# Rule of 78s

Also known as the "Sum of the Digits" method, the Rule of 78s is a term used in lending that refers to a method of yearly interest calculation. The name comes from the total number of months' interest that is being calculated in a year (the first month is 1 month's interest, whereas the second month contains 2 months' interest, etc.). This is an accurate interest model only based on the assumption that the borrower pays only the amount due each month. The outcome is that more of the interest is apportioned to the first part or early repayments than the later repayments. As such, the borrower pays a larger part of the total interest earlier in the term.[1]

If the borrower pays off the loan early, this method maximizes the interest paid by applying funds to the interest before principal. The Rule of 78 is designed so that borrowers pay the same interest charges over the life of a loan as they would with a loan that uses the simple interest method. But because of some mathematical quirks, you end up paying a greater share of the interest upfront. That means if you pay off the loan early, you’ll end up paying more overall for a Rule of 78s loan compared with a simple-interest loan.[2]

A simple fraction (as with 12/78) consists of a numerator (the top number, 12 in the example) and a denominator (the bottom number, 78 in the example). The denominator of a Rule of 78s loan is the sum of the digits, the sum of the number of monthly payments in the loan. For a twelve-month loan, the sum of numbers from 1 to 12 is 78 (1 + 2 + 3 + . . . +12 = 78). For a 24-month loan, the denominator is 300. The sum of the numbers from 1 to n is given by the equation n * (n+1) / 2. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300,[3] which is the loan's denominator, D.

For a 12-month loan, 12/78s of the finance charge is assessed as the first month's portion of the finance charge, 11/78s of the finance charge is assessed as the second month's portion of the finance charge and so on until the 12th month at which time 1/78s of the finance charge is assessed as that month's portion of the finance charge. Following the same pattern, 24/300 of the finance charge is assessed as the first month's portion of a 24-month precomputed loan.

Formula for calculating the earned interest at payment n:

${\displaystyle EarnedInterest(n)=f\times {\frac {2(k-n+1)}{k(k+1)}}}$

where ${\displaystyle f}$ is the total agreed finance charges, ${\displaystyle k}$ is the length of the loan ${\displaystyle n}$ is current payment number.

Formula for calculating the cumulative earned interest at payment n:

${\displaystyle CumulativeEarnedInterest(n)=f\times {\frac {n(2k-n+1)}{k(k+1)}}}$

where ${\displaystyle f}$ is the total agreed finance charges, ${\displaystyle k}$ is the length of the loan ${\displaystyle n}$ is current payment number.

If a borrower plans on repaying the loan early, the formula below can be used to calculate the unearned interest.

${\displaystyle UnearnedInterest(u)={\frac {f\times k(k+1)}{n(n+1)}}}$

where ${\displaystyle u}$ is the unearned interest for the lender, ${\displaystyle k}$ is the number of repayments remaining (not including current payment) and ${\displaystyle n}$ is the original number of repayments.

Figure 1 is an amortized table for gradual repayment of a loan with $500 in interest fees. ${\displaystyle Figure1}$ Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4%$77.00
2 11 78 14.1% $70.50 3 10 78 12.8%$64.00
4 9 78 11.5% $57.50 5 8 78 10.3%$51.50
6 7 78 9.0% $45.00 7 6 78 7.7%$38.50
8 5 78 6.4% $32.00 9 4 78 5.1%$25.50
10 3 78 3.8% $19.00 11 2 78 2.6%$13.00

## References

1. ^ "The Rule of 78 – What is it and why does it matter? | Jade FInance Blog". jade.finance. Retrieved 2021-03-09.
2. ^ "What Is the Rule of 78 and Can It Cost You?". Credit Karma. 2019-06-18. Retrieved 2021-03-09.
3. ^ "Solve 24*left(24+1right)/2". Microsoft Math Solver. Retrieved 2021-03-08.
4. ^ "What Is the Rule of 78 and Can It Cost You?". Credit Karma. 2019-06-18. Retrieved 2021-03-09.
5. ^ JOHNSON, ALONZO F. (1988). "The Rule of 78: A Rule That Outlived Its Useful Life". The Mathematics Teacher. 81 (6): 450–480. ISSN 0025-5769.
6. ^ 15 U.S.C. § 1615
7. ^ H.R. 1054, 107th Cong., A Bill to amend the Truth in Lending Act to expand protections for consumers by adjusting statutory exemptions and civil penalties to reflect inflation, to eliminate the Rule of 78s accounting for interest rebates in consumer credit transactions, and for other purposes
8. ^ a b Search Results - THOMAS (Library of Congress)