University of California Berkeley

Transcription

1 Univesity of Califonia Bekeley

2 Contingent Convetible Bonds and Capital Stuctue Decisions Bois Albul Dwight M. Jaffee Alexei Tchistyi Mach 26, 2010 Abstact This pape povides a fomal model of contingent convetible bonds CCBs, a new instument offeing potential value as a component of copoate capital stuctues fo all types of fims, as well as being consideed fo the efom of pudential bank egulation following the ecent financial cisis. CCBs ae debt instuments that automatically convet to equity if and when the issuing fim o bank eaches a specified level of financial distess. CCBs have the potential to avoid bank bailouts of the type that occued duing the subpime motgage cisis when banks could not aise sufficient new capital and bank egulatos feaed the consequences if systemically impotant banks failed. While qualitative discussions of CCBs ae available in the liteatue, this is the fist pape to develop a fomal model of thei popeties. The pape povides analytic popositions concening CCB attibutes and develops implications fo stuctuing CCBs to maximize thei geneal benefits fo copoations and thei specific benefits fo pudential bank egulation. Keywods: Contingent Convetible Bond, Banking Regulation, Subpime Motgage Cisis, Stuctual Model, Copoate Finance 1 Intoduction This pape povides a fomal model of contingent convetible bonds CCBs, a new instument offeing potential value as a component of copoate capital stuctues fo all types of fims, as well as being consideed fo the efom of pudential bank egulation following the ecent financial cisis. CCBs ae debt instuments that automatically convet to equity if and when the issuing fim o bank eaches a specified level of financial distess. While qualitative discussions of CCBs Haas School of Business, Univesity of Califonia at Bekeley, balbul@haas.bekeley.edu Haas School of Business, Univesity of Califonia at Bekeley, jaffee@haas.bekeley.edu Haas School of Business, Univesity of Califonia at Bekeley, tchistyi@haas.bekeley.edu 1

3 ae available in the liteatue, this is the fist pape to develop a fomal model of thei popeties. The pape povides analytic popositions concening CCB attibutes and develops implications fo stuctuing CCBs to maximize thei geneal benefits fo copoations and thei specific benefits fo pudential bank egulation. CCBs ae eceiving attention as a new instument fo pudential banking egulation because they have the potential to avoid the bank bailouts that occued duing the subpime motgage cisis when banks could not aise sufficient new capital and bank egulatos feaed the consequences if systemically impotant banks failed 1. A moe standad poposal fo bank egulatoy efom is to aise capital equiements since, if set high enough, they can achieve any desied level of bank safety. Vey high capital atios, howeve, impose significant costs on banks and theeby inhibit financial intemediation; and/o the capital equiements will be cicumvented though egulatoy abitage 2. Thee have also been poposals to focus a component of the capital equiements on systemic isk Adian and Bunnemeie 2009, o to pohibit banks outight fom isky activities that ae not fundamental to thei ole as financial intemediaies Volcke While these poposals could well impove pudential bank egulation, they do not diectly addess the issue of how distessed banks can aise new capital in ode to peclude the need fo govenment bailouts. In this setting, CCBs have been poposed by academics Flanney 2002, 2009a, 2009b, Duffie 2009, Squam Lake Woking Goup on Financial Regulation 2009, and McDonald 2010 and endosed fo futhe study by bank egulatos Benanke 2009 and Dudley In fact, 1 The bank bailouts duing the subpime cisis eflect a failue of the egulatoy pinciples ceated unde the 1991 Fedeal Deposit Insuance Copoation Impovement Act FDICIA and specifically of its equiement that bank egulatos take pompt coective action PCA in esponse to declining bank capital atios. Eisenbeis and Wall 2002 povide a detailed discussion of FDICIA and its PCA equiements. As examples of the PCA equiements, significantly undecapitalized banks ae to aise new equity o pomptly mege into a well capitalized bank, and citically undecapitalized banks ae to be placed unde a eceive within 90 days of attaining that status. The subpime cisis also evealed that subodinated debt holdes failed to discipline the banks, while the govenment bank bailouts potected these debt holdes fom the majo losses they would have othewise faced in a bankuptcy. 2 High capital equiements limit the use of debt tax shields, impose a high cost on existing shaeholdes when aising new capital due to the debt ovehang poblem, and accentuate impotant pincipal-agent inefficiencies within the banks; see Kashyap, Rajan, and Stein 2008, Squam Lake Woking Goup on Financial Regulation 2009, Dudley 2009, and Flanney 2009 fo futhe discussion of these issues. 3 Thee have also been poposals fo contingent capital instuments that ae not bonds. Kashyap, Rajan, and Stein 2008 popose an insuance contact that povides banks with capital when cetain tiggeing events occu and Zingales and Hat 2009 focus on the use of cedit swaps. Wall 2009 povides a suvey of this evolving liteatue. 2

4 Lloyd s bank issued the fist 7 billion $11.6 billion CCBs CoCo bonds 4 in CCBs initially ente a bank s capital stuctue as debt instuments, thus poviding the debt-instument benefits of a tax shield and a contol on pincipal-agent conflicts between bank management and shaeholdes. If and when the bank eaches the specified degee of financial distess, howeve, the debt is automatically conveted to equity. The convesion ecapitalizes he bank without equiing any ex-post action by banks to aise new equity o the govenment to bail them out. The automatic ecapitalization featue of CCBs thus offes a elatively low-cost mechanism to avoid the costs that othewise aise with the theatened bankuptcy of systemically impotant banking fims. The existing CCB poposals-see especially Flanney 2009a and McDonald 2010-povide a list of issues that must be settled in fomulating any specific plan fo implementation: The tigge must be designed to avoid accounting manipulation, and the esulting convesion of CCB to equity must be automatic and inviolable. In fact, an accounting tigge in the Lloyd s bank 2009 CCB issue has aleady aised seious concen; see Duffie Most poposals instead ecommend a tigge based on a maket measue of each bank s solvency, such as a minimum atio of maket equity to asset value 5. This is the case we model. We also analyze the issue of maket manipulation of the equity value that may aise with a maket-value tigge. The CCB to equity convesion tems applied afte the tigge is activated must be specified. A key question is how the value of the equity shaes eceived at convesion compae to the value of the conveting bonds; see Flanney 2009a and 2009b and McDonald We conside the geneal case in which the atio of the equity convesion value to the CCB value is a contact paamete λ to be chosen. Among othe effects, we analyze the impact this contact paamete may have on the incentive to manipulate the maket value of the bank s 4 Souce: Financial Times fom Novembe 5, The Squam Lake Woking Goup and McDonald poposals equie two tigges to be activated befoe convesion occus. One tigge is based on each bank s own financial condition while the second tigge is based on an aggegate measue of banking system distess. This means that individual banks can become insolvent pio to CCB convesion if the aggegate tigge is not activated. Fo this eason, Flanney 2009a agues fo a single, bank-based, tigge. In this pape, we fomally model only this single tigge case. 3

