# Numbers 101: Taxes, Investment, and Depreciation

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1 The Anderson School at UCLA POL Numbers 101: Taxes, Investment, and Depreciation Copyright 2002 by Richard P. Rumelt. In the Note on Cost and Value over Time (POL ), we introduced the basic mechanics of discounting, taking present values, and showed how to compare simple situations with differing durations. Unfortunately, the real world also contains corporate income taxes and taxes complicate matters. The complication arises because all cash inflows and outflows are not treated alike by the tax code. Put differently, taxes are levied on income, not on cash flow. Consequently, just adding up the present values of pre-tax numbers can lead you far astray in comparing alternatives. Before-Tax and After-Tax Numbers The average U.S. corporation pays federal and state income taxes at about a 40% marginal rate. 1 Make sure you understand this simple equation (T is the tax rate): {After-Tax Income} = (1 T) {Pre-Tax Income}. Anything that increase pre-tax income by \$1 will increase after-tax income by (1-T). Any expense that reduces pre-tax income by \$1 will reduce after-tax income by (1-T). Independent business-people often think about pre-tax dollars as less expensive than after-tax dollars. If they can lease a car as a business expense, a \$1000 lease payment reduces after-tax income by only \$600. If the alternative was a personal lease, the business lease is cheaper. It isn t that the lease is cheaper, it s that the government has helped out, chipping in \$400 each month towards the lease in the form of a reduced tax bill. The two main culprits in the tax story are capital expenditures and depreciation. Capital expenditures are expenditures for long-lived assets. Such money outflows are NOT expenses. They are not deducted from revenue to arrive at income. Therefore they do not reduce income or taxes on income. They are not tax deductible. Cash Flow and Discount Rates In business decision-making, what should be discount and at what rate? This is a very complex question and we only hope to introduce it here. We will put forward several rules that are guidelines for solving many business problems. They are not the most sophisticated rules, but they are the basic rules that most MBA s are expected to know and use. More sophisticated rules require specialized training in finance. 1 U.S. federal marginal corporate income tax rates start at 15%, rise to 39% for corporate incomes between \$100K and \$335K, then fall to 35%, which is the marginal rate for most large companies. State corporate tax rates vary by state. In California, for example, the marginal corporate income tax rate is 8.8%. State income taxes are a deductible expense in computing federal income tax liability.

2 Taxes, Investment, and Depreciation 2 POL Rule 1: Work with after-tax numbers. If the situation involves capital expenditures, or other items that are not revenues or expenses, it is very difficult to perform a correct analysis working only with before-tax numbers. Rule 2: Work with cash flows, not profits. The value of the firm is governed by cash flow, not by accounting profit. The first adjustment we make is to add depreciation and amortization back to Net Income to get cash flow. Depreciation and amortization are fictitious expenses Net Income understates cash flow by the amount of these fake expenses. Secondly, cash flow is reduced by any increases in working capital. 2 Thirdly, cash flow is reduced by investment (capital expenditures less capital sales). Cash Flow = Net Income + Depreciation + Amortization - Increases in Working Capital - Investments Oftentimes (as in the next section) we look at a project by itself, not having a clue as to the company s total Net Income. We then work with the incremental cash flows generated by the project. Any incremental benefits b have an after-tax impact on cash flow of (1 T)b. Any incremental expenses have an after-tax impact on cash flow of (1 T)e. Incremental investment I has an impact I. The only tricky part is depreciation. It has an impact on both Net Income and then we add it back in. So, incremental depreciation in the amount d has a total impact on cash flow of (1-T)d + d = Td. That amount, Td, is called the depreciation tax shield (DTS). It is the reduction in the tax bill caused by depreciation it is the impact on cash flow of depreciation. Rule 3: Don t include financing in your cash flows. Leave out interest charges and dividends. This is, perhaps, the trickiest rule to understand. The basic idea is that we work with the cash flows as if the firm had no debt. We include the effects of debt in the discount rate (Rule 4). If the cost of debt is already included in the discount rate, then putting debt charges into the cash flows is double counting. It is possible to do analyses looking at after-interest after-tax cash flows, but it is quite hard to get the discount rate right in these cases. Rule 4: Discount at the weighted average cost of capital. If the firm has total capital C and debt D, it is said to have a leverage ratio of L = D/C (the proportion of debt). Suppose the rate of interest on debt is k d, and the cost of equity (non-debt capital) is k e. Then the weighted average cost of capital is k = (1 T) Lk + (1 L) k. (1) Notice that we use the after-tax cost of debt in the weighted average. What is the cost of equity? It depends on risk. For the average firm it is the risk free rate (T-Bills) plus the equity risk premium perhaps 6%. So, if the risk-free rate is 6.5%, the cost of equity would be 12.5%. The cost of debt might be 8% (debt is not riskless). Taking these numbers, and assuming that a firm s leverage ratio is ¼, we compute the weighted average cost of capital to be k = 0.25(1 0.4) (.125) = 10.6%. There are complex economic arguments behind each of these numbers and relationships. For example, the cost of equity is also a function of the amount of leverage. So you can t just set L=1 d e 2 Working capital is Current Assets less Current Liabilities. It is the amount of Current Assets that has been, or has to be, financed by long-term sources. It requires investment, but not capital investment.

