portfolios or factor-mimicking portfolios. To keep things simple and to facilitate our example, we assume that there is only one factor that can explain returns, and that factor is market capitalization. The cross-sectional return model, when there is only one factor, is given by Ri(t) = bf,(t-l)Fi(t) + ut(t) (20.36) where brn{t - 1) is an N X 1 vector of exposures and Rf{t) is an N-vector of a cross section of asset returns. The weighted least squares estimator of F*{t)-where the weights are market capitalizations-can be written as \tft+ 1\ T>*t+\\ y c{t-l)bi{t-l)Rf-{t) covanance £?(£-1), R (t)\ ^-t n n FUt) =--------------- L-r---------- -,---- i = ^L---------------------------- (20-37) variance[^-l)] ^ _1)bi{t _lf "=1 where cjt - 1) represents the market weight on the Kth asset and we have imposed the assumptions that the exposures are standardized to have a cap-weighted mean of zero.14 Now we can write the estimate of the factor returns as the weighted average of the original N asset returns. N Fft^wjt-W'it) (20.38; n=\ cn(t-l)bj(t-l) where w{t-l)----------------------------- ^cjt-Dbiit-if Equation (20.38) shows that the return to the market capitalization factor is essentially the return on a portfolio consisting of the N assets used in the cross-sectional regression. This portfolio has interesting properties that we now summarize. II The return, Ff(t), represents the return on a portfolio that follows a zero net-investment strategy. This follows from the fact that the portfolio weights sum to zero, that is, N 2>^-i)=o 71=1 In practice, such a strategy can be approximated by constructing a long-short portfolio. 14Note that both the mean and standard deviation are computed on a cross-sectional basis.