In short: A loan payment is calculated to many decimal places and then charged in whole cents. On a $300,000 mortgage at 6.5% over 30 years, the exact payment is $1,896.20407, charged as $1,896.20. That missing 0.407 of a cent every month earns interest for the lender for up to 30 years, and the last payment collects it: $1,900.91 instead of $1,896.20.
The payment that cannot be charged
The standard loan formula produces the payment that brings a balance to exactly zero after the final period. For $300,000 at 6.5% over 360 months, that payment is $1,896.20407048.
No one can be charged $1,896.20407048. The payment is rounded to $1,896.20, which is 0.407 of a cent less than the mathematics requires. Each month the borrower underpays by that sliver, and the balance ends the month 0.407 of a cent higher than the formula's path.
There is a second rounding in the same month. Interest is the balance times the monthly rate, and that product rarely lands on a whole cent either. A lender rounds it to the cent before working out how much of the payment is left for principal.
Why four tenths of a cent becomes $4.71
A shortfall of 0.407 of a cent a month for 360 months is only $1.47 in total. The final payment is $4.71 higher than the rest, which is more than three times that.
The reason is that each shortfall stays in the balance and is charged interest every month afterward. The sliver left unpaid in month one accrues interest for 359 more months at 6.5%. It is the same compounding that drives the rest of the schedule, applied to the rounding error.
Worked through cent by cent, with the payment fixed at $1,896.20 and each month's interest rounded to the nearest cent, the balance after payment 359 is $1,890.67. The final month's interest on that is $10.24, so the amount needed to clear the loan is $1,900.91.
The direction depends on which way the payment was rounded
Had the payment been rounded up to $1,896.21 instead, the borrower would overpay by 0.593 of a cent a month, the balance would run slightly ahead of schedule, and the final payment would be $1,889.51, which is $6.70 less than the regular amount.
| Payment charged | Balance after payment 359 | Final payment | Difference from regular payment |
|---|---|---|---|
| $1,896.20 (rounded down) | $1,890.67 | $1,900.91 | $4.71 more |
| $1,896.21 (rounded up) | $1,879.33 | $1,889.51 | $6.70 less |
Lenders differ on this. Some round to the nearest cent, some always round up so that the final payment is never larger than the borrower expects. Either way the schedule has to end at exactly zero, and the last payment is the only place the difference can go.
How the schedules on this site handle it
The calculators here follow the same cent-by-cent convention described above. The payment is rounded to the nearest cent, each month's interest is rounded to the cent before the principal is worked out, and the balance carried forward is always a whole number of cents. The final payment is the remaining balance plus one month of interest on it.
On this loan that gives a regular payment of $1,896.20, a final payment of $1,900.91, and total interest of $382,636.71. The principal column sums to exactly $300,000.00 and the final balance is exactly $0.00, with no adjustment needed to make it so.
One choice is built in and worth knowing: the payment is rounded to the nearest cent. A lender that always rounds the payment up would show the second row of the table above, with a slightly smaller final payment. The regular payment and the shape of the schedule are the same either way, but the exact final payment on your own loan comes from your lender's rounding rule, and a payoff statement is the only authoritative figure.
Why floating point makes this worse
Most software stores decimals in binary floating point, which cannot represent 0.1 exactly. Over 360 periods of multiplication and subtraction, that representation error accumulates and can surface in the cents, on top of the legitimate rounding described above. A spreadsheet built from the loan formula can show a final balance of a few cents rather than zero for this reason alone.
The calculators here use exact decimal arithmetic, so the only rounding in a result is rounding that was chosen, and it can be stated, as it is in this guide.
What this means for a borrower
For a regular monthly payment, nothing. The difference is a few dollars, once, at the end.
It matters in two situations. When paying a loan off early, the payoff amount comes from the lender's actual balance and the interest accrued to the day, not from multiplying remaining payments by the regular amount. And when comparing a lender's schedule with a calculator's, a small difference in the final payment or in total interest usually reflects a different rounding rule, not an error in either.
See the full schedule
The mortgage calculator and loan calculator show every period, including the last. The mortgage payoff calculator recomputes the schedule with extra payments. The reasoning behind the schedule's shape is in why early loan payments are mostly interest, and the arithmetic is documented on the methodology page.
Sources
- Regulation Z (Truth in Lending), 12 CFR Part 1026, Appendix J, the federal method for computing the annual percentage rate on closed-end credit, including its treatment of payment schedules. ecfr.gov/current/title-12/chapter-X/part-1026/appendix-Appendix%20J%20to%20Part%201026