Comparing this with Equation (2) , we see that the return r for the total of the n periods satisfies
This is sometimes called the cumulative return.
Suppose you invest money in an investment vehicle that pays a fixed interest rate r per period, like every month, or every year. Assume further that at the end of each period, the interest is added to the amount invested (often referred to as the principal sum, principal amount, or principal for short). Then the principal increases by a fraction of r every period. Therefore, by definition, r is the return for each period. Equation (3) above then becomes:
This is the fundamental formula for compounding interest at a constant rate. It describes what happens when at the end of each period, interest is paid at a fixed rate and added to the principal. Such a fixed rate per period is also called a period rate of return. If the period is a month, quarter, or year, then, rather obviously, the rate is called the monthly, quarterly, or annual rate of return, respectively.
In real life, pure examples of compounding at a fixed period rate are rare. Just about the only thing that does it is the certificate of deposit (CD). For example, suppose you bought a 5-year CD with a face value of $10,000 and a fixed annual interest rate of 4.5%. According to Equation (5) above, at the end of the 5-year period, you can expect to be paid
Even with CDs, you have probably seen options such as having the interest paid out to you at the end of each year, or even more frequently, rather than let it compound. Also, you may have seen annual interest rates that come with qualifications such as "compounded quarterly," or "compounded monthly." Explaining these options and how they relate to the annual rate of return does not belong in this article, which is strictly about the foundations of compounding and return. In case you were wondering why this article does not explain terms such as "APY" and "APR," that's your answer.
You have probably heard the expression "the power of compounding." It alludes to the fact that the human mind tends to underestimate the growth of an investment that is subject to compounding of interest, or, for that matter, the increase of any quantity that is subject to epxonential growth. Suppose that we invest some amount of money in an investment vehicle with a fixed period rate r. At the end of each period, the value of the investment, that is, the principal, increases by the same factor, namely, 1 + r. But the amount to which this factor is applied increases with every period that goes by. As a consequence, the increment by which the principal goes up becomes larger with every period. The result is a kind of snowball effect, which causes the principal to grow in a manner that our intuition tends to underestimate, often dramatically so.
Let us look at an example. Suppose you invest $100,000 at an annual interest rate of 5% for 35 years. If you have the $5,000 of annual interest paid out to you at the end of each year, leaving the principal constant at $100,000, your total payout will be 35 × $5,000 = $175,000. If, on the other hand, you add the interest to the principal every year and let it compound, you will earn $425,334.80 of interest for a total ending balance of $525,334.80, as shown in this chart: