Involving copula functions in Conditional Tail Expectation

Brahim Brahimi111E-mail<EMAIL_ADDRESS>Tel.:+213-7 73 54 60
63; fax:+213-33 74 77 88.

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria

Abstract

Our goal in this paper is to propose an alternative risk measure which takes
into account the fluctuations of losses and possible correlations between
random variables. This new notion of risk measures, that we call Copula
Conditional Tail Expectation describes the expected amount of risk that can be
experienced given that a potential bivariate risk exceeds a bivariate
threshold value, and provides an important measure for right-tail risk. An
application to real financial data is given.

Keywords: Conditional tail expectation; Positive quadrant dependence; Copulas;
Dependence measure; Risk management; Market models.
AMS 2010 Subject Classification: 62P05; 62H20; 91B26; 91B30.

##  1\. Introduction

In actuarial science literature a several risk measures have been proposed,
namely: the Value-at-Risk (VaR), the expected shortfall or the conditional
tail expectation (CTE), the distorted risk measures (DRM), and recently the
copula distorted risk measure (CDRM) as risk measure which takes into account
the fluctuations dependence between random variables (rv). See Brahimi et al.
(2010).

The CTE in risk analysis represents the conditional expected loss given that
the loss exceeds its VaR and provides an important measure for right-tail
risk. In this paper we will always consider random variables with finite mean.
For a real number $s\ $in $\left(0,1\right),$ the CTE of a risk $X$ is given
by

$\mathbb{CTE}\left(s\right):=\mathbb{E}\left[\left.X\right|X>VaR_{X}\left(s\right)\right],$
(1.1)

where $VaR_{X}\left(s\right):=\inf\left\\{x:F\left(x\right)\geq s\right\\}$ is
the quantile of order $s$ pertaining to distribution function (df) $F.$

One of the strategy of an Insurance companies is to set aside amounts of
capital from which it can draw from in the event that premium revenues become
insufficient to pay out claims. Of course, determining these amounts is not a
simple calculation. It has to determine the best risk measure that can be used
to determine the amount of loss to cover with a high degree of confidence.

Suppose now that we deal with a couple of random losses $(X_{1},X_{2}).$ It’s
clear that the CTE of $X_{1}$ is unrelated with $X_{2}.$ If we had to control
the overflow of the two risks $X_{1}$ and $X_{2}$ at the same time, CTE does
not answer the problem, then we need another formulation of CTE which takes
into account the excess of the two risks $X_{1}$ and $X_{2}.$ Then we deal
with the amount

$\mathbb{E}\left[\left.X_{1}\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right].$
(1.2)

If the couple of random losses $(X_{1},X_{2})$ are independents rv’s then the
amount (1.2) defined only the CTE of $X_{1}.$ Therefore the case of
independence is not important.

In the recent years dependence is beginning to play an important role in the
world of risk. The increasing complexity of insurance and financial activities
products has led to increased actuarial and financial interest in the modeling
of dependent risks. While independence can be defined in only one way, but
dependence can be formulated in an infinite ways. Therefore, the assumption of
independence it makes the treatment easy. Nevertheless, in applications
dependence is the rule and independence is the exception.

The copulas is a function that completely describes the dependence structure,
it contains all the information to link the marginal distributions to their
joint distribution. To obtain a valid multivariate distribution function, we
combines several marginal distribution functions, or a different
distributional families, with any copula function. Using Sklar’s theorem
(Sklar, 1959), we can construct a bivariate distributions with arbitrary
marginal distributions. Thus, for the purposes of statistical modeling, it is
desirable to have a large collection of copulas at one’s disposal. A great
many examples of copulas can be found in the literature, most are members of
families with one or more real parameters. For a formal treatment of copulas
and their properties, see the monographs by Hutchinson and Lai (1990),
Dall’Aglio et al. (1991), Joe (1997), the conference proceedings edited by
Benes̆ and S̆tĕpán (1997), Cuadras et al. (2002), Dhaene et al. (2003) and the
textbook of Nelsen (2006).

Recently in finance, insurance and risk management has emphasized the
importance of positive or negative quadrant dependence notions (PQD or NQD)
introduced by Lehmann (1966), in different areas of applied probability and
statistics, as an example, see; Dhaene and Goovaerts (1997), Denuit et al.
(2001). Two rv’s are said to be PQD when the probability that they are
simultaneously large (or small) is at least as great as it would be were they
are independent. In terms of copula, if their copula is greater than their
product, i.e., $C(u_{1},u_{2})>u_{1}u_{2}$ or, simply $C>C^{\perp},$ where
$C^{\perp}$ denotes the product copula. For the sake of brevity, we will
restrict ourselves to concepts of positive dependence.

