# Rule of 78

If you normally sell your car before it is paid off, you are going to want to stay away from lenders that still use the Rule of 78 (also known as sum-of-the-digits) to calculate the interest you have accrued. This method was used before the technology was available to calculate a prepayment on the spot. At a time when everyone in the U.S. that deals in consumer finance has access to computers and financial calculators, there is no excuse for using this rule today, except to stick it to the borrower. Nationally, it is against the law to use the rule for loans over 61 months (the longer the loan, the bigger the interest penalty). Currently, 33 states still allow the rule (typically auto loans) for 61 months or less.

The alternate name for the Rule of 78, the sum of the digits, gets it name from the need to calculate the sum of the digits for each month of the loan. The sum of the digits for a one year loan, with monthly payments, is 1+2+3+4+5+6+7+8+9+10+11+12 = 78, thus the name. The formula for the sum of the digits is:

=Payments*(Payments+1)/2

or

12*(12+1)/2 =78

I set up two payment schedules, one for an amortizing loan (30/360 day count) and a loan that uses the Rule of 78. The amounts needed to pay off the loans at the end of each month are calculated, and the two are compared. In all months an early payoff using the Rule of 78 is higher than the standard amortized loan, with the highest penalty around a third of the way through the loan (chart below). I had to split the two side-by-side loans in half to fit the comparison on this post:

On the Rule of 78 side there is a column called “Total Interest Refund”. That term seems like a positive thing, but it is included only because lenders using this calculation might say you are receiving an interest refund, when it is actually just the part of the unused total interest.

Besides using the amortizing loan as a comparison to the Rule of 78 loan schedule, it has another use. It is necessary to compute the total interest for the loan, to use in the Rule of 78 loan schedule.

The interest for each month on the Rule of 78 schedule is calculated as follows:

Remaining Months / Sum of the Digits * Total Interest

Due to the interest being higher on the Rule of 78 schedule, there is less principal paid down each month. You can see the remaining math, by looking at the Excel spreadsheet. You can change the comparison by entering new loan data in the yellow cells.

Rule of 72

I added the rule of 72, only because it can come in handy to estimate the time it will take for a certain interest rate to double your investment. I have two table in the workbook. By simply dividing the interest rate into 72, you get the estimated years to double. I also included the a second table for continuous compounding. Next to the estimate for each interest rate is the actual time. The actual time formulas are below the tables.

Download “78“

Don

How can I calculate cumulative Pay-Off Difference for a certain period? I can run cumipmt and cumprinc to sum principal, interest and end balance of amortized rule, but not in rule 72. Thank you.

Marwan,

I am not clear as to what you are asking. The rule of 72 an estimate of the time it takes to double your money. I’m not sure what it has to do with a cumulative payoff. If you could send me an example spreadsheet, possibly I could help.

Don

Hey great post! I hope it’s ok that I shared this on my Twitter, if not, no issues just

tell me and I’ll remove it. Either way keep up the great

work.

Ellis,

Please do. I Tweet all my posts. Follow me at https://twitter.com/CFOExcelBlog

Thanks

Don