Suppose you have something worth \(V_0\) right now, and that it grows in value by \(r\) percent on some fixed schedule — it increases in value by 1% every month, for example. A table of its values looks like
| At the end of period \(n\), | the value is |
|---|---|
| 0 | \(V_0\) |
| 1 | \(V_1 = (1+r) V_0\) |
| 2 | \(V_2 = (1+r) V_1 = (1+r)^2 V_0\) |
| ⋮ | ⋮ |
| \(n\) | \(V_n = (1+r)^n V_0\) |
After \(n\) compounding periods with rate of return \(r\) per period, the new value is \((1+r)^n\) times the original value. The overall rate of return, \(R\), over the entire \(n\) periods is different: this is the percent change between the initial and final values: \(R = \frac{V_n - V_0}{V_0} = \frac{(1+r)^n V_0 - V_0}{V_0}\), so
\[R = (1+r)^n - 1\]
and
\[r = (1+R)^{1/n} - 1\]
These formulas are useful for answering questions like, “If home values have risen by about 5% over the past 10 years, how much should the value of my home go up every year on average?” About \((1+0.05)^{1 / 10} - 1\), or about 0.5%.
It's suspicious that 5% over 10 years equals 0.5% over 1 year — is that true in general? Yes: the Taylor series for the second equation above about \(R=0\) is \[0 + \frac{R}{n} + \frac{1}{2} \left(\frac{R}{n}\right)^2 + O(R^3)\] In other words, although the first two equations above are exactly correct, the intuitive and much simpler relationship \[r = \frac{R}{n}\] is nearly correct. For interest rates \(R<10\%\), the simple relationship is never off by more than half a percent, and its accuracy increases rapidly with the number of compounding periods.
These small errors and details can sometimes be significant, however. For some financial instruments, lenders will quote an annual interest rate, although the interest is actually compounded monthly. (Mortgage loans and credit cards are the two examples I see the most.) For example, a bank may offer a mortgage loan with an annual interest rate of 4.7%, although compound interest and payments to the bank will accrue monthly. These kinds of interest rates seem to be called “nominal interest rates,” and you get the actual, real periodic interest rate just by dividing by 12. So a mortgage loan that charges you 4.7% annual (“nominal”) interest has a monthly interest rate of about 0.3917%. The actual interest rate you pay annually is \((1 + 0.047/12)^{12} - 1\), or about 4.8%, slightly higher than the advertised rate. This difference is small, but significant over a long loan. The difference in total payments over the lifetime of the loan between one that charges 4.7% “nominal” annually versus one that charges 4.7% “true” annually works out to 2.12% of the principal, for example about $4,000 on a $200,000 principal.