Compound Interest Calculator

How compound interest is calculated

Compound interest is interest on interest. Simple interest is worked out on the amount you started with, every time; compound interest is worked out on the balance as it currently stands. Because each round of interest joins the balance, it becomes part of the amount the next round is calculated on. That one difference is the whole idea, and it is why the growth curves rather than running straight.

The rate the formula wants is the nominal annual rate, the one quoted on the account. It is not the APY, and dividing it by n to get the periodic rate is a convention rather than a derivation: 0.4167% a month applied twelve times does not come to 5% a year, it comes to 5.116%. That difference is not a rounding error. It is the entire subject of why compounding frequency matters, and it is why a savings account advertises an APY rather than the rate the formula uses.

Percentages go in as percentages and are converted once, at the edge of the calculation: what you type as 5 becomes 0.05 before anything else happens, and nothing downstream divides by a hundred again. The parameter table above prints both forms for that reason.

What your numbers mean

Why compounding frequency matters

Adding interest more often is worth more, because each addition starts earning sooner. That much is intuitive. What is not intuitive is how little it is worth, and how quickly the gains run out.

Going from annual to monthly compounding is worth $18.12 on these figures. Going from monthly all the way to continuous — past daily, past hourly, past every subdivision there is — is worth another $1.71. Continuous compounding beats daily by $0.06.

There is no chart of this above, and the absence is deliberate. Six curves spanning about one per cent would render as a single line at any size this page could draw them, and a picture that cannot show its own data is worse than the numbers it was drawn from. That the difference is too small to see is the finding.

Nominal, periodic, and effective are three different numbers

All three describe the same account. The nominal annual rate is what is quoted, and it is what goes into the formula as r. The periodic rate is r / n, the share applied each time interest is added. The effective annual rate is what a year of that process actually comes to, once each period’s interest has joined the balance: (1 + r/n)ⁿ − 1, or e^r − 1 under continuous compounding.

On the default figures those three are 5%, 0.4167% a month, and 5.116%. The first is what the account advertises as a rate, the second is what it does twelve times, and the third is what you actually get. The mortgage calculator makes the same distinction from the borrowing side, where the convention runs the other way and costs you rather than pays you.

APR and APY are not the same thing

APY, annual percentage yield, is the effective annual rate under another name. It is the figure a savings account is required to advertise, precisely because it makes accounts with different compounding frequencies comparable. APR, annual percentage rate, is the nominal rate — and on a loan it usually means something broader still, because United States lenders must fold certain fees into it. So an APR on a mortgage quote and an APY on a savings account are not two words for one idea, and neither is the number this formula takes.

If you have an APY and want to know what to type here, the relationship runs backwards as well: the nominal rate that produces a given effective rate is n[(1 + EAR)^1/n − 1]. For a 5% APY compounded monthly that is 4.889%, not 5%.

How the balance grows

Two pictures, both drawn from the same figures the tables below them carry. The first separates what you put in from what the interest added; the second sets compound interest against simple interest on identical terms.

Understanding the result

Where “interest on interest” actually shows up

On the default figures the first month earns $4.17 and the second earns $4.18. A single cent, on the same rate and the same deposit, and it is the entire mechanism: the second month is calculated on a balance that includes the first month’s interest. Nothing else changed.

A cent a month sounds like nothing, and over one month it is. Run it for ten years and the same effect is the difference between the first year, which earns $51.16, and the tenth, which earns $80.16 — on an unchanged rate and without another dollar going in. That is what the widening band in the first chart above is showing.

Compound interest against simple interest

Simple interest is always worked out on the amount you started with. At 5% on $1,000 that is a flat $50 a year, forever, and after ten years the balance is $1,500 — the straight line in the second chart. Compound interest reaches $1,647.01 over the same ten years at the same rate. The gap, $147.01, is precisely the interest that earned interest of its own.

The two are not always in that order. Over a period shorter than one compounding period, compound interest is worth slightly less than simple interest — see a partial compounding period. It is a genuine reversal rather than a rounding artefact, and it catches people out.

How long it takes to double

The exact answer is ln 2 / [n ln(1 + r/n)], which on your figures is 13.89 years.

