The Index Model and Realized Returns
We have said that the CAPM is a statement about ex ante or expected returns, whereas in practice all anyone can observe directly are ex post or realized returns. To make the leap from expected to realized returns, we can employ the index model, which we will use in excess return form as
We saw in Section 10.1 how to apply standard regression analysis to estimate equation 10.9 using observable realized returns over some sample period. Let us now see how this framework for statistically decomposing actual stock returns meshes with the CAPM.
We start by deriving the covariance between the returns on stock i and the market index. By definition, the firm-specific or nonsystematic component is independent of the mar-ketwide or systematic component, that is, Cov(RM,e;) = 0. From this relationship, it follows that the covariance of the excess rate of return on security i with that of the market index is
Note that we can drop « from the covariance terms because « is a constant and thus has zero covariance with all variables.
302 PART III Equilibrium in Capital Markets
Because Cov(R,, RM) = p, oM, the sensitivity coefficient, p,-, in equation 10.9, which is the slope of the regression line representing the index model, equals p = Cov(R;, Rm) pi ^M
The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship, except that we replace the (theoretical) market portfolio of the CAPM with the well-specified and observable market index.
|
Stock |
Capitalization |
Excess Return |
Standard Deviation | |
|
A |
$3,000 |
1.0 |
10% |
40% |
|
B |
$1,940 |
0.2 |
2 |
30 |
|
C |
$1,360 |
1.7 |
17 |
50 |
The single factor in this economy is perfectly correlated with the value-weighted index of the stock market. The standard deviation of the market index portfolio is 25%.
a. What is the mean excess return of the index portfolio?
b. What is the covariance between stock B and the index?
c. Break down the variance of stock B into its systematic and firm-specific components.
The single factor in this economy is perfectly correlated with the value-weighted index of the stock market. The standard deviation of the market index portfolio is 25%.
a. What is the mean excess return of the index portfolio?
b. What is the covariance between stock B and the index?
c. Break down the variance of stock B into its systematic and firm-specific components.
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