Optimal Capital Structure with Endogenous Bankruptcy:

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1 Univesity of Pisa Ph.D. Pogam in Mathematics fo Economic Decisions Leonado Fibonacci School cotutelle with Institut de Mathématique de Toulouse Ph.D. Dissetation Optimal Capital Stuctue with Endogenous Bankuptcy: Payouts, Tax Benefits Asymmety and Volatility Risk Flavia Basotti Ph.D. Supevisos: Pof. Maia Elvia Mancino Pof. Monique Pontie

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5 CONTENTS Intoduction Pape 1: An Endogenous Bankuptcy Model with Fim s Net Cash Payouts by Flavia Basotti, Maia Elvia Mancino, Monique Pontie. Woking pape, Extended vesion of: Capital stuctue with fim s net cash payouts by Flavia Basotti, Maia Elvia Mancino, Monique Pontie (June 2010). Accepted fo publication on a Special Volume edited by Spinge, Quantitative Finance Seies. Editos: Cia Pena, Mailena Sibillo. Fothcoming Pape 2: Copoate Debt Value with Switching Tax Benefits and Payouts by Flavia Basotti, Maia Elvia Mancino, Monique Pontie. Woking pape, Pape 3: Optimal Capital Stuctue with Endogenous Default and Volatility Risk by Flavia Basotti. Woking pape, Conclusions

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7 INTRODUCTION The dissetation deals with modeling cedit isk though a stuctual model appoach. The thesis consists of thee papes ([3], [4], [1]) in which we build on the capital stuctue of a fim poposed by Leland and we study diffeent extensions of his seminal pape [17] with the pupose of obtaining esults moe in line with histoical noms and empiical evidence, studying in details all mathematical aspects. Modeling cedit isk aims at developing and applying option picing techniques in ode to study copoate liabilities and analyzing which is the maket peception about the cedit quality of a fim. The theoy is focused on the capital stuctue of a fim subject to default isk. In ode to study the default pocess, two diffeent appoaches exist in cedit isk liteatue: stuctual and educed fom models. Fo each fim the decision about its own copoate capital stuctue is a vey complex choice since it is affected by a lage numbe of economic and financial factos. Stuctual models of cedit isk epesent an analytical famewok in which the capital stuctue of a fim is analyzed in tems of deivatives contacts. This idea has been poposed at fist in Meton s wok [23] consideing Black and Scholes [6] option picing theoy to model the debt issued by a fim. Stuctual models conside the dynamic of fim s activities value as a deteminant of the default time, poviding a link between the cedit quality of a fim and its financial and economic conditions. The main idea is that default is stictly elated to the evolution of fim s activities value. The stuctual appoach consides the value of the fim as the state vaiable and assume its dynamic being descibed by a stochastic pocess in ode to detemine the time of default. Fom an economic point of view default is detemined by the inability of the fim to cove its debt obligations. Fom a mathematical point of view default is tiggeed by fim s value cossing a specified level. Depending on the model, this theshold is assumed to be exogenous o endogenously detemined, poviding diffeent economic and mathematical implications. Default is endogenously elated to fim s paametes and deived within the model following a stuctual appoach, while it is completely exogenous in educed fom models: this is the eason why stuctual models ae also defined fim value models while educed fom models ae named intensity models. While a diect link between the evolution of fim s value and the time of default is assumed in stuctual models, the educed fom appoach does not conside an explicit elation between these two factos, specifying a default pocess govening bankuptcy and thus poviding diffeent insights fom both economic and mathematical point of view. Reduced fom appoach teats default as the fist jump of an exogenously given jump pocess (independent fom fim s value), usually a Poisson-like pocess, fom no-default to default and the pobability of a jump in a given time inteval is govened by the default intensity (o hazad ate) (e.g. see [14], [8]): default aives as a supise, it is a totally inaccessible hitting time. Assuming a continuous stochastic pocess descibing fim s assets i

8 value and complete infomation about asset value and default baie, stuctual models teat the bankuptcy time as a pedictable stopping time. Among fim value models, the fist stuctual model is [23]. Meton assumes that the capital stuctue of the fim is composed by equity and a single liability with pomised final payoff B. The ability of the fim to cove its obligations depends on the total value of its assets V. Debt can be seen as a claim on V : it is a zeo-coupon bond with matuity T and face value B. As suggested in [6], by issuing debt, equity holdes ae selling the fim s assets to bond holdes and keeping a call option to buy back the assets. Equivalently we can see the same poblem as: equity holdes own fim s assets and buy a put option fom bond holdes. In such a famewok equity can be epesented as a Euopean call option with undelying asset epesented by fim s value V and stike pice equal to face value of debt B. This model allows fo default only at matuity: at time T the fim will default if its value V is lowe that the baie B. The economic eason is that in such a case the fim is not able to pay its obligations to bond holdes since its assets ae below the value of outstanding debt. The citicism towads this appoach elies on the fact that default can occu only at matuity, which is not quite ealistic. A natual extension ae the so called fist passage models allowing fo possible default befoe the matuity of debt. This appoach is fistly poposed in [5], whee default is tiggeed by the fist time a cetain theshold is eached by fim value. Fist passage models define default as the fist time the fim s assets value cosses a lowe baie, letting default occu at any time. The default baie can be exogenously fixed o endogenously detemined: when it is an exogenous level, as in [5], [21], it woks as a covenant potecting bond holdes, othewise it is endogenously detemined as a consequence of equity holdes maximizing behavio as in [17]. In such a case the the fim is liquidated immediately afte the default tiggeing event. Ou eseach focus on stuctual models of cedit isk with endogenous default in the spiit of Leland s model [17] 1. As stating point the autho claims: The value of copoate debt and capital stuctue ae intelinked vaiables. Debt values (and theefoe yield speads) cannot be detemined without knowing the fim s capital stuctue, which affects the potential fo default and bankuptcy. But capital stuctue cannot be optimized without knowing the effect of leveage on debt value. In the classical Leland [17] famewok fim s assets value evolves as a geometic Bownian motion and an infinite time hoizon is consideed. The fim ealizes its capital fom both debt and equity. Moeove, the fim has only one pepetual debt outstanding, which pays a constant coupon steam C pe instant of time and this detemines tax benefits popotional to coupon payments. This assumption of pepetual debt can be justified, as Leland suggests, thinking about two altenative scenaios: a debt with vey long matuity 1 [17] has been awaded with the fist Stephen A. Ross Pize in Financial Economics:...the pize committee chose this pape because of the substantial influence it has given on eseach about capital stuctue and copoate debt valuation..., see[16]. ii

9 (in this case the etun of pincipal has no value) o a debt which is continuously olled ove at a fixed inteest ate (as in [19]). Assuming an infinite time hoizon is a easonable fist appoximation fo long tem copoate debt and enables to have an analytic famewok whee all copoate secuities depending on the undelying vaiable (fim value) ae time independent, thus obtaining closed fom solutions. Bankuptcy is tiggeed endogenously by the inability of the fim to aise sufficient capital to meet its cuent obligations. On the failue time T, agents which hold debt claims will get the esidual value of the fim (because of bankuptcy costs), and those who hold equity will get nothing (meaning the stict pioity ule holds). The iskiness of the fim is assumed to be constant: given limited liability of equity, as [15] suggest, equity holdes may have incentives to incease the iskiness of the fim, while the opposite happens fo debt holdes, since a highe volatility deceases debt value (asset substitution poblem, see also [18]). While a huge theoetical liteatue on isky copoate debt picing exists, less attention has been paid on empiical tests of these models. The main empiical esults in cedit isk liteatue emphasize a poo job of stuctual models in pedicting cedit speads fo shot matuities. To simplify the discussion two main motivations of stuctual models failue in pedicting bond speads could be: i) failue in pedicting the cedit exposue; ii) influence of othe non cedit elated vaiables on copoate debt speads. Moeove, stuctual models usually oveestimates leveage atios. We depat fom Leland s wok genealizing the model in diffeent diections with the aim at obtaining esults moe in line with empiical evidence, futhe poviding detailed mathematical poofs of all esults. We extend it by intoducing payouts in [2, 3], then, keeping this assumption, in [4] we conside an even moe ealistic famewok unde an asymmetic copoate tax schedule. As Leland suggests in [20], a possible way to follow in ode to impove empiical pedictions of stuctual models is to modify some citical hypothesis, mainly in the diection of intoducing jumps and/o emoving the assumption of constant volatility in the undelying fim s assets value stochastic evolution. This latte case is what motivates [1]. In [2], [3] (Pape 1) we extend Leland model [17] to the case whee the fim has net cash outflows esulting fom payments to bondholdes o stockholdes, fo instance if dividends ae paid to equity holdes, and we study its effect on all financial vaiables and on the choice of optimal capital stuctue. The inteest in this poblem is posed in [17] section VI-B, nevetheless the esulting optimal capital stuctue is not analyzed in detail. Ou aim is twofold: fom one hand we complete the study of copoate debt and optimal leveage in the pesence of payouts in all analytical aspects, fom the othe hand we study numeically the effects of this vaiation on the capital stuctue. Moeove in [3] we conduct a quantitative analysis on the effects of payouts on the pobability of default, on expected time to default and on agency costs fom both an economic and mathematical point of view. We follow Leland [17] by consideing only cash outflows which ae popotional to fim s assets value but ou analysis diffes fom Leland s one since we solve the optimal contol poblem as an optimal stopping poblem (see also [7] fo a simila iii

10 appoach) and not with odinay diffeential equations. We find that the incease of the payout ate paamete δ affects not only the level of endogenous bankuptcy, but modifies the magnitude of a change on the endogenous failue level as a consequence of an incease in isk fee ate, copoate tax ate, iskiness of the fim and coupon payments. Futhe the intoduction of payouts allows to obtain lowe optimal leveage atios and highe yield speads, compaed to Leland s [17] esults, thus making these empiical pedictions moe in line with histoical noms. Ou analysis suggests that adding payouts has an actual influence on all financial vaiables: fo an abitay coupon level C, a positive payout ate δ inceases equity value and deceases both debt and total value of the fim, making bankuptcy moe likely. In line with [19] suggestion, in [3] we show that the pobability of default is quite dependent on the dift of the pocess descibing fim s activities value, thus on payouts. Analyzing cumulative pobability of going bankuptcy ove a peiod longe than 10 yeas suggests that intoducing δ makes debt iskie, stongly inceasing the likelihood of default. Studying the influence of payouts on the asset substitution poblem we find that this poblem still exists and its magnitude is inceased by payouts, making highe potential agency costs aising fom the model. In [4] (Pape 2) we keep the intoduction of a company s assets payout atio δ as in [2],[3] and extend the model poposed by [17] in the diection of a switching (even debt dependent) in tax savings. Taxes ae a cucial economic vaiable affecting optimal capital stuctue, as ealy ecognized by [24] and obseved by [25]. While stuctual models assume constant copoate tax ates, Leland agues that default and leveage decisions might be affected by non constant copoate tax ates, because a loss of tax advantages is possible fo low fim values. The empiical analysis of [11] confims that the copoate tax schedule is asymmetic, in most cases it is convex, and [25] suggests that tax convexity cannot be ignoed in copoate financing decisions. We assume the copoate tax schedule based on two diffeent copoate tax ates: the switching fom a copoate tax ate to the othe is detemined by fim s activities value cossing a citical baie. We conside two diffeent famewoks: at fist, the switching baie is assumed to be a constant exogenous level; secondly, we analyze an even moe ealistic scenaio in which this level depends upon the amount of debt the fim has issued. We obtain an explicit fom fo the tax benefit claim, which allows us to study monotonicity and convexity of the equity function, to find the endogenous failue level in closed fom in case of no-payouts and to pove its existence and uniqueness in the geneal case with both asymmetic tax scheme and payouts, while liteatue in this field usually gives only numeical esults. Ou appoach diffes fom [17] since we solve the optimal contol poblem as an optimal poblem in the set of passage times; the key method is the Laplace tansfom of the stopping failue time. Ou esults show that tax asymmety inceases inceases the optimal failue level educes the optimal leveage atio, and that this last effect is moe ponounced, thus confiming the esults in [25]. Nevetheless, as fa as the magnitude is concened, intoducing payouts inceases this negative effect on both leveage and debt. Optimal leveage atios ae smalle than in case of a flat tax schedule since the potential loss in tax benefits makes debt less attactive iv

11 and this effect is moe ponounced when we deal with a debt dependent switching baie (i.e. depending on coupon payments). The joint influence of payouts and copoate tax asymmety poduces a significant impact on copoate financing decisions: it dops down leveage atios to empiically epesentative values suggesting a possible way to explain diffeence in obseved leveage acoss fims facing diffeent tax-code povisions. In [1] (Pape 3) the aim is to analyze the capital stuctue of a fim unde an infinite time hoizon emoving the assumption of constant volatility and consideing default as endogenously tiggeed. We intoduce a pocess descibing the dynamic of the diffusion coefficient diven by a one facto mean-eveting pocess of Ostein-Uhlenbeck type, negatively coelated with fim s assets value evolution. Diffeently fom a pue Leland famewok and even fom the moe geneal context with payouts and asymmetic copoate tax ates studied in [2, 3, 4], inside this famewok we cannot obtain explicit expessions fo all the vaiables involved in the capital stuctue by means of the Laplace tansfom of the stopping failue time. The key point elies in the fact that this tansfom, which was the key tool used in [2, 3, 4], is not available in closed fom unde ou stochastic volatility famewok. Nevetheless debt, equity, bankuptcy costs and tax benefits ae claims on the fim s assets, thus we apply ideas and techniques developed in [9] fo the picing of deivatives secuities whose undelying asset pice s volatility is chaacteized by means of its time scales fluctuations. This appoach has been applied in [10] to pice a defaultable zeo coupon bond. Hee a one-facto stochastic volatility model is consideed and single petubation theoy as in [9] is applied in ode to find appoximate closed fom solutions fo deivatives involved in ou economic poblem. Each financial vaiable is analyzed in tems of its appoximate value though an asymptotic expansion depending on the volatility mean-evesion speed. Moeove, we study the effects of the stochastic volatility assumption on the endogenous failue level detemined by equity holdes maximizing behavio. Unde ou appoach, the failue level deived fom standad smooth-fit pinciple is not the solution of the optimal stopping poblem, but only epesents a lowe bound which has to be satisfied due to limited liability of equity. Choosing that failue level would mean a non-optimal execise of the option to default. A coected smooth-pasting condition must be applied in ode to find the endogenous failue level solution of the optimal stopping poblem. Moeove, we show the convegence of ou esults to Leland case [17] as the paticula case of zeo-petubation. By taking into account the stochastic volatility isk component of the fim s asset dynamic, ou aim is to bette captue exteme etuns behavio which could be a obust way to impove empiical pedictions about speads and leveage. Intoducing andomness in volatility allows to deal with a stuctual model in which the distibution of stock pices etuns is not symmetic: in ou mind this seems to be the ight way fo captuing what stuctual models ae not able to explain with a constant diffusion coefficient. The numeical esults show that the assumption of stochastic volatility model poduces elevant effects on the optimal capital stuctue in tems of highe cedit speads and lowe leveage atios, if compaed with the oiginal Leland case. v

12 Refeences [1] BARSOTTI F. (2011), Optimal Capital Stuctue with Endogenous Bankuptcy and Volatility Risk, Woking pape. [2] BARSOTTI F., MANCINO M.E., PONTIER M. (2010), Capital Stuctue with Fim s Net Cash Payouts, accepted fo publication on a Special Volume edited by Spinge, Quantitative Finance Seies, editos: Cia Pena, Mailena Sibillo. [3] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), An Endogenous Bankuptcy Model with Fim s Net Cash Payouts, Woking pape. [4] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), Copoate Debt Value with Switching Tax Benefits and Payouts, Woking pape. [5] BLACK, F., COX, J. (1976), Valuing Copoate Secuities: Some Effects of Bond Indentue Povisions, Jounal of Finance, 31, [6] BLACK F., SCHOLES M. (1973), The Picing of Options and Copoate Liabilities, Jounal of Political Economy, 81, [7] DOROBANTU D., MANCINO M., PONTIER M. (2009), Optimal Stategies in a Risky Debt Context, Stochastics An Intenational Jounal of Pobability and Stochastic Pocesses, 81(3), [8] DUFFIE D., LANDO D. (2001), Tem Stuctues of Cedit Speads with Incomplete Accounting Infomation, Econometica, 69, [9] FOUQUE J.P., PAPANICOLAOU G., RONNIE K.R. (2000), Deivatives in Financial Makets with Stochastic Volatility, Cambidge Univesity Pess. [10] FOUQUE J.P., PAPANICOLAOU G., SOLNA K. (2005), Stochastic Volatility Effects on Defaultable Bonds, Woking Pape. [11] GRAHAM J.R., SMITH C.W. (1999), Tax incentives to hedge, Jounal of Finance, 54, [12] HILBERINK B. and ROGERS L.C.G. (2002), Optimal Capital Stuctue and Endogenous Default, Finance and Stochastics, 6, [13] HUANG J.Z., HUANG M., (2003), How Much of the Copoate-teasuy Yield Spead is Due to Cedit Risk?, Woking pape. [14] JARROW R.A., TURNBULL S. (1995), Picing Deivatives on Financial Secuities Subject to Cedit Risk, Jounal of Finance, 50. vi

13 [15] JENSEN M., MECKLING M. (1976), Theoy of The Fim: Manageial Behavio, Agency Costs, and Owneship Stuctue, Jounal of Financial Economics 4, [16] LELAND H., The Impact of Copoate Debt Value, Bond Covenants, and Optimal Capital Stuctue, Foundation fo the Advancement of Reseach in Financial Economics, available on-line at [17] LELAND H.E. (1994), Copoate Debt Value, Bond Covenant, and Optimal Capital Stuctue, The Jounal of Finance, 49, [18] LELAND H.E. (1998), Agency Costs, Risk Management and Capital Stuctue, The Jounal of Finance, Vol.53, 4, [19] LELAND H.E., TOFT K.B. (1996), Optimal capital stuctue, Endogenous Bankuptcy and the Tem Stuctue of Cedit Speads, The Jounal of Finance, 51, [20] LELAND H.E. (2009), Stuctual Models and the Cedit Cisis, China Intenational Confeence in Finance [21] LONGSTAFF F., SCHWARTZ E. (1995), A simple Appoach to Valuing Risky Fixed and Floating Rate Debt and Detemining Swap Speads, The Jounal of Finance, 50, [22] MERTON R. C. (1973), A Rational Theoy of Option Picing, Bell Jounal of Economics and Management Science, 4, [23] MERTON R. C. (1974), On the Picing of Copoate Debt: The Risk Stuctue of Inteest Rates, The Jounal of Finance, 29, [24] MODIGLIANI F., MILLER M. (1958) The Cost of Capital, Copoation Finance and the Theoy of Investment, Ameican Economic Review, 48, [25] SARKAR S. (2008), Can Tax Convexity Be Ignoed in Copoate Financing Decisions?, Jounal of Banking & Finance, 32, vii

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17 An Endogenous Bankuptcy Model with Fim s Net Cash Payouts Flavia Basotti Maia Elvia Mancino Monique Pontie Dept. Stat. and Applied Math. Dept. Math. fo Decisions Inst. Math. de Toulouse (IMT) Univesity of Pisa, Italy Univesity of Fienze, Italy Univesity of Toulouse, Fance [email protected] [email protected] [email protected] Abstact In this pape a stuctual model of copoate debt is analyzed following an appoach of optimal stopping poblem. We extend Leland model [7] intoducing a payout δ paid to equity holdes and studying its effect on copoate debt and optimal capital stuctue. Vaying the paamete δ affects not only the level of endogenous bankuptcy, which is deceased, but modifies the magnitude of a change on the endogenous failue level as a consequence of an incease in isk fee ate, copoate tax ate, iskiness of the fim and coupon payments. Concening the optimal capital stuctue, the intoduction of this payout allows to obtain esults moe in line with histoical noms: lowe optimal leveage atios and highe yield speads, compaed to Leland s [7] esults. 1 Intoduction Many fim value models have been poposed since Meton s wok [11] which povides an analytical famewok in which the capital stuctue of a fim is analyzed in tems of deivatives contacts. We focus on the copoate model poposed by Leland [7] assuming that the fim s assets value evolves as a geometic Bownian motion. The fim ealizes its capital fom both debt and equity. Debt is pepetual, it pays a constant coupon C pe instant of time and this detemines tax benefits popotional to coupon payments. Bankuptcy is detemined endogenously by the inability of the fim to aise sufficient equity capital to cove its debt obligations. On the failue time T, agents which hold debt claims will get the esidual value of the fim (because of bankuptcy costs), and those who hold equity will get nothing (the stict pioity ule holds). This pape examines the case whee the fim has net cash outflows esulting fom payments to bondholdes o stockholdes, fo instance if dividends ae paid to equity holdes. The inteest in this poblem is posed in [7] section VI-B, nevetheless the esulting optimal capital stuctue is not analyzed in detail. The aim of this pape is twofold: fom one hand we complete the study of copoate 1

18 debt and optimal leveage in the pesence of payouts in all analytical aspects, fom the othe hand we study numeically the effects of this vaiation on optimal capital stuctue. We will follow Leland [7] by consideing only cash outflows which ae popotional to fim s assets value but ou analysis diffes fom Leland s one since we solve the optimal contol poblem as an optimal stopping poblem (see also [3] fo a simila appoach). As in Leland model we assume that capital stuctue decisions, once made, ae not changed though time. This means, fo example, that the face value of debt is supposed to be constant and ou analysis appoaches the poblem of detemining the optimal amount of debt and the optimal endogenous failue level as a two-stage optimization poblem (see also [1]). Equity holdes have to chose both: i) the optimal amount of coupon payments C, ii) the optimal endogenous tiggeing failue level V B. These decisions ae stongly inteelated and not easily consideed as sepaable ones. This inteconnection is stictly elated to the conflict between equity and debt holdes, since they have diffeent inteests on the fim, leading also to potential agency costs aising fom the model (see [8]). When equity holdes have to chose the optimal amount of debt, i.e. coupon maximizing the total value of the fim, this will obviously depend on the endogenous failue level. At the same time, when choosing the optimal stopping time of the option to default, meaning finding the endogenous failue level, this will obviously depend on the amount of debt issued. An appoximation to solve these complicated issues aising fom equity and debt holdes conflict, will be to poceed in a two-step analysis as follows: i) the fist-stage optimization poblem is to detemine the endogenous failue level; ii) in the second stage we detemine the amount of debt which maximizes the total value of the fim, given the esult about the default tiggeing level of the fist stage. Ou findings show that the incease of the payout paamete δ affects not only the level of endogenous bankuptcy, which is deceased, but modifies the magnitude of a change on the endogenous failue level as a consequence of an incease in isk fee ate, copoate tax ate, iskiness of the fim and coupon payments. Futhe the intoduction of payouts allows to obtain lowe optimal leveage atios and highe yield speads, compaed to Leland s [7] esults, which ae moe in line with histoical noms. The pape is oganized as follows: Section 2 intoduces the model and detemines the optimal failue time as an optimal stopping time, getting the endogenous failue level. Then, in Section 3 the influence of coupon, payouts and copoate tax ate on all financial vaiables is studied. Section 4 descibes optimal capital stuctue as a consequence of optimal coupon choice. Once detemined the endogenous failue level, equity holdes aim is to find the coupon payment which allows to maximize the total value of the fim. 2 A Fim s Capital Stuctue with Payouts In this section we intoduce the model, which is vey close to Leland s [7], but we modify the dift with a paamete δ, which might epesent a constant popotional cash flow 2

19 geneated by the assets and distibuted to secuity holdes. We conside a fim ealizing its capital fom both debt and equity. Debt is pepetual and pays a constant coupon C pe instant of time. On the failue time T, agents which hold debt claims will get the esidual value of the fim, and those who hold equity will get nothing. We assume that the fim activities value is descibed by the pocess V t = Ve Xt, whee X t evolves, unde the isk neutal pobability measue, as dx t = δ 12 σ2 dt + σdw t, X 0 =0, (1) whee W is a standad Bownian motion, the constant isk-fee ate,, δ and σ > 0. Following [9] we assume δ being a constant faction of value V paid out to secuity holdes. In line with [8], paamete δ epesents the total payout ate to all secuity holdes, thus V epesents the value of the net cash flows geneated by the fim s activities (excluding cash flows elated to debt issuance). Moeove we assume that δ is not affected by changes in leveage (see [9] footnote 4). When bankuptcy occus at andom time T, a faction α (0 α < 1) of fim value is lost (fo instance payed because of bankuptcy pocedues), debt holdes eceive the est and stockholdes nothing, meaning that the stict pioity ule holds. We suppose that the failue time T is a stopping time. Thus, applying contingent claim analysis in a Black-Scholes setting, fo a given stopping (failue) time T, debt value is D(V,C,T)=E V T 0 e s Cds +(1 α)e T V T, (2) whee the expectation is taken with espect to the isk neutal pobability and we denote E V [ ] :=E[ V 0 = V ]. We assume that fom paying coupons the fim obtains tax deductions, namely τ,0 τ < 1, popotionally to coupon payments at a ate τc until default 1, so we get equity value as T E(V,C,T)=V E V (1 τ) e s Cds + e T V T. (3) 0 The total value of the (leveed) fim can always be expessed as sum of equity and debt value: this leads to wite the total value of the leveed fim as the fim s asset value (unleveed) plus tax deductions on debt payments C less the value of bankuptcy costs: v(v,c,t)=v + E V τ T 0 e s Cds αe T V T. (4) 1 Since we conside only tax benefits in this model, δv can be intepeted as the afte-tax net cash flow befoe inteest, see also [13] footnote 3. 3

20 2.1 Endogenous Failue Level In this subsection we analyze in detail the fist-stage optimization poblem faced by equity holdes, i.e. we find the endogenous failue level which maximizes equity value, fo a fixed coupon level. Recall that equity claim in an option-like contact since thee is an option to default embodied in it. The best fo equity holdes will be to execise this option when V will each a failue level V B endogenously deived (i.e. satisfying a smooth-fit pinciple). Fo this analysis, we suppose the coupon level C being fixed. On the set of stopping times we maximize the equity value T E(V,C,T), fo an abitay level of the coupon ate C. Fom optimal stopping theoy (see [4]) and following [2, 3], the failue time, optimal stopping time, is known to be a constant level hitting time. Hence default happens at passage time T when the value V. falls to some constant level V B. The value of V B is endogenously deived and will be detemined with an optimal ule late. Futhe we note that, given (1), it holds that T =inf{t 0: V t V B } =inf{t 0: X t log V B }. (5) V Moeove it holds V T = V B, as the pocess V t is continuous. Thus, the optimal stopping poblem of equity holdes is tuned to maximize the equity function defined in (3) as a function of V B : E : V B V (1 τ)c 1 EV [e T ] V B E V [e T ]. (6) In ode to compute the equity value (6) it emains to detemine E V e T. To this hand we use the following fomula fo the Laplace tansfom of a constant level hitting time by a Bownian motion with dift ([5] p ): Poposition 2.1 Let X t = µt + σw t and T b =inf{s : X s = b}, then, fo all γ > 0, it holds E[e γt b µb ]=exp σ 2 b µ 2 σ σ 2 +2γ. Since V t = V exp[( δ 1 2 σ2 )t + σw t ] by (1), we get E V [e T ]= VB V λ(δ) whee λ(δ) = δ 1 2 σ2 + ( δ 1 2 σ2 ) 2 +2σ 2 σ 2. (7) Remak 2.2 As a function of δ, thecoefficient λ(δ) in (7) is deceasing and convex. In ode to simplify the notation, we will denote λ(δ) as λ in the sequel. 4

21 Finally the equity function to be optimized w..t. V B is E : V B V (1 τ)c and the following popeties must be satisfied: (1 τ)c + λ VB V B, (8) V E(V,C,T) E(V,C, ) and E(V,C,T) 0 fo all V V B. (9) Consideing equity function in (8), the fist popety in (9) is equivalent to (1 τ)c λ VB E(V,C,T) E(V,C, ) = V B 0. V In fact this tem is the option to default embodied in equity. Since this is an option to be execised by the fim, this must have positive value, so it must be (1 τ)c V B 0. Finally we ae led to the constaint: V B (1 τ)c. (10) As fo the second popety in (9), we obseve that if V B was chosen by the fim, then the total value of the fim v would be maximized by setting V B as low as possible. Nevetheless, because equity has limited liability, then V B cannot be abitay small, but E(V,C,T)must be nonnegative. Note that E(V,C, ) =V (1 τ)c 0 unde the following constaint V (1 τ)c. (11) A natual constaint on V B is V B <V,indeed, if not, the optimal stopping time would necessaily be T = 0 and then (1 τ)c (1 τ)c E(V,C,T)=V + V B = V V B < 0. Finally E(V,C,T) 0 fo all V V B and the poblem faced by equity holdes is: which is equivalent to max V B 0, (1 τ)c E(V,V B,C), max V B 0, (1 τ)c (1 τ)c λ VB V B, (12) V since the isk less component of equity value V (1 τ)c does not depend on V B. This last fomulation (12) epesents exactly the optimal stopping poblem we want to solve: the economic meaning is that equity holdes have to chose which is the optimal execise time of thei option to default, o, equivalently, the endogenous failue level solution of (12). 5

