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:
FVN = PV(1+rs/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 FVN = PVersN
where e is the transcendental number e = 2.718… Most financial calculators have a function for ex.
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.
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