Question

Assume a bank offers an effective annual rate of 5.60%. If compounding is quarterly what is the APR?

Answer 1

To convert the effective annual rate (EAR) to the annual percentage rate (APR) when compounding is quarterly, you can use the following formula:

\[
\text{EAR} = \left(1 + \frac{\text{APR}}{n}\right)^n - 1
\]

where:
- \( n \) is the number of compounding periods per year (in this case, \( n = 4 \) for quarterly compounding).
- \( \text{APR} \) is what we want to find.

We are given that the EAR is 5.60%, which we will convert to decimal form: \( \text{EAR} = 0.056 \).

Now we can rearrange the formula to solve for APR:

\[
1 + \frac{\text{APR}}{4} = (1 + \text{EAR})^{1/4}
\]

First, rearranging gives us:

\[
\frac{\text{APR}}{4} = (1 + 0.056)^{1/4} - 1
\]

Calculating \( (1 + 0.056)^{1/4} \):

\[
(1.056)^{1/4} \approx 1.013669
\]

Now substitute back into the equation:

\[
\frac{\text{APR}}{4} = 1.013669 - 1 \approx 0.013669
\]

Finally, multiplying by 4 to get APR:

\[
\text{APR} \approx 0.013669 \cdot 4 \approx 0.054676
\]

Now, convert this to a percentage:

\[
\text{APR} \approx 5.47\%
\]

Thus, the annual percentage rate (APR) is approximately **5.47%**.

Answer 2

can you give theAPR in 4 decimal?

Answer 3

Sure! From the calculation we previously did, we found:

\[
\text{APR} \approx 0.054676
\]

To express this as a percentage rounded to four decimal places, we multiply by 100:

\[
\text{APR} \approx 0.054676 \times 100 = 5.4676\%
\]

Therefore, the APR, expressed in four decimal places, is approximately **5.4676%**.