WACC, constant cost of capital, constant leverage, cash flows. Clasificación JEL: D61, G31, H43

Transcription

1 A typcal approach for valung fnte cash flows s to assume that leverage s constant (usually as target leverage) and the cost of equty, Ke and the Weghted Average Cost of Captal, are also assumed to be constant. For cash flows n perpetuty, and wth the cost of debt, Kd as the dscount rate for the tax sheld, t s ndeed the case that the Ke and appled to the FCF are constant f the leverage s constant. However ths does not hold true for fnte cash flows. In ths document we show that for fnte cash flows, Ke and hence depend on the dscount rate that s used to value the tax sheld, and as expected, Ke and are not constant wth Kd as the dscount rate for the tax sheld, even f the leverage s constant. We llustrate ths stuaton wth a smple example. We analyze fve methods: DCF usng AP, FCF and tradtonal and general formulaton for, present value of CF plus debt and Captal Cash Flow, CCF., constant cost of captal, constant leverage, cash flows. Clasfcacón J: D61, G31, H43

2 Un enfoque típco para valorar flujos de caja fntos es suponer que el endeudamento es constante (generalmente como un endeudamento objetvo o deseado) y que por tanto, el costo del patrmono, Ke y el costo promedo ponderado de captal CPPC, tambén son constantes. Para los flujos de caja perpetuos, y con el costo de la deuda, Kd como la tasa de descuento para el ahorro en mpuestos o escudo fscal, Ke y el CPPC aplcado al flujo de caja lbre FC son constantes s el endeudamento es constante. Sn embargo esto no es verdad para los flujos de caja fntos. n este documento mostramos que para flujos de caja fntos, Ke y por lo tanto el CPPC dependen de la tasa de descuento que se utlza para valorar el ahorro en mpuestos, AI y según lo esperado, Ke y el CPPC no son constantes con Kd como la tasa de descuento para el ahorro en mpuestos, aunque el endeudamento sea constante. Ilustramos esta stuacón con un ejemplo smple. Analzamos cnco métodos: el flujo de caja descontado, FCD, usando AP, el FCD y la formulacón tradconal y general del CPPC, el valor presente del flujo de caja del acconsta, FCA más deuda y el flujo de caja de captal, FCC. Costo promedo ponderado de captal, CPPC, costo de captal constante, endeudamento constante, flujos de caja.

3 In ths document we show that usng the fndngs of Tham and elez- Pareja (2002), for fnte cash flows, Ke (cost of levered equty) and hence (Weghted Average Cost of Captal), depend on the dscount rate that s used to value the tax sheld (), and as expected, Ke and are not constant wth Kd (cost of debt) as the dscount rate for the tax sheld, even f the leverage s constant. We llustrate ths stuaton wth a smple example. We analyze fve methods: DCF Dscounted Cash flows (the Free Cash Flow, FCF) usng AP, FCF and tradtonal and general formulaton for, present value of Cash Flow to equty (CF) plus debt and Captal Cash Flow (CCF). A typcal approach for project or frm valuaton whch could be found n practce (See for example World Bank (2002), Bennnga (1997, 2006), Brealey and Myers (2000, 2003), Brealey, Myers and Allen (2006), Copeland, Koller and Murrn (1995, 2000) s to dscount cash flows expected wthn the fnte tme horzon at constant cost of captal, (usually as a target leverage) assumng that target leverage s mantaned throughout the lfe of the project, and thus ts cost of levered equty Ke and the are constant. Though t mght be convenent to perform calculatons under such assumpton, t s not n fact always true that Ke and are constant under the constant leverage fnancng polcy. As could be seen from the fndngs and example of Inselbag and Kaufold (1997), and as a general expresson for Ke and derved by Tham and elez-pareja (2002), both the cost of levered equty and the Weghted Average Cost of Captal depend on the value of the nterest tax sheld ( ), and n the case of fnte cash flows valuaton, they could be changng from perod to perod f certan choce s made for the rate to dscount expected tax shelds. The case of varable leverage has been studed elsewhere by Man and elez Pareja (2008), elez-pareja (2004, 2005), elez-pareja and Burbano (2006), elez-pareja and Tham (2001, 2004, 2006a, 2006b), and Tham and elez Pareja (2002, 2004). In these cases, they fnd complete consstency between all methods and wth dfferent assumptons about the dscount rate for the tax shelds. Practtoners frequently assume that the rsk (and correspondng dscount rate, ψ) of the nterest tax sheld s the cost of debt, Kd. Ths s done explctly when, for example, the AP method s appled, or mplctly, f popular formula Ke= Ku + (Ku Kd) (1 T) D/ (Ku, the cost of unlevered equty; T, corporate tax rate; D and are market values of debt and equty, respectvely) s used to estmate the cost of equty captal. As Taggart (1991) and Tham and elez-pareja (2002, 2004) prove, ths formulaton s vald only for a fxed (n perpetuty) dollar amount of debt, thus under constant leverage assumpton t could be appled only to perpetual cash flows. However, ths formula s used by Fernandez (2002), Shapro (2005) and others even wthn fnte plannng horzon and when dollar amount of debt s changng from perod to perod. Another example of mplct ψ = Kd assumpton s applyng Hamada s formulaton to unlever and relever betas. Intally developed