5 equity shaes. The CCB contact could impose a dynamic sequence in which specified amounts of CCB convet at diffeent thesholds. Flanney 2009a, futhemoe, poposes a egulatoy equiement wheeby conveted CCB must be pomptly eplaced in a bank s capital stuctue. While we comment on the possible advantages of such dynamic contact featues, ou fomal model coves only the case of a one-time and complete convesion. The adoption of CCBs by banks could be voluntay o a equied component of thei capital equiements 6. We conside both possibilities. The key contibution of the cuent pape is to povide a fomal financial model in which the effects of altenative CCB contact povisions can be analytically evaluated. We develop closed fom solutions fo CCB value by adapting the Leland 2004 model 7. Ou esults apply equally well to the addition of CCBs to the capital stuctue of copoations geneally, as well as fo thei specific application as a tool fo pudential bank egulation. We make thee assumptions thoughout the pape egading a fim s use of CCBs: 1 The fim is allowed a tax deduction on its CCB inteest payments as long as the secuity emains outstanding as a bond. This would be the likely case fo banks if CCBs wee to become a fomal and established component of pudential banking egulation. the same time, this means that the public cost of the CCB tax shield must be included when evaluating the possible ole of CCBs fo pudential bank egulation. Fo copoations moe geneally, we acknowledge that 6 Fo example, Flanney 2009a povides an illustative example in which banks ae equied to choose between satisfying thei capital equiements by i holding equity equal to 6% of an asset aggegate o ii holding equity equal to 4% of the asset aggegate and CCBs equal to 4% of the asset aggegate. This suggests a egulatoy tadeoff in which 4 pecentage points of CCBs ae the equivalent of 2 pecentage points of equity. 7 The Leland model has been successful applied in ecent studies of othe fixed-income debt secuity innovations, although none analyzes the case of a bond convesion tiggeed by financial distess. Bhanot and Mello 2006 study copoate debt that includes a ating tigge such that a ating downgade equies the equity holdes to compensate the bondholdes with ealy debt edemption o othe benefits. Manso, Stulovici, and Tchistyi 2009 study a class of debt obligations whee the equied inteest payments depend on some measue of the boowe s pefomance. This could include the exteme case in which the debt inteest coupons each zeo at some level of financial distess. This case would povide some of the same benefits in educing o eliminating bankuptcy costs as povided by CCBs in the cuent pape. 4

6 the tax deductibility of CCB inteest payment will likely equie futhe IRS ulings, including possible legal challenges and new legislative actions. 2 In all cases, we assume that adding CCBs to a fim s capital stuctue has no impact on the level of the fim s asset holdings A. In othe wods, we assume the addition of CCBs must take the fom of eithe a CCB fo equity swap with the CCB poceeds paid out as a dividend to equity holdes o as a CCB fo staight debt swap with the CCB poceeds used to etie existing staight debt. The CCB fo staight debt swap appeas to be the most impotant case fo egulatoy applications, while we acknowledge that the futue study of assets effects could be impotant fo moe geneal copoate finance applications. 3 Ou analysis is caied out unde the condition-condition 1 below-that CCBs must convet to equity at a time pio to any possible default by the fim on its staight debt. This condition constains the set of feasible CCB contacts that ae consideed in ou analysis. This constaint is a necessay, and sensible, equiement if CCBs ae to have the desied popety of educing the bankuptcy costs associated with a bond default. The following is a summay of ou main questions and esults. We fist conside a fim that has a new oppotunity to include CCBs within its existing capital stuctue in a setting whee no egulatoy estictions ae imposed on CCB issuance with the exception of the contact constaints ceated by Condition 1 as just descibed. Q1. Will a fim include CCBs in its capital stuctue if it is feely allowed to do so? A1. A fim will always gain fom including CCB in its capital stuctue as a esult of the tax shield benefit. This is tue whethe o not the fim also includes staight debt in its optimal capital stuctue; in fact, the optimal amount of staight debt is unaffected by the addition of CCB. Given that total assets ae unchanged by assumption, in effect the CCB ae being swapped fo, i.e. eplacing, equity in the capital stuctue. Since the asset-value default theshold on any existing staight debt is unchanged, adding CCB in this manne povides no benefits fo egulatoy safety, while taxpayes pay the cost of the additional tax shield. The addition of CCB in this fom 5

7 may also magnify the fim s incentive fo asset substitution to expand its asset isk. We next conside a fim that opeates unde the egulatoy constaint that it may issue CCB only as a pat of a swap that eties an equal amount of staight debt. This constaint is implicit o explicit in vaious poposals to use CCB fo pudential bank egulation; see Flanney 2009a. The esult depends on whethe the fim is ceating a de novo optimal capital stuctue o is adding CCB to an aleady existing capital stuctue. Q2. Will a fim add CCBs to a de novo optimal capital stuctue, assuming it faces the egulatoy constaint that the CCB can only eplace a pat of what would have been the optimal amount of staight debt? A2. A bank ceating a de novo capital stuctue unde the egulatoy constaint will always include at least a small amount of CCB in its optimal capital stuctue. The eduction in expected bankuptcy costs ensues a net gain, even if the tax shield benefits ae educed, at least fo small additions of CCB. The addition of CCBs also has the effect of educing the incentive fo asset substitution. The bottom line is that CCBs in this fom povide an unambiguous benefit fo egulatoy safety. Q3. Will a fim add CCBs to an existing capital stuctue, assuming it faces the egulatoy constaint that the CCB can only be intoduced as pat of a swap fo a pat of the outstanding staight debt? A3. Assuming the initial amount of staight debt equals o exceeds the optimal amount, the existing equity holdes will not voluntay ente into the poposed swap of CCB fo staight debt. While the swap may incease the fim s value the value of educed bankuptcy costs may exceed any loss of tax shield benefits the gain accues only to the holdes of the existing staight debt. This is thus a classic debt ovehang poblem in which the equity holdes will not act to enhance the oveall fim value. To be clea, this esult depends in pat on ou assumption that the staight debt has the fom of a consol with indefinite matuity. If the staight debt has finite matuities, then the CCB could be swapped only fo matuing debt, thus educing the debt ovehang cost. Q4. How can CCBs be designed to povide a useful egulatoy instument fo expanding the 6

8 safety and soundness of banks that ae acknowledged to be too big to fail TBTF? A4. We assume a TBTF bank is one fo which its staight debt is isk fee because the bond holdes coectly assume they will potected fom any potential insolvency. We also assume a egulatoy limitation on the amount of debt such a bank may issue. Unde this limitation, a CCB fo staight debt swap educes the value of the govenment subsidy because it educes the expected cost of bondholde bailouts. While this has a taxpaye benefit, the equity holdes of such a bank would not voluntaily paticipate in such a swap. Q5. May CCBs ceate an incentive fo maket manipulation? A5. CCB may potentially ceate an incentive fo eithe the CCB holdes o the bank s equity holdes to manipulate the bank s stock pice to a lowe value in ode to foce a CCB fo equity convesion. The incentive fo CCB holdes to manipulate the equity pice exists only if the atio of equity convesion value to CCB value λ in the model is sufficiently high to make the convesion pofitable fo the CCB holdes. The incentive fo bank equity holdes to manipulate the equity pice exists, compaably, only if the atio of equity convesion value to CCB value λ is sufficiently low to make the foced convesion pofitable fo the equity holdes. Q6. May estictions on CCB contact and issuance tems be useful in maximizing the egulatoy benefits of bank safety? A6. The egulatoy benefits of CCB issuance will geneally depend on the CCB contact and issuance tems. Pehaps most impotantly, the egulatoy benefits vanish if banks simply substitute CCBs fo capital, leaving the amount of staight debt unchanged. It is thus essential to equie CCB issuance to substitute fo staight debt and not fo equity. In addition, the highe the theshold fo the convesion tigge the geate the egulatoy safety benefits. The convesion atio of equity fo CCBs may also detemine the incentive fo CCB holdes o equity holdes to manipulate the stock pice. The stuctue of the pape is as follows. Pat 2 develops the fomal model. Pat 3 applies the model to detemine the ole CCBs play in a bank s optimal capital stuctue. Pat 4 analyzes bank issuance of CCBs when egulatos equie that the CCBs povide a net addition to bank 7

9 safety. Pat 5 applies the model to the ole of CCBs when banks ae too big to fail TBTF. Pat 6 povides ou discussion of maket manipulation involving CCBs. Pat 7 investigates the effects of CCBs on asset substitution efficiency. Pat 8 povides a summay and policy conclusions. 2 Model We use the taditional capital stuctue modeling famewok based on Leland A fim has poductive assets that geneate afte-tax cash flows with the following dynamics dδ t δ t = µdt + σdb Q t, 1 whee µ and σ ae constant, and B Q defines a standad Bownian motion unde the isk-neutal measue. The isk-fee ate,, is constant and, by assumption, is such that µ <. The tax ate θ 0, 1. Inteest payments ae tax deductible. t the maket value of assets A t is defined as the value of all futue cash flows. Given 1, that is A t = E Q t [ t ] e s t δ s ds = δ t µ. The dynamics fo A t ae: da t = µa t dt + σa t db Q t. The fim can issue equity and eithe a staight bond staight debt o both a staight bond and a CCB. Both bonds ae consol type, meaning they ae annuities with infinitive matuity. Staight debt pays coupon c b, continually in time, until default. default, faction α [0, 1] of the fim assets is lost. CCB pays coupon c c, continually in time, until stopping time τ = inf{s : A s }. τ CCB fully convets into equity - bond holdes eceive equity valued at its maket pice in the amount of λ cc. The coefficient λ is the CCB contact tem that detemines the atio of the maket value of equity elative to the maket value of debt at the point of convesion. With λ = 1, CCBs convet into a maket value of equity equal to the maket value of the CCB debt. 8

10 With λ < 1 λ > 1, the maket value of equity eceived is at a discount pemium elative to the maket value of the CCB deliveed fo convesion. The following Condition 1 is assumed to hold always. Condition 1: c b, c c, and λ ae such that the fim does not default befoe o at CCB convesion. One way to motivate Condition 1 is by consideing the banking system. Regulatos might look at CCB as a way to cushion individual banks and help them maintain capital atios above pedetemined levels in the event of a financial cisis. Having a bank default befoe o at convesion would obviate this ole fo CCB. Thee ae two esults fom the existing financial stuctue liteatue that will be used late in the pape. Fist, as in Duffie 2001, fo a given constant K 0, A t, the maket value of a secuity that claims one unit of account at the hitting time τk = inf{s : A s K} is, at t < τk, E Q t [ e τk t] =, 2 K whee γ = m+ m 2 +2σ 2 σ 2 and m = µ σ2 2. Second, also as in Duffie 2001, the default-tiggeing asset level that coesponds to the optimal default time τa B fo the case when the capital stuctue of the fim includes only equity and staight debt is, at t < τa B, A B = β θc b, 3 whee β = γ 1+γ. Lemma 1. Let the capital stuctue of the fim include equity and staight debt. If at t befoe default the fim decides to issue CCB without changing the existing amount of staight debt, the optimal default bounday A B will emain the same. 9

