A pension fund manager is con- sidering three mutual funds. The first is a stock

Question

A pension fund manager is con- sidering three mutual funds. The first is a stock fund, the second is a long-term govern- ment and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:

Expected Return

Stock fund (S) 20% Bond fund (B) 12

The correlation between the fund returns is .10.

Standard Deviation

30% 15

You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL.

What is the standard deviation of your portfolio?

What is the proportion invested in the T-bill fund and each of the two risky funds?

Explanation / Answer

Answer:

The parameters of the opportunity set are:

E(rS) = 20%, E(rB) = 12%

S = 30%, B = 15%

AB = 0.10

From the standard deviations and the correlation coefficient we can generate the “covariance matrix” (note that SB = SB x S X B):

Bonds

Stocks

Bonds

Stocks

Bonds

   1.00

   0.10

Bonds

   .0225

   .0045

Stocks

   0.10

   1.00

00

Stocks

   .0045

   .0900

For formula for the minimum variance weight is on page 213 in the text.

(I did not derive this in class.)

WSMin =

WSMin = 1 - 0.1739 = 0.8261

Expected Return Standard Deviation of the minimum variance portfolio are:

E(rMin) = (0.1739)(0.20) + (0.8261)(0.12) = 13.39%

Min =

          = [(0.1739)2(0.30) 2 + (0.8261)2(0.15) 2 + 2(0.1739)(0.8261)(0.0045)]1/2 = 13.92%

The optimal risky portfolio is the portfolio along the frontier that maximizes the slope of the CAL. The formula for the weight of an asset in the optimal risky portfolio is on page 217.

NOTE the change from E(r) – rf = E(R):

E(RS) = 0.20 – 0.08 = 0.12;   E(RB) = 0.12 – 0.08 = 0.04

S2 = 0.090; B2 = 0.0225; SB = 0.0045.

WB* = 1 - 0.4516 = 0.5484

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

E(rP) = (0.4516)(0.20) + (0.5484)(0.12) = 15.61%

P=

    = [(0.4516)2(0.30) 2 + (0.5484)2(0.15) 2 + 2(0.4616)(0.5484)(0.0045)]1/2 = 16.54%

Answer: E (rp) =0.548*12 + 0.452*20 = 15.6 %

15.6% =8%*w + 15.6%*(1-w)

w:weight in the risk free asset= 0.21

.the rest in stock and bond funds

Bonds

Stocks

Bonds

Stocks

Bonds

   1.00

   0.10

Bonds

   .0225

   .0045

Stocks

   0.10

   1.00

00

Stocks

   .0045

   .0900

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