MATT BRIGIDA
Associate Professor of Finance (SUNY Polytechnic Institute) & Financial Education Advisor, Milken Institute
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."
Because of this distinction, 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.
We can calculate the EAR with:
This is the actual interest rate over a year.
The idea in this equation is that 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 $m$ periods in a year. You can think of this as the future value of \$1 invested over the year: $FV = PV\left(1 + r\right)^n = \$1\left(1 + \frac{Q}{m}\right)^m$.
We then subtract the original \$1, to get the amount earned over the year (the EAR).
Use the following app to get an idea of how the quoted interest rate and compounding period affect the effective annual rate.