Question

You are constructing a portfolio of two assets, Asset A and Asset B. The expected returns of the assets are 7 percent and 13 percent, respectively. The standard deviations of the assets are 33 percent and 41 percent, respectively. The correlation between the two assets is .49 and the risk-free rate is 5.8 percent. What is the optimal Sharpe ratio in a portfolio of the two assets? What is the smallest expected loss for this portfolio over the coming year with a probability of 2.5 percent? (Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round your Sharpe ratio answer to 4 decimal places and Probability answer to 2 decimal places. Omit the "%" sign in your response.)

Sharpe ratio

Smallest expected loss

%

Answer 1

Expected Return of A   - 7%

Expected Return of B – 13%

Standard Deviation of A – 33%

Standard Deviation of B – 41%

Correlation – 0.49

Risk-free rate – 5.8%

W_A = [(0.07-0.058) (0.41²) – (0.13-0.058) (0.33) (0.41) (0.49)] / [(0.07-0.058) (0.41²) + (0.13-0.058) (0.33)² – (0.07-0.058+0.13-0.058) [(0.33) (0.41) (0.49)]

W_A = 0.6666

W_B = 0.3334

E(Rp) = (0.6666) (0.07) + (0.3334) (0.13) = 0.090

SD = [(0.6666²) (0.33²) + (0.3334²) (0.41²) + 2(0.6666) (0.3334) (0.33) (0.41) (0.49)]^0.5 = 0.3657

Note:

W_A – Weight of Asset A

W_B – Weight of Asset B

E(Rp) – Expected return of Portfolio

SD – Standard Deviation

Sharpe Ratio = Expected Return (E(Rp)) – Risk-free rate / Standard Deviation (SD)

                        = (0.090 – 0.058) / 0.3657

                        = 0.0875

Sharpe Ratio = 0.0875

Prob (R < 0.090 – 1.6449(0.3657)) = 5%

Prob (R < -0.5115) = 5 %

Smallest Expected Loss = -51.15%