Investment manager Max Gaines has several clients who wish t

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. nnual Returns (Planning Scenarios Year 1 26.73 18.61 18.04 11.33 8.05 21.00 Year 2 22.37 14.88 19.45 13.79 7.29 19.00 Year 3 6.46 10.52 15.91 -2.07 9.18 12.00 Year 4 -3.19 5.25 International Stock Large-Cap Blend Mid-Cap Blend Small-Cap Blend Intermediate Bond S&P; 500 Index 6.85 3.92 4.00

Solution

Let,

X1 = proportion of funds invested in International stock
X2 = proportion of funds invested in large-cap blend
X3 = proportion of funds invested in mid-cap blend
X4 = proportion of funds invested in small-cap blend
X5 = proportion of funds invested in intermediate bond

For the clarity of model exposition, we define five retrun variables R1, R2, R3, and R4 so that,

R1 = 26.73X1 + 18.61X2 + 18.04X3 + 11.33X4 + 8.05X5 (Year 1 return)
R2 = 22.37X1 + 14.88X2 + 19.45X3 + 13.79X4 + 7.29X5 (Year 2 return)
R3 = 6.46X1 + 10.52X2 + 15.91X3 - 2.07X4 + 9.18X5 (Year 3 return)
R4 = -3.19X1 + 5.25X2 - 1.94X3 + 6.85X4 + 3.92X5 (Year 4 return)

Formulation

Minimize. Z = Sum of square deviations = (R1 - 21)² + (R2 - 19)² + (R3 - 12)² + (R4 - 4)²
Subject to,
R1 = 26.73X1 + 18.61X2 + 18.04X3 + 11.33X4 + 8.05X5
R2 = 22.37X1 + 14.88X2 + 19.45X3 + 13.79X4 + 7.29X5
R3 = 6.46X1 + 10.52X2 + 15.91X3 - 2.07X4 + 9.18X5
R4 = -3.19X1 + 5.25X2 - 1.94X3 + 6.85X4 + 3.92X5

X1 + X2 + X3 + X4 + X5 = 1

X1, X2, X3, X4, X5 >= 0