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DYNAMIC ECONOMETRIC MODELS 
Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 

Ryszard Doman 
Adam Mickiewicz University in Poznań 

Modeling the Dependence Structure of the WIG20  
Portfolio Using a Pair-copula Construction†  

A b s t r a c t. Elliptical distributions commonly applied to modeling the returns of stocks in high-
dimensional portfolio are not capable of adequate describing the dependence between the compo-
nents when their statistical properties are very diverse. The MGARCH and standard dynamic 
copula models are often of little usefulness in such cases. In this paper, we apply a methodology 
called the pair-copula decomposition to model the joint conditional distribution of the returns on 
stocks constituting the WIG20 index, and show some advantage of this construction over the 
approach using the t Student DCC model. 

K e y w o r d s: dependence, portfolio, copula, pair-copula construction.  

1. Introduction  
 Elliptical distributions commonly applied to modeling the returns of stocks 
in high-dimensional portfolio are not capable of adequate describing the depen-
dence between the components of the return vector when their statistical proper-
ties are very diverse. Except of a huge number of parameters necessary to esti-
mate, this is why multidimensional GARCH models are often of little practical 
usefulness in such cases. Also the standard dynamic copula models which are 
successfully applied in the bivariate case are not so useful  for large dimension 
because of a small number of classical multivariate copulas that are flexible 
enough to fit the data. In this context, a new approach to modeling multivariate 
data which exhibit complex patterns, proposed recently by Aas et al. (2009), has 
triggered off much interest. It was inspired by the work of Joe (1996), Bedford 
and Cooke (2002), and Kurowicka and Cooke (2006).  The main idea of this 
methodology is to model dependence by decomposing higher-dimensional co-
pula densities into bivariate ones (pair-copula densities) arising from the condi-

                                                 
† This work was financed from the Polish science budget resources in the years 2007-2010 as 

the research project NN 111 1256 33. 



Ryszard Doman 32

tional and unconditional distribution functions of the modeled variables. A fur-
ther simplification of the  decomposition can be then obtained basing on condi-
tional independence. 
 In this paper, we apply the pair-copula decomposition approach to model 
the joint conditional distribution of the returns on stocks constituting the WIG20 
index. Results of our investigation support  a view that  this construction not 
only is superior over the approach that uses multidimensional t copula model 
but also represents a promising technique of building flexible and accessible 
multivariate extensions of classical bivariate copulas which can be of great im-
portance for optimal portfolio allocation and quantitative risk management. 

2. Dependence and Copulas 

 Consider a multivariate return series 1, , ( , , )t t n tr r ′=r … decomposed as 

t t t= +r μ y , where 1( | )t t tE −= Ωμ r and 1t−Ω is the set of information available 
up to time t . In standard multivariate GARCH models it is assumed that 

1/2
t t t=y H ε , ~ ( , )t niidε 0 I , and thus tH is the conditional covariance matrix of 

tr . A specific parameterization for the dynamics of  tH defines an element of 
the family of MGARCH models (Bauwens et al., 2006). One of the main diffi-
culties when dealing with these models is a problem of dimensionality because 
the number of parameters to be estimated increases very fast with the number k. 
Moreover, it is usually postulated that 1| ~ ( , )t t tN−Ωy 0 H or, slightly generally, 
that the conditional distribution of ty  is elliptical. The dynamic linear correla-
tion which can be obtained from MGARCH models still plays a central role in 
financial theory. One should realize, however, that this tool for measuring de-
pendence is appropriate only in the case of elliptical distributions. An alterna-
tive concept that allows for modeling the dependence in general situation is 
copula. Copulas were initially introduced by Sklar (1959). Formally, an  
n-dimensional copula is a distribution function C on n-cube n]1 ,0[ with standard 
uniform marginal distributions (Nelsen, 2006). Assume that X  is an  
n-dimensional random vector with joint distribution F and univariate marginal 
distributions iF . The importance of copulas in studying of multivariate distribu-
tion functions is summarized by Sklar’s theorem which states that the F can be 
written as 

