A new type of contingent capital with jump risk


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1 A new type of contingent capital with jump risk Abstract We eamine a new type of contingent capital, called contingent convertible security (CCS), when asset value follows a jumpdiffusion process. The merit of CCS is that it can dynamically adjust capital structure almost without incurring adjustment costs. We obtain closedform epressions of the equilibrium prices of all corporate securities. Compared with standard capital structures, CCS can lead to as much as a 9.5 percent increase in the issuing firm s value but the number declines to 5.7 percent if CCB is issued instead of CCS. The larger the investment risk, the more pronounced the advantage of CCS over SB and CCB. In our model, CCS does not suffer the debt overhang problem and shareholders have no riskshifting incentive to increase the diffusive volatility of asset value, although they benefit from a higher jump risk. Keywords: contingent capital, debt overhang, risktaking incentive, jump risk JEL: G13, G32 1. Introduction It is wellknown that a firm generally issues both equity and debt, instead of all equity. This is because there are two market imperfections: a ta deduction on interest epense and bankruptcy costs. A firm may have a strict Preprint submitted to World Finance Conference Buenos Aires. December 15, 2014
2 but limited incentive to issue debt and the optimal capital structure must be based on the tradeoff between the values of ta shields and of bankruptcy costs. However, if the firm is too important to fail, we must give much more weight to bankruptcy costs. In particular, after the global financial crisis of 2007/2009, the new global bank regulatory standard, Basel III, introduces stricter capital requirements for banks in order to reduce default risk to an acceptable level. Naturally, ecessive debt financing is prohibited and so the benefits of ta shields are decreased. To increase ta shields while keeping default risk in a low level, a proposal that has recently received much attention is to issue contingent convertible bond (CCB, henceforth). CCB, also known as contingent capital, is a bond that can automatically convert into equity if the issuer s financial health deteriorates to a prespecified threshold or trigger. The first CCB was issued by Lloyds Banking Group in November 2009, which launched its $13.7 bn issue of Enhanced Capital Notes. Net in line was Rabobank making its first entry in the market for contingent debt with a 1.25 bn issue early The Tier2 capital instrument with writedown features is actually a new type of contingent capital. In China, for the first time, Tianjin Binhai rural commercial bank issued RMB 1.5 bn yuan of the writedown contingent capital in July After that, ICBC (Hong Kong) raised $0.5 bn in October 2013 and PingAn bank raised RMB 9 bn yuan in March 2014 in this way. By April 2014, more than 20 banks in China announced their capitalraising plans, including the issue of more than RMB 420 bn yuan of the writedown contingent capital. Contingent capital has several appealing properties. For eample, it in 2
3 creases a bank s capital when a bank is weak, which is precisely when it is hardest for the distressed bank to issue new equity. An inherent problem within banking finance is the risk of panic: when a bank needs to convert hybrid debt into equity, it sends a clear signal to investors that the bank is in trouble. These investors are then tempted to withdraw their investments, making the initial problem much worse. Contingent capital is emerging as the most concrete new idea for solving these inherent problems. On the design of contingent capital, a crucial issue is how to determine the conversion threshold for the activation of contingent capital, which is considered by many papers in the literature. For eample, specific definitions for triggering events are put forward in a consultative document issued by Basel Committee on Banking Supervision (2010). McDonald (2013) evaluates a form of contingent capital for financial institutions that converts from debt into equity if two conditions are met: the firm s stock price is at or below a trigger value and the value of a financial institution inde is also at or below a trigger value. Sundaresan and Wang (2014) consider the design of contingent capital with a stock price trigger for mandatory conversion and the problem on multiple or no equilibrium is discussed. Glasserman and Nouri (2012) analyze the case of contingent capital with a capitalratio trigger and partial and ongoing conversion. The capital ratio is based on accounting or book values to approimate the regulatory ratios that determine capital requirements for banks. Himmelberg et al. (2014) consider the soft trigger case in which the conversion does not necessarily occur at the first instant when the trigger is breached. Instead, the conversion event happens randomly in a probability. For further conversion mechanism discussions, please 3
4 refer to Metzeler and Reeser (2011), Barucci and Del (2012), Hilscher and Raviv (2014), among others. However, all the papers in the literature assume that once contingent capital or CCB converts into equity, it keeps the form of equity all the time until the firm goes bankrupt and in particular, it will never convert back into debt. We think that this assumption is worth being improved. Intuitively, if such form of contingent capital can be converted repeatedly between debt and equity, the firm must get much more ta shields while default risk does not increase. Motivated by this observation, we rela the assumption and develop a new type of contingent capital, called contingent convertible security (CCS, henceforth). CCS is like CCB but differently it can be repeatedly and automatically converted between debt and equity depending on two specified values of a financial situation inde, say the level of the cash flow generated by the issuing firm or the corresponding unlevered firm s value. It is well known in corporate finance that optimal capital structure depends on the firm s financial conditions, say the cash flow level generated by the firm or the unlevered firm s value, i.e. asset value. Generally speaking, the better the financial conditions, the higher the leverage in capital structure should be. On account of that the financial conditions usually change randomly all the time and more often than not, the optimal capital structure just established will become obsolete quickly. To keep capital structure optimal, we are able to do nothing but update the capital structure dynamically. For eample, if a firm gets into (grows out of) trouble, the firm should retire (issue) debt to dynamically adjust the firm s leverage. Unfortunately, such update usually incurs considerable adjustment costs, see Fischer et al. (1989) 4
5 and Titman and Tsyplakov (2007) among others. In sharp contrast, if a firm includes CCS into its capital structure, to some etent, such adjustment is automatically realized and roughly incurs no etra costs. It is true that a revolving line of credit would do so also without incurring etra costs. However, if the firm is an small and mediumsized enterprise (SME), let alone a bank, the adjustment by a revolving line of credit is generally unfeasible, see e.g. Yang and Zhang (2013). In addition, as reported by Leland (1994), in the absence of transactions costs, restructuring by continuous readjustments of debt coupon would seem to be desirable to maimize total firm value as cash flow changes. However, shareholders and debtholders are reluctant to readjust the debt coupon by debt repurchase (issuance). For eample, the increasing of the current debt capacity by selling a small amount of additional debt will be blocked by debtholders, because this market operation will increase the risk of original bond holders, and do harm to their benefits. However, if issuing CCS, we have no such concerns since the adjustment is determined in advance in the agreement, which precludes the conflict of different stakeholders. CCS is similar to the preferred stock (preferred shares) with noncumulative dividends in addition to CCB. Preferred shares sit between debt and common equity in the capital structure, and noncumulative dividends can be suspended if the cash flow of the issuing firm is insufficient. Like CCS, nonpayment of preferred dividends does not trigger default and consequently under the rules of Bank for International Settlements, a noncumulative preferred stock is included in Tier 1 capital. Just as the preferred stock, CCS is senior (i.e. higher ranking) to common stock, but subordinate to bonds in 5
