Answer:
The company could pay at most $ 187,083.73 for the machine
Missing information:
effective monthly interest rate j = 0.4%
Explanation:
We calculate the present value of the employee's salaries using the annuity formula
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C 5,000.00
time 48 (4 years x 12 month per year
rate 0.004 (0.4% = 0.4 / 100 = 0.004)
[tex]5000 \times \frac{1-(1+0.004)^{-48} }{0.004} = PV\\[/tex]
PV $217,971.2447
Now the PV factor of 0.20 of a dollar to represent the 20% of the machine cost being recovered after four years:
[tex]\frac{Maturity}{(1 + rate)^{time} } = PV[/tex]
Maturity $0.20
time 48.00
rate 0.00400
[tex]\frac{0.2}{(1 + 0.004)^{48} } = PV[/tex]
PV 0.1651
now, we construct the equation:
217,971,25 - 0.1651X = X
When X is the maximum amount we could purchase the machine.
217,971,25 = X ( 1 + 0.1651)
217,971,25 / 1.1651 = X
X = $ 187.083,7267
We verify this:
Cost less PV of the salaries plus PV of the residual value = 0
[tex]\frac{Maturity}{(1 + rate)^{time} } = PV[/tex]
Maturity $ 187,083.73 x 20% = $37,416.74
time 48.00
rate 0.00400
[tex]\frac{37416.741}{(1 + 0.004)^{48} } = PV[/tex]
PV of the residual value 30,892.1221
Net present value of the investment:
187.083,70- 217,971.25 + 30,892.1221 ≅ 0
as there is rounding involve there is a minimal difference but we can be satisfy with the answer.
