Pricing contingent convertible bond with … contingent convertible bond with idiosyncratic risk...

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Pricing contingent convertible bond with idiosyncratic risk Xiaolin Wang, Zhaojun Yang 1 School of Finance and Statistics, Hunan University, Changsha 410079, China Abstract We consider the optimal capital structure of a firm, including equity, a straight bond and a con- tingent convertible bond (CCB), under an incomplete market. The cash flow of the firm is an observable arithmetic Brownian motion, which is correlated with the market portfolio return. We derive semi-closed-form solutions of the utility-based prices of the equity and the CCB, while explicit equilibrium prices of all the securities are provided. Numerical calculations show that for a more risk-averse equity holder or a higher idiosyncratic risk of the cash flow, the agent chooses a higher leverage and increases the issued amount of the straight bond. However, for a more risk-averse CCB holder, the agent chooses a lower leverage and decreases the issued amount of the CCB. We analyze the risk premium of the CCB, which includes as a part the premium of the idiosyncratic risk. If the firm’s cash flow is not too strongly negatively correlated with the market portfolio return, the CCB enhances risk-prevention incentive of the equity holder. The more risk-averse the agent is, the stronger the risk-prevention incentive will be. Our model finds that the CCB can not only decease bankruptcy risk but also can significantly increase the total firm’s value. Keywords: Contingent convertible bond, Capital structure, Utility-based pricing, Idiosyncratic risk, Risk-prevention incentive JEL: G21, G32, G33 1. Introduction Motivation. In the recent financial crisis, many financial institutions/firms have experienced fi- nancial distress, under which financial institutions were not able to raise significant new capital from the market and had to depend instead on governments to provide capital, i.e. government bailouts. However, emergency-type government bailouts can be controversial since the essence of government bailout is to give the taxpayers’ money to the troubled financial firms. This action inevitably leads to serious moral hazard problems. In order to avoid future financial crises, one of the most prominent suggested solutions is to introduce a contingent convertible bond (CCB, henceforth) into the capital structure of a firm. For this reason, after Albul et al. (2010), who first provide a quantitative research on CCBs, there are many papers discussing the design and pricing of CCBs and the problem on the new capital structure, where a CCB is issued. 1 Corresponding author. Tel: +86 731 8864 9918; Fax: +86 731 8868 4772. E-mail address: [email protected] (Z.J. Yang). Preprint submitted to SSRN February 28, 2013

Transcript of Pricing contingent convertible bond with … contingent convertible bond with idiosyncratic risk...

Pricing contingent convertible bond with idiosyncratic risk

Xiaolin Wang, Zhaojun Yang1

School of Finance and Statistics, Hunan University, Changsha 410079, China

Abstract

We consider the optimal capital structure of a firm, including equity, a straight bond and a con-tingent convertible bond (CCB), under an incomplete market. The cash flow of the firm is anobservable arithmetic Brownian motion, which is correlated with the market portfolio return. Wederive semi-closed-form solutions of the utility-based prices of the equity and the CCB, whileexplicit equilibrium prices of all the securities are provided. Numerical calculations show that fora more risk-averse equity holder or a higher idiosyncratic risk of the cash flow, the agent choosesa higher leverage and increases the issued amount of the straight bond. However, for a morerisk-averse CCB holder, the agent chooses a lower leverage and decreases the issued amount ofthe CCB. We analyze the risk premium of the CCB, which includes as a part the premium ofthe idiosyncratic risk. If the firm’s cash flow is not too strongly negatively correlated with themarket portfolio return, the CCB enhances risk-prevention incentive of the equity holder. Themore risk-averse the agent is, the stronger the risk-prevention incentive will be. Our model findsthat the CCB can not only decease bankruptcy risk but also can significantly increase the totalfirm’s value.

Keywords: Contingent convertible bond, Capital structure, Utility-based pricing, Idiosyncraticrisk, Risk-prevention incentiveJEL: G21, G32, G33

1. Introduction

Motivation. In the recent financial crisis, many financial institutions/firms have experienced fi-nancial distress, under which financial institutions were not able to raise significant new capitalfrom the market and had to depend instead on governments to provide capital, i.e. governmentbailouts. However, emergency-type government bailouts can be controversial since the essenceof government bailout is to give the taxpayers’ money to the troubled financial firms. This actioninevitably leads to serious moral hazard problems. In order to avoid future financial crises, oneof the most prominent suggested solutions is to introduce a contingent convertible bond (CCB,henceforth) into the capital structure of a firm. For this reason, after Albul et al. (2010), whofirst provide a quantitative research on CCBs, there are many papers discussing the design andpricing of CCBs and the problem on the new capital structure, where a CCB is issued.

1Corresponding author. Tel: +86 731 8864 9918; Fax: +86 731 8868 4772. E-mail address: [email protected](Z.J. Yang).

Preprint submitted to SSRN February 28, 2013

However, to the best of our knowledge, all the papers in the literature assume that agents arerisk-neutral at least towards idiosyncratic risk. It is true that this assumption can considerablysimplify computation and in particular, get explicit results for analysis. And so it is good forus to take this assumption as our first step along this research line. But without any doubt, thisassumption is of course unrealistic and it will inevitably lead to an overestimated asset price andencourage an agent to invest in a too high-risk project since it completely ignore idiosyncraticrisk, i.e. non-systematic risk.

In fact, the CCB is a hybrid bond and has a significant idiosyncratic risk just like equity2,and therefore, the risk-neutral prices of the CCB might lead to very bad conclusions. To somedegree, we think the recent financial crisis was partly derived from the risk-neutral assumption.Therefore, a theory on asset pricing and capital structure, which takes into account not only thesystem risk but also idiosyncratic risk, is very important. To achieve this goal, utility-basedindifference pricing approach is getting more and more popular.

Our work. This paper considers the design and pricing of a CCB and optimal capital structureof a firm, which can issue equity and either a straight bond or both a straight bond and a CCBinvolving idiosyncratic risk. Following Chen et al (2012), we assume the straight bond holderis fully diversified, and so we get the equilibrium value (market value) of the straight bondaccording to the equilibrium martingale pricing model used in Ingersoll (2002), Goetzmann,Ingersoll and Ross (2003) and essentially also in Merton (1976). Since the cash flow is non-tradable and exposed to a significant idiosyncratic risk, we take a consumption utility indifferencepricing approach to value the equity and CCB.

We assume that the cash flow process evolves according to an arithmetic Brownian mo-tion3. We assume that the agent has access to one risk-free asset and market portfolio to smoothhis consumption. Based on consumption utility-based indifference pricing approach, we derivesemi-closed-form solutions of the implied values of the equity and CCB with both exogenouslyspecified conversion threshold and default threshold. We provide numerical sensitivity analysisby finite difference method with regard to the impact of the risk aversion index and the volatilityof the cash flow on the implied values of the equity, CCB and capital structure. We compare twocapital structures: One includes a CCB and the other does not.

For a better analysis for the risk sharing of the CCB, we discuss the case where the CCBholder has different risk aversion indexes while the risk aversion index of the equity holder keepsunchanged. We also analyze the impact of changes in idiosyncratic volatility. Numerical cal-culations show that the more risk-averse the equity holder is or the higher the idiosyncratic riskis, the higher the leverage and the greater the face value of the straight bond will be, due tothe significant diversification benefits of straight bond. But for a more risk averse CCB holder,the agent chooses a lower leverage and less face value of the CCB, ceteris paribus, due to lessdiversification benefits of the CCB. In addition, the more risk-averse the agent is or the higherthe idiosyncratic risk is, the stronger the incentives for conversion and default will be. However,a more risk-averse CCB holder induces a weaker incentive of the agent to convert and default,ceteris paribus. This is different from the case where we do not take into account idiosyncraticrisk, i.e. the security prices are given by equilibrium pricing method.

2The CCB could in principle be issued by any firm. If the firm is a non-financial firm, the idiosyncratic risk may geteven more serious.

3It follows Miao and Wang (2007) and Ammann and Genser (2005)

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In an incomplete market, the risk premium for the CCB is determined not only by systematicrisk but also by idiosyncratic risk. We derive the idiosyncratic risk premium formula whichdepends on the risk aversion index and the conditional idiosyncratic variance of the impliedvalue of the CCB.

In order to understand whether the new capital structure will enhance risk-taking incentiveof the equity holder. We assume that the equity holder has an option to increase the riskiness ofthe firm’s revenue by choosing different technologies. We find that only if the firm’s cash flow isstrongly negatively correlated with the market portfolio return, the equity holder has an incentiveto invest in a high-risk project. The more risk-averse the agent is, the weaker the incentive willbe.

We analyze the capital structure by taking into account the bankruptcy costs and tax shields.The research results present that the tax will induce less diversification benefits which tends to bemore obvious for a greater risk-averse agent. Comparing with the capital structure that does notinclude a CCB, we find that the CCB can not only decease bankruptcy risk but also significantlyincrease the total firm’s value.

Literature review. Contingent convertible bonds (CCBs), also known as contingent convertibles,CoCo notes or Cocos, are specific type of corporate securities similar to traditional convertiblebonds. But in contrast to traditional convertible bonds, of which the conversion threshold isfreely decided by the bond holder, the possibility of converting a CCB to equity is contingenton a specified event, that can be different depending on the particular need of an issuer. In casethat the issuer of a CCB is a bank, the contingent event might, for example, be related to thelevel of Tier 1 capital falling under a given threshold, as we discuss in the following text. Thereare at least two merits if firms, say banks, issue CCBs: First, the banks are relieved of servicingtheir debt obligations in bad states of the world, when costly financial distress may result ifcreditors continue to demand their interest and principal payments; Second, the equity buffercould increase, further bolstering the bank’s capital as a result of the forced conversion of CCBs.Clearly, if a firm takes a CCB as a debt financing instrument, the bankruptcy risk of the firm willdecrease considerably. Therefore, for a firm like a bank, which is too important to fail, it is verybeneficial to issue CCBs, as argued by many researchers after the recent global financial crisis.

