the Role of Contingent Convertible Bond in Capital Structure Decisions Weikeng Chen [401248]

Transcription

1 the Role of Contingent Convertible Bond in Capital Structure Decisions A master thesis by Weikeng Chen [41248] MSc. Finance and International Business Supervisor: Peter Løchte Jørgensen Department of Economics and Business Summer 212 Aarhus University Business and Social Science

2 The Role of Contingent Convertible Bond in Capital Structure Decisions Weikeng Chen Aarhus University Abstract The financial crisis has exposed flaws in the regulation of capital positions of large financial institutions. When they get financially distressed, government regulators are implicitly forced to provide extensive amount of liquidity infusion, which usually causes a lot of public controversy. This paper develops a new derivative security, Contingent Convertible Bond, which is a debt instrument that automatically converts to equity if the issuing institution reaches a pre-specified level of financial distress. This kind of debt-to-equity swap or automatic bail-in is highly advantageous to the institution when it gets distressed, since raising new equity at this time has very high cost and makes it unfeasible. In this paper, we derive close-form formula for the market value of this security when the institution s assets are modeled as a Geometric Brownian Motion process and its conversion trigger is set as a threshold of asset value. Our calibration results show that, the introduction of Contingent Convertible Bond into the capital structure can reduce the institution s default probability and effectively mitigate the management s risk shifting motivation. Keywords: Contingent Convertible Bond, Capital Structure, Financial Stability, Structural Model, Close-form Solution

3 CONTENTS i Contents 1 Introduction 1 2 Literatures Conversion trigger of contingent capital Existing research on contingent capital Comparison between CoCo Bond Models with asset trigger Similarities across 3 models Differences across 3 models Models Asset Dynamics Girsanov Theorem Real World and Risk-Neutral World Absence of Arbitrage and Asset Equation Definition of variables, functions and expectations Benchmark Model Security Design and Assumptions Asset Equation Valuation of claims Debt-Equity Model Security Design and Assumptions Asset Equation Valuation of claims Subordinate Debt Model Security Design and Assumptions Asset Equation Valuation of claims CoCo Bond Model Security Design and Assumptions Asset Equation Valuation of claims Calibration Default Probability Mathematical derivation Calibrating the models Risk Shifting Motivation Mathematical derivation Calibrating the models Extensions Endogenous coupon rates Endogenous value paid-out rate

4 CONTENTS ii 6 Summary 65 7 Reference 66 8 Appendix Prove the basic properties of GBM Derive the explicit formula of H function Derive an intermediate formula (4) Valuation of European call option Prove Equation (68) Valuation of decompositions in Debt-Equity Model Valuation of down-and-in call barrier option Valuation of decompositions in Subordinate Debt Model Valuation of decompositions in CoCo Bond Model

5 1 INTRODUCTION 1 1 Introduction The financial crisis exposed flaws in the regulation of capital positions of large financial institutions (LFIs), especially big banks. The architecture governing the financial insolvency of banks and other financial institutions needs a particular examination. The insolvency and bankruptcy of these institutions might cause socially vital disruptions in the overall financial system. In order to prevent the possible consequence, the government regulators are forced to provide extensive amount of implicit guarantees when they get financially distressed, which usually take the form of outright infusion of taxpayer money to enrich their capital position. This is the so called Too Big To Fail (TBTF) problem and the need for government bailout brings a huge social cost. Facing this problem, government regulators need to put forward some strict proposals to improve the prudential bank regulation. However, more importantly, banks themselves should ensure that they have enough loss-absorbing capital as buffer and they can internalize the losses when they get financially distressed, in order to preclude a big dependence on the public bailout. To solve this problem, enhancing the bank s financial stability by optimizing its capital structure would be a good perspective. Some innovative and well-designed derivative securities could help to internalize the bank s possible losses during the financial crisis. Among the possible choices, Contingent Convertible Bond (CoCo bond, CCB), one kind of contingent capitals, would be a good derivative security that is potentially beneficial to the enhancement of the financial stability. Contingent Convertible Bond is a debt instrument that automatically converts to equity if the issuing bank reaches a pre-specified level of financial distress (called conversion trigger). This derivative is issued as debt and it can enjoy the debt benefit of tax deduction before conversion. When a distress-related trigger is breached (usually due to the depletion of capital during a financial crisis), CoCo bond will mandatorily convert into common equity to enrich its capital position. This kind of debt-to-equity swap or automatic bail-in is highly advantageous to the bank when it gets distressed, because raising new equity to enrich its capital position, another bail-in procedure instead, has very high cost in distress times and makes it unfeasible. So CoCo bond can be viewed as another type of capital buffer for big banks with default risk. It precludes the need of external government bail-out, which usually causes a lot of public controversy. This innovative derivative security was proposed by Flannery (25) and the following researches extend the analysis and construct financial models for quantitative valuation. However, regarding CoCo bond s conversion design, equilibrium pricing and possible influence on bank s capital structure, it still remains at an early research stage and seems far from reaching a consensus. There are already some attempts in real financial world. For example, Lloyd s bank issued the first 7 billion ($11.6 billion) CoCo bonds in 29. However, as a newly emerging security, the number of real world cases is very limited and thus the real world data is far from sufficient to test the theoretical model. Facing this kind of data restriction, usually researchers will calibrate the model with some well approximated parameters to see its application

