Accounting Noise and the Pricing of CoCos
AAccounting Noise and the Pricing of CoCos
Mike Derksen ∗ Peter Spreij † Sweder van Wijnbergen ‡ April 20, 2018
Abstract
Contingent Convertible bonds (CoCos) are debt instruments that convert into equity or are writtendown in times of distress. Existing pricing models assume conversion triggers based on market pricesand on the assumption that markets can always observe all relevant firm information. But all Cocosissued so far have triggers based on accounting ratios and/or regulatory intervention. We incorporatethat markets receive information through noisy accounting reports issued at discrete time instants,which allows us to distinguish between market and accounting values, and between automatic trig-gers and regulator-mandated conversions. Our second contribution is to incorporate that couponpayments are contingent too: their payment is conditional on the Maximum Distributable Amountnot being exceeded. We examine the impact of CoCo design parameters, asset volatility and account-ing noise on the price of a CoCo; and investigate the interaction between CoCo design features, thecapital structure of the issuing bank and their implications for risk taking and investment incentives.Finally, we use our model to explain the crash in CoCo prices after Deutsche Bank’s profit warningin February 2016.
JEL codes:
G12, G13, G18, G21, G28, G32
AMS subject classification:
Key Words:
Contingent capital pricing, accounting noise, Coco triggers, Coco design, risk takingincentives, investment incentives
Contingent Capital instruments or Contingent Convertible bonds (CoCos) are debt instruments designedto convert into equity or to be written down in times of distress. They differ from regular convertibles inthat conversion is not an option to be exercized by the holder; conversion is either automatically triggeredin response to a particular accounting ratio falling below a specified threshold or at the discretion ofthe regulator when a so called Point of Non-Viability (PONV) has been reached. They were originallyproposed by Flannery (2005) as an alternative to raising equity in times of distress. Their use hasexploded since the Great Financial Crisis eroded the capital base of banks across the world and regulatorsresponded by actually raising capital requirements so as to increase the Loss Absorption Capacity ofthe banking system. In this paper we develop a valuation model that not only takes into account theirparticular contingent properties but also explicitly incorporates the fact that markets get only imperfectinformation about the underlying firm dynamics through noisy accounting reports issued at regulardiscretely spaced time points.An academic literature has quickly emerged since Flannery’s original advocacy of CoCos (Flannery2005), and unanimously recommends basing trigger ratios on market values. In line with that view,the pricing models proposed so far assume conversion triggers based on market prices. Moreover, theliterature is without exception built on the assumption that markets can always observe all relevant ∗ Korteweg-de Vries Institute for Mathematics, University of Amsterdam, [email protected] † Korteweg-de Vries Institute for Mathematics, University of Amsterdam, [email protected] , and Institute for Mathematics,Astrophysics and Particle Physics, Radboud University Nijmegen ‡ University of Amsterdam, Tinbergen Institute, CEPR, [email protected] a r X i v : . [ q -f i n . P R ] A p r rm information, so the existing literature in fact assumes there is no difference between accountingand market values, which makes the choice between them obviously irrelevant. Whatever the merit ofthat view (the preference for market based triggers), fact is that basing conversion on market pricesactually makes a CoCo ineligible for being counted against capital requirements under European law(cf. Capital Requirements Regulation 575/2013/EU (2013, art. 54), henceforth referred to as CRR)under the framework implementing Basel III capital requirements in the European Union. Accordingly,all CoCos issued so far have without exception triggers for conversion based on accounting ratio’s fallingbelow a particular ratio (the trigger ratio).In this paper we attempt to bridge this discrepancy between the academic valuation literature on theone hand and the actual capital market practice on the other by valuing CoCos under the assumption thatthe only information available is noisy accounting information, our first contribution, which in additionis only received at pre-specified discrete moments in time. The underlying processes are continuous,but markets only receive noisy information on those underlying fundamental processes at discrete timeinstants: accounting reports are only released at discretely spaced points in time, the accounting dates(for example at the end of each quarter). In this way it is possible to distinguish between marketand accounting values. This informational structure gives rise to potential discrepancies between trueunderlying values, accounting ratios and market prices.We make a second contribution towards a better pricing model for CoCos. The asset pricing literaturehas so far concentrated on the conversion contingency. But the possibility of a write down or conversioninto equity is not the only option-like characteristic embedded in CoCo designs. The coupon paymentsare contingent too, they can only be paid out if that payment does not exceed the so called MaximumDistributable Amount, a trigger that binds much earlier than the conversion trigger (see (Kiewiet et al.2017) for extensive details). The relevance of this contingency became very clear in the beginning of2016, when a profit warning of Deutsche Bank ahead of their first quarter accounting report set off anaccross the board crash in CoCo prices (cf. again Kiewiet et al. (2017)).The model developed in this paper thus contributes in two different ways to the existing literature;it distinguishes between market and book values of assets in the valuation of CoCos and it allows thecoupons of CoCos to be already cancelled at a moment before the conversion date. The model is basedon the approach used by Duffie and Lando (2001), in which debt is valued under the assumption thatthe only information available is noisy accounting information which is received at selected moments intime. This setting is particularly relevant for the pricing of CoCos since, as pointed out above, the non-discretionary conversion triggers are always based on imperfect accounting ratios observed at discretemoments in time, rather than on continuously observable market prices.We first set up a comprehensive description of the structural credit risk model proposed by Duffieand Lando, including the derivation of all the relevant formulas and proofs. We then go beyond thepaper by Duffie and Lando by using their framework to provide explicit formulas and algorithms forthe pricing of CoCos. The setting is applied to the valuation of different kinds of CoCo bonds, namelyCoCos with a (partial) principal write down and CoCos with a conversion into shares. Also a distinctionis made between CoCos with a discretionary regulatory trigger, for which conversion could happen atany moment in time, and CoCos that can only be triggered at one of the accounting dates. The modeldoes not lead to closed form solutions, but the expressions for CoCo prices involve integrals that arecomputed using MCMC-methods.We first use the model developed in this paper to examine the impact of several CoCo design pa-rameters on the price of a CoCo; we then investigate the interaction between CoCo design features, thecapital structure of the issuing bank and their implications for risk taking and investment incentives.Finally, the model is used to explain the big downward price jump that CoCos of Deutsche Bank sufferedat the beginning of 2016 after the release of a profit warning. In this particular case the added valueof the proposed model becomes clear as it allows for the announcement of a bad accounting report andexplicitly allows for the early cancellation of coupon payments (before conversion) when the paymentwould exceed the so called Maximum Distributable Amount (MDA) trigger, cf. Pitt and Dewji (2016).Market sources indeed indicated at the time that the sudden price drop was out of fear for the MDAtrigger more than for setting off the conversion trigger, as the conversion trigger was still far out of reach.2he remainder of this paper is as follows. Section 2 describes CoCos in detail, their design featuresand their regulatory treatment. Section 3 surveys the existing asset price literature on CoCos. Section 4sets up the pricing model, making a distinction between market values and accounting values and in-corporating the possibility of early cancelling of coupons triggered by the MDA regulations referred toearlier, Section 5 outlines the MCMC algorithms used for evaluating the integrals involved in the finalpricing expressions. We outline the derivation of the key pricing formulas, with full details in the Ap-pendix. Section 6 uses the model to analyse the sensitivity of CoCo valuation to various design features,changes in the firm’s capital structure and various external shocks. We also analyse the events followingDeutsche Bank’s profit warning of late February 2016, and show that our pricing model does quite wellin explaining the observed CoCo price response. Section 7 summarizes and concludes. Proofs of thetechnical results are collected in the Appendix. A Contingent Convertible bond is a bond which converts into equity or is (partially) written down at theconversion date. This means that the design of a CoCo contract is specified by two main characteristics: • The trigger event: when does conversion happen? • The conversion mechanism: what does happen at conversion?
The trigger event specifies at which moment the conversion takes place. We can distinguish three types oftrigger events; an accounting trigger, a market trigger and a regulatory trigger. In case of an accountingtrigger, the conversion is triggered by an accounting ratio, e.g. the Common Equity Tier 1 Ratio (definedas the fraction of common equity over (risk weighted) assets) falling below a certain barrier. This typeof trigger is typical in practice, although it is widely criticized in the academic world. For example, inFlannery (2005) it is argued that a book value will only be triggered long after the damage has alreadyoccurred, because book values are not up-to-date at any moment. Therefore, the academic literaturewidely supports the use of market price based triggers. In the case of a market trigger, the conversionhappens if a market value, e.g. the share price of the issuing bank, falls below a certain threshold. Amarket price is thought to better reflect the current situation of the issuing bank, because a marketprice is a forward looking parameter; it reflects the market’s opinion on the future of the bank. SeeHaldane (2011) and also Pennacchi and Tschistyi (2015) for a very articulate defense of this point ofview, to which we return in the next section. Against this point of view, in Sundaresan and Wang (2015)and Glasserman and Nouri (2016) it is argued that a market trigger could lead to a multiple equilibriaproblem for the pricing of a CoCo if the terms of conversion are beneficial to CoCo holders. In thiscase, a market trigger could also encourage CoCo holders to short-sell shares of the issuing bank, toprofit from a conversion, which could subsequently lead to a “death spiral”. These warnings may wellexplain why market based triggers are actually outlawed in the European Union, cf. CRR. Whateverthe EU’s reason for this outlawing of market based triggers, as a consequence no CoCos with marketprice based triggers have been issued so far. A third type of trigger is the regulatory trigger, whichallows the regulator to call for a conversion. All CoCos issued so far have a trigger mechanism which isa combination of an accounting trigger and a regulatory trigger, since that is required for the CoCo tocount as regulatory capital in the European Union. The regulatory trigger has not been discussed in theasset pricing literature yet (but see Chan and van Wijnbergen (2018) for a corporate finance perspectiveon CoCo triggers focusing on risk taking and regulatory forbearance, i.e. regulatory behavior).
The conversion mechanism specifies what happens at the moment of conversion. There are two pos-sibilities: a (partial) principal write-down or a conversion into shares. In case of a (partial) principal3rite-down (PWD) mechanism, the principal of the CoCo bond is (partially) written down at the mo-ment of conversion, to strengthen the capital position of the issuing bank. In case of a conversion intoshares, the principal of the CoCo bond is converted into a number of shares. Of course, it needs to bespecified how many shares a CoCo holder receives at conversion. This conversion rule can be designedin two different ways. One possibility is that the CoCo holder receives a fixed number of shares ∆ forevery monetary unit of principal. This corresponds to a pre-specified share price 1 / ∆. Another optionis a variable number of shares related to the market price prevailing at the moment of time conversiontakes place. In this case the CoCo holder would “buy” a number of shares against a market price basedconversion price. Some authors have warned for the possibility of a “death spiral” when CoCo conversionis advantageous to CoCo holders and thus leads to incentives to short sell the stock (cf. Sundaresan andWang (2015)). This possibly leads to an infinitely large dilution of the existing shareholders. A way toavoid this would be to place a floor under the conversion price, again a requirement for the CoCo tocount as capital under European law. As Contingent Convertible bonds qualify as a form of capital in the Basel III regulations, they are alsoaffected by the concept of the Maximum Distributable Amount (MDA), which requires regulators toblock earnings distributions when the bank’s capital becomes too low. An example of such earningsdistributions are dividends, but also CoCo coupon payments if the CoCos qualify as AT1 capital. Thismeans that when the bank’s capital falls below some threshold, always (much) higher than the CoCo’sconversion trigger, the payment of coupons is stopped until the bank’s capital is again above the MDAtrigger. See Kiewiet et al. (2017) for a detailed discussion of the MDA trigger for coupons. This triggerhas not been considered before in the asset pricing literature, but will be introduced explicitly in thispaper.
