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The Industry Version of the Index Model
Not surprisingly, the index model has attracted the attention of practitioners. To the extent
that it is approximately valid, it provides a convenient benchmark for security analysis.
A portfolio manager who has neither special information about a security nor insight
that is unavailable to the general public will take the security’s alpha value as zero, and,
according to Equation 8.9, will forecast a risk premium for the security equal to b
i
R
M
. If
we restate this forecast in terms of total returns, one would expect
E(r
i
)
5 r
f
1 b
i
3E(r
M
)
2 r
f
4
(8.25)
A portfolio manager who has a forecast for the market index, E ( r
M
), and observes the
risk-free T-bill rate, r
f
, can use the model to determine the benchmark expected return for
any stock. The beta coefficient, the market risk, s
M
2
, and the firm-specific risk, s
2
( e ), can
be estimated from historical SCLs, that is, from regressions of security excess returns on
market index excess returns.
There are several proprietary sources for such regression results, sometimes called “beta
books.” The Web sites for this chapter at the Online Learning Center ( www.mhhe.com/bkm )
also provide security betas. Table 8.3 is a sample of a typical page from a beta book. Beta
books typically use the S&P 500 as the proxy for the market portfolio. They commonly
employ the 60 most recent monthly observations to calculate regression parameters, and use
total returns, rather than excess returns (deviations from T-bill rates) in the regressions. In
this way they estimate a variant of our index model, which is
r
5 a 1 br
M
1 e
*
(8.26)
instead of
r
2 r
f
5 a 1 b(r
M
2 r
f
)
1 e
(8.27)
To see the effect of this departure, we can rewrite Equation 8.27 as
r
5 r
f
1 a 1 br
M
2 br
f
1 e 5 a 1 r
f
(1
2 b) 1 br
M
1 e
(8.28)
Comparing Equations 8.26 and 8.28, you can see that if r
f
is constant over the sample
period, both equations have the same independent variable, r
M
, and residual, e. Therefore,
the slope coefficient will be the same in the two regressions.
15
However, the intercept that beta books call ALPHA, as in Table 8.3 , is really an estimate
of a 1 r
f
(1 2 b ). The apparent justification for this procedure is that, on a monthly basis,
r
f
(1 2 b ) is small and is likely to be swamped by the volatility of actual stock returns. But
it is worth noting that for b
≠ 1, the regression intercept in Equation 8.26 will not equal the
index model a as it does when excess returns are used as in Equation 8.27.
15
Actually, r
f
does vary over time and so should not be grouped casually with the constant term in the regression.
However, variations in r
f
are tiny compared with the swings in the market return. The actual volatility in the T-bill
rate has only a small impact on the estimated value of b .
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P A R T I I
Portfolio Theory and Practice
Always remember as well that these alpha estimates are ex post (after the fact) mea-
sures. They do not mean that anyone could have forecast these alpha values ex ante (before
the fact). In fact, the name of the game in security analysis is to forecast alpha values
ahead of time. A well-constructed portfolio that includes long positions in future positive-
alpha stocks and short positions in future negative-alpha stocks will outperform the market
index. The key term here is “well constructed,” meaning that the portfolio has to balance
concentration on high-alpha stocks with the need for risk-reducing diversification as dis-
cussed earlier in the chapter.
Much of the other output in Table 8.3 is similar to the Excel output ( Table 8.1 ) that we
discussed when estimating the index model for Hewlett-Packard. The R -square statistic
is the ratio of systematic variance to total variance, the fraction of total volatility attribut-
able to market movements. For most firms, R -square is substantially below .5, indicating
that stocks have far more firm-specific than systematic risk. This highlights the practical
importance of diversification.
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