1 CREDIT DEFAULT SWAP SPREADS AND US ... - SSRN papers

Credit Default Swap Spreads And U.S. Financial Market: Investigating Some Dependence Structure

CREDIT DEFAULT SWAP SPREADS AND U.S. FINANCIAL MARKET: INVESTIGATING SOME DEPENDENCE STRUCTURE

Pr. Dr. Hayette Gatfaoui Tenured Associate Professor Rouen School of Management, Economics & Finance Department, 1 Rue du Maréchal Juin, BP 215, 76825 Mont-Saint-Aignan Cedex France; Phone: 00 33 2 32 82 58 21, Fax: 00 33 2 32 82 58 33; E-mail: [email protected]

Abstract: Under Basel II framework, credit risk assessment is of high significance in the light of correlation risk. Correlation risk is often envisioned along with business conditions and financial market’s impact. We employ copula methodology to identify the dependence structures that may exist between market risk fundamentals and credit risk fundamentals. Considering credit derivative spreads as credit risk fundamentals and market data as market risk determinants, we describe and quantify the asymmetric link prevailing between credit risk and market risk. Credit risk is negatively linked with market price risk whereas it becomes positively linked with market volatility risk. Such patterns give rise to interesting asymmetric dependence structures between both risk sources… Keywords: Archimedean copulas, concordance measures, credit risk, market risk, tail dependence. J.E.L. codes: C16, C32, D81.

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Conference prizes the author would like the paper to be considered for: Journal of Banking and Finance Sydney Futures Exchange Prize

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1 Introduction Credit derivatives markets are currently widely developed due to the respective hedging and speculative roles played by corresponding financial instruments. Indeed, credit derivatives allow for trading credit risk in a liquid manner and with sufficiently low transaction costs (e. g., diversification of credit portfolios). On a practical viewpoint, credit derivatives represent useful financial channels for transferring and customizing credit risk (i.e., adapted risk/return profiles). Incidentally, market participants resort to credit derivative instruments for many prospects such as industry diversification, issuer risk diversification, and potentially attractive risk/return profiles (as perceived by market participants). Most well known and liquid traded credit derivatives consist of credit default swaps (CDS). Specifically, CDS spreads are considered as determinants of default risk as well as related liquidity risk (Das and Hanouna, 2006; Hull et al., 2004; Longstaff et al., 2005; Norden and Weber, 2004). Moreover, a long stream of research establishes a bridge between credit risk market and equity market (Gatfaoui, 2003; Merton, 1974; Stivers et al., 2002; Tarashev and Zhu, 2006; Vassalou and Xing, 2004). Further, Abid and Naifar (2005), Carr and Wu (2005), and Lin and Shyy (2003) among others exhibit the bridge existing between equity market and CDS rates. More recently, Dupuis et al. (2007) highlight first the link prevailing between CDS prices and stock returns (i.e., equity market). Second, Abid and Naifar (2006) as well as Zhang et al. (2005) among others exhibit the link prevailing between CDS and equity volatility. In an analogous way, another stream of research uses non linear quantitative methods to assess the dependence structure between credit risk fundamentals (e. g., multivariate techniques). Under this setting, the Copula methodology represents one of the most used quantitative methods to price credit derivatives and assess related credit risk (Andersen and Sidenius, 2004/2005; Bielecki et al., 2006; Burtschell et al., 2007; Canela and Collazo, 2006; Dunbar and Edwards, 2007; ECB, 2006; Garcia et al., 2003; Gatfaoui, 2005; Hull and White, 2006; Jorion and Zhang, 2006; Jouanin et al., 2003; Laurent and Gregory, 2005; Torresetti et al., 2007). Indeed, Abid and Naifar (2006) show the non linear relationship prevailing between CDS rates and equity volatility. Differently, Wang et al. (2007) and Bee (2004) prove that credit derivatives pricing (e. g., CDS spreads or tranches) is enhanced while employing copula methodology since tail fatness features of market data can be taken into account (e. g., asymmetric credit loss distributions as reported by Lucas et al., 2001). In the light of current theoretical and applied research, we focus on the financial market’s impact on CDS spreads. Such an impact is of non linear nature and requires some specific appropriate quantitative tool to be soundly assessed. For this purpose, we select the well known copula methodology to study the link prevailing between CDS spreads and financial market. Therefore, we focus on the joint dependence structure of both financial market (i.e., market risk) and CDS spreads (i.e., credit/default risk). Moreover, we argue that the financial market’s impact requires to be split into two distinct components (Abid and Naifar, 2006; Scheicher, 2006; Vassalou and Xing, 2004; Zhang et al., 2005), namely a market price risk component (e. g., risk of change in equity returns) and a market volatility risk component (e. g., risk of magnitude of potential changes, which determines the severity of loss). Consequently, the dependence

