Debt Policy, Corporate Taxes, and Discount Rates

Transcription

1 Commens Welcome Deb Policy, Corporae Taxes, and Discoun Raes Mark Grinbla and Jun iu UCA Firs Version: December 25, 21 Curren Version: November 7, 22 Grinbla and iu are boh from he Anderson School a UCA. The auhors hank he UCA Academic Senae and he Harold Price Cener for Enrepreneurial Sudies for financial suppor and seminar paricipans a he Universiy of Arizona, he Universiy of Kenucky, and he 22 Financial Managemen Associaion meeing for commens on earlier drafs. Correspondence o: Jun iu, The Anderson School a UCA; 11 Weswood Plaza; os Angeles; CA [email protected].

2 Absrac Deb Policy, Corporae Taxes, and Discoun Raes This paper sudies he valuaion of asses wih deb ax shields when deb policy is a general ime-dependen funcion of he asse s unlevered cash flows, value, and hisory. In a coninuous-ime seing, i shows ha he value of a projec s deb ax shield saisfies a parial differenial equaion, which simplifies o an easily solved ordinary differenial equaion for mos plausible deb policies. A large class of cases exhibis closed-form soluions for he value of a levered asse, he value of is ax shield, and he appropriae ax-adjused cos of capial for discouning unlevered cash flows.

3 Perhaps he mos popular applicaion of financial heory is capial budgeing. Virually every suden of finance sars his educaion in he field by learning how o discoun fuure cash flows. By he end of a firs course, he suden has developed he basic ools o implemen a discouned cash flow analysis in a real world seing. Because he real world seing mus accoun for he relaive advanage of deb financing, arising from he deb ineres ax subsidy, sudens of finance generally learn ha such subsidies can be accouned for by discouning unlevered cash flows (also referred o as free cash flows a a ax-adjused weighed average cos of capial (or W ACC. Such ax adjusmens o he discoun rae generae a value for levered asses ha exceed he value hey would have if hey were no levered wih deb financing. 1 Despie he cenral imporance of his opic, research on how o do a proper valuaion for capial budgeing purposes is sparse and largely ancien, paricularly when i comes o deb ax shields. An inrinsic difficuly associaed wih he valuaion of deb ax shields is idenifying he risk of he ax deducions arising from he sream of fuure deb ineres expenses. The rae a which one discouns he fuure sream of ineres-relaed ax shields, and hence he value of hose ax shields, has eluded prior research, excep for he simples of cases. These cases impose sringen resricions on he cash flow process and deb policy o circumven he complex issue of risk and valuaion. Among hese are he models of Modigliani and Miller (1958 and Miles and Ezzell (1985. The Modigliani and Miller deb policy is one where he deb level is consan and deb is boh perpeual and defaul-free. This deb policy implies ha one can discoun he sream of fuure ineres-based ax shields a he risk-free rae. If he ax rae is consan, as hey assume, he deb ax shield s presen value is necessarily proporional o he presen value of he deb because he cash flow sream from deb and he ax shield are proporional o one anoher. Here, since he consan of proporionaliy is he corporae ax rae, he presen value of he deb ax shield is he produc of he corporae ax rae and he presen value of he deb. Modigliani and Miller (1958 also use his model o develop formulas for discoun raes ha accoun for he value of he ax shield when cash flows have no endency o grow. The ineresing case sudied by Miles and Ezzell focuses on he dynamic issuance of perpeual risk-free deb. This case assumes: 1 he unlevered cash flow realizaion a each dae follows a random walk wih no drif, which is paid ou upon is realizaion (hence here is no expeced growh, 2 he unlevered cash flow sream is valued by applying a consan discoun rae, and 3 he deb-o-asse raio is consan. Under hese assumpions, he cash flow from each dae s ax shield is of he same risk as he one period lagged unlevered cash flow. As Grinbla and Timan (1997, 22 and Brealey and Meyers (2 poin ou, 1 Despie aemps o inroduce he Adjused Presen Value mehod ino he classroom, he Weighed Average Cos of Capial approach sill vasly dominaes he praciioner landscape. For example, Graham and Harvey (21 observe ha of welve capial budgeing echniques, many of which are long ou of favor wih finance academics, he Adjused Presen Value mehod is he leas-used mehod. 1

4 in he coninuous ime limi of his model, he cash flow sream from fuure deb-relaed ax deducions is of he same risk as he sream of unlevered cash flows and hus can be discouned a he same rae as he unlevered cash flows. Miles and Ezzell, as well as he sandard exbooks, presen formulas analogous o hose in Modigliani and Miller for his alernaive deb policy. I would be an exraordinary coincidence if cash flow processes and deb policies mached hose of he Modigliani-Miller or Miles-Ezzell models. For his reason, a more general analysis is of grea imporance o he field of finance. In his paper we provide a comprehensive analysis of he value of he risky deb ax shield for he highly general class of Markovian deb adjusmen policies. For a large se of dynamic deb policies, which have sae-coningen (and hence risky issuance and reiremen of risk-free deb, we obain closed-form soluions. For a sill larger class of cases, we can poin o a sysem of ordinary differenial equaions, which are easily solved numerically, ha generae he ax shield s value. The discoun rae for unlevered cash flows ha accouns for he deb ax shield is also of criical imporance, boh o praciioners and researchers. We sudy he heoreical underpinning of such a discoun rae and relae i o he weighed average cos of capial. We can generally derive closed-form soluions for his discoun rae whenever we have closed-form soluions for he deb ax shield. However, we also are able o show wha adjusmens are needed o conver he W ACC o an appropriae discoun rae. Such adjusmens are almos always needed as he W ACC is an appropriae discoun rae only in he Modigliani-Miller and Miles-Ezzell cases, or in some linear hybrid of hese wo well-known cases. Finally, we derive a formula for more general deb policies ha characerizes he equiy bea as a funcion of he leverage raio and he unlevered asse bea. This formula generalizes he sandard exbook formulas of Hamada (1972 and Miles and Ezzell (1985, which are associaed wih he Modigliani-Miller and Miles-Ezzell models, respecively. Our approach differs from ha found in prior research on deb ax shield valuaion. In lieu of srong resricions on cash flows, projec values, asse values, discoun raes, or deb policy, we impose resricions on he informaion srucure. Using he opion pricing approach of Black-Scholes (1973 and Meron (1973, we assume ha informaion follows a Markov diffusion process. The advanage of his informaion srucure is ha i makes he marke dynamically complee. In our case, as long as he shor-erm risk-free rae and he discoun rae (which can be any funcion of he informaion se for an oherwise idenical unlevered asse are specified, we can use he sandard coninuous-ime valuaion mehodology o price any fuure payoff, be i a fuure cash flow generaed by he unlevered asse or a ax shield from a complex, ye realisic, deb policy. Essenially, we are viewing he ax shield as a derivaive of he underlying unlevered asse. We can wrie down a dynamic porfolio of he unlevered asse and a risk-free securiy ha racks he flow from he asse s deb ax shield for any reasonable dynamic deb policy. The no arbirage condiion, which is a parial differenial equaion, generaes he value of he ax shield as a funcion of he value of he 2

