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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 2008. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

Some ancillaries, including electronic and print components, may not be available to customers outside the United States.

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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 not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites.

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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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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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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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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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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 TH

E STR EET

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

The 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

eXce 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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at w

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/b km

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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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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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?

eXce l Please visit us at www.mhhe.com/bkm

eXce 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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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.

FOR THE STUDENT

• 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.

FOR THE INSTRUCTOR

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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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- diation for every concept in the textbook. LearnSmart’s intelligent software adapts to every student response and automatically delivers concepts that advance students’ understanding while reducing time devoted to the concepts already mastered. The result for every student is the fast- est path to mastery of the chapter concepts. LearnSmart:

• Applies an intelligent concept engine to identify the relationships between concepts and to serve new con- cepts to each student only when he or she is ready.

• 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.

• Collect data and generate reports required by many accreditation organizations, such as AACSB.

MCGRAW-HILL’S CONNECT FINANCE

Less Managing. More Teaching. Greater Learning.

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 studying, and higher retention of knowledge.

McGraw-Hill’s Connect Finance Features Connect Finance offers a number of powerful tools and features to make managing assignments easier, so faculty can spend more time teaching. With Connect Finance, students can engage with their coursework anytime and anywhere, making the learning process more accessible 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.

• Streamline lesson planning, student progress report- ing, and assignment grading to make classroom man- agement more efficient than ever.

• Go paperless with the eBook and online submission and grading of student assignments.

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:

• Have assignments scored automatically, giving stu- 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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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.

MCGRAW-HILL CUSTOMER CARE CONTACT INFORMATION

At McGraw-Hill, we understand that getting the most from new technology can be challenging. That’s why our services don’t stop after you purchase our products. You can e-mail our product specialists 24 hours a day to get product-training online. Or you can search our knowledge bank of Frequently Asked Questions on our support Web site. For Customer Support, call 800-331-5094 or visit www.mhhe.com/support . One of our technical sup- port analysts will be able to assist you in a timely fashion.

McGraw-Hill’s Connect Plus Finance McGraw- Hill reinvents the textbook learning experience for the modern student with Connect Plus Finance. A seamless integration of an eBook and Connect Finance, Connect Plus Finance provides all of the Connect Finance features plus the following:

• An integrated media-rich eBook, allowing for anytime, anywhere access to the textbook.

• Dynamic links between the problems or questions you assign to your students and the location in the eBook where that problem or question is covered.

• A powerful search function to pinpoint and connect key concepts in a snap.

For more information about Connect, go to www.connect. mcgraw-hill.com , or contact your local McGraw-Hill sales representative.

AACSB STATEMENT

The McGraw-Hill Companies is a proud corporate mem- ber of AACSB International. Understanding the impor- 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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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 Drexel University

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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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

Steve L. Slezak 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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(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.

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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

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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

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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

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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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km 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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km 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

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S FR O

M TH

E STR EET

0.0

0.5

1.0

1.5

2.5

3.5

4.5

2.0

3.0

Pe rc

en ta

g e

Po in

ts 4.0

5.0 OPEC I

OPEC lI Penn Square

Market Crash

LTCM

Credit Crisis

19 70

19 72

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19 76

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19 86

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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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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.

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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.

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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.

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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 .

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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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km 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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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

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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

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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.

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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.

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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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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.

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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.

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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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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

The 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

The 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

80

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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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

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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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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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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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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

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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 .

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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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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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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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km 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 APR 41/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.

eXce 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 ea

lt h

In d

ex

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

Pe rc

en ta

g e

Po in

ts

5

10

15

20

T-Bills Inflation

19 27

19 32

19 37

19 42

19 47

19 52

19 57

19 62

19 67

19 72

19 77

19 82

19 87

19 92

19 97

20 02

20 07

20 12

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) 5a 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 5a 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

s51 p(s)r (s) 5

1 na

n

s51 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

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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,

E 3Geometric average 4 5 E 3Arithmetic average 4 2 ½s2 (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 na

n

s51 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 na

n

s51 3r (s) 2 r 42 5 1

n 2 1a n

s51 3r (s) 2 r 42

ŝ 5 Å 1n 2 1a n

s51 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%)

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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

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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.

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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.

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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

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A: T-Bill Rates

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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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145

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

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146

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.

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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

19The 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.

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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.

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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

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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

g1½ s2 5 Wem, and hence the expected effective rate of return is

E(r) 5 eg1½ 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 erCCT 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

(g1½ s2)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 "TVar(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 eq

u en

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 eq

u en

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 eq

u en

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 eq

u en

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 eq

u en

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 eq

u en

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 ea

lt h

In d

ex

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 ST

R EE

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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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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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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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 II

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 ST

R EE

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

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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

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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 y 3E(rP) 2 rf 4 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 y 3E(rP) 2 rf 4 5 rf 1

sC sP

3E(rP) 2 rf 4 5 7 1 822 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 y 3E(rP) 2 rf 4 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

lit y

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 rf 4 sCsP 5 7 1 315 2 7 4 sC22 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 TH

E STR EET

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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km 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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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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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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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 y 3E(rP) 2 rf 4 sC 5 ysP

and therefore the reward-to-volatility ratio of this portfolio is

SC 5 E(rC) 2 rf

sC 5

y 3E(rP) 2 rf 4 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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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 `

n50 Pr(n)R(n) 5 1@2 1 1@2 1 c5 `

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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 `

n50 Pr(n) ln 3R(n) 4 5 a`

n50 (1/2)n11 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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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:

E 3U(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. Kelly11 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:

E 3U(y) 4 5 p ln(1 1 r 1 yb) 1 q ln(1 1 r 2 ay) (6.C.1) 11J.L. Kelly Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal 35 (1956), 917–56.

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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

12See, 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 Po

rt fo

lio S

ta n

d ar

d D

ev ia

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 ar

ed t

o a

O n

e- St

o ck

P o

rt fo

lio (

% )

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 wDrD 1 wErE (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 wEE(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 wEwE 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 2wXwZ Cov(rX , rZ) 1 2wYwZ 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 2wDwEsDsErDE (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 wEsE)

2 (7.10)

and the portfolio standard deviation is

sp 5 Absolute value (wDsD 2 wEsE) (7.11)

When r   5   2 1, a perfectly hedged position can be obtained by choosing the portfolio pro- portions to solve

wDsD 2 wEsE 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 ST

R EE

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

The 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 na

n

i51

1 n

si 2 1 a

n

j51 j2 i

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 na

n

i51 si

2 (7.18)

Cov 5 1

n(n 2 1)a n

j51 j2 i

a n

i51 Cov(ri, rj) (7.19)

we can express portfolio variance as

sp 2 5

1 n

s2 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 y2s2 1 y2s2 1 0 5 2y2s2 (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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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)rDE

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 rDE3E(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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km 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%.