11 Poof. We assume that Condition 1 holds. Theefoe, thee is no default befoe o at convesion. convesion CCB holdes become equity holdes. The value of assets does not change. Afte convesion the maximum-equity-valuation poblem of equity holdes including the ones that became equity holdes as the esult of convesion is the same as in the case when the capital stuctue includes only equity and staight bond. Hence, the same A B. 2.1 Closed-Fom Solutions Ou goal in this subsection is to deive closed-fom solutions fo the values of claims associated with the capital stuctue when the fim issues equity, staight debt and CCB. t the following budget equation holds: A t + T BA t ; c b, c c = W A t ; c b, c c + U B A t ; c b, c c + U C A t ; c b, c c + BCA t ; c b, c c, 4 whee T BA t ; c b, c c is the expected pesent value of tax benefits, W A t ; c b, c c is the value of equity, U B A t ; c b, c c is the value of staight debt, U C A t ; c b, c c is the value of CCB and BCA t ; c b, c c is the expected pesent value of bankuptcy costs 8. The total value of the fim, GA t ; c b, c c, is the sum of the maket values of equity and debt GA t ; c b, c c = W A t ; c b, c c + U B A t ; c b, c c + U C A t ; c b, c c. 5 Based on 4, this can be e-witten as GA t ; c b, c c = A t + T BA t ; c b, c c BCA t ; c b, c c. 6 Poposition 1. Let the capital stuctue of the fim include equity, staight debt and CCB. Then, 8 Regading ou notations, in the futue c c = 0 will mean that CCB is not used. Fo instance, U B A t; c b, 0 is the maket value of staight debt at time t fo a fim that issued only equity and staight debt with coupon c b. 10

12 fo t < τ GA t ; c b, c c = A t + θc γ b + θc c αa B, A B A B W A t ; c b, c c = A t c γ b θ c c θ A B A B λ c c A t, A B U B A t ; c b, c c = c b + αa B, A B A B U C A t ; c c = c c T BA t ; c b, c c = θc b A B BCA t ; c b, c c = αa B. A B + λ c c, γ + θc c, The value of staight debt, U B A t ; c b, c c, and the cost of bankuptcy, BCA t ; c b, c c, ae not affected by the pesence of CCB 9. The total value of tax benefits, T BA t ; c b, c c, includes two pats: 1. the benefits associated with staight bond T B B A t ; c b, c c = θc b A B 2. and the benefits associated with CCB T B C A t ; c b, c c = θc c. 7 9 Although U B A t; c b, c c and BCA t; c b, c c do not depend on c c, we use it in ou notations when the capital stuctue of the fim includes CCB. 11

13 2.2 CCB Paamete Choice unde Condition 1 In this subsection we look at how affects the values of diffeent claims associated with the capital stuctue of the fim. We define the lowest that satisfies Condition 1 given the values of c c and λ. Let the capital stuctue of the fim at time t consist of equity and staight debt. Assume that equity holdes ae planning to issue CCB without changing the existing amount of staight debt. The poceeds will be paid off as dividends. The fim has fixed c c and λ, and is the only paamete that can still be changed. Poposition 2. The effect of on the values of diffeent claims associated with the capital stuctue of the fim when c c and λ ae fixed is such that a the total value of the fim, GA t ; c b, c c, inceases as deceases b the value of equity, W A t ; c b, c c, inceases as deceases fo λ + θ > 1 deceases as deceases fo λ + θ < 1 and emains unaffected by fo λ + θ = 1 c the value of CCB, U C A t ; c c, inceases as deceases fo λ < 1 and emains unaffected by fo λ = 1 d the total amount of tax savings, T BA t ; c b, c c, incease as deceases e and, the value of staight debt, U B A t ; c b, c c, and the cost of bankuptcy, BCA t ; c b, c c, emain unaffected by. Coollay 1. Given that c c and λ ae fixed, as vaies the change in the total value of the fim, GA t ; c b, c c, equals the change in the total amount of tax savings, T BA t ; c b, c c. 12

14 As indicated in Poposition 2, the value λ + θ plays an impotant ole in the analysis, so it is useful to povide an intuitive undestanding of it. Thee factos combine to detemine the net benefit o loss fo the existing shaeholdes at CCB convesion: 1 The equity holdes ae elieved of the CCB obligation, the maket value of which is cc. 2 The equity holdes lose the CCB tax shield benefit, the pesent value of which is θcc. 3 The equity holdes suffe a loss of λcc with λ > 1 o discount with λ < 1. if the convesion atio of equity fo CCB is at pemium The condition fo the equity holdes to suffe a net loss at convesion is thus: θcc + λcc > cc, which is equivalent to λ + θ > 1. Compaably, the equity holdes eceive a net benefit at convesion if λ + θ < 1, and the net benefit is zeo if λ + θ = 1. Poposition 2 and its coollay indicate that equity holdes will benefit fom loweing the CCB convesion theshold making convesion less imminent wheneve CCB convesion ceates a net loss fo the equity holdes that is, λ + θ > 1 and the opposite when CCB convesion ceates a net gain fo equity holdes λ + θ < 1. Fo the same eason, the CCB value inceases as deceases as long as λ < 1. The tax savings always ise as deceases, while the value of the staight debt emains unaffected. We use the closed-fom solution fo the value of equity fom Poposition 1 and conside seveal numeical execises next. The focus is on. Let A t = $100.0, = 5.00%, µ = 1.00%, θ = 35.0% and α = 50.0%. Figue 1 shows how the value of equity, W A T ; c b, c c, depends on futue ealizations of the value of assets, A T T > t. We stat with the top thee subfigues: a-c. We fix c b = $5.24, c c = $0.5 and λ = 0.9, and conside thee diffeent values of : $60.0, $75.0 and $66.9. Based on 3, fo all thee cases the default-tiggeing asset level A B = $ In subfigue a W A T ; c b, c c becomes negative befoe A T declines to the convesion-tiggeing asset level of $60.0 and befoe A T hits A B = $ Negative equity fo A T > tanslates into equity holdes defaulting befoe convesion. This violates Condition 1. 13

