The Effective Annual Rate (EAR) gives you a clearer picture of the actual interest earned or paid on an investment or loan over a year, especially when compounding occurs more than once a year. In this example, we want to find the EAR for an interest rate of 10 percent compounded semiannually.
To calculate the EAR, we can use the following formula:
EAR = (1 + (i / n))nt – 1
Where:
- i = nominal interest rate (decimal)
- n = number of compounding periods per year
- t = number of years
For our calculation:
- i = 0.10 (10 percent as a decimal)
- n = 2 (since it’s compounded semiannually)
- t = 1 (we’re looking for the rate over one year)
Now we can plug these values into the formula:
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
Thus, the Effective Annual Rate is:
0.1025 or 10.25%
In conclusion, when you have a nominal interest rate of 10 percent compounded semiannually, the effective annual rate comes out to be approximately 10.25%. This means that, with compounding taken into account, your investment grows by about 10.25 percent over one year, rather than just the nominal 10 percent.