5 unlevered asse (or equivalenly, he unlevered cash flow. The no arbirage valuaion mehodology of asse pricing heory has been applied before in a corporae seing, perhaps mos eleganly by Ross (1978. However, ineresing funcional forms ha link he ineres ax deducion associaed wih an asse s financing mix o is unlevered cash flows provide economic insighs ha elude a more general framework. One example of his is a heurisic descripion of how he value of he deb ax shield, as well as he appropriae discoun rae, vary wih deb policies ha can be viewed as weighed averages of he Modigliani-Miller and Miles-Ezzell deb policies. Our paper also analyzes he discree seing, bu o a more limied exen. Here, when deb policies are linear funcions of cash flows, we obain closed-form soluions for he value of he deb ax shield. Secion I of he paper develops a general approach for valuing deb ax shields. I also analyzes an exraordinarily large class which has a closed-form soluion for he ax shield and presens wo larger classes of cases for which numerical compuaion of he value of he deb ax shield is rivial. Secion II examines he weighed average cos of capial and relaes i o valuaion. I also characerizes how he W ACC is affeced by dynamic deb policies and sudies when he W ACC can be used o obain valuaions ha properly accoun for he value of he deb ax shield. Finally, his secion derives closed-form soluions for ax-adjused discoun raes ha generae he correc valuaions of cash flow sreams. In mos cases ouside of he Modigliani-Miller and Miles-Ezzell frameworks, we show ha hese discoun raes differ from he W ACC. Secion III analyzes how o lever and unlever equiy beas and equiy risk premia for arbirary deb policies. Secion IV concludes he paper. I. The Valuaion of Deb Tax Shields In a dynamically complee marke, wo asses wih payoffs driven by he same source of uncerainy, and hus insananeously perfecly correlaed, can be valued in relaion o one anoher. Jus as an opion is valued in relaion o is underlying securiy, so oo can a deb ax shield be valued in relaion o he unlevered asse i is associaed wih. So long as he uncerainy behind he deb policy ha generaes he ax shield is ied only o uncerainy in he unlevered cash flows, deb ax shields are simply derivaives. For his reason, mos of he paper assumes ha he unlevered cash flow, he afer-ax cash flow ha would be generaed in he absence of deb financing, saisfies a general Markov diffusion process. An asse ha is levered wih risk-free deb has wo sources of afer-ax cash flow a dae : 1 he unlevered cash flow, X d, which is he afer-ax flow ha direcly sems from he real asse, which is assumed o be unaffeced by he asse s financing mix, and 2 he flow from he deb ineres ax shield, τ c D r f d, which is he produc of he ax rae τ c and he 3

6 deb ineres paymen D r f d. 2 Because valuaion is linear, he value of a levered asse is he sum of he values of is wo cash flow componens. For simpliciy (and wihou loss of generaliy, we le one Brownian moion, B, drive he uncerainy. Tha is, beween daes and T, wih T possibly infinie, dx = g(, XXd + σ(, XXdB. Wih his assumpion, he marke is dynamically complee wih wo asses. 3 This means ha a dynamic rading sraegy can ransform a levered asse ino an unlevered asse, and vice versa. Similarly, knowing he value of an asse for any given deb policy allows us o compue he value of is ax shield for all deb policies. Soluions can be found wih a variey of mahemaically equivalen approaches, bu he mos popular mehod involves he soluion of a differenial equaion generaed by Io s emma and he principle of no arbirage. The coninuous-ime seing, described above, allows valuaion of almos any derivaive, including ax shields, by applying he well-known no arbirage principle. Hence, he following assumpions are primarily used for exposiional clariy and explici soluions: The corporae ax rae, τ c, is consan. The risk-free ineres rae, r f, is consan. 4 There are no personal axes, 5 bankrupcy, or oher marke fricions associaed wih deb beyond he corporae ax (implying ha he deb ineres coupon rae equals he risk-free rae. 2 In order for he levered asse o have he same invesmen policy in he presence of deb, we assume, wihou loss of generaliy, ha he flow from he ineres-based ax shield is paid ou. I could be reained in a risk-free ineres bearing accoun and disribued laer, bu his has ax consequences for he firm. In essence, such reenion amouns o negaive deb and i is he ne deb policy for which we are compuing he ax shield. Given his definiion of how o accoun for deb, and appropriae care aken o avoid double couning when his cash is evenually disribued, our resuls apply irrespecive of wheher cash is reained or paid ou. 3 B can be a vecor provided ha he insananeous changes in deb are perfecly correlaed wih insananeous changes in X. If he unlevered cash flow has J Brownian moion componens and deb policy depends differenly on each of hem, our resuls sill go hrough whenever we can value he levered asses a J disinc deb levels. Alernaively, J disinc securiies can be used o value he ax shield. These could include equiy, equiy opions, or comparable asses a he same or disinc deb levels. 4 This implies ha he risk-free yield curve is fla and nonsochasic. Deb mauriy is irrelevan in our model. 5 Personal axes are clearly imporan for asse valuaion and deb policy, as Green and Hollifield (22 prove heoreically and documen empirically. Their paper analyzes he opimal capial srucure for a firm wih a Modigliani-Miller deb policy, bankrupcy coss, and a cash disribuion policy o equiy holders ha is sensiive o he economic effecs of he corporae and capial gains axes. 4

7 A. Deb Policy and evered Asse Valuaion: The General Case We begin by sudying he valuaion of a levered asse under a general class of deb policies. The deb policy D can be any differeniable funcion of ime, unlevered cash flow X, levered asse value V, and hisory H. 6 Tha is, D = D(, X, V, H. These general deb policies can depend, in a quie complicaed manner, on he hisory of he asse, such as pas cash flows, pas deb values, pas asse values, in addiion o curren cash flow and curren asse value. We only require ha he hisory dependence a a given dae be summarized by addiional dae sae variables. This allows us o mainain he Markovian seing. Wihou loss of generaliy, we simplify noaion by reaing hese sae variables as he single variable H. 7 As long as he uncerainy associaed wih he pah of X spans he relevan sae space for H, we will sill be able o value he deb ax shield as a funcion of he value of he unlevered asses. To mainain his desirable propery, we assume ha H saisfies he diffusion where and dh = µ H (, X, D, V d + σ H (, X, D, V dx, = µ h (, X, D, V d + σ h (, X, D, V db µ h (, X, D, V = µ H (, X, D, V + σ H (, X, D, V g(, X X σ h (, X, D, V = σ H (, X, D, V σ(, X X. Recognize ha he funcional form of he exogenously specified µ h and σ h can be quie general. I would be difficul o imagine any empirically relevan deb policy ha could no be capured wih his flexibiliy. Given his descripion of deb policy and hisory, i follows ha he dae value of he levered asse, V = V (, X, H, depends on he curren dae, cash flow, and hisory. Moreover, if he unlevered cash flow sream erminaes a dae T, (essenially, becomes zero a dae T and forever hereafer, he funcional form of he valuaion funcion will be influenced by he proximiy o he erminaion dae. 6 The opimal deb policy, while a criical issue boh in formulaing D and he value of he ax shield, is beyond he scope of his paper. A coninuous-ime model wih closed-form soluions for he opimal deb level when bankrupcy coss are raded off agains a Modigliani-Miller deb policy is found in eland s (1994 seminal research. In our paper, because deb can be adjused coninuously, firms can avoid bankrupcy wih cerainy. 7 We can also regard H and he coefficiens in is diffusion as vecors wih virually no change o any of our equaions. 5

8 In he absence of arbirage, dynamic compleeness implies ha his value necessarily saisfies he parial differenial equaion (PDE V + (g ηx V X σ2 X 2 2 V X r fv 2 +(µ h ησ h V H V 2 σ2 h H 2 + σσh 2 V H X X = (X + r fτ c D. (1 where we have dropped he argumens of g, σ, ec. for noaional simpliciy. If he asse has a finie life, he erminal condiion is V (T, X T, H T =. This parial differenial equaion, a familiar exension of he well-known Black-Scholes differenial equaion, is simply he no arbirage condiion associaed wih an asse whose uncerainy is spanned by he payoff o a dynamic rading sraegy in he unlevered asse and a risk-free securiy. The η(, X erm in equaion (1, (shorened o η for noaional simpliciy, is he premium per uni of risk generaed by changes in B. In a corporae seing, i would be radiional o hink of his parameer as being deermined by he insananeous discoun rae of he unlevered asse. However, η also can be inferred from he levered asse s value for any deb policy. To derive he parial differenial equaion, noe ha Io s emma implies ha he change in he value of a levered asse plus all disribuions of cash flow: dv + (X + r f τ c D d = V V V d + dx + X H dh ( 1 2 V + 2 X 2 σ2 X V H X σσ h ( V = + V V gx + X H µ h d ( 1 2 V + 2 X 2 σ2 X V H X σσ hx ( V + X + V H σ H σxdb. V U 2 V H 2 σ2 h d + (X + r f τ c D d 2 V H 2 σ2 h + X + r f τ c D d The analogous equaion for an oherwise idenical unlevered asse wih dae value = V U (, X is ( V dv U U + X d = + V U X gx V U 2 X 2 σ2 X 2 + X d + V U X XσdB = (V U r f + V U U V Xηd + X X XσdB. 6

