proposed to measure investors' utility as a function of return distributions. As noted earlier, while investors are generally very sensitive to losses, they often seem much less cognizant of the risk that can lead to losses. We will explore some of these issues in the next chapter. To be concrete, we will here use the statistical measure, volatility, to quantify risk. For example, suppose we have some relevant data that allows us to measure the volatilities and correlation of the returns of domestic and international equities. Let these quantities be o^, O,, and p, respectively. Then one example of a simple risk function would be the volatility of the portfolio, given by: Risk(i, f) = [d2 .o2d+f2 .oj +2.d.f.od.of .p)m (2-1) Let us use the notation Ad to refer to the marginal contribution to portfolio risk of domestic equities. This quantity is defined to be the derivative of the risk function with respect to the quantity of domestic equity, that is, the difference in the risk of portfolios that have the same amount of international equities, but a small difference, 5, in domestic equities, divided by 5. Thus, we can formalize this as an equation: Arf(5) = Risk(<i + 5, f)-Risk(i, f) (22) and let Ad be the limit of Ad (5) as 5 goes to zero. Similarly, the marginal contribution to risk of international equities is given by A,, which is defined as the limit of A, (5) as 5 goes to zero, where: A (S)_Riskfo f + 5)-Risk(i, f) (23) These marginal contributions to risk are the key to optimal portfolio allocations. As we shall see, a condition for a portfolio to be optimal is that the ratio of expected excess return to marginal contribution to risk is the same for all assets in the portfolio. Let us return to the question of whether we can improve the portfolio by selling domestic equity and buying international equity. The ratio of marginal contributions to risk is Adl A,. Let the expected excess returns on domestic and international equities be given by ed and ef, respectively. Now suppose eje, is less than AjA,. How much international equity must we purchase in order to keep risk constant if we sell a small amount of domestic equities? The rate of change in risk from the sale of domestic equity sales is -Ad per unit sold. In order to bring risk back up to its previous level, we need to purchase {Adl A.) units of international equity. The effect on expected return to the portfolio is -ed per unit sold of domestic equity and +{AJ Af)ef from the purchase of an amount of international equity that leaves risk unchanged. If, in this context, expected return is increased, then we should continue to increase the allocation to international equity. If expected return is decreased, then we should sell international equity and buy domestic equity. The only case in which the expected return of the portfolio cannot be increased while holding risk constant is if the following condition is true: