Crisis and Risk Dependencies - CiteSeerX

Crisis and Risk Dependencies

PETER GRUNDKE*

SIMONE DIECKMANN

University of Osnabrück Chair of Banking and Finance Katharinenstraße 7, 49069 Osnabrück Germany Phone: ++49 (0)541 969 4720 Fax: ++49 (0)541 969 6111 Email: [email protected]

First version: November, 29, 2009 This version: March, 11, 2010** Abstract: The knowledge of the multivariate stochastic dependence between the returns of asset classes is of importance for many finance applications, such as, e.g., asset allocation or risk management. By means of goodness-of-fit tests, it is analyzed for a multitude of portfolios consisting of different asset classes whether the stochastic dependence between the portfolios’ constituents can be adequately described by multivariate versions of some standard parametric copula functions. Furthermore, it is tested whether the stochastic dependence between the returns of different asset classes has changed during the recent financial crisis. The main findings are: First, whether a specific copula assumption can be rejected or not, crucially depends on the asset class and the time period considered. Second, different goodness-of-fit tests for copulas can yield very different results and these differences can vary for different asset classes. Third, even when using various goodness-of-fit tests for copulas, it is not always possible to differentiate between various copula assumptions. Fourth, during the financial crisis, copula assumptions are more frequently rejected.

Key words: copulas, crisis, goodness-of-fit test, stochastic dependence JEL classification: C12, G19

* Corresponding author ** We thank Daniel Berg for kindly providing his R-package copulaGOF.

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1. Introduction The knowledge of the multivariate stochastic dependence between random variables is of crucial importance for many finance applications. For example, the co-movement of world equity markets is frequently interpreted as a sign of global economic and financial integration. Portfolio managers have to know the stochastic dependence between different asset classes to exploit diversification effects between these asset classes by adequate asset allocation. Market risk managers have to know the stochastic dependence between the market risk-sensitive financial instruments in their portfolio to compute risk measures, such as value-at-risk or expected shortfall. Credit risk managers have to know the stochastic dependence between latent variables driving the credit quality of their obligors to compute risk measures for their credit portfolios. Financial engineers constructing multi-asset financial derivatives have to know the stochastic dependence between the different underlyings to correctly price the derivatives.

Copulas are a popular concept for measuring the stochastic dependence between random variables. They describe the way that the marginal distributions of the components of a random vector have to be coupled to yield the joint distribution function. In other words, copulas provide a way of isolating the description of the dependence structure from the joint distribution function. However, the parameters of an assumed parametric family of copulas have to be estimated and, afterwards, the adequacy of the copula assumption has to be proved.1 One way of doing this adequacy check is by performing goodness-of-fit (gof) tests for copulas. In this paper, we employ gof tests based on the Rosenblatt transformation and based on the empirical copula for testing various copula assumptions with respect to stochastic dependence between the returns of different asset classes (credit, stocks, commodities, real estate and so on). The two main questions we want to answer are:

1. Is it possible to describe the multivariate stochastic dependence between the returns of different asset classes by some standard parametric copula?

2. If so, how did the stochastic dependence between the returns of different asset classes change during the recent financial crisis? A satisfying answer to this second question could be of importance for multivariate stress tests based on historical scenarios.

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An alternative to the usage of parametric families of copulas is to employ empirical copulas (see Deheuvels (1979, 1981)). However, the estimation of empirical copulas, for example combined with some kernel approach, usually requires a relatively long time series of data which in practice is not always available.

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Our main findings are: First, whether a specific copula assumption can be rejected or not, crucially depends on the asset class and the time period considered. Second, different gof tests for copulas can yield very different results and these differences can vary for different asset classes. This might be one explanation for the fact that we partly have different findings with respect to the adequacy of specific copula assumptions for specific asset classes compared to the literature. Third, even when using various gof tests for copulas, it is not always possible to differentiate between various copula assumptions. Thus, it is possible that several very different null hypotheses with respect to the copula are not rejected. Fourth, during the financial crisis, copula assumptions are more frequently rejected.

The remainder of the paper is structured as follows: In section 2, a short review of the relevant literature is given and the contributions of this paper are sketched. In section 3, the copulas employed in this paper and the gof tests are described. Section 4 contains the data description, the data processing and the empirical results. In section 5, a financial application is presented to analyze the potential economic significance of the results. Finally, in section 6, the main conclusions are summarized.

2. Literature review The idea of analyzing the stochastic dependence between the returns of different asset classes is not new. Early paper, such as Longin und Solnik (2001) and Ang und Chen (2002), focus on (conditional) correlations as dependence measure. Instead, recent papers use the copula concept for measuring stochastic dependencies. Mainly, these contributions deal with the stochastic dependence between market risk data, such as stocks or exchange rates. Breymann et al. (2003) deal with high-frequency data of two exchange rate returns (USD/DEM and USD/JPY). They consider bivariate Gaussian, t-, (survival) Gumbel and (survival) Claytoncopulas and find that the t-copula is the most adequate one (beside for short frequencies below 8 hours). Dias and Embrechts (2009) also analyze the stochastic dependence between the daily returns of the two exchange rates USD/DEM and USD/JPY. Among the various bivariate copulas and mixtures of copulas they consider, the t-copula and a mixture of the Gumbel and the survival Gumbel copula perform best. For the former one, they also consider a conditional version with change points. Chen et al. (2004) consider 30 daily US stock returns and 20 daily exchange rate returns (with USD as a base) and multivariate versions of the Gaussian and t-copula. They find that the Gaussian copula is always rejected (beside for the bivariate analysis of stock returns). The t-copula seems to be adequate for describing the bivariate and 2

multivariate stochastic dependence of stock returns (see similarly Mashal et al. (2003)), but not for exchange rates. Patton (2006) also deals with daily returns of the two exchange rates USD/DEM and USD/JPY and tests whether conditional and unconditional versions of the Gaussian copula and the so-called symmetrised Joe-Clayton copula are adequate for describing the stochastic dependence between them. He finds signs for an asymmetric dependence between these two exchange rates, but, nevertheless, the conditional and the unconditional Gaussian copula (as well as both versions of the symmetrised Joe-Clayton copula) cannot be rejected based on the goodness-of-fit test he employs. Malevergne and Sornette (2003) find that most pairs of exchange rates and pairs of major stocks quoted at the New York Stock Exchange are compatible with the Gaussian copula hypothesis, while this null hypothesis can be rejected for pairs of commodities (metal). However, the bivariate t-copula can also not be rejected for exchange rates and stocks if it has sufficiently many degrees of freedom. The stochastic dependence between various stock indices is analyzed, for example, by Ané and Kharoubi (2003), van den Goorbergh (2004), Jondeau and Rockinger (2006) and Rodriguez (2007). Ané and Kharoubi (2003) analyze whether unconditional versions of the Gaussian-, Frank-, Gumbel- or Cook-Johnson-copula are adequate for describing the stochastic dependence between pairs of five stock indices (DAX 30, Nikkei 225, FTSE 100, S&P 500, NASDAQ, Hang Seng). They find that the bivariate Cook-Johnson-copula, which exhibits lower tail dependence and upper tail independence, is most appropriate. Van den Goorbergh (2004) analyzes the bivariate stochastic dependence between the daily returns of three major stock indices (S&P 500, FTSE 100, CAC 40). He finds that in general a conditional version of the bivariate t-copula seems to be most adequate for describing the pairwise dependencies. Additionally, he finds that a copula-based approach is superior to a dynamic conditional correlation model of Engle (2002). Jondeau and Rockinger (2006), who additionally consider the DAX 30, also find that a bivariate conditional t-copula performs better than a conditional Gaussian copula. Rodriguez (2007) analyzes the bivariate stochastic dependence between daily returns from five East Asian stock indices during the Asian crisis and between the daily returns from four Latin American stock indices during the Mexican crisis. He considers a mixture of copulas consisting of the Frank, Gumbel, and Clayton copula with switching weights and a bivariate t-copula with switching correlations and switching degree of freedom parameter. He finds evidence for changing dependencies during the crisis, which, however, are different for the Asian and the Latin American countries. The dependence relation between short-term and long-term interest rates is analyzed by Junker et al. (2006). Having estimated a Gaussian two-factor generalized Vasicek model, they find a transformed Frank cop3

ula to be the most appropriate model for describing the stochastic dependence between term structure innovations at the short and the long end.2 In contrast, the Gaussian and the t-copula exhibit features that violate the observed complex dependence structure which is characterized by asymmetry and upper tail dependence. Studies that deal with default risk dependencies are, for example, by Das and Geng (2004), Das et al. (2007) and Aboura and Wagner (2008). Based on probability of default estimates of individual obligors provided by Moody’s econometric model, Das and Geng (2004) use copulas to describe the stochastic dependence between the obligors’ individual default intensity processes. Das et al. (2007) show that observable macroeconomic variables and latent common variables play a major role in explaining correlated defaults. Systematic credit risk is also the focus of Aboura and Wagner (2008). They analyse to which extent changes in Single-CDS spreads can be explained by a common factor which is represented by the DJ CDX.NA.IG index.

Typically, only bivariate problems are studied in academic research on copula estimation (see, e.g., Breymann et al. (2003), Malevergne and Sornette (2003), van den Goorbergh (2004), Jondeau and Rockinger (2006), Dobrić and Schmid (2007), Dias and Embrechts (2009)) and the results are taken as being representative for higher dimensional problems. However, those exceptions that deal with bivariate as well as multivariate problems (see, e.g., Chen et al. (2004), Grundke (2009)) show that the results can be quite different. That is why we test the null hypothesis of multivariate copulas (up to six dimensions) as an adequate modeling approach.

