### Capital words, CCFs equal FCFs plus the

Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows Richard S.

Ruback* This paper presents the Capital Cash Flow (CCF) method for valuing risky cash flows. I show that the CCF method is equivalent to discounting Free Cash Flows (FCF) by the weighted average cost of capital. Because the interest tax shields are included in the cash flows, the CCF approach is easier to apply whenever debt is forecasted in levels instead of as a percent of total enterprise value.The CCF method retains its simplicity when the forecasted debt levels and the implicit debt-to-value ratios change throughout forecast period.

The paper also compares the CCF method to the Adjusted Present Value (APV) method and provides consistent leverage adjustment formulas for both methods. The most common technique for valuing risky cash flows is the Free Cash Flow (FCF) method. In that method, interest tax shields are excluded from the FCFs and the tax deductibility of interest is treated as a decrease in the cost of capital using the after-tax weighted average cost of capital (WACC).Because the WACC is affected by changes in capital structure, the FCF method poses several implementation problems in highly leveraged transactions, restructurings, project financings, and other instances in which capital structure changes over time. In these situations, the capital structure has to be estimated and those estimates have to be used to compute the appropriate WACC in each period. Under these circumstances, the FCF method can be used to correctly value the cash flows, but it is not straightforward.

This paper presents an alternative method for valuing risky cash flows. I call this method the Capital Cash Flow (CCF) method, because the cash flows include all of the cash available to capital providers, including the interest tax shields. In a capital structure with only ordinary debt and common equity, CCFs equal the flows available to equity—NI plus depreciation less capital expenditure and the increase in working capital—plus the interest paid to debtholders.The interest tax shields decrease taxable income, decrease taxes and, thereby, increase after-tax cash flows. In other words, CCFs equal FCFs plus the interest tax shields. Because the interest tax shields are included in the cash flows, the appropriate discount rate is before-tax and corresponds to the riskiness of the assets.

Although the FCF and CCF methods treat interest tax shields differently, the two methods are algebraically equivalent.In other words, the CCF method is a different way of valuing cash flows using the same assumptions and approach as the FCF method. The advantage of the CCF method is its simplicity. Whenever debt is forecasted in levels, instead of as a percent of total enterprise value, the CCF method is much easier to use, because the interest tax shields are easy to calculate and easy to include in the cash flows.

The CCF method retains its simplicity when the forecasted debt levels and the implicit debt-to-value ratios change throughout the forecast period.Also, the expected asset return depends on the riskiness of the asset and, therefore, I would like to thank Malcolm Baker, Ben Esty, Stuart Gilson, Paul Gompers, Bob Holthausen, Chris Noe, Paul Maleh, Scott Mayfield, Lisa Meulbroek, Stewart Myers, Denise Tambanis, Peter Tufano, the Editors, Lemma Senbet and Alexander Triantis, the referees, and seminar participants at Duke, Georgetown, and Harvard for comments on earlier drafts and helpful discussions. * Richard S.Ruback is the Willard Prescott Smith Professor of Corporate Finance at Harvard Business School in Boston, MA.

Financial Management • Summer 2002 • pages 5 – 30 6 Financial Management • Summer 2002 does not change when capital structure changes. As a result, the discount rate for the CCFs does not have to be re-estimated every period. In contrast, when using the FCF method, the after-tax WACC has to be re-estimated every period.

Because the WACC depends on valueweights, the value of the firm has to be estimated simultaneously.The CCF method avoids this complexity so that it is especially useful in valuing highly levered firms whose debt is usually forecasted in levels and whose capital structure changes substantially over time. The CCF method is closely related to my work on valuing riskless cash flows (Ruback, 1986) and to Stewart Myers’ work on the Adjusted Present Value (APV) method (Myers, 1974).

In my paper on riskless cash flows, I showed that the interest tax shields associated with riskless cash flows can either be equivalently treated as increasing cash flows by the interest tax shield, or as decreasing the discount rate to the after-tax riskless rate.The analysis in this paper presents similar results for risky cash flows; namely, risky cash flows can be equivalently valued by using the CCF method with the interest tax shields in the cash flows or by using the FCF method with the interest shields in the discount rate. The APV method is generally calculated as the sum of FCFs discounted by the cost of assets plus interest tax shields discounted at the cost of debt. It results in a higher value than the CCF method, because it assigns a higher value to interest tax shields.The interest tax shields that are discounted by the cost of debt in the APV method are discounted by the cost of assets explicitly in the CCF method and implicitly in the FCF method. Stewart Myers suggests the term “Compressed APV” to describe the CCF method, because the APV method is equivalent to CCF when the interest tax shields are discounted at the cost of assets. However, most descriptions of APV suggest discounting the interest tax shields at the cost of debt (Taggart, 1991 and Luehrman, 1997).

The APV method treats the interest tax shields as being less risky than the assets, because the level of debt is implicitly assumed to be a fixed dollar amount. The intuition is that interest tax shields are realized roughly when interest is paid so that the risk of the shields matches the risk of the payment. This matching of the risk of the tax shields and the interest payment only occurs when the level of debt is fixed. Otherwise, the risk of the shields depends on both the risk of the payment and systematic changes in the amount of debt.

Because the risk of a levered firm is a weighted average of the risk of an unlevered firm and the risk of the interest tax shields, the presence of less risky interest tax shields reduces the risk of the levered firm. As a result, a tax adjustment has to be made when unlevering an equity beta to calculate an asset beta. The CCF method, like the FCF method, assumes that debt is proportional to value. The higher the value of the firm, then the more debt the firm uses in its financial structure. The more debt used, then the higher the interest tax shields.The risk of the interest tax shields, therefore, depends on the risk of the debt as well as the changes in the level of the debt. When debt is a fixed proportion of value, the interest tax shields will have the same risk as the firm, even when the debt is riskless.

