Assume there is a risk-free asset, and assume that a factor model holds in which each factor ˜f1,...,˜fk is an excess return.

(a) Show that each return R˜ on the mean-variance frontier equals Rf +

k j=1

βj

˜fj for some β1,...,βk. In other words, show that the risk-free return and the factors span the mean-variance frontier.

(b) Show that a return of the form (6.33) is on the mean-variance frontier if and only if β = (β1 ··· βk) satisfies β = δ−1 F λ
for some δ, where F is the (assumed to be nonsingular) covariance matrix of F˜ = (˜f1 ··· ˜fk)
, and λ = E[F˜] ∈ Rk .