Let's go through the steps to figure out the value of the annuity:
1. Understand the parameters provided:
- Amount of payment ([tex]\(PMT\)[/tex]) = [tex]$11,800
- Payment frequency: Semiannually
- Number of years = 8
- Annual interest rate = 7%
2. Calculate the number of payments:
Since payments are made semiannually, there are 2 payments per year.
\[
\text{Total number of payments} = \text{Payments per year} \times \text{Number of years} = 2 \times 8 = 16
\]
3. Calculate the periodic (semiannual) interest rate:
The annual interest rate is 7%, so the semiannual interest rate is half of that:
\[
\text{Periodic (semiannual) interest rate} = \frac{7\%}{2} = \frac{7}{100 \times 2} = 0.035
\]
4. Apply the annuity formula to find the future value of the annuity:
The future value (\(FV\)) of an ordinary annuity can be calculated using the annuity formula:
\[
FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \(PMT\) = 11,800 (payment amount)
- \(r\) = 0.035 (periodic interest rate)
- \(n\) = 16 (total number of payments)
5. Plug in the values:
\[
FV = 11,800 \times \left( \frac{(1 + 0.035)^{16} - 1}{0.035} \right)
\]
6. Calculate the future value:
By computing it step-by-step:
- First, calculate \( (1 + 0.035)^{16} \)
- Then subtract 1 from the result
- Next, divide by the periodic interest rate (0.035)
- Finally, multiply by the payment amount (11,800)
After performing these calculations, the future value of the annuity comes out to be:
\[
FV = 247,458.15
\]
Therefore, the completed table will have the value of the annuity as:
\[
\begin{array}{|r|c|c|c|c|c|}
\hline
\text{Amount of payment} & \text{Payment payable} & \text{Years} & \text{Interest rate} & \text{Value of annuity} \\
\hline
\$[/tex]11,800 & \text{Semiannually} & 8 & 7\% & \$247,458.15 \\
\hline
\end{array}
\]
