Question

Exercise 2 (12 points) Consider again the same data for your client as in exercise 1. Your client's degree of risk aversion is A= 2.5. a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return in your client's optimized portfolio Ton 2

Answer 1

Let us consider the Expected return of our porfolio (A) as ER(A) = 17% and Standard Deviation as SD(A)=27%

& T-Bill return (Risk Free -B) be ER(B) = 7% and SD(B)= 0

(Standard Deviation is used to measure risk & so for a Risk free security it is considered as Zero)

a. Expected Return of Portfolio be ER(P) = Weight of A x ER(A) + Weight of B x ER(B) = 70% * 17% + 30% * 7% = 14%

Standard Deviation of Portfolio be SD(P) =

Square root of [SD(A)² * W(A)² ] + [SD(B)² * W(B)² ] + 2 SD(A) * W(A) *SD(B) * W(B) * CovAB

= Sqrt of (0.27)² * (0.7)² + 0 + 0 = Sqrt of 0.0729 * 0.49 = 0.189 or 18.90%

b.Reward to Volatility Ratio (Reward shall be the return earned over and above risk free rate)

i. Our Portfolio = ER(A) - ER(R^f) divided by SD(A) = (17% - 7%) / 27% = 0.37

ii. Clients Portfolio = ER(P) - ER(R^f) divided by SD(P) = (14%-7%) / 18.9 = 0.37

c. Considering the same data above, the degree of risk aversion is 2.5, we will have to calculate utility score

= ER(P) - 0.5 * SD(P)² * Degree of risk aversion

= 0.14 - 0.5 * 0.189² * 2.5 = 0.14 -0.04465 = 0.09535 or 9.54%

Thus, the clients tolerance to risk can be measured by above number

Thus. the optimal portfolio will be one in which an expected return of 7.89% is earned

Hence, Let the weights be x and 1-x in our portfolio and T-Bills

0.0954 = x * 0.17 + (1 - x) * 0.07

W(A) = 25.40% & W(B) = 74.60%

d. Thus, ER(P) as above = 9.54%

e. SD(P) considering above ER(P) of 9.54% = Sq.root of (0.27)² * (0.254)² = Sq. root of (0.0729 * 0.0645)

= Sq Root of 0.0047025 = 0.0685 or 6.857%