The existing asset pricing literature on CoCos can be grouped in three categories (cf. Wilkens andBethkens (2014) for an early assessment following the same classification): structural models, equityderivative models and credit risk or reduced form models. In a structural model one starts by describingthe value of the assets of a firm by a stochastic process. Then the liabilities are introduced and equityis the difference between the assets and those liabilities. Conversion of CoCos occurs when the marketvalue of the firm’s assets or the firm’s capital ratio falls below a predetermined value (the conversiontrigger). In most papers, liquidation of the firm is also incorporated in the model by assuming that theequity holders liquidate the firm when the value of assets falls below some optimal threshold, chosen bythe shareholders to maximize equity value. Furthermore, it is assumed that default cannot occur beforeconversion. An early example of such a structural model is Albul et al. (2012); there the firm’s value A t is described by a geometric Brownian motion (GBM) process under the Risk Neutral measure given byd A t = µA t d t + σA t d W t , with µ and σ constants and W a standard Brownian motion. The risk-free rate is assumed constant.The firm also issues two types of debt: a straight bond and a CoCo, both with perpetual maturities andboth paying coupons at a constant rate. The CoCo converts into equity the first time the asset valuefalls below some threshold α c , so the conversion time is τ ( α c ) = inf { t ≥ A t ≤ α c } . In their set up, theCoCo converts into equity valued at market prices at a specified conversion ratio λ , where λ = 1 meansthat the CoCo holder receives equity with a market value equal to the face value of the CoCo at issue.Like in all other papers surveyed here, the value of the various claims (including the CoCos) is given bythe risk-neutral expectation of the discounted future cashflows regarding the claim. The simplicity ofthe model allows for closed form expressions. In Pennacchi (2011) a similar model is introduced, but alsoproportional jump processes are added to the firm’s dynamics by adding a compound Poisson process;4sset values are governed by the following stochastic process (also under the risk neutral measure)d A t = ( r − λ t k t ) A t d t + σA t d W t + ( Y q t −
1) d q t . Here λ t is the risk neutral jump intensity of the Poisson process q t , k t = E Q ( Y −
1) is the expectedproportional jump under the risk neutral measure in case a Poisson jump occurs. The model does notyield closed form solutions so Monte Carlo simulation is used to sketch the solution structure. In Chenet al. (2013) an equally involved model is proposed in which the asset value process also involves aGBM process with Poisson jumps added in, with a distinction between market wide and firm specificjumps. They are mainly interested in downside shocks and, for tractability, they assume that minusthe log of the jump sizes have exponential distributions, which allows the authors to derive closed formsolutions. Conversion of CoCos into equity is triggered the first time the value of assets falls below somespecified threshold. In contrast to the variable conversion share price featured in Albul et al. (2012)and Pennacchi (2011), the CoCo holders receive a fixed number of shares for every dollar of principalwhen the CoCo converts, which is the way all Cocos with a conversion into shares are set up in practice,cf. Avdiev et al. (2017). An interesting innovation is their introduction of finite maturity debt and theassociated potential debt roll over problems. This feature has a significant effect on risk taking behaviorbefore conversion: even when the share conversion takes place at a rate favorable to the old shareholders,conversion leads to higher roll over costs of short term debt, which mitigates risk taking incentives exante. The model in Pennacchi and Tschistyi (2015) reverts to a straight GBM process driving assetvalues, and the focus is on the uniqueness and in fact existence of a price equilibrium when conversioninvolves a wealth transfer favoring either the CoCo holder (dilutive CoCos) or the old equity holder(non-dilutive CoCos). Academics widely favor conversion triggers based on market prices and dilutiveconversion ratios, but in Sundaresan and Wang (2015) it has been argued that stipulating triggers basedon market prices leads to multiple price equilibria in the case of dilutive CoCos and in fact non-existencein the case of non-dilutive CoCos (i.e. conversion at terms favoring the old shareholders). Both inGlasserman and Nouri (2016) and Pennacchi and Tschistyi (2015) it is shown that price equilibria will infact be unique in the case of dilutive CoCos. In Pennacchi and Tschistyi (2015) it is furthermore shownthat for perpetual CoCos (which is the structure most seen in practice) non-existence only occurs forimplausible parameter values even when they are non-dilutive.All these models have in common that a conversion trigger based on market values is used, as is widelyrecommended in the academic literature (cf. in particular Haldane (2011) and Pennacchi and Tschistyi(2015) for an extensive discussion of why market prices should be used for conversion triggers). Theproblem with basing one’s analysis on that view, whatever its merit, is that there is literally no singleCoCo ever issued, at least within the European Union, that follows such a trigger definition. Withoutexception, within the European Union, where the bulk of all CoCos have been issued, trigger ratios arebased on accounting values. In fact in the EU, market based triggers are illegal under European law,cf. CRR, or at least cannot be counted as capital .Moreover, none of the models discussed so far actually distinguishes between market and accountingbased valuation. The single exception in the literature is Glasserman and Nouri (2012), where it isassumed that markets and accountants agree on whether a firm is solvent (if the value of the assetsexceeds the value of debt based on market prices, accounting values are assumed to do likewise). Butthe ratio between the market value of equity and the accounting value of equity, roughly similar to themarket-to-book (M-to-B) ratio, follows a GBM process. This approach gives two additional parametersto be used in calibration: the volatility of the M-to-B value process and its correlation to the marketprocess. This is an imaginative attempt to endogenize the M-to-B value process, but there are problemswith this first attempt at endogenizing the difference between market values and accounting ratios. Firstof all, Haldane (2011) doubted casts on their key assumption, that market and accounting valuationsalways agree on whether the firm is solvent. A less fundamental but practically speaking equally serious cf. Shreve (2004, Chapter 11) for a discussion of the various types of Poisson processes. European Law explicitly states that in order to qualify as an Additional Tier 1 instrument for capital purposes, aCoCo instrument should have a mechanical book-value based trigger which needs to have been mentioned explicitly in theprospectus.
This section starts with the model description. After that we derive the density of asset values, condi-tional on accounting information. Finally we present results for the valuation of CoCos for the differenttrigger events: regulatory triggers, possibly with conversion into shares, and accounting triggers.
The value of assets of the firm, denoted by V t , is modeled by a geometric Brownian motion, that isd V t V t = µ d t + σ d W t , (4.1)for some µ ∈ R , σ >
0. We will not explicitly include jumps in the asset value process, because this doesnot make sense in the noisy accounting information framework, as it would not be possible to distinguish6 big price movement caused by the dynamics of the asset process from a reaction to the accountinginformation. Define Z t = log V t and m = µ − σ /
2, then Z is a drifted Brownian motion with drift m and volatility σ , that is Z t = Z + mt + σW t . As mentioned before, we will consider a framework in which investors do not observe the real assetvalue, instead they receive imperfect accounting information at known observation times t < t < . . . (typically every three months). At every observation date t i there arrives an imperfect accounting reportof the real asset value V t i , denoted by ˆ V t i , where log ˆ V t i and log V t i are assumed to be joint normal. Thismeans that we can write Y t i := log ˆ V t i = Z t i + U t i , (4.2)where U t i is normally distributed and independent of Z t i . In the following we will use the notation Y i := Y t i and similar notations for Z and U . Of course, it is reasonable that there exists some correlationbetween the accounting noise U , U , . . . . To be more specific, following Duffie and Lando (2001), it isassumed that U i = κU i − + (cid:15) i , for some fixed κ ∈ R and independent and indentically distributed (cid:15) , (cid:15) , . . . , which have a normaldistribution with mean µ (cid:15) ∈ R and variance σ (cid:15) >
0, and are independent of Z .It is assumed that the firm issues two types of debt; straight debt and contingent convertible debt. Thetotal par value of straight debt outstanding is denoted by P , over which coupons are paid continuouslyat rate c . Furthermore, the straight bonds have a perpetual maturity and it is assumed that defaultoccurs the first time the log-value of assets falls below some threshold z b , such that the default time isdefined by τ b = inf { t ≥ Z t ≤ z b } . At the moment of default a fraction (1 − α ), for α ∈ (0 , α of the asset value is recovered and distributed among the senior debt holders.The total par value of CoCos outstanding is denoted by P , over which coupons are paid continuouslyat rate c . Furthermore, the maturity of the contingent convertible bonds is denoted by T . In ouraccounting report framework, we will consider two different types of conversion triggers. The first typeof conversion trigger that will be looked into is the regulatory trigger. Banks have the obligation toreport it to their supervisor at the moment they are approaching a trigger. Then the regulator will callfor conversion, this is called a Point of Non-Viability. Of course, this type of conversion can also happenin between accounting report dates. This type of conversion thus is triggered when the log-value of assetsfalls for the first time below a conversion threshold z c , i.e. the conversion time is given by τ c = inf { t ≥ Z t ≤ z c } . We will always assume that z b < z c , such that conversion will always happen before default, i.e. τ c < τ b .There are also also CoCos whose conversion trigger solely depends on accounting reports. An exampleis the Coco issued by Barclays on March 3, 2017 (cf. Barclays (2017)). This means that conversionhappens when the reported value of the capital ratio falls below some threshold and hence conversioncan only happen at one of the accounting report dates t , t , . . . . This corresponds to a setting in whichthe conversion time is defined as τ Ac = inf { t i ≥ Y t i ≤ y c } , for some threshold y c ≥ t isdescribed by the filtration H t , where H t = σ ( { Y t , . . . Y t n , { τ c ≤ s } , { τ b ≤ s } : s ≤ t } ) , for t n ≤ t < t n +1 . Here, the indicators are included to ensure that it is also observed in the market whether conversion hasalready occured or the firm is liquidated before time t . In case we deal with CoCos with an accountingtrigger, the market information is described by the filtration H t = σ ( Y t , . . . Y t n ) , for t n ≤ t < t n +1 . .2 The Density of Asset Value, Conditional on Accounting Information In the previous subsection it was explained that we will consider two different types of conversion triggers.For the first one, the regulatory trigger, the conversion time is determined by the process Z falling belowsome threshold. In order to compute the market value of CoCos with such a trigger, we need to be ableto compute the probability of conversion, conditional on the market information H t . In order to do so,we will need the conditional density of Z , given the market information H t . In this subsection, followingDuffie and Lando (2001), we will derive an expression for this conditional density, which is intensivelyused in the remainder of this article.Consider t > t n ≤ t < t n +1 and conversion did not happen until time t , that is τ c > t .The goal in this section is to find an expression for the conditional distribution of Z t , given H t , whichwe will denote by f ( t, · ). Most of the results in this section can be found in the article by Duffie andLando (2001), but we will consider them shortly, to illustrate how the particular density is derived andwe will provide some additional explicit formulas.Consider the following notation for the relevant random vectors and its realisations: Z ( n ) = ( Z , Z , . . . , Z n ) and its realisation z ( n ) = ( z , z , . . . , z n ) ,Y ( n ) = ( Y , Y , . . . , Y n ) and its realisation y ( n ) = ( y , y , . . . , y n ) ,U ( n ) = Y ( n ) − Z ( n ) and its realisation u ( n ) = y ( n ) − z ( n ) . As already mentioned, the goal is to compute f ( t, · ), the conditional density of Z t given Y ( n ) and τ c > t . In order to do so, we will first compute the conditional density of Z t n at the report time t n , whichwe will denote by g t n ( ·| Y ( n ) , τ c > t n ). To this end, we will first introduce some functions. Firstly, weneed an expression for the probability ψ ( z , x, σ √ t ) that min { Z s : s ≤ t } >
0, conditional on Z = z > Z t = x >
0. This expression is stated in the following lemma and can also be found in the paper byDuffie and Lando (2001).
Lemma 4.1
The probability ψ ( z , x, σ √ t ) that min { Z s : s ≤ t } > , conditional on Z = z > and Z t = x > , is given by ψ ( z , x, σ √ t ) = 1 − exp (cid:18) − z xσ t (cid:19) . Consider the conditional probability of the intersection { Z ( n ) ≤ z ( n ) } ∩ { τ c > t n } given Y ( n ) . We denoteby b n ( ·| Y ( n ) ) its partial derivative w.r.t. z ( n ) . Note that ( Z n ) n ∈ N and ( U n ) n ∈ N are Markov processes anddenote by p Z ( z n | z n − ) and p U ( u n | u n − ) their respective transition densities for realisations z ( n ) , u ( n ) .Furthermore, denote by p Y ( y n | y ( n − ) the conditional density of Y n given Y ( n − = y ( n − . It is thenpossible (see Duffie and Lando (2001)) to write b n ( z ( n ) | y ( n ) ) in a recursive way, b n ( z ( n ) | Y ( n ) ) = ψ ( z n − − z c , z n − z c , σ √ t n − t n − ) p Z ( z n | z n − ) p U ( y n − z n | y n − − z n − ) b n − ( z ( n − | y ( n − ) p Y ( y n | y ( n − ) . (4.3)It now follows that the conditional density g t n ( ·| Y ( n ) , τ c > t n ) of Z ( n ) is given by g t n ( z ( n ) | y ( n ) , τ c > t n ) = b n ( z ( n ) | y ( n ) ) (cid:82) ( z c , ∞ ) n b n ( z ( n ) | y ( n ) )d z ( n ) . (4.4)It should be noted that there is no explicit expression for the integral in the denominator of Equa-tion (4.4), but note the important fact that we know the density up to a normalizing constant. Now themarginal conditional density of Z n at time t n is given by g t n ( z n | y ( n ) , τ c > t n ) = (cid:90) ( z c , ∞ ) n − g t n ( z ( n ) | y ( n ) , τ c > t n )d z ( n − . (4.5)8ow that we found the conditional density for a report time t n , we can use this to find the conditionaldensity f ( t, · ) for a general time t >
0. For this we will need the H t -conditional density of Z t , at a timebefore the first accounting report has arrived. Complementing Duffie and Lando (2001), we will nowgive an explicit expression for this density. Lemma 4.2 ˜ f ( t, · , z ) , the H t -conditional density of Z t , at a time t < τ c before the first accountingreport has arrived, given that Z started in z , is given by ˜ f ( t, x, z ) = 1 σ √ t exp (cid:16) − m ( z − x ) σ − m t σ (cid:17) (cid:16) φ (cid:16) z − xσ √ t (cid:17) − φ (cid:16) − z − x +2 z c σ √ t (cid:17)(cid:17) Φ (cid:16) z − z c + mtσ √ t (cid:17) − e − m ( z − z c ) /σ Φ (cid:16) z c − z + mtσ √ t (cid:17) . (4.6) Proof.
The proof of this lemma can be found in the Appendix.Finally, we are now able time to compute the conditional density f ( t, · ) for a general time t > t n < t < t n +1 such that τ c > t . Using the stationarity of Z , the H t -conditional density of Z t can bewritten as f ( t, x ) = (cid:90) ∞ z c ˜ f ( t − t n , x, z n ) g t n ( z n | Y ( n ) , τ c > t n )d z n . (4.7)Equation (4.7) should be read as follows; until time t n the process Z has stayed above z c and ended in z n , then on the time interval ( t n , t ), in which no new accounting reports arrive, the process has to movefrom z n to x and stay above z c . Although we do not have an analytical expression for the density f ( t, · ),it is important to note at this point that f ( t, · ) is written as the integral of g t n , which is known up tonormalizing constant, as can be seen from Equation (4.4). This makes it possible to compute integralswith respect to f ( t, · ), using Monte Carlo Markov Chain simulations, which means that results that arestated as an integral weighted by the density f ( t, · ) can actually be computed. The necessary algorithmsare described in Section 5.As a first use of the density f ( t, · ), we can for a time s > t , where t < τ c , define the H t -(CoCo) survivalprobability p c ( t, s ) = P ( τ c > s |H t ). This probability is then given by p c ( t, s ) = (cid:90) ∞ z c (1 − π ( s − t, x − z c )) f ( t, x )d x, (4.8)where, as in Duffie and Lando (2001), π ( t, x ) denotes the probability that Z hits 0 before time t , startingfrom x >
0. This probability is given by the following lemma, which follows from the well knownexpression for the distribution of a Brownian motion’s running minimum (see e.g. Harrison (1985),Section 1.8, equation (11)).
