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 5.4%. The probability distributions of the risky funds are:

    Expected Return   Standard Deviation
Stock fund (S)   15%   44%
Bond fund (B)   8%   38%

The correlation between the fund returns is 0.0684.

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

Answer 1

First we will have to solve the mix for the risky optimal portfolio. Let W_s be the proportion of stock fund in the risky optimal portfolio. Then

Numerator = (15% - 5.4%) x 38%² - (8% - 5.4%) x 0.0684 x 44% x 38% = 0.0136

Denominator = (15% - 5.4%) x 38%² + (8% - 5.4%) x 44%² - (15% - 5.4% + 8% - 5.4%) x 0.0684 x 44% x 38% =  0.0175

Hence, W_S = 0.0136 / 0.0175 = 77.51%

Hence, portfolio invested in bond = W_B = 1 - W_S = 1 - 77.51% = 22.49%

Expected return, E(r_p) = W_S x E(r_S) + W_B x E(r_B)] = 77.51% x 15% + 22.49% x 8% = 13.43%

Variance = (Standard deviation)² = (W_Sσ_S)² + (W_Bσ_B)² + 2 x ρ_S,B x (W_Sσ_S) x (W_Bσ_B) = (77.51% x 44%)² + (22.49% x 38%)² + 2 x 0.0684 x (77.51% x 44%) x (22.49% x 38%) = 0.1276

Hence, standard deviation, σ_p = Variance^1/2 = 0.1276^1/2 = 35.72%

Sharpe Ratio = [E(r_c) - r_f] / σ_c

The equation for CAL is:

From this equation,

Sharpe ratio = [E(r_c) - r_f] / σ_c = [E(r_p) - r_f] / σ_p = (13.43% - 5.4%) / 35.72% = 0.2247