CoCo Bonds Valuation with Equity- and Credit-Calibrated First Passage Structural Models
aa r X i v : . [ q -f i n . P R ] F e b CoCo Bonds Valuation with Equity- andCredit-Calibrated First Passage Structural Models
Damiano Brigo ∗ Jo˜ao Garcia † Nicola Pede ‡ First version: Jan 31st, 2011Second version: May 4th, 2012. This version: February 28, 2013
Abstract
After the beginning of the credit and liquidity crisis, financial institutions have beenconsidering creating a convertible-bond type contract focusing on Capital. Under the termsof this contract, a bond is converted into equity if the authorities deem the institution to beunder-capitalized. This paper discusses this
Contingent Capital (or Coco) bond instrumentand presents a pricing methodology based on firm value models. The model is calibrated toreadily available market data. A stress test of model parameters is illustrated to account forpotential model risk. Finally, a brief overview of how the instrument performs is presented.
AMS Classification Code: 91B70, JEL Classification Code: G13Keywords: Contingent Capital, CoCo Bonds, AT1P model, Firm Value Models, Credit DefaultSwap Calibration, Conversion Time, Default Time, Hybrid Credit-Equity Products, Basel III,Systemic Risk
The credit crisis that began in Summer 2007 has led the financial industry to reconsider deeplyingrained concepts. One of those ideas is the role of capital and leverage in financial institutions.The term capital is related to the amount of assets and cash an institution is supposed to set asideto prevent its net asset value from falling below the level that could affect its business or hinder ∗ Imperial College, London, U.K. ( [email protected] ) † The opinions expressed in this paper are those of the authors and do not necessarily reflect those of theiremployers. The authors would also like to thank and acknowledge the comments of Thomas Aubrey (formerlyManaging Director in Fitch Solutions) that improved significantly the quality of this work. Any possible remainingerrors are those of the authors. ‡ Imperial College, London, U.K. ( [email protected] ) .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels regulatory (RC) and economic (ECAP) Capital. The former is the capitalmandated by the regulatory authorities for the financial institution to be considered safe. Thelatter is an institution’s internal estimate of the needed capital and is supposed to reflect moreclosely the market and economic conditions in which the company operates. Regulatory Capital assumptions: Granularity and Additivity
Up until the credit crisis, the regulatory capital requirement for a position in the portfolio of afinancial institution was based on two key technical assumptions (see Gordy (2003)): portfolioinvariance and additivity . The first means that the cost of capital should depend on the position that is added to a portfolio and not on the portfolio composition. This requirement implied a ratings based approach to capital management. The second requirement is that the total cost ofcapital is given by the sum of the capital costs of the individual positions. The second requirementhas been traditionally implemented by resorting to an assumption of infinite-granularity and tothe use of the 1-factor Gaussian copula. This approach features a global single systemic riskfactor. Within this framework the regulatory capital of a certain position depends ultimately onthe rating of the position and on the capital cost associated with the rating class. Since the latterhas been time invariant, the Basel regulatory capital of a portfolio has been a quite static measureby construction. For a critic see for example Blundell-Wignall and Atkinson (2010).
Economic Capital
The economic capital of a portfolio, on the other hand, is supposed to depend on assumptionsconcerning the economic activity and conditions at the time of calculation. Additionally, thecapital associated to a certain position depends on the specific portfolio in which the positionis located. This point becomes much more relevant when the instrument in the portfolio is asecuritization note.
Asset backed securities (ABS) have been in the market for at least 25 years,and the origin of that asset class has its roots in the US, where it has appeared as an attemptto address part of the economic problems caused by mortgages (see e.g. Kothari (2006)). Theeconomic capital of a portfolio depends on the shape of the tail of the loss distribution, and thisshape is very much affected by the portfolio assets correlations (or, better, dependence; herewe will use the word ”correlation” also to denote the more general and well-posed notions ofstatistical dependence). The concept of correlation and its calibration depends significantly onthe adopted assumptions for the determination of the loss distribution . The details are out of the scope of this paper. We refer to Garcia and Goossens (2009) and Brigo et al. (2010)for details..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Contingent Capital Bonds (CoCo’s)
As a consequence of the credit crisis, very large financial institutions are still under-capitalizedat a time when economic development is most needed. To solve the issues of low capital levels the industry has recently proposed a new concept of capital called Contingent Capital ( Coco ’s).This instrument has a hybrid nature and is similar to a convertible bond, with some relevantcaveats. In general a Coco is a form of capital that has a hybrid format: the Coco is a bond thatmay be converted into equity when or if a certain event happens. Typically, the event is related toa capital ratio falling below a predetermined threshold. The event, always related to the capitalsituation of the issuing institution, triggers the conversion of the bond into equity. In order tocompensate the bond-holder for the risk of a conversion at a possible distressed time, the Cocobond would pay a more generous coupon than a similar bond without Coco’s features. A crucialquestion is ”how much more generous”?