1 1 1( , , ) ( ( ), , ( ))n n nF x x C F x F x=… … ’                                      (1) 
for some copula C. If the marginal distribution functions are continuous then C 
is unique, and is called the copula of F or X . Conversely, if C is a copula and 

nFF ,,1 …  are univariate distribution functions, then the function F defined in (1) 



Modeling the Dependence Structure of the WIG20 Portfolio Using a Pair-copula… 33

is a joint distribution function with margins nFF ,,1 … . An explicit representa-
tion of C in terms of F and its margins is given by 

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

1
11 nnn uFuFFuuC −−= …… ’ 

(2)
  

where })(  :inf{)(1
iiiiii uxFxuF ≥=− . Since the marginals and the dependence 

structure in (1) can be separated, it makes sense to interpret the copula C as the 
dependence structure of the random vector X. The simplest copula is defined by 

∏
=

=
n

i
in uuuC

1
1 ),,( … ,

    
(3)

  
and it corresponds to independence of marginal distributions. The next impor-
tant example is the comonotonicity copula, +C , which takes the form 

},,min{),,( 11 nn uuuuC …… =+
.  (4) 

It corresponds to perfect dependence between the components of a random vec-
tor 1( , , )nX X ′=X … in the sense that iX is  the image of 1X under some strictly 
increasing transformation for ni ,,2 …= . 
 In the empirical part of this paper we will use the Student t copula. It is de-
fined as follows: 

Student 1 1
, 1 , 1( , , ) ( ( ), , ( ))P n P nC u u t t u t uη η η η

− −=… … ,                       (5) 
where tη is the distribution function of a standard Student t distribution withη
degrees of freedom, and ,Ptη is the joint distribution function of a multivariate 
Student t distribution with η degrees of freedom and the correlation matrix P.  

   If a copula C is absolutely continuous, its density c is, as usual, given by 

n

n
n uu

uuCuuc
∂∂

∂
=

…
……

1

1
1

),,(),,( . (6)
 

For a copula C of absolutely continuous joint distribution function F with mar-
ginal distribution functions nFF ,,1 … , joint density f, and marginal densities 

nff ,,1 … , the following representation holds 

)()())(,),((),,( 11111 nnnnn xfxfxFxFcxxf …… = ,  (7) 
  



Ryszard Doman 34

3. Pair-copula Decompositions 
 The density of a vector  1( , , )nX X ′=X … can be factorized as 

1

1 2 1 1 2

( , , )
( ) ( | ) ( | , ) ( | , )

n

n n n n n n n n

f x x
f x f x x f x x x f x x x− − −=
…

… .
       (8) 

The idea of a cascade of bivariate copulas or  a pair-copula decomposition  
(Aas et al. 2009) comes from the fact that each conditional density in (8) can be 
further  decomposed into a product of the appropriate bivariate copula (pair-
copula) density times a conditional marginal density. For example,  

1 2 12 1 1 2 2 1 1( | ) ( ( ), ( )) ( )f x x c F x F x f x= ,   (9) 
1 2 3 12|3 1 1 3 2 2 3 1 3( | , ) ( ( | ), ( | )) ( | )f x x x c F x x F x x f x x= .  (10) 

More generally, it holds that 

|( | ) ( ( | ), ( | )) ( | )
j jxv j j j jf x c F x F v f x

− − − −= vv v v v ,     (11)      
where v is a d-dimensional vector and j−v denotes the vector obtained from v 
by excluding the j-th component. As concerns the marginal conditional distribu-
tions of the form ( | )F x v , it was shown by Joe (1996) that, for every j, the fol-
lowing holds 

, | ( ( , ), ( , ))
( | )

( | )
j jx v j j j

j j

C F x F v
F x

F v
− − −

−

∂
=

∂
v v v

v
v  

.                          (12)
 

In particular, if x and v are observations of variables uniform on [0,1] then 

, ( , )
( | ) x vC x v

F x v
v

∂
=

∂  
.                                        (13)

 
It follows from (7) and (11) that by applying iteration one can express a multi-
variate density as a product of the form 

1
1

1

, |1, , 1 1 1 1 1
1 1

( , , ) ( ),

( ( | , , ), ( | , , ))

=

−−

+ − − + −
= =

=∏

∏∏ …

…

i … …

n

n k
k

n jn

j j i j j j j i j
j i

f x x f x

c F x x x F x x x ,
       (14) 

or 

1
1

1

, | 1, , 1 1 1 1 1
1 1

( , , ) ( ),

( ( | , , ), ( | , , )) .

=

−−

+ + + − + + − + + + −
= =

=∏

∏∏ …

…

i … …

n

n k
k

n jn

i i j i i j i i i j i j i i j
j i

f x x f x

c F x x x F x x x
 (15) 



Modeling the Dependence Structure of the WIG20 Portfolio Using a Pair-copula… 35

Decompositions such as (14) and (15) are called pair-copula constructions. 
In fact, for high-dimensional distributions there are many possible pair-copula 
decompositions. Some methods that help organize them are described by Bed-
ford and Cooke (2001, 2002) in the language of the so-called regular vines.  
   In what follows, we use some very basic notions concerning graphs, which 
can be found, for instance, in (Kurowicka, Cooke, 2006). We start with the de-
finition of a regular vine on n variables. It is the structure composed of 1n −  
trees 1( , , )nT T… in which 1T  is a tree with the set of nodes 1 {1, , }N n= … and the 
set of edges 1E , and for 2, , 1i n= −… , the iT  is a tree with the set of nodes 

1i iN E −= . Moreover, it should hold that if some nodes 1 2{ , }a a a= , 1 2{ , }b b b=
are connected by an edge then exactly one ia is equal to exactly one ib . In fi-
nancial applications, the most important are two special cases of regular vines: 
canonical vines and the D-vines. A regular vine is called a canonical vine  
(or C-vine) if in each  tree iT  ( 1)i n< − there exists exactly one node with de-
gree n i− . The node in 1T  that has maximal degree is called the root. A regular 
vine is called a D-vine if each node in 1T  has a degree of at most 2. Examples of 
C- and D-vines are shown in Figures 1 and 2. 
 

 

 
 

Figure 1. A canonical vine on 5 variables  

13 

T1 

T2 

T3 

T4 

14 

15 12 

1 

2 
3 

4 

5 

24|1 

25|1 
23|1 

12

13

14

15

45|123 
34|12 35|12 

35|12 

34|12 

23|1 
24|1 

25|1 



Ryszard Doman 36

 
Figure 2. A D-vine on 5 variables 

 It is not very hard to observe that formula (14) gives a pair-copula construc-
tion corresponding to a canonical vine, and formula (15) defines a pair-copula 
construction that can be described by a D-vine. It should be mention here that 
starting from n nodes one can construct !/ 2n  different canonical vines and

!/ 2n different D-vines (Aas et al., 2009). When some components iX and jX  
of the vector X are conditionally independent given a subvector V of X then 

, | ( ( | ), ( | )) 1i j v i jc F x v F x v = , and thus the pair-copula decomposition in (14) or 
(15) simplifies. This property is of great importance from a practical point of 
view. It shows how a careful selection of variables and a proper choice of their 
ordering can affect the model complexity.  
   The canonical vines and D-vines can be estimated by maximum likelihood 
method. If we assume that the data 1, ,( , , )t t n tx x x= … , 1, ,t T= … , are observa-
tions of variables that are independent over time then the log-likelihood for the 
canonical vine is given by 

1

, |1, , 1 , 1, 1,
1 1 1

, 1, 1, ,

( ; ) log[ ( ( | , , ),

( | , , ); )] ,

n jn T

j j i j j t t j t
j i t

j i t t j t j i

L x c F x x x

F x x x

−−

+ − −
= = =

+ −

Θ =

Θ

∑∑∑ … …

…         (16)

