V. SYMPOSIUM ZUR GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION GEABA. Optimal Debt Service: Straight vs. Convertible Debt.

Transcription

1 . SYMPOSIUM ZUR ÖKONOMISCHEN ANALYSE DER UNTERNEHMUNG Optimal Debt Sevice: Staight vs. Convetible Debt Chistian Koziol Session A3 GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION GEABA

2 Optimal Debt Sevice: Staight vs. Convetible Debt Chistian Koziol May 2004 JEL Classification: G12, G32 D. Chistian Koziol, Chai of Finance, Univesity of Mannheim, L5, 2, D Mannheim, Gemany, Phone: , Fax: , 0

3 Optimal Debt Sevice: Staight vs. Convetible Debt May 2004 Abstact In this pape, we analyze the optimal default stategy of a fim when debt is convetible into equity. Fo this pupose, we conside a convetible consol bond in a time-independent model in the pesence of bankuptcy costs and tax deductability. The optimal default and convesion stategy esult fom a game between equity and debt holdes. We show that an optimal default of convetible debt occus ealie than a default of othewise identical staight debt. Although the value of convetible debt is lowe when the fim follows the optimal debt stategy athe than the stategy fo staight debt, the values of the investment and option component of the convetible debt can be highe. Futhemoe, we find that the impotant diffeence between the default baie fo convetible and identical but non-convetible debt ises with the convesion atio, the pomised coupon, a lowe tax ate, and a lowe payoff ate. JEL Classification: G12, G32 1 Intoduction The detemination of the optimal default point is a complex decision fo evey fim. Fims not only default when thei fim value falls below an exogenous baie such as in the case of oveindebtedness but they also have the possibility to stop payments to the debt holdes which esults in a default due to the inability to pay. The possibility fo an insolvency without an oveindebted balance sheet is a valuable option fo the equity holdes because it can pevent fom unning the fim despite negative equity values. As a consequence, the detemination of the optimal time to default is an impotant task fo evey fim, because it affects the wealth of the equity holdes. Black/Cox (1976) and Leland (1994) among many othes deal with the poblem of stategic default by deiving an endogenous default stategy that maximizes the 1

4 equity value. The esult is that inteests should be paid as long as the equity value exceeds the size of the equied capital injections by the equity holdes. Othe appoaches extend the standad model fom Leland by elaxing the assumption that a default necessaily esults in a costly liquidation of the fim but allow fo eoganization (see e.g. Andeson/Sundaesan (1996), Mella-Baal/Peaudin (1997), Achaya/Huang/Subahmanyam/Sundaam (2002), and Fan/Sundaesan (2000)). The intoduction of stategic debt enegotiation pevents the liquidation of the fim and esults in a highe value of the fim due to the elimination of bankuptcy costs. All these models have in common that they all conside a simple capital stuctue with equity and staight debt only. Howeve, copoate debt is often equipped with additional ights such as a convesion featue fo the investos. The standad view to convetible bonds is to decompose it into an investment component and an option component (see e.g. Ingesoll (1977)). The value of the investment component is that of a non-convetible but othewise identical bond. The idea behind this decomposition is that both components can be sepaately piced, whee the investment value is typically obtained as the value of staight debt of an appopiate benchmak fim. In the ideal case, the benchmak fim has identical assets and the same debt obligation but without any convesion ights. Accoding to this view, the big vaiety on staight debt models can be applied to pice the investment value by egading the debt value of the benchmak fim as a poxy fo the investment value. This pactical appoach to picing convetible bonds, implicitly assumes that the convesion ight does not affect the default stategy. At fist glance, this assumption seems to be plausible because a default typically occus when the fim is unde pessue and a convesion is not desiable, while a convesion is optimal if the dange of a default is low. Howeve, to deive the optimal default stategy fo fims with convetible debt, a game-theoetical analysis is equied which fundamentally diffes fom simply assuming that a convesion does not affect the default stategy. Stictly speaking, the optimal default stategy of convetible debt is the esult of a game between the fim, which epesents the equity holdes, and the convetible debt holdes. In this dynamic game, the fim must continuously decide about a default, whee the debt holdes have to decide upon convesion. It is not clea whethe the optimal default stategy esulting fom this complex game diffes fom the stategy fo staight debt. Howeve, the elation between these two stategies is impotant when analyzing the default isk of convetible debt and especially when picing the investment component of a convetible bond by egading the identical but othewise non-convetible bond value. 2

5 The goal of this pape is to detemine optimal default stategies when debt is convetible and to analyze the factos that cause a diffeence between the optimal stategy and the stategy fo staight debt. To illustate the impotance of the optimal default stategy, we compae the optimal stategy with the stategy of staight debt by egading the pice impact on the debt value, the investment value, and the option value. Fo this pupose, we conside a time-independent model with a pepetual bond paying a continual coupon pe unit time in the pesence of bankuptcy costs and tax deductability. As a esult, we obtain that a default of convetible debt always occus befoe a default would be tiggeed when the debt was not convetible. The default stategy has a consideable effect on the values of convetible debt, the investment value and the option component. We can show that this elation between the optimal default stategy and the stategy of staight debt esults in a highe value of convetible debt when the fim follows the staight debt stategy athe than the optimal stategy. On the contay, the two components of a convetible bond, the investment value and the option value, can be highe o lowe when the staight debt athe than the optimal stategy is chosen by the fim. Theefoe, the pactical appoach to picing the investment value of a convetible bond by egading an othewise identical but non-convetible debt is not justified. When examining the impotant diffeence between the optimal default baie and the baie fo staight debt, we find that the diffeence ises with the convesion atio, the pomised coupon, a lowe tax ate, and a lowe payoff ate. The pape is oganized as follows: In Section 2, we descibe the model famewok and show how to detemine the optimal default stategy and the elated values of convetible debt and equity. Popeties of the optimal default stategy ae deived in Section 3. Section 4 povides a compaative static analysis of the diffeence between the optimal default baie and the default baie fo staight debt. Section 5 concludes. Technical developments ae in the appendix. 3

