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 3.0%. The probability distributions of the risky funds are:   

	Expected Return		Standard Deviation
Stock fund (S)	12	%	41	%
Bond fund (B)	5	%	30	%

The correlation between the fund returns is .0667.

Suppose now that your portfolio must yield an expected return of 9% 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.)

Stocks: ???%

Bonds: ???%

Answer 1

Solution: E(Rs) = 12%, E(Rb) = 5%,

Standard Deviation of Stock = 41%, Standard Deviation of Bond = 30%

Cov(B,S) = Correlation(B,S)*Standard Deviation of Stock*Standard Deviation of Bond

Cov(B,S) = 0.0667*0.41*0.30 = 0.008204

The proportion of stocks in the optimal risky portfolio is given by:

= [(12%-3%)*(30%)^2 - (5%-3%)*0.008204] / [(12%-3%)*(30%)^2+(5%-3%)*(41%)^2 - (12%-3%+5%-3%)*0.008204]

=(0.0081 - 0.000164) / (0.0081 + 0.003362 - 0.000902) = 0.751515

Weight of Bond (Wb) = 1 - Weight of Stock (Ws) = 1 - 0.751515 = 0.248485

The mean and standard deviation of optimal risk portfolio are:

E(Rp) = We*E(Rs) + Wb*E(Rb) = 0.751515*12% + 0.248485*5% = 0.102606 = 10.2606%

Standard deviation of Portfolio is calculated as:

Standard deviation of portfolio = {(0.751515)^2*(0.41)^2 + (0.248485)^2*(0.3)^2 + 2*0.751515*0.248485*0.0667*0.41*0.3}^(1/2)

= 0.321807 = 32.1807%

a) If we require our portfolio to yield a mean return of 9%, we can find the corresponding standard deviation from the optimum Capital Allocation Line (CAL). The formula for CAL is:

Using, E(Rc) = 9%, Rf = 3%, Standard deviation of Optimal Portfolio = 32.1807%

9% = 3% + Standard deviation of portfolio*(10.2606% - 3%)/32.1807%

6% = Standard deviation of portfolio*0.22562

Hence, Standard deviation of portfolio = 6%/0.22562 = 26.59%

b) Let 1-y be the proportion invested in T-bills and y be the proportion invested in the optimal portfolio of stocks and bond. Since, the mean of the complete portfolio is 9%, thus the proportion y is calculated as:

9% = (1-y)*3% + y*10.2606%

9% = 3% - 3%*y + 10.2606%*y

6% = 7.2606%*y

hence, y = 6%/7.2606% = 0.826378

Proportion of stocks in complete portfolio = 0.826378*0.751515 = 0.621035

Proportion of bonds in complete portfolio = 0.826378*0.248485 = 0.205343

b-2) Usiing only stock and bond funds to achieve a portfolio mean of 9%, the appropriate proportion in stock and bond is calculated as:

9% = Ws*12% + (1-Ws)*5%

9% = Ws*7% + 5%

4% = Ws*7%

Ws = 0.571429 = 57.14%

Thus, weight of bond = 1 - Ws = 1 - 0.571429 = 0.428571 = 42.86%