Compound Interest Calculator

$10,000 at various rates & time periods

The compound interest formula, term by term

Every figure this calculator produces comes from one equation, the standard formula for periodic compounding:

A = P(1 + r/n)^nt

• P — principal. The amount you start with. It is the seed the rest of the formula multiplies.
• r — the annual interest rate, written as a decimal. A quoted rate of 7% enters the formula as 0.07, not 7. Forgetting to divide by 100 is the single most common arithmetic slip in compound-interest math.
• n — the number of compounding periods per year. 1 for annual, 2 for semi-annual, 4 for quarterly, 12 for monthly, 365 for daily. This is how often earned interest is rolled back into the balance so it can begin earning interest of its own.
• t — the time in years the money is left to grow.
• A — the final amount: your principal plus every dollar of interest it accumulated.

The machinery sits in two pieces. The per-period growth factor (1 + r/n) is how much the balance is multiplied each compounding step; the exponent nt is the total number of those steps. Because the factor is applied again and again, interest credited in one period is included in the base for the next — the feedback loop that makes growth exponential rather than linear. When you also add a fixed deposit C every period, this tool layers on the future value of an ordinary annuity, C × ((1 + r/n)^nt − 1) ÷ (r/n), which sums the separately-compounded growth of each contribution.

Why compounding frequency matters — up to a ceiling

Hold the rate, principal and time fixed and increase n, and the final amount rises — but by less and less each time, climbing toward a hard limit it can never cross. Here is $10,000 at a 7% annual rate over 10 years, compounded at different frequencies:

Moving from annual to monthly compounding adds about $425; pushing all the way from daily to the theoretical limit of continuous compounding adds barely a dollar more. Frequency genuinely helps, but it suffers sharp diminishing returns. The effect is larger at higher rates and over longer horizons, and it is exactly why a savings account advertising "compounded daily" is only marginally better than one compounded monthly at the same stated rate.

Continuous compounding and the number e

Push the frequency to its logical extreme — compounding not daily or hourly but every instant — and the formula collapses into something far simpler:

A = P·e^rt

Here e is Euler's number, the mathematical constant roughly equal to 2.71828. It is precisely the value that (1 + 1/n)ⁿ approaches as n grows without bound, which is why it appears the moment compounding becomes continuous. That is not a coincidence of finance — it is the constant's origin story. In 1683 the Swiss mathematician Jacob Bernoulli was studying exactly this question: if an account paid 100% annual interest, how much more would you earn by compounding it more and more often? He worked out that compounding quarterly turned $1 into about $2.44 and monthly into about $2.61, and proved the sequence was bounded, settling somewhere between 2 and 3 no matter how finely you sliced the year. That ceiling is e ≈ 2.718281828, and Bernoulli's analysis is generally credited as the first time a number was defined as the limit of an expression. A constant that now turns up across calculus, probability and physics was first cornered by someone chasing the outer edge of compound interest.

The Rule of 72 (and 70, 114 and 144)

The Rule of 72 is a mental shortcut for doubling time: divide 72 by the interest rate written as a whole-number percent, and the result is roughly how many years your money takes to double. At 8%, that is 72 ÷ 8 = 9 years; at 6%, 12 years. The shortcut is centuries old — Luca Pacioli described it in his 1494 mathematics treatise, long before anyone could derive why it works.

The exact answer comes from logarithms. Money doubles when the growth factor reaches 2, so the precise doubling time is ln(2) ÷ ln(1 + r). For the small rates typical in finance, ln(1 + r) ≈ r, which makes the doubling time roughly ln(2) ÷ r. Since ln(2) ≈ 0.693, the mathematically honest numerator is about 69.3 (some people round it to 70, which is popular for inflation and population math because it divides cleanly). So why 72? Because 72 is far friendlier for mental arithmetic — it divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12 — and it happens to be most accurate in the 6–10% range where everyday rates cluster. The table below shows just how close the rough rule stays to the real, annually-compounded answer:

The same trick scales to bigger targets. To estimate tripling time, divide into 114 (the exact figure is ln(3) × 100 ≈ 109.9, rounded up for easy division); for quadrupling, use 144 (from ln(4) × 100 ≈ 138.6). For continuously compounded rates, swap 72 for 69.3, which is exact rather than approximate.

A surprisingly ancient idea

Compound interest is not a modern invention — it is one of the oldest pieces of applied mathematics we have a record of. Babylonian scribes were posing and solving interest-on-interest problems on clay tablets during the Old Babylonian period, roughly 2000–1700 BC; their Akkadian phrase şibāt şibtim translates literally as "interest on interest." The Code of Hammurabi, inscribed around 1750 BC, already regulated lending, capping interest at 20% per year on loans of silver and 33⅓% on loans of barley or grain.