6 2 Model Famewok and alues of Equity and Convetible Debt unde the optimal Stategy 2.1 Model Famewok We conside a standad model of time-independent secuity values with bankuptcy costs and tax deductability to analyze the optimal debt sevice of convetible debt. Time-independent models ae widely-accepted to detemining optimal default stategies (see e.g. Black/Cox (1976), Leland (1994), Mella-Baal/Peaudin (1997), and Fan/Sundaesan (2000)), because these models captue the elevant deteminants fo a default and esult in tactable fomulae fo asset pices. As an extension to these appoaches, we conside a convetible athe than a staight consol bond. A convetible consol bond is a claim of the fim that continuously pays a coupon C, pe instant of time as long as the fim is solvent and no convesion takes place. A convesion occus upon the notice of the debt holdes into a faction γ of total equity. 1 Since a convesion is associated with a loss of the coupon claim, the fim afte convesion is not leveaged anymoe and theefoe the convesion value equals γ, whee denotes the asset value of the fim. Befoe convesion when the fim is still leveaged, the asset value has the chaacte of the value of an identical but othewise non-leveaged fim afte tax. In the case of a default, which occus if the fim does not pay the coupon to the debt holdes, the fim is liquidated. The debt holde eceive the whole poceeds fom a liquidation, which equal the asset value minus bankuptcy costs α, while the stock holdes ae left with nothing. The elative bankuptcy costs α ange fom zeo to one. In the case of a default, we do not allow fo any enegotiations as pesented by Fan/Sundaesan (2000) to avoid bankuptcy costs, because these mechanisms would unnecessaily complicate the analysis without poviding us with futhe insights about the default stategy of convetible debt. If desied, one could easily incopoate these possibilities of enegotiation to pevent the occuence of bankuptcy costs. 1 We assume that the whole debt issue must be conveted at the same point in time in one block o not at all. Depending on the fact whethe pefect competition exists among the convetible bond holdes o a monopolist contols the whole convesions othe stategies athe than the block stategy can be optimal (see e.g. Constantinides (1984), Spatt/Stebenz (1988), and Bühle/Koziol (2002)). Since the focus of ou study lies on the optimal default decision, which is typically elevant if a convesion is elatively unlikely, it is appopiate to wok with this standad assumption. Moeove, the convetible debt value and theefoe also the default stategy ae unaffected by the convesion assumption if makets ae fictionless and fims have no additional debt outstanding. 4

7 In the special case of vey high elative bankuptcy costs such that (1 α) < γ, a convesion which esults in γ is beneficial fo the convetible debt holdes elative to a liquidation which amounts to (1 α). In this situation, the fim can foce a convesion pomptly by stopping the coupon payment, because the debt holdes have an incentive to pevent a liquidation by a convesion. To avoid a discussion of this tivial and untypical situation, we focus on cases with (1 α) > γ, in which debt holdes can obtain a highe value by taking ove the fim and liquidate it than by eceiving a faction of the fim value though convesion. The asset value follows a geometic Bownian motion d/ = (µ β) dt + σ dz, whee z is the value of a standad Wiene pocess, µ denotes the instantaneous expected ate of etun on the fim goss of all payout, and σ is the standad deviation of the instantaneous etun of the asset value. The fim s payout atio is β 0. Since taxes ae deductable, the equity holdes obtain a tax benefit equal to τ C when paying the coupon, whee τ indicates the tax ate. Hence, befoe a convesion and a default, the instantaneous net payment to the equity holdes amounts to (β C (1 τ)) dt, which is the whole payout minus coupon payment plus tax benefit. We note that β C (1 τ) can have a negative sign which means that the fim s payout β is not sufficient fo the afte-tax coupon payment. Then, the equity holdes must satisfy the esidual claim by an exta payment fom thei pivate wealth o declae a default. 2 In the special case without any net payout, β = 0, and non-convetible debt, γ = 0, ou model is consistent with that pesented by Leland (1994). Sibu/Pikovsky/Sheve (2002) also conside a model to picing convetible consol bonds. The diffeence to ou famewok is that they conside a diffeent payoff stuctue and an exogenous default baie. To achieve ou goal to analyze the effect of a convesion ight on the default baie, a model like ous with an endogenous default baie is equied athe than assuming an exogenous default baie. A futhe advantage of this appoach is that we obtain analytical solutions fo given default and convesion stategies which ae convenient fo a futhe analysis. 2 In pinciple an equity issue would be also possible to satisfy the coupon payment. Howeve, the issuance of new stocks sometimes affects the convesion atio of the convetible debt via anti dilution clauses in a complex way. Since it is not necessay to pesent a discussion of diffeent sophisticated anti dilution clauses to see the main factos fo the optimal debt sevice, we think of the case of an exta payment of the equity holdes, when the fim s payoffs do not suffice fo the coupon payment, to deal with a constant convesion atio γ. 5

8 2.2 Picing of Equity and Convetible Debt The values of equity S ( ) and convetible debt D ( ) aise fom the following two well-known diffeential equations as long as no pio convesion o default has occued yet 1 2 σ2 S, + ( β) S S + β C (1 τ) = 0, 1 2 σ2 D, + ( β) D D + C = 0. denotes the time-constant inteest ate p.a. fo all matuities. Othe appoaches such as those pesented by e.g. Kim/Ramaswany/Sundaesan (1993) and Longstaff/Schwatz (1995) intoduce stochastic inteest ates as a second souce of uncetainty. Howeve, this extension does not povide futhe insights fo ou goal to analyze the default stategy of convetible debt, but would substantially complicate the futhe deivations. We denote the two citical fim values at which the fim goes bankupt and at which a convesion occus by and C. Since unde the optimal default stategy by the fim and unde the convesion stategy by the debt holdes, both the equity value S ( ) and the debt value D ( ) ae continuous in the asset value, we can wite the following fou bounday conditions fo the values of the secuities lim S ( ) = 0, lim D ( ) = (1 α), lim C S ( ) = (1 γ) C, lim D ( ) = γ C. C The bounday conditions in the case of default ( ) eflect that debt holdes get the whole liquidation poceeds and equity holdes give up any ights o obligations. 3 Afte a convesion the fim is not leveed anymoe such that the fim value equals the asset value. Hence, the convesion value of the debt holdes is γ while the claim of the equity holdes compises of the emaining fim value (1 γ). 3 This liquidation ule is consistent with those typically used in the liteatue (see e.g. Leland (1994)). Unde pactical bankuptcy ules, howeve, thee can be a positive liquidation value fo the equity holdes in the case that the liquidation poceeds exceed the notional value of the consol bond. This possibility can be incopoated in the bounday conditions in a staightfowad way. As this extension would esult in seveal diffeent cases without poviding futhe insights about default baies of convetible debt, we abstain fom this feasible extension in what follows. 6

9 These two diffeential equations togethe with the bounday conditions allow us to wite the values of equity S (,, C ) and debt D (,, C ) fo abitay stategies and C. The unique solutions fo S (,, C ) and D (,, C ), as long as no default o convesion has occued yet, ae given by S (,, C ) = C (1 τ) + P B (,, C ) ( + C ) (1 τ) + P C (,, C ) ( γ C + C ) (1 τ), D (,, C ) = C ( + P B (,, C ) (1 α) C ) ( + P C (,, C ) γ C C ), whee ( ) Y X C X P B (,, C ) = B X, C X ( ) Y X B X P C (,, C ) = C C X, B X 4 ( β) ( + β) σ 2 + σ 4 X =, σ 2 Y = 1 X β. 2 σ 2 One can easily veify these solutions fo S (,, C ) and D (,, C ) by inseting them into the diffeential equations and checking the bounday conditions. Since the equity value S (,, C ) is negative fo below, we see that the fim can neve be alive fo < such that the optimal default stategy is to stop paying coupons when the asset value intesects fom above. On the contay, if no convesion has occued despite of > C, the debt value D (,, C ) is lowe than the convesion value γ. Theefoe, the optimal convesion stategy fo the debt holdes is to convet when hits C fom below. 2.3 Optimal Default and Convesion Stategies The optimal baies B and C fo default and convesion esult fom a Nash equilibium of a game between the equity and debt holdes. The equity holdes choose the default stategy in ode to maximize the equity value. The debt holdes select a convesion stategy C with egad to the debt value. A Nash equilibium is chaacteized by the popety that, given the optimal stategy of the 7

10 debt holdes, C, the equity holdes have no incentive to deviate fom thei stategy B and vice vesa. Fomally, we obtain the size of and C by applying the two smooth-pasting conditions fo the equity and debt value S (, B, C ) D (, B, C ) = 0, = B = γ. = C The fist condition means that the equity holdes tigge a default when the change of the equity value in the asset value of an alive fim coincides with that of a defaulted fim which is zeo. In addition, the equity holdes account fo the optimal behavio C of the convetible debt holdes. Accoding to the second condition, the debt holdes convet when the incease of thei holdings with is as high as the incease with the convesion value γ. We note that this solution is not necessaily paeto optimal, because a coopeation of both equity and debt holdes can esult in a highe fim value which might be advantageous fo both of these two goups. Then, one goup, howeve, could be bette off by deviating fom the coopeating stategy. The equied values B and C can be numeically obtained fom these equations. At this point, we emphasize that an equilibium solution with C > B always exists. The existence of a stategy and the validity of this elation follows fom popeties pesented in the next section. If and only if the payout ate β equals zeo, a convesion cannot be optimal fo the debt holdes. Fomally, this is because D(,, C) is below γ fo evey C but = C it conveges to γ fo C. Thus, we must evaluate the limit C when β is zeo. Convesely, when β is positive, the deivative D(,, C) is highe = C than γ fo a sufficiently high value of C such that the optimal convesion baie must be finite. This outcome is consistent with optimal execise stategies of C Ameican call options on stock. When the stock pays a popotional dividend, a pematue execise is optimal fo sufficiently high stock values, whee in the absence of dividends a pematue execise is not advantageous. We note that even though both the convetible debt and an Ameican call on a non-dividend-paying stock with infinite lifetime ae not execised, the call value and the convesion ight have a stictly positive value. 8

11 As a esult, we can wite the value of equity S ( ) = S (, B, C ) and debt D ( ) = D (, B, C ) as follows: 0, B C S ( ) = (1 τ) + P B (, B, C ) ( B + C (1 τ)) +P C (, B, C ) ( γ C + C (1 τ)) B, < < C (1 γ), C (1) (1 α), B C D ( ) = + P B (, B, C ) ((1 α) B ) C +P C (, B, C ) (γ C ) C B < < C (2), γ, C In the case of β = 0, the convesion baie is not finite and the evaluation of the limit C esults in a simple epesentation: 0, B S ( ) = ( ) Y ( ) C(1 τ) + B B + C(1 τ) γ ( Y X B), B < (1 α), B D ( ) = ( ) Y C + ( (1 α) B ) ( C + γ Y X B), B < B Given that the fim is solvent and no convesion has occued yet, B < < C, we can intepet P B (, B, C ) as the cuent value of binay up-and-out put that pays one unit at a futue point in time when the asset value hits the default baie B without being in the convesion egion, C, befoe. Othewise, this claim pays nothing. Accodingly, P C (, B, C ) can be seen as the pesent value of a binay down-and-out call that pays one unit only if hits C and no default has occued yet. Using this view, we see that the convetible debt value consists of the value of a default-fee consol bond C/ plus a claim that swaps this consol bond into the liquidation value at the default baie plus a futhe claim that swaps the consol bond into the convesion value at the convesion baie. Accodingly, the equity value S ( ) compises of the asset value of the fim minus the value of afte-tax coupon payments of a default-fee consol bond plus a claim that swaps the asset value into the value of the default-fee (afte-tax coupon) consol bond at the default baie and a futhe claim that swaps the convesion value of the convetible debt into the afte-tax consol bond at the convesion baie. We note that the fim value v ( ) consisting of the sum of the values of equity S ( ) and debt D ( ) geneally deviates fom the asset value. Fo the fim value v ( ), 9

12 we obtain: v ( ) = S ( ) + D ( ) = P B (, B, C) α B }{{} pesent value of bankuptcy costs + τ C (1 (P B (,, C) + P C (, B, C))). }{{} pesent value of tax shield In line with othe models consideing tax deductability and bankuptcy costs, the fim value v ( ) is the value of the fim s assets minus the pesent value of bankuptcy costs plus the pesent value of the tax shield. Repesentations (1) and (2) indicate that fo a given default and convesion stategy (B, C ) the elative bankuptcy costs α ae only at the costs of the debt value, whee the tax benefits only contibute to the equity value S ( ). Since the tax shield is pimaily in favo of the equity holdes athe than the debt holdes, it is not clea whethe the equity holdes can benefit fom the tax deductability in a such sevee way that the value of one stock is highe than the value of a numbe of convetible bonds that allows to eceive one stock in total. At fist glance, we would expect the value of one stock to be lowe than the coesponding numbe of convetible bonds, because the convetible debt holdes have the possibility to convet anytime. To answe this question, we compae the debt value, which is convetible into a faction of γ of the asset value, with a faction of γ 1 γ of the equity value, which also epesents a faction of the asset value equal to γ afte a convesion. Thus, the equity popotion of D ( ) coesponds to that of two values amounts to: γ S ( ). The diffeence of these 1 γ D ( ) γ 1 γ S ( ) = (1 α), B P B (, B, C ) (1 α) + (P C (, B, C ) C ) γ 1 γ B < < C + C (1 (P B(,, C)+P C(,B, C))) 1 γτ 1 γ 0, C In fact the diffeence D ( ) γ S ( ) can become negative, fo example if α = 1 and 1 γ τ is close to one which means that the debt holdes have to cay high bankuptcy costs, while equity holdes enomously benefit fom the tax shield. Then, fo B < < C, the tem (P C (,, C ) C ) γ is always negative and can have a 1 γ bigge size than the positive tem C (1 (P B (, B, C ) + P C (,, C 1 γτ ))). 1 γ 10

13 This finding is a fundamental diffeence to the standad famewok without fictions such as taxes and bankuptcy costs, in which convetible bonds ae typically examined. The value of a convetible bond in a fictionless maket must exceed the value of stocks times the convesion atio, as long as the fim has no additional claims outstanding alues of Investment and Option Component To analyze and pice convetible bonds, seveal appoaches decompose the convetible debt value D ( ) into a staight bond and an option component. 5 The bond component, which is typically denoted by the investment value of the convetible bond, is equal to the value of the convetible debt if the debt holdes do not make use of the convesion option, but the fim still follows its default stategy B.6 Thus, the investment value I ( ) of the convetible debt is given by { I ( ) = = (1 α), B lim C C + P B (, B, C) ((1 α) B C ) B < ( C + B (1 α), B ) Y ( (1 α) B ) C B <. (3) The idea behind this decomposition is to pice the investment value by egading an appopiate benchmak fim with identical assets and coupon obligations but with staight athe than convetible debt outstanding. Thus, the investment value is obtained by standad pocedues fo staight debt due to this pactical appoach. Then, the additional option component can be sepaately piced. We will analyze how seveely the investment value depends on the fact whethe the fim follows the optimal default stategy fo convetible debt o the fim follows the optimal stategy fo staight debt which is the stategy of the benchmak fim. The fist stategy will be denoted by optimal stategy while the othe stategy, that ignoes the convesion featue, is the staight debt stategy. Accodingly, the esidual amount D ( ) I ( ) is the option value O ( ), which 4 See e.g. Bühle/Koziol (2002). 5 See e.g. Ingesoll (1977), Tsiveiotis/Fenandes (1998), Takahashi/Kobayashi/Nakagawa (2001), and Hung/Wang (2002). 6 Of couse, if the fim was awae of the fact that no convesion took place, it could incease the equity value by adjusting the default level. 11

14 amounts to O ( ) = 0, B (P ( ) ) Y B (, B, C ) ((1 α) B B ) C +P C (, B, C ) (γ C ) C B < < C., ( ) Y γ C ( (1 α) B ) C, C B Figue 1: alues of Debt and Equity The diagam shows the value of convetible debt D ( ) and equity S ( ) as a function of the asset value. Moeove, it plots the debt components, the investment value I ( ) and the option value O ( ). The paamete values ae C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = D(), S(), I(), O() S() D() O() I() Figue 1 shows the values of equity S ( ), convetible debt D ( ), and the debt components I ( ) and O ( ) as a function of the asset value fo a typical example. We can see that the equity value is zeo in the case of default, but fo highe asset values it stictly inceases fist with a ecognizable convex shape and it is almost a linea function aftewads. The convetible debt value linealy inceases with fist because it equals the liquidation value. At = B, D ( ) has a kink and it futhe inceases with a concave shape. Afte a tuning point, D ( ) is a convex function which tends to the convesion value γ. The investment value of the convetible debt I ( ) coincides with D ( ) in the case of default. Then, it inceases less shaply than D ( ) such that I ( ) is a concave function that appoaches the value of a non-defaultable bond C/. The option value of the convetible debt O ( ) is a stictly convex and inceasing function of when the fim is alive and zeo othewise. 12

15 The functional behavio of these asset values is simila to the typical cases pesented e.g. in Bennan/Schwatz (1977). Fo low asset values the convetible debt is pimaily chaacteized by its investment value because it behaves like staight debt, wheeas fo high asset values the convetible debt athe behaves like a popotion of equity. Moeove, we can see that the cuvatue of the option value is moe ponounced than that of equity which is a typical popety of option values. Since the goal of this pape is to analyze the effect fom the convesion featue on the default stategy B, we need the stategy, if the debt was not convetible, as the efeence case (staight debt stategy). In the case of staight debt, the debt value ( ) Y unde the stategy fo is C + ( (1 α) B B ) C which diectly follows fom the epesentation of the investment value. Accodingly, ( the equity ( ) ) Y C value amounts fom the diffeence between the fim value + 1 τ ( ) Y ( ) Y α B B and the debt value C + ( (1 α) B B ) C. Evaluating the smooth-pasting condition of the equity holdes, we obtain the following epesentation fo the optimal default stategy of non-convetible debt B = C (1 τ) 2 ( β) σ2 σ 4 + X 2 ( β) σ 2 + σ 4 + X. Now, we egad the consequences if the fim follows the stategy B fo nonconvetible debt, but debt is convetible. The coesponding optimal stategy of the convetible debt holdes C follows fom the smooth-pasting condition D(,, C) = γ. Having detemined and C once, the investment I ( ) = C and the option value O ( ) unde this diffeent stategy is (1 α), I B ( ) = ( ) Y C + ( (1 α) B B ) C B <, (1 α), ( ) ) Y (P B (, B O ( ) =, C ) ((1 α) B ) C +P C (,, C ) (γ C ) C B < < C, ( ) Y γ C ( (1 α) B ) C, C B As befoe, if β = 0 holds, we have to evaluate the limit C, since it is always bette to wait with a convesion. 13

16 3 Analysis of optimal Debt Sevice In this section, we compae the optimal default stategy of the fim B with that of an identical fim with staight debt, i.e. the convesion featue is not taken into account. Similaly, we conside the convesion stategies C and C. Then, we egad the consequences fo the debt value, the investment value and the option value when the fim follows the optimal stategy athe than the optimal stategy fo non-convetible debt. As a fist popety of the default and convesion stategy, it is impotant to contol whethe equilibium solutions fo B and C as well as and C, espectively, can be always obtained. Popety 1 (Existence of Equilibium Stategies) An optimal default and convesion stategy, B and C, always exist. Also unde the optimal staight debt stategy, thee is an optimal convesion stategy C. The poof is shown in Appendix A.1. Fo the elation between the optimal and the staight debt stategy, we obtain the following popeties: Popety 2 (Default Baie) The optimal default baie B if debt is convetible is highe than fo non-convetible debt. Moeove, the default baie is below C (1 τ). The intuition fo the fist pat of this assetion, which is poven in Appendix A.2, is that the equity holdes suffe fom the convesion ight of the debt holdes because the convesion ight means an additional claim on the fim. Theefoe, the incentive of the equity holdes to keep the fim alive by paying the coupon is lowe when debt is convetible. Theefoe, the citical default baie B of convetible debt is highe than unde the stategy fo staight debt. Popety 3 (Convesion Baie) The convesion baie C when the fim optimally defaults is below the optimal convesion baie C when the fim follows the default stategy fo non-convetible debt. Moeove, the convesion baie C is above C γ. The poof of this popety is in Appendix A.3. The idea behind this assetion is that since the fim follows a suboptimal default stategy B athe than B, the 14

17 convetible debt holdes can longe benefit fom it by choosing a convesion baie C highe than C. As a consequence of the second pat of this popety, we can see that B must be below C. This is because the uppe bound fo the default baie is below the lowe bound fo the convesion baie. We can study the impotance of the default stategy by egading the consequences of it on the debt value, the investment value, and the option value. Popety 4 (Relation between Debt alues) If the fim follows the stategy fo staight debt B athe than convetible debt B, then the debt value D (, B, C ) is highe than D (, B, C ) fo evey asset value. We pove this popety in Appendix A.4. At fist glance, it is compehensible that the debt value is highe when the fim does not follow the optimal stategy. Howeve, the pimay eason fo this popety is that the non-optimal stategy of the fim esults in a highe fim value v ( ) due to lowe bankuptcy costs. Figue 2: Diffeence between Debt alues The left diagam shows the diffeence between the convetible debt value unde the staight debt and the optimal stategy D (,, C ) D (,, C ) as a function of the asset value. The ight diagam plots the diffeence elated to the debt value D (,, C ). The paamete values ae C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = fo the optimal stategy and = and C = fo the staight debt stategy absolute diffeence D 1 elative diffeence D Figue 2 shows the diffeence between the debt value D (,, C ) unde the staight debt stategy and the debt value D (, B, C ) unde the optimal stategy. If the asset value is low such that in both cases a default occus, the debt values unde both 15

18 stategies coincide. Then, the diffeence inceases with the asset value if only a default unde the optimal but not unde the staight debt stategy is optimal. When no default occus unde both stategies the diffeence D (, B, C ) D (, B, C ) declines with because the default stategy is less elevant the highe the asset value. Fo high asset values unde which a convesion is optimal, both debt values coincide again and theefoe D (,, C ) D (,, C ) is zeo. The diffeence as a pecentage of the debt value D (,, C ) has a simila stuctue. We can see that the diffeence can lie above ten pecent and it has a consideable magnitude ove a boad ange of asset values. Especially fo low asset values, when a default is elatively pobable, the diffeence is impotant, but it is less stiking fo asset values close to a convesion. Although the value of debt is highe when the fim follows the staight debt stategy, the investment value I (,, C ) as well as the option value O (, B, C ) of debt unde the optimal stategy can be highe than I (,, C ) and O (,, C ), espectively. Popety 5 (Relation between Investment alues) The investment value unde the staight debt stategy I (,, C ) is highe than unde the optimal stategy I (, B, C ) fo all if I (,, C ) > (1 α). Othewise, I (, B, C ) is highe than I (,, C ) fo >, but it is lowe fo fim values slightly above B. This poof is shown in Appendix A.5. Intuitively, we would expect that the investment value I (,, C ) unde the suboptimal stategy exceeds the value I (, B, C ) unde the optimal stategy as it is tue fo the convetible debt value. This elation nevetheless holds fo the investment value fo all, as long as the liquidation value unde the optimal stategy is not too high. Howeve, if the poceeds fom a liquidation at the baie unde the optimal stategy B (1 α) ae vey high and lie even above the investment value I (B,, C ) fo this fim value when the fim follows the suboptimal stategy, we see that a liquidation is bette fo the debt holdes than to keep the fim alive. This is the eason why the investment value unde the optimal stategy can also exceed the value unde the stategy fo staight debt. Figue 3 shows the diffeence between the investment values unde the staight debt stategy and the optimal stategy. In this case, the impotant deteminant I (B,, C ) (1 α) is positive. Fo low asset values, I (,, C ) I (, B, C ) behaves like the diffeence of the debt values pesented in Figue 2. 16

19 Figue 3: Diffeence between Investment alues The left diagam shows the diffeence between the investment value unde the staight debt and the optimal stategy I (,, C ) I (,, C ) as a function of the asset value. The ight diagam plots the diffeence elated to the investment value I (,, C ). The paamete values ae C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = fo the optimal stategy and = and C = fo the staight debt stategy. 2 absolute diffeence I elative diffeence I Fo highe asset values unde which no default occus, the diffeence deceases with and tends to zeo, as both investment values tend to the value C/ of the nondefaultable consol bond. A compaison of the diffeences of the debt values and the investment values, howeve, eveals that the diffeence of the investment value is much moe affected by the default stategy than the debt value itself. Even fo high asset values unde which a convesion is optimal, the diffeence is consideable. The eason why the investment value is much moe affected by the default stategy than the convetible debt value fo high asset values is that the convetible debt athe exhibits an equity chaacte fo high values of which is almost independent of the default stategy. Figue 4 displays the diffeence I (,, C ) I (,, C ) in the case in which the impotant deteminant I (B,, C ) (1 α) has the opposite sign than in Figue 3. To have a liquidation value (1 α) B highe than the investment value I (B,, C ) at, we need low elative bankuptcy costs α and a high value of B elative to. Fo this pupose, we set α = 0 and conside an especially high equity popotion of the debt value γ = In addition, we lowe β to 0.01 which also esults in a highe diffeence between B and B. Figue 4 shows that in the case of default unde the staight debt stategy but not unde the optimal stategy, the diffeence I (,, C ) I (,, C ) is fist 17

20 Figue 4: Diffeence between Investment alues The left diagam shows the diffeence between the investment value unde the staight debt and the optimal stategy I (,, C ) I (,, C ) as a function of the asset value. The ight diagam plots the diffeence elated to the investment value I (,, C ). The paamete values ae C = 5, γ = 0.75, α = 0, β = 0.01, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = fo the optimal stategy and = and C = fo the staight debt stategy. 2 absolute 0 diffeence I elative 0 diffeence I positive but tends to a negative local minimum aftewads. The eason is that the liquidation value (1 α) is such high that it even exceeds the investment value I (, B, C ). Then, if no default occus, the diffeence I (, B, C ) I (, B, C ) emains negative and tends to zeo with the asset value. Theefoe, the fact that the fim seves the debt too long can esult in a lowe investment value of the convetible debt. The diffeence in this case is consideable and lies between two and twelve pecent. Now, we econside the pactical appoach to picing the investment value of a convetible bond by valuing the debt like staight debt of an othewise identical fim. Accoding to this appoach, the used investment value is I (, B, C ) athe than the coect one I (, B, C ). Howeve, the optimal default stategy of a fim is fundamentally affected by the existence of a convesion featue. As seen in the consideed examples the magnitude of the diffeence between the eal investment value I (, B, C ) and the investment value I (,, C ) fom the pactical appoach is consideable and can be even much highe than the diffeence between the debt values itself. Popety 6 (Relation between Option alues) The option value fo low asset values unde the staight debt stategy O (,, C ) is highe than unde the optimal stategy O (, B, C ). Fo high asset values, O (,, C ) is highe than 18

