Question

Based on this week's readings (from Chapter 12 in the course text), address the following scenario and answer the questions: Lymann Brothers has a substantial number of clients who wish to own a mutual fund portfolio that closely matches the performance of the S&P 500 stock index. A manager at Lymann Brothers has selected five mutual funds that will be considered for inclusion in the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund. Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year3 Year 4 International Stock 25.64 27.62 5.80 −3.13 Large-Cap Blend 15.31 18.77 11.06 4.75 Mid-Cap Blend 18.74 18.43 6.28 −1.04 Small-Cap Blend 14.19 12.37 −1.92 7.32 Intermediate Bond 7.88 9.45 10.56 3.31 S&P 500 13.00 12.00 7.00 2.00 Minimize the variance of the portfolio subject to constraints on the expected return, assuming that Lymann Brothers' client requires the expected portfolio return to be at least 9 percent.

Answer 1

We are given the following information,

Mutual Fund Type	Year 1	Year 2	Year 3	Year 4
International Stock	0.2564	0.2762	0.058	-0.0313
Large Cap Blend	0.1531	0.1877	0.1106	0.0475
Mid Cap Blend	0.1874	0.1843	0.0628	-0.0104
Small Cap Blend	0.1419	0.1237	-0.0192	0.0732
Intermediate Bond	0.0788	0.0945	0.1056	0.0331
S&P 500 Stock Index	0.13	0.12	0.07	0.02

We are required to find out the optimal portfolio of these mutual funds which minimizes the standard deviation/variance as well as has a minimum return of 9%.

First we build the co-variance table for the data provided. Note that we mentioned the weights for different mutual funds on the sides of the table. Co-Variance can be found by doing a sum product of the returns of the various mutual funds. The bottom most row would contain the sum product of the co-variance and weights for each of the assets.

CO-VARIANCE TABLE
Weights		100%	0%	0%	0%	0%
		Int Stock	LC Blend	MC Blend	SC Blend	Int Bond
100%	Int Stock	0.017041742	0.006567	0.010905	0.005614	0.001942
0%	LC Blend	0.006566734	0.002734	0.00421	0.001609	0.001035
0%	MC Blend	0.010905289	0.00421	0.007043	0.003384	0.001348
0%	SC Blend	0.005614068	0.001609	0.003384	0.003907	-0.00042
0%	Int Bond	0.001942148	0.001035	0.001348	-0.00042	0.000763
100%		0.017041742	0	0	0	0

Then we add fields which specify the minimum expected return as well as the standard deviation of the portfolio which is basically the square root of the sum of the fields in the bottom most row of the co-variance table.

Also, based on the data provided we consider the average returns of the mutual funds as given below.

RETURNS
Int Stock	0.139825
LC Blend	0.124725
MC Blend	0.106025
SC Blend	0.0799
Int Bond	0.078

Finally upon providing the objectives, variables and constraints in the solver tool of excel, we get the following weights for the different mutual funds classes.

PARAMETER	VALUE
Return	9%
Standard Deviation	5.594%
International Stock	0%
Large Cap Blend	26%
Mid Cap Blend	0%
Small Cap Blend	0%
Intermediate Bond	74%