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structure between credit risk and market risk is envisioned in two dimensions, namely one market price risk and one market volatility risk. The remainder of our paper is organized as follows. Section 2 introduces the data set under consideration as well as key empirical features. As an extension, section 3 undertakes a follow-up preliminary investigation whereas section 4 introduces copula methodology and handles the related estimation of well chosen copulae as well as key corresponding features. As a final extension, section 5 solves the issue of selecting the optimal copula among a given set of copulae, and computes the probability of occurrence of some stressed situations (i.e., optimality and tail indexes). Finally, section 6 draws some concluding remarks and possible future extensions. 2 Data and empirical facts We introduce the data set under consideration as well as related statistical properties, which are important for the rest of the study. 2.1 Data Our data set consists of daily closing quotes ranging from September 20th, 2005 to August 14th, 2006, namely a total of 225 observations per time series. We consider two global sets of data, whose nature is different. The first set relates to the U.S. financial market whereas the second set relates to corporate and sovereign credit markets. With regard to the first set of data, we first consider Dow Jones Composite Index (DJCI) return in basis points as a proxy of market price risk factor. In particular, market price risk drives potentially the direction of CDS spread changes. Second, we select CBOE DJIA Volatility Index (VXD) as a proxy of market volatility risk. Specifically, market volatility risk drives eventually the magnitude of CDS spread changes. Incidentally, VXD index is extracted from option prices on the Dow Jones Industrial Average (DJIA) and reflects DJIA volatility while measuring the implied volatility of corresponding near-term options. Moreover, CBOE DJIA Volatility Index illustrates investors’ expectation about future one-month stock market volatility. Third, we build a global market indicator DJCI_N = DJCI / VXD, which consists of the ratio of DJCI return to VXD implied volatility index. Indicator DJCI_N summarizes the global market risk and represents basically some normalized DJCI return. With regard to the second set of data, we consider the spreads of eight Dow Jones credit derivative indexes (CDX indexes) against appropriate LIBOR rates (see www.markit.com). CDX spreads are expressed in basis points and correspond to closing mid-market quotes on individual reference entities (i.e., mid-market quotes on individual issuers). We view CDX spreads as proxies of related CDS spreads insofar as Dow Jones CDX indexes are designed to track the CDS market as well as related liquidity (Das and Hanouna, 2006; Longstaff et al., 2005). Namely, two groups of credit derivatives data are considered. The first group consists of six Dow Jones North America credit derivative indexes whereas the second group deals with two Dow Jones Emerging Markets credit derivative indexes. Incidentally, the first group handles reference entities domiciled in North America, and is composed of the investment grade (NA_IG), investment grade high volatility (NA_IG_HVOL), high yield (NA_HY), BB rated high yield (NA_HY_BB), B rated high yield (NA_HY_B) and crossover

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(NA_XO) Dow Jones CDX indexes. Specifically, crossover NA_XO index accounts for the potential divergences between the rating grades that Standard & Poor’s and Moody’s rating agencies assign to reference entities in the BB/Ba-BBB/Baa rating classes. Differently, the second group handles entities domiciled either in Latin America, Eastern Europe, Middle East, Africa or Asia, and encompasses emerging markets (EM) and emerging markets diversified (EM_DIV) Dow Jones CDX indexes. In particular, EM index deals with sovereign entities whereas EM_DIV deals with both sovereign and corporate entities. Finally, Dow Jones North America credit derivative indexes are equal-weighted indexes whereas CDS IndexCo LLC members establish the weights of Dow Jones Emerging Markets credit derivative index. All Dow Jones credit derivative indexes satisfy a semi-annual reviewing and updating process. 2.2 Empirical facts Our main focus being to study the joint behaviour of CDS spreads and financial market, we consider daily changes of our data, or equivalently, corresponding first order differences (e. g., for both a given CDX spread S and a given day t, the first order difference writes ∆St = St – St-1). We therefore analyze time series with a total of 224 daily observations. As a first glance, we investigate the statistical properties of such daily changes (see table 1). [Insert table 1 about here] Table 1 shows a positive excess kurtosis for all our data except for normalized DJCI return (DJCI_N), underlining then a leptokurtic feature (i.e., peaked probability distribution relative to the Gaussian one). Moreover, the skewness view is mitigated. Indeed, data exhibit a general positive skewness (i.e., long right tail) except for NA_IG_HVOL, NA_XO and VXD indexes as well as DJCI return (i.e., long left tail). Accordingly, data are asymmetric and fat-tailed. Further, we investigate their respective behaviours over time while achieving a one percent Phillips-Perron test on daily changes (see table 2). [Insert table 2 about here] Table 2 shows clearly that Dow Jones CDX spread daily changes as well as financial market data’s daily changes (i.e., DJCI, DJCI_N and VXD) are stationary. In the light of observed individual behaviours, we address now the question of how market and credit fundamentals may evolve jointly and simultaneously over time. 3 Preliminary investigations We investigate in a very simple way the kind of dependence structure describing the joint evolution of both the U.S. financial market and CDX spreads. 3.1 Non parametric correlation coefficients

