Continuous Compounding of Interest

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*}

xxxxxxxxxx
 
var('t')
@interact
def _(rate = slider(1,20,1/10,5, label="$$\\% \\text{ Rate}$$"),
      m = slider(1,365,1,label="$$\\text{Yearly Periods}$$"),
      end_time=slider(1,20,1/2,label="$$ \\text{Years}$$") ):
    rate = rate/100
    n = m*t
    discrete_compounding = (1+rate/m)^n
    continuous_compounding = e^(rate*t)
    G = plot(discrete_compounding,t,0,end_time)
    G += plot(continuous_compounding,t,0,end_time,color='green')
    G.show(title='Discrete vs Continuous Growth of 1 dollar',figsize=[5,3])
    H = plot(((continuous_compounding-discrete_compounding)/discrete_compounding),t,0,end_time,color='red')
    H.show(title='Relative Difference between the two',figsize=(5,3))