Question
A bank offers 8.00% on savings accounts. What is the effective annual rate if interest is compounded semi-annually?
Answers
GPT-4o mini
To calculate the effective annual rate (EAR) when interest is compounded semi-annually, you can use the following formula:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) = nominal annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
Given:
- \( r = 8.00\% = 0.08 \)
- \( n = 2 \) (since interest is compounded semi-annually)
Now, substitute these values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.08}{2}\right)^2 - 1
\]
\[
= \left(1 + 0.04\right)^2 - 1
\]
\[
= (1.04)^2 - 1
\]
\[
= 1.0816 - 1
\]
\[
= 0.0816
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} = 0.0816 \times 100 = 8.16\%
\]
Thus, the effective annual rate when interest is compounded semi-annually is **8.16%**.
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) = nominal annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
Given:
- \( r = 8.00\% = 0.08 \)
- \( n = 2 \) (since interest is compounded semi-annually)
Now, substitute these values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.08}{2}\right)^2 - 1
\]
\[
= \left(1 + 0.04\right)^2 - 1
\]
\[
= (1.04)^2 - 1
\]
\[
= 1.0816 - 1
\]
\[
= 0.0816
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} = 0.0816 \times 100 = 8.16\%
\]
Thus, the effective annual rate when interest is compounded semi-annually is **8.16%**.