Lemma 4.3
The probability π ( t, x ) that Z hits before time t , starting from x > , is given by π ( t, x ) = 1 − Φ (cid:18) x + mtσ √ t (cid:19) + e − mx/σ Φ (cid:18) − x + mtσ √ t (cid:19) . In this subsection we will provide formulas for the market values of the different type of CoCos. Firstly,in subsection 4.3.1, CoCos with a regulatory trigger which suffer a principal write down at conversion,are valued. Then, it is also shown how to incorporate early cancelling of coupons, due to the MDA-regulations. In subsection 4.3.2, we then extend the PWD-assumption to CoCos with a conversion intoshares. These first two cases are all for the regulatory trigger and the results are all in the form of anintegral weighted by the the above derived conditional density f ( t, · ). It is postponed to Section 5 toprovide the necessary Algorithms to compute this integrals. Then, in Section 4.3.3, PWD CoCos withonly an accounting trigger are valued. 9 .3.1 Valuation of PWD CoCos with a regulatory trigger In this section we will value CoCos with a regulatory trigger and a principal write down at conversion.At the end of the section we will also incorporate the MDA-trigger. Recall that in case of a regulatorytrigger, the conversion date was defined as τ c = inf { t ≥ Z t ≤ z c } . Also, recall that the firm pays coupons continuously at rate c until either maturity or conversion. Weconsider a principal write down CoCo, which means a fraction 1 − R of the principal value is writtendown at conversion, while a fraction R is recovered to the bond holder, for R ∈ [0 , r .Now the value at time t < τ c of the CoCos, given the imperfect accounting information H t , is given by C ( t ) = E (cid:16) P e − r ( T − t ) { τ c >T } |H t (cid:17) + E (cid:32)(cid:90) Tt c P e − r ( u − t ) { τ c >u } d u |H t (cid:33) + E (cid:16) RP e − r ( τ c − t ) { τ c ≤ T } |H t (cid:17) = P e − r ( T − t ) p c ( t, T ) + c P (cid:90) Tt e − r ( u − t ) p c ( t, u )d u − RP (cid:90) Tt e − r ( u − t ) p c ( t, d u ) . (4.9)Here the first term represents the payment of the principal, in case conversion does not happen beforematurity, while the second term accounts for the payment of coupons until either conversion or maturity.The last term values the recovery of the principal at conversion. Note that every term is written in termsof the CoCo survival probability p c ( t, s ), which was given as an integral, weighted by the density f ( t, · ).Unsurprisingly, it turns out that these three terms together can be written as one integral weighted bythe conditional density f ( t, · ), which was derived in the previous section. This leads to the main resultof this subsection, which is proved in the Appendix. Theorem 4.4 (Price of a PWD CoCo with a regulatory trigger)
The secondary market price ofthe CoCo at time t < τ c is given by C ( t ) = (cid:90) ∞ z c h ( x ) f ( t, x )d x, (4.10) where h ( x ) is defined as h ( x ) := r − cr P e − r ( T − t ) (1 − π ( T − t, x − z c )) + c r P + (cid:18) c P r − RP (cid:19) I ( x ) , (4.11) in which I ( x ) is given by I ( x ) := exp (cid:32) − m ( x − z c ) + ( x − z c ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z c − √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) + exp (cid:32) − m ( x − z c ) − ( x − z c ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z c + √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) , (4.12) where Φ denotes the normal cumulative distribution function. In the valuation of the firm’s convertible debt in Equation (4.9), it is assumed that coupons are paiduntil conversion. However, as pointed out before, CoCos are affected by the Maximum DistributableAmount (MDA), which requires regulators to stop earnings distributions when the firm’s total capitalfalls below some trigger, higher than the conversion trigger. This we will incorporate in the model by10ntroducing a trigger z cc > z c . If Z is below z cc the firm will not pay coupons, while if Z is above z cc thefirm still pays coupons. To value the CoCo in this case, only the second term in Equation (4.9) needs tobe adjusted. In this case, coupons are only paid at time u if Z u > z cc , so the term E (cid:32)(cid:90) Tt c P e − r ( u − t ) { τ c >u } d u |H t (cid:33) , needs to be replaced with E (cid:32)(cid:90) Tt c P e − r ( u − t ) { τ c >u,Z u >z cc } d u |H t (cid:33) . (4.13)For τ c > t and t n ≤ t < t n +1 this term equals c P (cid:90) Tt e − r ( u − t ) P (cid:16) τ c > u, Z u > z cc | Y ( n ) , τ c > t (cid:17) d u. Thus, to value the CoCos while including the effects of the MDA-trigger, the quantity we need to computeis P (cid:0) τ c > u, Z u > z cc | Y ( n ) , τ c > t (cid:1) , which can be written in a similar way as the CoCo survival probability p c ( t, s ). In order to compute this conditional probability, we first need the following well known result :the joint distribution of a drifted Brownian motion and its running minimum (see e.g. Harrison (1985),Section 1.8, Corollary 7). Lemma 4.5
The joint probability ˜ π ( t, x, y ) that Z , starting from x > , does not hit 0 before time t and that Z t > y is given by ˜ π ( t, x, y ) := P ( inf ≤ s ≤ t Z s > , Z t > y ) = Φ (cid:18) x − y + mtσ √ t (cid:19) − e − mx/σ Φ (cid:18) − x − y + mtσ √ t (cid:19) . (4.14)Now, similarly to Equation (4.8), we can write P (cid:16) τ c > u, Z u > z cc | Y ( n ) , τ c > t (cid:17) = (cid:90) ∞ z c ˜ π ( u − t, x − z c , z cc − z c ) f ( t, x )d x, (4.15)such that we again found the solution as an integral weighted by the density f ( t, · ). The other two termsin Equation (4.9) do not change, so the CoCo price at time t < τ c is given by the sum of the new termin (4.13) and the unchanged part P e − r ( T − t ) p c ( t, T ) − RP (cid:90) Tt e − r ( u − t ) p c ( t, d u ) . By an adaption of Equation (4.10) it is seen that this unchanged part can be written as (cid:90) ∞ z c ˜ h ( x ) f ( t, x )d x, where ˜ h ( x ) = P e − r ( T − t ) (1 − π ( T − t, x − z c )) − RP I ( x ) , in which I ( x ) is given by Equation (4.12). In this section we consider the valuation of contingent convertible bonds which convert into equity at theconversion date. To recall, we assumed the firm issues two types of debt; straight debt and contingentconvertible debt. The total par value of straight debt outstanding is denoted by P , over which coupons11re paid continuously at rate c . Furthermore, the straight bonds have a perpetual maturity and defaultoccurs at τ b = inf { t ≥ Z t ≤ z b } . At the moment of default a fraction (1 − α ), for α ∈ (0 , α of the asset value is recovered and distributed among the senior debt holders.The total par value of CoCos outstanding is denoted by P , over which coupons are paid continuouslyat rate c . Furthermore, the maturity of the contingent convertible bonds is denoted by T . We considera regulatory trigger, which means the conversion date is defined as τ c = inf { t ≥ Z t ≤ z c } , where z c > z b , to ensure that conversion happens before default. Following Chen et al. (2013), we willassume the CoCo holders receive ∆ shares for every dollar of principal at the moment of conversion.This means that, if we normalize the number of shares before conversion to 1, the CoCo holders own afraction ρ = ∆ P ∆ P +1 of the firm’s equity after conversion.To recall, the information in the market at time t is described by the filtration H t = σ ( { Y t , . . . Y t n , { τ c ≤ s } , { τ b ≤ s } : s ≤ t } ) , for t n ≤ t < t n +1 . In analogy to Equation (4.9), the market price of the CoCos is given by C ( t ) = E (cid:16) P e − r ( T − t ) { τ c >T } |H t (cid:17) + E (cid:32)(cid:90) Tt c P e − r ( u − t ) { τ c >u } d u |H t (cid:33) + E (cid:18) ∆ P ∆ P + 1 E P C ( τ c ) e − r ( τ c − t ) { τ c ≤ T } |H t (cid:19) . (4.16)Of course only the third term has changed compared to Equation (4.9), because this term describeswhat happens at the moment of conversion (note that the second term needs to be replaced by thecorresponding term in Equation (4.13), if we want to include early cancelling of coupons). The thirdterm now describes that the CoCo holders obtain a fraction ∆ P ∆ P +1 of the firms post-conversion equity,denoted by E P C ( τ c ). This post conversion equity satisfies E P C ( τ c ) = V τ c − D ( τ c ) − E (cid:16) e − r ( τ b − τ c ) (1 − α ) V τ b |H τ c (cid:17) . That is, the firm’s value of assets minus the value of straight debt, denoted by D ( τ c ), and bankruptcycosts, described by the last term. Note that the value of straight debt at conversion is given by D ( τ c ) = E (cid:18)(cid:90) ∞ τ c c P e − r ( u − τ c ) { τ b >u } d u |H τ c (cid:19) + E (cid:16) αV τ b e − r ( τ b − τ c ) |H τ c (cid:17) , where the first term accounts for the continuous payment of coupons and the second term describes thepayment at default. It follows that the post-conversion equity value is given by E P C ( τ c ) = V τ c − E (cid:18)(cid:90) ∞ τ c c P e − r ( u − τ c ) { τ b >u } d u |H τ c (cid:19) − E (cid:16) e − r ( τ b − τ c ) V τ b |H τ c (cid:17) = e z c − E (cid:18)(cid:90) ∞ t c P e − r ( u − τ c ) { τ c ≤ u,τ b >u } d u |H τ c (cid:19) − e z b E (cid:16) e − r ( τ b − τ c ) |H τ c (cid:17) . So for τ c > t , the third term in Equation (4.16) can be written as E (cid:18) ∆ P ∆ P + 1 E P C ( τ c ) e − r ( τ c − t ) { τ c ≤ T } |H t (cid:19)
12 ∆ P ∆ P + 1 e z c (cid:90) Tt e − r ( u − t ) P ( τ c ∈ d u | τ c > t, Y ( n ) ) − ∆ P c P ∆ P + 1 (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T ∧ u, τ b > u | τ c > t, Y ( n ) )d u − ∆ P ∆ P + 1 e z b (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T, τ b ∈ d u | τ c > t, Y ( n ) ) . (4.17)So in this case, the key to valuation is finding an expression for the joint conditional distribution of τ c and τ b , as needed in the above integrals. These expressions can again be written as (double) integrals,weighted by the density f ( t, · ). Which leads to the following theorem, for which a proof is provided inthe Appendix. Theorem 4.6 (Price of a CoCo with a regulatory trigger and a conversion into shares)
Thesecondary market price at time t < τ c of the CoCo with a regulatory trigger and a conversion into sharesis given by C ( t ) = (cid:90) ∞ z c ( h ( x ) + h ( x )) f ( t, x )d x + (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) h (˜ z )d˜ z d x, (4.18) where ˆ f ( x, y, ˜ z, t ) is given by ˆ f ( x, y, ˜ z, t ) = 1 σ √ t exp (cid:18) − m ( x − ˜ z ) σ − m t σ (cid:19) (cid:18) φ (cid:18) x − ˜ zσ √ t (cid:19) − φ (cid:18) − x − ˜ z + 2 yσ √ t (cid:19)(cid:19) (4.19) and where h ( x ) = r − c r P e − r ( T − t ) (1 − π ( T − t, x − z c )) + c P r + c P r I ( x ) ,h ( x ) = ∆ P ∆ P + 1 (cid:16) e z b J b ( x ) + c P ˜ I ( x ) − c P ˜ J b ( x ) − e z c I ( x ) (cid:17) ,h (˜ z ) = ∆ P ∆ P + 1 e − r ( T − t ) ( c P ˜ J b (˜ z ) − e z b J b (˜ z )) , in which I ( x ) is given by Equation (4.12), ˜ I ( x ) equals ˜ I ( x ) = − r e − r ( T − t ) (1 − π ( T − t, x − z c )) + 1 r + 1 r I ( x ) ,J b ( x ) is given by J b ( x ) = − exp (cid:32) − m ( x − z b ) + ( x − z b ) √ m + 2 rσ σ (cid:33) and ˜ J b ( x ) = r + r J b ( x ) . In this section we will consider PWD CoCos which conversion trigger solely depends on accountingreports, for example the CoCos issued by Barclays. This means that conversion happens when thereported value of the capital ratio falls below some threshold and hence conversion can only happen atone of the accounting report dates t , t , . . . . This corresponds to a setting in which the conversion timeis defined as τ Ac = inf { t i ≥ Y t i ≤ y c } , for some threshold y c ≥
0. In this case the available information at time t would reduce to H t = σ ( Y t , . . . , Y t n ) , n such that t n ≤ t . This means that we are interested in the probability that, given n accounting reports and τ Ac > t n , the ( n + i )th accounting report will cause a trigger event, for i = 1 , , . . . .This probability is given in the next proposition, of which the proof can be found in the Appendix. Proposition 4.7
