What is the Value of the Tax Shield of Debt?

Transcription

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

2 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-

3 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

4 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 Trier iese Schrifenreihe is im Inerne in elekronischer Form uner wiwischrifenreihe.uni-rier.de erreichbar.

5 Table of onens Table of Abbreviaions... IV 1 Inroducion Basic Model and Valuaion Mehods Basic Valuaion Model: Modigliani and Miller (1958) Inroducing orporae Taxes and he Mechanics of he Tax Shield of eb Valuaion Mehods The Adjused Presen Value Mehod The Free ash Flow Mehod The apial ash Flow Mehod Valuaion Theories for he Tax Shield of eb The Value of he Tax Shield if he Level of Risk-Free eb is onsan and Perpeual The Value of he Tax Shield if eb is Risky and eb Policy Fixed Oher Effecs on he Tax Shield of eb Inroducing oss of Financial isress Inroducing Personal Taxes and he Miller Model The Value of he Tax Shield if he Leverage Raio is onsan Which Valuaion Mehod Should Be Used wih Which Theory? Is he Value of Tax Shields equal o he Presen Value of Tax Shields or no? The Value of Tax Shields is no equal o he Presen Value of Tax Shields The General Idea The Value of Tax Shields for onsan Perpeuiies The Value of Tax Shields for Growing Perpeuiies The Value of Tax Shields is equal o he Presen Value of Tax Shields The Presen Value of Tax Shields for Growing Perpeuiies...31 II

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

7 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

8 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

9 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

10 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

11 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

12 (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

13 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

14 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

15 Table 1 The Tax-educibiliy of Ineres Expenses Microsof orporaion 2007 ($ millions) No eb eb ($20bn 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

16 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 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

17 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) 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

18 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 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

19 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

20 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

21 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

22 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

23 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 Oher Effecs on he Tax Shield of eb 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

24 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 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

25 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

26 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

27 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

28 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

29 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 = and r P = 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

30 Figure 2 The Relaionship Beween he Gain from Leverage and he Asse Bea dv/d 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) 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