Next: Present Values of Annuities Up: Basic Mathematics of Finance Previous: Notation

Compound Interest

We find the amount which accumulates when a principle P is put in an account with an annual interest rate r compounded m times a year for t years is given by

$\begin{displaymath}P_n = P\left( 1+ \frac{r}{m}\right)^{mt}\end{displaymath}$

This can be visualized using a timeline as follows:

$\begin{picture}(380,65)(-20,-65)\usebox{\timeline}\usebox{\timelabelsB}\put(-2,-......put(280,-62){$\displaystyle{=P\left(1+\frac{r}{m}\right)^{mt}}$ }\end{picture}$

Using this formula we can find the amount of money which results when an investment is left to collect interest or find out how much money would need to be invested to reach a goal.

Example:

For instance, suppose I deposit $2000 now at 5% annual interest compounded monthly and ask how much I will have in that account 15 years from now (approximately when I might retire). Here $P=\$2000, r=.05, m=12,\mbox{ and }t=15$ . I will end up with

$\begin{displaymath}\$2000\left(1+\frac{.05}{12}\right)^{12\cdot 15}=\$4227.41 \end{displaymath}$

Doing the same calculation for someone who will be retiring in 45 years instead gives

$\begin{displaymath}\$2000\left(1+\frac{.05}{12}\right)^{12\cdot 45}=\$18,886.98 \end{displaymath}$

With compound interest it pays to wait longer! $\diamondsuit$

Example:

A related calculation lets me calculate how much I would need to deposit now to have $1,000,000 in 15 years, again assuming 5% annual interest compounded monthly. Here I take

$\begin{displaymath}1000000 = P \left( 1+\frac{.05}{12}\right)^{12\cdot 15}\end{displaymath}$

and solve for P:

$\begin{displaymath}P=\frac{1000000}{ \left( 1+\frac{.05}{12}\right)^{12\cdot 15}}=473103.16\end{displaymath}$

$\diamondsuit$

Example:

We can also see how long it takes my $2000 to grow into $10000 by solving

$\begin{displaymath}10000 = 2000 \left( 1+\frac{.05}{12}\right)^{12\cdot t}\end{displaymath}$

for t using logarithms:

$\begin{eqnarray*}5=\frac{10000}{2000}&=& \left( 1+\frac{.05}{12}\right)^{12\cdot......eft( 1+\frac{.05}{12}\right)}&=&t\\t & = & 32.26 \mbox{ years }\end{eqnarray*}$

$\diamondsuit$

The compound interest formula can also help you evaluate consurmer finance situations:

Example:

A credit card which advertises that the rate of interest is .078% per day ends up turning a $100 bill into

$\begin{displaymath}\$100 (1+.00078)^{365}=132.92\end{displaymath}$

in one year, so the effective rate of interest is 32.92% per year. $\diamondsuit$

Next: Present Values of Annuities Up: Basic Mathematics of Finance Previous: Notation

Larry Stout
2000-10-09