Mortgage Payment Formula
Reference for mortgage formula M = P[r(1+r)^n]/[(1+r)^n-1].
Covers amortization, total interest paid, and a 30-year vs 15-year fixed mortgage comparison.
The Formula
The mortgage payment formula calculates the fixed monthly payment for a fully amortizing loan. Each payment covers both interest and principal, with the interest portion decreasing over time.
Variables
| Symbol | Meaning |
|---|---|
| M | Monthly payment |
| P | Principal (loan amount) |
| r | Monthly interest rate (annual rate / 12) |
| n | Total number of payments (years × 12) |
Example 1
$300,000 mortgage at 6.5% annual rate for 30 years
r = 0.065 / 12 = 0.005417, n = 30 × 12 = 360
M = 300,000 × [0.005417 × (1.005417)³⁶⁰] / [(1.005417)³⁶⁰ - 1]
M = 300,000 × [0.005417 × 6.9916] / [6.9916 - 1]
= $1,896.20 per month (total paid: $682,632 — that is $382,632 in interest)
Example 2
Same $300,000 at 6.5% but for 15 years
r = 0.005417, n = 15 × 12 = 180
M = 300,000 × [0.005417 × (1.005417)¹⁸⁰] / [(1.005417)¹⁸⁰ - 1]
= $2,613.32 per month (total paid: $470,398 — saves $212,234 in interest vs 30-year)
When to Use It
Use the mortgage formula when:
- Estimating monthly payments before buying a home
- Comparing different loan terms (15 vs 30 years)
- Calculating how much total interest you will pay
- Determining the maximum loan you can afford based on a target payment
Key Notes
- Monthly rate r = annual rate / 12: A 6% annual rate becomes r = 0.06/12 = 0.005 per month. Never use the annual rate directly in the mortgage formula — always convert to the monthly rate first.
- Front-loaded interest (amortization): Early payments are mostly interest; late payments are mostly principal. In the first payment on a 30-year mortgage, the majority of the payment goes to interest. This reverses slowly over the loan's life.
- Extra principal payments have large effects: Paying even small extra amounts toward principal early in the loan reduces total interest significantly because subsequent interest is calculated on the lower remaining balance.
- Total interest calculation: Total interest paid = (M × n) − P, where M is the monthly payment, n is the total number of payments, and P is the original principal. For a 30-year loan, total paid can be 2× the principal.
- Fixed vs adjustable rate: The formula above applies to fixed-rate mortgages. Adjustable-rate mortgages (ARMs) recalculate M each adjustment period using the remaining balance and the new interest rate.