For this reason, CCBs and corresponding capital structure problems have recently been at-tracting increasing research interest. Flannery (2005), who first suggests the idea of CCBs inthe context of financial institutions, proposes a new financial instrument, ”reverse convertibledebentures”, that forestalls financial distress without distorting bank shareholders’ risk-takingincentives, assuming a conversion threshold based on the bank’s equity price. Flannery (2009)gets that a CCB would convert into common stock if the issuer’s capital ratio fell below somecritical, pre-specified value. Contingent convertibles can decrease default probability of a firm.Then CCB has been mentioned favorably in policy recommendations by many researchers, sayDuffie (2009). Sundaresan and Wang (2010) provide the condition that the conversion ratio mustsatisfy in order for a unique equilibrium to exist and presents a design that mitigates the problemof multiple equilibria. McDonald (2011) proposes a form of contingent capital for financial in-stitutions that converts from debt to equity if two conditions are met: the firm’s stock price is ator below a trigger value and the value of a financial institutions index is also at or below a triggervalue. Pennacchi (2010) supposes the return on the bank’s assets follows a jump-diffusion pro-cess, and default-free interest rates are stochastic and studies the equilibrium pricing of the bank’sdeposits, contingent capital, and shareholders’ equity for various parameter values characteriz-ing the bank’s risk and the contractual terms of its contingent capital. Metaler and Reesor (2011)

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attempt to identify fundamental properties of CCBs in the context of the seminal Merton modelon capital structure. The following papers make a detailed discussion on the optimal capitalstructure contained CCBs. Albul et al. (2010) first provide a formal model of CCB, and analyticpropositions concerning CCB attributes. Albul et al. (2010) also develops implications for struc-turing CCBs to maximize their general benefits for corporations and their specific benefits forprudential bank regulation. Barucci and Del Viva (2012a) study the optimal capital structure of acompany issuing perpetual CCB, equity and straight bond with a two-period model. Barucci andDel Viva (2012b) analyze the optimal capital structure of a bank issuing countercyclical CCB,i.e. notes to be converted in common shares in case of a bad state for the economy.

Up until now, to the best of our knowledge, all the theories about the CCB uniformly sup-pose that the investors/agents are risk-neutral at least towards idiosyncratic risk. This assumptiongreatly simplifies the complexity of the problem. As a starting point of studying CCB, this as-sumption is desirable and essential. Especially, by risk-neutral pricing method we can recover theonly reasonable no-arbitrage prices in a complete market. However, in real market environment,the credit risks and business risks exposed to a firm, generally cannot be completely eliminated.The investor of the firm will take on some unhedgeable business risk (idiosyncratic risk) eventhough the firm is publicly listed4.

With regard to the problems we discuss here, all the risks generated from the cash flow of afirm are clearly not able to be hedged away. Especially, the equity holder and CCB holder bearmore risk of the cash flow or unhedgeable risk and the straight bond holder is minimally affected.According to Heaton and Lucas (2004) and Chen et al (2012), we assume that the straight bondis priced by fully diversified lenders. That is, the straight bond provides significant diversifi-cation benefits. For these reasons, in this paper we take consumption utility-based indifferencepricing approach on equity and the CCB and pricing straight bond using the equilibrium pricingapproach.

Unlike the risk-neutral pricing, the consumption utility-based indifference price is a nonlinearprice. It is a dynamic extension of the static concept of certainty equivalence from economics.Moreover, implicit in the calculation of the consumption utility-based indifference price is theconstruction of investment strategies. Recently, this method has been applied in some fields. Forexample, Henderson and Hobson (2002), Miao and Wang (2007), Henderson (2007), Ewald andYang (2008), Yang and Yang (2012), Chen et al (2012) among others, study real options problemsunder incomplete market. Especially, Chen et al (2012) characterize the optimal capital structureof the entrepreneurial firm where risky debt provides significant diversification benefits, whilethe firm only issues risky debt in addition to equity for a risk-averse entrepreneur. Leung, Sircarand Zariphopoulou (2008) and Liang and Jiang (2012) derive the price of a defaultable corporatebond by utility-based indifference pricing method.

In the literature, most papers assume that the cash flow of a firm follows a geometric Brow-nian motion. It means that the cash flow must be positive all the time. But actually, it is not, saythe cash flow of a temporally distressed firm. So, to describe a cash flow, an arithmetic Brow-nian motion appears more natural than a geometric Brownian motion. For example, Miao andWang (2007) assume that the cash flow of a project is governed by an arithmetic Brownian mo-tion. Ammann and Genser (2005) propose an EBIT-based capital structure model and the EBITfollows an arithmetic Brownian motion.

4Chen et al (2012) says that the lack of investor protection in some countries and (concentrated) ownership structureentrench under-diversified controlling shareholders and managers

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By now, Wang and Yang (2012) is most closely connected with our study. However, incontrast to this paper, Wang and Yang (2012) neither consider the bankruptcy cost and tax shieldnor take into account the market portfolio in order for partially hedging the business risk of thefirm.

The remainder of the paper proceeds as follows. Section 2 sets up the model and showsthe exogenous bankruptcy condition and conversion condition decided by the level of Tier 1capital falling under a given threshold. Section 3 presents the implied values of the contingentconvertible bond and equity. Section 4 provides the equilibrium prices of the securities. Sec-tion 5 discusses optimal capital structure. Section 6 presents numerical simulations. Section 7concludes. Proofs of the theorems and propositions are relegated to appendices.

2. Model setup

Consider a firm that has invested in projects, of which the total cash flow δ is observable andgoverned by the following arithmetic Brownian motion:

dδt = µdt + ρσdZ1t +

√1 − ρ2σdZ2

t , δ0 given, (1)

where µ is the rate of growth, σ is the volatility (standard deviation) and Z is a standard Brownianmotion on a complete probability space (Ω,F ,P). We denote by F ≡ Ft : t ≥ 0 the P-augmentation of the filtration σ(Zs; 0 ≤ s ≤ t) generated by process Z.

The firm can issue equity, a straight bond and a CCB. Both bonds are consol type, meaningthey are annuities with infinitive maturity. We denote by L1 the face value of the straight bondand by u1 the coupon rate. That is, the straight bond pays coupon b1 ≡ u1L1 per year, contin-ually in time, until default. The face value and coupon rate of the CCB are denoted by L2 andu2 respectively, i.e. the CCB pays coupon b2 ≡ u2L2 per year, continually in time, until it isconverted into equity. At conversion time τ2, the CCB is fully converted into equity of the samefirm according to the terms specified when the CCB was issued.

We assume the firm has taken a highly concentrated ownership structure and so the firminvolves a large idiosyncratic risk. The investors (shareholders and CCB holders) are all riskaverse. But, similar to Chen et al (2012), the straight bond investors are fully diversified. Thedefault threshold and the conversion rule are both pre-specified.

In addition, the investors have standard liquid financial opportunities which involves a risk-free asset and a risky market portfolio. Let Mt : t ≥ 0 denote the value of the market portfolio,which is governed by the following equation:

dMt/Mt = µedt + σedZ1t , M0 given, (2)

where µe and σe are the expected return and volatility of the market. So, the parameter ρ in Eq.(1) represents the correlation coefficient between the firm’s cash flow and the return of the marketportfolio. The parameters ρσ and

√1 − ρ2σ are the systematic and idiosyncratic volatility of the

firm’s cash flow. Thus, if |ρ| < 1, the agent faces undiversifiable idiosyncratic risk.Denote the wealth process of an agent by W = (Wt)t≥0, which is F-adapted. Let C be the

space of Ft : t ≥ 0-progressively measurable process C, taking value on [0,∞), such that∫ t0 |Cs|ds < ∞ for any t ≥ 0. In this paper, Cs represents the consumption rate selected by the

agent at time s. We call a consumption process C is admissible, if C ∈ C. Let θt denote the

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amount allocated to the market portfolio at time t, and let Θ denote the set of F-adapted processθ which satisfies the integrability condition

∫ s0 σ

2eθ

2t dt < ∞ (a.s.) for each s > 0.

Following many researchers, say Miao and Wang (2007) among others, we consider theCARA utility, i.e. exponential utility given by

U(c) = − exp(−γc)/γ, c ∈ ℜ, (3)

where γ > 0 is the absolution risk aversion parameter.Given the risk-free rate r and the market portfolio defined by Eq. (2) with sharpe ratio

η = (µe − r)/σe, we get the equilibrium stochastic discount factor5 which corresponds to therisk-neutral probability measure Q, see Duffie (2001). We can rewrite the cash flow process δ inEq. (1) as follows:

dδt = (µ − ρση)dt + ρσdZQt +

√1 − ρ2σdZ2

t , (4)

where µ − ρση is the risk-adjust drift, and ZQt is a standard Brownian motion under Q satisfying

dZQt = dZ1

t + ηdt.We get the equilibrium value or market value of an unlevered firm under the risk-neutral

probability measure Q:A0(δt) = EQ

t [∫ ∞

t e−r(s−t)δsds]=

δtr +

µ−ρσηr2 .

(5)

In this paper, we assume that the liquidation value of the firm is equal to a fraction α of themarket value of unlevered firm after tax (1 − τe)A0(δt), where τe is the tax rate. The remainingfraction (1−α) is lost due to bankruptcy costs. And the firm must be liquidated upon first passageof the market value A0 of the unlevered firm given by Eq. (5) to some pre-specified thresholdπL1, under a protective covenant for the holder of the straight bond. To this end, we introducethe hitting time

τ1 ≡ inft ≥ 0 : A0(δt) ≤ πL1, (6)

or equivalently

τ1 = inft ≥ 0 : δt ≤ x1 ≡ πrL1 −1r

(µ − ρση), (7)

which is the defaulting time while x1 is the default threshold.And we assume the conversion threshold is determined by the regulator and the conversion

takes place if and only if the firm’s Tier 1 capital ratio falls below level d, here d ∈ (0, 1), say4%, is specified by the regulator in advance. Naturally, the Tier 1 capital ratio in our model isdefined via the following fraction

ρ(δt) =A0(δt) − L1 − L2

A0(δt), (8)

5To determine a linear pricing rule, we must specify a stochastic discount factor. However, the market we considerhere is incomplete, i.e. there are infinite stochastic discount factors. To fix one, we can solve a single-agent optimizationproblem and take the marginal utility of the agent as the special stochastic discount factor, see Duffie (2001). If the agentselected is the representative agent, then we recover the equilibrium stochastic discount factor used in Ingersoll (2002),Goetzmann, Ingersoll and Ross (2003) and essentially also in Merton (1976).

