Question

The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue-producing investments together with annual rates of return are as follows: Type of Loan/Investment Annual Rate of Return (%) Automobile loans Furniture loans Other secured loans Signature loans Risk-free securities 10 12 8 The credit union will have $2.3 million available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments Risk-free securities may not exceed 30% of the total funds available for investment. . Signature loans may not exceed 10% of the funds invested in ai oans automobile furniture, other secured, and si na reloans Furniture loans plus other secured loans may not exceed the automobile loans. Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the $2.3 million be allocated to each of the loan/investment alternatives to maximize total annual return? Type of Loan/Investment Automobile loans Furniture loans Other secured loans Signature loans Risk-free securities What is the projected total annual return? Annual Return$ Fund Allocation

Answer 1

Let x₁, x₂, x₃, x₄ and x₅ are the investments in loans types automobiles, furniture,other secured loans, signature and risk free securities loans as listed above.

Then we derive following linear programming problem:

Maximize

Total_return = 0.07x₁ + 0.10x₂ + 0.11x₃ + 0.12x₄ + 0.08x₅

subject to constraints: x₁ ≥ 0, x₂ ≥ 0, x₃ ≥ 0, x₄ ≥ 0, x₅ ≥ 0,                 

x₁ + x₂ + x₃ + x₄ + x₅ ≤ 2,300,000 (Availability constraint),

x₅ ≤ 690,000 (30% of 2300000),

x₄ ≤ 0.1(x₁ + x₂ + x₃ + x₄),

x₂ + x₃ ≤ x₁   and

x₃ + x₄ ≤ x₅

Working notes:

Solving

x₄ ≤ 0.1(x₁ + x₂ + x₃ + x₄)

or, x₄ ≤ 0.1(2300,000 - x₅)

or, x₄ ≤ 0.1(2300000-690,000)

or, x₄ ≤ 161000

x₃ ≤ 529000 ( 690000-161000)

Using simplex, we get

x₁ ≤ 724500

x₂ ≤ 195500

Automobile loans	724500	7%	50715
Furniture loans	195500	10%	19550
Other secured loans	529000	11%	58190
Signature loans	161000	12%	19320
Risk free securities	690000	8%	55200
Total	2,300,000		202975

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