Continuous Compounding

Continuous Compounding

Continuous compounding is the limit of compounding $n$ times per year as $n \to \infty$:

$\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt} = e^{rt}$

Continuous Future Value & Present Value

$FV = PV \cdot e^{rt}$ $PV = FV \cdot e^{-rt}$

This is widely used in derivatives pricing, stochastic calculus, and option theory.

Converting Between Rates

A discrete rate $r_d$ compounded $n$ times per year is equivalent to continuous rate $r_c$:

$r_c = n \cdot \ln\left(1 + \frac{r_d}{n}\right)$

For annual compounding ($n = 1$): $r_c = \ln(1 + r_d)$

Example

1000 invested at 5% continuously for 2 years: $$FV = 1000 \cdot e^{0.05 \times 2} = 1000 \cdot e^{0.1} \approx \1105.17$$

Use Python's math.exp() and math.log() functions.