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Interest Rate Calculator: Solve the Rate from the Payment
If you know how much you borrowed, what you pay each month and for how long, this tool works out the annual interest rate the lender is charging, plus the effective annual rate and the total interest.
Why the rate has to be solved numerically
The payment formula for a standard loan is Payment = P × r ÷ (1 − (1 + r)−n). It lets you compute the payment when you know r, but there is no neat algebraic way to isolate r when the payment is known. The rate appears both as a multiplier and inside the exponent.
Calculators therefore use a search. This one uses bisection: it knows that a higher monthly rate always gives a higher payment, so it starts with a range from zero to 100% a month, tests the middle, discards the half that cannot contain the answer, and repeats two hundred times. The remaining range is far narrower than any rounding you would see on a statement.
When this is useful
- Checking a quote. A dealer or shop may advertise only a monthly figure. Entering the cash price, the payment and the term reveals the implied rate.
- Comparing offers. Two loans with different terms are easier to judge once each is reduced to a rate.
- Spotting zero-percent claims. If the payments add up to more than the amount borrowed, the interest is not zero, whatever the label says.
Nominal versus effective rate
The headline figure is the monthly rate multiplied by twelve, which is how most lenders quote loan rates. Interest is charged every month, so after a full year the real growth is slightly higher. The effective annual rate, (1 + r)12 − 1, shows that. At 1% a month, the nominal rate is 12% and the effective rate is about 12.68%. Use whichever your lender or contract states when comparing.
If the loan carried upfront charges, the rate you pay is higher than this result because you received less than the stated amount. In that case, enter the figures into the APR calculator.
Limits and tips
The tool assumes equal monthly payments, the first one a month after the loan starts, and no balloon payment or payment holiday. Loans with irregular payments, variable rates or interest calculated on daily balances will give an approximation only. Enter the exact payment from your agreement rather than a rounded one, because small payment differences move the rate noticeably on short loans. As a quick sense check, a payment that is only slightly above the amount divided by the number of payments implies a low rate, while a payment far above it implies a high one. The result is for information and does not replace the disclosure your lender must provide.
A worked example: a loan of 12,000 repaid in 12 equal payments of 1,000 has total payments equal to the amount borrowed, so the rate is exactly zero. If the payment is 1,100 instead, the payments total 13,200 and the search settles near a monthly rate of about 1.5 percent (roughly 18% a year), because the interest is charged on a balance that shrinks through the year rather than on the full 12,000.
Worked example
A 25,000 loan repaid with 60 payments of 520 each. What annual rate does that imply?
Inputs
| Amount borrowed | 25000 |
|---|---|
| Monthly payment | 520 |
| Number of payments | 60 months |
Result
Nominal annual rate, compounded monthly.
| Rate per month | 0.7571% |
|---|---|
| Effective annual rate | 9.474% |
| Total of all payments | 31,200.00 |
| Total interest | 6,200.00 |
| Interest as a share of the amount borrowed | 24.8% |
- Fees paid at the start are not included. If the lender charged any, the true cost is higher; the APR calculator handles that case.
Financial disclaimer: Results are estimates for planning and education. They are not financial, tax or legal advice. Fees, taxes, rates and rules vary by lender, product and country, so confirm figures with your provider or a qualified adviser before making decisions.
Frequently asked questions
How do I find the interest rate on a loan?
You need the amount borrowed, the regular payment and the number of payments. Because the rate cannot be isolated algebraically, it is found by repeatedly narrowing a range, as this calculator does.
Why do I get an error saying the payments are too small?
If the payments add up to less than the amount borrowed, the loan would have a negative rate. Usually a figure was mistyped.
Is the rate shown the APR?
Not quite. It is the nominal rate implied by the payment. APR also reflects upfront fees, so it can be higher.
What if my payment is every two weeks?
This tool assumes monthly payments. Convert the situation to a monthly payment first, or treat the rate shown as approximate.
Does a longer term change the rate?
The rate is determined by the amount, payment and number of payments you enter. A longer term at the same payment implies a lower rate.
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References and sources
- Numerical root finding by bisection, as described in standard numerical analysis texts.
- Consumer Financial Protection Bureau: educational guides on loan costs. Link