From Simple To Compound Interest

Imagine you loan a bank the principal P = 10000 $ at an interest rate of i = 5 %. This is the amount of interest you would receive with simple interest, given the duration t of the loan:

t = 1 year ? I = 10000 $ * 0.05 * 1 = 500 $

t = 2 years ? I = 10000 $ * 0.05 * 2 = 1000 $

t = 3 years ? I = 10000 $ * 0.05 * 3 = 1500 $

As you can see, the interest grows linearly with the duration of the loan. For each additional year, you get an additional 500 $, which is just 5 % of the principal 10000 $. In other words: each year the interest rate is applied to the principal. How could that be any different?

Consider this: At the end of the first year, you'll receive an interest payment in the amount of 500 $. This means that your bank statement will now read 10000 $ + 500 $ = 10500 $. So why not apply the interest rate to this updated value? This would lead to an interest payment of 10500 $ * 0.05 = 525 $ for the second year instead of just 500 $.

Continuing this train of thought, at the end of the second year your bank statement would read 10000 $ + 500 $ + 525 $ = 11025 $. Again we would rather have the interest rate applied to this updated value instead of the unchanging principal. This would result in an interest payment of 11025 $ * 0.05 = 551.25 $ for the third year.

For comparison, here's what the final pay out would be for the simple interest plan:

10000 $ + 500 $ + 500 $ + 500 $ = 11500 $

And this is what we would get with the "not simple" interest plan, where we apply the interest rate to the updated amounts instead of the principal:

10000 $ + 500 $ + 525 $ + 551.25 $ = 11576.25 $

The latter is called compound interest. It means that we include already paid interests in the calculation of next year's interest, which leads to the amount received growing exponentially instead of linearly.