Loan Amortization Formula and Monthly Payment Calculator

The loan amortization formula calculates the fixed periodic payment needed to repay a loan, including principal and interest, over a set term. Enter the loan amount, annual APR, and term in years to see the estimated monthly payment and total interest.

Formula

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

M = P × r ÷ [1 − (1 + r)⁻ⁿ]

• M is the monthly payment in dollars.
• P is the starting principal, or loan amount, 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 number of monthly payments. For a term in years, n = years × 12.

The calculator rounds the monthly payment to cents, then calculates total interest as the rounded payment multiplied by the number of payments, minus the original principal. This makes the displayed total consistent with the displayed payment. Actual final payments can differ slightly if a lender adjusts the last installment or uses different rounding.

Worked Example

Using the calculator defaults, the principal is $20,000, the APR is 8%, and the term is 5 years.

1. Convert the APR to a monthly decimal rate: r = 8 ÷ 1,200 = 0.0066667.
2. Convert the term to monthly payments: n = 5 × 12 = 60.
3. Substitute: M = 20,000 × 0.0066667 ÷ [1 − (1.0066667)⁻⁶⁰].
4. The unrounded payment is about $405.53, so the monthly payment rounded to cents is $405.53.
5. Total interest using that rounded payment is $405.53 × 60 − $20,000 = $4,331.80.

Assumptions

• The interest rate is fixed for the full term.
• Payments are equal, monthly, and made at the end of each payment period.
• Interest accrues monthly at APR divided by 12.
• Fees, taxes, insurance, PMI, and escrow are not included unless separately represented in the loan amount or payment.

How Amortization Works

Amortization is the process of repaying a debt through scheduled payments. Each payment covers interest accrued since the preceding payment, and the remaining amount reduces principal. Early in a standard fixed-payment schedule, the outstanding balance is larger, so more of the payment goes to interest. As principal declines, the interest portion generally shrinks and more of each payment reduces the balance.

The payment formula finds one level payment that, under its assumptions, brings the balance to zero after n payments. It does not mean the same dollar amount of interest is paid each month. Interest is recalculated on the remaining balance each period.

When to Use the Formula and Read the Results

Use the formula to estimate the scheduled payment and interest for a fully amortizing, fixed-rate loan with regular monthly installments. It can help explain how the amount borrowed, APR, and term shape the payment. Enter the loan’s actual principal, quoted APR, and repayment term when those figures are available; the example values are only calculator defaults.

A longer term generally spreads repayment across more installments: the scheduled payment is lower in many cases, but more months can mean more total interest. A shorter term has fewer payments, while the monthly payment is typically higher. These are mathematical effects under the same principal and rate, not a recommendation about what term to use.

Total interest here is an estimate for the stated schedule. It is not the same as APR: APR is an annualized rate measure, while total interest is a dollar amount accumulated across the assumed payments. A lender’s APR calculation or disclosure may account for details not present in this basic formula.

Common Mistakes and Formula Limits

• Using the wrong rate units: The formula needs a monthly decimal rate. For 8% APR, use 0.08 ÷ 12, equivalently 8 ÷ 1,200—not 8 as the monthly rate.
• Mixing years and months: A five-year monthly loan has 60 payments, not five. Match the periodic rate to the payment frequency.
• Rounding too early: Keep precision in intermediate calculations and round the payment only to cents. Small rounding differences can affect totals or the final payoff amount.
• Assuming every loan is fixed and fully amortizing: Variable rates, interest-only periods, balloon payments, irregular payment dates, and fees can change the schedule or make this formula unsuitable.
• Confusing payment with full housing or borrowing cost: The result excludes charges such as taxes, insurance, escrow, and lender fees unless they are explicitly included elsewhere.

For a more detailed schedule, calculate each month’s interest as the outstanding balance multiplied by the monthly rate, subtract that interest from the payment to find principal repaid, and subtract principal repaid from the balance. Repeat for each payment period. The last payment may need adjustment to clear the remaining balance exactly.

Frequently Asked Questions

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

How do I calculate a monthly loan payment from APR?
Convert APR to a monthly decimal by dividing the percentage by 1,200, convert the term to months, then use the payment formula. For example, 8% APR becomes 0.08 ÷ 12 per month.

Does an amortization payment include interest?
Yes. Each scheduled payment includes interest for the period and a principal repayment amount. The split changes over time as the balance falls.

Why is my lender’s payment different?
The lender may use a different payment schedule, APR or rate convention, rounding method, first-payment date, loan amount, or included charges. Check the loan disclosure and compare the figures and assumptions.

Does the formula work for a variable-rate loan?
Not as a single fixed schedule when the rate changes. The formula assumes one constant rate for the entire term; a rate change can alter future interest and payments.

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

Related

• 30-Year Loan Amortization Schedule Calculator
• 10-Year Business Loan Amortization Schedule Calculator
• 10 Year Loan Amortization Schedule Calculator

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