Question

Compute the payback period for each of these two separate investments (round the payback period to two decimals):

a. A new operating system for an existing machine is expected to cost \(520,000 and have a useful life of six years. The system yields an incremental after-tax income of \)150,000 each year after deducting its straight-line depreciation. The predicted salvage value of the system is \(10,000.

b. A machine costs \)380,000, has a \(20,000 salvage value, is expected to last eight years, and will generate an after-tax income of \)60,000 per year after straight-line depreciation.

Short answer

The payback period for the new operating system is 2.21 years and the payback period for the machine is 3.62 years.

Step-by-step solution

Step-by-Step SolutionStep 1: Computation of payback period of new operating system

$$\begin{gathered} \mathbf{Depreciation}\mathbf{}\mathbf{expense}\mathbf{=}\frac{\mathrm{Cost}-\mathrm{Salvage}\mathrm{value}}{\mathrm{Useful}\mathrm{life}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{520,000-10,000}{6} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{85}\mathbf{,}\mathbf{000} \end{gathered}$$

$$\begin{gathered} \mathbf{Annual}\mathbf{}\mathbf{cash}\mathbf{}\mathbf{flow}\mathbf{=}\mathbf{After}\mathbf{}\mathbf{tax}\mathbf{}\mathbf{income}\mathbf{+}\mathbf{Depreciation}\mathbf{}\mathbf{expense} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{150}\mathbf{,}\mathbf{000}\mathbf{+}\mathbf{85}\mathbf{,}\mathbf{000} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{235}\mathbf{,}\mathbf{000} \end{gathered}$$

$$\begin{gathered} \mathbf{Pay}\mathbf{}\mathbf{back}\mathbf{}\mathbf{Period}\mathbf{=}\frac{\mathrm{Initial}\mathrm{Investment}}{\mathrm{Annual}\mathrm{cash}\mathrm{flow}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{520,000}{235,000} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{21}\mathbf{years} \end{gathered}$$

Computation of payback period of the machine

$$\begin{gathered} \mathbf{Depreciation}\mathbf{}\mathbf{expense}\mathbf{=}\frac{\mathrm{Cost}-\mathrm{Salvage}\mathrm{value}}{\mathrm{Useful}\mathrm{life}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{380,000-20,000}{8} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{45}\mathbf{,}\mathbf{000} \end{gathered}$$

role="math" localid="1653455242828" $$\begin{gathered} \mathbf{Annual}\mathbf{}\mathbf{cash}\mathbf{}\mathbf{flow}\mathbf{=}\mathbf{After}\mathbf{}\mathbf{tax}\mathbf{}\mathbf{income}\mathbf{+}\mathbf{Depreciation}\mathbf{}\mathbf{expense} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{60}\mathbf{,}\mathbf{000}\mathbf{+}\mathbf{45}\mathbf{,}\mathbf{000} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{105}\mathbf{,}\mathbf{000} \end{gathered}$$

role="math" localid="1653455756153" $$\begin{gathered} \mathbf{Payback}\mathbf{}\mathbf{Period}\mathbf{=}\frac{\mathrm{Initial}\mathrm{Investment}}{\mathrm{Annual}\mathrm{cash}\mathrm{flow}} \\ \mathbf{=}\mathbf{}\frac{380,000}{105,000} \\ \mathbf{=}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{62}\mathbf{years} \end{gathered}$$