A good way of improving one's intuition for the power of compounding is to note that if interest is compounding at a constant rate, the principal will double every x number of years, where x is a number that depends on the interest rate. If the rate is relatively small, say within the 0 to 10 percent range, a decent rule of thumb is that the principal doubles every .7/r years, where r is the interest rate as a fraction. For example, at 7% interest, the principal doubles roughly every .7/.07 = 10 years, and at 3.5%, it doubles every .7/.035 = 20 years. A proof of this is given in the appendix at the end of this article.
Suppose that five years ago, you invested $10,000 in some mutual fund, and the current value of that investment is $13,069.60. According to Equation (1), the return for those five years is
Furthermore, assume that six years ago, you invested $10,000 somewhere else, and those $10,000 have now turned into $13,560.07. The return for the six years is
Which investment has performed better? The return is greater for the second investment. But that's for six years instead of five, so it must be greater even if the two investments performed equally well. But how much greater?
The way to answer this question is to assume, hypothetically, that the money had been invested in an account with a fixed annual interest rate, like a (somewhat idealized) savings account. In reality, the way the $10,000 became $13,069.60, or $13,560.07 in the second case, probably involved a lot of ups and downs. But that's all water under the bridge now. We can ask, had the two deposits of $10,000 each been placed in savings accounts with fixed interest rates, then which annual interest rates would have resulted in the ending balances of $13,069.60 and $13,560.07, respectively? In the first case, the answer is 5.5%, while in the second, it's 5.2%. We'll show how to get those results in a moment. The point here is that from hindsight, the first investment behaved like a savings account with a 5.5% interest rate, while the second one behaved like a savings account with a 5.2% interest rate. That's a fair and intuitive way of comparing the two investments: 5.5% annually is better than 5.2%. This rate is called the annual rate of return, or annualized return, of the investment. It is the standard measure of the performance of an investment over time.