Rule Of 78's
When lenders use the Rule of 78's, they distribute the
total finance charge over all payments, but charge more
interest early in the loan period and less later, compared
to other methods. The Rule of 78's is also called the "sum
of digits" because it gets its name from summing the
digits 1 through 12  the number of months in a one year
loan. Yes, this is 78.
Here is how the Rule of 78's is applied to a Amortized
loan. Interest is calculated as described in the topic
Amortized Loan  Normal. (The formulas are repeated
below but you will have to refer to the aforementioned
topic for details.)
Payment Amount
P = Payment
A = Amount borrowed
R = Annual Interest Rate
n = number of compound payments per year.
T = time in years.
nT = Term, the total number of payments in the loan schedule. This
is the number of periods per year multiplied by the number of years.
This is the exponent. In the formula, 1 + the Interest Rate(R/n) is
multiplied by itself as many times as the value of the Term(nT).
Total Plan Payments
TP = P x T
TP = Total Plan Payments
P = Payment Amount
T = number of Payments (Term)
Total Interest
TI = TP  A
TI = Total Interest
TP = Total Plan Payments
A = Amount Borrowed
When you know the Total Interest through the calculations
above, you can calculate interest rebate using Rule of 78's.
Interest Payments
First, you need the Sum of the number of payments.
For instance, if the Term is 12 monthly payments, the
summation of 12 payments is
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78
The amount of interest to be paid per payment is calculated
as follows.
12/78 x Total Interest = First Payment
11/78 x Total Interest = Second Payment
 and so on until...
1/78 x Total Interest = Final Payment
This rule applies to long term loans the same way.
For example, a 24month loan  where the sum of
the digits 1 through 24 is 300  would have a first
month's interest payment that is 24/300 of the
total interest. The second month would be 23/300
and so on.
A three year loan, 36 months  where the sum of
the digits 1 through 36 is 666  would have a first
month's interest payment that is 36/666 of the total
interest, second month would be 35/666 and so on.
The above fractions are applied to the Total
Interest amount. The remainder of the Payment amount
would be applied to outstanding principal. As the months
go by, the interest portion decreases and the principal
is paid off faster. However, the interest portion is a
fixed amount determined by the Rule of 78's. It is not
recalculated each time from outstanding principal.
The amount applied to Principal is
Payment Amount  Interest for Payment = Applied to Principal.
Early Payout Amount
If the loan is paid off before the full Term has expired, there
is usually an interest rebate. The borrower did not have the
money as long as originally planned and did not accumulate
as much interest on the loan. However, the lender depends
on the interest as payment for lending money. This is the
financial charge of the loan. If a loan is paid off early, the
portion of interest that is left is less with Rule of 78's and
this benefits the lender.
For instance, if a 12month loan is paid in 10 months,
2/78 of the Total Interest  the 11th payment and
1/78 of the Total Interest  the 12th and final payment
is all the remaining financial charge left unpaid. So 75/78 of the
Total Interest has already been paid. Only 3/78 of the Total Interest
is saved by paying the loan early.
To calculate the Early Payout Amount, first determine the
number of payments remaining. Then calculate the Interest
amount saved as shown above.
It is very straightforward:
1. Each payment is numbered from 1 to the total number in the
Term. (12 or 24 or 36 or whatever)
2. The bottom part of the fraction is figured by "summing the
digits." (78 or 300 or 666 or whatever)
3. Interest left to pay is calculated as described above.
If a 36month loan is paid in 30 months, for instance, the amount
of interest left to be refunded or deducted from the total outstanding
debt is the following.
1/666 of the Total Interest +
2/666 of the Total Interest +
3/666 of the Total Interest +
4/666 of the Total Interest +
5/666 of the Total Interest +
6/666 of the Total Interest = 21/666 of the Total Interest that must
be rebated to the borrower.
This is the Interest Amount Saved and is used to calculate
Early Payout Amount.
Early Payout Amount = OP + TI  IAS  PP  IP
OP = Original Principal
TI = Total Interest
IAS = Interest Amount Saved
PP = Principal Paid
IP = Interest Paid
Another formula which gives the Interest Amount Saved is
the following.
(No. of months remaining + 1) x (No. of months remaining)
divided by (No. of months in Term + 1) x (No. of months in Term)
This gives the same result as above.
Early Payoff Penalty
When a loan is paid off a great deal sooner than expected, the
borrower can actually experience a penalty. This is because
Rule of 78's calculates higher interest percentages in earlier
months than other types of calculating. This interest is still
due according to the agreement made, even though the
actual calculation of interest would not be as high.
For instance, borrowing $10,000 for two years at 12% would
amount to $1297.65 in Total Interest (using a 365 day year.)
If this loan is paid after only four months, the actual interest on the
amount and the length of time would be $377.61.
Using Rule of 78's, the interest charge for the first four months amounts
to $389.30. (Using the formulas given above.)
This is an additional 3% interest.
As a rule:
1. The higher the interest rate, the greater the penalty amount.
2. The earlier the prepayment in relation to the term, the greater
the penalty amount.
So, if you are a lender, you benefit from Rule of 78's. If you are a
borrower, you should try to avoid it.
WARNING: Some states have usury and other laws that may
limit the use of the Rule of 78's.
Rate Basis
For "Rule of 78" accounts, the Rate Basis is only a factor if the
first payment is deferred. A deferred payment is a payment where
the First Payment Date is more than one Period away from the
Calculate Interest From Date. Interest is calculated on this
difference and the current Rate Basis is a factor in this calculation.
See Also
 Amortized Loan  Normal
 Interest Detail
 How To Make An Interest Adjustment
 How To Setup Amortized Interest
 How To Use The Amortization Schedule

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