To calculate the value of the annuity due, we will use the formula for the present value of an annuity due. The formula is:
[tex]\[ \text{PV} = \text{Pmt} \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r) \][/tex]
Where:
- [tex]\(\text{PV}\)[/tex] is the present value of the annuity due
- [tex]\(\text{Pmt}\)[/tex] is the amount of the payment
- [tex]\(r\)[/tex] is the interest rate per period
- [tex]\(n\)[/tex] is the number of periods
Let's break down the calculation step-by-step:
1. Identify the known variables:
- Amount of payment (Pmt): \[tex]$4,000
- Number of years (n): 3
- Interest rate (r): 7% or 0.07
2. Calculate the present value factor for annuity due:
\[ \text{Present Value Factor} = \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r) \]
3. Plug in the values and solve for the present value factor:
\[ \text{Present Value Factor} = \left[ \frac{1 - (1 + 0.07)^{-3}}{0.07} \right] \times (1 + 0.07) \]
- First, calculate \(1 + r\):
\[ 1 + 0.07 = 1.07 \]
- Then raise to the power of \(-n\):
\[ 1.07^{-3} = \frac{1}{1.07^3} \approx 0.816297876 \]
- Subtract this value from 1:
\[ 1 - 0.816297876 \approx 0.183702124 \]
- Divide by the interest rate \(r\):
\[ \frac{0.183702124}{0.07} \approx 2.624317486 \]
- Finally, multiply by \((1 + r)\):
\[ 2.624317486 \times 1.07 \approx 2.808018168 \]
4. Calculate the value of the annuity due:
\[ \text{Annuity Value} = \text{Pmt} \times \text{Present Value Factor} \]
\[ \text{Annuity Value} = 4,000 \times 2.808018168 \approx 11,232.07267 \]
5. Round to the nearest cent:
\[ \text{Annuity Value} \approx \$[/tex]11,232.07 \]
So, the value of the annuity due is \$11,232.07.
