The basics of compounding
Before turning to the math, it's useful to go through the basic concept behind compound returns. The most important aspect of compounding is the idea that when you reinvest the investment income and gains that you earn back into your investment account, those reinvested funds also earn a return going forward. The further into the future you go, the more of your total account balance represents money that you earned as a return at some point along the way.
For instance, take a simple example in which you have $100 and an opportunity to invest it at 10% for the long term. At the end of the first year, your $100 will have generated a 10% return, rising in value by $10 and totaling up to $110. The next year, however, you won't just earn $10. Instead, you'll earn $11 because you'll get the 10% return not just on your original $100 investment but also on the $10 in returns from the first year. Similarly, by the third year, you'll be earning 10% on $121 rather than $100. As time goes by, your account balance will go up exponentially rather than linearly.