APR vs APY calculator and formula
APR is the yearly rate before compounding. APY is the same rate after compounding. APY = (1 + APR/n) to the power n, minus 1. A 5 percent APR compounded monthly is a 5.116 percent APY, so $10,000 earns $511.62 in a year rather than $500.
APY, the rate you actually get
5.116%
5.00% APR compounded monthly works out at 5.116% over a year.
- APR (nominal yearly rate)
- 5.000%
- APY (effective yearly rate)
- 5.116%
- Gap
- 0.116 points
- Interest on $10,000.00 in year one
- $511.62
Same 5.00% APR at every compounding frequency
| Compounding | APY | On $10,000.00 |
|---|---|---|
| Annually | 5.000% | $500.00 |
| Quarterly | 5.095% | $509.45 |
| Monthly | 5.116% | $511.62 |
| Daily | 5.127% | $512.67 |
APR is the quoted yearly rate. APY is what you actually earn or owe.
The formula
is the nominal annual rate as a decimal and is how many times a year interest is added. The result is the effective annual rate.
The difference in one line
APR is a rate you quote. APY is a rate you get.
An APR describes one period's rate multiplied up to a year. If a card charges 2 percent a month, its APR is 24 percent, and that calculation ignores the fact that unpaid interest starts earning interest. APY accounts for that, which is why it is always the higher number whenever interest is added more than once a year.
They are also the terms a lender and a bank are each required to show you, which is why the same account can advertise both.
The formula, both directions
To go from a quoted rate to the rate you actually get:
To go back the other way, when you know the yield and want the nominal rate behind it:
Both take , the compounding frequency, and neither means anything without it. An APY quoted with no frequency is complete on its own, because compounding is already inside it. An APR quoted with no frequency is not.
The gap between the two widens as rises, but with a ceiling. At 5 percent, monthly compounding gives 5.116 percent and daily gives 5.127 percent. Compounding every instant, the mathematical limit, gives 5.127 percent as well to three decimal places. There is far less in the frequency than the marketing suggests.
What APR includes on a loan
On savings the two terms differ only by compounding. On borrowing there is a second difference, and it runs the other way.
In the United States, the APR a lender quotes on a mortgage or a personal loan is required to fold in certain fees and charges, not just the interest rate on the note. That makes the APR higher than the note rate, and it is the reason a loan can advertise one rate in large type and a higher APR in small type. The APR is the more honest comparison of two offers with different fee structures.
What that APR still does not do is compound. So on a loan you have two adjustments pulling in opposite directions: fees pushing the APR above the note rate, and compounding pushing the true annual cost above the APR. The loan payment calculator works in note-rate terms, which is what actually drives the payment schedule.
Where the difference shows up
On a savings balance the gap is small and in your favour. On a revolving debt it is larger and against you, because card rates are high and the compounding is monthly or daily.
- A savings account quoting APY has already done this calculation for you.
- A card quoting APR has not.
- Two accounts with the same APR and different compounding are not the same account.
The compound interest calculator runs the same mechanism forward over many years, where these fractions of a point stop being fractions.
Worked examples
A 5 percent savings rate compounded monthly
An account quotes 5 percent APR and adds interest monthly. What is the APY, and what does $10,000 earn in a year?
- Find the period rate: a month.
- Grow one dollar for twelve periods: .
- Subtract the dollar you started with: , so the APY is 5.116 percent.
- The gap over the quoted rate is 0.116 percentage points.
- Apply it to the balance: $511.62.
- Compare with the headline rate applied once: $500.
The APY is 5.116 percent, and $10,000 earns $511.62 over the year instead of $500. The extra $11.62 is interest that was itself earned by interest.
A credit card at 24.99 percent
A card advertises 24.99 percent APR and compounds monthly. What does carrying $5,000 for a year actually cost?
- The monthly rate is , just over 2 percent a month.
- Compound it for twelve months: .
- The effective annual rate is 28.0606 percent, not 24.99 percent.
- Apply it to the balance: $1,403.03.
Carrying $5,000 for a year costs $1,403.03, which is 28.0606 percent of the balance rather than the 24.99 percent on the offer. On a card, the compounding works against you and the gap is three full percentage points.
Going backwards from a quoted yield
A bank advertises a 4.5 percent APY and compounds monthly. What nominal rate is behind it?
- Rearrange the formula: .
- The twelfth root of 1.045 is 1.0036748.
- Subtract one and multiply by twelve: .
The nominal rate is 4.4098 percent. A bank advertising 4.5 percent APY is paying 4.4098 percent APR, and both numbers describe the same account. This is why comparing one bank's APY against another's APR tells you nothing useful.
The mistake that costs the most
Comparing one product's APR against another's APY.
They are different units. An account advertising 5.05 percent APY loses to one advertising 5 percent APR compounded monthly, because that 5 percent APR is worth 5.116 percent APY. The larger headline number is the worse account, and nothing on either page tells you so.
The fix is to convert everything to APY before comparing anything, including debts. Two offers are comparable only once both are expressed as what you would actually earn or actually pay over a year.
Common questions
Which one is bigger, APR or APY?
APY, whenever interest is added more than once a year. They are equal only when compounding happens exactly once a year. The more often interest is added, the wider the gap, though it approaches a ceiling rather than growing without limit.
Why does my card show an APR and my savings account an APY?
In the United States, lenders disclose credit costs as an APR under the Truth in Lending Act, and deposit accounts disclose returns as an APY under the Truth in Savings Act. The two rules were written for different products, which is why the same idea arrives under two names.
Does a higher compounding frequency ever stop helping?
It approaches a limit. At 24.99 percent, monthly compounding gives 28.0606 percent and daily gives 28.3787 percent, and no frequency can push past the continuous-compounding ceiling just above that. The rate matters far more than the frequency.
Keep reading
This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.