Fifth Edition Theory

232 Part TWO Portfolio Theory

TA B L E 7.6 A. Market index

Expected excess return over T-bill rate, E(RM) 8%

True parameters

Standard deviation of excess return, (RM) 20%

of securities

B. Individual stocks

Standard Deviation Total Standard

Beta of Residual, (e) Deviation of Returns*

Stock A 1.30 54.07% 60%

Stock B 0.70 37.47 40

22 2

(e)]1/2

*Standard deviation [ M

Stock A: [1.32 202 54.072]1/2 60%

2 2 2 1/2

Stock B: [0.7 20 37.47 ] 40%

C. T-bills

Average value in sample period 5%

Month-to-month variation results in a standard deviation across months of 1.5%

These residuals are estimates of the monthly unexpected firm-specific component of the

rate of return on GM stock. Hence we can estimate the firm-specific variance by3

1 12 2

2

(eGM) et 12.60

10 t 1

Therefore, the standard deviation of the firm-specific component of GM™s return, (eGM),

equals 3.55% per month.

The CAPM and the Index Model

We have introduced the CAPM and shown how the model can be made operational and how

beta can be estimated with the additional simplification of the index model of security returns.

Of course, when we estimate the statistical properties of security returns (e.g., betas or vari-

ances) using historical data, we are subject to sampling error. Regression parameters are only

estimates and necessarily are subject to some imprecision.

In this section, we put together much of the preceding material in an extended example. We

show how historical data can be used in conjunction with the CAPM, but we also highlight

some pitfalls to be avoided.

Suppose that the true parameters for two stocks, A and B, and the market index portfolio

are given in Table 7.6. However, investors cannot observe this information directly. They must

estimate these parameters using historical returns.

To illustrate the investor™s problem, we first produce 24 possible observations for the risk-

free rate and the market index. Using the random number generator from a spreadsheet pack-

age (e.g., you can use “data analysis tools” in Microsoft Excel), we draw 24 observations from

a normal distribution. These random numbers capture the phenomenon that actual returns will

differ from expected returns: This is the “statistical noise” that accompanies all real-world re-

turn data. For the risk-free rate we set a mean of 5% and a standard deviation of 1.5% and

3

Because the mean of et is zero, e2 is the squared deviation from its mean. The average value of et2 is therefore the

t

estimate of the variance of the firm-specific component. We divide the sum of squared residuals by the degrees of

freedom of the regression, n 2 12 2 10, to obtain an unbiased estimate of 2(e).

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7 Capital Asset Pricing and Arbitrage Pricing Theory

TA B L E 7.7

Simulated data for estimation of security characteristic line (raw data from random number generator)

Residuals for

Each Stock Excess Returns

Excess Return

T-Bill Rate on Index Stock A Stock B Stock A Stock B

5.97 3.75 7.52 44.13 2.64 41.50

4.45 9.46 26.14 38.79 13.85 45.41

3.24 26.33 18.09 65.43 52.32 46.99

5.70 6.06 0.88 69.24 7.00 73.49

3.89 38.97 48.37 61.51 99.03 88.78

5.56 1.35 30.80 26.25 32.56 25.30

5.03 24.18 10.74 0.93 42.18 16.00

2.70 15.20 68.91 18.53 88.66 7.89

5.57 39.52 14.09 16.80 37.29 44.46

5.94 2.84 0.43 36.15 3.26 38.14

4.41 0.97 73.75 20.33 72.48 21.01

4.43 29.82 25.31 68.88 64.08 89.76

2.88 0.73 83.07 10.82 82.13 10.31

5.77 16.54 33.45 43.85 11.95 55.43

2.85 39.43 60.21 11.82 8.95 39.42

5.11 4.94 3.84 2.95 2.59 0.51

5.89 3.01 47.37 12.80 51.29 14.91

7.96 36.98 32.91 30.88 15.16 4.99

7.13 42.22 58.15 58.68 3.26 29.12

3.46 24.67 77.05 3.89 109.11 21.15

4.72 11.64 51.49 16.87 66.62 25.02

4.21 19.15 14.06 18.79 38.95 5.39

5.27 19.13 80.44 59.07 105.31 45.69

6.05 5.05 91.90 67.83 85.33 64.29

True mean 5.00 8.00 0.00 0.00 10.40 5.60

True standard deviation 1.50 20.00 54.07 37.47 60.00 40.00

Sample average 4.93 7.77 0.70 0.64 9.40 6.08

Sample standard deviation 1.34 21.56 50.02 41.48 58.31 43.95

record the results in the first column of Table 7.7. We then generate 24 observations for excess

returns of the market index, using a mean of 8% and a standard deviation of 20%. We record

these observations in the second column of Table 7.7.

The bottom four rows in Table 7.7 show the true values for the means and standard devia-

tions as well as the actual sample averages and standard deviations. As you would expect, the

sample averages and standard deviations are close but not precisely equal to the true parame-

ters of the probability distribution. This is a reflection of the statistical variation that gives rise

to sampling error.

