Continuous Compounding Calculator
Grow a principal under continuous compounding with A = Pe^{rt}, and see the effective annual rate that continuous compounding implies.
How it works
A = P · e^(r·t) · Effective annual rate = e^r − 1
Continuous compounding is the mathematical limit of compounding ever more frequently — every instant, interest is added and immediately begins earning more interest. It uses Euler’s number e ≈ 2.71828 and gives the largest possible balance for a given nominal rate. Real bank products compound daily at most, so continuous compounding acts as a clean theoretical ceiling and appears throughout finance (for example in the Black–Scholes model).
Worked example
Investing $10,000 at 5% compounded continuously for 10 years gives A = 10,000 · e^(0.05·10) = 10,000 · e^0.5 = 10,000 · 1.64872 = $16,487.21. The 5% nominal rate implies an effective annual rate of e^0.05 − 1 = 5.1271%.
Sources: Compound interest — Wikipedia
Frequently asked questions
How is continuous compounding different from daily compounding?
Daily compounding adds interest 365 times a year; continuous compounding adds it infinitely often. The difference is tiny in practice — at 5%, continuous gives an EAR of 5.1271% versus about 5.1267% for daily — but continuous compounding is easier to work with mathematically.
Why does finance use e for compounding?
As the number of compounding periods per year grows without bound, (1 + r/m)^m converges to e^r. That limit makes e the natural base for growth that happens continuously, which is why it appears in interest, option pricing and population models.
What effective rate does continuous compounding produce?
A nominal rate r compounded continuously has an effective annual rate of e^r − 1. This is always slightly higher than the same nominal rate compounded a finite number of times.
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Educational financial-math tool, not investment or financial advice.