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As a first investigation and given the asymmetric nature of data’s daily changes, we compute both Kendall (i.e., tau statistic or τ) and Spearman (i.e., tau statistic or ρ) correlation coefficients. Those correlation statistics are also called rank correlations, and correspond to association measures assessing the degree of concordance between two random variables (i.e., two time series). We compute therefore non parametric correlation coefficients between financial market fundamentals (i.e., VXD index, DJCI and normalized DJCI returns) and credit market determinants (i.e., CDX spreads). Related results are displayed in table 3 below. [Insert table 3 about here] First, table 3 exhibits a negative link between daily changes in CDX spreads and daily changes in both DJCI and normalized DJCI returns over our studied time horizon. Such a conclusion holds whatever the correlation coefficient under consideration. Arguably, a negative link prevails between CDX spreads and market price risk over time. Second, a positive link prevails between daily changes in CDX spreads and daily changes in VXD index, meaning a positive relationship between CDX spreads and market volatility risk. Moreover, all correlation coefficients are significant at a one percent bilateral test level except for NA_HY_BB index case (five percent significance level). By the way, table 3 underlines also a negative link between VXD index and DJCI returns (i.e., market prices are negatively correlated with market volatility). Finally, we would like to stress that Kendall and Spearman correlation coefficients consist of a special case of copulae employment. 3.2 Empirical dependence structure As a rough guide, we also plot the observed bivariate dependence structures between the U.S. financial market and CDX spreads. We process in two steps. First, we consider the interaction of DJCI return changes with CDX spread changes (see figures 1 to 8). Then, we focus on the dependence structure of both VXD index changes and CDX spread changes (see figures 9 to 16). [Insert figures 1 to 16 about here] With regard to the link between CDX spread changes and market price change (figures 1 to 8), all plots exhibit an almost horizontal elliptical dependence structure (i.e., oriented along the x axis). Indeed, observations are mainly laid out along the x axis. Generally speaking, corresponding dependence structures seem apparently leftasymmetric. Namely, unreported results show that the proportions of joint negative daily DJCI return changes and either negative or positive daily CDX spread changes (i.e., average value of 52.3437 percent across all CDX indexes) lie clearly above the proportions of joint positive daily DJCI return changes and either positive or negative CDX spread changes (i.e., average value of 42.2563 percent across all CDX indexes). Moreover, extreme values are clearly distinguishable and numerous. We therefore have a view about extreme values’ distribution and magnitude, which determine tail fatness features of observed joint dependence structures. With regard to the link between CDX spread changes and market volatility change (figures 9 to 16), all plots exhibit more spherical but asymmetric dependence structures. Indeed, observations are gathered

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around each plot’s origin (i.e., distributed around the centre of each plot). Moreover, extreme values are still numerous but more clearly visible and distinct. In unreported results and plots, we also considered the dependence structures of both CDX spread changes and normalized DJCI return change (i.e., link between CDX spread changes and market risk change). We found results and profiles similar to the ones observed for the joint behaviours of DJCI return change and CDX spread changes. Corresponding results remain available from the author upon request. Previous plots give interesting insights about potential dependence structures between CDX spreads (i.e., credit risk fundamentals) and financial market data. However, such dependence structures can be soundly and appropriately investigated with the well known copula methodology among others. 4 Copula analysis The copula methodology is a cornerstone quantitative method for assessing risk. Basically, one just needs to focus on the bivariate dependence structure of two random variables as a whole (McNeil et al., 2005). Consequently, such an approach reduces highly model risk since respective univariate distributions’ marginals are not required to be specified. 4.1 Archimedean copulae and properties A copula function can describe any bivariate (or multivariate) probability distribution according to Sklar (1959, 1973), and in a unique way when we lie under a continuous framework. For a given vector (X,Y) of two random variables, Sklar (1959) defines a (two-dimension) copula function C as follows: Sklar’s theorem: Let F be a joint distribution function with margins FX and FY. Then there exists a copula C such that for all x, y in R , F ( x, y ) = C (FX ( x ), FY ( y ))

(1)

If FX and FY are continuous, then C is unique; otherwise, C is uniquely determined on RanFX × RanFY. Conversely, if C is a copula and FX and FY are distribution functions, then the function F defined by (1) is a joint distribution function with margins FX and FY. Copula function C is a continuous and non-decreasing function, whose values lie between zero and unity. Moreover, both arguments of copula function C take realvalues and lie also in the [0,1] interval. Such a framework allows for defining easily concordance and association measures such as Kendall and Spearman statistics (i.e., τ and ρ correlations respectively). Indeed, previous non parametric correlation measures rewrite simply as:

τ = 4 ∫∫

[0,1]2

C (u, v ) dC (u , v ) − 1

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(2)


