2.2 Compound interest

And if we take that money out, then invest it for another year, the balance becomes:

We can see that the power of the term $(1+i)$ is the same as the number of years. Therefore,

Fractions add up to the whole

It is easy to show that this formula works even when $n$ is a fraction of a year. For example, let us invest the money for 1 month, then for 4 months, then for 7 months.

which is the same as investing the money for a year.

Look carefully at the long equation above. It is not as complicated as it looks! All we are doing is taking the opening amount ( $P$ ), then adding interest for just 1 month. Then we are taking that new balance and adding interest for a further 4 months, and then finally we are taking the new balance after a total of 5 months, and adding interest for 7 more months. Take a look again, and check how easy it really is.

Does the final formula look familiar? Correct - it is the same result as you would get for simply investing $P$ for one full year. This is exactly what we would expect, because:

1 month + 4 months + 7 months = 12 months,

which is a year. Can you see that? Do not move on until you have understood this point.

The power of compound interest

To see how important this “interest on interest" is, we shall compare the difference in closing balances for money earning simple interest and money earning compound interest. Consider an amount of R10 000 that you have to invest for 10 years, and assume we can earn interest of 9%. How much would that be worth after 10 years?

The closing balance for the money earning simple interest is:

The closing balance for the money earning compound interest is:

So next time someone talks about the “magic of compound interest", not only will you know what they mean - but you will be able to prove it mathematically yourself!

Again, keep in mind that this is good news and bad news. When you are earning interest on money you have invested, compound interest helps that amount to increase exponentially. But if you have borrowed money, the build up of the amount you owe will grow exponentially too.