box offers further refinements on the use of the
binomial model.
In the table in Example 21.3, u and d both get closer to 1 ( u is smaller and d is larger) as the time interval Δ t shrinks. Why does this make sense? Does the fact that u and d are each closer to 1 mean that the total volatility of the stock over the remaining life of the option is lower?
CONCEPT CHECK 21.5
21.4 Black-Scholes Option Valuation
While the binomial model is extremely flexible, a computer is needed for it to be useful in
actual trading. An option-pricing formula would be far easier to use than the tedious algo- rithm involved in the binomial model. It turns out that such a formula can be derived if one
is willing to make just two more assumptions: that both the risk-free interest rate and stock
price volatility are constant over the life of the option. In this case, as the time to expira-
tion is divided into ever-more subperiods, the distribution of the stock price at expiration
progressively approaches the lognormal distribution, as suggested by Figure 21.5 . When
the stock price distribution is actually lognormal, we can derive an exact option-pricing
formula.
The Black-Scholes Formula Financial economists searched for years for a workable option-pricing model before
Black and Scholes 7 and Merton
8 derived a formula for the value of a call option. Scholes
and Merton shared the 1997 Nobel Prize in Economics for their accomplishment. 9 Now
widely used by options market participants, the Black-Scholes pricing formula for a call option is
C0 5 S0N(d1) 2 Xe 2rTN(d2) (21.1)
where
d1 5 ln(S0/X) 1 (r 1 s
2/2)T
s"T d2 5 d1 2 s"T
7 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political
Economy 81 (May–June 1973).
8 Robert C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science 4
(Spring 1973).
9 Fischer Black died in 1995.
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738 P A R T V I Options, Futures, and Other Derivatives
and
C 0 5 Current call option value.
S 0 5 Current stock price.
N ( d ) 5 The probability that a random draw from a standard normal distribution will be less than d. This equals the area under the normal curve up to d, as in the shaded area of Figure 21.6 . In Excel, this function is called NORMSDIST( ).
X 5 Exercise price.
e 5 The base of the natural log function, approximately 2.71828. In Excel, e x can be evaluated using the function EXP( x ).
r 5 Risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the expiration date of the option, which is to be
distinguished from r f , the discrete period interest rate).
T 5 Time to expiration of option, in years.
ln 5 Natural logarithm function. In Excel, ln( x ) can be calculated as LN( x ).
s 5 Standard deviation of the annualized continuously compounded rate of return
of the stock.
Notice a surprising feature of Equation 21.1: The option value does not depend on the expected rate of return on the stock. In a sense, this information is already built into the
formula with the inclusion of the stock price, which itself depends on the stock’s risk
and return characteristics. This version of the Black-Scholes formula is predicated on the
assumption that the stock pays no dividends.
Although you may find the Black-Scholes formula intimidating, we can explain it at
a somewhat intuitive level. The trick is to view the N ( d ) terms (loosely) as risk-adjusted probabilities that the call option will expire in the money. First, look at Equation 21.1
assuming both N ( d ) terms are close to 1.0, that is, when there is a very high probability the option will be exercised. Then the call option value is equal to S 0 2 Xe
2 rT , which
is what we called earlier the adjusted intrinsic value, S 0 2 PV( X ). This makes sense; if exercise is certain, we have a claim on a stock with current value S 0 , and an obligation
with present value PV( X ), or, with continu- ous compounding, Xe 2 rT .
Now look at Equation 21.1 assuming the
N ( d ) terms are close to zero, meaning the option almost certainly will not be exercised.
Then the equation confirms that the call is
worth nothing. For middle-range values of
N ( d ) between 0 and 1, Equation 21.1 tells us that the call value can be viewed as the present
value of the call’s potential payoff adjusting
for the probability of in-the-money expiration.
How do the N ( d ) terms serve as risk- adjusted probabilities? This question quickly
leads us into advanced statistics. Notice,
however, that ln( S 0 / X ), which appears in the numerator of d 1 and d 2 , is approximately the percentage amount by which the option is cur-
rently in or out of the money. For example, if
S 0 5 105 and X 5 100, the option is 5% in the
N(d) = Shaded area
d 0
Figure 21.6 A standard normal curve
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C H A P T E R 2 1 Option Valuation 739
money, and ln(105/100) 5 .049. Similarly, if S 0 5 95, the option is 5% out of the money, and ln(95/100) 5 2 .051. The denominator, s"T, adjusts the amount by which the option is in or out of the money for the volatility of the stock price over the remaining life of the
option. An option in the money by a given percent is more likely to stay in the money
if both stock price volatility and time to expiration are low. Therefore, N ( d 1 ) and N ( d 2 ) increase with the probability that the option will expire in the money.
You can use the Black-Scholes formula fairly easily. Suppose you want to value a call option under the following circumstances:
Stock price: S 0 5 100
Exercise price: X 5 95
Interest rate: r 5 .10 (10% per year)
Time to expiration: T 5 .25 (3 months or one-quarter of a year)
Standard deviation: s 5 .50 (50% per year)
First calculate
ln(100 / 95) 1 (.10 1 .52 / 2).25 .43
.5√.25 d1 5 5
d2 5 .43 2 .5".25 5 .18 Next find N ( d 1 ) and N ( d 2 ). The values of the normal distribution are tabulated and may be found in many statistics textbooks. A table of N ( d ) is provided here as Table 21.2 . The normal distribution function, N ( d ), is also provided in any spreadsheet program. In Microsoft Excel, for example, the function name is NORMSDIST. Using either Excel or Table 21.2 we find that
N(.43) 5 .6664
N(.18) 5 .5714
Thus the value of the call option is
C 5 100 3 .6664 2 95e2.10 3 .25 3 .5714
5 66.64 2 52.94 5 $13.70
Example 21.4 Black-Scholes Valuation
Recalculate the value of the call option in Example 21.4 using a standard deviation of .6 instead of .5. Confirm that the option is worth more using the higher stock-return volatility.
CONCEPT CHECK 21.6
What if the option price in Example 21.4 were $15 rather than $13.70? Is the option
mispriced? Maybe, but before betting your fortune on that, you may want to reconsider
the valuation analysis. First, like all models, the Black-Scholes formula is based on some
simplifying abstractions that make the formula only approximately valid.
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740 P A R T V I Options, Futures, and Other Derivatives
d N ( d ) d N ( d ) d N ( d ) d N ( d ) d N ( d ) d N ( d )
2 3.00 .0013 2 1.58 .0571 2 0.76 .2236 0.06 .5239 0.86 .8051 1.66 .9515 2 2.95 .0016 2 1.56 .0594 2 0.74 .2297 0.08 .5319 0.88 .8106 1.68 .9535 2 2.90 .0019 2 1.54 .0618 2 0.72 .2358 0.10 .5398 0.90 .8159 1.70 .9554 2 2.85 .0022 2 1.52 .0643 2 0.70 .2420 0.12 .5478 0.92 .8212 1.72 .9573 2 2.80 .0026 2 1.50 .0668 2 0.68 .2483 0.14 .5557 0.94 .8264 1.74 .9591 2 2.75 .0030 2 1.48 .0694 2 0.66 .2546 0.16 .5636 0.96 .8315 1.76 .9608 2 2.70 .0035 2 1.46 .0721 2 0.64 .2611 0.18 .5714 0.98 .8365 1.78 .9625 2 2.65 .0040 2 1.44 .0749 2 0.62 .2676 0.20 .5793 1.00 .8414 1.80 .9641 2 2.60 .0047 2 1.42 .0778 2 0.60 .2743 0.22 .5871 1.02 .8461 1.82 .9656 2 2.55 .0054 2 1.40 .0808 2 0.58 .2810 0.24 .5948 1.04 .8508 1.84 .9671 2 2.50 .0062 2 1.38 .0838 2 0.56 .2877 0.26 .6026 1.06 .8554 1.86 .9686 2 2.45 .0071 2 1.36 .0869 2 0.54 .2946 0.28 .6103 1.08 .8599 1.88 .9699 2 2.40 .0082 2 1.34 .0901 2 0.52 .3015 0.30 .6179 1.10 .8643 1.90 .9713 2 2.35 .0094 2 1.32 .0934 2 0.50 .3085 0.32 .6255 1.12 .8686 1.92 .9726 2 2.30 .0107 2 1.30 .0968 2 0.48 .3156 0.34 .6331 1.14 .8729 1.94 .9738 2 2.25 .0122 2 1.28 .1003 2 0.46 .3228 0.36 .6406 1.16 .8770 1.96 .9750 2 2.20 .0139 2 1.26 .1038 2 0.44 .3300 0.38 .6480 1.18 .8810 1.98 .9761 2 2.15 .0158 2 1.24 .1075 2 0.42 .3373 0.40 .6554 1.20 .8849 2.00 .9772 2 2.10 .0179 2 1.22 .1112 2 0.40 .3446 0.42 .6628 1.22 .8888 2.05 .9798 2 2.05 .0202 2 1.20 .1151 2 0.38 .3520 0.44 .6700 1.24 .8925 2.10 .9821 2 2.00 .0228 2 1.18 .1190 2 0.36 .3594 0.46 .6773 1.26 .8962 2.15 .9842 2 1.98 .0239 2 1.16 .1230 2 0.34 .3669 0.48 .6844 1.28 .8997 2.20 .9861 2 1.96 .0250 2 1.14 .1271 2 0.32 .3745 0.50 .6915 1.30 .9032 2.25 .9878 2 1.94 .0262 2 1.12 .1314 2 0.30 .3821 0.52 .6985 1.32 .9066 2.30 .9893 2 1.92 .0274 2 1.10 .1357 2 0.28 .3897 0.54 .7054 1.34 .9099 2.35 .9906 2 1.90 .0287 2 1.08 .1401 2 0.26 .3974 0.56 .7123 1.36 .9131 2.40 .9918 2 1.88 .0301 2 1.06 .1446 2 0.24 .4052 0.58 .7191 1.38 .9162 2.45 .9929 2 1.86 .0314 2 1.04 .1492 2 0.22 .4129 0.60 .7258 1.40 .9192 2.50 .9938 2 1.84 .0329 2 1.02 .1539 2 0.20 .4207 0.62 .7324 1.42 .9222 2.55 .9946 2 1.82 .0344 2 1.00 .1587 2 0.18 .4286 0.64 .7389 1.44 .9251 2.60 .9953 2 1.80 .0359 2 0.98 .1635 2 0.16 .4365 0.66 .7454 1.46 .9279 2.65 .9960 2 1.78 .0375 2 0.96 .1685 2 0.14 .4443 0.68 .7518 1.48 .9306 2.70 .9965 2 1.76 .0392 2 0.94 .1736 2 0.12 .4523 0.70 .7580 1.50 .9332 2.75 .9970 2 1.74 .0409 2 0.92 .1788 2 0.10 .4602 0.72 .7642 1.52 .9357 2.80 .9974 2 1.72 .0427 2 0.90 .1841 2 0.08 .4681 0.74 .7704 1.54 .9382 2.85 .9978 2 1.70 .0446 2 0.88 .1894 2 0.06 .4761 0.76 .7764 1.56 .9406 2.90 .9981 2 1.68 .0465 2 0.86 .1949 2 0.04 .4841 0.78 .7823 1.58 .9429 2.95 .9984 2 1.66 .0485 2 0.84 .2005 2 0.02 .4920 0.80 .7882 1.60 .9452 3.00 .9986 2 1.64 .0505 2 0.82 .2061 0.00 .5000 0.82 .7939 1.62 .9474 3.05 .9989 2 1.62 .0526 2 0.80 .2119 0.02 .5080 0.84 .7996 1.64 .9495 2 1.60 .0548 2 0.78 .2177 0.04 .5160
Table 21.2
Cumulative normal distribution
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C H A P T E R 2 1 Option Valuation 741
Some of the important assumptions underlying the formula are the following:
1. The stock will pay no dividends until after the option expiration date.
2. Both the interest rate, r, and variance rate, s 2 , of the stock are constant (or in slightly more general versions of the formula, both are known functions of time— any changes are perfectly predictable).
3. Stock prices are continuous, meaning that sudden extreme jumps such as those in
the aftermath of an announcement of a takeover attempt are ruled out.
Variants of the Black-Scholes formula have been developed to deal with many of these
limitations.
Second, even within the context of the Black-Scholes model, you must be sure of
the accuracy of the parameters used in the formula. Four of these— S 0 , X, T, and r —are straightforward. The stock price, exercise price, and time to expiration are readily deter-
mined. The interest rate used is the money market rate for a maturity equal to that of the
option, and the dividend payout is reasonably predictable, at least over short horizons.
The last input, though, the standard deviation of the stock return, is not directly observ-
able. It must be estimated from historical data, from scenario analysis, or from the prices
of other options, as we will describe momentarily.
We saw in Chapter 5 that the historical variance of stock market returns can be calcu-
lated from n observations as follows:
s2 5 n
n 2 1 a n
t 5 1
(rt 2 r) 2
n
where r is the average return over the sample period. The rate of return on day t is defined to be consistent with continuous compounding as r t 5 ln( S t / S t 2 1 ). [We note again that the natural logarithm of a ratio is approximately the percentage difference between the numer-
ator and denominator so that ln( S t / S t 2 1 ) is a measure of the rate of return of the stock from time t 2 1 to time t. ] Historical variance commonly is computed using daily returns over periods of several months. Because the volatility of stock returns must be estimated, how-
ever, it is always possible that discrepancies between an option price and its Black-Scholes
value are simply artifacts of error in the estimation of the stock’s volatility.
In fact, market participants often give the option-valuation problem a different twist.
Rather than calculating a Black-Scholes option value for a given stock’s standard devia-
tion, they ask instead: What standard deviation would be necessary for the option price that
I observe to be consistent with the Black-Scholes formula? This is called the implied vola- tility of the option, the volatility level for the stock implied by the option price. 10 Investors can then judge whether they think the actual stock standard deviation exceeds the implied
volatility. If it does, the option is considered a good buy; if actual volatility seems greater
than the implied volatility, its fair price would exceed the observed price.
Another variation is to compare two options on the same stock with equal expiration
dates but different exercise prices. The option with the higher implied volatility would be
considered relatively expensive, because a higher standard deviation is required to justify
its price. The analyst might consider buying the option with the lower implied volatility
and writing the option with the higher implied volatility.
The Black-Scholes valuation formula, as well as the implied volatility, is easily calcu-
lated using an Excel spreadsheet like Spreadsheet 21.1 . The model inputs are provided in
10
This concept was introduced in Richard E. Schmalensee and Robert R. Trippi, “Common Stock Volatility
Expectations Implied by Option Premia,” Journal of Finance 33 (March 1978), pp. 129–47.
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742 P A R T V I Options, Futures, and Other Derivatives
column B, and the outputs are given in column E. The formulas for d 1 and d 2 are provided in the spreadsheet, and the Excel formula NORMSDIST( d 1 ) is used to calculate N ( d 1 ).
11
Cell E6 contains the Black-Scholes formula. (The formula in the spreadsheet actually
includes an adjustment for dividends, as described in the next section.)
To compute an implied volatility, we can use the Goal Seek command from the What-If
Analysis menu (which can be found under the Data tab) in Excel. See Figure 21.7 for an
illustration. Goal Seek asks us to change the value of one cell to make the value of another
cell (called the target cell ) equal to a specific value. For example, if we observe a call option selling for $7 with other inputs as given in the spreadsheet, we can use Goal Seek
to change the value in cell B2 (the standard deviation of the stock) to set the option value
in cell E6 equal to $7. The target cell, E6, is the call price, and the spreadsheet manipulates
cell B2. When you click OK, the spreadsheet finds that a standard deviation equal to .2783 is consistent with a call price of $7; this would be the option’s implied volatility if it were
selling at $7.
Spreadsheet 21.1
Spreadsheet to calculate Black-Scholes call option values
e X c e l Please visit us at www.mhhe.com/bkm
A C D E
1
2
3
4
5
6
7
B
INPUTS
Standard deviation (annual)
Maturity (in years)
Risk-free rate (annual)
Stock price
Exercise price
Dividend yield (annual)
FORMULA FOR OUTPUT IN COLUMN E
(LN(B5/B6)+(B4–B7+.5*B2^2)*B3)/(B2*SQRT(B3))
E2–B2*SQRT(B3)
NORMSDIST(E2)
NORMSDIST(E3)
B5*EXP(–B7*B3)*E4–B6*EXP(–B4*B3)*E5
B6*EXP(–B4*B3)*(1–E5)–B5*EXP(–B7*B3)*(1–E4)
OUTPUTS
d1
d2
N(d1)
N(d2)
B/S call value
B/S put value
0.2783
0.5
0.06
100
105
0
F G H I J
0.0029
–0.1939
0.5012
0.4231
7.0000
8.8968
11 In some versions of Excel, the function is NORM.S.DIST( d, TRUE).
A
Standard deviation (annual)
Maturity (in years)
Risk-free rate (annual)
Stock price
Exercise price
Dividend yield (annual)
0.2783
0.5
0.06
100
105
0
0.0029
20.1939
0.5012
0.4231
7.0000
8.8968
(LN(B5/B6)1(B42B71.5*B2^2)*B3)/(B2*SQRT(B3))
E22B2*SQRT(B3)
NORMSDIST(E2)
NORMSDIST(E3)
B5*EXP(2B7*B3)*E42B6*EXP(2B4*B3)*E5
B6*EXP(2B4*B3)*(12E5) 2 B5*EXP(2B7*B3)*(12E4)
d1
d2
N(d1)
N(d2)
B/S call value
B/S put value
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
B C D E F G H I J K
OUTPUTS FORMULA FOR OUTPUT IN COLUMN EINPUTS
Figure 21.7 Using Goal Seek to find implied volatility e X c e l Please visit us at www.mhhe.com/bkm
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C H A P T E R 2 1 Option Valuation 743
The Chicago Board Options Exchange
regularly computes the implied volatil-
ity of major stock indexes. Figure 21.8 is
a graph of the implied (30-day) volatility
of the S&P 500 since 1990. During peri-
ods of turmoil, implied volatility can spike
quickly. Notice the peaks in January 1991
(Gulf War), August 1998 (collapse of Long-
Term Capital Management), September 11,
2001, 2002 (build-up to invasion of Iraq),
and, most dramatically, during the credit cri-
sis of 2008. Because implied volatility cor-
relates with crisis, it is sometimes called an
“investor fear gauge.”
A futures contract on the 30-day implied
volatility of the S&P 500 has traded on the
CBOE Futures Exchange since 2004. The
payoff of the contract depends on market
implied volatility at the expiration of the contract. The ticker symbol of the contract is VIX.
As the nearby box makes clear, observers use it to infer the market’s assessment of possible
stock price swings in coming months. In this case, the article questioned the relatively low level
of the VIX in light of tense political negotiations at the end of 2012 over the so-called fiscal
cliff. The question was whether the price of the VIX contract indicated that investors were being
too complacent about the potential for market disruption if those negotiations were to fail.
Figure 21.8 reveals an awkward empirical fact. While the Black-Scholes formula is
derived assuming that stock volatility is constant, the time series of implied volatilities
derived from that formula is in fact far from constant. This contradiction reminds us that
the Black-Scholes model (like all models) is a simplification that does not capture all
aspects of real markets. In this particular context, extensions of the pricing model that
allow stock volatility to evolve randomly over time would be desirable, and, in fact, many
extensions of the model along these lines have been developed. 12
The fact that volatility changes unpredictably means that it can be difficult to choose
the proper volatility input to use in any option-pricing model. A considerable amount of
recent research has been devoted to techniques to predict changes in volatility. These tech-
niques, known as ARCH and stochastic volatility models, posit that changes in volatility are partially predictable and that by analyzing recent levels and trends in volatility, one can
improve predictions of future volatility. 13
Gulf War
LTCM
9/11
Iraq
Subprime and Credit Crises
U.S. Debt Downgrade
0
10
20
30
40
50
60
70
Ja n
-9 1
Ja n
-9 0
Ja n
-9 2
Ja n
-9 3
Ja n
-9 4
Ja n
-9 5
Ja n
-9 6
Ja n
-9 7
Ja n
-9 8
Ja n
-9 9
Ja n
-0 0
Ja n
-0 1
Ja n
-0 2
Ja n
-0 3
Ja n
-0 4
Ja n
-0 5
Ja n
-0 6
Ja n
-0 7
Ja n
-0 8
Ja n
-0 9
Ja n
-1 0
Ja n
-1 1
Ja n
-1 2
Ja n
-1 3
Im p
lie d
V o
la ti
lit y
(% )
Figure 21.8 Implied volatility of the S&P 500 (VIX index) Source: Chicago Board Options Exchange, www.cboe.com .
Suppose the call option in Spreadsheet 21.1 actually is selling for $8. Is its implied volatility more or less than 27.83%? Use the spreadsheet (available at the Online Learning Center) and Goal Seek to find its implied volatility at this price.
CONCEPT CHECK 21.7
12 Influential articles on this topic are J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Vola-
tilities,” Journal of Finance (June 1987), pp. 281–300; J. Wiggins, “Option Values under Stochastic Volatility,” Jour- nal of Financial Economics (December 1987), pp. 351–72; and S. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bonds and Currency Options,” Review of Financial Studies 6 (1993), pp. 327–43. For a more recent review, see E. Ghysels, A. Harvey, and E. Renault, “Stochastic Volatility,” in Hand- book of Statistics, Vol. 14: Statistical Methods in Finance, ed. G. S. Maddala (Amsterdam: North Holland, 1996). 13
For an introduction to these models see C. Alexander, Market Models (Chichester, England: Wiley, 2001).
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744
Dividends and Call Option Valuation We noted earlier that the Black-Scholes call option formula applies to stocks that do not
pay dividends. When dividends are to be paid before the option expires, we need to adjust
the formula. The payment of dividends raises the possibility of early exercise, and for most
realistic dividend payout schemes the valuation formula becomes significantly more com-
plex than the Black-Scholes equation.
We can apply some simple rules of thumb to approximate the option value, however.
One popular approach, originally suggested by Black, calls for adjusting the stock price
downward by the present value of any dividends that are to be paid before option expi-
ration. 14
Therefore, we would simply replace S 0 with S 0 2 PV(dividends) in the Black- Scholes formula. Such an adjustment will take dividends into account by reflecting their
eventual impact on the stock price. The option value then may be computed as before,
assuming that the option will be held to expiration.
In one special case, the dividend adjustment takes a simple form. Suppose the underlying
asset pays a continuous flow of income. This might be a reasonable assumption for options
on a stock index, where different stocks in the index pay dividends on different days, so that
dividend income arrives in a more or less continuous flow. If the dividend yield, denoted d , is
constant, one can show that the present value of that dividend flow accruing until the option
expiration date is S 0 (1 2 e 2 d T
). (For intuition, notice that e 2 d T approximately equals 1 2 d T, so the value of the dividend is approximately d TS 0 .) In this case, S 0 2 PV(Div) 5 S 0 e
2 d T , and
we can derive a Black-Scholes call option formula on the dividend-paying asset simply by
substituting S 0 e 2 d T
for S 0 in the original formula. This approach is used in Spreadsheet 21.1. These procedures yield a very good approximation of option value for European call
options that must be held until expiration, but they do not allow for the fact that the holder
of an American call option might choose to exercise the option just before a dividend. The
current value of a call option, assuming that it will be exercised just before the ex-dividend
date, might be greater than the value of the option assuming it will be held until expiration.
Although holding the option until expiration allows greater effective time to expiration,
14
Fischer Black, “Fact and Fantasy in the Use of Options,” Financial Analysts Journal 31 (July–August 1975).
“Fear” Gauge Showing Little of It
On the floor of the Chicago Board Options Exchange (CBOE), those who trade in fear have seen little but calm.
Measures of volatility in U.S. markets are pointing to relative calm, but some investors say the low readings are a sign of complacency. Unlike the rocky ride in the stock market since the U.S. presidential election, the CBOE’s Volatility Index, or VIX, has registered tranquility. For four months, the so-called fear gauge of financial markets has traded below its two-decade historical average of 20, its longest such streak in more than 5 years.
Some investors worry that the low readings are a sign of complacency, and that the potential for further declines in response to unexpected bad news isn’t reflected in stock prices. These investors say various barometers of skittish- ness could tick higher as the year-end deadline nears for an agreement on taxes and spending. That anxiety also would likely be reflected in increased volatility in the stock and commodities markets.
The VIX is an index calculated from the prices investors are willing to pay for options tied to the Standard & Poor’s
500-stock index. As investors become nervous, they are will- ing to pay more for options, driving up the value of the VIX.
As market watchers search for clues about whether the relative calm can last, some of them are looking back to the early summer of 2011. Back then, the VIX was trading at levels close to today’s, even though market pundits wor- ried that lawmakers wouldn’t agree to raise the debt ceil- ing. That scenario could have led the U.S. government to default on its debt.
Within a few weeks, as the debt-ceiling wrangling was going down to the wire, Standard & Poor’s cut the U.S.’s long-term triple-A credit rating. The VIX nearly tripled to 48 in a span of two weeks.
“The whole situation leads me to believe that this lack of a cushion in the market will lead to wilder shocks” to mar- kets if negative events happen, says Michael Palmer of Group One Trading who trades the VIX on the floor of the CBOE.
Source: Steven Russolillo and Kaitlyn Kiernan, The Wall Street Journal, November 26, 2012. Reprinted with permission. © 2012 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 2 1 Option Valuation 745
which increases the option value, it also entails more dividend payments, lowering the
expected stock price at expiration and thereby lowering the current option value.
For example, suppose that a stock selling at $20 will pay a $1 dividend in 4 months,
whereas the call option on the stock does not expire for 6 months. The effective annual
interest rate is 10%, so that the present value of the dividend is $1/(1.10) 1/3
5 $0.97. Black
suggests that we can compute the option value in one of two ways:
1. Apply the Black-Scholes formula assuming early exercise, thus using the actual stock
price of $20 and a time to expiration of 4 months (the time until the dividend payment).
2. Apply the Black-Scholes formula assuming no early exercise, using the dividend-
adjusted stock price of $20 2 $.97 5 $19.03 and a time to expiration of 6 months.
The greater of the two values is the estimate of the option value, recognizing that early
exercise might be optimal. In other words, the so-called pseudo-American call option value is the maximum of the value derived by assuming that the option will be held until expiration and the value derived by assuming that the option will be exercised just before
an ex-dividend date. Even this technique is not exact, however, for it assumes that the
option holder makes an irrevocable decision now on when to exercise, when in fact the
decision is not binding until exercise notice is given. 15
Put Option Valuation We have concentrated so far on call option valuation. We can derive Black-Scholes
European put option values from call option values using the put-call parity theorem.
To value the put option, we simply calculate the value of the corresponding call option in
Equation 21.1 from the Black-Scholes formula, and solve for the put option value as
P 5 C 1 PV(X) 2 S0
5 C 1 Xe2rT 2 S0 (21.2)
We calculate the present value of the exercise price using continuous compounding to be
consistent with the Black-Scholes formula.
Sometimes, it is easier to work with a put option valuation formula directly. If we sub-
stitute the Black-Scholes formula for a call in Equation 21.2, we obtain the value of a
European put option as
P 5 Xe2rT 31 2 N(d2)4 2 S0 31 2 N(d1)4 (21.3)
Using data from Example 21.4 ( C 5 $13.70, X 5 $95, S 5 $100, r 5 .10, s 5 .50, and T 5 .25), Equation 21.3 implies that a European put option on that stock with identical exercise price and time to expiration is worth
$95e2.10 3 .25(1 2 .5714) 2 $100(1 2 .6664) 5 $6.35
Example 21.5 Black-Scholes Put Valuation
15
An exact formula for American call valuation on dividend-paying stocks has been developed in Richard Roll,
“An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends,”
Journal of Financial Economics 5 (November 1977). The technique has been discussed and revised in Robert Geske, “A Note on an Analytical Formula for Unprotected American Call Options on Stocks with Known
Dividends,” Journal of Financial Economics 7 (December 1979), and Robert E. Whaley, “On the Valuation of American Call Options on Stocks with Known Dividends,” Journal of Financial Economics 9 (June 1981).
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746 P A R T V I Options, Futures, and Other Derivatives
Dividends and Put Option Valuation Equation 21.2 or 21.3 is valid for European puts on non-dividend-paying stocks. As we did
for call options, if the underlying asset pays a dividend, we can find European put values
by substituting S 0 2 PV(Div) for S 0 . Cell E7 in Spreadsheet 21.1 allows for a continuous dividend flow with a dividend yield of d . In that case S 0 2 PV(Div) 5 S 0 e
2 d T .
However, listed put options on stocks are American options that offer the opportunity of
early exercise, and we have seen that the right to exercise puts early can turn out to be valu-
able. This means that an American put option must be worth more than the corresponding
European option. Therefore, Equation 21.2 or 21.3 describes only the lower bound on the
true value of the American put. However, in many applications the approximation is very
accurate. 16
Notice that this value is consistent with put-call parity:
P 5 C 1 PV(X ) 2 S0 5 13.70 1 95e 2.10 3 .25 2 100 5 6.35
As we noted traders can do, we might then compare this formula value to the actual put price as one step in formulating a trading strategy.
16
For a more complete treatment of American put valuation, see R. Geske and H. E. Johnson, “The American Put
Valued Analytically,” Journal of Finance 39 (December 1984), pp. 1511–24.
21.5 Using the Black-Scholes Formula
Hedge Ratios and the Black-Scholes Formula In the last chapter, we considered two investments in FinCorp stock: 100 shares or 1,000
call options. We saw that the call option position was more sensitive to swings in the stock
price than was the all-stock position. To analyze the overall exposure to a stock price more
precisely, however, it is necessary to quantify these relative sensitivities. We can summa-
rize the overall exposure of portfolios of options with various exercise prices and times to
expiration using the hedge ratio , the change in option price for a $1 increase in the stock price. A call option, therefore, has a positive hedge ratio and a put option a negative hedge
ratio. The hedge ratio is commonly called the option’s delta . If you were to graph the option value as a function of the stock value, as we have done
for a call option in Figure 21.9 , the hedge ratio is simply the slope of the curve evaluated
at the current stock price. For example, suppose the slope of the curve at S 0 5 $120 equals .60. As the stock increases in value by $1, the option increases by approximately $.60, as the
figure shows.
For every call option written, .60 share of stock would be needed to hedge the investor’s
portfolio. If one writes 10 options and holds six shares of stock, according to the hedge ratio
of .6, a $1 increase in stock price will result in a gain of $6 on the stock holdings, whereas
the loss on the 10 options written will be 10 3 $.60, an equivalent $6. The stock price
movement leaves total wealth unaltered, which is what a hedged position is intended to do.
Black-Scholes hedge ratios are particularly easy to compute. The hedge ratio for a call is
N ( d 1 ), whereas the hedge ratio for a put is N ( d 1 ) 2 1. We defined N ( d 1 ) as part of the Black- Scholes formula in Equation 21.1. Recall that N ( d ) stands for the area under the standard normal curve up to d. Therefore, the call option hedge ratio must be positive and less than 1.0, whereas the put option hedge ratio is negative and of smaller absolute value than 1.0.
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C H A P T E R 2 1 Option Valuation 747
Figure 21.9 verifies that the slope of the
call option valuation function is less than
1.0, approaching 1.0 only as the stock price
becomes much greater than the exercise price.
This tells us that option values change less
than one-for-one with changes in stock prices.
Why should this be? Suppose an option is so
far in the money that you are absolutely cer-
tain it will be exercised. In that case, every
dollar increase in the stock price would
increase the option value by $1. But if there is
a reasonable chance the call option will expire
out of the money, even after a moderate stock
price gain, a $1 increase in the stock price will
not necessarily increase the ultimate payoff
to the call; therefore, the call price will not
respond by a full dollar.
The fact that hedge ratios are less than 1.0
does not contradict our earlier observation
that options offer leverage and disproportion-
ate sensitivity to stock price movements. Although dollar movements in option prices are less than dollar movements in the stock price, the rate of return volatility of options remains greater than stock return volatility because options sell at lower prices. In our
example, with the stock selling at $120, and a hedge ratio of .6, an option with exercise
price $120 may sell for $5. If the stock price increases to $121, the call price would be
expected to increase by only $.60 to $5.60. The percentage increase in the option value is
$.60/$5.00 5 12%, however, whereas the stock price increase is only $1/$120 5 .83%. The
ratio of the percentage changes is 12%/.83% 5 14.4. For every 1% increase in the stock
price, the option price increases by 14.4%. This ratio, the percentage change in option
price per percentage change in stock price, is called the option elasticity. The hedge ratio is an essential tool in portfolio management and control. An example
will show why.
Value of a Call (C)
S0
40
20
0
120
Slope 5 .6
Figure 21.9 Call option value and hedge ratio
Consider two portfolios, one holding 750 FinCorp calls and 200 shares of FinCorp and the other holding 800 shares of FinCorp. Which portfolio has greater dollar exposure to FinCorp price movements? You can answer this question easily by using the hedge ratio.
Each option changes in value by H dollars for each dollar change in stock price, where H stands for the hedge ratio. Thus, if H equals .6, the 750 options are equivalent to .6 3 750 5 450 shares in terms of the response of their market value to FinCorp stock price movements. The first portfolio has less dollar sensitivity to stock price change because the 450 share-equivalents of the options plus the 200 shares actually held are less than the 800 shares held in the second portfolio.
This is not to say, however, that the first portfolio is less sensitive to the stock’s rate of return. As we noted in discussing option elasticities, the first portfolio may have lower total value than the second, so despite its lower sensitivity in terms of total market value, it might have greater rate of return sensitivity. Because a call option has a lower market value than the stock, its price changes more than proportionally with stock price changes, even though its hedge ratio is less than 1.0.
Example 21.6 Hedge Ratios
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748
Portfolio Insurance In Chapter 20, we showed that protective put strate-
gies offer a sort of insurance policy on an asset. The
protective put has proven to be extremely popular with
investors. Even if the asset price falls, the put conveys
the right to sell the asset for the exercise price, which
is a way to lock in a minimum portfolio value. With an
at-the-money put ( X 5 S 0 ), the maximum loss that can be realized is the cost of the put. The asset can be sold for X, which equals its original value, so even if the asset price falls, the investor’s net loss over the period is just the cost of the put. If the asset value
increases, however, upside potential is unlimited. Figure 21.10 graphs the profit or loss
on a protective put position as a function of the change in the value of the underlying
asset, P.
What is the elasticity of a put option currently selling for $4 with exercise price $120 and hedge ratio 2 .4 if the stock price is currently $122?
CONCEPT CHECK 21.8
eXcel APPLICATIONS: Black-Scholes Option Valuation
T he spreadsheet below can be used to determine option values using the Black-Scholes model. The inputs are
the stock price, standard deviation, expiration of the option, exercise price, risk-free rate, and dividend yield. The call option is valued using Equation 21.1 and the put is valued using Equation 21.3. For both calls and puts, the dividend-adjusted Black-Scholes formula substitutes Se 2 d T for S, as outlined on page 744. The model also calculates the intrinsic and time value for both puts and calls.
Further, the model presents sensitivity analysis using the one-way data table. The first workbook presents the
analysis of calls while the second workbook presents simi- lar analysis for puts. You can find these spreadsheets at the Online Learning Center at www.mhhe.com/bkm .
Excel Questions 1. Find the value of the call and put options using the param-
eters given in this box but changing the standard deviation to .25. What happens to the value of each option?
2. What is implied volatility if the call option is selling for $9?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Chapter 21- Black-Scholes Option Pricing Call Valuation & Call Time Premiums
Standard deviation (σ) Variance (annual, σ2) Time to expiration (years, T) Risk-free rate (annual, r) Current stock price (S0) Exercise price (X) Dividend yield (annual, δ)
d1 d2 N(d1) N(d2) Black-Scholes call value Black-Scholes put value
Intrinsic value of call Time value of call
Intrinsic value of put Time value of put
0.27830 0.07745
0.50 6.00%
$100.00 $105.00
0.00%
0.0029095 −0.193878
0.50116 0.42314
$6.99992 $8.89670
$0.00000 6.99992
$5.00000 3.89670
A B C D
Standard Deviation
Call Option Value
0.15 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.35 0.38 0.40 0.43 0.45 0.48 0.50
E
7.000 3.388 4.089 4.792 5.497 6.202 6.907 7.612 8.317 9.022 9.726
10.429 11.132 11.834 12.536 13.236
F G
Standard Deviation
Call Time Value
H I J
Stock Price
Call Option Value
$60 $65 $70 $75 $80 $85 $90 $95
$100 $105 $110 $115 $120 $125 $130
$135.00
K
7.000 0.017 0.061 0.179 0.440 0.935 1.763 3.014 4.750 7.000 9.754
12.974 16.602 20.572 24.817 29.275 33.893
L M
Stock Price
Call Time Value
$60 $65 $70 $75 $80 $85 $90 $95
$100 $105 $110 $115 $120 $125 $130 $135
N
7.000 0.017 0.061 0.179 0.440 0.935 1.763 3.014 4.750 7.000 9.754 7.974 6.602 5.572 4.817 4.275 3.893
0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500
7.000 3.388 4.089 4.792 5.497 6.202 6.907 7.612 8.317 9.022 9.726
10.429 11.132 11.834 12.536 13.236
LEGEND: Enter data
Value calculated See comment
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C H A P T E R 2 1 Option Valuation 749
While the protective put is
a simple and convenient way
to achieve portfolio insurance , that is, to limit the worst-case
portfolio rate of return, there are
practical difficulties in trying
to insure a portfolio of stocks.
First, unless the investor’s port-
folio corresponds to a standard
market index for which puts are
traded, a put option on the port-
folio will not be available for
purchase. And if index puts are
used to protect a non-indexed
portfolio, tracking error can
result. For example, if the port-
folio falls in value while the
market index rises, the put will
fail to provide the intended pro-
tection. Moreover, the maturi-
ties of traded options may not match the investor’s horizon. Therefore, rather than using
option strategies, investors may use trading strategies that mimic the payoff to a protective
put option.
Here is the general idea. Even if a put option on the desired portfolio does not exist, a
theoretical option-pricing model (such as the Black-Scholes model) can be used to deter-
mine how that option’s price would respond to the portfolio’s value if it did trade. For
example, if stock prices were to fall, the put option would increase in value. The option
model could quantify this relationship. The net exposure of the (hypothetical) protective
put portfolio to swings in stock prices is the sum of the exposures of the two components
of the portfolio, the stock and the put. The net exposure of the portfolio equals the equity
exposure less the (offsetting) put option exposure.
We can create “synthetic” protective put positions by holding a quantity of stocks with
the same net exposure to market swings as the hypothetical protective put position. The
key to this strategy is the option delta, or hedge ratio, that is, the change in the price of the
protective put option per change in the value of the underlying stock portfolio.
Change in Value of Protected Position
Change in Value of Underlying Asset
2P
0 0
Cost of Put
Figure 21.10 Profit on a protective put strategy
Suppose a portfolio is currently valued at $100 million. An at-the-money put option on the portfolio might have a hedge ratio or delta of 2 .6, meaning the option’s value swings $.60 for every dollar change in portfolio value, but in an opposite direction. Sup- pose the stock portfolio falls in value by 2%. The profit on a hypothetical protective put position (if the put existed) would be as follows (in millions of dollars):
Loss on stocks: 2% of $100 5 $2.00
Gain on put: .6 3 $2.00 5 1.20
Net loss 5 $ .80
Example 21.7 Synthetic Protective Put Options
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750 P A R T V I Options, Futures, and Other Derivatives
The challenge with this procedure is that deltas constantly change. Figure 21.11
shows that as the stock price falls, the magnitude of the appropriate hedge ratio increases.
Therefore, market declines require extra hedging, that is, additional conversion of equity
into cash. This constant updating of the hedge ratio is called dynamic hedging (alterna- tively, delta hedging).
Dynamic hedging is one reason portfolio insurance has been said to contribute to mar-
ket volatility. Market declines trigger additional sales of stock as portfolio insurers strive to
increase their hedging. These additional sales
are seen as reinforcing or exaggerating mar-
ket downturns.
In practice, portfolio insurers often do
not actually buy or sell stocks directly when
they update their hedge positions. Instead,
they minimize trading costs by buying or
selling stock index futures as a substitute
for sale of the stocks themselves. As you
will see in the next chapter, stock prices and
index futures prices usually are very tightly
linked by cross-market arbitrageurs so that
futures transactions can be used as reliable
proxies for stock transactions. Instead of
selling equities based on the put option’s
delta, insurers will sell an equivalent number
of futures contracts. 17
Several portfolio insurers suffered great set-
backs during the market crash of October 19,
1987, when the market suffered an unprece-
dented 1-day loss of about 20%. A description
We create the synthetic option position by selling a proportion of shares equal to the put option’s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free T-bills. The rationale is that the hypothetical put option would have offset 60% of any change in the stock portfolio’s value, so one must reduce portfolio risk directly by sell- ing 60% of the equity and putting the proceeds into a risk-free asset. Total return on a synthetic protective put position with $60 million in risk-free investments such as T-bills and $40 million in equity is
Loss on stocks: 2% of $40 5 $.80 1 Loss on bills: 5 0 Net loss 5 $.80
The synthetic and actual protective put positions have equal returns. We conclude that if you sell a proportion of shares equal to the put option’s delta and place the proceeds in cash equivalents, your exposure to the stock market will equal that of the desired protective put position.
0
Value of a Put (P)
S0
Low Slope 5 Low Hedge Ratio
Higher Slope 5 High Hedge Ratio
Figure 21.11 Hedge ratios change as the stock price fluctuates
17 Notice, however, that the use of index futures reintroduces the problem of tracking error between the portfolio
and the market index.
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C H A P T E R 2 1 Option Valuation 751
of what happened then should let you appreciate the complexities of applying a seemingly
straightforward hedging concept.
1. Market volatility at the crash was much greater than ever encountered before. Put
option deltas based on historical experience were too low; insurers underhedged,
held too much equity, and suffered excessive losses.
2. Prices moved so fast that insurers could not keep up with the necessary rebalancing.
They were “chasing deltas” that kept getting away from them. The futures market
also saw a “gap” opening, where the opening price was nearly 10% below the previ-
ous day’s close. The price dropped before insurers could update their hedge ratios.
3. Execution problems were severe. First, current market prices were unavailable, with
trade execution and the price quotation system hours behind, which made computa-
tion of correct hedge ratios impossible. Moreover, trading in stocks and stock futures
ceased during some periods. The continuous rebalancing capability that is essential
for a viable insurance program vanished during the precipitous market collapse.
4. Futures prices traded at steep discounts to their proper levels compared to reported
stock prices, thereby making the sale of futures (as a proxy for equity sales) seem
expensive. Although you will see in the next chapter that stock index futures prices
normally exceed the value of the stock index, Figure 21.12 shows that on October 19,
futures sold far below the stock index level. When some insurers gambled that the
futures price would recover to its usual premium over the stock index, and chose to
defer sales, they remained underhedged. As the market fell farther, their portfolios
experienced substantial losses.
Although most observers at the time believed that the portfolio insurance industry
would never recover from the market crash, delta hedging is still alive and well on Wall
Street. Dynamic hedges are widely used by large firms to hedge potential losses from
options positions. For example, the nearby box notes that when Microsoft ended its
employee stock option program and J. P. Morgan purchased many already-issued options
0
10
210
220
230
240 10 11 12 1 2 3 4 10 11 12 1 2 3 4
October 19 October 20
Figure 21.12 S&P 500 cash-to-futures spread in points at 15-minute intervals Note: Trading in futures contracts halted between 12:15 and 1:05.
Source: The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1987 Dow Jones & Company, Inc. All rights reserved worldwide.
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752
of Microsoft employees, it was widely expected that Morgan would protect its options
position by selling shares in Microsoft in accord with a delta hedging strategy.
Option Pricing and the Crisis of 2008–2009 Merton
18 shows how option pricing models can provide insight into the financial crisis of
2008–2009. The key to understanding his argument is to remember that when banks lend
to or buy the debt of firms with limited liability, they implicitly write a put option to the
borrower (see Chapter 20, Section 20.5). If the borrower has sufficient assets to pay off
the loan when it comes due, it will do so, and the lender will be fully repaid. But if the
borrower has insufficient assets, it can declare bankruptcy and discharge its obligations by
transferring ownership of the firm to its creditors. The borrower’s ability to satisfy the loan
by transferring ownership is equivalent to the right to “sell” itself to the creditor for the
face value of the loan. This arrangement is therefore just like a put option on the firm with
exercise price equal to the stipulated loan repayment.
Consider the payoff to the lender at loan maturity (time T ) as a function of the value of the borrowing firm, V T , when the loan, with face value L, comes due. If V T $ L, the lender is paid off in full. But if V T , L, the lender gets the firm, which is worth less than the promised payment L.
We can write the payoff in a way that emphasizes the implicit put option:
Payoff 5 bL VT
5 L 2 b0 L 2 VT
if VT $ L
if VT , L (21.4)
Equation 21.4 shows that the payoff on the loan equals L (when the firm has sufficient assets to pay off the debt), minus the payoff of a put option on the value of the firm ( V T ) with an exercise price of L. Therefore, we may view risky lending as a combination of safe lending, with a guaranteed payoff of L, combined with a short position in a put option on the borrower.
18
This material is based on a lecture given by Robert Merton at MIT in March 2009. You can find the lecture
online at http://mitworld.mit.edu/video/659 .
J. P. Morgan Rolls Dice on Microsoft Options
Microsoft, in a shift that could be copied throughout the technology business, said yesterday that it plans to stop issuing stock options to its employees, and instead will pro- vide them with restricted stock.
Though details of the plan still aren’t clear, J. P. Morgan effectively plans to buy the options from Microsoft employ- ees who opt for restricted stock instead. Employee stock options are granted as a form of compensation and allow employees the right to exchange the options for shares of company stock.
The price offered to employees for the options pre- sumably will be lower than the current value, giving J. P. Morgan a chance to make a profit on the deal. Rather than holding the options, and thus betting Microsoft’s stock will rise, people familiar with the bank’s strategy say J. P. Morgan probably will match each option it buys from the company’s employees with a separate trade in the
stock market that both hedges the bet and gives itself a margin of profit.
For Wall Street’s so-called rocket scientists who do compli- cated financial transactions such as this one, the strategy behind J. P. Morgan’s deal with Microsoft isn’t particularly unique or sophisticated. They add that the bank has several ways to deal with the millions of Microsoft options that could come its way.
The bank, for instance, could hedge the options by short- ing, or betting against, Microsoft stock. Microsoft has the largest market capitalization of any stock in the market, and its shares are among the most liquid, meaning it would be easy to hedge the risk of holding those options. J. P. Morgan also could sell the options to investors, much as they would do with a syndicated loan, thereby spreading the risk.
Source: Jathon Sapsford and Ken Brown, The Wall Street Journal, July 9, 2003. Reprinted with permission. © 2003 Dow Jones & Com- pany, Inc. All rights reserved.
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 2 1 Option Valuation 753
When firms sell credit default swaps
(see Chapter 14, Section 14.5), the
implicit put option is even clearer. Here,
the CDS seller agrees to make up any
losses due to the insolvency of a bond
issuer. If the issuer goes bankrupt, leav-
ing assets of only V T for the creditors, the CDS seller is obligated to make up the
difference, L 2 V T . This is in essence a pure put option.
Now think about the exposure of these
implicit put writers to changes in the
financial health of the underlying firm.
The value of a put option on V T appears in Figure 21.13 . When the firm is finan-
cially strong (i.e., V is far greater than L ), the slope of the curve is nearly zero,
implying that there is little exposure of the
implicit put writer (either the bank or the
CDS writer) to the value of the borrowing firm. For example, when firm value is 1.75 times
the value of the debt, the dashed line drawn tangent to the put value curve has a slope of only
2 .040. But if there is a big shock to the economy, and firm value falls, not only does the
value of the implicit put rise, but its slope is now steeper, implying that exposure to further
shocks is now far greater. When firm value is only 75% of the value of the loan, the slope of
the line tangent to the put value valuation curve is far steeper, 2 .644. You can see how as you
get closer to the edge of the cliff, it gets easier and easier to slide right off.
We often hear people say that a shock to asset values of the magnitude of the financial
crisis was a 10-sigma event, by which they mean that such an event was so extreme that
it would be 10 standard deviations away from an expected outcome, making it virtually
inconceivable. But this analysis shows that standard deviation may be a moving target,
increasing dramatically as the firm weakens. As the economy falters and put options go
further into the money, their sensitivity to further shocks increases, increasing the risk
that even worse losses may be around the corner. The built-in instability of risk exposures
makes a scenario like the crisis more plausible and should give us pause when we discount
an extreme scenario as “almost impossible.”
Option Pricing and Portfolio Theory We’ve just seen that the option pricing model predicts that security risk characteristics can
be unstable. For example, as the firm weakens, the risk of its debt can quickly accelerate.
So, too, can equity risk change dramatically as the firm’s financial condition deteriorates.
We know from the last chapter (Section 20.5) that equity in a levered firm is like a call
option on the value of the firm. If firm value exceeds the value of the firm’s maturing debt,
the firm can choose to pay off the debt, retaining the difference between firm value and the
face value of its debt. If not, the firm can default on the loan, turning the firm over to its
creditors, and the equity holders get nothing. In this sense, equity is a call option, and the
firm’s total value is the underlying asset.
In Section 21.5, we saw that the elasticity of an option measures the sensitivity of its rate of return to the rate of return on the underlying asset. For example, if a call option’s elasticity
is five, its rate of return will swing five times as widely as the rate of return on the underlying
asset. This would imply that both the option’s beta and its standard deviation are five times the
210
0
10
20
30
40
50
50 75 100 125 150 175 200 225 250
Firm Value as Percent of Face Value of Debt
V a lu
e o
f P u
t a s
% o
f Fa
ce V
a lu
e o
f D
e b
t
Figure 21.13 Value of implicit put option on a loan guarantee as a percentage of the face value of debt (Debt maturity 5 1 year; Standard deviation of value of firm 5 40%; Risk free rate 5 6%)
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754 P A R T V I Options, Futures, and Other Derivatives
beta and standard deviation of
the underlying asset.
Therefore, when compil-
ing the “input list” for creating
an efficient portfolio, we may
wish to think of equity as an
implicit call option and com-
pute its elasticity with respect
to the total value of the firm.
For example, if the covariance
of the firm’s assets with other securities is stable, then we
can use elasticity to find the
covariance of the firm’s equity with those securities. This will
allow us to calculate beta and
standard deviation.
Unfortunately, elasticity can
itself be a moving target. As
the firm gets weaker, its elas-
ticity will increase, potentially
very quickly. Figure 21.14 uses
the Black-Scholes model to plot call option elasticity as a function of the value of the
underlying stock. Notice that as the option goes out of the money (the stock price falls
below 100), elasticity increases rapidly and without limit. Similarly, as the firm gets closer
to insolvency (the value of firm assets falls below the face value of debt), equity elasticity
shoots up, and even small changes in financial condition can lead to major changes in risk.
Elasticity is far more stable (and closer to 1) when the firm is healthy, i.e., the implicit call
option is deep in the money. Similarly, equity risk characteristics will be far more stable for
healthy firms than for precarious ones.
Hedging Bets on Mispriced Options Suppose you believe that the standard deviation of IBM stock returns will be 35% over the
next few weeks, but IBM put options are selling at a price consistent with a volatility of
33%. Because the put’s implied volatility is less than your forecast of the stock volatility,
you believe the option is underpriced. Using your assessment of volatility in an option-
pricing model like the Black-Scholes formula, you would estimate that the fair price for
the puts exceeds the actual price.
Does this mean that you ought to buy put options? Perhaps it does, but by doing so,
you risk losses if IBM stock performs well, even if you are correct about the volatility. You would like to separate your bet on volatility from the “attached” bet inherent in purchasing
a put that IBM’s stock price will fall. In other words, you would like to speculate on the
option mispricing by purchasing the put option, but hedge the resulting exposure to the
performance of IBM stock.
The option delta can be interpreted as a hedge ratio that can be used for this purpose. The delta was defined as
Delta 5 Change in value of option
Change in value of stock (21.5)
Therefore, delta is the slope of the option-pricing curve.
35
25
30
20
15E la
st ic
it y
5
0 25 50 75 100 125
Stock Price
175150 200
10
Figure 21.14 Call option elasticity as a function of stock price (Parameters: s 5 .25; T 5 .5, r 5 .06; X 5 100)
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C H A P T E R 2 1 Option Valuation 755
This ratio tells us precisely how many shares of stock we must hold to offset our
exposure to IBM. For example, if the delta is 2 .6, then the put will fall by $.60 in value
for every one-point increase in IBM stock, and we need to hold .6 share of stock to hedge
each put. If we purchase 10 option contracts, each for 100 shares, we would need to buy
600 shares of stock. If the stock price rises by $1, each put option will decrease in value by
$.60, resulting in a loss of $600. However, the loss on the puts will be offset by a gain on
the stock holdings of $1 per share 3 600 shares.
To see how the profits on this strategy might develop, let’s use the following example.
Suppose option expiration T is 60 days; put price P is $4.495; exercise price X is $90; stock price S is $90; and the risk-free rate r is 4%. We assume that the stock will not pay a dividend in the next 60 days. Given these data, the implied volatility on the option is 33%, as we posited. However, you believe the true volatility is 35%, implying that the fair put price is $4.785. Therefore, if the market assessment of volatility is revised to the value you believe is correct, your profit will be $.29 per put purchased.
Recall that the hedge ratio, or delta, of a put option equals N ( d 1 ) 2 1, where N (•) is the cumulative normal distribution function and
ln(S/X) 1 (r 1 s2 / 2)T
s√T d1 5
Using your estimate of s 5 .35, you find that the hedge ratio N ( d 1 ) 2 1 5 2 .453. Suppose, therefore, that you purchase 10 option contracts (1,000 puts) and purchase
453 shares of stock. Once the market “catches up” to your presumably better volatil- ity estimate, the put options purchased will increase in value. If the market assessment of volatility changes as soon as you purchase the options, your profits should equal 1,000 3 $.29 5 $290. The option price will be affected as well by any change in the stock price, but this part of your exposure will be eliminated if the hedge ratio is chosen properly. Your profit should be based solely on the effect of the change in the implied volatility of the put, with the impact of the stock price hedged away.
Table 21.3 illustrates your profits as a function of the stock price assuming that the put price changes to reflect your estimate of volatility. Panel B shows that the put option alone can provide profits or losses depending on whether the stock price falls or rises. We see in panel C, however, that each hedged put option provides profits nearly equal to the original mispricing, regardless of the change in the stock price.
Example 21.8 Speculating on Mispriced Options
Notice in Example 21.8 that the profit is not exactly independent of the stock price. This
is because as the stock price changes, so do the deltas used to calculate the hedge ratio. The
hedge ratio in principle would need to be continually adjusted as deltas evolve. The sensi-
tivity of the delta to the stock price is called the gamma of the option. Option gammas are analogous to bond convexity. In both cases, the curvature of the value function means that
hedge ratios or durations change with market conditions, making rebalancing a necessary
part of hedging strategies.
Suppose you bet on volatility by purchasing calls instead of puts. How would you hedge your exposure to stock-price fluctuations? What is the hedge ratio?
CONCEPT CHECK 21.9
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756 P A R T V I Options, Futures, and Other Derivatives
A variant of the strategy in Example 21.8 involves cross-option speculation. Suppose
you observe a 45-day expiration call option on IBM with strike price 95 selling at a price
consistent with a volatility of s 5 33% while another 45-day call with strike price 90 has
an implied volatility of only 27%. Because the underlying asset and expiration date are
identical, you conclude that the call with the higher implied volatility is relatively over-
priced. To exploit the mispricing, you might buy the cheap calls (with strike price 90 and
implied volatility of 27%) and write the expensive calls (with strike price 95 and implied
volatility 33%). If the risk-free rate is 4% and IBM is selling at $90 per share, the calls
purchased will be priced at $3.6202 and the calls written will be priced at $2.3735.
Despite the fact that you are long one call and short another, your exposure to IBM
stock-price uncertainty will not be hedged using this strategy. This is because calls with
different strike prices have different sensitivities to the price of the underlying asset. The
lower-strike-price call has a higher delta and therefore greater exposure to the price of
IBM. If you take an equal number of positions in these two options, you will inadvertently
establish a bullish position in IBM, as the calls you purchase have higher deltas than the
calls you write. In fact, you may recall from Chapter 20 that this portfolio (long call with
low exercise price and short call with high exercise price) is called a bullish spread. To establish a hedged position, we can use the hedge ratio approach as follows.
Consider the 95-strike-price options you write as the asset that hedges your exposure to the
90-strike-price options you purchase. Then the hedge ratio is
H 5 Change in value of 90-strike-price call for $1 change in IBM
Change in value of 95-strike-price call for $1 change in IBM
5 Delta of 90-strike-price call
Delta of 95-strike-price call . 1
You need to write more than one call with the higher strike price to hedge the purchase of each call with the lower strike price. Because the prices of higher-strike-price calls are less
sensitive to IBM prices, more of them are required to offset the exposure.
Table 21.3
Profit on hedged put portfolio
A. Cost to establish hedged position
1,000 put options @ $4.495/option $ 4,495 453 shares @ $90/share 40,770 TOTAL outlay $45,265
B. Value of put option as a function of the stock price at implied volatility of 35%
Stock Price: 89 90 91 Put price $ 5.254 $ 4.785 $ 4.347 Profit (loss) on each put 0.759 0.290 (0.148)
C. Value of and profit on hedged put portfolio
Stock Price: 89 90 91 Value of 1,000 put options $ 5,254 $ 4,785 $ 4,347 Value of 453 shares 40,317 40,770 41,223
TOTAL $45,571 $45,555 $45,570 Profit (5 Value 2 Cost from panel A) 306 290 305
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C H A P T E R 2 1 Option Valuation 757
Suppose the true annual volatility of the stock is midway between the two implied vola-
tilities, so s 5 30%. We know that the delta of a call option is N ( d 1 ). Therefore, the deltas of the two options and the hedge ratio are computed as follows:
Option with strike price 90:
d1 5 ln(90/90) 1 (.04 1 .302/2) 3 45/365
.30"45/365 5 .0995 N(d1) 5 .5396
Option with strike price 95:
d1 5 ln(90/95) 1 (.04 1 .302/2) 3 45/365
.30"45/365 5 2.4138 N(d1) 5 .3395
Hedge ratio:
.5396
.3395 5 1.589
Therefore, for every 1,000 call options purchased with strike price 90, we need to write
1,589 call options with strike price 95. Following this strategy enables us to bet on the rela-
tive mispricing of the two options without taking a position on IBM. Panel A of Table 21.4
shows that the position will result in a cash inflow of $151.30. The premium income on the
calls written exceeds the cost of the calls purchased.
When you establish a position in stocks and options that is hedged with respect to fluc-
tuations in the price of the underlying asset, your portfolio is said to be delta neutral , meaning that the portfolio has no tendency to either increase or decrease in value when the
stock price fluctuates.
Let’s check that our options position is in fact delta neutral. Suppose that the implied
volatilities of the two options come back into alignment just after you establish your
Table 21.4
Profits on delta- neutral options portfolio
A. Cost flow when portfolio is established
Purchase 1,000 calls (X 5 90) @ $3.6202 (option priced at implied volatility of 27%) $ 3,620.20 cash outflow
Write 1,589 calls (X 5 95) @ $2.3735 (option priced at implied volatility of 33%) 3,771.50 cash inflow TOTAL $ 151.30 net cash inflow
B. Option prices at implied volatility of 30%
Stock Price: 89 90 91
90-strike-price calls $3.478 $3.997 $4.557 95-strike-price calls 1.703 2.023 2.382
C. Value of portfolio after implied volatilities converge to 30%
Stock Price: 89 90 91 Value of 1,000 calls held $3,478 $3,997 $4,557
2 Value of 1,589 calls written 2,705 3,214 3,785
TOTAL $ 773 $ 782 $ 772
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758 P A R T V I Options, Futures, and Other Derivatives
position, so that both options are priced at implied volatilities of 30%. You expect to
profit from the increase in the value of the call purchased as well as from the decrease
in the value of the call written. The option prices at 30% volatility are given in panel
B of Table 21.4 and the values of your position for various stock prices are presented
in panel C. Although the profit or loss on each option is affected by the stock price,
the value of the delta-neutral option portfolio is positive and essentially independent
of the price of IBM. Moreover, we saw in panel A that the portfolio would have been
established without ever requiring a cash outlay. You would have cash inflows both
when you establish the portfolio and when you liquidate it after the implied volatilities converge to 30%.
This unusual profit opportunity arises because you have identified prices out of align-
ment. Such opportunities could not arise if prices were at equilibrium levels. By exploiting
the pricing discrepancy using a delta-neutral strategy, you should earn profits regardless of
the price movement in IBM stock.
Delta-neutral hedging strategies are also subject to practical problems, the most impor-
tant of which is the difficulty in assessing the proper volatility for the coming period. If the
volatility estimate is incorrect, so will be the deltas, and the overall position will not truly
be hedged. Moreover, option or option-plus-stock positions generally will not be neutral
with respect to changes in volatility. For example, a put option hedged by a stock might be
delta neutral, but it is not volatility neutral. Changes in the market assessments of volatility
will affect the option price even if the stock price is unchanged.
These problems can be serious, because volatility estimates are never fully reliable.
First, volatility cannot be observed directly and must be estimated from past data which
imparts measurement error to the forecast. Second, we’ve seen that both historical and
implied volatilities fluctuate over time. Therefore, we are always shooting at a moving
target. Although delta-neutral positions are hedged against changes in the price of the
underlying asset, they still are subject to volatility risk, the risk incurred from unpre- dictable changes in volatility. The sensitivity of an option price to changes in volatility
is called the option’s vega . Thus, although delta-neutral option hedges might eliminate exposure to risk from fluctuations in the value of the underlying asset, they do not elimi-
nate volatility risk.
21.6 Empirical Evidence on Option Pricing
The Black-Scholes option-pricing model has been subject to an enormous number of
empirical tests. For the most part, the results of the studies have been positive in that the
Black-Scholes model generates option values fairly close to the actual prices at which
options trade. At the same time, some regular empirical failures of the model have been
noted.
The biggest problem concerns volatility. If the model were accurate, the implied volatil-
ity of all options on a particular stock with the same expiration date would be equal—after
all, the underlying asset and expiration date are the same for each option, so the volatility
inferred from each also ought to be the same. But in fact, when one actually plots implied
volatility as a function of exercise price, the typical results appear as in Figure 21.15 , which
treats S&P 500 index options as the underlying asset. Implied volatility steadily falls as the
exercise price rises. Clearly, the Black-Scholes model is missing something.
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C H A P T E R 2 1 Option Valuation 759
Rubinstein 19
suggests that the prob-
lem with the model has to do with fears of
a market crash like that of October 1987.
The idea is that deep out-of-the-money puts
would be nearly worthless if stock prices
evolve smoothly, because the probabil-
ity of the stock falling by a large amount
(and the put option thereby moving into
the money) in a short time would be very
small. But a possibility of a sudden large
downward jump that could move the puts
into the money, as in a market crash, would
impart greater value to these options. Thus,
the market might price these options as
though there is a bigger chance of a large
drop in the stock price than would be sug-
gested by the Black-Scholes assumptions.
The result of the higher option price is a
greater implied volatility derived from the
Black-Scholes model.
Interestingly, Rubinstein points out
that prior to the 1987 market crash, plots
of implied volatility like the one in Figure 21.15 were relatively flat, consistent with the
notion that the market was then less attuned to fears of a crash. However, postcrash plots
have been consistently downward sloping, exhibiting a shape often called the option smirk. When we use option-pricing models that allow for more general stock price distributions,
including crash risk and random changes in volatility, they generate downward-sloping
implied volatility curves similar to the one observed in Figure 21.15 . 20
0.84 0.89 0.94 0.99 1.04 1.09
Im p
li e d
V o
la ti
li ty
( %
)
Ratio of Exercise Price to Current Value of Index
25
20
15
10
5
0
Figure 21.15 Implied volatility of the S&P 500 index as a function of exercise price
Source: Mark Rubinstein, “Implied Binomial Trees,” Journal of Finance (July 1994), pp. 771–818.
1. Option values may be viewed as the sum of intrinsic value plus time or “volatility” value. The volatility value is the right to choose not to exercise if the stock price moves against the holder.
Thus the option holder cannot lose more than the cost of the option regardless of stock price
performance.
2. Call options are more valuable when the exercise price is lower, when the stock price is higher, when the interest rate is higher, when the time to expiration is greater, when the stock’s volatility
is greater, and when dividends are lower.
3. Call options must sell for at least the stock price less the present value of the exercise price and dividends to be paid before expiration. This implies that a call option on a non-dividend-paying
stock may be sold for more than the proceeds from immediate exercise. Thus European calls are
worth as much as American calls on stocks that pay no dividends, because the right to exercise the
American call early has no value.
SUMMARY
19
Mark Rubinstein, “Implied Binomial Trees,” Journal of Finance 49 (July 1994), pp. 771–818.
20
For an extensive discussion of these more general models, see R. L. McDonald, Derivatives Markets, 3rd ed. (Boston: Pearson Education [Addison-Wesley], 2013).
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760 P A R T V I Options, Futures, and Other Derivatives
4. Options may be priced relative to the underlying stock price using a simple two-period, two- state pricing model. As the number of periods increases, the binomial model can approximate
more realistic stock price distributions. The Black-Scholes formula may be seen as a limiting
case of the binomial option model, as the holding period is divided into progressively smaller
subperiods when the interest rate and stock volatility are constant.
5. The Black-Scholes formula applies to options on stocks that pay no dividends. Dividend adjustments may be adequate to price European calls on dividend-paying stocks, but the proper
treatment of American calls on dividend-paying stocks requires more complex formulas.
6. Put options may be exercised early, whether the stock pays dividends or not. Therefore, American puts generally are worth more than European puts.
7. European put values can be derived from the call value and the put-call parity relationship. This technique cannot be applied to American puts for which early exercise is a possibility.
8. The implied volatility of an option is the standard deviation of stock returns consistent with an option’s market price. It can be backed out of an option-pricing model by finding the stock
volatility that makes the option’s value equal to its observed price.
9. The hedge ratio is the number of shares of stock required to hedge the price risk involved in writing one option. Hedge ratios are near zero for deep out-of-the-money call options and
approach 1.0 for deep in-the-money calls.
10. Although their hedge ratios are less than 1.0, call options have elasticities greater than 1.0. The rate of return on a call (as opposed to the dollar return) responds more than one-for-one with
stock price movements.
11. Portfolio insurance can be obtained by purchasing a protective put option on an equity position. When the appropriate put is not traded, portfolio insurance entails a dynamic hedge strategy
where a fraction of the equity portfolio equal to the desired put option’s delta is sold and placed
in risk-free securities.
12. The option delta is used to determine the hedge ratio for options positions. Delta-neutral portfo- lios are independent of price changes in the underlying asset. Even delta-neutral option portfo-
lios are subject to volatility risk, however.
13. Empirically, implied volatilities derived from the Black-Scholes formula tend to be lower on options with higher exercise prices. This may be evidence that the option prices reflect the pos-
sibility of a sudden dramatic decline in stock prices. Such “crashes” are inconsistent with the
Black-Scholes assumptions.
Related Web sites for this chapter are available at www. mhhe.com/bkm
intrinsic value
time value
binomial model
Black-Scholes pricing formula
implied volatility
pseudo-American call option
value
hedge ratio
delta
option elasticity
portfolio insurance
dynamic hedging
gamma
delta neutral
vega
KEY TERMS
Binomial model: u 5 exp(s"Dt); d 5 exp(2s"Dt); p 5 exp(rDt) 2 d u 2 d
Put-call parity: P 5 C 1 PV(X) 2 S0 1 PV(Dividends)
Black-Scholes formula (no dividend case): SN(d1) 2 Xe 2rTN(d2)
where d1 5 ln(S/X) 1 (r 1 ½ s2)T
s"T ; d2 5 d1 2 s"T Delta (or hedge ratio): H 5
Change in option value
Change in stock value
KEY EQUATIONS
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C H A P T E R 2 1 Option Valuation 761
1. We showed in the text that the value of a call option increases with the volatility of the stock. Is
this also true of put option values? Use the put-call parity theorem as well as a numerical example
to prove your answer.
2. W ould you expect a $1 increase in a call option’s exercise price to lead to a decrease in the
option’s value of more or less than $1?
3. Is a put option on a high-beta stock worth more than one on a low-beta stock? The stocks have
identical firm-specific risk.
4. All else equal, is a call option on a stock with a lot of firm-specific risk worth more than one on a
stock with little firm-specific risk? The betas of the two stocks are equal.
5. All else equal, will a call option with a high exercise price have a higher or lower hedge ratio than
one with a low exercise price?
6. I n each of the following questions, you are asked to compare two options with parameters as
given. The risk-free interest rate for all cases should be assumed to be 6%. Assume the stocks on which these options are written pay no dividends.
a. Put T X s Price of Option
A .5 50 .20 $10
B .5 50 .25 $10
Which put option is written on the stock with the lower price?
i. A.
ii. B.
iii. Not enough information.
b. Put T X s Price of Option
A .5 50 .2 $10
B .5 50 .2 $12
Which put option must be written on the stock with the lower price?
i. A.
ii. B.
iii. Not enough information.
c. Call S X s Price of Option
A 50 50 .20 $12
B 55 50 .20 $10
Which call option must have the lower time to expiration?
i. A.
ii. B.
iii. Not enough information.
d. Call T X S Price of Option
A .5 50 55 $10
B .5 50 55 $12
Which call option is written on the stock with higher volatility?
i. A.
ii. B.
iii. Not enough information.
e. Call T X S Price of Option
A .5 50 55 $10
B .5 50 55 $ 7
Basic
Intermediate
PROBLEM SETS
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762 P A R T V I Options, Futures, and Other Derivatives
Which call option is written on the stock with higher volatility?
i. A.
ii. B.
iii. Not enough information.
7. Reconsider the determination of the hedge ratio in the two-state model (see page 730), where
we showed that one-third share of stock would hedge one option. What would be the hedge ratio
for the following exercise prices: 120, 110, 100, 90? What do you conclude about the hedge
ratio as the option becomes progressively more in the money?
8. Show that Black-Scholes call option hedge ratios also increase as the stock price increases.
Consider a 1-year option with exercise price $50, on a stock with annual standard deviation
20%. The T-bill rate is 3% per year. Find N ( d 1 ) for stock prices $45, $50, and $55.
9. We will derive a two-state put option value in this problem. Data: S 0 5 100; X 5 110; 1 1 r 5 1.10. The two possibilities for S T are 130 and 80.
a. Show that the range of S is 50, whereas that of P is 30 across the two states. What is the hedge ratio of the put?
b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio? What is the present value of the portfolio?
c. Given that the stock currently is selling at 100, solve for the value of the put.
10. Calculate the value of a call option on the stock in the previous problem with an exercise price
of 110. Verify that the put-call parity theorem is satisfied by your answers to Problems 9 and 10.
(Do not use continuous compounding to calculate the present value of X in this example because we are using a two-state model here, not a continuous-time Black-Scholes model.)
11. Use the Black-Scholes formula to find the value of a call option on the following stock:
Time to expiration 6 months
Standard deviation 50% per year
Exercise price $50
Stock price $50
Interest rate 3%
12. Find the Black-Scholes value of a put option on the stock in the previous problem with the same
exercise price and expiration as the call option.
13. Recalculate the value of the call option in Problem 11, successively substituting one of the
changes below while keeping the other parameters as in Problem 11:
a. Time to expiration 5 3 months. b. Standard deviation 5 25% per year. c. Exercise price 5 $55. d. Stock price 5 $55. e. Interest rate 5 5%.
Consider each scenario independently. Confirm that the option value changes in accordance
with the prediction of Table 21.1 .
14. A call option with X 5 $50 on a stock currently priced at S 5 $55 is selling for $10. Using a volatility estimate of s 5 .30, you find that N ( d 1 ) 5 .6 and N ( d 2 ) 5 .5. The risk-free interest rate is zero. Is the implied volatility based on the option price more or less than .30? Explain.
15. What would be the Excel formula in Spreadsheet 21.1 for the Black-Scholes value of a straddle
position?
Use the following case in answering Problems 16–21: Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a
slight downturn and suggested delta-hedging the BIC portfolio. As predicted, the U.S. equity markets
did indeed experience a downturn of approximately 4% over a 12-month period. However, portfolio
performance for BIC was disappointing, lagging its peer group by nearly 10%. Washington has been
told to review the options strategy to determine why the hedged portfolio did not perform as expected.
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C H A P T E R 2 1 Option Valuation 763
16. W hich of the following best explains a delta-neutral portfolio? A delta-neutral portfolio is perfectly hedged against:
a. Small price changes in the underlying asset. b. Small price decreases in the underlying asset. c. All price changes in the underlying asset.
17. After discussing the concept of a delta-neutral portfolio, Washington determines that he needs
to further explain the concept of delta. Washington draws the value of an option as a function of
the underlying stock price. Using this diagram, indicate how delta is interpreted. Delta is the:
a. Slope in the option price diagram. b. Curvature of the option price graph. c. Level in the option price diagram.
18. Washington considers a put option that has a delta of 2 0.65. If the price of the underlying asset
decreases by $6, then what is the best estimate of the change in option price?
19. BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option
on Smith & Oates with a strike price of $70 is selling at $3.50 and has a delta of .69. What is the
number of call options necessary to create a delta-neutral hedge?
20. Return to the previous problem. Will the number of call options written for a delta-neutral
hedge increase or decrease if the stock price falls?
21. Which of the following statements regarding the goal of a delta-neutral portfolio is most accu- rate? One example of a delta-neutral portfolio is to combine a:
a. Long position in a stock with a short position in call options so that the value of the portfolio does not change with changes in the value of the stock.
b. Long position in a stock with a short position in a call option so that the value of the portfo- lio changes with changes in the value of the stock.
c. Long position in a stock with a long position in call options so that the value of the portfolio does not change with changes in the value of the stock.
22. Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive
to changes in interest rates than is the rate of return of the underlying bond?
23. If the stock price falls and the call price rises, then what has happened to the call option’s
implied volatility?
24. If the time to expiration falls and the put price rises, then what has happened to the put option’s
implied volatility?
25. According to the Black-Scholes formula, what will be the value of the hedge ratio of a call
option as the stock price becomes infinitely large? Explain briefly.
26. According to the Black-Scholes formula, what will be the value of the hedge ratio of a put
option for a very small exercise price?
27. The hedge ratio of an at-the-money call option on IBM is .4. The hedge ratio of an at-the-money
put option is 2 .6. What is the hedge ratio of an at-the-money straddle position on IBM?
28. Consider a 6-month expiration European call option with exercise price $105. The underlying
stock sells for $100 a share and pays no dividends. The risk-free rate is 5%. What is the implied
volatility of the option if the option currently sells for $8? Use Spreadsheet 21.1 (available at
www.mhhe.com/bkm ; link to Chapter 21 material) to answer this question.
a. Go to the Data tab of the spreadsheet and select Goal Seek from the What-If menu. The dia- log box will ask you for three pieces of information. In that dialog box, you should set cell E6 to value 8 by changing cell B2. In other words, you ask the spreadsheet to find the value of standard deviation (which appears in cell B2) that forces the value of the option (in cell
E6) equal to $8. Then click OK, and you should find that the call is now worth $8, and the
entry for standard deviation has been changed to a level consistent with this value. This is the
call’s implied standard deviation at a price of $8.
b. What happens to implied volatility if the option is selling at $9? Why has implied volatility increased?
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764 P A R T V I Options, Futures, and Other Derivatives
c. What happens to implied volatility if the option price is unchanged at $8, but option expira- tion is lower, say, only 4 months? Why?
d. What happens to implied volatility if the option price is unchanged at $8, but the exercise price is lower, say, only $100? Why?
e. What happens to implied volatility if the option price is unchanged at $8, but the stock price is lower, say, only $98? Why?
29. A collar is established by buying a share of stock for $50, buying a 6-month put option with
exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of
the volatility of the stock, you calculate that for a strike price of $45 and expiration of 6 months,
N ( d 1 ) 5 .60, whereas for the exercise price of $55, N ( d 1 ) 5 .35.
a. What will be the gain or loss on the collar if the stock price increases by $1? b. What happens to the delta of the portfolio if the stock price becomes very large? Very small?
30. These three put options are all written on the same stock. One has a delta of 2 .9, one a delta of
2 .5, and one a delta of 2 .1. Assign deltas to the three puts by filling in this table.
Put X Delta
A 10 B 20 C 30
31. You are very bullish (optimistic) on stock EFG, much more so than the rest of the market. In each question, choose the portfolio strategy that will give you the biggest dollar profit if your
bullish forecast turns out to be correct. Explain your answer.
a. Choice A: $10,000 invested in calls with X 5 50. Choice B: $10,000 invested in EFG stock. b. Choice A: 10 call option contracts (for 100 shares each), with X 5 50. Choice B: 1,000 shares of EFG stock.
32. You would like to be holding a protective put position on the stock of XYZ Co. to lock in a guar-
anteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year
the stock price will increase by 10% or decrease by 10%. The T-bill rate is 5%. Unfortunately,
no put options are traded on XYZ Co.
a. Suppose the desired put option were traded. How much would it cost to purchase? b. What would have been the cost of the protective put portfolio? c. What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that
would be provided by a protective put with X 5 100? Show that the payoff to this portfolio and the cost of establishing the portfolio matches that of the desired protective put.
33. Return to Example 21.1. Use the binomial model to value a 1-year European put option with
exercise price $110 on the stock in that example. Does your solution for the put price satisfy
put-call parity?
34. Suppose that the risk-free interest rate is zero. Would an American put option ever be exercised
early? Explain.
35. Let p ( S, T, X ) denote the value of a European put on a stock selling at S dollars, with time to maturity T, and with exercise price X, and let P ( S, T, X ) be the value of an American put.
a. Evaluate p (0, T, X ). b. Evaluate P (0, T, X ). c. Evaluate p ( S, T, 0). d. Evaluate P ( S, T, 0). e. What does your answer to ( b ) tell you about the possibility that American puts may be exer-
cised early?
36. You are attempting to value a call option with an exercise price of $100 and 1 year to expiration.
The underlying stock pays no dividends, its current price is $100, and you believe it has a 50%
chance of increasing to $120 and a 50% chance of decreasing to $80. The risk-free rate of inter-
est is 10%. Calculate the call option’s value using the two-state stock price model.
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C H A P T E R 2 1 Option Valuation 765
37. Consider an increase in the volatility of the stock in the previous problem. Suppose that if the
stock increases in price, it will increase to $130, and that if it falls, it will fall to $70. Show that
the value of the call option is now higher than the value derived in the previous problem.
38. Calculate the value of a put option with exercise price $100 using the data in Problem 36. Show
that put-call parity is satisfied by your solution.
39. XYZ Corp. will pay a $2 per share dividend in 2 months. Its stock price currently is $60 per
share. A call option on XYZ has an exercise price of $55 and 3-month time to expiration. The
risk-free interest rate is .5% per month, and the stock’s volatility (standard deviation) 5 7% per
month. Find the pseudo-American option value. (Hint: Try defining one “period” as a month,
rather than as a year.)
40. “The beta of a call option on General Electric is greater than the beta of a share of General
Electric.” True or false?
41. “The beta of a call option on the S&P 500 index with an exercise price of 1,330 is greater than
the beta of a call on the index with an exercise price of 1,340.” True or false?
42. What will happen to the hedge ratio of a convertible bond as the stock price becomes very large?
43. Goldman Sachs believes that market volatility will be 20% annually for the next 3 years. Three-
year at-the-money call and put options on the market index sell at an implied volatility of 22%.
What options portfolio can Goldman establish to speculate on its volatility belief without taking
a bullish or bearish position on the market? Using Goldman’s estimate of volatility, 3-year at-
the-money options have N ( d 1 ) 5 .6.
44. You are holding call options on a stock. The stock’s beta is .75, and you are concerned that the
stock market is about to fall. The stock is currently selling for $5 and you hold 1 million options
on the stock (i.e., you hold 10,000 contracts for 100 shares each). The option delta is .8. How
much of the market-index portfolio must you buy or sell to hedge your market exposure?
45. Imagine you are a provider of portfolio insurance. You are establishing a 4-year program. The
portfolio you manage is currently worth $100 million, and you hope to provide a minimum return
of 0%. The equity portfolio has a standard deviation of 25% per year, and T-bills pay 5% per year.
Assume for simplicity that the portfolio pays no dividends (or that all dividends are reinvested).
a. How much should be placed in bills? How much in equity? b. What should the manager do if the stock portfolio falls by 3% on the first day of trading?
46. Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike
price $90 are selling at an implied volatility of 30%. ExxonMobil stock currently is $90 per
share, and the risk-free rate is 4%. If you believe the true volatility of the stock is 32%, how can
you trade on your belief without taking on exposure to the performance of ExxonMobil? How
many shares of stock will you hold for each option contract purchased or sold?
47. Using the data in the previous problem, suppose that 3-month put options with a strike price of
$90 are selling at an implied volatility of 34%. Construct a delta-neutral portfolio comprising
positions in calls and puts that will profit when the option prices come back into alignment.
48. Suppose that JPMorgan Chase sells call options on $1.25 million worth of a stock portfolio
with beta 5 1.5. The option delta is .8. It wishes to hedge out its resultant exposure to a market
advance by buying a market-index portfolio.
a. How many dollars’ worth of the market-index portfolio should it purchase to hedge its position? b. What if it instead uses market index puts to hedge its exposure? Should it buy or sell puts?
Each put option is on 100 units of the index, and the index at current prices represents
$1,000 worth of stock.
49. Suppose you are attempting to value a 1-year maturity option on a stock with volatility (i.e.,
annualized standard deviation) of s 5 .40. What would be the appropriate values for u and d if your binomial model is set up using:
a. 1 period of 1 year. b. 4 subperiods, each 3 months. c. 12 subperiods, each 1 month.
Challenge
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766 P A R T V I Options, Futures, and Other Derivatives
50. You build a binomial model with one period and assert that over the course of a year, the stock
price will either rise by a factor of 1.5 or fall by a factor of 2/3. What is your implicit assumption
about the volatility of the stock’s rate of return over the next year?
51. Use the put-call parity relationship to demonstrate that an at-the-money call option on a
nondividend-paying stock must cost more than an at-the-money put option. Show that the prices
of the put and call will be equal if S 5 (1 1 r ) T .
52. Return to Problem 36. Value the call option using the risk-neutral shortcut described in the
box on page 736. Confirm that your answer matches the value you get using the two-state
approach.
53. Return to Problem 38. What will be the payoff to the put, P u , if the stock goes up? What will be the payoff, P d , if the stock price falls? Value the put option using the risk-neutral shortcut described in the box on page 736. Confirm that your answer matches the value you get using the
two-state approach.
1. T h e board of directors of Abco Company is concerned about the downside risk of a $100
million equity portfolio in its pension plan. The board’s consultant has proposed temporarily (for
1 month) hedging the portfolio with either futures or options. Referring to the following table, the
consultant states:
a. “The $100 million equity portfolio can be fully protected on the downside by selling (shorting) 4,000 futures contracts.”
b. “The cost of this protection is that the portfolio’s expected rate of return will be zero percent.”
Market, Portfolio, and Contract Data
Equity index level 99.00
Equity futures price 100.00
Futures contract multiplier $250
Portfolio beta 1.20
Contract expiration (months) 3
Critique the accuracy of each of the consultant’s two statements.
2. Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants
of the value of an option, the characteristics of various models used to value options, and the
potential for divergence of calculated option values from observed market prices.
a. What is the expected effect on the value of a call option on common stock if the volatility of the underlying stock price decreases? If the time to expiration of the option increases?
b. Using the Black-Scholes option-pricing model, Weber calculates the price of a 3-month call option and notices the option’s calculated value is different from its market price. With respect
to Weber’s use of the Black-Scholes option-pricing model,
i. Discuss why the calculated value of an out-of-the-money European option may differ
from its market price.
ii. Discuss why the calculated value of an American option may differ from its market
price.
3. Joel Franklin is a portfolio manager responsible for derivatives. Franklin observes an American-
style option and a European-style option with the same strike price, expiration, and underly-
ing stock. Franklin believes that the European-style option will have a higher premium than the
American-style option.
a. Critique Franklin’s belief that the European-style option will have a higher premium. Franklin is asked to value a 1-year European-style call option for Abaco Ltd. common stock, which last
traded at $43.00. He has collected the information in the following table.
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C H A P T E R 2 1 Option Valuation 767
Closing stock price $43.00
Call and put option exercise price 45.00
1-year put option price 4.00
1-year Treasury bill rate 5.50%
Time to expiration One year
b. Calculate, using put-call parity and the information provided in the table, the European-style call option value.
c. State the effect, if any, of each of the following three variables on the value of a call option. (No calculations required.)
i. An increase in short-term interest rate.
ii. An increase in stock price volatility.
iii. A decrease in time to option expiration.
4. A stock index is currently trading at 50. Paul Tripp, CFA, wants to value 2-year index options using
the binomial model. The stock will either increase in value by 20% or fall in value by 20%. The annual
risk-free interest rate is 6%. No dividends are paid on any of the underlying securities in the index.
a. Construct a two-period binomial tree for the value of the stock index. b. Calculate the value of a European call option on the index with an exercise price of 60. c. Calculate the value of a European put option on the index with an exercise price of 60. d. Confirm that your solutions for the values of the call and the put satisfy put-call parity.
5. Ken Webster manages a $200 million equity portfolio benchmarked to the S&P 500 index.
Webster believes the market is overvalued when measured by several traditional fundamental/
economic indicators. He is concerned about potential losses but recognizes that the S&P 500
index could nevertheless move above its current 1136 level.
Webster is considering the following option collar strategy:
• Protection for the portfolio can be attained by purchasing an S&P 500 index put with a strike
price of 1130 (just out of the money).
• The put can be financed by selling two 1150 calls (farther out-of-the-money) for every put purchased.
• Because the combined delta of the two calls (see following table) is less than 1 (that is,
2 3 .36 5 .72), the options will not lose more than the underlying portfolio will gain if the
market advances.
The information in the following table describes the two options used to create the collar.
Characteristics 1150 Call 1130 Put
Option price $8.60 $16.10 Option implied volatility 22% 24% Option’s delta 0.36 20.44 Contracts needed for collar 602 301
Notes: • Ignore transaction costs. • S&P 500 historical 30-day volatility 5 23%. • Time to option expiration 5 30 days.
a. Describe the potential returns of the combined portfolio (the underlying portfolio plus the option collar) if after 30 days the S&P 500 index has:
i. risen approximately 5% to 1193.
ii. remained at 1136 (no change).
iii. declined by approximately 5% to 1080.
(No calculations are necessary.)
b. Discuss the effect on the hedge ratio (delta) of each option as the S&P 500 approaches the level for each of the potential outcomes listed in part ( a ).
c. Evaluate the pricing of each of the following in relation to the volatility data provided:
i. the put
ii. the call
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768 P A R T V I Options, Futures, and Other Derivatives
E-INVESTMENTS EXERCISES Select a stock for which options are listed on the CBOE Web site ( www.cboe.com ). The price data for captions can be found on the “delayed quotes” menu option. Enter a ticker symbol for a stock of your choice and pull up its option price data.
Using daily price data from finance.yahoo.com calculate the annualized standard deviation of the daily percentage change in the stock price. Create a Black-Scholes option-pricing model in a spreadsheet, or use our Spreadsheet 21.1, available at www. mhhe.com/bkm (link to the Chapter 21 material). Using the standard deviation and a risk-free rate found at www.bloomberg.com/markets/rates/index.html , calculate the value of the call options.
How do the calculated values compare to the market prices of the options? On the basis of the difference between the price you calculated using historical volatility and the actual price of the option, what do you conclude about expected trends in market volatility?
SOLUTIONS TO CONCEPT CHECKS
1. If This Variable Increases . . . The Value of a Put Option
S Decreases
X Increases
s Increases
T Increases*
rf Decreases
Dividend payouts Increases
*For American puts, increase in time to expiration must increase value. One can always choose to exercise early if this is optimal; the longer expiration date simply expands the range of alternatives open to the option holder which must make the option more valuable. For a European put, where early exercise is not allowed, longer time to expiration can have an indeterminate effect. Longer expiration increases volatility value because the final stock price is more uncertain, but it reduces the present value of the exercise price that will be received if the put is exercised. The net effect on put value can be positive or negative.
To understand the impact of higher volatility, consider the same scenarios as for the call. The low-
volatility scenario yields a lower expected payoff.
High Stock price $10 $20 $30 $40 $50 volatility Put payoff $20 $10 $ 0 $ 0 $ 0 Low Stock price $20 $25 $30 $35 $40 volatility Put payoff $10 $ 5 $ 0 $ 0 $ 0
2. The parity relationship assumes that all options are held until expiration and that there are no cash
flows until expiration. These assumptions are valid only in the special case of European options
on non-dividend-paying stocks. If the stock pays no dividends, the American and European calls
are equally valuable, whereas the American put is worth more than the European put. Therefore,
although the parity theorem for European options states that
P 5 C 2 S0 1 PV(X)
in fact, P will be greater than this value if the put is American.
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C H A P T E R 2 1 Option Valuation 769
3. Because the option now is underpriced, we want to reverse our previous strategy.
Cash Flow in 1 Year for Each Possible Stock Price
Initial Cash Flow S 5 90 S 5 120
Buy 3 options 216.50 0 30 Short-sell 1 share; repay in 1 year 100 290 2120 Lend $83.50 at 10% interest rate 283.50 91.85 91.85 TOTAL 0 1.85 1.85
The riskless cash flow in 1 year per option is $1.85/3 5 $.6167, and the present value is
$.6167/1.10 5 $.56, precisely the amount by which the option is underpriced.
4. a. C u 2 C d 5 $6.984 2 0
b. uS 0 2 dS 0 5 $110 2 $95 5 $15
c. 6.984/15 5 .4656
d.
The portfolio must have a market value equal to the present value of $44.232.
e. $44.232/1.05 5 $42.126
f. .4656 3 $100 2 C 0 5 $42.126
C 0 5 $46.56 2 $42.126 5 $4.434
5. When Δ t shrinks, there should be lower possible dispersion in the stock price by the end of the subperiod because each shorter subperiod offers less time in which new information can
move stock prices. However, as the time interval shrinks, there will be a correspondingly greater
number of these subperiods until option expiration. Thus, total volatility over the remaining life of the option will be unaffected. In fact, take another look at Figure 21.2 . There, despite the fact
that u and d each get closer to 1 as the number of subintervals increases and the length of each subinterval falls, the total volatility of the stock return until option expiration is unaffected.
6. Because s 5 .6, s 2 5 .36.
d1 5 ln(100/95) 1 (.10 1 .36/2).25
.6".25 5 .4043 d2 5 d1 2 .6".25 5 .1043
Using Table 21.2 and interpolation, or from a spreadsheet function:
N(d1) 5 .6570
N(d2) 5 .5415
C 5 100 3 .6570 2 95e2.10 3 .25 3 .5415 5 15.53
7. Implied volatility exceeds .2783. Given a standard deviation of .2783, the option value is $7. A
higher volatility is needed to justify an $8 price. Using Spreadsheet 21.1 and Goal Seek, you can
confirm that implied volatility at an option price of $8 is .3138.
8. A $1 increase in stock price is a percentage increase of 1/122 5 .82%. The put option will fall by
(.4 3 $1) 5 $.40, a percentage decrease of $.40/$4 5 10%. Elasticity is 2 10/.82 5 2 12.2.
9. The delta for a call option is N ( d 1 ), which is positive, and in this case is .547. Therefore, for every 10 option contracts purchased, you would need to short 547 shares of stock.
Value in Next Period as Function of Stock Price
Action Today (time 0) dS0 5 $95 uS0 5 $110
Buy .4656 shares at price S0 5 $100 $44.232 $51.216
Write 1 call at price C0 0 26.984
TOTAL $44.232 $44.232
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22 FUTURES AND FORWARD contracts are like options in that they specify purchase or sale of some underlying security at some future date. The key difference is that the holder of an option is not compelled to buy or sell, and will not do so if the trade is unprofitable. A futures or forward contract, however, car- ries the obligation to go through with the agreed-upon transaction.
A forward contract is not an investment in the strict sense that funds are paid for an asset. It is only a commitment today to transact in the future. Forward arrangements are part of our study of investments, however, because they offer powerful means to hedge other investments and generally modify portfolio characteristics.
Forward markets for future delivery of various commodities go back at least to ancient Greece. Organized futures markets, though, are a relatively modern develop- ment, dating only to the 19th century. Futures markets replace informal forward contracts
with highly standardized, exchange-traded securities.
While futures markets have their roots in agricultural products and commodities, the markets today are dominated by trading in financial futures such as those on stock indexes, interest-rate-dependent securities such as gov- ernment bonds, and foreign exchange. The markets themselves also have changed, with trading today largely taking place in electronic markets.
This chapter describes the workings of futures markets and the mechanics of trading in these markets. We show how futures con- tracts are useful investment vehicles for both hedgers and speculators and how the futures price relates to the spot price of an asset. We also show how futures can be used in several risk-management applications. This chapter deals with general principles of future mar- kets. Chapter 23 describes specific futures markets in greater detail.
CHAPTER TWENTY-TWO
Futures Markets 2 PA
R T V
I
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C H A P T E R 2 2 Futures Markets 771
To see how futures and forwards work and how they might be useful, consider the port-
folio diversification problem facing a farmer growing a single crop, let us say wheat. The
entire planting season’s revenue depends critically on the highly volatile crop price. The
farmer can’t easily diversify his position because virtually his entire wealth is tied up in
the crop.
The miller who must purchase wheat for processing faces a risk management problem
that is the mirror image of the farmer’s. He is subject to profit uncertainty because of the
unpredictable cost of the wheat.
Both parties can hedge their risk by entering into a forward contract requiring the farmer to deliver the wheat when harvested at a price agreed upon now, regardless of the
market price at harvest time. No money need change hands at this time. A forward contract
is simply a deferred-delivery sale of some asset with the sales price agreed on now. All that
is required is that each party be willing to lock in the ultimate delivery price. The contract
protects each party from future price fluctuations.
Futures markets formalize and standardize forward contracting. Buyers and sellers
trade in a centralized futures exchange. The exchange standardizes the types of con-
tracts that may be traded: It establishes contract size, the acceptable grade of commodity,
contract delivery dates, and so forth. Although standardization eliminates much of the
flexibility available in forward contracting, it has the offsetting advantage of liquidity
because many traders will concentrate on the same small set of contracts. Futures con-
tracts also differ from forward contracts in that they call for a daily settling up of any
gains or losses on the contract. By contrast, no money changes hands in forward contracts
until the delivery date.
The centralized market, standardization of contracts, and depth of trading in each con-
tract allows futures positions to be liquidated easily rather than renegotiated with the other
party to the contract. Because the exchange guarantees the performance of each party,
costly credit checks on other traders are not necessary. Instead, each trader simply posts a
good-faith deposit, called the margin, to guarantee contract performance.
The Basics of Futures Contracts The futures contract calls for delivery of a commodity at a specified delivery or maturity
date, for an agreed-upon price, called the futures price, to be paid at contract maturity. The contract specifies precise requirements for the commodity. For agricultural commodities,
the exchange sets allowable grades (e.g., No. 2 hard winter wheat or No. 1 soft red wheat).
The place or means of delivery of the commodity is specified as well. Delivery of agri-
cultural commodities is made by transfer of warehouse receipts issued by approved ware-
houses. For financial futures, delivery may be made by wire transfer; for index futures,
delivery may be accomplished by a cash settlement procedure such as those for index
options. Although the futures contract technically calls for delivery of an asset, delivery
rarely occurs. Instead, parties to the contract much more commonly close out their posi-
tions before contract maturity, taking gains or losses in cash.
Because the futures exchange specifies all the terms of the contract, the traders need
bargain only over the futures price. The trader taking the long position commits to pur- chasing the commodity on the delivery date. The trader who takes the short position com- mits to delivering the commodity at contract maturity. The trader in the long position is
said to “buy” a contract; the short-side trader “sells” a contract. The words buy and sell
22.1 The Futures Contract
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772 P A R T V I Options, Futures, and Other Derivatives
are figurative only, because a contract is not really bought or sold like a stock or bond; it
is entered into by mutual agreement. At the time the contract is entered into, no money
changes hands.
Figure 22.1 shows prices for several futures contracts as they appear in The Wall Street Journal. The boldface heading lists in each case the commodity, the exchange where the futures contract is traded, the contract size, and the pricing unit. The first agricultural con-
tract listed is for corn, traded on the Chicago Board of Trade (CBT). (The CBT merged
with the Chicago Mercantile Exchange in 2007 but, for now, maintains a separate identity.)
Each contract calls for delivery of 5,000 bushels, and prices in the entry are quoted in cents
per bushel.
The next several rows detail price data for contracts expiring on various dates. The
March 2013 maturity corn contract, for example, opened during the day at a futures price
of 720.25 cents per bushel. The highest futures price during the day was 726, the lowest
was 714.50, and the settlement price (a representative trading price during the last few
minutes of trading) was 724.25. The settlement price increased by 3.50 cents from the
previous trading day. Finally, open interest, or the number of outstanding contracts, was
494,588. Similar information is given for each maturity date.
The trader holding the long position, that is, the person who will purchase the good,
profits from price increases. Suppose that when the contract matures in March, the price
of corn turns out to be 729.25 cents per bushel. The long-position trader who entered the
contract at the futures price of 724.25 cents therefore earns a profit of 5 cents per bushel.
As each contract calls for delivery of 5,000 bushels, the profit to the long position equals
5,000 3 $.05 5 $250 per contract. Conversely, the short position loses 5 cents per bushel.
The short position’s loss equals the long position’s gain.
To summarize, at maturity:
Profit to long 5 Spot price at maturity 2 Original futures price
Profit to short 5 Original futures price 2 Spot price at maturity
where the spot price is the actual market price of the commodity at the time of the delivery.
The futures contract, therefore, is a zero-sum game, with losses and gains netting out to zero. Every long position is offset by a short position. The aggregate profits to futures trad-
ing, summing over all investors, also must be zero, as is the net exposure to changes in the
commodity price. For this reason, the establishment of a futures market in a commodity
should not have a major impact on prices in the spot market for that commodity.
Figure 22.2 , panel A is a plot of the profits realized by an investor who enters the long
side of a futures contract as a function of the price of the asset on the maturity date. Notice
that profit is zero when the ultimate spot price, P T , equals the initial futures price, F 0 . Profit per unit of the underlying asset rises or falls one-for-one with changes in the final
spot price. Unlike the payoff of a call option, the payoff of the long futures position can
be negative: This will be the case if the spot price falls below the original futures price.
Unlike the holder of a call, who has an option to buy, the long futures position trader can- not simply walk away from the contract. Also unlike options, in the case of futures there is
no need to distinguish gross payoffs from net profits. This is because the futures contract
is not purchased; it is simply a contract that is agreed to by two parties. The futures price
adjusts to make the present value of entering into a new contract equal to zero.
The distinction between futures and options is highlighted by comparing panel A of
Figure 22.2 to the payoff and profit diagrams for an investor in a call option with exercise
price, X, chosen equal to the futures price F 0 (see panel C). The futures investor is exposed to considerable losses if the asset price falls. In contrast, the investor in the call cannot lose
more than the cost of the option.
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773
Copper-High (CMX)-25,000 Ibs.; $ per Ib. Jan 3.6580 3.6760 3.6520 3.6645 20.0065 411 March 3.6725 3.6965 3.6610 3.6765 20.0080 103,266 Gold (CMX)-100 troy oz.; $ per troy oz. Feb 1683.80 1685.80 1664.20 1669.90 216.80 160,918 April 1686.40 1688.10 1666.40 1672.10 216.80 149,804 June 1688.10 1689.60 1669.50 1674.20 216.80 38,688 Aug 1689.00 1689.00 1670.60 1676.00 216.80 24,841 Oct 1689.50 1690.20 1673.10 1677.70 216.90 11,949 Dec 1694.70 1694.70 1674.40 1679.60 216.90 26,055 miNY Gold (CMX)-50 troy oz.; $ per troy oz. Feb 1684.25 1684.50 1664.75 1670.00 216.70 2,282 April 1683.50 1684.00 1667.75 1672.00 216.90 96 June 1684.00 1684.00 1674.50 1674.25 216.75 12 Palladium (NYM)-50 troy oz.; $ per troy oz. Feb 718.80 718.80 718.70 726.30 0.50 4 March 725.20 728.80 716.60 726.70 0.50 30,654 June 725.55 729.90 719.20 728.15 0.65 1,480 Platinum (NYM)-50 troy oz.; $ per troy oz. Jan 1681.80 1681.80 1674.00 1682.80 28.90 47 Feb 1673.00 1680.40 1673.00 1680.80 28.00 13 Silver (CMX)-5,000 troy oz.; $ per troy oz. Jan 31.735 31.750 31.735 31.695 20.714 17 March 32.240 32.245 31.605 31.722 20.717 78,766 miNY Silver (CMX)-2500 troy oz.; $ per troy oz. March 32.138 32.438 31.613 31.725 20.714 295 May 31.863 31.900 31.838 31.775 20.722 11 Sept 32.038 32.038 32.038 31.875 20.717 1 Crude Oil, Light Sweet (NYM)-1,000 bbls.; $ per bbl. March 95.50 96.68 95.12 95.95 0.72 295,685 April 95.89 97.16 95.60 96.45 0.70 102,709 May 96.38 97.58 96.00 96.90 0.69 86,130 Aug 96.42 97.57 96.32 97.31 0.73 38,272 Sept 96.08 97.58 96.07 97.07 0.71 51,513 Oct 96.10 96.96 96.04 96.77 0.69 41,212 Heating Oil No.2 (NYM)-42,000 gal.; $ per gal. Feb 3.0795 3.0958 3.0693 3.0864 .0083 46,296 March 3.0674 3.0848 3.0561 3.0764 .0112 83,625 Gasoline-NY RBOB (NYM)-42,000 gal.; $ per gal. Feb 2.8335 2.8762 2.8171 2.8629 .0291 39,099 March 2.8495 2.8882 2.8316 2.8767 .0278 121,519 Natural Gas (NYM)-10,000 MMBtu.; $ per MMBtu. Feb 3.570 3.592 3.441 3.446 2.108 47,921 March 3.569 3.588 3.447 3.454 2.099 320,563 April 3.597 3.613 3.482 3.490 2.091 159,548 May 3.653 3.667 3.540 3.549 2.088 94,821 Oct 3.804 3.826 3.703 3.715 2.080 104,406 Jan’14 4.218 4.219 4.123 4.135 2.068 78,522
Agriculture Futures Corn (CBT)-5,000 bu.; cents per bu. March 720.25 726.00 714.50 724.25 3.50 494,588 Dec 589.75 591.00 583.75 585.50 24.75 223,643 Ethanol (CBT)-29,000 gal.; $ per gal. Feb 2.367 2.383 2.367 2.382 .01 1,131 March 2.378 2.397 2.374 2.396 .01 1,605 Oats (CBT)-5,000 bu.; cents per bu. March 360.00 361.25 355.50 361.00 1.25 7,404 May 365.50 367.75 362.75 367.75 ... 2,438 Soybeans (CBT)-5,000 bu.; cents per bu. March 1437.00 1439.00 1415.00 1435.25 21.75 226,757 May 1426.75 1428.25 1404.50 1423.75 23.00 140,830 Soybean Meal (CBT)-100 tons.; $ per ton. March 416.50 417.00 408.00 414.70 21.80 127,823 MAy 412.10 412.50 404.20 410.60 21.50 56,552 Soybean Oil (CBT)-60,000 lbs.; cents per lb. March 52.04 52.20 51.50 52.11 .08 155,915 May 52.40 52.60 51.87 52.49 .00 59,010 Rough Rice (CBT)-2,000 cwt.; cents per cwt. March 1530.50 1537.00 1523.00 1528.50 22.00 12,472 May 1562.00 1562.00 1554.50 1560.50 22.00 1,490 Wheat (CBT)-5,000 bu.; cents per bu. March 775.50 775.75 763.00 768.50 26.25 226.800 July 788.25 788.75 777.00 782.25 26.50 83,615 Wheat (KC)-5,000 bu.; cents per bu. March 819.00 820.00 816.75 821.50 28.75 91,904 July n.a. n.a. 835.25 840.50 28.00 35,595 Wheat (MPLS)-5,000 bu.; cents per bu. March 860.00 860.75 852.25 855.50 25.00 20,338 May 872.25 872.25 864.50 867.50 24.75 9,776 Cattle-Feeder (CME)-50,000 lbs.; cents per lb. Jan 144.425 145.000 144.100 144.600 .400 2,205 March 147.300 148.575 146.725 147.950 .800 14,972 Cattle-Live (CME)-40,000 lbs.; cents per lb. Feb 126.100 126.875 125.625 125.875 .100 47,761 April 130.650 131.425 130.175 130.350 2.100 155,046
Coffee (ICE-US)-37,500 lbs.; cents per Ib. March 150.65 151.30 146.10 146.55 23.85 90,658 May 153.75 154.20 149.10 149.50 23.85 29,315 Sugar-World (ICE-US)-112,000 lbs.; cents per Ib. March 18.54 18.59 18.31 18.49 2.01 341,806 May 18.59 18.59 18.30 18.46 2.07 167,304 Sugar-Domestic (ICE-US)-112,000 lbs.; cents per Ib. March 21.00 21.00 20.55 20.88 2.67 958 May 21.45 21.45 21.10 21.23 2.43 4,057 Cotton (ICE-US)-50,000 Ibs.; cents per Ib. March 80.10 84.00 80.00 82.89 2.41 132,025 May 80.35 82.95 80.15 82.20 1.53 34,978 Orange Juice (ICE-US)-15,000 Ibs.; cents per Ib. March 115.60 116.80 113.00 113.20 22.40 14,247 May 116.55 117.40 114.30 114.30 22.20 4,127
Interest Rate Futures Treasury Bonds (CBT)-$100,000; pts 32nds of 100% March 146-050 146-150 145-090 145-270 24.0 548,615 June 144-170 144-300 143-300 144-120 24.0 747 Treasury Notes (CBT)-$100,000; pts 32nds of 100% March 132-120 132-175 131-315 132-075 22.0 1,865,707 June 131-125 131-130 131-000 131-060 22.0 24,269 5 Yr. Treasury Notes (CBT)-$100,000; pts 32nds of 100% March 124-082 124-102 124-020 124-060 21.0 1,527,948 June 123-255 123-267 123-207 123-245 21.0 3,958 2 Yr. Treasury Notes (CBT)-$200,000; pts 32nds of 100% March 110-077 110-080 110-070 110-075 ... 1,007,447 June 110-070 110-075 110-065 110-067 ... 1,956 30 Day Federal Funds (CBT)-$5,00,000; 100-daily avg. Jan 99.858 99.858 99.855 99.858 ... 57,169 March ... 99.875 99.865 99,870 ... 36,813 10 Yr. Int. Rate Swaps (CBT)-$100,000; pts 32nds of 100% March 118.922 118.922 118.422 118.609 2.078 9,349 1 Month Libor (CME)-$3,000,000; pts of 100% Feb ... ... ... 99.7975 ... 3,622 Eurodollar (CME)-$1,000,000; pts of 100% Feb 99.7025 99.7050 99.7000 99.7025 .0025 53,143 March 99.7000 99.7050 99.7000 99.7000 ... 835,707 June 99.6850 99.6850 99.6800 99.6800 ... 706,194 Dec 99.6400 99.6400 99.6250 99.6300 2.0050 728,161
Currency Futures Japanese Yen (CME)-¥12,500,000; $ per 100¥ March 1.1296 1.1313 1.1045 1.1119 2.0163 204,630 June 1.1310 1.1310 1.1056 1.1127 2.0164 991 Canadian Dollar (CME)-CAD 100,000; $ per CAD March .9992 .9997 .9952 .9958 2.0036 139,041 June .9970 .9978 .9934 .9938 2.0036 2,204 British Pound (CME)-£62,500; $ per £ March 1.5828 1.5848 1.5752 1.5787 2.0049 158,952 June 1.5830 1.5837 1.5750 1.5781 2.0049 236 Swiss Franc (CME)-CHF 125,000; $ per CHF March 1.0758 1.0782 1.0730 1.0779 .0016 40,548 Australian Dollar (CME)-AUD 100,000; $ per AUD March 1.0495 1.0509 1.0405 1.0433 2.0074 204,989 June 1.0405 1.0441 1.0355 1.0366 2.0073 331 Sept 1.0320 1.0320 1.0310 1.0303 2.0073 1 Mexican Peso (CME)-MXN 500,000; $ per 10MXN March .07830 .07893 .07820 .07865 .00015 187,352 June .07758 .07783 .07758 .07798 .00015 122 Euro (CME)-€125,000; $ per € March 1.3317 1.3398 1.3291 1.3378 .0053 216,667 June 1.3329 1.3400 1.3300 1.3385 .0052 1,777 Euro/Japanese Yen (ICE-US)-€125,000; ¥ per € March 117.675 11.6750 11.6750 120.3200 2.2100 4,186 Euro/British Pound (ICE-US)-€125,000; £ per € March ... ... ... 0.8474 .0060 2,992 Euro/Swiss France (ICE-US)-€125,000; CHF per € March 1.2376 1.2445 1.2360 1.2411 .0031 8,588
Index Futures DJ Industrial Average (CBT)-$10 3 index March 13706 13815 13703 13781 62 10,749 Mini DJ Industrial Average (CBT)-$5 3 index March 13702 13818 13701 13781 62 112,004 S&P 500 Index (CME)-$250 3 index March 1483.80 1497.50 1483.30 1491.80 1.50 194,350 June 1482.90 1491.60 1480.60 1485.10 1.50 4,188 Mini S&P 500 (CME)-$50 3 index March 1483.25 1497.75 1482.75 1491.75 1.50 2,972,476 June 1477.25 1491.25 1476.25 1485.00 1.50 18,475 Mini S&P Midcap 400 (CME)-$100 3 index March 1077.50 1089.20 1075.70 1086.30 4.70 116,953 Nasdaq 100 (CME)-$100 3 index March 2716.75 2743.50 2709.75 2718.25 240.75 10,891
Open Settle Chg Open
interestHigh hi lo low Contract
Open Settle Chg Open
interestHigh hi lo low Contract
Metal & Petroleum Futures
Futures Contracts | WSJ.com/commodities
Figure 22.1 Futures listings Source: The Wall Street Journal, January 25, 2013. Reprinted by permission of The Wall Street Journal, © 2013 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
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774 P A R T V I Options, Futures, and Other Derivatives
Figure 22.2 , panel B is a plot of the profits realized by an investor who enters the short
side of a futures contract. It is the mirror image of the profit diagram for the long position.
Payoff
Profit
C: Buy a Call Option
X PT
Profit
B: Short Futures Profit = F0 – PT
F0
PT
Profit
A: Long Futures Profit = PT – F0
F0
PT
Figure 22.2 Profits to buyers and sellers of futures and options contracts
a. Compare the profit diagram in Figure 22.2 , panel B to the payoff diagram for a long position in a put option. Assume the exercise price of the option equals the initial futures price.
b. Compare the profit diagram in Figure 22.2 , panel B to the payoff diagram for an investor who writes a call option.
CONCEPT CHECK 22.1
Existing Contracts Futures and forward contracts are traded on a wide variety of goods in four broad catego-
ries: agricultural commodities, metals and minerals (including energy commodities), for-
eign currencies, and financial futures (fixed-income securities and stock market indexes).
In addition to indexes on broad stock indexes, one can now trade single-stock futures on individual stocks and narrowly based indexes. OneChicago has operated an entirely elec-
tronic market in single-stock futures since 2002. The exchange maintains futures markets
in actively traded stocks with the most liquidity as well as in some popular ETFs such as
those on the S&P 500 (ticker SPY), the NASDAQ-100 (QQQ), and the Dow Jones Indus-
trial Average (DIA). However, trading volume in single-stock futures has to date been
somewhat disappointing.
Table 22.1 offers a sample of the various contracts trading in 2013. Contracts now trade
on items that would not have been considered possible only a few years ago, such as elec-
tricity as well as weather futures and options contracts. Weather derivatives (which trade
on the Chicago Mercantile Exchange) have payoffs that depend on average weather con-
ditions, for example, the number of degree-days by which the temperature in a region
exceeds or falls short of 65 degrees Fahrenheit. The potential use of these derivatives in
managing the risk surrounding electricity or oil and natural gas use should be evident.
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C H A P T E R 2 2 Futures Markets 775
While Table 22.1 includes many contracts, the large and ever-growing array of markets
makes this list necessarily incomplete. The nearby box discusses some comparatively fan-
ciful futures markets, sometimes called prediction markets, in which payoffs may be tied to the winner of presidential elections, the box office receipts of a particular movie, or
anything else in which participants are willing to take positions.
Outside the futures markets, a well-developed network of banks and brokers has estab-
lished a forward market in foreign exchange. This forward market is not a formal exchange
in the sense that the exchange specifies the terms of the traded contract. Instead, partici-
pants in a forward contract may negotiate for delivery of any quantity of goods, whereas
in the formal futures markets contract size and delivery dates are set by the exchange. In
forward arrangements, banks and brokers simply negotiate contracts for clients (or them-
selves) as needed. This market is huge. In London alone, the largest currency market,
around $2 trillion of currency trades each day.
Foreign Currencies Agricultural
Metals and Energy Interest Rate Futures Equity Indexes
British pound Corn Copper Eurodollars S&P 500 index Canadian dollar Oats Aluminum Euroyen Dow Jones Industrials Japanese yen Soybeans Gold Euro-denominated bond S&P Midcap 400 Euro Soybean meal Platinum Euroswiss NASDAQ 100 Swiss franc Soybean oil Palladium Sterling NYSE index Australian dollar Wheat Silver British government bond Russell 2000 index Mexican peso Barley Crude oil German government bond Nikkei 225 (Japanese) Brazilian real Flaxseed Heating oil Italian government bond FTSE index (British)
Canola Gas oil Canadian government bond CAC-40 (French) Rye Natural gas Treasury bonds DAX-30 (German) Cattle Gasoline Treasury notes All ordinary (Australian) Hogs Propane Treasury bills Toronto 35 (Canadian) Pork bellies Commodity index LIBOR Dow Jones Euro STOXX 50 Cocoa Electricity EURIBOR Industry indexes, e.g., Coffee Weather Municipal bond index Banking Cotton Federal funds rate Telecom Milk Bankers’ acceptance Utilities Orange juice Interest rate swaps Health care Sugar Technology Lumber Rice
Table 22.1
Sample of futures contracts
The Clearinghouse and Open Interest Until about 10 years ago, most futures trades in the United States occurred among floor
traders in the “trading pit” for each contract. Today, however, trading is overwhelmingly
conducted through electronic networks, particularly for financial futures.
