53.8
8. Your company is considering a new
machine. The machine will generate an extra $13 million per year
starting next year, but will require $2 million in maintenance
every two years (starting 2 years from now and with the last
payment in year 28). The machine has a 30-year life and will cost
$100 million today. The correct discount rate for this investment
is 14% per year. Is your recommendation to buy the machine or
not? Justify your recommendation with the appropriate analysis.
[8 pts]
The cash flows look like this:
0 1 2 … 26 27 28 29 30 -100 13 13 13 13 13 13 13 -2 -2 -2 There are 2 annuities here. The first is a straight-up 30-year annuity of $13M per year. The second is an annuity with 14 payments of $2M, where the payments are made every other year. If the correct annual rate is 14%, then the effective two-year rate is ((1.14)^2)-1=0.2996 or 29.96%. We answer this problem by figuring PVs of the 2 annuities and then subtracting the initial cost of the machine.
\( \Large PV(Annuity1)= \frac{13}{.14} \left [ 1 - \frac{1}{\left(1+.14 \right )^{30} } \right ] = 91.0346\)
\( \Large PV(Annuity2)= \frac{-2}{.2996} \left [ 1 - \frac{1}{\left(1+.2996 \right )^{14} } \right ] = -6.5053\)
NPV= -100 + 91.0346 - 6.5053 = -15.471, so we should NOT buy the machine because doing so would reduce the value of the company by over $15 million.