Compound interest is one of the foundations of modern finance. The basic idea is that your investment will earn interest on the amount invested (the principal) as well as the interest itself. There are two primary versions of compound interest that we will explore in this section. The first is the easiest to make sense of, the case where there are a fixed number of times each year when the interest is computed and then added to the account. The other version is harder to understand intuitively because it involves interest being computed an infinite number of times. While it may seem that this would give you an infinite amount of money, since the interest rate for each period is infinitesimally small (it is the annual percentage rate divided by the number of compounding periods, so it is extremely small) the total amount reaches a fixed limit related to the number e.
Once we understand the basics of compound interest it can be applied to many other economic and financial concepts, such as present value and future value of an investment. The present value of an investment is the amount you would need to invest today in order to achieve a fixed level at the end of the investment period. This situation is most easily understood through the modern day phenomenon of the lottery. Most lotteries offer the winner two choices of payment: a lump sum now or small payments made over a longer period of time, say 20 years. If the winner ”won” $1 million, she would, for example, have to choose between monthly payments of $50,000 each year for twenty years (a total of $1 million) or a lump sum payment of $548,811.64 right now. Ignoring all taxes, of course, which substantially change the problem under consideration, the reason the lump sum payment is so much less than the actual winnings is that you are getting it now. If you were to invest it at 3% for 20 years, you would have about $1 million at the end, the same amount as the lottery winnings. Since the lottery company would have access to the money in the 20-year payment version, they would be earning interest on the $1 million over that entire 20 years. But if they have to pay you all right now, they lose that interest. Thus, the present value of the $1 million lottery winning is about $550,000, assuming a 3% interest rate annually. We will further explore the idea of present value in the problems for this section.