Contingent capital with epeated inteconvesion between debt and equity Zhaojun Yang 1, Zhiming Zhao School of Finance and Statistics, Hunan Univesity, Changsha 410079, China Abstact We develop a new type of contingent capital, called contingent convetible secuity (CCS, which is like a contingent convetible bond (CC but diffeently can be inteconveted epeatedly and automatically between debt and equity depending on two specified levels of the cash flow geneated by the fim that issues the CCS. We deive closed-fom epessions of the equilibium pices of copoate secuities and optimal capital stuctue when the cash flow of the fim is modeled as geometic ownian motion. specially, we povide vey simple fomulas on optimal capital stuctue including a CC. We show that the CCS can not only decease default isk like a CC, but also can significantly incease the fim s value. In paticula, the CCS can be used to solve financing poblems of small- and medium-sized entepises as well as banks. Keywods: Contingent convetible secuity, Contingent convetible bond, Capital stuctue, quilibium picing JL: G12, G23 1. Intoduction In geneal, a fim will have both equity and debt in its capital stuctue instead of all equity. This is because thee ae two maket impefections: a ta deduction on inteest epense and bankuptcy costs, such as loss fom selling eal assets, asset fie-sale losses, legal fees, etc, so that, with default, a faction of the value of the fim is lost. Consequently, a fim may have a 1 Coesponding autho. Tel: +86 731 8864 9918; Fa: +86 731 8868 4772. -mail addess: zjyang@hnu.edu.cn (Z.J. Yang. Pepint submitted to 2013 China Intenational Confeence in Finance Januay 15, 2013
stict but limited incentive to issue debt and optimal capital stuctue must be based on the tade-off between the values of ta shields and of bankuptcy costs. With egad to fims (e.g. banks that ae too impotant to fail, howeve, a new consideation aises, i.e. we must give much moe weight to bankuptcy costs. In paticula, afte the global financial cisis of 2007/2009, the new global bank egulatoy standad, asel III, intoduces sticte capital equiements fo banks in ode to educe default isk to an acceptable level. Natually, ecessive debt financing is pohibited and so the benefits of ta shields ae deceased. In ode to incease ta shields while keeping default isk in a low level, a poposal that has ecently eceived much attention is to induce a bank to have a contingent convetible bond (CC, hencefoth in the capital stuctue. The CC, also known as contingent capital, is a bond that can be automatically conveted into equity if the issue s financial health deteioates to a pe-specified theshold o tigge. The fist CCs wee issued by Lloyds anking Goup in Novembe 2009. On the design of contingent capital, a key poblem is how to detemine the convesion theshold fo the activation of contingent capital, which is consideed by a vaiety of papes in the liteatue. Fo eample, specific definitions fo tiggeing events ae put fowad in a consultative document issued by asel Committee on anking Supevision (2010. Mconald (2011 evaluates a fom of contingent capital fo financial institutions that convets fom debt to equity if two conditions ae met: the fim s stock pice is at o below a tigge value and the value of a financial institution inde is also at o below a tigge value. Sundaesan and Wang (2011 conside the design of contingent capital with a stock pice tigge fo mandatoy convesion and the poblem on multiple o no equilibium is discussed. Glasseman and Noui (2012 analyze the case of contingent capital with a capital-atio tigge and patial and on-going convesion. The capital atio is based on accounting o book values to appoimate the egulatoy atios that detemine capital equiements fo banks. Himmelbeg and Tsyplakov (2012 conside the soft tigge case in which the convesion does not necessaily occu at the fist instant when the tigge is beached. Instead, the convesion event happens andomly in a pobability. Fo futhe convesion mechanism discussions, please efe to Metzele and Reese (2011, aucci and Viva (2012a, Hilsche and Raviv (2011, among othes. Howeve, to the best of ou knowledge, all the papes in the liteatue 2
assume that once the CC is conveted into equity, it will keep the fom of equity all the time until the fim goes bankupt and in paticula, it will neve be conveted back into debt. We think that this assumption is not necessay and moe impotantly, it is woth being impoved. Intuitively, if such fom of contingent capital can be inteconveted epeatedly between debt and equity, the fim must get much moe ta shields while the default isk will not incease. In othe wods, the fim s value will be emakably inceased while the egulatoy standad is met still. Fo this eason, in this pape we ela the assumption and develop a new fom of contingent capital, called contingent convetible secuity (CCS, hencefoth, which can be inteconveted between debt and equity epeatedly only if the fim does not go bankupt. Without any doubt, an asset can be defined uniquely by its cash flow, which is in common descibed by a stochastic pocess fo a isk asset. Theefoe, to define the new fom of contingent capital, CCS, what we should do is just to define its cash flow. Fo this end, it is key to specify two convesion thesholds and the faction