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Chapter Five

Risk, Return, and the Historical Record

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No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

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A nominal interest rate is the growth rate of your money

A real interest rate is the growth rate of your purchasing power

Real and Nominal Rates of Interest

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Tax liabilities are based on nominal income and the tax rate determined by the investor’s tax bracket

After-tax return falls by the tax rate times the inflation rate

Taxes and the Real Interest Rate

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T-Bill Rates, Inflation Rates, and Real Rates, 1926-2018

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Interest Rates and Inflation, 1926-2018

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Sources of investment risk

Macroeconomic fluctuations

Changing fortunes of various industries

Firm-specific unexpected developments

Holding period return (HPR), or realized rate of return, is based on the price per share at year’s end and any cash dividends collected

Risk and Risk Premiums: Holding Period Returns

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Note the percent return from dividends is called the dividend yield, and dividend yield plus the rate of capital gains equals HPR.

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Expected returns

p(s) = probability of each scenario

r(s) = HPR in each scenario

s = scenario

Risk and Risk Premiums: Expected Return and Standard Deviation (1 of 2)

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Variance (VAR):

Standard Deviation (STD):

Risk and Risk Premiums: Expected Return and Standard Deviation (2 of 2)

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Risk premium is the difference between the expected HPR and the risk-free rate

Provides compensation for the risk of an investment

Risk-free rate is the rate of interest that can be earned with certainty

Commonly taken to be the rate on short-term T-bills

Difference between actual rate of return and risk-free rate is called excess return

Risk aversion dictates the degree to which investors are willing to commit funds to stocks

Risk and Risk Premiums: Excess Returns and Risk Premiums

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Risk premium is the expected value of the excess return, and the standard deviation of the excess return is a measure of its risk.

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Investors price risky assets so that the risk premium will be commensurate with the risk of expected excess returns

Best to measure risk by the standard deviation of excess, not total, returns

Sharpe ratio

Evaluates performance of investment managers

Learning from Historical Returns: The Reward-to-Volatility (Sharpe) Ratio

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The Normal Distribution (1 of 2)

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Historic Returns on Risky Portfolios

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Century-plus-long history (1900 – 2017) of average excess returns in 20 stock markets

Mean annual excess return across these countries was 7.4% and the median was 6.6%

U.S. performance consistent with international experience

Tremendous variability in year-by-year returns

Historic Returns on Risky Portfolios: A Global View of the Historical Record

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Average Excess Returns in 20 Countries, 1900 - 2017

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Chapter Six

Capital Allocation to Risky Assets

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No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

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Risk-averse investors consider only risk-free or speculative prospects with positive risk premiums

Portfolio is more attractive when its expected return is higher and its risk is lower

What happens when risk increases along with return?

Risk and Risk Aversion: Risk Aversion and Utility Values

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We assume each investor can assign a welfare, or utility, score to competing portfolios

Utility function

U = Utility value

E(r) = Expected return

A = Index of the investor’s risk aversion

σ2 = Variance of returns

½ = Scaling factor

Risk and Risk Aversion: Risk Aversion and Utility Values (Continued)

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Utility increases with expected return, but falls with volatility.

The utility score of risky portfolios can be interpreted as a certainty equivalent rate of return, where the certainty equivalent is the rate that a risk-free investment would need to offer to provide the same utility as the risky portfolio.

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Risk-averse investors consider risky portfolios only if they provide compensation for risk via a risk premium

A > 0

Risk-neutral investors find the level of risk irrelevant and consider only the expected return of risk prospects

A = 0

Risk lovers are willing to accept lower expected returns on prospects with higher amounts of risk

A < 0

Investor Types

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How can we estimate the levels of risk aversion of individual investors?

Questionnaires

Varying degrees of complexity

Observations of how portfolio composition changes over time

Average degrees of risk aversion from groups of individuals

Estimating Risk Aversion

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Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4

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Chapter Seven

Efficient Diversification

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No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

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The investment decision is a top-down process

Capital allocation (risky vs. risk-free)

Asset allocation

Security selection

Optimal risky portfolio construction

Efficient diversification

Long-term vs. short-term investment horizons

Chapter Overview

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Market risk

Attributable to market wide risk sources

Remains even after diversification

Also called systematic or nondiversifiable risk

Firm-specific risk

Risk that can be eliminated by diversification

Also called diversifiable or nonsystematic risk

Diversification and Portfolio Risk

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Portfolio Diversification

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Expected return

Weighted average of expected returns on the component securities

Portfolio risk

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

Portfolios of Two Risky Assets

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Expected Return: The expected return on the portfolio is a weighted average of expected returns on the component securities with portfolio proportions as weights: E(rp) = wD E(rD) + wE E(rE)

Portfolio Risk: (variance) depends on the correlation between the returns of the assets in the portfolio

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Portfolios of Two Risky Assets: Expected Return

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Consider a portfolio made up of equity (stocks) and debt (bonds)…

where rP = rate of return on portfolio

wD = proportion invested in the bond fund

wE = proportion invested in the stock fund

rD = rate of return on the debt fund

rE = rate of return on the equity fund

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Variance of rP

Bond variance

Equity variance

Covariance of returns for bond and equity

Portfolios of Two Risky Assets: Risk

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Covariance of returns on bond and equity

DE = Correlation coefficient of returns

D = Standard deviation of bond returns

E = Standard deviation of equity returns

Portfolios of Two Risky Assets: Covariance

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Portfolios of Two Risky Assets: Example — 50%/50% Split

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Expected Return:

Variance:

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Portfolio Expected Return as a Function of Standard Deviation

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Objective is to find the weights wD and wE that result in the highest slope of the CAL

Thus, our objective function is the Sharpe ratio:

The Sharpe Ratio

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Security selection

Determine the risk-return opportunities available

Minimum-variance frontier of risky assets

All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

Efficient frontier of risky assets is the portion of the frontier that lies above the global minimum-variance portfolio

Markowitz Portfolio Optimization Model (1 of 6)

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The Minimum-Variance Frontier of Risky Assets

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The Efficient Frontier of Risky Assets with the Optimal CAL

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Capital Allocation Lines with Various Portfolios from the Efficient Set

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