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EAR Calculator

Range: 0 – 50

Result

12.683%

Effective annual rate (APY, %)

Nominal annual rate (%)
12.000%
Gap to the effective rate (%)
0.683%

An EAR calculator starts where a statement leaves you: with the rate that is actually charged each period. Enter 1 percent a month, or 0.5 percent a month, or 0.05 percent a day, and say how many such periods a year holds, and the page returns three things — the effective annual rate that the periodic rate compounds to over a year, the nominal annual rate you get by multiplying the periodic rate by the number of periods, and the gap between the two. The last of those is the reason to look at all. A card that charges 1 percent a month charges 12 percent a year in the linear sense that a disclosure uses, and 12.683 percent in the sense that describes what a year of it actually costs, because each month's interest joins the balance that the next month is charged on. The 0.683 of a percentage point between those two figures is compounding, and it grows faster than most people expect as the period shortens: hold the nominal figure at 6 percent and 0.5 percent a month compounds to 6.168 percent, while 0.05 percent a day — the same 18.25 percent a year in linear terms — compounds to 20.016 percent, a gap of 1.766 points. Both of those are on the panel, and the second one is the one that makes the point: the shorter the period, the more often interest is charged on interest, and at small periodic rates the effect is small in absolute terms but large relative to the rate. The formula is the one deposit disclosures use for the annual percentage yield, which is the same quantity under a different name: the periodic rate plus one, raised to the number of periods in the year, minus one. Two things about the inputs are worth stating plainly. This page does not offer continuous compounding, and that is not an omission — a periodic rate presupposes a period, so there is no such thing as the rate charged per period when the compounding never stops; the continuous limit appears as an output on the sibling page that starts from a nominal annual rate instead. And the frequency list here stops at five rather than six for the same reason. Where this page sits among its neighbours comes down to what you already know. If you have a statement, you have a periodic rate and this is the page. If you have an advertisement, you have a nominal annual rate with a compounding frequency, and the annual percentage yield page is the one to use. If what you have is a compounding count that a dropdown cannot offer — 26, 52 — the effective interest rate page takes any integer from 1 to 365.

Monthly compounding, from 0.1% to 2% a month

Periodic rateNominal annual rateEffective annual rateGap to the effective rate
0.11.21.2070.007
0.2533.0420.042
0.566.1680.168
11212.6830.683
22426.8242.824

Every row here is monthly compounding, so the second column is simply the first times twelve and the table can be read as a single question: at this periodic rate, how much is the annualisation hiding? At 0.1 percent a month the answer is 0.007 of a percentage point, and at 2 percent a month it is 2.824 — the gap grows roughly with the square of the rate, which is why the last row is nine times the second-to-last rather than twice it. That is the useful thing about reading the table rather than a single calculation: the compounding term is invisible at rates people think of as small and dominant at rates people think of as large, and the point where it stops being ignorable is somewhere in the middle of these five rows. The first column is the input and the fourth is the answer to the question the page asks, so a reader who only looks at two columns should look at those.

Formula

Effective annual rate = (1 + periodic rate)^(periods per year) − 1 Nominal annual rate = periodic rate × periods per year

Periodic rate
The rate charged or credited in one period, which is what a statement shows and what a compounding step uses — 1 percent a month, 0.5 percent a month, 0.05 percent a day. This is the input that makes the page different from its neighbours: they start from an annual figure and divide, this one starts from the period and multiplies
Compounding frequency
How many of those periods a year holds: 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly and 365 for daily. It is both the exponent in the compounding and the multiplier in the nominal figure
Effective annual rate
What a year of the periodic rate actually costs or earns once each period's interest is itself charged interest — always at least the nominal figure, and the gap widens as the period shortens
Nominal annual rate
The periodic rate multiplied by the number of periods, which is the annualisation a credit disclosure uses and the figure most people would quote if asked what the account charges. It ignores compounding entirely, which is why it is printed beside the effective rate rather than instead of it
Rate gap
The effective rate minus the nominal rate, which is the compounding itself expressed in percentage points. It is the number that says how much the convention of quoting an annual rate is hiding

Use it when the rate you have is per period rather than per year, which in practice means when you are reading a statement rather than an advertisement. A credit card that says 1 percent a month, a store card that says 1.5 percent a month, an account that credits 0.25 percent a quarter — all of those are periodic rates, and multiplying by twelve or four gives a nominal annual figure that understates what a year costs. The gap column is the size of that understatement, and it is worth knowing for two reasons. It tells you what to quote back: if a card charges 1 percent a month, saying 12 percent is the convention and saying 12.683 percent is the truth, and the second number is the one that belongs in a comparison against a loan quoted annually. And it tells you when the compounding stops being a rounding detail. At a nominal 6 percent the gap is 0.168 of a point; at a nominal 18.25 percent charged daily it is 1.766 points; at a nominal 182.5 percent charged daily — half a percent a day, which is the shape of a payday-style product — the effective rate is 517.465 percent, and a gap of 334.965 points is no longer a detail but the whole cost. The page is also the natural first step in the other direction: the periodic interest rate page takes a nominal annual rate and splits it, this one takes the split and puts it back together, and running both on the same number is a quick way to confirm that the split and the recompounding are not inverses of each other — 6 percent split to 0.5 percent a month recompounds to 6.168 percent, not to 6.

