Compound Interst Formula

Suppose that on January 1st 2000 you invest \(P\) dollars in a retirement account that has a fixed return of \(R\) percent each year. On January 1st 2001 you will have \(P(1+r)\) dollars in your account, where \(r = R/100\). For example, if you invest \(\$100\) in an account that has a return of \(R=5\%\) each year, then after one year your account will have \(100(1+0.05) = 100\times1.05 = 105\) dollars in it.

Now suppose that instead of giving a return of \(R\) percent each year, the account gives a return of \(R/N\) percent \(N\) times per year. Then after \(\frac{1}{N}\) years have passed (yes this is a weird way to say that a few weeks has gone by) you will have \[ \begin{equation} A\left(\frac{1}{N}\right) = P\left(1 + \frac{r}{N}\right) \end{equation} \] dollars in your account where, recall, \(r = R/100\), \(P\) is the initial amount you invested at time zero, and \(A(t)\) is the amount in your account after \(t\) years has passed. After \(\frac{2}{N}\) years have passed you will earn another \(R\) percent on top of the amount you earned in the first \(\frac{1}{N}\) years, and so \[ \begin{aligned} A\left(\frac{2}{N}\right) & = A\left(\frac{1}{N}\right)\left(1 + \frac{r}{N}\right) \\ & = P\left(1 + \frac{r}{N}\right)\left(1 + \frac{r}{N}\right) \\ & = P\left(1 + \frac{r}{N}\right)^2. \end{aligned} \] We could continue in this way and see that in general, after \(k/N\) years have passed, we would have \[ \begin{aligned} A\left(\frac{k}{N}\right) = P\left(1+ \frac{r}{N}\right)^k. \end{aligned} \] If we let \(t = \frac{k}{N}\) then this formula becomes \[ \begin{aligned} A(t) = P\left(\left(1+ \frac{r}{N}\right)^N\right)^t. \end{aligned} \] This is the formula that describes the amount in your account when the initial investment is compounded \(N\) times per year. It can be shown that compounding interest in this way earns more than just compounding once at the end of the year. That is, earning \(R/N\) percent \(N\) time per year leads to more money in your account at the end of the year as compared to earning \(R\) percent once a year.

If we keep increasing \(N\) so that you earn interest on your investment more and more frequently you approach continuously compounding interest. Mathematically this can be represented with a limit: \[ \begin{aligned} A_{cont}(t) = \lim\limits_{N\to\infty} P\left(\left(1+ \frac{r}{N}\right)^N\right)^t. \end{aligned} \] Here we can use the fact that \[ \begin{aligned} e^r = \lim\limits_{N \to \infty}\left(1+ \frac{r}{N}\right)^N \end{aligned} \] to write \[ \begin{aligned} A_{cont}(t) = Pe^{rt}. \end{aligned} \]