What is the Value of the Tax Shield of Debt?

Trierer Beiräge zur Beriebsund Volkswirschafslehre Nr. 1 Wha is he Value of he Tax Shield of eb? hrisoph Theis Mai 2009

Geleiwor in eigener Sache (1) Uner dem Tiel Trierer Beiräge zur Beriebs- und Volkswirschafslehre werden herausragende Arbeien von Sudierenden, Arbeien mi besonders engem Bezug zur Wirschafspoliik sowie akuelle Forschungsarbeien einem breieren Publikum zugänglich gemach. ie Beiräge sind künfig ausschließlich über das Inerne uner wiwischrifenreihe.uni-rier.de zugänglich. (2) ie vorliegende Publikaion wird ersmalig in ihrem neuen Erscheinungsbild dieser universiären Schrifenreihe veröffenlich. ie Schrifenreihe der Sudienschwerpunke Finanzwissenschaf, Beriebswirschafliche Seuerlehre sowie Wirschafsprüfung und onrolling im Fachbereich IV der Universiä Trier wurde von den Professoren r. ierich ickermann (Volkswirschafslehre/Finanzwissenschaf) und r. Mahias Lehmann (Beriebswirschafslehre/Beriebswirschafliche Seuerlehre) im Juli 1982 gegründe. Im ezember 1998 kam Professor r. ieer Rückle (Beriebswirschafslehre/Wirschafsprüfung und onrolling) als drier Herausgeber hinzu. Absich der Herausgeber war es seinerzei, eine Plaform für die wissenschafliche Bearbeiung akueller Themen bei zügiger Publikaionsmöglichkei, für die Präsenaion umfangreicher Forschungsberiche und nich zulez auch für die Veröffenlichung hervorragender iplomarbeien zu schaffen. as is wohl überwiegend, wenngleich aufgrund gelegenlicher finanzieller Engpässe nich immer in koninuierlicher Abfolge gelungen. (3) Bis zum April 2009 umfass die Schrifenreihe in ihren drei Abeilungen 57 Arbeispapiere, 16 okumenaionen und 11 iplomarbeien mi einem brei angelegen Themenspekrum. Ein Gesamverzeichnis der bisher publizieren Beiräge finde sich auf der Websie wiwischrifenreihe.uni-rier.de. ie Veröffenlichungen sind über die Universiä Trier hinaus auch einem weien Fachpublikum zugänglich gemach worden. Sie haben bei ihren Leserinnen und Lesern eine ineressiere, of zusimmende Resonanz und eine wachsende Nachfrage gefunden. as gil nich zulez für die auf diesem Wege publizieren iplomarbeien, welche ansonsen häufig in den Regalen der Bereuer und der Prüfungsämer verschwinden und dami der Öffenlichkei vorenhalen werden. (4) Nach dem Ende der akiven ienszei der bisherigen Herausgeber haben vier jüngere Kollegen mi fachlich vergleichbaren Themenschwerpunken die Schrifenreihe übernommen und werden diese uner genereller Beibehalung der Zwecksezung zukünfig bereuen. ies sind die Professoren r. Axel Adam-Müller (Beriebswirschafslehre/Unernehmens-

finanzierung und Kapialmärke), r. Ludwig von Auer (Volkswirschafslehre/Finanzwissenschaf), r. Georg Müller-Fürsenberger (Volkswirschafslehre/Kommunal- und Umwelökonomie) und r. Michael Olbrich (Beriebswirschafslehre/Wirschafsprüfung und onrolling). (5) ie neuen Herausgeber danken den bisherigen Herausgebern für die langjährige, erfolgreiche Bereuung der Schrifenreihe und hoffen auf posiive Resonanz der Schrifenreihe. ie bisherigen und die neuen Herausgeber: Prof. r.. ickermann Prof. r. M. Lehmann Prof. r.. Rückle Prof. r. A. Adam-Müller Prof. r. L. von Auer Prof. r. G. Müller-Fürsenberger Prof. r. M. Olbrich Trier, im Mai 2009

Tiel: Verfasser: Wha is he Value of he Tax Shield of eb? (iplomarbei Universiä Trier 2008) hrisoph Theis (Bereuer: Prof. r. Axel Adam-Müller) Absrac: More han a quarer-cenury ago, Modigliani and Miller (1963), Myers (1974) as well as Miles and Ezzell (1980) suggesed ha he value of ax shields be compued by discouning he deb ax savings o he presen using a discoun rae ha reflecs he risk of he deb ax savings. The difference beween he various heories lies primarily in he assumed deb policy. In conras o his lieraure, Fernandez (2004) derives he value of ax shields by compuing he difference beween he presen value of axes for he unlevered company and he presen value of axes for he levered company. Fernandez resuls challenge exising valuaion heories, as he claims ha he value of ax shields he derives is generally valid and in fac he only way o compue he correc value of ax shields. This diploma hesis demonsraes ha Fernandez claim is no consisen wih his se of assumpions. Furher, i is shown ha Fernandez value of ax shields expands on he exisen valuaion heories by using an addiional se of assumpions ha includes a new deb policy, bu i does no correc or conradic he sandard valuaion heories. Herausgeber: Adresse: Prof. r. Axel Adam-Müller (adam-mueller@uni-rier.de) Prof. r. Ludwig von Auer (vonauer@uni-rier.de) Prof. r. Georg Müller-Fürsenberger (mueller-fuersenberger@uni-rier.de) Prof. r. Michael Olbrich (olbrich@uni-rier.de) Fachbereich IV Universiä Trier Universiäsring 15 54295 Trier iese Schrifenreihe is im Inerne in elekronischer Form uner wiwischrifenreihe.uni-rier.de erreichbar.

Table of onens Table of Abbreviaions... IV 1 Inroducion...1 2 Basic Model and Valuaion Mehods...3 2.1 Basic Valuaion Model: Modigliani and Miller (1958)...3 2.2 Inroducing orporae Taxes and he Mechanics of he Tax Shield of eb...6 2.3 Valuaion Mehods...8 2.3.1 The Adjused Presen Value Mehod...8 2.3.2 The Free ash Flow Mehod...9 2.3.3 The apial ash Flow Mehod...10 3 Valuaion Theories for he Tax Shield of eb...12 3.1 The Value of he Tax Shield if he Level of Risk-Free eb is onsan and Perpeual...12 3.2 The Value of he Tax Shield if eb is Risky and eb Policy Fixed...13 3.3 Oher Effecs on he Tax Shield of eb...15 3.3.1 Inroducing oss of Financial isress...15 3.3.2 Inroducing Personal Taxes and he Miller Model...16 3.4 The Value of he Tax Shield if he Leverage Raio is onsan...19 3.5 Which Valuaion Mehod Should Be Used wih Which Theory?...22 4 Is he Value of Tax Shields equal o he Presen Value of Tax Shields or no?...26 4.1 The Value of Tax Shields is no equal o he Presen Value of Tax Shields...26 4.1.1 The General Idea...26 4.1.2 The Value of Tax Shields for onsan Perpeuiies...28 4.1.3 The Value of Tax Shields for Growing Perpeuiies...30 4.2 The Value of Tax Shields is equal o he Presen Value of Tax Shields...31 4.2.1 The Presen Value of Tax Shields for Growing Perpeuiies...31 II

4.2.2 The Presen Value of Tax Shields for onsan Perpeuiies...33 5 iscussion...34 5.1 The Miles and Ezzell Framework...34 5.2 The Modigliani and Miller wih Growh Framework...37 5.3 The Value of he Tax Shield if he Leverage Raio is onsan in Book Values 40 6 onclusion...45 Appendix...49 References...56 III

Table of Abbreviaions α APV β β TS β E β L β U B APM F F EP L P discoun rae reflecing he risk of he increase in ne asses adjused presen value deb bea bea of he deb ax shield equiy bea bea of he levered firm unlevered asse bea dollar amoun of deb capial asse pricing model capial cash flow coss of financial disress marke value of deb depreciaion marke value of he levered firm s deb marke value of personal borrowed deb ε +1 random variable of he free cash flow E marke value of equiy EBIT earnings before ineres and ax Ebv book value of equiy EF equiy cash flow E L marke value of he levered firm s equiy E U marke value of he unlevered firm s equiy E( ) expeced value operaor FF free cash flow g growh rae GL gain from leverage G L presen value of axes paid by he levered company G U presen value of axes paid by he unlevered company INV invesmen (change in ne working capial plus capial expendiures) K a consan λ percenage share L consan leverage raio (/V=L) LBO leveraged buyou M pricing kernel ME Miles and Ezzell IV

MM ф PV PVTS ρ r A r Modigliani and Miller random variable of he increase of ne asses presen value presen value of he ax shield of deb unlevered cos of capial expeced asse reurn coss of deb/ expeced reurn of deb r fixed yield r emand demand ineres rae r E coss of equiy/ expeced reurn o equiy r f risk-free ineres rae marke risk-premium r P r Supply supply ineres rae τ τ PB marginal corporae ax rae personal income ax rae applicable o income from bonds i τ PB individual invesor s marginal personal ax rae on income from bonds τ PS personal income ax rae applicable o income from common socks Taxes L axes paid by he levered consan perpeuiy Taxes U axes paid by he unlevered consan perpeuiy TS ax shield from deb V firm value V L value of he levered firm VTS value of ax shields V U value of he unlevered firm WA afer-ax weighed average cos of capial WA BT before-ax WA X random amoun V

1 Inroducion The value of he ax shield of deb has gained considerable aenion in recen years in real world applicaions as well as in he academic lieraure. The recen boom in he privae equiy indusry has subsanially increased he number of highly leveraged ransacions, such as leveraged- or managemen buyous, in order o finance acquisiions. In hese ransacions, leverage is used as a significan source of value added. Kaplan (1989) as well as Newbould, hafield and Anderson (1992) show ha he ax benefis generaed by he use of deb are an imporan source of wealh gain in highly leveraged ransacions. Kaplan (1989), for insance, esimaes he median value of he ax benefis associaed wih managemen buyous o be in he range of 21 percen o 143 percen of he premium paid o pre-buyou shareholders. In addiion, Graham (2000) esimaes ax savings of deb, for a ypical firm in his sample, o be 20 percen of preax income annually and he value of he ax shield of deb o be abou 10 percen of firm value. The deb ax shield herefore represens a significan componen of firm value and hence i is imporan o calculae he correc value of he ax shield of deb. In he financial lieraure, he ongoing debae abou he value of deb ax shields has been reinforced in recen years by Fernandez (2004a), who claims ha his resuls are generally valid and in conflic wih he resuls of exising lieraure. If Fernandez were correc, his paper would be of grea imporance and would have far-reaching consequences in he field of corporae finance. This paper ries o shed ligh on how o correcly value he ax shield of deb. Specifically, I examine he assumpions ha Fernandez (2004a) makes and analyze wheher Fernandez (2004a) approach of valuing he deb ax shield is correc under broad condiions as he claims. Furhermore, I look a which valuaion mehod should be used o find he correc value of he deb ax shield and wheher all valuaion mehods lead o he same resul when appropriaely applied. I should be noed ha his paper does no address he debae on he opimal capial srucure. Tha is, I do no raise he quesion of which capial srucure will maximize he value of he deb ax shield; raher, I focus on wha he value of he deb ax shield will be if a cerain deb policy is given. The remainder of he paper proceeds as follows. haper 2 provides he building blocks by inroducing he basic valuaion model of Modigliani and Miller (1958), explaining he mechanics of he deb ax shield and reviewing hree commonly used discouned cash flow 1

mehods. haper 3 discusses in deail he differen heories on how o value he ax shield of deb. In addiion, he hree valuaion mehods are compared and analyzed when used ogeher wih he differen valuaion heories. haper 4 presens he approach of Fernandez (2004a) followed by he counersaemen of ooper and Nyborg (2006b). In haper 5 he assumpions of Fernandez (2004a) are hen analyzed in deail. haper 6 concludes. 2

2 Basic Model and Valuaion Mehods This chaper provides he building blocks for he paper s analyses on he value of he ax shield of deb. These building blocks are hen used in hapers 3 hrough 5 in he paper s main analyses on he value of he ax shield of deb. 2.1 Basic Valuaion Model: Modigliani and Miller (1958) Modigliani and Miller (1958), hereafer MM (1958), demonsrae in heir seminal paper on capial srucure ha corporae financial decisions are irrelevan for firm valuaion in perfec capial markes. To derive his resul, hey make, eiher explicily or implicily, he following nine assumpions (opeland, Weson and Shasri 2005, 559): (A1) The invesmen policy of he firm is given and consan: The asses of he firm are expeced o generae consan operaing cash flows in perpeuiy. I is imporan o noice ha he cash flow sream is compleely unaffeced by changes in capial srucure. (A2) Firms can be divided ino risk classes: Firms in he same risk class are assumed o have perfecly correlaed cash flows. Hence, he cash flow sreams of wo firms in he same risk class differ a mos by a scale facor. Therefore, invesors will require he same expeced reurn on any wo asses wihin a given risk class. (A3) No axes: There are neiher corporae income nor wealh axes on corporaions, nor personal axes on individual invesors. (A4) No ransacion and bankrupcy coss: apial markes are fricionless and no coss of financial disress occur in he even of bankrupcy. (A5) Symmeric informaion: orporae insiders and ousiders have he same informaion. (A6) No agency coss: Managers always ry o maximize shareholders wealh and hence only inves in value-increasing projecs. 3