Its reputation has not always been good. Charging interest on accumulated interest — historically called anatocism — was condemned under Roman law and treated as among the worst forms of usury, and religious prohibitions on interest shaped lending across the medieval world. The first work in English devoted entirely to the subject came much later: Richard Witt's 1613 book Arithmeticall Questions, which packed in roughly 124 worked examples and detailed compound-interest tables built around the 10% legal maximum of the day, covering both lump sums and streams of annual, half-yearly and quarterly payments. Earlier merchants' manuals, such as Francesco Pegolotti's around 1340, had included compound-interest tables, but Witt's was the first whole book on it.

Did Einstein really call it the eighth wonder?

You have almost certainly seen the line: "Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it" — attributed, confidently, to Albert Einstein. It makes for a great poster. There is just one problem: there is no credible evidence Einstein ever said or wrote it. Researchers at Quote Investigator and elsewhere have traced the saying and found no contemporary source, no letter, no interview — nothing tying it to Einstein. The quotation only began appearing in print around 1983, nearly three decades after his death in 1955. The "eighth wonder" label was being pinned on compound interest by advertising copywriters as far back as a 1925 bank advertisement, with no famous name attached; over time it was simply reassigned to Einstein, and sometimes to financiers like Rothschild or Rockefeller, to make it sound more authoritative.

We flag this because accuracy matters more than a tidy quote. The mathematics of compounding really is remarkable — but the awe is earned by the numbers, not by a misattributed celebrity endorsement.

Simple vs compound interest

Simple interest is paid only on the original principal: each year you earn the same flat amount, and the balance grows in a straight line. Compound interest is paid on the principal and on interest already earned, so the balance curves upward and accelerates. Over short periods the two look almost identical; over long ones they diverge dramatically. Here is $10,000 at 6% per year, simple versus annually compounded:

After 40 years the compounded balance is roughly three times the simple-interest result from the same deposit at the same rate — the entire gap is interest earning interest. This is the engine behind the time value of money: a dollar today is worth more than a dollar later precisely because it can be put to work compounding in the interim. It is also why starting early can matter more than investing more. A single $10,000 deposit left to compound at 7% for 40 years grows to about $149,745, while the same $10,000 invested for "only" 30 years reaches about $76,123 — nearly half, from an identical deposit, purely because the first version enjoyed ten extra years of compounding. (These are illustrations of the math, not predictions or investment advice; real returns vary and are never guaranteed.)

Nominal rate vs effective rate (APR vs APY)

Two accounts can quote the same headline rate and still pay differently, because the headline often ignores compounding frequency. The nominal annual rate (frequently labelled APR) is the stated rate before accounting for how often it compounds. The effective annual rate — EAR, or APY on a savings product — bakes the compounding in, and is what you can actually compare apples-to-apples. The conversion is EAR = (1 + r/n)ⁿ − 1. A nominal 12% compounded monthly is an effective 12.68%; if you are instead being charged 12%, more frequent compounding works against you. By law in many places, deposit accounts must advertise APY and loans must disclose APR, precisely so the compounding is not hidden in the fine print.

When compounding works against you

Compounding is direction-blind: it grows whatever it is applied to, including what you owe. On a credit card at 24% APR compounded monthly, the effective rate is about 26.82% a year, and any unpaid balance is recapitalised every month so that next month's interest is charged on this month's interest. The same exponential curve that builds a retirement balance can bury a borrower who pays only the minimum.

Inflation is best understood as compounding in reverse — it erodes the purchasing power of money at a compounding rate. At a steady 3% inflation, the Rule of 72 says prices roughly double, and a currency's buying power roughly halves, in about 24 years (72 ÷ 3). That is why a "safe" return that merely matches inflation leaves you no better off in real terms, and why long-term savers watch the real (after-inflation) rate, not just the nominal one. On the constructive side, reinvesting payouts — dividend reinvestment, rolling interest back in rather than spending it — is what keeps the compounding machine fed; spend the interest and you quietly convert compound growth into simple growth.

Common compound-interest mistakes

• Confusing the nominal rate with the effective rate. Comparing a "monthly compounded" rate to an "annual" one without converting to EAR/APY compares two different things.
• Ignoring compounding frequency entirely. The same nominal rate at daily vs annual compounding produces different totals; small at first, meaningful over decades.
• Expecting linear growth. People routinely underestimate long-horizon outcomes because intuition extrapolates in a straight line while compounding curves upward.
• Forgetting fees, taxes and inflation. A headline return is gross; what compounds for you is the return left after costs are skimmed off and after inflation is subtracted in real terms.
• Entering the rate wrong. Using 7 instead of 0.07, or applying an annual rate without matching the period count to the compounding frequency, throws the answer off by orders of magnitude.
• Treating projections as promises. A compound-interest model assumes a fixed rate forever; real-world rates fluctuate, so use the output to understand the mechanics, not to predict a guaranteed balance.