# Compound Interest

Imagine you put $\$100$ in a savings account with a yearly interest rate of $6\%$ .

After one year, you have
$100+6=\$106$
. After two years, if the interest is
**
simple
**
, you will have
$106+6=\$112$
(adding
$6\%$
of the original principal amount each year.) But if it is
**
compound interest
**
, then in the second year you will earn
$6\%$
of the
**
new
**
amount:

$1.06\times \$106=\$112.36$

## Yearly Compound Interest Formula

If you put $P$ dollars in a savings account with an annual interest rate $r$ , and the interest is compounded yearly, then the amount $A$ you have after $t$ years is given by the formula:

$A=P{\left(1+r\right)}^{t}$

**
Example:
**

Suppose you invest $\$4000$ at $7\%$ interest, compounded yearly. Find the amount you have after $5$ years.

Here, $P=4000$ , $r=0.07$ , and $t=5$ . Substituting the values in the formula, we get:

$\begin{array}{l}A=4000{\left(1+0.07\right)}^{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx 4000\left(1.40255\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=5610.2\end{array}$

Therefore, the amount after $5$ years would be about $\$5610.20$ .

## General Compound Interest Formula

If interest is compounded more frequently than once a year, you get an even better deal. In this case you have to divide the interest rate by the number of periods of compounding.

If you invest $P$ dollars at an annual interest rate $r$ , compounded $n$ times a year, then the amount $A$ you have after $t$ years is given by the formula:

$A=P{\left(1+\frac{r}{n}\right)}^{nt}$

**
Example:
**

Suppose you invest $\$1000$ at $9\%$ interest, compounded monthly. Find the amount you have after $18$ months.

Here $P=1000$ , $r=0.09$ , $n=12$ , and $t=1.5$ (since $18$ months = one and a half years).

Substituting the values, we get:

$\begin{array}{l}A=1000{\left(1+\frac{0.09}{12}\right)}^{12\left(1.5\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx 1000\left(1.143960\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1143.960\end{array}$

Rounding to the nearest cent, you have $\$1143.96$ .