6 terms of claim (or rights to their share of the assets of the firm). Even if CCS converts into equity, it is still different from common stock since like preferred stock, we assume that CCS carries no voting rights no matter whether it has converted into equity or not. For eample, in our model, the default timing and the technology choice of a firm are completely decided by the holders of common stock. However, there is a fundamental difference between CCS and preferred stock (including that with noncumulative dividends): CCS payments are tadeductible whereas preferred dividends are not. Because of this distinction, the preferred stock is unable to adjust capital structure while CCS can do so automatically to some degree and in contrast to preferred stock, CCS is able to increase the value of ta shields while keeping bankruptcy risk unchanged. Banks are generally the issuers of CCSs/CCBs, which, however, could be issued by any firms. For eample, Song and Yang (2013) discuss how CCB as a debt financing instrument might affect the timing and pricing of the option to invest, agency cost of debt and optimal capital structure and find that CCB has a new merit: There is an optimal fraction of equity allocated to CCB holders upon conversion, such that the agency cost reaches the minimum value zero. In fact, our conclusions in this paper are applicable to any firm, no mater whether it is a bank or not, and even no matter whether it is a small or a large firm. In particular, CCSs are suitable for SMEs to solve the financing problems they face, which are becoming increasingly serious all over the world. An SME usually has to default just because it cannot afford to pay coupons, even though the value of equity of the SME is very valuable. If an SME is a listed company, default can be avoided by dilution of equity in 6
7 common way but many SMEs are not listed. More often than not, they have to default too earlier incurring a net substantial social cost while thanks to CCSs, such default would be effectively avoided. In fact, a new contingent capital with our CCS feature was already issued by the Spanish bank BBVA in May 2013, through the introduction of coupons that are allowed to be deferred. A coupon deferral had already been accepted as a common feature in corporate or financial hybrids, but has been now found its way to contingent capital as well. The investor in a BBVA contingent convertible can see an upcoming coupon payment cancelled at the sole discretion of this bank or its regulator. The full anatomy of this bond has been given in Corcuera et al. (2014). 1 CCS is first developed by Yang and Zhao (2014), which assume the cash flow of a firm follows a geometric Brownian motion. By contrast, we directly model the value of an unlevered firm, which is assumed to follow a jumpdiffusion process. This assumption leads to a more complicated and realistic model. Incorporating jumps into the dynamics of a firm s value is important for three reasons. First it can solve the predictability problem of the conversion and default event, since it creates nonzero credit spreads for short maturities, as argued by Pelger (2012). Second, it can discuss the impact of market jump risk on the evaluation of contingent capital. Third, a limitation of many 1 Corcuera et al. (2014) introduce Coupon Cancellable CCBs, a new type of CCB where coupons can be cancelled during the lifetime of the note. They provide closedform pricing formulas for their CCBs and show that deathspiral effect is reduced. By contrast, our CCS will not generate deathspiral effect at all since CCS takes a fi amount of the issuer s shares when it converts into equity, as discussed by Pelger (2012). 7
