To determine the present value (PV) of an ordinary annuity with the given parameters, we follow these steps:
1. Identify the parameters:
- [tex]\( P = \$1900 \)[/tex] (annual payment)
- [tex]\( r = 0.06 \)[/tex] (annual interest rate)
- [tex]\( n = 16 \)[/tex] (number of years or periods)
2. Understand the formula:
The formula to find the present value of an ordinary annuity is:
[tex]\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\][/tex]
where:
- [tex]\( PV \)[/tex] is the present value of the annuity
- [tex]\( P \)[/tex] is the periodic payment
- [tex]\( r \)[/tex] is the interest rate per period
- [tex]\( n \)[/tex] is the number of periods
3. Substitute the known values into the formula:
[tex]\[
PV = 1900 \times \left( \frac{1 - (1 + 0.06)^{-16}}{0.06} \right)
\][/tex]
4. Perform the calculations:
- First calculate [tex]\( 1 + r \)[/tex]:
[tex]\[
1 + 0.06 = 1.06
\][/tex]
- Next calculate [tex]\( (1.06)^{-16} \)[/tex]:
[tex]\[
(1.06)^{-16} \approx 0.393646
\][/tex]
- Subtract this value from 1:
[tex]\[
1 - 0.393646 \approx 0.606354
\][/tex]
- Divide by [tex]\( r \)[/tex]:
[tex]\[
\frac{0.606354}{0.06} \approx 10.1059
\][/tex]
- Finally, multiply by [tex]\( P \)[/tex]:
[tex]\[
1900 \times 10.1059 \approx 19201.21
\][/tex]
5. Round to the nearest cent:
[tex]\[
19201.21 \approx 19201.20
\][/tex]
The present value of the ordinary annuity is $19,201.20.
