基于效用的或有可转换债券定价及公司资本结构
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摘要
本文针对市场的不完备性，根据经济理论分析，对或有可转换债券进行设计和定价，并研究包含或有可转换债券的公司最优资本结构。本文假设收益流服从算术布朗运动，投资者可以通过无风险资产或者市场组合部分对冲公司收益流风险和平滑消费。由于公司收益流的不可交易性而给公司股权带来较大的非系统风险，基于消费效用无差别定价推导出股权的半解析解，并针对不同情况给出或有可转换债券和普通债券的消费效用无差别价格或均衡价格。最后，利用数值方法，对不同情况下的或有可转换债券价格和包含或有可转换债券的公司资本结构进行比较静态分析，分析投资者保护水平对或有可转换债券价格的影响，研究包含或有可转换债券融资的可延迟和不可延迟的投资问题。
首先，为了说明不完备市场条件对或有可转换债券价值以及包含其在内的公司资本结构的影响，我们不考虑税收和破产成本，并且投资者只能通过无风险投资进行平滑消费，并在外生的破产条件和转换条件下对或有可转换债券、股权和普通债券进行消费效用无差别定价，结果显示：在风险厌恶情况下，即使没有税收和破产成本，公司总价值仍然与融资结构有关，与Modigliani-Miller定理是完全不同的；在最优资本结构下，发行或有可转换债券不仅可以减少普通债券的面值并降低破产风险，还可以增加公司总价值，收益流波动率或投资者的风险厌恶系数越大，或有可转换债券（普通债券）的发行量越多（少）；除此之外，本文模型明确指出普通债券会通过增强股东的预防性储蓄动机来降低股权隐含价值，而与此相反，或有可转换债券还会通过削弱股东的预防性储蓄动机来提升股权隐含价值，这是或有可转换债券的一个潜在优点，而在风险中性条件下是没有的。
第二，我们考虑税收和和破产成本，并且投资者可以通过无风险投资和市场组合来平滑消费，对于股权流动性较差或股权高度集中的公司，考虑到或有可转换债券的混合型特点，转换后可能发生控制权转移，转换的股权可能会与原始股权一样伴随不可分散的非系统风险，因此在外生的转换触发点和内生的破产触发点的基础上，我们对公司股权和或有可转换债券进行消费效用无差别定价。相反，由于普通债券具有较强的流动性，普通债券持有人通常可以完全分散非系统风险，因此我们对普通债券进行均衡定价，本文发现：或有可转换债券仍可以降低破产风险而且在没有税收的情况下，也可以增加公司总价值；公司总价值是关于转换比率的凹函数，由此存在唯一的最优转换比率，并且投资者越厌恶风险，最优转换比率越大；如果投资者足够的厌恶风险，那么非系统风险越大，公司应该发行越少的股权和或有可转换债券而发行更多的普通债券，然而，如果投资者是风险中性的或者市场是完备的，会得出相反的结论；当且仅当公司收益流与市场组合负相关性很强时，股权持有人才有可能进行冒险投资，在某种程度上，加入或有可转换债券对风险中性的股权投资者会有冒险激励，而对风险厌恶的股权投资者一般不会，转换比率越高或者风险厌恶系数越大，股权投资者的冒险激励就会越弱；或有可转换债券的风险溢价不但依赖于系统风险溢价，还依赖于非系统风险溢价。
第三，在投资者的保护水平不完善的情况下。假设一般的企业在进行或有可转换债券融资时，选择内生的转换触发点，并且会选择一个不会丧失控制权的转换比率，可以假设或有可转换债券持有人可以完全分散非系统风险，比如持有人可能是一个较大规模的投资公司，同时进行多向不同类型的投资项目，从而分散了各自的非系统风险，普通债券持有人也可以完全分散非系统风险，由此对股权进行消费效用无差别定价，对或有可转换债券和普通债券进行均衡定价，之后，我们讨论企业的不可延迟的投资问题，主要结论是：在投资者保护不完善的情况下，当普通债券息票支付较大时，企业家越厌恶风险，则最优转换比率越大；当普通债券息票支付较小时，确得到相反的结果；相较外部股权融资，或有可转换债券融资对企业家股权的稀释效果较弱；即使在投资者保护极不完善的情况下，加入或有可转换融资也可以增加企业总价值。
最后，研究包含或有可转换债券融资的企业的可延迟投资问题。数值结果显示：随着企业家风险厌恶系数的增大，投资期权的隐含价值（消费效用无差别价格）和投资触发水平都在减小；随着非系统风险波动率的增大，投资期权的隐含价值在减小而投资触发水平却在升高；一般情况下，期权的隐含价值会随着随着转换比率的增大而增大；加入或有可转换债券融资可以增加投资期权的隐含价值，并且降低了投资触发水平；在约束融资情况下，企业家的冒险动机比较明显，而在只含或有可转换债券的最优融资情况下，企业家的冒险动机较小；若以单纯的或有可转换债券方式融资，在同等条件下与其他债券相比，企业获得的融资金额偏低。
This paper considers the design and pricing of a contingent convertible bond(CCB) and the optimal capital structure with a CCB. We assume the cash flow evolvesaccording to an arithmetic Brownian motion. The investor has access to one risk-freeasset or the market portfolio to smooth his consumption. The cash flow isnon-tradable and the equity takes on the high idiosyncratic risk. Therefore, we derivethe implied values of equity by a consumption utility indifference pricing approachand derive the consumption utility indifference prices or equilibrium prices ofCCB/straight bond. We provide numerical sensitivity analysis for the value of aCCB and the optimal capital structure of a firm by numerical method. We also analysethe impact of investor protection on the value of CCB and discuss the deferred andtake-it-or-leave-it investment problems that involve CCB financing.
