Effective annual rate

The effective annual rate, often shortened to EAR, converts a quoted rate into what it comes to over a full year once interest starts earning interest. It is the number that puts two offers with different compounding schedules on the same footing, which a nominal rate on its own cannot do.

$$EAR = \left(1 + \frac{r}{n}\right)^{n} - 1$$

Here $r$ is the nominal annual rate and $n$ is the number of compounding periods in a year. When $n$ is 1 the effective rate equals the nominal rate, and for a positive rate every frequency above that is higher. Send $n$ towards infinity and the expression converges on $e^{r} - 1$, the continuous compounding limit, so the gap has a ceiling rather than growing without end. The compound interest calculator runs the same mechanism forward over many years.

EAR and APY are the same arithmetic wearing different labels: APY is the term used for deposit accounts in the United States, while effective annual rate is the general term and the one you meet on the borrowing side. What neither of them includes is fees. In the United States a mortgage APR folds certain charges into the quoted rate but does not compound, while an effective annual rate compounds but ignores every charge outside the interest, so neither figure is the full cost of a loan on its own. The APR against APY calculator handles the conversion.