Invesmen Under Uncerainy, Deb and Taxes Absrac

Transcription

1 Invesmen under Uncerainy, Deb and Taxes curren version, Ocober 16, 2006 Andrea Gamba Deparmen of Economics, Universiy of Verona Verona, Ialy Gordon A. Sick Haskayne School of Business, Universiy of Calgary Calgary, Canada Carmen Aranda León Universiy of Navarra Pamplona, Spain Corresponding auhor: Gordon A. Sick Haskayne School of Business Universiy of Calgary Calgary, Albera Canada T2N 1N4

2 Invesmen under Uncerainy, Deb and Taxes Absrac We presen a capial budgeing valuaion framework ha akes ino accoun boh personal and corporae axaion. We show broad circumsances under which axes do no affec he maringale expecaions operaor. Tha is, he maringale operaor is he same before and afer personal axes, which we call valuaion neuraliy. The appropriae discoun rae for riskless equiy-financed flows (maringale expecaions or cerainy-equivalens) is an equiy rae ha differs from he riskless deb rae by a ax wedge. This ax wedge facor is he afer-ax reenion rae for he corporae ax rae ha corresponds o ax neuraliy in he Miller equilibrium. We exend his resul o he valuaion of he ineres ax shield for exogenous deb policy wih defaul risk. Ineres ax shields accrue a a ne rae corresponding o he difference beween he corporae ax rae ha will be faced by he projec and he Miller equilibrium ax rae. Depending on he financing sysem, ineres ax shields can be incorporaed by using a ax-adjused discoun rae or by implemening an APV-like approach wih addiive ineres ax shields. We also provide an illusraive real opions applicaion of our valuaion approach o he case of an opion o delay invesmen in a projec, showing ha he applicaion of Black and Scholes formula may be incorrec in presence of personal and corporae axes. Keywords: Invesmen under uncerainy, real opions, capial srucure, risk-neural valuaion, corporae and personal axaion, defaul risk, ineres ax shields, cos of capial, ax-adjused discoun raes. JEL classificaions: G31, G32, C61 1

3 1 Inroducion Many models for valuing or opimizing capial invesmen decisions under uncerainy are based on cash flows discouned a a risk-adjused rae. However, he presence of leverage, discreion and asymmery forces a valuaion approach based on compuaion of cerainyequivalen, raher han expeced, cash flows. This includes he real opions approach. The marke maker who ses he relaive prices of financial derivaives and heir underlying insrumens is ypically axed a he same rae on all financial insrumens. Thus, he common pracice of discouning he cerainy equivalen payoff of financial derivaives (pus and calls on socks, for example) a he riskless deb rae is appropriae. However, we show ha i is incorrec o carry his pracice over o a real opions seing because he marginal invesor of a capial invesmen projec is likely o face a differenial axaion according o he ype of insrumen (i.e, equiy or deb). In fac, he cerainy equivalen rae of reurn for equiy funds is ypically lower han ha of deb funds. 1 In addiion, he ineracion of personal and corporae axes on various financing insrumens generaes a ne ineres ax shield ha may have posiive or negaive value. This ineracion is so ofen mishandled ha few people recognize ha i could jus as likely have a negaive value as a posiive value. However, he following feaures commonly found in capial budgeing problems sugges ha he bes valuaion approach is o use cerainy equivalen operaors, ax-adjused 1 The fac ha a financial marke maker does no use he same discoun rae o value an invesmen as a long-erm capial invesor would use does seem o lead o some arbirage opporuniies ha we do no explore. I may be difficul o ake advanage of hese arbirage opporuniies because a dynamic hedge based on he long erm capial value may generae adverse ax consequences, or may be hard o form because of incomplee markes. On he oher hand, i migh be possible o synhesize he cash flows of a commodiy producer, such as an oil company wih very predicable reserves and producion by a srip of forward or fuures conracs on oil and a srip of bonds. Boh of hese insrumens are financial insrumens and would be valued by discouning a he bond rae. They could be sold o an invesor o replace an equiy posiion in such a firm, and ha invesor may discoun he flows a he riskless equiy rae, which is less han he riskless deb rae. This could represen a simple ax arbirage. 2

4 discoun raes for equiy and a more sophisicaed analysis of ineres ax shields: 1. Projecs are financed parially by equiy and deb. 2. Personal and corporae axes compound, and axaion of deb income differs from ha of equiy income. 3. Some cash flows from a projec, such as he ax advanage o deb are coningen on he cash flows of he projec and are los in he even of defaul. In his paper we firs inroduce a coninuous-ime valuaion framework for cash flows emerging in capial budgeing ha akes ino accoun personal and corporae axes. We show broad valuaion neuraliy circumsances under which axes do no affec he equivalen maringale measure. Second, he paper shows how o adjus he value of asymmeric cash flows o compue he value of deb and axes under wo differen ypes of seings: defaul-free and defaulable deb. We can assume ha deb is defaul-free if managemen consanly revises he deb level of he firm o mainain a consan deb raio. 2 This assumpion, firs due o Miles and Ezzel (1980, 1985), is popular in capial budgeing (e.g., Grinbla and Liu (2002)), and is considered as he benchmark case here. We find ha he value of he ineres ax shield can be calculaed eiher by an addiive erm ha separaes he value under differen financing scenarios (an adjused presen value or APV reamen), or by adjusing he discoun rae o reflec he ax wedge ha separaes he riskless marke reurns for insrumens of 2 Sewar Myers has poined ou in privae communicaion ha if deb is coninuously rebalanced o keep a consan deb raio, here can be no defaul on he deb. As he firm s cash flows fall, is value falls, so he firm mus repurchase deb and replace i wih equiy o keep he deb raio consan. By he ime he firm approaches bankrupcy, here is no deb upon which o defaul. In discree ime, here could be defaul even wih a policy of rebalancing deb o keep a consan deb raio a he rebalance daes. 3

