APR and APY: not the same number.
APR is what's quoted on borrowing. APY is what's quoted on saving. They differ by exactly the amount that compounding contributes — and that gap is the reason they are quoted separately.
The definitions
- APR (Annual Percentage Rate). A nominal annual rate that does not include the effect of compounding. Required by US Truth in Lending regulation (Regulation Z) on consumer credit disclosures. A loan APR is supposed to include certain fees but not intra-period compounding.
- APY (Annual Percentage Yield). The effective annual yield after compounding. Required by US Truth in Savings regulation (Regulation DD) on deposit account disclosures. The APY is what you actually earn over a year if you make no withdrawals.
Why lenders quote APR and savers see APY
The framing reflects each institution's marketing interest:
- For a borrower, APR understates the true annual cost (because compounding adds to it). Showing APR makes the loan look cheaper than the realised interest charge.
- For a saver, APY includes compounding, so the headline number is the larger of the two. Showing APY makes the deposit account look more attractive.
The two regulations are deliberately asymmetric for this reason: they force each side of the market to disclose the figure that is least flattering to the institution.
The conversion
For a nominal rate compounded n times per year:
APY = (1 + APR/n)n − 1
And the inverse:
APR = n × ((1 + APY)1/n − 1)
Conversion table at common rates
| APR | APY (monthly) | APY (daily) | APY (continuous) |
|---|---|---|---|
| 1.00% | 1.0046% | 1.0050% | 1.0050% |
| 3.00% | 3.0416% | 3.0453% | 3.0455% |
| 5.00% | 5.1162% | 5.1267% | 5.1271% |
| 7.00% | 7.2290% | 7.2501% | 7.2508% |
| 10.00% | 10.4713% | 10.5156% | 10.5171% |
| 15.00% | 16.0755% | 16.1798% | 16.1834% |
| 20.00% | 21.9391% | 22.1336% | 22.1403% |
| 25.00% | 28.0732% | 28.3916% | 28.4025% |
| 30.00% | 34.4889% | 34.9692% | 34.9859% |
Where this matters in real decisions
- Comparing a credit card APR to a savings APY. Convert both to the same basis before comparing. A 22 % credit-card APR with daily compounding is an APY of roughly 24.6 % — that is the rate the balance is actually growing at. A 4.5 % savings APY converts to an APR of 4.41 % with daily compounding.
- Mortgage shopping. US mortgages are quoted in APR, which includes certain origination costs but is computed monthly. The realised APY for the borrower is slightly higher; the APR is the right number for comparison shopping.
- Promotional accounts. A “5 % APY for the first 90 days” is annualised over a 90-day window. The interest you actually earn on $10,000 over those 90 days is roughly $123, not $500.
How to use this with the calculator
The main calculator takes a nominal annual rate (which is the APR if quoted on borrowing). To enter an APY into the calculator, first convert it to a nominal rate using the formula above, then select the matching compounding frequency. The future-value figure will be identical either way; the conversion just keeps the inputs internally consistent.