Interest Rate Calculator
Calculate the implied annual interest rate (APR), total payments, and total interest for any fixed amortizing loan from the loan amount, payment, and term.
How It's Calculated
Formula
P = \text{PMT} \times \frac{1 - (1 + r)^{-N}}{r}, \quad N = \text{termYears} \times 12, \quad \text{APR} = r \times 12 \times 100\%When evaluating a financing offer, lenders often quote only the vehicle price and monthly payment, leaving the true underlying interest rate opaque. This calculator reverses the standard loan amortization formula to solve for the exact nominal Annual Percentage Rate (APR). By analyzing the initial loan balance, recurring monthly payment, and repayment duration in years, the calculator computes the periodic rate using a robust monotonic root solver, and determines total lifetime payments and financing charges.
Worked Examples
5-Year Auto Loan: $20,000 borrowed, $400/month, 5-year term
- Total repayment periods: N = 5 years × 12 months/year = 60 monthly payments.
- Total payments made: 60 × $400.00 = $24,000.00.
- Total interest charges: $24,000.00 − $20,000.00 = $4,000.00.
- Solve for periodic monthly rate r in: $20,000 = $400 × [1 − (1 + r)^(-60)] / r.
- Numerical bisection yields periodic rate r ≈ 0.0061834 (0.6183% per month).
- Annual Percentage Rate (APR) = 0.0061834 × 12 × 100% ≈ 7.42%.
0% Promotional Loan: $12,000 borrowed, $1,000/month, 1-year term
- Total repayment periods: N = 1 year × 12 months/year = 12 monthly payments.
- Total payments made: 12 × $1,000.00 = $12,000.00.
- Total interest charges: $12,000.00 − $12,000.00 = $0.00.
- Because total payments match the borrowed principal exactly, the loan has 0% interest with no financing charges.
30-Year Fixed Mortgage: $300,000 borrowed, $1,798.65/month, 30-year term
- Total repayment periods: N = 30 years × 12 months/year = 360 monthly payments.
- Total payments made: 360 × $1,798.65 = $647,514.00.
- Total interest charges: $647,514.00 − $300,000.00 = $347,514.00.
- Numerical solving converges to periodic rate r = 0.0050 (0.50% per month).
- Annual Percentage Rate (APR) = 0.0050 × 12 × 100% = 6.00%.
Frequently Asked Questions
Why does calculating the interest rate require numerical root solving?
In the standard loan amortization equation, the interest rate variable r appears both in the denominator and inside the exponent (1 + r)^(-N). This forms a transcendental equation that cannot be isolated algebraically. High-precision financial engines use deterministic numerical algorithms, such as bisection and Brent's method, to converge on the exact rate.
What does 'Payment is too low to amortize loan' mean?
To fully repay any amortizing loan, the total sum of all scheduled payments (Monthly Payment × Number of Months) must be at least equal to the initial loan principal (which corresponds to 0% interest). If the total payments are less than the principal borrowed, the loan cannot amortize, meaning the monthly payment entered is mathematically insufficient.
Is the calculated rate an APR or an effective annual rate (APY/EAR)?
The calculated rate is the nominal Annual Percentage Rate (APR) with monthly compounding, which equals the periodic monthly rate multiplied by 12. To compute the Effective Annual Rate (EAR) accounting for compound interest over the full year, use the formula EAR = (1 + r)^12 − 1.
Can this calculator be used for mortgages, auto loans, and personal loans?
Yes. Any fixed-rate loan that uses uniform, fully-amortizing monthly payments follows this identical financial relationship. You can use it to determine the true interest rate for auto loans, mortgages, student loans, or personal financing.