6 1 INTRODUCTION 2 in the capital structure decision, which is also what we conduct in our empirical part after introducing the models. Undoubtedly, different designs of the underlying assets and the contractual terms of CoCo bond would lead to totally different models with different valuation methods, sometimes even with slightly different conclusions. In this paper, we construct a Co- Co Bond Model and present a close-form solution for the CoCo price and all other bank claims (senior debt, subordinate debt, equity). It utilizes the outcomes from some of the existing models as well as makes some important improvements in the design of contractual terms. The existing research and our improvements will be explained more in detail in Section 2. For a brief description of our CoCo Bond Model, the asset dynamics follow the common Geometric Brownian Motion (GBM) process. We assume both the corporate senior debt and CoCo bond have continuous coupon payment and finite maturity, a design character that makes CoCo bond feasible and implementable in real financial market. The market values of all claims in our model have been successfully derived with close-form formula. Some other model elements such as dividend payment, possible bankruptcy cost, default probability, risk shifting motivation, etc. are also covered in this paper. The derivation of close-form formula is very important in derivative valuation. Without explicit solutions, the common data simulation methods (such as Monte Carlo simulation) have very low efficiency in converging to the real number if the derivative is complicatedly designed. Besides, close-form formula makes our comparative static analysis much easier and time-efficient. Models construction is the core of this paper. We begin from a Benchmark Model, the classic Black-Scholes-Merton model of debt and equity valuation. The model framework and research methodology inspire us to construct other models, either by adding another security into the capital structure, or by loosening some strict assumptions. Besides, conclusion from Benchmark Model, especially the equity-volatility relationship, is where the traditional and classical theory comes from and thus it is worth our attention. In order to see the role of CoCo bond in enhancing the financial stability, under the framework of Benchmark Model, we construct another 2 supporting models. Debt- Equity Model considers a capital structure with corporate debt and equity, the same as Benchmark Model. However, many assumptions from the Benchmark Model have been loosened. This model is the template and basis for models with more securities in the capital structure. Its conclusion can be used to compare with our core model, CoCo Bond Model, to see whether it is beneficial to add CoCo bond into the capital structure. Another supporting model is Subordinate Debt Model. Also for comparison purpose, we construct this model with another debt instrument, subordinate debt, which is used to directly compete with CoCo bond. So there are 4 models in total in our paper, 3 supporting models and a core model, CoCo Bond Model. Quantitative analysis by data calibration is performed for all 4 models, and the role of CoCo bond is clearly displayed after the comparison. The rest of this paper is organized as follows. Section 2 examines the recent research

7 2 LITERATURES 3 findings and compares our CoCo Bond Model with models in 2 published papers. Section 3 presents our 4 models, each beginning from the security assumptions, then the valuation of all claims, and ending with the mathematical formulas. Section 4 tests the advantage of CoCo bond quantitatively from 2 perspectives, default probability and risk shifting motivation, by data calibration. Some extensions for the CoCo Bond Model are provided in Section 5. It provides more choices and inspiring when designing CoCo bond in real world. Section 6 summarizes. Detailed calculations leading to our valuation formulas are deferred to Appendix. 2 Literatures 2.1 Conversion trigger of contingent capital Regarding the design of contingent capital, there exist a list of issues that need to be settled before implementation. Detailed proposals can refer to Flannery (29) and McDonald (21). Here we just provide the most important design characteristic of the contingent capital: the setting of distress-related conversion trigger. Conversion trigger is widely discussed in the literatures of contingent capital. Type of conversion trigger should be clearly determined no later than the issuance of contingent capital. When the trigger is reached, the conversion should be conducted mandatorily and automatically. Until now, it does not have a consensus regarding which should be chose as conversion trigger. For different considerations, there exist different types of triggers. We briefly list the conversion triggers that have been discussed and utilized in existing literatures. We will not give a detailed explanation about the advantage and disadvantage for each type of trigger as many papers have covered. Interested readers can refer to the corresponding literatures for a deeper study. Conversion triggers in the current research can be classified into 3 types: 1. A systemic event which will make a big influence on the banking system as a whole, for example, the financial crisis which can be observed and measured by some kind of market index, some big changes of banking supervision regulation, etc. Related research can refer to Kashyap, Rajan and Stein (28). 2. The trigger related to the individual LFI. This is the most common type in literature. This type of trigger includes the following financial indicators. - Capital ratio of LFI. This is one of the most important risk ratios of LFI. It can take the form of either equity to debt value, or equity to asset value. Debt value and asset value are both unique; however we have 2 choices for equity value: book value of equity and market value of equity. Accordingly, there exist both book capital ratio and market capital ratio. Their advantages and disadvantages will be analyzed more in detail below. In literature, for example, Flannery (25) suggested a capital ratio based on the market value of the bank s equity. However, Glasserman and Nouri (21) developed a model with a capital-ratio trigger based on book value.

