Assume there is a risk-free asset in zero net supply. Let = ( 1

Assume there is a risk-free asset in zero net supply. Let θ = (θ 1 ··· θ n) denote the vector of supplies of the n risky assets. Let μ denote the mean and  the covariance matrix of the vector X˜ = (x˜1 ···˜xn) of asset payoffs. Assume  is nonsingular. Suppose the utility functions of investor h are u0

(c) = −e−αhc and u1

(c) = −δhe−αhc

.

Let c0 denote the aggregate endowment H h=1 yh0 at date 0.

(a) Use the result of Exercise 2.2 on the optimal demands for the risky assets to show that the equilibrium price vector is p

def

= 1 Rf

(μ−αθ ). (4.24)

(b) Interpret the risk adjustment vector αθ in (4.24), explaining in economic terms why a large element of this vector implies an asset has a low price relative to its expected payoff.

(c) Assume δh is the same for all h (denote the common value by δ). Use the market-clearing condition for the date–0 consumption good to deduce that the equilibrium risk-free return is Rf def

= 1

δ

exp

α

*

θ



μ− c0

+

− 1 2

α2

θ



θ



. (4.25)

(d) Explain in economic terms why the risk-free return (4.25) is higher when θ



μ is higher and lower when δ, c0, or θ



θ is higher.

(e) Assume different investors may have different discount factors δh. Set

τh = 1/αh and τ = H h=1 τh (which is aggregate risk tolerance). Show that the equilibrium risk-free return is (4.25) when we define

δ =,H h=1

δ

τh/τ

h ⇔ logδ = H h=1

τh

τ

logδh .