of equity distibuted to the CCS holde while the CCS is conveted into equity. Geneally speaking, ecept fo the capital equiements intoduced in asel III, the convesion thesholds and the faction can be chosen abitaily, only if the faction of equity distibuted to the CCS holde is not so small that the equity holde will benefit fom the convesion of the CCS fom debt to equity o suffe a loss due to the convesion of the CCS fom equity to debt. 2 In ode to deive the pices of copoate secuities, we utilize the equilibium picing appoach. Specifically, we assume thee is a maket whee two assets ae taded: One is the isk-fee asset; The othe is a isk asset (e.g. maket potfolio, which is coelated with the cash flow of the fim. In this way, we get an matingale measue unde which, the equilibium pice of an asset is the epectation of the sum of the discounted cash flow geneated by the asset. Following this idea, in this pape, we develop a new type of contingent capital, CCS, which is like a contingent convetible bond (CC but diffeently can be inteconveted epeatedly and automatically between debt and equity depending on two specified values of a financial situation inde, i.e. 2 That is, the design of the faction must punish ( awad equity holdes if the financial situation gets bad (good. Othewise, the design must be inefficient since the fim is managed in geneal on behalf of equity holdes. 3
the level of the cash flow geneated by the fim that issues the secuity. We deive closed-fom epessions of the equilibium pices of the copoate secuities and optimal capital stuctue when the cash flow of the fim is modeled as geometic ownian motion. specially, we povide vey simple fomulas on optimal capital stuctue including a CC. We show that while the secuity can decease default isk like a CC, it can also significantly incease the fim s value. In addition, all the asset pices in ou model ae continuous and each convesion will not lead to a value tansfe between the equity holde and the CCS holde. Accodingly, thee eists no multiple equilibia poblem and pice manipulation is limited. The most common issues of CCs/CCSs ae banks, but without any doubt, CCs/CCSs could be used by all fims. Theefoe, ou conclusions in this pape ae applicable to any fim, no mate whethe it is a bank o not, and essentially, no matte whethe it is a small o a lage company. In paticula, CCSs ae etemely suitable fo small- and medium-sized entepises (SMs to solve financing poblems. An SM usually has to default just because it cannot affod to pay loans, even though the value of equity of the SM is valuable. If an SM was a listed company, the default would be avoided by dilution of equity. Unfotunately, an SM cannot do so and consequently default is inevitable. Howeve, if SMs issue CCSs athe than taking othe debt financing, the default poblem will be solved automatically to some degee. The pape is oganized as follows. The net section sets up the model and copoate secuities ae defined. In section 3, we eview on equilibium picing theoy in a fom, which facilitates picing all assets we discuss in the pape. Section 4 consides optimal capital stuctue. Section 5 discusses the advantages of CCSs. Section 6 pesents numeical esults. Section 7 concludes. 2. Model setup Conside a fim (e.g. bank o SM that has invested in pojects, of which the total (afte-ta cash flow δ is obsevable and independent of the capital stuctue of the fim. We assume the cash flow is govened by the following geometic ownian motion: dδ t = µ δ δ t dt + ρσδ t dz 1 t + 1 ρ 2 σδ t dz 2 t, δ 0 given, t [0,, (1 4
whee µ δ is the gowth ate, σ is the volatility ate and Z (Z 1, Z 2 is a 2-dimensional standad ownian motion on a complete pobability space (Ω, F, P. In addition, investos have standad liquid financial oppotunities which involves a isk-fee asset and a isky maket potfolio. enote by {M t : t 0} the value of the maket potfolio, which is govened by the following equation: dm t /M t = µ m dt + σ m dz 1 t, M 0 given, (2 whee µ m and σ m ae espectively the epected etun and the volatility ate of the maket. Clealy, the paamete ρ in (1 epesents the coelation coefficient between the fim s cash flow and the etun of the maket potfolio. The values ρσ and 1 ρ 2 σ ae the systematic and idiosyncatic volatility of the fim s cash flow espectively. Geneally, ρ < 1 as we assume in this pape, and so the investos of the fim face undivesifiable idiosyncatic isk. We denote by F {F t : t 0} the P-augmentation of the filtation σ(z s ; 0 s t geneated by pocess Z. The filtation F descibes the flow of infomation available to investos. We conside a fim that has a staight bond (S, hencefoth and a fom of contingent capital, CCS, in its capital stuctue. The S is consol type, meaning it equies continuous coupon payment at the constant ate of b pe unit of time, until default. The CCS is a hybid secuity, which takes a fom of eithe debt o equity and can be inteconveted epeatedly between debt and equity, as long as the fim emains solvent. If the CCS takes the fom of debt (debt secuity, it equies continuous coupon payment at the constant ate of c pe unit of time. If the CCS takes the fom of equity (equity secuity, its holde eceives a faction (β of the esidual cash flow in the fom of dividends at the ate β[δ t (1 θb] at time t, whee 0 < θ < 1 is a constant ta ate. In ode to implement easily, we should pevent the CCS fom being conveted fequently and hence we select two diffeent convesion thesholds of the inteconvesion: One is the convesion theshold, denoted by δ, at which once the level of the fim s cash flow gets fom above, the CCS is automatically conveted fom a debt secuity into an equity secuity; The othe is the convesion theshold, denoted by δ, at which once the level of the fim s cash flow gets fom below, the CCS is automatically conveted fom an equity secuity to a debt secuity. Natually, we assume δ < δ. On account of that the time is homogenous in ou model, we specify the default theshold, denoted by δ, i.e. once the level of the fim s cash 5
