APR Formula Breakdown: APR vs APY Explained
APR (Annual Percentage Rate) is the yearly cost of borrowing money, expressed as a percentage. It includes both the interest rate and certain fees, making it more comprehensive than the stated interest rate alone. APY (Annual Percentage Yield) is the effective annual return on savings, incorporating compounding.
For loans: APR helps borrowers compare offers on equal footing. For savings: APY shows true annual yield including compounding. The confusion arises because APR can refer to different things in different contexts — this guide clarifies all of them.
Formula
Loan APR ≈ (Total Interest + Fees) / Principal / Time × 100 | APY = (1 + APR/n)^n − 1
What Is APR?
APR (Annual Percentage Rate) is a standardised measure of borrowing cost. In the US, the Truth in Lending Act (TILA) requires lenders to disclose APR on all consumer loans. APR includes the interest rate plus certain fees (origination fees, mortgage points, PMI), annualised over the loan term.
What APR includes (for mortgages): Interest rate. Discount points. Origination fees. Mortgage broker fees. Certain closing costs. PMI (private mortgage insurance) if applicable.
What APR excludes: Appraisal fees. Title insurance. Attorney fees. Escrow deposits. Most voluntary charges.
Purpose of APR: Two mortgages with the same interest rate but different fees have different APRs. The higher-APR loan costs more overall. APR lets you compare the true cost across lenders offering different structures.
Key limitation: APR assumes you keep the loan for its full term. If you refinance or sell early, the annualised fee impact is higher, making the real cost greater than the APR suggests.
APR vs APY — The Critical Difference
APR (Annual Percentage Rate): Used for loans. The nominal annual rate — does not reflect compounding within the year. A 12% APR loan with monthly compounding has 12 monthly payments at 1% each.
APY (Annual Percentage Yield): Used for savings and investments. The effective annual rate — does reflect compounding. APY = (1 + APR/n)^n − 1, where n = compounding periods per year.
Same account at 12% APR, compounded monthly: APY = (1 + 0.12/12)^12 − 1 = (1.01)^12 − 1 = 1.12683 − 1 = 12.683%. The APY (12.683%) is higher than the APR (12%) because you earn interest on interest each month.
Borrower vs Saver perspective: Banks advertise APY for savings (higher number looks better to savers) and APR for loans (lower number looks better to borrowers). The math: a 6% APR mortgage compounded monthly has an EAR of 6.168% — you actually pay 6.168% effective annually, not 6%.
| Nominal APR | Compounding | Effective APY (EAR) | APY − APR |
|---|---|---|---|
| 3% | Monthly | 3.042% | +0.042% |
| 5% | Monthly | 5.116% | +0.116% |
| 6% | Monthly | 6.168% | +0.168% |
| 8% | Monthly | 8.300% | +0.300% |
| 10% | Monthly | 10.471% | +0.471% |
| 12% | Monthly | 12.683% | +0.683% |
| 20% | Monthly | 21.939% | +1.939% |
| 24% | Daily | 27.111% | +3.111% |
How Lenders Calculate Loan APR
The exact APR calculation for loans uses the Internal Rate of Return (IRR) method. The APR is the discount rate that makes the present value of all scheduled payments equal to the loan amount minus certain fees.
Simplified approximation: APR ≈ (2 × n × I) / (P × (N + 1)), where I = total interest paid, n = number of payments per year, P = principal, N = total number of payments. This is the "n-ratio" method used in the Rule of 78s and older consumer loan disclosures.
Modern (more accurate) method: Set the loan principal equal to the present value of all monthly payments discounted at APR/12 per month. Solve for APR using Newton-Raphson iteration — this is what mortgage calculators do.
Mortgage example: $300,000 loan at 7% interest rate, 30 years. Monthly payment: $1,995.91. Total interest over 30 years: $418,527. Now add $3,000 origination fee: effective loan amount received = $297,000 (fee taken upfront). Recalculate the rate that makes PV of payments = $297,000. Result: APR ≈ 7.13%.
Credit Card APR — Daily Periodic Rate
Credit card APR is almost always stated as a nominal annual rate, but interest is calculated daily. Daily Periodic Rate (DPR) = APR / 365.
Daily interest calculation: If you carry a $2,000 balance at 22% APR: DPR = 22% / 365 = 0.06027% per day. Daily interest = $2,000 × 0.0006027 = $1.21. Over a 30-day billing cycle: $36.30 interest.
The daily balance method: Most cards calculate interest on the average daily balance for the billing cycle. If your balance changes mid-cycle, each day's balance is weighted proportionally.
Variable vs fixed APR: Most credit cards have variable APR tied to the prime rate (a benchmark). Fixed APR is less common and can still change with 45 days' notice. Always check whether your rate is variable.
Penalty APR: If you miss a payment or violate terms, many cards apply a penalty APR (often 29.99%) — permanently until you make 6 consecutive on-time payments (US law). This is the highest APR most consumers will encounter.
Using APR to Compare Loan Offers
Rule: When comparing two loans with similar terms, choose the one with the lower APR — it costs less overall. This is the primary purpose of the APR disclosure requirement.
However, APR can mislead when: Comparing loans with different terms (15-year vs 30-year mortgage). When you plan to pay off the loan early (fees amortised differently). When the loan includes optional features (credit insurance). When fees are excluded from the APR calculation.
Example — Low rate vs low fee: Loan A: 6.5% rate, $5,000 fees, APR = 6.8%. Loan B: 6.8% rate, $500 fees, APR = 6.85%. Loan A has lower APR but more upfront fees. If you stay in the home 30 years, Loan A wins. If you sell in 3 years, Loan B wins (less upfront cost).
Break-even analysis: Calculate how many months of interest savings from the lower rate offset the higher upfront fees. This "break-even point" determines whether the lower-APR loan is actually better for your situation.
Nominal vs Effective Interest Rate
Nominal rate: The stated annual rate, not accounting for compounding within the year. Also called the "contractual rate" or "coupon rate." Example: 6% APR compounded monthly.
Effective rate (EAR/APY): The true annual rate after accounting for compounding. EAR = (1 + nominal/n)^n − 1. For 6% nominal monthly: EAR = (1.005)^12 − 1 = 6.168%.
When they're equal: For annual compounding (n=1), nominal rate = effective rate. EAR = (1 + r/1)^1 − 1 = r. This is why annually-compounded rates are unambiguous.
Continuous compounding: EAR = e^r − 1. For 6% continuously: EAR = e^0.06 − 1 = 6.184% — slightly higher than monthly.
For consumers: Savings: APY = effective rate (disclosed for deposits). Mortgages: APR = nominal rate + fees (effective cost is slightly higher). Credit cards: APR = nominal rate (daily compounding makes effective rate ≈ e^APR − 1 for continuously-carried balances).