This paper is most closely related to Kole et al. (2007) and Grundke (2009). Kole et al. (2007) analyze the stochastic dependence between the daily returns of three asset classes: stocks (represented by the S&P 500), bonds (represented by the JP Morgan’s US Government Bond Index), and real estate (represented by the FTSE EPRA/Nareit Global index). They test 3dimensional versions of the Gaussian, t-, and (survival) Gumbel copula.3 They find that only the null hypothesis of the t-copula is not rejected. Compared to Kole et al. (2007), first, we extend the inter-asset-class analysis by additionally considering credit, commodities and currencies as asset classes. Second, we add intra-asset class analyses. Third, we adequately filter the

2

3

The transformed Frank copula has been proposed by Junker and May (2005). They employ this copula for modelling the bivariate stochastic dependence between the returns of single stocks as well as between euro swap rate returns with different time horizons. They find that a linear convex combination of the transformed Frank copula and the respective survival copula is superior to the Gaussian copula, the t-copula and a linear convex combination of the Clayton copula and the survival Clayton copula. For the Gumbel copula, the extension of Bouyé (2002) is taken.

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returns so that the gof tests get i.i.d. data as input. Fourth, we also test whether the stochastic dependencies have changed during the recent financial crisis. Grundke (2009) only deals with an intra-asset-class analysis because he only tests the stochastic dependence between sectorspecific subindices of the DJ iTraxx Credit Default Swap (CDS) index for Europe. Thus, we extend his work by an extensive inter-asset-class analysis. Additionally, we employ a second gof test for copulas (based on the empirical copula) so that we can check the robustness of his results with respect to the chosen gof test.

As already mentioned above, some papers on gof tests for copulas employ conditional versions of copulas where the parameters of the chosen copula are time-varying (see, e.g., Chen et al. (2004), van den Goorbergh (2004), Jondeau and Rockinger (2006), Patton (2006), Rodriguez (2007)). However, all these contributions are restricted to an (admittedly very detailed) bivariate analysis, whereas this paper deals with multivariate dependencies. Unfortunately, testing multivariate conditional copulas is beyond the scope of this paper.

3. Copulas and goodness-of-fit tests 3.1 Copulas Any joint distribution function of random variables contains information on the marginal behavior of the single random variables and on the dependence structure of all random variables. Copula functions only deal with the stochastic dependence and provide a way of isolating the description of the dependence structure from the joint distribution function. According to Sklar’s Theorem, any joint cumulative density function (cdf) FX1 ,..., X D ( x1 , dom variables X1 ,..., X D can be written in terms of a copula function C(u1 , marginal cdf’s FX1 ( x1 ),

FX1 ,..., X D ( x1 ,

, xD ) of D ran, uD ) and the

, FX D ( xD ) :4 , xD ) C( FX1 ( x1 ),

, FX D ( xD )) .

(1)

The corresponding density representation is:

f X1 ,..., X D ( x1 , where c(u1 ,

f X1 ( x1 ),

, uD ) (

, xD ) D

/ u1

f X1 ( x1 ) uD )C(u1,

, FX D ( xD ))

(2)

, uD ) is the copula density function, and

, f X D ( xD ) are the marginal density functions. For recovering the copula function of

a multivariate distribution FX1 ,..., X D ( x1 , C (u1 , 4

f X D ( xD ) c( FX1 ( x1 ),

, uD )

, xD ) , the method of inversion can be applied:

FX1 ,..., X D ( FX11 (u1 ),

, FX D1 (uD ))

(3)

Standard references for copulas are Joe (1997) and Nelsen (2006).

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, FX D1 ( ) are the inverse marginal cdf’s. For continuous random variables

where FX11 ( ),

X1 ,..., X D , the copula function is unique.5 In our context, FX1 ,..., X D ( x1 ,

, xD ) are the marginal

cdf’s of the daily returns of indices that represent specific asset classes or the daily returns of the constituents of single asset classes.

In this paper, we employ the following copulas:

Gaussian copula: The D -dimensional Gaussian copula is defined as

C Gauss u1 , where

D

(u1 ),

1

,

(u D );

(4)

denotes the cdf for the D -dimensional standard normal distribution with

;

D

1

, uD ;

correlation matrix

, and

1

is the inverse of the cdf of a univariate standard normally

distributed random variable. The corresponding normal copula density is: cGauss u1 ,

, uD ;

1 exp det( )

(

1

(u1 ),

,

1

(uD )) (

1

1

I D )(

(u1 ),

,

1

(uD ))

(5)

2

where I D is the D -dimensional identity matrix. The Gaussian copula is a tail independent elliptical copula (for a formal definition of pairwise tail dependence see, e.g., Schönbucher (2003, p. 332)).

t-copula: An elliptical copula that exhibits symmetric tail dependence is the t -copula which is given by

C t u1 , where TD

; ;

lation matrix

, uD ; ;

TD T 1 (u1 ; ),

, T 1 (u D ; ); ;

denotes the cdf for the D -dimensional standard t -distribution with correand degree of freedom (dof)

2 , and T

a univariate t -distributed random variable with dof

5

(6)

1

;

is the inverse of the cdf of

. The t -copula density is:

However, in empirical applications, random variables and the respective empirical marginal distribution functions are rarely continuous, because usually only a finite number of observations of realizations of these random variables are available.

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D 1

D t

c u1 ,

2

, uD ; ;

2

det( )

where x ( x1 ,

, xD )

(T 1 (u1; ),

copula for increasing dof

D

1

x

1

1

x

D 2

1 D

1

xi2

2

(7)

i 1

2

, T 1 (uD ; )) . As a t -copula converges to a Gaussian

, both copulas are nested. For fixed correlation parameters, the

strength of the pairwise tail dependencies increase as the dof

decreases, and for fixed dof

, the pairwise tail dependencies increase as the correlation parameters increase.

Gumbel copula: The Gumbel copula which belongs to the class of so-called Archimedean copulas is defined by 1

D

C

Gumbel

u1 ,

, uD ;

exp

(8)

( ln ui ) i 1

with parameter

1 . The Gumbel copula is asymmetrically tail dependent (upper tail de-

pendence and lower tail independence). For pendence copula, and for

1 , the Gumbel copula converges to the inde-

, it goes to the comonotonicity copula. As all Archimedean

copulas, the Gumbel copula is completely determined by its generator

Gu

(t ) ( ln t ) .6 By

means of its generator, every Archimedean copula can be represented in the following way: C u1 ,

1

, uD

( (u1 )

(uD )) .

For a strict Archimedean copula generator

(9)

, the representation (9) yields a copula in any

dimension D if and only if the inverse of the generator

1

is completely monotonic (see

Kimberling (1974)).7

Clayton copula: Another asymmetrically tail dependent (lower tail dependence and upper tail independence) Archimedean copula is the Clayton copula (also known as Cook and Johnson’s (1981) family) that is defined as

6

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:[0,1] [0, ] that satisfies (1) 0 is called a generator of an Archimedean copula. It is called a strict generator if (0) (see McNeil et al. (2005, p. 221)). A decreasing function f (t ) is called completely monotonic on an interval [a, b] if it satisfies A continuous, strictly decreasing, convex function

( 1) k

dk f (t ) dt k

0, k

, t

(a, b) (see McNeil et al. (2005, p. 222)).

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1

D

C

Clayton

u1 ,

, uD ;

ui

D 1

(10)

i 1

with parameter

0 . The generator of the Clayton copula is

Cl

(t ) (t

1)

0 corresponds to the D -dimensional independence copula. For

ing case

. The limit, the

Clayton copula goes the comonotonicity copula.

Generalized Clayton copula: A two-parameter Archimedean copula is the generalized Clayton copula. Its generator is

(t ) (t

1)

with parameters

0,

1 (see McNeil et al. (2005, p. 220)). In contrast

to the Gumbel and Clayton copula, the generalized Clayton copula exhibits upper and lower tail dependence. In contrast to the t -copula, this tail dependence is asymmetric. For stock returns, Longin and Solnik (2001) and Ang and Chen (2002) find statistical evidence for asymmetric dependencies, and for currencies, Patton (2006) gets similar results. For generalized Clayton copula converges to the Gumbel copula, and

0 , the

1 yields the Clayton

copula.

Due to the asymmetric tail dependence of the Gumbel, Clayton and generalized Clayton copula, we also consider the survival copula that is induced by the respective copulas. In the bivariate case, this is Cˆ (u, v) u v 1 C (1 u,1 v) . In general, the survival copula Cˆ of a copula C is the distribution function of the transformed variables U * 1 U and V * 1 V when U and V have the distribution function C . For the survival copula, the asymmetric tail dependence is reversed compared to the original copula.