Because the interest tax shields have the same risk as the firm, leverage does not alter the asset beta of the firm. As a result, no tax adjustment has to be made when calculating asset betas. The primary contributions of this paper are to introduce the CCF method of valuation, to demonstrate its equivalency to the FCF method, and to show its relation to the APV method.The CCF method has been used in teaching materials to value cash flow forecasts, in Kaplan and Ruback (1995) to value highly levered transactions, and in Gilson, Hotchkiss, and Ruback (2000) to value firms emerging from Chapter 11 reorganizations. 1 Also, finance textbooks contain some of the ideas about the relation between the discount rate for interest tax shields, unlevering 1 Teaching materials include Ruback (1989, 1995a, 1995b) and Holthausen and Zmijewski (1996). Ruback • Capital Cash Flows 7 ormulas, and financial policy.

This paper provides the basis for the applications of CCFs and highlights the linkages between the three methods of cash flow valuation. Although the focus of this paper is on cash flow methods that yield estimates of total enterprise value, the results have implications for the Equity Cash Flow (ECF) and Residual Income (RI) methods. In the ECF method, the FCFs to equity are discounted at the cost of equity capital. That method is equivalent to the FCF method and has the same drawbacks. The cost of equity capital, like the WACC, changes as leverage changes; it requires a simultaneous estimation of the equity-to-value ratios and the value whenever the debt is forecasted in levels.

In those situations, the CCF method is a simpler approach. Similarly, as Lundholm and O’Keefe (2001) stress, the RI approach is equivalent to the FCF method as long as consistent assumptions are used—including the assumption that the discount rate consistently incorporates the assumed debt policy. Thus, the RI approach does not mitigate the valuation issues addressed in this paper. Section I describes the mechanics of the CCF method, including the alculation of the cash flows and the discount rate. Section II shows that the CCF method is equivalent to the FCF method through an example and, then, with a more general proof. Section III relates the CCF and the APV methods and shows that the difference between the two methods depends on the implicit assumption about the financial policy of the firm. I also show that the assumption about financial policy has implications regarding the impact of taxes on risk and, thereby, determines the approach used to transform equity betas into asset betas.

Section IV concludes. I. Mechanics of Capital Cash Flow ValuationThe present value of CCFs is calculated by discounting them by the expected asset return, KA. This section details the calculation of the CCFs in subsection A and explains the calculations of KA in subsection B. An example is presented in subsection C. A.

Calculating Capital Cash Flows CCFs include all of the cash flows that are paid or could be paid to any capital provider. By including cash flows to all security holders, CCFs measure all of the after-tax cash generated by the assets. Since CCFs measure the after-tax cash flows from the enterprise, the present value of these cash flows equals the value of the enterprise.

Figure I summarizes the calculation of CCFs. The calculations depend on whether the cash flow forecasts begin with net income (NI) or earnings before interest and taxes (EBIT). 1. The NI Path NI includes any tax benefit from debt financing because interest is deducted before computing taxes. The interest tax shields, therefore, increase NI.

Available cash flow is NI plus cash flow adjustments and non-cash interest. Cash flow adjustments include those adjustments required to transform the accounting data into cash flow data. Typical adjustments include adding depreciation and amortization, because these are non-cash subtractions from NI.Non-cash interest occurs when the interest is paid in kind by issuing additional debt, instead of paying the interest in cash. These non-cash interest payments are deducted from NI like cash interest but are not a cash outflow and, therefore, must be 2 See Ruback (1995b) and Esty (1999) for a discussion of the relation between the FCF and ECF methods.

8 Financial Management • Summer 2002 Figure I. Calculating Capital Cash Flows NI Does Cash Flow begin with EBIT or NI? EBIT Cash Flow Adjustments Depreciation Amortization Capital Expenditures Change in Working Capital Deferred Taxes Estimate of Corporate TaxAdd Non-Cash Interest EBIAT Available Cash Flow Cash Flow Adjustments Add Cash Interest Capital Cash Flow Free Cash Flow Add Interest Tax Shields added to NI to calculate cash flow available. Capital expenditures are subtracted from NI because these cash outflows do not appear on the income statement and, thus, are not deducted from NI. Subtracting the increases in working capital transforms the recognized accounting revenues and costs into cash revenues and costs. The label “available cash flow” often appears in projections and measures the funds available for debt repayments or other corporate uses.CCF is computed by adding cash interest to available cash flow so that cash flows represent the after-tax cash available to all cash providers. 2.

The EBIT Path When cash flow forecasts present EBIT, instead of NI, corporate taxes have to be estimated to calculate earnings before interest and after taxes (EBIAT). Typically the taxes are estimated by multiplying EBIT by a historical marginal tax rate. EBIAT is then adjusted using the cash flow adjustments that transform the accounting data into cash flow data. EBIAT plus cash flow adjustments equals FCF, which is used to compute value using the after-tax WACC (WACC).FCFs equal CCFs less the interest tax shields. Interest tax shields, therefore, have to be added to the FCFs to arrive at the CCFs.

The interest tax shields on both cash and non-cash debt are added because both types of interest tax shields reduce taxes and, thereby, increase after-tax cash flow. The EBIT path should yield the same CCFs as the NI path. In practice, however, the NI path is usually easier and more accurate than the EBIT path. The primary advantage of the NI path is that it uses the corporate forecast of taxes, which should include any special Ruback • Capital Cash Flows circumstances of the firm. Taxes are rarely equal to the marginal tax rate times taxable income.

The EBIT path involves estimating taxes, usually by assuming a constant average tax rate. This ignores the special circumstances of the firm and adds a likely source of error. B.

Calculating the Expected Asset Return The appropriate discount rate to value CCFs is a before-tax rate because the tax benefits of debt financing are included in the CCFs. The pre-tax rate should correspond to the riskiness of the CCFs. One such discount rate is the pre-tax WACC: Pre ? tax WACC = Pre-D E KD + KE V V (1) where D/V is the debt-to-value ratio; E/V is the equity-to-value ratio, and KD and KE are the respective expected debt and equity returns.