22.2 Trading Mechanics
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Prediction Markets
If you find S&P 500 or T-bond contracts a bit dry, perhaps you’d be interested in futures contracts with payoffs that depend on the winner of the next presidential election, or the severity of the next influenza season, or the host city of the 2024 Olympics. You can now find “futures markets” in these events and many others.
For example, both Intrade ( www.intrade.com ) and Iowa Electronic Markets ( www.biz.uiowa.edu/iem ) maintain presidential futures markets. In July 2011, you could have purchased a contract that would pay off $1 in November 2012 if the Republican candidate won the presidential race but nothing if he lost. The contract price (expressed as a percent- age of face value) therefore may be viewed as the probability of a Republican victory, at least according to the consensus view of market participants at the time. If you believed in July that the probability of a Republican victory was 55%, you would have been prepared to pay up to $.55 for the contract. Alternatively, if you had wished to bet against the Republi- cans, you could have sold the contract. Similarly, you could
bet on (or against) a Democrat victory using the Democrat contract. (When there are only two relevant parties, betting on one is equivalent to betting against the other, but in other elections, such as primaries where there are several viable candidates, selling one candidate’s contract is not the same as buying another’s.)
The accompanying figure shows the price of each con- tract from July 2011 through Election Day 2012. The price clearly tracks each candidate’s perceived prospects. You can see Obama’s price rise dramatically in the days shortly before the election as it became ever clearer that he would win the election.
Interpreting prediction market prices as probabilities actually requires a caveat. Because the contract payoff is risky, the price of the contract may reflect a risk premium. Therefore, to be precise, these probabilities are actually risk-neutral probabilities (see Chapter 21). In practice, how- ever, it seems unlikely that the risk premium associated with these contracts is substantial. W
O R
D S
FR O
M T
H E S
T R
E E T
Prediction markets for the 2012 presidential election. Contract on each party pays $1 if the party wins the election. Price is in cents.
0
Date
10
20
30
40
P ri
ce 50
60
70
80
90
100
0 1 -J
u l-
1 1
1 9 -J
u l-
1 1
0 6 -A
u g
-1 1
2 4 -A
u g
-1 1
1 1 -S
e p
-1 1
2 9 -S
e p
-1 1
1 7 -O
ct -1
1
0 4 -N
o v-
1 1
2 2 -N
o v-
1 1
1 0 -D
e c-
1 1
2 8 -D
e c-
1 1
1 5 -J
a n
-1 2
0 2 -F
e b
-1 2
2 0 -F
e b
-1 2
0 9 -M
a r-
1 2
2 7 -M
a r-
1 2
1 4 -A
p r-
1 2
0 2 -M
a y-
1 2
2 0 -M
a y-
1 2
0 7 -J
u n
-1 2
2 5 -J
u n
-1 2
1 3 -J
u l-
1 2
3 1 -J
u l-
1 2
1 8 -A
u g
-1 2
0 5 -S
e p
-1 2
2 3 -S
e p
-1 2
1 1 -O
ct -1
2
0 8 -N
o v-
1 2
Democrat
Republican
Source: Iowa Electronic Markets, downloaded January 25, 2013.
776
The impetus for this shift originated in Europe, where electronic trading is the norm.
Eurex, which is jointly owned by the Deutsche Börse and Swiss exchange, is among the
world’s largest derivatives exchanges. It operates a fully electronic trading and clearing
platform and, in 2004, received clearance from regulators to list contracts in the U.S. In
response, the Chicago Board of Trade adopted an electronic platform provided by Eurex’s
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C H A P T E R 2 2 Futures Markets 777
European rival Euronext.liffe, 1 and the CBOT’s Treasury contracts are now traded elec-
tronically. The Chicago Mercantile Exchange maintains another electronic trading system
called Globex. The electronic markets enable trading around the clock.
The CBOT and CME merged in 2007 into one combined company, named the CME
Group, with all electronic trading from both exchanges moving onto Globex. It seems
inevitable that electronic trading will continue to displace floor trading.
Once a trade is agreed to, the clearinghouse enters the picture. Rather than having the long and short traders hold contracts with each other, the clearinghouse becomes the
seller of the contract for the long position and the buyer of the contract for the short posi-
tion. The clearinghouse is obligated to deliver the commodity to the long position and to
pay for delivery from the short; consequently, the clearinghouse’s position nets to zero.
This arrangement makes the clearinghouse the trading partner of each trader, both long
and short. The clearinghouse, bound to perform on its side of each contract, is the only
party that can be hurt by the failure of any trader to observe the obligations of the futures
contract. This arrangement is necessary because a futures contract calls for future perfor-
mance, which cannot be as easily guaranteed as an immediate stock transaction.
Figure 22.3 illustrates the role of the clearinghouse. Panel A shows what would happen
in the absence of the clearinghouse. The trader in the long position would be obligated to
pay the futures price to the short-position trader, and the trader in the short position would
be obligated to deliver the commodity. Panel B shows how the clearinghouse becomes
an intermediary, acting as the trading partner for each side of the contract. The clearing-
house’s position is neutral, as it takes a long and a short position for each transaction.
The clearinghouse makes it possible for traders to liquidate positions easily. If you are
currently long in a contract and want to undo your position, you simply instruct your bro-
ker to enter the short side of a contract to close out your position. This is called a reversing trade. The exchange nets out your long and short positions, reducing your net position to zero. Your zero net position with the clearinghouse eliminates the need to fulfill at maturity
either the original long or reversing
short position.
The open interest on the con- tract is the number of contracts out-
standing. (Long and short positions
are not counted separately, meaning
that open interest can be defined
either as the number of long or short
contracts outstanding.) The clear-
inghouse’s position nets out to zero,
and so is not counted in the compu-
tation of open interest. When con-
tracts begin trading, open interest
is zero. As time passes, open inter-
est increases as progressively more
contracts are entered.
There are many apocryphal sto-
ries about futures traders who wake
up to discover a small mountain of
1 Euronext.liffe is the international derivatives market of Euronext. It resulted from Euronext’s purchase of LIFFE
(the London International Financial Futures and Options Exchange) and a merger with the Lisbon exchange in
2002. Euronext was itself the result of a 2000 merger of the exchanges of Amsterdam, Brussels, and Paris.
A Money
Commodity
Long Position
B Money
Commodity
Money
Commodity
Long Position
Clearinghouse
Short Position
Short Position
Figure 22.3 Panel A, Trading without a clearinghouse. Panel B, Trading with a clearinghouse.
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778 P A R T V I Options, Futures, and Other Derivatives
wheat or corn on their front lawn. But the truth is that futures contracts rarely result in
actual delivery of the underlying asset. Traders establish long or short positions in con-
tracts that will benefit from a rise or fall in the futures price and almost always close out,
or reverse, those positions before the contract expires. The fraction of contracts that result
in actual delivery is estimated to range from less than 1% to 3%, depending on the com-
modity and activity in the contract. In the unusual case of actual deliveries of commodities,
they occur via regular channels of supply, most often warehouse receipts.
You can see the typical pattern of open interest in Figure 22.1 . In the copper contract,
for example, the January delivery contract is approaching maturity, and open interest is
small; most contracts have been reversed already. The greatest open interest is in the March
contract. For other contracts, for example, gold, for which the nearest maturity date isn’t
until February, open interest is typically highest in the nearest contract.
The Margin Account and Marking to Market The total profit or loss realized by the long trader who buys a contract at time 0 and closes,
or reverses, it at time t is just the change in the futures price over the period, F t 2 F 0 . Symmetrically, the short trader earns F 0 2 F t .
The process by which profits or losses accrue to traders is called marking to market. At initial execution of a trade, each trader establishes a margin account. The margin is a secu-
rity account consisting of cash or near-cash securities, such as Treasury bills, that ensures
the trader is able to satisfy the obligations of the futures contract. Because both parties to
a futures contract are exposed to losses, both must post margin. To illustrate, return to the
first corn contract listed in Figure 22.1 . If the initial required margin on corn, for example,
is 10%, then the trader must post $3,620 per contract of the margin account. This is 10% of
the value of the contract, $7.24 per bushel 3 5,000 bushels per contract.
Because the initial margin may be satisfied by posting interest-earning securities, the
requirement does not impose a significant opportunity cost of funds on the trader. The ini-
tial margin is usually set between 5% and 15% of the total value of the contract. Contracts
written on assets with more volatile prices require higher margins.
On any day that futures contracts trade, futures prices may rise or may fall. Instead of wait-
ing until the maturity date for traders to realize all gains and losses, the clearinghouse requires
all positions to recognize profits as they accrue daily. If the futures price of corn rises from
724 to 726 cents per bushel, the clearinghouse credits the margin account of the long position
for 5,000 bushels times 2 cents per bushel, or $100 per contract. Conversely, for the short
position, the clearinghouse takes this amount from the margin account for each contract held.
This daily settling is called marking to market. It means the maturity date of the con- tract does not govern realization of profit or loss. Instead, as futures prices change, the pro-
ceeds accrue to the trader’s margin account immediately. We will provide a more detailed
example of this process shortly.
Marking to market is the major way in which futures
and forward contracts differ, besides contract standard-
ization. Futures follow this pay-(or receive-)as-you-go
method. Forward contracts are simply held until maturity,
and no funds are transferred until that date, although the
contracts may be traded.
If a trader accrues sustained losses from daily marking
to market, the margin account may fall below a critical
value called the maintenance margin. If the value of the account falls below this value, the trader receives a margin call, requiring that the margin account be replenished or the
position be reduced to a size commensurate with the remaining funds. Margins and
What must be the net inflow or outlay from marking to market for the clearinghouse?
CONCEPT CHECK 22.2
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C H A P T E R 2 2 Futures Markets 779
On the contract maturity date, the futures price will equal the spot price of the com-
modity. As a maturing contract calls for immediate delivery, the futures price on that
day must equal the spot price—the cost of the commodity from the two competing
sources is equalized in a competitive market. 2 You may obtain delivery of the commod-
ity either by purchasing it directly in the spot market or by entering the long side of a
futures contract.
A commodity available from two sources (spot or futures market) must be priced identi-
cally, or else investors will rush to purchase it from the cheap source in order to sell it in
the high-priced market. Such arbitrage activity could not persist without prices adjusting
to eliminate the arbitrage opportunity. Therefore, the futures price and the spot price must
converge at maturity. This is called the convergence property. For an investor who establishes a long position in a contract now (time 0) and holds that
position until maturity (time T ), the sum of all daily settlements will equal F T 2 F 0 , where F T stands for the futures price at contract maturity. Because of convergence, however, the futures price at maturity, F T , equals the spot price, P T , so total futures profits also may be expressed as P T 2 F 0 . Thus we see that profits on a futures contract held to maturity per- fectly track changes in the value of the underlying asset.
Suppose the maintenance margin is 5% while the initial margin was 10% of the value of the corn, or $3,620. Then a margin call will go out when the original margin account has fallen in half, by about $1,810. Each 1-cent decline in the corn price results in a $50 loss to the long position. Therefore, the futures price need only fall by 37 cents (or 5% of its current value) to trigger a margin call.
Example 22.1 Maintenance Margin
margin calls safeguard the position of the clearinghouse. Positions are closed out before
the margin account is exhausted—the trader’s losses are covered, and the clearinghouse is
not put at risk.
2 Small differences between the spot and futures price at maturity may persist because of transportation costs, but
this is a minor factor.
Assume the current futures price for silver for delivery 5 days from today is $30.10 per ounce. Suppose that over the next 5 days, the futures price evolves as follows:
Example 22.2 Marking to Market
Day Futures Price
0 (today) $30.10 1 30.20 2 30.25 3 30.18 4 30.18 5 (delivery) 30.21
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780 P A R T V I Options, Futures, and Other Derivatives
Cash versus Actual Delivery Most futures contracts call for delivery of an actual commodity such as a particular grade
of wheat or a specified amount of foreign currency if the contract is not reversed before
maturity. For agricultural commodities, where quality of the delivered good may vary, the
exchange sets quality standards as part of the futures contract. In some cases, contracts
may be settled with higher- or lower-grade commodities. In these cases, a premium or dis-
count is applied to the delivered commodity to adjust for the quality difference.
Some futures contracts call for cash settlement. An example is a stock index futures contract where the underlying asset is an index such as the Standard & Poor’s 500 or the
New York Stock Exchange Index. Delivery of every stock in the index clearly would be
impractical. Hence the contract calls for “delivery” of a cash amount equal to the value
that the index attains on the maturity date of the contract. The sum of all the daily settle-
ments from marking to market results in the long position realizing total profits or losses of
S T 2 F 0 , where S T is the value of the stock index on the maturity date T and F 0 is the origi- nal futures price. Cash settlement closely mimics actual delivery, except the cash value of
the asset rather than the asset itself is delivered.
More concretely, the S&P 500 index contract calls for delivery of $250 times the value
of the index. At maturity, the index might list at 1,200, the market-value-weighted index
of the prices of all 500 stocks in the index. The cash settlement contract calls for delivery
of $250 3 1,200, or $300,000 cash in return for $250 times the futures price. This yields
exactly the same profit as would result from directly purchasing 250 units of the index for
$300,000 and then delivering it for $250 times the original futures price.
Regulations Futures markets are regulated by the federal Commodities Futures Trading Commission.
The CFTC sets capital requirements for member firms of the futures exchanges, authorizes
trading in new contracts, and oversees maintenance of daily trading records.
The daily mark-to-market settlements for each contract held by the long position will be as follows:
The profit on Day 1 is the increase in the futures price from the previous day, or ($30.20 2 $30.10) per ounce. Because each silver contract on the Commodity Exchange (CMX) calls for purchase and delivery of 5,000 ounces, the total profit per contract is 5,000 times $.10, or $500. On Day 3, when the futures price falls, the long position’s margin account will be debited by $350. By Day 5, the sum of all daily proceeds is $550. This is exactly equal to 5,000 times the difference between the final futures price of $30.21 and original futures price of $30.10. Because the final futures price equals the spot price on that date, the sum of all the daily proceeds (per ounce of silver held long) also equals P T 2 F 0 .
Day Profit (Loss) per Ounce 3 5,000 Ounces/Contract 5 Daily Proceeds
1 30.20 2 30.10 5 .10 $500 2 30.25 2 30.20 5 .05 250 3 30.18 2 30.25 5 2.07 2350 4 30.18 2 30.18 5 0 0 5 30.21 2 30.18 5 .03 150
Sum 5 $550
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C H A P T E R 2 2 Futures Markets 781
The futures exchange may set limits on the amount by which futures prices may
change from one day to the next. For example, if the price limit on silver contracts were
set at $1 and silver futures close today at $22.10 per ounce, then trades in silver tomor-
row may vary only between $21.10 and $23.10 per ounce. The exchanges may increase
or reduce price limits in response to perceived changes in price volatility of the contract.
Price limits are often eliminated as contracts approach maturity, usually in the last month
of trading.
Price limits traditionally are viewed as a means to limit violent price fluctuations. This
reasoning seems dubious. Suppose an international monetary crisis overnight drives up the
spot price of silver to $30. No one would sell silver futures at prices for future delivery as
low as $22.10. Instead, the futures price would rise each day by the $1 limit, although the
quoted price would represent only an unfilled bid order—no contracts would trade at the
low quoted price. After several days of limit moves of $1 per day, the futures price would
finally reach its equilibrium level, and trading would occur again. This process means no
one could unload a position until the price reached its equilibrium level. We conclude that
price limits offer no real protection against fluctuations in equilibrium prices.
Taxation Because of the mark-to-market procedure, investors do not have control over the tax year
in which they realize gains or losses. Instead, price changes are realized gradually, with
each daily settlement. Therefore, taxes are paid at year-end on cumulated profits or losses
regardless of whether the position has been closed out. As a general rule, 60% of futures
gains or losses are treated as long term, and 40% as short term.
Hedging and Speculation Hedging and speculating are two polar uses of futures markets. A speculator uses a futures
contract to profit from movements in futures prices, a hedger to protect against price
movement.
If speculators believe prices will increase, they will take a long position for expected
profits. Conversely, they exploit expected price declines by taking a short position.
22.3 Futures Markets Strategies
Suppose you believe that crude oil prices are going to increase and therefore decide to purchase crude oil futures. Each contract calls for delivery of 1,000 barrels of oil, so for every dollar increase in the futures price of crude, the long position gains $1,000 and the short position loses that amount.
Conversely, suppose you think that prices are heading lower and therefore sell a contract. If crude oil prices do in fact fall, then you will gain $1,000 per contract for every dollar that prices decline.
If the futures price for delivery in February is $91.86 and crude oil is selling at the contract maturity date for $93.86, the long side will profit by $2,000 per contract pur- chased. The short side will lose an identical amount on each contract sold. On the other hand, if oil has fallen to $89.86, the long side will lose, and the short side will gain, $2,000 per contract.
Example 22.3 Speculating with Oil Futures
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782 P A R T V I Options, Futures, and Other Derivatives
Why does a speculator buy a futures contract? Why not buy the underlying asset
directly? One reason lies in transaction costs, which are far smaller in futures markets.
Another important reason is the leverage that futures trading provides. Recall that
futures contracts require traders to post margin considerably less than the value of the asset
underlying the contract. Therefore, they allow speculators to achieve much greater lever-
age than is available from direct trading in a commodity.
Suppose the initial margin requirement for the oil contract is 10%. At a current futures price of $91.86, and contract size of 1,000 barrels, this would require margin of .10 3 91.86 3 1,000 5 $9,186. A $2 jump in oil prices represents an increase of 2.18%, and results in a $2,000 gain on the contract for the long position. This is a percentage gain of 21.8% in the $9,186 posted as margin, precisely 10 times the per- centage increase in the oil price. The 10-to-1 ratio of percentage changes reflects the leverage inherent in the futures position, because the contract was established with an initial margin of one-tenth the value of the underlying asset.
Example 22.4 Futures and Leverage
Hedgers, by contrast, use futures to insulate themselves against price movements. A firm
planning to sell oil, for example, might anticipate a period of market volatility and wish to
protect its revenue against price fluctuations. To hedge the total revenue derived from the sale,
the firm enters a short position in oil futures. As the following example illustrates, this locks in
its total proceeds (i.e., revenue from the sale of the oil plus proceeds from its futures position).
Consider an oil distributor planning to sell 100,000 barrels of oil in February that wishes to hedge against a possible decline in oil prices. Because each contract calls for delivery of 1,000 barrels, it would sell 100 contracts that mature in February. Any decrease in prices would then generate a profit on the contracts that would offset the lower sales revenue from the oil.
To illustrate, suppose that the only three possible prices for oil in February are $89.86, $91.86, and $93.86 per barrel. The revenue from the oil sale will be 100,000 times the price per barrel. The profit on each contract sold will be 1,000 times any decline in the futures price. At maturity, the convergence property ensures that the final futures price will equal the spot price of oil. Therefore, the profit on the 100 contracts sold will equal 100,000 3 ( F 0 2 P T ), where P T is the oil price on the delivery date, and F 0 is the original futures price, $91.86.
Now consider the firm’s overall position. The total revenue in February can be com- puted as follows:
Example 22.5 Hedging with Oil Futures
Oil Price in February, PT
$89.86 $91.86 $93.86
Revenue from oil sale: 100,000 3 PT $8,986,000 $9,186,000 $9,386,000 1 Profit on futures: 100,000 3 (F0 2 PT) 200,000 0 2200,000 TOTAL PROCEEDS $9,186,000 $9,186,000 $9,186,000
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C H A P T E R 2 2 Futures Markets 783
Figure 22.4 illustrates the nature of the hedge in Example 22.5. The upward-sloping
line is the revenue from the sale of oil. The downward-sloping line is the profit on the
futures contract. The horizontal line is the sum of sales revenue plus futures profits. This
line is flat, as the hedged position is independent of oil prices.
To generalize Example 22.5, note that oil will sell for P T per barrel at the maturity of the contract. The profit per barrel on the futures will be F 0 2 P T . Therefore, total revenue is P T 1 ( F 0 2 P T ) 5 F 0 , which is independent of the eventual oil price.
The oil distributor in Example 22.5 engaged in a short hedge, taking a short futures position to offset risk in the sales price of a particular asset. A long hedge is the analogous hedge for someone who wishes to eliminate the risk of an uncertain purchase price. For
example, a power supplier planning to purchase oil may be afraid that prices might rise by
the time of the purchase. As the following Concept Check illustrates, the supplier might
buy oil futures to lock in the net purchase price at the time of the transaction.
The revenue from the oil sale plus the proceeds from the contracts equals the current futures price, $91.86 per barrel. The variation in the price of the oil is precisely offset by the profits or losses on the futures position. For example, if oil falls to $89.86 a barrel, the short futures position generates $200,000 profit, just enough to bring total revenues to $9,186,000. The total is the same as if one were to arrange today to sell the oil in February at the futures price.
Suppose as in Example 22.5 that oil will be selling in February for $89.86, $91.86, or $93.86 per barrel. Con- sider a firm that plans to buy 100,000 barrels of oil in February. Show that if the firm buys 100 oil contracts today, its net expenditures in February will be hedged and equal to $9,186,000.
CONCEPT CHECK 22.3
–20
0
20
40
60
80
100
120
85 90 95 91.86
Hedged revenues are constant at $91.86 per barrel, equal to the futures price
Sales revenue increases with oil price
Profit on short futures position falls with oil price
P ro
ce e d
s (p
e r
B a rr
e l)
Oil Price
Sales Revenue per Barrel Futures Profits per Barrel Total Proceeds
100 105
Figure 22.4 Hedging revenues using futures, Example 22.5 (Futures price 5 $91.86)
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784 P A R T V I Options, Futures, and Other Derivatives
Exact futures hedging may be impossible for some goods because the necessary futures
contract is not traded. For example, a portfolio manager might want to hedge the value
of a diversified, actively managed portfolio for a period
of time. However, futures contracts are listed only on
indexed portfolios. Nevertheless, because returns on the
manager’s diversified portfolio will have a high corre-
lation with returns on broad-based indexed portfolios,
an effective hedge may be established by selling index
futures contracts. Hedging a position using futures on
another asset is called cross-hedging.
Basis Risk and Hedging The basis is the difference between the futures price and the spot price. 3 As we have noted, on the maturity date of a contract, the basis must be zero: The convergence property implies
that F T 2 P T 5 0. Before maturity, however, the futures price for later delivery may differ substantially from the current spot price.
In Example 22.5 we discussed the case of a short hedger who manages risk by entering
a short position to deliver oil in the future. If the asset and futures contract are held until
maturity, the hedger bears no risk. Risk is eliminated because the futures price and spot
price at contract maturity must be equal: Gains and losses on the futures and the com-
modity position will exactly cancel. However, if the contract and asset are to be liquidated
early, before contract maturity, the hedger bears basis risk, because the futures price and spot price need not move in perfect lockstep at all times before the delivery date. In this
case, gains and losses on the contract and the asset may not exactly offset each other.
Some speculators try to profit from movements in the basis. Rather than betting on
the direction of the futures or spot prices per se, they bet on the changes in the difference
between the two. A long spot–short futures position will profit when the basis narrows.
What are the sources of risk to an investor who uses stock index futures to hedge an actively managed stock portfolio? How might you esti- mate the magnitude of that risk?
CONCEPT CHECK 22.4
3 Usage of the word basis is somewhat loose. It sometimes is used to refer to the futures-spot difference F 2 P, and
sometimes to the spot-futures difference P 2 F. We will consistently call the basis F 2 P.
4 Yet another strategy is an intercommodity spread, in which the investor buys a contract on one commodity and
sells a contract on a different commodity.
Consider an investor holding 100 ounces of gold, who is short one gold-futures con- tract. Suppose that gold today sells for $1,591 an ounce, and the futures price for June delivery is $1,596 an ounce. Therefore, the basis is currently $5. Tomorrow, the spot price might increase to $1,595, while the futures price increases to $1,599, so the basis narrows to $4.
The investor’s gains and losses are as follows:
Gain on holdings of gold (per ounce): $1,595 2 $1,591 5 $4 Loss on gold futures position (per ounce): $1,599 2 $1,596 5 $3
The net gain is the decrease in the basis, or $1 per ounce.
Example 22.6 Speculating on the Basis
A related strategy is a calendar spread position, where the investor takes a long posi- tion in a futures contract of one maturity and a short position in a contract on the same
commodity, but with a different maturity. 4 Profits accrue if the difference in futures prices
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C H A P T E R 2 2 Futures Markets 785
between the two contracts changes in the hoped-for direction, that is, if the futures price
on the contract held long increases by more (or decreases by less) than the futures price on
the contract held short.
Consider an investor who holds a September maturity contract long and a June contract short. If the September futures price increases by 5 cents while the June futures price increases by 4 cents, the net gain will be 5 cents 2 4 cents, or 1 cent. Like basis strate- gies, spread positions aim to exploit movements in relative price structures rather than to profit from movements in the general level of prices.
Example 22.7 Speculating on the Spread
The Spot-Futures Parity Theorem We have seen that a futures contract can be used to hedge changes in the value of the
underlying asset. If the hedge is perfect, meaning that the asset-plus-futures portfolio has
no risk, then the hedged position must provide a rate of return equal to the rate on other
risk-free investments. Otherwise, there will be arbitrage opportunities that investors will
exploit until prices are brought back into line. This insight can be used to derive the theo-
retical relationship between a futures price and the price of its underlying asset.
Suppose for simplicity that the S&P 500 index currently is at 1,000 and an investor
who holds $1,000 in a mutual fund indexed to the S&P 500 wishes to temporarily hedge
her exposure to market risk. Assume that the indexed portfolio pays dividends totaling
$20 over the course of the year, and for simplicity, that all dividends are paid at year-end.
Finally, assume that the futures price for year-end delivery of the S&P 500 contract is
1,010. 5 Let’s examine the end-of-year proceeds for various values of the stock index if the
investor hedges her portfolio by entering the short side of the futures contract.
22.4 Futures Prices
Final value of stock portfolio, ST $ 970 $ 990 $1,010 $1,030 $1,050 $1,070 Payoff from short futures position (equals F0 2 FT 5 $1,010 2 ST)
40 20 0 220 240 260
Dividend income 20 20 20 20 20 20 TOTAL $1,030 $1,030 $1,030 $1,030 $1,030 $1,030
5 Actually, the futures contract calls for delivery of $250 times the value of the S&P 500 index, so that each
contract would be settled for $250 times the index. With the index at 1,000, each contract would hedge about
$250 3 1,000 5 $250,000 worth of stock. Of course, institutional investors would consider a stock portfolio of
this size to be quite small. We will simplify by assuming that you can buy a contract for one unit rather than 250
units of the index.
The payoff from the short futures position equals the difference between the original
futures price, $1,010, and the year-end stock price. This is because of convergence: The
futures price at contract maturity will equal the stock price at that time.
Notice that the overall position is perfectly hedged. Any increase in the value of the
indexed stock portfolio is offset by an equal decrease in the payoff of the short futures
position, resulting in a final value independent of the stock price. The $1,030 total payoff is
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786 P A R T V I Options, Futures, and Other Derivatives
the sum of the current futures price, F 0 5 $1,010, and the $20 dividend. It is as though the investor arranged to sell the stock at year-end for the current futures price, thereby elimi-
nating price risk and locking in total proceeds equal to the futures price plus dividends paid
before the sale.
What rate of return is earned on this riskless position? The stock investment requires an
initial outlay of $1,000, whereas the futures position is established without an initial cash
outflow. Therefore, the $1,000 portfolio grows to a year-end value of $1,030, providing a
rate of return of 3%. More generally, a total investment of S 0 , the current stock price, grows to a final value of F 0 1 D, where D is the dividend payout on the portfolio. The rate of return is therefore
Rate of return on hedged stock portfolio 5 (F0 1 D) 2 S0
S0
This return is essentially riskless. We observe F 0 at the beginning of the period when we enter the futures contract. While dividend payouts are not perfectly riskless, they are
highly predictable over short periods, especially for diversified portfolios. Any uncertainty
is extremely small compared to the uncertainty in stock prices. Presumably, 3% must be the rate of return available on other riskless investments. If
not, then investors would face two competing risk-free strategies with different rates of
return, a situation that could not last. Therefore, we conclude that
(F0 1 D) 2 S0
S0 5 rf
Rearranging, we find that the futures price must be
F0 5 S0(1 1 rf) 2 D 5 S0(1 1 rf 2 d) (22.1)
where d is the dividend yield on the stock portfolio, defined as D / S 0 . This result is called the spot-futures parity theorem. It gives the normal or theoretically correct relationship between spot and futures prices. Any deviation from parity would give rise to risk-free
arbitrage opportunities.
Suppose that parity were violated. For example, suppose the risk-free interest rate were only 1% so that according to Equation 22.1, the futures price should be $1,000(1.01) 2 $20 5 $990. The actual futures price, F 0 5 $1,010, is $20 higher than its “appropriate” value. This implies that an investor can make arbitrage profits by short- ing the relatively overpriced futures contract and buying the relatively underpriced stock portfolio using money borrowed at the 1% market interest rate. The proceeds from this strategy would be as follows:
Example 22.8 Futures Market Arbitrage
Action Initial
Cash Flow Cash Flow in
1 Year
Borrow $1,000, repay with interest in 1 year 11,000 21,000(1.01) 5 2$1,010 Buy stock for $1,000 21,000 ST 1 $20 dividend Enter short futures position (F0 5 $1,010) 0 $1,010 2 ST TOTAL 0 $20
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C H A P T E R 2 2 Futures Markets 787
The net initial investment of the strategy is zero. But its cash flow in 1 year is $20 regard- less of the stock price. In other words, it is riskless. This payoff is precisely equal to the mispricing of the futures contract relative to its parity value, 1,010 2 990.
When parity is violated, the strategy to exploit the mispricing produces an arbitrage profit—a riskless profit requiring no initial net investment. If such an opportunity existed, all market participants would rush to take advantage of it. The results? The stock price would be bid up, and/or the futures price offered down until Equation 22.1 is satisfied. A similar analysis applies to the possibility that F 0 is less than $990. In this case, you simply reverse the strategy above to earn riskless profits. We conclude, therefore, that in a well- functioning market in which arbitrage opportunities are competed away, F 0 5 S 0 (1 1 r f ) 2 D.
The arbitrage strategy of Example 22.8 can be represented more generally as follows:
Action Initial Cash Flow Cash Flow in 1 Year
1. Borrow S0 dollars S0 2S0 (1 1 rf) 2. Buy stock for S0 2S0 ST 1 D 3. Enter short futures position 0 F0 2 ST TOTAL 0 F0 2 S0(1 1 rf) 1 D
The initial cash flow is zero by construction: The money necessary to purchase the stock
in step 2 is borrowed in step 1, and the futures position in step 3, which is used to hedge
the value of the stock position, does not require an initial outlay. Moreover, the total cash
flow at year-end is riskless because it involves only terms that are already known when the
contract is entered. If the final cash flow were not zero, all investors would try to cash in
on the arbitrage opportunity. Ultimately prices would change until the year-end cash flow
is reduced to zero, at which point F 0 would equal S 0 (1 1 r f ) 2 D. The parity relationship also is called the cost-of-carry relationship because it asserts
that the futures price is determined by the relative costs of buying a stock with deferred
delivery in the futures market versus buying it in the spot market with immediate delivery
and “carrying” it in inventory. If you buy stock now, you tie up your funds and incur a
time-value-of-money cost of r f per period. On the other hand, you receive dividend pay- ments with a current yield of d. The net carrying cost advantage of deferring delivery of the stock is therefore r f 2 d per period. This advantage must be offset by a differential between the futures price and the spot price. The price differential just offsets the cost-of-carry
advantage when F 0 5 S 0 (1 1 r f 2 d ). The parity relationship is easily generalized to multiperiod applications. We simply rec-
ognize that the difference between the futures and spot price will be larger as the maturity
of the contract is longer. This reflects the longer period to which we apply the net cost of
carry. For contract maturity of T periods, the parity relationship is
F0 5 S0(1 1 rf 2 d) T (22.2)
Return to the arbitrage strategy laid out in Example 22.8. What would be the three steps of the strategy if F 0 were too low, say, $980? Work out the cash flows of the strategy now and in 1 year in a table like the one in the example. Confirm that your profits equal the mispricing of the contract.
CONCEPT CHECK 22.5
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788 P A R T V I Options, Futures, and Other Derivatives
Notice that when the dividend yield is less than the risk-free rate, Equation 22.2 implies
that futures prices will exceed spot prices, and by greater amounts for longer times to contract
maturity. But when d > r f , as is the case today, the income yield on the stock exceeds the for- gone (risk-free) interest that could be earned on the money invested; in this event, the futures
price will be less than the current stock price, again by greater amounts for longer maturities.
You can confirm that this is so by examining the S&P 500 contract listings in Figure 22.1 .
Although dividends of individual securities may fluctuate unpredictably, the annualized
dividend yield of a broad-based index such as the S&P 500 is fairly stable, recently in the
neighborhood of a bit more than 2% per year. The yield is seasonal, however, with regular
peaks and troughs, so the dividend yield for the relevant months must be the one used.
Figure 22.5 illustrates the yield pattern for the S&P 500. Some months, such as January or
April, have consistently low yields, while others, such as May, have consistently high ones. 6
We have described parity in terms of stocks and stock index futures, but it should be
clear that the logic applies as well to any financial futures contract. For gold futures, for
example, we would simply set the dividend yield to zero. For bond contracts, we would
let the coupon income on the bond play the role of dividend payments. In both cases, the
parity relationship would be essentially the same as Equation 22.2.
The arbitrage strategy described above should convince you that these parity relation-
ships are more than just theoretical results. Any violations of the parity relationship give
rise to arbitrage opportunities that can provide large profits to traders. We will see in the
next chapter that index arbitrage in the stock market is a tool to exploit violations of the
parity relationship for stock index futures contracts.
Spreads Just as we can predict the relationship between spot and futures prices, there are similar
ways to determine the proper relationships among futures prices for contracts of different
6 The very high dividend yield in November 2004 was due to Microsoft’s special one-time dividend of $3 per share.
0
1
2
3
4
5
6
7
Ja n
-0 1
Ju l-
0 1
Ja n
-0 2
Ju l-
0 2
Ja n
-0 3
Ju l-
0 3
Ja n
-0 4
Ju l-
0 4
Ja n
-0 5
Ju l-
0 5
Ja n
-0 6
Ju l-
0 6
Ja n
-0 7
Ju l-
0 7
Ja n
-0 8
Ju l-
0 8
Ja n
-0 9
Ju l-
0 9
Ja n
-1 0
Ju l-
1 0
Ja n
-1 1
Ju l-
1 1
Ja n
-1 2
Ju l-
1 2
Ja n
-1 3
D iv
id en
d Y
ie ld
( A
n n
u a li
ze d
, %
)
Figure 22.5 S&P 500 monthly dividend yield
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C H A P T E R 2 2 Futures Markets 789
maturity dates. Equation 22.2 shows that the futures price is in part determined by time
to maturity. If the risk-free rate is greater than the dividend yield (i.e., r f > d ), then the futures price will be higher on longer-maturity contracts and if r f < d, longer-maturity futures prices will be lower. You can confirm from Figure 22.1 that in 2013, when the
risk-free rate was below the dividend yield, the longer-maturity S&P 500 contract did
have a lower futures price than the shorter term contract. For futures on assets like gold,
which pay no “dividend yield,” we can set d 5 0 and conclude that F must increase as time to maturity increases.
To be more precise about spread pricing, call F ( T 1 ) the current futures price for delivery at date T 1 , and F ( T 2 ) the futures price for delivery at T 2 . Let d be the dividend yield of the stock. We know from the parity Equation 22.2 that
F(T1) 5 S0(1 1 rf 2 d) T1
F(T2) 5 S0(1 1 rf 2 d) T2
As a result,
F(T2)/F(T1) 5 (1 1 rf 2 d) (T2 2 T1)
Therefore, the basic parity relationship for spreads is
F(T2) 5 F(T1)(1 1 rf 2 d) (T2 2 T1) (22.3)
Equation 22.3 should remind you of the spot-futures parity relationship. The major difference
is in the substitution of F ( T 1 ) for the current spot price. The intuition is also similar. Delaying delivery from T 1 to T 2 assures the long position that the stock will be purchased for F ( T 2 ) dol- lars at T 2 but does not require that money be tied up in the stock until T 2 . The savings realized are the net cost of carry between T 1 and T 2 . Delaying delivery from T 1 until T 2 frees up F ( T 1 ) dollars, which earn risk-free interest at r f . The delayed delivery of the stock also results in the lost dividend yield between T 1 and T 2 . The net cost of carry saved by delaying the delivery is thus r f 2 d. This gives the proportional increase in the futures price that is required to com- pensate market participants for the delayed delivery of the stock and postponement of the pay-
ment of the futures price. If the parity condition for spreads is violated, arbitrage opportunities
will arise. (Problem 19 at the end of the chapter explores this possibility.)
To see how to use Equation 22.3, consider the following data for a hypothetical contract:
Contract Maturity Data Futures Price
January 15 $105.00 March 15 104.75
Suppose that the effective annual T-bill rate is 1% and that the dividend yield is 2% per year. The “correct” March futures price relative to the January price is, according to Equation 22.3,
105(1 1 .01 2 .02)1/6 5 104.82
The actual March futures price is 104.75, meaning that the March futures price is slightly underpriced compared to the January futures and that, aside from transaction costs, an arbitrage opportunity seems to be present.
Example 22.9 Spread Pricing
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790
Equation 22.3 shows that futures prices with different maturities should all move
together. This is not surprising because all are linked to the same spot price through the
parity relationship. Figure 22.6 plots futures prices on gold for three maturity dates. It is
apparent that the prices move in virtual lockstep and that the more distant delivery dates
command higher futures prices, as Equation 22.3 predicts.
eXcel APPLICATIONS: Parity and Spreads
T he parity spreadsheet allows you to calculate futures prices corresponding to a spot price for different
maturities, interest rates, and income yields. You can use the spreadsheet to see how prices of more distant con- tracts will fluctuate with spot prices and the cost of carry. You can learn more about this spreadsheet by using the version available on our Web site at www.mhhe.com/bkm .
Excel Questions
1. Experiment with different values for both income yield and interest rate. What happens to the size of the time spread
(the difference in futures prices for the long versus short maturity contracts) if the interest rate increases by 2%?
2. What happens to the time spread if the income yield increases by 2%?
3. What happens to the spread if the income yield equals the interest rate?
Spot Futures Parity and Time Spreads
Spot price Income yield (%) Interest rate (%) Today’s date Maturity date 1 Maturity date 2 Maturity date 3
Time to maturity 1 Time to maturity 2 Time to maturity 3
100
100.00 101.26 101.58 102.66
2 4.5
5/14/09 11/17/09
1/2/10 6/7/10
0.51 0.63 1.06
Futures prices versus maturity
Spot price Futures 1 Futures 2 Futures 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A B C D E
1500
1520
1540
1560
1580
1600
1620
1640
1660
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Date in July 2012
Fu tu
re s
P ri
ce ,
$ /o
u n
ce
Dec ‘12 delivery June ‘13 delivery Dec ‘13 delivery
Figure 22.6 Gold futures prices
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C H A P T E R 2 2 Futures Markets 791
Forward versus Futures Pricing Until now we have paid little attention to the differing time profile of returns of futures and
forward contracts. Instead, we have taken the sum of daily mark-to-market proceeds to the
long position as P T 2 F 0 and assumed for convenience that the entire profit accrues on the delivery date. Our parity theorems apply only to forward pricing because they assume that
contract proceeds are in fact realized only on delivery. In contrast, the actual timing of cash
flows conceivably might affect the futures price.
Futures prices will deviate from parity when marking to market gives a systematic
advantage to either the long or short position. If marking to market tends to favor the long
position, for example, the futures price should exceed the forward price, because the long
position will be willing to pay a premium for the advantage of marking to market.
When will marking to market favor either a long or short trader? A trader will benefit
if daily settlements are received (and can be invested) when the interest rate is high and
are paid (and can be financed) when the interest rate is low. Because long positions will
benefit if futures prices tend to rise when interest rates are high, they will be willing to
accept a higher futures price. Therefore, a positive correlation between interest rates and
changes in futures prices implies that the “fair” futures price will exceed the forward price.
Conversely, a negative correlation means that marking to market favors the short position
and implies that the equilibrium futures price should be below the forward price.
For most contracts, the covariance between futures prices and interest rates is so low
that the difference between futures and forward prices will be negligible. However, con-
tracts on long-term fixed-income securities are an important exception to this rule. In this
case, because prices have a high correlation with interest rates, the covariance can be large
enough to generate a meaningful spread between forward and future prices.
So far we have considered the relationship between futures prices and the current spot price. What about the relationship between the futures price and the expected value of the spot price? In other words, how well does the futures price forecast the ultimate spot
price? Three traditional theories have been put forth: the expectations hypothesis, normal
backwardation, and contango. Today’s consensus is that all of these traditional hypotheses
are subsumed by modern portfolio theory. Figure 22.7 shows the expected path of futures
under the three traditional hypotheses.
Expectations Hypothesis The expectations hypothesis is the simplest theory of futures pricing. It states that the futures price equals the expected value of the future spot price: F 0 5 E ( P T ). Under this theory the expected profit to either position of a futures contract would equal zero: The
short position’s expected profit is F 0 2 E ( P T ), whereas the long’s is E ( P T ) 2 F 0 . With F 0 5 E ( P T ), the expected profit to either side is zero. This hypothesis relies on a notion of risk neutrality. If all market participants are risk neutral, they should agree on a futures
price that provides an expected profit of zero to all parties.
The expectations hypothesis resembles market equilibrium in a world with no uncertainty;
that is, if prices of goods at all future dates were currently known, then the futures price for
delivery at any particular date would equal the currently known future spot price for that date.
It is a tempting but incorrect leap to then assert that under uncertainty the futures price should
equal the currently expected spot price. This view ignores the risk premiums that must be built into futures prices when ultimate spot prices are uncertain.
22.5 Futures Prices versus Expected Spot Prices
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792 P A R T V I Options, Futures, and Other Derivatives
Normal Backwardation This theory is associated with the
famous British economists John
Maynard Keynes and John Hicks.
They argued that for most com-
modities there are natural hedgers
who wish to shed risk. For exam-
ple, wheat farmers desire to shed
the risk of uncertain wheat prices.
These farmers will take short posi-
tions to deliver wheat at a guaran-
teed price; they will short hedge.
To induce speculators to take the
corresponding long positions, the
farmers need to offer them an
expectation of profit. They will
enter the long side of the contract
only if the futures price is below the
expected spot price of wheat, for
an expected profit of E ( P T ) 2 F 0 . The speculators’ expected profit
is the farmers’ expected loss, but
farmers are willing to bear this
expected loss to avoid the risk of uncertain wheat prices. The theory of normal backward- ation thus suggests that the futures price will be bid down to a level below the expected spot price and will rise over the life of the contract until the maturity date, at which point F T 5 P T .
Although this theory recognizes the important role of risk premiums in futures markets,
it is based on total variability rather than on systematic risk. (This is not surprising, as
Keynes wrote almost 40 years before the development of modern portfolio theory.) The
modern view refines the measure of risk used to determine appropriate risk premiums.
Contango The polar hypothesis to backwardation holds that the natural hedgers are the purchasers of
a commodity, rather than the suppliers. In the case of wheat, for example, we would view
grain processors as willing to pay a premium to lock in the price that they must pay for
wheat. These processors hedge by taking a long position in the futures market; therefore,
they are called long hedgers, whereas farmers are short hedgers. Because long hedgers
will agree to pay high futures prices to shed risk, and because speculators must be paid a
premium to enter the short position, the contango theory holds that F 0 must exceed E ( P T ). It is clear that any commodity will have both natural long hedgers and short hedgers.
The compromise traditional view, called the “net hedging hypothesis,” is that F 0 will be less than E ( P T ) when short hedgers outnumber long hedgers and vice versa. The strong side of the market will be the side (short or long) that has more natural hedgers. The strong
side must pay a premium to induce speculators to enter into enough contracts to balance
the “natural” supply of long and short hedgers.
Modern Portfolio Theory The three traditional hypotheses all envision a mass of speculators willing to enter either
side of the futures market if they are sufficiently compensated for the risk they incur.
Future Prices
Expectations Hypothesis
Contango
Normal Backwardation
Time
E(PT)
Delivery Date
Figure 22.7 Futures price over time, in the special case that the expected spot price remains unchanged
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Modern portfolio theory fine-tunes this approach by refining the notion of risk used in the
determination of risk premiums. Simply put, if commodity prices pose positive systematic
risk, futures prices must be lower than expected spot prices.
To illustrate this approach, consider once again a stock paying no dividends. If E ( P T ) denotes the expected time- T stock price and k denotes the required rate of return on the stock, then the price of the stock today must equal the present value of its expected future
payoff as follows:
P0 5 E(PT)
(1 1 k)T (22.4)
We also know from the spot-futures parity relationship that
P0 5 F0
(1 1 rf) T (22.5)
Therefore, the right-hand sides of Equations 22.4 and 22.5 must be equal. Equating these
terms allows us to solve for F 0 :
F0 5 E(PT)¢ 1 1 rf 1 1 k
≤T (22.6) You can see immediately from Equation 22.6 that F 0 will be less than the expectation of P T whenever k is greater than r f , which will be the case for any positive-beta asset. This means that the long side of the contract will make an expected profit [ F 0 will be lower than E ( P T )] when the commodity exhibits positive systematic risk ( k is greater than r f ).
Why should this be? A long futures position will provide a profit (or loss) of P T 2 F 0 . If the ultimate value of P T entails positive systematic risk, so will the profit to the long posi- tion. Speculators with well-diversified portfolios will be willing to enter long futures posi-
tions only if they receive compensation for bearing that risk in the form of positive expected
profits. Their expected profits will be positive only if E ( P T ) is greater than F 0 . The short position’s profit is the negative of the long’s and will have
negative systematic risk. Diversified investors in the short
position will be willing to suffer that expected loss to
lower portfolio risk and will be willing to enter the con-
tract even when F 0 is less than E ( P T ). Therefore, if P T has positive beta, F 0 must be less than the expectation of P T . The analysis is reversed for negative-beta commodities.
What must be true of the risk of the spot price of an asset if the futures price is an unbiased esti- mate of the ultimate spot price?
CONCEPT CHECK 22.6
1. Forward contracts call for future delivery of an asset at a currently agreed-on price. The long trader purchases the good, and the short trader delivers it. If the price of the asset at the maturity
of the contract exceeds the forward price, the long side benefits by virtue of acquiring the good at
the contract price.
2. A futures contract is similar to a forward contract, differing most importantly in the aspects of standardization and marking to market, which is the process by which gains and losses on futures
contract positions are settled daily. In contrast, forward contracts call for no cash transfers until
contract maturity.
3. Futures contracts are traded on organized exchanges that standardize the size of the contract, the grade of the deliverable asset, the delivery date, and the delivery location. Traders negotiate only
over the contract price. This standardization increases liquidity and means that buyers and sellers
can easily find many traders for a desired purchase or sale.
SUMMARY
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794 P A R T V I Options, Futures, and Other Derivatives
4. The clearinghouse steps in between each pair of traders, acting as the short position for each long and as the long position for each short. In this way traders need not be concerned about the perfor-
mance of the trader on the opposite side of the contract. In turn, traders post margins to guarantee
their own performance.
5. The long position’s gain or loss between time 0 and time t is F t 2 F 0 . Because F T 5 P T , the long’s profit if the contract is held until maturity is P T 2 F 0 , where P T is the spot price at time T and F 0 is the original futures price. The gain or loss to the short position is F 0 2 P T .
6. Futures contracts may be used for hedging or speculating. Speculators use the contracts to take a stand on the ultimate price of an asset. Short hedgers take short positions in contracts to offset any
gains or losses on the value of an asset already held in inventory. Long hedgers take long posi-
tions to offset gains or losses in the purchase price of a good.
7. The spot-futures parity relationship states that the equilibrium futures price on an asset providing no service or payments (such as dividends) is F 0 5 P 0 (1 1 r f )
T . If the futures price deviates from
this value, then market participants can earn arbitrage profits.
8. If the asset provides services or payments with yield d, the parity relationship becomes F 0 5 P 0 (1 1 r f 2 d )
T . This model is also called the cost-of-carry model, because it states that
futures price must exceed the spot price by the net cost of carrying the asset until maturity date T.
9. The equilibrium futures price will be less than the currently expected time T spot price if the spot price exhibits systematic risk. This provides an expected profit for the long position who bears the
risk and imposes an expected loss on the short position who is willing to accept that expected loss
as a means to shed systematic risk.
Related Web sites for this chapter are available at www. mhhe.com/bkm
forward contract
futures price
long position
short position
single-stock futures
clearinghouse
open interest
marking to market
maintenance margin
convergence property
cash settlement
basis
basis risk
calendar spread
spot-futures parity theorem
cost-of-carry relationship
KEY TERMS
KEY EQUATIONS Spot-futures parity: F 0 ( T ) 5 S 0 (1 1 r 2 d ) T
Futures spread parity: F0(T2) 5 F0(T1) (1 1 r 2 d) (T2 2 T1)
Futures vs. expected spot prices: F0 5 E(PT) ¢1 1 rf 1 1 k
≤T
PROBLEM SETS 1. Why is there no futures market in cement? 2. Why might individuals purchase futures contracts rather than the underlying asset?
3. What is t h e difference in cash flow between short-selling an asset and entering a short futures
position?
4. Are the following statements true or false? Why?
a. All else equal, the futures price on a stock index with a high dividend yield should be higher than the futures price on an index with a low dividend yield.
b. All else equal, the futures price on a high-beta stock should be higher than the futures price on a low-beta stock.
c. The beta of a short position in the S&P 500 futures contract is negative.
5. What is the difference between the futures price and the value of the futures contract?
Basic
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6. Evaluate the criticism that futures markets siphon off capital from more productive uses.
7. a. Turn to the S&P 500 contract in Figure 22.1 . If the margin requirement is 10% of t he futures price times the multiplier of $250, how much must you deposit with your broker to trade the
March maturity contract?
b. If the March futures price were to increase to 1,498, what percentage return would you earn on your net investment if you entered the long side of the contract at the price shown in the
figure?
c. If the March futures price falls by 1%, what is your percentage return?
8. a. A single-stock futures contract on a non-dividend-paying stock with current price $150 has a maturity of 1 year. If the T-bill rate is 3%, what should the futures price be?
b. What should the futures price be if the maturity of the contract is 3 years? c. What if the interest rate is 6% and the maturity of the contract is 3 years?
9. How might a portfolio manager use financial futures to hedge risk in each of the following
circumstances:
a. You own a large position in a relatively illiquid bond that you want to sell. b. You have a large gain on one of your Treasuries and want to sell it, but you would like to
defer the gain until the next tax year.
c. You will receive your annual bonus next month that you hope to invest in long-term corpo-
rate bonds. You believe that bonds today are selling at quite attractive yields, and you are
concerned that bond prices will rise over the next few weeks.
10. Suppose the value of the S&P 500 stock index is currently 1,400. If the 1-year T-bill rate is 3%
and the expected dividend yield on the S&P 500 is 2%, what should the 1-year maturity futures
price be? What if the T-bill rate is less than the dividend yield, for example, 1%?
11. Consider a stock that pays no dividends on which a futures contract, a call option, and a put
option trade. The maturity date for all three contracts is T, the exercise price of both the put and the call is X, and the futures price is F. Show that if X 5 F, then the call price equals the put price. Use parity conditions to guide your demonstration.
12. It is now January. The current interest rate is 2%. The June futures price for gold is $1,500,
whereas the December futures price is $1,510. Is there an arbitrage opportunity here? If so, how
would you exploit it?
13. OneChicago has just introduced a single-stock futures contract on Brandex stock, a company
that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in
1 year. The T-bill rate is 6% per year.
a. If Brandex stock now sells at $120 per share, what should the futures price be? b. If the Brandex price drops by 3%, what will be the change in the futures price and the change
in the investor’s margin account?
c. If the margin on the contract is $12,000, what is the percentage return on the investor’s position?
14. The multiplier for a futures contract on a stock market index is $250. The maturity of the
contract is 1 year, the current level of the index is 1,300, and the risk-free interest rate is .5% per
month. The dividend yield on the index is .2% per month. Suppose that after 1 month, the stock
index is at 1,320.
a. Find the cash flow from the mark-to-market proceeds on the contract. Assume that the parity condition always holds exactly.
b. Find the holding-period return if the initial margin on the contract is $13,000.
15. You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in
3 months. You believe rates will soon fall, and you would like to repurchase the company’s
sinking fund bonds (which currently are selling below par) in advance of requirements. Unfor-
tunately, you must obtain approval from the board of directors for such a purchase, and this can
take up to 2 months. What action can you take in the futures market to hedge any adverse move-
ments in bond yields and prices until you can actually buy the bonds? Will you be long or short?
Why? A qualitative answer is fine.
Intermediate
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796 P A R T V I Options, Futures, and Other Derivatives
16. The S&P portfolio pays a dividend yield of 1% annually. Its current value is 1,500. The T-bill
rate is 4%. Suppose the S&P futures price for delivery in 1 year is 1,550. Construct an arbitrage
strategy to exploit the mispricing and show that your profits 1 year hence will equal the mispric-
ing in the futures market.
17. The Excel Application box in the chapter (available at www.mhhe.com/bkm ; link to Chapter 22 material) shows how to use the spot-futures parity relationship to find a “term structure of
futures prices,” that is, futures prices for various maturity dates.
a. Suppose that today is January 1, 2013. Assume the interest rate is 3% per year and a stock index currently at 1,500 pays a dividend yield of 1.5%. Find the futures price for contract
maturity dates of February 14, 2013, May 21, 2013, and November 18, 2013.
b. What happens to the term structure of futures prices if the dividend yield is higher than the risk-free rate? For example, what if the dividend yield is 4%?
18. a. How should t he parity condition (Equation 22.2) for stocks be modified for futures contracts on Treasury bonds? What should play the role of the dividend yield in that equation?
b. In an environment with an upward-sloping yield curve, should T-bond futures prices on more-distant contracts be higher or lower than those on near-term contracts?
c. Confirm your intuition by examining Figure 22.1 .
19. Consider this arbitrage strategy to derive the parity relationship for spreads: (1) enter a long
futures position with maturity date T 1 and futures price F ( T 1 ); (2) enter a short position with maturity T 2 and futures price F ( T 2 ); (3) at T 1 , when the first contract expires, buy the asset and borrow F ( T 1 ) dollars at rate r f ; (4) pay back the loan with interest at time T 2 .
a. What are the total cash flows to this strategy at times 0, T 1 , and T 2 ? b. Why must profits at time T 2 be zero if no arbitrage opportunities are present? c. What must the relationship between F ( T 1 ) and F ( T 2 ) be for the profits at T 2 to be equal to
zero? This relationship is the parity relationship for spreads.
e X c e l Please visit us at www.mhhe.com/bkm
Challenge
1. Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indi-
cated by the following information:
Spot price for commodity $120
Futures price for commodity expiring in 1 year $125
Interest rate for 1 year 8%
a. Describe the transactions necessary to take advantage of this specific arbitrage opportunity. b. Calculate the arbitrage profit.
2. Michelle Industries issued a Swiss franc–denominated 5-year discount note for SFr200 million.
The proceeds were converted to U.S. dollars to purchase capital equipment in the United States.
The company wants to hedge this currency exposure and is considering the following alternatives:
• At-the-money Swiss franc call options.
• Swiss franc forwards.
• Swiss franc futures.
a. Contrast the essential characteristics of each of these three derivative instruments. b. Evaluate the suitability of each in relation to Michelle’s hedging objective, including both
advantages and disadvantages.
3. Identify the fundamental distinction between a futures contract and an option contract, and
briefly explain the difference in the manner that futures and options modify portfolio risk.
4. Maria VanHusen, CFA, suggests that using forward contracts on fixed-income securities can
be used to protect the value of the Star Hospital Pension Plan’s bond portfolio against the pos-
sibility of rising interest rates. VanHusen prepares the following example to illustrate how such
protection would work:
• A 10-year bond with a face value of $1,000 is issued today at par value. The bond pays
an annual coupon.
• An investor intends to buy this bond today and sell it in 6 months.
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C H A P T E R 2 2 Futures Markets 797
• The 6-month risk-free interest rate today is 5% (annualized).
• A 6-month forward contract on this bond is available, with a forward price of $1,024.70.
• In 6 months, the price of the bond, including accrued interest, is forecast to fall to
$978.40 as a result of a rise in interest rates.
a. Should the investor buy or sell the forward contract to protect the value of the bond against rising interest rates during the holding period?
b. Calculate the value of the forward contract for the investor at the maturity of the forward con- tract if VanHusen’s bond-price forecast turns out to be accurate.
c. Calculate the change in value of the combined portfolio (the underlying bond and the appro- priate forward contract position) 6 months after contract initiation.
5. Sandra Kapple asks Maria VanHusen about using futures contracts to protect the value of the Star
Hospital Pension Plan’s bond portfolio if interest rates rise. VanHusen states:
a. “Selling a bond futures contract will generate positive cash flow in a rising interest rate envi- ronment prior to the maturity of the futures contract.”
b. “The cost of carry causes bond futures contracts to trade for a higher price than the spot price of the underlying bond prior to the maturity of the futures contract.”
Comment on the accuracy of each of VanHusen’s two statements.
E-INVESTMENTS EXERCISES Go to the Chicago Mercantile Exchange site at www.cme.com . From the Products & Trading tab, select the link to Equity Index, and then link to the NASDAQ-100 E-mini contract. Now find the tab for Contract Specifications.
1. What is the contract size for the futures contract?
2. What is the settlement method for the futures contract?
3. For what months are the futures contracts available?
4. Click the link to View Price Limits and then U.S. Equity Price Limits. What is the current value of the 10% price limit for this contract?
5. Click on View Calendar. What is the settlement date of the shortest-maturity outstand- ing contract? The longest-maturity contract?
SOLUTIONS TO CONCEPT CHECKS
Payoff, Profit
Short Futures
F0 PT
Payoff
Buy (Long) Put
X PT
Payoff
Write (Sell) Call
X PT
1.
2. The clearinghouse has a zero net position in all contracts. Its long and short positions are
offsetting, so that net cash flow from marking to market must be zero.
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798 P A R T V I Options, Futures, and Other Derivatives
3. Oil Price in February, PT
$89.86 $91.86 $93.86
Cash flow to purchase oil: 2100,000 3 PT 2$8,986,000 2$9,186,000 2$9,386,000
1 Profit on long futures: 100,000 3 (PT 2 F0) 2200,000 0 1200,000
TOTAL CASH FLOW 2$9,186,000 2$9,186,000 2$9,186,000
4. The risk would be that the index and the portfolio do not move perfectly together. Thus basis
risk involving the spread between the futures price and the portfolio value could persist even
if the index futures price were set perfectly relative to the index itself. You can measure this
risk using the index model. If you regress the return of the active portfolio on the return of the
index portfolio, the regression R-square will equal the proportion of the variance of the active portfolio’s return that could have been hedged using the index futures contract. You can also
measure the risk of the imperfectly hedged position using the standard error of the regression,
which tells you the standard deviation of the residuals in the index model. Because these are the
components of the risky returns that are independent of the market index, this standard deviation
measures the variability of the portion of the active portfolio’s return that cannot be hedged using the index futures contract.
5. The futures price, $980, is $10 below the parity value of $990. The cash flow to the following
strategy is riskless and equal to this mispricing.
Action Initial Cash Flow Cash Flow in 1 Year
Lend S0 dollars 21,000 1,000(1.01) 5 1,010
Sell stock short 11,000 2ST 2 20
Long futures 0 ST 2 980
TOTAL 0 $10 risklessly
6. It must have zero beta. If the futures price is an unbiased estimator, then we infer that it has a zero
risk premium, which means that beta must be zero.
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233 PA
R T V
I
THE PREVIOUS CHAPTER provided a basic introduction to the operation of futures mar- kets and the principles of futures pricing. This chapter explores both pricing and risk man- agement in selected futures markets in more depth. Most of the growth has been in finan- cial futures, which dominate trading, so we emphasize these contracts.
Hedging refers to techniques that offset particular sources of risk. Hedging activities therefore are more limited and more focused than more ambitious strategies seeking an optimal risk-return profile for an entire port- folio. Because futures contracts are written on specific quantities such as stock index values, foreign exchange rates, commodity prices, and so on, they are ideally suited for these applications. In this chapter we will con- sider several hedging applications, illustrating
general principles using a variety of contracts. We also show how hedging strategies can be used to isolate bets on perceived profit opportunities.
We begin with foreign exchange futures, where we show how forward exchange rates are determined by interest rate differentials across countries and examine how firms can use futures to manage exchange rate risk. We then move on to stock-index futures, where we focus on program trading and index arbitrage. Next we turn to the most actively traded markets, those for interest rate futures. We also examine commodity futures pricing. Finally, we turn to swaps markets in foreign exchange and fixed-income securities. We will see that swaps can be interpreted as portfolios of forward contracts and valued accordingly.
Futures, Swaps, and Risk
Management
CHAPTER TWENTY-THREE
The Markets Exchange rates between currencies vary continually and often substantially. This
variability can be a source of concern for anyone involved in international business. A U.S.
exporter who sells goods in England, for example, will be paid in British pounds, and the
dollar value of those pounds depends on the exchange rate at the time payment is made.
Until that date, the U.S. exporter is exposed to foreign exchange rate risk. This risk can
23.1 Foreign Exchange Futures
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800 P A R T V I Options, Futures, and Other Derivatives
be hedged through currency futures or forward markets. For example, if you know you
will receive £100,000 in 90 days, you can sell those pounds forward today in the forward
market and lock in an exchange rate equal to today’s forward price.
The forward market in foreign exchange is fairly informal. It is simply a network of
banks and brokers that allows customers to enter forward contracts to purchase or sell
currency in the future at a currently agreed-upon rate of exchange. The bank market in
currencies is among the largest in the world, and most large traders with sufficient credit-
worthiness execute their trades here rather than in futures markets. Unlike those in futures
markets, contracts in forward markets are not standardized in a formal market setting.
Instead, each is negotiated separately. Moreover, there is no marking to market, as would
occur in futures markets. Cur-
rency forward contracts call for
execution only at the maturity
date. Participants need to con-
sider counterparty risk, the pos- sibility that a trading partner may
not be able to make good on its
obligations under the contract if
prices move against it. For this
reason, traders who participate in
forward markets must have solid
creditworthiness.
Currency futures, however, trade in formal exchanges such as
the Chicago Mercantile Exchange
(International Monetary Market)
or the London International Finan-
cial Futures Exchange (LIFFE).
Here contracts are standardized by
size, and daily marking to market
is observed. Moreover, standard
clearing arrangements allow trad-
ers to enter or reverse positions
easily. Margin positions are used
to ensure contract performance,
which is in turn guaranteed by the
exchange’s clearinghouse, so the
identity and creditworthiness of
the counterparty to a trade are less
of a concern.
Figure 23.1 reproduces The Wall Street Journal listing of foreign exchange spot and for-
ward rates. The listing gives the
number of U.S. dollars required
to purchase some unit of foreign
currency and then the amount of
foreign currency needed to pur-
chase $1. Figure 23.2 reproduces
futures listings, which show the
Americas Argentina peso .2012 4.9709 1.1 Brazil real .5012 1.9954 –2.6 Canada dollar .9939 1.0062 1.4 Chile peso .002112 473.60 –1.2 Colombia peso .0005618 1780.00 0.7 Ecuador US dollar 1 1 unch Mexico peso .0783 12.7679 –0.7 Peru new sol .3905 2.561 0.6 Uruguay peso .05182 19.2990 1.0 Venezuela b. fuerte .229885 4.3500 unch
Middle East/Africa Bahrain dinar 2.6531 .3769 –0.1 Eqypt pound .1502 6.6558 4.7 Israel shekel .2684 3.7259 –0.2 Jordan dinar 1.4111 .7087 –0.2 Kuwait dinar 3.5503 .2817 0.1 Lebanon pound .0006649 1503.95 0.1 Saudi Arabia riyal .2666 3.7504 unch South Africa rand .1099 9.1021 7.6 UAE dirham .2722 3.6732 unch
WSJ Dollar Index 71.04 0.07 0.09 0.82
Country/currency
U.S.-dollar foreign-exchange rates in late New York trading
Currencies
Sources: ICAP plc., WSJ Market Data Group
Asia-Pacific Australian dollar 1.0417 .9600 –0.2 1-mos forward 1.0395 .9620 –0.3 3-mos forward 1.0349 .9663 –0.2 6-mos forward 1.0282 .9726 –0.2 China yuan .1607 6.2226 –0.1 Hong Kong dollar .1289 7.7572 0.1 India rupee .01850 54.055 –1.7 Indonesia rupiah .0001033 9676 0.4 Japan yen .011007 90.85 4.7 1-mos forward .011009 90.83 4.5 3-mos forward .011014 90.79 4.5 6-mos forward .011024 90.71 4.5 Malaysia ringgit .3284 3.0450 –0.4 New Zealand dollar .8340 1.1990 –0.7 Pakistan rupee .01022 97.845 0.6 Philippines peso .0244 40.936 –0.2 Singapore dollar .8077 1.2380 1.4 South Korea won .0009158 1091.92 2.6 Taiwan dollar .03390 29.500 1.6 Thailand baht .03345 29.896 –2.2 Vietnam dong .00004798 20840 0.3
Europe Czech Rep. koruna .05238 19.091 0.4 Denmark krone .1803 5.5448 –1.9 Euro area euro 1.3455 .7432 –1.9 Hungary forint .004521 221.21 0.2 Norway krone .1807 5.5328 –0.6 Poland zloty .3204 3.1207 0.9 Russia ruble .03314 30.178 –1.0 Sweden krona .1558 6.4197 –1.3 Switzerland franc 1.0796 .9263 1.2 1-mos forward 1.0799 .9260 1.2 3-mos forward 1.0806 .9254 1.3 6-mos forward 1.0819 .9243 1.3 Turkey lira .5640 1.7729 –0.6 UK pound 1.5694 .6372 3.5
1-mos forward 1.5692 .6373 3.4 3-mos forward 1.5687 .6375 3.4 6-mos forward 1.5681 .6377 3.4
in US$ per US$
US$ vs. YTD chg
(%) Mon
Country/currency in US$ per US$
US$ vs. YTD chg
(%) Mon
Close Net Chg %ChgYTD%Chg
January 25, 2013
Figure 23.1 Spot and forward exchange rates Source: The Wall Street Journal online, January 25, 2013. Reprinted by permission of The Wall Street Journal, © 2013 Dow Jones & Company, Inc. All rights reserved worldwide.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 801
number of dollars needed to purchase a given unit of
foreign currency. In Figure 23.1 , both spot and forward
exchange rates are listed for various delivery dates.
The forward quotations listed in Figure 23.1 apply to
rolling delivery in 30, 90, or 180 days. Thus tomorrow’s
forward listings will apply to a maturity date 1 day later
than today’s listing. In contrast, the futures contracts in
Figure 23.2 mature on only four dates each year, in March,
June, September, and December.
Interest Rate Parity As is true of stocks and stock futures, there is a spot-
futures exchange rate relationship that will prevail in well-
functioning markets. Should this so-called interest rate parity relationship be violated, arbitrageurs will be able to make risk-free profits in foreign exchange markets with
zero-net-investment. Their actions will force futures and spot
exchange rate back into alignment. Another term for interest
rate parity is the covered interest arbitrage relationship. We can illustrate the interest rate parity theorem by using two currencies, the U.S. dollar
and the British (U.K.) pound. Call E 0 the current exchange rate between the two curren- cies, that is, E 0 dollars are required to purchase one pound. F 0 , the forward price, is the number of dollars agreed to today for purchase of one pound at time T. Call the risk-free rates in the United States and United Kingdom r US and r UK , respectively.
The interest rate parity theorem then states that the proper relationship between E 0 and F 0 is
F0 5 E0 ¢ 1 1 rUS 1 1 rUK
≤ T
(23.1)
For example, if r US 5 .04 and r UK 5 .05 annually, while E 0 5 $2 per pound, then the proper futures price for a 1-year contract would be
$2.00a1.04 1.05
b 5 $1.981 per pound Consider the intuition behind Equation 23.1. If r US is less than r UK , money invested in
the United States will grow at a slower rate than money invested in the United Kingdom.
If this is so, why wouldn’t all investors decide to invest their money in the United Kingdom?
One important reason why not is that the dollar may be appreciating relative to the pound.
Although dollar investments in the United States grow slower than pound investments in
the United Kingdom, each dollar may be worth more pounds in the forward market than
in the spot market. Such a forward premium can exactly offset the advantage of the higher
U.K. interest rate.
To complete the argument, we ask how an appreciating dollar would show up in Equation
23.1. If the dollar is appreciating, fewer dollars are required to purchase each pound, and the
forward exchange rate F 0 (in dollars per pound) will be less than E 0 , the current exchange rate. This is exactly what Equation 23.1 tells us: When r US is less than r UK , F 0 must be less than E 0 . The forward premium of the dollar embodied in the ratio of F 0 to E 0 exactly compensates for the difference in interest rates available in the two countries. Of course, the
argument also works in reverse: If r US is greater than r UK , then F 0 is greater than E 0 .
Japanese Yen(CME) - ¥12,500,000; $ per 100¥ March 1.0675 1.0854 1.0672 1.0777 .0086 217,367 June 1.0687 1.0861 1.0683 1.0785 .0086 2,320 Canadian Dollar (CME) - CAD 100,000; $ per CAD March 1.0016 1.0021 .9954 .9964 2.0047 131,433 June .9994 .9998 .9939 .9944 2.0047 4,071 British Pound (CME) - £62,500; $ per £ March 1.5711 1.5842 1.5707 1.5794 .0086 165,688 June 1.5699 1.5817 1.5697 1.5787 .0086 457 Swiss Franc (CME) - CHF125,000; $ per CHF March 1.0888 1.0929 1.0886 1.0907 .0008 41,574 June 1.0913 1.0924 1.0912 1.0919 .0008 73
Open Settle Chg Open
interestHigh hi lo low Contract
Figure 23.2 Foreign exchange futures Source: The Wall Street Journal, February 11, 2013. Reprinted by permission of The Wall Street Journal, © 2013 Dow Jones & Company, Inc. All rights reserved worldwide.
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802 P A R T V I Options, Futures, and Other Derivatives
To generalize the strategy in Example 23.1:
Action Initial CF ($) CF in 1 Year ($)
1. Borrow 1 U.K. pound in London. Convert to dollars. $E 0 2$E1(1 1 rUK) 2. Use proceeds of borrowing in London to lend in
the U.S. 2$E 0 $E 0(1 1 rUS)
3. Enter (1 1 rUK) futures positions to purchase 1 pound for F0 dollars. 0 (1 1 rUK)(E1 2 F0)
TOTAL 0 E0(1 1 rUS) 2 F0(1 1 rUK)
Let us again review the stages of the arbitrage operation. The first step requires borrow-
ing one pound in the United Kingdom. With a current exchange rate of E 0 , the one pound is converted into E 0 dollars, which is a cash inflow. In 1 year the British loan must be paid off with interest, requiring a payment in pounds of (1 1 r UK ), or in dollars of E 1 (1 1 r UK ). In the second step the proceeds of the British loan are invested in the United States. This
involves an initial cash outflow of $ E 0 , and a cash inflow of $ E 0 (1 1 r US ) in 1 year. Finally, the exchange risk involved in the British borrowing is hedged in step 3. Here, the (1 1 r UK ) pounds necessary to satisfy the British loan are purchased ahead in the futures contract.
The net proceeds to the arbitrage portfolio are risk-free and given by E 0 (1 1 r US ) 2 F 0 (1 1 r UK ). If this value is positive, borrow in the United Kingdom, lend in the United States, and enter a long futures position to eliminate foreign exchange risk. If the
value is negative, borrow in the United States, lend in the United Kingdom, and take a
What if the interest rate parity relationship were violated? For example, suppose the futures price is $1.97/£ instead of $1.981/£. You could adopt the following strategy. In this example let E 1 denote the exchange rate ($/£) that will prevail in 1 year. E 1 is, of course, a random variable from the perspective of today’s investors.
Action Initial Cash Flow ($) CF in 1 Year ($)
1. Borrow 1 U.K. pound in London. Convert to dollars. Repay £1.05 at year-end.
2.00 2E1(£1.05)
2. Lend $2.00 in the United States. 22.00 $2.00(1.04) 3. Enter a contract to purchase £1.05 at
a (futures) price of F0 5 $1.97/£. 0 £1.05(E1 2 $1.97/£) TOTAL 0 $.0115
In step 1, you exchange the one pound borrowed in the United Kingdom for $2 at the current exchange rate. After 1 year you must repay the pound borrowed with interest. Because the loan is made in the United Kingdom at the U.K. interest rate, you would repay £1.05, which would be worth E 1 (1.05) dollars. The U.S. loan in step 2 is made at the U.S. interest rate of 4%. The futures position in step 3 results in receipt of £1.05, for which you would pay $1.97 each, and then convert into dollars at exchange rate E 1 .
Note that the exchange rate risk here is exactly offset between the pound obligation in step 1 and the futures position in step 3. The profit from the strategy is therefore risk- free and requires no net investment.
Example 23.1 Covered Interest Arbitrage
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 803
short position in pound futures. When prices preclude arbitrage opportunities, the expres-
sion must equal zero. This no-arbitrage condition implies that
F0 5 1 1 rUS 1 1 rUK
E0 (23.2)
which is the interest rate parity theorem for a 1-year horizon.
What would be the arbitrage strategy and associated profits in Example 23.1 if the initial futures price were F 0 5 $2.01/pound?
CONCEPT CHECK 23.1
Direct versus Indirect Quotes The exchange rate in Examples 23.1 and 23.2 is expressed as dollars per pound. This is an
example of a direct exchange rate quote. The euro-dollar exchange rate is also typically expressed as a direct quote. In contrast, exchange rates for other currencies such as the
Japanese yen or Swiss franc are typically expressed as indirect quotes, that is, as units of foreign currency per dollar, for example, 92 yen per dollar. For currencies expressed as
indirect quotes, depreciation of the dollar would result in a decrease in the quoted exchange rate ($1 buys fewer yen); in contrast, dollar depreciation versus the pound would show up
as a higher exchange rate (more dollars are required to buy £1). When the exchange rate is quoted as foreign currency per dollar, the domestic and foreign exchange rates in Equation
23.2 must be switched: in this case the equation becomes
F0 (foreign currency/$) 5 1 1 rforeign
1 1 rUS 3 E0 (foreign currency/$)
If the interest rate in the U.S. is higher than in Japan, the dollar will sell in the forward
market at a lower price (will buy fewer yen) than in the spot market.
Using Futures to Manage Exchange Rate Risk Consider a U.S. firm that exports most of its product to Great Britain. The firm is vulner-
able to fluctuations in the dollar/pound exchange rate for several reasons. First, the dollar
value of the pound-denominated revenue derived from its customers will fluctuate with
the exchange rate. Second, the pound price that the firm can charge its customers in the
United Kingdom will itself be affected by the exchange rate. For example, if the pound
Ample evidence bears out the interest-rate parity relation. For example, on February 3, 2013, the dollar-denominated LIBOR interest rate with maturity of 6 months was .48% while the comparable U.K. pound-denominated rate was higher, at .65%. Therefore, we should have expected the forward exchange rate (in $/£) to be lower than the spot rate. This is exactly what we observed: While the spot rate was $1.5694/£, the forward rate was $1.5681/£. More specifically, interest-rate parity would imply that the forward rate should have been 1.5694 3 (1.0048/1.0065) 1/2 5 1.5681, just equal to the actual rate.
Example 23.2 Covered Interest Arbitrage
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804 P A R T V I Options, Futures, and Other Derivatives
depreciates by 10% relative to the dollar, the firm would need to increase the pound price
of its goods by 10% in order to maintain the dollar-equivalent price. However, the firm
might not be able to raise the price by 10% if it faces competition from British produc-
ers, or if it believes the higher pound-denominated price would reduce demand for its
product.
To offset its foreign exchange exposure, the firm might engage in transactions that bring
it profits when the pound depreciates. The lost profits from business operations resulting
from a depreciation will then be offset by gains on its financial transactions. For example,
if the firm enters a futures contract to deliver pounds for dollars at an exchange rate agreed
to today, then if the pound depreciates, the futures position will yield a profit.
To illustrate, suppose that the futures price is currently $2 per pound for delivery in
3 months. If the firm enters a futures contract with a futures price of $2 per pound, and
the exchange rate in 3 months is $1.90 per pound, then the profit to the short position is
F 0 2 F T 5 $2.00 2 $1.90 5 $.10 per pound. How many pounds should be sold in the futures market to most fully offset the exposure
to exchange rate fluctuations? Suppose the dollar value of profits in the next quarter will
fall by $200,000 for every $.10 depreciation of the pound. To hedge, we need a futures
position that provides $200,000 extra profit for every $.10 that the pound depreciates. Therefore, we need a futures position to deliver £2,000,000. As we have just seen, the
profit per pound on the futures contract equals the difference in the current futures price
and the ultimate exchange rate; therefore, the foreign exchange profits resulting from a
$.10 depreciation 1 will equal $.10 3 2,000,000 5 $200,000.
The proper hedge position in pound futures is independent of the actual depreciation in
the pound as long as the relationship between profits and exchange rates is approximately
linear. For example, if the pound depreciates by only half as much, $.05, the firm would
lose only $100,000 in operating profits. The futures position would also return half
the profits: $.05 3 2,000,000 5 $100,000, again just offsetting the operating exposure.
If the pound appreciates, the hedge position still (unfortunately in this case) offsets the operating exposure. If the pound appreciates by $.05, the firm might gain $100,000 from
the enhanced value of the pound; however, it will lose that amount on its obligation to
deliver the pounds for the original futures price.
The hedge ratio is the number of futures positions necessary to hedge the risk of the
unprotected portfolio, in this case the firm’s export business. In general, we can think of
the hedge ratio as the number of hedging vehicles (e.g., futures contracts) one would establish to offset the risk of a particular unprotected position. The hedge ratio, H, in this case is
H 5 Change in value of unprotected position for a given change in exchange rate
Profit derived from one futures position for the same change in exchange rate
5 $200,000 per $.10 change in $/£ exchange rate
$.10 profit per pound delivered per $.10 change in $/£ exchange rate
5 2,000,000 pounds to be delivered
Because each pound-futures contract on the Chicago Mercantile Exchange calls for
delivery of 62,500 pounds, you would sell 2,000,000/62,500 per contract 5 32 contracts.
1 Actually, the profit on the contract depends on the changes in the futures price, not the spot exchange rate.
For simplicity, we call the decline in the futures price the depreciation in the pound.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 805
One interpretation of the hedge ratio is as a ratio of sensitivities to the underlying source
of uncertainty. The sensitivity of operating profits is $200,000 per swing of $.10 in the
exchange rate. The sensitivity of futures profits is $.10 per pound to be delivered per swing
of $.10 in the exchange rate. Therefore, the hedge ratio is 200,000/.10 5 2,000,000 pounds.
We could just as easily have defined the hedge ratio in terms of futures contracts.
Because each contract calls for delivery of 62,500 pounds, the profit on each contract per
swing of $.10 in the exchange rate is $6,250. Therefore, the hedge ratio defined in units of
futures contracts is $200,000/$6,250 5 32 contracts, as we found above.
Suppose a U.S. investor is harmed when the dollar depreciates. Specifically, suppose that its profits decrease by $200,000 for every $.05 rise in the dollar/pound exchange rate. How many contracts should the firm enter? Should it take the long side or the short side of the contracts?
CONCEPT CHECK 23.2
Given the sensitivity of the unhedged position to changes in the exchange rate, calculat-
ing the risk-minimizing hedge position is easy. Estimating that sensitivity is much harder.
For the exporting firm, for example, a naive view might focus only on the expected pound-
denominated revenue, and then contract to deliver that number of pounds in the futures or
forward market. This approach, however, fails to recognize that pound revenue is itself a
function of the exchange rate because the U.S. firm’s competitive position in the U.K. is
determined in part by the exchange rate.
One approach relies, in part, on historical relationships. Suppose, for example, that the
firm prepares a scatter diagram as in Figure 23.3 that relates its business profits (mea-
sured in dollars) in each of the last 40 quarters to the dollar/pound exchange rate in that
Profits per Quarter
$2.2 million
$2.0 million
$2.00/£$1.90/£ Exchange Rate
Slope = 2 million
Figure 23.3 Profits as a function of the exchange rate
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806 P A R T V I Options, Futures, and Other Derivatives
quarter. Profits generally are lower when the exchange rate is lower, that is, when the
pound depreciates. To quantify that sensitivity, we might estimate the following regres-
sion equation:
Profits 5 a 1 b($/£ exchange rate)
The slope of the regression, the estimate of b, is the sensitivity of quarterly profits to the exchange rate. For example, if the estimate of b turns out to be 2,000,000, as in Figure 23.3 , then on average, a $1 increase in the value of the pound results in a $2,000,000 increase in quarterly profits. This, of course, is the sensitivity we posited when we asserted that a $.10
drop in the dollar/pound exchange rate would decrease profits by $200,000.
Of course, one must interpret regression output with care. For example, one would not
want to extrapolate the historical relationship between profitability and exchange rates
exhibited in a period when the exchange rate hovered between $1.80 and $2.10 per pound
to scenarios in which the exchange rate might be forecast at below $1.40 per pound or
above $2.50 per pound.
In addition, extrapolating past relationships into the future can be dangerous. We saw
in Chapter 8 that regression betas from the index model tend to vary over time; such prob-
lems are not unique to the index model. Moreover, regression estimates are just that—
estimates. Parameters of a regression equation are sometimes measured with considerable
imprecision.
Still, historical relationships are often a good place to start when looking for the average
sensitivity of one variable to another. These slope coefficients are not perfect, but they are
still useful indicators of hedge ratios.
United Millers purchases corn to make cornflakes. When the price of corn increases, the cost of making cereal increases, resulting in lower profits. Historically, profits per quarter have been related to the price of corn according to the equation: Profits 5 $8 million 2 1 million 3 price per bushel. How many bushels of corn should United Millers purchase in the corn futures market to hedge its corn-price risk?
CONCEPT CHECK 23.3
The Contracts In contrast to most futures contracts, which call for delivery of a specified commodity,
stock-index contracts are settled by a cash amount equal to the value of the index on the
contract maturity date times a multiplier that scales the size of the contract. The total profit
to the long position is S T 2 F 0 , where S T is the value of the stock index on the maturity date. Cash settlement avoids the costs that would be incurred if the short trader had to pur-
chase the stocks in the index and deliver them to the long position, and if the long position
then had to sell the stocks for cash. Instead, the long trader receives S T 2 F 0 dollars, and the short trader F 0 2 S T dollars. These profits duplicate those that would arise with actual delivery.
There are several stock-index futures contracts currently traded. Table 23.1 lists some
of the major ones, showing under contract size the multiplier used to calculate contract
settlements. An S&P 500 contract, for example, with a futures price of 1,400 and a final
23.2 Stock-Index Futures
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 807
index value of 1,405 would result in a profit for the long side of $250 3 (1,405 2 1,400) 5
$1,250. The S&P contract by far dominates the market in U.S. stock index futures. 2
The broad-based U.S. stock market indexes are all highly correlated. Table 23.2 is a
correlation matrix for four well-known indexes: the S&P 500, the Dow Jones Industrial
Average, the Russell 2000 index of small capitalization stocks, and the NASDAQ 100.
The highest correlation, .979, is between the two large-cap indexes, the S&P 500 and the
DJIA. The NASDAQ 100, which is dominated by technology firms, and the Russell 2000
index of small-cap firms have smaller correlations with the large-cap indexes and with
each other, but even these are above .85.
2 We should point out that while these multipliers may make the resulting positions too large for small investors,
there are many effectively equivalent futures with smaller multipliers (typically one-fifth the value of the standard
contract) called E-Minis.
Contract Underlying Market Index Contract Size Exchange
S&P 500 Standard & Poor’s 500 index. $250 times Chicago Mercantile Exchange A value-weighted arithmetic average of 500 stocks.
S&P 500 index
Dow Jones Industrial Dow Jones Industrial Average. $10 times index Chicago Board of Trade Average (DJIA) Price-weighted average of 30
firms. Russell 2000 Index of 2,000 smaller firms. $100 times index Intercontinental Exchange (ICE) NASDAQ 100 Value-weighted arithmetic
average of 100 of the largest over-the-counter stocks.
$100 times index Chicago Mercantile Exchange
Nikkei 225 Nikkei 225 stock average. $5 times Nikkei Index Chicago Mercantile Exchange FTSE 100 Financial Times Stock Exchange
Index of 100 U.K. firms. £10 times FTSE Index London International Financial
Futures Exchange DAX-30 Index of 30 German stocks. 25 euros times index Eurex CAC-40 Index of 40 French stocks. 10 euros times index Euronext Paris Hang Seng Value-weighted index of
largest firms in Hong Kong. 50 Hong Kong dollars times index
Hong Kong Exchange
Table 23.1
Sample of stock-index futures
Table 23.2
Correlation coefficients using monthly returns, 2008–2012
S&P DJIA Russell NASDAQ
S&P 500 1 0.979 0.948 0.928 DJIA 1 0.908 0.876 Russell 2000 1 0.898 NASDAQ 100 1
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808 P A R T V I Options, Futures, and Other Derivatives
Creating Synthetic Stock Positions: An Asset Allocation Tool One reason stock-index futures are so popular is that they can substitute for holdings in
the underlying stocks themselves. Index futures let investors participate in broad market
movements without actually buying or selling large amounts of stock.
Because of this, we say futures represent “synthetic” holdings of the market portfo-
lio. Instead of holding the market directly, the investor takes a long futures position in
the index. The transaction costs involved in establishing and liquidating futures posi-
tions are much lower than taking actual spot positions. “Market timers,” who speculate
on broad market moves rather than on individual securities, are large players in stock-
index futures for this reason.
One means to market time, for example, is to shift between Treasury bills and broad-
based stock market holdings. Timers attempt to shift from bills into the market before
market upturns, and to shift back into bills to avoid market downturns, thereby profiting
from broad market movements. Market timing of this sort, however, can result in huge
trading costs. An attractive alternative is to invest in Treasury bills and hold varying
amounts of market-index futures contracts, which are far cheaper to trade.
The strategy works like this. When timers are bullish, they will establish many long
futures positions that they can liquidate quickly and cheaply when expectations turn
bearish. Rather than shifting back and forth between T-bills and stocks, they buy and hold
T-bills and adjust only the futures position.
You can construct a T-bill plus index futures position that duplicates the payoff to
holding the stock index itself. Here is how:
1. Purchase as many market-index futures contracts as you need to establish your
desired stock position. A desired holding of $1,000 multiplied by the S&P 500
index, for example, would require the purchase of four contracts because each
contract calls for delivery of $250 multiplied by the index.
2. Invest enough money in T-bills to cover the payment of the futures price at the
contract’s maturity date. The necessary investment is simply the present value of
the futures price.
Suppose that an institutional investor wants to invest $140 million in the market for 1 month and, to minimize trading costs, chooses to buy the S&P 500 futures contracts as a substitute for actual stock holdings. If the index is now at 1,400, the 1-month delivery futures price is 1,414, and the T-bill rate is 1% per month, it would buy 400 contracts. (Each contract controls $250 3 1,400 5 $350,000 worth of stock, and $140 million/$350,000 5 400.) The institution thus has a long position on $100,000 times the S&P 500 index (400 contracts times the contract multiplier of $250). To cover payment of the futures price, it must buy bills with 100,000 times the present value of the futures price. This equals 100,000 3 (1,414/1.01) 5 $140 million market value of bills. Notice that the $140 million outlay in bills is precisely equal to the amount that would have been needed to buy the stock directly. (The face value of the bills will be 100,000 3 1,414 5 $141.4 million.)
Example 23.3 Synthetic Positions Using Stock-Index Futures
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 809
The bills-plus-futures contracts strategy in Example 23.3 may be viewed as a 100%
stock strategy. At the other extreme, investing in zero futures results in a 100% bills
position. Moreover, a short futures position will result in a portfolio equivalent to that
obtained by short-selling the stock market index, because in both cases the investor gains
from decreases in the stock price. Bills-plus-futures mixtures clearly allow for a flexible
and low-transaction-cost approach to market timing. The futures positions may be estab-
lished or reversed quickly and cheaply. Also, because the short futures position allows the
investor to earn interest on T-bills, it is superior to a conventional short sale of the stock,
where the investor may earn little or no interest on the proceeds of the short sale.
The nearby box illustrates that it is now commonplace for money managers to use
futures contracts to create synthetic equity positions in stock markets. The article notes
that futures positions can be particularly helpful in establishing synthetic positions in
foreign equities, where trading costs tend to be greater and markets tend to be less liquid.
This is an artificial, or synthetic, stock position. What is the value of this portfolio at the maturity date? Call S T the value of the stock index on the maturity date T and, as usual, let F 0 be the original futures price:
In General (Per Unit of the Index) Our Numbers
1. Profits from contract S T 2 F0 $100,000(S T 2 1,414) 2. Face value of T-bills F0 141,400,000 TOTAL S T 100,000S T
The total payoff on the contract maturity date is exactly proportional to the value of the stock index. In other words, adopting this portfolio strategy is equivalent to hold- ing the stock index itself, aside from the issue of interim dividend distributions and tax treatment.
The market timing strategy of Example 23.3 also can be achieved by an investor who holds an indexed stock portfolio and “synthetically exits” the position using futures if and when he turns pessimistic con- cerning the market. Suppose the investor holds $140 million of stock. What futures position added to the stock holdings would create a synthetic T-bill exposure when he is bearish on the market? Confirm that the profits are effectively risk-free using a table like that in Example 23.3.
CONCEPT CHECK 23.4
Index Arbitrage Whenever the actual futures price falls outside the no-arbitrage band, there is an opportu-
nity for profit. This is why the parity relationships are so important. Far from being theo-
retical academic constructs, they are in fact a guide to trading rules that can generate large
profits. Index arbitrage is an investment strategy that exploits divergences between the actual futures price and its theoretically correct parity value.
In principle, index arbitrage is simple. If the futures price is too high, short the futures
contract and buy the stocks in the index. If it is too low, go long in futures and short the
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810
stocks. You can perfectly hedge your position and should earn arbitrage profits equal to the
mispricing of the contract.
In practice, however, index arbitrage presents challenges in implementation. The prob-
lem lies in buying “the stocks in the index.” Selling or purchasing shares in all 500 stocks
in the S&P 500 is impractical for two reasons. The first is transaction costs, which may
outweigh any profits to be made from the arbitrage. Second, index arbitrage calls for the
purchase or sale of shares of 500 different firms simultaneously, and any lags in the execu-
tion of such a strategy can destroy the effectiveness of a plan to exploit temporary price
discrepancies. Don’t forget that others also will be trying to exploit any deviations from
parity, and if they trade first, they may move prices before your trade is executed.
Arbitrageurs need to trade an entire portfolio of stocks quickly and simultaneously
if they hope to exploit disparities between the futures price and its corresponding stock
index. For this they need a coordinated trading program; hence the term program trading, which refers to purchases or sales of entire portfolios of stocks. Electronic trading enables
traders to submit coordinated buy or sell programs to the stock market at once. 3
The success of these arbitrage positions and associated program trades depends on
only two things: the relative levels of spot and futures prices and synchronized trading in
the two markets. Because arbitrageurs exploit disparities in futures and spot prices,
absolute price levels are unimportant.
Using Index Futures to Hedge Market Risk How might a portfolio manager use futures to hedge market exposure? Suppose, for
example, that you manage a $30 million portfolio with a beta of .8. You are bullish on the
Got a Bundle to Invest Fast? Think Stock-Index Futures
As investors go increasingly global and market turbulence grows, stock-index futures are emerging as the favorite way for nimble money managers to deploy their funds. Indeed, in most major markets, trading in stock futures now exceeds the buying and selling of actual shares.
What’s the big appeal? Speed, ease and cheapness. For most major markets, stock futures not only boast greater liquidity but also lower transaction costs than traditional trading methods.
“When I decide it’s time to move into France, Germany or Britain, I don’t necessarily want to wait around until I find exactly the right stocks,” says Fabrizio Pierallini, manager of New York–based Vontobel Ltd.’s Euro Pacific Fund.
Mr. Pierallini says he later fine-tunes his market picks by gradually shifting out of futures into favorite stocks. To the extent Mr. Pierallini’s stocks outperform the mar- ket, futures provide a means to preserve those gains, even while hedging against market declines.
For instance, by selling futures equal to the value of the underlying portfolio, a manager can almost completely insulate a portfolio from market moves. Say a manager suc- ceeds in outperforming the market, but still loses 3% while the market as a whole falls 10%. Hedging with futures
would capture that margin of out-performance, transform- ing the loss into a profit of roughly 7%.
Among futures-intensive strategies is “global tactical asset allocation,” which involves trading whole markets worldwide as traditional managers might trade stocks. The growing popularity of such asset-allocation strategies has given futures a big boost in recent years.
To capitalize on global market swings, “futures do the job for us better than stocks, and they’re cheaper,” said Jarrod Wilcox, director of global investments at PanAgora Asset Management, a Boston-based asset allocator. Even when PanAgora does take positions in individual stocks, it often employs futures to modify its position, such as by hedging part of its exposure to that particular stock market.
When it comes to investing overseas, Mr. Wilcox noted, futures are often the only vehicle that makes sense from a cost standpoint. Abroad, transaction taxes and sky-high commissions can wipe out more than 1% of the money deployed on each trade. By contrast, a comparable trade in futures costs as little as 0.05%.
Source: Abridged from Suzanne McGee, “Got a Bundle to Invest Fast? Think Stock-Index Futures,” The Wall Street Journal, February 21, 1995. Reprinted by permission of The Wall Street Journal, © 1995 Dow Jones & Company, Inc. All rights reserved worldwide.
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3 One might also attempt to exploit violations of parity using ETFs linked to the market index, but ETFs may trade
in less liquid markets where it can be difficult to trade large quantities without moving prices.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 811
market over the long term, but you are afraid that over the next 2 months, the market is
vulnerable to a sharp downturn. If trading were costless, you could sell your portfolio,
place the proceeds in T-bills for 2 months, and then reestablish your position after you
perceive that the risk of the downturn has passed. In practice, however, this strategy would
result in unacceptable trading costs, not to mention tax problems resulting from the real-
ization of capital gains or losses on the portfolio. An alternative approach would be to use
stock index futures to hedge your market exposure.
Suppose that the S&P 500 index currently is at 1,000. A decrease in the index to 975 would represent a drop of 2.5%. With a portfolio beta of .8, you would expect a loss of .8 3 2.5% 5 2%, or in dollar terms, .02 3 $30 million 5 $600,000. Therefore, the sensitivity of your portfolio value to market movements is $600,000 per 25-point move- ment in the S&P 500 index.
To hedge this risk, you could sell stock index futures. When your portfolio falls in value along with declines in the broad market, the futures contract will provide an offsetting profit.
The sensitivity of a futures contract to market movements is easy to determine. With its contract multiplier of $250, the profit on the S&P 500 futures contract varies by $6,250 for every 25-point swing in the index. Therefore, to hedge your market exposure for 2 months, you could calculate the hedge ratio as follows:
H 5 Change in portfolio value
Profit on one futures contract 5
$600,000 $6,250
5 96 contracts (short)
You would enter the short side of the contracts, because you want profits from the con- tract to offset the exposure of your portfolio to the market. Because your portfolio does poorly when the market falls, you need a position that will do well when the market falls.
Example 23.4 Hedging Market Risk
We also could approach the hedging problem in Example 23.4 using a similar regres-
sion procedure as that illustrated in Figure 23.3 for foreign exchange risk. The predicted
value of the portfolio is graphed in Figure 23.4 as a function of the value of the S&P 500
index. With a beta of .8, the slope of the relationship is 24,000: A 2.5% increase in the
index, from 1,000 to 1,025, results in a capital gain of 2% of $30 million, or $600,000.
Therefore, your portfolio will increase in value by $24,000 for each increase of one point
in the index. As a result, you should enter a short position on 24,000 units of the S&P 500
index to fully offset your exposure to marketwide movements. Because the contract multi-
plier is $250 times the index, you need to sell 24,000/250 5 96 contracts.
Notice that when the slope of the regression line relating your unprotected position to
the value of an asset is positive, your hedge strategy calls for a short position in that asset. The hedge ratio is the negative of the regression slope. This is because the hedge position
should offset your initial exposure. If you do poorly when the asset value falls, you need
a hedge vehicle that will do well when the asset value falls. This calls for a short position
in the asset.
Active managers sometimes believe that a particular asset is underpriced, but that the
market as a whole is about to fall. Even if the asset is a good buy relative to other stocks
in the market, it still might perform poorly in a broad market downturn. To solve this
problem, the manager would like to separate the bet on the firm from the bet on the
market: The bet on the company must be offset with a hedge against the market exposure
that normally would accompany a purchase of the stock. In other words, the manager
seeks a market-neutral bet on the stock, by which we mean that a position on the stock
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812 P A R T V I Options, Futures, and Other Derivatives
is taken to capture its alpha (its abnormal risk-adjusted expected return), but that market
exposure is fully hedged, resulting in a position beta of zero.
By allowing investors to hedge market performance, the futures contract allows the port-
folio manager to make stock picks without concern for the market exposure of the stocks
chosen. After the stocks are chosen, the resulting market risk of the portfolio can be modu-
lated to any degree using the stock futures contracts. Here again, the stock’s beta is the key
to the hedging strategy. We discuss market-neutral strategies in more detail in Chapter 26.
Predicted Value of Portfolio
Slope = 24,000
S&P 500 Index
$30.6 million
$30 million
1,000 1,025
Figure 23.4 Predicted value of the portfolio as a function of the market index
Suppose the beta of the stock is 2⁄3, and the manager purchases $375,000 worth of the stock. For every 3% drop in the broad market, the stock would be expected to respond with a drop of 2⁄3 3 3% 5 2%, or $7,500. The S&P 500 contract will fall by 30 points from a current value of 1,000 if the market drops 3%. With the contract multiplier of $250, this would entail a profit to a short futures position of 30 3 $250 5 $7,500 per contract. Therefore, the market risk of the stock can be offset by shorting one S&P contract. More formally, we could calculate the hedge ratio as
H 5 Expected change in stock value per 3% market drop
Profit on one short contract per 3% market drop
5 $7,500 swing in unprotected position
$7,500 profit per contract
5 1 contract
Now that market risk is hedged, the only source of variability in the performance of the stock-plus-futures portfolio will be the firm-specific performance of the stock.
Example 23.5 Market-Neutral Active Stock Selection
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 813
Hedging Interest Rate Risk Like equity managers, fixed-income managers also sometimes desire to hedge market risk,
in this case resulting from movements in the entire structure of interest rates. Consider, for
example, these problems:
1. A fixed-income manager holds a bond portfolio on which considerable capital gains
have been earned. She foresees an increase in interest rates but is reluctant to sell
her portfolio and replace it with a lower-duration mix of bonds because such rebal-
ancing would result in large trading costs as well as realization of capital gains for
tax purposes. Still, she would like to hedge her exposure to interest rate increases.
2. A corporation plans to issue bonds to the public. It believes that now is a good time
to act, but it cannot issue the bonds for another 3 months because of the lags inher-
ent in SEC registration. It would like to hedge the uncertainty surrounding the yield
at which it eventually will be able to sell the bonds.
3. A pension fund will receive a large cash inflow next month that it plans to invest in
long-term bonds. It is concerned that interest rates may fall by the time it can make the
investment and would like to lock in the yield currently available on long-term issues.
In each of these cases, the investment manager wishes to hedge interest rate uncertainty.
To illustrate the procedures that might be followed, we will focus on the first example,
and suppose that the portfolio manager has a $10 million bond portfolio with a modified
duration of 9 years. 4 If, as feared, market interest rates increase and the bond portfolio’s
yield also rises, say, by 10 basis points (.10%), the fund will suffer a capital loss. Recall
from Chapter 16 that the capital loss in percentage terms will be the product of modified
duration, D *, and the change in the portfolio yield. Therefore, the loss will be
D* 3 Dy 5 9 3 .10% 5 .90%
or $90,000. This establishes that the sensitivity of the value of the unprotected portfolio
to changes in market yields is $9,000 per 1 basis point change in the yield. Market prac-
titioners call this ratio the price value of a basis point, or PVBP. The PVBP represents the sensitivity of the dollar value of the portfolio to changes in interest rates. Here, we’ve
shown that
PVBP 5 Change in portfolio value
Predicted change in yield 5
$90,000
10 basis points 5 $9,000 per basis point
One way to hedge this risk is to take an offsetting position in an interest rate futures
contract, for example, the Treasury bond contract. The bond nominally calls for delivery
of $100,000 par value T-bonds with 6% coupons and 20-year maturity. In practice, the
contract delivery terms are fairly complicated because many bonds with different coupon
rates and maturities may be substituted to settle the contract. However, we will assume that
the bond to be delivered already is known and has a modified duration of 10 years. Finally,
suppose that the futures price currently is $90 per $100 par value. Because the contract
requires delivery of $100,000 par value of bonds, the contract multiplier is $1,000.
23.3 Interest Rate Futures
4 Recall that modified duration, D *, is related to duration, D, by the formula D * 5 D /(1 1 y ), where y is the bond’s
yield to maturity. If the bond pays coupons semiannually, then y should be measured as a semiannual yield. For simplicity, we will assume annual coupon payments, and treat y as the effective annual yield to maturity.
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814 P A R T V I Options, Futures, and Other Derivatives
Given these data, we can calculate the PVBP for the futures contract. If the yield on the
delivery bond increases by 10 basis points, the bond value will fall by D * 3 .1% 5 10 3 .1% 5 1%. The futures price also will decline 1%, from 90 to 89.10.
5 Because the
contract multiplier is $1,000, the gain on each short contract will be $1,000 3 .90 5 $900.
Therefore, the PVBP for one futures contract is $900/10-basis-point change, or $90 for
a change in yield of 1 basis point.
Now we can easily calculate the hedge ratio as follows:
H 5 PVBP of portfolio
PVBP of hedge vehicle 5
$9,000
$90 per contract 5 100 contracts
Therefore, 100 T-bond futures contracts will offset the portfolio’s exposure to interest rate
fluctuations.
Notice that this is another example of a market-neutral strategy. In Example 23.5, which
illustrated an equity-hedging strategy, stock-index futures were used to drive a portfolio
beta to zero. In this application, we used a T-bond contract to drive the interest rate expo-
sure of a bond position to zero. The hedged fixed-income position has a duration (or a
PVBP) of zero. The source of risk differs, but the hedging strategy is essentially the same.
5 This assumes the futures price will be exactly proportional to the bond price, which ought to be nearly true.
7
5
4
3
2
1
0 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006 2010 2014
Y ie
ld S
p re
a d
( %
)
6
Figure 23.5 Yield spread between 10-year Treasury and Baa-rated corporate bonds
Suppose the bond portfolio is twice as large, $20 million, but that its modified duration is only 4.5 years. Show that the proper hedge position in T-bond futures is the same as the value just calculated, 100 contracts.
CONCEPT CHECK 23.5
Although the hedge ratio is easy to compute, the hedging problem in practice is more dif-
ficult. We assumed in our example that the yields on the T-bond contract and the bond portfolio
would move perfectly in unison. Although interest rates on various fixed-income instruments
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 815
do tend to vary in tandem, there is considerable slippage across sectors of the fixed-income mar-
ket. For example, Figure 23.5 shows that the spread between long-term corporate and 10-year
Treasury bond yields has fluctuated considerably over time. Our hedging strategy would be
fully effective only if the yield spread across the two sectors of the fixed-income market were
constant (or at least perfectly predictable) so that yield changes in both sectors were equal.
This problem highlights the fact that most hedging activity is in fact cross-hedging, meaning that the hedge vehicle is a different asset than the one to be hedged. To the extent
that there is slippage between prices or yields of the two assets, the hedge will not be
perfect. Cross-hedges can eliminate a large fraction of the total risk of the unprotected
portfolio, but you should be aware that they typically are far from risk-free positions.
6 Interest rate swaps have nothing to do with the Homer-Liebowitz bond swap taxonomy described in Chapter 16.
7 The participants to the swap do not loan each other money. They agree only to exchange a fixed cash flow for a
variable cash flow that depends on the short-term interest rate. This is why the principal is described as notional. The notional principal is simply a way to describe the size of the swap agreement. In this example, a 7% fixed
rate is exchanged for the LIBOR rate; the difference between LIBOR and 7% is multiplied by notional principal
to determine the net cash flow.
Swaps are multiperiod extensions of forward contracts. For example, rather than agreeing
to exchange British pounds for U.S. dollars at an agreed-upon forward price at one single
date, a foreign exchange swap would call for an exchange of currencies on several future dates. The parties might exchange $1.6 million for £1 million in each of the next 5 years.
Similarly, interest rate swaps call for the exchange of a series of cash flows proportional to a given interest rate for a corresponding series of cash flows proportional to a floating inter-
est rate. 6 One party might exchange a variable cash flow equal to $1 million times a short-
term interest rate for $1 million times a fixed interest rate of 5% for each of the next 7 years.
The swap market is a huge component of the derivatives market, with well over $500
trillion in swap agreements outstanding. We will illustrate how these contracts work by
using a simple interest rate swap as an example.
23.4 Swaps
Consider the manager of a large portfolio that currently includes $100 million par value of long-term bonds paying an average coupon rate of 7%. The manager believes inter- est rates are about to rise. As a result, he would like to sell the bonds and replace them with either short-term or floating-rate issues. However, it would be exceedingly expensive in terms of transaction costs to replace the portfolio every time the forecast for interest rates is updated. A cheaper and more flexible approach is to “swap” the $7 million a year in interest income the portfolio currently generates for an amount of money tied to the short-term interest rate. That way, if rates do rise, so will the portfolio’s interest income.
A swap dealer might advertise its willingness to exchange, or “swap,” a cash flow based on the 6-month LIBOR rate for one based on a fixed rate of 7%. (The LIBOR, or London Interbank Offered Rate, is the interest rate at which banks borrow from each other in the Eurodollar market. It is the most commonly used short-term interest rate in the swap market.) The portfolio manager would then enter into a swap agreement with the dealer to pay 7% on notional principal of $100 million and receive payment of the LIBOR rate on that amount of notional principal. 7 In other words, the manager swaps a
Example 23.6 Interest Rate Swap
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816 P A R T V I Options, Futures, and Other Derivatives
Swaps and Balance Sheet Restructuring Example 23.6 illustrates why swaps have tremendous appeal to fixed-income managers.
These contracts provide a means to quickly, cheaply, and anonymously restructure the
balance sheet. Suppose a corporation that has issued fixed-rate debt believes that interest
rates are likely to fall; it might prefer to have issued floating-rate debt. In principle, it could
issue floating-rate debt and use the proceeds to buy back the outstanding fixed-rate debt.
But it is faster and easier to convert the outstanding fixed-rate debt into synthetic floating-
rate debt by entering a swap to receive a fixed interest rate (offsetting its fixed-rate coupon
obligation) and paying a floating rate.
Conversely, a bank that pays current market interest rates to its depositors, and thus is
exposed to increases in rates, might wish to convert some of its financing to a fixed-rate
basis. It would enter a swap to receive a floating rate and pay a fixed rate on some amount
of notional principal. This swap position, added to its floating-rate deposit liability, would
result in a net liability of a fixed stream of cash. The bank might then be able to invest in
long-term fixed-rate loans without encountering interest rate risk.
For another example, consider a fixed-income portfolio manager. Swaps enable the
manager to switch back and forth between a fixed- or floating-rate profile quickly and
cheaply as the forecast for the interest rate changes. A manager who holds a fixed-rate
portfolio can transform it into a synthetic floating-rate portfolio by entering a pay fixed–
receive floating swap and can later transform it back by entering the opposite side of a
similar swap.
Foreign exchange swaps also enable the firm to quickly and cheaply restructure its bal-
ance sheet. Suppose, for example, that a firm issues $10 million in debt at an 8% coupon
rate, but actually prefers that its interest obligations be denominated in British pounds.
For example, the issuing firm might be a British corporation that perceives advantageous
financing opportunities in the United States but prefers pound-denominated liabilities.
Then the firm, whose debt currently obliges it to make dollar-denominated payments
payment of .07 3 $100 million for a payment of LIBOR 3 $100 million. The manager’s net cash flow from the swap agreement is therefore (LIBOR 2 .07) 3 $100 million. Note that the swap arrangement does not mean that a loan has been made. The participants have agreed only to exchange a fixed cash flow for a variable one.
Now consider the net cash flow to the manager’s portfolio in three interest rate scenarios:
LIBOR Rate 6.5% 7.0% 7.5%
Interest income from bond portfolio (5 7% of $100 million bond portfolio) $7,000,000 $7,000,000 $7,000,000 Cash flow from swap [5 (LIBOR 2 7%) 3 notional principal of $100 million] (500,000) 0 500,000 Total (5 LIBOR 3 $100 million) $6,500,000 $7,000,000 $7,500,000
Notice that the total income on the overall position—bonds plus swap agreement— equals the LIBOR rate in each scenario times $100 million. The manager has, in effect, converted a fixed-rate bond portfolio into a synthetic floating-rate portfolio.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 817
of $800,000, can agree to swap a given number of pounds each year for $800,000. By
so doing, it effectively covers its dollar obligation and replaces it with a new pound-
denominated obligation.
Show how a firm that has issued a floating-rate bond with a coupon equal to the LIBOR rate can use swaps to convert that bond into synthetic fixed-rate debt. Assume the terms of the swap allow an exchange of LIBOR for a fixed rate of 8%.
CONCEPT CHECK 23.6
8 Actually, things are a bit more complicated. The dealer is more than just an intermediary because it bears the
credit risk that one or the other of the parties to the swap might default on the obligation. Referring to Figure 23.6 ,
if firm A defaults on its obligation, for example, the swap dealer still must maintain its commitment to firm B.
In this sense, the dealer does more than simply pass through cash flows to the other swap participants.
A pension fund holds a portfolio of money market securities that the manager believes are paying excel- lent yields compared to other comparable-risk short-term securities. However, the manager believes that interest rates are about to fall. What type of swap will allow the fund to continue to hold its portfolio of short-term securities while at the same time benefiting from a decline in rates?
CONCEPT CHECK 23.7
The Swap Dealer What about the swap dealer? Why is the dealer, which is typically a financial intermediary
such as a bank, willing to take on the opposite side of the swaps desired by these partici-
pants in these hypothetical swaps?
Consider a dealer who takes on one side of a swap, let’s say paying LIBOR and receiv-
ing a fixed rate. The dealer will search for another trader in the swap market who wishes to
receive a fixed rate and pay LIBOR. For example, Company A may have issued a 7% coupon
fixed-rate bond that it wishes to convert into synthetic floating-rate debt, while Company B
may have issued a floating-rate bond tied to LIBOR that it wishes to convert into synthetic
fixed-rate debt. The dealer will enter a swap with Company A in which it pays a fixed rate
and receives LIBOR, and will enter another swap with Company B in which it pays LIBOR
and receives a fixed rate. When the two swaps are combined, the dealer’s position is effec-
tively neutral on interest rates, paying LIBOR on one swap and receiving it on another. Simi-
larly, the dealer pays a fixed rate on one swap and receives it on another. The dealer becomes
little more than an intermediary, funneling payments from one party to the other. 8 The dealer
finds this activity profitable because it will charge a bid-ask spread on the transaction.
This rearrangement is illustrated in Figure 23.6 . Company A has issued 7% fixed-rate
debt (the leftmost arrow in the figure) but enters a swap to pay the dealer LIBOR and receive
a 6.95% fixed rate. Therefore, the company’s net payment is 7% 1 (LIBOR 2 6.95%) 5
LIBOR 1 .05%. It has thus transformed its fixed-rate debt into synthetic floating-rate debt.
Conversely, Company B has issued floating-rate debt paying LIBOR (the rightmost arrow),
but enters a swap to pay a 7.05% fixed rate in return for LIBOR. Therefore, its net payment
is LIBOR 1 (7.05% 2 LIBOR) 5 7.05%. It has thus transformed its floating-rate debt
into synthetic fixed-rate debt. The bid-ask spread, the source of the dealer’s profit, in the
example illustrated in Figure 23.6 is .10% of notional principal each year.
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818 P A R T V I Options, Futures, and Other Derivatives
Other Interest Rate Contracts Swaps are multiperiod forward contracts that trade over the counter. There are also
exchange-listed contracts that trade on interest rates. The biggest of these in terms of
trading activity is the Eurodollar contract, the listing for which we show in Figure 23.7 .
The profit on this contract is proportional to the difference between the LIBOR rate at
contract maturity and the contract rate entered into at contract inception. There are analogous
rates on interbank loans in other currencies. For example, one close cousin of LIBOR is
EURIBOR, which is the rate at which euro-denominated interbank loans within the euro
zone are offered by one prime bank to another.
The listing conventions for the Eurodollar contract are a bit peculiar. Consider, for
example, the first contract listed, which matures in February 2013. The settlement price
is presented as F 0 5 99.7075, or approximately 99.71. However, this value is not really a price. In effect, participants in the contract negotiate over the contract interest rate, and
the so-called futures price is actually set equal to 100 2 contract rate. Because the futures
price is 99.71, the contract rate is 100 2 99.71, or .29%. Similarly, the final futures price
on contract maturity date will be marked to F T 5 100 2 LIBOR T . Thus, profits to the buyer of the contract will be proportional to
FT 2 F0 5 (100 2 LIBORT) 2 (100 2 Contract rate) 5 Contract rate 2 LIBORT
Thus, the contract design allows participants to
trade directly on the LIBOR rate. The contract
multiplier is $1 million, but the LIBOR rate on
which the contract is written is a 3-month (quar-
terly) rate; for each basis point that the (annu-
alized) LIBOR increases, the quarterly interest
rate increases by only ¼ of a basis point, and the
profit to the buyer decreases by
.0001 3 ¼ 3 $1,000,000 5 $25
Examine the payoff on the contract, and you
will see that, in effect, the Eurodollar contract
allows traders to “swap” a fixed interest rate
(the contract rate) for a floating rate (LIBOR).
Company A Swap Dealer Company B
LIBOR
6.95% 7.05%
7% Coupon
LIBOR
LIBOR
Figure 23.6 Interest rate swap. Company B pays a fixed rate of 7.05% to the swap dealer in return for LIBOR. Company A receives 6.95% from the dealer in return for LIBOR. The swap dealer realizes a cash flow each period equal to .10% of notional principal.
