Compound Interest Calculator
See how compound interest grows your money exponentially over time.
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About
Compound Interest Calculator
This compound interest calculator converts a rate from one compounding frequency to another, so you can compare offers that are quoted on different schedules. Enter a rate, say what it compounds on, and pick the frequency you want it expressed in. A 6% rate compounded monthly comes out as 6.168% compounded annually, which means the two cost or earn exactly the same over a year. The panel also shows the effective annual rate and what $1,000 grows to in twelve months.
What compound interest is
Interest is the price of borrowed money: what a lender charges, or what a saver earns for lending it. It comes in two forms, and the difference between them is where all the money is.
Simple interest applies only to the original principal. Borrow $100 at 10% simple interest for two years and the interest is $100 × 10% × 2 = $20, full stop. Compound interest applies to the principal plus whatever interest has already piled up. On the same $100 at 10%, year one adds $10, which makes the balance $110. Year two charges 10% on that $110 instead of the original $100, so it adds $11. Total interest: $21 rather than $20.
One dollar of difference over two years sounds trivial, and it is. What matters is that the gap widens every period, because each round of interest earns interest of its own from then on. Simple interest is rare in practice; nearly every loan and savings product you will meet compounds.
Why time matters more than rate
Compounding rewards patience out of all proportion. Put $1,000 into the market at 20 and leave it alone at a 10% annual return, roughly what the S&P 500 has averaged since the 1920s, and at 65 it is worth about $72,890. Nobody added a cent along the way. The entire gain is 45 years of interest earning interest.
The same mechanism runs in reverse on money you owe. A balance left to sit accrues interest on interest exactly as an investment does, which is why deferring credit card debt or letting a loan drag on gets expensive faster than the headline rate suggests. Compound interest does not care which side of the ledger you are on.
Compounding frequency changes the real rate
A quoted rate means little until you know how often it compounds. Take 10% compounded semiannually. That is 5% applied twice. On $100, the first half-year adds $5, and the second half-year charges 5% on $105, adding $5.25. The year's total is $10.25, so 10% compounded semiannually is really 10.25% compounded annually.
Push the frequency higher and the effective rate climbs further. A 6% mortgage rate compounded monthly, which is 0.5% a month, works out to 6.168% compounded annually. This is exactly why lenders like quoting the monthly figure and savings institutions like quoting the annual one: the same money looks cheaper or richer depending on the schedule it is expressed on.
In practice the conventions are fairly settled. Savings accounts and CDs tend to be quoted as an annual yield, while mortgages, home equity loans, and credit cards compound monthly. This calculator converts between daily, biweekly, semimonthly, monthly, quarterly, semiannual, annual, and continuous compounding, so two offers on different schedules can be compared honestly. For the yield on a specific deposit, see the CD Calculator; for the interest-only version, the Simple Interest Calculator.
APR and APY are the same idea
The two acronyms you meet most often are just the two ends of this conversion. APR is a nominal rate, quoted per year but applied on a shorter cycle, usually monthly. APY is the effective rate, the figure after compounding has been counted. A card advertising 24% APR charges 2% a month, which compounds to an APY of about 26.82%, so the balance grows nearly three points faster than the headline number implies.
U.S. rules push the two into different places on purpose: lenders disclose APR on loans, banks advertise APY on deposits. Both are legitimate, and each flatters the party quoting it. Converting one into the other, which is what the tool above does, is the only way to see whether a 5.9% loan is genuinely cheaper than a 6% one or just quoted on a friendlier schedule.
Where compounding shows up
Nearly every product with a rate attached compounds, and the schedule varies more than people expect. Savings accounts and money market accounts usually compound daily and credit it monthly. CDs commonly compound daily or monthly and pay at maturity. Credit cards compound daily on the average balance, which is why carrying a balance costs more than the APR suggests, and mortgages and home equity loans compound monthly.
Investments compound in a looser sense. A stock does not pay interest, but reinvested dividends buy shares that pay dividends of their own, and a fund's returns build on the returns already banked. That is the mechanism behind the $1,000-to-$72,890 figure above, and it is why leaving money invested tends to beat timing it.
The formulas
The basic compound interest formula, where interest is added once per period:
At = A0(1 + r)n
where:
A0 : principal amount, or initial investment
At : amount after time t
r : interest rate
n : number of compounding periods, usually expressed in years
A depositor opens a $1,000 savings account paying 6% compounded once a year for two years. Put those figures into the equation and the balance at maturity is:
At = $1,000 × (1 + 6%)2 = $1,123.60
When interest is added more than once a year, the rate is divided by the number of periods and the exponent counts them all:
At = A0 × (1 + r/n)nt
where:
A0 : principal amount, or initial investment
At : amount after time t
n : number of compounding periods in a year
r : interest rate
t : number of years
Take the same $1,000 at 6%, this time compounded daily. The daily rate is 6% ÷ 365 = 0.0164384%, and across two years that is 730 compounding periods:
At = $1,000 × (1 + 0.0164384%)(365 × 2) = $1,127.49
So a two-year account holding $1,000 at 6% compounded daily grows to $1,127.49, beating the annually compounded version by $3.89.
