Question

You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The t-bill rate is 7%.

a) One of your clients chooses to invest 70% of a portfolio in your risky fund and 30% in t-bills. What is the expected return and standard deviation of your client’s portfolio?

b) What is the Sharpe Ratio of the risky portfolio you offer? What is the Sharpe Ratio of your client’s portfolio?

c) If another client wants a 15% expected return on a portfolio that invests in t-bills and the risky portfolio, what percentage of her assets will she have to put in the risky portfolio? What will be the standard deviation of the rate of return of this complete portfolio?

d) A third client wants the highest expected return on a complete portfolio that does not have a standard deviation of returns greater than 20%. What would be this expected return?

e) A fourth client is thinking about investing in a passive risky portfolio, the S&P 500 stock index, instead of your risky portfolio. Assume that the S&P 500 stock index has an expected return of 13% and a standard deviation of 25%. If the client is thinking about investing 70% of his portfolio in the passive index, then what is his expected return and standard deviation? Can you show that this client can do better using your risky portfolio instead of the passive index?

f) What fee (as a percentage of assets invested) could you charge clients to invest in your risky portfolio so that they would be indifferent between using your portfolio or the S&P 500 stock index described above? (Hint: Think about how the fee would change the Sharpe Ratio of your risky portfolio.)

Answer 1

a) As the return of the portfolio is the weighted average return of the constituent securities. and

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient beltween returns of security i and security j

Expected return of the client's portfolio = 0.7*17%+0.3*7% =0.14 or 14%

Expected standard deviation of the client's portfolio = 0.7*27% =0.189 or 18.90%

b) Sharpe Ratio of the Risky portfolio = (17%-7%)/27% =0.37

Sharpe Ratio of the Client's portfolio = (14%-7%)/18.9%= 0.37

Sharpe Ratio of both portfolios are same

c) Let w be the weight invested in Risky portfolio and (1-w) in T bills

then w*17%+ (1-w)*7% =15%

=> w*10% =8%

w= 0.8

So, to achieve a return of 15% , 0.8 or 80% must be invested in risky portfolio and 20% in T bills

standard deviation of the rate of return of this complete portfolio =0.8*27% = 21.6%

d) To achieve maximum returns with a standard deviation of no more than 20%, the standard deviation has to be 20%

Let w be the weight invested in Risky portfolio and (1-w) in T bills

w*27% = 20%

=> w =0.74

So, 74% must be invested in Risky portfolio and 26% in T bills

Expected return = 0.74*17%+0.26*7% =14.40%