8 2 LITERATURES 4 - Underlying asset value of LFI. Utilizing this trigger will make the analysis tractable and remove the possibility of multiple equilibriums. Related research can refer to Raviv and Hilscher (211) and Albul, Jaffee and Tchitsyi (21). In our paper, we also choose it as the conversion trigger. - Share price of LFI. Obviously this is another market value indicator. Related research can refer to Glasserman and Wang (29). 3. The combination of 2 or more indicators. One example is a trigger based on the health of both an individual bank and the financial system, suggested by Squam Lake Working Group (29). Another example can be referred to McDonald (21). As above, according to which kind of equity value we use, there are 2 types of capital ratios for LFI: book capital ratio and market capital ratio. Market value of equity is based on the market price of the LFI s stock. Book value of equity is based on regulatory accounting measures of debt and capital. It is also called regulatory value, or accounting value. In research, it will be the residual of asset value after deducting the value of all kinds of debt. In current literatures, there are many discussions regarding the advantage and disadvantage of using the book value or market value indicators. Since the discussion appears in many literatures, here we just list the general points in the literatures without detailedly pointing out their origin. Note that, the disadvantage of one type of indicator may construct part of the reason why we may use the other type, vice versa. Using the book value indicators: - Advantage: Existing regulatory capital requirements for LFIs are based primarily on book values. - Advantage: Existing issuances of contingent capital to date all use triggers based on regulatory values rather than market prices. Using the market value indicators: - Advantage: It is continuously updated with the newest information and reacts sensitively to any market shock. - Advantage: It is forward-looking and thus it can reveal any potential shock and show the market and the investors expectation. - Disadvantage: Market values could potentially be manipulated to trigger conversion. - Disadvantage: Market value indicators may result in multiple solutions or no solution for the market price of the contingent capital. This leads to the problem of the viability of contracts designed with market-based triggers. Sundaresan and Wang (21) have a wonderful explanation for this point.

9 2 LITERATURES Existing research on contingent capital Before the model section, we provide a brief overview of the literatures on contingent capital until now. The original proposal and idea of contingent capital was pioneered by Flannery. Flannery (25) proposed a new financial security for LFIs, called reverse convertible debentures. It is a form of debt that converts to equity if the institution s capital ratio falls below a threshold. His proposal utilized a capital ratio based on the market value of the bank s equity, a feature that may cause multiple equilibriums or even no equilibrium, proved later by Sundaresan and Wang (21). Flannery (29) extent this proposal and considered how it could be implemented in real financial market. In this proposal, he renamed it as Contingent Capital Certificates (CCC). Kashyap, Rajan and Stein (28) proposed a custodial account, i.e. a lock box to hold bank funds that would be released if an event, usually, one kind of crisis, happens over the life of the policy. It would resemble an investment in a defaultable catastrophe bond. The trigger is a systemic event, rather than the risk of the individual institution. Instead, the Squam Lake Working Group (29) recommended a trigger that based on the risk condition of both an individual bank and the banking system as a whole. Glasserman and Wang (29) studied the convertible securities designed by the US Treasury for its Capital Assistance Program. They suggested this kind of security can be viewed as a type of contingent capital in which banks hold the option to convert preferred shares to common equity and the trigger can be set as their share price. Other literatures regarding the design of contingent capital lead to model construction and quantitative valuation. Some models can achieve close-form solution, i.e. explicit formula for the pricing of contingent capital and other banking claims, while others need to utilize some kinds of data simulation to do the valuation. McDonald (21) got the value of contingent capital with a dual trigger through joint simulation of a bank s share price and one kind of market index. Pennacchi (21) developed a structural credit risk model of a bank that issued fixed or floating coupon bonds in the form of contingent capital. The return on the bank s assets is simulated to follow a jump-diffusion process, and default-free interest rates are stochastic. Raviv and Hilscher (211) obtained closed-form pricing formula under the assumption that the conversion trigger is set by a threshold level of assets and both debts take the form of zero-coupon deposit. They priced each banking claim by replicating its payoff using a combination of different barrier options that all have closed form solution. Albul, Jaffee and Tchitsyi (21) also used an asset-level trigger and obtained closed-form formula by assuming that all debts have infinite maturity. Glasserman and Nouri (21) developed a model with a capital-ratio trigger based on book value of equity. Different from the conversion process in other literatures, which is one-time and complete conversion, their model suggested a conversion mechanism that converts just enough contingent capital to meet the capital requirement each time a bank s capital ratio reaches the threshold. Sundaresan and Wang (21) proved that setting the conversion trigger at a level of share price may lead to multiple solutions or no