Open High Settle Chg Open
interest Contract
hi lo Low
Eurodollar(CME) - $1,000,000; pts of 100% Feb 99.7100 99.7100 99.7075 99.7075 ... 78,171 March 99.7000 99.7100 99.6800 99.7050 .0050 918,700 June 99.6850 99.6950 99.6650 99.6850 ... 789,137 Dec 99.6250 99.6450 99.6050 99.6250 ... 736,966
Figure 23.7 Eurodollar futures Source: The Wall Street Journal, February 9, 2013. Reprinted by permis- sion of The Wall Street Journal, © 2013 Dow Jones & Company, Inc. All rights reserved worldwide.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 819
Thus, this is in effect a one-period interest rate swap. Notice in Figure 23.7 that the total
open interest on this contract is enormous—almost 3 million contracts just for maturi-
ties extending to 1 year. Moreover, while not presented in The Wall Street Journal, significant trading in Eurodollars takes place for contract maturities extending out to
10 years. Contracts with such long-term maturities are quite unusual. They reflect the
fact that the Eurodollar contract is used by dealers in long-term interest rate swaps as a
hedging tool.
Swap Pricing How can the fair swap rate be determined? For example, how would we know that an
exchange of LIBOR is a fair trade for a fixed rate of 6%? Or, what is the fair swap
rate between dollars and pounds for a foreign exchange swap? To answer these ques-
tions we can exploit the analogy between a swap agreement and forward or futures
contract.
Consider a swap agreement to exchange dollars for pounds for one period only. Next
year, for example, one might exchange $1 million for £.5 million. This is no more than a
simple forward contract in foreign exchange. The dollar-paying party is contracting to buy
British pounds in 1 year for a number of dollars agreed to today. The forward exchange
rate for 1-year delivery is F 1 5 $2.00/pound. We know from the interest rate parity rela- tionship that this forward price should be related to the spot exchange rate, E 0 , by the for- mula F 1 5 E 0 (1 1 r US )/(1 1 r UK ). Because a one-period swap is in fact a forward contract, the fair swap rate is also given by the parity relationship.
Now consider an agreement to trade foreign exchange for two periods. This agreement
could be structured as a portfolio of two separate forward contracts. If so, the forward price
for the exchange of currencies in 1 year would be F 1 5 E 0 (1 1 r US )/(1 1 r UK ), while the forward price for the exchange in the second year would be F 2 5 E 0 [(1 1 r US )/(1 1 r UK )]
2 .
As an example, suppose that E 0 5 $2.03/pound, r US 5 5%, and r UK 5 7%. Then, using the parity relationship, prices for forward delivery would be F 1 5 $2.03/£ 3 (1.05/1.07) 5 $1.992/£ and F 2 5 $2.03/£ 3 (1.05/1.07)
2 5 $1.955/£. Figure 23.8, panel A illustrates this
sequence of cash exchanges assuming that the swap calls for delivery of one pound in each
year. Although the dollars to be paid in each of the 2 years are known today, they differ
from year to year.
In contrast, a swap agreement to exchange currency for 2 years would call for a fixed
exchange rate to be used for the duration of the swap. This means that the same number of
dollars would be paid per pound in each year, as illustrated in Figure 23.8, panel B . Because
the forward prices for delivery in each of the next 2 years are $1.992/£ and $1.955/£, the
fixed exchange rate that makes the two-period swap a fair deal must be between these two
values. Therefore, the dollar payer underpays for the pound in the first year (compared to
the forward exchange rate) and overpays in the second year. Thus, the swap can be viewed
as a portfolio of forward transactions, but instead of each transaction being priced indepen-
dently, one forward price is applied to all of the transactions.
Given this insight, it is easy to determine the fair swap price. If we were to purchase
one pound per year for 2 years using two independent forward agreements, we would
pay F 1 dollars in 1 year and F 2 dollars in 2 years. If instead we enter a swap, we pay a constant rate of F * dollars per pound. Because both strategies must be equally costly, we conclude that
F1 1 1 y1
1 F2
(1 1 y2) 2 5
F*
1 1 y1 1
F*
(1 1 y2) 2
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A: Two forward contracts, each priced independently
$1.992 $1.955
£1 £1
B: Two-year swap agreement
$1.974 $1.974
£1 £1
Figure 23.8 Forward contracts versus swaps
820 P A R T V I Options, Futures, and Other Derivatives
where y 1 and y 2 are the appropriate yields from the yield curve for discounting dollar cash flows of 1- and 2-year maturities, respectively. In our example, where we have assumed a
flat U.S. yield curve at 5%, we would solve
1.992
1.05 1
1.955
1.052 5
F*
1.05 1
F*
1.052
which implies that F * 5 1.974. The same principle would apply to a foreign exchange swap of any other maturity. In essence, we need to find the level annuity, F *, with the same present value as the stream of annual cash flows that would be incurred in a sequence of
forward rate agreements.
Interest rate swaps can be subjected to precisely the same analysis. Here, the forward
contract is on an interest rate. For example, if you swap LIBOR for a 7% fixed rate with
notional principal of $100, then you have entered a forward contract for delivery of $100
times LIBOR for a fixed “forward” price of $7. If the swap agreement is for many periods,
the fair spread will be determined by the entire sequence of interest rate forward prices
over the life of the swap.
Credit Risk in the Swap Market The rapid growth of the swap market has given rise to increasing concern about credit risk
in these markets and the possibility of a default by a major swap trader. Actually, although
credit risk in the swap market certainly is not trivial, it is not nearly as large as the magni-
tude of notional principal in these markets would suggest. To see why, consider a simple
interest rate swap of LIBOR for a fixed rate.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 821
At the time the transaction is initiated, it has zero net present value to both parties for the
same reason that a futures contract has zero value at inception: Both are simply contracts
to exchange cash in the future at terms established today that make both parties willing to
enter into the deal. Even if one party were to back out of the deal at this moment, it would
not cost the counterparty anything, because another trader could be found to take its place.
Once interest or exchange rates change, however, the situation is not as simple. Sup-
pose, for example, that interest rates increase shortly after an interest-rate swap agree-
ment has begun. The floating-rate payer therefore suffers a loss, while the fixed-rate payer
enjoys a gain. If the floating-rate payer reneges on its commitment at this point, the fixed-
rate payer suffers a loss. However, that loss is not as large as the notional principal of
the swap, for the default of the floating-rate payer relieves the fixed-rate payer from its
obligation as well. The loss is only the difference between the values of the fixed-rate and floating-rate obligations, not the total value of the payments that the floating-rate payer was obligated to make.
Consider a swap written on $1 million of notional principal that calls for exchange of LIBOR for a fixed rate of 8% for 5 years. Suppose, for simplicity, that the yield curve is currently flat at 8%. With LIBOR thus equal to 8%, no cash flows will be exchanged unless interest rates change. But now suppose that the yield curve imme- diately shifts up to 9%. The floating-rate payer now is obligated to pay a cash flow of (.09 2 .08) 3 $1 million 5 $10,000 each year to the fixed-rate payer (as long as rates remain at 9%). If the floating-rate payer defaults on the swap, the fixed-rate payer loses the prospect of that 5-year annuity. The present value of that annuity is $10,000 3 Annuity factor(9%, 5 years) 5 $38,897. This loss may not be trivial, but it is less than 4% of notional principal. We conclude that the credit risk of the swap is far less than notional principal.
Example 23.7 Credit Risk in Swaps
Credit Default Swaps Despite the similarity in names, a credit default swap, or CDS, is not the same type of instrument as interest rate or currency swaps. As we saw in Chapter 14, payment on a CDS
is tied to the financial status of one or more reference firms; the CDS therefore allows two
counterparties to take positions on the credit risk of those firms. When a particular “credit
event” is triggered, say, default on an outstanding bond or failure to pay interest, the seller
of protection is expected to cover the loss in the market value of the bond. For example,
the swap seller may be obligated to pay par value to take delivery of the defaulted bond (in
which case the swap is said to entail physical settlement) or may instead pay the swap buyer
the difference between the par value and market value of the bond (termed cash settlement).
The swap purchaser pays a periodic fee to the seller for this protection against credit events.
Unlike interest rate swaps, credit default swaps do not entail periodic netting of one
reference rate against another. They are in fact more like insurance policies written on par-
ticular credit events. Bondholders may buy these swaps to transfer their credit risk expo-
sure to the swap seller, effectively enhancing the credit quality of their portfolios. Unlike
insurance policies, however, the swap holder need not hold the bonds underlying the CDS
contract; therefore, credit default swaps can be used purely to speculate on changes in the
credit standing of the reference firms.
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822 P A R T V I Options, Futures, and Other Derivatives
9 Robert A. Jarrow and George S. Oldfield, “Forward Contracts and Futures Contracts,” Journal of Financial
Economics 9 (1981).
Commodity futures prices are governed by the same general considerations as stock futures.
One difference, however, is that the cost of “carrying” commodities, especially those subject
to spoilage, is greater than the cost of carrying financial assets. The underlying asset for
some contracts, such as electricity futures, simply cannot be “carried” or held in portfolio.
Finally, spot prices for some commodities demonstrate marked seasonal patterns that can
affect futures pricing.
Pricing with Storage Costs The cost of carrying commodities includes, in addition to interest costs, storage costs,
insurance costs, and an allowance for spoilage of goods in storage. To price commodity
futures, reconsider the earlier arbitrage strategy that calls for holding both the asset and a
short position in the futures contract on the asset. In this case we will denote the price of
the commodity at time T as P T , and assume for simplicity that all noninterest carrying costs ( C ) are paid in one lump sum at time T, the contract maturity. Carrying costs appear in the final cash flow.
Action Initial Cash Flow CF at Time T
Buy asset; pay carrying costs at T 2P0 PT 2 C Borrow P0; repay with interest at time T P0 2P0(1 1 rf) Short futures position 0 F0 2 PT TOTAL 0 F0 2 P0(1 1 rf) 2 C
Because market prices should not allow for arbitrage opportunities, the terminal cash flow
of this zero-net-investment, risk-free strategy should be zero.
If the cash flow were positive, this strategy would yield guaranteed profits for no invest-
ment. If the cash flow were negative, the reverse of this strategy also would yield profits. In
practice, the reverse strategy would involve a short sale of the commodity. This is unusual
but may be done as long as the short sale contract appropriately accounts for storage costs.
Thus, 9 we conclude that
F0 5 P0 (1 1 rf ) 1 C
Finally, if we define c 5 C / P 0 , and interpret c as the percentage “rate” of carrying costs, we may write
F0 5 P0 (1 1 rf 1 c ) (23.3)
which is a (1-year) parity relationship for futures involving storage costs. Compare
Equation 23.3 to the parity relation for stocks, Equation 22.1 from the previous chapter, and
you will see that they are extremely similar. In fact, if we think of carrying costs as a “nega-
tive dividend,” the equations are identical. This result makes intuitive sense because, instead
of receiving a dividend yield of d, the storer of the commodity must pay a storage cost of c. Obviously, this parity relationship is simply an extension of those we have seen already.
23.5 Commodity Futures Pricing
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 823
Although we have called c the carrying cost of the commodity, we may interpret it more gener-
ally as the net carrying cost, that is, the carrying cost net of the benefits derived from holding the
commodity in inventory. For example, part of the
“convenience yield” of goods held in inventory
is the protection against stocking out, which may
result in lost production or sales.
It is vital to note that we derive Equation 23.3
assuming that the asset will be bought and stored;
it therefore applies only to goods that currently
are being stored. Two kinds of commodities cannot be expected to be stored. The first kind
is commodities for which storage is technologi-
cally not feasible, such as electricity. The second
includes goods that are not stored for economic
reasons. For example, it would be foolish to buy
an agricultural commodity now, planning to store
it for ultimate use in 3 years. Instead, it is clearly
preferable to delay the purchase until after the
harvest of the third year, and avoid paying stor-
age costs. Moreover, if the crop in the third year
is comparable to this year’s, you could obtain it
at roughly the same price as you would pay this
year. By waiting to purchase, you avoid both interest and storage costs.
Because storage across harvests is costly, Equation 23.3 should not be expected to apply
for holding periods that span harvest times, nor should it apply to perishable goods that are
available only “in season.” Whereas the futures price for gold, which is a stored commodity,
increases steadily with the maturity of the contract, the futures price for wheat is seasonal;
its futures price typically falls across harvests between March and July as new supplies
become available.
Figure 23.9 is a stylized version of the seasonal price pattern for an agricultural product.
Clearly this pattern differs from financial assets such as stocks or gold for which there is
no seasonal price movement. Financial assets are priced so that holding them in portfo-
lio produces a fair expected return. Agricultural prices, in contrast, are subject to steep
periodic drops as each crop is harvested, which makes storage across harvests generally
unprofitable.
Futures pricing across seasons therefore requires a different approach that is not based
on storage across harvest periods. In place of general no-arbitrage restrictions we rely
instead on risk premium theory and discounted cash flow (DCF) analysis.
Figure 23.9 Typical agricultural price pattern over the season. Prices adjusted for inflation.
Price
Time
First Harvest
Second Harvest
Third Harvest
People are willing to buy and “store” shares of stock despite the fact that their purchase ties up capital. Most people, however, are not willing to buy and store soybeans. What is the difference in the properties of the expected evolution of stock prices versus soybean prices that accounts for this result?
CONCEPT CHECK 23.8
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824 P A R T V I Options, Futures, and Other Derivatives
Table 23.3 , which presents betas on a variety of commodities, shows that the beta of orange juice, for example, was estimated to be .117 over the period. If the T-bill rate is currently 5% and the historical market risk premium is about 8%, the appropriate discount rate for orange juice would be given by the CAPM as
5% 1 .117 3 8% 5 5.94%
If the expected spot price for orange juice 6 months from now is $1.45 per pound, the present value of a 6-month deferred claim to a pound of orange juice is simply
$1.45 / (1.0594)1/ 2 5 $1.409
What would the proper futures price for orange juice be? The contract calls for the ultimate exchange of orange juice for the futures price. We have just shown that the present value of the juice is $1.409. This should equal the present value of the futures price that will be paid for the juice. A commitment to a payment of F 0 dollars in 6 months has a present value of F 0 /(1.05)
1/2 5 .976 3 F 0 . (Note that the discount rate is the risk-free rate of 5%, because the promised payment is fixed and therefore indepen- dent of market conditions.)
To equate the present values of the promised payment of F 0 and the promised receipt of orange juice, we would set
.976F0 5 $1.409
or F 0 5 $1.444.
Example 23.8 Commodity Futures Pricing
Discounted Cash Flow Analysis for Commodity Futures Given the current expectation of the spot price of the commodity at some future date and
a measure of the risk characteristics of that price, we can measure the present value of
a claim to receive the commodity at that future date. We simply calculate the appropri-
ate risk premium from a model such as the CAPM or APT and discount the expected
spot price at the appropriate risk-adjusted interest rate, as illustrated in the following
example.
The general rule, then, to determine the appropriate futures price is to equate the present
value of the future payment of F 0 and the present value of the commodity to be received. This implies
F0 (1 1 rf)
T 5 E(PT)
(1 1 k)T
or
F0 5 E(PT)¢ 1 1 rf 1 1 k
≤ T
(23.4)
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 825
Table 23.3
Commodity betas Commodity Beta Commodity Beta
Wheat 20.370 Orange juice 0.117 Corn 20.429 Propane 23.851 Oats 0.000 Cocoa 20.291 Soybeans 20.266 Silver 20.272 Soybean oil 20.650 Copper 0.005 Soybean meal 0.239 Cattle 0.365 Broilers 21.692 Hogs 20.148 Plywood 0.660 Pork bellies 20.062 Potatoes 20.610 Egg 20.293 Platinum 0.221 Lumber 20.131 Wool 0.307 Sugar 22.403 Cotton 20.015
Source: Zvi Bodie and Victor Rosansky, “Risk and Return in Commodity Futures,” Financial Analysts Journal 36 (May–June 1980). Copyright 1980, CFA Institute. Reproduced from the Financial Analysts Journal with permission from the CFA Institute. All rights reserved.
Suppose that the systematic risk of orange juice were to increase, holding the expected time T price of juice constant. If the expected spot price is unchanged, would the futures price change? In what direction? What is the intuition behind your answer?
CONCEPT CHECK 23.9
1. Foreign exchange futures trade on several foreign currencies, as well as on a European currency index. The interest rate parity relationship for foreign exchange futures is
F0 5 E0 ¢ 1 1 rUS 1 1 rforeign
≤ T
with exchange rates quoted as dollars per foreign currency. Deviations of the futures price from this value imply an arbitrage opportunity. Empirical evidence, however, suggests that generally
the parity relationship is satisfied.
SUMMARY
where k is the required rate of return on the commodity, which may be obtained from a model of asset market equilibrium such as the CAPM.
Note that Equation 23.4 is perfectly consistent with the spot-futures parity relationship.
For example, apply Equation 23.4 to the futures price for a stock paying no dividends.
Because the entire return on the stock is in the form of capital gains, the expected rate
of capital gains must equal k, the required rate of return on the stock. Consequently, the expected price of the stock is its current price times (1 1 k ) T , or E ( P T ) 5 P 0 (1 1 k )
T .
Substituting this expression into Equation 23.4 results in F 0 5 P 0 (1 1 r f ) T , which is exactly
the parity relationship.
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826 P A R T V I Options, Futures, and Other Derivatives
2. Futures contracts calling for cash settlement are traded on various stock market indexes. The contracts may be mixed with Treasury bills to construct artificial equity positions,
which makes them potentially valuable tools for market timers. Market index contracts are
used also by arbitrageurs who attempt to profit from violations of the stock-futures parity
rela tionship.
3. Hedging requires investors to purchase assets that will offset the sensitivity of their portfolios to particular sources of risk. A hedged position requires that the hedging vehicle provide profits that
vary inversely with the value of the position to be protected.
4. The hedge ratio is the number of hedging vehicles such as futures contracts required to offset the risk of the unprotected position. The hedge ratio for systematic market risk is proportional
to the size and beta of the underlying stock portfolio. The hedge ratio for fixed-income portfolios
is proportional to the price value of a basis point, which in turn is proportional to modified dura-
tion and the size of the portfolio.
5. Many investors such as hedge funds use hedging strategies to create market-neutral bets on per- ceived instances of relative mispricing between two or more securities. They are not arbitrage
strategies, but pure plays on a particular perceived profit opportunity.
6. Interest rate futures contracts may be written on the prices of debt securities (as in the case of Treasury-bond futures contracts) or on interest rates directly (as in the case of Eurodollar
contracts).
7. Swaps, which call for the exchange of a series of cash flows, may be viewed as portfolios of forward contracts. Each transaction may be viewed as a separate forward agreement. However,
instead of pricing each exchange independently, the swap sets one “forward price” that applies
to all of the transactions. Therefore, the swap price will be an average of the forward prices that
would prevail if each exchange were priced separately.
8. Commodity futures pricing is complicated by costs for storage of the underlying commodity. When the asset is willingly stored by investors, the storage costs net of convenience yield enter
the futures pricing equation as follows:
F0 5 P0 (1 1 rf 1 c) T
The non–interest net carrying costs, c, play the role of a “negative dividend” in this context.
9. When commodities are not stored for investment purposes, the correct futures price must be determined using general risk–return principles. In this event,
F0 5 E(PT) ¢1 1 rf 1 1 k
≤ T
The equilibrium (risk–return) and the no-arbitrage predictions of the proper futures price are consistent with one another for stored commodities.
Related Web sites for this chapter are available at www. mhhe.com/bkm
hedging
interest rate parity relationship
covered interest arbitrage
relationship
hedge ratio
index arbitrage
program trading
market-neutral bet
price value of a basis point
cross-hedging
foreign exchange swap
interest rate swap
notional principal
credit default swap
KEY TERMS
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 827
KEY EQUATIONS Interest rate parity (covered interest arbitrage): F0 5 E0 ¢ 1 1 rUS
1 1 rUK ≤
T
Hedging with futures: Hedge ratio 5 Change in portfolio value
Profit on one futures contract
Parity for stored commodities: F 0 5 P 0 (1 1 r f 1 c )
Futures price versus expected spot price: F0 5 E (PT) ¢1 1 rf 1 1 k
≤ T
PROBLEM SETS 1. A stock’s beta is a key input to hedging in the equity market. A bond’s duration is key in fixed- income hedging. How are they used similarly? Are there any differences in the calculations
necessary to formulate a hedge position in each market?
2. A U.S. exporting firm may use foreign exchange futures to hedge its exposure to exchange
rate risk. Its position in futures will depend in part on anticipated payments from its customers
denominated in foreign currency. In general, however, should its position in futures be more or
less than the number of contracts necessary to hedge these anticipated cash flows? What other
considerations might enter into the hedging strategy?
3. Both gold-mining firms and oil-producing firms might choose to use futures to hedge uncertainty
in future revenues due to price fluctuations. But trading activity sharply tails off for maturities
beyond 1 year. Suppose a firm wishes to use available (short maturity) contracts to hedge
commodity prices at a more distant horizon, say, 4 years from now. Do you think the hedge will
be more effective for the oil- or the gold-producing firm?
4. You believe that the spread between municipal bond yields and U.S. Treasury bond yields
is going to narrow in the coming month. How can you profit from such a change using the
municipal bond and T-bond futures contracts?
5. Consider the futures contract written on the S&P 500 index and maturing in 6 months. The
interest rate is 3% per 6-month period, and the future value of dividends expected to be paid
over the next 6 months is $15. The current index level is 1,425. Assume that you can short sell
the S&P index.
a. Suppose the expected rate of return on the market is 6% per 6-month period. What is the expected level of the index in 6 months?
b. What is the theoretical no-arbitrage price for a 6-month futures contract on the S&P 500 stock index?
c. Suppose the futures price is 1,422. Is there an arbitrage opportunity here? If so, how would you exploit it?
6. Suppose that the value of the S&P 500 stock index is 1,600.
a. If each futures contract costs $25 to trade with a discount broker, how much is the transaction cost per dollar of stock controlled by the futures contract?
b. If the average price of a share on the NYSE is about $40, how much is the transaction cost per “typical share” controlled by one futures contract?
c. For small investors, a typical transaction cost per share in stocks directly is about 10 cents per share. How many times the transactions costs in futures markets is this?
7. You manage a $16.5 million portfolio, currently all invested in equities, and believe that the mar-
ket is on the verge of a big but short-lived downturn. You would move your portfolio temporarily
Basic
Intermediate
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828 P A R T V I Options, Futures, and Other Derivatives
into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing
your equity position. Instead, you decide to temporarily hedge your equity holdings with S&P
500 index futures contracts.
a. Should you be long or short the contracts? Why? b. If your equity holdings are invested in a market-index fund, into how many contracts should
you enter? The S&P 500 index is now at 1,650 and the contract multiplier is $250.
c. How does your answer to ( b ) change if the beta of your portfolio is .6?
8. A manager is holding a $1 million stock portfolio with a beta of 1.25. She would like to hedge
the risk of the portfolio using the S&P 500 stock index futures contract. How many dollars’
worth of the index should she sell in the futures market to minimize the volatility of her position?
9. Suppose that the relationship between the rate of return on IBM stock, the market index, and a
computer industry index can be described by the following regression equation: r IBM 5 .5 r M 1 .75 r Industry . If a futures contract on the computer industry is traded, how would you hedge the exposure to the systematic and industry factors affecting the performance of IBM stock? How
many dollars’ worth of the market and industry index contracts would you buy or sell for each
dollar held in IBM?
10. Suppose that the spot price of the euro is currently $1.30. The 1-year futures price is $1.35.
Is the interest rate higher in the United States or the euro zone?
11. a. The spot price of the British pound is currently $2.00. If the risk-free interest rate on 1-year government bonds is 4% in the United States and 6% in the United Kingdom, what must be
the forward price of the pound for delivery 1 year from now?
b. How could an investor make risk-free arbitrage profits if the forward price were higher than the price you gave in answer to ( a )? Give a numerical example.
12. Consider the following information:
rUS 5 4%; rUK 5 7%
E0 5 2.00 dollars per pound
F0 5 1.98(1-year delivery)
where the interest rates are annual yields on U.S. or U.K. bills. Given this information:
a. Where would you lend? b. Where would you borrow? c. How could you arbitrage?
13. Farmer Brown grows Number 1 red corn and would like to hedge the value of the coming har-
vest. However, the futures contract is traded on the Number 2 yellow grade of corn. Suppose
that yellow corn typically sells for 90% of the price of red corn. If he grows 100,000 bushels,
and each futures contract calls for delivery of 5,000 bushels, how many contracts should Farmer
Brown buy or sell to hedge his position?
14. Return to Figure 23.7 . Suppose the LIBOR rate when the first listed Eurodollar contract matures
in January is .40%. What will be the profit or loss to each side of the Eurodollar contract?
15. Yields on short-term bonds tend to be more volatile than yields on long-term bonds. Suppose
that you have estimated that the yield on 20-year bonds changes by 10 basis points for every
15-basis-point move in the yield on 5-year bonds. You hold a $1 million portfolio of 5-year
maturity bonds with modified duration 4 years and desire to hedge your interest rate exposure
with T-bond futures, which currently have modified duration 9 years and sell at F 0 5 $95. How many futures contracts should you sell?
16. A manager is holding a $1 million bond portfolio with a modified duration of 8 years. She
would like to hedge the risk of the portfolio by short-selling Treasury bonds. The modified dura-
tion of T-bonds is 10 years. How many dollars’ worth of T-bonds should she sell to minimize the
variance of her position?
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 829
17. A corporation plans to issue $10 million of 10-year bonds in 3 months. At current yields
the bonds would have modified duration of 8 years. The T-note futures contract is selling at
F 0 5 100 and has modified duration of 6 years. How can the firm use this futures contract to hedge the risk surrounding the yield at which it will be able to sell its bonds? Both the bond and
the contract are at par value.
18. If the spot price of gold is $1,500 per troy ounce, the risk-free interest rate is 2%, and stor-
age and insurance costs are zero, what should be the forward price of gold for delivery in
1 year? Use an arbitrage argument to prove your answer. Include a numerical example show-
ing how you could make risk-free arbitrage profits if the forward price exceeded its upper
bound value.
19. If the corn harvest today is poor, would you expect this fact to have any effect on today’s futures
prices for corn to be delivered (postharvest) 2 years from today? Under what circumstances will
there be no effect?
20. Suppose that the price of corn is risky, with a beta of .5. The monthly storage cost is $.03, and
the current spot price is $5.50, with an expected spot price in 3 months of $5.88. If the expected
rate of return on the market is 0.9% per month, with a risk-free rate of 0.5% per month, would
you store corn for 3 months?
21. Suppose the U.S. yield curve is flat at 4% and the euro yield curve is flat at 3%. The current
exchange rate is $1.50 per euro. What will be the swap rate on an agreement to exchange cur-
rency over a 3-year period? The swap will call for the exchange of 1 million euros for a given
number of dollars in each year.
22. Desert Trading Company has issued $100 million worth of long-term bonds at a fixed rate of
7%. The firm then enters into an interest rate swap where it pays LIBOR and receives a fixed
6% on notional principal of $100 million. What is the firm’s overall cost of funds?
23. Firm ABC enters a 5-year swap with firm XYZ to pay LIBOR in return for a fixed 6% rate
on notional principal of $10 million. Two years from now, the market rate on 3-year swaps is
LIBOR for 5%; at this time, firm XYZ goes bankrupt and defaults on its swap obligation.
a. Why is firm ABC harmed by the default? b. What is the market value of the loss incurred by ABC as a result of the default? c. Suppose instead that ABC had gone bankrupt. How do you think the swap would be treated
in the reorganization of the firm?
24. Suppose that at the present time, one can enter 5-year swaps that exchange LIBOR for 8%.
An off-market swap would then be defined as a swap of LIBOR for a fixed rate other than 8%. For example, a firm with 10% coupon debt outstanding might like to convert to synthetic
floating-rate debt by entering a swap in which it pays LIBOR and receives a fixed rate of 10%.
What up-front payment will be required to induce a counterparty to take the other side of this
swap? Assume notional principal is $10 million.
25. Suppose the 1-year futures price on a stock-index portfolio is 1,624, the stock index currently is
1,600, the 1-year risk-free interest rate is 3%, and the year-end dividend that will be paid on a
$1,600 investment in the market index portfolio is $20.
a. By how much is the contract mispriced? b. Formulate a zero-net-investment arbitrage portfolio and show that you can lock in riskless
profits equal to the futures mispricing.
c. Now assume (as is true for small investors) that if you short sell the stocks in the market index, the proceeds of the short sale are kept with the broker, and you do not receive any
interest income on the funds. Is there still an arbitrage opportunity (assuming that you don’t
already own the shares in the index)? Explain.
d. Given the short-sale rules, what is the no-arbitrage band for the stock-futures price relation- ship? That is, given a stock index of 1,600, how high and how low can the futures price be
without giving rise to arbitrage opportunities?
Challenge
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830 P A R T V I Options, Futures, and Other Derivatives
26. Consider these futures market data for the June delivery S&P 500 contract, exactly 6 months
hence. The S&P 500 index is at 1,350, and the June maturity contract is at F 0 5 1,351.
a. If the current interest rate is 2.2% semiannually, and the average dividend rate of the stocks in the index is 1.2% semiannually, what fraction of the proceeds of stock short sales would
need to be available to you to earn arbitrage profits?
b. Suppose that you in fact have access to 90% of the proceeds from a short sale. What is the lower bound on the futures price that rules out arbitrage opportunities? By how much does
the actual futures price fall below the no-arbitrage bound? Formulate the appropriate arbi-
trage strategy, and calculate the profits to that strategy.
1. Donna Doni, CFA, wants to explore potential inefficiencies in the futures market. The TOBEC
stock index has a spot value of 185. TOBEC futures contracts are settled in cash and underlying
contract values are determined by multiplying $100 times the index value. The current annual
risk-free interest rate is 6.0%.
a. Calculate the theoretical price of the futures contract expiring 6 months from now, using the cost-of-carry model. The index pays no dividends.
The total (round-trip) transaction cost for trading a futures contract is $15.
b. Calculate the lower bound for the price of the futures contract expiring 6 months from now.
2. Suppose your client says, “I am invested in Japanese stocks but want to eliminate my exposure
to this market for a period of time. Can I accomplish this without the cost and inconvenience of
selling out and buying back in again if my expectations change?”
a. Briefly describe a strategy to hedge both the local market risk and the currency risk of investing in Japanese stocks.
b. Briefly explain why the hedge strategy you described in part ( a ) might not be fully effective.
3. René Michaels, CFA, plans to invest $1 million in U.S. government cash equivalents for the next
90 days. Michaels’s client has authorized her to use non–U.S. government cash equivalents, but
only if the currency risk is hedged to U.S. dollars by using forward currency contracts.
a. Calculate the U.S. dollar value of the hedged investment at the end of 90 days for each of the two cash equivalents in the table below. Show all calculations.
b. Briefly explain the theory that best accounts for your results. c. On the basis of this theory, estimate the implied interest rate for a 90-day U.S. government
cash equivalent.
Interest Rates 90-Day Cash Equivalents
Japanese government 7.6%
Swiss government 8.6%
Exchange Rates Currency Units per U.S. Dollar
Spot 90-Day Forward
Japanese yen 133.05 133.47
Swiss franc 1.5260 1.5348
4. After studying Iris Hamson’s credit analysis, George Davies is considering whether he can
increase the holding-period return on Yucatan Resort’s excess cash holdings (which are held in
pesos) by investing those cash holdings in the Mexican bond market. Although Davies would
be investing in a peso-denominated bond, the investment goal is to achieve the highest holding-
period return, measured in U.S. dollars, on the investment.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 831
Davies finds the higher yield on the Mexican 1-year bond, which is considered to be free
of credit risk, to be attractive, but he is concerned that depreciation of the peso will reduce the
holding-period return, measured in U.S. dollars. Hamson has prepared the following selected
financial data to help Davies make the decision:
Selected Economic and Financial Data
U.S. 1-year Treasury bond yield 2.5% Mexican 1-year bond yield 6.5%
Nominal Exchange Rates
Spot 9.5000 Pesos 5 U.S. $1.00
1-year forward 9.8707 Pesos 5 U.S. $1.00
Hamson recommends buying the Mexican 1-year bond and hedging the foreign currency expo-
sure using the 1-year forward exchange rate. Calculate the U.S. dollar holding-period return that
would result from the transaction recommended by Hamson. Is the U.S. dollar holding-period
return resulting from the transaction more or less than that available in the U.S.?
5. a. Pamela Itsuji, a currency trader for a Japanese bank, is evaluating the price of a 6-month Japanese yen/U.S. dollar currency futures contract. She gathers the following currency and
interest rate data:
Japanese yen/U.S. dollar spot currency exchange rate ¥124.30/$1.00 6-month Japanese interest rate 0.10% 6-month U.S. interest rate 3.80%
Calculate the theoretical price for a 6-month Japanese yen/U.S. dollar currency futures contract,
using the data above.
b. Itsuji is also reviewing the price of a 3-month Japanese yen/U.S. dollar currency futures con- tract, using the currency and interest rate data shown below. Because the 3-month Japanese
interest rate has just increased to .50%, Itsuji recognizes that an arbitrage opportunity exists
and decides to borrow $1 million U.S. dollars to purchase Japanese yen. Calculate the yen
arbitrage profit from Itsuji’s strategy, using the following data:
Japanese yen/U.S. dollar spot currency exchange rate ¥124.30/$1.00 New 3-month Japanese interest rate 0.50% 3-month U.S. interest rate 3.50% 3-month currency futures contract value ¥123.2605/$1.00
6. Janice Delsing, a U.S.-based portfolio manager, manages an $800 million portfolio ($600 million
in stocks and $200 million in bonds). In reaction to anticipated short-term market events,
Delsing wishes to adjust the allocation to 50% stock and 50% bonds through the use of futures.
Her position will be held only until “the time is right to restore the original asset allocation.”
Delsing determines a financial futures–based asset allocation strategy is appropriate. The stock
futures index multiplier is $250 and the denomination of the bond futures contract is $100,000.
Other information relevant to a futures-based strategy is as follows:
Bond portfolio modified duration 5 years Bond portfolio yield to maturity 7% Price value of a basis point of bond futures $97.85 Stock-index futures price 1378 Stock portfolio beta 1.0
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832 P A R T V I Options, Futures, and Other Derivatives
a. Describe the financial futures–based strategy needed and explain how the strategy allows Delsing to implement her allocation adjustment. No calculations are necessary.
b. Compute the number of each of the following needed to implement Delsing’s asset allocation strategy:
i. Bond futures contracts.
ii. Stock-index futures contracts.
7. You are provided the information outlined as follows to be used in solving this problem.
Issue Price Yield to Maturity
Modified Duration*
U.S. Treasury bond 11¾% maturing Nov. 15, 2029 100 11.75% 7.6 years
U.S. Treasury long bond futures contract (contract expiration in 6 months) 63.33 11.85% 8.0 years
XYZ Corporation bond 12½% maturing June 1, 2020 (sinking fund debenture, rated AAA) 93 13.50% 7.2 years
Volatility of AAA corporate bond yields relative to U.S. Treasury bond yields 5 1.25 to 1.0 (1.25 times)
Assume no commission and no margin requirements on U.S. Treasury long bond futures contracts. Assume no taxes.
One U.S. Treasury bond futures contract is a claim on $100,000 par value long-term U.S. Treasury bonds.
*Modified duration 5 Duration/(1 1 y).
Situation A A fixed-income manager holding a $20 million market value position of U.S. Treasury 11¾% bonds maturing November 15, 2029, expects the economic growth rate and the
inflation rate to be above market expectations in the near future. Institutional rigidities prevent
any existing bonds in the portfolio from being sold in the cash market.
Situation B The treasurer of XYZ Corporation has recently become convinced that interest rates will decline in the near future. He believes it is an opportune time to purchase his company’s
sinking fund bonds in advance of requirements because these bonds are trading at a discount from
par value. He is preparing to purchase in the open market $20 million par value XYZ Corporation
12½% bonds maturing June 1, 2020. A $20 million par value position of these bonds is currently
offered in the open market at 93. Unfortunately, the treasurer must obtain approval from the board
of directors for such a purchase, and this approval process can take up to 2 months. The board of
directors’ approval in this instance is only a formality.
For each of these two situations, demonstrate how interest rate risk can be hedged using the
Treasury bond futures contract. Show all calculations, including the number of futures contracts
used.
8. You ran a regression of the yield of KC Company’s 10-year bond on the 10-year U.S. Treasury
benchmark’s yield using month-end data for the past year. You found the following result:
YieldKC 5 0.54 1 1.22 YieldTreasury
where Yield KC is the yield on the KC bond and Yield Treasury is the yield on the U.S. Treasury
bond. The modified duration on the 10-year U.S. Treasury is 7.0 years, and modified duration on
the KC bond is 6.93 years.
a. Calculate the percentage change in the price of the 10-year U.S. Treasury, assuming a 50-basis- point change in the yield on the 10-year U.S. Treasury.
b. Calculate the percentage change in the price of the KC bond, using the regression equation above, assuming a 50-basis-point change in the yield on the 10-year U.S. Treasury.
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C H A P T E R 2 3 Futures, Swaps, and Risk Management 833
E-INVESTMENTS EXERCISES Go to the Chicago Mercantile Exchange Web site ( www.cme.com ) and link to the tab for CME Products, then Foreign Exchange (FX). Link to the Canadian Dollar contracts and answer the following questions about the futures contract (see Contract Specifications ):
What is the size (units of $CD) of each contract?
What is the tick size (minimum price increment) for the contract?
What time period during the day is the contract traded?
If the delivery option is exercised, when and where does delivery take place?
SOLUTIONS TO CONCEPT CHECKS 1. According to interest rate parity, F 0 should be $1.981. Because the futures price is too high, we
should reverse the arbitrage strategy just considered.
CF Now ($) CF in 1 Year
1. Borrow $2.00 in the U.S. Convert to 1 U.K. pound. 12.00 22.00(1.04)
2. Lend the 1 pound in the U.K. 22.00 1.05E1 3. Enter a contract to sell 1.05 pounds at a futures
price of $2.01/£. 0 (£1.05)($2.01/£ 2 E1)
TOTAL 0 $.0305
2. Because the firm does poorly when the dollar depreciates, it hedges with a futures contract that
will provide profits in that scenario. It needs to enter a long position in pound futures, which means that it will earn profits on the contract when the futures price increases, that is, when more
dollars are required to purchase one pound. The specific hedge ratio is determined by noting that
if the number of dollars required to buy one pound rises by $.05, profits decrease by $200,000 at
the same time that the profit on a long future contract would increase by $.05 3 62,500 5 $3,125.
The hedge ratio is
$200,000 per $.05 depreciation in the dollar
$3,125 per contract per $.05 depreciation 5 64 contracts long
3. Each $1 increase in the price of corn reduces profits by $1 million. Therefore, the firm needs to
enter futures contracts to purchase 1 million bushels at a price stipulated today. The futures posi-
tion will profit by $1 million for each increase of $1 in the price of corn. The profit on the contract
will offset the lost profits on operations.
4. In General (per unit of index) Our Numbers
Hold 100,000 units of indexed stock portfolio with S0 5 1,400. ST 100,000 ST Sell 400 contracts. F0 2 ST 400 3 $250 3 (1,414 2 ST)
TOTAL F0 $141,400,000
The net cash flow is riskless, and provides a 1% monthly rate of return, equal to the risk-free rate.
5. The price value of a basis point is still $9,000, as a 1-basis-point change in the interest rate
reduces the value of the $20 million portfolio by .01% 3 4.5 5 .045%. Therefore, the number
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834 P A R T V I Options, Futures, and Other Derivatives
of futures needed to hedge the interest rate risk is the same as for a portfolio half the size with
double the modified duration.
6. LIBOR
7% 8% 9%
As debt payer (LIBOR 3 $10 million) 2700,000 2800,000 2900,000
As fixed payer receives $10 million 3 (LIBOR 2 .08) 2100,000 0 1100,000
Net cash flow 2800,000 2800,000 2800,000
Regardless of the LIBOR rate, the firm’s net cash outflow equals .08 3 principal, just as if it had
issued a fixed-rate bond with a coupon of 8%.
7. The manager would like to hold on to the money market securities because of their attractive rela-
tive pricing compared to other short-term assets. However, there is an expectation that rates will
fall. The manager can hold this particular portfolio of short-term assets and still benefit from the drop in interest rates by entering a swap to pay a short-term interest rate and receive a fixed inter-
est rate. The resulting synthetic fixed-rate portfolio will increase in value if rates do fall.
8. Stocks offer a total return (capital gain plus dividends) large enough to compensate investors for
the time value of the money tied up in the stock. Agricultural prices do not necessarily increase
over time. In fact, across a harvest, crop prices will fall. The returns necessary to make storage
economically attractive are lacking.
9. If systematic risk were higher, the appropriate discount rate, k, would increase. Referring to Equation 23.4, we conclude that F 0 would fall. Intuitively, the claim to 1 pound of orange juice is worth less today if its expected price is unchanged, while the risk associated with the value of
that claim increases. Therefore, the amount investors are willing to pay today for future delivery
is lower.
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244 MOST FINANCIAL ASSETS are managed by professional investors, who thus at least indirectly allocate the lion’s share of capital across firms. Efficient allocation therefore depends on the quality of these profes- sionals and the propensity of financial mar- kets to direct capital to the best stewards. Therefore, if capital markets are to be rea- sonably efficient, investors must be able to measure the performance of their asset managers. And such measurement must be sufficiently accurate to allow a proper rank ordering of their ability. At a minimum, effi- ciency demands that performance evaluation be accurate enough to distinguish managers who (net of fees) are inferior to a randomly chosen, diversified portfolio. Thus, the social benefit of a reasonably reliable performance
evaluation method is as large as that of mar- ket efficiency.
How can we evaluate the performance of a portfolio manager? It turns out that even an average portfolio return is not as straight- forward to measure as it might seem. In addition, adjusting average returns for risk presents a host of other problems. In the end, performance evaluation is far from trivial.
We begin with the measurement of port- folio returns. From there we move on to con- ventional approaches to risk adjustment. We identify the problems with these approaches when applied in various real-life situations. We then turn to some practical procedures for performance evaluation in the field such as style analysis, the Morningstar Star Ratings, and in-house performance attribution.
PA R
T V
II
Portfolio Performance
Evaluation
CHAPTER TWENTY-FOUR
24.1 The Conventional Theory of Performance Evaluation
Average Rates of Return We defined the holding-period return (HPR) in Section 5.1 of Chapter 5 and explained
the differences between arithmetic and geometric averages. Suppose we evaluate the per-
formance of a portfolio over a period of 5 years from 20 quarterly rates of return. The
arithmetic average of this sample of returns would be the best estimate of the expected
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836 P A R T V I I Applied Portfolio Management
rate of return of the portfolio for the next quarter. In contrast, the geometric average is the
constant quarterly return over the 20 quarters that would yield the same total cumulative
return. Therefore, the geometric average, r G , is defined by
(1 1 rG) 20 5 (1 1 r1)(1 1 r2) c (1 1 r20)
The right-hand side of this equation is the compounded final value of a $1 investment earn-
ing the 20 quarterly rates of return over the 5-year observation period. The left-hand side is
the compounded value of a $1 investment earning r G each quarter. We solve for 1 1 r G as 1
1 1 rG 5 3(1 1 r1)(1 1 r2) c (1 1 r20)41/20 Each return has an equal weight in the geometric average. For this reason, the geometric
average is referred to as a time-weighted average . To set the stage for discussing the more subtle issues that follow, let us start with a
trivial example. Consider a stock paying a dividend of $2 annually that currently sells for
$50. You purchase the stock today, collect the $2 dividend, and then sell the stock for $53
at year-end. Your rate of return is
Total proceeds
Initial investment 5
Income 1 Capital gain
50 5
2 1 3
50 5 .10, or 10%
Another way to derive the rate of return that is useful in the more difficult multiperiod
case is to set up the investment as a discounted cash flow problem. Call r the rate of return that equates the present value of all cash flows from the investment with the initial outlay.
In our example the stock is purchased for $50 and generates cash flows at year-end of $2
(dividend) plus $53 (sale of stock). Therefore, we solve 50 5 (2 1 53)/(1 1 r ) to find again that r 5 10%.
Time-Weighted Returns versus Dollar-Weighted Returns When we consider investments over a period during which cash was added to or with-
drawn from the portfolio, measuring the rate of return becomes more difficult. To continue
our example, suppose that you were to purchase a second share of the same stock at the
end of the first year, and hold both shares until the end of year 2, at which point you sell
each share for $54.
Total cash outlays are
Time Outlay
0 $50 to purchase first share 1 $53 to purchase second share a year later Proceeds 1 $2 dividend from initially purchased share 2 $4 dividend from the 2 shares held in the second year, plus $108 received from selling both shares at $54 each
1 This formula gives the geometric average as a quarterly rate of return, consistent with the quarterly rates used to
compute it. When the observation period is of length h years (1/4 in this example), the annualized compounded rate is defined by 1 1 r GA 5 (1 1 r Gh )
1/ h . In general, the annualized geometric average of T observations, each of
length h, is 1 1 rGA 5 aqT t51
(1 1 rt)b1/hT where ∏ is the product operator. In our example with T 5 20 quarterly observations, each of length h 5 ¼ year, 1/ hT 5 1/5, so to find the annualized geometric average, we would take the fifth root of the cumulative return over the 5-year investment period.
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C H A P T E R 2 4 Portfolio Performance Evaluation 837
Using the discounted cash flow (DCF) approach, we can solve for the average return
over the 2 years by equating the present values of the cash inflows and outflows:
50 1 53
1 1 r 5
2
1 1 r 1
112
(1 1 r)2
resulting in r 5 7.117%. This value is called the internal rate of return, or the dollar-weighted rate of return on
the investment. It is “dollar weighted” because the stock’s performance in the second year,
when two shares of stock are held, has a greater influence on the average overall return
than the first-year return, when only one share is held.
The time-weighted (geometric average) return is 7.81%:
r1 5 53 1 2 2 50
50 5 .10 5 10% r2 5
54 1 2 2 53
53 5 .0566 5 5.66%
rG 5 (1.10 3 1.0566) 1/2 2 1 5 .0781 5 7.81%
The dollar-weighted average is less than the time-weighted average in this example because
the return in the second year, when more money was invested, is lower.
Shares of XYZ Corp. pay a $2 dividend at the end of every year on December 31. An investor buys two shares of the stock on January 1 at a price of $20 each, sells one of those shares for $22 a year later on the next January 1, and sells the second share an additional year later for $19. Find the dollar- and time- weighted rates of return on the 2-year investment.
CONCEPT CHECK 24.1
Dollar-Weighted Return and Investment Performance Every household faces several daunting saving goals, for example, the education of chil-
dren and retirement. Many of these goals allow for tax-sheltered savings, for example,
IRAs or 401(k) plans for retirement and 529 plans for college expenses. These accounts
are, by their nature, separated from other household assets.
Households have considerable latitude in choices of investment venues and will want to
check results from time to time. How should they do this? The answer here is quite simple.
First, the household must maintain a spreadsheet of time-dated cash inflows and outflows.
It is a simple task to record the current value of the investment account. In this setting, the
dollar-weighted average over any investment period will yield the effective rate of return
earned for the period. 2
To guarantee that you can accomplish this important task, be sure to solve Problem 1 at
the end of the chapter. See also the nearby Excel Application box.
Adjusting Returns for Risk Evaluating performance based on average return alone is not very useful. Returns must be
adjusted for risk before they can be compared meaningfully. The simplest and most popu-
lar way to adjust returns for portfolio risk is to compare rates of return with those of other
investment funds with similar risk characteristics. For example, high-yield bond portfolios
2 Excel’s function XIRR allows you to input sums at any date. The function provides the IRR between any two
dates given a starting value, cash flows at various dates in between (with additions given as negative numbers, and
withdrawals as positive values), and a final value on the closing date.
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838
are grouped into one “universe,” growth stock equity funds are grouped into another uni-
verse, and so on. Then the (usually time-weighted) average returns of each fund within the
universe are ordered, and each portfolio manager receives a percentile ranking depending
on relative performance with the comparison universe . For example, the manager with the ninth-best performance in a universe of 100 funds would be the 90th percentile man-
ager: Her performance was better than 90% of all competing funds over the evaluation
period. 3 The nearby box reports on Vanguard’s recent revamp of its benchmark indexes for
several asset classes.
These relative rankings are usually displayed in a chart such as that in Figure 24.1 . The
chart summarizes performance rankings over four periods: 1 quarter, 1 year, 3 years, and
5 years. The top and bottom lines of each box are drawn at the rate of return of the 95th
and 5th percentile managers. The three dashed lines correspond to the rates of return of the
75th, 50th (median), and 25th percentile managers. The diamond is drawn at the average
return of a particular fund and the square is drawn at the return of a benchmark index such
as the S&P 500. The placement of the diamond within the box is an easy-to-read represen-
tation of the performance of the fund relative to the comparison universe.
This comparison of performance with other managers of similar investment style is a use-
ful first step in evaluating performance. However, such rankings can be misleading. Within
a particular universe, some managers may concentrate on particular subgroups, so that
Initial contribution 10,000.00 Data
$ Initial Price
# of Shares
Monthly Dividend
Total Dividend
End Price
(S1) Dividend-yielding security 5,000.00$ 9.50$ 526 0.30$ 157.89$ 13.00$ (S2) Non-dividend-yielding security 5,000.00$ 9.50$ 526 0 0 11.00$ (S3) Non-div yielding security (03/12) 8,000.00$ 8.80$ 909 0 0 13.50$
1-Mar-11 (10,000.00) 1-Apr-11 157.89
2-May-11 157.89 1-Jun-11 157.89 1-Jul-11 157.89
1-Aug-11 157.89 1-Sep-11 157.89 3-Oct-11 (5,631.58) 1-Nov-11 157.89
$ $ $ $ $ $ $ $ $
LEGEND:
Given data Value calculated
See comment
In 10/11, investor withdraws all of security S2 from the account at the current price.
2 1
3
4 5 6 7 8 9 10 11 12 13 14 15 16 17
A B DC E F G H I J
eXcel APPLICATIONS: Simple Investment Account
A n investment account starts with an initial contribution of $10,000 dollars. Over a time period of 2 years, the
account experiences both inflows and outflows of cash in the form of additional contributions, withdrawals, and dividends (not reinvested). Using Excel’s XIRR function, this spreadsheet shows the dollar-weighted average return of the account.
Excel Questions
1. What would happen to the rate of return if, instead of with- drawing security S2 in October, the investor holds onto it? Explain the difference in returns.
2. How much would the investor need to contribute to the account in 03/12 to bring the dollar-weighted average return up to a value of zero?
3 In previous chapters (particularly in Chapter 11 on the efficient market hypothesis), we have examined whether
actively managed portfolios can outperform a passive index. For this purpose we looked at the distribution of
alpha values for samples of mutual funds. We noted that any conclusion from such samples was subject to error
due to survivorship bias if funds that failed during the sample period were excluded from the sample. In this
chapter, we are interested in how to assess the performance of individual funds (or other portfolios) of interest.
When a particular portfolio is chosen today for inspection of its returns going forward, survivorship bias is not
an issue. However, comparison groups must be free of survivorship bias. A sample comprised only of surviving
funds will bias the return of the benchmark group upward and the relative performance of any particular fund downward.
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839
portfolio characteristics are not truly compa-
rable. For example, within the equity universe,
one manager may concentrate on high-beta or
aggressive growth stocks. Similarly, within
fixed-income universes, durations can vary
across managers. These considerations suggest
that a more precise means for risk adjustment
is desirable.
Methods of risk-adjusted performance evalu-
ation using mean-variance criteria came on
stage simultaneously with the capital asset
pricing model. Jack Treynor, 4 William Sharpe,
5
and Michael Jensen 6 recognized immediately
the implications of the CAPM for rating the
performance of managers. Within a short time,
academicians were in command of a battery of
performance measures, and a bounty of schol-
arly investigation of mutual fund performance
was pouring from ivory towers. Shortly thereaf-
ter, agents emerged who were willing to supply
rating services to portfolio managers and their
clients.
Vanguard to Change Target Benchmarks for 22 Index Funds
Vanguard plans to transition six international stock index funds to FTSE benchmarks and 16 U.S. stock and balanced index funds to new benchmarks developed by the Uni- versity of Chicago’s Center for Research in Security Prices (CRSP). The transition from the current MSCI benchmarks for the 22 funds is expected to result in considerable sav- ings for the funds’ shareholders over time.
“The indexes from FTSE and CRSP are well constructed, offer comprehensive coverage of their respective markets, and meet Vanguard’s ‘best practice’ standards for market bench- marks,” said Vanguard Chief Investment Officer Gus Sauter. “Equally important, and with our clients’ best interests in mind, we negotiated licensing agreements for these benchmarks that we expect will enable us to deliver significant value to our index fund and ETF shareholders and lower expense ratios over time.” In an environment in which index licensing fees, in gen- eral, have represented a growing portion of the expenses that investors pay to own index funds and ETFs, Mr. Sauter noted that the long-term agreements with FTSE and CRSP will provide cost certainty going forward with these two index providers.
In 2009, CRSP engaged with Vanguard to create a new series of investable indexes, the CRSP Indexes. Vanguard will be the first investment management firm to track CRSP’s
broadly diversified benchmarks that cover the broad U.S. market, market capitalization segments, and styles. CRSP’s capitalization-weighted methodology introduces the unique concept of “packeting,” which cushions the movement of stocks between adjacent indexes and allows holdings to be shared between two indexes of the same family. This approach maximizes style purity while minimizing index turnover.
Sixteen Vanguard stock and balanced index funds, with aggregate assets of $367 billion, will track CRSP bench- marks, including Vanguard’s largest index fund, the $197 billion Vanguard Total Stock Market Index Fund. The fund and its ETF Shares (ticker: VTI) will transition from the MSCI U.S. Broad Market Index to the CRSP US Total Market Index.
The benchmark changes will encompass all share classes of the 22 funds, including ETFs. The transitions will be stag- gered and are expected to occur collectively over a num- ber of months. No changes are planned for Vanguard U.S. stock index funds seeking to track Russell and Standard & Poor’s benchmarks, or the 11 Vanguard sector equity funds currently seeking to track MSCI benchmarks.
Source: October 2, 2012 © The Vanguard Group, Inc., used with permission.
W O
R D
S FR O
M T
H E ST
R E E T
30
1 Quarter 1 Year 3 Years 5 Years
Rate of Return (%)
5
10
15
20
25
S&P 500 The Markowill Group
Figure 24.1 Universe comparison, periods ending December 31, 2010
4 Jack L. Treynor, “How to Rate Management Investment Funds,” Harvard Business Review 43 (January– February
1966).
5 William F. Sharpe, “Mutual Fund Performance,” Journal of Business 39 (January 1966).
6 Michael C. Jensen, “The Performance of Mutual Funds in the Period 1945–1964,” Journal of Finance,
May 1968; and “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios,” Journal of Business, April 1969.
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840 P A R T V I I Applied Portfolio Management
But while widely used, risk-adjusted performance measures each have their own limita-
tions. Moreover, their reliability requires quite a long history of consistent management
with a steady level of performance and a representative sample of investment environ-
ments: bull as well as bear markets.
We start by cataloging some possible risk-adjusted performance measures for a portfo-
lio, P, and examine the circumstances in which each measure might be most relevant.
1. Sharpe ratio: (rP 2 rf)/sP
Sharpe’s ratio divides average portfolio excess return over the sample period by the standard deviation of returns over that period. It measures the reward to (total)
volatility trade-off. 7
2. Treynor measure: (rP 2 rf)/bP
Like the Sharpe ratio, Treynor’s measure gives excess return per unit of risk, but it uses systematic risk instead of total risk.
3. Jensen’s alpha: aP 5 rP 2 3rf 1 bP (rM 2 rf)4 Jensen’s alpha is the average return on the portfolio over and above that predicted
by the CAPM, given the portfolio’s beta and the average market return. 8
4. Information ratio: a P / s ( e P )
The information ratio divides the alpha of the portfolio by the nonsystematic risk of the portfolio, called “tracking error” in the industry. It measures abnormal return
per unit of risk that in principle could be diversified away by holding a market
index portfolio.
5. Morningstar risk-adjusted return: MRAR(g) 5 B 1 T a
T
t51 ¢ 1 1 rt
1 1 rft ≤2gR
12
g
2 1
The Morningstar rating is a sort of harmonic average of excess returns, where
t 5 1, . . . , T are monthly observations, 9 and g measures risk aversion. Higher g means greater punishment for risk. For mutual funds, Morningstar uses g 5 2,
which is considered a reasonable coefficient for an average retail client. 10
MRAR
can be interpreted as the risk-free equivalent excess return of the portfolio for an
investor with risk aversion measured by g .
Each performance measure has some appeal. But each does not necessarily provide
consistent assessments of performance, because the risk measures used to adjust returns
differ substantially.
7 We place bars over r f as well as r P to denote the fact that because the risk-free rate may not be constant over the
measurement period, we are taking a sample average, just as we do for r P . Equivalently, we may simply compute sample average excess returns.
8 In many cases performance evaluation assumes a multifactor market. For example, when the Fama-French
3- factor model is used, Jensen’s alpha will be: aP 5 rP 2 rf 2 bPM (rM 2 rf) 2 sPrSMB 2 hPrHML where s P is the loading on the SMB portfolio and h P is the loading on the HML portfolio. A multifactor version of the Treynor measure also exists. See footnote 13.
9 The fraction (1 1 r t )/(1 1 r ft ) is well approximated by 1 plus the excess return, R t .
10
The MRAR measure is the certainty-equivalent geometric average excess return derived from a more sophis- ticated utility function than the mean-variance function we used in Chapter 6. The utility function is called con- stant relative risk aversion (CRRA). When investors have CRRA, their capital allocation (the fraction of the portfolio placed in risk-free versus risky assets) does not change with wealth. The coefficient of risk aversion is:
A 5 1 1 g . When g 5 0 (equivalently, A 5 1), the utility function is just the geometric average of gross excess returns:
MRAR(0) 5 c qT t51
(1 1 Rt)d 12T 2 1
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C H A P T E R 2 4 Portfolio Performance Evaluation 841
The M 2 Measure of Performance While the Sharpe ratio can be used to rank portfolio performance, its numerical value is
not easy to interpret. Comparing the ratios for portfolios M and P in Concept Check 2, you should have found that S P 5 .69 and S M 5 .73. This suggests that portfolio P under- performed the market index. But is a difference of .04 in the Sharpe ratio economically
meaningful? We often compare rates of return, but these pure numbers are difficult to
interpret.
An equivalent representation of Sharpe’s ratio was proposed by Graham and Harvey,
and later popularized by Leah Modigliani of Morgan Stanley and her grandfather Franco
Modigliani, past winner of the Nobel Prize in Economics. 11
Their approach has been
dubbed the M 2 measure (for Modigliani-squared). Like the Sharpe ratio, M 2 focuses on total volatility as a measure of risk, but its risk adjustment leads to an easy-to-interpret dif-
ferential return relative to the benchmark index.
To compute M 2 , we imagine that a managed portfolio, P, is mixed with a position in T-bills so that the complete, or “adjusted,” portfolio matches the volatility of a market
index such as the S&P 500. If the managed portfolio has 1.5 times the standard deviation
of the index, the adjusted portfolio would be two-thirds invested in the managed portfolio
and one-third in bills. The adjusted portfolio, which we call P *, would then have the same standard deviation as the index. (If the managed portfolio had lower standard deviation than the index, it would be leveraged by borrowing money and investing the proceeds in
the portfolio.) Because the market index and portfolio P * have the same standard devia- tion, we may compare their performance simply by comparing returns. This is the M 2 measure for portfolio P:
MP 2 5 rP* 2 rM (24.1)
Consider the following data for a particular sample period:
Portfolio P Market M
Average return 35% 28% Beta 1.20 1.00
Standard deviation 42% 30% Tracking error (nonsystematic risk), s (e) 18% 0
Calculate the following performance measures for portfolio P and the market: Sharpe, Jensen (alpha), Treynor, information ratio. The T-bill rate during the period was 6%. By which measures did portfolio P outperform the market?
CONCEPT CHECK 24.2
11
John R. Graham and Campbell R. Harvey, “Market Timing Ability and Volatility Implied in Investment Advi-
sors’ Asset Allocation Recommendations,” National Bureau of Economic Research Working Paper 4890, October
1994. The part of this paper dealing with volatility-adjusted returns was ultimately published as “Grading the
Performance of Market Timing Newsletters,” Financial Analysts Journal 53 (November/December 1997), pp. 54–66. Franco Modigliani and Leah Modigliani, “Risk-Adjusted Performance,” Journal of Portfolio Manage- ment, Winter 1997, pp. 45–54.
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842 P A R T V I I Applied Portfolio Management
Sharpe’s Ratio Is the Criterion for Overall Portfolios Suppose that Jane Close constructs a portfolio and holds it for a considerable period of
time. She makes no changes in portfolio composition during the period. In addition, sup-
pose that the daily rates of return on all securities have constant means, variances, and
covariances. These assumptions are unrealistic, and the need for them highlights the short-
coming of conventional applications of performance measurement.
Now we want to evaluate the performance of Jane’s portfolio. Has she made a good
choice of securities? This is really a three-pronged question. First, “good choice” com-
pared with what alternatives? Second, in choosing between two dis-
tinct alternative portfolios, what are the appropriate criteria to evaluate
performance? Finally, the performance criteria having been identified,
is there a rule that will separate basic ability from the random luck of
the draw?
Earlier chapters of this text help to determine portfolio choice cri-
teria. If investor preferences can be summarized by a mean-variance
utility function such as that introduced in Chapter 6, we can arrive at
a relatively simple criterion. The particular utility function that we
used is
U 5 E(rP) 2 1⁄2 AsP 2
where A is the coefficient of risk aversion. With mean-variance pref- erences, Jane wants to maximize the Sharpe ratio [ E ( r P ) 2 r f ]/ s P . Recall that this criterion led to the selection of the tangency portfolio in
Chapter 7. Jane’s problem reduces to the search for the portfolio with
the highest possible Sharpe ratio.
Using the data of Concept Check 2, P has a standard deviation of 42% versus a market standard deviation of 30%. Therefore, the adjusted portfolio P * would be formed by mixing bills and portfolio P with weights 30/42 5 .714 in P and 1 2 .714 5 .286 in bills. The return on this portfolio would be (.286 3 6%) 1 (.714 3 35%) 5 26.7%, which is 1.3% less than the market return. Thus portfolio P has an M2P measure of 2 1.3%.
A graphical representation of M 2 appears in Figure 24.2 . We move down the capital allocation line corresponding to portfolio P (by mixing P with T-bills) until we reduce the standard deviation of the adjusted portfolio to match that of the market index. M2P is then the vertical distance (the difference in expected returns) between portfolios P * and M. You can see from Figure 24.2 that P will have a negative M 2 when its capital allocation line is less steep than the capital market line, that is, when its Sharpe ratio is less than that of the market index. 12
Example 24.1 M 2 Measure
12 M 2 is positive when the portfolio’s Sharpe ratio exceeds the market’s. Letting R denote excess returns and
S denote Sharpe measures, the geometry of Figure 24.2 implies that R P * 5 S P s M , and therefore that
M2 5 rp* 2 rM 5 Rp* 2 RM 5 SPsM 2 SMsM 5 (SP 2 SM)sM
M 2 and the Sharpe ratio will therefore rank order portfolios identically.
Figure 24.2 M 2 of portfolio P
E(r)
CAL(P)
CML
P* P
M
F
M2
σM σP σ
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C H A P T E R 2 4 Portfolio Performance Evaluation 843
Appropriate Performance Measures in Two Scenarios To evaluate Jane’s portfolio choice, we first ask whether this portfolio is her exclusive
investment vehicle. If the answer is no, we need to know her “complementary” port-
folio. The appropriate measure of portfolio performance depends critically on whether
the portfolio is the entire investment fund or only a portion of the investor’s overall
wealth.
Jane’s Portfolio Represents Her Entire Risky Investment Fund In this simplest case we need to ascertain only whether Jane’s portfolio has the highest Sharpe
measure. We can proceed in three steps:
1. Assume that past security performance is representative of expected performance,
meaning that realized security returns over Jane’s holding period exhibit averages
and covariances similar to those that Jane had anticipated.
2. Determine the benchmark (alternative) portfolio that Jane would have held if she
had chosen a passive strategy, such as the S&P 500.
3. Compare Jane’s Sharpe measure or M 2 to that of the best portfolio.
In sum, when Jane’s portfolio represents her entire investment fund, the benchmark is
the market index or another specific portfolio. The performance criterion is the Sharpe
measure of the actual portfolio versus the benchmark.
Jane’s Choice Portfolio Is One of Many Portfolios Combined into a Large Investment Fund This case might describe a situation where Jane, as a cor- porate financial officer, manages the corporate pension fund. She parcels out the entire
fund to a number of portfolio managers. Then she evaluates the performance of individual
managers to reallocate the fund to improve future performance. What is the correct perfor-
mance measure?
The Sharpe ratio is based on average excess return (the reward) against total SD
(total portfolio risk). It measures the slope of the CAL. However, when Jane employs
a number of managers, nonsystematic risk will be largely diversified away, so system-
atic risk becomes the relevant measure of risk. The appropriate performance metric is
now Treynor’s, which takes the ratio of average excess return to beta (because systematic
SD 5 b 3 market SD).
Consider portfolios P and Q in Table 24.1 and the graph in Figure 24.3 . We plot P and Q in the expected return–beta (rather than the expected return–standard deviation) plane, because we assume that P and Q are two of many subportfolios in the fund, and thus that
Table 24.1
Portfolio performance Portfolio P Portfolio Q Market
Beta .90 1.60 1.0 Excess return (r 2 rf) 11% 19% 10% Alpha* 2% 3% 0
* Alpha 5 Excess return 2 (Beta 3 Market excess return) 5 (r 2 rf ) 2 b(rM 2 rf ) 5 r 2 [r f 1 b(rM 2 rf )]
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844 P A R T V I I Applied Portfolio Management
nonsystematic risk will be largely diversified
away. The security market line (SML) shows
the value of a P and a Q as the distance of P and Q above the SML.
If we invest w Q in Q and w F 5 1 2 w Q in T-bills, the resulting portfolio, Q *, will have alpha and beta values proportional to Q ’s alpha and beta scaled down by w Q :
aQ* 5 wQaQ bQ* 5 wQbQ
Thus all portfolios such as Q *, generated by mixing Q with T-bills, plot on a straight line from the origin through Q. We call it the T-line for the Treynor measure, which is the slope of this line.
Figure 24.3 shows the T -line for portfo- lio P as well. P has a steeper T -line; despite its lower alpha, P is a better portfolio after all. For any given beta, a mixture of P with T-bills will give a better alpha than a mixture
of Q with T-bills. Figure 24.3 Treynor’s measure
.9 1.0 1.6
19
16
11 10 9
Q SML
Tp Line TQ Line
Excess Return (%) r − rf
P
M
β
αQ = 3%
αp = 2%
Suppose we choose to mix Q with T-bills to create a portfolio Q * with a beta equal to that of P. We find the necessary proportion by solving for w Q :
bQ* 5 wQbQ 5 1.6wQ 5 bP 5 .9
wQ 5 9⁄16
Portfolio Q * therefore has an alpha of aQ* 5 9⁄16 3 3% 5 1.69%
which is less than that of P.
Example 24.2 Equalizing Beta
The slope of the T -line, giving the trade-off between excess return and beta, is the appropriate performance criterion in this case. The slope for P, denoted by T P , is given by
TP 5 rP 2 rf
bP
Like M 2 , Treynor’s measure is a percentage. If you subtract the market excess return from Treynor’s measure, you will obtain the difference between the return on the T P line in Figure 24.3 and the SML, at the point where b 5 1. We might dub this difference T2 , analogous to M 2 . Be aware though that M 2 and T 2 are as different as Sharpe’s measure is from Treynor’s measure. They may well rank portfolios differently.
The Role of Alpha in Performance Measures With some algebra we can derive the relationship between the three performance measures
discussed so far. The following table shows these relationships.
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845
Treynor ( T p ) Sharpe* ( S p )
Relation to alpha E(rp) 2 rf
bp 5
ap
bp 1 TM
E(rp) 2 rf sp
5 ap
sp 1 rSM
Deviation from market performance Tp
2 5 Tp 2 TM 5 ap
bp Sp 2 SM 5
ap
sp 2 (1 2 r)SM
* r denotes the correlation coefficient between portfolio P and the market, and is less than 1.
All of these measures are consistent in that superior performance requires a positive
alpha. Hence, alpha is the most widely used performance measure. However, positive
alpha alone cannot guarantee a better Sharpe ratio for a portfolio. Taking advantage of
mispricing means departing from full diversification, which entails a cost in terms of non-
systematic risk. A mutual fund can achieve a positive alpha, yet, at the same time, increase
its SD enough that its Sharpe ratio will actually fall. 13
13
With a multifactor model, alpha must be adjusted for the additional factors. When you have K factors, k 5 1, . . . , K (the first of which, k 5 1, is the market index M ), a portfolio P ’s average realized excess return
is given by: RP 5 aP 1 a K
k51 bPkRk, where Rk is the average return on the zero-investment factor portfolio, or
the average excess rate when the direct factor growth rate is used. Hence, the generalization of Jensen’s
alpha is aP K 5 aP 2 a
K
k52 bPkRk. The generalized Treynor measure that accounts for all K factors is given by:
GTP 5 aP K
akbk MRk
akbP kRk , where b kM is the beta of factor k on the index M, and b Pk is the beta of P on factor k. [This
measure was developed by Georges Hubner (HEC School of Management, yet unpublished]). Notice that with
just one factor, the alpha reduces to the original Jensen’s alpha and GT to the single-index Treynor measure.
eXcel APPLICATIONS: Performance Measurement
T he following performance measurement spreadsheet computes all the performance measures discussed
in this section. You can see how relative ranking differs according to the criterion selected. This Excel model is avail- able at the Online Learning Center ( www.mhhe.com/bkm ).
Excel Questions
1. Examine the performance measures of the funds included in the spreadsheet. Rank performance and determine whether
the rankings are consistent using each measure. What explains these results?
2. Which fund would you choose if you were considering investing the entire risky portion of your portfolio? What if you were considering adding a small position in one of these funds to a portfolio currently invested in the market index?
2 1
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A B C D E F G H I J K
Non- Average Beta systematic Sharpe's Treynor's Jensen's M2 T2 Appraisal
Fund Return Deviation Coefficient Risk Measure Measure Measure Measure Measure Ratio Alpha 28.00% 27.00% 1.7000 5.00% 0.8148 0.1294 −0.0180 −0.0015 −0.0106 −0.3600 Omega 31.00% 26.00% 1.6200 6.00% 0.9615 0.1543 0.0232 0.0235 0.0143 0.3867 Omicron 22.00% 21.00% 0.8500 2.00% 0.7619 0.1882 0.0410 −0.0105 0.0482 2.0500 Millennium 40.00% 33.00% 2.5000 27.00% 1.0303 0.1360 −0.0100 0.0352 −0.0040 −0.0370 Big Value 15.00% 13.00% 0.9000 3.00% 0.6923 0.1000 −0.0360 −0.0223 −0.0400 −1.2000 Momentum Watcher 29.00% 24.00% 1.4000 16.00% 0.9583 0.1643 0.0340 0.0229 0.0243 0.2125 Big Potential 15.00% 11.00% 0.5500 1.50% 0.8182 0.1636 0.0130 −0.0009 0.0236 0.8667 S & P Index Return 20.00% 17.00% 1.0000 0.00% 0.8235 0.1400 0.0000 0.0000 0.0000 0.0000 T-Bill Return 6.00% 0.0000
Non- Average Beta systematic Sharpe's Treynor's Jensen's M2 T2 Appraisal
Fund Return Deviation Coefficient Risk Measure Measure Measure Measure Measure Ratio
Enter data LEGEND
Value calculated See comment
Ranking By Sharpe's Measure
Performance Measurement
Standard
Standard
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846 P A R T V I I Applied Portfolio Management
Actual Performance Measurement: An Example Now that we have examined possible criteria for performance evaluation, we need to deal
with a statistical issue: Can we assess the quality of ex ante decisions using ex post data?
Before we plunge into a discussion of this problem, let us look at the rate of return on
Jane’s portfolio over the last 12 months. Table 24.2 shows the excess return recorded each
month for Jane’s portfolio P, one of her alternative portfolios Q, and the benchmark index portfolio M. The last rows in Table 24.2 give sample average and standard deviations. From these, and regressions of P and Q on M, we obtain the necessary performance statistics.
The performance statistics in Table 24.3 show that portfolio Q is more aggressive than P, in the sense that its beta is significantly higher (1.40 vs. .70). At the same time, from its
residual standard deviation, P is better diversified (2.02% vs. 9.81%). Both portfolios out- performed the benchmark market index, as is evident from their larger Sharpe ratios (and
thus positive M 2 ), their positive alphas, and better Morningstar RAR.
Month Jane’s Portfolio P Alternative Q Benchmark M
1 3.58% 2.81% 2.20% 2 2 4.91 2 1.15 2 8.41 3 6.51 2.53 3.27 4 11.13 37.09 14.41 5 8.78 12.88 7.71 6 9.38 39.08 14.36 7 2 3.66 2 8.84 2 6.15 8 5.56 0.83 2.74 9 2 7.72 0.85 2 15.27 10 7.76 12.09 6.49 11 2 4.01 2 5.68 2 3.13 12 0.78 2 1.77 1.41 Average 2.77 7.56 1.64 Standard deviation 6.45 15.55 8.84
Table 24.2
Excess returns for port- folios P and Q and the benchmark M over 12 months
Portfolio P Portfolio Q Portfolio M
Sharpe ratio 0.43 0.49 0.19 M 2 2.16 2.66 0.00 Morningstar RAR 0.30 0.80 0.07 SCL regression statistics Alpha 1.63 5.26 0.00 Beta 0.70 1.40 1.00 Treynor 3.97 5.38 1.64 T 2 2.34 3.74 0.00 s ( e ) 2.02 9.81 0.00 Information ratio 0.81 0.54 0.00 R-SQR 0.91 0.64 1.00
Table 24.3
Performance statistics
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C H A P T E R 2 4 Portfolio Performance Evaluation 847
Which portfolio is more attractive based on reported performance? If P or Q represents the entire investment fund, Q would be preferable on the basis of its higher Sharpe measure (.49 vs. .43) and better M 2 (2.66% vs. 2.16%). For the second scenario, where P and Q are competing for a role as one of a number of subportfolios, Q also dominates because its Treynor measure is higher (5.38 vs. 3.97). However, as an active portfolio to be mixed with
the index portfolio, P is preferred because its information ratio (IR 5 a / s ( e )) is larger (.81 vs. .54), as discussed in Chapter 8 and restated in the next section. Thus, the example
illustrates that the right way to evaluate a portfolio depends in large part on how the port-
folio fits into the investor’s overall wealth.
This analysis is based on 12 months of data only, a period too short to lend statistical
significance to the conclusions. Even longer observation intervals may not be enough to
make the decision clear-cut, which represents a further problem. A model that calculates
these performance measures is available on the Online Learning Center ( www.mhhe. com/bkm ).
Performance Manipulation and the Morningstar Risk-Adjusted Rating Performance evaluation so far has been based on this assumption: Rates of return in each
period are independent and drawn from the same distribution; in statistical jargon, returns
are independent and identically distributed. This assumption can crumble in an insidi-
ous way when managers, whose compensation depends on performance, try to game the
system. They may employ strategies designed to improve measured performance even if they harm investors. Managers’ compensation may then lose its anchor to beneficial
performance.
Managers can affect performance measures over a given evaluation period because they
observe how returns unfold over the course of the period and can adjust portfolios accord-
ingly. Once they do so, rates of return in the later part of the evaluation period come to
depend on rates in the beginning of the period.
Ingersoll, Spiegel, Goetzmann, and Welch 14
show how all but one of the performance
measures covered in this chapter can be manipulated. The sole exception is the Morningstar
RAR, which is in fact a manipulation-proof performance measure (MPPM). While the
details of their model are challenging, the logic is straightforward, as we now illustrate
using the Sharpe ratio.
As we saw when analyzing capital allocation (Chapter 6), investment in the risk-free
asset (lending or borrowing) will not affect the Sharpe ratio of the portfolio. Put differently,
the Sharpe ratio is invariant to the fraction y in the risky portfolio (leverage occurs when y . 1). The reason is that excess returns are proportional to y and therefore so are both the risk premium and SD, leaving the Sharpe ratio unchanged. But what if y is changed during a period? If the decision to change leverage in mid-stream is made before any performance
is observed, then again, the Sharpe measure will not be affected because rates in the two
portions of the period will still be uncorrelated.
But imagine a manager already partway into an evaluation period. While realized excess
returns (average return, SD, and Sharpe ratio) are now known for the first part of the evalu-
ation period, the distribution of the remaining future rates is still the same as before. The
overall Sharpe ratio will be some (complicated) average of the known Sharpe ratio in the
first leg and the yet unknown ratio in the second leg of the evaluation period. Increasing
leverage during the second leg will increase the weight of that performance in the average
14
Jonathan Ingersoll, Matthew Spiegel, William Goetzmann, and Ivo Welch, “Portfolio Performance Manipula-
tion and Manipulation Proof Performance Measures,” Review of Financial Studies 20 (2007).
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848 P A R T V I I Applied Portfolio Management
because leverage will amplify returns, both good and bad. Therefore, managers will wish
to increase leverage in the latter part of the period if early returns are poor. 15
Conversely,
good first-part performance calls for deleveraging to increase the weight on the initial
period. With an extremely good first leg, a manager will shift almost the entire portfolio
to the risk-free asset. This strategy induces a (negative) correlation between returns in the
first and second legs of the evaluation period.
Investors lose, on average, from this strategy. Arbitrary variation in leverage (and there-
fore risk) is utility-reducing. It benefits managers only because it allows them to adjust
the weighting scheme of the two subperiods over the full evaluation/compensation period
after observing their initial performance. 16
Hence investors would like to prohibit or at
least eliminate the incentive to pursue this strategy. Unfortunately, only one performance
measure is impossible to manipulate.
A manipulation-proof performance measure (MPPM) must fulfill four requirements:
1. The measure should produce a single-value score to rank a portfolio.
2. The score should not depend on the dollar value of the portfolio.
3. An uninformed investor should not expect to improve the expected score by
deviating from the benchmark portfolio.
4. The measure should be consistent with standard financial market equilibrium
conditions.
Ingersoll et al. prove that the Morningstar RAR fulfills these requirements and is in fact
a manipulation-proof performance measure (MPPM). Interestingly, Morningstar was not
aiming at an MPPM when it developed the MRAR—it was simply attempting to accom-
modate investors with constant relative risk aversion.
Panel A of Figure 24.4 shows a scatter of Sharpe ratios vs. MRAR of 100 portfolios
based on statistical simulation. Thirty-six excess returns were randomly generated for each
portfolio, all with an annual expected return of 7% and SDs varying from 10% to 30%.
Thus the true Sharpe ratios of these simulated “mutual funds” are in the range of 0.7 to
0.23, with a mean of .39. Because of sampling variation, the actual 100 Sharpe ratios in the
simulation differ quite a bit from these population parameters; they range from 2 1.02 to
2.46 and average .32. The 100 MRARs range from 2 28% to 37% and average 0.7%. The
correlation between the measures was .94, suggesting that Sharpe ratios track MRAR quite
well. Indeed the scatter is pretty tight along a line with a slope of 0.19.
Panel B of Figure 24.4 (drawn on the same scale as panel A) illustrates the effect of
manipulation when one leverage change is allowed after initial performance is observed,
specifically in the middle of the 36-month evaluation period. 17
The effect of manipulation
is evident from the extreme-value portfolios. For high-positive initial MRARs, the switch
toward risk-free investments preserves the first-half high Sharpe ratios that might other-
wise be diluted or possibly even reversed in the second half. For the high-negative initial
MRARs, when leverage ratios are increased, we see two effects. First, MRARs look worse
because of cases where the high leverage backfired and worsened the MRARs compared
to panel A (points move to the left). In contrast, Sharpe ratios look better than in panel A
15
Managers who are precluded from increasing leverage will instead shift to high-beta stocks. If this is a wide-
spread phenomenon, it could help explain why high-beta stocks appear, on average, to be overpriced relative to
low-beta ones.
16
One way to reduce the potency of manipulation is to evaluate performance more frequently. This will reduce the
statistical precision of the measure, however.
17
To keep the exercise realistic, leverage ratios were capped at 2 (a debt-to-equity ratio of 1.0).
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C H A P T E R 2 4 Portfolio Performance Evaluation 849
Figure 24.4 MRAR scores and Sharpe ratios with and without manipulation
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
−0.5 −0.4 −0.3 0 0.1 0.2 0.3−0.2 −0.1
A: No manipulation: Sharpe vs. MRAR
MRAR
Sh a rp
e R
a ti
o
Average Sharpe Ratio .32 SD .66 Average MRAR .077% SD .13 Slope of Sharpe on MRAR .19 R-Square .90 Correlation between Halves .14
B: Manipulation: Sharpe vs. MRAR
–1.5
–1
–0.5
0.5
1
1.5
2
2.5
3
–0.5 –0.4 –0.3 0.1 0.3–0.2 –0.1 0 0.2 0.4
MRAR
Sh a rp
e R
a ti
o
Average Sharpe Ratio .40 SD .66 Average MRAR –2 .74% SD .18 Slope of Sharpe on MRAR .25 R-Square .84 Correlation between Halves –.06
0
(they move upward). Some Sharpe ratios move from negative to positive values, while oth-
ers do not look worse (because the increased SD in the second period reduced the absolute
value of the negative Sharpe ratios).
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850 P A R T V I I Applied Portfolio Management
The statistics in the box of panel B quantify the improvement of measured Sharpe
ratios; in contrast, MRARs clearly deteriorated from a slight positive value to a certainty-
equivalent of 2 2.74% per year! As predicted, the correlation between average returns in
the first and second legs of the period changes from positive to negative. All this happened
because of an average increase in leverage from 1.0 to 1.39. 18
Morningstar introduced the MRAR in 2002. It is particularly relevant to hedge funds,
where managers have great latitude and incentive to manipulate. See Chapter 26 for further
discussion. Given its immunity to manipulation, we would expect the MRAR measure to
become a standard performance statistic sometime in the future, required especially of
managers who have the most discretion over investment policy.
Realized Returns versus Expected Returns When evaluating a portfolio, the evaluator knows neither the portfolio manager’s original
expectations nor whether those expectations made sense. One can only observe perfor-
mance after the fact and hope that random results are neither taken for, nor hide, true under-
lying ability. But risky asset returns are “noisy,” which complicates the inference problem.
To avoid making mistakes, we have to determine the “significance level” of a performance
measure to know whether it reliably indicates ability.
Consider Joe Dart, a portfolio manager. Suppose that his portfolio has an alpha of 20
basis points per month, which makes for a hefty 2.4% per year before compounding. Let us
assume that the return distribution of Joe’s portfolio has constant mean, beta, and alpha, a
heroic assumption, but one that is in line with the usual treatment of performance measure-
ment. Suppose that for the measurement period Joe’s portfolio beta is 1.2 and the monthly
standard deviation of the residual (nonsystematic risk) is .02 (2%). With a market index stan-
dard deviation of 6.5% per month (22.5% per year), Joe’s portfolio systematic variance is
b2sM 2 5 1.22 3 6.52 5 60.84
and hence the correlation coefficient between his portfolio and the market index is
r 5 B b2sM2 b2sM
2 1 s2(e) R
1/2
5 c 60.84 60.84 1 4
d 1/2 5 .97 which shows that his portfolio is quite well diversified.
To estimate Joe’s portfolio alpha from the security characteristic line (SCL), we regress
the portfolio excess returns on the market index. Suppose that we are in luck and the regres-
sion estimates yield precisely the true parameters. That means that our SCL estimates for
the N months are
â 5 .2%, b̂ 5 1.2, ŝ(e) 5 2%
The evaluator who runs such a regression, however, does not know the true values, and
hence must compute the t -statistic of the alpha estimate to determine whether to reject the hypothesis that Joe’s alpha is zero, that is, that he has no superior ability.
The standard error of the alpha estimate in the SCL regression is approximately
ŝ(a) 5 ŝ(e)
"N where N is the number of observations and ŝ(e) is the sample estimate of nonsystematic risk. The t -statistic for the alpha estimate is then
t(â) 5 â
ŝ(a) 5
â"N ŝ(e)
(24.2)
18
Out of 100 funds, the leverage ratio was decreased in 38 portfolios, was increased to less than 2 in 14 portfolios,
and was increased to 2 (and would have been increased even more absent the cap) in 48 portfolios.
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C H A P T E R 2 4 Portfolio Performance Evaluation 851
Suppose an analyst has a measured alpha of .2% with a standard error of 2%, as in our example. What is the probability that the positive alpha is due to luck of the draw and that true ability is zero?
CONCEPT CHECK 24.3
Suppose that we require a significance level of 5%. This requires a t(â) value of 1.96 if N is large. With â 5 .2 and ŝ(e) 5 2, we solve Equation 24.2 for N and find that
1.96 5 .2"N
2
N 5 384 months
or 32 years!
What have we shown? Here is an analyst who has very substantial ability. The example
is biased in his favor in the sense that we have assumed away statistical complications.
Nothing changes in the parameters over a long period of time. Furthermore, the sample
period “behaves” perfectly. Regression estimates are all perfect. Still, it will take Joe’s
entire working career to get to the point where statistics will confirm his true ability. We
have to conclude that the problem of statistical inference makes performance evaluation
extremely difficult in practice.
Now add to the imprecision of performance estimates the fact that the average tenure of
a fund manager is only about 4.5 years. By the time you are lucky enough to find a fund
whose historic superior performance you are confident of, its manager is likely to be about
to move, or has already moved elsewhere. The nearby box explores this topic further.
24.2 Performance Measurement for Hedge Funds
In describing Jane’s portfolio performance evaluation we left out one scenario that may
well be relevant.
Suppose Jane has been satisfied with her well-diversified mutual fund, but now she
stumbles upon information on hedge funds. Hedge funds are rarely designed as candidates
for an investor’s overall portfolio. Rather than focusing on Sharpe ratios, which would
entail establishing an attractive trade-off between expected return and overall volatility,
these funds tend to concentrate on opportunities offered by temporarily mispriced securi-
ties, and show far less concern for broad diversification. In other words, these funds are
alpha driven, and best thought of as possible additions to core positions in more traditional portfolios established with concerns of diversification in mind.
In Chapter 8, we considered precisely the question of how best to mix an actively man-
aged portfolio with a broadly diversified core position. We saw that the key statistic for
this mixture is the information ratio of the actively managed portfolio; this ratio, therefore,
becomes the active fund’s appropriate performance measure.
To briefly review, call the active portfolio established by the hedge fund H, and the investor’s baseline passive portfolio M. Then the optimal position of H in the overall port- folio, denoted P *, would be
wH 5 wH
0
1 1 (1 2 bH)wH 0
; wH 0 5
aH
s2(eH) E(RM)
sM 2
(24.3)
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852
As we saw in Chapter 8, when the hedge fund is optimally combined with the baseline
portfolio using Equation 24.3, the improvement in the Sharpe measure will be determined
by its information ratio a H / s ( e H ), according to
SP* 2 5 SM
2 1 B aH s(eH)
R 2
(24.4)
Equation 24.4 tells us that the appropriate performance measure for the hedge fund is its
information ratio (IR).
Looking back at Table 24.3 , we can calculate the IR of portfolios P and Q as
IRP 5 aP
s(eP) 5
1.63
2.02 5 .81; IRQ 5
5.38
9.81 5 .54 (24.5)
If we were to interpret P and Q as hedge funds, the low beta of P, .70, could result from short positions the fund holds in some assets. The relatively high beta of Q, 1.40, might result from leverage that would also increase the firm-specific risk of the fund,
s ( e Q ). Using these calculations, Jane would favor hedge fund P with the higher infor- mation ratio.
Should You Follow Your Fund Manager?
The whole idea of investing in a mutual fund is to leave the stock and bond picking to the professionals. But fre- quently, events don’t turn out quite as expected—the manager resigns, gets transferred or dies. A big part of the investor’s decision to buy a managed fund is based on the manager’s record, so changes like these can come as an unsettling surprise.
There are no rules about what happens in the wake of a manager’s departure. It turns out, however, that there is strong evidence to suggest that the managers’ real con- tribution to fund performance is highly overrated. For example, research company Morningstar compared funds that experienced management changes between 1990 and 1995 with those that kept the same managers. In the five years ending in June 2000, the top-performing funds of the previous five years tended to keep beating their peers— despite losing any fund managers. Those funds that per- formed badly in the first half of the 1990s continued to do badly, regardless of management changes. While mutual fund management companies will undoubtedly continue to create star managers and tout their past records, inves- tors should stay focused on fund performance.
Funds are promoted on their managers’ track records, which normally span a three- to five-year period. But per- formance data that goes back only a few years is hardly a valid measure of talent. To be statistically sound, evidence of a manager’s track record needs to span, at a minimum, 10 years or more.
The mutual fund industry may look like a merry-go- round of managers, but that shouldn’t worry most inves- tors. Many mutual funds are designed to go through little or no change when a manager leaves. That is because,
according to a strategy designed to reduce volatility and succession worries, mutual funds are managed by teams of stock pickers, who each run a portion of the assets, rather than by a solo manager with co-captains. Meanwhile, even so-called star managers are nearly always surrounded by researchers and analysts, who can play as much of a role in performance as the manager who gets the headlines.
Don’t forget that if a manager does leave, the invest- ment is still there. The holdings in the fund haven’t changed. It is not the same as a chief executive leaving a company whose share price subsequently falls. The best thing to do is to monitor the fund more closely to be on top of any changes that hurt its fundamental investment qualities.
In addition, don’t underestimate the breadth and depth of a fund company’s “managerial bench.” The larger, established investment companies generally have a large pool of talent to draw on. They are also well aware that investors are prone to depart from a fund when a manage- rial change occurs.
Lastly, for investors who worry about management changes, there is a solution: index funds. These mutual funds buy stocks and bonds that track a benchmark index like the S&P 500 rather than relying on star managers to actively pick securities. In this case, it doesn’t really matter if the manager leaves. At the same time, index investors don’t have to pay tax bills that come from switching out of funds when managers leave. Most importantly, index fund investors are not charged the steep fees that are needed to pay star management salaries.
Source: Shauna Carther, “Should You Follow Your Fund Manager?” Investopedia.com , March 3, 2010. Provided by Forbes.
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853
In practice, evaluating hedge funds poses considerable practical challenges. We will
discuss many of these in Chapter 26, which is devoted to these funds. But for now we can
briefly mention a few of the difficulties:
1. The risk profile of hedge funds (both total volatility and exposure to relevant
systematic factors) may change rapidly. Hedge funds have far greater leeway than
mutual funds to change investment strategy opportunistically. This instability makes
it hard to measure exposure at any given time.
2. Hedge funds tend to invest in illiquid assets. We therefore must disentangle liquidity
premiums from true alpha to properly assess their performance. Moreover, it can be
difficult to accurately price inactively traded assets, and correspondingly difficult to
measure rates of return.
3. Many hedge funds pursue strategies that may provide apparent profits over long
periods of time, but expose the fund to infrequent but severe losses. Therefore, very
long time periods may be required to formulate a realistic picture of their true risk–
return trade-off.
4. Hedge funds have ample latitude to change their risk profiles and therefore consid-
erable ability to manipulate conventional performance measures. Only the MRAR
is manipulation-proof, and investors should urge these funds to use them.
5. When hedge funds are evaluated as a group, survivorship bias can be a major
consideration, because turnover in this industry is far higher than for investment
companies such as mutual funds.
The nearby box discusses some of the misuses of conventional performance measures
in evaluating hedge funds.
Sharpe Point: Risk Gauge is Misused
William F. Sharpe was probably the biggest expert in the room when economists from around the world gathered to hash out a pressing problem: How to gauge hedge-fund risk. About 40 years ago, Dr. Sharpe created a simple cal- culation for measuring the return that investors should expect for the level of volatility they are accepting. In other words: How much money do they stand to make compared with the size of the up-and-down swings they will lose sleep over?
The so-called Sharpe ratio became a cornerstone of modern finance, as investors used it to help select money managers and mutual funds. But the use of the ratio has been criticized by many prominent academics—including Dr. Sharpe himself.
The ratio is commonly used—“misused,” Dr. Sharpe says—for promotional purposes by hedge funds. Hedge funds, loosely regulated private investment pools, often use complex strategies that are vulnerable to surprise events and elude any simple formula for measuring risk. “Past average experience may be a terrible predictor of future performance,” Dr. Sharpe says.
Dr. Sharpe designed the ratio to evaluate portfolios of stocks, bonds, and mutual funds. The higher the Sharpe
ratio, the better a fund is expected to perform over the long term. However, at a time when smaller investors and pension funds are pouring money into hedge funds, the ratio can foster a false sense of security.
Dr. Sharpe says the ratio doesn’t foreshadow hedge- fund woes because “no number can.” The formula can’t predict such troubles as the inability to sell off investments quickly if they start to head south, nor can it account for extreme unexpected events. Long-Term Capital Manage- ment, a huge hedge fund in Connecticut, had a glowing Sharpe ratio before it abruptly collapsed in 1998 when Russia devalued its currency and defaulted on debt. Plus, hedge funds are generally secretive about their strategies, making it difficult for investors to get an accurate picture of risk.
Another problem with the Sharpe ratio is that it is designed to evaluate the risk-reward profile of an inves- tor’s entire portfolio, not small pieces of it. This shortcom- ing is particularly telling for hedge funds.
Source: Ianthe Jeanne Dugan, “Sharpe Point: Risk Gauge is Misused,” The Wall Street Journal, August 31, 2005, p. C1. © 2005 Dow Jones & Company, Inc. All rights reserved worldwide.
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854 P A R T V I I Applied Portfolio Management
24.3 Performance Measurement with Changing Portfolio Composition
We have seen already that the volatility of stock returns requires a very long observation
period to determine performance levels with any precision, even if portfolio returns are
distributed with constant mean and variance. Imagine how this problem is compounded
when portfolio return distributions are constantly changing.
It is acceptable to assume that the return distributions of passive strategies have constant
mean and variance when the measurement interval is not too long. However, under an
active strategy return distributions change by design, as the portfolio manager updates the
portfolio in accordance with the dictates of financial analysis. In such a case, estimating
various statistics from a sample period assuming a constant mean and variance may lead to
substantial errors. Let us look at an example.
Suppose that the Sharpe measure of the market index is .4. Over an initial period of 52 weeks, the portfolio manager executes a low-risk strategy with an annualized mean excess return of 1% and standard deviation of 2%. This makes for a Sharpe measure of .5, which beats the passive strategy. Over the next 52-week period this manager finds that a high -risk strategy is optimal, with an annual mean excess return of 9% and stan- dard deviation of 18%. Here, again, the Sharpe measure is .5. Over the 2-year period our manager maintains a better-than-passive Sharpe measure.
Figure 24.5 shows a pattern of (annualized) quarterly returns that are consistent with our description of the manager’s strategy of 2 years. In the first four quarters the excess returns are 2 1%, 3%, 2 1%, and 3%, making for an average of 1% and standard deviation of 2%. In the next four quarters the returns are 2 9%, 27%, 2 9%, 27%, making for an average of 9% and standard deviation of 18%. Thus both years exhibit a Sharpe measure of .5. However, over the eight-quarter sequence the mean and standard deviation are 5% and 13.42%, respectively, making for a Sharpe measure of only .37, apparently inferior to the passive strategy!
Example 24.3 Changing Portfolio Risk
What happened in Example 24.3? The
shift of the mean from the first four quar-
ters to the next was not recognized as a
shift in strategy. Instead, the difference
in mean returns in the 2 years added to
the appearance of volatility in portfolio returns. The active strategy with shift-
ing means appears riskier than it really
is and biases the estimate of the Sharpe
measure downward. We conclude that for
actively managed portfolios it is helpful
to keep track of portfolio composition and
changes in portfolio mean and risk. We
will see another example of this problem
in the next section, which deals with mar-
ket timing.
Rate of Return (%)
Quarter
27
3 −1
−9
Figure 24.5 Portfolio returns. Returns in last four quarters are more variable than in the first four.
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C H A P T E R 2 4 Portfolio Performance Evaluation 855
24.4 Market Timing
In its pure form, market timing involves shifting funds between a market-index portfolio
and a safe asset, depending on whether the market index is expected to outperform the safe
asset. In practice, most managers do not shift fully between T-bills and the market. How
can we account for partial shifts into the market when it is expected to perform well?
To simplify, suppose that an investor holds only the market-index portfolio and T-bills.
If the weight of the market were constant, say, .6, then portfolio beta would also be con-
stant, and the SCL would plot as a straight line with slope .6, as in Figure 24.6, panel A .
If, however, the investor could correctly time the market and shift funds into it in periods
when the market does well, the SCL would plot as in Figure 24.6, panel B . If bull and bear
markets can be predicted, the investor will shift more into the market when the market
is about to go up. The portfolio beta and the slope of the SCL will be higher when r M is higher, resulting in the curved line that appears in Figure 24.6, panel B .
Treynor and Mazuy were the first to propose estimating such a line by adding a squared
term to the usual linear index model: 19
rP 2 rf 5 a 1 b(rM 2 rf) 1 c(rM 2 rf) 2 1 eP
where r P is the portfolio return, and a, b, and c are estimated by regression analysis. If c turns out to be positive, we have evidence of timing ability, because this last term will make the characteristic line steeper as r M 2 r f is larger. Treynor and Mazuy estimated this equation for a number of mutual funds, but found little evidence of timing ability.
A similar but simpler methodology was proposed by Henriksson and Merton. 20
These authors suggested that the beta of the portfolio take only two values: a large value
if the market is expected to do well and a small value otherwise. Under this scheme
the portfolio characteristic line appears as Figure 24.6, panel C . Such a line appears in
regression form as
rP 2 rf 5 a 1 b(rM 2 rf) 1 c(rM 2 rf)D 1 eP
where D is a dummy variable that equals 1 for r M . r f and zero otherwise. Hence the beta of the portfolio is b in bear markets and b 1 c in bull markets. Again, a positive value of c implies market timing ability.
Henriksson 21
estimated this equation for 116 mutual funds. He found that the average
value of c for the funds was negative, and equal to 2.07. In sum, the results showed little evidence of market timing ability. Perhaps this should be expected; given the tremendous
values to be reaped by a successful market timer, it would be surprising in nearly efficient
markets to uncover clear-cut evidence of such skills.
To illustrate a test for market timing, return to Table 24.2 . Regressing the excess returns
of portfolios P and Q on the excess returns of M and the square of these returns,
rP 2 rf 5 aP 1 bP (rM 2 rf) 1 cP (rM 2 rf) 2 1 eP
rQ 2 rf 5 aQ 1 bQ (rM 2 rf) 1 cQ (rM 2 rf) 2 1 eQ
19
Jack L. Treynor and Kay Mazuy, “Can Mutual Funds Outguess the Market?” Harvard Business Review 43 (July–August 1966).
20
Roy D. Henriksson and R. C. Merton, “On Market Timing and Investment Performance. II. Statistical Proce-
dures for Evaluating Forecast Skills,” Journal of Business 54 (October 1981).
21
Roy D. Henriksson, “Market Timing and Mutual Fund Performance: An Empirical Investigation,” Journal of Business 57 (January 1984).
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856 P A R T V I I Applied Portfolio Management
we derive the following statistics:
Figure 24.6 Characteristic lines. Panel A: No market timing, beta is constant. Panel B: Market timing, beta increases with expected market excess return. Panel C: Market timing with only two values of beta.
Slope = b rM − rf
Slope = b + c
rP − rfSteadily Increasing Slope
rP − rf
rM − rf
rM − rf
Slope = .6rP − rf
A
B C
The numbers in parentheses are the regression estimates from the single variable regres-
sion reported in Table 24.3 . The results reveal that portfolio P shows no timing. It is not clear whether this is a result of Jane’s making no attempt at timing or that the effort to time
the market was in vain and served only to increase portfolio variance unnecessarily.
The results for portfolio Q, however, reveal that timing has, in all likelihood, success- fully been attempted. The timing coefficient, c, is estimated at .10. The evidence thus sug- gests successful timing (positive c ) offset by unsuccessful stock selection (negative a ). Note that the alpha estimate, a, is now 22.29% as opposed to the 5.28% estimate derived from the regression equation that did not allow for the possibility of timing activity.
This example illustrates the inadequacy of conventional performance evaluation tech-
niques that assume constant mean returns and constant risk. The market timer constantly
shifts beta and mean return, moving into and out of the market. Whereas the expanded
regression captures this phenomenon, the simple SCL does not. The relative desirability
Portfolio
Estimate P Q
Alpha (a) 1.77 (1.63) –2.29 (5.28) Beta (b) 0.70 (0.69) 1.10 (1.40) Timing (c) 0.00 0.10 R-SQR 0.91 (0.91) 0.98 (0.64)
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C H A P T E R 2 4 Portfolio Performance Evaluation 857
of portfolios P and Q remains unclear in the sense that the value of the timing success and selectivity failure of Q compared with P has yet to be evaluated. The important point for performance evaluation, however, is that expanded regressions can capture many of the
effects of portfolio composition change that would confound the more conventional mean-
variance measures.
The Potential Value of Market Timing Suppose we define perfect market timing as the ability to tell (with certainty) at the begin-
ning of each year whether the S&P 500 portfolio will outperform the strategy of rolling
over 1-month T-bills throughout the year. Accordingly, at the beginning of each year, the
market timer shifts all funds into either cash equivalents (T-bills) or equities (the all U.S.
stock portfolio), whichever is predicted to do better. Beginning with $1 on January 1, 1927,
how would the perfect timer end an 86-year experiment on December 31, 2012, in com-
parison with investors who kept their funds in either equity or T-bills for the entire period?
Table 24.4 , columns 1–3, presents summary statistics for each of the three passive strat-
egies, computed from the historical annual returns of bills and equities. From the returns
on stocks and bills, we calculate wealth indexes of the all-bills and all-equity investments
and show terminal values for these investors at the end of 2012. The return for the perfect
timer in each year is the maximum of the return on stocks and the return on bills. The first row in Table 24.4 tells all. The terminal value of investing $1 in bills over the
86 years (1927–2012) is $20, while the terminal value of the same initial investment in
equities is about $2,652. We saw a similar pattern for a 25-year investment in Chapter 5;
the much larger terminal values (and difference between them) when extending the hori-
zon from 25 to 86 years is just another manifestation of the power of compounding. We
argued in Chapter 5 that as impressive as the difference in terminal values is, it is best
interpreted as no more than fair compensation for the risk borne by equity investors.
Notice that the standard deviation of the all-equity investor was a hefty 20.39%. This is
also why the geometric average of stocks for the period is “only” 9.60%, compared with
the arithmetic average of 11.63%. (The difference between the two averages increases
with volatility.)
Strategy Bills Equities Perfect Timer Imperfect Timer *
Terminal Value 20 2,652 352,796 8,859 Arithmetic Average 3.59 11.63 16.75 11.98 Standard Deviation 3.12 20.39 13.49 14.36 Geometric Average 3.54 9.60 16.01 11.09 LPSD (relative to bills) 0 21.18 0 17.15 Minimum † 2 0.04 2 44.00 2 0.02 2 27.09 Maximum 14.72 57.42 57.42 57.42 Skew 0.99 2 0.42 0.72 0.71 Kurtosis 0.98 0.02 2 0.13 1.50
Table 24.4
Performance of bills, equities, and (annual) timers—perfect and imperfect
*The imperfect timer has P 1 5 .7, and P 2 5 .7. Therefore, P 1 1 P 2 2 1 5 .4. †A negative rate on “bills” was observed in 1940. The Treasury security used in the data series in these early years was actually not a T-bill but a T-bond with 30 days to maturity.
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858 P A R T V I I Applied Portfolio Management
Now observe that the terminal value of the perfect timer is about $353,000, a 133-fold
increase over the already large terminal value of the all-equity strategy! In fact, this result
is even better than it looks, because the return to the market timer is truly risk-free. This
is the classic case where a large standard deviation (13.49%) has nothing to do with risk.
Because the timer never delivers a return below the risk-free rate, the standard deviation is
a measure of good surprises only. The positive skew of the distribution (compared with the negative skew of equities) is a manifestation of the fact that the extreme values are all
positive. Another indication of this stellar performance is the minimum and maximum
returns—the minimum return equals the minimum return on bills (in 1940) and the maxi-
mum return is that of equities (in 1933)—so that all negative returns on equities (as low as
244% in 1931) were avoided by the timer. Finally, the best indication of the performance
of the timer is a lower partial standard deviation, LPSD. 22
The LPSD of the all-equity port-
folio is only slightly greater than the conventional standard deviation, but it is necessarily
zero for the perfect timer.
If we interpret the terminal value of the all-equity portfolio in excess of the value of the
T-bill portfolio entirely as a risk premium commensurate with investment risk, we must
conclude that the risk-adjusted equivalent value of the all-equity terminal value is the same
as that of the T-bill portfolio, $20. 23
In contrast, the perfect timer’s portfolio has no risk,
and so receives no discount for risk. Hence, it is fair to say that the forecasting ability of the
perfect timer converts a $20 final value to a value of $352,796.
Valuing Market Timing as a Call Option The key to valuing market timing ability is to recognize that perfect foresight is equivalent
to holding a call option on the equity portfolio. The perfect timer invests 100% in either
the safe asset or the equity portfolio, whichever will provide the higher return. The rate of
return is at least the risk-free rate. This is shown in Figure 24.7 . To see the value of information as an option, suppose that the market index currently
is at S 0 and that a call option on the index has an exercise price of X 5 S 0 (1 1 r f ). If the market outperforms bills over the coming period, S T will exceed X, whereas it will be less than X otherwise. Now look at the payoff to a portfolio consisting of this option and S 0 dollars invested in bills:
22
The conventional LPSD is based on the average squared deviation below the mean. Because the threshold
performance in this application is the risk-free rate, we modify the LPSD for this discussion by taking squared
deviations from that rate and the observations are conditional on being below that threshold. It ignores the num-
ber of such events.
23
It may seem hard to attribute such a big difference in final outcome solely to risk aversion. But think of it
this way: the final value of the equity position is 133 times that of the bills position ($2,652 versus $20). Over
86 years, this implies a reasonable annualized risk premium of 5.85%: 133 1/86
5 1.0585.
24
The analogy between market timing and call options, and the valuation formulas that follow from it, were devel-
oped in Robert C. Merton, “On Market Timing and Investment Performance: An Equilibrium Theory of Value for
Market Forecasts,” Journal of Business, July 1981.
S T , X S T $ X
Bills S 0 (1 1 r f ) S 0 (1 1 r f ) Call 0 S T 2 X Total S 0 (1 1 r f ) S T
The portfolio pays the risk-free return when the market is bearish (i.e., the market return
is less than the risk-free rate), and it pays the market return when the market is bullish and
beats bills. Such a portfolio is a perfect market timer. 24
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C H A P T E R 2 4 Portfolio Performance Evaluation 859
Because the ability to predict the better- performing investment is equivalent to holding
a call option on the market, we can use option-pricing models to assign a dollar value to
perfect timing ability. This value would constitute the fair fee that a perfect timer could
charge investors for its services. Placing a value on perfect timing also enables us to assign
value to less-than-perfect timers.
The exercise price of the perfect-timer call option on $1 of the equity portfolio is the
final value of the T-bill investment. Using continuous compounding, this is $1 3 e rT . When you use this exercise price in the Black-Scholes formula for the value of the call option, the
formula simplifies considerably to 25
MV(Perfect timer per $ of assets) 5 C 5 2N (½ sM"T) 2 1 (24.6) We have so far assumed annual forecasts, that is, T 5 1 year. Using T 5 1, and the stan-
dard deviation of stocks from Table 24.4 , 20.39%, we compute the value of this call option
as 8.12 cents, or 8.12% of the value of the equity portfolio. This is less than the historical-
average return of perfect timing shown in Table 24.5 , reflecting the fact that actual timing
value is sensitive to fat tails in the distribution of returns, whereas Black-Scholes presumes
a log-normal distribution.
Equation 24.6 tells us that perfect market timing would be equivalent to enhancing the
annual equity return by .0812 (or 8.12% per year). Since the average equity return over the
last 86 years has been 11.63%, this would be similar in value to enjoying an annual return
of 1.1162 3 1.0812 2 1 5 .2069, or 20.69%.
If a timer could make the correct choice every month instead of every year, the value of
the forecasts would dramatically increase. Of course, making perfect forecasts more fre-
quently requires even better powers of prediction. As the frequency of such perfect predic-
tions increases without bound, the value of the
services will increase without bound as well.
Suppose the perfect timer could make perfect
forecasts every month. In this case, each forecast
would be for a shorter interval, and the value
of each individual forecast would be lower, but
there would be 12 times as many forecasts, each
of which could be valued as another call option.
The net result is a big increase in total value.
With monthly predictions, the value of the call
will be 2N(½ 3 .2039 3 "1/12) 2 1 5 .0235. Using a monthly T-bill rate of 3.6%/12, the pres-
ent value of a 1-year string of such monthly
calls, each worth $.0235, is $.28. Thus, the
annual value of the monthly perfect timer is
28 cents on the dollar, compared to 8.12 cents
for an annual timer. For an investment period
of 86 years, the forecast future value of a $1
investment would be a far greater [(1 1 .28)
(1 1 .1163)] 86
5 $2.1 3 10 13
. This value sug-
gests the otherworldly power of these forecasts.
Figure 24.7 Rate of return of a perfect market timer as a function of the rate of return on the market index.
rf
rf rM
P e rf
e ct
T im
e r’
s R
e tu
rn
25 Substitute S 0 5 $1 for the current value of the equity portfolio and X 5 $1 3 e
rT in Equation 21.1 of Chapter 21,
and you will obtain Equation 24.6.
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860 P A R T V I I Applied Portfolio Management
The Value of Imperfect Forecasting A weather forecaster in Tucson, Arizona, who always predicts no rain may be right 90% of the time. But a high success rate for a “stopped-clock” strategy is not evidence of fore-
casting ability. Similarly, the appropriate measure of market forecasting ability is not the
overall proportion of correct forecasts. If the market is up 2 days out of 3 and a forecaster
always predicts market advance, the two-thirds success rate is not a measure of forecasting
ability. We need to examine the proportion of bull markets ( r M , r f ) correctly forecast and the proportion of bear markets ( r M . r f ) correctly forecast.
If we call P 1 the proportion of the correct forecasts of bull markets and P 2 the propor- tion for bear markets, then P 1 1 P 2 2 1 is the correct measure of timing ability. For exam- ple, a forecaster who always guesses correctly will have P 1 5 P 2 5 1, and will show ability of P 1 1 P 2 2 1 5 1 (100%). An analyst who always bets on a bear market will mispredict all bull markets ( P 1 5 0), will correctly “predict” all bear markets ( P 2 5 1), and will end up with timing ability of P 1 1 P 2 2 1 5 0.
What is the market timing score of someone who flips a fair coin to predict the market?
CONCEPT CHECK 24.4
When timing is imperfect, Merton shows that if we measure overall accuracy by the
statistic P 1 1 P 2 2 1, the market value of the services of an imperfect timer is simply
MV(Imperfect timer) 5 (P1 1 P2 2 1) 3 C 5 (P1 1 P2 2 1) 32N(½ sM"T) 2 14 (24.7) The last column in Table 24.4 provides an assessment of the imperfect market-timer. To
simulate the performance of an imperfect timer, we drew random numbers to capture the
possibility that the timer will sometimes issue an incorrect forecast (we assumed here both
P 1 and P 2 5 .7) and compiled results for the 86 years of history. 26
The statistics of this
exercise resulted in an average terminal value for the imperfect timer of “only” $8,859,
compared with the perfect timer’s $352,796, but still considerably superior to the $2,562
for the all-equity investments. 27
A further variation on the valuation of market timing is a case in which the timer does
not shift fully from one asset to the other. In particular, if the timer knows her forecasts are
imperfect, one would not expect her to shift fully between markets. She presumably would
moderate her positions. Suppose that she shifts a fraction v of the portfolio between T-bills
and equities. In that case, Equation 24.7 can be generalized as follows:
MV(Imperfect timer) 5 v(P1 1 P2 2 1)32N(sM"T) 2 14 If the shift is v 5 .50 (50% of the portfolio), the timer’s value will be one-half of the value
we would obtain for full shifting, for which v 5 1.0.
26
In each year, we started with the correct forecast, but then used a random number generator to occasionally
change the timer’s forecast to an incorrect prediction. We set the probability that the timer’s forecast would be
correct equal to .70 for both up and down markets.
27
Notice that Equation 24.7 implies that an investor with a value of P 5 0 who attempts to time the market would add zero value. The shifts across markets would be no better than a random decision concerning asset
allocation.
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C H A P T E R 2 4 Portfolio Performance Evaluation 861
24.5 Style Analysis
Style analysis was introduced by Nobel laureate William Sharpe. 28 The popularity of the concept was aided by a well-known study
29 concluding that 91.5% of the variation in returns
of 82 mutual funds could be explained by the funds’ asset allocation to bills, bonds, and
stocks. Later studies that considered asset allocation across a broader range of asset classes
found that as much as 97% of fund returns can be explained by asset allocation alone.
Sharpe’s idea was to regress fund returns on indexes representing a range of asset
classes. The regression coefficient on each index would then measure the fund’s implicit
allocation to that “style.” Because funds are barred from short positions, the regression
coefficients are constrained to be either zero or positive and to sum to 100%, so as to rep-
resent a complete asset allocation. The R -square of the regression would then measure the percentage of return variability attributable to style or asset allocation, while the remainder
of return variability would be attributable either to security selection or to market timing
by periodic changes in the asset-class weights.
To illustrate Sharpe’s approach, we use monthly returns on Fidelity Magellan’s Fund
during the famous manager Peter Lynch’s tenure between October 1986 and September
1991, with results shown in Table 24.5 . While seven asset classes are included in this
analysis (of which six are represented by stock indexes and one is the T-bill alternative),
the regression coefficients are positive for only three, namely, large capitalization stocks,
medium cap stocks, and high P/E (growth) stocks. These portfolios alone explain 97.5%
of the variance of Magellan’s returns. In other words, a tracking portfolio made up of the
three style portfolios, with weights as given in Table 24.5 , would explain the vast majority
of Magellan’s variation in monthly performance. We conclude that the fund returns are
well represented by three style portfolios.