First, in order to illustrate the behavior about the consumption utilityindifference price of CCB and the capital structure in the incomplete market, weignore the tax and bankruptcy cost. And the investor only has access to one risk-freeasset to smooth his consumption. We derive the implied values of equity, CCB andstraight bond by a consumption utility indifference pricing approach under anexogenous bankruptcy triggering level, an exogenously specified conversion rule.Numerical calculations show that the value of a firm is determined by how it isfinanced which is different from Modigliani-Miller theory. The CCB can not onlydirectly decrease the coupon of straight bond and the bankrupt risk, but also cansignificantly increase the total firm value. The higher the business risk or the morerisk-averse the agent is, the more (less) the CCB (straight bond) should be issued. Ourmodel explicitly finds that a straight bond will decrease the implied value of theequity by strengthening precautionary savings motive of the equity holder, but on thecontrary, a CCB will increase the implied value by weakening the savings motive.This is a hidden merit of a CCB, which is overlooked in a risk-neutral world.
Second, we consider the tax and bankruptcy cost. The investor has access to onerisk-free asset and the market portfolio to smooth his consumption. For the firm thatlack of liquidity or high stock concentration, considering the mixed characteristic ofCCB, it can induce the transfer of control rights. And the equity after conversion maytake on non-diversifiable idiosyncratic risk just as the original equity. So, We derive the implied values of equity and CCB by a consumption utility indifference pricingapproach under an endogenous bankruptcy triggering level, an exogenously specifiedconversion ratio. Because of the higher liquidity of straight bond, the straight bondholder is fully diversified. We provide the equilibrium price of straight bond. Theresults show that the CCB not only decreases bankruptcy risk but also considerablyincreases the total firm value without tax. The total firm value is a convex function ofconversion ratio and there is a unique global maximum point. The optimal conversionratio increases with the risk aversion index. If investors are risk-averse enough, asidiosyncratic risk rises, the firm should sell less CCB and equity but more straightbond. However, if the investors are risk-neutral, the opposite holds true. If the cashflow is sufficiently negatively correlated with the market portfolio return, the equityholder might have risk-taking incentives. In general, there are risk taking incentivesfor a risk-neutral investor but there are not if the investor is a little more risk-averse.The higher the conversion ratio or the larger the risk aversion index, the weaker arethe incentives.The risk premium for the CCB is determined not only by systematicrisk but also by idiosyncratic risk.
Third, under the poor investor protection, we assume the firm with CCBfinancing chooses the endogenous conversion trigger and the converson ratio thatkeeping control right. The CCB holder is a large-scale investment company. There aredifferent types of investment. So, the CCB holder is fully diversified. And the straightbond holder is also fully diversified. So, we provide implied values of equity by aconsumption utility indifference pricing approach and the equilibrium prices of CCB,straight bond. After that we discuss the take-it-or-leave-it investment problem. Theresults show that under the poor investor protection and for a higher coupon ofstraight bond, the higher the risk aversion, the bigger the optimal conversion ratio.But it presents the opposite result for a lower coupon of straight bond. Comparingwith the external equity financing, it presents the weaker dilution effect toentrepreneur’s equity. Even under the highly poor investor protection, a CCB stillcan improve the total firm value.
