Suppose there is a risk-free asset and n risky assets with payoffs x˜i and prices pi. Assume the vector x˜ = (x˜1 ···˜xn) is normally distributed with mean μx and nonsingular covariance matrix x. Let p = (p1 ···pn)

. Suppose there is consumption at date 0 and consider an investor with initial wealth w0 and CARA utility at date 1:

u1

(c) = −e−αc

.

Let θi denote the number of shares the investor considers holding of asset i and set θ = (θ1 ··· θn)

. The investor chooses consumption c0 at date 0 and a portfolio θ, producing wealth (w0 −c0 − θ

p)Rf +θ

x˜ at date 1.

(a) Show that the optimal vector of share holdings is

θ = 1

α

−1 x (μx − Rf p). (2.33)

(b) Suppose all of the asset prices are positive, so we can define returns x˜i/pi. Explain why (2.33) implies (2.22). Note: This is another illustration of the absence of wealth effects. Neither date-0 wealth nor date-0 consumption affects the optimal portfolio for a CARA investor.