(A7) Absence of arbirage opporuniies: Two asses wih an idenical payoff srucure mus sell in equilibrium a he same price. (A8) Individuals can borrow and lend a he risk-free rae. (A9) The capial srucure of a firm consiss only of risk-free deb and risky equiy. Under assumpions (A1) o (A9), MM s (1958) famous Proposiion 1 saes ha he marke value of he levered firm is he sum of he marke value of he firm s deb and he marke value of is common equiy, which is equal o he marke value of he unlevered firm. Thus, in equilibrium, i mus hold ha V = E + = V = E, (1) L L L U U where V L is he marke value of he levered firm, E L is he marke value of he levered firm s equiy, L is he marke value of he levered firm s deb, V U is he marke value of he unlevered firm and E U is he marke value of he unlevered firm s equiy. Noe from equaion (1) ha he value of a firm is independen of how he firm is financed. Thus, i follows ha corporae financial policies do no add value in equilibrium. To esablish heir Proposiion 1, MM (1958) use one of he very firs arbirage pricing argumens in finance heory. They assume he presence of wo firms in he same risk class and wih idenical cash flows. Firm U is capialized wih equiy only and firm L has boh deb and equiy ousanding. Suppose ha he value of he levered firm (V L ) is larger han ha of he unlevered firm (V U ). Then, insead of buying he levered firm s equiy (E L ), an invesor could combine he unlevered equiy (E U ) wih borrowing on personal accoun ( P ) in he same proporion as he capial srucure of he levered firm. 1 In oher words, insead of buying E = V ), he invesor buys E ) = ( V ). Since boh firms have L ( L L ( U P U P idenical cash flows, boh sraegies offer he same payoff in every sae of he world. Hence, he invesor could duplicae he reurn of he levered equiy a lower cos. Hence, in he absence of arbirage opporuniies, V L mus equal V U in equilibrium. onversely, if he levered firm is relaively undervalued, purchasing an equal percenage share (λ) of he levered firm s equiy and deb, λ E + ) = λ( V ), would cos ( L L L less han he same percenage of he all-equiy firm, λ E ) = λ( V ). However, he wo ( U U 1 Assumpions (A8) and (A9) imply ha invesors can borrow on heir own accoun as cheaply as he levered firm, and hus P equals L for a given amoun of deb. 4

sraegies would enile an invesor o exacly he same cash flow. Thus, in equilibrium, he wo firms mus sell for he same price, ha is, V L mus equal V U. These wo arbirage argumens lead o he conclusion ha capial srucure does no influence firm value, because invesors can undo he effec of any changes in capial srucure (Modigliani and Miller 1958). An example ha is ofen used o illusrae he inuiion behind Proposiion 1 is o compare he size of a pie wih he value of a firm. The size of he pie is independen of how he pie is sliced. 2 So oo, he value of he firm is independen of how he firm s financial policy divides is operaing cash flows among differen claimans (debholders and equiyholders). Ulimaely, firm value equals he presen value of operaing cash flows (Brealey, Myers and Allen 2006, 471). The basic principle underlying Proposiion 1 is he Value Addiiviy Theorem. Under he assumpion ha capial markes are complee, his heorem says: Assume no arbirage possibiliies exis. Then he price of a securiy whose payoffs are a linear combinaion of oher asses mus be given by he same linear combinaion of he prices of he oher asses (Varian 1987). Tha is, he value of he whole is equal o he sum of he values of is pars. Hence, he value of a firm in a given risk class should depend only on is operaing cash flows and his value equals he sum of he values of he firm s equiy and deb. The value does no depend on wheher he firm is financed more heavily by deb or equiy (Varian 1987). A firs sigh, MM (1958) seems o be no more han a heoreical economic model wih unrealisic assumpions ha is irrelevan for real-world applicaions. oncerning his skepicism, Miller (1988) replied: Looking back now, perhaps we should have pu more emphasis on he oher, upbea side of he nohing maers coin: showing wha doesn maer can also show, by implicaion, wha does. uring he course of his paper I relax several of he assumpions above in order o examine heir effec on he ax shield of deb. Righ away, by applying he modern asse-pricing approach, (A2) can be relaxed wihou changing he conclusion of he model. The modern asse-pricing approach requires firms wihin a given risk class o have only he same sysemaic risk, no perfecly correlaed cash flows. Since he cos of capial depends on he sysemaic risk of a firm, firms wih he same cos of capial mus be placed in he same risk class (Ross 1988). 2 Meron Miller usually compared he firm value wih he price of a pizza, which is also unchanged by how he pizza is sliced (opeland, Weson and Shasri 2005, 563). This example has he advanage ha he price of a pizza can increase, while he size of a pie can hardly grow. I is lef o he ase of he reader, which illusraion he prefers o choose. 5

2.2 Inroducing orporae Taxes and he Mechanics of he Tax Shield of eb I now urn o he quesion of how corporae axes affec he implicaions of he MM (1958) model, hereby relaxing assumpion (A3). A simple example will illusrae he general mechanics, as well as he calculaion, of he ax shield of deb. For ease of discussion and o mainain consisency wih he bulk of he lieraure, a classical ax sysem (such as ha found in he Unied Saes) is assumed. A classical sysem axes corporae and personal income separaely. The key feaure of a classical sysem is he ax-deducibiliy of ineres paymens a he corporae level, so ha ineres is paid ou of income before axes. In conras, equiy payous are no ax-exemp and are paid from residual corporae income afer axes. orporae income is axed a he marginal corporae ax rae τ (hereafer corporae ax rae), which is assumed o be consan over ime. Personal income on dividends, capial gains, and ineres is axed upon receip by invesors (Graham 2003). For now, however, I assume individual invesors are no axed. Furher, all axes hroughou are assumed o be proporional, if no differenly saed. Table 1, which shows a simplified income saemen of Microsof orporaion, illusraes he advanage of deb financing when axes are inroduced. The lef column is calculaed by applying he de faco capial srucure of Microsof in 2007 wih no deb ousanding and a corporae ax rae of 30 percen. In he righ column, I assume ha $20 billion of equiy is reired and replaced wih a perpeual risk-free bond issued a par value wih an ineres rae of 5 percen. The change in capial srucure reduces he ax bill of Microsof by $300 million and increases he oal cash flow o deb- and equiyholders by he same amoun, as he governmen subsidizes ineres paymens by allowing he firm o deduc ineres paymens as an expense. 6

Table 1 The Tax-educibiliy of Ineres Expenses Microsof orporaion 2007 ($ millions) No eb eb ($20bn bond @ 5%) EBIT 20,101 20,101 Ineres Expense 0 1,000 EBT 20,101 19,101 Taxes(30%) 6,030 5,730 Ne Income 14,071 13,371 ash flow o debholders 0 1,000 ash flow o equiyholders 14,071 13,371 Toal cash flow o deband equiyholders 14,071 14,371 Under he assumpion ha Microsof earns enough axable income o cover ineres paymens 3, he ax shield from deb (TS) generaes a cash flow sream ha is equal o he corporae ax rae imes he risk-free ineres paymen, ash flow from TS = τ *r *, (2) f where r f is he risk-free ineres rae and is he marke value of deb. Recall ha a issuance, he marke value of deb is no necessarily equal o he amoun borrowed or o he nominal value. Moreover, he marke value of deb migh change during he bond s life. This is imporan o noe, because he lieraure does no consisenly use he same variable o compue he ax savings from deb. 4 For simpliciy and in order o remain consisency among he differen approaches in he lieraure, i is assumed hroughou, if no differenly saed, ha he marke value of deb is equal o he nominal value and herefore also equal o he book value of deb. If he cash flow sream from equaion (2) could be expeced o occur every year in perpeuiy, a new valuable asse would resul. The quesion hen becomes: Wha would he value of his asse be, ha is, wha is he value of he ax shield of deb? The value can be found by discouning he cash flows from he fuure annual ax savings. In order o derive a reasonable value, one needs o know wo hings: Firs, he characerisics of he disribuion of he cash flows creaed by he ax shield (depending on he variables corporae ax rae, 3 If he curren ineres deducions canno be offse agains axable income in any given year, US ax law allows a firm o carry hem backward or forward agains pas or fuure axable income or o merge wih anoher firm ha can uilize he deducions. 4 Brealey, Myers and Allen (2006, 470) and Ruback (2002) use he amoun borrowed, opeland, Weson and Shasri (2005, 560) use he principal value and Modigliani and Miller (1963), Myers (1974), Miller (1977) and Taggar (1991), for insance, use he marke value of deb. 7

ineres expense and axable income), and second, he appropriae discoun rae reflecing he ime value and he riskiness of he cash flows. 2.3 Valuaion Mehods In his secion I review hree basic discouned cash flow mehods for valuing companies ha are commonly used in pracice and which explicily or implicily include he value of he ax shield of deb. 5 I is imporan o noe, as Beroneche and Federici (2006) and Fernandez (2007a) show, ha he differen valuaion mehods give he same resul for oal firm value as well as for he value of he deb ax shield, as long as he valuaion mehods rely on he same hypoheses and do no implicily include any addiional assumpions. Indeed, Fernandez (2007a) noes: This resul is logical, as all he mehods analyze he same realiy under he same hypoheses; hey differ only in he cash flows aken as a saring poin for he valuaion. 2.3.1 The Adjused Presen Value Mehod All valuaion mehods ha ry o capure he ax advanage of deb can be represened by he Adjused Presen Value (APV) mehod (ooper and Nyborg 2007). The erm adjused presen value applies because V U --he value of an unlevered firm, which implies ha he firm is all-equiy-financed-- is adjused for he presen value (PV) of he side effecs of oher invesmen and financing opions in order o derive he value of he levered firm (Myers 1974). Possible side effecs are he value of he deb ax shield, subsidized financing, coss of financial disress and issue coss. The APV mehod relies on he principle of value addiiviy, since i splis a company ino pieces, values each piece and hen adds hem up. The basic APV formula is as follows (Brealey, Myers and Allen 2006, 521): APV = V U + sum of PVs of side effecs = V L. (3) Noe ha all valuaion mehods hroughou his chaper assume he only side effec o be he ax shield of deb. I refer o his as he world of MM wih corporae axes, since Modigliani and Miller were he firs o assume ha all he effecs of he financing decision perain o he ax shield of deb. Thus, we have 5 The Equiy ash Flow (EF) mehod, which compues he value of equiy by discouning he expeced cash flows o equiyholders (EF) a he cos of equiy, r E, is no discussed in deail since he focus is on cash flow mehods ha yield esimaes of oal firm value. 8

V L = V U + PVTS = APV, (4) where PVTS is he presen value of he ax shield of deb. In his model, V U represens he effec of invesmen decisions while PVTS capures he effec of financing decisions. Since by assumpion (A9) he firm is financed only by deb and equiy, he following equaion mus hold: E + = V L = V U + PVTS, (5) where and E are he marke values of deb and equiy, which sum up o he firm value. I is imporan o noe ha expressions (4) and (5) are compleely general wih respec o he values of V U and PVTS, and hus furher assumpions are needed o specify hese values (Miles and Ezzell 1980). 2.3.2 The Free ash Flow Mehod The Free ash Flow (FF) mehod derives he value of a firm by discouning he expeced Free ash Flows, period by period, a he afer-ax weighed average cos of capial (variables wihou ime subscrips () are daed ime 0): V = E( FF ) T = 1 (1 + WA ), (6) where V is he firm value, E( ) he expeced value operaor, FF he Free ash Flow and WA he afer-ax weighed average cos of capial. WA is defined as: E 1 1, (1 ) WA = r τ + re,, (7) V1 V1 and r and r E are he coss of deb and equiy, 6 respecively. Usually, hese coss are derived using he capial asse pricing model (APM), r = r f + β * r P and (8) r E = r f + β E * r P, (9) where r P is he marke risk-premium, β he deb bea and β E he equiy bea (Fernandez 2007a). 6 All formulas hroughou assume ha he expeced reurn on deb and equiy, he cos of deb and equiy and he deb bea and equiy bea are defined consisenly. 9

Free cash flow (FF) is he cash flow in excess of he amoun required o fund all projecs ha have posiive ne presen values, and is defined in he sandard way: FF = EBIT ( 1 τ ) + EP INV, (10) where EBIT is earnings before ineres and ax, EP is depreciaion and INV is invesmen (change in ne working capial plus capial expendiures). The ax is calculaed by assuming an all-equiy-financed company. Thus, FF does no include he ax shield of deb (opeland, Weson and Shasri 2005, 510). Since he deb ax shield is excluded from FF calculaions, he ax deducibiliy of ineres is reaed as a decrease in he cos of capial using he afer-ax weighed average cos of capial. Thus, he FF mehod incorporaes he benefi of he ax shield ino he axadjused discoun rae, which is refleced by he afer-ax cos of deb r 1 τ ) ( (Luehrman 1997). The value of he deb ax shield is auomaically included wihou adding a separae componen of value as in he APV mehod. Thus, he FF mehod does no give an explici value for PVTS. However, under he assumpion ha capial markes are complee and he Value Addiiviy Theorem holds, he PVTS can be inferred by he following calculaion (ooper and Nyborg 2006a): PVTS = V L - V U. (11) Hence, PVTS is he difference beween he levered and he unlevered firm s value. 2.3.3 The apial ash Flow Mehod According o Ruback (1995, 2002), he apial ash Flow (F) mehod derives he firm value by discouning he expeced Fs, period by period, a he unlevered cos of capial ρ: V = T E( F ), (12) = 1 (1 + ρ) where he unlevered cos of capial is equal o he expeced asse reurn (r A ): ρ = r = r + β r. (13) A f U P 10

