Finance & Investment
Loan Balance Formula After a Stated Number of Payments
Calculate the remaining balance after k equal payments, verify it against a schedule, and distinguish principal balance from a dated payoff quote.
Specify when the balance is measured
A remaining-loan-balance calculation needs a timing point. In this example, the balance is measured immediately after the kth regular monthly payment, with no additional accrued interest, fees or extra repayments. It differs from the balance immediately before that payment. The model assumes a fixed rate, equal payments at month ends and a loan that amortizes fully over its original payment count. A generic Loan Calculator estimates the scheduled payment, but its three original-loan inputs do not by themselves specify how many payments have already occurred.
Write the balance formula
Let P be original principal, i the monthly rate, A the unrounded monthly payment and k the number of payments made. For nonzero i, remaining balance is Bk=P×(1+i)^k−A×[((1+i)^k−1)/i]. The first term grows the original balance forward; the second subtracts the accumulated effect of the payments. Use the same rate and timing convention that generated A. If the rate is zero, the formula reduces to Bk=P−kA. Entering an annual percentage directly as i would make the calculation fundamentally wrong.
Work through a twelve-payment example
Assume original principal of 100,000, a nominal annual rate of 6% divided into monthly periods, and a five-year repayment term. The monthly rate is 0.005 and there are 60 scheduled payments. The unrounded payment is approximately 1,933.280153. After 12 regular payments, the formula gives remaining principal of approximately 82,319.683276, displayed as 82,319.68. This is not 100,000 minus twelve times the full payment, because part of each payment covers interest. The year-one principal reduction is approximately 17,680.32.
Verify against the amortization schedule
Enter principal 100000, rate 6 and term 5 in the Loan Amortization Calculator. The tool displays the first twelve monthly rows, so the month-12 balance directly checks this example. Month one begins with interest of 100,000×0.005=500. Subtracting that from the payment leaves approximately 1,433.280153 of principal reduction and a closing balance of 98,566.719847. Repeating the calculation from each new opening balance reaches the same month-12 result as the formula. This row-by-row reconciliation helps locate a misplaced payment or rounding convention.
Explain how rounded payments affect the result
The formula and local schedule use the unrounded payment internally, then round displayed currency values. A lender may instead collect a payment rounded to cents each month and adjust the final payment. If actual payments are 1,933.28, using that exact collected amount in the balance formula gives a slightly different result from using all digits. Neither should be silently substituted for the other when reconciling a statement. Preserve the actual payment amounts when accuracy to the currency's smallest unit matters, especially over a long term.
Separate principal balance from a payoff amount
A payoff quote for a particular date can include interest accrued since the last payment and applicable charges or adjustments. The immediately-after-payment principal balance does not include those items by definition. If the quoted payoff date is between scheduled payments, identify the contract's accrual method rather than automatically adding one full month of interest. A balance calculation also does not determine whether a prepayment charge applies. Obtain the dated payoff terms from the lender and use the formula to check its principal component within the stated assumptions.
Adapt the method when the schedule changes
An extra principal payment, a missed payment, a rate reset or an irregular payment date requires an updated cash-flow schedule. One approach is to calculate the balance at the change point, apply the change and continue with the revised terms. Do not use the original equal-payment formula unchanged after a different payment history and expect an exact match. The local amortization tool shows only its first twelve rows for longer loans; use the explicit balance formula or a complete worksheet for later checkpoints, retaining the original inputs and the timing of every adjustment.
Use the Loan Amortization Calculator
Source and scope
Related guides
Derive the fixed-payment formula from present value, check a twelve-month loan and trace how each payment reduces principal.
Calculate a balloon payment from principal, rate and regular payments, then distinguish the maturity date from the amortization schedule.
Separate loan maturity from the amortization period, calculate a commercial-property loan’s scheduled payment and estimate the remaining balance at maturity.