EXAMPLE 5.3

E(r) 0.25 44% 0.50 14% 0.25 ( 16%) 14%

Expected Return

We use Equation 5.4 to find the variance. First we take the difference between the holding pe-

and Standard

riod return in each scenario and the mean return, then we square that difference, and finally

Deviation

we multiply by the probability of each scenario to find the average of the squared deviations.

The result is

2

14)2 14)2 14)2

0.25(44 0.50(14 0.25( 16 450

and so the standard deviation is

450 21.21%

<

2. A share of stock of A-Star Inc. is now selling for $23.50. A financial analyst sum- Concept

marizes the uncertainty about the rate of return on the stock by specifying three

CHECK

possible scenarios:

Business End-of-Year Annual

Conditions Scenario, s Probability, p Price Dividend

High growth 1 0.35 $35 $4.40

Normal growth 2 0.30 27 4.00

No growth 3 0.35 15 4.00

What are the holding-period returns for a one-year investment in the stock of

A-Star Inc. for each of the three scenarios? Calculate the expected HPR and stan-

dard deviation of the HPR.

Risk Premiums and Risk Aversion risk-free rate

How much, if anything, should you invest in an index stock fund such as the one described in The rate of return

Table 5.2? First, you must ask how much of an expected reward is offered to compensate for that can be earned

the risk involved in investing money in stocks. with certainty.

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 risk premium

such as Treasury bills, money market funds, or the bank. We call this difference the risk pre- An expected return in

mium on common stocks. For example, if the risk-free rate in the example is 6% per year, and excess of that on risk-

the expected index fund return is 14%, then the risk premium on stocks is 8% per year. free securities.

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Fifth Edition

138 Part TWO Portfolio Theory

The rate of return on Treasury bills also varies over time. However, we know the rate of re-

turn we will earn on T-bills at the beginning of the holding period, while we can™t know the

return we will earn on risky assets until the end of the holding period. Therefore, to study the

risk premium available on risky assets we compile a series of excess returns, that is, returns

excess return

in excess of the T-bill rate in each period. One possible estimate of the risk premium of any as-

Rate of return in

set is the average of its historical excess returns.

excess of the

The degree to which investors are willing to commit funds to stocks depends on risk aver-

Treasury-bill rate.

sion. It seems obvious that investors are risk averse in the sense that, if the risk premium were

zero, people would not be willing to invest any money in stocks. In theory then, there must al-

risk aversion

ways be a positive risk premium on stocks in order to induce risk-averse investors to hold the

Reluctance to

existing supply of stocks instead of placing all their money in risk-free assets.

accept risk.

In fact, the risk premium is what distinguishes gambling from speculation. Investors who

are willing to take on risk because they expect to earn a risk premium are speculating. Specu-

lation is undertaken despite the risk because the speculator sees a favorable risk-return trade-

off. In contrast, gambling is the assumption of risk for no purpose beyond the enjoyment of

the risk itself. Gamblers take on risk even without the prospect of a risk premium.1

It occasionally will be useful to quantify an investor™s degree of risk aversion. To do so, sup-

pose that investors choose portfolios based on both expected return, E(rP), and the volatility of

2

returns as measured by the variance, P. If we denote the risk-free rate on Treasury bills as rf ,

then the risk premium of a portfolio is E(rP ) rf . Risk-averse investors will demand higher ex-

pected returns to place their wealth in portfolios with higher volatility; that risk premium will

be greater the greater their risk aversion. Therefore, if we quantify the degree of risk aversion

with the parameter A, it makes sense to assert that the risk premium an investor demands of a

2

portfolio will be dependent on both risk aversion A and the risk of the portfolio, P.

We will write the risk premium that an investor demands of a portfolio as a function of its

risk as

2

E(rP ) rf 1

„2A (5.6)

P

Equation 5.6 describes how investors are willing to trade off risk against expected return.

