To determine whether the sign’s claim about the Annual Percentage Yield (APY) is correct, we need to calculate the actual APY based on the given interest rate of 3.6% compounded semiannually and compare it to the claimed 3.9%.
Here’s a step-by-step calculation:
1. Identify the given values:
- Annual interest rate ([tex]\( r \)[/tex]): 3.6% or 0.036 in decimal form.
- Number of compounding periods per year ([tex]\( n \)[/tex]): Semiannually, so [tex]\( n = 2 \)[/tex].
2. APY Formula:
To calculate the APY, we use the formula:
[tex]\[
\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1
\][/tex]
3. Plug in the values:
[tex]\[
\text{APY} = \left(1 + \frac{0.036}{2}\right)^2 - 1
\][/tex]
4. Calculations:
[tex]\[
\text{APY} = \left(1 + 0.018\right)^2 - 1
\][/tex]
[tex]\[
\text{APY} = \left(1.018\right)^2 - 1
\][/tex]
[tex]\[
\text{APY} = 1.036324 - 1
\][/tex]
[tex]\[
\text{APY} \approx 0.036324
\][/tex]
5. Convert APY to percentage:
[tex]\[
\text{APY percentage} \approx 0.036324 \times 100 \approx 3.63\%
\][/tex]
6. Compare with the claimed APY:
The calculated APY is approximately 3.63%, while the sign claims an APY of 3.9%. There is a discrepancy between these values.
So:
Select the option:
OB. Yes. The APY is incorrect. The correct APY is 3.63% (rounded to the nearest hundredth).
Thus, the sign is indeed incorrect because the actual APY calculated for 3.6% interest compounded semiannually is 3.63%, not the 3.9% claimed.
