236 Part TWO Portfolio Theory

rA - rf (%)

150

100

50

rM - rf (%)

0

“50 “30 “10 10 30 50

“50

“100

F I G U R E 7.7

Security characteristic line for stock A

rB - rf (%)

150

100

50

rM - rf (%)

0

“50 “30 “10 10 30 50

“50

“100

F I G U R E 7.8

Security characteristic line for stock B

means, standard deviations, and correlation coefficients of the full set of risky assets in the

universe of securities. (This additional information is not shown here.) Stocks A and B plot far

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Fifth Edition Theory

237

7 Capital Asset Pricing and Arbitrage Pricing Theory

20

SML

15

Expected return (%)

A

B

M

10

5

rf

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Beta

F I G U R E 7.9

Security market line

20

CML

15 Efficient frontier

A

M of risky assets

Expected return (%)

B

10

5

rf

0

0 10 20 30 40 50 60

Standard deviation (%)

F I G U R E 7.10

Capital market line

below the CML and below the efficient frontier, demonstrating that undiversified individual

securities are dominated by efficiently diversified portfolios.

Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

E XC E L Applications www.mhhe.com/bkm

> Beta

This Excel model contains a data set that can be used to estimate beta coefficients for a number

of stocks. The data contain individual stock and index returns and returns on Treasury bills. These

were obtained from the Standard & Poor™s Educational Version of Market Insight, available at

www.mhhe.com/edumarketinsight.com, and are analyzed using the regression function in Excel

that was discussed in section 6.5, “A Single-Factor Asset Market.”

You can learn more about this spreadsheet model by using the interactive version available on

our website at www.mhhe.com/bkm.

A B C D E F G H I

1 SUMMARY OUTPUT AXP

2

Regression Statistics

3

4 Multiple R 0.69288601

5 R Square 0.48009103

6 Adjusted R Square 0.47112708

7 Standard Error 0.05887426

8 Observations 60

9

10 ANOVA

df SS MS F Significance F

11

12 Regression 1 0.185641557 0.1856416 53.55799 8.55186E-10

13 Residual 58 0.201038358 0.0034662

14 Total 59 0.386679915

15

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

16

17 Intercept 0.01181687 0.00776211 1.522379 0.133348 0.003720666 0.027354414 0.0037207 0.02735441

18 X Variable 1 1.20877413 0.165170705 7.3183324 8.55E-10 0.878149288 1.539398969 0.87814929 1.53939897

Predicting Betas

Even if a single-index model representation is not fully consistent with the CAPM, the

concept of systematic versus diversifiable risk is still useful. Systematic risk is approximated

well by the regression equation beta and nonsystematic risk by the residual variance of the

regression.

Often, we estimate betas in order to forecast the rate of return of an asset. The beta from the

regression equation is an estimate based on past history; it will not reveal possible changes in

future beta. As an empirical rule, it appears that betas exhibit a statistical property called “re-

gression toward the mean.” This means that high (that is, 1) securities in one period

tend to exhibit a lower in the future, while low (that is, 1) securities exhibit a higher

in future periods. Researchers who desire predictions of future betas often adjust beta esti-

mates derived from historical data to account for regression toward the mean. For this reason,

it is necessary to verify whether the estimates are already “adjusted betas.”

A simple way to account for the tendency of future betas to “regress” toward the average

value of 1.0 is to use as your forecast of beta a weighted average of the sample estimate with

the value 1.0.

238

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

239

7 Capital Asset Pricing and Arbitrage Pricing Theory

Suppose that past data yield a beta estimate of 0.65. A common weighting scheme is 2„3 on the

sample estimate and 1„3 on the value 1.0. Thus, the final forecast of beta will be

EXAMPLE 7.7

Adjusted beta 2„3 1„3

0.65 1.0 0.77

Forecast of Beta

The final forecast of beta is in fact closer to 1.0 than the sample estimate.

A more sophisticated technique would base the weight assigned to the sample estimate of

beta on its statistical reliability. That is, if we have a more precise estimate of beta from his-

torical data, we increase the weight placed on the sample estimate.

However, obtaining a precise statistical estimate of beta from past data on individual stocks

is a formidable task, because the volatility of rates of return is so large. In other words, there

is a lot of “noise” in the data due to the impact of firm-specific events. The problem is less se-

vere with diversified portfolios because diversification reduces the effect of firm-specific

events.

One might hope that more precise estimates of beta could be obtained by using more data,

that is, by using a long time series of the returns on the stock. Unfortunately, this is not a so-

lution, because regression analysis presumes that the regression coefficient (the beta) is con-

stant over the sample period. If betas change over time, old data could provide a misleading

guide to current betas. More complicated regression techniques that allow for time-varying co-

efficients also have not proved to be very successful.

One promising avenue is an application of a technique that goes by the name of ARCH

models.4 An ARCH model posits that changes in stock volatility, and covariance with other

stocks, are partially predictable and analyzes recent levels and trends in volatility and covari-

ance. This technique has penetrated the industry only recently and so has not yet produced

truly reliable betas. Thus, the problem of estimating the critical parameters of the CAPM and

index models has been a stick in the wheels of testing and applying the theory.

7.4 THE CAPM AND THE REAL WORLD

In limited ways, portfolio theory and the CAPM have become accepted tools in the practi-