The i -step conditional survival probability (concerning τ Ac ), conditional on n previousaccounting reports, is given by P (cid:16) τ Ac > t n + i | Y ( n ) = y ( n ) (cid:17) = (cid:90) R n P ( ξ ( z n ) ∈ ( y c , ∞ ) i ) p Z ( z ( n ) | y ( n ) )d z ( n ) , (4.20) where p Z ( z ( n ) | y ( n ) ) = (cid:81) ni =1 p Z ( z i | z i − ) p U ( y i − z i | y i − − z i − ) p Y ( y n | y ( n − ) , (4.21) and where ξ ( z n ) denotes a multivariate normal distributed random variable with mean vector ˆ µ i andcovariance matrix Σ i , for which formulas, depending on z n , are provided in the Appendix. Note that that p Z ( z i | z i − ) is a Gaussian density with mean z i − + m ∆ t and variance σ ∆ t and that p U ( u i | u i − ) is a Gaussian density with mean κu i − + µ (cid:15) and variance σ (cid:15) . We did not provide a formulafor p Y ( y n | y n − ), but it turns out in Section 5 that we do not need this to compute the integral ofEquation (4.20).Using this proposition, it is now possible to value the contingent convertible bond with, as before,principal P , continuous coupon rate c , maturity T and a principal write-down with recovery rate R .As in Equation (4.9), this CoCo has secondary market price C (cid:48) ( t ) = E (cid:16) P e − r ( T − t ) { τ Ac >T } |H t (cid:17) + E (cid:32)(cid:90) Tt cP e − r ( u − t ) { τ Ac >u } d u |H t (cid:33) + E (cid:16) RP e − r ( τ Ac − t ) { τ Ac ≤ T } |H t (cid:17) , (4.22)which can be written in terms of the above derived i -step survival probability, as is stated in the nextresult, which is proved in the Appendix. Theorem 4.8 (Price of a PWD CoCo with a sole accounting trigger)
For t n ≤ t < t n +1 , T = t n + m for some m ∈ N and Y ( n ) = y ( n ) , where y i > y c , ≤ i ≤ n , the market price C (cid:48) ( t ) of the CoCos isgiven by C (cid:48) ( t ) = (1 − R ) P e − r ( T − t ) P ( τ Ac > t n + m | Y ( n ) = y ( n ) )+ m − (cid:88) i =1 (cid:18) cPr − RP (cid:19) ( e − r ( t n + i − t ) − e − r ( t n + i +1 − t ) ) P ( τ Ac > t n + i | Y ( n ) = y ( n ) )+ cPr (1 − e − r ( t n +1 − t ) ) + RP e − r ( t n +1 − t ) , (4.23)It should be noted that the only things left to compute are the i -step survival probabilities P ( τ Ac >t n + i | Y ( n ) = y ( n ) ), for which the formula is provided in Proposition 4.7 in terms of an integral, whichcan be evaluated using the method described in Section 5.As in the case of the regulatory trigger, we can also incorporate the MDA-regulations, which implythat coupons are already cancelled at a moment before the conversion date. It is now assumed thatcoupons over the time interval [ t i , t i +1 ) are only paid if Y i > y cc , for some trigger level y cc > y c . Tovalue the CoCo in this case, the second term in Equation (4.22) needs to be changed to E (cid:32) m − (cid:88) i =1 (cid:90) t n + i +1 t n + i cP e − r ( u − t ) { τ Ac >u,Y n + i >y cc } d u + { Y n >y cc } (cid:90) t n +1 t cP e − r ( u − t ) d u (cid:12)(cid:12)(cid:12) H t (cid:33) , t n ≤ t < t n +1 , T = t n + m for some m ∈ N .This leads us to the next result, stating the value of PWD CoCo with an trigger, when we also take intoaccount the early cancelling of coupons, due to the MDA regulations. Theorem 4.9 (Price of PWD CoCo with a sole accounting trigger, including MDA regulations)
When we include the MDA trigger, the CoCo price of Equation (4.23) modifies into C (cid:48) ( t ) = P e − r ( T − t ) P ( τ Ac > t n + m | Y ( n ) = y ( n ) ) + { Y n >y cc } cPr (1 − e − r ( t n +1 − t ) )+ m − (cid:88) i =1 cPr ( e − r ( t n + i − t ) − e − r ( t n + i +1 − t ) ) P ( τ Ac > t n + i , Y n + i > y cc | Y ( n ) = y ( n ) )+ RP m (cid:88) i =1 e − r ( t n + j − t ) (cid:16) P ( τ Ac > t n + i − | Y ( n ) = y ( n ) ) − P ( τ Ac > t n + i | Y ( n ) = y ( n ) ) (cid:17) , (4.24) where, similar to Equation (4.20), P ( τ Ac > t n + i , Y n + i > y cc | Y ( n ) = y ( n ) ) = (cid:90) R n P ( ξ ( z n ) ∈ ( y c , ∞ ) i − × ( y cc , ∞ )) p Z ( z ( n ) | y ( n ) )d z ( n ) . (4.25) In this section the algorithms that are necessary to compute all the derived CoCo values, are provided.The results in the previous section contain three kind of expressions, for which three different algorithmsare proposed in this subsection. The first expressions we will consider are those of the form (cid:90) ∞ z c h ( x ) f ( t, x )d x. That is, integrals of a function h , weighted by the density f ( t, · ). This type of expression is needed inthe valuation of a PWD CoCo with a regulatory trigger (cf. Theorem 4.4), when we include the MDAtrigger (cf. Equation (4.15)) and in the first part of the formula for the value of a CoCo with a conversioninto shares (cf. Theorem 4.6).First note that we can write (cid:90) ∞ z c h ( x ) f ( t, x )d x = (cid:90) ∞ z c h ( x ) (cid:90) ∞ z c ˜ f ( t − t n , x, z n ) g t n ( z n | Y ( n ) , τ c > t n )d z n d x = (cid:90) ∞ z c (cid:90) ( z c , ∞ ) n h ( x ) ˜ f ( t − t n , x, z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n )d z ( n ) d x = (cid:90) ( z c , ∞ ) n +1 h ( z n +1 ) ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n )d z ( n +1) . (5.1)So we will need a sample (( z ( n +1) ) , . . . , ( z n +1 ) G ) from the ( n +1)-dimensional distribution on ( z c , ∞ ) n +1 with density ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n ), in order to approximate this integral as C ( t ) ≈ G G (cid:88) g =1 h ( z gn +1 ) . (5.2)The algorithm used to obtain the sample, is the following MCMC-algorithm.15 lgorithm 5.1
1. In each iteration g , g = 1 , . . . n + G , given the current value ( z ( n +1) ) g , the proposal ( z n +1 ) (cid:48) isdrawn according to ( z ( n +1) ) (cid:48) = ( z ( n +1) ) g + X, for X ∼ N n +1 (0 , Σ) , where the ( n + 1) × ( n + 1)-covariance matrix Σ is chosen to reach some desired acceptance rate.2. Set ( z ( n +1) ) ( g +1) = (cid:26) ( z ( n +1) ) (cid:48) with prob. α (( z ( n +1) ) g , ( z ( n +1) ) (cid:48) ) z ( n +1) with prob. 1 − α (( z ( n +1) ) g , ( z ( n +1) ) (cid:48) ) , where the acceptance-probability α ( z ( n +1) , ( z ( n +1) ) (cid:48) ) is given by α ( z ( n +1) , ( z ( n +1) ) (cid:48) ) = min (cid:40) , ˜ f ( t − t n , z (cid:48) n +1 , z (cid:48) n ) g t n (( z ( n ) ) (cid:48) | y ( n ) , τ c > t n )˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | y ( n ) , τ c > t n ) (cid:41) = min (cid:40) , ˜ f ( t − t n , z (cid:48) n +1 , z (cid:48) n ) b n (( z ( n ) ) (cid:48) | y ( n ) )˜ f ( t − t n , z n +1 , z n ) b n ( z ( n ) | y ( n ) ) (cid:41) .
3. Discard the draws from the first n iterations and save the sample ( z ( n +1) ) n +1 , . . . , ( z ( n +1) ) n + G .The acceptance probability involves the term b n (( z ( n ) ) (cid:48) | y ( n ) ) b n ( z ( n ) | y ( n ) ) . It follows from Equation (4.3) that thisfraction is explicitly given by b n (( z ( n ) ) (cid:48) | y ( n ) ) b n ( z ( n ) | y ( n ) ) = (cid:81) ni =1 ψ ( z (cid:48) i − − z c , z (cid:48) i − z c , σ √ t i − t i − ) p Z ( z (cid:48) i | z (cid:48) i − ) p U ( y i − z (cid:48) i | y i − − z (cid:48) i − ) (cid:81) ni =1 ψ ( z i − − z c , z i − z c , σ √ t i − t i − ) p Z ( z i | z i − ) p U ( y i − z i | y i − − z i − ) , under the convention that t = 0 and p U ( ·| u ) = p U ( · ) is a Gaussian density with mean µ (cid:15) and variance σ (cid:15) . Note that p Z ( z i | z i − ) is a Gaussian density with mean z i − + m ( t i − t i − ) and variance σ ( t i − t i − ),that p U ( u i | u i − ) is a Gaussian density with mean κu i − + µ (cid:15) and variance σ (cid:15) and that an expression for ψ is provided in Lemma 4.1. Algorithm 5.1, in combination with Equations (5.2) and (5.1), allows ustwo compute all expressions which are of the form of an integral of a function, weighted by the density f ( t, · ).The second expression that occurs in the valuation of CoCos in the previous section, is the expressionwe see in the second part of the solution for a CoCo with a regulatory trigger and a conversion intoshares, as in Theorem 4.6. Which is the following double integral (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) h (˜ z )d˜ z d x. Note that this integral can be, similarly to the above, written as (cid:90) ∞ z c (cid:90) ( z c , ∞ ) n +1 h (˜ z ) ˆ f ( z n +1 , z c , ˜ z, T − t ) ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n )d z ( n +1) d˜ z = (cid:90) ( z c , ∞ ) n +2 h ( z n +2 ) ˆ f ( z n +1 , z c , z n +2 , T − t ) ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n )d z ( n +2) . (5.3)Now note that, by definition of ˆ f , it holds that (cid:90) ∞ z c ˆ f ( z n +1 , z c , z n +2 , T − t )d z n +2 = (cid:90) ∞ z c P (cid:18) inf ≤ s ≤ T − t Z s > z c , Z T − t ∈ d z n +2 (cid:12)(cid:12)(cid:12) Z = z n +1 (cid:19) P (cid:18) inf ≤ s ≤ T − t Z s > z c | Z = z n +1 (cid:19) = 1 − π ( T − t, z n +1 − z c ) . Hence, ˆ f ( z n +1 , z c , z n +2 , T − t ) ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n ) is not a density function on( z c , ∞ ) n +2 , so it is not possible to proceed in the same way as in the previous case. However, bythe above we know thatˆ f ( z n +1 , z c , z n +2 , T − t ) ˜ f ( t − t n , z n +1 , z n ) g t n ( z ( n ) | Y ( n ) , τ c > t n )1 − π ( T − t, z n +1 − z c )is a density function on ( z c , ∞ ) n +2 .So if we have a sample (( z ( n +2) ) , . . . , ( z n +2 ) G ) from the ( n + 2)-dimensional distribution with thisdensity, we can approximate the integral in Equation (5.3) by1 G G (cid:88) g =1 h ( z gn +2 )(1 − π ( T − t, z gn +1 − z c )) . (5.4)This sample is, in analogy to Algorithm 5.1, obtained by the following MCMC-algorithm. Algorithm 5.2
1. In each iteration g , g = 1 , . . . n + G , given the current value ( z ( n +2) ) g , the proposal ( z n +2 ) (cid:48) isdrawn according to ( z ( n +2) ) (cid:48) = ( z ( n +2) ) g + X, for X ∼ N n +2 (0 , Σ) , where the ( n + 2) × ( n + 2)-covariance matrix Σ is chosen to reach some desired acceptance rate.2. Set ( z ( n +2) ) ( g +1) = (cid:26) ( z ( n +2) ) (cid:48) with prob. α (( z ( n +2) ) g , ( z ( n +2) ) (cid:48) ) z ( n +2) with prob. 1 − α (( z ( n +2) ) g , ( z ( n +2) ) (cid:48) ) , where the acceptance-probability α ( z ( n +2) , ( z ( n +2) ) (cid:48) ) is given bymin (cid:40) , ˆ f ( z (cid:48) n +1 , z c , z (cid:48) n +2 , T − t ) ˜ f ( t − t n , z (cid:48) n +1 , z (cid:48) n ) b n (( z ( n ) ) (cid:48) | y ( n ) )(1 − π ( T − t, z n +1 − z c ))ˆ f ( z n +1 , z c , z n +2 , T − t ) ˜ f ( t − t n , z n +1 , z n ) b n ( z ( n ) | y ( n ) )(1 − π ( T − t, z (cid:48) n +1 − z c )) (cid:41) .
3. Discard the draws from the first n iterations and save the sample ( z ( n +2) ) n +1 , . . . , ( z ( n +2) ) n + G .The last type of expression that occurs in the valuation of CoCos in the previous section, are the i -stepsurvival probabilities in the valuation of a CoCo with an accounting report trigger, as in Theorem 4.8and Theorem 4.9. This expressions are in Equation (4.20) and Equation (4.25) given in the form (cid:90) R n P ( ξ ( z n ) ∈ Ξ i ) p Z ( z ( n ) | y ( n ) )d z ( n ) , for a set Ξ i which equals ( y c , ∞ ) i or ( y c , ∞ ) i − × ( y cc , ∞ ) and a multivariate normally distributedrandom variable ξ ( z n ). For a sample (( z ( n ) ) , . . . , ( z ( n ) ) G ) from p Z ( z ( n ) | y ( n ) ), this type of integral canbe approximated by 1 G G (cid:88) g =1 P ( ξ ( z gn ) ∈ Ξ i ) . (5.5)The necessary sample is again obtained using a MCMC-algorithm, as follows.17 lgorithm 5.3
1. In each iteration g , g = 1 , . . . , n + G , given the current value ( z ( n ) ) g , the proposal ( z n ) (cid:48) is drawnaccording to ( z ( n ) ) (cid:48) = ( z ( n ) ) g + X, for X ∼ N n (0 , Σ) , where the n × n -covariance matrix Σ is chosen to reach some desired acceptance rate.2. Set ( z ( n ) ) ( g +1) = (cid:26) ( z ( n ) ) (cid:48) with prob. α (( z ( n ) ) g , ( z ( n ) ) (cid:48) ) z ( n ) with prob. 1 − α (( z ( n ) ) g , ( z ( n ) ) (cid:48) ) , where the acceptance-probability α ( z ( n ) , ( z ( n ) ) (cid:48) ) is given by α ( z ( n ) , ( z ( n ) ) (cid:48) ) = min (cid:26) , p Z (( z ( n ) ) (cid:48) | y ( n ) ) p Z ( z ( n ) | y ( n ) ) (cid:27) = min (cid:26) , (cid:81) ni =1 p Z ( z (cid:48) i | z (cid:48) i − ) p U ( y i − z (cid:48) i | y i − − z (cid:48) i − ) (cid:81) ni =1 p Z ( z i | z i − ) p U ( y i − z i | y i − − z i − ) (cid:27) .
