# Rule of 78s

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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.

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.

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, 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:

$EarnedInterest(n)=f\times {\frac {2(k-n+1)}{k(k+1)}}$ where $f$ is the total agreed finance charges, $k$ is the length of the loan $n$ is current payment number.

Formula for calculating the cumulative earned interest at payment n:

$CumulativeEarnedInterest(n)=f\times {\frac {n(2k-n+1)}{k(k+1)}}$ where $f$ is the total agreed finance charges, $k$ is the length of the loan $n$ is current payment number.

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

$UnearnedInterest(u)={\frac {f\times k(k+1)}{n(n+1)}}$ where $u$ is the unearned interest for the lender, $k$ is the number of repayments remaining (not including current payment) and $n$ is the original number of repayments.

Figure 1 is an amortized table for gradual repayment of a loan with $500 in interest fees. $Figure1$ 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