Question

This question has two sub-questions.

Assume that t-bills generates a return of 4 %. Which of the following risky portfolios is the optimal risky portfolio?

Now assume that you wish to have a total expected return on your portfolio of 6 %, how much of your funds should be invested in the optimal risky portfolio, and how much in t-bills, respectively?

	E(r)	σ
Portfolio A	12 %	27 %
Portfolio B	13 %	32 %
Portfolio C	10 %	26 %

Select one:

a. Portfolio C is the optimal risky portfolio and you should have 33 % in the risky portfolio and the rest in t-bills.

b. I cannot tell which risky portfolio that is optimal with the given information.

c. Portfolio A is the optimal risky portfolio and you should have 25 % in the risky portfolio and the rest in t-bills.

d. Portfolio A is the optimal risky portfolio and you should have 35 % in the risky portfolio and the rest in t-bills.

e. At least two of the risky portfolios are equally preferable.

f. Portfolio C is the optimal risky portfolio and you should have 41 % in the risky portfolio and the rest in t-bills.

g. Portfolio B is the optimal risky portfolio and you should have 27 % in the risky portfolio and the rest in t-bills.

h. Portfolio B is the optimal risky portfolio and you should have 22 % in the risky portfolio and the rest in t-bills.

Answer 1

Answer is c. Portfolio A is the optimal risky portfolio and you should have 25 % in the risky portfolio and the rest in t-bills.

We need to calculate Sharpe ratio of which portfolio. the portfolio which has the highest Sharpe ratio is the optimal risky portfolio.

Sharpe ratio = (Portfolio return - risk-free rate or t-bills rate)/portfolio standard deviation

Sharpe ratio of Portfolio A = (12% - 4%)/27% = 8%/27% = 0.296

Sharpe ratio of Portfolio B = (13% - 4%)/32% = 9%/32% = 0.281

Sharpe ratio of Portfolio C = (10% - 4%)/26% = 6%/26% = 0.231

Portfolio A has the highest Sharpe ratio. so Portfolio A is the optimal risk portfolio.

Let's assume weight of Portfolio is x and weight of t-bills is 1-x.

Total portfolio return = weight of optimal risky portfolio*return of optimal risky portfolio + weight of t-bills*return of t-bills

6% = 12%*x + 4%*(1-x)

6% = 12%x + 4% - 4%x

6% - 4% = 12%x - 4%x

2% = 8%x

x = 2%/8% = 0.25 or 25%

25% funds should be invested in the optimal risky portfolio i.e. Portfolio A, and rest i.e. 75% in t-bills, respectively.