of K + 1 compounded sources of returns where each source of return, S^Jit), is defined in terms of log returns. The key to generating multiperiod sources of return that are additive was the conversion of percent returns to log returns. Step 3: Transform (19.41) back to percent returns. Originally, we defined all returns as percent returns. Therefore, step 3 is to transform (19.41) back to percent returns. To do this, define the new adjustment factor: t+T / \ Multiperiod portfolio percent return _ rt+ (t + j) Multiperiod portfolio log return r{+T (t + j) ]J[l+rt+T(t + j)]-l _ ;=0 (19.42) X nog,p (' + ;') ;=0 The T + 1 period cumulative attribution effect for the &th source, based on percent returns, is given by T YK(t + j)s, (t + j) sr(t)= Kt+T(t) Kt+T(t) Applying these transformations to (19.43) we are left with the result for cumulative percent returns: -it+T u\ ct+T, t]\l + rJt + j)]-l = ^^ = ^^ + ^^ + --- + ^^ (19.44) }il P J Kt+T(t) Kt+T(t) Kt+T(t) Kt+T(t) which yields T K Y[[l+rp(t + j)]-l = ^S{+T(t) (19.45) ;'=0 k =0 Note that all we have done in the preceding analysis is convert log returns back to percent returns. Equation (19.45) shows that the cumulative, multiperiod percent return is equal to the sum of cumulative, multiperiod sources of return (defined as percent returns). These results extend directly to the case where our focus is on active returns. In this case, the multiperiod active return is T T K Y[[l+rp(t + j)}-Y[[l+rb(t + j)] = ^St+T(t) (19.46) ;=0 ;'=0 k =0