An application of copulas in the Value-at-Risk estimation
Marcin Dudziński and Konrad Furmańczyk
Faculty of Applied Informatics and Mathematics
Department of Applied Mathematics
Warsaw University of Life Sciences
ul. Nowoursynowska 159, 02-776 Warsaw, Poland
E-mails: [email protected], [email protected]
Abstract. Value-at-Risk (VaR) is one of the most widely used risk measures in the field of risk
management. It defines a change in value of a portfolio of financial assets as the minimum amount of
money that one could expect to lose with a given probability (tolerance level) over a specific time
horizon. VaR of a portfolio is determined by the multivariate distribution of risk factors increments. This
distribution may be modeled by copulas. In our paper, we show some ideas of the estimation of VaR
by using of the copula approach. We apply these ideas to calculate VaR for a portfolio composed of
the stock prices of Boeing Co. and General Motors Corp., available from the Dow Jones Industrial
Average (DJIA).
Keywords and phrases: Value-at-risk, Copulas, Monte Carlo simulation, GARCH models, R package.
1. Introduction
Value-at-Risk (VaR) has become a key measure of market risk in the past decade.
Nowadays, it is probably the most popular measure of the risk of loss on a specific portfolio
of financial assets. For a given portfolio, the confidence level 1   and time period, VaR is
defined as a threshold value, such that the probability that the market loss on the portfolio
over the given time period exceeds or equals this value is equal to the tolerance level  .
Suppose that at time t a portfolio consists of d assets with the prices
S t  S1,t ,..., S d ,t . Assume in addition that the parts of individual assets in the entire portfolio
are constant in time and described by the vector of weights   1 ,...,  d  . Then, the value
of a portfolio at t is given by
d
Vt    i S i ,t .
i 1
Put Z i ,t : ln S i ,t , i  1,..., d . Clearly, we have
Vt    i exp Z i ,t  .
d
i 1
We define the Profit and Loss (P&L) function from period t  T
to t  T  1 as
LT 1  VT 1  VT . Furthermore, we denote by X i ,T 1 : Z i ,T 1  Z i ,T , i  1,..., d , the time
increment in the i -th risk factor from period T to T  1 . Then, we can write the P&L function
as follows
LT 1  L X 1,T 1 ,..., X d ,T 1     i S i ,T exp  X i ,T 1   1 .
d
(1)
i 1
Thus, in the case, when the distribution function of the P&L function is continous and strictly
increasing, VaRT 1   - the VaR, at time T  1 , of a specific portfolio of assets over a time
horizon from T to T  1 , with the confidence level 1   - satisfies
1
PLT 1  VaRT 1    1  .
Therefore,
PLT 1  VaRT 1     ,
and it is easily seen that VaRT 1   is the  -th quantile from the distribution of LT 1 , i. e.,
VaRT 1    FLT11  ,
where FLT11 stands for the inverse function of the distribution function FLT 1 of LT 1 .
Observe that the 1 -dimensional distribution function FLt of the P&L function Lt depends on


the d-dimensional distribution function FX t of the vector X t  X 1,t ,..., X d ,t . Such a
multivariate distribution function can be modeled by the functions called copulas. The theory
of copulas is a fundamental tool in modeling the multivariate distributions. By using of
copulas, we can split the distribution of random vector into individual components
(marginals). Copulas enable to define the joint distributions through the marginal ones and
provide the idea of modeling the dependence structure among marginals. They also allow to
couple 1 -dimensional distributions into multivariate ones. Thus, we may link any group of
univariate distributions with any copula. In particular, by appropriate facts from the copula
theory, each marginal distribution function FX i ,t of X i ,t , i  1,..., d , can be separately
modeled from their dependence structure and subsequently coupled together in order to
obtain the joint distribution function of FX t .
Our goal is to estimate VaR by applying of the copula approach. Several authors
have become increasingly concerned over the estimation of VaR by copulas in recent years.
There has been a lot of work on this issue throughout the past decade - one can refer to the
papers of: Dias and Embrechts (2003), Fantazzini (2008), Giacomini and Hardle (2005),
Hotta and Palaro (2006), Hotta et al. (2008), Hurlimann (2004), Jajuga (2008), and Martinelli
and Meyfredi (2007) in this context.
Our database consists of the log return series of 2520 daily stock prices of Boeing
Co. and General Motors Corp. in the period between August 30th 1993 and August 29th 2003,
available from the Dow Jones Industrial Average (DJIA). We model the dependence between
the both of the considered log return series by the four families of copulas.
The paper is organized as follows. Section 2 gives a brief theoretical background on
copulas and presents the examples of copula functions, we make extensive use of in our
investigations. Section 3 contains the details of our estimation procedure. Our simulation
results are summarized in Section 4. The performance of our models is assessed in
Section 5, by using of some bactesting procedure. Finally, Section 6 concludes our research.
All the figures and simulations in our paper were carried out by applying of the
R package, together with the copula, fGarch and VaR packages.
2. Copulas
At the beginning of this section, we give the general definition of the copula.
Definition 1. A d -dimensional copula is a multivariate cumulative distribution function
C : 0, 1  0, 1 , whose margins have the uniform distribution on the interval 0, 1 .
d
The following theorem is a very significant result in the copula theory.
2
Theorem 1 (Sklar's theorem). Let F denote a d -dimensional distribution function with
marginal distribution functions FX 1 ,..., FX d . Then, there exists a copula C , such that