5 differen ax classes in wha we will call a ax-adjused discoun rae (TADR) approach. Laer, we assume ha deb is defaulable and incorporae he effec of he probabiliy of defaul ino he valuaion. The paper concludes by discussing an applicaion of our resuls o a real opions seing. For clariy sake, we have seleced he basic case of an opion o delay an invesmen decision. We demonsrae ha he he projec can be evaluaed according o he Black, Scholes, and Meron formula. However, using he riskless deb rae as he discoun facor would be a misake, because he personal ax rae for bond invesmen income and ha for sock invesmen income are usually differen. The valuaion of asymmeric cash flow is ubiquious in corporae finance. To price corporae securiies, Meron (1974) sough o develop a heory for pricing bonds along Black-Scholes lines wih a significan probabiliy of defaul. However, he did no include axes of any kind in his analysis, so he Black-Scholes seing was appropriae. Laer on, Brennan and Schwarz (1978), Leland (1994) and Leland and Tof (1996) became aware of he imporance of corporae income axes and, accordingly, included corporae axaion in heir deb pricing and opimal capial srucure models. Our work exends his previous research by considering personal as well as corporae axaion. Kane e al. (1984) did analyze he value of ineres ax shields wih personal axes for risky deb where he unlevered firm value could jump o zero according o a Poisson process. They did no incorporae flexibiliy, so heir valuaion is a european opion model of defaul. They have a riskless rae for equiy-financed flows ha differs from he deb rae, bu hey find ha ineres ax shields have a posiive value. Our paper shows ha ineres ax shields can have a negaive or a posiive value. More recenly, Cooper and Nyborg (2004) model ax- and risk-adjused discoun raes when he company follows he Miles-Ezzell leverage policy bu deb is risky, and boh corporae and personal axes are considered. Likewise, 4

6 Fernandez (2004, 2005), Fieen e al. (2005) and Cooper and Nyborg (2006) debae he issue of wheher or no he value of he deb ax saving is equal o he presen value of ax shields. They use a risk-adjused discoun rae seings, and a Miles-Ezzell leverage policy while only considering corporae axes. In conras, we value general cash flows emerging in capial budgeing using a cerainy equivalen approach. Our analysis goes beyond hese papers in he following ways. I incorporaes personal and corporae axes, a variey of deb policies and uses a cerainy-equivalen approach o risk. I discusses he appropriaeness of various ax models. I also considers non-linear cash flows ha are no accuraely valued wih risk-adjused discoun raes, such as cash flows wih opionaliy, like real opions, or asymmeric cash flows, like ax shields. An ouline of our work is as follows. In Secions 2 and Secion 3, we provide an equilibrium valuaion approach in coninuous-ime for real and financial asses in an economy wih personal and corporae axes, where ax raes on bonds and socks are differen. This is he generalized Miller model. In Secion 4 we presen a coninuous-ime capial budgeing valuaion approach for levered and unlevered real asses. Using coninuous ime allows us o apply he resuls direcly o such valuaion problems (including real opions), bu he resuls have naural inerpreaions in discree ime, as well. We sar wih he case of no defaul risk, and hen we consider defaulable deb wih exogenous defaul. Deb policy is aken as exogenous in he sense ha i is no subjec o opimizaion. 3 In Secion 5 we inroduce he basic real opion o delay an invesmen under he assumpion ha he incremenal deb o finance he real asse is issued condiional on he decision o inves. The deb policy is sill exogenous. Secion 6 provides concluding remarks. 3 Foonoes 14 and 17 discuss how our approach can be exended o endogenous opimizaion of capial srucure and defaul decisions. 5

7 2 Asse valuaion in a generalized Miller economy 2.1 Tax equilibrium We assume a coninuous-ime Miller (1977) economy ha is generalized o allow for crosssecional variaion in corporae ax raes. In general, he personal ax rae for bond invesmen income is differen from he personal ax rae for sock invesmen income. Miller assumes ha here is cross-secional variaion in personal ax raes, bu no corporae ax raes. Thus, in his ax equilibrium, all corporaions are indifferen abou capial srucure, bu invesors have a ax-induced preference for deb or equiy, leading o ax clieneles. By allowing cross-secional variaion in corporae ax raes, as in Sick (1990), only firms a he margin are indifferen (on a ax basis) beween issuing deb and equiy, and firms wih a marginal ax rae τ c below he marginal firm s rae rae (τ c τ m = τ < 0) prefer o issue equiy and firms wih a ax rae above he marginal firm s rae of he marginal firm (τ > 0) prefer o issue deb. We assume ha capial gains and coupons in bond markes are axed a he same rae τ b, and capial gains and dividends in equiy markes are axed a he same rae τ e, for he marginal invesor. The operaors for invesor or personal ax are assumed o be linear a any dae, in he sense ha income and losses from a paricular invesmen are axed, or generae ax relief, a he same rae. 4 On he oher hand, he ax scheme for corporaions need no be linear. We allow τ c, τ b and τ e o be F-adaped sochasic processes, i.e. hey are deermined as a funcion of he (sochasic) facors underlying he economy, bu hey 4 In general, individual invesors do have a progressive ax srucure, wih increasing raes a increasing levels of income. Wha we are assuming here is ha a paricular invesmen ha he invesor is pricing does no have income variaions so large ha i moves he invesor o higher or lower ax brackes. The key assumpion for his paper is ha he marginal invesmen gains and losses (a he invesor level) for a paricular asse are axed a he same rae and ha here is no kink in he ax curve as an asse goes from a gain o a loss. Noe, also, ha we are speaking of he marginal ax raes of he marginal invesor or firm, and assume he reader can disinguish which marginal o use in any paricular conex. 6

8 mus have coninuous sample pahs, almos surely. In general, τ b and τ e are differen. Ross (1987) esablished he exisence of equilibrium and of an EMM for an economy wih personal axes and a convex ax schedule. These assumpions are saisfied in our seing. Consider a firm wih ax rae τ c ha is deciding wheher o issue deb or equiy o an invesor wih ax raes τ e and τ b on equiy and deb, respecively. If (1 τ b ) < (1 τ c )(1 τ e ), (2.1) hen he firm has he abiliy and incenives o issue equiy on erms ha would be more favourable o he marginal invesor han deb a issue. To see his, suppose riskless deb has a yield of r f. Then, if equaion (2.1) holds, i is possible o choose a rae of reurn r z for equiy, 5 so ha he afer-corporae-ax cos of equiy paid by he company is less han ha of deb cos and, simulaneously, he afer-all-ax rae of reurn for equiy received by he invesor is higher han ha of deb. Tha is, we can choose any value of r z such ha 1 τ b 1 τ e rf < r z < (1 τ c )r f. On he one hand, his implies ha (1 τ b )r f < (1 τ e )r z, so ha he invesor achieves a higher afer-all-ax reurn on equiy han deb. On he oher hand, i implies ha r z < (1 τ c )r f, so ha he cos of equiy o he firm is lower han he afer-ax cos of deb. Thus, if (2.1) canno hold for he marginal invesor and he marginal firm wih an equilibrium economy, since he firm and he invesor can boh be beer off by swiching some deb o equiy. Similarly, he marginal firm has incenives o issue (and he marginal invesor has an 5 We se aside he risk premium for a momen, so ha r z is a riskless yield. We will address his poin laer on. 7

