Sure, let's break down the solution to find the present value of the given ordinary annuity step by step. Here's the scenario:
- The payment amount is [tex]$9347.
- Payments are made semiannually.
- The duration is 8 years.
- The annual interest rate is 10%, compounded semiannually.
We'll use the present value formula for an ordinary annuity, which is:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \(PV\) is the present value we need to find.
- \(PMT\) is the payment amount.
- \(r\) is the interest rate per period.
- \(n\) is the total number of periods.
Step-by-Step Solution:
1. Determine the interest rate per period:
Since the interest is compounded semiannually:
\[
r = \frac{10\%}{2} = 5\% = 0.05
\]
2. Calculate the total number of periods:
Given that payments are made semiannually for 8 years:
\[
n = 2 \times 8 = 16
\]
3. Plug the values into the present value formula:
\[
PV = 9347 \times \left( \frac{1 - (1 + 0.05)^{-16}}{0.05} \right)
\]
4. Solve the expression inside the parenthesis:
\[
(1 + 0.05)^{-16} = (1.05)^{-16}
\]
Now calculate \(1 - (1.05)^{-16}\):
\[
1 - (1.05)^{-16} \approx 1 - 0.4581 \approx 0.5419
\]
5. Divide by the interest rate per period:
\[
\frac{0.5419}{0.05} \approx 10.838
\]
6. Multiply by the payment amount:
\[
PV = 9347 \times 10.838 \approx 101300.63
\]
So, the present value of the annuity is approximately \$[/tex]101,300.63.
