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 0.06 for the client. The firm's risk indicators show the risk of the growth fund at 0.10, the income fund at 0.07, and the money market fund at 0.01. 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 as shown here:
(average risk) = (total risk)/(total investment).
For example, to calculate the risk of a $100,000 portfolio with 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 rather than the average risk to avoid round-off errors.)
Additionally, Williams is currently forecasting annual yields of 8% for the growth fund, 6% 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? A linear program has been developed to help with this. However, this may or may not be totally correct. You should verify that it is the correct formulation before solving the problem. If it is not correct, make any necessary changes to the linear program before solving it on the computer.
G = dollars invested in the growth fund
I = dollars invested in the income fund
M= dollars invested in the money market fund
Maximizeyield (return) = 0.08G + 0.06I + 0.02M
Subject to:
G + I + M = 300000 Total investment
0.10G + 0.07I + 0.01M <18000 Total risk (based on 6% average risk)
G >60000 Minimum in growth fund
G < 120000 Maximum in growth fund
I >60000 Minimum in income fund
I < 150000 Maximum in income fund
M >90000 Minimum in money market fund
G, I, M > 0
NOTE: The equality for the total investment constraint simplifies the other constraints. While the total amount invested is G + I + M, this can be replaced by 300000 due to the equality condition. For example, 20% of (G + I + M) becomes 20% of $300,000 or simply $60,000. If this were a less-than-or-equal-to constraint, this could not be done and that constraint would be G > 0.20(G + I + M).