15 In figue b W A T ; c b, c c is stictly positive befoe and when A T hits = $75. Condition 1 is not violated but, accoding to Poposition 2, equity holdes could have set lowe in ode to incease the total value of the fim. Finally, in subfigue c, W A T ; c b, c c touches zeo ight at the convesion point and is positive fo all A T >. The fim is not going to default befoe convesion. convesion, since the value of equity is zeo, equity holdes ae indiffeent between defaulting and continuing to hold equity until τa B. We assume that they continue to hold equity. Condition 1 is not violated. Given c b = $5.24, c c = $0.5 and λ = 0.9, = $66.9 is the lowest convesion-tiggeing asset level that satisfies Condition 1. a =$60.0, λ=0.9, c c=$0.5, c b =$5.24, A B = $45.85 b =$75.0, λ=0.9, c c=$0.5, c b =$5.24, A B = $45.85 c =$66.9, λ=0.9, c c=$0.5, c b =$5.24, A B = $45.85 d =$37.0, λ=0, c c=$3.0, c b =$3.0, A B = $26.23 e =$40.0, λ=0.05, c c=$2.5, c b =$3.0, A B = $26.23 Figue 1: Equity value with diffeent convesion-tiggeing asset levels. It is impotant to note that in subfigues a-c W A T ; c b, c c stictly inceases in A T fo A T. One could have found the optimal that satisfies Condition 1 by solving equation 14

16 W ; c b, c c = 0 fo, such that > A B. We tun to the lowe two subfigues. Hee, = $37.0, λ = 0, c c = $3.0 and c b = $3.0 fo subfigue d, and = $40.0, λ = 0.05, c c = $2.5 and c b = $3.0 fo subfigue e. One impotant chaacteistic that makes these two subfigues diffeent fom the ones above is that hee fo A T the value of equity as a function of A T is non-monotone and declines immediately afte the values. In subfigue d W A T ; c b, c c becomes negative befoe A T eaches the convesion-tiggeing asset level of $37.0. As befoe, negative equity fo A T > tanslates into equity holdes defaulting befoe convesion which violates Condition 1. Note, that, although the value of equity in subfigue d is negative fo an inteval of values of A T, it becomes positive as A T appoach fom the ight-hand side. Fom the point of view of old equity holdes i.e., not including those who become equity holdes as the esult of convesion conveting CCB into equity tanslates into getting id of the obligation to pay c c. If λ is low as in subfigue d the cost of eliminating this obligation, λ cc, is also low and, theefoe, as the chance of such an event inceases i.e., A T get close to it tanslates into highe values of equity. If λ is high as in subfigue a the cost of eliminating the obligation to pay c c is also high and, theefoe, equity value continues to decline as A T appoaches. The same intuition stands behind the shapes of the equity value functions in the est of the subfigues. In subfigue e the value of equity is non-monotone but only touches zeo fo some A T > without becoming negative. With the assumption that at zeo equity holdes pefe to continue holding equity to defaulting we get Condition 1 satisfied. Given c b = $3.00, c c = $2.5 and λ = 0.05, = $40.0 is the lowest convesion-tiggeing asset level that satisfies Condition 1. Since in subfigue e W A T = $40.0; c b, c c > 0, we could not have found the above optimal convesion-tiggeing asset level by solving equation W ; c b, c c = 0 fo as in the case of subfigue c. Theefoe, in geneal, we define the lowest level of that satisfies Condition 1 as L = inf{ : W A s ; c b, c c 0, s τ }. 8 15

17 Lemma 2. If λ + θ > 1, then the value of equity, W A T ; c b, c c, is a stictly inceasing function of A T fo A T. Independent of the values of λ and θ, the lowest asset level L that satisfies definition 8 is such that L A B + λ c c, λc c + c b θ. One obsevation is that, if λ + θ > 1, the ule fo finding L that satisfies definition 8 is clea - L solves equation W ; c b, c c = 0 subject to > A B. If, howeve, this condition does not hold, one can only specify a elatively wide inteval in which L needs to be in. The second obsevation is that fo λ + θ > 1, since W A T ; c b, c c is stictly inceasing in A T, with no asymmetic infomation thee is a one-to-one coespondence between equity and assets values fo A T. Theefoe, the convesion condition fo CCB can be fomulated in tems of equity values. The debt convets into equity if the value of equity dops to W C = W ; c b, c c. This becomes impotant since equity pices ae obsevable while asset values ae not. Note that the analytical condition λ + θ > 1 fo the equity value to be stictly inceasing in A T is too stong. Based on the above numeical execises, the value of equity is non-monotone in A T only fo vey small values of λ. The main economic take-away of this section as a whole is that, if egulatos do not want the fim to default befoe o at convesion, they might need to egulate how CCB paametes c b, c c, λ and ae set. Ou analysis does not exclude the possibility that equity holdes might have an incentive to incease the total value of the fim by choosing paametes that violate Condition 1. 3 Optimal Capital Stuctue Assume that at time t the fim has no debt but is planning to leveage up by issuing both staight and CCB. The ownes eithe equity holdes o the oiginal ownes of the pivate fim fix the 16

18 amount of CCB they plan to issue by setting, c c and λ fist. Then, they maximize the total value of the fim by finding an optimal amount of staight debt. We look at how the esulting capital stuctue compaes to the optimal capital stuctue without CCB. Theoem 1. The optimal amount of staight debt in a capital stuctue that includes CCB, equity and staight debt equals the amount of staight debt in the optimal capital stuctue that includes only equity and staight debt. The coupon on staight debt is the same fo both cases c b = A t β θ 1 [ θ γ γ + 1 θ ] 1 + αβ θ γ. 9 Given that the amount of CCB is exogenously low and paametes c b, c c, and λ satisfy Condition 1, the above esult is intuitive. The fim does not default befoe o at convesion and, theefoe, as in the poof of Lemma 1, afte convesion the maximum-equity-valuation poblem of equity holde is the same as in the case when the capital stuctue includes only equity and staight debt. This leads to the same optimal amount of staight debt. Note, that c b depends neithe on c c no on o λ. Poposition 3. If the fim chooses a capital stuctue that includes CCB, equity and the optimal amount of staight debt then, compaed to the case when it chooses the optimal capital stuctue that includes only equity and staight debt, i the total value of the fim will be highe by the amount of tax savings associated with c c GA t ; c b, c c = GA t ; c b, 0 + T B C A t ; c b, c c ii adjusted fo tax benefits, equity will be cowded by CCB one-to-one W A t ; c b, c c = W A t ; c b, 0 U C A t ; c b, c c + T B C A t ; c b, c c 10 17

