Quantifying the impact of different copulas in a generalized CreditRisk+ framework An empirical study

Abstract Without any doubt, credit risk is one of the most important risk types in the classical banking industry. Consequently, banks are required by supervisory audits to allocate economic capital to cover unexpected future credit losses. Typically, the amount of economical capital is determined with a credit portfolio model, e.g. using the popular CreditRisk+ framework (1997) or one of its recent generalizations (e.g. [8] or [15]). Relying on specific distributional assumptions, the credit loss distribution of the CreditRisk+ class can be determined analytically and in real time. With respect to the current regulatory requirements (see, e.g. [4, p. 9-16] or [2]), banks are also required to quantify how sensitive their models (and the resulting risk figures) are if fundamental assumptions are modified. Against this background, we focus on the impact of different dependence structures (between the counterparties of the bank’s portfolio) within a (generalized) CreditRisk+ framework which can be represented in terms of copulas. Concretely, we present some results on the unknown (implicit) copula of generalized CreditRisk+ models and quantify the effect of the choice of the copula (between economic sectors) on the risk figures for a hypothetical loan portfolio and a variety of parametric copulas.


Introduction
Financial institutions are allowed to use their own internal models in order to manage di erent types of risk, such as market, liquidity, credit or operational risk, on an economic (in contrast to a regulatory) basis. In this context, banks are requested by national supervisors to quantify the accompanied amount of model risk.
If methods and processes, underlying assumptions, parameters or in uent data are rather complex, a qualitative and quantitative validation of these components as well as the useability of the resulting risk gures is necessary (see [2]).
Regarding credit risk, portfolio models always assume a speci c dependency structure between economic sectors and obligors. Mathematically, dependency can be described with the help of copula functions, originally introduced by [33]. Recently, several authors such as [20], [11], [6] or [24] have addressed the topic of copulas in credit portfolio models. However, they have always concentrated on only one copula class. By contrast, we take a wide range of copulas into account, analyze which ts best to default data and how risk gures change for several copulas. Because in practical applications the number of counterparties is simply too high to model their dependencies directly, we focus on the dependency structure between sector variables. In this regard the article can be seen as an extension of [6], who exchanged the copula of the compound gamma model¹ with a Gaussian copula and a t-copula. Our analysis incorporates representatives from the elliptical, the generalized hyperbolic and the Archimedean copula classes together with two copulas, implicitly de ned by the corresponding portfolio model.
The outline of this article is as follows: Section 2 gives a brief introduction to copulas and de nes some notation used throughout the article. In the third section we shortly introduce the CreditRisk + model together with two model enhancements covering exible sector dependencies. Here we also analyze the implicit copulas induced by sector models in order to compare them with other dependency models. In section 4 the credit portfolio under consideration and the default data used for parameter estimation are described. Afterwards we analyze the change in risk gures concerning di erent copulas. The article concludes with a summary.

Copulas
The notion of copulas dates back to [33]. Technically, a copula C is a multivariate distribution function on the d-dimensional unit hypercube I d with uniform margins. With the help of copulas one can separate the problem of nding a suitable multivariate distribution function into two parts. In a rst step, one can concentrate on the univariate margins independently from each other. In a second step one analyzes the dependency structure regardless of the marginal distribution. The justi cation for this procedure is given by Sklar's theorem.

