0.8 Mathematics of finance  (Page 2/9)

Discounts and proceeds

Compound interest

Section overview

In this section you will learn to:

1. Find the future value of a lump-sum.
2. Find the present value of a lump-sum.
3. Find the effective interest rate.

In the [link] , we did problems involving simple interest. Simple interest is charged when the lending period is short and often less than a year. When the money is loaned or borrowed for a longer time period, the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded .

Suppose we deposit $200 in an account that pays 8% interest. At the end of one year, we will have $\$\text{200}+\$\text{200}\left(\text{.}\text{08}\right)=\$\text{200}\left(1+\text{.}\text{08}\right)=\$\text{216}$ .

Now suppose we put this amount, $216, in the same account. After another year, we will have $\$\text{216}+\$\text{216}\left(\text{.}\text{08}\right)=\$\text{216}\left(1+\text{.}\text{08}\right)=\$\text{233}\text{.}\text{28}$ .

So an initial deposit of $200 has accumulated to $233.28 in two years. Further note that had it been simple interest, this amount would have accumulated to only $232. The reason the amount is slightly higher is because the interest ($16) we earned the first year, was put back into the account. And this $16 amount itself earned for one year an interest of $\$\text{16}\left(\text{.}\text{08}\right)=\$1\text{.}\text{28}$ , thus resulting in the increase. So we have earned interest on the principal as well as on the past interest, and that is why we call it compound interest.

Now suppose we leave this amount, $233.28, in the bank for another year, the final amount will be $\$\text{233}\text{.}\text{28}+\$\text{233}\text{.}\text{28}\left(\text{.}\text{08}\right)=\$\text{233}\text{.}\text{28}\left(1+\text{.}\text{08}\right)=\$\text{251}\text{.}\text{94}$ .

Now let us look at the mathematical part of this problem so that we can devise an easier way to solve these problems.