flow is less than δ, the fim goes bankupt. The default theshold will be detemined endogenously o default is tiggeed once the unleveled value of the fim is less than a given level, to be defined late. Natually, we assume δ < δ < δ. At default, the value of the fim s futue cash flow is assigned to the S holde but a faction, denoted by α, of the value of the futue cash flow will lost due to bankuptcy costs, whee 0 α 1 is constant and called bankuptcy loss ate in the liteatue. 3. Picing of copoate secuities In this section, we fist eview on the equilibium picing theoy and then based on the theoy, we eplicitly obtain the equilibium pices of all the copoate secuities defined in pevious section. 3.1. quilibium picing theoy In finance, most asset pices ae deived fom a linea picing schedule. To detemine a linea picing ule, we must specify a stochastic discount facto. Howeve, the maket we conside hee is incomplete, i.e. thee ae infinite stochastic discount factos. To fi one, we can solve a single-agent optimization poblem and take the maginal utility of the agent as the special stochastic discount facto, see uffie (2001. If the agent selected is a epesentative agent, then we ecove the equilibium stochastic discount facto used in Ingesoll (2002, Goetzmann, Ingesoll and Ross (2003, Yang and Zhang (2013 and essentially also in Meton (1976. Fo the poblem we discuss hee, we get the equilibium stochastic discount facto, which coesponds to an matingale measue Q. 3 Unde this matingale measue Q, we can ewite the cash flow pocess δ in (1 as follows: dδ t = µδ t dt + ρσδ t dz Q t + 1 ρ 2 σδ t dz 2 t, (3 whee η (µ m /σ m is the Shape atio of the maket, µ µ δ ρση is the isk-adjust dift, and (Z Q, Z 2 is a 2-dimensional standad ownian motion unde Q satisfying dz Q t = dzt 1 + ηdt. 3 Moe eactly, the estictions of the two pobability measue P and Q to F t ae equivalent fo any time t 0. See fo eample Poposition 6F in uffie (2001 fo the elationship between the stochastic discount facto (state-pice deflato and the matingale measue. 6
ased on what we discuss above, accoding to the dynamic asset picing theoy, see uffie (2001 among othes, we deive the following equilibium pice [ ] = Q ep ( (s t η s ds F t, t [0, (4 V η t t fo any asset defined by an F-adapted stochastic pocess η, which is the cash flow geneated by the asset. If the asset is a deivative poduct undelying the cash flow of the fim, meaning that η t = f(δ t, t [0,, then the value of the asset is independent of time and uniquely detemined by the cuent level of the cash flow of the fim. We denote by V f ( the value function, whee epesents the cuent level of the cash flow of the fim. Fom Ito s fomula, the function V f ( satisfies the following odinay diffeential equation (O: µv f ( + 1 2 σ2 2 V f ( + f( V f ( = 0, (5 whee and thoughout the tet the subscipt of a function, say V f hee, epesents the diffeentiation with espect to that vaiable. In copoate finance, we often need to conside the picing of the following asset: The cash flow ate of the asset is a linea function of the cash flow of the fim, i.e. η t = f(δ t = aδ t + K, t [0,, up to a stopping time τ inf{t : δ t / }, which is the time of fist depatue of δ t fom a domain. At the stopping time τ, the asset geneates a lump-sum dividend, which is a function, denoted by g(, of the cash flow ate δ τ, whee epesents the bounday of the domain. Afte time τ, the asset disappeas o its cash flow ate is zeo foeve. Fo the same eason with (4, the equilibium pice (value of the asset is independent of time and given by [ τ ] Q ep ( (s t (aδ s + Kds F t = V (δ t, t τ (6 t fo some function V (. Thanks to Ito s fomula, the function V ( satisfies µv ( + 1 2 σ2 2 V ( + (a + K V ( = 0, (7 with bounday condition V ( = g(,. (8 7
y a standad appoach, the geneal solution of (7 is given by V ( = A 1 γ 1 + A 2 γ 2 + a µ + K, (9 whee A 1 and A 2 ae constants to be detemined by bounday condition (8, and γ 1 and γ 2 ae solutions of the quadatic equation 1 2 σ2 y 2 + (µ 1 2 σ2 y = 0, (10 that is (µ 1 2 σ2 ± (µ 1 2 σ2 2 + 2σ 2 γ 1/2 =. (11 σ 2 Clealy, we have γ 1 > 0 and γ 2 < 0. In paticula, fo any given constant κ (0, δ 0, at any time t < τ κ, the pice, denoted by V κ ( with = δ t, of a secuity that claims one unit of account at the stopping time τ κ = inf{t : δ t κ} is a solution of the following O: µv κ ( + 1 2 σ2 2 V( κ V κ ( = 0, (12 with bounday condition and anothe obvious bounday condition V κ (κ = 1 (13 V κ ( 0,. (14 Noting that a = K = 0 and utilizing (9 with (13 and (14, we get ( V κ γ1 ( =. (15 κ Amed with the equilibium picing theoy in this section, we eplicitly deive the pices of all the copoate secuities defined in Section 2 in the following tet. 3.2. Picing of staight bond On account of that the time is ielevant to the answes of ou poblems, without loss of geneality, we assume the cuent time is zeo. Fom a ealistic 8