CoCo pricing, default models and conversion
A very important issue is then what would be the price that would cover that risk and how itwould be determined. Coco’s are driven by conversion, but conversion in turn may be drivenby credit quality and default risk, so that we need to consider a model for the credit and thedefault time of the issuing institution. Generally speaking, there are two distinct ways to modelthe default process.The first default modeling area is the one we will adopt in the present work. This is knownas the firm value models area and it is generally attributed to Merton (1974) and Black and Cox(1976), and underpins the Black-Scholes formula. In these models a company defaults when alatent variable, the firm asset value, breaches some barrier, typically the debt value and safetycovenants. One then needs assumptions for the asset value process and for the capital structureof the company. Since the original work of Merton and Black and Cox there have been manyextensions to it, and we refer to Bielecki and Rutkowski (2002) and the references therein fordetails. In this paper, we will use the analytically tractable first passage model (AT1P) devel-oped by Brigo and Tarenghi (2004) to price a contingent capital bond. The model is calibratedto the implied risk neutral probabilities extracted from the CDS market by the use of analyticalformulas for barrier options.The second default modeling area is termed ”reduced form models” or ”intensity models”,see for example the original work of Duffie and Singleton (1999). In this case one renouncesmodeling the default from an economic point of view and just assumes that the default time isthe first jump of a time-inhomogeneous Poisson process, where the intensities of default (or in-stantaneous credit spreads) are usually calibrated to prices of the CDS market. This has been thepreferred approach by market practitioners interested in fitting (by construction) the implied riskneutral probability of default from the CDS market. In this context, the market is often assuming The use of taxpayer money to bail out the financial system has become, expectedly, a matter of very intensedebate. The model has been developed by Brigo and Tarenghi (2004) for pricing Equity Swap under counterpart riskand by Brigo et al. (2010) for the same problem while analyzing Lehman Brothers..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels
This paper and earlier literature
Given Capital concerns in banks, there has been much interest in CoCos in the literature. Wemay roughly classify previous research in two areas. The first area deals with the pricing andrisk management of CoCos instruments while the second area deals with investigating the impactof the introduction of CoCos on the optimal capital structure of the issuer. This paper lies in thefirst area.Concerning the second area of research, Barucci and Del Viva (2011) extends an approachoriginally proposed by Goldstein et al. (2001) where the underlying state variable is the claimto earnings before interest and taxes (EBIT) and where straight debt, contingent capital, equityand bankruptcy costs are modeled as perpetual assets, thus allowing for closed-form valuationformulas. The default barrier level, the trigger level and the optimal capital structure are then cal-culated such as to maximize the equity value and the net value of the company. Again, the paperBarucci and Del Viva (2011) shows as the optimal capital structure strategy and the bankruptcycosts can change depending on the introduction of CoCos and on the type of trigger event andrule for equity conversion that are adopted.Going back to the first area of pricing and risk management of CoCos, we refer the readerto Wilkens and Bethke (2012) and references therein for a fairly comprehensive review of thethree main previous approaches to CoCo bonds valuation, which we may briefly describe asstructural-default based, reduced-form based and equity modeling based.The Structural-default approach models the issuer default via the first passage time of thefirm value process through some threshold barrier associated with debt and safety covenants.The application of such models to CoCos pricing relies on the same idea and considers twodifferent thresholds, the first one triggering equity conversion and the second one triggering .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels • Bond-Equity Conversion time, • Equity price at Conversion, and • Default time.Without one or more of these three features it is not possible to calibrate a model to creditmarkets and equity market at the same time. This implies that one is discarding important infor-mation. The set of instruments to which the CoCo pricing model is calibrated affects the set ofconsidered risk factors and, hence, hedging strategies. Models able to only capture the credit-component of this hybrid instrument, will only allow to hedge it through credit derivatives, whilemodels able only to handle the equity component will only allow to use equity instruments. ACoCo bond, however, could react to information coming from both markets.This paper adds to the literature in important ways. Firstly, we show how to use the AT1Pfirm value model to price Coco’s. Secondly, we describe a new calibration procedure, how toimplement it, and finally we show the numerical results of our pricing approach, comparingour outputs with market quotes for a traded instrument. Thirdly, we analyze stress tests on ourframework to see the impact on prices during periods of financial distress.The paper is organized as follows. In section 2 we describe the Coco bond instrument andits use in practice, inclusive of the motivation underlying its creation. The AT1P model used inour pricing approach is summarized in section 3 while the discussion on its calibration is shownin section 4. Results and a related discussion are presented in section 5. Finally, in section 6 wesummarize our conclusions.
As mentioned in Section 1, Coco’s is a new form of capital. They have been created with the mainobjective of increasing capital levels of systemically important large banks. These instruments’equity like feature may be attractive for banks in their effort to comply with the new regulatorycapital rules required by the Basel III framework. .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels
Indeed, an important feature of a Coco is its conversion price (CP). There are two important andcontrasting possibilities underlying the CP. In the first case the conversion price is determinedat the conversion date. If so, the price might be far too low, possibly resulting in significantdilution for the shareholders. In that case, shareholders may certainly be tempted to sell theirequity holdings as soon as they sense an increase in risk of the capital ratio reaching the barrier(triggering conversion). This rush to sell may put the stock into a downward spiral that willpotentially be self fulfilling. This approach, however, is quite convenient to the bond holders,given that they will be acquiring the shares at distressed prices. One can say that the approachwill affect bond holders less at the expense of the equity holders. Depending on the equityprice at the time of the conversion, the bond holders might become a quite large portion of theinstitution shareholders. Furthermore, depending on the created amount of equity, this might beseen as good news for the stock.
