The Loan Payment Formula, Worked by Hand
October 21, 2026
The standard payment on a fixed-rate loan is M = P × r ÷ (1 − (1 + r)^−n), where P is the amount borrowed, r is the periodic (monthly) rate, and n is the number of payments. For $10,000 at 5% over 36 months, r = 0.05 ÷ 12 = 0.0041667, and M = $299.71. Total paid is $10,789.52, so interest costs $789.52.
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Open the Loan Calculator →The Steps
First, find r: 5% ÷ 12 = 0.0041667. Next, compute (1 + r)^−36 = 1 ÷ 1.0041667^36 = 1 ÷ 1.16147 = 0.86096. Then 1 − 0.86096 = 0.13904. Finally, M = 10,000 × 0.0041667 ÷ 0.13904 = $299.71.
Any calculator with an exponent key can follow these steps, and spreadsheets have the PMT function that does it all at once.
Where Each Payment Goes
In month one, interest is $10,000 × 0.0041667 = $41.67, so $258.04 of the $299.71 payment reduces principal and the balance falls to $9,741.96. In month two, interest is $40.59, and principal reduction is slightly larger at $259.12.
Each month the interest share shrinks and the principal share grows, which is the signature of an amortizing loan.
| Month | Interest | Principal | Balance after |
|---|---|---|---|
| 1 | $41.67 | $258.04 | $9,741.96 |
| 2 | $40.59 | $259.12 | $9,482.84 |
| 36 | $1.24 | $298.47 | $0.00 |
Sensitivity
Changing the term or the rate changes the payment in predictable ways. The same loan over 48 months costs $230.29 a month and $1,054 in interest. At 7% over 36 months, the payment is $308.77. Use the formula to compare offers on equal footing, and always check for fees that increase the true cost.
Frequently Asked Questions
What is the loan payment formula?
M = P × r ÷ (1 − (1 + r)^−n), with monthly rate r and n monthly payments.
How do I find the monthly rate?
Divide the annual rate by 12.
Why does interest shrink over time?
Because it is charged on the remaining balance, which falls with each payment.
What spreadsheet function calculates this?
PMT(rate, nper, pv).