10 2 LITERATURES 6 solution for the market price of contingent capital, raising questions about the feasibility of contracts designed with market-based triggers. Their research cast doubt on the proposal suggested by Flannery, who recommended a capital ratio based on the market value of the bank s equity. Remaining researches proposed other methods to improve LFI s capital position during a financial crisis. Those proposals usually take the form of either some guiding regulations, or suggestions of some innovative securities or security combination. For example, Duffie (21) proposed the mandatory offering of new equity by banks when they face financial crisis and their capital position deteriorates. As opposed to the conversion of debt to equity, a mandatory rights offering provides new cash that may reduce the risk of a liquidity crisis. Hart and Zingales (21) designed a new, implementable capital requirement for LFIs which ensures that LFIs are always solvent, while preserving some of the disciplinary effects of debt. Their mechanism required that LFIs should maintain a sufficiently large equity cushion. If the CDS (Credit Default Swap) price goes above the threshold, the LFI regulator forces the LFI to issue equity until the CDS price moves back down. Pennacchi, Vermaelen and Wolff (21) proposed a new security, the Call Option Enhanced Reverse Convertible (COERC). The security is a form of contingent capital, but at the same time, equity holders have the option to buy back the shares from the bondholders at the conversion price. Compared to other forms of contingent capital proposed in the literature, the COER- C is less risky in a world where bank assets can experience sudden and large declines in value. Among the existing literatures, our paper can be categorized into the type of literatures which lead to model construction and quantitative valuation. Similar to Raviv and Hilscher (211) and Albul, Jaffee and Tchitsyi (21), the conversion trigger in our model is also set as a threshold level of asset value. We make improvements by loosening some calculation-simplified assumptions in the above models, which makes the security more implementable in real financial world. Our model also leads to close-form solution for the valuation of each claim. In the next subsection, we will introduce the similarities and differences between our model and the above 2 models more in detail, from each model s security design, the reason why they design in this way, to the logic of mathematical derivation under each model. The comparison also guides the readers into our model construction gradually and implicitly explains why we design the model in this way and its significance. 2.3 Comparison between CoCo Bond Models with asset trigger To introduce our model gradually, we describe and compare 2 CoCo Bond Models with our model. 3 models all choose the asset value as conversion trigger. As is explained before, using asset value as conversion trigger makes the analysis tractable and removes the possibility of multiple equilibriums. The comparison of 3 models can be clearly shown in Table 1.

11 2 LITERATURES 7 Table 1: Compare 3 Models using Asset Value as Conversion Trigger Models RH Model AJT Model New Model Capital Structure Senior debt, CoCo bond and Equity Conversion Trigger Asset Value Default Trigger Asset Value Asset Dynamics GBM Conversion Dynamics One-time and Complete Conversion Close-form Solution available Yes Coupon paid No Yes Yes Maturity Finite Infinite Finite Dividiend paid No No Yes Bankruptcy Cost considered No Yes Yes Tax Benefit considered No Yes No Similarities across 3 models All 3 models share the same capital structure. The bank issues senior debts, CoCo bond and a residual equity. The function and role of CoCo bond in LFI s capital structure is the same across 3 models. Most importantly, all 3 models choose asset value as its conversion trigger and default trigger. For the dynamics of the underlying asset, all 3 models choose it to be the common stochastic process of Geometric Brownian Motion. From the beginning of Section 3, GBM process is one of the very few cases of stochastic process that can be solved with explicit solution. So the assumption of asset dynamics makes it possible that the market value of each claim can be solved out explicitly. Until now, there are not any papers that give explicit solution of the price of contingent capital with another type of asset dynamics (such as jump diffusion process, or stochastic volatility process). For example, Pennacchi (21) has chosen the jump diffusion process as the dynamics of the underlying assets. Close-form solution is not available and instead he utilized Monte Carlo simulation in his paper. Since asset value is chose as the conversion trigger and the asset dynamics meet the GBM process in all 3 models, the conversion trigger, i.e. asset value, is decided outside the model, so it is one kind of exogenous triggers. Generally speaking, choosing an exogenous trigger guarantees that the equilibrium price of contingent capital will exist and be unique, so it avoids the problem of multiple equilibriums or no equilibrium, as again by Sundaresan and Wang (21). All 3 models also have the same conversion dynamics: one-time and complete conversion. This assumption makes it easier to construct a model, as well as easier to implement the conversion in real world, although it has the potential problem of converting too much or converting not at the best time for the issuing institution. The opposite is partial and ongoing conversion, which converts just enough contingent capital to meet the capital requirement each time the trigger is breached. One example of this kind of dynamic conversion is provided by Glasserman and Nouri

12 2 LITERATURES 8 (21). Different from the attempt of using other stochastic process than GBM process, which makes it nearly impossible to get a close-form solution, the assumption of partly conversion can lead to close-form solution after a slightly more complicated calculation. The model by Glasserman and Nouri has successfully done it Differences across 3 models More importantly, we need to overview the different designs across 3 models, which show the improvements we have made in our model. In brief, the improvement work focuses mainly on the loosening of 2 assumptions of bond design, zero-coupon assumption for RH Model and infinite maturity assumption for AJT Model. Naturally, coupon payment and bond maturity will be the concentration of the following analysis. RH Model We call the CoCo Bond Model developed by Raviv and Hilscher (211) as RH Model for short. In RH Model, the senior debt takes the form of zero-coupon deposit. No dividend is distributed to the equity holder during the maturity. This means that there are no cash outflows before the maturity. If we use Expression (13) in Section 3 to describe the dynamics of underlying assets under Q-measure, in this model we have δ = and the drift of the underlying assets is equal to the risk free interest rate r. Without coupon payment, the only way for senior debt holder (deposit holder here) to realize value from their investment is the payment at maturity, so in this model, the maturity should be finite. Otherwise the senior debt holders get nothing if their life span is smaller than infinite. Without coupon payment, it is not possible to consider the tax-saving benefit. Tax benefit takes a form of tax saving for the debt interest the banks pay out during the year. In RH Model, banks do not pay out any coupons during the maturity, so there is no tax benefit and thus it is not considered. In RH Model, bankruptcy cost is also not covered. This simplification does not in line with most structural models in literature, and we will make the correction in our model. Because of the assumption of no coupon payment and no bankruptcy cost, the valuation of 3 banking claims (senior debts, CoCo bond and equity) is simplified. They price each claim by replicating its payoff using a combination of different barrier options that all have closed form solution. So the close-form solution of each claim is easily calculated. Note that all the barrier options they use to replicate the banking claims are the type of down barrier option (down-and-in option or down-and-out option), because by assuming the initial asset value larger than the conversion trigger, the CoCo bond will converse only when the asset value falls down and hits the trigger. To save space, we will not list their pricing formulas here. It is easily checked in their

