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 × Aσ2 , where E(r) is the expected return, A is the risk aversion coefficient and σ 2 is the variance of returns.

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? 

(ii) Comparing your answers to d)(i)=81.8% and e)(i)=40.09%, what do you conclude about the relationship between the proportion y invested in the your fund and your client’s attitude toward risk? 

(iii) Given that the optimal proportion of the risky asset in the complete portfolio is given by the equation y ∗ = E(rp)−rf /Aσ2 p , where rf is the risk-free rate, E(rp) is the expected return of the risky portfolio, σ 2 p is variance of returns, and A is the risk aversion coefficient. For each of the variables on the right side of the equation, discuss 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.