ρ = 12∫∫

[0,1]2

uv dC (u , v ) − 3

(3)

where u = FX(x) and v = FY(y) are uniform transformations of x and y on the [0,1] real subset. Further, copula functions allow for defining upper tail (λU) and lower tail (λL) dependence measures, which are significant for describing distributions’ fat tails. Those tail dependence measures lie between zero and unity, and express as follows for any given vector (U,V) of random variables distributed in [0,1]²:

1 − 2u + C (u , u ) u →1 u →1 1− u C (u , u ) λL = lim+ Prob(V < u U < u ) = lim+ u →0 u →0 u

λU = lim Prob(V > u U > u ) = lim −

−

(4) (5)

Specifically, tail dependence measures assess whether an extreme of one of the two random variables under consideration impacts an extreme of the other random variable. Basically, when a tail dependence measure is zero/one, there is no/a perfect tail dependence respectively (i.e., extremes are uncorrelated/perfectly correlated). For convenience and measurability reasons, we focus on Archimedean copulae (i.e., parametric copulae), which depend on one parameter (Cherubini et al., 2004; Joe, 1997; Nelsen, 1999). Archimedean copulae are also called explicit copulae. Indeed, in numerous cases, tail dependence measures and concordance measures correspond simply to parametric functions (Genest and MacKay, 1986). For example, Kendall statistic τ reduces to a function of one parameter θ so that τ = f(θ) for any given τ value (i.e., Kendall correlation is assumed to be known). Consequently, the Archimedean copula under consideration is described by corresponding parameter value θ so that either related function is inverted (i.e., θ = f -1(τ)) or previous relationship is solved numerically. Moreover, θ parameter belongs to the real numbers’ set and is defined so that Kendall as well as Spearman statistics lie effectively between -1 and 1 range. 4.2 Selected copulae and parameter estimates

Given observed asymmetric dependence structures and for flexibility prospects, we focus on a set of three specific Archimedean copulae as well as a non Archimedean parametric copula. Specifically, we concentrate our study on Gumbel, Clayton and Frank Archimedean copulae as well as Farlie-Gumbel-Morgenstern (FGM) copula. Indeed, Gumbel copula accounts for upper tail dependence whereas Clayton copula accounts for lower tail dependence (e. g., contagion phenomena). Both copulae illustrate asymmetric dependence structures, which is the case of our financial data. However, Clayton copula is appropriate when negative changes in both studied random variables are more strongly correlated than corresponding positive changes (i.e., stronger negative relationship between both variables). Conversely, Gumbel copula is adequate when positive changes in the two considered random variables are more significantly correlated than corresponding negative changes. Further, Frank copula is symmetric and handles positive as well as negative correlations. We display in table 4 below the

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analytical expressions of corresponding copulae, Kendall statistics as functions of θ parameter, and related θ parameter as a function of Kendall correlation τ. [Insert table 4 about here] We stress that λU = 2 – 21/θ and λL = 0 (i.e., no lower tail dependence) for Gumbel copula whereas λL = 2-1/θ and λU = 0 (i.e., no upper tail dependence) for Clayton copula. Conversely, Frank copula exhibits no tail dependence since λL = λU = 0 (i.e., no dependence between extremes). As regards copulae description, estimating θ parameter is straightforward for Clayton, FGM and Gumbel copulae. On the other hand, estimating θ parameter for Frank copula requires solving numerically the relationship linking Kendall correlation estimate to θ parameter. For this purpose, we employ a scaled Broyden, Fletcher, Goldfarb, Shanno-type optimization algorithm with a convergence criterion of 10-6 for both estimation errors and gradient constraints. We compute related estimation errors while considering the dependence structures between both CDX spreads and successively DJCI return, VXD index and finally normalized DJCI return (i.e., market variables). Namely, estimation errors range from -1.4970E-07/-1.4366E-07/-9.9677E-08 for NA_IG_HVOL CDX index to 2.2277E-07/1.4716E-07/1.6260E-07 for EM/NA_HY/EM CDX indexes respectively, and relative to each market variable under consideration. Related parameter estimates are displayed in tables 5 to 7 below as a function of the financial market variable under consideration (i.e., DJCI return, VXD index or normalized DJCI return). Recall that Clayton copula handles lower tail dependence and positive association patterns whereas Gumbel copula handles upper tail dependence and non negative association patterns. Differently, Frank copula handles no tail dependence behaviour and negative as well as positive association. Given such features and previous observed Kendall statistics, the dependence structure between CDX spread changes and DJCI return change as well as the dependence structure between CDX spread changes and normalized DJCI return change are investigated in the light of FGM and Frank copulae. However, FGM copula is incompatible with observed Kendall statistics of EM index (i.e., observed Kendall’s tau (τ) is unattainable by this copula function). Conversely, the dependence structure between CDX spread changes and VXD index change is investigated in the light of Clayton, FGM, Frank and Gumbel copulae. [Insert tables 5 to 7 about here] Generally speaking, parameter estimates are respectively below -0.5 and -1 levels for FGM and Frank copulae (except for NA_HY_BB index case) with regard to DJCI return and normalized DJCI return as market variables (see tables 5 and 7). With regard to VXD implied volatility index (see table 6), parameter estimates are respectively above 0.3, 0.6 and 1 levels for Clayton, FGM, and both Frank and Gumbel copulae. Knowing corresponding parameter estimates, we have a complete description of the copulae under consideration. As a natural extension, we need now to select the copula function, which describes at best the observed dependence structures between CDX spreads and considered market fundamentals (i.e., DJCI, VXD and DJCI_N).

