excess return by .055. The purchase of 1.442 units of international equity increases expected excess return by: 1.442-.05 = .07215 (2.14) Thus, at the margin, selling domestic equity and purchasing international equity at a rate that keeps risk constant raises expected excess by .07215-.055 = .01715 (2.15) per unit of domestic equity sold. The signal provided by this marginal analysis is clear and intuitive. The investor should continue to sell domestic equity as long as the effect on portfolio expected excess returns is positive and the risk is unchanged. Unfortunately, of course, this increasing of expected return cannot go on indefinitely. As soon as the investor sells domestic equity and purchases international equity, the marginal contribution to risk of domestic equity begins to fall and that of international equity begins to rise. This effect is why the marginal analysis is only an approximation, valid for small changes in portfolio weights. Before we investigate what happens as the investor moves from domestic to international equities, however, we might consider what is the expected excess return on international equities for which the investor would be indifferent to such a transaction. Clearly, from the preceding analysis this point of indifference is given by the value, e,, such that: (1.442 *^ -.055) = 0 (2.16) In other words, the hurdle rate, or point of indifference for expected return, such that expected returns beyond that level justify moving from domestic to foreign equity, is e,= 3.8%. To put it differently, if the expected excess return on foreign equity is less than this value, then we would not have any incentive to purchase international equities. If we were to look only at the risks and not expected excess returns, we might suppose that because of the diversification benefit we would always want to hold some international equity, at least at the margin. In fact, when assets are positively correlated, as they are in this example, even the first marginal allocation creates marginal risk and requires an expected excess return hurdle in order to justify a purchase. Now suppose the investor has sold 10 percent of the domestic equity. In order to keep risk constant the investor can purchase 13.18 percent of international equity. Using the new values d = .5667 and f = .1318 in the above formulas we can confirm that the volatility of the portfolio remains 10 percent and that Ad = .148 and Af = .122. The impact on expected excess return of the portfolio per unit sold at this point is given by: rA " A/ ?f -ed= (1.212*.05-.055)=.01667 (2.17)