22 Poposition 2.3 The endogenous failue level solution of (12) is V B (C; δ, τ) = C(1 τ) λ λ +1, (13) whee λ is given by (7). Poof In ode to obtain the endogenous failue level V B we maximize the function (8), which tuns in maximizing the option to default g(v B ) embodied in equity given by (1 τ)c g(v B ):V B λ VB V B. V We have g (V B )= VB V λ f(vb ) V B, g (V B )= λ VB V λ (f(vb )+C(1 τ)) V B 2, with f(v B )=V B (1 + λ) λc(1 τ). Function g(v B ) is inceasing fo 0 <V B < V B, then deceasing fo V B <V B < (1 τ)c, with V B solution of f(v B ) = 0: V B = (1 τ)c λ 1+λ. Moeove g(v B ) is convex fo V B < V B,with V B solution of g (V B )=0 and concave fo V B <V B < (1 τ)c. V (1 τ)c λ 1 B = 1+λ, Obseve what follows: i) (1 τ)c > V B > V B, λ > 0; i) in case λ < 1, VB < 0, meaning g(v B ) concave inside [0, C(1 τ) ]. The maximum exists and is unique and it is achieved at point V B ; ii) in case λ > 1, 0 < V B < V B, meaning g(v B ) convex fo 0 <V B < V B and concave fo V B <V B < C(1 τ). 6

23 The sign of g (V B ) is the same of f(v B )= V B (1 + λ) +λc(1 τ). Moeove, we have g ( V (1 τ)c B ) > 0, g < 0, since f ( V B )=(1 τ)c >0, (1 τ)c f = (1 τ)(1 + λ)c <0, As a consequence the maximum exists and is unique inside [0, (1 τ)c ], fo any value of λ > 0 and it is achieved fo V B such that: g (V B )=0 (1 τ)cλ =(λ + 1)V B. Remak 2.4 Note that (13) satisfies the smooth pasting condition (see [1], [10] footnote 60): E V V =V B =0. This condition holds since the endogenous failue level (13) is the lowest admissible failue level equity holdes can consistent with both i) limited liability of equity, ii) equity being a non-negative and inceasing function of cuent fim s assets value V. Obseve that equity is an inceasing function of V when the following constaint is satisfied: E V =1 λ V VB V λ (1 τ)c V B 0. (14) Moeove, the lowest value V can assume is V B and at point V = V B we have E(V B,V B )=0, thus in ode to have equity inceasing fo V V B it is sufficient Solving fo V B gives V B with the ight hand side being exactly (13). E V V =V B 0. (15) C(1 τ) λ λ +1, Accoding to [1] we obseve that the smooth pasting conditions gives an endogenous failue level which is also solution of the optimal stopping poblem (12), since ˆ V B being (13). E(V,V B,C) V B 0, ˆ V B <V B < (1 τ)c, 7

24 We obseve that the equity function is convex w..t. fim s cuent assets value V, as constaint (10) is satisfied: (1 τ)c V B (C; δ, τ) <. Futhe (13) has to satisfy V B V, theefoe the following inequality holds: C(1 τ) λ λ +1 V. (16) Remak 2.5 As a paticula case when δ =0we obtain Leland [7], whee λ = 2 σ 2 E(V,C,V B )=V (1 τ)c (1 τ)c 2/σ 2 VB + V B, V and the failue level is Since the application δ λ λ+1 V B (C;0, τ) = C(1 τ) + 1. (17) 2σ2 is deceasing, (17) is geate than (13) fo any value of τ: V B (C; δ, τ) = C(1 τ) λ λ +1 <V B(C;0, τ) = C(1 τ) + 1. (18) 2σ2 The failue level V B (C; δ, τ) is deceasing with espect to τ,,σ 2 and popotional to the coupon C, fo any value of δ. We note that the dependence of V B (C; δ, τ) on all paametes τ,,σ 2,C is affected by the choice of the paamete δ. In fact the application δ V B(C;δ,τ) τ is negative and inceasing, while δ V B(C;δ,τ) C is positive and deceasing: thus intoducing a payout δ > 0 implies a lowe eduction (incease) of the endogenous failue level as a consequence of a highe tax ate (coupon), if compaed to the case δ = 0. Similaly a change in the isk fee ate o in the iskiness σ 2 of the fim has a diffeent impact on V B (C; δ, τ) depending on the choice of δ. Figue 1 shows that depending on both the payout level and the iskiness of the fim, a change in σ can poduce an incease (decease) on the endogenous failue level with a vey diffeent magnitude depending on the payout. As shown in the plot, consideing V B (C; δ, τ) as function of the volatility fo diffeent values of payout means facing diffeent functions, i.e. the shape and the slope is not constant fo each level of σ when δ changes. The distance between two cuves (efeing to a diffeent δ) is not constant, meaning that payouts do not simply geneate a taslation. In line with the esults in [7] the endogenous failue level V B (C; δ, τ) in (13) is independent of both fim s assets value V and α, the faction of fim value which is lost in the event of bankuptcy (since the stict pioity ule holds). The choice of the endogenous failue level V B (C; δ, τ) is a consequence of equity holdes maximizing behavio: this is 8

25 70 VB(C ; δ, τ ) δ =0 δ =0.01 δ = σ Figue 1: Endogenous failue level as function of σ. This plot shows the behavio of the endogenous failue level V B(C; δ, τ) given by (13) as function of σ, fo a fixed level of coupon C =6.5 satisfying (11). We assume V =100, =6%,τ =0.35, α =0.5. why it is independent of α. The economic eason behind is elated to the option to default embodied in equity: ecall that in ode to find the optimal failue level equity holdes face the poblem of maximizing V B g(v,c,v B ) given by (12) and equity value is not affected by bankuptcy costs since the stict pioity ule holds. The endogenous failue level will always be lowe than face value of debt, hence equity holdes will get nothing at bankuptcy, and in this sense thei decision does not depend on paamete α. Only debt holdes will bea bankuptcy costs. 2.2 Expected Time to Default We have poved that intoducing payouts has an actual influence on the endogenous failue level V B (C; δ, τ): in paticula we showed that δ lowes the failue bounday chosen by equity holdes when we conside the coupon being fixed. Also when the coupon is chosen to maximize the total value of the fim, the optimal failue level VB (V ; δ, τ) educes as a consequence of a highe payout, as we will show in subsection 4. Consistent with ou base case paametes values, Table 3 gives an idea of the magnitude of this eduction in tems of new optimal default tiggeing level. An inteesting point should be to analyze not only the influence of payouts on the failue level, but also on expected time to default. How long does it take fo V to each the failue level V B? Do payouts have a quantitatively significant effect on it o not? Since we ae consideing a famewok with infinite hoizon, it should be inteesting to analyze whethe intoducing payouts will have a significant influence on the expected time to default. We know that fim s activities value evolves as a log-nomal vaiable, thus the 9

26 expected time fo pocess V t to each the constant failue level V B can be studied as shown in the following Poposition. Poposition 2.6 Let T b defined in Poposition 2.1. b := log V B V, with V V B.Thefollowingholds: 2µ if µ>0, E V [T b ]= VB σ 2 log V B V V µ, Conside µ := δ 1 2 σ2 and if µ<0, E V [T b ]= log V B V µ. Poof The esult follows by Poposition 2.1 and E V [T b ]= E[e γt b] γ=0. γ Poposition 2.6 is useful to show that payouts have an actual influence on the expected time to default and this impact must be analyzed fom two diffeent points of view: i) payouts influence on the dift µ of pocess X t ; ii) payouts influence on the endogenous failue level. Fom ou pevious analysis we know that payouts decease both µ and the endogenous (and also optimal) failue level chosen by equity holdes. But what eally mattes is the inteaction between these effects. We think that fom an empiical point of view studying the impact of these two combined effects on the expected time to default could be an altenative way to measue the isk of default associated to the fim s capital stuctue also fom both a qualitative and quantitative point of view. 3 Compaative Statics of Financial Vaiables In this section we aim at analyzing the dependence of all financial vaiables on C, δ, τ at the endogenous failue level V B (C; δ, τ) obtained fom the fist-stage optimization poblem (12). By substituting its expession (13) into equity, debt and total value of the fim, we obtain the following functions: (1 τ)c (1 τ)c 1 C(1 τ) λ λ E :(C; δ, τ) V + (19) λ +1 V λ +1 D :(C; δ, τ) C C λ C(1 τ) λ λ 1 (1 α)(1 τ) (20) λ +1 V λ +1 v :(C; δ, τ) V + τc C λ(1 τ) C(1 τ) λ λ τ + α. (21) λ +1 V λ +1 Figue 1 shows the behavio of equity, debt and total value of the fim given by (19)- (21) as function of the payout ate δ. 10

27 Limit as Behaviou w..t. Vaiable V V V B C δ τ E V (1 τ)c 0 Convex, Convex, C λc(1 τ)(1 α) D (1+λ) Concave, -Shaped a v V + τc λc(1 τ)(1 α) (1+λ) Concave b (1+λ) R λ(1 α)(1 τ) Concave (1+λ(α+τ ατ)) R 0 λ(1 α)(1 τ) Concave a See Poposition 3.5. b See Poposition 3.7. Table 1: Compaative statics of financial vaiables. The table shows the behaviou of all financial vaiables at V B (C; δ, τ) unde constaint (11). 3.1 Equity We analyze equity s behaviou with espect to δ. Poposition 3.1 The equity function (19) is deceasing and convex as function of λ. Poof Equity s behaviou w..t. λ is summaized by The logaithmic deivative of (22) is log( C(1 τ) V f (λ) = f(λ) = 1 C(1 τ) λ λ. (22) λ +1 V 1+λ λ λ+1 ) which is negative by (16). Moeove 1 λ(1 + λ) f(λ)+ C(1 τ) λ log + log f (λ) > 0, (23) V 1+λ thus equity is deceasing and convex w..t. λ. As a consequence of Poposition 3.1 and Remak 2.2, equity is inceasing w..t. δ. Concening equity s convexity w..t. δ, the following esult holds. Poposition 3.2 The equity function (19) is convex w..t. δ if whee V B (C; δ, τ) is given by (13). 2 V > V B (C; δ, τ)e λ 2 µ 2 +2σ 2 (24) 11

28 Poof In ode to study equity s convexity w..t. δ, we evaluate δ 2 E using (22) and (23), obtaining: δ 2 E = 1 λ(1 + λ) f(λ)+ + f(λ) log C(1 τ) V log 2 C(1 τ) λ V 1+λ λ 1+λ f(λ) λ 2 µ 2 +2σ 2 2 (µ 2 +2σ 2 ) µ 2 +2σ 2 (25) Substituting V B (C; δ, τ) =C(1 τ)λ/((1 + λ)) and e-aanging tems gives: δ 2 E = f(λ) λ µ 2 +2σ 2 1+λ + log V B λ 2 log V B V V + 2 µ 2 +2σ 2 (26) As V V B then log V B V < 0 fo each paametes choice, then a sufficient condition fo equity s convexity w..t. δ is which is equivalent to (24). λ 2 log V B V + 2 µ 2 +2σ 2 < 0, Remak 3.3 Equity s dependence on the payout ate δ is stictly elated to equity s dependence on µ. Obseve that δ 2E = 2 µe, since µ λ = δ λ and µλ 2 = δ 2 λ. Intoducing payouts has a positive effect on equity value. Studying equity s behavio w..t. C and τ, we can obseve that E in (19) is a function of the poduct C(1 τ); thus E(C; δ, τ) is: i) deceasing and convex w..t. coupon C, ii) inceasing and convex w..t. the copoate tax ate τ. We obseve that also in the pesence of a payout ate δ > 0, equity holdes have incentives to incease the iskiness of the fim, since λ deceases with highe volatility. These incentives ae highe as the payout ate inceases: Figue 3 shows the behavio of E E σ as function of V, fo thee diffeent levels of δ. Fo each value of V, σ inceases with δ, meaning that a highe payout poduces geate incentives fo shaeholdes to incease the iskiness of the fim, once debt is issued, thus ising potential agency costs due to incentive compatibility poblem between equity and debt holdes. The payout influence on a potential ise in agency costs is stictly elated to the actual distance to default of the fim, as shown in Figue 3. Obseve that we epot E σ as function of cuent assets value V : if it is tue that δ E σ is an inceasing function V V B, this incease in equity sensitivity to fim iskiness is stongly elated to the distance to default. Recall that the endogenous failue level is independent of V, thus the plot in Figue 3 is built on an endogenous failue level which changes only with δ (the othe paametes being fixed). The potential incease in agency costs due to a highe E σ is moe ponounced when V is small o vey high, meaning fo exteme values of the distance to default, since V B is endogenously given. 12

29 E(C;δ,τ), D(C;δ,τ), v(c;δ,τ) E(C;δ,τ) D(C;δ,τ) v(c;δ,τ) δ Figue 2: Equity, debt and total value of the fim as function of δ. This plot shows the behavio of equity (19), debt (20) and total value of the fim (21) as function of δ, fo a fixed level of coupon C =6.5 satisfying (11). We assume V =100, =6%,σ =0.2, τ =0.35, α = δ =0 δ =0.01 δ = E σ V Figue 3: Effect of a change in σ on equity value. This plot shows the behavio of E σ as function of fim s cuent assets value V, fo a fixed level of coupon C =6.5 anddiffeent values of δ =0, We conside =0.06, σ =0.2, α =0.5, τ =0.35. Fo each level of the payout ate δ, equity function has adiffeent suppot, i.e. V V B(C; δ, τ), with V B(C; δ, τ) given by (13). As a consequence, each cuve epesenting equity sensitivity w..t. σ is plotted in its own suppot. 3.2 Debt and Yield Spead We conside now the debt function D(C; δ, τ) in (20). The application D(C; δ, τ) is concave w..t. coupon C, allowing to analyze the maximum capacity of debt of the fim 13

30 as it is shown in the following Poposition. Poposition 3.4 The application C D(C; δ, τ) is concave and achieves a maximum at C max (V,δ, τ) = V (1 + λ) λ(1 τ) 1 1 λ. (27) λ(τ + α(1 τ)) + 1 C max (V,δ, τ) epesents the maximum capacity of the fim s debt. Substituting this value fo the coupon into debt function D(C; δ, τ) and simplifying yields: D max (V,δ, τ) = V 1 τ 1 λ(τ + α(1 τ)) λ. (28) δ =0 δ =0.01 δ =0.04 D(C ; δ, τ ) C Figue 4: Debt value as function of the coupon. This plot shows the behaviou of debt value given in (20) as function of coupon payments C, fo diffeent levels of δ. WeassumeV =100, =0.06, σ =0.2, τ =0.35, α =0.5. We conside thee diffeent levels of δ =0, 0.01, The value C = V is the (1 τ) maximum value that coupon C can assume due to constaint C(1 τ) V < 0. With ou base case it is appoximately C =9.23. Equation (28) epesents the debt capacity of the fim: the maximum value that debt can achieve by choosing the coupon C. Not supisingly the debt capacity of the fim is popotional to fim s cuent assets value V, deceases with highe bankuptcy costs α and inceases if the copoate tax ate ises. In the pesence of payouts, if τ changes, its effect on debt capacity is lowe than in case δ = 0, since δ Dmax(V ;δ,τ) τ is deceasing. Unde constaint (11), as δ inceases, debt deceases as shown in the following Poposition. 14

31 Poposition 3.5 Debt value D(C; δ, τ) defined in (20) is a deceasing function of δ fo with V B (C; δ, τ) given in (13). τ+α(1 τ) V > V B (C, δ, τ)e 1+λ(τ+α(1 τ)), (29) Poof It is enough to study the monotonicity of debt function with espect to λ. Debt s dependence on λ is the opposite of its log-deivative being λ C(1 τ) λ λ g(λ) = 1 (1 α)(1 τ), λ +1 V λ +1 h(λ) = g g (λ) = log λ λ +1 + α + τ ατ C(1 τ) + log, 1+λ(α + τ ατ) V thus h(λ) = g g (λ) = α + τ ατ 1+λ(α + τ ατ) + log V B(C, δ, τ). V We have h(λ) > 0 fo with V B (C; δ, τ) given in (13). V < V B (C, δ, τ)e τ+α(1 τ) 1+λ(τ+α(1 τ)), As the payout ate δ inceases, the maximum capacity of debt educes (the application λ D max (V ; δ, τ) is inceasing) and C max inceases. The economic eason is that with a highe payout ate less assets emain in the fim, thus a lowe debt issuance can be suppoted, giving an insight fo a potential influence on agency costs. And ou analysis shows that inceasing payouts poduces impotant effects on agency costs. As [6] suggest, afte debt is issued, equity holdes can potentially extact value fom debt holdes by inceasing the iskiness of the fim, since E σ > 0, while the opposite happens fo debt holdes, D σ < 0 : a highe volatility deceases debt value unde (29). The asset substitution poblem still exists with payouts. Moeove, payouts can stongly modify the magnitude of potential agency costs aising fo the model. Thee is an incentive compatibility poblem between debt holdes and equity holdes: once debt is issued, shaeholdes will benefit fom an incease in the iskiness of the fim (i.e. deciding to invest in iskie activities/pojects), though tansfeing value fom debt to equity (also if equity is not exactly an odinay call option, see [9] footnote 29), and these incentives ae highe fo equity holdes as δ ises, fo each fim s cuent assets value V V B. Analyzing in detail fom both analytic and economic point of view the asset substitution poblem is beyond the scope of this pape, nevetheless we found some inteesting insights. 15

32 E σ, D σ E σ, δ =0 D σ, δ =0 E σ, δ =0.04 D σ, δ = V Figue 5: Effect of a change in σ on equity and debt values. This plot shows the behavio of E σ (dashed line) and D (solid line) as function of fim s cuent assets value V, fo a fixed level of coupon σ C =6.5. We conside =0.06, σ =0.2, α =0.5, τ =0.35 and two diffeent levels of δ =0, Equity value is given by (19), debt value by (20). Fo each level of the payout ate δ, equity and debt functions have a diffeent suppot, i.e. V V B(C; δ, τ), with V B(C; δ, τ) given by (13). As a consequence, each cuve epesenting equity and debt sensitivity w..t. σ is plotted in its own suppot. In Figue 5 we analyze which is the effect of a change in the volatility level σ on both equity and debt sensitivity to σ as payouts incease. Following [9] we study the magnitude of this effect as function of V. Consideing two diffeent levels of the payout ate δ =0, 0.04 we compute E σ, D σ and analyze them fo diffeent values of V V B.The fim will bea potential agency costs fo the ange of values V such that E D σ > 0, σ < 0, meaning unde (29). Altenative measues of potential agency costs ae: i) how wide is the ange of V such that the poblem exists; ii) the magnitude of the gap between E σ, D σ (see also [9] footnote 30). We efe to agency costs only as potential, since we ae assuming that capital stuctue decisions, once made, ae not subsequently changed. Obseve that δ D σ is not a monotonic function (diffeently fom equity sensitivity), meaning that once again, the influence of payouts is stongly connected with the actual distance to default faced by the fim. Payouts have an influence also on the shape of V D σ.inline with [9], the incentives fo inceasing isk ae positive fo both equity and debt holdes when bankuptcy is imminent, i.e. when the distance to default appoaches zeo. Figue 5 shows that intoducing payouts inceases the ange of V fo which potential agency costs exist, thus ising the poblem of advese incentives between debt holdes and equity holdes and so potential agency costs fo the fim. This ange is V > 66 in case δ = 0 and V > 60 fo a 4% payout, meaning an incease of this ange aound 6% of cuent asset s value in ou base case. The ange always stats when cuent assets value is such that debt sensitivity w..t. σ is null. And this will happen at point V satisfying constaint (29) 16

33 as equality, i.e. fo V := V B (C, δ, τ)e τ+α(1 τ) 1+λ(τ+α(1 τ)), with V B (C; δ, τ) given in (13). Notice that V deceases with payouts in ou base case, but we stess that the magnitude of this eduction can be quantitatively diffeent depending on all paametes involved in the capital stuctue (i.e. bankuptcy costs, copoate tax ate, isk fee ate, volatility), since they ae all inteelated vaiables. To give an idea of this, consideing a lowe copoate tax ate τ =0.15 and a payout δ =0.05 the incease (w..t. the case δ = 0) in the ange of V fo which potential agency costs exist is about 10% of initial assets value V = δ =0 δ = E σ D σ V Figue 6: Effect of a change in σ on equity and debt values. This plot shows the magnitude of the gap between E D and as function of fim s cuent assets value V, fo a fixed level of coupon C =6.5. σ σ We conside =0.06, σ =0.2, α =0.5, τ =0.35 and two diffeent levels of δ =0, Equity value is given by (19), debt value by (20). Fo each level of the payout ate δ, equity and debt functions have adiffeent suppot, i.e. V V B(C; δ, τ), with V B(C; δ, τ) given by (13). As a consequence, each cuve epesenting equity and debt sensitivity w..t. σ is plotted in its own suppot. Figue 6 analyses the magnitude of the gap between E D σ > 0 and σ < 0 inside the ange whee the conflict exists. What is inteesting is that agency costs ae elatively flat (o slightly deceasing) only fo V being between appoximately As the distance to default inceases (o deceases), the magnitude of the gap between the two sensibilities hadly inceases, i.e. the incentive compatibility poblem becomes moe difficult to solve with a geate payout δ. And this is still tue also consideing a pue Modigliani-Mille famewok with zeo tax benefits and bankuptcy costs, whee E σ = D σ. In such a case the conflict appoaches a zeo sum game as found in [9] and the intoduction of payouts inceases its magnitude (see Figue 7). As obseved peviously, the payout influence on 17

34 δ = δ =0 E σ, D σ 0 50 δ =0 100 δ = V Figue 7: Effect of a change in σ on equity and debt values when α = τ =0. This plot shows the behaviou of E D (dashed line) and (solid line) as function of fim s cuent assets value V, fo a σ σ fixed level of coupon C =6.5. We conside =0.06, σ =0.2, α =0.5, τ =0.35 and two diffeent levels of payout ate δ =0, Equity value is given by (19), debt value by (20). Fo each level of the payout ate δ, equity function has a diffeent suppot, i.e. V V B(C; δ, τ), with V B(C; δ, τ) givenby(13). Asa consequence, each cuve epesenting equity sensitivity w..t. σ is plotted in its own suppot. agency costs stongly depends on fim s activities value V, thus on the distance to default. Ou base case allows to show that as the distance to default inceases o deceases, the payout can stongly contibute to ise aveage fim isk, by ising the magnitude of the gap between equity and debt sensitivity w..t. σ, inceasing the incentive incompatibility poblem between equity and debt holdes. This will poduce a diect effect on cedit speads, inceasing them when aveage fim isk is highe in ode to compensate debt holdes. Finally, a highe coupon C has a positive effect on the inteest ate paid by isky debt, yield, defined as C R(C; δ, τ) := D(C; δ, τ), (30) with D(C; δ, τ) given in (20). Actually yield R(C; δ, τ) is inceasing as function of C and deceasing as function of τ. A highe copoate tax ate τ will educe both yield R(C; δ, τ) and yield spead R(C; δ, τ) by ising debt (loweing the endogenous failue level V B (C; δ, τ), see also [7] τ D(C;δ,τ) footnote 22): this follows by the elation τ R(C; δ, τ) = C 18 (D(C;δ,τ)) 2.

35 Poposition 3.6 The function yield R(C; δ, τ) defined in (30) is inceasing w..t. δ. Poof As D is an inceasing function of λ and λ R = C λd, we obtain that R is a D 2 deceasing function of λ. Thus by Remak 2.2 R is inceasing w..t. δ. While the intoduction of δ educes debt, the opposite happens fo yield speads, which ae highe. This is due to two main easons: fist, as δ inceases, less assets emains in the fim, thus inceasing the likelihood of default. Secondly, intoducing the payout ate poduces a diect effect of ising the aveage fim isk, as shown befoe. As a consequence, debt holdes must be compensated with highe yield speads. Obseve that R can be expessed as: R :(C; δ, τ) R C, (31) V with R C = 1 V C λ (1 τ) V λ λ +1 λ 1 λ 1 (1 α)(1 τ). (32) λ +1 As inceasing function of the atio C V,thetem R C V epesents the isk-adjustment facto paid to debt holdes. Intoducing payouts ises debt s volatility as Figue 8 shows: as a consequence, the compensation paid by the fim to debt holdes fo the isk assumed must be highe, and this is why R C V inceases σ E δ σ D δ Figue 8: Volatility of Equity and Debt. This plot shows the behavio of equity and debt volatility σ E, σ D as function of payout ate, fo a fixed level of coupon C =6.5. We conside V =100, =0.06, σ =0.2, α =0.5, τ =0.35. By Ito calculus fomula we deive the behavio of equity and debt volatility σ E, σ D. 19

36 When V, yield spead R appoaches to, D σ 0 and debt becomes isk fee: this is exactly as in [7], since in such a case, the hypothesis of debt being edeemed in full becomes quite cetain and this is not affected by the choice of the payout ate δ. Intoducing payouts will instead ise R in case V V B, consideing a pue Modigliani- Mille [12] famewok: if α = τ = 0, as V appoaches the failue level V B, R (1 + 1 λ ), while in case δ =0wehaveR σ2. If thee ae no bankuptcy costs o tax benefits of debt, intoducing payouts allows to have yield exceeding the isk fee ate by moe than 1 2 σ2,since λ > 1 2 σ2, thus poviding to bondholdes a highe compensation fo isk, if compaed to the case δ = Total Value The total value of the fim v(c; δ, τ) in (21) is a concave function of coupon C and an inceasing function of copoate tax ate τ. The following poposition shows the behavio of the total value of the fim with espect to the payout ate δ. Poposition 3.7 The total value v(c; δ, τ) defined in (21) is deceasing w..t. δ if with V B (C; δ, τ) given in (13). V > V B (C; δ, τ)e Poof The behavio of v(c; δ, τ) is the one of the following: G : λ (τ + λ(τ + α(1 τ)) τ+α(1 τ) τ+λ(τ+α(1 τ)) (33) 1 C(1 τ) λ λ, λ +1 V λ +1 which satisfies: G G (λ) =h(λ) = τ + α(1 τ) C(1 τ) τ + λ(τ + α(1 τ)) + log λ = V λ +1 (34) = τ + α(1 τ) τ + λ(τ + α(1 τ)) + log V B(C; δ, τ). V (35) Actually, the behaviou of G is given by the sign of h(λ), with log V B(C;δ,τ) V < 0. Thus, in case V > V B (C; δ, τ)e τ+α(1 τ) τ+λ(τ+α(1 τ)) we have h(λ) < 0 and the total value of the fim is deceasing w..t. δ. τ+α(1 τ) τ+λ(τ+α(1 τ)) > 0 and 20

37 The economic intuition behind constaint (33) being satisfied, is that if the initial value of the fim V is sufficiently geate 2 than the failue level, δ v(c; δ, τ) is deceasing due to the fact that intoducing payouts makes bankuptcy moe likely, since less assets emain in the fim. 3.4 Pobability of Default An impotant issue to take into account in this famewok is the pobability of the fim of going bankuptcy. In ode to analyze the payout influence on default ates, we now conduct this study unde the histoical measue as follows. The cumulative pobability F (s) of going bankuptcy in the inteval [0,s) is given by: b γs b + γs F (s) =N σ + e 2bγ/σ2 N s σ, (36) s whee N( ) is the nomal cumulative pobability function, b := log V B V and γ := µ A δ 1 2 σ2,wheeµ A := +π A and π A being the histoical asset isk pemium (see also Equation (15) in [9]). As obseved in [9], footnote 27, the pobability of default is quite dependent on the dift assumed fo the pocess V t. As a consequence, the payout ate δ has an actual influence on the pobability of going bankuptcy F (s). We show qualitative behavio of F (s) afte the intoduction of payouts in Figue 9, while Table 2 povides numeical esult. We study cumulative default pobability fo π A =7.5% (as in [9]) and π A = 5%, fo a ange of payout values. Looking at cumulative pobability ove a peiod between 10 and 25 yeas shows that a highe payout makes debt iskie, though ising the likelihood of default. This effect is moe ponounced as the hoizon we conside is longe: going fom 10 to 25 yeas the incease in the pobability of default has a geate magnitude. And this effect is highe if the asset isk pemium educes. As exteme cases we conside that when π A =7.5%, looking at a 10-yeas hoizon the pobability of default is 2.249% in case δ = 0, while it becomes 2.405% when δ = 4%, thus the incease is 7% of 2.249%. When we conside F (25), the pobability of default is 2.537% with no cash payouts. The case δ = 4% shows a ising in the default pobability up to 3.857%, meaning an incease of aound 52% of its stating value 2.537%. A eduction in the isk pemium futhe inceases this gap: when π A = 5%, fo instance, the 25-yeas pobability of default goes fom 3.857% to 9.686%, meaning a ising of aound 150% of the 10-yeas value. τ+α(1 τ) τ+λ(τ+α(1 τ)) 2 Obseve that at point V = V B(C; δ, τ)e satisfying (33) as equality, we have E = D. λ λ Moeove unde (33), constaint (29) is always satisfied, meaning we ae inside the ange in which agency costs exist. 21

38 0.04 F(s) δ =0 δ =0.01 δ =0.02 δ =0.03 δ = Yeas(s) Figue 9: Cumulative Pobability of Default. This plot shows the cumulative pobability of default F (s) ove the peiod (0,s], consideing Equation (36). The plot shows F (s) fo diffeent values of the payout ate δ =0, 0.01, 0.02, 0.03, Base case paametes values ae V =100, =0.06, σ =0.2, α =0.5, τ =0.35; the coupon is chosen optimally. We conside an histoical asset isk pemium π A =7.5%. π A =7.5% F (10) F (15) F (20) F (25) δ = δ = δ = δ = δ = π A = 5% F (10) F (15) F (20) F (25) δ = δ = δ = δ = δ = Table 2: Cumulative Default Pobability. The table shows cumulative default pobabilities fo two diffeent values of histoical assets isk pemium π A analyzing the influence of the payout ate δ on the pobability of going bankuptcy F (s) given by Equation(36). 4 Optimal Capital Stuctue We now conside the second-stage optimization poblem, meaning equity holdes have to find the optimal amount of debt (coupon payments) which maximizes the total value of 22

39 the fim. This will be done, taking into account the esult of the fist-stage optimization poblem as a elation between coupon payments C and endogenous failue level V B. The second stage-optimization poblem is: max C v(v,v B(C; δ, τ),c), (37) whee the failue level V B is eplaced by its endogenous value V B (C; δ, τ) given by Equation (13). Finally, solving (37) is equivalent to optimizing the total value of the fim v(c; δ, τ) given by Equation (21) v :(C; δ, τ) V + τc with espect to the coupon C. C τ + α λ(1 τ) C(1 τ) λ +1 V λ λ +1 The application C (C; δ, τ) is concave since A := τ + α λ(1 τ) (λ+1) > 0 and λ > 0, theefoe the following esult holds. λ, Poposition 4.1 Fo any fixed δ, τ, theoptimalcouponis: C (V ; δ, τ) = V (λ + 1) λ(1 τ) τ λ(τ + α(1 τ)) + τ 1 λ. (38) We obseve that C (V ; δ, τ) <C max (V ; δ, τ), whee C max is defined in (27). Moeove, this max-coupon satisfies V > (1 τ)cmax λ λ+1. The optimal coupon C (V ; δ, τ) is an inceasing function of τ. In fact C (V,δ, τ) 1 = τ 1 τ + α C (V,δ, τ) > 0. τ(τ(1 + λ)+αλ(1 τ)) Replacing (38) in (13) yields the optimal failue level VB(V ; δ, τ) =V τ λ(τ + α(1 τ)) + τ 1 λ. (39) Remak 4.2 In case δ =0, we have λ = 2 σ 2 and we get the same esults as in [7]: V B(V ;0, τ) =V τσ 2 σ 2 2. (40) 2(τ + α(1 τ)) + τσ 2 23

40 7 6 5 C * τ δ Figue 10: Optimal Coupon. This plot shows the behavio of optimal coupon C (V ; δ, τ) as function of the payout ate δ and copoate tax ate τ. WeconsideV =100, =0.06, σ =0.2, α =0.5. Poposition 4.3 Conside the optimal failue level (39). The following esults hold: i) δ VB (V ; δ, τ) is a deceasing function; ii) τ VB (V ; δ, τ) is an inceasing function. Poof i) Using Remak 2.2, it is enough to study the following function F : λ 1 τ + λ(τ + α(1 τ)) λ log. τ Taking the deivative w..t. λ, we obtain: F (λ) = 1 λ 2 log (1 + z) z, 1+z with z := λ 1+α 1 τ τ.itissufficient to study the sign of G : z log (z + 1) z 1+z, with z 0, 2 σ 1+α 1 τ 2 τ.sinceg(0) = 0 and G (z) 0, fo any z the function F is inceasing. Finally δ VB (V ; δ, τ) is deceasing. ii) The esult follows by: VB (V ; δ, τ) := V B (C (V ; δ, τ)) = V B (C (V ; δ, τ)) C (V ; δ, τ) τ τ C > 0. τ 24

41 We now completely descibe the optimal capital stuctue of the fim. Let y := λ(τ + α(1 τ)), the following holds: C = V (1 + λ) λ(1 τ) VB = 1 τ λ V y + τ D V τ = λ(1 τ) y + τ τ E = V 1 y + τ R = (1 + λ) λ + y(1 τ) y+τ τ y + τ 1 λ 1 λ λ + 1 λ 1 λ y(1 τ) y + τ 1+λ + τ y + τ (41) (42) (43) (44). (45) Intoducing a payouts into (39) has an actual influence: δ VB (V ; δ, τ) is a deceasing function fo any value of τ, while τ VB (V ; δ, τ) is inceasing fo any value of δ. Similaly optimal coupon C (V ; δ, τ) given by (38) will benefit fom a highe copoate tax ate and decease w..t. δ, as Figue 10 shows. δ C D R R E V B v L % % % % % % % % % % % % Table 3: Effect of payout ate δ on all financial vaiables at the optimal leveage atio. Base case paametes values: V =100,σ =0.2, τ =0.35, =0.06, α =0.5. The fist ow of the table shows Leland s famewok with his base case paametes values, in paticula with δ =0. L is in pecentage (%), R,R ae in basis points (bps). 25

42 The capital stuctue of the fim is stongly affected by payouts: Tables 3 and 4 show the behavio of all financial vaiables at optimal leveage atio, when the paamete δ moves away fom zeo. Consistent with ou base case, these tables epot both numeical esults and a qualitative analysis. Columns 6 and 7 of Table 3 show equity and debt values when the coupon C is chosen to maximize the total value of the fim. Optimal equity value inceases with a highe payout, while optimal debt deceases (ecall that δ does not include cash flows elated to debt financing). These two effects have a diffeent magnitude: δ influence on debt is in fact geate than δ influence on equity, as a consequence the optimal total value of the fim, v := D + E, educes. Now conside optimal leveage atio, defined as L := D v. The last column of Table 3 shows that inceasing payouts deceases optimal leveage atio L, and this effect is moe ponounced as δ is highe. Consideing ou base case, optimal leveage can dop fom appoximately 75% to 66.75% passing fom δ =0toδ = A highe payout δ will bing to a lowe total value of the fim v, since a lowe leveage atio can be suppoted when less assets emain in the fim, as Leland suggests [8]. Optimal leveage atios can be stongly affected by payouts, but thei influence is elated to the iskiness of the fim. Figue 9 shows optimal leveage as function of δ fo thee diffeent values of σ. Fo each level of δ, optimal leveage atio deceases as σ ises. Obseve also that L is deceasing w..t. δ fo each level of σ, but this eduction in optimal leveage is lowe as the iskiness of the fim ises σ=0.1 σ=0.2 σ= L * δ Figue 11: Optimal leveage atio as function of δ. This plot show optimal leveage atio L as function of δ fo thee diffeent levels of volatility σ. WeconsideV =100, =0.06, α =0.5, τ =0.35. Recall fom Section 2 that highe payouts ise the pobability of going bankuptcy given by (36): as a consequence, when δ ises, optimal yield R and optimal yield spead 26

43 α=0.2 α=0.5 α=0.8 R * δ Figue 12: Optimal speads as function of δ. This plot show optimal speads R as function of δ fo thee diffeent levels of α. WeconsideV =100, =0.06, σ =0.2, τ =0.35. R incease. Two ae the main easons: fist, less assets emain in the fim, thus bankuptcy is moe likely; secondly, the aveage fim isk is highe, thus debt holdes must be compensated fo the highe isk assumed. Leland [7] obseves that as bankuptcy costs ise, supisingly optimal yield spead educes when the coupon is chosen optimally. This is due to the fact that a highe α will decease the optimal coupon. Ou analysis shows that when payouts ae intoduced, optimal yield speads ae still deceasing w..t. α fo each level of δ (see Figue 10). Moeove, payouts influence on optimal yield speads is highe as α educes. Consideing as exteme cases δ = 0 and δ =0.055, optimal yield spead ises fom bps (basis points) to bps with α =0.8, fom bps to bps with α =0.5 and fom bps to bps with α =0.2. Financial Vaiables C D R R E V B v L Sign of change as δ < 0 < 0 > 0 > 0 > 0 < 0 < 0 < 0 Table 4: Effect of payout ate δ on all financial vaiables at the optimal leveage atio. The table shows fo each financial vaiable the effect of inceasing δ. Consideing ou base case, we epot the sign of change in each vaiable as the payout moves away fom 0. We now tun to the study of tax deduction τ influence on all financial vaiables. Tables 5 and 6 show the behavio of all financial vaiables at optimal coupon level C fo diffeent values of the copoate tax ate τ when a payout δ =0.01 is intoduced. As the tax deduction inceases, all financial vaiables, except equity, will benefit fom this. This esult extends Table II in [7] since it allows fo a payout δ > 0. Concening the optimal failue level VB (V ; δ, τ) we obseve that by (39) the copoate tax ate τ has no influence 27

44 on the optimal failue level at optimal leveage atio in case α = 0. The same holds fo equity value E at optimal leveage atio: a change in the copoate tax ate has no effect on equity value in the absence of bankuptcy costs. τ C D R R E V B v L % % % % % Table 5: Effect of a change in the copoate tax ate τ on all financial vaiables at the optimal leveage atio. This table consides a case in which a payout δ is intoduced (δ =0.01) and studies the effect of a change in the copoate tax τ. R,L ae in pecentage (%), R is in basis points (bps). Financial Vaiables C D R R E V B v L Sign of change as τ > 0 > 0 > 0 > 0 < 0 a > 0 a > 0 > 0 a No effect if α =0. Table 6: Effect of copoate tax ate τ on all financial vaiables at the optimal leveage atio. The table shows fo each financial vaiable the effect of inceasing τ fo fixed δ = Consideing ou base case, we epot the sign of change in each vaiable as the copoate tax ate inceases. 5 Conclusions Intoducing the payout ate has an actual influence on all financial vaiables detemining the capital stuctue of the fim. We ae consideing δ as the total payout ate to all secuity holdes excluding cash flows elated to debt issuance, see [7]. Fo an abitay coupon level C, an inceasing payout will ise equity and educe both debt and total value of the fim, making bankuptcy moe likely since less assets will emain in the fim. Payouts will have a diect and indiect influence on the endogenous failue level V B (C; δ, τ) chosen by equity holdes: this default bounday will be lowe as payouts incease (diect effect). Adding to this, ou esults suggest that payouts stongly modify the influence of all paametes, τ,c,σ 2 on the endogenous failue level, though affecting the magnitude of thei influence (indiect effect). Potential agency costs aising fom the model due to the asset substitution poblem ae not independent of the choice of δ. They ae affected in two main diections. Equity holdes will always have geate incentives to incease the iskiness of the fim when the payout ises, fo each value V bigge than the endogenous 28

45 failue level, meaning that equity sensitivity to σ is an inceasing and positive function of δ. At fist, the ange of values V fo which the conflict of inteests between equity and debt holdes exists inceases with payouts. Secondly, the gap between equity and debt sensitivity to σ inceases damatically as function of the distance to default when δ is intoduced (only fo a small ange of values agency costs ae elatively flat). Concening optimal capital stuctue Leland s [7] esults show too high leveage atios (and/o too low yield speads): assuming payouts δ > 0 allows to ovecome this, poviding lowe optimal leveage atios and highe yield speads. Leveage atio educes because when payouts incease, less assets is available inside the fim, thus only a lowe leveage can be suppoted by the fim. Yield speads inceases with payouts due to two main easons: i) bankuptcy is moe likely; ii) aveage fim isk ises, since both debt and equity sensitivity to σ ise (absolute value). Debt becomes iskie and moe sensible to a change in the volatility of the fim, consequently debt holdes must be compensated fo this. In this pape the payout δ is consideed exogenous and constant though time: elaxing this assumption and making it dependent on the coupon level will allow to analyze the payout decision as an endogenous one, tying to study it as the solution of an optimal payout policy. Moeove, making the payout ate endogenous will povide an inteesting famewok in which asymmetic infomation between equity holdes and debt holdes can be intoduced as long as a detailed analysis of the asset substitution poblem, as idea fo futue eseach. Refeences [1] CHEN N., KOU S.G. (2009), Cedit Speads, Optimal Capital Stuctue, and implied volatility with endogenous default and jump isk, Mathematical Finance, 19:3, [2] DOROBANTU D. (2007), Modélisation du Risque de Défaut en Entepise, Ph.D. Thesis, Univesity of Toulouse, Diana.pdf. [3] DOROBANTU D., MANCINO M.E., PONTIER M. (2009), Optimal Stategies in a Risky Debt Context, Stochastics An Intenational Jounal of Pobability and Stochastic Pocesses, 81:3, [4] El KAROUI N. (1981), Les Aspects Pobabilistes du Contôle Stochastique, Lectue Notes in Mathematics 876, pp Spinge-Velag, Belin [5] KARATZAS I., SHREVE S. (1988), Bownian Motion and Stochastic Calculus, Spinge, Belin, Heidelbeg, New Yok [6] JENSEN M., MECKLING W. (1976), Theoy of the Fim: Manageial Behavio, Agency Costs, and Owneship Stuctue, Jounal of Financial Economics 4,

46 [7] LELAND H.E. (1994), Copoate Debt Value, Bond Covenant, and Optimal Capital Stuctue, The Jounal of Finance, 49, [8] LELAND H.E. (1998), Agency Costs, Risk Management, and Capital Stuctue, Woking Pape. [9] LELAND H.E., TOFT K.B. (1996), Optimal Capital Stuctue, Endogenous Bankuptcy and the Tem Stuctue of Cedit Speads, The Jounal of Finance, 51, [10] MERTON R. C. (1973), A Rational Theoy of Option Picing, Bell Jounal of Economics and Management Science, 4, [11] MERTON R. C. (1974), On the picing of Copoate Debt: the Risk Stuctue of Inteest Rates, The Jounal of Finance, 29, [12] MODIGLIANI F., MILLER M. (1958), The Cost of Capital, Copoation Finance and the Theoy of Investment, Ameican Economic Review, 48, [13] UHRIG-HOMBURG M. (2005), Cash Flow Shotage as an Endogenous Bankuptcy Reason, Jounal of Banking and Finance, 29,

47 II Tax Benefits Asymmety

48

49 Copoate Debt Value with Switching Tax Benefits and Payouts Flavia Basotti Maia Elvia Mancino Monique Pontie Dept. Stat. and Applied Math. Dept. Math. fo Decisions Inst. Math. de Toulouse (IMT) Univesity of Pisa, Italy Univesity of Fienze, Italy Univesity of Toulouse, Fance Abstact This pape analyzes a stuctual model of copoate debt in the spiit of Leland model [17] within a moe ealistic geneal context whee payouts and asymmetic taxcode povisions ae intoduced. We analytically deive the value of the tax benefit claim in this context and study the joint effect of tax asymmety and payouts on optimal copoate financing decisions. Results show a quantitatively significant impact on both optimal debt issuance and leveage atios, thus poviding a way to explain diffeences in obseved leveage acoss fims. Keywods: stuctual model; copoate debt; default; optimal stopping; tax benefits of debt. 1 Intoduction The capital stuctue of a fim has been analyzed in tems of deivatives contacts since Meton s wok [23]. The capital stuctue decision is a complex issue due to many factos enteing in the deteminacy of copoate financing policy. Riskiness of the fim, bankuptcy costs, payouts, inteest ates and taxes ae impotant factos to take into account when defining the capital stuctue of a fim. Accoding to Modigliani - Mille theoem [24] the activities of the fim ae assumed to be independent fom the financial stuctue when no taxes and no bankuptcy costs ae assumed. When we conside a fim subject to default isk in a famewok with bankuptcy costs and taxes, the ownes of the fim can choose optimally the capital stuctue, whee optimally means the capital stuctue which maximizes the total value of the fim. The economic intuition is that deciding the optimal capital stuctue means choosing which will be the best allocation of values (effective and potential) belonging to the fim. All this can poduce vey diffeent fim s values due fo example to the tade-off existing between some inteacting vaiables (i.e. debt and tax 1

50 benefits). In paticula copoate tax ate ae an impotant deteminant of optimal capital stuctue, as ealy ecognized by [24] and obseved in moe ecent empiical studies (see [14, 27]). In this pape the model poposed by [17] is extended by means of a switching (even debt dependent) in tax savings and the intoduction of a company s assets payout atio (as in [3, 4]). We pefom a quantitative study of the effects of both components on the optimal capital stuctue of the fim, obtaining analytical esults in most cases. Ou findings show that the combined effect of tax asymmety and payouts poduces pedicted optimal leveages atios which ae moe in line with histoical noms (significantly educed w..t. the ones in [17]) and empiical evidence. As a matte of fact, tax code povisions can vay acoss nations, acoss industies, acoss activity secto in which the fim is opeating in, and also acoss time. Suppose fo example the tax code being modified to encouage investments (see [27] footnote 1, [26]). The main empiical insight when analyzing tax influence on capital stuctue decisions is that leveage atios ae highe fo fims facing highe copoate tax ates (as it is shown in [30]). The economic insight we want to analyze is how asymmety in tax-code povisions ae incopoated in fim s financing decisions and moeove the quantitative impact of this on optimal debt issuance and leveage decisions. The total value of the fim is ealized fom both equity and debt. Equivalently, it can be achieved by consideing fim s activities value in the unleveed case (meaning when no debt is issued) plus tax benefits of debt, less bankuptcy costs. We assume bankuptcy being detemined endogenously by the inability of the fim to aise sufficient equity capital to cove its debt obligations. Following [17] we conside an infinite hoizon assuming that the fim issues debt and debt is pepetual. Debt pays a constant coupon pe instant of time and this detemines tax benefits popotional to coupon payments. A payout ate δ is also intoduced as in [3, 4]. On the failue time, agents which hold debt claims will get the esidual value of the fim (because of bankuptcy costs), and those who hold equity will get nothing (the stict pioity ule holds). Stuctual models assume constant copoate tax ates, thus tax benefits of debt ae constant though time. The oiginal assumption in [17] is that the fim has deductibility of inteest payments fo all fim s assets values above the failue level, poducing a constant tax-shelteing value. Leland agues that default and leveage decisions might be affected by non constant copoate tax ates, because a loss of tax advantages is possible fo low fim values. Thus in [17] section VI.A the autho suggests that, when assets value deceases, it is moe likely that pofits will be lowe than coupon payments and the fim will not be able to fully benefit tax savings. The empiical analysis of [14] confims that the copoate tax schedule is asymmetic, in most cases it is convex. The quantitative impact of this asymmety on the optimal default bounday and the leveage atio is consideed in [27] unde the hypothesis of a piecewise linea tax function when the state vaiable is the opeating income; the simulation study theein shows that the effect of tax asymmety on the optimal leveage atio is quantitatively significant, while it is lowe on the optimal default bounday. Futhe [26] examines the elation between tax convexity and investment in the pesence of a stictly convex (quadatic) tax function. 2

51 In this pape we extend Leland model by incopoating the possibility of two diffeent copoate tax ates, namely τ 1, τ 2 and net cash outflows as in [3, 4]. We conside as state vaiable fim s cuent assets value. The switching fom a copoate tax ate to the othe is detemined by the fim value cossing a citical level. We conside two altenative famewoks: at fist, the switching baie is assumed to be a constant exogenous level; then we analyze a moe ealistic scenaio in which this level depends upon the amount of debt the fim has issued. In fact, as pointed out in [17], unde U.S. tax codes, a necessay condition equied to fully benefit tax savings, is that the fim s EBIT (eanings befoe inteest and taxes) must cove payments equied fo coupons. We obtain an explicit fom fo the tax benefit claim, which allows us to study monotonicity and convexity of equity function, to find the endogenous failue level analytically in the geneal case with payouts and to pove its existence and uniqueness in the geneal case. Futhe, exploiting the lineaity of the smooth pasting condition with espect to the coupon, we ae able to study the optimal capital stuctue of the fim. Ou appoach diffes fom [17] since we solve the optimal contol poblem as an optimal poblem in the set of passage times; the key method is the Laplace tansfom of the stopping failue time [1], [13], [16]. We intoduce paamete θ := τ 2 τ 1 as a measue of the degee of asymmety. Ou study shows that tax asymmety inceases the optimal failue level and educes the optimal leveage atio, with a moe ponounced effect on optimal leveage atios, thus confiming esults in [27]. Nevetheless, we find that, as fa as the magnitude is concened, intoducing a payout poduces an even moe significant eduction in optimal leveage atios. Thus the joint effect of tax asymmeties and payouts dops down optimal leveage to empiically epesentative values and moeove seems to be a flexible way to captue diffeences among fims facing diffeent tax-code povisions. Fo example obseving fims belonging to diffeent activity sectos, this could be a way to explain diffeences in obseved capital stuctue decisions, mainly in leveage atios. The analysis developed in [3, 4] showed that intoducing payouts in a stuctual model with a unique copoate tax ate has the effect of educing both optimal leveage atio and optimal failue level. In the pesent pape we find that this eduction in both optimal leveage and optimal failue level inceases as the degee of asymmety of the tax schedule ises, meaning as the diffeence between the two copoate tax ates is highe. We study both the impact of asymmety in tax benefits on optimal capital stuctue compaed to what happens unde a flat tax schedule (i.e. a unique constant tax ate) as benchmak model, but also how these decisions change as the asymmety vaies (i.e. fo example if the switching baie moves), showing two altenative appoaches to measue the impact of asymmety on copoate decisions. Finally, the maximum total value of the fim value, as fa as optimal debt issuance decease both with asymmety and payouts, since less assets emain in the fim (due to payouts) and debt becomes less attactive (due to a potential loss of value). Futhe we study optimal capital stuctue when the switching baie consideed an inceasing and linea function of the coupon level, in ode to epesent a moe ealistic famewok in which EBIT is consideed as a baie detemining a potential loss in coupon payments deductibility. In 3

52 such a case a highe pofit is needed in ode to fully benefit fom tax savings. Given a payout ate, as the optimal coupon deceases fo highe degees of asymmety, then also the optimal switching baie deceases: this tade-off concening the potential tax benefits loss leads to empiically epesentative value in pedicted leveage atios. One may wonde why payouts and asymmety ae consideed in the same model and why thei joint effect is quantitatively significant. Notice that δ and θ have a deeply diffeent natue fom an economic point of view: if it is tue that in ou model δ is exogenously given, we can also ecognize that it can be patly modified o chosen by the fim, even when it is not a esult of an endogenous decision (i.e. even if it does not depend on coupon payments, as in ou case). What we mean is that, opposite to this, the copoate tax schedule is instead imposed to the fim by an extenal authoity. Moeove, the copoate tax schedule could be vey diffeent depending fo example on the secto in which the fim is opeating in. Thus, we think that analyzing the joint effect of these two factos could be an inteesting and flexible way to analyze and impove empiical findings inside a stuctual model of cedit isk allowing to explain why a high dispesion in obseved leveage atios exists. We think that the joint effect in quantitatively significant since this model captues insights which ae intenal (payouts) and extenal to the fim (tax asymmety). Factos appaently vey fa can poduce a quantitative joint influence, as in ou case. The pape is oganized as follows. In Section 2 we intoduce the model whee a payout ate δ is intoduced and tax benefits ae not constant though time, allowing fo a possible switching in tax savings. We compute the tax benefit claim. In Section 3 the endogenous failue level is deived and the influence of tax asymmety on it is analyzed. The optimal capital stuctue when the switching baie is fixed and with debt dependent switching baie is achieved in Section 4. Section 5 concludes. Poofs ae in Appendix. 2 The Model In this section we intoduce a stuctual model of copoate debt in the spiit of [17]; nevetheless, ou model exhibits two diffeences: the model fo the fim s activities includes a paamete δ which epesents a constant faction of value paid to secuity holdes (e.g. dividends, see also [19, 4]), futhe, we conside a copoate tax schedule which is not flat, meaning we suppose the copoate tax ate being not unique and constant though time. We deive the value of the tax benefit claim in this famewok, following Leland [17] in modeling tax benefits of debt: asymmety in copoate tax code povisions becomes also asymmety in tax benefits of debt. The switching in tax benefits is due to asymmety in the copoate tax schedule which we suppose being based on the existence of two diffeent copoate tax ates. We assume the switching being detemined by fim s assets value cossing a specified baie, thus depending on cuent activities value a fim can face a diffeent copoate tax ate. We assume an infinite time hoizon, as in [17]. This is a easonable fist appoxima- 4

53 tion fo long tem copoate debt and enables us to have an analytic famewok whee all copoate secuities depending on the undelying vaiable ae time independent, thus obtaining closed foms. We conside a fim ealizing its capital fom both debt and equity. The fim has only one pepetual debt outstanding, which pays a constant coupon steam C pe instant of time 1. This assumption can be justified, as Leland suggests, thinking about two altenative scenaios: a debt with vey long matuity (in this case the etun of pincipal has no value) o a debt which is continuously olled ove at a fixed inteest ate (as in [19]). Bankuptcy is tiggeed endogenously by the inability of the fim to aise sufficient capital to meet its cuent obligations. On the failue time T, agents which hold debt claims will get the esidual value of the fim, and those who hold equity will get nothing. Following Leland [17] we do not conside pesonal taxes, thus we model the tax benefits claim as a deivative depending diectly on copoate tax ate povisions. Suppose that fim s activities value is descibed by pocess V t = Ve Xt, whee X t evolves, unde the isk neutal pobability measue, as dx t = δ 12 σ2 dt + σdw t, X 0 =0, (1) whee W is a standad Bownian motion, the constant isk-fee ate,, δ and σ > 0. The tem δv t epesents the fim s cash flow: we can think of it as an afte-tax net cash flow befoe inteest, since we only conside tax benefits of debt. When bankuptcy occus at stopping time T, a faction α (0 α < 1) of fim value is lost (fo instance payed to who takes cae of the bankuptcy pocedues), the debt holdes eceive the est and the stockholdes nothing, meaning that the stict pioity ule holds. The failue passage time is detemined when the fim value falls to some constant level V B. The value of V B is endogenously deived and will be detemined with an optimal ule late. Define T VB := inf{t 0: V t = V B } =inf{t 0: X t = log V B V }. In the spiit of Leland we assume that fom paying coupons the fim obtains tax deductions. Most stuctual models of cedit isk assume a setting in which tax benefits ae constant though time: fom paying coupons a fim obtains tax deductions popotionally to the coupon payment. In [17] the copoate tax ate τ is assumed to be constant; nevetheless, in Appendix A the autho deives the endogenous failue level in the case when thee ae no tax benefits fo the assets value going unde an exogenously specified level. The empiical analysis of [14] confims that the copoate tax schedule is asymmetic. Moeove [27] assumes the hypothesis of a piecewise linea tax function and epots a quite significant impact of this asymmety on the optimal leveage atio. These studies motivate ou extension of Leland s setting in the diection of a stuctual model with endogenous default bounday pesenting a moe geneal (even debt dependent) copoate tax schedule. 1 Instantaneous coupon payments can be witten as C := cf, whee F is face value of debt, supposed to be constant though time, as in Leland [17]. 5

54 Following [17, 23] we can always expess tax benefits as a claim on the undelying asset epesented by the unleveed value of the fim V. We now descibe the geneal scheme to detemine the value of this claim fo a given copoate tax schedule. On R + let τ be the copoate tax function and F the tax benefits function. Tax benefits of debt can be seen as the value of a claim on V t paying a continuous instantaneous dividend τ(v t )C if thee is no default and 0 in the event of bankuptcy. Let G t := e t F (V t )+ t 0 e s τ(v s )Cds, epesent the value of tax benefits at time t, discounted at the isk fee ate, plus cumulated tax-shelteing value of inteest payment τ(v t )C up to time t, discounted at. To avoid abitage oppotunities we impose such a pocess to be a local matingale unde the isk neutal pobability (see [23]). Assuming that F belongs to Cb 2,thenG is a tue matingale since the matingale pat of the pocess e t F (V t )is t 0 e s F (V s )σv s dw s, F is bounded and e t V t is squae integable on Ω R +. Then it follows using Doob theoem: Lemma 2.1 Fo any stopping time T the value of the tax benefits of debt is equal to: T F (V )=E V e T F (V T )+ e s τ(v s )Cds, (2) 0 whee the expectation is taken with espect to the isk neutal pobability and we denote E V [ ] :=E[ V 0 = V ]. In this section the asymmetic tax benefits schedule is specified though the intoduction of an exogenously given level of fim s assets value at which the tax deduction changes. We modify Leland [17] assumption about a unique constant level τ, consideing a piecewise linea model in which two diffeent copoate tax ates τ 1, τ 2 ae in foce. We assume that the deductibility of coupon payment geneates tax benefits fo all value V V B,butthese tax savings ae educed when V falls to a specified constant value. As the fim appoaches bankuptcy, it will lose tax benefits. The copoate tax ate switches fom τ 1 to τ 2 when the value of the fim eaches a cetain (exogenous) baie V S, theefoe the copoate tax function is equal to τ = τ 1 1 (VS, ) + τ 2 1 (VB,V S ) (3) depending on the fim s activities value V t staying upon the pescibed level V S. Obviously we assume V S >V B. The tax-shelteing value of inteest payments will not be constant though time: it will be τ 1 C fo V V S, and τ 2 C in case V B V < V S. We assume τ 2 τ 1, meaning loss of tax benefits below V S. 6

55 The fist passage time at V S is defined T VS =inf{t 0:V t = V S } =inf{t 0:X t = log V S }. (4) V Note that V TVB = V B and V TVS = V S, as the pocess V t is continuous. Using integal epesentation of tax benefits we can wite (2) as: TVS F (V )=E V e T T VB V S T VB F (V TVS T VB )+ e s Cτ(V s )ds. (5) 0 whee τ(v s ) is specified by (3). It is easily seen that in ode to compute (5), it is enough to have explicit fomulas fo the Laplace tansfom of a double boundaies passage time. The fomulas fo the Laplace tansfoms ae obtained in Appendix A, Popositions 6.2, 6.3. As V B is a failue level, we impose that tax benefits ae completely lost at failue, then the equied bounday condition is F (V B ) = 0. Finally we can state the following esult. Poposition 2.2 Suppose that the deduction tax function τ( ) is defined by (3), then the tax benefits claim F (V ) in (2) is equal to: F (V )= A 0 + A 1 V λ 1 + A 2 V λ 2 1 (VB,V S )(V )+ B 0 + B 2 V λ 2 1 (VS, )(V ), (6) whee and λ 1, λ 2 ae defined as A 0 = τ 2C A 1 = Cλ 2V λ 1 S (τ 2 τ 1 ), (8) (λ 1 λ 2 ) A 2 = A 1 V λ 1 B + τ 2C V λ 2 B, (9) B 0 = τ 1C, (10) λ 1 V λ 2 λ 1 S B 2 = A 2 + A 1. (11) λ 2 (7) with λ 1 = µ µ 2 +2σ 2 σ 2, λ 2 = µ + µ 2 +2σ 2 σ 2 (12) µ := δ 1 2 σ2. (13) 7

56 We comment the above esult in ode to undeline the effects of the tax asymmety assumption on the value of the tax benefit claim F (V ). Unde the hypothesis τ 2 τ 1, it holds A 1 > 0, A 2 < 0, B 2 < 0. Theefoe in both segments V V S and V B V < V S, the function F (V ) is stictly inceasing w..t. fim s cuent assets value V. Futhe, we note that the tax benefits value in the segment V V S is a stictly concave function of V, since B 2 is negative. If the tax ate wee always τ 2, both above and below V S, we would have F L (V,τ 2 )= τ 2C τ 2C V B λ 2 V λ 2, which coincides with the esult obtained in [17]. We now compae this value with F (V ) in ou famewok in case V B V < V S : F (V )=A 0 + A 1 V λ 1 + A 2 V λ 2, with A 0,A 1,A 2 given by Equations (7)-(9). Notice that now, fo all assets value below the switching baie (but obviously above the failue level V B ) the value of the claim F (V ) exhibits thee tems instead of two: while the constant tem τ 2C appeas in both models, in Leland famewok the tem depending on V λ 1 does not appea. The pesence of A 1 V λ 1 captues the effect of: i) payouts, since in case δ =0wehaveλ 1 = 1; ii) most impotant, it captues the possible switching fom τ 2 to a highe level τ 1, thus epesenting the value of a possible gain in tax savings, though coefficient A 1. This is why it is positive and inceasing w..t. V. Coefficient A 1 eflects exactly the asymmety in the copoate tax schedule; it is inceasing w..t. both τ 2 τ 1 and the switching baie V S.Coefficient A 2 is instead negative and depends on both the asymmety of the copoate tax schedule and the default event. Remak 2.3 In Appendix A [17] the autho poposes a stuctual model in which the instantaneous tax benefit is zeo, if the fim s value V falls unde a pescibed level. His appoach to the poblem is that of odinay diffeential equations with bounday conditions. We obseve that consideing the paticula case of δ =0and τ 2 =0, we ecove the same esult as in [17]. We ae now eady to complete the desciption of the copoate capital stuctue model. Applying contingent claim analysis in a Black-Scholes setting, given the stopping (failue) time T VB, the expession fo debt value is given by: D(V,V B,C)=E V TVB 0 e s Cds +(1 α)e T V B V B. (14) It is impotant to stess that the copoate tax schedule does not affect diectly debt value, thus equation (14) holds fo whateve copoate tax schedule. What is impotant is knowing that copoate tax povisions have an influence on capital stuctue decisions since issuing debt allows to have some tax savings, thus a potential incease in fim s value. 8

57 But they do not affect diectly debt value, which depends only on the coupon level and obviously, on the default event though bankuptcy costs α and the failue level V B. The effect of asymmety in the tax scheme will poduce an impact on debt value only though the choice of the endogenous failue level and thus on the optimal coupon equity holdes will choose. The total value of the fim v(v ) consists of thee tems: fim s assets value (unleveed), plus the value of the tax benefits claim F (V ) given in (6), less the value of the claim on bankuptcy costs: v(v,v B,C)=V + F (V ) E V [e T V B αv B ]. (15) Since an altenative but equivalent fomulation fo the total value of the fim is the sum of equity and debt values, finally it is possible to wite equity value as: E(V,V B,C)=v(V,V B,C) D(V,V B,C). (16) Equity holdes have to define the capital stuctue of the fim. In ode to do this, they have to chose both the endogenous failue level and the optimal amount of debt to issue. As stessed in [4] these ae inteelated decisions which can hadly be sepaated. Ou appoach to the poblem is to conduct the analysis in two stages: i) at fist we detemine the endogenous failue level, ii) then we find the optimal coupon, given the esult about the default bounday. 3 Endogenous Failue Level The aim of this section is to investigate the effects of intoducing a diffeent asymmetic copoate tax ate schedule on the endogenous failue level chosen by equity holdes. We will conduct a detailed analytical study consideing the influence of the copoate tax function in (3) on the fim s capital stuctue. The analysis is conducted fo a given and fixed level of coupon payments, namely C. 3.1 Failue Level with exogenous switching baie Given the value of the tax benefits claim F (V ) in (6) we can wite debt, equity and total value of the fim. Fist conside the debt function, which is not diectly affected by (3), since fo the moment we ae consideing V B as a constant level and C is fixed. Debt value can be seen as the sum of a isk-fee debt C/ plus a positive tem depending on the isk of default. Using Poposition 6.1, (14) becomes: D(V,V B,C)= C + (1 α)v B C VB V λ2. (17) 9

58 Futhe using Poposition 2.2, the total value of the fim defined in (15) is equal to τ2 C v(v,v B,C)=V + + A 1 V λ 1 + A 2 V λ 2 1 (VB,V S )(V ) (18) τ1 C λ2 + + B 2 V λ 2 VB 1 (VS, )(V ) αv B. V Finally fom (16) we obtain τ2 C E(V,V B,C)=V + + A 1 V λ 1 + A 2 V λ 2 1 (VB,V S )(V ) (19) τ1 C + + B 2 V λ 2 1 (VS, )(V ) C (V B C λ2 ) VB. V Equity function must eflect its natue of an option-like contact. Fo any C, wehave E(V B,V B,C) = 0 meaning that when V falls to V B thee is no equity to cove the fim s debt obligations, thus equity holdes will chose to default. We fist analyze the equity value fo V B V < V S : E(V,V B,C)=V (1 τ 2 ) C + A 1V λ 1 + A 1 V λ 1 B + C V λ2 (1 τ 2) V B, V B (20) whee A 1 defined in (8) is positive if τ 2 < τ 1. Obseve that the fist tem V (1 τ 2 ) C is nothing but equity value consideing a constant tax-shelteing value of inteest payments τ 2 C, unless limit of time (when thee is no isk of default). The tem A 1 V λ 1 is stictly elated to ou tax benefits assumption: since in this model the copoate tax ate can switch fom τ 1 to τ 2,coefficient A 1 captues this effect. It depends on both the switching baie V S and the diffeence between the two tax levels, it disappeas when consideing a unique copoate tax ate (i.e. τ 2 = τ 1, as in [17]), and achieves its maximum in case τ 2 = 0, meaning full loss of tax benefits below V S.Coefficient A 1 epesents the oppotunity-cost of V being in [V B,V S ) instead of [V S, ). In fact, consideing Leland [17] famewok, equity value is inceasing with espect to τ: ou assumption about the tax deductibility scheme modifies the unique constant τ intoducing an asymmety. Suppose V being in [V B,V S ]: this asymmety becomes an oppotunity, since τ 2 < τ 1. As a consequence, coefficient A 1 is positive and inceasing w..t. τ 2 τ 1, deceasing w..t. the switching baie V S. As the diffeence τ 2 τ 1 inceases, A 1 inceases too: the possible gain in tax benefits in the event of V = V S is highe. As V S becomes highe, coeteis paibus, the pobability of V eaching V S befoe eaching V B is educed, thus obtaining a gain in tax benefits becomes less likely and A 1 will be lowe. The last tem is exactly the option to default which is embodied in equity. Obseve that in this case, the option to default will compensate equity holdes also fo the tax deductibility asymmety, though the tem A 1 V λ 1 B. Theefoe it must hold A 1 V λ 1 B + C (1 τ 2) V B > 0. (21) 10

59 Analogously we conside the equity value fo V V S : E(V,V B,C)=V (1 τ 1)C λ 1 +A 1 V λ 2 λ 1 S V λ 2 + A 1 V λ 1 B + C V λ2 λ 2 (1 τ 2) V B. V B (22) The fist tem V (1 τ 1 ) C epesents equity value when thee is no isk of default, with a constant copoate tax ate τ 1. The second tem captues two diffeent effects: one due to the switching in tax benefits, the othe elated to default, both aising in the event λ of V falling to V S. Obseve that the second tem A 1 1 λ 2 V λ 2 λ 1 S is negative, eflecting the possible patial loss of tax benefits below V S. Its negative effect on equity inceases with an incease in the switching baie V S and also as a consequence of a highe diffeence τ 2 τ 1. What emains is the option to default, which will be activated only if V eaches V S. As peviously, the option to default must have positive value, meaning constaint (21) being satisfied. In the following poposition we analyze the veification of constaint (21) to get equity convex in V. Poposition 3.1 Suppose that τ 2 < τ 1, then condition (21) is satisfied fo V B < V B such that 1 C(1 τ 2 ) < V B < C(1 τ 2). 1+A 1 Remak 3.2 If δ =0, then condition (21) becomes A 1 V B + C (1 τ 2) V B > 0, theefoe V B < 1 C(1 τ 2 ). (23) 1+A 1 The convexity condition equied in [17] is V B < C(1 τ 2) ;notethat,asa 1 > 0, 1 C(1 τ 2 ) 1+A 1 < C(1 τ 2). On the othe side, unde the hypothesis δ > 0 and a unique constant copoate tax ate τ 1 = τ 2, then A 1 =0and the condition (21) becomes V B < C(1 τ 2),asfoundin[3,4]. This emphasizes the fact that the diffeence between these two convexity constaints is due to asymmety in tax benefits: intoducing asymmety in tax benefits makes the convexity constaint on V B moe tight if compaed to the case of a unique constant tax-shelteing value of τ 2 C. This esult does not depend on payouts, it is tue fo each level of δ. Poposition 3.1 gives an uppe bound fo V B ; nevetheless, due to limited liability of equity, the failue level V B cannot be chosen abitaily small, but E(V,V B,C)mustbe non negative fo all values V V B. To this end we wite equity function (20) as: E(V,V B,C)=f(V,C)+g(V,V B,C), 11

60 whee and g(v,v B,C)= f(v,c)=v (1 τ 2 ) C + A 1V λ 1, (24) A 1 V λ 1 B + C (1 τ 2) V B V V B λ 2. (25) Constaint (21) is needed in ode to make the option embodied in equity having positive value and its value g(v,v B,C) in (25) being a convex and deceasing function of V.The value g(v,v B,C) inceases as V appoaches V B, theefoe compensating equity holdes fo the eduction in equity due to a lowe V. The function f(v,c) in (24) epesents equity value with no isk of default unless limit of time, when asymmety in tax benefits is assumed. Unde constaint (21) f is inceasing and convex when V [V B,V S ). At point V = V B, g(v,v B,C) and f(v,c) have exactly the same value, with opposite sign. Thus the following poposition povides a condition such that an incement in V must poduce an impact on equity value without default isk f(v,c) highe than its effect on the option to default g(v,v B,C). This allows to keep E(V,V B,C) 0 when the option to default appoaches its execise instant, i.e. V V B. Poposition 3.3 The function V E(V,V B,C) is inceasing and stictly convex in V B V < V S if V B satisfies constaints (21) and Moeove E(V,V B,C) 0 fo V V B. λ V B (1 + λ 2 )+A 1 V 1 B (λ 2 λ 1 ) C(1 τ 2) λ 2, (26) Remak 3.4 If δ =0, constaint (26) becomes V B 2(1 τ 2 )V S C V S (σ 2 +2)+2C(τ 1 τ 2 ). (27) Recalling also constaint (23), in ode to have equity inceasing and convex w..t. V, the endogenous failue level has to satisfy: 2(1 τ 2 )V S C V S (σ 2 +2)+2C(τ 1 τ 2 ) V B 2+ σ2 (1 τ 2 )V S C V S (σ 2 +2)+2C(τ 1 τ 2 ). We conside the coupon ate C being fixed and maximize equity value in ode to find the endogenous failue level. To this end we impose the smooth-pasting condition (see [17] footnote 20 and [22] footnote 60): E V V =V B =0. (28) 12

61 Theoem 3.5 Suppose constaint (21) holds, then the endogenous failue level V B (C; τ 1, τ 2 ; δ) which satisfies (28) exists and is unique, unde the condition V S (1 τ 1)C We note that condition (28) is equivalent to f V V =V B = g V V =V B, λ 2 1+λ 2. (29) whee f and g ae defined in (24) and (25). Thus a solution of (28) is an implicit solution of 2 (1 + λ 2 )= λ λ1 2C V 1 VS B (τ 2 τ 1 )+(1 τ 2 ). (31) V B Remak 3.6 Note that this choice of bankuptcy level also optimizes V B E(V,V B,C). Unde constaints (21) and (29), equity is inceasing (and convex) w..t. assets value V if V B V B (C; τ 1, τ 2 ; δ), whee V B (C; τ 1, τ 2 ; δ) satisfies the smooth pasting condition, being the minimum failue level that equity holdes can choose due to limited liability of equity. Conside the function g in (25) which is the option to default embodied in equity. The endogenous failue level V B (C; τ 1, τ 2 ; δ) satisfies E(V,V B,C) V B =0if and only if g(v,v B,C) V B =0. Futhe g(v,v B,C) V B =0is equivalent to equation (31), which is exactly the optimal stopping poblem equation. It gives the optimal execise time of the option to default embodied in equity. Diffeentiating equity value w..t. V B, we have VB E = VB g which has the same sign as the deceasing function V B A 1 V λ 1 B (λ C 1 λ 2 )+λ 2 (1 τ 2) V B (1 + λ 2 ). This function is positive then negative and cancel at point V B exactly solution to Equation (31). If δ = 0, explicit solution can be obtained by solving equation (31) with espect to V B V B (C; τ 1, τ 2 ; 0) = 2CV S (1 τ 2 ) V S (σ 2 +2)+2C(τ 1 τ 2 ), (32) 2 In paticula if we assume a switch to zeo tax level, i.e. τ 2 = 0, we obtain the optimal failue V B(C; τ 1, 0; δ) asimplicitsolutionof (1 + λ λ2c 2)= V 1 B λ1 VS 1 τ 1. (30) V B This esult completes the analysis in [19] Appendix B, in the case whee we let the matuity T since the authos analyze the switch to zeo tax level only in the no-dividend case. 13

62 thus extending [17] Appendix A (case τ 2 = 0). Finally in case τ 2 = τ 1 =: τ we obtain: V B (C; τ, τ; δ) = λ 2C(1 τ) (1 + λ 2 ), (33) which is the endogenous failue level in case of a unique constant tax-shelteing value of inteest payment τc with a payout ate δ. This esult extends that found in [17] equation (14) to the case when the fim s assets value model accounts fo a non zeo payout ate. See [4] fo a detailed analysis of this case. Remak 3.7 Obseve that unde constaint (29), the endogenous failue level V B (C; τ 1, τ 2 ; 0) is lowe than the switching baie exogenously given V S. The following inequality 2(1 τ 2 )V S C V S (σ 2 +2)+2C(τ 1 τ 2 ) V S (34) holds unde V S 2C(1 τ 1) (σ 2 +2) which is exactly constaint (29) in case δ =0. (35) Effect of copoate tax ate asymmety on the endogenous failue level In this paagaph we analyze the impact of the copoate tax function τ( ) defined in (3) on the endogenous failue level. We fix τ 1 and V S, and we study the influence of τ 2 on the endogenous failue level, given the coupon C. Intoducing asymmety makes debt moe o less attactive hence it should incease o decease the optimal leveage atio. Asymmety also makes moe o less attactive to keep a loss-making fim alive, hence it should aise o decease the endogenous failue level and bing default close o fae. In ode to wok with explicit fomulas we conside the no-payout case δ = 0: theefoe λ 1 = 1 and λ 2 = 2 σ 2. In the geneal case we will esot to numeical compaisons. Let us conside: the endogenous failue level obtained with the constant instantaneous tax benefits in [17] V BL (C; τ 1 ; 0) = 2C(1 τ 1) σ 2 +2, (36) the level (32) obtained in the case of switching between two tax levels and, as a paticula case of (32) with τ 2 = 0, the switch to zeo tax benefits (as in [17] Appendix A) V B (C; τ 1, 0; 0) = 2CV S V S (σ 2 +2)+2τ 1 C. (37) 14

63 Poposition 3.8 The function τ 2 V B (C; τ 1, τ 2 ; 0) defined in (32) is deceasing. In paticula V B (C; τ 1, 0; 0) in (37) is geate than V B (C; τ 1, τ 2 ; 0) in (32). Futhe fo any τ 1 > τ 2 V B (C; τ 1, τ 2 ; 0) >V BL (C; τ 1 ; 0), V BL (C; τ 2 ; 0) >V B (C; τ 1, τ 2 ; 0). We can obseve that unde ou asymmetic copoate tax schedule, a highe τ 2 will incease equity value (fo each coupon level C), and educe the endogenous failue level V B (C; τ 1, τ 2 ; 0).Thus, inceasing tax deductions could be a way to suppot fims in cisis times. Finally we conclude that in the model whee δ = 0 a highe asymmety in the tax deductibility will incease the failue level endogenously chosen. We can intoduce θ := τ 2 τ 1 as a measue of the degee of asymmety of the copoate tax schedule: θ =1epesents Leland famewok (no asymmety), θ = 0 is the maximum asymmety case, meaning full loss of tax shelte below V S. Any othe case 0 θ 1 epesent nothing but an intemediate asymmetic scenaio. As asymmety inceases, meaning θ := τ 2 τ 1 close to 0, the endogenous failue level V B (C; τ 1, τ 2 ; 0) inceases fo any value of the exogenous switching baie. In such a case, in fact, below V S the fim will have less tax benefits, due to the lowe τ 2, binging the endogenous failue level highe. Remak 3.9 Suppose fo a moment τ 2 being fixed. Note that the application τ 1 V B (C; τ 1, τ 2 ; 0) is deceasing. This is in line with a eduction in the degee of asymmety of the copoate tax schedule (i.e. θ close to 1). A highe τ 1 allows the fim to have geate tax savings above V S, binging equity value highe both above and below (coefficient A 1 will be highe) the switching baie V S, thus binging down the endogenous failue level. The impact of the deductibility asymmety affects the endogenous failue level also though the exogenous switching baie V S. In the no-payout case it is easily seen the following. Poposition 3.10 The following esult holds: the failue level V B (C; τ 1, τ 2 ; 0) in (32): i) is deceasing with espect to the exogenous baie V S if τ 2 > τ 1, ii) is inceasing with espect to the exogenous baie V S if τ 2 < τ 1. Assume that τ 2 < τ 1. Given a cetain degee of asymmety, meaning θ := τ 2 τ 1 being fixed, suppose to change the exogenous switching baie V S. This means that a highe V S will incease the endogenous default bounday. Stating fom V V S,theswitching fom τ 1 to τ 2 will be moe likely, thus it will be moe likely losing some tax benefits. As 15

64 exteme case we can conside what happens as V S inceases: the model will appoach Leland famewok with a unique constant tax-shelteing value on inteest payments τ 2 C. Equity holdes will lose the oppotunity to switch fom τ 2 to a highe tax savings egion, thus equity will be lowe and the endogenous failue level highe, as shown in Poposition 3.8. All this holds since fo the moment we ae conducting an analysis supposing the coupon C being fixed. Notice that these esults can also be seen as those ones obtained in a limit-model with constant τ 2 : when the coupon is fixed, the endogenous failue level is deceasing w..t. the copoate tax ate (see [17]), confiming ou esults in this subsection. A change in θ, as fa as a change in V S, should be an altenative way to analyze how the impact of a vaiation in the asymmety of the copoate tax schedule can affect optimal capital stuctue decisions, as we will do in Section 4. Obseve that a change in θ o a change in V S poduce a diffeent effect on the asymmety of the tax schedule: we popose to intepet θ as a vetical measue of asymmety, V S as hoizontal measue. What we mean is that θ modifies the degee of asymmety, by acting on the distance between the two copoate tax ates, thus measuing the potential instantaneous loss of tax benefits at point V = V S. A change in V S epesents an hoizontal measue of asymmety since it modifies the ange of fim s values fo which the fim faces a highe (lowe) deductibility. When θ 1, o V S V B, the limit-model is a famewok with a flat copoate tax schedule with a constant copoate tax ate τ 1, but the economic intuition behind is completely diffeent. Conside a coeteis paibus analysis in which all vaiables except θ ae constant: as θ moves, what is changing is only the measue of the potential loss in tax benefits, meaning the distance between the two levels. As opposite case, when V S is the only vaiable to move, the potential loss in tax benefits is still the same, what changes is the pobability of eaching the baie, thus the likelihood of the potential loss. 3.2 Failue level with debt dependent switching baie In section 3.1 we assumed that the switching baie V S being exogenously given. Nevetheless this hypothesis is not completely ealistic, and we expect that V S will depend on the amount of debt issued by the fim (see [17] section VI.A). If assets value falls, it is moe likely that pofits will be lowe than coupon payments, thus the fim will not fully benefit tax savings. Unde U.S. tax codes, a necessay condition equied to fully benefit tax savings, is that the fim s EBIT (eanings befoe inteest and taxes) must cove payments equied fo coupons (see [17]). We now intoduce the ate of EBIT and suppose it is elated 3 to assets value in the following way: EBIT := av k, (38) with 0 <a<1,k > 0, whee k epesents costs and a is a faction of fim s cuent assets value. In this case the goss pofit falls to 0 when V equals k a. We assume that the 3 In [17] EBIT is modeled as a linea function of V,andalsoin[18]issupposedtobeequaltoaconstant faction of fim s assets value. 16

65 copoate tax ate is τ 1 in case EBIT C 0, and τ 2 othewise, with τ 2 τ 1. Unde this specification the switching baie V S depends upon the amount of debt issued by the fim in the following way: V S = k + 1 C. (39) a In this scenaio the switching baie is not exogenously given: it depends on a constant tem k due to costs needed to detemine the ate of EBIT, then it is a linea function of the coupon level. The switching baie V S inceases with both k, C: a highe pofit is equied to cove highe costs k and/o highe inteest payments, in ode to benefit tax savings fom issuing debt. We now analyze how this diffeent definition of the switching baie V S affects the endogenous failue level. The endogenous failue level is optimally chosen by equity holdes by applying the smooth pasting condition; when applying the smooth pasting condition, we diffeentiate equity w..t. V and then evaluate this deivative at point V B. We stess that definition (39) makes the switching baie dependent and linea on C, butv S does not depend on fim s cuent assets value V. Thus we can use esults fom Section 3.1 about equity value in ode to find the default bounday in this case. In the case δ =0we conside equation (32) and modify it accoding to the debt dependent switching baie (39), the endogenous failue level becomes: V B c (C; τ 1, τ 2 ; 0; k, a) = 2C(ak + C)(1 τ 2 ) (ak + C)(σ 2 +2)+2aC(τ 1 τ 2 ). (40) Look at inceasing costs k o educing a, the faction of fim s value necessay to detemine the ate of EBIT: this will bing default close, ising the endogenous failue level in (40) optimally chosen by equity holdes. Poposition 3.11 The endogenous failue level V B c (C; τ 1, τ 2 ; 0; k, a) defined in (40) is: i) inceasing and concave w..t. k; ii) deceasing and convex w..t. a; iii) inceasing and concave w..t. C. Conside a compaative static analysis: if k inceases and/o a educes, EBIT is lowe fo each fim s assets value V, thus the default bounday is highe since debt has a geate likelihood of losing its tax benefits, meaning fo the fim is moe likely to lose potential value. We analyze the elation between total coupon payments suppoted by the fim and the endogenous failue level chosen by equity holdes unde this debt dependent asymmety famewok. A compaison between (32), (36) and (40) shows that unde the assumption of tax benefits asymmety, the endogenous failue level is an inceasing and concave function of the coupon level, instead of being a linea inceasing function of C (in case of a unique constant copoate tax ate). It is still tue that the endogenous failue level is independent 17

66 of fim s cuent assets value V and the faction (of fim s value) α which is lost because of bankuptcy pocedues 4. When the copoate tax ate is unique, a change in the coupon level affects the optimal equity holdes choice in the same way fo all coupon levels. A debt-dependent asymmety, meaning the switching baie depending on coupon, intoduces a diffeent effect though modifying the shape of the endogenous failue level as function of C. As a consequence, a change in C modifies the endogenous failue level with diffeent magnitudes, depending on the value of outstanding debt. If the fim is suppoting vey high inteest payments, a eduction (incease) in the coupon level will poduce a small effect on the failue level, while in case of low coupon payments, a vaiation in C will stongly affect the endogenous default bounday, poducing a bigge impact on it V B (C;τ 1,τ 2 ;0) V B (C;τ 1,τ 2 ;0) c V (C;τ1 B,τ 2 ;k,a) V B (C;τ 1,τ 1 ;0):=V BL (C;τ 1 ) V B (C;τ 2,τ 2 ;0):=V BL (C;τ 2 ) C Figue 1: Endogenous failue level. This plot shows the behavio of the endogenous failue level w..t. coupon level C. We conside fou altenative scenaios: two with a constant copoate tax ate epesenting Leland famewok espectively with τ 1, τ 2 altenatively. Then we conside asymmetic tax benefits cases: V B(C; τ 1, τ 2; 0) when the switching baie is exogenous, V B(C; τ 1, τ 2;0;k, a) whenitisdebt dependent. Paametes values ae: σ =0.2, =0.05, δ =0, τ 1 =0.35, θ =0.4,V S = 90. We then conside k =60,a = 1 6,givingVS c =60+6C. Recall that θ := τ 2 τ 1 benefits. epesents the degee of asymmety in tax Figue 1 shows the behavio of the endogenous failue level when diffeent famewoks ae consideed: two constant tax benefits cases (altenatively a unique constant τ 1 o τ 2 ), two switching scenaios, one with V S exogenous (V S = 90), the othe with the switching baie debt dependent (V S c = 60+6C). Leland famewoks with constant τ 1, τ 2 epesent 4 The independence w..t. α means that bankuptcy costs does not diectly affect the endogenous failue level, since the stict pioity ule holds. Bankuptcy costs will instead affect the optimal failue level though the choice of the optimal coupon C which maximizes total fim value. 18

67 two exteme boundaies between which both the endogenous failue levels obtained unde asymmetic tax benefits lie. We now compae (32) and (40): they ae both inceasing and concave w..t. coupon level C. When coupon payments ae low, (32) is geate than (40), though thei diffeence is vey small. As coupon inceases, the behavio completely changes: the debt dependent switching baie causes the failue level to be highe than in case V S constant, and the diffeence between the two levels inceases too. The eason is that the debt dependent switching baie (39) inceases with coupon level, so a fim paying a high coupon C is facing a highe switching baie, thus a geate pobability of losing tax benefits, since now EBIT must cove a geate value of inteest payments. Poposition 3.12 Conside V B (C; τ 1, τ 2 ; 0) in (32) and V B c (C; τ 1, τ 2 ; 0; k, a) in (40). The following holds: V B (C; τ 1, τ 2 ; 0) >V B c (C; τ 1, τ 2 ; 0; k, a), if V B (C; τ 1, τ 2 ; 0) = V B c (C; τ 1, τ 2 ; 0; k, a), if V S > C a + k V S = C a + k V B (C; τ 1, τ 2 ; 0) <V B c (C; τ 1, τ 2 ; 0; k, a), if V S < C a + k. 4 Optimal Capital Stuctue In this section we detemine the optimal capital stuctue within the model assuming the copoate tax function (3) in both cases of exogenous and debt dependent switching baie. In both cases we give numeical esults in the geneal famewok with both asymmety and payouts. As paticula case, we show also what happen fo δ = 0 in ode to isolate the asymmety effect on copoate financing decisions. Befoe consideing optimal capital stuctue, the coupon was supposed to be fixed. We now tun to the optimization of the total value of the fim depending on the endogenous failue level solution of the optimal stopping poblem faced by equity holdes. Once detemined this default bounday as en endogenous one, equity holdes will incopoate this decision into the total value of the fim. Then maximize it w..t. C in ode to find the optimal amount of debt to issue, that one which guaantees the maximum total value of the fim due to the limited liability constaint. Thus the optimal coupon, namely C, maximize total fim value. Once found, we eplace C in all expessions of pevious subsections in ode to fully descibe the optimal capital stuctue. We then conside and show numeical esults analyzing each financial vaiable at its optimal level and study effects of both i) copoate tax asymmety and ii) payout ate on optimal coupon C, optimal debt value D, optimal equity value E, optimal default bounday V B and optimal total value of the fim v. We also analyze the optimal yield spead R whee R := C D, and the optimal leveage atio, defined as the atio between optimal debt and optimal total value L := D v (when coupon is at its optimal level C ). 19