9 where η, defined by is placemen above, is simply a convenien symbol for a scaling of he risk premium aached o db. We can express η in erms of he insananeous unlevered cos of capial, r U as η(, X = ru r f. (2 ln(v U ln(x Clearly, insananeous changes in V and V U are perfecly correlaed. Thus, o preven arbirage, he raios of he risk premia per dollar invesed in he levered and unlevered asses mus be proporional o he risk born per dollar invesed in each of he asses. This implies V + V V gx + X H µ h V 2 X 2 σ2 X V H X σσ hx + 1 ( 2 V = X + V H σ H Xη 2 V H 2 σ2 h + X + r f τ c D r f V which, when rearranged, gives us equaion (1. In analogous fashion, he value of he axshield = V V U saisfies he PDE + (g ηx X σ2 X 2 X r f 2 +(µ h ησ H H σ2 h H + σσ 2 2 h H X X = r fτ c D. In principle, hese parial differenial equaions can be solved. However, wihou furher resricions, hese differenial equaions are difficul o solve, even numerically. Hence, he remainder of his secion explores cases where soluions are insighful or, from a numerical perspecive, quickly aainable. Essenially, whenever we can ransform he PDE ino an ordinary differenial equaion (ODE, numerical soluions are easily found. We explore wo such classes of cases. In he firs, all of he model s parameers depend only on he conemporaneous level of he unlevered cash flow, X. Here, because here is no ime dependence, equaion (1 reduces o an ordinary differenial equaion in X. In he second class of cases, which we refer o as Addiively Separable Asses, he levered asse value is addiively separable in a se of argumens, which consis of H and a finie collecion of real powers of X: X λ 1, X λ 2,..., X λ N. The coefficiens of hese argumens may be ime dependen. The addiively separable class of cases is paricularly ineresing for is abiliy o generae remarkably general closed-form soluions for he value of he deb ax shield. These apply o boh finie-lived and perpeual asses. They arise whenever he N cash flow coefficiens of he value addiive funcions for deb and hisory are growing a consan exponenial raes and he remaining coefficiens are consan. Given his level of generaliy, i appears as if our closed-form soluions could generae fairly good approximaions for he value of a deb ax shield for mos conceivable deb policies. 7

10 B. Cash Flows and Deb wih Time Independen Parameers When he parameers of he dynamic process for X do no depend explicily on ime (ha is, g(, X = g(x, σ(, X = σ(x, and η(, X = η(x and he deb policy D does no depend on eiher ime or he hisory H, he asse value V depends only on he conemporaneous cash flow level, X. In his case, he PDE for he value of he levered asse (1 becomes he second order ODE: (g(x η(x V X σ2 (XX 2 2 V X 2 r fv = (X + r f τ c D(X. which is rivial o solve numerically for any specificaion of g(x, σ(x, and η(x. 8 Special cases wih closed-form soluions include he coninuous-ime versions of he Modigliani- Miller deb policy (g(x =, η(x = η and D(X = D implying V U = X/(r f + η and V = V U + τ c D and he Miles-Ezzell deb policy, (g(x =, η(x = η and D(X = d x X, wih d x consan, implying V U = X/(r f +η and V = V U +τ c Dr f /(r f +η. We defer furher discussion of his as he class of deb policies analyzed nex also includes he Modigliani- Miller and Miles-Ezzell models as special cases. C. Addiively Separable Asses: Numerical Soluions Addiively separable asses have ax shields wih values ha are addiively separable linear funcions of hisory, H, and any se of real powers of he cash flow, X λ. Simple examples of addiively separable asses include he consan coefficien quadraic case, V = V U + c x + c x 1X + c x 2X 2, which is generaed by he consan coefficien hisory-independen quadraic deb policy and he consan coefficien square roo case, D = d x + d x 1X + d x 2X 2, V = V U + c x + c x 1/2 X, which is generaed by he consan coefficien hisory-independen square roo deb policy D = d x + d x 1/2 X. 8 Any wo boundary condiions, which implicily deermine he deb level in all saes of he world, deermine a unique soluion o he differenial equaion. Hence, specifying he deb policy is clearly sufficien for obaining he levered asse s value. 8

11 A more complicaed case arises when he deb level is hisory dependen wih hisory given by H = H e mh 1 2 (lh 2 l h B + m x e mh ( s l 2 (lh 2 ( s l h (B s B X s ds + m d e mh ( s 1 2 (lh 2 ( s l h (B s B D s ds + m v e mh ( s 1 2 (lh 2 ( s l h (B s B V s ds + l x e mh ( s 1 2 (lh 2 ( s l h (B s B X s db s + l d e mh ( s 1 2 (lh 2 ( s l h (B s B D s db s + l v e mh ( s 1 2 (lh 2 ( s l h (B s B V s db s. In his special case, he diffusion process for H saisfies dh = (m x X + m d D + m v V and hus has drif and volailiy of m h H d + (l x X + l d D + l v V l h H db, µ h = m x X + m d D + m v V m h H, σ h = l x X + l d D + l v V l h H. This hisory process, wih he ms and ls consan, when combined wih an analogous funcional form for he deb process, leads o a closed-form addiively separable soluion for he value of he deb ax shield, as we show in he nex subsecion. The mos general class of addiively separable asses has hisory diffusion and deb policy of he form: ( dh = m x λ(x λ + m v (V + m d (D m h (H d and + λ ( λ l x λ(x λ + l v (V + l d (D l h (H db, D = λ d x λ(x λ + d v (V + d h (H, along wih risk premia, η(, unlevered cash flow growh rae, g(, and volailiies, σ( and σ H (, ha depend only on ime. An implicaion of g( and η( depending only on ime is ha he price-earnings raio for an unlevered asse, y U = V U /X, depends only on ime. 9 9 To prove his, noe ha X drops ou of he raio V U T = X e s g(ωdω e s ru ω dω ds. 9

12 One can show ha he value of a levered asse wih deb policy and hisory saisfying hese properies is of he addiively separable form V = V U + λ c x λ(x λ + c h (H and he PDE, equaion (1, is of he form 1 c x λ Xλ + ch H + (g η λc x λx λ σ2 c x λλ(λ 1X λ λ λ λ [ + (k h k d d h (k v + k d d v c h H + ] ( kλ x + k d d x λ + (k v + k d d v c x λ X λ + (k v + k d d v V U c h λ ( [ ( r f c x λx λ + c h H = r f τ c d x λx λ + d v V U + ] c x λx λ + c h H + d h H λ λ λ wih k q = m q ηl q for q {x, d, v, h}. Equaing he coefficiens of H and X λ on each side produces N + 1 ordinary differenial equaions wih N being he number of powers of X ha appear in he deb and hisory equaions: dc x λ d dc h d + (g ηλc x λ σ2 c x λλ(λ 1 + ( k x λ + k d d x λ + (k v + k d d v c x λ + (k v + k d d v y U δ λ,1 c h ( r f c x λ + r f τ c d x λ + d v c x λ + d v y U δ λ,1 = ( (k h k d d h (k v + k d d v c h c h r f c h + r f τ c d v c h + d h = wih δ 1,λ a binary variable ha akes on he value 1 if λ = 1 and oherwise, 11 and wih he erminal condiion given by c x λ(t = c h (T =. This sysem of Riccai equaions is easily solved numerically. However, here are large classes of cases ha have closed-form soluions. We explore hese below. D. Addiively Separable Asses wih Closed-Form Soluions Suppose ha each of he coefficiens k d (, k v (, k h (, d v (, d h ( are consan and k x λ( = k x λ(e gk λ d x λ( = d x λ(e gd λ 1 Noe ha many of erms involving V U cancel because of he no arbirage PDE for V U. 11 Wihou loss of generaliy, and only for noaional simpliciy, we assume ha one of he powers of λ is λ = 1 if one of d v, k v, d x 1, or k x 1 is non-zero. Also, noe ha if λ i =, we have an exponenially growing consan erm. We explore a special case wih his feaure laer. 1

