Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.0%. The probability distributions of the risky funds are: (Total 20 points) Expected Return of Market Standard Deviation Stock funds (S) 15% 32% Bond funds (B) 9% 23% The correlation between the fund returns is 0.15.

a. Use the formula below to compute the optimal portfolio weights.

B. You manage $2000 for your client and your client wants an expected return of 10%. Using the optimal weights you got from part a, how much money do you invest in each of the three funds? (10 points) Hint: you need to compute the risky portfolio return using optimal weights from part a first.

Answer 1

Corer = 0.15, Cave = 0.15 × 0.32 × 0.23 = 0.01104

Compute risk premium, R1 = 15 — 5 = 10%; R2 = 9 — 5 = 4%; and R1 +R2 = 14%

Following weights distribution for optimal Sharpe ratio

WI = (0.10 × 0.23^2 — 0.04 × 0.01104) / (0.10 × 0.23^2 + 0.04 × 0.32^2 — 0.14 × 0.01104)

WI = 0.6184, W2= 1 - 0.6184 = 0.3816

Return on risky portfolio = 0.6184 × 0.15 + 0.3816 × 0.09 = 0.1271 or 12.71%

Variance of risky portfolio = 0.6184^2 × 0.32^2 + 0.3816^2 × 0.23^2 + 2 × 0.6184 × 0.3816 × 0.01104

Variance of risky portfolio = 0.052073353 = (0.228195864) ^2

SD of risky portfolio = 0.2282 or 22.82%

Sharpe ratio = (0.1271— 0.05) / 0.2282 = 033786

Given, return on complete portfolio is 10% and compute distribution of $2,000.

Suppose, y is invested in optimal risky portfolio and 1 — y is invested in risk-free asset.

Weight of investment in risk-free asset I — y

Weight of investment in asset 1: yW1

Weight of investment in asset 2: yW2

Return on complete portfolio = 0.05 × (1 — y) + y × 0.1271 = 0.10

y = (0.10 — 0.05) / (0.1271 — 0.05) = 0.6485; and I — y = 0.3515

Weight of investment in risk-free asset: I — y = 0.3515

Weight of investment in asset 1: yW1 = 0.6485 × 0.6184 = 0.4010

Weight of investment in asset 2: yW2 = 0.6485 × 0.3816 = 0.2475

Now, distribution of $2000 as below

For risk-free asset: 0.3515.2000 = $703

For asset 1: 0.4010.2000 = $802

For asset 2: 0.2475.2000 = $495