The Rule of 72. Divide 72 by the rate as a percentage and you get an approximate doubling time: here 72 ÷ 5 = 14.4 years, against the exact 13.89 years. It is a mental shortcut, it is accurate enough for rates in the low single and double digits, and it is not used anywhere in this calculator — every figure on this page comes from the formula, not from the rule.

What happens when you change one input

Both tables below are the same calculation run again against figures you did not enter, so that “what if the rate were a point higher” is answered by the formula rather than by a rule of thumb. Your own row is marked in each.

Changing the rate

Changing the length of time

Rate and time do not act alike. The rate sits inside the base of an exponent and time sits in the exponent itself, which is why doubling the years does far more than doubling the rate. It is also why the two tables above are worth reading together: the same balance can be reached by a higher rate you may not be able to find, or by a longer wait you can simply decide to take.

What the balance will be worth

The formula returns a number of future dollars. It says nothing at all about what those dollars will buy, and the difference is not small. Growing $1,000 to $1,647.01 over ten years is a 5% nominal return; if prices rise 3% a year over the same decade, the amount that actually matters — what the balance buys — has grown by about 1.94% a year, not 5%.

The relationship is exact rather than a subtraction: real = (1 + nominal) / (1 + inflation) − 1. Subtracting inflation from the nominal rate gets 2.00% where the true figure is 1.94%, which is close enough for conversation and not close enough for a page that is in the middle of explaining compounding.

The optional adjustment in the calculator applies A / (1 + i)^t to the answer and shows the result under a separate heading, in lighter type, below a dashed rule. That presentation is deliberate: it is not a peer of the balance, because the formula did not produce it.

What the adjustment does not do: forecast inflation, or know anything about what you personally buy. It applies one constant rate that you supplied. Published inflation measures track a basket of goods that is nobody’s actual basket, and future inflation is not a quantity any calculator can know.

What if you add money every month?

Then this is not the calculation you want, and the difference is worth being precise about. Compound interest describes what happens to an amount that is already there. A stream of equal payments made at regular intervals is an annuity, it has a formula of its own, and the two are added together rather than being one idea.

For a payment PMT made at the end of every period, the payments alone grow to PMT × [((1 + r/n)^nt − 1) / (r/n)] — an ordinary annuity. Paid at the start of each period instead, every payment compounds one period longer, so the whole thing is multiplied by (1 + r/n); that is an annuity due. The timing is not a detail: $100 a year for five years at 6% comes to $563.71 paid at year end and $597.53 paid at year start, on identical money.

This page does not model any of that, on purpose. The formula printed above the calculator is the formula that produced the answer, and that is the promise the whole page rests on. Folding contributions in would break it: the number on screen would no longer be A = P(1 + r/n)^nt, and the reader would have no way of telling which part of the total came from growth and which from their own deposits. A savings growth calculator that owns the annuity formula outright is a better answer than a drawer bolted onto this one, and it is listed under related calculators.

In the meantime, the honest workaround: run this calculator on the amount you already have to see what compounding does to it, and treat anything you pay in later as separate. What you must not do is enter the total of all your future deposits as the starting amount — money paid in five years from now has not been compounding for those five years, and the formula would credit it as though it had.

What moves the number, and what this model can see

The four tags are the whole point of the table. In the formula means the figure is one of the four symbols above. You supply it means the page will carry it, separately labelled, if you enter it, and will not guess it if you do not. Outside the model means it is real and this arithmetic does not describe it. Not estimable means no calculator could produce it honestly from what this one is given, and a figure offered anyway would be invented rather than computed.

Two of the “outside the model” rows have a workaround worth knowing. Tax and fees both act, to a first approximation, as a reduction in the rate: if interest is taxed at 30%, entering 3.5% instead of 5% models it, and if a fund charges 0.5% a year, entering 4.5% instead of 5% does the same. Neither is offered as an input here, because there is no universal tax treatment to assume — the right rate depends on your jurisdiction, your bracket, and whether the account is sheltered — and a calculator that picked one for you would be inventing a figure rather than computing one.

Common misconceptions

“5% interest means 5% more money every year”

Only when interest is added once a year. Under monthly compounding the account applies 0.4167% a month twelve times, and because each application joins the balance, the year comes to 5.116%. The quoted rate and the yearly growth are two different numbers whenever interest is added more than once a year, and the gap between them is exactly what the effective annual rate measures.

