The following data is for question 1-7. 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 8%. The probability distribution of the risky funds is as follows:

The correlation between the fund returns is .10.

1. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?

Here is the solution.

Two assets
	Expected return	Std dev	Cor(1,2)
Asset 1	0.20	0.30	0.10
Asset 2	0.12	0.15
Riskfree	0.08
Covariance matrix
0.0900	0.0045
0.0045	0.0225
Minimum variance portfolio
Weight 1	0.174
Weight 2	0.826
Exp ret	0.134
Std dev	0.139

So weights are 17.4% for S and 82.6% for B

Expected return is13.4% and SD is 13.9%

Here is the list of formula used and the steps.

First calculate the Covariance Matrix

1. Square the return of S and B

2. Multiply the correlation of S and B with SDs of S and B

Now come the real formula part

for weight of S

=MMULT(MINVERSE(A11:B12),A28:A29)/SUM(MINVERSE(A11:B12))

for weight of B

=MMULT(MINVERSE(A11:B12),A28:A29)/SUM(MINVERSE(A11:B12))

A11 to B12 is covariance matrix which we calculated earlier

A28 is Return of S - Risk free rate

A29 is Return of B - Risk free rate

Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of zero to 100% in increments of 20%.

Proportion	Proportion	Expected	SD of the
S	B	Return	Portfolio
0.0	1.0	12%	15.00
0.2	0.8	14%	13.94
0.4	0.6	15%	15.70
0.6	0.4	17%	19.53
0.2	0.8	14%	13.94
1.0	0.0	20%	30.00
Note:
Expected return = 20%*proportion invested in S+12%*proportion invested in B
Standard deviation of the portfolio = (Ws^2*SDs^2+Wb^2*SDb^2+2*Ws*Wb*SDs*SDb*Cor(b,s)^0.5
Ws and Wb = respective weights
SDs and SDb respective standard deviations

3. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio?

4. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.