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where A0(δt), given by Eq. (5), is the current market value of the firm. For this reason, theconversion of the CCB occurs at the time

τ2 ≡ inft ≥ 0 : ρ(δt) ≤ d= inft ≥ 0 : A0(δt) ≤ K ≡ 1

1−d (L1 + L2)= inft ≥ 0 : δt ≤ x2,

(9)

where x2 =r

1−d (L1 + L2) − 1r (µ − ρση).

Last, the ownership stake λ, i.e. the ratio of the equity allocated to the CCB holder at theconversion time is defined according to the market value of total equity, which is given by

Etotal(x, b1) = (1 − τe)EQt [

∫ ∞t e−r(s−t)(δs − b1)ds]

= (1 − τe)( xr +

µ−ρσηr2 − b1

r (1 − eκ(x−x1))−( x1

r +µ−ρση

r2 )eκ(x−x1)), x1 ≤ x ≤ x2.

(10)

For easy operation, the ownership stake λ is then defined by

λ ≡ min

1,L2

Etotal(x2, b1)

. (11)

In the following text, we discuss the problem of optimal capital structure, i.e. how much thecoupons b1 and b2 should be to maximize the firm’s value, including the pricing of the equity,straight bond and CCB. Since the market is incomplete and equity holders and CCB holdersare both under-diversified, we value the equity and CCB by the consumption utility indifferencepricing method. Taking into account that straight bond holders are much more diversified, inthe same way with (5), we price the straight bond using risk-neutral technique under the risk-neutral probability measure Q. We take as the target function the total value of the firm, i.e.the sum of equity value, straight bond value and CCB value. We maximize the target functionover all admissible coupons b1 and b2 for the given exogenous conversion terms and bankruptcyconditions.

3. Utility-based pricing of the contingent convertible bond and equity

3.1. PreliminariesIn order to derive the consumption utility indifference prices of the CCB and equity in the new

capital structure, we first consider the following standard investment and consumption problem.An agent is characterized by his initial wealth W0 and his preference U(·) defined by (3). He

can invest in a risk-free asset and a risky market portfolio to smooth his consumption. The agentchooses a consumption process C ∈ C and a portfolio θ ∈ Θ in order to maximize his lifetimetime-additive consumption utility:

V0(w) = supc∈C

∫ ∞

0exp (−βs) U(Cs)ds, C ∈ C, (12)

subject todWs = (θs(µe − r) + rWs −Cs)ds + θsσedZ1

S , s ≥ t, W0 = w, (13)

where β > 0 is a time-discount rate. Similar to Merton (1971), we easily obtain by dynamicprograming that

V0(w) = − 1γr

exp(1 − β/r − γr

(w +

η2

2r2γ

)). (14)

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3.2. Implied value of the contingent convertible bondIn order to price a claim, the first thing we must do is to make clear what the cash flow of the

claim is. We note that the cash flow of the CCB has three different expressions corresponding todifferent periods: The first is the coupon b2 if 0 ≤ t ≤ τ2, i.e. the conversion has not taken place,where τ2 is the conversion time of the CCB and given by Eq. (9); The second is λ(1−τe)(δt −b1)if τ2 ≤ t < τ1, where λ is a constant ratio, given at (11), of the equity allocated to the CCB holderafter conversion; The last is 0 if t ≥ τ1, i.e. the firm has been liquidated. For this reason, thewealth of the CCB holder evolves in the following way:

dWt =

(θt(µe − r) + rWt + b2 −Ct)dt + θtσedZ1

t , 0 ≤ t ≤ τ2;[θt(µe − r) + rWt + λ(1 − τe)(δt − b1) −Ct]dt+θtσedZ1

t , τ2 ≤ t < τ1;(θt(µe − r) + rWt −Ct)dt + θtσedZ1

t , t ≥ τ1.

(15)

The CCB holder seeks to choose a consumption process C ∈ C and a portfolio allocate rule θ ∈ Θso as to maximize the expected lifetime time-additive consumption utility:

G(w, x) = sup(c,θ)∈C×Θ

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (16)

subject to wealth accumulation equation (15). We solve this optimization problem backward bydynamic programming.

First, after the firm is liquidated, the optimization problem is clearly a standard investmentand consumption problem (12) and (13), so the optimal solution is given by (14).

Second, if the conversion has taken place but the firm is not liquidated, we denote by G1(w, x)the value function of the problem (16) subject to (15). The Hamilton-Jacobi-Bellman (HJB)equation has the form

supc≥0,θ(rw + λ(1 − τe)(x − b1) − c)G1

w + U(c) + θ(µe − r)G1w

+θσeσρG1wx +

θ2

2 σ2eG1

ww + µG1x +

σ2

2 G1xx − βG1 = 0,

(17)

where and throughout the text, the subscript of G1 represents the differentiation with respectto that variable. In order to find a solution G1(w, x) of (17), we must specify the following twoconditions. The first is the value matching condition, see e.g. Krylov (1980) or Dixit and Pindyck(1994), shown below

G1(w, x1) = V0(w) = − 1γr

exp(1 − β/r − γr

(w +

η2

2r2γ

)), (18)

where x1 is the default threshold of the firm.In addition, if δ → +∞, the bankruptcy will not happen, i.e. τ1 → +∞. Consequently, the

optimization problem defined by (16) and (15) is just the same as the following optimizationproblem

G2(w, x) = supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (19)

subject todWt = [θt(µe − r) + rWt + λ(1 − τe)(δt − b1) −Ct]dt + θtσedZ1

t ,τ2 ≤ t < +∞. (20)

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Therefore, let δ→ +∞, then we get another boundary condition

limx→+∞

G1(w, x) = G2(w, x). (21)

We now solve G1(w, x) with two boundary conditions (18) and (21). At first, thanks to (17), theoptimal consumption rate is evidently given by

U′(c) = G1w(w, x), (22)

which, as we expect, says that at the optimal solutions, the marginal utility of current consump-tion is equal to the marginal utility of wealth increased if we consume less at present. And theportfolio is given by

θ =−G1

w

G1ww

η

σe+−G1

wx

G1ww

σρ

σe. (23)

According to consumption utility indifference pricing principle, we define the consumption util-ity indifference price, i.e. the implied value, denoted by CB1(x, b1), of the equity as follows:

G1(w, x) = V0(w +CB1(x, b1)), (24)

where x is the current rate of the cash flow of the firm. If the current rate x is large enough, itfollows from (21), that

G2(w, x) = V0(w +CB1(x, b1)). (25)

Substituting (22) and (24) into (17) and noting that similar to Miao and Wang (2007), we areable to get an explicit solution to the optimization problem defined by (19) and (20), we obtainthe following proposition:

Proposition 3.1. If the CCB has been converted into 100α percent of the equity and the defaultof the firm has not taken place, then the consumption utility indifference price CB1(x, b1) of theCCB is given by the following ordinary differential equation (ODE):

λ(1 − τe)(x − b1) + (µ − ρση)CB1x +

σ2

2 CB1xx

−σ2

2 (1 − ρ2)rγ(CB1x)2 = rCB1,

(26)

where the independent variable and parameter of the function CB1(·, ·) are suppressed, subjectto the first boundary condition

CB1(x1, b1) = 0, (27)

where x1 is given by (7), and the second boundary condition

limx→+∞

CB1(x, b1) = λ(1 − τe) x−b1r +

1r2 [µ − ρση

− 12σ

2(1 − ρ2)γλ(1 − τe)].(28)

The optimal consumption rate is given by c∗t =β−rγr + r[Wt +

η2

2r2γ+CB1(δt, b1)], τ2 ≤ t < τ1;

c∗t =β−rγr + r(Wt +

η2

2r2γ), t ≥ τ1.

(29)

And the optimal portfolio rule is given by θ∗t = ηγσe

1r −

ρσσe

CB1x, τ2 ≤ t < τ1;

θ∗t =ηγσe

1r .