9 incenive o buy) deb raher han equiy if (1 τ b ) > (1 τ c )(1 τ e ). In equilibrium, here will be no furher incenives for he firm o issue deb o reire equiy, or o issue equiy o refund deb. Denoing he marginal firm s ax rae by τ m, we have he Miller (1977) equilibrium relaionship amongs he marginal ax raes of he marginal invesor and marginal firm: (1 τ b ) = (1 τ m )(1 τ e ). Thus, we have esablished he following proposiion: Proposiion 1. If he marginal firm has ax rae τ m, and he marginal invesor has marginal ax raes τ e in equiy and τ b on deb, hen: 1 τ m = 1 τ b 1 τ e. (2.2) Moreover, he marke yields on riskless deb and equiy are relaed by he following ax wedge: r z = (1 τ m )r f. (2.3) Thus, i is appropriae o refer o r z as a ax-adjused discoun rae. Equaion (2.3) provides a way o esimae he marginal ax rae τ m, since r z and r f can eiher be observed or derived from securiy prices. The equilibrium raes of reurn on he money marke and sock marke are he same afer all axes for he marginal invesor. Defining his common afer-all-ax reurn o be 8

10 r z,a r f,a, we have r f,a = r f (1 τ b ) = r z (1 τ e ) = r z,a. (2.4) 2.2 Tax clieneles This model uses he marginal invesor and her ax raes o derive he marginal firm s ax rae, which is no oherwise idenified. We can say ha he marginal firm s ax rae τ m 0, because he marginal invesor s deb ax rae usually exceeds her equiy ax rae, τ b τ e. In his siuaion, he riskless deb reurn exceeds he riskless equiy reurn: r f r z. If τ b = τ e, hen τ m = 0 and hus r z = r f. The assumpion ha τ b = τ e is common in he lieraure. I was used by Modigliani and Miller (1963) and underlies mos of he sandard exposiion of he CAPM, such as Sharpe (1964) and APT as in Ross (1976). Ohers, such as Sick (1990) and Taggar (1991) have aken he reurn differenials implici in Miller (1977) o ge r f r z. We choose he noaion r z o bring he analogy wih he zero-bea rae of reurn in he Black (1972) version of he CAPM. There may be no riskless equiy securiy, bu in many circumsances i is he inercep erm in such a CAPM. I is he appropriae discoun rae for cerainy-equivalens ha are all-equiy financed. We shall use i as he discoun rae in maringale and PDE valuaion models for all-equiy financed cash flows. There may be clienele effecs whereby he marginal invesors for differen ypes of securiies have differen ax classes. Thus, i may be appropriae o use differen discoun raes o value hese securiies. For example, in he financial derivaives lieraure and acual pracice, here is rarely any consideraion ha a ax wedge should be applied o he riskless deb rae in valuaion models. This is jusified, since he marginal invesor for a derivaive is likely an incorporaed marke maker ha rades frequenly and hus akes capial gains and dividends as ordinary income. For such a marginal invesor, τ b = τ e and 9

11 hence r z = r f even if hey finance heir posiions enirely wih equiy. On he oher hand, for real asses, he marginal invesor is likely a axable individual who holds shares in a corporaion. If he real opion is enirely financed wih equiy, hen i is appropriae o use a riskless rae r z < r f. In wha follows, we only require ha all ax raes be beween 0 and 1: 0 τ b, τ e, τ c, τ m < 1. Tax arbirage has a endency o make all firms behave as if hey have he same ax rae. A he corporae level, an arbirage scheme could involve a highly axed firm, wih τ c > τ m or (1 τ b ) > (1 τ c )(1 τ e ), issuing deb o acquire is own equiy, for example. Or, i could involve a low-ax firm, wih τ c < τ m, issuing equiy o buy back deb. We assume ha here are ax laws and agency coss 6 ha preven a firm from underaking such an arbirage ransacion. Thus, here will generally be firms wih τ > 0 and firms wih τ < 0 in his generalized Miller ax equilibrium. I is more difficul o generae ax arbirage opporuniies ha would have all invesors facing he same effecive personal ax rae as ha held by he marginal invesor. This is because personal ax laws idenify he individual and generally change when a financial eniy (such as a corporaion, muual fund or rus) is insered beween he invesor and he invesmen vehicle. There will generally be invesors who prefer deb o equiy and invesors who prefer equiy o deb in his generalized Miller ax equilibrium. Indeed, Miller (1977) also assumed his. For example, suppose an invesor pays lile or no ax on any invesmen (e.g. a pension fund), bu he marginal invesor pays higher ax on deb han on equiy. The unaxed invesor would prefer he ax benefis of deb, bu his will preven he invesor from earning risk premia paid on equiy invesmens. We assume ha any aemp o conver an equiy invesmen wih a risk premium o a deb insrumen for 6 See for insance, Myers and Majluf (1984). 10

12 ax purposes is prevened by ax law. 3 Sochasic model We assume an economy wih complee financial markes ha has personal and corporae axes. The ime horizon is T, he underlying complee probabiliy space is (Ω, F, P), where he se of possible realizaions of he economy is Ω, he σ-field of disinguishable evens a T is F, and he acual probabiliy on F is P. We denoe by F = {F, [0, T ]} he augmened filraion or informaion generaed by he process of securiy prices, wih F T = F. 3.1 Sochasic ax raes Under many ax sysems, such as he US ax code, he ax benefi of a bond coupon paymen is fully allowed only if he firm s EBIT is greaer han he coupon. If i is less, a ax loss occurs ha can be carried forward or backwards agains posiive axable income. If i is carried forward, he effecive marginal ax rae is reduced by a loss of ime value of money, as observed by Leland (1994, p. 1220). We assume ha he uncerainy in he effecive ax rae is caused by oher cash flows of he firm ha are independen of he projec s ax rae. This allows us o model he carryforwards wih a sochasic ax rae τ c. Because he ax rae is independen of he projec under consideraion, all of our proofs can be done condiional on he value of τ c, and hen an expecaion ake over his risk. I is quie likely ha many of he valuaion resuls can be exended o he siuaion where ax raes are no independen of projec cash flows. We consider a firm ha may have many projecs ha are independenly financed wih non-recourse or projec financing. Thus, defaul and bankrupcy evens are confined o each projec and don affec he res of he firm. The shocks o earnings of projecs can affec he ax rae faced by oher projecs. We assume ha hese shocks o he ax rae 11