Using the pre-tax WACC as a discount rate is correct, but there is a much simpler approach. If expected returns in equation (1) are determined by the Capital Asset Pricing Model (CAPM): K D = RF + ? D RP K E = RF + ? E RP (2) (3) where RF is the risk-free rate, RP is the risk premium, and bD and bE are the debt and equity betas, respectively. Substituting equations (2) and (3) into equation (1) yields: Pr e ? tax WACC = D( E R F + ?D R P ) + (R F + ? E R P ) V V (4) Simplifying: ?D ? E Pr e ? tax WACC = R F + ? ? D + ? E ? R P V ? V ? The beta of the assets, bU, is a weighted average of the debt and equity beta: (5) ?U = D E ? D + ? E V V (6) Substituting equation (6) into equation (5) provides a simple formula for the pre-tax WACC which is also labeled as the Expected Asset Return, KA: K A = Pr e ? tax WACC = R F + ? U R P (7) Note that the pre-tax expected asset return depends only on the market-wide parameters for the risk-free rate, RF, the risk premium, RP, and on the unlevered asset beta, bU.The debt-to-value and equity-to-value ratios are not in equation (7). KA, therefore, does not depend on capital structure and does not have to be recomputed as capital structure changes. This means that the debt-tovalue and equity-to-value ratios do not have to be estimated to use the CCF valuation method.

This eliminates much of the complexity encountered when applying the FCF method. The discount rate for the CCFs is simple to calculate regardless of the capital structure. It takes two steps. First, estimate the asset beta, bU.

The asset beta is usually estimated using 10 Financial Management • Summer 2002 quation (6) by multiplying the equity beta, bE, by the equity-to-value ratio and adding an estimate of the debt beta multiplied by the debt-to-value ratio. Second, use bU, together with the risk-free rate, RF, and the risk premium, RP, to compute the expected asset return, KA. For example, if the equity beta is 1. 25, the debt beta is 0. 25, and the equity-to-value ratio is 0. 75, then the asset beta is 0. 75 X 1.

25 + 0. 25 X 0. 25 = 1.

0. With an asset beta of 1. 0, an assumed risk free rate of 10%, and an assumed risk premium of 8%, the expected asset return is 10% + 1. 0 X 8% = 18%. C.

Numerical Example Table I contains a numerical example that demonstrates the CCF method. The example assumes an initial investment of $100,000 to be depreciated equally over three years. Panel A details the assumptions. The asset beta is assumed to equal 1. 0 in all three years. 3 The forecasted expected pre-tax operating profits are $50,000 in year one, $60,000 in year two, and $70,000 in year three.

The risk-free rate is assumed to be 10%, the risk premium is assumed to be 8%, and the tax rate is assumed to be 33%. The debt is assumed to be risky, with a debt beta of 0. 35 in the first year, 0. 0 in the second year, and 0. 25 in the third year. The project is financed with debt so that the initial debt is $100,000 at the beginning of year one, $50,000 at the beginning of year two, and $20,000 at the beginning of year three. The CCF is calculated by following the NI path.

The cash flow available is equal to NI plus noncash adjustments. CCF is calculated by adding the expected interest to the cash flow available. The value of the CCFs is calculated using the expected asset return. The easiest way to calculate the asset return is to use the asset beta in the CAPM. Using a risk-free rate of 10%, an asset beta of 1. , and a risk premium of 8% yields an expected asset return of 18%. The asset return does not depend on leverage because it is a pre-tax cost of capital.

It remains constant even though the leverage changes through time. As Panel B of Table I shows, discounting the CCFs at the expected asset return results in a value of $136,996. II.

The Relation Between Capital Cash Flow and Free Cash Flow Valuation The FCF and CCF methods are equivalent. I demonstrate this equivalency in subsection A by extending the numerical example of Table I and showing that the FCF valuation is the same as the CCF valuation.Subsection B presents a more general proof of the equivalence of the CCF and FCF methods. Because the two methods are equivalent, the choice between them is governed by the ease of use. Subsection C presents some suggestions on choosing between the methods. A. Numerical Example Panel C of Table I presents a FCF valuation of the same cash flows valued using CCFs in Panel B.

The FCFs are calculated from EBIT, which is reduced by the hypothetical taxes on EBIT to determine EBIAT. Adding the non-cash adjustments to EBIAT results in FCFs. The FCFs are valued using the after-tax WACC.The WACC has two components: the aftertax cost of debt and the levered cost of equity. The after-tax cost of debt depends on the assumed riskiness of the debt with the cost of debt calculated as its CAPM expected return using equation (2). The levered cost of equity is calculated by levering the asset beta to 3 The equivalency of the CCF and FCF methods does not depend on a constant asset beta.

The asset beta could change each year. In practice, however, the asset beta is related to the assets of the company and, thus, is assumed to be constant unless the asset composition changes.Ruback • Capital Cash Flows 11 Table I. An Example of Capital Cash Flow and Free Cash Flow Panel A.

Assumptions Market Parameters: Riskless Debt Rate = 10% Risk Premium = 8% Tax Rate = 33% Year 1 Asset Beta Debt Beta Expected Cash Flows: Operating Profit Less: Depreciation EBIT Less: Expected Interesta Pre-Tax Income Less: Taxes Net Income Non-Cash Adjustmentsb Cash Flow Available Beginning Debt Cash Flow Available Plus: Expected Interesta Capital Cash Flow Cost of Assetsc Discount Factor Present Value of CCFs Total Enterprise Value a Year 2 1. 0 0. 30 60,000 33,333 26,667 6,200 20,467 6,754 13,713 43,333 57,046 50,000 Year 3 1. 00 0. 25 70,000 33,333 36,667 2,400 34,267 11,308 22,959 43,333 66,292 20,000 66,292 2,400 68,692 18. 0% 0. 6086 41,808 1.

00 0. 35 50,000 33,333 16,667 12,800 3,867 1,276 2,591 43,333 45,924 100,000 Panel B. Capital Cash Flow Valuation 45,924 57,046 12,800 6,200 58,724 63,246 18. 0% 18. 0% 0. 8475 0.

7182 49,766 45,422 136,996 Expected Interest is calculated using the Expected Cost of Debt from the CAPM (riskfree rate plus the debt beta times the risk premium). Noncash adjustments include depreciation plus $10,000 of other adjustments. c Expected asset return is calculated using the assumed asset beta in the CAPM with the assumed riskless debt rate and risk premium.

determine the levered equity beta. Because the fraction of debt is not the same each year, the WACC and its components need to be recomputed each year. The formula for levering the asset or unlevered beta is: D ? ? E ? E = ? ?U ? ? D ? / V ? ? V (8) which requires information on the value of the firm to compute the percentage of debt and equity in the capital structure. Generally, an iterative or dynamic programming approach is used to solve for a consistent estimate of enterprise value. 5 However, because the value is already computed in Panel B, that value can be used to compute the debt and equity proportions.

Based on the implied equity-to-value ratio of 27. 0% in the first year, the asset beta of 1. 0, and the debt beta of 0. 35, the implied equity beta is 2. 76.

Using the CAPM and the 4 5 This formula is derived in Section III. B of this paper. See Esty (1999) an explanation of the iterative technique and a project finance application of that approach. 2 Financial Management • Summer 2002 Table I. An Example of Capital Cash Flow and Free Cash Flow (Continued) Panel C.

Free Cash Flow Valuation Year 1 Year 2 EBIT Less: Tax on EBIT EBIAT Non-Cash Adjustmentsb Free Cash Flows Capitalization: Total Enterprise Valued Debt WACC Calculations: Debt Percent After-Tax Coste Contributionf Equity Percent Equity Beta g Costh Contributioni WACC Discount Factor Present Value of FCFs Total Enterprise Value b d Year 3 36,667 12,100 24,567 43,333 67,900 58,214 20,000 16,667 5,500 11,167 43,333 54,500 136,996 100,000 6,667 8,800 17,867 43,333 61,200 102,932 50,000 73. 0% 8. 6% 6. 3% 27. 0% 2. 76 32. 1% 8.

7% 14. 9% 0. 8702 47,426 136,996 48.

6% 8. 3% 4. 0% 51. 4% 1. 66 23. 3% 12.

0% 16. 0% 0. 7501 45,905 34. 4% 8. 0% 2.

8% 65. 6% 1. 39 21. 1% 13. 9% 16. 6% 0. 6431 43,665 Noncash adjustments include depreciation plus $10,000 of other adjustments.

Total Enterprise Value is the present value of the remaining cash flows. e After-tax cost of debt is the Expected Cost of Debt times (1-tax rate). f Debt contribution is the After-tax Expected Cost of Debt times the percent debt. Equity is determined by levering the asset beta ((asset beta – debt beta contribution)/percent equity). h Cost of equity is calculated using the CAPM as the riskfree rate plus the equity beta times the risk premium. i Equity contribution is the cost of equity times the percent equity.

assumed market parameters, the expected cost of equity is 32. 1% in the first year. Weighting the expected after-tax cost of debt and the expected cost of equity by their proportions in the capital structure results in a WACC of 14.

9% for the first year.The capital structure changes in each period, because the ratio of the value of the remaining cash flows, and the amount of debt outstanding does not remain constant throughout the life of the project. Repeating the process of valuing the enterprise, determining the debt and equity proportions, unlevering the asset beta, and estimating the equity cost of capital according to the CAPM, results in a WACC of 16. 0% for the second year and 16. 6% for the third year. These after-tax WACCs rise as the percentage of debt in the capital structure, and the corresponding amount of the interest tax shields, fall.

