To find the present value of an ordinary annuity, we can use the present value formula for an ordinary annuity. Here is a detailed, step-by-step solution:
1. Identify the parameters:
- Amount of annuity expected (Payment, [tex]\( PMT \)[/tex]): \[tex]$760
- Time (number of years, \( n \)): 3 years
- Interest rate (\(r\)): 7% per year
2. Understand the Present Value Factor for an Ordinary Annuity:
The present value factor for an ordinary annuity can be calculated using the following formula:
\[
PV \text{ Factor} = \frac{1 - (1 + r)^{-n}}{r}
\]
3. Substitute the values into the Present Value Factor formula:
- Interest rate (\( r \)) = 0.07
- Number of years (\( n \)) = 3
\[
PV \text{ Factor} = \frac{1 - (1 + 0.07)^{-3}}{0.07}
\]
4. Calculate the Present Value Factor:
\[
PV \text{ Factor} = \frac{1 - (1.07)^{-3}}{0.07}
\]
Calculating this step-by-step:
- Calculate \( (1.07)^{-3} \):
\[
(1.07)^{-3} \approx 0.816297876
\]
- Subtract this value from 1:
\[
1 - 0.816297876 \approx 0.183702124
\]
- Divide by the interest rate:
\[
\frac{0.183702124}{0.07} \approx 2.624316044
\]
Thus, the present value factor is approximately \( 2.624316044 \).
5. Calculate the present value of the annuity:
The present value (\( PV \)) is calculated by multiplying the payment by the present value factor:
\[
PV = PMT \times PV \text{ Factor}
\]
Substituting the values:
\[
PV = 760 \times 2.624316044
\]
Performing the multiplication:
\[
PV \approx 1994.4782048
\]
6. Round the result to the nearest cent:
\[
PV \approx 1994.48
\]
So, the present value (amount needed now to invest to receive the annuity) is \$[/tex]1994.48.
