Question

Zerox Copying Company plans to borrow $172,000. New Jersey National Bank will lend the money at one-half percentage point over the prime rate at the time of 17½ percent (18 percent total) and requires a compensating balance of 21 percent. The principal in this case will be funds that the firm can effectively use in the business. This loan is for one year. What is the effective rate of interest? What would the effective rate be if Zerox were required to make four quarterly payments to retire the loan?

Short answer

The effective interest rate is 22.78% and the effective interest rate on an instalment loan is 28.8%.

Step-by-step solution

Information provided in the question

Loan amount = $172,000

Loan term = 1 year

Interest rate = 18%

Compensating balance = 21%

Calculation of interest payable

The interest payable is $30,960.

$\begin{array}{rcl}\mathrm{Interest}\mathrm{payable}& =& \mathrm{Interest}\times \mathrm{Principal}\\ & =& 18\%\times \$172,000\\ & =& \$30,960\end{array}$

Calculation of compensating balance in monetary terms

The compensating balance is $36,120.

$\begin{array}{rcl}\mathrm{Compensating}\mathrm{balance}& =& \mathrm{Compensating}\mathrm{balance}\mathrm{rate}\times \mathrm{Principal}\\ & =& 21\%\times \$172,000\\ & =& \$36,120\end{array}$

Calculation of effective interest rate on loan

The effective interest rate with a 21% compensating rate is 22.78%.

$\begin{array}{rcl}\mathrm{Effective}\mathrm{rate}& =& \frac{\mathrm{Interest}}{\mathrm{Principal}-\mathrm{Compensating}\mathrm{balance}}\times \frac{\mathrm{Days}\mathrm{in}\mathrm{year}}{\mathrm{Days}\mathrm{loan}\mathrm{is}\mathrm{outstanding}}\\ & =& \frac{\$30,960}{\$172,000-\$36,120}\times \frac{360}{360}\\ & =& 22.78\%\end{array}$

Calculation of effective interest rate when quarterly instalments have to be made

The effective interest rate is 28.8%.

$\begin{array}{rcl}\mathrm{Effective}\mathrm{rate}\mathrm{on}\mathrm{instalment}& =& \frac{2\times \mathrm{Annual}\mathrm{number}\mathrm{of}\mathrm{payments}\times \mathrm{Interest}}{\left(\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{payments}+1\right)\times \mathrm{Principal}}\\ & =& \frac{2\times 4\times \$30,960}{\left(4+1\right)\times \$172,000}\\ & =& \frac{\$247,680}{\$860,000}\\ & =& 28.8\%\end{array}$