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Abstract: Most finance textbooks (See Benninga and Sarig, 1997, Brealey, Myers and Marcus, 1996, Copeland, Koller and Murrin, 1994, Damodaran, 1996, Gallagher and Andrew, 2000, Van Horne, 1998, Weston and Copeland, 1992) present the Weighted Average Cost of Capital WACC calculation as:
WACC = d(1-T)D% eE% (1)
Where d is the cost of debt before taxes, T is the tax rate, D% is the percentage of debt on total value, e is the cost of equity and E% is the percentage of equity on total value. All of them precise (but not with enough emphasis) that the values to calculate D% y E% are market values. Although they devote special space and thought to calculate d and e, little effort is made to the correct calculation of market values. This means that there are several points that are not sufficiently dealt with: Market values, location in time, occurrence of tax payments, WACC changes in time and the circularity in calculating WACC. The purpose of this note is to clear up these ideas and emphasize in some ideas that usually are looked over.
Also, some suggestions are presented on how to calculate, or estimate, the equity cost of capital.
Published as "Market Value Calculation and the Solution of Circularity between Value and the Weighted Average Cost of Capital".
Weighted Average Cost of Capital, WACC, firm valuation, capital budgeting, equity cost of capital
Abstract: In Velez-Pareja and Tham (2001), we presented several different ways to value cash flows. First, we apply the standard after-tax Weighted Average Cost of Capital, WACC to the free cash flow (FCF). Second, we apply the adjusted WACC to the FCF, and third we apply the WACC to the capital cash flow. In addition, we discount the cash flow to equity (CFE) with the appropriate returns to levered equity. We refer to these four ways as the "discounted cash flow (DCF)" methods. In recent years, two new approaches, the Residual Income Method (RIM) and the Economic Value Added (EVA) have become very popular. Supporters claim the RIM and EVA are superior to the DCF methods. It may be case that the RIM and EVA approaches are useful tools for assessing managerial performance and providing proper incentives. However, from a valuation point of view, the RIM and EVA are problematic because they use book values from the balance sheet. It is easy to show that under certain conditions, the results from the RIM and EVA exactly match the results from the DCF methods. Velez-Pareja 1999 reported that when using relatively complex examples and book values to calculate Economic Value Added (EVA), the results were inconsistent with Net Present Value (NPV). Tham 2001, reported consistency between the Residual Income Model (RIM) and the Discounted Cash Flow model (DCF) with a very simple example. Fernandez 2002 shows examples where there is consistency between DCF, RIM and EVA. He uses a constant value for the cost of levered equity capital and in another example constant debt. Young and O'Byrne, 2001, show simple examples for EVA but do not show the equivalence between DCF and EVA. Ehrbar (1998) uses a very simple example with perpetuities and shows the equivalence between EVA and DCF. Lundholm and O'Keefe, 2001, show this equivalence with an example with constant Ke. Tham 2001, commented on their paper. Stewart, 1999, shows the equivalence between DCF and EVA with an example using a constant discount rate. Copeland, et al, show an example with constant WACC and constant cost of equity even with varying debt and assuming a target leverage that is different to the actual leverage. In general, textbooks do not specify clearly how EVA should be used to give consistent results. In this teaching note using a complex example with varying debt, varying leverage and terminal (or continuing value), we show the consistency between DCF, RIM and EVA. We stress what Velez-Pareja 1999 and Fernandez 2002 said: for a single period, RI or EVA does not measure value. We have to include expectations and market values in the calculation of discount rates and hence values.
Economic Value Added, EVA, Market Value Added, MVA, Net Present Value, NPV, cash flows, free cash flows, market value of equity, market value of the firm
Abstract: Recently, the residual income (RI) model has become very popular in valuation because it purports to measure "value added" by explicitly taking into account the cost for capital in the income statement. Some proponents of the residual income approach have even suggested that the RI model is superior to the discounted cash flow (DCF) method and consequently, the DCF model should be abandoned in favor of the RI model. The residual income model is seductive because it purports to provide assessments of performance at any given point in time. The claim that the RI model is superior to the DCF model in valuation is puzzling because the RI model is simply an interesting algebraic rearrangement of the DCF model. Since the same information is used in both models, it is not unexpected that both models should give the same valuation results. In this paper, I examine the idea that the residual income model is superior to the discounted cash flow model. Using a simple numerical example, I show that in a M & M world, the two approaches to valuation are equivalent. In practice, the choice between the two valuation methods will be determined by the ease with which the relevant information is available.
Economic Value Added, Residual Income Model, Free Cash Flow
Abstract: In a typical market-based valuation, the standard procedure is to discount the expected free cash flow (EFCF) at the weighted average cost of capital (WACC) and the effect of financing is taken into account by adjusting the WACC. However, in many cases, it may be difficult to capture the various impacts of financing by simply adjusting the WACC. In other cases, as a component of the analysis, the construction of the cash flows to the equity holder may be necessary. In this paper, I present a simple spreadsheet model for the direct valuation of the cash flows to the equity holder without the usual simplifying assumptions. In particular, I specify the following flexible conditions: a. Multiperiod investments and reinvestments, b. Finite cash flows, with variable growth rates, c. Variable debt-equity ratios, d. Losses carried forward. With the inclusion of these conditions, the analysis is more realistic. In the model, the present value of the tax shield is discounted at rho, the required return with all-equity financing. In addition, the impacts of inflation are directly incorporated into the analysis. In particular, the model shows that the NPV of the free cash flow (including the tax savings from debt financing), discounted at the WACC is equal to the NPV of the free cash flow to the equity holder, discounted at the annually adjusted return to equity. The general approach presented here can be easily modified to take into account the varied circumstances and special complexities that are often encountered by practitioners.
Multiperiod WACC, Equity Valuation, Cost of Capital
Abstract: In the arena of valuation, the fanciful claims about the dethronement of the champion (a.k.a. NPV) by the concept of economic value added (EVA) have been greatly exaggerated, and it would be premature and unwise to abandon our reliable and trusted NPV. EVA is simply an interesting algebraic rearrangement of the standard cash flow model in terms of parameters from financial statements. In this non-technical note, I use simple numerical examples to illustrate the games that people can play with EVA. There are two major flaws with EVA. First, in year n, the equity charge for calculating the residual income is based on the book equity value at the beginning of year n, and second, the residual income profile is dependent on the schedule for accounting depreciation. Consequently, it is problematic to interpret the meaning of the economic value added in any particular year, and furthermore, it is difficult to compare two different residual income profiles because the same cash flow stream can generate multiple profiles for the residual income. The champion NPV is unfazed by the new arrival in the arena of valuation. In spite of all the hype in the media, the new arrival is simply an alter-ego.
Economic value added, residual income model, discounted cash flow
Abstract: In project finance, the viability of the project is based on the expected cash flows generated by the project rather than on the strength of the company's balance sheet. Thus, it is relevant to construct the annual cash flow from the equity point of view and estimate the annual returns to the equity holder but the usual simplifications for calculating the cost of capital do not permit the explicit estimation of the annual returns to the equity holder. In this paper, I relax many of the assumptions in the typical analysis, and provide a simple and practical way to estimate directly the annual returns to the equity holder. This approach requires the calculation of the annual present values of the future cash flows from the point of view of the equity holder. Two equivalent ways for calculating the annual equity values are shown. Most importantly, the construction of the cash flow statement from the equity point of view permits the analysis of the likely impacts of contracts on the risk profile of the project for the equity holder.
Abstract: In cash flow valuation (CFV), there are two main categories of mistakes: derivation of the appropriate cash flows and estimation of the cost of capital. A simple-minded view of the world would suggest that with near perfect capital markets, the presence of arbitrage would severely punish wrong valuations and eradicate such mistakes in the derivations of cash flows and estimations of the cost of capital. Nonetheless, to the dismay of academics, such mistakes continue to exist and thrive. It is not clear why such mistakes persist in practice. In this paper we present our list of the top nine mistakes in cash flow valuation. In the age of the computer these mistakes are both unnecessary and avoidable. In the usual triumph of hope over experience, we are attempting to persuade analysts that they would benefit from paying attention to these mistakes. Ultimately, the (un)importance of the mistakes is an empirical question and depends on the considered judgment of practitioners.
Cost of capital, WACC, valuation
Abstract: Risk-neutral valuation is simple, elegant and central in option pricing theory. However, in teaching risk-neutral valuation, it is not easy to explain the concept of 'risk-neutral' probabilities. Beginners who are new to risk-neutral valuation always have lingering doubts about the validity of the probabilities. What do the probabilities really mean? Are they real or fictional? Where do they come from? What is the relationship between the risk-neutral probabilities and the actual probabilities? Does it mean that all investors are risk-neutral? When is it appropriate to use the risk-free rate as the discount rate? From a pedagogical point of view, in the beginning it is best to avoid the use of probabilities because probabilities can be a barrier to understanding. Instead, it is far preferable to introduce the idea of state prices and then show that the approach with risk-neutral probabilities is equivalent to the use of state prices. In this teaching note, we use simple one-period examples to explain the intuitive ideas behind risk-neutral valuation. It is a gentle introduction to risk-neutral valuation, with a minimum requirement of mathematics and prior knowledge. We will provide the motivation and the rationale for calculating state prices and we will show that the risk-neutral approach is simply another way of looking at the issue of state prices.
Risk-neutral valuation
Abstract: Vélez-Pareja and Tham (2001), presented several different ways to valor cash flows. First, we apply the standard after-tax Weighted Average Cost of Capital, WACC to the free cash flow (FCF). Second, we apply the adjusted WACC to the FCF, and third we apply the WACC to the capital cash flow. In addition, we discount the cash flow to equity (FCA) with the appropriate returns to levered equity. We refer to these four ways as the "discounted cash flow (DCF)" methods. In recent years, two new approaches, the Residual Income Method (RIM) and the Economic Valor Added (EVA) have become very popular. Supporters claim the RIM and EVA are superior to the DCF methods. It may be case that the RIM and EVA approaches are useful tools for assessing managerial performance and providing proper incentives. However, from a valuation point of view, the RIM and EVA are problematic because they use book values from the balance general. We refer to these methods as valor added methods. It is easy to show that under certain conditions, the results from the RIM and EVA exactly match the results from the DCF methods. Velez-Pareja 1999 reported that when using relatively complex examples and book values to calculate Economic Valor Added (EVA), the results were inconsistent with Net Present Valor (NPV). Tham 2001, reported consistency between the Residual Income Model (RIM) and the Discounted Cash Flow model (DCF) with a very simple example. Fernandez 2002 shows examples where there is consistency between DCF, RIM and EVA. He uses a constant valor for the cost of levered equity capital and in another example constant debt. Young and O'Byrne, 2001, show simple examples for EVA but do not show the equivalence between DCF and EVA. Ehrbar (1998) uses a very simple example with perpetuities and shows the equivalence between EVA and DCF. Lundholm and O'Keefe, 2001, show this equivalence with an example with constant Ke. Tham 2001, commented on their paper. Stewart, 1999, shows the equivalence between DCF and EVA with an example using a constant discount rate. Copeland, et al, show an example with constant WACC and constant cost of equity even with varying debt and assuming a target leverage that is different to the actual leverage. In general, textbooks do not specify clearly how EVA should be used to give consistent results. In this teaching note using a complex example with varying debt, varying leverage and terminal (or continuing valor), we show the consistency between DCF, RIM and EVA. We stress what Velez-Pareja 1999 and Fernandez 2002 have said: for a single period, RI or EVA does not measure valor. We have to include expectations and market values in the calculation of discount rates and hence values.
Abstract: In the financial appraisal of a project, the cashflow statements are constructed from two points of view: the Total Investment (TI) Point of View and the Equity Point of View. One of the most important issues is the estimation of the correct financial discount rates for the two points of view. In the presence of taxes, the benefit of the tax shield from the interest deduction may be excluded or included in the free cashflow (FCF) of the project. Depending on whether the tax shield is included or excluded, the formulas for the weighted average cost of capital (WACC) will be different. In this paper, using some basic ideas of valuation from corporate finance, the estimation of the financial discount rates for cashflows in perpetuity and single-period cashflows will be illustrated with simple numerical examples.
Abstract: Velez-Pareja and Tham, 2003a, Velez-Pareja and Tham, 2003b and Tham and Velez-Pareja, 2004 showed the matching between discounted cash flow (DCF) methods and value added methods. They departed from the net operating profit less adjusted taxes NOPLAT and net income when using market values to calculate the weighted average cost of capital (WACC) and the cost of levered equity, Ke. In those previous works they assumed that the proper discount rate for the tax savings is the unlevered cost of equity, Ku. We assume the same discount rate in this paper. The previous procedures implied circularity between the cost of capital and the levered values. In this paper we show that the same firm values can be obtained using the cost of unlevered equity, Ku and the net income and the interest charges. No circularity is found using this procedure.
EVA, economic value added, cash flows, free cash flow, cash flow to equity, valuation, levered value, levered equity value, cost of levered equity, cost of unlevered equity, tax savings
Abstract: We discuss some ideas useful when forecasting financial statements that are based on historical data.
The chapter is organized as follows: First we discuss the relevance of prospective analysis for non traded firms. In a second section we a basic reviews of subjects that will be needed for forecasting financial statements. We discuss the use of plugs for financial forecasting. We show an alternate approach to avoid such popular practice. The approach we propose follows the Double Entry Principle. This principle guarantees consistent and error free financial statements. We show with a simple example how the plug works and its limitations and problems that arise when using it.
Next, the reader will find what information is needed for the forecasting of financial statements and where and how to find it. We present the procedure to identify policies that govern the ongoing of a firm such as accounts receivable and payable, inventories, dividend payout, and identify price increases and other basic variables. We also deal with the real life problem of a firm with multiple products and/or services.
We start with historical financial statements. We include inflation rates, real increases in prices and volume and policies in order to construct intermediate tables that make very easy the construction of the pro forma financial statements. We use a detailed example to illustrate the method.
We derive the cash flows that will be used in the book to value a firm. This type of models might be used by non traded firm for a permanent assessment of the value creation. Finally we show some tools to perform sensitivity analysis for financial management and analysis.
Financial statements forecasting, sensitivity analysis, cash flows, plug, financial statement balancing
Abstract: This paper presents a critical review of the conceptual issues involved in accounting for financial risk in project appraisal. It begins by examining three of the main approaches to assessing risk: the use of the probability distributions of project outcomes, such as the NPV, the use of a single risk-adjusted discount rate for the life of the project, and the use of certainty equivalents. The first two approaches are very common, while the third is used less often. Next, it proposes an approach based on annual "certainty equivalents" that is conceptually similar to using multiple risk-adjusted discount rates and which involves specifying the risk profile of a project over its lifetime. Finally, this approach is illustrated with a simple numerical example. The certainty equivalent approach is compelling because it clearly separates the time value of money from the issue of risk valuation. While the authors point out the analytical challenges of the certainty equivalent approach, they note that its informational requirements are no greater than those posed by the older, more traditional approaches, while avoiding the numerous inadequacies of the latter.
Risk analysis; Discount rates; Criteria for assessing risk; Certainty equivalents; Multiple risk-adjusted discount rates
Abstract: Recently, Lundholm and O'Keefe (2000) identified the estimation of the WACC as an important reason for the discrepancy between the value estimates obtained from the Discounted Cash Flow (DCF) and Residual Income (RI) models. In this paper, I discuss how we can obtain consistent value estimates from the two models in M & M worlds without and with taxes. It is common to assume that the return to unlevered equity is constant. Additionally, in practice, one assumes that the return to levered equity is constant although the debt-equity ratio is changing. In M & M worlds without and with taxes, it would be inconsistent to assume that the returns to unlevered and levered equity are constant and the debt-equity ratio is changing.
Cost of capital, weighted average cost of capital (WACC), cash flow to equity, residual income model, discounted cash flow model, valuation, return to equity
Abstract: In a recent paper, Ruback (2000) assumes that the discount rate for the tax shield in the Adjusted Present Value (APV) approach is the cost of debt and shows that the Capital Cash Flow (CCF) method and the Adjusted Present Value (APV) approach give different answers for the levered value. In this paper, using a simple numerical example with the Miles and Ezzell formulation for the tax shield, we show that the Free Cash Flow (FCF) method, the Capital Cash Flow (CCF) method and the APV approach give the same answers. The FCF method captures the benefit of the tax shield by "lowering" the cost of capital, the CCF method adds the tax shield directly to the FCF and the APV approach calculates the value of the tax shield separately. We organize the paper as follows. In Section One, we briefly review the assumptions underlying the three methods. In Section Two, we use a simple five-period binomial process to represent a finite stream of free cash flow and calculate the unlevered value with the risk-neutral method. In Section Three, we introduce risk-free debt financing with risk-free tax shields, and calculate the levered value with the three ways, the Free Cash Flow (FCF) method, the Capital Cash Flow (CCF) method and the APV approach. In addition, we verify that the APV approach is consistent with the answer from the risk-neutral method.
