6.1. Prove that the portfolio-weighted average of a stock’s sensitivity to a particular factor is the same as the covariance between the return of the portfolio and the factor divided by the variance of the factor if the factors are uncorrelated with each other. Do this with the following steps: (1) Write out the factor equation for the portfolio by multiplying the factor equations of the individual stocks by the portfolio weights and adding.
(2) Group terms that multiply the same factor.
(3) Replace the factor betas of the individual stock returns by the covariance of the stock return with the factor divided by the variance of the factor.
(4) Show that the portfolio-weighted average of the covariances that multiply each factor is the portfolio return’s covariance with the factor. The rest is easy.