Thus, he unlevered cos of capial only depends on he risk-free rae, he risk premium and he unlevered asse bea (β U ), which capures operaing risk only, ha is, ρ depends only on operaing risk, wih he effecs of financial leverage removed. Hence, ρ is he firm s cos of capial given all-equiy financing and herefore when applying he F mehod he discoun rae does no have o be recompued as capial srucure changes (Ruback 1995). 7 F is defined as all afer-ax cash flows available o capial providers. In a capial srucure wih only ordinary deb and common equiy, F equals he cash flow available o equiyholders plus he cash flow o debholders. In oher words, F equals FF plus he deb ax shield. Since he deb ax shield is included in he F, he discoun rae is before-ax and corresponds o he riskiness of he asses (Ruback 2002). 7 Noe ha Ruback (1995, 2002) claims ha equaion (12) holds in general. This is no correc as will be shown in secion 3.5, because he F mehod uses implici assumpions. 11

3 Valuaion Theories for he Tax Shield of eb In his chaper, he main heories for valuing he firm value and he deb ax shield are presened and analyzed. In addiion, he resuls of he hree valuaion mehods according o he differen valuaion heories are compared and discussed. 3.1 The Value of he Tax Shield if he Level of Risk-Free eb is onsan and Perpeual As I noe above, addiional assumpions are needed o derive specific values for V U, PVTS and V L. Modigliani and Miller (1963), hereafer MM (1963), inroduce he firs heory for deriving he firm value and he value of he ax shield of deb. MM (1963) assume he expeced free cash flow o be consan and permanen. This implies ha alhough he acual cash flow each period is risky, once each period s cash flow has been received he value of he firm is always he same, hence he company has zero expeced growh (ooper and Nyborg 2006a). Noe ha FF is equal o EBIT 1 τ ) ( under MM (1963) since he firm is assumed o be a perpeuiy and herefore depreciaion equals invesmen each year o keep he same amoun of capial in place (opeland, Weson and Shasri 2005, 561). The firm s risk-free deb is perpeual and consan a he level. This implies ha he fuure levels of are known wih cerainy a ime = 0. Furher, he company always earns enough axable income o obain he full ax benefi of ineres deducions and he corporae ax rae remains consan. Based upon he above, he expeced capial cash flow o deb- and equiyholders is E( F) = ( E( EBIT) r )(1 τ ) r, (14) f + which can be wrien as he sum of wo componens, namely, a sochasic sream E EBIT)(1 τ ), and a non-sochasic sream, τ. The firs componen is he expeced ( FF, which is equal o he cash flow o he unlevered firm. Therefore, i is discouned a ρ, a rae ha appropriaely reflecs he risk of he expeced cash flow o he unlevered firm. The second componen, he ax savings from deb, is riskless and hus i is discouned a he risk-free rae, r f. Therefore, he value of a levered firm can be given by (Modigliani and Miller 1963): r f f 12

V L E( FF) τ r f = + = V ρ r f U + τ. (15) Equaion (15) implies ha firms should finance heir operaions wih 100 percen deb, because he marginal benefi of deb is τ, which is ofen assumed o be posiive. Addiionally, since τ is assumed o be consan, firm value increases linearly wih (Graham 2003). Noe ha equaion (15) capures only he ax benefi of deb bu conains no offseing coss of deb. Equaion (15) correcs an inaccuracy in MM (1958), where he ax advanage of deb was due solely o he fac ha he deducibiliy of ineres paymens implied a higher level of afer-ax income for any given level of before earnings (Modigliani and Miller 1963). Under formula (15), an addiional gain appears, since he non-sochasic ax savings from deb is discouned a a lower rae han he sochasic sream E EBIT)(1 τ ). Given ha leverage creaes he non-sochasic cash flow sream, τ r f (, he variabiliy of oal cash flows is reduced by leverage (Modigliani and Miller 1963). The approach of MM (1963) has fairly resricive and unrealisic assumpions, which lead o he implicaions above. However, he purpose of heir paper was o show he ax advanage of deb financing. MM did no claim ha heir assumpions were realisic (Koller, Goedhar and Wessels 2005, 720). Indeed, MM (1963) poin ou ha equaion (15) gives only an upper bound on he value of he firm. 3.2 The Value of he Tax Shield if eb is Risky and eb Policy Fixed In his secion, I addiionally relax assumpion (A9), which assers ha only risk-free deb exiss, by inroducing risky deb. eb is defined as risky in he sense ha he value of deb can change over ime if he yield is fixed and he cos of deb changes. Noe ha his assumes ineres rae risk bu no defaul risk of a bond. Furher, deb is perpeual and fixed a a dollar amoun (B) wih a fixed yield (Ruback 2002). This implies ha he fuure dollar amouns of deb ousanding are known wih cerainy and hus deb policy is fixed in = 0. Noe ha he dollar amoun of deb ousanding (B) is no assumed equal o he marke value of deb () over ime. Ruback (2002) compues he cash flow sream from deb ax savings as follows: ash flow from TS = τ r B, (16) 13

where r is he fixed yield on he risky deb. The quesion ha arises is: wha is he appropriae discoun rae for his cash flow sream? In a APM framework, he discoun rae for deb ax shields (r TS ) should depend on he bea of deb ax shields (β TS ) (Ruback 2002): TS = r f + β TS rp (17) r * Ruback (2002) shows ha, under he assumpions given, he bea of he deb ax shield equals he bea of he deb. 8 The deb ax shield has he same amoun of sysemaic risk as he deb. 9 This implies ha he appropriae discoun rae for deb ax shields is he expeced reurn on he deb. Thus, he following equaion holds: V L E( EBIT)(1 τ ) τ r B = + = V ρ r U + τ, (18) where r is he expeced reurn on deb and he value of deb is defined as r =. (19) r B Noe ha he cash flow sream from equaion (16) is non-sochasic. However, he PVTS is subjec o risk since i varies as he cos of deb or equivalenly he value of deb changes. Noe also ha equaion (18) is perfecly consisen wih equaion (15): if deb is assumed o be riskless, hen he deb ax shield will also be riskless and should be discouned a he risk-free rae. Thus, equaion (18) is also called he exended MM approach, where he ax savings from deb have he same bea as he deb and are herefore discouned a r (ooper and Nyborg 2006a). Myers (1974) proposes a generalized approach o equaions (15) and (18) o allow for finie and uneven expeced operaing cash flow sreams. In paricular, if fuure deb levels are currenly known wih cerainy, Myers suggess discouning he ax savings, τ r, a he cos of deb (r ), since he argues (similar o Ruback 2002) ha he risk of he ax savings arising from he use of deb is he same as he risk of he deb. Hence, he value of a levered firm whose useful life ends a ime T can be given by: 8 For he formal proof, see Appendix A. 9 If deb is assumed o be fixed in value, hen, as Ruback (2002) poins ou, he bea of deb ax shields is zero and has no sysemaic risk regardless of he deb bea. 14

V L = E( FF ) τ r T T 1 + = 1 (1 + ρ) = 1 (1 + r ), (20) Equaions (18) and (20) coninue o imply ha a firm should ake up as much deb as possible. However, empirical evidence conradics his analysis, since firms are seldom highly leveraged. Miller (1988) pus i his way: We seemed o face an unhappy dilemma: eiher corporae managers did no know (or perhaps care) ha hey were paying oo much in axes; or somehing major was being lef ou of he model. To solve his empirical paradox, eiher oher offseing coss mus be associaed wih issuing deb, or he ax benefi of deb mus be lower han expeced. 10 3.3 Oher Effecs on he Tax Shield of eb 3.3.1 Inroducing oss of Financial isress The implicaions in secions 3.1 and 3.2 ha a firm should be financed wih 100 percen deb are somewha exreme. In his secion, I addiionally inroduce an offseing cos of deb erm by relaxing assumpion (A4), which assers ha no coss of financial disress (F) exis. Financial disress occurs when cash flow is no sufficien o cover curren obligaions o crediors. F include direc coss like legal and adminisraive coss of bankrupcy as well as indirec coss like moral hazard, agency, monioring and conracing coss, which can desroy firm value even if formal defaul is avoided (Myers 1984). 11 To deermine expeced F, which are imporan o he calculaion of V L, he coss menioned above need o be muliplied by he probabiliy of disress, which increases wih addiional borrowing. Thus, V L can be broken down ino hree componens (Brealey, Myers and Allen 2006, 476): V L = V U + PVTS - PV (expeced F). (21) Expression (21) is also known as he Saic Tradeoff Hypohesis, in which he incenive o finance wih deb increases wih he corporae ax rae and firm value increases wih he use of deb up o he poin (he opimal deb raio) where he marginal cos equals he marginal benefi of deb (Graham 2003). However, empirically he expeced F appear 10 Under he assumpion of asymmeric informaion and coss of financial disress, Myers (1984) suggess a hird way o explain he empirical paradox: he Pecking Order Theory, according o which he firm prefers inernal o exernal financing and deb o equiy if i issues securiies. The Pecking Order Theory assumes he deb ax shield o be a second-order effec. For reasons of space, however, I do no rea his heory in deail. 11 For a comprehensive analysis of hese indirec coss, see Jensen and Meckling (1976) and Myers (1977). 15

o be very small compared o he apparenly large ax benefis derived from equaions (18) or (20) (Andrade and Kaplan 1998). Thus, he PVTS may no lose is imporance by he offseing cos of deb erm and he quesion remains why corporaions do no use he PVTS more aggressively. Empirical sudies by Wald (1998) and Rajan and Zingales (1995) reinforce his quesion even furher. They show across a range of counries including Japan, Germany, he Unied Kingdom and he Unied Saes ha he mos profiable firms wih high amouns of axable income o shield borrow he leas. Since he F do no appear o fully explain he empirical paradox, I urn now o he second explanaion for why he ax benefi of deb migh be lower han expeced by inroducing personal axes. 3.3.2 Inroducing Personal Taxes and he Miller Model In a classical ax sysem, including personal axes, he afer-personal-ax value of $1 o deb invesors is $1(1-τ PB ) and o equiy invesors is $1(1-τ )(1-τ PS ), where τ PB is he personal income ax rae applicable o income from bonds and τ PS is he personal income ax rae applicable o income from common sock. 12 The ne ax advanage of $1 of deb payou, relaive o $1 of equiy payou, is (1- τ PB ) (1-τ )(1-τ PS ). (22) If expression (22) is posiive, deb ineres is he ax-favored way o reurn capial o invesors and he firm has a ax incenive o issue deb insead of equiy (Graham 2003). In his well-known model, Miller (1977) claims ha since he empirical paradox canno be convincingly explained by inroducing coss of financial disress he ax advanage of deb financing mus be subsanially less han he convenional wisdom suggess. Miller (1977) argues ha in a world wih fully deducible ineres expenses, personal axes can eliminae he ax advanage of deb wihou he need for coss of financial disress. Miller shows ha when personal and corporae income axes are aken ino accoun, he value of a levered firm is given by: V L = V U (1 τ )(1 τ + 1 (1 τ PB ) PS ), (23) 12 τ PS is assumed o be a blended dividend and capial gains ax rae. 16

where (1 τ )(1 τ 1 (1 τ PB ) PS ) = GL is he gain from leverage. The inuiion of he erm in brackes is ha as long as he afer-personal-ax income from bonds, 1 τ ), is higher ( PB han ha from common sock, 1 τ )(1 τ ), here is a gain from corporae leverage ( PS (Miller 1977). Noe ha whenτ = τ, or when here are no personal income axes a all, PB PS he gain from leverage is τ, he same resul as in equaion (15) under he original MM (1963) model wih corporae axes. Thus, Miller (1977) expands he model of MM (1963) by inroducing personal income axes. Miller assumes in his model ha τ PS equals zero, all firms face he same corporae ax rae, τ, and personal income ax is progressive. Moreover, regulaions by ax auhoriies preven axable invesors from eliminaing heir ax liabiliies. 13 In such a world, he marke equilibrium for corporae deb would be ha picured in Figure 1: Figure 1 Equilibrium in he Marke for orporae eb r Supply = r f 1 1 τ Supply r emand = r f 1 1 τ i PB r f emand B* Quaniy of Bonds ousanding Source: Miller (1977) where r Supply is he supply ineres rae, r emand he demand ineres rae, B* he equilibrium quaniy of aggregae corporae deb and τ i PB he individual invesor s marginal personal ax rae on ineres income (Miller 1977). The horizonal supply curve of bonds is perfecly elasic because he marginal ax benefi of deb is consan for all firms and hus all offer he same pre-ax ineres rae, r supply. The demand curve is iniially horizonal, represening demand by ax-free invesors, bu evenually slopes upward since personal income ax is progressive and he demand 13 To mainain coninuiy wih MM (1963), Miller (1977) assumes perpeual and riskless deb. 17

