Question

A bank offers 4.00% 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 following formula:

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

Where:
- \( r \) is the nominal interest rate (as a decimal)
- \( n \) is the number of compounding periods per year
- \( t \) is the number of years (for EAR, we typically set \( t = 1 \))

For your case:
- \( r = 0.04 \) (4.00% expressed as a decimal)
- \( n = 365 \) (compounding daily)
- \( t = 1 \)

Plugging in the values:

\[
EAR = \left(1 + \frac{0.04}{365}\right)^{365 \cdot 1} - 1
\]

Calculating \( \frac{0.04}{365} \):

\[
\frac{0.04}{365} \approx 0.00010958904
\]

Now, substituting this back into the formula:

\[
EAR = \left(1 + 0.00010958904\right)^{365} - 1
\]

Calculating \( \left(1 + 0.00010958904\right)^{365} \):

\[
EAR \approx (1.00010958904)^{365} - 1
\]

Now we can approximate:

\[
\approx 1.040808 - 1
\]

Therefore:

\[
EAR \approx 0.040808
\]

Converting this back to a percentage gives us:

\[
EAR \approx 4.08\%
\]

So, the Effective Annual Rate (EAR) for a nominal rate of 4.00% compounded daily is approximately **4.08%**.