              

 
 

and for the D-vine it has the form 

, | 1, , 1 , 1, 1,
1 1 1

, 1, 1, ,

( , ) log[ ( ( | , , ), 

( | , ); ))] .

n jn T

i i j i i j i t i t i j t
j i t

i j t i t i j t i j

L x c F x x x

F x x x

−

+ + + − + + −
= = =

+ + + −

Θ =

… Θ

∑∑∑ … …

 
(17)

           

 

12 34 23 45 

13|2 34|5 24|3 

1 3 2 4 5 T1 

T2 

T3 

T4 
15|234 

14|23 25|34 



Modeling the Dependence Structure of the WIG20 Portfolio Using a Pair-copula… 37

 The number of parameters depends on copula types used in the model speci-
fication. In the presence of temporal dependence, as is in the case of real data, 
usually some ARMA-GARCH models are fitted to the margins, and the estima-
tion is performed for the standardized residuals. Thus in fact, the estimation 
method is that of maximum pseudo-likelihood. The consistency and asymptotic 
normality of the estimators obtained in such a way is discussed by Genest et al. 
(1995) and Joe (1997).  

4. The Data and Model Specification 
 The data we use in this paper consist of daily returns on the stocks of the 
companies that constituted the WIG20 index of the Warsaw Stock Exchange 
during the period from September 23, 2005 to May 29, 2009.The tickers of the 
securities under scrutiny are as follows: ACP, BHW, BIO, BRE, BSK, BZW, 
CST, GTC, GTN, KGH, LTS, ORB, PBG, PEO, PGN, PKN, PKO, PXM, TPS, 
TVN. The returns are calculated as 1100(ln ln )t t tr P P−= − , where tP  is the clos-
ing quotation on day t.  
       The return series showed some autocorrelation and in all cases conditional 
homoskedasticity was strongly rejected by the Engle test. Following a common-
ly accepted approach, we first estimated the ARMA-GJR-GARCH models for 
the marginal returns. In each case a standardized skewed Student’s t distribution 
(Lambert, Laurent, 2001) was applied as the error distribution. Next, the stan-
dardized residuals series were transformed into uniform on [0,1] by using the 
corresponding probability integral transforms.   
      For the transformed data, we estimated a D-vine, described by the decompo-
sition  (15). Prior to choosing an ordering of the univariate series we estimated 
a bivariate Student’s t copula for each of the possible 190 pairs. Next, we ana-
lyzed  the pairs with respect the estimated number of degrees of freedom,  
which was assumed  to be a risk factor. We decided to apply bivariate Student’s 
t copulas for all pairs in estimated pair-copula decomposition. The final order-
ing of the marginal univariate series was chosen in such a way that the numbers 
of degrees of freedom of copulas connecting  the consecutive series in tree 1 of 
the D-vine formed a non-decreasing sequence. 

5. Empirical Results 
 In applications of pair-copula constructions it is of great importance to care-
fully consider the selection of specific factorization, and the choice of bivariate 
copula types. Thus in the first stage of our investigation we tried to fit bivariate 
copulas of diverse type to each of the possible pairs of the series from our data-
set. Finally we decided to use Student’s t copulas, and focus on their numbers of 
degrees of freedom considered as risk factors. The obtained estimates of the 
numbers of degrees of freedom are presented in Table 1. 



