Continuous Compounding Calculator

Compute the future value under continuous compounding (A = Pe^rt) and compare it with daily, monthly, and annual compounding.

Inputs

Initial amount.

Nominal annual interest rate.

Time in years.

Result

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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 − 1

As compounding frequency n grows, the discrete formula converges to the continuous one. The effective annual rate captures the true yearly growth.

Source: Investopedia — Continuous Compounding

Worked examples

Example 1
$1,000, 5%, 10 yr
Inputs:
principal=1000, rate=5, years=10
Output:
continuous $1,648.72; monthly $1,647.01
Example 2
$5,000, 8%, 20 yr
Inputs:
principal=5000, rate=8, years=20
Output:
continuous $24,765.16
Example 3
$10,000, 3%, 5 yr
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.

Frequently asked

What is continuous compounding?+
It is compounding at every instant — the limit of A = P(1 + r/n)^(nt) as the number of periods n approaches infinity. That limit equals A = Pe^(rt), using Euler's number e. It represents the maximum value compounding can reach for a given rate.
Why use e in the formula?+
As you compound more and more frequently, the growth factor (1 + r/n)^(nt) converges to e^(rt). The constant e (≈2.71828) is precisely the base that makes continuous growth work out cleanly, which is why it appears throughout finance and natural growth models.
How much more does continuous compounding earn?+
Only slightly more than daily compounding. At 5% over 10 years on $1,000, continuous gives about $1,648.72 versus $1,648.66 daily and $1,647.01 monthly. The gap widens with higher rates and longer terms but stays modest.
What is the effective annual rate under continuous compounding?+
It is e^r − 1. For a 5% nominal rate, the effective annual rate is e^0.05 − 1 ≈ 5.127%. This is the simple annual rate that would produce the same one-year growth.
Where is continuous compounding actually used?+
In quantitative finance: the Black-Scholes option-pricing model, bond and derivative math, and continuous-time growth models all use e^(rt). It simplifies the calculus even though real cash flows are discrete.
Is continuous compounding better for savers or borrowers?+
For a given nominal rate, more frequent compounding benefits whoever is earning interest (savers) and costs whoever is paying it (borrowers) slightly more. That is why loan disclosures use APR/EAR to make rates comparable across compounding frequencies.

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