Calculate APY from APR for any compounding frequency, and convert APY back to APR, to compare the true yearly return on savings or the real cost of a loan.
APR to APY Formula
APR is the nominal annual rate before compounding. APY is the effective annual rate after compounding is applied. To convert an APR to an APY for a given number of compounding periods per year, use:
APY = (1 + APR/n)^n - 1
When interest compounds continuously, the number of periods becomes infinite and the formula becomes:
APY = e^(APR) - 1
To go the other way and find the APR that produces a known APY at a set compounding frequency, rearrange the first formula:
APR = n * ((1 + APY)^(1/n) - 1)
For continuous compounding the reverse formula is:
APR = ln(1 + APY)
- APR = nominal annual percentage rate, written as a decimal in the formula (5% is 0.05)
- APY = annual percentage yield, the effective yearly rate after compounding
- n = number of times interest compounds per year (12 for monthly, 365 for daily)
- e = the constant 2.71828, used for continuous compounding
The solve-for selector controls which pair of formulas runs. Choosing “APY from APR” takes the rate you enter as the APR and returns the APY. Choosing “APR from APY” treats your entry as the APY and returns the equivalent nominal APR. The compounding frequency you pick sets n, which is the only other input the conversion needs. The optional balance field does not change the rate; it multiplies the resulting APY by your starting balance to show the interest earned and the ending balance after one year.
APY by Compounding Frequency
The more often interest compounds, the higher the APY for the same APR. This table shows the APY produced by a 5% APR at several common frequencies.
| Compounding frequency | Periods per year (n) | APY on a 5% APR |
|---|---|---|
| Annually | 1 | 5.0000% |
| Semiannually | 2 | 5.0625% |
| Quarterly | 4 | 5.0945% |
| Monthly | 12 | 5.1162% |
| Daily | 365 | 5.1267% |
| Continuously | infinite | 5.1271% |
Use the next table to decide which number matters for your situation.
| Situation | Rate to use | Why |
|---|---|---|
| Saving or investing | APY | It shows the full yearly return after compounding is added in, so it is the number to compare across accounts. |
| Borrowing | APR plus APY | APR is the quoted rate, but the APY shows the true yearly cost once interest compounds on the balance. |
| Comparing two offers | APY | Convert each quoted APR to APY first so the compounding schedules are put on equal footing. |
Example Problems
Example 1. A savings account quotes a 5% APR that compounds monthly. Set the solve-for to “APY from APR”, enter 5, and choose monthly. The calculator divides 0.05 by 12, adds 1, raises the result to the 12th power, and subtracts 1: (1 + 0.05/12)^12 – 1 = 0.051162, or an APY of 5.1162%.
Example 2. A credit card states an 18% APR that compounds monthly. Using the same steps, (1 + 0.18/12)^12 – 1 = 0.195618, so the true yearly cost is an APY of about 19.56%. The gap between the quoted 18% and the 19.56% you actually pay comes from interest compounding on the balance each month.
Frequently Asked Questions
Is APY always higher than APR? APY is higher than APR whenever interest compounds more than once per year. If interest compounds only once a year, the two rates are equal. The more frequent the compounding, the larger the gap, though the increases get smaller as you move from monthly to daily to continuous.
Which rate should I compare between accounts? Compare APY. Two accounts can quote the same APR but pay different amounts if they compound on different schedules. APY folds the compounding schedule into a single number, so comparing APY to APY is an apples-to-apples comparison.
Why does my loan cost more than the APR suggests? The quoted APR does not include the effect of compounding within the year. When interest is charged monthly or daily, you pay interest on previously added interest, which raises the effective yearly cost to the APY. For loans the APR also may exclude certain fees, so check the lender terms for the full picture.