Theorem 1. Let F denote a d-dimensional distribution function on
It should be noted that, especially if all F i are continuous on I, the copula is unique on I d . All information about the dependency structure are contained in the copula. In this article we will use Sklar's theorem in reverse. In order to analyze the impact of the copula on the risk gures of the CreditRisk + model, we exchange the copula function and create new multivariate distribution functions with the same margins as in the original case.
Our analysis includes the following copulas: -Independence copula de ned as C(u) := d i= u i . The copula is the one of d stochastically independent random variables U i . -Elliptical copulas are implicitly given by the class of elliptical distributions, see [7]. Famous representatives are the normal or Gaussian copula and the t-copula Here, Φ Σ / t Σ,ν denotes the multivariate normal / t-distribution with zero mean, dispersion matrix Σ (and ν degrees of freedom). Φ − and t − ν are the corresponding quantile functions. -Generalized hyperbolic copulas (ghyp) arise implicitly from the corresponding multivariate distribution rst studied by [3]. In general, ghyp copulas are not elliptically symmetric. Under certain parameter restrictions, ghyp copulas belong to the elliptical copula class. Therefore, the class of ghyp copulas also contains the normal and the t-copula as limiting cases. A detailed parametrization of the ghyp copula is given in the appendix.
-Archimedean copulas (AC) are de ned by a d-monotone² generator function ψ : [ , ∞) → I with ψ( ) = and lim x→∞ ψ(x) = . In contrast to the former classes, these copulas have an explicit representation. For dimension d, an AC is de ned by: (1) -Hierarchical Archimedean copulas (HAC) are an extension of the Archimedean class where one can combine di erent generator functions. For example, consider a generator function ψ θ with parameter θ ∈ Θ, the ordinary AC is given by Under suitable conditions³ one can exchange the inner AC denoted by (#) with another AC arising from a di erent generator function. This nesting procedure can be repeated until only two variables are left. The resulting copula is called a fully nested HAC (left panel of gure 1). Another nesting procedure could be to exclude not only one variable at each nesting level, but two or more, which again can be grouped via an AC. The resulting copula is called a partially nested HAC (right panel of gure 1). The technical de nition of a general HAC depends on a rather complex notation. Since, for our purposes, the general de nition is not necessary, we refer to [32].
In case of the Frank, Clayton or Gumbel copula, one can alternate the parameter values θ for di erent nesting levels. Thus, one can create hierarchical structures with stronger dependencies at the ground level (e.g. between industry groups of speci c countries) and weaker dependencies at the top level (e.g. between countries themselves). Figure (1) shows two possible trees of a ve dimensional HAC where the generator function is of the Clayton type, i.e. ψ Cl θ (x) := (θx + ) − /θ for θ ≥ . For a detailed discussion of HAC we refer to [32] and [29] as well as [23] and [16], where the su ciency of the nesting condition has been proven and rst examples were presented. Please note that the partially nested structure is just a special case of the fully nested one.

The CreditRisk + model and extensions
In this section, after a short primer on credit risk in general, where we explain the reasons for using credit portfolio models in practice, the CreditRisk + model is introduced. At the end we discuss the topic of sector dependencies and present two extensions of the basic model to incorporate correlated sectors.

. A short primer on credit risk in general
In the banking industry, credit portfolio models are used to quantify the risk arising from a portfolio of obligors within a xed time horizon (e.g. one year). Thinking of the overall portfolio loss, one distinguishes between expected and unexpected losses. The former one corresponds to the expectation value of the portfolio loss (e.g. caused by defaults). Since this component is not a ected by relationships between counterparties, it can be calculated separately for each obligor. Typically, the expected loss is incorporated within the credit pricing process. By contrast, the unexpected loss corresponds to the possible loss exceeding the expected one. With the help of so called economic capital, nancial institutions measure the amount of capital needed to cover unexpected losses. In order to quantify the necessary amount, the so called value at risk approach is used. For a given level α ∈ [ , ] the corresponding value at risk (VaRα) is de ned as the α−quantile of the portfolio loss distribution. The economic capital is de ned as where EL denotes the expected loss (over one year) of the portfolio.
The major task of credit portfolio models is to calculate the distribution function of the portfolio loss, such that one can extract the VaRα and other characteristic values we de ne later.
In general, credit risk consists of two di erent parts: -default risk arising from the loss caused by counterparties' defaults and migration risk occurring if the creditworthiness of counterparties decreases. Please note that within this article we only concentrate on the default risk component. For a more detailed introduction to credit risk we refer to [24].