13 τ c are independen of he shocks o he cash flows o he projec we analyze. This would happen for an arbirarily small projec or for one where he corporae ax rae is consan. Similarly, we can have he marginal firm s ax rae τ m o be sochasic, bu we assume i o be independen of he projec cash flows. The sochasic naure of ax raes is no essenial o achieve he resuls of his paper and readers may find i more convenien o hink of he ax raes as being deerminisic or even consan. Taxes inroduce a risk-sharing mechanism beween governmen and axed invesors. This could cause a difference beween he equilibrium EMM pricing measure wih and wihou personal axes. Or, he EMM could differ for differen ax clieneles. We will esablish a valuaion neuraliy principle, which says ha he equilibrium pricing measure is unchanged by he presence of axaion. For valuaion purposes, his means ha he expecaion (under he EMM) of he pre-ax cash flow sream of a securiy, discouned a a pre-ax rae, is equal o he expecaion (under he same EMM) of he afer-ax cash flow sream, discouned a an afer-ax rae. A second imporan issue inroduced by axaion of securiy reurns is he presence of iming opions relaed o axaion of capial gains, as poined ou by Consaninides (1983). Taxaion of capial gains produces iming opions due o a (raional) delay of liquidaion of posiions in financial securiies, unil a dae of forced liquidaion, if he accrued capial gain is posiive and he anicipaion of liquidaion of he posiion if he capial gain is negaive, o ake advanage of he ax credi. We avoid hese iming opions by assuming ha capial gains are axed as accrued, as if he invesor were o have a mark-o-marke cash flow ha induces he capial gain. Taxaion of accrued capial gains is one example of a holding-period neural ax scheme in which he ax scheme does no inroduce any iming opions relaed o axaion of capial gains. Auerbach (1991), Auerbach and Bradford (2001) and Jensen (2003) describe axa- 12

14 ion schemes ha provide holding-period neuraliy. I may be ha our resuls generalize o hese oher ax schemes as well, bu ha is a opic for fuure research. 3.2 EBIT and asse-price dynamics Suppose he process for EBIT, under he acual probabiliy measure is dx = g(x, )d + σ(x, )dz. (3.1) Consider a projec of value V = V (X, ) ha pays equiy holders he afer-corporaeax cash flow a he rae Y (X, τ c, ), 0 θ p, (3.2) where he projec life is θ p. For an unlevered firm, Y (X, τ c, ) = X (1 τ c ). For a levered firm, we will break ou ineres paymens and ineres ax shields by having a more general form for Y in equaion (4.2). Since he EBIT process is no raded, we will value he claim o he equiy-financed sream Y by using an afer-all-ax version of he consumpion CAPM, as developed in Consaninides (1983, Theorem 6) and used by ohers, such as Kane e al. (1984). Ross (2005) also uses a general equilibrium afer-all-ax model o ge he marke price of risk. 7 Then we will calculae a sum of values of individual cash flows, as in Consaninides (1978). To keep he noaion convenien, suppose he marke risk premium for he diffusion dz is λ in an afer-ax consumpion CAPM. 8 7 We would ge he same resuls wih replicaion argumens using a raded securiy. Ineresingly, he hedge raios used for he replicaion do no need a ax adjusmen. 8 Tha is, suppose a securiy has price P following dp = µ(p )d + σ(p )dz and pays a dividend a he rae P δ(p ). Then, by he afer-ax consumpion CAPM, we have he insana- 13

15 By Iô s lemma, he afer-all-ax wealh gain o he equiy invesor follows he process ( σ 2 (X, ) 2 V V + g(x, ) 2 X2 X + V ) + Y (X, τ c, ) (1 τ e )d + σ(x, ) V X (1 τ e )dz. Here we ake advanage of he fac ha he capial gains are axed as accrued o he invesor and a he same rae as he dividend disribuion Y. Taking expecaions under he acual probabiliy measure P, he expeced afer-all-ax growh rae in wealh from he invesmen is ( σ 2 (X, ) 2 V V + g(x, ) 2 X2 X + V ) + Y (X, τ c, ) (1 τ e ). (3.3) In he afer-ax consumpion CAPM equilibrium, his mus equal he riskless rae afer ax plus he risk premium ( (1 τ e ) r z + λ V X ) σ(x, ) V. (3.4) V Equaing (3.3) and (3.4) and simplifying yields he following proposiion: Proposiion 2. Under he assumpions of his secion, including personal axaion, he neous required rae of reurn: «µ(p ) P + δ(p ) (1 τ e ) = r z (1 τ e ) + λ σ(p ) P (1 τ e ). Since dividends and capial gains are axed a he same rae in our model, he facor (1 τ e ) facors ou and his becomes he same as a preax consumpion CAPM: µ(p ) P + δ(p ) = rz + λ σ(p ) P. 14

16 value of an equiy-financed real asse saisfies he fundamenal afer-all-ax PDE: ( 1 2 σ2 (X, ) 2 V V + ĝ(x, ) X2 X + V ) (1 τ e ) + Y (X, τ c, )(1 τ e ) = r z,a V (3.5) where he risk-neural growh rae is ĝ(x, ) g(x, ) λσ(x, ), (3.6) whenever 0 θ p. We also have he equivalen preax PDE: 1 2 σ2 (X, ) 2 V V + ĝ(x, ) X2 X + V + Y (X, τ c, ) = r z V, (3.7) where he risk-neural growh rae ĝ is he same as defined in (3.6). Proposiion 3. If he assumpions of Proposiion 2 hold and he projec has residual erminal value a ime θ p of V θ p = V (X θ p, θ p ) = TV(X θ p, θ p ), (3.8) hen V (X, ) = B z Ê [ θ p Y (X s, s) Bs z ds + TV(X θ p, ] θp ) Bθ z p (3.9) 15

17 where he money marke accoun for riskless equiy is ( u ) Bu z = exp rsds z 0. (3.10) Proof. Since he capial gains on changes in V are axed as accrued, here is no personal ax on he final disribuion of TV(X θ p, θ p ) o he invesor. We can use he Feynman-Kac soluion 9 o he PDE (3.7) o obain he resul. Characerizing he value as he expecaion of a discouned sream of payoffs is very useful, because i allows us o value projecs by Mone Carlo simulaion, even if hey are american real opions where he manager can adjus sraegies in response o he arrival of informaion. 3.3 Valuaion neuraliy Noe ha he risk-neural growh rae ĝ and hence he risk-neural expecaion operaor Ê is he same for he pre-personal-ax and afer-all-ax valuaion models. This means ha axaion affecs he valuaion only hrough he cashflow process and he discoun rae, bu i does no affec he risk premium or cerainy-equivalen calculaion. 10 We refer o his principle as valuaion neuraliy. 9 See, for example, Duffie (2001, pp ). 10 Ross (1987) shows ha here is a maringale pricing operaor in he presence of axes. His resuls are general and, in our seing, esablish a risk-neural expecaion operaor (or, equivalenly, a cerainyequivalen operaor) afer all axes, as well as risk-neural expecaion operaors for deb flows before personal ax and for equiy flows before personal ax. There is no immediae guaranee ha hese pricing operaors are relaed or equivalen. Sick (1990) raised his quesion in a discree-ime seing and showed ha he cerainy equivalen operaors associaed wih hese pricing operaors are all idenical o each oher. Tha is, axes and ax shields do no generae a risk premium. Proposiion 2 exends his resul o coninuous ime. 16