13 wih he consan growh parameers gλ k and gd λ possibly. For exposiional clariy, we also assume ha he mean and volailiy of he unlevered cash flow growh rae, as well as he marke price of risk, are consan. Tha is, g(, X = g, σ(, X = σ, and η(, X = η. This allows us o express he value of an unlevered asse wih he growing annuiy formula, as in he Gordon Growh Model: V U = X r U g ( 1 e (ru g(t. (3 The Gordon growh assumpions imply ha ln(v U ln(x = 1 and ha he risk premium on he unlevered asse r U r f = η. We could allow g, σ, and η o be deerminisic funcions of ime and sill achieve soluions similar o hose developed below bu a he cos of expressions wih confusing ses of inegrals in hem. As his discussion is abou he valuaion of ax shields for complex deb policies, and no abou he complexiies of valuaion in a no-ax seing, we op for an approach ha makes he laer valuaion as uncomplicaed as possible. Under hese assumpions, he sysem of Riccai equaions is solved by c h ( = b(t 1 e 1 ch c h +b/ae b(t ch, (4 b = c h = a = k v + k d d v, (k h k d d h + r f (1 τ c d v 2 4aτ c r f d h, b2 + 4aτ c r f d h b 2a and for λ = λ 1,..., λ N c x λ( = r f τ c d x λ(c 1 (gλ d + r fτ c d v ( r U g δ λ,1 C 1 ( e (ru g(t C 1 (r U g + kλ(c x 2 (gλ k + k d d x λ(c 2 (gλ d + a ( r U g δ λ,1 C 2 ( e (ru g(t C 2 (r U g, (5 where C 1 (z = C 2 (z = 1 1 ch c h +b/ae b(t c h 1 ch c h +b/ae b(t [ 1 e (gλ x z(t ( gλ x z e b(t c h 1 e (gλ x c h + b/a gλ x z b ( ] b(t 1 e (gλ x z b(t e [ 1 e (gλ x z(t g x λ z g x λ z b z b(t 12 For he ax shield of a finie-lived asse o have a finie value, b, given below, has o be a real number. Also, for hisory o be sable, k h k d d h + r f (1 τ c d v has o be posiive. Throughou he paper, we assume ha parameers saisfy he ransversaliy condiions so ha he deb ax shield is finie. 11 ]

14 wih gλ x defined by g x λ = r f (1 τ c d v (k v + k d d v c h + (η gλ 1 2 σ2 λ(λ 1. Case 1: Perpeual Asses As T, equaion (4 becomes c h ( = c h and he N cash flow coefficiens given by equaion (5 simplify o c x λ( = kx λ (ch g x λ gk λ since, as T, + (r fτ c + k d c h d x λ ( g x λ gd λ + ac h + r f τ c d v (r U g(r f (1 τ c d v + η g δ λ,1, C 1 (z = C 2 (z = c h gλ x z 1 gλ x z. This implies ( V = 1 + ac h + r f τ c d v r f (1 τ c d v + η g V U + λ ( k x λ (c h g x λ gk λ + (r fτ c + k d c h d x λ ( gλ x X gd λ + c h H. λ Case 2: Perpeual Deb as a Funcion of Cash Flows Only When T and d v = d h =, c h =. In his case, he soluion for Case 1 simplifies o c x λ( = r fτ c d x λ ( g x λ gd λ implying V = V U + λ r f τ c r f + (η gλ 1 d x 2 σ2 λ(λ 1 g λ(x λ λ d. Case 3: Deb ha is a inear Funcion of Asse Value Plus Consan Growh For his special case, D = d x (e gd + d v V wih he sensiiviy of deb o asse value consan; ha is, d v ( = d v. The remaining coefficiens are zero. Noe ha he consan growh rae componen in deb, g, d may differ 12

15 from g, he expeced growh rae in cash flows. When d v is zero, deb grows a he consan geomeric rae of g d (possibly zero. When d x ( =, he deb o asse raio is consan over he life of he asse. Hence, his policy, as well as he saionary model described in he prior subsecion, ness boh he Modigliani-Miller and Miles-Ezzell deb policies. (Noe ha if d v is nonzero, he expeced growh rae in deb is influenced boh by he expeced growh rae in V as well as g d. This is a case where c h ( = c h = C 2 (z = and he soluion reduces o V = V U + c x ( + c x 1(X, wih he values for c x λ from equaion (5 becoming ( c x ( = r f τ c d x 1 e (r f (1 τ cd v g d (T ( r f (1 τ c d v g d c x 1( = r ( fτ c d v 1 e (r f (1 τ cd v +η g(t r U g r f (1 τ c d v + η g A paricularly simple expression exiss for a perpeual levered asse in Case 3. Here, he coefficiens above imply 13 ( V = V U r f + τ c d x r f r f (1 τ c d v g ( + d r f (1 τ c d v + η g dv V U (6 Noe ha when g = g d = d v =, equaion (6 is he Modigliani-Miller value, V = V U + τ c d x ( = V U + τ c D. When d v is zero bu g d and g are nonzero, we have an exension of he Modigliani-Miller deb policy ha allows for growing deb and unlevered cash flows ha are expeced o grow. As we will learn in he nex secion of he paper on he W ACC, he iniial weighed average cos of capial, used as a discoun rae, does no generae his value as he value of he levered asse when g d g. However, here is a simple adjusmen o he W ACC ha generaes he correc levered asse value. 13 One can also map D ino V. A small amoun of algebraic manipulaion reveals. V = V U + r f r f g d τ c (D + g η g d r f (1 τ c d v + η g dv V U Thus, he value of he deb ax shield is he sum of he Modigliani-Miller deb ax shield (wih consanly growing deb and a erm which may be posiive or negaive. The sign of he final erm in parenheses depends on wheher g η, he risk-adjused growh rae of he unlevered cash flows, is larger han g d, he growh rae for he nonsochasic deb componen. 13

16 When d x ( =, deb policy is an exension of he coninuous-ime Miles and Ezzell deb policy ha allows for cash flows wih nonzero expeced growh. In his case, equaion (6 indicaes ha here is a proporional relaionship beween he value of a levered asse and is oherwise idenical unlevered counerpar: implying V = V r f + η g r f (1 τ c d v + η g V U = V U r f + r f + η g τ cd. If g d = g, he deb ax shield can be wrien as a simple weighed average of he ax shields for he Miles-Ezzell consan leverage raio deb policy and he exended Modigliani-Miller deb policy (wih possibly growing deb. In his case, equaion (6 reduces o V = V U r f + w r f g τ cd + (1 w r f r f + η g τ cd where he weigh r f g w = (1 dv. r f (1 τ c d v g D /V This weigh is monoonically decreasing in d v, he sensiiviy of deb o he value of he asse, holding D fixed. 14 The Modigliani-Miller deb ax shield, which muliplies w above, has a smaller value han is Miles-Ezzell counerpar, which muliplies (1 w above. 15 Hence, for he same iniial deb level, increasing he deb sensiiviy coefficien, d v, while holding he iniial deb level fixed, reduces he value of he ax shield. This is because he value of he deb ax shield falls when is risk increases, oher hings equal. If he deb sensiiviy coefficien, d v, exceeds D /V, so ha w is negaive, he leverage raio will rise as he asse s value increases and fall when i decreases. In his case, he value of he asse will be below ha obained wih he consan leverage raio Miles-Ezzell deb policy. Conversely, if d v is negaive, so ha some of he exising deb is reired when he asse s value rises, 16 and deb is issued when he asse s value declines, he value of he deb ax shield will be above he τ c D value proposed by Modigliani and Miller. This confirms he inuiion in Grinbla and Timan (1997, 22 and suggess ha he appropriae discoun rae for unlevered cash flows will be 14 To see his, i is necessary o obain an equaion for he weigh wihou V. This is accomplished by subsiuing he former equaion ino he laer and solving for w. 15 This and he saemens ha follow from i assume ha η, he risk premium for he unlevered cash flow, is posiive. If he unlevered cash flow has a negaive risk premium, he Miles-Ezzell value exceeds he Modigliani-Miller value. 16 This deb paydown paern has been esimaed in empirical work by Kaplan and Sein (

