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CalcMax

Effective Interest Rate Calculator

Range: 0 – 50

Range: 1 – 365

Result

5.122%

Effective annual rate (APY, %)

Periodic interest rate
0.20833%
Continuous compounding limit (%)
5.127%
Gain over the nominal rate (%)
0.122%

An effective interest rate calculator does one thing its neighbours cannot: it takes any whole number of compounding periods a year, from one to 365. Enter 26 for a biweekly schedule, 52 for weekly, 3 for a count that follows no calendar at all, and the page returns the periodic rate that count implies, the effective annual rate it compounds to, the gain over the nominal rate you entered, and the continuous compounding limit that all of them approach. The last figure is the one that makes the page worth visiting. A nominal 5 percent compounded once a year is 5 percent; compounded monthly it is 5.116 percent; weekly 5.125; daily 5.127; and the continuous limit is 5.1271 percent — which is to say that the last stretch, from 365 periods a year to infinitely many, is worth four ten-thousandths of a percentage point. That is the shape of the whole relationship, and it is only visible when the frequency is a number you can set freely rather than one of six labels on a dropdown. The count matters most at the low end, which is also where the fixed lists are thinnest: annual compounding gives 5 percent, semiannual 5.062, quarterly 5.095, monthly 5.116 — steps of 0.062, 0.033 and 0.021 between neighbours that a dropdown offers. The counts a dropdown cannot offer are the ones people actually live with. A biweekly mortgage or a weekly-paid savings product compounds 26 or 52 times a year, and neither is on the standard list. Entering 26 gives 5.122 percent and 52 gives 5.125, which is between monthly and daily, as it should be. Above 365 the answer stops moving: 500 periods a year gives 5.1270 percent and continuous compounding gives 5.1271, so a count beyond a year's worth of days is a curiosity rather than a product feature, and the page caps the input there. Two things are worth knowing before you use the panel. The gain column is the compounding itself, in percentage points, and it grows roughly with the square of the rate: at 5 percent the gain from monthly compounding is 0.116 of a point, while at 36 percent the same monthly schedule gains 6.576 points — the same frequency, a rate seven times larger, a gain fifty-seven times larger. And the periodic rate column is a plain division, the rate divided by the count, which is the convention credit disclosures annualise rather than the equivalent rate; the equivalent one is the effective annual rate. Where this page sits relative to its two closest relatives is a matter of what you already know: the annual percentage yield page starts from a balance and offers six named frequencies, and the effective annual rate page starts from a rate charged per period. This one starts from a nominal annual rate and a count that you supply.

5% a year compounded at seven different counts

Compounding periods per yearPeriodic rateEffective annual rateGain over the nominal rate
1550
22.55.0620.062
41.255.0950.095
120.416675.1160.116
240.208335.1220.122
520.096155.1250.125
3650.01375.1270.127

The first column is a count rather than a name, which is what lets the table include 24 and 52 — the semimonthly and weekly schedules that a six-item frequency list cannot reach. The second column falls away quickly: 5 percent split once is 5, split twelve ways is 0.41667, split 365 ways is 0.0137, and the split is why the third column rises. Read the third column against the 5 that heads it and the increments are 0.062, 0.033, 0.021, 0.006, 0.003, 0.002 — each step smaller than the one before it, which is the shape of the whole table. By the time the count reaches 52, almost all of what compounding can ever do at this rate has already been done; the 365 row is worth 0.002 more than the 52 row and the continuous limit is worth 0.0001 more than that. The fourth column is the same information as the third, expressed as a gain rather than as a level, and it is the one to read if you are deciding whether a frequency change is worth pursuing: at 5 percent, no frequency change is worth much at all.