13 2 LITERATURES 9 paper. AJT Model We call the CoCo Bond Model developed by Albul, Jaffee and Tchitsyi (21) as AJT Model for short. AJT Model calls senior debt as straight bond or straight debt. It is just a matter of denomination and its role is exactly the same as senior debt in our analysis. In AJT Model, both straight bond and CoCo bond are consol type, meaning they are annuities with infinitive maturity. Straight bond pays coupon continually in time until default. At default, fraction α of the bank s assets is lost. CoCo bond pays coupon continually in time until it hits the conversion trigger. The amounts of both coupon payments are constants and they are decided exogenously. Since both bonds pay coupon continuously in time before the maturity, it is possible to assume that the bonds maturity is infinite (not necessarily though, such as our model), i.e. one kind of annuities. The payoffs of both bonds are taking the form of continuously paying coupons. They have a final payment only when it defaults or converses. Although not common in real world, this assumption also simplifies the pricing calculation. To calculate the present value of coupon payment, it is always easier when the maturity is infinite. As in Section 3, both stopping times (conversion time and default time) are always finite (P(τ = + ) = ). So when the maturity is infinite, CoCo bond will always covert before the maturity, and the bank will always default before the maturity, meaning that there is just one possibility to be considered in the valuation. If the maturity is finite as in our model, we need to figure out different possibilities regarding the comparison of stopping time and maturity. The calculation of the present value of coupon payment with infinite maturity is somewhat similar to what we calculate for the present value of the annuity (interest paid out each period divided by interest rate). Besides, we always need to consider the final payment when default or conversion happens, which is a certain event because the maturity is infinite. This kind of u- nique possibility also simplifies the calculation. Because of the payment of continuous coupon, the parameter of cash outflow δ is not equal to in this model, meaning that the drift of underlying assets under Q-measure will be smaller than r (it will be r δ). They call it µ in their paper. The exact value of δ (and thus µ) is needed because it decides both stopping times. In AJT Model, its value is exogenously decided. In their paper, they do not consider dividend of equity. Different from RH Model, AJT Model considers bankruptcy cost. As in the general literature, it is set as a constant proposition of the asset value when it hits the default trigger. In their model, bankruptcy cost will always be positive because, as before, default trigger will always be reached with an infinite maturity. Besides, they also consider tax benefit in their model. As before, tax benefit is the tax deduction for paid debt interest during the year. In general literature, it is always set as a constant fraction of the coupon paid out (it is the exogenously decided θ in their paper). So the amount of tax benefit is easily got after the valuation of both

14 2 LITERATURES 1 coupons is calculated. From the above analysis, we can get the conclusion that, in order to get the closeform solution, both models have made some strict assumptions. In RH Model, they assume that the senior debt takes the form of zero-coupon deposit. However, in general structural models, we always assume the bonds pay coupons, continuously or discreetly. Actually, coupon payment is one important element of the design of bonds and its amount will greatly affect the value of bonds. In AJT Model, even though coupon payment is considered, they assume that both bonds are consol type, meaning they are annuities with infinitive maturity. This simplification assumption is also not in line with most cases of bonds in real financial world. In one word, both designs make CoCo bond easier for theoretical valuation, but not implementable in real financial market. What about a new model to loosen both assumptions, making the design of CoCo bond more realistic and implementable even though the valuation may be a bit more complicated? This is our motivation and inspiration to design a new CoCo Bond Model. New Model The biggest difference in our model is the loose of some assumptions of both bonds. We assume that both bonds have continuously coupon payment, with finite maturity. The assumption is more in compliance with the general assumption of bonds in literature, as well as the feasible design of real world bonds. So our correction will be an improvement for the existing CoCo Bond Models which use asset level as its conversion trigger. A positive coupon payment, different from RH Model s zero-coupon deposit, means the bank will have cash outflow before the maturity. As in AJT Model, the drift of underlying asset under Q-measure will be smaller than r (we set it as r δ in our model). A finite maturity, different from AJT Model, means that both stopping times can be either before or after the maturity. Now we need to consider both possibilities when we do the valuation for each claim. In other aspects of the model design, there also exist some differences. In AJT Model, value paid-out rate δ is decided exogenously (via µ). In the main content of this paper, δ is exogenously decided when dividend payment is included. In the Extension section, δ is endogenously decided when dividend payment is not considered. Besides, regarding the amount of coupon payment, our model is slightly different from that of AJT Model. In their model, the amounts of both coupon payments are constants and they are decided exogenously. In our paper, coupon payment is a constant fraction (we call it coupon rate) of the face value of bond. Usually the face value of bond is decided at the initial time when the bond is issued. So if the coupon rate is decided exogenously, the coupon design will actually be the same as AJT Model (both kinds are constant). This is the case in our main content. However, in one of our extensions, we also consider the case of endogenously coupon rates. So at this setting, the coupon payment is still fixed continuously across the maturity, however

