Compound Interest Calculator

Enter a starting amount, an annual rate, a time horizon, and how often interest is added, and the calculator returns the future value, the interest earned along the way, and the effective annual rate. That last figure collapses any compounding schedule into a single comparable number, the honest way to line up accounts that quote the same nominal rate. Everything recalculates in your browser as you type.

How the future value is calculated

FV = P × (1 + r ÷ (100 × m))^{m × t}

Here P is the principal, r the annual rate in percent, m the number of compounding periods per year, and t the time in years. It is the standard future-value formula from introductory finance, and the effective annual rate uses its companion, (1 + r ÷ (100 × m))^m − 1. That is the number a US bank advertises as APY. Regulation DD, the rule implementing the Truth in Savings Act, writes it a different way — 100 × ((1 + interest ÷ principal)^{365 ÷ days in term} − 1), built from the interest actually earned and annualized to a 365-day year — but for a fixed rate compounded a whole number of times a year the two land on the same figure.

With the defaults — 10,000 at 5% compounded monthly for 10 years — the monthly rate is 5 ÷ 12 = 0.41667% per month, applied 120 times. Raising 1.0041667 to the 120th power gives 1.647009, so the balance grows to 16,470.09. Interest earned is 6,470.09, and the effective annual rate works out to 5.116%.

A four-thousand-year-old idea

Charging interest, and calculating how fast a debt grows, is one of the oldest recorded pieces of financial mathematics. Scribal students in Old Babylonian schools, roughly 2000 to 1600 BC, were set exercises on exactly this question. At the standard silver rate of one-sixtieth of the principal each month — 20 percent a year — clay tablets from the period ask how long a loan takes to double. Left as simple interest the answer is a tidy five years; fold each year's interest back into the balance and it drops to just under four. The same corpus works through exponential growth, constrained growth and doubling time, the last of these squarely in the setting of loan interest. The arithmetic of a balance that grows on itself predates most of what we would recognize as banking.

The same civilization worried about where it could lead. The Code of Hammurabi, set down in Babylon around 1750 BC, capped interest at 20 percent on loans of silver and 33 and a third percent on loans of grain, and made a written, witnessed contract the condition of any claim: law 122 tells anyone handing over silver or gold for safekeeping to show it to a witness and draw up a contract, and law 123 leaves whoever skipped that step with no legitimate claim at all. Ceilings on interest, and the suspicion that a balance can run away from a borrower, are as old as the technique itself.

One of the earliest printed treatments belongs to Luca Pacioli, the Franciscan friar whose Summa de arithmetica, published in Venice in 1494, carried one of the first published descriptions of the double-entry bookkeeping Renaissance merchants were already using. On folio 181 Pacioli states the rule of 72 — divide 72 by the rate to estimate the doubling time — but never derives or explains it, which is generally read as a sign that the shortcut predated him among Italian merchants. A book given over entirely to the subject took another century: Richard Witt's Arithmeticall Questions of 1613, which ran 124 worked examples and tables built around 10 percent, then the maximum rate lawfully chargeable on an English loan.

The deepest result came from Basel. In 1683 Jacob Bernoulli asked what happens to compounding taken to its limit: an account of one unit paying 100 percent a year returns 2 if interest is added once, but more if the year is split into smaller and more frequent periods. Bernoulli proved the sequence stays between 2 and 3 without ever reaching the top of that range, though he could not fix its exact value — arguably the first important number defined by a limiting process rather than by measurement. That number is the constant Leonhard Euler later wrote as e. The notation first surfaces in a letter he sent Christian Goldbach in 1731, reaches print in his Mechanica of 1736, and by the time of the 1748 Introductio in analysin infinitorum he quotes the value to 18 decimal places, 2.718281828459045235, without ever saying where he got it. That constant, 2.71828 to five places, is the ceiling on how much frequency alone can add, which is why the continuous-compounding figure in the next section sits so close to daily.

