To determine the table factor for an annuity due of [tex]$2,000 per year at an 8% interest rate for 4 years, follow these steps:
1. Understand the Annuity Due: An annuity due involves making payments at the beginning of each period. Therefore, compared to an ordinary annuity (where payments are made at the end of each period), the annuity due will yield slightly higher values because each payment is invested for an additional period.
2. Use the Annuity Due Formula:
- The formula for the present value of an annuity due is:
\[
PV = PMT \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r)
\]
- Here, \( PV \) stands for present value, \( PMT \) is the payment amount per period, \( r \) is the interest rate per period, and \( n \) is the number of periods.
3. Plug in the Values:
- The payment amount \( PMT \) is $[/tex]2,000.
- The interest rate [tex]\( r \)[/tex] is 8% or 0.08.
- The number of periods [tex]\( n \)[/tex] is 4 years.
4. Calculate the Table Factor:
- First, find the factor without considering the annuity due adjustment:
[tex]\[
\text{Factor without annuity due adjustment} = \frac{1 - (1 + 0.08)^{-4}}{0.08}
\][/tex]
- Next, the factor needs to be adjusted for annuity due by multiplying with [tex]\( (1 + r) \)[/tex]:
[tex]\[
\text{Adjusted factor for annuity due} = \left[ \frac{1 - (1 + 0.08)^{-4}}{0.08} \right] \times (1 + 0.08)
\][/tex]
5. Determine the Calculated Table Factor:
- After plugging in the values and performing the calculations, the table factor approximately equals [tex]\( 3.577 \)[/tex].
6. Compare to Given Options:
- The provided options are:
- 4.5061
- 5.8666
- 4.1216
- 8.5829
- The value closest to our calculated table factor [tex]\( 3.577 \)[/tex] is [tex]\( 4.1216 \)[/tex].
Therefore, the closest matching table factor for the given annuity due is [tex]\( \boxed{4.1216} \)[/tex].
