Question

You manage a risky portfolio with an expected rate of return of 20% and a standard deviation of 36%. The T-bill rate is 5%. Your client’s degree of risk aversion is A = 1.6, assuming a utility function U = E(r) - ½Aσ².

a. What proportion, y, of the total investment should be invested in your fund? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

a.

U = E(r) - 1/2*A*σ^2

U = rf+y(E(rp)-rf)-1/2*A*σ^2

dU/dy = E(rp) - rf -A*σ^2*y = 0

where y is weight of portfolio in risky portfolio.

So by putting value in the above formula,

y = (E(rp)- rf) / A * σp^2

y = (0.2 - 0.05 )/(1.6*0.36^2)

y= 72.34 % in risky portfolio

Weight in risky portfolio = 72.34%

Weight in T-Bills = 100 - 72.34 = 27.66%

b.

Expected return on portfolio = Weight in risky portfolio*Expected return in risky portfolio + Weight in T-Bills *Expected return in T-Bills

= 72.34% *0.2 +27.66%*0.05= 15.85%

Standard Deviation of portfolio = ((Weight in risky portfolio*Std deviation of risky portfolio)^2 + (Weight in T-Bills *Std deviation of T-Bills)^2+( 2 *Weight in risky portfolio*Std deviation of risky portfolio*Weight in T-Bills *Std deviation of T-Bills*correlation between two asset))^0.5

As std deviation of T-Bills = 0

Standard Deviation of portfolio = Weight in risky portfolio*Std deviation of risky portfolio = 0.7234*0.36 = 26.04%