Compounding taken to its limit
Every time interest is added to a balance, the new interest immediately starts earning interest of its own. Compound monthly instead of yearly and you get a little more; compound daily and a little more again. Keep chopping that interval into ever-smaller slices — hourly, by the second, by the instant — and the future value does not run away to infinity. Instead it settles on a precise ceiling described by Euler’s number e ≈ 2.71828, the irrational constant that is the limit of (1 + 1/n)n as n grows without bound. That limit is continuous compounding: the smoothest, fastest version of growth a given annual rate can produce.
The continuous compounding formula
A = P · e^(r·t), where e ≈ 2.71828
where P = the principal you start with, r = the annual rate as a decimal, t = the time in years, andA = the final amount. The constant e is the base of the natural logarithm; raising it to the power r·t is what captures interest being reinvested an infinite number of times over the term.
What makes this calculator different
- Shows the effective annual rate. Beyond the final balance, it computes the true yearly growth, er − 1, so you can compare a continuously compounded rate against the APY quoted on an ordinary account.
- Quantifies the gap vs annual compounding. It puts the continuous result side by side with plain once-a-year compounding, so you see in dollars exactly how much the infinite-frequency limit actually adds.
- Built on the exact P·e^(rt) limit. No frequency-by-frequency approximation — the result is the precise mathematical ceiling, the same form used in finance theory and options pricing.
Frequently asked questions
What is continuous compounding?+
Continuous compounding is the theoretical limit of compounding interest as the number of compounding periods grows toward infinity. Ordinary accounts add interest at fixed intervals — once a year, monthly, or daily — and each time, the new interest starts earning interest of its own. If you keep shrinking that interval until interest is credited at every instant, the result converges on a single value described by Euler’s number e. It is the smoothest, fastest version of compound growth for a given annual rate.
What is the continuous compound interest formula, and what is e?+
The formula is A = P · e^(r·t), where P is the principal, r is the annual rate as a decimal, t is the time in years, and A is the final amount. The constant e ≈ 2.71828 is Euler’s number, an irrational mathematical constant that arises naturally as the limit of (1 + 1/n)^n as n grows without bound. It is the base of the natural logarithm and shows up wherever something grows in proportion to its own current size. In this formula, e captures the effect of interest being reinvested an infinite number of times.
How much more does continuous compounding earn than monthly or annual compounding?+
Less than most people expect. At a 5% annual rate, compounding once a year turns $10,000 into $10,500 after one year, monthly compounding gives about $10,511.62, and continuous compounding gives $10,512.71 — a difference of roughly a dollar over annual once monthly is involved. The gap between annual and continuous widens with higher rates and longer terms, but the bulk of the benefit is already captured by monthly or daily compounding. Continuous compounding is the ceiling, not a dramatic jump beyond frequent compounding.
What is the effective annual rate under continuous compounding?+
The effective annual rate (EAR) is the actual percentage your balance grows in one year, and under continuous compounding it equals e^r − 1. For a 5% nominal rate that is e^0.05 − 1 ≈ 5.127%, and for 10% it is e^0.10 − 1 ≈ 10.517%. The EAR is always slightly higher than the stated nominal rate because the interest is being reinvested continuously. This calculator shows the EAR so you can compare a continuously compounded rate against an APY quoted on an ordinary account.
Where is continuous compounding actually used?+
Almost nowhere on everyday bank accounts — savings accounts and CDs compound daily or monthly, not continuously. Its real home is finance theory and quantitative modeling, where the smoothness of e^(rt) makes the calculus far cleaner. It underpins the Black-Scholes options-pricing model, bond and derivative valuation, and the way economists model exponential growth and decay. Think of it as the mathematician’s idealization that simplifies the equations, rather than something your bank literally does to your balance.
Disclaimer: This calculator is for educational purposes only. Continuous compounding is a mathematical idealization — most real accounts compound daily or monthly, not continuously, so a bank’s actual figures may differ slightly. It is not financial, investment, or tax advice. Consider speaking with a qualified professional before making decisions.