Continuous Compound Interest Calculator
Using FV = P·e^(rt), compute the future value, interest and equivalent effective annual rate under continuous compounding.
Input Data
Results
At a glance:Continuous compounding is the limit as frequency → infinity; FV = P x e^(r·t), e ≈ 2.71828; interest = FV - P; equivalent EAR = e^r - 1. It is the theoretical upper bound of compounding, used in mathematical finance (Black-Scholes, rate models). Example: 1,000 at 12%, 1y → ≈1,127.50. WARNING: No real deposit compounds truly every instant; daily is the practical max and is nearly identical. Theoretical estimate, not advice.
Formula
FV = P × e^(r×t), e ≈ 2.71828.
Interest = FV − P.
Equivalent EAR = e^r − 1.
$$$Interest = FV - P$$$$$$EAR = e^{r}-1$$$$$(12%1 )$1000\\times e^{0.12}\\approx1127.50$$$How to Use
- Enter the principal.
- Enter the annual rate and years.
- View the future value, interest and equivalent EAR.
FAQ
What is continuous compounding?
It assumes interest is reinvested at every instant — the limit as frequency → infinity. At the same nominal rate, it produces the highest possible future value.
How much does it differ from daily compounding?
Very little. At 12%, HK$1,000 daily for a year ≈ HK$1,127.47; continuous ≈ HK$1,127.50 — cents apart. Continuous is mainly for theoretical analysis.
Does continuous compounding really exist, and where is it used?
Strictly, no bank compounds truly every instant; the most frequent is daily, already near the continuous limit. Continuous compounding is used in mathematical finance (Black-Scholes option pricing, interest-rate and bond models), academia, and as a benchmark to convert rates (force of interest) for fair comparison.
What is the constant e, and why does it appear here?
e ≈ 2.71828 is the natural constant, the limit of (1 + 1/n)^n as n → infinity — exactly the continuous-compounding limit. Generalised, FV = P x e^(r·t). It also appears in natural growth/decay, probability and calculus. The tool has e built in; just enter P, r, t.
What is the equivalent EAR for?
It converts continuous compounding to an 'equivalent annual rate compounded once' (e^r - 1) so different compounding bases can be compared fairly. Example: '8% continuous' → EAR ≈ 8.33%, which beats '8.2% annual'. Like APY for deposits or the true annual rate for loans, EAR strips out frequency to reveal the real yearly rate. This tool outputs the equivalent EAR for easy comparison.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.