18 3.4 Afer-all-ax valuaion The mark-o-marke axaion of capial gains resuls in a cash-flow sream ha is similar o a dividend sream, and his has o be considered if we wan o compue values from afer-all-ax cash flows. Proposiion 4. Under he assumpions of Proposiions 2 and 3, he value of he real asse saisfies he following expecaion: [ θ p V = B z,a Ê (1 τu)y e (X u, u) θ p Bu z,a du τu e Bu z,a where he afer-ax money marke accoun for riskless equiy is B z,a u ( u = exp 0 dv u + V θ p B z,a θ p ], (3.11) ) rs z,a ds. (3.12) Proof. The proof is an exension of he proof of he Feynman-Kac resul, as discussed, for example in Duffie (1988, Secion 21). We will show ha he soluion o he maringale valuaion (3.11), will also saisfy he fundamenal valuaion PDE (3.7). Then we appeal o he uniqueness of he soluion of he PDE subjec o he erminal condiion (3.8). Define he differenial operaor D such ha if a real-valued funcion f C 2,1, hen Df(x, ) = 1 2 σ2 (x, ) 2 f(x, ) f(x, ) f(x, ) x 2 + ĝ(x, ) +. (3.13) x where 0 θ p. Define he afer-all-ax discouned value of V as W (x, θ) V (x, θ) B z,a θ, (3.14) 17

19 for θ θ p. By equaion (3.11), Thus W (X, ) = Ê = Ê [ θ [ θ (1 τu)y e (X u, u) θ Bu z,a du (1 τu)y e (X u, u) Bu z,a du τu e Bu z,a θ τu e Bu z,a dv u + V θ B z,a θ ] dv u ] + Ê [W (X θ, θ)]. (3.15) [ θ (1 τ Ê [W (X θ, θ)] W (X, ) = u)y e (X u, u) Ê Bu z,a [ θ = Ê θ du (1 τu)y e (X u, u) Bu z,a du θ τu e Bu z,a τu e B z,a u dv u ] ] DV du (3.16) The second equaliy comes from evaluaing dv u by Iô s lemma and noing ha he diffusion erm has an expecaion of zero. Applying Iô s formula o W gives W (X θ, θ) W (X, ) = θ θ W DW du + x σ(x u)dz u. (3.17) Taking risk-neural expecaions, he second erm on he righ also becomes zero. The lef side is he same as he lef side of equaion (3.16), so we can equae he righ sides, geing: Ê [ θ ( DW + (1 τ u)y e (X u, u) Bu z,a τ u e Bu z,a ) ] DV du = 0. This holds for all θ [, θ p ], including values arbirarily close o, and his can only happen if he inegrand is idenically zero. Thus DW + (1 τ e )Y (X, ) B z,a τ e B z,a DV = 0. (3.18) 18

20 Now, we can evaluae DW by subsiuing he definiion (3.14) for W, expanding he differenial operaor and applying he produc rule and chain rule: DW = 1 2 σ2 (x, ) 2 W x 2 = σ2 (x, ) 2B z,a = 1 B z,a 2 V x + ĝ(x, ) W x + W V B z,a x + 1 B z,a 2 + ĝ(x, ) DV rz (1 τ e ) B z,a V. V rz (1 τ e )B z,a ( ) 2 V B z,a Subsiuing for DW in equaion (3.18) and simplifying gives equaion (3.7), which is he desired resul. The second erm in he expecaion of he maringale valuaion (3.11) is he presen value of he mark-o-marke capial gains axaion where he iniial capial gains basis is V. 11 If we ake he same iniial basis, bu assume ha capial gains are only axed when realized, hen we have he following varian of he valuaion (3.11), which is consrucive: [ θ p V = B z,a Ê (1 τ e u)y (X u, u) B z,a u du + (1 τ e T )V θ p + τ e T V B z,a θ p In his case, here is no simple pre-personal-ax equivalen valuaion comparable o equaion (3.9). ]. 4 Valuaion of levered and unlevered projecs We consider a projec wih life θ p and EBIT following (3.1). We assume here is no residual value a he end of he projec, so TV(X θ p, θ p ) = I may be more naural o ake he basis o be V 0, since we allow o vary. This jus adds anoher erm o (3.11) for he valuaion of he capial gains from 0 o, and we omi i for breviy. 19

21 4.1 Valuaion of an unlevered projec For his secion, suppose Y (X, τ c, ) = X (1 τ c ) is he unlevered earnings before ineres bu afer corporae ax. The risk-neural drif ĝ(x, ) is given in (3.6). The value of he unlevered projec U(x, ) is given by Proposiion 3: U(x, ) = B z Ê [ θ p X s (1 τs c ] ) Bs z ds (4.1) for 0 θ p. 4.2 Valuaion of a levered projec The sum of he values of he deb and equiy claims agains a projec is called is Adjused Presen Value or APV. Suppose he firm is financed wih deb paying a coupon a he rae R = R(X, ). This can be a fixed coupon, or a growing or amorizing coupon. I can also be a risky coupon, where paymens are a funcion of EBIT, X. We could have he coupon depend on anoher random variable, bu for simpliciy, we keep he dimensionaliy of he risk o one. The deb maures a ime θ d θ p wih a promised principal repaymen of P o he bondholders. For θ d, he cash flow o equiy-holders is a he rae (X R )(1 τ c )(1 τ e ) and ha o deb-holders is a he rae R (1 τ b ). Afer he deb is reired, he projec is financed compleely wih equiy and we can keep he same represenaion of cash flow o equiy holders by seing R = 0 for θ d < θ p. Thus, he oal afer-all-ax cash flow from he projec o he equiy and deb invesors is (dropping he common subscrip ): X(1 τ c )(1 τ e ) + R(1 τ b ) = (X(1 τ c ) + τ R) (1 τ e ) (4.2) Y (X, τ c )(1 τ e ), 20