WACC, Free cash flow, Capital cash flow, Adjusted present value, Risk-neutral valuation
Abstract: Unquestionably, before the advent of the personal computer, modeling the impacts of inflation in investment appraisal was an enormous task. Currently, with the widespread availability of personal computers, conducting investment appraisal by constructing financial statements with nominal prices is a straightforward and simple task. In this paper, we would like to persuade the reader (if indeed there is need for persuasion) that conducting investment appraisal based on financial statements with real prices is potentially misleading and under certain circumstances, the adverse effects of inflation could result in the selection of 'bad' projects. The paper is organized as follows. In Section One, we discuss some of the apparent reasons why the real prices approach persists in investment appraisal. In Section Two, we review briefly some of the main impacts of inflation and use simple numerical examples to illustrate the ideas. In Section Three, we combine all of the previous examples into a single numerical example and use sensitivity and scenario analyses to examine the impacts of inflation on the NPV of the project. First, we conduct a simple sensitivity analysis of the NPV of the project with the expected inflation rate. Second, we conduct a detailed sensitivity analysis of the PV of each line item in the FCF statement and identify the specific effects of inflation. We show clearly why the results from the real prices approach are incorrect and explain the reasons for the inadequacy of the real prices method. Note that the sensitivity analysis is unrealistic because it assumes that the same inflation rate will occur for all the years. In Section Four, we redo the analysis with different scenarios for the expected inflation rates. Scenario analysis is extremely flexible. For example, for one scenario, we can specify that the expected inflation rate is 8% for two years and 10% for the next three years.
Project evaluation; Impacts of inflation
Abstract: Debt is rarely risk-free. Yet, on grounds of simplicity, in most discussions on the weighted average cost of capital (WACC), we assume that the debt is risk-free. At the same, in the calculation of the WACC, we may use a value for the cost of debt d that is higher than the risk-free rate rf. In this teaching note, using simple binomial models, we examine the weighted average cost of capital (WACC) with risky debt and no taxes. Taxes raise additional complications. In a subsequent note, we analyze the case with taxes. With risky debt, we have to use the expected rate of return on the debt rather than the promised rate of return on the debt in the formula for the WACC. Furthermore, we model the expected cost of risky debt as an increasing function of the amount of debt.
multi-period WACC, cost of capital, risky debt
Abstract: In the recent writings on valuation, there is no consensus about the correct formulas for calculating the relevant cost of capital in an M & M world. The proliferation of alpha number of methods and omega number of theories for the calculation of the cost of capital is puzzling because in the derivation of the original M & M result, the use of the no-arbitrage argument would suggest that there should only be a single result. In this non-technical, introductory teaching note, we would like to present a simple, general and unified approach to valuation in an M & M world. The Holy Grail in the Quest for Value (with alpha methods and omega theories) is rho, the required return to unlevered equity.
Cost-Benefit Analysis, Capital Budgeting, Project evaluation
Abstract: Using no-arbitrage arguments in an M & M world, we show that in the N-period case, the appropriate discount rate for the tax shield is rho, the return to unlevered equity. We make no assumption about the appropriate discount rate for the tax shield. Instead, the appropriate discount rate for the tax shield is deduced from the no-arbitrage arguments. Furthermore, it is shown that the appropriate discount rate for the tax shield does not depend on whether the value of the debt is a fixed amount or is a fixed proportion of the levered value of the firm. The analysis begins at the end of the penultimate period N-1. First, we assume that the value of the levered cash flow is higher than the sum of the value of the unlevered cash flow and the value of the tax shield. It is shown that the inequality cannot hold because arbitrage opportunities exist. The equality only holds if the discount rate for the tax shield is rho, the return to unlevered equity. Second, we assume that the value of the levered cash flow is lower than the sum of the value of the unlevered cash flow and the value of the tax shield. Again, it is shown that the inequality cannot hold because arbitrage opportunities exist. The equality only holds if the discount rate for the tax shield is rho, the return to unlevered equity. Using an iterative process, the argument can be extended period by period backwards to period zero. In conclusion, in the N-period case, the appropriate discount rate for the tax shield is rho, the return to unlevered equity.
Value of Tax Shield
Abstract: If the forecast period is short, then the specification of the assumption for the calculation of the terminal may be an important element of the valuation exercise. To be specific, with respect to the reference year 0, the (present) value of the terminal value may be more than fifty percent of the total levered value. In this teaching note, we present a numerical example that values consistently a finite stream of cash flows with a terminal value from three different points of view: the Adjusted Present Value (APV) approach, the Capital Cash Flow (CCF) method and the traditional after-tax Weighted Average Cost of Capital that is applied to the Free Cash Flow (FCF). We assume an M & M world.
Abstract: For the practitioner, making sense of the bewildering number of theories on the cost of capital must be a truly challenging and daunting task. In a perfect world without taxes, the cost of capital formula for a finite stream of free cash flows, with debt and equity financing, is elegant, simple and eminently sensible. The cost of capital is a weighted average of the cost of debt and the cost of equity, where the weights are the market values of debt and equity as percentages of the levered market value. In a perfect world with taxes, complications abound. What criteria should we use to select the best expression for the cost of capital from all the available formulations? Fundamentally, the cost of capital is a question about how to properly account for the tax benefits (if any) from the interest deduction with debt financing. In other words, what are the appropriate risk-adjusted discount rates for the tax shield? At last count, there were 23 theories! In this note, we briefly describe two methods for estimating the cost of capital: the traditional after-tax WACC applied to the free cash flow (FCF) and the alternative WACC applied to the capital cash flow (CCF). Using three criteria, simplicity, flexibility and correctness, we assess the strengths and weaknesses of the two different methods for calculating the cost of capital. Based on these criteria, we select the best expression for the cost of capital for a finite stream of cash flows.
WACC; Cost of capital; Free cash flow (FCF); Capital cash flow (CCF); Cash flow to equity (CFE)
Abstract: There are many different ways to calculate the Weighted Average Cost of Capital (WACC) and for the beginner the plethora of possibilities may be very confusing. We present a general framework for classifying the WACCs that are applied to the FCF and the CCF. For the moment, we avoid complexities. To facilitate the discussion, we classify the menagerie of WACCs along three dimensions. We hope that the structured framework assists the reader in making the correct decision with respect to the calculation of the cost of capital in practice. First, we present a qualitative discussion on the dimensions of the framework. Second, we specify the appropriate formulas and calculations for the cells in the framework.
At the outset, it is important to stress that this paper is concerned only with finite cash flows. In our judgment, it is best to discuss expressions for the cost of capital that are relevant to finite cash flows and this needs no further justification. In valuation, the continued use of cost of capital formulas (derived from cash flows in perpetuity), which are inappropriate for finite cash flows, is inexplicable and incomprehensible.
From a practical point of view, free cash flows are derived from financial statements, which are not constructed in perpetuity. At best, expressions for the cost of capital that are derived from cash flows in perpetuity may be reasonable approximations for finite cash flows. At worst, the results could be misleading.
multi-period WACC, cost of capital
Abstract: In a recent paper, Loeffler (2001) showed that the Miles & Ezzell (M & E) WACC allows arbitrage if the cash flow process does not have a "certain growth rate". To be specific, for a particular period, the set of up and down coefficients must be the same at all the nodes in a binomial cash flow process. In this teaching note, we use simple three period numerical examples to illustrate the calculations of the WACC for cash flow processes that satisfy and violate the growth rate assumption. The note is organized as follows. In Section One, we present a simple three period binomial cash flow process S with multiplicative coefficients that satisfy the growth rate assumption. We calculate the unlevered value with two equivalent methods. One approach uses the risk-neutral probabilities and the risk-free rate. The other approach uses the objective probabilities that are consistent with the risk-neutral probabilities and the return to unlevered equity. In Section Two, we introduce debt and calculate the levered values with the same two methods. Since process S satisfies the growth rate assumption, we verify that the WACCME does not allow arbitrage. In Section Three, with a simple modification, we create a new binomial cash flow process that violates the growth rate assumption and show that the WACCME allows arbitrage.
Risk-neutral valuation, WACC Theory
Abstract: In this note, the present value of the tax shield is reconsidered. In most corporate finance textbooks, it is commonly assumed that the discount rate for the tax shield is d, the risk-free rate for debt. Here, it is shown that the correct discount rate for the tax shield is rho, the required rate of return with all equity financing.
Abstract: In the traditional formulation of the WACC, the debt and the tax shield are risk-free. However, even though the debt is risk-free, the tax shield can be risky. Furthermore, both the debt and the tax shield can be risky. In this paper, we present the non-conventional weighted average cost of capital (WACC) for the single period binomial process with risky debt and risky tax shield, and derive the relevant formulas for the returns to the levered equity holder and the debt holder. Risk-free debt does not imply that the tax shield is risk-free. Unlike the traditional formulation, with the non-conventional WACC, the discount rate for the tax shield is not be restricted to two values, the risk-free rate rf and the return to unlevered equity rho. We make no prior assumption about the value of the discount rate for the tax shield psi. The value of psi depends on the riskiness of the tax shield. If the tax shield is risk-free, then the value of psi is equal to the risk-free rate rf. If the tax shield is risky, then the value of psi is simply higher than the risk-free rate rf. In particular, if the payoff structure of the tax shield is the same as the payoff structure for the levered equity holder, the value of psi is equal to the return to the levered equity holder. In Section One, we present the non-conventional WACC for the single period with the capital cash flow (CCF) rather than the free cash flow (FCF) and examine the traditional position, where both the debt and the tax shield are risk-free. In Section Two, we assume that both the debt and the tax shield are risky.