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5 Empirical study

We target now to identify the optimal copula describing the observed bivariate dependence structures between both CDX spread changes and successively DJCI return change, VXD implied volatility index change and normalized DJCI return change respectively. Once this task is achieved, we also propose a scenario analysis while studying worst case situations (i.e., stress testing). 5.1 Optimal copula

Considering a set E of possible (theoretical) copulae, we target to extract the copula function, which describes at best observed empirical dependence structures (i.e., in the most faithful way). For this, purpose, we identify the optimal copula as the theoretical copula function, which minimizes the mean ‘absolute’ error relative to the observed empirical copula function (i.e., observed dependence structure). Labelling CObs the observed empirical copula function, which is estimated from observed financial data, we compute the corresponding mean ‘absolute’ error d C , C Obs (i.e., average value of the square root of observed squared errors’ sum) as follows for any copula function C in E copula set:

(

(

d C, C

Obs

)

1 = T

⎧ ⎛i j ⎞ j ⎞⎫ Obs ⎛ i ⎨C ⎜ , ,θ ⎟ − C ⎜ , ,θ ⎟⎬ ∑∑ ⎠ ⎝ T T ⎠⎭ i =1 j =1 ⎩ ⎝ T T T

T

)

2

(6)

where T = 224 is the number of observations. Therefore, the optimal copula function C* satisfies the following constraint:

(

C * = min d C , C Obs C∈E

)

(7)

Consequently, we look for minimizing the observed distance between a possible theoretical copulae and observed dependence structures (i.e., observed empirical copulae, see Durrleman et al., 2000). Corresponding results are displayed in tables 8 to 10 as a function of the market variable under consideration (i.e., DJCI, VXD, and DJCI_N). [Insert tables 8 to 10 about here] With regard to dependence structures relative to DJCI return (see table 8), 25% of CDX spreads are optimally represented by FGM copula (i.e., EM_DIV and NA_HY index cases) copula whereas remaining 75% are optimally described by Frank copula. With regard to dependence structures relative to normalized DJCI return (see table 10), 37.5000% of CDX spreads are optimally represented by FGM copula (i.e., EM_DIV, NA_HY_BB and NA_HY index cases) whereas remaining 62.5000% are optimally described by Frank copula. With regard to dependence structures relative to VXD implied volatility index (see table 9), 25% of CDX spreads are optimally represented by Clayton copula (i.e., EM_DIV and NA_IG index cases) whereas remaining 37.5000%

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and 37.5000% are respectively optimally described by Frank copula (i.e., EM, NA_HY and NA_IG_HVOL index cases) and Gumbel copula (i.e., NA_HY_BB, NA_HY_B and NA_XO index cases). 5.2 Tail dependence and scenario analysis

We can now describe optimally observed bivariate dependence structures between CDX spread changes and some specific market variable changes. Such a setting allows for customizing some scenario analysis, or equivalently, stress testing. Namely, we attempt to quantify risky situations where a degradation of market risk triggers a deterioration of credit risk (as represented by CDX spreads). Such contagion phenomenon and spillover effect are important in the light of the close link prevailing between credit markets and financial/equity markets. Accordingly, we need to account for the degree of association between credit risk fundamentals and market risk determinants. Given the negative correlation between CDX spread changes and both DJCI return and normalized DJCI return changes, an unfavourable scenario would consist of a widening of CDX spreads given a decrease or at most a limited potential increase in either DJCI return or normalized DJCI return (scenario 1). Conversely, given the positive correlation between CDX spread changes and VXD implied volatility index change, an unfavourable situation is so that CDX spreads widen given that VXD implied volatility index increases (scenario 2). Furthermore, stress testing allows for analyzing the sensitivity of CDX spreads to market risk changes (e.g., Value-at-Risk setting). For example, we can address the following question: Are CDX spread changes more sensitive to market price risk or to market volatility risk? Assessing risk under (scenario 1) and (scenario 2) settings requires considering the following related conditional probabilities (8) and (9) respectively: u − C * (u , v,θ ) u 1 − u − v − C * (u, v, θ ) Prob(V > v U ≥ u ) = 1− u Prob(V > v U ≤ u ) =

