Effective Annual Rate Calculator
Convert a nominal rate into the effective annual rate (EAR / APY)
Last updated September 2026
Method: The effective annual rate uses the standard identity EAR = (1 + r/n)n − 1, and er − 1 for continuous compounding. The reverse mode solves the same equation for r: r = n × ((1 + EAR)1/n − 1).
Included: Forward conversion from a nominal rate to the effective rate, reverse conversion from an effective rate back to a nominal rate, seven compounding frequencies from annual to continuous, the periodic rate, the compounding boost in percentage points, first-year interest on a balance, a multi-year projection, and two comparison tables.
Not included: Taxes, account fees, deposits or withdrawals during the year, rate changes, and bank-specific day-count or rounding rules. This is a rate conversion, not a financial offer.
Balance & time horizon (optional)
📊 Effective annual rate (EAR)
💰 What it means on $10,000
Assumes one deposit, no withdrawals, a constant rate and no taxes. Over 10 years, compounding adds $285 compared with the same nominal rate credited just once a year.
🔁 6.00% nominal at every compounding frequency
More frequent compounding raises the effective rate, but the gains shrink fast.
| Compounding | Periods / yr | Effective rate | Interest on $10,000 |
|---|---|---|---|
| Annually | 1 | 6.0000% | $600.00 |
| Semiannually | 2 | 6.0900% | $609.00 |
| Quarterly | 4 | 6.1364% | $613.64 |
| Monthly | 12 | 6.1678% | $616.78 |
| Weekly | 52 | 6.1800% | $618.00 |
| Daily | 365 | 6.1831% | $618.31 |
| Continuously | ∞ | 6.1837% | $618.37 |
📈 Effective annual rate for nominal rates of 1% to 12%
The gap between nominal and effective widens as the rate rises.
| Nominal | Quarterly | Monthly | Daily | Continuous |
|---|---|---|---|---|
| 1% | 1.004% | 1.005% | 1.005% | 1.005% |
| 2% | 2.015% | 2.018% | 2.020% | 2.020% |
| 3% | 3.034% | 3.042% | 3.045% | 3.045% |
| 4% | 4.060% | 4.074% | 4.081% | 4.081% |
| 5% | 5.095% | 5.116% | 5.127% | 5.127% |
| 6% | 6.136% | 6.168% | 6.183% | 6.184% |
| 7% | 7.186% | 7.229% | 7.250% | 7.251% |
| 8% | 8.243% | 8.300% | 8.328% | 8.329% |
| 9% | 9.308% | 9.381% | 9.416% | 9.417% |
| 10% | 10.381% | 10.471% | 10.516% | 10.517% |
| 11% | 11.462% | 11.572% | 11.626% | 11.628% |
| 12% | 12.551% | 12.683% | 12.747% | 12.750% |
Math only, not financial advice. EAR = (1 + r/n)n − 1, and er − 1 for continuous compounding. Banks may round the posted rate or use a different day count, so a quoted APY can differ by a few hundredths of a percentage point.
Effective annual rate: what it is and how to calculate it
The effective annual rate (EAR) is the interest you truly earn or pay in one year once compounding is counted. A stated 6% compounded monthly is really 6.168% per year, because every month's interest starts earning interest of its own. This effective interest rate calculator converts any nominal rate into its EAR, and reverses the math when a bank quotes only the effective figure.
Two neighboring tools cover the same idea from different angles. The APY Calculator uses the same formula but speaks the language banks use on deposit accounts, so reach for it when you are shopping for a savings account or a CD. The Compound Interest Calculator takes the rate as given and projects a balance forward with regular contributions. Use this page when the question is about the rate itself: converting nominal to effective, comparing compounding schedules, or working backwards from a quoted yield.
How the effective annual rate is calculated
A nominal rate is never a complete description of an account. It has to be paired with a compounding schedule before it means anything. The effective annual rate folds both pieces into one number:
EAR = (1 + r ÷ n)n − 1 where r is the nominal annual rate written as a decimal (6% becomes 0.06) and n is the number of compounding periods per year: 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 52 for weekly and 365 for daily. The quantity r ÷ n is the periodic rate, the slice of interest actually added each period. Raising (1 + periodic rate) to the power n compounds those slices across the year, and subtracting 1 strips out the original principal to leave the growth.
Two properties follow immediately. First, when n = 1 the formula collapses to EAR = r, so a rate compounded once a year is already effective. Second, EAR is always greater than or equal to r for a positive rate, and it rises with n. That rise is not linear: it is steep from annual to monthly and nearly flat from monthly to daily.
Continuous compounding, the upper limit
If you keep increasing n, the effective rate does not run away. It converges on a ceiling given by the exponential function:
EARcontinuous = er − 1 Here e is Euler's number, roughly 2.71828. At a 6% nominal rate, continuous compounding gives 6.184% against 6.183% for daily compounding: an extra six ten-thousandths of a percentage point for compounding infinitely often instead of once a day. That tiny gap is the practical lesson of the whole page. Continuous compounding is standard in options pricing and academic finance, and it is almost never used on a consumer account, but it is useful as a sanity check because no schedule can beat it.
Worked example: 12% compounded monthly
Take a nominal rate of 12% compounded monthly on a $5,000 balance. Work it through step by step:
- Write the rate as a decimal: r = 0.12, and set n = 12 for monthly compounding.
- Find the periodic rate: r ÷ n = 0.12 ÷ 12 = 0.01, so 1.0% is credited every month.
- Compound it across the year: (1 + 0.01)12 = 1.0112 = 1.126825.
- Subtract 1: 1.126825 − 1 = 0.126825, so the effective annual rate is 12.6825%.
- Turn it into money: $5,000 × 0.126825 = $634.13 of interest in the first year, against $600.00 if the same 12% were credited once at year end.
The extra $34.13 is what compounding bought. Push the same 12% to daily compounding and the effective rate becomes 12.7475% ($637.37 on $5,000); compound it continuously and it reaches 12.7497% ($637.48). The step from annual to monthly is worth 0.6825 percentage points, the step from monthly to daily only 0.065, and the step from daily to continuous 0.0022. Frequency matters, but nothing like the headline rate does.
One nominal rate, seven compounding schedules
The table below takes a single 6% nominal rate and runs it through every compounding frequency, with the resulting first-year interest on a $10,000 balance. Every figure comes from EAR = (1 + r/n)n − 1, with the last row from er − 1.
| Compounding | Periods / yr | Effective rate | Boost over nominal | Interest on $10,000 |
|---|---|---|---|---|
| Annually | 1 | 6.0000% | 0.0000 pp | $600.00 |
| Semiannually | 2 | 6.0900% | 0.0900 pp | $609.00 |
| Quarterly | 4 | 6.1364% | 0.1364 pp | $613.64 |
| Monthly | 12 | 6.1678% | 0.1678 pp | $616.78 |
| Weekly | 52 | 6.1800% | 0.1800 pp | $618.00 |
| Daily | 365 | 6.1831% | 0.1831 pp | $618.31 |
| Continuously | ∞ | 6.1837% | 0.1837 pp | $618.37 |
Read the last column as a single sentence: moving from annual to monthly compounding is worth $16.78 a year on $10,000, and moving from monthly all the way to continuous is worth $1.59. A savings account paying 0.05 percentage points more will beat any compounding upgrade.
Effective annual rate for nominal rates of 1% to 12%
The gap between nominal and effective is not a fixed markup. It grows roughly with the square of the rate, so it is a rounding detail on a 1% savings account and a serious number on a 12% loan. This table shows the effective rate for each whole nominal rate from 1% to 12%.