15 3 MODELS 11 it is decided endogenously. Bankruptcy cost is also considered in our model. The setting is the same as AJT Model, which is in line with most literatures. Different from their model, we do not consider the tax benefit. As before, this claim is easy to get because it is just a constant fraction of the coupon payment. About the dividend payment, we take it into consideration in our main model, but do not consider it in our Extension. As is shown in the model section, this makes a big difference about the setting of value paid out rate δ. In conclusion, our models make some improvements about the design of CoCo bond (also the normal senior debt) by loosening some calculation-orientation assumptions. It makes the calculation a little more complicated, but its advantage in theoretical design and practical application overwhelms. More detail of the design of our model is provided in the model section. In the following model section, 3 supporting models are introduced before our core model, the CoCo Bond Model. We begin from a Benchmark Model, the classic Black-Scholes-Merton model of debt and equity valuation. Based on the framework of Benchmark Model, we construct Debt-Equity Model in which many assumptions of Benchmark Model have been loosened. Subordinate Debt Model adds subordinate debt, another debt instrument, into the bank s capital structure, which is used to directly compete with CoCo bond in our analysis. Finally, CoCo Bond Model will be introduced. 3 Models 3.1 Asset Dynamics All of our models utilize standard option pricing method, which is based on a long line of research on capital structure that includes Black and Scholes (1973), Merton (1973; 1974), Black and Cox (1976), Leland (1994; 1996), and numerous subsequent papers. This approach starts by modeling the dynamics of a firm s assets and then prices debt and equity as claims on those assets. We will begin with the dynamics analysis of the firm s underlying assets. Before our model, we will introduce Girsanov Theorem briefly since it is frequently applied in our paper Girsanov Theorem Let {Z t } t be a standard Brownian motion, defined on a probability space (Ω, F, P), and let {F t } t be the associated Brownian filtration. Let {θ t } t be an adapted process satisfying the hypotheses of Novikov s Proposition: [ ( t )] E exp θs 2 ds < + t (1) Define ( t M t = exp θ s dz s 1 2 t ) θs 2 ds (2)

16 3 MODELS 12 Now we can define a new probability measure Q on the measurable space (Ω, F) as follows. For each T > and any event F F T, Q(F ) = E P [ M T 1 F ] (3) If we define Z t = Z t t θ s ds, t [, T ] (4) Under the probability measure Q, the stochastic process { Z t } t T is a standard Brownian motion. This is the famous Girsanov Theorem. It is commonly used for the transformation of probability measures in stochastic process. Now we move to a special application of Girsanov Theorem. If we set θ t = θ R t, it simplifies to the Cameron-Martin Theorem, which is viewed as the most important special case of Girsanov Theorem. When θ t = θ, the hypotheses of Novikov s Proposition (1) is easily met. We can write (2) as: ( t M t (θ) = exp θ dz s 1 t ) θ 2 ds = e θzt θ2 t/2 (5) 2 As before, {Z t } t is a standard Brownian motion under the probability measure P (we also write it as P, with the corresponding expectation operator E ). We can define a new probability measure Q (we also write it as P θ, with the corresponding expectation operator E θ ) on (Ω, F) as follows. For each T > and any event F F T, [ ] [ P θ (F ) = E MT (θ)1 F or P (F ) = E θ MT (θ) 1 ] 1 F (6) For each T > and any nonnegative random variable Y, Similar to (4), if we define E θ [Y ] = E [M T (θ)y ] or E [Y ] = E θ [ MT (θ) 1 Y ] (7) Z t = Z t t θ ds = Z t θt, t [, T ] (8) Under the probability measure Q = P θ, the stochastic process { Z t } t T is a standard Brownian motion. From Cameron-Martin Theorem, we know that {Z t } t T is a Brownian motion with drift θ under the probability measure Q = P θ. Cameron-Martin Theorem deals only with special probability measures under which paths are distributed as Brownian motion with constant drift. However, Girsanov Theorem applies to nearly all probability measures. For the mathematical derivation in this paper, we need to transform the Brownian motion (with or without drift) under P-measure into another Brownian motion (with or without drift) under Q-measure. So Cameron-Martin Theorem is well enough for our calculation.

