APR Calculator — Annual Percentage Rate
Calculate the Annual Percentage Rate (APR) of a loan from the loan amount, monthly payment, and term. Compare the true cost of borrowing.
What Is the APR Calculator — Annual Percentage Rate?
The Annual Percentage Rate (APR) is the true annualised cost of borrowing, expressed as a percentage. Unlike a nominal interest rate, APR accounts for fees and the compounding effect, allowing accurate comparison between loan products. Lenders are legally required to disclose APR in many jurisdictions. This calculator finds APR from loan amount, regular payment, and term using numerical root-finding.
Formula
How to Use
Enter the loan amount (principal), monthly payment, and loan term in months. If there are upfront fees, enter them to see how they inflate the effective APR. The calculator solves for the monthly interest rate numerically, then annualises it.
Example Calculation
Loan of $10,000, monthly payment $200, 60-month term: APR ≈ 7.42%. If the lender also charges $300 upfront fees, the effective APR rises to ≈ 9.05% because you received less money for the same payment stream.
Understanding APR — Annual Percentage Rate
APR was standardised in consumer protection legislation (US: Truth in Lending Act 1968; EU: Consumer Credit Directive) to give borrowers a single comparable number. Before APR disclosure requirements, lenders could obscure loan costs with complex fee structures.
The numerical solution for APR uses Newton-Raphson iteration: starting with an initial guess, the algorithm refines the root of f(r) = PMT × [1−(1+r)^−n]/r − P = 0 until convergence to within 10^−8. This typically converges in 5–15 iterations.
For mortgages, APR includes points, origination fees, mortgage insurance, and certain closing costs. For credit cards, APR is the daily periodic rate times 365. For personal loans, it reflects origination fees amortised over the loan term.
A small difference in APR over a long loan has significant absolute cost implications. On a 30-year $300,000 mortgage, the difference between 4.5% and 5.0% APR is approximately $30,000 in total interest.
Frequently Asked Questions
What is the difference between APR and interest rate?
The stated interest rate only reflects the cost of the principal. APR also includes fees like origination fees, mortgage points, and insurance, making it a more complete measure of borrowing cost. Two loans can have the same interest rate but different APRs.
How is APR calculated numerically?
APR requires solving for the rate r in the loan payment equation P = PMT × [1−(1+r)^−n]/r. This has no closed-form inverse, so the calculator uses the Newton-Raphson method to iteratively converge on r, then multiplies by 12 to annualise.
Is APR the same as EAR (Effective Annual Rate)?
No. APR is a nominal annualised rate (monthly rate × 12). The EAR accounts for compounding: EAR = (1 + APR/12)^12 − 1. For monthly-compounded loans, EAR is always slightly higher than APR.
Why does my lender quote a lower rate than the APR I calculate?
Lenders often advertise the nominal annual rate (base interest rate) without fees. Your calculated APR includes all costs and represents the true cost of the loan. Always compare APRs, not nominal rates.
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