In the last, we discuss the deferred investment problem that involves CCBfinancing. The numerical results show that the higher the risk aversion, the lower arethe implied value (the consumption utility indifference price) of real option and theinvestment trigger level. The higher the idiosyncratic risk, the lower the implied valueof real option but the higher the investment trigger level. In general, the implied valueof real option will increase with the conversion ratio. A CCB financing can improve the implied value of real option, and decrease the investment trigger level. Under thefinancing constraint, the entrepreneur might have higher risk-taking incentives.However, under the optimal financing condition only with CCB financing, theentrepreneur might have lower risk-taking incentives. If the firm only has CCBfinancing, the enterprise will obtain less amount of the financing comparing with thesame condition of other bonds.
首先，为了说明不完备市场条件对或有可转换债券价值以及包含其在内的公司资本结构的影响，我们不考虑税收和破产成本，并且投资者只能通过无风险投资进行平滑消费，并在外生的破产条件和转换条件下对或有可转换债券、股权和普通债券进行消费效用无差别定价，结果显示：在风险厌恶情况下，即使没有税收和破产成本，公司总价值仍然与融资结构有关，与Modigliani-Miller定理是完全不同的；在最优资本结构下，发行或有可转换债券不仅可以减少普通债券的面值并降低破产风险，还可以增加公司总价值，收益流波动率或投资者的风险厌恶系数越大，或有可转换债券（普通债券）的发行量越多（少）；除此之外，本文模型明确指出普通债券会通过增强股东的预防性储蓄动机来降低股权隐含价值，而与此相反，或有可转换债券还会通过削弱股东的预防性储蓄动机来提升股权隐含价值，这是或有可转换债券的一个潜在优点，而在风险中性条件下是没有的。
第二，我们考虑税收和和破产成本，并且投资者可以通过无风险投资和市场组合来平滑消费，对于股权流动性较差或股权高度集中的公司，考虑到或有可转换债券的混合型特点，转换后可能发生控制权转移，转换的股权可能会与原始股权一样伴随不可分散的非系统风险，因此在外生的转换触发点和内生的破产触发点的基础上，我们对公司股权和或有可转换债券进行消费效用无差别定价。相反，由于普通债券具有较强的流动性，普通债券持有人通常可以完全分散非系统风险，因此我们对普通债券进行均衡定价，本文发现：或有可转换债券仍可以降低破产风险而且在没有税收的情况下，也可以增加公司总价值；公司总价值是关于转换比率的凹函数，由此存在唯一的最优转换比率，并且投资者越厌恶风险，最优转换比率越大；如果投资者足够的厌恶风险，那么非系统风险越大，公司应该发行越少的股权和或有可转换债券而发行更多的普通债券，然而，如果投资者是风险中性的或者市场是完备的，会得出相反的结论；当且仅当公司收益流与市场组合负相关性很强时，股权持有人才有可能进行冒险投资，在某种程度上，加入或有可转换债券对风险中性的股权投资者会有冒险激励，而对风险厌恶的股权投资者一般不会，转换比率越高或者风险厌恶系数越大，股权投资者的冒险激励就会越弱；或有可转换债券的风险溢价不但依赖于系统风险溢价，还依赖于非系统风险溢价。
第三，在投资者的保护水平不完善的情况下。假设一般的企业在进行或有可转换债券融资时，选择内生的转换触发点，并且会选择一个不会丧失控制权的转换比率，可以假设或有可转换债券持有人可以完全分散非系统风险，比如持有人可能是一个较大规模的投资公司，同时进行多向不同类型的投资项目，从而分散了各自的非系统风险，普通债券持有人也可以完全分散非系统风险，由此对股权进行消费效用无差别定价，对或有可转换债券和普通债券进行均衡定价，之后，我们讨论企业的不可延迟的投资问题，主要结论是：在投资者保护不完善的情况下，当普通债券息票支付较大时，企业家越厌恶风险，则最优转换比率越大；当普通债券息票支付较小时，确得到相反的结果；相较外部股权融资，或有可转换债券融资对企业家股权的稀释效果较弱；即使在投资者保护极不完善的情况下，加入或有可转换融资也可以增加企业总价值。
最后，研究包含或有可转换债券融资的企业的可延迟投资问题。数值结果显示：随着企业家风险厌恶系数的增大，投资期权的隐含价值（消费效用无差别价格）和投资触发水平都在减小；随着非系统风险波动率的增大，投资期权的隐含价值在减小而投资触发水平却在升高；一般情况下，期权的隐含价值会随着随着转换比率的增大而增大；加入或有可转换债券融资可以增加投资期权的隐含价值，并且降低了投资触发水平；在约束融资情况下，企业家的冒险动机比较明显，而在只含或有可转换债券的最优融资情况下，企业家的冒险动机较小；若以单纯的或有可转换债券方式融资，在同等条件下与其他债券相比，企业获得的融资金额偏低。
This paper considers the design and pricing of a contingent convertible bond(CCB) and the optimal capital structure with a CCB. We assume the cash flow evolvesaccording to an arithmetic Brownian motion. The investor has access to one risk-freeasset or the market portfolio to smooth his consumption. The cash flow isnon-tradable and the equity takes on the high idiosyncratic risk. Therefore, we derivethe implied values of equity by a consumption utility indifference pricing approachand derive the consumption utility indifference prices or equilibrium prices ofCCB/straight bond. We provide numerical sensitivity analysis for the value of aCCB and the optimal capital structure of a firm by numerical method. We also analysethe impact of investor protection on the value of CCB and discuss the deferred andtake-it-or-leave-it investment problems that involve CCB financing.