| Nominal | Quarterly | Monthly | Daily | Continuous |
|---|---|---|---|---|
| 1% | 1.004% | 1.005% | 1.005% | 1.005% |
| 2% | 2.015% | 2.018% | 2.020% | 2.020% |
| 3% | 3.034% | 3.042% | 3.045% | 3.045% |
| 4% | 4.060% | 4.074% | 4.081% | 4.081% |
| 5% | 5.095% | 5.116% | 5.127% | 5.127% |
| 6% | 6.136% | 6.168% | 6.183% | 6.184% |
| 7% | 7.186% | 7.229% | 7.250% | 7.251% |
| 8% | 8.243% | 8.300% | 8.328% | 8.329% |
| 9% | 9.308% | 9.381% | 9.416% | 9.417% |
| 10% | 10.381% | 10.471% | 10.516% | 10.517% |
| 11% | 11.462% | 11.572% | 11.626% | 11.628% |
| 12% | 12.551% | 12.683% | 12.747% | 12.750% |
At 2% nominal, monthly compounding adds 0.018 percentage points. At 12% it adds 0.683, and at an 18% credit-card rate it adds 1.562, lifting the true annual cost to 19.562%. That asymmetry is why the effective rate matters far more on debt than on a modest savings balance.
Reverse: turning an effective rate back into a nominal rate
Banks advertise the effective figure, but many contracts, spreadsheets and loan schedules need the nominal rate that produced it. Solve the same equation for r:
r = n × ((1 + EAR)1 ÷ n − 1) For continuous compounding the inverse is simply r = ln(1 + EAR). Worked example: a bank posts a 5.25% APY and credits interest monthly. The underlying nominal rate is 12 × ((1.0525)1/12 − 1) = 5.1278%. If the same 5.25% were credited daily, the nominal rate behind it would be 5.1172%. The table below gives the nominal rate you need at each schedule to land on a given effective rate.
| Target EAR | Semiannual | Quarterly | Monthly | Daily | Continuous |
|---|---|---|---|---|---|
| 4.00% | 3.9608% | 3.9414% | 3.9285% | 3.9223% | 3.9221% |
| 5.00% | 4.9390% | 4.9089% | 4.8889% | 4.8793% | 4.8790% |
| 6.00% | 5.9126% | 5.8695% | 5.8411% | 5.8274% | 5.8269% |
| 8.00% | 7.8461% | 7.7706% | 7.7208% | 7.6969% | 7.6961% |
How to use this calculator
The tool updates as you type, so there is nothing to submit. Work through it in this order:
- Pick a direction. Leave it on "Nominal to EAR" if you know the stated rate and want the true one. Switch to "EAR to Nominal" if a bank or a contract gave you the effective figure and you need the rate underneath.
- Enter the rate. Type the annual percentage, not the monthly one. Use the quick buttons for common values if you just want to see the pattern.
- Choose the compounding frequency. Check the account disclosure or the loan agreement. Most U.S. savings accounts and credit cards compound daily; CDs and bonds often compound monthly, quarterly or semiannually.
- Add a balance and a horizon. Open the optional panel to turn the percentage into dollars for a specific amount, and to project the balance forward.
The headline number is the answer. Below it, the compounding boost tile shows how many percentage points the schedule added, the rate per period tile shows what actually gets credited each period, and the frequency table lets you see instantly what a different schedule would have produced at the same nominal rate.
Who this calculator is for
- Savers comparing accounts where one bank quotes a nominal rate and another quotes APY, so the two are not directly comparable as printed.
- Borrowers who want the real annual cost of a card or a line of credit that compounds daily rather than the quoted APR.
- Finance and accounting students working through EAR, effective annual yield and continuous compounding problems and needing a check on their answers.
- Small-business owners pricing invoice financing, a merchant advance or a short-term credit line, where the stated rate and the true annual cost diverge sharply.
- Anyone reading a disclosure that mentions a periodic rate and a compounding schedule but never states the annual number in plain form.
Key terms explained
- Nominal rate: the headline annual rate, sometimes called the stated or quoted rate. It is meaningless without a compounding schedule attached.
- Periodic rate: the nominal rate divided by the number of periods, which is what is actually credited each period. A 12% nominal rate compounded monthly has a 1.0% periodic rate.
- Effective annual rate (EAR): the single annual rate that produces the same one-year result as the nominal rate plus its compounding schedule.