What compounding frequency is actually worth

Running the default inputs through each frequency option:

Two things stand out. Frequency matters — the gap between annual and daily compounding is 197.70 over the decade — but the benefit flattens fast: moving from annual to monthly captures 181.14 of that gap, while monthly to daily adds only 16.56. The theoretical ceiling is continuous compounding, which multiplies the principal by e raised to the rate times the years: 10,000 × e^0.5 = 16,487.21, less than a dollar above the daily figure. Bernoulli's limit is not an abstraction here; it is the point of diminishing returns visible in the last column of the table. The practical lesson for a saver is to weight the rate itself heavily and treat compounding frequency as a tie-breaker between accounts that are otherwise identical.

The rule of 72

Pacioli's shortcut still works. Divide 72 by the interest rate to estimate how many years doubling takes: at 5% that gives 72 ÷ 5 = 14.4 years, against an exact 14.21 years with annual compounding, or 13.89 years if interest is added monthly. Judged against annual compounding the estimate stays within roughly four months of the truth right across the 4% to 10% band, which makes it a quick sanity check on any pitch that promises to double your money suspiciously fast. The reason it works is that doubling means the growth factor has to reach 2, so the true answer is the natural logarithm of 2 — about 0.693 — divided by the rate. A rule of 69.3 would therefore be exact under continuous compounding. The number 72 wins out partly because it divides cleanly by 1, 2, 3, 4, 6, 8, 9 and 12, and partly because the slightly larger numerator offsets the drag of compounding at yearly rather than continuous intervals.

Assumptions and country notes

The model is a single lump sum with no further deposits or withdrawals, a rate that never changes, and no tax, fees, or inflation. For plans with monthly contributions, use the savings goal calculator instead. Terminology shifts by market: the US quotes the effective rate as APY, standardized by the Truth in Savings Act that Congress passed on 19 December 1991 as part of the Federal Deposit Insurance Corporation Improvement Act; UK savings products carry the same figure under the label AER, the annual equivalent rate; and in the EU, Article 4 of the Consumer Credit Directive requires credit advertising that quotes any rate to show the annual percentage rate of charge with it. Canada is a special case — section 6 of the Interest Act says a mortgage with blended payments of principal and interest has to state its rate calculated yearly or half-yearly, not in advance, and section 7 makes the penalty for omitting it the loss of any interest above the stated rate. Lenders settle on the half-yearly basis, so a 5% Canadian fixed mortgage compounds semi-annually and is effectively 5.063%, not the 5.116% a monthly quote would imply. The same formula runs in reverse on what you owe: card and loan balances compound too, usually daily and at far higher rates, which is exactly the outcome Hammurabi's ceilings were written to restrain.

Projections assume a constant rate and are illustrations, not financial advice or a guarantee of returns. See the site disclaimer.

Frequently asked questions

What is the difference between simple and compound interest?

Simple interest is paid on the principal only, so 10,000 at 5% earns a flat 500 every year — 5,000 over ten years. Compound interest is paid on principal plus accumulated interest, which turns the same inputs into 6,470.09 with monthly compounding. The gap widens every year, which is why long horizons reward compounding so heavily.

Is daily compounding much better than monthly?

Barely. On 10,000 at 5% over ten years, daily compounding produces 16,486.65 against 16,470.09 for monthly — a difference of 16.56, or about 1.66 a year. Frequency helps, but the rate itself matters far more than how often it is applied.

What does APY mean and how is it different from the interest rate?

APY, or annual percentage yield, is the effective annual rate this calculator reports: the nominal rate adjusted for compounding. A 5% rate compounded monthly is a 5.116% APY. In the UK and Australia the same concept on savings products is labeled AER. When comparing accounts, compare APY to APY, never a nominal rate to an effective one.

How long will it take to double my money at 5 percent?

About 14 years. The rule of 72 estimates 72 ÷ 5 = 14.4 years, and the exact figure is 14.21 years with annual compounding or 13.89 years with monthly. At 7% the doubling time drops to roughly 10 years, which is why small rate differences grow into large ones over decades.

Does compound interest apply to debt as well as savings?

Yes, and usually at higher frequency. Most credit cards compound daily: carrying a 3,000 balance at 22% APR accrues about 1.81 in interest on day one, and the balance grows to roughly 3,738 after a year if nothing is paid. The same math that builds savings works against borrowers.