The main idea of this paper is to use the information of dependence between
PQD or NQD risks to quantifying insurance losses and measuring financial risk
assessments, we propose a risk measure defined by:

$\mathbb{CCTE}_{X_{1}}\left(s;t\right):=\mathbb{E}\left[\left.X_{1}\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right].$

We will call this new risk measure by the Copula Conditional Tail Expectation
(CCTE), like a risk measure which measure the conditional expectation given
the two dependents losses exceeds $VaR_{X_{1}}\left(s\right)$ and
$VaR_{X_{2}}\left(t\right)$ for $0<s,t<1$ and usually with $s,t>0.9.$ Again,
CCTE satisfies all the desirable properties of a coherent risk measure
(Artzner et al., 1999). The notion of copula in risk measure filed has
recently been considered by several authors, see for instance Embrechts et al.
(2003a), Di Clemente and Romano (2004), Dalla Valle (2009), Brahimi et al.
(2010) and the references therein.

This risk measures can give a good quantifying of losses when we have a
combined dependents risks, this dependence can influence in the losses of
interested risks. Therefore, quantify the riskiness of our position is useful
to decide if it acceptable or not. For this reason we use the all informations
a bout this interest risk. The dependence of our risk with other risks is one
of important information that we must take it in consideration.

This paper is organized as follows. In section 2, we give an explicit
formulations of the new notion copula conditional tail expectation risk
measure in bivariate case. The relationship of this new concept and tail
dependence measure, given in section 3. In section 4 we presents an
illustration examples to explain how to use the new CCTE measure. Application
in real financial data is given in section 5. Concluding notes are given in
Section 6. Proofs are relegated to the Appendix.

##  2\. Copula conditional tail expectation

A risk measure quantifies the risk exposure in a way that is meaningful for
the problem at hand. The most commonly used risk measure in finance and
insurance are: VaR and CTE (also known as Tail-VaR or expected shortfall). The
risk measure is simply the loss size for which there is a small (e.g. $1\%$)
probability of exceeding. For some time, it has been recognized that this
measure suffers from serious deficiencies if losses are not normally
distributed.

According to Artzner et al. (1999) and Wirch and Hardy (1999), the conditional
tail expectation of a random variable $X_{1}$ at its
$VaR_{X_{1}}\left(s\right)$ is defined by:

$\mathbb{CTE}_{X_{1}}\left(s\right)=\frac{1}{1-F_{X_{1}}(VaR_{X_{1}}\left(s\right))}\int_{VaR_{X_{1}}\left(s\right)}^{\infty}xdF_{X_{1}}(x),$

where $F_{X_{1}}$ is the df of $X_{1}$.

Since $X_{1}$ is continuous, then $F_{X_{1}}(VaR_{X_{1}}\left(s\right))=s,$ it
follows that for all $0<s<1$

$\mathbb{CTE}_{X_{1}}\left(s\right)=\frac{1}{1-s}\int_{s}^{1}VaR_{X_{1}}\left(u\right)du.$
(2.3)

The CTE can be larger that the VaR measure for the same value of level $s$
described above since it can be thought of as the sum of the quantile
$VaR_{X_{1}}\left(s\right)$ and the expected excess loss. Tail-VaR is a
coherent measure in the sense of Artzner et al. (1999). For the application of
this kind of coherent risk measures we refer to the papers Artzner et al.
(1999) and Wirch and Hardy (1999).

Thus the CTE is nothing, see Overbeck and Sokolova (2008), but the
mathematical transcription of the concept of ”average loss in the worst
$100(1-s)\%$ case”, defining by $\upsilon=VaR_{X_{1}}(s)$ a critical loss
threshold corresponding to some confidence level $s,$
$\mathbb{CTE}_{X_{1}}(s)$ provides a cushion against the mean value of losses
exceeding the critical threshold $\upsilon.$

Now, assume that $X_{1}$ and $X_{2}$ are dependent with joint df $H$ and
continuous margins $F_{X_{i}},$ $i=1,2,$ respectively. Through this paper we
calls $X_{1}$ the target risk and $X_{2}$ the associated risk. In this case,
the problem becomes different and its resolution requires more than the usual
background.

Our contribution is to introduce the copula notion to provide more flexibility
to the CTE of risk of rv’s in terms of loss and dependence structure. For
comprehensive details on copulas one may consult the textbook of Nelsen
(2006).