3. Discard the draws from the first n iterations (because the Markov chain needs a burn-in periodto converge to the target distribution) and save the sample ( z ( n ) ) n +1 , . . . , ( z ( n ) ) n + G . In this section we use the model to shed light on a variety of questions related to the basic valuationmodel itself and its sensitivity to design and “environmental” variables such as volatility shocks. Wethen explore the interaction between CoCos and other elements of the capital structure, and in particularlook at risk taking and investment incentives when CoCos are used instead of other types of funding,like straight debt or equity. Finally we use the fact that we incorporate the MDA trigger and the couponpayment contingency by comparing the Deutsche Bank profit scare and its impact on CoCo prices withmodel predictions we obtain with our valuation model.Parameter ValueInitial asset value V n , the number of accounting reports until time t v c v b α P P c T t+5Drift asset process m σ µ (cid:15) σ (cid:15) .1 Parametrization of the base case Table 6 lists the values of the base case parameters. For the choice of the base case parameters, somerestraints should be taken into account. For example, the conversion trigger should be higher than thedefault trigger. Also, a CoCo should pay a higher coupon than straight debt, to compensate for thehigher risk. Furthermore, we have no empirical evidence for a reasonable level of accounting noise, sowe set the volatility of accounting noise equal to the base case parameter chosen by Duffie and Lando(2001), where the accounting noise variance is chosen to match short run default probabilities implicitin short run CDS spreads.In the base case, we will assume the CoCo has a regulatory trigger, i.e. the regulator has access to thetrue state of the bank and conversion can take place at any time, not just at accounting dates, but themarket has to evaluate conversion probabilities given this trigger rule using accounting information only(cf. Section 4.3.2 for the mathematics of this trigger). We will also explore other trigger mechanisms.Furthermore, we define the dilution ratio ρ (cf. Section 4.3.2) as the fraction shares owned by the CoCoholder post-conversion: ρ = ∆ P ∆ P + 1 (6.1)where P is the face value of the CoCo before conversion, and ∆ equals the number of shares the CoCoholder receives at conversion. The number of old shares is normalized to 1. A dilution ratio of ρ = 0means that the CoCo suffers a principal write-down (PWD) at conversion, while ρ = 1 corresponds tothe extreme case that the original shareholders are completely wiped out at conversion.To compute prices for PWD CoCos, we make use of Theorem 4.4. The integral involved is approximatedas in Equation (5.2), for which the necessary sample is obtained by using Algorithm 5.1. To computeprices for CoCos with a conversion into shares, we make use of Theorem 4.6, where the first term inthe pricing formula follows again by using Algorithm 5.1 and the second term is approximated as inEquation (5.4), for which the necessary sample is obtained by execution of Algorithm 5.2. Then thefigures are produced by repeatedly following this procedures for different values of the parameters. Volatility of Assets C o C o p r i c e ρ = 0 (PWD) ρ = 0.5 ρ = 1 Figure 1: CoCo Prices and asset volatility for different CoCo design parameters19 .2 Asset Volatility, Accounting Noise and CoCo design parameters
Volatility accounting noise C o C o p r i c e Regulatory triggerAccounting trigger
Figure 2: Accounting noise and trigger designIn this subsection we study the impact of changes in asset volatility and accounting noise on CoCoprices as a function of different design parameters.
We first look at the price impact of changes in volatility of the underlying asset value process fordifferent CoCo designs. In Figure 1 several CoCo prices are plotted against the volatility of assets σ , seeEquation (4.1). The solid line corresponds to a PWD CoCo. Clearly, the price of a PWD CoCo decreaseswhen assets become more volatile. This is of course as one would expect, as a higher σ increases theprobability of the principal write-down happening, causing the CoCo price to decrease. The dashed line,corresponding to ρ = 0 .
5, shows already that this negative effect from volatility on the CoCo price isweaker when terms of conversion are more favorable to the CoCo investor in that her loss is lower, atleast some shares are received after conversion, although not yet enough to compensate for the loss ofprincipal. In the extreme case that shareholders are completely wiped out at conversion, correspondingto the dashed-dotted line, this negative effect is even partially reversed. In this case, the price firstincreases with volatility as the (now favorable) conversion becomes more likely. However, for highervolatility levels the increasing probability of default and associated costs of bankruptcy push the pricedown again.
We next consider the relationship between accounting noise σ (cid:15) and the price of a CoCo. In Figure 2,CoCos with a regulatory trigger and CoCos with an accounting based trigger are considered. Thebook value CoCo is priced by the formula given in Equation (4.23); This value is computed using theapproximation in Equation (5.5), for which the necessary samples are obtained by using 5.3.Figure 2 shows the importance of taking into account the trigger design for the pricing of the CoCo.The increase in accounting volatility has almost no impact on the value of the CoCo with a regulatorytrigger (the solid line in Figure 2); but the dashed line shows that the value of CoCos with a trigger20hat depends on accounting reports, the CoCo price is seriously (and obviously negatively) affected byaccounting noise. This is in line with the results of Duffie and Lando (2001): they find that the defaultprobability increases when the reports become more noisy. In our CoCo setting, this means that theprobability of conversion increases when σ (cid:15) increases, causing the CoCo price to go down. Figure 2 showsthat the price of a CoCo with the trigger depending on accounting reports (slotted line) is much moresensitive to accounting noise than the price of a CoCo with a regulatory (PONV) trigger (solid line). Consider next the impact of the correlation coefficient κ in the accounting noise error term. In Figure 3we show the price response of a PWD CoCo to a bad news accounting report. The set up is as follows.After the first report ( Y = log 100), a second report is issued: Y = log 85. The conversion trigger is setat log 80, with a PONV trigger type. The plots show a clear and immediate price response to the arrivalof the bad news. Days after release accounting report -2 -1 0 1 2 3 4 5 6 7 C o C o p r i c e =0.01 =0.3 =0.99 Figure 3: Price response to “bad accounting news” for different values of the autocorrelation parameter κ . Interestingly, a clear pattern emerges if the exercise is repeated for different values of the autocorre-lation parameter κ : although the pattern is similar over the entire range from almost no correlation inaccounting noise ( κ = 0 .
01) to almost complete persistence of accounting noise innovations ( κ = 0 . κ means that more of the past noise arrivals survive in the current one, while at the same time thevariance of the accounting noise term U i increases with κ , as it is, in a stationary regime, proportionalto 1 / (1 − κ ). This in turn lowers the information value of accounting news and explains why a bad (i.e.worse than the previous one) report leads to a smaller negative price response for higher κ : the signalis less informative so triggers a smaller price response. In Figure 4 we report on a different experiment: we show how different CoCo designs are influenced bytime lapsed since the last accounting report. The plot shows the value of three differently structured21oCo’s, each with a different degree of shareholder dilution after conversion as a function of time lapsedsince the last accounting report. The black line represents a PWD CoCo where the CoCo is writtenoff upon conversion and no subsequent dilution of the old shareholder takes place; the other two linesrepresent equity converters, one with partial dilution of the old shareholder ( ρ = 0 . ρ = 1), the dashed line.The plots show very little impact on the PWD CoCo while the two equity converters decline in value Months since last accounting report C o C o p r i c e ; =0 (PWD CoCo) ; =0.5 ; = 1 Figure 4: Time since last accounting report.as the time since the last accounting report increases. A longer time lapse does not change the assetprice dynamics but leads to a higher uncertainty as to where the asset value is at the time of valuation.This is similar to moving more weights in the tails and thus a larger probability of bankruptcy. Sincebankruptcy follows conversion, a higher probability of bankruptcy does not influence the PWD, theywill then already have lost everything because conversion precedes bankruptcy. But the more shares theCoCo holder receives upon conversion, the more she loses from a subsequent bankruptcy, so the pricedecline increases more for higher values of the dilution parameter ρ . Consider next the impact on pricing of the main characteristics of the CoCo design: the trigger leveland the number of shares received upon conversion.
In Figure 5, the CoCo price is plotted against the conversion trigger for different degrees of dilution.The solid line corresponds to a PWD CoCo, the other lines to CoCos with varying degrees of dilution ofthe original shareholders upon conversion as specified in the legend.As one would expect, the price of a PWD CoCo (the solid line) is lower for a higher conversion trigger : Note that the trigger is defined as a percentage of the asset value with the losses coming from the top (i.e. equityabove debt on the liability side), so a higher trigger value means a higher probability of conversion, as is done in the restof the academic literature. In the banking and supervision literature, it is more conventional to define the trigger valuealso as a percentage of (risk weighted) assets, but with the losses coming from the bottom, with equity below debt; in thatdefinition a higher trigger ratio leads to a lower probability of conversion. onversion trigger
75 80 85 90 95 100 C o C o p r i c e ρ = 1 ρ = 0.5 ρ = 0.1 ρ = 0 (PWD) Figure 5: Conversion Triggera higher conversion trigger increases the probability of a principal write-down and its associated loss ofprincipal. However, the other lines show that if conversion terms are more favorable to the CoCo investor,the impact of the trigger level changes: price will increase with the conversion trigger if the dilution ratiofavors the CoCo holder enough. In the extreme case that the dilution ratio ρ equals one (the dashed linein Figure 5), the CoCo price goes up with the conversion trigger. For less extreme dilution parameters,this positive effect is weaker, and the corresponding lines are in between the two extremes (no dilutionversus complete dilution). In Figure 6, the price of a CoCo is plotted against ∆, the number of shares received at conversion perunit of principal, for different values of straight debt in the firm’s capital structure. The case ∆ = 0corresponds to a principal write-down CoCo, while ∆ = ∞ corresponds to the case in which all of theoriginal shareholders are wiped out at conversion and the CoCo investors are then the only shareholdersleft. Figure 6 clearly shows that the CoCo price increases with ∆. This is of course as expected, asa higher ∆ means a higher payout at conversion. Furthermore, the figure shows that a CoCo with aconversion into shares has a higher price when there is a lower amount of straight debt issued. Hencethe CoCo is more valuable when the firm has a lower leverage. This can also easily be explained, as theCoCo investors receive a fraction of the firm’s equity value at conversion and the equity value is higherin case there are less liabilities.The lines for different leverage converge to the same point on the vertical axis as ∆ →
0; for a PWDCoCo, leverage has no impact on the price since both the CoCo and equity are junior to debt. Thisresult does depend on the assumption that the variance of the asset value process is exogenously chosen;if it would be endogenously chosen, higher leverage would lead to more risk taking and a higher variance,which would have an impact on the value of the CoCo even if it has a PWD structure (this point is madein Chan and van Wijnbergen (2016; revised November 2017)).23
100 200 300 400 500 600 700 800 900 1000 C o C o p r i c e Debt: 40, CoCo: 5Debt: 45, CoCo: 5Debt: 50, CoCo: 5
Figure 6: CoCo prices against dilution, for different leverage ratios.
