El papel de modelizar la severidad implícita de mercado para calcular el valor en riesgo de crédito

EL PAPEL DE MODELIZAR LA SEVERIDAD IMPLÍCITA DE MERCADO PARA CALCULAR EL VALOR EN RIESGO DE CRÉDITO* J. Samuel Baixauli Soler, sbaixau[email protected], Universidad de Murcia Susana Alvarez Diez, [email protected], Universidad de Murcia RESUMEN En este trabajo proponemos el uso de mixturas de distribciones beta para modelizar la severidad impícita en el mercado. En nuestro análisis extraemos las tasas de recuperación de la cotización de los credit default swaps (CDS) en lugar de utilizar bonos que han impagado. La principal ventaja de obtener la estructura temporal implícita de tasas de recuperación de los CDS en lugar de utilizar los bonos impagados es que nos permite identificar tasas de recuperación para empresas con baja probabilidad de impago. Del análisis empírico se obtienen resultados que no permiten aceptar que una única distribución beta sea capaz de representar las tasas de recuperación implícitas mientras que la propuesta basada en mixturas de distribuciones beta si que es aceptada. Además, hay que destacar la importancia de utilizar esta metodología de ajuste por su importancia para una correcta estimación del Valor en Riesgo de crédito. PALABRAS CLAVE: Credit default swap, tasa de recuperación implícita, mixtura de distribuciones beta, Valor en Riesgo de crédito * Los autors agraden la financiación recibida a través del proyecto ECO2008-02846. 1 THE ROLE OF MARKET-IMPLIED SEVERITY MODELING FOR CREDIT VaR ABSTRACT In this paper we propose to use beta-component mixtures to model the market-implied severity. In our analysis we extract and identify recovery rates from credit default swaps instead of using defaulted bonds. The main advantage of extracting implied, endogenous and dynamic functions of recovery rates from credit default swaps versus using defaulted bonds is that it allows to identify recovery rates of low probability of default companies. We carry out an empirical analysis and our results show that a single beta distribution is rejected as a correct specification for implied recovery rates while a beta-component mixture is accepted. Futhermore, we highlight the importance of this modeling approach by focusing on its role for credit VaR. KEY WORDS Credit default swaps, implicit recovery rates, beta-component mixture, Credit VaR 2 1. INTRODUCTION At present there is a growing interest in modeling severity, which is defined as one minus the recovery rate. It is essential to approximate the severity distribution because risk quantities, such as expected credit loss, loss given default and credit VaR, rely on it. Usually in credit risk practice, it has been approximated by the analysis of recovery rates on defaulted bond issues (Altman et al. 2005; Acharya et al., 2007), eventhough there exists a lack of data on recoveries. The main weakness of this approach is that it does not allow to estimate the severity distribution of low probability of default companies, which have not enough defaulted bonds to estimate recovery rates accurately. Most of the industry-sponsored models, such as Portfolio Manager, CreditMetrics and Moody's KMV model, treat recovery rates as stochastic variables modeled through a beta distribution. Although there is no theoretical reason that this is the right shape, it has been widely used in practice to describe the observed behavior of recovery rates since beta distribution is one of the few common “named” distributions that give probability 1 to a finite interval. Nevertheless, in the related literature, there is strong evidence that the recovery rate distribution may exhibit several local modes (Asarnow and Edwards, 1995; Gourieroux and Monfort, 2006; Hagmann et al., 2005; Renault and Scaillet, 2004; Schuermann, 2005, among others). The several modes can arise from different periods (recession and expansion), different types of collateral securing the instruments or from various seniority levels in the same data set (senior secured, senior unsecured, subordinated and junior subordinated). Using data from Moody's Default Risk Service Database, Schuermann (2005) illustrates that recovery rate distributions conditioned to the stage of the business are clearly multimodal. Renault and Scaillet (2004) shows that nonparametric plots of the recovery function frequently exhibit more than two local modes using data from Standard & Poor's PMD classified by seniority and by industry. The presence of multimodality can be suggestive of more than one underlying unimodal distribution, each referring to a certain group of recovery rates. These groups can be estimated by means of beta-component mixtures. They have simple tractability for modeling