- APY: annual percentage yield, the term U.S. banks must use on deposit accounts. It is the same calculation as EAR.
- APR: annual percentage rate, a nominal rate used on credit that folds in certain fees but does not itself compound.
- Compounding frequency: how often accrued interest is added to the balance so that it begins earning interest itself.
- Percentage point (pp): the unit for the difference between two rates. Going from 6.000% to 6.168% is a rise of 0.168 percentage points, not 0.168 percent.
Second worked example: two savings accounts
Bank A offers 4.90% compounded monthly. Bank B offers 4.95% compounded annually. The headline favors B, but the effective rates tell a different story: A works out to (1 + 0.049/12)12 − 1 = 5.0116%, while B is already effective at 4.9500%. On a $50,000 balance, A earns $2,505.78 in the first year and B earns $2,475.00, so the "lower" rate is worth $30.78 more. Leave both balances untouched for five years and A reaches $63,849.21 against B's $63,662.28, a difference of $186.92. Comparing headline numbers would have picked the worse account every single year.
The effective rate on the borrowing side
Compounding cuts both ways, and the effect is much larger at credit-card rates. A card with a 19.99% APR that compounds daily has an effective annual rate of 22.1214%; at 24.99% APR the effective cost is 28.3787%, and at 29.99% it reaches 34.9558%. Those are gaps of 2.1, 3.4 and 5.0 percentage points respectively, versus the 0.18 percentage points that daily compounding added to a 6% savings rate. If you carry a balance, the number to compare against an investment return is the effective rate, not the APR on the statement. The same logic applies to short-term business borrowing, where a rate quoted per week or per month can look modest and annualize into something very different.
What changes the result the most
- The nominal rate. It dominates everything. A 0.25 percentage point better rate beats any change of compounding schedule at ordinary savings rates.
- The first jump in frequency. Going from annual to quarterly or monthly captures most of the available benefit. At 6%, annual to monthly is worth 0.168 pp; monthly to continuous adds only 0.016 pp.
- The rate level. The compounding boost grows roughly with the square of the rate, so it is negligible at 1% and substantial above 15%.
- The time horizon. The effective rate is a one-year figure, but it compounds over the years. $10,000 at 6% grows to $17,908 over a decade with annual compounding and $18,194 with monthly compounding.
- Day-count conventions. A 360-day year, leap-year handling and rounding rules can shift the last decimal place of a posted rate.
Practical tips
- Convert every offer to an effective rate before comparing. Two quotes are only comparable once they share the same compounding assumption.
- On deposit accounts, read the posted APY and skip the conversion entirely. U.S. institutions are required to disclose it, and it is already the effective figure.
- Do not pay for a compounding upgrade. An account promoting daily compounding at a lower rate is almost always the worse deal.
- Compare after-tax where it matters. Interest is generally taxed as ordinary income, so a 5.13% effective rate is roughly 3.85% after a 25% marginal bite, whereas a tax-advantaged account keeps the full figure.
- Watch for rates quoted per period. A "1.5% monthly" charge is 19.562% effective, not 18%; multiply by 12 first, then convert.
- Fees are not part of the EAR. A $5 monthly maintenance fee on a $2,000 balance costs 3% a year and can erase the entire yield.
Limitations and assumptions
- The calculator assumes a constant rate for the whole year. Most savings and money market rates are variable and can change at any time.
- It assumes no deposits or withdrawals. Money moving in or out changes the dollar interest, though not the rate itself.
- It ignores taxes and fees, both of which reduce your real return and are not part of the compounding formula.
- It uses a 365-day year for daily compounding. Institutions using a 360-day basis or handling leap years differently will post a slightly different figure.
- It does not model promotional or tiered rates, introductory periods, or balance caps above which a lower rate applies.
- Continuous compounding is included for completeness. It is a mathematical limit and is not used on ordinary consumer accounts.
How it compares to related calculators
This page answers "what is this rate really worth once it compounds?" Other questions have better-fitting tools:
- Shopping for a deposit account and want the bank's own language? Use the APY Calculator.