19 iii the total value of tax benefits will be highe by the amount of savings associated with c c T BA t ; c b, c c = T BA t ; c b, 0 + T B C A t ; c b, c c iv and, the values of staight debt and bankuptcy costs will be the same U B A t ; c b, c c = U B A t ; c b, 0, BCA t ; c b, c c = BCA t ; c b, 0. The ownes of the fim will issue CCB as it inceases the total value of the fim by the amount of additional tax savings. The amount of staight debt does not change as CCB is issued on top of the optimal amount of staight debt. Theefoe, allowing fims to intoduce CCB to thei capital stuctues in the way descibed above will ceate exta social costs in the fom of additional tax subsidies. The cost of bankuptcy and timing of default will emain the same so the quality of staight debt will not impove. 3.1 Leveaged Fim with Suboptimal Amount of Staight Debt We stated this section with the fim being unleveed. Assume instead that the fim has a capital stuctue that includes equity and staight debt paying c b not necessaily equal to c b and decides to issue CCB without changing the amount of staight debt. Based to Lemma 1, the default bounday does not change and, theefoe, issuing CCB does not affect the values of staight debt and bankuptcy costs. The total value of the fim inceases by the value of tax benefits associated with CCB, GA t ; c b, c c = GA t ; c b, 0 + T B C A t ; c b, c c. And, based on budget equation 5, the new value of equity is W A t ; c b, c c = W A t ; c b, 0 [ U C A t ; c b, c c T B C A t ; c b, c c ]. 18

20 These esults ae simila to the ones fom Poposition 3. Equity holdes ae willing to issue CCB as it inceases thei oveall value. Although they expeience a dop in the value of thei holdings in the amount of U C A t ; c b, c c T B C A t ; c b, c c, they collect dividends in the amount of U C A t ; c b, c c. Exta social costs ae ceated. The quality of staight debt emains the same. The main economic esult of this section as a whole is that allowing fims to issue CCB on top of staight debt would ceate additional social costs in the fom of exta tax subsidies without impoving the quality of staight debt. 4 CCB Instead of Staight Debt In Section 3 CCB was issued on top of staight debt. We move now to cases when CCB eplaces a potion of staight debt that is eithe to be newly issued in the optimal amount by the fim when it has no debt o is aleady pat of a capital stuctue that includes equity and staight debt not necessaily in the optimal amount. We study the effect of debt eplacement on the values of diffeent claims associated with the capital stuctue of the fim. 4.1 Initial Choice of Capital Stuctue Unde Regulatoy Constaint We stat with the case when CCB eplaces a potion of the optimal amount of staight debt that is to be newly issued. Assume that at time t the fim has no debt but is planning to leveage up. Instead of issuing an optimal amount of staight debt, U B A t, A B ; c b, 010, it has an option to issue both staight and CCB unde a egulatoy constaint. Regulatos fix the amount of staight debt, U B A t, ĀB; c b, c c, and the amount of CCB, U C A t, ĀB; c b, c c, so that U B A t, ĀB; c b, c c + U C A t, ĀB; c b, c c = U B A t, A B; c b, We slightly change notations in ode to keep tack of the coesponding default boundaies. Fo instance, in the case of U B A t, A B ; c b, 0 the default bounday is A B. 19

21 The total amount of debt equals the optimal amount of staight debt when the capital stuctue of the fim includes only equity and staight debt. The same amount of CCB can be issued with diffeent coupons and convesion-tiggeing asset levels. The fim, fo instance, can pick and find c c by solving 11 as shown below. c c U C A t, ĀB; c b, c c = U B A t, A B; c b, 0 U B A t, ĀB; c b, c c + λ c c = U B A t, A B; c b, 0 U B A t, ĀB; c b, c c c c = U B A t, A B ; c b, 0 U B A t, ĀB; c b, c c A λ t We investigate if the fim would pefe the optimal capital stuctue that includes only equity and staight debt to the one that includes equity, staight debt and CCB but is subject to egulatoy constaint 11. Poposition 4. If instead of the optimal capital stuctue that includes equity and staight debt an unleveed fim chooses its capital stuctue based on egulatoy constaint 11 then a the change in the total value of the fim will equal the diffeence in the coesponding values of equity GA t, ĀB; c b, c c GA t, A B; c b, 0 = W A t, ĀB; c b, c c W A t, A B; c b, 0 = γ θ + α θα A B ĀB θ λ c c A B 13 b if coupon c c is elatively small, the total value of the fim will be highe: c 1 such that GA t, ĀB; c b, c c > GA t, A B ; c b, 0 fo c c 0, c 1 c and, the cost of bankuptcy will be lowe, BCA t, ĀB; c b < BCA t, A B ; c b. The key esult is that the ownes of the fim gain fom eplacing a small amount c c 0, c 1 of staight debt with CCB. The intuition is that the tax savings associated with coupon payments 20

22 decease due to τ < τa B, but the fim benefits fom educing its bankuptcy costs due to a smalle amount of staight debt afte the eplacement. Fo small amounts of CCB the benefits exceed the lost tax savings. The amount of CCB that the fim can issue is set by egulatos exogenously, via constaint 11. Theefoe, fo the fim to be willing to eplace staight debt with CCB, egulatos need know how to set the constaint so that c c does not exceed c 1. We continue with analyzing the effects of issuing CCB instead of staight debt numeically. We denote GA t, ĀB; c b, c c GA t, A B ; c b, 0 by G and use 13 to show how the total value of the fim changes depending on how much of the optimal amount of staight debt is being eplaced with CCB. Figue 2: G. A t = $100, = 0.05, µ = 0.01, σ = 15%, θ = 35%, α = 50%, λ = 1; c b = $5.24 and U B A t, A B ; c b, 0 = $ Conside an unleveed fim with paametes in the desciption of Figue 2. The optimal amount of staight debt that can be issued without the egulatoy constaint is U B A t, A B ; c b, 0 = $ The amount of staight debt that is being eplaced subject to constaint 11 anges fom $1 to $25. As mentioned above, the same amount of CCB can be issued with diffeent convesion- 21