Consider next the impact of the issuance of CoCos on the capital structure of the bank and, deducedfrom that, on incentives for shareholders . First consider the case in which straight debt is replacedwith CoCos. In Figure 7 we show the change in equity value (on the vertical axis) as a consequenceof replacing 5 units of straight debt with 5 units of CoCos, set off against different trigger prices. Thedifferent lines correspond to different degrees of the dilution parameter ρ , again ranging from 0 to 1(from no dilution at all to infinite dilution). The solid line and the dotted line indicate that shareholdersonly benefit from replacing debt with CoCos when the terms of conversion are favorable enough to theshareholders and the trigger is high enough. For low trigger ratios, the conversion possibility becomesvery small and the exercise comes down to swapping debt for debt. That actually turns out to have anegative impact on equity values because CoCos then are just a more expensive form of debt so replacingdebt with CoCos then actually destroys equity value. As the trigger ratio goes up (move to the right inFigure 7), the probability of getting the benefit of wiping out the CoCo debt at favorable terms becomesmore likely and starts to dominate, hence the positive sign for high trigger ratios. Of course that secondeffect does not take place for highly dilutive CoCos, for low probability of conversion the impact of thedebt for CoCo swap is negative, as with non-dilutive CoCo’s. But as the conversion trigger rises andwith it the conversion probability, the negative impact of a highly dilutive conversion comes closer, so theprice impact turns even more negative. So the dashed line and the dashed-dotted line (highly dilutivecases) show that shareholders have no incentive to swap debt for highly dilutive CoCos, and increasinglyless so as the probability of conversion increases with higher trigger levels (academic convention). Consider next a change in capital structure in the other direction, where equity instead of debt is replacedby CoCos. Specifically, we assume a CoCo is issued and the proceeds are used to buy back equity at The computation of the prices and the production of the figures is performed following the same procedures as inSection 6.3. onversion trigger
75 80 85 90 95 100 C hange i n equ i t y v a l ue -20-15-10-505 ρ = 0 (PWD) ρ = 0.1 ρ = 0.5 ρ = 1 Figure 7: Change in Equity Value when 5 units of debt are replaced with 5 units of CoCo (in marketvalue terms).market value. The consequences on equity values are shown in Figure 8, again for different trigger levels(on the horizontal axis with the different lines representing different degrees of dilution after conversion).The pattern is very similar to the debt for CoCo swaps analyzed in Figure 7. Equity holders havea strong incentive to issue PWD CoCos with ρ = 0 instead of new equity (or even issue PWD CoCosto buy back debt as is done in this policy experiment), since they actually gain on conversion. Howeverfor lower trigger ratios the probability of conversion becomes too small, turning CoCos de facto intoexpensive debt, so the impact on equity value turns negative for low values of the trigger ratio. Andequity holders will never want to issue dilutive CoCos ( ρ = 1 is the extreme case with infinite dilution)for any level of the trigger ratio: before conversion these CoCos are an expensive form of debt and afterconversion or rather at conversion time equity holders will actually loose out when conversion takes place,making the instrument unambiguously unattractive to shareholders when structured this way. Theseresults may well explain why some 60% of all CoCos are PWD CoCos, cf. Avdiev et al. (2017), insteadof the dilutive CoCos favored by the academic literature (Calomiris and Herring (2013) is an early andeloquent example of what is a widely shared view in the academic literature arguing CoCos should behighly dilutive). Debt overhang arises when the firm’s loss absorption capacity has become too low to protect the debthold-ers from fluctuations in asset values (cf. Merton (1974), Myers (1977)), possibly to the point of arrearshaving emerged already. One consequence of debt overhang is that investment incentives are reducedfor equity holders, since part of the benefits of a new project will in effect have to be shared with thecreditors. Even if there are no actual arrears yet, but debt is trading under par, part of the asset valueincrease will go into increased market value of the debt, at the (partial) expense of a higher marketvalue for equity. In a structural model without CoCos, the shareholders then do not have an incentiveto invest exactly at the moment the firm most needs an increase in asset values, i.e. when the firm isnear bankruptcy. Almost all of the value of the investment will then be captured by the debt holders,as the value of debt increases when the probability of a bankruptcy is reduced. In which way CoCos25 onversion trigger
75 80 85 90 95 100 C hange i n equ i t y v a l ue -16-14-12-10-8-6-4-2024 ρ = 0 (PWD) ρ = 0.1 ρ = 0.5 ρ = 1 Figure 8: Change in equity value when 5 units of equity are replaced with 5 units of CoCos (in marketvalue)interact with a situation with debt overhang is an interesting question; CoCos introduce additional lossabsorption capacity which is good for debt holders, but CoCos may also have their own impact on equityvalues. Debt holders may also profit in another way in that, depending on the design of the CoCos,shareholders may have an increased incentive to make an investment to avoid conversion.The debt overhang and incentive issue can be looked at within the context of our model by lookingat what happens when assets are increased by one unit, financed through one unit of equity (issued atmarket value). If the total market value of equity goes up by more than one unit, the shareholders wouldmake a profit when they invest, giving them an incentive to do so. However, when equity increases byless than one unit, the investment is not beneficial to shareholders to offset the expense, all or part ofthe benefits are apparently captured by debt holders. We therefore consider the case in which a newaccounting report has just be released, with an asset value, see Equation (4.2), of Y t n = 100; we canthen examine what happens when this asset value increases by one unit. The profit of this investmentof one unit is plotted against the conversion trigger in Figure 9.The solid black line is our benchmark case with only straight debt in addition to equity. Thesimulation shows the impact of debt overhang: without CoCos (the solid line) the shareholders do notmake a profit when they invest, they actually suffer a small loss. The dashed-dotted lines show that whenthe terms of conversion are favorable to shareholders (i.e. CoCo holders loose out upon conversion), theshareholders have even less of an incentive to engage in additional investment, actually worsening thedebt overhang problem. The black dashed-dotted line corresponds to the existence of a PWD CoCo inthe capital structure of the firm and shows that the PWD CoCo indeed makes the investment incentivefor shareholders more negative, especially close to the conversion trigger. So the strongest increase inDebt Overhang is with the CoCos that most favors shareholders, the CoCos with a principal write-down. The same happens to a somewhat lesser degree with CoCos at slightly less non-dilutive termsbut still favorable to shareholders. Thus PWD or insufficiently dilutive CoCos are not capable of solvingthe problem of debt overhang. However, highly dilutive CoCos do strengthen shareholders’ incentivesto invest because they want to avoid conversion. See in particular the dashed lines in Figure 9, whichcorrespond to highly dilutive CoCos; clearly such CoCos improve the shareholders’ investment incentivesbecause they wish to avoid conversion. Especially close to the conversion trigger, the shareholders have26 onversion trigger
75 80 85 90 95 100 P r o f i t o f i n v e s t m en t -1012345 No CoCos ρ = 0 (PWD) ρ = 0.1 ρ = 1 ρ = 0.5 Figure 9: Debt overhang, CoCos and investment incentivesin this case a substantive incentive to invest in a last attempt to avoid the unfavorable conversion. Tosummarize, when terms of conversion are beneficial enough to CoCo investors instead of favoring theold shareholders, CoCos are capable of creating more of an investment incentive for the shareholders.However, PWD CoCos and in general less dilutive CoCos actually lead to lower investment incentivesand worsen the debt overhang problem when compared to straight debt.
In the literature it is generally assumed that coupons are paid until conversion. However coupon paymentsare affected by the so called Maximum Distributable Amount trigger, under which regulators stop thepayment of coupons (and dividends) when the firm’s capital value falls below some trigger that is higherthan the conversion trigger. Coupon payments can start again when the capital value goes back up andexceeds the trigger value again. This means that in the valuation of a CoCo, we can apply Theorem 4.6and Algorithm 5.2, but with the coupon term defined as in Equation (4.13). To demonstrate the relevanceof the inclusion of this trigger in the valuation of CoCos, we will look at the big price drop that theCoCos of Deutsche Bank suffered at the beginning of 2016. On January 28 Deutsche Bank reported anet loss of 2.1 Billion EUR over the last quarter of 2015. The relevant report furthermore reported for itsRisk-Weighted Assets a value of 397 Billion EUR, down from 408 Billion EUR in the previous accountingreport. Also, the Common Equity Tier 1 (CET1) ratio, defined as the fraction of the common equityand the risk weighted asset (RWA), fell from 11.5% to 11.1%, primarily reflecting the net loss over thequarter. The information is taken from the Financial Data Supplement 4Q2015, Deutsche Bank (2016),the report that caused a big downward move in the price of the CoCos of Deutsche Bank.At this time, Deutsche Bank had four different CoCos issued (two in USD, one in EUR, one in GBP,all PWD CoCos). To avoid having to deal with an additional exchange rate risk factor, we will onlyconsider the EUR CoCo. This CoCo’s write-down is triggered when the CET1-ratio hits the level of5.125% and it pays a coupon of 6%. As is clear from the above, the CET1-ratio did not even comeclose to the low trigger level. Still, the CoCo price tumbled 19.5% percent within the week after theannouncement of the report. Market publications at the time widely argued that this happened out27f fear for reaching the MDA trigger and the subsequent cancelling of coupon payments. The modeldeveloped in this paper is particularly relevant to analyze this case, as we can include the announcementof a bad accounting report in the valuation, as well as the early cancelling of coupons when the MDAtrigger is hit. The precise value of the MDA trigger is not publicly known, so it is not possible to usethe real value of the MDA trigger. However, it is still interesting to examine how much of a price dropthe model can explain by taking the MDA trigger close to the reported values. Unless stated otherwise,we use the same parameters as in Table 6. Before the bad accounting report arrives, we assume thereis one accounting report, with a value Y t = EUR 408 bn. Then the new accounting report arrives, sowe now have two accounting reports with values Y t =EUR 408 bn and Y t =EUR 397 bn. The triggersare chosen such that they correspond with CET1 ratios at the moment of the accounting report. Thatis, we choose v c such that it corresponds to a CET1 ratio of 5.125%. We know the CET1 ratio is 11.1%where RWA is EUR 397 bn, so the total amount of debt (only CoCos and straight debt in the model) isEUR 397 bn × v c .The value of the MDA trigger v cc can be chosen in the same way, a MDA trigger at a CET1 ratio of 10%would correspond to a RWA value of EUR 352.93/(1-0.1) bn = EUR 392 bn. The coupon of the CoCo is c = 0.06. As the relevant CoCo has a perpetual maturity, we choose the first call date, 10/10/18, as thematurity. Because we assumed that the second accounting report arrives at 01/28/16, t=0 correspondsto 07/28/15. Hence T = 3 + 2/12 + 13/365. In Figure 10 the price change after the announcement ofa bad accounting report is illustrated for different choices of the MDA trigger. Weeks after release of accounting report -3 -2 -1 0 1 2 3 C o C o p r i c e No MDA trigger10%11%12%
Figure 10: CoCo price after the release of the bad accounting report for different values of the MDAtrigger.The solid line corresponds to the case where the MDA-trigger is not included in the model, in thiscase only a drop of 14.3% in the CoCo price occurs, when looking at the price just before the releaseof the accounting report and afterwards. However, if we add the MDA trigger to the model, a strongernegative price change follows. The dashed line corresponds to the case that we take the MDA triggerat 11%, i.e. just beneath the reported CET1 value. This gives a price drop of 18.7%. If we take theMDA trigger to be 10%, the price drops by 17.3%, which is illustrated by the dotted line. However, wehave to take the MDA trigger above the reported CET1 ratio of 11.1% to create a price drop of 19.5%,cf. the dashed-dotted line. That is, a price drop of 19.5% corresponds in the model to the situation that28he MDA trigger is already breached, which was not the case. However, it is clear that a significant partof the price change is driven by the MDA trigger, not by the conversion trigger. The above illustratesthe added value of explicitly incorporating accounting reports into the analysis and taking the MDAtrigger into account in the valuation of a CoCo, especially when the MDA trigger is coming close, butthe conversion trigger is still far away.
CoCos are debt instruments that are written down or converted into equity when the value of the issuingbank becomes too low. CoCos have taken European capital markets by storm. Over 560 bn Euro hasbeen issued over the past five years, with more likely to come. Apparently banks see CoCos as anattractive alternative to issuing new equity when faced with a capital shortage. The academic literaturehas rapidly developed attempting to analyse and price the new debt instruments, but at the same timea remarkable divergence has opened up between this academic literature and the type of CoCos issuedin actual practice. Without exception, the academic literature argues for conversion triggers based onmarket values instead of accounting ratios. Accordingly, with the exception of Glasserman and Nouri(2012), the asset pricing literature on CoCos has analysed market based conversion triggers only. Yet, atleast in the European Union and Switzerland, market based triggers disqualify the instrument as capitalunder EU regulation, so without a single exception all CoCos issued so far base their conversion triggeron accounting ratios. In addition, they have to keep open the possibility of regulatory intervention whena so called Point of Non-Viability is reached. Moreover, the literature has paid no attention to thetriggers in place for suspension of coupon payments, although those triggers have most likely causedmost of the recent volatility in CoCo prices. In this paper we bridge the gap between the academicliterature and actual practice by explicitly introducing coupon suspension triggers and, arguably moreimportant, explicitly introducing key features of the accounting process in the model which allows us toanalyse accounting ratio based conversion triggers and PONV interventions in a meaningful way.In order to do so we model the basic stochastic process driving asset values as a standard geometricBrownian Motion, as does most of the literature. Where we diverge is in our assumption that theprocess is not directly observable. Instead information reaches the market based on noisy accountingreports issued at regular but discrete time epochs only. In this way we can take into account differencesbetween accounting values and market values. Our model is based on the premise that the market cannotobserve the true asset value process, it only has access to noisy accounting reports which moreover, areonly published at discrete moments in time. In this way, the price of CoCos can only be based on theinformation from the accounting reports, not on the underlying true asset process as this is not observeddirectly. The model does not lead to closed form solutions for CoCo prices, but Markov Chain MonteCarlo methods are used to compute prices.The model was remarkably successful in reproducing the price response of CoCos to a widely reportedadverse profit warning issued by Deutsche Bank in February 2016. This exercise has shown the impor-tance of incorporating the so called MDA trigger in valuation models, the trigger that governs suspensionof coupon payments. Using the model as a tool of analysis yields a rich set of results on the relationbetween valuation, CoCo design and environment variables such as asset volatility and accounting noise.Moreover, we have shown that CoCos depending on their design have a significant impact on share-holder incentives to take on additional risk or on investment incentives in situations of debt overhang,and interact in interesting ways with the capital structure of the bank issuing them. The various results,such as the attractiveness of PWD CoCos for equity holders, can help explain the design choices madein practice where about 60% of all CoCos issued are of that variety. The explicit incorporation of theaccounting process as providing noisy reports on the underlying unobservable firm fundamentals, whichare issued at regular but discrete time instants, allows us to analyse CoCo designs based on accountingratios as well as triggers based on reaching a so called Point of Non-Viability. The relation between risktaking incentives, leverage and CoCo design should be of interest to regulators. We show that CoCos,again depending on their design features, may significantly change the sensitivity of equity values to risk,thereby possibly opening the door to risk arbitrage for given capital requirements, taking into account29sset characteristics only, as is the case under the BIS based capital regime.
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A Appendix
In this Appendix, all the mathematical details and proofs that are left out in the main text, are provided.
