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 revenueproducing investments together with annual rates of return are as follows:

Type of Loan/Investment	Annual Rate of Return (%)
Automobile loans	8
Furniture loans	10
Other secured loans	11
Signature loans	12
Risk-free securities	9

The credit union will have $2.1 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 all loans (automobile, furniture, other secured, and signature loans).
• 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 $2.1 million be allocated to each of the loan/investment alternatives to maximize total annual return? Round your answers to the nearest dollar.

Automobile Loans	$
Furniture Loans	$
Other Secured Loans	$
Signature Loans	$
Risk Free Loans	$

What is the projected total annual return? Round your answer to the nearest dollar.

Answer 1

Given table:

Type of Loan/Investment	Annual Rate of Return (%)
Automobile loans	8
Furniture loans	10
Other secured loans	11
Signature loans	12
Risk-free securities	9

Let the invested amount in Automobile loans be X1

Let the invested amount in Furniture loans be X2

Let the invested amount in other secured loans be X3

Let the invested amount in Signature loans be X4

Let the invested amount in Risk free loans be X5

Objective Function

Maximize Z = 0.08 X1 + 0.10 X2 + 0.11 X3 + 0.12 X4 + 0.09 X5

X1 + X2 + X3 + X4 + X5 = 2100000 (Investment Constraints)

X5 < 30% of 2300000 (risk free)

X4 < 10% (X1+X2+X3+X4) [Signature loan constraint]

X2 +X3 < X1 [furniture and other secured loan constraint]

X3+X4 < X5 [other secured loan and signature loan constraints]

X1 ,X2, X3, X4, X5 > 0 [Positive constraints]

Solving constraints

X1 + X2 + X3 + X4 + X5 = 2100000 ..............................(1)

X5 = 0.3*2100000

X5= 630000.......................(2)

X4 = 0.1(X1+X2+X3+X4)......................(3)

X2+X3 = X1..................................(4)

X3+X4 = X5................................................(5)

Substitute X5 value in equation (5)

X3+X4 = 630000.....................................(6)

Substitute equation 2,6 in equation (1)

X1 + X2 + 630000 + 630000 = 2100000

X1+X2 = 2100000 – (630000 + 630000)

X1+X2 = 840000.....................................(7)

Substitute equation 6, 7 in equation (3)

X4 = 0.1(840000+630000)

X4 = 147000

From equation (5)

X3+X4 = 630000

X3+147000 = 630000

X3 = 630000 - 147000

X3 = 483000.............................................(8)

Substitute equation 8 in equation (4)

X2 + X3 = X1

X2 + 483000 = X1

X2 = X1 - 483000

X2 - X1 = - 483000

-X1+X2 = -483000.....................................(9)

By Adding(7) and (9)

X1+X2 = 840000

-X1+X2 = -483000

2 X2 = 357000

X2 = 357000 / 2

X2 = 178500....................(10)

Substitute equation (10) in equation (7)

X1+X2 = 840000

X1 = 840000 - 178500

X1 = 661500

we obtain

X1 = $661500

X2 = $178500

X3 =$483000

X4 = $147000

X5= $630000

From maximum return, amount will be

Automobile Loans	$661500
Furniture Loans	$178500
Other Secured Loans	$483000
Signature Loans	$147000
Risk Free Loans	$630000

Projected annual return

from

0.08 X1 + 0.10 X2 + 0.11 X3 + 0.12 X4 + 0.09 X5

= 0.08 * 661500 + 0.10 *178500 + 0.11 * 483000 + 0.12 *147000 + 0.09 * 630000

= 52920 + 17850 + 53130 + 17640 + 56700

Projected annual return = $198240

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