To determine the present value of an ordinary annuity given the following parameters, follow these steps:
1. Identify the Given Data:
- Amount of the annuity payment: [tex]\( \$15,100 \)[/tex]
- Payment frequency: Quarterly
- Time period: 5 years
- Annual interest rate: [tex]\( 8\% \)[/tex]
2. Convert the Annual Interest Rate to Quarterly:
Since payments are made quarterly, we need to adjust the annual interest rate accordingly.
- Quarterly interest rate = [tex]\( \frac{8\%}{4} = 2\% \)[/tex] per quarter or [tex]\( 0.02 \)[/tex]
3. Calculate the Total Number of Payment Periods:
Since the annuity is paid quarterly over 5 years:
- Number of periods = [tex]\( 4 \times 5 = 20 \)[/tex] quarters
4. Use the Formula for Present Value of an Ordinary Annuity:
The formula to calculate the present value (PV) of an ordinary annuity is:
[tex]\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\][/tex]
Where:
- [tex]\( PMT \)[/tex] is the payment amount per period (\[tex]$15,100)
- \( r \) is the interest rate per period (0.02)
- \( n \) is the total number of periods (20)
5. Plug in the Values into the Formula:
\[
PV = 15{,}100 \times \left( \frac{1 - (1 + 0.02)^{-20}}{0.02} \right)
\]
6. Calculate the Factor Inside the Parentheses:
\[
(1 + 0.02)^{-20} = (1.02)^{-20}
\]
\[
(1.02)^{-20} \approx 0.6730 \quad \text{(using a calculator)}
\]
\[
1 - 0.6730 = 0.327
\]
\[
\frac{0.327}{0.02} = 16.35
\]
7. Calculate the Present Value:
\[
PV = 15{,}100 \times 16.35 \approx 246,906.64
\]
Therefore, the present value of the annuity, or the amount needed now to invest to receive the annuity, is approximately \( \$[/tex]246,906.64 \).
