Ignacio VélezPareja Politécnico Grancolombiano
Joseph Tham Duke University
Reference: Received 23rd November 2005; Published 20th February 2006. ISSN 15791475
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Resumen
 Existen diversas maneras de calcular el Coste Promedio Ponderado de Capital (CPPC) o WACC (Weighted Average Cost of Capital) y para el principiante, la plétora de posibilidades puede ser confusa. Presentamos, pues, un marco general para la clasificación de los CPPC que se aplican al FCF y al CCF. Por el momento, evitamos las complejidades. Para facilitar el debate, clasificamos los diversos CPPC en tres dimensiones. Esperamos que dicho marco estructural ayude al lector a hacer la correcta elección con respecto al cálculo del coste de capital en la práctica. En primer lugar, mostramos un debate cualitativo sobre las dimensiones de dicho marco. En segundo lugar, especificamos las fórmulas y cálculos apropiados para las celdas del cuadro. Al principio, es importante subrayar que este artículo trata únicamente con flujos de caja finitos o finite cash flows. A nuestro parecer, es mejor analizar expresiones para el coste de capital que sean relevantes para los finite cash flow, para lo que no se necesita mayor justificación. Seguir usando fórmulas de coste de capital apropiadas para los cash flows a perpetuidad resulta inexplicable e incomprensible. Desde un punto de vista práctico, los free cash flows se derivan de los balances financieros, que no se construyen a perpetuidad. En el mejor de los casos, las expresiones para el coste de capital derivadas de los cash flows a perpetuidad pueden ser aproximaciones razonables para los finite cash flows. En el peor de los casos, los resultados pueden ser equívocos.
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. The continued use of cost of capital formulas that are appropriate for cash flows in perpetuity 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.

Introduction
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. The continued use of cost of capital formulas that are appropriate for cash flows in perpetuity 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, under ideal and simplistic circumstances, 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.
It may be useful to briefly discuss some of the important aspects in which finite cash flows differ from cash flows in perpetuity. First, the growth rates for the stream of finite cash flows are not constant in perpetuity and may be unequal over time. Second, the leverage is also not constant and may vary over time. Thirdly, the risk profile of the tax shield is complex. With the provision for losses carried forward (LCF), the tax shields may not be realized in the years that they occur.
With a clear understanding of the assumptions, the reader can select the formula that is most convenient and suitable. Next, we list and discuss the WACCs along three dimensions. The three dimensions of the framework are as follows.
1. Type of cash flow 2. Risk of tax shield 3. Formulation for the cost of capital.
First dimension
On the first dimension we specify the appropriate cash flow for the cost of capital: the free cash flow (FCF) or the capital cash flow (CCF). The basic cash flow relationship is as follows. In any year i, the sum of the FCF and the tax shield (TS) is equal to the sum of the cash flow to equity (CFE) and the cash flow to debt (CFD).
FCFi + TSi = CFEi + CFDi (1)
Solving for the FCF in equation 1, we obtain,
FCFi = CFEi + CFDi  TSi (2)
The FCF is exclusive of the tax shield (TS).
For the WACC that is applied to the FCF, we "lower" the WACC to take account of the tax benefits from the interest deduction with debt financing. This assumes that the tax shields are fully realized in the year that they occur. In other words, there are no losses carried forward (LCF).
The CCF is simply the sum of the CFE and the CFD.
CCFi = CFEi + CFDi (3)
Alternatively, the CCF is the sum of the FCF and the TS. Unlike the FCF, the CCF is inclusive of the TS.
CCFi = FCFi + TSi (4)
For the WACC that is applied to the CCF, there is no need to "lower" the WACC because the CCF already includes the TS. Furthermore, if there are losses carried forward, the tax shields are only taken into account in the years that the tax shields are actually realized.
If there are no losses carried forward, the WACC applied to the FCF and the WACC applied to the CCF gives the same answer. If there are losses carried forward, the WACC applied to the FCF overstates the levered value, relative to the WACC applied to the CCF, because the WACC applied to the FCF assumes that the tax shields are realized in the years that they occur even if the tax shields are not actually utilized.
Second dimension
The second dimension of the matrix is the risk of the tax shield. The risk of the tax shield determines the appropriate value that we should use to discount the tax shield. Typically, we assume that the debt is riskfree even though the debt may not be actually riskfree. With this assumption we avoid making the distinction between the contractual or "promised" return on the debt versus the expected return on the debt. From a conceptual point of view, in the calculation of the tax shield, we use the contractual return on the debt, whereas in the calculation of the WACC, we use the expected return on the debt. In practice, the difference between the promised and the expected may be small, especially if the default risk is slow.
Furthermore, we also assume that the tax shields are riskfree. In other words, the tax shields are always realized in the year that they occur. Thus, if the tax shields are riskfree, then the appropriate discount rate for the tax shield is equal to the cost of debt d, which in turn is equal to the riskfree rate rf.
For completeness and without justification or demonstration, we state that the value for the discount rate applied to the tax shield affects the formula for the calculation of the return to levered equity ei. Later we show that the return to levered equity is a function of the discount rate for the tax shield.
