Amortization formula explained (with examples)

The amortization formula is the standard way to calculate a fixed payment for a loan that is paid back in equal installments. You see it everywhere: mortgages, auto loans, personal loans, and many student loans use the same basic math. The goal is simple: find a payment that will reduce the loan balance to zero after a known number of payments, while charging interest on the outstanding balance along the way.

The formula is usually written as:

PMT = P × r / (1 − (1 + r)⁻ⁿ)

where PMT is your payment each period (most commonly monthly), P is the loan amount, r is the interest rate per period, and n is the number of payments. The magic is in that (1 + r)⁻ⁿ term: it captures the time value of money by discounting future payments back to today.

Your lender might advertise an annual interest rate (APR), but the amortization formula needs the interest rate per payment period. If you pay monthly, there are 12 payments per year, so:

r = APR / 12

For example, a 6% APR means r = 0.06 / 12 = 0.005 per month (0.5% per month). If you pay biweekly, there are usually 26 payments per year, so r = APR / 26. This calculator lets you choose the frequency so the same formula works for monthly, weekly, quarterly, and more.

Next, compute n, the total number of payments. If your term is in years and you pay monthly:

n = years × 12

For a 30-year mortgage, n = 30 × 12 = 360 payments. For a 5-year car loan, n = 5 × 12 = 60 payments. If you choose a different payment frequency, the calculator uses:

n = years × paymentsPerYear

Once you have P, r, and n, plug them into the amortization formula. The denominator, 1 − (1 + r)⁻ⁿ, prevents the payment from being “too small.” If you tried to pay only the interest each period, the balance would never decrease. The denominator forces the payment to be large enough to pay down principal over time.

Suppose you borrow $10,000 at 6% APR for 3 years, paid monthly.

P = 10,000
r = 0.06 / 12 = 0.005
n = 3 × 12 = 36

PMT = 10,000 × 0.005 / (1 − (1.005)⁻³⁶)

The result is a fixed monthly payment. Each period, interest = balance × r, and principal = payment − interest. Because the balance shrinks over time, interest shrinks, and more of the payment goes to principal.

Many people are surprised that early payments barely reduce the balance. That’s not a trick; it’s how interest works on a large balance. In the beginning, the balance is highest, so interest = balance × r is also high. As the balance falls, interest falls too. Over the life of the loan, the same payment “tilts” from mostly interest toward mostly principal.

If you pay extra toward principal, you reduce the balance faster, which reduces future interest. Even a modest extra payment can save a surprising amount of interest and shorten the payoff time. In this calculator, the “extra payment” is added to the normal payment each period (but never beyond what would pay the balance to zero).

• Compute r: convert APR to the per-period rate.
• Compute n: term × payments per year.
• Compute PMT: use PMT = P×r / (1 − (1+r)^−n) (or P/n if r = 0).
• Build schedule: for each period: interest = balance×r, principal = payment−interest, new balance = old balance−principal.

Note: Some real loans have fees, daily compounding, escrow, balloon payments, or irregular schedules. The classic PMT formula assumes a fixed rate and regular payments.

Frequently Asked Questions

• What does “PMT” mean in the amortization formula?

  PMT stands for “payment.” It’s the fixed amount you pay each period (usually monthly) so the loan is fully paid off after n payments at interest rate r per period.

• Is this the same formula used in Excel or Google Sheets?

  Yes. Spreadsheet functions like PMT(rate, nper, pv) are based on the same amortization math. This calculator shows the underlying formula and generates the schedule step-by-step.

• Why is my lender’s payment slightly different?

  Lenders may include fees, insurance/escrow, daily interest rules, rounding policies, or different timing conventions. Use this as a clean baseline, then compare to your official loan disclosure.

• What if the APR is 0%?

  If the interest rate is 0, the payment is simply P / n. This calculator handles that case automatically.

• Does paying biweekly reduce interest?

  Often, yes—because 26 biweekly payments is equivalent to 13 monthly payments per year, which can accelerate payoff. But it depends on how your lender applies the payments. This calculator models a consistent period-based schedule.

• Is amortization only for loans?

  Amortization also shows up in accounting (spreading an intangible asset’s cost over time), but the PMT formula here is specifically for loan repayment schedules.