Horses and Rabbits? Optimal Dynamic Capital Structure from Shareholder and Manager Perspectives

Transcription

1 Horses and Rabbits? Optimal Dynamic Capital trctre from hareholder and Manager Perspectives Nengji J University of Maryland Robert Parrino University of exas at Astin Allen M. Poteshman University of Illinois Michael. Weisbach University of Illinois and National Brea of Economic Research March 15, 22 J is in the Department of Finance, mith chool of Bsiness, University of Maryland, College Park, MD 2742 (Phone: (31) , nj@rhsmith.md.ed). Parrino is in the Department of Finance, McCombs chool of Bsiness, University of exas at Astin, Astin, X (Phone: (512) , Parrino@mail.texas.ed). Poteshman and Weisbach are in the Department of Finance, College of Commerce and Bsiness Administration, University of Illinois, Champaign, IL 6182 (Phone: (217) , Poteshma@ic.ed and Phone: (217) , Weisbach@ic.ed).

2 Horses and Rabbits? Optimal Dynamic Capital trctre from hareholder and Manager Perspectives Abstract his paper examines optimal capital strctre choice sing a dynamic capital strctre model that is calibrated to reflect actal firm characteristics. We examine capital strctre choices that maximize either the tility of managers or the vale of a share. he model ses contingent-claims methods to vale interest tax shields, allows for bankrptcy if firm vale reaches a pre-specified bondary, and explicitly models the firm maintaining a long-rn target debt/eqity ratio by refinancing matring debt at that ratio. ince the model s major forces favoring debt or eqity are tax shields and bankrptcy costs, this model provides a way to calclate optimal capital strctres in a realistic representation of the traditional tradeoff model. he model s predicted optimal capital strctres are mch more consistent with actal capital strctres than have been presmed by the literatre dating at least to Miller (1977). One important reason for this difference is that the vale of a firm s existing tax shields declines with incremental leverage as bankrptcy costs increase, a relation not commonly recognized by the existing literatre.

3 Horses and Rabbits? Optimal Dynamic Capital trctre from hareholder and Manager Perspectives 1. Introdction. It is not mch of an overstatement to say that since Modigliani and Miller s (1963) tax correction paper, the central isse in corporate finance research has been the qestion of why, despite the large tax advantage enjoyed by debt, actal firms have fairly low leverage ratios. his qestion has stimlated mch of the early research on agency theory (Jensen/Meckling, 1976; Myers, 1977), some of the bestknown work on information asymmetries (Myers and Majlf, 1984), three American Finance Association presidential addresses (Miller, 1977; Myers, 1984; and Leland, 1998), and contines to motivate wellregarded research (Graham, 2). he consenss view nderlying this vast literatre is that bankrptcy costs alone are too small to offset the vale of tax shields, and that other factors, sch as agency costs, mst be introdced into the cost-benefit analysis to explain actal capital strctres. Miller (1977, p. 264) perhaps pts it best: the spposed trade-off between tax gains and bankrptcy costs looks like the recipe for the fabled horse-and-rabbit stew one horse and one rabbit. he nderlying logic of this widespread view is that while tax shields are relatively large (abot 9.7% of firm vale according to Graham (2)), bankrptcy costs are incrred infreqently and are relatively small when they are incrred. Yet, the tradeoff theory does not contain any predictions abot the level of tax shields and bankrptcy costs; rather, it states that at the margin, adding an additional dollar of debt will not change firm vale. An important consideration, largely neglected by the literatre, is that tax shields themselves are options, and that adding additional debt decreases the vale of existing tax shields. 1 Incremental debt increases the probability of bankrptcy, and tax shields lose vale if firms go bankrpt. 2 In other words, in the tradeoff framework, a firm considering adding leverage shold trade off 1 Exceptions are Majd and Myers (1987), J (1998), and Parrino, Poteshman, and Weisbach (22). 2 he vale of tax shields in bankrptcy is severely limited by the ax Reform Act of 1986, which restricts the ability of firms to sell net operating losses. ee Ginsbrg and Levin (21), Chapter 12, for details on how these restrictions work.

4 the vale coming from the tax shields created by the new leverage against the increase in expected bankrptcy costs, pls the decrease in the vale of existing tax shields. his paper estimates optimal capital strctre sing a dynamic model of capital strctre. We view the model as a representation of the traditional tradeoff model becase the primary determinants of the optimal capital strctre are tax shields and bankrptcy costs. However, the model incorporates a nmber of elements not typically seen in the capital strctre literatre. First, tax shields are valed sing contingent-claims methods. econd, as in Black and Cox (1976), the firm defalts if the vale of its assets reaches a pre-specified bankrptcy bondary. hird, the firm refinances its debt when it matres at a pre-specified target debt/eqity ratio. he vale of a share today reflects both tax shields and expected bankrptcy costs, both directly and throgh their impact on sbseqent financing costs. In the model, the manager of an nlevered firm ndertakes a fairly priced debt/eqity swap, in which he selects the fraction of eqity to be exchanged for debt, with the objective of maximizing either the per-share vale of the firm s eqity or his own tility. he swap that maximizes the per-share vale of eqity is, from the shareholders perspective, the optimal capital strctre. he model is calibrated to be representative of the typical pblic firm in the U.. and U.. capital markets. he vale of the firm s assets is assmed to follow geometric Brownian motion with a drift of 5% and a standard deviation of 32%. imilarly, the stock and option holdings of the firm s senior manager are selected to reflect the typical levels for senior U.. exectives. Finally, the firm is assmed to isse 1-year, copon-bearing debt, which is priced at par. he bankrptcy bondary and costs incrred in bankrptcy are selected so that the spread between the copon rate and the reasry rate and recovery rates in bankrptcy are consistent with vales observed in practice. We find that the optimal debt-eqity swap leads to a market debt/total capital ratio of approximately 2%. In comparison, the median firm in the Compstat database in 1999 had a debt/total capital ratio of 21.9%. hese two ratios are far closer to one another than are implied by the previos literatre. 2