F x1 ,..., xd   C FX1 x1 ,..., FX d xd 
x1 ,..., xd   R d .
for any
(2)
In addition, we have that, if FX 1 ,..., FX d are continous, then the copula C is a unique one.
Conversely, if C is a copula and FX 1 ,..., FX d are distribution functions, then the function F ,
defined by (2), is the joint distribution function with marginal distribution functions FX 1 ,..., FX d .
In our considerations, we restrict ourselves to the case of 2 -dimensional (bivariate)
copulas. Below, we present the four families of copulas used in our paper, namely: the
bivariate normal copula, the bivariate Student t-copula, the bivariate Plackett copula, and the
bivariate Clayton copula.
2.1. The bivariate normal copula
The bivariate normal copula is the function of the form
C u1 , u 2 ;   
 1 u1   1 u 2 




 r 2  2 rs  s 2 
exp 
drds,
2 1  2
2 1   2


1


where  is the linear correlation coefficient between the two random variables and  1
stands for the inverse of the univariate standard normal distribution function.
2.2. The bivariate Student t-copula
The bivariate Student t-copula is the function given by
C u1 , u 2 ;  , v  
t v1 u1  t v1 u 2 




 r 2  2 rs  s 2 
1 

v 1  2
2 1   2 

1

 v  2  / 2

drds,
where  is the linear correlation coefficient between the two random variables and t v1
denotes the inverse of the univariate Student- t distribution function with v degrees of
freedom.
2.3. The bivariate Plackett copula
The bivariate Plackett copula is defined by



1/ 2
 1
2
1    1u1  u 2   1    1u1  u 2   4u1u 2   1

C u1 , u 2 ;    2  1

u1u 2


for   1,
for   1,
where  stands for the given parameter value.
2.4. The bivariate Clayton copula
The following function is called the bivariate Clayton (or Cook-Johnson) copula


C u1 , u 2 ;   max u1  u 2  1
1 / 

,0 ,
where  denotes the fixed parameter value.
3. Estimation procedure
We calculate VaR by applying of the GARCH model together with the copula theory.
 
The GARCH(1,1) model is used to the log return series x1,t
2520
t 1
 
and x 2,t
2520
t 1
of 2520 daily
stock prices of Boeing Co. and General Motors Corp., respectively, in the period between
August 30th 1993 and August 29th 2003. Thus, the models for our margins are given by:
3
x i ,t   i   i ,t ;
 i ,t   i ,t i ,t ;
(3)
 i2,t   i   i  i2,t 1   i i2,t 1 ; i  1,2,
where:  i,t are the white noise processes with zero mean and unit variance, and
 i2,t  E  i2,t xi ,1 ,..., xi ,t 1 .
At the beginning, we estimate the parameters of the models, separately for each asset - we
model the conditional distribution of the standardized innovations  i ,t   i ,t /  i ,t by Student’s




t -distribution and calculate the maximum likelihood (ML) estimates  i ,  i ,  i ,  i of the
parameters  i , i ,  i ,  i of the models for the marginal variables X i ,t , i  1,2 , as well as, the

estimates  i,t of the conditional standard deviations  i ,t , i  1,2 . Then, we fit the four
chosen families of copulas - the bivariate normal copula, the bivariate Student t -copula, the


bivariate Plackett copula, the bivariate Clayton copula - to the vector of residuals 1,t , 2,t .
For each kind of copula, we estimate copula’s parameters by using of the ML method. In the
5000



next step, we generate 5000 Monte Carlo data  1, 2521, 2, 2521 
from the previously

 k 1
estimated copulas. Subsequently, by applying of the estimated GARCH(1,1) model (see (3)),
5000



we obtain a sample of 5000 Monte Carlo data  x 1, 2521, x 2, 2521 
according to the relations:

 k 1
5000
5000






x




1
,
2521
1
,
2521


 1, 2521 
1

 k 1

 k 1
5000
5000






and  x 2, 2521 
  2   2, 2521 2, 2521  .

 k 1

 k 1



Then, due to (1), we get a Monte Carlo sample of 5000 losses L 2521  L( x1, 2521, x 2, 2521 ) .


