The Effective Annual Interest Rate

We described an interest rate (r) is the rate of return that equates the value of different cash flows on different dates.

If you could pay $975 today to receive $1,000 in 1 year, you can consider the value of waiting 1 year to be $25 ($1,000 – $975.) It can be expressed as an interest rate, r = 25/975 = 0.02564 or 2.56%.

In many cases, interest is paid more frequently than annually. In such cases the future value formula can be expressed as:
FV_N = PV(1+r_s/m)^^mN
where m is the number of compounding periods and N is the number of years.

Interest can also be compounded continuously, in which case FV_N = PV_e^r_sN where e is the transcendental number e = 2.718… Most financial calculators have a function for e^x.

Consider a stated interest rate of 6% annually. For different compounding periods, the effective annual interest rate the investor would earn is summarized below.

m	mN	rs/m	Future value per $
Annual	1	6%/1 = 6%	1(1.06) = 1.06
Semiannual	2	6%/2 = 3%	1(1.03)2 = 1.0609
Quarterly	4	6%/4 = 1.5%	1(1.015)4 = 1.0614
Monthly	12	6%/12 = 0.5%	1(1.005)12 = 1.0616
Daily	365	6%/365 = 0.0164%	1(1.0000164)365 = 1.0618
Continuous			1e0.06 = 1.06184

The more frequently a given rate is compounded, the higher the ending value for the investor.