Loan Payoff Calculator with Extra Payments

Extra money sent to a loan goes entirely to principal. It does not reduce the next payment; it reduces the balance, and therefore every interest charge from that point on. The scheduled payment is

$M = P\cdot\frac{r(1+r)^n}{(1+r)^n - 1}$

with $r$ the periodic rate and $n$ the number of payments. Raise the amount you actually pay to $M' = M + X$ and the term is no longer $n$. Invert the same formula for the number of payments:

$n' = \frac{-\ln\!\left(1 - \dfrac{rP}{M'}\right)}{\ln(1+r)}$

The interest saved is the difference of two totals:

$\text{saving} = (nM - P) - (n'M' - P) = nM - n'M'$

Two conditions make the formula behave. If $M' \le rP$ the payment does not even cover the period's interest and the logarithm is undefined — the balance never falls. And $n'$ is rarely a whole number: the fractional part is a final smaller payment, so round up to get the count of payments actually made.

The payoff quote today

A payoff is not the statement balance. It is the balance plus interest accrued since the last payment:

$\text{Payoff} = B + \text{per diem} \times \text{days}, \qquad \text{per diem} = \frac{B \cdot r_{\text{annual}}}{365}$

The balance itself, $k$ payments in, is

$B_k = P(1+r)^k - M\cdot\frac{(1+r)^k - 1}{r}$

One lump sum instead of monthly extras

A single payment $L$ made after $k$ periods reduces the balance to $B_k - L$; feed that into the $n'$ formula as the new principal with $k$ periods already elapsed.

Why timing dominates

Interest each period is $B_{k-1}\cdot r$, and the balance is largest at the start. The same dollar paid in month 6 removes far more interest than in month 50 — which is also why the saving is never proportional to the extra amount.

Some loans carry prepayment penalties or apply extra funds to future instalments rather than to principal by default. Those are contract terms, not arithmetic; check yours before assuming the numbers below apply.

• Assuming the extra reduces interest proportionally: paying $40\%$ more does not cut interest by $40\%$. The relationship runs through a logarithm.
• Using the annual rate as $r$: $r$ must match the payment period. $6.5\%$ annual is $0.065/12 \approx 0.0054167$ monthly.
• Rounding $n'$ down: $n' = 39.96$ means 39 full payments and a smaller fortieth, not 39 payments total.
• Comparing $nM$ against $n'M'$ with different payments and calling the difference savings without checking the totals: the saving is $nM - n'M'$, both totals computed to the same precision.
• Treating the statement balance as the payoff figure: it omits accrued per-diem interest, so a payment for exactly the balance leaves the loan open for a few dollars.
• Forgetting the extra must be earmarked as principal: otherwise a lender may simply credit it toward the next scheduled instalment, which saves almost nothing.