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Finance Calculator

APY Calculator

Convert rates, compare compounding, and annualize term interest.

Live calculator

APY conversion inputs

Amount in USD. Term annualization requires a positive balance.

Percentage before compounding; this is not a loan APR including fees.

Fixed equal periods; daily uses 365. Continuous is a mathematical limit.

Changes labels only; no currency conversion.

Annual percentage yield

5.11618978817%5.11618978817\%
APY
5.11618978817%5.11618978817\%
Starting balance
$10,000.00
One-year equivalent interest
$511.62
One-year equivalent balance
$10,511.62
Difference from simple annual interest
$11.62

Your APY calculation, step by step

Convert the nominal annual percentage to a decimal, apply the compounding schedule, then convert the effective yield to a percentage.

r=5/100≈0.05r=5/100\approx 0.05y=(1+0.0512)12−1≈0.0511618978817y=\left(1+\frac{0.05}{12}\right)^{12}-1\approx 0.0511618978817APY=(0.0511618978817)×100%≈5.11618978817%\mathrm{APY}=(0.0511618978817)\times100\%\approx 5.11618978817\%I1yr=Py=(10000)(0.0511618978817)≈511.618978817I_{1\mathrm{yr}}=P y=(10000)(0.0511618978817)\approx 511.618978817A1yr=P(1+y)≈10511.6189788A_{1\mathrm{yr}}=P(1+y)\approx 10511.6189788

Constant rate, retained interest, no cash flows, fees, taxes, or per-period cent rounding. Currency selection changes display only. Amounts and substitutions are rounded for display; CSV retains numeric precision. One-year equivalents are not a term payout quote.

Same nominal rate, different compounding

Each row keeps the entered nominal annual rate unchanged and applies a different fixed schedule. These are mathematical scenarios, not bank offers.

APY compounding comparison
SchedulePeriods / yearNominal rateAPYOne-year interestOne-year balance
Annually15%5\%5%5\%$500.00$10,500.00
Semi-annually25%5\%5.0625%5.0625\%$506.25$10,506.25
Quarterly45%5\%5.09453369141%5.09453369141\%$509.45$10,509.45
Monthly (selected)125%5\%5.11618978817%5.11618978817\%$511.62$10,511.62
Twice monthly245%5\%5.12164200232%5.12164200232\%$512.16$10,512.16
Every two weeks265%5\%5.12206204122%5.12206204122\%$512.21$10,512.21
Weekly525%5\%5.12458419272%5.12458419272\%$512.46$10,512.46
Daily (365)3655%5\%5.12674964675%5.12674964675\%$512.67$10,512.67
ContinuousContinuous5%5\%5.1271096376%5.1271096376\%$512.71$10,512.71
What Can You Create?

Compare yields across savings products

Rate conversion

Convert a stated nominal annual rate to APY, or start with an effective annual yield and solve for the equivalent nominal rate under a chosen schedule.

Interest estimates

Translate a rate into one-year money amounts, or annualize a fixed principal and an entered term interest amount. The actual term balance remains distinct from a one-year equivalent.

Compounding checks

Compare eight fixed schedules and the continuous limit, with a custom period count available when needed. Twice monthly and every two weeks remain separate choices.

Formula

APY formulas, inverse rates, and annualized term returns

Use decimal rates inside the formulas, then multiply by one hundred percent for the displayed rate. Each relationship assumes retained interest and no intervening cash flows.

Working formulas

Periodic compounding

y=(1+rn)n−1y=\left(1+\frac{r}{n}\right)^n-1

The nominal annual decimal rate is divided into equal compounding periods.

Inverse periodic conversion

r=n[(1+y)1/n−1]r=n\left[(1+y)^{1/n}-1\right]

An effective annual yield determines a nominal rate only after the schedule is specified.

Continuous limit and inverse

y=er−1,r=ln⁡(1+y)y=e^r-1,\qquad r=\ln(1+y)

Continuous compounding is an ideal mathematical limit, not a posting calendar.

Fixed-principal term annualization

y=(1+IP)365/d−1y=\left(1+\frac{I}{P}\right)^{365/d}-1

This model requires a positive principal and term ending balance. It uses an entered day count and a fixed 365-day year.

One-year equivalents

I1yr=Py,A1yr=P(1+y)I_{1\mathrm{yr}}=Py,\qquad A_{1\mathrm{yr}}=P(1+y)

These amounts describe a one-year equivalent and should not be mistaken for a shorter term payout.

