0.8 Mathematics of finance  (Page 3/9)

After one year, we had

After two years, we had

But $\$\text{216}=\$\text{200}\left(1+\text{.}\text{08}\right)$ , therefore, the above expression becomes

$\begin{array}{|c|}\hline \$\text{200}\left(1+\text{.}\text{08}\right)\\ \hline\end{array}\left(1+\text{.}\text{08}\right)=\$\text{233}\text{.}\text{28}$

After three years, we get

Which can be written as

Suppose we are asked to find the total amount at the end of 5 years, we will get

We summarize as follows:

Banks often compound interest more than one time a year. Consider a bank that pays 8% interest but compounds it four times a year, or quarterly. This means that every quarter the bank will pay an interest equal to one-fourth of 8%, or 2%.

Now if we deposit $200 in the bank, after one quarter we will have $\$\text{200}\left(1+\frac{\text{.}\text{08}}{4}\right)$ or $204.

After two quarters, we will have $\$\text{200}{\left(1+\frac{\text{.}\text{08}}{4}\right)}^2$ or $208.08.

After one year, we will have $\$\text{200}{\left(1+\frac{\text{.}\text{08}}{4}\right)}^4$ or $216.49.

After three years, we will have $\$\text{200}{\left(1+\frac{\text{.}\text{08}}{4}\right)}^{\text{12}}$ or $253.65, etc.

Therefore, if we invest a lump-sum amount of $P$ dollars at an interest rate $r$ , compounded $n$ times a year, then after $t$ years the final amount is given by

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

Interest can be compounded yearly, semiannually, quarterly, monthly, daily, hourly, minutely, and even every second. But what do we mean when we say the interest is compounded continuously, and how do we compute such amounts. When interest is compounded "infinitely many times", we say that the interest is compounded continuously . Our next objective is to derive a formula to solve such problems, and at the same time put things in proper perspective.