What an extra payment buys in time and interest

Where the extra money actually goes

The scheduled payment already covers the interest each period accrues; that is what makes it a workable payment at all. So an extra amount added on top has nowhere to go but the principal — none of it is consumed by interest. A smaller principal accrues less interest the very next period, which frees more of the unchanged scheduled payment to attack the principal in turn. Acceleration compounds, in the borrower’s favour, by the same mechanism that made the debt expensive in the first place.

The method is a comparison, not a new formula: the payoff-time relation evaluated at the scheduled payment gives the baseline horizon, the same relation at the boosted payment gives the revised one, and the interest on each path is the total money paid along it minus the principal. The difference between the two interest totals is the saving — money that under the baseline schedule would have been accrued and paid, and under the accelerated one simply never comes into existence.

The reward is not linear in the extra amount. The first increment of overpayment does disproportionate work, especially when the scheduled payment sits close to the interest-only level, and each further increment buys a little less time than the one before. The chart on this page sweeps the revised horizon across a range of extra amounts so that bend is visible before any commitment is made — the point where more acceleration stops buying much is a fact worth seeing, not guessing.

What is modelled is a recurring extra: the same additional amount every period, from now until the debt ends. A one-off lump sum is a different action — it resets the balance once rather than boosting every payment — and answering it honestly means re-running the baseline with a reduced principal instead. The distinction matters because a windfall and a raise feel similar and behave completely differently over a schedule.

The workspace also offers the reverse direction: name the horizon the debt should have — the revised payoff target — and it solves for the recurring extra that achieves it. That turns a finish date into a required top-up, the acceleration question asked from the calendar’s side instead of the wallet’s.

When this calculation is used

• Pricing a recurring overpayment before committing to it — what this much extra per period buys in months removed and interest never accrued.
• Comparing the accelerated schedule against the baseline when deciding what a monthly surplus should do.
• Using the reverse direction to find the extra amount a chosen finish date requires.
• Checking a claim that a small top-up saves years — the sweep chart shows exactly when that is true and when it flattens.
• Separating the effect of a recurring top-up from a one-off lump sum before mixing the two up.

Worked example

The pack’s anchor vector takes a long-dated home loan whose scheduled payment exactly amortises it over the full term, then adds a modest recurring extra to every monthly payment; companion vectors do the same for a mid-sized balance and a small one.

Two outputs come back. The revised payoff time lands years inside the original term — the schedule the extra amount buys. The interest saved is the gap between what the baseline schedule would have paid in interest and what the accelerated one does, and on the anchor it amounts to a substantial slice of the whole interest bill. Read together they are the price tag of acceleration, stated in the two currencies it pays out in.

Every figure is produced by the certified engine when the calculator loads; the prose holds none. The pack also declares a refusal: a scheduled payment at or below the period’s interest-only amount is declined before any extra is considered, because a baseline that never ends has no horizon to improve.

What each input represents

Assumptions and limits

• The extra amount is paid every period without fail and is applied to the principal immediately, in the same period it is paid.
• The lender levies no prepayment penalty or overpayment charge — where one exists, it comes straight out of the saving reported here.
• The scheduled payment stays unchanged after overpaying: the loan is not recast onto a new, lower payment, which would trade the time saving for cash-flow relief.
• The rate is fixed across both schedules, and the periodic rate follows the nominal convention of the annual rate divided by the payments per year.
• The comparison models a recurring top-up, not a one-off lump sum — a windfall applied once needs the baseline re-run at a reduced balance instead.

What the guards protect against

• A scheduled payment at or below the interest-only amount for the period is refused: the baseline schedule would never end, so neither the revised horizon nor the saving against it can be stated.
• The balance and the scheduled payment must both be greater than nought, and the extra amount cannot be negative — a withdrawal from the payment is not an acceleration.
• The rate is bounded to a realistic range and the payments per year must be at least one, so both schedules describe actual repayment arrangements.

Provenance

Extra-payment comparison of the payoff-time relation

The logarithmic payoff-time relation evaluated at the scheduled payment and again at the scheduled payment plus the recurring extra; interest on each path as total payments less principal, and the saving as the difference between the two.

Educational reference, not financial advice, and silent on any prepayment terms a specific agreement may impose. The signed pack carries its own citation — a direct application of the annuity-inversion relation — which displays from the verified leaf once the calculator loads; the page reports the verification state of the release it mounted rather than asserting one.