point of view, we also assume δ 0 > δ and the CCS takes the fom of debt at the vey beginning. We note fom Section 2 that all the copoate secuities can be consideed as the deivative poducts undelying the cash flow of the fim and specifically, (7, (9 and (8 can be utilized to deive the pices of the copoate secuities. In ode to pice the staight bond (S, we must in advance obtain the equilibium value o maket value of the fim unde the matingale measue Q when the fim is unleveed, i.e. it is financed by equity only. Accoding to (4 and (3, at any time t [0,, the value of the unleveed fim is independent of time and accoding to the qualities of geometic ownian motion, it is immediately given by [ ] A( = Q ep ( (s t δ s ds δ t = = = = t t t ep ( (s t Q [δ s δ t = ] ds ep ((µ (s t ds µ, (16 if cuent cash flow ate is and µ <. 4 Accoding to the definition of the S given in Section 2, it is a deivative with the cash flow defined by f(δ s = b up to the stopping time τ inf{t 0 : δ t δ } (default time, at which, the bond holde gets lump-sum dividend (1 αa(δ, whee α is the bankuptcy loss ate and A(δ is the equilibium value of the fim s futue cash flow that is given by (16. That is, a = 0, K = b, = (δ, fo the S. Theefoe, utilizing (7 and (8, we conclude that the value, denoted by V (, of the S satisfies the following O: µv ( + 1 2 σ2 2 V ( + b V ( = 0, (δ, (17 with bounday condition V (δ = (1 αa(δ = (1 αδ µ, (18 4 If µ, then the value of the fim that is unleveled is infinite. Fo this eason, we assume µ < thoughout the tet. 9
and anothe obvious bounday condition given by V ( b/,. (19 Utilizing the geneal solution (9 with the bounday conditions, we get the following eplicit solution of (17 with (18 and (19 as follows: [ V ( = b ( ] γ1 1 + (1 α δ ( γ1, (δ,. (20 δ µ δ Remak 1. In fact, the conclusion of (20 is eactly what we epect since accoding to (15, it says that the value of the S is equivalent to the value of coupon payments to the bond holde befoe default, plus the value of the claim of the bond holde to the salvage value of (1 αδ /( µ at the default time τ. It is clea that the value of the S is ielevant to how the CCS is defined. 3.3. Picing of contingent convetible secuity To deive the value (equilibium pice of the contingent convetible secuity (CCS, we must paticulaly identify whethe it takes the fom of debt (debt secuity o the fom of equity (equity secuity if the cuent cash flow ate [δ, δ ]. Fo this eason, we denote by V C ( (V C ( the value function of the CCS if it is a debt (equity secuity. We fist deive the equilibium pice of the CCS when it is a debt secuity, i.e. the value function V C (. Fo this end, accoding to the cash flow of the CCS defined in Section 2, we note that a = 0, K = c, = (δ,, = δ fo the CCS unde this situation. Then, utilizing (7 and (8, we obtain that function V C ( satisfies: µv C ( + 1 2 σ2 2 V C ( + c V C ( = 0, (δ, (21 with bounday condition and anothe obvious bounday condition V C (δ = V C (δ, (22 V C ( c/,. (23 10
Thanks to (9 with (22 and (23, the solution is easily given by [ V C ( = c ( ] γ1 ( γ1 1 + V C (δ, (δ, (24 δ δ whee V C (δ will be deived late. Now we tun to the value, V C (, of the CCS when it is an equity secuity. In the same way, we conclude that function V C ( satisfies: µv C ( + 1 2 σ2 2 V C ( + β[ (1 θb] V C ( = 0, (δ, δ (25 with bounday conditions { V C (δ = V C (δ, V C (26 (δ = 0. Utilizing (9 with a = β, K = β(1 θb and (26, the solution is given by whee V C ( = A 1 γ 1 + A 2 γ 2 + β µ β(1 θb, (δ, δ, (27 β(1 θb (δ γ 2 A 1 = δ γ 2 + β (δ µ δ γ 2 δ γ 2 δ δ γ 2 V C (δ δ γ 1 δ γ 1 (δγ 1 γ 2, (28 and β(1 θb (δ γ 1 A 2 = δγ 1 + β µ (δγ 1+1 δ γ 1+1 + δγ 1 V C (δ δ γ 1 γ 2. (29 Combining (24 and (27 with (28 and (29, afte some tedious algeba, we conclude that V C (δ = c (δ γ 1 δ γ 1 1(δγ 1 γ 2 δ γ 1 γ 2 + β(1 θb + β δ γ1 µ δ γ 1 δ γ 1 δγ 1 (δ γ 2 δ γ 2 + δ γ 2 (δγ 1 δγ 1 + δγ 1 γ 2 δ γ1 δγ1 (δ δ γ2 δ γ2 δ γ 1 γ 2 δ + δ γ2 (δγ1+1 δ γ1+1 δ γ 1 γ 2 11 + δ (δ γ1 γ2 δ γ1 γ2, (30
and V C (δ = c + (1 δ γ1 δγ1 (δγ1 γ2 δ γ1 γ2 β(1 θb δ γ 1 γ 2 δ γ1 (δ γ2 δ γ2 + δγ1 γ2 (1 δ γ1 δγ1 δ γ1 γ2 + δ γ1 δ γ1 γ2 δγ1 (δγ1 γ2 δ γ1 γ2 + β δ γ 1 (δ δ γ 2 δ γ 2 δ + δ γ 1 δγ 1 γ 2 (δ γ 1+1 δ γ 1+1 + δ γ 1 δγ 1+1 (δ γ 1 γ 2 µ δ γ1 γ2 δ γ1 γ2 (31 Net, we tun to the picing of equity. 3.4. Picing of equity Fo the same eason with the CCS, the value (equilibium pice of equity also depends on whethe the CCS is a debt secuity o an equity secuity. If the cuent cash flow ate is, we denote by V ( (V ( the value of equity when the CCS is a debt (equity secuity. Fistly, we conside V (. Clealy, we have a = 1, K = (1 θ(b + c and = (δ, unde this case. Simila with the picing of the CCS, it follows fom (7 and (8, that µv ( + 1 2 σ2 2 V ( + [ (1 θ(b + c] V ( = 0, (δ, (32 with bounday condition and anothe bounday condition V ( = V (δ = V (δ, (33 µ + c (1 θb,. (34 Accoding to the geneal solution (9 with the above bounday conditions, we obtain the eplicit pice as follows: V ( = + c (1 θb + µ [ (1 θ b + c 12 δ ] ( γ1 µ + V (δ δ (35.
