p *, between the exchange rate and the U.S. and Japanese equities, respectively, from a U.S. dollar perspective. Note that the correlations between each asset and the exchange rate from the Japanese yen perspective are simply -1 times the correlation from the U.S. dollar perspective; that is: PYX}=-P% (6-8) Beyond Siegel's paradox, which relates expected excess returns on currencies from different country perspectives to the variance of the exchange rate, there are similarly derived relationships between the expected excess returns on investment assets and currencies from different currency perspectives that involve covari-ances. One example comes up only when there are more than two currencies. Consider the expected excess return on holding the euro from the perspective of a yen investor. It turns out that this expectation is equal to the sum of the expected excess return to holding dollars from a yen perspective and the expected excess return to holding euros from a dollar perspective, less the covariance of returns to holding yen and returns to holding euros from a dollar perspective. Of course, this covariance term doesn't enter our two-country example because we have only two currencies. There is also a relationship between the expected excess returns on U.S. equity from a dollar versus a yen perspective, that is, between |xj^ and jxj. In this case again, it is not the variance of the exchange rate that relates the two expectations, but rather the covariance between the exchange rate and the stock return that comes into play. A similar relationship exists between jx* and |xY. Notice that we are considering currency-hedged stock returns in both cases, so the covariance that drives this expected return difference is not due to the currency effect directly entering one of the returns, but rather is a function of the expectation being taken from different currency perspectives.9 To gain an intuition about this covariance term in the expected return relationship, consider the hedged return on U.S. equity from a Japanese perspective. Suppose there is a positive correlation between currency hedged U.S. equity returns from a yen perspective and the returns to a yen investor holding dollars. When this correlation, p^ is positive, returns on U.S. equity will have a component that moves with the dollar when viewed from a yen perspective. Recall that expected returns on dollar holdings, from a yen perspective, have a positive component, oj, due to Siegel's paradox. To the extent U.S. equity returns mirror those of the dollar, this effect similarly increases the expected return on U.S. equity from a yen perspective relative to expected returns from a dollar perspective. If we form a projection of U.S. equity returns on dollar currency returns from a yen perspective, we can decompose the equity returns into a component that is a multiple of the dollar returns and an uncorrelated component. This projection on the dollar return has a coefficient that is the ratio of the above-mentioned covariance (between U.S. equity returns and returns on holding dollars from a yen perspective) and the variance to a sThe existence of these covariance terms in the expected excess return relationships can be derived using a multivariate version of Ito's lemma.