Homework SourseThe Portfolio Manager of Pension Raiders Inc has been asked
The Portfolio Manager of Pension Raiders, Inc. has been asked to invest $1,000,000 of a large pension fund. The Investment Research Department has identified six mutual funds with varying investment strategies, resulting in different expected returns and associated risks, as summarized in the following table. Mutual Fund
1
2
3
4
5
6
Price ($/share)
45
76
110
17
23
22
Expected Return (%)
30
20
15
12
10
7
Risk Category
High
High
High
Medium
Medium
Low
One way to control the risk is to limit the amount of money invested in the various funds. To that end, the management of Pension Planners, Inc. 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, Inc., 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 in the ratio 1:2.
With these guidelines, formulate a single LP model to maximize the expected rate of return.
To determine what portfolio you, the Portfolio Manager, should recommend so as to maximize the expected rate of return, obtain the formulation and answer the following questions.
a. Use Excel’s Solver to solve the linear programming formulation. Prepare a table indicating the amount of money invested in each fund.
b. Indicate the highest return that can be achieved if no money is invested in Fund 6.
c. Provide the optimal investment strategy if the amount invested in Fund 4 is to be the same as that invested in Fund 5.
d. Analyze the effect that increasing the amount that must be invested in the low-risk fund would have on the optimal rate of return.
4. A winner of the Florida Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical company with an estimated return on investment of 12%, and a public utility with an estimated return on investment of 6%. Although a long-range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1-10 (with 10 being the most risky) is assigned to each of the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The petrochemical company’s stock has a risk index of 9 while the public utility’s stock has a risk index of 4. The average risk index of the portfolio should not be higher than 6. The following LP model is proposed, where C is the $ amount invested in the petrochemical company’s stock and U is the $ amount invested in the public utility’s stock:
Maximize 0.12 C + 0.06 U
Subject to
C + U < 50000
9 C + 4 U < 6 (C + U)
C, U > 0
Solve the problem using Excel’s Solver and determine the average risk for this investment. Explain the optimal solution and objective function value in the context of the problem.
One way to control the risk is to limit the amount of money invested in the various funds. To that end, the management of Pension Planners, Inc. 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, Inc., 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 in the ratio 1:2. With these guidelines, formulate a single LP model to maximize the expected rate of return. To determine what portfolio you, the Portfolio Manager, should recommend so as to maximize the expected rate of return, obtain the formulation and answer the following questions. a. Use Excel’s Solver to solve the linear programming formulation. Prepare a table indicating the amount of money invested in each fund. b. Indicate the highest return that can be achieved if no money is invested in Fund 6. c. Provide the optimal investment strategy if the amount invested in Fund 4 is to be the same as that invested in Fund 5. d. Analyze the effect that increasing the amount that must be invested in the low-risk fund would have on the optimal rate of return. 4. A winner of the Florida Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical company with an estimated return on investment of 12%, and a public utility with an estimated return on investment of 6%. Although a long-range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1-10 (with 10 being the most risky) is assigned to each of the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The petrochemical company’s stock has a risk index of 9 while the public utility’s stock has a risk index of 4. The average risk index of the portfolio should not be higher than 6. The following LP model is proposed, where C is the $ amount invested in the petrochemical company’s stock and U is the $ amount invested in the public utility’s stock: Maximize 0.12 C + 0.06 U Subject to C + U < 50000 9 C + 4 U < 6 (C + U) C, U > 0 Solve the problem using Excel’s Solver and determine the average risk for this investment. Explain the optimal solution and objective function value in the context of the problem. |
Solution
The excel solution to the given problem is:
It satisfies all the given condition in part A, by investing 75% in High risk and 20% and 5% resp. in medium risk and the low risk assets.
Considering the second way, where the assets have to be in a given ratio,
the investment should be as follows :
The return in this case is 16.85 %
b) If no money can be invested in fund 6, we must relax the assumption that Money invested in low risk funds must be greater than 5%
This is obtained by keeping other constraints intact.
c) we now make fund 4 = fund 5 and remove all other constaints on other funds,
Note that the weight constraints haven\'t been removed yet.
Increasing the amount of money invested in the low risk firm will decrease the total return obtained.
P.S : The assumptions that are made can be changed to get different answers. The assupmtions made are clearly stated. Incase of a difficulty, kindly comment with the constraints that you wish to take and I\'ll provide the answer for that as well.
| Amount Invested | 1000000 | |||||
| Company | Share Price | Return | Amt. Inv | % wt | Total Return | |
| 1 | 45 | 30 | 750000 | 0.75 | 22.5 | |
| 2 | 76 | 20 | 0 | 0 | 0 | |
| 3 | 110 | 15 | 0 | 0 | 0 | |
| 4 | 17 | 12 | 200000 | 0.2 | 2.4 | |
| 5 | 22 | 10 | 0 | 0 | 0 | |
| 6 | 23 | 7 | 50000 | 0.05 | 0.35 | |
| sum | 1000000 | 1 | 25.25 | |||
| High | 0.75 | 0 | ||||
| Medium | 0.2 | |||||
| Low | 0.05 |