fo (δ,, whee V (δ will be given soon. Secondly, we fi the value V ( of equity when the CCS is an equity secuity. Accoding to the definition of each copoate secuity, and noting (7 and (8, it is immediate to deive that µv (+ 1 2 σ2 2 V (+(1 β[ (1 θb] V ( = 0, (δ, δ (36 with bounday conditions { V (δ = V (δ, V (37 (δ = 0. Thanks to (9 with (37, we get the following solution: V ( = Ā1 γ 1 +Ā2 γ 2 + whee (1 β µ (1 β(1 θb, (δ, δ, (38 and Ā 1 = (1 β(1 θb (δ γ 2 δ γ 2 + 1 β µ (δ δ γ 2 δ γ 2 δ δ γ 2 δ γ 1 δ γ 1 (δγ 1 γ 2 V (δ, (39 Ā 2 = (1 β(1 θb (δ γ 1 δγ 1 + 1 β µ (δγ 1+1 δ γ 1+1 + δγ 1 V (δ δ γ 1 γ 2. (40 Taking into account (35 and (38 simultaneously with (39 and (40, afte some tedious algeba, we obtain V (δ = (1 θb (δ γ1 δγ1 1 + δ δ γ 1 δγ 1+1 µ + (1 θc (1 δ γ 1 δ γ 1 (δγ 1 γ 2 δ γ1 γ2 δ γ1 γ2 β(1 θb δ γ 1 δ γ 1 δγ 1 (δ γ 2 δ γ 2 + δ γ 2 (δγ 1 δγ 1 + δγ 1 γ 2 δ γ1 γ2 δ γ1 γ2 β δ γ 1 δ γ 1 δγ 1 (δ δ γ 2 δ γ 2 δ + δ γ 2 (δγ 1+1 δ γ 1+1 + δ (δ γ 1 γ 2 µ δ γ 1 γ 2, (41 13
and V (δ = (1 θb (δ γ 1 δγ 1 1 + δ δ γ 1 µ δγ 1+1 + (1 θc (δ γ 1 δγ 1 1(δγ 1 γ 2 δ γ 1 γ 2 β(1 θb δ γ 1 (δ γ 2 δ γ 2 + δ γ 1 δγ 1 γ 2 (δ γ 1 δγ 1 + δ γ 1 δγ 1 (δγ 1 γ 2 δ γ 1 γ 2 β δ γ1 (δ δ γ2 µ δ γ2 δ + δ γ1 δγ1 γ2 (δ γ1+1 δ γ1 γ2 δ γ1+1 δ γ1 γ2 + δ γ1 δγ1+1 (δ γ1 γ2 δ γ1 γ2 Now, we finish the wok of picing copoate secuities. Since all the conclusions ae eplicit, it is convenient fo us to discuss the poblem of optimal capital stuctue in the following section. 4. Optimal capital stuctue In this section, we take it that the oiginal ownes of the fim have chosen a capital stuctue including an S, a CCS and pue equity. We analyze the optimal capital stuctue unde some eogenous easonable conditions. efoe doing so, we fist povide an equation to detemine an endogenous bankuptcy theshold in the following. 4.1. ndogenous default timing We conside the case whee the bond holde has no potective covenant. Theefoe, fo the given CCS and S, which ae completely defined by paamete vecto (b, c, β, δ, δ up to default. The equity holde will declae default at a stopping time that solves the maimum-equity-valuation poblem [ τ ]} V ( ma { Q ep( tη t dt δ 0 = (43 τ T 0 whee T is the set of all F stopping times, and η t = δ t (1 θ(b + c o η t = (1 β[δ t (1 θb] depending on the fom of the CCS at that time (debt o equity. Since the cash flow of the fim, stochastic pocess δ, is time-homogeneous, the maimization poblem (43 is solved by a hitting time of the fom τ = inf{t : δ t δ }, fo some default-tiggeing level δ (bankuptcy theshold of the fim s cash flow, to be detemined. It is clea that, fo any candidate 14 (42.
bankuptcy theshold δ, the taget function of the maimization poblem (43 is just V ( given by (35 since we assume > δ and the CCS takes the fom of debt at cuent time fom a ealistic point of view as pointed out befoe. Fo this eason, the maimization poblem (43 is equivalent to finding that solves δ ma δ >0 { + c (1 θb + µ whee V (δ is given by (41. following poblem: [ (1 θ b + c δ ] ( } γ1 µ + V (δ δ (44 Obviously, it is equivalent to solve the { ma V (δ }. δ >0 Afte some tedious algeba, we conclude that the optimal bankuptcy theshold δ is a oot of the following equation: (1 β(1 θbγ 1 (1 β(1+γ 1 δ µ + (γ 1 γ 2 [ (1 θ(βb+c (δ γ 1 δγ 1 β µ (δγ 1 +1 δ γ 1 +1 ] δ γ 1 γ 2 δ γ 1 γ 2 δ γ 2 = 0. (45 It is easy to veify that if c = β = 0 in (45, i.e. the CCS is not effectively included in the capital stuctue, we ecove indeed fom (45 a standad conclusion in copoate finance theoy, see Chapte 11 of uffie (2001 fo eample. Let δ, we immediately deive fom (45 the bankuptcy theshold fo a capital stuctue including a CC, which is given by aucci and Viva (2012b among othes. 4.2. Optimal capital stuctue Fo the capital stuctue, we must detemine default-tiggeing condition at the vey beginning. Fo this aim, we conside two cases: One is the endogenous default timing we have just discussed and the othe is the case whee the S is potected. Fistly, if the default timing is endogenous, the optimal capital stuctue poblem is to choose vecto (b, c, β, δ, δ that solves the following optimization poblem: { V ( + V C ( + V ( }, (46 sup (b,c,β,δ,δ A 15
whee = δ 0 is the cuent cash flow ate of the fim and A is the admissible set defined by A {(b, c, β, δ, δ b 0, c 0, 0 β < 1, δ δ < δ }. Accoding to (46, (20, (24 and (35 with some tedious algeba, it is enough to solve the following nonlinea pogamming poblem: sup (b,c,β,δ,δ A { θb [1 ( δ γ 1 ] + θc[1 ( δ γ 1 ] αδ µ ( + θc (δ γ 1 δ γ 1 1(δγ 1 γ 2 δ γ 1 γ 2 δ γ 1 γ 2 δ γ 1 γ 2 } γ1 δ, ( γ1 δ (47 whee δ is a function of (b, c, β given by (45. We can eamine that as we epect, the capital stuctue (b, c, β, δ, δ that solves (47 is what maimizes the diffeence between the values of ta shields and of financial distess costs. Fom (47, it is evident that as we epect, δ = δ. Theefoe, noting that sup (b,c,β,δ A lim δ δ δ γ1 δ γ 1 1 δ γ 1 γ 2 = γ 1 γ 1 γ 2 δ γ 2 γ 1, we conclude that (47 can be equivalently witten as { θb [1 ( δ γ 1 ] + θc[1 ( δ γ 1 ] αδ µ + θcγ 1 (γ 1 γ 2 (δγ 1 γ 2 γ 1 δ γ 2 }, ( γ1 δ whee admissible set A {(b, c, β, δ b 0, c 0, 0 β < 1, δ δ }. (48 Remak 2. Clealy, in any case, we should let δ = δ in ode to maimize the fim s value but in fact, we had bette choose δ lage enough fo the pupose of easy account management. Hence, we always assume δ > δ in the following tet. Secondly, we tun to the case whee the S is potected. We suppose that the bond contact conveys to the S holde a potective covenant allowing liquidation wheneve the value of the unleveed fim is less than L(b, whee L( is a given inceasing function of b and 0 L(b b/. Theefoe, δ = L(b( µ. Unde this situation, fo the optimal capital stuctue, it is enough to find (b, c, δ, δ that solves the following nonlinea pogamming 16