ineres rae has o keep rising o arac invesors from higher and higher ax brackes. The inuiion is ha he higher personal ax o income from bonds relaive o equiy causes invesors o demand higher pre-ax reurns on deb, oher hings being equal (including risk), in order o equalize afer-ax reurns from deb and equiy (Graham 2003). The marke equilibrium, defined by he inersecion of he wo curves, has an equilibrium ineres rae of 1 rf 1 τ 1 = rf 1 τ i PB. Observe ha he ax benefi for corporaions vanishes enirely because he higher pre-ax reurn for corporae deb required by he marginal invesor jus offses he advanage of corporae deb financing. Thus, in he Miller model, he gain from leverage, GL, equals zero and a firm s value should be independen of is capial srucure (Miller 1977). This explanaion works only if all firms face he same corporae ax rae, which is an unrealisic assumpion given he large non-deb ax shields (e.g., depreciaion and invesmen ax credis) ha differ across firms (Myers 1984). 14 Even if one disagrees wih Miller s resul ha GL = 0, equaion (23) can be seen as a general version of Miller s argumen. As long as τ PS is less han τ PB, he gain from leverage will be less han τ in he original version of MM (1963) (Miller 1977). Several empirical sudies (e.g., Graham 2000, Kemsley and Nissim 2002) find evidence of a posiive ax benefi when firms use corporae deb, whereas Fama and French (1998) fail o find any increase in firm value for deb ax savings. The empirical sudy by Kemsley and Nissim (2002) esimaes for he US ha he ax savings from deb ne of personal axes is approximaely 40 percen of deb balances, which was close o he marginal corporae ax rae a ha ime. 15 Recenly, yreng and Graham (2007) find in an even sudy of he anadian income rus marke ha each addiional dollar of deb increases firm value by $0.38, which is no saisically differen from he sauory corporae ax rae of anadian firms. espie hese findings, he empirical value of he deb ax shield remains an open quesion since i is difficul empirically o disinguish beween he impac of leverage on value and he impac of facors such as profiabiliy, wih which leverage is associaed (ooper and Nyborg 2004). ooper and Nyborg (2007) sugges using he full corporae ax rae, τ, as a reasonable assumpion for he ne ax savings from 14 eangelo and Masulis (1980) presen a model where he deb ax shield decreases in nondeb ax shields. This implies ha he supply of deb curve can become downward sloping. Under his assumpion, he ax benefi of deb sill adds value for some high ax rae firms. 15 Noe ha he US ongress passed in mid-2003 a law ha largely reduced he ax rae on boh dividends and capial gains o 15 percen for individual invesors (Graham 2003). Hence, he ax savings from deb ne of personal axes migh be lower oday for he US han a he ime when Kemsley and Nissim (2002) conduced heir empirical sudy. 18

deb when valuing he deb ax shield, as long as he ax sysem is a classical ax sysem in which τ PS equals approximaely τ PB and he company can use all is fuure ineres expenses o save ax. Therefore, i is reasonable o use τ as a good approximaion in he sandard valuaion mehods in order o calculae he ax savings from deb under a classical ax sysem. 3.4 The Value of he Tax Shield if he Leverage Raio is onsan In his secion, i is assumed ha equaion (4) sill holds, ha is, he value of he levered firm is equal o he marke value of he unlevered firm plus he value of he ax shield of deb. Hence, no coss of financial disress exis and here are no personal axes. I inroduce he new assumpion ha he deb policy follows a consan leverage raio so ha he marke value of deb is a consan proporion of sochasic firm value. This is in conras o he previous secions where all fuure deb levels were assumed o be known wih cerainy so ha he firm had prese levels of deb. When we assume ha he value of a firm follows a random walk 16 over ime, because each period s expeced free cash flow also follows a random walk, one can argue ha i is inconsisen o assume ha he fuure deb levels are known wih cerainy in =0 (Taggar 1991). 17 Miles and Ezzell (1980) are he firs o address his inconsisency. They show ha he only assumpion ha is generally consisen wih he sandard use of a consan WA in he FF mehod, regardless of fuure paerns of FFs, is he assumpion ha he firm mainains a consan deb-o-value raio in marke values (/V). Hence, Miles and Ezzell (1980, 1985), herafer ME, assume he firm rebalances is capial srucure a he end of every year o mainain a consan leverage raio. This implies ha only he firs-period deb level is known and he firs-period deb ax shield is non-sochasic. Afer he firs period, if a consan leverage raio is assumed, fuure deb levels () are sochasic since hey are dependen on he sochasic fuure firm value (V) and consequenly ax savings from deb, τ r, canno be known wih cerainy. Thus, even hough he firm migh issue only riskless deb, he deb ax shields become a sochasic cash flow sream (Miles and Ezzell 1980). Harris and Pringle (1985) assume ha all deb ax shields, including ha of he firs period, are sochasic. Their approach is analogous o ME, as shown by Taggar (1991), if 16 A random walk is defined as a series of price changes ha are unpredicable and which are only subjec o new informaion (see, for insance, Bodie, Kane and Marcus 2008, 340). 17 Noe ha he assumpion ha firm value follows a random walk implies ha he expecaions abou fuure cash flows are revised on he arrival of new informaion (Arzac and Glosen 2005). 19

he firm s cash flows are coninuous and deb levels are adjused insananeously. As Harris and Pringle (1985) poin ou, heir assumpions yield o valuaions ha differ only slighly from he original ME approach wih discree cash flows and deb level adjusmens. Throughou he remainder of he paper, he Miles and Ezzell leverage policy wih coninuous rebalancing (hereafer, he ME deb policy) is assumed, which implies ha all deb ax shields are assumed o be sochasic. This is done in order o avoid unnecessary complexiy and o be consisen wih he lieraure in he following chapers. Taggar (1991) shows ha if he ME deb policy is assumed, he cash flow sreams from ax savings are as risky as he cash flow sreams o he unlevered firm and should herefore be discouned a he same rae, ρ. As equaion (11) shows, he FF mehod can be replaced by he basic APV mehod in order o derive an explici value for PVTS. The following expression for he value of a levered firm whose useful life ends a ime T and ha only issues risk-free deb hen obains: V L = E( FF ) T T + = 1 (1 + ρ) = 1 (1 + ρ) E( TS ), (24) where E(TS ) is he expecaion a ime 0 of he ax savings from deb a ime, which is equal o E TS ) τ r L[ E( V )], where L = / V is he consan leverage raio and E(V -1 ) ( = f 1 he expecaion a ime 0 of he firm value a ime -1. Expression (24) is also applicable o firms ha are level perpeuiies (Miles and Ezzell 1985 and Taggar 1991). The main difference beween he approach of MM (1963) or Myers (1974) wih a fixed deb policy and he ME (1980) approach wih a consan leverage raio policy is due o he assigned risk and magniude of PVTS. 18 For boh heories, he discoun rae for he FFs is ρ, he unlevered cos of capial. The heories differ in he discoun rae for he ax savings from deb. Since MM (1963) and Myers (1974) use he cos of deb, a higher value is assigned o PVTS han under he ME approach, which uses ρ. 19 Hence, he heories differ only in he assigned risk and value o he ax shield of deb (Ruback 2002). The difference in he risk of PVTS is caused by he differen assumpions abou deb policy: fuure deb levels are known wih cerainy in he MM (1963) and Myers (1974) 18 The assumpions in he MM (1963) approach lead o a consan leverage raio as well, since MM assume a saic amoun of deb and a saic firm value. However, he crucial poin is ha ax savings from deb are nonsochasic in he MM approach since fuure deb levels are fixed (ooper and Nyborg 2006a). 19 Noe ha under he ME deb policy he risk of he deb ax shield is equal o he risk of he FFs. This implies he special case ha if he FFs have no sysemaic risk, ρ is equal o he risk-free rae, which is also he discoun rae for he deb ax shields in his case. 20

approach. onsequenly, he risk of he non-sochasic ax savings from deb is equal o he risk of deb. By conras, he fuure deb levels are sochasic if a ME deb policy is assumed. Thus, he ax saving from deb in each period becomes sochasic and has risk equal o he operaing risk of he firm (ooper and Nyborg 2006a). The deb policy-specific effec on he gain from leverage can also be seen in Figure 2. onsider wo firms, boh of which are level perpeuiies; firm value is no expeced o change for hese firms. The firs firm, called MM, follows he MM (1963) approach wih permanen and risk-free deb, and hus is fuure deb levels are known wih cerainy. The firm called ME follows he ME approach and hence faces uncerain fuure deb levels. Now assume boh firms issue an addiional dollar of perpeual deb. The marginal gain from leverage is represened by he erm dv/d, which indicaes he value creaed a ime =0 by an addiional dollar of perpeual deb issued a ime =0. Asse bea represens he firm s operaing risk (Miles and Ezzell 1985). 20 The gain from leverage per addiional dollar of deb for he firm MM (yellow line) is derived from equaion (15) and equals dv/d = τ regardless of he asse bea. In conras, he firm ME wih a proporional deb policy (blue line) derives a gain from leverage from equaion (24) if V -1 is no expeced o change, ha is: dv d r f τ ρ =. 21 (25) Noe ha in he figure, i is assumed ha τ = 0.3, r f = 0.055 and r P = 0.05. 20 Operaing risk of he firm is equivalen o sysemaic risk of he firm s free cash flows in a APM world. The former is used o beer illusrae he fac ha his risk is only dependen on he firm s operaions and no on financing decisions. 21 The formula appears o be slighly more complex in Miles and Ezzell (1985), since hey assume he firsperiod deb ax shield o be non-sochasic. 21

Figure 2 The Relaionship Beween he Gain from Leverage and he Asse Bea dv/d 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0 0.5 1 1.5 asse-bea Gain from Leverage (ME) Gain from Leverage (MM) Based upon: Miles and Ezzell (1985) Figure 2 shows ha he marginal gain from leverage is only equal o τ if deb is assumed o be consan and non-sochasic under he MM (1963) approach or in he absence of operaing risk under he ME approach. However, if operaing risk is assumed o be posiive he marginal gain from leverage is less han τ under he ME approach even in he absence of personal axes. Moreover, dv/d appears o be a decreasing funcion of he firm s operaing risk, as measured by eiher he asse bea or ρ, if he ME deb policy is applied. This illusraes he dependency of he PVTS on he operaing risk of he firm if he ME deb policy is assumed (Miles and Ezzell 1985). 22 3.5 Which Valuaion Mehod Should Be Used wih Which Theory? As noed above, he hree valuaion mehods discussed in secion 2.3 lead o he same resuls as long as he same assumpions or heories are applied and he valuaion mehods hemselves do no make any implici assumpions excep heir general hypoheses (Fernandez 2007a). Given a paricular se of assumpions, however, i is of pracical imporance o know which mehod is easier o apply and less prone o error. The advanage of he FF mehod is is simpliciy: he FF mehod calculaes he value of a levered firm in one sep. The FF mehod is mos commonly applied in pracice when he leverage raio is consan and hus he risk of he deb ax shield is he same as he 22 Arzac and Glosen (2005) derive a similar conclusion by working in a pricing kernel framework and applying he modern asse pricing approach. 22

risk of he firm. However, according o Taggar (1991), he FF mehod and heir basic equaions (6) and (7) can be used under any assumpion abou he risk of he deb ax shield because he valuaion of he deb ax shield is capured in he cos of equiy, r E. This implies ha he FF mehod can be used under any assumpion abou deb policy. The only variable ha differs when he risk of he deb ax shield changes is he cos of equiy. Neverheless, he FF mehod works bes for firms ha have a long-run arge capial srucure, which is in line wih he assumpion of a consan leverage raio. In he case of a dynamic capial srucure, however, he WA mus be adjused each period o reflec changes in capial srucure. These adjusmens are prone o errors and he WA is easy o misesimae. Hence, he PVTS migh be incorrecly esimaed oo (Luehrman 1997). Ruback (2002) demonsraes ha when a consan leverage raio is assumed, he F and FF mehods are equivalen o each oher. The F mehod includes he deb ax shield in he cash flows whereas he FF mehod incorporaes he ax benefi of deb in he WA. Ruback (2002) furher poins ou ha he advanage of he F mehod, compared o he FF mehod, is is simpliciy whenever he capial srucure changes over ime and deb is forecased in levels insead of as a proporion of firm value. Ruback argues ha since he F mehod includes he deb ax shield in he cash flows, he discoun rae does no change when leverage raios change. 23 I is imporan o noice ha he F mehod discouns all cash flows, including he ax savings from deb, a he unlevered cos of capial, which is only dependen on operaing risk. However, since he ax savings from deb are discouned a ρ, he F mehod implicily makes he assumpion ha he risk of he deb ax shield maches he operaing risk of he firm (ooper and Nyborg 2006a). Ruback (1995, exhibi 1) demonsraes by way of example how o use he F mehod o value a firm ha has undergone a leveraged buyou (LBO) and for which he schedule of fuure deb levels is given. By applying he F mehod in his example, he ax saving from deb is implicily discouned by he unlevered cos of capial. Now, exending Ruback, assume furher ha he deb policy is fixed (i.e., fuure deb levels are known wih cerainy, which is very likely in an LBO ransacion) and he ax savings from deb are realized when ineres is paid. Then, we know from secion 3.2 ha he risk of he ax shields maches he risk of deb and consequenly he appropriae discoun rae mus be he cos of deb. Since he F mehod uses ρ as he consan discoun rae for valuing he deb ax shield, he F mehod is inconsisen wih a fixed deb policy when ineres expenses 23 Ruback (1995) saes: The inroducion of... a changing capial srucure has no impac on he expeced asse reurn which is used o discoun he apial ash Flows. 23

