Ignacio VélezPareja Politécnico Grancolombiano
Joseph Tham Duke University
Viviana Fernández Universidad de Chile
Reference: Received 30th May 2005; Published 10th October 2005. ISSN 15791475
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Resumen
 Al aplicar el coste de capital o coste promedio ponderado de capital (WACC) al free cash flow (FCF), suponemos que el coste de la deuda es la tasa de mercado no financiada. Con una deuda fijada a la tasa de mercado y con unos mercados de capitales perfectos, la deuda solo crea valor si se establecen impuestos por medio de una proteccion fiscal (tax shield). En algunos casos, la empresa puede obtener un préstamo a un interés inferior a la tasa de mercado. Al fijar una deuda y unos impuestos financiados, se obtendría un beneficio para la financiación de la deuda, pero los valores endeudados y no endeudados de los cash flows resultarían desiguales. El beneficio obtenido por un ahorro fiscal más bajo se compensa con el beneficio de la financiación. Estos dos beneficios se deben aplicar explicitamente. En el presente artículo exponemos los ajustes a la WACC con una deuda y unos impuestos financiados y el coste del patrimonio (equity) endeudado para múltiples periodos. Mostramos el analisis tanto del WACC aplicado al FCF, como del WACC aplicado al capital cash flow (CCF). Empleamos el cálculo del Valor Presente Ajustado (APV), para considerar tanto el ahorro fiscal como la financiación. Por último, mostramos como encajan todos los métodos.
Abstract

In the 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 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. In this paper we present the adjustments to the WACC with subsidized debt and taxes and the cost of leveraged equity for multiple periods. 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. 
INTRODUCTION
In the 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 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 TS, are offset by the benefit of the subsidy. These two benefits have to be introduced explicitly.
In this paper we present the adjustments to the WACC and the cost of leveraged equity for multiple periods 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 TS and the subsidy. We show how all the methods match.
The issue of the effect of subsidy in interest rate on the WACC is not widely dealt in the literature. Ross et al, 1999 mention the effect on value and propose to use the APV method and Damodaran 1996 suggests including the value of the subsidy in the cash flow. This paper is organized as follows: In Section One we derive the expressions for the cost of capital in the presence of subsidy and corporate taxes for multiple periods and illustrate it with an example. In Section Two we conclude.
SECTION ONE
First we derive the cost of leveraged equity, Ke. Let be the leveraged value, let be the unleveraged value, let the value of the TS, let T the corporate tax rate and let be the value of the interest subsidy. Then, with respect to the end of year 0, the leveraged value equals the sum of the unleveraged value, plus the value of the TS and the value of the interest subsidy.
Using the APV approach, it would be very easy to estimate the value of the subsidized debt. Let be the cost of the non subsidized debt, and let be the cost of the subsidized debt. The value of the debt at the end of year 0 is
Let be the interest subsidy at the end of year 1 and be the TS at the end of year 1. Then the interest subsidy equals the value of the debt times the difference between the two interest rates adjusted for taxes and the TS are the cost of unsubsidized debt times the debt, D0 and times the tax rate, T.
The expression for the value of the interest subsidy is as follows, where is the appropriate discount rate for the interest subsidy.
The expression for the value of the TS is as follows, where is the appropriate discount rate for the TS.
DERIVATION OF KE
Let CCF1 be the capital cash flow at the end of year 1 with financing. At the end of year 1, the capital cash flow equals the sum of the FCF, plus the TS and the interest subsidy.
Then,
Also, at the end of year 1, the capital cash flow equals the sum of the cash flow to equity (CFE) and the cash flow to debt (with the subsidized interest rate).
Substituting the appropriate value expressions for each of the cash flow items in equation 8, we obtain,
where Ke is the cost of leveraged equity and Ku is the cost of unleveraged equity.
Applying equation 9 to equation 10, we obtain,
If we assume that the appropriate discount rate for the interest subsidy and for the TS is equal to the cost of unleveraged equity, then the third and fourth terms in equation 12.2 are zero.
