The mortgage amortization formula
A fixed-rate, fully amortizing mortgage is repaid using the formula M = P[r(1+r)^n] / [(1+r)^n - 1], where M is the fixed monthly payment, P is the original loan principal, r is the monthly interest rate (the annual rate divided by 12), and n is the total number of monthly payments over the loan term (360 for a standard 30-year mortgage). This formula produces a single, unchanging monthly payment amount for the life of a fixed-rate loan, even though the proportion of that payment allocated to interest versus principal changes every month.
The formula is derived so that the fixed payment amount exactly pays off both the principal and all accrued interest by the final scheduled payment, assuming no extra payments are made and the interest rate does not change. Each month, interest is calculated on the current remaining balance, and whatever portion of the fixed payment is not consumed by that month's interest is applied to reduce the principal balance used in the following month's calculation.
Worked example: $300,000 at 4.5% over 30 years
For a $300,000 loan at a 4.5% annual interest rate amortized over 30 years (360 monthly payments), the monthly interest rate is 4.5% / 12 = 0.375%, or 0.00375 as a decimal. Applying the amortization formula produces a fixed monthly payment of $1,520.06. In the very first month, interest accrues on the full $300,000 balance: $300,000 x 0.00375 = $1,125.00 in interest, leaving only $395.06 of the $1,520.06 payment to reduce the principal balance.
The table below shows cumulative interest paid, cumulative principal paid, and the remaining balance at several points across the 30-year term for this example. Over the full loan term, this mortgage generates $247,220.13 in total interest, bringing the total of all payments to $547,220.13 on the original $300,000 borrowed.
| Time elapsed | Cumulative interest paid | Cumulative principal paid | Remaining balance |
|---|---|---|---|
| 1 year (12 payments) | $13,400.99 | $4,839.68 | $295,160.32 |
| 5 years (60 payments) | $64,677.11 | $26,526.25 | $273,473.75 |
| 10 years (120 payments) | $122,675.02 | $59,731.70 | $240,268.30 |
| 15 years (180 payments) | $172,311.93 | $101,298.14 | $198,701.86 |
| 30 years (360 payments) | $247,220.13 | $300,000.00 | $0.00 |
Why early payments are interest-heavy
Because interest is calculated each month on the remaining loan balance, and the balance is largest at the start of the loan, the earliest payments contain the largest interest components and the smallest principal components of the entire loan term. As the balance gradually declines with each payment, the interest portion of the fixed payment shrinks and the principal portion grows, even though the total payment amount itself never changes on a fixed-rate loan.
In the $300,000 example above, the crossover point -- the month at which the principal portion of the payment first exceeds the interest portion -- does not occur until month 176, close to the 14.7-year mark of the 30-year term. This means that for roughly the first half of the loan's life, the majority of every single payment goes toward interest rather than building home equity through principal reduction, which is why paying down a mortgage early in its term tends to save proportionally more interest than making the same extra payment later in the term.
How extra payments affect an amortizing loan
Because the interest charged each month is calculated on the current balance, any extra payment applied directly to principal reduces the balance immediately, which in turn reduces every subsequent month's interest calculation for the remaining life of the loan. This compounding effect means that extra principal payments made early in a loan's term -- when the balance and monthly interest charge are largest -- generally produce greater total interest savings than the same extra payment made later in the term, when the balance has already been substantially reduced.
Lenders and mortgage servicers typically require that borrowers specify an extra payment be applied to principal rather than to future scheduled payments, since simply sending more money without this instruction may be applied differently. Borrowers considering extra payments should also confirm whether their loan has a prepayment penalty, though prepayment penalties are uncommon on standard fixed-rate residential mortgages originated in the United States in recent years.
The 28/36 rule for mortgage affordability
The 28/36 rule is a conventional mortgage underwriting guideline stating that a borrower's total housing costs -- principal, interest, property taxes and insurance, sometimes abbreviated PITI -- should generally not exceed 28% of gross monthly income (the front-end ratio), and that total monthly debt obligations, including housing costs plus other debts such as car loans, student loans and credit card minimum payments, should generally not exceed 36% of gross monthly income (the back-end ratio). These thresholds have long been used as reference points by conventional mortgage lenders when assessing how much a borrower can reasonably afford to borrow.
