319 At time t = 1:

Contribution to geometric return = w^R^l) + w2R2(l) + w2R3(l) At time t = 2:

Contribution to geometric return = [1 + r(l)] X [w^R^l) + w2R2(2) + w3R3(2)] At time t = 3:

Contribution to geometric return = [1 + r(l)] X [1 + r(2)] X [u'1R1(3) + "/2R2(3) + "/3R3(3)]

At time t = 4:

Contribution to geometric return = [1 + r(l)] X [1 + r(2)] X [1 + r(3)] x[u'1R1(4)+W;2R2(4) + ^3R3(4)]

Next define T('-i)=n[i+f('')] where y(0) = 1. Using this notation, we can write asset l's contribution to the portfolio's geometric return as

^^(D + YtlKRi (2)+ 7(2^^(3)+ 7(3)^^(4) (19.54)

Generally, the nth asset's contribution to the portfolio return is

^y(t-l)ivnRJt) t=\ We can now rewrite (19.52) so that the portfolio's geometric return is T NT

Y[[l+r{j)]-l = J^j(t-l)wnRH(t)-l (19.55) ;'=1 w=l t=l

This concludes our description of Mirabelli's methodology. In summary, we've taken the cumulative product of returns (i.e., geometric returns) and expressed them as the sum of one-period returns. Each period's contribution to return (at time t) is scaled by the portfolio's geometric return from the start of the attribution period through t-1. Finally, note that although we can write the geometric return as the sum of one-period returns without using any approximations, cross terms are still involved.

This completes our description of the computations behind multiperiod return attribution. The results on linking hold both of the methods for generating sources of return, the factor model-based approach and the asset grouping methodology. Next, we turn our attention to international equity portfolios.

RETURN ATTRIBUTION OH INTERNATIONAL PORTFOLIOS

In this section we explain return attribution in the context of international equity portfolios. We assume that such portfolios may hold currency and equity futures as well as forwards, American depositary receipts (ADRs), cash, and similar instruments.