we must confront head-on some of the complexity that comes with foreign exchange risk. Consideration of foreign exchange risk adds a number of complexities in the real world, most of which we will safely ignore, but some of which we must address. We will ignore the complexity associated with different securities that can be used to add or hedge foreign exchange risk. One could use forward contracts, swaps, futures, or simply short-term borrowing and lending. We will also ignore the risk of depreciation of the profits earned during a finite period of time, a small effect sometimes referred to as the "cross product." If the time period is sufficiently short, the profit is arbitrarily small relative to the exposure, and so the risk of depreciation of the profit can be ignored. Finally, we will ignore the effects of inflation. We can think most simply of a foreign exchange hedge as any position that benefits when a foreign currency depreciates, but does not create any other risk exposures. One obvious such position is a forward contract. Another is a short-term loan denominated in the foreign currency and invested in domestic short rates. Think of the currency hedge as the amount of such a loan. If a dollar-based investor borrows in yen, exchanges the yen for dollars at the beginning of the period, and invests the dollars in the U.S. short-term deposits, then depreciation of the yen allows the investor to repay the loan with fewer dollars, and thus benefit from the depreciation. The profit on the loan would exactly offset the loss from currency depreciation of a similarly sized yen-denominated investment. Expected returns on such currency positions cause much confusion. Many investors have heard that currency is a zero-sum game, and thus assume the expected return on currency exposures is zero. This is not true, even in equilibrium. Currencies can have positive or negative expected returns. Consider the expected returns on currencies in our simple two-country world and focus on the relationship between (A and (j,^. The first term is the expected return to a dollar investor of holding yen. The second term is the expected return to a yen investor of holding dollars. Clearly, in a rational, efficient equilibrium these two different expectations should be consistent with each other. If one exchange rate is expected to go up, it would seem intuitive that the other must be expected to go down. The most natural intuition might seem to be that H£=-H£ (6-6) Interestingly, the relationship is not quite that simple. Consider that if the exchange rate for S/yen goes from 1 to 1.1, then there is a 10 percent appreciation of the yen from a dollar perspective and a .1/1.1 = 9.09% depreciation of the dollar from the yen perspective. Conversely, a move from 1 to .9-that is, a 10 percent depreciation of the yen from a dollar perspective-implies an appreciation of 11.1 percent of the dollar from a yen perspective. More generally, the percentage appreciation of one currency relative to another is always larger than the percentage depreciation of the second currency relative to the first. If one currency appreciates by x from a second currency perspective, then the second currency depreciates by x/(l + x) from the perspective of the first. This