- Projecting a balance forward with monthly contributions? Use the Compound Interest Calculator.
- Comparing CD terms and early-withdrawal trade-offs? Use the CD Calculator.
- Building a savings plan around a monthly deposit? Use the Savings Calculator.
- Need interest without compounding at all? Use the Simple Interest Calculator.
- Want a quick estimate of doubling time at a given rate? Use the Rule of 72 Calculator.
- Looking at the cost of borrowing rather than the return on saving? Use the APR Calculator.
Sources
- Consumer Financial Protection Bureau (CFPB) - Regulation DD (Truth in Savings), 12 CFR Part 1030, which requires depository institutions to disclose an annual percentage yield.
- Consumer Financial Protection Bureau (CFPB) - Appendix A to Part 1030: Annual Percentage Yield Calculation.
- Internal Revenue Service (IRS) - About Form 1099-INT, Interest Income, on the reporting of taxable interest.
- The compounding identities EAR = (1 + r/n)n − 1 and er − 1 are standard mathematics and need no external source.
⚠️ Common mistakes & edge cases
Comparing a nominal rate against an APY
A 4.95% rate compounded annually loses to a 4.90% rate compounded monthly, which is 5.0116% effective. Convert both sides before you decide; the headline number alone picks the wrong account surprisingly often.
Multiplying a periodic rate by the number of periods
A card that charges 1.5% per month is not 18% per year. Multiplying gives the nominal rate; compounding it gives 19.562%, which is the effective cost you actually carry.
Overrating daily compounding
At 6%, daily compounding beats monthly by 0.015 percentage points, worth $1.53 a year on $10,000. A marketing line about daily compounding is not a reason to accept a lower rate.
Confusing percent with percentage points
Going from 6.000% to 6.168% is a rise of 0.168 percentage points, which is a 2.8% relative increase in the interest earned. Mixing the two units produces answers that are off by orders of magnitude.
Treating the EAR as an after-tax, after-fee return
The formula covers compounding and nothing else. Taxes on interest income, monthly maintenance fees and early-withdrawal penalties all come off the top and can easily outweigh the compounding boost.
Reversing the formula by dividing
To get a nominal rate back from an EAR, take the nth root: r = n x ((1 + EAR)^(1/n) - 1). Dividing the EAR by n gives the periodic rate of the wrong quantity and overstates the answer.
❓ Frequently asked questions
What is the effective annual rate (EAR)?
The effective annual rate is the interest rate you actually earn or pay over one year once compounding is taken into account. A nominal rate is just a headline number attached to a compounding schedule; the EAR converts that schedule into a single figure you can compare across accounts. A 6% nominal rate compounded monthly is really 6.168% per year, because each month's interest starts earning interest of its own.
How do you calculate the effective annual rate?
EAR = (1 + r/n)^n - 1, where r is the nominal annual rate written as a decimal and n is the number of compounding periods per year. For 6% compounded monthly: (1 + 0.06/12)^12 - 1 = 1.005^12 - 1 = 0.061678, or 6.168%. For continuous compounding the formula becomes EAR = e^r - 1, which gives 6.184% at the same 6% nominal rate.
Is the effective annual rate the same as APY?
Mathematically, yes. APY (annual percentage yield) and EAR use the identical formula, (1 + r/n)^n - 1. U.S. banks are required to quote APY on deposit accounts under the Truth in Savings Act, while 'effective annual rate' is the term used in lending, corporate finance and textbooks. Some sources also call it the effective annual yield or the equivalent annual rate. Whatever the label, it means the true one-year return after compounding.
Is the effective annual rate the same as APR?
No. APR on a loan is a nominal rate that adds certain lender fees but does not compound the rate itself, so a 24.99% APR credit card that compounds daily has an effective annual rate of about 28.379%. The EAR is always at least as large as the nominal rate it comes from, and the gap grows with both the rate and the compounding frequency.
How do I convert an EAR back into a nominal rate?