WACC theory
Abstract: Velez-Pareja and Tham (2003) presented a method to match the value added approaches (Residual Income Method, RIM and Economic Valor Added, EVA) with the discounted cash flow, DCF methods. There they used a relatively complex example, but yet, far away from reality. In this note we use a real life case from an emerging country to illustrate the same procedure, but with additional and real life complexities such as unpaid taxes, losses carried forward, foreign exchange debt, presumptive income and inflation adjustments to the financial statements. In all methods we use market values to calculate the discount rates.
We stress what Velez-Pareja 1999 and Fernandez 2002 have said: for a single period, RI or EVA does not measure valor. We have to include expectations and market values in the calculation of discount rates and hence values.
Economic Value Added, EVA, Market Value Added, MVA, residual income model, utilidad, economica, valor presente neto (VPN), flujos de cja, flujos de caja libre, valor de Mercado del patrimonio, valor de la firma, perdidas amortizadas, losses carried forward, perdida en cambio, deuda en moneda etanjera, foreign exchange loss, foreign exchange debt, renta presuntiva, presumptive income, ajustes por inflacion a los estados financieros, inflation adjustments to the financial statements
Abstract: Although perpetuities are somewhat artificial in the sense that in practice they do not exist, they are relevant because no matter how detailed and complex a forecasted financial plan for a firm or project could be terminal value usually is calculated as perpetuity. This terminal value might be a growing or a non growing perpetuity. On the other hand, usually terminal value is a substantial part of the firm value. In this note we examine in detail the proper discount rate for cash flows in perpetuity, the present value of tax savings and the calculation of terminal value. The findings contradict what is generally accepted in the literature.
WACC, perpetuities, terminal value, tax savings
Abstract: The typical assumption about cashflows in perpetuity is not appropriate in practical project appraisal because the length of project life is always finite. In this paper, I discuss the calculation of multiperiod financial discount rates for a project with a finite life. The impact of taxes and inflation will also be included in the analysis. First, we may assume that a constant debt-equity ratio is maintained during the life of the project. The loan schedule is constructed to keep the debt-equity ratio constant for the life of the project. Second, the loan schedule may be fixed. In this case, the debt-equity ratio changes over the life of the project. By explicitly calculating the appropriate discount rate for each period, it is not necessary to assume that the debt-equity ratio is constant and the cashflows are in perpetuity.
Abstract: Recently, the residual income (RI) model has become very popular in valuation because it purports to measure "value added" by explicitly taking into account the cost for capital in the income statement. Some proponents of the residual income approach have even suggested that the RI model is superior to the discount cash flow (DCF) method and consequently, the DCF model should be abandoned in favor of the RI model. The residual income model is seductive because it purports to provide assessments of performance at any given point in time. The claim that the RI model is superior to the DCF model in valuation is puzzling because the RI model is simply an interesting algebraic rearrangement of the DCF model. Since the same information is used in both models, it is not unexpected that both models should give the same valuation results. In this paper, I examine the idea that the residual income model is superior to the discounted cash flow model. Using a simple numerical example, I show that in a M & M world, the two approaches to valuation are equivalent. In practice, the choice between the two valuation methods will be determined by the ease with which the relevant information is available.
Economic Value Added, Residual Income Model, Cash Flow Model
Abstract: In the latest edition of Principles of Corporate Finance (Brealey, Myers and Allen, 2006) the authors use a finite cash flow example to illustrate the valuation procedure for using the Discounted Cash Flow (DCF) method with the free cash flow (FCF) and the Adjusted Present Value (APV). The two firm values obtained are different. They say that the "... difference [...] is not a big deal considering all the lurking risks and pitfalls in forecasting [...] cash flows". In this teaching note we show that the two methods give identical values when the proper discount rates are used.
Cash flows, free cash flow, cash flow to equity, valuation, levered value, Adjusted Present Value, APV, Discounted Cash Flow, DCF, weighted average cost of capital, WACC, cost of unlevered equity, tax savings
Abstract: This teaching note is a continuation of the previous teaching note on risk-neutral valuation. In Section One, we estimate the value of levered equity in a levered company in an M & M world with risk-free debt and without taxes. The structure of the presentation will facilitate the subsequent analysis with taxes. We find the value of the levered firm in two ways. First, we use the replicating portfolio method. Second, we use the risk-neutral approach, which is equivalent to the replicating portfolio method. In Section Two, we analyze risky debt in a world without taxes. In Section Three, we introduce taxes. However, we continue to assume risk-free debt. The risk of the tax shield is indeterminate. In Section Four, we analyze risky debt in the presence of taxes and derive the relevant expressions for the returns to the equity and debt holders.
Risk-neutral valuation, Cost of Capital
Abstract: The objective of this teaching note is to present the basic principles of discrete option pricing in simple language, with detailed jargon-free explanation and rudimentary algebra. Simple numerical examples are used to illustrate the main ideas. The basic ideas for call and put options are explained in the familiar context of dealings between a coffee shop owner, a coffee dealer and a coffee farmer. At the beginning, the reader is informed that the actual probabilities of the future states of nature are irrelevant for calculating the values of the options. First, the valuation formulas are derived by replicating the payoff structure for the options with bags of coffee and cash. Second, the valuation formulas are derived with the construction of the risk-free hedge portfolio with bags of coffee and number of options. It is hoped that this introductory exposition of discrete option pricing will facilitate the reader's engagement with other materials in option pricing, both discrete and continuous.
Discrete Option Pricing, Risk-neutral valuation
Abstract: Researchers continue to "horse race" the Residual Income (RI) model and the Cash Flow (CF) model, with no regard for the underlying assumptions. Recently, Lundholm and O'Keefe (2000) asserted that they have identified an important reason for the discrepancy between the results obtained from the two models, the so-called "incorrect discount rate error". In their paper, it is unclear whether this error with the calculation of the weighted average cost of capital (WACC) can occur in a M & M world because the assumption about a M & M world is not explicitly stated. In other words, they are agnostic about the correspondence between the M & M world and reality. In this note, I illustrate the use of the WACC in a M & M world and show that the error that has been identified by Lundholm and O'Keefe cannot occur in a M & M world. Thus, it is reasonable to conclude that perhaps the error with the calculation of the WACC only occurs in a non-M & M world, whatever that may be. In addition, I show the estimation of the residual income and CF models with variable returns to levered equity.
Multi-period WACC, Residual Income (RI) model, Cash Flow to Equity (CFE), Cash Flow (CF) model
Abstract: Many firms have debt financing in a foreign currency. What are the tax implications of the foreign loan for the calculation of the Weighted Average Cost of Capital (WACC)? With a foreign loan, there are two effects. First, there is the standard tax savings from the interest deduction with the foreign loan. Second, we assume that changes in the value of the loan in the foreign currency can be listed in the income statement for tax purposes. In this teaching note, we examine how the WACC must be properly used to take into account both of the effects: the interest deduction and the change in the value of the foreign debt.
WACC, foreign loan, cost of debt
Abstract: In this note we analyze the tutorial based on the McKinsey methodology for valuing companies. We have found that the McKinsey methodology has one of the most common mistakes mentioned in Tham and Vélez-Pareja (2004a and b): valuing cash flows with a constant cost of capital when the leverage is not constant. We calculate the proper firm and equity values by assuming the free cash flow, FCF calculated in the tutorial, and deriving the cash flow to equity, CFE. We develop the proper calculations of firm and equity values for constant leverage as well. For both calculations we calculate the deviations from the values calculated in the tutorial.
Valuation, free cash flow, discounting, accounting data
Abstract: This is a continuation of the simplified exposition of Discrete Option Pricing. In this teaching note, I discuss some of the determinants of the values for the call and put options. Second, I derive the put-call parity relationship. Third, as a practical application, the ideas of put and call options are used to value the position of the equity holder in a levered firm. Finally, I extend the discrete option pricing model to two and three periods.