Total Enterprise Value (TEV) is calculated by discounting the FCFs by the after-tax WACCs. Since the after-tax WACCs change, the discount rate for each period is the compounded rate that uses the preceding after-tax WACCs. The resulting value of the FCFs is $136,996, exactly the same value as obtained in the CCF calculations in Panel B of Table I. Ruback • Capital Cash Flows 13 B. Proof of Equivalency This section shows that the CCF method is equivalent to the FCF method. To keep the analysis simple, assume the asset being valued generates a constant pre-tax operating cash flow.This cash flow is before tax, but after cash adjustments such as depreciation, capital expenditures, and changes in working capital. The after-tax operating cash flow equals earnings before interest and after-tax plus the cash flow adjustments and equals FCF, which measures the cash flow of the firm if it were all equity financed. The value, VFCF, is calculated using the FCF method by discounting the FCFs by the after-tax WACC: V FCF = FCF WACC (9) where V is the value of the project being valued. WACC, the after-tax WACC, is defined as: WACC = E D K D (1 ? ) + K E V V (10) with D and E equal to the market value of debt and equity, respectively; t is the tax rate; KD(1-t) is the after-tax expected cost of debt; and KE is the expected cost of equity. The CCF is the expected cash flow to all capital providers with its projected financing policy, including any benefits of interest tax shields from its financial structure. Since FCF measures the cash flow assuming a hypothetical all equity capital structure, then CCF is equal to FCF plus interest tax shields: CCF = FCF + Interest Tax Shield = FCF + ? K D D (11) here KDD is the interest tax shield calculated as the tax rate, t, times the interest rate on the debt, KD, times the amount of debt outstanding, D. Value is calculated using the CCF method, VCCF, by discounting the CCFs by the expected return on assets. The expected asset return is measured using the Capital Asset Pricing Model (CAPM) and the asset beta (bU) of the project being valued: FCF + ? K D D VCCF = (12) RF + ? U RP where RF is the risk-free rate and RP is the risk premium. The goal is to show that the value obtained using FCFs and WACC is the same as the value obtained using CCFs and K A.In other words, the goal is to show that equation (9) is identical to equation (12). By combining equations (9) and (10): D (13) (1 ? ? ) + E K E V V In equation (13), KE and KD are measured using the CAPM according to equations (2) and (3). By substituting the equality between the pre-tax WACC and the cost of assets from equation (7): KD V FCF = FCF = FCF K A ? ?K D D V V FCF = FCF (R F + ? U R P ) ? ?K D D V (14) 14 Financial Management • Summer 2002 Multiplying both sides by the denominator on the right-hand-side of equation (14) yields: V FCF (K A ) ? ?K D D = FCF (15)Rearranging terms by adding tKDD to both sides and dividing by the cost of assets shows that: VFCF = FCF+? KDD = VCCF KA (16) which is identical to equation (12). Thus, this proof shows that the FCF approach in equation (9) and the CCF approach of equation (12) will, when correctly applied, result in identical present values for risky cash flows. 6 C. Choosing Between Capital Cash Flows and Free Cash Flow Methods The proof in subsection II. B shows that the CCF method and the FCF method are equivalent because they make the same assumptions about cash flows, capital structure, and taxes.When applied correctly using the same information and assumptions, the two methods provide identical answers. The choice between the two methods, therefore, is governed by ease of use. The ease of use, of course, is determined by the complexity of applying the method and the likelihood of error. The form of the cash flow projections generally dictates the choice of method. In the simplest valuation exercise, when the cash flows do not include the interest tax shields and the financing strategy is specified as broad ratios, the FCF method is easier than the CCF method.To apply the FCF method, the discount rate can be calculated in a straightforward manner using prevailing capital market data and information on the target capital structure. Because that target structure does not (by assumption) change over time, a single WACC can be used to value the cash flows. This type of valuation often occurs in the early stages of a project valuation before the detailed financial plan is developed. When the goal is to get a simplified ‘back-of-the-envelope’ value, the FCF method is usually the best approach.When the cash flow projections include detailed information about the financing plan, the CCF method is generally the more direct valuation approach. Because such plans typically include the forecasted interest payments and NI, the CCFs are simply computed by adding the interest payments to the NI and making the appropriate non-cash adjustments. These cash flows are valued by discounting them at the expected cost of assets. This process is simple and straightforward even if the capital structure changes through time.In contrast, applying the FCF method is more complex and more prone to error because, as illustrated in subsection II. A and Panel C of Table I, firm and the equity values have to be inferred to apply the FCF method. Also, the CCF method can easily incorporate complex tax situations. Therefore, in most transactions, restructurings, leverage buyouts, and bankruptcies, the CCF method will be the easier to apply. 6 Taggart (1991) analyzes the impact of personal taxes on the FCF approach and shows that the corporate tax rate is the only tax ate that explicitly enters the valuation equation. The algebraic equivalence of the CCF and FCF implies that the corporate tax rate is also the only tax rate in the CCF valuation equation. Nevertheless, personal taxes may affect the expected debt and equity returns and, thereby, affect the FCF and CCF value. Ruback • Capital Cash Flows 15 III. Capital Cash Flows and Adjusted Present Value Both CCF and APV methods can be expressed as: Value = Free Cash Flows Discounted at K A + Interest Tax Shields Discounted at K ITS where KITS is the discount rate for interest tax shields.For both methods, the discount rate for the FCFs is the cost of assets, KA, which is generally computed using the CAPM with the beta of an unlevered firm. The methods differ in KITS, the discount rate for interest tax shields: the APV method generally uses the debt rate and the CCF method uses the cost of assets, KA. APV assigns a higher value to the interest tax shields so that values calculated with APV will be higher than CCF valuations. 7 To gauge how much higher APV valuations are relative to CCF valuations, Table II calculates the difference in values assuming perpetual cash flows and interest tax shields.I define the value of the interest tax shields in the CCF valuation as a proportion, g, of the all equity value. The ratio of V APV to VCCF becomes: V APV VCCF ? K 1+ ? ? A ? K ? D = 1+ ? ? ? ? ? (17) Table II presents the percentage differences between the APV and CCF valuations. For example, if KD =10% and KA =15%, then the ratio of the expected asset return to the expected debt return is 1. 