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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.

##### Solution Summary

A risk management problem is solved using linear programming methods.

##### 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 ...

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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• "Thank you very much for your valuable time and assistance!"

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