Table 1.  Estimates for the number of degrees of freedom in bivariate Student’s t copu-
las fitted to the pairs of the return series, period Sept. 23, 2005 – May 29, 2009   

 ACP BHW BIO BRE BSK BZW CST GTC GTN 
BHW 300.0         
BIO 36.7 40.9        
BRE 8.9 13.5 13.6       
BSK 17.4 24.6 83.4 21.6      
BZW 6.9 25.7 15.1 11.1 10.5     
CST 21.5 300.0 286.7 272.1 49.3 14.7    
GTC 10.3 24.1 33.6 19.4 43.4 16.1 23.2   
GTN 13.7 19.4 19.0 21.4 12.5 10.9 153.6 14.1  
KGH 8.0 11.4 22.2 11.2 19.3 13.2 44.3 25.3 17.6 
LTS 7.0 54.3 300.0 15.7 29.0 15.1 67.2 24.0 15.6 
ORB 7.9 300.0 233.1 300.0 300.0 17.2 32.1 18.8 25.7 
PBG 13.0 300.0 19.6 11.5 28.6 13.4 17.2 10.2 10.6 
PEO 9.6 300.0 11.0 7.8 21.5 53.2 85.8 16.1 27.1 
PGN 13.1 11.4 26.5 18.4 48.1 13.0 7.1 18.5 13.0 
PKN 10.9 22.3 300.0 20.7 68.0 13.4 300.0 84.9 8.1 
PKO 12.5 13.5 36.7 12.0 13.8 30.2 63.8 194.6 12.2 
PXM 9.4 11.8 12.3 9.2 21.3 8.4 13.5 7.5 11.4 
TPS 15.0 23.1 21.5 70.9 148.8 11.2 42.6 21.8 300.0 
TVN 30.9 49.9 158.0 19.0 300.0 15.9 23.0 19.2 17.3 

 
 KGH LTS ORB PBG PEO PGN PKN PKO PXM TPS 

LTS 24.2          
ORB 20.5 10.1         
PBG 8.8 16.4 29.5        
PEO 17.6 16.3 24.4 300.0       
PGN 15.1 10.6 15.8 13.7 11.5      
PKN 27.7 5.1 14.5 85.0 13.0 13.6     
PKO 14.6 10.0 11.8 22.6 7.3 6.2 22.7    
PXM 8.7 33.7 8.8 58.7 10.2 18.7 28.2 14.2   
TPS 56.6 13.2 9.5 47.4 25.3 17.3 26.3 25.2 17.0  
TVN 21.4 9.6 119.1 18.1 12.8 21.5 13.3 15.7 12.3 108.4 

Note: We estimated the number of degrees of freedom parameter subject to upper bound equal to 300. 
In practice, the value 300 means that the Gaussian copula is the proper one.  



Modeling the Dependence Structure of the WIG20 Portfolio Using a Pair-copula… 39

Table 2.  Estimates for the number of degrees of freedom in the fitted D-vine 

 1 2 3 4 5 6 7 8 9 
1 5.1 26.4 25.1 38.6 300.0 300.0 300.0 20.6 14.3 
2 13.6 54.2 53.8 13.7 300.0 35.9 300.0 53.2 300.0 
3 6.2 22.1 32.5 14.0 300.0 272.2 45.0 46.8 300.0 
4 12.5 17.4 300.0 8.6 21.1 22.8 36.7 223.1 24.2 
5 6.9 300.0 18.8 13.3 15.1 31.0 77.2 31.8 43.2 
6 14.7 66.5 13.0 23.3 300.0 37.3 28.5 20.1 8.1 
7 85.8 300.0 27.9 31.6 14.5 74.9 18.0 300.0 46.2 
8 7.8 300.0 38.5 21.1 19.8 25.6 36.1 17.1 15.0 
9 300.0 20.5 14.1 31.9 12.2 294.0 260.6 44.8 39.1 

10 18.8 11.2 29.1 300.0 15.6 187.2 300.0 300.0 300.0 
11 7.5 65.0 11.7 20.0 37.8 35.9 63.3 59.4 36.6 
12 8.7 279.5 29.8 20.8 33.4 300.0 19.3 38.1  
13 8.8 300.0 247.0 300.0 285.1 13.3 34.5   
14 47.4 16.6 23.2 16.3 153.7 42.6    
15 111.6 20.8 300.0 38.5 300.0     
16 158.0 13.4 78.1 300.0      
17 19.0 50.4 300.0       
18 19.4 15.6        
19 24.6         

