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 4.5%. The probability distribution of the risky funds is as follows:

Expected Return | Standard Deviation | |

Stock fund (S) |
15% | 35% |

Bond fund (B) |
6 | 29 |

The correlation between the fund returns is 0.15.

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 | % |

Answer #1

*Cov(Rs,Rb)* = correlation coefficient * SDs * SDb

= 0.15 x 35 x 29 = 152.25

Ws = [{(ERs - Rf)*SDb^{2}} - {(ERb - Rf)*Cov(Rs,Rb)}] /
[{(ERs - Rf)*SDb^{2}} + {(ERb - Rf)*SDs^{2}}]

= [{(15 - 4.5)*(29)^{2}} - {(6 - 4.5)*152.25}] / [{(15 -
4.5)*(29)^{2}} - {(6 - 4.5)*(35)^{2}}]

= [8,830.50 - 228.375] / [8,830.50 + 1,837.50] = 8,602.125 / 10,668.00 = 0.8063, or 80.63%

So, **Portfolio invested in the stock =
80.63%**

Wb = 1 - Ws = 1 - 0.8063 = 0.1937, or 19.37%

So, **Portfolio invested in the bond = 19.37%**

Expected return = (Ws * ERs) + (Wb * ERb)

= (0.8063 * 15%) + (0.1937 * 6%] = 12.10% + 1.16% = 13.26%

So, **Expected return = 13.26%**

S.D. = [(SDs^{2} * Ws^{2}) + (SDb^{2} *
Wb^{2}) + (2*Ws*Wb*Cov(Rs,Rb)]^{1/2}

= [(35^{2} * 0.8063^{2}) + (29^{2} *
0.1937^{2}) + (2 * 0.8063 * 0.1937 *
152.25)]^{1/2}

= [796.49 + 31.54 + 47.55]^{1/2} =
[875.58]^{1/2} = 29.59%

So, **Standard Deviation = 29.59%**

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