5 We next introdce agency costs into this framework by calclating the vale of the swap that maximizes the tility of a potential manager. We assme the manager has a constant relative risk aversion tility fnction with a risk-aversion parameter of 2, owns.32% of the company s stock, has atthe-money options on.38% of the company s stock, and has non-firm wealth eqal to the vale of his shares. When the swap is chosen to maximize this tility fnction, the optimal leverage drops to approximately 15% of firm vale. We perform nmerical comparative statics to evalate the impact of sensitivity analysis on the major parameters on optimal capital strctre. Not srprisingly, corporate tax rates, bankrptcy costs, and the ability of debtholders to force the firm into bankrptcy all impact optimal capital strctre ratios. We also calibrate the model to reflect 15 actal firms. For these 15 firms, the predicted stockprice maximizing leverage ratio is less than the firm s actal leverage ratio for 13 of them and the predicted tility-maximizing leverage is less than actal leverage for 14 of them. In general, the model is able to predict, within a reasonable degree of error, the leverage observed at firms that have relatively small to typical levels of debt, bt sbstantially nderestimates the level of debt observed at highly levered firms. Overall, the reslts in this paper sggest that the tradeoff model performs reasonably well in predicting capital strctres for firms with typical levels of debt. Certainly, the horse and rabbit stew analogy seems inappropriate actal capital strctres appear to be of the same order of magnitde as those predicted by this model. If anything, or reslts sggest that the tradeoff model predicts leverage levels that are lower than those observed empirically, especially when it is adjsted for managerial preferences. As sch, these reslts highlight the importance of theories sch as Jensen (1986), in which debt provides advantages from an incentive perspective. Or model implies that important factors determining capital strctre are the nderlying risk of the firm s assets, the ability of bondholders to force defalt for a given level of firm vale, as well as the incremental costs conditional on defalt. Or ability to measre these variables is qite limited sing 3

6 crrent econometric methods; a better nderstanding of their relative importance can advance or nderstanding of capital strctre choices, and potentially improve the financing choices of actal firms. Or model also sggests several policy isses that have not been emphasized by the literatre. First, we assme that firms cannot sell their tax shields when they enter bankrptcy. his assmption seems plasible given the restrictions on doing so imposed by the ax Reform Act of However, or analysis sggests that easing this restriction cold have important implications for capital strctre and make debt mch more attractive. econd, it sggests that the ability of bondholders to force defalt throgh covenants has an important effect on both the vale of a firm s tax shields, and, conseqently, its optimal leverage ratio. Reforms that increase the rights of bondholders to force defalt may have a meaningfl impact on the ex ante financing decisions that firms make. he rest of this paper is organized as follows: ection 2 describes the model in detail. ection 3 explains how we calibrate the model to reflect crrent market data. ection 4 discsses the implications of the calibrated model, while ection 5 contains a short conclsion. 2. A Dynamic Model of Capital trctre he models that we se are based on J (1998). In these models, the firm isses debt with a matrity of, which pays a continos, constant (tax-dedctible) copon. he manager s wealth at time zero is divided between non-firm wealth and his stake in the firm, which consists of eqity shares and standard Eropean call options, which expire at time. he manager cannot sell or hedge his shares or options. For simplicity, it is assmed that the manager s non-firm wealth grows at the risk-free rate, r, and is therefore ncorrelated with the vale of the manager s stake in the firm. he manager s tility is given by a CRRA tility fnction defined over his entire wealth. he vale process of the firm s assets (e.g., the vale of the cash flows from operations) follows geometric Brownian motion. he model is in continos time with < <. At time zero the vale of the firm s assets is V (. ) Before the swap, the firm s capital consists of N N shares of stock with a total market vale of 4

7 EN by: ( ) = V ( ). 3 he vale of the firm s assets, V (), t follows geometric Brownian motion described () () dv t V t ( µ δ) dt σdz ( t) = + (1) where µ and σ > are constants and dz () t is a standard Weiner process. he firm liqidates assets at a rate of δ of the total vale of the firm s assets, so that δ V () t dt is eqal to a time varying dividend div() t dt paid to eqity holders over the time interval dt : () div() t dt. he vale of δ is specified exogenosly as a model parameter. δ V t dt = (2) We will consider a fair eqity for debt swap at time zero that either (1) maximizes the vale of a share of eqity or (2) maximizes the manager s expected tility at time. he swap is fair in the sense that the debt which is issed is done so at its correct market vale. he debt has a face vale of has a market vale when it is issed at time zero of ( ) D. he debt pays a copon at a constant F and annalized rate C which is set so that the bond is priced at par, that is, so that F D ( ) =. he firm dedcts its copon payments from its taxes at an effective rate τ, and the tax benefit of the debt at time zero has a vale ( ) B. he bond has a protective covenant which specifies that if the firm vale at anytime dring the life of the bond [, ] decreases to an exponential bondary, the firm is forced into bankrptcy. 4 When this occrs, the stock becomes worthless and the debtholders recover 1 α BC of the vale of the assets. he fraction of the vale of the assets not recovered by the debtholders is assmed to 3 he sbscript N refers to qantities before the swap and the sbscript refers to qantities after the swap is performed. 4 Following Black and Cox (1976), we are implicitly assming that this covenant acts somewhat like the actal covenants seen in bond indentres. he idea is that actal bond covenants are set p for the prpose of allowing bondholders to seize assets when they are in danger of being lost this assmption models this process explicitly. 5