 
 10 11 12 13 14 15 16 17 18 19 
1 300.0 300.0 300.0 42.5 42.4 14.6 300.0 108.1 300.0 300.0 
2 97.3 19.3 300.0 36.5 19.3 300.0 9.2 300.0 300.0  
3 44.1 30.4 76.6 300.0 300.0 100.4 30.6 300.0   
4 14.2 82.3 113.4 300.0 24.5 22.1 300.0    
5 300.0 73.7 275.0 39.5 300.0 27.7     
6 22.9 15.2 36.9 300.0 11.8      
7 300.0 29.8 300.0 300.0       
8 16.2 300.0 151.2        
9 23.7 44.8         
10 300.0          

Note: The numbers of degrees of freedom for the copulas appearing in trees 1-19 of the estimated D-vine are 
presented in columns. We estimated the number of degrees of freedom parameter subject to upper bound 
equal to 300. In practice, the value 300 means that the Gaussian copula is the proper one.   

 We used the obtained estimates of the number of degrees of freedom for the 
bivariate return series to chose an efficient ordering of the variables included in  



Ryszard Doman 40

the decomposition (15). Our choice was the following: 1. LTS, 2. PKN, 3. PGN, 
4. PKO, 5. ACP, 6. BZW, 7. CST, 8. PEO, 9. BRE, 10. ORB, 11. GTC, 12. 
PXM, 13. KGH, 14. PBG, 15. TPS, 16. TVN, 17. BIO, 18. GTN, 19. BHW, 20. 
BSK. The motivation was that in that case, for the 19 bivariate copulas fitted to  
the pairs of the returns in accordance to the formula of tree 1 of the estimated 
D-vine, the sequence of the corresponding numbers of degrees of freedom is 
non-decreasing. The D-vine estimation results are presented in table 2. 
 For a comparison we fitted to the investigated vector return series a standard 
20-dimensional Student’s t copula. As an estimate for the number of degrees of 
freedom we obtained 34.4265. Looking at the estimates for the bivariate copulas 
in tree 1 of the estimated D-vine, which vary from 5.1 to 300, we can state that 
the superiority of the pair-copula construction approach over the standard mul-
tidimensional copula approach is strongly supported. A significantly better fit of 
the D-vine model has been also indicated by the Akaike information criterion. 
 Our next objective was to use the estimated D-vine to compute in-sample 
VaR estimates for long and short positions (Giot, Laurent, 2003) for the portfo-
lio composed of the considered stocks, and compare them to the ones obtained 
by using Engle’s (2002) DCC model with multivariate Student’s t distribution. 
In the DCC model case we could use the well-known formulas for a portfolio 
VaR (see e.g. Giot, Laurent, 2003), having the estimates of the conditional co-
variances and the degree of freedom for the conditional Student’s t distribution, 
which was estimated as 13.7881. For the approach using the pair-copula con-
struction, we simulated for each day 1000 20-dimensional vectors from the fit-
ted D-vine. Then we transformed them into the one-dimensional standardized 
residuals, and, finally, into the daily returns of the portfolio components. After 
that we obtained the daily VaR estimates as the corresponding quantiles. The 
VaR  calculation was performed for significance levels 0.01, 0.025, and 0.05. 
 An algorithm for sampling from an n-dimensional D-vine proceeds as fol-
lows (see Aas et al., 2009). Start with sampling variates nuu ,,1 …  independent 
uniform on [0, 1]. Then set 
 11 uw = , 

 )|( 12
1

2 wuFw −= , 

 ),|( 213
1

3 wwuFw −= , 

  

 ),,|( 11
1

−
−= nnn wwuFw … . 

  The general formula for the functions ),,|( 11 −jj xxxF … involves (12) and 
(13), and it is given in the paper by Aas et al. (2009), where one can also find 
explicit formulas for the inverse in the case in which all the bivariate copulas of 
a pair copula construction are Student’s t copulas.  