poblem: sup (b,c,δ,δ Ā { θb ( [1 ( L(b( µ γ 1 ] + θc[1 ( δ γ 1 ] αl(b ( } [L(b( µ] γ 1 γ 2 γ1 δ, + θc (δ γ 1 δ γ 1 1 (δ γ 1 γ 2 δ γ 1 γ 2 δ γ 1 γ 2 L(b( µ γ1 (49 whee Ā is the admissible set defined by Ā {(b, c, δ, δ b 0, c 0, L(b( µ δ < δ }. Remak 3. Obviously, the solution of (49 is ielevant to the vaiable β. Howeve, we must choose the β so lage that the equity holde will not benefit fom such convesion of the CCS. Othewise, the fim may find it optimal to bun money to push its cash flow below the tigge, see Himmelbeg and Tsyplakov (2012. At the same time, the vaiable β had bette be as small as possible since the fim is managed by the equity holde, who should have a sufficient incentive to manage the fim well. Fom a ealistic point of view, in detemining the bounday δ and δ, simila with Koziol and Lochen (2012, it is easonable to epect that these values will depend on the etent of the fim s debt liabilities accoding to a egulatoy standad. Fo eample, we can take δ = ϕ 1 (b + c and δ = ϕ 2 (b + c fo some 0 < ϕ 1 < ϕ 2. In paticula, paamete ϕ 1 allows us to adjust the egulatoy standad. Clealy, the bigge the paamete ϕ 1 is, the sticte the egulatoy standad will be. Moe eplicit esults. Taking into consideation a possible egulatoy standad, a convenient account management and that the S had bette be well potected, now we tun to a special situation whee we eogenously specify L(b = b/, δ = ϕ 1 (b + c and δ = ϕ 2 (b + c fo some given 0 < ϕ 1 < ϕ 2. Unde this case, accoding to (49, the optimal capital stuctue is actually detemined by (b, c that solves the following system of equations: θ γ 1 + θγ 1 ϕ γ 1 γ 2 1 ( 1 ϕ γ 2 1 c(b + c γ1 1 (γ 1 + 1 3 (θ + αb γ 1 θ 1 2 ((γ 1 γ 2 (b + c + γ 2 bcb γ 1 γ 2 1 (b + c γ2 1 = 0 θ γ 1 + θϕ γ 1 γ 2 1 ( 1 ϕ γ 2 1 (b + c γ 1 + θγ 1 ϕ γ 1 γ 2 1 ( 1 ϕ γ 2 1 c(b + c γ 1 1 θ 1 2 (b + c + γ 2 cb γ 1 γ 2 (b + c γ 2 1 = 0 (50 17
whee ϕ γ 1 1 ϕ γ ( 1 γ1 γ 2 µ 2 ( γ1 µ 1 = ϕ γ 1 γ 2 1 ϕ γ 1 γ 2, 2 =, 3 =. (51 2 In paticula, let ϕ 2, we get the optimal capital stuctue including the CC instead of a CCS, (b, c, that solves the following simple system of equations: { θ γ 1 θγ 1 ϕ γ 1 1 c(b + c γ1 1 (γ 1 + 1 3 (θ + αb γ 1 = 0, γ 1 ϕ γ 1 1 (b + c γ 1 γ 1 ϕ γ 1 1 c(b + c γ1 1 (52 = 0, whee 3 is given by (51. Theefoe, afte some tedious algeba, we get the following solution: { ( b = 1 1 ϕ 1 1 + + γ1 γ 1 (1 + γ 1 1, (53 c = b, whee is given by [ ] 1 = µ (θ+α(γ1 +1 γ 1 ϕ 1 1. (54 θ On the othe hand, letting c = 0, i.e. the CCS is absent, we diectly obtain fom (49 that the optimal capital stuctue is uniquely detemined by b, which is eplicitly given by b = [ µ θ (θ + α(γ 1 + 1 ] 1 γ 1. (55 5. What ae the advantages of contingent convetible secuity? With egad to this question, as a stating point, we popose an intuitive answe in the following. Fist, a CCS has the appealing popeties of CCs, which ae impotant to pevent financial cisis. Fo eample, it inceases a bank s capital when a bank is weak, which is pecisely when it is hadest fo a bank to issue new equity. Second, this new fom of contingent capital can geatly incease the value of ta shields while keeping default isk in a low level. 18
Last but not least, a lot of the small- and medium-sized entepises (SMs, especially in China, have to default only because they epeience negative cash flows afte paying inteest epense on a debt, even though the values of equity ae valuable still. This happens because an SM is diffeent fom lage companies, which can sell newly issued equity into the maket and continually finance the negative potion of the cash flow afte inteest and taes by dilution of equity only if the value of equity does not each zeo. Howeve, if an SM issue a CCS instead of taking othe debt financing, then to a geat etent, the SM can solve the poblem of negative cash flows like a lage company by dilution of equity, since a CCS can automatically be conveted into equity once the SM epeiences financial distess. Following the peceding intuitive discussion, we now quantitatively compae the values of a fim that issues a CCS and an S, of a fim that issues a CC and an S and of a fim that issues an S only. Fo this aim, we note that the capital stuctue of the latte two fims ae just special cases of ou model and so it is immediate to deive the following conclusions. Capital stuctue including equity, CC and staight bond. Clealy, a fim that issues a CC and an S is actually equivalent to a fim in ou model when the convesion theshold of the CCS conveting fom equity into debt eaches, i.e. δ. Hence, the values of the S is still given by (20. If the CC has not been conveted into equity, accoding to (24, the value of the CC is given by [ V CC ( = c ( ] γ1 ( γ1 1 + V CC (δ, (δ, (56 δ whee V CC (δ is given fom (30 by ( [ V CC (δ = lim V C δ (1 θb (δ = β 1 δ µ To deive (57, we note that γ 2 = (µ + 1 2 σ2 δ (µ + 1 2 σ2 2 + 2( µσ 2 since we assume µ < thoughout the tet. ( δ δ σ 2 1 < 1, 19 γ1 ]. (57
