Problem 5 (40 marks). Assume that you manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate (risk-free rate) is 7%. Your client chooses to invest 70% in the risky portfolio in your fund and 30% in a T-bill money market fund. We assume that investors use mean-variance utility: U = E(r) - 0.5 x Ao, where E(r) is the expected return, A is the risk aversion coefficient and o is the variance of returns. a) What is the expected value and standard deviation of the rate of return on your client's portfolio? [4 marks] b) What is the reward-to-volatility ratio (Sharpe ratio) of your risky portfolio? What is the reward-to-volatility ratio (Sharpe ratio) of your client's risky portfolio? Comment on the relationship between these two Sharpe ratio calculated and explain the intuition behind. [8 marks] c) Draw the Capital Allocation Line (CAL) of your portfolio on an expected return- standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL. [6 marks] d) Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16% (i) What is the proportion y? [2 marks] (ii) What is the standard deviation of the rate of return on your client's portfolio? [3 marks] e) Your client's degree of risk aversion is A= 3.5. (i) What proportion, y, of the total investment should be invested in your risky fund? [2 marks] (ii) What is the expected value and standard deviation of the rate of return on your client's optimized portfolio? [4 marks] f) If your client's degree of risk aversion increases from A = 3.5 to A= 4.5. (i) What proportion, y, of the total investment should be invested in your risky fund? [2 marks] (ii) Comparing your answers to d)(i) and e) (i), what do you conclude about the relationship between the proportion y invested in the your fund and your client's attitude toward risk? [3 marks] (iii) Given that the optimal proportion of the risky asset in the complete portfolio is given by the equation y* E(rp);", where rg is the risk-free rate, E(rp) is the expected return of the risky portfolio, o, is variance of returns, and A is the risk aversion coefficient. For each of the variables on the right side of the equation, dis- cuss the impact of the variable's effect on y* and why the nature of the relationship makes sense intuitively. Assume the investor is risk averse. [6 marks]