Banking
How To Calculate APY
The annual percentage yield is what a deposit account actually returns in a year once compounding is counted. Comparing two accounts by their headline rates instead of their APYs is the most common way savers pick the slower one.
Quick Answer
APY = (1 + r/n)^n - 1
- APY
- Annual percentage yield — the effective annual return
- r
- Nominal annual rate as a decimal (APR)
- n
- Number of compounding periods per year
- e
- Euler's number, about 2.71828, used for continuous compounding
APY equals one plus the nominal annual rate divided by the number of compounding periods per year, raised to the power of that number of periods, minus one. Interest compounded monthly at a nominal 5% gives an APY of 5.11619%, so $10,000 grows to $10,511.62 over one year. Continuous compounding uses e^r - 1 instead.
What Is APY?
A nominal rate is a statement about units, not about outcomes. An APR of 5% compounded monthly means one-twelfth of 5% is applied twelve times; because each application lands on a slightly larger balance than the last, the year produces more than 5%. The APY is the single number that captures that entire process — what one dollar actually becomes over twelve months.
The conversion is straightforward and worth doing by hand once. Divide the nominal rate by the number of compounding periods, add one, raise to the number of periods, subtract one. At 5% nominal with monthly compounding: (1 + 0.05/12)^12 - 1 = 0.0511619, an APY of 5.11619%. Compounded daily instead: (1 + 0.05/365)^365 - 1 = 0.0512675. Compounded annually, the APY is simply 5.00000%, because there is nothing to compound within the year.
The pattern is that more frequent compounding raises APY, but with sharply diminishing returns. Going from annual to monthly adds 0.11619 percentage points on a 5% rate; going from monthly to daily adds another 0.01056 points, which is $1.05 a year on a $10,000 deposit. Adding continuous compounding pushes the limit to e^0.05 - 1 = 5.12711%, only 0.00036 points above daily. Once you are at daily, the frequency question is settled — look at the rate instead.
This diminishing curve produces a genuinely counterintuitive result that trips up comparison shopping. Take two deposit offers: 5.00% compounded annually, and 4.75% compounded daily. The daily account sounds more aggressive, but its APY is 4.86430% versus the annual account's clean 5.00000%. On $10,000 over one year that is $10,486.43 versus $10,500.00 — the nominally lower rate wins by $13.57, because a 0.25 percentage point rate advantage is worth roughly twenty times the entire benefit of daily over annual compounding.
Practically, this is why the Truth in Savings Act requires deposit accounts to disclose APY alongside the interest rate. Regulation DD specifies formulas precisely so that two institutions cannot make identical offers look different. The published APY is the comparable figure; two offers with the same APY produce identical balances, whatever their nominal rates and whatever their advertised compounding frequency.
Two things sit outside the number. Promotional and tiered rates are not captured — an APY assumes the rate holds all year, and introductory periods expire. Fees are also excluded by construction: an account paying 5.116% APY with a monthly maintenance charge of $12 on a $10,000 balance nets closer to 3.7% in practice. Yield measures interest, not total return to the depositor.
Formula
APY = (1 + r/n)^n - 1
Divide the nominal rate by the compounding frequency, add one, raise to that frequency, and subtract one. The result is the true one-year return.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Nominal annual rate as a decimal | decimal | The quoted APR. Divide percentages by 100: 5% becomes 0.05. |
| n | Compounding periods per year | periods per year | 1 annually, 2 semiannually, 4 quarterly, 12 monthly, 365 daily. Check the account agreement for the actual convention. |
| APY | Effective annual yield | decimal | Multiply by 100 for the quoted percentage. This is the figure deposit disclosures must publish. |
APY = e^r - 1
The upper bound as compounding frequency approaches infinity. On a 5% nominal rate this is 5.12711% — only marginally above daily compounding.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| e | Euler's number | constant | Approximately 2.71828. Use exp(r) - 1 rather than typing the constant. |
Ending Balance = Principal x (1 + APY)
Once APY is known, one year of growth is a single multiplication. This is what makes two offers directly comparable.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Principal deposited | currency | Assumes a single deposit held the full year with no withdrawals and no fees. |
How To Calculate APY
- 1
Find the nominal rate and the compounding frequency
Both are in the account agreement or disclosure. The rate alone is not enough — a 5% rate compounds differently if credited annually, monthly or daily.
- 2
Convert the percentage to a decimal
Divide by 100. A 5% nominal rate becomes 0.05. Leaving it as 5 produces a nonsense APY above one hundred thousand percent.
- 3
Divide the rate by the number of periods
Monthly compounding means twelve periods: 0.05 / 12 = 0.00416667. This is the periodic rate actually applied each time interest is credited.
- 4
Add one, raise to the nth power, subtract one
(1 + 0.00416667)^12 = 1.0511619, so the APY is 0.0511619. Expressed as a percentage that is 5.11619%. The exponent is doing all the compounding work.