Reverse the formula: r = n x ((1 + EAR)^(1/n) - 1). If a bank advertises a 5.25% APY compounded monthly, the underlying nominal rate is 12 x ((1.0525)^(1/12) - 1) = 5.1278%. For continuous compounding the reverse is r = ln(1 + EAR). Switch the calculator to 'EAR to Nominal' mode and it does this for every frequency.
Does daily compounding really beat monthly compounding?
It helps, but far less than people expect. At a 6% nominal rate, monthly compounding gives 6.168% and daily gives 6.183%, a difference of 0.015 percentage points, or $1.53 a year on a $10,000 balance. At 20% the same gap widens to 0.195 percentage points. Compare posted effective rates rather than chasing compounding frequency.
What is continuous compounding?
Continuous compounding is the mathematical limit of compounding more and more often: hourly, then every second, then constantly. Its effective annual rate is e^r - 1, where e is about 2.71828. It sets the ceiling on what any compounding schedule can produce, so a 6% nominal rate can never exceed 6.184% no matter how often it compounds. Almost no consumer account uses it, but it is standard in options pricing and academic finance.
Why is my bank's posted APY slightly different from this result?
Small differences come from rounding and day-count conventions. Some institutions compound on a 360-day basis, exclude or include leap days, credit interest monthly on a 365/365 daily accrual, or round the posted figure to two decimals. This calculator applies the standard (1 + r/n)^n - 1 formula to the periods you choose, so expect agreement within a few hundredths of a percentage point.
Does the effective annual rate depend on my balance?
No. EAR is a pure rate conversion, so a 6.168% effective rate is 6.168% on $500 or on $500,000. The dollar amount changes but the percentage does not. The balance field in this calculator exists only to translate the rate into money so the difference between two accounts is easier to see.
Which rate should I use to compare savings accounts?
Always compare effective rates, meaning APY on deposit accounts. A 4.90% nominal rate compounded monthly (5.0116% EAR) beats a 4.95% rate compounded once a year, even though the headline number looks smaller. On $50,000 that is $2,505.78 of first-year interest versus $2,475.00, a $30.78 advantage that grows to $186.92 over five years.
What is the effective annual rate of 12% compounded monthly?
12% compounded monthly gives an effective annual rate of 12.6825%. The monthly periodic rate is 12% / 12 = 1%, and 1.01 raised to the 12th power is 1.126825, so the yearly growth factor is 12.6825%. Compounded daily the same 12% nominal rate produces 12.7475%, and compounded continuously it reaches 12.7497%.
Can the effective annual rate ever be lower than the nominal rate?
Not with positive rates and standard compounding. When interest compounds once a year the EAR equals the nominal rate exactly, and any more frequent compounding pushes it higher. An effective cost can look lower than a quoted APR only when the quoted number includes fees that are not really interest, which is a different calculation.
Does the effective annual rate account for taxes or fees?
No. The EAR is a compounding conversion only. Interest on savings accounts, money market accounts and CDs is generally taxable as ordinary income and is reported on Form 1099-INT when it reaches $10 in a year, so your after-tax yield is lower than the number shown here. Monthly maintenance fees, early-withdrawal penalties and minimum-balance requirements also reduce your real return and are not part of the formula.
Is this effective interest rate calculator free?
Yes. There is no sign-up, no fee and no limit on how many rates you can run. Everything is computed in your browser, so no numbers you type are sent anywhere. Switch between the forward and reverse modes as often as you like to compare offers.
💡 Good to know
EAR, APY and effective annual yield are the same number
The three names come from different fields, not different formulas. Banks say APY because Regulation DD requires it, lenders and textbooks say EAR. If a page offers you both, you can use either one.
The compounding boost has a hard ceiling
No matter how often interest compounds, the effective rate can never exceed e raised to the power of the nominal rate, minus one. At 6% that ceiling is 6.184%, only 0.184 percentage points above the stated rate.
On debt, the gap is where the money is
Daily compounding adds 0.18 percentage points to a 6% savings rate but 3.39 percentage points to a 24.99% credit-card APR, lifting the true cost to 28.379%. Paying down high-rate debt is the highest-certainty return most households can get.