. The basic CreditRisk + model
The CreditRisk + portfolio model was introduced by the Financial Products division of Credit Suisse in 1997. A detailed description of the model is given in [35]. It belongs to the class of mixture models. This model type is characterized by the assumption that defaults are conditionally independent given the state of the economy or a speci c sector. As a consequence of certain distributional assumptions the pdf of the portfolio loss distribution can be calculated analytically. Therefore, it enjoys great popularity in practice. In a rst step, the exposures are discretized with respect to a common loss unit L > . With the help of the discretization, the model becomes numerically tractable. Afterwards, the original potential loss ead i · lgd i and the PDp i are then replaced by ν i := max ead i · lgd i L , and p i := ead i · lgd i ·p i ν i · L respectively. Here x denotes the nearest integer to x. Since the original PDp i is replaced by p i the discretization of the potential loss does not a ect the expected loss of the portfolio, because: with L := M i= ν i · L · D i representing the discretized portfolio loss and D i ∼ Bernoulli (pi). In a second step, the Bernoulli distribution is replaced by a Poisson distribution. This replacement ensures, that the density function of the overall portfolio loss can be calculated analytically. The intensity of defaults depends on the modi ed PD p i and some sector variables S k ∼ Γ (θ k , δ k ) representing the state of the economy or speci c sectors.⁴ Every counterparty is mapped onto one or more of K sectors via sector weights w i,k ∈ [ , ] for i = , .., M and k = , ..., K. The idiosyncratic risk is represented by The default intensity of CP i is de ned by In the basic model, the sector variables S k are assumed to be independent from each other. Extensions to correlated sectors will be discussed in the following section. In combination with the sector weights w i,k the default intensities of two counterparties CP i and CP j sharing at least one sector, ⁵ are correlated with where σ k denotes the variance of S k . In order to ensure an unchanged expected loss, E (S k ) = has to be assured, which means that δ k = θ k for all sectors k = , ..., K. As a second condition for the parametrization of the sector distribution one can estimate the variances σ k either from historical default data or use one out of several approximation formulas given in [14].
With the help of the probability generating function (pgf) of the Poisson distribution and the independence of sector variables and default events, the pgf of the total portfolio loss is given by The pgf can be evaluated with the help of a nested evaluation algorithm. A detailed derivation of the pgf as well as a numerically stable algorithm to calculate the (discrete) density of L from its pgf is given in [14].

. Modeling dependent sectors
Modeling dependencies between several counterparties in a credit portfolio is a crucial issue. Typically, credit portfolios consist of thousands of obligors. Modeling the dependencies directly between counterparties is not manageable because the dimension of the copula or multivariate distribution function is just too high. Instead, the counterparties are categorized by industry, country and other attributes. Then each group is mapped onto a sector speci c variable a ecting the default probability (PD) of the group members. The dependency between sector variables translates into a dependency between counterparties. Since dependencies between counterparties have a major impact on the portfolio loss distribution, the way they are modeled directly in uences the amount of economic capital which must be provided by nancial institutions to cover unexpected losses.
Modeling the dependencies between sectors rather then counterparties reduces the dimension of the copula dramatically. In our empirical analysis (section 4) we work with a ten dimensional copula. In the following we introduce two extensions of the basic CreditRisk + model, in order to overcome the assumption of independent sectors.

. . The Common-Background-Vector model
In practice, economic sectors are not independent. Therefore, [8] proposed the so called common-backgroundvector (CBV) model, which is a generalization of the model from [15]. The main idea is to replace each sector variable by a linear combination of L independent gamma distributed variablesŜ ∼ Γ θ , and an independent sector speci c variable S k ∼ Γ (θ k , δ k ). I.e. we de ne with non-negative weights γ k, for k = , ..., K and = , ..., L. The vectorŜ := Ŝ , ...,Ŝ L T is called common-background-vector.Ŝ is equal for all sectors k. How much the background factorŜ l in uences sector k is determined by γ k,l . De ning additional counterparty sector weights w i,K+ := K k= w i,k γ k, for = , ..., L and plugging (4) into (2) we can rewrite the formula for the default intensity as This setting is equal to the basic model with K + L independent sectors. Therefore, the pgf can be expressed analytically as in case of the basic model.
The variance covariance (VCV) of the original sectors S k can be derived as The model parameters γ k, , δ k , θ k andθ should be chosen such that the theoretical VCV structure given by equations (5) and (6) meets an empirical one. This can be achieved by solving a high dimensional optimization problem with K + L( + K) variables. The dimension of the optimization problem should not be mixed up with the dimension K of the corresponding copula (equation 7). In order to guarantee that E S k = we have to restrict the parameter space such that θ k δ k + L = γ k, θ k = for all sectors k = , ..., K. The multivariate distribution of the sector variables is determined by a linear combination given in equation (4). Writing this in a more general way with vectors X ∈ R d and Y ∈ R K , where each single X i= ,...,d is independently gamma distributed with an individual shape and scale parameter and a matrix A ∈ R K×d , we have Y = AX.
The univariate distribution of the single Y k has already been studied by [26]. The copula corresponds to a multi-factor copula, investigated by [28], with gamma distributed factors. Figure (2) shows 10,000 realizations of the bivariate distribution ( rst row) and the implicit copula (second row) for the bivariate case with A = .
. The shape and scale parameters for the gamma distributed variables X and X are given in the header. As the plot in the left-hand column shows, the support for the bivariate copula function is not the whole unit square. Instead we have a concave and a convex zero curve. In the bivariate distribution above, these curves occur as linear bounds. For illustrative issues we have added the red lines representing them. One can show that the slope of these lines is given by the minimum and the maximum of the ratios between the matrix entries in the several columns of A. In particular the following theorem holds. Theorem 2. Given a vector X of non-negative random variables X i with arbitrary copula and a matrix A ∈ R K×d with entries a j,i for some K, d ∈ N > , then for the vector Y = AX it holds: Proof. The proof is given in the appendix.
Because of the nonlinear probability transformation, for the copula the linear bounds of the bivariate distribution transform into concave and convex curves. One can also observe that the number of realizations in the area around the zero curves grows with the variance of the corresponding X i . In general, this copula belongs to none of the copula classes presented in section 2. Because of its shape and the bounded support it is neither elliptical nor generalized hyperbolic. The copula is also not an Archimedean one because those are unable to form concave zero curves (see [27] Theorem 4.3.2). For some parameter setting, the scatter plots and especially the zero curves look very similar to those of a two parameter extreme value copula class named BC , introduced by [22]. Also the coe cients of lower and upper tail dependence (see section 3) of these two copula classes are similar. Since the density and distribution function of the copula given by (7) cannot be stated explicitly, we leave the question in which cases these two copula classes coincide as an open one for further research.
In the special case of the CBV model, equation (7) takes the form where Γ ∈ R K×L with entries γ k, . So the rst columns of matrix A contain the K dimensional identity matrix. Therefore the lower and upper bounds of Theorem (2) disappear because they are given by and ∞.

