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%.

1. What is the expected return and the standard deviation of the investor?

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

3. 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%.

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

                       

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

Answer 1

Expected return and standard deviation of the portfolio, sharpe ratio of the risky portfolio and the investor's overall portfolio are computed as follows:

Formula sheet:

Part (3-a)

Portfolio's overall expected return = (Weight of T-bills portfolio * Expected return of T-bills portfolio) + (Weight of risky portfolio * Expected return the risky portfolio)

Suppose, weight of the risky portfolio be y, and weight of the T-bills portfolio be (1 - y).

Portfolio's overall expected return = ((1 - y) * Expected return of T-bills portfolio) + (y * Expected return the risky portfolio)

0.10 = ((1 - y) * 0.07) + (y * 0.17)

0.10 = 0.07 - 0.07y + 0.17y

0.17y - 0.07y = 0.10 - 0.07

0.10y = 0.03

   y = 0.03 / 0.10

= 0.30 or 30%

Hence,

Weight of the T-bills portfolio = 1 - y

= 1 - 0.30

= 0.70 or 70%

Therefore, proportion that should be invested in T-bills = 70%

Part (3-b)

Standard deviation on the investor's portfolio = (Weight of T-bills portfolio * Standard deviation of the T-bills portfolio) + (Weight of risky portfolio * Standard deviation of the risky portfolio)

   = (0.70 * 0%) + (0.30 * 0.27)

   = 0 + 0.0810

   = 0.0810 or 8.10%   

A B 1 Weight of risky fund 2 Weight of T-bill fund 3 Expected return of the risky portfolio 4 Standard deviation of the risky portfolio 5 Expected return on T-Bill (Risk-free rate) 6 Standard deviation of the T-Bill (Risk-free) 60% 40% 17% 27% 7% 0% 7 8 Expected return on the investor's overall portfolio 13.00% 9 Standard deviation on the investor's overall portfolio 16.20% 10 11 Sharpe ratio of the risky portfolio 12 Sharpe ratio of the overall portfolio 0.37 0.37

A B 1 Weight of risky fund 0.6 2 Weight of T-bill fund 0.4 3 Expected return of the risky portfolio 0.17 4 Standard deviation of the risky portfolio 0.27 5 Expected return on T-Bill (Risk-free rate) 0.07 6 Standard deviation of the T-Bill (Risk-free) 0 7 8 Expected return on the investor's overall portfo=(B1*B3)+(B2*B5) 9 Standard deviation on the investor's overall portfolio 10 11 Sharpe ratio of the risky portfolio =(B3-B5)/B4 12 Sharpe ratio of the overall portfolio =(B8-B5)/B9