The Providential Insurance Company has developed a list of seven investment alternatives, with corresponding financial factors, for a 10-year investment horizon. These investments and their corresponding financial factors are presented in the following table. With this table, the meaning of the various financial factors is as follows:

**The length of investment **– the expected number of years required for the annual rate of return to be realized, taking into account the possibility of reinvestment;

**The annual rate of return** – the expected rate of return over the 10-year investment horizon;

**The risk coefficient** – a subjective, dimensionless estimate representing the portfolio manager’s appraisal of the relative safety of each alternative, based on an ordinal scale of 10;

**The growth potential**– a subjective estimate representing the portfolio manager’s appraisal of the potential increase in the value of the investment alternative for the 10-year period.

The Providential Insurance Company seeks to maximize the return on its portfolio of investments, subject to the following restrictions on the selection of the portfolio:

The average length of the investment for the portfolio should not exceed 7 years

The average risk for the portfolio should not exceed 5

The average growth potential for the portfolio should be at least 10%

At least 10% of all available funds must be retained in the form of cash at all times in order to maintain working capital liquidity

The proportions of funds invested in the various alternatives must sum to 1.0.

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**Task**

Using clear notation, Identify the decision variables for this problem.

Use these decision variables to formulate the firm’s objective function, clearly stating what the objective is.

Use the information in Tables 1 and 2 above to identify the constraints for this problem.

Using what you have identified in 1-3 above formulate and present the complete linear programming model for this problem.

Enter the linear programming model you have derived in 4 above into Solver to obtain the optimal solution to the problem.

Extract the optimal solution from Solver and state the value of the objective function, and the values of the decision variables then summarise the results, making clear the contribution to the Total Rate of Return made by each investment alternative.

Using Solver extract a Sensitivity Report for the problem.

From the Sensitivity Report extract the slack and surplus variables and interpret their meaning.

Using the results of the Sensitivity Report indicates whether the optimal solution to the problem is unique or if there are alternative optimal solutions.

Present a brief interpretation of the ranging information on the objective function coefficients presented in the Sensitivity Report for this problem.

Identify the shadow prices from the Sensitivity Report and interpret their meaning.

Using the shadow price on the average length of investment indicates the impact on the objective function of increasing the length by 1 year. What is the upper limit of the availability of component 1 for which this shadow price is valid?

What would be the impact of increasing the liquidity requirement by 1 percent?

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