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% 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 that 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 $800,000 to invest. 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, 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, 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% for the growth fund, 12.5% for the income fund, and 7.5% for the money market fund. Based on the information provided, how should the new client be advised to allocate the $800,000 among the growth, income, and money market funds? Develop a linear programming 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, in which 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% or even to 14%?
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

Part A:

The investment advisory of JD Williams firm that manages $120 million funds for its clients. To achieve optimal portfolio returns, the company uses several financial approaches.

They are listed as follow:

1. An asset allocation model:-individual clients are provided separate strategies for optimal investment combinations.

2. A risk tolerance analysis:-To adjust the portfolio of individual clients based on their risk tolerance analysis.

3.Percentage limitations:- Protection on investors assets as an investment diversity.

(Q)How should the new client be advised to allocate the $800,000 in available funds for optimal growth, income, and money market funds?

Ans: (i) Model formulation

Decision variables

GF: Investment in growth stock fund

IF: investment in an income fund

MMF: investment in the money market fund

(ii)Objective function definition:

Maximize the total return of the portfolio (maximum)

= 18% *GF + 12.5% *IF +7.5%*MMF

= .18*GF + .125*IF + .075*MMF

(iii) Constraint definition :

GF+IF+MMF <=$800,000 this is the total amount which is available for investment.

.80GF - .20IF - .20 MMF >=0

As per the directives given i the question, the amount invested in the growth fund should be atleast 20 % of the portfolio.

= .60 GF --.40IF- .40MMF <=0

This is the amount investment in the growth fund should be 40% of the portfolio.

Subsequently, Amount investment in the income fund should be 20% and 50 %, in the money market fund should be at least 30 %, of the total portfolio.

And, the investor's risk tolerance is,

0.05GF + .21F -.04MMF <=0, investor's risk tolerance index.

Therefore the optimal portfolio allocation will be

Growth fund : $248,889

Income fund: $160,000

money market Fund: $391,111

Total of all three funds are $800,000

(1) Now, the anticipated annual yield is ,

Growth fund: $248,889*0.18 = $44,800

Income fund: $160,000 *0.125 = $20,000

Money market fund : $391,111*0.075= $29,333

Total = $94,133

Therefore , total anticipated annual yield is = (94133/800000)*100 = 11.77 %

(2)In terms of risk tolerance, it is increased to .055, from 0.5 which is a one-half increase. So, the annual yield consequently will increase by $4,667

= $94,133+ $ 4,667 = $98,800

Therefore, the modified asset allocation and the projected annual returns are,

Fund allocation	Project Annual Yield
Growth Fund= $293,333	Growth Fund= $293,333*0.18 = $52,800
Income Fund = $160,000	Income Fund = $160,000*0.12 5= $20,000
MMF = $346,667	MMF = $346,667 *$0.075 =$26,000
Total = $800,000	Total = $98,800

Total anticipated annual yield percent= ($98800/$800,000)*100 = 12.357%

(3) There would be no change in the investment recommendation as long as the yield is above the lower limit. But, it downgrades to 14% or even so, it will be outside the lower limit and the values are bound to chnge as the allocation of resources will change at that point.

If the yields hit lower limits, there would be a need for a new investment strategy as the optimal projected valued would not be in line with the results.

(4) Choosing an appropriate constraint is an impactful method to balance between risk and return according to the portfolio theory.

The risk indices given are 0.10 and .07 of the growth fund and income fund, so, if the client may choose to invest less aggressively, but that would result in a less annual yield of $86,607 than the projected yield of $94,133 by the original most risky recommendation.

(5) The usage of the new model is only recommended when the outlined criteria are met by the model potential new clients.

The mission of the company is to provide the best financial advice to its investors' needs, therefore the company will not be recommending the use of this asset allocation model as a casual guide to financial investment.