68 4.1 Exogenous switching baie Hee we show how optimal capital stuctue is affected by both asymmety and payouts when the switching baie is exogenously given. The optimal coupon C must be chosen in ode to maximize C v(v,v B (C; τ 1, τ 2 ; δ),c), whee v(v,v B (C; τ 1, τ 2 ; δ),c) is defined in (18). The optimal failue level is not given in closed fom, nevetheless the following esult allows us to study the optimal capital stuctue. Poposition 4.1 The function V B C(V B ; τ 1, τ 2 ; δ) is inceasing, whee V B is implicitly given by equation (31). It follows that the study of the influence of the coupon C on the total value of the fim v, is equivalent to study v as function of V B. Thus, optimizing C v(v,v B (C; τ 1, τ 2 ; δ),c) is equivalent to optimize V B v(v,v B,C(V B ; τ 1, τ 2 ; δ)). Remak 4.2 Even if we ae not able to get an analytical expession of the optimal coupon C, we obtain that it has to satisfy constaints (21) and (26). We numeically detemine the optimal coupon and veify that these constaints ae satisfied fo ou case studies. In the case δ = 0 the endogenous failue level is given in closed fom by equation (32). Then we study the optimal capital stuctue by maximizing the application C v(v,v B (C; τ 1, τ 2 ; 0),C). Poposition 4.3 The function C v(v,v B (C; τ 1, τ 2 ; 0),C) is a concave function achieving a maximum at point C as solution of v(v,v B(C;τ 1,τ 2 ;0),C) C =0, unde the condition 5 τ 1 < τ 2 3. Thus an optimal capital stuctue exists and is unique. Finally the optimal failue level V B (V ; τ 1, τ 2 ; δ) is obtained by plugging the optimal coupon C into the endogenous failue level given by Theoem 3.5, that is V B (C (V ); τ 1, τ 2 ; δ). In ode to study the asymmety effect on these vaiables we conside θ := τ 2 τ 1, θ 0, measuing the vetical degee of asymmety of the copoate tax schedule. Notice that θ = 1 epesents a non asymmetic case: moeove, consideing also δ =0 will lead exactly to esults obtained by Leland [17]. The case δ = 0, θ = 1 is the one noted in Remak (33), which is compaable with ou analysis in [4], whee a detailed analysis 5 This conditions is always satisfied with ou paamete values, since we always conside τ 1 <

69 V B * δ 0 0 θ Figue 2: Optimal failue level as function of δ, θ. This plot shows the behavio of the optimal failue level V B as function of dividend δ and degee of asymmety θ. The switching baie V S is exogenous and paametes values ae: σ =0.2, =0.06, τ 1 =0.35, α =0.5,V S =90,V =100. Recallthatθ := τ 2 τ 1 epesents the degee of asymmety in tax benefits. about the influence of payouts on optimal capital stuctue decisions is conducted fom both a qualitative and quantitative point of view. Asymmety inceases as θ goes to 0, achieving its maximum fo θ = 0 (maximum asymmety case). This scenaio epesents a switching fom a tax level τ 1 to zeo-tax benefits: this happens when tax benefits ae completely lost fo V < V S. Fist notice that tax asymmety aises the optimal failue level V B (V ; τ 1, τ 2 ; δ): fo any value of δ (and fo values of V S <V), as the tax asymmety inceases then the optimal failue level V B (V ; τ 1, τ 2 ; δ) inceases. The opposite happens when consideing the payout influence, given a degee of asymmety θ. Fo any fixed value of θ, the optimal failue level deceases as δ inceases fom 0 to Results ae in Table 1, while Figue 2 shows the behavio of the optimal failue level as function of both δ, θ. Fom [4] we know that only intoducing payouts will bing to a lowe optimal failue level when the copoate tax ate is unique and constant though time. This moe geneal scenaio, whee both payouts and asymmety in tax benefits ae in foce, shows though ou numeical analysis that the final joint effect can be quantitatively significant. Conside as exteme cases θ =1, δ =0 21

70 and θ =0, δ =0.04: passing fom no asymmety and no payouts, to a payout ate equal to a 4% of cuent assets value, bings to a eduction in optimal failue level of aound 8.8%, passing fom to L*(%) δ θ Figue 3: Optimal Leveage as function of δ, θ. This plot shows the behavio of optimal leveage atio L as function of dividend δ and degee of asymmety θ. The switching baie V S is exogenous and paametes values ae: σ =0.2, =0.06, τ 1 =0.35, α =0.5,V S =90,V =100. Recallthatθ := τ 2 τ 1 epesents the degee of asymmety in tax benefits. Futhe optimal leveage atio is stongly affected by asymmety in the copoate tax schedule as shown in Figue 3. In ode to isolate the asymmety effect, conside Table 1 in case δ = 0: esults ae due only to the switching in tax benefits and bing to a eduction in optimal debt, optimal total value of the fim and also optimal leveage atios. Extending the analysis by consideing the geneal case in which both δ > 0, 0 θ < 1 shows that fo each level of payouts, inceasing the degee of asymmety educes optimal leveage and this effect is stonge when the the payout ate is highe. Conside the last column of Table 1: compaing the two exteme cases θ = 1 and θ = 0, the diffeence in optimal leveage atio is 4.5% when δ = 0, 6% when δ =0.01 and 12% when δ =0.04. Tax asymmety has a negative effect on optimal leveage atios L : fo any value of δ consideed, L decease as the degee of asymmety inceases, that is as θ 0. The decease of the optimal leveage is quantitatively moe significant as the payout ate ises. Analogously, as obseved in [3, 4] the capital stuctue of a fim is stongly affected by payouts. Fom [3, 4] we know that intoducing payouts in a stuctual model with a unique copoate tax ate τ has the effect of educing optimal leveage atios. Table 1 allows us to 22

71 confim this esult also when the tax schedule is asymmetic. We conside as exteme cases to compae δ = 0 and δ =0.04. Consideing a unique copoate tax ate means θ = 1: in such a case we know fom [3, 4] that the diffeence in optimal leveage atio is quite 6%. We now intoduce convexity and look at θ =0.8, θ =0.4, θ = 0: this diffeence in optimal leveage becomes espectively 7%, 11% and 13%. A highe payout will lowe the optimal total value of the fim v, since a lowe debt issuance can be suppoted because less assets emain in the fim. As a consequence, this will bing down leveage atios. But also the asymmety effect is to educe leveage atios, since the potential loss in tax benefits due to the existence of the switching baie makes debt less attactive: consideing a scenaio whee both effects exist, will bing to a stong eduction in pedicted optimal total value of the fim, optimal debt and optimal leveage atios. We now conside Leland [17] case, i.e. θ =1, δ = 0, and compae it with a scenaio in which both payouts and asymmety exists, meaning θ =0, δ =0.04 in ode to captue the joint effect of these two ealistic genealizations. Obseve that passing fom no asymmety and no payouts, to a payout ate equal to a 4% of cuent assets value V, bings to a damatic eduction in optimal leveage: in such a case, this joint influence of δ, θ bings to an optimal leveage atio of 57.36%, with a significant eduction of 17% fom Leland esult of a 75%-leveaged fim, leading to a value which is moe in line with histoical noms 6. This stong impact on optimal leveage atios suggests that asymmety and payouts seem to be impotant factos involved in the deteminacy of copoate capital stuctue decisions. Figue 4 shows the behavio of optimal coupon C as function of δ and θ. Obseve that fo each degee of convexity 0 θ < 1 the optimal coupon is deceasing w..t. δ, extending esults in [3, 4] to the case of asymmetic copoate tax schedule. What we stess now in this geneal famewok whee both payouts and an asymmetic tax scheme inteact, is that this negative effect of payouts on C is geate as the asymmety in tax benefits inceases, i.e. as θ 0. Fom Figue 4 we can also obseve that the optimal coupon C deceases as θ 0 fo each level of the payout ate δ. The economic eason is that intoducing asymmety in tax benefits makes debt less attactive fo the fim, thus leading to a not negligible eduction in the optimal coupon level choice. The decease in C due to the asymmetic tax benefits scheme will be highe as payouts incease, as we can note consideing the slope in Figue 4 w..t. θ fo each level of δ. Payouts and asymmety in tax benefits influence each othe by inceasing the magnitude of thei own effects on the optimal coupon, binging to a joint influence on optimal coupon which is quantitative significant. To analyze the inteaction between δ and θ on C conside fo example thee altenative scenaios θ =1, 0.4, 0: when δ goes fom 0 to 0.04, the optimal coupon educes fom 6.5% to 6.23% in case θ = 1, fom 6.03% to 5.2% in case θ =0.4, fom 5.78% to 4.32% in case θ = 0. The eduction in C due to an inceased payout is consideably highe as asymmety in tax benefits inceases: when tax benefits ae completely lost unde V S the eduction of optimal coupon is moe than 5 times the eduction in case of a constant τ. Ou analysis in this pape confims ou esults in [3, 4] and moeove extend thei 6 Leland [17] in his Section D obseves that a leveage of 52% is quite in line with histoical noms. 23

72 δ =0 θ C D R E VB v L % % % % % % % % % % % δ =0.01 θ C D R E VB v L % % % % % % % % % % % δ =0.04 θ C D R E VB v L % % % % % % % % % % % Table 1: Effect of payouts and asymmety in the tax schedule on all financial vaiables at thei optimal level when the switching baie V S is exogenous. Base case paamete values: V 0 =100,σ =0.2, τ 1 =0.35, =6%,α =0.5. This table epots esults consideing an exogenous baie V S = 90 and thee diffeent cases: δ =0,δ =0.01, δ =0.04. Remembe that asymmety inceases (deceases) as θ deceases (inceases). The last ow of the table shows the case θ = 1 fo each level of δ: the tax ate is unique (no asymmety). This ow epesents the model pesented in [3, 4]. In case δ =0it epesents Leland [17] esults. Leveage is in pecentage (%), speads in basis points (bps). 24

73 C* δ θ Figue 4: Optimal Coupon as function of δ, θ. This plot shows the behavio of the optimal coupon C as function of payout ate δ and degee of asymmety θ. The switching baie V S is exogenous and paametes values ae: σ =0.2, =0.05, τ 1 =0.35, α =0.5,V S =90,V =100. Recallthatθ := τ 2 τ 1 epesents the degee of asymmety in tax benefits. validity unde asymmety in tax benefits. Adding to this, the contibution of the pesent wok is also to show how optimal capital stuctue is much moe affected by payouts when consideing a moe ealistic famewok allowing also fo asymmety in the copoate tax schedule. As it concens optimal equity value and optimal speads we note that the joint effect of asymmety and payouts aises both optimal equity and optimal speads. We can explain this as a consequence of two main insights aising fom the model. i) Fist, when payouts ae intoduced, less assets emain in the fim, thus making possible only a lowe optimal debt issuance. Adding to this, asymmety makes debt less attactive, due to a possible switching to lowe tax benefits, thus a potential loss of value has to be taken into account. As a consequence, the joint effect is to educe both the optimal coupon C and the optimal amount 7 of debt D.Equity value inceases at its optimal level due to the joint effect of δ, θ on both C,VB : ecall that the optimal (endogenous) failue level inceases as the degee of asymmety is highe. 7 As noted in [17] fo δ = 0, and also suppoted by esults in [4] fo δ > 0, the fim will always chose a coupon level which is lowe than that one coesponding to the maximum capacity of debt. As a consequence, a lowe coupon means a lowe debt value. Moeove, as in [17] we ae assuming the face value of debt being constant. 25

74 ii) Secondly, we can also think about the joint effect of payouts and asymmety in tax benefits as something which contibutes to incease the aveage iskiness of the fim and moeove makes bankuptcy moe likely. This is why, despite lowe optimal leveage atios, optimal speads incease, in line with [17] suggestions. When θ 0, the potential loss in tax benefits due to passing fom τ 1 to τ 2 inceases. Optimal debt will be lowe, equity highe but the fist effect will always dominate the second one, binging to lowe optimal total values of the fim. This is because payouts and copoate tax asymmety incease the likelihood of default. Debt holdes must be compensated, this is why optimal speads pedicted in this model ae consideably highe w..t. the case θ =1, δ = 0, captuing all these economic insights. Table 2 epots the qualitative behavio of financial vaiables at thei optimal level when the exogenous switching baie V S and the payout ate δ ae fixed, while the vetical degee of asymmety inceases, i.e. θ 0, aiming at isolating and captuing only this asymmety influence on optimal capital stuctue decisions made by the fim in a compaative static analysis. Fin. Va. C D R E V B v L θ 0 < 0 < 0 > 0 > 0 > 0 < 0 < 0 Table 2: Effect of a change in the vetical degee of asymmety of the copoate tax schedule on financial vaiables at optimal leveage atio as θ 0 when V S is exogenous. The table shows fo each financial vaiable the effect of inceasing asymmety in the copoate tax schedule, i.e. fo θ 0, given the payout ate δ and the exogenous switching baie V S. We epot the sign of change in each vaiable as the degee of asymmety inceases. Concening the asymmety of the copoate tax schedule a simila analysis could be done analyzing how optimal capital stuctue decisions ae affected when the exogenous switching baie V S moves, meaning when the hoizontal degee of asymmety changes, fixing both θ and the payout ate δ. Numeical esults show that diffeent values of the baie can significantly modify optimal choices, meaning the copoate tax schedule is an impotant deteminant in leveage decisions. Table 3 shows numeical esult fo this case, Table 4 epots only the qualitative behavio of all financial vaiables at thei optimal level as V S inceases. The switching baie being exogenously given, esults show in which diection a highe V S will move optimal capital stuctue decisions. A possible explanation of what we obseve in Table 4 could be that, coeteis paibus, as V S ises (deceases), the hoizontal degee of asymmety changes. As exteme case, ou famewok tends to a limit-model in which the tax shelteing value of inteest payments is constant and equal to the lowe τ 2 C (highe τ 1 C). And this epesents the limit-model fo each degee of asymmety θ and payout level δ. A eduction (incease) in the copoate tax ate poduces exactly the effects shown by ou esults: each vaiable at its own optimal level deceases (inceases), except equity value which instead ises (educes). And this 26

75 δ =0, θ =0.8 V S C D R E V B v L % % % % % % % % % % % δ =0.01, θ =0.4 V S C D R E V B v L % % % % % % % % % % % δ =0.04, θ =0 V S C D R E V B v L % % % % % % % % % % % Table 3: Effect of asymmety in the tax schedule on all financial vaiables at optimal level. Base case paamete values: V 0 =100,σ =0.2, τ 1 =0.35, =6%,α =0.5. This table epots esults consideing an exogenous baie V S vaying fom 80 to 95 and diffeent scenaios fo payouts and degee of asymmety θ: δ =0, θ =0.8, δ =0.01, θ =0.4, δ =0.04, θ = 0. Leveage is in pecentage (%), speads in basis points (bps). esult is obust w..t. each payout level. The behavio we find in this limit-model is in line with [17] Table II, whee δ = 0: all vaiables except equity ae inceasing w..t. the 27

76 constant copoate tax ate τ. Moeove, when payouts ae intoduced in a flat copoate tax schedule model, esults ae still in line with [4] Table 5. This behavio of financial vaiables holds fo each level of payout δ and each vetical degee of asymmety θ: what is diffeent among diffeent scenaios is only the magnitude of the effect, obviously depending on the joint influence. And the joint influence is highe as payouts incease and θ 0. Conside as an example a switching baie of 95: in case δ =0.04 and θ =0themodel pedicts a 56% optimal leveage, while in case δ = 0 and θ =0.8 this optimal atio is aound 73%. This second case is vey close to Leland s esults [17] with constant τ 2, and the diffeence in leveage is quite negligible, i.e. 2%, while the fist case bings to a huge 19%-eduction in leveage atios. Fin. Va. C D R E V B v L V S < 0 < 0 < 0 > 0 < 0 < 0 < 0 Table 4: Effect of a change in the asymmety of the copoate tax schedule on financial vaiables at optimal leveage atio as V S inceases. The table shows fo each financial vaiable the effect of changing the asymmety in the copoate tax schedule when the exogenous switching baie V S inceases. These esults ae given fo a fixed θ and payout ate δ. We epot the sign of change in each vaiable as V S inceases. 4.2 Debt dependent switching baie In this section we study optimal capital stuctue when the switching baie consideed is inceasing with coupon C, being defined as V S := k + 1 ac. We will show not only the case δ = 0 fo which we have closed fom solution fo the endogenous failue level, but also numeical esults fo the geneal case δ > 0 whee a payout ate is consideed. In case δ = 0, we define the optimal capital stuctue by substituting the endogenous failue level V B c (C; τ 1, τ 2 ; 0; k, a) obtained in (40) into the total value of the fim v and then maximizing it w..t. C. This will give the optimal coupon C, allowing to analyze optimal leveage and optimal capital stuctue decisions as epoted in Table 5. In the geneal case δ > 0 we do not have a closed fom fo the endogenous failue level, and the smooth pasting condition is not linea w..t. C, since also the switching baie depends on C. Thus we numeically analyze the existence of an optimal capital stuctue and show in this subsection ou esults. The peculiaity of this model is that as the optimal coupon deceases fo highe vetical degee of asymmety θ 0, then also the optimal switching baie VS deceases, meaning that also the hoizontal degee of asymmety is changed. Fom the opposite point of view, we obseve that as θ 1 the debt dependent switching baie appoaches cuent fim s activities value V : in the limit, fo θ = 1, asymmety disappeas, meaning the copoate tax schedule tends to a flat one. 28

77 δ =0 θ C D R E VB v L V S % % % % % % % % % % % δ =0.01 θ C D R E VB v L V S % % % % % % % % % % % δ =0.04 θ C D R E VB v L V S % % % % % % % % % % % Table 5: Effect of payouts and asymmety in the tax schedule on all financial vaiables at thei optimal level when the switching baie V S is debt dependent. Base case paamete values: V 0 =100,σ =0.2, τ 1 =0.35, =6%,α =0.5, k =60,a =1/6. This table epots esults consideing an exogenous baie V S = 90 and thee diffeent cases: δ =0,δ =0.01, δ =0.04. Remembe that asymmety inceases (deceases) as θ deceases (inceases). The last ow of the table shows the case θ = 1 fo each level of δ: the tax ate is unique (no asymmety). This ow epesents the model pesented in [3, 4]. In case δ = 0 it epesents Leland [17] esults. 29

78 If compaed to the case consideed in pevious subsection, whee V S is exogenously given, this moe ealistic famewok allows to analyze the joint effect of a change in both the vetical and hoizontal degees of asymmety of the copoate tax schedule. As θ moves, the optimal coupon changes, and this in tuns modifies the optimal switching baie V S. A change in the vetical degee of asymmety will affect optimal capital stuctue decisions, both diectly and indiectly, in this last case by changing the ange of fim s values fo which tax benefits depend altenatively on τ 1, τ 2. Fin. Va. C D R E V B v L θ 0 < 0 < 0 < 0 > 0 < 0 < 0 < 0 Table 6: Effect of a change in the degee of asymmety of the copoate tax schedule on financial vaiables at optimal leveage atio as θ 0 when V S is debt dependent. The table shows fo each financial vaiable the effect of changing the degee of asymmety in the copoate tax schedule as θ 0 and the switching baie is debt dependent. Results hold fo each level of the payout ate δ 0. We epot the sign of change in each vaiable as θ 0. When the switching baie depends on the amount of debt issued, a highe pofit is needed in ode to have highe coupon paymentsfully deductible. Recall that we ae assuming EBIT has to cove coupon payments in ode to benefit fom tax savings. In this even simplified but moe ealistic famewok, geate debt has a geate likelihood of losing its tax benefits, and optimal leveage dops significantly. This eduction is quantitative highe than what we found in case V S being exogenously given, fo each level of payout. The decease in optimal leveage is a 19%-eduction in case δ =0.04, much moe than 12% in case of V S fixed. Leveage can each a 52% in line with histoical noms (see [17]). Table 6 shows that optimal cedit speads deceases in this scenaio, eflecting the lesse leveage, in line with suggestions in [17]. In this simplified famewok we model EBIT as a linea function of V and this allows to show that opeational costs could be anothe vaiable to analyze in ode to explain obseved leveage atios. As k and/o a ise, this will affect the optimal amount of debt issued, since a highe pofit is necessay to fully benefit fom coupon deductibility. An incease in k and/o a will dop pedicted leveage. The economic insight we want to give is that this simple model is flexible to analyze the impact of many factos on optimal capital stuctue decisions, poviding a famewok to develop in the diection of a moe empiical eseach, allowing to explain diffeences in obseved leveages among fims facing diffeent tax-code povisions. What we find is in line with what [14] obseve: when the copoate tax ate is highe, obseved leveage atios ae highe. Nevetheless ou model also analyzes how a change in tax-code povisions can affect a single fim s copoate decisions. The geneal esult is that asymmety always lowes optimal debt, optimal leveage atios and the maximum total value of the fim since less tax savings (actual and/o potential) ae available, meaning thee is always a loss of potential value fo the fim 8. 8 As in Leland [17] this model does not conside tax loss cayfowads which could be an inteesting 30

79 5 Conclusions Copoate capital stuctue is a complex decision since it is affected by a lage numbe of factos. We analyze a stuctual model with endogenous bankuptcy stating fom Leland famewok [17] and extending it in two main diections. We conside a moe geneal model in which we intoduce a payout ate δ and an asymmetic copoate tax schedule. Rathe than consideing a flat tax scheme, i.e. a unique copoate tax ate, we analyze asymmetic tax code povisions allowing fo a switching in copoate tax ates. The switching fom a copoate tax ate to the othe is detemined by the fim value cossing i) an exogenous baie, ii) a debt dependent switching baie (allowing to model EBIT). We investigate the joint effects of this copoate tax scheme and payouts on optimal default level and optimal capital stuctue. We popose altenative ways to measue the degee of asymmety of the copoate tax schedule: i) the degee of vetical asymmety, elated to the gap between tax ates; ii) the degee of hoizontal asymmety, elated to the ange of fim s values above and/o below the switching baie. We deive the value of the tax benefit claim in this famewok, following Leland [17] about how to model tax benefits of debt. Asymmety in copoate tax code povisions becomes also asymmety in tax benefits of debt. Optimal capital stuctue in analyzed though the deivation of the endogenous failue level chosen by equity holdes, and optimal coupon (maximizing total fim s value). Ou esults suppot [14], [27] suggestion that tax-code povisions should be consideed when studying copoate financing decisions. We obseve that all financial vaiables at thei optimal level ae affected by this asymmetic tax schedule, if compaed to Leland [17] esults with a flat tax-code (a unique constant copoate tax ate, meaning no asymmety). Moeove, we analyze the joint effect on optimal capital stuctue of payouts and asymmety in tax benefits. Result show that payouts and asymmety always lowes optimal debt, optimal leveage atios and the maximum total value of the fim. This because less assets emain in the fim (due to payouts), in line with [3, 4], and also less tax savings (actual and/o potential) ae available fo the fim, meaning a loss of potential value fo the fim. Leveage atios ae significantly affected fom a quantitative point of view by both factos: they dop down fom 75% in Leland case to a 52% in case of a 4% payout ate and maximum asymmety case (when we conside the deb dependent switching baie). These ae two exteme cases, but we find a huge impact on leveage atios fo most cases between these two scenaios. Ou analysis in this pape confims ou esults in [4] and shows how optimal capital stuctue is much moe affected by the intoduction of a payout ate δ > 0 inside a moe ealistic famewok allowing fo asymmety in tax-code povisions. These two factos influence each othe with a esulting quantitative huge joint effect on optimal debt and leveage. The degee of vetical asymmety in the copoate tax schedule (θ := τ 2 τ 1 ) is a paamete imposed to the fim by extenal authoities. Moeove, this paamete can vay stongly depending on the secto in which the fim is opeating in, and can also vay in time, fo example to encouage point to develop, since they will intoduce path dependence, making the model even moe ealistic. 31

80 investments. Thus, ou analysis povides a way to measue the economic influence (fom both a qualitative and quantitative point of view) of such an impotant and extenal facto on intenal and endogenous optimal choices made by the fim. Moeove, we also analyze effects on copoate decisions poduced by its joint influence with payout ates, which ae supposed to be constant (but we know that they should be even patly modified by the fim). We think that this famewok should be a flexible way to follow in ode to bette explain diffeences in obseved leveage atios acoss fims belonging to diffeent activity sectos, facing diffeent tax-code povisions, as an idea to apply fo a futue moe empiical eseach. 6 Appendix A The following esult contains the fomula fo the Laplace tansfom of a constant level hitting time by a Bownian motion with dift ([15] p ): Poposition 6.1 Let X t = µt + σw t and T b =inf{s : X s = b}, then fo all γ > 0, E[e γt b µb ]=exp σ 2 b µ 2 σ σ 2 +2γ. is con- The computation of the Laplace tansfom of double passage times, T VS T VB tained in the following esult ([15] p ). Poposition 6.2 Let T b =inf{t, W t = b}. Then if b<0 <c, E 0 [e γt b 1 {Tb <T c}] = sinh(c 2γ) sinh((c b) 2γ) ; E 0[e γtc 1 {Tc<T b }]= sinh( b 2γ) sinh((c b) 2γ). We now tun to compute such expectation fo a geometic Bownian motion with dift. Poposition 6.3 Let T =inf{t, log V + µt + σw t = log V B } and T S =inf{t, log V + µt + σw t = log V S }. Then if V B <V <V S, E V [e T 1 {T<TS }]= E V [e T S 1 {TS <T }] = VB V VS V µ/σ 2 2+µ sinh(log[(vs /V ) 2 /σ 2 σ ]) 2+µ sinh(log[(v S /V B ) 2 /σ 2 σ ]) µ/σ 2 2+µ sinh(log[(v/vb ) 2 /σ 2 σ ]) 2+µ sinh(log[(v S /V B ) 2 /σ 2 σ ]) noted as g(v,v S,V B ), noted as f(v,v S,V B ). 32

81 Poof It follows by Poposition 6.2 using a change of pobability measue. The pocess t W t = µt+σw t is a Q Bownian motion with Q = LP, with the P matingale defined as dl t = L t µ σ dw t. Equivalently, P = Z.Q, with dz t = Z t µ σ d W t. Remak that Z t =exp( µ σ W t 1 2 ( µ σ )2 t). Thus E V [e T 1 {T<TS }]=E Q 0 [Z T e T 1 {T<TS }]withnow So we get T =inf{t, W t = log( V B V )1/σ }. E V [e T 1 {T<TS }]=E 0 [exp( µ σ W T 1 2 (µ σ )2 T T)1 {T<TS }]. We can use the fact that, by continuity, WT = log( V B V ) 1/σ, so E V [e T 1 {T<TS }]=( V B V )µ/σ2 E 0 [exp( ( 1 2 (µ σ )2 + )T )1 {T<TS }]. Finally, we use Poposition 6.2 with γ = 1 2 ( µ σ )2 +, b = log( V B V ) 1/σ,c= log( V S V ) 1/σ to conclude: E V [e T 1 {T<TS }]=( V V sinh((log S B V ) ( 1/σ µ σ )2 +2) )µ/σ2. V sinh((log V S V B ) ( 1/σ µ σ )2 +2) The second poof is quite simila. Coollay 6.4 Using sinh log(x β )= 1 2 (xβ x β ), we get: E V [e T 1 {T<TS }]= V B λ 2 (V S ) λ 2 λ 1 V λ 2 V B λ 2 V λ 1 (V S ) λ 2 λ 1 V λ 2 λ 1 B (41) with and E V [e T S 1 {TS <T }] = (V S) λ 2 (V B ) λ 2 λ 1 V λ 2 (V S ) λ 2 V λ 1 V B λ 2 λ 1 (V S ) λ 2 λ 1 (42) λ 1 = µ µ 2 +2σ 2 σ 2, λ 2 = µ + µ 2 +2σ 2 σ 2. µ = δ 1 2 σ2. 33

82 7 Appendix B Poof (of Poposition 2.2) Using integal epesentation of tax benefits we get: - if V V S TVS F (V )=E V e T V S F (V TVS )+ e s τ 1 C1 (VS, )(V s )ds if V B V < V S : F (V ) = E V e T V S T VB F (V TVS T VB )+ = τ 2C TVS T VB +(F (V S ) τ 2C )E V e T V S 1 TVS T VB + e T V B 1 TVB <T VS + (F (V B ) τ 2C )E V Fom bounday condition F (V B ) = 0 we get F (V )= τ 2C Theefoe we obtain τ 2 C 0 e s τ 2 C1 (VB,V S )(V s )ds 1 E V e T V B 1 TVB <T VS +(F (V S ) τ 2C )E V e T V S 1 TVS T VB F (V )= + (F (V S) τ 2C )V λ 2 S + τ 2C V λ 2 λ 1 S V λ 2 λ 1 B τ 1 C + F (V S ) τ λ2 1C VS 1 V (VS, ) V λ 2 B V λ 1 V λ 1 B (F (V S) τ 2C )+V λ 1 S V λ 2 λ 1 S V λ 2 λ 1 B τ 2 C (V B V S ) λ 2 V λ 2 Futhe F is a C 1 function, thus imposing continuity of the deivative at point V S yields: λ 2 V 1 S (F (V S) τ 1C )= λ 1V 1 (F (V S ) τ 2C )V λ 2 λ 1 S S V λ 2 λ 1 S V λ 2 λ 1 B + λ 1 V 1 S τ 2C V λ 2 B V λ 1 S V λ 2 λ 1 S V λ 2 λ 1 B 1 (VB,V S ) Finally +λ 2 V 1 (F (V S ) τ 2C )V λ 2 λ 1 B S V λ 2 λ 1 S V λ 2 λ 1 B λ 2 V 1 S τ 2C V λ 1 S V λ 2 B V λ 2 λ 1 S V λ 2 λ 1 B. F (V S )= C(λ 2τ 1 λ 1 τ 2 ) (λ 2 λ 1 ) + λ 2C(τ 2 τ 1 ) (λ 2 λ 1 ) V λ 1 λ 2 S V λ 2 λ 1 B τ 2C V λ 2 S V λ 2 B. 34

