Continuous Compounding

Builds onEffective Annual Rate (EAR) — if this page feels steep, start there.

FV = PV \cdot e^{r t}

$FV$: future value after growing continuously
$e$: Euler's number, about 2.718 — the universal constant of continuous growth
$e^{rt}$: e raised to r×t: the growth factor when compounding never pauses
$r$: the continuously compounded annual rate, as a decimal
$t$: time in years (can be fractional — 6 months is 0.5)

Play with it

Continuous compounding is the limit curve — push the rate up and watch it pull away from annual compounding.

Rate8.0%Horizon20y

2.6×

Simple @ 20y

4.7×

Compound @ 20y

2.1×

Interest on interest

The gap between the straight line and the curves is interest earning interest. Continuous compounding is the ceiling — barely above annual at low rates, decisive at high ones.

Why it must be true

Compound more and more often — monthly, daily, every second — and the growth factor $(1 + r/m)^{mt}$ climbs toward a ceiling. That ceiling is $e^{rt}$ . Continuous compounding is compounding with the brakes fully off, and $e$ is simply the one-year growth of $1 at 100% compounded continuously.

It matters beyond banking: option pricing and much of quantitative finance quote continuously compounded (log) returns because they add across time instead of multiplying.

The derivation

Start from discrete compounding $m$ times per year for $t$ years:

FV = PV\left(1 + \frac{r}{m}\right)^{mt}

Let the frequency explode. The definition of $e$ is $\lim_{k\to\infty}(1 + \tfrac{1}{k})^k = e$ . Substituting $k = m/r$ :

\lim_{m \to \infty}\left(1 + \frac{r}{m}\right)^{mt} = \left[\lim_{k\to\infty}\left(1 + \frac{1}{k}\right)^{k}\right]^{rt} = e^{rt}

FV = PV \cdot e^{rt}

When to reach for it

The rate is explicitly quoted as continuously compounded, the problem involves log returns, or you are in a derivatives/term-structure setting where continuous time is the convention.

Listen for

“continuously compounded”“log return / force of interest”“e^{rt}”“quant / option-pricing context”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

≈
Rule of 69: with continuous compounding, doubling time is exactly $0.693 / r$ periods (ln 2 ≈ 0.693). At 7%: about 9.9 years.
≈
$e^x \approx 1 + x + \frac{x^2}{2}$ for small $x$ : $e^{0.06} \approx 1.0618$ . The continuous answer sits just above annual compounding at the same rate.
≈
Total exponent first: $r \times t = 0.05 \times 10 = 0.5$ , and $e^{0.5} \approx 1.65$ — memorize $e^{0.5} \approx 1.65$ , $e^1 \approx 2.72$ , $e^{0.7} \approx 2$ .

Traps in applying it

✗Mixing a continuously compounded rate with discrete-compounding formulas in the same calculation.
✗Forgetting the natural log when solving backwards for the rate or time.
✗Leaving t in months while r is per year — the exponent needs consistent units.

Limits & criticisms

No bank actually compounds continuously — it is a modeling idealization chosen because the calculus is cleaner. The numerical difference from daily compounding is tiny, which tempts people to ignore convention mismatches; at institutional scale, and in rate conversions for derivatives, those mismatches are real money.

Where it came from

This formula is where the number $e$ was discovered. In 1683 Jacob Bernoulli asked what a deposit earns as compounding becomes infinitely frequent and found the limit 2.718… — the first constant in mathematics discovered by asking a question about money. Leonhard Euler later named it $e$ and built the calculus around it.

Three centuries on, quantitative finance runs on it: Black, Scholes and Merton (1973) discount continuously in option pricing, term-structure models are written in continuous time, and quants quote log returns because they add across periods. A banking curiosity became the native language of modern finance.

One identity, 3 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Grow at the compounding limit

FV = PV\,e^{rt}

Compounding with the brakes off — the ceiling that more frequent compounding approaches but never exceeds.

Drill this face →

Continuous discounting

PV = FV\,e^{-rt}

The same limit run backwards: multiply by e^(−rt). Option-pricing models discount exactly this way.

Drill this face →

Log (continuously compounded) return

r = \frac{\ln(FV/PV)}{t}

Logs turn multiplicative growth into addition — which is why quants quote log returns: they sum cleanly across time.

Drill this face →

On the BA II Plus

Worked example: A payoff of $30,758.87 arrives in 7.5 years. Discounting continuously at 2.5% per year, what is it worth now?

1.0.025 [×] 7.5 [=] [+|-]the exponent −r×t
2.[2ND] [eˣ]the continuous discount factor
3.[×] 30,758.87 [=]times the future amount

→ $25,500.00

Drill this formula Back to the library