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Precomputed interest · early settlement
Early-settlement interest rebate under the Rule of 78s
What a Rule of 78s rebate returns on early settlement of a precomputed loan — and why it is less than the pro-rata share you might expect.
Verified engine
What the engine returns
The rebate is the unearned share the method hands back; the net payoff is the lender’s gross figure less that rebate. Hold the rebate against the halfway intuition: roughly half the payments remain, yet the rebate is well under half the interest, because the payments already made carried the heaviest weights in the schedule. That distance is the cost of the method itself.
Inputs the result is built from
Total precomputed interest
Total scheduled payments
Payments already made
Gross payoff before rebate
How the result is derived
MethodThe count of remaining payments multiplied by one more than itself, over the total payment count multiplied by one more than itself, gives the unearned fraction; that fraction of the precomputed interest is the rebate, and the gross payoff less the rebate is the net settlement figure.
StandardRule of 78s sum-of-digits unearned-interest rebate for precomputed instalment loans
GuardMore payments made than were ever scheduled is refused — the settlement point must lie inside the schedule for a remaining count to mean anything.
How the rebate moves with payments made
How sum-of-digits weighting shrinks the rebate
A precomputed loan fixes its total interest at signing and folds it into the scheduled payments, so settling early raises a question an ordinary declining-balance loan never asks: how much of that fixed interest was never earned, and comes back? The Rule of 78s answers by weighting each instalment with the count of payments then remaining — the first instalment carries the heaviest share of the interest and the last the lightest.
The consequence is the shortfall this page exists to show. The rebate is the sum of the remaining weights over the sum of all of them, and that fraction is always smaller than the simple remaining-payments-over-total intuition once the loan is genuinely under way. The gap is widest near the middle of the term — precisely where most early settlements happen — and closes toward either end.
The name is an artefact of the arithmetic: for a loan of a year of monthly instalments, the payment-count weights sum to seventy-eight, and the method kept the name even for schedules of other lengths. Nothing about the name changes the behaviour — it is sum-of-digits weighting at every term.
The method carries a legal asterisk the page must state plainly: many jurisdictions restrict or prohibit the Rule of 78s for consumer credit — in the United States it was phased out for most longer-term consumer loans by federal law. This page computes what the method yields when a contract names it; whether the method may lawfully be applied at all is a question for the governing law, not for the arithmetic.
The count of remaining payments multiplied by one more than itself, over the total payment count multiplied by one more than itself, gives the unearned fraction; that fraction of the precomputed interest is the rebate, and the gross payoff less the rebate is the net settlement figure.
When this calculation is used
- Checking a settlement quote on a precomputed instalment loan against the rebate the named method actually implies.
- Seeing how far the sum-of-digits rebate falls short of the pro-rata intuition at the current point in the term, before agreeing to settle.
- Reading an older or foreign loan agreement that names the Rule of 78s and putting a figure on what walking away early would return.
- Weighing whether settling now or a few instalments later changes the rebate enough to matter, given the weights left in the schedule.
Worked example
A short precomputed loan settled near the halfway mark of its schedule — the point where the gap between the sum-of-digits rebate and the pro-rata intuition is at its widest.
The rebate is the unearned share the method hands back; the net payoff is the lender’s gross figure less that rebate. Hold the rebate against the halfway intuition: roughly half the payments remain, yet the rebate is well under half the interest, because the payments already made carried the heaviest weights in the schedule. That distance is the cost of the method itself.
Move the settlement one instalment earlier or later and re-run: the rebate shifts by exactly the weight of that single instalment, largest early in the term and smallest at the end. The shortfall against pro-rata peaks near mid-term and vanishes at either boundary — settled at signing, everything returns; settled at the last payment, nothing was left to return.
What each input represents
Total precomputed interest
The interest fixed for the whole term at signing — the pool the rebate is carved from. It is stated in the loan agreement of a precomputed loan; on an ordinary declining-balance loan no such fixed figure exists, and this method does not apply.
Total scheduled payments
The count of instalments the loan was written for — the whole schedule, whether or not it will run to the end. It sets the denominator of the sum-of-digits weighting, so the contract’s own count is the one that belongs here, not the count now expected.
Payments already made
How many instalments have been paid when the settlement happens. Every payment made surrenders the heaviest remaining weight in the schedule, which is why the rebate shrinks fastest in the early months and why waiting to settle costs more than the pro-rata view suggests. Settling before any payment returns the whole interest pool.
Gross payoff before rebate
The payoff figure the lender states before the rebate is applied — typically the remaining scheduled payments taken at face value. This page does not derive it; it takes the lender’s figure and subtracts the rebate the method yields to reach the net.
Assumptions and limits
- The loan is genuinely precomputed: its total interest was fixed at signing rather than accruing on a declining balance.
- The gross payoff amount is as the lender states it; this page subtracts the rebate from that figure and does not audit it.
- No settlement fees or penalties are modelled beyond whatever the gross payoff already contains.
- Instalments are equal and on schedule; the sum-of-digits weights assume the payment count alone tells the story of timing.
- Whether the Rule of 78s may be applied at all is a matter of the governing law — this page computes the method, not the entitlement to use it.
What the guards protect against
- More payments made than were ever scheduled is refused — the settlement point must lie inside the schedule for a remaining count to mean anything.
- A schedule of no payments is refused: the sum-of-digits has nothing to sum, and the total payment count must be a whole, positive count within a realistic term.
- Negative amounts for the precomputed interest or the gross payoff are refused — both are sums of money, and the rebate is carved from one and subtracted from the other.
Provenance
Rule of 78s sum-of-digits unearned-interest rebate for precomputed instalment loans
The count of remaining payments multiplied by one more than itself, over the total payment count multiplied by one more than itself, gives the unearned fraction; that fraction of the precomputed interest is the rebate, and the gross payoff less the rebate is the net settlement figure. · Labelled reference only: many jurisdictions restrict or prohibit the Rule of 78s for consumer credit, and the governing law decides whether the method may be applied at all.
Educational reference, not financial advice, and not a statement of any jurisdiction’s consumer-credit law. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.