83 Poof (of Poposition 3.1) Condition (21) can be witten as f(v B ) >g(v B ) with f(v B )= A 1 V λ 1 B and g(v B )=V B C (1 τ 2) and we study the poblem fo V B 0. Obseve that: f(0) = 0, f(v B ) 0, V B 0 since A 1 > 0 fo τ 2 < τ 1, f (V B )=λ 1 A 1 V λ 1 1 B < 0, g(0) = C (1 τ 2) < 0, g =1. Since f(v B ) is negative, f is deceasing function of V B, g is an inceasing function of V B and g(0) <f(0) = 0, we can state that a solution V B of f( V B )=g( V B ) exists and it is unique and V B < C (1 τ 2), since g( C (1 τ 2)) = 0. Futhe we obtain a lowe bound fo V B,bystudyingf as function of δ. As f(v B,δ) f(v B,δ) λ 1 λ 1 δ and f(v B, δ) λ 1 = 1 λ 2 λ 1 + log V S V B f(v B, δ) < 0; λ 1 δ = λ 1 µ 2 +2σ 2 < 0, δ = thus δ f(v B, δ) is inceasing. This implies that the intecept V B of f and g is inceasing, so that V B > V B (0), whee we denoted by V B (0) the bound esulting fom condition (21) in the special case δ = 0. In fact in this case (21) solves explicitly giving V B (0) = 1 C(1 τ 2 ) 1+A 1 (see also Remak 3.2). Poof (of Poposition 3.3) Unde constaint (21) the option to default has positive value. In ode to pove that equity be is inceasing, we split the equity function (20) as the sum of two tems f(v,c)=v (1 τ 2 ) C + A 1V λ 1, (43) and g(v,v B,C)= A 1 V λ 1 B + C (1 τ 2) V B V V B λ 2. (44) Obseve that: i) the application V f(v,c) is inceasing and convex fo V V B ; ii) the application V g(v,v B,C) is deceasing and convex fo V V B. As a consequence, in ode to ensue V E(V,V B,C) to be inceasing fo V V B,we need that f(v,c) V V =VB g(v,v B,C) V =VB (45) V 35

84 which leads to constaint (26). In such a case, by i) and ii), V E(V,V B,C) is also convex. Poof (of Theoem 3.5) Since the optimal failue level, if it exists, has to be lowe than the switching baie V S, we study the existence and uniqueness of the solution V B (C; τ 1, τ 2 ; δ) of Equation (31) in [0,V S ]. Imposing the smooth pasting condition gives that V B (C; τ 1, τ 2 ; δ) has to satisfy f(v B )=g(v B ) (46) whee f(v B )=(1+λ 2 )V B and g(v B )=A + BV λ 1 B, with the constants A and B given by A = λ 2C (1 τ 2) (47) B = V λ λ 1 2 C S (τ 2 τ 1 ). (48) Obseve that f(v B ) is a linea and inceasing function of V B since λ 2 > 0, going fom 0 to (1 + λ 2 )V S in [0,V S ]. On the othe side g(v B ) is deceasing as g (V B )= λ 1 BV λ 1 B < 0 since λ 1 < 0 and B<0 fo τ 2 < τ 1. Obseve that g(v S )= λ 2C(1 τ 1 ). Theefoe thee exists a solution to (46) if and only if (1+λ 2 )V S > λ 2C(1 τ 1 ), o equivalently V S > λ 2C(1 τ 1 ) (1+λ 2 ). Similaly a solution V B (C; τ 1, τ 2 ; δ) exists and is unique in case τ 2 > τ 1. Poof (of Poposition 3.8) Compute V B (C; τ 1, τ 2 ; 0) = λ 2CV S (V S (1 + λ 2 ) Cλ 2 (1 τ 1 )) τ 2 ( V S (1 + λ 2 ) Cλ 2 (τ 1 τ 2 )) 2 (49) = 2CV S VS (σ 2 +2)+2C(τ 1 1) (V S (σ 2 +2)+2C(τ 1 τ 2 )) 2, (50) then, by condition (35), it follows that V B(C;τ 1,τ 2 ;0) τ 2 < 0. This poves both the monotonicity of τ 2 V B (C; τ 1, τ 2 ; 0) and the inequality V B (C; τ 1, 0; 0) >V B (C; τ 1, τ 2 ; 0). Poof (of Poposition 3.10) Conside Equation (32); the deivative of this endogenous failue level V B (C; τ 1, τ 2 ; 0) with espect to V S is: V B (C; τ 1, τ 2 ; 0) V S = 4C 2 (1 τ 2 )(τ 1 τ 2 ) (V S (σ 2 +2)+2C(τ 1 τ 2 )) 2. Poof (of Poposition 3.11) i) It is sufficient to conside: V B c (C; τ 1, τ 2 ; 0; k, a) k = 4C 2 a 2 (τ 2 1)(τ 2 τ 1 ) [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 2 > 0 36

85 2 V B c (C; τ 1, τ 2 ; 0; k, a) k 2 = 8C2 a 3 (1 τ 2 )(τ 2 τ 1 )(σ 2 +2) [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 3 < 0. ii) Evaluating the fist and second deivative of the failue level w..t. a: V B c (C; τ 1, τ 2 ; 0; k, a) a = 4C 3 (τ 2 1)(τ 1 τ 2 ) [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 2 < 0, 2 V B c (C; τ 1, τ 2 ; 0; k, a) a 2 iii) The following holds: V B c (C; τ 1, τ 2 ; 0; k, a) C = 8C3 (1 τ 2 )(τ 1 τ 2 )(2C(τ 1 τ 2 )+k(σ 2 +2)) [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 3 > 0. = 2(1 τ 2)[(σ 2 +2)(C + ak) 2 +2aC 2 (τ 1 τ 2 )] [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 2 > 0, 2 V B c (C; τ 1, τ 2 ; 0; k, a) C 2 = 8a3 k 2 (1 τ 2 )(σ 2 +2)(τ 2 τ 1 ) [(ak + C)(σ 2 +2)+2aC(τ 1 τ 2 )] 3 < 0. Poof (of Poposition 3.12) Fom Poposition 3.10 when an asymmetic tax benefits scheme is intoduced, the endogenous failue level inceases with the switching baie. Thus, it is sufficient to compae the two switching baies, since they ae both independent on fim s cuent assets value V. Poof (of Poposition 4.1) Solving Equation (31) with espect to C we have: C(V B ; τ 1, τ 2 ; δ) = V B (1 + λ 2 ) λ 2 VS λ 1 V B λ 1 (τ 2 τ 1 )+(1 τ 2 ) (51) In this case, taking the deivative of C with espect to the failue level V B we get: C(V B ; τ 1, τ 2 ; δ) = (1 + λ λ 2) VS 1 λ V 1 B (τ 2 τ 1 )(1 + λ 1 )+(1 τ 2 ). V B λ 2 λ VS 1 λ V 1 B (τ 2 τ 1 )+(1 τ 2 ) 2 (52) Notice that V S λ 1 V B λ 1 (τ 2 τ 1 )(1 + λ 1 )+1 τ 2 is geate than inf (1 τ 2, (τ 2 τ 1 )(1 + λ 1 )+(1 τ 2 )) since V S λ 1 V B λ 1 [0, 1]. Obviously 1 τ 2 > 0, and (τ 2 τ 1 )(1 + λ 1 )+1 τ 2 > 0 since it is a linea deceasing function of λ 1,withλ 1 < 0. This function deceases fom + to 1 τ 1 > 0, as λ 1 and λ 1 0 espectively. 37

86 Poof (of Poposition 4.3) Hee we conside the case δ = 0. Fom the smooth pasting condition we have: V B (C; τ 1, τ 2 ; 0) = 2CV S (1 τ 2 ) V S (σ 2 +2)+(τ 1 τ 2 )C. The application C V B (C; τ 1 ; τ 2 ; 0) is inceasing and concave. Substituting this expession fo V B (C; τ 1, τ 2 ; 0) and setting λ 2 = 2 we have that v is the sum of a linea function σ 2 of C and the following non linea function of C: 2C(τ1 τ 2 ) f(c) := V S (σ 2 +2) V B(C; τ 1 ; τ 2 ; 0) + τ 2 2C VB (C; τ 1 ; τ 2 ; 0) σ + αv B (C; τ 1 ; τ 2 ; 0) 2. V (53) We want to pove is that the application C v(v B (C; τ 1 ; τ 2 ; 0)) is a concave function. It is sufficient to pove that the above function f is concave. It is useful to ewite Equation (53) in the following way, studying sepaately the thee tems: f(c) = (f 1 (C)+f 2 (C)+f 3 (C)) V 2 σ 2 with f 1 (C) = 2(τ 1 τ 2 ) V S (σ 2 +2) CV B(C; τ 1 ; τ 2 ; 0) 2 σ 2 +1 (54) f 2 (C) = τ 2C V B(C; τ 1 ; τ 2 ; 0) 2 σ 2 f 3 (C) = αv B (C; τ 1 ; τ 2 ; 0) 2 σ 2 +1 Applications C f 1 (C) and C f 2 (C) ae convex, since: 2 f 1 (C) C 2 = 2(τ 1 τ 2 ) V S (σ 2 +2) 2 f 2 (C) C 2 = τ 2 The application C f 3 (C) is convex: 2V B (C; τ 1 ; τ 2 ; 0) 2 σ 2 +1 (σ 2 +2) 2 VS 2(σ4 +3σ ) σ 4 (V S (σ 2 +2)+(τ 1 τ 2 )C) 2 > 0(55) C 2V B (C; τ 1, τ 2 ; 0) 2 σ 2 VS 2(σ2 +2)(σ 4 +4σ ) σ 4 (V S (σ 2 +2)+(τ 1 τ 2 )C) 2 > 0. (56) C if 2 f 3 (C) C 2 = 2αV 2 B(C; τ 1, τ 2 ; 0) σ 2 +1 (σ 2 +2) 2 (V S (σ 2 +2)+(τ 2 τ 1 )C) σ 4 (V S (σ 2 +2)+(τ 1 τ 2 )C) 2 C 2 > 0 V S > (τ 1 τ 2 )C σ 2 +2 On the othe side by (35) it holds that V S > 2(1 τ 1)C (σ 2 +2), (57) 38

87 theefoe constaint (57) holds if 2 2τ 1 > τ 1 τ 2, which is equivalent to τ 1 < τ 2 3. Unde this last condition, constaint (57) is always satisfied fo 0 τ 2 τ 1, meaning that f 3 (C) is convex τ 2 :0 τ 2 τ 1. Finally the application f(c) :C (f 1 (C)+f 2 (C)+f 3 (C)) is concave since it is the sum of thee concave functions. Refeences [1] ALVAREZ L.H.R. (2001), Rewad Functionals, Salvage Values, and Optimal Stopping, Math. Meth. Ope. Res. 54, [2] AMMANN M. (2001), Cedit Risk Valuation. Methods, Models, and Applications, Spinge, Belin, Heidelbeg, New Yok. [3] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), Capital Stuctue With Fim s Net Cash Payouts, accepted fo publication on a Special Volume edited by Spinge, Quantitative Finance Seies, editos: Cia Pena, Mailena Sibillo. [4] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), An Endogenous Bankuptcy Model with Fim s Net Cash Payouts, Woking Pape. [5] BLACK F., COX, J. (1976), Valuing Copoate Secuities: Some Effects of Bond Indentue Povisions, Jounal of Finance, 31, [6] DECAMPS J.P. and VILLENEUVE S. (2007), Optimal Dividends Policy and Gowth Option, Finance and Stochastics, 11(1), [7] DIXIT, A. (1993), The At of Smooth Pasting, Fundamentals of pue and Applied Economics, 55. [8] DOROBANTU D. (2008), Aêt Optimal pou les Pocessus de Makov Fots et les Fonctions Affines, submitted to CRAS, [9] DOROBANTU D., MANCINO M., PONTIER M. (2009), Optimal Stategies in a Risky Debt Context, Stochastics An Intenational Jounal of Pobability and Stochastic Pocesses, 81:3, [10] DOROBANTU D. (2007), Modélisation du Risque de Défaut en Entepise, Ph.D. Thesis Univesity of Toulouse, [11] DUFFIE D. and SINGLETON K.J. (2003), Cedit isk: Picing, Measuement, and Management, Pinceton : Pinceton Univesity Pess. [12] EL KAROUI N. (1981), Les Aspects Pobabilistes du Contôle Stochastique, Lectue Notes in Mathematics 876, p , Spinge-Velag, Belin. 39

88 [13] GERBER H.U. and SHIU E.S.W. (1994), Matingale Appoach to Picing Pepetual Ameican Options, ASTIN Bulletin 24, [14] GRAHAM, J.R. and SMITH, C.W., (1999), Tax Incentives to Hedge, Jounal of Finance, 54, [15] KARATZAS I. and SHREVE S. (1988), Bownian Motion and Stochastic Calculus, Spinge, Belin, Heidelbeg, New Yok. [16] KYPRIANOU A.E. and PISTORIUS M.R. (2003), Pepetual Options and Canadization though Fluctuation Theoy, The Annals of Applied Pobability 13(3), [17] LELAND H.E. (1994), Copoate Debt Value, Bond Covenant, and Optimal Capital Stuctue, The Jounal of Finance, 49, [18] LELAND H.E. (1998), Agency Costs, Risk Management, and Capital Stuctue, Woking Pape. [19] LELAND H.E. and TOFT K.B. (1996), Optimal Capital Stuctue, Endogenous Bankuptcy and the Tem Stuctue of Cedit Speads, The Jounal of Finance, 51, [20] LONGSTAFF F. and SCHWARTZ E. (1995), A Simple Appoach to Valuing Risky Fixed and Floating Rate Debt and Detemining Swap Speads, The Jounal of Finance, 50, [21] MAUER, D. and SARKAR, S., (2005), Real options, agency conflicts, and optimal capital stuctue, Jounal of Banking & Finance, 29, [22] MERTON R. C. (1973), A Rational Theoy of Option Picing, Bell Jounal of Economics and Management Science, 4, [23] MERTON R. C. (1974), On the Picing of Copoate Debt: The Risk Stuctue of Inteest Rates, The Jounal of Finance, 29, [24] MODIGLIANI F. and MILLER M. (1958) The Cost of Capital, Copoation Finance and the Theoy of Investment, Ameican Economic Review, 48, [25] PHAM H. (2007), Optimisation et Contôle Stochastique Appliqués à la Finance, Spinge, SMAI, Belin. [26] SARKAR S. and GOUKASIAN L. (2006), The Effect of Tax Convexity on Copoate Investment Decisions and Tax Budens, Jounal of Public Economic Theoy, 8 (2),

89 [27] SARKAR S. (2008), Can Tax Convexity be Ignoed in Copoate Financing Decisions?, Jounal of Banking & Finance, 32, [28] SHIRYAEV A. (1978), Optimal Stopping Rules, Spinge-Velag, New-Yok. [29] VILLENEUVE S. (2007), On the Theshold Stategies and Smooth-fit Pinciple fo Optimal Stopping Poblem, Jounal of Applied Pob. 44, [30] ECKBO B.E. (2007), Handbook of Copoate Finance: Empiical Copoate Finance, Elsevie. 41

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91 III Volatility Risk

92

93 Optimal Capital Stuctue with Endogenous Default and Volatility Risk Flavia Basotti Dept. of Statistics and Applied Mathematics Univesity of Pisa, Italy Abstact This pape analyzes the capital stuctue of a fim in an infinite time hoizon famewok following Leland [12] unde the moe geneal hypothesis that the fim s assets value pocess belongs to a faily lage class of stochastic volatility models. In such a scheme, we descibe and analyze the effects of stochastic volatility on all vaiables which constitute the capital stuctue. The endogenous failue level is deived in ode to exploit the optimal amount of debt chosen by the fim. To this aim we deive and popose a coected vesion of the smooth-fit pinciple unde volatility isk in ode to detemine the optimal stopping poblem solution. Exploiting optimal capital stuctue we found that the stochastic volatility famewok seems to be a obust way to impove esults in the diection of both highe speads and lowe leveage atios in a quantitatively significant way. Keywods: stuctual model; volatility isk; volatility time scales; endogenous default; optimal stopping. 1 Intoduction In this pape we extend the study of the optimal capital stuctue with endogenous default poposed by Leland [12] assuming that the fim value pocess belongs to a faily lage class of stochastic volatility models. The main empiical esults in cedit isk liteatue have emphasized a poo job of stuctual models in pedicting cedit speads [5]; this weakness of the modeling could be elated to the diffusion assumptions made in the papes by [12, 13]. Thus Leland suggests in [14] that a possible impovement to these esults could ensue fom intoducing jumps and/o emoving the assumption of constant volatility in the undelying fim value stochastic evolution. The fome extension has been addessed 1

94 in [10] who extend the analysis to allow the value of the fim s assets to make downwad jumps, in paticula they suppose that the dynamics of the fim s assets is diven by the exponential of a Levy pocess; the authos find explicit expession fo the bankuptcy level, while fo the value of the fim and the value of its debt they no longe have closed fom expessions. In [4] models the fim s asset as a double exponential jump-diffusion pocess and [11] study Black-Cox cedit famewok unde the assumption that the log-leveage atio is a time changed Bownian motion. Futhe [6] conside a pue jump pocess of the Vaiance-Gamma type. To the best of ou knowledge, the latte extension of [12] to a geneal class of stochastic volatility models has not be addessed. Thus the aim of this pape is to study the optimal capital stuctue of a fim inside a stuctual model with endogenous bankuptcy in the spiit of [12], but assuming a stochastic volatility fo the fim s assets. We intoduce a pocess descibing the dynamic of the diffusion coefficient diven by a one facto mean-eveting pocess of Ostein-Uhlenbeck type, negatively coelated with fim s assets value evolution. Diffeently fom the classical Leland famewok and even fom the moe geneal context with payouts and asymmetic copoate tax ates studied in [1, 2, 3], the key point elies in the fact that inside this famewok we cannot obtain explicit expessions fo all the vaiables involved in the capital stuctue by means of the Laplace tansfom of the stopping failue time, because this tansfom, which was the key tool, is not available in closed fom in ou stochastic volatility famewok. Nevetheless debt, equity, bankuptcy costs and tax benefits ae claims on the fim s assets, thus we apply ideas and techniques developed in [7] fo the picing of deivatives secuities whose undelying asset pice s volatility is chaacteize by means of its time scales fluctuations. This appoach has been applied in [8] to the picing of a defaultable zeo coupon bond. Hee we conside a one-facto stochastic volatility model and apply single petubation theoy as in [7] in ode to find appoximate closed fom solutions fo vaiables involved in ou poblem. We analyze all financial vaiables and study the effects of the stochastic volatility assumption on the endogenous failue level detemined by equity holdes maximizing behavio. All claims have a moe complicated expession with espect to the constant volatility case, depending not only on the pocess descibing fim s activities value, but also on the pocess diving the diffusion coefficient. Equity holdes still face the poblem of optimizing equity value w..t. the failue level. Nevetheless unde ou appoach, the failue level deived fom standad smooth pasting pinciple is not the solution of the optimal stopping poblem, but only epesents a lowe bound which has to be satisfied due to limited liability of equity. Choosing that failue level would mean an ealy execise of the option to default. A coected smooth pasting condition must be applied in ode to find the endogenous failue level solution of the optimal stopping poblem. Moeove, we show the convegence of ou esults to Leland case [12] as the paticula case of zeo-petubation. Picing and hedging poblems elated to equity makets suggest that a picing model with stochastic volatility is seen a fundamental featue in modeling the undelying assets 2

95 dynamic. Empiical findings about stuctual models of cedit isk show that this kind of models usually undeestimates speads and default pobabilities, while pedicted leveage atios ae too high. In [1, 2, 3], we showed how to modify a pue Leland model in ode to obtain a moe ealistic famewok and also empiical pedictions about default ates, leveage and speads moe in line with histoical noms. In this pape by taking into account the stochastic volatility isk component of the fim s asset dynamic, ou aim is to bette captue exteme etuns behavio which could be a obust way to impove empiical pedictions about speads and leveage. Intoducing andomness in volatility allows to deal with a stuctual model in which the distibution of stock pices etuns is not symmetic: in ou mind this seems to be the ight way fo captuing what stuctual models ae not able to explain with a constant diffusion coefficient. The numeical esults show that the assumption of stochastic volatility model poduces elevant effects on the optimal capital stuctue in tems of highe cedit speads and lowe leveage atios, if compaed with the oiginal Leland case. Moeove the coected smooth-fit pinciple seems to be an issue to develop in ode to analyze the optimal execise time of Ameican-style options unde a stochastic volatility picing model (see also [15]). The pape is oganized as follows. Section 2 descibes the stochastic volatility picing model. Section 3 povides a detailed analysis of defaultable claims valuation in all mathematical aspects. In Section 4 we fully exploit the capital stuctue of the fim unde volatility isk poviding appoximate values fo all deivatives depending on fim s cuent assets value. Section 5 shows numeical esults about optimal capital stuctue, then Section 6 gives some concluding emaks. 2 The model We intoduce a pocess descibing the dynamic of the diffusion coefficient diven by a one facto mean-eveting pocess of Ostein-Uhlenbeck type, negatively coelated with fim s assets value evolution. Fom an economic point of view how to chose the diffusion coefficient, i.e. how to model volatility is a fundamental issue. Assuming the diffusion coefficient being constant means assuming that the iskiness of the fim does not change though time. While volatility is not an obsevable vaiable, maket data suggest that the iskiness of the fim can deeply vay in time. Theefoe the economic intuition is that analyzing the capital stuctue of a fim in a stochastic volatility picing famewok should be a obust and flexible way to impove stuctual models pedictions binging them close to empiical evidence. We stess obust and flexible since we will suppose fim s activities value belonging to a faily lage class of stochastic volatility models. We conside a fim whose (unleveed) activities value dynamic is descibed by pocess V t. Pocess Y t is intoduced to descibe the evolution of the diffusion coefficient. Unde 3

96 the physical measue P the dynamics of the model ae descibed by the following SDEs in R 2 : dv t = µv t dt + f(y t )V t dw t, (1) dy t = η(m Y t )dt + βd W t, (2) whee dw, W t = ρdt and β 2 =2ν 2 η, η =1/, and f( ) is supposed to be bounded and Lipschitz. Paamete µ epesents the expected ate of etun of fim s assets value. We also suppose f(y ) L 2 (Ω [0, ]) and that thee exist two constants a and A such that 0 <a f A to avoid explosion. The above SDE admits a unique stong solution. Y t is a Gaussian pocess, which is explicitly known given the initial condition Y 0 = y: Y t = m +(y m)e ηt + β t 0 e η(t s) d W s, (3) The invaiant distibution of Y t, obtained as t is N(m, β2 2η ) and the impotant featue to stess is that it does not depend on the initial condition y. Substituting β 2 =2ν 2 η and η = 1,wehave: Y t = m +(y m)e t + ν 2 t 0 e (t s) d W s. If we conside 0, then almost suely Yt m, so does diffusion paamete f(y t ) f(m). Moeove, following [7] (pg ) we assume ρ < 0. Analyzing financial data suggests the existence of a negative leveage effect between stock pices and volatility, i.e. ρ < 0, since eal data shows that pices tend to decease when volatility ises. In ou model we conside this coelation ρ being constant, also if we know that it could be vaying though time. Fom stuctual models theoy we know that each component of the capital stuctue of the fim can be seen as a claim on the undelying assets epesented by fim s activities value V. Thus, in ode to find the values of these claims, the picing poblem is addessed unde a isk neutal pobability measue Q, whee the asset s evolution follows the SDEs (cf. [7] p. 31): dv t = V t dt + f(y t )V t dw t, (4) dy t = (η(m Y t ) βλ(y t )) dt + βd W t, (5) whee is the constant isk fee ate and Λ(Y t ) is defined as: Λ(Y t )=ρ µ f(y t ) + γ(y t) 1 ρ 2, (6) µ whee f(y is the excess etun-to-isk atio, and γ(y t) t)istheisk pemium facto o maket pice of volatility isk which allows to take into account the second souce of andomness W t diving the volatility pocess. Following [8] we assume γ( ) being bounded and a function of y only 1. 1 As paticula case we will also conside γ being a constant when facing the hedging poblem. 4

97 Fom now on the aim is to develop a picing model fo the capital stuctue of a fim whose undelying assets value is V t, without specifying a paticula function f( ). This will allow to pesent esults which ae not stictly dependent on a specific volatility pocess but instead elated to a faily geneal class of one-facto pocesses and in this sense they ae obust, i.e. model-independent. Following stuctual models appoach the aim is to analyze the capital stuctue of a fim in tems of deivative contacts. In the spiit of [12] we conside an infinite time hoizon and a fim which is issuing both equity and debt. The fim issues debt and debt is pepetual. Debt holdes eceive a constant coupon C pe instant of time. We assume that fom issuing debt the fim obtains tax deductions popotional to coupon payments. The copoate tax ate τ is assumed to be constant, thus the fim will benefit of a taxshelteing value of inteest payments τc. The fim is subject to the isk of default, thus: when coupon payments ae low, the total value of the fim ises with an incease in C due to tax benefits of debt, but as C eaches a cetain level, the total value of the fim deceases, due to bankuptcy costs. This is the tade-off between taxes and bankuptcy costs. Default is endogenously tiggeed.the economic intuition is that default aives when the fim is not able to cove its debt obligations, meaning when equity value is null (due to limited liability). The mathematical tool to teat default is that the failue passage time is detemined when fim s activities value falls to some constant level V B. The value of V B is endogenously deived by equity holdes in ode to maximize equity value. We define the stopping time T VB =inf{t 0: V t = V B }, moeove, since pocess V is ight continuous, it holds V TB = V B. Following [12, 17] contingent claim valuation can be used, so it is possible to expess each component of the capital stuctue as a claim on the undelying assets epesenting fim activities value (see also [1, 2, 3]). 3 Picing Defaultable Claims unde Volatility Risk In this section we apply ideas and techniques developed in [7] fo the picing of deivatives secuities whose undelying asset pice s volatility is chaacteized by means of its time scales fluctuations. We conside a one-facto stochastic volatility model and apply single petubation theoy as in [7] in ode to find appoximate values fo defaultable deivatives involved in ou poblem. This appoach has been applied in [8] to the picing of a defaultable zeo coupon bond. We will conduct the analysis consideing the moe geneal case of defaultable paying-dividend deivatives. We ae assuming that fim s activities value belongs to a faily lage class of stochastic volatility models. In this famewok we obtain vey geneal expessions fo all claims 5

98 descibing the components of the capital stuctue, each one of them depending on both pocesses V and Y. Actually both pocesses V and Y depend on the paamete but we will omit it to have a simple notation. The aim is to detemine how the capital stuctue is affected by volatility isk, meaning both a qualitative and quantitative analysis of such an influence. Let the pice of a geneal claim be F (V t,y t ), function F being supposed to be Cb 2, depending on the paamete. We conside this geneal claim as a dividend-paying contact, thus by no abitage hypothesis, the pocess t e t F (V t,y t )+d t 0 e s ds which epesents the value of the claim at time t, discounted at the isk fee ate, plus cumulated dividends up to time t, discounted at and d is the constant dividend paid by the claim, has to be a local matingale. Moeove, being a tue matingale T stopping time, yields: T E e T F (V T,Y T )+d e s ds V 0 = x, Y 0 = y = F (x, y), (7) 0 whee the expectation is taken in the isk neutal measue (see [17]). Thus, the function F (x, y) has to satisfy the following patial deivatives equation: d F (x, y)+x x F 1 ν 2 (x, y)+ (m y) Λ(y) y F (x, y)+ (8) x2 f 2 (y) x 2 F (x, y)+ρν 2 f(y)x 2 xyf (x, y)+ν y F (x, y) =0. 2 Re-aanging tems we have: d F (x, y)+x x F (x, y)+ 1 2 x2 f 2 (y) x 2 F (x, y) (9) (m y) y F (x, y)+ν 2 y 2 F (x, y) 2 2 +ρν f(y)x 2 xyf (x, y) ν 2 Λ(y) y F (x, y) =0. Since diffeential equation (9) involves tems of ode 1, 1, 1, we intoduce the following notation fo any C 2 b whee L 0 g(x, y) function g(x, y): = (m y) y g(x, y)+ν 2 y 2g(x, y), L 1 g(x, y) = ρ 2νf(y)x xy g(x, y) 2νΛ(y) y g(x, y), (10) (L BS (f))g(x, y) = d g(x, y)+x x g(x, y)+ 1 2 x2 f 2 (y) x 2g(x, y). 6

