A bank offers 4.00% on savings accounts. What is the effective annual rate if interest is compounded monthly?

Answer 1

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%.