Individual lenders and loan programs apply their own specific debt-to-income (DTI) requirements, which can differ from the traditional 28/36 guideline; for example, many government-backed and conventional loan programs permit back-end DTI ratios above 36%, sometimes up to 43% to 50%, depending on other factors in a borrower's financial profile such as credit score, down payment size and cash reserves. The 28/36 rule remains a useful conservative reference point for estimating affordability even where actual underwriting guidelines are more flexible.
Reading an amortization schedule
A full amortization schedule lists every scheduled payment over the life of a loan, showing for each payment the interest charged, the principal applied, and the remaining balance afterward. Reviewing a full schedule -- rather than only the fixed monthly payment amount -- makes visible how slowly principal is reduced in the early years of a long-term, low-extra-payment mortgage, and how quickly the principal portion accelerates in the final years of the term.
Amortization schedules are also useful for evaluating refinancing decisions, since they show the remaining balance and the proportion of future payments still allocated to interest at any point in the loan. A borrower considering a refinance partway through a loan term can compare the remaining balance and interest trajectory of the current loan against the terms of a new loan to evaluate whether refinancing is likely to reduce total interest paid over the remaining time horizon.
Questions fréquentes
What is the formula for a mortgage payment?
The standard formula for a fixed-rate, fully amortizing mortgage payment is M = P[r(1+r)^n] / [(1+r)^n - 1], where M is the fixed monthly payment, P is the loan principal, r is the monthly interest rate (annual rate divided by 12), and n is the total number of monthly payments. For a $300,000 loan at 4.5% annual interest over 30 years (360 payments), this formula produces a fixed monthly payment of $1,520.06.
Why do early mortgage payments go mostly toward interest?
Interest on an amortizing loan is calculated each month on the current outstanding balance, which is at its largest at the very start of the loan. Because the balance declines only gradually, the interest portion of each payment starts large and shrinks slowly, while the principal portion starts small and grows over time. In a $300,000, 4.5%, 30-year example, the first monthly payment of $1,520.06 includes $1,125.00 in interest and only $395.06 in principal reduction.
What is the 28/36 rule in mortgage lending?
The 28/36 rule is a conventional guideline used by mortgage lenders stating that housing costs (principal, interest, property taxes and insurance) should generally not exceed 28% of a borrower's gross monthly income, and that total monthly debt payments, including housing costs, should generally not exceed 36% of gross monthly income. Individual loan programs may permit higher debt-to-income ratios depending on other factors in a borrower's financial profile, but the 28/36 rule remains a widely referenced conservative benchmark for affordability.
Does making extra principal payments really save money?
Yes. Because interest on an amortizing loan is calculated on the current outstanding balance, any extra payment applied directly to principal reduces that balance immediately, lowering every subsequent month's interest charge for the remainder of the loan. Extra payments made earlier in the loan term, when the balance is larger, generally produce greater total interest savings than the same extra payment made later in the term, when the balance -- and therefore the interest calculated on it -- has already been substantially reduced.
How much total interest is paid over the life of a 30-year mortgage?
The total interest paid depends on the loan amount and interest rate. For a $300,000 loan at 4.5% annual interest over 30 years, total interest across all 360 payments comes to $247,220.13, bringing total payments (principal plus interest) to $547,220.13, meaning the total interest paid is close to 82% of the original amount borrowed. Higher interest rates, longer loan terms, or larger loan amounts all increase total lifetime interest, while extra principal payments or a shorter loan term reduce it.
Références
- Consumer Financial Protection Bureau. "Explainer: How mortgage amortization works." Consumerfinance.gov.
- Consumer Financial Protection Bureau. "What is a debt-to-income ratio?" Consumerfinance.gov.
- Brueggeman WB, Fisher JD. Real Estate Finance and Investments. 15th ed. McGraw-Hill Education, 2019.
- Fannie Mae. "Selling Guide: Debt-to-Income Ratios." Fanniemae.com.