(8) (9)

where u and v lie between zero and unity, and C* is the optimal copula describing the dependence structure under consideration. Specifically, we consider in a simpler way the two following quantile-quantile dependence measures relative to scenarii 1 and 2 respectively: Prob(V > qα U ≤ qα ) = Prob(V > qα U ≥ qα ) =

qα − C * (qα , qα , θ ) =α qα

1 − 2qα − C * (qα , qα ,θ ) =α 1 − qα

(10) (11)

where α illustrates the critical risk level under consideration, and qα represents the corresponding quantile level (i.e., qα = 1 - α). Of course, as a back testing means, we checked empirically whether the natural relation qαObs = (1 - α) holds whatever the CDX spread and market variable under consideration, qαObs being the observed

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empirical quantile. With regard to DJCI return, VXD implied volatility index and normalized DJCI return market variables, unreported results exhibit an average error (i.e., average value of qαObs - qα) level of 0.1373%, 0.0702% and 0.1363% respectively when α is 5%, and an average error level of 0.0058%, 0.6549% and 0.0058% respectively when α is 1%. Such errors illustrate the granularity issue in data (i.e., how far we may lie from the continuous framework assumption). Summarized results are available from the author upon request. Our stress testing study focuses on two extreme scenarii where α is either 5% or 1% (i.e., stressed/disturbed situation and crisis/crash framework respectively). As a rough guide, we estimate the empirical critical values of CDX spread changes and related market determinants (i.e., DJCI, VXD, DJCI_N) corresponding to previous observed quantiles qαObs (i.e., estimated empirical values in the neighbourhood of 95% and 99% respectively). Related results are displayed in tables 11 to 13 as a function of relevant market fundamentals. [Insert tables 11 to 13 about here] Therefore, we can assess the sensitivity of credit risk fundamentals to market risk determinants. Indeed, with regard to table 12 and considering EM index case for instance, there is a 5% probability level to face an increase of at least 9.6900 basis points in the CDX spread given an increase of at least 1.2700 points in VXD implied volatility index. Arguably, we can identify and quantify stressed and crisis scenarii along with critical risk level α (i.e., probability of occurrence of unfavourable scenarii). The way market risk (i.e., market fundamentals as representatives of market price and market volatility risks) impairs credit risk is of high significance in the light of (credit) correlation risk among others. Consequently, our stress testing framework is useful to credit portfolio managers insofar as it allows for establishing credit Value-atRisk in the light of correlation risk. This methodology provides then a helpful benchmark for establishing credit portfolio allocation target and policy. 6. Conclusion

Under Basel II framework, market risk is all the more important that it drives somehow credit/default correlations between corporate issuers. Specifically, it represents (to some large extent) the common systematic risk factor (as represented by business cycle conditions), which drives credit risk fundamentals such as CDS spreads (i.e., general co-movements or correlation risk, see ECB, 2006). Indeed, it is now common knowledge that default correlations drive joint credit loss distributions (across different corporate issuers) and ultimately any portfolio’s credit risk level and related credit risk allocation policy. Our study exhibited the impact of market risk on credit risk in the light of two risk dimensions and along with a two-step study. Namely, we first considered and described the bivariate dependence structures between CDX spread changes and both DJCI return change (as an indicator of market price risk) and VXD implied volatility index change (as an indicator of market volatility risk). As a summary, we also

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considered and characterized the dependence structure between CDX spread changes and normalized DJCI return change (as an indicator of global market risk). All dependence structures are asymmetric and exhibit sometimes some tail dependence patterns depending on the CDX spread under consideration. Second, we achieved a risk analysis along with a stress testing methodology and quantified the impact of both market price and market volatility risks on CDX spread changes. Market price risk impacts credit risk along with a non linear negative relationship whereas market volatility risk impacts credit risk along with a non linear positive relationship. However, our study was handled over the whole time horizon under consideration (i.e., in sample overview). A natural extension should be to update continuously such an overview over time. One means to achieve such an instantaneous risk assessment consists of the conditional copula methodology as proposed by Cherubini et al. (2004), Kolev et al. (2006), and Patton (2006) for example. The conditional copula methodology providing a useful tool for credit risk forecast prospects… References

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16


Table 1: Descriptive statistics for daily changes in data CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO DJCI VXD DJCI_N

Mean -0.1726 -0.1160 -0.1601 -0.1953 -0.1789 -0.0725 -0.0295 -0.1247 0.3718 0.0071 0.0291

Median Std. Dev. Excess kurtosis Skewness -0.7000 6.3847 9.3285 1.5816 -0.3800 3.5815 4.2827 0.3250 -0.4650 4.9126 7.9344 1.2028 -0.6700 5.8542 19.9748 2.7278 -0.8150 7.5362 33.9309 3.7808 -0.1150 1.6155 2.7238 -0.1790 -0.0550 0.6582 3.9450 0.2486 -0.1600 4.4644 9.1206 -1.2259 -8.3259 113.9735 0.0459 -0.0180 -0.0100 0.8945 8.1376 -0.0842 -0.5945 8.7927 -0.3207 0.0214

Table 2: Phillips-Perron test statistics (test without trend and/or constant) CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO DJCI VXD DJCI_N