8 be consmed in the bankrptcy process. he bankrptcy bondary is an exponential crve that increases at a rate g and is eqal to the face vale of debt at time. Conseqently, the bankrptcy bondary is described by gt ( Fe ). he bankrptcy costs for the firm are the present vale of the expected losses in bankrptcy, and are denoted by ( ) BC. After the swap the firm still liqidates assets at a rate of δ of the total vale of the firm s assets, so that δ V() t dt eqals the sm of the after-tax copon paid to bond holders [( τ ) Cdt] and a time varying dividend () interval dt : 1 () = () + ( τ) div t dt paid to eqity holders over the time δv t dt div t 1 C dt. (3) In order to garantee that the dividend rate is non-negative, we reqire that () ( τ) δv t 1 C. (4) We assme that the swap is flly transparent so that the post-swap vales of the debt and eqity exchanged are eqal in magnitde and opposite in sign. hat is, D N N =. N N ( ) E ( ) At time zero there is an infinite nmber of fair eqity for debt swaps available to the firm. We will analyze two of these. he first type of swap that we consider maximizes the vale of a share of eqity. hat is, it maximizes the qantity, ( ) E N. he second type of swap that we consider maximizes the manager s expected tility at time. hat is, it maximizes the expected vale of the manager s CRRA tility fnction (which is defined over his total wealth) at time. At time zero the manager s stake in the firm consists of N ( N ) Man < N shares and Calls Eropean call options with strike price K that expire at time. For prposes of comptational tractability, we assme that the firm bys the manager s calls from a third party. Hence, if the manager N 6

9 exercises the calls at time, he bys N Calls shares from the third party at a price of NCallsK dollars. We assme that the manager cannot sell or hedge either his shares or his options. In addition, at time zero the manager has NFW ( ) dollars of non-firm wealth. For simplicity, this wealth is assmed to grow at the risk-free rate. When the swap is performed in order to maximize the manager s expected tility at time, this tility is described by ( ) U Wealth = ( ) 1 γ Wealth 1 1 γ (5) where γ is a risk-aversion parameter and Wealth is the manager s total wealth at time. he vale of the debt, the bankrptcy costs, and the tax benefit of debt are compted from the probability density fnction for first hitting the exponential bankrptcy bondary. Let ( * ; ( ),,,,, ) f t V time * A g r δσ be the probability density for first hitting a bondary described by t, where A is a constant, if the variable V initially has a vale V ( ) Brownian motion with drift r gt Ae at a > A and follows geometric δ and volatility σ. In or model, A is the vale of the bankrptcy g bondary at time zero, so that A is eqal to Fe. An explicit expression for ( * ; ( ),,,,, ) f t V A g r δσ is provided in the Appendix. Next define: ( ( ) δσ) ( ) * * (6) ( δσ) G, V, A, g, r,, f t ; V, A, g, r,, dt and rt (, ( ),,,, δσ, ) ( ; ( ),,,, δσ, ) H V A g r e f t V A g r dt (7) ( r ) ( ( ) δσ) * g t * *,,,,,, ;,,,,,. ( ( ) δσ) I V A g r e f t V A g r dt (8) Closed form soltions for these expressions are derived in the Appendix. 7

10 Following Leland and oft (1996), the vale of the debt at time zero is the sm of a contribtion from the copon, a contribtion from the payment to debtholders if bankrptcy occrs, and the repayment of the face vale at time if bankrptcy does not occr: or g ( ( δσ) ) * rt * ( ) = 1, ( ),,,,, * D C e G t V F e g r dt * * rt g ( t) * g * + e ( 1 αbc ) F e f ( t, V( ), F e, g, r, δ, σ) dt (9) g ( 1 (, ( ),,,, δσ, )) + F G V F e g r e g r g ( ( ( δσ) ) ( ( ) δσ) ) g g ( 1 αbc ) Fe I( V, ( ), Fe, gr,, δ, σ) g r F 1 G(, V( ), Fe, g, r, δσ, ) e C D( ) = 1 1 G V, ( ), Fe, gr,,, e H V,, Fe, g, r,, r + + ( ) r (1) Another modeling decision involves the qestion of whether the firm shold refinance the debt obtained in the swap when it matres. We consider two alternative models: he first is a static model, in which the firm does not refinance debt, and becomes an all-eqity firm sbseqent to the time the debt matres. he second is a dynamic model, in which new debt is reissed at the time of matrity. ince the dynamic framework seems a priori more appealing, and in fact J (1998) shows that the refinancing assmption can affect corporate financing decisions ex ante, we analyze the dynamic model. Nonetheless, it is convenient to present the soltion of the dynamic model in terms of that for the static model that we develop now. In the static model, when the firm is forced into bankrptcy at time t *, the bankrptcy costs are * α BCV( t ). Hence, at time zero the vale of the bankrptcy costs are or * g( t ) * rt * g * BC = α F e e f t ; V, F e, g, r, δ, σ dt (11) ( ) BC ( ( ) ) 8

11 g ( ) α ( ) g ( δ σ) BC = F e I, V, F e, g, r,, (12) BC he tax benefits of debt accre to the firm as long as it has not gone bankrpt. Conseqently, the tax benefits of debt in the static model can be compted by or g ( ( δ σ) ) * rt * ( ) = τ 1, ( ),,,,, * B C e G t V F e g r dt (13) g r g ( ( ( δσ) ) ( ( ) δσ) ) ( τc B ) 1 1, ( ),,,,,,,,,,, = G V Fe g r e H V Fe g r (14) r he vale of the eqity is eqal to the vale of the assets pls the tax benefits of debt mins the bankrptcy costs mins the vale of the debt: ( ) ( ) ( ) ( ) ( ) E = V + B BC D (15) In order to compte the manager s time zero expectation of his tility at time, K let ( ) the vale of the firm s assets at time that makes a share of stock worth K at time. hen the manager s time zero expectation of his tility at time is the sm of three components. he first V be component is a fnction of the probability density for the vale of the firm s assets being at varios levels V at time withot having toched the bankrptcy bondary between time zero and time K above ( ). he second component is a fnction of the probability density for the vale of the firm s assets being K at varios levels below ( ) V at time withot having toched the bankrptcy bondary between time zero and time. he third component is the tility derived from his non-firm wealth if the bankrptcy bondary is hit. Let ( ) ( ) a vale V ( ) A the interval t [, ] > and being at ( ) g (,,,,,,, ) gv V Agµδσ be the density fnction for starting at V > Ae at time > withot ever hitting the bondary when the V process follows geometric Brownian motion with drift µ δ gt Ae in and 9