can be fully deduced in every period. The reason is ha, as noed earlier, unlike he oher valuaion mehods he F mehod makes he implici assumpion ha he risk of he ax shield is always equal o he firm s operaing risk. As Fernandez (2007a) shows, he F mehod holds in general if a discoun rae is used ha is no always equal o he unlevered cos of capial bu consisen wih a fixed deb policy. Fernandez (2007a) uses he beforeax WA, which he defines as: E 1 1 WA BT, = r, + re,. (26) V1 V1 However, he WA BT varies over ime when a fixed deb policy is assumed. Therefore, he F mehod would lose is simpliciy. In order o jusify he F mehod inroduced by Ruback (2002), even when he deb policy is fixed, proponens claim ha he fuure ax savings from deb should depend on he level of fuure operaing income. If he company does no have enough axable income o shield, ineres canno be used o save ax immediaely and hence he risk of he ax shield will be higher han he risk of deb. In such cases, proponens assume, as a simple approximaion, ha he risk of he ax shield of deb is equal o he operaing risk of he firm. Given his assumpion, he ax savings may be discouned a he unlevered discoun rae wihou assuming a specific deb policy. Therefore, if he fuure ax saus of he firm is uncerain because he firm could become non-axpaying, he F mehod migh be a good approximaion in pracice (ooper and Nyborg 2007). Noe ha he reason for discouning he deb ax shield a he unlevered cos of capial in he F mehod is hen he uncerain fuure ax saus of he firm, which is independen of he firm s deb policy. In he FF and APV mehods, however, a proporional deb policy is assumed, which forces he deb ax savings o have he same risk as he firm s operaing risk. I is furher assumed ha he firm will always be paying axes. Hence, he deb ax shield is discouned a he unlevered cos of capial (ooper and Nyborg 2006a). If he leverage raio is no assumed o be consan and he firm will remain axpaying in he fuure, he approach ha is less prone o error han he oher mehods is he APV mehod, using a discoun rae for he deb ax shield ha correcly reflecs he firm s risk (ooper and Nyborg 2007). Since he APV mehod lays ou each componen of value separaely, i is excepionally ransparen. In paricular, if he schedule of fuure deb levels is given iniially, he APV mehod provides a echnique o value each period s deb ax shield separaely (Luehrman 1997). onsisen wih his resul, Koller, Goedhar and 24

Wessels (2005, 104) sugges he use of he APV mehod when capial srucure changes significanly over ime, as in an LBO or in he valuaion of disressed companies. 25

4 Is he Value of Tax Shields equal o he Presen Value of Tax Shields or no? The previous secions have shown ha each valuaion heory relies on a specific se of assumpions abou he firm s deb policy, assumpions ha affec he value of he ax shield of deb. The debae abou he value of ax shields is sill ongoing, however, and has been reinforced by he developmen of a new approach by Fernandez (2004a). Fernandez (2004a) claims ha The value of ax shields is NOT equal o he presen value of ax shields and inroduces a new heory on how o calculae he value of deb ax shields. As a reply o Fernandez (2004a), ooper and Nyborg (2006b) argue, The value of ax shields IS equal o he presen value of ax shields and claim ha he sandard valuaion heories are sill valid. Boh approaches and heir underlying assumpions are presened in his chaper. 4.1 The Value of Tax Shields is NOT equal o he Presen Value of Tax Shields Fernandez (2004a) derives he value of ax shields for wo differen cases: for perpeuiies wih no growh and for perpeuiies wih consan growh. Fernandez hen argues ha he value of ax shields depends only on growh characerisics of he firm and is independen of deb policies. 4.1.1 The General Idea Fernandez (2004a) claims ha he only way o obain he correc value for he deb ax shield ha holds under broad condiions is o compue he difference beween he presen values of wo separae cash flows, each wih is own risk. Tha is, he difference beween he presen value of axes paid by he unlevered firm minus he presen value of axes paid by he levered firm. The general idea underlying he model of Fernandez (2004a) can be explained by hree simple expressions, where coss of financial disress are explicily ignored. From equaion (5), we know ha if he Value Addiiviy Theorem holds, hen he afer-ax value of a levered firm is: E L + = V L = V U + VTS. (27) 26

Hence, he sum of he marke values of deb () and levered equiy (E L ) is equal o he value of he unlevered firm (V U ) plus he value of ax shields from deb (VTS). Noe ha he erm PVTS (Presen value of deb ax shields) from previous secions is replaced by he erm VTS (value of ax shields) o be consisen wih he claim ha The value of ax shields is NOT equal o he presen value of ax shields. In addiion o shareholders and bondholders, here is a hird sakeholder in he company, namely, he governmen, which has a claim on axes. If here are no coss of financial disress and FFs are unaffeced by changes in capial srucure, he before-ax value of he unlevered firm mus be equal o he before-ax value of he levered firm. Therefore: V U + G U = V L + G L, (28) where G U is he presen value of axes paid by he unlevered company and G L he presen value of axes paid by he levered company. Equaion (28) means ha he oal value of he firm, which is he afer-ax value of he firm plus he presen value of axes, is independen of capial srucure (Fernandez 2004a). Knowing from (27) ha VTS = V L - V U, we ge: VTS = G U - G L. (29) Hence, VTS is he difference beween wo presen values of wo cash flows wih differen risk (Fernandez 2004a). Fernandez (2004a) admis ha calculaing he presen value of ax savings from deb, as suggesed by he sandard lieraure, is no necessarily wrong if he appropriae discoun rae is used. 24 However, he claims ha i is no feasible o evaluae he riskiness of he difference beween wo expeced cash flows wih differen risk in order o derive he appropriae discoun rae. Thus, Fernandez (2004a) claims ha perhaps he mos imporan issue... is ha he erm discouned value of ax shields in iself is senseless. Hence, Fernandez (2004a) approach challenges he convenional resuls in he lieraure. In paricular, he argues ha he sandard valuaion heories by MM (1963), Myers (1974), ME (1980), Harris and Pringle (1985) and Ruback (2002) resul in inconsisen valuaions of he ax shield or inconsisen relaion beween he cos of capial of he unlevered and levered firm (Fernandez 2004a). 24 Noe ha he sandard lieraure, as shown in he previous chaper, compues firs he difference beween he wo cash flows, which is he deb ax saving, and hen his difference is discouned a he appropriae rae in order o derive he presen value of ax savings from deb. 27

4.1.2 The Value of Tax Shields for onsan Perpeuiies Fernandez (2004a) only explici assumpion is a consan perpeuiy, which implies an expeced growh rae of zero (g=0), which is analogous o assuming ha he firm is following a random walk wih no drif indefiniely as in he MM (1963) seing. Fernandez (2004a) does no assume a specific deb policy since he claims ha his resul for he VTS is generally valid for consan perpeuiies independen of deb policy. In order o derive he VTS for a consan perpeuiy from equaion (29), one has o know he axes paid by he unlevered and he levered company and he appropriae discoun raes reflecing he risk of boh of he cash flows. The axable income for an unlevered firm a ime can be derived from equaion (10) by solving for EBIT: EBIT = FF EP + INV ) /(1 τ ). (30) ( Since he firm is assumed o be a consan perpeuiy, i is known from secion 3.1 ha depreciaion (EP) equals invesmen (INV) o keep he same amoun of capial in place. Thus, for a consan perpeuiy, equaion (30) simplifies o: EBIT = ( FF ) /(1 τ ). (31) Hence, he axes paid every year by he unlevered consan perpeuiy, Taxes U,, are: Taxes = EBIT τ = ( FF ) τ /(1 τ ). (32) U, In expecaions, he axes o be paid every year by he unlevered company, E(Taxes U, ), are: 25 E ( TaxesU, ) = E( FF ) τ /(1 τ ). (33) On he oher hand, he axable income of he levered firm a ime can be inferred from he equiy cash flow (EF) formula: EF = ( EBIT r 1)(1 τ ) + EP INV +, (34) 25 Fernandez (2004b) saes ha he negleced o use expeced value noaions for he sake of simpliciy in Fernandez (2004a). I will include however he expeced value operaor E( ) ino he formulas o be as specific as possible. 28

where = 1is he change in he amoun of deb ousanding a ime. Solving equaion (34) for he axable income of a levered firm, we ge: EBIT r = ( EF EP + INV ) /(1 τ ). (35) 1 For he levered consan perpeuiy, Fernandez (2004a) claims ha..axes paid by he levered company are proporional o EF... Therefore, he obains: Taxes = ( EBIT r 1 ) τ = ( EF ) τ /(1 τ ), (36) L, where Taxes L, are he axes paid by he levered consan perpeuiy a ime. Hence, in expecaions, he axes o be paid every year by he levered company, E(Taxes L, ), are: E ( TaxesL, ) = E( EF ) τ /(1 τ ). (37) In order o ge he presen values of he wo expeced cash flow sreams from equaions (33) and (37), one has o derive he appropriae discoun rae reflecing he risk of each cash flow sream. As can be seen from equaions (32) and (36), he axes paid by he unlevered firm are proporional o FF and he axes paid by he levered firm are proporional o EF under he assumpions ha Fernandez (2004a) makes. Hence, Fernandez (2004a) concludes ha in he case of consan perpeuiies, he axes of he unlevered firm have he same risk as he FF and he axes of he levered firm have he same risk as he EF. Thus, he axes of he unlevered firm mus be discouned a he unlevered cos of capial, ρ, and he axes of he levered firm mus be discouned a he expeced reurn o equiy, r E (Fernandez 2004a). Applying hese findings i follows ha: VTS E( Taxes ) E( Taxes ) E( FF) τ /(1 τ ) E( EF) τ /(1 τ ) U L = GU GL = = (38) ρ re ρ re and by knowing ha E = company in his case), we obain: ( FF) / ρ VU and E EF) / r E = EL ( (he equiy of he levered VTS = ( V U E L ) τ /(1 τ ). (39) As from equaion (27), since V U E L = VTS, i follows ha (Fernandez 2004a): VTS = τ. (F16) 29

Thus, Fernandez (2004a) inroduces a new approach for deriving a resul ha was found, for insance, by MM (1963) or Myers (1974) (see secions 3.1 and 3.2). The difference, however, is ha Fernandez claims ha his resul is generally valid for consan perpeuiies independen of he deb policy of he firm. 4.1.3 The Value of Tax Shields for Growing Perpeuiies Fernandez (2004a) furher invesigaes a growing perpeuiy wih an expeced consan growh rae of g (g>0), which is equal o a firm following a random walk wih drif indefiniely. Noe ha for growing perpeuiies he expeced values of annual depreciaion (EP), invesmen (INV) and he change in he amoun of deb ( ) are no equal o zero as in he previous case wih consan perpeuiies. Hence, i follows from equaion (30) ha he axes expeced o be paid every year by he unlevered growing perpeuiy are: E ( TaxesU, ) = [ E( FF ) E( EP ) + E( INV )] τ /(1 τ ), (40) whereas i follows from (35) ha he levered growing perpeuiy expecs o pay: E ( TaxesL, ) = [ E( EF ) E( EP ) + E( INV ) E( )] τ /(1 τ ). (41) As can be seen from equaions (40) and (41), i can no longer be assumed ha he axes paid by he unlevered or he levered company are proporional o FF or EF, respecively. Hence, he appropriae discoun raes for he wo cash flow sreams above canno be equal o he discoun raes, ρ and r E, from he previous secion (Fernandez 2004a). Fernandez (2004a) derives he following expression for he value of ax shields for consan growing perpeuiies: 26 τ ρ VTS( g) =. (F28 ) ρ g Fernandez (2004a) claims ha (F28 ) is generally valid for growing perpeuiies and he only formula in he lieraure ha leads o correc resuls. (F28 ) is he cenral claim in Fernandez (2004a) and is obained by subracing (F38) from (F37): [ ρ r (1 τ )] ( E / )( r ρ) = ( VTS / )( ρ g), (F37) L E 26 For he enire formal proof by Fernandez (2004a), see Appendix B. 30