Mean -13.7581 -15.6304 -13.6334 -14.4605 -13.4548 -12.8133 -12.7232 -11.3686 -71.7621 -15.0188 -80.2453

Table 3: Correlations between daily changes in market fundamentals and CDX spreads CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO DJCI VXD DJCI_N

DJCI -0.2451 -0.1593 -0.1053 -0.1240 -0.1574 -0.1754 -0.1611 -0.1557 1.0000 -0.1343 0.9372

Kendall’s tau VXD DJCI_N 0.2070 -0.2435 0.1526 -0.1565 0.2056 -0.0998 0.1343 -0.1213 0.2047 -0.1555 0.2218 -0.1690 0.1766 -0.1592 0.1913 -0.1512 -0.1343 0.9372 1.0000 -0.1351 -0.1351 1.0000

17

Spearman’s rho DJCI VXD DJCI_N -0.3584 0.2950 -0.3599 -0.2363 0.2227 -0.2337 -0.1541 0.2986 -0.1438 -0.1827 0.2011 -0.1803 -0.2315 0.2992 -0.2286 -0.2535 0.3204 -0.2452 -0.2375 0.2632 -0.2346 -0.2282 0.2742 -0.2226 1.0000 -0.1810 0.9935 -0.1810 1.0000 -0.1854 0.9935 -0.1854 1.0000


Table 4: Parametric copulae and properties Copula

(u

Clayton

C(u,v,θ) −θ

−θ

)

− θ1

+ v −1 u v [1 + θ (1 − u )(1 − v )]

FGM

[

−θ u −θ v − θ1 ln 1 + (e e−1−θ)(−e1 −1)

(a)

Frank

[

]

⎧ Gumbel exp⎨− (− ln u ) + (− ln v ) ⎩ (a)

D(x ) = 1x ∫0

x

t e t −1

dt

θ

θ

τ = f(θ)

θ = f -1(τ)

θ

θ

θ +2

2τ 1−τ

>0

2θ 9

9τ 2

1 − θ4 [1 − D(θ )]

] ⎫⎬⎭ 1

θ

τ ]0,1]

[-1,1] [-2/9,2/9]

Solution of * τ + θ [1 − D(θ )] − 1 = 0 R

[-1,1]

≥1

[0,1]

4

θ −1 θ

1 1−τ

is a Debye function.

Table 5: Parameter estimates relative to DJCI return CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

FGM -0.7167 -0.4736 -0.5579 -0.7084 -0.7891 -0.7251 -0.7008

Frank -2.3201 -1.4637 -0.9559 -1.1300 -1.4460 -1.6188 -1.4815 -1.4298

Table 6: Parameter estimates relative to VXD implied volatility index CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

Clayton 0.5221 0.3602 0.5176 0.3102 0.5146 0.5702 0.4288 0.4733

FGM 0.9316 0.6868 0.9252 0.6043 0.9210 0.9983 0.7945 0.8611

Frank 1.9308 1.4001 1.9167 1.2266 1.9073 2.0806 1.6305 1.7753

Table 7: Parameter estimates relative to normalized DJCI return (DJCI_N) CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

FGM -0.7041 -0.4491 -0.5457 -0.6997 -0.7606 -0.7164 -0.6802

Frank -2,3042 -1.4369 -0.9056 -1.1045 -1.4276 -1.5575 -1.4630 -1.3863

18

Gumbel 1.2611 1.1801 1.2588 1.1551 1.2573 1.2851 1.2144 1.2366


Table 8: Average observed distance relative to DJCI return CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

Frank 7.7536E-03 9.5958E-03 9.6139E-03 7.5972E-03 8.3381E-03 1.0985E-02 9.9544E-03 9.1989E-03

FGM 9.4965E-03 9.6231E-03 7.7318E-03 8.2129E-03 1.1303E-02 1.0267E-02 9.3248E-03

Table 9: Average observed distance relative to VXD implied volatility index CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

Clayton 9.7081E-03 8.4215E-03 1.4376E-02 9.1264E-03 1.0669E-02 1.1081E-02 9.7201E-03 1.1939E-02

FGM 9.1147E-03 8.7780E-03 9.5506E-03 8.4040E-03 8.7707E-03 9.1772E-03 1.1403E-02 9.7243E-03

Frank 8.4409E-03 8.6525E-03 9.1616E-03 8.1916E-03 8.2531E-03 8.8270E-03 1.1173E-02 9.2989E-03

Gumbel 9.1615E-03 8.9988E-03 7.2184E-03 7.5396E-03 8.4902E-03 9.6125E-03 1.2411E-02 9.0764E-03

Table 10: Average observed distance relative to normalized DJCI return (DJCI_N) CDX Index EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

FGM 9.2216E-03 8.4228E-03 7.0518E-03 8.2263E-03 1.0466E-02 9.3547E-03 8.9171E-03

Frank 8.2791E-03 9.3809E-03 8.4379E-03 6.9804E-03 8.4721E-03 1.0213E-02 9.0873E-03 8.8636E-03