12 volatility. σ An explicit expression for ( ) ( ) (,,,,,,, ) gv V Agµδσ is presented in the Appendix. hen at time zero, the manager s expectation of his tility at time after the swap is given by Utility U NFW Man Calls ( ) = ( ) + ( ) ( ) ( ) ( ) Calls K V ( ) K where ( ) V K ( ) N + N V B BC D N K N + g V V F e g dv g ( (, ) ( ),,,, µδσ,, ) ( ) N + U NFW + g F ( ) N e g V V F e g dv Man ( ) V( ) + B ( ) BC ( ) D ( ) g ( (, ) ( ),,,, µδσ,, ) ( ) ( ( ) ) ( ) V satisfies the following eqation: g ( ;,,, µδσ,, ) + U NFW f t V F e g dt (16) K ( ) + ( ) ( ) ( ). K V B BC D = (17) N Note that all terms on the right hand side of eqation (17) are a fnction of V K ( ). Next we extend the model to a more realistic dynamic setting. As in the static case, after the swap at time zero the firm has a bond otstanding with years to matrity. Now, however, if the firm has not gone bankrpt at the end of years, the firm isses a new year bond at time. he new bond has a copon of ( ) ( ) CV V. imilarly, as shown in the Appendix, all other secrities will be scaled by a factor of V( ) V (, ) becase at time the firm is identical to itself at time zero except that it is V( ) V ( ) as large. he process of issing a new -year bond each time that a bond expires contines indefinitely ntil the firm goes bankrpt. In this dynamic setting, the price of the debt is still given by eqation (1). he firm vale, however, will reflect the costs and benefits of the debt issed in the ftre ntil the firm goes bankrpt. In 1

13 order to determine the total tax benefit and total bankrptcy cost of the crrent and potential ftre isses of debt, the following qantity will be sefl: ( ) ( ) r Q V φ e E V 1 { Firm does not go bankrpt over [,] } (18) he indicator fnction 1 { Firm does not go bankrpt over [,] } is eqal to one if the firm does not go bankrpt over the interval [, ] and zero otherwise. he expectation is taken over the risk-netral Q measre. In the Appendix, we show that φ is given by the following expression: where and ( + ( r δ g σ 2 g ) σ ) Fe δ φ = e N( d ) N d V ( ) ( ) 1 2 g 2 ( ( )) ( δ σ ) (19) log Fe V + r g+ 2 1 d = (2) σ g 2 ( ( )) + ( δ + σ ) log Fe V r g 2 2 d =. (21) σ We also show in the Appendix that the total tax benefit of debt and the total bankrptcy costs are given by and B Dynamic ( ) ( ) B = 1 φ (22) BC Dynamic ( ) = BC ( ) 1 φ (23) imilar to eqation (15), the vale of the eqity is eqal to the vale of the assets pls the tax benefits of debt mins the bankrptcy costs mins the vale of the debt: ( ) ( ) ( ) ( ) ( ) Dynamic Dynamic Dynamic E = V + B BC D. (24) 11

14 Finally, the manager s tility after the swap in the dynamic model is given by Dynamic NMan + NCalls Dynamic Dynamic ( ) = ( ) + ( ) ( ) ( ) ( ) Calls K N + V ( ) Utility U NFW V B BC D N K K V ( ) g ( F ) e g ( (, ) ( ),,,, µδσ,, ) ( ) g V V F e g dv N + U NFW + N Man Dynamic Dynamic ( ) V( ) + B ( ) BC ( ) D ( ) g ( (, ) ( ),,,, µδσ,, ) ( ) g V V F e g dv ( ( ) ) ( ) g ( µδσ) + U NFW f t; V, F e, g,,, dt. 3. Calibrating the Model In choosing the amont of debt that will be swapped for otstanding eqity, the manager is assmed to select the face vale of 1-year debt (i.e., = 1 years) sch that he maximizes his expected tility one year in the ftre (i.e., = 1). he total vale of the firm s assets before the swap, V( ), is normalized to $1. ince the firm has no leverage before the swap, the vale of the firm s assets eqals the vale of the eqity, which is divided among a total of 1 shares. We assme that the manager of the firm owns a.32 share of stock and a 1-year exchange traded Eropean call option on an additional.38 shares. 5 he strike price for the call option is set eqal to the time zero vale of a share of eqity of the firm before the swap, $1. For the base-case, the manager s non-firm wealth is assmed to eqal the timezero vale of the shares that the manager owns withot the project, $.32. Consistent with the literatre, we assme the manager s risk aversion parameter γ eqals 2 (see pp of Ljngqvist and argent (2) for a discssion of the interpretation of this vale and other vales of γ sed in the sensitivity analysis). 5 he manager s stock and option holdings represent the median vales for managers at 1,45 firms for which sfficient data to estimate these figres are available for 1999 on the ExecComp database. 12

