At 5%, ten thousand dollars earns either $500 or $511.62 over a year. Both are correct. Which one you get depends on a single word in the paperwork: compounding.
APR and APY describe the same underlying rate and arrive at different numbers because of it.
- APR, the annual percentage rate, is the nominal rate. It is the periodic rate multiplied out to a year, and it ignores the fact that interest earns interest.
- APY, the annual percentage yield, is what you actually end up with after compounding has been applied for a year.
Whenever interest compounds more than once a year, APY is the larger number.
The formula
APY = (1 + r/n)^n - 1
where r is the nominal annual rate as a decimal and n is the number of
compounding periods in a year.
At 5% compounded monthly:
(1 + 0.05/12)^12 - 1
= (1.00416666...)^12 - 1
= 1.0511618... - 1
= 0.0511618
= 5.12% APY
So a 5% APR account compounding monthly yields 5.12% over a year. On $10,000 that is $511.62 rather than $500. The extra $11.62 exists purely because each month’s interest went on to earn interest of its own.
What each frequency is worth
The same 5% nominal rate, compounded differently:
| Compounding | APY |
|---|---|
| Annually | 5.000% |
| Quarterly | 5.095% |
| Monthly | 5.116% |
| Daily | 5.127% |
The pattern is worth internalising: the first step matters most and each one after matters less. Going from annual to monthly adds about 0.12 points. Going from monthly to daily adds about 0.01.
A bank advertising “compounded daily” is offering something real but small. A bank offering 0.25 points more on the headline rate is offering roughly twenty times as much. When comparing accounts, the rate is almost always the term that decides it.
Why you get quoted different ones
This is not marketing preference. Two different laws are at work.
Deposit accounts fall under the Truth in Savings Act, implemented as Regulation DD. It requires institutions to disclose the annual percentage yield, and it specifies the calculation precisely, in Appendix A to Part 1030. The point of mandating one basis is comparability: without it, a bank could quote a nominal rate with an unusual compounding frequency and look better than it is.
Consumer credit falls under the Truth in Lending Act, implemented as Regulation Z, 12 CFR 1026, which requires the annual percentage rate.
So the asymmetry a saver notices, that deposits are quoted in the number that flatters them and loans in the number that does not, follows from two statutes written decades apart for different purposes.
What this means when you borrow
On a credit card, the quoted APR is divided down to a periodic rate. A 22.9% APR becomes a monthly rate of:
22.9 / 100 / 12 = 0.0190833... (1.90833% a month)
Because that division ignores compounding, carrying a balance for a full year costs slightly more than 22.9%. Compounding the monthly rate gives:
(1 + 0.0190833)^12 - 1 = 25.463%
The card is quoted at 22.9% and the effective annual cost of carrying the balance is about 25.46%. Nothing improper has happened; APR is defined as the nominal rate and the issuer disclosed it correctly. It is simply a different measure from the one your savings account uses.
Our APY calculator converts in both directions, and every page on this site states its compounding assumption next to the result, because a yield figure with an unstated frequency is not checkable.
Questions
- Which number is bigger, APR or APY?
- APY, whenever interest compounds more often than once a year, because APY includes the effect of earning interest on interest and APR does not. At a 5% nominal rate compounded monthly the APY is about 5.12%. The two are equal only when compounding is annual.
- Why does my savings account quote APY but my credit card quotes APR?
- Because different rules govern each. Deposit accounts fall under the Truth in Savings Act and Regulation DD, which requires the annual percentage yield so savers can compare accounts on one basis. Consumer credit falls under Truth in Lending and Regulation Z, which requires the annual percentage rate.
- Does a higher compounding frequency make a large difference?
- Less than most people expect, and it shrinks as you go. Moving from annual to monthly compounding at 5% adds about 0.12 percentage points. Moving from monthly to daily adds roughly another 0.01. The gap between two banks' rates almost always matters more than the gap between their compounding frequencies.
- Is APR on a credit card the same as the rate I am charged monthly?
- Not exactly. US issuers typically divide the APR by twelve to get a monthly periodic rate, or by 365 for a daily one. Because that division ignores compounding, the effective annual cost of carrying a balance is slightly higher than the quoted APR.