The proportion of return variability not explained by asset allocation can be attrib- uted to security selection within asset classes, as well as timing that shows up as periodic
changes in allocation. For Magellan, residual variability was 100 2 97.5 5 2.5%. This sort
Style Portfolio Regression Coefficient
T-Bill 0 Small Cap 0 Medium Cap 35 Large Cap 61 High P/E (growth) 5 Medium P/E 0 Low P/E (value) 0 Total 100 R -square 97.5
Source: Authors’ calculations. Return data for Magellan obtained from finance.yahoo.com/funds and return data for style portfolios obtained from the Web page of Professor Kenneth French: mba. tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html .
Table 24.5
Style analysis for Fidelity’s Magellan Fund
28
William F. Sharpe, “Asset Allocation: Management Style and Performance Evaluation,” Journal of Portfolio Management, Winter 1992, pp. 7–19.
29
Gary Brinson, Brian Singer, and Gilbert Beebower, “Determinants of Portfolio Performance,” Financial Analysts Journal, May/June 1991.
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862 P A R T V I I Applied Portfolio Management
of result is commonly used to play down the importance of security selection and timing in
fund performance, but such a conclusion misses the important role of the intercept in this
regression. (The R -square of the regression can be 100%, and yet the intercept can be non- zero due to a superior risk-adjusted abnormal return.) For Magellan, the intercept was 32
basis points per month, resulting in a cumulative abnormal return over the 5-year period of
19.19%. The superior performance of Magellan is displayed in Figure 24.8 , which plots the
cumulative impact of the intercept plus monthly residuals relative to the tracking portfolio
composed of the style portfolios. Except for the period surrounding the crash of October
1987, Magellan’s return consistently increased relative to the benchmark portfolio.
Style analysis provides an alternative to performance evaluation based on the security
market line (SML) of the CAPM. The SML uses only one comparison portfolio, the broad
market index, whereas style analysis more freely constructs a tracking portfolio from a
number of specialized indexes. To compare the two approaches, the security characteristic
line (SCL) of Magellan was estimated by regressing its excess return on the excess return
of a market index composed of all NYSE, Amex, and NASDAQ stocks. The beta estimate
of Magellan was 1.11 and the R -square of the regression was .99. The alpha value (inter- cept) of this regression was “only” 25 basis points per month, reflected in a cumulative
abnormal return of 15.19% for the period.
How can we explain the higher R -square of the regression with only one factor (the market index) relative to the style regression, which deploys six stock indexes? The answer
is that style analysis imposes extra constraints on the regression coefficients: It forces them
to be positive and to sum to 1.0. This “neat” representation may not be consistent with
actual portfolio weights that are constantly changing over time. So which representation
better gauges Magellan’s performance over the period? There is no clear-cut answer. The
SML benchmark is a better representation of performance relative to the theoretically pre-
scribed passive portfolio, that is, the broadest market index available. On the other hand,
Figure 24.8 Fidelity Magellan Fund cumulative return difference: Fund versus style benchmark and fund versus SML benchmark
Source: Authors’ calculations.
Oct-86 May-87 Dec-87 Jun-88 Jan-89 Jul-89 Feb-90 Aug-90 Mar-91 Oct-91 −3
2
7
12
17
Cumulative Differential Performance (%)
Cumulative Residuals from Style Analysis Cumulative Residuals from SML
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C H A P T E R 2 4 Portfolio Performance Evaluation 863
style analysis reveals the strategy
that most closely tracks the fund’s
activity and measures perfor-
mance relative to this strategy. If
the strategy revealed by the style
analysis method is consistent with
the one stated in the fund prospec-
tus, then the performance relative
to this strategy is the correct mea-
sure of the fund’s success.
Figure 24.9 shows the fre-
quency distribution of average
residuals across 636 mutual funds
from Sharpe’s style analysis. The
distribution has the familiar bell
shape with a slightly negative
mean of 2 .074% per month. This
should remind you of Figure 11.7,
where we presented the frequency
distribution of CAPM alphas for
a large sample of mutual funds.
As in Sharpe’s study, these
risk-adjusted returns plot as a
bell-shaped curve with slightly
negative mean.
Style Analysis and Multifactor Benchmarks Style analysis raises an interesting question for performance evaluation. Suppose a growth-
index portfolio exhibited superior performance relative to a mutual fund benchmark such
as the S&P 500 over some measurement period. Including this growth index in a style
analysis would eliminate this superior performance from the portfolio’s estimated alpha
value. Is this proper? Quite plausibly, the fund’s analysts predicted that an active portfolio
of growth stocks was underpriced and tilted the portfolio to take advantage of it. Clearly,
the contribution of this decision to an alpha value relative to the benchmark is a legitimate
part of the overall alpha value of the fund, and should not be eliminated by style analysis.
This brings up a related question.
Chapter 11 pointed out that the conventional performance benchmark today is a four-
factor model, which employs the three Fama-French factors (the return on the market
index, and returns to portfolios based on size and book-to-market ratio) augmented by
a momentum factor (a portfolio constructed based on prior-year stock return). Alphas
estimated from these four factor portfolios control for a wide range of style choices that
may affect average returns. But using alpha values from a multifactor model presupposes
that a passive strategy would include the aforementioned factor portfolios. When is this
reasonable?
Use of any benchmark other than the fund’s single-index benchmark is legitimate only
if we assume that the factor portfolios in question are part of the fund’s alternative passive
strategy. This assumption may be unrealistic in many cases where a single-index bench-
mark is used for performance evaluation even if research shows a multifactor model better
explains asset returns. In Section 24.8 on performance attribution we show how portfolio
Figure 24.9 Average tracking error for 636 mutual funds, 1985–1989 Source: William F. Sharpe, “Asset Allocation: Management Style and Performance Evaluation,” Journal of Portfolio Management, Winter 1992, pp. 7–19. Used with per- mission of Institutional Investor, Inc., www.iijournals.com. All Rights Reserved.
−1 .0
0
−0 .5
0
0 .0
0
0 .5
0
1 .0
0
Average Tracking Error (%/month)
0
10
20
30
40
50
60
70
80
90
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864 P A R T V I I Applied Portfolio Management
managers attempt to uncover which decisions contributed to superior performance. This
performance attribution procedure starts with benchmark allocations to various indexes
and attributes performance to asset allocation on the basis of deviation of actual from
benchmark allocations. The performance benchmark may be and often is specified in
advance without regard to any particular style portfolio.
Style Analysis in Excel Style analysis has become very popular in the investment management industry and has
spawned quite a few variations on Sharpe’s methodology. Many portfolio managers utilize
Web sites that help investors identify their style and stock selection performance.
You can do style analysis with Excel’s Solver. The strategy is to regress a fund’s rate
of return on those of a number of style portfolios (as in Table 24.5 ). The style portfolios
are passive (index) funds that represent a style alternative to asset allocation. Suppose you
choose three style portfolios, labeled 1–3. Then the coefficients in your style regression
are alpha (the intercept that measures abnormal performance) and three slope coefficients,
one for each style index. The slope coefficients reveal how sensitively the performance of
the fund follows the return of each passive style portfolio. The residuals from this regres-
sion, e ( t ), represent “noise,” that is, fund performance at each date, t, that is independent of any of the style portfolios. We cannot use a standard regression package in this analysis,
however, because we wish to constrain each coefficient to be nonnegative and sum to 1.0,
representing a portfolio of styles.
To do style analysis using Solver, start with arbitrary coefficients (e.g., you can set
a 5 0 and set each b 5 1/3). Use these to compute the time series of residuals from the
style regression according to
e(t) 5 R(t) 2 3a 1 b1R1(t) 1 b2R2(t) 1 b3R3(t)4 (24.8) where
R ( t ) 5 Excess return on the measured fund for date t
R i ( t ) 5 Excess return on the i th style portfolio ( i 5 1, 2, 3)
a 5 Abnormal performance of the fund over the sample period
b i 5 Beta of the fund on the i th style portfolio
Equation 24.8 yields the time series of residuals from your “regression equation” with
those arbitrary coefficients. Now square each residual and sum the squares. At this point,
you call on the Solver to minimize the sum of squares by changing the value of the four
coefficients. You will use the “by changing variables” command. You also add four con-
straints to the optimization: three that force the betas to be nonnegative and one that forces
them to sum to 1.0.
Solver’s output will give you the three style coefficients, as well as the estimate of the
fund’s unique, abnormal performance as measured by the intercept. The sum of squares also
allows you to calculate the R -square of the regression and p -values as explained in Chapter 8.
24.6 Performance Attribution Procedures
Rather than focus on risk-adjusted returns, practitioners often want simply to ascertain
which decisions resulted in superior or inferior performance. Superior investment perfor-
mance depends on an ability to be in the “right” securities at the right time. Such timing
and selection ability may be considered broadly, such as being in equities as opposed to
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C H A P T E R 2 4 Portfolio Performance Evaluation 865
fixed-income securities when the stock market is performing well. Or it may be defined
at a more detailed level, such as choosing the relatively better-performing stocks within a
particular industry.
Portfolio managers constantly make broad-brush asset allocation decisions as well as
more detailed sector and security allocation decisions within asset classes. Performance
attribution studies attempt to decompose overall performance into discrete components
that may be identified with a particular level of the portfolio selection process.
Attribution studies start from the broadest asset allocation choices and progressively
focus on ever-finer details of portfolio choice. The difference between a managed port-
folio’s performance and that of a benchmark portfolio then may be expressed as the sum
of the contributions to performance of a series of decisions made at the various levels of
the portfolio construction process. For example, one common attribution system decom-
poses performance into three components: (1) broad asset market allocation choices across
equity, fixed-income, and money markets; (2) industry (sector) choice within each market;
and (3) security choice within each sector.
The attribution method explains the difference in returns between a managed portfo-
lio, P, and a selected benchmark portfolio, B, called the bogey . Suppose that the universe of assets for P and B includes n asset classes such as equities, bonds, and bills. For each asset class, a benchmark index portfolio is determined. For example, the S&P 500 may
be chosen as a benchmark for equities. The bogey portfolio is set to have fixed weights in
each asset class, and its rate of return is given by
rB 5 a n
i51 wBirBi
where w Bi is the weight of the bogey in asset class i, and r Bi is the return on the benchmark portfolio of that class over the evaluation period. The portfolio managers choose weights
in each class, w Pi , based on their capital market expectations, and they choose a portfolio of the securities within each class based on their security analysis, which earns r Pi over the evaluation period. Thus the return of the managed portfolio will be
rP 5 a n
i51 wPirPi
The difference between the two rates of return, therefore, is
rP 2 rB 5 a n
i51 wPirPi 2 a
n
i51 wBirBi 5 a
n
i51
(wPirPi 2 wBirBi) (24.9)
Each term in the summation of Equation 24.9 can be rewritten in a way that shows how
asset allocation decisions versus security selection decisions for each asset class contrib-
uted to overall performance. We decompose each term of the summation into a sum of two
terms as follows. Note that the two terms we label as contribution from asset allocation and
contribution from security selection in the following decomposition do in fact sum to the
total contribution of each asset class to overall performance.
Contribution from asset allocation (wPi 2 wBi)rBi 1 Contribution from security selection wPi (rPi 2 rBi)
5 Total contribution from asset class i wPirPi 2 wBirBi
The first term of the sum measures the impact of asset allocation because it shows how
deviations of the actual weight from the benchmark weight for that asset class multiplied
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866 P A R T V I I Applied Portfolio Management
by the index return for the
asset class added to or sub-
tracted from total perfor-
mance. The second term
of the sum measures the
impact of security selec-
tion because it shows how
the manager’s excess return
within the asset class com- pared to the benchmark
return for that class mul-
tiplied by the portfolio
weight for that class added
to or subtracted from total
performance. Figure 24.10
presents a graphical inter-
pretation of the attribution
of overall performance into
security selection versus
asset allocation.
To illustrate this method,
consider the attribution
results for a hypothetical port-
folio. The portfolio invests
in stocks, bonds, and money
market securities. An attribution analysis appears in Tables 24.6 through 24.9 . The portfo-
lio return over the month is 5.34%.
The first step is to establish a benchmark level of performance against which perfor-
mance ought to be compared. This benchmark, again, is called the bogey. It is designed to
measure the returns the portfolio manager would earn if he or she were to follow a com-
pletely passive strategy. “Passive” in this context has two attributes. First, it means that
the allocation of funds across broad asset classes is set in accord with a notion of “usual,”
or neutral, allocation across sectors. This would be considered a passive asset-market
allocation. Second, it means that within each asset class, the portfolio manager holds an indexed portfolio such as the S&P 500 index for the equity sector. In such a manner,
the passive strategy used as a performance benchmark rules out asset allocation as well
as security selection decisions. Any departure of the manager’s return from the passive
benchmark must be due to either asset allocation bets (departures from the neutral alloca-
tion across markets) or security selection bets (departures from the passive index within
asset classes).
While we have already discussed in earlier chapters the justification for indexing
within sectors, it is worth briefly explaining the determination of the neutral alloca-
tion of funds across the broad asset classes. Weights that are designated as “neutral”
will depend on the risk tolerance of the investor and must be determined in consulta-
tion with the client. For example, risk-tolerant clients may place a large fraction of
their portfolio in the equity market, perhaps directing the fund manager to set neutral
weights of 75% equity, 15% bonds, and 10% cash equivalents. Any deviation from
these weights must be justified by a belief that one or another market will either
Figure 24.10 Performance attribution of i th asset class. Enclosed area indicates total rate of return.
rPi
rBi
wPiwBi wi
Return in Asset Class
Mixed Origin (attributed to selection)
Weight in Asset Class
Bogey Return from ith Asset Class = rBiwBi
A llo
ca tio
n
Added by Selection
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C H A P T E R 2 4 Portfolio Performance Evaluation 867
over- or underperform its usual risk–return profile. In contrast, more risk-averse cli-
ents may set neutral weights of 45%/35%/20% for the three markets. Therefore, their
portfolios in normal circumstances will be exposed to less risk than that of the risk-
tolerant client. Only intentional bets on market performance will result in departures
from this profile.
In Table 24.6 , the neutral weights have been set at 60% equity, 30% fixed income,
and 10% cash (money market securities). The bogey portfolio, comprised of investments
in each index with the 60/30/10 weights, returned 3.97%. The managed portfolio’s mea-
sure of performance is positive and equal to its actual return less the return of the bogey:
5.34 2 3.97 5 1.37%. The next step is to allocate the 1.37% excess return to the separate
decisions that contributed to it.
Asset Allocation Decisions Our hypothetical managed portfolio is invested in the equity, fixed-income, and money
markets with weights of 70%, 7%, and 23%, respectively. The portfolio’s performance
could have to do with the departure of this weighting scheme from the benchmark
60/30/10 weights and/or to superior or inferior results within each of the three broad markets.
To isolate the effect of the manager’s asset allocation choice, we measure the perfor-
mance of a hypothetical portfolio that would have been invested in the indexes for each
market with weights 70/7/23. This return measures the effect of the shift away from the
benchmark 60/30/10 weights without allowing for any effects attributable to active man-
agement of the securities selected within each market.
Superior performance relative to the bogey is achieved by overweighting investments in
markets that turn out to perform well and by underweighting those in poorly performing
markets. The contribution of asset allocation to superior performance equals the sum over
all markets of the excess weight (sometimes called the active weight in the industry) in each market times the return of the market index.
Panel A of Table 24.7 demonstrates that asset allocation contributed 31 basis points to
the portfolio’s overall excess return of 137 basis points. The major factor contributing to
superior performance in this month is the heavy weighting of the equity market in a month
when the equity market has an excellent return of 5.81%.
Bogey Performance and Excess Return
Component Benchmark
Weight Return of Index
during Month (%)
Equity (S&P 500) .60 5.81 Bonds (Barclays Aggregate Index) .30 1.45 Cash (money market) .10 0.48 Bogey 5 (.60 3 5.81) 1 (.30 3 1.45) 1 (.10 3 0.48) 5 3.97%
Return of managed portfolio 5.34% 2 Return of bogey portfolio 3.97
Excess return of managed portfolio 1.37%
Table 24.6
Performance of the managed portfolio
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868 P A R T V I I Applied Portfolio Management
Sector and Security Selection Decisions If .31% of the excess performance ( Table 24.7, panel A) can be attributed to advantageous
asset allocation across markets, the remaining 1.06% then must be attributable to sector
selection and security selection within each market. Table 24.7, panel B, details the contri-
bution of the managed portfolio’s sector and security selection to total performance.
Panel B shows that the equity component of the managed portfolio has a return of
7.28% versus a return of 5.81% for the S&P 500. The fixed-income return is 1.89% versus
1.45% for the Barclays Aggregate Bond Index. The superior performance in both equity
and fixed-income markets weighted by the portfolio proportions invested in each market
sums to the 1.06% contribution to performance attributable to sector and security selection.
Table 24.8 documents the sources of the equity market performance by each sector
within the market. The first three columns detail the allocation of funds within the equity
market compared to their representation in the S&P 500. Column (4) shows the rate of
return of each sector. The contribution of each sector’s allocation presented in column (5)
equals the product of the difference in the sector weight and the sector’s performance.
Note that good performance (a positive contribution) derives from overweighting well-
performing sectors such as consumer noncyclicals, as well as underweighting poorly per-
forming sectors such as technology. The excess return of the equity component of the
portfolio attributable to sector allocation alone is 1.29%. Table 24.7 , panel B, column (3),
shows that the equity component of the portfolio outperformed the S&P 500 by 1.47%. We
conclude that the effect of security selection within sectors must have contributed an addi- tional 1.47% 2 1.29%, or .18%, to the performance of the equity component of the portfolio.
A similar sector analysis can be applied to the fixed-income portion of the portfolio, but
we do not show those results here.
A. Contribution of Asset Allocation to Performance
(1) (2) (3) (4) (5) 5 (3) 3 (4)
Market
Actual Weight in
Market
Benchmark Weight
in Market
Active or Excess Weight
Market Return
(%)
Contribution to Performance
(%)
Equity .70 .60 .10 5.81 .5810 Fixed-income .07 .30 2.23 1.45 2.3335 Cash .23 .10 .13 .48 .0624 Contribution of asset allocation .3099
B. Contribution of Selection to Total Performance
(1) (2) (3) (4) (5) 5 (3) 3 (4)
Market
Portfolio Performance
(%)
Index Performance
(%)
Excess Performance
(%) Portfolio Weight
Contribution (%)
Equity 7.28 5.81 1.47 .70 1.03 Fixed-income 1.89 1.45 0.44 .07 0.03 Contribution of selection within markets 1.06
Table 24.7
Performance attribution
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(1) (2) (3) (4) (5) 5 (3) 3 (4)
Beginning of Month Weights (%) Active
Weights (%)
Sector Return
(%)
Sector Allocation
Contribution Sector Portfolio S&P 500
Basic materials 1.96 8.3 2 6.34 6.9 2 0.4375 Business services 7.84 4.1 3.74 7.0 0.2618 Capital goods 1.87 7.8 2 5.93 4.1 2 0.2431 Consumer cyclical 8.47 12.5 2 4.03 8.8 0.3546 Consumer noncyclical 40.37 20.4 19.97 10.0 1.9970 Credit sensitive 24.01 21.8 2.21 5.0 0.1105 Energy 13.53 14.2 2 0.67 2.6 2 0.0174 Technology 1.95 10.9 2 8.95 0.3 2 0.0269 TOTAL 1.2898
Table 24.8
Sector selection within the equity market
eXcel APPLICATIONS: Performance Attribution
T he performance attribution spreadsheet develops the attribution analysis that is presented in this section.
Additional data can be used in the analysis of performance for other sets of portfolios. The model can be used to ana- lyze performance of mutual funds and other managed portfolios.
You can find this Excel model on the Online Learning Center at www.mhhe.com/bkm .
Excel Questions
1. What would happen to the contribution of asset allocation to overall performance if the actual weights had been 75/12/13 instead of 70/7/23? Explain your result.
2. What would happen to the contribution of security selection to overall performance if the actual return on the equity portfolio had been 6.81% instead of 5.81% and the return on the bond portfolio had been 0.45% instead of 1.45%? Explain your result.
A B C D E F
Bogey Portfolio
Component Equity Bonds Cash
Index S&P 500
Barclays Index Money Market
Benchmark Weight
0.60 0.30 0.10
Return on Index
5.8100% 1.4500% 0.4800%
Portfolio Return
Portfolio Weight
0.70 0.07 0.23
Actual Return 5.8100% 1.4500% 0.4800%
Portfolio Return
Managed Portfolio
Component Equity Bonds Cash
Performance Attribution
Return on Bogey
Return on Managed Excess Return
5.0960% 0.1323% 0.1104% 5.3387% 1.3697%
3.4860% 0.4350% 0.0480% 3.9690%
19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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Contribution (basis points)
1. Asset allocation 31 2. Selection a. Equity excess return (basis points) i. Sector allocation 129 ii. Security selection 18
147 3 .70 (portfolio weight) 5 102.9 b. Fixed-income excess return 44 3 .07 (portfolio weight) 5 3.1
Total excess return of portfolio 137.0
Table 24.9
Portfolio attribution: summary
Summing Up Component Contributions In this particular month, all facets of the portfolio selection process were successful.
Table 24.9 details the contribution of each aspect of performance. Asset allocation across
the major security markets contributes 31 basis points. Sector and security allocation
within those markets contributes 106 basis points, for total excess portfolio performance
of 137 basis points.
The sector and security allocation of 106 basis points can be partitioned further. Sector
allocation within the equity market results in excess performance of 129 basis points, and
security selection within sectors contributes 18 basis points. (The total equity excess per-
formance of 147 basis points is multiplied by the 70% weight in equity to obtain contribu-
tion to portfolio performance.) Similar partitioning could be done for the fixed-income
sector.
a. Suppose the benchmark weights in Table 24.7 had been set at 70% equity, 25% fixed-income, and 5% cash equivalents. What would have been the contributions of the manager’s asset allocation choices?
b. Suppose the S&P 500 return is 5%. Compute the new value of the manager’s security selection choices.
CONCEPT CHECK 24.5
1. The appropriate performance measure depends on the role of the portfolio to be evaluated. Appro- priate performance measures are as follows:
a. Sharpe: when the portfolio represents the entire investment fund.
b. Information ratio: when the portfolio represents the active portfolio to be optimally mixed with the passive portfolio.
c. Treynor or Jensen: when the portfolio represents one subportfolio of many.
2. Many observations are required to eliminate the effect of the “luck of the draw” from the evalu- ation process because portfolio returns commonly are very “noisy.”
3. Hedge funds or other active positions meant to be mixed with a passive indexed portfolio should be evaluated based on their information ratio.
SUMMARY
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C H A P T E R 2 4 Portfolio Performance Evaluation 871
4. The shifting mean and variance of actively managed portfolios make it even harder to assess per- formance. A typical example is the attempt of portfolio managers to time the market, resulting in
ever-changing portfolio betas.
5. A simple way to measure timing and selection success simultaneously is to estimate an expanded security characteristic line, with a quadratic term added to the usual index model. Another way to
evaluate timers is based on the implicit call option embedded in their performance.
6. Style analysis uses a multiple regression model where the factors are category (style) portfolios such as bills, bonds, and stocks. A regression of fund returns on the style portfolio returns gener-
ates residuals that represent the value added of stock selection in each period. These residuals can
be used to gauge fund performance relative to similar-style funds.
7. The Morningstar Star Rating method compares each fund to a peer group represented by a style portfolio within four asset classes. Risk-adjusted ratings (RAR) are based on fund returns relative
to the peer group and used to award each fund one to five stars based on the rank of its RAR. The
MRAR is the only manipulation-proof performance measure.
8. Common attribution procedures partition performance improvements to asset allocation, sector selection, and security selection. Performance is assessed by calculating departures of portfolio
composition from a benchmark or neutral portfolio.
Related Web sites for this chapter are available at www. mhhe.com/bkm
time-weighted average
dollar-weighted rate of return
comparison universe
Sharpe’s ratio
Treynor’s measure
Jensen’s alpha
information ratio
bogey KEY TERMS
Sharpe ratio: S 5 rP 2 rf
s
M 2 of portfolio P relative to its Sharpe ratio: M 2 5 s M ( S P 2 S M )
Treynor measure: T 5 rP 2 rf
b
Jensen’s alpha: aP 5 rP 2 3rf 1 bP (rM 2 rf) 4 Information ratio:
aP
s(eP)
Morningstar risk-adjusted return: MRAR(g) 5 B 1 T a
T
t51 ¢ 1 1 rt
1 1 rft ≤ 2g
R 212
g
2 1
KEY EQUATIONS
1. A household (HH) savings-account spreadsheet shows the following entries:
Date Additions Withdrawals Value
1/1/10 148,000 1/3/10 2,500 3/20/10 4,000 7/5/10 1,500 12/2/10 13,460 3/10/11 23,000 4/7/11 3,000 5/3/11 198,000
Calculate the dollar-weighted average return on the HH savings account between the first and
final dates.
Basic
PROBLEM SETS
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2. Is it possible that a positive alpha will be associated with inferior performance? Explain.
3. We know that the geometric average (time-weighted return) on a risky investment is always lower
than the corresponding arithmetic average. Can the IRR (the dollar-weighted return) similarly be
ranked relative to these other two averages?
4. W e have seen that market timing has tremendous potential value. Would it therefore be wise to
shift resources to timing at the expense of security selection?
5. Consider the rate of return of stocks ABC and XYZ.
Year r ABC r XYZ
1 20% 30% 2 12 12 3 14 18 4 3 0 5 1 2 10
a. Calculate the arithmetic average return on these stocks over the sample period. b. Which stock has greater dispersion around the mean? c. Calculate the geometric average returns of each stock. What do you conclude? d. If you were equally likely to earn a return of 20%, 12%, 14%, 3%, or 1%, in each year (these
are the five annual returns for stock ABC), what would be your expected rate of return? What
if the five possible outcomes were those of stock XYZ?
6. XYZ stock price and dividend history are as follows:
Year Beginning-of-Year Price Dividend Paid at Year-End
2013 $100 $4 2014 120 4 2015 90 4 2016 100 4
An investor buys three shares of XYZ at the beginning of 2013, buys another two shares at the
beginning of 2014, sells one share at the beginning of 2015, and sells all four remaining shares at
the beginning of 2016.
a. What are the arithmetic and geometric average time-weighted rates of return for the investor?
b. What is the dollar-weighted rate of return? ( Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2013, to January 1, 2016. If your calculator cannot calculate internal rate of return, you will have to use trial
and error.)
7. A manager buys three shares of stock today, and then sells one of those shares each year for the
next 3 years. His actions and the price history of the stock are summarized below. The stock pays
no dividends.
Time Price Action
0 $ 90 Buy 3 shares 1 100 Sell 1 share 2 100 Sell 1 share 3 100 Sell 1 share
a. Calculate the time-weighted geometric average return on this “portfolio.” b. Calculate the time-weighted arithmetic average return on this portfolio. c. Calculate the dollar-weighted average return on this portfolio.
Intermediate
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C
rP rf
A
rP rf
D rM rf
rP rf
B
rM rf
rP rf
rM rf
rM rf
C H A P T E R 2 4 Portfolio Performance Evaluation 873
8. Based on current dividend yields and expected capital gains, the expected rates of return on port-
folios A and B are 12% and 16%, respectively. The beta of A is .7, while that of B is 1.4. The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%. The standard
deviation of portfolio A is 12% annually, that of B is 31%, and that of the S&P 500 index is 18%.
a. If you currently hold a market-index portfolio, would you choose to add either of these port- folios to your holdings? Explain.
b. If instead you could invest only in T-bills and one of these portfolios, which would you choose?
9. Consider the two (excess return) index-model regression results for stocks A and B. The risk- free rate over the period was 6%, and the market’s average return was 14%. Performance is
measured using an index model regression on excess returns.
Stock A Stock B
Index model regression estimates 1% 1 1.2(rM – rf) 2% 1 .8(rM – rf) R-square .576 .436
Residual standard deviation, s(e) 10.3% 19.1%
Standard deviation of excess returns 21.6% 24.9%
a. Calculate the following statistics for each stock:
i. Alpha
ii. Information ratio
iii. Sharpe ratio
iv. Treynor measure
b. Which stock is the best choice under the following circumstances?
i. This is the only risky asset to be held by the investor.
ii. This stock will be mixed with the rest of the investor’s portfolio, currently composed
solely of holdings in the market-index fund.
iii. This is one of many stocks that the investor is analyzing to form an actively managed
stock portfolio.
10. Evaluate the market timing and security selection abilities of four managers whose perfor-
mances are plotted in the accompanying diagrams.
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11. Consider the following information regarding the performance of a money manager in a recent
month. The table represents the actual return of each sector of the manager’s portfolio in col-
umn 1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neu-
tral sector allocations in column 3, and the returns of sector indices in column 4.
Actual Return Actual Weight Benchmark Weight Index Return
Equity 2% .70 .60 2.5% (S&P 500) Bonds 1 .20 .30 1.2 (Salomon Index) Cash 0.5 .10 .10 0.5
a. What was the manager’s return in the month? What was her overperformance or under performance?
b. What was the contribution of security selection to relative performance? c. What was the contribution of asset allocation to relative performance? Confirm that the sum of
selection and allocation contributions equals her total “excess” return relative to the bogey.
12. A global equity manager is assigned to select stocks from a universe of large stocks through-
out the world. The manager will be evaluated by comparing her returns to the return on the
MSCI World Market Portfolio, but she is free to hold stocks from various countries in whatever
proportions she finds desirable. Results for a given month are contained in the following table:
Country Weight In
MSCI Index Manager’s
Weight Manager’s Return
in Country Return of Stock Index
for That Country
U.K. .15 .30 20% 12% Japan .30 .10 15 15 U.S. .45 .40 10 14 Germany .10 .20 5 12
a. Calculate the total value added of all the manager’s decisions this period. b. Calculate the value added (or subtracted) by her country allocation decisions. c. Calculate the value added from her stock selection ability within countries. Confirm that the
sum of the contributions to value added from her country allocation plus security selection
decisions equals total over- or underperformance.
13. Conventional wisdom says that one should measure a manager’s investment performance over
an entire market cycle. What arguments support this convention? What arguments contradict it?
14. Does the use of universes of managers with similar investment styles to evaluate relative invest-
ment performance overcome the statistical problems associated with instability of beta or total
variability?
15. During a particular year, the T-bill rate was 6%, the market return was 14%, and a portfolio
manager with beta of .5 realized a return of 10%.
a. Evaluate the manager based on the portfolio alpha. b. Reconsider your answer to part ( a ) in view of the Black-Jensen-Scholes finding that the
security market line is too flat. Now how do you assess the manager’s performance?
16. Bill Smith is evaluating the performance of four large-cap equity portfolios: Funds A, B, C, and D. As part of his analysis, Smith computed the Sharpe ratio and the Treynor measure for all four funds. Based on his finding, the ranks assigned to the four funds are as follows:
Fund Treynor Measure Rank Sharpe Ratio Rank
A 1 4 B 2 3 C 3 2 D 4 1
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C H A P T E R 2 4 Portfolio Performance Evaluation 875
The difference in rankings for Funds A and D is most likely due to:
a. A lack of diversification in Fund A as compared to Fund D. b. Different benchmarks used to evaluate each fund’s performance. c. A difference in risk premiums.
Use the following information to answer Problems 17–20: Primo Management Co. is look- ing at how best to evaluate the performance of its managers. Primo has been hearing more and
more about benchmark portfolios and is interested in trying this approach. As such, the com-
pany hired Sally Jones, CFA, as a consultant to educate the managers on the best methods for
constructing a benchmark portfolio, how best to choose a benchmark, whether the style of the
fund under management matters, and what they should do with their global funds in terms of
benchmarking.
For the sake of discussion, Jones put together some comparative 2-year performance numbers
that relate to Primo’s current domestic funds under management and a potential benchmark.
Weight Return
Style Category Primo Benchmark Primo Benchmark
Large-cap growth .60 .50 17% 16% Mid-cap growth .15 .40 24 26 Small-cap growth .25 .10 20 18
As part of her analysis, Jones also takes a look at one of Primo’s global funds. In this particular
portfolio, Primo is invested 75% in Dutch stocks and 25% in British stocks. The benchmark
invested 50% in each—Dutch and British stocks. On average, the British stocks outperformed
the Dutch stocks. The euro appreciated 6% versus the U.S. dollar over the holding period while
the pound depreciated 2% versus the dollar. In terms of the local return, Primo outperformed
the benchmark with the Dutch investments, but underperformed the index with respect to the
British stocks.
17. What is the within-sector selection effect for each individual sector?
18. Calculate the amount by which the Primo portfolio out- (or under-)performed the market over
the period, as well as the contribution to performance of the pure sector allocation and security
selection decisions.
19. If Primo decides to use return-based style analysis, will the R 2 of the regression equation of a passively managed fund be higher or lower than that of an actively managed fund?
20. Which of the following statements about Primo’s global fund is most correct? Primo appears to
have a positive currency allocation effect as well as
a. A negative market allocation effect and a positive security allocation effect. b. A negative market allocation effect and a negative security allocation effect. c. A positive market allocation effect and a negative security allocation effect.
21. Kelli Blakely is a portfolio manager for the Miranda Fund (Miranda), a core large-cap equity
fund. The market proxy and benchmark for performance measurement purposes is the S&P
500. Although the Miranda portfolio generally mirrors the asset class and sector weightings of
the S&P, Blakely is allowed a significant amount of leeway in managing the fund. Her portfolio
holds only stocks found in the S&P 500 and cash.
Blakely was able to produce exceptional returns last year (as outlined in the table below)
through her market timing and security selection skills. At the outset of the year, she became
extremely concerned that the combination of a weak economy and geopolitical uncertainties
would negatively impact the market. Taking a bold step, she changed her market allocation. For
the entire year her asset class exposures averaged 50% in stocks and 50% in cash. The S&P’s
allocation between stocks and cash during the period was a constant 97% and 3%, respectively.
The risk-free rate of return was 2%.
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One-Year Trailing Returns
Miranda Fund S&P 500
Return 10.2% –22.5%
Standard deviation 37% 44%
Beta 1.10 1.00
a. What are the Sharpe ratios for the Miranda Fund and the S&P 500? b. What are the M 2 measures for Miranda and the S&P 500? c. What is the Treynor measure for the Miranda Fund and the S&P 500? d. What is the Jensen measure for the Miranda Fund?
22. G o to Kenneth French’s data library site at http://mba.tuck.dartmouth.edu/pages/faculty/ ken.french/data_library.html . Select two industry portfolios of your choice and download 36 months of data. Download other data from the site as needed to perform the following
tasks.
a. Compare the portfolio’s performance to that of the market index on the basis of the vari- ous performance measures discussed in the chapter. Plot the monthly values of alpha plus
residual return.
b. Now use the Fama-French three-factor model as the return benchmark. Compute plots of alpha plus residual return using the FF model. How does performance change using this
benchmark instead of the market index?
Challenge
e X c e l Please visit us at www.mhhe.com/bkm
1. You and a prospective client are considering the measurement of investment performance, par-
ticularly with respect to international portfolios for the past 5 years. The data you discussed are
presented in the following table:
International Manager or Index Total Return
Country and Security Return Currency Return
Manager A 2 6.0% 2.0% 2 8.0%
Manager B 2 2.0 2 1.0 2 1.0
International Index 2 5.0 0.2 2 5.2
a. Assume that the data for manager A and manager B accurately reflect their investment skills and that both managers actively manage currency exposure. Briefly describe one strength and
one weakness for each manager.
b. Recommend and justify a strategy that would enable your fund to take advantage of the strengths of each of the two managers while minimizing their weaknesses.
2. Carl Karl, a portfolio manager for the Alpine Trust Company, has been responsible since 2015
for the City of Alpine’s Employee Retirement Plan, a municipal pension fund. Alpine is a
growing community, and city services and employee payrolls have expanded in each of the past
10 years. Contributions to the plan in fiscal 2020 exceeded benefit payments by a three-to-one
ratio.
The plan board of trustees directed Karl 5 years ago to invest for total return over the long
term. However, as trustees of this highly visible public fund, they cautioned him that volatile or
erratic results could cause them embarrassment. They also noted a state statute that mandated that
not more than 25% of the plan’s assets (at cost) be invested in common stocks.
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C H A P T E R 2 4 Portfolio Performance Evaluation 877
At the annual meeting of the trustees in November 2020, Karl presented the following portfo-
lio and performance report to the board:
Alpine Employee Retirement Plan
Asset Mix as of 9/30/20 At Cost (millions) At Market (millions)
Fixed-income assets: Short-term securities $ 4.5 11.0% $ 4.5 11.4% Long-term bonds and mortgages 26.5 64.7 23.5 59.5 Common stocks 10.0 24.3 11.5 29.1
$41.0 100.0% $39.5 100.0%
Investment Performance
Annual Rates of Return for P eriods Ending 9/30/20
5 Years 1 Year
Total Alpine Fund: Time-weighted 8.2% 5.2% Dollar-weighted (internal) 7.7% 4.8% Assumed actuarial return 6.0% 6.0% U.S. Treasury bills 7.5% 11.3%
Large sample of pension funds (average 60% equities, 40% fixed income)
10.1% 14.3%
Common stocks—Alpine Fund 13.3% 14.3% Alpine portfolio beta coefficient 0.90 0.89 Standard & Poor’s 500 stock index 13.8% 21.1% Fixed-income securities—Alpine Fund 6.7% 1.0% Salomon Brothers’ bond index 4.0% 211.4%
Karl was proud of his performance and was chagrined when a trustee made the following critical
observations:
a. “Our 1-year results were terrible, and it’s what you’ve done for us lately that counts most.” b. “Our total fund performance was clearly inferior compared to the large sample of other
pension funds for the last 5 years. What else could this reflect except poor management
judgment?”
c. “Our common stock performance was especially poor for the 5-year period.” d. “Why bother to compare your returns to the return from Treasury bills and the actuarial
assumption rate? What your competition could have earned for us or how we would have fared
if invested in a passive index (which doesn’t charge a fee) are the only relevant measures of
performance.”
e. “Who cares about time-weighted return? If it can’t pay pensions, what good is it!”
Appraise the merits of each of these statements and give counterarguments that Mr. Karl
can use.
3. The Retired Fund is an open-ended mutual fund composed of $500 million in U.S. bonds and
U.S. Treasury bills. This fund has had a portfolio duration (including T-bills) of between 3 and
9 years. Retired has shown first-quartile performance over the past 5 years, as measured by an
independent fixed-income measurement service. However, the directors of the fund would like to
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measure the market timing skill of the fund’s sole bond investor manager. An external consulting
firm has suggested the following three methods:
a. Method I examines the value of the bond portfolio at the beginning of every year, then calcu- lates the return that would have been achieved had that same portfolio been held throughout
the year. This return would then be compared with the return actually obtained by the fund.
b. Method II calculates the average weighting of the portfolio in bonds and T-bills for each year. Instead of using the actual bond portfolio, the return on a long-bond market index and T-bill
index would be used. For example, if the portfolio on average was 65% in bonds and 35% in
T-bills, the annual return on a portfolio invested 65% in a long-bond index and 35% in T-bills
would be calculated. This return is compared with the annual return that would have been
generated using the indexes and the manager’s actual bond/T-bill weighting for each quarter
of the year.
c. Method III examines the net bond purchase activity (market value of purchases less sales) for each quarter of the year. If net purchases were positive (negative) in any quarter, the perfor-
mance of the bonds would be evaluated until the net purchase activity became negative (posi-
tive). Positive (negative) net purchases would be viewed as a bullish (bearish) view taken by
the manager. The correctness of this view would be measured.
Critique each method with regard to market timing measurement problems.
Use the following data to solve CFA Problems 4–6: The administrator of a large pension fund wants to evaluate the performance of four portfolio managers. Each portfolio manager
invests only in U.S. common stocks. Assume that during the most recent 5-year period, the aver-
age annual total rate of return including dividends on the S&P 500 was 14%, and the average
nominal rate of return on government Treasury bills was 8%. The following table shows risk and
return measures for each portfolio:
Portfolio Average Annual Rate of Return Standard Deviation Beta
P 17% 20% 1.1
Q 24 18 2.1
R 11 10 0.5
S 16 14 1.5
S&P 500 14 12 1.0
4. What is the Treynor performance measure for portfolio P ?
5. What is the Sharpe performance measure for portfolio Q ?
6. An analyst wants to evaluate portfolio X, consisting entirely of U.S. common stocks, using both the Treynor and Sharpe measures of portfolio performance. The following table provides the
average annual rate of return for portfolio X, the market portfolio (as measured by the S&P 500), and U.S. Treasury bills during the past 8 years:
Average Annual Rate of Return Standard Deviation of Return Beta
Portfolio X 10% 18% 0.60
S&P 500 12 13 1.00
T-bills 6 N/A N/A
a. Calculate the Treynor and Sharpe measures for both portfolio X and the S&P 500. Briefly explain whether portfolio X underperformed, equaled, or outperformed the S&P 500 on a risk- adjusted basis using both the Treynor measure and the Sharpe ratio.
b. On the basis of the performance of portfolio X relative to the S&P 500 calculated in part ( a ), briefly explain the reason for the conflicting results when using the Treynor measure versus
the Sharpe ratio.
7. Assume you invested in an asset for 2 years. The first year you earned a 15% return, and the sec-
ond year you earned a negative 10% return. What was your annual geometric return?
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8. A portfolio of stocks generates a 2 9% return in 2013, a 23% return in 2014, and a 17% return
in 2015. What is the annualized return (geometric mean) for the entire period?
9. A 2-year investment of $2,000 results in a cash flow of $150 at the end of the first year and
another cash flow of $150 at the end of the second year, in addition to the return of the original
investment. What is the internal rate of return on the investment?
10. In measuring the performance of a portfolio, the time-weighted rate of return is superior to the
dollar-weighted rate of return because:
a. When the rate of return varies, the time-weighted return is higher. b. The dollar-weighted return assumes all portfolio deposits are made on day 1. c. The dollar-weighted return can only be estimated. d. The time-weighted return is unaffected by the timing of portfolio contributions and
withdrawals.
11. A pension fund portfolio begins with $500,000 and earns 15% the first year and 10% the second
year. At the beginning of the second year, the sponsor contributes another $500,000. What were
the time-weighted and dollar-weighted rates of return?
12. During the annual review of Acme’s pension plan, several trustees questioned their investment
consultant about various aspects of performance measurement and risk assessment.
a. Comment on the appropriateness of using each of the following benchmarks for perfor- mance evaluation:
• Market index.
• Benchmark normal portfolio.
• Median of the manager universe.
b. Distinguish among the following performance measures:
• The Sharpe ratio.
• The Treynor measure.
• Jensen’s alpha.
i. Describe how each of the three performance measures is calculated.
ii. State whether each measure assumes that the relevant risk is systematic, unsystematic, or
total. Explain how each measure relates excess return and the relevant risk.
13. Trustees of the Pallor Corp. pension plan ask consultant Donald Millip to comment on the fol-
lowing statements. What should his response be?
a. Median manager benchmarks are statistically unbiased measures of performance over long periods of time.
b. Median manager benchmarks are unambiguous and are therefore easily replicated by man- agers wishing to adopt a passive/indexed approach.
c. Median manager benchmarks are not appropriate in all circumstances because the median manager universe encompasses many investment styles.
14. James Chan is reviewing the performance of the global equity managers of the Jarvis University
endowment fund. Williamson Capital is currently the endowment fund’s only large-capitaliza-
tion global equity manager. Performance data for Williamson Capital are shown in Table 24A .
Chan also presents the endowment fund’s investment committee with performance informa-
tion for Joyner Asset Management, which is another large-capitalization global equity manager.
Performance data for Joyner Asset Management are shown in Table 24B . Performance data for
the relevant risk-free asset and market index are shown in Table 24C .
a. Calculate the Sharpe ratio and Treynor measure for both Williamson Capital and Joyner Asset Management.
b. The Investment Committee notices that using the Sharpe ratio versus the Treynor measure produces different performance rankings of Williamson and Joyner. Explain why these crite-
ria may result in different rankings.
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Table 24C
Relevant risk-free asset and market index performance data, 1999–2010
Risk-Free Asset
Average annual rate of return 5.0%
Market Index
Average annual rate of return 18.9%
Standard deviation of returns 13.8%
E-INVESTMENTS EXERCISES Several popular finance-related Web sites offer mutual fund screeners. Go to moneycentral. msn.com and click on the Investing link on the top menu. Choose Funds from the submenu, then look for the Easy Screener link on the left-side menu. Before you start to specify your preferences using the drop-down boxes, look for the Show More Options link toward the bottom of the page and select it. When all of the options are shown, devise a screen for funds that meet the following criteria: 5-star Morningstar Overall Rating, a Minimum Initial Investment as low as possible, Low Morningstar Risk, No Load, Manager Tenure of at least 5 years, Morningstar Overall Return high, 12b-1 fees as low as possible, and Expense Ratio as low as possible. Click on the Find Funds link to run the screen.
When you get the list of results, you can sort them according to any one criterion that interests you by clicking on its column heading. Are there any funds you would rule out based on what you see? If you want to rerun the screen with different choices click on the Change Criteria link toward the top of the page and make the changes. Click on Find Funds again to run the new screen. You can click on any fund symbol to get more informa- tion about it.
Are any of these funds of interest to you? How might your screening choices differ if you were choosing funds for various clients?
SOLUTIONS TO CONCEPT CHECKS 1.
a. Dollar-weighted return:
240 1 26
1 1 r 1
21
(1 1 r)2 5 0
r 5 .1191, or 11.91%
Time Action Cash Flow
0 Buy two shares 240
1 Collect dividends; then sell one of the shares 4 1 22
2 Collect dividend on remaining share, then sell it 2 1 19
Table 24A
Williamson capital performance data, 1999–2010
Average annual rate of return 22.1%
Beta 1.2
Standard deviation of returns 16.8%
Table 24B
Joyner asset management performance data, 1999–2010
Average annual rate of return 24.2%
Beta 0.8
Standard deviation of returns 20.2%
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b. Time-weighted return: The rates of return on the stock in the 2 years were:
r1 5 2 1 (22 2 20)
20 5 .20
r2 5 2 1 (19 2 22)
22 5 2.045
(r1 1 r2)/2 5 .077, or 7.7%
2. Sharpe: (r 2 rf)/s
SP 5 (35 2 6)/42 5 .69
SM 5 (28 2 6)/30 5 .733
Alpha: r 2 3rf 1 b(rM 2 rf) 4 aP 5 35 2 36 1 1.2(28 2 6) 4 5 2.6 aM 5 0
Treynor: (r 2 rf)/b
TP 5 (35 2 6)/1.2 5 24.2
TM 5 (28 2 6)/1.0 5 22
Information ratio: a / s ( e )
IP 5 2.6/18 5 .144
IM 5 0
3. The alpha exceeds zero by .2/2 5 .1 standard deviations. A table of the normal distribution (or,
somewhat more appropriately, the distribution of the t- statistic) indicates that the probability of such an event, if the analyst actually has no skill, is approximately 46%.
4. The timer will guess bear or bull markets completely randomly. One-half of all bull
markets will be preceded by a correct forecast, and similarly for bear markets. Hence
P 1 1 P 2 2 1 5 ½ 1 ½ 2 1 5 0.
5. First compute the new bogey performance as (.70 3 5.81) 1 (.25 3 1.45) 1 (.05 3 .48) 5 4.45.
a. Contribution of asset allocation to performance:
Market
(1) Actual
Weight in Market
(2) Benchmark Weight in
Market
(3) Active
or Excess Weight
(4) Market Return
(%)
(5) 5 (3) 3 (4)
Contribution to Performance (%)
Equity .70 .70 .00 5.81 .00
Fixed-income .07 .25 2.18 1.45 2.26
Cash .23 .05 .18 0.48 .09
Contribution of asset allocation 2.17
b. Contribution of selection to total performance:
Market
(1) Portfolio
Performance (%)
(2) Index
Performance (%)
(3) Excess
Performance (%)
(4)
Portfolio Weight
(5) 5 (3) 3 (4)
Contribution (%)
Equity 7.28 5.00 2.28 .70 1.60
Fixed-income 1.89 1.45 0.44 .07 0.03
Contribution of selection within markets 1.63
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25 2 ALTHOUGH WE IN the United States cus- tomarily use a broad index of U.S. equities as the market-index portfolio, the practice is increasingly inappropriate. U.S. equities rep- resent less than 40% of world equities and a far smaller fraction of total world wealth. In this chapter, we look beyond domestic markets to survey issues of international and extended diversification. In one sense, international investing may be viewed as no more than a straightforward generaliza- tion of our earlier treatment of portfolio selection with a larger menu of assets from which to construct a portfolio. Similar issues of diversification, security analysis, security selection, and asset allocation face the inves- tor. On the other hand, international invest- ments pose some problems not encountered in domestic markets. Among these are the presence of exchange rate risk, restrictions on capital flows across national boundar- ies, an added dimension of political risk and
country-specific regulations, and differing accounting practices in different countries. Therefore, in this chapter we review the major topics covered in the rest of the book, emphasizing their international aspects. We start with the central concept of portfo- lio theory—diversification. We will see that global diversification offers opportunities for improving portfolio risk–return trade- offs. We also will see how exchange rate fluctuations and political risk affect the risk of international investments. We next turn to passive and active investment styles in the international context. We will consider some of the special problems involved in the inter- pretation of passive index portfolios, and we will show how active asset allocation can be generalized to incorporate country and currency choices in addition to traditional domestic asset class choices. Finally, we dem- onstrate performance attribution for inter- national investments.
CHAPTER TWENTY-FIVE
International
Diversification
PA R
T V
II
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C H A P T E R 2 5 International Diversification 883
25.1 Global Markets for Equities
1 FTSE Index Co. [the sponsor of the British FTSE (Financial Times Share Exchange) stock market index] uses
14 specific criteria to divide countries into “developed” and “emerging” lists. Our list of developed countries
includes all 25 countries that appear on FTSE’s list.
You can easily invest today in capital markets of nearly 100 countries and obtain
up-to-date data about your investments in each of them. By 2011, 52 countries had stock
markets with aggregate market capitalization above $1 billion. The data and discussion in
this chapter are based on these countries.
The investments industry commonly distinguishes between “developed” and “emerg-
ing” markets. A typical emerging economy still is undergoing industrialization, is growing
faster than developed economies, and has capital markets that usually entail greater risk.
We use the FTSE 1 criteria, which emphasize capital market conditions, to classify markets
as emerging or developed.
Developed Countries To appreciate the myopia of an exclusive investment focus on U.S. stocks and bonds, consider
the data in Table 25.1 . The U.S. accounts for less than 40% of world stock market capitaliza-
tion. Clearly, active investors can attain better risk–return trade-offs by extending their search
for attractive securities to both developed and emerging markets. Developed countries made
up 68% of world gross domestic product in 2010, and 85% of the world market capitalization.
The first two columns of Table 25.1 show market capitalization in 2000 and 2011. The first
line shows capitalization for all world exchanges, showing total capitalization of corporate equity
in 2011 as $38.2 trillion, of which U.S. stock exchanges made up $13.9 trillion, or 36.4%. The
year-to-year changes in the figures in these columns demonstrate the volatility of these markets.
The next three columns of Table 25.1 compare country equity capitalization as a per-
centage of the world’s in 2000 and 2011, as well as the growth in capitalization over those
12 years. The two crises of the first dozen years of the 21st century, the bursting of the
tech bubble in 2000–2001 and the financial crisis of 2008–2009, hit the developed coun-
tries hardest. Average growth of developed-country equity markets over these years was an
anemic 1.7%, compared with a world average of 2.8% and 16.3% for emerging markets.
The last three columns of Table 25.1 show GDP, per capita GDP, and the equity capi-
talization as a percentage of GDP in 2010. Although per capita GDP in developed coun-
tries is not as variable across countries as total GDP, which is determined in part by total
population, market capitalization as a percentage of GDP is quite variable. This suggests
widespread differences in economic structure even across developed countries.
Emerging Markets For a passive strategy, one could argue that a portfolio of equities of just the six countries
with the largest capitalization would make up 64% (in 2011) of the world portfolio and
may be sufficiently diversified. However, this argument will not hold for active portfo-
lios that seek to tilt investments toward promising assets. Active portfolios will naturally
include many stocks or indexes of emerging markets.
Table 25.2 makes the point. Surely, active portfolio managers must prudently scour
stocks in markets such as China and Russia with annual growth rates so far in the 21st cen-
tury in excess of 33% (1⁄3 per year!). Table 25.2 shows data from the 20 largest emerging
markets. But managers also would not want to have missed other markets that exhibited
marked, if not quite so dramatic, growth over the same years.
These 20 emerging markets make up 24% of the world GDP and, together with the 32
developed markets in Table 25.1 , make up 92% of the world GDP. Per capita GDP in these
emerging markets was quite variable, ranging from $1,019 (Pakistan) to $41,122 (Singapore).
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8 8 4
Market Capitalization
Billions of
U.S. Dollars Percent of
World
Annual Growth
(%) GDP
GDP per
Capita
Market Capitalization as % of GDP
2000 2011 2000 2011 2000–2011 2010 2010 2010
World 27,473 38,200 100% 100% 2.8 63,124 9,228 68 U.S. 12,900 13,917 47.0 36.4 0.6 14,587 47,199 98 Japan 3,140 3,289 11.4 8.6 0.4 5,459 42,831 69 U.K. 2,566 2,794 9.3 7.3 0.7 2,249 36,144 133 Canada 615 1,581 2.2 4.1 8.2 1,577 46,236 114 France 1,278 1,455 4.7 3.8 1.1 2,560 39,460 70 Hong Kong 564 1,369 2.1 3.6 7.7 225 31,758 701 Germany 1,061 1,177 3.9 3.1 0.9 3,281 40,152 43 Switzerland 783 1,062 2.9 2.8 2.6 528 67,464 224 Australia 349 1,039 1.3 2.7 9.5 925 42,131 132 Korea 123 763 0.4 2.0 16.4 1,015 20,757 86 Spain 331 546 1.2 1.4 4.2 1,407 30,542 44 Italy 716 460 2.6 1.2 2 3.6 2,051 33,917 28 Sweden 274 440 1.0 1.2 4.0 459 48,936 118 Netherlands 680 376 2.5 1.0 2 4.8 779 46,915 60 Mexico 112 372 0.4 1.0 10.5 1,035 9,123 39 Norway 52 238 0.2 0.6 13.5 413 84,538 61 Chile 44 229 0.2 0.6 14.7 213 12,431 136 Belgium 159 216 0.6 0.6 2.6 469 43,144 54 Denmark 99 176 0.4 0.5 4.9 310 55,891 67 Turkey 50 164 0.2 0.4 10.4 734 10,094 34 Finland 280 139 1.0 0.4 2 5.7 239 44,512 86 Israel 46 119 0.2 0.3 8.3 217 28,504 80 Poland 27 112 0.1 0.3 12.5 469 12,293 34 Austria 28 85 0.1 0.2 9.8 379 45,209 33 Ireland 82 65 0.3 0.2 2 1.9 211 47,170 30 Portugal 64 59 0.2 0.2 2 0.6 229 21,505 35 Czech Rep. 12 39 0.0 0.1 10.4 192 18,245 23 New Zealand 20 35 0.1 0.1 5.0 136 31,067 35 Luxemburg 28 34 0.1 0.1 1.7 53 105,438 79 Greece 72 29 0.3 0.1 2 7.1 301 26,600 21 Hungary 12 19 0.0 0.0 4.1 129 12,852 22 Slovenia 2 6 0.0 0.0 11.2 47 22,851 18
Table 25.1
Market capitalization of stock exchanges in developed countries
Source: Market capitalization: Datastream, online.thomsonreuters.com/datastream ; GDP and GDP per capita: The World Bank, data.worldbank.org .
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Market Capitalization
Billions of
U.S. Dollars Percent of
World Growth
(%) GDP
GDP per
Capita
Market Capitalization as % of GDP
2000 2011 2000 2011 2000–2011 2010 2010 2010
Brazil 180 1,056 0.7 2.8 15.9 2,088 10,710 66 India 107 868 0.4 2.3 19.0 1,727 1,475 69 Russia 19 694 0.1 1.8 34.9 1,480 10,440 58 China 13 499 0.0 1.3 35.2 5,927 4,428 11 Taiwan 177 455 0.6 1.2 8.2 430 18,300 134 Singapore 136 428 0.5 1.1 10.1 209 41,122 241 South Africa 104 405 0.4 1.1 12.0 364 7,275 134 Malaysia 83 330 0.3 0.9 12.1 238 8,373 135 Indonesia 21 301 0.1 0.8 24.8 707 2,946 41 Thailand 23 219 0.1 0.6 20.7 319 4,608 70 Colombia 4 191 0.0 0.5 37.3 288 6,225 70 Philippines 20 141 0.1 0.4 17.6 200 2,140 67 Peru 5 77 0.0 0.2 25.9 157 5,401 64 Argentina 24 36 0.1 0.1 3.6 369 9,124 15 Pakistan 5 26 0.0 0.1 15.2 177 1,019 17 Sri Lanka 1 14 0.0 0.0 27.5 50 2,375 31 Romania 0 14 0.0 0.0 36.9 162 7,538 9 Venezuela 6 6 0.0 0.0 2 0.2 392 13,590 3 Cyprus 9 3 0.0 0.0 2 9.7 23 28,779 28 Bulgaria 0 2 0.0 0.0 29.6 48 6,325 4
Table 25.2
Market capitalization of stock exchanges in emerging markets
Source: Market capitalization: Datastream, online.thomsonreuters.com/datastream ; GDP and GDP per capita: The World Bank, data.worldbank.org .
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Market capitalization as a percent of GDP of the BRICS countries (Brazil, Russia, India,
China, and South Africa) is still below 70% (only 11% in China!), suggesting that these
emerging markets are expected to show significant growth over the coming years, even with-
out spectacular growth in GDP.
The growth of capitalization in emerging markets over this period was much more vola-
tile than growth in developed countries, implying that both risk and rewards in this seg-
ment of the globe may be substantial.
Market Capitalization and GDP A contemporary view of economic development (rigorously stated in de Soto, 2000) holds
that an important requirement for economic advancement is a developed code of business
laws, institutions, and regulations that allows citizens to legally own, capitalize, and trade
capital assets. As a corollary, we expect that development of equity markets will serve as
a catalyst for enrichment of the population, that is, that countries with larger relative capi-
talization of equities will tend to be richer. For rich countries, with already-large equity
markets, this relationship will be weaker.
Figure 25.1 depicts the relationship 2 between per capita GDP and market capitalization
(where both variables have been transformed to log 10 scale). Figure 25.1 , panel A shows a scat-
ter diagram and regression line for 2000, while the situation in 2011 is shown in Figure 25.1 ,
panel B. While developed markets are mostly above the line and emerging markets mostly
below it, the latter dramatically moved up in relative market capitalization over these years.
This move was sufficient to greatly moderate the slope of the line. One can also easily see the
upward shift of the whole world on the vertical axis that measures per capita GDP.
The regression slope coefficient measures the average percent change in per capita
income when market capitalization increases by 1%. In 2000, this value was .64, but it fell
to .35 in 2011. The scatter around the regression line has also visibly grown, as reflected in
an R-square of .52 in 2000 but only .10 in 2011.
Home-Country Bias Home-country bias refers to the common tendency for investors to underweight foreign
equities in their portfolio of risky assets. If investors allocated their stock investments
across countries in proportion to outstanding equity, U.S. investors in 2011 would have
placed only 36.4% of their equity in U.S. firms ( Table 25.1 ) with the remaining 63.6% held
in foreign markets. Non-U.S. investors would have held a greater share of U.S. equities
than domestic investors. But in fact, most investors show a pronounced bias toward hold-
ing stock in their home countries.
U.S. investors’ holdings of foreign stocks and long-term bonds and foreigners’ holdings
of U.S stocks and long-term bonds in 2001 and 2011 were:
Year U.S. Investor
Holdings Abroad a Foreign Investor Holdings in U.S. b
2001 2,170 3,932 2011 6,481 11,870
a Billions of dollars. b About 2/3 of these holding are in equities. Source: Report on U.S. portfolio holdings of foreign securities as of 12/31/2011. Department of the Treasury, October 2012.
Home-country bias remains in force but is far less pronounced today than it was 10 years ago.
2 This simple single-variable regression is put forward not as a causal model but simply as a way to describe the
relation between per capita GDP and the size of markets.
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C H A P T E R 2 5 International Diversification 887
Figure 25.1 Per capita GDP and market capitalization as percent of GDP. Panel A: Log scale, 2000 data; Panel B: Log scale, 2010 data.
U.S.Japan
U.K.
Canada
France
Germany
Brazil
India
Russia
China
100
1,000
10,000
100,000
A
1 10 100 1000
P e r
C a p
it a G
D P (
$ )
Market Capitalization as % of GDP
Developed Countries Emerging Markets Regression Line
Developed Countries Emerging Markets Regression Line
B
U.S. Japan Canada
France Germany
Brazil
India
Russia
Taiwan
China
100
1,000
10,000
100,000
1 10 100 1000
P e r
C a p
it a G
D P (
$ )
Market Capitalization as % of GDP
25.2 Risk Factors in International Investing
Opportunities in international investments do not come free of risk or of the cost of special-
ized analysis. Two risk factors that are unique to international investments are exchange
rate risk and political risk, discussed in the next two sections.
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Exchange Rate Risk It is best to begin with a simple example.
Using the data in Example 25.1, calculate the rate of return in dollars to a U.S. investor hold- ing the British bill if the year-end exchange rate is: ( a ) E 1 5 $2.00/£; ( b ) E 1 5 $2.20/£.
CONCEPT CHECK 25.1
We can generalize from Example 25.1 . The $20,000 is exchanged for $20,000/ E 0 pounds, where E 0 denotes the original exchange rate ($2/£). The U.K. investment grows to (20,000/ E 0 )[1 1 r f (UK)] British pounds, where r f (UK) is the risk-free rate in the United Kingdom. The pound proceeds ultimately are converted back to dollars at the subsequent
exchange rate E 1 , for total dollar proceeds of 20,000( E 1 / E 0 )[1 1 r f (UK)]. The dollar- denominated return on the investment in British bills, therefore, is
1 1 r (US) 5 31 1 rf (UK)4E1/E0 (25.1) We see in Equation 25.1 that the dollar-denominated return for a U.S. investor equals
the pound-denominated return times the exchange rate “return.” For a U.S. investor, the
investment in British bills is a combination of a safe investment in the United Kingdom
and a risky investment in the performance of the pound relative to the dollar. Here, the
pound fared poorly, falling from a value of $2 to only $1.80. The loss on the pound more
than offset the earnings on the British bill.
Figure 25.2 illustrates this point. It presents rates of
returns on stock market indexes in several countries for
2010. The colored bars depict returns in local currencies,
while the dark bars depict returns in dollars, adjusted for
exchange rate movements. It’s clear that exchange rate
fluctuations over this period had large effects on dollar-
denominated returns in several countries. Pure exchange rate risk is the risk borne by invest-
ments in foreign safe assets. The investor in U.K. bills of
Example 25.1 bears the risk of the U.K./U.S. exchange rate only. We can assess the mag-
nitude of exchange rate risk by examining historical rates of change in various exchange
rates and their correlations.
Table 25.3 , panel A shows historical exchange rate risk measured by the standard
deviation of monthly percent changes in the exchange rates of major currencies against
the U.S. dollar over the period 2001–2011. The data show that currency risk is quite
high. The annualized standard deviation of the percent changes in the exchange rate
ranged from 9.13% (Japanese yen) to 13.87% (Australian dollar). The standard deviation
Consider an investment in risk-free British government bills paying 10% annual interest in British pounds. While these U.K. bills would be the risk-free asset to a British inves- tor, this is not the case for a U.S. investor. Suppose, for example, the current exchange rate is $2 per pound, and the U.S. investor starts with $20,000. That amount can be exchanged for £10,000 and invested at a riskless 10% rate in the United Kingdom to provide £11,000 in 1 year.
What happens if the dollar–pound exchange rate varies over the year? Say that dur- ing the year, the pound depreciates relative to the dollar, so that by year-end only $1.80 is required to purchase £1. The £11,000 can be exchanged at the year-end exchange rate for only $19,800 ( 5 £11,000 3 $1.80/£), resulting in a loss of $200 relative to the initial $20,000 investment. Despite the positive 10% pound-denominated return, the dollar-denominated return is negative 1%.
Example 25.1 Exchange Rate Risk
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0 5% 10% 15% 20% 25% 30%
Japan
Australia
Norway
Germany
U.K.
Pakistan
India
Russia
Brazil
China
Return (in Local Currencies)Return (in U.S. $)
15.59 0.71
14.73 0.66
11.82 12.51
9.32 16.91
8.80 12.22
27.06 29.05
20.95 16.22
19.40 20.30
6.81 1.72
4.83 5.10
C H A P T E R 2 5 International Diversification 889
Figure 25.2 Stock market returns in U.S. dollars and local currencies for 2010 Source: Datastream, online.thomsonreuters.com/datastream .
of returns on U.S. large stocks for the same period was 16%. Hence, exchange rate
volatility was roughly 70% that of the volatility on stocks. Clearly, an active investor
who believes that a foreign stock is underpriced but has no information about any mis-
pricing of its currency should consider hedging the currency risk exposure when tilt-
ing the portfolio toward the stock. Exchange rate risk of the major currencies has been
relatively high so far in this century. For example, a study by Solnik (1999) for the
period 1971–1998 finds lower standard deviations, ranging from 4.8% (Canadian dollar)
to 12% (Japanese yen).
In the context of international portfolios, exchange rate risk may be mostly diversifi-
able. This is evident from the low correlation coefficients in Table 25.3 , panel B. (There
are notable exceptions in the table, though, and this observation will be reinforced when
we compare the risk of hedged and unhedged country portfolios in a later section.) Thus,
passive investors with well-diversified international portfolios need not be concerned with
hedging exposure to foreign currencies.
The effect of exchange rate fluctuations also shows up in Table 25.3 , panel C, which
presents the returns on money market investments in different countries. While these invest-
ments are virtually risk-free in local currency, they are risky in dollar terms because of
exchange rate risk. International investment flows by currency speculators should roughly
equalize the expected dollar returns in various currencies, adjusted for risk. Moreover,
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890 P A R T V I I Applied Portfolio Management
A. Standard deviation (annualized %)
Euro (€) U.K. (£) Switzerland
(SF) Japan (¥) Australia
(A$) Canada
(C$)
11.04 9.32 11.94 9.13 13.87 10.04
B. Correlation matrix
Euro (€) U.K. (£) Switzerland
(SF) Japan (¥) Australia
(A$) Canada
(C$)
U.K. (£) 0.63 1 Switzerland (SF) 0.83 0.51 1 Japan (¥) 0.27 0.08 0.42 1 Australia (A$) 0.75 0.6 0.61 0.05 1 Canada (C$) 0.51 0.49 0.37 20.02 0.72 1
C. Average annual returns from rolling over one-month LIBOR rates (%)
Country Currency
Return in Local
Currency
Expected Gain from Currency
Actual Gain from Currency
Actual Return in
U.S. Dollars
Surprise Component of Return
SD of Annual Return
U.S. $ 2.18 2.18 Euro € 2.38 20.20 4.38 6.77 4.58 11.04 U.K. £ 3.51 21.32 1.09 4.60 2.41 9.32 Switzerland SF 0.90 1.28 6.46 7.36 5.17 11.94 Japan ¥ 0.24 1.94 5.75 5.99 3.81 9.13 Australia A$ 5.25 23.07 7.94 13.19 11.01 13.87 Canada C$ 2.50 20.31 5.01 7.51 5.32 10.04
Table 25.3
Rates of change in major currencies against the U.S. dollar, 2002–2011 (annualized monthly rate)
Source: Exchange rates: Datastream, online.thomsonreuters.com/datastream; LIBOR rates: www.economagic.com.
exchange rate risk is largely diversifiable, as Table 25.3 , panel B shows, and hence we
would expect similar dollar returns from cash investments in major currencies.
We can illustrate exchange rate risk using a yen-denominated investment during this
period. The low yen-denominated LIBOR rate, .24%, compared to the U.S.-dollar LIBOR
rate, 2.18%, suggests that investors expected the yen to appreciate against the dollar by
around 1.94%, the interest rate differential across the two countries. But those expectations
were not realized; in fact, the yen actually appreciated against the dollar at an annual rate
of 5.75%, leading to an annual dollar-denominated return on a yen investment of 5.99%
(the .24% yen interest rate together with the realized exchange rate appreciation of 5.75%).
However, such deviations between prior expectations and actual returns of this magnitude
are not shocking. The “surprise” return in a yen LIBOR investment (converted into dollars)
was the difference between the actual return in dollars, 5.99%, and the dollar-denominated
LIBOR rate of 2.18%, amounting to 3.81%. This surprise is actually considerably less than
the yen standard deviation of 9.13%. In fact, none of the six surprises exceeded the stan-
dard deviation, which is actually the surprising event here.
Investors can hedge exchange rate risk using a forward or futures contract in foreign
exchange. Recall that such contracts entail delivery or acceptance of one currency for
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C H A P T E R 2 5 International Diversification 891
How many pounds would the investor in Example 25.2 need to sell forward to hedge exchange rate risk if: ( a ) r (UK) 5 20%; and ( b ) r (UK) 5 30%?
CONCEPT CHECK 25.2
another at a stipulated exchange rate. To illustrate, recall Example 25.1. In this case, to
hedge her exposure to the British pound, the U.S. investor would agree to deliver pounds
for dollars at a fixed exchange rate, thereby eliminating the future risk involved with con-
version of the pound investment back into dollars.
You may recall that the hedge underlying Example 25.2 is the same type of hedging
strategy at the heart of the spot-futures parity relationship first discussed in Chapter 22.
In both instances, futures or forward markets are used to eliminate the risk of holding
another asset. The U.S. investor can lock in a riskless dollar-denominated return either by
investing in United Kingdom bills and hedging exchange rate risk or by investing in risk-
less U.S. assets. Because investments in two riskless strategies must provide equal returns,
we conclude that [1 1 r f (UK)] F 0 / E 0 5 1 1 r f (US), which can be rearranged to
F0 E0 5
1 1 rf (US)
1 1 rf (UK) (25.2)
This relationship is called the interest rate parity relationship or covered interest arbitrage relationship , which we first encountered in Chapter 23.
Unfortunately, such perfect exchange rate hedging usually is not so easy. In our exam-
ple, we knew exactly how many pounds to sell in the forward or futures market because the
pound-denominated return in the United Kingdom was riskless. If the U.K. investment had
not been in bills, but instead had been in risky U.K. equity, we would have known neither
the ultimate value in pounds of our U.K. investment nor how many pounds to sell forward.
The hedging opportunity offered by foreign exchange forward contracts would thus be
imperfect.
To summarize, the generalization of Equation 25.1 for unhedged investments is that
1 1 r(US) 5 31 1 r(foreign)4E1/E0 (25.3) where r (foreign) is the possibly risky return earned in the currency of the foreign investment. You can set up a per-
fect hedge only in the special case that r (foreign) is itself a known number. In that case, you know you must sell in
the forward or futures market an amount of foreign cur-
rency equal to [1 1 r (foreign)] for each unit of that cur- rency you purchase today.
Political Risk In principle, security analysis at the macroeconomic, industry, and firm-specific level is
similar in all countries. Such analysis aims to provide estimates of expected returns and
risk of individual assets and portfolios. However, to achieve the same quality of informa-
tion about assets in a foreign country is by nature more difficult and hence more expensive.
Moreover, the risk of coming by false or misleading information is greater.
If the forward exchange rate in Example 25.1 had been F 0 5 $1.93/£ when the invest- ment was made, the U.S. investor could have assured a riskless dollar-denominated return by arranging to deliver the £11,000 at the forward exchange rate of $1.93/£. In this case, the riskless U.S. return would then have been 6.15%:
[1 1 rf (UK)]F0/E0 5 (1.10)1.93/2.00 5 1.0615
Example 25.2 Hedging Exchange Rate Risk
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892 P A R T V I I Applied Portfolio Management
Rank in January 2011 Country
Composite Risk Rating
January 2011
Composite Risk Rating
February 2012
January 2011 versus
February 2010 Rank in
February 2010
Very low risk 1 Norway 90.5 90.00 0.50 1
11 Germany 83.5 83.50 0.00 5 13 Canada 82.8 82.75 0.00 6 16 Qatar 82.0 81.25 0.75 11 19 Japan 81.0 80.00 1.00 17
Low risk 31 United Kingdom 77.3 73.75 3.50 39 32 United States 77.0 77.25 2 0.25 26 39 China, People’s Rep. 75.0 76.25 2 1.25 30 44 Brazil 74.5 72.75 1.75 46 68 Spain 70.0 71.00 2 1.00 58
Moderate risk 78 Indonesia 68.5 67.25 1.25 81 86 India 67.3 70.50 2 3.25 62
104 Egypt 64.5 66.50 2 2.00 84 111 Turkey 63.3 63.50 2 0.25 100
High risk 124 Venezuela 59.5 53.75 5.75 133 127 Iraq 58.5 59.25 2 0.75 119 129 Pakistan 57.3 57.00 0.25 125
Very high risk 138 Haiti 48.5 49.75 2 1.25 137 140 Somalia 41.5 36.75 4.75 140
Table 25.4
Composite risk ratings for January 2011 versus February 2010
Source: International Country Risk Guide, January 2011, Table 1, The PRS Group, Inc. Used with permission.
3 You can find more information on the Web site: www.prsgroup.com . We are grateful to the PRS Group for
supplying us data and guidance.
Consider two investors: an American wishing to invest in Indonesian stocks and an
Indonesian wishing to invest in U.S. stocks. While each would have to consider macro-
economic analysis of the foreign country, the task would be much more difficult for the
American investor. The reason is not that investment in Indonesia is necessarily riskier
than investment in the U.S. You can easily find many U.S. stocks that are, in the final
analysis, riskier than a number of Indonesian stocks. The difference lies in the fact that
U.S. financial markets are more transparent than those of Indonesia.
In the past, when international investing was novel, the added risk was referred to as
political risk and its assessment was an art. As cross-border investment has increased and more resources have been utilized, the quality of related analysis has improved. A lead-
ing organization in the field (which is quite competitive) is the PRS Group (Political Risk
Services) and the presentation here follows the PRS methodology. 3
PRS’s country risk analysis results in a country composite risk rating on a scale of 0
(most risky) to 100 (least risky). Countries are then ranked by the composite risk measure
and divided into five categories: very low risk (100–80), low risk (79.9–70), moderate risk
(69.9–60), high risk (59.9–50), and very high risk (less than 50). To illustrate, Table 25.4
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C H A P T E R 2 5 International Diversification 893
Political Risk Variables Financial Risk Variables Economic Risk Variables
Government stability Foreign debt (% of GDP) GDP per capita Socioeconomic conditions Foreign debt service (% of GDP) Real annual GDP growth Investment profile Current account (% of exports) Annual inflation rate Internal conflicts Net liquidity in months of imports Budget balance (% of GDP) External conflicts Exchange rate stability Current account balance (% GDP) Corruption Military in politics Religious tensions Law and order Ethnic tensions Democratic accountability Bureaucracy quality
Table 25.5
Variables used in PRS’s political risk score
shows the placement of countries in the January 2011 issue of the PRS International Country Risk Guide. It is not surprising to find Norway at the top of the very-low-risk list, and small emerging markets at the bottom, with Somalia (ranked 140) closing the list.
What may be surprising is the fairly mediocre ranking of the U.S. (ranked 32), comparable
to Libya (20) and Bahrain (29), all three in the low-risk category.
The composite risk rating is a weighted average of three measures: political risk, finan-
cial risk, and economic risk. Political risk is measured on a scale of 100–0, while financial
risk and economic risk are measured on a scale of 50–0. The three measures are added and
divided by 2 to obtain the composite rating. The variables used by PRS to determine the
composite risk rating from the three measures are shown in Table 25.5 .
Table 25.6 shows the three risk measures for seven of the countries in Table 25.4 , in
order of the January 2011 ranking of the composite risk ratings. The table shows that
by political risk, the United States ranked third among these seven countries. But in the
financial risk measure, the U.S. ranked sixth among the seven. The surprisingly poor
Composite Ratings Current Ratings
Country Year Ago
February 2010 Current
January 2011 Political Risk January 2011
Financial Risk January 2011
Economic Risk January 2011
Norway 90.00 90.50 88.5 46.5 46.0 Canada 82.75 82.75 86.5 40.0 39.0 Japan 80.00 81.00 78.5 44.0 39.5 United States 77.25 77.00 81.5 37.0 35.5 China, People’s Rep. 76.25 75.00 62.5 48.0 39.5 India 70.50 67.25 58.5 43.5 32.5 Turkey 63.50 63.25 57.0 34.5 35.0
Table 25.6
Current risk ratings and composite risk forecasts
Source: International Country Risk Guide, January 2011, Table 2B, The PRS Group, Inc. Used with permission.
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894 P A R T V I I Applied Portfolio Management
A. Composite risk forecasts
One Year Ahead Five Years Ahead
Country Current Rating January 2011
Worst Case
Best Case
Risk Stability
Worst Case
Best Case
Risk Stability
Norway 90.5 88.3 93.3 5.0 83.3 92.8 9.5 Canada 82.8 78.3 84.3 6.0 75.3 86.5 11.3 Japan 81.0 77.0 84.3 7.3 72.5 87.5 15.0 United States 77.0 73.3 80.3 7.0 69.5 83.0 13.5 China, People’s Rep. 75.0 70.8 79.0 8.3 61.3 82.0 20.8 India 67.3 64.0 72.3 8.3 57.5 77.0 19.5 Turkey 63.3 57.8 67.5 9.8 53.8 71.5 17.8
B. Political risk forecasts
One Year Ahead Five Years Ahead
Country Current Rating January 2011
Worst Case
Best Case
Risk Stability
Worst Case
Best Case
Risk Stability
Norway 88.5 88.0 92.0 4.0 86.0 89.5 3.5 Canada 86.5 83.0 88.5 5.5 81.5 89.5 8.0 Japan 78.5 75.5 84.0 8.5 72.0 88.0 16.0 United States 81.5 77.5 85.5 8.0 76.0 87.0 11.0 China, People’s Rep. 62.5 58.5 68.5 10.0 55.0 73.0 18.0 India 58.5 55.0 64.0 9.0 53.5 71.0 17.5 Turkey 57.0 52.5 63.5 11.0 51.5 69.0 17.5
Table 25.7
Composite and political risk forecasts
Source: A: International Country Risk Guide, January 2011, Table 2C; B: International Country Risk Guide, January 2011, Table 3C. The PRS Group, Inc. Used with permission.
performance of the U.S. in this dimension was probably due to its exceedingly large gov-
ernment and balance-of-trade deficits, which put considerable pressure on its exchange
rate. Exchange rate stability, foreign trade imbalance, and foreign indebtedness all enter
PRS’s computation of financial risk. The financial crisis that began in August of 2008 was
a striking vindication of PRS’s judgment of assigning relatively low financial scores to the
U.S. and other major markets.
Country risk is captured in greater depth by scenario analysis for the composite mea-
sure and each of its components. Table 25.7 (panels A and B) shows 1- and 5-year worst-
case and best-case scenarios for the composite ratings and for the political risk measure.
Risk stability is based on the difference in the rating between the best- and worst-case
scenarios and is quite large in most cases. The worst-case scenario can move a country to a
higher risk category. For example, Table 25.7 , panel B, shows that in the worst-case 5-year
scenario, China and Turkey were particularly vulnerable to deterioration in the political
environment.
Finally, Table 25.8 shows ratings of political risk by each of its 12 components.
Corruption (variable F) in Japan is rated better than in the U.S. In democratic accountability
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C H A P T E R 2 5 International Diversification 895
(variable K), China ranked worst, and the United States, Canada, and India best, while
China ranked best in government stability (variable A).
Each monthly issue of the International Country Risk Guide of the PRS Group includes great detail and holds some 250 pages. Other organizations compete in supplying such
evaluations. The result is that today’s investor can become well equipped to properly assess
the risk involved in international investing.
This table lists the total points for each of the following political risk components out of the maximum points indi- cated. The final columns in the table show the overall political risk rating (the sum of the points awarded to each component) and the change from 2010.
A Government stability 12 G Military in politics 6 B Socioeconomic conditions 12 H Religious tensions 6 C Investment profile 12 I Law and order 6 D Internal conflict 12 J Ethnic tensions 6 E External conflict 12 K Democratic accountability 6 F Corruption 6 L Bureaucracy quality 4
Country A B C D E F G H I J K L
Risk Rating
January 2011
Change from
December 2010
Canada 8.5 9.0 11.5 11.0 11.0 5.0 6.0 6.0 5.5 3.5 5.5 4.0 86.5 0.5 China, People’s Rep. 9.0 8.0 6.5 9.0 9.0 2.0 3.0 5.0 4.0 3.5 1.5 2.0 62.5 0.0 India 6.0 4.5 8.5 6.0 9.5 2.0 4.0 2.5 4.0 2.5 6.0 3.0 58.5 21.5 Japan 5.0 8.5 11.5 10.0 9.0 4.5 5.0 5.5 5.0 5.5 5.0 4.0 78.5 20.5 Norway 7.5 10.5 11.5 11.0 11.0 5.0 6.0 5.5 6.0 4.5 6.0 4.0 88.5 0.0 Turkey 8.5 5.5 7.5 7.5 7.5 2.5 2.0 4.0 3.5 2.0 4.5 2.0 57.0 0.0 United States 8.0 8.5 12.0 10.0 9.5 4.0 4.0 5.5 5.0 5.0 6.0 4.0 81.5 0.5
Table 25.8
Political risk points by component, January 2011
Source: International Country Risk Guide, January 2011, Table 3B, The PRS Group, Inc. Used with permission.
25.3 International Investing: Risk, Return, and Benefits from Diversification
U.S. investors have several avenues through which they can invest internationally. The
most obvious method, which is available in practice primarily to larger institutional inves-
tors, is to purchase securities directly in the capital markets of other countries. However,
even small investors now can take advantage of several investment vehicles with an inter-
national focus.
Shares of several foreign firms are traded in U.S. markets either directly or in the form
of American depository receipts, or ADRs. A U.S. financial institution such as a bank will
purchase shares of a foreign firm in that firm’s country, then issue claims to those shares in
the United States. Each ADR is then a claim on a given number of the shares of stock held
by the bank. Some stocks trade in the U.S. both directly and as ADRs.
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896 P A R T V I I Applied Portfolio Management
There is also a wide array of mutual funds with an international focus. In addition to
single-country funds, there are several open-end mutual funds with an international focus.
For example, Fidelity offers funds with investments concentrated overseas, generally in
Europe, in the Pacific Basin, and in developing economies in an emerging opportunities
fund. Vanguard, consistent with its indexing philosophy, offers separate index funds for
Europe, the Pacific Basin, and emerging markets. Finally, as noted in Chapter 4, there
are many exchange-traded funds known as iShares or WEBS (World Equity Benchmark
Shares) that are country-specific index products.
U.S. investors also can trade derivative securities based on prices in foreign security
markets. For example, they can trade options and futures on the Nikkei stock index of 225
stocks traded on the Tokyo stock exchange, or on FTSE (Financial Times Share Exchange)
indexes of U.K. and European stocks.
Risk and Return: Summary Statistics Illustrations for most of our discussions in the remaining part of this chapter derive
from a database of country market-index returns. We use 10 years of monthly returns
over 2002–2011 for 48 non-U.S. country market indexes as well as the U.S. S&P 500.
This decade stretches from the beginning of the recovery from the bursting of the tech
bubble in 2001, through the low–interest rate boom period that followed and the ensu-
ing financial crisis of 2008, and, finally, to the beginning of the slow recovery from
that crisis.
Analysis of risky assets typically focuses on excess returns over the risk-free rate. This alone adds a perplexing aspect to international investing, since the appropriate risk-free
rate varies around the globe. Rates of return on identical indexes (as well as individual
assets) will generate different excess returns when safe bonds are denominated in different
currencies. Although our perspective is U.S.-based, our methodology would serve inves-
tors in any country, yet the numbers may differ when applied to risk-free rates denomi-
nated in other currencies.
The tumultuous period we analyze resulted in unexpected low average excess returns,
primarily in developed markets, while most emerging markets continued unabated growth.
This fact alone conveys an important lesson. It provides an extreme example of the general
observation that realized returns are very noisy reflections of investor expectations and
may not provide accurate forecasts of future returns. Past returns do, however, provide an
indication of risk, at least for the near future. While the near-efficient market hypothesis
applies to expected returns (to wit: future returns cannot be forecast from past returns), it does not apply to forecasting risk. Thus our exercise will allow us to demonstrate the distinction between what you can and cannot learn from historical returns that evidently
departed from prior expectations.
While active-strategy managers engage in both individual-market asset allocation and
security selection, we will restrict our international diversification to country market-index
portfolios, keeping us on the side of an enhanced passive strategy. Nevertheless, our analy-
sis illustrates the essential features of extended active management as well.
We begin with an investigation of the characteristics of individual markets and then
proceed to analyze the benefits of diversification, using portfolios constructed from
these individual markets. The market capitalization of individual-country indexes can be
found in Tables 25.1 and 25.2 , and the aggregated results for the portfolios are shown in
Table 25.9A . This table also displays the performance of two types of portfolios: portfo-
lios aggregated from country indexes and regional-index portfolios. The performances of
individual-country-index portfolios are shown in Table 25.9B .
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897
For the aggregated country-index portfolios, we examine a strategy that constructs
value-weighted portfolios of developed and emerging markets based on market capital-
izations at the beginning of 2002. These portfolios are rebalanced after 5 years, in 2007,
based on capitalizations at the end of 2006, and held for another 5 years. (Dividends
are reinvested throughout the 10-year experiment.) Such a strategy is feasible to a large
degree, since many (although, admittedly, not all) country-index portfolios are investable
as index funds or ETFs. Because not all country indexes are investable, this hypothetical
strategy perhaps generates a bit more efficient diversification than is actually possible.
On the other hand, if actually held, these value-weighted portfolios would automatically
be rebalanced to value weights continually. In contrast, we rebalance only once after five
years, which slightly attenuates diversification benefits. On balance, then, we expect these
hypothesized portfolios to perform about as well as a feasible country-based, passive
international strategy.
Index Funds Called WEBS Reduce the Cost of
Investing Abroad
With foreign markets generally stronger this year, a new way to invest abroad has appeared at a good time. WEBS, an acronym for World Equity Benchmark Shares, represents an investment in a portfolio of publicly traded foreign stocks in a selected country. Each WEBS Index Series seeks to generate investment results that generally correspond to the price and yield performance of a specific Morgan Stanley Capital International (MSCI) index.
You sell these shares rather than redeeming them, but there the similarity to closed-end country funds ends. WEBS are equity securities, not mutual funds. WEBS shares trade continuously like any other publicly traded U.S. stock. In contrast, mutual fund shares do not trade in the secondary market, and are normally bought and sold from the issu- ing mutual fund at prices determined only at the end of the day. The new funds create and redeem shares in large blocks as needed, thus preventing the big premiums or discounts to net asset value typical of closed-end country funds. As index portfolios, WEBS are passively managed, so their expenses run much lower than for current open- or closed-end country funds.
WEBS shares offer U.S. investors portfolio exposure to country-specific equity markets, in a single, listed security you can easily buy, sell, or short. Unlike American Deposi- tory Receipts (ADRs) that give you an investment in just one company, WEBS shares enable you to gain exposure to a broad portfolio of a desired foreign country’s stocks. You gain broad exposure in the country or countries of your choice without the complications usually associated with buying, owning, or monitoring direct investments in for- eign countries. You also have the conveniences of trading on a U.S. exchange and dealing in U.S. dollars.
Some investors may prefer the active management, diversity, and flexibility of open-end international equity
index funds as a way to limit currency and political risks of investing in foreign markets. As conventional open-end funds, however, the international funds are sometimes forced by net redemptions to sell stocks at inopportune times, which can be a particular problem in foreign mar- kets with highly volatile stocks.
You pay brokerage commissions on the purchase and sale of WEBS, but since their portfolios are passively man- aged, their management and administrative fees are relatively low and they eliminate most of the transaction charges typical of managed funds.
FOREIGN INDEX BASKETS
WEBS Ticker Symbol WEBS
Ticker Symbol
Australia EWA Malaysia EWM Austria EWO Mexico EWW Belgium EWK Netherlands EWN Canada EWC Singapore EWS France EWQ Spain EWP Germany EWG Sweden EWD Hong Kong EWH Switzerland EWL Italy EWI U.K. EWU Japan EWJ
Sources: Modified from The Outlook, May 22, 1996, published by Standard and Poor’s; and Amex website, www.amex.com, February 2000.
Reprinted by permission of Standard & Poor’s, a division of The McGraw-Hill Companies, Inc.
W O
R D
S FR O
M T
H E ST
R E E T
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898 P A R T V I I Applied Portfolio Management
Market Value ($ billion) Performance 2002–2011
($ - denom. returns)
Country-Index Portfolios 2001 2006 2011 Average SD Sharpe Ratio
U.S. 11,850 15,520 13,917 0.21 4.63 0.0444 Developed Ex U.S 10,756 22,065 18,487 0.48 5.48 0.0869 Emerging 1,230 5,319 5,765 1.35 6.86 0.1971 Developed 1 EM 11,987 27,384 24,251 0.61 5.63 0.1077 U.S. 1 Developed 22,606 37,585 32,403 0.34 4.99 0.0680 U.S. 1 EM 13,080 20,839 19,681 0.36 4.94 0.0736 World 23,836 42,904 38,168 0.50 5.32 0.0948
Regional-Index Portfolios
MSCI EAFE (Europe 1 Australia 1 Far East) 5 Developed 0.42 5.41 0.0771 MSCI emerging markets 1.21 7.00 0.1734 MSCI World excluding U.S. 0.46 5.41 0.0841 World 0.31 4.90 0.0637
Table 25.9A
Market value and performance of country portfolio combinations compared with regional-index portfolios
Source: Datastream.
Table 25.9B
Performance of individual-country-index portfolios, monthly excess returns in U.S. dollars over 2002–2011
Regression against U.S.
Individual-Country- Index Portfolios Average SD
Sharpe Ratio Corr. w/U.S. Beta Alpha Residual SD Info Ratio
Developed U.S. 0.21 4.63 0.0444 1 1 0 0 0 Australia 1.17 6.75 0.1728 0.82 1.20 0.92 3.84 0.24 Austria 0.78 8.87 0.0876 0.74 1.42 0.49 6.00 0.08 Belgium 0.32 7.47 0.0434 0.79 1.28 0.06 4.56 0.01 Canada 0.99 6.33 0.1557 0.80 1.10 0.76 3.77 0.20 Denmark 0.97 6.48 0.1496 0.78 1.10 0.74 4.04 0.18 Finland 0.09 8.93 0.0101 0.77 1.49 2 0.22 5.72 2 0.04 France 0.37 6.76 0.0548 0.88 1.29 0.11 3.19 0.03 Germany 0.58 7.80 0.0743 0.89 1.49 0.27 3.64 0.08 Greece 2 0.51 10.10 2 0.0506 0.66 1.43 2 0.81 7.65 2 0.11 Hong Kong 0.74 6.33 0.1175 0.69 0.94 0.55 4.61 0.12 Ireland 2 0.37 7.69 2 0.0477 0.79 1.31 2 0.64 4.73 2 0.13 Israel 0.56 6.19 0.0904 0.64 0.86 0.38 4.76 0.08
continued
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C H A P T E R 2 5 International Diversification 899
Regression against U.S.
Individual-Country- Index Portfolios Average SD
Sharpe Ratio Corr. w/U.S. Beta Alpha Residual SD Info Ratio
Italy 0.21 7.18 0.0291 0.82 1.27 2 0.05 4.11 2 0.01 Japan 0.23 4.95 0.0463 0.56 0.60 0.11 4.11 0.03 Netherlands 0.41 6.95 0.0589 0.88 1.32 0.14 3.32 0.04 New Zealand 0.97 6.22 0.1554 0.67 0.90 0.78 4.64 0.17 Norway 1.35 8.98 0.1507 0.78 1.51 1.04 5.68 0.18 Portugal 0.35 6.74 0.0524 0.73 1.06 0.13 4.63 0.03 Singapore 0.99 6.71 0.1475 0.75 1.09 0.77 4.43 0.17 Spain 0.76 7.53 0.1009 0.79 1.29 0.49 4.62 0.11 Sweden 0.94 8.01 0.1177 0.86 1.49 0.64 4.11 0.16 Switzerland 0.60 5.04 0.1194 0.79 0.86 0.42 3.10 0.14 UK 0.38 5.29 0.0725 0.87 1.00 0.18 2.60 0.07 Emerging Argentina 1.32 11.05 0.1191 0.44 0.44 1.05 1.10 0.96 Brazil 2.10 10.86 0.1931 0.71 0.71 1.66 1.76 0.94 Chile 1.46 6.69 0.2179 0.63 0.63 0.91 1.27 0.72 China 1.37 8.13 0.1683 0.62 0.62 1.10 1.14 0.96 Colombia 2.95 8.76 0.3367 0.53 0.53 1.00 2.74 0.36 Czech Republic 2.06 7.93 0.2593 0.62 0.62 1.06 1.84 0.58 Egypt 2.17 10.29 0.2111 0.40 0.40 0.89 1.99 0.45 Hungary 1.25 10.78 0.1155 0.70 0.70 1.64 0.91 1.80 India 1.53 9.13 0.1676 0.62 0.62 1.21 1.28 0.95 Indonesia 2.77 10.01 0.2766 0.54 0.54 1.17 2.53 0.46 Jordan 0.59 6.32 0.0932 0.28 0.28 0.38 0.51 0.75 Malaysia 1.07 5.27 0.2033 0.59 0.59 0.67 0.93 0.71 Mexico 1.28 7.09 0.1809 0.83 0.83 1.27 1.02 1.24 Morocco 1.18 6.01 0.1970 0.31 0.31 0.41 1.10 0.37 Pakistan 1.87 10.09 0.1849 0.14 0.14 0.31 1.80 0.17 Peru 2.53 9.54 0.2655 0.52 0.52 1.08 2.31 0.47 Philippines 1.27 7.37 0.1725 0.48 0.48 0.77 1.11 0.69 Poland 1.19 10.24 0.1162 0.75 0.75 1.65 0.85 1.94 Russia 1.50 10.30 0.1457 0.61 0.61 1.35 1.22 1.10 South Africa 1.61 7.93 0.2031 0.68 0.68 1.16 1.37 0.84 Korea 1.34 8.76 0.1526 0.75 0.75 1.41 1.05 1.35 Sri Lanka 1.82 10.92 0.1663 0.19 0.19 0.45 1.72 0.26 Taiwan 0.52 7.44 0.0697 0.70 0.70 1.12 0.29 3.86 Thailand 1.92 8.34 0.2298 0.57 0.57 1.03 1.70 0.61 Turkey 1.66 13.14 0.1267 0.59 0.59 1.67 1.32 1.27
Table 25.9B Concluded
Performance of individual-country-index portfolios, monthly excess returns in U.S. dollars over 2002–2011
Source: Datastream, online.thomsonreuters.com/datastream.
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900 P A R T V I I Applied Portfolio Management
The performance of these portfolios is contrasted with that of completely feasible
international diversification by investing in regional-index funds, which are also shown
in Table 25.9A . This comparison is based on standard performance statistics, namely,
average and standard deviation of excess returns (denominated in U.S. dollars), as well
as beta and alpha estimated against a U.S.-only portfolio. We will assess these perfor-
mance statistics after we first take a broad-brush look at the return behavior of developed
versus emerging market indexes.
Are Investments in Emerging Markets Riskier? In Figure 25.3 , developed countries and emerging markets are separately ordered from
lowest to highest standard deviation. The standard deviations of investments in emerg-
ing markets are charted with those in developed countries. The graphs clearly show that,
considered as total portfolios, emerging markets are generally riskier than developed coun-
tries, at least when risk is measured by total volatility of returns. Still, you can find emerg-
ing markets that appear safer than some developed countries. However, if one considers
adding a foreign country index to an indexed U.S. portfolio, the relevant risk measure is
the country’s beta against the U.S. 4
Figure 25.4 ranks and charts the betas of country returns (in U.S. dollars) against the
U.S. index. It shows that betas of six developed and eight emerging markets were estimated
to be below 1. Notice, however, that this is only about one-third of all 48 foreign markets.
Thus we can foresee the conclusion: A well-diversified international portfolio may well be
riskier than the U.S. alone, which has consistently exhibited the lowest standard deviation
0
2
4
6
8
10
12
14
0 5 10 15 20 25
St a n
d a rd
D e vi
a ti
o n
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/m o
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)
Rank Markets are ranked by SD of return, from low to high
25
24232221 2019181716
15 14
1312 111098
76 5
4 3
2 1
24
232221
2019181716 15141312111098765
432 1
Developed + World Emerging
1 U.S. Malaysia 2 Japan Morocco 3 Switzerland Jordan 4 U.K. Chile 5 Israel Mexico 6 New Zealand Philippines 7 Canada Taiwan 8 Hong Kong Czech Republic 9 Denmark South Africa 10 Singapore China 11 Portugal Thailand 12 Australia Colombia 13 France Korea 14 Netherlands India 15 Italy Peru 16 Belgium Indonesia 17 Spain Pakistan 18 Ireland Poland 19 Germany Egypt 20 Sweden Russia 21 Austria Hungary 22 Finland Brazil 23 Norway Sri Lanka 24 Greece Argentina 25 Turkey
Rank Developed Emerging Rank Legend
Figure 25.3 Monthly standard deviation of excess returns in developed and emerging markets, 2002–2011
4 A sufficient condition to reduce the standard deviation of a portfolio by adding an asset is that the beta of the
asset on the portfolio be less than 1.
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C H A P T E R 2 5 International Diversification 901
25242322
21 20
19 18
1716 15141312
109 87
6
5
4 32
1
24232221 2019
181716151413 12
11109 8
76 5
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11
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0.70
0.90
1.10
1.30
1.50
1.70
B e ta
o n
U .S
.
0 5 10 15 20 25 Rank
Markets are sorted by beta, from low to high
Developed + World Emerging
1 Japan Pakistan 2 Switzerland Jordan 3 Israel Morocco 4 New Zealand Sri Lanka 5 Hong Kong Malaysia 6 U.K. Philippines 7 U.S. Egypt 8 Portugal Chile 9 Singapore Colombia 10 Denmark Thailand 11 Canada Argentina 12 Australia Czech Republic 13 Italy Peru 14 Belgium China 15 Spain Taiwan 16 France South Africa 17 Ireland Indonesia 18 Netherlands India 19 Austria Mexico 20 Greece Russia 21 Finland Korea 22 Sweden Hungary 23 Germany Poland 24 Norway Brazil 25 Turkey
Rank Developed Emerging Rank Legend
Figure 25.4 Beta against the U.S. market of developed and emerging markets, 2002–2011
of all countries. This is not to say, however, that an international portfolio with higher vari-
ance would necessarily be inferior. In fact, when a risk-free asset is available, minimum-
variance portfolios are never efficient (they are dominated by the maximum Sharpe ratio,
or tangency, portfolio on the efficient frontier). But, then, the international portfolio must
show a sufficiently larger average return to provide a larger Sharpe ratio.
Comparison between developed and emerging markets’ betas in Figure 25.4 shows that,
in contrast to the picture painted by standard deviation, emerging markets are not meaning-
fully riskier to U.S. investors than developed markets. This is the most important lesson
from this exercise.
Are Average Returns Higher in Emerging Markets? Figure 25.5 repeats the previous exercise for average excess returns. The graph shows
that emerging markets generally provided higher average returns than developed markets
over the period 2002–2011. The fact that only 2 (developed) of the 49 markets aver-
aged a lower rate than the risk-free alternative is quite unusual given the volatility of
these markets. However, this result is partially due to the weak U.S. dollar over these
years. When measured in local currencies, returns in eight countries, all developed, aver-
aged below U.S. T-bills over the 10-year period. Beyond that, we see that countries with
relatively low betas (e.g., Pakistan) earned higher returns than countries with relatively
high betas, even the highest-beta country, Turkey. Further, average returns in emerging
markets were generally higher than those in developed countries despite the fact that
emerging market betas were not higher, implying that emerging markets provided better
diversification opportunities than developed markets in this period.
We shouldn’t be too surprised by these results. Remember again that the SD of an average esti-
mated over 120 months is approximately SD(10-year average) 5 SD(1-month average)/ "120 .
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902 P A R T V I I Applied Portfolio Management
Thus, the SD of the 10-year average monthly return for Pakistan would be about .92%, and that of
Turkey about 1.20%. A departure from mean return of one SD in opposite directions for these two
portfolios would span a distance of about 2.12%, while the difference in average returns is only
.21%. The conclusion is one we’ve noted before: We cannot read too much into realized averages
even over periods as long as 10 years.
Instinct calls for estimating alpha or information ratios for individual markets, to see
whether they are distributed around zero. Recall from our discussion of performance eval-
uation in Chapter 18 that, without positive alpha, we cannot conclude that an asset has
shown superior performance on any measure. The information ratio measures the potential
increase in the Sharpe ratio if the country index were to be added in an optimal proportion
to the U.S index.
Figure 25.6 verifies that information ratios in emerging markets were, on the whole,
clearly better than those in developed markets. This is a result of inferior performance of
the eight markets most affected by the financial crisis, all developed countries, and four
stellar emerging market performers. The performance of the other 36 markets cannot be
distinguished in terms of emerging versus developed. Here again, given the high volatili-
ties, finding four outperformers and eight underperformers in a group of 48 countries is not
surprising or significant.
One striking result is the inferior performance of the U.S. We see this in Table 25.9 A:
Although the U.S. has the lowest standard deviation among all countries, it still ranks near
the bottom in terms of the Sharpe ratio. This may be explained by the financial crisis and/
or by the steady decline in the international economic position of the U.S, as reflected
by the steady decline in the value of the dollar. 5 To investigate the latter possibility,
Figure 25.5 Average dollar-denominated excess returns of developed and emerging markets, 2002–2011
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 5 10 15 20 25
A ve
ra g
e E
xc e ss
R e tu
rn (
% /m
o n
th )
Rank Markets are sorted by average returns, from low to high
25
24 23
2221201918 171615
141312 1110987
654
2 1
24
23
222120 191817
16151413121110987654 3
21
Developed + World Emerging
1 Greece Taiwan 2 Ireland Jordan 3 Finland Malaysia 4 U.S. Morocco 5 Italy Poland 6 Japan Hungary 7 Belgium Philippines 8 Portugal Mexico 9 France Argentina 10 U.K. Korea 11 Netherlands China 12 Israel Chile 13 Germany Russia 14 Switzerland India 15 Hong Kong South Africa 16 Spain Turkey 17 Austria Sri Lanka 18 Sweden Pakistan 19 New Zealand Thailand 20 Denmark Czech Republic 21 Canada Brazil 22 Singapore Egypt 23 Australia Peru 24 Norway Indonesia 25 Colombia
Rank Developed Emerging Rank Legend
3
5 The decline of the U.S dollar has so far been sporadically interrupted by international crises. A regular feature of
those crises has been a flight to the safety of dollar instruments.
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C H A P T E R 2 5 International Diversification 903
Figure 25.6 Information ratios of developed and emerging markets against U.S. dollar-denominated returns, 2002–2011
–0.2
–0.1
0.0
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0.5
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0 5 10 15 20 25 In
fo rm
a ti
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R a ti
o
Rank Markets are sorted by information ratio from low to high
25
2423 21201918171615
1413121110987 6543
2 1
22
23 22
2120191817 16
1514 13121110
98765 4
3
2 1
Developed + World Emerging
1 Ireland Taiwan 2 Greece Jordan 3 Finland Argentina 4 Italy Hungary 5 Belgium Turkey 6 Japan Poland 7 Portugal Russia 8 France Sri Lanka 9 Netherlands Philippines 10 U.K. India 11 Germany Korea 12 Israel China 13 Austria Pakistan 14 Spain Morocco 15 Hong Kong Egypt 16 Switzerland Malaysia 17 Sweden Brazil 18 New Zealand South Africa 19 Singapore Chile 20 Norway Thailand 21 Denmark Mexico 22 Canada Peru 23 Australia Czech Republic 24 Indonesia 25 Colombia
Rank Developed Emerging Rank Legend
we compare emerging and developed markets using returns in local currencies. Recall that
U.S. investors can achieve such returns by hedging the currencies of the country portfolios
in which they invest.
Is Exchange Rate Risk Important in International Portfolios? Table 25.3 revealed that changes in exchange rates vary widely across country pairs.
In Figures 25.7 to 25.10 we compare results for SD, beta coefficients, average excess
return, and information ratios for both developed and emerging markets using dollar and
local-currency returns. Figures 25.7 and 25.8 look at the issue of risk. Both measures
show that returns in local currency are convincingly less risky than dollar-denominated
returns. The difference (at least over the recent 10 years) is greater when comparing
beta. Remember, however, that this result applies only to adding one country to the U.S.
portfolio; relative contributions to risk could change if one were to consider broader
diversification.
Hedging currency risk when investing internationally is often undertaken to reduce
overall portfolio risk. However, the decision of whether to hedge foreign currencies in
an internationally diversified portfolio can also be made as part of active management. If
a portfolio manager believes the U.S. dollar is overvalued against a given currency, then
hedging the exposure to that currency would, if correct, enhance the portfolio return in U.S.
dollars. The potential gain from this decision depends on the weight of that currency in the
overall portfolio. Such a decision applied to investments in only one country would have a
small effect on overall risk. But what if the manager estimates that the dollar is generally
overvalued against most or all currencies? In this case, hedging the entire exposure would
constitute a bet with significant effect on total risk. At the same time, if the decision is
correct, such a large position can provide handsome gains.
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904 P A R T V I I Applied Portfolio Management
Figure 25.8 Market beta against U.S. using dollar-denominated and local-currency excess returns, 2002–2011
1 2 3
4
5 6
7
8 9 10 11 12
13 14 15
16 17 18 19 20 21
22 23 24 25 26 27
28 29 30 31 32 33 34
35 36 37
38 39 40 41 42 43 44
45 46 47 48
1
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31323334 3536
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0
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5 10 15 20 25 30 35 40 45 50
B e ta
o f
D o
ll a r-
D e n
o m
in a te
d a
n d
L o
ca l-
C u
rr e n
cy E
xc e ss
R e tu
rn s
Rank Betas are ranked from low to high
Beta for Local Currency Beta for U.S. $
Rank Country Rank Country Rank Legend
1 Pakistan 27 Indonesia 2 Jordan 28 Australia 3 Morocco 29 India 4 Sri Lanka 30 Mexico 5 Japan 31 Italy 6 Malaysia 32 Belgium 7 Philippines 33 Spain 8 Switzerland 34 France 9 Israel 35 Ireland 10 Egypt 36 Netherlands 11 New Zealand 37 Russia 12 Chile 38 Korea 13 Hong Kong 39 Austria 14 U.K. 40 Greece 15 Colombia 41 Finland 16 Thailand 42 Sweden 17 Argentina 43 Germany 18 Portugal 44 Norway 19 Czech Republic 45 Hungary 20 Peru 46 Poland 21 Singapore 47 Brazil 22 China 48 Turkey 23 Denmark 24 Canada 25 Taiwan 26 South Africa
Figures 25.9 and 25.10 show that dollar-denominated average excess returns and
information ratios are a bit better than those denominated in local currencies. Since the
risk-adjusted returns are no better in local currency than in dollars, we must conclude that
the superior performance of emerging market portfolios is due to surprises about their eco-
nomic performance and not solely due to the decline in the U.S. dollar.
Figure 25.7 Standard deviation of excess returns in dollar-denominated and local currencies, 2002–2011
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30 35 40 45 50
SD o
f D
o ll
a r-
D e
n o
m in
a te
d a
n d
L
o ca
l- C
u rr
e n
cy E
xc e
ss R
e tu
rn s
Rank Standard deviations of excess returns are ranked from low to high
SD of U.S. $-Denominated Returns SD of Return in Local Currency
49
48474645 444342414039
38 373635343332
31 3029282726252423222120191817161514131211109876
543 1
2 1
2
3 4 5
6 7
8
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2122 2324
25 26
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49
Rank Country Rank Country Rank Legend
1 U.S. 27 Czech Republic 2 Japan 28 South Africa 3 Switzerland 29 Sweden 4 Malaysia 30 China 5 U.K. 31 Thailand 6 Morocco 32 Colombia 7 Israel 33 Korea 8 New Zealand 34 Austria 9 Jordan 35 Finland 10 Canada 36 Norway 11 Hong Kong 37 India 12 Denmark 38 Peru 13 Chile 39 Indonesia 14 Singapore 40 Pakistan 15 Portugal 41 Greece 16 Australia 42 Poland 17 France 43 Egypt 18 Netherlands 44 Russia 19 Mexico 45 Hungary 20 Italy 46 Brazil 21 Philippines 47 Sri Lanka 22 Taiwan 48 Argentina 23 Belgium 49 Turkey 24 Spain 25 Ireland 26 Germany
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C H A P T E R 2 5 International Diversification 905
Figure 25.10 Information ratios against U.S. computed from dollar-denominated and local-currency returns, 2002–2011
1 2
3 4
5 6 7 8 9
10 11 12 1314
15 16 17 18 1920
21 22 23 24 25
26 27 28 29 30 31 32 33 34 35 36
37 38 39 40
41 42 43 44
45 46 47
48
–0.30
–0.20
–0.10
0.00
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0.20
0.30
0.40
0.50
0.60
5 10 15 20 25 30 35 40 45
In fo
rm a ti
o n
R a ti
o o
f M
a rk
e ts
a g
a in
st U
.S .
Rank Information ratios are ranked from low to high
U.S.-Dollar Returns Local-Currency Returns
1 2
3 4
5 6 7
8 9
10 11
12
13 14
15
16
17
18 19 21
22
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27
28
29 3031
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39 40
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42 43
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0 50
Rank Country Rank Country Rank Legend
senippilihP72dnalerI1 eropagniS82eceerG2
aidnI92dnalniF3 aeroK03ylatI4 anihC13muigleB5
natsikaP23napaJ6 yawroN33lagutroP7
kramneD43ecnarF8 occoroM53sdnalrehteN9
adanaC63nawiaT01 tpygE73.K.U11
aisyalaM83ynamreG21 lizarB93learsI31
acirfAhtuoS04airtsuA41 ailartsuA14nadroJ51
elihC24niapS61 dnaliahT34anitnegrA71
ocixeM44yragnuH81 ureP54gnoKgnoH91
cilbupeRhcezC64yekruT02 aisenodnI74dnaloP12 aibmoloC84dnalreztiwS22
23 Russia 24 Sweden 25 Sri Lanka 26 New Zealand
Figure 25.9 Average dollar-denominated and local-currency excess returns, 2002–2011
–1.5
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
5 10 15 20 25 30 35 40 45
A ve
ra g
e E
xc e ss
R e tu
rn s
% /m
o n
th
Rank Average excess returns are ranked from low to high
U.S.-Dollar Excess Returns Local-Currency Excess Returns
1 2
3 4 5 6
7 8 9 10 11
12 13 14 15 16
1718 19 20 21 22 23 24
25 26 27 28
29 30 31 32 33 34 35
3637 38
39 40 41 42
43 44 45
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0
Rank Country Rank Country Rank Legend
1 Greece 27 Morocco 2 Ireland 28 Poland 3 Finland 29 Hungary 4 U.S. 30 Philippines 5 Italy 31 Mexico 6 Japan 32 Argentina 7 Belgium 33 Korea 8 Portugal 34 Norway 9 France 35 China 10 U.K. 36 Chile 11 Netherlands 37 Russia 12 Taiwan 38 India 13 Israel 39 South Africa 14 Germany 40 Turkey 15 Jordan 41 Sri Lanka 16 Switzerland 42 Pakistan 17 Hong Kong 43 Thailand 18 Spain 44 Czech Republic 19 Austria 45 Brazil 20 Sweden 46 Egypt 21 New Zealand 47 Peru 22 Denmark 48 Indonesia 23 Canada 49 Colombia 24 Singapore 25 Malaysia 26 Australia
50
Benefits from International Diversification Table 25.10 tells the story of international diversification. First, notice the strong trend
of increasing correlation. Of the 16 countries, only four show stable correlation from the
late 1960s through 2011. The rest show significant increases. In the most recent decade,
the correlation of the portfolio of tradeable world country portfolios excluding the U.S.
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906 P A R T V I I Applied Portfolio Management
with that of the U.S. is .90 (see also the nearby box). Hence, the benefits of diversification
can be expected to emanate from countries with relatively lower correlations (admittedly,
among themselves, as well as with the U.S.). Table 25.9 B shows that such low correlations
are found mostly in the emerging markets.
To assess the value of international diversification, we return to Table 25.9 A, where
we first viewed portfolio standard deviation. We find that whether we diversify from
U.S.-only investments either to the entire world or to emerging markets only, the stan-
dard deviation of the portfolio increases. This results from higher standard deviations
of foreign markets that are not offset by the relatively low correlations of emerging
markets.
Nevertheless, the objective of diversification is not merely risk reduction. Rather, it is to increase the Sharpe ratio. Here we see that, in any configuration, the Sharpe ratio
of internationally diversified portfolios is higher than that of the U.S. alone. Even with-
out any notion about emerging versus developed markets, holding the world portfolio
results in a significantly higher Sharpe ratio. Using the more revealing M2 measure (see Chapter 24), the advantage of the world countries portfolio amounts to 284 basis points
annually. Even the less diversified world ETF portfolio earned an annual risk-adjusted
premium of 107 basis points.
Thus the data clearly indicate that despite increasing correlations, even a passive
strategy of holding a world ETF is still superior to the U.S.-only portfolio. Figure 25.11
shows diversification benefits based on 1995 correlations and diversification into ran-
domly selected stocks from around the world compared with a one-stock portfolio.
This representation no longer serves the purpose of analyzing the current benefits from
2002–2011 1967–2001
World 0.97 0.95 World (excluding U.S.) 0.90 NA Germany 0.89 0.75 Netherlands 0.88 0.71 France 0.88 0.70 United Kingdom 0.87 0.83 Sweden 0.86 0.73 Australia 0.82 0.71 Italy 0.82 0.55 Canada 0.80 0.82 Spain 0.79 0.65 Switzerland 0.79 0.65 Belgium 0.79 0.46 Denmark 0.78 0.67 Norway 0.78 0.63 Austria 0.74 0.46 Hong Kong 0.69 0.67 Japan 0.56 0.58
NA 5 not available.
Source: Datastream online.thomsonreuters.com/datastream ; 1970–1989: Campbell R. Harvey, “The World Price of Covariance Risk,” Journal of Finance 46 (March 1991), issue 1, pp. 111–58.
Table 25.10
Correlation of foreign investments with U.S. returns over time (dollar-denominated excess returns)
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907
international diversification. These benefits
accrue not from reduction in standard devia-
tion but rather from improved risk-adjusted
performance.
