To calculate the value of an annuity due, we use the formula for the future value of an annuity due. The formula is:
[tex]\[ \text{FV} = P \times \left( \left(1 + r \right)^n - 1 \right) \times \frac{\left(1 + r \right)}{r} \][/tex]
where:
- [tex]\( P \)[/tex] is the annuity payment per period,
- [tex]\( r \)[/tex] is the interest rate per period,
- [tex]\( n \)[/tex] is the number of periods.
Given:
- [tex]\( P = \$2000 \)[/tex]
- [tex]\( r = 0.08 \)[/tex] (8% per year)
- [tex]\( n = 4 \)[/tex] years
Let's plug these values into the formula step-by-step:
1. Calculate [tex]\((1 + r)^n\)[/tex]:
[tex]\[ (1 + 0.08)^4 = 1.08^4 \approx 1.36048896 \][/tex]
2. Subtract 1 from [tex]\((1 + r)^n\)[/tex]:
[tex]\[ 1.36048896 - 1 = 0.36048896 \][/tex]
3. Multiply the result by [tex]\(\frac{(1 + r)}{r}\)[/tex]:
[tex]\[ 0.36048896 \times \frac{1.08}{0.08} = 0.36048896 \times 13.5 \approx 4.86660176 \][/tex]
4. Multiply by the annuity payment [tex]\( P \)[/tex]:
[tex]\[ 2000 \times 4.86660176 \approx 9733.20 \][/tex]
Thus, the value of the investment at the end of Year 4 is:
[tex]\[ \$9733.20 \][/tex]
Among the given options, the correct answer is:
[tex]\[ \$9733.20 \][/tex]