17 a weighed average of he W ACCs proposed by Miles and Ezzell and Modigliani and Miller. For d v > D /V, he weigh on he Modigliani-Miller W ACC mus be negaive, for d v <, i is above 1, and oherwise, i lies beween and 1. As we will show laer, he weighing on he W ACC formulas of Modigliani-Miller and Miles-Ezzell is idenical o he weighing of he respecive ax shields given here. The linear deb policy in Case 3 easily exends o discree ime. The lineariy implies ha over any discree inerval, he levered asse is a fixed-weigh porfolio of an oherwise idenical unlevered asse and a risk-free securiy. In his case, he values of he levered and unlevered asses are perfecly correlaed, as boh are linear funcions of he cash flow. Solving he difference equaions ha generae he no arbirage value of he levered asse in an analogous manner yields he discree ime analogue o equaion (6: 17 ( V = V U r f + τ c d x r f (1 + r f + η r f (1 τ c d v g ( + d (1 + r f (r f (1 τ c d v + η g ηr f τ c d v dv V U. This valuaion soluion ness boh he discree-ime Modigliani-Miller and Miles-Ezzell deb policies as special cases. The discree case valuaion formula, provided above, applies only o an infiniely-lived asse. A similar approach generaes a discree ime closed-form soluion for a finie-lived asse. I is omied for he sake of breviy. II. Tax-Adjused Discoun Raes for Unlevered Cash Flows and he Weighed Average Cos of Capial For finance praciioners, discouning expeced unlevered cash flows a a ax-adjused discoun rae is he mos popular way o value an asse. This secion sudies he relaion beween his discoun rae and he afer-ax weighed average cos of capial. I develops formulas for hese discoun raes for a variey of deb policies and shows when and why naive applicaion of he weighed average cos of capial as he appropriae discoun rae can generae erroneous valuaions. We ake he perspecive of an invesor a dae. This invesor would like o know he discoun rae, applied o expeced fuure unlevered cash flows, ha generaes. Our analysis will show ha his discoun rae is rarely he W ACC compued a dae. Before we do his, i is imporan o sudy he W ACC and how i evolves hrough ime. 17 Simple algebraic manipulaion indicaes ha he mapping from D o V associaed wih he equaion below is given by V = V U + r f (1 + r f (g η g r f g d τ c (D d + + η(r f g d (1 + r f (r f (1 τ c d v + η g ηr f τ c d v dv V U. 15

18 A. Risk, Expeced Reurn, and he W ACC In coninuous ime, he weighed average cos of capial of an asse is defined o be he asse s insananeous expeced reurn, r, less he reurn componen due o he deb ax shield: W ACC = r D r f τ c. (7 A levered asse s dae insananeous expeced reurn, equivalen o is pre-ax weighed average cos of capial, is defined by 18 V r d = E (dv + X d + r f τ c D d. (8 This expeced reurn has hree componens: he expeced capial gain, he unlevered cash flow, and he cash flow from he deb ax shield. 19 One can readily show from he no arbirage condiion ha ( ln V r = r f + + σ H (, X, D, V ln V η(, X X. X H Combining his equaion wih equaion (7 provides a direc formula for compuing a W ACC given he value of he levered asse: ( D ln V W ACC = r f (1 τ c + + σ H (, X, D, V ln V η(, X X. (9 X H V Equaion (9 is a generalizaion of he Modigliani-Miller adjused cos of capial formula. I is convenien formula for compuing he W ACC given he exensive closed-form soluions compued in he las secion. For example, in he case of consan deb for a finie-lived asse wih zero expeced growh and a consan risk premium for unlevered cash flows, and ln V X In his case, he levered asse has a value of V V ln V H = = V U 1. V X = V U + τ c D (1 e r f (T, 18 This expeced reurn is he appropriae discoun rae for he capial cash flow sream, which is he ne payou o all cash flow claimans. See Ruback (22 for a lucid discussion of he advanages of his approach. 19 As menioned earlier, he laer flow mus be paid ou o mainain he same invesmen policy and capial gains appreciaion as an oherwise idenical unlevered asse. 16

19 he formula for he W ACC reduces o ( [ W ACC = r U D 1 τ c τ V c η D ] (1 e r V f (T. Noe ha he erm in brackes is decreasing in T and converges o zero for perpeual asses. Hence, his formula generaes a smaller W ACC han ha generaed by he Modigliani and Miller formula for perpeual asses. I is also possible o wrie down a differenial equaion for he W ACC. Subsiuing he expeced reurn formula, equaion (8, ino equaion (7 implies W ACC d = E [ ] dv + X d. (1 V Applying Io s lemma o his equaion, we find ha he W ACC saisfies he PDE ( V W ACC V = + gx V X σ2 X 2 2 V V + X 2 µh H V 2 σ2 h H + σσ 2 V 2 h H X V + X. To undersand he relaion beween he W ACC and sochasic discoun raes for unlevered cash flows, observe ha ρ, he dae sochasic insananeous discoun rae for X ha generaes he value of he levered asse, saisfies he sochasic inegral equaion V = E [ e s ρωdω X s ] ds. By he Feynman-Kac heorem, any ρ ha saisfies his sochasic inegral equaion also saisfies he PDE ( V ρv = + gx V X σ2 X 2 2 V V + X 2 µh H V 2 σ2 h H + σσ 2 V 2 h V + X. H X Since he PDE ha ρ has o solve is idenical o he PDE ha he W ACC solves (by Io s lemma, he following proposiion mus hold: 2 Proposiion 1 The W ACC is idenical o he sochasic insananeous discoun rae ha generaes he levered asse value when applied o he unlevered cash flows. This insigh is no very useful for capial budgeing praciioners. As equaion (9 indicaes, he W ACC a a fuure dae depends on he sae variables, X and H, a ha dae. Hence, fuure W ACCs are generally sochasic when viewed from dae, he relevan dae of he valuaion. 2 Obviously, here is no reason for he boundary condiions o differ or for eiher of he parial differenial equaions o be ill-behaved. 17

20 B. The Appropriae Tax-Adjused Discoun Raes Even hough he W ACC is generally sochasic, i may be ha some consruc relaed o he W ACC can be used o discoun expeced fuure unlevered cash flows o dae in a manner ha accouns for he ax shield. This subsecion explores his issue. We firs begin by analyzing a discoun rae, known a dae, ha ranslaes he expeced unlevered cash flow a dae s ino is value an insan earlier. We call his he ax-adjused forward rae. Once having developed an undersanding wha his ax-adjused forward rae is, and how o compue i, we prove ha he W ACC is an appropriae ax-adjused discoun rae whenever he erm srucure of ax-adjused forward raes is fla. These forward raes, while compuable, are fairly impracical for capial budgeing purposes. However, hey are consisen wih a single ax-adjused discoun rae for unlevered cash flows he axadjused hurdle rae for he IRR which generaes he same presen value. Moreover, when he unlevered asses are perpeual and he Gordon-Growh assumpions apply, his hurdle rae, denoed ρ, is easily obained wih a simple formula. Define dae s ax-adjused forward discoun rae for cash flows a dae s, f s, by f s ds = E [ dv s + X s ds ]. (11 E [Vs ] This is clearly an appropriae discoun rae. I is known a dae, and he recursive relaionship expressed in equaion (11, applied ieraively, implies V = e fsds E [X ] d. (12 How does his series of forward raes relae o he W ACC? A mos horizons, he comparison is meaningless because he fuure W ACCs are sochasic when viewed from dae. As a general maer, he raio of dae expecaions in equaion (11, used o compue he dae s forward rae, differs from he dae expecaion of W ACC s. Moreover, he expecaions in equaion (11, while obainable, do no lend hemselves o simple expressions. Despie his, developing an undersanding of ax-adjused forward raes is useful for undersanding when he W ACC can be used for discouning. Trivially, f s converges o W ACC as s approaches zero. Because f = W ACC, and f s is an appropriae insananeous discoun rae for dae s cash flows, i follows ha whenever f s = f for all s, W ACC is an appropriae discoun rae for unlevered cash flows. Proposiion 2 Whenever he erm srucure of ax-adjused forward raes is fla (f s = f, s, he W ACC is an appropriae ax-adjused discoun rae for unlevered cash flows. Two cases where his siuaion arises are he no-growh Modigliani-Miller model and he Miles-Ezzell model (wih or wihou cash flow growh. The only oher siuaion where he 18

