Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%.

The probability distributions of the risky funds are:

Expected Return Stand. Dev

Stock fund (S) E.R 15 % S.D. 32 %

Bond fund (B) E.R. 9 % S.D. 23 %

The correlation between the fund returns is 0.15.

What is the Sharpe ratio of the best feasible CAL? (Do not round intermediate calculations. Round your answer to 4 decimal places.)

Answer 1

The best feasible CAL (Capital Allocation Line) is the one which is just tangent to the efficient frontier of risky assets. This point of tangency can be determined by finding the optimal risky portfolio for the given combination of stock and bond fund.

Let the weight of stocks and bonds in the optimal risky portfolio be Wa and Wb respectively.

Now, Ra = 15%, Rb = 9%, Sa = 32 % and Sb = 23 %, Risk-Free Rate = Rf = T-Bill Rate = 5.5 %

Correlation Between A and B = 0.15 = p

Wa = [(Ra - Rf) x (Sb)^(2) - (Rb - Rf) x p x Sa x Sb] / [ (Ra - Rf) x (Sb)^(2) + (Rb - Rf) x (Sa)^(2) - (Ra - Rf + Rb - Rf) x p x Sa x Sb] = [(15 - 5.5) x (23)^(2) - (9 - 5.5) x 0.15 x 32 x 23] / [(15-5.5) x (23)^(2) + (9-5.5) x (32)^(2) - (15 - 5.5 + 9 - 5,5) x 0.15 x 32 x 23] = 0.85249

Wb = (1-0.85249) = 0.14751

Expected Return of Optimal Portfolio = Rp = Ra x Wa + Rb x Wb = 15 x 0.85249 + 9 x 0.14751 = 14.1149 %

Standard Deviation of Portfolio = Sp = [{Sa x Wa}^(2) + {Sb x Wb}^(2) + {2 x Wa x Wb x p x Sa x Sb}]^(1/2) = [{32 x 0.85249}^(2) + {23 x 0.14751}^(2) + {2 x 0.85249 x 0.14751 x 0.15 x 32 x 23}]^(1/2) = 13.894 %

Sharpe's Ratio of Optimal Risky Portfolio (Best Possible CAL) = [Rp - Rf] / Sp = [14.1149 - 5.5] / 13.894 = 0.62004 ~ 0.62