18. Matching the S&P 500 Return. Let us consider again the investment data from Hauck Financial Services used in Section 14.4 to illustrate the Markowitz portfolio model. The data are shown below, along with the return of the S&P 500 Index. Hauck would like to create a portfolio using the funds listed, so that the resulting portfolio matches the return of the S&P 500 index as closely as possible.

Mutual Fund Year 1 Year 2 Year 3 Year 4 Year 5 Foreign Stock 10.060 13.120 13.470 45.420 221.930 Intermediate-Term Bond 17.640 3.250 7.510 21.330 7.360 Large-Cap Growth 32.410 18.710 33.280 41.460 223.260 Large-Cap Value 32.360 20.610 12.930 7.060 25.370 Small-Cap Growth 33.440 19.400 3.850 58.680 29.020 Small-Cap Value 24.560 25.320 26.700 5.430 17.310 S&P 500 Return 25.000 20.000 8.000 30.000 210.000

a. Develop an optimization model that will give the fraction of the portfolio to invest in each of the funds so that the return of the resulting portfolio matches as closely as possible the return of the S&P 500 Index. (Hint: Minimize the sum of the squared deviations between the portfolio’s return and the S&P 500 Index return for each year in the data set.)

b. Solve the model developed in part (a).