. . The Multi Compound Gamma model
Another approach to model correlated sectors within a CreditRisk + framework goes back to [13]. The basis of the (multi) compound gamma (MCG) is a mixture approach as in the CreditRisk + model itself. In the standard model, the default distribution (Poisson) is mixed with a Gamma distribution or a linear combination of several independent ones. In the MCG model the gamma distribution of the sectors again is mixed with one or more gamma distributions. In more detail, again L background variablesŜ ∼ Γ σ − ,σ are introduced. The shape parameter θ k of the original sector variables is assumed to follow a linear combination of the background variables with weights α k, > . The scale parameter is xed. Summarizing this we have: Again to ensure that E S * k = we have the additional parameter condition β k = L = α k, − . The VCV structure of the MCG model is given by: Cov The pgf of the portfolio loss L reads as with P k (t) := M i= w i,k p i (t νi − ). Again, the pgf can be calculated with the help of a nested evaluation algorithm.
Having a set of CBV-or MCG-parameters, one can easily switch between the two models with the help of the following lemmas.
Proof. A simple algebraical calculation shows that the equations (5) and (6) with the parameters stated in the lemma are equivalent with (8) and (9) which proves the lemma.

Lemma 4.
Let δ k , θ k ,θ , γ k, denote the parametrization of a CBV model. The identical VCV structure is asymptotically generated by an MCG setup with K sectors andL = L + background sectors and the following parameters It holds: lim Proof. Again, simply plugging in the stated parametrization into (8) and (9) and taking the limit leads to the formulas (5) and (6).
In both models, nding a suitable parametrization for an empirical VCV structure -especially for a higher number of sectors -is crucial. Hence, the last two lemmas are very helpful because now one has to nd the parametrization for just one model. Furthermore, we do not have to consider any e ects of unequal parametrization caused by di erent algorithms when we switch from one model to another.

Empirical analysis . Portfolio and data
The underlying portfolio consists of 5000 counterparties. For reasons of simplicity, we assume a constant loss given default (LGD) of one for each counterparty. Since we only want to concentrate on the relative changes of the economic capital for di erent copulas under consideration, this assumption will not restrict our results. Assuming counterparty speci c constant LGDs (not necessarily 1) is equal to a scaling of the corresponding exposure, resulting in a shift / reduction of the value at risk. The same applies to the case of stochastic LGDs. For those who are interessted in the e ect of stochastic LGDs, we refer to [14, section 7]. Each counterparty is mapped onto a single sector out of the ten sectors. To keep the analysis manageable, we do not distinguish between di erent regions or countries. The distribution of potential losses (PL) and counterparties (CP) across sectors is shown in table 1.
For parameter estimation a data pool with more than 30,000 corporates around the world is used. Based on a Merton model (see [25]), the data represent the one-year PD of exchange traded corporates between 2003 and 2013. The PDs are estimated monthly over the last ten years and then aggregated on sector level via the median. Finally, in order to take time dependencies into account, we have tted a univariate auto-regressive process to every sector time series and use the residuals for the maximum likelihood estimation instead.

