Investment manager Max Gaines has several clients who wish to own a mutual fund

Question

Investment manager Max Gaines has several clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the S&P 500 stock index. His task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the S&P 500 index. Formulate the appropriate nonlinear program.

Annual Returns (Planning Scenarios)

Mutual Fund Year 1 Year 2 Year 3 Year 4

International Stock 26.73 22.37 6.46 3.19

Large-Cap Blend 18.61 14.88 10.52 5.25

Mid-Cap Blend 18.04 19.45 15.91 -1.94

Small-Cap Blend 11.33 13.79 -2.07 6.85

Intermediate Bond 8.05 7.29 9.18 3.92

S&P 500 Index 21 19 12 4

Explanation / Answer

Options

Variables

a: % of the portfolio invested in International stock

b: % of the portfolio invested in large cap blend

c: % of the portfolio invested in mid cap blend

d: % of the portfolio invested in small cap blend

e: % of the portfolio invested in Intermediate bond

Constraints

1. a+b+c+d+e = 1 = 100%

2. a,b,c,d,e are all non-negative

Objective

Minimize the difference between the returns of all 5 options combined (let's call it portfolio) and S&P 500 index

For this we will take absolute difference of return of every year of portfolio and S&P 500 index and add it

and try to minimize the sum of difference

Year 1 difference (d1) = 26.73a + 18.61b+18.04c+11.33d + 8.05e - 21

Year 2 difference (d2) = 22.37a + 14.88b+19.45c+13.79d + 7.29e - 19

Year 3 difference (d3) = 6.46a + 10.52b +15.91c-2.07d + 9.18e - 12

Year 4 difference (d4) = -3.19a + 5.25b +-1.94c-6.85d -3.92e - 4

minimize z = |d1|+|d2|+|d3|+|d4|

Since the question only requires to formulate the problem, I am not solving it here.

Please upvote the answer.

Mutual fund y1 y2 y3 y4 International stock 26.73 22.37 6.46 3.19 Large-Cap blend 18.61 14.88 10.52 5.25 Mid-Cap blend 18.04 19.45 15.91 -1.94 Small-Cap blend 11.33 13.79 -2.07 6.85 Intermediate bond 8.05 7.29 9.18 3.92

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