Assume there is a risk-free asset, and let m˜ be an SDF.

(a) Show that each return R˜ satisfies E[R˜] −Rf = var∗(R˜)

Rf

− cov(m˜ R˜,R˜), where var∗ denotes variance under the risk-neutral probability corresponding to m˜ .

(b) Assume there is a representative investor with constant relative risk aversion ρ, so γ u

(R˜ m) def

= γ R˜ −ρ

m is an SDF for some constant γ . Show that u

(x)x is a decreasing function of x if and only if ρ > 1.

(c) If f(x) is a decreasing function of x, then cov(f(x˜),x˜) < 0 for any random variable x˜. Using this fact and the above results, explain why E[R˜ m] −Rf ≥

var∗(R˜ m)

Rf when ρ > 1.