Faircross Farms harvests its crops four times annually and receives payment for its crop 90 days after it is picked and shipped.However, planting, irrigating, and harvesting must be done on a nearly continual schedule.The firm uses 90-day bank notes to finance its operations.The firm arranges an 11 percent discount interest loan with a 20 percent compensating balance four times annually.What is the effective annual interest rate of these discount loans?
A) 11.00%
B) 15.94%
C) 11.46%
D) 13.75%
E) 12.72%

Question

Quizplus · Accepted Answer

To calculate the effective annual interest rate of the discount loan, we need to use the following formula:

Effective annual interest rate = [(1 - discount)(1 + i/n)]^n - 1

Where i is the stated annual interest rate, n is the number of discount periods per year, and discount is the percentage discount given by the bank.

In this case, i = 11%, n = 1 (since there are four discount periods per year), and discount = 20%.

So, plugging in the numbers:

Effective annual interest rate = [(1 - 0.20)(1 + 0.11/1)]^1 - 1
Effective annual interest rate = (0.80)(1.11) - 1
Effective annual interest rate = 0.894 - 1
Effective annual interest rate = -0.106

Since the result is negative, we need to adjust the formula by taking the absolute value of the effective annual interest rate and multiplying it by -1:

Effective annual interest rate = |-0.106| x -1
Effective annual interest rate = 0.106 x -1
Effective annual interest rate = -0.106

So, the effective annual interest rate of the discount loan is -10.6%.

However, this result doesn't make sense, as the effective annual interest rate cannot be negative. The issue is that we didn't account for the compensating balance requirement.

Since the bank requires a 20% compensating balance, only 80% of the loan proceeds are available to the borrower. Therefore, we need to adjust the formula again to take this into account:

Effective annual interest rate = [(1 - discount)(1 + i/n)/(1 - balance)]^n - 1

Where balance is the decimal equivalent of the percentage compensating balance.

Plugging in the numbers:

Effective annual interest rate = [(1 - 0.20)(1 + 0.11/1)/(1 - 0.20)]^1 - 1
Effective annual interest rate = (0.80)(1.11)/0.80 - 1
Effective annual interest rate = 0.994 - 1
Effective annual interest rate = -0.006

Again, we need to adjust for the negative result:

Effective annual interest rate = |-0.006| x -1
Effective annual interest rate = 0.006 x -1
Effective annual interest rate = -0.006

The effective annual interest rate of the discount loan taking into account the compensating balance is -0.6%. However, this still doesn't make sense, as it implies that the borrower is actually earning money on the loan.

The issue is that the discount loan is not a true loan, as the borrower is not receiving the full amount of the loan proceeds. Instead, the borrower is paying a fee (the discount) to receive the funds early.

Therefore, the effective annual interest rate is not meaningful in this case, and we cannot calculate it.

Faircross Farms Harvests Its Crops Four Times Annually and Receives