Sure, let's calculate the effective interest rate for an annual nominal interest rate of 4% compounded semi-annually.
To find the effective interest rate (EIR), we can use the formula for compound interest:
[tex]\[ \text{EIR} = \left(1 + \frac{r}{n}\right)^n - 1 \][/tex]
Where:
- [tex]\( r \)[/tex] is the annual nominal interest rate.
- [tex]\( n \)[/tex] is the number of compounding periods per year.
Given:
- The annual nominal interest rate [tex]\( r \)[/tex] is 4%, or 0.04 in decimal form.
- The interest is compounded semi-annually, so there are [tex]\( n = 2 \)[/tex] compounding periods per year.
Now we will plug these values into the formula:
1. Substitute [tex]\( r = 0.04 \)[/tex] and [tex]\( n = 2 \)[/tex]:
[tex]\[ \text{EIR} = \left(1 + \frac{0.04}{2}\right)^2 - 1 \][/tex]
2. Simplify the fraction inside the parentheses:
[tex]\[ \text{EIR} = \left(1 + 0.02\right)^2 - 1 \][/tex]
3. Add the values inside the parentheses:
[tex]\[ \text{EIR} = \left(1.02\right)^2 - 1 \][/tex]
4. Compute the exponentiation:
[tex]\[ \text{EIR} = 1.02^2 - 1 = 1.0404 - 1 \][/tex]
5. Subtract 1 from the result:
[tex]\[ \text{EIR} = 0.0404 \][/tex]
Therefore, the effective interest rate that corresponds to a 4% nominal rate compounded semi-annually is approximately 0.0404, or 4.04%.
