APR (Annual Percentage Rate)
APY (Annual Percentage Yield)
Nominal Rate
Effective Difference
Table of Contents
APR vs APY: same rate, different number
APR strips out compounding; APY includes it — which is why "5% APR" and "5% APY" describe different actual returns. This calculator converts between APR, APY, and the underlying nominal rate at any compounding frequency so you can compare loans, CDs, and credit cards on equal footing.
How APR converts to APY
The core conversion formula:
APY = (1 + APR ÷ n)n − 1
And the reverse:
APR = n × ((1 + APY)1÷n − 1)
- APR — nominal/stated annual rate (decimal, e.g. 0.06)
- APY — effective annual yield (decimal)
- n — number of compounding periods per year (12 monthly, 365 daily, etc.)
For continuous compounding (the mathematical limit), the formulas collapse to:
APY = eAPR − 1 | APR = ln(1 + APY)
Worked example using the calculator's defaults (6% APR, monthly compounding):
- APR = 0.06, n = 12
- APR ÷ n = 0.06 ÷ 12 = 0.005
- (1 + 0.005)12 = 1.0617
- APY = 1.0617 − 1 = 0.0617 = 6.17%
- Effective difference: 6.17% − 6.00% = 17 basis points
That's the extra you actually earn (or pay) because interest gets added to the balance 12 times per year instead of once.
APR → APY at different compounding frequencies
Holding the APR constant at three common levels — 5% (typical savings/CD), 10% (typical auto loan or unsecured personal loan), and 22% (typical credit card) — here's how the APY shifts with compounding frequency:
| Compounding | 5% APR → APY | 10% APR → APY | 22% APR → APY |
|---|---|---|---|
| Annual | 5.000% | 10.000% | 22.000% |
| Semi-annual | 5.063% | 10.250% | 23.210% |
| Quarterly | 5.095% | 10.381% | 23.882% |
| Monthly | 5.116% | 10.471% | 24.360% |
| Daily | 5.127% | 10.516% | 24.603% |
| Continuous | 5.127% | 10.517% | 24.608% |
Two patterns to notice. First, the gap between APR and APY grows non-linearly with the rate — nearly invisible at savings-account rates, very meaningful at credit-card rates. Second, the gap between daily and continuous compounding is tiny at any rate — if a bank advertises "compounded daily," there's no practical reason to chase "continuous."
When to compare by APR vs APY
| Product | Disclosed as | Compare by | Why |
|---|---|---|---|
| Mortgage | APR (TILA-required) | APR | Includes origination fees & points |
| Personal/auto loan | APR | APR | Includes lender fees |
| Credit card | APR | APR (then convert) | Daily compounding makes effective rate much higher |
| Savings account | APY (TISA-required) | APY | Already includes compounding |
| CD | APY | APY | Lets you compare different compounding schedules |
| Money market | APY | APY | Same reason as CDs/savings |
The general rule: when comparing borrowing costs use APR (fees are usually the bigger factor than compounding frequency); when comparing savings yields use APY (compounding is usually the bigger factor than fees). If a product crosses categories — e.g. a no-fee credit card balance — convert to APY to see the actual annual cost.
Why this distinction matters in practice
The dollar impact is small on small balances over short periods, and significant on large balances over long periods. On a $10,000 deposit:
- 5% APR vs 5.127% APY (daily compounding), 1 year: difference of $12.71.
- Same scenario over 10 years: difference of $169 between using the wrong rate and the right one.
- $300,000 mortgage at 6.5% note rate vs 6.85% APR (with fees): the 0.35% APR-rate gap reflects roughly $1,050/year in additional cost over the loan's life when amortized.
- $5,000 credit card balance at 22% APR vs 24.36% APY (the true daily-compounded annual cost): a $118/year difference in real interest accrued.
Limitations of this calculator
- Conversion only. This tool converts between rate metrics — it doesn't include fees, taxes, or balance-dependent effects (introductory rates, tiered savings APYs, late fees on credit cards).
- Single rate. Some real products use multiple rates (intro APR + go-to APR, tiered savings APY by balance). Convert each tier separately.
- Doesn't capture loan APR fees. The "APR" in mortgage disclosures includes loan fees amortized over the term, which this calculator can't reproduce from the rate alone. Get the lender's official disclosure for that figure.
Sources & references
- Consumer Financial Protection Bureau — APR — federal consumer guidance on APR disclosure and what it includes.
- US Federal Reserve — current rate environment and Truth in Lending regulatory background.
- FDIC National Rates and Rate Caps — reference APY data for savings products and CDs.
- Regulation Z — Truth in Lending — the federal regulation requiring APR disclosure on consumer credit.
FAQs
APR (Annual Percentage Rate) is the simple annual rate without compounding. APY (Annual Percentage Yield) is the effective annual rate including compounding. For the same nominal rate, APY is always equal to or higher than APR — and the gap widens with more frequent compounding. By US law: loans must disclose APR (Truth in Lending Act), savings must disclose APY (Truth in Savings Act). The asymmetry isn't accidental — APR makes loans look cheaper, APY makes savings look richer.
Smaller than people expect at typical rates. A 5% APR compounded monthly produces a 5.116% APY — about 12 basis points of difference. At 10% APR monthly, the gap is roughly 47 basis points (10% APR → 10.471% APY). At 22% APR monthly (typical credit card), the gap balloons to about 144 basis points (22% APR → 24.36% APY). The gap grows non-linearly with the rate — meaningful on high-rate debt, almost trivial on typical savings.
Two reasons. First, credit card rates are high — and the APR-APY gap grows with the rate. Second, most cards compound interest daily on the average daily balance, the most aggressive compounding schedule. A 24.99% purchase APR compounded daily produces an APY around 28.4%. That's the actual annual cost of carrying a balance — not the APR figure printed on your statement. The Federal Reserve's data confirms this is the structural design of most US revolving credit.
Continuous compounding is the mathematical limit as compounding frequency approaches infinity. The formula simplifies to APY = eAPR − 1. Practically, it's used in finance models (Black-Scholes options pricing, bond math) but almost never in retail products — banks use daily, monthly, or quarterly. The gap between daily and continuous compounding is about one basis point at typical rates, so for consumer purposes daily and continuous produce essentially the same number.
Yes — this is the whole point of APR disclosure on loans. The Truth in Lending Act requires mortgage APR to include most lender fees (origination fees, discount points, mortgage insurance) amortized over the loan term. This is why a mortgage's APR is almost always higher than its quoted “note rate” or “interest rate.” A loan with a 6.5% note rate and $5,000 in fees might disclose a 6.85% APR. When shopping mortgages, compare APRs — not headline rates — for an apples-to-apples cost comparison.