21 forward erm srucure is fla is when deb policy is any weighed average of deb policies in hese wo models, bu only for g = g d. We explore his shorly. The rariy of an equivalence beween he iniial W ACC or expeced fuure W ACCs and he corresponding forward raes should no be surprising. Alhough equaions (1 and (11 look similar, Jensen s inequaliy alone prevens he expecaion of he former from equalling he laer when V, as well as he overall raio in equaion (1, are sochasic. Dae s consan ax-adjused discoun rae, ρ, is defined by he equaion V = e ρ E [X ]d. In cases where he unlevered cash flows have perpeual consan growh g, his ax-adjused discoun (hurdle rae is mos easily compued as ρ = g + X. (13 Equaion (13, an algebraic manipulaion of he growing perpeuiy formula, allows us o obain ρ from he formulas for V developed in he prior secion. 21 C. Classes of Cases wih Easy Numerical Soluions Recall from he las secion ha he values of asses wih ime-independen cash flows and deb policy, as well as asses wih addiively separable hisory diffusion and deb equaions could easily be obained numerically. In he former class of cases, he fundamenal valuaion equaion reduces o an ordinary differenial equaion in X. In he laer class, i reduces o a sysem of Riccai equaions. In hese cases, he W ACC and appropriae discoun rae, ρ are similarly solved. Jus as numerical soluions for V are easily obained, so oo are he W ACC using equaion (9. For ρ, he formula in equaion (13 generaes he discoun rae direcly from he numerically solved V. 22 Similarly, he forward raes are he soluions o ordinary differenial equaions which can be solved numerically. 23 Obviously, i is more illuminaing o analyze closed-form soluions for hese ineresing variables. We urn our aenion o his nex. 21 Wih a finie-lived asse, ρ is easily idenified implicily as he parameer ha solves = X ( ρ 1 e (ρ g(t. g 22 For finie-lived asses, his discoun rae can be easily solved implicily, as described in he prior foonoe. 23 The ODEs are available upon reques. 19

22 D. Addiively Separable Asses wih Closed-Form Soluions Consider, as in he las secion, he case where hisory follows he diffusion dh = (m x (X + m d D + m v V and deb policy has he funcional form m h H d + (l x (X + l d D + l v V l h H db, D = λ d x λ(x λ + d v V + d h H. Recall ha g, η, and σ are consan, d x λ ( = dx λ (egd λ, m x λ ( ηlx λ ( = kx λ ( = kx λ (egk λ, and T =. Here, he parial derivaives in equaion (9 have closed-form soluions, allowing us o express he W ACC as an explici funcion of he unlevered cash flows, hisory sae variable, and leverage raio, D /V, as follows: ( D W ACC = r f 1 τ c + 1/(r f + η g + λ cx λ (λxλ 1 + c h (σ H V V U + ηx λ cx λ (Xλ + c h, (14 (H where he c coefficiens are explicily given in he prior secion of he paper and wih σ H = λ l x λ(x + l d D + l v V (l h + σh. Case 1: Perpeual Asses Here, equaion (14 simplifies o he same expression, bu wih and c x λ( = kx λ (ch g x λ gk λ + (r fτ c + k d c h d x λ ( g x λ gd λ c h ( = c h + (k v + k d d v c h + r f τ c d v (r U g(r f (1 τ c d v + η g δ λ,1, and where he consans g x λ and ch are given in he previous secion of he paper. By conras, he consan ax-adjused discoun rae is ρ = g + X λ cx λ (Xλ + c h H. Case 2: Perpeual Deb as a Funcion of Cash Flows Only Consider, as in he las secion, he case where D = λ d x λ(x λ, 2

23 g, η, and σ are consan, d x λ ( = dx λ (egd λ, and T =. For his case, we showed ha V = V U + λ r f τ c r f + (η gλ 1 d x 2 σ2 λ(λ 1 g λ(x λ λ d. This valuaion soluion simplifies equaion (14 o where W ACC = r f (1 τ c D c x λ( = V while he consan ax-adjused discoun rae + 1/(r f + η g + λ cx λ (λxλ 1 V U + λ cx λ (Xλ r f τ c d x λ ( r f + (η gλ 1, 2 σ2 λ(λ 1 gλ d X ρ = g + U + r f τ c. λ d x r f +(η gλ 1 2 σ2 λ(λ 1 gλ d λ (Xλ This happens o be a case where he presenaion of he erm srucure of ax-adjused forward raes compued from he dae valuaion dae will no significanly lenghen he paper. To do his, we need o ake dae expecaions of Vs. Given he lognormal process for X s, he formula for he condiional λ h momen of X s is E [ X λ s ] = X λ e λ(g 1 2 σ2 s+ 1 2 λ2 σ 2 s which can be subsiued ino he expeced value of equaion (11. Thus, ηx, E [V s ] = V U e gs + λ τ c r f d x X λ e λ(g 1 2 σ2 s+ 1 2 λ2 σ 2 s λ r f + (η gλ 1 2 σ2 λ(λ 1. I follows ha de [V s ] ds = V U ge gs + λ τ c r f d x (λ(g 1 2 σ λ2 σ 2 X λ e λ(g 1 2 σ2 s+ 1 2 λ2 σ 2 s λ r f + (η gλ 1. 2 σ2 λ(λ 1 We now have all he ingrediens o compue he dae s ax-adjused forward rae. I is given by he formula: f s = U (r f + ηe gs + λ τ cr f d x (λ(g 1 2 σ λ2 σ 2 X λ 1 eλ(g 2 σ2 s+ 1 2 λ2 σ 2 s λ r f +(η gλ 1 2 σ2 λ(λ 1 V U e gs + λ τ cr f d x λ X λ 1 eλ(g 2 σ2 s+ 1 2 λ2 σ 2 s r f +(η gλ 1 2 σ2 λ(λ 1. I is easily verified ha W ACC = f, bu obviously, he dae s W ACC depends on X s and hus canno be idenical o f s. Moreover, he dae s forward rae is no he dae 21

24 expecaion of he dae s W ACC, irrespecive of wheher expecaions are aken wih respec o he acual probabiliy densiy funcion or he probabiliy densiy funcion generaed by he risk-neural measure. Case 3: Perpeual Deb ha is a inear Funcion of Asse Value Plus Consan Growh Recall ha for his special case, explored in he las secion, deb policy is given by D = d x (e gd + d v V and a perpeual levered asse has a paricularly simple funcional form for is valuaion: ( V = V U r f + τ c d x r f r f (1 τ c d v g ( + d r f (1 τ c d v + η g dv V U. This valuaion soluion simplifies equaion (14 o D W ACC = r f (1 τ c + V ηx (r f (1 τ c d v + η gv wih V given by he equaion immediaely above, while he appropriae ax-adjused consan discoun rae, ρ, is given by ρ = g + V U + τ c ( r f X r f (1 τ cd v g d d x ( +. r f r f (1 τ cd v +η g dv U Using equaion (11, he simple valuaion formula for his case can be used o show ha he ax-adjused forward discoun rae is given by he weighed average f s = where he coefficiens and γ (s γ 1 (s γ (s + γ 1 (s gd + γ (s + γ 1 (s (r f(1 τ c d v + η γ (s = γ 1 (s = r f τ c d x ( e gd r f (1 τ c d v g d s, X r f (1 τ c d v + η g egs, V = γ ( + γ 1 (. I is possible o compare forward raes, W ACCs, and appropriae consan ax-adjused discoun raes here bu he discussion is more illuminaing if we focus on several special cases of his example. 22,

25 Case 3a: Consanly Growing Deb wih No Sochasic Componen and No Expeced Cash Flow Growh In his exension of he Modigliani-Miller deb policy ha accouns for he possibiliy of growing deb, g = d v =, and r f > g d >. The γ (s and γ 1 (s coefficiens above simplify o implying γ (s = r fτ c D e gd r f g d s = e gd s, X γ 1 (s = r f + η = V U, f s = e gd s g d U U + e + gd s U + e gd s ru. The iniial W ACC, which is idenical o f, is W ACC = g d + V U r U ( = r U D η 1 τ c r f g d However, he ax-adjused consan discoun rae is ρ = X V = r U V U which he iniial W ACC exceeds by he amoun = r U ( 1 r f r f g d g d τ c D V D τ c, g d = r f g d D r f g τ d c. Case 3b: Modigliani-Miller Saic Deb Policy wih Consan Expeced Growh in Cash Flows Here we assume ha g >, bu ha d v = g d =. In his case γ (s = τ c D γ 1 (s = X r f + η g egs = U e gs, implying f s = V U e gs r U. 23