Similaly, if the CC has not been conveted into equity, thanks to (35, the value of the equity is given by V ( = + c (1 θb + µ [ (1 θ b + c fo (δ,, whee V (δ is given fom (41 by V (δ = lim δ V (δ = (1 βδ µ (1 βδ µ ( γ1 δ δ (1 β(1 θb δ ] ( γ1 µ + V (δ δ (58 [ 1 ( δ δ γ1 ]. Theefoe, keeping all othe conditions the same, we immediately get the diffeence, denoted by V 1(, between the values of the fim that issues a CCS and of the fim that issues instead a CC as follows: V 1( = θc (δ γ 1 δ γ 1 1(δ γ 1 γ 2 δ γ 1 γ 2 ( γ1. (59 Capital stuctue including only equity and staight bond. Last, if a fim issues only equity and an S, we can easily deive the values of copoate secuities by letting c = β = 0. Thus, the value of the S is obviously given by (20 and the value of the equity is easily given fom (35 and (41 by V S ( = µ δ ( γ1 + µ δ (1 θb δ [ ( ] γ1 1, (60 fo any (δ,. Clealy, the conclusions accod with classical copoate finance theoy, see Leland (1994 o uffie (2001 among othes. ased on these conclusions, we still keep all othe conditions the same and make a simple compaison with a fim that issue a CCS. We conclude that the inceased value, denoted by V 2(, of the fim that issues a CCS instead of issuing an S only, is even moe than that shown in (59. Specifically, the inceased value is given fo (δ, by V 2( = θc [ ( ] γ1 1 δ δ + θc (δ γ 1 δ γ 1 1(δ γ 1 γ 2 δ γ 1 γ 2 20 ( γ1. δ (61
Fom the inceased values given by (59 and (61, it is clea that a capital stuctue including a CCS is much supeio to that including a CC instead of the CCS, let alone the common capital stuctue that includes only an S. The fomulas also eplicitly tell us how the inceased values ae elated to the model paametes. Howeve, (59 and (61 do not take into account optimal capital stuctue and a egulatoy standad and so, the conclusions ae not fully effective. Fo this aim, we pesent futhemoe numeical simulations based on the esults of Section 4 in the following. 6. Compaative statics and numeical simulations Fom economic intuition along with (59 and (61, it is clea that a CCS can geatly incease the value of a fim and how the inceased value is elated to model paametes. Going one step futhe, in this section we povide numeical eamples fo the models of Section 4 unde the assumptions that L(b = b/, δ = ϕ 1 (b+c and δ = ϕ 2 (b+c fo some given 0 < ϕ 1 < ϕ 2. As we pointed befoe, these assumptions ae easonable fom an application s point of view although it is pobable that the eample does not lead to an optimal capital stuctue accoding to (49. Howeve, unde these eogenous conditions, we will see that the inceased value of the fim that issues the CCS is still significant. In othe wods, if we select the optimal capital stuctue following (49, the inceased value must be even moe. Moeove, to measue the isk of the S, we pesent yield speads of the S unde diffeent volatility ates of the cash flow and diffeent capital stuctues. To make an effective compaison, we select the baseline paamete values caefully following Albul et al. (2010 and Himmelbeg and Tsyplakov (2012 among othes. Specifically, the baseline paamete values ae as follows: = 0.05, µ = 0.01, θ = 0.35, σ = 0.15, α = 0.5, ϕ 1 = 1, ϕ 2 = 1.1, = δ 0 = 15. Tables 1 and 2 pesent the values of the fim and optimal capital stuctue (b, c o b (if CCS and CC ae not included with diffeent copoate secuities and volatility ates of the cash flow of the fim unde baseline paamete values but ϕ 1 being 0.8 and 1 espectively. The optimal capital stuctues ae deived fom (50 (55. The esults shown in the second ow of both Tables 1 and 2 coespond to a capital stuctue including equity, a CCS and an S. The esults in the thid ow coespond to a capital stuctue including equity, a CC and an S and those in the last ow coespond to a 21
Table 1: Fim s value and optimal capital stuctue with diffeent copoate secuities and volatility ates of the cash flow. aseline paamete values ae set as = 0.05, µ = 0.01, θ = 0.35, α = 0.5, ϕ 1 = 0.8, ϕ 2 = 1.1, δ 0 = 15. σ 0.1 0.15 0.2 0.25 0.3 V CCS 460.8 444.5 432.5 423.3 416.2 (b, c (7.18,8.40 (4.53,10.24 (2.94,11.35 (1.93,12.09 (1.26,12.62 V CC 453.3 434.1 420.5 410.7 403.6 (b, c (7.36,6.85 (4.73,7.82 (3.12,8.20 (2.06,8.30 (1.36,8.27 V S 428.5 408.3 395.7 387.8 282.8 (b (9.71 (7.07 (5.16 (3.73 (2.65 capital stuctue including equity and an S only. The esults show futhe that the CCS can incease significantly the value of a fim and the coupon ate of a CCS is alway geate than that of a CC. The loose the financial constaint is, i.e. the less the paamete ϕ 1, the values of the fim that issues a CCS o a CC ae bigge as we epect. If the business isk of the fim inceases, i.e. paamete σ gets lage, the fim should decease quickly the amount of an S and at the same time incease the amount of the CCS o CC. Table 3 shows the yield speads of an S with egad to diffeent capital stuctues and diffeent business isks (σ of the fim. The second ow descibes the speads of the S of a fim that issues equity, a CCS and an S. The thid ow coesponds to a fim that issues equity, a CC and an S. The last ow epesents the speads of the S of a fim which includes only equity and an S in its capital stuctue. The esults eplain that the CC and paticulaly CCS decease significantly the isk of the S. To sum up, we find fom the numeical esults that as we epect, the CCS can not only stengthen financial stability of a fim but also can significantly incease the fim s value. 