8 structural models is that they do not incorporate asymmetric information between shareholders and creditors. This is partly mitigated by the inclusion of jumps in a firm s value, which could effectively reflect a sudden release of information, as pointed out by Duffie and Lando (2001). The goal of this paper is to develop CCS, investigate the incentives it creates for shareholders when the value of the issuing firm follows the double eponential jump diffusion process. We develop a structural model of the type introduced by Leland (1994), and etended by Kou (2002) and Chen and Kou (2009) to include jumps. We derive closedform epressions of the equilibrium prices of all corporate securities and the optimal capital structure when the cash flow of a firm or asset value is modeled as a jumpdiffusion process, namely a Kou process. We analyze the incentives created by CCS and investigate how CCS affects debt overhang, asset substitution and the sensitivity of shareholders to various types of risk. We show that while CCS decreases default risk like CCB, it also significantly increase the value of the issuing firm by dynamically adjusting capital structure without incurring etra adjustment costs. The rest of the paper is organized as follows. Section 2 sets up the model and corporate securities are defined. Section 3 discusses the equilibrium pricing of all corporate securities and all the security prices are eplicitly derived. Section 4 provides numerical simulations and comparative statics. Section 5 concludes. Some proofs and epressions are shown in Appendices. 8
9 2. The model In this section, we develop a structural model of CCS. In general, it is assumed in the literature that the cash flow or value of an unlevered firm follows a pure diffusion process. However, it might be unrealistic for many firms since the cash flow or asset value would sometimes undergo a sudden decline or sudden increase. Hence, following Leland (1994), Pelger (2012) and Pennacchi et al. (2014), we model firm s value directly but we assume the unlevered firm s value (i.e. asset value) follows a jumpdiffusion process. We assume that asset value is affected by infrequent jumps due to the arrival of important events that have more than a marginal effect on the value of the unlevered firm. According to Merton (1976), jumps are described by a Poisson counting process. The dynamics of asset value is described as the resultant of two components: the continuous part which is a reflection of new information which has a marginal impact on the firm and the jump part which is a reflection of important new information that has an instantaneous, big impact on the firm. Following Merton (1976), we assume the latter type of information is firmspecific, i.e. its impact is restricted to the firm and so, the jump component of asset value represents idiosyncratic risk. Accordingly, the jump component is uncorrelated with the market portfolio. The arrival time and size J of the jump are independent. Specifically, the probability that the jump arrives during time interval t ( t 0) is λ t, where λ is arrival intensity or jump intensity that describes the epected number of jumps per year. If the ith jump arrives at time t with size J i, the value A of the unlevered firm will change from A t to A t+ = J i A t in the net instant t+, where J i > 0 is realization of a random variable J at the ith 9
10 jump. Variable J i, i = 1, 2,, is independently and identically distributed (i.i.d.). In addition, all investors have standard liquid financial opportunities which involve a riskfree asset with interest rate r > 0 and a risky asset (the market portfolio). Formally, we assume directly that under the riskneutral measure Q, 2 the risky asset price M and the firm s value A are governed by the following stochastic differential equations: dm t /M t = rdt + σ m dz 1 t, da t A t M 0 given, = (µ λk)dt + σ(ρdz 1 t + 1 ρ 2 dz 2 t ) + d ( Nt i=1 J i ), A 0 given, for any t [0, ), where N is a standard Poisson process with jump intensity λ, k is the epected percentage change if a jump occurs, i.e. k = E Q (J 1), σ m and σ are the strictly positive volatility rates, and Z (Z 1, Z 2 ) is a 2dimensional standard Brownian motion on a complete probability space (Ω, F, Q). The parameter ρ in (1) represents the correlation coefficient between the firm s value and the risky asset, while ρσ is the systematic volatility of the firm s value. Naturally, we assume ρ < 1. The parameter µ in (1) is the riskadjusted epected growth rate, i.e. µ = µ A ρση, where µ A is the epected growth rate under the physical probability measure and η is the market Sharpe ratio. We denote by F {F t (1) : t 0} the Qaugmentation of the filtration 2 According to a standard pricing method, we get the equilibrium stochastic discount factor, which determines the measure Q. 10
11 σ(z s ; 0 s t) generated by process Z. The filtration F describes the information available to investors. Clearly, according to the dynamic asset pricing theory, (1) means that the cash flow, denoted by δ t, at any time t 0 generated by the firm is (r µ)a t. I.e. we have the relation, δ t = (r µ)a t for all t 0. Naturally, we assume µ < r, which is well known. We take it that the firm has chosen a capital structure including a straight bond (SB, henceforth), CCS and equity. SB is consol type, meaning that it requires continuous coupon payment at the constant rate of b per unit of time (year), until default. CCS is a hybrid security, which takes the form of either debt or equity and can interconvert repeatedly between them, as long as the firm remains solvent. If CCS takes the form of debt (debt security), it requires continuous coupon payment at the constant rate of c per unit of time. If CCS takes the form of equity (equity security), its holder receives a fraction β, called ownership stake, of the residual cash flow in the form of dividends at the rate β[δ t (1 θ)b] at time t, where 0 < θ < 1 is a constant ta rate. In order to implement easily, we should prevent CCS from converting too frequently and hence we select two different conversion thresholds of the interconversion: One is the conversion threshold, denoted by A E, which once asset value hits from above, CCS automatically converts from a debt security into an equity security; The other is the conversion threshold, denoted by A D, which once asset value hits from below, CCS automatically converts from an equity security to a debt security. On account of that the time is homogenous in our model, we specify the 11
12 default threshold, denoted by, which is independent of time. I.e., once asset value is less than, the firm goes bankrupt. The default threshold can be eogenously given or endogenously determined. Naturally, we have < A E < A D, i.e. the default threshold should be less than the two conversion thresholds of CCS. At default, asset value is assigned to the SB holder but a fraction, denoted by α, of the value will lost due to bankruptcy costs, where 0 α 1 is constant and called bankruptcy loss rate in the literature. Remark 1. One may worry about that asset value is unsuitable to determine a trigger since it is unobservable. However, Pelger (2012) has shown that the unobservability of asset value process can be circumvented by using credit spreads or the CDS risk premium. Credit spreads have the same advantages as stock prices as they constantly adjust to new information in contrast to accounting triggers. They are a sufficient statistic for asset value process. Thus, defining the conversion event in terms of credit spreads is equivalent to using asset value process. The same holds for CDS risk premiums. After a transformation, our CCS can be therefore applied in practice with a trigger event based on observable market prices, see Metzeler and Reeser (2011), Koziol and Lochen (2012), Chen et al. (2013), and Pennacchi et al. (2014). 3. Pricing corporate securities including CCS with jump risk 3.1. Preliminaries It is well known that the unique solution of the second equation in (1) is given by A t = A 0 ep(x t ) 12
13 where X t = µt + σ(ρzt 1 + N t 1 ρ 2 Zt 2 ) + ln J i i=1 and µ = µ 1 2 σ2 λk. The process X is a Lévy process, i.e., a process with stationary and independent increments. The Laplace eponent of X is the function G( ) such that E Q [ep(ξx t )] = ep (G(ξ)t) for an any time t with E Q [ep(ξx t )] <. According to Cont and Tankov (2003), we have G(ξ) = 1 2 σ2 ξ 2 + µξ + λk. (2) In particular, we now consider the double eponential jumpdiffusion model, developed by Kou (2002). To simplify the formula and model jump sizes, we introduce the variables Y i ln J i, i = 1, 2,. Clearly, these variables are i.i.d. and valued in R. Following Kou (2002), we assume the jump size Y i has an asymmetric double eponential distribution with density f(y) = f + (y) + f (y), where f + (y) = pη 1 e η 1y 1 [y 0] and f (y) = (1 p)η 2 e η 2y 1 [y<0] for η 1, η 2 > 0 and 0 p 1, represent the probability densities of upward and downward jumps respectively. Therefore, we obtain E Q [e ξy i ] = e ξy f(y)dy = pη 1 η 1 ξ + (1 p)η 2 η 2 +ξ <, ξ ( η 2, η 1 );, ξ / ( η 2, η 1 ). 13
