A pension fund manager is considering three mutual funds. The first is a stock f

Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a longterm government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 5.7%. The probability distribution of the risky funds is as follows: The correlation between the fund returns is 0.17. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. (Do not round intermediate calculations and round your final answers to 2 decimal places. Omit the "%" sign in your response.) Portfolio invested in the stock Portfolio invested in the bond Expected return Standard deviation

Explanation / Answer

Portfolio Invested in Stock = 106.06%

Portfolio invested in Debt = - 6.06%

Expected Return                 = 18.67%

Standard Deviation            = 49.49%

working

For Stock Fund - Expected Return Re = 18%   Standard Deviation SDe = 47%

For Debt Fund - Edpected Return Rd = 7%   Standard Deviation SDd = 41%

Correlation p = 0.17

Risk-free rate on T-Bill Fund = 5.70%

Covariance Cov(e,d) = correlation * SD of Stock Fund * SD of Bond Fund

                                      = 0.17 * 47 * 41 = 327.59

Weight of debt in an optimal portfolio

Wd = A/B

Where

A= {(Rd - Rf) * SDe^2 - (Re - Rf) * Cov (e,d)}

B = {(Re - Rf) * SDd^2 + (Rd-Rf)* SDe^2 - (Re - Rf + Rd - Rf) * Cov (e,d)}

A = (7-5.7) * 47^2 - (18-5.7) * 327.59 = 2871.70 – 4029.357 = - 1157.657

B = (18-5.7) * 41^2 + (7-5.7)*47^2 - (18-5.7+7-5.7)*327.59

= 20676.30 + 2871.7 – 4455.224

= 19092.776

Portfolio invested in debt Wd = -1157.657 / 19092.776 = -0.06063

Portfolio invested in Stock We = 1-(-0.06063) = 1.06063

Expected return Rp = We*Re + Wd * Rd = 1.06063 * 18 + (-0.06063)*7

                   = 19.0914 – 0.42441 = 18.66699 or 18.67% (rounded off)

Standard Deviation of Portfolio

SD = [We^2 * SDe^2 + Wd^2 * SDd^2 + 2 * We * Wd * Cov (e,d)]^1/2

   = [1.06063^2 * 47^2 + (-0.06063)^2 * 41^2 + 2 * 1.06063 * (-0.06063) * 327.59]^1/2

   = [1.124936*2209 + 0.003676 * 1681 – 42.132]^1/2

   = [2484.983624+6.179356-42.132]^1/2 = [2449.03098]^1/2 = 49.4877% or 49.49% (rounded off)

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