proportional to both the risk of the market, as measured by the variance of its returns, and to

the degree of risk aversion of the average investor, denoted by A* in Equation 7.1.

Suppose the risk-free rate is 5%, the average investor has a risk-aversion coefficient of A* 2,

and the standard deviation of the market portfolio is 20%. Then, from Equation 7.1, we esti-

7.1 EXAMPLE mate the equilibrium value of the market risk premium1 as 2 0.202 0.08. So the expected

rate of return on the market must be

Market Risk,

E(rM) rf

the Risk Equilibrium risk premium

Premium, and 0.05 0.08 0.13 13%

Risk Aversion

If investors were more risk averse, it would take a higher risk premium to induce them to hold

shares. For example, if the average degree of risk aversion were 3, the market risk premium

would be 3 0.202 0.12, or 12%, and the expected return would be 17%.

>

2. Historical data for the S&P 500 index show an average excess return over Treasury

Concept

bills of about 8.5% with standard deviation of about 20%. To the extent that these

CHECK averages approximate investor expectations for the sample period, what must have

been the coefficient of risk aversion of the average investor? If the coefficient of

risk aversion were 3.5, what risk premium would have been consistent with the

market™s historical standard deviation?

Expected Returns on Individual Securities

The CAPM is built on the insight that the appropriate risk premium on an asset will be deter-

mined by its contribution to the risk of investors™ overall portfolios. Portfolio risk is what mat-

ters to investors, and portfolio risk is what governs the risk premiums they demand.

We know that nonsystematic risk can be reduced to an arbitrarily low level through diver-

sification (Chapter 6); therefore, investors do not require a risk premium as compensation for

bearing nonsystematic risk. They need to be compensated only for bearing systematic risk,

which cannot be diversified. We know also that the contribution of a single security to the risk

of a large diversified portfolio depends only on the systematic risk of the security as measured

by its beta.2 Therefore, it should not be surprising that the risk premium of an asset is propor-

tional to its beta; for example, if you double a security™s systematic risk, you must double its

risk premium for investors still to be willing to hold the security. Thus, the ratio of risk pre-

mium to beta should be the same for any two securities or portfolios.

1

To use Equation 7.1, we must express returns in decimal form rather than as percentages.

2

See Section 6.5. This is literally true with a sufficient number of securities so that all nonsystematic risk is diversi-

fied away. In a market as diversified as the U.S. stock market, this would be true for all practical purposes.

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7 Capital Asset Pricing and Arbitrage Pricing Theory

For example, if we were to compare the ratio of risk premium to systematic risk for the

market portfolio, which has a beta of 1.0, with the corresponding ratio for Dell stock, we

would conclude that

E(rM) rf E(rD) rf

1 D

Rearranging this relationship results in the CAPM™s expected return“beta relationship

E(rD) rf [E(rM) rf] (7.2)

D

expected

In words, the rate of return on any asset exceeds the risk-free rate by a risk premium equal return“beta

to the asset™s systematic risk measure (its beta) times the risk premium of the (benchmark) relationship

market portfolio. This expected return“beta relationship is the most familiar expression of Implication of the

the CAPM. CAPM that security

The expected return“beta relationship of the CAPM makes a powerful economic statement. risk premiums

(expected excess

It implies, for example, that a security with a high variance but a relatively low beta of 0.5 will

returns) will be

carry one-third the risk premium of a low-variance security with a beta of 1.5. Thus, Equa-

proportional to beta.

tion 7.2 quantifies the conclusion we reached in Chapter 6 that only systematic risk matters to

investors who can diversify and that systematic risk is measured by the beta of the security.

Suppose the risk premium of the market portfolio is 9%, and we estimate the beta of Dell as

1.3. The risk premium predicted for the stock is therefore 1.3 times the market risk pre-

D

EXAMPLE 7.2

mium, or 1.3 9% 11.7%. The expected rate of return on Dell is the risk-free rate plus the

risk premium. For example, if the T-bill rate were 5%, the expected rate of return would be Expected

5% 11.7% 16.7%, or using Equation 7.2 directly, Returns and

E(rD) rf D[Market risk premium] Risk Premiums

5% 1.3 9% 16.7%

If the estimate of the beta of Dell were only 1.2, the required risk premium for Dell would fall

to 10.8%. Similarly, if the market risk premium were only 8% and D 1.3, Dell™s risk premium

would be only 10.4%.

The fact that few real-life investors actually hold the market portfolio does not necessarily

invalidate the CAPM. Recall from Chapter 6 that reasonably well-diversified portfolios shed

(for practical purposes) firm-specific risk and are subject only to systematic or market risk.

Even if one does not hold the precise market portfolio, a well-diversified portfolio will be so

highly correlated with the market that a stock™s beta relative to the market still will be a use-

ful risk measure.

In fact, several researchers have shown that modified versions of the CAPM will hold de-

spite differences among individuals that may cause them to hold different portfolios. A study

by Brennan (1970) examines the impact of differences in investors™ personal tax rates on mar-

ket equilibrium. Another study by Mayers (1972) looks at the impact of nontraded assets such

as human capital (earning power). Both find that while the market portfolio is no longer each

investor™s optimal risky portfolio, a modified version of the expected return“beta relationship

still holds.