[ ρ r (1 τ )] ( E / )( r ρ) = τ ρ. (F38) L (F37) is derived for perpeuiies wih g>0. (F38) is obained by subsiuing VTS = τ (F16), he formula for g = 0 E, in (F37). The ransformaion from (F37) o (F38) is only valid, however, if he lef-hand side of (F37) does no depend on he growh rae g. Therefore, Fernandez (2004a) makes he imporan assumpion ha if (E L /) is consan, he lef-hand side of equaion (F37) does no depend on growh (g) because for any growh rae (E L /), ρ, r and r E are consan. These and oher assumpions by Fernandez (2004a) are examined in chaper 5. 4.2 The Value of Tax Shields IS equal o he Presen Value of Tax Shields ooper and Nyborg (2006b) derive he value of deb ax shields for consan perpeuiies and growing perpeuiies, he wo cases discussed above, by compuing he presen value of ax savings as suggesed by he sandard lieraure. Therefore, he VTS 27 is obained by applying he valuaion heories of ME (1980) and MM (1963) for each of he wo cases. As opposed o Fernandez (2004a), ooper and Nyborg (2006b) claim ha The value of ax shields is equal o he presen value of ax shields and ha he VTS does depend on boh growh characerisics and he specific deb policy of he firm. 4.2.1 The Presen Value of Tax Shields for Growing Perpeuiies ooper and Nyborg (2006b) assume in line wih Fernandez (2004a) - he exisence of a growing perpeuiy ha generaes sochasic cash flows and ha is expeced o grow a a consan rae g. Furher, no personal axes and no coss of financial disress are assumed. Then, he value of he unlevered growing perpeuiy is: V U E( FF) = ρ g (42) and he value of he levered growing perpeuiy is: E( FF) VL = EL + = + VTS. (43) ρ g 27 For consisency wih he previous secion, I coninue o use VTS (value of ax shields) insead of PVTS (presen value of ax shields), which is in line wih boh Fernandez (2004a) and ooper and Nyborg (2006b). 31

If he ME deb policy (see secion 3.4) is assumed, he levered firm value for a growing perpeuiy can also be derived by applying he FF mehod and by using he consan WA as he discoun rae (ooper and Nyborg 2006b). Given he ME deb policy, he consan WA can be wrien as: WA = ρ τ r /( E ). (44) L + Equaion (44) can be decomposed ino wo pars: The firs par, ρ, he unlevered cos of capial, represens he required reurn due o operaing risk only, as if no financial leverage had been used. The second par, τ r /( E ), represens he benefi of deb financing L + o he firm. Hence, he ax shield of deb is included in he WA by adjusing he discoun rae downward (Harris and Pringle 1985). Then, he value of he levered firm is: V L = E L E( FF) + =. (45) ( ρ τ r /( E + ) g) L As shown by equaion (11), he VTS can generally be derived by subracing he unlevered value (V U ) from he levered firm value (V L ).onsequenly, wih he ME deb policy assumpion, he VTS for a growing perpeuiy is given by: VTS ME ( g) = V L V U E( FF) E( FF) τ r = =. (46) ( ρ τ r /( E + ) g) ( ρ g) ( ρ g) L This is he VTS obained under he ME deb policy assumpion if i is direcly valued by compuing he presen value of ax savings (ooper and Nyborg 2006b). By conras, now assume ha he same firm follows a fixed deb policy. This is similar o he MM (1963) approach, where fuure deb levels are known wih cerainy a ime =0. In his case, however, as opposed o assuming ha deb levels are consan as in MM (1963), he amoun of deb consanly grows a he same rae as he firm, g. To faciliae comparison wih he ME deb policy, his policy is called he MM deb policy wih growh. As shown in secion 3.2, if fuure deb levels are fixed iniially, he deb ax shield has he same risk as he deb and hus he deb ax shield mus be discouned a he cos of deb, r. If he ax shield is evaluaed direcly by compuing he presen value of ax savings, under he assumpion of a MM deb policy wih growh, one obains: 28 28 For he formal proof, see Appendix. 32

VTS MM r ( g) = τ. (47) ( r g) The VTS is higher in his case han ha for he ME deb policy since deb is non-sochasic and consequenly he deb ax shield has lower risk (ooper and Nyborg 2006b). 4.2.2 The Presen Value of Tax Shields for onsan Perpeuiies The VTS for consan perpeuiies can be easily inferred from eiher equaion (46) or (47) by seing he consan growh rae equal o zero (g=0). For he consan perpeuiy following he ME deb policy, he VTS is (ooper and Nyborg 2006b): τ r VTS ME =. (48) ρ In conras, wih he MM deb policy (wihou growh), VTS is given by: VTS MM τ r = = τ. (49) r Equaions (48) and (49) are already known from chaper 3, where he underlying valuaion heories are inroduced. As can be seen from equaions (48) and (49), he VTS differs by a raio of ρ / r even if boh equaions assume a consan perpeuiy. The difference arises since he discoun rae under a ME deb policy is he unlevered cos of capial, whereas i is he cos of deb under he MM deb policy. Hence, ooper and Nyborg (2006b) draw he conclusion ha he VTS is no only dependen on he growh rae, g, bu also on he deb policy of he firm. Under he ME deb policy, deb is sochasic and consequenly he ax saving is sochasic oo, rising and falling wih firm value, which follows a random walk. In conras, he MM deb policy assumes a non-sochasic amoun of deb regardless of he firm s value, and accordingly he ax saving is also non-sochasic (ooper and Nyborg 2006b). 33

5 iscussion In his secion I analyze he assumpions made by Fernandez (2004a) o derive he VTS for growing perpeuiies, which is he cenral case in his paper. 29 In doing so, I also examine he assumpions in he case of consan perpeuiies. ooper and Nyborg (2006b) argue (see he previous chaper) ha he value of he deb ax shield is a funcion of he deb policy of he firm ha is assumed. However, if Fernandez (2004a) is righ, and his approach for valuing he deb ax shield, which is independen of a specific deb policy, is generally valid, hen Fernandez approach and underlying assumpions should be consisen wih he ME and MM deb policy. 5.1 The Miles and Ezzell Framework In his secion, a growing perpeuiy following he ME deb policy is assumed. This seup should be consisen wih he VTS for growing perpeuiies derived by Fernandez (2004a) if his approach is generally valid. However, he VTS ha he derives, given by equaion (F28 ), differs from he value derived by ooper and Nyborg (2006b) in equaion (46) under he ME deb policy assumpion. Equaion (F28 ) is equaion (46) imes a facor of ρ / r. In he following, i is shown which assumpions of Fernandez (2004a) are inconsisen wih he ME deb policy ha in urn lead o resuls ha are no valid in a ME seing. In order o compleely undersand he assumpions of he ME deb policy, which implies sochasic ax shields, i is helpful o know he assumed underlying free cash flow process of he firm. Arzac and Glosen (2005) give he following expression, which describes he explici sochasic free cash flow process in a ME seing when g = 0 : wih [ 1 FF + 1 = FF ( 1 + ε + 1), (50) E FF (1 + ε + )] = FF, whereε + 1 is a sochasically independen random variable such ha E ( ε 1 ) = 0. The inuiion is ha expecaions abou fuure free cash flows + (FF +1 ) are revised a ime on he basis of new informaion, in paricular, he informaion included by he realized free cash flow a ime (FF ) (Arzac and Glosen 2005). 29 Fieen a al. (2005) poin ou ha Fernandez (2004a) makes several unsaed assumpions o derive his resuls such as zero personal axes and he absence of non-deb ax shields. Furher, he does no specify he sochasic free cash flow process of he firm and i is also no clear from he ouse wheher he assumed deb is perpeual or finie bu perpeually rolled-over. 34

This sochasic free cash flow process gives rise o firm value following a random walk. Given he assumpion of a consan leverage raio implied by he ME deb policy, deb is sochasic because i adjuss over ime as a funcion of he value of he firm and is herefore a funcion of random cash flow realizaions. This in urn leads o sochasic ax shields. Hence, he process of he sochasic ax shields is dependen no only on he assumed consan leverage raio policy bu also on he sochasic process of he firm s free cash flows (Arzac and Glosen 2005). The expression describing he sochasic free cash flow process of a consan growing perpeuiy in a ME seing is (Arzac and Glosen 2005): FF FF 1 + g)(1 ε ), (51) + 1 = ( + + 1 wih E [ FF (1 g)(1 + ε 1)] = FF (1 g) for E ( ε 1 ) = 0. + + + If i is furher assumed ha he free cash flows are posiively correlaed wih he marke porfolio, which implies sysemaic risk in a APM world, he firm s value has sysemaic risk oo. This implies, under he assumpion of a consan leverage raio, ha he deb and he deb ax shields have sysemaic risk oo (Arzac and Glosen 2005). Le us now assume a consan perpeuiy, subjec o sysemaic risk, following he ME deb policy, which implies he free cash flow process described by equaion (50). The effec of he assumpions on he value of he deb ax shield is illusraed by Figure 2 in secion 3.4 by he blue line (ME). All firms following he ME deb policy and wih a posiive asse bea, which implies sysemaic risk of he free cash flows, have a marginal gain from leverage per addiional dollar of deb ha is less han he corporae ax rae, τ. Hence, he value of ax shields for hese firms mus be + VTS < (Arzac and Glosen 2005). However, his resul is no consisen wih he resul by Fernandez (2004a). He finds ha he value of ax shields for consan perpeuiies is τ VTS =, which he claims is independen of a firm s specific deb policy. This is an imporan resul since Fernandez (2004a) uses VTS = for g = 0 as a subsiuion for VTS in equaion (F37) o derive he τ VTS for growing perpeuiies (ooper and Nyborg 2006b). As shown in secion 4.2.2, he expression τ VTS = is correc only if he MM deb policy is assumed, where fuure deb levels are fixed iniially. Under he ME deb policy, VTS is given by equaion (48) for consan perpeuiies. The error ha Fernandez (2004a) makes, and which leads o he resuls he derives, can be raced back o he assumpion ha he makes o derive he axes paid by he levered consan perpeuiy. In paricular, he τ 35

assumes ha he axes paid by he levered consan perpeuiy are proporional o he EF, ha is, based upon he proporionaliy assumpion, he assumes ha he appropriae discoun rae for he expeced axes paid by he levered consan perpeuiy (E(Taxes L, ) mus be he expeced reurn o equiy, r E (ooper and Nyborg 2006b). This relaion does no hold in general, however, as can be illusraed by he following wo equaions (ooper and Nyborg 2006b): Taxes = ( EBIT r 1) τ and (52) L, EF = ( EBIT r 1)(1 τ ) +. (53) Equaion (53) compues he EF for consan perpeuiies. If a consan perpeuiy is assumed, EP INV = 0. However, = 0 only if = 1. This is only given if = = all deb levels are consan over ime (MM deb policy). If a ME deb policy were assumed insead, 1 since deb is sochasic and herefore 0. onsequenly, in a ME seup, EF is no proporional o axes paid by he levered consan perpeuiy and hence axes have a differen risk and discoun rae han EF (ooper and Nyborg 2006b). Fieen e al. (2005) poin ou ha if deb is assumed o be sochasic and posiively correlaed wih equiy value (which is he case in a ME seing); he axes of he levered consan perpeuiy are of lower risk han EF and consequenly have a lower discoun rae. ooper and Nyborg (2006b) reconcile he case for consan perpeuiies in Fernandez (2004a) when he ME deb policy is assumed. They argue ha under a ME deb policy, he presen value of he axes paid by he levered consan perpeuiy mus be: G L = V τ /( 1 τ ) ( τ r ) / ρ. (54) U The firs par of he equaion, VUτ /( 1 τ ), is he presen value of axes paid by he unlevered firm (see equaions (38) and (39) for he derivaion) and he second par, ( τ ) / ρ, is he value of ax shields for a consan perpeuiy derived under he r assumpion of he ME deb policy. Then, he VTS for a consan perpeuiy in a ME seing is: VTS ME = G U G L = V τ /( 1 τ ) [ V τ /(1 τ ) ( τ r ) / ρ] = ( τ r ) / ρ, (55) U U which is equal o equaion (48), he presen value of ax shields. Furher, i can be shown ha if Fernandez had used he VTS from equaion (55) for g = 0 o derive he VTS for 36

growing perpeuiies (F28 ), he would have found he same resul as equaion (46), he VTS in he ME seing for growing perpeuiies (ooper and Nyborg 2006b). 30 To summarize, his secion has shown ha Fernandez (2004a) erroneously uses an assumpion ha is rue in a MM seup bu no in a ME seup in deriving his VTS for growing perpeuiies. Therefore, Fernandez (2004a) resul lacks generaliy (ooper and Nyborg 2006b). 5.2 The Modigliani and Miller wih Growh Framework In a reply o a working paper version of ooper and Nyborg (2006b), Fernandez (2004b) clarifies some aspecs of he assumpions used in Fernandez (2004a) by he following saemen: There is also a suble difference beween he Miles-Ezzell (1980) assumpion abou he capial srucure, namely, = K E, and he assumpion ha I use in my paper: E{} = K E{E}, being E{ } he expeced value operaor, he value of deb and E he equiy value. K is a consan. Miles-Ezzell (1980) assumpion requires coninuous rebalancing, while my assumpion does no. ooper and Nyborg s (2006b) inerpreaion of his saemen is ha Fernandez (2004b) assumes a deb policy where fuure deb levels are fixed iniially as under a MM deb policy bu a he same ime follows a consan leverage raio. Therefore, in conras o he ME seup, deb levels are independen of he free cash flow process of he firm and he deb policy is consisen wih he resul ha VTS = for g = 0, which makes i a valid subsiuion for VTS in equaion (F37) (ooper and Nyborg 2006b). However, since he firm is a growing perpeuiy and Fernandez (2004a) assumes a consan leverage raio, deb has o grow a he same expeced rae g as he levered firm s value, so ha LV L, τ = is saisfied a every ime. The deb policy is called MM deb policy wih growh since deb is deerminisic as under a MM deb policy bu grows a rae g (ooper and Nyborg 2006b). Noe ha since all fuure deb levels are fixed iniially and he leverage raio remains consan over ime, V L, mus be known a ime 0. Hence, he following free cash flow process is required given he MM deb policy wih growh (Arzac and Glosen 2005): 30 For he derivaion of his resul (ha ooper and Nyborg 2006b do no show), see Appendix. 37