3 Taxes, Investment, and Depreciation 3 POL and get a nice low discount rate. Making L high makes the equity stream very risky and pushes k e through the roof! A Simple Capital Project NOT! Capital projects involve making capital expenditures in the expectation of future gains. For example, consider a situation in which management will invest \$1 million for a plant expansion that will produce \$450K annually in benefits (income less expenses). Now consider a different problem: there are two machines being compared, and one costs \$1 million more than the other, but has maintenance expenses Table 1 The Capital Project Capital Expenditure (\$K) Useful Life (years) that are \$450K less per year. If the two ma- Pre-Tax Benefits (\$K/Year) 450 chines have the same life, the problem is just like that of the factory. In both cases we are comparing spending Tax Rate (%) 40 \$1 million (or \$1 million more) in order to get benefits of \$450K per year. Discount Rate (%) 10 Assume that the useful life of the equipment is 7 years, that the marginal corporate tax rate is 40%, and that the appropriate discount rate is 10%. The data about our project are summed up in Table 1. Let s first do an incorrect pre-tax analysis. The present value of \$450 received each year for 7 years at 10% is PV(0.1,7,-450) = \$1,191K. The net present value of the project, then, is = \$191K. So it looks OK. The problem with the above analysis is that the benefits are taxed each year as they occur, but that the tax effects of the capital expenditure are spread-out over time, through the mechanism of depreciation accounting. Working correctly with taxes is often confusing. The problem is made worse because many people prefer to ignore taxes and just work with the before tax numbers. This bad habit is reinforced by divisional accounting (a business inside a corporation) that stops at Operating Income taxes are handled by corporate. But correct decisions require that the tax implications of the decision be taken into account. Another place you will see improper tax treatment is in consulting reports aimed at senior management. Often, to keep things simple, the consultants just deal with before tax numbers. But the results of such analyses can be misleading. Although capital expenditures are not deductible, U.S. tax law lets you depreciate an asset over its useful life and deduct the depreciation. As discussed above in Rule 2, depreciation is a fictitious expense but it is tax deductible. Because it is fictitious, depreciation only affects aftertax profits through its impact on tax payments. That is, if you have profits, depreciation of amount d reduces your tax bill by dt, where T is your marginal tax rate. That is like having Uncle Sam make an after-tax payment into net income of dt. Let s look at the after-tax cash flows. The before-tax present value of the benefit stream is \$1191. Therefore, the after-tax value of the benefit stream is simply (1 T)(1191) = (.6)(1191) = \$714K. The before-tax cost of the equipment is \$1,000K, spent immediately. But the after-tax cost is not (1 T)(1000). The \$1 million capital expenditure is not an expense and it isn t tax deductible. What is tax deductible is depreciation expense, which is the accounting convention of spreading the \$1 million over the useful life of the equipment. If we use straight-line depreciation, we divide the \$1 million into 7 equal chunks of \$142.9K. Each year we declare this (fictitious) expense until the equipment is fully depreciated after 7 years. Each year the \$142.9K in depreciation reduces taxable income by that amount, reducing taxes by T(142.9) = (.4)(142.9) = \$57.1. After 7 years of doing this we have obtained 7(57.1) = \$400K in tax savings. That is ex-

