Find Interest Rate Calculator
Know the payment and balance but not the rate? This calculator works backwards to find the implied interest rate in any loan or investment.
If a dealer offers you a $20,000 car for $450/month for 48 months, the implied interest rate is 6.4% APR. This calculator finds the hidden rate in any loan, investment, or financing offer — using numerical root-finding on the payment formula.
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Find Interest Rate Calculator
Loan: principal + payment + months → rate | Investment: start + end + years → rate
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Mode
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Loan details
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months
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Implied rate
Annual interest rate (APR)
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Monthly rate
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Total paid
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Total interest cost
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How it's calculated
Numerical root-finding: Newton-Raphson
For loans, we solve for the monthly rate r that satisfies the payment formula. Since the equation can't be rearranged algebraically, the calculator uses iterative numerical approximation:
Loan payment formula:
PMT = P × r / (1 − (1+r)^-n)
Solve for r: Newton-Raphson iteration
r_(k+1) = r_k − f(r_k) / f'(r_k)
where f(r) = P × r − PMT × (1 − (1+r)^-n)
Investment (lump sum, no contributions):
CAGR = (Final / Initial)^(1/years) − 1
(exact closed-form solution)
- 1Rate per period—
- 2Annualized rate—
- APR (Annual Percentage Rate)
- The annualized interest rate for a loan, calculated as monthly rate × 12. Different from APY which includes compounding.
- CAGR (Compound Annual Growth Rate)
- The year-over-year growth rate that brings an investment from its starting value to its ending value. Smooths out volatility to show the implied annual return.
- Newton-Raphson
- An iterative numerical method for finding roots of equations. Used here because the loan rate equation has no closed-form algebraic solution.
Disclaimer: the found rate is the mathematical implied rate. Actual loan APR may differ if fees are included in payments or excluded. Always verify with your lender's official disclosure.
Frequently asked questions
How do I find the interest rate of a loan?
Enter the principal (amount borrowed), monthly payment, and number of payments. The calculator finds the monthly rate that satisfies the standard amortization formula using Newton-Raphson iteration, then converts to annual APR.
What is CAGR?
Compound Annual Growth Rate — the steady annual rate that would take an investment from its initial value to its final value over the holding period. CAGR = (Final ÷ Initial)^(1/years) − 1. It smooths out year-to-year volatility to show the overall implied return.
When would I use this calculator?
Whenever a loan or financing offer gives you payments and amounts but not the rate; or when you want to verify that a stated rate matches the actual payments. Also useful for finding the implied return of an investment that went from amount A to amount B over a known period.
What if the calculator can't find a rate?
This happens if the numbers are inconsistent — for example, a monthly payment less than the first month's interest means the loan can never be paid off. Or if the payment is exactly equal to the principal divided by months (0% rate). Check that your payment exceeds the interest portion.