Symbols

yy - effective annual decimal yield
The displayed annual percentage yield is 100y%100y\%.
rr - nominal annual decimal rate
For example, a nominal rate of 4.8%4.8\% means r=0.048r=0.048.
nn - compounding periods per year
A positive whole number of equally spaced periods for a periodic schedule.
PP - fixed starting principal
The monetary amount used to translate the yield into interest and an ending balance.
I,dI,d - term interest and term days
Interest excludes principal; days describe the actual entered term, without inferring calendar dates.
Worked examples

Check an APY conversion step by step

Start with P=12,000P=12{,}000 and a nominal annual rate of 4.8%4.8\%, compounded monthly. Interest stays invested and there are no fees, taxes, or additional deposits.

  1. 1. Convert the percentage and identify the periods

    r=4.8/100=0.048,n=12r=4.8/100=0.048,\qquad n=12

  2. 2. Compute the effective annual yield

    y=(1+0.048/12)12−1≈0.0490702075348y=(1+0.048/12)^{12}-1\approx0.0490702075348APY≈4.90702075348%\mathrm{APY}\approx4.90702075348\%

  3. 3. Translate yield into money and check the inverse

    I1yr=12,000y≈588.84,A1yr≈12,588.84I_{1\mathrm{yr}}=12{,}000y\approx588.84,\quad A_{1\mathrm{yr}}\approx12{,}588.84r=12[(1+y)1/12−1]=0.048r=12[(1+y)^{1/12}-1]=0.048

For a separate fixed-balance term example, 125125 interest on 10,00010{,}000 over 9090 days gives y=(1+125/10,000)365/90−1≈0.0516708866655y=(1+125/10{,}000)^{365/90}-1\approx0.0516708866655, or about 5.16708866655%5.16708866655\% annualized. The term balance is only 10,12510{,}125; the larger one-year equivalent assumes the growth pattern repeats.

Why Users Love This Tool

A clearer way to read savings rates

APY math made visible

  • The formula is shown directly so users can audit how compounding frequency changes the yield.
  • One-year interest and ending balance translate the rate into the selected display currency. Balance size changes money results while the yield stays unchanged.
  • The compounding difference compares one-year compounded interest against simple annual interest at the same nominal rate. It does not measure fees, inflation, or a risk premium.
  • Small rates retain their yield through numerically stable calculations. A zero starting balance is valid for rate conversion; term annualization needs a positive balance.

Comparison context

  • The forward comparison holds nominal rate constant. Inverse and term modes hold APY constant and show equivalent nominal rates, so the comparison question is explicit.
  • Fixed rates, retained interest, and no cash flows define the model. Bank rounding, changing rates, minimum balances, withdrawal restrictions, taxes, and fees can change actual account earnings.
  • Related links connect APY math to CD interest, savings goals, and the APR vs APY guide.
  • Copy, sharing, CSV, and print actions retain known inputs and assumptions. Invalid fields disable exports until corrected, instead of saving a stale result.
Perfect For

APY support for savers, students, and rate shoppers

Account comparisons

Compare savings accounts, CDs, and money market rates when compounding schedules differ.

Savings projections

Estimate how much a balance could earn in one year after compounding is included.

Finance education

Demonstrate how nominal rates, compounding periods, and effective yield relate.

How It Works

How it works in three quick steps.

1

Choose the conversion you need

Choose nominal rate to APY, APY to nominal rate, or annualize term interest. Enter percentages as percentages, and keep interest separate from the starting balance.

2

Choose compounding frequency

Choose a fixed compounding schedule, a custom whole number of periods, or continuous compounding. For term interest, enter the actual term days; the annualization year is fixed at 365 days.

3

Review APY and interest impact

Review the typeset substitutions, one-year equivalents, and comparison table. Check its heading to see whether nominal rate or APY is held constant, then copy, share, export CSV, or print a focused PDF.

Download & Print

Save, share, and print your APY result

Copy the APY summary

Copy the calculation mode, known inputs, solved rates, one-year equivalents, and model assumptions. Share a link to restore those inputs; its financial values are readable by anyone receiving it.

Print the comparison

Choose Print / Save PDF, then select a printer or the browser PDF destination. The focused layout keeps inputs, typeset steps, assumptions, and every comparison row while hiding navigation and supporting articles.

Export the full comparison

Download CSV to keep every comparison row and unrounded numeric values. Its summary identifies which rate is held constant, the display currency, and the day convention used in term mode.