7. Conclusion This pape poposes a new fom of contingent capital, contingent convetible secuity (CCS, which can be epeatedly inteconveted between debt and equity depending on the financial situation of the fim: Once the fim slips into ecession, the secuity is automatically conveted fom debt into equity; Convesely, once the fim ecoves fom a ecession, the secuity is 22
Table 2: Fim s value and optimal capital stuctue with diffeent copoate secuities and volatility ates of the cash flow. aseline paamete values ae set as = 0.05, µ = 0.01, θ = 0.35, α = 0.5, ϕ 1 = 1, ϕ 2 = 1.1, δ 0 = 15. σ 0.1 0.15 0.2 0.25 0.3 V CCS 451.1 436.8 426.2 417.9 411.6 (b, c (7.37,6.45 (4.62,8.52 (2.99,9.72 (1.96,10.51 (1.28,11.06 V CC 440.9 424.3 412.7 404.4 398.4 (b, c (7.75,4.22 (4.94,5.54 (3.23,6.15 (2.12,6.41 (1.39,6.48 V S 428.5 408.3 395.7 387.8 282.8 (b (9.71 (7.07 (5.16 (3.73 (2.65 Table 3: The yield spead (asis Point changes with business isk (σ and capital stuctue. aseline paamete values ae set as = 0.05, µ = 0.01, θ = 0.35, α = 0.5, ϕ 1 = 1, ϕ 2 = 1.1, δ 0 = 15. σ 0.1 0.15 0.2 0.25 0.3 CCS 8.01 14.5 21.9 29.6 37.4 CC 9.68 16.7 24.4 32.1 39.9 S 22.8 36.2 47.9 58.3 67.4 automatically conveted fom equity into debt again. This happens again and again until default occus. Obviously, the new secuity is simila with the ecently invented contingent convetible bond (CC, which, howeve, will be conveted into equity at most once. This diffeence makes the new secuity much supeio to the CC, let alone to a staight bond (S. A CC is impotant because it can help egulatos/banks to pevent financial cisis. The CCS is an impovement of the CC. ut in fact, without a doubt, the CCS can be used by any fim. In paticula, we think the new secuity will play an indispensable ole in solving the seious financing poblems faced by the small- and medium-sized entepises (SMs, especially in China among othe counties. An SM is in geneal unable to sell equity into maket like a lage company and thus, it usually has to default if an SM epeiences negative cash flow, even the value of equity of the SM is still vey valuable. Howeve, if an SM issues a CCS athe than othe bonds, then the SM can cheaply solve the negative cash flow poblem by automatically conveting the CCS fom a debt secuity into an equity secuity. On the othe hand, once the financial situation of the fim ecoves fom ecession, the CCS can be automatically conveted fom 23
an equity secuity into a debt secuity and consequently, it gets the values of ta shields again. The CCS we develop in this pape is actually a hybid secuity but it is fundamentally diffeent fom the pefeence shaes and convetible bonds. The CCS can take the fom of debt o equity anytime depending only on which of debt and equity is moe valuable to fims (o egulatos unde a special financial situation. We deals with the picing of copoate secuities and optimal capital stuctue fo fims (e.g. banks o SMs, which issue the new secuity. Most of the esults ae given eplicitly while numeical analysis is povided. The esults eplain that the CCS is a vey valuable contingent capital in inceasing fims values while keeping default isk almost unchangeable. In addition, thee might be two shotcomings of CCSs: Fist, CCSs will incease management costs to some degee; Second, CCSs might involve isks in tems of moal hazad and advese selection. In paticula, when the cash flow ate of a fim is close to the convesion thesholds of CCSs, manages of the fim may manipulate the eal cash flow level on behalf of the equity holde although these isks ae much less than those involved by CCs. Fo this eason, we should incease the diffeence between the two convesion thesholds and in the same time stengthen the ovesight and egulation of fims that issue CCSs. We suppose that the cash flow of the fim is ielevant to the capital stuctue in this pape but actually, fims may switch ove time among a set of altenative cash-flow ate pocesses. Unde this case, inefficiencies of asset substitution o debt ovehang may aise due to taking CCSs into capital stuctue. We leave these questions fo futue eseach. Acknowledgments The authos ae gateful to. Xiaolin Wang of Hunan univesity fo he help. The eseach fo this pape was suppoted by National Natual Science Foundation of China (Poject Nos. 70971037, 71171078 and 71221001. Refeences Albul,., Jaffee,.M., Tchistyi, A., 2010. Contingent convetible bonds and capital stuctue decisions. Woking Pape, Univesity of Califonia ekeley Coleman Fung Risk Management Reseach Cente. 24
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