Assume there is a representative investor with utility function u. The first-order condition E[u

(R˜ m)(R˜ 1 − R˜ 2)] = 0 must hold for all returns R˜ 1 and R˜ 2. Assume there is a risk-free asset. Consider any return R˜. By orthogonal projection, we have R˜ −Rf = α + β(R˜ m −Rf) + ˜ε

for some α and β, where E[˜ε] = E[R˜ mε˜] = 0.

(a) Use the first-order condition in conjunction with the returns R˜ m and R˜ ∗

def

= R˜ +(1− β)(R˜ m −Rf) = R˜ m +α + ˜ε

to show that

α = −E[u

(R˜ m)ε)˜ ]

E[u

(R˜ m)] .

(b) Use the result of the previous part to derive the CAPM when there is a representative investor and the residual ε˜ of each asset return is mean independent of the market return (which is true, for example, when returns are joint elliptically distributed; Chu, 1973).