Proof of Lemma 4.2 ˜ f ( t, · , z ) is defined by P ( Z t ∈ d x | τ b > t ) = ˜ f ( t, x, z )d x. By Bayes’ rule we can write P ( Z t ∈ d x | τ b > t ) = P ( Z t ∈ d x, τ b > t ) P ( τ b > t ) . The denominator of this expression is given by P ( τ b > t ) = 1 − π ( t, z − z b ) = Φ (cid:18) z − z b + mtσ √ t (cid:19) − e − m ( z − z b ) /σ Φ (cid:18) z b − z + mtσ √ t (cid:19) . In order to compute the numerator, we will rely on the following result by Harrison (1985), which canbe found in Section 1.8, Proposition 1. Denote by X t a Brownian motion with drift µ , variance σ and X = 0. Furthermore define M t := max { X s : 0 ≤ s ≤ t } . Then the joint distribution of X t and M t satisfies P ( X t ∈ d x, M t ≤ y ) = 1 σ √ t exp (cid:18) µxσ − µ t σ (cid:19) (cid:18) φ (cid:18) xσ √ t (cid:19) − φ (cid:18) x − yσ √ t (cid:19)(cid:19) d x, (A.1)where φ denotes the standard normal density function. Now, denote X t = − Z t + z , which is a Brownianmotion with drift − m , variance σ and X = 0. Furthermore, denote M t = max { X s : 0 ≤ s ≤ t } . ThenEquation (A.1) implies that P ( Z t ∈ d x, τ b > t ) = P (cid:18) Z t ∈ d x, inf ≤ s ≤ t Z s > z b (cid:19) P ( X t ∈ d( z − x ) , M t ≤ z − z b )= 1 σ √ t exp (cid:18) − m ( z − x ) σ − m t σ (cid:19) (cid:18) φ (cid:18) z − xσ √ t (cid:19) − φ (cid:18) − z − x + 2 z b σ √ t (cid:19)(cid:19) d x. (A.2)So we conclude that˜ f ( t, x, z ) = 1 σ √ t exp (cid:16) − m ( z − x ) σ − m t σ (cid:17) (cid:16) φ (cid:16) z − xσ √ t (cid:17) − φ (cid:16) − z − x +2 z b σ √ t (cid:17)(cid:17) Φ (cid:16) z − z b + mtσ √ t (cid:17) − e − m ( z − z b ) /σ Φ (cid:16) z b − z + mtσ √ t (cid:17) . (cid:3) Proof of Theorem 4.4
Recall that the CoCo price was written as C ( t ) = P e − r ( T − t ) p c ( t, T ) + cP (cid:90) Tt e − r ( u − t ) p c ( t, u )d u − RP (cid:90) Tt e − r ( u − t ) p c ( t, d u ) . (A.3)The integral in this last term can be written as (cid:90) Tt e − r ( u − t ) p c ( t, d u ) = (cid:90) Tt e − r ( u − t ) ∂∂u p c ( t, u )d u = (cid:90) Tt e − r ( u − t ) (cid:90) ∞ z c ∂∂u (1 − π ( u − t, x − z c )) f ( t, x, ω )d x d u = (cid:90) ∞ z c f ( t, x, ω ) (cid:90) Tt e − r ( u − t ) ∂∂u (1 − π ( u − t, x − z c ))d u d x = (cid:90) ∞ z c f ( t, x, ω ) I ( x )d x, where I ( x ) = (cid:90) Tt e − r ( u − t ) ∂∂u (1 − π ( u − t, x − z c ))d u. Furthermore, the integral in the second term of Equation (A.3) can be written as (cid:90) Tt e − r ( u − t ) p c ( t, u )d u = (cid:90) Tt e − r ( u − t ) (cid:90) ∞ z c (1 − π ( u − t, x − z c )) f ( t, x, ω )d x d u = (cid:90) ∞ z c f ( t, x, ω ) (cid:90) Tt e − r ( u − t ) (1 − π ( u − t, x − z c ))d u d x = (cid:90) ∞ z c f ( t, x, ω ) ˜ I ( x )d x, where ˜ I ( x ) = (cid:90) Tt e − r ( u − t ) (1 − π ( u − t, x − z c ))d u = (cid:20) − r e − r ( u − t ) (1 − π ( u − t, x − z c )) (cid:21) Tu = t + 1 r I ( x )= − r e − r ( T − t ) (1 − π ( T − t, x − z c )) + 1 r + 1 r I ( x ) . Putting the above together allows us to write the CoCo price C ( t ) as a single integral, weighted by thedensity f ( t, x ), as follows C ( t ) = (cid:90) ∞ z c (cid:16) P e − r ( T − t ) (1 − π ( T − t, x − z c )) + cP ˜ I ( x ) − RP I ( x ) (cid:17) f ( t, xa )d x (cid:90) ∞ z c (cid:18) r − cr P e − r ( T − t ) (1 − π ( T − t, x − z c )) + cr P + (cid:18) cPr − RP (cid:19) I ( x ) (cid:19) f ( t, x )d x. It now remains to find an analytical expression for I ( x ). First consider ∂∂u (1 − π ( u − t, x − z c ))= ∂∂u (cid:18) Φ (cid:18) x − z c + m ( u − t ) σ √ u − t (cid:19) − e − m ( x − z c ) /σ Φ (cid:18) − ( x − z c ) + m ( u − t ) σ √ u − t (cid:19)(cid:19) = φ (cid:18) x − z c + m ( u − t ) σ √ u − t (cid:19) (cid:18) m σ √ u − t − x − z c σ ( u − t ) / (cid:19) − e − m ( x − z c ) /σ φ (cid:18) − ( x − z c ) + m ( u − t ) σ √ u − t (cid:19) (cid:18) m σ √ u − t + x − z c σ ( u − t ) / (cid:19) = z c − xσ ( u − t ) / φ (cid:18) x − z c + m ( u − t ) σ √ u − t (cid:19) , which implies I ( x ) = (cid:90) Tt e − r ( u − t ) z c − xσ ( u − t ) / √ π exp (cid:18) − ( x − z c + m ( u − t )) σ ( u − t ) (cid:19) d u = z c − x √ πσ exp (cid:18) − m ( x − z c ) σ (cid:19) (cid:90) T − t e − ru u / exp (cid:18) − ( x − z c ) σ u − m u σ (cid:19) d u = z c − x √ πσ exp (cid:18) − m ( x − z c ) σ (cid:19) (cid:90) T − t u / exp (cid:18) − ( x − z c ) σ u − (cid:18) m σ + r (cid:19) u (cid:19) d u = 2 z c − x √ πσ exp (cid:18) − m ( x − z c ) σ (cid:19) (cid:90) ∞ ( T − t ) − / exp (cid:18) − Av − B v (cid:19) d v, (A.4)where the last line follows by substitution of v = u − / and by setting A = ( x − z c ) σ , B = m σ + r. Now, by noting that ( Av + B/v ) = ( √ Av − √ B/v ) + 2 √ AB , as well as ( Av + B/v ) = ( √ Av + √ B/v ) − √ AB , the remaining integral can be evaluated, by doing the substitutions u = √ Av − √ B/v and u = √ Av + √ B/v , as follows (cid:90) ∞ ( T − t ) − / exp (cid:18) − Av − B v (cid:19) d v = 12 √ A (cid:90) ∞ ( T − t ) − / exp (cid:16) − ( √ Av − √ B/v ) − √ AB (cid:17) ( √ A + √ B v )d v + 12 √ A (cid:90) ∞ ( T − t ) − / exp (cid:16) − ( √ Av + √ B/v ) + 2 √ AB (cid:17) ( √ A − √ B v )d v = 12 √ A e − √ AB (cid:90) ∞ √ A/ ( T − t ) − √ B ( T − t ) e − u d u + 12 √ A e √ AB (cid:90) ∞ √ A/ ( T − t )+ √ B ( T − t ) e − u d u = √ π √ A (cid:16) e − √ AB erfc (cid:16)(cid:112) A/ ( T − t ) − (cid:112) B ( T − t ) (cid:17) + e √ AB erfc (cid:16)(cid:112) A/ ( T − t ) + (cid:112) B ( T − t ) (cid:17)(cid:17) , (A.5)33here erfc( x ) is the complementary error function, which is defined byerfc( x ) := 2 √ π (cid:90) ∞ x e − u d u and satisfies 12 erfc( x/ √
2) = 1 − Φ( x ) . Combining Equations (A.4) and (A.5) and substituting back the expressions for A and B , finally leadsto the expression for I ( x ): I ( x ) = 2 z c − x √ πσ exp (cid:18) − m ( x − z c ) σ (cid:19) (cid:90) ∞ ( T − t ) − / exp (cid:18) − Av − B v (cid:19) d v = exp (cid:18) − m ( x − z c ) σ (cid:19) (cid:18) − e − √ AB
12 erfc (cid:16)(cid:112) A/ ( T − t ) − (cid:112) B ( T − t ) (cid:17) − e √ AB
12 erfc (cid:16)(cid:112) A/ ( T − t ) + (cid:112) B ( T − t ) (cid:17)(cid:19) = exp (cid:32) − m ( x − z c ) + ( x − z c ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z c − √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) + exp (cid:32) − m ( x − z c ) − ( x − z c ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z c + √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) . (A.6) (cid:3) Proof of Theorem 4.6
The market price of the CoCos is given by C ( t ) = E (cid:16) P e − r ( T − t ) { τ c >T } |H t (cid:17) + E (cid:32)(cid:90) Tt c P e − r ( u − t ) { τ c >u } d u |H t (cid:33) + E (cid:18) ∆ P ∆ P + 1 E P C ( τ c ) e − r ( τ c − t ) { τ c ≤ T } |H t (cid:19) . (A.7)The first two terms together, are captured in the integral (cid:90) ∞ z c h ( x ) f ( t, x )d x, as is clear from taking R = 0 in the PWD case, cf. Theorem 4.4.Recall that the third term in Equation (A.7) was written as E (cid:18) ∆ P ∆ P + 1 E P C ( τ c ) e − r ( τ c − t ) { τ c ≤ T } |H t (cid:19) = ∆ P ∆ P + 1 e z c (cid:90) Tt e − r ( u − t ) P ( τ c ∈ d u | τ c > t, Y ( n ) ) − ∆ P c P ∆ P + 1 (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T ∧ u, τ b > u | τ c > t, Y ( n ) )d u − ∆ P ∆ P + 1 e z b (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T, τ b ∈ d u | τ c > t, Y ( n ) ) . (A.8)Note that the first integral in this equation is already computed in the proof of Theorem 4.4 and givenby e z c (cid:90) Tt e − r ( u − t ) P ( τ c ∈ d u | τ c > t, Y ( n ) ) = − e z c (cid:90) ∞ z c f ( t, x ) I ( x )d x, (A.9)34here I ( x ) is given by Equation (A.6).To compute the other integrals in Equation (A.8), it is sufficient to find expressions for P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) = y ( n ) ) and P ( τ c ≤ u, τ b > u | τ c > t, Y ( n ) = y ( n ) ) . In order to find expressions for this joint probabilities, we first need the following lemma.
