Skip to main content
\(\require{cancel}\newcommand\degree[0]{^{\circ}} \newcommand\Ccancel[2][black]{\renewcommand\CancelColor{\color{#1}}\cancel{#2}} \newcommand{\alert}[1]{\boldsymbol{\color{magenta}{#1}}} \newcommand{\blert}[1]{\boldsymbol{\color{blue}{#1}}} \newcommand{\bluetext}[1]{\color{blue}{#1}} \delimitershortfall-1sp \newcommand\abs[1]{\left|#1\right|} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

SectionContinuous Growth

Recall that in section \(3.4 \) we found that with a principal \(P \text{,}\) and interest rate \(r\text{,}\) if we compound this interest \(n\)-times a year the value, \(A(t)\text{,}\) of the investment after \(t\) years is given by the formula

\begin{equation*} A(t)=P(1+\dfrac{r}{n})^{nt}. \end{equation*}

As \(n \) gets large (i.e. as \(n\) approaches infinity) the growth factor in the above formula, \((1+\dfrac{r}{n})^n\text{,}\) approaches \(e^r \) where \(e \approx 2.71828\text{.}\) So, for very large values of \(n\text{,}\)

\begin{equation*} P(1+\dfrac{r}{n})^{nt}\approx Pe^{rt}. \end{equation*}

In general, we have the following definition:

Compounded Continuously

The amount, A(t), accumulated (principal plus interest) in an account bearing interest compounded continuously is

\begin{equation*} A(t) = P(e)^{rt} \end{equation*}

where \begin{align*} \amp P \amp\amp \text{is the principal invested,} \\ \amp r \amp\amp \text{is the continuous interest rate,} \\ \amp t \amp\amp \text{is the time period, in years}. \end{align*}

SubsectionToo Many Rates!

Throughout this chapter we have discussed several different rates which can be very confusing, especially since some of the different names mean the same thing! First, let's talk about effective interest rate and annual interest rate.

Effective interest rate and annual interest rate mean the exact same thing. If we label this interest rate as \(r\text{,}\) then this would be the \(r \) appearing in the formula

\begin{equation*} A(t)=a(1+r)^t \end{equation*}

for an interst which is compounded once a year (or annually).

Nominal rate is synonymous with stated (or given) rate. If we label this rate is \(r \text{,}\) then this would be the \(r \) appearing in each of the following formulas:

  • \(a(1+r)^t\text{,}\)
  • \(a(1+\dfrac{r}{n})^{nt}\text{,}\)
  • \(P(e)^{rt}. \)

Continuous interest rate is simply the interest rate appearing in the formula for interest which is compounded continuously. In other words, if we label this interest rate as \(r \) then this would be the \(r \) appearing in the formula

\begin{equation*} A(t)=P(e)^{rt}. \end{equation*}