APY Calculator

What this calculator answers

What an interest rate is worth as an annual percentage yield (APY), and the reverse: which interest rate an advertised APY stands for with a given compounding. A third mode works out the APY from the interest an account actually paid, by the formula Regulation DD gives banks. Add a balance to see a year’s interest and the first month’s in dollars.

How to use it

• Convert: Interest rate to APY, APY to interest rate, or Interest earned to APY.
• Interest rate (nominal, per year): the rate before compounding, in percent. Account disclosures list it next to the APY. Credit unions call it the dividend rate, and a deposit disclosure may also call it the annual percentage rate. Type 3.95 for 3.95%.
• APY (annual percentage yield): the yield in ads and account disclosures, in percent.
• Compounding: how often interest is added to the balance, from the account terms. Choose Daily, rate ÷ 360 when the terms say the daily rate is the interest rate divided by 360.
• Balance (optional): adds a year’s interest, the balance after a year and the first month’s interest.
• Principal, Interest earned and Days in term (Interest earned to APY): the deposit, the dollars of interest it earned and the number of days. For the “APY earned” on a statement, enter the average daily balance and the days in the statement period.

Results update as you type. The table under the steps shows every compounding frequency at once, with your choice marked, and downloads as a CSV file. The Try buttons load three cases from this page: 5% APY on $1,000, Regulation DD’s six-month CD example and the ÷ 360 daily rate. To see which account pays more, press Save for comparison for each offer, whether it quotes an interest rate or an APY: the saved APYs line up, and each later offer shows its difference from the first. Continue in carries the rate, compounding and balance to the compound interest or CD calculator.

How to calculate APY from an interest rate

Divide the rate by the number of compounding periods in a year, add 1, raise the result to that number of periods and subtract 1:

• $r$ is the interest rate as a decimal (3.95% is 0.0395).
• $m$ is the number of compounding periods a year: 1, 2, 4, 12 or 365.
• Continuous compounding is the limit as $m$ grows: $\text{APY} = e^{r} - 1$.
• With a daily rate of $r/360$ paid on all 365 days, $\text{APY} = (1 + r/360)^{365} - 1$.

With compounding once a year, the APY equals the rate. In Excel, =EFFECT(3.95%, 365) gives the same APY. EFFECT returns an error for a rate of 0% or less, so use =(1+r/m)^m-1 for those.

Worked example: 3.95% compounded daily on $12,000

A savings account pays a 3.95% interest rate, compounded daily, on a $12,000 balance.

1. Rate as a decimal: r = 3.95 ÷ 100 = 0.0395.
2. Daily rate: 0.0395 ÷ 365 = 0.00010821918.
3. APY: $(1 + 0.00010821918)^{365} - 1$ = 1.04028828 − 1 = 0.04028828, or 4.0288%. Compounding adds 0.0788 percentage point to the rate.
4. Rounded to two decimals, as the bank must disclose it: 4.03%.
5. A year’s interest: $12,000 × 0.04028828 = $483.46, for a balance of $12,483.46.
6. The first month: the equivalent monthly rate is $1.04028828^{1/12} - 1$ = 0.3297%, so the first month earns $39.56. That is a little under the $40.29 monthly average for the year, because each month starts from a larger balance.

Daily vs. monthly compounding: how much difference does it make?

At a rate like 3.95%, very little once interest compounds monthly. Most of the gain over annual compounding is already there with quarterly or monthly compounding:

Moving from monthly to daily compounding adds 0.0065 percentage point, or $0.78 a year on $12,000. Continuous compounding adds only 0.0002 point more ($0.03). The day count matters more: Regulation DD lets a bank use a daily rate of 1/360 of the interest rate as long as it pays that rate on all 365 days. At 3.95% that raises the APY to 4.0859%, 0.0571 point above the ÷ 365 figure and $6.85 a year more on $12,000.

How to convert APY back to an interest rate

Take the root of 1 + APY for the number of periods, subtract 1, and multiply by the number of periods:

For continuous compounding, $r = \ln(1 + \text{APY})$. For example, an account advertising a 3.80% APY compounded monthly:

1. $1.038^{1/12} = 1.00311282$.
2. Rate per month: 1.00311282 − 1 = 0.00311282.
3. Interest rate: 12 × 0.00311282 = 0.0373538, or 3.7354%. Compounded daily, the same APY would need 3.7298%.

Convert first whenever a calculator asks for an interest rate and a compounding frequency but the account quotes only an APY. Entering 3.80 as a rate compounded monthly turns it into a 3.8669% APY and overstates a year’s interest on $12,000 by $8.03. In Excel, =NOMINAL(3.8%, 12) gives the rate.