Misleading Representation of Diversification Benefits The baseline technique for constructing effi-
cient portfolios is the efficient frontier. A useful
efficient frontier is constructed from expected returns and an estimate of the covariance matrix
of returns. This frontier combined with cash
assets generates the capital allocation line, the
set of efficient complete portfolios, as elabo-
rated in Chapter 7. The benefit from this effi-
cient diversification is reflected in the curvature
of the efficient frontier. Other things equal, the
lower the covariance across stocks, the greater
the curvature of the efficient frontier and
the greater the risk reduction for any desired
expected return. So far, so good. But suppose we replace expected returns with realized aver- age returns from a sample period to construct
an efficient frontier; what is the possible use of
this graph?
The ex post efficient frontier (derived from
realized returns) describes the portfolio of only
one investor—the clairvoyant who actually predicted the precise averages of realized
returns on all assets and estimated a covariance matrix that materialized, precisely, in the
Investors’ Challenge: Markets Seem Too Linked
It’s one of the golden rules of investing: Reduce risk by diversifying your money into a variety of holdings—stock funds, bonds, commodities—that don’t move in lockstep with one another. And it’s a rule that’s getting tougher to obey.
According to recent research, an array of investments whose prices used to rise and fall independently are now increasingly correlated. For an example, look no fur- ther than the roller coaster in emerging-markets stocks of recent weeks. The MSCI EAFE index, which measures emerging markets, now shows .96 correlation to the S&P, up from just .32 six years ago.
For investors, that poses a troubling issue: how to main- tain a portfolio diversified enough so all the pieces don’t tank at once.
The current correlation trend doesn’t mean investors should go out and ditch their existing investments. It’s just that they may not be “getting the same diversification”
they thought if the investment decisions were made some time ago, says Mr. Ezrati, chief economist at money- management firm Lord Abbett & Co. He adds that over long periods of time, going back decades, sometimes varied asset classes tend to converge.
One explanation for today’s higher correlation is increased globalization, which has made the economies of various countries more interdependent. International stocks, even with their higher correlations at present, deserve some allocation in a long-term investor’s hold- ings, says Jeff Tjornehoj, an analyst at data firm Lipper Inc. Mr. Tjornehoj is among those who believe these correla- tions are a temporary phenomenon, and expects that the diversity will return some time down the line—a year or few years.
Source: Shefali Anand, “Investors Challenge: Markets Seem Too Linked,” The Wall Street Journal, June 2, 2006, p. C1. © 2006 Dow Jones & Company, Inc. All rights reserved worldwide.
W O
R D
S FR O
M T
H E ST
R E E T
27
11.7
1 10 20 30 40 50
U.S. Stocks
Global Stocks
Number of Stocks
P e rc
e n
t R
is k
0
20
40
60
80
100
Figure 25.11 International diversification. Portfolio standard deviation as a percentage of the average standard deviation of a one-stock portfolio
Source: B. Solnik, “Why Not Diversify Internationally Rather Than Domestically.” Financial Analysts Journal, July/August 1974, pp. 48–54. Copyright 1995, CFA Institute. Reproduced and repub- lished from Financial Analysts Journal with permission from the CFA Institute.
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908 P A R T V I I Applied Portfolio Management
actual realizations of the sample period returns on all assets. Obviously, we are talking
about a slim to empty set of investors. For all other, less-than-clairvoyant investors, such a
frontier may have value only for purposes of performance evaluation.
In the world of volatile stocks, some stocks are bound to realize large, unexpected average returns. This will be reflected in ex post efficient frontiers of enormous apparent
“potential.” They will, however, suggest exaggerated diversification benefits. Such (elu-
sive) potential was enumerated in Chapter 24 on performance evaluation. It has no mean-
ing as a tool to discuss the potential for future investments for real-life investors.
Realistic Benefits from International Diversification While recent realized returns can be highly misleading estimates of expected future returns,
they are more useful for measuring prospective risk. There are two compelling reasons for
this. First, market efficiency (or even near efficiency) implies that stock prices will be
difficult to predict with any accuracy, but no such implication applies to risk measures.
Second, it is a statistical fact that errors in estimates of standard deviation and correlation
from realized data are of a lower order of magnitude than estimates of expected returns.
For these reasons, using risk estimates from realized returns does not bias assessments of
the potential benefits from diversification.
Figure 25.12 shows the efficient frontier using realized average monthly returns on the
stock indexes of the 25 developed countries, with and without short sales. Even when
the (ex post) efficient frontier is constrained to preclude short sales, it greatly exagger-
ates the benefits from diversification. Unfortunately, such misleading efficient frontiers are
still presented in articles and texts on the benefits of diversification.
A more realistic description of diversification is achievable only when we input rea-
sonable equilibrium expected returns. Absent superior information, such expected returns
are best based on appropriate risk measures of the assets. The capital asset pricing model
(CAPM) suggests using the beta of the stock against the world portfolio. To generate
Figure 25.12 Ex post efficient frontier of country portfolios, 2001–2005
World
Austria Korea
New Zealand
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 2 4 6
U.S.
8 10 12
Standard Deviation (% per month)
R e a li ze
d A
ve ra
g e M
o n
th ly
E xc
e ss
R e tu
rn (
% )
Efficient Frontier with Short Sales Efficient Frontier w/ No Short Sales Country Portfolios
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C H A P T E R 2 5 International Diversification 909
expected excess returns (over the risk-free rate) for all assets, we specify the expected
excess return on the world portfolio. We obtain the expected excess return on each asset
by multiplying the beta of the asset by the world portfolio expected excess return. This
procedure presupposes that the world portfolio will lie on the efficient frontier, at the point
of tangency with the world capital market line. The curvature of the efficient frontier will
not be affected by the estimate of the world portfolio excess return. A higher estimate will
simply shift the curve upward.
We perform this procedure with risk measures estimated from actual returns and further
impose the likely applicable constraint on short sales. We use the betas to compute the
expected return on individual markets, assuming the expected excess return on the world
portfolio is .6% per month. This excess return is in line with the average return over the
previous 50 years. Varying this estimate would not qualitatively affect the results shown
in Figure 25.13 (which is drawn on the same scale as Figure 25.12 ). The figure shows a
realistic assessment that reveals modest but significant benefits from international diver-
sification using only developed markets. Incorporating emerging markets would further
increase these benefits.
Are Benefits from International Diversification Preserved in Bear Markets? Some studies suggest that correlation in country portfolio returns increases during peri-
ods of turbulence in capital markets. 6 If so, benefits from diversification would be lost
6 F. Longin and B. Solnik, “Is the Correlation in International Equity Returns Constant: 1960–1990?” Journal of
International Money and Finance 14 (1995), pp. 3–26; and Eric Jacquier and Alan Marcus, “Asset Allocation Models and Market Volatility,” Financial Analysts Journal 57 (March/April 2001), pp. 16–30.
World
U.S.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6 8 10 12 Standard Deviation (% per month)
E xp
e ct
e d
M o
n th
ly E
xc e ss
R e tu
rn (
% ) World CML
Efficient Frontier w/ No Short Sales Efficient Frontier w/ Short Sales Country Portfolios
Figure 25.13 Efficient frontier of country portfolios (world expected excess return 5 .6% per month)
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910 P A R T V I I Applied Portfolio Management
Figure 25.14 Regional indexes around the crash, October 14–October 26, 1987 Source: From Richard Roll, “The International Crash of October 1987,” Financial Analysts Journal, September–October 1988. Copyright 1995, CFA Institute. Reproduced from Financial Analysts Journal with permission from the CFA Institute.
Value of One Currency Unit
1.05
Tick Marks on October Date, 4:00 P.M., U.S. Eastern Standard Time
12 14 16 18 20 22 24 26
Symbols Positioned at Market Close Local Time
North America
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Australia/New Zealand Asia Small Europe Large Europe Ireland, So. Africa, U.K.
exactly when they are needed the most. For example, a study by Roll of the crash of
October 1987 shows that all 23 country indexes studied declined over the crash period
of October 12–26. 7 This correlation is reflected in the movements of regional indexes
depicted in Figure 25.14 . Roll found that the beta of a country index on the world index
(estimated prior to the crash) was the best predictor of that index’s response to the
October crash of the U.S. stock market. This suggests a common factor underlying the
movement of stocks around the world. This model predicts that a macroeconomic shock
would affect all countries and that diversification can only mitigate country-specific
events.
The 2008 crash of stock markets around the world allows us to test Roll’s prediction.
The data in Figure 25.15 include average monthly rates of return for both the 10-year
period 1999–2008 and the crisis period corresponding to the last 4 months of 2008, as well
as the beta on the U.S. market and monthly standard deviation for several portfolios. The
graph shows that both beta against the U.S. and the country-index standard deviation help
explain the difference between crisis period returns and overall period averages. Market
behavior during the 1987 crisis, that is, larger correlations in extreme bad times, repeated
itself in the crisis of 2008, vindicating Roll’s prediction.
7 Richard Roll, “The International Crash of October 1987,” Financial Analysts Journal, September–October 1988.
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C H A P T E R 2 5 International Diversification 911
25.4 Assessing the Potential of International Diversification
Figure 25.15 Beta and SD of portfolios against deviation of monthly return over September–December 2008 from average return over 1999–2008
Source: Authors’ calculations.
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0 217.0
SD o
f M
o n
th ly
R e tu
rn s,
1 9 9 9 –2
0 0 8
215.0 213.0
Deviation from Average Return
211.0 29.0 27.0 0
0.2
B e ta
a g
a in
st U
.S .
0.4
0.6
0.8
1
1.2
1.4
1.6
Market
USA 2.47 28.31 27.84 1 4.81 World largest six (non–U.S.) markets 2.16 27.51 27.35 0.77 4.71 EU developed markets 2.05 210.34 210.29 1.06 6.08 Other Europe developed markets .14 27.59 27.73 0.82 4.95 Australia 1 Far East 01. 29.29 29.38 1.04 6.21 Emerging Far East 1 South Africa .20 29.70 29.90 1.01 7.10 Emerging Latin America 08. 211.72 212.52 1.27 7.83 Emerging markets in Europe .90 215.43 216.32 1.38 9.54 World minus U.S. (48 countries by cap) .01 28.79 28.81 0.91 5.19 World portfolio (by country cap) 2.15 28.60 28.45 0.94 4.88
1999–2008 2008: Sept.–Dec. Average Monthly Return Deviation
from Average Beta on U.S. SD
SD Beta on U.S.
We focus first on investors who wish to hold largely passive portfolios. Their objective is
to maximize diversification with limited expense and effort. Passive investment is simple:
Rely on market efficiency to guarantee that a broad stock portfolio will yield the best pos-
sible Sharpe ratio. Estimate the mean and standard deviation of the optimal risky portfolio,
and select a capital allocation to achieve the highest expected return at a level of risk you
are willing to bear. But now, a passive investor must also decide whether to add an interna-
tional component to the more convenient home-country index portfolio.
Suppose the passive investor could rely on efficient markets as well as a world
CAPM. Then the world capitalization–weighted portfolio would be optimal. Abiding
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912 P A R T V I I Applied Portfolio Management
by this theoretically simple solution is also practical. A world index fund would do
the trick. Over the decade ending in 2011, the performance of the world portfolio and
that of a U.S.-only portfolio can be summarized (using statistics from Table 25.9 ) as
follows:
Portfolio World U.S. Only
Average monthly return (%) .31 .21 Standard deviation (%) 4.90 4.63
These results are instructive. First, we see that U.S. stocks make for a relatively low-risk
portfolio. While the U.S. portfolio may lie inside the world efficient frontier, and thus may
offer a lower Sharpe ratio than the world portfolio, it nevertheless may have lower volatil-
ity than the better-diversified world portfolio.
Things are more complicated when we recognize that the data do not support the
validity of the world CAPM, and hence we cannot be certain that the world portfolio
is the most efficient risky portfolio. We do observe that higher country standard devia-
tions tend to be rewarded with higher average returns. A passive investor may therefore
wish to examine simple rules of thumb for including a small number of countries (via
international index funds of various combinations) in an attempt to dull the effect of high
individual-country standard deviations and yet improve the Sharpe ratio of the overall
portfolio. In all three of these rules, we assume the perspective of a U.S. investor, using
dollar-denominated returns. We include countries on the basis of market capitalization
for two reasons: (1) the resultant portfolio will be at least reasonably close to the theo-
retically efficient portfolio, and (2) the weights of any foreign country will not be too
large. We estimate the risk of progressively more diversified portfolios relative to the
number of foreign countries included, and the total portfolio weight of the international
component.
The three rules of thumb are to include country indexes in order of:
1. Market capitalization (from high to low). This rule is motivated by a world CAPM consideration in which the optimal portfolio is capitalization weighted.
2. Beta against the U.S. (from low to high). This rule concentrates on diversifying the risk associated with investments in higher-risk countries.
3. Country index standard deviation (from high to low). This rule is motivated by the observation that higher country standard deviations (SDs) are correlated
with higher average returns. It relies on diversification to mitigate individual-
country risk.
These alternatives illustrate the potential risks and rewards of international diversifi-
cation. Results of this exercise appear in Table 25.11 and Figure 25.16 . First turn to panel
A of Figure 25.16 , which vividly shows how portfolio SD progresses as we diversify the
U.S. portfolio using the three rules. Clearly, adding countries in order of beta (or covari-
ance with the U.S. market), from low to high, quickly reduces portfolio risk despite the
fact that the standard deviations of all 12 included countries are higher than those of
the U.S. However, once we have adequate diversification, adding these higher-volatility
indexes eventually begins to increase portfolio standard deviation. Adding countries in
order of standard deviation (but this time, from high to low to improve expected returns,
which are correlated with volatility) incurs the greatest increase in portfolio SD, as we
would expect.
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9 1 3
Portfolio Composition Weight in World Portfolio Weight of U.S. in Portfolio Std Dev Average Return
A. Inclusion based on capitalization 1 U.S. only 0.33 1 5.17 2 0.20 2 Portfolio 1 plus Japan * 0.42 0.79 4.95 2 0.24 3 Portfolio 2 plus U.K. * 0.49 0.67 4.97 2 0.20 4 Portfolio 3 plus France * 0.54 0.61 5.02 2 0.16 5 Portfolio 4 plus Canada * 0.58 0.57 5.07 2 0.10 6 Portfolio 5 plus Hong Kong * 0.62 0.54 5.06 2 0.07 7 Portfolio 6 plus Germany * 0.65 0.51 5.11 2 0.06 8 Portfolio 7 plus Brazil * 0.68 0.49 5.19 0.03 9 Portfolio 8 plus Australia * 0.71 0.46 5.19 0.07 10 Portfolio 9 plus Switzerland * 0.74 0.45 5.18 0.08 11 Portfolio 10 plus China * 0.76 0.44 5.19 0.10 12 Portfolio 11 plus Taiwan * 0.77 0.43 5.19 0.10 13 Portfolio 12 plus Netherlands * 0.78 0.42 5.20 0.10 B. Inclusion based on beta 1 U.S. only 0.33 1 5.17 2 0.20 2 Portfolio 1 plus Pakistan * 0.33 1.00 5.16 2 0.20 3 Portfolio 2 plus Malaysia * 0.34 0.98 5.12 2 0.18 4 Portfolio 3 plus Japan * 0.43 0.78 4.85 2 0.22 5 Portfolio 4 plus Philippines * 0.43 0.77 4.84 2 0.22 6 Portfolio 5 plus Portugal * 0.43 0.77 4.84 2 0.22 7 Portfolio 6 plus Chile * 0.44 0.76 4.83 2 0.20 8 Portfolio 7 plus Israel * 0.44 0.75 4.83 2 0.19 9 Portfolio 8 plus Hong Kong * 0.48 0.70 4.83 2 0.15 10 Portfolio 9 plus Switzerland * 0.50 0.66 4.81 2 0.12 11 Portfolio 10 plus Colombia * 0.51 0.65 4.82 2 0.10 12 Portfolio 11 plus U.K. * 0.58 0.57 4.84 2 0.09 13 Portfolio 12 plus New Zealand * 0.58 0.57 4.84 2 0.09
Table 25.11
Standard deviation of international portfolios by degree of diversification continued
b o d 6 1 6 7 1 _ ch
2 5 _ 8 8 2 -9
2 5 .in
d d 9
1 3
b o d 6 1 6 7 1 _ ch
2 5 _ 8 8 2 -9
2 5 .in
d d 9
1 3
7 /2
5 /1
3 2
:0 4 A
M 7 /2
5 /1
3 2
:0 4 A
M
F inal P
D F
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9 1 4
Portfolio Composition Weight in World Portfolio Weight of U.S. in Portfolio Std Dev Average Return
C. Inclusion based on standard deviation 1 U.S. only 0.33 1 5.17 2 0.20 2 Portfolio 1 plus Turkey * 0.34 0.98 5.25 2 0.18 3 Portfolio 2 plus Argentina * 0.34 0.98 5.25 2 0.17 4 Portfolio 3 plus Russia * 0.36 0.93 5.39 2 0.08 5 Portfolio 4 plus Indonesia * 0.36 0.92 5.41 2 0.05 6 Portfolio 5 plus Pakistan * 0.36 0.92 5.40 2 0.05 7 Portfolio 6 plus Brazil * 0.39 0.84 5.66 0.10 8 Portfolio 7 plus Finland * 0.40 0.83 5.69 0.10 9 Portfolio 8 plus Poland * 0.40 0.83 5.70 0.11 10 Portfolio 9 plus Hungary * 0.40 0.83 5.70 0.11 11 Portfolio 10 plus Korea * 0.42 0.79 5.80 0.15 12 Portfolio 11 plus India * 0.44 0.74 5.87 0.22 13 Portfolio 12 plus Thailand * 0.45 0.74 5.87 0.23 D. All countries with various weighting schemes Equally weighted 0.99 0.33 6.14 0.76 By capitalization 0.99 0.33 5.60 0.27 World portfolio actual return † 1.00 0.33 5.34 2 0.01 Minimum variance portfolio—no short sales 0.99 0.33 4.14 0.02 Minimum variance portfolio—no restrictions 0.99 0.33 2.21 0.32
Table 25.11 concluded
Standard deviation of international portfolios by degree of diversification
* Portfolio weighted by capitalization of included countries. † All countries (including five omitted here) capitalization-weighted.
b o d 6 1 6 7 1 _ ch
2 5 _ 8 8 2 -9
2 5 .in
d d 9
1 4
b o d 6 1 6 7 1 _ ch
2 5 _ 8 8 2 -9
2 5 .in
d d 9
1 4
7 /2
5 /1
3 2
:0 4 A
M 7 /2
5 /1
3 2
:0 4 A
M
F inal P
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4.60
4.80
5.00
5.20
5.40
5.60
5.80
5.85
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.700.65 0.75 0.80 Fraction of World Portfolio Capitalization
St a n
d a rd
D e vi
a ti
o n
( %
p e r
m o
n th
) Inclusion in Order of Market Capitalization Inclusion in Order of Beta Inclusion in Order of SD
A
20.20
20.30
20.10
0.00
0.10
0.20
0.30
0.40
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.700.65 0.75 0.80 Fraction of World Portfolio Capitalization
A ve
ra g
e R
e tu
rn (
% p
e r
m o
n th
)
Inclusion in Order of Market Capitalization Inclusion in Order of Beta Inclusion in Order of Standard Deviation
B
C H A P T E R 2 5 International Diversification 915
Figure 25.16 , panel B shows that average returns increase along with the standard devi-
ation of returns. Average returns also increase with beta, at least for low-beta countries,
suggesting that at a qualitative level, world-systematic risk affects asset pricing, consistent
with an international CAPM.
Broadly speaking, these results are consistent with logic of the previous chapters. First,
diversification pays, and risk is rewarded. Second, even with strong home-country bias,
Figure 25.16 Risks and rewards of international portfolios, 2000–2009. Panel A, Standard deviations for international portfolios; Panel B, Average return of international portfolios.
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916 P A R T V I I Applied Portfolio Management
covariance risk still plays a role internationally. We also see that when confined to domes-
tic markets, risk aversion across the world is not too different: Higher country standard
deviations match up with higher average returns.
In panel D of Table 25.11 , we examine risk and reward from fuller international diversi-
fication. Observe first that an equally weighted portfolio of all countries is the riskiest in the
group. At the same time, because this portfolio assigns much larger weights to the smaller,
high-volatility–high-return countries, it also provides a higher average return. At the other
extreme, consider the minimum-variance portfolios, with and without short-sale constraints.
Without the short-sale restriction, the minimum-variance portfolio attains the amazingly low
SD of 2.21%, less than half that of the lowest-SD country (the U.S.). However, this portfolio
is probably not practical, including 22 short positions, the largest being 2 15% (in Sweden).
When short sales are disallowed, the minimum SD is far higher, 4.14%, offering much less
improvement over the capitalization-weighted portfolio. Moreover, these portfolio weights
also would be impractical, with the largest weight in Malaysia (29%), and only 7% in the U.S.
One puzzling and instructive feature of the results in Table 25.11 is the lower average
return on the actual world portfolio (ACWI) compared with the 44-country portfolios. The
difference arises because MSCI country-index portfolios are not capitalization-weighted
portfolios. MSCI uses industry-weighted portfolios, which places greater weights on the
larger stocks in each country. Since small stocks performed better over 2000–2009, the
ACWI portfolio had a lower average return. This pattern is not guaranteed, or necessarily
even likely, to apply to future returns.
25.5 International Investing and Performance Attribution
The benefits from international diversification may be modest for passive investors, but for
active managers international investing offers greater opportunities. International investing
calls for specialization in additional fields of analysis: currency, country and worldwide
industry, as well as a greater universe for stock selection.
Constructing a Benchmark Portfolio of Foreign Assets Active international investing, as well as passive, requires a benchmark portfolio (the
bogey). One widely used index of non-U.S. stocks is the European, Australasia, Far East (EAFE) index computed by Morgan Stanley. Additional indexes of world equity performance are published by Capital International Indices, Salomon Brothers, Credit
Suisse First Boston, and Goldman Sachs. Portfolios designed to mirror or even replicate
the country, currency, and company representation of these indexes would be the obvious
generalization of the purely domestic passive equity strategy.
An issue that sometimes arises in the international context is the appropriateness of
market-capitalization weighting schemes in the construction of international indexes.
Capitalization weighting is far and away the most common approach. However, some
argue that it might not be the best weighting scheme in an international context. This is in
part because different countries have differing proportions of their corporate sector orga-
nized as publicly traded firms.
Table 25.12 shows 1998 and 2011 data for market-capitalization weights versus GDP
weights for countries in the EAFE index. These data reveal substantial disparities between
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C H A P T E R 2 5 International Diversification 917
2011 1998
Country % of EAFE Market
Capitalization % of EAFE GDP % of EAFE Market
Capitalization % of EAFE GDP
Japan 21.1% 23.7% 26.8% 29.1% United Kingdom 17.9 9.8 22.4 10.5 France 9.3 11.1 7.2 10.7 Germany 7.5 14.2 8.9 15.8 Switzerland 6.8 2.3 6.0 1.9 Italy 2.9 8.9 3.9 8.8 Netherlands 2.4 3.4 5.9 2.9 Hong Kong 8.8 1.0 4.0 1.2 Australia 6.7 4.0 2.9 2.7 Spain 3.5 6.1 2.7 4.3 Sweden 2.8 2.0 2.4 1.8 Finland 0.9 1.0 0.7 1.0 Belgium 1.4 2.0 1.4 1.8 Singapore 2.7 0.9 1.1 0.6 Denmark 1.1 1.3 0.9 1.3 Norway 1.5 1.8 0.6 1.1 Ireland 0.4 0.9 0.5 0.6 Greece 0.2 1.3 0.3 0.9 Portugal 0.4 1.0 0.6 0.8 Austria 0.5 1.6 0.4 1.6 New Zealand 0.2 0.6 0.4 0.4
Table 25.12
Weighting schemes for EAFE countries
Source: Datastream, online.thomsonreuters.com/datastream .
the relative sizes of market capitalization and GDP. Since market capitalization is a stock
figure (the value of equity at one point in time), while GDP is a flow figure (production
of goods and services during the entire year), we expect capitalization to be more volatile
and the relative shares to be more variable over time. Some discrepancies are persistent,
however. For example, Hong Kong’s share of capitalization in 2011 is about eight times its
share of GDP, while Germany’s share of capitalization is much less than its share of GDP.
These disparities indicate that a greater proportion of economic activity is conducted by
publicly traded firms in Hong Kong than in Germany.
Some argue that it would be more appropriate to weight international indexes by GDP
rather than market capitalization. The justification for this view is that an internationally
diversified portfolio should purchase shares in proportion to the broad asset base of each
country, and GDP might be a better measure of the importance of a country in the inter-
national economy than the value of its outstanding stocks. Others have even suggested
weights proportional to the import share of various countries. The argument is that inves-
tors who wish to hedge the price of imported goods might choose to hold securities in
foreign firms in proportion to the goods imported from those countries.
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918
International Investing Raises Questions
As Yogi Berra might say, the problem with international investing is that it’s so darn foreign.
Currency swings? Hedging? International diversifica- tion? What’s that?
Here are answers to five questions that I’m often asked:
• Foreign stocks account for some 60% of world stock market value, so shouldn’t you have 60% of your stock market money overseas?
The main reason to invest abroad isn’t to replicate the global market or to boost returns. Instead, “what we’re trying to do by adding foreign stocks is to reduce volatil- ity,” explains Robert Ludwig, chief investment officer at money manager SEI Investments.
Foreign stocks don’t move in sync with U.S. shares and, thus, they may provide offsetting gains when the U.S. mar- ket is falling. But to get the resulting risk reduction, you don’t need anything like 60% of your money abroad.
• So, how much foreign exposure do you need to get decent diversification?
“Based on the volatility of foreign markets and the cor- relation between markets, we think an optimal portfolio is 70% in the U.S., 20% in developed foreign markets, and 10% in emerging markets,” Mr. Ludwig says.
Even with a third of your stock market money in foreign issues, you may find that the risk-reduction benefits aren’t all that reliable. Unfortunately, when U.S. stocks get really pounded, it seems foreign shares also tend to tumble.
• Can U.S. companies with global operations give you international diversification?
“When you look at these multinationals, the factor that drives their performance is their home market,” says Mark Riepe, a vice president with Ibbotson Associates, a Chicago research firm.
How come? U.S. multinationals tend to be owned by U.S. investors, who will be swayed by the ups and downs of the U.S. market. In addition, Mr. Riepe notes that while multinationals may derive substantial profits and revenue abroad, most of their costs—especially labor costs—will be incurred in the U.S.
• Does international diversification come from the for- eign stocks or the foreign currency?
“It comes from both in roughly equal pieces,” Mr. Riepe says. “Those who choose to hedge their foreign currency raise the correlation with U.S. stocks, and so the diversifica- tion benefit won’t be nearly as great.”
Indeed, you may want to think twice before investing in a foreign-stock fund that frequently hedges its currency exposure in an effort to mute the impact of—and make money from—changes in foreign-exchange rates.
“The studies that we’ve done show that stock managers have hurt themselves more than they’ve helped themselves by actively managing currencies,” Mr. Ludwig says.
• Should you divvy up your money among foreign coun- tries depending on the size of each national stock market? At issue is the nagging question of how much to put in
Japan. If you replicated the market weightings of Morgan Stanley Capital International’s Europe, Australasia and Far East index, you would currently have around a third of your overseas money in Japan.
That’s the sort of weighting you find in interna- tional funds, which seek to track the performance of the EAFE or similar international indexes. Actively managed foreign-stock funds, by contrast, pay less attention to market weights and on average, these days have just 14% in Japan.
If your focus is risk reduction rather than performance, the index—and the funds that track it—are the clear win- ners. Japan performs quite unlike the U.S. market, so it provides good diversification for U.S. investors, says Tricia Rothschild, international editor at Morningstar Mutual Funds, a Chicago newsletter.
“But correlations aren’t static,” she adds. “There’s always a problem with taking what happened over the past 20 years and projecting it out over the next 20 years.”
Source: Jonathan Clements, “International Investing Raises Questions on Allocation, Diversification, Hedging,” The Wall Street Journal, July 29, 1997. Excerpted by permission of The Wall Street Journal. © 1997 Dow Jones & Company, Inc. All rights reserved worldwide.
W O
R D
S FR
O M
T H
E S
T R
E E T
Performance Attribution We can measure the contribution of each of these factors following a manner similar to the
performance attribution techniques introduced in Chapter 24.
1. Currency selection measures the contribution to total portfolio performance attrib- utable to exchange rate fluctuations relative to the investor’s benchmark currency,
which we will take to be the U.S. dollar. We might use a benchmark like the EAFE
index to compare a portfolio’s currency selection for a particular period to a passive
benchmark. EAFE currency selection would be computed as the weighted average
of the currency appreciation of the currencies represented in the EAFE portfolio
using as weights the fraction of the EAFE portfolio invested in each currency.
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eXcel APPLICATIONS: International Portfolios
T his Excel model provides an efficient frontier analysis similar to that in Chapter 6. In Chapter 6 the frontier was based on individual securities, whereas this model examines the returns on international exchange–traded funds and enables us to analyze the benefits of interna- tional diversification. Go to the Online Learning Center at www.mhhe.com/bkm .
Excel Questions
1. Find three points on the efficient frontier corresponding to three different expected returns. What are the portfolio standard deviations corresponding to each expected return?
2. Now assume that the correlation between the S&P 500 and the other country indexes is cut in half. Find the new stan- dard deviations corresponding to each of the three expected returns. Are they higher or lower? Why?
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
A B C D E F G H I J Bordered Covariance Matrix for Target Return Portfolio
EWD EWH EWI EWJ EWL EWP EWW SP 500 Weights 0.00 0.00 0.08 0.38 0.02 0.00 0.00 0.52
0.0000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0826 0.00 0.00 4.63 3.21 0.55 0.00 0.00 7.69 0.3805 0.00 0.00 3.21 98.41 1.82 0.00 0.00 53.79 0.0171 0.00 0.00 0.55 1.82 0.14 0.00 0.00 2.09 0.0000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.5198 0.00 0.00 7.69 53.79 2.09 0.00 0.00 79.90 1.0000 0.00 0.00 16.07 157.23 4.59 0.00 0.00 143.47
Port Via 321.36 Port S.D. 17.93 Port Mean 12.00
Weights Mean St. Dev EWD EWH EWI EWJ EWL EWP EWW SP 500
6 21.89 0.02 0.00 0.00 0.71 0.00 0.02 0.00 0.26 9 19.66 0.02 0.00 0.02 0.53 0.02 0.00 0.00 0.41
12 17.93 0.00 0.00 0.08 0.38 0.02 0.00 0.00 0.52 15 16.81 0.00 0.00 0.14 0.22 0.02 0.00 0.00 0.62 18 16.46 0.00 0.00 0.19 0.07 0.02 0.00 0.00 0.73 21 17.37 0.00 0.00 0.40 0.00 0.00 0.00 0.00 0.60 24 21.19 0.00 0.00 0.72 0.00 0.00 0.00 0.00 0.28 27 26.05 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00
919
2. Country selection measures the contribution to performance attributable to invest- ing in the better-performing stock markets of the world. It can be measured as the
weighted average of the equity index returns of each country using as weights the share of the manager’s portfolio in each country. We use index returns to abstract
from the effect of security selection within countries. To measure a manager’s
contribution relative to a passive strategy, we might compare country selection to
the weighted average across countries of equity index returns using as weights the
share of the EAFE portfolio in each country.
3. Stock selection ability may, as in Chapter 24, be measured as the weighted aver- age of equity returns in excess of the equity index in each country. Here, we would use local currency returns and use as weights the
investments in each country.
4. Cash/bond selection may be measured as the excess return derived from weighting bonds and
bills differently from some benchmark weights.
Table 25.13 gives an example of how to measure the
contribution of the decisions an international portfolio
manager might make.
Using the data in Table 25.13 , compute the man- ager’s country and currency selection if portfo- lio weights had been 40% in Europe, 20% in Australia, and 40% in the Far East.
CONCEPT CHECK 25.3
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920 P A R T V I I Applied Portfolio Management
EAFE Weight
Return on Equity Index
Currency Appreciation
E1/E0 2 1 Manager’s
Weight Manager’s
Return
Europe 0.30 10% 10% 0.35 8% Australia 0.10 5 210 0.10 7 Far East 0.60 15 30 0.55 18
Overall performance (dollar return 5 return on index 1 currency appreciation)
EAFE: .30(10 1 10) 1 .10(5 2 10) 1 .60(15 1 30) 5 32.5% Manager: .35(8 1 10) 1 .10(7 2 10) 1 .55(18 1 30) 5 32.4%
Loss of .10% relative to EAFE
Currency selection
EAFE: (0.30 3 10%) 1 (0.10 3 (210%)) 1 (0.60 3 30%) 5 20% appreciation Manager: (0.35 3 10%) 1 (0.10 3 (210%)) 1 (0.55 3 30%) 5 19% appreciation Loss of 1% relative to EAFE
Country selection
EAFE: (0.30 3 10%) 1 (0.10 3 5%) 1 (0.60 3 15%) 5 12.5% Manager: (0.35 3 10%) 1 (0.10 3 5%) 1 (0.55 3 15%) 5 12.25% Loss of 0.25% relative to EAFE
Stock selection
(8% 2 10%)0.35 1 (7% 2 5%)0.10 1 (18% 2 15%)0.55 5 1.15% Contribution of 1.15% relative to EAFE
Sum of attributions (equal to overall performance) Currency (21%) 1 country (2.25%) 1 selection (1.15%) 5 2.10%
Table 25.13
Example of performance attribu- tion: international
1. U.S. assets are only a part of the world portfolio. International capital markets offer important opportunities for portfolio diversification with enhanced risk–return characteristics.
2. Exchange rate risk imparts an extra source of uncertainty to investments denominated in foreign currencies. Much of that risk can be hedged in foreign exchange futures or forward markets, but
a perfect hedge is not feasible unless the foreign currency rate of return is known.
3. Several world market indexes can form a basis for passive international investing. Active inter- national management can be partitioned into currency selection, country selection, stock selec-
tion, and cash/bond selection.
SUMMARY
exchange rate risk
interest rate parity relationship
covered interest arbitrage
relationship
political risk
Europe, Australasia, Far East
(EAFE) index
currency selection
country selection
stock selection
cash/bond selection
KEY TERMS
Related Web sites for this chapter are available at www. mhhe.com/bkm
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C H A P T E R 2 5 International Diversification 921
Interest rate parity (covered interest arbitrage) for direct ($/foreign currency) exchange rates:
F0 5 E0 1 1 rf (U.S.)
1 1 rf (foreign)
Interest rate parity for indirect (foreign currency/$) exchange rates:
F0 5 E0 1 1 rf (foreign)
1 1 rf (U.S.)
KEY EQUATIONS
1. Return to the box “International Investing Raises Questions” on page 918. The article was writ-
ten several years ago. Do you agree with its response to the question, “Can U.S. companies with
global operations give you international diversification?”
2. I n Figure 25.2 , we provide stock market returns in both local and dollar-denominated terms.
Which of these is more relevant? What does this have to do with whether the foreign exchange
risk of an investment has been hedged?
3. S uppose a U.S. investor wishes to invest in a British firm currently selling for £40 per share. The
investor has $10,000 to invest, and the current exchange rate is $2/£.
a. How many shares can the investor purchase? b. Fill in the table below for rates of return after 1 year in each of the nine scenarios (three pos-
sible prices per share in pounds times three possible exchange rates).
Dollar-Denominated Return for Year-End Exchange Rate
Price per Share (£) Pound-Denominated
Return (%) $1.80/£ $2/£ $2.20/£
£35
£40
£45
c. When is the dollar-denominated return equal to the pound-denominated return?
4. If each of the nine outcomes in Problem 3 is equally likely, find the standard deviation of both the
pound- and dollar-denominated rates of return.
5. Now suppose the investor in Problem 3 also sells forward £5,000 at a forward exchange rate of
$2.10/£.
a. Recalculate the dollar-denominated returns for each scenario. b. What happens to the standard deviation of the dollar-denominated return? Compare it to both
its old value and the standard deviation of the pound-denominated return.
6. Calculate the contribution to total performance from currency, country, and stock selection for the
manager in the example below. All exchange rates are expressed as units of foreign currency that
can be purchased with 1 U.S. dollar.
EAFE Weight Return on
Equity Index E1/E0 Manager’s
Weight Manager’s
Return
Europe 0.30 20% 0.9 0.35 18%
Australasia 0.10 15 1.0 0.15 20
Far East 0.60 25 1.1 0.50 20
7. If the current exchange rate is $1.75/£, the 1-year forward exchange rate is $1.85/£, and the inter-
est rate on British government bills is 8% per year, what risk-free dollar-denominated return can
be locked in by investing in the British bills?
Basic
Intermediate
PROBLEM SETS
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922 P A R T V I I Applied Portfolio Management
8. If you were to invest $10,000 in the British bills of Problem 7, how would you lock in the dollar-
denominated return?
9. M uch of this chapter was written from the perspective of a U.S. investor. But suppose you are
advising an investor living in a small country (choose one to be concrete). How might the lessons
of this chapter need to be modified for such an investor?
Challenge
1. You are a U.S. investor who purchased British securities for £2,000 one year ago when the British
pound cost U.S.$1.50. What is your total return (based on U.S. dollars) if the value of the securities
is now £2,400 and the pound is worth $1.75? No dividends or interest were paid during this period.
2. The correlation coefficient between the returns on a broad index of U.S. stocks and the returns on
indexes of the stocks of other industrialized countries is mostly _____ , and the correlation coef-
ficient between the returns on various diversified portfolios of U.S. stocks is mostly _____ .
a. less than .8; greater than .8. b. greater than .8; less than .8. c. less than 0; greater than 0. d. greater than 0; less than 0.
3. An investor in the common stock of companies in a foreign country may wish to hedge against
the _____ of the investor’s home currency and can do so by _____ the foreign currency in the
forward market.
a. depreciation; selling. b. appreciation; purchasing. c. appreciation; selling. d. depreciation; purchasing.
4. John Irish, CFA, is an independent investment adviser who is assisting Alfred Darwin, the head of
the Investment Committee of General Technology Corporation, to establish a new pension fund.
Darwin asks Irish about international equities and whether the Investment Committee should
consider them as an additional asset for the pension fund.
a. Explain the rationale for including international equities in General’s equity portfolio. Identify and describe three relevant considerations in formulating your answer.
b. List three possible arguments against international equity investment and briefly discuss the significance of each.
c. To illustrate several aspects of the performance of international securities over time, Irish shows Darwin the accompanying graph of investment results experienced by a U.S. pension
fund in the recent past. Compare the performance of the U.S. dollar and non-U.S. dollar equity
and fixed-income asset categories, and explain the significance of the result of the account
performance index relative to the results of the four individual asset class indexes.
10 20 30 40
6
0
1
2
3
4
5
Annualized Historical Performance Data (%)
Variability (standard deviation)
U.S. $ Bonds Non-U.S. $ Bonds
EAFE Index Account Performance Index
S&P Index
Real Returns (%)
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C H A P T E R 2 5 International Diversification 923
5. You are a U.S. investor considering purchase of one of the following securities. Assume that the
currency risk of the Canadian government bond will be hedged, and the 6-month discount on
Canadian dollar forward contracts is 2 .75% versus the U.S. dollar.
Bond Maturity Coupon Price
U.S. government 6 months 6.50% 100.00
Canadian government 6 months 7.50% 100.00
Calculate the expected price change required in the Canadian government bond that would result
in the two bonds having equal total returns in U.S. dollars over a 6-month horizon. Assume that
the yield on the U.S. bond is expected to remain unchanged.
6. A global manager plans to invest $1 million in U.S. government cash equivalents for the next
90 days. However, she is also authorized to use non-U.S. government cash equivalents, as long as
the currency risk is hedged to U.S. dollars using forward currency contracts.
a. What rate of return will the manager earn if she invests in money market instruments in either Canada or Japan and hedges the dollar value of her investment? Use the data in the following
tables.
b. What must be the approximate value of the 90-day interest rate available on U.S. government securities?
Interest Rates (APR) 90-Day Cash Equivalents
Japanese government 2.52%
Canadian government 6.74%
Exchange Rates Dollars per Unit of Foreign Currency
Spot 90-Day Forward
Japanese yen .0119 .0120
Canadian dollar .7284 .7269
7. The Windsor Foundation, a U.S.-based, not-for-profit charitable organization, has a diversified
investment portfolio of $100 million. Windsor’s board of directors is considering an initial invest-
ment in emerging market equities. Robert Houston, treasurer of the foundation, has made the
following four comments:
a. “For an investor holding only developed market equities, the existence of stable emerging market currencies is one of several preconditions necessary for that investor to realize strong
emerging market performance.”
b. “Local currency depreciation against the dollar has been a frequent occurrence for U.S. inves- tors in emerging markets. U.S. investors have consistently seen large percentages of their
returns erased by currency depreciation. This is true even for long-term investors.”
c. “Historically, the addition of emerging market stocks to a U.S. equity portfolio such as the S&P 500 index has reduced volatility; volatility has also been reduced when emerging market
stocks are combined with an international portfolio such as the MSCI EAFE index.”
d. “Although correlations among emerging markets can change over the short term, such correla- tions show evidence of stability over the long term. Thus, an emerging markets portfolio that
lies on the efficient frontier in one period tends to remain close to the frontier in subsequent
periods.”
Discuss whether each of Houston’s four comments is correct or incorrect.
8. After much research on the developing economy and capital markets of the country of Otunia,
your firm, GAC, has decided to include an investment in the Otunia stock market in its Emerging
Markets Commingled Fund. However, GAC has not yet decided whether to invest actively or by
indexing. Your opinion on the active versus indexing decision has been solicited. The following is
a summary of the research findings:
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924 P A R T V I I Applied Portfolio Management
Otunia’s economy is fairly well diversified across agricultural and natural resources, manufacturing (both consumer and durable goods), and a grow- ing finance sector. Transaction costs in securities markets are relatively large in Otunia because of high commissions and government “stamp taxes” on securi- ties trades. Accounting standards and disclosure regulations are quite detailed, resulting in wide public availability of reliable information about companies’ financial performance.
Capital flows into and out of Otunia, and foreign ownership of Otunia secu- rities is strictly regulated by an agency of the national government. The settle- ment procedures under these ownership rules often cause long delays in settling trades made by nonresidents. Senior finance officials in the government are working to deregulate capital flows and foreign ownership, but GAC’s political consultant believes that isolationist sentiment may prevent much real progress in the short run.
a. Briefly discuss aspects of the Otunia environment that favor investing actively, and aspects that favor indexing.
b. Recommend whether GAC should invest in Otunia actively or by indexing. Justify your rec- ommendation based on the factors identified in part ( a ).
E-INVESTMENTS EXERCISES A common misconception is that investors can earn excess returns by investing in foreign bonds with higher interest rates than are available in the U.S. Interest rate parity implies that any such interest rate differentials will be offset by premiums or discounts in the for- ward or futures market for foreign currency.
Interest rates on government bonds in the U.S., U.K., Japan, Germany, Brazil, and Australia can be found at www.bloomberg.com/markets/rates/index.html .
Spot exchange rates on international currencies can be found at www.bloomberg. com/markets/currencies/fxc.html .
Forward exchange rates on currency futures contracts can be found at www.cmegroup. com/trading/fx/index.html .
1. Select one of these countries and record the yield on a short-term government security from the Bloomberg Web site. Also make note of the U.S. Treasury yield on an instru- ment with the same maturity.
2. Record the spot exchange rate from the Bloomberg site and the futures contract exchange rate from the CME Web site for the date closest to the maturity of the invest- ment you chose in the previous step.
3. Calculate the rate of return available on the foreign government security, converting the foreign currency transactions into dollars at the current and forward exchange rates.
4. How well does interest rate parity seem to hold? Are there bargains to be found in other currencies? What factors might account for interest rate parity violation?
SOLUTIONS TO CONCEPT CHECKS 1. 1 1 r (US) 5 [1 1 r f (UK)] 3 ( E 1 / E 0 )
a. 1 1 r (US) 5 1.1 3 1.0 5 1.10. Therefore, r (US) 5 10%. b. 1 1 r (US) 5 1.1 3 1.1 5 1.21. Therefore, r (US) 5 21%.
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C H A P T E R 2 5 International Diversification 925
2. You must sell forward the number of pounds you will end up with at the end of the year. This value
cannot be known with certainty, however, unless the rate of return of the pound-denominated
investment is known.
a. 10,000 3 1.20 5 12,000 pounds. b. 10,000 3 1.30 5 13,000 pounds.
3. Country selection:
(0.40 3 10%) 1 (0.20 3 5%) 1 (0.40 3 15%) 5 11%
This is a loss of 1.5% (11% versus 12.5%) relative to the EAFE passive benchmark.
Currency selection:
(0.40 3 10%) 1 (0.20 3 (210%)) 1 (0.40 3 30%) 5 14%
This is a loss of 6% (14% versus 20%) relative to the EAFE benchmark.
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26 26 22 PA
R T V
II
WHILE MUTUAL FUNDS are still the domi- nant form of investing in securities markets for most individuals, hedge funds enjoyed far greater growth rates in the last decade, with assets under management increasing from $200 billion in 1997 to about $2 trillion in 2012. Like mutual funds, hedge funds allow private investors to pool assets to be invested by a fund manager. Unlike mutual funds, however, they are commonly organized as private partnerships and thus not subject to many SEC regulations. They typically are open only to wealthy or institutional investors.
Hedge funds touch on virtually every issue discussed in the earlier chapters of the text, including liquidity, security analysis, market efficiency, portfolio analysis, hedg- ing, and option pricing. For example, these funds often bet on relative mispricing of specific securities, but hedge broad market exposure. This sort of pure “alpha seeking” behavior requires a procedure for optimally mixing a hedge fund position with a more
traditional portfolio. Other funds engage in aggressive market timing; their risk profiles can shift rapidly and substantially, raising dif- ficult questions for performance evaluation. Many hedge funds take extensive deriva- tives positions. Even those funds that do not trade derivatives charge incentive fees that resemble the payoff to a call option; an option-pricing background therefore is neces- sary to interpret both hedge fund strategies and costs. In short, hedge funds raise the full range of issues that one might confront in active portfolio management.
We begin with a survey of various hedge fund orientations. We devote considerable attention to the classic “market-neutral” or hedged strategies that historically gave hedge funds their name. We move on to evidence on hedge fund performance, and the difficul- ties in evaluating that performance. Finally, we consider the implications of their unusual fee structure for investors in and managers of such funds.
CHAPTER TWENTY-SIX
Hedge Funds
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C H A P T E R 2 6 Hedge Funds 927
Like mutual funds, the basic idea behind hedge funds is investment pooling. Investors buy shares in these funds, which then invest the pooled assets on their behalf. The net
asset value of each share represents the value of the investor’s stake in the portfolio. In this
regard, hedge funds operate much like mutual funds. However, there are important differ-
ences between the two.
Transparency Mutual funds are subject to the Securities Act of 1933 and the Invest- ment Company Act of 1940 (designed to protect unsophisticated investors), which require
transparency and predictability of strategy. They periodically must provide the public with
information on portfolio composition. In contrast, hedge funds usually are set up as limited
liability partnerships, and provide minimal information about portfolio composition and
strategy to their investors only.
Investors Hedge funds traditionally have no more than 100 “sophisticated” investors, in practice usually defined by minimum net worth and income requirements. They gener-
ally do not advertise to the general public, and minimum investments usually are between
$250,000 and $1 million.
Investment Strategies Mutual funds lay out their general investment approach (e.g., large, value stock orientation versus small-cap growth orientation) in their prospectus. They
face pressure to avoid style drift (departures from their stated investment orientation), espe- cially given the importance of retirement funds such as 401(k) plans to the industry, and the
demand of such plans for predictable strategies. Most mutual funds promise to limit their
use of short-selling and leverage, and their use of derivatives is highly restricted. In recent
years, some so-called 130/30 mutual funds 1 have opened, primarily for institutional clients,
with prospectuses that explicitly allow for more active short-selling and derivatives posi-
tions, but even these have less flexibility than hedge funds. In contrast, hedge funds may
effectively partake in any investment strategy and may act opportunistically as conditions
evolve. For this reason, viewing hedge funds as anything remotely like a uniform asset class
would be a mistake. Hedge funds by design are empowered to invest in a wide range of
investments, with various funds focusing on derivatives, distressed firms, currency specula-
tion, convertible bonds, emerging markets, merger arbitrage, and so on. Other funds may
jump from one asset class to another as perceived investment opportunities shift.
Liquidity Hedge funds often impose lock-up periods, that is, periods as long as several years in which investments cannot be withdrawn. Many also employ redemption notices
that require investors to provide notice weeks or months in advance of their desire to
redeem funds. These restrictions limit the liquidity of investors but in turn enable the funds
to invest in illiquid assets where returns may be higher, without worrying about meeting
unanticipated demands for redemptions.
Compensation Structure Hedge funds also differ from mutual funds in their fee struc- ture. Whereas mutual funds assess management fees equal to a fixed percentage of assets,
for example, between .5% and 1.5% annually for typical equity funds, hedge funds charge
a management fee, usually between 1% and 2% of assets, plus a substantial incentive fee
26.1 Hedge Funds versus Mutual Funds
1 These are funds that may sell short up to 30% of the value of their portfolios, using the proceeds of the sale to
increase their positions in invested assets. So for every $100 in net assets, the fund could sell short $30, investing
the proceeds to increase its long positions to $130. This gives rise to the 130/30 moniker.
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928 P A R T V I I Applied Portfolio Management
equal to a fraction of any investment profits beyond some benchmark. The incentive fee
is often 20%. The threshold return to earn the incentive fee is often a money market rate
such as LIBOR. Indeed, some observers only half-jokingly characterize hedge funds as “a
compensation scheme masquerading as an asset class.”
Convertible arbitrage Hedged investing in convertible securities, typically long convertible bonds and short stock.
Dedicated short bias Net short position, usually in equities, as opposed to pure short exposure.
Emerging markets Goal is to exploit market inefficiencies in emerging markets. Typically long-only because short-selling is not feasible in many of these markets.
Equity market neutral Commonly uses long/short hedges. Typically controls for industry, sector, size, and other exposures, and establishes market-neutral positions designed to exploit some market inefficiency. Commonly involves leverage.
Event driven Attempts to profit from situations such as mergers, acquisitions, restructuring, bankruptcy, or reorganization.
Fixed-income arbitrage Attempts to profit from price anomalies in related interest rate securities. Includes interest rate swap arbitrage, U.S. versus non-U.S. government bond arbitrage, yield-curve arbitrage, and mortgage-backed arbitrage.
Global macro Involves long and short positions in capital or derivative markets across the world. Port- folio positions reflect views on broad market conditions and major economic trends.
Long/short equity hedge Equity-oriented positions on either side of the market (i.e., long or short), depend- ing on outlook. Not meant to be market neutral. May establish a concentrated focus regionally (e.g., U.S. or Europe) or on a specific sector (e.g., tech or health care stocks). Derivatives may be used to hedge positions.
Managed futures Uses financial, currency, or commodity futures. May make use of technical trading rules or a less structured judgmental approach.
Multistrategy Opportunistic choice of strategy depending on outlook.
Fund of funds Fund allocates its cash to several other hedge funds to be managed.
Table 26.1
Hedge fund styles CS/TASS (Credit Suisse/Tremont Advisors Shareholder Services) maintains one of the most comprehensive databases on hedge fund performance. It categorizes hedge funds into these 11 different investment styles.
Table 26.1 lists most of the common investment themes found in the hedge fund industry.
The list contains a wide diversity of styles and suggests how hard it can be to speak generi-
cally about hedge funds as a group. We can, however, divide hedge fund strategies into two
general categories: directional and nondirectional.
Directional and Nondirectional Strategies Directional strategies are easy to understand. They are simply bets that one sector or another will outperform other sectors of the market.
In contrast, nondirectional strategies are usually designed to exploit temporary misalign- ments in security valuations. For example, if the yield on corporate bonds seems abnormally
high compared to that on Treasury bonds, the hedge fund would buy corporates and short sell
26.2 Hedge Fund Strategies
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C H A P T E R 2 6 Hedge Funds 929
Treasury securities. Notice that the fund is not betting on broad movements in the entire bond market: It buys one type of bond and sells another. By taking a long corporate–short Treasury
position, the fund hedges its interest rate exposure while making a bet on the relative valua- tion across the two sectors. The idea is that when yield spreads revert back to their “normal”
relationship, the fund will profit from the realignment regardless of the general trend in the
level of interest rates. In this respect, it strives to be market neutral, or hedged with respect to the direction of interest rates, which gives rise to the term “hedge fund.”
Nondirectional strategies are sometimes further divided into convergence or relative value positions. The difference between convergence and relative value is a time horizon at which one can say with confidence that any mispricing ought to be resolved. An example
of a convergence strategy would entail mispricing of a futures contract that must be cor-
rected by the time the contract matures. In contrast, the corporate versus Treasury spread
we just discussed would be a relative value strategy, because there is no obvious horizon
during which the yield spread would “correct” from unusual levels.
We can illustrate a market-neutral position with a strategy used extensively by several hedge funds, which observed that newly issued or “on-the-run” 30-year Treasury bonds regularly sell at higher prices (lower yields) than 29½-year bonds with almost identi- cal duration. The yield spread presumably is a premium due to the greater liquidity of the on-the-run bonds. Hedge funds, which have relatively low liquidity needs, therefore buy the 29½-year bond and sell the 30-year bond. This is a hedged, or market-neutral, position that will generate a profit whenever the yields on the two bonds converge, as typically happens when the 30-year bonds age, are no longer the most liquid on-the-run bond, and are no longer priced at a premium.
Notice that this strategy should generate profits regardless of the general direction of interest rates. The long-short position will return a profit as long as the 30-year bonds underperform the 29½-year bonds, as they should when the liquidity premium dissi- pates. Because the pricing discrepancies between these two securities almost necessarily must disappear at a given date, this strategy is an example of convergence arbitrage. While the convergence date in this application is not quite as definite as the maturity of a futures contract, one can be sure that the currently on-the-run T-bonds will lose that status by the time the Treasury next issues 30-year bonds.
Example 26.1 Market-Neutral Positions
Long-short positions such as in Example 26.1 are characteristic of hedged strategies. They
are designed to isolate a bet on some mispricing without taking on market exposure. Profits are made regardless of broad market movements once prices “converge” or return to their
“proper” levels. Hence, use of short positions and derivatives is part and parcel of the industry.
A more complex long-short strategy is convertible bond arbitrage, one of the more prominent sectors of the hedge-fund universe. Noting that a convertible bond may be
viewed as a straight bond plus a call option on the underlying stock, the market-neutral
strategy in this case involves a position in the bond offset by an opposite position in the
stock. For example, if the convertible is viewed as underpriced, the fund will buy it and
offset its resultant exposure to declines in the stock price by shorting the stock.
Although these market-neutral positions are hedged, they are not risk-free arbitrage strategies. Rather they should be viewed as pure plays, that is, bets on particular (perceived) mispricing between two sectors or securities, with extraneous sources of risk
such as general market exposure hedged away. Moreover, because the funds often operate
with considerable leverage, returns can be quite volatile.
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930 P A R T V I I Applied Portfolio Management
Statistical Arbitrage Statistical arbitrage is a version of a market-neutral strategy, but one that merits its own discussion. It differs from pure arbitrage in that it does not exploit risk-free positions
based on unambiguous mispricing (such as index arbitrage). Instead, it uses quantitative
and often automated trading systems that seek out many temporary and modest misalign-
ments in prices among securities. By taking relatively small positions in many of these
opportunities, the law of averages would make the probability of profiting from the col-
lection of ostensibly positive-value bets very high, ideally almost a “statistical certainty.”
Of course, this strategy presumes that the fund’s modeling techniques can actually identify
reliable, if small, market inefficiencies. The law of averages will work for the fund only if
the expected return is positive!
Statistical arbitrage often involves trading in hundreds of securities a day with holding
periods that can be measured in minutes or less. Such rapid and heavy trading requires
extensive use of quantitative tools such as automated trading and mathematical algorithms
to identify profit opportunities and efficient diversification across positions. These strategies
try to profit from the smallest of perceived mispricing opportunities, and require the fastest
trading technology and the lowest possible trading costs. They would not be possible with-
out the electronic communication networks discussed in Chapter 3.
A particular form of statistical arbitrage is pairs trading, in which stocks are paired up based on an analysis of either fundamental similarities or market exposures (betas). The gen-
eral approach is to pair up similar companies whose returns are highly correlated but where
one company seems to be priced more aggressively than the other. 2 Market-neutral positions
can be formed by buying the relatively cheap firm and selling the expensive one. Many such
pairs comprise the hedge fund’s overall portfolio. Each pair may have an uncertain outcome,
but with many such matched pairs, the presumption is that the large number of long-short
bets will provide a very high probability of a positive abnormal return. More general versions
of pairs trading allow for positions in clusters of stocks that may be relatively mispriced.
Statistical arbitrage is commonly associated with data mining, which refers to sort- ing through huge amounts of historical data to uncover systematic patterns in returns that
can be exploited by traders. The risk of data mining, and statistical arbitrage in general,
is that historical relationships may break down when fundamental economic conditions
change or, indeed, that the apparent patterns in the data may be due to pure chance. Enough
analysis applied to enough data is sure to produce apparent patterns that do not reflect real
relationships that can be counted on to persist in the future.
Classify each of the following strategies as directional or nondirectional.
a. The fund buys shares in the India Investment Fund, a closed-end fund that is selling at a discount to net asset value, and sells the MSCI India Index Swap.
b. The fund buys shares in Petrie Stores and sells Toys “R” Us, which is a major component of Petrie’s balance sheet.
c. The fund buys shares in Generic Pharmaceuticals betting that it will be acquired at a premium by Pfizer.
CONCEPT CHECK 26.1
2 Rules for deciding relative “aggressiveness” of pricing may vary. In one approach, a computer scans for stocks
whose prices historically have tracked very closely but have recently diverged. If the differential in cumulative
return typically dissipates, the fund will buy the recently underperforming stock and sell the outperforming one.
In other variants, pricing aggressiveness may be determined by evaluating the stocks based on some measure of
price to intrinsic value.
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C H A P T E R 2 6 Hedge Funds 931
An important implication of the market-neutral pure play is the notion of portable alpha. Suppose that you wish to speculate on a stock that you think is underpriced, but you think
that the market is about to fall. Even if you are right about the stock being relatively under- priced, it still might decline in response to declines in the broad market. You would like
to separate the stock-specific bet from the implicit asset allocation bet on market perfor-
mance that arises because the stock’s beta is positive. The solution is to buy the stock
and eliminate the resultant market exposure by selling enough index futures to drive beta
to zero. This long stock–short futures strategy gives you a pure play or, equivalently, a
market-neutral position on the stock. More generally, you might wish to separate asset allocation from security selection. The
idea is to invest wherever you can “find alpha.” You would then hedge the systematic risk
of that investment to isolate its alpha from the asset market where it was found. Finally,
you establish exposure to desired market sectors by using passive products such as indexed
mutual funds, ETFs, or index futures. In other words, you have created portable alpha that
can be mixed with an exposure to whatever sector of the market you choose. This proce-
dure is also called alpha transfer, because you transfer alpha from the sector where you find it to the asset class in which you ultimately establish exposure. Finding alpha requires
skill. By contrast, beta, or market exposure, is a “commodity” that can be supplied cheaply
through index products and offers little value added.
An Example of a Pure Play Suppose you manage a $1.4 million portfolio. You believe that the alpha of the portfolio
is positive, a . 0, but also that the market is about to fall, that is, that r M , 0. You would therefore try to establish a pure play on the perceived mispricing.
The return on portfolio over the next month may be described by Equation 26.1, which
states that the portfolio return will equal its “fair” CAPM return (the first two terms on the
right-hand side), plus firm-specific risk reflected in the “residual,” e, plus an alpha that reflects perceived mispricing:
rportfolio 5 rf 1 b(rM 2 rf) 1 e 1 a (26.1)
To be concrete, suppose that b 5 1.20, a 5 .02, r f 5 .01, the current value of the S&P 500 index is S 0 5 1,344, and, for simplicity, that the portfolio pays no dividends. You want to capture the positive alpha of 2% per month, but you don’t want the positive beta
that the stock entails because you are worried about a market decline. So you choose to
hedge your exposure by selling S&P 500 futures contracts.
Because the S&P contracts have a multiplier of $250, and the portfolio has a beta of
1.20, your stock position can be hedged for 1 month by selling five futures contracts: 3
Hedge ratio 5 $1,400,000
1,344 3 $250 3 1.20 5 5 contracts
26.3 Portable Alpha
3 We simplify here by assuming that the maturity of the futures contract precisely equals the hedging horizon, in
this case, 1 month. If the contract maturity were longer, one would have to slightly reduce the hedge ratio in a
process called “tailing the hedge.”
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932 P A R T V I I Applied Portfolio Management
A warning: Even market-neutral positions are still bets, and they can go wrong. This
is not true arbitrage because your profits still depend on whether your analysis (your per-
ceived alpha) is correct. Moreover, you can be done in by simple bad luck, that is, your
analysis may be correct but a bad realization of idiosyncratic risk (negative values of e in Equation 26.1 or 26.2) can still result in losses.
The dollar value of the stock portfolio after 1 month will be
$1,400,000 3 (1 1 rportfolio) 5 $1,400,000 31 1 .01 1 1.20 (rM 2 .01) 1 .02 1 e4 5 $1,425,200 1 $1,680,000 3 rM 1 $1,400,000 3 e
The dollar proceeds from your futures position will be:
5 3 $250 3 (F0 2 F1) Mark to market on 5 contracts sold
5 $1,250 3 [S0 (1.01) 2 S1] Substitute for futures prices from parity relationship
5 $1,250 3 S0 [1.01 2 (1 1 rM)] Because S1 5 S0 (1 1 rM) when no dividends are paid
5 $1,250 3 [S0 (.01 2 rM)] Simplify
5 $16,800 2 $1,680,000 3 rM Because S0 5 1,344
The total value of the stock plus futures position at month’s end will be the sum of the
portfolio value plus the futures proceeds, which equals
Hedged proceeds 5 $1,442,000 1 $1,400,000 3 e (26.2)
Notice that the exposure to the market from your futures position precisely offsets your
exposure from the stock portfolio. In other words, you have reduced beta to zero. Your
investment is $1.4 million, so your total monthly rate of return is 3% plus the remain-
ing nonsystematic risk (the second term of Equation 26.2). The fair or equilibrium
expected rate of return on such a zero-beta position is the risk-free rate, 1%, so you
have preserved your alpha of 2%, while eliminating the market exposure of the stock
portfolio.
This is an idealized example of a pure play. In particular, it simplifies by assuming a
known and fixed portfolio beta, but it illustrates that the goal is to speculate on the stock
while hedging out the undesired market exposure. Once this is accomplished, you can
establish any desired exposure to other sources of systematic risk by buying indexes or
entering index futures contracts in those markets. Thus, you have made alpha portable.
Figure 26.1 is a graphical analysis of this pure play. Panel A shows the excess returns to
betting on a positive-alpha stock portfolio “naked,” that is, unhedged. Your expected return is better than an equilibrium return given your risk, but because of your market exposure
you still can lose if the market declines. Panel B shows the characteristic line for the posi-
tion with systematic risk hedged out. There is no market exposure.
What would be the dollar value and rate of return on the market-neutral posi- tion if the value of the residual turns out to be 2 4%? If the market return in that month is 5%, where would the plot of the strategy return lie in each panel of Figure 26.1 ?
CONCEPT CHECK 26.2
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C H A P T E R 2 6 Hedge Funds 933
Even market-neutral bets can result in considerable volatility because most hedge funds
use considerable leverage. Most incidents of relative mispricing are fairly minor, and
the hedged nature of long-short strategies makes overall volatility low. The hedge funds
respond by scaling up their bets. This amplifies gains when their bets work out, but also
amplifies losses. In the end, the volatility of the funds is not small.
Return on Positive Alpha Portfolio
Return for Fairly Priced Assets
Alpha = 2%
Excess Market Return, rM − rf
Excess Rate of Return, rp − rf
α = 2%
Characteristic line of your hedged (β = 0) portfolio is flat
Characteristic line of fairly priced zero-beta asset Total Market Return, rM
Total Return on Hedged Portfolio
← α = 2%
rf = 1%
3%
A
B
Figure 26.1 A pure play. Panel A, unhedged position. Panel B, hedged position.
An apparently market-neutral bet misfired badly in 1998. While the 30- versus 29½-year maturity T-bond strategy (see Example 26.1) worked well over several years, it blew up when Russia defaulted on its debt, triggering massive investment demand for the safest, most liquid assets that drove up the price of the 30-year Treasury relative to its 29½-year counterpart. The big losses that ensued illustrate that even the safest bet—one based on convergence arbitrage—carries risks. Although the T-bond spread had to converge eventually, and in fact it did several weeks later, Long Term Capital Management and other hedge funds suffered large losses on their positions when the spread widened temporarily. The ultimate convergence came too late for LTCM, which was also facing massive losses on its other positions and had to be bailed out. 4
Example 26.2 The Risks of Pure Plays
4 This timing problem is a common one for active managers. We saw other examples of this issue when we dis-
cussed limits to arbitrage in Chapter 12. More generally, when security analysts think they have found a mispriced
stock, they usually acknowledge that it is hard to know how long it will take for price to converge to intrinsic value.
While the classic hedge fund strategy may have focused on market-neutral opportunities,
as the market has evolved, the freedom to use derivatives contracts and short positions
means that hedge funds can in effect follow any investment strategy. While many hedge
26.4 Style Analysis for Hedge Funds
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934 P A R T V I I Applied Portfolio Management
funds pursue market-neutral strategies, a quick glance at the range of investment styles in
Table 26.1 should convince you that many, if not most, funds pursue directional strategies.
In these cases, the fund makes an outright bet, for example, on currency movements, the
outcome of a takeover attempt, or the performance of an investment sector. These funds are
most certainly not hedged, despite their name.
In Chapter 24, we introduced you to style analysis, which uses regression analysis
to measure the exposure of a portfolio to various factors or asset classes. The analysis
thus measures the implicit asset class exposure of a portfolio. The betas on a series of
factors measure the fund’s exposure to each source of systematic risk. A market-neutral
fund will have no sensitivity to an index for that market. In contrast, directional funds
will exhibit significant betas, often called loadings in this context, on whatever factors the fund tends to bet on. Observers attempting to measure investment style can use these
factor loadings to impute exposures to a range of variables.
We present a simple style analysis for the hedge fund indexes in Table 26.2 . The four
systematic factors we consider are:
• Interest rates: the return on long-term U.S. Treasury bonds.
• Equity markets: the return on the S&P 500.
• Credit conditions: the difference in the return on Baa-rated bonds over Treasury bonds.
• Foreign exchange: the percentage change in the value of the U.S. dollar against a
basket of foreign currencies.
The returns on hedge fund index i in month t may be statistically described by 5
Rit 5 ai 1 bi1 Factor1t 1 c1 bi4 Factor 4t 1 eit (26.3)
The betas (equivalently, factor loadings) measure the sensitivity to each factor. As usual,
the residual, e it , measures nonsystematic risk that is uncorrelated with the set of explana- tory factors, and the intercept, a i , measures average performance of fund i net of the impact of these systematic factors.
Table 26.2 presents factor exposure estimates for 13 hedge fund indexes. The results
confirm that most funds are in fact directional with very clear exposures to one or more
of the four factors. Moreover, the estimated factor betas seem reasonable in terms of the
funds’ stated style. For example:
• The equity market–neutral funds have uniformly low and statistically insignificant
factor betas, as one would expect of a market-neutral posture.
• Dedicated short bias funds exhibit substantial negative betas on the S&P index.
• Distressed-firm funds have significant exposure to credit conditions (more positive
credit spreads in this table indicate better economic conditions) as well as to the
S&P 500. This exposure arises because restructuring activities often depend on
access to borrowing, and successful restructuring depends on the state of the
economy.
• Global macro funds show negative exposure to a stronger U.S. dollar, which would
make the dollar value of foreign investments less valuable.
We conclude that, by and large, most hedge funds are making very explicit directional bets
on a wide array of economic factors.
5 This analysis differs in two important respects from style analysis for mutual funds introduced in Chapter 24.
First, in this application, factor loadings are not constrained to be non-negative. This is because, unlike mutual
funds, hedge funds easily can take on short positions in various asset classes. Second, portfolio weights are not
constrained to sum to 1.0. Again, unlike mutual funds, hedge funds can operate with considerable leverage.
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C H A P T E R 2 6 Hedge Funds 935
Fund Group* Alpha S&P 500 Long T-Bond Credit Premium U.S. Dollar
All funds 0.0052 3.3487
0.2718 5.0113
0.0189 0.3064
0.1755 2.0462
20.1897 22.1270
Market neutral 0.0014 0.1990
0.1677 0.6917
20.0163 20.0589
0.3308 0.8631
20.5097 21.2790
Short bias 0.0058 1.3381
20.9723 26.3684
0.1310 0.7527
0.3890 1.6113
20.2630 21.0476
Event driven 0.0071 5.1155
0.2335 4.7858
0.0000 20.0002
0.2056 2.6642
20.1165 0.1520
Risk arbitrage 0.0034 3.0678
0.1498 3.8620
0.0130 0.0442
20.0006 20.0097
20.2130 23.3394
Distressed 0.0068 5.7697
0.2080 4.9985
0.0032 0.0679
0.2521 3.8318
20.1156 21.6901
Emerging markets 0.0082 2.8867
0.3750 3.7452
0.2624 2.2995
0.4551 2.8748
20.2169 21.3173
Fixed Income 0.0018 1.0149
0.1719 2.8139
0.2284 3.2806
0.5703 5.9032
20.1714 21.7063
Convertible arb 0.0005 0.2197
0.2477 3.1066
0.2109 2.3214
0.5021 3.9825
20.0972 20.7414
Global macro 0.0079 3.5217
0.0746 0.9437
0.0593 0.6587
0.1492 1.1938
20.2539 21.9533
Long-short equity 0.0053 2.5693
0.4442 6.1425
20.0070 20.0850
0.0672 0.5874
20.1471 21.2372
Managed futures 0.0041 0.8853
0.2565 1.5944
20.2991 21.6310
20.5223 22.0528
20.2703 21.0217
Multistrategy 0.0075 4.2180
0.2566 4.1284
20.0048 20.0684
0.1781 1.8116
20.1172 21.1471
Table 26.2
Style analysis for a sample of hedge fund indexes *Fund definitions are given in Table 26.1 . Note: Top line of each entry is the estimate of the factor beta. Lower line is the t -statistic for that estimate. Source: Authors’ calculations. Hedge fund returns are on indexes computed by Credit Suisse/Tremont Index, LLC, available at www.hedgeindex.com .
Analyze the betas of the fixed-income arbitrage index in Table 26.2 . On the basis of these results, are these funds typically market neutral? If not, do their factor exposures make sense in terms of the markets in which they operate?
CONCEPT CHECK 26.3
Table 26.3 shows basic performance data for a collection of hedge fund indexes computed
from the standard index model with the S&P 500 used as the market benchmark. The
model is estimated using monthly excess returns over the period January 2005 through
26.5 Performance Measurement for Hedge Funds
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936 P A R T V I I Applied Portfolio Management
November 2011. We report for each index the beta relative to the S&P 500, the serial cor-
relation of returns, the Sharpe ratio, and the alpha. Betas tend to be considerably less than
one; not surprisingly, the beta of the short bias index is large and negative. The market-
neutral index has a beta near zero.
By and large, hedge fund performance is impressive. Most of the alpha estimates are
positive, and the average alpha is substantial, 0.17% per month. Similarly, most Sharpe
ratios exceed that of the S&P 500, and the average Sharpe ratio across hedge fund groups,
0.123, is four times that of the S&P 500. What might be the source of such performance?
One possibility, of course, is the obvious one: These results may reflect a high degree of
skill among hedge fund managers. Another possibility is that funds maintain some expo-
sure to omitted risk factors that convey a positive risk premium. One likely candidate
for such a factor would be liquidity, and we will see shortly that liquidity and liquidity
risk are associated with average returns. Moreover, several other factors make hedge fund
performance difficult to evaluate, and these too are worth considering.
Liquidity and Hedge Fund Performance One explanation for apparently attractive hedge fund performance is liquidity. Recall from
Chapter 9 that one of the more important extensions of the CAPM is a version that allows
for the possibility of a return premium for investors willing to hold less liquid assets.
Hedge funds tend to hold more illiquid assets than other institutional investors such as
mutual funds. They can do so because of restrictions such as the lock-up provisions that
commit investors to keep their investment in the fund for some period of time. Therefore,
it is important to control for liquidity when evaluating performance. If it is ignored, what
Table 26.3
Index model regressions for hedge fund indexes Estimation period: January 2005–November 2011 Source: Authors’ calculations using data downloaded from Hedge Fund Research Inc., www.hedgefundresearch.com , March 2012.
Beta Serial Correlation Alpha Sharpe Ratio
Hedge fund composite index 0.355 0.321 0.200 0.123 Event-driven: Distressed 0.324 0.570 0.206 0.120 Event-driven: Merger arbitrage 0.153 0.254 0.273 0.287 Event-driven: All 0.368 0.466 0.216 0.128 Market neutral 0.090 0.133 2 0.007 0.007 Short bias 2 0.668 0.147 2 0.169 2 0.076 Emerging markets 0.618 0.357 0.415 0.133 Long/short hedge 0.506 0.306 0.097 0.061 Fund of funds 0.261 0.361 2 0.016 0.012 Relative value 0.245 0.576 0.300 0.204 Fixed income: Asset backed 0.088 0.570 0.468 0.468 Fixed income: Convertible arbitrage 0.435 0.597 0.163 0.072 Fixed income: Corporate 0.322 0.585 0.113 0.075 Multi-strategy 0.241 0.565 0.136 0.100 S&P 500 1.000 0.218 0.000 0.031 Average across hedge funds 0.238 0.415 0.171 0.123
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C H A P T E R 2 6 Hedge Funds 937
may be no more than compensation for illiquidity may appear to be true alpha, that is, risk-
adjusted abnormal returns.
Aragon demonstrates that hedge funds with lock-up restrictions do tend to hold less
liquid portfolios. 6 Moreover, once he controlled for lock-ups or other share restrictions
(such as redemption notice periods), the apparently positive average alpha of those
funds turned insignificant. Aragon’s work suggests that the typical “alpha” exhibited by
hedge funds may be interpreted as an equilibrium liquidity premium rather than a sign
of stock-picking ability, in other words a “fair” reward for providing liquidity to other
investors.
One symptom of illiquid assets is serial correlation in returns. Positive serial correla-
tion means that positive returns are more likely to be followed by positive than by negative
returns. Such a pattern is often taken as an indicator of less liquid markets for the following
reason. When prices are not available because an asset is not actively traded, the hedge
fund must estimate its value to calculate net asset value and rates of return. But such proce-
dures are at best imperfect and, as demonstrated by Getmansky, Lo, and Makarov, tend to
result in serial correlation in prices as firms either smooth out their value estimates or only
gradually mark prices to true market values. 7 Positive serial correlation is therefore often
interpreted as evidence of liquidity problems; in nearly efficient markets with frictionless
trading, we would expect serial correlation or other predictable patterns in prices to be
minimal. Most mutual funds show almost no evidence of such correlation in their returns,
and the serial correlation of the S&P 500 in most periods is just about zero. 8
Hasanhodzic and Lo find that hedge fund returns in fact exhibit significant serial cor-
relation. This suggestion of smoothed prices has two important implications. First, it lends
further support to the hypothesis that hedge funds are holding less liquid assets and that
their apparent alphas may in fact be liquidity premiums. Second, it implies that their risk-
adjusted performance measures are upward-biased, because any smoothing in the estimates
of portfolio value will reduce total volatility (increasing the Sharpe ratio) as well as covari-
ances and therefore betas with systematic factors (increasing risk-adjusted alphas). In fact,
Figure 26.2 shows that both the alphas and Sharpe ratios of the hedge fund indexes in
Table 26.3 increase with the serial correlation of returns. These results are consistent with
the fund-specific results of Hasanhodzic and Lo and suggest that price smoothing may
account for some part of the apparently superior average hedge fund performance.
Whereas Aragon focuses on the average level of liquidity, Sadka addresses the liquidity risk of hedge funds. 9 He shows that exposure to unexpected declines in market liquidity is an important determinant of average hedge fund returns, and that the spread in average
returns across the funds with the highest and lowest liquidity exposure may be as much
6 George O. Aragon, “Share Restrictions and Asset Pricing: Evidence from the Hedge Fund Industry,” Journal of
Financial Economics 83 (2007), pp. 33–58.
7 Mila Getmansky, Andrew W. Lo, and Igor Makarov, “An Econometric Model of Serial Correlation and Illiquid-
ity in Hedge Fund Returns,” Journal of Financial Economics 74 (2004), pp. 529–609.
8 The 2005–2011 period, in which the serial correlation of monthly excess returns for the S&P 500 was 0.218
(see Table 26.3 ), is a striking exception to this general rule. This aberration arises from the period of the financial
crash, when the return on the S&P 500 was strongly negative in sequential months (September through November
2008, and then again in January and February of 2009). These sequences of large, consecutive negative returns
resulted in positive serial correlation over the sample period, a highly unusual outcome for the index. Note, how-
ever, that even in this period, the average serial correlation of the hedge fund indexes is nearly twice that of the
S&P 500.
9 Ronnie Sadka, “Liquidity Risk and the Cross-Section of Hedge-Fund Returns,” Journal of Financial Economics,
98 (October, 2010), 54–71.
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938 P A R T V I I Applied Portfolio Management
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Serial Correlation of Returns
A lp
h a
A
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Serial Correlation of Returns
Sh a rp
e R
a ti
o
B
Figure 26.2 Hedge funds with higher serial correlation in returns, an indicator of illiquid portfolio holdings, exhibit higher alphas ( panel A) and higher Sharpe ratios ( panel B).
Source: Plotted from data in Table 26.3 .
as 6% annually. Hedge fund performance may therefore reflect significant compensation
for liquidity risk. Figure 26.3 , constructed from data reported in his study, is a scatter dia-
gram relating average return for the hedge funds in each style group of Table 26.2 to the
liquidity-risk beta for that group. Average return clearly rises with exposure to changes in
market liquidity.
Returns can be even more difficult to interpret if a hedge fund takes advantage of illiquid
markets to manipulate returns by purposely misvaluing illiquid assets. In this regard, it is
worth noting that, on average, reported hedge fund returns in December are substantially
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C H A P T E R 2 6 Hedge Funds 939
greater than average returns
in other months. 10
The pattern
is stronger for lower-liquidity
funds and funds that are near
or beyond the threshold return
at which performance incen-
tive fees kick in. It appears that
some funds use their discre-
tion in valuing assets to move
returns to December when that
will enhance their annual incen-
tive fees. It also appears that
some hedge funds attempt to
manipulate their measured per-
formance by buying additional
shares in stocks they already
own in an effort to push up their
prices. 11
The buying takes place
just before market close at the
end of the month when hedge
fund performance is reported.
Moreover, the effort is concen-
trated in less liquid stocks where the price impact would be expected to be greater. If, as
these papers suggest, funds take advantage of illiquid markets to manage returns, then accu-
rate performance measurement becomes almost impossible.
Hedge Fund Performance and Survivorship Bias We already know that survivorship bias (when only successful funds are included in a
database) can affect the estimated performance of a sample of mutual funds. The same
problems, as well as related ones, apply to hedge funds. Backfill bias arises because hedge funds report returns to database publishers only if they choose to. Funds started with seed
capital will open to the public and therefore enter standard databases only if their past
performance is deemed sufficiently successful to attract clients. Therefore, the prior per-
formance of funds that are eventually included in the sample may not be representative of
typical performance. Survivorship bias arises when unsuccessful funds that cease opera- tion stop reporting returns and leave a database, leaving behind only the successful funds.
Malkiel and Saha find that attrition rates for hedge funds are far higher than for mutual
funds—in fact, commonly more than double the attrition rate of mutual funds—making
this an important issue to address. 12
Estimates of survivorship bias in various studies are
typically substantial, in the range of 2%–4%. 13
10 Vikas Agarwal, Naveen D. Daniel, and Narayan Y. Naik, “Do Hedge Funds Manage Their Reported Returns?”
Review of Financial Studies 24 (2011), 3281–3320. 11
Itzhak Ben-David, Francesco Franzoni, Augustin Landier, Rabih Moussawi, “Do Hedge Funds Manipulate
Stock Prices?” Journal of Finance, forthcoming, 2013. 12
Burton G. Malkiel and Atanu Saha, “Hedge Funds: Risk and Return,” Financial Analysts Journal 61 (2005), pp. 80–88. 13
For example, Malkiel and Saha estimate the bias at 4.4%; G. Amin and H. Kat, “Stocks, Bonds and Hedge Funds:
Not a Free Lunch!” Journal of Portfolio Management 29 (Summer 2003), pp. 113–20, find a bias of about 2%; and William Fung and David Hsieh, “Performance Characteristics of Hedge Funds and CTA Funds: Natural versus
Spurious Biases,” Journal of Financial and Quantitative Analysis 35 (2000), pp. 291–307, find a bias of about 3.6%.
0 0.5
Liquidity Beta
1 1.5
0.1
0.2
0.3
0.4
0.5
A ve
ra g
e E
xc e ss
R e tu
rn (
% /m
o n
th )
−0.5
0.6
0
Figure 26.3 Average hedge fund returns as a function of liquidity risk Source: Plotted from data in Sadka, “Liquidity Risk and the Cross-Section of Hedge-Fund Returns.”
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940 P A R T V I I Applied Portfolio Management
Hedge Fund Performance and Changing Factor Loadings In Chapter 24, we pointed out that an impor-
tant assumption underlying conventional
performance evaluation is that the portfolio
manager maintains a stable risk profile over
time. But hedge funds are designed to be
opportunistic and have considerable flexibil-
ity to change that profile. This too can make
performance evaluation tricky. If risk is not
constant, then estimated alphas will be biased
if we use a standard, linear index model. And
if the risk profile changes in systematic man-
ner with the expected return on the market,
performance evaluation is even more difficult.
To see why, look at Figure 26.4 , which
illustrates the characteristic line of a per-
fect market timer (see Chapter 24, Section
24.4) who engages in no security selection
but moves funds from T-bills into the market
portfolio only when the market will outper-
form bills. The characteristic line is non-
linear, with a slope of 0 when the market’s
excess return is negative, and a slope of 1
when it is positive. But a naïve attempt to
estimate a regression equation from this pat-
tern would result in a fitted line with a slope
between 0 and 1, and a positive alpha. Nei-
ther statistic accurately describes the fund.
As we noted in Chapter 24, and as is evi-
dent from Figure 26.4 , an ability to conduct
perfect market timing is much like obtaining
a call option on the underlying portfolio with-
out having to pay for it. Similar nonlinearities
would arise if the fund actually buys or writes
options. Figure 26.5 , panel A illustrates the
case of a fund that holds a stock portfolio and
writes put options on it, and panel B illustrates
the case of a fund that holds a stock portfo-
lio and writes call options. In both cases, the
characteristic line is steeper when portfolio
returns are poor—in other words, the fund
has greater sensitivity to the market when it is
falling than when it is rising. This is the oppo-
site profile that would arise from timing abil-
ity, which is much like acquiring rather than
writing options, and therefore would give the
fund greater sensitivity to market advances. 14
Return to Perfect Market Timer
Fitted Regression Line
Portfolio Return
Market Returnrf
Figure 26.4 Characteristic line of a perfect market timer. The true characteristic line is kinked, with a shape like that of a call option. Fitting a straight line to the relationship will result in misestimated slope and intercept.
Stock Alone
Stock Alone
A
B
Exercise Price
Stock Value
Exercise Price
Stock Value
Stock with Written Put
Stock with Written Call
Figure 26.5 Characteristic lines of stock portfolio with writ- ten options. Panel A, Buy stock, write put. Here, the fund writes fewer puts than the number of shares it holds. Panel B, Buy stock, write calls. Here, the fund writes fewer calls than the number of shares it holds.
14
But the fund that writes options would at least receive fair compensation for the unattractive shape of its char-
acteristic line in the form of the premium received when it writes the options.
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C H A P T E R 2 6 Hedge Funds 941
15
Not all the hedge fund categories exhibited this sort of pattern. Many showed effectively symmetric up- and
down-market betas. However, Figure 26.6 , panel A shows that the asymmetry affects hedge funds taken as group.
16
Nassim N. Taleb, Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets (New York: TEXERE (Thomson), 2004); Nassim N. Taleb, The Black Swan: The Impact of the Highly Improbable (New York: Random House, 2007).
Figure 26.6 presents evidence on these sorts of nonlinearities. A nonlinear regression
line is fitted to the scatter diagram of returns on hedge fund indexes plotted against returns
on the S&P 500. The fitted lines in each panel suggest that these funds have higher down-
market betas (higher slopes) than up-market betas. 15
This is precisely what investors do not want: higher market sensitivity when the market is weak. This is evidence that funds may be writing options, either explicitly or implicitly through dynamic trading strategies (see Chapter 21, Section 21.5, for a discussion of such
dynamic strategies).
Just as hedge fund betas may be unstable, so may be other aspects of their risk profile, for
example, total volatility of returns. Because they have great discretion to use leverage and
to trade in derivatives, these funds have tremendous capacity to alter their risk exposures.
Recall from Chapter 24 that when portfolio managers can change risk within any measurement
period, they can also manipulate standard measures of risk-adjusted return. Thus, one would
like them to compute and report manipulation-proof performance measures such as Morning-
star’s risk-adjusted return.
Tail Events and Hedge Fund Performance Imagine a hedge fund whose entire investment strategy is to hold an S&P 500 index fund
and write deep out-of-the-money put options on the index. Clearly the fund manager brings
no skill to his job. But if you knew only his investment results over limited periods, and
not his underlying strategy, you might be fooled into thinking that he is extremely talented.
For if the put options are written sufficiently out-of-the-money, they will only rarely end up
imposing a loss, and such a strategy can appear over long periods—even over many years—to
be consistently profitable. In most periods, the strategy brings in a modest premium from
the written puts and therefore outperforms the S&P 500, yielding the impression of con-
sistently superior performance. The huge loss that might be incurred in an extreme market
decline might not be experienced even over periods as long as years. Every so often, such
as in the market crash of October 1987, the strategy may lose multiples of its entire gain
over the last decade. But if you are lucky enough to avoid these rare but extreme tail events (so named because they fall in the far-left tail of the probability distribution), the strategy
might appear to be gilded.
The evidence in Figure 26.6 indicating that hedge funds are at least implicitly option
writers should make us nervous about taking their measured performance at face value.
The problem in interpreting strategies with exposure to extreme tail events (such as short
options positions) is that these events by definition occur very infrequently, so decades of results may be needed to fully appreciate their true risk and reward attributes. In two
influential books, Nassim Taleb, who is a hedge fund operator himself, argues that many
hedge funds are analogous to our hypothetical manager, racking up fame and fortune
through strategies that make money most of the time but expose investors to rare but extreme losses.
16
Taleb uses the metaphor of the black swan to discuss the importance of highly improb-
able, but highly impactful, events. Until the discovery of Australia, Europeans believed that
all swans were white: they had never encountered swans that were not white. In their experi-
ence, the black swan was outside the realm of reasonable possibility, in statistical jargon, an
extreme outlier relative to their sample of observations. Taleb argues that the world is filled
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942942
Figure 26.6 Monthly return on hedge fund indexes versus return on the S&P 500, 2008–2012. Panel A, hedge fund index. Panel B, market-neutral funds. Panel C, short-bias funds.
Source: Constructed from data downloaded from www.hedgeindex.com and finance.yahoo.com .
−12%
−10%
−8%
−6%
−4%
−2%
0%
2%
4%
6%
S&P 500 Return
R e tu
rn o
n B
ro a d
H e d
g e F
u n
d I n
d e x
A
−20% −15% −10% −5% 0% 5% 10% 15%
−20%
−15%
−10%
−5%
0%
5%
10%
15%
−20% −15% −10% −5% 0% 5% 10% 15%
S&P 500 Return
R e tu
rn o
n M
a rk
e t-
N e u
tr a l Fu
n d
s
B %1555
−20%
−15%
−10%
−5%
0%
5%
10%
15%
−20% −15% −10% −5% 0% 5% 10% 15%
S&P 500 Return
R e tu
rn o
n S
h o
rt B
ia s
Fu n
d s
C
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C H A P T E R 2 6 Hedge Funds 943
with black swans, deeply important developments that simply could not have been predicted
from the range of accumulated experience to date. While we can’t predict which black swans
to expect, we nevertheless know that some black swan may be making an appearance at any
moment. The October 1987 crash, when the market fell by more than 20% in 1 day, might be
viewed as a black swan—an event that had never taken place before, one that most market
observers would have dismissed as impossible and certainly not worth modeling, but with
high impact. These sorts of events seemingly come out of the blue, and they caution us to
show great humility when we use past experience to evaluate the future risk of our actions.
With this in mind, consider again the example of Long Term Capital Management.
In the late 1990s, Long Term Capital Management was widely viewed as the most successful hedge fund in history. It had consistently provided double-digit returns to its investors, and it had earned hundreds of millions of dollars in incentive fees for its managers. The firm used sophisticated computer models to estimate correlations across assets and believed that its capital was almost 10 times the annual standard deviation of its portfolio returns, presumably enough to withstand any “possible” shock to capital (at least, assuming normal distributions!). But in the summer of 1998, things went badly. On August 17, 1998, Russia defaulted on its sovereign debt and threw capital markets into chaos. LTCM’s 1-day loss on August 21 was $550 million (approximately nine times its estimated monthly standard deviation). Total losses in August were about $1.3 bil- lion, despite the fact that LTCM believed that most of its positions were market-neutral relative-value trades. Losses accrued on virtually all of its positions, flying in the face of the presumed diversification of the overall portfolio.
How did this happen? The answer lies in the massive flight to quality and, even more so, to liquidity that was set off by the Russian default. LTCM was typically a seller of liquidity (holding less liquid assets, selling more liquid assets with lower yields, and earning the yield spread) and suffered huge losses. This was a different type of shock from those that appeared in its historical sample/modeling period. In the liquidity crisis that engulfed asset markets, the unexpected commonality of liquidity risk across usually uncorrelated asset classes became obvious. Losses that seemed statistically impossible on past experience had in fact come to pass; LTCM fell victim to a black swan.
Example 26.3 Tail Events and Long-Term Capital Management
However, Figure 26.6 shows that the broad hedge fund index did not exhibit notice-
ably greater tail risk than other stock investments during the financial crisis of 2008–2009.
While equity returns were generally dismal in that period, typical hedge fund returns were
actually less negative than those of the S&P 500. This, of course, is consistent with the
generally low betas of these funds.
The typical hedge fund fee structure is a management fee of 1% to 2% of assets plus an
incentive fee equal to 20% of investment profits beyond a stipulated benchmark perfor- mance, annually. Incentive fees are effectively call options on the portfolio with a strike
price equal to current portfolio value times 1 1 benchmark return. The manager gets the
fee if the portfolio value rises sufficiently, but loses nothing if it falls. Figure 26.7 illustrates
the incentive fee for a fund with a 20% incentive fee and a hurdle rate equal to the money mar-
ket rate, r f . The current value of the portfolio is denoted S 0 and the year-end value is S T . The incentive fee is equivalent to .20 call options on the portfolio with exercise price S 0 (1 1 r f ).
26.6 Fee Structure in Hedge Funds
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944 P A R T V I I Applied Portfolio Management
Suppose the standard deviation of a hedge fund’s annual rate of return is 30% and the incentive fee is 20% of any investment return over the risk-free money market rate. If the portfolio currently has a net asset value of $100 per share, and the effective annual risk-free rate is 5% (or 4.88% expressed as a continuously compounded rate), then the implicit exercise price on the incentive fee is $105. The Black-Scholes value of a call option with S 0 5 100, X 5 105, s 5 .30, r 5 .0488, T 5 1 year is $11.92, just a shade below 12% of net asset value. Because the incentive fee is worth 20% of the call option, its value is just about 2.4% of net asset value. Together with a typical management fee of 2% of net asset value, the investor in the fund pays fees with a total value of 4.4%.
Example 26.4 Black-Scholes Valuation of Incentive Fees
The major complication to this description of the typical
compensation structure is the high water mark. If a fund experiences losses, it may not be able to charge an incentive
fee unless and until it recovers to its previous higher value.
With large losses, this may be difficult. High water marks
therefore give managers an incentive to shut down funds that
have performed poorly, and likely are a cause of the high attri-
tion rate for funds noted above.
One of the fastest-growing sectors in the hedge fund uni-
verse has been in funds of funds, which are hedge funds that invest in one or more other hedge funds. Funds of funds
are also called feeder funds, because they feed assets from the original investor to the ultimate hedge fund. They are
marketed as providing investors the capability to diver-
sify across funds, and also as providing the due diligence
involved in screening funds for investment worthiness. In
principle, this can be a valuable service because many hedge
funds are opaque and feeder funds may have greater insight
than typical outsiders.
However, when Bernard Madoff was arrested in December 2008 after admitting to
a massive Ponzi scheme, many large feeder funds were found to be among his biggest
clients, and their “due diligence” may have been, to put it mildly, lacking. At the head
of the list was Fairfield Greenwich Advisors, with exposure reported at $7.5 billion, but
several other feeder funds and asset management firms around the world were also on
the hook for amounts greater than $1 billion. In the end, it appears that some funds had
in effect become little more than marketing agents for Madoff. The nearby box presents
further discussion of the Madoff affair.
The option-like nature of compensation can have a big impact on expected fees in funds
of funds. This is because the fund of funds pays an incentive fee to each underlying fund
that outperforms its benchmark, even if the aggregate performance of the fund of funds is
poor. In this case, diversification can hurt you! 17
S0(1 + rf) ST
Incentive Fee
Slope = .20
Figure 26.7 Incentive fees as a call option. The current value of the portfolio is denoted S 0 and its year-end value is S T . The incentive fee is equivalent to .20 call options on the portfolio with exercise price S 0 (1 1 r f ).
17
S. J. Brown, W. N. Goetzmann, and B. Liang, “Fees on Fees in Funds of Funds,” Journal of Investment Management 2 (2004), pp. 39–56.
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945
Suppose a fund of funds is established with $1 million invested in each of three hedge funds. For simplicity, we will ignore the asset-value-based portion of fees (the manage- ment fee) and focus only on the incentive fee. Suppose that the hurdle rate for the incentive fee is a zero return, so each fund charges an incentive fee of 20% of total return. The following table shows the performance of each underlying fund over a year, the gross rate of return, and the return realized by the fund of funds net of the incentive fee. Funds 1 and 2 have positive returns, and therefore earn an incentive fee, but Fund 3 has terrible performance, so its incentive fee is zero.
Example 26.5 Incentive Fees in Funds of Funds
The Bernard Madoff Scandal
Bernard Madoff seemed like one of the great success stories in the annals of finance. His asset management firm, Bernard L. Madoff Investment Securities, reported to its clients that their investments of around $20 billion were worth about $65 billion in 2008. But that Decem- ber, Madoff reportedly confessed to his two sons that he had for years been operating a Ponzi scheme. A Ponzi scheme is an investment fraud in which a manager col- lects funds from clients, claims to invest those funds on the clients’ behalf, reports extremely favorable investment returns, but in fact uses the funds for his own purposes. (The schemes are named after Charles Ponzi, whose suc- cess with this scheme in the early 1900s made him notori- ous throughout the United States.) Early investors who ask to redeem their investments are paid back with the funds coming in from new investors rather than with true earn- ings. The scheme can continue as long as new investors provide enough funds to cover the redemption requests of the earlier ones—and these inflows are attracted by the superior returns “earned” by early investors and their apparent ability to redeem funds as requested.
As a highly respected member of the Wall Street estab- lishment, Madoff was in a perfect position to perpetrate such a fraud. He was a pioneer in electronic trading and had served as chairman of the NASDAQ Stock Market. Aside from its trading operations, Bernard L. Madoff Investment Securities LLC also acted as a money manager, and it claimed to achieve highly consistent annual returns, between 10% and 12% in good markets as well as bad. Its strategy was supposedly based on option hedging strate- gies, but Madoff was never precise about his approach. Still, his stature on Wall Street and the prestige of his cli- ent list seemed to testify to his legitimacy. Moreover, he played hard to get, and the appearance that one needed connections to join the fund only increased its appeal. The scheme seems to have operated for decades, but in
the 2008 stock market downturn, several large clients requested redemptions totaling around $7 billion. With less than $1 billion of assets left in the firm, the scheme collapsed.
Not everyone was fooled, and in retrospect, several red flags should have aroused suspicion. For example, some institutional investors shied away from the fund, objecting to its unusual opacity. Given the magnitude of the assets supposedly under management, the option hedging trades purportedly at the heart of Madoff’s investment strategy should have dominated options market trading volume, yet there was no evidence of their execution. Moreover, Madoff’s auditor, a small firm with only three employees (including only one active accountant!), seemed grossly inadequate to audit such a large and complex operation. In addition, Madoff’s fee structure was highly unusual. Rather than acting as a hedge fund that would charge a per- centage of assets plus incentive fees, he claimed to profit instead through trading commissions on the account—if true, this would have been a colossal price break to clients. Finally, rather than placing assets under management with a custodial bank as most funds do, Madoff claimed to keep the funds in house, which meant that no one could inde- pendently verify their existence. In 2000, the SEC received a letter from an industry professional named Harry Mar- kopolos concluding that “Madoff Securities is the world’s largest Ponzi scheme,” but Madoff continued to operate unimpeded.
Even today, several questions remain unanswered. How much help did Madoff receive from others? Exactly how much money was lost? Most of the “lost” funds represented fictitious profits that had never actually been earned, but some money was returned to early investors. How much was skimmed off to support Madoff’s lifestyle? And most important, why didn’t the red flags and early warnings prompt a more aggressive response from regulators?
W O
R D
S FR O
M T
H E ST
R E E T
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946 P A R T V I I Applied Portfolio Management
The idea behind funds of funds is to spread risk across several different funds. However,
investors need to be aware that these funds of funds operate with considerable leverage,
on top of the leverage of the primary funds in which they invest, which can make returns
highly volatile. Moreover, if the various hedge funds in which these funds of funds invest
have similar investment styles, the diversification benefits of spreading investments across
several funds may be illusory—but the extra layer of steep management fees paid to the
manager of the fund of funds certainly is not. 18
Fund 1 Fund 2 Fund 3 Fund of Funds
Start of year (millions) $1.00 $1.00 $1.00 $3.00 End of year (millions) $1.20 $1.40 $0.25 $2.85 Gross rate of return 20% 40% 275% 25% Incentive fee (millions) $0.04 $0.08 $0.00 $0.12 End of year, net of fee $1.16 $1.32 $0.25 $2.73 Net rate of return 16% 32% 275% 29%
Even though the return on the aggregate portfolio of the fund of funds is negative 5%, it still pays incentive fees of $.12 for every $3 invested because incentive fees are paid on the first two well-performing funds. The incentive fees amount to 4% of net asset value. As demonstrated in the last column, this reduces the rate of return earned by the fund of funds from 2 5% to 2 9%.
18
One small silver lining: while funds of funds pay incentive fees to each of the underlying funds, the incentive
fees they charge their own investors tend to be lower, typically around 10% rather than 20%.
1. Like mutual funds, hedge funds pool the assets of several clients and manage the pooled assets on their behalf. However, hedge funds differ from mutual funds with respect to disclosure, investor
base, flexibility and predictability of investment orientation, regulation, and fee structure.
2. Directional funds take a stance on the performance of broad market sectors. Nondirectional funds establish market-neutral positions on relative mispricing. However, even these hedged positions
still present idiosyncratic risk.
3. Statistical arbitrage is the use of quantitative systems to uncover many perceived misalignments in relative pricing and ensure profits by averaging over all of these small bets. It often uses
data-mining methods to uncover past patterns that form the basis for the established investment
positions.
4. Portable alpha is a strategy in which one invests in positive-alpha positions, then hedges the sys- tematic risk of that investment, and, finally, establishes market exposure where desired by using
passive indexes or futures contracts.
5. Performance evaluation of hedge funds is complicated by survivorship bias, by the potential instability of risk attributes, by the existence of liquidity premiums, and by unreliable market
valuations of infrequently traded assets. Performance evaluation is particularly difficult when the
fund engages in option positions. Tail events make it hard to assess the true performance of posi-
tions involving options without extremely long histories of returns.
SUMMARY
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C H A P T E R 2 6 Hedge Funds 947
hedge funds
lock-up periods
directional strategy
nondirectional strategy
market neutral
pure plays
statistical arbitrage
pairs trading
data mining
portable alpha
alpha transfer
backfill bias
survivorship bias
incentive fee
high water mark
funds of funds
KEY TERMS
PROBLEM SETS 1. Would a market-neutral hedge fund be a good candidate for an investor’s entire retirement port- folio? If not, would there be a role for the hedge fund in the overall portfolio of such an investor?
2. How might the incentive fee of a hedge fund affect the manager’s proclivity to take on high-risk
assets in the portfolio?
3. Why is it harder to assess the performance of a hedge fund portfolio manager than that of a typi-
cal mutual fund manager?
4. Which of the following is most accurate in describing the problems of survivorship bias and back- fill bias in the performance evaluation of hedge funds?
a. Survivorship bias and backfill bias both result in upwardly biased hedge fund index returns. b. Survivorship bias and backfill bias both result in downwardly biased hedge fund index returns. c. Survivorship bias results in upwardly biased hedge fund index returns, but backfill bias results
in downwardly biased hedge fund index returns.
5. Which of the following would be the most appropriate benchmark to use for hedge fund evaluation?
a. A multifactor model. b. The S&P 500. c. The risk-free rate.
6. With respect to hedge fund investing, the net return to an investor in a fund of funds would be
lower than that earned from an individual hedge fund because of:
a. Both the extra layer of fees and the higher liquidity offered. b. No reason; fund of funds earn returns that are equal to those of individual hedge funds. c. The extra layer of fees only.
7. Which of the following hedge fund types is most likely to have a return that is closest to risk-free?
a. A market-neutral hedge fund. b. An event-driven hedge fund. c. A long/short hedge fund.
8. Is statistical arbitrage true arbitrage? Explain.
9. A hedge fund with $1 billion of assets charges a management fee of 2% and an incentive fee of
20% of returns over a money market rate, which currently is 5%. Calculate total fees, both in
dollars and as a percent of assets under management, for portfolio returns of:
a. 2 5% b. 0 c. 5% d. 10%
Basic
Intermediate
6. Hedge funds typically charge investors both a management fee and an incentive fee equal to a percentage of profits beyond some threshold value. The incentive fee is akin to a call option on
the portfolio. Funds of hedge funds pay the incentive fee to each underlying fund that beats its
hurdle rate, even if the overall performance of the portfolio is poor.
Related Web sites for this chapter are available at www. mhhe.com/bkm
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10. A hedge fund with net asset value of $62 per share currently has a high water mark of $66. Is the
value of its incentive fee more or less than it would be if the high water mark were $67?
11. Reconsider the hedge fund in the previous problem. Suppose it is January 1, the standard devia-
tion of the fund’s annual returns is 50%, and the risk-free rate is 4%. The fund has an incentive
fee of 20%, but its current high water mark is $66, and net asset value is $62.
a. What is the value of the annual incentive fee according to the Black-Scholes formula? b. What would the annual incentive fee be worth if the fund had no high water mark and it
earned its incentive fee on its total return?
c. What would the annual incentive fee be worth if the fund had no high water mark and it earned its incentive fee on its return in excess of the risk-free rate? (Treat the risk-free
rate as a continuously compounded value to maintain consistency with the Black-Scholes
formula.)
d. Recalculate the incentive fee value for part ( b ) now assuming that an increase in fund lever- age increases volatility to 60%.
12. Go to the Online Learning Center at www.mhhe.com/bkm , link to Chapter 26, and find there a spreadsheet containing monthly values of the S&P 500 index. Suppose that in each month you
had written an out-of-the-money put option on one unit of the index with an exercise price 5%
lower than the current value of the index.
a. What would have been the average value of your gross monthly payouts on the puts over the 10-year period October 1977–September 1987? The standard deviation?
b. Now extend your sample by 1 month to include October 1987, and recalculate the average payout and standard deviation of the put-writing strategy. What do you conclude about tail
risk in naked put writing?
13. Suppose a hedge fund follows the following strategy. Each month it holds $100 million of an
S&P 500 index fund and writes out-of-the-money put options on $100 million of the index with
exercise price 5% lower than the current value of the index. Suppose the premium it receives for
writing each put is $.25 million, roughly in line with the actual value of the puts.
a. Calculate the Sharpe ratio the fund would have realized in the period October 1982–September 1987. Compare its Sharpe ratio to that of the S&P 500. Use the data from the previous prob-
lem, available at the Online Learning Center, and assume the monthly risk-free interest rate
over this period was .7%.
b. Now calculate the Sharpe ratio the fund would have realized if we extend the sample period by 1 month to include October 1987. What do you conclude about performance evaluation
and tail risk for funds pursuing option-like strategies?
14. The following is part of the computer output from a regression of monthly returns on Water-
works stock against the S&P 500 index. A hedge fund manager believes that Waterworks is
underpriced, with an alpha of 2% over the coming month.
Beta R-square Standard Deviation
of Residuals
.75 .65 .06 (i.e., 6% monthly)
a. If he holds a $2 million portfolio of Waterworks stock, and wishes to hedge market exposure for the next month using 1-month maturity S&P 500 futures contracts, how many contracts
should he enter? Should he buy or sell contracts? The S&P 500 currently is at 1,000 and the
contract multiplier is $250.
b. What is the standard deviation of the monthly return of the hedged portfolio? c. Assuming that monthly returns are approximately normally distributed, what is the prob-
ability that this market-neutral strategy will lose money over the next month? Assume the
risk-free rate is .5% per month.
15. Return to the previous problem.
a. Suppose you hold an equally weighted portfolio of 100 stocks with the same alpha, beta, and residual standard deviation as Waterworks. Assume the residual returns (the e terms in
e X c e l Please visit us at www.mhhe.com/bkm
e X c e l Please visit us at www.mhhe.com/bkm
Challenge
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C H A P T E R 2 6 Hedge Funds 949
Equations 26.1 and 26.2) on each of these stocks are independent of each other. What is the
residual standard deviation of the portfolio?
b. Recalculate the probability of a loss on a market-neutral strategy involving equally weighted, market-hedged positions in the 100 stocks over the next month.
16. Return again to Problem 14. Now suppose that the manager misestimates the beta of Water-
works stock, believing it to be .50 instead of .75. The standard deviation of the monthly market
rate of return is 5%.
a. What is the standard deviation of the (now improperly) hedged portfolio? b. What is the probability of incurring a loss over the next month if the monthly market return
has an expected value of 1% and a standard deviation of 5%? Compare your answer to the
probability you found in Problem 14.
c. What would be the probability of a loss using the data in Problem 15 if the manager simi- larly misestimated beta as .50 instead of .75? Compare your answer to the probability you
found in Problem 14.
d. Why does the misestimation of beta matter so much more for the 100-stock portfolio than it does for the 1-stock portfolio?
17. Here are data on three hedge funds. Each fund charges its investors an incentive fee of 20% of
total returns. Suppose initially that a fund of funds (FF) manager buys equal amounts of each of
these funds, and also charges its investors a 20% incentive fee. For simplicity, assume also that
management fees other than incentive fees are zero for all funds.
Hedge Fund 1
Hedge Fund 2
Hedge Fund 3
Start of year value (millions) $100 $100 $100 Gross portfolio rate of return 20% 10% 30%
a. Compute the rate of return after incentive fees to an investor in the fund of funds. b. Suppose that instead of buying shares in each of the three hedge funds, a stand-alone (SA)
hedge fund purchases the same portfolio as the three underlying funds. The total value and composition of the SA fund is therefore identical to the one that would result from aggregat-
ing the three hedge funds. Consider an investor in the SA fund. After paying 20% incentive
fees, what would be the value of the investor’s portfolio at the end of the year?
c. Confirm that the investor’s rate of return in SA is higher than in FF by an amount equal to the extra layer of fees charged by the fund of funds.
d. Now suppose that the return on the portfolio held by hedge fund 3 were 2 30% rather than 1 30%. Recalculate your answers to parts ( a ) and ( b ). Will either FF or SA charge an incentive fee in this scenario? Why then does the investor in FF still do worse than the investor in SA?
E-INVESTMENTS EXERCISES Log on to www.hedgeindex.com , a site run by Credit Suisse/Tremont, which maintains the TASS Hedge Funds Data Base of the performance of more than 2,000 hedge funds, and produces indexes of investment performance for several hedge fund classes. Click the Downloads tab (free registration is required for access to this part of the Web site). From the Downloads page, you can access historical rates of return on each of the hedge fund subclasses (e.g., market neutral, event-driven, dedicated short bias, and so on). Download 5 years of monthly returns for each subclass and download returns on the S&P 500 for the same period from finance.yahoo.com . Calculate the beta of the equity-market-neutral and dedicated short bias funds. Do the results seem reasonable in terms of the orienta- tion of these funds? Next, look at the year-by-year performance of each hedge fund class. How does the variability of performance results in different years compare to that of the S&P 500?
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SOLUTIONS TO CONCEPT CHECKS 1. a. Nondirectional. The shares in the fund and the short position in the index swap constitute a
hedged position. The hedge fund is betting that the discount on the closed-end fund will shrink
and that it will profit regardless of the general movements in the Indian market.
b. Nondirectional. The value of both positions is driven by the value of Toys “R” Us. The hedge fund is betting that the market is undervaluing Petri relative to Toys “R” Us, and that as the
relative values of the two positions come back into alignment, it will profit regardless of the movements in the underlying shares.
c. Directional. This is an outright bet on the price that Generic Pharmaceuticals will eventually command at the conclusion of the predicted takeover attempt.
2. The expected rate of return on the position (in the absence of any knowledge about idiosyncratic risk
reflected in the residual) is 3%. If the residual turns out to be 2 4%, then the position will lose 1%
of its value over the month and fall to $1.188 million. The excess return on the market in this month
over T-bills would be 5% 2 1% 5 4%, while the excess return on the hedged strategy would be
2 1% 2 1% 5 2 2%, so the strategy would plot in panel A as the point (4%, 2 2%). In panel B, which
plots total returns on the market and the hedge position, the strategy would plot as the point (5%, 2 1%).
3. Fixed-income arbitrage portfolios show positive exposure to the long bond and to the credit
spread. This pattern suggests that these are not hedged arbitrage portfolios, but in fact are directional portfolios.
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277 THIS CHAPTER CONSIDERS practical com- plexities in the process of constructing opti- mal portfolios. It might seem that the word “theory” in the chapter title is inconsistent with this practical goal, and that the forego- ing chapters must already have exhausted the insight that theory can impart to portfo- lio management in the field—the rest surely must rely on learning by doing.
We will see, however, that theory has a significant contribution to offer even when it comes to the daily grind of putting it all together. We begin with the Treynor-Black model that we first encountered in Chapter 8, now showing how to handle limited precision
in the forecasts of alpha values and the extreme portfolio positions often prescribed by the model. Armed with these insights, we present a prototype organizational chart and discuss the efficacy of fitting the organization to the theoretical underpinning of portfolio management.
In the next section, we present the Black- Litterman model that allows flexible views about the expected returns of asset classes to improve asset allocation. Finally, we look into the potential profitability of security analysis and end with concluding remarks. An appen- dix to the chapter presents the mathematics underlying the Black-Litterman model.
PA R
T V
II
The Theory of Active
Portfolio Management
CHAPTER TWENTY-SEVEN
27.1 Optimal Portfolios and Alpha Values
In Chapter 8 we showed how to form an optimal risky portfolio with a single-index model.
Table 27.1 summarizes the steps in this optimization, commonly known as the Treynor-
Black model. 1 The outlined procedure uses the index model that ignores nonzero cova-
riance values across residuals. This is sometimes called the diagonal model, because it assumes that the covariance matrix of residuals has nonzero entries only on the diagonals.
Moreover, we saw that despite significant correlation between some pairs of residuals in
the portfolio construction example we used in Chapter 8, for example, between Shell and
BP, the efficient frontiers formed from the index model and the Markowitz model were
barely distinguishable (see Figure 8.5 of Chapter 8).
1 We know from Chapter 10 that a multiple-index model such as that of Fama and French may better describe
security returns. In that case, the passive market-index portfolio will be augmented with positions in the addi-
tional factor portfolios (for example, the size and value portfolios in the FF model). However, the rest of the
Treynor-Black procedure will remain unchanged.
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952 P A R T V I I Applied Portfolio Management
For illustration, in this chapter we continue with the example employed in Chapter 8.
Spreadsheet 27.1 recaps the data and results of this exercise. Table D in the spreadsheet
shows the improvement in the Sharpe ratio over the passive market-index portfolio offered by adding the active portfolio to the mix. To better appreciate this improvement we have included the M -square measure of performance. M -square is the incremental expected return of the optimized portfolio compared to the passive alternative once the
active portfolio is mixed with bills to provide the same total volatility as the index portfolio
(for a review, see Chapter 24).
Forecasts of Alpha Values and Extreme Portfolio Weights The overriding impression from Spreadsheet 27.1 is the apparently meager performance
improvement: Table D of the spreadsheet shows that M -square increases by only 19 basis points (equivalent to an improvement of .0136 in the Sharpe ratio). Notice that the Sharpe
ratio of the active portfolio is inferior to that of the passive portfolio (due to its large
standard deviation) and hence its M -square is actually negative. But the active portfolio is mixed with the passive portfolio, so total volatility is not its appropriate measure of
risk. When combined with the passive portfolio, it does offer some improvement in per-
formance, albeit quite modest. This is the best that can be had given the alpha values
Table 27.1
Construction and properties of the optimal risky portfolio
1. Initial position of security i in the active portfolio
wi 0 5
ai
s2(ei)
2. Scaled initial positions wi 5
w i 0
a n
i51
ai
s2(ei)
3. Alpha of the active portfolio aA 5 a
n
i51 wi ai
4. Residual variance of the active portfolio s2(eA) 5 a
n
i51 wi
2s2(ei)
5. Initial position in the active portfolio
wA 0 5
aA
s2(eA) E(RM)
sM 2
6. Beta of the active portfolio bA 5 a
n
i51 wibi
7. Adjusted (for beta) position in the active portfolio w
* A 5
wA 0
1 1 (1 2 bA)wA 0
8. Final weights in passive portfolio and in security i
w*M 5 1 2 w*A; w*i 5 w*Awi
9. The beta of the optimal risky portfolio and its risk premium
bP 5 w*M 1 w*AbA 5 1 2 w*A(1 2 bA)
E(RP) 5 bP E(RM) 1 w * AaA
10. The variance of the optimal risky portfolio sP 2 5 bP
2sM 2 1 3w*As(eA)42
11. Sharpe ratio of the risky portfolio SP
2 5 SM 2 1 a
n
i51 ¢ ai
s(ei) ≤
2
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C H A P T E R 2 7 The Theory of Active Portfolio Management 953
uncovered by the security analysts (see Table C). Notice that the position in the active port-
folio amounts to 17%, financed in part by a combined short position in Dell and Walmart
of about 10%. Because the figures in Spreadsheet 27.1 are annualized, this performance is
equivalent to a 1-year holding-period return (HPR).