Now we derive the WACC to be applied to the FCF. As before, from (8) we can write the following
In summary we have
From this summary, we can obtain simpler formulations depending on the assumptions regarding the discount rate for TS and subsidy. For instance, if we assume that and are equal to Ku, then the formulae for the different costs are
The formula for Ke resembles the typical formulation of Ke when is Ku, except that Kd is replaced by . For the CCF we have equal to Ku; this is what is expected when we use the CCF and assume Ku as the discount rate for TS. Finally, for discounting the FCF we have and this resembles the adjusted WACC. (See Tham and VelezPareja 2004).
We illustrate these ideas with a three period numerical example. The values of the various parameters are shown below. We present the input variables and the final tables after solving the circularity . The input variables are shown in table 2.
Next we calculate the CFD with , the TS, the subsidy and the CFE. These values will be needed to calculate Ke and WACC for FCF and CFE.
Now we can calculate the value of Ke for every year and we calculate the market value of equity.
For instance, for year 1, in the previous table we apply equation
(allow for rounding errors if the reader tries to replicate this calculation).
In the case of we have for year 1,
The figures from this table are taken from previous tables except the unleveraged value that is calculated as the present value of the FCF at Ku.
The CCF is derived from data from table 2. The is derived using the next equation. For year 1 we have.
Now we calculate the leveraged value assuming what is the current practice: to include the in the traditional formula for WACC for the FCF. First we calculate the leveraged value without subsidy. This is what is shown in the next table.
Now we calculate the value using the traditional WACC for the FCF and including as the cost of debt.
Observe that the leveraged value has been reduced compared when we use the traditional WACC and include the . A lower cost of debt destroys value! This is counter evident. This occurs because we have lost part of the value generated by the TS and because the Ke calculation absorbs the reduction of the cost of debt. This means that the subsidy has to be explicitly included in the analysis.
In the next table we present a summary of the different calculations for values:
In the numerical example, we assume that the appropriate discount rate for the interest subsidy is the market rate of interest. However, we could also use the subsidized rate or the Ku. For completeness, in the next table we show the consistent results for the two other values for , namely and Ku.
It might be argued that the differences in this example are irrelevant. However, we think that it is not a matter of precision; it is a matter of correctness that can be reached without extra cost. More, it is usual to assume that differences are assigned to rounding errors or that the magnitude is negligible or that practical approaches are more important than theoretical and precise ones. However, while errors could cancel out, sometimes errors cumulate. See for instance Vélez Pareja 2004 and 2005.
CONCLUSION
In this paper, we show the adjustments that have to be made to the WACC in the presence of a subsidized loan and taxes. It is interesting to observe that when obtaining a subsidy in the cost of debt, using that lower cost in the WACC is not the correct approach to measure the increase in value due to the subsidy. The adjustments to the WACC and the explicit introduction of the subsidy in the analysis, give the proper result.
References
Damodaran, Aswath, 1996, Investment Valuation, John Wiley.
Ross, S. A., R. W. Westerfield and J. Jaffe, 1999, Corporate Finance, IrwinMcGrawHill.
Tham, Joseph and Ignacio VélezPareja, 2004, Principles of Cash Flow Valuation, Academic Press.
Tham, Joseph and VelezPareja, Ignacio, 2005, "With subsidized debt how do we adjust the WACC?" Social Science Research Network.
VelezPareja, Ignacio, 2004, "Modeling the Financial Impact of Regulatory Policy: Practical Recommendations and Suggestions. The Case of World Bank" Social Science Research Network.
VelezPareja, Ignacio, 2005, "Cash Flow Valuation in an Inflationary World: The Case of World Bank for Regulated Firms" Social Science Research Network.
VélezPareja, Ignacio and Joseph Tham, 2000, "A Note on the Weighted Average Cost of Capital WACC", (last version October, 2002). Social Science Research Network.
VelezPareja, Ignacio and Tham, Joseph, 2005, "Proper Solution of Circularity in the Interactions of Corporate Financing and Investment Decisions: A Reply to the Financing Present Value Approach" Social Science Research Network.
About the Authors
Autor: Ignacio VélezPareja Dirección: Politécnico Grancolombiano, Bogotá, Colombia Correo electrónico: ivelez@cable.net.co
Autor: Joseph Tham Dirección: Duke University Correo electrónico: ThamJx@duke.edu
Autor: Viviana Fernández Dirección: Universidad de Chile Correo electrónico: vfernand@dii.uchile.cl

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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) 