The formula that lenders use
A mortgage payment is calculated using a fixed formula that accounts for three things: how much you borrowed, the interest rate, and how many months you have to pay it back. The formula is called the amortization formula, and it produces the same number that a lender's computer produces. You can work through it with a calculator and paper.
The formula is: M = P × [r(1 + r)^n] / [(1 + r)^n − 1]
In this formula, M is your monthly payment, P is the principal (the amount borrowed), r is the monthly interest rate (the annual rate divided by 12), and n is the total number of payments (the loan term in years multiplied by 12). The ^ symbol means "to the power of"—you multiply a number by itself that many times.
This formula works for any fixed-rate mortgage, whether it is a 15-year loan, a 30-year loan, or any other term. The payment stays the same every month for the life of the loan.
Key Takeaways
- The amortization formula uses your loan amount, monthly interest rate, and total number of payments to calculate what you owe each month.
- You must convert the annual interest rate to a monthly rate by dividing by 12, and convert years to months by multiplying by 12.
- The exponent (the ^ symbol) means you multiply the base number by itself that many times—a 30-year loan requires raising numbers to the 360th power.
- Working through the formula step by step with a basic calculator takes about five minutes and produces the exact monthly payment amount.
- The monthly payment covers both principal and interest, but the split between them changes each month—early payments are mostly interest.
Breaking the formula into steps
The formula looks intimidating because it is all one line, but you can solve it by working through it in pieces. Start with a real example: a $300,000 loan at 6.5% annual interest over 30 years.
First, convert the annual interest rate to a monthly rate. Divide 6.5% by 12: 6.5 ÷ 100 = 0.065, then 0.065 ÷ 12 = 0.00542 (rounded). This is your monthly rate, or r.
Next, calculate the total number of payments. Multiply 30 years by 12: 30 × 12 = 360. This is n.
Now you have P = 300,000, r = 0.00542, and n = 360. The hardest part comes next: you need to calculate (1 + r)^n, which is (1.00542)^360. This means multiplying 1.00542 by itself 360 times. Use a scientific calculator or a spreadsheet for this step. The result is approximately 6.898.
Finishing the calculation
Once you have (1 + r)^n = 6.898, plug it into the rest of the formula. The numerator (top part) is r × (1 + r)^n, which is 0.00542 × 6.898 = 0.0374. The denominator (bottom part) is (1 + r)^n − 1, which is 6.898 − 1 = 5.898.
Divide the numerator by the denominator: 0.0374 ÷ 5.898 = 0.00634. Multiply this by the principal: 0.00634 × 300,000 = 1,902. Your monthly payment is approximately $1,902.
This payment covers both principal and interest. In the first month, most of it goes toward interest. As you pay down the loan, more of each payment goes toward principal. By the end of the loan, almost all of it is principal.
Why the exponent matters so much
The (1 + r)^n part of the formula is what makes the calculation work. It accounts for the fact that interest compounds—you pay interest on the interest you already owe. The longer the loan term, the higher this number gets, and the more total interest you pay.
For a 15-year loan, n = 180, so (1.00542)^180 ≈ 2.454. For a 30-year loan, n = 360, so (1.00542)^360 ≈ 6.898. The difference is huge. A longer loan spreads the payment over more months, which lowers your monthly payment but increases the total interest you pay over the life of the loan.
If you change the interest rate, the monthly rate r changes, and the whole calculation shifts. A higher rate makes (1 + r)^n larger, which increases your monthly payment. This is why even a small difference in interest rate—say, 6% versus 6.5%—produces a noticeably different monthly payment.
Common mistakes when working by hand
The most common error is forgetting to convert the annual rate to a monthly rate. If you use 6.5 instead of 0.00542, your answer will be wildly wrong. Always divide the annual percentage by 100 first (to get the decimal), then divide by 12.
The second common mistake is miscalculating the exponent. If you are raising a number to the 360th power, a small rounding error early on gets magnified. Use a calculator with a power function (usually labeled ^ or x^y) rather than multiplying by hand. Most scientific calculators and spreadsheets have this built in.
A third mistake is using the wrong loan amount. Make sure you are using the principal—the amount you actually borrowed—not the sale price of the home. If you put down 20%, your principal is 80% of the sale price.
When to use this formula versus a calculator
Calculating by hand teaches you how mortgages actually work and why small changes in rate or term produce big changes in payment. It is useful if you want to understand the mechanics or if you are comparing different loan scenarios and want to see the math yourself.
For practical purposes—figuring out what you can afford or comparing actual loan offers—an online mortgage calculator or a spreadsheet is faster and less error-prone. But if you understand the formula, you can spot when a lender's number seems wrong, and you know what to ask about.
A spreadsheet is a middle ground. You can set up the formula once in Excel or Google Sheets, plug in different numbers, and see the results when ready. This lets you run scenarios without doing the arithmetic yourself.
How the payment splits between principal and interest
Your monthly payment stays the same, but what you are paying for changes every month. In month one, you owe interest on the full loan amount. As you pay down the principal, the interest portion shrinks and the principal portion grows.
To find the interest portion of any payment, multiply the remaining balance by the monthly interest rate. In month one with a $300,000 loan at 0.00542 monthly rate, the interest is 300,000 × 0.00542 = $1,626. Your $1,902 payment minus $1,626 interest leaves $276 toward principal.
In month two, your balance is now $299,724. The interest is 299,724 × 0.00542 = $1,624. Now $278 goes to principal. This continues for 360 months. By month 360, almost the entire payment is principal because the balance is nearly zero.
Frequently Asked Questions
Do I need a scientific calculator to do this by hand?
Yes, you need a calculator with a power function (^ or x^y) to calculate (1 + r)^n. A basic four-function calculator will not work. A scientific calculator, a spreadsheet, or an online calculator with a power function will all work.
What if my interest rate changes during the loan?
This formula only works for fixed-rate mortgages, where the rate stays the same for the entire loan. If you have an adjustable-rate mortgage (ARM), the rate changes on a set schedule, and the payment recalculates at each change. You would need to recalculate the payment using the new rate and the remaining balance and term.
Why is my calculated payment slightly different from what the lender quoted?
Small differences come from rounding at each step. Lenders use more decimal places than you can reasonably track by hand. They also may include property taxes, insurance, and HOA fees in the quoted payment, which are not part of the mortgage formula itself. Ask the lender for the principal-and-interest payment separately.
Can I use this formula for a loan that is not a mortgage?
Yes. The amortization formula works for any fixed-rate loan: car loans, personal loans, student loans. As long as you know the principal, annual interest rate, and loan term in months, you can calculate the monthly payment the same way.
What happens if I pay extra toward principal each month?
This formula calculates the standard payment that keeps you on schedule. If you pay extra, you reduce the balance faster, which means you pay less total interest and finish the loan early. The formula does not account for extra payments—you would need to recalculate the remaining balance and term after each extra payment to see the new timeline.