The formula
M = P · r(1+r)n / ((1+r)n − 1)
- P — principal (amount borrowed)
- r — monthly rate = annual rate ÷ 12, in decimal
- n — number of monthly payments = years × 12
Worked example: $300,000 at 7% for 30 years
| Step | Computation | Result |
|---|---|---|
| 1. Monthly rate | 0.07 ÷ 12 | 0.0058333 |
| 2. Payment count | 30 × 12 | 360 |
| 3. Growth factor | (1.0058333)360 | 8.1165 |
| 4. Numerator | 300,000 × 0.0058333 × 8.1165 | 14,203.87 |
| 5. Denominator | 8.1165 − 1 | 7.1165 |
| 6. Payment | 14,203.87 ÷ 7.1165 | $1,995.91 |
Monthly principal & interest: $1,995.91. Over 360 payments that's $718,527 repaid — $418,527 of it interest.
Calculator tip: the exponent is the step people fumble. On any scientific calculator: 1.0058333, then xy, then 360. On a phone, rotate to landscape for the xy key.
Check your arithmetic against the full amortization engine.Mortgage calculator →
Where the first payment goes
Interest for any month is simply balance × r. Month one: 300,000 × 0.0058333 = $1,750.00 of interest — leaving only $245.91 of your $1,995.91 to reduce the balance. Month two charges interest on the slightly smaller balance, so a few more cents hit principal. Repeat 360 times: that slow crossover is the whole story of compounding, and why extra payments early in the loan punch so far above their weight.
Sanity checks for your result
- A rough bound: at 7%/30yr, payments run ≈ $6.65 per $1,000 borrowed. 300 × 6.65 ≈ $1,996 ✓
- Your P&I should exceed month-one interest (P × r) — if not, the loan never amortizes.
- Cross-check on the loan calculator; the same formula runs every fixed loan.
Run your own numbers
FAQ
What does each symbol in the formula mean?
M is the monthly payment, P the amount borrowed, r the monthly interest rate (annual rate ÷ 12, as a decimal), and n the total number of monthly payments (years × 12).
Why divide the annual rate by 12?
Interest accrues monthly on a mortgage, so the annual rate is split into 12 periodic rates. 7% a year becomes 0.07 ÷ 12 ≈ 0.005833 per month.
Does this formula include taxes and insurance?
No — it produces principal and interest (P&I) only. Lenders add property tax, homeowners insurance, and any PMI or HOA on top to reach your full monthly bill.
Why does (1+r)^n appear twice?
It's the compound growth factor over the whole term. The numerator scales the payment to cover compounded interest; the denominator normalizes so exactly n payments retire the balance to zero.
Can I use the same formula for car or personal loans?
Yes — any fixed-rate, fully amortizing loan follows it. Only P, r, and n change. Try it against our loan calculator to confirm your arithmetic.
Sources
Primary references used for the figures and rules on this page.
- What is included in a monthly mortgage payment? — Consumer Financial Protection Bureau
- Loan Estimate explainer — Consumer Financial Protection Bureau