Key Takeaways
A mortgage payment is solved, not guessed — it comes from the amortization formula that makes the present value of all payments equal the loan amount. Interest dominates early: on a 30-year loan at today's rates, roughly two-thirds of year-one payments is interest. Rate beats amount: shaving 0.5% off the rate usually saves more than negotiating $10,000 off the price. And term length is the biggest lever on total interest you will ever pull.
The formula, and where it comes from
Every fixed-rate mortgage payment comes from one equation:
M = L × c × (1+c)n ÷ ((1+c)n − 1)
where L is the loan principal, c is the monthly interest rate (annual rate ÷ 12, as a decimal), and n is the number of monthly payments (years × 12).
The logic is cleaner than it looks. Each month, interest accrues on the remaining balance, and whatever you pay above that interest retires principal. A payment is "correct" when the balance hits exactly zero after the last one. Work that requirement backwards — sum the discounted value of every payment and set it equal to L — and the closed form above pops out. This is the same present-value annuity formula taught in finance courses; banks did not invent it, they just industrialised it.
Two practical consequences fall straight out of the algebra. First, payment is not proportional to rate: at higher rates the (1+c)n terms compound, so payment rises faster than linear. Second, because n is in the exponent, stretching the term lowers payment sub-linearly while multiplying total interest — the cheapest-looking option on a monthly basis is usually the most expensive in total.
Worked example: $400,000 at 6.5% for 30 years
L = 400,000. Monthly rate c = 0.065 ÷ 12 = 0.0054167. n = 360.
(1+c)n = 1.0054167360 ≈ 7.0336. So:
M = 400,000 × 0.0054167 × 7.0336 ÷ (7.0336 − 1) = 400,000 × 0.038108 ÷ 6.0336… compute it directly: M ≈ $2,528.27 principal and interest.
Now the part that surprises first-time buyers. Month one interest is L × c = 400,000 × 0.0054167 = $2,166.67. Of that first $2,528 payment, only about $361 touches principal. The crossover — when principal overtakes interest — lands around year 21 on this loan. Total paid over 30 years: about $910,000, of which $510,000 is interest: more than the house's financed price.
Run the same loan at 6.0%: M ≈ $2,398.20. That half-point rate cut saves $130 per month and about $46,800 over the term — compare that to haggling a $10,000 price cut, which saves only about $63/month on the same loan. Rate shopping pays.
What the formula deliberately leaves out
M above is principal and interest only. The payment your servicer quotes (PITI) adds property taxes (annual bill ÷ 12), homeowner's insurance, and PMI if your down payment is under 20% — typically 0.4–1.5% of the loan per year until you reach 20% equity. On our $400k example with 10% down, PMI at 0.8% adds roughly $267/month. None of these change the amortization math; they ride on top of it.
Extra principal payments, by contrast, change everything downstream: an extra $200/month on this loan cuts roughly 5.5 years and $80,000+ of interest, because every early dollar retired stops accruing for decades. That is why biweekly payment plans work — you make 13 full payments a year without feeling a single big one.
Try it yourself
The Mortgage Calculator implements exactly this formula and adds taxes, insurance, PMI and HOA on top, with the full year-by-year amortization table. The Amortization Calculator shows the principal/interest split for every single payment, and Mortgage Payoff quantifies what extra payments buy you. For the affordability side of the same decision, read how much house you can afford.