Loan and mortgage payment calculator | Calculator of Alexandria

The formula

M = \frac{P\,i}{1 - (1 + i)^{-n}}

$M$ is the monthly payment, $P$ the amount borrowed, $i$ the monthly interest rate (the annual rate divided by 12) and $n$ the number of monthly payments.

Where it comes from

Each month, interest is charged on whatever you still owe, and your payment covers that interest plus some of the balance. The payment $M$ is the one fixed amount that brings the balance to exactly zero after $n$ months.

Think of it from the lender’s side. A payment you make $k$ months from now is worth less than money today, because the lender could have earned interest on it in the meantime. Its value today is $M/(1+i)^k$ . The loan is fair when the present value of all the payments equals the amount lent:

P = \frac{M}{1+i} + \frac{M}{(1+i)^2} + \cdots + \frac{M}{(1+i)^n}

That’s a geometric series with first term $M/(1+i)$ and ratio $1/(1+i)$ . Summing it gives

P = \frac{M}{i}\left[1 - (1+i)^{-n}\right]

and solving for $M$ gives the formula above.

Worked example

A $300,000 mortgage at 6.5% over 30 years means $i = 0.065/12 = 0.0054167$ and $n = 360$ :

M = \frac{300000 \times 0.0054167}{1 - 1.0054167^{-360}} = 1896.20

Over 360 payments you pay $682,633 in total, so $382,633 of it is interest. The table shows why it feels slow at first: in year one, most of each payment is interest.

Watch out for

Payment here means principal and interest only. Mortgage bills often add property tax, insurance and PMI on top.
APR vs interest rate. APR folds in fees, so it’s a little higher. Use the note rate for the payment.
Extra payments go straight to principal and shorten the loan, which saves a lot of interest early on.

Sources

Consumer Financial Protection Bureau, "What is amortization and how could it affect my auto loan?"

Last checked November 11, 2026