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		Standard Deviation
Stock fund (S)	16	%	45	%
Bond fund (B)	7	%	39	%

The correlation between the fund returns is .0385.

Suppose now that your portfolio must yield an expected return of 14% and be efficient, that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b-1. What is the proportion invested in the T-bill fund? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b-2. What is the proportion invested in each of the two risky funds? (Do not round intermediate calculations. Round your answers to 2 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 = (16% - 5.5%) x 39%² - (7% - 5.5%) x 0.0385 x 39% x 45% = 0.0159

Denominator = (16% - 5.5%) x 39%² + (7% - 5.5%) x 45%² - (16% - 5.5% + 7% - 5.5%) x 0.0385 x 39% x 45% =  0.0182

Hence, W_S = 0.0159 / 0.0182 = 87.21%

Hence, portfolio invested in bond = W_B = 1 - W_S = 1 - 87.21% = 12.79%

Expected return, E(r_p) = W_S x E(r_S) + W_B x E(r_B)] = 87.21% x 16% + 12.79% x 7% = 14.85%

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) = (87.21% x 45%)² + (12.79% x 39%)² + 2 x 0.0385 x (87.21% x 45%) x (12.79% x 39%) = 0.1580

Hence, standard deviation, σ_p = Variance^1/2 = 0.1580^1/2 = 39.75%

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Part (a)

The formula for CAL is:

Or, 14% = 5.5% + (14.85% - 5.5%)/39.75% x σ_c = 5.5% + 0.2352 x σ_c

Hence, the standard deviation of your portfolio σ_c = (14% - 5.5%) / 0.2352 = 36.14%

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Part b - 1

Let y be the proportion invested in the T bill fund

Hence, E(r_c) = 14% = y x r_f + (1 - y) x E(r_p) = y x 5.5% + (1 - y) x 14.85%

Hence, the proportion invested in the T bill fund = y = (14.85% - 14%) / (14.85% - 5.5%) = 9.08%

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Part b - 2

Proportion invested in risk portfolio = 1- y = 1 - 9.08% = 90.92%

Proportion invested in stock fund = 90.92% x W_s = 90.92% x 87.21% = 79.29%

Proportion invested in bond funds = 90.92% - 79.29% = 11.63%