According to Sklar’s Theorem Sklar (1959), there exists a unique copula
$C:\left[0,1\right]^{d}\rightarrow\left[0,1\right]$ such that

$H\left(x_{1},x_{2}\right)=C\left(F_{1}\left(x_{1}\right),F_{2}\left(x_{2}\right)\right).$
(2.4)

The formula of CTE focuses only on the average of loss. For this you should
think of an other formula to be more inclusive, this formula must take in
consideration the dependence structure and the behavior of margin tails. These
two aspects have an important influence when quantifying risks. On the other
hand if the correlation factor is neglected, the calculation of the CTE
follows from formula (2.3), which only focuses on the target risk.

Now by considering the correlation between the target and the associated
risks, we define a new notion of CTE called Copula Conditional Tail
Expectation (CCTE) given in (1.2), this notion led to give a risk measurement
focused in the target risk and the link between target and associated risk.

Let’s denote the survival functions by
$\overline{F}_{i}(x_{i})=\mathbb{P}(X_{i}>x_{i}),$ $i=1,2,$ and the joint
survival function by
$\overline{H}(x_{1},x_{2})=\mathbb{P}(X_{1}>x_{1},X_{2}>x_{2}).$ The function
$\overline{C}$ which couples $\overline{H}$ to $\overline{F}_{i},$ $i=1,2$ via
$\overline{H}(x_{1},x_{2})=\overline{C}(\overline{F}_{1}(x_{1}),\overline{F}_{2}(x_{2}))$
is called the survival copula of $\left(X_{1},X_{2}\right).$ Furthermore,
$\overline{C}$ is a copula, and

$\overline{C}(u_{1},u_{2})=u_{1}+u_{2}-1+C(1-u_{1},1-u_{2}),$ (2.5)

where $C$ is the (ordinary) copula of $X_{1}$ and $X_{2}.$ For more details on
the survival copula function see, Section 2.6 in Nelsen (2006).

The CCTE of the target risk $X_{1}$ with respect to the associated risk
$X_{2}$ is given in the following proposition.

######  Proposition 2.1.

Let $\left(X_{1},X_{2}\right)$ a bivariate rv with joint df represented by the
copula $C.$ Assume that $X_{1}$ have a finite mean and df $F_{X_{1}}.$ Then
for all $s$ and $t$ in $\left(0,1\right)$ the copula conditional tail expected
of $X_{1}$ with respect to the bivariate thresholds $(s,t)$ is given by

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)\left(1-C_{u}(u,t)\right)du}{\overline{C}\left(1-s,1-t\right)},$
(2.6)

where $F_{X_{1}}^{-1}$ is the quantile function of $F_{X_{1}}$and
$C_{u}(u,v):=\partial C(u,v)/\partial u.$

This notion does not depend on the df of the associated risk, but it depend
only by the copula function and the df of target risk.

By definition of PQD risks we have that $C(u,v)>uv,$ then it easy to check
that

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)\leq\frac{\mathbb{CTE}_{X_{1}}\left(s\right)}{\left(1-t\right)}\text{
for }s,t<1,$

next, in Section 4, we will proved that the risk when we consider the
correlation between PQD risks is greater than in the case of a single one.
That means, for all $s\leq t$ and $s,$ $t$ in $\left(0,1\right)$ then

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)\geq\mathbb{CTE}_{X_{1}}\left(s\right).$
(2.7)

Notice that in the NQD rv’s we have the reverse inequality of (2.7) and the
CCTE coincide with CTE measures in the non-dependence case, i.e. the copula
$C=C^{\bot}.$

##  3\. CCTE and tail dependence

The concept of tail dependence is an asymptotic measure of the dependence
between two random variables in the tail of their joint distribution function.
Specifically, tail dependence is the probability that a random variable
$X_{1}$ and $X_{2}$ takes a values in the extreme tail of its distributions
simultaneously, for example we consider $X_{1}$ and $X_{2}$ which measure
bankruptcy for two companies and both companies simultaneously go bankrupt.

We describes the joint upper tail dependence of the random variables $X_{1}$
and $X_{2}:$

$\lim_{\begin{subarray}{c}t\rightarrow 1^{-}\\\ s\rightarrow
1^{-}\end{subarray}}\mathbb{P}\left(\left.X_{1}>F_{X_{1}}^{-1}\left(s\right)\right|X_{2}>F_{X_{2}}^{-1}\left(t\right)\right)$

However, it can be seen as a good indicator of systemic risk (for $s=t$). If
we considering the tail dependence as a dependence measure in the extreme
tails of the joint distribution, it is possible for two rv’s to be dependent,
but for there to be no dependence in the tail of the distributions, this is
the case described for example by the Gaussian copula, hyperbolic copula or
Farlie-Gumbel-Morgenstern copula (tail dependence is zero). Furthermore, the
Clayton copula puts the entire tail dependence in the lower tail unlike Gumbel
copula in the upper tail and the Student copula behave identically in the
lower as in the upper tail. However, it is not suitable to model extreme
negative outcomes similarly as with extreme positive outcomes.