4 by Hamada (1972) for flat perpetuty and rsk free debt, hs formula s persstently used n conjuncton wth dscountng at constant under constant leverage assumpton, 1 potentally producng sgnfcant valuaton errors as can be seen from the comprehensve example analyzed by Man and elez-pareja (2008). elez-pareja and Tham (2004, 2006a, 2006b) repeatedly show that, f assumptons and formulae are msmatched, nconsstences arse when calculatng value wth dfferent methods. So analysts should be very careful dealng wth fnte cash flows. To obtan correct and consstent valuaton results one should specfy assumpton for the rsk of the tax sheld (ψ) frst, and from that assumpton choose the proper formulaton for Ke and. Dfferent values one proposes for ψ, the rsk or dscount rate for the mght be questoned based on the partcular debt polcy and underlyng expectatons for the cash flow profle. However, when selectng the assumpton or approach for ψ, we have to be consstent n the use of the formulaton for the cost of captal. If under the constant leverage fnancng polcy the rsk ψ of the nterest tax sheld s assumed 2 to be equal to Kd, then Ke and could not be assumed constant. Put t another way, ψ = Kd and constant Ke and are ncompatble assumptons wthn the constant leverage set up. To llustrate the scenaro of non constant cost of captal wth constant leverage 3 we present a smple example, and analyze fve DCF methods: 1. Adjusted present value (AP); 2. Dscountng FCF at calculated from the tradtonal formulaton; 3. Dscountng FCF at calculated from the general formulaton; 4 4. Cash flow to equty (CF) dscounted at the cost of levered equty plus the value of debt, and 5. Captal Cash Flow (CCF) dscounted at the correspondng cost of captal. The rest of the document s organzed as follows: n Secton Two, we present the generalzed formulaton for the cost of captal for the fnte cash flow valuaton, and n partcular formulae under the assumpton that the dscount rate for the tax sheld () s Kd. In Secton Three we show a smple example. In Secton Four we conclude. Taggart (1991) presents a revson of the set of formulatons for the cost of levered Ke and for perpetutes and fnte cash flows. He ntroduces the formulaton wth and 1 Here we can menton very dfferent texts from practtoners and academcs: Pratt, Relly and Schwehs (2000), Abrams (2001), Damodaran (2002), 2 Ths assumpton s by tself debatable 3 Whch for example could be acheved through debt rebalancng at the end of every perod to keep constant ts percentage of the estmated project value) 4 See Tham and elez-pareja (2002, 2004).