Discrete Option Pricing, Risk Neutral Valuation
Abstract: The typical assumption about cashflows in perpetuity is not appropriate in practical project appraisal because the length of project life is always finite. In this paper, I discuss the calculation of multiperiod financial discount rates for a project with a finite life. For simplicity, I assume that there are no taxes. The analysis is conducted without and with inflation. There are two possibilities for calculating the discount rates. First, we may assume that a constant debt-equity ratio is maintained during the life of the project. In this case, the loan schedule is constructed to keep the debt-equity ratio constant for the life of the project. Since the debt-equity ratio is constant, the Weighted Average Cost of Capital (WACC) and the return to equity e will be constant. Second, the loan schedule may be fixed for the life of the project. In this case, as the loan is repaid, the debt-equity ratio changes because the principal is being repaid over the life of the project. In each period, the return to equity e will be a function of the debt-equity ratio for the period. The return to equity e varies with the change in the debt-equity ratios. The return to equity e will be adjusted to keep the WACC constant. By explicitly calculating the appropriate discount rate for each period of the project with a finite life, it is not necessary to assume that the debt-equity ratio is constant and the cashflows are in perpetuity.
Abstract: Vélez-Pareja and Tham, 2003a, Vélez-Pareja and Tham, 2003b and Tham and Vélez-Pareja, 2004 showed the matching between discounted cash flow (DCF) methods and value added methods. They departed from the net operating profit less adjusted taxes NOPLAT and net income when using market values to calculate the weighted average cost of capital (WACC) and the cost of levered equity, Ke. In those previous works they assumed that the proper discount rate for the tax savings is the unlevered cost of equity, Ku. We assume the same discount rate in this paper. The previous procedures implied circularity between the cost of capital and the levered values. In this paper we show that the same firm values can be obtained using the cost of unlevered equity, Ku and the net income and the interest charges. No circularity is found using this procedure. This article is based upon Vélez-Pareja and Tham 2004.
Abstract: In the standard construction of the free cash flow (FCF) in the M & M world without taxes, it is assumed that ALL of the generated cash flow is distributed to the debt holder and the equity holder, and there are no surplus funds that are invested in short-term marketable securities. Under these conditions, the FCF is equal to the capital cash flow (CCF), which in turn is equal to the sum of the cash flow to equity (CFE) and the cash flow to debt (CFD). However, the firm may have surplus funds that are not returned to the equity holder and these surplus funds are invested in short-term marketable securities. The availability of surplus funds that are invested in short-term marketable securities creates an ambiguity in the correct interpretation of the FCF.
free cash flow, investment of surplus funds
Abstract: It is a well known problem the interactions between the market value of cash flows and the discount rate (usually the weighted average cost of capital, WACC) to calculate that value. This is mentioned in almost all textbooks in corporate finance. However, the solution adopted by most authors is to assume a constant leverage D%, and hence assume that the leverage gives raise to an optimal capital structure and the discount rate is constant. On the other hand, most authors use the definition of the Ke, the cost of leveraged equity for perpetuities even if the planning horizon is finite. Among these authors we find the work of Wood and Leitch W&L 2004. In this paper we wish to analyze the claim made by W&L 2004 in the sense to have found an iterative solution to the problem of circularity that results in a near matching with the Adjusted Present Value APV, proposed by Myers, 1974. They use as the basic principle the fact that there is a near constant relation between Ke the cost of equity and Kd the cost of debt. They consider as well that the cost of debt Kd is not constant and changes proportionately with the leverage D%. We propose a very simple and precise approach to solve the above mentioned circularity problem.
Adjusted Present Value, APV, weighted average cost of capital, circularity problem, discounted cash flow, DCF equity value, cost of equity
Abstract: Using illustrative examples, this paper shows that the Net Present Value for project evaluation should be based on estimates of free cash flows at nominal prices. It is a widespread practice to evaluate projects at constant or real prices. These days, the use of constant or real prices is an unnecessary oversimplification. In particular, we present an example where the results from the constant and real price methodologies are biased upwards and there is a risk that in practice, bad projects will be accepted as good projects.
Project evaluation, engineering economy, NPV, Net Present Value, inflation, constant price methodology, current price methodology, nominal price methodology, free cash flows, pro-forma financial statements, relative prices
Abstract: In this teaching note, we present an integrated, consistent market-based framework for valuing finite cash flows. We derive the relevant cash flows from integrated financial statements, and based on Modigliani and Miller's (M & M) theories, we estimate the appropriate cost of capital and value the cash flows in seven different ways. The first five methods are variations of the Discounted Cash Flow (DCF) method.
The last two methods, the RIM and EVA, are interesting because they differ from the DCF methods. In particular, they apply a charge for equity (based on the book value of equity) to the net income or a charge for invested capital (based on the book value of invested capital) to the Net Operating Profit after tax (NOPLAT), roughly speaking. Happily, the results from the DCF methods are fully consistent with the RIM and EVA.
Since the results from the seven methods must always match, calculating the (present) values with the seven methods is a powerful check on the consistency of the valuation exercise. With the availability of computing resources, it is easy to implement the seven methods on a routine basis.
In Principles of Cash Flow Valuation, 2004, Academic Press we present and explain the valuation methods in detail.
Multiperiod WACC, firm valuation, levered equity, valuation with EVA, cost of equity
Abstract: In a forthcoming paper, Fernandez (2002) claims to derive a formula for the valuation of debt tax shields for firms with cash flows that grow perpetually at a constant rate. We show that his formula is incorrect and provide an example where his valuation would admit arbitrage.
Present value of tax shield, perpetuities
Abstract: Fernandez (2004) claims to derive a formula for the valuation of debt tax shields for firms with cash flows that grow perpetually at a constant rate. We show that his formula is incorrect.
Abstract: In a recent paper, Pablo Fernandez (2002) makes the unusual and paradoxical sounding claim that for cash flows in perpetuity with a constant growth rate g, the value of the tax shields VTS is NOT equal to the present value of the tax shields. To be specific, Fernandez purportedly shows that the formula for the present value of the tax shields is as follows. VTS = TDKu/(Ku - g) Where Ku is the return to unlevered equity, g is the constant growth rate, T is the tax rate and D is the market value of debt. Fernandez (2002) asserts that the value of the tax shield, as given in equation, should be properly interpreted as the difference in the taxes paid by the unlevered and levered firms, where the taxes have different risk profiles. Let VTxU be the present value of the taxes paid by the unlevered firm, discounted by KTxU, which is the appropriate risk-adjusted discount, and let VTxL be the present value of the taxes paid by the levered firm, discounted by KTxL, which is the appropriate risk-adjusted discount. In this note, we assess the validity of the proposed expression for the value of the tax shield. The note is organized is as follows. In Section One, we review and discuss the assumptions underlying the model that Fernandez uses to derive equation 1. In Section Two, we examine critically the derivation of equation 1 and its general relevance and applicability.
Present value of tax shield
Abstract: It is widely accepted that the correct discount rate for the tax shield depends on whether the value of the debt is a fixed amount or is a proportion of the value of the firm. In this pedagogical note, using a simple two period numerical example, I assume a fixed amount of debt and demonstrate that if the cost of debt is used to discount the tax shield to determine the value of the levered firm, arbitrage opportunities exist. I propose that it is always correct to use rho, the return on unlevered equity, to discount the tax shield. Consequently, the correct discount rate for the tax shield does not depend on whether the value of the debt is a fixed amount or is a proportion of the value of the firm. It is not difficult to extend the analysis to the multiperiod case. Happily, all the inconsistencies are resolved if rho, the return on unlevered equity, is used to discount the tax shield.
Valuation, WACC, Cost of Capital, Tax shield
Abstract: It is widely known that if the leverage is constant over time, then the after-tax Weighted Average Cost of Capital (WACC) is constant over time. In other words, it is inappropriate to use a constant after-tax WACC to discount the free cash flow (FCF) if the leverage changes over time. However, it is common to find analysts who inconsistently use a constant after-tax WACC even if the leverage is not constant. In this teaching note, we use a simple numerical example to illustrate how to model cash flows that are consistent with constant leverage. We verify the consistency of the example with two basic principles: conservation of cash flows and conservation of values.
WACC, constant leverage, cash flows
Abstract: The discount rate for the tax shield depends on the risk of the tax shield. If the tax shield is risk-free, then the appropriate discount rate for the tax shield is the risk-free rate rf. If the debt is risky, then we must make the distinction between the contractual return and the expected return on the debt. In this paper, using a simple numerical example, we illustrate the calculation of the present value of the tax shield (PVTS) for a free cash flow (FCF) in perpetuity with a constant growth rate g. We assume that the tax shield is risk-free and the debt is risky. Most importantly, we model explicitly the risk of the tax shield and the debt with a stochastic process. In addition, the net incomes for the unlevered and levered firms are not equal to the corresponding cash flows for the unlevered and levered firms. Consequently, the discount rates for the taxes paid by the unlevered and levered firms are not equal to the return to unlevered equity Ku and the return to levered equity Ke, respectively. Without a specific stochastic process, it would not be possible to calculate the discount rates for the taxes paid by the unlevered and levered firms.
