Problem 04-02
What is the APY if the interest rate is 6%, compounded quarterly? (Round your answer to 2 decimal places.)
APY
6.00%



Answer :

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%.