“Daily compounding makes a big difference”

It makes a small one. On $1,000 at 5% over ten years, daily compounding is worth $19.77 more than annual — and monthly already collects $18.12 of that. The returns diminish sharply, because each further split of the year applies a smaller rate over a shorter interval. An account advertising daily compounding at a lower rate is almost always worse than one compounding monthly at a higher one; compare the APYs, not the frequencies.

“APR and APY are the same thing”

They are related and they are not equal. APY is the effective annual rate, which includes the effect of compounding; APR is the nominal rate, which does not, and on a loan it also folds in certain fees. A 5% APY and a 5% APR compounded monthly are different accounts. The formula on this page takes the nominal rate — see APR and APY are not the same thing for converting between them.

“Compound interest includes the money I add each month”

It does not. P is what you start with, and nothing in A = P(1 + r/n)^nt represents a later deposit. Regular payments have their own formula and are added to this one rather than being part of it. The mistake matters because it runs in the expensive direction: entering the total of your future deposits as the starting amount credits every one of them with growth it has not had.

“This is what my investment will be worth”

It is what a constant rate would produce. Savings accounts and certificates come close to that for as long as the rate holds. Shares, funds, and property do not: their returns vary year to year, and a variable series does not compound at its own average — a year of −20% followed by a year of +20% averages zero and leaves you down 4%. Treat the output as arithmetic about a rate, not a forecast about a market.

“Compound interest already accounts for inflation”

Nothing in the formula knows what money buys. The result is a count of future dollars, and future dollars are worth less than today’s. This is the single most common way a compound interest figure misleads someone, which is why the page offers an explicit adjustment and keeps its answer visibly separate from the balance rather than blending the two.

What this calculator does not include

Being explicit about this is more useful than quietly modelling a guess. None of the following is accounted for:

• Money paid in or taken out. Deposits after the first and withdrawals of any kind. Both change the balance the rate is applied to, and neither appears in the formula.
• Tax on the interest. Interest is generally taxable income, and tax paid each year is money that does not compound in the years after. There is no universal treatment to assume.
• Fees, expense ratios, and penalties. Including early withdrawal penalties on certificates, which can exceed the interest earned.
• Rate changes. One rate, held for the whole period, is an assumption and rarely a fact.
• Rounding of each interest credit. Real accounts round to the cent every period; the formula does not.
• Day-count conventions. 365 days, no leap years, no 30/360.
• Minimum balances, tiered rates, and promotional periods. Common on real accounts, and all of them make the rate a function of the balance or the date.
• Currency and jurisdiction. All figures are United States dollars, and no deposit insurance limit, reporting rule, or account type is modelled.
• Any judgement about whether this is a good idea. This page computes arithmetic. It has no view on what you should do with your money.

Edge cases

A zero rate

The balance is the amount you started with, at every frequency and over any length of time: (1 + 0)^nt is 1 whatever the exponent. The effective annual rate is zero, the growth multiple is 1.00, and there is no doubling time to report, so the page does not offer one. Both charts flatten and say so rather than drawing a line that implies growth.

A negative rate

Accepted, down to −20%. Negative deposit rates were policy across much of Europe and in Japan between 2014 and 2022, the arithmetic describes them perfectly well, and refusing them would be the calculator declining to describe something that existed. The balance decays instead of growing, and the page reports a decline rather than calling a loss “interest earned”.

One genuine surprise here: at a negative rate, compounding more often leaves you with more money, not less. Taking 5% off once gets you to 0.95 of the balance; taking 2.5% off twice gets you to 0.9506, because the second deduction comes off an amount the first one already reduced. Continuous compounding is the best case in both directions rather than the best in one and the worst in the other.

Zero time, and fractions of a year

At t = 0 the balance is the starting amount, whatever the rate: no period has elapsed. Fractional years are allowed and are not rounded to whole periods — eighteen months is t = 1.5, and the Years/Months toggle exists so you can type whichever you actually have.

A partial compounding period

If n × t is not a whole number — annual compounding over ten and a half years, say — the formula raises the base to a fractional power, which prorates that last part-period. It is the standard reading of the equation and it is what this page does, but it is a modelling choice rather than a fact about accounts: many real accounts credit only completed periods and would pay nothing for the extra six months. The calculator says so under the time field whenever the situation arises, rather than leaving it in the assumptions where nobody would look.