FF ( 1 + g + ε, wih (56) = FF ) = E ( FF ) = FF(1 g (57) E ( FF ) 1 + ) for E( ε ) 1 ( ε ) = 0 where E ( ) is he uncondiional expeced value and E ( ) he = E expeced value condiional on he informaion available a ime -1 (ooper and Nyborg 2006b). The inuiion of he free cash flow process is ha expecaions are consan over ime and no revised on he basis of new informaion and hence firm value does no follow a random walk. This is a less realisic scenario han he assumed free cash flow process in he ME seup, however, because firm value is predicable and hence he efficien marke hypohesis does no hold. Noe ha alhough he expecaions are no revised and are independen of free cash flow realizaions, he free cash flows are sill assumed o be sochasic (Arzac and Glosen 2005). Given his inerpreaion by ooper and Nyborg of Fernandez (2004b) saemen, Fernandez (2004a) assumpions for a growing perpeuiy are as follows (ooper and Nyborg 2006b): 1 1. Expeced free cash flows grow a rae g > 0. 2. Fuure deb levels are known wih cerainy a ime 0 (Inerpreaion by ooper and Nyborg (2006b)). 3. L = / V is a consan and independen of g and ime. 4. ρ, r E and r are consans and independen of g and ime. 5. The firm has sysemaic risk greaer han zero in a APM world and hence: ρ > r f. These assumpions given, i is imporan o noe ha he independence beween he variables (L, ρ, r E and r ) and g is criical, since his allows Fernandez (2004a) o subsiue VTS = τ for g = 0 ino equaion (F37) o derive he VTS for growing perpeuiies (ooper and Nyborg 2006b). ooper and Nyborg (2006b) show ha if g < r f and deb is risk-free, all of he assumpions above are saisfied, excep possibly one, namely, r E being independen of g. Below I show ha r E mus be dependen on g if all oher assumpions above hold simulaneously. Therefore, he VTS for growing perpeuiies derived by Fernandez (2004a) is no valid under he assumpion of a MM deb policy wih growh seup. 38

The following reasoning uses risk insead of expeced reurn expressions. This is consisen because expeced reurns are a funcion of bea in a APM world (assuming he risk premium and risk-free rae are consan). 31 According o Ruback (2002), equaion (58) shows he differen risk posiions of a marke-value balance shee wih asses and he deb ax shield on he lef and deb and equiy on he righ, where he sysemaic risk of he levered firm (β L ) is a value-weighed average: V V U L VTS β + E U + β TS = β L = β β E. (58) VL VL VL We know ha ρ and r are consans independen of g. This implies ha he risk of he unlevered firm (β U ) and deb (β ) is consan. Furhermore, since and independen of g; V U / VL, VL L = / V is also consan / and E / VL mus be consan oo. onsequenly, he equiy bea (β E ) can only be independen of g if β L is consan and independen of g. I can be seen from equaion (58) ha if VTS is a funcion of g and is fracion of oal firm value increases over ime, β L is independen of g if β = β, which is he case in he ME seup. However, from secion 3.2 we know ha if deb levels are fixed iniially, as in he case of a MM deb policy wih growh, he deb ax shields are as risky as he deb. Hence, β TS = β. In our case, deb is assumed o be risk free and herefore we obain β TS = β TS = 0. Thus, β L and herefore β E decrease over ime and are dependen on g since he VTS of a growing perpeuiy is dependen on g and is fracion of oal value increases over ime. onsequenly, r E mus also be dependen on g since expeced reurns are a funcion of bea. 32 Since he deb is risk-free in our case he ax savings per period are: U TS = r. In addiion, we know from above ha he discoun rae of he deb ax shield mus be r f because β = 0. Hence, he VTS for a growing perpeuiy following he MM deb policy TS wih growh mus be: τ f VTS MM r f ( g) = τ. (59) r g f 31 The approach I use is differen from ooper and Nyborg (2006b), who use expeced reurns insead of bea. Their approach is misleading since hey assume ha he discoun rae of ax shields is equal o he ineres rae used in he numeraor o compue deb ax savings, which is no generally valid, as for insance in he ME seup. 32 Fieen e al. (2005) also idenify his inconsisency in he assumpions by arguing ha subracing a fixed deb from a firm value ha follows a geomeric random walk resuls in a levered equiy of non-consan risk. 39

This is equivalen o equaion (47), he presen value of ax shields derived by ooper and Nyborg (2006b). Noe furher ha if Fernandez (2004a) would allow for he dependence of r E on g in he derivaion of (F28 ), he VTS would give he same value as in equaion (47) (ooper and Nyborg 2006b). 33 I conclude ha Fernandez (2004a, 2004b) assumpions for deriving his VTS for growing perpeuiies are neiher consisen wih he MM deb policy wih growh nor he ME deb policy and heir required specific cash flow processes. 5.3 The Value of he Tax Shield if he Leverage Raio is onsan in Book Values In a series of papers, Fernandez (for insance, 2004b, 2005a, 2005b and 2007b) replies o he inconsisencies menioned above. In doing so, he derives he VTS by applying a differen approach, which allows him o obain he VTS wihou specifying he exac values of G U and G L (he presen values of axes paid by he unlevered firm and he levered firm, respecively). Alhough he approach is differen, he VTS is he same as in Fernandez (2004a) for growing perpeuiies (equaion F28 ) if a cerain se of assumpions is used. In his secion, his new approach is derived by applying a concep developed by Arzac and Glosen (2005) and I show under wha circumsances he VTS of Fernandez (2004a) is appropriae o use and how i expands he exisen heories. We know from equaion (29) ha he VTS is he difference beween G U and G L, which are he presen values of he wo following cash flows: Taxes = ( FF + A ) τ /(1 τ ) and (60) U, Taxes = ( EF + A ) τ /(1 τ ), (61) L, where A = EP + INV, which is he increase in ne asses (in book values) in period. In order o derive he correc presen value of he cash flows, we mus know he appropriae discoun rae. To avoid argumens abou he appropriae discoun rae, a pricing kernel is applied (also called sochasic discoun facor). I is assumed ha he price of an asse ha pays a random amoun x a ime is he sum of he expecaion of he produc of he random amoun (x ) and he pricing kernel for ime cash flows (M ) (Arzac and Glosen 2005). Thus: 33 For he derivaion of his resul (ha ooper and Nyborg 2006b do no show), see Appendix E. 40

= P x E[ M x ]. (62) =1 I can hen derive he VTS as he difference beween he presen values of equaion (60) and (61) by applying he pricing kernel, wihou worrying abou he correc discoun rae for each cash flow. The general resul for he VTS is hen: which is equal o: E[ M TaxesU, ] E[ M TaxesL, ] = 1 = 1 VTS =, (63) τ VTS = E[ M FF ] E[ M EF ] + E[ M 1 τ = 1 = 1 = 1 Equaion (64) can be rewrien as: = τ + VTS VU E E[ M 1 τ =1 ]. (64) ]. (65) Knowing from equaion (5) ha V U E = VTS, we ge: Simplifying he above leads o: = τ + VTS VTS E[ M 1 τ =1 ]. (66) =1 VTS = τ + τ E[ M ]. (67) Equaion (67) is a general resul for he VTS and i can be seen ha he VTS is only dependen on he naure of he sochasic process of he ne change in deb (Arzac and Glosen 2005 and Fernandez 2005a). I is imporan o noe ha he value of he ne change in deb, E[ M ], depends 1 = on wheher is correlaed wih he marke porfolio and how srong he correlaion is. In oher words, he value of he ne change in deb depends on he sysemaic risk of in a APM world. If is posiively correlaed wih he marke, he value of he ne change in deb is less han if i is uncorrelaed and has no sysemaic (or priced) risk so ha = 1 E [ M ] < E [ ]/(1 + r ) = 1 f. onsequenly, if a consan perpeuiy wih g = 0 41

and E [ ] = 0 is assumed and if is posiively correlaed wih he marke, he value of = 1 he ne change in deb is E [ M ] < 0. Hence, he VTS of his consan perpeuiy inferred from equaion (67) mus be VTS < (Arzac and Glosen 2005). This is exacly τ he resul I derived in secion 5.1 for a consan perpeuiy in he ME seing and ha is shown in Figure 2 in secion 3.4 for a firm following he ME deb policy. The resuls mus be idenical, because he assumpions in secions 3.4 and 5.1 implied a ne change in deb ha is sochasic and has sysemaic risk, since I assumed a consan leverage raio (which implies perfec correlaion wih firm value) and firm value wih sysemaic risk following a random walk. Given equaion (67), Fernandez (2005b) specifies he sochasic cash flow process he assumes o derive he VTS for consan growing perpeuiies: FF FF 1 + g)(1 ε ), (68) + 1 = ( + + 1 + [ M, + 1 + 1 wih E ( ε 1 ) = 0 and E ε ] < 0, which implies sysemaic risk. M,+1 is he oneperiod pricing kernel a ime for cash flows a ime +1. Noe ha he assumed cash flow process by Fernandez (2005b) is equivalen o equaion (51), he cash flow process under he ME assumpion. Furher, Fernandez (2005b, 2007b) assumes a deb policy where he firm follows a consan leverage raio in book values, insead of a consan leverage raio in marke values as in he ME seing. Thus, he assumes he following relaionship, which describes he assumed deb policy ha he leverage raio is consan in book values. = K * Ebv, (69) where Ebv is he book value of equiy and K a consan. Equaion (69) implies ha he increase in deb 34 is equal o he increase of book value of equiy: = K * Ebv. (70) In addiion, we know ha in erms of book values, he increase in ne asses is equal o he increase in equiy plus he increase in deb: A = Ebv +. (71) 34 Since a consan growing perpeuiy is assumed, he ne change in deb ( ) mus be an increase in deb. 42

Solving equaions (70) and (71) for Ebv and seing hem equal o one anoher, we ge: A = K. (72) / Solving for, i can be seen ha he increase in deb is proporional o he increase in ne asses and hence hey mus boh have he same risk (Fernandez 2005b). = A /( 1 1/ K). (73) + I is assumed ha he increase in ne asses follows he sochasic process defined by: A = A 1 + g)(1 φ ), (74) + 1 ( + + 1 where φ+ 1 is a random variable wih E ( φ + 1 ) = 0 bu E ( φ 1 ) 0, which implies M, + 1 + < sysemaic or priced risk (Fernandez 2005b). Then, by assuming ha he firm follows a consan leverage raio in book values, which implies ha he risk of is equal o he risk of A, Fernandez (2005a, 2005b) derives he following resul for he value of he ne change in deb for consan growing perpeuiies: = 1 g E[ M ] =, (75) α g where α is he appropriae discoun rae for he expeced increase in ne asses and under a consan book-value leverage raio equal o he appropriae discoun rae for. Subsiuing equaion (75) in (67), he VTS we obain is: τ α VTS =. (76) α g If i is now furher assumed ha he risk of A is equal o he risk of he FF, which implies ha φ + 1 = ε + 1 and hence A is proporional o FF, hen he appropriae discoun rae is follow: α = ρ for he expeced increase in ne asses (Fernandez 2005b). Then i mus = 1 g E[ M ] = ρ g and (77) 43

τ ρ VTS =. (78) ρ g Equaion (78) is equal o equaion (F28 ) of Fernandez (2004a). Noe ha equaion (78) only holds if he increase in deb is as risky as he free cash flows, which is a special case and depends no only on he assumpion of a consan bookvalue leverage raio bu also on he assumpion ha he increase in ne asses is as risky as he free cash flows. If a consan book-value leverage raio is assumed and he risk of A is differen from he risk of he free cash flows, equaion (76) holds wih a discoun rae of α, reflecing he risk of A (Fernandez 2005a). Noe ha α is difficul o apply in pracice since he risk of he increase in ne asses ( A ) is difficul o measure. I should be noed furher ha equaion (78) is only valid for consan growing perpeuiies. If consan perpeuiies wih g = 0 are assumed, we would obain VTS = τ. However, his is no correc because equaion (68) assumes a sochasic FF process wih sysemaic risk, which implies ha mus have sysemaic risk oo, since is as risky = 1 as he free cash flows. We know from above, however, ha E [ M ] < 0 when has sysemaic risk (posiively correlaed wih he marke). Subsiuing his resul in equaion (67), i can be seen ha VTS < for perpeuiies wih g = 0, under he τ assumpions made by Fernandez (2005a, 2005b). I conclude ha equaion (F28 ) is only valid for consan growing perpeuiies wih a sochasic cash flow process subjec o sysemaic risk as in he ME seing and wih a consan book-value leverage raio deb policy. Furher, he risk of he increase in ne asses mus be equal o he risk of he free cash flows. If he las assumpion is no given, formula (76) should be used, alhough i is difficul o apply in pracice. Finally, noe ha he VTS for consan growing perpeuiies derived by Fernandez (2005a, 2005b, 2007b) under he assumpions given in his secion expands he exisen valuaion heories bu i does no conradic or correc hem. 44

