Question

A bank offers 5.00% on savings accounts. What is the effective annual rate if interest is compounded semi-annually?

Answer 1

To find the effective annual rate (EAR) when interest is compounded semi-annually, you can use the following formula:

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

Where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year,
- \( t \) is the number of years.

In this case:
- The nominal interest rate \( r = 0.0500 \) (5.00%),
- The number of compounding periods per year \( n = 2 \) (since the interest is compounded semi-annually),
- We will calculate for \( t = 1 \) year.

Plugging in the values:

\[
\text{EAR} = \left(1 + \frac{0.0500}{2}\right)^{2 \cdot 1} - 1
\]

Now we calculate \( \frac{0.0500}{2} \):

\[
\frac{0.0500}{2} = 0.0250
\]

Now we compute:

\[
\text{EAR} = \left(1 + 0.0250\right)^{2} - 1 = \left(1.0250\right)^{2} - 1
\]

Next, we calculate \( (1.0250)^{2} \):

\[
(1.0250)^{2} = 1.050625
\]

So,

\[
\text{EAR} = 1.050625 - 1 = 0.050625
\]

Finally, converting back to percentage:

\[
\text{EAR} = 0.050625 \times 100 = 5.0625\%
\]

Thus, the effective annual rate when interest is compounded semi-annually at a nominal rate of 5.00% is **5.0625%**.

Answer 2

thank you