14 Especially, if η 1 > 1, then E Q [e Y i ] < and E Q [e Y i ] = pη 1 η (1 p)η 2 η Thus, from (2), for ξ ( η 2, η 1 ) the Laplace eponent function is given by G(ξ) = 1 ( 2 σ2 ξ 2 pη1 + µξ + λ η 1 ξ + (1 p)η ) 2 1. η 2 + ξ According to Kou and Wang (2003), the equation G(ξ) = r has four roots denoted by β 1, β 2, β 3 and β 4, which satisfy 0 < β 1 < η 1 < β 2 < and 0 < β 3 < η 2 < β 4 <. These parameters are important in the pricing of corporate securities as shown in the following tet Equilibrium pricing theory with jump We first develop an equilibrium pricing theory under the jumpdiffusion model to derive the main conclusions of this section. Suppose that an asset is a derivative product underlying the value of the unlevered firm, which generates the cash flow ζ t = f(a t ), t [0, ). We denote its equilibrium price by V f (), where represents the current level of the value of the unlevered firm. From Ito s formula, the function V f ( ) satisfies the following relationship: (µ λk)v f ()+ 1 2 σ2 2 V f ()+f()+λe Q [V f (e Y i ) V f ()] rv f () = 0. 14
15 In particular, if the cash flow is a linear function of the value of the unlevered firm up to a stopping time, i.e. ζ t = f(a t ) = aa t + K, t [0, τ D ), where the stopping time τ D inf{t 0 : A t / D} represents the first departure of A from a given domain D and at the stopping time τ D, the holder of the asset gets nothing but a lumpsum dividend or the value of the remaining cash flow, which is a given function, denoted by g( ), of the value of the unlevered firm, then thanks to Ito s formula, for D, the value function V ( ) of the specified asset must satisfy (µ λk)v ()+ 1 2 σ2 2 V ()+a+k+λe Q [V (e Y i ) V ()] rv () = 0 (3) with the following condition V () = g(), / D. (4) Motivated by a result in Yang and Zhao (2014) and applying the guessandverify method, we derive that the general solution of (3) is given by V () = A 1 β 1 + A 2 β 2 + A 3 β 3 + A 4 β 4 + a r µ + K, D, (5) r where A 1, A 2, A 3 and A 4 are constants to be determined from boundary condition (4). Remark 2. The pricing formula (5) we derive here is powerful to obtain the eplicit prices of the corporate securities including CCS under the jumpdiffusion model, as shown in the following tet. 15
16 3.3. The pricing of corporate securities As pointed out before, we denote by A D (A E ) the conversion threshold of CCS from equity (debt) into debt (equity) and denote the bankruptcy threshold by. The pricing of SB. According to (3) and (4), we obtain that if the firm is solvent, i.e. (, ), the value V B () of SB must satisfy (µ λk)v B () σ2 2 V B () + b + λe Q [V B (e Y i )] (λ + r)v B () = 0 (6) with the condition V B () = (1 α), and the condition V B () = b/r,, since default will never happen if the value of the unlevered firm approaches infinity. Thanks to (5) with these conditions, we easily get the following eplicit value of SB: V B () = C B 1 ( ) β3 + C B 2 ( ) β4 + b r, (, ), (7) where and C1 B = (1 α) (β 3 η 2 )(1 + β 4 ) (β 3 β 4 )(1 + η 2 ) β 4(β 3 η 2 ) b η 2 (β 3 β 4 ) r C2 B = (1 α) (η 2 β 4 )(1 + β 3 ) (β 3 β 4 )(1 + η 2 ) β 3(η 2 β 4 ) b η 2 (β 3 β 4 ) r. The proof is presented in Appendi A. The pricing of CCS. We first derive the price V CD ( ) of CCS when it is a debt security. Utilizing (3) and (4), we derive that for (A E, ), the value 16
17 V CD () satisfies (µ λk)v CD +λ 0 ln A E () σ2 2 V CD () + c + λ ln AE ln V CE (e y )f (y)dy V CD (e y )f (y)dy + λ 0 V CD (e y )f + (y)dy (λ + r)v CD () = 0 (8) with the conditions: V CD () = c/r if ; V CD () = V CE () if (, A E ]; V CD () = 0 if. According to (5), the solution is given by ( ) β3 ( ) β4 V CD () = C 1 + C 2 + c A E A E r, (A E, ), (9) where the constants C 1 and C 2 are to be determined later. presented in Appendi B. The proof is Now, we turn to the price V CE ( ) of CCS when it is an equity security. In the same way, the price V CE ( ) of CCS is a solution to the following equation: (µ λk)v CE +λ 0 ln () σ2 2 V CE () + β[(r µ) (1 θ)b] V CE (e y )f (y)dy + λ ln AD 0 V CE (e y )f + (y)dy +λ ln A D V CD (e y )f + (y)dy (λ + r)v CE () = 0, < < A D, (10) with the conditions: V CE () = V CD () if A D ; V CD () = 0 if. Thanks to (5), after a careful computation, we obtain that for (, A D ), 17
