The effective annual rate (EAR) is a way to measure the actual interest earned or paid on an investment or loan over a one-year period, taking into account the effects of compounding.
To calculate the EAR for a nominal interest rate that is compounded semiannually, you can use the formula:
EAR = (1 + (i/n))^(n*t) – 1
Where:
- i = nominal interest rate (10% or 0.10),
- n = number of compounding periods per year (2 for semiannual),
- t = number of years (1 year).
Plugging in the values:
EAR = (1 + (0.10/2))^(2*1) – 1
This simplifies to:
EAR = (1 + 0.05)^2 – 1
EAR = (1.05)^2 – 1
EAR = 1.1025 – 1
EAR = 0.1025
So, the effective annual rate is 0.1025, or 10.25%.
This means that with a nominal rate of 10% compounded semiannually, the effective rate you would realize by the end of the year is 10.25%. This illustrates how compounding more frequently than once a year can yield higher returns overall.