Question

An investor chooses to invest 60% of a portfolio in a risky fund and 40% in a T-bill fund. The expected return of the risky portfolio is 17% and the standard deviation is 27%. The T-bill rate is 7%.

What is the Sharpe ratio of the risky portfolio and the investor’s overall portfolio?

Suppose the investor decides to invest a proportion (y) of his total budget in the risky portfolio so that his overall portfolio will have an expected return of 10%.

a. What is the proportion that must be invested in T-bills?

b. What is the standard deviation of the rate of return on the investor’s portfolio?

Answer 1

Sharpe ratio = ( Expected return - risk free rate ) / standard deviation

Sharpe ratio of risky portfolio = ( 17 -7) / 27 = 10 /27 = 0.37

expected return on the portfolio = 0.6 * 17 + 0.4 * 7 = 13

standard deviation of overall portfolio = weight of risky fund * standard deviation of risky fund = 0.6* 27 = 16.20%

sharpe ratio of the overall portfolio = ( 13 - 7 ) / 16.20 = 0.37

a)

10 = y * 17 + (1-y) 7

10 = 17y + 7 - 7y

3 = 10 y

y = 0.30

(1-y) = 1 - 0.3 = 0.7

proportion that must be invested in T-bills = 70%

b)

standard deviation of the rate of return on the investor’s portfolio = 0.3 * 27 = 8.1%