6 onclusion The debae in he finance lieraure abou he correc value of deb ax shields (VTS) is ongoing. One hread of he lieraure suggess ha he VTS be compued by discouning he deb ax savings o he presen using a discoun rae ha reflecs he risk of he deb ax savings. The difference across he various heories in his lieraure lies primarily in he assumed deb policy. MM (1963) and Myers (1974) sugges ha he deb ax savings be discouned a he cos of deb when fuure deb levels are fixed iniially, whereas ME (1980) and Harris and Pringle (1985) propose o discoun he deb ax savings a he unlevered cos of capial when he firm mainains a consan leverage raio in erms of marke values. 35 In conras o his lieraure, Fernandez (2004a) derives he value of ax shields by compuing he difference beween he presen value of axes for he unlevered company and he presen value of axes for he levered company. Fernandez (2004a) resuls challenge exan valuaion heories as he claims ha he VTS he derives is generally valid and in fac he only way o compue he correc VTS. Fernandez (2004a) claim is no consisen wih his se of assumpions, however. I has been shown in his paper ha Fernandez VTS for growing perpeuiies is no consisen wih he assumpion of a firm following a ME- or MM-syle deb policy. In addiion, he VTS for perpeuiies wih zero growh mus be less han he value ha Fernandez derives if he firm is following a ME deb policy and is free cash flows have sysemaic risk. onsequenly, Fernandez VTS is no generally valid. In addiion, I have shown ha he following se of assumpions mus hold if he VTS for growing perpeuiies by Fernandez is o be valid: he expeced growh rae of he firm is greaer han zero, he firm follows a sochasic free cash flow process subjec o sysemaic risk and he firm applies a deb policy ha follows a consan leverage raio in erms of book values. Furher, he increase in ne asses mus be as risky as he free cash flows of he firm. In his paper, i has also been shown ha Fernandez approach of compuing he VTS as he difference beween wo presen values does no lead o resuls ha are differen from hose obained when direcly compuing he presen value of he deb ax savings, as suggesed by he sandard lieraure, if he same assumpions are made. onsequenly, 35 Noe ha Harris and Pringle (1985) discoun all deb ax shields a he unlevered cos of capial, whereas ME (1980) discoun he firs-period deb ax shield a he cos of deb and all subsequen periods ax shields a he unlevered cos of capial. 45

Fernandez VTS for growing perpeuiies expands he exisen valuaion heories by using an addiional se of assumpions ha includes a new deb policy, bu i does no correc or conradic he sandard valuaion heories of MM (1963), Myers (1974), ME (1980) or Harris and Pringle (1985), for insance. Taken ogeher, hese findings indicae ha i is crucial o explicily specify a he ouse he assumpions one is using in valuing a company and is ax shields, be i in heory or in pracice. In paricular, he deb policy applied by he firm mus be specified in order o correcly choose from among he differen valuaion heories. This paper s findings also poin o he imporance of he choice of valuaion mehod used. I is essenial o choose he mehod ha will be less prone o error and more likely o provide a correc esimae of he value of he firm (including he VTS). In general, he differen valuaion mehods should give he same resul provided ha he same assumpions are used, which makes sense since hey analyze he same realiy under he same hypoheses. However, as I have shown, he capial cash flow mehod does no lead always o resuls ha are consisen wih he oher mehods, since i includes implici assumpions abou he risk of he deb ax shield in is valuaion. The following able summarizes on he verical he differen valuaion heories as well as he assumpions made by he capial cash flow mehod and on he horizonal he hree differen valuaion mehods analyzed in his paper. A indicaes which mehod is likely o work bes under a given se of assumpions, a indicaes wheher a valuaion mehod is consisen wih a cerain se of assumpions bu difficul o apply in pracice and an X indicaes which mehods are inconsisen wih he given assumpions. 46

Table 2 Which Valuaion Mehod Should Be Used Under Which Se of Assumpions? Assumpions Mehods Assumpion name Growh characerisics Leverage policy Fuure ax saus APV FF F MM (1963) zero growh consan and permanen amoun of deb always ax-paying X X Exended MM (Myers 1974) any growh paern fuure deb levels are fixed iniially always ax-paying ME (1980) wih coninuous rebalancing any growh paern consan markevalue leverage raio always ax-paying Fernandez (2004a) consanly growing consan book - value leverage raio apial cash flow (Ruback 2002) any growh paern any deb policy Based upon: ooper and Nyborg (2007) always ax-paying uncerain; i is assumed ha he risk of ax savings is equal o risk of free cash flows X The able shows ha he APV and FF mehods can be used wih every se of assumpions if hey are applied correcly and boh should give he same resuls as long as he same assumpions are used. The FF mehod works bes for firms ha have a long-run arge capial srucure, which is in line wih he assumpion of a consan leverage raio in erms of marke values. 36 The FF mehod is difficul o apply however, and likely o produce errors if he capial srucure changes over ime, since he WA mus be adjused each period o reflec he changes in capial srucure. In he case ha he fuure ax saus of he firm is uncerain and he company could become non-axpaying, he F mehod migh be reasonably used as an approximaion, if 36 If Fernandez (2004a) is applied and he increase in ne asses is as risky as he free cash flows and he firm is assumed o grow a a consan rae, hen he WA says consan and he FF mehod is also simple o apply as shown by Fernandez (2007a). 47

one is willing o assume ha he risk of he ax savings is equal o he risk of he free cash flows of he firm. For all oher ses of assumpions, he APV mehod is he bes choice since i lays ou each componen of value separaely and is hus highly ransparen. I is paricularly useful when fuure deb levels are fixed iniially and he capial srucure changes significanly over ime, as i is likely o be he case in an LBO, for example. To conclude he above discussion, one should ask wo fundamenal quesions before saring o value a company: 1. Which se of assumpions (valuaion heory) is closes o he acual siuaion he company is facing? 2. Which valuaion mehod works bes given he chosen se of assumpions? The difficul ask is hen o idenify he se of assumpions ha is closes o he acual siuaion of a specific company. 48

Appendix Appendix A: Bea of eb Tax Shields equals Bea of eb (based upon Ruback 2002) Assumpion: eb is perpeual and fixed as a dollar amoun, B, which does no change as he value of he firm changes. Then he value of he deb ax shield is PVTS τ r =, (A1) r, B where r is he fixed yield on he deb and r, is he cos of deb in period, which is deermined by equaion (8). The value of he deb can change hrough ime since r is fixed and he cos of deb migh change. The value of deb in period ( ) is r =. (A2) r B, By subsiuing equaion (A2) ino equaion (A1), he value of he deb ax shield a ime can be expressed as PVTS = τ. (A3) Using a modified definiion of bea, he bea of he deb ax shield, β TS, equals ov( PVTS, RM ) β TS =. (A4) V Var( R ) TS, 1 M By subsiuing equaion (A3) ino equaion (A4) and simplifying (τ disappears since consan) ov(, R ) β TS = = β V M, 1Var( RM ) eb. (A5) 49

Appendix B: The derivaion of he value of ax shields (VTS) for growing perpeuiies by Fernandez (2004a) For comparison, he numbering of he equaions in Appendix B is aken from Fernandez (2004a). Knowing ha: E + = V VTS. (F1) L U + Noice furher ha he unlevered value of a firm wih an expeced growh rae of g is derived by V U = FF /( ρ g). (F29) By subsiuing equaion (F29) in (F1), we ge E L + = FF /(ρ g) + VTS. (F30) The relaion beween he EF and he FF is FF = EF + r ( 1 τ ) g. (F31) By subsiuing (F31) in (F30), we ge E + = [ EF + r (1 τ ) g]/( ρ g) VTS. (F32) L + As for a levered company EF = E ( r g), (F32) may be rewrien as L E E + = [ E( r g) + r (1 τ ) g /( ρ g) VTS. (F33) L E + Muliplying boh sides of equaion (F33) by ( ρ g), we ge ( E + )( ρ g) = [ E( r g) + r (1 τ ) g] + VTS( ρ g). (F34) L E Eliminaing g( EL + ) on boh sides of equaion (F34), we have ( E + ) ρ = [ r E + r (1 τ ) + VTS( ρ g). (F35) L E 50

Equaion (F35) can be rewrien as [ρ r ( 1 τ )] E (r ρ) = VTS(ρ g). (F36) L E ividing boh sides of equaion (F36) by (deb value), we ge, [ ρ r (1 τ )] ( E / )( r ρ) = ( VTS / )( ρ g). (F37) L E If ( E L / ) is consan, he lef-hand side of equaion (F37) does no depend on growh (g) because for any growh rae ( E L / ), ρ, r and r E are consan. We know ha for g = 0, VTS = τ. Then, equaion (F37) applied o consan perpeuiies ( g = 0) is [ ρ r (1 τ )] ( E / )( r ρ) = τ ρ. (F38) L E Subracing (F38) from (F37), we ge 0 = ( VTS / )( ρ g) τ ρ. Solving for VTS, we ge τ ρ VTS =. (F28 ) ( ρ g) 51

Appendix : The derivaion of he value of ax shields for growing perpeuiies under he assumpion of a MM deb policy wih growh The proof I show is based upon Bodie, Kane and Marcus (2008) where i is used o derive he consan-growh dividend discoun model. I is shown in secion 3.2, ha if fuure deb levels are fixed iniially, he deb ax shield has he same risk as he deb. Hence, he deb ax shields mus be discouned a he cos of deb. τ r τ r (1 + g) τ r (1 + g) VTS = + + 2 3 1 + r (1 + r ) (1 + r ) 2 +... (A6) Muliplying boh sides of equaion (A6) by ( 1 + r ) /(1 + g), we ge (1 + r ) τ r τ r τ r VTS = + + (1 + g) (1 + g) (1 + r ) (1 + r ) 2... (A7) Subracing equaion (A6) from equaion (A7), we obain (1 + r ) τ r VTS VTS =, (A8) (1 + g) (1 + g) which implies ( r g) τ r VTS =. (A9) (1 + g) (1 + g) Solving for VTS, we find ha r VTS = τ. (A10) ( r g) 52

Appendix : The derivaion of he value of ax shields for growing perpeuiies in a ME seup when VTS = τ r / ρ for g = 0 If a ME seing is assumed, Fernandez (2004a) analysis is correc up o equaion (F37). Thereafer, he mus use VTS = τ r / ρ for g = 0, as explained in secion 5.1. The resul ha mus follow for growing perpeuiies in a ME seup is shown below. [ ρ r (1 τ )] ( E / )( r ρ) = ( VTS / )( ρ g) (F37) L E Assuming a ME seing, perpeuiies ( g = 0 ) is VTS = τ r / ρ for g = 0. Then, equaion (F37) applied o [ ρ r (1 τ )] ( E / )( r ρ) = τ r (A11) L E Subracing (A 11) from (F 37), we ge 0 = ( VTS / )( ρ g) τ, solving for VTS, we find ha r τ r VTS =. (A12) ρ g This is equal o equaion (46) ha is he presen value of ax shields for growing perpeuiies in a ME seing as shown in secion 4.2.1. 53

54 Appendix E: The derivaion of he value of ax shields for growing perpeuiies when he dependence of r E on g is allowed for. The following formula from opeland, Koller and Murrin (2000, 475) is used when 0 > g so ha he formula allows for he dependence of r E on g. + = L E g r r E r g r τ ρ ρ 1 ) ( ) (. (A13) Hence, if 0 = g, (A13) simplifies o [ ] L E E r r τ ρ ρ + = 1 ) (. (A14) Since equaion (F37) assumes 0 > g, I subsiue (A13) for r E in (F37) and I obain ) )( / ( 1 ) ( ) (1 ( g VTS g r r r r = ρ τ ρ τ ρ. (A15) And since equaion (F38) assumes 0 = g, I subsiue (A14) for r E in (F38) and I obain [ ] ρ τ τ ρ τ ρ r r = 1 ) ( ) (1 (. (A16) As in Fernandez (2004a), subracing (A16) from (A15), I ge ρ τ ρ τ ρ τ ρ g VTS r g r r r = + ) )( / ( ) )(1 ( 1 ) (. (A17) (A17) can be rewrien as ρ τ ρ τ ρ τ ρ g VTS r g r r r = ) )( / ( ) ( ) (. (A18) (A18) can be rewrien as ρ τ ρ τ ρ g VTS g r r r = ) )( / ( 1 ) (. (A19)

By subsiuing r r g g for 1 in (A19), I ge g ( ρ r ) τ = VTS ρ g τ ρ r g ( / )( ). (A20) (A20) can be rewrien as ( ρ * g r * g) τ /( r g) = ( VTS / )( ρ g) τ ρ. (A21) Adding τ ρ on boh sides of equaion (A21) and rearranging, I ge (( * g r * g + ρ * r ρ * g) /( r g) ) τ = ( VTS / )( ρ g) ρ. (A22) (A22) can be rewrien as τ r ( ρ g) /( r g) = ( VTS / )( ρ g). (A23) Solving for VTS, I obain r VTS = τ. (A24) r g Equaion (A24) is equivalen o equaion (47), which is he value of ax shields, derived by ooper and Nyborg (2006b), for growing perpeuiies if he MM deb policy wih growh is assumed. Therefore, I have shown ha if he dependence of r E on g is allowed for in he derivaion of Fernandez (2004a), we obain he same resul as ooper and Nyborg (2006b). 55

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