Worked examples

  1. 1% a month, the card case

    1. Periods per year: 12
    2. Nominal annual rate: 1% × 12 = 12%
    3. Effective annual rate: 1.01¹² − 1 = 12.6825%, printed as 12.683
    4. Gap: 12.683% − 12% = 0.683 of a percentage point

    The default case and the one everybody has seen: a card that quotes a monthly rate. Twelve percent a year is the figure the disclosure convention produces, and 12.683 percent is what a year of unpaid balance actually costs. The 0.683 is not a fee and not a penalty — it is the interest on the interest, and it appears in any account where the balance is left to compound rather than paid off each month.

  2. 0.05% a day — the same nominal rate, a different animal

    1. Periods per year: 365
    2. Nominal annual rate: 0.05% × 365 = 18.25%
    3. Effective annual rate: 1.0005³⁶⁵ − 1 = 20.0156%, printed as 20.016
    4. Gap: 20.016% − 18.25% = 1.766 of a percentage point

    A twentieth of a percent a day sounds smaller than 18.25 percent a year, and it is the same number: 0.05 times 365 is 18.25. But compounding daily turns it into 20.016 percent, which is 1.766 points above the nominal figure — nearly three times the gap that the monthly card case produced at a similar rate. The lesson is not that daily compounding is dangerous in itself; it is that the gap depends on both the rate and the period, and both of them got larger here.

  3. 2% a year charged semiannually

    1. Periods per year: 2
    2. Nominal annual rate: 2% × 2 = 4%
    3. Effective annual rate: 1.02² − 1 = 4.04%
    4. Gap: 4.04% − 4% = 0.04 of a percentage point

    Bonds are the usual place this input comes from: a bond paying a 4 percent coupon semiannually pays 2 percent twice a year, and if the coupon is reinvested the year returns 4.04 percent rather than 4. Four hundredths of a point is small, and it is the reason a semiannual-pay bond and an annual-pay bond with the same coupon are not the same investment — the second one pays 4 percent and compounds once.

Limitations

It assumes the periodic rate holds for the whole year. A variable rate, a promotional monthly rate that resets, a card whose rate is tied to a benchmark — each of those makes the effective annual rate a statement about today's rate rather than about the year, and on a card whose rate can change the gap column describes a year that will not happen. It assumes the periods are equal in length. Monthly compounding on a 365-day year does not divide evenly, and an account that compounds on actual days rather than on twelve equal months will differ slightly; so will an account credited on a 360-day basis, as banks in some countries compute. It says nothing about how the balance got there. Compounding only happens on a balance that is carried: charge a card and pay it off in full each month and neither the effective rate nor the gap ever applies to you, because there is nothing for the interest to compound into. It is not an annual percentage rate, which annualises a periodic rate without compounding and folds in fees; this page annualises the same periodic rate twice, once linearly and once with compounding, and adds no fees to either figure. And it is not a disclosure: the effective annual rate and the annual percentage yield are the same quantity when the term is 365 days, so a deposit disclosure's yield should match this page's first output on the same rate and frequency, but a credit disclosure's annual percentage rate will not, because it is built on the linear annualisation that is printed here as the second output.

Frequently asked questions

What is the effective annual rate?
It is what a rate charged each period comes to over a year once each period's interest is itself charged interest. One percent a month is 12.683 percent a year, not 12 percent, because the second month is charged on the balance plus the first month's interest. The effective annual rate is the figure that accounts for that; the nominal annual rate is the periodic rate multiplied by twelve, which is the figure a disclosure convention produces and the one that ignores compounding.
How do I calculate the effective annual rate from a periodic rate?
Add one to the periodic rate, raise it to the number of periods in the year, and subtract one. One percent a month is 1.01 to the twelfth power minus one, which is 0.126825, or 12.683 percent. This is the same formula that produces the annual percentage yield on a deposit account — the two are the same quantity when the term is a year, and this page's inputs are the ones you have when you are reading a statement instead of an advertisement.
Why is there no continuous compounding option?
Because a periodic rate needs a period. Continuous compounding is the limit as the period shrinks toward zero, so there is no rate charged per period to start from — the effective annual rate still exists, and it appears as the continuous limit on the page that starts from a nominal annual rate. On this page the frequency list stops at five, and the absence of the sixth is a property of the input rather than a missing feature.
Is the nominal annual rate the same as the annual percentage rate?
They are computed the same way — a periodic rate multiplied by the number of periods in a year — but the annual percentage rate also includes fees, and this page adds none. On a loan with no fees the two agree, which is why the nominal figure here is a useful check on a credit disclosure. On a loan with an origination fee the annual percentage rate will be higher than the nominal annual rate this page prints, and the difference is the fee.
Does the gap between the two rates grow with the rate?
Yes, and sharply. At a nominal 6 percent charged monthly the gap is 0.168 of a percentage point; at 12 percent monthly it is 0.683; at 18.25 percent charged daily it is 1.766; at half a percent a day — a nominal 182.5 percent — the effective rate is 517.465 percent and the gap is 334.965 points. Compounding is proportional to the square of the rate at these levels, so a rate that doubles roughly quadruples the gap rather than doubling it.
Which rate should I compare between two loans?
The effective annual rate, if neither loan charges fees, because it is the only one of the two that accounts for how often the interest compounds — and it is the same quantity as the annual percentage yield on a deposit. If either loan charges fees, the annual percentage rate is the better comparison, since it is built on the linear annualisation printed here as the nominal figure and also includes the fees. Comparing a monthly-quoted loan against an annually-quoted one by their nominal rates is the mistake this page exists to prevent.

References

Related calculators

Periodic Interest Rate CalculatorThe rate charged each period, from a nominal annual rate.Effective Interest Rate CalculatorEffective annual rate for any compounding frequency.APY CalculatorAnnual percentage yield from a nominal rate.Compound Interest CalculatorSee how savings grow with compounding at any frequency.