######  Remark 3.1.

Negative outcomes can be treated in the same way that the extremes positive
outcomes by replacing their copula by the survival copula.

The tail dependence can be also expressed through copula

$\lambda_{U}=\lim_{u\rightarrow 1^{-}}\frac{1-2u+C\left(u,u\right)}{1-u}\text{
and }\lambda_{L}=\lim_{u\rightarrow 0^{+}}\frac{C\left(u,u\right)}{u}.$

Now, let’s denote by

$\tilde{\lambda}_{U}\left(u,v\right):=\frac{1-u-v+C\left(u,v\right)}{1-v}\text{
and }\tilde{\lambda}_{L}\left(u,v\right):=\frac{C\left(u,v\right)}{v}.$

Note that $\lim_{u,v\rightarrow
1^{-}}\tilde{\lambda}_{U}\left(u,v\right)=\lambda_{U}$ and
$\lim_{u,v\rightarrow 0^{+}}\tilde{\lambda}_{L}\left(u,v\right)=\lambda_{L}.$
We can rewrite CCTE of according to $\tilde{\lambda}_{U}$ as

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)\left(1-C_{u}(u,t)\right)du}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)},$

this has no impact on the limiting value at $0$ for PQD risks. Then we have

$\lim_{\begin{subarray}{c}s\rightarrow 1^{-}\\\ t\rightarrow
1^{-}\end{subarray}}\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)=0.$

From Theorem 2.2.7 in (Nelsen, 2006, page 13) we have $0\leq C_{u}(u,t)\leq 1$
for such $u$ and $t,$ then

$\left|\mathbb{CCTE}_{X_{1}}\left(s;t\right)\right|\leq\left|\frac{\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)}\right|\leq\left|\frac{\mathbb{E}\left(X_{1}\right)}{\left(1-t\right)\tilde{\lambda}_{U}\left(s,t\right)}\right|.$

In the next section, we give an example to describe the impact of the upper
tail dependence nearly $1$ and the lower tail dependence near $0$ in CCTE, and
we discuss the relationship between the properties of the dependence of copula
model with upper and lower tail dependence and how to derive the degree of
risk in each case.

##  4\. Illustration examples

###  4.1. CCTE via Farlie-Gumbel-Morgenstern Copulas

One of the most important parametric family of copulas is the Farlie-Gumbel-
Morgenstern (FGM) family defined as

$C_{\theta}^{FGM}(u,v)=uv+\theta uv(1-u)(1-v),\ \ \ u,v\in[0,1],$ (4.8)

where $\theta\in[-1,1].$ The family was discussed by Morgenstern (1956),
Gumbel (1958) and Farlie (1960).

The copula given in (4.8) is PQD for $\theta\in(0,1]$ and NQD for
$\theta\in[-1,0).$ In practical applications this copula has been shown to be
somewhat limited, for copula dependence parameter
$\theta\in\left[-1,1\right],$ Spearman’s correlation
$\rho\in\left[-1/3,1/3\right]$ and Kendall’s $\tau\in\left[-2/9,2/9\right],$
for more details on copulas see, for example, Nelsen (2006).

Members of the FGM family are symmetric, i.e.,
$C_{\theta}^{FGM}(u,v)=C_{\theta}^{FGM}(v,u)$ for all $(u,v)$ in
$\left[0,1\right]^{2}$ and have the lower and upper tail dependence
coefficients equal to $0.$

A pair $\left(X,Y\right)$ of rv’s is said to be exchangeable if the vectors
$\left(X,Y\right)$ and $\left(Y,X\right)$ are identically distributed. Note
that, in applications, exchangeability may not always be a realistic
assumption. For identically distributed continuous random variables,
exchangeability is equivalent to the symmetry of the FGM copula.

For practical purposes we consider a copula families with only positive
dependence. Furthermore, risk models are often designed to model positive
dependence, since in some sense it is the “dangerous” dependence: assets (or
risks) move in the same direction in periods of extreme events, see Embrechts
et al. (2003b).

Consider the bivariate loss PQD rv’s $(X_{i},Y),$ $i=1,2,3,$ having continuous
marginal df’s $F_{X_{i}}(x)$ and $F_{Y}(y)$ and joint df $H_{X_{i},Y}(x,y)$
represented by FGM copula of parameters $\theta_{i}$, respectively for
$i=1,2,3$

$H_{X_{i},Y}(x,y)=C_{\theta_{i}}^{FGM}(F_{X_{i}}\left(x\right),F_{Y}\left(y\right)).$

The marginal survival functions $\overline{F}_{X_{i}}(x),$ $i=1,2,3$ and
$\overline{F}_{Y}(y)$ are given by

$\overline{F}_{X_{i}}\left(x\right)=\left\\{\begin{tabular}[]{ll}$\left(1+x\right)^{-\alpha},$&$x\geq
0,$\\\ $1,$&$x<0,$\end{tabular}\ \ \ \right.\text{ and }\ \
\overline{F}_{Y}\left(y\right)=\left\\{\begin{tabular}[]{ll}$\left(1+y\right)^{-\alpha},$&$y\geq
0,$\\\ $1,$&$y<0.$\end{tabular}\ \ \ \ \ \ \ \right.$ (4.9)

where $\alpha>0$ called the Pareto index, the case $\alpha\in(1,2)$ means that
$X_{i}$ have a heavy-tailed distributions. So that $X_{i}$ and $Y$ have
identical Pareto df’s.

For each couple $\left(X_{i},Y\right),$ $i=1,2,3,$ we propose
$\theta_{1}=0.01,$ $\theta_{2}=0.5$ and $\theta_{3}=1,$ respectively. The
choice of parameters $\theta_{i},i=1,2,3$ correspond respectively to the weak,
medium and the high dependence.

In this example, the target risks are $X_{i}$ and the associated risk is $Y.$
The $\mathbb{CTE}$’s and the VaR’s of $X_{i}$ are the same and are given
respectively by

$\mathbb{CTE}_{X_{i}}\left(s\right)=\frac{\alpha\left(1-s\right)^{-1/\alpha}}{\alpha-1}$
(4.10)

and

$VaR_{X_{i}}\left(s\right)=(1-s)^{-1/\alpha},$ (4.11)

for $i=1,2,3.$

We have that

$\overline{C}\left(1-s,1-t\right)=(1-s)(1-t)\left(st\theta_{i}+1\right).$
(4.12)

Now, we calculate

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{s}^{1}(1-u)^{-1/\alpha}\left(1-t\right)\left(2tu\theta_{i}-t\theta_{i}+1\right)du$

by substitution (4.12) we get

$\mathbb{CCTE}_{X_{i}}\left(s;t\right)=\frac{\alpha\left(2\alpha+t\theta_{i}-2st\theta_{i}+2st\alpha\theta_{i}-1\right)}{\left(2\alpha^{2}-3\alpha+1\right)\left(st\theta_{i}+1\right)}\left(1-s\right)^{-1/\alpha}.$
(4.13)

We have in Table 4.1 and Figures 4.1 the comparison of the riskiness of
$X_{1},$ $X_{2}$ and $X_{3}.$ Recall that, the $\mathbb{CTE}$’s risk measure
of $X_{i}$ at level $s$ are the same in all cases. Note that $\mathbb{CCTE}$
coincide with $\mathbb{CTE}$ in the independence case $(\theta_{1}=0).$ The
$\mathbb{CCTE}$ of the loss $X_{3}$ is riskier than $X_{2}$ and $X_{1}$ but
not very significant, in the 6th column of Table 4.1, the relative difference
between $64.7946$ and $64.633$ is only about $0.025\%$. This is due to that
FGM copula does not take into account the dependence in upper and lower tail
$(\lambda_{L}=\lambda_{U}=0).$ In this case we can not clearly confirm which
is the risk the more dangerous.

