When my bank offered a savings account at "5.00% APY," I assumed that was the same as 5% interest. It's not. And when a credit card advertises "24.99% APR," that's not the real cost either. The difference between APR and APY is compounding, and it's the single most important concept in personal finance that most people never learn.
What APR actually means
APR stands for Annual Percentage Rate. It's the simple interest rate over a year, without accounting for compounding. If you borrow $1,000 at 12% APR and interest is calculated once at the end of the year, you owe $1,120. Simple.
But interest is almost never calculated once a year. Credit cards compound daily. Mortgages compound monthly. The APR doesn't tell you how often compounding happens, and that omission is where the real cost hides.
What APY actually means
APY stands for Annual Percentage Yield. It includes the effect of compounding. It tells you the actual amount you'll earn (or owe) over a year when interest is applied to previously accumulated interest.
The formula:
APY = (1 + r/n)^n - 1
Where r is the nominal annual rate (APR as a decimal) and n is the number of compounding periods per year.
Let's run the numbers for a 12% APR:
| Compounding | n | APY |
|---|---|---|
| Annually | 1 | 12.000% |
| Semi-annually | 2 | 12.360% |
| Quarterly | 4 | 12.551% |
| Monthly | 12 | 12.683% |
| Daily | 365 | 12.747% |
| Continuously | inf | 12.750% |
That 12% APR becomes 12.747% APY when compounded daily. On a $10,000 balance, that's the difference between $1,200 and $1,274.70 in interest. On larger balances or longer time periods, the gap grows substantially.
The rule banks play by
Banks are required by the Truth in Savings Act to quote APY on deposit accounts (savings accounts, CDs). This is consumer-friendly because APY shows you what you'll actually earn.
For loans, however, the Truth in Lending Act requires APR. This is less helpful because APR understates the true cost when compounding is frequent. A credit card with 24.99% APR compounded daily has an effective APY of 28.35%. That's the rate you're really paying.
This asymmetry isn't an accident. When selling you a savings product, the bank advertises the higher-looking number (APY). When selling you a loan, they advertise the lower-looking number (APR). Both are technically honest. Neither tells the full story without the other.
Compounding frequency matters more than you think
Here's a concrete example. You put $10,000 in a savings account at 5% nominal rate. After 10 years:
- Compounded annually: $16,288.95
- Compounded monthly: $16,470.09
- Compounded daily: $16,486.65
The difference between annual and daily compounding on $10,000 over 10 years is about $198. Not life-changing. But scale that to $100,000 or extend the time horizon to 30 years, and the numbers diverge significantly.
After 30 years at 5% on $100,000:
- Compounded annually: $432,194
- Compounded daily: $448,169
That's nearly $16,000 more from daily compounding on the same nominal rate. Over a lifetime of investing, compounding frequency is one of the variables that separates good outcomes from great ones.
The continuous compounding limit
As n approaches infinity, the APY formula converges to:
APY = e^r - 1
Where e is Euler's number (approximately 2.71828). For a 5% nominal rate:
APY = e^0.05 - 1 = 0.05127 = 5.127%
Continuous compounding is the mathematical upper bound. No finite compounding frequency can exceed it. In practice, daily compounding gets you 99.97% of the way to continuous compounding, which is why most financial institutions don't bother with anything more frequent than daily.
This is also why e shows up everywhere in finance. It's the natural limit of the compounding process.
Common mistakes when comparing rates
Comparing APR to APY directly. When one bank offers 4.90% APY and another offers 5.00% APR compounded monthly, the second bank's APY is actually 5.116%. You need to convert everything to the same basis before comparing.
Ignoring the compounding frequency. Two accounts can have the same APR and different APYs. A 5% APR compounded monthly gives you more than 5% APR compounded quarterly. Always ask how often interest compounds.
Forgetting that APY works against you on debt. The same compounding effect that grows your savings also grows your debt. A 24.99% APR credit card compounded daily costs you more than 24.99% per year. If you're carrying a balance, the effective rate is what matters.
Assuming rates are fixed. Many savings accounts advertise a high APY that's variable. The bank can change it at any time. A 5.00% APY today could be 3.50% next quarter. CDs lock in the rate, but savings accounts typically don't.
Calculating APY yourself
If you want to do this in code:
function calculateAPY(apr, compoundingPeriodsPerYear) {
return Math.pow(1 + apr / compoundingPeriodsPerYear,
compoundingPeriodsPerYear) - 1;
}
// 5% APR compounded monthly
console.log(calculateAPY(0.05, 12)); // 0.05116 = 5.116% APY
// 24.99% APR compounded daily
console.log(calculateAPY(0.2499, 365)); // 0.28355 = 28.355% APY
For quick calculations without writing code, I built an APY calculator at zovo.one/free-tools/apy-calculator that converts between APR and APY and lets you compare different compounding frequencies side by side.
Understanding the math behind interest rates isn't optional if you're making financial decisions. The difference between APR and APY is the difference between what the bank tells you and what actually happens to your money.
I'm Michael Lip. I build free developer tools at zovo.one. 350+ tools, all private, all free.
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