15 Given these assmptions, calibration of the model reqires estimates of (1) the risk-free rate, r, (2) the effective tax rate, τ, (3) the drift parameter for the total vale of the firm, µ, (4) the volatility of the total vale of the firm, σ, (5) the level of dividends, DivRate, paid by the firm, (6) the debtholder 1 BC bankrptcy recovery rate, ( α ) estimate these parameters sing data from the end of Janary 21., and (7) the bankrptcy bondary s exponential growth rate, g. We As or estimate of the risk-free rate, we se the rate on 1-year reasry bonds as of Janary 3, 21 as reported in the Febrary 7, 21 edition of tandard & Poor s he Otlook. his rate eqals 5.22 percent. We estimate the tax rate sed to calclate the tax shields from the debt sing data on estimated marginal tax rates (before interest expense) provided by John Graham, who constrcted these estimates sing the approach described in Graham (1996). In particlar, for the base case, we assme that the tax rate eqals the median marginal tax rate of 34 percent for the 5,519 firms for which 1999 estimates are available. We set the drift parameter of the firm, µ, eqal to 5 percent. his vale is consistent with an expected long-term inflation rate of 2.5 percent and 2.5 percent real growth. he 2.5 percent long-term inflation rate is consistent with five-year estimates pblished by WEFA (formerly Wharton Econometric Forecasting Associates) for the Consmer Price Index in its U Otlook report for December 2. o estimate the volatility of the total vale of the firm s assets, σ, we examine the sample of 1,43 firms for which the necessary data are available on Compstat for the entire 198 to 1999 period. he median vale of the annal standard deviation of the percentage change in firm vale for the 1,43 firms,.2852, provides a lower bond for or estimate of σ. 6 his vale is a lower bond becase there is a srvivorship bias in the sample. We se.32 as or estimate of the vale of σ for the niverse of 6 his estimate is only an approximation, as it does not incorporate bankrptcy costs, which are not observable. It is relatively insensitive to the sample and period. Estimates of σ range from.2513 to.3333 for different time periods (ten and 2 years) and samples (firms for which all data are available for the fll 2 year period and for which data are only available for ten years). 13

16 firms. Becase we are interested in examining how the volatility of the firm s assets affects the financing decision, we examine the impact of σ vales ranging from.15 to.5 on the swap decision in the sensitivity analyses below. We set the dividend rate, DivRate, eqal to 1.5 percent in the base case. Becase this rate is stated as a percentage of the nlevered vale of the firm, we se a nmber that is on the lower end of the 1.5 to 2. percent dividend yield paid by pblic firms at the beginning of 21. he debtholder bankrptcy recovery rate and the exponential growth rate for the bankrptcy bondary are selected to yield an expected recovery rate of 45 percent and a spread over the 1-year reasry bond rate for the firm s debt eqal to 1.9 percent when the firm has a debt to total capital ratio of 21.9 percent (the median vale for all firms in the Compstat database in 1999). he 45 percent recovery rate is broadly consistent with recovery rates pblished by Hamilton, Gpton, and Berhalt (21). For the 1981 to 2 period, Hamilton, Gpton, and Berhalt estimate the mean defalt recovery rates for senior secred bonds, senior nsecred bonds, and sbordinated bonds of all ratings to eqal 53.9 percent, 47.4 percent, and 32.3 percent, respectively. he 1.9 percent spread over the reasry bond rate eqals the spread for 1-year A-rated corporate debt as of Janary 3, 21, as reported in the Febrary 7, 21 edition of tandard & Poor s he Otlook. he bankrptcy recovery and bankrptcy bondary growth rates for or base case eqal.5194 (a BC =.486) and 5.19 percent, respectively. Panel A of able I smmarizes or parameter choices. hese choices are sed to derive the set of parameters that are presented in Panel B of able I. 4. Optimal Capital trctre in this Model In this model, each potential capital strctre leads to a different vale of tax shields and bankrptcy costs, and ltimately different price distribtions for the firm s secrities. o determine the optimal capital strctre, we begin with an all eqity firm, and consider the range of potential debt/eqity swaps. Potential swaps are assmed to be fair in the sense that the new debt is issed at its 14

17 fair market vale. We also assme that the swap is flly transparent so that the post-swap vales of the debt and eqity that are exchanged in the swap are eqal in magnitde and opposite in sign. hat is, D N N =. N N ( ) E ( ) he optimal capital strctre from a particlar viewpoint is defined as the swap that maximizes the objective fnction in qestion. o, from the shareholders perspective, the optimal capital strctre is defined by the swap that maximizes the post-swap vale of each share of eqity, and from the managers viewpoint, the optimal capital strctre maximizes the manager s expected (post-swap) tility at time Optimal Capital trctres for a Representative Firm he hareholders Perspective able II presents calclations of shareholders optimal capital strctre, assming the model is calibrated as discssed above. Each colmn represents a different level of asset volatility; all other parameters are the base-case ones discssed above. he optimal capital strctre is shown in Row 1. It is clearly very sensitive to asset volatility, eqaling 42.2% when asset volatility is 12% and 9.9% when asset volatility is 52%. his relation is consistent with casal empiricism, as well as with more formal stdies sggesting that riskier firms do in fact se less leverage (see for example itman and Wessels, 1988; or Rajan and Zingales, 1995). Yet, to evalate the model s qantitative prediction reqires a precise estimate of asset volatility. As stated above, we se.32 as or point estimate for the volatility of the firm s asset vale. Or estimates of volatility are likely to be somewhat crde, given that they rely on book vales of debt, and are net of the nobservable level of bankrptcy costs, which presmably vary over time. Yet, despite their crdeness, the model does srprisingly well at predicting capital strctres. For the median firm, the model predicts a debt/total capital ratio of 19.7%. Using the same sample of 1,43 Compstat firms, the actal debt/total capital ratio is 21.9%. his comparison sggests that asset risk is an important factor in 15

