Loan Interest Rate Finder
Reverse-Calculate Your True APR

The standard loan payment formula M = P × r × (1+r)^n / ((1+r)^n − 1) can be solved for M, P, or n algebraically, but not for r. The variable r appears in both the base and the exponent, creating a transcendental equation with no closed-form solution. The only practical approach is numerical iteration: start with an initial guess for r, evaluate how close the resulting payment is to M, and adjust r according to the derivative of the error function until convergence. Newton-Raphson is the standard method because it converges quadratically, meaning the number of correct decimal places roughly doubles with each iteration.

APR (Annual Percentage Rate) is the monthly rate multiplied by 12. EAR (Effective Annual Rate) compounds the monthly rate over 12 periods: EAR = (1 + r_monthly)^12 − 1. On a 7% APR loan the monthly rate is 0.5833% and the EAR is 7.229%. EAR is always higher than APR for the same loan because it captures the effect of compounding. When comparing loans, EAR is the more accurate cost measure. Some EU regulations require lenders to disclose the EAR as the APRC (Annual Percentage Rate of Charge).

Common uses: verifying that the rate charged on a loan matches what was agreed; finding the implied rate on a buy-now-pay-later arrangement where only the payment is stated; checking the rate on an existing loan when you have lost the paperwork; comparing two loan offers where one states a rate and the other states a payment; and understanding the true APR on hire purchase or lease agreements where the headline rate may differ from the effective rate. Any time you know three of the four standard loan variables (principal, payment, term, rate) and need the fourth, use this calculator.

The solver fails to converge in two situations: when the monthly payment is less than or equal to the monthly interest at even a very low rate (meaning the balance would never decrease), or when the numbers are mathematically inconsistent. For example, if you enter a $10,000 loan with a $1 monthly payment over 5 years, no interest rate can produce this because you are repaying $60 total on a $10,000 loan. The calculator detects non-convergence and displays an error rather than a misleading result.