J$.$ D$.$ Williams, Inc. is an investment advisory firm that manages more than $$120$ millionin funds for its numerous clients. The company uses an asset allocation model that rec$-$ommends the portion of each client's portfolio to be invested in a growth stock fund, anincome fund, and a money market fund. To maintain diversity in each client's portfolio,the firm places limits on the percentage of each portfolio that may be invested in eachof the three funds. General guidelines indicate that the amount invested in the growthfund must be between $20\%$ and $40\%$ of the total portfolio value. Similar percentagesfor the other two funds stipulate that between $20\%$ and $50\%$ of the total portfolio valuemust be in the income fund and that at least $30\%$ of the total portfolio value must be inthe money market fund.In addition, the company attempts to assess the risk tolerance of each client and adjustthe portfolio to meet the needs of the individual investor. For example, Williams just con$-$tracted with a new client who has S$800,000$ to invest. Based on an evaluation of the client'srisk tolerance, Williams assigned a maximum risk index of $0\mathrm{.}05$ for the client. The firm'srisk indicators show the risk of the growth fund at $0\mathrm{.}10,$ the income fund at $0\mathrm{.}07,$ and themoney market fund at $0\mathrm{.}01.$ An overall portfolio risk index is computed as a weighted aver$-$age of the risk rating for the three funds, where the weights are the fraction of the client'sportfolio invested in each of the funds.Additionally, Williams is currently forecasting annual yields of $18\%$ for the growthfund, $12\mathrm{.}5\%$ for the income fund, and $7\mathrm{.}5\%$ for the money market fund. Based on the infor$-$mation provided, how should the new client be advised to allocate the $$800,000$ amongthe growth, income, and money market funds? Develop a linear programming model thatwill provide the maximum yield for the portfolio. Use your model to develop a managerialreport.

Questions:

Managerial Report

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

$2.$ Assume that the client's risk index could be increased to $0\mathrm{.}055.$ How much wouldthe yield increase, and how would the investment recommendation change?

$3.$ Refer again to the original situation, in which the client's risk index was assessedto be $0\mathrm{.}05.$ How would your investment recommendation change if the annual yieldfor 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 inthe growth fund. How would the original recommendation change if the amountinvested in the growth fund is not allowed to exceed the amount invested in theincome fund?

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