Amortization Equation With Extra Payments

This calculator estimates how a fixed-rate loan balance changes when you make scheduled monthly payments plus a fixed extra amount each month. Enter the loan amount, APR, term, and extra payment to see the payoff time, total interest, and interest saved compared with making only the scheduled payment.

Formula

The standard amortization equation calculates a level monthly payment. With an extra payment, the recurring amount applied to the loan is that scheduled payment plus the extra amount:

r = APR / 1200
n = 12 × years
PMT = P × r / [1 − (1 + r)⁻ⁿ]
M = PMT + E
k = ceil[−ln(1 − rP/M) / ln(1 + r)]

P is the starting principal in dollars; APR is the annual percentage rate entered as a percentage; r is the monthly rate as a decimal; n is the original number of monthly payments; PMT is the scheduled monthly payment in dollars; E is the additional monthly payment in dollars; M is the combined monthly amount; and k is the estimated number of months to repay. The payoff-time expression assumes the same combined amount each month, including a final payment that may be smaller. Here, the scheduled payment is calculated without rounding to cents; real billing schedules may differ slightly.

The remaining balance after a particular payment is another useful form of the amortization equation:

B_k = P(1 + r)^k − M[(1 + r)^k − 1] / r

In a real payoff schedule, the final payment is capped at the balance due rather than exceeding it. The calculator’s interest estimate uses combined monthly amount multiplied by rounded-up payoff months, less principal. It is a simplified estimate because the last payment is normally smaller; see the worked schedule and limitations below.

Worked Example

Using the calculator’s defaults: principal is $20,000, APR is 8%, term is 5 years, and the extra monthly payment is $100.

• Convert the annual APR to a monthly decimal rate: r = 8 / 1200 = 0.0066667.
• Convert the term to scheduled payments: n = 5 × 12 = 60 months.
• Calculate the scheduled payment: PMT = 20,000 × 0.0066667 / [1 − (1.0066667)⁻⁶⁰] ≈ $405.53.
• Add the extra amount: M = $405.53 + $100 = $505.53 per month.
• Calculate and round up the repayment time: k ≈ 45.5, so the estimate is 46 months.
• Estimate interest using the calculator’s displayed method: $505.53 × 46 − $20,000 = $3,254.38.
• Scheduled-payment interest is $405.53 × 60 − $20,000 = $4,331.80; estimated interest saved is $4,331.80 − $3,254.38 = $1,077.42.

The interest and savings figures above use the calculator’s rounded monthly payment and rounded-up month count. They are estimates, not a lender’s exact payoff schedule: actual final-payment size and interest can change the totals.

Assumptions

• The interest rate remains fixed throughout repayment.
• Payments occur monthly, with the extra amount paid alongside each scheduled payment.
• Interest is modelled monthly at APR divided by 12.
• Fees, taxes, insurance, PMI, escrow, late charges, and changes in payment timing are not included.

How Extra Payments Change Amortization

Each payment first covers interest accrued for the period; the remainder reduces principal. Because interest in later periods is calculated on the remaining balance, reducing principal sooner generally reduces the interest that accrues afterward under the stated fixed-rate assumptions. An extra payment does not change the original contract’s scheduled payment in this model; it shortens the estimated payoff period.

To build a month-by-month schedule, start with the opening balance, multiply it by the monthly rate to estimate that month’s interest, and subtract interest from the payment. Apply the extra amount to principal as well, then carry the resulting balance into the next month. Repeat until the balance reaches zero, adjusting the last payment to the amount actually due. A spreadsheet can make this process clearer when extra payments vary over time.

When to Use This Estimate

This equation is useful for understanding the mathematical effect of a consistent extra monthly payment on a fixed-rate, fully amortizing loan. It can help illustrate a payoff timeline or compare modeled payment scenarios using the same starting balance, rate, and term. Enter the APR and payment terms from the relevant loan disclosure; defaults are illustrative inputs, not a claim about available rates or costs.

The estimate is less suited to loans with variable rates, irregular payment dates, skipped or changing payments, interest-only periods, balloon payments, or fees added to the balance. Those features require additional inputs and a more detailed schedule. For a one-time extra payment, the balance equation can be applied at the month it occurs, then the schedule recalculated from that new balance.

Common Mistakes and Limitations

• Confusing APR with the monthly rate: APR entered as 8 means 8% per year; the formula converts it to 8 / 1200 per month. Do not enter 0.08 in the APR field.
• Assuming every extra payment is applied to principal: Confirm how a servicer applies additional money and whether it is treated as principal reduction or as a future scheduled payment.
• Ignoring timing: Interest may be calculated based on actual payment dates or daily balances. A monthly approximation will not reproduce every lender’s accounting method.
• Expecting exact cents: Payment rounding, daily accrual, posting dates, and a smaller final payment affect the precise payoff amount and date.

Extra payments may also be restricted or handled under terms that are not represented by this equation. Check the loan documents or ask the servicer how additional amounts are credited and how to request a payoff figure. The calculator does not determine fees, tax effects, or whether a particular payment strategy is appropriate.

Frequently Asked Questions

What is the amortization equation with extra payments?
Calculate the scheduled payment from principal, monthly rate, and term, then add the extra amount to get the modeled monthly payment. The remaining-balance equation is B_k = P(1 + r)^k − M[(1 + r)^k − 1] / r.

Do extra payments reduce the monthly payment?
Not necessarily. In this calculation, the scheduled payment stays the same and the added amount reduces the balance faster, shortening the estimated repayment period. A lender’s process or a loan recast may work differently.

Does this formula work for a one-time extra payment?
Yes, as part of a month-by-month schedule: apply the one-time amount to the balance in the period it is made, then calculate future interest and payments from the adjusted balance. The fixed-extra calculator here assumes the same additional amount every month.

Why can my lender’s payoff amount differ?
The estimate simplifies payment timing and interest accrual, and it excludes fees and other loan-specific amounts. Lenders may calculate interest using actual dates and balances, and the final payment is usually adjusted to the exact amount due.

Is the interest saved exact?
No. It is a mathematical estimate based on the entered APR, term, and extra payment. The exact difference depends on the lender’s payment application, accrual method, and final payoff amount.

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

Related

• LIC HFL EMI Calculator: Estimate Your Monthly Payment
• EMI Calculator With Down Payment: Monthly Payment
• EMI Calculator With Extra Payment

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
in extra | Extra payment each month | $ | 100 | 0 | 100000000
out months | Payoff time with extra payments (months) | months | ceil(round(-ln(1 - (apr / 1200) * p / (p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) + extra)) / ln(1 + apr / 1200), 6)) | 0
out int | Interest with extra payments | $ | (p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) + extra) * months - p | 2
out saved | Interest saved vs scheduled payments | $ | pmt * yrs * 12 - p - int | 2