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	9
Furniture loans	11
Other secured loans	12
Signature loans	13
Risk-free securities	10

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 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 $2.3 million be allocated to each of the loan/investment alternatives to maximize total annual return?

Type of Loan/Investment	Fund Allocation
Automobile loans	$
Furniture loans	$
Other secured loans	$
Signature loans	$
Risk-free securities	$

What is the projected total annual return?

Annual Return = $

Answer 1

Answer:

Given Table

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 = 0.09X1 + 0.11 X2 + 0.12 X3 + 0.13 X4 + 0.1 X5

X1 + X2 + X3 + X4 + X5 = 2300000 (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 = 2300000 ..............................(1)

X5 = 0.3*2300000

X5= 690000....................................................................(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 = 690000.................................................................(6)

Substitute equation 2,6 in equation (1)

X1 + X2 + 690000 + 690000 = 2300000

X1+X2 = 2300000 – (690000 + 690000)

X1+X2 = 920000...................................................................(7)

Substitute equation 6, 7 in equation (3)

X4 = 0.1(920000+690000)

X4 = 0.1*1610000

X4 = 161000

From equation (5)

X3+X4=690000

X3+161000=690000

X3= 690000-161000

X3= 529000.............................................(8)

Substitute equation 8 in equation (4)

X2+X3 = X1

X2+529000=X1

-X1+X2 = -529000.....................................(9)

By Adding(7) and (9)

X1+X2 = 920000

-X1+X2 = -529000

2 X2 = 391000

X2 = 391000/2

X2 = 195500

X1 = 920000-195500

X1 = 724500

X1 = $724500

X2 =$ 195500

X3= $529000

X4 = $161000

X5 = $690000

From maximum return, amount will be

Projected annual return

0.09*724500 + 0.11 *195500 + 0.12 *529000 + 0.13 *161000 + 0.1 *690000

= 65205 + 21505 +63480+20930+69000

Projected annual return =$ 240120