Suppose there is a risk-free asset and suppose Jensen’s alpha in (6.22) is positive. Consider an investor with initial wealth w0 who holds the benchmark portfolio and therefore has terminal wealth w0R˜

b. Assume E[u

(w0R˜ b)] > 0.

Consider the return R˜ 1 = R˜ +(1− β)(R˜ b −Rf) = R˜ b +α + ˜ε .

Show that E[u

(w0R˜ b)(R˜ 1 −R˜ b)] = αE[u

(w0R˜ b)] > 0 (6.34)

if utility is quadratic or ifR˜ andR˜ b are joint normally distributed. Note: Condition

(6.34) implies that the expected utility of a convex combination λR˜ 1 +(1−λ)R˜ b is greater than the expected utility of R˜ b for sufficiently small λ > 0. Thus, this exercise shows that a positive Jensen’s alpha implies that utility improvements are possible if utility is quadratic or returns are joint normal.