If the expected return“beta relationship holds for any individual asset, it must hold for any

combination of assets. The beta of a portfolio is simply the weighted average of the betas of

the stocks in the portfolio, using as weights the portfolio proportions. Thus, beta also predicts

the portfolio™s risk premium in accordance with Equation 7.2.

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226 Part TWO Portfolio Theory

Consider the following portfolio:

7.3 EXAMPLE Asset Beta Risk Premium Portfolio Weight

Portfolio Beta

Microsoft 1.2 9.0% 0.5

and Risk

Con Edison 0.8 6.0 0.3

Premium

Gold 0.0 0.0 0.2

Portfolio 0.84 ? 1.0

If the market risk premium is 7.5%, the CAPM predicts that the risk premium on each stock is

its beta times 7.5%, and the risk premium on the portfolio is 0.84 7.5% 6.3%. This is the

same result that is obtained by taking the weighted average of the risk premiums of the indi-

vidual stocks. (Verify this for yourself.)

A word of caution: We often hear that well-managed firms will provide high rates of return.

We agree this is true if one measures the firm™s return on investments in plant and equipment.

The CAPM, however, predicts returns on investments in the securities of the firm.

Say that everyone knows a firm is well run. Its stock price should, therefore, be bid up, and

returns to stockholders who buy at those high prices will not be extreme. Security prices re-

flect public information about a firm™s prospects, but only the risk of the company (as mea-

sured by beta in the context of the CAPM) should affect expected returns. In a rational market,

investors receive high expected returns only if they are willing to bear risk.

>

3. Suppose the risk premium on the market portfolio is estimated at 8% with a stan-

Concept

dard deviation of 22%. What is the risk premium on a portfolio invested 25% in GM

CHECK with a beta of 1.15 and 75% in Ford with a beta of 1.25?

The Security Market Line

We can view the expected return“beta relationship as a reward-risk equation. The beta of a se-

curity is the appropriate measure of its risk because beta is proportional to the risk the security

contributes to the optimal risky portfolio.

Risk-averse investors measure the risk of the optimal risky portfolio by its standard devia-

tion. In this world, we would expect the reward, or the risk premium on individual assets, to

depend on the risk an individual asset contributes to the overall portfolio. Because the beta of

a stock measures the stock™s contribution to the standard deviation of the market portfolio, we

expect the required risk premium to be a function of beta. The CAPM confirms this intuition,

stating further that the security™s risk premium is directly proportional to both the beta and the

risk premium of the market portfolio; that is, the risk premium equals [E(rM ) rf ].

The expected return“beta relationship is graphed as the security market line (SML) in

security market

Figure 7.5. Its slope is the risk premium of the market portfolio. At the point where 1.0

line (SML)

(which is the beta of the market portfolio) on the horizontal axis, we can read off the vertical

Graphical

axis the expected return on the market portfolio.

representation

It is useful to compare the security market line to the capital market line. The CML graphs

of the expected

return“beta the risk premiums of efficient portfolios (that is, complete portfolios made up of the risky

relationship market portfolio and the risk-free asset) as a function of portfolio standard deviation. This is

of the CAPM.

appropriate because standard deviation is a valid measure of risk for portfolios that are candi-

dates for an investor™s complete (overall) portfolio.

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7 Capital Asset Pricing and Arbitrage Pricing Theory

F I G U R E 7.5

E(r) (%)

The security market

SML

line and a positive-

alpha stock

Stock

17 ±

15.6

M

14

6

β

1.0 1.2

The SML, in contrast, graphs individual asset risk premiums as a function of asset risk. The

relevant measure of risk for individual assets (which are held as parts of a well-diversified

portfolio) is not the asset™s standard deviation; it is, instead, the contribution of the asset to the

portfolio standard deviation as measured by the asset™s beta. The SML is valid both for port-

folios and individual assets.

The security market line provides a benchmark for evaluation of investment performance.

Given the risk of an investment as measured by its beta, the SML provides the required rate of

return that will compensate investors for the risk of that investment, as well as for the time

value of money.

Because the security market line is the graphical representation of the expected return“beta

alpha

relationship, “fairly priced” assets plot exactly on the SML. The expected returns of such as-

The abnormal rate of

sets are commensurate with their risk. Whenever the CAPM holds, all securities must lie on

return on a security in

the SML in market equilibrium. Underpriced stocks plot above the SML: Given their betas,

excess of what would

their expected returns are greater than is indicated by the CAPM. Overpriced stocks plot be- be predicted by an

low the SML. The difference between the fair and actually expected rate of return on a stock equilibrium model

is called the stock™s alpha, denoted . such as the CAPM.

Suppose the return on the market is expected to be 14%, a stock has a beta of 1.2, and the

T-bill rate is 6%. The SML would predict an expected return on the stock of

EXAMPLE 7.4

E(r) rf rf ]

[E(rM)

The Alpha of

6 1.2 (14 6) 15.6%

a Security

If one believes the stock will provide instead a return of 17%, its implied alpha would be 1.4%,

as shown in Figure 7.5.

Applications of the CAPM

One place the CAPM may be used is in the investment management industry. Suppose the

SML is taken as a benchmark to assess the fair expected return on a risky asset. Then an

analyst calculates the return he or she actually expects. Notice that we depart here from the

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