calculate-loan

01 Loan payments

Loan Payment Equation: Formula and Calculator

Calculator No. 01-6465 · Updated · Formula and assumptions shown below

Monthly payment $405.53
Total interest
$4,331.67
= 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 loan payment equation calculates the fixed payment needed to repay a loan over a set number of monthly periods, assuming a constant interest rate. Enter the loan amount, annual percentage rate (APR), and term in years to estimate the monthly payment and total interest; the results exclude fees and other charges.

Formula

For a fixed-rate loan with equal monthly payments, the payment equation is:

Payment = P × r ÷ [1 − (1 + r)−n]

  • P is the amount borrowed, in dollars.
  • r is the monthly interest rate as a decimal. With an APR entered as a percentage, r = APR ÷ 1,200. For example, 8% gives 0.08 ÷ 12 = 0.0066667 per month.
  • n is the number of monthly payments. For a term in years, n = years × 12.

The second result is estimated total interest over the full term: monthly payment × number of payments − original loan amount. It uses the payment rounded to cents, as displayed. The equation works by setting the present value of all scheduled payments equal to the amount borrowed: each future payment is discounted for the interest accumulated before it is made.

Worked Example

Using the calculator’s default inputs—$20,000 principal, 8% APR, and 5 years—the calculation proceeds as follows:

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

Assumptions

  • The interest rate is fixed for the entire modeled term.
  • Payments are equal, made monthly, and occur at the end of each period.
  • Interest is modeled monthly at APR ÷ 12; this is a standard loan-payment approximation, not a statement of every lender’s interest method.
  • The calculation excludes fees, taxes, insurance, mortgage insurance, escrow, and any other charges or changes not entered as inputs.

How the Equation Works

A loan balance generates interest while it remains unpaid. Each payment covers the interest due for that period, and the remainder reduces principal. Early in a typical amortizing loan, the balance is relatively high, so a larger share of the payment goes to interest. As principal falls, the interest portion generally shrinks and more of the same payment reduces the balance.

The equation combines those changing portions into one level payment. It does not mean that the borrower pays the same amount of interest every month. It means the scheduled payment stays level under the stated assumptions, while its allocation between interest and principal changes over time.

When to Use It and How to Read Results

Use the equation to estimate a scheduled payment when you know the principal, APR, and repayment term. It can help explain how changing one input affects the arithmetic. For a given rate and term, a larger principal produces a larger payment. For a given principal and rate, a longer term generally lowers the scheduled monthly amount but increases the number of payments, and therefore can increase total interest.

The displayed monthly payment is not necessarily the full amount due on a bill. Some loans include charges or separate payments not represented by principal and interest. Enter the actual loan amount, rate, and term shown in the relevant lender disclosure for a closer mathematical estimate. APR can include certain borrowing costs in its definition, but this calculator uses it as the annual rate input in a monthly-rate formula; it does not separately add fees to the payment.

Common Mistakes and Limits

  • Mixing percent and decimal rates: 8% is 0.08 annually, not 8. The calculator converts a percentage input to a monthly decimal by dividing by 1,200.
  • Using the wrong period count: A five-year monthly loan has 60 payments, not five. The formula’s rate and number of periods must use matching monthly periods.
  • Rounding too early: Keep full precision in intermediate steps, then round the payment to cents. Small differences may occur if an outside calculation rounds the monthly rate or intermediate values differently.
  • Treating payment as total cost: The payment result covers modeled principal and interest only. It does not automatically include fees, insurance, taxes, or other costs.
  • Applying it to a changing-rate or irregular loan: The equation assumes a fixed rate and regular equal monthly payments. Variable rates, skipped or extra payments, interest-only periods, and unusual payment schedules require a different calculation or a period-by-period model.

A zero-rate loan is a special case: its payment is principal divided by the number of payments. This calculator’s APR input starts above zero because its displayed equation divides by the monthly rate. The result is an estimate; the lender’s disclosure gives the exact figures.

Frequently Asked Questions

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

How do I convert APR to a monthly rate?
For this calculation, divide the APR percentage by 1,200. For example, 6% ÷ 1,200 = 0.005, or 0.5% per month. The method assumes monthly rate conversion and may not match every contract’s exact interest calculation.

Does the monthly payment include interest?
Yes. The calculated level payment includes both interest and principal repayment. The share assigned to each changes over the loan term, even when the scheduled payment stays constant.

Why does a longer loan term reduce the payment?
It spreads repayment over more monthly payments. Because interest accrues for more periods, the total interest can be higher even if the monthly payment is lower.

Does this payment equation include fees or insurance?
No. It models principal and interest under its assumptions. Check the lender’s figures for fees, insurance, taxes, escrow, or other amounts that may affect the payment due.

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 | $ | p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) | 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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