Loan Payment Formula: A Fixed-Rate Reference

The model being described

A fully amortizing fixed-rate loan is repaid through scheduled payments that cover interest and reduce principal until the balance reaches zero. In the standard monthly model, the rate and regular payment remain constant, payments occur at the end of each month, and there are no additional advances or missed payments. The formula describes this arrangement, not every product marketed as a loan.

Let P be the opening principal, i the monthly decimal interest rate, n the total number of monthly payments, and M the payment. The equation is M = Pi/[1−(1+i)^−n]. If i is zero, use M = P/n instead. That separate zero-rate expression avoids an indeterminate fraction and states the intuitive result: divide the amount borrowed evenly across the payments.

Define the inputs precisely

For a nominal annual percentage R in a monthly model, i = R/1200. For a term of y years, n = 12y. A displayed annual percentage of 6 therefore gives i = 0.005. Entering 6 directly as a monthly decimal rate would be a severe unit error. An effective annual rate requires a different conversion, and a fee-inclusive APR should not be substituted without understanding its definition.

Principal is the amount financed. It may differ from a purchase price because of a down payment or financed charges. Do not deduct a down payment twice if the lender already gives a net loan amount. The mortgage basics reference illustrates the relationship between price, deposit and principal, while the loan calculator works from the loan amount itself.

Derive the balance recurrence

After one period, interest adds Pi to the balance and payment M is subtracted, leaving B1 = P(1+i)−M. Repeating gives B2 = P(1+i)²−M(1+i)−M. After n periods, the balance is P(1+i)^n−M[(1+i)^n−1]/i for nonzero i. Setting that final balance equal to zero and solving for M produces the payment formula.

This derivation explains the assumptions rather than merely presenting a button result. The rate acts on the opening balance each period, and the payment arrives afterward. If payments arrive at another time or balances change between scheduled dates, the recurrence must change. The simple interest guide describes interval-based interest and why a fixed original principal is insufficient for a repaying balance.

Worked monthly example

Borrow 12,000 for 24 months at a nominal annual 6%. With i = 0.005, the formula gives M approximately 531.8473, normally displayed as 531.85. The first interest amount is 60. Subtracting it from the unrounded payment gives about 471.8473 of principal reduction, leaving approximately 11,528.1527. The second interest amount is approximately 57.6408, so the second principal reduction is slightly larger.

Using the unrounded payment throughout, total scheduled payments are about 12,764.34 and total interest about 764.34. Multiplying a displayed two-decimal payment by 24 may differ by a few cents. A real lender may adjust the last payment to reconcile its rounding and accrual rules. An educational schedule should state that limitation rather than promise a cent-exact contractual payoff.

Read an amortization schedule

Each row normally identifies opening balance, interest, payment, principal reduction and closing balance. Interest equals opening balance times the periodic rate. Principal reduction equals payment minus interest. Closing balance equals opening balance minus principal reduction. Those identities let you check a row and connect it to the next row without recalculating the full closed-form expression.

For a positive fixed rate and a normally amortizing balance, the interest portion falls as principal is repaid. The CFPB explanation describes this changing split in a mortgage context. If a payment does not cover the period's interest, the balance behavior is different; do not assume the standard fully amortizing formula represents an interest-only or negatively amortizing arrangement.

Compare terms and total cost

Holding principal and rate constant, extending the term reduces the scheduled payment but increases the time over which interest accrues. Compare total payments and total interest as well as the monthly figure. A smaller payment is not itself evidence of a cheaper loan. The monthly payment article works through a two-year versus four-year comparison with the same initial balance.

Additional charges can also change the wider cost. For a mortgage, the principal-and-interest amount may exclude insurance and taxes. The CFPB payment breakdown makes that distinction explicit. Do not sum only M × n and call it every possible expense unless the scope truly excludes nothing else relevant.

When to use another calculation

An adjustable rate, a balloon balance, interest-only months, irregular payments or daily accrual requires an appropriate schedule. The fixed-payment formula can sometimes describe one segment, but not necessarily the entire contract. Extra payments also need explicit timing and allocation rules; the formula does not infer whether a lender shortens the term or recalculates the payment.

Use the result as an explanation or scenario, not as a credit decision or official statement. Verify actual terms with the provider. Simple versus compound interest helps interpret interest labels, but neither that comparison nor this reference replaces product documentation. The numerical examples are hypothetical and do not quote a currently available borrowing rate.

Verify the final balance identity

For a scheduled fixed payment M and a nonzero periodic rate i, the balance after k payments is Bk = P(1+i)^k − M[(1+i)^k−1]/i. Setting k = 0 recovers P because the payment-series term is zero. Setting k = n with the unrounded payment derived earlier produces zero, apart from numerical computation error. These endpoint checks test the recurrence's timing and exponent together.

At k = 1, the expression reduces to P(1+i)−M, matching the first-row calculation. If a purported schedule instead subtracts the payment before applying interest, it follows a different timing convention and will generally produce a different balance. Neither sequence should be silently substituted for the other when comparing the result with a monthly end-of-period model.

Display rounding introduces a practical distinction. A schedule that subtracts a rounded payment every month can finish with a small residual, while a symbolic calculation using the exact payment finishes at zero. A real schedule may adjust the final payment, round interest each period or use actual-day accrual. State which procedure the illustration uses. A discrepancy of a few cents deserves reconciliation, but it is not the same as a substantial balloon balance caused by an incorrect rate or payment count. This algebraic check verifies the stated model only; it does not establish the legally payable settlement amount for an actual account.