Introduction
Every fixed-rate mortgage payment is calculated using the same compounded-interest formula. Understanding it lets you estimate payments in seconds and verify any lender quote.
The Formula
$$ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} $$
Variables:
- M = monthly principal + interest payment
- P = loan amount
- r = monthly interest rate (annual rate ÷ 12)
- n = number of monthly payments
When the rate is 0%, the formula collapses to M = P ÷ n.
Why It Matters
- Same loan amount + different rate = vastly different payments
- Knowing the math lets you spot mispriced offers
- Helps you decide between 15- and 30-year terms
Step-by-Step Calculation
Loan: $300,000 at 7% for 30 years.
- Monthly rate (r) = 0.07 ÷ 12 = 0.0058333
- Total payments (n) = 30 × 12 = 360
- (1 + r)ⁿ = (1.0058333)³⁶⁰ ≈ 8.1165
- Numerator = 0.0058333 × 8.1165 ≈ 0.047346
- Denominator = 8.1165 − 1 = 7.1165
- Payment factor = 0.047346 ÷ 7.1165 ≈ 0.006653
- Monthly payment = $300,000 × 0.006653 ≈ $1,995.91
Amortization
Each payment splits between interest and principal:
| Month | Payment | Interest | Principal | Balance |
|---|---|---|---|---|
| 1 | $1,995.91 | $1,750.00 | $245.91 | $299,754.09 |
| 12 | $1,995.91 | $1,733.59 | $262.32 | $296,887.21 |
| 180 | $1,995.91 | $1,255.51 | $740.40 | $214,610.36 |
| 360 | $1,995.91 | $11.57 | $1,984.34 | $0.00 |
Early payments are mostly interest; later payments are mostly principal.
What Drives the Payment
| Variable | Effect when it rises |
|---|---|
| Principal | Payment rises proportionally |
| Interest rate | Payment rises non-linearly — small rate jumps hit hard |
| Term | Longer term lowers payment but raises total interest |
Worked Comparison
$300,000 at 7%:
- 30-year: $1,996/mo, total interest $418,527
- 15-year: $2,696/mo, total interest $185,367
The 15-year saves $233,000 in interest for ~$700 more per month.
Common Mistakes
- Forgetting taxes, insurance, and PMI when budgeting
- Using annual rate directly without dividing by 12
- Comparing rates across different terms without comparing total interest
- Ignoring closing costs in the effective rate
Conclusion
The formula is universal — only the inputs change. Model your scenario with the Mortgage Calculator below to see your full amortization schedule.