19


Table 11: CDX spread changes’ critical values (bps) relative to DJCI return CDX Index DJCI EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

q5% 177.1126 9.6900 5.5800 7.5500 8.4100 8.7300 2.4300 1.0200 6.6300

q1% 283.1951 19.1400 11.3900 13.5100 19.5300 19.4300 4.7800 1.7600 9.8100

Table 12: CDX spread changes’ critical values (bps) relative to VXD volatility index CDX Index VXD EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

q5% 1.2700 9.6900 5.5800 7.5500 8.4100 8.7300 2.4300 1.0200 6.6300

q1% 2.8100 19.1400 11.3900 13.5100 19.5300 19.4300 4.7800 1.7600 9.8100

Table 13: CDX spread changes’ critical values (bps) relative to normalized DJCI return CDX Index DJCI_N EM EM_DIV NA_HY_BB NA_HY_B NA_HY NA_IG_HVOL NA_IG NA_XO

q5% 14.2148 9.6900 5.5800 7.5500 8.4100 8.7300 2.4300 1.0200 6.6300

q1% 19.5759 19.1400 11.3900 13.5100 19.5300 19.4300 4.7800 1.7600 9.8100

20


50

40

EM spread

30

20

10

0

-10

-20

-400

-300

-200

-100

0

100

200

300

-30 400

DJCI return

Figure 1: Dependence structure between daily changes in EM CDX spread and DJCI return 20

15

EM_DIV spread

10

5

0

-5

-10

-15

-400

-300

-200

-100

0

100

200

300

DJCI return

Figure 2: Dependence structure between changes in EM_DIV CDX spread and DJCI return

21

-20 400


40

NA_HY_BB spread

30

20

10

0

-10

-400

-300

-200

-100

0

100

200

300

-20 400

DJCI return

Figure 3: Dependence structure between changes in NA_HY_BB CDX spread and DJCI return 60

NA_HY_B spread

50 40 30 20 10 0 -10 -20

-400

-300

-200

-100

0

100

200

300

DJCI return

Figure 4: Dependence structure between changes in NA_HY_B CDX spread and DJCI return

22

-30 400


80 70

NA_HY spread

60 50 40 30 20 10 0 -10 -20

-400

-300

-200

-100

0

100

200

300

-30 400

DJCI return

Figure 5: Dependence structure between changes in NA_HY CDX spread and DJCI return 6

NA_IG_HVOL spread

4

2

0

-2

-4

-6

-400

-300

-200

-100

0

100

200

300

DJCI return

Figure 6: Dependence structure between changes in NA_HY_HVOL CDX spread and DJCI return

23

-8 400


4

3

NA_IG spread

2

1

0

-1

-2

-3

-400

-300

-200

-100

0

100

200

300

-4 400

DJCI return

Figure 7: Dependence structure between changes in NA_IG CDX spread and DJCI return 20 15

NA_XO spread

10 5 0 -5 -10 -15 -20 -25

-400

-300

-200

-100

0

100

200

300

DJCI return

Figure 8: Dependence structure between changes in NA_XO CDX spread and DJCI return

24

-30 400


50

40

EM spread

30

20

10

0

-10

-20

-30 -6

-4

-2

0

2

4

6

VXD volatility index

Figure 9: Dependence structure between changes in EM CDX spread and VXD index 20

15

EM_DIV spread

10

5

0

-5

-10

-15

-20 -6

-4

-2

0

2

4


Figure 10: Dependence structure between changes in EM_DIV CDX spread and VXD index

25

6


40

NA_HY_BB spread

30

20

10

0

-10

-20 -6

-4

-2

0

2

4

6


Figure 11: Dependence structure between changes in NA_HY_BB CDX spread and VXD index

60

NA_HY_B spread

50 40 30 20 10 0 -10 -20 -30 -6

-4

-2

0

2

4


Figure 12: Dependence structure between changes in NA_HY_B CDX spread and VXD index

26

6


80 70

NA_HY spread

60 50 40 30 20 10 0 -10 -20 -30 -6

-4

-2

0

2

4

6


Figure 13: Dependence structure between changes in NA_HY CDX spread and VXD index 6

NA_IG_HVOL spread

4

2

0

-2

-4

-6

-8 -6

-4

-2

0

2

4


Figure 14: Dependence structure between changes in NA_IG_HVOL CDX spread and VXD index

27

6


4

3

NA_IG spread

2

1

0

-1

-2

-3

-4 -6

-4

-2

0

2

4

6


Figure 15: Dependence structure between changes in NA_IG CDX spread and VXD index 20 15

NA_XO spread

10 5 0 -5 -10 -15 -20 -25 -30 -6

-4

-2

0

2

4


Figure 16: Dependence structure between changes in NA_XO CDX spread and VXD index

28

6