Question

J.D. Williams, Inc. is an investment advisory firm that manages more than $120 million in 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, an 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 percent of the total portfolio value. Similar percentages for the other two funds stipulate that between 20 and 50 percent of the total portfolio value must be in the income fund and that at least 30 percent 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 $800,000 to invest. Based on an evaluation of the client's risk tolerance. Williams assigned a maximun risk index of 0.05 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 morney market fund at 0.01. An overall portfolio risk index is computed as a weighted average of the risk rating for the three funds, where the weights are the fraction of the client's portfolio invested in each of the funds.

     Additionally, Williams is currently forecasting annual yields of 18 percent for the growth fund, 12.5 percent for the income fund, and 7.5 percent for the money market fund. Based on the information provided, how should the new client be advised to allocate the $800,000 amont the growth, income, and money market funds? Develop a linear programing model that will provide the maximum yield for the portfolio. Use your model to develop a managerial report.

MANAGERIAL REPORT

1. Recommend how much of the $800,000 should be invested in each of the three funds. What is the annual yield you anticipate for the investment recommendation?

2. Assume that the client's risk index could be increased to 0.055. How much would the yield increase, and how would the investment recommendation change?

3. Refer again to the original situation where the client's risk index was assessed to be 0.05. How would your investment recommendation change if the annual yield for the growth fund were revised downward to 16 percent or even to 14 pecent?

4. Assume that the client expressed some concern about having too much money in the growth fund. How would the original recommendation change if the amount invested in the growth fund is not allowed to exceed the amount invested in the income fund?

5. The asset allocation model you developed may be useful in modifying the portfolios for all of the firm's clients whenever the anticipated yields for the three funds are periodically revised. What is your recommendation as to whether use of this model is possible?

Answer 1

Solution:

1) Recommending how much of the $800,000 should be invested in each of the three funds. and Calculation of the annual yield you anticipate for the investment recommendation:

Developing a Linear Programming model for maximizing return subject to constraints for funds available, diversity, and risk tolerance,

The LP Formulation is as follows:

The optimal objective function value is 94,133.336. The sensitivity report is as follows,

Variable Cells
Model Variable	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
G	Growth Fund	248,888.889	0.000	0.180	1E+30	0.030
I	Income Fund	160,000.000	0.000	0.125	0.020	0.588
M	Money market fund	391,111.111	0.000	0.075	0.105	0.060

Constraints
Constraint Number	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
1	Funds Available	800,000.000	0.118	800,000.000	1E+30	800,000.000
2	Min Growth Fund	88,888.889	0.000	0.000	88,888.889	1E+30
3	Max Growth Fund	-71,111.111	0.000	0.000	1E+30	-71,111.111
4	Min Income Fund	0.000	-0.020	0.000	133,333.333	106,666.667
5	Max Income Fund	-195,555.556	0.000	0.000	1E+30	195,555.556
6	Min Money Market Fund	151,111.111	0.000	0.000	151,111.111	1E+30
7	Max Risk	0.000	1.167	0.000	6,400.000	8,000.00

The portfolio recommendation for Langford is as follows:

Fund	Amount Invested
Growth	$248,889
Income	$160,000
Money Market	$391,111
Total	$800,000

Therefore, the Portfolio Yield is 0.118 or 11.8%

2) The value has increased as predicted, the new optimal allocation is as follows:

Fund	Amount Invested
Growth	$293,333
Income	$160,000
Money Market	$346,667
Total	$800,000

Optimal Solution:

Variable	Value	Reduced Costs
G	293,333.313	0.000
I	160,000.00	0.000
M	346,666.656	0.000

3) Since 16% is in the objective coefficient range for the growth fund return, there would be no change in allocation. However, the return would decrease by (0.02) ($248,889)=$4,978.

The new formulation and optimal solution is as follows,

Optimal Solution:

Variable	Value	Reduced Costs
G	160,000.016	0.000
I	293,333.313	0.000
M	346,666.688	0.000

4) We have to expect a decrease in return as is shown in the following:

Optimal Solution:

Variable	Value	Reduced Costs
G	213,333.313	0.000
I	213,333.313	0.000
M	373,333.313	0.000

5) It is possible a model that can be developed for each client. The changed yield estimates requires a change in the objective function coefficients, if the change was outside the objective coefficient range.