Continuous Compounding: A = Pe^(rt)

The idea

Compounding more frequently gives more growth, but the gain flattens out as $n \to \infty$. The limit is continuous compounding: $A = P e^{rt}$

where $e \approx 2.71828$ is Euler's number. This is the theoretical "max speed" for a given annual rate.

Why the limit is $e^{rt}$

Start from discrete compounding $A = P (1 + r/n)^{nt}$. Let $m = n/r$, so $r/n = 1/m$ and $nt = mrt$: $A = P\left[\left(1 + \dfrac{1}{m}\right)^m\right]^{rt}$

As $n \to \infty$, $m \to \infty$, and $\left(1 + \dfrac{1}{m}\right)^m \to e$. So $A \to P e^{rt}$.

Step-by-step comparison

Setup: $P = 1000$, $r = 0.10$, $t = 5$.

Compute for several $n$:

Annually ($n = 1$): $A = 1000 \cdot (1.10)^5 = 1610.51$

Quarterly ($n = 4$): $A = 1000 \cdot (1.025)^{20} = 1638.62$

Monthly ($n = 12$): $A = 1000 \cdot (1.00833)^{60} \approx 1645.31$

Daily ($n = 365$): $A = 1000 \cdot (1 + 0.10/365)^{1825} \approx 1648.66$

Continuous: $A = 1000 \cdot e^{0.5} \approx \boxed{1648.72}$

The jump from annual to daily is ~$38; the jump from daily to continuous is only ~$0.06. The curve flattens fast past daily.

Rearrangements

Given any three of $A, P, r, t$ you can solve for the fourth:

• $P = A e^{-rt}$ (present value)
• $t = \dfrac{\ln(A/P)}{r}$ (time to a target)
• $r = \dfrac{\ln(A/P)}{t}$ (required rate)

These are the log-rate versions, convenient for analytical work.

Where continuous compounding lives

• Derivative pricing (Black–Scholes, options): quote rates as continuously compounded.
• Physics-style growth/decay: radioactive decay, population models, Newton's law of cooling.
• Effective-rate conversions: convert any compounding convention to continuous and back.

Converting between conventions

Discrete nominal rate $r_n$ compounded $n$ times per year ⟷ continuous rate $r_c$: $r_c = n \ln(1 + r_n/n)$ $r_n = n\left(e^{r_c/n} - 1\right)$

For small rates these are nearly equal; for large rates they diverge meaningfully.

Common mistakes

• Using $e^r$ instead of $e^{rt}$. The exponent is the rate times time, not the rate alone.
• Confusing continuous rates with APY. APY = effective annual yield; continuous rate is a different convention, related by $APY = e^{r_c} - 1$.
• Thinking continuous always crushes monthly. Over a 30-year horizon at 6%, monthly $\approx$ continuous to within 0.1%. The extra is often negligible.

Try it in the visualization

A growth-factor curve shows $(1 + r/n)^{nt}$ approaching $e^{rt}$ from below as $n$ increases. A side panel prints the dollar values so you can see the shrinking gap.