distributions of returns are known (that is, if r, jie, |ife, 0E, 0&, and p are given), then it is a relatively easy mathematical optimization exercise to choose asset weights that maximize utility. As was discussed in Chapter 2, the optimal weights must be ones for which the ratio of the marginal contribution to portfolio expected return to contribution to portfolio risk is the same for equity and bonds. The contributions to portfolio expected returns for equity and for bonds are given by (jj,e - r) and (|ife - r), respectively. Given a set of weights we and wb> the marginal contribution to portfolio risk for an increase in the weight in equity is given by: wr .0? +wh .0, (4.5; Similarly, the contribution to portfolio risk for a marginal increase in the weight in bonds is given by: wh .0? +uis .o. (4.< Thus, one condition for the portfolio weights to be optimal is that: The risk aversion parameter, X, determines how much risk is desired given the available expected returns. Given the form of the utility function, it is clear that for the portfolio to be optimal it must be the case that marginal changes in any portfolio weights must create a change in expected return that is equal to .5 X times the marginal change in portfolio variance. In particular, for a marginal change in the weight in equity, wg, it must be the case that: He -r = .5.X.(2.we .c? +2.wb .o^) (4.8) The quantity in parentheses is the marginal change in portfolio variance given a small change in the weight w . The analogous condition must hold for bonds. Thus, we have the additional condition: y _ Vc-r _ Vb-r A"-------------- 2------------------- "------------- 2------------------- 4- Wr .0, +tVh *Grh Wh .Oh +W, .O.i,