If the conversion price, on the other hand, is pre-determined, there are different considerationsto make. To begin with, it is important to understand what a reasonable initial value for theconversion price would be, in order to fix it at inception. In this sense, if the conversion priceis determined as the spot price at the issuance date of the Coco, then the bond holder will beacquiring newly issued shares for a price that might be much higher than the actual equity priceat the moment the conversion trigger kicks in. In this case, the dilution in shares will be relativelysmall while the losses for the bond holders will be large. The bond holder might request a highercoupon, given the risk of having to acquire equity for a considerably higher value than the marketprice (in case of conversion). This might seem a good solution for the equity holders, at the costof penalizing the bond holders. The latter might be tempted to short the stock or buy put options.Bond holders will do so especially if they sense conversion as unavoidable and intend to hedgetheir losses at that event. Again there is a high possibility of downside pressure on the stock andof a self fulfilling ”downward spiral”.A different possibility would be to align the conversion price with the percentage in fallassociated with the conversion trigger. That is, if the reduction trigger in capital amounts to say40%, then the equity conversion trigger would also amounts to a 40% reduction in the spot equityprice with respect to the known equity price at inception. From the equity holder perspective this .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels
A second important point on Coco’s has to do with transparency . The instrument is stronglydependent on the capital ratios of a financial institution. Essentially, for pricing purposes, oneneeds to have access to the portfolio composition of the issuing institution. That is, one needs tohave access to information on the portfolio compositions of the financial institution, or at leastits aggregate percentage distribution in terms of sectors (and regions) and types of instruments.A portfolio composition however is a very hard information to obtain. In the next section we willpresent a model that does not depend on the portfolio composition while depending on the pathof the equity return of the company.
There are currently two main approaches to model default. The first approach is known as firmvalue, structural or equity based models and is inspired on the classic work of Merton (1974)and Black and Cox (1976), both building on the original framework of Black and Scholes (1973).The second approach is known as intensity or reduced form models, and is inspired by the semi-nal work of Duffie and Singleton (1999). In this work we will concentrate on Merton like mod-els and we refer to Garcia and Goossens (2009) and the references therein for details, see alsoBielecki and Rutkowski (2002) and Brigo et al. (Forthcoming, 2013).In the following we will use different notations for functions of time and stochastic processes,the former being denoted by X ( t ) and the latter by X t . Vectors will be indicated by boldfaceletters.In structural models default occurs when a latent variable, the asset value, breaches a barrier,typically the debt book value of the company. One needs an assumption for the asset valueprocess and an assumption for the capital structure of a company. Let us consider a probabilityspace (Ω , F , P ) with a standard Brownian Motion defined on it, W Pt , and the filtration generatedby the Brownian motion, {F t } t ≥ . Denote also by V t the value of the company, S t its equity priceand B t the value of its outstanding debt at time t. Additionally D is the par or notional value ofthe debt at maturity T . The Merton model only checks for default at the final debt maturity T . The default time is thematurity if the firm value is below the debt par level D , and is time infinity (meaning no default)in the other case.Formally, the default time is defined as τ = T { V T
1Y 0.3019 0.0 0.2316 -0.3775 0.1737 0.3775 0.1330 -0.42192Y 0.1171 0.1554 0.0938 -0.0222 0.0729 -0.0888 0.0572 -0.06663Y 0.1992 0.1110 0.1558 -0.0999 0.1189 -0.0777 0.0922 -0.19984Y 0.1886 -0.0222 0.1495 -0.1110 0.1154 0.0666 0.0902 -0.08885Y 0.2088 -0.0666 0.1663 - 0.0333 0.1289 0.0444 0.1011 -0.03337Y 0.1946 -0.0222 0.1565 -0.0666 0.1225 0.0444 0.0967 0.022210Y 0.2182 0.0222 0.1780 -0.1221 0.1653 0.0222 0.1126 -0.0333Table 2: Calibrated volatilities and correspondent CDS spread relative errors under differentinitial guesses for H . Here ∆ S is the difference between the model spreads and the marketspreads S M . H (0) = 0 . H (0) = 0 . H (0) = 0 . H (0) = 0 . Calibrated H H correspondent to four different starting points.Table 2 shows that the four calibration problems converged with similarly good fit to themarket observed spreads. That is the four calibrations result in a model satisfactorily near to thebenchmark market quantities. However, different starting points for the barrier level resulted indifferent calibrated values both for sigmas and for H (see Table 3 for values for calibrated H ),due to the over-parametrization for the model. This issue will be addressed later on, when weuse the AT1P model to price a CoCo bond. When dealing with a contingent conversion bond, in addition to the default time τ we need tomodel a second random time. This second time accounts for the conversion of the CoCo bond.Assume τ c to be the (random) conversion time that is defined in the contract as τ c = inf { t ≥ c t ≤ ¯ c } , inf Φ := ∞ where c t is the regulatory capital at time t and ¯ c is the threshold level below which the CoCobond is converted into equity. In the current set up, c t is an exogenous variable whose proxy isshown in section 4.2. An alternative approach would involve the direct modeling of c t . This isleft for further research. .