18 financing decisions. Improved estimates of asset risk are likely to lead to better capital strctre decisions in practice. hese findings are conter to the conventional wisdom that the tradeoff approach to capital strctre implies sbstantially more leverage than is observed in the data. o nderstand why or model leads to different findings from the sal intition, one shold examine Rows 1 and 11 of able II, showing the vales of bankrptcy costs and tax shields predicted by or model. For each level of volatility, the vale of tax shields is sbstantially higher than the vale of bankrptcy costs. hese differences exist despite the fact that the leverage levels shown in able II maximize share vale. he reason for these differences is that adding leverage beyond that shown in able II wold decrease the vale of tax shields sfficiently to lower share vales. he tre marginal cost of additional leverage incldes not jst bankrptcy costs, bt a decline in the vale of existing tax shields. his potential decline in the vale of tax shields is one reason the predicted vales from or model are lower than those prodced by the common intition abot the tradeoff theories of capital strctre (Miller, 1977) he Manager s Perspective Mch of the capital strctre literatre has concerned the implications of agency problems for financing decisions. We next evalate or model in an agency framework by replacing the assmption that capital strctre is chosen to maximize the per share vale of eqity with the assmption that it is chosen to maximize the manager s tility fnction. 7 We first assme that the manager maximizes a constant relative risk aversion (CRRA) tility fnction with a risk aversion parameter of 2 and has 5 percent of his non-option wealth invested in 7 Mch of the attention on agency problems and capital strctre has historically focsed on conflicts between stockholder and bondholders [Jensen and Meckling, 1976 and Myers, 1977]. While researchers have spent mch time on the impact of these conflicts elsewhere (Mello and Parsons (1992), Leland (1998), Parrino and Weisbach, 1999; Parrino, Poteshman, and Weisbach, 22), we ignore these conflicts here. o incorporate stockholder/bondholder conflicts wold reqire specifying a distribtion of potential projects, integrating over this distribtion given a particlar manager s preferences, and then seeing how the vale of the firm s secrities varies with the investment opportnity set. ch an exercise is beyond the scope of this paper, althogh or opinion is that it wold be an excellent direction for ftre research. 16

19 shares of the firm, with the remainder invested in risk-free assets. As noted in ection 3, we assme that this stake in the firm eqals.32% of the firm s eqity and at the money call options to prchase.38% more of the firm s eqity. We calclate the optimal capital strctre from the manager s perspective by choosing the swap that maximizes the vale of this tility fnction rather than the vale of a share of common stock. he reslts from this managerial model are presented in able III. ince the manager is assmed to be risk-averse and the risk of the firm s eqity increases with leverage, it is not srprising that the manager prefers less leverage than the shareholders. Comparing ables II and III, for each level of risk, the optimal leverage from manager s perspective appears to be abot 4 percentage points lower than from the shareholders perspective ensitivity of Optimal Capital trctre to Model Parameters ax Rates ince the major factor leading to a preference for debt is its tax-dedctibility, we expect the model s reslts to be especially sensitive to tax rates. We compte optimal capital strctres (from the shareholder s perspective) as a fnction of corporate tax rates in able IV. able IV indicates, nsrprisingly, that optimal leverage ratios are negatively related to the firm s tax rate. However, this relation appears to be nonlinear and is not as strong as one might expect. With a corporate tax rate of jst one percent, the optimal leverage ratio eqals 4.5%. his ratio rises 7.3 percentage points to 11.8% when the tax rate rises to 12%. In contrast, at higher tax rates the same 11 percentage point increase in tax rates (from 23% to 34%) leads to only a 3.6 percentage point increase in leverage, from 16.1% to 19.7%. Only when tax rates are twice as high as the crrent median rate estimated by Graham (1996) does the estimated debt to total capital ratio exceed 3%. 17

20 Bankrptcy Bondary An important element of or model is that the firm is assmed to defalt if it hits a pre-specified bankrptcy bondary. he idea nderlying this assmption is that most pblicly traded debt contains covenants enabling debtholders to force defalt when the vale of the firm is sfficiently low. In or model, the parameter g represents the steepness of this bondary, so that a lower g increases the likelihood that the firm defalts given poor performance. Intitively, g can be thoght of as a negative fnction of the strength of the bond s covenants. It is not clear conceptally how we expect this variable to be related to the shareholders optimal leverage: tronger bondholder rights make debt more attractive allowing debt to be issed at lower interest rates. Whether these lower interest rates are sfficient to compensate shareholders for the increased bankrptcy probabilities is not obvios. able V presents estimates of the optimal capital strctre as a fnction of g. he reslts in this able indicate that optimal leverage is a positive fnction of g. As the rights of debtholders to force defalt increase, firms find it optimal to se less leverage. hs, it appears that the direct effect of a lower g throgh increased bankrptcy probabilities is more than sfficient to offset the indirect effect of lower interest rates Costs Conditional on Reaching Bankrptcy he bankrptcy cost parameter in or model, α BC, represents the proportional vale lost to bankrptcy costs conditional on hitting the defalt bondary. We examine the sensitivity of optimal capital strctre to this parameter in able VI. Not srprisingly, leverage is negatively related to bankrptcy costs. With optimal leverage ratio is 31.3%, compared to 19.4% with relatively weak. While one might expect that as highly leveraged, the reslts in able VI sggest that when α BC eqal to 1%, the α BC of 5%. However, the relation is α BC approaches zero, the firm will become extremely α BC declines to 1%, the leverage ratio only increases to 31.3%. hese reslts from ables V and VI sggest that the threshold at which the 18