Continuous compounding
Compound often enough and the gains stop growing. Continuous compounding is the mathematical ceiling, where interest is added at every instant rather than at set intervals:
At = A0ert
where:
A0 : principal amount, or initial investment
At : amount after time t
r : interest rate
t : number of years
e : mathematical constant e, ~2.718
Run the same $1,000 at 6% for two years through it:
At = $1,000 × e(6% × 2) = $1,127.50
That is exactly one cent more than daily compounding produced. That is the whole prize for going from 365 periods a year to infinitely many. Frequency matters a great deal moving from annual to monthly, and almost not at all past daily, particularly on small balances.
The Rule of 72
The Rule of 72 estimates how long money takes to double at a fixed annually compounding rate: divide 72 by the rate. At 8%, $100 doubles to $200 in roughly 72 ÷ 8 = 9 years. At 6% it takes about 12 years, and at 3% about 24.
Use the whole number, not the decimal, so 8 rather than 0.08. It is an approximation, most accurate for rates in the middle single digits, and it drifts at very high or very low rates. As mental arithmetic for judging an investment or a debt, it is hard to beat. Our Investment Calculator runs the exact figure with contributions included.
Where compound interest came from
The practice is roughly 4,400 years old. Babylonian and Sumerian records show interest accruing at 20% of the principal until the accumulated interest equalled the principal itself, at which point it was folded in and the process restarted. The mechanics differ from modern compounding, but the idea of interest earning interest is plainly there.
It was not always welcome. Simple interest was widely legal, while compound interest was often condemned as usury: Roman law prohibited it, and both Christian and Islamic texts treated it as sinful. Lenders used it anyway through the medieval period, and it entered ordinary commercial use once compound interest tables were published in the 1600s.
The mathematics caught up shortly after. Jacob Bernoulli, studying compounding in 1683, noticed that adding more periods within a fixed span produced faster growth, but that the gains approached a ceiling rather than running away. Leonhard Euler later pinned that limit at about 2.71828 and gave it the name it still carries, e.
How this calculator works
It converts your input to an effective annual rate first, then re-expresses that as a nominal rate on the output schedule. For a finite frequency it applies (1 + r/n)n − 1 to get the effective rate, and the reverse for the output; for continuous compounding it uses er − 1 and the natural logarithm. Both rates it reports are financially identical over a year, which is what makes them comparable.
Use it whenever two quotes are not on the same footing: a savings account advertising an annual yield against a bond quoted semiannually, or a card rate quoted monthly against a loan quoted annually. To project a balance forward with deposits over time, use the Savings Calculator, and to compare a rate that includes fees, the APR Calculator.
Common questions
Frequently asked questions
Compound interest is interest charged on the principal plus the interest already accumulated. Borrow $100 at 10% for two years and compound interest totals $21, because year two charges 10% on the $110 balance rather than the original $100. Simple interest, charged only on the principal, would total $20.
Simple interest applies only to the original principal, so $100 at 10% for two years earns $20. Compound interest applies to the principal plus accrued interest, earning $21 over the same period. The gap widens every period, which is why it matters far more over decades than over months.
More frequent compounding raises the effective rate. A 10% rate compounded semiannually is really 10.25% a year, and 6% compounded monthly is 6.168%. The effect shrinks fast at high frequencies: on $1,000 over two years at 6%, daily compounding earns $1,127.49 and continuous compounding earns $1,127.50.
For interest added once per period it is At = A0(1 + r)^n. When interest compounds several times a year, use At = A0(1 + r/n)^nt, where n is the periods per year and t the number of years. Continuous compounding uses At = A0 x e^(rt), with e about 2.718.
Divide 72 by the annual rate to estimate the years it takes money to double. At 8%, $100 doubles in roughly 9 years; at 6% it takes about 12. Use the whole number, so 8 rather than 0.08. It is an approximation that works best for mid single-digit rates.
It is what a rate actually earns or costs over a year once compounding is counted, which makes it the fair basis for comparing offers quoted on different schedules. A nominal 6% compounded monthly has an effective annual rate of 6.168%. The calculator above shows it alongside the converted rate.
Barely. On $1,000 at 6% over two years, monthly compounding produces about $1,127.16 and daily about $1,127.49, a difference of roughly 33 cents. The meaningful jump is from annual to monthly; beyond daily the gain is effectively nil, so judge a savings account on its rate rather than its compounding frequency.
About $72,890 at a 10% annual return, roughly the long-run average of the S&P 500 since the 1920s, with nothing added along the way. Invested at 20 and left untouched to 65, that is nearly 73 times the original amount, and all of it comes from interest earning interest.