Present value of the tax shield, risk neutral valuation
Abstract: In the standard Weighted Average Cost of Capital (WACC) applied to the free cash flow (FCF), we assume that the cost of debt is the market, unsubsidized rate. With debt at the market rate and perfect capital markets, debt only creates value in the presence of taxes through the tax shield. In some cases, the firm may be able to obtain a loan at a rate that is below the market rate. In a previous work, we showed how to adjust the WACC in the presence of a subsidy and no taxes. There, we showed that plugging the lower (subsidized) cost of debt into the WACC formula is not the correct approach to measuring the value creation due to the subsidy. With subsidized debt and taxes, there would be a benefit to debt financing, and the unleveraged and leveraged values of the cash flows would be unequal. The benefit of lower tax savings are offset by the benefit of the subsidy. These two benefits have to be introduced explicitly. How would we adjust the WACC to take account of the subsidized debt? And how would we adjust the expression for the required return to leveraged equity?
In this paper, using a multiple period example we present the adjustments to the WACC with subsidized debt and taxes. We demonstrate the analysis for both the WACC applied to the FCF and the WACC applied to the capital cash flow (CCF). We use the calculation of the Adjusted Present Value, APV, to consider both, the tax savings and the subsidy. We show how all the methods match.
Adjusted Present Value, APV, weighted average cost of capital, discounted cash flow, DCF equity value, cost of equity, WACC, subsidized debt with taxes, valuation of cash flows, project evaluation, project appraisal, firm valuation, cost of capital, cash flows, free cash flow, capital cash flow
Abstract: For cash flows in perpetuity without growth, analysts typically use the following formula for the return to levered equity Ke. Ke = Ku + (Ku Kd)(1 T)D/E (1) where Ku is the return to unlevered equity, Kd is the cost of debt, T is the tax rate, D is the market value of debt and E is the market value of equity. What is the corresponding formula for finite cash flows? Is it the same as equation 1? In other words, is equation 1 appropriate for both finite and infinite cash flows? One may be tempted to believe that equation 1 is the general formulation for the return to levered equity and applies to both cash flows in perpetuity and finite cash flows. However, this conclusion is misleading. In this short note, using simple algebra, we derive the general formulation for the return to levered equity for finite cash flows, and show that equation 1 is not the general formulation for finite cash flows.
Present value of the tax shield, formulation for Ke, cost of levered equity, cash flows, free cash flow, cash flow to equity, valuation, levered value, levered equity value, terminal value, cost of levered equity, cost of unlevered equity, tax savings, growth rate for the free cash flow
Abstract: In the standard Weighted Average Cost of Capital (WACC) applied to the free cash flow (FCF), we assume that the cost of debt is the market, unsubsidized rate. With debt at the market rate and perfect capital markets, debt only creates value in the presence of taxes through the tax shield. In some cases, the firm may be able to obtain a loan at a rate that is below the market rate. With subsidized debt and no taxes, there would be a benefit to debt financing, and the unlevered and levered values of the cash flows would be unequal. How would we adjust the WACC to take account of the subsidized debt? And how would we adjust the expression for the required return to levered equity? In this paper, using a single period example we present the adjustments to the WACC with subsidized debt. We demonstrate the analysis for both the WACC applied to the FCF and the WACC applied to the capital cash flow (CCF). For simplicity, we assume that there are no taxes. The analysis can be extended easily to multiple periods in the presence of taxes.
WACC, cost of capital, subsidized debt, valuation of cash flows, project evaluation, project appraisal, firm valuation, cost of capital, cash flows, free cash flow, capital cash flow
Abstract: In cash flow valuation, on grounds of simplicity, it is common to assume that the leverage is constant over time. With constant leverage, the return to levered equity is constant and consequently, the Weighted Average Cost of Capital (WACC) applied to the Free Cash Flow is constant. However, typically the constant leverage is not reflected in the financial statements. Specifically, the values of the annual debt (as listed in the balance sheet) as percentages of the annual levered values are not constant. The Hershey case study in the popular book on valuation by Copeland et al. (2nd edition, 1995) is a good illustration of this common and widespread inconsistency. Distressingly, readers may not realize or recognize the inconsistency between the cost of capital and the financial statements and authors of textbooks make no attempt to mention it. The consistency between the leverage assumption in the WACC applied to the FCF and the values for the debt in the balance sheet can be resolved if the debt is rebalanced each year to maintain the constant leverage. In this paper, we demonstrate the inconsistency. First, we calculate the annual leverage and show that it is not constant. Second, we calculate the annual equity by subtracting the annual debt values from the annual levered values and demonstrate the discrepancies with the present value of the CFE. This paper is aimed to those who have learnt valuation with that edition (1995).
Cash flows, Copeland, Hershey, free cash flow, cash flow to equity, valuation, levered value, levered equity value, terminal value, cost of levered equity, cost of unlevered equity, tax savings
Abstract: In Consistency in Chocolate: A Fresh Look at Copeland's Hershey Foods & Co Case we showed the inconsistencies regarding the assumption of constant leverage and the inconsistency in the values for equity calculated with different approaches. In this second part we show the differences in the calculated values using an approach consistent with the assumptions implicit in the calculation of Copeland et al. (1995)'s Hershey example. In particular, we show the calculation of the levered value for the firm using the proper calculations for WACC and cost of levered equity assuming that the discount rate for the tax savings is Kd, the cost of debt and using finite cash flows. In this paper, we use the terminal value calculated in the original example. We also calculate the levered values assuming that the discount rate for the tax savings is Ku, the cost of the unlevered equity and using finite cash flows. We calculate the differences in values and show the consistency of our approach regarding the calculated values for equity. This paper is aimed to those who have learnt valuation with that edition (1995).
Cash flows, free cash flow, cash flow to equity, valuation, levered value, levered equity value, terminal value, cost of levered equity, cost of unlevered equity, tax savings, cash flows, Copeland, Hershey
Abstract: It is widely known that if the leverage is constant over time, then the cost of equity and the Weighted Average Cost of Capital (WACC) for the free cash flow, FCF, is constant over time. In other words, it is inappropriate to use a constant WACCFCF to discount the free cash flow (FCF) if the leverage changes over time and some conditions are not satisfied. However, it is common to find analysts who inconsistently use a constant WACCFCF even if the leverage is not constant and the proper conditions are not satisfied. In this teaching note, we use a simple numerical example to illustrate how to model cash flows that are consistent with constant leverage. We verify the consistency of the example with two basic principles: conservation of cash flows and conservation of values. The note is based on a previous one and includes the procedure to value with constant leverage when some restrictive conditions are not satisfied.
Abstract: In the financial appraisal of a project, the cashflow statements are constructed from two points of view: The Total Investment (TI) Point of View and Equity Point of View. One of the most important issues is the estimation of the correct financial discount rates for the two points of view. In the presence of taxes, the benefit of the tax shield from the interest deduction may be excluded or included in the free cashflow (FCF) of the project. Depending on whether the tax shield is included or excluded, the formulas for the weighted average cost of capital (WACC) will be different. In this paper, using some basic ideas of valuation from corporate finance, the estimation of financial discount rates for cashflow in perpetuity and single-period cashflows will be illustrated with simple numerical examples.
cost-benefit analysis, capital budgeting, project evaluation
Abstract: In this note, we show that with respect to the Miles and Ezzell (M&E) Weighted Average Cost of Capital (WACC), the return to levered equity for finite cash flows is constant if the debt-equity ratio is constant. We assume that the reader is familiar with the M&E WACC. The expression that we derive is not new. We hope that our straightforward derivation with simple algebra makes the M&E WACC more widely known.
M&E WACC, tax shields
Abstract: In this teaching note we show that using the findings of Tham and Velez-Pareja 2002, for finite cash flows, Ke and hence WACC depend on the discount rate that is used to value the tax shield, TS and as expected, Ke and WACC are not constant with Kd as the discount rate for the tax shield, even if the leverage is constant. We illustrate this situation with a simple example. We analyze five methods: DCF using APV, FCF and traditional and general formulation for WACC, present value of CFE plus debt and Capital Cash Flow, CCF.
In Tham and Velez-Pareja 2002, they derive a general expression for Ke, the cost of levered equity and for the Weighted Average Cost of Capital (WACC) applied to the Free Cash Flow (FCF) and Capital Cash Flow (CCF). For finite cash flows and perpetuities, the derivation presents the analysis for different levels of risk with respect to discounting the tax shields (TS). Taggart 1991 presents a revision of the set of formulations for the cost of levered Ke and WACC. He introduces the formulation with and without personal taxes and for different level of risk for discounting the TS, including the proposal by Miles and Ezzel 1980. However, Taggart does not include the case of Kd, the cost of debt as the level of risk for the TS and finite cash flows.