${\small s}$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.6415}$ | ${\small 5.5013}$ | ${\small 6.9144}$ | ${\small 9.8192}$ | ${\small 21.5443}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.9247}$ | ${\small 16.5039}$ | ${\small 20.7433}$ | ${\small 29.4577}$ | ${\small 64.6330}$
${\small t}$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=0.01}$
${\small 0.9000}$ | ${\small 13.9309}$ | ${\small 13.9311}$ | ${\small 13.9312}$ | ${\small 13.9314}$ | ${\small 13.9316}$
${\small 0.9225}$ | ${\small 16.5096}$ | ${\small 16.5097}$ | ${\small 16.5099}$ | ${\small 16.5100}$ | ${\small 16.5101}$
${\small 0.9450}$ | ${\small 20.7484}$ | ${\small 20.7485}$ | ${\small 20.7487}$ | ${\small 20.7488}$ | ${\small 20.7489}$
${\small 0.9675}$ | ${\small 29.4619}$ | ${\small 29.4620}$ | ${\small 29.4621}$ | ${\small 29.4623}$ | ${\small 29.4624}$
${\small 0.9900}$ | ${\small 64.6359}$ | ${\small 64.6359}$ | ${\small 64.6360}$ | ${\small 64.6361}$ | ${\small 64.6362}$
${\small t}$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ \ {\small\theta=0.5}$
${\small 0.9000}$ | ${\small 14.1477}$ | ${\small 14.1517}$ | ${\small 14.1555}$ | ${\small 14.1594}$ | ${\small 14.1631}$
${\small 0.9225}$ | ${\small 16.7072}$ | ${\small 16.7108}$ | ${\small 16.7143}$ | ${\small 16.7178}$ | ${\small 16.7212}$
${\small 0.9450}$ | ${\small 20.9234}$ | ${\small 20.9266}$ | ${\small 20.9297}$ | ${\small 20.9327}$ | ${\small 20.9357}$
${\small 0.9675}$ | ${\small 29.6077}$ | ${\small 29.6103}$ | ${\small 29.6129}$ | ${\small 29.6154}$ | ${\small 29.6179}$
${\small 0.9900}$ | ${\small 64.7336}$ | ${\small 64.7353}$ | ${\small 64.7370}$ | ${\small 64.7387}$ | ${\small 64.7404}$
${\small t}$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=1}$
${\small 0.9000}$ | ${\small 14.2709}$ | ${\small 14.2756}$ | ${\small 14.2803}$ | ${\small 14.2848}$ | ${\small 14.2892}$
${\small 0.9225}$ | ${\small 16.8183}$ | ${\small 16.8226}$ | ${\small 16.8267}$ | ${\small 16.8308}$ | ${\small 16.8348}$
${\small 0.9450}$ | ${\small 21.0208}$ | ${\small 21.0245}$ | ${\small 21.0281}$ | ${\small 21.0316}$ | ${\small 21.0351}$
${\small 0.9675}$ | ${\small 29.6880}$ | ${\small 29.6910}$ | ${\small 29.6940}$ | ${\small 29.6969}$ | ${\small 29.6997}$
${\small 0.9900}$ | ${\small 64.7868}$ | ${\small 64.7888}$ | ${\small 64.7908}$ | ${\small 64.7927}$ | ${\small 64.7946}$
Table 4.1. Risk measures of dependent pareto(1.5) rv’s with FGM copula. Figure
4.1. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD pareto
(1.5) rv’s with FGM copula and $0.9\leq s=t\leq 0.99$

###  4.2. CCTE via Archimedean Copulas

A bivariate copula is said to be Archimedean (see, Genest and MacKay, 1986) if
it can be expressed by

$C(u_{1},u_{2})=\psi^{[-1]}\left(\psi(u_{1})+\psi(u_{2})\right),$

where $\psi,$ called the generator of $C,$ is a continuous strictly decreasing
convex function from $\left[0,1\right]$ to $[0,\infty]$ such that $\psi(1)=0$
with $\psi^{[-1]}$ denotes the pseudo-inverse of $\psi,$ that is

$\psi^{[-1]}\left(t\right)=\left\\{\begin{tabular}[]{lll}$\psi^{-1}\left(t\right),$&for&$t\in\left[0,\psi\left(0\right)\right],$\\\
$0,$&for&$t\geq\psi\left(0\right).$\end{tabular}\ \right.$

When $\psi(0)=\infty,$ the generator $\psi$ and $C$ are said to be strict and
therefore $\psi^{[-1]}=\psi^{-1}.$ All notions of positive dependence that
appeared in the literature, including the weakest one of PQD as defined by
Lehmann (1966), require the generator to be strict.

Archimedean copulas are widely used in applications due to their simple form,
a variety of dependence structures and other “nice” properties. For example,
in the Actuarial field: the idea arose indirectly in Clayton (1978) and was
developed in Oakes (1982), Cook and Johnson (1981). A survey of Actuarial
applications is in Frees and Valdez (998).