17 3 MODELS Real World and Risk-Neutral World The stochastic process of asset dynamics is the same for all the models we develop in our paper. So we display it at the beginning of all models and will refer to it every time we begin a new model. We suppose that the firm s future cash flows have a total market value at time t given by A t. We always refer to A t as the assets of the firm. In order to justify this valuation of the firm, we could assume that there is some other security whose market value at any time t is A t. Assume that we have a real world probability measure P (P-measure for short) defined on the measurable space (Ω, F), then we have a real world probability space (Ω, F, P). Let {Z P t } t be a standard Brownian motion under (Ω, F, P). For the dynamics of assets in real world, we assume that A t meets a general Geometric Brownian Motion (GBM) process. It satisfies: da t = (ϕ δ)a t dt + σa t dz P t (9) where ϕ is the real world rate of return of assets, δ is the constant fraction of value paid out to security holders continuously, σ is the diffusion coefficient, and Zt P is a standard Brownian motion under P-measure. 1 The initial market value of assets is set to be A, and maturity is T. For the valuation, the most important part in our paper, we will utilize the Risk Neutral Valuation Method, which values the market price of all claims under the risk-neutral world. We define a new probability measure Q (Q-measure for short) for the risk neutral world and thus have a new probability space (Ω, F, Q). In order to apply Girsanov Theorem, as before we write it as P P and Q P θ. 2 For Q-measure to be a risk-neutral measure, we should set θ = ϕ r σ Where r is the risk free interest rate. From Girsanov Theorem, we can get a new stochastic process {Z Q t } t which is a standard Brownian motion under Q-measure. From (11), we can get its differential form: (1) Z Q t = Z P t θt = Z P t + ϕ r σ t (11) dz Q t = dz P t + ϕ r σ dt (12) From (9) and (12), we can get the dynamics of A t under the risk-neutral world (Qmeasure): da t = (r δ)a t dt + σa t dz Q t (13) 1 Parameters ϕ, δ, σ and r have already been annualized. 2 θ has the same definition as in Section

18 3 MODELS 14 So we can see that, under the risk-neutral world, the dynamics of assets follow a GBM process with drift (r δ) and diffusion coefficient σ. For the pricing and valuation in the rest of our paper, except for some explicit notes, all the calculation and derivation is under the risk-neutral world. For the convenience of the mathematic derivation, we can set: µ = r δ σ2 2 drift = r δ = µ + σ2 2 (14) From the basic property of GBM and the lognormal distribution of A t 3, we have: A t = A e (r δ σ2 /2)t+σZ Q t = A e µt+σzq t (15) E Q [A t ] = A e (r δ)t σ2 (µ+ = A e 2 )t (16) log A t N (log A + µt, σ 2 t) (under Q-measure) (17) ( ) log m (log P Q A + µt) (A t < m) = Φ σ, for m > (18) t where Φ( ) is the Cumulative Distribution Function (CDF) of standard normal distribution Absence of Arbitrage and Asset Equation Market values of securities are viewed as the claims on the asset A. The absence of well-behaved arbitrage implies that at any time t [, T ], the market value of assets A t is equal to the market values of all claims. It consists of all the securities, as well as other claims on asset value such as bankruptcy cost, etc. Then we have the following asset equation. A t = t the market value of each claim, t [, T ] (19) Usually, we define the firm value as the market value of all securities. Then we have: F t = t the market value of each security, t [, T ] (2) Then the asset equation can also be written as: A t = F t + t the market value of other claim than security, t [, T ] (21) 3 Note that, because of the continuous path property of Brownian Motion and Geometric Brownian Motion, for related variables all across this paper, inequality relation less than (<) and less or equal to ( ) can be regarded as the same. The same logic applys to the inequality realtion of greater than (>) and greater or equal to ( ). 4 A detailed derivation for (15)-(18) is provided in Appendix Section 8.1.

19 3 MODELS 15 Especially, at time t =, the sum of the initial market value of all claims is equal to the initial asset value A. Asset Equation is the most important equality relationship in our paper. It will be verified in every model we develop. The fulfillment of asset equation guarantees that our models are internally consistent. Besides, it can also be used to verify whether our close-form solution is correctly derived Definition of variables, functions and expectations For the valuation of the securities in the following models, we need to define some variables, functions and expectations. All the definition and description below is under the risk-neutral world (Q-measure). Let s define: W t = log A t (22) A From (15), we have W t = µt + σz Q t (23) Since Z Q t is a standard Brownian motion and meets the normal distribution with Z Q t N (, t), W t is a general Brownian motion and meets the normal distribution with W t N (µt, σ 2 t). Let s also define: m t = min W s (24) s t We can see that for a specific t >, m t is the minimum value of general Brownian motion until time t. We can get the CDF of m t as below. 5 ( ) m µt P Q (m t m) = Φ σ + exp (2µ m ) t σ 2 Φ ( m + µt σ t ), for m (25) As before, Φ( ) is the CDF of standard normal distribution. For a special case, we have P Q (m t ) = 1. We set a threshold K(K < A ) and define the time τ as the first time that the asset value drops and hits K. Obviously τ is one kind of stopping time. 6 τ = inf{s : A s K} (26) With maturity T, we have the following relationship. They are mutually derivable and thus equivalent. τ T min A s K A e m T K m T log K (27) s T A 5 We provide the proof in Step 3, Appendix Section Stopping time τ is an important random time in stochastic process, meaning that the decision about when to stop is based solely on information available up to time τ. A formal definition can refer to any textbook of stochastic process, such as Protter (199) or Duffie (21).

