\o% The total log return, including currency, is given by K,Jt) = <At) + E^fi) (19-33) Now, we consider cumulative returns. The T + 1-period percent return, denoted by R^T(t)-from t to t + T-is the product of T + 1 one-period returns, that is, T K+T(t) = Y[[i + K(t + ;)]-i (19.34) ;=0 The T + 1 period cumulative log return, R£T (t)-again, from t to t + T-is the sum of T + 1 one-period log returns, that is, O') = i>io6sw(' + ;) (19"35) ;=0 Equation (19.35) shows the time aggregation property of log returns. Namely, the sum of one-period returns is equal to the multiperiod return. This is a very convenient property that is not shared by percent returns. Suppose that instead of using percent returns, we assume that all returns are computed using log returns. In this case, we write the portfolio log return as a function of K + 1 sources of return. Since log returns are additive over time, one may think that we should work with log returns since time aggregation would be easier (i.e., additive and, therefore, no cross terms to worry about). However, at a particular point in time log returns are not additive across assets. That is to say, when using log returns on individual assets, the return on the portfolio is no longer equal to the weighted average of individual asset returns. This leads to an obvious dilemma about how to compute returns. We can summarize our dilemma of choosing log versus percent returns as follows: II Percent returns are additive when dealing with cross sections. That is, a one-period portfolio return using percent returns is a weighted average of one-period asset level percent returns. Multiperiod percent returns are multiplicative. II Log returns are additive across time but not in cross sections. That is, multi-period log returns are the sum of successive one-period returns. However, one-