Finding the rate behind a payment
Dealers, stores and some lenders quote a loan as a monthly payment — “$500 a month for five years” — and leave the rate in the small print. Three numbers are enough to recover it:
- Loan amount — what you borrow, after any down payment or trade-in.
- Monthly payment — principal and interest only. If the quote bundles insurance, a service plan or other add-ons into the payment, take them out first, or the rate comes out too high.
- Term — how many monthly payments you make.
The calculator finds the one interest rate at which those payments pay the loan off exactly, then shows it three ways: the annual rate (monthly rate × 12, the way US loan rates and APRs are quoted), the monthly rate itself and the effective annual rate, which includes compounding. It also shows the total interest and how the balance falls year by year at that rate.
If you already know the rate and want the payment, use the loan calculator. If the lender charges an origination fee or points, the rate found here is the note rate; the APR calculator adds those fees on top.
The formula, and why the rate has to be solved for
A fixed-rate loan with level monthly payments satisfies the annuity equation:
P = M × (1 − (1 + i)⁻ⁿ) ÷ i
- P
- loan amount
- M
- monthly payment
- n
- number of monthly payments
- i
- monthly interest rate, as a decimal
Given i, the payment follows directly. Going the other way there is no formula that isolates i, so it is found numerically. The calculator uses Newton’s method — each step follows the slope of the equation towards the answer — inside a bracket that is known to contain the answer. Any step that would leave the bracket is replaced by halving it (bisection), so the search always converges, even for very high or very low rates.
Annual rate = 12 × i · Effective annual rate = (1 + i)¹² − 1
When the payments add up to exactly the loan amount, the rate is 0%. When they add up to less, only a negative rate would balance the equation. The calculator reports that as an error instead of a result: ordinary loans don’t charge negative interest, so it almost always means a mistyped payment or term.
The yearly table is a cent-rounded schedule at the solved rate: each month’s interest is the balance times i, rounded to the cent. Those roundings leave the last month a few cents off, and the last month’s interest absorbs the difference, so the table adds up to exactly your payments.
Worked example
You are offered $25,000, repaid at $500 a month for 5 years.
- Total of payments: 60 × $500 = $30,000, so the loan costs $5,000 in interest, 20% of the amount borrowed
- Solving $25,000 = $500 × (1 − (1 + i)⁻⁶⁰) ÷ i gives i = 0.00618341, a monthly rate of about 0.6183%
- Annual rate: 0.00618341 × 12 = 7.420%
- Effective annual rate: (1.00618341)¹² − 1 = 7.678%
- The first payment is $154.59 interest and $345.41 principal. After one year you have repaid $4,288.90 and owe $20,711.10.
As a check the other way: a $20,000 loan at exactly 6% over 5 years has a payment of $386.66 (rounded to the cent). Entering $20,000, $386.66 and 5 years here gives 6.000%.
Reading the result
- A rounded payment gives a slightly rounded rate. Payments are quoted to the cent, so the rate recovered from one can differ from the lender’s rate in the third or fourth decimal. $386.66 on $20,000 over 60 months comes out at 6.0004%, not 6% exactly.
- Bundled extras inflate the rate. A payment that includes credit insurance, a warranty or gap cover is not a loan payment alone. Subtract those amounts, or ask for the principal-and-interest payment.
- Annual rate vs. effective rate. The annual rate (12 × monthly) is how loan rates are quoted in the US. The effective annual rate is always a little higher because it compounds the monthly rate; it is the figure to compare with a savings APY.
- 0% offers. If the payments add up to exactly the price, the rate is 0%. Some 0% deals are only available instead of a cash rebate; the rebate you give up is a real cost the rate doesn’t show.
Related calculators
Compare terms and payments with the loan calculator, see a car loan with trade-in and sales tax in the auto loan calculator, or print the full month-by-month schedule with the amortization calculator.
Limits of this calculator
- Payments are monthly and level. Balloon payments, interest-only periods and adjustable rates are not modelled.
- The first payment is assumed one month after the loan starts. A longer first period (common on car loans) means slightly more interest in the first payment and a slightly lower true rate.
- Upfront fees are not included. The result is the rate on the loan balance, not the APR; add fees with the APR calculator.
- Rates above 1,000% a year are not reported; inputs that imply them are almost always a typo.
Frequently asked questions
How do I find the interest rate if I only know the monthly payment?
Enter the amount you borrow, the monthly principal-and-interest payment and the number of payments. The calculator finds the monthly rate at which those payments repay the loan exactly and multiplies it by 12 for the annual rate. For $25,000 at $500 a month over 60 months, that is 7.420%.
Why can’t the rate be worked out with a simple formula?
In the loan equation the rate appears both inside a power and in a denominator, so it cannot be rearranged to stand alone. It has to be found by trial: the calculator uses Newton’s method with a bisection fallback, which pins it down to many decimal places in a few steps.
What does it mean if the payments add up to less than the loan?
Then you would repay less than you borrowed, which only a negative interest rate could explain. Ordinary loans do not work that way, so the calculator shows an error. Check that the payment is the full monthly amount and that the term is right.
Is this the same as the APR?
Only when there are no upfront fees. This calculator finds the rate on the loan balance. The APR also counts finance charges you pay to get the loan, such as an origination fee or points, and is higher whenever there are any. Use the APR calculator to add them.
What is the difference between the annual rate and the effective annual rate?
The annual rate is the monthly rate times 12, which is how US lenders quote loan rates and APRs. The effective annual rate compounds the monthly rate over a year, (1 + i)^12 − 1, so it is slightly higher: 7.678% versus 7.420% in the $25,000 example.
Sources
Last reviewed September 19, 2026