23 tiggeing asset levels. Fo each value of U C we conside fou values: $80, $85, $90, and $95. Thee ae thee main obsevations based on Figue 2. Fist, only a potion of the optimal amount of staight debt can be eplaced with CCB without loweing the total value of the fim. This is consistent with pat b of Poposition 4. As U C gets above oughly $20, G becomes negative and keeps deceasing fo all values. Losses in tax benefits due to significant eductions in the amount of staight debt lead to lowe total values of the fim. Second, lowe values esult in highe total values of the fim. This is consistent with pat a of Poposition 2. Cuves that coespond to lowe values lie stictly above the ones that coespond to highe values. Lowe values tanslate into late convesions and lead to highe tax savings associated with coupon c c. Finally, changes in the total value of the fim ae non-monotonic in U C. Fo lowe values they fist incease and then decease. By gadually eplacing staight debt with CCB stating with vey small amounts, the fim educes its bankuptcy costs and inceases its total value. But, as the amount of CCB keeps inceasing, educed tax savings stat dominating the benefits of lowe bankuptcy costs which causes the total value of the fim to go down. In summay, the fim in the above example would pefe a capital stuctue based on egulatoy constaint 11 to the optimal capital stuctue that includes only equity and staight debt. The amount of CCB, howeve, would have to be elatively small. The main economic esult of this section is that letting unleveed fims eplace staight debt with CCB in thei new, leveaged capital stuctues ceates benefits without additional costs. Total fim values incease and bankuptcy costs decease. The total amount of debt in the economy emains the same, so thee ae no exta costs in the fom of additional tax subsidies. The only equiement is that egulatos need to know how to set thei constaint so that fims ae incentivize to issue CCB. 22

24 4.2 Patially Replacing Existing Staight Debt We continue with the case when CCB eplaces a potion of aleady existing not necessaily in the optimal amount staight debt. Assume that at time t the capital stuctue of the fim consists of equity and staight debt paying coupon ĉ b not necessaily equal to c b. The fim wants to issue CCB and swap it fo a potion of staight debt in ode to educe ĉ b to c b, whee c b < ĉ b. Once the announcement is made, the maket value of staight debt, that is still paying ĉ b, will ise fom U B A t, ÂB; ĉ b, 0 to U B A t, ĀB; ĉ b, 0 to eflect a lowe default bounday due to a lesse amount of staight debt afte the swap. Fo the staight debt holdes to be indiffeent between exchanging thei holdings fo CCB and continuing to hold staight debt the following budget equation should be tue U B A t, ĀB; ĉ b, 0 = U C A t, ĀB; c b, c c + U B A t, ĀB; c b, c c. 14 The value of existing staight debt post announcement should equal the value of CCB plus the value of staight debt that emains afte the swap. Coupon c b is set exogenously and, as befoe, the same amount of CCB can be issued with diffeent coupons and convesion-tiggeing asset levels. The fim, fo example, could pick c b and fist and then find c c by solving 14 as shown below. c c c c U C A t, ĀB; c b, c c = U B A t, ĀB; ĉ b, 0 U B A t, ĀB; c b, c c γ + λ c c = ĉb + αāb + λ c c c c = c b = ĉ b c b ĉ b c b λ A t A t αāb 15 23

25 We ty to undestand if equity holdes would be willing to eplace some of the existing staight debt with CCB and what effect this eplacement would have on the total value of the fim. Poposition 5. If a leveaged fim with a capital stuctue that includes equity and staight debt eplaces a potion of staight debt with CCB then i the value of equity deceases, W A t, ĀB; c b, c c W A t, ÂB; ĉ b, 0 < 0 ii the change in the total value of the fim is such that a GA t, ĀB; c b, c c GA t, ÂB; ĉ b, 0 = ĉbθ αāb θ + αâb λ c c b GA t, ĀB; c b, c c GA t, ÂB; ĉ b, 0 > W A t, ĀB; c b, c c W A t, ÂB; ĉ b, 0 c if ĉ b c b then c 1 such that GA t, ĀB; c b, c c > GA t, ÂB; ĉ b, 0 fo λ 2 c c 0, c 1 iii and, the cost of bankuptcy deceases, BCA t, ĀB; c b < BCA t, ÂB; ĉ b. A t γ and 16 If the fim is leveaged optimally o ove-leveaged compaed to its optimal capital stuctue ĉ b c b it could benefit in tems of its total value fom eplacing a cetain amount c c 0, c 1 of γ A staight debt with CCB issued with λ 2 t 11. Thee ae two things at play hee. Fist, as befoe, eplacing staight debt with CCB pushes the tax savings down, but the fim benefits fom educing the cost of bankuptcy. Fo cetain amounts of CCB the benefits will dominate the lost tax savings. Second, although debt becomes less isky due to ĀB < ÂB the total amount of debt inceases by the diffeence between the value of staight debt post announcement, U B A t, ĀB; ĉ b, 0, and the value of staight debt pe announcement, U B A t, ÂB; ĉ b, 0. By inceasing the total amount of debt while educing the cost of bankuptcy the fim benefits fom elatively highe compaed 11 Note, that 2 γ 1 fo. 24

