Assume there are H investors with CARA utility and the same absolute risk aversion . Assume there

Assume there are H investors with CARA utility and the same absolute risk aversion α. Assume there is a risk-free asset. Assume there are two risky assets with payoffs x˜i that are joint normally distributed with mean vector μ

and nonsingular covariance matrix . Assume HU investors are unaware of the second asset and invest only in the risk-free asset and the first risky asset. If all investors invested in both assets (HU = 0), then the equilibrium price vector would be p∗ = 1 Rf

μ− α

HRf

θ , where θ is the vector of supplies of the risky assets (Exercise 4.1). Assume 0 < HU < H, and set HI = H −HU.

(a) Show that the equilibrium price of the first asset is p1 = p∗

1, and the equilibrium price of the second asset is p2 = p∗

2 − α

HRf

HU HI

var(x˜2) − cov(x˜1,x˜2)2 var(x˜1)



< p∗

2 .

(b) Show that there exist A > 0 and λ such that E[R˜ 1] = Rf + λ.
cov(R˜ 1,R˜ m)
var(R˜ m) , (6.35a)
E[R˜ 2] = A+ Rf + λ
cov(R˜ 2,R˜ m)
var(R˜ m) , (6.35b)
λ = E[R˜ m] −Rf − Aπ2 , (6.35c)
where π2 = p2θ 2/(p1θ 1 +p2θ 2)is the relative date–0 market capitalization of the second risky asset. (Note that λ is less than in the CAPM, and the second risky asset has a positive alpha, relative to λ.)