Back out the true interest rate on a loan from its amount, payment and term.
The monthly payment formula PMT = P·r/(1−(1+r)^−n) cannot be solved for r algebraically — it is transcendental in r. This calculator uses Newton-Raphson iteration with a bisection fallback to find the monthly rate that exactly reproduces your payment, then multiplies by 12 for the nominal annual rate. The result is the contractual rate: no fees are included. Use the APR Calculator if you have origination fees.
Solve PMT for r (no closed form)
Find r where P × r / (1 − (1+r)^−n) = monthly payment M
Annualise
Annual rate = monthly rate × 12
This is the nominal interest rate only, based on the payment, principal and term. APR includes fees (origination, points, etc.) and is always higher than the nominal rate. Use the APR Calculator to fold in those costs.
The relationship between payment, principal, term and rate is not linear, so a $400 payment on a $20,000, 60-month loan almost never corresponds to a round rate like 7.000%. The solver gives you the precise rate the lender is charging.
If the sum of all your payments is less than the loan amount, there is no positive interest rate that makes the math work — you would have received a subsidy. The calculator will tell you this and ask you to reconsider the inputs.