_ ef ^7"V (2-5) Thus, in this simple two-asset example we have derived a simple version of the general condition that the expected return divided by the marginal contribution to portfolio risk should be the same for all assets in order for a portfolio to be optimal. If this condition is not met, then we can increase the expected return of the portfolio without affecting its risk. More generally, we can consider sales and purchases of any pair of assets in a multiple asset portfolio. The above analysis must hold, where in this context let the risk function, Risk(zf), give the risk for a vector iv, which gives the weights for all assets. Let Riskm("/, 5) give the risk of the portfolio with weights w and a small increment, 5, to the weight for asset m. Define the marginal contribution to portfolio risk for asset m as Am, the limit as 5 goes to zero of: Am(5)=Rlsk-K5)"RlskM (2.6) 5 Then, as earlier, in an optimal portfolio it must be the case that for every pair of assets, m and n, in a portfolio the condition is true. If not, the prescription for portfolio improvement is to buy the asset for which the ratio is higher and sell the asset for which the ratio is lower and to continue to do so until the ratios are equalized. Note, by the way, that if the expected return of an asset is zero then the optimal portfolio position must be one in which the A is also zero. Readers familiar with calculus will recognize that this condition-that the derivative of the risk function is zero-implies that the risk function is at a minimum with respect to changes in the asset weight. Let us consider how this approach might lead us to the optimal allocation to international equities. To be specific, let us assume the values shown in Table 2.1 for the volatilities and expected excess returns for domestic and international equity, and for cash. Assume the correlation between domestic and international equity is .65. We will use as the risk function the volatility of the portfolio: Riskfi, f) = id2 .o2d+f2.o2+2.d.f.od.Of -p)112 (2-8) In order to make the analysis simple, let us assume that the investor wants to maximize expected return for a total portfolio volatility of 10 percent. Consider an