Finally, we calculate the empirical quantile of L 2521 at level 0.05 , which is VaR 25210.05 - the
desired estimate of the Value-at-Risk measure of our portfolio at time T  1  2521, over a
time horizon from T  2520 to T  1  2521, with the confidence level 1 0.05 .
4. Empirical studies
We start our empirical analysis with the graphical presentation of our dataset. For
recollection, we consider a portfolio composed of Boeing Co. and General Motors Corp.
assets. Figures 1a), 1b) present the daily stock prices of our assets from August 30th 1993 to
August 29th 2003. Figures 2a), 2b) present their daily log-returns in the same time period (we
denote these log-returns by x1,t , x 2 ,t ).
4
Figure 1a)
Figure 1b)
Figure 2a)
Figure 2b)
The table below presents the maximum likelihood estimates of the two fitted
GARCH(1,1) models (with the t -Student innovations), together with the Akaike’s information
criteria (AIC), applied for the log-return series x1,t , x 2 ,t .
Parameters
Estimates
Std. Errors
p-values
μ1
-4.199e-04
3.389e-04
0.2153
α1
5.683e-06
2.491e-06
0.0225
β1
5.706e-02
1.423e-02
6.1e-05
γ1
9.319e-01
1.696e-02
< 2e-16
v1
5.589e+00
< 2e-16
Log Likelihood1
6.181e-01
-6357.18
AIC1
5.051354
μ2
-6.007e-05
3.662e-04
5
0.8697
α2
1.316e-05
4.464e-06
0.0032
β2
6.448e-02
1.374e-02
2.71e-06
γ2
9.062e-01
2.053e-02
< 2e-16
v2
7.606e+00
1.047e+00
-6304.96
5.009893
< 3.82e-13
Log Likelihood2
AIC2
Table 1: The parameter estimates of GARCH(1,1) models and standard errors
We notice that, in almost all cases, the Ljung-Box test does not reject the null
hypothesis of null autocorrelations of the residuals from lag 1 to 15 for both series at the 5%
significance level. The exceptions concern only the case of lag 2, for the first of our series,
and the case of lag 1, for the second one. The corresponding p-values are as follows:
for the first series (relating to Boeing Co.): 0.3792, 0.04081, 0.08453, 0.0849, 0.08299,
0.09862, 0.1014, 0.1440, 0.1684, 0.09945, 0.1013, 0.1392, 0.1314, 0.1218, 0.1521,
for the second series (relating to General Motors Corp.): 0.04136, 0.08907, 0.1091, 0.09876,
0.05587, 0.07591, 0.1184, 0.1322, 0.1874, 0.2309, 0.2952, 0.3711, 0.4418, 0.3546, 0.3598.
Therefore, we may consider our models to be adequate.





Figure 3 gives the scatterplot of  1,t , 2 ,t  - the estimates of the standardized
innovations.
Figure 3
Figures 4a), 4b) below confirm that our innovations may have Student’s t distributions.
6
Figure 4a
Figure 4b)
In Table 2 below, we give the obtained values of VaR25210.05 estimates, together
with the observed returns of the Profit and Loss (P&L) function L2521 .
Normal
copula
Copula’s
parameters

VaR 2521 0.05
  0.27
Student’s
t-copula
  0.28,
v  12.64
Realized
Plackett’s Clayton’s returns
of the
copula
copula
P&L
function
  2.34
  0.27
 1.30
 1.27
 1.24
 1.32
0.04
 1.03
 0.99
 0.98
 1.03
 0.05
 0.89
 0.86
 0.85
 0.90
 0.09
 0.77
 0.74
 0.72
 0.77
 0.14
 0.54
 0.53
 0.52
 0.55
 0.23
1  0.2,  2  0.8