99 L 0 is the infinitesimal geneato of an egodic Makov pocess, involving only y vaiable; L 1 is the opeato depending on the mixed deivative xy, since we ae supposing a coelation ρ between the Bownian motions of pocesses V and Y ; t f + L BS (f) is the Black-Scholes opeato when the volatility level is f(y) and the claim pays a constant dividend d. It is now possible to wite diffeential equation (9) in the following way: 1 L L 1 + L BS (f) F (x, y) =0, x R +, y R. (11) We now discuss the bounday conditions of this poblem. Since we ae consideing an infinite hoizon, a teminal condition equies F (x,y) x being bounded when x in ode to avoid bubbles (see [4]). Futhe a bounday condition at default is given by F (x B,y)=f (x B )wheef (x B ) depends on the specific claim we ae consideing and x B is the failue level. This last bounday condition is stictly elated to the economic meaning of each specific claim (i.e. fo equity we have f E (x B ) = 0, while fo debt f D (x B )= (1 α)x B since the stict pioity ule holds). Fom an economic point of view this last bounday condition simply means that tax benefits ae completely lost in the event of bankuptcy. The solution F (x, y) of Equation (11) exists and is unique > 0. Following [7], we expand the solution F in powes of : F = P 0 + P 1 + P 2 + P , (12) whee P 0,P 1,P 2,... ae functions of (x, y) to be detemined such that P 0(x,y) x bounded as x and P 0 (x B,y) = 0. Substituting Equation (12) into Equation (11) we have 1 L P0 L 1 + L BS (f) + P 1 + P 2 + P =0. So we obtain: 1 L 0P L 1 P 0 + L BS (f)p 0 + L BS (f)p L 0 P 1 + L 1 P 1 L 0 P 2 + L 1 P 2 + L BS (f)p 2 + L 0 P 3 + L 1 P 3 + L BS (f)p =0 Re-aanging tems (ef. Eq. (5.16) pag. 90): will be 1 L 0P (L 1 P 0 + L 0 P 1 )+ (13) + (L BS (f)p 0 + L 1 P 1 + L 0 P 2 ) + (L 0 P 3 + L 1 P 2 + L BS (f)p 1 ) +... (14) = 0. 7

100 Equating tems of ode 1,wemusthave L 0 P 0 =0. Since the opeato L 0 in Equation (10) is the infinitesimal geneato of an egodic Makov pocess acting only on y vaiable, (and P 0 has to be bounded) P 0 must depend only on x vaiable. Similaly, to eliminate the tem of ode 1,wemusthave: L 1 P 0 + L 0 P 1 =0. Obseve that the opeato L 1 in Equation (10) involves only the deivative w..t. y vaiable. Since P 0 only depends on x vaiable, as a consequence we have L 1 P 0 = 0 and what emains is L 0 P 1 = 0. Using the same agument as above fo the opeato L 0,wehavetofindP 1 as function of x vaiable only. Remak 3.1 Obseve that if we conside only P 0 and P 1 tems, this implies that we ae looking fo an appoximate value of the claim F (x) P 0 (x)+ P 1 (x), and this solution does not depend on y vaiable, meaning that it does not depend on the pesent volatility f 2 (y). Now ecall Equation (13). What we have to do is continuing to eliminate tems of ode 1,,... and so on. The idea is to study asymptotic appoximations fo F which become moe accuate as 0. At this point the poblem is to solve the following equation: L BS (f)p 0 + L 1 P 1 + L 0 P 2 =0. (15) Since L 1 involves the mixed deivative xy and P 1 must depend only on x vaiable, we ae sue that L 1 P 1 =0. What emains is: L BS (f)p 0 + L 0 P 2 = 0 (16) The vaiable x being fixed, focusing on the dependence of L BS (f) on y, Equation (16) is a Poisson equation (cf.[7], p. 91) which admits a unique solution P 2 only if L BS (f)p 0 =0, whee g := g(y)φ(y)dy, (17) R 8

101 and Φ(y) denotes the density function of the Gaussian distibution N (m, ν 2 ). Following [7] we define the effective volatility as σ 2 = f 2. (18) As a consequence, L BS (f) = L BS ( σ), and the zeo-ode tem P 0 will be the solution of as shown in the following Poposition. L BS ( σ)p 0 = 0 (19) Poposition 3.2 Equation (19) admits as unique solution the zeo-ode tem P 0 with the following fom: P 0 (x) =k + l ( x B x )λ (20) with λ = 2 σ 2, (21) and k,l depend on the specific dividend and bounday conditions of each claim. Poof We ae looking fo a solution P 0 satisfying the following ODE P 0 (x) L BS ( σ)p 0 =0, with bounday conditions: x to be bounded as x, and P 0 (x B )=f (x B ), whee f (x B ) depends on the specific claim we ae consideing. Recalling Equation (10) fo Black-Scholes opeato L BS ( σ), we have to find P 0 (x) depending only on x vaiable, as solution of d P 0 (x)+xp 0(x)+ 1 2 σ2 x 2 P 0 (x) =0, (22) whee d is the dividend paid by the claim consideed. Equation (22) is exactly what we have to solve when consideing Leland famewok assuming a constant volatility σ. Bounday conditions fo each claim ae needed to detemine k,l as we will show late. The following Poposition povides the value of each specific claim descibing the capital stuctue of the fim, i.e. equity value, debt, tax benefits, bankuptcy costs and total value of the fim, unde a picing model with constant volatility equal to σ. 9

102 Poposition 3.3 Unde ou assumption of stochastic volatility, the capital stuctue of the fim has the following P 0 tems: P TB 0 (x, x B )= τc τc xb λ, (23) x P D 0 (x, x B )= C + (1 α)x B C xb λ, (24) x P BC xb λ 0 (x, x B )=αx B, (25) x P E (1 τ)c (1 τ)c xb λ 0 (x, x B )=x + x B, (26) x P V (1 τ)c (1 τ)c xb λ 0 (x, x B )=x + x B. (27) x Poof Fom Poposition 3.2 we know that each claim will be of the fom with P 0 (x) =k + l ( x B x )λ λ = 2 σ 2. We will define a claim on x fo each component of the capital stuctue: equity (E), tax benefits (TB), debt (D), bankuptcy costs (BC) and the total value of the fim (V). Recall that d denote the dividend paid by each claim. Bounday conditions specific fo each claim will give k and l as shown in the table below. Claim TB D BC E V P 0 (x B,x B ) 0 (1 α)x B αx B 0 (1 α)x B k =lim x P 0 (x,x B ) x τc C 0 x (1 τ)c x + τc l τc (1 α)x B C αx B (1 τ)c x B αx B + τc d τc C 0 (1 τ)c τc Table 1: This table shows bounday conditions fo each specific claim: ow 1 descibes bounday conditions at default, i.e. fo x x B; ow 2 descibes bounday conditions as x. We now seach fo the the second ode coection tem P 2, in ode to be able to define an equation allowing us to find P 1. The following poposition holds. 10

103 Poposition 3.4 The solution of Equation (16) is the second-ode coection tem P 2 depending on both x and y vaiables as follows: P 2 (x, y) = 1 2 φ(y)x2 2 P 0 (x) x 2, (28) with P 0 (x) given by Equation (20) and φ(y) being solution of ν 2 φ (y)+(m y)φ (y) =f(y) 2 σ 2, (29) o equivalently ν 2 Φ(y)φ (y) = y (f 2 (z) σ 2 )Φ(z)dz. (30) Poof We ae seaching the second ode coection tem P 2 as solution of Equation (16). Since L BS ( σ)p 0 =0, we can wite L BS (f)p 0 (x) =L BS (f)p 0 (x) L BS ( σ)p 0 (x) = 1 2 So, fom Equation (16), it emains to find P 2 such that L 0 P 2 = 1 2 f(y) 2 σ 2 x 2 2 P 0 (x) x 2, f(y) 2 σ 2 x 2 2 P 0 (x) x 2. whee L 0 is given by Equation (10) being an opeato involving only the deivative w..t. y vaiable. Thus P 2 (x, y) = 1 2 φ(y)x2 2 P 0 (x) x 2, with φ(y) being solution of L 0 φ(y) = f(y) 2 σ 2. Following the same methodology as befoe, we have to impose the coefficient of in Equation (13) being null: L 0 P 3 + L 1 P 2 + L BS (f)p 1 =0. (31) We now have a Poisson equation which admits a unique solution P 3 only if L 1 P 2 + L BS (f)p 1 =0. (32) Equation (32) and (28) will lead us to find the fist ode coection tem P 1 (x) fo each claim defining a specific component of fim s capital stuctue (i.e. equity, debt, tax benefits of debt, bankuptcy costs, total value) as we show in the following Poposition. 11

104 Poposition 3.5 Unde ou assumption of stochastic volatility, assuming bounday conditions P 1 (x B,x B,C)=0and lim x P 1 (x, x B,C)=0, the capital stuctue of the fim has the following component fo fist coection tems P 1 (x, x B ): whee P TB 1 (x, x B )= l TB H ( x B x )λ log x B x P D 1 (x, x B )= l D H ( x B x )λ log x B x P BC 1 (x, x B )= l BC H ( x B x )λ log x B x P E 1 (x, x B )= l E H ( x B x )λ log x B x P V 1 (x, x B )= l V H ( x B x )λ log x B x (33) (34) (35) (36) (37) H = 4 2 σ 4 2 νλφ σ 2 v 3, v 3 = ρ 2 νfφ, (38) and l TB = τc, ld =(1 α)x B C, lbc = αx B, (39) l E (1 τ)c = x B, l V = αx B + τc. Poof We can ewite Equation (32) in this way (L BS ( σ)p 1 )(x) = R L 1P 2 (x, y)φ(y)dy. (40) Recall equation (28) fo the second ode coection tem P 2 : P 2 (x, y) = 1 2 φ(y)x2 2 P 0 (x) x 2 and Definition (10) fo the opeato L 1.Wehave L 1 P 2 (x, y) = ρ 2νf(y)x 2 xy 1 2 x2 2 P 0 = ρ 2νf(y)x x 1 2 x2 2 P 0 x 2 x 2 φ(y) 2νΛ(y) y φ (y) 2νΛ(y)φ (y) 1 2 x2 2 P 0 x 2 φ(y) 1 2 x2 2 P 0 x 2 (41) (42) 12

105 In ode to find P 1 we have to solve the following equation L BS ( σ)p 1 (x, x B,C) = v 3 x 3 3 P 0 x 3 + v 2x 2 2 P 0 meaning equation L BS ( σ)p 1 (x, x B,C)=ρ with v 2 = 2ρν fφ v 3 = ρ x 2, (43) 2 2 ν Λφ 2 =2v 3 2 ν Λφ, (44) 2 2 ν fφ, (45) 2 2 ν fφ x 3 3 P 0 2ρν x 3 + fφ 2 2 ν Λφ x 2 2 P 0 x 2. (46) Recall the geneal stuctue given by Equations (20) fo the zeo ode tems P 0 (x): P 0 (x) =k + l ( x B x )λ, with λ = 2 σ 2, whee k,l ae given in Table 3. Obseve that k is constant w..t. x fo all claims except equity and fo equity value it is linea w..t. x. The following holds fo each claim, namely : Finally P 0 (x) = k x x l λx λ B x λ 1 (47) 2 P 0 (x) x 2 = l λ(λ + 1)x λ B x λ 2 (48) 3 P 0 (x) x 3 = l λ(λ + 1)(λ + 2)x λ B x λ 3 (49) 2 L BS ( σ)p 1 (x, x B,C) = l λ 2 (λ + 1) 2 ρν fφ x B λ 2 x 2 ν Λφ l xb λ λ(λ + 1) x = l xb λ λ(λ + 1) (v 2 (λ + 2)v 3 ) x 2 = l λ(λ + 1) 2v 3 2 ν Λφ xb λ (λ + 2)v 3 x 2 L BS ( σ)p 1 (x, x B,C) = l λ(λ + 1) 2 ν Λφ xb λ + λv 3. (50) x 13

106 Recall Definition (10) of the Black-Scholes opeato: L BS ( σ)p 1 (x, x B,C)=d P 1 (x, x B,C)+xP 1(x)+ 1 2 σ2 x 2 P 1 (x) Thus P 1 is the solution of: with d P 1 (x)+xp 1(x)+ 1 2 σ2 x 2 P 1 (x) =Bx λ (51) B = l λ(λ + 1) (v 2 (λ + 2)v 3 ) x λ B, (52) 2 = l λ(λ + 1) 2 ν Λφ + λv 3 x λ B. (53) This second ode ODE admits P 1 given by Equation (36) as unique solution, taking into account bounday conditions lim P 1(x, x B )=0, (54) x The solution is P 1 (x B,x B )=0. (55) P 1 (x, x B )=A 0 + A 1 x + A 2 x λ + A 3 x λ log x, Fom bounday condition (54)-(55) we have espectively : A 0 =0,A 1 = 0; A 2 = A 3 log x B, thus the P 1 component of the fist coection tem is of the fom with A 3 = 2B 2+ σ 2, meaning : P 1 (x, x B )= A 3 x λ log x B x, A 3 = 2l λ(λ + 1) (v 2 (λ + 2)v 3 ) x λ B 2 + σ 2 2 2l λ(λ + 1) 2 νλφ + λv 3 x λ B = 2 + σ 2 = l 4 2 σ 4 2 ν Λφ + 2 σ 2 v 3 x λ B. (56) P 1 = l λ 2 (λ + 1) 2ρν fφ x λ B 2 + σ 2 x λ log x B x. 14

107 Finally we can wite P 1 in tems of v 3 as: P 1 = l 4 σ 4 The final esult with this method is: Theefoe we can wite: whee which simplifies to 2 2 ν Λφ + 2 σ 2 v 3 P 1 (x, x B )= l ρ 2νfφ 42 σ 6 xb x xb λ x B log x x. λ log xb x. (57) P 1 (x) = l H ( x B x )λ log x B x, (58) H = 2λ(λ + 1)(v 2 (λ + 2)v 3 ) 2 + σ 2, (59) H = 4 2 σ 4 2 ν Λφ + 2 σ 2 v 3, (60) and fo each specific claim, the tem l is given in Table 3, ow 3: l TB = τc, ld =(1 α)x B C, lbc = αx B, l E = (1 τ)c x B, l V = αx B + τc. 4 Capital Stuctue of the Fim unde Volatility Risk We now analyze in detail all financial vaiables defining the capital stuctue of the fim by using pevious esults in ode to detemine how thei values ae affected by volatility isk. We can intepet each financial vaiable as a deivative contact, thus the aim is to undestand how the pice of each claim must be coected due to volatility isk. We ae assuming the diving Onstein-Uhlenbeck pocess fo the diffusion coefficient being a fast mean eveting pocess, with η := 1 as speed of mean-evesion. By applying singula petubation theoy, each component of the capital stuctue will be defined as a claim whose value can be found though an asymptotic expansion of the pice in powe of. One of the main poblems concening petubation theoy is that, once we have explicit expessions fo the appoximate value of ou claims, we have to calibate paametes. The main issue is how to estimate σ fom maket data since a volatility pocess is not obsevable in financial makets. Usually the appoach is to estimate σ fom histoical data and then 15

108 to calibate the small paametes V 2,V 3. In this pape we do not addess the issue of how to calibate paametes, but we take esults fom [7], [8], [9] whee a specific and detailed analysis of this topic is conducted. The value of each claim F (x, x B ) will be appoximate with the following coected picing fomula: F (x, x B )=P 0 (x, x B )+ P 1 (x, x B ), (61) whee P 0 (x, x B ) and P 1 (x, x B ) ae obtained, espectively, as solutions of Equations (19), (46). Remak 4.1 i) Let P1 (x, x B ):= P 1 (x, x B ). Obseve that an equivalent fomulation fo F (x, x B ) is to wite diectly each appoximate claim as solution of with L BS ( σ)(p 0 (x, x B )+ P 1 (x, x B )) = V 2 x 2 2 P 0 x 2 + V 3x 3 3 P 0 x 3, (62) V 2 = v 2, V 3 = v 3, and v 2,v 3 ae given by Equations (44)-(45). Coefficients V 2,V 3 coespond to the notation used in [7] (Equations ( ), page 95 whee paamete α in the book coesponds to ou η = 1 ). This esult follows by applying L BS ( σ) to function P 0 + P 1 (fo P 1 look at Equation (46)). Following [7] we can intepet the ight hand side of Equation (62) as a path dependent payment steam which coects the pice accounting fo volatility andomness duing the path befoe the default time. Notice that it may be positive o negative depending on the specific P 0 claim we ae dealing with, since it involves its second and thid deivatives w..t. x, meaning that it will be stictly elated to the value of each claim in a pue Leland [12] model. Moe specifically, this expession involves Geeks of the coesponding pice in a Black and Scholes setting with constant volatility σ. While the second deivative is well known to be the Γ of the P 0 deivative, [7] popose to name the thid deivative Epsilon defined as Γ x. ii) As a consequence of i), we can obseve what follows about the sign of coefficient H given in Equation (59): H = 2λ(λ + 1)(v 2 (λ + 2)v 3 ) 2 + σ 2 = 2 2ν σ 4 Λφ + 2 σ 2 ρfφ, (63) thus the sign of H is the same as Λφ + 2 ρfφ. σ 2 iii) Following [8] we will assume ρ < 0, V 2 < 0 meaning ρfφ < Λφ and V 3 > 0 meaning ρfφ > 0 (see [7] and [9] fo a detailed analysis about paametes calibation), thus obtaining H>0. 16

109 Fom pevious esults we can now wite each appoximate claim, namely, on x with the following stuctue: F (x, x B )=P 0 (x, x B ) xb l H x λ log x B x, (64) with H = 2 2ν σ 4 Λφ + 2 σ 2 ρfφ > 0, and l given in Table 3 depending fo each specific claim on its own bounday conditions. Notice that: i) assuming H>0, the sign of the coection tem P 1 (x, x B ) is the same as l, thus can be positive o negative depending only the specific bounday conditions of each deivative contact. Moe specifically, fom an economic point of view, it will depend on the final payoff of each deivative. An impotant featue to stess is that this tem allows to coect the pice accounting fo andomness in volatility duing the path of pocess x t, i.e. fo x>x B in a model-independent way. Results hold fo a lage class of pocesses, since we ae not specifying a paticula function f( ) fo the diffusion coefficient. ii) H-dependence on V 2,V 3 is fundamental in ode to analyze the stochastic volatility effects on all defaultable claims defining the capital stuctue of the fim, since both coefficients V 2 and V 3 involved in ou picing poblem have a pecise economic intepetation. Fom [8] we know that calibating V 2,V 3 fom maket data suggests to assume these small paametes being espectively: V 2 < 0, V 3 > 0. Typically coefficient V 2 < 0 since it epesents a coection fo the pice in tems of volatility level, while V 3 is the skew effect elated to the thid moment of stock pices etuns. Assuming a negative coelation ρ < 0 in this picing model with stochastic volatility will poduce its effect on the distibution of stock pices etuns, making it asymmetic. In paticula it will stongly modify the left tail of etuns distibutions, making it fatte. An impotant issue to conside is the accuacy of these appoximations. Let F (x, x B ) be the tue unknown value of the claim unde stochastic volatility with teminal condition F (x, x B )=z(x). As analyzed in detail in [7] (see Chapte 5), it can be shown that when z(x) is smooth and bounded, we have F (x, x B ) (P 0 (x, x B )+ P 1 (x, x B )) k, whee k is a constant which does not depend on the speed of mean evesion of the volatility pocess but may depend on its cuent level y. Remak 4.2 It is impotant to stess that as 0 the model becomes a pue Leland model with constant volatility σ, whilev 3 in Equation (62) vanishes as ρ becomes 0, meaning the uncoelated case becomes a pue Leland model with constant volatility equal to the 17

110 coected effective volatility σ = σ 2 2V 2 = σ. The diffeence between the two cases will be only a volatility level coection due to the maket pice of isk which exists also if we assume ρ =0, Λ = 0: V 2 will still coect claims values fo the maket pice of volatility isk. Typically, the effective coected volatility σ is highe that the histoical aveage volatility σ and this is why maket data suggest to assume V 2 < 0. These two cases coincide only if we assume ρ =0, Λ =0. All these featues will poduce impotant economic implications on the analysis of the capital stuctue of the fim, as shown in what follows. 4.1 Equity We now conside equity value unde stochastic volatility. Using singula petubation theoy we have appoximate equity value Ẽ(x, x B)with the following fom: Ẽ(x, x B )=P 0 E (x, x B )+ P 1 E (x, x B ), with P E 0 and P E 1 given by Equations (26), (36). Its explicit expession is: (1 τ)c (1 τ)c xb λ Ẽ(x, x B )=x + x B 1 H log x B x x, (65) with H = 2 2ν σ 4 Λφ + 2 σ 2 ρfφ > 0. Conside now Equation (65): e-aanging tems we can always expess appoximate equity value as the sum of two diffeent components by isolating equity value without isk of default and the option to default embodied in equity unde this picing model with stochastic volatility: Ẽ(x, x B )=f(x, C)+ g(x, x B ), with and (1 τ)c f(x, C) =x, (1 τ)c xb λ g(x, x B )= x B h (x, x B ), (66) x h (x, x B ):= 1+ H log x. (67) x B 18

111 Since H>0, we have h (x, x B ) > 1 x x B, x B [0,x]. (68) We can intepet appoximate equity Ẽ(x, x B) as a new deivative contact whose value deives fom two diffeent souces. Fom an economic point of view f(x, C) is equity value without isk of default and unless limit of time, and it is exactly the same value we have in a pue Leland [12] famewok because this function doesn t depend diectly on the failue level x B. As a consequence, the stochastic volatility assumption does not poduce any effect on it, since this tem is independent of the pobability of x eaching the baie x B. Its value is always positive unde x> (1 τ)c. (69) Function g(x, x B ) depends diectly on fim s cuent assets value x, on coupon payments C, on the failue level x B and also on all paametes descibing the volatility mean eveting pocess. And this because g(x, x B ) epesents nothing but a defaultable contact: since it is an option, it must have positive value, thus we must impose x B < (1 τ)c. (70) We can intepet g(x, x B ) as a coected option to default embodied in equity having positive value x x B, with x B [0, (1 τ)c ]. Notice that constaint (69) and (70) ae the same we have in a pue Leland model. We have no moe constaints consideing sepaately the two components f(x, C), g(x, x B ) since the andomness in volatility does not poduce any effect when i) thee is no isk of default, i.e. on f(x, C) value; ii) at default, i.e. when x = x B, thus on the payoff l E = (1 τ)c x B taken by equity holdes at the execise time. Obseve that we can wite the option to default as g(x, x B )=g(x, x B )h (x, x B ), whee g(x, x B ):= (1 τ)c x B xbx λ is the value of the option to default in a pue Leland famewok with constant volatility σ. As a consequence we can intepet h (x, x B ) as a path-dependent coection fo the pice due to ou stochastic volatility assumption. The idea is to intepet it as a coection fo the pice due to andomness in volatility when x>x B, meaning in each instant befoe the default event: only when the option is alive but still not execised. In this case it will be a coection fo the option value only befoe default, while at failue x = x B the option value is the same as in Leland, since the coection tem h (x, x B ) becomes 1 and the fist ode coection tem P 1 disappeas at bounday. This intepetation seems to be analogous to the Euopean case teated in [7]: the stochastic volatility effect is null at matuity. Results pesented in the book, also fo baie options (pg. 128), seem to be in the diection of constucting and studying a new claim with the 19

112 same stuctue of the P 0 tem, only coected fo stochastic volatility befoe the execise time, in ou case meaning befoe the baie is touched, thus befoe default is tiggeed. Let s think about a single bond unde a stochastic volatility picing model. What happens is that as the matuity date appoaches, bond pice volatility tends to zeo, becoming exactly zeo at expiation date. And this must hold: othewise, to keep bond pice equal to its (fixed) face value at matuity, the undelying assets pocess should have an indefinitely inceasing dift (i.e. tending to + ) in ode to compensate a potential volatility effect at expiation date. And this why the coection tem h (x, x B ) becomes 1 fo x = x B. In this famewok with volatility isk what is stongly affected by this assumption is the pobability of x eaching x B. Let s think fo example about the pesent value of one unit of money obtained at failue. Unde volatility isk, this value is diffeent fom its pesent value in a constant volatility case. In a pue Leland setting with constant volatility σ, such a value is given by xb λ f 1 (x, x B )=, (71) x while in this model the coected pesent value of 1 unit of money at default is f 2 (x, x B )=f 1 (x, x B )h (x, x B ) >f 1 (x, x B ), (72) since the holde of the contact has to be compensated fo andomness of volatility, i.e. the uncetainty of the iskiness of his contact. Figue 1 shows how stochastic volatility affects these values duing the path. f1(x, xb), f2(x, xb) x B Figue 1: Behavio of Functions f 1(x, x B),f 2(x, x B). The plot shows f 1(x, x B),f 2(x, x B)givenby (71)-(72) as functions of the failue level x B [0,x]. Base case paametes values ae: Λ =0,=0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, ρ < 0. Remak 4.3 Analyzing this option to default embodied in equity we can also obseve what 20

113 follows: (1 τ)c xb λ g(x, x B,C,) = x B h (x, x B,C), x which is equal to (1 τ)c xb g(x, x B,C,) = x B x λ 1+ H log x. x B Indicating with P 0 g (x, x B ):=g(x, x B ) the option to default in Leland famewok with volatility σ, we can wite the appoximate option to default as: g(x, x B,C,) =P 0 g (x, x B )+ P 1 g (x, xb ), whee g P 1 (x, xb )= (1 τ)c H x B log x B xb x x λ. As expected, the fist coection tem P 1 g (x, xb ) is the same as P 1 E (x, xb ),sincethisis exactly the option to default embodied in equity, meaning the defaultable contact. Fom Remak 4.1 we know that H > 0. As a consequence, obseve that the fist coection tem P E 1 (x, x B ) fo the pice of equity claim due to stochastic volatility has positive value x >x B, meaning that it always has the effect of inceasing equity claim value. The economic eason is that holding equity claim is now iskie since andomness in volatility is intoduced, thus modifying the iskiness of the fim. Equity holdes have to be compensated fo this. Fom now on, we will use the following fomulation fo appoximate equity: (1 τ)c (1 τ)c xb λ Ẽ(x, x B )=x + x B h (x, x B ), (73) x with h (x, x B )= 1 H log x B x Fist Ode Coection We now want to analyze which is the effect of the fist-ode coection tem P 1 E (x, xb ) on the behavio of appoximate equity w..t. cuent fim s assets value x, as shown in the following Poposition. Poposition 4.4 The fist coection tem E P 1 (x, xb )= (1 τ)c H x B log x B xb x x inceases equity value fo x>x B. The maximum coection effect is achieved when the distance to default is log x B x = 1 λ. 21 λ,

114 35 P0 E (x, xb), Ẽ(x, x B) P 0 E (x, x B ) Ẽ(x, x B ) x B Figue 2: Appoximate Equity. The plot shows appoximate equity value Ẽ(x, xb)and P0E (x, x B) tem as function of the failue level x B [0,x]. Base case paametes values ae: Λ =0,=0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, ρ < 0, x =100. E Poof We now conside the behavio of P1 (x, xb ) > 0 studying its patial deivative w..t. x, as follows: x B P 1 E (x, xb ) x = H (1 τ)c x xb x λ λ log x B x +1, The sign of P E 1 (x,xb ) x is the same as f(x, x B ):=λ log x B x +1. The application x f(x, x B ) is a deceasing function going fom f(x B,x B ) = 1 to. So the sign of P 1 E (x,xb ) x is : P 1 E (x, xb ) x P 1 E (x, xb ) x 0 fo x x B e 1 λ, < 0 fo x>x B e 1 λ. Thus, the application x P 1 E (x, xb ) admits a unique maximum at point x = x B e 1 λ Endogenous Failue Level We now tun to analyze the endogenous failue level chosen by equity holdes. The economic poblem is that equity holdes want to maximize equity value, but due to limited 22

115 liability of equity they cannot chose an abitay small failue level. A natual constaint on x B can be found by imposing: Ẽ(x, x B) x=xb 0 x x B, x which guaantees equity being a non-negative and inceasing function of fim s cuent assets value x fo x x B. The following Poposition detemines the failue level which satisfies this condition. Poposition 4.5 Unde λ > H, the endogenous failue level chosen by equity holdes in ode to maximize x B Ẽ(x, x B) has to belong to the following inteval: (1 τ)c x B,, (74) whee (1 τ)c λ H x B := 1+λ H > 0 (75) is solution of the taditional smooth-pasting condition: Poof Obseve that Ẽ(x, x B) x Ẽ(x, x B) x=xb =0. x x=xb = (P 0 E (x, x B )+ P E 1 (x, xb )) x=xb, x thus in ode to have equity a non-negative and inceasing function of cuent assets value x it is sufficient to impose: This in tuns lead to with l E = (1 τ)c x B > 0 due to (70). Ẽ(x, x B) x=xb 0 x x B. x 1 le x B (λ H) 0, (76) Finally, unde λ H>0, (76) is satisfied fo x B x B with x B solution of: 1 le (λ H) =0. (77) x B This solution x B satisfies also x B < (1 τ)c, thus the inteval x B, (1 τ)c is not empty. 23

116 Remak 4.6 Notice that this lowe bound x B > 0 fo the endogenous failue level depends on all paametes involved in the volatility diffusion pocess, since it depends on, andh, thus on V 2,V 3. To be moe pecise, this lowe bound is stictly elated to the maket pice of isk, to the skew effect and also to the speed of convegence chosen fo the volatility mean eveting pocess. Obseve that as paticula case, fo 0, this lowe bound is exactly the lowe bound aising fom a pue Leland famewok x BL with constant volatility σ: x BL := (1 τ)c λ 1+λ. In ou moe geneal case, we stess that x B educes as the speed of mean evesion inceases, since the application x B is deceasing and the speed of mean evesion is 1. This means that unde ou stochastic volatility famewok equity holdes can chose a failue level which is lowe that in a pue Leland model. The lowe bound x B is also deceasing w..t. H>0, thus its dependence on the maket pice of volatility and on the leveage effect ρ has the same sign of H-dependence on the same paametes. Remak 4.6 is useful to fomulate the optimal stopping poblem faced by equity holdes in this famewok of stochastic volatility: with x B given in (75). max Ẽ(x, x B,C,), (78) x B [ x B, (1 τ)c ] Fom taditional optimal stopping theoy we know that, unde appopiate hypothesis, applying the smooth pasting condition to the function which has to be optimized, will give a failue level which is exactly the solution of the optimal stopping poblem. What we showed in all analytical aspects in [2, 3], is an example of a famewok in which applying the smooth pasting condition (which is a low contact condition) gives the same failue level we obtain by maximizing equity value w..t. the constant default baie. An analogous elation exists when we analyze Ameican-style options unde Black and Scholes model (see also [7]). Woking inside a Black and Scholes picing famewok means that the smooth pasting condition applied to equity value gives not only a lowe bound fo the failue level chosen by equity holdes, but diectly the endogenous failue level solution of the optimal stopping poblem. In this stochastic volatility picing model the taditional smooth pasting condition applied to appoximate equity Ẽ(x,x B) x x=xb = 0 gives only a lowe bound x B > 0 fo the endogenous failue level which has to be satisfied (due to limited liability of equity) in ode to have equity as an inceasing function of fim s assets value. In the constant volatility case, this lowe bound is also solution of the optimal stopping 24

