APR Calculator

Understanding Annual Percentage Rate (APR)

When shopping for a mortgage, auto loan, or personal loan, the stated interest rate only tells half the story. The Annual Percentage Rate (APR) measures the total annualized cost of borrowing, incorporating both the nominal interest rate and any upfront lender fees, closing costs, points, or administrative charges required to secure financing.

Under the United States Truth in Lending Act (TILA) and Regulation Z, lenders are legally mandated to disclose the APR before loan finalization. This standard allows consumers to compare financing offers with differing fee structures on an equal, transparent footing. If you want to compute standard monthly installment amounts without closing fees, check our EMI calculator or explore comprehensive repayment features in our advanced loan calculator. For short-term commercial debt and promissory notes where interest is deducted upfront from maturity value, calculate net proceeds and discount yields with our bank discount calculator.

Interest rate vs. APR: What is the difference?

While people often use the terms interchangeably, interest rate and APR represent distinct financial metrics:

• Nominal Interest Rate: The percentage charged solely on the unpaid principal balance to calculate your regular scheduled installment payment.
• APR: The comprehensive effective borrowing rate that spreads all prepaid finance charges and closing fees across the full repayment term.

Whenever a loan includes origination fees, discount points, or mandatory underwriting fees, the APR will always exceed the stated interest rate. If a loan advertises zero upfront fees, the nominal interest rate and the nominal APR will be identical.

Mathematical formula for APR

APR calculation relies on finding the internal rate of return (IRR). First, the regular scheduled monthly payment $M$ is determined from the base principal $P$, nominal monthly interest rate $r = \text{Annual Rate} / 12$, and total periods $n$:

Next, the net amount financed $P_{\text{net}}$ is established by subtracting all prepaid finance charges $F$ from the original principal:

The periodic monthly rate $r_{\text{APR}}$ is the discount rate that equates the net amount financed to the discounted stream of scheduled monthly payments:

Because this equation cannot be solved algebraically for $r_{\text{APR}}$, numerical methods like the Newton-Raphson algorithm are applied iteratively. Once converged, the nominal annual APR is:

Published worked example

Consider a $250,000 fixed-rate mortgage over 30 years (360 monthly payments) with a stated interest rate of 6.50% and $4,500 in upfront closing fees (origination points and processing charges):

• Monthly Payment ($M$): At 6.50% annual interest ($0.065 / 12 = 0.0054167$ per month), the monthly payment is $1,580.17.
• Net Amount Financed ($P_{\text{net}}$): $250,000 - $4,500 = $245,500.
• Solving for $r_{\text{APR}}$: Finding the discount rate that equates $245,500 to 360 payments of $1,580.17 yields a monthly rate of $0.0055629.
• Calculated APR: $0.0055629 \times 12 = 6.676\%$ (displayed as 6.68%).

The borrower pays an effective annualized borrowing cost of 6.68% despite the face rate being 6.50%. For loans with full principal and interest amortizations, you can inspect complete year-by-year balance progression in our amortization calculator.

Nominal APR vs. Effective APR (EAR)

In consumer lending disclosures in the US and Canada, APR is customarily reported as a nominal simple annualized rate ($r_{\text{APR}} \times 12$). In Europe and certain financial contexts, regulations require the Effective Annual Rate (EAR or Annual Percentage Rate of Charge), which accounts for intra-year compound interest:

For the example above, a nominal APR of 6.676% corresponds to an effective annual rate of 6.883%. To convert nominal rates to effective annual yields across daily, weekly, or quarterly compounding frequencies, use our APR to APY calculator, or compute annual compound earnings on deposit accounts with our APY calculator. When converting financing APR into lease money factors for vehicle negotiations, our auto lease calculator handles those rate conversions automatically.

Which fees are included in APR?

Under federal lending guidelines, not every expense associated with buying a home or taking a loan is included in the APR calculation:

• Included Fees (Finance Charges): Loan origination fees, discount points, processing and underwriting fees, private mortgage insurance (PMI), and prepaid interest.
• Excluded Fees: Third-party closing fees that would be incurred even in an all-cash purchase, such as title search, property appraisal, home inspection, recording fees, and document preparation charges.

Frequently asked questions