Fifth Edition Theory

228 Part TWO Portfolio Theory

simple CAPM world in that some investors apply their own analysis to derive an “input list”

that may differ from their competitors™. If a stock is perceived to be a good buy, or under-

priced, it will provide a positive alpha, that is, an expected return in excess of the fair return

stipulated by the SML.

The CAPM also is useful in capital budgeting decisions. If a firm is considering a new proj-

ect, the CAPM can provide the return the project needs to yield to be acceptable to investors.

Managers can use the CAPM to obtain this cutoff internal rate of return (IRR) or “hurdle rate”

for the project.

Suppose Silverado Springs Inc. is considering a new spring-water bottling plant. The business

plan forecasts an internal rate of return of 14% on the investment. Research shows the beta of

7.5 EXAMPLE similar products is 1.3. Thus, if the risk-free rate is 4%, and the market risk premium is esti-

mated at 8%, the hurdle rate for the project should be 4 1.3 8 14.4%. Because the IRR

The CAPM

is less than the risk-adjusted discount or hurdle rate, the project has a negative net present

and Capital

value and ought to be rejected.

Budgeting

Yet another use of the CAPM is in utility rate-making cases. Here the issue is the rate of re-

turn a regulated utility should be allowed to earn on its investment in plant and equipment.

Suppose equityholders™ investment in the firm is $100 million, and the beta of the equity is 0.6.

If the T-bill rate is 6%, and the market risk premium is 8%, then a fair annual profit will be

7.6 EXAMPLE 6 (0.6 8) 10.8% of $100 million, or $10.8 million. Since regulators accept the CAPM,

they will allow the utility to set prices at a level expected to generate these profits.

The CAPM and

Regulation

>

4. a. Stock XYZ has an expected return of 12% and risk of 1.0. Stock ABC is ex-

Concept

pected to return 13% with a beta of 1.5. The market™s expected return is 11%

CHECK and rf 5%. According to the CAPM, which stock is a better buy? What is the

alpha of each stock? Plot the SML and the two stocks and show the alphas of

each on the graph.

b. The risk-free rate is 8% and the expected return on the market portfolio is

16%. A firm considers a project with an estimated beta of 1.3. What is the re-

quired rate of return on the project? If the IRR of the project is 19%, what is the

project alpha?

7.3 THE CAPM AND INDEX MODELS

The CAPM has two limitations: It relies on the theoretical market portfolio, which includes all

assets (such as real estate, foreign stocks, etc.), and it deals with expected as opposed to actual

returns. To implement the CAPM, we cast it in the form of an index model and use realized,

not expected, returns.

An index model uses actual portfolios, such as the S&P 500, rather than the theoretical

market portfolio to represent the relevant systematic factors in the economy. The important ad-

vantage of index models is that the composition and rate of return of the index is easily meas-

ured and unambiguous.

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In contrast to an index model, the CAPM revolves around the “market portfolio.” However,

because many assets are not traded, investors would not have full access to the market port-

folio even if they could exactly identify it. Thus, the theory behind the CAPM rests on a shaky

real-world foundation. But, as in all science, a theory may be viewed as legitimate if its pre-

dictions approximate real-world outcomes with a sufficient degree of accuracy. In particular,

the reliance on the market portfolio shouldn™t faze us if we can verify that the predictions of

the CAPM are sufficiently accurate when the index portfolio is substituted for the market.

We can start with one central prediction of the CAPM: The market portfolio is mean-

variance efficient. An index model can be used to test this hypothesis by verifying that an

index chosen to be representative of the full market is a mean-variance efficient portfolio.

Another aspect of the CAPM is that it predicts relationships among expected returns, while

all we can observe are realized (historical) holding-period returns; actual returns in a particu-

lar holding period seldom, if ever, match our initial expectations. To test the mean-variance

efficiency of an index portfolio, we would have to show that the reward-to-variability ratio of

the index is not surpassed by any other portfolio. The reward-to-variability ratio, however, is

set in terms of expectations, and we can measure it only in terms of realizations.

The Index Model, Realized Returns,

and the Expected Return“Beta Relationship

To move from a model cast in expectations to a realized-return framework, we start with a

form of the single-index regression equation in realized excess returns, similar to that of Equa-

tion 6.6 in Chapter 6:

ri rf i (rM rf ) ei (7.3)

i

where ri is the holding-period return (HPR) on asset i, and i and i are the intercept and slope

of the line that relates asset i™s realized excess return to the realized excess return of the index.

We denote the index return by rM to emphasize that the index portfolio is proxying for the

market. The ei measures firm-specific effects during the holding period; it is the deviation of

security i™s realized HPR from the regression line, that is, the deviation from the forecast that

accounts for the index™s HPR. We set the relationship in terms of excess returns (over the risk-

free rate, rf ), for consistency with the CAPM™s logic of risk premiums.

Given that the CAPM is a statement about the expectation of asset returns, we look at the

expected return of security i predicted by Equation 7.3. Recall that the expectation of ei is zero

(the firm-specific surprise is expected to average zero over time), so the relationship expressed

in terms of expectations is

E(ri) rf i [E(rM ) rf ] (7.4)

i

Comparing this relationship to the expected return“beta relationship (Equation 7.2) of the

CAPM reveals that the CAPM predicts i 0. Thus, we have converted the CAPM predic-

tion about unobserved expectations of security returns relative to an unobserved market port-

folio into a prediction about the intercept in a regression of observed variables: realized excess

returns of a security relative to those of a specified index.