5, locating it in the middle column of Table II. If the tax rate is 36% and the debt is 42% of the all equity value, then the value of the interest tax shield is about 15% of he all equity value, locating it in the middle row of Table II. In this example, therefore, the APV approach would provide a discounted cash flow value that is 7% higher than the CCF value. In the CAPM framework, the discount rate for the interest tax shields should depend on the beta of the interest tax shields: K ITS = Risk free rate + ? ITS ? Risk Pre mium (18) When debt is assumed fixed, subsection A shows that the beta of the interest tax shields equals the beta of the debt. This implies that the appropriate discount rate for the interest tax shields is the debt rate, which is the rate used in the APV method.It also implies that the interest tax shields reduce risk so that a tax effect should appear when unlevering equity betas. When debt is assumed proportional to value, subsection B shows that the beta of the interest tax shields is equal to the unlevered or asset beta. This implies that the appropriate discount rate is the cost of assets, which is the rate used in the CCF method. It also implies that taxes have no effect on the transformation of equity betas into asset betas. Debt could also be a linear function of firm value with both fixed and proportional components.Subsection C shows that the beta of the interest tax shields with a linear debt policy is a value-weighted average of the interest tax shield betas for the fixed and proportional debt policies described 7 Inselbag and Kaufold (1997) present examples of FCF and APV valuations that result in identical values for debt policies with both fixed debt and proportional debt. This occurs because they infer the equity costs that result in equivalence in their FCF valuations instead of obtaining discount rates from the CAPM. 16 Financial Management • Summer 2002 Table II.Percentage Differences Between APV Values and CCF Values (VAPV/VCCF) Tax Shield/ All Equity Value 10% 15% 20% a b b Ratio of Expected Asset Return to Debt Rate (KA/KD)a 1. 25 1. 50 1. 75 2% 3% 4% 5% 7% 8% 7% 10% 13% Calculations assume perpetual cash flows and interest tax shields. All Equity Value is the FCFs discounted at the cost of assets. in subsections A and B, respectively. A. Fixed Debt Policy When debt is perpetual and fixed as a dollar amount, D, which does not change as the value of the firm changes, then the value of the interest tax shields is: V ITS ,t = ?K DD K D ,t (19) here K D is the fixed yield on the debt, K D,t is the cost of debt in period t from equation (2), and t is the tax rate. The value of the debt can change through time if K D is fixed and the cost of debt changes. Assuming K D is the fixed yield, VD ,t = DK D K D ,t (20) By substituting equation (20) into equation (19), the value of the interest tax shield at time t can, therefore, be expressed as: V ITS ,t = ? V D,t The beta of the interest tax shields, bITS, equals: (21) ? ITS = V ITS ,t ? 1Var (R M Cov (V ITS ,t , R M ) ) (22) By substituting equation (21) into equation (22) and simplifying, = ?D (23) V D,t ? 1Var (R M ) The beta of the interest tax shields is, therefore, equal to the beta of the debt when the debt is assumed to be a fixed dollar amount. 8,9 If the debt is assumed to be riskless, then the interest tax shields will also be riskless. If the debt is risky, then the interest tax shields will ? ITS = Cov (V D ,t , R M ) Ruback • Capital Cash Flows 17 have the same amount of systematic risk as the debt. This result shows that the practice of discounting interest tax shields by the expected return on the debt is appropriate when the debt is assumed to be a fixed dollar amount.The assumption of fixed debt and the result that the beta of interest tax shields equals the debt beta implies that leverage reduces the systematic risk of the levered assets. The value of a levered firm, VL, exceeds the value of an unlevered or all equity firm, V U, by value of the interest tax shields from the debt of the levered firm, V ITS: V L = VU + V ITS (24) Equation (24) holds in each time period and abstracts from differences between levered and unlevered firms other than taxes. Also, the analysis assumes strictly proportional taxes.I assume that interest is deductible and that interest tax shields are realized when interest is paid. The beta of the levered firm, bL, is a value-weighted average of the unlevered beta, bU, and the beta of the interest tax shields, bITS: ?L = VU V ? U + ITS ? ITS VL VL (25) When the beta of the interest tax shields equals the debt beta, equation (25) simplifies to: ? L = ? U ? ? D (? U ? ? D ) VL (26) The beta of a levered firm, bL, can also be expressed as a value weighted average of the debt and equity of the levered firm: ?L = E D ? E + ? D VL VL (27)Where E is the equity of the levered firm, bE is the equity beta and bD is the debt beta. By setting equation (26) equal to equation (27): E D D (? U ? ? D ) ? E + ? D = ? L = ? U ? ? VL VL VL (28) which can be simplified as: ? ? D ? ? E = ? ?U ? (? D + ? (? U ? ? D ))? / E (29) ? V VL Thus, ? the equity beta is equal to? theLasset beta less the proportion of debt borne by the debt holder and the reduction due to the tax effect and scaled by leverage. The equity beta is reduced by the tax effect, because the government absorbs some of the risk of the cash 8When debt is assumed to be fixed in value instead of a fixed dollar amount, then the beta of the interest tax shields is zero regardless of the debt beta. 9 If the debt is not principal, then the beta of the interest tax shields would equal the beta of the debt when the interest payment and principle payments have the same beta. 18 Financial Management • Summer 2002 flows. With fixed debt, the interest tax shields portion of the cash flows are insulated from fluctuations in the market value of the firm. When the debt is riskless, the beta of the debt is zero. Therefore, equation (29) simplifies to: E + D(1 ? ) ? U E ?E = (30) Equation (30) is the standard unlevering formula that correctly includes tax effects when the debt is assumed to be fixed and assumes a zero debt beta. In the next subsection, I show that when debt is assumed to be proportional to firm value, taxes do not appear in the unlevering formula. B. Proportional Debt Policy When the value of debt is assumed to be proportional to total enterprise value, the firm varies the amount of debt outstanding in each period so that: V D = ? VU (31) where d is the proportionality coefficient and VU is the value of the unlevered firm.The value of the interest tax shields is the tax rate times the value of the debt so that: V ITS = ? V D = ?? VU (32) By substituting equation (32) into the definition of the beta of the interest tax shields from equation (22): ? ITS = = = V ITS , t ? 1Var (R M Cov(V ITS ,t , R M ) ) Cov(?? VU,t, R M ) ?? VU , t ? 1Var (R M ) Cov (VU ,t , R M ,t ) VU , t ? 1Var (R M ) = ? U (33) The equality between the beta of the interest tax shields and the beta of the unlevered firm implies that the rate used to discount the interest tax shields is equal to KA, the unlevered or asset cost of capital. 