Question: What is the Effective Annual Rate (EAR) if the Annual Percentage Rate (APR) is $5.66$ percent compounded quarterly? Enter the answer to 4 decimals. It should look something like $0.1234$, NOT $12.34$
1. **State the problem:** We need to find the Effective Annual Rate (EAR) given an Annual Percentage Rate (APR) of $5.66\%$ compounded quarterly.
2. **Formula:** The formula to calculate EAR when APR is compounded $n$ times per year is:
$$EAR = \left(1 + \frac{APR}{n}\right)^n - 1$$
where $APR$ is in decimal form and $n$ is the number of compounding periods per year.
3. **Convert APR to decimal:**
$$APR = \frac{5.66}{100} = 0.0566$$
4. **Identify compounding periods:** Quarterly compounding means $n = 4$.
5. **Plug values into the formula:**
$$EAR = \left(1 + \frac{0.0566}{4}\right)^4 - 1$$
6. **Calculate inside the parentheses:**
$$1 + \frac{0.0566}{4} = 1 + 0.01415 = 1.01415$$
7. **Raise to the power of 4:**
$$EAR = 1.01415^4 - 1$$
8. **Calculate $1.01415^4$:**
$$1.01415^4 \approx 1.0579$$
9. **Subtract 1:**
$$EAR = 1.0579 - 1 = 0.0579$$
10. **Final answer:** The Effective Annual Rate (EAR) is approximately $0.0579$ or $5.79\%$.
**Answer:** $0.0579$