Question

A bank offers 9.00% on savings accounts. What is the effective annual rate if interest is compounded daily?

Answer 1

To calculate the effective annual rate (EAR) when interest is compounded daily, we can use the formula for EAR:

\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^{n \cdot t} - 1
\]

where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year (for daily compounding, \( n = 365 \)),
- \( t \) is the number of years (for one year, \( t = 1 \)).

Given:
- \( r = 0.09 \) (9.00% expressed as a decimal)
- \( n = 365 \)
- \( t = 1 \)

Substituting these values into the formula, we get:

\[
\text{EAR} = \left(1 + \frac{0.09}{365}\right)^{365 \cdot 1} - 1
\]

Calculating it step by step:

1. Calculate \( \frac{0.09}{365} \):
\[
\frac{0.09}{365} \approx 0.00024657534
\]

2. Calculate \( 1 + \frac{0.09}{365} \):
\[
1 + 0.00024657534 \approx 1.00024657534
\]

3. Raise it to the power of 365:
\[
\left(1.00024657534\right)^{365} \approx 1.094171
\]

4. Subtract 1:
\[
1.094171 - 1 \approx 0.094171
\]

5. Convert to percentage:
\[
0.094171 \times 100 \approx 9.42\%
\]

So, the effective annual rate (EAR) when interest is compounded daily is approximately **9.42%**.