Question
A bank offers 4.00% on savings accounts. What is the effective annual rate if interest is compounded monthly?
Answers
GPT-4o mini
To calculate the effective annual rate (EAR) when interest is compounded monthly, you can use the formula:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) is the nominal interest rate (as a decimal)
- \( n \) is the number of compounding periods per year
- \( t \) is the number of years
For this problem:
- \( r = 0.04 \) (4.00% as a decimal)
- \( n = 12 \) (monthly compounding)
- \( t = 1 \) (for one year)
Substituting the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.04}{12}\right)^{12 \cdot 1} - 1
\]
Calculating it step-by-step:
1. Calculate \( \frac{0.04}{12} \):
\[
\frac{0.04}{12} = 0.0033333
\]
2. Add 1:
\[
1 + 0.0033333 = 1.0033333
\]
3. Raise to the power of 12:
\[
(1.0033333)^{12} \approx 1.040742
\]
4. Subtract 1:
\[
1.040742 - 1 = 0.040742
\]
5. Convert to a percentage:
\[
0.040742 \times 100 \approx 4.0742\%
\]
Thus, the effective annual rate (EAR) when the nominal interest rate is 4.00% compounded monthly is approximately **4.07%**.
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) is the nominal interest rate (as a decimal)
- \( n \) is the number of compounding periods per year
- \( t \) is the number of years
For this problem:
- \( r = 0.04 \) (4.00% as a decimal)
- \( n = 12 \) (monthly compounding)
- \( t = 1 \) (for one year)
Substituting the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.04}{12}\right)^{12 \cdot 1} - 1
\]
Calculating it step-by-step:
1. Calculate \( \frac{0.04}{12} \):
\[
\frac{0.04}{12} = 0.0033333
\]
2. Add 1:
\[
1 + 0.0033333 = 1.0033333
\]
3. Raise to the power of 12:
\[
(1.0033333)^{12} \approx 1.040742
\]
4. Subtract 1:
\[
1.040742 - 1 = 0.040742
\]
5. Convert to a percentage:
\[
0.040742 \times 100 \approx 4.0742\%
\]
Thus, the effective annual rate (EAR) when the nominal interest rate is 4.00% compounded monthly is approximately **4.07%**.