Scope and assumptions
APR = nominal annual rate; APY = effective rate after compounding n times/yr. Illustrative.
Two savings accounts advertise 5%. One credits interest annually, the other monthly. Over a decade they produce noticeably different balances, and neither bank is lying — they're quoting different things. The first number is a nominal annual rate; the second product's real yield depends on how often it compounds.
Understanding which number you're looking at is worth more than any single rate comparison.
Two rates, one relationship
| Term | What it is | Typical use |
|---|---|---|
| APR | Nominal annual rate, before compounding | Loan and card marketing |
| APY | Effective annual rate, after compounding | Savings and deposit marketing |
APY = (1 + APR ÷ n)^n − 1
n is the number of compounding periods per year. When n = 1, APY equals APR exactly; as n rises, APY pulls away.
The conversion, at realistic rates
| APR | Compounding | APY |
|---|---|---|
| 12% | Monthly | 12.68% |
| 12% | Daily | 12.75% |
| 6% | Monthly | 6.17% |
| 3% | Monthly | 3.04% |
A 12% APR compounded monthly is genuinely a 12.68% annual cost. The extra 0.68 points come from interest being charged on interest twelve times a year.
Why the gap grows with frequency
Each compounding period adds interest to the balance, and the next period charges interest on that larger balance. More periods means more opportunities for that feedback loop.
The growth is diminishing, though. Moving from annual to monthly adds 0.68 points at 12%; monthly to daily adds only 0.07. There's a ceiling — continuous compounding — and daily already sits within a hair of it.
Borrowers and savers face opposite directions
The same mechanics cut both ways:
- Borrowing: the higher APY is what you actually pay. A "low" APR compounded daily can cost more than a slightly higher APR compounded annually.
- Saving: the higher APY is what you actually earn, so APY is the number to compare across accounts.
The formula doesn't change sides; the sign of the outcome does.
What the headline doesn't tell you
APR for regulated credit products is meant to include certain fees, not just the interest rate, which makes it broader than a bare percentage. Deposit APYs usually exclude account fees and promotional expiry. In both cases, the advertised number is a starting point, not the full cost.
Why APR and APY Get Confused
Treating APR as the real annual rate; forgetting the "− 1" after compounding; comparing a savings APY against a loan APR as if they were the same measure; ignoring compounding frequency when two APRs match; assuming a promotional APY lasts; and mixing monthly periods into a formula that expects years.
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Compound Growth Percentage and Doubling Time Percentage.
FAQ
Is APY always higher than APR? Whenever compounding occurs more than once a year, yes. With annual compounding the two are identical; anything more frequent pushes APY above APR.
How do I compare a 5.1% APY account with a 5% APR account? Convert both to APY. A 5% APR compounded monthly is about 5.12% APY, so it edges out the 5.1% APY offer — the opposite of what the headline numbers suggest.
Does APR include fees as well as interest? For mortgages and loans, the regulated APR is meant to fold in certain fees, making it broader than the interest rate alone. Store cards and informal quotes often don't.
What's the difference between nominal and effective rate? Nominal is the quoted annual rate before compounding (APR); effective is after compounding (APY). Same relationship, different vocabulary depending on the product.
Sources: CFPB, APR; Investopedia, APY.
Disclaimer: Illustrative. Actual terms vary by product and jurisdiction. Informational only.
Sources and references
- CFPB — APRhttps://www.consumerfinance.gov/ask-cfpb/what-is-a-loan-annual-percentage-rate-apr-en-135/
- Investopedia — APYhttps://www.investopedia.com/terms/a/apy.asp
Maintenance and corrections
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