Question

You are a portfolio manager of a risky portfolio with an expected rate of return of 14% and a standard deviation of 28%. The T-bill rate is 4%. Suppose your client decides to invest in your risky portfolio a proportion (y) of his total investment budget so that his overall portfolio will have an expected return of 10%.

a. What is the proportion y ?
b. What will be the standard deviation of your client’s portfolio ?
c. What is the Sharpe ratio ?
d. Suppose your client is wondering if he should switch his money in your fund to a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 9% with a standard deviation of 25%. Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of your client’s CAL by reducing the expected return net of the fee.) ?

Answer 1

1) Let's assume that the rest of the client's money is invested in T-bills (risk-free asset) which yield 4%.

Hence, 10%=y*14% + (1-y)*4%. This should yield y=60%.

2) Variance of client portfolio = 0.6²x0.28² = 2.82%

Hence, standard deviation of client portfolio = 16.8%.

3) Sharpe ratio of client's overall portfolio = (10%-4%)/(16.8%) = 0.357

4) Let's assume that the client uses Sharpe Ratio as the metric when choosing between any two portfolio.

If the client invested 60% of his investment budget into the S&P 500 index, his expected return would be 0.6*9% + 0.4*4% = 7%.

At the same time his portfolio's standard deviation = 0.6*25% = 15%.

Hence, his Sharpe Ratio would be = (7%-4%)/15% = 0.2

Thus, the portfolio with S&P 500 index has a Sharpe Ratio which is lower than that of my portfolio. This implies I can charge a fee for my portfolio management services. My fee can be high enough so that the Sharpe Ratio on my portfolio just exceeds the Sharpe Ratio on the S&P 500 index. Given that standard deviation of my portfolio is 28%, I should provide an excess return of at least 5.6%, which means an overall return of 5.6%+4%=9.6%. Thus, I can charge a maximum fee of 4.4% for my portfolio management services.