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:

$$\text{EAR}={(1+\frac{r}{n})}^{nt}-1$$

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:

$$\text{EAR}={(1+\frac{0.04}{12})}^{12\cdot 1}-1$$

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%.