Convert a nominal annual interest rate to APY, or APY back to an equivalent nominal rate, using the selected compounding frequency. Optionally illustrate one year of growth on a fixed USD balance.
APR to APY Formula
In this calculator, APR means the nominal annual interest rate before compounding. APY is the effective annual rate after compounding. Disclosed loan APR may include certain fees and can differ from the nominal interest rate; see the CFPB explanation of mortgage APR. To convert a nominal annual rate 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 interest rate for this model, 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 | Disclosed APR and loan terms | Review fees, payments and balance changes. This interest-only conversion does not determine the full loan cost. |
| 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 hypothetical fixed balance accrues an 18% nominal annual interest rate with monthly compounding and no payments or fees. Then (1 + 0.18/12)^12 – 1 = 0.195618, or an effective annual rate of about 19.56%. Actual credit-card costs depend on the contract, timing, payments and fees.
Frequently Asked Questions
Is APY always higher than APR? For a positive nominal rate in this model, APY is higher than APR whenever interest compounds more than once per year. At zero interest, both rates are zero. 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.
Does this calculate my total loan cost? No. It converts interest rates under a constant-rate compounding model. Loan cost also depends on fees, the amount and timing of payments, and balance changes. Use the lender disclosures and repayment schedule to assess an actual loan.