26 The iniial W ACC, which is idenical o f, is W ACC = V U V However, he ax-adjused consan discoun rae is ρ = X V = V U D which exceeds he iniial W ACC by gτ c. r U = r U ( 1 τ c D V + g = V U V r U + gτ c D V (r U g + g Case 3c: The Modigliani-Miller Model wih Equal Growh in Deb and Expeced Cash Flows The las wo cases illusraed ha he W ACC was no an appropriae ax-adjused consan discoun rae. This is because i changed over ime. In Case 3a, he iniial W ACC was oo large because i was expeced o decrease. In Case 3b, he iniial W ACC was oo small because i was expeced o increase. I is naural o hink ha if d v = bu g d = g >, so ha deb growh keeps pace wih he expeced growh of he cash flows, he iniial W ACC will sill work. This indeed is he case. To prove his, noe ha here W ACC = f = g d + V ( U r U = r U D η 1 τ c g d D r f g τ d c Thus, W ACC = ρ whenever which only occurs when g d = g. g d + V U r U = g + (ru g U This happens o be a case in which he forward rae is independen of s. By Proposiion 2, he iniial W ACC has o work as an appropriae ax-adjused discoun rae under his condiion. Case 3d: A Combinaion of he Exended Modigliani-Miller and Miles-Ezzell Deb Models wih Equal Growh in Deb and Expeced Cash Flow Here, we generalize Case 3c and once again show ha he iniial W ACC is an appropriae discoun rae for unlevered cash flows. In his example, g d = g bu d v may be nonzero. These assumpions imply: γ (s γ (s + γ 1 (s γ 1 (s γ (s + γ 1 (s = 1 X = X. 1 r f (1 τ c d v + η g 1 r f (1 τ c d v + η g. 24,

27 To compue f s, he former raio muliplies g, while he laer muliplies r f (1 τ c d v + η, and hen he wo producs are summed. Thus, f s = f = ρ = W ACC = g + X. When g d g, f s depends on s. Hence, wihin Case 3, i is only by imposing he condiion g d = g ha one can use he iniial W ACC as an appropriae discoun rae. This discoun rae can be hough of as a weighed average of he W ACC from a Miles and Ezzell deb policy a an iniial deb of D and he W ACC from a Modigliani and Miller deb policy a an iniial deb of D. Specifically, denoe he W ACCs of Modigliani-Miller and Miles-Ezzell as W ACC MM = r U D (1 τ c gη r f g τ c W ACC ME = r U D τ c r f, respecively. Then he discoun rae given above D W ACC = ρ = f = w W ACC MM + (1 w W ACC ME, where w = r f g (1 dv. r f (1 τ c d v g D / Case 3e: The Growh Exension of he Miles-Ezzell model Consider he case wih d v = D /V, d x ( =. This is a coninuous-ime version of he Miles-Ezzell model, bu wih he exension of expeced growh in cash flows. Here, and V V U = X r f + η g = V U r f + r f + η g τ cd. In his case, since γ (s is zero, he dae s forward rae is independen of s, which has he value f = ρ = r U r f τ c D V This ax-adjused discoun rae is idenical o he weighed average cos of capial. 25

28 To undersand his from anoher perspecive, recognize ha everyhing is saionary here, and ha he flow from he deb ax shield, being proporional o he unlevered cash flow, shares is discoun rae, so ha r = r f + η. The W ACC, from equaion (7, is herefore he consan W ACC = r f + η r f τ c D V where, by assumpion, D/V is he same a all poins in ime. Noe ha he growh rae of he cash flows never appears when he W ACC is wrien as a funcion of he leverage raio, bu ha i affecs V above. E. Discussion To develop an inuiive undersanding of he resuls above i is useful o firs review why discouning unlevered cash flows a he iniial afer-ax weighed average cos of capial someimes accouns for he value of he deb ax shield. e s begin wih he perpeual level deb model of Modigliani and Miller. Because he marke value of he deb financing never changes in his model, i is useful o hink of he Modigliani and Miller analysis as he valuaion of he deb ax shield of a zero bea deb sraegy. In such a model, he familiar equaion V = V U + τ c D reflecs he separae valuaions of he wo componens of he asse: he unlevered afer-ax cash flow sream, wih sochasic value X a dae, and he sream of cash flows from he ineres-based ax shield, Dr f τ c, which is consan and idenical a every dae. Noe ha he gross (as opposed o ne pre-ax weighed average cos of capial, which is also he gross expeced reurn on he levered asse, is 1 + r = 1 + (D/V r f + (E/V r E. This can be viewed as he expeced cash flow per dollar of asses o invesors who buy an asse, hold i for an insan, and hen liquidae i. Boh he asse value and he flow include he deb ax shield. Hence, he gross afer-ax W ACC 1 + (D/V r f (1 τ c + (E/V r E is jus he expeced flow o invesors per dollar of levered asses, less he expeced flow per dollar of asses from he deb ax shield, (D/V r f τ c. This ne flow is idenical o he unlevered cash flow per dollar of levered asses. Viewed wih he arrow of ime in reverse, his insigh implies ha if we discoun he unlevered cash flows a he afer-ax W ACC, we end up back where we sared, wih he one dollar value of he asses, including ha componen of value generaed by he deb ax shield. 26

29 I is easy o see ha his insigh abou he zero bea deb sraegy does no easily exend o he case of an asse wih a value ha ends o grow. Here, he value of he unlevered asse is affeced by he growh rae, bu he deb ax shield, having a perpeual value of τ c D is no affeced by i. Because he value of such a consan deb ax shield ends o decline over ime as a proporion of oal asse value, he iniial afer-ax weighed average cos of capial is an inappropriae discoun rae for all fuure cash flows. Essenially, he weighed average cos of capial is changing over ime. In his simple case, we learned ha here is a single discoun rae ha can be used o discoun all fuure unlevered cash flows: he sum of he iniial afer-ax W ACC and gτ c D/V, he laer being he produc of he growh rae of he unlevered cash flows, he ax rae, and he deb o value raio. For a firm wih a 4% growh rae, and a 5% ax rae and leverage, his alone represens a 1 basis poin error in he discoun rae, even given he rue unlevered cos of capial. We can see ha if he risk of he levered asse is no expeced o change, as in he Modigliani-Miller model of deb adjusmen, which assumes no growh, discouning he expeced unlevered cash flow sream a he iniial W ACC generaes he value of he levered asses, including he deb ax shield. The oher case where he risk of he levered asse is no expeced o change arises in an exension of he Miles and Ezzell (1985 model. Their case is one where he deb level is adjused over ime o mainain a consan deb o value raio. This is a posiive bea deb sraegy even hough he deb, a issue, is risk-free. The perpeually consan W ACC is an appropriae discoun rae in his case, bu he valuaion is never ou of peril if he acual deb policy associaed wih he asse differs from he policy implici in he formulas used for he valuaion. For example, if he unlevered asse s risk premium is wice he risk-free rae, he Miles-Ezzell ax shield is a mere one-hird ha of he Modigliani-Miller deb ax shield for he same deb level. If he risk-free rae is 4%, he bias in he W ACC from applying a Miles-Ezzell formula o Modigliani-Miller deb policy becomes 2 basis poins, even if he unlevered cos of capial for he asse is esimaed perfecly. These problems can become much more severe when he unlevered cos of capial is from comparison asses wih deb policies ha are also mismached o he formulas. For example, an asse wih a nonsochasic rend o grow is deb bu pay i down if he cash flows urn ou o be surprisingly good, has a very low W ACC oher hings equal. Unless his is recognized, unlevering such an asse wih a sandard formula ends o produce oo low an unlevered cos of capial. Similarly, obaining he W ACC of such an asse a a arge deb level using he sandard formulas ends o produce oo large a W ACC. By conras, consider an asse for which deb has a nonsochasic reiremen rend, bu for which deb ends o increase rapidly when cash flows are surprisingly good. Applying he sandard formulas o such an asse ends o generae unlevered coss of capial ha are oo high given a known W ACC and a W ACC a a arge deb level ha is oo low given he asse s unlevered cos of capial. If he former asse has a known W ACC and is used o generae he W ACC for he laer 27

30 asse, (or vice versa, he bias can easily run o 5 basis poins or more. 24 III. evering and Unlevering Equiy Beas The relaionship beween equiy beas and unlevered asse beas is criical for valuaion. Equiy beas, unlike asse beas, are observed. Hence, o obain he necessary discoun raes, valuaions analyze he raded equiy of asses ha are deemed similar o he asse being valued. The problem is ha he risk of he raded equiy of comparison asses is affeced by he leverage policy associaed wih he comparison asse. I is generally necessary o undo he leverage-induced disorion on he equiy bea of he comparison asse(s in order o obain he criical valuaion inpus: he unlevered asse bea or unlevered cos of capial. This secion explores how o do his. The reurn on an unlevered asse is given by dv U + X d = (r V U f + η ln V U ln X d + ln V U ln X σdb. while he reurn on a levered asse is given by ( ( ( dv + Xd + r f τ c Dd ln V = r V f + X + σ ln V ln V H ηx d+ H X + σ H ln V The raio of he erms ha muliply db in he wo equaions above represen he effec of leverage on volailiy. Wih risk-free deb, he proporional effec on bea is he same. Moreover, wih risk-free deb, he equiy bea of a levered asse always is 1 + D/E imes he bea of he underlying asse (including he asse componen from he ax shield. 25 I follows ha he formula for levering and unlevering equiy beas involves muliplying 1 + D/E by he raio of he erms ha muliply db above. Tha is, β E = wih he derivaives aken a dae. ( 1 + D E ln V X + σ H ln V H ln V U X The above equaion suggess ha whenever here is a closed-form soluion for he levered asse value, here is a closed-form soluion for he equiy bea formula, obainable afer firs aking parial derivaives. For example, he exension of he Modigliani-Miller deb policy wih g and g d has ln V H = 24 Moreover, his analysis assumes ha he WACC is an appropriae discoun rae, which i rarely is. 25 This is jus an inversion of he porfolio formula ha generaes he bea of he levered asse as a porfolio-weighed average of he bea of is deb (zero and equiy (β E. 28 β U H σxdb.