22 for 0 θ d, where he ne rae of ineres ax shield benefis is τ = τ c τ m and τ m is he marginal ax rae defined in (2.2). This formula has he following imporan feaures: 1. The erm τ R is he ineres ax shield. 2. The facor (1 τ e ) emerges on he righ side of he firs equaliy. In effec, he ax shield accrues o equiy invesors and is axed a heir marginal rae We can define he afer-corporae-ax levered cash flow Y = X(1 τ c ) + τ R by he las equaliy, since he facor (1 τ e ) is common in he second equaliy. Noe ha τ = τ c τ m is he rae a which ne ax shields accrue o invesors. The radiional lieraure is based on ineres ax shields having posiive value, which requires τ > 0. However, since τ is based on he difference beween he firm s ax rae τ c and he marginal firm s ax rae τ m, ineres ax shields could jus as easily have negaive value as hey have posiive value. For high-ax firms, hey have posiive value, and for low-ax firms, hey have negaive value. Since we have an afer-all-ax cash flow ha is axed a he personal equiy rae, we can apply Proposiion 3 o obain he following valuaion resul: Proposiion 5. The APV, incorporaing he value of he ax shield, saisfies he fundamenal PDE: 1 2 σ2 (X, ) 2 V V + ĝ(x, ) X2 X + V + X (1 τ c ) + τ R = r z V. (4.3) 12 Sick (1990) was he firs o poin ou his fac, bu i has been observed by ohers, such as Ross (2005). One inerpreaion of (4.2) is he following: he ineres ax shield can be inerpreed as a swap of deb for equiy. The swap is valued by he marginal invesor, who is indifferen beween receiving cash flow from equiies and cash flows from bonds. A = 0, he firm swaps B dollars of riskless or zero-bea equiy claims for an equivalen amoun (a he marke value) of deb, so, he iniial ne change in invesmen for invesors is equal o 0. A every coupon dae, he firm avoids having o remi he required rae of reurn r z B o equiy holders, bu mus pay r f (1 τ c )B o deb-holders. The ne gain for equiy-holders is r z B r f (1 τ c )B, which can be rewrien using he generalized Miller equilibrium (2.3) as r f B(τ c τ m ). 21

23 wih boundary condiions V (θ d, X θ d) = U(θ d, X θ d) and U(θ p, X θ p) = 0, and is [ ] θ d V (x, ) = B z X s (1 τ c Ê s ) + τs R s ds + U(θd, X θ d) B z θ d = U(x, ) + B z Ê [ θ d B z s τs R s Bs z ds ], (4.4) for 0 θ d. The second erm in he righ side is he value of he sream of ineres ax shields. In line wih Cooper and Nyborg (2006) and in conras o Fernandez (2004), equaion (4.4) implies ha he value of ineres ax shield can be calculaed by an addiive erm in an adjused presen value or APV reamen. Equaion (4.4) also implies ha when compuing he APV from he oal cash flows generaed by a company, he appropriae ax-adjused discoun rae (TADR) is he riskless equiy rae of reurn, r z, no maer wha is capial srucure is. In paricular, i is inappropriae o discoun he ineres ax shield a a deb rae, which is an error commonly seen in he lieraure APV wih specific deb policies Now we assume he corporaion has an exogenous capial srucure policy. Models wih endogenous capial srucure include Fisher e al. (1989), Mauer and Trianis (1994), Goldsein e al. (2001), Chrisensen e al. (2002), Dangl and Zechner (2003), Timan and Tsyplakov (2006), Srebulaev (2006), Hennessy and Whied (2005, 2006) bu we will no consider hese here. 14 They do no esablish any relaionship beween pre-ax and pos-ax 13 Even under he assumpion on personal and corporae axaion Mello and Parsons (1992), Mauer and Trianis (1994), Goldsein e al. (2001), and Timan and Tsyplakov (2006) use r f as he ax-adjused discoun rae (TADR). 14 Capial budgeing valuaion wih endogenous financial decisions enails he soluion of a dynamic programming problem, where deb is a conrol variable. In his case, our approach could be exended o he use of a Hamilon-Jacobi-Bellman equaion for opimizaion (as opposed o he parial differenial valuaion equaion) of he problem, wihou modifying our model of personal and corporae axaion. A 22

24 valuaion operaors as we do here. One exogenous policy is he consan deb policy of Modigliani and Miller (1958), which can generae bankrupcy. We will ake he bankrupcy hreshold o be exogenous. Anoher is he consan proporional deb policy of Miles and Ezzel (1985), which does no have defaul risk if deb is revised coninuously and he unlevered cash flow is non-negaive, as in a lognormal diffusion. Only in he case of a proporional deb policy, can we find a ax-adjused discoun rae ρ ha can be applied o unlevered flows o correcly compue he value of levered flows. 4.4 APV wih a riskless deb policy In his secion we assume ha managemen mainains a riskless deb policy by coninuously buying and selling bonds in he open marke as he firm value rises and falls. If he firm value falls, he bondholders can ender heir bonds o managemen a a marke price ha raionally anicipaes no defaul. Thus, hey do no suffer a loss and he coupon rae for corporae bonds would be he risk free rae. One policy ha achieves his is he policy of Miles and Ezzel (1980, 1985), which is ha he firm mainains a consan deb raio L. We assume he deb consiss of floaing rae coupon bonds for a erm θ d θ p, a he end of which he deb is reired a par. If he projec and deb are perpeual θ d = θ p =, he firm issues consol bonds, which are never reired. The coupon is se a he insananeous riskless rae, so he bonds are always priced a par. Tha is, he coupon flow is R = r f D(x, ), which is an adaped process, and D(x, ) only varies because of he bonds issues or repurchases by managemen in response o changes in x, and no because of ineres-rae or defaul risk. When he firm mainains a consan deb raio, we always have D(x, ) = LV (x, ). similar siuaion applies o endogenous defaul, as noed in foonoe

