Question

You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 24%. You manage an active portfolio with expected return 18% and standard deviation 29%. The risk-free rate is 5%. Your client's degree of risk aversion is A = 2.5.

a. If he chose to invest in the passive portfolio, what proportion, y, would he select? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. What is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of y)? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

a)
Utility Score = Expected Return - 0.5 x σ² A
E(S&P) = 13%, σ(S&P) = 24%

In the portfolio,
E(P) = (1-y)*5% + y*13%
σ(S&P) = y24%

​​​​​​​Utility Score = (1-y)*5% + y*13% - 0.5*(y24%)²*2.5
To maximize utility, dU/dY = 0
0.08 - 0.0576*2.5y = 0
y = 0.5556 = 55.56%

b) Utility from active fund = y*18% - 0.5*(y29%)²*2.5
Now, equating the two utilites, considering after fee return to be x
y*x - 0.5*(y29%)²*2.5 ​​​​​​​= y*13% - 0.5*(y24%)²*2.5 (y = 0.5556)
x = 13% + 0.5*0.5556*(29%)²*2.5 - 0.5*0.5556*(24%)²*2.5
x = 14.84%
Fee that can be charged = 18%-14.84% = 3.16%