A typical approach for valuing finite cash flows is to assume that leverage is constant (usually as target leverage) and the Ke and WACC are also assumed to be constant. For cash flows in perpetuity, and with Kd as the discount rate for the tax shield, it is indeed the case that the Ke and WACC applied to the FCF are constant if the leverage is constant. However this does not hold true for finite cash flows. Though it might be convenient to perform calculations under such assumption, it is not in fact always true that Ke and WACC are constant under the constant leverage financing policy. As could be seen from the findings and example of Inselbag and Kaufold (1997), and as a general expression for Ke and WACC derived by Tham and Velez-Pareja (2002) shows, both the cost of levered equity and the Weighted Average Cost of Capital depend on the value of the interest tax shield (VTS), and in the case of finite cash flows valuation they could be changing from period to period if certain choice is made for the rate to discount for the expected tax shields.
The teaching note is organized as follows: An Introduction to state the problem; in Section Two we present the generalized formulation for the cost of capital for the finite cash flow valuation, and in particular formulae under the assumption that the discount rate for the tax shield (TS) is Kd. In Section Three we show a simple example. In Section Four we conclude.
WACC, constant cost of capital, constant leverage, cash flows
Abstract: In the latest edition of Principles of Corporate Finance (Brealey, Myers and Allen, 2006) the authors use a finite cash flow example to illustrate the valuation procedure for using the Discounted Cash Flow (DCF) method with the free cash flow (FCL) and the Adjusted Present Value (APV). The two firm values obtained are different. They say that the "... difference [...] is not a big deal considering all the lurking risks and pitfalls in forecasting [...] cash flows". In this teaching note we show that the two methods give identical values when the proper discount rates are used.
Abstract: The conventional wisdom about psi, the appropriate discount rate for the tax shield, is as follows. If the tax shield is risk-free, that is, the revenue is sufficient to ensure that the interest deduction will be used with full certainty in the relevant period, then the appropriate discount rate for the tax shield is d, which is the cost of the risk-free debt. On the other hand, if the revenue is stochastic, there may be a finite probability that the revenue will be insufficient to allow for the use of the tax shield. In such a case, whether the debt is risk-free or not, the tax shield is no longer risk-free. In the single period case, some authors suggest that if the tax shield is risky, then the risk of the tax shield is the same as the risk of the free cash flow (FCF) and consequently, the appropriate discount rate is rho, which is the return to unlevered equity. In this teaching note, using a single period binomial example, we show that the conventional wisdom on the appropriate discount rate for the tax shield is incorrect when the tax shield is risky. The appropriate discount rate for the tax shield depends on the risk of the tax shield and the value of psi may be higher than e, the return to levered equity. If the tax shield is correlated with the cash flow to equity, that is, the payoff structure for the tax shield is similar to the payoff structure for the cash flow to equity, then the value of psi is equal to e, which is higher than rho.
Discount Rate for Tax Shield
Abstract: In a forthcoming paper, Fernandez (2002) claims to derive a formula for the valuation of debt tax shields for firms with cash flows that grow perpetually at a constant rate. We show that his formula is incorrect.
Abstract: In theory, different valuation methods, with consistent assumptions, must give identical results. Numerical examples that purport to illustrate the theory should demonstrate the identical results. Unfortunately, in popular textbooks it is all too easy to find numerical examples that are at odds with the theory. There are several possible explanations for the discrepancies. First, there might be some conceptual confusion about the underlying assumptions. Second, it could simply be "rounding errors." It is intellectual laziness to ascribe the discrepancies to the tyranny of rounding errors when in fact it is easy to show that rounding errors are not the reasons for the discrepancies. It is common to read that different valuation methods give different results. For instance, Brealey and Myers (2000, 2003) say: "If the company's debt ratio is constant over time, the flow-to-equity method should give the same answer as discounting company cash flows at the WACC and subtracting debt." On the other hand, they say, "If financial leverage will change significantly discounting flows to equity at today's cost of equity will not give the right answer." Inselbag and Kaufold, 1997, conclude that the APV is better than the DCF when the debt schedule is given. This is misleading in two senses: one, they mix methods because they disregard the possibility to solve the circularity posed by the relationship between value and discount rates and second, as a consequence, they say that "one must already have calculated the firm's value" in order to know the WACC. In the latest edition of Principles of Corporate Finance (Brealey, Myers and Allen, 2006) the authors use a finite cash flow example to illustrate the valuation procedure for using the Discounted Cash Flow (DCF) method with the free cash flow (FCF) and the Adjusted Present Value (APV). The two firm values obtained are different. They say that the "... difference [...] is not a big deal considering all the lurking risks and pitfalls in forecasting [...] cash flows". Once more, in this teaching note we show that the two methods give identical values when the proper discount rates are used.
Abstract: It is widely known that if the leverage is constant over time, then the cost of equity and the Weighted Average Cost of Capital (WACC) for the free cash flow, FCF, is constant over time. In other words, it is inappropriate to use a constant WACCFCF to discount the free cash flow (FCF) if the leverage changes over time. However, it is common to find analysts who inconsistently use a constant WACCFCF even if the leverage is not constant. In this teaching note, we use a simple numerical example to illustrate how to model cash flows that are consistent with constant leverage. We verify the consistency of the example with two basic principles: conservation of cash flows and conservation of values.
Abstract: This paper discusses the calculation of financial discount rates in the presence of taxes and inflation. With respect to financing, there are two options. The debt-equity ratio may be constant or variable over the life of the project. If it is assumed that the debt-equity ratio is constant, then the loan schedule should be adjusted to ensure that the constant debt-equity ratio is maintained. On the other hand, if the debt-equity ratio changes over the life of the project, then the return to equity in each period should take into account the debt-equity ratio in that period.
Abstract: There are two ways to define the present value of the tax shield (PVTS). First, the PVTS is simply the tax shield (TS), discounted by the appropriate discount rate for the tax shield. Second, the PVTS is the difference in the taxes paid by the unlevered and levered firms. In his recent book, Fernandez (2002) claims that these definitions are not equivalent. It can be shown that both of these definitions are equivalent. Using non-technical language, we briefly comment on the reasons why one may mistakenly believe that there is non-equivalence between the two definitions
risk of the tax shield, present value of tax shield
Abstract: Risk-neutral valuation is simple, elegant and central in option pricing theory. However, in teaching risk-neutral valuation, it is not easy to explain the concept of "risk-neutral" probabilities. Beginners who are new to risk-neutral valuation always have lingering doubts about the validity of the probabilities. What do the probabilities really mean? Are they real or fictional? Where do they come from? What is the relationship between the risk-neutral probabilities and the actual probabilities? Does it mean that all investors are risk-neutral? When is it appropriate to use the risk-neutral rate as the discount rate? From a pedagogical point of view, in the beginning it is best to avoid the use of probabilities because probabilities can be a barrier to understanding. Instead, it is far preferable to introduce the idea of state prices and then show that the approach with risk-neutral probabilities is equivalent to the use of state prices. In this teaching note, we use simple one-period examples to explain the intuitive ideas behind risk-neutral valuation. It is a gentle introduction to risk-neutral valuation, with a minimum requirement of mathematics and prior knowledge. We will provide the motivation and the rationale for calculating state prices and we will show that the risk-neutral approach is simply another way of looking at the issue of state prices.
Cost-Benefit Analysis, Capital Budgeting, Project Evaluation
Abstract: In this teaching note, we present a clear and detailed exposition of Harberger's approach to economic appraisal. The Harberger framework consists of three basic postulates. The first postulate states that the demand curve measures the benefits of a project. The second postulate states that the supply curve measures the costs of a project. The third postulate ("summing up") states that we simply use the values from the demand and supply curves without any weighting. We assume that there are no market distortions such as taxes and subsidies. In a subsequent teaching note, we will introduce the impact of market distortions, such as taxes and subsidies, on the estimation of economic prices.
Project appraisal, economic price, consumer surplus, producer surplus, Harberger Triangle
Abstract: There are two ways to define the present value of the tax shield (PVTS). First, the (present) value of the tax shield VTS is simply the tax shield, discounted by y, which is the appropriate discount rate for the tax shield. Second, the (present) value of the tax shield VTS is the difference in the present values of the tax payments for the unlevered firm and the levered firm, where the risk profiles of the two tax payments are different. Happily, in this note, we show that both definitions of the present value of the tax shield are equivalent. To reconcile the two definitions of VTS, it is important to specify the risk profile of the free cash flow (FCF) and estimate the appropriate discount rates for the different cash flow profiles. We use a simple binomial model to specify the risk profile of the FCF and present simple numerical examples to illustrate the equivalence between the two definitions of the present value of the tax shield. The growth rate of the FCF and the leverage cost are zero. We assume that the reader is familiar with risk-neutral valuation. In Section One, we present a numerical example with risk-free debt. In Section Two, we present a numerical example with risky debt.
