What is the Effective Annual Rate (EAR)?
The Effective Annual Rate (EAR) is the true annual interest rate that accounts for the effects of compounding. Unlike the nominal interest rate (also called the stated or annual percentage rate), the EAR shows the actual rate of return or cost of borrowing over a year.
Formula
The formula for calculating the Effective Annual Rate is:
$$\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1$$
Where:
- r = nominal interest rate (as a decimal)
- n = number of compounding periods per year
Example Calculation
Problem: A savings account offers a nominal interest rate of 4.5% compounded quarterly. What is the effective annual rate?
Given:
- Nominal rate (r) = 4.5% = 0.045
- Compounding periods (n) = 4 (quarterly)
Solution:
$$\text{EAR} = \left(1 + \frac{0.045}{4}\right)^4 - 1$$
$$\text{EAR} = (1 + 0.01125)^4 - 1$$
$$\text{EAR} = 1.0456 - 1 = 0.0456$$
The effective annual rate is 4.56%, slightly higher than the nominal rate of 4.5% due to quarterly compounding.
Common Compounding Frequencies
| Frequency | Periods per Year (n) |
|---|---|
| Annual | 1 |
| Semi-annual | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Weekly | 52 |
| Daily | 365 |
Why is EAR Important?
- Accurate Comparison: Allows you to compare financial products with different compounding frequencies
- True Cost/Return: Shows the actual cost of borrowing or return on investment
- Informed Decisions: Helps you make better financial decisions based on real rates
The more frequently interest compounds, the higher the effective annual rate compared to the nominal rate.