Question

8. A university student painter is considering the purchase of a new air compressor and paint gun to replace an old paint sprayer. Suppose the CCA rates are 20% and ent, respectively. The two new items cost $8,500 and have a useful life of four years, at which time they can be sold for $1,100. The old paint sprayer can be sold now for $275 and be scraped for $150 in four years. The student is very confident that operating revenue will increase 25% for the old equipment and new equipm 2 annually by $7,600. If tax rate is 24% and the required rate of return is 19%, should the purchase be made? What is the maximum price of the new equipment that makes the student to replace the old equipment?

Answer 1

Let's calculate this in four different parts and then pool together the result to get the final output.

Part -1

Initial outlay at t = 0

A_N = Cost of new equipment = 8,500

A_O = Sale Price of old equipment = 275

Hence, Net Initial Investment = NII = net cash outflow at t=0 will be = A_N - A_O = 8,500 - 275 = 8,225

Part -2

Salvage Value at the end of year 4,

S_N = Salvage Value of new equipment = 1,100

S_O = Salvage value of old equipment = 150

Hence, Net salvage value, NSV = net cash inflow at t=4 will be = S_N - S_O = 1,100 - 150 = 950

PV(NSV) = PV of Net Salvage Value = NSV / (1 + r)^N where r = discount rate = 19% and N = life of the project = 4 years

Hence, PV(NSV) = 950 / (1 + 19%)⁴ = 473.74

Part - 3

Post tax incremental operating profit = (1 - T) x Incremental operating profit

Incremental operating profit = incremental revenue in this case = 7,600

Tax rate = t = 24%

Hence, post tax incremental operating profit = (1-24%) x 7,600 = 5,776 per year for next four years

This is an annuity for N = 4 years as this incremental profit occurs every year for next four years

Hence, PV (Incremental post tax income) = Incremental post tax income x Sum of PV factors @ 19% over 4 years

$=5,776\times \frac{(1 - (1+r)^{-N})}{r} = 5,776\times \frac{(1 - (1+0.19)^{-4})}{0.19} = 15,240.47$

Part - 4

PV(NCCATS) = Present value of Net CCA tax shield

PV (CCATS) = present value of CCA tax shield for an asset is given by the formula:

$PV(CCTS)=\frac{T\times d\times A\times (1+0.5r)}{(r+d)(1+r)}-\frac{T\times d\times S}{(r+d)(1+r)^{N}}$

where T = tax rate, d = CCA depreciation rate, A = original asset cost, S = Salvage value, N = life, r= discount rate

For the new equipment,

$PV(CCTS)=\frac{T\times d\times A\times (1+0.5r)}{(r+d)(1+r)}-\frac{T\times d\times S}{(r+d)(1+r)^{N}}$

$=\frac{0.24\times 0.25\times 8,500\times (1+0.5\times 0.19)}{(0.19+0.25)(1+0.19)}-\frac{0.24\times 0.25\times 1,100}{(0.19+0.25)(1+0.19)^{4}} = 991.76$

For the old equipment,

$PV(CCTS)=\frac{T\times d\times A\times (1+0.5r)}{(r+d)(1+r)}-\frac{T\times d\times S}{(r+d)(1+r)^{N}}$

$=\frac{0.24\times 0.20\times 275\times (1+0.5\times 0.19)}{(0.19+0.20)(1+0.19)}-\frac{0.24\times 0.20\times 150}{(0.19+0.20)(1+0.19)^{4}} = 21.94$

PV (NCCTS) = PV(CCTS) of new equipment - PV (CCTS) of old equipment = 991.76 - 21.94 = 969.82

NPV of the replacement decision = - NII + PV(NSV) + PV(Post tax incremental profit) + PV(NCCTS) = - 8,225 + 473.74 + 15,240.47 + 969.82 = $ 8,459.03

Since it's positive NPV, the purchase of new equipment should be undertaken.

Maximum price that one should pay = Cost of new equipment that makes NPV equal to zero. Let's say the incremental cost is C. Then S should be such that

C - PV (CCATS) on C = NPV

Hence, we need to solve the following equation:

$C - \frac{T\times d\times C\times (1+0.5r)}{(r+d)(1+r)} = NPV$

$C - \frac{0.24\times 0.25\times C\times (1+0.5\times 0.19)}{(0.19+0.25)(1+0.19)} = 8,459.03$

Hence C = 9,672.74

Hence the maximum price = Original price + incremental price = A_N + C = 8,500 + 9,672.74 =  $18,172.74