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# Investment Linear Programming model: Risk Management

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The portfolio Manager of Pension planners has been asked to invest \$1,000,000 of a large pension fund. The investment research development department has identified six mutual funds with varying investment strategies, resulting in different potential returns and associated risks, as summarized in the following table:

1 2 3 4 5 6
Price(\$/share) 455 76 110 17 23 22
ExpectedReturn(%) 30 20 15 12 10 7
Risk Category High High High Medium Medium low

One way to control the risk is to limit the amont of money invested in the various funds. To that end, the management of Pension Planners has specified the following guidelines:
1- the total amount invested in high risk funds must be between 50 and 75% of the portfolio.
2- The total amount invested in medium risk funds must be between 20 and 30% of the portfolio
3- The total amount invested in low risk funds must be at least 5 % of the portfolio

A second way to control risk is to diversify- that is, to spread the risk by investing in many different alternatives. The management of Pension Planners, has specified that the amount invested in the high-risk funds 1,2 and 3 should be in the ratio 1:2:3, respectively. The amount invested in the medium risk funds 4 and 5 should be 1:2.
With these guidelines, formulate a single LP model to maximize the expected rate of return.

https://brainmass.com/math/linear-programming/investment-linear-programming-model-risk-management-7956

#### Solution Preview

Solution. From the hypothesis, we can assume that we invest x,2x,3x,y,2y and z in "fund 1", "fund 2", "fund 3", "fund 4" , "fund 5" and "fund 6", respectively. So the expected return is

R(x,y,z)=0.30*x+0.20*2x+0.15*3x+0.12*y+0.10*2y+0.07z
=1.15x+0.32y+0.07z
Now we form the constraints as follows.
1. The total is 1000,000, so
x+2x+3x+y+2y+z<=1000,000
that is,
6x+3y+z<=1000,000 (1)
2. the total amount invested in high risk funds must be between 50 and 75% of the portfolio, so
0.50*1000,000<=x+2x+3x<=0.75*1000,000,

that ...

#### Solution Summary

A risk management problem is solved using linear programming methods.

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