20 3 MODELS 16 Oppositely it holds: τ > T m T > log K A (28) Now we define the functions of H( ). We call it H function in the rest of the paper. It is very important because most of the expectations in our paper are calculated in some forms of H function. Let s define: H µ,σ (t, v, k, y) H(t, v, k, y) = E [ exp(vw t + km t )1 {mt y}], t, y, v, k (, ) (29) 1 { } is an indicator function that is equal to 1 if the condition within the curly brace meets and otherwise equal to. If we set y =, the indicator function degenerates to a constant 1 and disappear from the expectation operator. Note that H function is based on the general Brownian motion W t with drift µ and diffusion σ. For Brownian motion with other drift or diffusion coefficients, we need to do some transformation (usually by applying the introduced Girsanov Theorem) before using the explicit formula of H function. For the convenience of our calculation, based on H function, we also define the function of H i ( ): [ ] H i (t, v, k) = E exp(vw t + km t )1 {log K d = H <m A t log Ki } A (t, v, k, log Ki A ) H (3) K d and K i are exogenous thresholds and will be defined in the model section. 7 The explicit formula of H function (and thus H function) is available, meaning that the right-hand-side expression of (29) and (3) can be achieved explicitly. We show the result as below. 8 H µ,σ (t, v, k, y) H(t, v, k, y) = e µvt+v2 σ 2 t/2 h µ,σ (t, k, y) (31) ) (t, v, k, log Kd with ( ( 2γ y + tγ h µ,σ (t, k, y) = 2γ + kσ 2 eky+2yγ/σ2 Φ )+ σ 2γ + 2kσ2 σ 2 y (γ + kσ t/2 2 ) )t t 2γ + kσ 2 ekγt+k2 Φ σ t (32) where γ = µ + vσ 2, and Φ( ) is the CDF of standard normal distribution. The explicit formula of H i ( ) can be get from (3) and the result of (31). To save some space, we do not display it here. From (22) and (24), we have: A t = A e Wt (33) min s t A s = A e mt (34) 7 In Section 3, K d is the default threshold for the last 3 models. K s (i = s) is the depression threshold for Subordinate Debt Model. K c (i = c) is the conversion threshold for CoCo Bond Model. 8 We provide the detailed derivation in Appendix Section 8.2. A

21 3 MODELS 17 m t y min A s A e y (35) s t After some transformation, we can rewrite H( ) and H i ( ) as: ) ( v min H(t, v, k, y) = E ( A )k s At s t 1 A A { min A s A e y } (36) s t H i (t, v, k) = E ) ( v min ( A s At s t A A )k 1 {K d < min A s K i (37) } s t With the threshold K and stopping time τ defined in (26), it is easy to get the following risk-neutral probabilities: ) [ ] ) P Q (τ T ) = P (m T log KA = E 1 {mt log K } = H (T,,, log KA (38) A ) [ ] ) P Q (τ > T ) = P (m T > log KA = 1 E 1 {mt log K } = 1 H (T,,, log KA A (39) We have an important intermediate result, which will be used many times in the following derivation. We display it as below. 9 E [ ( e rτ ] 1 {τ T } = e rt K A ) θ σ H ( T, θ σ,, log K A ) where θ = µ σ + µ 2 σ 2 + 2r for all the models in our paper. (4) 3.2 Benchmark Model To begin our models, we outline the classic Black-Scholes-Merton model of debt and equity valuation as our Benchmark Model. It is the first one of our 3 supporting models. We suppose that the original owners of the firm choose a capital structure consisting of debt and equity Security Design and Assumptions For a simple start, we assume that the corporate debt is in the form of a single zerocoupon bond maturing at time T, with face value D. Another security is pure equity with no dividend. In the event that the total value A T of the firm at maturity is less than the contractual payment D due on the debt, the firm defaults, giving its future cash flows, worth A T, to debt holders. Without default the debt holders will receive the face value D. At maturity T, equity holders receive the residual after 9 The detailed derivation is provided in Appendix Section 8.3.

22 3 MODELS 18 debt holders. An important assumption is the absence of early default and no bankruptcy cost. The firm will not go bankruptcy before maturity, even when its asset value at some time before the maturity is less than the face value of the debt D. The only possible default time is at maturity T, when its asset value is less than D. Besides, we assume it does not have any bankruptcy cost when it defaults at the maturity. In order to make it clearer and easier to compare with other models, we will repeat the security setting and assumptions in Table 2 and Table 3. Table 2: Security Design of the Benchmark Model Corporate Debt Equity Face Value D Dividend div = Coupon Rate c 1 = Maturity T Default Threshold K d = D Default Time τ d = T, if A T < D Bankruptcy Cost ω = Table 3: Important Assumptions of the Benchmark Model Assumptions 1. The firm just issues 2 types of securities: corporate debt and equity. 2. No coupon payment for debt (c 1 = ) 3. No dividend payment for equity (div = ) 4. Absence of early default, i.e. τ d / (, T ) 5. No bankruptcy cost (ω = ) Asset Equation According to Section 3.1.3, the total market value of debt and equity must be the market value of the assets at time t [, T ]. In this model, both claims are securities (due to the absence of bankruptcy cost), so the firm value is equal to the asset value. A t = F t = D t + W t (41) D t and W t are the market values of corporate debt and equity at time t. Especially, at time t =, we have: A = F = D + W (42) This is what we will verify after we get the market value expressions of both claims.