APY Formula (Annual Percentage Yield)
Reference for APY = (1 + r/n)^n - 1.
Calculates annual yield from nominal rate and compounding frequency — daily, monthly, or quarterly — for savings and CDs.
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The Formula
APY = (1 + r/n)^n - 1
Variables
| Symbol | Meaning |
|---|---|
| APY | Annual Percentage Yield (actual yearly return) |
| r | Nominal annual interest rate (as a decimal) |
| n | Number of compounding periods per year |
Compounding Frequencies
| Frequency | n | APY at 5% rate |
|---|---|---|
| Annually | 1 | 5.000% |
| Semi-annually | 2 | 5.063% |
| Quarterly | 4 | 5.095% |
| Monthly | 12 | 5.116% |
| Daily | 365 | 5.127% |
| Continuously | ∞ | 5.127% (e^r - 1) |
APY vs APR
- APY — what you earn on savings and investments
- APR — what you pay on loans and credit cards
- Higher APY = better for savings
- Lower APR = better for borrowing
Example
A savings account pays 5% interest, compounded monthly. What is the APY?
APY = (1 + 0.05/12)^12 - 1
APY = (1.004167)^12 - 1
APY = 1.05116 - 1 = 5.116%
The actual yield is 5.116%, not 5%, thanks to monthly compounding.
On a $10,000 deposit, that is an extra $11.60 per year from compounding alone.
Limitations
- APY assumes the stated interest rate remains constant for the full year — variable-rate accounts will have a different actual yield
- Account maintenance fees and taxes on earned interest reduce your real return below the advertised APY
- APY describes earnings on savings and CDs only — for the cost of borrowing, use APR (Annual Percentage Rate) instead
- Continuously compounded APY (e^r - 1) is the theoretical maximum; daily compounding is nearly identical in practice
Key Notes
- Formula: APY = (1 + r/n)^n − 1: r is the nominal annual interest rate and n is the number of compounding periods per year (12 for monthly, 365 for daily). APY represents the true annual return including the effect of compounding.
- APY vs APR: APY (Annual Percentage Yield) always equals or exceeds APR (nominal rate). The difference grows with both the rate and compounding frequency. For 6% APR: monthly compounding gives APY ≈ 6.168%; daily gives APY ≈ 6.183%.
- Continuous compounding limit: APY = e^r − 1: As n → ∞, the compounding frequency reaches the continuous limit. For 6% nominal rate, continuous APY ≈ 6.184% — only marginally more than daily compounding.
- Legal disclosure requirement: U.S. banks must disclose APY (not just the nominal rate) under the Truth in Savings Act. This lets consumers directly compare accounts with different compounding frequencies on equal terms.
- Applications: APY is used to compare savings accounts, CDs, and money market accounts. For borrowing (credit cards, loans), the analogous measure is APR — which is disclosed by law but does not include compounding effects the way APY does for savings.