an alternative methodology for linking sources of return that is described as "simply additive, yet formally exact." We present this methodology in three parts. First, we show that the geometrically compounded returns can be written as the sum of variables that are functions of the portfolio returns. We refer to the values of these variables at time t as diff(£), which are defined as follows: diff(l) = [l+R(l)] dif f (2) = [1 + R(l)] x [l + R(2)] - [l + R(l)] diff(3) = [l + R(l)] x [l + R{2)] x [l + R(3)] - [l +R(T)] x [l + R(2)] diff(4) = [l+R(l)]x[l + R(2)]x[l + R(3)]x[l + R(4)]-[l+R(l)]x[l+R(2)]x[l+R(3)] and so on. In general we can write t t-\ diff(f) = f[[l+R(;)]-f[[l+R(;)] (19.47) ;=1 ;=1 It follows from these definitions that the geometric return can be written as the sum of diffs, that is, ;=1 ,=1 Equation (19.48) is important because it allows us to write the geometric return as a sum. Second, we rewrite the diffs as follows. Consider diff(2). Let's expand it so that we have diff (2) = [l + R(T)] x [l + R(2)] - [l + R(T)] = l+R{2) + R(l) + R{l)xR(2)-l-R(T) = [l+R(T)]xR(2) Similarly, working with diff(4), we get diff(4) = [l+R(l)]x[l+R(2)]x[l+R(3)]x[l+R(4)]-[l + R(l)]x[l+R(2)]x[l + R(3)] = [l+R(l)]x[l+R(2)]x[l+R(3)] + [l+R(l)]x[l + R(2)]x[l + R(3)]xR(4) -[l + R(l)]x[l+R(2)]x[l+R(3)\ = {[l + R(l)]x[l+R(2)]x[l+R(3)]}xR(4) Generally, we have