is equal to the &th element of bp(t - l)F(t). Let SQ(t) represent the contribution from the total specific return.9 This implies that there are K + 1 sources of return. Using these definitions we write equation (19.24) as rp(t) = YSk(t) + S0(t) (1^.25) k=i where the returns in (19.25) are defined in terms of percent changes. The T-period (T > 0) portfolio total return (cumulative return over T periods) is defined as t+T-l ■(t)=f[[l+rt(t + h-l)]-l = f[ 1 + Ysk(t + b-i) -1 (19.26) ;=0 When h = 1, the one-period return is rHt) = r'At), by definition. Our goal is to determine the multiperiod attribution from a particular source. A natural definition of the T-period attribution from the kxh source is the cumulative return from that source, i.e., T stk+T-1(t) = Y[[i+sk(t + b-i)] (19>27) h=\ Note that the definition of portfolio return in (19.26) and source of return in (19.27) are incompatible-that is, you cannot identify (19.27) by using (19.26) due to the presence of cross terms between sources. Upon closer inspection, (19.26) shows that the multiperiod portfolio return is the product of sums of sources of return. This product of sums results in cross terms, which makes it impossible to isolate the source of any one return. For example, suppose that T = 2 (two periods) and K = 2 (two sources). In this case, the two-period return (from t- 1 to t + 1) is l + r;+1(t) = [l + r;(t)][l + r;+1(t+l)] = [l+S0(t) + S1(t) + S2(t)\ x[l + SQ(t + l) + S1(t + l) + S2(t + l)] = i+s0(t)+s1(t)+s2(t)+s0(t+i)+s1(t+i)+s2(t+i) (1928) +S0(t)S0(t+T) + S0(t)S1(t + T) + S0(t)S2(t + T) +S1 (t)S0 (t + l) + S1 (f )5j (t + l)+ S1 (t)S2 (t +1) +S2(t)S0(t + 1) + S2(t)S1(t +1) + S2(t)S2(t +1) 'Earlier we decomposed the specific return into N components.