Interest Rate Calculator
Back out the true interest rate on a loan from its amount, payment and term.
About Interest Rate Calculator
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.
Formulas
Solve PMT for r (no closed form)
Find r where P × r / (1 − (1+r)^−n) = monthly payment M
- P
- Loan principal
- r
- Monthly interest rate (unknown)
- n
- Number of monthly payments
- M
- Known monthly payment
Annualise
Annual rate = monthly rate × 12
Worked example
With the default values below, Interest Rate Calculator returns Annual Interest Rate of 7.4201%. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Loan Amountaccepts 1–10,000,000 $
- 20,000 $
- Monthly Paymentaccepts 1–1,000,000 $
- 400 $
- Loan Termaccepts 1–600 months
- 60 months
Results
- Annual Interest RateNominal annual rate, monthly compounding.
- 7.4201%
- Monthly Rate
- 0.618341%
- Total Interest Paid
- $4,000
- Total Amount Paid
- $24,000
Step by step
- 1Solve PMT(P, r, n) = payment for r= Monthly rate r = 0.618341%
- 2Annualize (nominal, monthly compounding)= 7.4201%
- 3Verification — PMT at solved rate= $400.00Exact match.
- 4Total interest= $4,000.00
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
- This is an estimate for general information and not professional financial, tax or legal advice. See the full disclaimer for the limits that apply to this kind of calculator.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Is this the APR or just the nominal rate?
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.
Why can't I get an exact round number?
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.
What if my payments don't cover the principal?
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.