How much interest will I earn in a year or a month at this APY?

Multiply the balance by the APY: an APY is a year’s interest as a share of the balance, so $12,000 at 4.0288% earns $483.46 in a year. That assumes the rate stays the same and the interest stays in the account.

For one month, don’t divide the APY by 12. Take the twelfth root of the year’s growth instead: $(1 + \text{APY})^{1/12} - 1$. At 4.0288% that is 0.3297% a month, while 4.0288% ÷ 12 would be 0.3357%. The difference is small, but the root is what makes twelve months of growth add up to the APY.

APY from the interest you earned (Truth in Savings)

Regulation DD’s Appendix A defines the APY from dollars, which is how banks calculate the figure they disclose:

$I$ is the interest earned in dollars, $P$ the principal and $d$ the days in the term: the actual number of days. For a savings account with no maturity date, the term is taken to be 365 days, so the APY is simply interest ÷ principal. The formula assumes the principal and interest stay on deposit for the whole term, with no deposits or withdrawals.

The appendix’s own example: $30.37 of interest on a $1,000 six-month CD whose term has 182 days. The return over the term is 3.037%, and $1.03037^{365/182} - 1$ = 6.1837%, disclosed as 6.18%. Converted back, the same APY comes from a 6.0005% rate compounded daily. Interest is paid in whole cents, so a rate found from it can be off in the fourth decimal.

The “APY earned” on a periodic statement uses the same formula with the average daily balance and the days in the statement period. For example, $17.84 of interest on an average daily balance of $6,200 over a 31-day statement: $(1 + 17.84 \div 6{,}200)^{365/31} - 1$ = $1.00287742^{11.7742} - 1$ = 3.4409%, shown as 3.44%.

Banks round the APY to two decimals, and a disclosed APY within 0.05 percentage point of the exact figure counts as accurate. A bank may also use a 366-day year in a leap year. So a bank’s figure can differ from this calculator’s in the second decimal.

APY vs. interest rate vs. APR: which number to compare

Compare APYs. The interest rate leaves compounding out, so a 3.95% rate compounded daily and a 3.95% rate compounded annually earn different amounts; their APYs, 4.0288% and 3.9500%, show it. A deposit disclosure may also call the interest rate the “annual percentage rate”, but that is still the rate before compounding. A loan’s APR is a different measure: it expresses the cost of credit as a yearly rate, and that cost includes loan fees and points, so it answers a borrower’s question rather than a saver’s.

APY for common interest rates

The same conversion for a range of rates, at monthly and daily compounding (plain math, not current market rates):

Reading the result

• The headline is the APY (or, converting the other way, the interest rate) to four decimals.
• APY as banks show it is the same figure rounded to two decimals. In the APY-to-rate mode, the first tile is the interest rate rounded the same way.
• Compounding adds is the APY minus the interest rate, in percentage points.
• Equivalent monthly rate is the growth in a twelfth of a year at this APY.
• Interest in one year, Balance after one year and Interest in the first month appear when you enter a balance, or the principal in the interest-earned mode.
• The steps substitute your numbers into the formula and end with the spreadsheet version.
• The table repeats the conversion for every compounding frequency, with your choice marked, and shows each row’s difference from it in percentage points. Converting an APY to a rate, it lists the rate each frequency needs for the same APY: the more often interest compounds, the lower that rate.

Assumptions and limitations

• The rate stays the same for a year. A variable rate can change after you open the account.
• Interest stays in the account and compounds, with no deposits or withdrawals.
• The APY counts interest only. Account fees, bonuses, taxes and inflation are not in it.
• One rate at a time. On a tiered account each balance tier has its own APY. For a CD whose rate steps up during the term, enter the total interest over the term in the interest-earned mode.
• The result is an educational estimate, not financial advice or a quote.

Common mistakes

• Comparing one account’s interest rate with another’s APY. A 3.95% rate compounded daily is a 4.0288% APY. Compare APY with APY.
• Entering an APY where a calculator asks for the interest rate. A 3.80% APY entered as a monthly-compounded rate becomes a 3.8669% APY.
• Dividing the APY by 12 for a month. It slightly overstates the monthly growth; use the twelfth root.
• Typing the rate as a decimal. The fields are in percent, so 0.04 means 0.04%, not 4%. The calculator points this out.
• Reading a CD’s interest over its term as its APY. $30.37 on $1,000 over 182 days is a 3.037% return but a 6.18% APY.
• Entering months as days. Days in term wants days: a six-month CD is about 182, not 6.
• Treating a loan’s APR like a deposit rate. A loan’s APR builds in loan fees, so converting it here doesn’t give what the loan costs.