Question

Truck-Or-Treat specializes in leasing trucks to delivery companies. It is considering adding 25 more trucks to its available stock. Doing so will not change the risk of the company's business. The trucks depreciate over five years under the straight-line depreciation method, all the way to zero. Truck-Or-Treat believes that these newly added trucks would be able to bring the company $220,000 in annual earnings before taxes and depreciation (i.e., sales revenue minus costs of goods sold) for five years. The company is unlevered. It is in 21 percent tax rate bracket. The required annual rate of return on Truck-Or-Treat's unlevered equity is 15 percent. The risk-free rate, e.g., the Treasury bill rate, is 6 percent per year.

Required:
Calculate the maximum price that Truck-or-Treat should be willing to pay for the purchase of the new trucks if it remains an unlevered company. (In other words, what should be the "initial investment" of this unlevered truck project such that the project's NPV equals $0?

Answer 1

The maximum price that Truck-or-Treat should be willing to pay for the purchase of the new trucks if it remains an unlevered company is $510,702.49.

Step-by-step explanation:

Let:

x = Maximum price for the new truck = initial investment = ?

AEBTD = Annual earnings before taxes and depreciation = $220,000

T = Tax rate = 21%, or 0.21

n = Number of years = 5

Since the it is assumed that Truck-or-Treat remains an unlevered company, this implies the required annual rate of return on Truck-Or-Treat's unlevered equity of 15 percent is the relevant rate of return to use.

Therefore, we have:

r = required annual rate of return = 15%, or 0.15

D = Annual depreciation = Maximum price for the new truck / Number of useful years = x / 5 = 0.2x

P = Annual cash flow = ((AEDTD - D) * (1 - T)) + D = ((220000 - 0.2x) * (1 - 0.21)) + 0.2x = ((220000 - 0.2x) * 0.79) + 0.2x = 173,800 - 0.158x + 0.2x = 173,800 - 0.042x

Using the formula for calculating the present value (PV) of an ordinary annuity, we have:

PVP = Present value of annual cash flow = P * ((1 - (1/(1 + r))^n) / r) = (173,800 - 0.042x) * ((1 - (1/(1 + 0.15))^5) / 0.15) = (173,800 - 0.042x) * 3.3521550980114 = 582,604.56 - 0.140790514116479x

For the NPV of this unlevered truck project to be equal to $0, we must have:

x = PVP

That is:

x = 582,604.56 - 0.140790514116479x

Solving for x, we have:

x + 0.140790514116479x = 582,604.56

x(1 + 0.140790514116479) = 582,604.56

x1.140790514116479 = 582,604.56

x = 582,604.56 / 1.140790514116479 = $510,702.49

Therefore, the maximum price that Truck-or-Treat should be willing to pay for the purchase of the new trucks if it remains an unlevered company is $510,702.49.