Continuous Compounding Calculator
Compute the future value under continuous compounding (A = Pe^rt) and compare it with daily, monthly, and annual compounding.
Result
- Continuous (e^rt)$1,648.72
- Daily compounding$1,648.66
- Monthly compounding$1,647.01
- Annual compounding$1,628.89
- Interest earned (continuous)$648.72
- Effective annual rate (continuous)5.1271%
Step-by-step
- Continuous: A = P·e^(rt) = $1,000.00 × e^(0.05×10) = $1,648.72.
- Compare discrete: monthly $1,647.01, daily $1,648.66 — both approach the continuous limit.
- Effective annual rate under continuous compounding = e^r − 1 = 5.1271%.
How to use this calculator
- Enter the principal amount.
- Enter the annual interest rate and the number of years.
- Read the continuous-compounding future value.
- Compare it with daily, monthly, and annual compounding in the breakdown.
About this calculator
Continuous compounding is the mathematical limit of compound interest as the compounding frequency increases without bound — instead of adding interest yearly, monthly, or daily, interest is effectively added at every instant. The future value is given by the elegant formula A = Pe^(rt), where e is Euler's number (≈2.71828), r is the annual rate, and t is the time in years. While no real bank compounds literally continuously, the formula is the theoretical ceiling and is widely used in finance — for pricing options (Black-Scholes), modeling growth, and converting between rates. In practice the difference between continuous and daily compounding is tiny, but it grows with higher rates and longer horizons. This calculator shows the continuous result alongside annual, monthly, and daily compounding so you can see how they converge.
How it works — the formula
Continuous: A = P · e^(r·t)
Discrete: A = P · (1 + r/n)^(n·t)
EAR (cont): e^r − 1As compounding frequency n grows, the discrete formula converges to the continuous one. The effective annual rate captures the true yearly growth.
Worked examples
- Inputs:
- principal=1000, rate=5, years=10
- Output:
- continuous $1,648.72; monthly $1,647.01
- Inputs:
- principal=5000, rate=8, years=20
- Output:
- continuous $24,765.16
- Inputs:
- principal=10000, rate=3, years=5
- Output:
- continuous $11,618.34
Limitations
- Theoretical limit; real accounts compound at finite intervals.
- Assumes a constant rate over the whole period.
- Pre-tax and pre-fee.
Educational/financial-math tool, not an account projection.