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 5.9%. The probability distributions of the risky funds are:

    Expected Return   Standard Deviation
Stock fund (S)   20%   49%
Bond fund (B)   9%   43%

The correlation between the fund returns is 0.0721.

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 = (20% - 5.9%) x 43%² - (9% - 5.9%) x 0.0721 x 43% x 49% = 0.0256

Denominator = (20% - 5.9%) x 43%² + (9% - 5.9%) x 49%² - (20% - 5.9% + 9% - 5.9%) x 0.0721 x 43% x 49% =  0.0309

Hence, W_S = 0.0256 / 0.0309 = 82.84%

Hence, portfolio invested in bond = W_B = 1 - W_S = 1 - 82.84% = 17.16%

Expected return, E(r_p) = W_S x E(r_S) + W_B x E(r_B)] = 82.84% x 20% + 17.16% x 9% = 18.11%

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) = (82.84% x 49%)² + (17.16% x 43%)² + 2 x 0.0721 x (82.84% x 49%) x (17.16% x 43%) = 0.1745

Hence, standard deviation, σ_p = Variance^1/2 = 0.1745^1/2 = 41.78%

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 = (18.11% - 5.9%) / 41.78% = 0.2923