Sunk costs
Sunk costs are costs that have already been incurred in the past and that nothing we do now or in the future can affect.
These costs won’t affect the decision making and economic analysis at present and in the future. A typical example for sunk cost in the oil and gas industry is the cost that has been spent on drilling a well. That well may have been producing for many years by the time a decision must be made for whether the well should be abandoned, but in this situation, drilling cost is sunk cost and it’s irrelevant for the analysis. A similar concept is applicable to revenues from previous years and all its tax and commitments that have been paid.
Opportunity cost
Opportunity cost is hidden or implied cost that is incurred when a person or organization forgoes the opportunity to realize positive cash flow from an investment in order to take a different investment course of action. A typical opportunity cost example is to sell a property or keep and develop it. If an investor forgoes realizing a sale value positive cash flow in order to keep and develop a property, an opportunity cost equal to the positive cash flow that could be realized from selling must be included in the analysis of development economics.
Also, as explained before, minimum rate of return used to analyze a project is actually the opportunity cost of capital (not the cost of borrowing money). Minimum rate of return is the return on capital that could be invested in other projects. Consequently, minimum rate of return is equivalent to opportunity cost of capital.
Breakeven analysis
Breakeven analysis includes calculating one unknown parameter (such as annual revenues, product selling prices, project selling prices, and breakeven acquisition costs) based on all other known parameters under the condition that costs break even the profits. When calculating and analyzing the unknown parameters for after tax considerations, it is very important to apply the after tax values. For example, minimum rate of return applied to calculate after tax NPV should be the rate corresponding to after tax analysis of the project.
Example 94
Consider a fairly old producing machine. As a manager you have two alternatives:
A) Sell the machine: You can sell the machine in the market now for $500,000 with zero book value and pay the tax of 35%.
B) Keep the machine: You can decide to keep the machine but an overhaul cost of $800,000 is required to repair and improve the machine. The overhaul cost is depreciable from time 0 to year 3 (over four years) based on MACRS 7year life depreciation with the half year convention (Table A1 at IRS). After overhaul, the machine would be able to produce and generate equal annual revenue for three years (year 1 to 3). In the end of year 3, salvage value of the machine will be 100,000 dollars (zero book value). The operating cost of the machine for year 1, 2, and 3 will be $300,000, $400,000 and $500,000.
Assuming 35% income tax rate and aftertax minimum ROR of 18%, calculate the minimum annual revenue that the machine has to generate to breakeven the selling with NPV of keeping the machine.
A) Selling the machine
Revenue of selling  500,000 
 Book value  0 


Taxable income  500,000 
 income tax 35%  175,000 


Net Income  325,000 
Book value  0 


ATCF  325,000 
Note that because in this case the machine would be sold at time zero the NPV_{selling the machine} = $325,000.
B) Keeping the machine
In this case, the annual revenue is the unknown variable (X).
Depreciation rate based on method MACRS 7year life with the half year convention for year 0, 1, and 2 will be 0.1429, 0.2449, and 0.1749. And for year 3 we apply the remaining: $1\left(0.1429+0.2449+0.1749\right)=0.4373$
$$\text{Year0depreciation}:\text{}0.1429*800,000=\$114,320$$ $$\text{Year1depreciation}:\text{}0.2449*800,000=\$195,920$$ $$\text{Year2depreciation:}0.1749*800,000=\$139,920$$ $$\text{Year3depreciation}:\text{}0.4373*800,000=\$349,840$$Year  0  1  2  3 


Revenue  X  X  X  
+ Salvage  100,000  
 Operating cost  300,000  400,000  500,000  
 Depreciation  114,320  195,920  139,920  349,840 


Taxable income  114,320  X  495,920  X  539,920  X  749,840 
 Income tax 35% 
+40,012

0.35X + 173,572

0.35X + 188,972  0.35X + 262,444 


Net Income 
74,308

0.65X  322,348

0.65X  350,948

0.65X  487,396

+ Depreciation  114,320  195,920  139,920  349,840 
 Repair Cost  800,000  


ATCF 
759,988

0.65X  126,428

0.65X  211,028

0.65X  137,556

NPV for this After Tax Cash Flow can be calculated as:
$$\begin{array}{l}NP{V}_{Keeping\text{}the\text{}machine}=759,988\text{}+\left(0.65X126,428\right)*\left(P/{F}_{18\%,1}\right)\\ +\left(0.65X211,028\right)*\left(P/{F}_{18\%,2}\right)+\left(0.65X137,556\right)*\left(P/{F}_{18\%,3}\right)\end{array}$$ $$\begin{array}{l}NP{V}_{Keeping\text{}the\text{}machine}=759,988+\left(0.65X126,428\right)/\left(1+0.18\right)\\ +\left(0.65X211,028\right)/{\left(1+0.18\right)}^{2}+\left(0.65X137,556\right)/{\left(1+0.18\right)}^{3}\end{array}$$ $$\begin{array}{l}NP{V}_{Keeping\text{}the\text{}machine}=759,988+0.551X107,142+0.467X151,557\text{}\\ +0.396X83,721=1.413X1,102,408\end{array}$$In order to calculate the minimum annual income of X, we have to equate the NPV_{Keeping the machine} and NPV_{selling the machine}.
$$\begin{array}{l}NP{V}_{Keeping\text{}the\text{}machine}=NP{V}_{selling\text{}the\text{}machine}\\ 1.413X1,102,408=325,000\\ X=\$1,010,000\end{array}$$So, the minimum revenue equals $1,010,000 for year 1 to year 3.
Italicized sections are from Stermole, F.J., Stermole, J.M. (2014) Economic Evaluation and Investment Decision Methods, 14 edition. Lakewood, Colorado: Investment Evaluations Co.