21 debtholders can force the firm into bankrptcy is likely to be as important as the magnitde of the vale that is consmed in the bankrptcy process. Perhaps this finding shold not be srprising since the relevant measre of bankrptcy costs in the capital strctre decision is the expected bankrptcy costs at the time the financing decision is made. Yet, in most textbook discssions of the effect of bankrptcy on capital strctre, mch is discssed abot bankrptcy costs while the rights of bondholders to force bankrptcy are not sally emphasized Model Estimates for Individal Firms In addition to estimating the model sing parameters for a typical firm, we examine its ability to predict the capital strctres observed in a sample of 15 actal firms, five firms from each of three indstries wholesale distribtion, beer and wine manfactring, and paper and allied prodcts. o calibrate the model for individal firms and yet retain comparability across them, the natre of the debt contract is assmed to be similar across all firms. hat is, we assme that each firm is financed with 1- year debt, and that the bankrptcy bondary and bankrptcy costs are the same for all firms. We allow the volatility of the firm s assets and the CEO s stock and option holdings to vary across firms. he volatility for each firm is estimated, sing the model, by compting the volatility that yields the observed spread between each firm s actal crrent cost of debt and the yield on reasry Bonds. hese vales range from 12.9% to 56.4% with a median vale of 27.3%. he stock and option holdings for the individal CEO s are from the 2 proxy statements filed by the sample firms with the EC. able VII reports the estimated asset volatility, actal leverage, and estimated leverage, both vale maximizing and tility maximizing, for each of the 15 sample firms. he striking featre of these data is that, while the model appears to do a good job of predicting leverage for firms with relatively little to typical levels of debt, sch as tewart & tevenson, Ravenswood, Kimberly-Clark, P H Glatfelter, and Wasa-Mosinee, it sbstantially nderestimates leverage for firms with large amonts of debt. he fact that the model tends to nderestimate rather than overestimate leverage is conter to the sal intition that tax shields are far too large to be offset by bankrptcy costs. Or model emphasizes that bankrptcy 19

22 costs are not the only factor limiting leverage in the tradeoff framework; rather, the fact that tax shields lose vale in bankrptcy limits the leverage ratio that maximizes shareholder vale. 5. Conclsions his paper considers a model of optimal capital strctre in which the major forces affecting firms financing decisions are corporate taxes and bankrptcy costs. As sch, this model incorporates the effects that have been discssed at great length in the corporate finance literatre since Modigliani and Miller (1963). he model contains a nmber of featres designed to captre key elements of the capital strctre decision as realistically as possible, inclding contingent-claims valation of tax shields, a bankrptcy bondary on firm vale below which firms defalt, and a target capital strctre at which the firm refinances its debt at matrity. We calclate closed-form soltions for the important variables in this model, calibrate it sing recent market data, and solve for the optimal capital strctres from both the shareholders and manager s perspectives. In contrast to most of the literatre since at least Miller (1977), we find that the tradeoff model does not predict that firms are nderlevered. For a hypothetical firm constrcted to be typical of large, pblicly-traded companies, the model predicts a leverage ratio close to actal sample median the predicted debt to total capital ratio is 19.7% compared to a sample median of 21.9%. When we calibrate the model to reflect actal firms, the model performs less well. However, the model s failre goes in the opposite direction from what it sally presmed. In contrast to the sal intition, the model sggests that a nmber of firms appear to be overlevered, at least when only taxes and bankrptcy costs are considered. hese findings occr in a model in which the average tax benefits are sbstantially larger than the average expected bankrptcy costs. he explanation for the difference between or reslts and the sal intition probably lies in the fact that we explicitly vale a firm s tax shields. hese shields lose their vale when a firm defalts, so any decision that increases the probability of defalt, lowers the vale of a firm s tax shields. herefore, when a firm adds additional leverage, the additional debt creates new tax 2

23 shields; however, in doing so it increases bankrptcy probabilities and decreases the vale of the tax shields from the existing debt. his last effect has, to or knowledge, been ignored by the existing literatre. We also perform comparative statics on the model s nderlying parameters, to determine their impact on capital strctre choice. One parameter that appears to be particlarly important is g, the slope of the bankrptcy bondary, which we interpret as measring the strength of a firm s bond covenants. Or model assmes that this parameter is set exogenosly; in a more realistic model of capital strctre the strength of these covenants shold be thoght of as an important decision variable in a firm s financing decisions. By focsing on the tradeoff between taxes and capital strctre, we do not mean to downplay the importance of other factors. Clearly, the literatre has identified agency and information isses as key isses that mst be considered in financing decisions. An interesting recent paper applying methods similar to ors that incorporates some of these factors is itman and syplacov (21). In addition, we do not address the srprising lack of evidence on adjstments that managers make to capital strctre in order to keep capital strctre ratios eqal to some long-rn target (Welch 22). Rather, or message is that the simple tradeoff framework actally does mch better at predicting average leverage levels than has typically been spposed, and shold not be dismissed lightly as at least a first-pass way of nderstanding a firm s financing choices. We also want to emphasize the seflness of the approach of taking models seriosly and calibrating them sing market data. his qantitative approach has been seflly applied in other branches of economics, notably macroeconomics. Its main appeal is that it allows for qantitative comparisons between alternative theories. Given the mltitde of theories in corporate finance together with the general lack of exogenos variation across firms facing any researcher attempting to do traditional empirical work, it seems likely that sbseqent advances are likely to come from taking some of these models seriosly and applying nmerical methods to them. 21

