Example 7.7.5. Proportional growth and continuous reinvestment.
I put $20,000 in a retirement account that earns 4% interest compounded continuously. I reinvest all my earnings from the account back into the retirement account. I would like a simple formula for the principal at sometime in the future.
Solution.
We already have the formula for continuous growth, but it is worthwhile to derive it again. We are told
\begin{equation*}
\Principal' (\Time)=.05*\Principal(\Time)\text{.}
\end{equation*}
or
\begin{equation*}
\frac{\Principal' (\Time)}{\Principal(\Time)}=.05\text{.}
\end{equation*}
Integrating both sides with respect to time, we get
\begin{equation*}
\ln (\Principal(\Time))=.05*\Time+C\text{.}
\end{equation*}
Exponentiating both sides gives
\begin{equation*}
\Principal(\Time)=\exp(0.05*\Time)*\exp(C)=e^C e^{.05*\Time}\text{.}
\end{equation*}
Since we know that the Principal is $20,000 at time 0, we see that \(e^C=20000\) and our equation simplifies to
\begin{equation*}
\Principal(\Time)=20000e^{.05*\Time}\text{.}
\end{equation*}
This is the formula we took on faith in the last chapter.