Continuous Compound Interest — The Formula A = Pe^(rt) Explained

What Is Continuous Compounding?

Normal compounding happens at intervals — daily (365 times/year), monthly (12 times), or annually (once). Continuous compounding is the theoretical limit: compounding happens infinitely often.

The formula uses Euler’s number (e ≈ 2.71828):

A = P × e^(r × t)

Worked Example

$10,000 at 5% for 10 years, continuous compounding:

• A = 10,000 × e^(0.05 × 10)
• A = 10,000 × e^0.5
• A = 10,000 × 1.64872
• A = $16,487.21

Compare: daily compounding gives $16,486.65 — a difference of just $0.56.

Continuous vs Other Frequencies

$10,000 at 5%:

Period	Continuous	Daily	Monthly	Annual
1 year	$10,512.71	$10,512.67	$10,511.62	$10,500.00
10 years	$16,487.21	$16,486.65	$16,470.09	$16,288.95
30 years	$44,816.89	$44,812.69	$44,677.44	$43,219.42

Continuous compounding is essentially identical to daily for practical purposes.

When Is This Formula Used?

Continuous compounding is mostly theoretical, but it’s important in:

1. Black-Scholes model — options pricing uses continuous compounding
2. Academic finance — simplifies calculus-based derivations
3. Population growth — bacteria and populations grow continuously
4. Physics — radioactive decay, temperature change
5. Upper-bound estimates — the maximum possible growth at a given rate

Continuous Compounding APY

To find the effective annual rate (APY) for continuous compounding:

APY = e^r − 1

Nominal Rate	Continuous APY
3%	3.045%
5%	5.127%
7%	7.251%
10%	10.517%
12%	12.750%

The Practical Takeaway

For real-world investing, continuous compounding is irrelevant — the difference from daily compounding is cents on thousands. It matters when:

• You’re studying financial mathematics
• You’re pricing derivatives
• You need the simplest formula for calculus operations

For everything else, use our compound interest calculator with daily or monthly compounding — the results will match reality.