0 The equality between the betas for the interest tax shields and the assets also implies that there is no levering/unlevering tax effect. From equation (25), the beta of a levered firm is a weighted average of the beta of the unlevered firm and the beta of the interest tax shields. Since the asset beta equals the interest tax shield beta, the beta of the levered firm equals the beta of the 10 Harris and Pringle (1985) also show that the interest tax shields should be discounted by the pre-tax weighted average cost of capital when debt is assumed to be proportional to value.Ruback • Capital Cash Flows 19 unlevered firm. To calculate the beta of levered equity, equation (29) can be restated as: D ? ? E ? E = ? ?U ? ? D ? / V ? ? V (34) This result means that tax terms should not be include when applying the CCF or FCF methods. 11 C. Choosing Between Capital Cash Flows and Adjusted Present Value Methods Subsection B shows that the difference between the CCF and the APV methods is the implicit assumption about the determinants of leverage. CCF (and equivalently FCF) assumes that debt is proportional to value; APV assumes that debt is fixed and independent of value.Debt cannot literally be strictly proportional to value at all levels of firm value. For example, when a firm is in financial distress, the option component of risky debt increases, thereby, distorting the proportionality. Nevertheless, Graham and Harvey (2001) report that about 80% of firms have some form of target debt-to-value ratio, and that the range around the target is tighter for larger firms. That suggests that the CCF approach is more appropriate than the APV approach when valuing corporations.There are circumstances when the fixed debt assumption is more accurate. These cases typically involve some tax or regulatory restriction on debt, such as industrial revenue bonds that are fixed in dollar amounts. Luehrman (1997) presents an example of APV valuation in which debt is assumed to be a constant fraction of book value. To the extent that book value does not respond to market forces, a fraction of book value is a fixed dollar amount. In practice, valuations are often performed on forecasts that make assumptions about debt policy.When that policy is characterized as a target debt-to-value ratio, the proportional policy seems more accurate. In project finance or leveraged buyout situations, however, the forecasts typically are characterized as a changing dollar amount of debt in each year. These amounts can, of course, be characterized as a changing percentage of value or as a changing dollar amount through time. It is not obvious from the forecasts themselves whether the assumption of proportional debt or fixed debt is the better description of debt policy.The answer in these circumstances depends on the likely dynamic behavior. If debt policy adheres to the forecasts regardless of the evolution of value through time, the fixed assumption is probably better. Alternatively, if debt is likely to increase as the firm expands and value increases, then the proportional assumption is probably better. Debt policy can, of course, be more complex than either an exclusively fixed debt or proportional debt policy; whatever the debt policy, valuation depends on that policy.For example, debt policy can include a fixed component and a component that is proportional to value: V D = V F + ? VU (35) Such a linear debt policy could occur in a project finance application where a fixed amount of debt is subsidized or guaranteed by a government agency and the remaining debt is roughly proportional to the value of the project. 11 Kaplan and Ruback (1995) incorrectly uses tax adjustments to unlever observed equity betas to obtain asset betas when applying the CCF method. Correcting this error does not meaningfully change the results of Kaplan and Ruback (1995). 0 Financial Management • Summer 2002 The valuation of cash flows from a project with a linear debt policy such as equation (35) will combine features of the fixed debt and proportional debt policies. The beta of the interest tax shields for the linear debt policy, for example, is a value-weighted average of the beta with a fixed debt policy (equation 23) and the beta with a proportional debt policy (equation 33) with the value-weights equal to the relative values of the fixed and proportional debt components: ? ITS = Cov (? V D ,t R M ) ? V D,t ? Var (R M ) Cov(? V Ft + ? VUt , R M) = ? V D,t-1Var (R M ) = ?V V F,t ? 1 ? D + U,t ? 1 ? A V D,t ? 1 V D,t ? 1 (36) The value of a project with a linear capital structure could be valued by valuing the interest tax shields using the pre-tax expected return implied by the beta of the interest tax shields from equation (36) and adding that value to the value of the FCFs: V APV = FCF ? K D D + KA K ITS (37) where D is the amount of debt including the fixed and proportional component, and K ITS is calculated using the CAPM with bITS from equation (37).The value of the project can also be valued more simply by adding the value of the fixed interest tax shields to the CCF value: VCCF = FCF + ? K D D P ? K D D F + KA KD (38) where DP and D F are the amount of proportional debt and fixed debt, respectively. The first term on the right-hand-side of equation (38) is the formula for the CCF value when the debt is proportional to value (equation 16). The discount rate for the interest tax shields from the proportional debt is the expected asset return for the same reasons it is the correct rate for the proportional interest tax shields discussed in subsection B.In short, interest tax shields are proportional to the value of the debt so that when debt is proportional to value, the interest tax shields will have the same risk as the value of the firm. The second term on the right-hand side of equation (38) is the value of the interest tax shields associated with the fixed portion of the linear debt policy. The discount rate for the interest tax shields from the fixed portion is the expected debt rate for the same reason that it is the correct rate for the fixed interest tax shields in discussed in subsection A.When the amount of debt is fixed, the interest tax shields are also fixed, and the value of the interest tax shields will vary as the value of the debt varies. The fixed interest tax shields, therefore, will have the same risk as the fixed debt. Equation (38) is consistent with the generally accepted approach of identifying project cash flows with different risk characteristics and valuing those components at an expected return that reflects their risk. In Gilson, Hotchkiss, and Ruback (2000), for example, the valueRuback • Capital Cash Flows 21 of firms emerging from bankruptcy are valued as the CCF value of their continuing operations plus the value of their fixed net operating losses discounted at a debt rate. The different risk characteristics of the interest tax shields in equation (38) arise because of the combination of fixed and proportional debt in the linear debt policy. Similarly, Miles and Ezzell (1980, 1983) model the debt policy as mixed through time