5 wthout personal taxes and for dfferent level of rsk for dscountng the, ncludng the Mles and zzell (1980). However, Taggart does not nclude the case of ψ = Kd for fnte cash flows. Inselbag and Kaufold (1997) nclude the formulaton of Ke and for the case of Kd, the cost of debt as the level of rsk for the and fnte cash flows, but nether Taggart (1991) nor Inselbag and Kaufold (1997) show the formulaton for the cost of captal approprate to dscount captal cash flow (CCF) under ψ = Kd and fnte cash flows scenaro. Tham and elez-pareja (2002) present a dervaton of the general expresson for Ke, the cost of levered equty for dfferent levels of ψ correspondng to the rsk of the tax shelds, and resultng formulatons for the general, whch should be appled to dscount the Free Cash Flow (FCF) and Captal Cash Flow (CCF) both for fnte tme horzon valuaton and for perpetutes. The general formulaton for Ke s, D Ke Ku (Ku - Kd ) (Ku - ) -1-1 Where Ku s the unlevered cost of equty, ψ s the rsk (dscount rate) of the, D s market value of debt, s market value of equty and s the market value of ; s the perod of analyss. The general formulaton for FCF s, FCF 1 Ku - -(Ku ) -1 1 where s tax savngs, s the market value of the levered frm and the other varables were defned above. Followng the path of the classc dervaton, we can easly show that general expresson for the classc s FCF -1 D-1 Ke Kd - -1 and we obtan tradtonal formula FCF % -1 Ke D% -1 Kd (1 T ) when = Kd D -1 T. The general formulaton for the CCF s CCF 1 Ku -[(Ku ) 1 The general formula for the value of s N j 1 j (1 ) j When the rsk of, ψ, s Ku, then Ke smplfes to D-1 Ke Ku (Ku - Kd ) Ths formulaton s vald for fnte cash flows or perpetutes. The for the FCF smplfes to FCF Ku - When taxes are pad when accrued and there s enough BIT to earn the, then FCF s FCF T D-1 Kd Ku - Ku T D% -1-1 Kd And the for the CCF smplfes to CCF Ku t

6 The value of the s 1 N (1 Ku) When the rsk of, ψ, s Kd, then Ke smplfes to Ke Ku D -1 1 Ku - Kd for fnte cash flows and Ke Ku D Ku - Kd 1- T for perpetutes. The for the FCF smplfes to FCF 1 Ku - -[(Ku Kd ) for fnte cash flows. And the for the CCF smplfes to CCF Ku - Ku The value of the s 1 N 1 (1 Kd) - Kd -1-1 Observe that n the case of ψ = Ku, Ke does not depend on and does not depend on the value of. Instead, when ψ = Kd, Ke depends on and the value of. On the other hand, when ψ = Ku, depends on and t wll be constant when taxes are pad when accrued and there s enough BIT to earn the. Instead, when ψ = Kd, depends on and the value of. From these formulatons we can conclude that for fnte cash flows leverage and cost of captal are constant when: 1. There s enough BIT to fully earn the. 2. Taxes are pad when accrued. 3. The rsk of s Ku. 4. Tax rate T, s constant. 5. Interest rate on debt s equal to the (market) cost of debt, Kd. Wth ths set of formulatons we can llustrate wth a smple numercal example that Ke and are not constant when leverage s mantaned constant f one assumes Kd (or any other value dfferent from Ku) to be the approprate dscount rate for the expected nterest tax shelds. Ths can be seen n the formulaton for Ke and. Full consstency n valuaton results could be obtaned for all fve methods we mentoned n the Introducton f proper formulaton for the cost of captal s appled to dscount correspondng cash flow. A typcal approach for valung fnte cash flows s to assume that leverage s constant and hence, Ke and are ntroduced as constant. In ths document, we show that usng the fndngs of Tham and elez-pareja (2002), Ke, and hence, depends on the value of the tax sheld,, and as expected, Ke and are not constant when we assume the rsk of as Kd. We llustrate ths stuaton wth a smple example. We analyze fve methods: 1. AP; 2. DCF usng FCF and tradtonal formulaton for ; 3. DCF usng FCF and general formulaton for ;

