APY Calculator
Turns a nominal interest rate into the effective annual yield, so two quoted rates can be compared honestly.
Last updated
Your details
Optional — shows what the yield is worth in money after a year.
Result
- Effective annual yield (APY)
- 12.68%
- Nominal rate (APR)
- 12.00%
- Compounding gains you
- 0.68%
- Balance after one year
- $11,268.25
- Interest earned
- $1,268.25
Compounded 12 times a year
The gap between the quoted rate and what you actually earn
12.00% compounded 12 times a year…
APY = (1 + r ÷ n)ⁿ − 1
r = 12.00% = 0.1200, n = 12
1 + 0.1200 ÷ 12 = 1.01000000
(1.01000000)^12 = 1.126825
APY = 12.68%
Formula — written step by step
12% nominal, compounded monthly…
APY = (1 + r ÷ n)ⁿ − 1
r = 12% = 0.12, n = 12
0.12 ÷ 12 = 0.01
(1.01)¹² = 1.126825
APY = 12.6825%
r is the nominal annual rate as a decimal and n the number of times interest is added each year. The extra 0.6825 points over the quoted 12% is interest earning interest — the whole reason compounding frequency matters.
A worked example
Two accounts, same headline rate
- Account A
- 5.00% compounded annually
- Account B
- 4.95% compounded daily
- Deposit
- 10,000
- A: APY = (1 + 0.05 ÷ 1)¹ − 1 = 5.0000%
- B: APY = (1 + 0.0495 ÷ 365)³⁶⁵ − 1
- = (1.00013562)³⁶⁵ = 1.050745
- B: APY = 5.0745%
Account B wins, despite advertising the lower rate
B pays 507.45 in a year against A’s 500.00. The lower nominal rate compounds often enough to overtake the higher one — which is exactly why comparing quoted rates rather than yields leads people to the worse account.
How to use this calculator
- Enter the nominal annual rate — the headline percentage the bank advertises, sometimes labelled APR or "interest rate".
- Choose how often interest is added to the balance. If the account documents say "compounded daily", pick daily; the difference from monthly is small but real.
- Read the APY. This is the figure to compare between accounts, because it already accounts for compounding.
- Add an opening deposit to see what the difference is worth in money over a year rather than in percentage points.
Why APY and APR are different numbers
A nominal rate — often quoted as APR on a savings product — is the yearly rate before compounding is taken into account. The annual percentage yield is what you actually earn once interest starts earning interest. The two are equal only when interest is added exactly once a year.
The gap grows with frequency. Twelve percent compounded monthly is 12.6825%; compounded daily it is 12.7475%. Both accounts advertise "12%". Only the APY tells you which one leaves more in your account in December.
The ceiling on compounding
It is tempting to assume that compounding more often keeps producing more, without limit. It does not. As the number of periods rises, the yield converges on a fixed value: e raised to the power of the rate, minus one. For a 12% nominal rate that limit is 12.7497%.
This is why the jump from annual to monthly compounding is worth two thirds of a percentage point, while the jump from daily to continuous is worth about two thousandths. Once an account compounds daily, the compounding frequency has essentially stopped being a differentiator, and the nominal rate is what matters again.
Reading the number a bank gives you
In the United States, deposit accounts must disclose APY under Regulation DD, calculated on a standard basis so that advertised yields are genuinely comparable. Loans, by contrast, disclose APR under Regulation Z — which is why a savings account and a credit card quote different kinds of number and cannot be compared directly.
Watch for introductory rates that revert after a few months, minimum balances below which the headline rate does not apply, and caps that pay the advertised yield only on the first portion of the balance. None of these appear in the formula, and all of them change what you receive.
Formula & methodology
What this calculator assumes
- Assumes the rate is fixed for the whole year and that interest is left to compound rather than withdrawn.
- Ignores fees, minimum-balance conditions and introductory rates that revert after a promotional period — all of which change what you actually earn.
- In the United States, APY on deposit accounts is defined by Regulation DD; the formula here matches that definition but a bank’s disclosed figure may differ where its compounding convention does.
Sources
- US Consumer Financial Protection Bureau, Regulation DD (12 CFR Part 1030, 2011), Appendix A — Annual Percentage Yield calculation.
A 12% nominal rate at different compounding frequencies
| Compounded | Periods a year | APY |
|---|---|---|
| Annually | 1 | 12.0000% |
| Semi-annually | 2 | 12.3600% |
| Quarterly | 4 | 12.5509% |
| Monthly | 12 | 12.6825% |
| Weekly | 52 | 12.7341% |
| Daily | 365 | 12.7475% |
| Continuously | ∞ | 12.7497% |
The gain from compounding more often shrinks rapidly. Moving from annual to monthly adds 0.68 points; moving from daily to continuous adds 0.0022. There is a hard ceiling at eʳ − 1, and daily compounding is already almost exactly at it.
Frequently asked questions
Is APY the same as effective annual rate?
Yes. APY, EAR and effective annual yield are three names for the same calculation. APY is the term used for deposit accounts in the United States; EAR is more common in textbooks and in corporate finance.
Why does my bank quote a rate lower than the APY?
Because the quoted rate is nominal. The bank adds interest several times a year, and each addition earns interest for the rest of the year, so the yield ends up higher than the headline. Both figures are correct; they measure different things.
What does continuous compounding mean?
It is the mathematical limit of compounding infinitely often, calculated as e raised to the power of the rate. No retail account works this way, but it sets the ceiling: no compounding frequency can ever beat it.
Does APY account for fees?
No. It describes interest only. A monthly account fee can easily exceed the interest earned on a small balance, turning a positive yield into a net loss.