The Pfeiffer Company manages approximately $15 million for clients. For each client, Pfeiffer chooses a mix of three investment vehicles: a growth stock fund, an income fund, and a money market fund. Each client has different investment objectives and different tolerances for risk. To accommodate these differences, Pfeiffer places limits on the percentage of each portfolio that may be invested in the three funds and assigns a portfolio risk index to each client. Here’s how the system works for Dennis Hartmann, one of Pfeiffer’s clients. Based on an evaluation of Hartmann’s risk tolerance, Pfeiffer has assigned Hartmann’s portfolio a risk index of 0.05. Furthermore, to maintain diversity, the fraction of Hartmann’s portfolio invested in the growth and income funds must be at least 10% for each, and at least 20% must be in the money market fund. The risk ratings for the growth, income, and money market funds are 0.10, 0.05, and 0.01, respectively. A portfolio risk index is computed as a weighted average of the risk ratings for the three funds, where the weights are the fraction of the portfolio invested in each of the funds. Hartmann has given Pfeiffer $300,000 to manage. Pfeiffer is currently forecasting a yield of 20% on the growth fund, 10% on the income fund, and 6% on the money market fund.
a. Develop a linear programming model to select the best mix of investments for Hartmann’s portfolio.
b. Solve the model you developed in part (a).
c. How much may the yields on the three funds vary before it will be necessary for Pfeiffer to modify Hartmann’s portfolio?
d. If Hartmann were more risk tolerant, how much of a yield increase could he expect? For instance, what if his portfolio risk index is increased to 0.06?
e. If Pfeiffer revised the yield estimate for the growth fund downward to 0.10, how would you recommend modifying Hartmann’s portfolio?
f. What information must Pfeiffer maintain on each client in order to use this system to manage client portfolios?
g. On a weekly basis Pfeiffer revises the yield estimates for the three funds. Suppose Pfeiffer has 50 clients. Describe how you would envision Pfeiffer making weekly modifications in each client’s portfolio and allocating the total funds managed among the three investment funds.