7 4. Present value of CF plus debt, and 5. Captal Cash Flow, CCF dscounted at the for CCF. In the Appendx, the reader wll fnd the complete nformaton and the fnancal statements. Assume a project (or the frm) wth the followng nformaton: 1. Some nput data. 2. Income Statement, Cash Budget and Balance Sheet. 3. Cash flows derved from the fnancal statements. Assume that the nput data s as n Table 1. Sales start wth $10 n year 1. The fxed assets cost $4. The expected endng balance for the ntal debt s shown n Table 2. The complete fnancal statements are shown n the Appendx. From the fnancal statements we derve the followng cash flows: 1. Free Cash Flow, FCF. 2. Cash Flow to Debt, CFD. 3. Cash Flow to quty, CF. 4. Tax savngs.

8 Now we calculate the workng captal for each year. Wth the change n workng captal we can construct the FCF usng the ndrect method. We show the dfferent cash flows n the next table. Wth ths nformaton we can perform the valuaton of the cash flows. In the next tables we show the valuaton for each method after solvng the teraton process to solve the crcularty between value and dscount rate, where necessary.

9 Usng the AP and assumng that the dscount rate for the s Kd: Usng the DCF, the tradtonal, and assumng that the dscount rate for the s Kd: Observe that Ke and are not constant. Ths occurs because the Ke s a functon of the value of. Usng the DCF, the general, and assumng that the dscount rate for the s Kd:

10 FCF Ku [(Ku Kd ) 1 1 Usng the CF and assumng that the dscount rate for the s Kd: And fnally, usng the CCF wth the CCF, and assumng that the dscount rate for the s Kd:

11 CCF Ku - -1 Ku - Kd -1 Observe that the CCF s not constant even f we assume that Ku s constant. CCF s constant and equal to Ku (assumng no change n the operatng rsk for the frm and constant nflaton) when we assume that the dscount rate for the s Ku. The reason s dentcal to the one that makes Ke and FCF non constant when leverage s constant and we assume the rsk of equal to Kd: Ke, FCF and CCF depend on the value of. As we have shown, frst of all, all methods match 5 when we use the proper formulaton for the cost of captal (Ke and ); second, we have shown that the constant leverage does not mean that Ke and are constant. Tham and elez-pareja (2002, 2005), and elez-pareja and Tham (2006a, 2006b) have shown that when usng Ku as the rsk for the and some condtons regardng the payment of taxes, the exstence of enough BIT to earn the and the source of the, the cost of captal s constant. 6 Observe that the value calculated assumng ψ equal to Kd s hgher than the value when we assume that ψ equal to Ku. A queston arses here: s t reasonable to thnk that, changng the fnancng polcy from constant leverage to predetermned debt schedule (non constant leverage), the frm wll ncrease ts value? We leave the answer to ths queston for another work. Now we can check the dfference between the ntal debt schedule and the new debt schedule based on the market value of debt. 5 Ths s a matchng of dentcal results. We have tested t for more than 10 decmals and the dfference s strctly, zero. The nterested reader mght receve the spreadsheet upon request to the authors. 6 We are assumng that BIT 0. When ths condton s not met, the tradtonal expresson for s no longer vald. When ths happens we should use equaton (4) and as can be seen from t, mght not be constant (magne that there s not earned durng some perod). When s not earned, for the FCF s just Ku.