As all parametric families of copulas that we test in the empirical part of the paper belong to different classes of copulas (elliptical and Archimedean) each with a different extent of tail dependence, we believe that we test a reasonable coverage of parametric families of copulas. This is why we restrict ourselves to (low-) parametric families of copulas instead of estimating non-parametric (empirical) copulas. Table 1 shows the different coefficients of upper and lower tail dependence of the copulas considered. - insert table 1 about here -

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3.2 Estimation of the parameters For estimating a multivariate model consisting of the marginal distributions and the copula, several approaches exist. Basically, the parameters of the marginal distributions and the parameters of the copula, combined in the parameter vector θ out of the parameter space can be estimated simultaneously by maximizing the log-likelihood function: θˆ MLE

,

8

arg max l (θ)

(11)

θ

with T

l (θ) ln

f X1 ,..., X D ( x1t ,

, xDt ;θ)

t 1 T

ln c( FX1 ( x1t ;θ),

, FX D ( xDt ;θ);θ)

ln f X1 ( x1t ;θ)

f X D ( xDt ;θ) .

t 1

Assuming that the usual regularity conditions for asymptotic maximum likelihood theory (see Serfling (1980)) are fulfilled for the marginal distributions as well as for the copulas, the maximum likelihood estimator (MLE) θˆ MLE exists, is consistent, asymptotically efficient and asymptotically normally distributed (see Cherubini et al. (2004, p. 154)). Of course, to apply (11), an à priori choice for the type of the marginal distribution and for the parametric copula family is necessary. To reduce the computational costs for solving the optimization problem (11), which results from the necessity to estimate jointly the parameters of the marginal distributions and the copula, the method of inference-functions for margins (IFM) can be applied (see Joe (1997, pp. 299), Cherubini et al. (2004, pp. 156)). The IFM method is a two-stepmethod where, first, the parameters θ1

(θ1,1 ,

,θ1,D ) of the marginal distributions are esti-

mated and, second, given the parameters of the marginal distributions, the parameters θ 2 of the copula are determined: θˆ 1,IFM i

θˆ IFM 2

arg max θ1,i

arg max θ2

T

ln f X i ( xit ;θ1,i )

(i {1,

, D})

(12)

t 1

T

IFM ln c( FX1 ( x1t ;θˆ 1,1 ),

. , FX D ( xDt ;θˆ IFM D ,1 );θ 2 )

(13)

t 1

The complete IFM estimator is defined as: θˆ IFM

IFM (θˆ 1,1 ,

ˆ IFM ) ' . ,θˆ 1,IFM D ,θ 2

(14)

In general, an equivalence of the two estimators θˆ MLE and θˆ IFM does not hold, but, as for the MLE estimator, it can be proved that, under regularity conditions, the IFM estimator is as-

8

It is assumed that the copula function and its parameters do not vary within the data period.

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ymptotic normally distributed (see Joe (1997, p. 301)). A further possibility for estimating the parameters of the copula function is the canonical maximum likelihood (CML) estimation (also called pseudo-maximum likelihood). For this method, there is no need to specify the parametric form of the marginal distributions because these are replaced by the empirical marginal distributions. Thus, only the parameters of the copula function have to be estimated by MLE (see Cherubini et al. (2004, p. 160)). In the empirical part of the paper, this approach is employed, however, partly in a modified version to further reduce the computational burden. For example, for estimating the parameters of a t -copula, first the correlation parameters are computed by the method-of-moments inverting the relationship

( X i , X j ) (2 / ) arcsin

ij

between Kendall’s tau and the correlation parameters of the t -copula (see McNeil (2005, p. 231)). Afterwards, the remaining degree of freedom is estimated by maximum (pseudo-) likelihood. Similarly, the correlation parameters of a Gaussian copula can be computed using their relationship

S

( X i , X j ) (6 / )arcsin 0.5

ij

to Spearman’s rho.

3.3 Goodness-of-fit tests for copulas In recent years, many goodness-of-fit (gof) tests for copulas have been proposed and applied (e.g., Breymann et al. (2003), Malevergne and Sornette (2003), Mashal et al. (2003), Chen et al. (2004), Savu and Trede (2008), Dobrić and Schmid (2005, 2007), Fermanian (2005), Genest et al. (2006), Kole et al. (2007)). From the proposed gof tests, we choose one test based on the Rosenblatt transformation (see, e.g., Breymann et al. (2003), Dobrić and Schmid (2007)) and one test based on the empirical copula process (see Genest and Rémillard (2008)). Both tests exhibited a relatively good performance in power comparisons of different gof tests (see Berg (2009) and Genest et al. (2009)).

3.3.1 Goodness-of-fit test based on the Rosenblatt transformation Let

X

FX ( x1 , x2 ,

( X1 , X 2 ,

, X D ) be a random vector with absolutely continuous joint cdf

, xD ) and marginal cdf’s FX i ( xi ) ( i {1,

, D} ). Assuming that C(u1 ,

, uD ) is

the correct copula describing the stochastic dependence between the components of the random vector X , the multivariate distribution function has the representation (1). Denoting the joint i -marginal distribution of (U1,

Ci (u1 ,

, ui ) C(u1,

,Ui ) by

, ui ,1,

,1) for i {2,

, D 1} ,

(15)

10

with C1 (u1 ) u1 and CD (u1, the realizations of U1 ,

, uD ) , the conditional distribution of U i , given

, uD ) C(u1,

,Ui 1 , is for i {2,

, D}

i 1

Ci ui u1 ,

, ui

Ci (u1 , , ui ) u1 ui 1

1

Under the assumption that C(u1 ,

Ci 1 (u1 , , ui 1 ) . u1 ui 1

(16)

, uD ) is the correct copula, the random variables Z i (the

so-called Rosenblatt transforms) that are defined for i {1, Z1

FX1 ( X 1 ),

Z2

C2 FX 2 ( X 2 ) FX1 ( X 1 ) ,

ZD

CD FX D ( X D ) FX1 ( X 1 ),

, D} in the following way

, FX D 1 ( X D 1 ) ,

are uniformly and independently distributed on [0,1]d (see Breymann et al. (2003)). From this follows that the random variable D

S ( X1 , X 2 ,

1

, XD)

( Zi )

2

(17)

i 1

is chi square distributed with D degrees of freedom. Thus, the validity of the null hypothesis of interest H0 : “ X

( X1 , X 2 ,

null hypothesis H 0* : “ S ( X1 , X 2 ,

, X D ) has copula C(u1 ,

, uD ) ” implies the validity of the

, X D ) is chi square distributed with D degrees of freedom”

and H0 can be rejected if H 0* is rejected.9 The validity of the latter hypothesis can be tested, for example, by the Anderson-Darling (AD) test statistic.10 However, as the true marginal distribution functions FX i ( xi ) for i {1,

, D} are typically not known in empirical applications,

but have to be substituted by their empirical counterparts when computing the Rosenblatt transformations, the theoretical distribution of the AD test statistic does not hold any longer, as argued by Dobrić and Schmid (2007). Instead, the empirical distribution function of the AD test statistic under the null hypothesis has to be estimated by bootstrapping. During the bootstrap procedure, repeatedly realisations of D -dimensional random vectors are simulated from the D -dimensional copula under the null hypothesis, given its estimated parameters. Based on each simulated D -dimensional time series, the parameters of the copula are estimated and the Rosenblatt transforms and the AD test statistic are computed. After a sufficiently large 9

10

Basically, a disadvantage of the gof test based on the Rosenblatt transformation is that the test statistic for a given data set may be different depending on the permutation order when computing the Rosenblatt transforms. However, as a simulation study of Berg (2009) shows, this problem seems to be negligible in practical applications. In the empirical part of the paper, the gof test based on the Rosenblatt transformation is also combined with the Cramér-von Mises test statistic.

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number of simulated time series11, this procedure yields the empirical distribution function of the AD test statistic under the null hypothesis, which can be used for computing critical values and the p -value (see, e.g., Dobrić and Schmid (2007) for details). For performing the gof test based on the Rosenblatt transformation, an experimental software written in R (“copulaGOF”) that has been kindly provided by Daniel Berg is employed. Only for the generalized Clayton copula, the computations were done with self-made programs in MATLAB.

3.3.2 Goodness-of-fit test based on the empirical copula As a second test procedure, a gof test based on the empirical copula is employed. The multivariate empirical copula is defined as (see, e.g., Genest et al. (2009)):

Cˆ (u ) where u

(u1 ,

with t {1,

T

1 T 1t

I FˆX1 ( xt1 ) u1 ,

, FˆX D ( xtD ) uD

(18)

1

, uD ) [0,1]D , I { ,

, } is the multivariate indicator function, ( xt1 ,

, T } is a realization of the random vector X

empirical cdf of X i with i {1,

( X1 , X 2 ,

, xtD )

, X D ) and FˆX i ( ) is the

, D} . The idea of this gof test is to compare the empirical

copula with the parametric copula C ( ) under the null hypothesis (given its estimated parameters). A Cramér-von Mises test statistic for this approach is: T

CvM

Cˆ ( zt ) C ( zt )

2

(19)

t 1

with zt

( FˆX1 ( xt1 ),

, FˆX D ( xtD )) ( t {1,

, T } ). Detailed implementation issues for this gof

test can be found in Genest and Rémillard (2008). As for the gof test based on the Rosenblatt transformation, approximate p-values are obtained by a bootstrap procedure. The validity of the bootstrap procedure and the properties of the underlying empirical process are discussed by Genest and Rémillard (2008). A Monte Carlo simulation study of Genest et al. (2009) shows that this gof test performs very well compared with several other gof test procedures. For carrying out the gof test based on the empirical copula, the R-package “copula” recently written by Jun Yan and Ivan Kojadinovic is used. Again, for the generalized Clayton copula, the computations were done with self-made programs in MATLAB.

11

For all gof tests and all copulas, the number of bootstrap simulations is set equal to 1,000 in the empirical part of the paper.

12

4. Empirical analysis This section contains the empirical part of the paper. First, the data and its processing are described. Afterwards, the ARMA-GARCH filtering of the data is sketched which shall ensure that the gof tests get i.i.d. data as input. Finally, the results of the gof tests are presented for portfolios of different asset classes.