with fixed debt in the first period and proportional debt in subsequent periods. The est approach to estimating the value of interest tax shields is to model the debt policy, and then appropriately value resulting interest tax shields using the corresponding discount rate. As an example, Arzac (1996) recognizes that excess available cash flow is typically used to repay senior debt after a leveraged buyout and suggests a “recursive APV approach” to value the transactions. In most corporate circumstances, however, the valuation, at least at its initial stage, will not have the information to model the debt policy in detail and with precision.The practical alternatives may be to simply choose between APV approach with its assumed fixed debt policy and the CCF (or equivalent FCF) approach with its assumed proportional debt policy. Beyond the Graham and Harvey (2001) evidence that most corporations have target debt ratio, theories of debt policy generally suggest that debt changes as value changes. For example, in the static tradeoff between tax benefits and bankruptcy costs, doubling the operations of a firm would double its value, which, in turn, doubles the tax benefits of debt financing and bankruptcy costs so that the amount of debt would also double.Thus, for most applications, the proportional debt assumption appears to be a more accurate description of corporate behavior. That means that the CCF or the equivalent FCF method of valuation will generally be preferred to APV and that asset beta calculations should not include tax adjustments. IV. Conclusions This paper presents the Capital Cash Flow (CCF) method of valuing risky cash flows. The CCF method is simple and intuitive. The after-tax CCFs are just the before-tax cash flows to both debt and equity, reduced by taxes including interest tax shields.The discount rate is the same expected return on assets that is used in the before-tax valuation. Because the benefit of tax deductible interest is included in the cash flows, the discount rate does not change when leverage changes. The CCF method is algebraically equivalent to the popular method of discounting FCFs by the after-tax WACC. But in many instances, the CCF method is substantially easier to apply and, as a result, is less prone to error. The ease of use occurs because the CCF method puts the interest tax shields in the cash flows and discounts by a before-tax cost of assets. The cash flow alculations can generally rely on the projected taxes, and the cost of assets does not generally change through time even when the amount of debt changes. In contrast, when applying the FCF method, taxes need to be inferred, and the cost of capital changes as the amount of debt changes. The CCF method is closely related to the APV method. APV is generally calculated as the sum of operating cash flows discounted by the cost of assets plus interest tax shields discounted at the cost of debt. The interest tax shields that are discounted by the cost of debt in the APV method are discounted by the cost of assets in the CCF method.The APV method results in a higher value than the CCF method, because it treats the interest tax shields as being less risky than the firm as a whole, because the level of debt is implicitly assumed to be a fixed dollar amount. As a result, a tax adjustment is made when unlevering 22 Financial Management • Summer 2002 an equity beta to calculate an asset beta. In contrast, the CCF method, like the FCF method, makes the more economically plausible assumption that debt is proportional to value. The risk of the interest tax shields, therefore, matches the risk of the assets.Beyond introducing the CCF method, demonstrating its conceptual equivalence to the FCF method, and showing its relation to the APV method, this paper makes the more general point that the financial policy affects the choice of valuation technique. A proportional debt policy, for example, implies that interest tax shields are valued at the cost of assets and that taxes do not affect the measure of risk that goes into calculating the discount rate. In contrast, when the amount of debt is fixed, interest tax shields are valued at the expected return of debt and taxes do affect the measure of risk.Furthermore, the debt policy need not be exclusively proportional or fixed, and, as an example, I provide a CCF valuation for a linear debt policy that has both fixed and proportional components. Whatever the debt policy, valuation depends on that policy and the challenge is to value the cash flows using an approach that consistently incorporates the assumption about debt policy. n References Arzac, E. R. , 1996, “Valuation of Highly Leveraged Firms,” Financial Analysts Journal 52, 42-49. Esty, B. C. , 1999, “Improved Techniques for Valuing Large-Scale Projects,” Journal of Project Finance 5, 9-25. Gilson, S. C. E. S. Hotchkiss, and R. S. Ruback, 2000, “Valuation of Bankrupt Firms,” Review of Financial Studies 13, 43-74. Graham, J. R. and C. R. Harvey, 2001, “The Theory and Practice of Corporate Finance: Evidence from the Field,” Journal of Financial Economics 60, 187-243. Harris, R. S. and J. J. Pringle, 1985, “Risk-Adjusted Discount Rates-Extensions from the Average-Risk Case,” Journal of Financial Research 8, 237-244. Holthausen, R. W. and M. E. Zmijewski, 1996, “Security Analysis: How to Analyze Accounting and Market Data to Value Securities,” The Wharton School and University of Chicago, Working Paper. Inselbag, I. nd H. Kaufold, 1997, “Two DCF Approaches for Valuing Companies Under Alternative Financing Strategies and How to Choose Between Them,” Journal of Applied Corporate Finance 10, 114-122. Kaplan, S. N. and R. S. Ruback, 1995, “The Valuation of Cash Flow Forecasts: An Empirical Analysis,” Journal of Finance 50, 1059-1093. Luehrman, T. A. , 1997, “Using APV: A Better Tool for Valuing Operations,” Harvard Business Review 75, 145-154. Lundholm, R. J. and T. O’Keefe, 2001, “Reconciling Value Estimates from the Discounted Cash Flow Model and the Residual Income Model,” Contemporary Accounting Research 18, 311-335.Miles, J. A. and J. R. Ezzell, 1980, “The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification,” Journal of Financial and Quantitative Analysis 15, 719-730. Miles, J. A. and J. R. Ezzell, 1985, “Reformulating Tax Shield Valuation: A Note,” Journal of Finance 40, 1485-1492. Myers, S. C. , 1974, “Interactions of Corporate Financing and Investment Decisions-Implications for Capital Budgeting,” Journal of Finance 29, 1-25. Ruback • Capital Cash Flows 23 Ruback, R. S. , 1986, “Calculating the Market Value of Riskless Cash Flows,”Journal of Financial Economics 15, 323-339. Ruback, R. S. , 1989,“Teaching Note for RJR Nabisco,” Harvard Business School, Case No. 289-057. Ruback, R. S. , 1995a, “Technical Note for Capital Cash Flow Valuation,” Harvard Business School, Case No. 295-069. Ruback, R. S. , 1995b, “Technical Note for Introduction to Cash Flow Valuation Methods,” Harvard Business School, Case No. 295-155. Taggart, R. A. , 1991, “Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes,” Financial Management 20, 8-20.

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