For an Archimedean copula, the Kendall’s tau can be evaluated directly from
the generator of the copula, as shown in Genest and MacKay (1986)

$\tau=4{\displaystyle\int_{0}^{1}}\frac{\psi\left(u\right)}{\psi^{\prime}\left(u\right)}du+1.$
(4.14)

where $\psi^{\prime}\left(u\right)$ exists a.e., since the generator is
convex. This is another “nice” feature of Archimedean copulas. As for tail
dependency, as shown in (Joe, 1997, page 105) the coefficient of upper tail
dependency is

$\lambda_{U}=2-2\lim_{s\rightarrow
0^{+}}\frac{\psi^{-1\prime}\left(2s\right)}{\psi^{-1\prime}\left(s\right)}$

and the coefficient of lower tail dependency is

$\lambda_{L}=2\lim_{s\rightarrow+\infty}\frac{\psi^{-1\prime}\left(2s\right)}{\psi^{-1\prime}\left(s\right)}.$

A collection of twenty-two one-parameter families of Archimedean copulas can
be found in Table 4.1 of Nelsen (2006).

Notice that in the case of Archimedean copula the copula conditional tail
expectation has not an explicit formula, so we give by the following
Proposition the expression of CCTE in terms of generator.

######  Proposition 4.1.

Let $C$ be an Archimedean copula absolutely continuous with generator $\psi,$
the CCTE of the target risk in terms of generator with respect to the
bivariate thresholds $(s,t),$ $0<s,t<1,$ is given by

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\left(1-s\right)\mathbb{CTE}_{X_{1}}\left(s\right)-\int_{s}^{1}\frac{\psi^{\prime}(u)F_{X_{1}}^{-1}\left(u\right)}{\psi^{\prime}\left(C\left(u,t\right)\right)}du\right).$
(4.15)

Note that in practice we can easily fit copula-based models with the maximum
likelihood method or with estimate the dependence parameter by the
relationship between Kendall’s tau of the data and the generator of the
Archimedean copula given in (4.14) under the specified copula model.

In the following section we give same examples to explain how to calculate and
compare the CCTE with other risk measure such VaR and CTE.

####  4.2.1. CCTE via Gumbel Copula

The Gumbel family has been introduced by Gumbel (1960). Since it has been
discussed in Hougaard (1986), it is also known as the Gumbel-Hougaard family.
The Gumbel copula is an asymmetric Archimedean copula. This copula is given by

$C_{\theta}^{G}\left(u,v\right)=\exp\left\\{-\left[\left(-\ln
u\right)^{\theta}+\left(-\ln v\right)^{\theta}\right]^{1/\theta}\right\\},$

its generator is

$\psi_{\theta}\left(t\right)=\left(-\ln t\right)^{\theta}.$

The dependence parameter is restricted to the interval $[1,\infty).$ It
follows that the Gumbel family can represent independence and “positive”
dependence only, since the lower and upper bound for its parameter correspond
to the product copula and the upper Fréchet bound. The Gumbel copula families
is often used for modeling heavy dependencies in right tail. It exhibits
strong upper tail dependence $\lambda_{U}=2-2^{1/\theta}$ and relatively weak
lower tail dependence $\lambda_{L}=0.$ If outcomes are known to be strongly
correlated at high values but less correlated at low values, then the Gumbel
copula will be an appropriate choice.

We give the CCTE of rv’s $X_{i},$ $i=1,2,3$ in terms of Gumbel copula by

$\displaystyle\mathbb{CCTE}_{X_{i}}\left(s;t\right)$
$\displaystyle=\frac{1}{\overline{C}_{\theta_{i}}^{G}\left(1-s,1-t\right)}\left(\frac{\alpha\left(1-s\right)^{1-1/\alpha}}{\alpha-1}\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\left.-\int_{s}^{1}u^{-1}\left(1-u\right)^{-1/\alpha}\left(-\ln
u\right)^{\theta_{i}-1}C_{\theta_{i}}^{G}\left(u,t\right)\left(-\ln\left(C_{\theta_{i}}^{G}\left(u,t\right)\right)\right)^{1-\theta_{i}}du\right),$
(4.16)

where
$\overline{C}_{\theta_{i}}^{G}\left(s,t\right)=s+t-1+C_{\theta_{i}}^{G}(1-s,1-t).$

The CTE’s and VaR’s of $X_{i}$ is the same and it’s given respectively by
(4.10) and (4.11), for $i=1,2,3.$

By the relationship between Kendall’s tau $\tau$ and the Gumbel copula
parameter $\theta_{i}$ given by:

$\tau=\left(\theta_{i}-1\right)/\theta_{i},$

we select the values of $\theta_{i}$ corresponding respectively to a weak, a
moderate and a strong positive association witch summarized in Table 4.2.

$\lambda_{U}$ | $\theta_{i}$ | $\tau$
---|---|---
$0.013$ | $1.01$ | $0.009$
$0.585$ | $2$ | $0.500$
$0.928$ | $10$ | $0.900$
Table 4.2. Upper tail, Kendall’s tau and Gumbel copula parameters used in
calculate of risk measures.

