Question

Hi there, please solve these questions. These type of questions might be in my final exam this weekend.

Thank you

2. 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 8%. 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×Aσ 2, where E(r) is the expected return, A is the risk aversion coefficient and σ 2 is the variance of returns.

a) What is the expected value and standard deviation of the rate of return on your client’s portfolio?

b) 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?

(ii) What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio?

c) Prove 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.

Answer 1

a) What is the expected value and standard deviation of the rate of return on your client’s portfolio?
Expected return=70%*18%+30%*8%=15.00%

Standard deviation=70%*28%=19.60%

b) 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?
=(rp-rf)/(A*variance)
=(18%-8%)/(3.5*28%*28%)=0.364431487

(ii) What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio?
Expected returns=(18%-8%)/(3.5*28%*28%)*18%+(1-(18%-8%)/(3.5*28%*28%))*8%=11.6443149%

Standard deviation=(18%-8%)/(3.5*28%*28%)*28%=10.2040816%

c) Prove 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.

If rp increases, y increases because return is more so it is appropraite to buy more
rf increases, y decreases because risk free rate is becoming more attractive
If variance increases, y decreases becuase it is becoming more risky