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Annual Rates of Return

Trying to solve below problem to figure out what is going to be the total annual return (percentage)?
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The employee credit union at State University is planning the allocation of funds for the coming year. The credit union make 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 8

Furniture loans 10

Other secured loans 11

Signature loans 12

Risk-free securities 9

The credit union will have $2,000,000 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).
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 $2,000,000 be allocated to each of the loan/investment alternatives to maximize total annual return? What is the projected annual return?
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I come up with the equation as:

max .08x1 + .1x2 + .11x3 + .12x4 + .09x5
st
x1 + x2 + x3 + x4 + x5 < 2000000
x5 < 600000
x4 < .1
x2 + x3 < .08
x3 + x4 < .09

.

Solution Summary

The employee credit union at State University is planning the allocation of funds for the coming year. The credit union make 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 8

Furniture loans 10

Other secured loans 11

Signature loans 12

Risk-free securities 9

The credit union will have $2,000,000 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).
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 $2,000,000 be allocated to each of the loan/investment alternatives to maximize total annual return? What is the projected annual return?
---
I come up with the equation as:

max .08x1 + .1x2 + .11x3 + .12x4 + .09x5
st
x1 + x2 + x3 + x4 + x5 < 2000000
x5 < 600000
x4 < .1
x2 + x3 < .08
x3 + x4 < .09

$2.19