Interest Compounding and EAR


Quoted Rates

Interest rates are often quoted as annual rates, even though the payment and interest period is actually less than a year. For example, you might see an interest rate quoted as, '8% compounded semiannually'.

  • What this actually means is the interest rate is 4% over 6-months.

  • Importantly, 8% over a year is not the same return as 4% over 6 months.

Because of this the annual 8% rate is referred to as the quoted rate, to make it clear it is not the actual rate earned. Quoted rates are calculated, by convention, as the rate per period multiplied by the number of periods in a year.

  • In the above example, this is `4\% * 2 = 8\%` where the `2` denotes two 6 month periods.

Effective Annual Rates (EAR)

We can calculate the EAR with: `EAR = \left(1 + \frac{Q}{m}\)^m - 1`. This is the actual interest rate over a year.

  • The idea in this equation is we first take the quoted rate and calculate the rate per period `\frac{Q}{m}`. This is the actual interest rate (which in the previous example was 4% over 6-months).

  • We then calculate the future value at the end of the year, given an interest rate of `\frac{Q}{m}` per period, and mm periods in a year. You can think of this as the future value of $1 invested over the year: `FV = \$1\left(1 + \frac{Q}{m}\)^m`.

  • We then subtract the original $1, to get the amount earned over the year (the EAR).

Interactive App

Use the following app to get an idea of how the quoted interest rate and compounding period affect the effective annual rate.

  • Notice the greater the quoted rate, the larger the effect of increasing the compounding period.

Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

Learn more about how to contribute here.