For starters, one central prediction of the CAPM is that the market portfolio is a meanvariance efficient portfolio. Consider that the CAPM treats all traded risky assets. To test the efficiency of the CAPM market portfolio, we would need to construct a value-weighted portfolio of a huge size and test its efficiency. So far, this task has not been feasible. An even more difficult problem, however, is that the CAPM implies relationships among expected returns, whereas all we can observe are actual or realized holding period returns, and these need not equal prior expectations. Even supposing we could construct a portfolio to repre- sent the CAPM market portfolio satisfactorily, how would we test its mean-variance efficiency? We would have to show that the reward-to-variability ratio of the market portfolio is higher than that of any other portfolio. However, this reward-to-variability ratio is set in terms of expectations, and we have no way to observe these expectations directly.

The problem of measuring expectations haunts us as well when we try to establish the validity of the second central set of CAPM predictions, the expected return beta relation- ship. This relationship is also defined in terms of expected returns E(ri) and E(rM):

E(ri) rf i[E(rM) rf] (10.8) The upshot is that, as elegant and insightful as the CAPM is, we must make additional assumptions to make it implementable and testable.

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 ex- cess return form as Ri i iRM ei (10.9) 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 frame- work 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,ei) 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

Cov(Ri, RM) Cov( i RM ei, RM) i Cov(RM, RM) Cov(ei, RM) i 2

Note that we can drop i from the covariance terms because i is a constant and thus has zero covariance with all variables.

III. Equilibrium In Capital Markets

10. Single−Index and Multifactor Models The McGraw−Hill Companies, 2001