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

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Answer:

Corer = 0.15, Cove = 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 W1 = (0.10*0.23^2 -0.04*0.01104)/(0.10*0.23^2 + 0.04*0.32^2 -0.14*0.01104) W1 = 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*0.3242 +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 = 0.33786

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: 1 - 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 1-y=0.3515 Weight of investment in risk-free asset: 1 - 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