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% to 40% of the total portfolio value. Similar percentages for the other two funds stipulate that between 20% to 50% of the total portfolio 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 $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, William’s is currently forecasting annual yields of 18% for the growth fund, 12.5% for the income fund and 7.5% fir the money market fund. Based on the information provided, how should the new client be advised to allocate $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.
a.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 change?
b.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?
c.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% or even to 14%?
d.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?
e.The asset allocation model you developed may be useful in modifying the portfolios for all 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?
J.D. Williams Inc.
J.D. Williams is an investment advisory firm that manages $120 million in funds for its clients. The company utilizes several financial approaches in advising their clients how to achieve optimal portfolio returns. They are as follows:
· An Asset Allocation Model – An asset allocation model, which provides individual clients with an investment strategy in order to obtain optimal investment combinations.
· Percentage Limitations – The Company strongly recommends investment diversity as a protection of the investors’ assets.
· A Risk Tolerance Analysis – The Company conducts an analysis of the individual investor’s risk tolerance and adjusts their portfolios accordingly.
J.D. Williams has recently contracted with a new client and would like to determine the best way to allocate the client’s $800,000 in available funds for optimal growth. The subsequent sections of this report provide an outline of the investment recommendation provided to the client.
a. Decision Variables
GF = $ amount of investment in growth stock fund
IF = $ amount of investment in income fund
MMF $ amount of investment in money market fund
b. Objective Function Definition
Maximize the total return of the portfolio Max 0.18GF + 0.125 +0.075MMF
3. Constraint Definition
s.t.1GF + 1IF + 1MMF <= 800,000 $ amount available to invest
.80GF -.20IF -.20MMF >= 0
$ amount invested in the growth fund should be at least 20% of total portfolio.
.60GF -.40IF -.40MMF <= 0 $ amount invested in the growth fund
should be at most 40% of total portfolio
-.20GF +.80IF -.20MMF > 0$ amount invested in the income fund should be at least 20% of total portfolio
-.50&F +.50IF -.50MMF <= 0$ amount invested in the income fund should be at most 50% of total portfolio
-.30GF -.30IF +.70MMF >= 0$ amount invested in the money market fund that should be least 30% of total portfolio
.05GF +.021F -.04MMF <= 0 Investor’s risk tolerance index
III. Key Assumptions
The following table provides information that stipulates the key assumptions taken into consideration in the development of the investment recommendation.
PortfolioRisk IndicatorsForecasted Annual Yields
Growth Stock Fund0.100.18
Money Market Fund0.010.075
This means that, we assume that the risk indicators and forecasted yields are given as true above. We also assume that the client does not want to consider other investment options. A maximum risk index of 0.05 has been assigned for the new client.
The optimal portfolio allocation J.D. Williams recommends is as follows:
Growth Fund = $248,889
Income Fund = $160,000
Money Market Fund = $391,111
Total = $800,000
The anticipated annual yield is:
Growth Fund = $248,889 x 0.18 = $44,800
Income Fund = $160,000 x 0.125= $20,000
Money Market Fund = $391,111 x .075= $29,333
Total = $94,133
Total Anticipated Annual Yield$ 94,133 = 11.77%
In regards to risk tolerance index, if the client’s index were raised by one half of a percentage point, from .05 to .055, the annual yield on investment consequently would increase by $4,667, from $94,133 to $98,800.
The modified asset allocation recommendation and its corresponding projected annual return are as follows:
Fund Allocation Projected Annual Yield
Growth Fund =$ 293,333 Growth Fund = $293,333 x 0.18 = $52,800
Income Fund =$ 160,000 Income Fund = $160,000 x 0.125 = $20,000
M Market Fund =$ 346,667 M Market Fund=$346,667 x 0.075 =$26,000
Total =$ 800,000 Total = $98,800
Total Anticipated Annual Yield = $ 98,800/$800,000 = 12.35%
I would not propose a change in the investment recommendation if the annual yield is revised downward to 16% as it is within the Range of Optimality. However, if it were to change to 14%, it will be outside the lower limit. As such, the value will change from $94,133. to $85,067, therefore, we would not recommend going below 15%.
Any lower index values that fall outside the growth fund’s range of optimality under the origins recommendation would warrant a new investment strategy for the client, because these values results in a change of the original projected total annual yield value.
However, the present recommendation of investment allocation in the growth fund is based on the fund’s given range of optimality, which measures from 15% to an infinite number of annual yield values. This range indicates that any possible index increase of this fund, i.e., 16% and higher, would not have an impact in the portfolio’s optimal asset allocation yet, it would positively affect the objective function’s total annual yield value from the original projected value of $94,133.
Conversely, potential downward fluctuations, falling below the growth fund’s annual yield lower limit index of 15%, would constitute a deviation from the originally recommended asset allocation and it’s corresponding projected value of $94,133, as the new value falls outside the fund’s range of optimality.
Financial portfolio theory stresses obtaining a proper balance between risk and return. Choosing the appropriate constraints is an effective method that ensures this balance.
Given the risk indexes of .10 and .07 of the growth fund and income fund, respectively, the client may opt to choose a less aggressive approach by limiting the growth fund’s investment amount to equal, yet not surpass, the amount invested in the income fund This change in investment strategy, however, would generate a lower annual yield of $85,067, than the projected annual return of $94,133 by the original, more risky recommendation.
J. D. Williams would recommend the use of this model only when the model potential new clients meet the present outlined criteria. The company’s mission, however, is to the professional, financial advice that best meets the individual investors’ needs. The company would therefore, not recommend the use of this asset allocation model as a general guide to financial investment.
Please provide formulas and excel sheets including functions.
Can you please explain how to get these following six values if they are correct: