The Amortization Formula
The standard mortgage payment formula is: M = P × [r(1+r)^n] / [(1+r)^n − 1] where M is your monthly payment, P is the principal (loan amount), r is the monthly interest rate (annual rate ÷ 12), and n is the total number of payments (years × 12). This formula was not invented for mortgages — it is the standard compound interest annuity formula applied to loans. Every fixed-rate mortgage payment you will ever make comes from this equation.
P = $350,000. Annual rate = 7%, so monthly rate r = 0.07 ÷ 12 = 0.005833. n = 30 × 12 = 360 payments. M = 350,000 × [0.005833 × (1.005833)^360] ÷ [(1.005833)^360 − 1] = 350,000 × [0.005833 × 8.1165] ÷ [8.1165 − 1] = 350,000 × 0.04734 ÷ 7.1165 = $2,329/month.
Why Each Variable Behaves Differently
The formula has three variables: principal (P), monthly rate (r), and number of payments (n). Each one affects your payment and total interest differently. Principal changes the result linearly — double the loan, double the payment. The rate appears in both the numerator and the exponent, creating a nonlinear effect that amplifies as rates rise. The term (n) affects both the payment amount and the total interest paid in complex ways because of how the exponent interacts with both.
Sensitivity analysis — how each formula variable affects payment and total interest
| Variable Change | Example | Payment Impact | Total Interest Impact |
|---|---|---|---|
| Principal +$50,000 | $350K → $400K, 7%, 30yr | +$333/month | +$119,880 |
| Rate +0.5% | $350K, 7% → 7.5%, 30yr | +$118/month | +$42,480 |
| Rate +1.0% | $350K, 7% → 8%, 30yr | +$239/month | +$86,040 |
| Term -5 years | $350K, 7%, 30yr → 25yr | +$178/month | -$71,400 saved |
| Term -15 years | $350K, 7%, 30yr → 15yr | +$691/month | -$295,440 saved |
Why Early Payments Are Mostly Interest
The amortization formula front-loads interest — not because of lender preference, but because of mathematics. Each month's interest is calculated as the remaining balance multiplied by the monthly rate. When the balance is $349,713 in Month 2, the interest charge is 0.005833 × $349,713 = $2,040. Your $2,329 payment covers that $2,040 first, with only $289 remaining to reduce your balance. This is not a trick — it is arithmetic. As the balance falls (very slowly at first), each month's interest shrinks and principal grows.
Full amortization progression — $350,000 at 7%, 30-year fixed
| Payment # | Monthly Payment | Interest Portion | Principal Portion | Balance After |
|---|---|---|---|---|
| 1 | $2,329 | $2,042 (87.7%) | $287 (12.3%) | $349,713 |
| 12 | $2,329 | $2,040 | $289 | $349,399 |
| 60 (Year 5) | $2,329 | $2,006 | $323 | $343,456 |
| 120 (Year 10) | $2,329 | $1,953 | $376 | $334,174 |
| 180 (Year 15) | $2,329 | $1,886 | $443 | $322,226 |
| 240 (Year 20) | $2,329 | $1,790 | $539 | $306,042 |
| 300 (Year 25) | $2,329 | $1,651 | $678 | $283,074 |
| 360 (final) | $2,329 | $14 | $2,315 | $0 |
In Year 1 of a $350,000 mortgage at 7%, you make 12 payments totaling $27,948 — but only $3,444 reduces your balance. The other $24,504 goes to interest. After 5 years of perfect payments, your balance is still $343,456. This front-loading is why refinancing into a new 30-year loan after 10 years resets your equity building to near-zero.
How Rate Changes Affect the Formula Nonlinearly
Because the rate appears in the exponent of the formula, rate changes have a nonlinear effect. The jump from 6% to 7% costs more in total interest than the jump from 5% to 6%, even though both are 1 percentage point increases. This accelerating cost makes rate shopping increasingly valuable at higher rate levels.
Full rate sensitivity table — $350,000 loan, 30-year fixed
| Rate | Monthly Payment ($350K, 30yr) | Total Paid | Total Interest |
|---|---|---|---|
| 5.00% | $1,879 | $676,440 | $326,440 |
| 5.50% | $1,987 | $714,920 | $364,920 |
| 6.00% | $2,098 | $755,280 | $405,280 |
| 6.50% | $2,212 | $796,320 | $446,320 |
| 7.00% | $2,329 | $838,440 | $488,440 |
| 7.50% | $2,447 | $880,920 | $530,920 |
| 8.00% | $2,568 | $924,480 | $574,480 |
| 8.50% | $2,692 | $969,120 | $619,120 |
How Refinancing Resets the Formula
When you refinance, the formula recalculates from scratch with your current balance as the new P, your new rate as r, and your chosen new term as n. If you are 10 years into a $350,000 30-year mortgage and refinance your $334,000 remaining balance into a new 30-year, the n resets to 360 — meaning you have now committed to 40 total years of payments. The monthly payment drops (because your rate may be lower and you have reset the amortization clock), but total interest paid often increases unless the rate improvement is substantial.
When refinancing, consider keeping approximately the same monthly payment by refinancing into a shorter term. If you are 10 years into a 30-year and refinance into a 20-year at a lower rate, your payment stays similar but you eliminate 10 years of future interest and pay off on the original timeline.
Calculating How Extra Payments Affect the Formula
When you make extra principal payments, you effectively reduce P in all future formula calculations — but the payment M stays the same (because your lender does not recalculate unless you recast). The result: a larger share of each subsequent payment attacks principal, accelerating the formula's convergence to $0 balance. This is why extra payments are more powerful than they initially appear.
See the Formula in Action
Enter your loan details and get the full amortization schedule — watch exactly how your balance falls over time.