The alpha values we used in Spreadsheet 27.1 are actually small by the standard of
typical analysts’ forecasts. On June 1, we downloaded the current prices of the six stocks
in the example, as well as analysts’ 1-year target prices for each firm. These data and the
implied annual alpha values are shown in Table 27.2 . Notice that all alphas are positive,
indicating an optimistic view for this group of stocks. Figure 27.1 shows the graphs of the
stock prices, as well as the S&P 500 index (ticker 5 GSPC), for the previous year. The
graph shows that the optimistic views in Table 27.2 are not a result of extrapolating rates
from the past.
Spreadsheet 27.1
Active portfolio management with a universe of six stocks
σ2(e) α/σ2(e) w0(i)
[w0(i)] 2
αA σ2(eA) w0 w*
1 2 3 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
37 38 39
40
41
42 43 44 45 46 47 48 49
A C D E F G H I J
Table A: Risk Parameters of the Investable Universe (annualized)
SD of Excess Return Beta
SD of Systematic Component
SD of Residual
Correlation with the S&P
500 S&P 500 0.1358 1.00 0.1358 0 1
HP 0.3817 2.03 0.2762 0.2656 0.72 DELL 0.2901 1.23 0.1672 0.2392 0.58 WMT 0.1935 0.62 0.0841 0.1757 0.43
TARGET 0.2611 1.27 0.1720 0.1981 0.66 BP 0.1822 0.47 0.0634 0.1722 0.35
SHELL 0.1988 0.67 0.0914 0.1780 0.46
Table B: The Index Model Covariance Matrix
SP 500 HP DELL WMT TARGET BP SHELL Beta 1.00 2.03 1.23 0.62 1.27 0.47 0.67
S&P 500 1.00 0.0184 0.0375 0.0227 0.0114 0.0234 0.0086 0.0124 HP 2.03 0.0375 0.1457 0.0462 0.0232 0.0475 0.0175 0.0253
DELL 1.23 0.0227 0.0462 0.0842 0.0141 0.0288 0.0106 0.0153 WMT 0.62 0.0114 0.0232 0.0141 0.0374 0.0145 0.0053 0.0077
TARGET 1.27 0.0234 0.0475 0.0288 0.0145 0.0682 0.0109 0.0157 BP 0.47 0.0086 0.0175 0.0106 0.0053 0.0109 0.0332 0.0058
SHELL 0.67 0.0124 0.0253 0.0153 0.0077 0.0157 0.0058 0.0395
Table C: Macro Forecast (S&P 500) and Forecasts of Alpha Values
SP 500 HP DELL WMT TARGET BP SHELL Alpha 0 0.0150 −0.0100 −0.0050 0.0075 0.012 0.0025 Risk premium 0.0600 0.1371 0.0639 0.0322 0.0835 0.0400 0.0429
Table D: Computation of the Optimal Risky Portfolio
S&P 500 Active Pf A HP DELL WMT TARGET BP SHELL 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317
0.5505 0.2126 −0.1748 −0.1619 0.1911 0.4045 0.0789 1.0000 0.3863 −0.3176 −0.2941 0.3472 0.7349 0.1433
0.1492 0.1009 0.0865 0.1205 0.5400 0.0205
0.0222
0.0404
0.1691 Overall 0.8282 0.1718 Portfolio 0.0663 −0.0546 −0.0505 0.0596 0.1262 0.0246
Beta 1 1.0922 1.0158 0.0663 −0.0546 −0.0505 0.0596 0.1262 0.0246 Risk premium 0.06 0.0878 0.0648 0.0750 0.1121 0.0689 0.0447 0.0880 0.0305 SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988 Sharpe ratio 0.44 0.35 0.4556 M-square 0 −0.0123 0.0019 Benchmark risk 0.0346
B
e X c e l Please visit us at www.mhhe.com/bkm
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954 P A R T V I I Applied Portfolio Management
Table 27.3 shows the optimal portfolio using the analysts’ forecasts rather than the
original alpha values in Table D in Spreadsheet 27.1 . The difference in performance is
striking. The Sharpe ratio of the new optimal portfolio has increased from the bench-
mark’s .44 to 2.32, amounting to a huge risk-adjusted return advantage. This shows up
in an M -square of 25.53%! However, these results also expose a potential major problem with the Treynor-Black model. The optimal portfolio calls for extreme long/short positions
that may be infeasible for a real-world portfolio manager. For example, the model calls for
a position of 5.79 (579%) in the active portfolio, largely financed by a short position of
2 4.79 in the S&P 500 index. Moreover, the standard deviation of this optimal portfolio
is 52.24%, a level of risk that only extremely aggressive hedge funds would be willing to
bear. It is important to notice that this risk is largely nonsystematic because the beta of the
active portfolio, at .95, is less than 1.0, and the beta of the overall risky portfolio is even
lower, only .73, because of the short position in the passive portfolio. Only hedge funds
may still be interested in this portfolio.
One approach to this problem is to restrict extreme portfolio positions, beginning with
short sales. When the short position in the S&P 500 index is eliminated, forcing us to con-
strain the position in the active portfolio to be no more than 1.0, the position in the passive
portfolio (the S&P 500) is zero, and the active portfolio comprises the entire risky posi-
tion. Table 27.4 shows that the optimal portfolio now has a standard deviation of 15.68%,
not overwhelmingly greater than the SD of the passive portfolio (13.58%). The beta of
the overall risky portfolio is now that of the active portfolio (.95), still a slightly defensive
portfolio in terms of systematic risk. Despite this severe restriction, the optimization pro-
cedure is still powerful, and the M -square of the optimal risky portfolio (now the active portfolio) is a very large 16.42%.
Stock HP Dell WMT Target BP Shell
Current price 32.15 25.39 48.14 49.01 70.8 68.7 Target price 36.88 29.84 57.44 62.8 83.52 71.15 Implied alpha 0.1471 0.1753 0.1932 0.2814 0.1797 0.0357
Table 27.2
Stock prices and analysts’ target prices
Figure 27.1 Rates of return on the S&P 500 (GSPC) and the six stocks
+60
HP
BP RDS-B GSPC WMT TGT
DELL
+40
+20
−20
−40
−60 July September
R a te
o f
R e tu
rn (
% )
November January March May
0
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C H A P T E R 2 7 The Theory of Active Portfolio Management 955
S&P 500 Active Pf A HP Dell WMT Target BP Shell
s 2 ( e ) 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317 25.7562 a / s 2 ( e ) 2.0855 3.0641 6.2544 7.1701 6.0566 1.1255 1.0000 w 0 ( i ) 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437 [ w 0 ( i )]
2 0.0066 0.0142 0.0590 0.0775 0.0553 0.0019 a A 0.2018
s 2 ( e A ) 0.0078 w 0 7.9116 w * 0.0000 1.0000 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437
Overall Portfolio
Beta 1 0.9538 0.9538 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437 Risk premium 0.06 0.2590 0.2590 0.2692 0.2492 0.2304 0.3574 0.2077 0.0761 SD 0.1358 0.1568 0.1568 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988 Sharpe ratio 0.44 1.65 1.6515 M -square 0 0.1642 0.1642 Benchmark risk 0.0887
Table 27.4
The optimal risky portfolio with constraint on the active portfolio ( w A # 1)
Table 27.3
The optimal risky portfolio with the analysts’ new forecasts
S&P 500
Active Pf A HP Dell WMT Target BP Shell
s 2 ( e ) 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317 25.7562 a / s 2 ( e ) 2.0855 3.0641 6.2544 7.1701 6.0566 1.1255 1.0000 w 0 ( i ) 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437 [ w 0 ( i )]
2 0.0066 0.0142 0.0590 0.0775 0.0553 0.0019 a A 0.2018
s 2 ( e A ) 0.0078 w 0 7.9116 w * 2 4.7937 5.7937 0.4691163 0.6892459 1.4069035 1.6128803 1.3624061 0.2531855
Overall Portfolio
Beta 1 0.9538 0.7323 0.4691 0.6892 1.4069 1.6129 1.3624 0.2532 Risk premium 0.06 0.2590 1.2132 0.2692 0.2492 0.2304 0.3574 0.2077 0.0761 SD 0.1358 0.1568 0.5224 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988 Sharpe ratio 0.44 1.65 2.3223 M -square 0 0.1642 0.2553 Benchmark risk 0.5146
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956 P A R T V I I Applied Portfolio Management
Is this a satisfactory solution? This would depend on the organization. For hedge funds,
this may be a dream portfolio. For most mutual funds, however, the lack of diversifica-
tion would rule it out. Notice the positions in the six stocks: Walmart, Target, and British
Petroleum alone account for 76% of the portfolio.
Here we have to acknowledge the limitations of our example. Surely, when the invest-
ment company covers more securities, the problem of lack of diversification would largely
vanish. But it turns out that the problem with extreme long/short positions typically per-
sists even when we consider a larger number of firms, and this can gut the practical value
of the optimization model. Consider this conclusion from an important article by Black
and Litterman 2 (whose model we will present in Section 27.3):
the mean-variance optimization used in standard asset allocation models is extremely
sensitive to expected return assumptions the investor must provide . . . The optimal port-
folio, given its sensitivity to the expected returns, often appears to bear little or no rela-
tion to the views the investor wishes to express. In practice, therefore, despite obvious
conceptual attractions of a quantitative approach, few global investment managers regu-
larly allow quantitative models to play a major role in their asset allocation decisions.
This statement is more complex than it reads at first blush, and we will analyze it in
depth in Section 27.3. We bring it up in this section, however, to point out the general con-
clusion that “few global investment managers regularly allow quantitative models to play
a major role in their asset allocation decisions.” In fact, this statement also applies to many
portfolio managers who avoid the mean-variance optimization process altogether for other
reasons. We return to this issue in Section 27.4.
Restriction of Benchmark Risk Black and Litterman point out a related important practical issue. Many investment man-
agers are judged against the performance of a benchmark, and a benchmark index is pro- vided in the mutual fund prospectus. Implied in our analysis so far is that the passive
portfolio, the S&P 500, is that benchmark. Such commitment raises the importance of
tracking error. Tracking error is estimated from the time series of differences between the returns on the overall risky portfolio and the benchmark return, that is, T E 5 R P 2 R M . The portfolio manager must be mindful of benchmark risk, that is, the standard deviation
of the tracking error.
The tracking error of the optimized risky portfolio can be expressed in terms of the beta
of the portfolio and thus reveals the benchmark risk:
Tracking error 5 TE 5 RP 2 RM RP 5 w
* AaA 1 31 2 w*A (1 2 bA)4RM 1 w*AeA
TE 5 w * AaA 2 w
* A (1 2 bA)RM 1 w
* AeA
Var (TE) 5 3w*A(1 2 bA)42 Var (RM) 1 Var (w*AeA) 5 3w*A (1 2 bA)42sM2 1 3w*As(eA)42 Benchmark risk 5 s(TE) 5 w
* A"(1 2 bA)2sM2 1 3s(eA)42
(27.1)
Equation 27.1 shows us how to calculate the volatility of tracking error and how to set the
position in the active portfolio, w*A, to restrict tracking risk to any desired level. For a unit investment in the active portfolio, that is, for w*A 5 1, benchmark risk is
s(TE; w * A 5 1) 5 "(1 2 bA)2sM2 1 3s(eA)42 (27.2)
2 Fischer Black and Robert Litterman, “Global Portfolio Optimization,” Financial Analysts Journal, September/
October 1992. © 1992, CFA Institute. Reprinted with permission from the CFA Institute.
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C H A P T E R 2 7 The Theory of Active Portfolio Management 957
For a desired benchmark risk of s0 ( TE ) we would restrict the weight of the active port- folio to
wA (TE) 5 s0(TE)
s (TE; w * A 5 1)
(27.3)
Obviously, introducing a constraint on tracking risk entails a cost. We must shift weight
from the active to the passive portfolio. Figure 27.2 illustrates the cost. The portfolio opti-
mization would lead us to portfolio T, the tangency of the capital allocation line (CAL), which is the ray from the risk-free rate to the efficient frontier formed from A and M. Reducing risk by shifting weight from T to M takes us down the efficient frontier, instead of along the CAL, to a lower risk position, reducing the Sharpe ratio and M -square of the constrained portfolio.
Notice that the standard deviation of tracking error using the “meager” alpha fore-
casts in Spreadsheet 27.1 is only 3.46% because the weight in the active portfolio is only
17%. Using the larger alphas based on analysts’ forecasts with no restriction on portfolio
weights, the standard deviation of tracking error is 51.46% (see Table 27.3 ), more than
any real-life manager who is evaluated against a benchmark would be willing to bear.
However, with weight of 1.0 on the active portfolio, the benchmark risk falls to 8.87%
( Table 27.4 ).
Finally, suppose a manager wishes to restrict benchmark risk to the same level as it was
using the original forecasts, that is, to 3.46%. Equations 27.2 and 27.3 instruct us to invest
WA 5 .43 in the active portfolio. We obtain the results in Table 27.5 . This portfolio is mod- erate, yet superior in performance: (1) its standard deviation is only slightly higher than
that of the passive portfolio, 13.85%; (2) its beta is .98; (3) the standard deviation of track-
ing error that we specified is extremely low, 3.85%; (4) given that we have only six securi-
ties, the largest position of 12% (in Target) is quite low and would be lower still if more
securities were covered; yet (5) the Sharpe ratio is a whopping 1.06, and the M -square is an impressive 8.35%. Thus, by controlling benchmark risk we can avoid the flaws of the
unconstrained portfolio and still maintain superior performance.
Figure 27.2 Reduced efficiency when benchmark risk is lowered
M
CAL
T
A
5
6
7
8
9
10
11
12
13
12 14 16 18 20 22 24 26
Pf Standard Deviation
P o
rt fo
li o
M e a n
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958 P A R T V I I Applied Portfolio Management
27.2 The Treynor-Black Model and Forecast Precision
Suppose the risky portfolio of your 401(k) retirement fund is currently in an S&P 500 index
fund, and you are pondering whether you should take some extra risk and allocate some
funds to Target’s stock, the high-performing discounter. You know that, absent research
analysis, you should assume the alpha of any stock is zero. Hence, the mean of your prior distribution of Target’s alpha is zero. Downloading return data for Target and the S&P 500 reveals a residual standard deviation of 19.8%. Given this volatility, the prior mean of zero,
and an assumption of normality, you now have the entire prior distribution of Target’s alpha.
One can make a decision using a prior distribution, or refine that distribution by expend-
ing effort to obtain additional data. In jargon, this effort is called the experiment. The experiment as a stand-alone venture would yield a probability distribution of possible out-
comes. The optimal statistical procedure is to combine one’s prior distribution for alpha
with the information derived from the experiment to form a posterior distribution that reflects both. This posterior distribution is then used for decision making.
A “tight” prior, that is, a distribution with a small standard deviation, implies a high
degree of confidence in the likely range of possible alpha values even before looking at
the data. In this case, the experiment may not be sufficiently convincing to affect your
beliefs, meaning that the posterior will be little changed from the prior. 3 In the context of
the present discussion, an active forecast of alpha and its precision provides the experiment
that may induce you to update your prior beliefs about its value. The role of the portfolio
manager is to form a posterior distribution of alpha that serves portfolio construction.
S&P 500 Active Pf A HP Dell WMT Target BP Shell
s 2 ( e ) 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317 25.7562 a / s 2 ( e ) 2.0855 3.0641 6.2544 7.1701 6.0566 1.1255 1.0000 w 0 ( i ) 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437 [ w 0 ( i )]
2 0.0066 0.0142 0.0590 0.0775 0.0553 0.0019 a A 0.2018
s 2 ( e A ) 0.0078 w 0 7.9116 w * 0.5661 0.4339 0.0351 0.0516 0.1054 0.1208 0.1020 0.0190
Overall Portfolio
Beta 1 0.9538 0.9800 0.0351 0.0516 0.1054 0.1208 0.1020 0.0190 Risk premium 0.06 0.2590 0.1464 0.0750 0.1121 0.0689 0.0447 0.0880 0.0305 Standard deviation
0.1358 0.1568 0.1385 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988
Sharpe ratio 0.44 1.65 1.0569 M -square 0 0.1642 0.0835 Benchmark risk 0.0385
Table 27.5
The optimal risky portfolio with the analysts’ new forecasts (benchmark risk constrained to 3.85%)
3 In application to debates about social issues, you might define a fanatic as one who enters the debate with a prior
that is so tight that no argument will influence his posterior, making the debate altogether a waste of time.
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C H A P T E R 2 7 The Theory of Active Portfolio Management 959
Adjusting Forecasts for the Precision of Alpha Imagine that you have just downloaded from Yahoo! Finance the analysts’ forecasts we
used in the previous section, implying that Target’s alpha is 28.1%. Should you conclude
that the optimal position in Target, before adjusting for beta, is a / s 2 ( e ) 5 .281/.198 2 5
7.17 (717%)? Naturally, before committing to such an extreme position, any reasonable
manager would first ask: “How accurate is this forecast?” and “How should I adjust my
position to take account of forecast imprecision?”
Treynor and Black 4 asked this question and supplied an answer. The logic of the answer
is quite straightforward; you must quantify the uncertainty about this forecast, just as you
would the risk of the underlying asset or portfolio. A Web surfer may not have a way to
assess the precision of a downloaded forecast, but the employer of the analyst who issued
the forecast does. How? By examining the forecasting record of previous forecasts issued by the same forecaster.
Suppose that a security analyst provides the portfolio manager with forecasts of
alpha at regular intervals, say, the beginning of each month. The investor portfolio is
updated using the forecast and held until the update of next month’s forecast. At the end
of each month, T, the realized abnormal return of Target’s stock is the sum of alpha plus a residual:
u(T) 5 RTGT(T) 2 bTGT RM(T) 5 a(T) 1 e(T) (27.4)
where beta is estimated from Target’s security characteristic line (SCL) using data for peri-
ods prior to T,
SCL: RTGT(t) 5 a 1 bTGT RM(t) 1 e(t), t , T (27.5)
The 1-month, forward-looking forecast a f ( T ) issued by the analyst at the beginning of
month T is aimed at the abnormal return, u ( T ), in Equation 27.4. In order to decide how to use the forecast for month T, the portfolio manager uses the analyst’s forecasting record. The analyst’s record is the paired time series of all past forecasts, a
f ( t ), and realizations,
u ( t ). To assess forecast accuracy, that is, the relationship between forecast and realized alphas, the manager uses this record to estimate the regression:
u(t) 5 a0 1 a1a f(t) 1 e(t) (27.6)
Our goal is to adjust alpha forecasts to properly account for their imprecision. We will
form an adjusted alpha forecast a ( T ) for the coming month by using the original fore- casts a
f ( T ) and applying the estimates from the regression Equation 27.6, that is,
a(T) 5 a0 1 a1a f(T) (27.7)
The properties of the regression estimates assure us that the adjusted forecast is the
“best linear unbiased estimator” of the abnormal return on Target in the coming month, T. “Best” in this context means it has the lowest possible variance among unbiased fore-
casts that are linear functions of the original forecast. We show in Appendix A that the
value we should use for a 1 in Equation 27.7 is the R -square of the regression Equation 27.6. Because R -square is less than 1, this implies that we “shrink” the forecast toward zero. The lower the precision of the original forecast (the lower its R -square), the more we shrink the adjusted alpha back toward zero. The coefficient a 0 adjusts the forecast upward if the forecaster has been consistently pessimistic, and downward for consistent
optimism.
4 Jack Treynor and Fischer Black, “How to Use Security Analysis to Improve Portfolio Selection,” Journal of
Business, January 1973.
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960 P A R T V I I Applied Portfolio Management
Distribution of Alpha Values Equation 27.7 implies that the quality
of security analysts’ forecasts, as mea-
sured by the R -square in regressions of realized abnormal returns on their fore-
casts, is a critical issue for construction
of optimal portfolios and resultant per-
formance. Unfortunately, these num-
bers are usually impossible to come by.
Kane, Kim, and White 5 obtained a
unique database of analysts’ forecasts
from an investment company specializ-
ing in large stocks with the S&P 500 as
a benchmark portfolio. Their database
includes a set of 37 monthly pairs of
forecasts of alpha and beta values for
between 646 and 771 stocks over the
period December 1992 to December
1995—in all, 23,902 forecasts. The
investment company policy was to
truncate alpha forecasts at 1 14% and
2 12% per month. 6 The histogram of
these forecasts is shown in Figure 27.3 . Returns of large stocks over these years were about
average, as shown in the following table, including one average year (1993), one bad year
(1994), and one good year (1995):
The histogram shows that the distribution of alpha forecasts was positively skewed,
with a larger number of pessimistic forecasts. The adjusted R -square in a regression of these forecasts with actual alphas was .001134, implying a tiny correlation coefficient of
.0337. As it turned out, the optimistic forecasts were of superior quality to the pessimistic
ones. When the regression allowed separate coefficients for positive and negative fore-
casts, the R -square increased to .001536, and the correlation coefficient to .0392. These results contain “good” and “bad” news. The “good” news is that after adjusting
even the wildest forecast, say, an alpha of 12% for the next month, the value to be used by
a forecaster when R -square is .001 would be .012%, just 1.2 basis points per month. On an annual basis, this would amount to .14%, which is of the order of the alpha forecasts
of the example in Spreadsheet 27.1 . With forecasts of this small magnitude, the problem
of extreme portfolio weights would never arise. The bad news arises from the same data:
the performance of the active portfolio will be no better than in our example—implying an
M -square of only 19 basis points. An investment company that delivers such limited performance will not be able to cover
its cost. However, this performance is based on an active portfolio that includes only six
Figure 27.3 Histogram of alpha forecast
−15 −10 −5 0 5 10 15 0
4,000
3,000
2,000
1,000
6,000
5,000
7,000
1993 1994 1995 1926–1999 Average SD (%)
Rate of return, % 9.87 1.29 37.71 12.50 20.39
5 Alex Kane, Tae-Hwan Kim, and Halbert White, “Active Portfolio Management: The Power of the Treynor-Black
Model,” in Progress in Financial Market Research, ed. C. Kyrtsou (New York: Nova, 2004). 6 These constraints on forecasts make sense because on an annual basis they imply a stock would rise by more
than 380% or fall below 22% of its beginning-of-year value.
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C H A P T E R 2 7 The Theory of Active Portfolio Management 961
stocks. As we show in Section 27.5, even small information ratios of individual stocks can
add up (see line 11 in Table 27.1 ). Thus, when many forecasts of even low precision are
used to form a large active portfolio, large profits can be made.
So far we have assumed that forecast errors of various stocks are independent, an
assumption that may not be valid. When forecasts are correlated across stocks, precision
is measured by a covariance matrix of forecasting errors, which can be estimated from
past forecasts. While the necessary adjustment to the forecasts in this case is algebraically
messy, it is just a technical detail. As we might guess, correlations among forecast errors
will call for us to further shrink the adjusted forecasts toward zero.
Organizational Structure and Performance The mathematical property of the optimal risky portfolio reveals a central feature of invest-
ment companies, namely, economies of scale. From the Sharpe measure of the optimized
portfolio shown in Table 27.1 , it is evident that performance as measured by the Sharpe
ratio and M -square grows monotonically with the squared information ratio of the active portfolio (see Equation 8.22, Chapter 8, for a review), which in turn is the sum of the
squared information ratios of the covered securities (see Equation 8.24). Hence, a larger
force of security analysts is sure to improve performance, at least before adjustment for
cost. Moreover, a larger universe will also improve the diversification of the active port-
folio and mitigate the need to hold positions in the neutral passive portfolio, perhaps even
allowing a profitable short position in it. Additionally, a larger universe allows for an
increase in the size of the fund without the need to trade larger blocks of single securities.
Finally, as we will show in some detail in Section 27.5, increasing the universe of securi-
ties creates another diversification effect, that of forecasting errors by analysts.
The increases in the universe of the active portfolio in pursuit of better performance
naturally come at a cost, because security analysts of quality do not come cheap. However,
the other units of the organization can handle increased activity with little increase in cost.
All this suggests economies of scale for larger investment companies provided the organi-
zational structure is efficient.
Optimizing the risky portfolio entails a number of tasks of different nature in terms of
expertise and need for independence. As a result, the organizational chart of the portfolio
management outfit requires a degree of decentralization and proper controls. Figure 27.4
shows an organizational chart designed to achieve these goals. The figure is largely self-
explanatory and the structure is consistent with the theoretical considerations worked out
in previous chapters. It can go a long way in forging sound underpinnings to the daily work
of portfolio management. A few comments are in order, though.
The control units responsible for forecasting records and determining forecast adjust-
ments will directly affect the advancement and bonuses of security analysts and estimation
experts. This implies that these units must be independent and insulated from organiza-
tional pressures.
An important issue is the conflict between independence of security analysts’ opinions
and the need for cooperation and coordination in the use of resources and contacts with
corporate and government personnel. The relative size of the security analysis unit will
further complicate the solution to this conflict. In contrast, the macro forecast unit might
become too insulated from the security analysis unit. An effort to create an interface and channels of communications between these units is warranted.
Finally, econometric techniques that are invaluable to the organization have seen a
quantum leap in sophistication in recent years, and this process seems still to be accelerat-
ing. It is critical to keep the units that deal with estimation updated and on top of the latest
developments.
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962 P A R T V I I Applied Portfolio Management
27.3 The Black-Litterman Model
Product
Final Risky Portfolio
Customers
Control Is final portfolio super-efficient?
Product
Active Portfolio
Product
Passive Portfolio
Data
Feedback
Feedback
Feedback
Control Is alpha positive?
Data
Feedback Data
Passive Portfolio Manager Form passive portfolio at minimum cost
Feedback
Control Is correlation with market 1?
Feedback
Feedback
Data
Macro Analyst Estimate E(RM), σ(M)
Forecasts
Quality Control 1. Keep forecast records 2. Estimate quality 3. Adjust forecasts
Micro Analyst Estimate alpha valuesEconometric Unit
1. Estimate betas 2. Estimate σ(e) 3. Help other units
Forecasts
Forecasts
Data
Forecasts Estimates
Estimates and Help with Statistics
Feedback Feedback
Overall Portfolio Manager 1. Mix active and passive portfolios 2. Monitor performance of active portfolio 3. Monitor performance of passive portfolio
1. Form active portfolio 2. How good are alpha forecasts? 3. How good are beta + residual variance estimates?
Active Portfolio Manager
Figure 27.4 Organizational chart for portfolio management Source: Adapted from Robert C. Merton, Finance Theory, Chapter 12, Harvard Business School.
Fischer Black, famous for the Black-Scholes option-pricing formula as well as the Treynor-
Black model, teamed up with Robert Litterman to produce another useful model of port-
folio construction. The Black-Litterman (BL) model allows portfolio managers to quantify
complex forecasts (which they call views ) and apply these views to portfolio construction. 7
We begin the discussion of the BL model with an illustration of a simple problem of asset
allocation. Although we devote the next section to a comparison of the two models, some
comments on commonalities of the models will help us better understand the BL model.
Black-Litterman Asset Allocation Decision Consider a portfolio manager laboring over asset allocation to bills, bonds, and stocks for the next month. The risky portfolio will be constructed from bonds and stocks so as to
maximize the Sharpe ratio. So far this is no more than the problem described in Section 7.3
7 Black and Litterman, “Global Portfolio Optimization.”
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C H A P T E R 2 7 The Theory of Active Portfolio Management 963
of Chapter 7. There, we were concerned with optimizing the portfolio given a set of data
inputs. In real life, however, optimization using a given dataset is the least of the manager’s
problems. The real issue that dogs any portfolio manager is how to come by that input data.
Black and Litterman propose an approach that uses past data, equilibrium considerations,
and the private views of the portfolio manager about the near future.
Data enters the BL model from two sources: history and forecasts, called views, about the future. The historical sample is used to estimate the covariance matrix of the asset
classes involved in the asset allocation decision. The estimated covariance matrix, com-
bined with a model of equilibrium returns (for example, the CAPM) is used to produce
baseline forecasts that would be the basis of a passive strategy. In the next step, views are
introduced and quantified. The views represent a departure from the baseline forecast and
result in a revised set of expected returns. With the new set of inputs (just as with alpha
forecasts in the Treynor-Black model), an optimal risky portfolio is designed to replace the
(no-longer-efficient) passive portfolio.
Step 1: The Covariance Matrix from Historical Data This straightforward task is the first in the chain that makes up the BL model. Suppose
step 1 results in the following annualized covariance matrix, estimated from recent his-
torical excess returns:
Notice that step 1 is common to both the BL and the Treynor-Black (TB) models. This
activity appears in the organizational chart in Figure 27.4 .
Step 2: Determination of a Baseline Forecast Because past data are of such limited use in inferring expected returns for the next month,
BL propose an alternative approach. They start with a baseline forecast derived from the assumption that the market is in equilibrium where current prices of stocks and bonds
reflect all available information and, as a result, the theoretical market portfolio with
weights equal to market-value proportions is efficient. Suppose that current market values
of outstanding bonds and stocks imply that the weight of bonds in the baseline portfolio is
w B 5 .25, and the weight of stocks is w S 5 .75. When we apply these portfolio weights to the covariance matrix from step 1, the variance of the baseline portfolio emerges as
Var(RM) 5 wB 2 Var(RB) 1 wS
2 Var(RS) 1 2wBwS Cov(RB, RS) (27.8)
5 .252 3 .0064 3 .752 3 .0289 1 2 3 .25 3 .75 3 .00408 5 .018186
The CAPM equation (Equation 9.2 in Chapter 9) gives the relationship between the
market portfolio risk (variance) and its risk premium (expected excess return) as
E(RM) 5 A 3 Var(RM) (27.9)
where A is the average coefficient of risk aversion. Assuming A 5 3 yields the equilibrium risk premium of the baseline portfolio as: E ( R M ) 5 3 3 .018186 5 .0546 5 5.46%. The
Bonds ( B ) Stocks ( S )
Standard deviation .08 .17 Correlation (bonds/stocks) .3 Covariance Bonds .0064 .00408 Stocks .00408 .0289
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964 P A R T V I I Applied Portfolio Management
equilibrium risk premiums on bonds and stocks can be inferred from their betas on the
baseline portfolio:
E(RB) 5 Cov(RB, RM)
Var(RM) E(RM)
Cov(RB, RM) 5 Cov(RB, wBRB 1 wSRS) 5 .25 3 .0064 1 .75 3 .00408 5 .00466
E(RB) 5 .00466
.018186 3 5.46% 5 1.40% (bond beta 5 0.26)
E(RS) 5 .75 3 .0289 1 .25 3 .00408
.018186 3 5.46% 5 6.81% (stock beta 5 1.25)
(27.10)
Thus, step 2 ends up with baseline forecasts of a risk premium for bonds of 1.40% and for
stocks of 6.81%.
The final element in step 2 is to determine the covariance matrix of the baseline fore-
casts. This is a statement about the precision of these forecasts, which is different from the covariance matrix of realized excess returns on the bond and stock portfolios. We are
looking for the precision of the estimate of expected return, as opposed to the volatility of actual returns. A conventional rule of thumb in this application is to use a standard devia-
tion that is 10% of the standard deviation of returns (or equivalently, a variance that is 1%
of the return variance). To illustrate, imagine that the covariance matrix of actual return
was estimated from the returns of the last 100 months. The variance of the average return
(which is the forecast of the expected return) would then be 1% of the variance of the
actual return. Hence in this case it would be correct to use .01 times the covariance matrix
of returns for the expected return. Thus step 2 ends with a forecast and covariance matrix:
Now that we have backed out market expectations, it is time to integrate the manager’s
private views into our analysis.
Step 3: Integrating the Manager’s Private Views The BL model allows the manager to introduce any number of views about the baseline
forecasts into the optimization process. Appended to the views, the manager specifies his
degree of confidence in them. Views in the BL model are expressed as values of various
linear combinations of excess returns, and confidence in them as a covariance matrix of
errors in these values.
Bonds ( B ) Stocks ( S )
Expected return (%) .0140 .0681 Covariance matrix of baseline forecasts
Bonds .000064 .0000408 Stocks .0000408 .000289
Suppose the manager takes a contrarian’s view concerning the baseline forecasts, that is, he believes that in the next month bonds will outperform stocks by .5%. The follow- ing equation expresses this view:
1 3 RB 1 (21) 3 RS 5 .5%
Example 27.1 Views in the Black-Litterman Model
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C H A P T E R 2 7 The Theory of Active Portfolio Management 965
A view must come with a degree of confidence, that is, a standard deviation to measure
the precision of Q. The manager’s view is really Q 1 á, where á represents zero-mean error in the view with a standard deviation that reflects the manager’s less than perfect
confidence. Noticing that the standard deviation of the difference between the expected
rates on stocks and bonds is 1.65% (calculated below in Equation 27.13), suppose that the
manager assigns a value of s (á) 5 1.73%. 9 To summarize, if we denote the array of returns
by R 5 ( R B , R S ), then the manager’s view, P, applied to these returns is 10
PRT 5 Q 1 e
P 5 (1, 21)
R 5 (RB, RS)
Q 5 .5% 5 .005
s2(e) 5 .01732 5 .0003
(27.11)
Step 4: Revised (Posterior) Expectations The baseline forecasts of expected returns derived from market values and their covari-
ance matrix comprise the prior distribution of the rates of return on bonds and stocks. The
manager’s view, together with its confidence measure, provides the probability distribution
arising from the “experiment,” that is, the additional information that must be optimally
integrated with the prior distribution. The result is the posterior: a new set of expected
returns, conditioned on the manager’s views.
To acquire intuition about the solution, consider what the baseline expected returns
imply about the view. The expectations derived from market data were that the expected
return on bonds is 1.40% and on stocks 6.81%. Therefore, the baseline view is that E ( R B ) 2 E ( R S ) 5 2 5.41%. In contrast, the manager thinks this difference is Q 5 R B 2 R S 5 .5%. Using the BL linear-equation notation for market expectations:
QE 5 PRE T
P 5 (1, 21)
RE 5 3E(RB), E(RS)4 5 (1.40%, 6.81%) QE 5 1.40 2 6.81 5 25.41%
(27.12)
8 A simpler view that bonds will return 3% is also legitimate. In that case P 5 (1, 0) and the view is really like an
alpha forecast in the Treynor-Black model. If all views were like this simple one, there would be no difference
between the TB and BL models.
9 Absent specific information shedding light on the SD of the view, for example, the track record of the source of
the view, the SD calculated from the covariance matrix of the baseline forecasts is commonly used. In that case,
the SD would be that of Q E in Equation 27.13: s(QE) 5 ".0002714 5 .0165 (1.65%). 10
Notice that the view is expressed as a row vector with as many elements as there are risky assets (here, two)
applied to the row vector of returns. The manager’s view ( Q ) is obtained from the vector, P, marking the assets included in the view, times their actual returns. We denote the return row vector, R, with a superscript “ T ” (for transpose—turning a row vector into a column), and therefore compute the “sumproduct” of the two vectors.
More generally, any view that is a linear combination of the relevant excess returns can be presented as an array (in Excel, an array would be a column of numbers) that multiplies another array (column) of excess returns. In this case, the array of weights is P 5 (1, 2 1) and the array of excess returns is ( R B , R S ). The value of this linear combina- tion, denoted Q, reflects the manager’s view. In this case, Q 5 .5% must be taken into account in optimizing the portfolio. 8
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966 P A R T V I I Applied Portfolio Management
Thus, the baseline “view” is 2 5.41% (i.e., stocks will outperform bonds), which is vastly
different from the manager’s view. The difference, D, and its variance are
D 5 Q 2 QE 5 .005 2 (2.0541) 5 .0591
s2(D) 5 s2(e) 1 s2(QE) 5 .0003 1 s2(QE)
s2(QE) 5 Var 3E(RB) 2 E(RS)4 5 sE(RB)2 1 sE(RS)2 2 2Cov3E(RB), E(RS)4 5 .000064 1 .000289 2 2 3 .0000408 5 .0002714
s2(D) 5 .0003 1 .0002714 5 .0005714
(27.13)
Given the large difference between the manager’s and the baseline views, we would expect
a significant change in the conditional expected returns from those of the baseline and, as
a result, a very different optimal portfolio.
The expected returns conditional on the view is a function of four elements: the baseline
expectations, E ( R ); the difference, D, between the manager’s view and the baseline view (see Equation 27.13); the contribution of the asset return to the variance of D; and the total variance of D. The result is:
E(R 0view) 5 R 1 D Contribution of asset to sD2 sD
2
E(RB 0 P) 5 E(RB) 1 D5sE(RB) 2 2 Cov3E(RB), E(RS)46
sD 2
5 .0140 1 .0591(.000064 2 .0000408)
.0005714 5 .0140 1 .0024 5 .0164
E(RS 0 P) 5 E(RS) 1 D5Cov3E(RB), E(RS)4 2 sE(RS) 2 6
sD 2
(27.14)
5 .0681 1 .0591(.0000408 2 .000289)
.0005714 5 .0681 2 .0257 5 .0424
We see that the manager increases his expected returns on bonds by .24% to 1.64%, and
reduces his expected return on stocks by 2.57% to 4.24%. The difference between the
expected returns on stocks and bonds is reduced from 5.41% to 2.60%. While this is a very
large change, we also realize that the manager’s private view that Q 5 .5% has been greatly tempered by the prior distribution to a value roughly halfway between his private view and
the baseline view. In general, the degree of compromise between views will depend on the
precision assigned to them.
The example we have described contains only two assets and one view. It can easily be
generalized to any number of assets with any number of views about future returns. The
views can be more complex than a simple difference between a pair of returns. Views can
assign a value to any linear combination of the assets in the universe, and the confidence level (the covariance matrix of the set of á values of the views) can allow for dependence
across views. This flexibility gives the model great potential by quantifying a rich set of
information that is unique to a portfolio manager. Appendix B to the chapter presents the
general BL model.
Step 5: Portfolio Optimization At this point, the portfolio optimization follows the Markowitz procedure of Chapter 7,
with an input list that replaces baseline expectations with the conditional expectations
arising from the manager’s view.
Spreadsheet 27.2 presents the calculations of the BL model. Table 1 of the spreadsheet
shows the calculation of the benchmark forecasts and Table 2 incorporates a view to arrive
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C H A P T E R 2 7 The Theory of Active Portfolio Management 967
at the revised (conditional) expectations. Figure 27.5 shows portfolio performance mea-
sured by M -square for various levels of confidence in the view when the view is correct
Spreadsheet 27.2
Sensitivity of the Black-Litterman portfolio to confidence in views
1
2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
29 30 31 32 33 34 35 36
Table 1: Bordered Covariance Matrix from Historical Excess Returns and Market-Value Weights and Calculation of Baseline Forecasts
Bonds Stocks Weights 0.25 0.75
Bonds Stocks
sumproduct 11.65 170.21 Market portfolio variance V(M) = sum(c11,d11) = Coefficient of risk aversion of representative investor = Baseline market portfolio risk premium = 0.01A*V(M) =
Covariance with RM Baseline risk premiums
Proportion of covariance attributed to expected returns 0.01 1.247920819 0.256237542
Covariance matrix of expected returns Bonds Stocks
Bonds Stocks
Table 2: Views, Confidence, and Revised (Posterior) Expectations
View: Difference between returns on bonds and stocks, Q = View embedded in baseline forecasts QE = Variance of QE = Var(RB − RS) Var[E(RB)] − Cov[E(RB),E(RS)] = Cov[E(RB),E(RS)] − Var[E(RB)] = Difference between view and baseline data, D = Confidence measured by standard deviation of view Q Possible SD 0 1 1.73 3.00 6.00 Variance 0 1.5 3 9 36 Baseline E(RB|P) 1.90 1.72 1.64 1.52 1.43 1.40 E(RS|P) 1.40 3.33 4.24 5.56 6.43 6.81
A C D E F G H IB
0.25 64 40.8 0.75 40.8 289
46.6 226.95 1.40 6.81
0.64 0.408 0.408 2.89
0.5 −5.41 2.71
0.23 −2.48 5.91
181.86 3
5.46
e X c e l Please visit us at www.mhhe.com/bkm
Figure 27.5 Sensitivity of Black-Litterman portfolio performance to confidence level
−3.0
−2.0
−1.0
0.0
1.0
2.0
3.0
0 1 2 3 4 5 6 0.0
0.2
0.4
0.6
0.8
1.0
1.2
M-square–View is False M-square–View is Correct
Weight in Bonds
Confidence (SD)
SD of View = 1.73%
R is
k A
d ju
st e d
P e rf
o rm
a n
ce (
% )
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968 P A R T V I I Applied Portfolio Management
and incorrect. The weight in bonds declines as the confidence in the view falls (the SD of
the view increases). With no confidence in the view (SD very large), the weight in bonds
falls to 0.3, determined by the baseline forecast. At this point, the portfolio is passive; its
M -square is zero. Notice that the M -square profile is asymmetric. With great confidence in the view and the
resultant large position in bonds, the gain in M -square when the view is correct is smaller than the loss in M -square when the view is incorrect. With less confidence and therefore a smaller position in bonds, the “game” becomes more symmetric between a correct and
incorrect view. Since determination of the SD of a view is quite abstract, the graph tells us
that to err on the side of skepticism may well be the prudent choice.
27.4 Treynor-Black versus Black-Litterman: Complements, Not Substitutes
Treynor, Black, and Litterman have earned a place among the important innovators of the
investments industry. Wide implementation of their models could contribute much to the
industry. The comparative analysis of their models presented here is not aimed at elevating
one at the expense of the other—in any case, we find them complementary—but rather to
clarify the relative merits of each.
First and foremost, once you reach the optimization stage, the models are identical. Put
differently, if users of either model arrive at identical input lists, they will choose identical
portfolios and realize identical performance measures. In Section 27.6, we show that these
levels of performance should be far superior to passive strategies, as well as to active strat-
egies that do not take advantage of the quantitative techniques of these models. The models
differ primarily in the way they arrive at the input list, and analysis of these differences
shows that the models are true complements and are best used in tandem.
The BL Model as Icing on the TB Cake The Treynor-Black (TB) model is really oriented to individual security analysis. This can
be seen from the way the active portfolio is constructed. The alpha values assigned to
securities must be determined relative to the passive portfolio. This portfolio is the one that
would be held if all alpha values turned out to be zero. Now suppose an investment com-
pany prospectus mandates a portfolio invested 70% in a U.S. universe of large stocks, say,
the S&P 500, and 30% in a well-defined universe of large European stocks. In that case,
the macro analysis of the organization would have to be split, and the TB model would
have to be run as two separate divisions. In each division, security analysts would compile
values of alpha relative to their own passive portfolio. The product of this organization
would thus include four portfolios, two passive and two active. This scheme is workable
only when the portfolios are optimized separately. That is, the parameters (alpha, beta, and
residual variance) of U.S. securities are estimated relative to the U.S. benchmark, while the
parameters of European stocks are estimated relative to the European benchmark. Then the
final portfolio would be constructed as a standard problem in asset allocation.
The resulting portfolio could be improved using the BL approach. First, views about the
relative performance of the U.S. and European markets can be expected to add informa-
tion to the independent macro forecasts for the two economies. For reasons of specializa-
tion, the U.S. and European macro analysts must focus on their respective economies.
Obviously, when more country or regional portfolios are added to the company’s universe,
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C H A P T E R 2 7 The Theory of Active Portfolio Management 969
the need for decentralization becomes more compelling, and the potential of applying the
BL model to the TB product greater. Moreover, the foreign-stock portfolios will result in
various positions in local currencies. This is a clear area of international finance and the
only way to import forecasts from this analysis is with the BL technique. 11
Why Not Replace the Entire TB Cake with the BL Icing? This question is raised by the need to use the BL technique if the overall portfolio is to
include forecasts from comparative economic and international finance analyses. It is
indeed possible to use the BL model for the entire process of constructing the efficient
portfolio. The reason is that the alpha compiled for the TB model can be replaced with BL
views. To take a simple example, suppose only one security makes up the active portfolio.
With the TB model, we have macro forecasts, E ( R M ) and s M , as well as alpha, beta, and residual variance for the active portfolio. This input list also can be represented in the
following form, along the lines of the BL framework:
R 5 3E(RM), E(RA) 5 bAE(RM)4 P 5 ¢0, 1 1 aA
bAE(RM) ≤
PRT 5 Q 1 e 5 aA 1 e
QE 5 0
D 5 aA s2(e) 5 Var(forecasting error) in Equation 27.6
s2(D) 5 s2(e) 1 s2(e)
(27.15)
where e is the residual in the SCL regression of Equation 27.5. Calculation of the condi- tional expectations from Equation 27.15 as in Equation 27.13 will bring us to the same
adjusted alpha as in Equation 27.7 of the TB model.
In this light, the BL model can be viewed as a generalization of the TB model. The BL
model allows you to adjust expected return from views about alpha values as in the TB
model, but it also allows you to express views about relative performance that cannot be incorporated in the TB model.
However, this conclusion might produce a false impression that is consequential to
investment management. To understand the point, we first discuss the degree of confi-
dence, which is essential to fully represent a view in the BL model. Spreadsheet 27.2 and
Figure 27.5 illustrate that the optimal portfolio weights and performance are highly sensi-
tive to the degree of confidence in the BL views. Thus, the validity of the model rests in
large part on the way the confidence about views is arrived at.
When a BL view is structured to replace a direct alpha estimate in a TB framework,
we must use the variance of the forecasting error taken from Equation 27.7 and applied to
Equation 27.15. This is how “confidence” is quantified in the BL model. Whereas in the
TB framework one can measure forecast accuracy by computing the correlation between
analysts’ alpha forecasts and subsequent realizations, such a procedure is not as easily
applied to BL views about relative performance. Managers’ views may be expressed about
different quantities in different time periods, and, therefore, we will not have long forecast
histories on a particular variable with which to assess accuracy. To our knowledge, no
promotion of how to quantify “confidence” appears in academic or industry publications
about the BL model.
11
The BL model can also be used to introduce views about relative performance of various U.S and foreign
corporations.
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970 P A R T V I I Applied Portfolio Management
This raises the issue of adjusting forecasts in the TB model. We are not aware of actual
results where analysts’ track records are systematically compiled and used to adjust alpha
forecasts, although we cannot assert that such effort is nowhere expended. However, indi-
rect evidence suggests that alphas are usually not adjusted, leading to the common “com-
plaint” that the TB model is not applied in the field because it results in “wild” portfolio
weights. Yet, as we saw in Section 27.3, those wild portfolio weights are a consequence of
failing to adjust alpha values to reflect forecast precision. Any realistic R -square that can be obtained even by excellent forecasters will result in moderate portfolio weights. Even
when “wild” weights do occasionally materialize, they can be “tamed” by a straightfor-
ward restriction on the variance of the tracking error.
It is therefore useful to keep the two models separate and distinct; the TB model for the
management of security analysis with proper adjustment of forecasts and the BL model for
asset allocation where views about relative performance are useful despite the fact that the degree of confidence must in practice be inaccurately estimated.
12
Alex Kane, Alan Marcus, and Robert R. Trippi, “The Valuation of Security Analysis,” Journal of Portfolio Management 25 (Spring 1999).
27.5 The Value of Active Management
We showed in Chapter 24 that the value of successful market timing is enormous. Even
a forecaster with far-from-perfect predictive power would contribute significant value.
Nevertheless, active portfolio management based on security analysis has even greater
potential. Even if each individual security analyst has only modest forecasting power, the
power of a portfolio of analysts is potentially unbounded.
A Model for the Estimation of Potential Fees The value of market timing was derived from the value of an equivalent number of call
options that mimic the return to the timer’s portfolio. Thus, we were able to derive an
unambiguous market value to timing ability, that is, we could price the implicit call in the
timer’s services. We cannot get quite that far with valuation of active portfolio manage-
ment, but we can do the next best thing, namely, we can calculate what a representative
investor would pay for such services.
Kane, Marcus, and Trippi 12
derive an annuitized value of portfolio performance mea-
sured as a percentage of funds under management. The percentage fee, f, that investors would be willing to pay for active services can be related to the difference between the
square of the portfolio Sharpe ratio and that of the passive portfolio as
f 5 (SP 2 2 SM
2 )/2A (27.16)
where A is the coefficient of the investor’s risk aversion. The source of the power of the active portfolio is the additive value of the squared infor-
mation ratios ¢ ai s(ei)
≤ and precision of individual analysts. Recall the expression for the square of the Sharpe ratio of the optimized risky portfolio:
SP 2 5 SM
2 1 a n
i51 B ai
s(ei) R
2
Therefore,
f 5 1
2A a n
i51 B ai
s(ei) R
2
(27.17)
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C H A P T E R 2 7 The Theory of Active Portfolio Management 971
Thus, the fee that can be charged, f, depends on three factors: (1) the coefficient of risk aversion, (2) the distribution of the squared information ratio in the universe of securities,
and (3) the precision of the security analysts. Notice that this fee is in excess of what an
index fund would charge. If an index fund charges about 20 basis points, the active manager
could charge incremental fees above that level by the percentage given in Equation 27.17.
Results from the Distribution of Actual Information Ratios Kane, Marcus, and Trippi investigated the distribution of the squared IR for all S&P 500
stocks over two 5-year periods and estimated that this (annualized) expectation, E (IR 2 ), is in the range of .845 to 1.122. With a coefficient of risk aversion of 3, a portfolio manager who
covers 100 stocks with security analysts whose R -square of forecasts with realized alpha is only .001 would still be able to charge an annual fee that is 4.88% higher than that of an index
fund. This fee is based on the lower end of the range of the expected squared information ratio.
One limitation of this study is that it assumes that the portfolio manager knows the
quality of the forecasts, however low they may be. As we have seen, portfolio weights are
sensitive to forecast quality, and when that quality is estimated with error, performance
will be further reduced.
Results from Distribution of Actual Forecasts A study of actual forecasts by Kane, Kim, and White (see footnote 5) found the distribution
of over 11,000 alpha forecasts for over 600 stocks over 37 months presented in Figure 27.3 .
The average forecast precision from this database of forecasts provided an R -square of .00108 using ordinary least squares (OLS) regressions and .00151 when allowing separate
coefficients for positive and negative forecasts. These are only marginally better than the
precision used to interpret the Kane, Marcus, and Trippi study of the distribution of real-
ized information value. Kane, Kim, and White use these R -squares to adjust the forecasts in their database and form optimal portfolios from 105 stocks selected randomly from the
646 covered by the investment company.
Kane, Kim, and White assume that forecast quality is the same each month for all
alpha forecasts for the 105 stocks, but act as though they do not know that quality. Thus,
the adjustment process is performed each month by using past forecasts. This introduces
another source of estimation error that compounds the difficulty of low forecast quality. To
dull the impact of this real-life difficulty, the estimation of forecast quality adopts improved
econometric technique. They find that least absolute deviation (LAD) regressions perform
uniformly better than OLS regressions. The optimization model used both the diagonal
index model (as in TB) as well as the full-covariance model (the Markowitz algorithm).
The annualized M -square measures of performance are shown in Table 27.6 . The M -square values, which range from 2.67% to 6.31%, are quite impressive. The results in Table 27.6 also show that using the residual covariance matrix can significantly improve
performance when many stocks are covered, contrary to the small difference when only six
stocks are covered, as in Spreadsheet 8.1 of Chapter 8.
Forecast Adjustment Diagonal Model Covariance Model
Line * 2.67 3.01 Kinked ** 4.25 6.31
*Same coefficients for positive and negative forecasts. **Different coefficients for positive and negative forecasts.
Table 27.6
M -square for the portfolio, actual forecasts
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972 P A R T V I I Applied Portfolio Management
Results with Reasonable Forecasting Records To investigate the role of the forecasting record in performance with low-quality forecasts,
Kane, Kim, and White simulate a market with the S&P 500 index portfolio as benchmark and
500 stocks with the same characteristics as the S&P 500 universe. 13
Various sizes of active
portfolios are constructed by selecting stocks randomly from this universe with available fore-
casting records of only 36 to 60 months. To avoid estimation techniques that may not be
available to portfolio managers, all estimates in this study are obtained from OLS regressions.
The portfolio manager in the simulation must deploy a full-blown “organizational struc-
ture” to capture performance under realistic conditions. At any point, the manager uses
only past returns and past forecast records to produce forward-looking estimates which
include (1) the benchmark risk premium and standard deviation, (2) beta coefficients for
the stocks in the active portfolio, and (3) the forecasting quality of each security analyst. At
this point, the manager receives a set of alpha forecasts from the security analysts and pro-
ceeds to construct the optimal portfolio. The portfolio is optimized on the basis of macro
forecasts for the benchmark portfolio, and alpha forecasts adjusted for quality using the
past record of performance for each analyst. Finally, the next month returns are simulated
and the performance of the portfolio is recorded.
Table 27.7 summarizes the results for portfolios when, unbeknownst to the portfolio
manager, security-analyst forecasts are generated with an R -square of .001. M -square clearly increases when performance records are longer. The results also show that, in gen-
eral, performance improves with the size of the portfolio.
The results of all three studies show that even the smallest forecast ability can result
in greatly improved performance. Moreover, with better estimation techniques, perfor-
mance can be further enhanced. We believe that one reason the proposed procedures are
not widely used in the industry is that security analysts believe that low individual correla-
tions imply low aggregate forecasting value and thus wish to avoid the estimation of their
abilities. We hope that results of studies of the type discussed here will lure investment
companies to adopt these techniques and move the industry to new levels of performance.
Forecast Record (months)
Stocks in Portfolio 36 48 60
100 0.96 3.12 6.36 300 0.60 5.88 12.72 500 3.00 5.88 15.12
Table 27.7
M -square of simulated portfolios
13 Alex Kane, Tae-Hwan Kim, and Halbert White, “Forecast Precision and Portfolio Performance,” Journal of
Financial Econometrics, (2010) 8 (3).
27.6 Concluding Remarks on Active Management
A common concern of students of investments who encounter a heavy dose of theory laced
with math and statistics is whether the analytical approach is necessary or even useful.
Here are some observations that should allay any such concern. Investment theory has
developed in recent decades at a galloping pace. Yet, perhaps surprisingly, the distance
between the basic science of investments and industry practice, one that exists in any field,
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C H A P T E R 2 7 The Theory of Active Portfolio Management 973
has actually narrowed in recent years. This satisfying trend is due at least in part to the
vigorous growth of the CFA Institute. The CFA designation has become nearly a prerequi-
site to success in the industry, and the number of individuals seeking it already exceeds that
of finance-major MBAs. They continuously contribute to the proximity between invest-
ments science and the industry.
Even more important is the zeal of the Institute in advancing and enriching the cur-
riculum of the CFA degree and taking it ever closer to contemporary investment theory.
Indeed, finance professors indirectly benefit from this curriculum, because they can argue
the practicality of the text material by pointing out that it is part of the body of knowledge
required of CFA candidates.
Yet there is one area in which practice still lags far behind theory, and that is the subject
of this chapter—this despite the fact that TB and BL models have been around since 1973
and 1992, respectively. Yet, as we have seen, these models have to date failed to materially
penetrate the industry. We speculated on the reason for this failure in the previous section.
We hope, however, that we will be forced to discard this paragraph from future editions
due to obsolescence.
Finally, there is little time in the already dense investments curriculum to discuss the
welfare implication of nearly efficient security prices. Prices can reach such levels only
when investors optimize portfolios with high-quality analysis and implementation. The
value of nearly efficient prices to the welfare of the economy is enormous, competing in
importance with advances in technology. High-quality active management therefore can
contribute to society even as it enriches its practitioners.
SUMMARY 1. Treynor-Black portfolio weights are sensitive to large alpha values, which can result in practically infeasible long/short portfolio positions.
2. Benchmark portfolio risk, the variance of the return difference between the portfolio and the benchmark, can be constrained to keep the TB portfolio within reasonable weights.
3. Alpha forecasts must be shrunk (adjusted toward zero) to account for less-than-perfect forecast- ing quality. Compiling past analyst forecasts and subsequent realizations allows one to estimate
the correlation between realizations and forecasts. Regression analysis can be used to measure
the forecast quality and guide the proper adjustment of future forecasts. When alpha forecasts are
scaled back to account for forecast imprecision, the resulting portfolio positions become far more
moderate.
4. The Black-Litterman model allows the private views of the portfolio manager to be incorporated with market data in the optimization procedure.
5. The Treynor-Black and Black-Litterman models are complementary tools. Both should be used: the TB model is more geared toward security analysis while the BL model more naturally fits
asset allocation problems.
6. Even low-quality forecasts are valuable. Imperceptible R -squares of only .001 in regressions of realizations on analysts’ forecasts can be used to substantially improve portfolio performance.
Related Web sites for this chapter are available at www. mhhe.com/bkm
passive market-index portfolio
active portfolio
alpha values
benchmark portfolio
tracking error
prior distribution
posterior distribution
forecasting record
adjusted alphas
views
asset allocation
baseline forecasts
information ratio
KEY TERMS
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974 P A R T V I I Applied Portfolio Management
APPENDIX A: Forecasts and Realizations of Alpha
1. How would the application of the BL model to a stock and bond portfolio (as the example in the
text) affect security analysis? What does this suggest about the hierarchy of use of the BL and TB
models?
2. Figure 27.4 includes a box for the econometrics unit. Item (3) is to “help other units.” What sorts
of specific tasks might this entail?
3. Make up new alpha forecasts and replace those in Spreadsheet 27.1 in Section 27.1. Find the
optimal portfolio and its expected performance.
4. Make up a view and replace the one in Spreadsheet 27.2 in Section 27.3. Recalculate the optimal
asset allocation and portfolio expected performance.
5. Suppose that sending an analyst to an executive education program will raise the precision of
the analyst’s forecasts as measured by R -square by .01. How might you put a dollar value on this improvement? Provide a numerical example.
Intermediate
Challenge
e X c e l Please visit us at www.mhhe.com/bkm
E-INVESTMENTS EXERCISES Visit www.jpmorganfunds.com . Enter “tracking error” in the Keyword Search box, and you will be directed to a discussion about the measurement of tracking error. What factors are mentioned as possible causes of high tracking error? What is the relationship between high tracking error and a manager’s generation of high positive alphas? How can the tracking error measurement and the Sharpe ratio be used to assess a manager’s performance?
PROBLEM SETS
Basic
A linear representation of the process that generates forecasts from the (yet unknown)
future values of alpha would be
a f(t) 5 b0 1 b1u(t) 1 h(t) (27A.1)
where h ( t ) is the forecasting error and is uncorrelated with the actual u ( t ). Notice that when the forecast is optimized as in Equation 27.7, the error of the adjusted forecast, á( t ) in Equation 27.6, is uncorrelated with the optimally adjusted forecast a ( T ). The coefficients b 0 and b 1 are shift and scale biases in the forecast. Unbiased forecasts would result in b 0 5 0 (no shift) and b 1 5 1 (no scale bias).
We can derive both the variance of the forecast and the covariance between the forecast
and realization from Equation 27A.1:
s2(a f ) 5 b1 2 3 s2(u) 1 s2(h)
Cov(a f, u) 5 b1 3 s 2(u) (27A.2)
Therefore the slope coefficient, a 1 , in Equation 27.6 is
a1 5 Cov(u, a f )
s2(a f ) 5
b1 3 s 2(u)
b1 2 3 s2(u) 1 s2(h)
(27A.3)
When the forecast has no scale bias, that is, when b 1 5 1, a 1 equals the R -square of the regression of forecasts on realizations in Equation 27A.1, which also equals the R -square
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C H A P T E R 2 7 The Theory of Active Portfolio Management 975
APPENDIX B: The General Black-Litterman Model
The BL model is easiest to write using matrix notation. We describe the model according
to the steps in Section 27.3.
Steps 1 and 2: The Covariance Matrix and Baseline Forecasts A sample of past excess returns of the universe of n assets is used to estimate the n 3 n cova- riance matrix, denoted by S. It is assumed that the excess returns are normally distributed.
Market values of the universe assets are obtained and used to compute the 1 3 n vector of weights w M in the baseline equilibrium portfolio. The variance of the baseline portfolio is calculated from
sM 2 5 wM gwM
T (27B.1)
A coefficient of risk aversion for the representative investor in the economy, A, is applied to the CAPM equation to obtain the baseline macro forecast for the market portfolio risk
premium,
E(RM) 5 AsM 2 (27B.2)
The 1 3 n vector of baseline forecasts for the universe securities risk premiums, R, is com- puted from the macro forecast and the covariance matrix by
E(R9) 5 E(RM)gwM T (27B.3)
The data so far describe the prior (baseline) distribution of the rates of return of the asset
universe by
R| , N3E(R), g 4 (27B.4) The n 3 n covariance matrix of the baseline expected returns, t S, is assumed proportional to the covariance matrix, S, by the scalar t.
Step 3: The Manager’s Private Views The k 3 n matrix of views, P, includes k views. The i th view is a 1 3 k vector that mul- tiplies the 1 3 n vector of returns, R|, to obtain the value of the view, Q i , with forecasting error á i . The entire vector of view values and their forecasting errors is given by
RP 5 Q 1 e (27B.5)
The confidence of the manager in the views is given by the k 3 k covariance matrix, Ω, of the vector of errors in views, á. The views embedded in the baseline forecast, R, are given by Q E ,
RP 5 QE
Thus, the 1 3 k vector of deviation of the view from the baseline view (forecasts) and its covariance matrix S D is
D 5 QE 2 Q
SD 5 tPgP T 1 V
(27B.6)
of the regression of realizations on forecasts in Equation 27.6. When b 1 is different from 1.0, we must adjust the coefficient a 1 to account for the scale bias. Notice also that with this adjustment, a 0 5 2 b 0 .
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V is
it u
s a t
w w
w .m
h h
e .c
o m
/b k m
976 P A R T V I I Applied Portfolio Management
Step 4: Revised (Posterior) Expectations The 1 3 n vector of posterior (revised) expectations conditional on the views, as well as the revised covariance matrix, is given by
R* 5 R 0 P 5 R 1 tDSD21gPT (27B.7)
g* 5 g 1 M M 5 tg 2 tgPT SD
21Ptg
Step 5: Portfolio Optimization The vector of revised expectations is used in conjunction with the covariance matrix of
excess returns to produce the optimal portfolio weights with the Markowitz algorithm.
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288 PA
R T
V II
TRANSLATING THE ASPIRATIONS and cir- cumstances of diverse households into appro- priate investment decisions is a daunting task. The task is equally difficult for institutions, most of which have many stakeholders and often are regulated by various authorities. The investment process is not easily reduced to a simple or mechanical algorithm.
While many principles of investments are quite general and apply to virtually all inves- tors, some issues are peculiar to the specific investor. For example, tax bracket, age, risk tolerance, wealth, job prospects, and uncer- tainties make each investor’s circumstances somewhat unique. In this chapter we focus on the process by which investors systematically review their particular objectives, constraints, and circumstances. Along the way, we sur- vey some of the major classes of institutional investors and examine the special issues they must confront.
Of course, there is no unique “correct” investment process. However, some approaches are better than others, and it can be helpful to take one high-quality approach as a useful case study. For this reason, we will examine
the systematic approach suggested by the CFA Institute. Among other things, the Institute administers examinations to certify investment professionals as Chartered Financial Analysts. Therefore, the approach we outline is also one that a highly respected professional group endorses through the curriculum that it requires investment practitioners to master.
The basic framework involves dividing the investment process into four stages: specify- ing objectives, specifying constraints, for- mulating policy, and later monitoring and updating the portfolio as needed. We will treat each of these activities in turn. We start with a description of the major types of inves- tors, both individual and institutional, as well as their special objectives. We turn next to the constraints or circumstances peculiar to each investor class, and we consider some of the investment policies that each can choose.
We will examine how the special circum- stances of both individuals as well as institu- tions such as pension funds affect investment decisions. We also will see how the tax system can impart a substantial effect on investment decisions.
Investment Policy and the
Framework of the CFA
Institute
CHAPTER TWENTY-EIGHT
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978 P A R T V I I Applied Portfolio Management
28.1 The Investment Management Process
Figure 28.1 CFA Institute investment management process
Portfolio policies
and strategies
Capital market expectations
Planning Execution Feedback
Specification and quantification of
investor objectives, constraints,
and preferences
Relevant economic,
social, political, and
sector considerations
Portfolio construction
and revision
Asset allocation,
portfolio optimization,
security selection,
implementation, and
execution
Monitoring economic
and market input
factors
Monitoring investor-
related input factors
Attainment of
investor
objectives
Performance
measurement
The CFA Institute divides the process of investment management into three main elements
that constitute a dynamic feedback loop: planning, execution, and feedback. Figure 28.1
and Table 28.1 describe the steps in that process. As shorthand, you might think of plan- ning as focused largely on establishing all the inputs necessary for decision making. These include data about the client as well as the capital market, resulting in very broad policy
guidelines (the strategic asset allocation). Execution fleshes out the details of optimal asset allocation and security selection. Finally, feedback is the process of adapting to changes in expectations and objectives as well as to changes in portfolio composition that result from
changes in market prices.
The result of this analysis can be summarized in an Investment Policy Statement
addressing the topics specified in Table 28.2 . In the next sections we elaborate on the steps
leading to such an Investment Policy Statement. We start with the planning phase, the first
panel of Table 28.1 .
Objectives Table 28.1 indicates that the management planning process starts off by analyzing one’s
investment clients—in particular, by considering the objectives and constraints that gov-
ern their decisions. Portfolio objectives center on the risk–return trade-off between the expected return the investors want ( return requirements in the first column of Table 28.3 ) and how much risk they are willing to assume ( risk tolerance ). Investment managers must know the level of risk that can be tolerated in the pursuit of a higher expected rate of return.
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Risk Tolerance Questionnaire
Here is an example of a short quiz that may be used by financial institutions to help estimate risk tolerance.
yourself a moderate investor. This means you are willing to accept some risk in exchange for a potential higher rate of return. If you score fewer than 15 points, consider yourself a conservative investor. If you have fewer than 10 points, you may consider yourself a very conservative investor.
Source: Securities Industry and Financial Markets Association.
W O
R D
S FR O
M T
H E ST
R E E T
Add the number of points for all seven questions. Add one point if you choose the first answer, two if you choose the second answer, and so on. If you score between 25 and 28 points, consider yourself an aggressive investor. If you score between 20 and 24 points, your risk tolerance is above average. If you score between 15 and 19 points, consider
Question 1 Point 2 Points 3 Points 4 Points
1. I plan on using the money I am investing:
Within 6 months. Within the next 3 years.
Between 3 and 6 years. No sooner than 7 years from now.
2. My investments make up this share of assets (excluding home):
More than 75%. 50% or more but less than 75%.
25% or more but less than 50%.
Less than 25%.
3. I expect my future income to:
Decrease. Remain the same or grow slowly.
Grow faster than the rate of infl ation.
Grow quickly.
4. I have emergency savings:
No. — Yes, but less than I’d like to have.
Yes.
5. I would risk this share in exchange for the same probability of doubling my money:
Zero. 10%. 25%. 50%.
6. I have invested in stocks and stock mu- tual funds:
— Yes, but I was uneasy about it.
No, but I look forward to it.
Yes, and I was c omfortable with it.
7. My most important investment goal is to:
Preserve my origi- nal investment.
Receive some growth and provide income.
Grow faster than in- fl ation but still provide some income.
Grow as fast as possible. Income is not important today.
979
The nearby box is an illustration of a questionnaire designed to assess an investor’s risk
tolerance. Table 28.4 lists factors governing return requirements and risk attitudes for each
of the seven major investor categories we will discuss.
Individual Investors The basic factors affecting individual investor return requirements and risk tolerance are
life-cycle stage and individual preferences. A middle-aged tenured professor will have a
different set of needs and preferences from a retired widow, for example. We will have
much more to say about individual investors later in this chapter.
Personal Trusts Personal trusts are established when an individual confers legal title to property to another person or institution (the trustee) to manage that property for one or more benefi-
ciaries. Beneficiaries customarily are divided into income beneficiaries , who receive the interest and dividend income from the trust during their lifetimes, and remaindermen , who receive the principal of the trust when the income beneficiary dies and the trust is
dissolved. The trustee is usually a bank, a savings and loan association, a lawyer, or an
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980 P A R T V I I Applied Portfolio Management
I. Planning A. Identifying and specifying the investor’s objectives and constraints B. Creating the Investment Policy Statement (See Table 28.2.) C. Forming capital market expectations D. Creating the strategic asset allocation (target minimum and maximum class weights)
II. Execution: Portfolio construction and revision A. Asset allocation (including tactical) and portfolio optimization (combining assets to meet risk and return
objectives) B. Security selection C. Implementation and execution
III. Feedback A. Monitoring (investor, economic, and market input factors) B. Rebalancing C. Performance evaluation
Table 28.1
Components of the investment management process
Source: John L. Maginn, Donald L. Tuttle, Dennis W. McLeavey, and Jerald E. Pinto, “The Portfolio Management Process and the Investment Policy Statement,” in Managing Investment Portfolios: A Dynamic Process, 3rd ed. (CFA Institute, 2007) and correspondence with Tom Robinson, head of educational content.
1. Brief client description 2. Purpose of establishing policies and guidelines 3. Duties and investment responsibilities of parties involved 4. Statement of investment goals, objectives, and constraints 5. Schedule for review of investment performance and the IPS 6. Performance measures and benchmarks 7. Any considerations in developing strategic asset allocation 8. Investment strategies and investment styles 9. Guidelines for rebalancing
Table 28.2
Components of the investment policy statement
Objectives Constraints Policies
Return requirements Liquidity Asset allocation Risk tolerance Horizon Diversification
Regulations Risk positioning Taxes Tax positioning Unique needs Income generation
Table 28.3
Determination of portfolio policies
investment professional. Investment of a trust is subject to trust laws, as well as “prudent
investor” rules that limit the types of allowable trust investment to those that a prudent
person would select.
Objectives for personal trusts normally are more limited in scope than those of the
individual investor. Because of their fiduciary responsibility, personal trust managers
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 981
typically are more risk averse than are individual investors. Certain asset classes such as
options and futures contracts, for example, and strategies such as short-selling or buying
on margin are ruled out.
Mutual Funds Mutual funds are pools of investors’ money. They invest in ways specified in their pro-
spectuses and issue shares to investors entitling them to a pro rata portion of the income
generated by the funds. The return requirement and risk tolerance across mutual funds are
highly variable because funds segment the investor market. Various funds appeal to distinct
investor groups and will adopt a return requirement and risk tolerance that fit a particular
market niche. For example, income funds cater to the conservative investor, while high-
growth funds seek out the more risk-tolerant ones. Tax-free bond funds segment the market
by tax bracket.
Pension Funds Pension fund objectives depend on the type of pension plan. There are two basic types:
defined contribution plans and defined benefit plans . Defined contribution plans are in effect tax-deferred retirement savings accounts established by the firm in trust for its
employees, with the employee bearing all the risk and receiving all the return from the
plan’s assets.
In defined benefit plans, by contrast, the employer has an obligation to provide a speci-
fied annual retirement benefit. That benefit is defined by a formula that typically takes
into account years of service and the level of salary or wages. For example, the employer
may pay the retired employee a yearly amount equal to 2% of the employee’s final annual
salary for each year of service. A 30-year employee would then receive an annual benefit
equal to 60% of his or her final salary. The payments are an obligation of the employer, and
the assets in the pension fund provide collateral for the promised benefits. If the investment
performance of the assets is poor, the firm is obligated to make up the shortfall by contrib-
uting additional assets to the fund. In contrast to defined contribution plans, the risk sur-
rounding investment performance in defined benefit plans is borne by the firm. We discuss
pension plans more fully later in this chapter.
Type of Investor Return Requirement Risk Tolerance
Individual and personal trusts Life cycle (education, children, retirement) Life cycle (younger are more risk tolerant)
Mutual funds Variable Variable Pension funds Assumed actuarial rate Depends on proximity of payouts Endowment funds Determined by current income needs and
need for asset growth to maintain real value Generally conservative
Life insurance companies Should exceed new money rate by sufficient margin to meet expenses and profit objectives; also actuarial rates important
Conservative
Non–life insurance companies No minimum Conservative Banks Interest spread Variable
Table 28.4
Matrix of objectives
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982 P A R T V I I Applied Portfolio Management
Endowment Funds Endowment funds are organizations chartered to use their money for specific nonprofit purposes. They are financed by gifts from one or more sponsors and are typically managed
by educational, cultural, and charitable organizations or by independent foundations estab-
lished solely to carry out the fund’s specific purposes. Generally, the investment objectives
of an endowment fund are to produce a steady flow of income subject to only a moderate
degree of risk. Trustees of an endowment fund, however, can specify other objectives as
dictated by the circumstances of the particular endowment fund.
Life Insurance Companies Life insurance companies generally try to invest so as to hedge their liabilities, which are
defined by the policies they write. Thus there are as many objectives as there are distinct
types of policies. Until the 1980s, there were for all practical purposes only two types of
life insurance policies available for individuals: whole-life and term.
A whole-life insurance policy combines a death benefit with a kind of savings plan that provides for a gradual buildup of cash value that the policyholder can withdraw at a
later point in life, usually at age 65. Term insurance , on the other hand, provides death benefits only, with no buildup of cash value.
The interest rate that is embedded in the schedule of cash value accumulation promised
under a whole-life policy is a fixed rate, and life insurance companies try to hedge this
liability by investing in long-term bonds. Often the insured individual has the right to bor-
row at a prespecified fixed interest rate against the cash value of the policy.
During the inflationary years of the 1970s and early 1980s, when many older whole-life
policies carried contractual borrowing rates far lower than those available in the capital
markets, policyholders borrowed heavily against the cash value to invest in money market
mutual funds paying double-digit yields. In response to these developments the insurance
industry came up with two new policy types: variable life and universal life . Under a variable life policy the insured’s premium buys a fixed death benefit plus a cash value that
can be invested in a variety of mutual funds from which the policyholder can choose. With
a universal life policy, policyholders can increase or reduce the premium or death ben-
efit according to their needs. Furthermore, the interest rate on the cash value component
changes with market interest rates. The great advantage of variable and universal life insur-
ance policies is that earnings on the cash value are not taxed until the money is withdrawn.
Non–Life Insurance Companies Non–life insurance companies such as property and casualty insurers have investable
funds primarily because they pay claims after they collect policy premiums. Typically, they are conservative in their attitude toward risk. A common thread in the objectives of
pension plans and insurance companies is the need to hedge predictable long-term liabili-
ties. Investment strategies typically call for hedging these liabilities with bonds of various
maturities.
Banks The defining characteristic of banks is that most of their investments are loans to
businesses and consumers and most of their liabilities are accounts of depositors. Banks
earn profit from the interest rate spread between loans extended (the bank’s assets)
and deposits and CDs (the bank’s liabilities), as well as from fees for services. Man-
aging bank assets calls for balancing the loan portfolio with the portfolio of deposits
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 983
and CDs. A bank can increase the interest rate spread by lending to riskier borrowers and
by increasing the proportion of longer-term loans. However, bank capital regulations are
risk-based, so higher-risk strategies will elicit higher capital requirements as well as the
possibility of greater regulatory interference in the bank’s affairs.
28.2 Constraints
Even with identical attitudes toward risk, different households and institutions might
choose different investment portfolios because of their differing circumstances. These cir-
cumstances include tax status, requirements for liquidity or a flow of income from the
portfolio, or various regulatory restrictions. These circumstances impose constraints on
investor choice. Together, objectives and constraints determine investment policy.
As noted, constraints usually have to do with investor circumstances. For example, if a
family has children about to enter college, there will be a high demand for liquidity since
cash will be needed to pay tuition bills. Other times, however, constraints are imposed
externally. For example, banks and trusts are subject to legal limitations on the types of
assets they may hold in their portfolios. Finally, some constraints are self-imposed. For
example, social investing means that investors will not hold shares of firms involved in ethically objectionable activities. Some criteria that have been used to judge firms as ineli-
gible for a portfolio are involvement in countries with human rights abuses, production of
tobacco or alcohol, and participation in polluting activities. Table 28.5 presents a matrix
summarizing the main constraints in each category for each of the seven types of investors.
Liquidity Liquidity is the ease (and speed) with which an asset can be sold and still fetch a fair price. It is a relationship between the time dimension (how long will it take to sell) and the price
dimension (any discount from fair market price) of an investment asset. (See the discussion
of liquidity in Chapter 9.)
When an actual concrete measure of liquidity is necessary, one thinks of the discount
when an immediate sale is unavoidable. Cash and money market instruments such as
Treasury bills and commercial paper, where the bid-ask spread is a small fraction of 1%,
are the most liquid assets, and real estate is among the least liquid. Office buildings and
manufacturing structures can potentially experience a 50% liquidity discount.
Table 28.5
Matrix of constraints
Type of Investor Liquidity Horizon Regulations Taxes
Individuals and personal trusts Variable Life cycle None Variable Mutual funds High Variable Few None Pension funds Young, low; mature, high Long ERISA None Endowment funds Low Long Few None Life insurance companies Low Long Complex Yes Non–life insurance companies High Short Few Yes Banks High Short Changing Yes
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984 P A R T V I I Applied Portfolio Management
Both individual and institutional investors must consider how likely they are to dispose
of assets at short notice. From this likelihood, they establish the minimum level of liquid
assets they want in the investment portfolio.
Investment Horizon This is the planned liquidation date of the investment or substantial part of it. Examples of an individual investment horizon could be the time to fund a child’s college education or the retirement date for a wage earner. For a university endowment, an investment horizon
could relate to the time to fund a major campus construction project. Horizon needs to be
considered when investors choose between assets of various maturities, such as bonds,
which pay off at specified future dates. For example, the maturity date of a bond might
make it a more attractive investment if it coincides with a date on which cash is needed.
This idea is analogous to the matching principle from corporate finance: Strive to match
financing maturity to the economic life of the financed asset.
Regulations Only professional and institutional investors are constrained by regulations. First and
foremost is the prudent investor rule . That is, professional investors who manage other people’s money have a fiduciary responsibility to restrict investment to assets that would
have been approved by a prudent investor. The law is purposefully nonspecific. Every pro-
fessional investor must stand ready to defend an investment policy in a court of law, and
interpretation may differ according to the standards of the times.
Also, specific regulations apply to various institutional investors. For instance, U.S. mutual
funds (institutions that pool individual investor money under professional management) may
not hold more than 5% of the shares of any publicly traded corporation. This regulation keeps
professional investors from getting involved in the actual management of corporations.
Tax Considerations Tax consequences are central to investment decisions. The performance of any investment
strategy is measured by how much it yields after taxes. For household and institutional
investors who face significant tax rates, tax sheltering and deferral of tax obligations may
be pivotal in their investment strategy.
Unique Needs Virtually every investor faces special circumstances. Imagine husband-and-wife aeronauti-
cal engineers holding high-paying jobs in the same aerospace corporation. The entire human
capital of that household is tied to a single player in a rather cyclical industry. This couple
would need to hedge the risk of a deterioration of the economic well-being of the aerospace
industry by investing in assets that will yield more if such deterioration materializes.
Similar issues would confront an executive on Wall Street who owns an apartment near
work. Because the value of the home in that part of Manhattan probably depends on the
vitality of the securities industry, the individual is doubly exposed to the vagaries of the
stock market. Because both job and home already depend on the fortunes of Wall Street,
the purchase of a typical diversified stock portfolio would actually increase the exposure
to the stock market.
These examples illustrate that the job, or more generally, human capital, is often an
individual’s biggest “asset,” and the unique risk profile that results from employment can
play a big role in determining a suitable investment portfolio.
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 985
Other unique needs of individuals often center around their stage in the life cycle, as
discussed below. Retirement, housing, and children’s education constitute three major
demands for funds, and investment policy will depend in part on the proximity of these
expenditures.
Institutional investors also face unique needs. For example, pension funds will differ in
their investment policy, depending on the average age of plan participants. Another exam-
ple of a unique need for an institutional investor would be a university whose trustees allow
the administration to use only cash income from the endowment fund. This constraint
would translate into a preference for high-dividend-paying assets.
28.3 Policy Statements 1
An investment policy statement (IPS) serves as a strategic guide to the planning and imple-
mentation of an investment program. When implemented successfully, the IPS anticipates
issues related to governance of the investment program, planning for appropriate asset
allocation, implementing an investment program with internal and/or external managers,
monitoring the results, risk management, and appropriate reporting. The IPS also estab-
lishes accountability for the various entities that may work on behalf of an investor. Per-
haps most important, the IPS serves as a policy guide that can offer an objective course of
action to be followed during periods of disruption when emotional or instinctive responses
might otherwise motivate less prudent actions.
The nearby box suggests desirable components of an Investment Policy Statement for
use with individual and/or high net worth investors. Not every component will be appropri-
ate for every investor or every situation, and there may be other components that are desir-
able for inclusion reflecting unique investor circumstances.
Sample Policy Statements for Individual Investors Perhaps the best way to get a concrete feel for deriving actual policy statements is to con-
sider a sample of such statements for a variety of investors. Therefore, we next present
several examples.
1. Scope and Purpose 1a. Define the context:
A preamble is often useful to relate information about the investor and/or the source of wealth as a way of establishing the context in which an investment program will be implemented.
Example: The assets of the Leveaux family trusts trace back to the establish-
ment of Leveaux Vintners in 1902 by Claude Leveaux. Over the course of the
next 77 years, three generations of the Leveaux family worked to grow the fam-
ily business to include distilled spirits, gourmet snack foods, and the LVX chain
of cafes in Europe and Canada. Each business line was committed to delivering
outstanding quality and value to consumers, as well as investing in the commu-
nities in which Leveaux did business. In 1979, LVX Industries was purchased
by the British conglomerate FoodCo for the equivalent of US$272 million.
Michelle Leveaux established the Leveaux Foundation with $100 million of the
sale proceeds, and much of the remainder constituted the Leveaux Family Trusts
which are the subject of this investment policy statement.
1 This section is adapted from documents of the CFA Institute that were made available to the authors in draft
form. They may differ from the final published documents.
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986
1b. Define the investor: Define who the investor is, be it a natural person or legal/corporate entity.
Example: “This Investment Policy Statement governs the personal investment
portfolios of Mr. Chen Guangping.”
Specify which of the investors’ assets are to be governed by the IPS. Example: “Investment portfolios governed by this Investment Policy Statement
include all portfolios established in Jorge Castillo’s name, in his name with joint
rights of survivorship with Maria Castillo, charitable remainder trusts estab-
lished by Jorge Castillo, and uniform gift to minors accounts established for the
benefit of Jorge Castillo, Jr. and Cynthia Castillo.”
1c. Define the structure: Set forth key responsibilities and actors.
Example: “Janice Jones, as financial advisor to Sam and Mary Smith, is respon-
sible for coordinating updates to the Investment Policy, including soliciting
input from the designated tax and legal advisors to Sam and Mary Smith. Ms.
Jones is also responsible for monitoring application of the Investment Policy
Statement, and shall promptly notify Sam and Mary Smith of the need for
updates to the Policy and/or violations of the Policy in implementation. Sam and
Mary Smith shall be responsible for approving the Investment Policy Statement
and all subsequent revisions to it.”
Set forth a “standard of care” for those serving as advisor. Regulations in differ- ent jurisdictions may allow for advisors to abide by different standards depending
on their preferences, business models, and client preferences. Fiduciary standards
generally require that advisors always hold client interests as foremost, whereas
suitability standards require recommendations that are suitable for an investor based
on the advisor’s knowledge of that investor’s circumstances. Investors may not per-
ceive or understand the distinction in the absence of addressing the issue in the
Investment Policy Statement.
Example: “Fuji Advisors acts as a fiduciary in its capacity as advisor to the
Takesumi Family Accounts, and acknowledges that all advice and decisions
Desirable Components of an Investment Policy Statement
for Individual Investors
SCOPE AND PURPOSE: Define the Context
Define the Investor
Define the Structure
GOVERNANCE: Specify responsibility for determining investment policy
Describe process for review of IPS
Describe responsibility for engaging/discharging external advisers
Assign responsibility for determination of asset allocation
Assign responsibility for risk management
INVESTMENT, RETURN, AND RISK OBJECTIVES:
Describe overall investment objective
State return, distribution, and risk requirements
Describe relevant constraints
Describe other relevant considerations
RISK MANAGEMENT Establish performance measurement accountabilities
Specify appropriate metrics for risk measurement
Define a process by which portfolios are rebalanced
W O
R D
S FR
O M
T H
E S
T R
E E T
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 987
rendered must reflect first and foremost the best interests of its clients. Fuji
Advisors also affirms its compliance as a firm with the CFA Institute Asset
Manager Code of Professional Conduct.”
Identify a risk management structure applicable to investing. Example: “Susan Smith, as investment advisor to Russell Roberts, is respon-
sible for monitoring investment risks and reporting them to Russell Roberts in
the reporting format that has been agreed to, a sample of which is presented in
Appendix XX.”
2. Governance 2a. Specify who is responsible for determining investment policy, executing invest-
ment policy, and monitoring the results of implementation of the policy. The IPS documents accountability for all stages of investment policy development and
implementation. It can reinforce the obligations of advisors to offer counsel and
principals to ultimately approve or disapprove of the policy.
Example: “As trustee for the Charitable Remainder Trust, Nigel Brown is
responsible for approval of the investment policy and any subsequent changes to
it. In their capacity as counselors to the Trust, Tower Advisors shall counsel the
trustee as to development of the Investment Policy, suggest appropriate revisions
to the Investment Policy on an ongoing basis, and monitor and report on results
achieved from implementation of the Policy on no less than a monthly basis.”
2b. Describe the process for review and updating of the IPS. A process for refreshing the IPS as investor circumstances and/or market conditions change should be clearly identified in advance.
Example: “Wanda Wood is responsible for monitoring the investing require-
ments of Sam and Susan Smith as well as investment and economic issues, and
for suggesting changes to the IPS as necessary. Wanda Wood shall review the
IPS no less frequently than annually with Sam and Susan Smith.”
2c. Describe the responsibilities for engaging or discharging external advisors. The IPS should set forth who is responsible for hiring and firing external money manag-
ers, consultants, or other vendors associated with the investment assets.
Example: “Marcel Perrold delegates exclusive authority to his financial advisor
Francois Finault to retain and dismiss individuals and/or firms to manage his
investment assets. Francois Finault shall, prior to hiring any external investment
manager, disclose in writing to Marcel Perrold any compensation or other con-
sideration received or due to be received from the external investment manager.”
2d. Assign responsibility for determination of asset allocation, including inputs used and criteria for development of input assumptions. An asset allocation framework provides strategic context to many of the more tactical investment decisions. Asset
allocation policies are likely to change over time as characteristics of the inves-
tor change and as market circumstances vary. Accordingly, the IPS may reference
an Asset Allocation Policy as an appendix, which can be revised without requir-
ing approval of an entirely new IPS. It is appropriate for the IPS to address the
assumptions used in developing and selecting inputs to the asset allocation decision
process.
Example: “At least annually, Tower Advisors shall review the asset allocation
of the Family Investment Accounts, and suggest revisions for final approval by
James and Jennifer Jensen. The asset allocation plan is incorporated as Appen-
dix A to this Investment Policy Statement, and shall consider the proportions of
investments in cash equivalents, municipal securities, US fixed income obliga-
tions, US large capitalization equities, US small capitalization equities, and
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988 P A R T V I I Applied Portfolio Management
American Depositary Receipts (ADRs). Tower Advisors shall consider expected
returns and correlation of returns for a broad representation of asset classes in
the US capital markets, as well as anticipated changes in the rate of inflation,
and changes in marginal tax rates.”
2e. Assign responsibility for risk management, monitoring, and reporting. The IPS should document who is responsible for setting risk policy, monitoring the risk pro-
file of the investment portfolio, and reporting on portfolio risk.
Example: “As investment advisor, Tower Capital is responsible for using
the statements prepared by CCC Brokerage as a basis for evaluating the risk
profile of the Jorge Luiz account, consistent with the risk management policies
approved and adopted by Jorge Luiz (see Appendix ZZZ). Tower Capital shall
be responsible for identifying variances in risk positions that exceed tolerable
limits as specified in the risk management policies, and taking prompt correc-
tive action. No less than quarterly, Tower Capital shall provide to Jorge Luiz a
reporting of all such variances in the prior quarter.”
3. Investment, Return, and Risk Objectives 3a. Describe overall investment objective. The IPS should relate the purpose of the
assets being invested to a broad investment objective.
Example: “The investment program governed by the IPS is intended to supple-
ment the earned income of Marcel Perrold in satisfying ongoing living expenses
as well as to provide for funds upon his retirement in 2026.”
3b. State the return, distribution, and risk requirements.
State the overall investment performance objective. Careful specification of the overall investment performance objective is likely to incorporate descriptions of
general funding needs as well as relations to key factors (such as inflation, spending
rate, etc.).
Example: “The financial plan developed for Margarita Mendez indicates a
required real growth rate of 4% to satisfy her future obligations and allow her
to retire in 2027 as planned.”
Identify performance objectives for each asset class eligible for investment. The Investment Policy Statement should set forth all permissible asset classes in
which the portfolio may be invested. Some investors may find benefit in employing
techniques to risk-adjust the benchmark return and portfolio return for purposes of
comparison. Note that some asset classes may not be employed at all times, but they
should still be identified in the IPS. For each asset class, a brief description of the
class should be provided, and a benchmark for performance identified. Within each
asset class, there may be sub-asset classes (for example, US Large Cap equity as a
sub-asset class of US equity). Descriptions and benchmarks for sub-asset classes
may be identified in the IPS, or reserved for an Asset Allocation Plan that may be
attached as an appendix.
Example: “The Family Trust accounts may invest in US equity, US fixed
income, US money market, and Developed Country international equity securi-
ties. The following benchmarks have been selected for comparison to each asset
class: US equity: Russell 3000 index. US fixed income: Barclays US Aggregate
Index. US money market: Lipper US Government Money Market Average.
Developed Country international equity securities: MSCI EAFE index.”
Define distribution/spending assumptions or policies. Spending or distributions from the portfolio should be defined. Often, a “spending calculus” that reconciles
investment return objectives, fees, taxes, inflation, and anticipated spending is useful
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 989
to document as a guide to realistic assumptions. Distributions may be characterized
as a percentage of portfolio market value or as a specific cash value.
Example: “Based on the overall expected portfolio return of 7.5%, fees of 1.2%,
inflation of 2.8%, and an effective tax rate of 32% of total appreciation, the Linzer
Trust Portfolio may support an annual spending rate of 1.2% of the portfolio mar-
ket value while retaining potential for capital preservation or nominal growth.”
Define a policy portfolio to serve as a basis for performance and risk assessments. An asset allocation policy should designate target allocations to each asset class, with
allowable ranges around the targets. Similar targets and ranges may be specified for
sub-asset classes. Overall fund returns, weighted according to strategic target alloca-
tions, may be constructed and compared to overall actual fund performance. Similarly,
some insight as to risk exposures may be developed from examining deviations from
target allocations and violations of acceptable ranges of deviation.
Example: “An asset allocation plan for the Mendez Charitable Trust is attached
as Exhibit ZZZ, and shall be subject to periodic review and change under the
sole authority of Jose Carrios as trustee. For each asset class, a target allocation
has been established that reflects the optimized asset allocation study con-
ducted by Hill Counsel as investment advisors, as well as allowable ranges from
which actual allocations to each asset class may vary. The investment advisor is
responsible for adhering to the asset allocation plan and for maintaining actual
allocations to asset classes within the ranges established.”
3c. Define the risk tolerance of the investor. Describe the investor’s general philosophy regarding tolerance of risk. The IPS should acknowledge the assumption of risk, and the potential for returns associ-
ated with risk to be both positive and negative over time. Relevant risks are usually
myriad, and may include liquidity, legal, political, regulatory, longevity, mortality,
business, and/or health risks. Beyond specifying relevant risks, defining acceptable
paths of risk may also be important: volatility as a descriptive measure of risk may
be irrelevant beyond an absolute level of loss that completely derails an investment
portfolio given personal (i.e., loss of job, disability, life cycle stage) risks.
For individuals, assessing risk tolerance may be difficult and subjective. Where
possible, the IPS should account for known liabilities to lend some quantitative basis
to the risk tolerance assessment. Individual investors may also require assessment
of intellectual and emotional tolerance for potential losses associated with risks,
using interviews or questionnaires. More nuanced approaches may attempt to define
multiple levels of risks associated with avoiding financial catastrophe, maintaining
a current standard of living, or developing significant further wealth. The results
of this sort of analysis may suggest boundaries for tolerance for risk and associ-
ated policies (for example, stop-loss or rebalancing policies). Such policies may be
incorporated by reference in an Appendix.
Example: “James and Jennifer Jensen seek to generate investment returns that
are proportionate to the risks assumed in the Family Trust portfolios, under-
standing that the very nature of risk is uncertainty about the future, and spe-
cifically, the uncertainty as to future investment returns. Tower Advisors, as
investment advisor, seeks to implement an investment strategy that balances the
need to grow the Family Trust assets consistent with the objectives identified in
the Financial Plan with the risks associated with that strategy. Tower Advisors
understands that an absolute loss in any 12 month period of more than 2 33% is
intolerable, and policies and procedures to minimize subsequent risk of further
loss should be implemented by Tower Advisors at that threshold.”
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990 P A R T V I I Applied Portfolio Management
3d. Describe relevant constraints. Investors must address a variety of constraints that affect their investment programs. Such constraints may reflect legal or regulatory
imperatives, or may reflect internal policies. Often, such constraints are closely
linked to particular risks that are relevant to the investor.
Define an evaluation horizon for achievement of performance objectives. Although relatively short time periods may be used for monitoring performance,
establishing a minimum time horizon for achievement of performance objectives
makes clearer when action may need to be taken to resolve underperformance issues.
Example: “The investment advisor will provide the Family Trust trustees with a
quarterly report that summarizes the performance of each investment manager,
each asset class, and the Family Trust in its entirety. The basis for evaluation of
relative success in achieving investment objectives will be on a rolling 8-quarter
basis.”
Identify any requirements for maintaining liquidity. Investors may have short or medium term needs for cash, which should be specified in the IPS if they are ongo-
ing requirements.
Example: “All dividend and interest income will be transferred to the James
Jensen checking account at the end of each month. In addition, up to 15% of the
market value of the portfolio should be invested such that it could be liquidated
upon 5 days’ notice without suffering capital depreciation.”
Identify to what extent, if any, tax considerations shall affect investment decision making. In some instances, the tax consequences of an investment deci- sion may significantly change the desirability of the proposed transaction. The
investor’s general tax situation as well as specific tax issues should be accounted for
in the Investment Policy Statement.
Example: “In general, it will be the investment policy for the Wen portfolios
to invest for appreciation in the taxable individual accounts, and invest for
dividend and interest income in the individual retirement accounts. In addition,
the investment advisor shall consider tax harvesting of existing high-basis hold-
ings as holdings in similar industries or sectors are considered, secondary to the
primary investment objective of the purchase/sale decision.”
Identify any relevant legal constraints. Example: “Management of the Aquilla Family Foundation account is subject to
the provisions of the Uniform Prudent Investor Act.”
Specify any policies related to leverage. The ability to leverage portfolios may be constrained by policy or relevant statute. Any such constraints should be identified.
In addition, to the extent that different manager portfolios and/or different asset
classes have different leverage allowances, accountability for monitoring overall
leverage should be defined.
Example: “At the discretion of Tower Advisors as investment manager, the Xie
Weng portfolio may be margined up to 50% of its value.”
3e. Describe other considerations relevant to investment strategy. State the investment philosophy. The IPS should document the investor’s philo- sophical approach to investing, which may include such dimensions as market
efficiency; the degree of opportunism anticipated; desirability for inclusion of envi-
ronmental, social, and/or governance factors in decision making; etc.
Example for an Individual Investor: “James and Judy Jensen have as a philo-
sophical basis for investment the conviction that markets are efficient, and thus
active management of assets is unlikely to add value beyond the short term
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 991
net of investment costs. Further, James and Judy Jensen believe in a long term
orientation to their investment program and do not intend to seek to exploit
investment opportunities that may exist in the very short term, given their belief
in their inability to consistently do so profitably.”
Identify special factors to be used in including or excluding potential invest- ments from the portfolio. Investors may choose to impose limits on certain invest- ments, consistent with their beliefs in extrafinancial factor effects on securities
prices, a desire to avoid concentrated risks in a particular industry, or to be con-
sistent with their philosophical or political orientation of the organization. In par-
ticular, use of Environmental, Social, or Governance (ESG) factors is increasingly
common, and such use should be explicitly allowed or disallowed in the Invest-
ment Policy Statement. Islamic clients may choose to restrict investment activity to
shariah-compliant investments.
Example for an Individual Investor: “Consistent with her personal beliefs, no
investments in companies that derive revenue from products or services that
are contrary to the teachings of the Catholic Church will be made for Jennifer
Jensen’s account. The investment advisor will be responsible for reviewing the
portfolio monthly to assure that this requirement is satisfied, and shall immedi-
ately dispose of any portfolio holding found to be in violation of this policy.”
4. Risk Management 4a. Establish performance measurement and reporting accountabilities. The IPS
should establish an objective, reliable mechanism for reporting on investment
performance.
Example for an Individual Investor: “Hill Counsel, as investment advisor to the
Charitable Remainder Trust, will calculate the performance of each investment
account under its supervision and report to the trustees by the 15th day of the
new quarter. Calculations will be performed consistent with the Global Invest-
ment Performance Standards published by CFA Institute.”
4b. Specify appropriate metrics for risk measurement and evaluation. Consistent use of metrics to assess and evaluate the risk profile of investment portfolios is impor-
tant to allow for meaningful comparisons over time and avoid inappropriate use of
different metrics to highlight or disguise certain risks. There is reasonable debate
over the suitability of various metrics, and continued review of the choice of metrics
is recommended as a strategic imperative.
Example for an Individual Investor: “In addition to performance reporting,
Tower Capital shall report to the Marcel Family Trust trustees on a quarterly
basis indicative risk metrics, calculated as the annualized standard deviation of
portfolio returns relative to each portfolio’s specified benchmark; and the infor-
mation ratio for each portfolio based on annualized returns for the portfolio and
benchmark as of the end of each quarter.”
4c. Define a process by which portfolios are rebalanced to target allocations. Outside boundaries of acceptable variations from target, or another target rebalancing point
should be documented in the Investment Policy Statement. The rebalancing mecha-
nism may be integrated with the risk management system in some cases, in which
case a brief description of the rebalancing policy with reference to the risk manage-
ment process in a separate Appendix is likely to be appropriate. If the policy is not
to rebalance, this should also be documented in the IPS.
Example for an Individual Investor: “On the first business day of each new
quarter, the investment advisor for the Jensen personal accounts will propose
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992 P A R T V I I Applied Portfolio Management
rebalancing transactions to return the accounts to their target allocations, and
shall execute these transactions within two business days of receiving authori-
zation from the Investment Committee, except that if the principal value of a
proposed rebalancing transaction is less than $50,000, that rebalancing transac-
tion will be deferred indefinitely.”
28.4 Asset Allocation
Consideration of their objectives and constraints leads investors to a set of investment poli-
cies. The policies column in Table 28.3 lists the various dimensions of portfolio manage-
ment policymaking—asset allocation, diversification, risk and tax positioning, and income
generation. By far the most important part of policy determination is asset allocation, that
is, deciding how much of the portfolio to invest in each major asset category.
We can view the process of asset allocation as consisting of the following steps:
1. Specify asset classes to be included in the portfolio. The major classes usually
considered are the following:
a. Money market instruments (usually called cash ).
b. Fixed-income securities (usually called bonds ).
c. Stocks.
d. Real estate.
e. Precious metals.
f. Other.
Institutional investors will rarely invest in more than the first four categories,
whereas individual investors may include precious metals and other more exotic
types of investments in their portfolios.
2. Specify capital market expectations. This step consists of using both historical data
and economic analysis to determine your expectations of future rates of return over the
relevant holding period on the assets to be considered for inclusion in the portfolio.
3. Derive the efficient portfolio frontier. This step consists of finding portfolios that
achieve the maximum expected return for any given degree of risk.
4. Find the optimal asset mix. This step consists of selecting the efficient portfolio that
best meets your risk and return objectives while satisfying the constraints you face.
Taxes and Asset Allocation Until this point we have glossed over the issue of income taxes in discussing asset alloca-
tion. Of course, to the extent that you are a tax-exempt investor such as a pension fund, or
if all of your investment portfolio is in a tax-sheltered account such as an individual retire-
ment account (IRA), then taxes are irrelevant to your portfolio decisions.
But let us say that at least some of your investment income is subject to income taxes at
the highest rate under current U.S. law. You are interested in the after-tax holding-period
return (HPR) on your portfolio. At first glance it might appear to be a simple matter to
figure out what the after-tax HPRs on stocks, bonds, and cash are if you know what they
are before taxes. However, there are several complicating factors.
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Looking to Lower Your Risk? Just Add More
If you’re like a lot of investors these days, you’re looking to make your portfolio “less risky.”
The way to do that is by adding more risk, or at least more types of risk. That strange twist—adding more to get less—is why risk is one of the hardest elements of investing to understand.
The first rule in risk is the hardest for many investors to accept: There is no such thing as a “risk-free investment.”
Avoiding one form of risk means embracing another; the safest of investments generally come with the lowest returns, while the biggest potential gainers bring larger potential losses.
The primary risks in fund investing include the following.
Market risk: This is the big one, also known as princi- pal risk, and it’s the chance that downturn chews up your money.
Purchasing power risk: Sometimes called “inflation risk,” this is the “risk of avoiding risk,” and it’s at the oppo- site end of the spectrum from market risk. In a nutshell, this is the possibility that you are too conservative and your money won’t grow fast enough to keep pace with inflation.
Interest-rate risk: This is a key factor in an environ- ment of declining rates, where you face potential income declines when a bond or certificate of deposit matures and you need to reinvest the money.
Goosing returns using higher-yielding, longer-term securities creates the potential to get stuck losing ground to inflation if the rate trend changes again.
Timing risk: This is another highly individual factor, revolving around your personal time horizon. Simply put, the chance of stock mutual funds making money over the
next 20 years is high; the prospects for the next 18 months are murky.
If you need money at a certain time, this risk must be factored into your asset allocation.
Liquidity risk: Another risk heightened by current ten- sions, it affects everything from junk bonds to foreign stocks. If world events were to alter the flow of money in credit markets or to close some foreign stock exchanges for an extended period, your holdings in those areas could be severely hurt.
Political risk: This is the prospect that government deci- sions will affect the value of your investments. Given the current environment, it is probably a factor in all forms of investing, whether you are looking at stocks or bonds.
Societal risk: Call this “world-event risk.” It was evident when the first anthrax scares sent markets reeling briefly. Some businesses are more susceptible (airlines, for exam- ple), though virtually all types of investing have some con- cerns here.
Even after all of those risks, some investments face cur- rency risk, credit risk, and more. Every type of risk deserves some consideration as you build your holdings.
Ultimately, by making sure that your portfolio addresses all types of risk—heavier on the ones you prefer and lighter on those that make you queasy—you ensure that no one type of risk can wipe you out.
That’s something that a “less risky” portfolio may not be able to achieve.
Source: Abridged from Charles A. Jaffee’s article of the same title, Boston Sunday Globe, October 21, 2001. BOSTON SUNDAY GLOBE (“GLOBE STAFF”/“CONTRIBUTING REPORTER” PRODUCED COPY ONLY) by CHARLES A. JAFFEE. Copyright 2001 by GLOBE NEWSPA- PER CO (MA). Reproduced with permission of GLOBE NEWSPAPER CO (MA) in the format Textbook via Copyright Clearance Center.
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The first is the fact that you can choose between tax-exempt and taxable bonds. We
discussed this issue in Chapter 2 and concluded there that you will choose to invest in tax-
exempt bonds (i.e., municipal bonds) if your personal tax rate is such that the after-tax rate
of interest on taxable bonds is less than the interest rate on “munis.”
The second complication is not quite so easy to deal with. It arises from the fact that
part of your HPR is in the form of a capital gain or loss. Under the current tax system you
pay income taxes on a capital gain only if you realize it by selling the asset during the holding period. This applies to bonds as well as stocks, and it makes the after-tax HPR
a function of whether the security will actually be sold at the end of the holding period.
Sophisticated investors time the realization of their sales of securities to minimize their tax
burden. This often calls for selling securities that are losing money at the end of the tax
year and holding on to those that are making money.
Furthermore, because cash dividends on stocks are fully taxable and capital gains
taxes can be deferred by not selling stocks that appreciate in value, the after-tax HPR
on stocks will depend on the dividend payout policies of the corporations that issued
the stock.
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994 P A R T V I I Applied Portfolio Management
28.5 Managing Portfolios of Individual Investors
The overriding consideration in individual investor goal-setting is one’s stage in the
life cycle. Most young people start their adult lives with only one asset—their earning
power. In this early stage of the life cycle an individual may not have much interest in
investing in stocks and bonds. The needs for liquidity and preserving safety of principal
dictate a conservative policy of putting savings in a bank or a money market fund. The
purchase of life and disability insurance will be required to protect the value of human
capital.
When labor income grows to the point at which insurance and housing needs are met,
saving for retirement may begin, especially if the government provides tax incentives for
such savings. Retirement savings typically constitute a family’s first pool of investable
funds. This is money that can be invested in stocks, bonds, and real estate (other than the
primary home).
Human Capital and Insurance The first significant investment decision for most individuals concerns education, building
up their human capital. The major asset most people have during their early working years
is the earning power that draws on their human capital. In these circumstances, the risk of
illness or injury is far greater than the risk associated with financial wealth.
The most direct way of hedging human capital risk is to purchase insurance. With the
combination of your labor income and a disability insurance policy viewed as a portfolio,
the rate of return on this portfolio is less risky than the labor income by itself. Life insur-
ance is a hedge against the complete loss of income as a result of death of any of the fam-
ily’s income earners.
Investment in Residence The first major economic asset many people acquire is their own house. Deciding to buy
rather than rent a residence qualifies as an investment decision.
An important consideration in assessing the risk and return aspects of this investment
is the value of a house as a hedge against two kinds of risk. The first kind is the risk of
increases in rental rates. If you own a house, any increase in rental rates will increase the
return on your investment.
The second kind of risk is that the particular house or apartment where you live may not
always be available to you. By buying, you guarantee its availability.
Saving for Retirement and the Assumption of Risk People save and invest money to provide for future consumption and leave an estate. The
primary aim of lifetime savings is to allow maintenance of the customary standard of liv-
ing after retirement. As Figure 28.2 suggests, your retirement consumption depends on
your life expectancy at that time. Life expectancy, when one makes it to retirement at
age 65, approximates 85 years, so the average retiree needs to prepare a 20-year nest egg
and sufficient savings to cover unexpected health care costs. Investment income may also
increase the welfare of one’s heirs, favorite charity, or both.
Questionnaires suggest that attitudes shift away from risk tolerance and toward risk
aversion as investors near retirement age. With age, individuals lose the potential to recover
from a disastrous investment performance. When they are young, investors can respond to
a loss by working harder and saving more of their income. But as retirement approaches,
investors realize there will be less time to recover. Hence the shift to safe assets.
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 995
Retirement Planning Models In recent years, investment companies and finan-
cial advisory firms have created a variety of “user-
friendly” interactive tools and models for retirement
planning. Although they vary in detail, the essential
structure behind most of them can be explained using
the American Saving Education Council’s “Ballpark
Estimate” worksheet (see Figure 28.3 ). The work-
sheet assumes you’ll need 70% of current income,
that you’ll live to age 87, and you’ll realize a constant
real rate of return of 3% after inflation. For example,
let’s say Jane is a 35-year-old working woman with
two children, earning $30,000 per year. Seventy
percent of Jane’s current annual income ($30,000)
is $21,000. Jane would then subtract the income she
expects to receive from Social Security ($12,000 in
her case) from $21,000, equaling $9,000. This is how
much Jane needs to make up for each retirement year.
Jane expects to retire at age 65, so (using panel 3 of
the worksheet) she multiplies $9,000 3 16.4 equal-
ing $147,600. Jane has already saved $2,000 in her
401(k) plan. She plans to retire in 30 years so (from panel 4) she multiplies $2,000 3 2.4
equaling $4,800. She subtracts that from her total, making her projected total savings needed
at retirement $142,800. Jane then multiplies $142,800 3 .020 5 $2,856 (panel 6). This is the
amount Jane will need to save annually for her retirement.
Figure 28.2 Long life expectancy is a double- edged sword
Source: www.glasbergen.com . Copyright 2000 by Randy Glasbergen. Reprinted by permission of Randy Glasbergen.
a. Think about the financial circumstances of your closest relative in your parents’ generation (preferably your parents’ household if you are fortunate enough to have them around). Write down the objec- tives and constraints for their investment decisions.
b. Now consider the financial situation of your closest relative who is in his or her 30s. Write down the objectives and constraints that would fit his or her investment decision.
c. How much of the difference between the two statements is due to the age of the investors?
CONCEPT CHECK 28.1
Manage Your Own Portfolio or Rely on Others? Lots of people have assets such as Social Security benefits, pension and group insurance
plans, and savings components of life insurance policies. Yet they exercise limited control,
if any, on the investment decisions of these plans. The funds that secure pension and life
insurance plans are managed by institutional investors.
Outside the “forced savings” plans, however, individuals can manage their own invest-
ment portfolios. As the population grows richer, more and more people face this decision.
Managing your own portfolio appears to be the lowest-cost solution. However, against the fees and charges that financial planners and professional investment managers impose,
you will want to offset the value of your time and energy expended on diligent portfolio
management. Most of all, you must recognize the potential difference in investment results. Besides the need to deliver better-performing investments, professional managers face
two added difficulties. First, getting clients to communicate their objectives and constraints
requires considerable skill. This is not a one-time task because objectives and constraints
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996 P A R T V I I Applied Portfolio Management
1. How much annual income will you want in retirement? (Figure 70% of your current annual income just to maintain your current standard of living. Really.) $ 21,000
2. Subtract the income you expect to receive annually from: • Social Security
If you make under $25,000, enter $8,000; between $25,000 2 $40,000, enter $12,000; over $40,000, enter $14,500 − $ 12,000
• Traditional Employer Pension—a plan that pays a set dollar amount for life, where the dollar amount depends on salary and years of service (in today’s dollars) − $
• Part-time income − $
• Other − $
This is how much you need to make up for each retirement year = $ 9,000
Now you want a ballpark estimate of how much money you’ll need in the bank the day you retire. So the accountants went to work and devised this simple formula. For the record, they figure you’ll realize a constant real rate of return of 3% after inflation, you’ll live to age 87, and you’ll begin to receive income from Social Security at age 65.
3. To determine the amount you’ll need to save, multiply the amount you need to make up by the factor below. $147,600
Age you expect to retire: 55 Your factor is: 21.0 60 18.9 65 16.4 70 13.6
4. If you expect to retire before age 65, multiply your Social Security benefit from line 2 by the factor below. + $
Age you expect to retire: 55 Your factor is: 8.8 60 4.7
5. Multiply your savings to date by the factor below (include money accumulated in a 401(k), IRA, or similar retirement plan): − $ 4,800
If you want to retire in: Your factor is: 1.3 1.6 1.8 2.1 2.4 2.8
10 years 15 years 20 years 25 years 30 years 35 years 40 years 3.3
Total additional savings needed at retirement: = $142,800
6. To determine the ANNUAL amount you’ll need to save, multiply the TOTAL amount by the factor below. $ 2,856
If you want to retire in: Your factor is: .085 .052 .036 .027 .020 .016
10 yrs. 15 yrs. 20 yrs. 25 yrs. 30 yrs. 35 yrs. 40 yrs. .013
BALLPARK ESTIMATE®
This worksheet simplifies several retirement planning issues such as projected Social Security benefits and earnings assumptions on savings. It also reflects today’s dollars; therefore you will need to re-calculate your retirement needs annually and as your salary and circumstances change. You may want to consider doing further analysis, either yourself using a more detailed worksheet or computer software or with the assistance of a financial professional.
Figure 28.3 Sample of American Saving Education Council worksheet Source: EBRI (Employee Benefit Research Institute)/American Saving Education Council.
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Check Out Your Investment Adviser
Before selecting an investment adviser, you should know what services you’re paying for, how much those services cost, how the adviser gets paid, and what conflicts of interest the adviser may have when giving you investment advice. If you have a registered investment adviser, review the firm’s brochure when you first receive it and when it is updated by the firm—there’s a wealth of valuable infor- mation in there! And if you don’t recall receiving the bro- chure, request it. You can also find the brochure on the SEC’s Investment Adviser Public Disclosure (IAPD) Web site.
Here are some of the questions to ask when evaluating an investment adviser:
1. Are you registered with the SEC, a state, or the Financial Industry Regulatory Authority (FINRA)?
2. Have you or your firm ever been disciplined by any reg- ulator? If yes, for what reasons and how was the matter resolved?
3. Have you ever been sued by a client who was not happy with your work, the services you provided, or the prod- ucts you recommended?
4. How are you paid for your services? What is your usual hourly rate, flat fee, or commission?
5. What experience do you have, especially with people in my circumstances?
6. Where did you go to school? What is your recent employment history?
7. What products and services do you offer? Are you only supposed to recommend a limited number of products or services to me? If so, why?
Source: http://investor.gov/researching-managing-investments/ working-investment-professionals/brokers-advisors/research- advisor .
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are forever changing. Second, the professional needs to articulate the financial plan and
keep the client abreast of outcomes. Professional management of large portfolios is com-
plicated further by the need to set up an efficient organization where decisions can be
decentralized and information properly disseminated. The nearby box presents some ques-
tions to consider when looking for an investment advisor.
The task of life cycle financial planning is a formidable one for most people. It is not
surprising that a whole industry has sprung up to provide personal financial advice.
Tax Sheltering In this section we explain three important tax sheltering options that can radically affect opti-
mal asset allocation for individual investors. The first is the tax-deferral option, which arises
from the fact that you do not have to pay tax on a capital gain until you choose to realize the
gain. The second is tax-deferred retirement plans such as individual retirement accounts, and
the third is tax-deferred annuities offered by life insurance companies. Not treated here at all
is the possibility of investing in the tax-exempt instruments discussed in Chapter 2.
