Question
A bank offers 6.00% on savings accounts. What is the effective annual rate if interest is compounded quarterly?
Answers
GPT-4o mini
To calculate the effective annual rate (EAR) when the interest is compounded quarterly, you can use the formula:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year.
In this case:
- \( r = 6.00\% = 0.06 \)
- \( n = 4 \) (since the interest is compounded quarterly)
Now, plug the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.06}{4}\right)^4 - 1
\]
Calculating \( \frac{0.06}{4} \):
\[
\frac{0.06}{4} = 0.015
\]
Now substitute back into the formula:
\[
\text{EAR} = \left(1 + 0.015\right)^4 - 1
\]
\[
= (1.015)^4 - 1
\]
Now calculate \( (1.015)^4 \):
\[
(1.015)^4 \approx 1.061364
\]
So,
\[
\text{EAR} \approx 1.061364 - 1
\]
\[
\text{EAR} \approx 0.061364
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} \approx 6.1364\%
\]
Therefore, the effective annual rate (EAR) is approximately **6.14%**.
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year.
In this case:
- \( r = 6.00\% = 0.06 \)
- \( n = 4 \) (since the interest is compounded quarterly)
Now, plug the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.06}{4}\right)^4 - 1
\]
Calculating \( \frac{0.06}{4} \):
\[
\frac{0.06}{4} = 0.015
\]
Now substitute back into the formula:
\[
\text{EAR} = \left(1 + 0.015\right)^4 - 1
\]
\[
= (1.015)^4 - 1
\]
Now calculate \( (1.015)^4 \):
\[
(1.015)^4 \approx 1.061364
\]
So,
\[
\text{EAR} \approx 1.061364 - 1
\]
\[
\text{EAR} \approx 0.061364
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} \approx 6.1364\%
\]
Therefore, the effective annual rate (EAR) is approximately **6.14%**.