Question: An investment advisory firm manages funds for its numerous clients. The company uses an asset allocation model that recommends the portion of each client’s portfolio to be invested in a growth stock fund, and income fund, and a money market fund. To maintain diversity in each client’s portfolio, the firm places limits on the percentage of each portfolio that may be invested in each of the three funds. General guidelines indicate that the amount invested in the growth fund must be between 20% and 40% of the total portfolio value. Similar percentages for the other two funds stipulate that between 20% and 50% of the total portfolio value must be in the income fund, and at least 30% of the total portfolio value must be in the money market fund.

In addition, the company attempts to assess the risk tolerance of each client and adjust the portfolio to meet the needs of the individual investor. For example, Williams just contracted with a new client who has $300,000 to invest, and all of it must be invested. Based on an evaluation of the client’s risk tolerance, Williams assigned a maximum risk index of 0.05 for the client. The firm’s risk indicators show the risk of the growth fund at 0.10 per dollar invested, the income fund at 0.06 per dollar invested, and the money market fund at 0.01 per dollar invested. An overall portfolio risk index is computed as a weighted average of the risk rating for the three funds. The average risk of the portfolio would be the total risk divided by the total investment. Thus,

(average risk) = (total risk)/(total investment).

For example, if a person invested 50,000 in the growth fund, 30,000 in the income fund, and 20,000 in the money market fund, the total risk would be 0.10(50,000) + 0.07(30,000) + 0.01(20,000) = 7,300; the average risk would be 7,300/100,000 = 0.073. (NOTE: When putting the risk measure into the linear program, it is better to work with the total risk to avoid round-off errors.)

Additionally, Williams is currently forecasting annual yields of 12% for the growth fund, 8% for the income fund, and 2% for the money market fund. Based on the information provided, how should the new client be advised to allocate the $300,000 among the growth, income, and money market funds? Develop a linear programming model that will provide the maximum yield for the portfolio.

a) Carefully define the decision variables.

b) Formulate the linear program.

c) Solve this linear program and fill in the table below for the optimal solution.