- 5
Apply it to the deposit
Multiply principal by (1 + APY). For $10,000 at 5.11619%, the year ends at $10,511.62 — compared with $10,500.00 if the same 5% had compounded annually.
Examples
Example 1: $10,000 at 5.00% compounded monthly
- Principal
- $10,000
- Nominal APR
- 5.00%
- Compounding
- Monthly (n = 12)
| Step | Calculation | Result |
|---|---|---|
| Convert the rate | 5.00% ÷ 100 | 0.05 |
| Periodic rate | 0.05 ÷ 12 | 0.00416667 |
| Add one and raise to twelve | (1 + 0.00416667)^12 | 1.0511619 |
| Subtract one for the APY | 1.0511619 - 1 | 0.0511619 (5.11619%) |
| Ending balance after one year | $10,000 x 1.0511619 | $10,511.62 |
Result: APY 5.11619%; ending balance $10,511.62, interest $511.62
Example 2: Comparing two offers — the lower rate can win
- Principal
- $10,000
- Offer A
- 5.00% compounded annually
- Offer B
- 4.75% compounded daily
| Step | Calculation | Result |
|---|---|---|
| Offer A APY — no compounding within the year | (1 + 0.05/1)^1 - 1 | 0.05 (5.00000%) |
| Offer B periodic rate | 0.0475 ÷ 365 | 0.00013014 |
| Offer B APY | (1 + 0.00013014)^365 - 1 | 0.0486430 (4.86430%) |
| Offer A ending balance | $10,000 x 1.05 | $10,500.00 |
| Offer B ending balance | $10,000 x 1.0486430 | $10,486.43 |
| Difference in favour of the lower nominal rate | $10,500.00 - $10,486.43 | $13.57 |
Result: $10,500.00 vs $10,486.43 — Offer A wins by $13.57 despite daily compounding elsewhere
Calculator
Annual percentage yield (APY)
5.12%
- Balance after one year
- $10,511.62
- Interest earned in one year
- $511.62
- Continuous-compounding ceiling
- 5.13%
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the APY calculator page.
Common Mistakes
Comparing two accounts by nominal rate alone
A 4.75% account compounding daily beats nothing, but a 5.00% account compounding annually still beats it by $13.57 on $10,000. Compare published APYs; the account agreement's frequency already is priced in.
Assuming daily compounding beats monthly by a meaningful amount
On $10,000 at 5%, monthly yields $511.62 of interest and daily $512.67 — a difference of $1.05 a year. Frequency beyond monthly is nearly irrelevant; spend attention on rate and fees instead.
Treating promotional rates as if they last the year
Introductory rates expire, and tiered accounts pay different rates on different slices of the balance. The APY calculation assumes one rate held all year, so a twelve-month projection based on a three-month teaser overstates the outcome.
Ignoring fees that sit outside the APY
Yield measures interest only. A monthly $12 maintenance fee costs $144 a year, which on a $10,000 balance is 1.44% — enough to erase the gap between competing offers entirely.
Using the APY formula across multiple years
APY is defined as a one-year measure. Over several years the same process compounds further, so use (1 + APY)^t for a t-year horizon rather than multiplying APY by t, which ignores compounding between years.
FAQ
What is the difference between APR and APY?
APR is the nominal annual rate before compounding is considered. APY includes the effect of compounding within the year and is always equal to or higher than the APR for the same rate. Deposit disclosures must show APY precisely because it is the comparable figure.
Is daily compounding worth seeking out?
Rarely by itself. Moving from monthly to daily compounding adds about $1.05 of annual interest per $10,000 at 5%. A rate difference of even 0.05 percentage points outweighs it by several times.
What does continuous compounding change?
It is the theoretical limit as compounding frequency goes to infinity, giving e^r - 1. At 5% that is 5.12711%, versus 5.12675% for daily — a difference well under a hundredth of a percentage point.
Are savings account APYs fixed?
Usually variable. Most deposit accounts reserve the right to change the rate at any time, so the published APY describes today's terms rather than a guaranteed annual return. Certificates differ: their rate is locked for the term.
Why do loans quote APR instead of APY?
Because APR on credit includes certain fees and origination costs in addition to interest, which makes borrowing costs comparable. Compounding-direction arguments run the other way for debt, and consumer lending rules require APR rather than APY.
References
- [1]Consumer Financial Protection Bureau, Truth in Savings Act — Regulation DD — https://www.consumerfinance.gov/rules-policy/regulations/1030/
- [2]Electronic Code of Federal Regulations, Title 12, Annual Percentage Yield Calculation, Appendix A to Part 1030 — https://www.ecfr.gov/current/title-12/chapter-II/part-1030/appendix-Appendix%20A%20to%20Part%201030
- [3]Federal Deposit Insurance Corporation, Deposit insurance and annual percentage yield disclosures — https://www.fdic.gov/resources/deposit-insurance/