Chapter 9 Risk and the Cost of Capital 227bre44380_ch09_221-248.indd 226 10/09/15 09:59 PM bre44380_ch09_221-248.indd 227 10/09/15 09:59 PM
diversifiable risk. The variance of the returns on Microsoft stock was 428.^6 So we could say
that the variance in stock returns that was due to the market was .37 × 428 = 158, and the vari-
ance of diversifiable returns was .63 × 428 = 270.
The estimates of beta shown in Figure 9.2 are just that. They are based on the stocks’
returns in 60 particular months. The noise in the returns can obscure the true beta.^7 Therefore,
statisticians calculate the standard error of the estimated beta to show the extent of possible
mismeasurement. Then they set up a confidence interval of the estimated value plus or minus
two standard errors. For example, the standard error of Campbell Soup’s estimated beta in the
most recent period is about .16. Thus the confidence interval for Campbell Soup’s beta is .39
plus or minus 2 × .16. If you state that the true beta for Campbell Soup is between .07 and .71,
you have a 95% chance of being right. Notice that we can be equally confident of our estimate
of the beta of Dow and Microsoft.
Usually you will have more information (and thus more confidence) than this simple, and
somewhat depressing, calculation suggests. For example, you know that Campbell Soup’s
estimated beta was well below 1 in two successive five-year periods. Dow Chemical’s esti-
mated beta was well above 1 in both periods. Nevertheless, there is always a large margin for
error when estimating the beta for individual stocks.
Fortunately, the estimation errors tend to cancel out when you estimate betas of portfolios.^8
That is why financial managers often turn to industry betas. For example, Table 9.1 shows
estimates of beta and the standard errors of these estimates for the common stocks of six rail-
road companies. Three of the standard errors are close to .2. However, the table also shows the
estimated beta for a portfolio of all six railroad stocks. Notice that the estimated industry beta
is somewhat more reliable. This shows up in the lower standard error.The Expected Return on Union Pacific Corporation’s Common Stock
Suppose that in December 2014 you had been asked to estimate the company cost of capital
of Union Pacific. Table 9.1 provides two clues about the true beta of Union Pacific’s stock: the
direct estimate of .98 and the average estimate for the industry of 1.24. We suspect that the direct
figure may underestimate Union Pacific’s beta.^9 We will use the industry estimate of 1.24.BEYOND THE PAGEmhhe.com/brealey12eTry It! Beta
estimates for U.S.
stocksBeta Standard Error
Canadian Pacific 1.34 0.19
CSX 1.34 0.14
Kansas City Southern 1.27 0.20
Genesee & Wyoming 1.34 0.19
Norfolk Southern 1.16 0.16
Union Pacific 0.98 0.12
Industry portfolio 1.24 0.12❱ TABLE 9.1 Estimates of betas
and standard errors for a sample of
railroad companies and for an equally
weighted portfolio of these companies,
based on monthly returns from
December 2009 to November 2014.
The portfolio beta is more reliable than
the betas of the individual companies.
Note the lower standard error for the
portfolio.(^6) This is an annual figure; we annualized the monthly variance by multiplying by 12 (see footnote 22 in Chapter 7). The standard
deviation was √
428 = 20.7%
(^7) Estimates of beta may be distorted if there are extreme returns in one or two months. This is a potential problem in our estimates for
2004–2009 because there was one month in 2008 when Dow Chemical stock rose by 90%. (This month is not shown in Figure 9.2. It is
off the top of the chart.) Dow’s performance that month has a large effect on the estimated beta. In such cases statisticians may prefer
to give less weight to the extreme observations or even to omit them entirely.
(^8) If the observations are independent, the standard error of the estimated mean beta declines in proportion to the square root of the
number of stocks in the portfolio.
(^9) One reason that Union Pacific’s beta may truly be lower than that of the average railroad is that the company has a below-average debt
ratio. Chapter 19 explains how to adjust betas for differences in debt ratios.