What is the difference between APR and APY?

APR is the nominal rate — the stated rate before compounding within the year, usually quoted with monthly compounding. APY is the effective annual rate: what you actually earn or pay after compounding does its work. A 6% APR compounded monthly is a 6.16778% APY. U.S. law makes deposit accounts advertise APY and loans advertise APR, which is not a coincidence: each side quotes the number that flatters it.

Does compounding frequency matter much?

Less and less as it gets finer. On $1,000 at 6% for two years: annual compounding gives $1,123.60, daily gives $1,127.49, and continuous — the mathematical limit — gives $1,127.50. The jump from annual to monthly is real money on large balances; the jump from daily to continuous is a cent. Frequency matters most when rates and principals are large.

What is the Rule of 72?

A shortcut: divide 72 by the annual return (as a percent) to estimate the years needed to double your money. At 8%, about 9 years; at 6%, about 12. It is an approximation — most accurate between roughly 6% and 10% — meant for quick estimates, not contracts.

Is compound interest good or bad?

It is a double-edged sword, and which edge you get depends on which side of the loan you stand. Savers and investors earn interest on interest, which is how small early deposits become retirements. Borrowers pay it: a credit-card balance left to compound monthly grows the same exponential way, against them.

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Compound Interest Calculator

First job of this page: convert a rate between compounding frequencies. A rate means nothing until you know how often it compounds — 6% compounded monthly and 6% compounded annually are different rates wearing the same number.

Input Interest

Compound

Output Compound

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Growth Projector

Second job: what a balance actually becomes. Add a starting amount, monthly contributions, and time. (For taxes on interest and inflation adjustments, use the Interest Calculator.)

Initial Investment

Monthly Contribution

Annual Interest Rate

Years to Grow

Compound

What Compound Interest Is

Interest is the price of using someone else's money, paid as a percentage of the principal. It comes in two species. Simple interest is charged only on the original principal: borrow $100 at 10% simple for two years and the interest is $100 × 10% × 2 = $20, end of story. Compound interest is charged on principal plus accumulated interest: the same $100 at 10% compounded annually earns $10 in year one (balance $110), then $110 × 10% = $11 in year two — $21 total. One extra dollar sounds trivial. It is the seed of everything below.

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Because each period's interest joins the principal, growth is exponential — a snowball, not a staircase. Start early and the effect is startling: $1,000 invested at 20 in a fund returning the stock market's long-run ~10% becomes $72,890 by age 65 — about 73× the deposit, from nothing but time. The same sword cuts the other way for borrowers: a card balance left to compound monthly grows on exactly the same curve, against its owner. Compounding rewards whoever owns the balance and punishes whoever owes it.

Frequency: why the converter exists

Interest can compound on any schedule, and the schedule changes the true rate. A 10% rate compounded semiannually charges 5% each half-year: $5 on $100 in the first half, then ($100 + $5) × 5% = $5.25 in the second — $10.25 for the year, identical to 10.25% compounded annually. In practice, savings accounts and CDs usually compound annually, while mortgages, home equity loans and credit cards compound monthly. And note the marketing asymmetry: the more often a rate compounds, the smaller the stated number can be for the same true cost — which is why a 6% mortgage is quoted as 6% (0.5% a month) rather than as the 6.17% it totals to over a year. The converter at the top of this page translates any rate between annually, semiannually, quarterly, monthly, semimonthly, biweekly, weekly, daily and continuous compounding.

The formulas

With one compounding period, growth is just:

A_t = A₀(1 + r)ⁿ

where A₀ is the principal, A_t the amount after time t, r the rate per period, and n the number of periods. Example: $1,000 at 6% compounded annually for two years → $1,000 × (1.06)² = $1,123.60. For n compounding periods per year over t years:

A_t = A₀ × (1 + r/n)^nt

Same $1,000 at 6% compounded daily: the daily rate is 6% ÷ 365 = 0.0164384%, and the formula gives $1,000 × (1.000164384)⁷³⁰ = $1,127.49. Push n toward infinity and compounding becomes continuous:

A_t = A₀ · e^rt

$1,000 × e^0.12 = $1,127.50. Daily compounding already lands within a cent of the theoretical maximum — past monthly, extra frequency buys diminishing gains, especially on small principals.

The Rule of 72

Divide 72 by the annual return (in percent, not decimal — use 8, not 0.08) and you get roughly how many years money takes to double: at 8%, about 9 years; at 6%, about 12; at 10%, about 7. It is a mental shortcut, most accurate in the 6–10% range — excellent for judging whether a pitch is plausible, useless as a contract term.

A short history of compounding

Compounding is older than coinage's widespread use: Babylonian and Sumerian texts from roughly 4,400 years ago already apply it, in a distinctive form — interest accrued at 20% of principal until the accumulated interest equaled the principal, and only then was it folded in. Later civilizations were ambivalent: Roman law condemned compound interest, and Christian and Islamic traditions both treated it as usury, a sin. Practice outran doctrine anyway — lenders compounded through the medieval period, and the printing of compound interest tables in the 1600s made it routine. Its mathematical soul arrived in 1683, when Jacob Bernoulli, studying exactly this problem, noticed that ever-finer compounding approaches a limit: the constant e ≈ 2.71828, which Leonhard Euler later named. Every "continuously compounded" option in the converter above is Bernoulli's limit, still working.

FAQ

APR vs APY — which is bigger?

APY. It includes intra-year compounding; APR doesn't. 6% compounded monthly (APR-style) = 6.16778% APY. Deposits must advertise APY, loans APR — each industry quotes its flattering number.

Is daily compounding worth chasing?

Barely. At 6% on $1,000 over two years, daily beats annual by $3.89 and loses to continuous by 1 cent. On a $400,000 mortgage balance the frequency matters; on a small savings account, the rate itself matters far more.

Where do I calculate growth with taxes or inflation?

The Interest Calculator — same compounding engine, with tax skimmed as interest accrues and a buying-power result after inflation.

Why do lenders quote monthly-compounded rates?

Because they look smaller. 6% compounded monthly and 6.17% compounded annually cost the same; the first number wins the advertisement.

Disclaimer: This calculator provides estimates for educational purposes only and does not constitute financial advice. Projections assume a constant rate of return.