Formula

Effective annual rate = (1 + nominal rate ÷ count)^(count) − 1 Periodic rate = nominal rate ÷ count Continuous limit = e^(nominal rate) − 1

Nominal annual rate
The stated rate per year before any compounding is applied — the input this page shares with the annual percentage yield page, and the figure the gain column measures itself against
Compounding periods per year
Any whole number from 1 to 365: 1 for annual, 12 for monthly, 26 for biweekly, 52 for weekly, 365 for daily. This is the field that makes the page different, because it is a number rather than a choice from a list — a biweekly schedule has no entry on a six-item dropdown
Periodic rate
The nominal rate divided by the count, which is the rate applied at each compounding step and the rate a credit disclosure would annualise by multiplying it back by the count
Effective annual rate
What a year of that periodic rate compounds to, which is the figure to compare two accounts by and the same quantity a deposit disclosure calls the annual percentage yield when the count is 365
Gain over the nominal rate
The effective rate minus the nominal rate — the compounding in percentage points. It rises with the count but much faster with the rate, and it is bounded by the continuous limit
Continuous compounding limit
The effective annual rate as the count grows without bound: the base of the natural logarithm raised to the nominal rate, minus one. It is the ceiling of the fourth column, and the reason a count above 365 buys nothing worth having

Use it when the compounding schedule is a number rather than a name. The two cases that come up most are weekly and biweekly: a savings product that credits interest every week compounds 52 times a year, and a loan or account on a biweekly cycle compounds 26, and neither of those is a choice a six-item frequency list can offer. The second case is checking someone else's arithmetic — a rate quoted as 5.116 percent is monthly compounding, and if the count that produces 5.116 is 12, then 12 is the frequency the account actually uses, whatever the marketing says. The third case is the one that makes the continuous limit worth having on the panel: it answers how much room is left above the frequency you are being offered. At a nominal 5 percent, the distance from daily compounding to the limit is 0.0004 of a percentage point, so an account advertising continuous compounding at that rate is offering almost exactly what a daily-compounded account pays. At 50 percent the gap is wider — 64.816 percent at 365 periods against 64.872 continuous — but even there it is six hundredths of a point, because the exponential flattens out quickly. The one thing to keep in mind is that this page annualises twice: once linearly, in the periodic rate column, and once with compounding, in the effective rate. The linear one is what a credit disclosure uses and it is not wrong — it is a convention — but it cannot tell you what a year in the account is worth, which is what the effective rate is for. Comparing a loan quoted at 12 percent a year with an account quoting 1 percent a month is exactly the comparison this page's fourth column settles.

Worked examples

  1. 5% compounded 24 times a year

    1. Periodic rate: 5% ÷ 24 = 0.20833% per period
    2. Effective annual rate: 1.0020833²⁴ − 1 = 5.1219%, printed as 5.122
    3. Gain over the nominal rate: 5.122% − 5% = 0.122 of a percentage point
    4. Continuous limit: e^0.05 − 1 = 5.1271%, printed as 5.127

    The default case, and a count no dropdown offers: twenty-four periods a year is a semimonthly schedule, and it sits between monthly (5.116) and biweekly (5.125) exactly where arithmetic says it should. The continuous limit is only 0.005 of a point above it, which is the page's recurring point — at ordinary rates, most of the compounding effect is already spent by the time you get to twice a month.

  2. The same 5% compounded weekly — count 52

    1. Periodic rate: 5% ÷ 52 = 0.09615% a week
    2. Effective annual rate: 1.0009615⁵² − 1 = 5.1246%, printed as 5.125
    3. Gain over the nominal rate: 5.125% − 5% = 0.125 of a percentage point
    4. Continuous limit: e^0.05 − 1 = 5.1271%, printed as 5.127

    Weekly compounding, which is what this page exists for: 52 is not a frequency any of the standard dropdowns offers, and the answer it produces is closer to daily (5.127) than to monthly (5.116) — 0.125 of a point of gain against 0.116, a difference of nine thousandths of a point. Anyone choosing between a weekly-compounded and a monthly-compounded account at the same rate is choosing between 5.125 and 5.116, and a tenth of a point on the rate itself is worth ten times that.