(30)

9

Last, if the conversion of the CCB has not taken place, the optimization problem is given by(16) subject to (15). In the same way, the HJB equation has the form

supc≥0,θ(rw + b2 − c)Gw + U(c) + θ(µe − r)Gw + θσeσρGwx

+ θ2

2 σ2eGww + µGx +

σ2

2 Gxx − βG = 0.(31)

In order to find a solution G(w, x) of (31), which is just the value function defined by (16),we must specify the following two conditions as before. On the one hand, the following valuematching condition must hold:

G(w, x2) = G1(w, x2), (32)

where x2 represents conversion threshold of the CCB. On the other hand, if δ → +∞, the con-version of the CCB will not happen. So the solution to the optimization problem (16) subject toEq. (15) must be the same solution to the following optimization problem

G3(w, x) = supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (33)

subject todWt = (θt(µe − r) + rWt + b2 −Ct)dt + θtσedZ1

t , t ≥ 0. (34)

So, when δ→ +∞, we get another boundary condition

limx→+∞

G(w, x) = G3(w, x). (35)

We now solve (31) with two boundary conditions (32) and (35). At first, thanks to (31), theoptimal consumption rate and portfolio rule are respectively given by

U′(c) = Gw(w, x) and θ =−Gw

Gww

η

σe+−Gwx

Gww

σρ

σe. (36)

The consumption utility indifference price CB(x, b1, b2) (i.e. implied value) of the CCB is givenby

G(w, x) = V0(w +CB(x, b1, b2)), (37)

and if the current rate x of the cash flow is large enough, we have

G3(w, x) = V0(w +CB(x, b1, b2)). (38)

Following a similar derivation with the Proposition 3.1, we obtain the following theorem:

Theorem 3.2. If the CCB is not converted, the consumption utility indifference price CB(x, b1, b2)of the CCB is given by the following ODE:

b2 + (µ − ρση)CBx +σ2

2 CBxx − σ2

2 (1 − ρ2)rγ(CBx)2 = rCB, (39)

subject to the following boundary conditions:

CB(x2, b1, b2) = CB1(x2, b1), and limx→+∞

CB(x, b1, b2) =b2

r, (40)

10

where the conversion threshold x2 is the solution of the equation (9) and the function CB1(·, b1)is given by Proposition 3.1. The optimal consumption rate is given by

c∗t =β−rγr + r[Wt +

η2

2r2γ+CB(δt, b1, b2)], 0 ≤ t < τ2;

c∗t =β−rγr + r[Wt +

η2

2r2γ+CB1(δt, b1)], τ2 ≤ t < τ1;

c∗t =β−rγr + r(Wt +

η2

2r2γ), t ≥ τ1.

(41)

And the corresponding optimal portfolio rule is given byθ∗t =

ηγσe

1r −

ρσσe

CBx, 0 ≤ t < τ2;θ∗t =

ηγσe

1r −

ρσσe

CB1x, τ2 ≤ t < τ1;

θ∗t =ηγσe

1r .

(42)

We next discuss the implications of this theorem in short. Generally speaking, the value ofa CCB may increase with the volatility σ of the cash flow in a complete market, since the CCBholder will harvest the growth of the cash flow but the loss suffered by the holder is limitedthanks to the bankruptcy protection. In fact, if γ → 0, which corresponds to the results fromthe equilibrium pricing approach, (39) of this theorem just tells us the same story. However, ifthe equity holder is risk averse, i.e. γ > 0, (39) says that a larger volatility may result in a lessvalue of the CCB. More specifically, if the holder is risk averse enough, i.e. γ is very large, thenthe value of the CCB will decrease with a growth of the volatility of the cash flow. Intuitively,this is because a larger volatility means a higher risk, so the value of the claim may be less for arisk-averse agent.

In particular, the last term σ2(1 − ρ2)rγ(CBx)2/2 on the left side of Eq.(39), , reflects theeffect of the risk attitude of the CCB holder and the idiosyncratic risk volatility on the impliedvalue of the CCB. It says clearly that under this incomplete market, the implied value of theCCB decreases with a growth of the risk-averse index (γ) or the idiosyncratic risk volatility(√

1 − ρ2σ). If the risk-averse index γ approaches 0 or the idiosyncratic risk is fully diversified,we can derive the explicit implied value given by (70). In this case, only the systematic riskdemands a risk premium. And we find that the second term on the left side of Eq. (39), (µ −ρση)CBx , presents the risk-adjusted expected growth rate of the cash flow through the systematicvolatility of the cash flow and the Sharpe ratio of market portfolio. This risk-adjustment can beobtained from the CAPM model. In an incomplete market, if the idiosyncratic risk is under-diversifiable, the last term σ2(1 − ρ2)rγ(CBx)2/2 on the left side of Eq. (39) plays an importantrole on the risk-adjustment due to idiosyncratic risk. So, this term implies idiosyncratic riskpremium.

The first equation of (40) says that the implied value of the CCB will not jump before and afterconversion. The second equation of (40) shows that the implied value of the CCB will convergeto the value of default-free bond with the same coupon if the conversion of the CCB will surelynever happen. (41) represents that optimal consumption just consists of a fixed consumption ,an excess return of the market portfolio investment and a part of the equivalent risk-free incomeof the claimant’ total wealth including the indifference price of the future cash flow. Constantβ−rγr is the fixed consumption, which says that the greater the time-discount rate, the more the

current consumption. (42) suggests that hedging demand increases with the absolute values of thecorrelation coefficient (|ρ|) and the marginal value of the CCB. After the default has taken place,the third equalities of (41) and (42) are the standard Merton-style consumption and portfolio rulerespectively.

11

3.3. Implied value of the equityIn this subsection, we compute the price (implied value) of the equity such that there is no

difference for an agent to hold the equity at the cost of the price or do nothing while the agentcan invest in a risk-free asset and a risky market portfolio to smooth his consumption in order tomaximize his expected total consumption utility.

For this aim, we note that the cash flow of the equity also has three different expressionscorresponding to different periods: The first is the residual cash flow (1 − τe)(δt − b1 − b2) if0 ≤ t ≤ τ2, i.e. the conversion has not taken place; The second is (1 − λ)(1 − τe)(δt − b1) ifτ2 ≤ t < τ1, where 1− λ represents the equity allocated to the equity holder after the conversion;The last is 0 if t ≥ τ1, i.e. the firm has been liquidated. So, his wealth dynamics is given by

dWt =

[θt(µe − r) + rWt + (1 − τe)(δt − b1 − b2) −Ct]dt+θtσedZ1

t , 0 ≤ t < τ2;[θt(µe − r) + rWt + (1 − τe)(1 − λ)(δt − b1) −Ct]dt+θtσedZ1

t , τ2 ≤ t < τ1;(θt(µe − r) + rWt −Ct)dt + θtσedZ1

t , t ≥ τ1.

(43)

We solve the following optimization problem

J(w, x) = supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (44)

subject to (43) backward by dynamic programming as before.First, after the firm goes bankrupt, the optimization problem is a standard investment con-

sumption problem defined by (12) and (13), and so the value function is given by (14).Second, after the CCB is converted into the equity and before the firm goes bankrupt, we

should solve the following optimization problem

J1(w, x) = supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (45)

subject to

dWt =

[θt(µe − r) + rWt + (1 − τe)(1 − λ)(δt − b1) −Ct]dt+θtσedZ1

t , τ2 ≤ t < τ1;(θt(µe − r) + rWt −Ct)dt + θtσedZ1

t , t ≥ τ1.(46)

The HJB equation has the form

supc≥0(rw + (1 − τe)(1 − λ)(x − b1) − c)J1

w + U(c) + θ(µe − r)J1w

+θσeσρJ1wx +

θ2

2 σ2e J1

ww + µJ1x +

σ2

2 J1xx − βJ1 = 0,

(47)

with the following boundary conditions:

J1(w, x1) = V0(w) and limx→+∞

J1(w, x) = J2(w, x), (48)

where J2(w, x) is an explicit solution, see Miao and Wang (2007) as before, to the followingoptimization problem

J2(w, x) = supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (49)

12

subject to

dWt = [θt(µe − r) + rWt + (1 − τe)(1 − λ)(δt − b1) −Ct]dt + θtσedZ1t , t ≥ 0. (50)

We now solve (47) with boundary conditions (48). At first, thanks to (3), the optimal consump-tion and portfolio rules are evidently given by

U′(c) = J1w(w, x) and θ =

−J1w

J1ww

η

σe+−J1

wx

J1ww

σρ

σe. (51)

Therefore, if the current rate of the cash flow is x, the consumption utility indifference price orimplied value E1(x, b1) of the equity is given by

J1(w, x) = V0(w + E1(x, b1)). (52)

Substituting (52) and (14) into (47), it is not difficult for us to obtain the following proposition:

Proposition 3.3. If the CCB is converted into the equity but the firm does not go bankrupt, theconsumption utility indifference price of the original equity, i.e. 100(1 − λ) percent of the equity,is a solution of the following ODE:

(1 − τe)(1 − λ)(x − b1) + (µ − ρση)E1x +

σ2

2 E1xx

−σ2

2 (1 − ρ2)rγ(E1x)2 = rE1(x, b1),

(53)

subject to the following boundary conditions

E1(x1, b1) = 0, (54)

andlim

x→+∞E1(x, b1) = (1 − τe)(1 − λ)[ x−b1

r +1r2 (µ − ρση

− 12σ

2(1 − ρ2)(1 − τe)γ(1 − λ))],(55)

where the default threshold x1 is given by (7). The optimal consumption rate is given by c∗t =β−rγr + r[Wt +

η2

2r2γ+ E1(δt, b1)], τ2 ≤ t < τ1;

c∗t =β−rγr + r(Wt +

η2

2r2γ), t ≥ τ1.

(56)

And the corresponding optimal portfolio rule is given by θ∗t = ηγσe

1r −

ρσσe

E1x , τ2 ≤ t < τ1;

θ∗t =ηγσe

1r .

(57)

Last, if the CCB is not converted yet, in the same way, the HJB equation has the form

supc≥0(rw + (1 − τe)(x − b1 − b2) − c)Jw + U(c) + θ(µe − r)Jw

+θσeσρJwx +θ2

2 σ2e Jww + µJx +

σ2

2 Jxx − βJ = 0,(58)

with the following boundary conditions:

J(w, x2) = J1(w, x2) and limx→+∞

J(w, x) = J3(w, x), (59)13

where J3(w, x) is given by an explicit solution of the following simple optimization problem:

J3(w, x) ≡ supc∈C

E[∫ ∞

0exp (−βs) U(Cs)ds |W0 = w, δ0 = x

], (60)

subject to

dWt = [θt(µe − r) + rWt + (1 − τe)(x − b1 − b2) −Ct]dt + θtσedZ1t , t ≥ 0. (61)

We therefore define the consumption utility indifference price E(x, b1, b2) of the equity, whichsatisfies

J(w, x) = V0(w + E(x, b1, b2)). (62)

Almost in the same way with the derivation of Proposition 3.3 and taking all the items together,we obtain the following theorem:

Theorem 3.4. If the CCB is not converted, the consumption utility indifference price E(x, b1, b2)is a solution of the following ODE:

(1 − τe)(x − b1 − b2) + (µ − ρση)Ex +σ2

2 Exx

−σ2

2 (1 − ρ2)rγ(Ex)2 = rE,(63)

subject to the boundary conditions:

E(x2, b1, b2) = E1(x2, b1) (64)

andlim

x→+∞E(x, b1, b2) = (1 − τe)[ x−b1−b2

r + 1r2 (µ − ρση

− 12σ

2(1 − ρ2)(1 − τe)γ)],(65)

where x2 is the conversion threshold, i.e. the solution of (7), and the function E1(·, b1) is givenby Proposition 3.3. The optimal consumption rate is given by

c∗t =β−rγr + r[Wt +

η2

2r2γ+ E(δt, b1, b2)], 0 ≤ t < τ2;

c∗t =β−rγr + r[Wt +

η2

2r2γ+ E1(δt, b1)], τ2 ≤ t < τ1;

c∗t =β−rγr + r(Wt +

η2

2r2γ), t ≥ τ1.