21 O (,, C ) if I (,, C ) (1 α) holds and vice vesa. The poof of the popety is in Appendix A.6. When a default only unde the optimal stategy athe than the staight debt stategy occus, the option value unde the staight debt stategy is highe because it is zeo unde the optimal stategy. Fo high asset values, the convetible debt value is hadly affected by the choice fo one of the two default stategies. Howeve, the value of the investment component still depends on the default stategy even fo high asset values. Since the option value is the esidual claim between the debt value and the investment value, the option value is highe (lowe) unde the optimal stategy when the investment value unde this stategy is lowe (highe) than unde the staight debt stategy. Figue 5: Diffeence between Option alues The left diagam shows the diffeence between the investment value unde the staight debt and the optimal stategy O (,, C ) O (,, C ) as a function of the asset value. The ight diagam plots the diffeence elated to the investment value O (,, C ). The paamete values ae C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = fo the optimal stategy and = and C = fo the staight debt stategy. 0.2 absolute 0 diffeence O elative 0.05 diffeence O Figue 5 shows the diffeence between the option values O (, B, C ) O (, B, C ) in the case that coesponds to Figue 3. As long as no default occus, the option value is stictly positive. Theefoe, the diffeence stictly inceases with when a default only unde the staight debt but not unde the optimal stategy occus. Then, O (,, C ) O (,, C ) declines and attains a local minimum. Aftewads, it appoaches zeo. Since the convetible debt values unde both stategies ae nealy equal fo high asset values but the investment value unde the staight debt stategy is highe, the option value unde this stategy must be lowe. This effect explains why the diffeence O (, B, C ) O (, B, C ) 19

22 changes the sign when is vaied. Moeove, we can see that the size of the diffeences in pecentage tems is much highe than those of the investment values when the asset values ae low. Convesely, fo high asset values, the option values deviate by less than 0.1 pecent such that the default stategy is less impotant fo the option value in this asset value egion. Figue 6: Diffeence between Option alues The left diagam shows the diffeence between the option value unde the staight debt and the optimal stategy O (,, C ) O (,, C ) as a function of the asset value. The ight diagam plots the diffeence elated to the option value O (,, C ). The paamete values ae C = 5, γ = 0.75, α = 0, β = 0.01, σ = 0.5, τ = 0.3, and = The citical asset values ae B = and C = fo the optimal stategy and = and C = fo the staight debt stategy. 20 absolute diffeence O elative diffeence O In the case of Figue 6, which coesponds to Figue 4, the diffeence of option values is always non-negative. It fist inceases with until B and declines to zeo aftewads. Ove a elatively boad ange of asset values the diffeence has a consideable size above thee pecent, but it is again neglectable close to convesion. When egading the diffeences between the investment and option values in Figues 3, 4, 5, and 6, we find that eithe the investment value unde the staight debt stategy dominates the investment value unde the optimal stategy and the diffeence of option values changes its sign o the diffeence of option values is always positive and the diffeence of investment values has a change of the sign. As a consequence of the impact of the default stategy on the value of the option component, we see that it is impotant to account fo the eal stategy not only when computing the investment value but also the option value. This is a futhe point which is not in line with the pactical appoach to decomposing the convetible debt value, because it suggests that the option value can be sepaately piced independent 20

23 of the default stategy. 4 Compaative Static Analysis of the optimal Default Baie The potential fo high diffeences between the convetible debt values, between the investment values, and between the option values unde the optimal and staight debt stategy pimaily depends on the diffeence between the optimal B and the staight debt stategy. In geneal, the diffeences of the asset values ae moe ponounced, the highe the diffeence of the stategies B. Theefoe, we povide a detailed compaison of the optimal default stategy B with the staight debt stategy in this section. Fo this pupose, we do a compaative static analysis of the diffeence B. The paametes, which ae vaied, ae the equity popotion γ, the size of the coupon C, the tax ate τ, the payout ate β, the liquidation costs α, the inteest ate, and the volatility σ of the etun of the asset value. The gaphical esults fo a typical example ae pesented in Figue 7. Equity Popotion γ In this figue, we can see that the optimal default baie B inceases with γ. Clealy, the staight debt stategy is not affected by γ as the staight debt value does not depend on the convesion featue. Thus, the diffeence, B, inceases with γ. The eason fo this behavio is that the equity holdes let the debt holdes to paticipate moe in the fim value the highe γ. Since this highe paticipation is at thei own costs, the incentive fo the equity holdes to save the fim fom a default declines with γ which is indicated by a B Coupon C inceasing with γ. A highe coupon esults in inceasing default baies B and because it is moe costly fo the equity holdes to save the fim. Theefoe, the incentive to default ises with C. Moeove, the impotant diffeence B between these two stategies also inceases with C. In geneal, the diffeence is moe ponounced, the moe valuable the convesion ight fo slightly above B is. Since the value of this ight inceases with such as the default boundaies B and, we can see why the diffeence B also inceases with C moe shaply than. Tax Rate τ The highe the tax ate the lowe the default baies B and B and also the lowe 21

24 Figue 7: aiation of Default Baies These diagams show the default baie B unde the optimal stategy and unde the staight debt stategy as a function of the equity popotion γ, the coupon C, the tax ate τ, the elative liquidation costs α, the payout ate β, the inteest ate, and the volatility σ of the etun of the asset value. The standad paamete values ae C = 5, γ = 0.75, α = 0, β = 0.01, σ = 0.5, τ = 0.3, and = *, ' *, ' γ ' * ' * τ * B, ' C 20 * * ' * B, ' 15 ' α * B, 25 ' 20 * ' *, ' 5 ' β * 60 * B, ' * ' σ 22