  3. 36% compounded monthly — where the gain stops being small

    1. Periodic rate: 36% ÷ 12 = 3% a month
    2. Effective annual rate: 1.03¹² − 1 = 42.576%
    3. Gain over the nominal rate: 42.576% − 36% = 6.576 percentage points
    4. Continuous limit: e^0.36 − 1 = 43.333%

    The same monthly frequency that gains 0.116 of a point at 5 percent gains 6.576 points at 36 percent — fifty-seven times as much from a rate that is seven times as large, which is the quadratic relationship in action. It is also why a monthly rate of 3 percent cannot be described as 36 percent a year without misleading someone: the year costs 42.576 percent, and the difference is larger than most of the fees people argue about.

Limitations

It assumes the nominal rate holds for the whole year and that the count is the one the account actually uses. A rate that resets, a promotional period, or a compounding schedule that changes with the balance will all make these figures a snapshot rather than a description of the year. It assumes the count is a whole number, which is why the input is capped at 365 and rejects fractions — but real accounts do not always compound on a fixed count: an account compounding on the actual days in each month compounds 365 times in a common year and 366 in a leap year, and a bank computing on a 360-day year, as institutions in some countries do, will not match this page at all. It does not model deposits or withdrawals, so it describes an untouched balance and not a saver's year. It does not include fees, which is the whole difference between this effective annual rate and an annual percentage rate on a loan. The continuous limit it prints is a mathematical ceiling rather than a rate anyone is paid, and no institution compounds continuously; it is on the panel because it is the number that says how much compounding is left, and at ordinary rates the answer is very little. And it is not a disclosure: a deposit account's advertised yield is computed under the regulation as written, including its rules for tiered rates, stepped rates and time accounts longer than a year, none of which are implemented here.

Frequently asked questions

What is the effective interest rate?
It is the annual rate that results once a nominal annual rate is compounded at some frequency. A nominal 5 percent compounded monthly is 5.116 percent effective, because each month's interest earns interest for the rest of the year. The effective rate is always at least the nominal rate, and the gap between them is the compounding — which is why the effective rate is the one to compare accounts by and the nominal rate is the one to quote.
Why enter a number of periods instead of choosing a frequency?
Because the frequencies people actually live with are not always on the list. Weekly compounding is 52 periods a year and biweekly is 26, and neither appears on a standard six-item frequency selector. Entering the count also lets you answer questions a list cannot: what a count of 3 would pay, what an odd schedule works out to, and how much is left between the count you have and the continuous limit.
What is the continuous compounding limit for?
It is the ceiling the effective rate approaches as the count grows: the base of the natural logarithm raised to the nominal rate, minus one. At 5 percent it is 5.1271 percent, and daily compounding at 365 periods already reaches 5.127 percent — four ten-thousandths of a point below it. So the limit is not a product anyone offers; it is how you find out that beyond daily compounding there is almost nothing left to gain, at least at ordinary rates.
How much does the compounding count matter?
Far less than the rate. At a nominal 5 percent, going from annual to monthly compounding gains 0.116 of a percentage point, from monthly to weekly 0.009, and from weekly to the continuous limit 0.002. But the gain grows with the square of the rate: at 36 percent compounded monthly the gain is 6.576 points. So a high-rate product is where frequency matters, and a low-rate one is where it does not — a quarter of a point on the rate is worth more than any frequency change at 5 percent.
Is the periodic rate the same as the rate charged each period?
It is the linear split: the nominal rate divided by the count. On a credit disclosure that is the convention, and multiplying the periodic rate back by the count returns the nominal rate exactly. What it is not is the equivalent rate — 5 percent split into twelve parts gives 0.41667 percent a month, and 0.41667 percent compounded twelve times gives 5.116 percent, not 5. The periodic rate column and the effective rate column are the two halves of that fact.
Can the count be higher than 365?
Mathematically yes, but there is no point and this page does not allow it. At 500 periods a year the effective rate is 5.1270 percent against the continuous limit of 5.1271 — less than a thousandth of a point above daily compounding. Nothing pays interest more often than daily, and the input stops at 365 because beyond it the answer stops moving.

References

Related calculators

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