VaR 2521 0.05
1  0.4,  2  0.6

VaR 2521 0.05
1  0.5,  2  0.5

VaR 2521 0.05
1  0.6,  2  0.4

VaR 2521 0.05
1  0.8,  2  0.2
Table 2: The estimation results for the different weights 1 , 2
7
5. Bactesting
We shall assess the quality of the adjustment of our model by using of the bactesting
procedure, which can be described stepwise as follows:
For i  1, ..., 50 , we proceed according to the algorithm below:
Step i.1 . We use the data from t  1 to t  2471  i  1 - we consider the daily stock

prices s1,1 , s1, 2 ,..., s1, 2471i 1
Corp.,

respectively,


and s 2,1 , s 2, 2 ,..., s 2, 2471i 1
and
calculate
the

of Boeing Co. and General Motors
log-returns

i.2 . We model the obtained log-return series

, x1,3 ,..., x1, 2471i 1 
x
, x1,3 ,..., x1, 2471i 1 
1, 2
and x 2, 2 , x 2,3 ,..., x 2, 2471i 1 ;
Step
x
1, 2

and x 2, 2 , x 2,3 ,..., x 2, 2471i 1 by the GARCH(1,1) model with Student’s- t innovations;
Step i.3 . By applying of Sklar’s theorem, we fit the four chosen families of copulas the bivariate normal copula, the bivariate Student t -copula, the bivariate Plackett copula, the


bivariate Clayton copula - to the vector of standardized residuals 1,t , 2,t ; for each kind of
copula, we estimate copula’s parameters by using of the maximum likelihood method;





Step i.4 . We simulate a sample of 5000 Monte Carlo data  1,2471i 11 , 2,2471i 11 
from the previously estimated copulas;
Step i.5 . By using of the estimated GARCH(1,1) model, we get a sample of 5000





Monte Carlo data  x1,2471i 11 , x 2,2471i 11  according to the relations:








x1,2471i 11  1   1,2471i 11  1,2471i 11 and x 2,2471i 11   2   2,2471i 11  2,2471i 11 ,




where 1 ,  2 ,  1,2471i 11 ,  2,2471i 11 are the previously obtained estimates;
Step
i.6 . Due to (1), we get a Monte Carlo sample of 5000 losses



L 2471i 11  L x1,2471i 11 , x 2,2471i 11  ; subsequently, we calculate, at level 0.05 , the








empirical quantile from the obtained sample  L 2471i 11  , which is VaR 2471i 11 0.05 - the
appropriate estimate of VaR;
Thus, after proceeding according to the steps i.1 - i.6 , i  1, ..., 50 , we obtain


(separately for each kind of the copula) 50 values: VaR 2472 0.05 , …, VaR 25210.05 . The
final steps of our bactesting procedure, which follow after the steps 1.1- 1.6 ,…, 50.1- 50.6 ,
are:
Step 51 . By using of (1), we calculate L2472 , …, L2521 - the observed returns of the


Profit and Loss (P&L) function - and compare them with VaR 2472 0.05 , …, VaR 25210.05 ,
respectively;
8
Step 52 . We count the number of exceedances of the observed portfolio losses


L2472 , …, L2521 over the corresponding VaR estimates VaR 2472 0.05 , …, VaR 25210.05 .
It is easily seen that our estimation procedure (presented in Section 3) is also
described by the steps 50.1-50.6 of our bactesting.
Figures 5a), 5b) present the graphical comparison between the values mentioned
in Step 52 . The points indicate the values of Li  : L2471i 11 , i  1, ..., 50 , i. e., the realized
returns of the P&L function. In turn, the solid lines connect the points representing the values


of VaR estimates VaRi 0.05 : VaR 2471i 11 0.05 , i  1, ..., 50 , separately for each of the
copulas taken into account.
We have restricted ourselves to the cases, when the P&L function is calculated with the
weights 1  2  0.5 (Figure 5a)) or 1  0.8, 2  0.2 (Figure 5b)). We observe that, for

each kind of the considered copulas, VaRi 0.05 exceeds Li  only once in the first case,
while it does not occur at all in the second one.
Figure 5a)
9
Figure 5b)
6. Conclusions
The calculated VaR estimates (see Table 2) are similar for all of the four considered
copulas. On average, the smallest values of VaR estimates have been obtained in the case
of GARCH+Clayton copula approach, while the largest ones have been observed in the case
of GARCH+Plackett copula model. Thus, although we must be aware of the small differences
between the results obtained for our copula models, we may expect the largest probabilities
of large losses, if we apply the GARCH+Clayton copula approach and the smallest ones, if
we apply the GARCH+Plackett copula model. Therefore, the GARCH+Clayton copula
approach seems to be the most pessimistic and the GARCH+Plackett one the most
optimistic among the models used in our VaR evaluations.
We believe that the procedure of VaR estimation presented in our paper may be
developed in further research concerning the area of risk management.
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10
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