First, in order to illustrate the behavior about the consumption utilityindifference price of CCB and the capital structure in the incomplete market, weignore the tax and bankruptcy cost. And the investor only has access to one risk-freeasset to smooth his consumption. We derive the implied values of equity, CCB andstraight bond by a consumption utility indifference pricing approach under anexogenous bankruptcy triggering level, an exogenously specified conversion rule.Numerical calculations show that the value of a firm is determined by how it isfinanced which is different from Modigliani-Miller theory. The CCB can not onlydirectly decrease the coupon of straight bond and the bankrupt risk, but also cansignificantly increase the total firm value. The higher the business risk or the morerisk-averse the agent is, the more (less) the CCB (straight bond) should be issued. Ourmodel explicitly finds that a straight bond will decrease the implied value of theequity by strengthening precautionary savings motive of the equity holder, but on thecontrary, a CCB will increase the implied value by weakening the savings motive.This is a hidden merit of a CCB, which is overlooked in a risk-neutral world.
Second, we consider the tax and bankruptcy cost. The investor has access to onerisk-free asset and the market portfolio to smooth his consumption. For the firm thatlack of liquidity or high stock concentration, considering the mixed characteristic ofCCB, it can induce the transfer of control rights. And the equity after conversion maytake on non-diversifiable idiosyncratic risk just as the original equity. So, We derive the implied values of equity and CCB by a consumption utility indifference pricingapproach under an endogenous bankruptcy triggering level, an exogenously specifiedconversion ratio. Because of the higher liquidity of straight bond, the straight bondholder is fully diversified. We provide the equilibrium price of straight bond. Theresults show that the CCB not only decreases bankruptcy risk but also considerablyincreases the total firm value without tax. The total firm value is a convex function ofconversion ratio and there is a unique global maximum point. The optimal conversionratio increases with the risk aversion index. If investors are risk-averse enough, asidiosyncratic risk rises, the firm should sell less CCB and equity but more straightbond. However, if the investors are risk-neutral, the opposite holds true. If the cashflow is sufficiently negatively correlated with the market portfolio return, the equityholder might have risk-taking incentives. In general, there are risk taking incentivesfor a risk-neutral investor but there are not if the investor is a little more risk-averse.The higher the conversion ratio or the larger the risk aversion index, the weaker arethe incentives.The risk premium for the CCB is determined not only by systematicrisk but also by idiosyncratic risk.
Third, under the poor investor protection, we assume the firm with CCBfinancing chooses the endogenous conversion trigger and the converson ratio thatkeeping control right. The CCB holder is a large-scale investment company. There aredifferent types of investment. So, the CCB holder is fully diversified. And the straightbond holder is also fully diversified. So, we provide implied values of equity by aconsumption utility indifference pricing approach and the equilibrium prices of CCB,straight bond. After that we discuss the take-it-or-leave-it investment problem. Theresults show that under the poor investor protection and for a higher coupon ofstraight bond, the higher the risk aversion, the bigger the optimal conversion ratio.But it presents the opposite result for a lower coupon of straight bond. Comparingwith the external equity financing, it presents the weaker dilution effect toentrepreneur’s equity. Even under the highly poor investor protection, a CCB stillcan improve the total firm value.
In the last, we discuss the deferred investment problem that involves CCBfinancing. The numerical results show that the higher the risk aversion, the lower arethe implied value (the consumption utility indifference price) of real option and theinvestment trigger level. The higher the idiosyncratic risk, the lower the implied valueof real option but the higher the investment trigger level. In general, the implied valueof real option will increase with the conversion ratio. A CCB financing can improve the implied value of real option, and decrease the investment trigger level. Under thefinancing constraint, the entrepreneur might have higher risk-taking incentives.However, under the optimal financing condition only with CCB financing, theentrepreneur might have lower risk-taking incentives. If the firm only has CCBfinancing, the enterprise will obtain less amount of the financing comparing with thesame condition of other bonds.
引文
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