31 and which implies β E = ln V X = V U ln V U V X [ ( ] r f D τ c β U r f g d E. When g d is nonzero, his formula differs from he leveraging-unleveraging formula proposed by Hamada (1972 for he Modigliani and Miller deb policy. I also differs from his formula when g = g d = bu he asse has a finie life. In he laer case, ln V H = and ln V X = V U ln V U V X. However, because he finie horizon deb ax shield has a differen value han he perpeual deb ax shield, he formula reduces o [ β E = 1 + ( 1 τ c (1 e r f (T ] D β U E. By conras, he exension of he Miles-Ezzell policy wih g has ln V H = and ln V X = ln V U X = 1/X implying ( β E = 1 + D β U E. This is idenical o he formula proposed by Miles and Ezzell and i applies o boh an infiniely-lived and finie-lived asse (in conras o he Hamada formula for he Modigliani- Miller deb policy. For a perpeual asse ha is a hybrid of he Modigliani-Miller and Miles-Ezzell deb policies, as given in Case 3 of he prior secions of he paper, he equiy bea formula is [ ( ] d β E v r f D = 1 + (1 + τ c 1 β D /E r f (1 τ c d v g d U. E 29

32 The formula above is a weighed average of he bea leveraging formulas of Hamada/Modigliani- Miller and Miles-Ezzell. Tha is ( ( ( β E r f D = [w τ c + (1 w r f g d E 1 + D ] β U E wih he weigh on Hamada/Modigliani-Miller formula r f g d w = (1 dv r f (1 τ c d v g d D /V There are several addiional cases wih simple closed-form soluions for he equiy bea. These parallel cases 1 and 2 in he prior wo secions of he paper. Since hese involve mere subsiuions and elemenary calculus, we omi hem for he sake of breviy. Finally, we noe ha since equiy risk premia are proporional o equiy beas, he formulas for leveraging and unleveraging equiy risk premia are he same as hose above, wih he levered and unlevered equiy risk premia subsiuing for heir respecive beas above.. IV. Conclusion This paper has underaken a comprehensive valuaion of deb ax shields. In as many cases as possible, we offered closed-form soluions for he values of levered asses and he associaed deb ax shields. Our approach for obaining hese presen values was he APV approach. The ax-adjused discoun raes ha generae he presen values were reverse engineered, in ha we needed o use he presen values o generae closed-form soluions for he discoun raes. In his sense, we are in he APV camp, raher han he W ACC camp, and like many academics fail o undersand why he APV approach is no viewed as he simpler echnique. Obviously, however, boh approaches are equivalen and i is possible o generae correc valuaions eiher way once he deb policy and cash flow process are known. The examples explored in his paper are paricularly useful in ha mos dynamic deb policies can be hough of as fiing ino one of hese examples. The examples demonsraed ha i is possible o develop inuiion from polar cases so ha a manager can heurisically assess how his discoun rae and deb ax shield value will change, given he dynamic naure of he policy. For insance, firms wih deb policies ha sluggishly bu posiively reac o changes in he value of he firm s unlevered asses migh be expeced o have a deb ax shield wih a value ha lies somewhere beween he Modigliani-Miller and Miles-Ezzell values. The appropriae ax-adjused consan discoun rae for he fuure cash flows of he unlevered asses, as well as he formulas for levering and unlevering equiy beas will also lie beween he values given by he polar cases. Moreover, when firms engage in wha we erm negaive bea deb policies, paying down deb as he cash flow prospecs brighen, 3

33 and vice versa, he formulas for he deb ax shield, discoun rae, and levered equiy beas are again weighed averages of he wo polar cases, wih a negaive weigh on he formula associaed wih he Miles-Ezzell posiive-bea deb sraegy and a weigh above one on he Modigliani-Miller zero-bea deb sraegy formula. The comprehensive reamen of deb ax shields presened here is essenial for praciioners. Confusion has proliferaed because he formulas ha previously had been developed for he simples of cases are generally reaed as black boxes wihou a clear undersanding of where hey come from. I is rare when pedagogy appropriaely links he formulas for levering and unlevering beas o he value of he ax shield. The Miles-Ezzell valuaion can easily be 1/3 he valuaion using he Modigliani-Miller approach. I is quie common, however, o observe boh sudens and finance professors mix he Miles and Ezzell formula for leveraging and unleveraging equiy wih Modigliani-Miller inpus, hinking ha hey are geing a valuaion of τ c D for he deb ax shield. In properly linking he values of ax shields for differen deb policies o discoun raes and laying ou he heory behind his linkage, we hope o remedy some of his confusion. This paper is imporan, however, no jus for hose doing corporae valuaions, bu for hose doing research on capial srucure and bankrupcy coss. In paricular, i provides a comprehensive se of benchmarks for he impac of deb on asse values in a marke ha is fricionless, excep for axes. Empirical research by Graham (2 and by Kemsley and Nissim (21 indicaes ha for he average U.S. firm, deb ax shields in he U.S. are abou he size of τ c D. Clearly, more research ha describes he cross-secional variaion in his esimae is coming, and is warraned. To properly esimae bankrupcy coss from exensions of his research, i is criical o know he value of he deb ax shield in a marke where he only fricion is axes. We believe ha research in his area has been hindered by a lack of undersanding of benchmark valuaions. In plugging his hole in he lieraure, we hope o simulae addiional research. 31

34 References [1] Brealey, Richard and Sewar Meyers (2, Principles of Corporae Finance, McGraw-Hill/Irwin. [2] Graham, John (2, How Big Are he Tax Benefis of Deb? Journal of Finance, Vol. 55, No. 5, pp [3] Graham, John, and Campbell Harvey (2, The Theory and Pracice of Corporae Finance: Evidence from he Field, Journal of Financial Economics, Vol. 6, No. 2-3, pp [4] Green, Richard C., and Buron Hollifield (22, The Personal-Tax Advanages of Equiy, Journal of Financial Economics, forhcoming, Vol. 67, No. 2. [5] Grinbla, Mark and Sheridan Timan (22, 2nd ed.; 1997, 1s ed., Financial Markes & Corporae Sraegy, McGraw-Hill/Irwin. [6] Hamada, Rober (1972, The Effec of he Firm s Capial Srucure on he Sysemaic Risk of Common Socks, Journal of Finance, Vol. 27, No. 2, pp [7] Kemsley, Deen and Doron Nissim (22, Valuaion of he Deb-Tax Shield, forhcoming, Journal of Finance. [8] eland, Hayne (1994, Corporae Deb Value, Bond Covenans, and Opimal Capial Srucure, Journal of Finance, Vol. 49, No. 4, pp [9] Miles, James and Russell Ezzell (1985, Reformulaing Tax Shield Valuaion, Journal of Finance, Vol. 4, No. 5, pp [1] Modigliani, Franco and Meron Miller (1958, The Cos of Capial, Corporaion Finance and he Theory of Invesmen, The American Economic Review, Vol. 48, No. 3, pp [11] Ross, Sephen (1978, A Simple Approach o he Valuaion of Risky Sreams, Journal of Business, Vol. 51, No. 3, pp [12] Ruback, Richard (22, Capial Cash Flows: A Simple Approach o Valuing Risky Cash Flows, Financial Managemen, Vol. 31, No. 2, pp