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Ryszard Doman 42

Value-at-Risk. Our results show that the VaR estimates for long trading posi-
tions at tolerance level 0.05 obtained in this way definitely outperform the ones 
calculated by using a fitted DCC model with Student’s t conditional distribu-
tion. 

References  
Aas, K., Czado, C., Frigessi, A., Bakken, H. (2009), Pair-Copula Constructions of Multiple De-

pendence, Insurance: Mathematics and Finance, 44, 182–198. 
Bauwens, E. Laurent, S. Rombouts, J.V.K. (2006), Multivariate GARCH Models: A Survey, 

Journal of Applied Econometrics, 21, 79–109. 
Bedford, T., Cooke, R.M. (2001), Probability Density Decomposition for Conditionally Depen-

dent Random Variables Modeled by Vines, Annals of Mathematics and Artificial Intelli-
gence, 32, 245–268. 

Bedford, T., Cooke, R.M. (2002), Vines – a New Graphical Model for Dependent Random Va-
riables, Annals of Statistics, 30, 1031–1068.  

Christoffersen, P. F. (1998), Evaluating Interval Forecasts, International Economic Review, 39, 
841–862. 

Engle, R. F., (2002), Dynamic Conditional Correlation: A Simple Class of Multivariate Genera-
lized Autoregressive Conditional Heteroskedasticity Models, Journal of Business & Eco-
nomic Statistics, 20, 339–350.   

Genest, C. Ghoudi K. Rivest, L.-P. (1995), A Semiparametric  Estimation Procedure of Depen-
dence Parameters in Multivariate  Families of Distributions, Biometrika, 82, 543–552. 

Giot, P., Laurent, S. (2003), Value-at-Risk for Long and Short Trading Positions, Journal of 
Applied Econometrics, 18, 641–664. 

Joe, H. (1996), Families of m-variate Distributions with Given Margins and m(m - 1)/2 Bivariate 
Dependence Parameters. In: Rüschendorf, L., Schweizer, B.,Taylor, M.D. (Eds.), Distribu-
tions with Fixed Marginals and Related Topics, IMS Lecture Notes Monograph Series 28, 
Institute of Mathematical Statistics, Hayward, CA, 120–141. 

Joe, H. (1997), Multivariate Models and Dependence Concepts, Chapman & Hall, London. 
Kurowicka, D., Cooke, R. M., (2006), Uncertainty Analysis with High Dimensional Dependence 

Modelling, Wiley, New York. 
Lambert, P., Laurent, S. (2001), Modelling Financial Time Series Using GARCH-type Models 

with a Skewed Student Distribution for the Innovations, Institut de Statistique, Université 
Catholique de Louvain, Discussion Paper 0125. 

Nelsen, R. B. (2006) An Introduction to Copulas (2nd ed.), Springer, New York 
Sklar, A. (1959), Fonctions de rérpartition à n dimensions et leurs marges, Publications de 

l’Institut  Statistique de l’Université de Paris, 8, 229–231.  

Modelowanie struktury zależności portfela indeksu WIG20  
za pomocą kaskady kopuli dwuwymiarowych  

Z a r y s  t r e ś c i. W artykule przedstawiono wyniki zastosowania nowej metodologii modelo-
wania zależności pomiędzy zwrotami składników portfela wysokowymiarowego. Idea tego po-
dejścia polega na dekompozycji gęstości rozkładu łącznego na iloczyn, w którym występują 
jedynie gęstości kopuli dwuwymiarowych pewnych rozkładów warunkowych wyznaczonych 
przez modelowane zmienne. Badania dotyczą stóp zwrotu z akcji wchodzących w skład indeksu 
WIG20 i potwierdzają pewną przewagę nowej metodologii nad podejściem, w którym stosowany 
jest model DCC Engle’a z wielowymiarowym rozkładem t Studenta.   

S ł o w a  k l u c z o w e: zależność, portfel, kopula, kaskada kopuli dwuwymiarowych.  