Operationalizing the CAPM in the form of an index model has a drawback, however. If in-

tercepts of regressions of returns on an index differ substantially from zero, you will not be

able to tell whether it is because you chose a bad index to proxy for the market or because the

theory is not useful.

In actuality, few instances of persistent, positive significant alpha values have been iden-

tified; these will be discussed in Chapter 8. Among these are: (1) small versus large stocks;

(2) stocks of companies that have recently announced unexpectedly good earnings; (3) stocks

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230 Part TWO Portfolio Theory

with high ratios of book value to market value; and (4) stocks that have experienced recent

sharp price declines. In general, however, future alphas are practically impossible to predict

from past values. The result is that index models are widely used to operationalize capital as-

set pricing theory.

Estimating the Index Model

Equation 7.3 also suggests how we might go about actually measuring market and firm-spe-

cific risk. Suppose that we observe the excess return on the market index and a specific asset

over a number of holding periods. We use as an example monthly excess returns on the S&P

500 index and GM stock for a particular year. We can summarize the results for a sample pe-

riod in a scatter diagram, as illustrated in Figure 7.6.

The horizontal axis in Figure 7.6 measures the excess return (over the risk-free rate) on the

market index; the vertical axis measures the excess return on the asset in question (GM stock

in our example). A pair of excess returns (one for the market index, one for GM stock) over a

holding period constitutes one point on this scatter diagram. The points are numbered

1 through 12, representing excess returns for the S&P 500 and GM for each month from

January through December. The single-index model states that the relationship between the

excess returns on GM and the S&P 500 is given by

RGMt GMRMt eGMt

GM

We have noted the resemblance of this relationship to a regression equation.

In a single-variable linear regression equation, the dependent variable plots around a

straight line with an intercept and a slope . The deviations from the line, ei , are assumed to

F I G U R E 7.6 8

5

Characteristic 7

11

line for GM

6

1

5

Excess rate of return on GM stock (%)

4

3 12

2

1

6

0

10

9

“1

“2 7

“3

2

8

“4

“5

“6

“7

4

“8

3

“9

“1 0 1

“5 “3 3 5 7

Excess rate of return on the market index (%)

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7 Capital Asset Pricing and Arbitrage Pricing Theory

be mutually independent and independent of the right-hand side variable. Because these as-

sumptions are identical to those of the index model, we can look at the index model as a re-

gression model. The sensitivity of GM to the market, measured by GM, is the slope of the

regression line. The intercept of the regression line is (which represents the average firm-

specific return), and deviations of particular observations from the regression line are de-

noted e. These residuals are the differences between the actual stock return and the return that

would be predicted from the regression equation describing the usual relationship between the

stock and the market; therefore, they measure the impact of firm-specific events during the

particular month. The parameters of interest, , , and Var(e), can be estimated using standard

regression techniques.

Estimating the regression equation of the single-index model gives us the security

characteristic line (SCL), which is plotted in Figure 7.6. (The regression results and raw data

appear in Table 7.5.) The SCL is a plot of the typical excess return on a security over the risk- security

free rate as a function of the excess return on the market. characteristic

This sample of 12 monthly holding-period returns is, of course, too small to yield reliable line (SCL)

statistics. We use it only for demonstration. For this sample period, we find that the beta coef- A plot of a security™s

ficient of GM stock, as estimated by the slope of the regression line, is 1.136, and that the in- expected excess

tercept for this SCL is 2.59% per month. return over the risk-

free rate as a function

For each month, our estimate of the residual, e, which is the deviation of GM™s excess re-

of the excess return

turn from the prediction of the SCL, equals

on the market.

Residual Actual Predicted return

eGMt RGMt ( GMRMt GM)

Monthly Excess Excess

TA B L E 7.5 GM Market T-Bill GM Market

Characteristic line for Month Return Return Rate Return Return

GM stock

January 6.06 7.89 0.65 5.41 7.24

February 2.86 1.51 0.58 3.44 0.93

March 8.17 0.23 0.62 8.79 0.39

April 7.36 0.29 0.72 8.08 1.01

May 7.76 5.58 0.66 7.10 4.92

June 0.52 1.73 0.55 0.03 1.18

July 1.74 0.21 0.62 2.36 0.83

August 3.00 0.36 0.55 3.55 0.91

September 0.56 3.58 0.60 1.16 4.18

October 0.37 4.62 0.65 1.02 3.97

November 6.93 6.85 0.61 6.32 6.24

December 3.08 4.55 0.65 2.43 3.90

Mean 0.02 2.38 0.62 0.60 1.76

Standard deviation 5.19 3.48 0.05 5.19 3.46

rGM rf rf)

Regression results (rM

Estimated coefficient 2.591 1.136

Standard error of estimate (1.59) (0.309)

Variance of residuals 12.585

Standard deviation of residuals 3.548

R-SQR 0.575

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