(66)

And the corresponding optimal portfolio rule is given byθ∗t =

ηγσe

1r −

ρσσe

Ex, 0 ≤ t < τ2;θ∗t =

ηγσe

1r −

ρσσe

E1x , τ2 ≤ t < τ1;

θ∗t =ηγσe

1r .

(67)

This theorem almost tells us the same story with Theorem 3.2, which explains how the riskattitude to impact on the price of the equity.

Remark 1. Suppose that the original owner of the firm chooses a capital structure consistingof pure equity and the owner can also invest in the risk-free asset and the market portfolio tosmooth his consumption, then thanks to Miao and Wang (2007), the implied value A after tax,

14

i.e. the consumption utility indifference price, of the firm, if current rate of the cash flow is x, isgiven by

A(x) = (1 − τe)( x

r+µ − ρση

r2

)− (1 − τe)2 (1 − ρ2)σ2γ

2r2 . (68)

The consumption utility indifference A(·) is a risk-adjust price, and the risk-adjustment on id-iosyncratic risk decreases in direct proportion to (1 − τe)2 after tax while other terms decreasesin direct proportion to (1 − τe). Specifically, as γ → 0, we get from (68) the equilibrium mar-tingale price or market value of the pure equity firm after tax, i.e. (1 − τe)A0(x), where A0(·) isgiven by Eq. (5).

4. Equilibrium prices of the corporate securities

Taking into account that the straight bond holder will be well diversified, we naturally take theprice of the straight bond as the equilibrium price, which is given by the risk-neutral techniqueunder the risk-neutral probability measure Q, similar to (5). It is well known that, such marketvalue includes only the systematic risk premium since the idiosyncratic risk is considered to befully diversified. Furthermore, in order to make a comparison, we also provide the equilibriumprices of the equity and CCB in this section.

We first solve the equilibrium prices of the straight bond and then we present the equilibriumprices of the CCB and equity which are the special case of Theorems 3.2 and 3.4 and Propositions3.1 and 3.3.

4.1. Equilibrium price of the straight bond

Since the straight bond is held by well diversified investors in a competitive capital market,we price the straight bond by computing the expectation of the sum of coupons from the bonddiscounted with the risk-free interest rate under the risk-neutral probability measure Q. There-fore, we easily derive the following theorem:

Theorem 4.1. Suppose the firm does not go bankrupt, then the equilibrium price of straight bondis given by:

B(x, b1) = b1r

(1 − eκ(x−x1)

)+ (1 − α)(1 − τe)

(x1r +

µ−ρσηr2

)eκ(x−x1), (69)

where κ = − (µ−ρση)+√

(µ−ρση)2+2rσ2

σ2 and x1 is the default threshold and given by (7).

Remark 2. The proof of Theorem 4.1 is simple since the equilibrium price of the straight bond ismerely the equilibrium price of the coupon payments obtained by the straight bond holder beforebankruptcy, plus a fraction 1 − α of the market value of unlevered firm at bankruptcy.

4.2. The equilibrium prices of the contingent convertible bond and equity

The equilibrium prices of contingent convertible bond and equity which are the special casesof Theorems 3.2 and 3.4 respectively when the agent’s risk aversion γ approaches 0. Hence, weget the following equilibrium prices immediately:

CB(x, b1, b2) = b2r (1 − eκ(x−x2)) + λEtotal(x2, b1)eκ(x−x2), x > x2, (70)

15

E(x, b1, b2) = (1 − τe)[ xr +

µ−ρσηr2 − b1

r (1 − eκ(x−x2)) − ( x2r

+µ−ρση

r2 )eκ(x−x2) − b2r (1 − eκ(x−x2))]

+(1 − λ)Etotal(x2, b1)eκ(x−x2), x > x2,

(71)

where Etotal(·, b1), i.e. the market value of all the equity after conversion but before bankruptcy,is given by Eq. (10), the default threshold x1 and the converting threshold x2 respectively satisfiesEq. (7) and Eq. (9), and the ownership stake λ is given by (11). Clearly, the special cases onlyinvolve systematic risk premium and the idiosyncratic risk premium is ignored.

Moreover, in the same way, after the conversion of the CCB but before bankruptcy, we getthe following equilibrium prices of the claims that belong to the original CCB holder and theoriginal equity holder respectively

CB1(x, b1) = λEtotal(x, b1) x1 ≤ x < x2, (72)

E1(x, b1) = (1 − λ)Etotal(x, b1), x1 ≤ x < x2, (73)

which are the special cases of Propositions 3.1 and 3.3 respectively as the agent’s risk aversioncoefficient γ approaches 0. Then the total firm value is given by:

E(x, b1, b2) +CB(x, b1, b2) + B(x, b1)= (1 − τe)( x

r +µ−ρση

r2 ) + τeb1r (1 − eκ(x−x1)) + τe

b2r (1 − eκ(x−x2))

−α(1 − τe)( x1r +

µ−ρσηr2 )eκ(x−x1), x1 ≤ x < x2.

(74)

As we expect, Eq. (74) shows that the equilibrium price of the levered firm is just the after taxequilibrium price of the corresponding unlevered firm plus the equilibrium price of tax shields(CCB and straight bond), less a deduction equal to the equilibrium price of the bankruptcy cost.

The special cases are similar to the previous literature in a complete market only involvingsystematic risk, e.g., Barucci and Del Viva (2012a). Naturally, we want to know the optimalcapital structure in a risk-averse world taking into account idiosyncratic risk premium and thedifference between the consumption utility indifference prices and the equilibrium prices. Wewill discuss this problem in the next section.

5. Optimal Capital Structure

In practice, there are seldom firms who take pure equity financing and on the contrary, a mixof debts and equity financing is common. A general explanation for this phenomenon is that debtfinancing can reduce the tax burden on firms. This explanation is reasonable, but not complete.In fact, another cause may be more important: Comparing to pure equity financing, a mix ofdebts and equity financing could obviously improve risk structure exposed to the agents. It’s justlike the common fact: Retail prices are generally higher than wholesale prices.

Specifically, following Duffie (2001), we assume the original owner of the firm will sellthe firm and distribute all the cash flow of the firm among an equity holder, a CCB holder atthe consumption utility indifference prices defined in this paper respectively, and a diversifiedstraight bond holder at the equilibrium price.

In fact, an agent, who is risk averse, may benefit very much from a mix financing insteadof pure equity financing. We can analyze the benefit of mixed financing as follows. In goodstates, the bond holder obtain fix coupons and provide tax benefits, while the equity holder bearsmost risk of the cash flow. In bad states but if the default does not happen, the equity holder can

16

benefit from risk sharing with the CCB holder due to the conversion terms which decrease defaultthreshold and the straight bond continues to provide tax benefits. In default states, the straightbond will provide diversification benefits. Naturally, we want to know what the best combinationis of the equity, CCB and straight bond, i.e. the optimal capital structure, for the firm.

Clearly, the optimal capital structure problem is effectively equivalent to the following op-timization problem: The original owner seeks to maximize the sum of the three prices of thecorporate securities (the equity, CCB and straight bond respectively) over all admissible couponsb1 and b2. More specifically, given the exogenous conversion and bankruptcy conditions, theoriginal owner divides the cash flow of the firm into three partial cash flows that are uniquelydetermined by the coupons b1 and b2. That is, to derive an optimal capital structure of the firm,we need only solve the following optimization problem:

maxb1,b2E(x, b1, b2) +CB(x, b1, b2) + B(x, b1) , (75)

where the function E(·, b1, b2), CB(·, b1, b2) and B(·, b1) are given by Theorems 3.2∼4.1 respec-tively. In subsequent discussions, we fix exogenously coupon rates u1 and u2. So, the optimalproblem Eq. (75) can be rewritten as

maxL1,L2E(x, u1L1, u2L2) +CB(x, u1L1, u2L2) + B(x, u1L1) . (76)

At the very beginning, we introduce a CCB into a capital structure just because the CCBwill reduce the default risk of a firm with the same leverage ratio of a firm. In particular, to afirm that is too important to fail, the CCB can prevent the firm from going bankrupt instead ofgovernment bailout, which will create a serious moral hazard problem. In fact, our research findsanother merit of the CCB in the next section. That is, if the capital structure includes a CCB,the owner of the firm will sell the firm at a better price. In addition, the tax system provides notonly tax benefits τeb1 and τeb2 but also reduces the agent’s risk produced by the cash flow (thegovernment takes a fraction τe of the cash flow as tax income).

Since an explicit solution to the optimization problem defined in (75) is not available, weprovide a numerical simulation in the paper.

6. Comparative statics and numerical simulations

In this section, we perform numerical simulation of the results obtained in this paper withregard to conversion thresholds, the implied values of the equity, CCB and straight bond, optimalcapital structure and optimal leverages. In order to make a comparison effectively, the baselineparameter values are carefully selected as r = 0.045, δ0 = x = 5, π = 0.9, d = 0.04, u1 =

0.055, u2 = 0.07, L1 = 60, L2 = 20, σ = 0.7, µ = 0.21, η = 0.33, γ = 1, ρ = 0.8, τe = 0.25,α = 0.5.

