in his work. What makes the CAPM interesting, however, is that it goes beyond this individual investor optimization problem for given expected excess returns. Rather than take jie and ji6 as given, as we did in the two-asset example, CAPM asks for what values of these mean returns will the demand for assets be equal to the outstanding supply. In our simple context of investors holding equity, bonds, and cash, CAPM asks what values for jj,e and ji6 will lead the sum of demands for equity and bonds of the optimizing investors to be equal to the market capitalization weights, e and b. In this simple world, we can easily develop an intuition of what the answer must be. First, since all investors have identical information, they must each hold the same expected excess returns. In optimizing portfolio allocations the only difference across investors will be the risk aversion parameter. One might expect investors with higher risk aversion to hold more bonds and less equity, remaining fully invested. In fact, we can see from the above equations that higher risk aversion will cause an investor to hold proportionally more cash and both less bonds and less equity. All investors, however, will hold the same ratio of bonds to equity. The intuition behind this result follows directly from the requirement that expected excess return be proportional to contribution to portfolio risk. If a more risk-averse investor decided to hold more bonds and less equity than other investors, then the marginal contribution to risk of bonds in that investor's portfolio would be higher than that of other investors. But in equilibrium expected excess returns are assumed to be the same across investors. Thus, following the example in Chapter 2, for the investor holding more bonds and less equity a higher-returning portfolio with the same risk could be obtained by selling bonds and adding a combination of equity and cash. If all investors hold the same ratio of bonds to equity, then the equilibrium ratio of bonds to equity must be b/e, the ratio of the outstanding market capitalizations. More generally, we see from the matrix version of the equation for optimal portfolio weights that when there are more than two assets the optimal portfolio weights of investors with different degrees of risk aversion will still be proportional. Thus, in the general case each investor must hold some fraction of the market capitalization weighted portfolio and some fraction in cash. Also notice that the marginal contribution to portfolio risk for each asset is proportional to the covariance of the returns of that asset with the portfolio. For example, the covariance of equity returns with portfolio returns, Covariance(re,rp) = oe5p =we .02e +wb .oefi = Op (Equity marginal contribution to risk) For optimal portfolios the expected excess returns for each asset are also proportional to the marginal contributions to risk. Thus, in optimal portfolios, the expected return of an asset is proportional to the covariance of that asset with the portfolio. That is, for each asset i and a constant proportionality k, the expected excess return, ji, is given by the following equation: M-, = * ,,P (4.18)