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To explain the concept of compound interest, the following example is discussed:
I deposit R1 000 into a special bank account which pays a Simple Interest of 7%. What if I empty the bank account after a year, and then take the principal and the interest and invest it back into the same account again. Then I take it all out at the end of the second year, and then put it all back in again? And then I take it all out at the end of 3 years?
We are required to find the closing balance at the end of three years.
We know that:
After the first year, we withdraw all the money and re-deposit it. The opening balance for the second year is therefore , because this is the balance after the first year.
After the second year, we withdraw all the money and re-deposit it. The opening balance for the third year is therefore , because this is the balance after the first year.
The closing balance after withdrawing all the money and re-depositing each year for 3 years of saving R1 000 at an interest rate of 7% is R1 225,04.
In the two worked examples using simple interest ( [link] and [link] ), we have basically the same problem because =R1 000, =7% and =3 years for both problems. Except in the second situation, we end up with R1 225,04 which is more than R1 210 from the first example. What has changed?
In the first example I earned R70 interest each year - the same in the first, second and third year. But in the second situation, when I took the money out and then re-invested it, I was actually earning interest in the second year on my interest (R70) from the first year. (And interest on the interest on my interest in the third year!)
This more realistically reflects what happens in the real world, and is known as Compound Interest. It is this concept which underlies just about everything we do - so we will look at it more closely next.
Compound interest is the interest payable on the principal and its accumulated interest.
Compound interest is a double-edged sword, though - great if you are earning interest on cash you have invested, but more serious if you are stuck having to pay interest on money you have borrowed!
In the same way that we developed a formula for Simple Interest, let us find one for Compound Interest.
If our opening balance is and we have an interest rate of then, the closing balance at the end of the first year is:
This is the same as Simple Interest because it only covers a single year. Then, if we take that out and re-invest it for another year - just as you saw us doing in the worked example above - then the balance after the second year will be:
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