The risk of the tax shield may not be riskfree and in fact, the tax shield may be correlated with the free cash flow. In general, the risk of the tax shield is complex. Here, for simplicity, we consider two possible values for : the cost of debt d or the return to unlevered equity . In principle, the range of possible values for is quite large. In brief, the value of depends on the payoff structure or risk profile of the tax shield.
Thus, if we assume that the risk of the tax shield is the same as the risk of the FCF then the appropriate value of is the return to unlevered equity . Since the value of is higher than the cost of debt d, the (present) value of the tax shield is lower and correspondingly, the levered value is lower.
Third dimension
The third dimension of the framework is concerned with the formulation of the expression for the WACC. There are two equivalent ways to write the formulas for the WACCs and each of these ways can be applied to the FCF and the CCF. The first way is the standard "weighted average" formulation. Roughly speaking, the "weighted average" formula for the WACC is a weighted average of the cost of debt di and the return to levered equity ei, where the weights are the market values of the debt and equity.
Recall that the formula for the return to levered equity ei depends on the value that we specify for the discount rate for the tax shield The "weighted average" formula is easiest to use if the leverage is constant over the life of the cash flow. If the leverage is not constant, then we must adjust the return to levered equity ei to take proper account of the variable leverage. Again, we remind the reader that the formula for the return to levered equity ei that is derived from cash flows in perpetuity is different from the formula for the return to levered equity ei that is derived for finite cash flows.
We name the second way the "adjusted value" formulation. The "adjusted value" formula defines the WACC relative to the return to unlevered equity and does not require the calculation of the return to levered equity ei.
Matrix for the WACC applied to the FCF
We present the matrix for the WACC applied to the FCF and the relevant formulas for the cells in the matrix. The first column lists the two different ways to formulate the expressions for the cost of capital, and the actual formulas are listed in the third column. The formulas depend on the assumptions that are made about the risk of the tax shield and the two possibilities are listed in the second column. In the fourth column, we list the expression for the return to levered equity ei that corresponds to the assumption made about the value of
Matrix for the WACC applied to the CCF
Next, we present the matrix for the WACC applied to the CCF and the relevant formulas for the cells in the matrix.
Now, we briefly derive the general algebraic expressions for the cost of capital that is applied to finite cash flows. First, we show that in general, the return to levered equity ei is a function of i, and this is a most important point. Second, we derive the general expressions for the WACCs.
First, we write the main equations as follows.
We emphasize that in equation 8, we make no assumption about the value of i, which depends on the risk profile of the tax shield.
General expression for the return to levered equity ei
From above, we know that,
The return to levered equity ei is a positive linear function of the debtequity ratio. It is also a linear function of the (present) value of the TS to equity ratio, where the direction (sign) of the relationship depends on the value of
If we assume that the appropriate discount rate for the tax shield is equal to the return to unlevered equity i, then we can simplify equation 14 as follows.
If we assume that the appropriate discount rate for the tax shield is equal to the cost of debt di, then we can simplify equation 14 as follows.
Standard WACC applied to the FCF
We know that,
Let be the standard WACC that is applied to the FCF in year i. With respect to the end of the year i1, the levered value is equal to the sum of the FCF in year i and discounted by the WACC for year i, where is the (present) value in year i of all the FCF in the years beyond year i
Adjusted WACC applied to the FCF
Let be the adjusted WACC that is applied to the FCF in year i. We follow the same steps that we outlined for the standard WACC applied to the FCF and obtain an equation that is similar to equation 22.
Standard WACC applied to the CCF
We know that the CCF is equal to the sum of the CFE and the CFD.
Adjusted WACC applied to the CCF
We know that the CCF is equal to the sum of the FCF and the TS.
Conclusion
In this teaching note, we have presented a simple framework for organizing the various formulations for the WACC along three dimensions. The first dimension is the type of the cash flow. There are two possibilities, the FCF or the CCF. The FCF is exclusive of the TS and the CCF is inclusive of the TS. The second dimension concerns the risk of the tax shield. The risk of the tax shield determines the appropriate discount rate for calculating the (present) value of the tax shield.
The third dimension of the framework specifies the formulation of the expression for the WACC. There are two formulations. The first formulation is the standard well known "weighted average" approach. The second formulation is named the "adjusted value WACC" approach. We hope that this threedimensional framework helps the reader in organizing and applying the menagerie of WACCs to the valuation of cash flows.

About the Authors
Autor: Ignacio VélezPareja Dirección: Politécnico Grancolombiano Correo electrónico: ivelez@poligran.edu.co
Autor: Joseph Tham Dirección: Duke University
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Editor: Fernando GonzálezLaxe. (Universidade da Coruña) Director: Venancio Salcines. (Universidade da Coruña) Subdirector: Andrés Blancas. Instituto de Investigaciones Económicas (UNAM) Editor Asociado para America Latina: Luis Miguel Galindo. Facultad de Ecomomía (UNAM) 