The Tax-Deferral Option A fundamental feature of the U.S. Internal Revenue Code is that tax on a capital gain on an asset is payable only when the asset is sold; this is its
tax-deferral option . The investor therefore can control the timing of the tax payment. This conveys a benefit to stock investments.
To see this, compare IBM stock with an IBM bond. Suppose both offer an expected
total return of 12%. The stock has a dividend yield of 4% and expected price appreciation
of 8%, whereas the bond pays an interest rate of 12%. The bond investor must pay tax on
the bond’s interest in the year it is earned, whereas the stockholder pays tax only on the
dividend and defers paying capital gains tax until the stock is sold.
Suppose one invests $1,000 for 5 years. Although in reality interest is taxed as ordinary
income while capital gains and dividends are taxed at a rate of only 15% for many inves-
tors, 2 to isolate the benefit of tax deferral, we will assume that all investment income is
2 As of 2013, the tax rate on capital gains and dividends is 15% for married couples earning between $72,500 and
$250,000. At incomes above $250,000, a 3.8% Medicare surtax on investment income is added, and for incomes
above $450,000, the tax rate is 20% (plus the 3.8% surtax).
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taxed at 15%. The bond will earn an after-tax return of 12% 3 (1 2 .15) 5 10.2%. The
after-tax accumulation at the end of 5 years is
$1,000 3 1.1025 5 $1,625.20
For the stock, the dividend yield after taxes is 4% 3 (1 2 .15) 5 3.4%. Because no taxes
are paid on the 8% annual capital gain until year 5, the before-tax accumulation will be
$1,000 3 (1 1 .034 1 .08)5 5 1,000(1.114)5 5 $1,715.64
In year 5, when the stock is sold, the (now-taxable) capital gain is
$1,715.64 2 $1,000(1.034)5 5 1,715.64 2 1,181.96 5 $533.68
Taxes due are $80.05, leaving $1,635.59, which is $10.39 more than the bond investment
yields. Deferral of the capital gains tax allows the investment to compound at a faster rate
until the tax is actually paid. Note that the more of one’s total return that is in the form of
price appreciation, the greater the value of the tax-deferral option.
Tax-Deferred Retirement Plans Recent years have seen increased use of tax- deferred retirement plans in which investors can choose how to allocate assets. Such plans include traditional IRAs, Keogh plans, and employer-sponsored “tax-qualified”
defined contribution plans such as 401(k) plans. A feature they have in common is that
contributions and earnings are not subject to federal income tax until the individual with-
draws them as benefits.
Typically, an individual may have some investment in the form of such qualified retire-
ment accounts and some in the form of ordinary taxable accounts. The basic investment
principle that applies is to hold whatever bonds you want to hold in the retirement account
while holding equities in the ordinary account. You maximize the tax advantage of the
retirement account by holding it in the security that is the least tax advantaged.
To see this point, consider an investor who has $200,000 of wealth, $100,000 of it in a
tax-qualified retirement account. She currently invests half of her wealth in bonds and half in
stocks, so she allocates half of her retirement account and half of her nonretirement funds to
each. She could reduce her tax bill with no change in before-tax returns simply by shifting her bonds into the retirement account and holding all her stocks outside the retirement account.
Suppose our investor earns a 10% per year rate of interest on bonds and 15% per year on stocks, all in the form of price appreciation. In 5 years she will withdraw all her funds and spend them. By how much will she increase her final accumulation if she shifts all bonds into the retirement account and holds all stocks outside the retirement account? She is in a 28% tax bracket for ordinary income, and her capital gains income is taxed at 15%.
CONCEPT CHECK 28.2
Deferred Annuities Deferred annuities are essentially tax-sheltered accounts offered by life insurance companies. They combine deferral of taxes with the option of
withdrawing one’s funds in the form of a life annuity. Variable annuity contracts offer the
additional advantage of mutual fund investing. One major difference between an IRA and
a variable annuity contract is that whereas the amount one can contribute to an IRA is tax-
deductible and extremely limited as to maximum amount, the amount one can contribute to
a deferred annuity is unlimited, but not tax-deductible.
The defining characteristic of a life annuity is that its payments continue as long as the
recipient is alive, although virtually all deferred annuity contracts have several withdrawal
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 999
options, including a lump sum of cash paid out at any time. You need not worry about
running out of money before you die. Like Social Security, therefore, life annuities offer
longevity insurance and thus would seem to be an ideal asset for someone in the retirement
years. Indeed, theory suggests that where there are no bequest motives, it would be optimal
for people to invest heavily in actuarially fair life annuities. 3
There are two types of life annuities, fixed annuities and variable annuities . A fixed annuity pays a fixed nominal sum of money per period (usually each month), whereas a
variable annuity pays a periodic amount linked to the investment performance of some
underlying portfolio.
Variable annuities are structured so that the investment risk of the underlying asset port-
folio is passed through to the recipient, much as shareholders bear the risk of a mutual
fund. There are two stages in a variable annuity contract: an accumulation phase and a
payout phase. During the accumulation phase, the investor contributes money periodically to one or more open-end mutual funds and accumulates shares. The second, or payout, stage usually starts at retirement, when the investor typically has several options, including
the following:
1. Taking the market value of the shares in a lump sum payment.
2. Receiving a fixed annuity until death.
3. Receiving a variable amount of money each period that depends on the investment
performance of the portfolio.
Variable and Universal Life Insurance Variable life insurance is another tax- deferred investment vehicle offered by the life insurance industry. A variable life insurance
policy combines life insurance with the tax-deferred annuities described earlier.
To invest in this product, you pay either a single premium or a series of premiums.
In each case there is a stated death benefit, and the policyholder can allocate the money
invested to several portfolios, which generally include a money market fund, a bond fund,
and at least one common stock fund. The allocation can be changed at any time.
Variable life insurance policies offer a death benefit that is the greater of the stated
face value or the market value of the investment base. In other words, the death benefit
may rise with favorable investment performance, but it will not go below the guaranteed
face value. Furthermore, the surviving beneficiary is not subject to income tax on the
death benefit.
The policyholder can choose from a number of income options to convert the policy
into a stream of income, either on surrender of the contract or as a partial withdrawal. In
all cases income taxes are payable on the part of any distribution representing investment
gains.
The insured can gain access to the investment without having to pay income tax by bor-
rowing against the cash surrender value. Policy loans of up to 90% of the cash value are
available at any time at a contractually specified interest rate.
A universal life insurance policy is similar to a variable life policy except that, instead
of having a choice of portfolios to invest in, the policyholder earns a rate of interest that is
set by the insurance company and changed periodically as market conditions change. The
disadvantage of universal life insurance is that the company controls the rate of return to
the policyholder, and, although companies may change the rate in response to competitive
pressures, changes are not automatic. Different companies offer different rates, so it often
pays to shop around for the best.
3 For an elaboration of this point see Laurence J. Kotlikoff and Avia Spivak, “The Family as an Incomplete
Annuities Market,” Journal of Political Economy 89 (April 1981).
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28.6 Pension Funds
Pension plans are defined by the terms specifying the “who,” “when,” and “how much,” for
both the plan benefits and the plan contributions used to pay for those benefits. The pen- sion fund of the plan is the cumulation of assets created from contributions and the invest- ment earnings on those contributions, less any payments of benefits from the fund. In the
United States, contributions to the fund by either employer or employee are tax-deductible,
and investment income of the fund is not taxed. Distributions from the fund, whether to
the employer or the employee, are taxed as ordinary income. There are two “pure” types of
pension plans: defined contribution and defined benefit.
Defined Contribution Plans In a defined contribution plan, a formula specifies contributions made by and on behalf of
employees but does not promise the benefits to which they will be entitled. Contribution
rules usually are specified as a predetermined fraction of salary (e.g., the employer con-
tributes 10% of the employee’s annual wages to the plan), although that fraction need not
be constant over the course of an employee’s career. The pension fund consists of a set of
individual investment accounts, one for each employee. Pension benefits are not specified,
other than that at retirement the employee may apply that total accumulated value of con-
tributions and earnings on those contributions to purchase an annuity. The employee often
has some choice over both the level of contributions and the way the account is invested.
In principle, contributions could be invested in any security, although in practice most
plans limit investment choices to bond, stock, and money market funds. The employee
bears all the investment risk, and the employer has no legal obligation beyond making its
periodic contributions.
For defined contribution plans, investment policy is essentially the same as for a tax-
qualified individual retirement account. Indeed, the main providers of investment products
for these plans are the same institutions such as mutual funds and insurance companies that
serve the general investment needs of individuals. Therefore, in a defined contribution plan
much of the task of setting and achieving the income-replacement goal falls on the employee.
An employee is 45 years old. Her salary is $40,000 per year, and she has $100,000 accumulated in her self- directed defined contribution pension plan. Each year she contributes 5% of her salary to the plan, and her employer matches it with another 5%. She plans to retire at age 65. The plan offers a choice of two funds: a guaranteed return fund that pays a risk-free real interest rate of 3% per year and a stock index fund that has an expected real rate of return of 6% per year and a standard deviation of 20%. Her current asset mix in the plan is $50,000 in the guaranteed fund and $50,000 in the stock index fund. She plans to reinvest all investment earnings in each fund in that same fund and to allocate her annual contribution equally between the two funds. If her salary grows at the same rate as the cost of living, how much can she expect to have at retirement? How much can she be sure of having?
CONCEPT CHECK 28.3
Defined Benefit Plans Whereas defined contribution plans specify the contributions made on behalf of employ-
ees, defined benefit plans specify the retirement benefits to which the employee is enti- tled. The firm is responsible for ensuring that funding will be adequate to provide those
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 1001
benefits. The benefit formula typically takes into account years of service for the employer
and level of wages or salary (e.g., an employer might pay an employee for life, beginning
at age 65, a yearly amount equal to 1% of his final annual wage for each year of service).
The employer (called the plan sponsor ) or an insurance company hired by the sponsor guarantees the benefits and thus absorbs the investment risk.
As measured both by number of plan participants and the value of total pension liabili-
ties, the defined benefit form still dominates in most countries around the world. However,
the strong trend since the mid-1970s has been for sponsors to choose the defined contri-
bution form when starting new plans. But the two plan types are not mutually exclusive.
Many sponsors adopt defined benefit plans as their primary plan, in which participation is
mandatory, and supplement them with voluntary defined contribution plans.
With defined benefit plans, there is an important distinction between the pension plan and the pension fund. The plan is the contractual arrangement setting out the rights and obligations of all parties; the fund is a separate pool of assets set aside to provide collateral
for the promised benefits. There may be no separate fund, in which case the plan is said to
be unfunded. When there is a separate fund with assets worth less than the present value of
the promised benefits, the plan is underfunded. And if the plan’s assets have a market value
that exceeds the present value of the plan’s liabilities, it is said to be overfunded.
An employee is 40 years old and has been working for the firm for 15 years. If normal retirement age is 65, the interest rate is 8%, and the employee’s life expectancy is 80, what is the present value of the accrued pension benefit?
CONCEPT CHECK 28.4
Pension Investment Strategies The special tax status of pension funds creates the same incentive for both defined contri-
bution and defined benefit plans to tilt their asset mix toward assets with the largest spread
between pretax and after-tax rates of return. In a defined contribution plan, because the
participant bears all the investment risk, the optimal asset mix also depends on the risk
tolerance of the participant.
In defined benefit plans, optimal investment policy may be different because the spon-
sor absorbs the investment risk. If the sponsor has to share some of the upside potential of
the pension assets with plan participants, there is an incentive to eliminate all investment
risk by investing in securities that match the promised benefits. If, for example, the plan
sponsor has to pay $100 per year for the next 5 years, it can provide this stream of benefit
payments by buying a set of five zero-coupon bonds each with a face value of $100 and
maturing sequentially. By so doing, the sponsor eliminates the risk of a shortfall. This is an
example of immunization of the pension liability. If the present value of promised pension benefits exceeds the market value of its
assets, FASB Statement 87 requires that the corporation recognize the unfunded liability
on its balance sheet. If, however, the pension assets exceed the present value of obliga-
tions, the corporation cannot include the surplus on its balance sheet. This asymmetric
accounting treatment expresses a deeply held view about defined benefit pension funds.
Representatives of organized labor, some politicians, and even a few pension professionals
believe that the sponsoring corporation, as guarantor of the accumulated pension benefits,
is liable for pension asset shortfalls but does not have a clear right to the entire surplus in
case of pension overfunding.
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Investing in Equities If the only goal guiding corporate pension policy were share- holder wealth maximization, it is hard to understand why a financially sound pension
sponsor would invest in equities at all. The tax advantage of a pension fund stems from
the ability of the sponsor to earn the pretax interest rate on pension investments. To maxi-
mize the value of this tax shelter, it is necessary to invest entirely in assets offering the
highest pretax interest rate. These will be the most tax dis advantaged assets, meaning that corporate pension funds should invest entirely in taxable bonds and other fixed-income
investments.
Yet we know that in general pension funds invest from 40% to 60% of their portfolios in
equity securities. Even a casual perusal of the practitioner literature suggests that they do so
for a variety of reasons—some right and some wrong. There are three possible correct reasons.
The first possibility is that corporate management views the pension plan as a trust for
the employees and manages fund assets as if it were a defined contribution plan. It believes
that a successful policy of investment in equities might allow it to pay extra benefits to
employees and is therefore worth taking the risk.
The second possible correct reason is that management believes that through supe-
rior market timing and security selection it is possible to create value in excess of man-
agement fees and expenses. Many executives in nonfinancial corporations are used
to creating value in excess of cost in their businesses. They assume that it can also
be done in the area of portfolio management. Of course, if that is true, then one must
ask why they do not do it on their corporate account rather than in the pension fund.
That way they could have their tax shelter “cake” and eat it too. It is important to real-
ize, however, that to accomplish this feat, the plan must beat the market, not merely
match it.
Note that a very weak form of the efficient markets hypothesis would imply that man-
agement cannot create shareholder value simply by shifting the pension portfolio out of
bonds and into stocks. Even when the entire pension surplus belongs to the shareholders,
investing in stocks just moves the shareholders along the capital market line (the market
trade-off between risk and return for passive investors) and does not create value. When the
net cost of providing plan beneficiaries with shortfall risk insurance is taken into account,
increasing the pension fund equity exposure reduces shareholder value unless the equity
investment can put the firm above the capital market line. This implies that it makes sense
for a pension fund to invest in equities only if it is able to pursue an active strategy that beats the market either through superior timing or security selection. A completely passive
strategy will add no value to shareholders.
For an underfunded plan of a corporation in financial distress there is another pos-
sible reason for investing in stocks and other risky assets—federal pension insurance.
Firms in financial distress have an incentive to invest pension fund money in the risk-
iest assets, just as troubled thrift institutions insured by the Federal Savings and Loan
Insurance Corporation (FSLIC) in the 1980s had similar motivation with respect to their
loan portfolios.
Wrong Reasons to Invest in Equities The wrong reasons for a pension fund to invest in equities stem from interrelated fallacies. The first is the notion that stocks are not
risky in the long run. This fallacy was discussed at length in Chapter 5. Another related
fallacy is the notion that stocks are a hedge against inflation. The reasoning behind this fal-
lacy is that stocks are an ownership claim over real physical capital. If real profits are either
unaffected or enhanced when there is unanticipated inflation, owners of real capital should
not be hurt by it. However, empirical studies show that stock returns have demonstrated
low or even negative correlation with inflation. Thus, the case for stocks as an inflation
hedge is weak.
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28.7 Investments for the Long Run
As the aged population around the world grows more rapidly than any other age group,
issues of saving for the long run, for the most part surrounding retirement, have come to the
fore of the investments industry. Traditionally, the advice for the long run could be summa-
rized by rules of thumb concerning various rates of gradual, age-determined shifts in asset
allocation from risky to safe assets. Implications of “modern” portfolio management, now
more than 30 years old, originated from Merton’s lifetime consumption/investment model
(ICAPM) suggesting that one consider hedge assets to account for extramarket sources of
risk, such as inflation, and needs emanating from uncertain longevity.
Target Investing and the Term Structure of Bonds Interest rates usually vary by maturity. For example, a person considering investing money
in an insured certificate of deposit or a Treasury security will observe that the interest rate
she can earn depends on its maturity. Thus, for any given target date there is a different
risk-free interest rate. Each investor, with a unique horizon, therefore has his or her own
risk-free asset. For Mr. Short it is bills and for Ms. Long it is bonds. Thus, to accommodate
investors with different time horizons, there must be a menu of choices that has a term
structure of risk-free investments. The principle of duration matching means matching
one’s assets to one’s objectives (liabilities) and is equivalent to the immunization strategy
for pension funds that we examined in Chapter 16.
In what unit of account should the risk-free term structure be denominated? This is a
critical issue because a bond is risk-free only in terms of a specified numeraire (unit of
account), such as dollars, yen, and so on. Thus, if a bond promises to pay $100 two years
from now, its payoff in terms of yen depends on the dollar price of the yen 2 years from
now, and vice versa. Thus even a zero-coupon bond with no default risk can still be very
risky if it is denominated in a unit of account (such as a foreign currency) that does not
match the investor’s goal. This type of risk is called “basis risk.”
To illustrate, assume the goal is retirement. If the goal is specified as a level of real wealth at
the retirement date, then the unit of account should be consumption units. The risk-free asset in
this case would be a bond with a payoff linked to an index of consumer prices, such as the CPI.
However, if the index chosen does not truly reflect the specific investor’s future cost of living,
there will be some risk. If the goal is to maintain a certain standard of living for the rest of one’s
life, then instead of a fixed level of retirement wealth in terms of consumption units, a more
appropriate unit of account would be a lifetime flow of real consumption. This can be computed by dividing dollar amounts by the market price of a lifetime real annuity that starts paying
benefits at the target retirement date. The term structure is then given by the prices of lifetime
real annuities with different starting dates. Similarly, education bonds that are linked to the cost
of college education would provide the appropriate unit of account for children’s college funds.
Making Simple Investment Choices A target-date retirement fund (TDRF) is a fund of funds diversified across stocks and
bonds with the feature that the proportion invested in stocks is automatically reduced
as time passes. 4 TDRFs are often advocated as the simple solution to the complex task
4 Vanguard describes its TDRFs as follows: “With Target Retirement Funds, you have only one decision to make:
when you plan to retire. Your Target Retirement Fund automatically grows more conservative as your retire-
ment date nears. When you are ready to draw income in your retirement, your Target Retirement Fund has a
stable, income-oriented mix of assets.” From “Choose a Simple Solution: Vanguard Target Retirement Funds,” www.vanguard.com/jumppage/retire .
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of determining the appropriate asset allocation among funds in 401(k) plans, IRAs, and
other personal investment accounts. TDRFs are marketed as enabling investors to put their
investment plans on autopilot. Once you choose a fund with a target year matching your
horizon, the life cycle manager moves some of your money out of stocks and into bonds as
your retirement date nears.
Inflation Risk and Long-Term Investors While inflation risk is usually low for short horizons, it is a first-order source of risk for
retirement planning, where horizons may be extremely long. An inflation “shock” may last
for many years, and impart substantial uncertainty to the purchasing power of any dollar
you (or your client) have saved for retirement.
A conventional answer to the problem of inflation risk is to invest in price-indexed
bonds such as TIPS (see Chapter 14 for a review). This is a good first step but is not a
full answer to inflation risk. A zero-coupon priced-indexed bond with maturity equal to
an investor’s horizon would be a riskless investment in terms of purchasing power. This
can be achieved with CPI-indexed savings bonds, but the government limits the amount
of such bonds one may buy in any year. Unfortunately, market-traded TIPS bonds are
not risk-free. As the (real) interest rate changes, the value of those bonds will fluctuate.
Moreover, these bonds pay coupons, so the accumulated (real) value of the portfolio is
subject to reinvestment-rate risk. These issues should remind you of our discussion of
bond risk in Chapter 16. In this context too, one must balance price risk with reinvestment
rate risk by tailoring the duration of the bond portfolio to the investment horizon. But in
this case, we need to calculate duration using the real interest rate and focus on real pay-
offs from our investments.
1. When the principles of portfolio management are discussed, it is useful to distinguish among seven classes of investors:
a. Individual investors and personal trusts. b. Mutual funds. c. Pension funds. d. Endowment funds. e. Life insurance companies. f. Non–life insurance companies. g. Banks.
In general, these groups have somewhat different investment objectives, constraints, and portfolio policies.
2. To some extent, most institutional investors seek to match the risk-and-return characteristics of their investment portfolios to the characteristics of their liabilities.
3. The process of asset allocation consists of the following steps:
a. Specifying the asset classes to be included. b. Defining capital market expectations. c. Finding the efficient portfolio frontier. d. Determining the optimal mix.
SUMMARY
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4. People living on money-fixed incomes are vulnerable to inflation risk and may want to hedge against it. The effectiveness of an asset as an inflation hedge is related to its correlation with
unanticipated inflation.
5. For investors who must pay taxes on their investment income, the process of asset allocation is complicated by the fact that they pay income taxes only on certain kinds of investment income.
Interest income on munis is exempt from tax, and high-tax-bracket investors will prefer to hold
them rather than short- and long-term taxable bonds. However, the really difficult part of the tax
effect to deal with is the fact that capital gains are taxable only if realized through the sale of an
asset during the holding period. Investment strategies designed to avoid taxes may conflict with
the principles of efficient diversification.
6. The life cycle approach to the management of an individual’s investment portfolio views the individual as passing through a series of stages, becoming more risk averse in later years. The
rationale underlying this approach is that as we age, we use up our human capital and have less
time remaining to recoup possible portfolio losses through increased labor supply.
7. People buy life and disability insurance during their prime earning years to hedge against the risk associated with loss of their human capital, that is, their future earning power.
8. There are three ways to shelter investment income from federal income taxes besides investing in tax-exempt bonds. The first is by investing in assets whose returns take the form of apprecia-
tion in value, such as common stocks or real estate. As long as capital gains taxes are not paid
until the asset is sold, the tax can be deferred indefinitely.
The second way of tax sheltering is through investing in tax-deferred retirement plans such
as IRAs. The general investment rule is to hold the least tax-advantaged assets in the plan and
the most tax-advantaged assets outside of it.
The third way of sheltering is to invest in the tax-advantaged products offered by the life
insurance industry—tax-deferred annuities and variable and universal life insurance. They com-
bine the flexibility of mutual fund investing with the tax advantages of tax deferral.
9. Pension plans are either defined contribution plans or defined benefit plans. Defined contribu- tion plans are in effect retirement funds held in trust for the employee by the employer. The
employees in such plans bear all the risk of the plan’s assets and often have some choice in the
allocation of those assets. Defined benefit plans give the employees a claim to a money-fixed
annuity at retirement. The annuity level is determined by a formula that takes into account years
of service and the employee’s wage or salary history.
10. If the only goal guiding corporate pension policy were shareholder wealth maximization, it would be hard to understand why a financially sound pension sponsor would invest in equities
at all. A policy of 100% bond investment would both maximize the tax advantage of funding the
pension plan and minimize the costs of guaranteeing the defined benefits.
11. If sponsors viewed their pension liabilities as indexed for inflation, then the appropriate way for them to minimize the cost of providing benefit guarantees would be to hedge using securities
whose returns are highly correlated with inflation. Common stocks would not be an appropriate
hedge because they have a low correlation with inflation.
Related Web sites for this chapter are available at www. mhhe.com/bkm
risk–return trade-off
personal trusts
income beneficiaries
remaindermen
defined contribution plans
defined benefit plans
endowment funds
whole-life insurance policy
term insurance
variable life
universal life
liquidity
investment horizon
prudent investor rule
tax-deferral option
tax-deferred retirement plans
deferred annuities
fixed annuities
variable annuities
immunization
KEY TERMS
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1006 P A R T V I I Applied Portfolio Management
1. Your neighbor has heard that you successfully completed a course in investments and has come
to seek your advice. She and her husband are both 50 years old. They just finished making their
last payments for their condominium and their children’s college education and are planning for
retirement. What advice on investing their retirement savings would you give them? If they are
very risk averse, what would you advise?
2. What is the least-risky a sset for each of the following investors?
a. A person investing for her 3-year-old child’s college tuition. b. A defined benefit pension fund with benefit obligations that have an average duration of
10 years. The benefits are not inflation-protected.
c. A defined benefit pension fund with benefit obligations that have an average duration of 10 years. The benefits are inflation-protected.
3. George More is a participant in a defined contribution pension plan that offers a fixed-income
fund and a common stock fund as investment choices. He is 40 years old and has an accumulation
of $100,000 in each of the funds. He currently contributes $1,500 per year to each. He plans to
retire at age 65, and his life expectancy is age 80.
a. Assuming a 3% per year real earnings rate for the fixed-income fund and 6% per year for common stocks, what will be George’s expected accumulation in each account at age 65?
b. What will be the expected real retirement annuity from each account, assuming these same real earnings rates?
c. If George wanted a retirement annuity of $30,000 per year from the fixed-income fund, by how much would he have to increase his annual contributions?
4. The difference between a Roth IRA and a conventional IRA is that in a Roth IRA taxes are paid
on the income that is contributed but the withdrawals at retirement are tax-free. In a conven-
tional IRA, however, the contributions reduce your taxable income, but the withdrawals at retire-
ment are taxable. Try using the Excel spreadsheet introduced in the Appendix to answer these
questions.
a. Which of these two types provides higher after-tax benefits? b. Which provides better protection against tax rate uncertainty?
Basic
Intermediate
PROBLEM SETS
1. Angus Walker, CFA, is reviewing the defined benefit pension plan of Acme Industries. Based
in London, Acme has operations in North America, Japan, and several European countries.
Next month, the retirement age for full benefits under the plan will be lowered from age 60
to age 55.
The median age of Acme’s workforce is 49 years. Walker is responsible for the pension
plan’s investment policy and strategic asset allocation decisions. The goals of the plan
include achieving a minimum expected return of 8.4% with expected standard deviation no
greater than 16.0%.
Walker is evaluating the current asset allocation ( Table 28A ) and selected financial informa-
tion for the company ( Table 28B ). There is an ongoing debate within Acme Industries about the
pension plan’s investment policy statement (IPS). Two investment policy statements under con-
sideration are shown in Table 28C .
a. Determine, for each of the following components, whether IPS X or IPS Y (see Table 28C ) has the appropriate language for the pension plan of Acme Industries. Justify each response with
one reason.
i. Return requirement
ii. Risk tolerance
iii. Time horizon
iv. Liquidity
Note: Some components of IPS X may be appropriate, while other components of IPS Y may be appropriate.
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 1007
IPS X IPS Y
Return requirement Plan’s objective is to outperform the relevant benchmark return by a substantial margin.
Plan’s objective is to match the relevant benchmark return.
Risk tolerance Plan has a high risk tolerance because of the long-term nature of the plan and its liabilities.
Plan has a low risk tolerance because of its limited ability to assume substantial risk.
Time horizon Plan has a very long time horizon because of the plan’s infinite life.
Plan has a shorter time horizon than in the past because of plan demographics.
Liquidity Plan needs moderate level of liquidity to fund monthly benefit payments.
Plan has minimal liquidity needs.
Table 28C
Investment policy statements
b. To assist Walker, Acme has hired two pension consultants, Lucy Graham and Robert Michael. Graham believes that the pension fund must be invested to reflect a low risk tolerance, but
Michael believes the pension fund must be invested to achieve the highest possible returns.
The fund’s current asset allocation and the allocations recommended by Graham and Michael
are shown in Table 28D . Select which of the three asset allocations in Table 28D is most
appropriate for Acme’s pension plan. Explain how your selection meets each of the following
objectives or constraints for the plan:
i. Return requirement
ii. Risk tolerance
iii. Liquidity
2. Your client says, “With the unrealized gains in my portfolio, I have almost saved enough money
for my daughter to go to college in 8 years, but educational costs keep going up.” On the basis
of this statement alone, which one of the following appears to be least important to your client’s
investment policy?
a. Time horizon. b. Purchasing power risk. c. Liquidity. d. Taxes.
Acme Industries total assets £16,000
Pension plan data:
Plan assets 6,040
Plan liabilities 9,850
Table 28B
Acme Industries selected financial information (in millions)
International Equities (MSCI World, excluding U.K.) 10%
U.K. bonds 42
U.K. small capitalization equities 13
U.K. large capitalization equities 30
Cash 5
Table 28A
Acme pension plan: Current asset allocation
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1008 P A R T V I I Applied Portfolio Management
3. The aspect least likely to be included in the portfolio management process is
a. Identifying an investor’s objectives, constraints, and preferences. b. Organizing the management process itself. c. Implementing strategies regarding the choice of assets to be used. d. Monitoring market conditions, relative values, and investor circumstances.
4. Sam Short, CFA, has recently joined the investment management firm of Green, Spence, and
Smith (GSS). For several years, GSS has worked for a broad array of clients, including employee
benefit plans, wealthy individuals, and charitable organizations. Also, the firm expresses expertise
in managing stocks, bonds, cash reserves, real estate, venture capital, and international securities.
To date, the firm has not utilized a formal asset allocation process but instead has relied on the
individual wishes of clients or the particular preferences of its portfolio managers. Short recom-
mends to GSS management that a formal asset allocation process would be beneficial and empha-
sizes that a large part of a portfolio’s ultimate return depends on asset allocation. He is asked to
take his conviction an additional step by making a proposal to executive management.
a. Recommend and justify an approach to asset allocation that could be used by GSS. b. Apply the approach to a middle-aged, wealthy individual characterized as a fairly conserva-
tive investor (sometimes referred to as a “guardian investor”).
5. Jarvis University (JU) is a private, multiprogram U.S. university with a $2 billion endowment fund
as of fiscal year-end May 31, 2019. With little government support, JU is heavily dependent on its
endowment fund to support ongoing expenditures, especially because the university’s enrollment
growth and tuition revenue have not met expectations in recent years. The endowment fund must
make a $126 million annual contribution, which is indexed to inflation, to JU’s general operating
budget. The U.S. Consumer Price Index is expected to rise 2.5% annually and the U.S. higher edu-
cation cost index is anticipated to rise 3% annually. The endowment has also budgeted $200 million
due on January 31, 2020, representing the final payment for construction of a new main library.
In a recent capital campaign, JU only met its fund-raising goal with the help of one very successful
alumna, Valerie Bremner, who donated $400 million of Bertocchi Oil and Gas common stock at fis-
cal year-end May 31, 2019. Bertocchi Oil and Gas is a large-capitalization, publicly traded U.S.
company. Bremner donated the stock on the condition that no more than 25% of the initial number of
shares may be sold in any fiscal year. No substantial additional donations are expected in the future.
Given the large contribution to and distributions from the endowment fund, the endowment
fund’s investment committee has decided to revise the fund’s investment policy statement. The
investment committee also recognizes that a revised asset allocation may be warranted. The asset
allocation in place for the JU endowment fund as of May 31, 2019, is given in Table 28E .
a. Prepare the components of an appropriate investment policy statement for the Jarvis University endowment fund as of June 1, 2019, based only on the information given.
Note: Each component in your response must specifically address circumstances of the JU endowment fund.
Current Graham Michael
U.K. large capitalization equities 30 20 40
U.K. small capitalization equities 13 8 20
International equities (MSCI World ex-U.K.) 10 10 18
U.K. bonds 42 52 17
Cash 5 10 5
Total 100 100 100
Expected portfolio return (%) 9.1 8.2 10.6
Expected portfolio volatility (standard deviation in %) 16.1 12.8 21.1
Table 28D
Asset allocations (in %)
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 1009
b. Determine the most appropriate revised allocation percentage for each asset in Table 28E as of June 1, 2019. Justify each revised allocation percentage.
6. Susan Fairfax is president of Reston Industries, a U.S.-based company whose sales are entirely
domestic and whose shares are listed on the New York Stock Exchange. The following are addi-
tional facts concerning her current situation:
• Fairfax is single, aged 58. She has no immediate family, no debts, and does not own a resi-
dence. She is in excellent health and covered by Reston-paid health insurance that continues
after her expected retirement at age 65.
• Her base salary of $500,000/year, infl ation-protected, is suffi cient to support her present
lifestyle but can no longer generate any excess for savings.
• She has $2,000,000 of savings from prior years held in the form of short-term instruments.
• Reston rewards key employees through a generous stock-bonus incentive plan but provides
no pension plan and pays no dividend.
• Fairfax’s incentive plan participation has resulted in her ownership of Reston stock worth
$10 million (current market value). The stock, received tax-free but subject to tax at a 35%
rate (on entire proceeds) if sold, is expected to be held at least until her retirement.
• Her present level of spending and the current annual infl ation rate of 4% are expected to
continue after her retirement.
• Fairfax is taxed at 35% on all salary, investment income, and realized capital gains. Assume
her composite tax rate will continue at this level indefi nitely.
Fairfax’s orientation is patient, careful, and conservative in all things. She has stated that an
annual after-tax real total return of 3% would be completely acceptable to her if it was achieved in
a context where an investment portfolio created from her accumulated savings was not subject to
a decline of more than 10% in nominal terms in any given 12-month period. To obtain the benefits
of professional assistance, she has approached two investment advisory firms—HH Counselors
(“HH”) and Coastal Advisors (“Coastal”)—for recommendations on allocation of the investment
portfolio to be created from her existing savings assets (the “Savings Portfolio”) as well as for
advice concerning investing in general.
a. Create and justify an investment policy statement for Fairfax based only on the information provided thus far. Be specific and complete in presenting objectives and constraints. (An asset
allocation is not required in answering this question.)
b. Coastal has proposed the asset allocation shown in Table 28F for investment of Fairfax’s $2 million of savings assets. Assume that only the current yield portion of projected total
return (comprised of both investment income and realized capital gains) is taxable to Fairfax
and that the municipal bond income is entirely tax-exempt.
Critique the Coastal proposal. Include in your answer three weaknesses in the Coastal
proposal from the standpoint of the investment policy statement you created for her in ( a ).
Asset
Current Allocation (millions)
Current Allocation Percentage
Current Yield
Expected Annual Return
Standard Deviation of Returns
U.S. money market bond fund $ 40 2% 4.0% 4.0% 2.0% Intermediate global bond fund 60 3 5.0 5.0 9.0 Global equity fund 300 15 1.0 10.0 15.0 Bertocchi Oil and Gas common stock 400 20 0.1 15.0 25.0 Direct real estate 700 35 3.0 11.5 16.5 Venture capital 500 25 0.0 20.0 35.0
TOTAL $2,000 100%
Table 28E
Jarvis University endowment fund asset allocation as of May 31, 2019
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1010 P A R T V I I Applied Portfolio Management
Asset Class Proposed
Allocation (%) Current
Yield (%) Projected Total
Return (%)
Cash equivalents 15.0 4.5 4.5
Corporate bonds 10.0 7.5 7.5
Municipal bonds 10.0 5.5 5.5
Large-cap U.S. stocks 0.0 3.5 11.0
Small-cap U.S. stocks 0.0 2.5 13.0
International stocks (EAFE) 35.0 2.0 13.5
Real estate investment trusts (REITs) 25.0 9.0 12.0
Venture capital 5.0 0.0 20.0
TOTAL 100.0 4.9 10.7
Inflation (CPI), projected 4.0
Table 28F
Susan Fairfax proposed asset allocation, prepared by Coastal Advisors
c. HH Counselors has developed five alternative asset allocations (shown in Table 28G ) for client portfolios. Answer the following questions based on Table 28G and the investment
policy statement you created for Fairfax in ( a ).
i. Determine which of the asset allocations in Table 28G meet or exceed Fairfax’s stated
return objective.
ii. Determine the three asset allocations in Table 28G that meet Fairfax’s risk tolerance crite-
rion. Assume a 95% confidence interval is required, with 2 standard deviations serving as
an approximation of that requirement.
Asset Class
Projected Total
Return
Expected Standard Deviation
Asset Allocation
A
Asset Allocation
B
Asset Allocation
C
Asset Allocation
D
Asset Allocation
E
Cash equivalents 4.5% 2.5% 10% 20% 25% 5% 10%
Corporate bonds 6.0 11.0 0 25 0 0 0
Municipal bonds 7.2 10.8 40 0 30 0 30
Large-cap U.S. stocks 13.0 17.0 20 15 35 25 5
Small-cap U.S. stocks 15.0 21.0 10 10 0 15 5
International stocks (EAFE) 15.0 21.0 10 10 0 15 10
Real estate investment trusts (REITs) 10.0 15.0 10 10 10 25 35
Venture capital 26.0 64.0 0 10 0 15 5
TOTAL 100 100 100 100 100
Summary Data
Asset Allocation
A
Asset Allocation
B
Asset Allocation
C
Asset Allocation
D
Asset Allocation
E
Projected total return 9.9% 11.0% 8.8% 14.4% 10.3%
Projected after-tax total return 7.4% 7.2% 6.5% 9.4% 7.4%
Expected standard deviation 9.4% 12.4% 8.5% 18.1% 10.1%
Sharpe ratio 0.574 0.524 0.506 — 0.574
Table 28G
Alternative asset allocations, prepared by HH Counselors
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 1011
d. Assume that the risk-free rate is 4.5%.
i. Calculate the Sharpe ratio for Asset Allocation D.
ii. Determine the two asset allocations in Table 28G having the best risk-adjusted returns,
based only on the Sharpe ratio measure.
e. Recommend and justify the one asset allocation in Table 28G you believe would be the best model for Fairfax’s savings portfolio.
7. John Franklin is a recent widower with some experience in investing for his own account. Fol-
lowing his wife’s recent death and settlement of the estate, Mr. Franklin owns a controlling inter-
est in a successful privately held manufacturing company in which Mrs. Franklin was formerly
active, a recently completed warehouse property, the family residence, and his personal holdings
of stocks and bonds. He has decided to retain the warehouse property as a diversifying investment
but intends to sell the private company interest, giving half of the proceeds to a medical research
foundation in memory of his deceased wife. Actual transfer of this gift is expected to take place
about 3 months from now. You have been engaged to assist him with the valuations, planning, and
portfolio building required to structure his investment program appropriately.
Mr. Franklin has introduced you to the finance committee of the medical research foundation
that is to receive his $45 million cash gift 3 months hence (and will eventually receive the assets of
his estate). This gift will greatly increase the size of the foundation’s endowment (from $10 million
to $55 million) as well as enable it to make larger grants to researchers. The foundation’s grant-
making (spending) policy has been to pay out virtually all of its annual net investment income. As
its investment approach has been very conservative, the endowment portfolio now consists almost
entirely of fixed-income assets. The finance committee understands that these actions are causing
the real value of foundation assets and the real value of future grants to decline due to the effects of
inflation. Until now, the finance committee has believed that it had no alternative to these actions,
given the large immediate cash needs of the research programs being funded and the small size of
the foundation’s capital base. The foundation’s annual grants must at least equal 5% of its assets’
market value to maintain its U.S. tax-exempt status, a requirement that is expected to continue
indefinitely. No additional gifts or fund-raising activities are expected over the foreseeable future.
Given the change in circumstances that Mr. Franklin’s gift will make, the finance committee
wishes to develop new grant-making and investment policies. Annual spending must at least meet
the level of 5% of market value that is required to maintain the foundation’s tax-exempt status,
but the committee is unsure about how much higher than 5% it can or should be. The committee
wants to pay out as much as possible because of the critical nature of the research being funded;
however, it understands that preserving the real value of the foundation’s assets is equally impor-
tant in order to preserve its future grant-making capabilities. You have been asked to assist the
committee in developing appropriate policies.
a. Identify and briefly discuss the three key elements that should determine the foundation’s grant-making (spending) policy.
b. Formulate and justify an investment policy statement for the foundation, taking into account the increased size of its assets arising from Mr. Franklin’s gift. Your policy statement must encom-
pass all relevant objectives, constraints, and the key elements identified in your answer to part ( a ). c. Recommend and justify a long-term asset allocation that is consistent with the investment
policy statement you created in part ( b ). Explain how your allocation’s expected return meets the requirements of a feasible grant-making (spending) policy for the foundation. ( Hint: Your allocation must sum to 100% and should use the economic/market data presented in
Table 28H and your knowledge of historical asset-class characteristics.)
8. Christopher Maclin, aged 40, is a supervisor at Barnett Co. and earns an annual salary of £80,000
before taxes. Louise Maclin, aged 38, stays home to care for their newborn twins. She recently
inherited £900,000 (after wealth-transfer taxes) in cash from her father’s estate. In addition, the
Maclins have accumulated the following assets (current market value):
• £5,000 in cash.
• £160,000 in stocks and bonds.
• £220,000 in Barnett common stock.
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1012 P A R T V I I Applied Portfolio Management
Historic Averages
Intermediate Term Consensus Forecast
U.S. Treasury bills 3.7% 4.2%
Intermediate-term U.S. T-bonds 5.2 5.8
Long-term U.S. T-bonds 4.8 7.7
U.S. corporate bonds (AAA) 5.5 8.8
Non-U.S. bonds (AAA) N/A 8.4
U.S. common stocks (all) 10.3 9.0
U.S. common stocks (small-cap) 12.2 12.0
Non-U.S. common stocks (all) N/A 10.1
U.S. inflation 3.1 3.5
Table 28H
Capital markets annualized return data
The value of their holdings in Barnett stock has appreciated substantially as a result of the com-
pany’s growth in sales and profits during the past 10 years. Christopher Maclin is confident that
the company and its stock will continue to perform well.
The Maclins need £30,000 for a down payment on the purchase of a house and plan to make
a £20,000 non–tax deductible donation to a local charity in memory of Louise Maclin’s father.
The Maclins’ annual living expenses are £74,000. After-tax salary increases will offset any future
increases in their living expenses.
During their discussions with Grant Webb, the Maclins’ express concern about achieving their
educational goals for their children and their own retirement goals. The Maclins tell Webb:
• They want to have suffi cient funds to retire in 18 years when their children begin their
4 years of university education.
• They have been unhappy with the portfolio volatility they have experienced in recent years
and they do not want to experience a loss greater than 12% in any one year.
• They do not want to invest in alcohol and tobacco stocks.
• They will not have any additional children.
After their discussions, Webb calculates that in 18 years the Maclins will need £2 million to meet
their educational and retirement goals. Webb suggests that their portfolio be structured to limit
shortfall risk (defined as expected total return minus two standard deviations) to no lower than
a 2 12% return in any one year. Maclin’s salary and all capital gains and investment income are
taxed at 40% and no tax-sheltering strategies are available. Webb’s next step is to formulate an
investment policy statement for the Maclins.
a. Formulate the risk objective of an investment policy statement for the Maclins. b. Formulate the return objective of an investment policy statement for the Maclins.
Calculate the pretax rate of return that is required to achieve this objective. Show your
calculations.
c. Formulate the constraints portion of an investment policy statement for the Maclins, address- ing each of the following:
i. Time horizon
ii. Liquidity requirements
iii. Tax concerns
iv. Unique circumstances
9. Louise and Christopher Maclin have purchased their house and made the donation to the local
charity. Now that an investment policy statement has been prepared for the Maclins, Grant Webb
recommends that they consider the strategic asset allocation described in Table 28I .
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C H A P T E R 2 8 Investment Policy and the Framework of the CFA Institute 1013
Asset Class Recommended
Allocation Current Yield
Projected Annualized Pretax
Total Return
Expected Standard Deviation
Cash 15.0% 1.0% 1.0% 2.5%
U.K. corporate bonds 55.0 4.0 5.0 11.0
U.K. small-capitalization equities 0.0 0.0 11.0 25.0
U.K. large-capitalization equities 10.0 2.0 9.0 21.0
U.S. equities* 5.0 1.5 10.0 20.0
Barnett Co. common stock 15.0 1.0 16.0 48.0
Total portfolio 100.0 — 6.7 12.4
Table 28I
Louise and Christopher Maclin’s recommended strategic asset allocation
*U.S. equity data are in British pound terms.
a. Identify aspects of the recommended asset allocation in Table 28I that are inconsistent with the Maclins’ investment objectives and constraints. Support your responses.
b. After further discussion, Webb and the Maclins agree that any suitable strategic asset alloca- tion will include 5 to 10% in U.K. small-capitalization equities and 10 to 15% in U.K. large-
capitalization equities. For the remainder of the portfolio, Webb is considering the asset class
ranges described in Table 28J .
Recommend the most appropriate allocation range for each of the asset classes in Table 28J . Justify each appropriate allocation range with a reason based on the Maclins’ investment
objectives and constraints.
Note: No calculations are required.
Asset Class Allocation Ranges
Cash 0%–3% 5%–10% 15%–20%
U.K. corporate bonds 10%–20% 30%–40% 50%–60%
U.S. equities 0%–5% 10%–15% 20%–25%
Barnett Co. common stock 0%–5% 10%–15% 20%–25%
Table 28J
Louise and Christopher Maclin’s asset class ranges
E-INVESTMENTS EXERCISES Visit the Asset Allocation Wizard site, which provides suggestions about portfolio asset proportions based on your time frame and attitude toward risk: http://cgi.money.cnn. com/tools/assetallocwizard/assetallocwizard.html . After you run the calculator with your preferences, change your inputs slightly to see what effect that would have on the results. For a comprehensive retirement planning calculator, go to http://cgi.money.cnn.com/ tools/retirementplanner/retirementplanner.jsp . After you specify your current income and savings habits, your attitude toward risk, and other relevant information, the calculator will tell you the probability of successfully meeting your goals. It also offers suggestions for future savings plans and a graph of probabilities for several pos- sible outcomes.
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1014 P A R T V I I Applied Portfolio Management
SOLUTIONS TO CONCEPT CHECKS 1. Identify the elements that are life cycle–driven in the two schemes of objectives and constraints.
2. If the investor keeps her present asset allocation, she will have the following amounts to spend
after taxes 5 years from now:
Tax-qualified account:
Bonds: $50,000(1.1) 5 3 .72 5 $ 57,978.36 Stocks: $50,000(1.15) 5 3 .72 5 $ 72,408.86 Subtotal $130,387.22
Nonretirement account:
Bonds: $50,000[1 1 (.10 3 .85)] 5 5 $ 75,182.83 Stocks: $50,000(1.15) 5 2 .15 3 [50,000(1.15) 5 2 50,000] 5 $ 92,982.68
Subtotal $ 168,165.51
Total $298,552.73
If she shifts all of the bonds into the retirement account and all of the stock into the nonretirement
account, she will have the following amounts to spend after taxes 5 years from now:
Tax-qualified account:
Bonds: $100,000(1.1) 5 3 .72 5 $115,956.72
Nonretirement account:
Stocks: $100,000(1.15) 5 2 .15 3 [100,000(1.15) 5 2 100,000] 5 $185,965.36
Total 5 $301,922.08
Her spending budget will increase by $3,369.35.
3. The contribution to each fund will be $2,000 per year (i.e., 5% of $40,000) in constant dollars. At
retirement she will have in her guaranteed return fund:
$50,000 3 1.0320 1 $2,000 3 Annuity factor(3%, 20 years) 5 $144,046
That is the amount she will have for sure.
In addition the expected future value of her stock account is:
$50,000 3 1.0620 1 $2,000 3 Annuity factor(6%, 20 years) 5 $233,928
4. He has accrued an annuity of .01 3 15 3 15,000 5 $2,250 per year for 15 years, starting in
25 years. The present value of this annuity is $2,812.13:
PV 5 2,250 3 Annuity factor(8%, 15) 3 PV factor(8%, 25) 5 2,812.13
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1015
9. 1994 Level I CFA Study Guide, © 1994
10. 1994 Level I CFA Study Guide, © 1994
11. 2002 Level II CFA Study Guide, © 2002
12. 2000 Level II CFA Study Guide, © 2000
Chapter 10
1. 2001 Level II CFA Study Guide, © 2001
2–8. 1991–1993 Level I CFA Study Guides
Chapter 11
1–5. 1993 Level I CFA Study Guide, © 1993
6. 1992 Level I CFA Study Guide, © 1992
7. 1992 Level I CFA Study Guide, © 1992
8–10. 1996 Level III CFA Study Guide,
© 1996
Chapter 12
1. 2000 Level III CFA Study Guide, © 2000
2. 2001 Level III CFA Study Guide, © 2001
3. 2004 Level III CFA Study Guide, © 2004
4. 2003 Level III CFA Study Guide, © 2003
5. 2002 Level III CFA Study Guide, © 2002
Chapter 13
1. 1993 Level I CFA Study Guide, © 1993
2. 1993 Level I CFA Study Guide, © 1993
3. 2002 Level II CFA Study Guide, © 2002
Chapter 14
1. 1993 Level I CFA Study Guide, © 1993
2. 1994 Level I CFA Study Guide, © 1994
3. 1999 Level II CFA Study Guide, © 1999
4. 1992 Level II CFA Study Guide, © 1992
5. 1993 Level I CFA Study Guide, © 1993
6. 1992 Level I CFA Study Guide, © 1992
Chapter 15
1. 1993 Level II CFA Study Guide, © 1993
2. 1993 Level I CFA Study Guide, © 1993
3. 1993 Level II CFA Study Guide, © 1993
4. 1994 Level I CFA Study Guide, © 1994
5. 1994 Level II CFA Study Guide, © 1994
6. 2004 Level II CFA Study Guide, © 2004
7. 1999 Level II CFA Study Guide, © 1999
8. 2000 Level II CFA Study Guide, © 2000
9. 1996 Level II CFA Study Guide, © 1996
10. 2000 Level II CFA Study Guide, © 2000
References to CFA Problems
Each end-of-chapter CFA problem is reprinted with permission from the CFA Institute,
Charlottesville, VA. Following is a list of the CFA problems in the end-of-chapter material
and the exams and study guides from which they were taken and updated.
Chapter 2
1–3. 1996 Level I CFA Study Guide, © 1996
4. 1994 Level I CFA Study Guide, © 1994
5. 1994 Level I CFA Study Guide, © 1994
Chapter 3
1. 1986 Level I CFA Study Guide, © 1986
2–3. 1986 Level I CFA Study Guide, © 1986
Chapter 5
1. 1992 Level I CFA Study Guide, © 1992
2. 1992 Level I CFA Study Guide, © 1992
3–7. 1993 Level I CFA Study Guide, © 1993
Chapter 6
1–3. 1991 Level I CFA Study Guide, © 1991
4–5. 1991 Level I CFA Study Guide, © 1991
6. 1991 Level I CFA Study Guide, © 1991
7–9. 1993 Level I CFA Study Guide, © 1993
Chapter 7
1–3. 1982 Level III CFA Study Guide,
© 1982
4. 1993 Level I CFA Study Guide, © 1993
5. 1993 Level I CFA Study Guide, © 1993
6. 1992 Level I CFA Study Guide, © 1992
7. 1992 Level I CFA Study Guide, © 1992
8–10. 1994 Level I CFA Study Guide,
© 1994
11. 2001 Level III CFA Study Guide, © 2001
12. 2001 Level II CFA Study Guide, © 2001
13. 2000 Level II CFA Study Guide, © 2000
Chapter 8
1. 1982 Level I CFA Study Guide, © 1982
2. 1993 Level I CFA Study Guide, © 1993
3. 1993 Level I CFA Study Guide, © 1993
4. 1993 Level I CFA Study Guide, © 1993
5. 1994 Level I CFA Study Guide, © 1994
Chapter 9
1. 2002 Level I CFA Study Guide, © 2002
2. 2002 Level I CFA Study Guide, © 2002
3–5. 1993 Level I CFA Study Guide, © 1993
6. 1992 Level I CFA Study Guide, © 1992
7. 1994 Level I CFA Study Guide, © 1994
8. 1993 Level I CFA Study Guide, © 1993
Chapter 16
1. 1993 Level II CFA Study Guide, © 1993
2. 1992–1994 Level I CFA study guides
3. 1993 Level I CFA Study Guide, © 1993
4. 1993 Level I CFA Study Guide, © 1993
5. 2004 Level II CFA Study Guide, © 2004
6. 1996 Level III CFA Study Guide, © 1996
7. 1998 Level II CFA Study Guide, © 1998
8. From various Level I study guides
9. 1994 Level III CFA Study Guide, © 1994
10. 2000 Level III CFA Study Guide, © 2000
11. 2003 Level II CFA Study Guide, © 2003
12. 2001 Level II CFA Study Guide, © 2001
13. 1992 Level II CFA Study Guide, © 1992
Chapter 17
1. 1993 Level I CFA Study Guide, © 1993
2. 1993 Level I CFA Study Guide, © 1993
3. 1993 Level II CFA Study Guide, © 1993
4. 1993 Level II CFA Study Guide, © 1993
5. 1998 Level II CFA Study Guide, © 1998
6. 1995 Level II CFA Study Guide, © 1995
7. 1993 Level I CFA Study Guide, © 1993
Chapter 18
1. 1995 Level II CFA Study Guide, © 1995
2. 2001 Level II CFA Study Guide, © 2001
3. 2001 Level II CFA Study Guide, © 2001
4. 2001 Level II CFA Study Guide, © 2001
5–6. 2004 Level II CFA Study Guide,
© 2004
7. 2001 Level II CFA Study Guide, © 2001
8. 1993 Level I CFA Study Guide, © 1993
9. 2003 Level I CFA Study Guide, © 2003
10. 2003 Level I CFA Study Guide, © 2003
11. 2003 Level II CFA Study Guide, ©
2003
Chapter 19
1. 1998 Level II CFA Study Guide, © 1998
2. 1999 Level II CFA Study Guide, © 1999
3. 1994 Level I CFA Study Guide, © 1994
4. 1998 Level II CFA Study Guide, © 1998
5–8. 1992 Level I CFA Study Guide,
© 1992
9. 1998 Level I CFA Study Guide, © 1998
10. 1994 Level I CFA Study Guide, © 1994
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References to CFA Problems
1016
11. 1992 Level I CFA Study Guide, © 1992
12. 2002 Level II CFA Study Guide, © 2002
13. 1990 Level II CFA Study Guide, © 1990
Chapter 20
1. 2002 Level II CFA Study Guide, © 2002
2. 2000 Level II CFA Study Guide, © 2000
3. 2001 Level II CFA Study Guide, © 2001
4. 2002 Level II CFA Study Guide, © 2002
5. From various Level I study guides
Chapter 21
1. 1998 Level II CFA Study Guide, © 1998
2. 2003 Level II CFA Study Guide, © 2003
3. 1998 Level II CFA Study Guide, © 1998
4. 2000 Level II CFA Study Guide, © 2000
5. 1997 Level III CFA Study Guide, © 1997
Chapter 22
1. 2000 Level II CFA Study Guide, © 2000
2. 1993 Level II CFA Study Guide, © 1993
13. 2000 Level III CFA Study Guide,
© 2000
14. 2002 Level III CFA Study Guide, © 2002
Chapter 25
1–3. From various Level I CFA study
guides
4. 1986 Level III CFA Study Guide, © 1986
5. 1991 Level II CFA Study Guide, © 1991
6. 1995 Level II CFA Study Guide, © 1995
7. 2003 Level III CFA Study Guide, © 2003
8. 1998 Level II CFA Study Guide, © 1998
Chapter 28
1. 2001 Level III CFA Study Guide, © 2001
2–4. 1988 Level I CFA Study Guide, © 1988
5. 2002 Level III CFA Study Guide, © 2002
6. 1996 Level III CFA Study Guide, © 1996
7. 1993 Level III CFA Study Guide, © 1993
8. 2004 Level III CFA Study Guide, © 2004
9. 2004 Level III CFA Study Guide, © 2004
3. 1986 Level III CFA Study Guide, © 1986
4. 2004 Level II CFA Study Guide, © 2004
5. 2004 Level II CFA Study Guide, © 2004
Chapter 23
1. 2001 Level II CFA Study Guide, © 2001
2. 1995 Level III CFA Study Guide, © 1995
3. 1991 Level III CFA Study Guide, © 1991
4–5. 2003 Level II CFA Study Guide, ©
2003
6. 2000 Level III CFA Study Guide, © 2000
7. 1985 Level III CFA Study Guide, © 1985
8. 1996 Level II CFA Study Guide, © 1996
Chapter 24
1. 1995 Level III CFA Study Guide, © 1995
2. 1981 Level I CFA Study Guide, © 1981
3. 1986 Level II CFA Study Guide, © 1986
4–11. From various Level I CFA study
guides
12. 2001 Level III CFA Study Guide, © 2001
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G-1
arbitrage pricing theory An asset pricing theory that is derived from a factor model, using diversification and arbitrage arguments. The theory describes the relationship between expected returns on securities, given that there are no opportunities to create wealth through risk-free arbitrage investments.
ask price The price at which a dealer will sell a security.
asset allocation Choosing among broad asset classes such as stocks versus bonds.
at the money When the exercise price and asset price of an option are equal.
auction market A market where all traders in a good meet at one place to buy or sell an asset. The NYSE is an example.
average collection period, or days’ receivables The ratio of accounts receivable to sales, or the total amount of credit extended per dollar of daily sales (average AR/sales 3 365).
B backfill bias Bias in the average returns of a sample of funds induced by including past returns on funds that entered the sample only if they happened to be successful.
balance sheet An accounting statement of a firm’s finan- cial position at a specified time.
bank discount yield An annualized interest rate assuming simple interest, a 360-day year, and using the face value of the security rather than purchase price to compute return per dollar invested.
banker’s acceptance A money market asset consisting of an order to a bank by a customer to pay a sum of money at a future date.
baseline forecasts Forecast of security returns derived from the assumption that the market is in equilibrium where current prices reflect all available information.
basis The difference between the futures price and the spot price.
basis risk Risk attributable to uncertain movements in the spread between a futures price and a spot price.
behavioral finance Models of financial markets that emphasize implications of psychological factors affecting investor behavior.
benchmark error Use of an inappropriate proxy for the true market portfolio.
benchmark portfolio Portfolio against which a manager is to be evaluated.
beta The measure of the systematic risk of a security. The tendency of a security’s returns to respond to swings in the broad market.
Glossary
A abnormal return Return on a stock beyond what would be predicted by market movements alone. Cumulative abnor- mal return (CAR) is the total abnormal return for the period surrounding an announcement or the release of information.
accounting earnings Earnings of a firm as reported on its income statement.
acid test ratio See quick ratio.
active management Attempts to achieve portfolio returns more than commensurate with risk, either by forecasting broad market trends or by identifying particular mispriced sectors of a market or securities in a market.
active portfolio In the context of the Treynor-Black model, the portfolio formed by mixing analyzed stocks of perceived nonzero alpha values. This portfolio is ultimately mixed with the passive market-index portfolio.
adjusted alphas Forecasts for alpha that are modulated to account for statistical imprecision in the analyst’s estimate.
agency problem Conflicts of interest among stockholders, bondholders, and managers.
algorithmic trading The use of computer programs to make trading decisions.
alpha The abnormal rate of return on a security in excess of what would be predicted by an equilibrium model like CAPM or APT.
alpha transfer A strategy in which you invest in positive alpha positions, hedge the systematic risk of the investment, and finally establish market exposure where you want it using passive indexes.
American depository receipts (ADRs) Domestically traded securities representing claims to shares of foreign stocks.
American option An American option can be exercised before and up to its expiration date. Compare with a European option, which can be exercised only on the expiration date.
announcement date Date on which particular news concerning a given company is announced to the public. Used in event studies, which researchers use to evaluate the economic impact of events of interest.
annual percentage rate (APR) Interest rate is annualized using simple rather than compound interest.
anomalies Patterns of returns that seem to contradict the efficient market hypothesis.
appraisal ratio The signal-to-noise ratio of an analyst’s forecasts. The ratio of alpha to residual standard deviation.
arbitrage A zero-risk, zero-net investment strategy that still generates profits.
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Glossary
G-2
bid–ask spread The difference between a dealer’s bid and ask price.
bid price The price at which a dealer is willing to purchase a security.
binomial model An option-valuation model predicated on the assumption that stock prices can move to only two values over any short time period.
Black-Scholes formula An equation to value a call option that uses the stock price, the exercise price, the risk-free interest rate, the time to maturity, and the standard deviation of the stock return.
blocks, block sale A transaction of more than 10,000 shares of stock.
bogey The return an investment manager is compared to for performance evaluation.
bond A security issued by a borrower that obligates the issuer to make specified payments to the holder over a spe- cific period. A coupon bond obligates the issuer to make interest payments called coupon payments over the life of the bond, then to repay the face value at maturity.
bond equivalent yield Bond yield calculated on an annual percentage rate method. Differs from effective annual yield.
bond indenture The contract between the issuer and the bondholder.
bond reconstitution Combining stripped Treasury securi- ties to re-create the original cash flows of a Treasury bond.
bond stripping Selling bond cash flows (either coupon or principal payments) as stand-alone zero-coupon securities.
book-to-market effect The tendency for stocks of firms with high ratios of book-to-market value to generate abnor- mal returns.
book value An accounting measure describing the net worth of common equity according to a firm’s balance sheet.
breadth The extent to which movements in the broad mar- ket index are reflected widely in movements of individual stock prices.
brokered market A market where an intermediary (a bro- ker) offers search services to buyers and sellers.
budget deficit The amount by which government spend- ing exceeds government revenues.
bull CD, bear CD A bull CD pays its holder a specified percentage of the increase in return on a specified market index while guaranteeing a minimum rate of return. A bear CD pays the holder a fraction of any fall in a given market index.
bullish, bearish Words used to describe investor attitudes. Bullish means optimistic; bearish means pessimistic. Also used in bull market and bear market.
bundling, unbundling A trend allowing creation of secu- rities either by combining primitive and derivative securities into one composite hybrid or by separating returns on an asset into classes.
business cycle Repetitive cycles of recession and recovery.
C calendar spread Buy one option and write another with a different expiration date.
callable bond A bond that the issuer may repurchase at a given price in some specified period.
call option The right to buy an asset at a specified exercise price on or before a specified expiration date.
call protection An initial period during which a callable bond may not be called.
capital allocation decision Allocation of invested funds between risk-free assets versus the risky portfolio.
capital allocation line (CAL) A graph showing all feasible risk–return combinations of a risky and risk-free asset.
capital gains The amount by which the sale price of a security exceeds the purchase price.
capital market line (CML) A capital allocation line pro- vided by the market-index portfolio.
capital markets Includes longer-term, relatively riskier securities.
cash/bond selection Asset allocation in which the choice is between short-term cash equivalents and longer-term bonds.
cash equivalents Short-term money-market securities.
cash flow matching A form of immunization, matching cash flows from a bond portfolio with an obligation.
cash ratio Measure of liquidity of a firm. Ratio of cash and marketable securities to current liabilities.
cash settlement The provision of some futures contracts that requires not delivery of the underlying assets (as in agri- cultural futures) but settlement according to the cash value of the asset.
certainty equivalent rate The certain return providing the same utility as a risky portfolio.
certificate of deposit A bank time deposit.
clearinghouse Established by exchanges to facilitate transfer of securities resulting from trades. For options and futures contracts, the clearinghouse may interpose itself as a middleman between two traders.
closed-end (mutual) fund A fund whose shares are traded through brokers at market prices; the fund will not redeem shares at their net asset value. The market price of the fund can differ from the net asset value.
collar An options strategy that brackets the value of a port- folio between two bounds.
collateral A specific asset pledged against possible default on a bond. Mortgage bonds are backed by claims on prop- erty. Collateral trust bonds are backed by claims on other securities. Equipment obligation bonds are backed by claims on equipment.
collateralized debt obligation (CDO) A pool of loans sliced into several tranches with different levels of risk.
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Glossary
G-3
collateralized mortgage obligation (CMO) A mortgage pass-through security that partitions cash flows from under- lying mortgages into classes called tranches that receive principal payments according to stipulated rules.
commercial paper Short-term unsecured debt issued by large corporations.
common stock Equities, or equity securities, issued as ownership shares in a publicly held corporation. Shareholders have voting rights and may receive dividends based on their proportionate ownership.
comparison universe The collection of money managers of similar investment style used for assessing relative perfor- mance of a portfolio manager.
complete portfolio The entire portfolio, including risky and risk-free assets.
conditional tail expectation Expectation of a random variable conditional on its falling below some threshold value. Often used as a measure of down side risk.
confidence index Ratio of the yield of top-rated corporate bonds to the yield on intermediate-grade bonds.
conservativism Notion that investors are too slow to update their beliefs in response to new evidence.
constant-growth DDM A form of the dividend discount model that assumes dividends will grow at a constant rate.
contango theory Holds that the futures price must exceed the expected future spot price.
contingent claim Claim whose value is directly depen- dent on or is contingent on the value of some underlying assets.
contingent immunization A mixed passive-active strategy that immunizes a portfolio if necessary to guarantee a minimum acceptable return but otherwise allows active management.
convergence arbitrage A bet that two or more prices are out of alignment and that profits can be made when the prices converge back to proper relationship.
convergence property The convergence of futures prices and spot prices at the maturity of the futures contract.
convertible bond A bond with an option allowing the bondholder to exchange the bond for a specified number of shares of common stock in the firm. A conversion ratio specifies the number of shares. The market conversion price is the current value of the shares for which the bond may be exchanged. The conversion premium is the excess of the bond’s value over the conversion price.
convexity The curvature of the price-yield relationship of a bond.
corporate bonds Long-term debt issued by private corpo- rations typically paying semiannual coupons and returning the face value of the bond at maturity.
correlation coefficient A statistic in which the covariance is scaled to a value between 2 1 (perfect negative correla- tion) and 11 (perfect positive correlation).
cost-of-carry relationship See spot-futures parity theorem.
country selection A type of active international manage- ment that measures the contribution to performance attribut- able to investing in the better-performing stock markets of the world.
coupon rate A bond’s interest payments per dollar of par value.
covariance A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covari- ance means they vary inversely.
covered call A combination of selling a call on a stock together with buying the stock.
covered interest arbitrage relation See interest rate parity relation.
credit default swap (CDS) A derivative contract in which one party sells insurance concerning the credit risk of another firm.
credit enhancement Purchase of the financial guarantee of a large insurance company to raise funds.
credit risk Default risk.
cross-hedge Hedging a position in one asset using futures on another commodity.
cumulative abnormal return See abnormal return.
currency selection Asset allocation in which the investor chooses among investments denominated in different currencies.
current ratio A ratio representing the ability of the firm to pay off its current liabilities by liquidating current assets (current assets/current liabilities).
current yield A bond’s annual coupon payment divided by its price. Differs from yield to maturity.
cyclical industries Industries with above-average sensitiv- ity to the state of the economy.
D dark pools Electronic trading networks where participants can anonymously buy or sell large blocks of securities.
data mining Sorting through large amounts of historical data to uncover systematic patterns that can be exploited.
day order A buy order or a sell order expiring at the close of the trading day.
days’ receivables See average collection period.
dealer market A market where traders specializing in particular commodities buy and sell assets for their own accounts. The OTC market is an example.
debenture or unsecured bond A bond not backed by spe- cific collateral.
debt securities Bonds; also called fixed-income securities.
dedication strategy Refers to multiperiod cash flow matching.
default premium A differential in promised yield that compensates the investor for the risk inherent in purchasing a corporate bond that entails some risk of default.
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Glossary
G-4
DuPont system Decomposition of a firm’s profitability measures into the underlying factors that determine such profitability.
duration A measure of the average life of a bond, defined as the weighted average of the times until each payment is made, with weights proportional to the present value of the payment.
dynamic hedging Constant updating of hedge positions as market conditions change.
E EAFE index The Europe, Australasia, Far East index, computed by Morgan Stanley, is a widely used index of non-U.S. stocks.
earnings management The practice of using flexibility in accounting rules to improve the apparent profitability of the firm.
earnings retention ratio Plowback ratio.
earnings yield The ratio of earnings to price, E/P.
economic earnings The real flow of cash that a firm could pay out forever in the absence of any change in the firm’s productive capacity.
economic value added (EVA) The spread between ROA and cost of capital multiplied by the capital invested in the firm. It measures the dollar value of the firm’s return in excess of its opportunity cost.
effective annual rate (EAR) Interest rate annualized using compound rather than simple interest.
effective annual yield Annualized interest rate on a security computed using compound interest techniques.
effective duration Percentage change in bond price per change in the level of market interest rates.
efficient diversification The organizing principle of mod- ern portfolio theory, which maintains that any risk-averse investor will search for the highest expected return for any level of portfolio risk.
efficient frontier Graph representing a set of portfolios that maximize expected return at each level of portfolio risk.
efficient frontier of risky assets The portion of the minimum-variance frontier that lies above the global minimum-variance portfolio.
efficient market hypothesis The prices of securities fully reflect available information. Investors buying securities in an efficient market should expect to obtain an equilibrium rate of return. Weak-form EMH asserts that stock prices already reflect all information contained in the history of past prices. The semistrong-form hypothesis asserts that stock prices already reflect all publicly available informa- tion. The strong-form hypothesis asserts that stock prices reflect all relevant information including insider information.
elasticity (of an option) Percentage change in the value of an option accompanying a 1% change in the value of a stock.
defensive industries Industries with little sensitivity to the state of the economy.
deferred annuities Tax-advantaged life insurance product. Deferred annuities offer deferral of taxes with the option of withdrawing one’s funds in the form of a life annuity.
defined benefit plans Pension plans in which retirement benefits are set according to a fixed formula.
defined contribution plans Pension plans in which the employer is committed to making contributions according to a fixed formula.
degree of operating leverage Percentage change in profits for a 1% change in sales.
delta (of option) See hedge ratio.
delta neutral The value of the portfolio is not affected by changes in the value of the asset on which the options are written.
demand shock An event that affects the demand for goods and services in the economy.
derivative asset/contingent claim Securities provid- ing payoffs that depend on or are contingent on the values of other assets such as commodity prices, bond and stock prices, or market-index values. Examples are futures and options.
derivative security A security whose payoff depends on the value of other financial variables such as stock prices, interest rates, or exchange rates.
direct search market Buyers and sellers seek each other directly and transact directly.
directional strategy Speculation that one sector or another will outperform other sectors of the market.
discount bonds Bonds selling below par value.
discretionary account An account of a customer who gives a broker the authority to make buy and sell decisions on the customer’s behalf.
disposition effect The tendency of investors to hold on to losing investments.
diversifiable risk Risk attributable to firm-specific risk, or nonmarket risk. Nondiversifiable risk refers to systematic or market risk.
diversification Spreading a portfolio over many invest- ments to avoid excessive exposure to any one source of risk.
dividend discount model (DDM) A formula stating that the intrinsic value of a firm is the present value of all expected future dividends.
dividend payout ratio Percentage of earnings paid out as dividends.
dividend yield The percent rate of return provided by a stock’s dividend payments.
dollar-weighted rate of return The internal rate of return on an investment.
doubling option A sinking fund provision that may allow repurchase of twice the required number of bonds at the sinking fund call price.
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Glossary
G-5
factor loading See factor beta.
factor model A way of decomposing the factors that influence a security’s rate of return into common and firm- specific influences.
factor portfolio A well-diversified portfolio constructed to have a beta of 1.0 on one factor and a beta of 0 on any other factor.
factor sensitivity See factor beta.
fair game An investment prospect that has a zero risk premium.
fair value accounting Use of current values rather than historic cost in the firm’s financial statements.
federal funds Funds in a bank’s reserve account.
FIFO The first-in first-out accounting method of inventory valuation.
financial assets Financial assets such as stocks and bonds are claims to the income generated by real assets or claims on income from the government.
financial engineering Creating and designing securities with custom-tailored characteristics.
financial intermediary An institution such as a bank, mutual fund, investment company, or insurance company that serves to connect the household and business sectors so households can invest and businesses can finance production.
firm-specific risk See diversifiable risk.
first-pass regression A time series regression to estimate the betas of securities or portfolios.
fiscal policy The use of government spending and taxing for the specific purpose of stabilizing the economy.
fixed annuities Annuity contracts in which the insurance company pays a fixed dollar amount of money per period.
fixed-charge coverage ratio Ratio of earnings to all fixed cash obligations, including lease payments and sinking fund payments.
fixed-income security A security such as a bond that pays a specified cash flow over a specific period.
flight to quality Describes the tendency of investors to require larger default premiums on investments under uncer- tain economic conditions.
floating-rate bond A bond whose interest rate is reset periodically according to a specified market rate.
forced conversion Use of a firm’s call option on a callable convertible bond when the firm knows that bondholders will exercise their option to convert.
forecasting record The historical record of the forecasting errors of a security analyst.
foreign exchange market An informal network of banks and brokers that allows customers to enter forward con- tracts to purchase or sell currencies in the future at a rate of exchange agreed upon now.
foreign exchange swap An agreement to exchange stipu- lated amounts of one currency for another at one or more future dates.
electronic communication network (ECN) A computer- operated trading network offering an alternative to formal stock exchanges or dealer markets for trading securities.
endowment funds Organizations chartered to invest money for specific purposes.
equities Ownership shares in a firm. equity Ownership in a firm. Also, the net worth of a margin account.
equivalent taxable yield The pretax yield on a taxable bond providing an after-tax yield equal to the rate on a tax-exempt municipal bond.
Eurodollars Dollar-denominated deposits at foreign banks or foreign branches of American banks.
Europe, Australasia, Far East (EAFE) index A widely used index of non-U.S. stocks computed by Morgan Stanley.
European option A European option can be exercised only on the expiration date. Compare with an American option, which can be exercised before, up to, and on its expiration date.
event study Research methodology designed to measure the impact of an event of interest on stock returns.
event tree Depicts all possible sequences of events. excess return Rate of return in excess of the risk-free rate. exchange rate Price of a unit of one country’s currency in terms of another country’s currency.
exchange rate risk The uncertainty in asset returns due to movements in the exchange rates between the dollar and foreign currencies.
exchange-traded funds (ETFs) Offshoots of mutual funds that allow investors to trade portfolios of securities just as they do shares of stock.
exchanges National or regional auction markets providing a facility for members to trade securities. A seat is a member- ship on an exchange.
exercise or strike price Price set for calling (buying) an asset or putting (selling) an asset.
expectations hypothesis (of interest rates) Theory that forward interest rates are unbiased estimates of expected future interest rates.
expected return The probability-weighted average of the possible outcomes.
expected return–beta (or mean-beta ) relationship Impli- cation of the CAPM that security risk premiums (expected excess returns) will be proportional to beta.
expected shortfall The expected loss on a security conditional on returns being in the left tail of the probability distribution.
F face value The maturity value of a bond.
factor beta Sensitivity of security returns to changes in a systematic factor. Alternatively, factor loading; factor sensitivity.
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Glossary
G-6
hedge ratio (for an option) The number of stocks required to hedge against the price risk of holding one option. Also called the option’s delta.
hedging Investing in an asset to reduce the overall risk of a portfolio.
hedging demands Demands for securities to hedge par- ticular sources of consumption risk, beyond the usual mean variance diversification motivation.
high-frequency trading A subset of algorithmic trading that relies on computer programs to make rapid trading decisions.
high water mark The previous value of a portfolio that must be reattained before a hedge fund can charge incentive fees.
holding-period return The rate of return over a given period.
home bias The tendency of investors to allocate a greater share of their portfolios to domestic securities than would be the case under neutral diversification.
homogenous expectations The assumption that all investors use the same expected returns and covariance matrix of security returns as inputs in security analysis.
horizon analysis Forecasting the realized compound yield over various holding periods or investment horizons.
I illiquidity Difficulty, cost, and/or delay in selling an asset on short notice without offering substantial price concessions.
illiquidity cost Costs due to imperfect liquidity of some security.
illiquidity premium Extra expected return as compensa- tion for limited liquidity.
immunization A strategy that matches durations of assets and liabilities so as to make net worth unaffected by interest rate movements.
implied volatility The standard deviation of stock returns that is consistent with an option’s market value.
incentive fee A fee charged by hedge funds equal to a share of any investment returns beyond a stipulated bench- mark performance.
income beneficiary One who receives income from a trust. income statement A financial statement showing a firm’s revenues and expenses during a specified period.
indenture The document defining the contract between the bond issuer and the bondholder.
index arbitrage An investment strategy that exploits divergences between actual futures prices and their theoreti- cally correct parity values to make a profit.
index fund A mutual fund holding shares in proportion to their representation in a market index such as the S&P 500.
index model A model of stock returns using a market index such as the S&P 500 to represent common or systematic risk factors.
forward contract An agreement calling for future delivery of an asset at an agreed-upon price. Also see futures contract.
forward interest rate Rate of interest for a future period that would equate the total return of a long-term bond with that of a strategy of rolling over shorter-term bonds. The forward rate is inferred from the term structure.
framing Decisions are affected by how choices are described, for example, whether uncertainty is posed as potential gains from a low baseline level, or as losses from a higher baseline value.
fully diluted earnings per share Earnings per share expressed as if all outstanding convertible securities and warrants have been exercised.
fundamental analysis Research to predict stock value that focuses on such determinants as earnings and dividends prospects, expectations for future interest rates, and risk evaluation of the firm.
fundamental risk Risk that even if an asset is mispriced, there is still no arbitrage opportunity, because the mispricing can widen before price eventually converges to intrinsic value.
funds of funds Hedge funds that invest in several other hedge funds.
futures contract Obliges traders to purchase or sell an asset at an agreed-upon price on a specified future date. The long position is held by the trader who commits to purchase. The short position is held by the trader who commits to sell. Futures differ from forward contracts in their standardiza- tion, exchange trading, margin requirements, and daily settling (marking to market).
futures option The right to enter a specified futures con- tract at a futures price equal to the stipulated exercise price.
futures price The price at which a futures trader commits to make or take delivery of the underlying asset.
G gamma The curvature of an option pricing function (as a function of the value of the underlying asset).
geometric average The n th root of the product of n num- bers. It is used to measure the compound rate of return over time.
globalization Tendency toward a worldwide investment environment, and the integration of national capital markets.
gross domestic product (GDP) The market value of goods and services produced over time including the income of foreign corporations and foreign residents working in the United States, but excluding the income of U.S. residents and corporations overseas.
H hedge fund A private investment pool, open to institu- tional or wealthy investors, that is largely exempt from SEC regulation and can pursue more speculative policies than mutual funds.
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Glossary
G-7
investment Commitment of current resources in the expectation of deriving greater resources in the future.
investment bankers Firms specializing in the sale of new securities to the public, typically by underwriting the issue.
investment company Firm managing funds for investors. An investment company may manage several mutual funds.
investment-grade bond Bond rated BBB and above or Baa and above. Lower-rated bonds are classified as speculative-grade or junk bonds.
investment horizon Time horizon for purposes of investment decisions.
investment portfolio Set of securities chosen by an investor.
J Jensen’s alpha The alpha of an investment.
junk bond See speculative-grade bond.
K kurtosis Measure of the fatness of the tails of a probability distribution. Indicates probability of observing extreme high or low values.
L latency The time it takes to accept, process, and deliver a trading order.
Law of One Price The rule stipulating that equivalent securities or bundles of securities must sell at equal prices to preclude arbitrage opportunities.
leading economic indicators Economic series that tend to rise or fall in advance of the rest of the economy.
leverage ratio Ratio of debt to total capitalization of a firm. LIFO The last-in first-out accounting method of valuing inventories.
limited liability The fact that shareholders have no per- sonal liability to the creditors of the corporation in the event of bankruptcy.
limit order An order specifying a price at which an inves- tor is willing to buy or sell a security.
liquidation value Net amount that could be realized by selling the assets of a firm after paying the debt.
liquidity Liquidity refers to the speed and ease with which an asset can be converted to cash.
liquidity preference theory Theory that the forward rate exceeds expected future interest rates.
liquidity premium Forward rate minus expected future short interest rate.
load Sales charge on the purchase of some mutual funds. load fund A mutual fund with a sales commission, or load. lock-up period Period in which investors cannot redeem investments in the hedge fund.
index option A call or put option based on a stock market index.
indifference curve A curve connecting all portfolios with the same utility according to their means and standard deviations.
industry life cycle Stages through which firms typically pass as they mature.
inflation The rate at which the general level of prices for goods and services is rising.
information ratio Ratio of alpha to the standard deviation of diversifiable risk.
initial public offering Stock issued to the public for the first time by a formerly privately owned company.
input list List of parameters such as expected returns, variances, and covariances necessary to determine the optimal risky portfolio.
inside information Nonpublic knowledge about a corpora- tion possessed by corporate officers, major owners, or other individuals with privileged access to information about a firm.
insider trading Trading by officers, directors, major stock- holders, or others who hold private inside information allow- ing them to benefit from buying or selling stock.
insurance principle The law of averages. The average outcome for many independent trials of an experiment will approach the expected value of the experiment.
interest coverage ratio Measure of financial leverage. Earnings before interest and taxes as a multiple of interest expense.
interest coverage ratio, or times interest earned A finan- cial leverage measure (EBIT divided by interest expense).
interest rate The number of dollars earned per dollar invested per period.
interest rate parity relation (theorem) The spot-futures exchange rate relationship that prevails in well-functioning markets.
interest rate swaps A method to manage interest rate risk where parties trade the cash flows corresponding to different securities without actually exchanging securities directly.
intermarket spread swap Switching from one segment of the bond market to another (from Treasuries to corporates, for example).
in the money In the money describes an option whose exercise would produce profits. Out of the money describes an option where exercise would not be profitable.
international financial reporting standards Accounting standards used in many non-U.S. markets that rely more on principles and less on rules than U.S. standards.
intrinsic value (of a firm) The present value of a firm’s expected future net cash flows discounted by the required rate of return.
intrinsic value of an option Stock price minus exercise price, or the profit that could be attained by immediate exercise of an in-the-money option.
inventory turnover ratio Cost of goods sold as a multiple of average inventory.
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Glossary
G-8
market-value-weighted index An index of a group of securities computed by calculating a weighted average of the returns of each security in the index, with weights proportional to outstanding market value.
marking to market Describes the daily settlement of obligations on futures positions.
mean-variance analysis Evaluation of risky prospects based on the expected value and variance of possible outcomes.
mean-variance criterion The selection of portfolios based on the means and variances of their returns. The choice of the higher expected return portfolio for a given level of variance or the lower variance portfolio for a given expected return.
mental accounting Individuals mentally segregate assets into independent accounts rather than viewing them as part of a unified portfolio.
minimum-variance frontier Graph of the lowest possible portfolio variance that is attainable for a given portfolio expected return.
minimum-variance portfolio The portfolio of risky assets with lowest variance.
modern portfolio theory (MPT) Principles underlying analysis and evaluation of rational portfolio choices based on risk–return trade-offs and efficient diversification.
modified duration Macaulay’s duration divided by 1 1 yield to maturity. Measures interest rate sensitivity of bonds.
momentum effect The tendency of poorly performing stocks and well-performing stocks in one period to continue that abnormal performance in following periods.
monetary policy Actions taken by the Board of Governors of the Federal Reserve System to influence the money supply or interest rates.
money market Includes short-term, highly liquid, and relatively low-risk debt instruments.
mortality tables Tables of probability that individuals of various ages will die within a year.
mortgage-backed security Ownership claim in a pool of mortgages or an obligation that is secured by such a pool. Also called a pass-through, because payments are passed along from the mortgage originator to the purchaser of the mortgage-backed security.
multifactor CAPM Generalization of the basic CAPM that accounts for extra-market hedging demands.
multifactor models Model of security returns positing that returns respond to several systematic factors.
municipal bonds Tax-exempt bonds issued by state and local governments, generally to finance capital improvement projects. General obligation bonds are backed by the general taxing power of the issuer. Revenue bonds are backed by the proceeds from the project or agency they are issued to finance.
mutual fund A firm pooling and managing funds of investors.
mutual fund theorem A result associated with the CAPM, asserting that investors will choose to invest their entire risky portfolio in a market-index mutual fund.
lognormal distribution The log of the variable has a normal (bell-shaped) distribution.
London Interbank Offered Rate (LIBOR) Rate that most creditworthy banks charge one another for large loans of Eurodollars in the London market.
long position hedge Hedging the future cost of a pur- chase by taking a long futures position to protect against changes in the price of the asset.
lower partial standard deviation Standard deviation computed using only the portion of the probability distribution below the mean of the variable.
M Macaulay’s duration Effective maturity of bond, equal to weighted average of the times until each payment, with weights proportional to the present value of the payment.
maintenance, or variation, margin An established value below which a trader’s margin cannot fall. Reaching the maintenance margin triggers a margin call.
margin Describes securities purchased with money borrowed from a broker. Current maximum margin is 50%.
mark-to-market accounting See fair value accounting .
market–book-value ratio Ratio of price per share to book value per share.
market capitalization rate The market-consensus estimate of the appropriate discount rate for a firm’s cash flows.
market model Another version of the index model that breaks down return uncertainty into systematic and nonsystematic components.
market neutral A strategy designed to exploit relative mispricing within a market, but which is hedged to avoid taking a stance on the direction of the broad market.
market order A buy or sell order to be executed immediately at current market prices.
market or systematic risk, firm-specific risk Market risk is risk attributable to common macroeconomic factors. Firm-specific risk reflects risk peculiar to an individual firm that is independent of market risk.
market portfolio The portfolio for which each security is held in proportion to its market value.
market price of risk A measure of the extra return, or risk premium, that investors demand to bear risk. The reward-to-risk ratio of the market portfolio.
market risk See systematic risk.
market segmentation The theory that long- and short-maturity bonds are traded in essentially distinct or segmented markets and that prices in one market do not affect those in the other.
market timer An investor who speculates on broad market moves rather than on specific securities.
market timing Asset allocation in which the investment in the market is increased if one forecasts that the market will outperform T-bills.
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Glossary
G-9
P pairs trading Stocks are paired up based on underlying similarities, and long/short positions are established to exploit any relative mispricing between each pair.
par value The face value of the bond.
passive investment strategy See passive management.
passive management Buying a well-diversified portfolio to represent a broad-based market index without attempting to search out mispriced securities.
passive market-index portfolio A market-index portfolio.
passive strategy See passive management.
pass-through security Pools of loans (such as home mort- gage loans) sold in one package. Owners of pass-throughs receive all principal and interest payments made by the borrowers.
peak The transition from the end of an expansion to the start of a contraction.
P/E effect That portfolios of low P/E stocks have exhibited higher average risk-adjusted returns than high P/E stocks.
personal trust An interest in an asset held by a trustee for the benefit of another person.
plowback ratio The proportion of the firm’s earnings that is reinvested in the business (and not paid out as dividends). The plowback ratio equals 1 minus the dividend payout ratio.
political risk Possibility of the expropriation of assets, changes in tax policy, restrictions on the exchange of foreign currency for domestic currency, or other changes in the business climate of a country.
portable alpha; alpha transfer A strategy in which you invest in positive alpha positions, then hedge the systematic risk of that investment, and, finally, establish market exposure where you want it by using passive indexes.
portfolio insurance The practice of using options or dynamic hedge strategies to provide protection against investment losses while maintaining upside potential.
portfolio management Process of combining securities in a portfolio tailored to the investor’s preferences and needs, monitoring that portfolio, and evaluating its performance.
portfolio opportunity set The expected return–standard deviation pairs of all portfolios that can be constructed from a given set of assets.
posterior distribution Probability distribution for a variable after adjustment for empirical evidence on its likely value.
preferred habitat theory Holds that investors prefer specific maturity ranges but can be induced to switch if risk premiums are sufficient.
preferred stock Nonvoting shares in a corporation, paying a fixed or variable stream of dividends.
premium The purchase price of an option.
premium bonds Bonds selling above par value.
N NAICS codes North American Industrial Classification System codes that use numerical values to identify industries.
naked option writing Writing an option without an offsetting stock position.
NASDAQ The automated quotation system for the OTC market, showing current bid ask prices for thousands of stocks.
neglected-firm effect That investments in stock of less well-known firms have generated abnormal returns.
net asset value (NAV) The value of each share expressed as assets minus liabilities on a per-share basis.
nominal interest rate The interest rate in terms of nominal (not adjusted for purchasing power) dollars.
nondirectional strategy A position designed to exploit temporary misalignments in relative pricing. Typically involves a long position in one security hedged with a short position in a related security.
nondiversifiable risk See systematic risk.
nonsystematic risk Nonmarket or firm-specific risk factors that can be eliminated by diversification. Also called unique risk or diversifiable risk. Systematic risk refers to risk factors common to the entire economy.
normal distribution Bell-shaped probability distribution that characterizes many natural phenomena.
notional principal Principal amount used to calculate swap payments.
O on the run Recently issued bond, selling at or near par value.
on-the-run yield curve Relationship between yield to maturity and time to maturity for newly issued bonds selling at par.
open-end (mutual) fund A fund that issues or redeems its own shares at their net asset value (NAV).
open interest The number of futures contracts outstanding.
optimal risky portfolio An investor’s best combination of risky assets to be mixed with safe assets to form the complete portfolio.
option elasticity The percentage increase in an option’s value given a 1% change in the value of the underlying security.
original issue discount bond A bond issued with a low coupon rate that sells at a discount from par value.
out of the money Out of the money describes an option where exercise would not be profitable. In the money describes an option where exercise would produce profits.
over-the-counter market An informal network of brokers and dealers who negotiate sales of securities (not a formal exchange).
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Glossary
G-10
pure plays Bets on particular mispricing across two or more securities, with extraneous sources of risk such as general market exposure hedged away.
pure yield curve Refers to the relationship between yield to maturity and time to maturity for zero-coupon bonds.
pure yield pickup swap Moving to higher-yield bonds.
put bond A bond that the holder may choose either to exchange for par value at some date or to extend for a given number of years.
put-call parity theorem An equation representing the proper relation between put and call prices. Violation of parity allows arbitrage opportunities.
put/call ratio Ratio of put options to call options outstanding on a stock.
put option The right to sell an asset at a specified exercise price on or before a specified expiration date.
Q quality of earnings The realism and conservatism of the earnings number and the extent to which we might expect the reported level of earnings to be sustained.
quick ratio A measure of liquidity similar to the current ratio except for exclusion of inventories (cash plus receiv- ables divided by current liabilities).
R random walk Describes the notion that stock price changes are random and unpredictable.
rate anticipation swap A switch made in response to forecasts of interest rates.
real assets, financial assets Real assets are land, buildings, and equipment that are used to produce goods and services. Financial assets are claims such as securities to the income generated by real assets.
real interest rate The excess of the interest rate over the inflation rate. The growth rate of purchasing power derived from an investment.
realized compound return Yield assuming that coupon payments are invested at the going market interest rate at the time of their receipt and rolled over until the bond matures.
rebalancing Realigning the proportions of assets in a portfolio as needed.
registered bond A bond whose issuer records ownership and interest payments. Differs from a bearer bond, which is traded without record of ownership and whose possession is its only evidence of ownership.
regression equation An equation that describes the average relationship between a dependent variable and a set of explanatory variables.
regret avoidance Notion from behavioral finance that individuals who make decisions that turn out badly will have more regret when that decision was more unconventional.
present value of growth opportunities (PVGO) Net present value of a firm’s future investments.
price–earnings multiple See price–earnings ratio.
price–earnings (P/E) ratio The ratio of a stock’s price to its earnings per share. Also referred to as the P/E multiple.
price value of a basis point The change in the value of a fixed-income asset resulting from a 1 basis point change in the asset’s yield to maturity.
price-weighted average Weighted average with weights proportional to security prices rather than total capitalization.
primary market New issues of securities are offered to the public here.
primitive security, derivative security A primitive security is an instrument such as a stock or bond for which payments depend only on the financial status of its issuer. A derivative security is created from the set of primitive securities to yield returns that depend on factors beyond the characteristics of the issuer and that may be related to prices of other assets.
principal The outstanding balance on a loan.
prior distribution Probability distribution for a variable before adjusting for empirical evidence on its likely value.
private equity Investment in a company that is not traded on a stock exchange.
private placement Primary offering in which shares are sold directly to a small group of institutional or wealthy investors.
profit margin See return on sales.
program trading Coordinated buy orders and sell orders of entire portfolios, usually with the aid of computers, often to achieve index arbitrage objectives.
prospect theory Behavioral (as opposed to rational) model of investor utility. Investor utility depends on changes in wealth rather than levels of wealth.
prospectus A final and approved registration statement including the price at which the security issue is offered.
protective covenant A provision specifying requirements of collateral, sinking fund, dividend policy, etc., designed to protect the interests of bondholders.
protective put Purchase of stock combined with a put option that guarantees minimum proceeds equal to the put’s exercise price.
proxy An instrument empowering an agent to vote in the name of the shareholder.
prudent investor rule An investment manager must act in accord with the actions of a hypothetical prudent investor.
pseudo-American call option value The maximum of the value derived by assuming that an option will be held until expiration and the value derived by assuming that the option will be exercised just before an ex-dividend date.
public offering, private placement A public offering consists of bonds sold in the primary market to the general public; a private placement is sold directly to a limited number of institutional investors.
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Glossary
G-11
risk-free rate The interest rate that can be earned with certainty.
risk lover See risk-averse.
risk-neutral See risk-averse.
risk pooling Investing the portfolio in many risky assets.
risk premium An expected return in excess of that on risk-free securities. The premium provides compensation for the risk of an investment.
risk–return trade-off Investors must take on greater risk if they want higher expected returns.
risk sharing Sharing the risk of a portfolio of given size among many investors.
risky asset An asset with an uncertain rate of return.
S scatter diagram Plot of returns of one security versus returns of another security. Each point represents one pair of returns for a given holding period.
seasoned new issue Stock issued by companies that already have stock on the market.
secondary market Already existing securities are bought and sold on the exchanges or in the OTC market.
second-pass regression A cross-sectional regression of portfolio returns on betas. The estimated slope is the measurement of the reward for bearing systematic risk during the period.
sector rotation An investment strategy which entails shifting the portfolio into industry sectors that are forecast to outperform others based on macroeconomic forecasts.
securitization Pooling loans for various purposes into standardized securities backed by those loans, which can then be traded like any other security.
security analysis Determining correct value of a security in the marketplace.
security characteristic line A plot of the excess return on a security over the risk-free rate as a function of the excess return on the market.
security market line (SML) Graphical representation of the expected return–beta relationship of the CAPM.
security selection See security selection decision.
security selection decision Choosing the particular securities to include in a portfolio.
semistrong-form EMH See efficient market hypothesis.
separation property The property that portfolio choice can be separated into two independent tasks: (1) determina- tion of the optimal risky portfolio, which is a purely techni- cal problem, and (2) the personal choice of the best mix of the risky portfolio and the risk-free asset.
Sharpe ratio Reward-to-volatility ratio; ratio of portfolio excess return to standard deviation.
shelf registration Advance registration of securities with the SEC for sale up to 2 years following initial registration.
reinvestment rate risk The uncertainty surrounding the cumulative future value of reinvested bond coupon payments.
REIT Real estate investment trust, which is similar to a closed-end mutual fund. REITs invest in real estate or loans secured by real estate and issue shares in such investments.
relative strength The extent to which a security has out- performed or underperformed either the market as a whole or its particular industry.
remainderman One who receives the principal of a trust when it is dissolved.
replacement cost Cost to replace a firm’s assets. “ Reproduction” cost.
representativeness bias People seem to believe that a small sample is just as representative of a broad population as a large one and therefore infer patterns too quickly.
repurchase agreements (repos) Short-term, often over- night, sales of government securities with an agreement to repurchase the securities at a slightly higher price. A reverse repo is a purchase with an agreement to resell at a specified price on a future date.
residual claim Refers to the fact that shareholders are at the bottom of the list of claimants to assets of a corporation in the event of failure or bankruptcy.
residual income See economic value added (EVA). residuals Parts of stock returns not explained by the explan- atory variable (the market-index return). They measure the impact of firm-specific events during a particular period.
resistance level A price level above which it is supposedly difficult for a stock or stock index to rise.
return on assets (ROA) A profitability ratio; earnings before interest and taxes divided by total assets.
return on equity (ROE) An accounting ratio of net profits divided by equity.
return on sales (ROS), or profit margin The ratio of oper- ating profits per dollar of sales (EBIT divided by sales).
reversal effect The tendency of poorly performing stocks and well-performing stocks in one period to experience reversals in following periods.
reversing trade Entering the opposite side of a currently held futures position to close out the position.
reward-to-volatility ratio Ratio of excess return to portfolio standard deviation.
riding the yield curve Buying long-term bonds in anticipation of capital gains as yields fall with the declining maturity of the bonds.
risk arbitrage Speculation on perceived mispriced securi- ties, usually in connection with merger and acquisition targets.
risk-averse, risk-neutral, risk lover A risk-averse investor will consider risky portfolios only if they provide compensa- tion for risk via a risk premium. A risk-neutral investor finds the level of risk irrelevant and considers only the expected return of risk prospects. A risk lover is willing to accept lower expected returns on prospects with higher amounts of risk.
risk-free asset An asset with a certain rate of return; often taken to be short-term T-bills.
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Glossary
G-12
statistical arbitrage Use of quantitative systems to uncover many perceived misalignments in relative pricing and ensure profit by averaging over all of these small bets.
stock exchanges Secondary markets where already-issued securities are bought and sold by members.
stock selection An active portfolio management technique that focuses on advantageous selection of particular stocks rather than on broad asset allocation choices.
stock split Issue by a corporation of a given number of shares in exchange for the current number of shares held by stockholders. Splits may go in either direction, either increasing or decreasing the number of shares outstanding. A reverse split decreases the number outstanding.
stop-loss order A sell order to be executed if the price of the stock falls below a stipulated level.
stop orders Order to trade contingent on security price designed to limit losses if price moves against the trader.
straddle A combination of buying both a call and a put on the same asset, each with the same exercise price and expiration date. The purpose is to profit from expected volatility.
straight bond A bond with no option features such as callability or convertibility.
street name Describes securities held by a broker on behalf of a client but registered in the name of the firm.
strike price See exercise price.
strip, strap Variants of a straddle. A strip is two puts and one call on a stock; a strap is two calls and one put, both with the same exercise price and expiration date.
stripped of coupons Describes the practice of some investment banks that sell “synthetic” zero-coupon bonds by marketing the rights to a single payment backed by a coupon-paying Treasury bond.
strong-form EMH See efficient market hypothesis.
subordination clause A provision in a bond indenture that restricts the issuer’s future borrowing by subordinating the new leaders’ claims on the firm to those of the existing bond holders. Claims of subordinated or junior debtholders are not paid until the prior debt is paid.
substitution swap Exchange of one bond for a bond with similar attributes but more attractively priced.
supply shock An event that influences production capacity and costs in the economy.
support level A price level below which it is supposedly difficult for a stock or stock index to fall.
survivorship bias Bias in the average returns of a sample of funds induced by excluding past returns on funds that left the sample because they happened to be unsuccessful.
swaption An option on a swap.
systemic risk Risk of breakdown in the financial system, particularly due to spillover effects from one market into others.
systematic risk Risk factors common to the whole economy, nondiversifiable risk; also called market risk.
short position or hedge Protecting the value of an asset held by taking a short position in a futures contract.
short rate A one-period interest rate.
short sale The sale of shares not owned by the investor but borrowed through a broker and later repurchased to replace the loan. Profit is earned if the initial sale is at a higher price than the repurchase price.
single-factor model A model of security returns that acknowledges only one common factor. See factor model.
single-index model A model of stock returns that decomposes influences on returns into a systematic factor, as measured by the return on a broad market index, and firm-specific factors.
single-stock futures Futures contracts on single stock rather than an index.
sinking fund A procedure that allows for the repayment of principal at maturity by calling for the bond issuer to repurchase some proportion of the outstanding bonds either in the open market or at a special call price associated with the sinking fund provision.
skew Measure of the asymmetry of a probability distribution.
small-firm effect That investments in stocks of small firms appear to have earned abnormal returns.
soft dollars The value of research services that brokerage houses supply to investment managers “free of charge” in exchange for the investment managers’ business.
Sortino ratio Excess return divided by lower partial standard deviation.
specialist A trader who makes a market in the shares of one or more firms and who maintains a “fair and orderly market” by dealing personally in the stock.
speculation Undertaking a risky investment with the objective of earning a greater profit than an investment in a risk-free alternative (a risk premium).
speculative-grade bond Bond rated Ba or lower by Moody’s, or BB or lower by Standard & Poor’s, or an unrated bond.
spot-futures parity theorem, or cost-of-carry relationship Describes the theoretically correct relationship between spot and futures prices. Violation of the parity relationship gives rise to arbitrage opportunities.
spot rate The current interest rate appropriate for discounting a cash flow of some given maturity.
spread (futures) Taking a long position in a futures contract of one maturity and a short position in a contract of different maturity, both on the same commodity.
spread (options) A combination of two or more call options or put options on the same stock with differing exer- cise prices or times to expiration. A money spread refers to a spread with different exercise price; a time spread refers to differing expiration date.
standard deviation Square root of the variance.
statement of cash flows A financial statement showing a firm’s cash receipts and cash payments during a specified period.
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Glossary
G-13
turnover The ratio of the trading activity of a portfolio to the assets of the portfolio.
12b-1 fees Annual fees charged by a mutual fund to pay for marketing and distribution costs.
U unbundling See bundling.
underwriters Investment bankers who help companies issue their securities to the public.
underwriting, underwriting syndicate Underwriters (investment bankers) purchase securities from the issuing company and resell them. Usually a syndicate of investment bankers is organized behind a lead firm.
unemployment rate The ratio of the number of people classified as unemployed to the total labor force.
unique risk See diversifiable risk.
unit investment trust Money invested in a portfolio whose composition is fixed for the life of the fund. Shares in a unit trust are called redeemable trust certificates, and they are sold at a premium above net asset value.
universal life policy An insurance policy that allows for a varying death benefit and premium level over the term of the policy, with an interest rate on the cash value that changes with market interest rates.
utility The measure of the welfare or satisfaction of an investor.
utility value The welfare a given investor assigns to an investment with a particular return and risk.
V value at risk (VaR) Measure of downside risk. The loss that will be incurred in the event of an extreme adverse price change with some given, typically low, probability.
variable annuities Annuity contracts in which the insur- ance company pays a periodic amount linked to the invest- ment performance of an underlying portfolio.
variable life policy An insurance policy that provides a fixed death benefit plus a cash value that can be invested in a variety of funds from which the policyholder can choose.
variance A measure of the dispersion of a random variable. Equals the expected value of the squared deviation from the mean.
variation margin See maintenance margin.
vega Response of option price to change in standard deviation of underlying asset.
venture capital Money invested to finance a new, not yet publicly-traded firm.
views An analyst’s opinion on the likely performance of a stock or sector compared to the market-consensus expectation.
volatility risk The risk in the value of options portfolios due to unpredictable changes in the volatility of the underlying asset.
T tax anticipation notes Short-term municipal debt to raise funds to pay for expenses before actual collection of taxes.
tax-deferral option The feature of the U.S. Internal Revenue Code that the capital gains tax on an asset is payable only when the gain is realized by selling the asset.
tax-deferred retirement plans Employer-sponsored and other plans that allow contributions and earnings to be made and accumulate tax-free until they are paid out as benefits.
tax swap Swapping two similar bonds to receive a tax benefit.
technical analysis Research to identify mispriced securi- ties that focuses on recurrent and predictable stock price pat- terns and on proxies for buy or sell pressure in the market.
tender offer An offer from an outside investor to share- holders of a company to purchase their shares at a stipulated price, usually substantially above the market price, so that the investor may amass enough shares to obtain control of the company.
term insurance Provides a death benefit only, no build-up of cash value.
term premiums Excess of the yields to maturity on long-term bonds over those of short-term bonds.
term structure of interest rates The pattern of interest rates appropriate for discounting cash flows of various maturities.
times interest earned Ratio of profits to interest expense. time value (of an option) The part of the value of an option that is due to its positive time to expiration. Not to be confused with present value or the time value of money.
time-weighted average An average of the period-by-period holding-period returns of an investment.
Tobin’s q. Ratio of market value of the firm to replacement cost.
total asset turnover The annual sales generated by each dollar of assets (sales/assets).
tracking error The difference between the return on a specified portfolio and that of a benchmark portfolio designed to mimic that portfolio.
tracking portfolio A portfolio constructed to have returns with the highest possible correlation with a systematic risk factor.
tranche See collateralized mortgage obligation. Treasury bill Short-term, highly liquid government securities issued at a discount from the face value and returning the face amount at maturity.
Treasury bond or note Debt obligations of the federal government that make semiannual coupon payments and are issued at or near par value.
Treynor’s measure Ratio of excess return to beta. trin statistic Ratio of average trading volume in declining stocks to average volume in advancing stocks. Used in technical analysis.
trough The transition point between recession and recovery.
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Glossary
G-14
Y yield curve A graph of yield to maturity as a function of time to maturity.
yield to maturity A measure of the average rate of return that will be earned on a bond if held to maturity.
Z zero-beta portfolio The minimum-variance portfolio uncorrelated with a chosen efficient portfolio.
zero-coupon bond A bond paying no coupons that sells at a discount and provides payment of face value only at maturity.
zero-investment portfolio A portfolio of zero net value, established by buying and shorting component securities, usually in the context of an arbitrage strategy.
W warrant An option issued by the firm to purchase shares of the firm’s stock.
weak-form EMH See efficient market hypothesis.
well-diversified portfolio A portfolio spread out over many securities in such a way that the weight in any security is close to zero.
whole-life insurance policy Provides a death benefit and a kind of savings plan that builds up cash value for possible future withdrawal.
workout period Realignment period of a temporary misaligned yield relationship.
world investable wealth The part of world wealth that is traded and is therefore accessible to investors.
writing a call Selling a call option.
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I-1
Fisher, Irving, 121
Fisher, Kenneth L., 393
Franzoni, Francesco, 939
French, Kenneth R., 142–143, 146, 187,
314, 340, 342, 365–369, 371, 377,
426–427, 437–438, 951
Friend, I., 190
Froot, K. A., 396
Fung, William, 939
G Gabelli, Mario, 379
Garber, Peter, 375
Gervais, S., 400
Geske, Robert, 745–746
Getmansky, Mila, 937
Ghysels, E., 743
Givoly, Dan, 371
Goetzmann, William N., 439–440, 847, 944
Goldberg, Lisa, 141
Goldgar, Anne, 375
Goyal, Amit, 428
Graham, Benjamin, 665
Graham, John R., 841
Green, R. C., 379
Green, T. C., 351–352
Greenspan, Alan, 398
Grinblatt, Mark, 400
Grind, Kristen, 189
Grossman, Sanford J., 190, 352
Gruber, Martin J., 341, 376, 379
Guy, J., 282
H Han, Bing, 400
Harris, Lawrence E., 65, 70
Harris, M., 389
Harvey, A., 743
Harvey, Campbell R., 422, 841
Hasbrouck, J., 313
Heaton, John, 306, 423–424
Henriksson, Roy D., 855
Heston, S., 743
Hicks, John, 792
Hlavka, M., 376
Homer, Sidney, 517, 543, 815
Hsieh, David, 939
Huang, Ming, 441
Hubner, Georges, 845
Hull, J., 743
Brown, P., 369
Brown, S. S., 944–945
Buffett, Warren, 379
Busse, Jeffrey A., 351–352, 378
C Campbell, John Y., 314, 366
Caniel, Naveen D., 939
Carhart, Mark M., 109, 378, 433
Carther, Shauna, 852
Chan, L. K. C., 431
Chen, Nai-Fu, 425
Chopra, Navin, 365, 391
Chordia, Tarun, 69, 313, 373
Churchill, Winston, 5
Clements, Jonathan, 216, 395, 918
Conrad, Jennifer, 364
Constantinides, George M., 389, 440
Cooper, M., 399
Coval, J. D., 394
Cox, John C., 736
D Dabora, E. M., 396
Das, S., 376
DeBondt, Werner F. M., 365, 390, 392
Del Guercio, Diane, 379
DeLong, J. B., 394
Desai, M., 661
Dimitrov, O., 399
Dimson, Elroy, 151–152
Dodd, David, 665
Droms, William, 216
Dugan, Ianthe Jeanne, 853
E Elton, Edwin J., 341, 376, 379
Engle, Robert, 314
Evans, Richard, 377
F Faboi, Frank J., 392
Fama, Eugene F., 142, 308, 314, 340, 342,
365–366, 368–369, 371, 377, 399,
420–421, 426–427, 437–438, 951
Ferson, Wayne, 422
Note: Page numbers followed by n indicate
material found in notes.
A Acharya, V. V., 313, 434
Agarwal, Vikas, 939
Aït-Sahalia, Yacine, 161
Akerlof, George A., 311
Alexander, C., 743
Allen, F., 398
Altman, Edward I., 472n
Amihud, Yakov, 310, 312–313, 368, 434
Amin, G., 939
Anand, Shefali, 907
Angel, James J., 65, 70
Angier, Deniz, 393
Aragon, George, 937
Arbel, Avner, 368
B Baginski, Stephen, 648
Ball, R., 369
Banz, Rolf, 367
Barber, Brad, 375, 390
Barberis, Nicholas, 389, 441
Basu, Sanjoy, 367
Batchelor, Roy, 401
Battalio, R. H., 369
Bavishi, Vinod, 664
Beaver, W. H., 472n
Beebower, Gilbert, 216, 861
Ben-David, Itzhak, 939
Bergstresser, D., 661
Berk, J. B., 379
Bernard, V., 369
Bernoulli, Daniel, 199–200
Bierwag, G. O., 543
Black, Fischer, 271, 306–307, 341–342, 419,
737, 744, 956, 959, 962
Blake, Christopher R., 379
Blume, Marshall E., 190, 368
Bodie, Zvi, 825
Bogle, John, 176
Bollen, Nicolas P. B., 378
Boudette, Neal, 559
Bradshaw, Mark, 648
Brealey, R. A., 398
Breeden, Douglas, 308
Brennan, Michael J., 304
Brinson, Gary, 216, 861
Name Index
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Name Index
I-2
O Odean, Terrance, 390, 392, 395, 400
Oldfield, George S., 822
P Palmon, Dan, 371
Pástor, L., 313, 433, 435
Patell, J. M., 351
Paulson, Henry, 661
Pedersen, Lasse Heje, 310, 313, 434
Perritt, Gerald, 216
Petkova, Ralista, 429–430
Pinto, Jerald E., 980
Pontiff, Jeffrey E., 95n, 373, 397
Ponzi, Charles, 945
Porter, Michael, 581–582
Poterba, James, 365
Prescott, Edward, 435
R Rajaratnam, Raj, 353
Ramsey, Mike, 536, 559
Ramyar, Richard, 401
Rapoport, Michael, 662
Rau, P. R., 399
Raub, J., 661
Redington, F. M., 536
Reinganum, Marc R., 368
Renault, E., 743
Rendleman, R. J., Jr., 370
Reuter, Jonathan, 379
Rhie, J-W., 472n
Ritter, Jay R., 62–63, 365, 391
Roberts, Harry, 405–407
Robinson, Tom, 980
Roll, Richard, 69, 305, 313, 417–418, 425,
745, 910
Rosansky, Victor, 825
Rosenberg, Barr, 282
Ross, Stephen A., 327, 397, 418, 425, 469,
736
Rothchild, John, 580
Rubinstein, Mark, 308, 759
Russolilo, Steven, 744
Rydqvist, K., 63
S Sadka, Ronnie, 434, 937
Saha, Atanu, 939
Samuelson, Paul, 380
Sauter, Gus, 839
Scheck, Justin, 663
Schleifer, A., 394
Leonard of Pisa (Fibonacci), 401
Liang, B., 944
Liebowitz, Martin L., 517, 543, 815
Liew, J., 372, 428–429
Lintner, John, 291, 417
Litterman, Robert, 956, 962
Lo, Andrew W., 364, 937
Longin, F., 909
Loughran, T., 63
Lucas, Deborah, 306, 423–424
Lucas, Robert, 308
Lynch, Peter, 379, 580–581, 611, 861
M Macaulay, Frederick, 519
MacBeth, James, 420–421
MacKinlay, A. Craig, 364
MacLean, L. C., 204
Maddala, G. S., 743
Madoff, Bernard, 944
Maginn, John L., 980
Makarov, Igor, 937
Malkiel, Burton G., 109, 315, 517, 522, 939
Malloy, C. J., 437
Malmendier, U., 390
Marcus, Alan J., 161, 909, 970
Markowitz, Harry, 10, 141, 222, 291
Marsh, Paul, 151–152
Mayers, David, 307, 421
Mazuy, Kay, 855
McDonald, Robert L., 733, 759
McGee, Suzanne, 810
McLean, David R., 373
McLeavey, Dennis W., 980
McNichols, M. F., 375, 472n
Mehra, Rajnish, 435, 440–441
Mei, Jianping, 341
Mendelson, Haim, 310, 312, 368
Mendenhall, R., 369
Merton, Robert C., 305, 307–308, 324, 368,
736–737, 752, 855, 858, 962
Milken, Michael, 470
Miller, Merton H., 314, 417, 419, 620
Mitchell, Mark, 361
Modigliani, Franco, 620, 841
Modigliani, Leah, 841
Monga, Vipal, 536
Moskowitz, T., 437
Mossin, Jan, 291
Moussawi, Rabih, 939
Myers, S. C., 398
N Naik, Narayan Y., 939
Netter, Jeffry, 361
Nofsinger, John, 395
I Ibbotson, Roger G., 439
Icahn, Carl, 7
Ingersoll, Jonathan, 847
J Jacod, Jean, 161
Jacquier, Eric, 161, 909
Jaffee, Charles A., 993
Jaffee, Jeffrey E., 86n, 371
Jagannathan, Ravi, 257, 310, 314, 422,
436–437
Jagedeesh, N., 364
Jarrow, Robert A., 822
Jegadeesh, Narasimhan, 376, 432
Jensen, Michael C., 307, 314, 419, 421, 839
Johnson, H. E., 746
Jones, C. P., 84, 370
Jurion, Philippe, 440
K Kahneman, D., 390, 393
Kandel, Schmuel, 418
Kane, Alex, 161, 960, 970–972
Kaplin, Andrew, 257
Karabell, Zachaary, 692
Karceski, J., 431
Kat, H., 939
Kaufman, G. C., 543
Kaul, Gautam, 364
Keim, Donald B., 366, 368
Kelly, J. L., Jr., 203n
Kendall, Maurice, 349
Keynes, John Maynard, 792
Kiernan, Kaitlyn, 744
Kim, J., 376
Kim, Tae-Hwan, 960, 971–972
Knight, Frank, 187
Korajczyk, Robert, 422
Kosowski, R., 379
Kothari, S. P., 369
Kotlikoff, Laurence J., 999
Krische, S. D., 376
L La Porta, Raphael, 372, 432
Laise, Eleanor, 141
Lakonishok, Josef, 365, 372, 391, 431–432
Lamont, O. A., 84, 396
Landier, Augustin, 939
Latané, H. A., 370
Lee, C. M., 376, 397
Lehavy, R., 375
Lehmann, B., 364
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Name Index
I-3
Volcker, Paul, 14, 22
Von Neumann, 200, 203
Vuolteenaho, Tuomo, 314
W Wahlen, James, 648
Waldmann, R., 394
Wang, Yong, 310, 436–437
Wang, Zhenyu, 314, 422
Welch, Ivo, 847
Wermers, R. R., 378–379
Westerfield, Randolph W., 469
Whaley, Robert E., 745
White, A., 743
White, Halbert, 379, 960, 971–972
Wiggins, J., 743
Wolfson, M. A., 351
Womack, K. L., 375
Worthen, Ben, 663
Y Yang, Jerry, 7
Z Zhang, Lu, 429–430
Ziemba, W. T., 204
Subrahmanyam, Avanidhar, 69, 313, 373
Summers, Lawrence, 365, 394
T Taleb, Nassim N., 158, 941
Tate, G., 390
Templeton, John, 379
Thaler, Richard H., 365, 389–390, 392,
396–397
Thomas, J., 369
Thorp, E. O., 204
Timmerman, A., 379
Titman, Sheridan, 364, 432
Tobin, James, 225, 593
Toevs, A., 543
Tong, W., 373
Train, John, 665
Treussard, Jonathan, 140
Treynor, Jack L., 271, 419, 839, 855, 959
Trippi, Robert R., 741, 970
Trueman, B., 375
Tuttle, Donald L., 980
Tversky, A., 390, 393
V Vasicek, Oldrich A., 281
Vassalou, M., 372, 429
Vishny, R. W., 372, 394, 432
Vissing-Jorgensen, A., 437
Schmalensee, Richard E., 741
Scholes, Myron, 307, 314, 417, 419, 737
Selling, Thomas I., 648
Seppi, D. H., 313
Seyhun, H., 371
Shanken, Jay, 369
Sharpe, William F., 10, 134, 262, 291, 294,
839, 853, 861, 863
Shefrin, H., 392
Shiller, Robert J., 190, 366, 692
Shleifer, Andrei, 372, 397, 432
Shumway, T., 394
Singer, Brian, 216, 861
Sloan, Richard G., 369
Smith, Jason A., 606
Smith, Randall, 613
Solnik, B., 907, 909
Sonne, Paul, 663
Spatt, Chester, 65, 70
Speidell, Lawrence S., 664
Spiegel, Matthew, 847
Spivak, Avia, 999
Stambaugh, Robert F., 313, 366, 368, 418,
433, 435
Statman, Meir, 206–207, 392–393, 395
Staunton, Mike, 151–152
Stein, Jeremy, 301
Stickney, Clyde P., 648
Stiglitz, Joseph E., 352
Stoval, Sam, 578
Strebel, Paul J., 368
Stulz, R., 389
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I-4
Note: Page numbers followed by n indicate
material found in notes.