(The equation requires that we put rates of return in decimal form.) As a benchmark, we note

that risk-free portfolios have zero variance, so the investor does not require a risk premium”

2

the return must be equal only to the risk-free rate. A risk premium of 1„2A P is required to in-

duce the investor to establish an overall portfolio that has positive volatility. The term 1„2 is

merely a scale factor chosen for convenience and has no real bearing on the analysis.

It turns out that if investors trade off risk against return in the manner specified by Equa-

tion 5.6, then we would be able to infer their risk aversion if we could observe risk premiums

and volatilities of their actual portfolios. We can solve Equation 5.6 for A as

E(rP ) rf

A (5.7)

2

1

„2 P

For example, if an investor believes the risk premium on her portfolio is 8%, and the standard

deviation is 20%, then we could infer risk aversion as A .08/(.5 .20 2) 4.

In practice, we cannot observe the risk premium investors expect to earn. We can observe

only actual returns after the fact. Moreover, different investors may have different expectations

about the risk and return of various assets. Finally, Equations 5.6 and 5.7 apply only to the vari-

ance of an investor™s overall portfolio, not to individual assets held in that portfolio. We can-

not observe an investor™s total portfolio of assets. While the exact relationship between risk and

1

Sometimes a gamble might seem like speculation to the participants. If two investors have different views about the

future, they might take opposite positions on a security, and both may have an expectation of earning a positive risk

premium. In such cases, only one party can, in fact, be correct.

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Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

139

5 Risk and Return: Past and Prologue

return in capital markets cannot be known exactly, many studies conclude that investors™ risk

aversion is likely in the range of 2“4.

<

3. A respected analyst forecasts that the return of the S&P 500 index portfolio over Concept

the coming year will be 10%. The one-year T-bill rate is 5%. Examination of recent

CHECK

returns of the S&P 500 Index suggest that the standard deviation of returns will be

18%. What does this information suggest about the degree of risk aversion of the

average investor, assuming that the average portfolio resembles the S&P 500?

WEBMA STER

Estimating the Equity Risk Premium

A research report by Stern Stewart & Company, the consulting firm that specializes in

Economic Value Added, presents some analysis on this subject. Go to http://www.eva.

com/content/evaluation/info/032001.pdf to read the report entitled “The Equity Risk

Measurement Handbook.”

After reading pages 1“8 of this report, answer the following questions:

1. What range of estimates does this report indicate are being used in the market?

2. What factors does the report indicate should lead toward using a shorter

estimation period over which to estimate the market risk premium?

3. What did the report show with respect to the convergence of stock and bonds

with respect to volatility and returns?

4. Does the estimate of market risk premium appear to be sensitive to the

measurement period?

5.3 THE HISTORICAL RECORD

Bills, Bonds, and Stocks, 1926“2001

The record of past rates of return is one possible source of information about risk premiums

and standard deviations. We can estimate the historical risk premium by taking an average of

the past differences between the HPRs on an asset class and the risk-free rate. Table 5.3 pre-

sents the annual HPRs on five asset classes for the period 1926“2001.

“Large Stocks” in Table 5.3 refer to Standard & Poor™s market-value-weighted portfolio of

500 U.S. common stocks selected from the largest market capitalization stocks. “Small Com-

pany Stocks” are represented by the Russell 2000 Index as of 1996. Stocks in this portfolio

rank below the 1,000 U.S. largest-capitalization stocks. Prior to 1996, “Small Stocks” were

represented by the smallest 20% of the stocks trading on the NYSE.

Until 1996, “Long-Term T-Bonds” were represented by government bonds with at least a

20-year maturity and approximately current-level coupon rate,2 and “Intermediate-Term

T-Bonds” were represented by government bonds with a seven-year maturity and a current-

level coupon rate. Since 1996, these two bond series have been measured by the Lehman

Brothers Long-Term or Intermediate-Term Bond Indexes, respectively.

