The formula and what each number means
You can calculate a mortgage payment using the standard amortization formula, which accounts for the loan amount, interest rate, and loan term. The formula is:
M = P × [r(1 + r)^n] / [(1 + r)^n − 1]
Here is what each letter represents:
- M is your monthly payment (principal plus interest only, not taxes or insurance).
- P is the principal — the amount you borrowed.
- r is the monthly interest rate, which is your annual rate divided by 12.
- n is the total number of monthly payments over the life of the loan.
The formula works because it spreads your interest across the loan term in a way that keeps your payment the same every month, even though the split between principal and interest changes over time.
Key Takeaways
- Convert your annual interest rate to a monthly rate by dividing by 12, and convert it to decimal form (5% becomes 0.05).
- The total number of payments is your loan term in years multiplied by 12 (a 30-year mortgage is 360 payments).
- The formula calculates principal and interest only; property taxes, homeowners insurance, and HOA fees are separate.
- A calculator or spreadsheet makes the math much faster, but working through the formula by hand shows you exactly how the payment is built.
Converting your interest rate to a monthly decimal
The interest rate in your loan documents is annual. To use it in the formula, you must convert it to a monthly rate and express it as a decimal.
Take your annual rate and divide by 12. If your rate is 5.5% per year, divide 5.5 by 12 to get 0.458333. Then convert that to decimal form by dividing by 100: 0.458333 ÷ 100 = 0.00458333. This is your monthly rate r.
Keep all decimal places during calculation — rounding too early introduces error. If you are working by hand, write down at least four decimal places for the monthly rate.
Finding your total number of payments
Multiply your loan term in years by 12 to get the total number of monthly payments. A 30-year mortgage has 360 payments. A 15-year mortgage has 180 payments. A 20-year mortgage has 240 payments.
This number is n in the formula. It stays the same for every calculation you do with that loan.
Working through the formula step by step
Once you have P (principal), r (monthly rate as a decimal), and n (number of payments), follow these steps in order:
- Add 1 to your monthly rate: 1 + r
- Raise that result to the power of n: (1 + r)^n
- Multiply r by the result from step 2: r × (1 + r)^n
- Subtract 1 from the result of step 2: (1 + r)^n − 1
- Divide the result of step 3 by the result of step 4
- Multiply by P to get your monthly payment
Example: You borrow $300,000 at 5.5% for 30 years. Your monthly rate is 0.00458333, and n is 360.
- 1 + 0.00458333 = 1.00458333
- (1.00458333)^360 = 5.5160 (approximately)
- 0.00458333 × 5.5160 = 0.025277
- 5.5160 − 1 = 4.5160
- 0.025277 ÷ 4.5160 = 0.005596
- $300,000 × 0.005596 = $1,678.57
Your monthly payment for principal and interest would be approximately $1,678.57.
What this payment does and does not include
The number you calculate using this formula covers only principal and interest. It does not include property taxes, homeowners insurance, or mortgage insurance (PMI), which are often bundled into your total monthly payment.
Your lender may also require you to pay an escrow amount each month for taxes and insurance. That amount varies by location and your insurance choices, so it is not part of the formula. Your loan documents will show your total monthly payment, which includes all of these.
Why you might calculate this by hand
Most people use a calculator or spreadsheet because the exponent step (raising 1 + r to the power of n) is tedious to do by hand. But working through the formula yourself shows you exactly how your payment is built and why it stays the same every month even though the split between principal and interest changes.
If you are comparing loan offers, you can use this formula to verify the payment amount your lender quoted. If the numbers do not match after you account for taxes and insurance, ask your lender to explain the difference.
For most daily use, a mortgage calculator or a spreadsheet with the formula built in is faster and less error-prone. But the formula itself is the same whether you calculate it by hand, on a calculator, or in software.
Frequently Asked Questions
Do I need to calculate the payment myself, or will my lender give me the number?
Your lender will provide the payment amount in your loan estimate and closing documents. You do not need to calculate it yourself to get a mortgage. Calculating it by hand is useful if you want to verify the number, compare different loan offers, or understand how the payment was built.
What if I want to know how much of my payment goes to principal versus interest?
That split changes every month. In the first payment, most of your money goes to interest. Over time, more goes to principal. You can find this breakdown in your amortization schedule, which your lender provides or which you can generate using a spreadsheet. The formula above does not break down that split — it only calculates the total payment.
Can I use this formula for a loan other than a mortgage?
Yes. The formula works for any fixed-rate loan where you make equal monthly payments: car loans, personal loans, student loans. You just need the principal amount, the annual interest rate, and the loan term in years. Convert the rate and term the same way.
What if my interest rate changes during the loan?
This formula only works for fixed-rate loans. If you have an adjustable-rate mortgage (ARM), your rate changes at set intervals, and your payment recalculates each time. You would use this formula to calculate the payment for each rate period separately, not for the entire loan at once.
Why does the formula have (1 + r)^n in both the numerator and denominator?
That structure is what makes the payment equal every month. The numerator accounts for how much interest compounds over the life of the loan. The denominator adjusts for the total number of payments. Together, they create a payment amount that covers all the interest and principal by the end of the term.