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Given that τ and τ c are not independent, a natural question is whether a CoCo bond can defaultbefore being triggered. In this regard we can make some considerations: • these instruments are designed specifically to avoid that, in case of troubles caused bycapital issues, banks will need to go through expensive new stocks issuances; • at default t = τ , it makes sense to have c τ = 0 ; • regulatory capital information is neither continuously (say daily in this context) updated,nor it is as transparent as liquidly traded quantities. This capital information is expected tobe updated at most twice a year, on the balance account dates of June and December.In practice the updating frequency of c t is quite low (usually twice a year) and it can certainlyhappen that default occurs between two updating dates. Let us assume for the time being thatissuers and regulators alike will use the conversion option as a last resort before any technicaldefault . The relation between τ and τ c , default and CoCo conversion time respectively, is thenconstrained by P ( τ < τ c | τ < T ) = Q ( τ < τ c | τ < T ) = 0 . (8)This implies that, before T , the default of the bond will always follow the conversion time.This way, since at conversion the payoff is converted into equity, the payoff of the CoCo bonddepends explicitly on τ c and equity only, with no explicit dependence on τ . Of course equity willdepend on τ implicitly.Consider now two times t and T , namely the evaluation and the maturity dates respectively.Assume Π c ( t, T ) and Π( t, T ) to be respectively the discounted payoff of the CoCo bond and thatof an identical risk-free bond without the conversion feature. Additionally, suppose D ( t, T ) = β ( t ) /β ( T ) to be the risk free discount factor at time t for the period [ t, T ] , and let E t be the valueof the stock price at time t . We can write Π c ( t, T ) = { τ c >T } Π( t, T ) + { τ c ≤ T } (cid:0) Π( t, τ c ) + N E τ c D ( t, τ c ) (cid:1) , (9)where the number of shares N obtained in case of conversion is set at the CoCo issuance date ina way that N E ⋆ = E (cid:2) Π c (0 , T ) (cid:3) . Here E ⋆ denotes the stock conversion price, and (9) can berewritten as Π c ( t, T ) = { τ c >T } Π( t, T ) + { τ c ≤ T } (cid:18) Π( t, τ c ) + E (cid:2) Π c (0 , T ) (cid:3) E τ c E ⋆ D ( t, τ c ) (cid:19) . (10)In passing we observe that the following analogy is possible with traditional CDS models.Usually the CoCo bond is issued at par, meaning that E (cid:2) Π c (0 , T ) (cid:3) = 1 , so that As it will become clear in section 4.2 one does not need this assumption as it is in fact a consequence of theproxy assumed for c t (see 13 for detail)..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Π c ( t, T ) = { τ c >T } Π( t, T ) + { τ c ≤ T } (cid:18) Π( t, τ c ) + E τ c E ⋆ D ( t, τ c ) (cid:19) . (11)If one interprets the conversion time as a default time, E τ c /E ⋆ can be thought as a recoveryrate. Formula (11) suggests an alternative approach to the valuation of these instruments via areduced-form approach. That is, we could use the first jump of a Poisson process as a model forthe conversion time. A full analogy with the reduced-form approach used to price CDS contracts,would mean thati) τ c would no longer be a {F t } t ≥ stopping-time;ii) the intensity of the underlying Poisson process could be made random and linked with othervariables (rates and equity/recovery), but the exponential component of the random timewould be independent of all Brownian motions. This may lead to a weak dependency be-tween conversion time and other market drivers.It is worth stressing that in the AT1P framework we can have strong dependency between τ c and E t as they are driven by the same factor. c t We estimated a proxy for c t in terms of total assets and total liabilities asTotal Assets A t ≈ V t ; Total Liabilities ≈ ˆ H ( t ) . Consider total equity to be given by the difference between total asset and total liabilities. Acommon indicator of the leverage of a company is the ratio between total assets and total equity(also known as Asset/Equity Ratio (AER)). We assume here that this ratio is the driver for thechange in capital ratios. To have an idea of how the capital ratio behaves with respect to theAER, we selected from
Fitch’s Financial Delivery Service (FDS) all the end-of-year balanceaccount dates. We filtered at these dates all the entities by their individual ratings and, for everyindividual rating class i and for every end-of-year balance account date, we went through a crosssectional ordinary least square (OLS) estimation of the following statistical model c i = α + βX i + ǫ i , (12) X i := A i A i − L i . where ǫ is the regression residual term. This calibration step is a determining aspect in themethodology. Important points to keep in mind are the following. FDS is a proprietary financial database that includes banks and insurance companies. The individual/viability rating is a concept developed by Fitch Ratings and we refer to the company’s websitefor details..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels C ap i t a l T R a t i o (a) Scatter plot and OLS estimation of Tier1 Capital Ratio against Asset/Equity ratio onDecember 2009 for class rating C. α β (b) Evolution of α and β for the C rating class Figure 1: Regression results for the model in Eq. (12) • First, individual ratings do not assume that financial institutions are inherently protectedby an underlying government support. • Second, individual ratings are based on a database spanning more than 20 years of obser-vations. This means at least three business cycles, and shows a history of the enormousimpact and relevance of the information technology in the industry. • Finally, although each financial institution might invest according to its own idiosyncraticstrategy, we assume that financial institutions may be aggregated in homogeneous groupsfor capital purposes. That is, we are assuming that the individual ratings capture the longterm aspect of the institutions that are supposed to be reflected in the price of the instru-ment.In Figure 1 we show some results for the OLS estimation. In particular, Figure 2a containsthe scatter plot and the simple linear regression for the balance account date of 31/12/09 and forthe individual rating class C. Figure 2b shows the evolution, year by year, of parameters α and β for the same rating class.For every individual rating class, γ , we chose a simple average of the evolution over time ofthe two parameters α and β . Let us call the two averaged values ¯ α γ and ¯ β γ . It is possible toexpress our proxy for the regulatory capital ratio as c t = ˆ c ( V t , ˆ H ( t ); γ ) , with ˆ c ( V t , ˆ H ( t ); γ ) = ( ¯ α γ + ¯ β γ V t V t − ˆ H ( t ) V t ≥ ˆ H ( t ) , V t < ˆ H ( t ) . (13)Notice that, once the model has been calibrated, Eq. (13) allows us to evaluate the price of aCoCo bond via a Monte Carlo simulation using the payoff as in Eq. (11).The following proposition specifies under which conditions the proxy in (13) implies thecondition in (8). .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Proposition.
