calculate-loan

01 Loan payments

Payment Amortization Formula and Monthly Calculator

Calculator No. 01-2326 · Updated · Formula and assumptions shown below

Monthly payment $405.53
Total interest
$4,331.80
= Monthly payment × loan term × 12 − loan amount
Loan amount
$20,000
Input
Interest rate (APR)
8 %
Input
You enter
Loan amount ($), Interest rate (APR) (%), Loan term (years)
You get
Monthly payment ($), Total interest ($)
Rounding
2 dp

Estimate for education, not financial advice. The result follows from the numbers you enter and the formula shown. It is not a quote or an offer, and it does not include fees, taxes or insurance unless the calculator asks for them. Ask your lender for the exact figures.

The payment amortization formula calculates the level monthly payment needed to repay a fixed-rate loan over a set number of months. Enter the loan amount, annual APR, and term in years to estimate the payment and total interest; the calculation assumes equal monthly payments and a constant rate.

Formula

The standard amortizing-loan payment formula is:

Payment = P × r ÷ [1 − (1 + r)−n]

  • P is the starting principal, or amount borrowed, in dollars.
  • r is the monthly interest rate as a decimal. For an APR entered as a percentage, r = APR ÷ 1,200.
  • n is the total number of monthly payments. For a term in years, n = years × 12.

The calculator rounds the monthly payment to the nearest cent, then estimates total interest as the rounded payment multiplied by the number of payments, minus the original principal. For the inputs above, r = 8 ÷ 1,200 = 0.0066667 and n = 5 × 12 = 60.

Worked Example

Using the calculator’s default inputs—$20,000 principal, 8% APR, and a 5-year term:

  1. Convert the APR to a monthly decimal rate: 8 ÷ 1,200 = 0.0066667.
  2. Find the number of payments: 5 × 12 = 60.
  3. Substitute into the formula: $20,000 × 0.0066667 ÷ [1 − (1.0066667)−60].
  4. The unrounded payment is about $405.53; rounded to cents, the monthly payment is $405.53.
  5. Using the rounded payment, total of payments = $405.53 × 60 = $24,331.80. Estimated total interest = $24,331.80 − $20,000 = $4,331.80.

Assumptions

  • The interest rate stays fixed for the full term.
  • Payments are equal, made monthly, and modeled at the end of each monthly period.
  • Interest compounds monthly at APR ÷ 12; the APR is treated as a nominal annual rate.
  • The estimate excludes fees, taxes, insurance, PMI, and escrow because none are calculator inputs.

How Amortization Works

Each scheduled payment covers interest accrued since the prior payment and reduces principal by whatever remains. Early in a loan, the balance is relatively large, so a larger share of the fixed payment goes to interest. As principal falls, the interest portion generally shrinks and more of each payment goes toward reducing the balance. This changing division is called amortization.

The formula works by finding a payment whose discounted value across all n months equals the original principal. In other words, it accounts for the fact that money repaid later has interest accruing in the meantime. It is not the same as dividing the principal by the number of months, nor does it calculate interest by multiplying the original amount by the APR for every year.

How to Read the Results

Monthly payment is the modeled amount required each month for principal and interest under the stated assumptions. Total interest is the estimated interest over the full schedule, based on that cent-rounded payment. The total of payments includes principal plus interest; it is not an extra charge to add to the original principal again.

An amortization schedule shows the detail month by month: beginning balance, interest charged, principal repaid, and ending balance. For a particular month, interest is approximately the starting balance for that period multiplied by the monthly rate. Subtracting that interest from the payment gives the principal portion. Actual statements may differ because of payment dates, daily-interest methods, rounding, or contract terms.

When This Formula Applies—and Its Limits

This formula is suited to a fully amortizing loan with a fixed rate, regular monthly payments, and no changing balance from additional borrowing. It can provide a baseline estimate for loans such as a fixed-rate mortgage, auto loan, or personal loan, but it does not reproduce every loan contract.

It does not model variable rates, interest-only periods, balloon payments, skipped or irregular payments, payment holidays, fees added to the balance, or extra principal payments. A loan with a changing rate may have its payment recalculated, while a balloon loan may leave a balance due at the end. For an exact schedule, use the lender’s disclosed rate, payment timing, compounding method, and any contract-specific charges.

Common Calculation Mistakes

  • Using APR as a monthly decimal directly: For an APR entered as 8%, the monthly rate in this formula is 8 ÷ 1,200, not 8 or 0.08.
  • Mixing years and months: Convert a year-based term to months before using it as n. Five years means 60 monthly payments.
  • Confusing APR with the periodic rate: APR is the annual input here; the formula uses APR divided by 12 as a percentage, then expressed as a decimal.
  • Rounding too early: Keep precision through the formula and round the payment to cents at the end. Small differences can accumulate over many payments.
  • Assuming the payment includes every housing or borrowing cost: The result covers modeled principal and interest only, not items such as taxes, insurance, or fees.

Frequently Asked Questions

What is the payment amortization formula?
The monthly payment is P × r ÷ [1 − (1 + r)−n], where P is principal, r is the monthly rate as a decimal, and n is the number of monthly payments.

How do I calculate a monthly payment from APR?
Convert APR to a monthly decimal rate by dividing the percentage value by 1,200, convert the term to months, and substitute both into the formula with the loan amount.

Does an amortized payment stay the same every month?
With a fixed rate and an unchanged schedule, the modeled principal-and-interest payment stays level. Its interest and principal portions change over time.

Why is my lender’s payment different from this estimate?
The quoted amount may reflect fees, a different rate or term, insurance or escrow, payment timing, rounding, or other contract details not represented here. Check the lender’s itemized disclosure.

Does total interest include the amount borrowed?
No. Total interest is separate from principal. Adding the two gives the scheduled total of payments under the estimate.

Can I use this for a variable-rate loan?
Only as a snapshot if you supply a rate for a particular period. A single fixed-rate calculation cannot predict later payments when the rate changes.

This result is an estimate, and the lender’s disclosure gives the exact figures.

Related

Formula used
in p | Loan amount | $ | 20000 | 1 | 100000000
in apr | Interest rate (APR) | % | 8 | 0.01 | 100
in yrs | Loan term | years | 5 | 0.0833 | 50
out pmt | Monthly payment | $ | round(p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)), 2) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard loan formula for this calculation, fixed rate and monthly compounding unless stated, rounded as each result states. Not financial advice; no lender, rate or product is recommended. See formulas and assumptions, the disclaimer or report an error.

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