FAQ

Frequently Asked Questions

What is the difference between a nominal rate and APY?
A nominal annual rate describes the rate before within-year compounding. APY expresses effective annual growth after that compounding is included. In this tool, rr is the nominal decimal rate and yy is the effective annual decimal yield; their displayed percentages are 100r%100r\% and 100y%100y\%. Keep the starting amount unchanged when comparing money results. A loan APR can include charges and should not automatically be treated as a deposit nominal rate.
How do I convert APY back to a nominal annual rate?
Choose APY to nominal rate, enter the effective annual percentage, and select a compounding schedule. The solver uses r=n[(1+y)1/n−1]r=n\left[(1+y)^{1/n}-1\right] for periodic compounding and r=ln⁡(1+y)r=\ln(1+y) for continuous compounding. The result depends on the schedule because many nominal rates can produce the same effective yield. The comparison table keeps APY fixed in this mode and shows those equivalent rates rather than treating them as different annual returns.
Which compounding schedules and custom values are supported?
The fixed schedules use 11, 22, 44, 1212, 2424, 2626, 5252, or 365365 equal periods per year. Twice monthly is therefore different from every two weeks. Custom compounding accepts a whole number from one to 365,000 periods. Continuous compounding uses the mathematical limit y=er−1y=e^r-1 and has no finite period count. These ideal schedules do not reproduce a bank’s holiday calendar, leap-year policy, or rounding procedure.
How does annualizing term interest work?
Enter a positive fixed starting balance PP, the interest alone II, and the actual number of term days dd. The model annualizes the return using y=(1+I/P)365/d−1y=(1+I/P)^{365/d}-1. The term ending balance stays separate from the one-year equivalent. No deposits or withdrawals are modeled. A statement’s annual percentage yield earned can require average daily balance and additional account rules, so this fixed-principal example does not replace that disclosure calculation.
Can I enter a zero balance, zero rate, or negative yield?
Rate conversions allow a zero balance because the rate relationship does not depend on the amount invested. Money results then remain zero. A zero rate gives zero APY. Negative rates and yields are supported within the displayed limits for educational loss scenarios; term interest must leave a positive ending balance. A zero starting balance cannot define a term return, and an extreme annualized value outside the supported range produces an explanation rather than an infinite result.
Why does the comparison table change between modes?
In nominal-rate mode, each row applies a different schedule to the same nominal annual rate, so APYs and interest amounts can change. In inverse and term modes, each row preserves the same APY and solves for its corresponding nominal rate. In that second comparison, the one-year money amounts stay equal. The table heading and CSV description identify the fixed quantity so that unlike comparisons are not accidentally mixed together.
Does selecting another currency change the APY?
Currency selection changes formatting and labels only. It does not apply an exchange rate, change the entered starting amount, or modify the yield. For example, selecting yen displays whole currency units while dollar amounts normally show two decimals. The unrounded calculation remains available in CSV. Use the same monetary unit for starting balance and term interest; converting only one of those inputs would change the term return incorrectly.
Why might a real account pay a different amount?
This tool assumes a constant rate, retained interest, and no intervening cash flows. Actual accounts may change rates, round interest at each posting, apply fees or minimum balances, restrict withdrawals, or use particular day-count rules. The displayed one-year equivalent also differs from a short-term maturity payout. Compare the mathematical result with the institution’s disclosures, and use a cash-flow-aware tool when deposits, withdrawals, or changing balances are part of the scenario.
About This Tool

Why APY is the rate savers should understand

Savings products often advertise rates with terms that sound similar but measure different things. A nominal annual rate describes the stated rate before compounding is applied. APY shows the effective annual yield after compounding. That distinction matters because a balance that earns interest monthly or daily can end the year with more interest than a simple annual calculation would suggest. Toolarithm's APY Calculator makes the conversion explicit and translates the result into one-year money amounts for a chosen deposit amount.

The calculator is useful for comparing savings accounts, money market accounts, CDs, and classroom examples where compounding frequency is part of the question. It does not replace official account disclosures. Real accounts may have variable rates, minimum balances, fees, promotional periods, tax treatment, or withdrawal restrictions. By pairing APY conversion with related CD, savings goal, and APR vs APY content, the page helps users compare interest-bearing options with a clearer understanding of what the quoted rate actually means.

A useful comparison starts by choosing what must stay fixed. In forward mode, the same nominal annual rate is tested against different schedules, so effective annual yields can differ. In inverse mode, the known APY remains fixed and each schedule requires its own equivalent nominal rate. Those rows describe the same effective annual growth in different ways. Reading the table heading prevents a common mistake: comparing unlike nominal rates as if the same percentage automatically implied the same account return.

Term-interest mode answers a separate question. It translates a fixed-balance return over a stated day count into an equivalent annual rate using a fixed 365-day year. That figure describes a hypothetical repetition of the same growth pattern; it does not promise that a short promotional return will continue. Accounts with deposits or withdrawals need additional balance information, and statement calculations can use average daily balances and specific disclosure rules. Use the result to understand the mathematics, then check the institution’s account terms before relying on an earnings estimate.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Formulas are checked with worked examples and boundary cases. Timing, compounding, payment, and rounding assumptions remain visible; results are educational estimates, not financial advice.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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