117 poblem, i.e. max xb E(x, x B ), meaning that equity holdes will always choose the lowest admissible failue level. But this is not still tue when volatility isk s intoduced in the model. The failue level x B is not the solution of the optimal stopping poblem (78). We show this in the following Poposition. Poposition 4.7 The endogenous failue level solution of the optimal stopping poblem max Ẽ(x, x B,C,), whee x B [ x B, (1 τ)c ] (1 τ)c λ H x B = 1+λ H, is x B,solutionof 1+ H log x λ(1 τ)c (λ + 1)x B H (1 τ)c x B =0. (79) x B Poof Equity holdes face the following optimal stopping poblem: max Ẽ(x, x B,C,). (80) x B [ x B, (1 τ)c ] Recall that Ẽ(x, x B,C,) =f(x, C)+ g(x, x B,C,), whee f(x, C) does not depend on the failue level x B. As a consequence, ou poblem (80) is equivalent to whee max g(x, x B,C,), x B [ x B, (1 τ)c ] (1 τ)c xb g(x, x B,C,) := x B x λ 1+ H log x x B is the coected option to default embodied in equity. To study g behavio, we tun to its deivative computation witing it in this compact fomuation: xb xb g(x, x B,C)= x λ h (x, x B ) 1+ λle x B le H, x B whee l E = (1 τ)c x B. Actually, the sign of patial deivative of g w.. t. x B is the one of function g 1 (x B )= x B h (x, x B )+l E λh (x, x B ) H, (81) explicitly given by g 1 (x B )= x B 1+ H log x (1 τ)c x B λ 1+ H log x H. x B x B 25

118 Remak that the smallest value of x B, i.e. the lowe bound x B solution of: 1 le (λ H) =0, x B thus, letting l E := (1 τ)c x B => 0, at point x B = x B we have As a consequence, in explicit fom H l E x B = 1+ λ l E. x B g 1 ( x B )= Hl E (h (x, x B ) 1) > 0, g 1 ( x B )=H log x (1 τ)c H x B 1+λ H = (1 τ)c > 0, λ H 1+λ H is Concening the biggest value x B := (1 τ)c,wehave g 1 (x B )= x B h (x, x B ) < 0. Now, we tun to: g 1(x B )= h (x, x B )(1+λ)+h (x, x B )(λl E x B )+ H, g1(x B )= h (x, x B ) 1+ λle +2λ > 0, x B since h (x, x B ) < 0. Respectively, thei explicit fom is: g1(x B )= 1+ H log x (1 + λ) λ(1 τ)c H (λ + 1) x B x B H g1(x B )= x B 1+λ 1+ (1 τ)c x B > 0. + H, This function g1 (x B) is inceasing, so g 1 (x B ) is convex, stating fom positive value to negative value, thus thee exists a unique solution x B which ealizes the maximum of equity and it is solution of g 1 (x B ) = 0, whose equation is explicitly given by 1+ H log x λ(1 τ)c (λ + 1)x B H (1 τ)c x B =0. (82) x B 26

119 Remak 4.8 i) Let x BL := (1 τ)c λ 1+λ being Leland endogenous failue level. Obseve that ou coected option to default embodied in equity has a negative fist deivative w..t. x B at point x BL since the sign of g 1 given in (81) at that point is: g 1 (x BL )= (1 τ)c H (1 + λ) < 0, meaning that the endogenous failue level x B satisfies (using g 1 convexity) x B < x B <x BL. Notice that these thee points coincides as 0. ii) We can ewite Equation (79) as: h (x, x B ) le x B λh (x, x B ) H =0, and obseve that the only tem which depends on coupon is l E = (1 τ)c linea w..t. C. Solve it w..t. the endogenous coupon C as: x B which is explicitly given by The application x B C is inceasing. C = x B (λ + 1)h (x, x B ) H 1 τ λh (x, x B ) H, (83) C = x B (λ + 1)(1 + H log x x B ) H 1 τ λ(1 + H log x x B ) H. We now analyze Equation (79) which gives the endogenous failue level x B in ode to define a coected smooth-pasting condition inside a picing model of stochastic volatility. Poposition 4.9 Coected Smooth-Pasting. At point x B implicit solution of (79), the following coected smooth-pasting condition holds: E P 0 x x=x B h (x, x B )+ P 1 E x x=x B =0, (84) whee P E 1 = P 1 E. Poof What we want to show is that finding the solution of g(x,x B,C,ε) x B = 0 is equivalent to find the solution of what we define a coected smooth-pasting condition. In this case also the smooth pasting condition must take into account the petubation given by the 27

120 stochastic volatility effect, and in this sense we call it coected. Conside Equation (79) and obseve that we can ewite it unde the following equivalent fomulation: h (x, x B ) 1+ λle le H =0. x B Moeove, we have: P 0 E x x=x B =1 λle x B, x B P E 1 x x=x B = H le x B. As in a pue Leland model, the endogenous failue level chosen by equity holdes x B does not depend on bankuptcy costs, since the stict pioity ule still holds, but instead depends on coupon, isk fee ate and tax ate. Coeteis paibus, it is inceasing w..t the coupon level and deceasing w..t. the copoate tax ate τ. When volatility isk is intoduced, equity holdes will choose the default baie which maximizes equity value depending on both the maket pice of volatility and the leveage effect, since coefficient H in Equation (79) captues both these effects. Moeove, what is completely new fo an endogenous failue level deived inside a stuctual model famewok is its dependence on the initial fim s assets value x. This dependence could be elated to the fact that also the standad smooth-pasting condition Ẽ(x,x B) x x=xb = 0 does not give the endogenous failue level, but only a lowe bound fo it. And this lowe bound is not the solution of the optimal stopping poblem. Equation (84) suggests that the endogenous failue level is the solution of a coected smoothpasting condition in the sense that we must take into account the andomness intoduced in volatility. As we obseved in Poposition 4.4 the coection fo the pice is not constant though the path, achieving a maximum coection effect when the distance to default is log x B x = 1 λ. Figue 2 shows the behavio of both appoximate equity Ẽ(x, x B) and P E 0 (x, x B ) w..t. the failue level x B. As we can see, Ẽ(x, x B ) achieves its maximum befoe P E 0 (x, x B ), meaning fo a lowe failue level. Remak 4.10 Recall Equation (84). Obseve that we can intepet its solution x B as the failue level at which the following conditions hold: h (x, x B )= P E 1 x P E 0 x x= x B x= x B = P E 1 x x= x B P0 E ( x B ), (85) whee denotes the Geek of the P E 0 deivative evaluated at point x = x B,o Ẽ x B H log x = 28 x x= x B P0 E ( x B ). (86)

121 P0 E (x, xb), P1 E (x, xb), Ẽ(x, xb) P 0 E (x, x B ) P 1 E (x, x B ) Ẽ(x, x B ) 5 40 x B x Figue 3:. The plot shows appoximate equity value Ẽ(x, xb), and P0E (x, x B), P E 1 (x, xb) tems as function of cuent assets value x. The suppot of each function is [ x B,x]. Base case paametes values ae: Λ =0,=0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, ρ < 0. The endogenous failue level x B is detemined fo x =100. i) Fist ecall that the lowe bound x B solution of Ẽ(x,x B) x x=xb =0is independent of cuent assets value x. Equivalently, it guaantees: 1= P E 1 x x= x B P0 E ( x B ) meaning the slope of P 0 E and P 1 E have the same absolute value, but opposite sign. This only guaantees equity being an inceasing and non-negative function of x as shown in Figue 3. ii) Secondly, obseve that the two equivalent fomulations (85)-(86) fo equation (84) ae useful to bette undestand the economic optimality of the endogenous failue level x B and its dependence on fim s activities value x. The left hand side of both equation is the only one depending on x. Choosing x B means choosing the endogenous level coesponding to the optimal execise time: due to cuent assets value x, befoe and afte x B the coection fo the pice h (x, x B ) does not exactly compensate the atio between the instantaneous vaiations (due to an instantaneous vaiation in x) in the fist ode coection tem and the of the coesponding Black and Scholes contact P E 0 (x, x B ). Equation (86) suggests the same dependence elating the distance to default and the atio between the instantaneous vaiations in appoximate claim Ẽ(x, x B) and, again, the - sensitivity of P E 0 (x, x B ). The endogenous failue level can be seen as an equilibium level which inceases as cuent assets value x ises., 29

122 In this sense, looking at Equation 85, the coection tem h (x, x B ) can be intepet as an elasticity measue at equilibium. By simply applying Ẽ(x,x B) x x=xb = 0 will give a failue level x B which is independent of fim s cuent activities value x. Choosing this failue baie coesponds to a nonoptimal execise of the option to default embodied in equity. The coected smoothpasting condition leads to an endogenous failue level x B which is geate than its lowe bound, and moeove depends on fim s assets value x. This is a new insight aising fom a stuctual modeling appoach. A possible explanation of this dependence is that unde volatility isk, assuming ρ < 0 makes the distibution of stock pice etuns not symmetic. In paticula, exteme etuns behavio is bette captued, since with ou assumptions the left tail of thei distibution is fatte: this is why it is not optimal to execise at the standad smooth pasting level but choosing a failue level geate than this. The stating point of pocess x t now mattes in the choice of the the optimal stopping time, since volatility is not constant. The coection fo the pice epesented by h (x, x B ) is a path-dependent coection depending on cuent assets value x: theefoe, the optimal execise time must conside this. Inside this famewok the iskiness of the fim is taken into account fom two diffeent points of view: i) its maket pice though Λ, ii) the leveage effect though ρ. Obseve that even in the uncoelated case ρ = 0, the endogenous failue level still depend on x and it is still diffeent fom the lowe bound x B. In such a case the coection fo the volatility level due to coefficient V 2 is still in foce. As 0 the dependence of the endogenous failue level on x disappeas (and also in case ρ =0, Λ = 0). 4.2 Debt, Tax Benefits, Bankuptcy Costs and Total Value of the Fim Using contingent claim valuation, debt value can be expessed as a claim on the undelying asset descibing fim s activities value x. Unde stochastic volatility assumption we have the following expessions fo appoximate values of debt, tax benefits and bankuptcy costs: D(x, x B )= C + (1 α)x B C xb λ h (x, x B ), (87) x TB(x, x B )= τc BC(x, x B )=αx B xb x with the coection tem h (x, x B )= τc xb λ h (x, x B ), (88) x λ h (x, x B ), (89) 1+ H log x x B, and H = 2 σ 2ν Λφ + 2 ρfφ. 4 σ 2 Due (only) to the assumption of infinite hoizon, in this setting debt holdes eceive C without limit of time in the event of no default and (1 α)x B C in case of x eaching 30

123 P 0 D (x, x B ) D(x, x B ) P0 D (x, xb), D(x, xb) x B Figue 4: Appoximate Debt. The plot shows appoximate debt value D(x, x B)andP 0 D (x, x B) tem as functions of the failue level x B [0,x]. Base case paametes values ae: Λ =0,=0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, x =100,ρ < 0. x B, meaning C the component of debt value without default isk. Obseve that this is the same as in Leland famewok, meaning that at boundaies stochastic volatility does not poduce effect. What is diffeent fom Leland famewok is that the coection tem P 1 modifies the claim value fo x>x B. The value of appoximate tax benefits of debt has a downwad coection: the fist-ode coection tem P TB 1 (x, x B ) is negative fo all values x>x B, and this is due to the volatility isk influence on the likelihood of default. In ode to completely descibe the capital stuctue we can wite the total value of the fim as the sum of equity and debt value o equivalently as cuent assets value plus tax benefits of debt, less bankuptcy costs. Altenative and equivalent definitions fo the total value of the fim ae: ṽ(x, x B,C,) :=Ẽ(x, x B,C,)+ D(x, x B,C,) =x + TB(x, x B ) BC(x, x B ). (90) Appoximate total value of the fim is given by: ṽ(x, x B,C)=x + τc αx B + τc xb h (x, x B ) x λ. (91) The coection tem h (x, x B )=1+ H log x x B is a fundamental quantity in this famewok of stochastic volatility. Its contibution is that one of a path dependent coection 31

124 P 0 TB (x, x B ) TB(x, x B ) P0 TB (x, xb), TB(x, xb) x B Figue 5: Appoximate Tax Benefits of Debt. The plot shows appoximate tax benefits of debt TB(x, x B)andP 0 TB (x, x B) tem as functions of the failue level x B [0,x]. Base case paametes values ae: Λ =0,=0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, x =100,ρ < 0. fo pices due to andomness in volatility. Ou focus to undestand if the stochastic volatility effect can incease cedit speads and educe leveage atios pedicted by the model. Recall that cedit speads ae defined as C R(x, x B ):= D(x, x B ), whee D(x, x B ):=P 0 D (x, x B ). We will denote with R(x, x B ) the cedit spead unde a pue Leland famewok and with R(x, x B ):= C D(x, x B ) appoximate cedit spead unde stochastic volatility. Figue 6 shows that andomness in volatility moves cedit speads exactly in the expected diection, ising them befoe the default time in ode to compensate investos fo the new souce of isk. 32

125 25 R(x, xb), R(x, xb) R(x, x B ) R(x, x B ) x B Figue 6: Appoximate Cedit Speads. The plot shows appoximate cedit speads R(x, x B) and R(x, x B) as functions of the failue level x B [0,x]. Base case paametes values ae: Λ =0, =0.06, σ =0.2, α =0.5, τ =0.35, C =6.5, V 3 =0.003, V 2 =2V 3, x =100,ρ < 0. 5 Optimal Capital Stuctue We now analyze the capital stuctue of the fim consideing that equity holdes will chose the coupon C in ode to maximize the total value of the fim. We ecall ou appoximate total value of the fim, given by Equation (91). ṽ(x, x B,C)=x + τc αx B + τc 1+ H log x x B xb λ. (92) x The poblem is now to optimize: g : C ṽ(x, x B (C),C) whee x B (C) is the endogenous failue level solution of (79). Since we do not have x B is explicit fom, by exploiting the lineaity of (79) w..t. coupon C as noted in Remak 4.8, we will poceed following the same idea poposed in [3]. Conside x B C(x B ), whee C is solution of (79). Thus, an equivalent poblem will be to optimize w..t. x B the following function: x B ṽ(x, x B, C(x B )). We now numeically compute the optimal capital stuctue of the fim, i.e. all financial vaiables at thei optimal level in ode to analyze the volatility isk influence. We conside Leland [12] esults as a benchmak in ode to undestand the effect poduced by volatility isk on all financial vaiables. The aim is to analyze both souces 33

126 of isk induced by the model: at fist only the skew effect, captued by ρ, then its joint influence with the coection fo the volatility level, elated to the diffeence between the effective volatility σ and the aveage volatility σ. Table 2 shows how optimal copoate ρ C D R R Ẽ x B ṽ L % % % % % % % Table 2: Skew effect on optimal capital stuctue. The table shows financial vaiables at thei optimal level when only the skew effect is consideed, i.e. ρ < 0, Λ = 0. The fist ow of the table epots Leland [12] esults as benchmak, as paticula case of ρ =0, Λ =0. Weconside =0.06, σ =0.2, α =0.5, τ =0.35. Recall V 3 := ρ 2 2 νfφ. We conside V 3 = 0.06ρ, V 2 =2V 3,seealso[8]. L, R ae in pecentage (%), R in basis points (bps). decisions ae influenced by the intoduction of a negative coelation ρ between assets value dynamic and the volatility pocess. We leave ρ vaying fom ρ = 0.05 to ρ = 0.1, in ode to captue its effects on copoate decisions. The fist step is to conduct the analysis by assuming Λ = 0, meaning the coection fo the volatility level being null. Numeical esults show that only the skew effect induced by ρ < 0 poduces a significant impact on copoate financing decisions. Skewness in the undelying dynamics makes debt less attactive. What emeges is that optimal coupon, debt, total value of the fim and leveage atios dop down. And in some cases, this eduction is significant. Only a slightly negative coelation ρ = 0.05 bing down leveage of aound 8%, while a 15%- eduction is achieved with ρ = 0.1. The maximum total value of the fim is also educed, since the incease in optimal equity value is always lowe that the eduction in optimal debt. The coupon level chosen to maximize total fim value is deceasing with the skew effect, geneating a downwad jump in optimal debt fom 96.3 in case ρ = 0 to 73.4 in case ρ = 0.1, which is absolutely a not negligible one. Despite lowe leveage atios, yield speads ae inceasing with ρ. This behavio can be explained thinking about the likelihood of default, which should be inceased by the skew effect. The maket peception about the cedit isk of the fim changes and the fim becomes a iskie activity. Thee is uncetainty about its volatility, and its iskiness moves in time. Investing in this fim equies a highe compensation. Letting ρ = 0, and intoducing the coection fo the volatility level will bing the model to a pue Black and Scholes setting with constant volatility σ. This is not the case we ae inteested in. As a second step we conside both souces of isk associated to fim 34

127 value. Table 3 shows how financial vaiables ae modified when also the coection fo the volatility level is consideed in a famewok with negative leveage effect, i.e. ρ < 0. The elation existing between the coected effective volatility σ and the aveage volatility σ is: σ = σ 2 2V 2, with V 2 = 2 2 ν Λφ < 0 unde ρ = 0. As noted in [7], makets data suggest that usually the coected effective volatility is highe than the aveage volatility, this is why we conside σ > σ. As example we conside a negative coelation ρ = 0.05 and a gap between σ and σ of 1%, 2%, espectively. The skew effect and the volatility level coection seem to epesent an inteesting featue to develop applied to cedit isk models. Optimal financing decisions move w..t. a pue Leland model, and whee both souces of isk ae consideed togethe, thei joint influence is stong. The optimal amount of debt is educed and leveage atios can dop down fom 75% to 62% only with a slightly negative coelation ρ = 0.05 and a volatility level coection of 2%. σ C D R R Ẽ x B ṽ L σ % σ % σ % Table 3: Skew effect and volatility level coection: influence on optimal capital stuctue. The table shows financial vaiables at thei optimal level when ρ = 0.05 and also a volatility coection is consideed. Recall that σ = σ 2 2V 2.Weconside =0.06, σ =0.2, α =0.5, τ =0.35, V 3 = L, R ae in pecentage (%), R in basis points (bps). Numeical esults emphasize inteesting insights aising fom a model whee a negative coelation among assets etuns and shocks in volatility and a volatility level coection coexist. Thus suggesting a possible diection to follow aiming at impoving empiical pedictions inside a stuctual model with endogenous bankuptcy. Intoducing a pocess descibing the evolution of assets volatility makes possible to captue how pices ae modified due to the maket s peception of fim s cedit isk: thee is uncetainty about the volatility level and its evolution in time. Due to possible shocks in volatility, the fim becomes a iskie activity fo the maket. Investos will equie highe compensations fo this, thus yield speads must be highe despite lowe leveage atios. 6 Conclusions The focus in this pape is to intoduce volatility isk inside a stuctual model of cedit isk with endogenous default. The capital stuctue of a fim is analyzed in a famewok of infinite time hoizon following Leland s idea [12] but assuming a stochastic volatility 35

128 model fo the fim s assets. In paticula we conside a stochastic volatility picing model whee a one-facto pocess of Ostein-Uhlenbeck type descibing the dynamic of the diffusion coefficient is intoduced. Moeove, following [7], assets value pocess belongs to a faily lage class of stochastic volatility models and fom this point of view esults ae model-independent. The idea of the pape is to bette captue exteme etuns behavio, making stock pices etuns distibution asymmetic and with fatte left tails. Inside this famewok we descibe and analyze the effects of volatility isk on all financial vaiables descibing the capital stuctue to undestand if intoducing volatility isk could be a way to impove empiical pedictions of stuctual models (i.e. highe speads, lowe leveage atios). We analyze each component of the capital stuctue as a deivative contact whose value can be deived by applying singula petubation techniques as in [7]. If compaed to the classical Leland model [12], the value of each claim must be coected to compensate the holde of each deivative contact fo volatility isk. This coection acts only duing the path of pocesses, i.e. not when the contact is iskless, not when default aives. This coection is due to two main souces of isk associated with the intoduced andomness in volatility: i) the skew effect, aising by assuming a coelation ρ between assets and volatility dynamics; ii) a volatility level coection. Equity holdes still face the poblem of optimizing equity value w..t. the failue level. Unde this appoach, the failue level deived fom standad smooth pasting pinciple is not the solution of the optimal stopping poblem, but only epesents a lowe bound which has to be satisfied due to limited liability of equity in ode to have equity an inceasing function of x. Choosing that failue level is not optimal since it would mean an ealy execise of the option to default embodied in equity. A coected smooth pasting condition must be applied in ode to find the endogenous failue level solution of the optimal stopping poblem. What is new in ou appoach, is that we ae dealing in a stuctual model famewok and the failue level solution of the optimal stopping poblem depends on cuent fim s activities value. An idea is to bette exploit this insight tying to undestand how it is affected by the mean-evesion speed, meaning studying the distance between the lowe bound x B and the optimal solution x B unde ou famewok. Unde volatility isk, thee is uncetainty about the cuent iskiness of the fim, thus the actual value of fim s activities mattes. Numeical esults obtained by exploiting optimal capital stuctue suggest that this stochastic volatility picing model seems to be a obust way to impove esults in the diection of both highe speads and lowe leveage atios in a quantitatively significant way. The maket peception of the cedit isk associated to the fim is captued by appoximate coected pices: yield speads ae highe, despite lowe leveage, since the equied compensation fo isk inceases. Refeences [1] BARSOTTI F., MANCINO M.E., PONTIER M. (2010), Capital Stuctue with Fim s Net Cash Payouts, accepted fo publication on a Special Volume edited by 36

129 Spinge, Quantitative Finance Seies, editos: Cia Pena, Mailena Sibillo. [2] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), An Endogenous Bankuptcy Model with Fim s Net Cash Payouts, Woking pape. [3] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), Copoate Debt Value with Switching Tax Benefits and Payouts, Woking pape. [4] CHEN N., KOU S.G. (2009), Cedit Speads, Optimal Capital Stuctue, and Implied Volatility with Endogenous Default and Jump Risk, Mathematical Finance, 19(3), [5] EOM Y.H, HELWEGE J., HUANG J.Z. (2004), Stuctual Models of Copoate Bond Picing: an Empiical Analysis, Review of Financial Studies, 17, [6] FIORANI L., LUCIANO E., SEMERARO P. (2010), Single and Joint Default in a Stuctual Model with Puely Discontinuous Assets, Quantitative Finance, 10(3), [7] FOUQUE J.P., PAPANICOLAOU G., RONNIE K.R. (2000), Deivatives in Financial Makets with Stochastic Volatility, Cambidge Univesity Pess. [8] FOUQUE J.P., PAPANICOLAOU G., SOLNA K. (2005), Stochastic Volatility Effects on Defaultable Bonds, Woking Pape. [9] FOUQUE J.P., PAPANICOLAOU G., SIRCAR R., SOLNA K. (2004), Timing the Smile, Wilmott Magazine. [10] HILBERINK B. and ROGERS L.C.G. (2002), Optimal Capital Stuctue and Endogenous Default, Finance and Stochastics, 6, [11] HURD T.R. (2009), Cedit Risk Modeling Using Time Changed Bownian Motion, Intenational Jounal of Theoetical and Applied Finance, 12(8), [12] LELAND H.E. (1994), Copoate Debt Value, Bond Covenant, and Optimal Capital Stuctue, The Jounal of Finance, 49, [13] LELAND H.E., TOFT K.B. (1996), Optimal Capital Stuctue, Endogenous Bankuptcy and the Tem Stuctue of Cedit Speads, The Jounal of Finance, 51, [14] LELAND H.E. (2009), Stuctual Models and the Cedit Cisis, China Intenational Confeence in Finance. [15] MEDVEDEV A., SCAILLET O. (2010), Picing Ameican Options Unde Stochastic Volatility and Stochastic Inteest Rates, Jounal of Financial Economics, 98(1),

130 [16] MERTON R. C. (1973), A Rational Theoy of Option Picing, Bell Jounal of Economics and Management Science, 4, [17] MERTON R. C. (1974), On the Picing of Copoate Debt: The Risk Stuctue of Inteest Rates, The Jounal of Finance, 29, [18] MODIGLIANI F., MILLER M. (1958) The Cost of Capital, Copoation Finance and the Theoy of Investment, Ameican Economic Review, 48,

131 CONCLUSIONS The thesis analyzes cedit isk modeling following a stuctual model appoach with endogenous default. We extend the classical Leland [5] famewok in thee main diections with the aim at obtaining esults moe in line with empiical evidence. We intoduce payouts in [2, 3], and then also conside copoate tax ate asymmety [4]: numeical esults show that these lead to pedicted leveage atios close to histoical noms, though thei joint influence on optimal capital stuctue. Finally, in [1], we intoduce volatility isk. Following Leland suggestions in [6], we conside a famewok in which the assumption of constant volatility in the undelying fim s assets value stochastic evolution is emoved. Analyzing defaultable claims involved in the capital stuctue of the fim we deive thei coected pices unde a faily lage class of stochastic volatility models by applying singula petubation theoy as in [9]. Exploiting optimal capital stuctue, the stochastic volatility famewok seems to be a obust way to impove esults in the diection of both highe speads and lowe leveage atios in a quantitatively significant way. Some ideas fo futue eseach deal with both theoetical and empiical issues. At fist, we would like to extend Leland [5] model in the diection of a dynamic capital stuctue famewok with endogenous default. Allowing fo vaiations in the coupon payments level should be a way to undestand how equity holdes can adjust the capital stuctue as a moe ealistic featue to analyze. The main empiical esults in cedit isk liteatue emphasize a poo job of stuctual models in pedicting cedit speads fo shot matuities. In ou minds the stochastic volatility famewok is able to bette captue the cedit exposue of the fim and this is why we would like to wok moe to develop ou application in [1]. Fo example, studying how volatility time scales (fast and slow factos) can inteact and affect speads and default pobability inside a stuctual model famewok with finite hoizon. In the same diection, but fom a diffeent point of view, we would like to impove esults obtained by calibating Meton-like stuctual models unde a stochastic volatility famewok (see in [12], [8]). In [7] the classical Meton model [10] is studied inside an empiical (egession) analysis showing that, suppoting patly findings in [11], even this simple setting is able to pedict bond etuns sensitivity with espect to changes in stock etuns (hedge atios). Nevetheless, volatility isk seems to be a fundamental issue to deal with. Among existing liteatue about cedit isk, [12] popose to conside cedit default swap (CDS) pemiums as a diect measue of cedit speads, suggesting that the volatility and jump isks of a fim ae able to pedict most of the vaiation in thei levels. Ou aim, following this idea, is to use high-fequency equity pices in ode to bette captue the volatility isk component though a volatility estimato obust with espect to the mico-stuctue noise.

132 Refeences [1] BARSOTTI F. (2011), Optimal Capital Stuctue with Endogenous Bankuptcy and Volatility Risk, Woking pape. [2] BARSOTTI F., MANCINO M.E., PONTIER M. (2010), Capital Stuctue with Fim s Net Cash Payouts, accepted fo publication on a Special Volume edited by Spinge, Quantitative Finance Seies, editos: Cia Pena, Mailena Sibillo. [3] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), An Endogenous Bankuptcy Model with Fim s Net Cash Payouts, Woking pape. [4] BARSOTTI F., MANCINO M.E., PONTIER M. (2011), Copoate Debt Value with Switching Tax Benefits and Payouts, Woking pape. [5] LELAND H.E. (1994), Copoate debt value, bond covenant, and optimal capital stuctue, The Jounal of Finance, 49, [6] LELAND H.E. (2009), Stuctual Models and the Cedit Cisis, China Intenational Confeence in Finance. [7] BARSOTTI F., DEL VIVA L. (2010), Meton Model of cedit isk: New Evidence fom Copoate Bond Retuns Sensitivity, Woking pape, available on-line at id= Submitted to Jounal of Empiical Finance. [8] BRUCHE M. (2009), Estimating Stuctual Models of Copoate Debt and Equity, Woking pape. [9] FOUQUE J.P., PAPANICOLAOU G., RONNIE K.R., Deivatives in Financial Makets with Stochastic Volatility, Cambidge Univesity Pess (2000). [10] MERTON R. C. (1974), On the Picing of Copoate Debt: The Risk Stuctue of Inteest Rates, The Jounal of Finance, 29, [11] SCHAEFER S.M., STREBULAEV I.A. (2008), Stuctual Models of Cedit Risk ae Useful: Evidence fom Hedge Ratios on Copoate Bonds, Jounal of Financial Economics, 90(1), [12] ZHANG B.Y., ZHOU H., ZHU H. (2008), Explaining Cedit Default Swap Speads with the Equity Volatility and Jump Risks of Individual Fims, Woking Pape.