Continuous compounding: formula, calculator and worked example

What the symbols mean

$A$: the final amount
$P$: the principal (starting amount)
$r$: the annual interest rate as a decimal (5% = 0.05)
$t$: the time in years (years)
$e$: Euler's number, about 2.71828

When to use it

Use it when a rate is quoted as compounded continuously, in finance theory (option pricing, continuously compounded returns), and as the upper limit of how much compounding frequency can help. In practice daily compounding gives almost the same answer.

Worked example

1,000 is invested at 5% compounded continuously for 10 years. What is it worth?

Start from the formula.
$A = P e^{r t}$
Substitute $P$ = $1,000.00$ , $r$ = $0.05$ , $t$ = $10$ and work it out.
$A = 1,000.00 e^{0.05 \cdot 10}$
$= 1,000.00 e^{0.5}$
$\approx 1,648.72$

Answer A ≈ 1,648.72

Why it works

As n grows, $(1 + r/n)^(nt)$ approaches $e^(rt)$ ; that limit is one way of defining e. Going from yearly to monthly compounding makes a noticeable difference, but going from daily to continuous adds almost nothing.

Common mistakes

Using the percentage in the exponent: $e^(5×10)$ instead of $e^(0.05×10)$ .
Expecting a big gain over monthly compounding; at ordinary rates the difference is small.
Solving for t with logs in base 10 but e in the formula; use ln, the natural log.

Continuous compounding

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes