Skip to content
Calcapedia
Menu
Money

APY Calculator

See what a savings rate really earns in a year (its APY) once interest earns interest, or turn an APY back into the stated rate.

Level: Grades 10–12

Printable worksheet

APY Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Convert

Result

Annual percentage yield (APY)

5.1162%

A 5% stated rate compounded monthly gives an APY of 5.1162%.

Extra yield from compounding
0.1162%
Interest on $10,000 over one year
$511.62

See the step-by-step solution ↓

Step-by-step solution

  1. Write the stated rate as a decimal, so 5% becomes 0.05

    5%100=0.05

  2. Divide by 12, because the bank adds a share of the yearly rate 12 times a year

    0.0512=0.00416667

  3. Grow by that rate 12 times in a row, then subtract 1 to leave just the interest earned

    (1+0.00416667)12−1=0.0511619

  4. Multiply by 100 to show it as a percent

    0.0511619×100=5.11619%

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

Please note: Results are estimates for planning and comparison. Lenders, banks, and tax authorities may round differently or include fees not modeled here, so confirm figures with your provider before deciding.

What this calculator does

Find out how much a savings rate really earns you in a year. Savings accounts and certificates often show two numbers. One is the stated (or "nominal") interest rate. The other is the annual percentage yield, or APY. The stated rate leaves out compounding, which is interest earning interest. The APY shows what you actually earn over a year once that happens. The more often the bank adds interest, the bigger the gap.

Enter a stated rate and how often interest is added, and the calculator gives you the APY. That makes it easy to compare accounts that add interest on different schedules. It also works backward. Give it an APY, and it finds the stated rate that produces it at the schedule you choose.

Where it's used

Why this math matters outside the classroom:

  • Bank account rates. A 5% rate compounded monthly earns an APY of about 5.116%, slightly more than the stated rate.
  • Compounding frequency. Savers see that daily compounding at 5% gives about 5.127%, barely more than monthly, and close to the continuous limit.
  • Comparing accounts. Converting every offer to APY lets a saver compare an account that compounds monthly with one that compounds quarterly.
  • Working backward. Finding the nominal rate behind an advertised APY shows the per-period rate a bank actually credits each month.
  • Precalculus lessons. Students see (1 + r/n)^n approach e^r as n grows, which connects compound interest to the number e.

Grade level and standards: Grades 10–12; Common Core State Standards for Mathematics HSA.SSE.B.3.c. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose the direction: stated rate to APY, or APY to stated rate.
  2. Enter the rate you know as a percent.
  3. Choose how often interest is added (compounds), from once a year to daily or continuously.
  4. Read the result. You'll also see the extra yield from compounding, or the rate for each compounding period.

What the inputs mean

Stated yearly rate (APR)
The stated yearly rate before compounding. It's sometimes called the APR, or just the interest rate on a savings account.
Annual percentage yield (APY)
What you really earn over one year, with compounding included.
Compounding
How many times a year interest is added to your balance. Daily uses 365 periods. Continuous compounding is the limit you reach as the number of periods keeps growing forever.

Formula and method

APY = (1 + r ÷ n)^n − 1
continuous: APY = e^r − 1
reverse: r = n × ((1 + APY)^(1 ÷ n) − 1)
continuous reverse: r = ln(1 + APY)

r is the stated yearly rate written as a decimal, so 5% is 0.05. n is how many times a year interest is added. Each period earns r ÷ n. Growing by that much n times in a row shows how much $1 becomes after a year. Subtracting 1 leaves just the interest, which is the APY.

e is a special math constant, about 2.71828, that appears when compounding happens nonstop. ln is the natural logarithm, the button that undoes e. The reverse formula undoes the first one. It finds the rate per period that grows to the APY over n periods, then multiplies by n. With yearly compounding (n = 1), the APY and the stated rate are the same.

Worked example

A savings account pays a 5% stated rate, compounded monthly.

  1. As a decimal, the rate is 0.05.
  2. Rate per month: 0.05 ÷ 12 ≈ 0.00416667.
  3. APY: (1.00416667)^12 − 1 ≈ 0.05116190.
  4. As a percent: 5.1162%.

Result: The APY is about 5.1162%. So $10,000 left for a year earns about $511.62, rather than $500. Going the other way, an APY of 5.12% with monthly compounding matches a stated rate of about 5.0036%.

Understanding the result

When you compare savings accounts, compare APYs, not stated rates. The APY already includes how often each account adds interest.

The jump from yearly to monthly compounding is noticeable. But the gain from monthly to daily or continuous is small. For typical rates, it's a few thousandths of a percentage point.

Assumptions and limitations

  • The stated rate stays the same for the whole year.
  • Interest stays in the account to compound; nothing is taken out or added.
  • Daily compounding uses 365 periods per year.
  • No rounding happens inside the calculation; results are shown to four decimal places.
  • Fees, tiered rates (rates that change with your balance) and promotional periods aren't included.
  • Some banks round the APY they publish, or work it out under specific disclosure rules. So a published figure can differ in the last decimal place.
  • On a loan, the APR may include fees. Converting it here gives the effective interest rate, not the full cost of borrowing.

Frequently asked questions

What is the difference between APY and APR?

APR is a stated yearly rate that leaves out compounding. APY is what you really earn in a year once compounding is included. For the same rate, APY equals APR with yearly compounding. With anything more frequent, APY is higher.

Is daily compounding much better than monthly?

Only slightly. At 5%, monthly compounding gives an APY of about 5.1162% and daily about 5.1267%. On $10,000 over a year, the difference is around a dollar.

What does continuous compounding mean?

It's what you get if interest is added more and more often, without end. The APY becomes e^r − 1, where e is about 2.71828. It's the most compounding can ever add for a given stated rate.

Why convert an APY back to a stated rate?

Some formulas, such as loan or savings projections that work period by period, need the rate for each compounding period. Converting the APY to a stated rate and dividing by n gives you that rate.

How this calculator is checked

Method: Standard effective annual rate definition: APY = (1 + r/n)^n − 1, with the continuous limit e^r − 1, and their algebraic inverses. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

Embed this calculator

Put this APY Calculator on your own website or blog for free. Paste this code where you want it to appear: