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 4.0%. The probability distributions of the risky funds are:   

	Expected Return		Standard Deviation
Stock fund (S)	10	%	32	%
Bond fund (B)	7	%	24	%

The correlation between the fund returns is .1250.

Suppose now that your portfolio must yield an expected return of 8% 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.)

Standard deviation             %

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

Proportion invested in the T-bill fund             %

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.)

	Proportion Invested
Stocks	%
Bonds	%

Answer 1

In case of a mixture of two risky assets, the equation for the weights of the individual assets would be given by:

where w(d) is the weight of the bond fund and w(e) is the weight of the stock fund. E(rd) is the expected return of the bond fund and E(re) is the expected return of the stock fund. Rf is the risk-free rate equal to the T-Bill yield of 4 %

E(re) = 10 %, s(e) = 32 % (standard deviation), E(rd) = 7 % and s(d) = 24 %

Covariance(E(re),E(rd)) = Correlation x s(e) x s(d) = 0.125 x 32 x 24 = 96

wd = [{(7-4) x (32)^(2)} - {(10-4) x 96}] / [{(7-4) x (32)^(2)} + {(10-4) x (24)^(2)} - {(10+7 - 2 x 4) x 96}] = 0.4407

and we = 1-wd = 1 - 0.4407 = 0.5593

Expected Return = r = wd x rd + we x re = 0.5593 x 10 + 0.4407 x 7 = 8.6779 % approximately

Standard Deviation = s = [{0.5593 x 32}^(2) + {0.4407 x 24}^(2) + 2 x 0.5593 x 0.4407 x 32 x 24 x 0.125]^(1/2) = 21.898 % approximately.

Let the proportion of the T-Bill and Optimal Risky portfolio be (1-y) and y in the complete portfolio.

Expected Return of Complete Portfolio = Rc = 8 %

Rc = Rf + y x (r-Rf) = 8

4 + y x (8.6779 - 4) = 8

y = 0.8551

(a) Standard Deviation of the complete portfolio = y x s = 0.8551 x 21.898 = 18.725 % approximately.

(b1) Proportion Invested in T-Bill Fund = 1-y = 1-0.8551 = 0.1449

(b2) Proportion of Stock = y x 0.5593 = 0.8551 x 0.5593 = 0.4783 and Proportion of Bond = y x 0.4407 = 0.8551 x 0.4407 = 0.3768