Loans and mortgages Updated November 2026

Extra payments and payoff time

Adding a fixed extra amount to each payment shortens an amortizing loan. The new number of payments is n = -ln(1 - P x r / PMT_new) / ln(1 + r), where PMT_new is the regular payment plus the extra. Interest saved is old total interest minus new total interest. In Excel: =NPER(rate, -PMT_new, P).

Syntax

n_new = -ln(1 - P x r / PMT_new) / ln(1 + r)   interest saved = (PMT x n - P) - (PMT_new x n_new - P)

Part	What it is
`P`	Current loan balance
`r`	Rate per period
`PMT_new`	Regular payment plus the extra amount

Examples

200,000 at 6%, 30 years, PMT 1,199.10, pay 200 extra monthly
n = -ln(1 - 200,000 x 0.005 / 1,399.10) / ln(1.005)

Result: n = 251.5 months (21.0 years instead of 30)

Excel: =NPER(6%/12, -1399.10, 200000)

Interest saved in that case
231,676 - (1,399.10 x 251.5 - 200,000)

Result: About 79,800 saved (151,875 paid instead of 231,676)

20,000 car loan at 7%, 60 months, PMT 396.02, pay 100 extra
n = -ln(1 - 20,000 x (0.07/12) / 496.02) / ln(1 + 0.07/12)

Result: n = 46.1 months instead of 60

Common mistakes

Assuming the extra payment is applied to principal; some lenders apply it to the next month's payment unless told otherwise, which saves nothing.
Reporting the fractional n as the full answer; the last payment is a partial one, so 251.5 means 251 full payments and a small final one.
Ignoring prepayment penalties or the alternative of investing the extra money at a higher after-tax return than the loan rate.

Practice Extra payments and payoff time

These drills check your answer on live data, free, no account needed.

Payoff time with extra paymentsA $150,000, 30-year mortgage at 4.25% APR has a required payment of $737.91 a month. If you pay $800 extra every month from the start, how many months until the loan is paid off? Give N to 2 decimals.
Total interest over the loanYou borrow $105,000 for 30 years at 4.25% APR with monthly payments. Using the payment rounded to the cent, how much interest do you pay over the life of the loan?

Questions

Does one extra payment a year make a difference?

Yes. One extra monthly payment a year is about 100 extra per month on the 200,000 example, which cuts a 30-year mortgage to roughly 24.5 years and saves about 49,000 in interest.

Is it better to pay extra or to refinance?

Paying extra saves interest at the loan's own rate with no fees. Refinancing helps when the new rate is meaningfully lower after closing costs. Many borrowers do both.

Total interest paid on a loanTotal interest on an amortizing loan is the payment times the number of payments minus the principal. Examples for 30-year and 15-year loans, Excel CUMIPMT.
Remaining loan balanceThe balance after k payments is the present value of the payments still to come: B_k = PMT x (1 - (1 + r)^-(n-k)) / r. Mortgage and car loan examples, Excel.
Mortgage payment formulaHow to calculate a monthly mortgage payment with M = P x r / (1 - (1 + r)^-n). Examples for 200,000 at 6% and 300,000 at 5%, Excel PMT and common mistakes.
Amortization scheduleAn amortization schedule splits each payment into interest and principal and tracks the balance. Worked schedule for a 10,000 loan, Excel IPMT and PPMT.