4.1 Data In total, we consider six different asset classes: credit (represented by the DJ iTraxx Credit Default Swap (CDS) master index for Europe and six sector-specific subindices which are described in more detail in the following), stocks (represented by six major stock indices: MSCI World, S&P 500, DAX 30, DJ Euro STOXX 50, FTSE 100, CAC 40), bonds (represented by the Datastream EMU Benchmark 10 year DC Govt. index), currencies (represented by the USD/EUR exchange rate), commodities (represented by the Reuters/Jefferies CRB index), real estate (represented by the FTSE EPRA/Nareit Global index).12

The DJ iTraxx CDS master index for Europe, usually referred to as the DJ iTraxx, contains the 125 most liquid credit default swaps of investment-grade names in Europe. These are equally weighted in the index and selected by a dealer poll based on the CDS volume traded over the previous six months. Every six months, the index is rolled over to the next series to consider the most liquid CDS of the last six months. Each of the 125 entities of the DJ iTraxx belongs to one of the following industry sectors (see www.indexco.com): Energy (20 entities), Telecommunications / Media / Information Technology (TMT; 20 entities), Autos (10 entities), Industrials (20 entities), Financials (25 entities), and Consumers (supermarkets, airlines, etc.; 30 entities). As in Grundke (2009), we have constructed a subindex of the DJ iTraxx for each of these six industry sectors based on series 9. Altogether, we had to exclude 14 entities from the total sample of 125 for which not enough reliable data was available for the whole period. All index spread quotes used in this study are based on five-year-maturity Single-CDS contracts because these are typically the most liquid ones.

12

All indices are price indices.

13

The daily data for all asset classes is gathered from Datastream and Bloomberg and covers the period January 2, 2006 to October 21, 2008. Based on daily midpoint quotes, log returns are computed. Each time series is divided into two subsamples covering the time periods January 2, 2006 to June 29, 2007 and July 2, 2007 to October 21, 2008, respectively. The first sample represents the pre-crisis sample whereas the second sample is considered to be the crisis sample. There has been a dramatic increase in CDS spreads since July 2007 which could be interpreted as a tightening of the financial crisis. At the end of July, for example, IKB Deutsche Industriebank AG announced their considerable financial problems. The length of the data sample is shortened because the time series are adjusted for holidays and filtered by ARMAGARCH models (see section 4.2). In detail, this means that twenty holidays in the pre-crisis and eighteen holidays in the crisis sample were excluded. The results of the ARMA-GARCH filter show different lags of the AR process element. To get time series with equal length, the first values of those time series with a lower lag are deleted. In consequence, the pre-crisis sample and the crisis sample contain N

365 and N

321 values, respectively.

For three different groups of portfolios, the null hypotheses that specific copulas adequately describe the stochastic dependence between the portfolios’ constituents are tested. The first group of portfolios consists of all possible two- to six-dimensional combinations of the six sector-specific subindices of the DJ iTraxx CDS index for Europe (“DJ iTraxx portfolios”). The second group of portfolios (“stock portfolios”) are formed by all two- to five-dimensional combinations of the five major stock indices (S&P 500, DAX 30, DJ Euro STOXX 50, FTSE 100, CAC 40). The third group of portfolios (“mixed portfolios”) consist of all possible twoto six-dimensional combinations of the following asset classes: credit (DJ iTraxx CDS master index13), stocks (MSCI World), bonds (EMU index), currencies (USD/EUR exchange rate), commodities (CRB index), and real estate (FTSE EPRA/Nareit index).

4.2 Filtering of the data The gof tests described above need independent and identically distributed (i.i.d.) data as input. However, plots of the autocorrelation function and the partial autocorrelation function of the returns and the squared returns show that in general, as expected, they exhibit autocorrelation and time-varying conditional volatilities. These visual results are also confirmed by formal statistical tests, such as the Ljung-Box test which rejects the null hypothesis of no autocorrelation and Engle’s Lagrange multiplier (LM) test which indicates that there are indeed 13

This index consists of the weighted average of the six synthetic sector-specific subindices.

14

ARCH effects. To remove autocorrelation and conditional heteroscedasticity in the univariate time series of returns, an ARMA model with GARCH errors is fitted to the raw index returns

(rn,t )t

of each subindex and asset class n {1,

, N } , respectively.14 First, an ARMA(pn,qn)-

model is fitted to the data:

rn,t

n ,t

n ,t

,

(20)

pn n ,t

qn

n

n ,i

rn ,t

i

n

n , j n ,t j

i 1

.

(21)

j 1

Afterwards, a GARCH(rn,sn)-model is fitted to the residuals ˆn,t n ,t

n ,t n ,t

(22) sn 2 n , i n ,t i

n ,0

n, j

i 1

where (

i 1,

)

n, j

0 , j 1,

148)). Finally, the

ˆn ,t

ˆn ,t ˆ n ,t

2 n ,t j

(23)

j 1

is strict white noise with mean zero and variance one, and

n ,t t

, rn ,

ˆ n,t :

, rn

2 n ,t

rn,t

rn ,t

rn

, sn , and

i 1

sn n ,i

j 1

n, j

0,

n ,i

0,

1 (see McNeil et al. (2005, pp.

gof tests for copulas are applied to

ˆ n ,t

n ,0

the filtered returns

ˆ n ,t . Table 2 shows the specifications of the ARMA(pn,qn)- and

GARCH(rn,sn)- models that are necessary to remove autocorrelation and conditional heteroskedasticity in the returns ( rn ,t )t

before and in the crisis: - insert table 2 about here -

The lags of the estimated ARMA-GARCH models vary between the different asset classes. In most of the cases, it is sufficient to employ processes with one or two lags. However, there are some exceptions for which larger lags are needed. For estimating the parameters of the GARCH-models by maximum likelihood, the innovations (

)

n ,t t

are assumed to have either

a standard normal distribution or a standardized t-distribution. The adequacy of these assumptions is proved by means of the Jarque-Bera, the Kolmogorow-Smirnov and the AndersonDarling test which are applied to ˆn ,t

14

t

.

There are also many papers on gof testing for copulas that ignore the fact that the raw return data is (presumably) not i.i.d. (see, e.g., Breymann et al. (2003), Malevergne and Sornette (2003), Mashal et al. (2003), Dobrić and Schmid (2007) or Kole et al. (2007)). The findings in Chen et al. (2004), Berg and Bakken (2007), and Grundke (2009) indicate that the distortions of the gof tests when applied to non-i.i.d. data are not too large and that the overall conclusions do not change when using filtered returns instead of raw returns. However, as there is no guarantee that this always the case, filtered returns are used in this paper.

15

Applying the Ljung-Box test and Engle’s LM test to the filtered returns

ˆn , t

t

, both up to

lag 10, the null hypothesis of no autocorrelation and no ARCH effects, respectively, cannot be rejected any more. The model fitting has been done by using the statistics toolbox of MATLAB and additionally some functions of Kevin Sheppard’s freely available GARCH toolbox.15

4.3 Results In the following, summary statistics and the correlation results for the filtered returns are shown. Furthermore, the results of the gof tests for various copula null hypotheses for DJ iTraxx portfolios, stock portfolios and mixed portfolios are presented and discussed.

4.3.1 Summary statistics Table 3 shows some summary statistics for the filtered daily returns of the asset classes considered. Most of the filtered returns clearly exhibit a non-normal distribution before as well as during the crisis as the Jarque-Bera test shows. The returns of the DJ iTraxx subindices are in general right-skewed whereas almost all other returns are left-skewed. Apart from the bond index EMU in the time period before the crisis, all returns are leptokurtic. Before the crisis, skewness and excess kurtosis of the returns of the DJ iTraxx subindices are particularly large (compared the values of the other returns), but they decrease during the crisis. - insert table 3 about here -

4.3.2 Results for portfolios of sector-specific subindices of the DJ iTraxx CDS master index for Europe The usual (Bravais-Pearson) correlation coefficients of the filtered daily subindex returns before the crisis, during the crisis, and for the whole period can be seen in table 4. Before the crisis, the correlation coefficients range from 23% to 57%. In contrast, during the crisis, all correlation coefficients increase considerably; in this time period they range from 77% to 88%. The relatively low correlations between the index return of the financial sector and all other sectors before the crisis are striking. During the crisis, these correlations sharply increase to the same (large) level that we also observe for pairs of index returns of non-financial sectors. - insert table 4 about here -

15

See http://www.kevinsheppard.com/wiki/UCSD_GARCH (date: 18.06.2009).

16

Table 7 shows that the null hypothesis of a Gaussian copula as an adequate approach for modelling the stochastic dependence between daily returns of subindices of the DJ iTraxx CDS index for Europe is mainly rejected when the test based on the Rosenblatt transformation is employed. This is in particular true for the crisis sample. Before the crisis, there are less rejections of the Gaussian copula. As expected, the number of rejections of the Gaussian copula increases with increasing dimension of the Gaussian copula (which cannot be seen from table 7). Using the Cramér-von Mises test statistic instead of the Anderson-Darling test statistic for the gof test based on the Rosenblatt transformation does not yield much differences in the number of rejections. However, employing the gof test based on the empirical copula, the number of rejections of the Gaussian copula decreases dramatically for the pre crisis sample. For the crisis sample, the number of rejections that are produced by this test also decreases (compared with the test based on the Rosenblatt transformation), but still the null hypothesis of a Gaussian copula is predominantly rejected. - insert table 7 about here -

As can be seen in table 8, the t-copula is almost never rejected for the pre crisis sample. This result holds irrespective of the chosen gof test and irrespective of the chosen test statistic. For the crisis sample, the number of rejections increases, in particular for higher-dimensional tcopulas (which again cannot be seen in table 8). These results are in line with Grundke (2009). Furthermore, the differences in the rejections rates that are produced by the different gof tests and the different test statistics get larger for the crisis sample. Contrary to the results for the null hypothesis of a Gaussian copula in the pre crisis sample, this time, the gof test based on the empirical copula yields many more rejections than the two gof tests based on the Rosenblatt transformation. As table 8 shows, the average degree of freedom of the t-copula slightly decreases during the crisis. Together with the increase in the correlations coefficients (see table 4), this result implies that, as expected, the tail dependence between the changes in the DJ iTraxx subindices increases during the crisis. Thus, the occurrence of joint extreme changes in the subindices gets more likely during the crisis. - insert table 8 about here -

The results for the Gumbel and the Clayton copula are disappointing. That is why the presentation of the rejection rates is omitted here. Irrespective of the chosen gof test and irrespective of the chosen test statistic, these two null hypotheses are predominantly rejected, both before and during the crisis. As expected, the rejection rates even increase during the crisis. Switch17

ing to the survival copulas has almost no effect on the rejection rates. Non-rejections are mainly observed for the pre crisis sample when the null hypothesis of a bivariate (survival) Gumbel copula is tested.16 There are also some non-rejections for the null hypothesis of a three-dimensional (survival) Gumbel copula in the pre crisis period. As the (survival) Gumbel and the (survival) Clayton copulas are one-parametric Archimedean copulas, the bad performance of these copulas, in particular in the higher-dimensional case, is not surprising.

For the two-parametric generalized Clayton (survival) copula, the gof tests have only been implemented for the bivariate case.17 Furthermore, for the gof test based on the Rosenblatt transformation, only the Anderson-Darling test statistic has been employed. As table 9 shows, only before the crisis, both gof tests do predominantly not reject the bivariate generalized Clayton (survival) copula. This result is in line with Grundke (2009). During the crisis, only the test based on the Rosenblatt transformation does also predominantly not reject the bivariate generalized Clayton survival copula. Again, this result shows that the two gof tests can yield significantly different evaluations with respect to the adequacy of specific copula assumptions. The non-survival copula is mainly rejected by both gof tests during the crisis. - insert table 9 about here -

4.3.3 Results for portfolios of stock indices The correlation coefficients of the filtered daily returns of the stock indices can be seen in table 5. Striking is that there is no significant variation in the correlation coefficients before and during the crisis, but all values remain more or less the same. - insert table 5 about here -

For the stock portfolios, the null hypothesis of a Gaussian copula is not rejected for almost all combinations in the time period before the crisis (see table 7). Even the five-dimensional Gaussian copula is not rejected at all considered significance levels. These results, which are in contrast to those of Chen et al. (2004), are produced by all gof tests and all employed test statistics. However, as table 8 shows, before the crisis, the null hypothesis of a t-copula is also not rejected for all possible combinations. As the average degree of freedom of the t-copula is 14.4, the t-copula and the Gaussian copula significantly differ. As none of the gof tests can 16

17

When the term „survival“ is inserted before the name of a copula, this means that the following statement is true for the original copula as well as for the respective survival copula. Some tests have also been done for the three-dimensional case. These tests indicate that testing for higher dimensional generalized Clayton copulas tends to result in an increase of the rejection rate. This seems to be true more or less for all considered portfolios.

18

differentiate between these two copulas, in principle, a considerable amount of model risk would remain for the stock portfolio. For the crisis sample, both copulas are predominantly rejected when the gof test based on the Rosenblatt transformation is used. However, employing the gof test based on the empirical copula yields the opposite result. With this test, we only find one rejection of a bivariate Gaussian copula (for the pair Euro Stoxx 50 and CAC 40) at a significance level of 10% and one rejection of the t-copula (for the same pair) at a significance level of 5%. Contrary to the DJ iTraxx portfolio, the average degree of freedom of the t-copula even slightly increases during the crisis. The non-rejection of the symmetric Gaussian- and t-copula for the portfolio of stock indices is contrary to the finding that there is an asymmetric tail dependence in stock markets, i.e. that in bear markets losses are more dependent than gains in bull markets (see, e.g., Longin and Solnik (2001), Ané and Kharoubi (2003)).

The Gumbel and the Clayton (survival) copulas are again predominantly rejected, both before and during the crisis and irrespective of the chosen gof test and irrespective of the chosen test statistic, whereby the Gumbel copula performs slightly better than the Clayton copula.

The bivariate generalized Clayton (survival) copula is predominantly not rejected in both time periods (see table 9). During the crisis, more rejections can be observed, but the increase in the rejection rate is not as large as for the DJ iTraxx portfolios.

4.3.4 Results for mixed portfolios The correlation coefficients of the filtered daily returns of the asset classes out of which the mixed portfolios consist can be seen in table 6. During the crisis, many (but not all) correlation coefficients considerably increase. In almost all cases, the sign of the correlation coefficients is the same in both time periods. Only for the two pairs MSCI World - USD/EUR and EMU - USD/EUR, the sign changes. - insert table 6 about here -

For mixed portfolios, the null hypotheses of a Gaussian and a t-copula are predominantly not rejected for the time period before the crisis. However, there are some more rejections of the respective copula than for the stock portfolios in this time period. For the pre crisis sample, the R-package in which the gof test based on the empirical copula is implemented frequently produced error messages for the null hypothesis of a t-copula. That is why the results are 19

omitted in table 8. The average degree of freedom of the t-copula is with 39.7 very large. During the crisis, this value sharply decreases to 11.92. Contrary to the results for the stock portfolios, both copula hypotheses, Gaussian and t-copula, are also not rejected in the crisis period. When the gof test based on the Rosenblatt transformation is employed, there is only a slight increase in the number of rejections. When the gof test based on the empirical copula is used, we do not observe any rejection of the Gaussian and the t-copula at all. At least with respect to the results for the t-copula, these findings are in line with those of Kole et al. (2007). However, they do not report a good performance of the Gaussian copula for their mixed portfolio consisting of the three asset classes stocks, bonds and real estate.

For the mixed portfolios, even the (survival) Gumbel copula, which poorly performed for the DJ iTraxx and stock portfolios, is not rejected for half of the possible combinations in the time period before the crisis. During the crisis, the (survival) Gumbel copula is again predominantly rejected. However, for testing the null hypothesis of a (survival) Gumbel copula only the gof test based on the Rosenblatt transformation could be used because the R-package in which the gof test based on the empirical copula is implemented again frequently produced error messages. In spite of fifty percent of non-rejections of the null hypothesis of a (survival) Gumbel copula for the pre crisis sample, this copula choice does not seem to be adequate for describing the stochastic dependencies between the returns of the asset classes in the mixed portfolios because the estimated values for the parameter

of the (survival) Gumbel copula

are in almost all cases at the lower boundary 1 or only slightly above it. This would correspond to the independence copula. The same can be observed for the (survival) Clayton copula, for which the parameter tends to 0, and for the generalized Clayton (survival) copula, for which both parameters tend to their lower boundaries 0 and 1, respectively.

4.4 Robustness check As the results in section 4.3 have shown, there are situations in which several very different copulas are not rejected by the data and the employed gof tests. For example, for the portfolios of stock indices before the crisis and for the mixed portfolios during the crisis, neither the null hypothesis of a Gaussian copula nor the null hypothesis of a t-copula could be clearly rejected (see tables 7, 8). That’s why in this section, we want to test whether this result is due to an insufficient length of the time series. Thus, as a robustness check, we extend both periods (pre-crisis and crisis period, respectively) and repeat the gof tests. Now, the pre-crisis subsample covers the period February 2, 2005 to June 29, 2007 and the crisis subsample covers 20

the period July 2, 2007 to November 30, 2009. First, the gof tests are repeated for all two- to five-dimensional combinations of the five major stock indices (S&P 500, DAX 30, DJ Euro STOXX 50, FTSE 100, CAC 40) before the crisis. Second, the gof tests are done again for the mixed portfolios with up to six asset classes (DJ iTraxx CDS index for Europe18, MSCI World, EMU index, USD/EUR exchange rate, CRB index, FTSE EPRA/Nareit index) during the crisis.

As can be seen in table 10, the results are quite different: For the stock portfolios before the crisis, a rejection of the Gaussian- or the t-copula is still not possible. However, for the mixed portfolios during the crisis, the extension of the time series has the consequence that (in contrast to the results in section 4.3.3; see table 7) the Gaussian copula is predominantly rejected, whereas (as before; see table 8) the t-copula is still predominantly not rejected. - insert table 10 about here -

5. Application: Dependence structure and risk measures In this section, we want to test within a risk measurement application whether the copula uncertainty discussed in the previous section also causes considerable uncertainty with respect to portfolio risk measures, such as value at risk and expected shortfall. For this purpose, on one hand, we construct an equally-weighted portfolio of the five-stock indices (S&P 500, DAX 30, DJ Euro STOXX 50, FTSE 100, CAC 40) before the crisis and, on the other hand, an equally-weighted mixed portfolio with six assets (DJ iTraxx CDS index for Europe, MSCI World, EMU index, USD/EUR exchange rate, CRB index, FTSE EPRA/Nareit index) during the crisis. For both portfolios, neither the null hypothesis of a multivariate Gaussian copula nor the null hypothesis of a multivariate t-copula can be clearly rejected (which can not be seen in the tables of the previous section).19 Using the specification and the estimated parameters of the ARMA-GARCH models and the estimated parameters of the Gaussian and the tcopula in the respective time periods, we simulate the portfolio returns over a 10-days risk horizon. The number of simulations is 100,000. From these simulated portfolio returns, the value at risk and expected shortfall over a risk horizon of 10 days and for confidence levels of 95%, 18

19

In contrast to section 4.3, this index does not consist of the weighted average of the six synthetic sectorspecific subindices, but now, it corresponds to the quoted iTraxx Europe 5Y (series 5 to 12). For the stocks portfolio before the crisis, the five-dimensional Gaussian and t-copula could not be rejected at a 10% significance level, irrespective of the chosen gof test and test statistic. For the mixed portfolio during the crisis, the six-dimensional t-copula could also not be rejected at a 10% significance level, irrespective of the chosen gof test and test statistic. For the six-dimensional Gaussian copula, the results were ambiguous: Employing the gof test based on the Rosenblatt transformation, the null hypothesis could be rejected at a 5% significance level; however, using the gof test based on the empirical copula, a rejection at a 10% significance level was not possible (p-value is 93.8%).

21

99% and 99.9% are computed. The value at risk numbers are defined as the difference between the 5%-, 1%-, and 0.1%-percentile, respectively, and the mean of the empirical return distribution. The expected shortfall is given as the conditional mean of those simulated portfolio returns that are smaller than the 5%-, 1%-, and 0.1%-percentile, respectively.

In detail, we proceed as follows: First, the daily returns of those indices that are part of the considered portfolios are predicted by the estimated ARMA-GARCH models. To minimise the influence of the starting values, the prediction is done for five hundred time steps and only the last ten daily returns are used for computing the 10-days index return. The starting values are chosen as follows: For

2 n ,t j

( j 1) , the variance of the unfiltered return series of each

index n is taken, the index-specific error terms to zero, and for the return rn,t

j

n ,t j

( j 1) of the ARMA part are set equal

( j 1) , the mean of the unfiltered return series of each index

n is chosen. The index-specific error terms

n ,t

of the GARCH part are simulated simultane-

ously according to the copula specification (Gaussian- and t-copula) and the estimated copula parameters and according to the specified error term distribution of the ARMA-GARCH model (normal and standardized t-distribution, respectively). The 10-days index returns are computed by just adding up the last ten elements rn,491 ,

, rn,500 of the five hundred day-to-day

index return forecasts. Finally, the 10-days portfolio return is calculated according to:

rP

ln

1 N

N

10

exp n 1

rn,490

t

(24)

t 1

where N {5, 6} is the number of indices in the considered portfolio. Repeating this procedure 100,000 times, we get an empirical distribution function for the 10-days portfolio return, from which the value at risk and expected shortfall can be computed. As can be seen in table 11, in our specific application, the risk measures are not very sensitive with respect to the chosen copula. In general, there is only a slight increase in the risk measures when switching from the Gaussian to the t-copula (although the degree of freedom is 18.3 (stocks) and 12.3 (mixed), respectively, and, hence, not particularly large). Thus, at least in this example, model risk with respect to the dependence structure is limited and the copula uncertainty does not cause considerable uncertainty with respect to the portfolio risk measures. - insert table 11 about here -

22

6. Conclusions For a multitude of portfolios consisting of different asset classes, it is tested whether the stochastic dependence between the portfolios’ constituents can be adequately described by multivariate versions of some standard parametric copula functions. The knowledge of the multivariate stochastic dependence between the returns of asset classes is of importance for many finance applications, such as, e.g., asset allocation or risk management. Furthermore, it is tested whether the stochastic dependence between the returns of different asset classes has changed during the recent financial crisis.

Several main results can be stated: First, whether a specific copula assumption can be rejected or not, crucially depends on the asset class and the time period considered. For example, the multivariate t-copula is predominantly rejected in the crisis sample for the higher-dimensional DJ iTraxx portfolios, but it is predominantly not rejected in the pre crisis period and it is also predominantly not rejected for stock portfolios (in the pre crisis period) and for mixed portfolios (in both periods). Second, different gof tests for copulas can yield very different results and these differences can be different for different asset classes. For example, employing the gof test based on the empirical copula, the number of rejections of the Gaussian copula in the pre crisis sample decreases dramatically for DJ iTraxx portfolios (compared to the rejection rates that the gof test based on the Rosenblatt transformation yields). However, for the null hypothesis of a t-copula for DJ iTraxx portfolios in the crisis period, the number of rejections is highest when the gof test based on the empirical copula is used. In consequence, these findings raise some doubt with respect to the validity of the results based on gof tests for copulas. Furthermore, they might be one explanation for different findings in the literature with respect to the adequacy of specific copula assumptions for specific asset classes. Third, even when using different gof tests for copulas, it is not always possible to differentiate between various copula assumptions. For example, for stock portfolios in the time period before the crisis, neither the null hypothesis of a Gaussian copula nor the null hypothesis of t-copula with a relatively low average degree of freedom can be rejected. With respect to the initial question raised in the introduction, whether risk dependencies changed during the recent financial crisis, it can be stated that there is a tendency for increased rejection rates during the crisis. This is true across different portfolios, different copulas of the null hypothesis and different test statistics. Thus, risk dependencies are seemingly more complex during the crisis and tend to be not adequately described by standard parametric copulas. As expected, correlations and the

23

degree of freedom of the t-copula increase on average (beside for the stocks portfolios). Both results imply a larger degree of tail dependence during the crisis than before the crisis.

Finally, we think that it would be promising in future research to consider also some nonstandard copula concepts (such as the grouped t-copula (see Demarta and McNeil (2005)) or pair-copula constructions (see, e.g., Joe (1996), Aas et al. (2009)) for modelling the multivariate stochastic dependence between returns of different asset classes, in particular in turbulent times on financial markets.

24

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25

Dias, A., P. Embrechts. 2009. Testing for structural changes in exchange rates dependence beyond linear correlation. Forthcoming: European Journal of Finance. Dobrić, J., F. Schmid. 2005. Testing goodness of fit for parametric families of copulas - application to financial data. Communications in Statistics – Simulation and Computation 34 10531068. Dobrić, J., F. Schmid. 2007. A goodness of fit test for copulas based on Rosenblatt’s transformation. Computational Statistics & Data Analysis 51 (9) 4633-4642. Engle, R.F. 2002. Dynamic conditional correlation – a simple class of multivariate GARCH models. Journal of Business and Economic Statistics 20 339-350. Fermanian, J.-D. 2005. Goodness of fit tests for copulas. Journal of Multivariate Analysis 95 (1) 119-152. Genest, C., J.-F. Quessy, B. Rémillard. 2006. Goodness-of-fit procedures for copula models based on the probability integral transformation. Scandinavian Journal of Statistics 33 337366. Genest, C., B. Rémillard. 2008. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 44 (6) 1096-1127. Genest, C., B. Rémillard, D. Beaudoin. 2009. Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Ecnomics 44 199-214. Grundke, P. (2009): Changing default risk dependencies during the subprime crisis: DJ iTraxx subindices and goodness-of-fit-testing for copulas, forthcoming in: Review of Managerial Science. Joe, H. 1996. Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In L. Rüschendorf, B. Schweizer and M.D. Taylor (Eds.), Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes-Monograph Series. Hayward, CA, pp. 120-141. Joe, H. 1997. Multivariate models and dependence concepts. Chapman & Hall, London. Jondeau, E., M. Rockinger. 2006. The copula-GARCH model of conditional dependencies: An international stock market application. Journal of International Money and Finance 25 827-853. Junker, M., A. May. 2005. Measurement of aggregate risk with copulas. Econometrics Journal 8 428-454. Junker, M., A. Szimayer, N. Wagner. 2006. Nonlinear term structure dependence: Copula functions, empirics, and risk implications. Journal of Banking & Finance 30 1171-1199. Kimberling, C.H. 1974. A probabilistic interpretation of complete monotonicity. Aequationes Mathematicae 10 152-164.

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Kole, E., K. Koedijk, M. Verbeek. 2007. Selecting copulas for risk management. Journal of Banking & Finance 31 2405–2423. Longin, F., B. Solnik. 2001. Extreme Correlation of International Equity Markets. Journal of Finance 56 (2) 649-676. Malevergne, Y., D. Sornette. 2003. Testing the Gaussian copula hypothesis for financial assets dependences. Quantitative Finance 3 231-250. Mashal, R., M. Naldi, A. Zeevi. 2003. On the dependence of equity and asset returns. Risk, October 83-87. McNeil, A.J., R. Frey, P. Embrechts. 2005. Quantitative Risk Management. Princeton University Press, Princeton and Oxford. Nelson, R.B. 2006. An introduction to copulas. 2nd edition, Springer Series in Statistics, Springer, New York. Patton, A. 2006. Modelling asymmetric exchange rate dependence. International Economic Review 47 (2) 527-556. Rodriguez, J. 2007. Measuring financial contagion: A Copula Approach. Journal of Empirical Finance 14 401-423. Savu, C., M. Trede. 2008. Goodness-of-fit tests for parametric families of Archimedean copulas. Quantitative Finance 8(2) 109-116. Schönbucher, P.J. 2003. Credit derivatives pricing models. Wiley, Chichester. Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. Van den Goorbergh, R. 2004. A copula-based autoregressive conditional dependence model of international stock markets. DNB Working Paper No. 22.