Lemma A.1
The joint probability γ ( x, y, z, t , t ) that Z , starting from x , does not hit z before time t but does hit y before time t , is for x > y > z given by γ ( x, y, z, t , t ) = P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s ≤ y )= (cid:26) π ( t , x − y ) − π ( t , x − z ) for t ≤ t , − π ( t , x − z ) − (cid:82) ∞ y (1 − π ( t − t , ˜ z − z )) ˆ f ( x, y, ˜ z, t )d˜ z for t > t , where ˆ f ( x, y, ˜ z, t ) = 1 σ √ t exp (cid:18) − m ( x − ˜ z ) σ − m t σ (cid:19) (cid:18) φ (cid:18) x − ˜ zσ √ t (cid:19) − φ (cid:18) − x − ˜ z + 2 yσ √ t (cid:19)(cid:19) (A.10) Proof. • For t ≤ t , we can write P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s ≤ y ) = P ( inf ≤ s ≤ t Z s ≤ y ) − P ( inf ≤ s ≤ t Z s ≤ z, inf ≤ s ≤ t Z s ≤ y )= P ( inf ≤ s ≤ t Z s ≤ y ) − P ( inf ≤ s ≤ t Z s ≤ z )= π ( t , x − y ) − π ( t , x − z ) . • For t > t , note that P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s ≤ y ) = P ( inf ≤ s ≤ t Z s > z ) − P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s > y )= 1 − π ( t , x − z ) − P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s > y ) , where P ( inf ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s > y )= (cid:90) ∞ y P (cid:18) inf t ≤ s ≤ t Z s > z, inf ≤ s ≤ t Z s > y | Z t = ˜ z (cid:19) P ( Z t ∈ d˜ z )= (cid:90) ∞ y P (cid:18) inf t ≤ s ≤ t Z s − Z t > z − ˜ z (cid:19) P ( inf ≤ s ≤ t Z s > y, Z t ∈ d˜ z )= (cid:90) ∞ y P (cid:18) inf ≤ s ≤ t − t Z s > z − ˜ z + x (cid:19) P ( inf ≤ s ≤ t Z s > y, Z t ∈ d˜ z )= (cid:90) ∞ y (1 − π ( t − t , ˜ z − z )) P ( inf ≤ s ≤ t Z s > y, Z t ∈ d˜ z ) , where is used that Z has independent and stationary increments.Now the result follows by noting that by a modification of Equation (A.2) to the current setting,it holds that P ( inf ≤ s ≤ t Z s > y, Z t ∈ d˜ z ) = ˆ f ( x, y, ˜ z, t )d˜ z. P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) = y ( n ) ) = (cid:90) ∞ z c γ ( x, z c , z b , u − t, T − t ) f ( t, x )d x and P ( τ c ≤ u, τ b > u | τ c > t, Y ( n ) = y ( n ) ) = (cid:90) ∞ z c γ ( x, z c , z b , u − t, u − t ) f ( t, x )d x. Recall that the objective was to compute the last two integrals in Equation (A.8). Let us first considerthe second one, that is − (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T, τ b ∈ d u | τ c > t, Y ( n ) ) = (cid:90) ∞ t e − r ( u − t ) ∂∂u P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) )d u = (I) + (II) , where (I) = (cid:90) Tt e − r ( u − t ) ∂∂u P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) )d u = (cid:90) ∞ z c f ( t, x ) (cid:90) Tt e − r ( u − t ) ∂∂u γ ( x, z c , z b , u − t, T − t )d u d x = (cid:90) ∞ z c f ( t, x ) (cid:90) Tt e − r ( u − t ) ∂∂u ( − π ( u − t, x − z b ))d u d x = (cid:90) ∞ z c f ( t, x ) I b ( x )d x, in which I b ( x ) = (cid:90) Tt e − r ( u − t ) ∂∂u ( − π ( u − t, x − z b ))d u = exp (cid:32) − m ( x − z b ) + ( x − z b ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z b − √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) + exp (cid:32) − m ( x − z b ) − ( x − z b ) √ m + 2 rσ σ (cid:33) (cid:32) Φ (cid:32) x − z b + √ m + 2 rσ ( T − t ) σ √ T − t (cid:33) − (cid:33) , (A.11)which follows from Equation (A.6), by replacing z c by z b . Furthermore, we have(II) = (cid:90) ∞ T e − r ( u − t ) ∂∂u P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) )d u = (cid:90) ∞ z c f ( t, x ) (cid:90) ∞ T e − r ( u − t ) ∂∂u γ ( x, z c , z b , u − t, T − t )d u d x = (cid:90) ∞ z c f ( t, x ) (cid:90) ∞ T e − r ( u − t ) ∂∂u ( − π ( u − t, x − z b ))d u d x − (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) (cid:90) ∞ T e − r ( u − t ) ∂∂u (1 − π ( u − T, ˜ z − z b ))d u d˜ z d x = (cid:90) ∞ z c f ( t, x )( J b ( x ) − I b ( x ))d x − (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) e − r ( T − t ) J b (˜ z, T )d˜ z d x, J b ( x ) = (cid:90) ∞ t e − r ( u − t ) ∂∂u (1 − π ( u − t, x − z b ))d u = − exp (cid:32) − m ( x − z b ) + ( x − z b ) √ m + 2 rσ σ (cid:33) , (A.12)where the last line follows by taking T → ∞ in Equation (A.11). This leaves us with an expression forthe last integral in Equation (A.8).Similarly, the other integral satisfies (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T ∧ u, τ b > u | τ c > t, Y ( n ) )d u = (III) + (IV) , where (III) = (cid:90) Tt e − r ( u − t ) P ( τ c ≤ u, τ b > u | τ c > t, Y ( n ) )d u = (cid:90) ∞ z c f ( t, x ) (cid:90) Tt e − r ( u − t ) γ ( x, z c , z b , u − t, u − t )d u d x = (cid:90) ∞ z c f ( t, x ) (cid:90) Tt e − r ( u − t ) ( π ( u − t, x − z c ) − π ( u − t, x − z b ))d u d x = (cid:90) ∞ z c f ( t, x )( ˜ I b ( x ) − ˜ I ( x ))d x in which ˜ I ( x ) is defined in the proof Theorem 4.4 and ˜ I b ( x ) is equivalently defined as˜ I b ( x ) = (cid:90) Tt e − r ( u − t ) (1 − π ( u − t, x − z b ))d u = (cid:20) − r e − r ( u − t ) (1 − π ( u − t, x − z b )) (cid:21) Tu = t + 1 r I b ( x )= − r e − r ( T − t ) (1 − π ( T − t, x − z b )) + 1 r + 1 r I b ( x ) . (A.13)Furthermore, we have(IV) = (cid:90) ∞ T e − r ( u − t ) P ( τ c ≤ T, τ b > u | τ c > t, Y ( n ) )d u = (cid:90) ∞ z c (cid:90) ∞ T e − r ( u − t ) γ ( x, z c , z b , u − t, T − t )d u d x = (cid:90) ∞ z c f ( t, x ) (cid:90) ∞ T e − r ( u − t ) (1 − π ( u − t, x − z b ))d u d x − (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) (cid:90) ∞ T e − r ( u − t ) (1 − π ( u − T, ˜ z − z b ))d u d˜ z d x = (cid:90) ∞ z c f ( t, x )( ˜ J b ( x ) − ˜ I b ( x ))d x − (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) e − r ( T − t ) ˜ J b (˜ z )d˜ z d x, in which ˜ J b ( x ) = (cid:90) ∞ t e − r ( u − t ) (1 − π ( u − t, x − z b ))d u (cid:20) − r (1 − π ( u − t, x − z b ) (cid:21) ∞ u = t + 1 r J b ( x )= 1 r + 1 r J b ( x ) . (A.14)Putting all the above together leads to an expression for the last two integrals in Equation (A.8), givenby − c P (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T ∧ u, τ b > u | τ c > t, Y ( n ) )d u − e z b (cid:90) ∞ t e − r ( u − t ) P ( τ c ≤ T, τ b ∈ d u | τ c > t, Y ( n ) )= e z b (( I ) + ( II )) − c P (( III ) + ( IV ))= (cid:90) ∞ z c f ( t, x ) (cid:16) e z b J b ( x ) + c P ˜ I ( x ) − c P ˜ J b ( x ) (cid:17) d x + (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) e − r ( T − t ) ( c P ˜ J b (˜ z ) − e z b J b (˜ z ))d˜ z d x. (A.15)Finally, by combining Equations (A.8), (A.9) and (A.15), it follows that the third term in Equa-tion (4.16), i.e. E (cid:18) ∆ P ∆ P + 1 E P C ( τ c ) e − r ( τ c − t ) { τ c ≤ T } |H t (cid:19) , is given by (cid:90) ∞ z c f ( t, x ) h ( x )d x + (cid:90) ∞ z c (cid:90) ∞ z c f ( t, x ) ˆ f ( x, z c , ˜ z, T − t ) h (˜ z )d˜ z d x, (A.16)where h ( x ) = ∆ P ∆ P + 1 (cid:16) e z b J b ( x ) + c P ˜ I ( x ) − c P ˜ J b ( x ) − e z c I ( x ) (cid:17) ,h (˜ z ) = ∆ P ∆ P + 1 e − r ( T − t ) ( c P ˜ J b (˜ z ) − e z b J b (˜ z )) . (cid:3) Proof of Proposition 4.7
Denote by ∆ t the time between two successive accounting reports and recallfrom section 4.1 the notation Y i = Y t i , Z i = Z t i , U i = U t i and that Y i = Z i + U i , where U i = κU i − + (cid:15) i , for some fixed κ ∈ R and independent and identically distributed (cid:15) , (cid:15) , . . . , which have a normal distri-bution with mean µ (cid:15) and variance σ (cid:15) , and are independent of Z . This allows us to write, for i = 1 , , . . . Y n + i ... Y n +1 = M Z n + i − Z n + i − ... Z n +1 − Z n (cid:15) n + i ... (cid:15) n +1 + κ i ... κ Y n + − κ i ...1 − κ Z n , M denotes the ( i × i )-matrix defined by M = i components (cid:122) (cid:125)(cid:124) (cid:123) · · · · · · · · · · · · i components (cid:122) (cid:125)(cid:124) (cid:123) κ κ · · · κ i − κ · · · κ i − ... . . . . . . . . . ...0 · · · κ · · · and where the vector ( Z n + i − Z n + i − , . . . , Z n +1 − Z n , (cid:15) n + i , . . . , (cid:15) n +1 ) follows a multivariate normal dis-tribution with 2 i -dimensional mean vector µ (cid:48) i and (2 i × i )-dimensional covariance matrix Σ (cid:48) i , definedby µ (cid:48) i = m ∆ t ... m ∆ tµ (cid:15) ... µ (cid:15) , Σ (cid:48) i = Diag( σ ∆ t, . . . σ ∆ t, σ (cid:15) , . . . , σ (cid:15) ) . Hence it follows that the conditional density p Y ( y n + i , . . . , y n +1 | y ( n ) , z ( n ) ) of y n + i , . . . , y n +1 given Y ( n ) = y ( n ) and Z ( n ) = z ( n ) , is the density of a multivariate normal distribution with mean vectorˆ µ i = M µ (cid:48) i + κ i ... κ y n + − κ i ...1 − κ z n and covariance matrix Σ i = M Σ (cid:48) i M (cid:62) . The conditional density of the next i accounting values, given Y ( n ) can be written as p Y ( y n + i , . . . , y n +1 | y ( n ) ) = (cid:90) R n p Y ( y n + i , . . . , y n +1 | y ( n ) , z ( n ) ) p Z ( z ( n ) | y ( n ) )d z ( n ) , where the conditional density p Z ( z ( n ) | y ( n ) ) of Z ( n ) , given Y ( n ) = y ( n ) , can be computed in the same wayas b n ( z ( n ) | y ( n ) ) in section 4.2, which leads to p Z ( z ( n ) | y ( n ) ) = p Z ( z n | z n − ) p U ( y n − z n | y n − − z n − ) p Z ( z ( n − | y ( n − ) p Y ( y n | y ( n − )= (cid:81) ni =1 p Z ( z i | z i − ) p U ( y i − z i | y i − − z i − ) p Y ( y n | y ( n − ) , (A.17)This leads to an expression for the survival probability until time t n + i , given survival until time t n ≤ t < t n +1 , that is P (cid:16) τ Ac > t n + i | Y ( n ) = y ( n ) (cid:17) = (cid:90) ( y c , ∞ ) i p Y ( y n + i , . . . , y n +1 | y ( n ) )d y n +1 , . . . , d y n + i = (cid:90) R n P ( ξ ( z n ) ∈ ( y c , ∞ ) i ) p Z ( z ( n ) | y ( n ) )d z ( n ) , (A.18)where ξ ( z n ) denotes a multivariate normal distributed random variable with mean vector ˆ µ i and covari-ance matrix Σ i . (cid:3) roof of Theorem 4.8 The CoCo’s market price is given by C (cid:48) ( t ) = E (cid:16) P e − r ( T − t ) { τ c >T } |H t (cid:17) + E (cid:32)(cid:90) Tt cP e − r ( u − t ) { τ c >u } d u |H t (cid:33) + E (cid:16) RP e − r ( τ c − t ) { τ c ≤ T } |H t (cid:17) . (A.19)For t n ≤ t < t n +1 , T = t n + m for some m ∈ N and Y ( n ) = y ( n ) , where y i > y c , ≤ i ≤ n , this can bewritten as C (cid:48) ( t ) = P e − r ( T − t ) P ( τ c > T | Y ( n ) = y ( n ) ) + (cid:90) Tt cP e − r ( u − t ) P ( τ c > u | Y ( n ) = y ( n ) )d u + RP m (cid:88) i =1 e − r ( t n + i − t ) P ( τ c = t n + i | Y ( n ) = y ( n ) )= P e − r ( T − t ) P ( τ c > t n + m | Y ( n ) = y ( n ) )+ cP (cid:32) m − (cid:88) i =1 (cid:90) t n + i +1 t n + i e − r ( u − t ) d u P ( τ c > t n + i | Y ( n ) = y ( n ) ) + (cid:90) t n + i t e − r ( u − t ) d u (cid:33) + RP m (cid:88) i =1 e − r ( t n + i − t ) (cid:16) P ( τ c > t n + i − | Y ( n ) = y ( n ) ) − P ( τ c > t n + i | Y ( n ) = y ( n ) ) (cid:17) = P e − r ( T − t ) P ( τ c > t n + m | Y ( n ) = y ( n ) )+ m − (cid:88) i =1 cPr ( e − r ( t n + i − t ) − e − r ( t n + i +1 − t ) ) P ( τ c > t n + i | Y ( n ) = y ( n ) )+ cPr (1 − e − r ( t n +1 − t ) )+ RP m (cid:88) i =1 e − r ( t n + i − t ) (cid:16) P ( τ c > t n + i − | Y ( n ) = y ( n ) ) − P ( τ c > t n + i | Y ( n ) = y ( n ) ) (cid:17) = (1 − R ) P e − r ( T − t ) P ( τ c > t n + m | Y ( n ) = y ( n ) )+ m − (cid:88) i =1 (cid:18) cPr − RP (cid:19) ( e − r ( t n + i − t ) − e − r ( t n + i +1 − t ) ) P ( τ c > t n + i | Y ( n ) = y ( n ) )+ cPr (1 − e − r ( t n +1 − t ) ) + RP e − r ( t n +1 − t ) , (cid:3) Proof of Theorem 4.9
This theorem is only a small adaption of Theorem 4.8. The second term inEquation (A.19) above, needs to be replaced by E (cid:32) m − (cid:88) i =1 (cid:90) t n + i +1 t n + i cP e − r ( u − t ) { τ Ac >u,Y n + i >y cc } d u + { Y n >y cc } (cid:90) t n +1 t cP e − r ( u − t ) d u (cid:12)(cid:12)(cid:12) H t (cid:33) , where t n ≤ t < t n +1 , T = t n + m for some m ∈ N .For Y ( n ) = y ( n ) , where y i > y c , ≤ i ≤ n , this can be written as m − (cid:88) i =1 (cid:90) t n + i +1 t n + i e − r ( u − t ) P ( τ Ac > u, Y n + i > y cc | Y ( n ) = y ( n ) )d u + { Y n >y cc } (cid:90) t n +1 t cP e − r ( u − t ) d u = m − (cid:88) i =1 cPr ( e − r ( t n + i − t ) − e − r ( t n + i +1 − t ) ) P ( τ Ac > t n + i , Y n + i > y cc | Y ( n ) = y ( n ) )40 { Y n >y cc } cPr (1 − e − r ( t n +1 − t ) ) , where, similar to Equation (A.18), P ( τ Ac > t n + i , Y n + i > y cc | Y ( n ) = y ( n ) ) = P ( Y n +1 > y c , . . . , Y n + i − > y c , Y n + i > y cc | Y ( n ) = y ( n ) )= (cid:90) R n P ( ξ ( z n ) ∈ ( y c , ∞ ) i − × ( y cc , ∞ )) p Z ( z ( n ) | y ( n ) )d z ( n ) . (cid:3)(cid:3)