Loan Repayment Formula: Monthly Payment Explained

The loan repayment formula calculates the fixed monthly payment needed to repay a loan over a set term, assuming a fixed annual percentage rate and monthly payments. Enter the loan amount, APR, and term to estimate the payment and total interest; the results are arithmetic estimates, not a lender quote.

Formula

For a fixed-rate loan repaid with equal monthly payments, the payment is the principal multiplied by the monthly interest rate, divided by one minus the accumulation factor raised to the negative number of payments:

PMT = P × r ÷ [1 − (1 + r)⁻ⁿ]

• PMT is the monthly payment, in dollars.
• P is the amount borrowed, or principal, in dollars.
• r is the monthly interest rate as a decimal: APR ÷ 12 ÷ 100.
• n is the total number of monthly payments: years × 12.

The calculator’s second result is total interest: PMT × n − P. It uses the displayed monthly payment rounded to the nearest cent, so the total reflects repeated payments of that rounded amount. The formula assumes the final scheduled payment is the same as the others; real servicing may adjust a final payment slightly because of rounding or payment dates.

Worked Example

Using the calculator’s defaults, the loan amount is $20,000, APR is 8%, and term is 5 years. First convert the annual rate to a monthly decimal and the term to months:

• Monthly rate: r = 8 ÷ 1,200 = 0.0066666667.
• Number of payments: n = 5 × 12 = 60.
• Substitute: PMT = 20,000 × 0.0066666667 ÷ [1 − (1 + 0.0066666667)⁻⁶⁰].
• Monthly payment, rounded to cents: $405.53.
• Total interest using the displayed payment: $405.53 × 60 − $20,000 = $4,331.80.

The payment calculation before cent rounding is about $405.53 per month. The total is based on the rounded payment, as shown, rather than on an unrounded mathematical payment.

Assumptions

• The interest rate is fixed for the full term.
• Payments are equal, made monthly, and treated as end-of-period payments.
• Interest compounds monthly using APR ÷ 12; APR is entered as a percentage per year.
• The calculation excludes fees, taxes, insurance, PMI, and escrow. It does not model extra payments, missed payments, rate changes, or lender-specific rounding.

How the Repayment Formula Works

Each payment covers interest accrued since the previous payment and reduces principal. At the beginning, the balance is larger, so more of a level payment goes toward interest. As the balance falls, the interest portion generally shrinks and the principal portion grows. This schedule is called amortization.

The equation sets the present value of all scheduled payments equal to the amount borrowed. Each future payment is discounted by the monthly rate; the formula solves for the payment that makes their combined present value equal the principal. If the rate is zero, the simpler calculation is principal divided by the number of payments. This calculator requires a positive APR because its displayed formula uses a nonzero monthly rate.

“Loan repayment formula” can also refer to the balance remaining after a particular number of payments. That is a related calculation, not the same output as the monthly payment. It depends on the original principal, monthly rate, payment amount, and number of payments already made. A statement balance may differ because of accrued interest, fees, payment timing, or other account activity.

How to Use and Interpret the Results

Enter the principal you expect to borrow, the APR supplied for the loan, and the repayment term in years. The term is converted to months, and the APR is divided by 12 and 100. For example, entering 8 means 8% per year, not 0.08% or a monthly rate of 8%.

The monthly payment is the scheduled amount under the assumptions above. Total interest estimates how much the payments exceed the original principal over the full term. It is not the total amount paid: the estimated total of payments is principal plus interest, or payment multiplied by the number of months. Fees and excluded costs can make the amount paid differ from that estimate.

A longer term generally spreads repayment over more installments, which changes the payment and usually increases the time during which interest accrues. A different APR also changes both the monthly interest charge and the payment required to amortize the balance. These are mathematical effects, not a recommendation about which terms to select. Use the actual figures and payment schedule in the lender’s disclosure for a loan-specific comparison.

Common Errors and Limits

• Mixing rate formats: The formula needs a monthly decimal rate. For an APR entered as a percentage, the conversion is APR ÷ 1,200. Do not divide by 12 and then treat a percentage number as a decimal.
• Using the wrong term: This calculator accepts years and converts them to months. If a term is stated in months, convert it to years before entering it, or use a calculator designed for month inputs.
• Confusing APR with a periodic rate: APR is an annual percentage input here. The formula applies APR ÷ 12; it does not accept a monthly interest rate in the APR field.
• Assuming all borrowing costs are included: The calculation includes principal and modeled interest only. It does not add fees or other loan-related charges.
• Expecting a changing-rate schedule: A fixed-rate amortization formula cannot represent an adjustable rate or a payment that changes over time.

Payments may be due on dates that do not match a simple monthly schedule, and some contracts calculate interest using daily balances or another method. Additional principal payments can shorten repayment and change total interest, but are not represented here. For those cases, a full amortization schedule based on the actual contract terms is more suitable.

Frequently Asked Questions

How do I calculate a loan payment by hand?
Convert APR to a monthly decimal by dividing by 1,200, convert the term to total monthly payments, and substitute both into PMT = P × r ÷ [1 − (1 + r)⁻ⁿ]. Round the resulting payment to cents for an estimate.

Does the loan repayment formula include interest?
Yes. Interest is built into the payment calculation through the monthly rate. The calculator separately estimates total interest by subtracting the original principal from the sum of the rounded scheduled payments.

Why does a longer loan term change the payment?
A longer term means more monthly installments, so the principal is repaid over a longer period. That changes the required payment and the amount of time interest can accrue; the exact result depends on principal and APR.

Is APR the same as the monthly interest rate?
No. APR is entered as an annual percentage here. The formula converts it to a monthly decimal by dividing by 1,200. A lender’s APR may also reflect costs differently from a simple interest rate, so check the disclosed calculation details.

Why might my lender’s payment differ from this estimate?
The lender may use different payment dates, interest accrual conventions, fees, rounding, or contract terms. A variable rate or other costs can also change the amount. The lender’s disclosure provides the figures for that specific loan.

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

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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