Calculate the future value of an investment under continuous compounding using the formula A = Pe^(rt).
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Time Value of Money
Continuous compounding is the theoretical limit of compounding frequency — interest compounds every instant. This calculator uses the exponential formula A = Pe^(rt) to show the maximum possible compound growth.
As compounding frequency increases from annual to monthly to daily, the future value converges to a limit defined by Euler’s number (e ≈ 2.71828). Continuous compounding represents this mathematical ceiling.
In practice, no bank compounds continuously, but the formula is widely used in derivatives pricing (Black-Scholes), theoretical finance, and as a benchmark for comparing different compounding frequencies.
The exponential growth formula.
A = P × e^(rt)
P = principal, r = annual rate (decimal), t = time in years, e = Euler’s number ≈ 2.71828.
No real-world instrument compounds continuously. The difference between daily and continuous compounding is negligible for typical rates and periods. Useful primarily as a theoretical benchmark.
Frequently asked questions
Very little. At 8% for 10 years on 10,000: monthly compounding gives 22,196; continuous gives 22,255 — a difference of about 59 or 0.27%. The gap widens slightly with higher rates or longer periods.
In the Black-Scholes option pricing model, in continuously compounded return calculations for academic finance, and in theoretical present value calculations. Banks and savings accounts use discrete compounding.
For continuous compounding: EAR = e^r − 1. At 8% nominal: EAR = e^0.08 − 1 = 8.33%. This is the highest possible EAR for a given nominal rate.
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