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 rate of 5%. The probability distribution of the risky funds is as follows:

expected return of the stock fund: 19%, standard deviation of the stock fund is 32 %

expected return of the bond fund: 12%, standard deviation of the bond fund is 15 %

The correlation between the fund returns is 0.11. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?

Answer 1

R_S = Expected return of the stock fund = 19%

R_B = Expected return of the bond fund = 12%

_S = Standard deviation of the return on stock fund = 32%

_B = Standard deviation of the return on bond fund = 15%

R_f = T-bill money market fund yield = 5%.

= 0.11

As a first step we will calculate the weights of the stock fund and bond fund in the optimal risky portfolio. For that purpose we will have to calculate the co-variance first.

Cov(R_S, R_B) = x _S x _B = 0.11 x 32% x 15% = 0.00528

Proportion of stock fund in the optimal risky portfolio

= 0.301915476 = 30.19%

Hence, w_B = 1 - w_S = 1 - 30.19% = 69.81%

R_P = Expected return of the optimal risky portfolio = w_S x R_S + w_B x R_B = 30.19% x 19% + 69.81% x 12% = 14.1134%

= Standard deviation of the optimal risky portfolio

= 0.150081511 = 15.01%

Part (a)

The expected return on the portfolio is 14%.

The equation for the CAL is

R_C = R_f + [(R_P - R_f) / _P ] x _C

Hence, 14% = 5% + [(14.1134% – 5%)/15.01%] x _C

Or, 14% - 5% = 9% = 0.607230584 x _C

Hence, _C = 0.148213879 = 14.82%

Hence, standard deviation of the portfolio = 14.82%

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Part (b)

Lets say w is the proportion invested in T bill fund.

R_C = 14% = w x R_f + (1 - w) x R_P = w x 5% + (1 - w) x 14.1134% = 14.1134% - 9.1134% x w

Hence, w = (14.1134% - 14%) / 9.1134% = 1.24%

Hence, the proportion invested in T bill fund = 1.24%

Proportion invested in optimal risky portfolio = 1 - w = 1 - 1.24% = 98.76%

Proportion invested in stock fund = (1 - w) x w_S = 98.76% x 30.19% = 29.82%

Proportion invested in bond fund = (1 - w) x w_B = 98.76% x 69.81%% = 69.81%