25 the diffeence B. The eason fo these elations is that the tax ate has an opposite effect on the default baies as the coupon. This is because we can think of a highe tax ate τ as a highe deductability of the inteest ate payments as the net payment by the equity holdes is not the full coupon payment but the afte tax payment C (1 τ). Thus, the net payment declines with τ. Since the diffeence B and the baies, and, decline with a lowe C they must also decline with a highe τ. Costs fo Liquidation α Since the elative liquidation costs α do not affect the wealth of the equity holdes, the staight debt stategy is invaiant of α. Howeve, the convetible debt holdes follow a convesion stategy that depends on α such that also the optimal default baie B is affected by the liquidation costs. Howeve, this effect is athe maginal that a dependency cannot be ecognized in Figue 7. Payout Rate β The default baies B and decease with the payout ate β. This effect stems fom a highe dividend payment to the equity holdes when β inceases. The value of equity typically inceases with the size of the dividend payment at the costs of the debt holdes and theefoe we can undestand why the incentive of the equity holdes to pevent a default by making coupon payments inceases with β. In othe wods, the highe β, the highe the faction of the asset value that is exclusively fo the equity holdes, while othewise the asset value is shaed with the debt holdes. Moeove, the diffeence B also deceases with β. This is due to the fact that the convesion ight becomes less inteesting fo the debt holdes when β is high. The eason fo this effect is that the gowth of the fim s asset value is mitigated by highe payoffs β such that the debt holdes suffe fom a lowe futue convesion value γ. As a consequence, the value of the convesion ight declines with β such that B appoaches B Inteest Rate with β. When the inteest ate ises, the default baies B and tend to zeo. This is due to the fact that the pesent value of pomised coupon payments C/ declines and tends to zeo if inceases. Thus, it is cheape fo the equity holdes to fulfil thei debt obligation when is high such that thei suvival incentive goes up. The diffeence B and, howeve, fist inceases with and then appoaches zeo as Figue 7 shows. This is a esult of two counte-effects. The fist effect is that the convesion pobability (unde the isk-neutal measue) inceases with such that 23

26 the convesion ight ises with. On the othe hand, both baies B and B tend to zeo with such that the diffeence B B olatility σ must decline fo high. In pinciple, the vaiation of the default baies, B and, and thei diffeence B with the volatility of the etun of the asset value is compaable with a vaiation of as seen befoe. Since the equity value typically benefits fom a highe volatility at the costs of the debt holdes, the default baies decline with σ and both appoach zeo when σ tends to infinity. This effect also explains why the diffeence B tends to zeo fo sufficiently lage volatilities σ. As the pesent value of the convesion ight, which the equity holdes ae shot, inceases with the volatility, we see why the diffeence B B inceases fist befoe it goes to zeo. 5 Conclusion This pape ests on the obsevation that a boad liteatue discusses optimal default stategies of staight bonds, but on eal makets debt is often equipped with additional ights such as a convesion featue. Theefoe, the goal of this pape is to deive and to analyze the optimal default stategy of convetible debt. Fo this pupose, we use a time-independent model famewok, simila to that pesented by Leland (1994), with bankuptcy costs and taxes. This model eveals how the default stategy of convetible debt is affected by the convesion featue. We can see why an optimal default of convetible debt occus ealie than a default of non-convetible debt. Moeove, the value of convetible debt is highe when the fim follows the optimal staight debt stategy, but the values of the investment and option component can be lowe. As a esult, the optimal default stategy is a elevant picing facto fo convetible debt. Hence, the pactical appoach to picing the investment value of a convetible bond, by egading the value of staight debt of an othewise identical fim, can esult in sevee inaccuacies of the investment value. Accodingly, the debt and option value ae also impacted by this poblematical appoach especially when a default is likely. The diffeences of the asset values pimaily aise fom the diffeence between the optimal default stategy and the staight debt stategy. As a compaative static analysis shows, the diffeence between the two default baies B ises with the convesion atio γ, the pomised coupon C, a lowe tax ate τ, and a lowe payoff ate β. These ae testable implications which can be empiically evaluated in 24

27 anothe pape. In addition, we can extact futhe insights of this pape when using the model famewok to analyze the default stategy of fims whose debt is equipped with othe non-staight featues. Analogously, we can show that the default stategy of a fim with debt that is attached with waants is in pinciple simila to that of convetible debt. Convesely, if debt is callable by the fim, we can ague that a default occus late due to the call featue. The intuition is that the equity holdes have an additional call ight when they save the fim by poviding the coupon payments which povides them with a highe incentive to fulfil the coupon obligation elative to the case of non-callable debt. A Poofs of the Popeties A.1 Poof of Popety 1 Fo evey abitay stategy of the equity holdes C, an optimal stategy of the convetible debt holdes C exists which satisfies the smooth-pasting condition D(,,C) = γ. If β = 0, the optimal stategy is infinite as discussed above. = C Othewise, a finite solution exists. This is because the deivative D(,,C) = C is negative fo C close to and it is above γ fo C. As a consequence of the intemediate value theoem, a finite solution C above exists fo the condition D(,,C) = γ. Since B is below C, we can see that unde the staight debt = C stategy = B an equilibium stategy C always exists. We will use the finding that an optimal convesion baie C () exists, given that the fim follows the stategy C, to pove the existence of the optimal stategy. The impotant tem of the smooth-pasting condition S(,,C ()) has the =B opposite sign at = than fo = C (1 τ). Since S(,,C ()) is a =B continuous function in, the smooth-pasting condition S(,,C ()) = 0 =B must be satisfied fo a between B and C (1 τ). Then, this point and C () is the Nash equilibium stategy. 25