6.1. Implied values of equity and CCB

Table 1 presents the impact of changes in risk aversion index γ on the implied values, i.e.consumption utility indifference prices, of equity and CCB. This table says that for a more risk-averse agent, i.e. a bigger risk aversion index γ, the implied values of the claims (equity, CCB)get less. As we expected, the equity value decreases quickly while the implied value of theCCB decreases slowly. This is because the equity bears more risk of the cash flow. At the same

17

time, we also see from Eq. (7) and (9) and Theorem 4.1, the default/conversion threshold andequilibrium price of straight bond are invariable with the changing levels of risk aversion.

Table 2 indicates that, implied values of equity and CCB as well as the market value ofstraight bond are globally decreasing functions of the volatility of the cash flow of the firm. Theintuition is as follows. The agent in our model is risk averse and thus a high volatility, whichmeans a high risk, will definitely decrease the implied values of the contingent claims (equityand CCB). Since a greater volatility implies a higher bankruptcy risk and so, according to therisk-neutral model, the market value of the straight bond is a decreasing function of the volatilityof the cash flow of the firm. We also see from Eq. (7) and (9), that both the default threshold andconversion threshold increase along with a growth of the volatility.

6.2. Risk premiumIn a complete market or in the case where the prices are given by equilibrium pricing ap-

proach, the risk premiums are completely determined by the systematic risk, i.e. the idiosyn-cratic risk premiums are considered to be zero. But in a incomplete market and for a risk-averseagent, both systematic and idiosyncratic risks affect the risk premiums. With regard to the CCB,we denote the systematic and idiosyncratic risk premium by S R(x) and IR(x) respectively, whichare immediately given by

S R(x) = ρσηCBx(x, b1, b2)/CB(x, b1, b2), x > x2,S R(x) = ρσηCB1

x(x, b1, b2)/CB1(x, b1, b2), x1 < x ≤ x2,(77)

and IR(x) = γr(1 − ρ2)σ2(CBx(x, b1, b2))2/[2CB(x, b1, b2)], x > x2,IR(x) = γr(1 − ρ2)σ2(CB1

x(x, b1, b2))2/[2CB1(x, b1, b2)], x1 < x ≤ x2,(78)

where x is the current cash flow rate.The systematic risk premium S R(x) defined in Eq. (77) takes the same form as in standard

asset pricing models, which is the product of the Sharpe ratio η, systematic volatility ρσ andthe relative change rate of implied value of the CCB, CBx(x, b1, b2)/CB(x, b1, b2), at the currentcash flow rate x. The idiosyncratic risk premium IR(x) defined in Eq. (78) depends on the riskaversion index γ and the conditional idiosyncratic variance (1 − ρ2)σ2(CBx(x, b1, b2))2 of theimplied value of the CCB.

With the baseline parameter values, if the current rate x of cash flow is less than the conver-sion trigger x2 = 3.19 but greater than the default trigger x1 = 1.87, the implied value of theCCB is given in Proposition 3.3. If the current rate of cash flow x is greater than the conversiontrigger x2 = 3.19, the implied value of the CCB is shown in Theorem 3.4. So, we get the riskpremium of the CCB described by Figure 1-4.

Figures 1 and 3 say that the systematic risk premium diverges to infinity when the currentrate x of cash flow approaches the default trigger. This is just the same with the conclusion undercomplete markets, which can be easily given explicitly. Because of the value matching condition,i.e. Eq. (40), at conversion, the systematic risk premium goes to zero at the conversion trigger.When the current cash flow is large enough, the systematic premium goes to zero, which canalso be proved strictly by using the second equation of (64). Figure 2 and 4 show that theidiosyncratic risk premium deceases with a growth of the current cash flow x no matter whetherthe conversion has taken place or not. In the same way, when the current cash flow is largeenough, the idiosyncratic risk premium goes to zero. In addition, the systematic and idiosyncraticrisk premium are both increasing in the risk aversion index γ or the idiosyncratic risk volatility.All the results are quite in agreement with intuition.

18

6.3. Risk-prevention incentiveIn this subsection, we assume that the CCB is included in the capital structure and in par-

ticular, the equity holder has an option to increase the volatility σ, i.e. riskiness, of the firm bychoosing different technologies. We want to know whether the new capital structure will enhancerisk-prevention incentive to the equity holder or not. To this end, we compute the implied valuesof the equity while the volatility changes from σ = 0.1 to σ = 0.9 for several different risk aver-sion indexes and correlation coefficients and the other parameters take the baseline parametervalues defined before. The numerical results are reported in Figures 5 and 6.

Figure 5 shows the impact of changes of the volatility σ on the implied value of the equitywith different levels of risk aversion. It says that the implied value of the equity decreases as theriskiness of the firm increases even under the risk-neutral world, and as we expected, the valuedeceases more quickly for a more risk-averse agent. This result explains that the CCB can notonly decrease directly the bankruptcy risk, but it also can prevent the equity holder from investingin a high-risk project and therefore, indirectly increase the financial safety of the firm.

Figure 6 shows the impact of changes in the volatility σ on the implied value of the equitywith several different levels of the correlation coefficient (ρ) between the cash flow of the firm andthe return of the market portfolio. We find that in common, the same story described in Figure6 happens again. However, on the contrary, if the cash flow of the firm is strongly negativelycorrelated with the return of the market portfolio6, the implied value of the equity will decreasewith a growth of the riskiness. It implies that the equity holder would like to invest in a high-riskproject. This is what we expected since a negatively correlated project is just like an insuranceproduct and can definitely decrease the total systematic risk exposed to the equity holder.

6.4. Comparison of optimal capital structureIn the following, we analyse the capital structure of a firm issuing CCB. In order to consider

the diversification benefits, we firstly ignore tax benefits, i.e. τe = 0, and analyze the optimalcapital structure. And then, we take into account the tax benefits and take τe = 0.25. Beyondthat, we compare the risk averse case with the risk-neutral one.

Tables 3∼6 present simulation results under optimal capital structure, while Tables 4 and 6do not include a CCB in the capital structure. All the results explain that the total firm value,i.e. the sum of the payments the original owner of the firm receives by selling all the claims,is much greater than the total firm value if the firm’s capital structure does not include a CCB.This evidence suggests that issuing CCB can raise the total firm value, although the primitiveaim of doing so is just for decreasing the bankrupt risk of a firm that is too important to fail. Ina risk-neutral world, this happens mainly because selling a CCB can reduce the amount of thestraight bond and decrease the bankruptcy cost significantly and so increase the total firm value.

In particular, Panel A of Table 3 shows the optimal capital structure under the case withouttax, i.e. τe = 0. It is shown that if the agent is risk averse, the CCB also provides diversificationbenefits. In a complete market or the values are given by the equilibrium pricing approach,the firm will not take a straight bond into the capital structure and the total value of the firmis independent of the amount of the CCB as implied by Eq. (74). On the contrary, under anincomplete market, the investors face idiosyncratic risk and so the firm will take both equity anddebt financing to diversify the idiosyncratic risk. The numerical results show that the optimal facevalue and implied value of the CCB decrease with a growth of the risk-averse index. In contrast,

6Generally speaking, the correlation coefficient is actually positive.19

the optimal face value and market value of straight bond increase with the index, which accordswith Chen et al (2012). This is because a more risk-averse agent can get a greater diversificationbenefit to issue straight bond but a higher risk-averse CCB holder gets less diversification benefit.

Panel B of Table 3 presents the optimal capital structure with tax benefits (τe = 0.25). Itshows that the optimal face value of the straight bond/CCB increases/decreases quickly with agrowth of the risk-averse index. This is true because as the risk-averse index increases, the agenthas a strong incentive to diverse idiosyncratic risk and so the the straight bond is issued morein place of the CCB. The CCB, though, can also reduce the bankruptcy risk, but to a more risk-averse agent, the CCB provides less diversification benefits. Beyond that, it is shown that themore risk-averse agent has a stronger incentive to take conversion and default. Comparing thefirst line of Panel A with that of Panel B, as we expect, the straight bond generates significant taxbenefits.

Panels A∼C of Table 5 show the optimal capital structure with different idiosyncratic riskvolatility levels. The optimal face value and implied value of the straight bond/CCB increase/decreasewith the idiosyncratic risk volatility level. The optimal face value of the CCB is much lower thanthe optimal face value of straight bond. A greater idiosyncratic risk volatility induces a higherconversion and default threshold. But in the case with equilibrium prices, we get the oppositeconclusion, as shown in Panel D of Table 5. As we expect, an increase in the loss rate leads to ahigher optimal face value of the CCB, a lower optimal face value and market value of the straightbond and naturally a lower default threshold.

Tables 3 and 5 also show that the more the idiosyncratic risk volatility√

1 − ρ2σ or the morethe risk aversion index γ, the more the optimal leverage ((CB+ B)/(E +CB+ B)). By comparingTable 3 with Table 4 and comparing Table 5 with Table 6 respectively, we find that the levels ofleverage with a CCB in the capital structure is greater than that in the classical model without aCCB. The conclusions shown in Panel D of Table 5 are very different from those with equilibriumprices but quite in agreement with intuition.

Table 7 considers the case where the equity holder takes the risk aversion index γ = 1 whilethe CCB holder takes different risk aversion indexes. We find that the face value of the straightbond and the default threshold increase with the risk aversion index of the CCB holder, while theface value and the implied value of the CCB, conversion threshold and optimal leverage decreasewith the risk aversion index, which implies that a more risk-averse CCB holder takes less risksharing. And the agent which is exposed to more idiosyncratic risk will sell more straight bondsfor diversification benefits. Naturally, the optimal leverage reduces along with the decline of theidiosyncratic risk.

