The Effective Annual Rate

The same quoted rate earns more when it compounds more often. The effective rate is (1 + rate ÷ periods)^periods − 1.

Explanation

A rate quoted as "5% a year" does not tell you how much you actually gain. That depends on how often it compounds.

\text{effective rate} = \left(1 + \dfrac{r}{m}\right)^m - 1

Here $r$ is the nominal annual rate and $m$ the number of compounding periods per year.

Example

Take 5% a year, compounded monthly.

\left(1 + \dfrac{0.05}{12}\right)^{12} - 1 = 0.051162 = 5.1162\ \%

The label says 5%, but the money grows by 5.1162%. The extra 0.1162% is interest earning interest.

Compounding more and more often

Hold the nominal rate at 5% and vary only the frequency.

Yearly — 5.0000%
Half-yearly — 5.0625%
Monthly — 5.1162%
Daily — 5.1267%
Continuously (infinitely often) — 5.1271%

It keeps rising, but it does not run away. It converges, and the limit is the continuous case, which takes the beautiful form $e^r - 1$ .

e^{0.05} - 1 = 0.051271

Compound every second, every nanosecond, and you still cannot beat 5.1271%. Euler's number appears here of its own accord.

Credit cards

A card advertising "15% APR" is charging 1.25% a month, compounded twelve times.

\left(1 + \dfrac{0.15}{12}\right)^{12} - 1 = 16.08\ \%

The headline says 15%; the debt actually grows by 16.08% over a year.

Watch out

The advantage flips depending on which side of the loan you stand. Frequent compounding is good when you are lending and bad when you are borrowing.

When comparing financial products, compare effective rates, never the headline ones. It is the only way to put them on the same footing.