. Copulas under consideration
In order to estimate the unknown parameters for the copulas from section 2, we have used the maximum likelihood approach. For this purpose, we used the R-packages "copula" by [17], "ghyp" from [21] and the "HAC" package from [30]. The implicit copula of the CBV model was estimated with the help of an optimization algorithm, minimizing the L −distance between the empirical and the theoretical VCV matrix. For the parameters of the MCG model we used lemma 4. A short overview of the estimation results is given over the next few pages. If we exchange the independence copula of the basic CreditRisk + model with some arbitrary one, in general the model can no longer be solved analytically.⁶ Therefore, we simulate the sector variables S , .., S k with a speci ed copula C, calculate the individual default intensity λ i via (2) and restrict the analytical calculation to the rst product in (3). Thus we just have to calculate the distribution of the portfolio loss caused by M independent obligors. Simulating the vector S = (S , .., S k ) T of sector variables N times and averaging each single exposure band over all N pdfs gives us a nal estimation for the overall pdf. This idea has already been mentioned in [6].

. . Tail dependence
During the following analysis we will also focus on the index of tail dependence. In contrast to dependency measures such as the linear correlation coe cient, Kendall's τ or Spearman's ρ, which measure dependency on an overall level, the index of tail dependence measures dependency only in extreme situations. Following [27], the coe cients of upper and lower tail dependence are de ned for a pair (X , X ) of random variables by where F i and F − i denotes the marginal distributions and the quantile functions, respectively. In the context of the nancial crisis, the Gaussian copula was blamed for failing to model economic dependencies correctly because it admits no tail dependencies. This means that, especially during the crisis, counterparties or economic sectors tend to behave independently from each other rather than dependently. Therefore, the amount of economic capital required to cover the loss in such situations, estimated with the help of the Gaussian copula, may be not su cient. Interestingly, this disadvantage of the Gaussian copula has already been pointed out by several mathematicians years before the nancial crisis started (see [34]).

. . Gaussian vs. Student-t copula
For reasons of clarity, we focus our discussion of the estimation results on just three of the ten sectors, namely the industrial sector, the nancial sector and the sector of cyclical consumer goods, which cover around 75% of the total exposure. Table 2 shows the estimated entries of the dispersion matrix as well as the index of tail dependence. Please note that, in contrast to the Student-t copula, the Gaussian copula admits no upper nor lower tail dependence. In the case of the Student-t copula, the coe cients of upper and lower tail dependence are equal because of the elliptical symmetry. For the t-copula we estimated .
degrees of freedom. The corresponding copula functions together with the empirical observations are illustrated in gure 3. The strongest dependency occurs between the industrial sector and the sector of cyclical consumer goods. Here, we observe the highest σ as well as the highest coe cient of tail dependence for the t-copula. Furthermore, the copula functions between these two sectors concentrate more mass on the main diagonal compared to the others. Table 3 summarizes the log-likelihood values of the Gaussian-and t copula as well as those from the estimated ghyp copulas, presented in the next section. The log-likelihood of the Gaussian copula is around 634 whereas the t-copula has a value of approximately 728. Based on a likelihood ratio test, we can conclude that the t-copula ts the data signi cantly better than the Gaussian one.