In the next step we wish to generate excess returns for stocks A and B that are consistent

with the CAPM. According to the CAPM, the rate of return on any security is given by

r rf (rM rf ) e

or using capital letters to denote excess returns,

R RM e

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Fifth Edition Theory

234 Part TWO Portfolio Theory

Coefficients Standard Error t Stat

TA B L E 7.8

Alpha”Stock A 0.46 11.12 0.04

Regression analysis

Beta”Stock A

for stock A 1.27 0.50 2.52

Residual Output”Stock A

Observation Predicted A Residuals Actual Returns

1 5.22 7.86 2.64

2 12.45 26.29 13.85

3 32.93 19.40 52.32

4 7.23 0.23 7.00

5 48.94 50.08 99.03

6 2.17 30.38 32.56

7 31.12 11.05 42.18

8 4.86 69.50 74.36

9 49.65 12.36 37.29

10 4.06 0.80 3.26

11 1.69 74.17 72.48

12 37.35 26.73 64.08

13 0.46 82.59 82.13

14 20.51 32.46 11.95

15 50.45 59.40 8.95

16 6.73 4.14 2.59

17 3.36 47.92 51.29

18 46.43 31.27 15.16

19 53.08 56.33 3.26

20 30.82 78.30 109.11

21 15.22 51.40 66.62

22 23.81 15.13 38.95

23 15.83 80.38 96.21

24 5.95 91.28 85.33

Therefore, the CAPM hypothesizes an alpha of zero in Equation 7.3. Given the values of

and RM, we need only random residuals, e, to generate a simulated sample of returns on each

stock. Using the random number generator once again, we generate 24 observations for the

residuals of stock A from a normal distribution with a mean of zero and a standard deviation

of 54.07%. These observations are recorded in the third column of Table 7.7. Similarly, the

randomly generated residuals for stock B use a standard deviation of 37.47% and are recorded

in the fourth column of Table 7.7.

The excess rates of return of stocks A and B are computed by multiplying the excess return

on the market index by beta and adding the residual. The results appear in the last two

columns of Table 7.7. Thus, the first two and last two columns of Table 7.7 correspond to the

type of historical data that we might observe if the CAPM adequately describes capital mar-

ket equilibrium. The numbers come from probability distributions consistent with the CAPM,

but, because of the residuals, the CAPM™s expected return“beta relationship will not hold ex-

actly due to sampling error.

We now use a regression program (again, from the “data analysis” menu of our spread-

sheet) to regress the excess return of each stock against the excess return of the index. The re-

gression routine allows us to save the predicted return for each stock, based on the market

return in that period, as well as the regression residuals. These values, and the regression sta-

tistics, are presented in Table 7.8 for stock A and Table 7.9 for stock B.

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Fifth Edition Theory

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7 Capital Asset Pricing and Arbitrage Pricing Theory

Coefficients Standard Error t Stat

TA B L E 7.9

Alpha”Stock B 0.39 9.22 0.04

Regression analysis

Beta”Stock B

for stock B 0.73 0.42 1.76

Residual Output”Stock B

Observation Predicted B Residuals Actual Returns

1 2.36 43.87 41.50

2 6.55 38.86 45.41

3 19.70 66.69 46.99

4 4.83 68.65 73.49

5 28.96 59.82 88.78

6 0.60 25.91 25.30

7 17.35 1.35 16.00

8 3.46 19.05 15.59

9 29.37 15.09 44.46

10 1.70 36.45 38.14

11 0.33 20.68 21.01

12 22.25 67.50 89.76

13 0.92 11.23 10.31

14 12.52 42.91 55.43

15 28.52 10.90 39.42

16 3.24 2.72 0.51

17 2.60 12.31 14.91

18 27.51 32.50 4.99

19 31.35 60.47 29.12

20 18.48 2.68 21.15

21 8.15 16.87 25.02

22 14.43 19.82 5.39

23 8.50 59.09 50.59

24 4.09 68.38 64.29

Observe from the regression statistics in Tables 7.8 and 7.9 that the beta of stock A is esti-

mated at 1.27 (versus the true value of 1.3) and the beta of stock B is estimated at 0.73 (versus

the true value of 0.7). The regression also shows estimates of alpha as 0.46% for A and

0.39% for B (versus a true value of zero for both stocks), but the standard error of these esti-

mates is large and their t-values are low, indicating that these are not statistically significant.

The regression estimates allow us to plot the security characteristic line (SCL) for both stocks,

shown in Figure 7.7 for stock A and Figure 7.8 for stock B.

The CAPM representation of the securities is shown in Figures 7.9 and 7.10. Figure 7.9

shows the security market line (SML) supported by the risk-free rate and the market index.

Stock A has a negative estimated alpha and is therefore below the line. This suggests that

stock A is overpriced, that is, its expected return is below that which can be obtained with

efficient portfolios and the risk-free rate. The negative estimated alpha is due to the effect

of the firm-specific residuals. Similarly, stock B plots above the SML. Here, it appears that

stock B is underpriced and has an expected return above that which can be obtained with the

market index and the risk-free asset (given by the SML).

Figure 7.10 shows the capital market line (CML) that is supported by the risk-free rate and

the market index. The efficient frontier is generated by the Markowitz algorithm applied to the

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

Essentials of Investments, and Arbitrage Pricing Companies, 2003