Theory of Finance

3. Consider an investor with initial wealth Wo who chooses a portfolio to max- imize her expected utility of wealth next period, where the utility function is power (CRRA) with coefficient of relative risk aversion : WI- U(W) (8) 7>0. Assume that the wealth next period, W, is uncertain and lognormaly dis- tributed, i.e. w = log(W) ~ N(,0). (9) (a) [20 marks) Find the expected utility function, E [U(W.)), expressed in terms of the model parameters. (b) (20 marks] Show that maximizing the expected utility in (a) is equivalent to maximising F: F = Hw+5(1 no (10) (Hint: Apply monotonic transformations to E,[U(W.)].] (c) [20 marks] Assume that there are two assets, a riskless asset with log return r; and a risky asset with log return r. Assume that the log risky return is normally distributed, N(1,0), and that the optimal portfolio places a weight a on the risky asset. It can be shown that the log rate of return on wealth can be approximated as 1. = log(1 + Rw) rs + a(n = rs) +ac ajo?. (11) Use this approximation to find the expected return and variance on the wealth portfolio (d) [20 marks] Using the results from (c), find the optimal share of the risky asset in the investor's portfolio, a", that maximize the expected utility (or its monotonic transformation Fin (b)). (Hint: Recall that W; = W.(1+ Rw).) (e) [20 marks] Define the Sharpe ratio of the log return on the risky asset as: E[n] ry + fo? (12) Find an expression for the derived utlity function (the investor's utility F maximized at a*) as a function of initial wealth, the riskless interest rate, the coefficient of relative risk aversion and the Sharpe ratio of the risky asset. 3. Consider an investor with initial wealth Wo who chooses a portfolio to max- imize her expected utility of wealth next period, where the utility function is power (CRRA) with coefficient of relative risk aversion : WI- U(W) (8) 7>0. Assume that the wealth next period, W, is uncertain and lognormaly dis- tributed, i.e. w = log(W) ~ N(,0). (9) (a) [20 marks) Find the expected utility function, E [U(W.)), expressed in terms of the model parameters. (b) (20 marks] Show that maximizing the expected utility in (a) is equivalent to maximising F: F = Hw+5(1 no (10) (Hint: Apply monotonic transformations to E,[U(W.)].] (c) [20 marks] Assume that there are two assets, a riskless asset with log return r; and a risky asset with log return r. Assume that the log risky return is normally distributed, N(1,0), and that the optimal portfolio places a weight a on the risky asset. It can be shown that the log rate of return on wealth can be approximated as 1. = log(1 + Rw) rs + a(n = rs) +ac ajo?. (11) Use this approximation to find the expected return and variance on the wealth portfolio (d) [20 marks] Using the results from (c), find the optimal share of the risky asset in the investor's portfolio, a", that maximize the expected utility (or its monotonic transformation Fin (b)). (Hint: Recall that W; = W.(1+ Rw).) (e) [20 marks] Define the Sharpe ratio of the log return on the risky asset as: E[n] ry + fo? (12) Find an expression for the derived utlity function (the investor's utility F maximized at a*) as a function of initial wealth, the riskless interest rate, the coefficient of relative risk aversion and the Sharpe ratio of the risky asset