equity asset.10 Let rt be a (In - l)-vector of returns of risky assets from the perspective of country 1, which we will designate the home, or base currency, country.11 We arrange to have the first element of rt be the return of the home country equity (which obviously has no currency risk), the next element be the currency-hedged return of the equity of country 2 (or, over a short time interval, equivalently the domestic return of the equity in country 2), and so on through the nxh element, which is the return of the currency-hedged equity from country n. The n + 1st element is the return on holding currency from country 2, the n + 2nd element is the return on holding currency from country 3, and so on through the last element, which is the return on holding currency from country n. Let S1 be the (In - 1) X (In - 1) covariance matrix of rv We define rt similarly as the returns on risky assets from the perspective of country i. The first n elements are currency-hedged returns on the equities of countries 1 through n. The n + 1st element is the return on holding currency of country 1, the n + 2nd is the return on holding currency of country 2, and so on through the n + (i - 1 )st element, which is the return on holding currency of country i - 1. The n + z'th element is the return on holding currency of country i + 1, and so on through the last element, which is the return on holding currency of country n. For example, in a four-country world including the United States, Japan, Europe, and the United Kingdom, the four return vectors, rt, r2 r2, r4, would include the following assets: rx = U.S. equity, Japan equity, Europe equity, U.K. equity, yen, euro, pound r2 = U.S. equity, Japan equity, Europe equity, U.K. equity, dollar, euro, pound r2 = U.S. equity, Japan equity, Europe equity, U.K. equity, dollar, yen, pound r4 = U.S. equity, Japan equity, Europe equity, U.K. equity, dollar, yen, euro Let Si be the (2n - 1) X (In - 1) covariance matrix of rc We will find it convenient to define a matrix, I(, which transforms r1 into rt. The elements of I are all O's, I's, and -I's, and it has a particularly simple structure. Of course, I1 is simply the identity; it transforms rx into rv If we partition each of I2 through I into four submatrices, an n X n upper-left corner, the n X (n - 1) upper-right corner, the (n - 1) X n lower-left corner, and the (n - 1) X (n - 1) lower-right corner, only the latter is interesting. The upper-left corner is always the identity; currency-hedged returns on equities are the same from each country perspective. The upper-right and lower-left corners are always identically 0. The (n - 1) X (n - 1) lower-right submatrix has a column of -I's in the (i - l)st 10The reader should think of our equity asset as an equity market index, or more generally as a market capitalization weighted basket of equities, bonds, and other assets. At the cost of slight notational complexity, one could easily include multiple assets in each country. "There is nothing special about the home country except that it establishes a basis for defining notation.