Question

Suppose that you are managing a portfolio with a standard deviation of 30% and an expected return of 16%. The Treasury bill rate is 9%. A client wants to invest 19% of their investment budget in a T-bill money market fund and 81% in your fund.

1. What is the expected rate of return on your client's complete portfolio?

2. What is the standard deviation for your client's complete portfolio?

3. What is the Sharpe ratio of your client's complete portfolio?

4. What is the Sharpe ratio of the portfolio of risky assets that you manage?

Answer 1

Solution 1:
When the client is investing 81% in the fund and 19% in the T-bill, the expected return of the client's complete portfolio will be equal to:
16*81% + 9*19%
=12.96+1.71=14.67%

Solution 2:
Standard deviation of t-bill is zero as it is a risk free asset. Standard deviation of the portfolio is 30%. The client has invested 81% of the total fund in the portfolio. As standard deviation of t-bill or risk free asset is zero, the standard deviation of client's portfolio is the percentage of fund invested in the risky portfolio times the standard deviation of the portfolio.
Standard deviation of the client's complete portfolio is given by .3*.81=.243=24.3%

Solution 3:
Sharpe ratio is given by:

(Expected return on the portfolio - Risk free rate of return)/Standard deviation of the portfolio return

Here, expected portfolio return of client's portfolio is 14.67%
Risk free rate of return is 9%, which is the treasury bill rate of return.
Standard deviation of the client's complete portfolio is 24.3%

So, Sharpe ratio of the client's complete portfolio = (14.67-9)/24.3=.2333=23.33%

Solution 4:
Sharpe ratio of the portfolio that gives 16% return with a standard deviation of 30% is given by (16-9)/30=.2333=23.33%