Let us define f ( X t ) = α + βX t and assume f ( X ) > ¯ c and β < . Then thetriggering event occurs before default, namely Q ( τ c < τ ) = 1 . Proof.
We have easily X → ∞ as V → ˆ H + and, given that β < , f → −∞ as X → ∞ .This means that in the limit of a default occurring we would have f → −∞ . Observing that f isa continuous function completes the proof, since f would have to pass by ¯ c before reaching thedefault barrier.The relation between c and X for the individual rating C is shown in Figure 2a. In terms ofour proxy we can infer from Figure 2b that the value of ¯ β γ is negative. Eventually, as the proxystarts from a value higher than the contractual trigger ¯ c it becomes relevant that we add it to thecalibration procedure requiring that ˆ c ( V , ˆ H (0); γ ) matches the last reported capital ratio . Wewill address this point in more detail in Section 4.4. We chose a setting which implies extreme values for the correlation between the capital ratio andthe process AER. Indeed a straightforward calculation shows that corr( c γt , X t ) = sign( β ) . (14)We can however obtain intermediate values for the correlation considering an alternative shapefor the capital ratio. We propose C γt = ¯ α γ + ¯ β γ std( X t ) (cid:18) η X t std( X t ) + p − η ǫ γt (cid:19) (15)where we’ve introduced • a second random shock, independent of the Brownian motion underlying the total asset ǫ γt ∼ N (0 , ∀ t • and a new parameter η ∈ [0 , .Model (15) satisfies the following two conditionsi) Conservation of variance. The new capital ratio has the same variance as the old one var( C γt ) = var( c γt ); ii) Generalization. The old capital ratio is a special case of the new one as C γt → c γt for η → . .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels corr( C γt , X t ) = η sign( β ) . (16)It is in view of the above equation that we will often refer to η as the correlation parameter.Finally, we will refer to the time step ∆ t for checking the capital ratios as to the ”samplingfrequency”. As explained in Section 4, in AT1P an entity can default at any time and not only at maturity.If we assume that the debt still has a clear single final maturity T and that early default is givenby safety covenants, it is not unreasonable to model equity as an option on the firm value withmaturity T that is killed if the default barrier is reached before T . See also the Equity chapter inBrigo et al. (Forthcoming, 2013). Namely, we calculate the stock price E t (in this framework) asa down-and-out European call option, that is E t = β ( t ) E t (cid:16) V T − ˆ H ( T ) (cid:17) + { τ>T } β ( T ) = f ( t, V t ) . (17)We can use (17) to calculate the stock price both inside a simulation of (11) and in the calibrationprocedure.A closed form solution for the price of the option is given for example in Rapisarda, F. (2003),see also the Equity chapter in Brigo et al. (Forthcoming, 2013).The formula is as follows f = β ( t ) β ( T ) V t e R Tt ( v ( s )+ σ ( s )22 ) d s (cid:0) − Φ( d ) (cid:1) − ˆ H ( T ) (cid:0) − Φ( d ) (cid:1) − ˆ H ( t ) ˆ H ( t ) V t ! B e R Tt ( v ( s )+ σ ( s )22 ) d s (cid:0) − Φ( d ) (cid:1) + ˆ H ( T ) ˆ H ( t ) V t ! B − (cid:0) − Φ( d ) (cid:1)! , (18) .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels v ( t ) = r − q − σ ( t ) ,d = (cid:16) log ˆ H ( T ) H (cid:17) + − log V t H − R Tt ( v ( s ) + σ ( s ) ) d s (cid:16)R Tt σ ( s ) d s (cid:17) / ,d = (cid:16) log ˆ H ( T ) H (cid:17) + − log V t H − R Tt ( v ( s )) d s (cid:16)R Tt σ ( s ) d s (cid:17) / ,d = (cid:16) log ˆ H ( T ) H (cid:17) + − log ˆ H ( t ) HV t − R Tt ( v ( s ) + σ ( s ) ) d s (cid:16)R Tt σ ( s ) d s (cid:17) / ,d = (cid:16) log ˆ H ( T ) H (cid:17) + − log ˆ H ( t ) HV t − R Tt ( v ( s )) d s (cid:16)R Tt σ ( s ) d s (cid:17) / . In this section we will try to capture the effects of some of the AT1P model features in the stockprice evaluation.We reported in Figure 2 a series of plots illustrating a comparison between the solution of (17)and the evaluation of an European call option under the classic Black and Scholes framework.We used a set of parameters calibrated to the relevant market prices and such that B = 0 .Observe from (4b) that, in case B = 0 , the growth rate of ˆ H ( · ) is the same as the one of themean value of V t .Main points we wanted to highlight by showing Figure 2 are the following ones:i) Comparison between the plots in Figures 2a and 2b shows the impact of considering a fixedstrike K against a time-dependent strike K ( t ) = ˆ H ( t ) that follows the safety covenantsbarrier. The latter results in a effect similar to considering a null drift for the underlyingprocess;ii) Comparison between plot in Figure 2b and the one in Figure 2d shows that the same behavioris attainable in the AT1P framework when we modify the payoff in (17) by considering aplain call option rather than a down-and-out call option, that is by removing the indicatorfunction inside the expectation ; We resorted to a MonteCarlo engine to calculate the call prices in AT1P..