Introduction to Financial Engineering
⎯
ISyE 4803
1. For the current year a company has income of R and cost of C for a net income before tax of
R-C for the year. The company’s marginal tax rate is
τ
. The company’s after-tax cash flow for the year is therefore (1-
τ
)[R – C].
Example 1.
Say R = 3,000 and C = 1,000 with
τ
= 40%. Then the company’s after-tax cash flow for the year is (1 – 0.40)[3,000 – 1,000] = 1,200. Note that this may be equivalently expressed as (0.60)[3,000] – (0.60)[1,000] = 1,800 – 600 = 1,200. That is, the company receives a net of 60 cents for each dollar of revenue it takes in but it only costs the company 60 cents for dollar of expense. On the expense side note that
-(1-
τ
)[C] = -C +
τ
C = -1000 + (0.40)(1000), and so the net effect of being able to deduct expenses it that the company “receives” a “tax benefit” of
τ
C = 400, i.e., it “wrote a check” for 1,000 during the year and receives back 400 due to the expense write-off.
2. Let’s assume the revenue R is sufficiently high to offset the expenses. In what follows we shall ignore the after-tax cash flow due to revenue, since it is a “constant” as far as the analysis of the after-tax cash flow due to cost.
Now let’s suppose for the current year a company spends I = 1000 on capital investment for equipment. The equipment is classified as 5-year property and straight-line depreciation is used.
If the company were able to write-off this capital investment as an expense, then the after-tax cash flow would be –600, exactly as in Example 1.
However, the government’s perspective on this is that the equipment is being used over a 5year period and provides economic value over its entire “book” life of 5 years. (The company may choose to use this equipment for more than 5 years.) Consequently, the government will only allow the company to expense each year an “appropriate” amount for the “wear and tear” or depreciation of the equipment. Since straight-line depreciation is being used, the depreciation expense each year is 1000/5 = 200.
There is NO out-of-pocket depreciation expense each year. That is, the 1000 is spent now; the 200 is computed only for tax purposes. As explained in #1 above, from the after-tax cash flow perspective, this 200 depreciation “allowance” yields a tax benefit of 0.40(200) = 80 each year for the next 5 years. That is, the after-tax cash flow due to the capital investment for the next 5 years (ignoring the salvage value due to possible sale of equipment) is:
0 1 2 3 4 5
Note how the undiscounted sum of the depreciation allowances over the 5-year horizon adds up to 400, the total tax benefit. Again, the total tax benefit of 400 would be added to the –1,000 if this were an ordinary expense (for materials, labor, etc.).

Suppose the company were to use a 6% cost of capital for purposes of determining the Net
Present Value of this cash flow stream, then the
NPV = -1,000 + {[1 – (1.06)
-5
]/(0.06)} 80 = -1,000 + 4.212(80) = 663.01, which translates to a net cost of 66.3% per dollar spent on this capital equipment. Note that if this equipment had been expensed, then the net cost would be 60% per dollar spent. The “loss” of
6.3% is due to the DELAY of receiving the benefit of 80 per year for 5 years, as above, as opposed to receiving the total of 400 right NOW.
Which cash flow would a company prefer? For the purpose of after-tax cash flow, the company would prefer to be able to “write-off” as much depreciation as quickly as it can. That is, it would prefer accelerated depreciation . For the purpose of reporting profits, however, the company would prefer to delay the expense! The IRS allows a company to keep “two sets of books,” one for tax purposes and the other for reporting to shareholders.
B. Accounting for Depreciation in the Income Statement
Example 2 . Suppose our company has a net operating income (R – C) for the year of 3,000 and a depreciation expense of 1,000.
According to what we have learned above, the taxes owed to the government this year would be 0.40(3,000 – 1,000) = 800. The “Flow In” is therefore 3,000 and the “Flow Out” is 800 for a net of 2,200. Here is the income statement approach to this calculation:
Income Statement Approach
Net Revenue 3,000
Flow In – Flow Out Approach
Net Revenue 3,000
Expenses
Total
Income tax
Net Income after tax
1,000
800
1,200
Adjustments
Taxes -800
Depreciation
After tax cash flow
1,000
2,200 After tax cash flow 2,200
In general notation, the two approaches to after tax cash flow are, respectively:
Income Statement Approach: = (1 -
τ
)[R – C – Dep] + Dep = (1-
τ
)[R – C] +
τ
Dep,
Flow In – Flow Out Approach: = [R – C] -
τ
[R – C – Dep] = (1-
τ
)[R – C] +
τ
Dep, which produce the same result, as it should. Note that if the depreciation adjustment “below the line” (i.e. below the Net Income after tax) is NOT made, then the income statement approach would incorrectly report an after tax cash flow of 1,200. The discrepancy would arise because the depreciation allowance is NOT an “out of pocket” expense.