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%.