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Depreciation allows a business to allocate the cost of an asset over its useful life. This is a gradual deduction in value, which means that companies do not have to expense the entire cost in the year of purchase. This is beneficial for maintaining financial stability.
In the context of the exercise, both purification systems are depreciated in a straight-line method, which is straightforward and typically easy to apply. For the Quick & Dirty system, the depreciation is calculated by taking the purchase price of 10 million dollars, dividing it by its useful life of 5 years, giving an annual depreciation of 2 million dollars. Similarly, the Do-It-Right system cost is 12 million dollars and has a useful life of 8 years, resulting in an annual depreciation of 1.5 million dollars.
Understanding the depreciation will help to analyze how much cost is absorbed each year by the business, impacting profits and taxes.

A tax shield is a reduction in taxable income resulting from the use of allowable deductions—such as depreciation—that can be offset against a company's income. The tax shield provides a valuable financial advantage, as these deductions reduce the company's overall tax liability.
In this exercise, the tax shield is calculated by multiplying the annual depreciation by the firm's tax rate of 35%. For the Quick & Dirty system, this comes out to 0.7 million dollars annually (2 million dollars * 0.35). Conversely, for the Do-It-Right system, the tax shield is 0.525 million dollars per year (1.5 million dollars * 0.35).
The tax shield effectively reduces the company's cash outlay for taxes, offering a notable benefit that should be considered when choosing between investments.

The discount rate is crucial in evaluating the present value of future cash flows. It reflects the opportunity cost of capital investment, which is the return rate that could be earned on an investment of equivalent risk. In net present value (NPV) analysis, the discount rate helps to determine how much these future cash flows are worth right now.
In this exercise, the discount rate is set at 12%. This means that when calculating the NPV, future cash flows are discounted back to their present value using 12% as the rate. This adjustment helps to account for the time value of money, suggesting that a dollar today is worth more than a dollar in the future.
Selecting an accurate discount rate is essential for making informed investment decisions, ensuring you're comparing potential projects on a like-for-like basis.

Operating costs are the expenses associated with the day-to-day functions of a business. These are regular expenses required to keep a project or system running effectively. In this exercise, both purification systems have an equivalent operating cost of 1 million dollars annually.
To calculate total costs, it's important to consider both the initial investment and these ongoing operating costs over the system's lifespan. After adjusting for taxes, these operating costs lessen the overall net cash inflows, reducing profitability.
The after-tax effect on operating costs is calculated by multiplying the expense by (1 - tax rate). In this example, the after-tax operating cost for both systems is 0.65 million dollars. This adjustment gives a more accurate picture of the true expense impacting the company's cash flow, vital for comparing these two investment options.