4 Taxes, Investment, and Depreciation 4 POL actly (1 T)(\$1000K) in absolute dollars. But the tax savings are spread over time. Take the present value of these depreciation tax savings (DTS) and find PV(0.1,7,-57.1) = \$278.2K, much less than \$400K. By forcing us to depreciate the equipment, rather than expense it, the tax law has reduced the value of the tax shield. So the after-tax cost of the equipment is = Table 2 After-Tax Analysis Straight-Line Depreciation a Annual Benefit (\$K) b Annual Benefit After-Tax (\$K) [(1 T)(a)] c Annual Depreciation Expense (\$K) [1000/7] d Depreciation Tax Savings (\$K) [Tc] 57.1 e Present Value of After-Tax Benefits (\$K) [PV(0.1,7,-270)] f Present Value of Depreciation Tax Savings (\$K) [PV(0.1,7,-54.17)] g Net Present Value of Project (\$K) [ e + f] -7.3 \$721.8K much more than (1 T)(1000). This series of calculations is displayed in Table 2. Whereas the erroneous pre-tax calculations showed a positive NPV, the after-tax calculations show a negative NPV of -\$7.3K. The difference is NOT taxes per se, but the tax-code s asymmetric treatment of CAPEX and benefits. Whereas the tax on benefits is immediate, depreciation rules defer the tax shield on CAPEX. Working with Real Depreciation Real depreciation (for tax purposes) is usually calculated on an accelerated basis. Doubledeclining balance works like this: if the un-depreciated balance is B, and the total useful life was L years, then you can deduct 2B/L in depreciation, or the remaining straight-line amount, whichever is larger. (150% depreciation works the same way but the factor is 1.5B/L). 3 Look at Table 3. The DDB depreciation charges \$1 million over 7 years are shown. 4 The first-year s depreciation is (1000)(2/7) and the second year it is 2/7 s of the remaining balance. In year 5 the depreciation switches over to straight-line because DDB is actually slower than straight-line from then on. Table 3 DDB Depreciation Year Depreciation (\$K) The life over which you can depreciate an asset is also controlled by law. Accelerated depreciation is allowed for 6 classes of assets with lives of 3, 5, 7, 10, 15, and 20 years. You can, for example, depreciate a race horse over 12 years old in 3 years. You can depreciate an automobile over 5 years. Most industrial machinery depreciates over 7 years. A tugboat depreciates over 10 years, and so on. 4 Double-declining balance method switching to straight-line in year 3. Check your accounting course and Excel function VDB for details.

5 Taxes, Investment, and Depreciation 5 POL Excel provides two functions to help with these calculations. DDB() function gives the double-declining balance depreciation charges for an asset in any period. However, it never switches to straight-line. The VDB() function takes the DDB or the straight-line, whichever is larger in each year. See Excel documentation for the details about how these functions work. Use Excel s NPV() function to take the present value of the stream of numbers in Table 3 and you get \$ Multiplying by 0.4 yields the present value of the depreciation tax savings (DTS) = \$ Using straight-line depreciation (Table 2) we found the present value of DTS to be \$278.2K. Using DDB depreciation increases it to \$302.6, an increase of \$24.4K. This is more than enough to swing the NPV of the project from -\$7.3 to +\$17K. To summarize, the pure pre-tax view of the project was much too rosy. The real NPV of the project is \$17K, not \$191K. If we had used only the simple straight-line depreciation analysis, the after-tax NPV would have been -\$7K. Making an Excel Macro One problem with these Excel functions is that you must produce a table, like Table 3, before you can calculate a present value. Wouldn t it be nice if there was a function that just spat out the present value of the depreciation stream in one swoop? The function is called PV_DDB() and is easily added as a macro VBA function. It takes 5 parameters, the last one being optional. The parameters are: Cost = the initial investment cost of the capital equipment Salvage = the salvage value of the equipment at the end of its useful life Life = the useful life in years Rate = the discount rate expressed as a fraction (not a percentage) Ddbf = the depreciation rate. The default value is 2 (double declining). To specify another rate, specify it as a number such as 1.5 for 150% depreciation rules. Example: The present value of the stream shown in Table 3 can be obtained as =PV_DDB(1000,0, 7, 0.1). The result is Multiply by 0.4 to get the DTS. To add this function to your system, type Alt-F11, add a module to your workbook and paste this function into it. If you don t understand any of this, get some help from someone who does. Public Function PV_DDB(Cost, Salvage, Life, Rate, _ Optional ddbf As Variant) As Double Dim f As Double, beta As Double, gamma As Double Dim i As Integer f = 2 If Not IsMissing(ddbf) Then f = ddbf End If PV_DDB = 0# beta = 1# / (1# + Rate) gamma = beta For i = 0 To Life - 1 PV_DDB = PV_DDB + gamma * _ Application.WorksheetFunction.Vdb_ (Cost, Salvage, Life, i, i + 1, f) gamma = gamma * beta Next i End Function

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