24 References Black, Fischer and John Cox, 1976, Valing corporate secrities: ome effects of bond indentre provisions, Jornal of Finance, 31, Ginsbrg, Martin and Jack Levin, 21, Mergers, acqisitions and leveraged byots - ransactional analysis, CCH Bond eries, Commerce Clearing Hose, Inc.: Chicago. Graham, John R., 1996, Proxies for the corporate marginal tax rate, Jornal of Financial Economics 42, Graham, John R., 2, How big are the tax benefits of debt?, Jornal of Finance 55, Hamilton, David., Greg Gpton, and Alexandra Berhalt, 21, Defalt and recovery rates of corporate bond issers: 2, (Moody s Investors ervice). Jensen, Michael C, 1986, Agency costs of free cash flow, corporate finance, and takeovers, American Economic Review 76, Jensen, Michael C., and William H. Meckling, 1976, heory of the firm: Managerial behavior, agency costs and ownership strctre, Jornal of Financial Economics 3, pp J, Nengji, 1998, Essays in corporate finance and derivatives pricing, Unpblished Ph.D. Dissertation, (University of California at Berkeley). Leland, Hayne E., Agency costs, risk management, and capital strctre, Jornal of Finance 53, Leland, Hayne E., and Klas oft (1996). Optimal capital strctre, endogenos bankrptcy, and the term strctre of credit spreads, Jornal of Finance 51, Ljngqvist, Lars, and homas J. argent, 2, Recrsive macroeconomic theory (he MI Press, Cambridge, MA). Majd, aman and tewart C. Myers, 1987, ax asymmetries and corporate income tax reform, from he Effects of axation on Capital Accmlation, Edited by Martin Feldstein, University of Chicago Press, Chicago Mello, Antonio., and John E. Parsons, 1992, Measring the agency cost of debt, Jornal of Finance 47, Miller, Merton, 1977, Debt and taxes, Jornal of Finance 32, Modigliani, Franco, and Merton H. Miller, 1963, Corporate income taxes and the cost of capital: A correction, American Economic Review 53, Myers, tewart C., 1977, Determinants of corporate borrowing, Jornal of Financial Economics 5,

25 Myers, tewart C., 1984, he capital strctre pzzle, Jornal of Finance 39, Myers, tewart C. and Nicholas. Majlf, 1984, Corporate financing and investment decisions when firms have information that investors do not have, Jornal of Financial Economics 13, Parrino, Robert, Allen M. Poteshman, and Michael. Weisbach, 22, Measring investment distortions when risk-averse managers decide whether to ndertake risky projects, NBER Working Paper No Parrino, Robert, and Michael. Weisbach, 1999, Measring investment distortions arising from stockholder-bondholder conflicts, Jornal of Financial Economics 53, Rajan, Raghram G. and Ligi Zingales, 1995, What do we really know abot capital strctre? Evidence from international data, Jornal of Finance 5, itman, heridan and ergei syplacov, 21, A dynamic model of optimal capital strctre, Working Paper, University of exas at Astin and University of oth Carolina. itman, heridan and R. Wessels, 1988, he determinants of capital strctre, Jornal of Finance, 43, Welch, Ivo, 22, Colmbs Egg: he real determinants of capital strctre, NBER Working Paper No

26 able I Model Parameters Panel A: Chosen Parameters Variable Calibrated Vale Variable Description 1 ime at which manager evalates tility and options matre 1 ime at which debt matres r.522 Annalized risk-free rate V ( ) $1 Vale of assets before swap µ.5 Drift of vale of firm assets σ.32 Volatility of vale of firm assets N N 1 otal shares otstanding before swap γ 2 Manager s risk aversion parameter N Man.32 Nmber of shares owned by manager N Calls.38 Nmber of exchange traded Eropean calls owned by manager K $1 trike price of calls NFW ( ) $.32 Manager s non-firm wealth in dollars at time zero α BC Debtholder bankrptcy recovery rate g.519 Bankrptcy bondary exponential growth rate τ.34 Effective tax rate for debt tax shield DivRate.15 Dividend payot rate to eqity holders as a percentage of the nlevered vale of the firm.

27 able I (contined) Panel B: Derived Variables Variable F Variable Description Face vale of debt after swap C ( ) s N Constant annalized copon rate paid on debt after swap. his is set to price the debt at par. D Initial total vale of debt after swap ( ) otal shares otstanding after swap E Initial total vale of eqity after swap ( ) BC Initial total vale of bankrptcy costs after swap B ( ) Initial total vale of tax benefits of debt after swap NFW ( ) Vale of manager s non-firm wealth at time Utility() Utility () φ Expected ftre vale of manager s tility before swap Expected ftre vale of manager s tility after swap Disconted risk-netral expected vale of the V V qantity ( ) ( ) Dynamic ( ) E Initial total vale of eqity before swap ( ) Dynamic BC Initial total vale of bankrptcy costs after swap ( ) Dynamic B Initial total vale of tax benefits of debt after swap δ After tax cash payot rate to both debtholders and eqity holders as a percentage of the nlevered vale of the firm. ( ) K V Vale of assets that makes a share of stock worth K dollars at time U.

28 able II Model Otpt for Firms with Different Firm Asset Volatilities Where Objective is to Maximize hare Vale Volatility of Firm Asset Vale Row Variable ) Debt/otal Capital After wap 42.2% 34.3% 28.2% 23.5% 19.7% 16.6% 14.% 11.8% 9.9% Eqity: 2) Vale of Eqity Before wap $1 $1 $1 $1 $1 $1 $1 $1 $1 3) Nmber of hares Before wap ) Vale of Eqity After wap $73.5 $79.76 $84.14 $87.9 $89.17 $9.72 $91.93 $92.94 $ ) Nmber of hares After wap ) Change in hare Price $.2642 $.2139 $.1724 $.1382 $.112 $.874 $.689 $.538 $.417 Debt: 7) Face Vale of Debt After wap $53.37 $41.63 $33.1 $26.73 $21.85 $18.2 $14.96 $12.44 $1.34 8) Vale of Debt After wap $53.37 $41.63 $33.1 $26.73 $21.85 $18.2 $14.96 $12.44 $1.34 9) Copon After wap $ $ $ $ $1.378 $1.121 $.9685 $.8397 $ ) Bankrptcy Costs After wap $ $ $ $ $ $4.836 $5.183 $5.94 $ ) ax Benefit After wap $ $ $ $ $ $ $ $ $ ) Change in Utility ) Firm Vale At Which E=K After wap $ $ $89.41 $ $ $ $ $ $