. . Generalized hyperbolic copulas
The class of generalized hyperbolic copulas contains the class of elliptical copulas if no asymmetries are taken into account. Therefore, as in the case of the t-and the Gaussian copula, we would expect a considerably higher log-likelihood than in the former cases. For the asymmetric ghyp copula and the underlying data set we obtain a log-likelihood value of 13566. Again, a likelihood ratio test indicates that the ghyp copula ts the data signi cantly better than the t-copula and, of course, better than the Gaussian copula. Besides this, we also considered a symmetric ghyp copula, where we restrict γ = ( , ..., ) T ∈ R d . In this case, the log-likelihood is still very much higher compared to the case of the Gaussian and the t-copula, i.e. the log-likelihood equals 8848. Performing likelihood ratio tests leads us to two results. On the one hand, the symmetric ghyp copula ts the data signi cantly better than the Gaussian and the t-copula. On the other hand, the asymmetric ghyp copula also has a better t compared to the symmetric one. Hence, we can state that the asymmetry in our empirical data is signi cant and therefore should not be neglected.
In the asymmetric case, all components ofγ are strictly positive. Therefore, the multivariate distribution and the copula are skewed towards higher values. Since higher values correspond to higher default rates, industry nance cyclical consumer goods Fig. 3. Fitted Gaussian and t-copula together with empirical observations. this supports the economic intuition of a higher dependency between corporates when economic conditions become worse. However, as gure 4 shows, the optical di erences to the elliptical copulas (lower triangle matrix) are not as high as expected. Nonetheless, in all three bivariate copulas of the upper triangle matrix of gure 4 there is slightly more probability mass concentrated in the upper right corner than in the lower left one.

. . (Hierarchical) Archimedean copulas
Another exible class of copulas is the Archimedean one. Since the ghyp copula has already indicated that a positive skewed distribution is needed to t the data, we only consider the Gumbel copula for our analysis. In fact, after tting other Archimedean copulas (Clayton and Frank) and the adoption of a goodness of t test as discussed by [12], we can clearly reject them. Besides the ordinary Gumbel copula, we also take a hierarchical  construction into account. To estimate the parameters and the nesting structure we used the package "HAC" by [30]. The package uses a stepwise maximum likelihood approach⁷. Since the estimation of a HAC is not in the scope of this paper, please refer to the mentioned article for further details on the estimation process. The estimated hierarchical structure is illustrated in gure 5. The Gumbel copula is positively ordered, which means that a higher parameter value implies a higher dependency between variables. Therefore, we have the strongest dependency on the lowest level between cyclical consumer goods and the industrial sector. Indicated by the lowest parameter value, diversi ed companies have the weakest dependency to all other sectors. The parameter range reaches fromθ= . toθ = . , which corresponds to values of Kendall's τ between . and . . Calculating the empirical values of Kendall's τ by means of the default data gives us a similar interval.
If we consider just an ordinary Gumbel copula instead of a hierarchical one, we obtainθ = .
. Since all of the bivariate copulas of a multivariate AC are identical,⁸ choosing the copula parameter is always a tradeo between stronger and weaker dependencies. The "average" value forθ corresponds to τ = .
. Again, gure 6 shows the bivariate copulas for sectors 3, 7 and 8 of the ordinary multivariate Archimedean (upper triangle matrix) and the hierarchical Archimedean copula (lower triangle matrix). In the ordinary case, all the bivariate copulas coincide. In case of the industrial and the nancial sector or the nancial sector and the one of cyclical consumer goods, the ordinary and the HAC are quite close θ HAC = .
But in the upper right and lower left case (industry vs. cycl. consumer goods) the choice of a common θ is a major disadvantage, because the dependency measured by the HAC (τ = . ) and the observations τ = .
are higher than the ordinary Gumbel copula suggests (τ = . ). Because of the speci c generator function, the copula function is not as symmetric as the Gaussian or the t-copula. Comparing the Gumbel copula to a symmetric Archimedean one (Frank) and one with inverted asymmetry (Clayton) by means of goodness of t tests shows that the Gumbel copula is the most suitable one for our data. The coe cients for implied upper tail dependency are summarized in table 4. Comparing these values to those of table 2 we can state that the HAC in all three cases generates a higher upper tail dependence than the t-copula does. The coe cient of lower tail dependency of the ordinary and the hierarchical copula is zero, by de nition. Table 4. Coe cients of upper tail dependency of ordinary Gumbel copula (upper triangle matrix) and HAC (lower triangle matrix).
cycl. consumer goods

