t) return on a portfolio. The geometric return over T periods can now be written as n[l+r(,)]-l = x(n[l+^)]xr(i-l (19-50) ;=1 t=\ [;=0 J where we define r (0) = 0. Equation (19.50) allows us to write the T-period geometric return as the sum of T one-period returns-the R(£)'s-which are scaled by one plus the geometric portfolio return from time 0 through time t-1. Consider the example where we compute a portfolio's return over four periods. In this case we have 4 fj[l+f(;)] = f(l) + [l + f(l)]xf(2) + [l + f(2)]xf(3) + [l+f(3)]xr(4) (19.51) ;=i The third part of the methodology involves writing the one-period portfolio return (at time t) in terms of its constituent level weights and returns. That is, N where we assume there are N assets in the portfolio and w represents the weight on the wth asset. Substituting the expression for the portfolio return into (19.50) yields n[i+^)]=z|n[i+^)]xZwA(4 <19-52) ;'=1 t=\ [;'=0 "=1 J Equation (19.52) forms the basis for return attribution and linking sources of return at the asset (and any subsequent grouping) level. To see this, let's take the example where we have a portfolio with three assets (N = 3) and the portfolio's return is computed over four periods (T = 4). 4 3 3 n[l+f(;]]-l = JtfA(l) + [l + ^)]xX"'A(2) + [l+'(l)Hl+K2)] ;=1 =1 n=l (1953) x^^^(3) + [l+r(l)]x[l+r(2)]x[l + r(2)] 3 x5>2U4)-l Let's break (19.53) down period by period (and ignore the minus ones).