25 By replacing his condiion in equaion (4.3) we obain 1 2 σ2 (X, ) 2 V V + g(x, ) X2 X + V + X (1 τ c ) = ρ V (4.5) where ρ = r z τ r f L = (1 L)rz + L(1 τ c )r f (4.6) is he ax-adjused riskless cos of capial. 15 I is he cos of capial under he equivalen maringale measure ha incorporaes he effec of ineres ax shield and ha does depend on he level of deb. This is a coninuous-ime exension of he ax-adjused rae of reurn in Sick (1990). Our resul is in accordance wih he common pracice of compuing he NPV by discouning he expeced free cash flows under he acual probabiliy a he weighed average cos of capial (a risk-adjused discoun rae). Here, we use risk-neural valuaion, which allows for sochasic beas, as in many real opion problems. Thus, we remove he risk premium from he discoun rae and use risk-neural expecaions insead. However, we can sill use he weighed average o capure ineres ax shields. Applying condiions V (θ d, X θ d) = U(θ d, X θ d) and U(θ p, X θ p) = 0, and considering ha, for > θ d, D 0 (since R 0), we can use Proposiion 2 o calculae he APV of he projec as: V (x, ) = B w Ê [ θ d ] [ ] X s (1 τs c ) Bs w ds + B z U(θ d, X θ d) Ê B z, (4.7) θ d for 0 θ d, where B w = exp 0 ρ udu is he ime- value of one dollar accrued a he ax-adjused riskless discoun rae. Comparing Equaion (4.7) o (4.4) clarifies he role of he ax-adjused CE cos of capial, ρ, as he sochasic insananeous discoun rae ha generaes he levered asse value when applied o he unlevered cash flow process. 16 In 15 Grinbla and Liu (2002) have an analogous definiion, alhough hey limi he analysis o he less general seing wih consan rae of reurns and wih τ m = Grinbla and Liu (2002) have an analogue definiion of WACC, alhough hey limi he analysis o a 24

26 paricular, he randomness of ρ derives from he randomness of r z and r f, bu no from he leverage raio. Hence, equaion (4.7) provides a ime-consisen pricing operaor for levered cash flows. 4.5 APV wih defaulable deb If deb is riskless, he value of he projec is monoonically increasing (if τ > 0) or decreasing (if τ < 0) in he deb raio L, so we would need some oher consrains such as agency coss or laws o preven he deb raio from going o is limis of 0 and 1. If defaul can happen, hen we have a radeoff beween he value of ax shields for a firm wih (τ > 0) and he deadweigh losses from bankrupcy, which can resul in an inernal opimal capial srucure. When τ < 0, boh he ax shields and he defaul coss draw down firm value as deb increases, and here is no inernal opimal capial srucure. We assume here is a given barrier of coupon coverage, x D, such ha defaul occurs when he EBIT process X firs his he barrier from above. 17 In his case, he bond-holders file for bankrupcy and receive an equiy claim o he projec asses, ne of bankrupcy coss, which are he fracion α of he unlevered asse value a he ime of defaul. As long as he projec is solven (i.e., X u > x D u ), equaion (4.2) gives he oal afer-all-ax cash of he projec. If he firm defauls a dae, he ax shield is los over he inerval u θ p. Moreover, he deadweigh bankrupcy cos αu(, x D ) is los. Proposiion 6. Define T D inf { s [, θ d ], X s = x D } o be he random ime of defaul and le χ A be he indicaor funcion for even A. 18 Denoe x y min{x, y}. The APV, incorporaing he value of he ineres ax shield, saisfies equaion (4.3) wih boundary less general seing wih consan rae of reurns and wih τ m The exension of our valuaion approach o he case of endogenous defaul would enail he applicaion of he valuaion principles presened in Secions 2 and 3 o he parial differenial equaion for equiy wih a free boundary, as opposed o a known x D. See Leland (1994) and Timan and Tsyplakov (2006) for a discussion abou plausible values for x D. 18 χ A(ω) = 1 if ω A, χ A(ω) = 0 oherwise. 25

27 condiions V (θ d, x) = U(θ d, x) and U(θ p, x) = 0 and V (s, x D ) = (1 α)u(s, x D ) for all s θ d θ p. For x > x D, we have V (x, ) = U(x, ) + B z Ê αb z Ê [ [ TD θ d χ { s [,θ d ],X s=x D} ] τs R s Bs z ds U(T D, x D ) B z T D ]. (4.8) Proof. The APV saisfies equaion (4.3) when he firm is solven (while X > x D ). The erminal dae T D θ d = min{t D, θ d } is now sochasic, so he sandard Feynman-Kac soluion for he PDE (4.3) has o be replaced by a generalized resul where he coninuaion region on which he PDE holds is an open bounded se. Such a resul is in Dynkin (1965, Theorem 13.14) or Lamberon and Lapeyre (1996, Theorem 5.1.9). The second erm on he righ-hand-side of equaion (4.8) is he value of he ineres ax shield, aking ino accoun he risk i is los afer he projec defauls on he deb. The hird erm is he value of he bankrupcy coss incurred a he dae of defaul. In he las secion we also need D, he marke value of corporae bonds, in he generalized Miller equilibrium economy. As above, R is he coupon paymen and P he principal paymen (face value) paid back a mauriy θ d. Under he assumpion of exogenous defaul a he hreshold x D, deb is a coningen claim on X, he EBIT of he firm. The nex proposiion deermines he value of defaulable deb in he generalized Miller equilibrium. Proposiion 7. The marke value of deb saisfies 1 2 σ2 (X, ) 2 D D + g(x, ) X2 X + D + R = r f D (4.9) wih boundary condiions D(θ d, X θ d) = P, D(s, x D ) = (1 α)u(s, x D ) for all s θ d 26

28 θ p and, assuming x > x D, is D(x, ) = B f Ê [ TD θ d R s B f s ] [ ] ds + B f P Ê χ { s [,θ d ],X s>x D} B f θ [ d ] + (1 α)b f U(T D, x D ) Ê χ { s [,θ d ],X s=x D} B f T D (4.10) where T D θ d is he firs ime X = x D. Proof. The valuaion PDE (4.9) for D comes from he same argumen we used in he proof of Proposiion 2. Tha is, he expeced afer-all-ax paymens o bondholders under he acual probabiliy measure P saisfies: ( 1 2 σ2 (X, ) 2 D D + g(x, ) X2 X + D ) + R (1 τ b ). (4.11) In he afer-ax consumpion CAPM equilibrium, his mus equal he riskless rae afer ax plus he risk premium ( r f + λ D X ) σ(x, ) (1 τ b )D. (4.12) D Equaing (4.11) and (4.12), cancelling he common facor and simplifying yields equaion (4.9). Using he same exension o he Feynman-Kac resul used in he proof of Proposiion 6, he soluion of his PDE subjec o he boundary condiions is (4.10). Noe ha he marke price of volailiy risk λ is he same in he deb marke as i is in he equiy marke, because we derived boh from an afer-all-ax equilibrium perspecive where deb and equiy cash flows are on an equal ax fooing. Thus, he risk neural probabiliy disribuion is he same for deb and equiy markes, so we can use he same 27

