Your monthly payment isn't an estimate or a negotiation — it's one formula, identical at every lender. Here it is, taken apart.
The formula
payment = P · r(1+r)ⁿ ÷ ((1+r)ⁿ − 1)
where P is the principal, r the monthly rate (annual ÷
12 ÷ 100), and n the number of monthly payments. It answers a precise question:
what fixed payment makes the balance hit exactly zero on payment n, given that interest
accrues on the remaining balance each month?
By hand: $300,000 at 6% for 30 years
- Monthly rate: 6 ÷ 12 ÷ 100 = 0.005; payments: 30 × 12 = 360.
- Growth factor: (1.005)³⁶⁰ ≈ 6.0226 — what $1 becomes after 360 months of compounding at 0.5%.
- Numerator: 0.005 × 6.0226 = 0.030113. Denominator: 6.0226 − 1 = 5.0226.
- Payment: 300,000 × 0.030113 ÷ 5.0226 ≈ $1,799.
Multiply out: 360 payments × $1,799 ≈ $647,514, of which $300,000 is your money coming back and $347,515 is interest.
Where the intuition lives
The denominator (1+r)ⁿ−1 is the future value of a stream of $1 payments; the numerator scales it to the loan. As n grows, the payment falls — but ever more slowly, while total interest grows relentlessly. That asymmetry is why stretching 15→30 years cuts the payment by only ~35% while roughly tripling the interest.
The fine print lenders apply
Real servicers compute in cents with explicit rounding (each month's interest rounded to the cent; the final payment absorbing the residual), which is why your last payment is slightly odd-sized. Two same-rate quotes can also differ via fees buried in APR — the rate drives this formula; the APR describes the whole cost. Every calculator on this site uses exactly this formula with documented rounding — see the methodology page, or test it against the total interest calculator.
Where the formula comes from
It's a geometric series in disguise. Each payment, discounted back to today at the
monthly rate, must sum to exactly the principal: P = pmt·(1/(1+r) + 1/(1+r)² + … +
1/(1+r)ⁿ). The bracket is a geometric series summing to (1−(1+r)⁻ⁿ)/r;
solve for pmt and you get the formula above. Nothing mystical — just "the present value of
your payments equals the loan."
Edge cases worth knowing
At exactly 0% the formula divides by zero; the payment is simply P ÷ n (our engine special-cases this). For very long terms, the payment approaches P × r — pure interest, the balance never meaningfully falling, which is the mathematical skeleton of the credit-card minimum trap. And compounding convention matters at the margins: US mortgages compound monthly, Canadian mortgages semi-annually, so the same quoted rate produces slightly different payments across the border.
Why this is worth an hour of your life
Once you can run this formula, every loan pitch becomes checkable: a dealer's "just $50 more a month" resolves into its actual rate and term; a quoted payment that doesn't match the formula means fees are hiding in the balance. Put any quote into the calculator — if the numbers disagree with the paperwork, ask why before signing, not after.
Verify it yourself
Take any real quote — yours, a dealer's, a refinance offer — and run its three numbers through the formula or the calculator. Agreement to the cent means the quote is clean. Disagreement means something extra is in the balance or the rate isn't what was said. Either way, you'll know before signing — which is precisely the position lenders assume you won't be in.
From payment to schedule
The formula gives the payment; iterating the two-step rule (interest on balance, then principal reduction) gives the entire schedule, month by month, with no further mathematics. That iteration is all our schedule generator does — 360 small subtractions, performed exactly, which is also why its totals reconcile to the cent with the closed-form payment above.