27

Tables

Table 1: Tail dependence coefficients This table shows the lower ( l) and upper ( u) tail dependence coefficients of the copulas that are tested in this paper. For the t-copula, T ; 1 denotes the cdf of a univariate tdistribution with

1 degrees of freedom and

Copula

is the correlation parameter.

l

u

Gaussian

0

0

Student’s t

2T

Gumbel

0

Clayton

2

1

Generalized Clayton

2

1(

(

1)(

1) (1

);

1

2T

(

1)(

1) (1

);

1

2 21

0 )

2 21

28

Table 2: Specification of the ARMA(pn,qn)-GARCH(rn,sn)-models This table shows the lags of the ARMA-GARCH-models by which the single time series of daily returns are filtered. Additionally, the distributional assumptions for the residuals can be seen. Pre-Crisis Portfolios of subindices of the DJ iTraxx

Energy TMT Autos Industrials Financials Consumers

ARMA ARMA(2,2) ARMA(1,0) ARMA(1,0) ARMA(1,1) ARMA(1,1) ARMA(1,1)

GARCH GARCH(1,0) GARCH(0,0) GARCH(1,1) GARCH(1,1) GARCH(7,0) GARCH(0,0)

Distribution Student’s t Student’s t Student’s t Student’s t Student’s t Student’s t

Portfolios of stock indices

S&P DAX STOXX FTSE CAC

ARMA(1,0) ARMA(4,0) ARMA(4,0) ARMA(2,1) ARMA(4,0)

GARCH(1,1) GARCH(1,1) GARCH(1,1) GARCH(1,1) GARCH(1,1)

Student’s t Student’s t Student’s t Student’s t Student’s t

Mixed portfolios

DJ iTraxx MSCI EMU USD/EUR CRB FTSE EPRA

ARMA(1,1) ARMA(4,0) ARMA(1,0) ARMA(2,0) ARMA(1,0) ARMA(1,0)

GARCH(1,0) GARCH(1,1) GARCH(4,0) GARCH(1,1) GARCH(3,2) GARCH(4,0)

Student’s t Student’s t Normal Student’s t Normal Student’s t

Crisis Portfolios of subindices of the DJ iTraxx

Energy TMT Autos Industrials Financials Consumers

ARMA ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(1,0)

GARCH GARCH(1,1) GARCH(1,1) GARCH(2,1) GARCH(1,1) GARCH(1,1) GARCH(1,1)

Distribution Student’s t Student’s t Student’s t Student’s t Student’s t Student’s t

Portfolios of stock indices

S&P DAX STOXX FTSE CAC

ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(1,0)

GARCH(2,1) GARCH(1,1) GARCH(1,1) GARCH(3,1) GARCH(4,1)

Student’s t Student’s t Student’s t Normal Normal

Mixed portfolios

DJ iTraxx MSCI EMU USD/EUR CRB FTSE EPRA

ARMA(1,0) ARMA(1,0) ARMA(1,0) ARMA(2,0) ARMA(1,0) ARMA(1,0)

GARCH(1,1) GARCH(2,1) GARCH(3,2) GARCH(1,1) GARCH(1,1) GARCH(1,1)

Student’s t Normal Normal Student’s t Student’s t Student’s t

29

Table 3: Summary statistics of the time series of returns This table shows the mean, standard deviation, skewness and excess kurtosis of the filtered daily returns of the asset classes before the crisis, during the crisis and for the whole period. Also, the p-value of the Jarque-Bera test is added to the table. Pre-Crisis Energy mean std. deviation. skewness ex. kurtosis J-B test

TMT

Autos

-0.0063

0.0026

-0.0068

Industrials 0.0182

Financials -0.0122

Consumers 0.0004

DJ iTraxx 0.0123

0.9403

1.0271

0.9263

0.9836

0.9693

0.9127

0.7583

2.0161

0.5509

1.6680

0.4038

3.5984

15.6810

2.2016

9.8744

0.10%

0.10%

0.10%

S&P

DAX

STOXX

FTSE

CAC

MSCI

EMU

-0.0098

-0.0034

-0.0056

0.0003

0.0004

-0.0021

-0.0105

USD / EUR 0.0023

0.9655

0.9804

1.0007

1.0010

1.0040

1.0026

1.0141

1.0008

0.9618

0.8085

-0.8720

-0.5801

-0.4285

-0.4562

-0.4137

-0.8329

2.9119

6.3055

2.0976

4.4881

1.0878

0.7749

0.7166

0.7431

0.10%

0.10%

0.10%

0.10%

0.10%

0.10%

0.17%

0.16%

0.21%

Financials 0.0145

Consumers 0.0111

DJ iTraxx 0.0224

CRB -0.0043

FTSE EPRA /Nareit -0.0078

1.0087

0.9958

1.0054

-0.0227

0.5810

-0.2023

-0.2639

4.0770

-0.2864

1.8129

0.1561

0.6143

0.10%

50.00%

0.10%

20.71%

1.34%

Crisis Energy mean std. deviation. skewness ex. kurtosis J-B test

0.0048

0.0224

0.0183

Industrials 0.0257

1.0033

1.0041

1.0000

0.9971

1.0044

1.0010

0.2961

0.2073

0.0704

0.1313

0.0682

1.1901

1.2858

0.7020

1.6511

0.11%

0.10%

3.39%

Whole Period Energy mean std. deviation. skewness ex. kurtosis J-B test

TMT

TMT

Autos

Autos

0.0196

0.0092

0.0017

-0.0024

-0.0104

0.0112

0.0123

USD / EUR 0.0257

1.0038

1.0018

1.0155

1.0059

1.0031

1.0041

1.0028

0.9941

-0.1416

0.1449

-0.2909

-0.7843

-0.2957

-0.1599

-0.1289

-0.1972

2.3996

0.6789

1.1376

0.3043

4.0538

1.4649

0.2257

0.1580

0.10%

0.10%

2.99%

0.25%

5.04%

0.10%

0.10%

32.27%

50.00%

Financials 0.0003

Consumers 0.0054

DJ iTraxx 0.0171

-0.0011

0.0119

0.0049

Industrials 0.0217

0.9696

1.0157

0.9609

0.9892

0.9852

0.9544

0.5215

1.1999

0.3021

0.9336

0.2394

2.3339

9.2392

1.3999

5.9058

0.10%

0.10%

0.10%

0.10%

S&P

S&P

DAX

DAX

STOXX

STOXX

FTSE

FTSE

CAC

CAC

MSCI

0.0362

FTSE EPRA /Nareit 0.0044

0.9993

0.9997

1.0006

0.0436

-0.4215

-0.4784

0.2218

0.0679

0.3293

1.1099

0.8329

0.5356

30.58%

42.86%

0.10%

0.15%

3.87%

MSCI

EMU

EMU

0.0039

0.0025

-0.0022

-0.0010

-0.0046

0.0041

0.0001

USD / EUR 0.0133

0.9829

0.9899

1.0070

1.0026

1.0028

1.0026

1.0081

0.9970

0.3720

0.4796

-0.5897

-0.6776

-0.3658

-0.3177

-0.2802

-0.5410

2.6518

3.1967

1.6119

2.4585

2.5166

1.1021

0.4866

0.46549

0.10%

0.10%

0.10%

0.10%

0.10%

0.10%

0.15%

0.29%

CRB

CRB 0.0147

FTSE EPRA /Nareit -0.0021

1.0036

0.9971

1.0025

0.0078

0.1187

-0.3318

-0.0385

2.2552

-0.0008

1.4673

0.4637

0.5838

0.10%

50.00%

0.10%

0.14%

1.17%

30

Table 4: Correlation coefficients between the credit index returns This table shows the Bravais-Pearson correlation coefficients between the daily filtered returns of the DJ iTraxx CDS subindices before the crisis, during the crisis and for the whole period. Pre-Crisis Energy Energy TMT Autos Industrials Financials Consumers

1.0000

Bravais-Pearson correlation coefficient TMT Autos Industrials Financials Consumers 0.4331 1.0000

0.4354 0.5088 1.0000

0.4630 0.5754 0.5390 1.0000

0.2318 0.2630 0.2970 0.2461 1.0000

0.4602 0.5605 0.5560 0.5580 0.3490 1.0000

Crisis Energy Energy TMT Autos Industrials Financials Consumers

1.0000

TMT 0.8526 1.0000

Autos 0.7897 0.8609 1.0000

Industrials Financials Consumers 0.8349 0.8771 0.8558 1.0000

0.8363 0.8417 0.7669 0.8062 1.0000

0.8089 0.8629 0.8281 0.8449 0.7764 1.0000

Whole Period Energy Energy TMT Autos Industrials Financials Consumers

1.0000

TMT 0.6332 1.0000

Autos 0.6137 0.6774 1.0000

Industrials Financials Consumers 0.6441 0.7158 0.6940 1.0000

0.5297 0.5354 0.5300 0.5150 1.0000

0.6370 0.7061 0.6947 0.6993 0.5624 1.0000

31

Table 5: Correlation coefficients between the stock index returns This table shows the Bravais-Pearson correlation coefficients between the daily filtered returns of the five stock indices considered before the crisis, during the crisis and for the whole period. Pre-Crisis S&P S&P DAX STOXX FTSE CAC

Bravais-Pearson correlation coefficient DAX STOXX FTSE CAC

1.0000

0.5862 1.0000

0.5813 0.9726 1.0000

0.5279 0.8513 0.8751 1.0000

0.5652 0.9398 0.9711 0.8656 1.0000

Crisis S&P S&P DAX STOXX FTSE CAC

1.0000

DAX 0.5842 1.0000

STOXX 0.5993 0.9614 1.0000

FTSE 0.5811 0.8679 0.9035 1.0000

CAC 0.5974 0.9134 0.9636 0.9156 1.0000

Whole Period S&P S&P DAX STOXX FTSE CAC

1.0000

DAX 0.5853 1.0000

STOXX 0.5898 0.9673 1.0000

FTSE 0.5530 0.8591 0.8884 1.0000

CAC 0.5802 0.9273 0.9675 0.8890 1.0000

32

Table 6: Correlation coefficients between the asset returns of the mixed portfolios This table shows the Bravais-Pearson correlation coefficients between the daily filtered returns of the indices that represent specific asset classes in the mixed portfolios before the crisis, during the crisis and for the whole period. Pre-Crisis DJ iTraxx DJ iTraxx MSCI EMU USD/EUR CRB FTSE EPRA/Nareit

1.0000

Bravais-Pearson correlation coefficient MSCI EMU USD/EUR CRB FTSE EPRA/ Nareit -0.2654 1.0000