A Abnormal return, 359
cumulative abnormal returns, 360
event studies, 359–361
market model to estimate, 359–360
to infer damages, 361
Accounting earnings, 636
economic earnings vs., 636–637
Accounting practices, 662–663
Accrued interest, 447
Acid test ratio, 651
Active bond management, 543–545
hedging interest rate risk, 813–815
horizon analysis, 544–545
intermarket spread swap, 544
pure yield pickup swap, 544
rate anticipation swap, 544
sources of profit, 543–545
substitution swap, 543–544
tax swap, 544
Active investment strategy, 515
Active management, 11
Active portfolio, 952
Active portfolio management, 274, 972–973;
see also Black-Litterman model; Treynor-Black model
alphas values, forecasts of, 952–956
extreme portfolio weights, 952–956
forecasts and, 971–972
information ratio and, 971
manager’s private views, 964–965
market timing, 855–857
optimal portfolios and alpha values,
951–958
passive portfolio management vs.,
356–357
potential fees estimation, 970–971
restriction of benchmark risk, 956–958
value of, 970–972
Active weight, 867
Actual delivery, 780
Adjustable-rate mortgages (ARMs), 19
Adjusted alpha, 959
ADRs (American Depository Receipts),
605
Affect, 392–393
Agency problems, 6–7, 20, 42
Aggregate stock market, 622–623
forecasting of, 622–623
Algorithmic trading, 71–72
Subject Index
Allowance for bad debt, 662
Alpha, 260, 299
adjusted beta, 281
alpha betting, 283
distribution of alpha values, 960–961
estimate of, 267–268
forecasts and extreme portfolio weights,
952–956
forecasts and realizations of, 974–975
mutual fund alphas (1972–1991), 315
optimal portfolios and, 951–958
performance measures and, 844–845
portable alpha, 931–933
positive-alpha stock, 299
security analysis, 271–272
Alpha transfer, 931
Alpha transport, 284
Alpha values, 952
Altman’s Z ratio, 471–472 American Depository Receipts (ADRs),
44, 895
American option, 682
Analysis of variance (ANOVA), 266–267
Angel investors, 15
Annual percentage rates (APRs), 35, 123
effective annual rate (EAR) vs., 123
Annuity factor, 453
Anomalies, 366, 371–374; see also Market anomalies
data mining, 373
over time, 373–374
Arbitrage, 324, 328, 700
convertible bond arbitrage, 929
covered interest arbitrage, 801–803
equity carve-outs, 396–397
execution of, 332
fundamental risk, 394
futures markets, 786–787
implementation costs, 395
index arbitrage, 809–810
law of one price and, 328, 395
limits to, 394–398
mispricing and, 339
model risk, 395
security market line and, 327
“Siamese twin” companies, 396
statistical arbitrage, 930
systematic factor and, 333
Arbitrage pricing theory (APT), 324,
327–334
betas and expected returns, 260, 297,
415
CAPM and, 334–335
cost of capital and, 341
early versions of, 424–425
equilibrium and, 328
Fama and French (FF) three-factor model,
340–342
index model and single-factor ATP,
415–421
individual assets and, 340
multifactor APT, 338–340
multifactor CAPM and, 421–426
no-arbitrage equation, 333–334
one-factor security market line, 340
portfolio optimization in single-index
market, 336–338
risk arbitrage and, 328
security market line, 327
well-diversified portfolios, 329–331
Arbitrageurs, 328
ARCH, 743
Arithmetic average
expected return and, 130–131
geometric vs., 132
Arthur Andersen, 8
Asian options, 709
Ask price, 64
Asked price, 29, 446
Asset allocation, 9, 28, 205, 216, 992–993;
see also Capital allocation Black-Litterman model, 962–968
flexible funds and, 98
international diversification; see Interna- tional diversification
performance attribution and, 867
policy statements, 985
risk tolerance and, 182–187
risky and risk-free portfolios, 178–181
security analysis, 8
security selection, 229
steps in, 962–968
stock, bonds, and bills, 215–219
successful investing and, 216
synthetic stock positions, 808–809
taxes and, 992–993
two risky asset classes, 215–219
Asset-backed bonds, 450–451
Asset-backed commercial paper, 31
Asset-backed securities, 40–41
amount outstanding (1996–2012), 41
Asset plays, 581
Asset pricing models; see Arbitrage pricing theory (APT); Capital asset pricing
model (CAPM)
Asset turnover, 648
Asset utilization, 654
Asset utilization ratios, 649–650
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Subject Index
I-5
Board of Governors of the Federal Reserve
System, 564
Bogey, 865
Bond-equivalent yield, 30, 35
Bond funds, 97
Bond indentures, 446, 471
collateral, 473
dividend restrictions, 473
sinking funds, 472–473
subordination clauses, 473
Bond market, 34–41
corporate bonds, 39
federal agency debt, 35–36
inflation-protected Treasury bonds, 35
innovations, 450–452
international bonds, 36
mortgages and mortgage-backed
securities, 39–40
municipal bonds, 36–39
tax-exempt debt outstanding (1979–
2012), 37
Treasury notes and bonds, 34–35
Bond market indicators, 50–51
Bond portfolios; see Active bond manage- ment; Passive bond management
Bond pricing, 445, 452–457, 488–490
between coupon dates, 456
callable and straight debt, 701
coupon bonds, 489
default risk and, 468–469
Excel application for, 457
financial calculator, 454
forward rates with interest rate risk, 496
interest rate sensitivity, 516–518
price/yield relationship, 455
prices over time, 463–468
quoted bond prices, 447
yield curve and, 488–490
Bond ratings, 469
Bond reconstitution, 489
Bond safety
Altman’s Z, 472 cash flow-to-debt ratio, 470–472
coverage ratios, 470
determinants of, 470–472
leverage ratio, 470
liquidity ratios, 470
profitability ratios, 470
Bond stripping, 489
Bond swap, 543; see also Active bond management
Bond trading, 73–74
Bond yields, 458–463
current yield, 459
effective annual yield, 468
realized compound yield vs. yield to
maturity, 461–463
yield to call, 460–461
yield to maturity (YTM), 458–460
vs. holding period return, 465–466
for mutual funds, 107
for ratio analysis, 653–655
style analysis and multifactor
benchmarks, 863–864
Beta, 297
adjusted beta, 281
commodity betas, 825
consumption betas, 437
equalizing beta, 844
estimate of, 268
expected return-beta relationship,
260, 297
expected returns and, 260, 415
factor betas, 326
as hurdle rate, 299
liquidity betas, 313
mean-beta relationship, 297
measurement error in, 419–421
predicting betas, 282
zero-beta model, 305–306
Bid-ask spread, 29, 64
Bid price, 29, 64, 446
Binomial model, 733
Binomial option pricing, 729–737
making the model practical, 733–737
two-state option pricing, 731–732
Binomial tree, 136
Black-Litterman model, 962–968,
975–976
asset allocation decision, 962–963
baseline forecast, 963–964
covariance matrix from historical data,
963
manager’s private views and, 964–965
portfolio optimization, 966–968
revised (posterior) expectations,
965–966
sensitivity of, 967
steps in, 962–968
Treynor-Black vs., 968–970
Black-Scholes pricing formula, 737–746
assumptions of, 741
dividends and call options, 744–745
dividends and put option valuation, 746
example of, 739
Excel model for, 742, 748
hedge ratios and, 746–747
hedging ratios, 746–747
implied volatility, 741–742
incentive fees, 944
portfolio insurance, 748–751
pseudo-American call option value, 745
put option valuation, 745
synthetic protective put options,
749–750
use of, 746–758
Black swan problem, 117, 158,
941n, 943
Blocks, 73
Blue sky laws, 83
Assets; see also Derivative assets; Financial assets
asset classes; see Financial instruments domestic net worth (2012), 3
individual assets and ATP, 340
nontraded assets, 306–307, 423–424
real assets vs. financial assets, 2–3
risk-free asset, 177–178
Assets in place, 142
Asymmetric information, 311
At the money, 680
Auction markets, 64
Average collection period, 650
B Back-end load, 100
Backfill bias, 939
Balance sheet, 637
commercial banks and savings
institutions (2012), 12
common-size balance sheets, 637
Home Depot example, 638
nonfinancial U.S. corporations (2012), 13
restructurings and swaps, 816–817
U.S. households (2012), 3
Balanced funds, 97
Bank-discount method, 30
Bankers’ acceptances, 31
Banks, 982–983
Barclays Capital U.S. Aggregate Bond
Index, 534
Bargaining power of buyers/suppliers, 582
Barrier options, 709
Baseline forecast, 963
Basis, 784
Basis risk, 784, 1003
Behavioral biases, 391–394
affect, 392–393
framing, 391
mental accounting, 391–392
prospect theory, 393–394
regret avoidance, 392
Behavioral economics, 398–399
Behavioral finance, 388–389, 431
bubbles and behavioral economics,
398–399
disposition effect, 400
equity premium pule, 436
evaluation of, 399–400
information processing, 389–391
limits to arbitrage, 394–395
mental accounting, 391–392
technical analysis and, 400–407
warnings on, 405–407
Benchmark error, 418
Benchmark risk, 956–957
Benchmarks, 956
choice of, 653–655
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Subject Index
I-6
consumption-based CAPM, 308–310
expected returns on individual securities,
295–297
extensions of, 302–305, 440–441
hedge portfolios, 307–308
homogeneous expectations, 303
human capital and, 423
hurdle rate, 299
ICAPM (intertemporal CAPM), 308
ideas and implications of, 300–301
investment industry and, 315–316
labor income and nontraded assets,
306–307
liquidity and, 310–313
market portfolio (M) and, 292–293
multifactor CAPM and APT, 421–426
multiperiod model, 307–308
passive strategy as efficient, 293
required rate of return, 299
risk premium of market portfolio,
293–294
Roll’s critique, 417–419
security market line (SML), 298–300, 307
single-index market, 301–302, 414
tests of, 417
utility rate-making cases and, 299
zero-beta model, 305–306
Capital gains, 42
Capital market line (CML), 188–189
efficient frontier and, 292
passive strategies and, 187–190
Capital markets, 28
CAPM; see Capital asset pricing model (CAPM)
Case-Shiller Index of U.S. housing prices,
17
Cash, 992
Cash cows, 579
Cash equivalents, 28
Cash flow matching, 542
Cash flow-to-debt ratio, 470
Cash ratio, 651, 654
Cash settlement, 780
Cash/bond selection, 919
Catastrophe bonds, 451
Center for Research in Security Prices,
University of Chicago, 125
Certainty equivalent rate of return, 171
Certificates of deposit (CDs), 4, 30
index-linked CDs, 707–708
3-mo CD/T-bill spread (1970–2013), 33
CFA (Chartered Financial Analysts)
Institute, 84, 973, 977
investment management process,
978–983
standards of professional conduct, 85
Chartists, 355
Chicago Board Options Exchange (CBOE),
678, 681, 743–744
Chicago Board of Trade (CBT), 5, 53, 772
Broker’s call, 32
Broker’s call loans, 76
Bubbles, 374–375, 398
Budget deficit, 562
Bulldog bonds, 450
Bullish spread, 695, 697
Business cycles, 566–571, 578, 614
cyclical indicators, 567
defensive industries, 566
degree of operating leverage, 577
economic indicators, 568–571
Great Moderation, 16
other indicators, 571
peaks/troughs, 566
sector rotation and, 578
sensitivity to, 575–577
Buying on margin, 76–79
C Calendar spread, 784
Call option, 4n, 51, 679
corporate bonds (callable), 38, 448
covered calls, 691–694
current stock price and, 727
determinants of value, 724–725
dividends and, 727, 744–745
early exercises and dividends, 727
hedge ratio and, 747
incentive fee as, 944
market timing valued as, 858–859
profits and losses on, 679, 774
pseudo-American call option value, 745
put/call ratio, 405
restrictions on value of, 725–727
values at expiration, 685–686
values before expiration, 724
Call price, 448
Call provisions, 448–449
Callable bonds, 39, 448, 701
duration and convexity of, 528–530
mortgage-backs vs., 532
negative convexity, 528
values vs. straight bonds, 702
yield to call, 460–461
Capacity utilization rate, 561
Capital allocation, 168, 183, 205; see also Asset allocation
risk tolerance and, 182–187
risky and risk-free portfolios, 175–177
separation property and, 223–226
Capital allocation line (CAL), 180–181, 186,
215, 248–249, 292
Capital asset pricing model (CAPM),
291–302
academic world and, 313–315
arbitrage pricing theory (APT), 335
assumptions of, 302–305
challenges and extensions to, 305
Bonds, 446, 992; see also Active bond management; Fixed-income capital
market; Interest rates; Term structure
of interest rates; Yield curve
accrued interest, 447
after-tax returns, 467–468
asset allocation with, 215–219
asset-backed bonds, 450–451
call provisions, 448
callable bonds, 701–702
catastrophe bonds, 451
characteristics of, 446–452
collateral, 473–474
convertible bonds, 39, 448–449, 701–704
convexity; see Convexity corporate bonds, 447–448
coupon rate, 446
current yield, 459
default risk; see Default risk dividend restrictions, 473
floating-rate bonds, 4, 449
indentures, 446, 472
indexed bonds, 451–452
inflation-linked bonds, 35
innovations in bond market, 450–452
international bonds, 36, 450
inverse floaters, 450
junk bonds, 470
other domestic issuers, 450
par/face value, 446
pricing; see Bond pricing protective covenants, 472
puttable bonds, 449
quoted bond prices, 447
ratings, 469
safety, determinants of, 470–472
secured bonds, 39
sinking funds, 472–473
subordination clauses, 473
trading of, 73–74
unsecured bonds, 473
yield to call, 460–461
yield to maturity; see Yield to maturity zero-coupon bonds; see Zero-coupon
bonds
Book, 62
Book-to-market effects, 301, 314, 369, 426,
428
Book value, 592
limitations of, 592–593
Bookbuilding, 62
Border-multiplied covariance, 209
Bordered covariance matrix, 209, 246
Bottom-up strategy, 9
Bracket creep, 121
Brane v. Roth, 691 Breadth, 403–404
BRIC countries (Brazil, Russia, India, and
China), 558
Brokered markets, 64
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Subject Index
I-7
Covering the short position, 80
Credit default swaps (CDS), 20,
475–477, 821
Credit enhancement, 21
Credit events, 475
Credit risk, 468
collateralized debt obligations and,
19–20, 477–478, 532
in swap market, 820–821
Cross-hedging, 784, 815
Crowd out, 562
Cumulative abnormal return (CAR), 360
Cumulative dividends, 43
Currency futures, 800; see also Foreign exchange futures
Currency selection, 918
Currency-translated options, 709–710
Current ratio, 470, 651, 654
Current yield, 459
Cyclical indicators, 567
Cyclical industries, 259, 566
Cyclicals, 581
D Dark pools, 72–73
Data mining, 373, 930
Days sales in receivables, 650, 654
Dealer markets, 64, 67
Debentures, 39, 473
Debt securities, 3–4, 445; see also Fixed- income securities
risky debt, 707
Debt-to-equity ratios, 470
Dedication strategy, 542
Default premium, 475
Default risk
bond pricing and, 468–478
corporate bonds, 468–469
default premium and, 475
financial ratios as predictors, 470–471
yield to maturity and, 474–475
Defensive industries, 566
Deferred annuities, 998
Deferred callable bonds, 448
Defined benefit plans, 981, 1000–1001
Defined contribution plans, 981, 1000
Degree of operating leverage (DOL), 577
Degrees of freedom bias, 133
Delta, 746, 754
Delta neutral, 757
Demand shocks, 562–563
Demand-side management, 563
Depository receipts, 44
Depreciation, 659
accounting vs. economic definition
of, 659
comparability problems of, 659
international accounting conventions, 663
Constant relative risk aversion (CRRA),
309n, 840
Constraints; see Investment constraints Consumer Price Index (CPI), 35
Consumption-based CAPM (CCAPM),
308–310, 435–450
market rates of return and, 435–437
Consumption-mimicking portfolio, 309
Consumption timing, 5–6
Consumption-tracking portfolio, 309
Contango, 792
Contingent liability, 663
Continuous compounding (CC), 124
Convergence property, 779
Conversion premium, 449
Conversion ratio, 448, 702
Conversion value, 702
Convertible bonds, 39, 448–449, 701–704
convertible bond arbitrage, 929
value as function of stock price, 703
Convertible securities, 701–704
Convexity, 455, 525–533
callable bonds and, 528–530
as desirable trait for investors, 528
mortgage-backed securities, 530–532
negative convexity, 528
Corporate bonds, 4, 39, 447–448; see also Bonds
callable bonds, 39, 448
convertible bonds, 39, 448–449
debentures, 39
default risk, 468–469
floating-rate bonds, 449
pricing of, 452–457
puttable bonds, 449
secured bonds, 39
subordinated debentures, 39
Corporate ethics, 8
Corporate governance, 8
Corrections, 400
Correlation, covariance matrix, 268–271
Correlation coefficient, 253–254
Cost of capital, using APT to find, 341
Cost-of-carry relationship, 787
Counterparty risk, 16, 800
Country selection, 919
Coupon payments, 35, 446
Coupon rate, 446
Covariance, 252–253
border-multiplied covariance, 209
bordered covariance matrix, 209
single-index model and, 260
Covariance matrix, 244–246
correlation and, 268–271
from historical data, 963
Coverage ratios, 470
Covered calls, 691–694
value at expiration, 693
Covered interest arbitrage relationship,
801–803, 891
Claw-back provisions, 23
Clearinghouse, 777
Clientele effect, 312
Closed-end funds, 94–95
Law of One Price, 397
CME Group, 777
Coincident indicators, 568
Collars, 695–698
Collateral, 473
Collateral trust bond, 473
Collateralized debt obligations (CDOs), 19,
478, 532
credit risk and, 477–478
Collateralized loans, 705–706
Commercial banking, 14
Commercial paper, 30–31
Commingled funds, 95
Commodities, 5
Commodity betas, 825
Commodity futures
discounted cash flow analysis, 824–825
pricing of, 822–825
pricing with storage costs, 822–823
Commodity Futures Trading Commission
(CFTC), 83, 780
Common-size balance sheets, 637
Common-size income statement, 636
Common stock, 4; see also Equity securities characteristics of, 42
limited liability of, 42
as ownership shares, 41–42
as residual claim, 42
Comparability problems, 658–664
depreciation, 659
fair value accounting, 660–662
financial statement analysis, 658–664
inflation and interest expense, 659–660
international accounting conventions,
663–664
inventory valuation, 658–659
quality of earnings, 662–663
Comparison universe, 838–839
Competition
in financial markets, 9–11
Porter’s five determinants of, 581–582
rivalry between competitors, 582
as source of efficiency, 351–353
Complete portfolio, 176
optimal complete portfolio, 218
proportions of, 219
steps to, 218
Compound leverage factor, 647, 654
Conditional tail expectation (CTE), 140
Conference Board, 568
Confidence index, 405
Conforming mortgages, 18
Conservatism, 390
Consolidation stage, 580
Constant-growth (Gordon model) DDM,
596–599
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Subject Index
I-8
financial markets and, 5–8
global economy, 558–560
separation of ownership and
management, 6–7
EDGAR Web site, 591
Effective annual rate (EAR), 122
annual percentage rates (APRs) vs., 123
total return vs., 123
Effective duration, 529–530
Efficient diversification, 205, 208
spreadsheet model for, 244–249
Efficient frontier of risky assets, 220, 222
capital market line and, 292
optimal risky portfolio, 248
spreadsheet model for, 221, 247–248
Efficient market hypothesis (EMH), 10–11,
350–354, 388, 414
abnormal returns, 359–361
active vs. passive portfolio management,
356–357
book-to-market ratios, 368–369
bubbles, 374–375
CAPM and, 366, 371
competition as source, 351–353
data mining, 373
event studies, 359–361
fundamental analysis, 356
implications of, 354–359
inside information, 370–371
liquidity effects, 368
lucky event issue, 363
magnitude issue, 362
market anomalies, 366–367
mutual fund managers, 376–380
neglected-firm effect, 368
portfolio management and, 357–358
post-earnings-announcement price drift,
369–370
predictors of broad market returns,
365–366
random walks and, 350–351
resource allocation, 358–359
returns over short/long horizons, 364–365
risk premiums, 371–373
selection bias issue, 362
semistrong form of, 354, 366–370
small-firm-in-January effect, 367–368
stock market analysts, 375–376
strong-form of, 354, 370–371
technical analysis and, 354–356
versions of, 353–354
weak-form of, 353, 364–365
Electronic communication networks
(ECNs), 67, 71
Electronic trading
algorithmic trading, 72
bond trading, 73–74
dark pools, 73
flash crash (2010), 74
high-frequency trading, 72–73
put option valuation, 746
Dodd-Frank Wall Street Reform and Consumer
Protection Act, 14, 22, 83, 477
Dollar-weighted rate of return, 837
Dollar-weighted returns, 836
Domestic macroeconomy, 560–562
budget deficit, 562
employment, 561
gross domestic product (GDP), 561
inflation, 561–563
interest rates, 562
sentiment, 562
Dow Jones Industrial Average (DJIA), 44–47
composition (1928/2013), 47
as price-weighted average, 44–45
splits and, 45–46
DuPont system, 645–648, 653
Duration, 519–525
bond maturity vs., 523
callable bonds, 528–539
determinants of, 522–525
effective duration, 529–530
Excel application for, 519–520, 524
interest rate risk, 521
as key concept, 521
Macaulay’s duration, 519
modified duration, 521
mortgage-backed securities, 530–533
rules for, 522–525
Dynamic hedging, 737, 750
E Earnings
accounting vs. economic earnings,
636–637
fully diluted earnings per share, 704
news content of, 370
post-earnings-announcement price drift,
369–370
quality of, 662–663
Earnings management, 613–614
Earnings retention ratio, 601
Earnings smoothing, 662
Earnings surprise, 370
Earnings yield, 622, 654
Economic calendar, 570
Economic earnings, 636
accounting earnings vs., 636–637
Economic indicators, 568–571
coincident/lagging indicators, 568–569
leading economic indicators, 568–569
other indicators, 568, 572
Economic value added, 621, 644–645
Economies of scale, 14
Economy; see also Business cycles; Macroeconomic analysis
consumption timing, 506
domestic macroeconomy, 560–562
Derivative assets, 4, 51–53; see also Options; Swaps
futures contracts, 53
options, 51–52
risk management and, 691–692
Derivative markets, 51–53
futures contracts, 53
mortgage derivatives and 2008 crisis, 19–20
options, 51–52
Derivative securities, 4, 678
Derivatives, 678
Detachable warrant, 704
Developed countries, 883
market capitalization of stock exchanges,
884
Diagonal model, 951
Digital options, 710
Direct quote, 803
Direct search markets, 63
Directional strategies, 928
Discount bonds, 460
Discount brokers, 76
Discounted cash flow (DCF), 599
commodity futures pricing, 824–825
problems with models, 621–622
Discretionary account, 76
Discriminant analysis, 471
Disposition effect, 400
Diversifiable risk, 206, 263
Diversification, 10, 93, 205–206, 210; see also International diversification
benefits from, 908–911
efficient diversification, 205
index model and, 262–264
insurance principle and, 206
misleading representation of benefits,
907–908
portfolio risk and, 206–207
power of, 226–228
residual risk in practice, 331
time diversification, 230
well-diversified portfolios, 329–331
Dividend discount models (DDM), 591,
595–608
constant-growth (Gordon model) DDM,
596–599
convergence of price to intrinsic value,
600
life cycles and, 603–607
multistage growth models, 608
P/E analysis and, 609–617
preferred stock and, 597
stock prices and investment opportunities,
600–603
Dividend payout ratio, 601
Dividend restrictions, 473
Dividend yield, 128
Dividends, 43
call option valuation, 727, 744–745
cumulative dividends, 43
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Subject Index
I-9
Fair game, 169, 201
Fair value accounting, 660–662
Fallen angels, 470
Fama and French (FF) three-factor model,
340–342, 426–428, 438–439
behavioral explanations, 431–432
CAPM vs., 428
momentum as fourth factor, 432–433
Fannie Mae; see Federal National Mortgage Association (FNMA or Fannie Mae)
Fast growers, 581
Fat-tailed distributions, 138, 141, 229
Fear gauge, 744
Fed funds (federal funds), 32
Federal agency debt, 35–36
Federal Deposit Insurance Corporation, 30
Federal funds, 32
Federal funds rate, 32, 565
Federal government policy, 563–566
fiscal policy, 563–564
monetary policy, 564–565
supply-side policies, 565–566
Federal Home Loan Bank (FHLB), 36
Federal Home Loan Mortgage Corporation
(FHLMC or Freddie Mac), 18, 36
Federal National Mortgage Association
(FNMA or Fannie Mae), 18, 36
Federal Reserve Board, monetary policy,
564–565
Feedback, 978
Feeder funds, 944
Fees
active management, 970–971
hedge funds, 943–946
mutual funds, 99, 101–102
Fibonacci numbers, 401
FIFO (first-in, first-out), 658
Financial Accounting Standards Board’s
Statement No. 157 (fair value
accounting), 660
Financial assets, 1–5
balance sheet of U.S. households, 2–3
derivatives, 3
domestic net worth, 3
efficient pricing of, 1
equity, 3
fixed-income or debt securities, 3–4
real assets vs., 2–3
separation of ownership/management,
6–7
Financial crisis of 2008, 2, 15–23
antecedents of, 15–18
changes in housing finance, 18–19
credit default swaps, 20, 477
Dodd-Frank Reform Act, 22–23
money market funds and, 34
mortgage derivatives, 19–20
option pricing and, 752–753
peak of crisis, 21–22
systemic risk, 20–21
European options, 682
Event study, 359–361
Event tree, 135
Excess returns, 129, 147, 330
risk premiums and, 129–130
Exchange rate, 559–560
Exchange rate risk, 888
futures and, 803–806
hedging exchange rate risk, 799–802
international investing and, 903–905
Exchange-traded funds (ETFs), 49, 103,
283, 357
advantages/disadvantages of, 106
growth of (1998–2011), 105
sponsors and products, 104–106
Exchange-traded notes (ETNs), 106
Execution, 978
Executive compensation, 7–8
Exercise, 51
Exercise price, 679
Exotic options, 709–710
Asian options, 709
barrier options, 709
currency-translated options, 709–710
digital options, 710
lookback options, 709
Expectations hypothesis, 497–498,
791–792
forward inflation rates, 498
futures pricing, 791
Expected holding-period return, 594
Expected return, 10n
arithmetic average and, 130–131
beta and, 260, 415
expected holding-period return, 594
expected vs. promised YTM, 474
expected return-beta relationship, 297
individual securities and, 295–297
portfolio expected return, investment
proportions, 212
realized return vs., 438–439
spreadsheet model for, 246, 249–250
standard deviation and, 128–129, 212
Expected shortfall (ES), 140, 186, 229
Expected spot prices, 791–793
Expected utility, 199–202
Expert network providers, 353
F Face value, 446
Facebook, 613
Factor betas, 326
Factor loadings, 326, 940–941
Factor models of security returns, 325–327;
see also Multifactor models Fama-French-type models, 426–433
Factor portfolio, 338
Fads hypothesis, 365
new trading strategies, 71–74
rise of, 68–69
Elliott wave theory, 403
Emerging market funds, 97
Emerging markets, 883–886
market capitalization of stock exchanges,
885
risk and return in, 900–903
Employment, 561
Endowment funds, 982
Enron, 8
Equally weighted indexes, 49
Equipment obligation bond, 473
Equities, 41
Equity, 4, 41–44; see also Equity securities levered equity, 707
Equity carve-outs, 396–397
Equity funds, 97
Equity premium, 436
behavioral explanations of, 441
consumption growth and market rates of
return, 435–437
expected vs. realized returns, 438–439
extensions to CAPM, 440–441
liquidity and, 441
survivorship bias, 439–440, 838n, 939
Equity premium pule, 436
Equity securities, 4, 41–44
asset allocation with, 215–219
characteristics of, 42
common stock as ownership shares, 41–42
depository receipts, 44
developed countries, 883
emerging markets, 883–886
global markets for, 883–886
home-country bias, 886
options vs. stock investments, 687–689, 691
pension investment strategies and, 1002
preferred stock, 43–44, 449–450
stock market listings, 42–43
synthetic stock positions, 808–809
Equity trusts, 95
Equity valuation models, 591–623
aggregate stock market, 622–623
comparison of models, 620–621
dividend discount model; see Dividend discount model (DDM)
free cash flow approaches, 617–620
intrinsic value vs. market price, 593–595
price-earnings analysis and DDM, 615
valuation by comparables, 591–593
Equivalent taxable yield, 37–39
ETFs; see Exchange-traded funds (ETFs) Ethics
CFA standards of professional conduct, 85
corporate ethics, 8
Eurobonds, 36, 450
Eurodollars, 31
European, Australasia, Far East (EAFE)
index, 916
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Subject Index
I-10
interest rate parity, 801–803
managing exchange rate risk, 803–806
the markets, 799–801
Foreign exchange swap, 815
Foreign and international stock market
indexes, 49–50
Foreign stocks; see Global equity markets Forward contract, 771
forward rates as, 504–506
futures pricing vs., 791
swaps vs., 820
synthetic forward loan, 505
Forward interest rate, 493–494
as forward contracts, 504
interest rate uncertainty and, 495–497
slopes of yield curve, 501
Framing, 391
Freddie Mac; see Federal Home Loan Mortgage Corporation (FHLMC or
Freddie Mac)
Free cash flow for the firm (FCFF), 617
Free cash flow to equityholders (FCFE), 617
Free cash flow valuation approaches,
617–622
comparisons of, 620–621
Free float, 48
Free-rider benefit, 189
Front-end load, 100
Full-covariance model; see also Markowitz portfolio optimization model
index model vs., 278–279
Full-service brokers, 76
Fully diluted earnings per share, 704
Fundamental analysis, 356, 557
Fundamental analysts, 591
Fundamental risk, 394
Funds of funds, 97, 944–946
Futures contracts, 53, 678, 771–775
basics of, 771–774
cash vs. actual delivery, 780
existing contracts, 774–775
leverage and, 782
maintenance margin, 778–779
margin account, 778–780
marking to market, 778–780
oil futures, 781
profit to buyers/sellers, 774
sample of, 775
as zero-sum game, 772
Futures markets, 770
arbitrage, 786–787
basis risk and hedging, 784–785
clearinghouse, 775–778
currency futures, 800
exchange rate risk, 803–806
foreign exchange futures, 799–806
forward vs. futures pricing, 791
future vs. expected spot price, 791–793
hedging and speculation, 781–784
return on equity, 642–644
statement of cash flows, 637–640
value investing (Graham technique), 665
Financial supermarkets, 99
Firm commitment, 61
Firm size, 426
Firm-specific risk, 206, 268
First-pass regression, 415
Fiscal cliff crisis, 564
Fiscal policy, 563–564
fiscal cliff and, 564
Fisher equation, 121
Fixed annuities, 999
Fixed-asset turnover, 649
Fixed asset turnover, 654
Fixed assets, 637
Fixed-income capital market, 4, 34; see also Bonds; Money market; Treasury bills
corporate bonds, 4, 39
federal agency debt, 35–36
inflation-protected Treasury bonds, 35
international bonds, 36
mortgages and mortgage-backed securi-
ties, 39–40
municipal bonds, 36–39
preferred stock, 43–44, 449–450
Treasury notes and bonds, 4, 34–35
U.S. market, 51
Fixed-income portfolio management; see Active bond management
Fixed-income securities, 3–4, 445; see also Bonds; Money market
collateralized loan, 705–706
convertible bonds, 448–449
corporate bonds, 447–448
floating-rate bonds, 449
innovations in, 450–452
international bonds, 450
levered equity and risky debt, 707
preferred stock, 449–450
puttable bonds, 449
Treasury bonds and notes, 446–447
Flat prices, 456
Flexible funds, 98
Floating-rate bonds, 449
Forced conversion, 704
Forecasting
active portfolio management, 958–962,
972
long horizons, 161
risk premium forecasts, 276–277
value of imperfect forecasting, 860
Forecasting errors, 390
Forecasting record, 959
Foreign bonds, 450
Foreign currency options, 684
Foreign exchange futures, 799–806
covered interest arbitrage, 801–803
direct vs. indirect quotes, 803
Financial engineering, 707–708; see also Derivative assets
Financial Industry Regulatory Authority
(FINRA), 84
Financial instruments
bond market, 34–41
derivative markets, 51–53
equity securities, 41–44
money market, 29–34
stock and bond market indexes, 44–51
Financial intermediaries, 11–14
investment bankers, 14–15
investment companies, 13
venture capital and private equity, 15
Financial leverage
business cycle sensitivity, 577
ROE and, 642–644
Financial markets
allocation of risk, 6
capital markets, 28
as competitive, 9–11
consumption timing, 5–6
corporate governance/ethics, 8
economy and, 5–8
as efficient, 10–11
financial intermediaries in, 12–14
firms as net borrowers, 11
government’s role in, 11
households as net savers, 11
information role of, 5
investment bankers and, 14–15
major players in, 11–15
money markets, 28033
risk-return trade-off, 10
separation of ownership and
management, 6–7
venture capital and private equity, 15
Financial ratio analysis; see Ratio analysis Financial Stability Oversight Council
(FSOC), 83
Financial statement analysis, 635–640
accounting vs. economic earnings,
636–637
balance sheet, 637
comparability problems, 658–664
depreciation, 659
economic value added, 644–645
fair value accounting, 660–662
financial leverage and ROE, 642–644
illustration of use, 655–657
income statement, 635–637
inflation and interest expense, 659–660
international accounting conventions,
663–664
inventory valuation, 658–659
measuring firm performance, 640–641
profitability measures, 641–645
quality of earnings, 662–663
ratio analysis; see Ratio analysis
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Subject Index
I-11
High-frequency trading, 71–73
High water mark, 944
High-yield bonds, 470
Historic returns on risky portfolios, 141–152
common stocks (1926–2012), 145–146
excess returns and risk premiums,
129–130
expected shortfall (ES), 140
global view of, 151–152
long-term investments, 152–157
portfolio returns, 148–151
U.S. T-bills, 125–127
value at risk (VaR), 139, 148–149
Holding-period return (HPR), 127–128,
168, 835
expected holding period return, 594
original-issues discount bonds, 468
simulation of long-term future rates of
return, 157–159
yield to maturity vs., 465–466
zero-coupon bonds, 492–493
Home-country bias, 886
Homogeneous expectations, 303
Honda Motor Co., 559
Horizon analysis, 463, 544–545
House money effect, 392
Housing finance, changes in, 18–19
Human capital, 994
insurance and, 994
Hurdle rate, 299
I Illiquidity, 310
asset pricing and, 433–434
average returns and, 312
Immunization, 535–541, 1001
cash flow matching and dedication, 542
constructing a portfolio, 540–541
holding period immunization, 539
pension liability, 536
problems of, 542–543
rebalancing portfolios, 539, 541
Implementation costs, 395
Implied volatility, 741, 743
In the money, 680
In-sample data, 266
Incentive fee, 927, 943
Black-Scholes valuation of, 944
funds of funds, 945
Income beneficiaries, 979
Income funds, 97
Income statement, 635–637
common-size income statement, 636
Home Depot example, 636
Indenture, 472; see Bond indentures Index arbitrage, 809–810
Index funds, 49, 98–99, 357
Growth opportunities, 142, 603
present value of growth opportunities
(PVGO), 603
H HealthSouth, 8
Hedge fund performance, 935–943
changing factor loadings, 940–941
liquidity and, 936–939
Long-Term Capital Management exam-
ple, 943
survivorship bias and, 939
tail events and, 941–943
Hedge fund strategies, 928–930
directional strategies, 928–929
long-short strategy, 928–929
market neutral, 929
market-neutral positions, 929
nondirectional strategies, 928–929
portable alpha, 931–933
pure play, 929–933
statistical arbitrage, 930
styles of, 928, 933–935
Hedge funds, 13, 96, 284, 926–927
compensation structure, 927–928
fee structure in, 943–946
investors in, 927
liquidity issues in, 927
mutual funds vs., 927–928
performance measurement for, 851–853
portable alpha, 931–933
risks of, 937
strategies of, 927
style analysis for, 933–935
transparency, 927
Hedge portfolios, 307–308
Hedge ratio, 746–747, 931
Black-Scholes formula and, 746–747
call option value, 747
dynamic hedging, 737
exchange rate risk, 804
Hedging, 799
basis risk and, 784
cross-hedging, 784, 815
dynamic hedging, 737
factor betas and, 326
futures markets and, 781–784
index futures and market risk, 810–812
interest rate risk, 813–815
long/short hedge, 783
mispriced options and, 754–758
oil futures, 782–783
speculation and, 755
Hewlett-Packard
explanatory power of SCL, 265–266
security characteristic line (SCL),
265–266
interest rate futures, 813–815
interest rate parity, 496
leverage and, 782
open interest, 775–778
presidential/other prediction futures, 776
regulations, 780–781
spot-futures parity theorem, 785–788
stock-index futures, 806–812
strategies for, 781–785
taxation, 781
trading mechanics, 775–781
Futures prices, 771, 785–791
contango, 792
cost-of-carry relationship, 787
expectations hypothesis, 791
expected spot prices vs., 791–793
forward vs., 791
future market arbitrage, 786–788
modern portfolio theory and, 792–793
normal backwardation, 792
parity and spreads (Excel), 790
spot-futures parity theorem, 785–788
spreads, 788–790
G Gambling, 169
Gamma, 759
General Electric, 6, 42
General obligation bonds, 36
Geometric (time-weighted) average return,
131–132
arithmetic average vs., 132
Ginnie Mae; see Government National Mortgage Association (GNMA or
Ginnie Mae)
Glass-Steagall Act, 14
Global Crossing, 8
Global economy, 558–560
Global equity markets, 74–76; see also International investing
developed countries, 883
domestic market capitalization
(2012), 75
emerging markets, 883–886
home-country bias, 886
market capitalization and GDP, 886
stock market return (2012), 558
Global funds, 97
Goodwill, 637
Government, in financial markets; see Federal government policy
Government National Mortgage Association
(GNMA or Ginnie Mae), 36
Great Moderation, 16
Gross domestic product (GDP), 561
market capitalization and, 886
Growth funds, 97
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Subject Index
I-12
Interest rate swap, 815–816
balance sheet restructuring, 816–817
other contracts, 818–819
swap dealer, 817
Interest rate swaps, 815
Interest rates; see also Term structure of interest rates
approximating the real rate, 119
determinants of level of, 118–121
equilibrium nominal rate, 120–121
equilibrium real rate, 119–120
inflation and (1926–2012), 127
macroeconomy and, 562
nominal rate, 118–119
real interest rate, 118–119
risk structure of interest rates, 475
taxes and, 121
yield curve and future rates, 490–492
Intermarket spread swap, 544
Internal rate of return (IRR), 299
International accounting conventions,
663–664
depreciation, 663
intangibles, 664
reserving practices, 663
International bonds, 36, 450
International diversification, 882
assessing potential of, 911–916
bear markets and, 909–911
benefits from, 905–911
developed countries, 883
efficient diversification, 205, 208
emerging markets, 883–886, 900–901
exchange rate risk, 888–891, 903–904
global equity markets, 883–887
home-country bias, 886–887
market capitalization and GDP, 886
misleading representation of benefits,
907–908
performance attribution and, 916–920
political risk, 891–895
risk factors in, 887–895
risk, return, and benefits from, 895–911
spreadsheet model for, 245
International financial reporting standards
(IRFS), 664
International funds, 97
International investing
cash/bond selection, 919
country selection, 919
currency selection, 918
emerging market risk, 900–903
Excel application, 919
exchange rate risk, 888–891
performance attribution and, 916–920
political risk, 891–895
risk factors in, 887–895
risk, return, and benefits of diversification,
895–911
stock selection, 919
Industry life cycles, 579–581
asset plays, 581
consolidation stage, 580
cyclicals, 581
fast growers, 581
maturity stage, 579–580
relative decline, 580–581
slow growers, 581
stalwarts, 581
start-up stage, 579–580
turnarounds, 581
Inflation, 561–562
interest expense and, 659–660
long-term investing and, 1004
real income and, 660
T-bills (1926–2012) and, 125–127
Inflation-protected Treasury bonds, 35
Inflation risk, 308, 993, 1004
Information
asymmetric information, 311
insider/outsider trading, 86, 353
leakage of, 360
on mutual funds, 110–112
role of financial markets, 5
Information processing, 389–391
conservatism, 390
forecasting errors, 390
overconfidence, 390
sample size neglect/representativeness,
391
Information ratio, 274–276, 840, 970
Initial public offerings (IPOs), 61–62
average first-day returns, 63
Input list, 223
Inside information, 86, 353, 370–371
Inside quotes, 66
Insider trading, 86
Insurance contracts, 203
Insurance principle, 206, 230–231
diversification and, 206
risk-pooling and, 230–231
risk-sharing and, 232–233
Intangible fixed assets, 637
Intangibles, 664
Intercommodity spread, 784n
Interest burden, 654
Interest-burden ratio, 646
Interest coverage, 654
Interest coverage ratio, 647
Interest expense, inflation and, 659–660
Interest rate futures, 813–815
hedging interest rate risk, 813–815
Interest rate options, 685
Interest rate parity relationship, 801–803,
891
Interest rate risk, 515–525, 993
bond prices and forward rates, 496
duration, 519–525
hedging of, 813–815
sensitivity to, 516–518
Index futures; see Stock-index futures Index model, 256, 415–421; see also Single-
index model
alpha betting, 283
betas and, 281–283, 419–421
CAPM and, 417
diversification and, 262–264
expected return-beta relationship,
415–416
full-covariance model vs., 278–279
industry version of, 279–282
as investment asset, 272–273
portfolio management with, 276–284
single-factor APT and, 415–421
single-factor security market, 257–259
tracking portfolios, 283–284
Index options, 684
Index portfolio, as investment asset,
272–273
Indexed bonds, 451–452
Indifference curve, 173, 183–185
Indirect quotes, 803
Individual investors, 979; see also Behav- ioral finance
deferred annuities, 998–999
human capital and insurance, 994
life cycles and, 97
managing portfolios of, 994–999
professional services or self-management,
995–997
residence as investment, 994
retirement planning models, 994–995
risk tolerance, 994
sample policy statements, 985–992
saving for retirement, 994
tax-deferral option, 997–998
tax-deferred retirement plans, 998
tax sheltering, 997
variable and universal life insurance, 999
Individual securities, expected returns on,
295–297
Industrial development bond, 36
Industrial production, 561
Industry, defined, 573–575
Industry analysis, 557, 571–582
bargaining power of buyers/suppliers, 582
business cycle sensitivity, 575–577
defining an industry, 573–575
degree of operating leverage (DOL),
576–577
industry cyclicality (example), 575
industry stock price performance (2012),
573
NAICS codes, 574
Porter’s competitive analysis, 581–582
rivalry between competitors, 582
sector rotation, 578
structure and performance, 581
substitute products, 52
threat of entry, 581–582
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Subject Index
I-13
“Siamese Twin” companies, 396
Leading economic indicators, 568–569
Leakage of information, 360
LEAPS (Long-Term Equity AnticiPation
Securities), 682
Lehman bankruptcy, 8, 21–22
Level of significance, 267
Leverage, 654
compound leverage factor, 647, 654
degree of operating leverage (DOL), 577
financial leverage and business cycle, 577
futures and, 782
operating leverage, 576
return on equity (ROE) and, 642–644
in risky asset, 180–181
Leverage ratios, 470, 647
Levered equity, 707
LIBOR; see London Interbank Offered Rate (LIBOR)
Life-cycles; see also Industry life cycles life-cycle funds, 97
multistage growth models, 603–607
LIFO (last-in, first-out), 658
Limit order book, 65
Limit orders, 65
Limited liability, 42
Liquidation value, 593
Liquidity, 21n, 60, 496n, 651, 654
asset pricing and, 433–435
CAPM and, 310–313
efficient market anomalies and, 368
equity premium pule, 436
hedge funds and, 927, 936–939
as investment constraint, 983
Liquidity beta, 313
Liquidity preference theory, 498–500
Liquidity premium, 496, 498
Liquidity ratios, 470, 651
Liquidity risk, 313, 993
Liquidity traders, 311
Load, 95
Loadings, 428, 934, 940–941
Lock-up periods, 927
Lock-ups, 96
Lognormal distribution, 153
London Interbank Offered Rate (LIBOR),
16, 32
LIBOR scandals, 33
TED spread (2000–2012), 16
Long hedge, 783
Long position, 53, 771
Long-Term Capital Management, 313, 943
Long-term investments, 152–161, 233–234
forecasts for, 161
inflation risk and, 1004
normal and lognormal distribution, 153–156
shortfall risk in short/long run, 155
simulation of future rates of return, 157–159
target date retirement fund (TDRF),
1003–1004
feedback, 978
individual investors, 979
investments for long run, 1003–1004
life insurance companies, 982
mutual funds, 981
non-life insurance companies, 982
objectives of, 978–979
pension funds, 981
personal trusts, 979–981
planning, 978
policy statements, 985–992
risk tolerance questionnaire, 979
Investment performance
business cycle sensitivity, 575–577
dollar-weighted return, 837
fees and mutual fund returns, 101–102
international investments, 916–920
mutual funds, 107–109
Investment policy statements, 985–992
components of, 986
governance, 987–991
investment, return, and risk, 986
risk management, 991–992
sample statement, 985–992
scope and purpose, 985–987
Investment process, 8–9; see also Portfolio management
asset allocation decisions, 9
bottom-up strategy, 9
security analysis, 9
security selection decision, 9
top-down portfolio construction, 9
Investor fear gauge, 743–744
Invoice price, 456
IPOs; see Initial public offerings Irrational exuberance, 374n
J Jensen’s alpha, 840
Joint normally distributed, 258
Junk bonds, 468, 470
K Kelly criterion, 203–204
Kondratieff waves, 403
Kurtosis, 139
L Labor income, 306–307, 421–423
Lagging indicators, 568
Latency, 71
Law of one price, 328, 395–398
closed-end funds, 397–398
equity carve-outs, 396–397
limits to arbitrate and, 395
International Securities Exchange, 681
Intertemporal capital asset pricing model
(ICAPM), 308, 435
Intrinsic value, 557, 594, 723
convergence of price to, 600
market price vs., 593–595
Inventory turnover, 654
Inventory turnover ratio, 650
Inventory valuation, 658–659
Inverse floaters, 450
Inverted yield curve, 488
Investable universe, 272
Investing; see Investment decisions; Invest- ment process
Investment, defined, 1
Investment adviser, 997
Investment bankers, 14–15
commercial banking vs., 14
Investment companies, 13, 92–93; see also Mutual funds
closed-end funds, 94–95
commingled funds, 95
functions performed, 92–93
hedge funds, 13, 96
managed investment companies, 94–95
open-end funds, 94
other organizations, 95–96
real estate investment trusts (REITs), 95
types of, 93–96
unit investment trusts, 93–94
Investment Company Act of 1940, 927
Investment Company Institute, 110
Investment constraints, 983–985
investment horizon, 984
liquidity, 983–984
matrix of constraints, 983
regulations, 984
tax considerations, 984
unique needs, 984–985
Investment decisions, 205; see also Long- term investments
banks, 9820983
endowment funds, 982
individual investors; see Individual investors
life insurance companies, 982
mutual funds, 96, 981
non-life insurance companies, 982
objectives, 978–979
pension funds, 981
personal trusts, 979–981
Investment environment, 1–2
Investment-grade bonds, 468
Investment horizon, 984
Investment management process, 978–983
banks, 982–983
components of, 980
constraints, 983–985
endowment funds, 982
execution, 978
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Subject Index
I-14
input list of, 257–258
optimal portfolios and nonnormal returns,
229
power of diversification, 226–228
security selection, 220–223
separation property, 223–226
Maturity stage, 580
Mean-beta relationship, 297
Mean-variance (M-V) criterion, 173
Memory bias, 390
Mental accounting, 391–392
Merrill Lynch Domestic Master index,
534
Microsoft, 7
Minimum-variance portfolio, 213, 225, 247
Model risk, 395
Modern portfolio theory, 10, 792–793
Modified duration, 521
Momentum, 401, 432–433
Momentum effect, 364
Monetary policy, 564–565
Money market, 4, 22n, 28–33
bankers’ acceptances, 31
brokers’ calls, 32
certificates of deposit (CD), 30
commercial paper, 30–31
credit crisis of 2008, 34
Eurodollars, 31
federal funds, 32
LIBOR market, 32
major components of, 29
repos and reverses, 31
Treasury bills, 29–30
yields on, 32–33
Money market funds, 96–97
Money spread, 695
Morningstar’s Mutual Fund Sourcebook, 110–111, 189
Morningstar’s risk-adjusted return (MRAR),
840, 847–850
Mortgage-backed securities; see Mortgages and mortgage-backed securities
Mortgage bond, 473
Mortgage trusts, 95
Mortgages and mortgage-backed securities,
39–40
adjustable-rate mortgage (ARMs), 19
callable corporate bonds vs., 532
cash flows in, 18
conforming/conventional mortgages, 18
derivatives, 19–20
duration and convexity of, 530–533
housing finance changes pre-2008 crisis,
18–19
nonconforming subprime mortgages, 19
price-yield curve, 531
securities outstanding (1979–2012), 40
Moving averages, 401–403
Multi-index model, 262
Multifactor benchmarks, 863–864
emerging markets, 885
Market capitalization rate, 595
Market conversion value, 449
Market efficiency; see also Efficient market hypothesis (EMH)
broad market returns, 365–366
bubbles and, 374–375
competition as source of, 351–353
lucky event issues, 363
magnitude issues, 362
market anomalies, 366–370
portfolio management in, 357–358
random walk and, 350–351, 414
returns over long horizons, 364–365
returns over short horizons, 364
selection bias issue, 362
semistrong tests, 366–370
strong-form tests, 354, 370–371
weak-form tests of, 364–365
Market index, 417–419
Market neutral, 929–931
Market-neutral bet, 811
Market portfolio ( M ), 292 risk premium of, 293–294
Market price ratios, 652–653
Market price of risk, 296
Market psychology; see Behavioral finance Market risk, 206, 993
hedging with index futures, 810–812
Market timing, 855–857
imperfect forecasting, 860
potential value of, 857–858
valuing as a call option, 858–859
Market-to-book, 654
Market-value-weighted index, 48
Markets
auction markets, 64
brokered markets, 64
as competitive, 9–11
dealer markets, 64
direct search markets, 63
efficient markets, 10–11
risk-return trade-off, 10
transparency of, 8
types of, 63–64
Markets and instruments
bond market, 34–41
derivative markets, 51–54
equity securities, 41–44
money market, 29–34
stock and bond market indexes, 44–51
Marking to market, 778–779
Markowitz portfolio optimization model,
220–229, 244, 256
asset allocation and security selection,
229
capital allocation and separation property,
223–226
drawbacks of, 256
index model vs., 278–279
Lookback options, 709
Lower partial standard deviation (LPSD),
140
Lucky event issue, 363
M M
2 measure of performance, 841–842
Macro factor model, 425–426
Macroeconomic analysis, 557
budget deficit, 562
business cycles, 566
demand shocks, 562–563
domestic macroeconomy, 560–562
economic indicators, 568–571
employment/unemployment rate, 562
federal government policy, 563–566
global economy, 558–560
gross domestic product (GDP), 561
inflation, 561–562
interest rates, 562
sentiment, 562
supply shocks, 562–563
Madoff scandal, 944–945
Magnitude issue, 362
Maintenance margin, 77, 181n, 778–779
Managed investment companies, 94–95
Manipulation-proof performance measure
(MPPM), 847–848
Margin, 21, 76–77, 181, 771, 778
buying on margin, 76–79
maintenance margin, 77, 181n
margin call, 77
options and, 683
turnover vs., 648
Margin account, 778
Margin call, 77, 181n
Mark-to-market accounting, 660–661
Market anomalies, 366–370; see also Efficient market hypothesis (EMH)
book-to-market ratios, 368–369
data mining, 373
inside information, 370–371
interpretation of, 371–374
mutual fund and analyst performance,
375–380
neglected-firm and liquidity effects, 368
P/E effect, 367
post-earnings-announcement price drift,
369–370
risk premiums vs. inefficiencies,
371–373
semistrong tests, 366–370
small-firm-in-January effect, 367–368
strong-form tests, 370–371
Market-book-value ratio (P/B), 652
Market cap, 141, 592
Market capitalization, 886
developed countries, 884
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Subject Index
I-15
Oil futures, 781–783
On-the-run yield curve, 490
Open-end funds, 94–95
Open-end investment companies, 92
Open interest, 777
Operating expenses, mutual funds, 99
Operating income, 636
Operating leverage, 576
Optimal complete portfolio, 218
determination of, 219
indifference curves and, 185
proportions of, 219
Optimal risky portfolio, 168, 205, 217, 277
alpha values and, 951–958
construction and properties of, 952
diversification and portfolio risk,
206–207
nonnormal returns, 229
optimization procedure, 276
short sales and, 249
single-index model, 273–274
spreadsheet (Excel) model, 224,
248–249
two risky assets, 208–219, 221
Optimization
portfolio optimization, 336–338,
966–968
summary of procedure, 276
Option contract, 679–685
Option elasticity, 747
Option-like securities, 701–707
callable bonds, 701
collateralized loans, 705–706
convertible securities, 701–704
levered equity and risky debt, 707
warrants, 704–705
Options, 51–52, 678
adjustments in contract terms, 682–683
American options, 682–683
call options; see Call options collars, 695–698
covered calls, 691–693
European option, 682–683
exercise (strike) price, 51
exotic options; see Exotic options foreign currency options, 684
futures options, 684
index options, 684
interest rate options, 685
levered equity and risk debt, 707–708
other listed options, 683–685
protective put, 689–691
put-call parity relationship, 698–700
put options; see Put options spreads, 694–695
stock investments vs., 687–689, 691
straddles, 694–695
strategies, 689–698
trading options, 681–682
values at expiration, 685–690, 693
survivorship bias, 439–440, 838n, 939
taxation of income on, 103
12b-1 charges, 100–101
N NAICS codes (North American Industry
Classification System), 574–575
Naked puts, 686
Naked short-selling, 80
NASDAQ Stock Market (NASDAQ), 67, 70
level of subscribers, 70
Negative convexity, 528
Negative correlation, 210
Neglected-firm effect, 368
Net asset value (NAV), 93
New York Mercantile Exchange, 5
New York Stock Exchange (NYSE), 71
market share of trading, 70
No-arbitrage equation, 333
Noise trades, 311
Nominal interest rate, 118
equilibrium nominal rate of interest,
120–121
Non-life insurance companies, 982
Non-normal distributions
conditional tail expectation (CTE), 140
expected shortfall, 140
lower partial standard deviation (LPSD),
139
relative frequency of large, negative
3-sigma returns, 140–141
Sortino ratio, 140
value at risk (VaR), 139, 141
Nonconforming ”subprime” loans, 19
Nondirectional strategies, 928
Nondiversifiable risk, 206
Nonnormal returns, 186–187, 229
Nonrecurring items, 662
Nonsystematic risk, 206
Nontraded assets, 306–307, 423–424
Normal backwardation, 792
Normal distribution, 135–138; see also Non- normal distributions
deviations from, 137–141
in Excel, 137
fat-tailed distributions, 138
skewed distributions and, 138
Normal and lognormal returns, 153–156
Normative testes, 302
North American Industry Classification Sys-
tem (NAICS) codes, 574–575
Notional principal, 815
O Objectives, 978–979
Off-balance-sheet assets and liabilities, 663
Multifactor CAPM
APT and, 421–426
early versions of, 424–425
labor income, 421–423
macro factor model, 425–426
test of, 421–426
Multifactor models, 324, 326
factor models of security returns,
325–327
Fama and French (FF) three-factor model,
340–342
mispricing and arbitrage, 339
multifactor ATP, 338–340
multifactor security market line, 339
overview of, 325–327
risk assessment using, 327
Multiplier, 708
Multistage growth models, 608
Excel application, 608
life cycles and, 603–607
Multivariate normal distribution, 258
Municipal bonds, 36–39
equivalent taxable yield, 37
general obligation bonds, 36
industrial development bond, 36
ratio of yields vs. corporate Baa-rated
debt (1953–2013), 38
revenue bonds, 36
tax anticipation notes, 36
tax-exempt debt outstanding (1979–
2012), 37
Mutual fund managers, 376–380
Mutual fund theorem, 293
Mutual funds, 13, 92, 96–99, 189, 981
analyst performance, 375–380
asset allocation and flexible funds, 98
back-end load, 100
balanced funds, 97
bond funds, 97
costs of investing in, 99–102
equity funds, 97
exchange-traded funds (ETFs), 103–107
fee structure, 99, 101–102
front-end load, 100
hedge funds vs., 926–928
how sold, 99
index funds, 98–99
information on, 110–112
international funds, 97
investment classification, 98
investment performance, 107–109
investment policies, 96–97
money market funds, 96–97
net asset value (NAV), 93
operating expenses, 99
returns and fees, 101–102
sector funds, 97
soft dollars, 102
stock pickers, 189
strategies of, 927
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Subject Index
I-16
Portfolio opportunity set, 214
Portfolio performance evaluation; see also Historic returns on risky portfolios
alpha and, 844–845
average rates of return, 835–836
changing portfolio composition, 854
dollar-weighted returns, 836–837
equalizing beta, 844
example of, 846–847
Excel example, 838, 845
hedge funds, 851–853, 935–943
information ratio, 840
international investing, 916–920
Jensen’s measure, 840
M 2
measure of performance, 841–842
market timing, 855–859
Morningstar’s risk adjusted return, 840,
847–850
mutual funds, 107–109
performance attribution procedures,
864–870
realized return vs. expected return,
850–851
risk-adjusted measures, 837–840
Sharpe’s measure, 840, 842
style analysis, 861–863
time-weighted returns, 836–837
Treynor’s measure, 840
value of imperfect forecasting, 860
Portfolio risk
diversification and, 206–207
number of stocks in portfolio, 207
performance measurement and, 854
return and, 211
Portfolio statistics, 249–255
correlation coefficient, 253–254
covariance, 252–253
expected returns, 249–250
portfolio rate of return, 250
portfolio variance, 254–255
review of, 249–255
variance and standard deviation, 132–133,
251–252
Portfolio theory; see also Capital allocation option pricing and, 753–754
risk and risk aversion, 168–174
Portfolio variance
border-multiplied covariance matrix, 209
bordered covariance matrix, 209, 246
spreadsheet model, 246, 254–255
Positive tests, 302
Post-earnings-announcement price drift,
369–370
Posterior distribution, 958
Prediction markets, 775–776
Preferred stock, 43–44, 449–450
cumulative dividends, 43
DDM and, 597
Premium, 679; see also Equity premium; Risk premiums
rebalancing, 441, 539
Passive investment strategy, 357, 515
Passive management, 11
Passive market index portfolio, 952
Passive portfolio, 272, 274
Passive portfolio management, 356–357
Passive strategy, 187
active vs., 356–357
capital market line and (CML), 187–190
as efficient, 293
index funds, 189
Peak, 566
PEG ratio, 611
Pension funds, 981, 1000–1002
defined benefit plans, 1000–1001
defined contribution plans, 1000
equity investments, 1002
immunization, 536
investment strategies, 1001
Pension investment strategies, 1001
Performance attribution procedures,
864–870
asset allocation decisions, 867
cash/bond selection, 919
component contributions, 870
country selection, 919
currency selection, 918
Excel application for, 869, 919
international investing and, 916–920
sector and security selection, 868
stock selection, 919
Performance measurement; see Portfolio performance evaluation
Personal trusts, 979–981
Planning, 978
Plowback ratio, 601
Political risk, 892, 993
Ponzi scheme, 945
Portable alpha, 931–933
Portfolio, 8
complete portfolio, 176
hedge portfolios, 307–308
market portfolio ( M ), 292–294 minimum-variance portfolio, 213, 225,
247
one risky and a risk-free asset, 178–181
portfolio returns, 148–151
replicating portfolio, 729
tracking portfolio, 283–284, 338
two risky assets, 208–214
well-diversified portfolio, 329–331
Portfolio insurance, 691, 748–752
Portfolio management; see also Active port- folio management
index model and, 278–284
individual investors, 994–999
investment decisions; see Investment decisions
organizational chart for, 961–962
role in efficient market, 357–358
Options Clearing Corporation (OCC), 683
Options valuation, 722–725
binomial option pricing, 729–737
Black-Scholes model, 737–746
assumptions of, 741
dividends and call/put options,
744–746
Excel model for, 742, 748
formula for, 737–743
hedge ratios and, 746–747, 750
implied volatility, 741
portfolio insurance, 748–751
pseudo-American call option value,
745
put option valuation, 745
synthetic protective put options,
749–750
call option, 744
determinants of, 723–725
empirical evidence on, 758–759
financial crisis of 2008–2009, 752
hedging bets on mispriced options,
754–758
intrinsic and time values, 722–723
portfolio theory, 753–754
protective put strategy, 749
restrictions on option values, 725–728
call option, 726–727
early exercise of American puts, 728
early exercise and dividends, 727
two-state option pricing, 729–731
Order types, 64–66
limit orders, 65
market orders, 64–65
price-contingent orders, 65–66
stop orders, 66
Organizational structure and performance,
961–962
Original-issue discount (OID) bonds, 467
OTC (over-the-counter) market, 67, 681
Out of the money, 680
Out-of-sample data, 266
Overconfidence, 390
Over-the-counter (OTC) market, 67
P P-value, 267
P/E effect, 367
P/E ratio; see Price-earnings (P/E) ratio Pairs trading, 72, 930
Par value, 446
Parmalat, 8
Participation rate, 708
Pass-throughs, 18, 530
Passive bond management, 533–543
bond-index funds, 534
cash flow matching and dedication, 542
immunization, 535–541
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Subject Index
I-17
long-term debt, 471
major industry groups, 655
margin vs. turnover, 648
market price ratios, 652–654
market valuation statistics (1955–2005),
617
price-to-book ratio, 616, 652
price-to-cash-flow ratio, 616
price-to-sales ratio, 616
profitability, 654
summary of key ratios, 654
turnover/other utilization ratios, 649–650
Real assets, 2
financial assets vs., 2–3
Real estate investment trusts (REITs), 95
Real interest rate, 118
equilibrium real rate of interest, 119–120
taxes and, 121
Realized compound return, 462
yield to maturity vs., 461–463
Realized return, expected return vs.,
438–439
Rebalancing, 8, 539, 541
Redeemable trust certificates, 94
Refunding, 448
Regional funds, 97
Regression equation, single-index model,
259–260
Regret avoidance, 392
Regulation of securities market, 83–86; see also Securities and Exchange Commission (SEC)
futures market, 780–781
insider trading, 86
as investment constraint, 984
prudent investor rule, 980, 984
Sarbanes-Oxley Act, 84–86
self-regulation, 84
Reinvestment rate risk, 463
Relative decline stage, 580
Relative strength, 355, 403
Remaindermen, 979
Replacement cost, 593
Replicating portfolio, 729
Repos (repurchase agreements), 31
Representativeness bias, 391
Repurchase agreements (repos), 31
Reserving practices, 663
Residual, 260
Residual claim, 42
Residual income, 644
Residual risk, 331
Resistance levels, 355
Resource allocation, 358–359
Retirement assets; see Individual investors; Pension funds
Retirement planning models, 995
Retirement savings, 994
Return on assets (ROA), 470, 641, 647, 654
Return on capital (ROC), 641
Put bond, 449
Put-call parity relationship, 698–700
Put-call parity theorem, 699
Put options, 51, 680
Black-Scholes put valuation, 745–746
dividends and, 746
early excise of American put, 728
naked puts, 686
profits and losses on, 680
protective put, 689–691
synthetic protective put, 749–750
values at expiration, 686
Put/call ratio, 405
Puttable bonds, 449
PV factor, 453
Q Quanto, 709
Quick ratio, 470, 651, 654
Quoted bond prices, 447
Qwest Communications, 8
R Random walk, 350–351, 414
Rate anticipation swap, 544
Rate of return; see also Historic returns on risky portfolio
annual percentage rates (APRs), 123
annualized rates of return, 122, 144
consumption growth and market rates of
return, 435–437
continuous compounding, 123–125
different holding periods, 122–125
effective annual rate (EAR), 122–123
expected returns and arithmetic average,
130–131
future research, 160–161
geometric (time-weighted) average
return, 131–132
hedged stock portfolio, 786
portfolio rate of return, 250
required rate of return, 299
reward-to-variability (Sharpe) ratio,
134–135, 180
time series analysis, 130–135
time series vs. scenario analysis, 130
Ratio analysis, 645–655; see also Price- earnings (P/E) ratio
asset utilization, 649–650, 654
benchmark for, 653–655
comparative valuation ratios, 615–617
decomposition of ROE, 645–648
as default risk predictors, 470–471
economic value added, 644–645
leverage, 647, 654
liquidity ratios, 651, 654
liquidity premium, 496
Premium bonds, 459
Present value of growth opportunities
(PVGO), 602
Price-contingent orders, 65–66
Price-earnings multiple, 609
Price-earnings (P/E) ratio, 43, 609–617,
652, 664
business cycle, 614
DDM and, 615
earnings management and, 613–614
effect of ROE and plowback, 610
growth opportunities and, 609–612
growth rate vs. (PEG ratio), 611
industry comparisons, 616
pitfalls in analysis, 612–615
stock risk and, 612
Price-earnings ratio, 654
Price-to-book ratio, 616, 652
growth options and, 652
Price-to-cash-flow ratio, 616
Price-to-sales ratio, 616
Price value of a basis point (PVBP), 813
Price-weighted average, 44
Primary market, 14, 59, 64
Prior distribution, 958
Private corporations, 59
Private equity, 15
Private (nontraded) business, 423–424
Private placement, 60
Privately held firms, 60
Pro forma earnings, 614
Probability of loss, 231
Profit margin, 646, 648, 654
Profitability, 654
Profitability measures, 641–645
economic value added, 644–645
financial leverage and ROE, 642–644
key financial ratios of, 470
return on assets (ROA), 641
return on capital (ROC), 641
return on equity (ROE), 642
Program trading, 810
Prospect theory, 393–394
Prospectus, 61
Protective covenants, 472
Protective put, 689–691
stock investment vs., 691
Proxy, 41
Proxy contest, 7
Prudent investor rule, 980, 984
Pseudo-American call option value, 745
Publicly listed firms, 59
Publicly traded companies, 60–61
Purchasing power risk, 993
Pure plays, 284, 929, 933
examples of, 931–933
risks of, 933
Pure yield curve, 489
Pure yield pickup swap, 544
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Subject Index
I-18
Salomon Broad Investment Grade (BIG)
Index, 534
Sample size neglect, 391
Samurai bonds, 450
Sarbanes-Oxley Act, 8, 84–85
Saving, 9
Scatter diagram, 265
Scenario analysis, time series vs., 130
Seasoned equity offering, 61
Second-pass regression, 415
Secondary market, 15, 59
Sector decisions, 868
Sector funds, 97
Sector rotation, 578
Secured bonds, 39
Securities; see also Convertible securities; Equity securities
how firms issue, 59–63
initial public offerings, 61–63
investment banking and, 61
private placements, 60
shelf registration, 61
trading; see Securities trading Securities Act of 1933, 83, 927
Securities Exchange Act of 1934, 83, 354
Securities and Exchange Commission
(SEC), 9n, 83
EDGAR website, 591
insider trading, 354, 371
Investment Adviser Public Disclosure
(IAPD), 997
Office of Credit Ratings, 23
Official Summary of Securities Transactions and Holdings, 86, 371
prospectus disclosure, 61
public offerings, 61
Regulation NMS (National Market
System), 68
Rule 10b-5, 354
Rule 144A (private placements), 60
Rule 415 (shelf registration), 61
12b-1 charges, 100–101
Securities Investor Protection Act of 1970, 83
Securities Investor Protection Corporation
(SIPC), 83
Securities markets; see also Markets and instruments
bond trading, 73–74
as competitive, 9–11
electronic communication networks
(ECNs), 67, 71
globalization and consolidation of,
74–76
insider trading, 86
Nasdaq, 67
New York Stock Exchange, 70–71
regulation of, 83–86
Sarbanes-Oxley Act, 84–86
self-regulation of, 84
unique risk, 206
value at risk (VaR), 139, 141, 148
volatility risk, 758
Risk arbitrage, 328
Risk averse, 170
Risk aversion, 129, 168–174, 199–202
estimation of, 174
expected utility, 199–204
speculation and, 169
utility values and, 170–174
Risk-free asset, 177–178
risky asset and, 178–181
Risk-free portfolios, capital allocation
across, 175–176
Risk-free rate, 129, 159–160
Risk lovers, 172
Risk management
derivatives and, 691–692
multifactor models in, 327
Risk-neutral, 172
Risk-neutral economy, 736
Risk-neutral investors, 172
Risk pooling, 230–234
insurance principle and, 230–231
investment for the long run,
233–234
Risk premiums, 129, 168
excess returns and, 129–130
forecasts of, 276–277
holding-period returns, 127–130
inefficiencies vs., 371–373
market portfolio, 293–294
Risk-return trade-off, 1, 978
emerging markets, 900–903
international investing, 896–900
Risk sharing, 232–233
Risk structure of interest rates, 475
Risk tolerance, 174, 978
asset allocation and, 182–187
questionnaire for, 174–175
Risk tolerance questionnaire, 979
Risky assets
capital allocation and, 168, 175–176
efficient frontier of, 220–222
levered equity and risky debt, 707
risk-free asset and, 178–181
two-security Excel model, 221
Risky debt, 707
Risky portfolios, 177; see also Optimal risky portfolio
Rite Aid, 8
Rivalry between existing competitors, 582
Road shows, 61–62
Roll’s critique, 417–418
S Safe investing; see Bond safety St. Petersburg Paradox, 199–202
Return on equity (ROE), 470, 642, 653–654
decomposition of, 645–648
financial leverage and, 642–644
industry comparison (2012), 572
major software development firms
(2012), 574
Return requirement, 978
Return on sales, 654
Return on sales (ROS), 646
Return on shareholder equity, 605
Revenue bonds, 36
Revenue recognition, 662
Revenue sharing, 99
Reversal effect, 365
Reverse repo, 31
Reward-to-volatility ratio, 134–135, 180; see also Sharpe ratio
Risk; see also Portfolio risk adjusting returns for, 837–840
allocation of, 6
basis risk, 1003
benchmark risk, 956–957
counterparty risk, 16, 800
credit risk, 468, 477–478
default risk; see Default risk deviations from normality, 137–141
diversifiable risk, 206, 263
exchange rate risk, 888
firm-specific, 206, 268
fundamental risk, 394
hedge funds, 937
inflation risk, 308
interest rate risk; see Interest rate risk international investing and, 887–895
liquidity risk, 313, 434, 993
market price of risk, 296
market risk, 206, 811, 993
model risk, 395
Morningstar’s risk-adjusted rating, 840,
847–850
nondiversifiable risk, 206
nonsystematic risk, 206
political risk, 993
portfolio risk; see Portfolio risk purchasing power risk, 993
pure play, 933
reinvestment rate risk, 463
risk aversion and, 168–174
risk premiums; see Risk premiums risk-return trade-off, 10
shortfall risk, 155
single-index model, 260
societal risk, 993
speculation and gambling, 169
stock risk, 612
systematic risk, 206, 258–259
systemic risk, 15, 20–21
tail risk, 148
time vs., 160
uncertainty vs., 187
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Subject Index
I-19
Societal risk, 993
Soft dollars, 102
Sortino ratio, 140
Special purpose entities, 8
Specialist, 67
Specialist markets, 67
Speculation, 169
on the basis, 784
hedging and, 781–784
mispriced options, 755
oil futures, 781
on the spread, 785
Speculative-grade bonds, 468
Spot-futures parity theorem, 785–788
Spot rate, 491, 493, 495
Spreads, 694–695
bid-asked spread, 29
bullish spread, 695, 697
calendar spread, 784
Excel application, 694, 790
futures, 788–790
parity and, 790
speculating on, 785
spread pricing, 789
Stalwarts, 581
Standard & Poor’s 500 index, 48–49
cumulative returns (1980–2013), 17
earnings per share vs. (1995–2011), 561
implied volatility, 743, 759
as market-value-weighted index, 48
monthly dividend yield (2001–2013), 788
Standard deviation
expected return and, 128–129
higher-frequency observations, 133–134
historical record of returns, 147, 150
lower partial standard deviation
(LPSD), 140
minimum-variance portfolio, 213, 225,
247
variance and, 132–133, 251–253
Start-up stage, 579–580
Statement of cash flows, 637–640
Home Depot example, 639
Stochastic volatility models, 743
Stock exchange, 71
Stock-index futures, 806–812
contracts, 806–807
hedging market risk, 810–812
index arbitrage, 809–810
major contracts, 807
synthetic stock positions, 808–809
Stock market
aggregate stock market, 622–623
globalization of, 74–76, 558
listings, 42
predictors of broad market returns,
365–366
as self-regulating, 84
Stock market analysts, 375–376
Stock market bubble, 398
Semistrong-form hypothesis, 354
Sentiment, 562
Sentiment/sentiment indicators, 404–405
confidence index, 405
put/call ratio, 405
trin statistic, 404
Separation of ownership and management,
6–7
Separation property, 223–226
Serial bond issue, 473
Serial correlation, 364
Shareholders’ equity, 637
Sharpe ratio, 134–135, 137, 180, 296,
840–841
hedge funds and, 853
historical record of, 147
overall portfolios, 215–216, 842
Shelf registration, 61
Short hedge, 783
Short position, 53, 771
Short rate, 491
Short sales, 66, 80–84
cash flows from purchasing vs., 81
example of, 81–82
Excel application, 80
margin calls on, 82
optimal risky portfolio and, 220, 249
“Siamese Twin” companies, 396
Significance level, 303n
Single-index model, 258–264, 325
analysis of variance, 266–267
correlation and covariance matrix,
268–271
diversification and, 262–264
estimate of alpha, 267–268
estimate of beta, 268
estimates needed for, 261–262
estimating the model (example),
264–271
expected return-beta relationship, 260
firm-specific risk, 268
portfolio construction and, 271–278
alpha and security analysis,
271–272
example of, 276–278
information ratio, 274–276
input list, 271, 273
optimal risky portfolio, 273–274, 277
optimization summary, 276
risk premium forecasts, 276–277
regression equation of, 259–260
risk and covariance in, 260
security characteristic line, 265–266
Single-stock futures, 774
Sinking funds, 472–473
Skew, 138
Skewed distributions, 138
Slow growers, 581
Small-firm effect, 367
Small-firm-in-January effect, 367–368
trading mechanisms, 66–67
types of markets, 63–64
types of orders, 64–66
U.S. securities markets, 69–71
Securities trading, 59, 63–67
algorithmic trading, 72
auction markets, 64
block sales, 73
bond trading, 73–74
brokered markets, 64
dark pools, 73
dealer markets, 64, 67
direct search markets, 63
electronic communication networks
(ECNs), 67
electronic trading, 68–69
globalization and consolidation, 74–76
high-frequency trading, 72–73
insider trading, 86
margin buying, 76–79
market orders, 64–65
new trading strategies, 71–74
over-the-counter (OTC) market, 67
price-contingent orders, 65–66
program trade, 810
regulation of markets, 83–86
self-regulation, 84
short sales, 66, 80–83
specialists markets, 67
stop orders, 66
trading costs, 76
trading mechanisms, 66
types of markets, 63–64
types of orders, 64–66
Securitization, 18
Security analysis, 9
alpha and, 271–272
Security Analysis (Graham & Dodd), 665 Security characteristic line (SCL)
analysis of variance, 266–267
estimation of, 415–416
explanatory power of, 265–266
Hewlett-Packard example, 265
Security market line (SML), 298–300
arbitrage and, 327
estimation of, 416
multifactor SML, 339
positive-alpha stock and, 299
Security returns
empirical evidence on, 414–415
factor models of, 325–327
Security selection, 9, 28, 205
asset allocation and, 229
efficient frontier of risky assets, 220
Markowitz portfolio selection model,
220–223
minimum-variance frontier, 220
Security selection decisions, 868
Selection bias issue, 362
Self-regulation, 84
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Subject Index
I-20
Term repo, 31
Term structure of interest rates, 467, 487,
497–500
expectations hypothesis, 497–498
forward inflation rates and, 498
interpretation of, 501–504
liquidity preference, 498–500
theories of, 497–499
uncertainty and forward rates, 495–497
Threat of entry, 581
Tick size, 35
Tick sizes, 68
Time, risk vs., 160
Time deposit, 30
Time diversification, 230
Time series analysis
expected returns and arithmetic average,
130–131
geometric (time-weighted) average
return, 131–132
mean and standard deviation from higher-
frequency observations, 133–134
past rates of return, 130–135
reward-to-volatility (Sharpe) ratio,
134–135
scenario analysis vs., 139
variance and standard deviation, 132–133
Time spread, 695
Time value, 723
Time-weighted average, 836
Time-weighted returns, 836
Times interest earned, 647, 654
Timing, 156
Timing problem, 933n
Timing risk, 993
TIPS (Treasury Inflation-Protected Securi-
ties), 35, 119, 451, 1004
Tobin’s q, 593
Top-down portfolio construction, 9
Total asset turnover (ATO), 646, 654
Tracking error, 840, 956
Tracking portfolios, 283–284, 338
Trading costs, 76
Trading mechanisms, 66–67; see also Secu- rities trading
dealer markets, 67
electronic communication networks
(ECNs), 67
futures markets, 771–781
options trading, 681–682
specialist markets, 67
Tranches, 19, 478, 532
Transparency, 927
Treasury bills (T-bills), 4, 29–30
asset allocation with, 215–219
bank-discount method, 30
bid-asked spread, 29
bond-equivalent yield, 30
inflation and (1926–2012), 125–127
intermarket spread swap, 544
other interest rate contracts, 818–819
pricing of, 819–820
pure yield pickup swap, 544
rate anticipation swap, 544
substitution swap, 543–544
swap dealer, 817–818
tax swap, 544
Synthetic protective put options, 749–750
Synthetic stock positions, 808–809
Systematic risk, 206, 260
normality of returns and, 258–259
Systemic risk, 15, 21
financial crisis of 2008, 20–21, 477
T Tail events, 941–943
Tail risk, 148
Tangible fixed assets, 637
Target-date retirement fund (TDRF), 1003
Targeted-maturity funds, 97
Tax anticipation notes, 26
Tax-burden ratio, 646
Tax-deferral option, 997
Tax-deferred retirement plans, 998
Tax Reform Act of 1986, 121
Tax sheltering, 997–999
deferred annuities, 998–999
tax-deferral option, 997–998
tax-deferred retirement plans, 998
variable and universal life insurance, 999
Tax swap, 544
Taxes
asset allocation and, 992–993
futures and, 781
investment considerations and, 984
mutual fund income, 103
OID bonds, 467
real rate of interest and, 121
supply-siders and, 566
tax sheltering, 997–999
taxable vs. tax-exempt yields, 99
Technical analysis, 354, 388, 400–406
behavioral finance and, 389, 400
breadth, 403–404
cautions on use of, 405–407
confidence index, 405
implications of EMH, 354–356
momentum and moving averages,
401–403
put/call ratio, 405
relative strength, 355, 403
resistance/support levels, 355
sentiment indicators, 404
trends and corrections, 400
trin statistic, 404–405
TED (Treasury-Eurodollar) spread, 16
Stock market indexes, 44–50
Dow Jones averages, 44–47
equally weighted indexes, 49
foreign and international indexes, 49–50
market-value-weighted indexes, 48
other U.S. market-value indexes, 49
price-weighted average, 44
Standard & Poor’s indexes, 48–49
value-weighted index, 48
Stock prices
industry stock price performance
(2012), 573
investment opportunities and, 600–603
random walk argument, 350–351, 414
Stock returns
over long horizons, 364–365
over short horizon, 364
Stock selection, 919
Stockholders’ equity, 637
Stop-buy orders, 66
Stop-loss orders, 66
Stop orders, 66
Storage costs, 822–823
Straddles, 694–696
Straight bond, 701
Straps, 695
Strike price, 51, 679
Strips, 695
Strong-form tests, 354, 370–371
Structured debt obligations (CDOs), 477
Structured Investment Vehicle (SIV), 477
Structured products, 19n
Style analysis, 861–864
in Excel, 864
Fidelity’s Magellan Fund (example),
861–862
hedge funds, 933–935
multifactor benchmarks, 863–864
Style drift, 927
Style portfolios, 428
Subordinate debentures, 39
Subordination clauses, 473
Subprime mortgages, 40
Substitute products, 582
Substitution swap, 543
Supply shocks, 562–563
Supply-side policies, 565–566
Support levels, 355
Survivorship bias, 439–440, 838n, 939
Swap dealer, 817
Swap pricing, 819–820
Swaps, 815–821
balance sheet restructuring and, 816–817
bond swap, 543
credit default swaps (CDS), 475–477, 821
credit risk in, 820–821
foreign exchange swap, 815
forward contract vs., 820
interest rate swap, 815–816
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Subject Index
I-21
Warrants, 704–705
Weak-form hypothesis, 353
Wealth index, 126
Well-diversified portfolios, 329–331
Whole-life insurance policy, 982
Workout period, 543
WorldCom, 8
Y Yahoo!, 7, 110, 571
Yankee bonds, 450
Yield curve, 487–490
bond pricing, 488–490
forward rates, 493–494
future interest rates and, 490–492
holding-period returns, 465–466,
492–493
inverted yield curve, 488
on-the-run yield curve, 490
pure yield curve, 489
slopes of, 501, 503
spot and forward yields, 495
Treasury yield curves, 488
under certainty, 490–492
Yield to call, 460–461
Yield to maturity (YTM), 35, 458–460
default risk and, 474–475
Excel function for, 459
expected vs. promised YTM, 474
holding period return vs., 465–466
realized compound return vs., 461–463
Yields; see also Bond yields bank-discount method, 30
bond-equivalent yield, 30, 35
equivalent taxable yield, 37–38
money market instruments, 32–33
spot and forward yields, 495
spread between 10-year vs. 90-day
Treasury, 504
taxable vs. tax-exempt, 39
Z Zero-beta model, 305–306
Zero-coupon bonds, 122, 446, 466–467,
492–493
Zero-sum game, 772
expected utility, 199–202
insurance contracts, 203
prospect theory and, 393
Utility scores, 170
evaluating investments by, 171
risk aversion and, 170–174
V Value at risk (VaR), 139, 141, 148, 186, 229
Value investing, Graham technique, 665
Value Line Investment Survey, 575 Value stocks, 112
Value-weighted indexes, 48
Vanguard Group, 897
Variable annuities, 999
Variable life, 982, 999
Variance
portfolio variance, 254–255
of risky asset in Excel, 251
standard deviation and, 132–133,
251–252
Vega, 758
Venture capital (VC), 15
Views, 962–963
Volatility risk, 758
Volcker Rule, 14, 22
W WACC (weighted average after-tax cost of
debt), 618–620
Wall Street Journal Online closed-end mutual funds, 94
corn futures on CBOT, 53
Eurodollar futures, 818
foreign exchange futures, 801
futures listings, 773
market diaries: volume, advancers,
decliners, 404
spot/forward prices in foreign exchange,
800
stock market listings, 43
stock options on IBM, 52, 681
Treasury bill yields, 30
Treasury bonds and notes, 35
Treasury issues, 488
Treasury yield curve, 488
nominal and real wealth indexes (1926–
2012), 126
yields on, 30
Treasury bonds; see Treasury notes and bonds
Treasury-Eurodollar (TED) spread, 16
Treasury Inflation-Protected Securities
(TIPs)), 35, 119, 451
Treasury notes and bonds, 34–35, 122,
446–447
inflation-protected Treasury bonds, 35,
119
yield to maturity, 35
Treasury strips, 466–467
Trends, 400
Treynor-Black model, 336
Black-Litterman vs., 968–970
distribution of alpha values, 960–961
forecast precision of alpha, 958–959
organizational structure and performance,
961–962
Treynor’s measure, 840
Trin statistic, 404
Trough, 566
Turnarounds, 581
Turnover, 103, 648, 650
12b-1 charges, 100–101
Two-state option pricing, 731–732
Type I error, 303n, 419n
Type II error, 303n, 419n
U Uncertainty, 187
Underwriters, 14, 61
Underwriting syndicate, 61
Unemployment rate, 561
Uniform Securities Act, 83
Unique needs, 984–985
Unique risk, 206
Unit investment trusts, 93–94
Units, 95
Universal banks, 14
Universal life, 982, 999
Unmanaged trusts, 94
Unsecured bonds, 473
Utility, 170
Utility function
allocation to risky asset, 183
bod61671_sidx_I4-I22.indd 21bod61671_sidx_I4-I22.indd 21 8/2/13 7:44 AM8/2/13 7:44 AM
Final PDF to printer
- Cover
- Title
- Copyright
- Contents
- Preface
- PART I: Introduction
- CHAPTER 1 The Investment Environment
- 1.1 Real Assets versus Financial Assets
- 1.2 Financial Assets
- 1.3 Financial Markets and the Economy
- The Informational Role of Financial Markets
- Consumption Timing
- Allocation of Risk
- Separation of Ownership and Management
- Corporate Governance and Corporate Ethics
- 1.4 The Investment Process
- 1.5 Markets Are Competitive
- The Risk–Return Trade-Off
- Efficient Markets
- 1.6 The Players
- Financial Intermediaries
- Investment Bankers
- Venture Capital and Private Equity
- 1.7 The Financial Crisis of 2008
- Antecedents of the Crisis
- Changes in Housing Finance
- Mortgage Derivatives
- Credit Default Swaps
- The Rise of Systemic Risk
- The Shoe Drops
- The Dodd-Frank Reform Act
- 1.8 Outline of the Text
- End of Chapter Material
- CHAPTER 2 Asset Classes and Financial Instruments
- 2.1 The Money Market
- Treasury Bills
- Certificates of Deposit
- Commercial Paper
- Bankers' Acceptances
- Eurodollars
- Repos and Reverses
- Federal Funds
- Brokers' Calls
- The LIBOR Market
- Yields on Money Market Instruments
- 2.2 The Bond Market
- Treasury Notes and Bonds
- Inflation-Protected Treasury Bonds
- Federal Agency Debt
- International Bonds
- Municipal Bonds
- Corporate Bonds
- Mortgages and Mortgage-Backed Securities
- 2.3 Equity Securities
- Common Stock as Ownership Shares
- Characteristics of Common Stock
- Stock Market Listings
- Preferred Stock
- Depository Receipts
- 2.4 Stock and Bond Market Indexes
- Stock Market Indexes
- Dow Jones Averages
- Standard & Poor's Indexes
- Other U. S. Market-Value Indexes
- Equally Weighted Indexes
- Foreign and International Stock Market Indexes
- Bond Market Indicators
- 2.5 Derivative Markets
- Options
- Futures Contracts
- End of Chapter Material
- CHAPTER 3 How Securities Are Traded
- 3.1 How Firms Issue Securities
- Privately Held Firms
- Publicly Traded Companies
- Shelf Registration
- Initial Public Offerings
- 3.2 How Securities Are Traded
- Types of Markets
- Direct Search Markets
- Brokered Markets
- Dealer Markets
- Auction Markets
- Types of Orders
- Market Orders
- Price-Contingent Orders
- Trading Mechanisms
- Dealer Markets
- Electronic Communication Networks (ECNs)
- Specialist Markets
- 3.3 The Rise of Electronic Trading
- 3.4 U. S. Markets
- NASDAQ
- The New York Stock Exchange
- ECNs
- 3.5 New Trading Strategies
- Algorithmic Trading
- High-Frequency Trading
- Dark Pools
- Bond Trading
- 3.6 Globalization of Stock Markets
- 3.7 Trading Costs
- 3.8 Buying on Margin
- 3.9 Short Sales
- 3.10 Regulation of Securities Markets
- Self-Regulation
- The Sarbanes-Oxley Act
- Insider Trading
- End of Chapter Material
- CHAPTER 4 Mutual Funds and Other Investment Companies
- 4.1 Investment Companies
- 4.2 Types of Investment Companies
- Unit Investment Trusts
- Managed Investment Companies
- Other Investment Organizations
- Commingled Funds
- Real Estate Investment Trusts (REITs)
- Hedge Funds
- 4.3 Mutual Funds
- Investment Policies
- Money Market Funds
- Equity Funds
- Sector Funds
- Bond Funds
- International Funds
- Balanced Funds
- Asset Allocation and Flexible Funds
- Index Funds
- How Funds Are Sold
- 4.4 Costs of Investing in Mutual Funds
- Fee Structure
- Operating Expenses
- Front-End Load
- Back-End Load
- 12b-1 Charges
- Fees and Mutual Fund Returns
- 4.5 Taxation of Mutual Fund Income
- 4.6 Exchange-Traded Funds
- 4.7 Mutual Fund Investment Performance: A First Look
- 4.8 Information on Mutual Funds
- End of Chapter Material
- PART II: Portfolio Theory and Practice
- CHAPTER 5 Risk, Return, and the Historical Record
- 5.1 Determinants of the Level of Interest Rates
- Real and Nominal Rates of Interest
- The Equilibrium Real Rate of Interest
- The Equilibrium Nominal Rate of Interest
- Taxes and the Real Rate of Interest
- 5.2 Comparing Rates of Return for Different Holding Periods
- Annual Percentage Rates
- Continuous Compounding
- 5.3 Bills and Inflation, 1926–2012
- 5.4 Risk and Risk Premiums
- Holding-Period Returns
- Expected Return and Standard Deviation
- Excess Returns and Risk Premiums
- 5.5 Time Series Analysis of Past Rates of Return
- Time Series versus Scenario Analysis
- Expected Returns and the Arithmetic Average
- The Geometric (Time-Weighted) Average Return
- Variance and Standard Deviation
- Mean and Standard Deviation Estimates from Higher-Frequency Observations
- The Reward-to-Volatility (Sharpe) Ratio
- 5.6 The Normal Distribution
- 5.7 Deviations from Normality and Risk Measures
- Value at Risk
- Expected Shortfall
- Lower Partial Standard Deviation and the Sortino Ratio
- Relative Frequency of Large, Negative 3-Sigma Returns
- 5.8 Historic Returns on Risky Portfolios
- Portfolio Returns
- A Global View of the Historical Record
- 5.9 Long-Term Investments
- Normal and Lognormal Returns
- Simulation of Long-Term Future Rates of Return
- The Risk-Free Rate Revisited
- Where Is Research on Rates of Return Headed?
- Forecasts for the Long Haul
- End of Chapter Material
- CHAPTER 6 Capital Allocation to Risky Assets
- 6.1 Risk and Risk Aversion
- Risk, Speculation, and Gambling
- Risk Aversion and Utility Values
- Estimating Risk Aversion
- 6.2 Capital Allocation across Risky and Risk-Free Portfolios
- 6.3 The Risk-Free Asset
- 6.4 Portfolios of One Risky Asset and a Risk-Free Asset
- 6.5 Risk Tolerance and Asset Allocation
- Nonnormal Returns
- 6.6 Passive Strategies: The Capital Market Line
- End of Chapter Material
- Appendix A: Risk Aversion, Expected Utility, and the St. Petersburg Paradox
- Appendix B: Utility Functions and Equilibrium Prices of Insurance Contracts
- Appendix C: The Kelly Criterion
- CHAPTER 7 Optimal Risky Portfolios
- 7.1 Diversification and Portfolio Risk
- 7.2 Portfolios of Two Risky Assets
- 7.3 Asset Allocation with Stocks, Bonds, and Bills
- Asset Allocation with Two Risky Asset Classes
- 7.4 The Markowitz Portfolio Optimization Model
- Security Selection
- Capital Allocation and the Separation Property
- The Power of Diversification
- Asset Allocation and Security Selection
- Optimal Portfolios and Nonnormal Returns
- 7.5 Risk Pooling, Risk Sharing, and the Risk of Long-Term Investments
- Risk Pooling and the Insurance Principle
- Risk Sharing
- Investment for the Long Run
- End of Chapter Material
- Appendix A: A Spreadsheet Model for Efficient Diversification
- Appendix B: Review of Portfolio Statistics
- CHAPTER 8 Index Models
- 8.1 A Single-Factor Security Market
- The Input List of the Markowitz Model
- Normality of Returns and Systematic Risk
- 8.2 The Single-Index Model
- The Regression Equation of the Single-Index Model
- The Expected Return–Beta Relationship
- Risk and Covariance in the Single-Index Model
- The Set of Estimates Needed for the Single-Index Model
- The Index Model and Diversification
- 8.3 Estimating the Single-Index Model
- The Security Characteristic Line for Hewlett-Packard
- The Explanatory Power of the SCL for HP
- Analysis of Variance
- The Estimate of Alpha
- The Estimate of Beta
- Firm-Specific Risk
- Correlation and Covariance Matrix
- 8.4 Portfolio Construction and the Single-Index Model
- Alpha and Security Analysis
- The Index Portfolio as an Investment Asset
- The Single-Index-Model Input List
- The Optimal Risky Portfolio in the Single-Index Model
- The Information Ratio
- Summary of Optimization Procedure
- An Example
- Risk Premium Forecasts
- The Optimal Risky Portfolio
- 8.5 Practical Aspects of Portfolio Management with the Index Model
- Is the Index Model Inferior to the Full-Covariance Model?
- The Industry Version of the Index Model
- Predicting Betas
- Index Models and Tracking Portfolios
- End of Chapter Material
- PART III: Equilibrium in Capital Markets
- CHAPTER 9 The Capital Asset Pricing Model
- 9.1 The Capital Asset Pricing Model
- Why Do All Investors Hold the Market Portfolio?
- The Passive Strategy Is Efficient
- The Risk Premium of the Market Portfolio
- Expected Returns on Individual Securities
- The Security Market Line
- The CAPM and the Single-Index Market
- 9.2 Assumptions and Extensions of the CAPM
- Assumptions of the CAPM
- Challenges and Extensions to the CAPM
- The Zero-Beta Model
- Labor Income and Nontraded Assets
- A Multiperiod Model and Hedge Portfolios
- A Consumption-Based CAPM
- Liquidity and the CAPM
- 9.3 The CAPM and the Academic World
- 9.4 The CAPM and the Investment Industry
- End of Chapter Material
- CHAPTER 10 Arbitrage Pricing Theory and Multifactor Models of Risk and Return
- 10.1 Multifactor Models: An Overview
- Factor Models of Security Returns
- 10.2 Arbitrage Pricing Theory
- Arbitrage, Risk Arbitrage, and Equilibrium
- Well-Diversified Portfolios
- Diversification and Residual Risk in Practice
- Executing Arbitrage
- The No-Arbitrage Equation of the APT
- 10.3 The APT, the CAPM, and the Index Model
- The APT and the CAPM
- The APT and Portfolio Optimization in a Single-Index Market
- 10.4 A Multifactor APT
- 10.5 The Fama-French (FF) Three-Factor Model
- End of Chapter Material
- CHAPTER 11 The Efficient Market Hypothesis
- 11.1 Random Walks and the Efficient Market Hypothesis
- Competition as the Source of Efficiency
- Versions of the Efficient Market Hypothesis
- 11.2 Implications of the EMH
- Technical Analysis
- Fundamental Analysis
- Active versus Passive Portfolio Management
- The Role of Portfolio Management in an Efficient Market
- Resource Allocation
- 11.3 Event Studies
- 11.4 Are Markets Efficient?
- The Issues
- The Magnitude Issue
- The Selection Bias Issue
- The Lucky Event Issue
- Weak-Form Tests: Patterns in Stock Returns
- Returns over Short Horizons
- Returns over Long Horizons
- Predictors of Broad Market Returns
- Semistrong Tests: Market Anomalies
- The Small-Firm-in-January Effect
- The Neglected-Firm Effect and Liquidity Effects
- Book-to-Market Ratios
- Post–Earnings-Announcement Price Drift
- Strong-Form Tests: Inside Information
- Interpreting the Anomalies
- Risk Premiums or Inefficiencies?
- Anomalies or Data Mining?
- Anomalies over Time
- Bubbles and Market Efficiency
- 11.5 Mutual Fund and Analyst Performance
- Stock Market Analysts
- Mutual Fund Managers
- So, Are Markets Efficient?
- End of Chapter Material
- CHAPTER 12 Behavioral Finance and Technical Analysis
- 12.1 The Behavioral Critique
- Information Processing
- Forecasting Errors
- Overconfidence
- Conservatism
- Behavioral Biases
- Framing
- Mental Accounting
- Regret Avoidance
- Affect
- Prospect Theory
- Limits to Arbitrage
- Fundamental Risk
- Implementation Costs
- Model Risk
- Limits to Arbitrage and the Law of One Price
- "Siamese Twin" Companies
- Equity Carve-Outs
- Closed-End Funds
- Bubbles and Behavioral Economics
- Evaluating the Behavioral Critique
- 12.2 Technical Analysis and Behavioral Finance
- Trends and Corrections
- Momentum and Moving Averages
- Relative Strength
- Breadth
- Sentiment Indicators
- Trin Statistic
- Confidence Index
- Put/Call Ratio
- A Warning
- End of Chapter Material
- CHAPTER 13 Empirical Evidence on Security Returns
- 13.1 The Index Model and the Single-Factor APT
- The Expected Return–Beta Relationship
- Setting Up the Sample Data
- Estimating the SCL
- Estimating the SML
- Tests of the CAPM
- The Market Index
- Measurement Error in Beta
- 13.2 Tests of the Multifactor CAPM and APT
- Labor Income
- Private (Nontraded) Business
- Early Versions of the Multifactor CAPM and APT
- A Macro Factor Model
- 13.3 Fama-French-Type Factor Models
- Size and B/M as Risk Factors
- Behavioral Explanations
- Momentum: A Fourth Factor
- 13.4 Liquidity and Asset Pricing
- 13.5 Consumption-Based Asset Pricing and the Equity Premium Puzzle
- Consumption Growth and Market Rates of Return
- Expected versus Realized Returns
- Survivorship Bias
- Extensions to the CAPM May Resolve the Equity Premium Puzzle
- Liquidity and the Equity Premium Puzzle
- Behavioral Explanations of the Equity Premium Puzzle
- End of Chapter Material
- PART IV: Fixed-Income Securities
- CHAPTER 14 Bond Prices and Yields
- 14.1 Bond Characteristics
- Treasury Bonds and Notes
- Accrued Interest and Quoted Bond Prices
- Corporate Bonds
- Call Provisions on Corporate Bonds
- Convertible Bonds
- Puttable Bonds
- Floating-Rate Bonds
- Preferred Stock
- Other Domestic Issuers
- International Bonds
- Innovation in the Bond Market
- Inverse Floaters
- Asset-Backed Bonds
- Catastrophe Bonds
- Indexed Bonds
- 14.2 Bond Pricing
- Bond Pricing between Coupon Dates
- 14.3 Bond Yields
- Yield to Maturity
- Yield to Call
- Realized Compound Return versus Yield to Maturity
- 14.4 Bond Prices over Time
- Yield to Maturity versus Holding-Period Return
- Zero-Coupon Bonds and Treasury Strips
- After-Tax Returns
- 14.5 Default Risk and Bond Pricing
- Junk Bonds
- Determinants of Bond Safety
- Bond Indentures
- Sinking Funds
- Subordination of Further Debt
- Dividend Restrictions
- Collateral
- Yield to Maturity and Default Risk
- Credit Default Swaps
- Credit Risk and Collateralized Debt Obligations
- End of Chapter Material
- CHAPTER 15 The Term Structure of Interest Rates
- 15.1 The Yield Curve
- Bond Pricing
- 15.2 The Yield Curve and Future Interest Rates
- The Yield Curve under Certainty
- Holding-Period Returns
- Forward Rates
- 15.3 Interest Rate Uncertainty and Forward Rates
- 15.4 Theories of the Term Structure
- The Expectations Hypothesis
- Liquidity Preference
- 15.5 Interpreting the Term Structure
- 15.6 Forward Rates as Forward Contracts
- End of Chapter Material
- CHAPTER 16 Managing Bond Portfolios
- 16.1 Interest Rate Risk
- Interest Rate Sensitivity
- Duration
- What Determines Duration?
- Rule 1 for Duration
- Rule 2 for Duration
- Rule 3 for Duration
- Rule 4 for Duration
- Rule 5 for Duration
- 16.2 Convexity
- Why Do Investors Like Convexity?
- Duration and Convexity of Callable Bonds
- Duration and Convexity of Mortgage-Backed Securities
- 16.3 Passive Bond Management
- Bond-Index Funds
- Immunization
- Cash Flow Matching and Dedication
- Other Problems with Conventional Immunization
- 16.4 Active Bond Management
- Sources of Potential Profit
- Horizon Analysis
- End of Chapter Material
- PART V: Security Analysis
- CHAPTER 17 Macroeconomic and Industry Analysis
- 17.1 The Global Economy
- 17.2 The Domestic Macroeconomy
- 17.3 Demand and Supply Shocks
- 17.4 Federal Government Policy
- Fiscal Policy
- Monetary Policy
- Supply-Side Policies
- 17.5 Business Cycles
- The Business Cycle
- Economic Indicators
- Other Indicators
- 17.6 Industry Analysis
- Defining an Industry
- Sensitivity to the Business Cycle
- Sector Rotation
- Industry Life Cycles
- Start-Up Stage
- Consolidation Stage
- Maturity Stage
- Relative Decline
- Industry Structure and Performance
- Threat of Entry
- Rivalry between Existing Competitors
- Pressure from Substitute Products
- Bargaining Power of Buyers
- Bargaining Power of Suppliers
- End of Chapter Material
- CHAPTER 18 Equity Valuation Models
- 18.1 Valuation by Comparables
- Limitations of Book Value
- 18.2 Intrinsic Value versus Market Price
- 18.3 Dividend Discount Models
- The Constant-Growth DDM
- Convergence of Price to Intrinsic Value
- Stock Prices and Investment Opportunities
- Life Cycles and Multistage Growth Models
- Multistage Growth Models
- 18.4 Price–Earnings Ratio
- The Price–Earnings Ratio and Growth Opportunities
- P/E Ratios and Stock Risk
- Pitfalls in P/E Analysis
- Combining P/E Analysis and the DDM
- Other Comparative Valuation Ratios
- Price-to-Book Ratio
- Price-to-Cash-Flow Ratio
- Price-to-Sales Ratio
- 18.5 Free Cash Flow Valuation Approaches
- Comparing the Valuation Models
- The Problem with DCF Models
- 18.6 The Aggregate Stock Market
- End of Chapter Material
- CHAPTER 19 Financial Statement Analysis
- 19.1 The Major Financial Statements
- The Income Statement
- The Balance Sheet
- The Statement of Cash Flows
- 19.2 Measuring Firm Performance
- 19.3 Profitability Measures
- Return on Assets, ROA
- Return on Capital, ROC
- Return on Equity, ROE
- Financial Leverage and ROE
- Economic Value Added
- 19.4 Ratio Analysis
- Decomposition of ROE
- Turnover and Other Asset Utilization Ratios
- Liquidity Ratios
- Market Price Ratios: Growth versus Value
- Choosing a Benchmark
- 19.5 An Illustration of Financial Statement Analysis
- 19.6 Comparability Problems
- Inventory Valuation
- Depreciation
- Inflation and Interest Expense
- Fair Value Accounting
- Quality of Earnings and Accounting Practices
- International Accounting Conventions
- 19.7 Value Investing: The Graham Technique
- End of Chapter Material
- PART VI: Options, Futures, and Other Derivatives
- CHAPTER 20 Options Markets: Introduction
- 20.1 The Option Contract
- Options Trading
- American and European Options
- Adjustments in Option Contract Terms
- The Options Clearing Corporation
- Other Listed Options
- Index Options
- Futures Options
- Foreign Currency Options
- Interest Rate Options
- 20.2 Values of Options at Expiration
- Call Options
- Put Options
- Option versus Stock Investments
- 20.3 Option Strategies
- Protective Put
- Covered Calls
- Straddle
- Spreads
- Collars
- 20.4 The Put-Call Parity Relationship
- 20.5 Option-Like Securities
- Callable Bonds
- Convertible Securities
- Warrants
- Collateralized Loans
- Levered Equity and Risky Debt
- 20.6 Financial Engineering
- 20.7 Exotic Options
- Asian Options
- Barrier Options
- Lookback Options
- Currency-Translated Options
- Digital Options
- End of Chapter Material
- CHAPTER 21 Option Valuation
- 21.1 Option Valuation: Introduction
- Intrinsic and Time Values
- Determinants of Option Values
- 21.2 Restrictions on Option Values
- Restrictions on the Value of a Call Option
- Early Exercise and Dividends
- Early Exercise of American Puts
- 21.3 Binomial Option Pricing
- Two-State Option Pricing
- Generalizing the Two-State Approach
- Making the Valuation Model Practical
- 21.4 Black-Scholes Option Valuation
- The Black-Scholes Formula
- Dividends and Call Option Valuation
- Put Option Valuation
- Dividends and Put Option Valuation
- 21.5 Using the Black-Scholes Formula
- Hedge Ratios and the Black-Scholes Formula
- Portfolio Insurance
- Option Pricing and the Crisis of 2008–2009
- Option Pricing and Portfolio Theory
- Hedging Bets on Mispriced Options
- 21.6 Empirical Evidence on Option Pricing
- End of Chapter Material
- CHAPTER 22 Futures Markets
- 22.1 The Futures Contract
- The Basics of Futures Contracts
- Existing Contracts
- 22.2 Trading Mechanics
- The Clearinghouse and Open Interest
- The Margin Account and Marking to Market
- Cash versus Actual Delivery
- Regulations
- Taxation
- 22.3 Futures Markets Strategies
- Hedging and Speculation
- Basis Risk and Hedging
- 22.4 Futures Prices
- The Spot-Futures Parity Theorem
- Spreads
- Forward versus Futures Pricing
- 22.5 Futures Prices versus Expected Spot Prices
- Expectations Hypothesis
- Normal Backwardation
- Contango
- Modern Portfolio Theory
- End of Chapter Material
- CHAPTER 23 Futures, Swaps, and Risk Management
- 23.1 Foreign Exchange Futures
- The Markets
- Interest Rate Parity
- Direct versus Indirect Quotes
- Using Futures to Manage Exchange Rate Risk
- 23.2 Stock-Index Futures
- The Contracts
- Creating Synthetic Stock Positions: An Asset Allocation Tool
- Index Arbitrage
- Using Index Futures to Hedge Market Risk
- 23.3 Interest Rate Futures
- Hedging Interest Rate Risk
- 23.4 Swaps
- Swaps and Balance Sheet Restructuring
- The Swap Dealer
- Other Interest Rate Contracts
- Swap Pricing
- Credit Risk in the Swap Market
- Credit Default Swaps
- 23.5 Commodity Futures Pricing
- Pricing with Storage Costs
- Discounted Cash Flow Analysis for Commodity Futures
- End of Chapter Material
- PART VII: Applied Portfolio Management
- CHAPTER 24 Portfolio Performance Evaluation
- 24.1 The Conventional Theory of Performance Evaluation
- Average Rates of Return
- Time-Weighted Returns versus Dollar-Weighted Returns
- Dollar-Weighted Return and Investment Performance
- Adjusting Returns for Risk
- The M[sup(2)] Measure of Performance
- Sharpe's Ratio Is the Criterion for Overall Portfolios
- Appropriate Performance Measures in Two Scenarios
- Jane's Portfolio Represents Her Entire Risky Investment Fund
- Jane's Choice Portfolio Is One of Many Portfolios Combined into a Large Investment Fund
- The Role of Alpha in Performance Measures
- Actual Performance Measurement: An Example
- Performance Manipulation and the Morningstar Risk-Adjusted Rating
- Realized Returns versus Expected Returns
- 24.2 Performance Measurement for Hedge Funds
- 24.3 Performance Measurement with Changing Portfolio Composition
- 24.4 Market Timing
- The Potential Value of Market Timing
- Valuing Market Timing as a Call Option
- The Value of Imperfect Forecasting
- 24.5 Style Analysis
- Style Analysis and Multifactor Benchmarks
- Style Analysis in Excel
- 24.6 Performance Attribution Procedures
- Asset Allocation Decisions
- Sector and Security Selection Decisions
- Summing Up Component Contributions
- End of Chapter Material
- CHAPTER 25 International Diversification
- 25.1 Global Markets for Equities
- Developed Countries
- Emerging Markets
- Market Capitalization and GDP
- Home-Country Bias
- 25.2 Risk Factors in International Investing
- Exchange Rate Risk
- Political Risk
- 25.3 International Investing: Risk, Return, and Benefits from Diversification
- Risk and Return: Summary Statistics
- Are Investments in Emerging Markets Riskier?
- Are Average Returns Higher in Emerging Markets?
- Is Exchange Rate Risk Important in International Portfolios?
- Benefits from International Diversification
- Misleading Representation of Diversification Benefits
- Realistic Benefits from International Diversification
- Are Benefits from International Diversification Preserved in Bear Markets?
- 25.4 Assessing the Potential of International Diversification
- 25.5 International Investing and Performance Attribution
- Constructing a Benchmark Portfolio of Foreign Assets
- Performance Attribution
- End of Chapter Material
- CHAPTER 26 Hedge Funds
- 26.1 Hedge Funds versus Mutual Funds
- 26.2 Hedge Fund Strategies
- Directional and Nondirectional Strategies
- Statistical Arbitrage
- 26.3 Portable Alpha
- An Example of a Pure Play
- 26.4 Style Analysis for Hedge Funds
- 26.5 Performance Measurement for Hedge Funds
- Liquidity and Hedge Fund Performance
- Hedge Fund Performance and Survivorship Bias
- Hedge Fund Performance and Changing Factor Loadings
- Tail Events and Hedge Fund Performance
- 26.6 Fee Structure in Hedge Funds
- End of Chapter Material
- CHAPTER 27 The Theory of Active Portfolio Management
- 27.1 Optimal Portfolios and Alpha Values
- Forecasts of Alpha Values and Extreme Portfolio Weights
- Restriction of Benchmark Risk
- 27.2 The Treynor-Black Model and Forecast Precision
- Adjusting Forecasts for the Precision of Alpha
- Distribution of Alpha Values
- Organizational Structure and Performance
- 27.3 The Black-Litterman Model
- Black-Litterman Asset Allocation Decision
- Step 1: The Covariance Matrix from Historical Data
- Step 2: Determination of a Baseline Forecast
- Step 3: Integrating the Manager's Private Views
- Step 4: Revised (Posterior) Expectations
- Step 5: Portfolio Optimization
- 27.4 Treynor-Black versus Black-Litterman: Complements, Not Substitutes
- The BL Model as Icing on the TB Cake
- Why Not Replace the Entire TB Cake with the BL Icing?
- 27.5 The Value of Active Management
- A Model for the Estimation of Potential Fees
- Results from the Distribution of Actual Information Ratios
- Results from Distribution of Actual Forecasts
- Results with Reasonable Forecasting Records
- 27.6 Concluding Remarks on Active Management
- End of Chapter Material
- Appendix A: Forecasts and Realizations of Alpha
- Appendix B: The General Black-Litterman Model
- CHAPTER 28 Investment Policy and the Framework of the CFA Institute
- 28.1 The Investment Management Process
- Objectives
- Individual Investors
- Personal Trusts
- Mutual Funds
- Pension Funds
- Endowment Funds
- Life Insurance Companies
- Non–Life Insurance Companies
- Banks
- 28.2 Constraints
- Liquidity
- Investment Horizon
- Regulations
- Tax Considerations
- Unique Needs
- 28.3 Policy Statements
- Sample Policy Statements for Individual Investors
- 28.4 Asset Allocation
- Taxes and Asset Allocation
- 28.5 Managing Portfolios of Individual Investors
- Human Capital and Insurance
- Investment in Residence
- Saving for Retirement and the Assumption of Risk
- Retirement Planning Models
- Manage Your Own Portfolio or Rely on Others?
- Tax Sheltering
- The Tax-Deferral Option
- Tax-Deferred Retirement Plans
- Deferred Annuities
- Variable and Universal Life Insurance
- 28.6 Pension Funds
- Defined Contribution Plans
- Defined Benefit Plans
- Pension Investment Strategies
- Investing in Equities
- Wrong Reasons to Invest in Equities
- 28.7 Investments for the Long Run
- Target Investing and the Term Structure of Bonds
- Making Simple Investment Choices
- Inflation Risk and Long-Term Investors
- End of Chapter Material
- REFERENCES TO CFA PROBLEMS
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