Assume the asset returns R˜i for i = 1,...,n satisfy R˜i = E[R˜i] +Cov(F˜,R˜i)



−1 F (F˜ − E[F˜])+ ˜εi , where each ε˜i is mean independent of the factors F˜, that is, E[˜εi |F˜] = 0 (note it is not being assumed that cov(ε˜i, ε˜j) = 0). Assume markets are complete and the market return is well diversified in the sense of having no idiosyncratic risk:

R˜ m = E[R˜ m] +Cov(F˜,R˜ m)



−1 F (F˜ −E[F˜]).

Show that there is a factor model with factors F˜. Hint: Pareto optimality implies sharing rules w˜ h = fh(w˜ m).