18 V CE () ( ) β1 ( = C 3 A D + C4 ) β2 ( A D + C5 ) β3 + C6 ( ) β4 ( + β (1 θ) b r), (11) where the constants C 3, C 4, C 5, C 6 together with C 1, C 2 in (9) constitute a column vector (C 1, C 2, C 3, C 4, C 5, C 6 ), which is the solution of the system of linear equations: ΓX = Φ, where Γ A β 3 DE β 3 +η A β4 DE 1 1 β 4 +η 1 β 1 η 1 β 2 η 1 A β 1 DB β 1 +η 2 A β3 DB β 3 +η 1 A β4 DB β 4 +η 1 A β2 DB 1 1 β 2 +η 2 β 3 η 2 β 4 η 2 A β 3 DE A β 4 DE 1 1 A β 3 DB A β 4 DE 0 0 A β 1 DB A β 2 DB η 2 β 3 1 η 2 β 4 A β1 DE β 1 +η 2 A β2 DE β 2 +η 2 A β3 EB η 2 β 3 A β4 EB η 2 β A β 1 DE A β 2 DE A β 3 EB A β 4 EB Φ βa D 1 η 1 β(1 θ)b+c rη 1 β( 1+η 2 + b(1 θ) rη 2 ) βa D + β(1 θ)b+c r β( b(1 θ) r ) βa E 1+η 2 + β(1 θ)b+c rη 2 βa E + β(1 θ)b+c r,, (12) and A DE A D /A E, A DB A D / and A EB A E /. The proof is also presented in Appendi B. The pricing of equity. Motivated by the derivation of the price of CCS, we fist consider the price V ED ( ) of equity when CCS is a debt security. We get 18
19 that V ED ( ) satisfies (µ λk)v ED () σ2 2 V ED () + (r µ) (1 θ)(b + c) +λ ln A E ln V EE (e y )f (y)dy + λ 0 ln A E V ED (e y )f (y)dy +λ 0 V ED (e y )f + (y)dy (λ + r)v ED () = 0, > A E with the conditions: V ED () = (1 θ)(b+c) r if ; V ED () = V EE () r if (, A E ]; V ED () = 0 if. Thanks to (5), after some more tedious algebra, we get the solution as follows: ( ) β3 ( ) β4 V ED () = C 7 + C 8 + A E A E where the constants C 7 and C 8 are determined later. (1 θ)(b + c), (A E, ), r (13) Net, we discuss the price V EE ( ) of equity when CCS is an equity security. In the same way, the price function V EE ( ) must satisfy (µ λk)v EE +λ 0 ln () σ2 2 V EE () + (1 β)((r µ) (1 θ)b) V EE (e y )f (y)dy + λ ln AD 0 V EE (e y )f + (y)dy +λ ln A D V ED (e y )f + (y)dy (λ + r)v EE () = 0, < < A D, with the conditions: V EE () = V ED () if A D ; V EE () = 0 if. According to the general solution (5) with the above conditions, in the same way, we obtain an eplicit price of equity when CCS is an equity security 19
20 as follows: V EE () = ( ) β1 ( ) β2 ( ) β3 ( ) β4 ( ) C 9 A D + C10 A D + C11 + C12 + (1 β) (1 θ)b r where (, A D ) and constants C 9, C 10, C 11, C 12 with C 7, C 8 in (13) constitute a column vector (C 7, C 8, C 9, C 10, C 11, C 12 ), which is the solution of the system of linear equations: ΓX = Ψ, where Γ is given by (12) and Ψ βa D 1 η 1 + (1 θ)(bβ+c) rη 1 (1 β)( 1+η 2 + (1 θ)b rη 2 ) βa D (1 θ)(βb+c) r (1 β)( (1 θ)b r ) βa E 1+η 2 (1 θ) βb+c rη 2 βa E (1 θ) βb+c r. Remark 3. All the results are eplicit even under the complicated jumpdiffusion case. The eplicit results are valuable to make a strict and general analysis. In addition, if we let the parameter λ 0, it is not difficult to recover the main results derived by Yang and Zhao (2014) The optimal capital structure To determine capital structure, we must first specify each issued corporate security. For eample, with regard to CCS, we assume its ownership stake β, its conversion barrier pair (A E, A D ) and bankruptcy barrier are given in advance. After that, the capital structure is determined by the coupon rate pair (b, c) and the optimal capital structure is given by the SB coupon rate b 20
21 and CCS coupon rate c, which solve the following firm s value maimisation problem: sup { V B () + V CD () + V ED () }, (14) (b,c) A where A represents the set of all admissible coupon rate pairs, and naturally we have A = [0, ) [0, ). Based on the previous eplicit prices of the SB, CCS and equity, the optimal capital structure can be derived by solving a nonlinear programming problem. It leads to a system of equations with two variables, of which the solution is tedious to write. Hence, we omit the details and directly provide numerical simulations in the net section. 4. Numerical simulations and comparative statics In this section, we conduct numerical simulations and derive comparative statics. By numerical computation, we make clear how much a firm s value is increased due to CCS. We eamine whether CCS will aggravate inefficiencies from asset substitution and debt overhang. We also discuss the relationship between investment risk and optimal capital structure if CCS is issued. For simplification but without loss of the effectiveness of our analysis, following Koziol and Lochen (2012), we assume the two conversion thresholds of CCS are given by A E = φ 1 (b + c)/(r µ), A D = φ 2 (b + c)/(r µ), (15) for some 0 < φ 1 < φ 2. In particular, parameter φ 1 allows us to easily adjust the regulatory standard. Clearly, the bigger the parameter φ 1, the stricter 21
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