and flexibility enough to describe unknown and multimodal distributional shapes which apparently can not be modeled by a beta distribution. The rapid growth of credit derivative market enables to make use of credit default swaps (CDS) as market indicators. As Düllmann and Sosinska (2007) pointed out credit default swaps are less limitated as market indicators than credit spreads of subordinated debt issues, since CDS represent insurance premia for default events and measure credit risk more directly. Most of studies on analyzing the usefulness of CDS as market indicators infer probability of default from CDS, imposing an exogenously constant recovery rate. The market convenction is to assume that the average recovery rate is around 50%. Under such assumption, the term structure of CDS spreads can be used to extract the term structure of risk-neutral default probabilities either using a structural model (Finger et al., 2002; Düllmann and Sosinska, 2007) or a reduced-form framework (Duffie and Singleton, 1999; Jarrow et al., 1997; Jarrow, 2001; Madan et al, 2003). However it is unrealistic to consider that recovery rates held fixed given that the pattern of recovery rate distribution can vary significantly across seniority level, industries, stages of business cycle, etc. At present recovery rate extraction from CDS is relatively scarce. The approaches which focus on extracting simultaneously both the probability of default and recovery rates may be classified into time-series dependent approaches, cross-sectional approaches and panel data approaches (Christensen, 2005; Pan and Singleton, 2008; Chava et al., 2006; Acharya et al., 2007). Das and Hanouna (2008) adopts a calibration approach for bootstrapping implied recovery rates from CDS spread curves at any single point in time. Their 3 procedure uses information from the equity market, the credit default swap market and it also uses the forward curve of riskless rates, thereby incorporating information from the interest rate market as well. In contrast to the above approaches only information on a given trading day is used, an entire forward term structure of recovery is delivered and a dynamic model of recovery is offered through a functional relation between recovery and state variables. Das and Hanouna (2008) model is flexible and robust. It is flexible in the sense that it can be used with different state variables, alternate recovery functional forms and calibrated to multiple debt tranches of the same issuer. It is robust because it evidences parameter stability over time, is stable to changes in inputs and provides similar recovery term structures for different functional specifications. Finally, their model is easy to calibrate. In this paper we approximate the severity distribution of a given company at any single point in time using recovery rates implicit in the term structure of CDS. In doing this, we implement the approach introduced by Das and Hanouna (2008). Our main objective is to model the market-implied severity as a mixture of beta components in order to capture the observed multimodality. Futhermore, we highlight the importance of this modeling approach by focusing on its role for credit VaR, which is a commonly used risk quantity. Specifically, simulation experiments are carried out to evaluate the implications of computing credit VaR in the case where a beta distribuion is wrongly assumed when the true underlying severity distribution is a beta-component mixture. The paper is organized as follows. Section 2 describes briefly the methodology developed by Das and Hanouna (2008) to extract and identify the implied forward curve of recovery rates. Section 3 describes our proposal of modeling market-implied severity by finite mixtures of beta components. This section reports the representation, interpretation and estimation of mixture distributions using the Expectation-Maximization (EM) algorithm. Section 4 reports empirical results based on four companies which belong to the European stock index EUROSTOXX 50. In Section 5 an application to credit VaR estimation for two sets of portfolios is carried out. Finally, conclusions are drawn in Section 6. 