The same fractional exponent produces the reversal mentioned earlier. Over less than one full period, compound interest is worth slightly less than simple interest: half a year of annual compounding at 5% turns $1,000 into $1,024.70, where simple interest would give $1,025.00. Both are correct; they are answers to different questions.

Very large numbers

The form accepts up to $100,000,000, up to 25%, and up to 100 years. All three at once produces a figure with nineteen digits in front of the decimal point, and at that size cents are no longer a fact about the money: a JavaScript number carries about fifteen significant digits, so a balance in the trillions cannot distinguish one cent from the next. Rather than print a fabricated “.00”, the page drops to whole dollars above a trillion — every figure below that line is exact to the cent, and no amount you can type is anywhere near it. Anything beyond the bounds themselves is refused outright, because past them a compound interest figure is theatre rather than information.

Frequently asked questions

What is compound interest?

Compound interest is interest calculated on a balance that already includes the interest added before it. Simple interest is always worked out on the amount you started with; compound interest is worked out on the balance as it stands, so each round of interest becomes part of the amount the next round is calculated on. That is the whole of it, and it is why the growth curves rather than running straight.

What is the compound interest formula?

A = P(1 + r/n)^(nt), where P is the amount you start with, r is the annual rate written as a decimal, n is the number of times a year interest is added, and t is the number of years. Under continuous compounding it becomes A = Pe^(rt), which is a different equation rather than a very large value of n. Both are set out symbol by symbol at the top of this page.

What does compounding frequency mean, and is monthly better than annual?

The frequency is how many times a year interest is added to the balance. More often is worth more, because interest starts earning sooner, but the effect is far smaller than most people expect. On $1,000 at 5% over ten years, monthly compounding beats annual by $18.12 — and everything beyond monthly, up to and including continuous compounding, is worth another $1.71.

What is the difference between APR and APY?

APY, the annual percentage yield, is the effective annual rate: what a year of the account’s compounding actually comes to. APR is the nominal rate, before compounding is accounted for. A 5% nominal rate compounded monthly has an APY of 5.116%. On a loan, APR usually means something broader again, because United States lenders must fold certain fees into it, so an APR on a mortgage quote and an APY on a savings account are not two names for one idea.

What is continuous compounding?

Continuous compounding is the limit you reach as interest is added more and more often: not monthly or daily but at every instant. It has its own formula, A = Pe^(rt), rather than being a very large n. It is the ceiling on what a given nominal rate can produce, and the gap between it and daily compounding is tiny — six cents on $1,000 at 5% over ten years.

Does this calculator include monthly contributions?

No, deliberately. This page works out what a single amount grows to on its own. Money paid in later is an annuity riding on top of compound interest, it has a formula of its own, and folding it in here would mean the equation printed above the calculator was no longer the equation that produced the answer. The section on adding money every month sets out the mathematics and the timing distinction between an ordinary annuity and an annuity due.

Does compound interest account for inflation?

No. The formula returns a number of future dollars and says nothing about what those dollars will buy. The optional adjustment on this page divides the answer by (1 + i)^t to show it in today’s money, which is a separate calculation applied afterwards rather than part of the formula — and it is kept visibly separate in the result panel for that reason.

Why does my bank or investment account show a different number?

Several ordinary reasons, and usually more than one at once. Real accounts round every interest credit to the cent and this page does not. Banks differ on day-count conventions. Rates change, and this page assumes one constant rate. Fees and tax are taken out along the way. And an investment account does not earn a fixed rate at all: its returns vary, and a variable series does not compound at its own average.

Does compound interest apply to loans and credit cards?

Yes, in the same way and against you. Credit card interest is typically compounded daily on the balance carried, which is why an unpaid balance grows faster than the quoted annual rate suggests. Mortgages work differently again, because the balance falls as you pay it down, so the arithmetic there is amortization rather than growth.

Assumptions

• One interest rate, constant for the whole period.
• Interest is added to the balance and compounds from then on; none of it is paid out.
• Nothing is paid in and nothing is withdrawn after the starting amount.
• Daily compounding means 365 periods a year. Leap years are ignored, and the 30/360 convention is not offered.
• Nothing is rounded per period. Each figure is the formula evaluated once, and rounding happens only for display.
• A part-period at the end is prorated by the formula rather than dropped.
• Continuous compounding uses A = Pe^rt, which is the limit of the discrete formula and not an approximation of it.
• All figures are in United States dollars.

