An individual with an initial wealth of w and a von-Neumann-Morgenstern utility function u(y) = lny has the opportunity to invest in a risky asset with a price of 1 and with a final payoff distributed as (1 + h; 0.5 ; 1 d; 0.5 ) with h > d. The risk-free return on bonds is 0.

(a) Suppose that the investor chooses to invest an amount of into the risky asset and the rest of his wealth into bonds. Write down the lottery associated with such a portfolio. Determine the expected utility of the individual for each value of [0;w].

(b) Formulate the individuals optimization problem. Derive the first order condition. Can =0 be an optimal choice?

(c) Derive the optimal choice of a portfolio, . Show that is a linear function of wealth. Give an intuition for this result.

(d) Show that when d increases, falls. What is the impact of an increase in h on ?

(e) Compare the exact solution in (a) with the following approximation: w 1 RRA, where RRA is the relative risk aversion and and 2 are the mean and variance of excess stock (risky asset) returns.