capitalization weighted portfolio, we have shown that in equilibrium the expected excess return of each asset must be proportional to the covariance of that asset's return with the returns of the market portfolio. That is, we can substitute the market portfolio for the optimal portfolio in equation (4.18) and obtain: |i = k o (4.19) In particular, in equilibrium assets whose returns are uncorrelated with the market portfolio have zero expected excess return. This is an important result, and we will return to its implications in Chapter 12. Switching to vector notation, let 0 be the vector of returns of all assets and m' be the returns of the market portfolio, then the vector of covariances of asset returns with the market portfolio returns is given by Cov((|), m') = £ m. And finally, we can write the formula for the vector of equilibrium expected excess returns for all assets as: |i = k Z m (4.20) Now, assume there are n investors with the proportion of wealth of the 2th investor given by W. In the general case, the total portfolio holdings are given by: Total portfolio holdings = X -=1 n (Wi). w   2-V = 1 i=\,n - ^ <'=!,"   - ^ <'=!," \X> J . rt . lm (4.2i; However, we know that in equilibrium the total portfolio holdings must equal the market capitalization weights, m. Thus, we can solve for k. 1 Z,-=i W; kK; (4.22) Substituting back into the formula for the equilibrium expected excess returns, we have for each asset Hi = ^('=l,H W kK; (4.23) The term in parentheses is the wealth of investor i divided by the investor's risk aversion. The inverse of risk aversion is risk tolerance. Thus, the greater the