This page is an explanation of arithmetic, not financial advice, a rate quote, or a forecast of investment returns. Curiosity Mapped does not track, publish, or offer savings rates. Use the figures your bank or account provider gives you.

Methodology

• Money is carried in double-precision dollars, not integer cents. This is the opposite of the decision the mortgage calculator made, and the reason is worth stating because a reader who checks will notice. That page accumulates a schedule three hundred and sixty rows deep, where binary fractions drift and a column stops adding up; integer cents fix it. Nothing here accumulates — every figure is one closed form, evaluated once — so there is no drift to prevent. And cents would actively break: at the largest inputs the form accepts, the balance in cents exceeds the largest integer JavaScript can represent exactly, and the arithmetic would quietly return a wrong whole number.
• Every row comes from the formula, not from the row above it. The year-by-year table evaluates A = P(1 + r/n)^nt at each year rather than carrying a running balance forward. The last row is therefore the headline figure itself, not a number that resembles it.
• The percentage becomes a decimal exactly once. At the edge of the calculation, and nowhere else. The parameter table prints both forms so the conversion is visible rather than assumed.
• Rounding to the cent happens only for display. With one deliberate exception: a difference between two figures already printed on the page is taken from the printed values. The gap between annual and monthly compounding is $18.1149; rounded once it is $18.11, but the two balances in the table read $1,628.89 and $1,647.01, and a reader who subtracts them gets $18.12. A difference column that cannot be checked against the rows beside it is worse than one that is a hundredth of a cent coarse.
• Per-period rounding is not modelled, and the size of that is known. A real account rounds every interest credit to the nearest cent. On the default scenario the difference between doing that and not is about three cents over ten years; compounded daily over the same ten years it is about thirty-two cents. Both directions are possible. The figure is asserted in the test suite rather than estimated here.
• The charts are drawn from the same functions that produced the markup. The paths in the page source are not a hand-drawn imitation of the live chart; they are the live chart, rendered ahead of time by the code that later updates it. That is what makes the page correct with JavaScript switched off.
• Everything runs in the browser, and nothing reaches the address bar. There is no server, no network request, and no stored state. There is also deliberately no shareable link that carries your figures: a URL is the one thing analytics records verbatim, so putting an amount into it would leak through a channel consent does not govern. The test suite fails if this file grows a reference to the network, to storage, or to the address bar.
• The arithmetic has unit tests. Known values for every frequency, cross-checks that reach the same balance by a second route, a sweep asserting that nothing the form accepts can produce NaN or infinity, and the rounding figure quoted above. The cross-checks are the point: a test that only confirms the implementation agrees with itself proves nothing.

Related calculators and concepts

• Calculator Mortgage Calculator The same arithmetic pointed the other way: what it takes to retire a balance rather than what a balance grows to. The monthly payment, the formula that produced it, and the full amortization schedule. Open the calculator →
• Calculator Savings Growth Compound interest with regular contributions, on the annuity formula this page deliberately leaves out. Ordinary annuity and annuity due, kept visibly apart. Coming soon
• Calculator Simple Interest Interest worked out on the starting amount only, for the cases where that is genuinely what is happening rather than a mistake. Coming soon
• Calculator APY and Effective Rate Converting between a nominal rate and an effective one in both directions, for comparing accounts that compound on different schedules. Coming soon

The tools above shown as plain text rather than as links are not yet published, because there is nowhere for a link to go.

Every tool on this site is listed at Tools.

Concepts this page relies on

Five ideas do most of the work above. None of them has a page of its own yet; until one does, each link goes to the place on this page where the idea is explained rather than to a definition that does not exist.

• Interest on interest. The whole of compounding, and visible in a single cent between one period and the next before it is visible in anything else.
• Nominal, periodic, and effective rates. Three numbers describing one account. Only the first goes into the formula, and only the third tells you what you get.
• APR and APY. The regulated names for two of those three, plus a complication on the lending side that makes them not quite mirror images.
• Simple interest. The straight line compound interest departs from — and, over less than one compounding period, briefly beats.
• Real against nominal returns. The difference between more dollars and more purchasing power, which the formula has no opinion about at all.