Dear Professor Cram:

Ellen borrowed $500 at 12% simple interest, to be repaid in 8 equal monthly installments of $70. If she pays off the loan when she makes her 4th payment, how much will she save in finance charges under the rule of 78s?

Charles, SFC USAR

Thanks for your question, Charles.

This a combination of Simple Interest and Rule of 78s. If you haven't already, you may want to review our tutorial on interest rates.

Simple interest is calculated on the original amount borrowed, and is therefore better suited for short-term borrowing and single payment loans than for long-term installment loans.

*Simple Interest = Principle x Rate x Time*

The Rule of 78 is a method of allocating interest charges for the life of a loan to the periods within the loan. Under this method, interest is calculated for the life of the loan, and allocated to each month by proportion using reverse sum of the digits methodology. You start by adding up the numbers of months for the note. In your example of an 8 month loan, counting month 1, plus month 2, and so forth through month 8 is:

*1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36*

With 12 months this totals 78, and hence the name, the Rule of 78s.

The interest total for the loan life (Interest = Rate x Time x Principle) is divided by the sum of the digits (78 in this example) and applied in reverse proportion across the life of the loan. Thus, 8/36 of the interest applies to the first month, 7/36 to the second month, and so forth down to 1/36th at the end of the 8th month.

To solve this problem you need to calculate the full amount of interest, then allocate it to the months, and add up the parts saved by paying off early.

We are told she is going to repay the $500 loan in 8 installments of $70:

*$560 total – $500 principal = $60 interest*

(That is 12% and it was NOT an annual rate! Wow!) So the $60 interest is what she is trying to save part of by paying in full with the 4th payment. The last four months count as 1/36 + 2/36 + 3/36 + 4/36 = 10/36 of the interest, and 10/36 x $60 = $16.67 of interest saved.

That is not very much savings from prepaying the loan. Hence – try to avoid rule of 78s loans!

I hope this helps. Let us know if you need anything else.

Good Studying,

Professor Cram