Tax shield, risk of tax shield, WACC, Cost of Capital
Abstract: Many firms have debt financing in a foreign currency. What are the tax implications of the foreign loan for the calculation of the Weighted Average Cost of Capital (WACC)? With a foreign loan, there are two effects. First, there is the standard tax savings from the interests deduction with the foreign loan. Second, we assume that changes in the value of the loan in the foreign currency can be listed in the income statement ofr tax purposes. In this teaching note, we examine how the WACC must be properly used to take into account both of the effects: The interest deduction and the change in the value of the foreign debt.
Abstract: In the traditional formulation of the WACC, the debt and the tax shield are risk-free. However, even though the debt is risk-free, the tax shield can be risky. Furthermore, both the debt and the tax shield can be risky. In this paper, we present the non-conventional weighted everage cost of capital (WACC) for the single period binomial process with risky debt and risky tax shield, and derive the relevant formulas for the returns to the levered equity holder and the debt holder. Risk-free debt does not imply that the tax shield is risk-free. Unlike the traditional formulation, with the non-conventional WACC, the discount rate for the tax shield is not be restricted to two values, the risk-free rate rf and the return to unlevered equity rho. We make no prior assumption about the value of the discount rate for the tax shield psi. The value of psi depends on the riskiness of the tax shield. If the tax shield is risk-free, then the value of psi is equal to risk-free rate rf. If the tax shield is risky, the the value of psi is simply higher than the risk-free rate rf. In particular, if the payoff structure of the tax shield is the same as the payoff structure for the levered equity holder, the value of psi is equal to the return to the levered equity holder. In Section One, we present the non-conventional WACC for the single period with the capital cash flow (CCF) rather than the free cash flow (FCF) and examine the traditional position, where both the debt and the tax shield are risk-free. In Section Two, we assume that both the debt and the tax shield are risky.
Abstract: In a typical market-based valuation, the standard procedure is to discount the expected free cash flow (EFCF) at the weighted average cost of capital (WACC) and the effect of financing is taken into account by adjusting the WACC. However, in many cases, it may be difficult to capture the various impacts of financing by simply adjusting the WACC. In others cases, as a component of the analysis, the construction of the cash flows to the equity holder may be necessary. In this paper, I present a simple spreadsheet model for the direct valuation of the cash flows to the equity holder without the usual simplifying assumptions. In particular, I specify the following flexible conditions: a. Multiperiod investments and reinvestments, b. Finite cash flows, with variable growth rates, c. Variable debt-equity ratios, d. Loses carried forward. With the inclusion of these conditions, the analysis is more realistic. In the model, the present value of the tax shield is discounted at rho, the required return with all-equity financing. In addition, the impacts of inflation are directly incorporated into the analysis. In particular, the model shows that the NPV of the free cash flow (including the tax saving from debt financing), discounted at the WACC is equal to the NPV of the free cash flow to the equity holder, discounted at the annually adjusted return to equity. The general approach presented here can be easily modified to take into account the varied circumstances and special complexities that are often encountered by practitioners.
Asset Pricing, Capital Budgeting
Abstract: There are two ways to view the inter-temporal risk profile of a finite stream of cash flows that is represented by a binomial process. We can examine the risk profile of the cash flow process or the value process that is derived from the cash flow process. First, with respect to a given year n, we can assess the individual risks of the annual cash flows by specifying a set of annual risk-adjusted discount rates, where rho(ni) is the discount rate for the cash flow in year i with reference to year n. In the second view on the inter-temporal risk profile for a finite stream of cash flows, we construct the value process from the cash flow process and with respect to a given point in time n, we assess the risk of the "present value" of the cash flow stream and summarize the collective risk of the cash flow stream with rho(nV) a single risk-adjusted discount rate for year n. In this paper, we illustrate and compare the two views on the inter-temporal risk profile by using a five-period binomial process to represent the finite stream of tax shields. In particular, we examine the inter-temporal evolution of the "cash flow discount rates" and the "present value discount rates" for the tax shields, and the relationships between them. In Section One, we briefly discuss the two different ways to characterize the risk profile. In Section Two, to illustrate the two descriptions of the risk profile, we present the tax shield process based on the M & E formulation.
Inter-temporal resolution of risk, tax shield
Abstract: It is widely accepted that the correct discount rate for the tax shield depends on whether the value of the debt is a fixed amount or is a proportion of the value of the firm. In this pedagogical note, using a simple two period numerical example, I assume a fixed amount of debt and demonstrate that if the cost of debt is used to discount the tax shield to determine the value of the levered firm, arbitrage opportunities exist. I propose that it is always correct to use the return on unlevered equity, to discount the tax shield. Consequently, the correct discount rate for the tax shield does not depend on whether the value of the debt is a fixed amount or is a proportion of the value of the firm. It is not difficult to extend the analysis to the multiperiod case. Happily, all the inconsistencies are resolved if the return on unlevered equity, is used to discount the tax shield.
capital budgeting
Abstract: In this teaching note, we discuss the basic principles for tariff setting. Tariff setting is very important for regulated industries, such as water and power. The tariff should provide an appropriate risk-adjusted return to the investor. If the tariff were too low, then the investors would not be willing to invest. On the other hand, if the tariff were too high, then it would reduce the consumers' welfare. We examine the Rate of Return method for calculating the tariff in a regulated firm. In the rate of return method, the tariff compensates the investor for all the costs that the investor incurs, including a fair return. We use the discounted cash flow approach to value the return that the investor receives. The results of both calculations must be consistent. In particular, using simple examples, we show that in the presence of a positive expected inflation rate, the typical tariff calculation, Rate of return method, is an overestimation of the required payment to the equity holder.
WACC, taxes, regulation, tariff regulation
Abstract: This is the first chapter of our book Principles of Cash Flow Valuation. It is an overview of what we present in the book. In this chapter, we present an informal introduction to the basic concepts and ideas in market-based cash flow valuation. The simplified exposition will provide sufficient background knowledge to understand the context of the materials that are presented in subsequent chapters. Later, in the appropriate chapters, we return to these ideas in valuation and explain them with detailed numerical examples. The reader will feel comfortable because she has already been exposed to the ideas informally in this chapter. For some readers, the concepts and ideas in this chapter will be a review. For other readers who find the explanations and discussions to be too terse, we assure them that the topics will be explored in greater detail and more formally in subsequent chapters. Most readers will be familiar with the standard after-tax Weighted Average Cost of Capital (WACC) that is applied to the free cash flow (FCF). However, for many readers the WACC applied to the capital cash flow (CCF), a term that Professor Richard Ruback has coined and popularized, will be new. Later we explain the CCF in greater detail. Now we are simply surveying the main ideas in the domain of cash flow valuation. We are providing an informal sketch of the territory that we will be covering and hope that all readers will find this introductory overview useful.
Cash flows, free cash flow, cash flow to equity, valuation, levered value, levered equity value, terminal value, cost of levered equity, weighted average cost of capital, WACC, cost of unlevered equity
Abstract: To value a firm or a project, it is necessary to construct estimated financial statements and free cash flows. In this introductory note we will present some basic principles for constructing the financial statements needed for valuation. We will illustrate the ideas with a concrete numerical example. The reader is encouraged to read actively by constructing the financial statements for themselves on a spreadsheet. The relevant financial statements are: the Balance Sheet (BS), the Income statement (IS) and the Cash Budget (CB). The construction of the financial statements starts from policies and/or targets (i.e. accounts receivable policy or target). With these targets or policies we can construct the financial statements. For valuation purposes, the balance sheet and the income statements are important but may be insufficient. For that reason we construct the CB and in future notes we will derive the FCF from the CB. The first table to be constructed is the table of parameters. This table organizes all of the relevant information. We have constructed the tables in EXCEL. The subsequent tables are linked to the table of parameters via formulas. Before constructing the financial statements, we will construct other supplementary tables that will be used in the construction of the main financial statements. In the main text, we describe the construction of all the tables and statements. The listing of all the tables is given in Appendix A. In these tables we show the table as seen in a spreadsheet with the columns and lines. In columns C and D the reader will find the formula as appears in the spreadsheet in columns E and F. There are a few exceptions, but they will be announced. If the reader wishes to construct the model exactly as we did, she will be able to do that following, step by step, not only the explanations in the body of the chapter but the appendix A as well. In a future note we will propose a way to construct the FCF from the CB because it is closer to the idea of free cash flows. In fact, the CB records all the cash movements of a firm. We prefer this approach because we can "see" most of the items that are considered as part of the FCF. With this approach the probability of mistakes in the construction of the FCF is reduced. The only item that is not seen in the CB is the tax adjustment or tax savings, as will be seen at the end of this chapter. Another advantage of using the CB to derive the FCF is that you do not disregard a very useful managerial tool such as the CB. We expect the reader will find this approach more intuitive and easy to follow than the traditional.
Project evaluation, Financial statements, Free cash flows, Cash budget, Income statement, Balance sheet
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