To calculate the effective annual rate (EAR) when the nominal interest rate is compounded monthly, you can use the following formula:
\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1 \]
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.
In this case:
- \( r = 0.05 \) (5.00% as a decimal),
- \( n = 12 \) (monthly compounding),
- \( t = 1 \) (we're calculating for 1 year).
Plugging in these values:
\[ \text{EAR} = \left(1 + \frac{0.05}{12}\right)^{12 \times 1} - 1 \]
First, calculate \( \frac{0.05}{12} \):
\[ \frac{0.05}{12} = 0.00416667 \]
Next, calculate \( 1 + 0.00416667 \):
\[ 1 + 0.00416667 = 1.00416667 \]
Now we raise this to the power of 12:
\[ (1.00416667)^{12} \approx 1.0511619 \]
Finally, subtract 1:
\[ \text{EAR} \approx 1.0511619 - 1 \approx 0.0511619 \]
Convert this back to a percentage:
\[ \text{EAR} \approx 0.0511619 \times 100 \approx 5.11619% \]
Thus, the effective annual rate is approximately 5.12% when rounded to two decimal places.