To determine the Annual Percentage Yield (APY) for an interest rate of 6% compounded quarterly, you can use the formula for APY. The formula is:
[tex]\[ \text{APY} = \left(1 + \frac{r}{n}\right)^{n} - 1 \][/tex]
where:
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( n \)[/tex] is the number of compounding periods per year.
Given that the annual interest rate [tex]\( r \)[/tex] is 6% (or 0.06 when expressed as a decimal) and the interest is compounded quarterly (which means [tex]\( n = 4 \)[/tex] as there are 4 quarters in a year), you can follow these steps:
1. Convert the annual interest rate and identify the number of compounding periods:
[tex]\[
r = 0.06 \quad \text{and} \quad n = 4
\][/tex]
2. Substitute these values into the APY formula:
[tex]\[
\text{APY} = \left(1 + \frac{0.06}{4}\right)^{4} - 1
\][/tex]
3. Calculate the term inside the parentheses:
[tex]\[
1 + \frac{0.06}{4} = 1 + 0.015 = 1.015
\][/tex]
4. Raise this term to the power of the number of compounding periods:
[tex]\[
(1.015)^4
\][/tex]
5. Subtract 1 to find the APY:
[tex]\[
(1.015)^4 - 1
\][/tex]
6. The result gives the APY as a decimal. To convert it to a percentage, multiply by 100 and round to two decimal places:
7. After calculating and rounding, we find:
[tex]\[
\text{APY} \approx 6.14\%
\][/tex]
Therefore, the APY for an interest rate of 6% compounded quarterly is approximately 6.14%.