26 to the case when the total amount of debt did not change as in Section 4.1 tax savings 12. The pesence of these elative benefits is independent of the amount of CCB. The new tax savings fo the fim, though, might if not compensated by the eduction in tax savings due to the use of CCB tanslate into additional social costs in the fom of exta tax subsidies. As the amount of CCB debt inceases c c > c 1 lost tax benefits become lage and can tun all the gains, including the ones fom lowe bankuptcy costs and the additional tax savings, into losses. Although the total value of the fim could incease, equity holdes will not eplace voluntaily any amount of existing staight debt with CCB as thei value deceases. All the potential gains in the total value of the fim plus a potion the value of equity ae passed on to debt holdes. The obseved effect is due to debt ovehang inefficiency. We analyze the esults of Poposition 5 numeically. We plot values of GA t, ĀB; c b, c c GA t, ÂB; ĉ b, 0, denoted by Ĝ and computed based on 16, fo a ange of CCB values and seveal convesiontiggeing asset levels. We use the fim fom Section 4.1. The assumption is that it has issued staight debt in the optimal amount and now is eplacing some of this debt with CCB. Coupon ĉ b is set equal to c b and values of the est of the paametes, including the maket value of assets, ae exactly the same as in Section 4.1 all shown in the desciption of Figue 3. Figues 2 and 3 ae compaable. Figue 2 shows how the total value of the fim changes depending on the amount of CCB the fim uses when it leveages up fo the fist time based on egulatoy constaint 11. Figue 3 shows how the total value of the fim changes depending on the same amounts of CCB when the fim is aleady leveaged and eplaces staight debt with CCB so that maket constaint 14 holds. In both cases the fim uses CCB to eplace potions of the same optimal amount of staight debt. The geneal logic of the obsevations based on Figue 2 applies to Figue 3, so we ae not going to epeat the elated discussions fom Section Based on this, one would expect that, eveything else being equal, c 1 > c 1. 25

27 Figue 3: Ĝ. A t = $100, = 0.05, µ = 0.01, σ = 15%, θ = 35%, α = 50%, λ = 1; ĉ b = c b = $5.24 and U B A t, ÂB; ĉ b, 0 = U B A t, A B ; c b, 0 = $ Thee is an impotant diffeence between the two figues, though. Notice that, if plotted togethe, the cuves fom Figue 3 would lie above thei countepats fom Figue 2. Changes in total values of the fim fo the cuent execise exceed the coesponding changes in total values fo the execise in Section 4.1. This is due to elatively lage total amounts of debt post eplacement and, coespondingly, highe tax savings. In summay, in the above execise all the benefits fo the fim and the economy oveall fom eplacing staight debt with CCB discussed in Section 4.1 emain. Howeve, the ealization of these benefits inceases the total amount of debt which tanslates into moe value fo the fim but equies additional tax subsidies. The key economic esult of Section 4.2 is that if the fim decided to patially eplaces existing staight debt with CCB the total value of the fim would incease and bankuptcy costs togethe with the total amount of isky staight debt would decease. No extensive egulation would be equied. Equity holdes, howeve, will neve initiate this kind of debt eplacement on thei own due to debt ovehang inefficiency. 26

28 5 TBTF Fims In this section we look at fims that ae too big fo the govenment to let them fail, as bankuptcy of such fims might cause majo disuptions in the oveall financial system/economy. the point of bankuptcy of a TBTF fim the govenment might assume contol ove its assets and take ove its obligation to make payments to debt holdes. Fom modeling pospective, a fim eaches bankuptcy when the value of assets, A t, hits the default bounday level, A B. that point, if the govenment decides to step in to pevent bankuptcy, it would obtain assets woth A B and an obligation to pay c b foeve woth c b. Theefoe, the value of govenment subsidy at the time of bankuptcy is c b A B. Given 2, at any time t befoe bankuptcy, the value of subsidy is 13 SA t ; c b, c c = cb A A t B. 17 A B Optimal time to default τa B solves the maximum-equity-valuation poblem of equity holdes. Govenment subsidy kicks in at time τa B and coves only staight debt obligations. Theefoe, it affects neithe the timing of default no the value of A B o the value of equity. Also, based on Lemma 1, the time of default and the value of assets at the time of default do not depend on whethe the capital stuctue of the fim includes equity and staight debt o equity, the same amount of staight debt and CCB. Theefoe, the value of subsidy is the same whethe staight debt is used on not: SA t ; c b, c c = SA t ; c b, Poposition 6. Let a fim have a capital stuctue that includes equity, staight debt and CCB. If at t the govenment issues a guaantee fo the staight debt of the fim, then 13 We etun to ou initial notations. The default bounday is tied to the coesponding coupon on staight debt based on 3. 27

29 1. the total value of the fim will incease and will become stictly inceasing in c b GA t ; c b, c c = A t + θc b A B + θc c cb + A A t B 19 A B 2. staight debt will become isk-fee, U B A t ; c b, c c = c b 3. bankuptcy will be eliminated, BCA t ; c b = 0 4. and, the values of equity, tax benefits and CCB will not change. 5.1 Staight Debt Instead of CCB: Initial Choice of Capital Stuctue Assume that a TBTF fim is cuently unleveed but is consideing leveaging up. It could be owned by eithe equity holdes deciding to issue debt o the oiginal ownes deciding to issue equity and debt. We ty to undestand how the value of govenment subsidy would depend on whethe the fim chooses to issue staight debt o both staight and CCB unde a egulatoy limit on how much debt the fim is allowed to issue. Equation 19 gives the closed-fom solution fo the value of the fim when it issues equity, staight debt and CCB. It is easy to see that thee is a simila solution fo the case when the fim issues only equity and staight debt. In both cases the total value of the fim is stictly inceasing in c b. Thee is no moe tade-off between tax benefits associated with debt and losses due to bankuptcy, as bankuptcy, fom the point of view of the total value of the fim, has been eliminated. Equity holdes o the oiginal ownes of the fim would ty to issue as much debt as possible and collect an amount of tax benefits as lage as possible and default immediately afte the issuance. Knowing this, the govenment could set limits on how much debt a TBTF fim could issue. It could set a maximum staight debt coupon c g b fo the capital stuctue that includes only equity and staight debt. It could also give the fim an altenative to issue staight debt with coupon c b and CCB with coupon c c such that the following egulatoy constaint holds U B A t ; c g b, 0 = U C A t ; c b, c c + U B A t ; c b, c c