2. IDENTIFICATION OF IMPLIED RECOVERY RATES FROM CDS The most important instrument in the credit derivative market is the credit default swap (CDS), which essentially provides insurance against the default of an issuer (the reference credit) or on a specific underlying bond (the reference security). In its most basic form the buyer of the protection pays an annual or semiannual premium until either the maturity of the contract or default on the reference entity, whichever comes first. If a default occurs, the seller of the protection compensates the buyer for the loss on the reference security by either paying the face value of the bond in exchange for the defaulted bond (physical settlement) or by paying an amount of cash which compensates the buyer of the protection for the difference between the post-default market value of the bond and the par value (cash settlement). Typically, the underlying credit of a default swap is a rated firm with publicly traded debt or a sovereign entity. More details on CDS may be found in Lando (2004). The steps of Das and Hanouna (2008) methodology to identify implied, endogenous, dynamic functions of the recovery rate and default probability from CDS can be sum up as follows: 1. The standard relationship of CDS spreads to default intensities and recovery rates is presented, considering that the fair pricing of a default swap must be such that the expected present value of payments made by buyer and seller are equal: 4 )1()1( 1 1 1 1jj N jj N jjjN DeSDSC j φ λ −−== − = − = −∑∑ (1) where N is the number of periods in the model, indexed by Nj ,...,2,1 = ; CN is the fair pricing of a default swap (is the premium); ∑ == − j k k eSj1 λ is the survival function of a firm and j λ denotes the default intensity, ),1( jj j−= λ λ , constant over forward period j (it is assumed that 1)0( =S; that is, a firm is solvent); j D is the discount function, written as functions of the forward rates, ∑ == − j k k f jeD 1; ),1( jj j−= φ φ is the recovery rate in the event of default (it is the recovery rate in the event that default occurs in period j; then, the loss payment on default is equal to )1( − j φ ); )1( j eSj λ − − is the probability of surviving until period 1 − j (.e., the expected loss payment in period j is based on the probability of default in period j conditional on no default in a prior period). 2. Default intensities are represented in terms of spreads and recovery rates. Through a process of bootstrapping, the general form of the intensity probability is (for all j): ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − −+− −= − − == −∑∑ )1( )1( ln)1( 1 1 11 1 jjj j k j kjjkjjj jDS HCGDS φ φ λ , )1()1( 1jjjj DeSG j φ λ −−≡ − −, jjj DSH 1− ≡ . (2) 3. A functional relationship of recovery rates to default intensities is choosen, which may generally be written as [ ] θ λ φ ;g=, where both n R∈ λφ , are term structure vectors, and θ is a parameter set. 4. An iterative fixed-point algorithm is begun using a starting value for 5.0)( =T φ , for all T. In the iteration process, (i) finding )(T λ from equation (2) and (ii) finding )(T φ from )(T λ using the loglinear regression relationship. The system stabilizes rapidly within a few iterations. The approach which is taken in this paper is to use information from the equity market through the Merton model (Merton, 1974). The identification function between recovery rate and default intensity for the iterative process is given by the following loglinear relationship: ))(ln())(ln( 10 TT λ θ θ φ + =. (3) The term structure of interest rates is estimated using the commonly used Nelson and Siegel model (Nelson, 1987), which uses a single exponential functional form over the entire maturity range. This model suggests a parsimonious parametrization of the instantaneous forward rate curve given as follows: 5 ττ τ ααα / 3 / 21 )( tt e t etf −− ++= , (4) The parameters α1, α2, α3 and τ can be interpreted as: α1+α2 is the instantaneous short rate; α1 is the consol rate; that is, 1 )(lim α = ∞→ tf t; -α2 is the slope of the term structure of forward rates; α3 affects the curvature of the term structure over the intermediate terms; τ>0 is the speed of convergence of the term structure toward the consol rate. These four parameters are estimated by minimizing the sum of squared errors: ∑ = k ii Min 1 2 ,,, 321 ε τααα (5) 3. FINITE MIXTURES OF BETA DISTRIBUTIONS IN MODELING IMPLIED RECOVERY RATES We propose to use finite mixtures of beta distributions in proportions π1,..., πg to model implied recovery rates. The mixing proportions represent the percentage of recovery rates belonging to each component of the mixture, are non-negative and sum to 1. Such distributions provide an extremely flexible method of modeling unknown and multimodal distributional shapes which apparently cannot be modelled by a single beta distribution. Each component represents a local area of support of the true distribution which may reflect the behaviour of recovery rates, for instance, belonging to a particular industry, with a specific seniority level or during a stage of the business cycle. The probability density function of the recovery rates is given by ),|(),,;( 1jjj g jjqpyfqpyf ∑ = = ππ , (6) in which ),...,( 1g π π π =, ),...,( 1g ppp =, ),...,( 1g qqq = and ),;( jjj qpyf , j=1,…,g, denotes the values of the univariate beta probability function specified by the parameters pj and qj, given by: 11 )1( ),( 1 ),;( −− −= jj qp jj jjj yy qpB qpyf , 10 < < y, 0> j p, 0> j q (7) where, ),( jj qpB denotes the beta function, pj is the shape parameter and qj is the scale parameter. For a given value of g the unknown parameters in the beta mixture model are estimated by the EM (Expectation- Maximization) algorithm (Dempster et al., 1977). Under the assumption that n yy ,..., 1 are independent and identically distributed random variables following a beta mixture distribution, the log-likelihood function is given by ),|(log)|,,(log 11 jjij n i g jjqpyfyqpL ∑∑ == = ππ , (8) With the maximum likelihood approach to the estimation of ),,( qp π ψ = , an estimate is provided by an appropriate root of the likelihood equation 6 0 )|(log = ∂ ∂ ψ ψ yL (9) The EM algorithm is used to find solutions of (9) corresponding to local maxima and it is guaranteed to converge to the MLE. Overall, it is based on the idea of replacing one difficult likelihood maximization with a sequence of easier maximizations whose limit is the answer to the original problem. In the EM framework, the observed univariate data vector ),...,( 1n YYY = is completed with a componentlabel vector ),...,( 1n ZZZ =. The label variable )( jZZ iij = , i=1,…,n, j=1,…,g, is 0 or 1 according to whether i corresponds to the component j. Hence, ),...,( 1n ZZZ = is an unobservable vector of componentindicator variables, and i Z, i=1,…,n, are assumed to be independent random variables from a multinomial distribution consisting of one draw on g categories with respective probabilities g π π ,..., 1. That is, n ZZ ,..., 1 ∼ Multg(1, π ), (10) where ),...,( 11 − =g π π π . The complete-data log-likelihood is {} ∑∑ == =n ijjijjij g j cqpyfzyqpL 11 ),;(log)|,,(log ππ . (11) The EM algorithm allows us to maximize )|,,( yqpL π by working with ),|,,( zyqpLc π . The EM algorithm is an iterative procedure. Each iteration comprises of the “E-step”, which calculates the expected log likelihood, and the “M-step”, which finds its maximum. Now, the algorithm starts: From an initial value ),,( )0()0()0()0( qp πψ =, a sequence is created according to = +)1(r ψ the value that maximizes [ ] );(,|),|(log )()( rr QyzyLE ψψψψ =, (12) which is the conditional expectation of the complete data log-likelihood ),|,,(log zyqpLc π , given the observed data y, using the current fit )(r ψ for ψ . On the (r+1) iteration, the E-step requires the calculation of );( )(r Q ψψ . Since ),...,( 1n ZZZ = is non observed data, the E-step is affected by replacing zij by its conditional expectation given yj, using )(r ψ for ψ . That is, zij is replaced by { } { } iijiij r ij yZyZEy rr |1Pr|);( )()( )( === ψψ ψτ . On the M-step, on the (r+1) iteration we choose the value of ψ , say )1( +r ψ , that maximizes );( )(r Q ψψ . Then, the vector )1( +r ψ is obtained as an appropriate root of 0 )|(log ),( )( 11 = ∂ ∂ ∑∑ == ψ ψ ψτ yL yj r i n ij g j (13) The hidden key to the algorithm is the application of the information inequality (Demspter et al., 1977, lemma 1), which states that )| ˆ ()| ˆ ()()1( yLyL rr ψψ ≥ +, with equality holding if and only if successive 7 iterations yield the same value of the maximized expected complete-data log-likelihood, that is, [ ] [ ] yzyLEyzyLE rrrr , ˆ |),| ˆ (log, ˆ |),| ˆ (log )()()()1( ψψψψ = +. 4. MARKET-IMPLIED SEVERITY: A MODEL BASED ON A BETA-COMPONENT MIXTURE This study uses data from Bloomberg Financials on CDS spreads for Spanish industrial quoted companies with liquid traded CDS, which belong to the EUROSTOXX 50 index, for the period from January 2004 to October 2007. They are low credit risk companies and are Repsol, Iberdrola, Telefonica and Endesa. The data consist on a CDS spread curve with maturities from 1 to 10 years for each company and each day. For each of the 935 days in the sample and each company we compute the term structure of forward recovery rates by applying (Das and Hanouna, 2008) methodology. Then, we obtain ten implied recovery rate distributions for each company, one for each maturity. To examine the shape of the implied recovery rate distribution we will restrict ourselves to the most frequently traded CDS maturity of 5 years, other maturities are considerably less liquid. Figure 1 shows the histograms of the 5-year maturity implied recovery rates which we have identified from (Das and Hanouna, 2008) methodology. Fig. 1 Histograms of implied recovery rates 9 Table 1: PURE method PURE method g=1 g=2 g=3 g=4 REPSOL 465.01546 77.371338 70.502056 72.312064 ENDESA 96.141265 98.535255 49.032340 49.447739 IBERDROLA 613.87152 81.406940 56.200952 58.182254 TELEFONICA 629.94422 451.27192 93.137999 93.681212