12 The new debt schedule s shown n next table. Ths means that management has to adjust debt from the begnnng n order to acheve the target leverage. The dfference n debt level s as follows. We have shown that a constant leverage does not grant that the cost of captal s constant when the rsk of the s Kd. Moreover, n order to acheve a proper valuaton of fnte cash flows wth a constant leverage when the rsk of s Kd, we have to use some formulatons that dffer from the tradtonal used by practtoners and textbooks. In other words, assumng constant leverage s not a suffcent condton to have constant cost of captal. We need to make explct assumptons on the rsk for the and use formulaton for the cost of captal that s consstent wth the assumed rsk of the tax sheld. Usng the proper formulaton n ths scenaro, we obtan full consstency n the calculaton of value. Ths means that there are no advantages of one method over another. All of them gve the same value (when properly done) and all of them (even the AP) requre teratons when the rsk of the s Kd. In short, we can conclude that for fnte cash flows leverage and cost of captal are constant when: 1. There s enough BIT to fully earn the. 2. Taxes are pad when accrued. 3. The rsk of s Ku. 4. Tax rate T, s constant. 5. Interest rate on debt s equal to the (market) cost of debt, Kd. In addton, we have to be aware that performng cash flow valuaton wth constant Ke and under constant leverage assumpton mples that partcular formulatons must be

13 used for the estmaton of Ke. Snce the possblty of constant leverage and constant cost of captal scenaro arses only when ψ = Ku, analysts should use formula Ke Ku (Ku - Kd D ) -1-1 to calculate the cost of levered equty drectly, and formula u or ev D d D-1 1 ev D u (u d ) for unleverng and leverng the beta n case they use the CAPM. Here βu and βev are the unlevered and levered β s and D t-1 and t-1 are the market values of debt and equty When we assume that the rsk of the s Kd, we cannot assume constant or Ke because leverage s constant. The formulatons for Ke and (ether for the Free Cash Flow, FCF or the Captal Cash Flow, CCF) depend not only on the constant leverage, but on the value of the and that refrans the cost of captal from beng constant even f leverage s constant. Summary of results assumng Kd as the rsk of the, Table 13. In the case of the rsk for the equal to Ku, we can observe the equatons for Ke, for the FCF and the CCF, as follows: The general formulaton for Ke s, D Ke Ku (Ku -Kd ) (Ku - ) -1-1 When the rsk of the s Ku, the thrd term of the RHS of the equaton FCF 1 Ku - -[(Ku Kd ) -1 1 CCF -1 Ku - Ku - Kd -1

14 vanshes and Ke depends only on Ku, Kd and leverage (constants). Hence, Ke s constant. The general formulaton for WAC- C FCF s, FCF Ku (Ku ) 1 1 When the rsk of the s Ku, the thrd term n the prevous equaton vanshes and the second term s T Kd D% and hence FCF depends only on leverage whch s constant. Hence, s constant. FCF -1-1 D Ke -1-1 Kd - -1 In ths case, f Ke s constant then FCF s constant. The general formulaton for the CCF s CCF 1 Ku -[(Ku ) 1 When the rsk of the s Ku, the second term of the RHS of the equaton vanshes and CCF = Ku whch s a constant.