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels V0 (USD) E qu i t y ( U S D ) Black&Scholes with K T = H T =0.1T =3.0667T =6.0333T =9 (a) V BS β ( t ) E t (cid:20) ( V BST − K ) + β ( T ) (cid:21) with K = ˆ H (0) . V0 (USD) E qu i t y ( U S D ) Black&Scholes with K T = barrier(T) T =0.1T =3.0667T =6.0333T =9 (b) V BS β ( t ) E t (cid:20) ( V BST − K ( T ) ) + β ( T ) (cid:21) with K ( t ) = ˆ H ( t ) . V0 (USD) E qu i t y ( U S D ) AT1P Option T =0.1T =3.0667T =6.0333T =9 (c) Equity spot price in AT1P. V0 (USD) E qu i t y ( U S D ) MC T =0.1T =3.0667T =6.0333T =9 (d) V β ( t ) E t (cid:20) ( V T − ˆ H ( T ) ) + β ( T ) (cid:21) Figure 2: Comparison between using the AT1P model and the (widely spread) Black and Scholesformula. Top charts shows B&S results, where d V BSt = rV BSt d t + σ V BSt d W t with W aBrownian motion under Q . Bottom charts show results where V t follows the process from (4a)with q = 0 . .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels We followed the calibration procedure described in Section 3.2 to determine the model’s param-eters. It is worth noticing that the model was calibrated not only to the quoted CDS spreads:indeed we used the results from Sections 4.2 and 4.3 to include the flexibility of calibrating themodel to the spot value of the stock price and to the most recent available regulatory capital ratio.The inclusion of these two additional market observables was needed to fix the indeterminationin the values of the remaining model parameters (e.g. H), shown by results in Tables 2 and 3.As explained in Definition 1, we used in sequence simulated annealing and Levenberg-Marquardt algorithms to perform the minimization needed for this calibration.In particular, considering n quoted CDS par spreads for different maturities ( S T , . . . , S T n ) ,the last published tier 1 capital ratio , c , and the spot price of the equity E , the calibration worksas follows:1. simulated annealing is used to set a starting point, x (0) , in the region X := (0 , × (0 , n +1 .2. starting from x (0) , we used Levenberg-Marquardt to find the optimum point p , in the statespace X := R + × (0 , × R n + , whereas we considered the set of market observables φ M = ( S Y , . . . , S Y , c, E ) . In this section we will show numerical results for the 2-step calibration procedure describedabove. Additionally, we will illustrate the algorithm by using it to price a Lloyds issued CoCobond.Before we move forward an important point to keep in mind is the following. As we haveseen in section 3, the AT1P model is a
Gaussian based model and the advantage of it is certainlymathematical tractability. This happens in the form of analytical formulas that make the modelvery handy for the calibration process. However, this model also suffers from a number ofproblems. The basic AT1P suffers from lack of short term credit spreads, common to more basicfirm value models, see for example Brigo et al. (Forthcoming, 2013). It happens however thatmany processes seen in finance not only have jumps (discontinuities, see for example Brigo et al.(2010)) but also present the feature of long term memory (see e.g. Garcia and Goossens (2009) .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels ISIN ID:
XS0459088877
Level of subordination
Lower Tier 2 (LT2)
Issue date
Dec – 09
Maturity date
Mar – 20
Coupon fixed, 11.04% per year paid twice per year, on Mar and Sept
Trigger event
Core Capital Tier 1 Ratio falling below 5%
Conversion Price fixed on 27th Sept 2009 at 59 GBp per shareTable 4: Financial features of the CoCo bond.for a more extensive discussion on these issues). In that case one might be tempted to addressthe issue of model risk when using AT1P. For this reason we have added a section on stresstests to get a feeling of what might happen with the model output in case input parameters havesignificant changes. Further analysis could consider the extensions of the AT1P model withrandom default barrier and possibly random volatility, see for example Brigo et al. (2010) andBrigo et al. (Forthcoming, 2013).