0.0767 -0.0636 1.0000

-0.0719 -0.0804 0.0308 1.0000

-0.1334 0.1897 -0.0259 0.1413 1.0000

-0.3474 0.6768 -0.0237 0.1358 0.1087 1.0000

Crisis DJ iTraxx DJ iTraxx MSCI EMU USD/EUR CRB FTSE EPRA/Nareit

1.0000

MSCI

-0.5850 1.0000

EMU

0.4633 -0.5433 1.0000

USD/EUR CRB

-0.0094 0.0228 -0.0778 1.0000

-0.1701 0.1747 -0.2086 0.2941 1.0000

FTSE EPRA/ Nareit -0.5434 0.8143 -0.4076 0.0962 0.0995 1.0000

Whole Period DJ iTraxx DJ iTraxx MSCI EMU USD/EUR CRB FTSE EPRA/Nareit

1.0000

MSCI

-0.4170 1.0000

EMU

0.2606 -0.2858 1.0000

USD/EUR CRB

-0.0421 -0.0326 -0.0195 1.0000

-0.1508 0.1828 -0.1110 0.2128 1.0000

FTSE EPRA/ Nareit -0.4406 0.7406 -0.2021 0.1175 0.1045 1.0000

33

Table 7: Gof tests for the null hypothesis of a Gaussian copula This table shows the number of rejections of the null hypothesis of a Gaussian copula at significance levels of 10%, 5% and 1% for various gof tests and for various test statistics. All possible two- to six-dimensional combinations of the six subindices of the DJ iTraxx CDS index for Europe (number of combinations: 57), the five stock indices (number of combinations: 26) and the six asset classes in the mixed portfolios (number of combinations: 57), respectively, have been tested. ADRB: Gof test based on the Rosenblatt transformation with usage of the Anderson-Darling test statistic; CvMRB: Gof test based on the Rosenblatt transformation with usage of the Cramér-von Mises test statistic; CvMempCop: Gof test based on the empirical copula with usage of the Cramér-von Mises test statistic. ADRB DJ iTraxx Stocks

10% 5% 1%

37/57 30/57 20/57

Crisis 10% 5% 1%

53/57 53/57 44/57

Pre-Crisis

Mixed

CvMRB DJ iTraxx

Mixed

CvMempCop DJ iTraxx Stocks

Stocks

Mixed

1/26 0/26 0/26

5/57 3/57 1/57

35/57 28/57 17/57

2/26 0/26 0/26

3/57 3/57 1/57

11/57 2/57 0/57

0/26 0/26 0/26

1/57 0/57 0/57

23/26 20/26 14/26

9/57 1/57 0/57

53/57 51/57 43/57

20/26 19/26 13/26

6/57 1/57 0/57

42/57 38/57 20/57

1/26 0/26 0/26

0/57 0/57 0/57

34

Table 8: Gof tests for the null hypothesis of a t-copula This table shows the number of rejections of the null hypothesis of a t-copula at significance levels of 10%, 5% and 1% for various gof tests and for various test statistics. All possible two- to six-dimensional combinations of the six subindices of the DJ iTraxx CDS index for Europe (number of combinations: 57), the five stock indices (number of combinations: 26) and the six asset classes in the mixed portfolios (number of combinations: 57), respectively, have been tested. For the pre crisis sample of returns of the mixed portfolios’ constituents, the R-package in which the gof test based on the empirical copula is implemented frequently produced error messages. That is why the results are omitted here. ADRB: Gof test based on the Rosenblatt transformation with usage of the Anderson-Darling test statistic; CvMRB: Gof test based on the Rosenblatt transformation with usage of the Cramér-von Mises test statistic; CvMempCop: Gof test based on the empirical copula with usage of the Cramér-von Mises test statistic; dofavg: average of the ML-estimates of the degree of freedom of the t-copulas. ADRB DJ iTraxx Stocks

Mixed

CvMRB DJ iTraxx

Stocks

10% 5% 1%

3/57 3/57 0/57

0/26 0/26 0/26

5/57 3/57 1/57

5/57 2/57 0/57

dofavg

8.98

14.4

39.70

Crisis 10% 5% 1%

31/57 3/57 1/57

19/26 17/26 10/26

7/57 2/57 2/57

26/57 16/57 3/57

dofavg

7.38

16.77

11.92

Pre-Crisis

Mixed

CvMempCop DJ iTraxx Stocks

Mixed

0/26 0/26 0/26

4/57 1/57 1/57

2/57 0/57 0/57

0/26 0/26 0/26

-

17/26 15/26 8/26

5/57 2/57 2/57

41/57 36/57 17/57

1/26 1/26 0/26

0/57 0/57 0/57

35

Table 9: Gof tests for the null hypothesis of a bivariate generalized Clayton (survival) copula This table shows the number of rejections of the null hypothesis of a bivariate generalized Clayton copula at significance levels of 10%, 5% and 1% for various gof tests. In brackets, the number of rejections for the bivariate generalized Clayton survival copula are shown. All possible pairwise combinations of the six subindices of the DJ iTraxx CDS index for Europe (number of combinations: 15), the five stock indices (number of combinations: 10) and the six asset classes in the mixed portfolios (number of combinations: 15), respectively, have been tested. ADRB: Gof test based on the Rosenblatt transformation with usage of the Anderson-Darling test statistic; CvMempCop: Gof test based on the empirical copula with usage of the Cramér-von Mises test statistic. Pre-Crisis

10% 5% 1%

Crisis 10% 5% 1%

ADRB DJ iTraxx Stocks

Mixed

CvMempCop DJ iTraxx Stocks

Mixed

4/15 (3/15) 2/15 (1/15) 0/15 (0/15)

2/10 (2/10) 2/10 (2/10) 1/10 (2/10)

4/15 (4/15) 3/15 (2/15) 2/15 (2/15)

1/15 (0/15) 1/15 (0/15) 0/15 (0/15)

4/10 (3/10) 2/10 (3/10) 2/10 (2/10)

7/15 (9/15) 6/15 (7/15) 5/15 (5/15)

10/15 (4/15) 10/15 (3/15) 4/15 (1/15)

7/10 (4/10) 7/10 (4/10) 5/10 (2/10)

4/15 (4/15) 4/15 (4/15) 3/15 (3/15)

14/15 (13/15) 13/15 (12/15) 9/15 (8/15)

5/10 (6/10) 5/10 (4/10) 3/10 (2/10)

8/15 (9/15) 8/15 (9/15) 6/15 (6/15)

36

Table 10: Robustness check with extended time series This table shows the number of rejections of the null hypothesis of a Gaussian copula and a t-copula, respectively, for the extended time series. Now, the pre-crisis subsample covers the period February 2, 2005 to June 29, 2007 and the crisis subsample covers the period July 2, 2007 to November 30, 2009. First, the gof tests are repeated for all two- to five-dimensional combinations of the five major stock indices (S&P 500, DAX 30, DJ Euro STOXX 50, FTSE 100, CAC 40) before the crisis. Second, the gof tests are done again for the mixed portfolios with up to six asset classes (DJ iTraxx CDS index for Europe, MSCI World, EMU index, USD/EUR exchange rate, CRB index, FTSE EPRA/Nareit index) during the crisis. ADRB: Gof test based on the Rosenblatt transformation with usage of the Anderson-Darling test statistic; CvMempCop: Gof test based on the empirical copula with usage of the Cramér-von Mises test statistic; dofavg: average of the ML-estimates of the degree of freedom of the t-copulas. Stocks (Pre-Crisis) GaussianADRB copula 10% 5% 1%

3/26 1/26 0/26

CvMRB

CvMempCop

3/26 1/26 0/26

1/26 0/26 0/26

t-copula

dof avg Mixed (Crisis) GaussianADRB copula 10% 5% 1%

34/57 29/57 17/57

CvMRB

CvMempCop

31/57 24/57 19/57

5/57 5/57 0/57

CvMRB

CvMempCop

3/26 2/26 1/26

2/26 1/26 1/26

0/26 0/26 0/26

ADRB

CvMRB

CvMempCop

4/57 2/57 0/57

3/57 2/57 0/57

6/57 2/57 0/57

11.8

t-copula

dof avg

ADRB

12.9

37

Table 11: Dependence structure and risk measures This table shows the value at risk and expected shortfall for the 10-days return of an equally-weighted portfolio of the five-stock indices before the crisis and an equally-weighted mixed portfolio with six asset classes during the crisis, respectively. The daily returns of those indices that are part of the considered portfolios are predicted by the estimated ARMA-GARCH models. To minimise the influence of the starting values, the prediction is done for five hundred time steps and only the last ten daily returns are used for computing the 10-days index returns. The index-specific error terms n ,t of the GARCH part are simulated simultaneously according to the copula specification (Gaussian- and t-copula) and the estimated copula parameters and according to the specified error term distribution of the ARMA-GARCH model (normal and standardized t-distribution, respectively). The degree of freedom of the t-copula is 18.3 (stocks) and 12.3 (mixed), respectively. Doing 100,000 simulations yields an empirical distribution function for the 10-days portfolio return, from which the value at risk and expected shortfall can be computed. The value at risk numbers are defined as the difference between the 5%-, 1%-, and 0.1%-percentile, respectively, and the mean of the empirical return distribution. The expected shortfall is given as the conditional mean of those simulated portfolio returns that are smaller than the 5%-, 1%-, and 0.1%-percentile, respectively.

Value at Risk 5% Value at Risk 1% Value at Risk 0.1% Expected Shortfall 5% Expected Shortfall 1% Expected Shortfall 0.1%

Stocks (Pre-Crisis) Mixed (Crisis) Gaussian-Copula -3.53% -5.12% -5.41% -7.73% -8.52% -12.09% -4.30% -6.13% -9.24%

-6.29% -8.85% -13.05% t-Copula

Value at Risk 5% Value at Risk 1% Value at Risk 0.1%

-3.55% -5.52% -8.57%

-5.09% -7.81% -12.19%

Expected Shortfall 5% Expected Shortfall 1% Expected Shortfall 0.1%

-4.41% -6.29% -9.63%

-6.30% -8.91% -13.46%

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