“T-Bills” in Table 5.3 are of approximately 30-day maturity, and the one-year HPR repre-

sents a policy of “rolling over” the bills as they mature. Because T-bill rates can change from

2

The importance of the coupon rate when comparing returns on bonds is discussed in Part Three.

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Fifth Edition

140 Part TWO Portfolio Theory

TA B L E 5.3

Rates of return 1926“2001

Small Large Long-Term Intermediate-

Year Stocks Stocks T-Bonds Term T-Bonds T-Bills Inflation

1926 8.91 12.21 4.54 4.96 3.19 1.12

1927 32.23 35.99 8.11 3.34 3.12 2.26

1928 45.02 39.29 0.93 0.96 3.21 1.16

1929 50.81 7.66 4.41 5.89 4.74 0.58

1930 45.69 25.90 6.22 5.51 2.35 6.40

1931 49.17 45.56 5.31 5.81 0.96 9.32

1932 10.95 9.14 11.89 8.44 1.16 10.27

1933 187.82 54.56 1.03 0.35 0.07 0.76

1934 25.13 2.32 10.15 9.00 0.60 1.52

1935 68.44 45.67 4.98 7.01 1.59 2.99

1936 84.47 33.55 6.52 3.77 0.95 1.45

1937 52.71 36.03 0.43 1.56 0.35 2.86

1938 24.69 29.42 5.25 5.64 0.09 2.78

1939 0.10 1.06 5.90 4.52 0.02 0.00

1940 11.81 9.65 6.54 2.03 0.00 0.71

1941 13.08 11.20 0.99 0.59 0.06 9.93

1942 51.01 20.80 5.39 1.81 0.26 9.03

1943 99.79 26.54 4.87 2.78 0.35 2.96

1944 60.53 20.96 3.59 1.98 0.07 2.30

1945 82.24 36.11 6.84 3.60 0.33 2.25

1946 12.80 9.26 0.15 0.69 0.57 18.13

1947 3.09 4.88 1.19 0.32 0.50 8.84

1948 6.15 5.29 3.07 2.21 0.81 2.99

1949 21.56 18.24 6.03 2.22 1.10 2.07

1950 45.48 32.68 0.96 0.25 1.20 5.93

1951 9.41 23.47 1.95 0.36 1.49 6.00

1952 6.36 18.91 1.93 1.63 1.66 0.75

1953 5.68 1.74 3.83 3.63 1.82 0.75

1954 65.13 52.55 4.88 1.73 0.86 0.74

1955 21.84 31.44 1.34 0.52 1.57 0.37

1956 3.82 6.45 5.12 0.90 2.46 2.99

1957 15.03 11.14 9.46 7.84 3.14 2.90

1958 70.63 43.78 3.71 1.29 1.54 1.76

1959 17.82 12.95 3.55 1.26 2.95 1.73

1960 5.16 0.19 13.78 11.98 2.66 1.36

1961 30.48 27.63 0.19 2.23 2.13 0.67

1962 16.41 8.79 6.81 7.38 2.72 1.33

1963 12.20 22.63 0.49 1.79 3.12 1.64

1964 18.75 16.67 4.51 4.45 3.54 0.97

1965 37.67 12.50 0.27 1.27 3.94 1.92

1966 8.08 10.25 3.70 5.14 4.77 3.46

1967 103.39 24.11 7.41 0.16 4.24 3.04

1968 50.61 11.00 1.20 2.48 5.24 4.72

1969 32.27 8.33 6.52 2.10 6.59 6.20

1970 16.54 4.10 12.69 13.93 6.50 5.57

1971 18.44 14.17 17.47 8.71 4.34 3.27

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Fifth Edition

141

5 Risk and Return: Past and Prologue

TA B L E 5.3

(Concluded)

Small Large Long-Term Intermediate-

Year Stocks Stocks T-Bonds Term T-Bonds T-Bills Inflation

1972 0.62 19.14 5.55 3.80 3.81 3.41