In addition, comparing the last line of Table 3 with the last line of Table 7, we find that theface values of the straight bond and CCB and the implied value of the CCB, the optimal leverage,conversion threshold and default threshold all increase with the risk aversion index of the equityholder, when fixing the risk aversion γ = 2 of the CCB holder. The fact shows once again thatthe CCB provides diversification benefits.

7. Conclusion

After the recent global financial crisis, one of the most common suggestions is to introducecontingent convertible bond (CCB) into the capital structure of a firm, which is too important tofail. To the best of our knowledge, all the current theories about CCBs are based on the followingassumptions: The market is complete or agents are risk-neutral. At least, agents are assumed to

20

1.87 3.17 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

current rate of cash flow x

syst

emat

ic r

isk

prem

ium

IR(x

)

Systematic risk premium with two levels of risk aversion

γ=1γ=2

Figure 1: Impact of changes in current cash flow rate(x) on the systematic risk premium of the CCB withtwo levels of risk aversion index (γ = 1, 2).

1.87 3.17 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

current rate of cash flow x

idio

sync

ratic

ris

k pr

emiu

m IR

(x)

Idiosyncratic risk premium with two levels of risk aversion

γ=1γ=2

Figure 2: Impact of changes in current cash flow rate(x) on the idiosyncratic risk premium of the CCB withtwo levels of risk aversion index (γ = 1, 2).

be risk-neutral to idiosyncratic risk. However, in practice, it is common for a risk-averse agentto invest under an incomplete market. For this reason, we relax these assumptions and discussthe pricing of the equity and CCB and optimal capital structure under an incomplete market for arisk-averse agent, based on consumption utility indifference pricing approach, while the straightbond is given by equilibrium pricing approach due to its low idiosyncratic risk.

We assume that the cash flow of a firm evolves according to an arithmetic Brownian motionwhich is observable. We derive the market value of the straight bond and semi-closed-form so-lutions for the implied value of the equity and CCB Based on consumption utility indifferencepricing method for both exogenously given conversion threshold and default threshold. Follow-ing that, an optimal capital structure is presented numerically.

We perform numerical simulation by finite difference methods and compare the results withthose obtained by equilibrium pricing approach. We also compare the optimal capital structurewith that only including the equity and straight bond. We find that

(i) for a more risk-averse agent or a higher business risk, i.e. a higher volatility of the cashflow of a firm, the implied values of the equity and CCB are smaller;

(ii) although idiosyncratic risk premium is considered to be zero under the risk-neutral assump-tion, the premium could actually be significant in a risk-averse world;

(iii) In general, i.e. only if the cash flow of the firm is not too strongly negatively correlated withthe market portfolio return, the capital structure including the CCB can enhance a strongrisk-prevention incentive of the equity holder;

(iv) in contrast to equilibrium pricing case, for a more risk averse equity holder or higher id-iosyncratic risk, the agent chooses a higher leverage and increases the issued amount of thestraight bond due to the significant diversification benefits of the straight bond and has astronger incentive for the conversion and default;

(v) for a more risk-averse CCB holder, the agent chooses a lower leverage, decrease the issuedamount of the CCB due to the less diversification benefits of the CCB and has a weakerincentive for conversion and default ;

21

1.87 3.17 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

current rate of cash flow x

syst

emat

ic r

isk

prem

ium

SR

(x)

Systematic risk premium with two levels of idiosyncratic risk volatility

ε=0.42ε=0.22

Figure 3: Impact of changes in current cash flow rate(x) on the systematic risk premium of the CCB withtwo levels of the idiosyncratic risk volatility (ϵ =0.22, 0.42). Note: For getting a clear figure, we set theagent’s risk aversion index γ = 2.

1.87 3.17 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

current rate of cash flow x

idio

sync

ratic

ris

k pr

emiu

m IR

(x)

Idiosyncratic risk premium with two levels of idiosyncratic risk premium

ε=0.42ε=0.22

Figure 4: Impact of changes in current cash flow rate(x) on the idiosyncratic risk premium of the CCB withtwo levels of the idiosyncratic risk volatility (ϵ =0.22, 0.42). Note: For getting a clear figure, we set theagent’s risk aversion index γ = 2.

(vi) the higher the idiosyncratic risk volatility is or the more risk-averse the agent is, the greaterthe optimal leverage in an optimal capital structure will be;

(vii) the level of the leverage is always greater than that in the classical model without issuing aCCB;

(viii) the CCB can not only decrease directly the bankruptcy risk, but it also can prevent theequity holder from investing in a high-risk project and therefore, indirectly increase thefinancial safety of the firm.

(ix) the CCB can not only decease bankruptcy risk but also significantly increase the total firm’svalue.

The conclusions show that the CCB is a valuable financial instrument and even a non-financialfirm might benefit significantly from taking a CCB into its capital structure.

Several opportunities exist for future research. For example, it is worth considering a firm,which has an option to invest in a project. The firm can take equity financing and debt financingby issuing a CCB. Thanks to the merits of the CCB, the pricing and timing of the option toinvest would be different from the classical real options theory. An interesting problem shouldbe whether the new capital structure increases the valuation of the real options. What is theoptimal capital structure with regard to investing in the project? In addition, we think the firmwould benefit even more if this new hybrid bond (CCB) could be convertible repeatedly betweenequity and debt depending on the cash flow rate of the firm, which should be studied seriously.

Acknowledgments

The research reported in this paper was supported by the National Natural Science Founda-tion of China (project nos. 70971037, 71171078 and 71221001), and the Doctoral Fund of theMinistry of Education of China (project No. 20100161110022).

22

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

volatility σ

impl

ied

valu

e E

of e

quity

Risk taking motivation with three levels of risk aversion

γ→0γ=1γ=2

Figure 5: Impact of changes in the volatility (σ) on theimplied value of the equity with three levels of risk aver-sion.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140

volatility σ

impl

ied

valu

e E

of e

quity

Risk taking motivation with four levels of correlation coefficient

ρ=−0.8

ρ=−0.4

ρ=0.4

ρ=0.8

Figure 6: Impact of changes in the volatility (σ) on theimplied value of the equity with four levels of correla-tion coefficient.

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AppendicesAppendix A Proof of Proposition 1

By Bellman principle, G1(w, x) can be equivalently written as

G1(w, x)= sup

C∈CE[

∫ τ1

0 exp (−βt) U(Ct)dt + exp(−βτ1)V0(Wτ1 )|W0 = w, δ0 = x], (A.1)

subject to (15). Therefore, by a standard computation, we derive (17).From the first-order condition (22) and exponential utility (3), the optimal consumption rate

is evidently given by

c = −1γ

ln G1w(w, x). (A.2)

Substitute (A.2) into (17), we immediately get

(rw + λ(1 − τe)(x − b1) + 1γ

ln G1w − 1

γ)G1

w + µG1w +

12η

2G1xx

− ((µe−r)G1w+σσeρG1

wx)2

2σ2eG1

ww− βG1 = 0.

(A.3)

According to (14) and (24), CB1(x, b1) satisfies

G1(w, x) = − 1γr

exp(1 − β/r − γr

(w +CB1(x, b1) +

η2

2r2γ

)). (A.4)

And if the current rate x of the cash flow is large enough, according to (25), CB1(x, b1) satisfies

G2(w, x) = − 1γr

exp(1 − β/r − γr

(w +CB1(x, b1) +

η2

2r2γ

)). (A.5)

24

Plugging (A.4) back into HJB Eq. (A.3) and boundary condition (18) respectively leads to (26)and boundary condition (27) of Proposition 1. Plugging (A.4) and (A.5) back into boundarycondition (21) we get boundary condition (28) of Proposition 1. Substituting (A.4) into (A.2)and (23) we derive the optimal consumption rate (29) and portfolio rule (30).

Appendix B Proof of Theorem 1

Thanks to Bellman principle, the optimization problem (16) can be equivalently written as

G(w, x)= sup

C∈CE[

∫ τ2

0 exp (−βt) U(Ct)dt + exp(−βτ2)G1(Wτ2 )|W0 = w, δ0 = x], (B.1)

subject to (15). By a standard computation, we then derive (31). From the firt term of the first-order condition (36) and exponential utility (3), the optimal consumption rate is at once givenby

c = −1γ

ln Gw(w, x). (B.2)

Thus, it follows from (B.2) and (31) that

(rw + b2 +1γ

ln Gw − 1γ)Gw + µGw +

12η

2Gxx − ((µe−r)Gw+σσeρGwx)2

2σ2eGww

− βG = 0, (B.3)

According to (14) and (37), CB(x, b1, b2) satisfies

G(w, x) = − 1γr

exp(1 − β/r − γr(w +CB(x, b1, b2) +η2

2r2γ)). (B.4)

Therefore, if the current rate x of the cash flow is large enough, according to (38), CB(x, b1, b2)satisfies

G3(w, x) = − 1γr

exp(1 − β/r − γr(w +CB(x, b1, b2) +η2

2r2γ)). (B.5)

Substituting (B.4) into HJB Eq. (B.3) and boundary condition (32) respectively gives (39) andthe first equation of boundary condition (40) in Theorem 1. Plugging (B.4) and (B.5) back intoboundary condition (35) gives the second equation of boundary condition (40) of Theorem 1.Plugging (B.4) back into (B.2) and the second equation of (36) gives the optimal consumptionrate (41) and portfolio rule (42).

The proofs of Proposition 3.3 and Theorems 3.4 are similar to the proofs of Proposition 3.1and Theorem 3.2 respectively and so we omit them here.

25

Table 1: Impact of changes in risk aversion (γ) on conversion threshold and impliedvalues. Baseline parameter values are set as r = 0.045, δ0 = x = 5, π = 0.9, d =0.04, u1 = 0.055, u2 = 0.07, L1 = 60, L2 = 20, σ = 0.7 and µ = 0.21.