. . The CBV and MCG copula
In a nal step we also analyze the implicit copulas of the CBV and the MCG model for the given data. Since we do not have an analytical representation of the copula densities, we have estimated them based on simulated values for each pair of variables. For the CBV model we chose ve background variables. For the calibration of the MCG model, we used lemma 4 with ϵ = − . In the upper triangle matrix of gure 7 we plotted the bivariate copulas of the CBV model and in the lower triangle matrix those of the MCG model. Please note that, in contrast to the former copulas, the parametrization of the CBV and the MCG model was executed on the basis of the empirical VCV structure rather than the likelihood of the observations. In general, the copulas of the two models are fairly similar. The copulas of the MCG model show a slightly higher concentration of probability mass around the ridge of the density function (black curve) whereas the implicit copula of the CBV model is wider. What di erentiates both copulas from their competitors (e.g. Gaussian or tcopula) is the large amount of asymmetry. As in the case of the Gumbel copula, mutual higher realizations are more likely than mutual lower ones. However, the magnitude of this skewness is much greater. In addition, industry nance cyclical consumer goods the copulas of both models are not symmetric with respect to the main diagonal. Since the estimation was done on the basis of the VCV structure, which in general does not cause asymmetry, this is quite an interesting observation.
Because of the signi cant asymmetries, we decided to numerically estimate the coe cients of lower and upper tail dependence with the help of methods discussed by [5] and [9]. The results, based on simulations, are stated in table 5. We used three di erent estimators for upper and lower tail dependence. However, the results are fairly equal. So we state only one value for lower and one for upper tail dependency. Irrespective of the model (CBV or MCG) the estimated upper tail dependence in all cases is considerably higher than the lower tail dependence⁹. Whereas for the MCG model λ L is clearly positive, the copula of the CBV model seems to be lower tail independent. Besides the signi cant di erence between λ L and λ U , the level of the upper tail dependence of all pictured copulas is also remarkably high. industryλ U = . λ U = . λ L = .
λ L = . goods industry nance cyclical consumer goods

. Impact on risk gures
After analyzing the estimation results, now we discuss the impact of the several dependency structures and, in a rst step, also of the marginal sector distributions on the risk gures within a CreditRisk + framework. As already mentioned in section 3, banks use internal models and the concept of economic capital to allocate equity capital necessary to cover unexpected losses. Against this background, the variations in risk gures can be interpreted as the model risk arising from the choice of a speci c dependency structure.
Besides the economic capital, we also state the values for expected shortfall. The expected shortfall on level α, denoted by ESα, is de ned as the conditional expectation value of the portfolio loss, once the corresponding value at risk VaRα is exceeded. It is an alternative risk measure to VaR ful lling the property of subadditivity, i.e. for two portfolios A and B it holds: ESα(A) + ESα(B) ≥ ESα(A + B), see [1].

. . Marginal sector distributions
Originally, the distribution of each single sector of the CreditRisk + model was assumed to be gamma. In order to handle correlated sectors, the distribution changed by construction to a compound gamma distribution (MCG) or a linear combination of independent gamma distributions (CBV), which in general is not gamma any more. Hence, before analyzing the impact of the copula, we have to quantify the e ect caused by a change from the model speci c margins to gamma distributed ones. Exemplarily, gure 8 shows the pdf of sector variable 3 in the CBV (green), and the MCG model (blue) as well as the pdf of an ordinary gamma distributed random variable (red) with equal mean and variance. The marginal sector distributions of the CBV and the MCG model are more heavily tailed than an ordinary gamma distribution. Hence, we could expect a remarkable decrease in risk gures when we switch the marginal sector distribution back to an ordinary gamma distribution. The e ect will be stronger for the CBV than for the MCG model.  Table 6 summarizes di erent risk gures,¹⁰ describing the portfolio loss distribution of the CBV and the MCG model with di erent marginal sector distributions. In both models, the loss distributions of the CBV/MCG models are more heavily tailed compared to the models with gamma distributed sectors. On a % loss level alone, the values for economic capital are lower in the case of the combined Γ distribution than in the case of an ordinary Γ distribution. The markups for higher quantiles are in the range of . % to % depending on the model and quantile level. Naturally, the impact is stronger for higher quantiles. The original CBV model also accounts for higher risk numbers compared to the original MCG model. This is consistent with former studies (e.g. [8]) and the observation of more heavily tailed sector distributions from gure 8.

