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 8 percent. The probability distribution of the risky funds is as follows:

	Expected Return	Standard Deviation
Stock fund (S)	.20	.30
Bond fund (B)	.12	.15

The correlation between the fund returns is 0.10.

1. What are the investment proportions of the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?
2. Solve numerically for the proportions of each of the assets and for the expected return and standard deviation of the optimal risky portfolio.

Answer 1

  The weight of stock in the minimum variance portfolio is calculated using the following formula

w_min(S) = [σ²_B - Cov(r_s, r_B)]/[σ²_{s +} σ²_B - 2 Cov(r_s, r_B)]

The parameters of the opportunity set are:

E(rS) = 20%, E(rB) = 12%, σS = 30%, σB = 15%, ρ = 0.10

From the standard deviations and the correlation coefficient we can calculate the

covariance matrix

Cov(rS, rB) = ρσSσB

Bonds Stocks

Bonds 225 45

Stocks 45 900

The minimum-variance portfolio is computed as follows:

w_min(S) = [225-45]/[900+225-(2x45)] = 0.1739

w_min(B) = 1 − 0.1739 = 0.8261

The minimum variance portfolio mean and standard deviation are:

E(rMin) = (0.1739 × 20) + (0.8261 × 12) = 13.39%

σMin = [w²_s σ²_s+ w²_B σ²_B + 2w_s w_B Cov(r_s ,r_B )]

= [(0.1739² × 900) + (0.8261² × 225) + (2 × 0.1739 × 0.8261 × 45)]^1/2

= 13.92%

Now,Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal

risky portfolio.

The optimal risky portfolio is the combination of stock and bond fund that gives investor the best risk-return trade-off when combining with T-bill

The proportion of the optimal risky portfolio invested in the stock fund is given by:

w_{s =} {[E(r_s) -r_f] σ²_B-[E(r_B)-r_f] Cov(r_s,r_B)}/[E(r_s) - r_f]σ²_B+[E(r_B)-r_f]σ²_s-[E(r_s)-r_f+E(r_B)-r_f]Cov(r_s,r_B)

= {[(20-8)x 225]-[(12-8)x45]}/[(20-8) X225]+[(12-8)x900-[(20-8+12-8)x45] = 0.4516

w_s= 1-0.4516 = 0.5484

The mean and standard deviation of the optimal risky portfolio are:

E(rP) = (0.4516 × 20) + (0.5484 × 12) = 15.61%

σp = [(0.4516²× 900) + (0.5484² × 225) + (2 × 0.4516 × 0.5484 × 45)]^1/2

= 16.54%