Section 2.3 Continuous Compounding of Interest
As interest is compounded more often, the effective interest rate tends to increase. Indeed, as m increases then the time between periods decreases and becomes infinitesimal. We would then say that interest is compounded continuously. As it turns out, we will see that there is a limit to the effective rate as m grows.
Theorem 2.3.1. Future Value when Compounding Continuously.
\begin{equation*}
\displaystyle A = P e^{it}
\end{equation*}
Proof.
\begin{align*}
A & = P \lim_{m \rightarrow \infty} (1+i/m)^{m t}\\
& = P \lim_{m \rightarrow \infty} \left ( (1+i/m)^m \right )^t\\
& = P \lim_{m \rightarrow \infty} \left ( e^{ \ln(1+i/m)^m} \right )^t\\
& = P \lim_{m \rightarrow \infty} \left ( e^{m \cdot \ln(1+i/m)} \right )^t\\
& = P \lim_{m \rightarrow \infty} \left ( e^{\frac{ \ln(1+i/m)}{1/m}} \right )^t\\
& = P \lim_{m \rightarrow \infty} \left ( e^{\frac{ \frac{-i/m^2}{(1+i/m)}}{-1/m^2}} \right )^t\\
& = P \lim_{m \rightarrow \infty} \left ( e^{ \frac{i}{(1+i/m)}} \right )^t\\
& = P \left ( e^{ \frac{i}{1+0}} \right )^t\\
& = P e^{i \cdot t}
\end{align*}