. . Copulas
Finally we analyze how the loss distribution and hence the risk gures change under copula assumptions. The results are summarized in table 7. In order to facilitate comparison, we state the values in percent of the corresponding CBV value. The right tail area of the loss distributions is illustrated in gure 9.
At rst we note that all alternative copulas (independence, Gaussian, t-, ghyp, AC, and HAC) imply a lower risk than the copulas of the CBV and the MCG model. The pdfs of the portfolio loss of the CBV and the MCG model are fairly equal, as gure 9 shows. The risk gures of the model with independent sectors show that 25% of the portfolio loss standard deviation and nearly one third of the required economic capital (99.9% level) are due to sector dependencies. The risk reduction e ect grows with the considered risk level.
Since the Gaussian copula is elliptically symmetric and admits no tail dependence, it produces the second lowest risk. In the tail area above the % loss quantile, the pdf is strictly dominated by all others. On average the values for economic capital are % to % lower compared to the CBV model with gamma distributed margins. Using a symmetric ghyp copula instead of a Gaussian one, the results are similar. Switching to a t-copula, the positive tail dependence causes an increase in risk of approximately % on the highest level. The asymmetric ghyp, Gumbel and hierarchical Archimedean copula imply the highest risk among the alternative copulas under consideration. In contrast to the Gaussian and t-copula, they generate an asymmetric dependency structure. Since the realization of multiple high default rates, in those cases, is more likely than the realization of multiple lower ones, high losses are also more likely. The resulting loss distribution is more heavily tailed than those of the elliptical models. Since the pdfs of the asymmetric ghyp, Gumbel and HAC model are very close, we have plotted just the density of the model with an asymmetric ghyp copula. Although having no tail dependency, the risk arising from this copula is fairly equal to that arising from the Archimedean copulas, which admits a positive upper tail dependence. Therefore, we can conclude that, the risk arising from an asymmetric dependency structure is higher than the risk implied by a copula with positive tail dependence for this data set.
The highest risk is observed for the copulas underlying the MCG and the CBV model. This is reasonable because they have the highest upper tail dependence and, in the case of the CBV copula, no lower tail dependence. Furthermore, the probability mass of these copulas is more concentrated around the main diagonal as gure 7 shows.

Conclusions
In a credit portfolio model such as CreditRisk + , the choice of a particular copula noticeably a ects the risk gures. Of course, using a Gaussian copula implies the lowest risk. Switching to a skewed copula with or without positive tail dependence increases the risk. Here the impact of an asymmetric dependency structure is stronger than the impact of a positive tail dependence. However, the implied copulas of the CBV and the MCG model account for the highest risk. Therefore, from a copula point of view, the MCG and the CBV model are rather conservative within the framework of CreditRisk + . With the help of likelihood ratio tests, we showed that for our data, the magnitude of asymmetry is signi cant. Di erent estimators suggest that these copulas admit a high upper tail dependence, while the lower one is considerably smaller. The di erence between the copulas of the CBV and the MCG model is negligible.
Aside from the copula, the marginal distributions of the sector variables also considerably a ect the risk gures. For the CreditRisk + model, this e ect is even stronger than the e ect of the copula. Hence, the di erence between the CBV and the MCG model mainly arises from the marginal sector distribution.
Since modeling the dependency directly on counterparty level is not manageable, an interesting question is what the resulting copula on counterparty level would be after the translation of the sector copula via the link function. But this is a more general topic which we leave open for further research.

A Proof of Theorem 2
Theorem. Let X be a vector of real random variables X i with arbitrary copula and P (Xi ≥ ) = for all i = , ..., d. Furthermore let a j,i denote the elements of a matrix A ∈ R K×d for some K, d ∈ N > . Then the ratio of any two components of the vector Y = AX is bounded below and above. If P (Y = ) = it holds: Proof. Without loss of generality we set k = and = . At rst we concentrate on the bivariate case of X, so d = . For two realizations (x , x ) of (X , X ) set q = a , x +a , x a , x +a , x . Taking derivatives yields: x (a , a , − a , a , ) (a , x + a , x ) and ∂q ∂x = − x (a , a , − a , a , ) (a , x + a , x ) . Now we can distinguish between three cases according to the sign of (a , a , − a , a , ). -(a , a , − a , a , ) > : q is increasing with x for all x > and decreasing with x for all x > . Therefore we get: a , a , = q ( , x ) < q (x , x ) < q (x , ) = a , a , .
The ghyp family contains a lot of special cases e.g. normal, (skewed) t, variance gamma or the normalized inverse Gaussian distribution. For more information on this topic as well as the GIG distribution we refer to [31]. The ghyp family also possesses several di erent representations. For other parametrizations and ways of switching between them, we refer to [21]. These authors also created the R-package "ghyp".
Finally, the ghyp copula is given via the multivariate distribution F and the quantile functions of the margins F i , i.e. C(u) = F F − (u ) , ..., F − d (u d ) .