15 Fnancal Statements and Cash Flows

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20 Abrams, J.B. (2001). Quanttatve Busness aluaton: A Mathematcal Approach for Today s Professonal. New York, NY: McGraw-Hll. Bennnga, S.Z. (2006). Prncples of Fnance wth xcel. New York, NY: Oxford Unversty Press. Bennnga, S.Z., & Oded, H.S. (1997). Corporate Fnance. A aluaton Approach. New York, NY: Mc- Graw-Hll. Brealey, R. & Myers, S.C. (2000). Prncples of Corporate Fnance (6 th ed.). New York, NY: McGraw Hll-Irwn. Brealey, R. & Myers, S.C. (2003). Prncples of Corporate Fnance (7 th ed.). New York, NY: McGraw Hll-Irwn. Brealey, R., Myers, S.C., & Allen, F. (2006). Prncples of Corporate Fnance (8 th ed.). New York, NY: McGraw Hll-Irwn. Copeland, T.., Koller, T. & Murrn, J. (1995). aluaton: Measurng and Managng the alue of Companes (2 nd ed.). New York, NY: John Wley & Sons. Copeland, T.., Koller, T. & Murrn, J. (2000). aluaton: Measurng and Managng the alue of Companes (3 rd ed.). New York, NY: John Wley & Sons. Damodaran, A. (2002). Investment aluaton: Tools and Technques for Determnng the alue of Any Asset (2nd ed.). New York, NY: John Wley & Sons. Fernandez, P. (2002). aluaton Methods and Shareholder alue Creaton. San Dego, CA: Academc Press. Hamada, R.S. (1972). The ffect of the Frm s Captal Structure on the Systematc Rsk of Common Stock. Journal of Fnance, 27(2), Inselbag, I. & Kaufold, H. (1997, Sprng). Two DCF Approaches n alung Companes under Alternatve Fnancng Strateges (and How to Choose between Them). Journal of Appled Corporate Fnance, 10(1), Internatonal Bank for Reconstructon and Development The World Bank. (2002). Fnancal Modelng of Regulatory Polcy [CD set]. Man, M.A. & elez-pareja, I. (2008). Applcablty of the Classc Concept n Practce. atn Amercan Busness Revew, 2(8). Avalable at SSRN: Mles, J. & zzell, J.R. (1980). The Weghted Average Cost of Captal, Perfect Captal Markets, and Project fe: A Clarfcaton. Journal of Fnancal and Quanttatve Analyss, 15, Pratt, S.P., Relly, R.F. & Schwehs, P. (2000). alung A Busness: The Analyss and Apprasal of Closely Held Companes (4 th ed.) New York, NY: McGraw-Hll. Shapro, A.C. (2005). Captal Budgetng and Investment Analyss. Upper Saddle Rver, NJ: Pearson Prentce Hall. Taggart, R.A., Jr. (1991). Consstent aluaton and Cost of Captal xpressons wth Corporate and Personal Taxes. Fnancal Management, 20(3), Tham, J. & elez Pareja, I. (2002). An mbarrassment of Rches: Wnnng Ways to alue wth the (SSRN Workng Paper ). Avalable at: com/sol3/papers.cfm?abstract_ d=352180

21 Tham, J. & elez-pareja, I. (2004). Prncples of Cash Flow aluaton. An Integrated Market Approach. ondon: Academc Press. Tham, J. & elez Pareja, I. (2005). Modelng Cash Flows wth Constant everage: A Note (SSRN Workng Paper ). Avalable at: elez-pareja, I. (2004). Modelng the Fnancal Impact of Regulatory Polcy: Practcal Recommendatons and Suggestons. The Case of World Bank (SSRN Workng Paper ). Avalable at: com/abstract= elez-pareja, I. (2005). Cash Flow aluaton n an Inflatonary World: The Case of World Bank for Regulated Frms. (SSRN Workng Paper ). Avalable at: com/abstract= elez Pareja, I. & Burbano, A. (2006). Consstency n aluaton: A Practcal Gude (SSRN Workng Paper ). Avalable at: papers.ssrn.com/sol3/papers. cfm?abstract_d= elez Pareja, I. & Tham, J. (2001). A Note on the Weghted Average Cost of Captal (SSRN Workng Paper ). Avalable at: elez-pareja, I. & Tham, J. (2004). Consstency n Chocolate. A Fresh ook at Copeland s Hershey Foods & Co Case (SSRN Workng Paper ). Avalable at: com/abstract= elez Pareja I. & Tham, J. (2006a). Constant everage Modelng: A Reply to A Tutoral on the McKnsey Model for aluaton of Companes (SSRN Workng Paper ). Avalable at: elez-pareja, I. & Tham, J. (2006b). The Msmatchng of AP and the DCF n Brealey, Myers and Allen 8th dton of Prncples of Corporate Fnance, 2006 (SSRN Workng Paper ). Avalable at: ssrn.com/abstract= elez Pareja I. & Tham, J. (2006c). aluaton of Cash Flows wth Constant everage: Further Insghts (SSRN Workng Paper ). Avalable at: com/sol3/papers.cfm?abstract_ d=879505