perspectives is given the name "Siegel's paradox."7 One consequence is that u^ and u,^ are not simply equal but opposite in sign as in equation (6.6). In fact, it is very possible for both li|- and Li^to be positive at the same time. This strange behavior of currency expected returns makes many people uncomfortable. It feels like a magic trick. How can investors in both countries rationally expect their foreign currency holdings to appreciate? In order to understand this phenomenon, consider a simple world in which the S/yen exchange rate starts at 1. At the end of a period a coin is flipped: If it comes up heads, the S/yen exchange rate is 2; if it comes up tails, the S/yen exchange rate is .5. From a dollar perspective, a person holding yen has two outcomes with equal probability, a return of 100 percent or a return of -50 percent. The expected return is positive, in fact is 25 percent. But consider the symmetry of the situation. The expected return to someone viewing the world from a yen perspective holding dollars is also positive 25 percent. How can this be? How can individuals from both perspectives and identical information and expectations have positive expected return from holding each other's currency? The simplest answer is that they could not both rationally expect to be better off if all wealth was measured in the same units-but as long as they each measure their wealth from their own different currency perspective, they can both expect to be better off holding some of the other's currency-at least as measured in their own units. The more volatility there is to the exchange rate, the more these currency expected returns are biased upward relative to each other. The relationship that must be true between li^. and u,^.is as follows: )4=-u|+oi (6.7) where ox is the volatility of the exchange rate.8 Why do we care about this curiosity of exchange rates? After all, we assume the time of our period is arbitrarily short so that the actual returns on yen and dollar during this period are arbitrarily close to equal, but of opposite sign. The answer to why we care is that this variance term in the expected excess returns relationship pins down the equilibrium returns on all assets in a world with multiple currencies. Consider again the portfolio optimization problem discussed earlier. In addition to the volatilities and correlation of the equities, o, Op and p^., and the 7The name Siegel's paradox" is widely used. The reference is to a paper: Siegel, J. J., 1972, "Risk, Interest Rates and the Foreign Exchange," Quarterly Journal of Economics 89, 173-175. 8The origin of this variance term is the positive curvature of the function, 1/x, relating yen/$ to $/yen. The more variance there is in the distribution of potential outcomes, the more this curvature increases the expected value of the foreign exchange holding. The mathematical theorem required to show that this is the correct formula involves taking a limit as the length of the time period goes to zero and is known as Ito's lemma. An intuitive derivation of Ito's lemma can be found in Robert C. Merton's text, Continuous-Time Finance (Black-well, 1990).