About

The stated interest rate on a loan does not reflect its true cost. Origination fees, discount points, closing costs, and other charges inflate the effective rate a borrower pays. The Annual Percentage Rate (APR) captures this reality by folding all mandatory charges into a single annualized rate, computed per U.S. Regulation Z (Truth in Lending Act). This calculator solves for APR using the Newton-Raphson iterative root-finding method applied to the present-value equation of cash flows. It does not approximate. Failing to compare APRs across lenders can cost thousands over the life of a mortgage or auto loan.

The tool assumes a fixed-rate, fully amortizing loan with equal periodic payments. It does not model variable-rate products, balloon payments, or interest-only periods. All fees are assumed financed (added to the loan balance). Pro tip: lender-quoted APRs sometimes exclude third-party fees like title insurance or appraisal. Always verify which charges are included before comparing.

Formulas

The APR is the annualized periodic rate r that satisfies the present-value equilibrium between the net amount financed and the stream of equal payments:

n∑k=1PMT(1 + r)^k = L − F

Where PMT is the periodic payment calculated at the nominal interest rate, L is the total loan principal, F is total upfront fees, n is the total number of payment periods, and r is the unknown periodic rate. The annual percentage rate is then:

APR = r × m × 100

Where m is the number of payment periods per year (12 for monthly, 26 for biweekly, 52 for weekly). The periodic payment uses the standard annuity formula:

PMT = L × i1 − (1 + i)⁻ⁿ

Where i = nominal annual rate ÷ m. The Newton-Raphson iteration updates r as:

r_k+1 = r_k − f(r_k)f′(r_k)

Convergence is typically achieved in 5 - 15 iterations with tolerance 10⁻¹⁰.

Reference Data

Fee Type	Typical Range	Included in APR?	Notes
Origination Fee	0.5% - 1.5%	Yes	Charged by lender for processing
Discount Points	0 - 3 points	Yes	1 point = 1% of loan amount
Closing Costs (lender)	$500 - $5,000	Yes	Underwriting, document prep
Private Mortgage Insurance	0.3% - 1.5%/yr	Yes (if required)	Applies when LTV > 80%
Title Insurance	$500 - $3,500	No (third-party)	Varies by state
Appraisal Fee	$300 - $700	No (third-party)	Required by most lenders
Credit Report Fee	$25 - $50	No	Minimal impact on APR
Survey Fee	$150 - $500	No	Property boundary verification
Recording Fee	$50 - $250	No	Government charge
Prepaid Interest	Varies	Yes	Interest from closing to first payment
Broker Fee	1% - 2%	Yes	If using a mortgage broker
FHA Upfront MIP	1.75%	Yes	FHA loans only
VA Funding Fee	1.25% - 3.3%	Yes	VA loans only
Loan Term	Typical APR Impact	Shorter terms have lower APR spread vs. rate
30-year fixed @ 6.5%	APR ≈ 6.7% - 6.9%	With typical fees
15-year fixed @ 5.8%	APR ≈ 6.1% - 6.3%	Fees amortized over fewer years
5-year auto @ 7.0%	APR ≈ 7.2% - 7.8%	Doc fees and dealer charges
3-year personal @ 10%	APR ≈ 11% - 14%	Origination fees are proportionally larger

Frequently Asked Questions

The APR incorporates mandatory fees (origination, points, broker charges) into the effective rate. Since these fees increase the true cost of borrowing without increasing the principal received, the APR is mathematically guaranteed to be ≥ the nominal rate. The only case where APR equals the rate is when total fees are zero.

Shorter loan terms amplify the APR spread. A $2,000 fee on a 5-year $20,000 loan increases APR far more than the same fee on a 30-year $200,000 mortgage. This is because the fixed fee is amortized over fewer periods, making each period's effective cost higher. Always compare APRs within the same term length.

No. This tool computes APR for fixed-rate, fully amortizing loans only. ARM APR calculations require assumptions about future rate adjustments per the index plus margin, which introduces significant uncertainty. Regulation Z requires ARM APR disclosures to use the fully-indexed rate for the initial period, but the methodology differs from the fixed-rate Newton-Raphson approach used here.

Include all lender-mandated charges: origination fees, discount points, underwriting fees, broker fees, and any required mortgage insurance premiums paid upfront. Exclude third-party fees (appraisal, title insurance, survey, recording) and prepaid items (property taxes, homeowner's insurance) unless you want a broader cost comparison. The inclusion criteria follow Regulation Z Section 1026.4.

Regulation Z permits a tolerance of ±0.125% for regular transactions and ±0.25% for irregular ones. Differences also arise from which fees the lender classifies as "finance charges" versus third-party costs. Additionally, odd-days interest (the gap between closing and first payment) affects the present-value equation. This calculator assumes the first payment occurs exactly one period after disbursement.

Yes, that is one of APR's primary purposes. Whether a loan is paid monthly, biweekly, or weekly, the APR normalizes the cost to an annual basis. However, note that a biweekly payment schedule on a 30-year mortgage effectively makes 26 half-payments per year (equivalent to 13 monthly payments), which accelerates principal paydown and reduces total interest independently of the APR.