We applied the 2-step algorithm described in Section 4.4 considering the market data as of 15–Dec–10.Results are as follows:1. The simulated annealing optimization set the output, to be used for the local optimizationstarting point, atPoint B H σ σ σ σ σ σ σ x (0) σ σ σ σ σ σ σ p For the pricing exercise of this paper we have used the Lloyds Bank issued Coco bond shown inTable 4. .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Y i e l d − t o − M a t u r i t y ( % ) Lloyds bond−yields term structure Lloyds LT2 bondsLloyds CoCo3.09%
Figure 3: Lloyds’ bond-yields term structure as of 15-Dec-10. ∆ t Estimated Confidence Annualized(Years) value interval
YtM1/2 1.04405 1.03488; 1.05321 0.1004241/10 1.03041 1.02196; 1.03887 0.1025671/20 1.01775 1.00972; 1.02578 0.1045891/500 1.02351 1.01599; 1.03102 0.103666Table 5: Price, 95%-confidence interval and annualized yield-to-maturity dependence on thesampling frequency ∆ t .The pricing date for this example is 15–Dec–10. The instrument is actively traded in themarket and on the trading date of our example had a bid-offer price per unit of nominal of (1 . . , corresponding to a yield-to-maturity (YTM) of (10 . . .Figure 3 shows the YTM term structure for Lower Tier 2 bonds issued by Lloyds on 15-Dec-10. Observe that the YTM of the CoCo bond is 3.09% above the LT2 term structure. Aninterpretation for it is the following: 3.09% is the additional spread that the market considers fairto pay to account for the additional risks embedded in a CoCo bond. In general capital ratios are supposed to be made public by companies twice per year. However,as soon as the sampling is not continuously performed, the condition (8) is not satisfied anymore.As trigger and default time can be observed only on discrete times, observing simultaneity be-tween the two becomes more likely. Simultaneity of trigger and default time impacts on the pricein two ways in the current framework: • it increases the number of coupons received by bondholders, since default is postponed tothe next discrete date; • it brings to zero the value of the “conversion recovery rate”. .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels V τ = ˆ H ( τ ) so that the value of the barrier option through which we calculate the stockprice is zero .In Table 5 we show how the choice of the sampling frequency impacts the price of the CoCobond. We fix the payout ratio q = 0 and the correlation parameter η = 1 . We resort to MonteCarlo simulation to calculate prices. We also report the 95%- confidence interval and the YTM.Observe that varying the sampling frequency causes a variation in price and YTM of about 2.5%and 4% respectively.The change in price that is due to the change of the sampling frequency points to the samesort of model risk that has been observed e.g. in Garcia and Goossens (2009) (Ch.15 and the ref-erences therein) when analyzing CPDO’s using the continuous Gaussian process. This suggeststhe necessity of additional research on the AT1P model this time using jump / Levy processes.Furthermore, a richer version of AT1P, called SBTV, with a random default barrier and possi-ble inclusion of misreporting or fraud is introduced in Brigo et al. (2010), see also Brigo et al.(Forthcoming, 2013) and references therein. Let us consider a bond paying a fixed rate c per annum on a time schedule T b , . . . , T bN . Assumea null recovery rate in case of default. We can calculate this bond price per unit notional at time t with t < T b < T b (where T b is the first reset date) as P ( t ) = c N X i =1 α bi D ( t, T i ) Q t ( τ > T bi ) + D ( t, T bN ) Q t ( τ > T bN ) , α bi := T bi − T bi − . (19)The AT1P model allows us to calculate analytically expressions as the one in equation (19)by using (4c). We can use this fact to establish a criteria to set the time step of the Monte Carlotime grid that we use to price the CoCo bond via simulation. We may request the grid to be suchthat the price of plain defaultable bonds obtained through the Monte Carlo engine agrees with theone calculated through the AT1P closed-form formula. Following this criteria, we found that byusing ∆ t = 1 / the price calculated through (19) always lies in the 95%-confidence intervalof the Monte Carlo engine. Therefore, our choice for the sampling frequency is 1/500. For the calculations in this subsection, we fixed the sampling frequency to ∆ t = 1 / . It isworth stressing the extreme difficulty in calibrating a crucial parameter: the payout ratio q . Sincea sound and robust estimation for this parameter does not appear to be immediate, we decided to Although the institution might review capital ratios in an annual or bi-annual basis, Rating Agency analystsand Regulators will certainly follow it much more closely. In that case either a rating agency might downgradean institution, thus triggering a chain of events that will call the conversion trigger, or a regulatory authority mightcome forward and request the conversion to happen. This sort of events have not been explicitly modeled in ourapproach..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels η . Wecalibrated our model to different values of the payout ratio q and for every model we obtainedin this way, we calculated the price of the CoCo bond under different correlation conditions.Results are shown in Table 6.On one hand, with maximum η and given that β < , when X grows then c decreases morethan with lower η . In the limit case where η = 0 and if we consider std( X t ) as constant, thecapital ratio does not depend on X t . In this case, it is more likely that the capital ratio does notcross the trigger ratio at all, even in scenarios where V t approaches ˆ H t . In those cases, however,conversion is triggered by the default of the bond as per (8). The effect of lowering η is anincrease in the duration of the bond together with a decrease of the equity price at conversion.Table 6 illustrates how, for low values of η , the price monotonically decreases as η increases.This holds until η reaches a level such that the trigger time is actually given by the capital ratiofalling below the contractual threshold rather than the firm value crossing the barrier. In thosecases one can notice that the duration is less sensitive to η , and thus the fact that the equity priceat conversion is increasing in η becomes more relevant. We inferred default probabilities from the CDS market. It is indeed possible to price a plaindefaultable bond using only such default probabilities and deterministic interest rates. As wehave pointed out in Section 4, the payoff Π d ( t, T ) of the PDB with unit notional 1 can be writtenas: Π d ( t, T ) = { τ>T } Π( t, T ) + { τ ≤ T } (cid:0) Π( t, τ ) + RD ( t, τ )1 (cid:1) , (20)where R is a deterministic recovery rate, and Π( t, T ) the discounted payoffs of an identical riskfree bond.We used Monte Carlo simulation to value E t (cid:2) Π d ( t, T ) (cid:3) and we chose a PDB with character-istics (except possibly for the conversion feature of the CoCo) that was as close as possible to theCoCo bond we valued in Section 5.2. Its features are shown in Table 7.In Table 8 we show our estimation for the price of this PDB (for convenience purposes wealso show the bid market price).The difference between our model price and the market price of this security is an example ofCDS-bond basis. There is a wide literature on this subject, and the basis itself can be measured indifferent ways. We refer the interested reader to Haworth et al. (2009) for an extensive discussionand for a survey on credit spread measures for bonds. We may consider a correction to our CoCobond price that takes into account the basis. If we translate the two prices in two yields tomaturity, we obtain a difference of annual rates given by 27 basis points. We could use thisadjustment in discount rates in order to account for the CDS/Bond basis when pricing the CoCobond. This will be done in further work, where we will also consider explicit models for thebasis. .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels η B ond P r i c e ( un i t no t i ona l ) η = 0 η = 0 . η = 0 . η = 0 . η = 1 q = 3 r q = 2 r q = r q = 0 q = − r q values and different η values. r = 0 . p.a. We recallthat the market traded price at the same date was (1 . . ISIN code
XS0497187640
Level of subordination
Lower Tier 2 (LT2)
Issue date
Mar – 10
Maturity date
Mar – 20
Coupon fixed, 6.5% per year paid once per year, on Mar
Trigger event n.a.
