A pension fund manager is considering three mutual funds. The first is a stock f

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 5.3%. The probability distributions of the risky funds are:

   

   

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

Expected Return Standard Deviation    Stock fund (S) 14%         43%             Bond fund (B) 7%         37%         

Explanation / Answer

Solution:

Let S be the stock fund and

B be the bond fund

First, we calculate the Covariance between stock fund and bond fund, COV (B, S)

Cov (B, S) = r*S*B

Cov (B, S) = 0.0459*43%*37%

Cov (B, S) = 0.0073 or 0.73%

The proportion of stock fund in the optimal risky portfolio, wS is given by:-

wS = [(E (rS) – rf)B^2 – (E (rB) – rf) Cov (B, S)]/[ (E (rS) – rf)B^2 + (E (rB) – rf)S^2 – (E (rS) – rf + E (rB) – rf) Cov (B, S)]

wS = [(14 – 5.3)(37)^2 – (7 – 5.3) (0.73)]/[(14 – 5.3)(37)^2 + (7 – 5.3) (43)^2 – (14 – 5.3 + 7 – 5.3) (0.73)]

wS = [11910.3 – 1.241]/[11910.3 + 3143.3 – 7.592]

wS = 11909.06/15046.1

ws = 0.79

The proportion of bond fund in the optimal risky portfolio, wB is given by:-

wB = 1 – wS

wB = 1 – 0.79

wB = 0.21

Hence, the investor should invest 64% in the stock fund and 36% in the bond fund.

The Expected value of the optimal risky portfolio, E (Rp) is given by:-

E (Rp) = wS*E (rS) + wB*E (rB)

E (Rp) = 0.79*14 + 0.21*7

E (Rp) = 12.53%

The standard deviation of the optimal risky portfolio, p is given by:-

p = wS^2*S^2 + wB^2*B^2 + 2*ws*wB*r*s*B

p = (0.79)^2*43^2 + (0.21)^2*37^2 + 2*0.79*0.21*0.0459*43*37

p = 1153.961 + 60.3729 + 24.2303

p = 1238.564

p = 35.19%

The reward-to-volatility ratio of the best feasible CAL

Sharpe ratio = [(E (Rp) – rf]/ p

Sharpe ratio = [12.53 – 5.3]/35.19

Sharpe ratio = 0.2054

Hence, the reward-to-volatility ratio of the best feasible CAL is 0.2054.

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