APR and APY measure the same interest rate from different angles. APR (Annual Percentage Rate) is the rate before compounding is applied — the number you usually see advertised. APY (Annual Percentage Yield) is what you actually earn or pay after compounding, and it is always higher than or equal to APR. The gap between them grows with more frequent compounding. At 6% APR compounded daily, the APY is 6.183% — a difference that compounds into real money over time.
The formula
APR to APY:
APY = (1 + APR/n)^n − 1
APY to APR:
APR = n × ((1 + APY)^(1/n) − 1)
Where:
- APR = Annual Percentage Rate (as a decimal, e.g., 6% = 0.06)
- APY = Annual Percentage Yield (as a decimal)
- n = number of compounding periods per year
Common values of n: 1 (annual), 2 (semi-annual), 4 (quarterly), 12 (monthly), 365 (daily).
Source: Consumer Financial Protection Bureau — APR vs APY.
Practical examples
Example 1: Monthly compounding
A savings account offers 5% APR compounded monthly. What is the APY?
APY = (1 + 0.05/12)^12 − 1
= (1.004167)^12 − 1
= 1.05116 − 1
= 0.05116 = 5.116%
The account earns 5.116% per year, not 5%.
Example 2: Daily compounding
Same 5% APR, compounded daily:
APY = (1 + 0.05/365)^365 − 1
= 1.05127 − 1
= 5.127%
Daily compounding adds another 0.011 percentage points over monthly.
Example 3: Converting APY back to APR
A CD advertises 5.25% APY with monthly compounding. What APR did they use?
APR = 12 × ((1 + 0.0525)^(1/12) − 1)
= 12 × (1.0042659 − 1)
= 12 × 0.0042659
= 5.119%
Common mistakes
Using APR directly as a yield. If a savings account compounds monthly at 5% APR, you do not earn exactly $500 on $10,000 in a year — you earn $511.60 (5.116% × $10,000). The difference seems small but compounds significantly over multiple years.
Comparing APR from one account to APY from another. Banks must disclose APY for savings products (required by the US Truth in Savings Act), but loan rates are often quoted as APR. Comparing them directly is apples-to-oranges.
Assuming the same APR means the same cost on any loan. Loan APR disclosures often include fees beyond the interest rate, making them not directly comparable to the APR/APY formula used here. Mortgage APR in the US includes origination fees; deposit APR/APY does not.
Forgetting that n is compounding periods, not payment periods. Some loans are paid monthly but interest accrues daily. The compounding frequency (n) for the formula refers to how often interest is calculated and added to the balance.
International and regional variations
| Country / Region | Equivalent term | Legal disclosure requirement |
|---|---|---|
| United States | APY (savings), APR (loans) | APY required for deposits (Truth in Savings Act); APR required for loans (TILA) |
| United Kingdom | AER (Annual Equivalent Rate) | AER = APY equivalent; required for savings by FCA. EAR used for loans. |
| European Union | EAR / APRC | EAR (Effective Annual Rate) for savings; APRC for consumer credit |
| Canada | EAR / EFR | Canadian mortgages use semi-annual compounding by law; EAR must be disclosed |
| Australia | Comparison Rate | Comparison Rate includes fees; different concept from APY/AER |
Quick reference: APR to APY at common rates
| APR | Annual (n=1) | Quarterly (n=4) | Monthly (n=12) | Daily (n=365) |
|---|---|---|---|---|
| 1% | 1.000% | 1.004% | 1.005% | 1.005% |
| 3% | 3.000% | 3.034% | 3.042% | 3.045% |
| 5% | 5.000% | 5.095% | 5.116% | 5.127% |
| 7% | 7.000% | 7.186% | 7.229% | 7.250% |
| 10% | 10.000% | 10.381% | 10.471% | 10.516% |
| 15% | 15.000% | 15.865% | 16.075% | 16.180% |
Annual compounding (n=1) produces APY = APR exactly. All other frequencies produce APY > APR.