Question

Roberts Investments is a financial services firm that manages funds for numerous clients. The company uses a proprietary asset allocation model that recommends the portion of each client’s portfolio to be invested in exchange traded fund, income fund and hedge fund.

To maintain diversity in each client’s portfolio, the firm places limits on the percentage of a portfolio that may be invested in each of the three funds, and assigns a limit on the portfolio’s risk index.

A new client, Boris Grey should be advised on allocation of $500,000. Based on an evaluation of the client’s risk tolerance, the Roberts Investments assigned a maximum risk index of .05 to the portfolio. The firm’s risk indicators show the risk rating for the exchange traded fund at .075, while the risk ratings for income fund and hedge fund are at .055 and .01 respectively. An overall portfolio’s risk index is computed as a weighted average of the risk rating for the three funds, where the weights are the fraction of the portfolio invested in each of the funds. Higher value of the risk index represents a more risky portfolio.

To maintain diversity the following investment guidelines are used:

• The amount invested in exchange traded fund should be between 15% and 40% of the total portfolio value.
• The amount invested in income fund should be between 25% and 40% of the total portfolio value.
• At least 30% of the total portfolio value must be in in hedge fund.

Furthermore, the amount invested in exchange traded fund should be at least as high (>=) as the amount invested in hedge fund.

The firm is forecasting annual yields (returns) of 13.5% for the exchange traded fund, 10.25% for income fund and 6.0% for the hedge fund.

Based on the information provided, how should the new client be advised to allocate the $500,000 among the three funds to maximize the overall yield from the portfolio?

Formulate a linear programming model for this portfolio selection problem (define decision variables, write down the objective function, develop constraints – show all steps/calculations) and solve it using Excel’s Solver. Generate Answer report and sensitivity reports.

Answer 1

Solution:

Lets assume the funds invested in Exchange traded funds is 'x'; Income fund is 'y'; and hedge fund is 'z'.

Objective Function:

Maximise the protfolio yield i.e.    .135x+.1025y + .06z

Subject to

- amount invested in exchange traded fund should be between 15% and 40% of the total portfolio value

x>=.15 and x<=.4

- amount invested in income fund should be between 25% and 40% of the total portfolio value

y>=.25 and y<=.4

- At least 30% of the total portfolio value must be in in hedge fund

z>=.3

- the amount invested in exchange traded fund should be at least as high (>=) as the amount invested in hedge fund

x>=z

- addition of all proportions will be one

x+y+z = 1

- Risk Rating below 0.05

0.075x + 0.055y + 0.01z <=0.05

The solver details are as under:

NOTE: I have named the cells C5 as 'X', C6 as 'Y' and C7 as 'Z'.

Answer Report:

Sensitivity Report

I hope you get the solution. Thanks.

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