29 expecaion Ê o value risky deb flows and risky equiy flows even a he marke level before personal ax is considered. 5 Valuaion of real opions wih deb financing and axes This secion addresses real opions valuaion under he general framework inroduced in Secion 2, assuming ha he ax shield may be valued according o he analysis of Secion 4. In paricular, we will differeniae our analysis according o he cases of defaul-free deb and defaulable deb. Here we resric ourselves o he basic real opion o delay an invesmen decision, alhough our approach can be applied o a broader class of real opions. 19 We hen assume ha we have he opporuniy o delay invesmen in a projec, under he EMM. The projec has life θ p (possibly, θ p = ), so ha he projec sars from he dae he opion is exercised, T I, and ends a T I + θ p. The cos o implemen he projec is I (an adaped process) and we have he opporuniy o delay he invesmen unil dae θ o (possibly, θ o = ). 20 We assume ha he capial expendiure o implemen he projec is also parially financed wih incremenal deb, and he incremenal deb is issued if and when he opion o inves is exercised. 21 This assumpion is realisic since here would be no reason o raise capial before invesmen. The opimal exercise policy depends on X, he EBIT process, 19 See Dixi and Pindyck (1994) and Trigeorgis (1996) for general discussions of real opions. 20 To simplify our analysis, we assume perfec informaion of shareholders and equiy-holders and he absence of agency coss beween shareholders and equiy-holders and equiy-holders and managemen. Hence, invesmen is implemened under a firs-bes invesmen policy (i.e., a policy aiming a maximizing he oal projec/firm value as opposed o a policy in he sole ineres of shareholders) by he managers. The role of agency coss of debs on invesmen decisions have been analyzed among ohers by Mello and Parsons (1992), Mauer and O (2000) and Childs e al. (2005). 21 Mauer and O (2000) and Childs e al. (2005) assume ha he incremenal invesmen is financed only wih equiy. Their assumpion is included in our framework by posing ha no incremenal deb is issued a he ime he invesmen decision is made. 28

30 and consequenly he dae when he firm will issue deb is a sopping ime. Issuance of incremenal deb is coningen on he decision o inves, so he financing decision is influenced by he invesmen decision. Conversely, he invesmen decision is influenced by he financing decision, since he former is made if he reurn from he projec compensaes is financing cos. The deb policy is alernaively he one in Secion 4.4 (defaul-free) or he one in Secion 4.5 (defaulable). The deb has erm θ d, so ha i is issued a T I and is paid-back a T I + θ d. The value of he levered projec, a he dae i is implemened, is V (, X ) from equaion (4.4) in he case wih defaul-free deb or from equaion (4.8) wih defaulable deb. In case defaul is possible, given he above assumpion ha deb is issued condiional on he invesmen decision, we assume ha defaul can happen only afer he invesmen dae. Le Π denoe he payoff of he opion a he exercise dae, Π(, X ) = max{v (, X ) I, 0}, (5.1) and le F (, X ) denoe he value of he invesmen projec including he ime-value of he opion o pospone he decision. Proposiion 8. The value of he opion o delay invesmen saisfies equaion 1 2 σ2 (X, ) 2 F X 2 + g(x, ) F X + F = rz F. (5.2) wih boundary condiion F (T I, X TI ) = Π(T I, X TI ), and is F (, x) = B z sup Ê T I [ ] Π(T I, X TI ) BT z, (5.3) I where T I θ o is he invesmen (sopping) ime, i.e., he firs ime F (, X ) = Π(, X ). 29

31 Proof. The proof ha F saisfies equaion (5.2) is he same as for he proof of Proposiion 5. The soluion in (5.3) is derived using sandard resuls (see Duffie (2001, pp )). From equaion (5.3) i is worh sressing ha, while he opion o delay invesmen is sill alive, he appropriae CE discoun facor o be applied o is expeced payoff is he one for equiy flows, B z. Obviously, his CE discoun facor is independen of boh he curren capial srucure, in case he invesmen opion is owned by an ongoing firm, and he capial srucure afer he projec is implemened. In addiion, equaion (5.3) suggess ha he opion o inves in a marginal projec (i.e., a projec wih corporae ax rae, τ c, equal o he marginal ax rae, τ m, and no ax shield τ = 0) is evaluaed according o Black, Scholes, and Meron s formula, bu using r z insead of r f. Noe ha in our seing, since a projec canno be all-deb financed, r f can only be used in he special case where τ e = τ b (which implies τ m = 0). 6 Concluding Remarks We have shown how o implemen he value of ineres ax shields in a real opions seing and invesigaed he implicaions of hese on he valuaion of an invesmen opion. Firs, we had o sudy ineres ax shields more rigourously han is common in he lieraure because he differenial axaion of deb and equiy income a he personal level has he effec of making ineres ax shields less valuable han is commonly believed. Indeed, for firms wih low ax raes, ineres ax shields can have negaive value. In a generalized Miller ax equilibrium, here will be marginal firms and invesors who are indifferen beween he ax implicaions of deb and equiy. Their ax raes can be used o deermine he differenial rae of reurn on riskless deb and riskless equiy ha is needed o susain such a ax equilibrium. 30

32 This allowed us o characerize ineres ax shields in erms of he difference beween a firm s ax rae and he marginal firm s ax rae. Having cash flows a a pre-corporae ax level, afer-corporae ax level and an aferall-ax level leads o naural quesions of he level a which cash flows should be valued. The mos appropriae valuaion occurs a he afer-all-ax level for he marginal invesor. Unforunaely, such cash flows are no readily observable. Forunaely, we have been able o show ha if here is linear axaion a he personal level, a linear valuaion operaor afer all axes leads o a linear valuaion operaor afer corporae ax bu prior o personal ax. Moreover, he cerainy-equivalen operaors or risk-neural expecaion operaors of hese preax and afer-ax cash flows are he same. The only difference in valuaion is he choice of discoun raes: a riskless equiy rae is correcly used o discoun risk-neural expeced equiy flows, a riskless deb rae is correcly used o discoun risk-neural expeced deb flows and an afer-all-ax rae is used o discoun risk-neural expeced flows afer all ax. In erms of he fundamenal pde for valuing risky payoffs or ineres ax shields, his means ha he only change needed o reflec ha he flows go o equiy or deb invesors is o change he riskless discoun rae in he risk-neural required reurn. The coefficiens corresponding o risk adjusmens are he same for he equiy and deb pde, despie he differenial axaion. This allows us o value ineres ax shields wih and wihou defaul risk on he deb and exend heses resuls o he real opion o inves. We showed ha he riskless equiy rae is he appropriae ax-adjused discoun rae when compuing he value of he company (APV) from oal cashflows. In paricular, i is inappropriae o discoun he ineres ax shield a a deb rae. We showed ha real opions should be evaluaed according o Black, Scholes, and Meron s formula bu applying he risless equiy rae as discoun facor. Noe ha using he general formula where he riskless deb rae is used o discoun he cerainy 31