γ market values 0.01 0.1 1 1.5 2 2.5

x2 3.19 3.19 3.19 3.19 3.19 3.19 3.19x1 1.87 1.87 1.87 1.87 1.87 1.87 1.87E 20.94 20.76 19.82 13.41 11.36 9.87 8.75CB 24.46 24.38 24.03 21.33 20.23 19.28 18.43B 61.63 61.63 61.63 61.63 61.63 61.63 61.63

Note: The table provides the market value B of the straight bond, the impliedvalue CB of the CCB, the implied value E of the equity, the conversion thresh-old x2 and default threshold x1.

Table 2: Impact of changes in volatility (σ) on conversion thresh-old and implied values. Baseline parameter values are set as r =0.045, δ0 = x = 5, π = 0.9, d = 0.04, u1 = 0.055, u2 = 0.07, L1 =

60, L2 = 20, µ = 0.21 and γ = 1.

σ 0.5 0.55 0.6 0.65 0.7 0.75

x2 2.02 2.31 2.60 2.90 3.19 3.48x1 0.70 0.99 1.28 1.58 1.87 2.16E 24.93 21.12 18.13 15.64 13.41 11.26CB 29.52 27.83 25.73 23.51 21.33 19.30B 72.58 71.19 68.82 65.56 61.63 57.29

Note: The table provides the market value B of the straightbond, the implied value CB of the CCB, the implied value Eof the equity, the conversion threshold x2 and default thresh-old x1.

26

Table 3: Impact of changes in risk aversion (γ) on the optimal capital structure including the equity,straight bond and CCB

γ L1 B L2 CB E T L(%) x2 x1

Panel A:τe = 0

γ → 0 0 0 – – – 117.60 – – -0.56γ = 0.5 2 2.3350 66 85.27 22.20 109.80 79.79 2.63 -0.48γ = 1 51 55.05 32 34.22 12.42 101.69 87.79 3.33 1.51γ = 1.5 67 68.81 17 16.89 11.68 97.38 88.00 3.38 2.15γ = 2 76 75.19 12 10.72 9.16 95.08 90.36 3.57 2.52Panel B:τe = 0.25

γ → 0 15 17.12 64 78.85 15.14 111.11 86.38 3.14 0.05γ = 0.5 41 44.60 43 46.66 11.31 102.57 88.98 3.38 1.10γ = 1 54 56.63 32 31.46 9.17 97.26 90.58 3.47 1.63γ = 1.5 63 63.97 26 22.95 6.92 93.84 92.63 3.61 1.99γ = 2 69 68.28 22 17.71 5.48 91.47 94.01 3.71 2.23

Note: This table reports the numerical results for the case where the firm issues theequity, straight bond and CCB. We consider two business income tax (τe = 0 or τe =

0.25) and five levels of risk aversion. The case ”γ → 0” corresponds to the resultsfrom the equilibrium pricing approach. In particular, the first line of Panel A representsthat the total value of the firm is independent of the amount of the CCB, as impliedby Eq. (74). The table provides the optimal face value L1 and market value B of thestraight bond, the optimal face value L2 and implied value CB of the CCB, the impliedvalue E of the equity, the total value T = E + CB + B of the firm, optimal leverageL ≡ (CB + B)/(E + CB + B), the optimal conversion threshold x2 and optimal defaultthreshold x1 under optimal capital structure.

27

Table 4: Impact of changes in risk aversion (γ) on the optimal capital structureincluding the equity and straight bond

γ L1 B E T L (%) x1

Panel A: τe = 0

γ → 0 0 0 117.60 117.60 0 0γ = 0.5 41 45.30 61.11 106.41 42.57 1.11γ = 1 65 67.24 31.18 98.42 68.32 2.07γ = 1.5 77 75.83 18.57 94.40 80.33 2.56γ = 2 82 78.75 13.25 92.00 85.60 2.76Panel B: τe = 0.25

γ → 0 60 61.63 40.86 102.49 60.13 1.87γ = 0.5 69 68.28 27.52 95.80 71.28 2.23γ = 1 74 71.46 20.23 91.69 77.94 2.44γ = 1.5 79 74.21 14.84 89.05 83.33 2.64γ = 2 82 75.62 11.63 87.25 86.67 2.76

Note: This table reports the numerical results for the case where the firmonly issues the equity and straight bond. We mainly compare the resultswith those reported in Table 3. The case ”γ → 0” corresponds to theresults from the equilibrium pricing approach. We discuss the optimalcapital structure about the optimal face value L1 and market value B ofthe straight bond, the implied value of the equity E, the total firm valueT = E + B, optimal leverage L = B/[E + B] and optimal default thresholdx1.

28

Table 5: Impact of changes in idiosyncratic risk volatility (ε =√

1 − ρ2σ) and loss rate (α) on the optimalcapital structure including the equity, straight bond and CCB

γ L1 B L2 CB E T L (%) x2 x1

Panel A: γ = 1,τe = 0, α = 0.5

ε = 0.33 36 40.77 42 50.63 15.92 107.32 85.17 3.10 0.90ε = 0.42 51 55.05 32 34.22 12.42 101.69 87.79 3.33 1.51ε = 0.50 65 65.89 18 18.14 11.49 95.52 87.97 3.34 2.07Panel B: γ = 1,τe = 0.25, α = 0.5

ε = 0.33 50 54.20 36 38.08 9.71 101.99 90.48 3.46 1.47ε = 0.42 54 56.63 32 31.46 9.17 97.26 90.58 3.47 1.63ε = 0.50 59 59.47 29 25.98 7.89 93.34 91.55 3.57 1.83Panel C: γ = 1,τe = 0.25, α = 0.6

ε = 0.33 45 48.98 39 42.20 10.35 101.53 89.80 3.37 1.26ε = 0.42 49 51.59 35 35.22 9.77 96.58 89.88 3.38 1.42ε = 0.50 54 54. 63 32 29.40 8.45 92.48 90.86 3.47 1.63Panel D: γ → 0,τe = 0.25, α = 0.5

ε = 0.33 18 20.74 63 78.05 13.25 112.04 88.17 3.24 0.17ε = 0.42 15 17.12 64 78.85 15.14 111.11 86.38 3.14 0.05ε = 0.50 11 12.45 66 80.96 16.84 110.25 84.73 3.05 -0.11

Note: This table reports the numerical results for the case where the firm issues the equity, straightbond and CCB. We consider two business income tax (τe = 0 or τe = 0.25) and three levels of theidiosyncratic risk volatility (ε = 0.33, ε = 0.42 or ε = 0.50). We also compare the consumptionutility indifference prices with the equilibrium prices in Panel B and Panel D. The case ”γ → 0”corresponds to the results from the equilibrium pricing approach. We discuss the optimal capitalstructure about the optimal face value L1 and market value B of the straight bond, optimal facevalue L2 and the implied value CB of the CCB, the implied value E of the equity, the total firmvalue T = E + CB + B, optimal leverage L = [CB + B]/[E + CB + B], the optimal conversionthreshold x2 and optimal default threshold x1.

29

Table 6: Impact of changes in idiosyncratic risk volatility (ε =√

1 − ρ2σ) and loss rate (α) on theoptimal capital structure including only the equity and straight bond

γ L1 B E T L(%) x1

Panel A: γ = 1, τe = 0, α = 0.5

ε = 0.33 54 58.90 45.20 104.10 56.58 1.63ε = 0.42 65 67.24 31.18 98.42 68.32 2.07ε = 0.50 73 71.63 22.70 94.33 75.93 2.40Panel B: γ = 1, τe = 0.25, α = 0.5

ε = 0.33 73 72.76 23.02 95.78 75.96 2.40ε = 0.42 74 71.46 20.23 91.69 77.94 2.44ε = 0.50 76 70.73 17.53 88.26 80.14 2.52Panel C: γ = 1, τe = 0.25, α = 0.6

ε = 0.33 69 68.95 25.65 94.60 72.88 2.23ε = 0.42 71 68.29 22.00 90.29 75.64 2.32ε = 0.50 72 66.99 19.68 86.67 77.29 2.36Panel D: γ → 0, τe = 0.25, α = 0.5

ε = 0.33 63 65.51 38.05 103.56 63.26 1.99ε = 0.42 60 61.63 40.86 102.49 60.13 1.87ε = 0.50 57 57.89 43.63 101.52 57.02 1.75

Note: This table reports the numerical results for the case where the firm only issues theequity and straight bond. We compare the results with those reported in Table 3. The case”γ → 0” corresponds to the results from the equilibrium pricing approach. We discussthe optimal capital structure about the optimal face value L1 and market value B of thestraight bond, the implied value E of the equity, the total firm value T = E + B, optimalleverage L = B/[E + B] and optimal default threshold x1.

Table 7: Impact of changes in risk aversion (γ) of CCB holder on the optimal capital structure including the equity,straight bond and CCB

γ L1 B L2 CB E T L(%) x2 x1

γ → 0 35 38.59 62 65.13 4.37 108.09 95.95 3.99 0.86γ = 0.5 47 50.33 45 44.67 6.43 101.43 93.67 3.75 1.34γ = 1 54 56.63 32 31.46 9.17 97.26 90.58 3.47 1.63γ = 1.5 59 60.83 25 23.79 10.84 95.46 88.64 3.38 1.83γ = 2 62 63.21 19 18.13 13.00 94.34 86.22 3.24 1.95

Note: This table reports the numerical results for the case where the firm issues the equity, straight bondand CCB. We provide business income tax (τe = 0.25), equity holder’s risk aversion ”γ = 1” and five levelsof risk aversion for the CCB holder. The case ”γ → 0” corresponds to the results from the equilibriumpricing approach. We discuss the optimal capital structure about the optimal face value L1 and market valueB of the straight bond, optimal face value L2 and the implied value B of the CCB, the implied value B ofthe equity, the total firm value T = E +CB + B, optimal leverage L = [CB + B]/[E +CB + B], the optimalconversion threshold x2 and optimal default threshold x1.

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