Conversion Price n.a.Table 7: Financial features of the PDB with ISIN code XS0497187640Market Price Estimated value Confidence interval .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Level of subordination
Lower Tier 2 (LT2)
Issue date
Dec – 09
Maturity date
Mar – 20
Coupon fixed, 11.04% per year paid twice per year, on Mar and Sept
Trigger event n.a.
Conversion Price n.a.Table 9: Financial features of the CoCo - PDB.Estimated value Confidence interval
Annualized YtM1.21666 1.20807; 1.22525 0.076Table 10: Price of the CoCo - PDB, case q = 0 , η = 1 . The analogous price with conversionfeatures in is 1.01775 In this section we estimate the impact of the conversion option embedded in the CoCo bond.We calculate this impact in terms of price/yield by pricing the same instrument but eliminatingthe conversion feature. That is, we price a plain defaultable bond (PDB) with the characteristicspresented in Table 9.With a slight abuse of notation, we can use (20) to write the discounted payoff of this instru-ment. Results are shown in Table 10. As expected the inclusion of the conversion option reducesthe price of the instrument. Conversely, removing the optionality from the CoCo, we get a higherprice for the instrument.Observe that an important difference between a PDB and a Coco is what the holder receivesin case of default (for the first) or conversion (for the second). The concept of recovery rate inthe case of the Coco is the ratio between equity at trigger time ( E τ ) and the contractually-statedconversion price ( E ⋆ ). In Figure 4a we show the distribution of the mentioned ratio (that is E τ E ⋆ )resulted of our Monte Carlo simulation. For comparison purposes we also plotted the widespreadassumption of recovery rate in the CDS market (40%). Notice that in our MC ( gaussian based)simulation the average price at conversion is 50% of the price set in the contract. The higherrecovery rate is compensated by a sensibly smaller value for the conversion time compared tothe default time as shown in Figure 4b. In order to address possible issues linked to the model risk mentioned in Section 5.2 we decidedto keep the model as is, but to stress its parameters .We have tested the price changes for variations of the CDS curve and the equity value ofthe company. The results are shown in Table 11. We moved CDS spreads up and equity prices One may certainly come up with jump diffusion models. In that case however, complexity increases signifi-cantly as to our knowledge there is no closed form solutions available for calibration purpose..Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels Density of E τ c /E ⋆ PDB recvery rateAverage value of E τ c /E ⋆ (a) Ratio between equity value at conversion ( E τ )and contractual conversion price ( E ⋆ ). q = 0 and η = 1 . Density of τ c E ( τ c | τ c < T )= 2.4839 years Density of τE ( τ | τ c < T )= 4.6548 years (b) Conversion and default times. Figure 4: Empirical distributions.Percentage changeStressed market data 10% 30%Price Confidence Interval Price Confidence IntervalSpot equity price 0.991562 0.983595; 0.999528 0.916089 0.908518; 0.92366CDS 1.00439 0.99638 ; 1.01241 0.97059 0.962817; 0.978362Table 11: Prices of the CoCo bond under stressed market data input. The percentage changehas to be intended as negative when referred to the Equity spot price and positive and uniformlyapplied to the whole term structure when referred to CDS. The unstressed price is 1.01775down by 10% and 30% for both cases. Observe that under the model the sensitivity to equityprice moves is significantly higher than to CDS spread moves. Or in another way, moves of 10%and 30% on equity would be equivalent to higher moves on CDS spreads. Indeed observe that amove of 10% in equity causes a larger impact than a 30% upper move on CDS spreads.
In this paper we have made a detailed study of an equity / Merton based approach to price aContingent Capital / Coco bond. The model is based on the seminal work of Brigo and Tarenghi(2004). As a Gaussian based model it has the advantage of analytical formulas readily availablefor calibration purposes. On the data side we have also used (for calibration purposes) CDSand Capital ratio estimations based on (the) proprietary database available from Fitch Solutions.Another advantage of the model is exactly the fact one does not need to have access to thewhole portfolio of the institution (to have an estimation of an instrument that is in fact capitaldependent). .Brigo,J.GarciaandN.Pede. CoCobondsvaluationwithfirmvaluemodels
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