Table 4.3 and Figure 4.2 shows that the loss $X_{1}$ is considerably riskier
than $X_{2}$ and $X_{3},$ in the 6th column of Table 4.3, the relative
difference between $112.1868$ and $69.6017$ is about $61.184\%.$

${\small s}$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.641}$ | ${\small 5.501}$ | ${\small 6.914}$ | ${\small 9.819}$ | ${\small 21.544}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.924}$ | ${\small 16.503}$ | ${\small 20.743}$ | ${\small 29.457}$ | ${\small 64.633}$
${\small t}$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=1.01}$
${\small 0.9000}$ | ${\small 15.937}$ | ${\small 16.485}$ | ${\small 17.410}$ | ${\small 19.365}$ | ${\small 25.007}$
${\small 0.9225}$ | ${\small 18.879}$ | ${\small 19.528}$ | ${\small 20.625}$ | ${\small 22.948}$ | ${\small 33.690}$
${\small 0.9450}$ | ${\small 23.699}$ | ${\small 24.507}$ | ${\small 25.873}$ | ${\small 28.760}$ | ${\small 40.588}$
${\small 0.9675}$ | ${\small 33.556}$ | ${\small 34.667}$ | ${\small 36.534}$ | ${\small 40.454}$ | ${\small 56.275}$
${\small 0.9900}$ | ${\small 72.992}$ | ${\small 75.133}$ | ${\small 78.645}$ | ${\small 85.726}$ | ${\small 112.18}$
${\small t}$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ {\small\ \theta=2}$
${\small 0.9000}$ | ${\small 18.158}$ | ${\small 19.769}$ | ${\small 22.691}$ | ${\small 28.950}$ | ${\small 52.929}$
${\small 0.9225}$ | ${\small 20.209}$ | ${\small 21.653}$ | ${\small 24.338}$ | ${\small 30.607}$ | ${\small 53.742}$
${\small 0.9450}$ | ${\small 23.842}$ | ${\small 25.059}$ | ${\small 27.383}$ | ${\small 33.070}$ | ${\small 55.276}$
${\small 0.9675}$ | ${\small 31.849}$ | ${\small 32.766}$ | ${\small 34.543}$ | ${\small 39.128}$ | ${\small 59.207}$
${\small 0.9900}$ | ${\small 66.087}$ | ${\small 66.606}$ | ${\small 67.583}$ | ${\small 70.074}$ | ${\small 86.385}$
${\small t}$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=10}$
${\small 0.9000}$ | ${\small 13.765}$ | ${\small 16.694}$ | ${\small 23.338}$ | ${\small 39.483}$ | ${\small 128.31}$
${\small 0.9225}$ | ${\small 15.612}$ | ${\small 16.626}$ | ${\small 21.902}$ | ${\small 36.924}$ | ${\small 120.00}$
${\small 0.9450}$ | ${\small 19.378}$ | ${\small 19.446}$ | ${\small 20.821}$ | ${\small 32.807}$ | ${\small 106.54}$
${\small 0.9675}$ | ${\small 29.457}$ | ${\small 29.458}$ | ${\small 29.480}$ | ${\small 31.692}$ | ${\small 95.737}$
${\small 0.9900}$ | ${\small 64.633}$ | ${\small 65.034}$ | ${\small 66.412}$ | ${\small 67.753}$ | ${\small 69.601}$
Table 4.3. Risk measures of PQD pareto (1.5) rv’s with Gumbel copula. Figure
4.2. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD pareto
(1.5) rv’s with Gumbel copula and $0.9\leq s=t\leq 0.99.$

By definition of our risk measurement, the risks concern the study is
necessarily comonotonic, then to have a good decision we must select a copula
model with upper tail dependence, we show in next example that the dependence
models with no upper tails dependence do not helps us to take a decision.

####  4.2.2. CCTE via Clayton Copula

In the following example, we consider the bivariate Clayton copula which is a
member of the class of Archimedean copula, with the dependence parameter
$\theta$ in $\left.\left[-1,\infty\right)\right\backslash\left\\{0\right\\}$.

The Clayton family was first proposed by Clayton (1978) and studied by Oakes,
(1982; 1986), Cox and Oakes, (1981; 1986). The Clayton copula has been used to
study correlated risks, it has the form

$C_{\theta}^{C}(u,v):=\left[\max\left(u^{-\theta}+v^{-\theta}-1,0\right)\right]^{-1/\theta}.$
(4.17)

For $\theta>0$ the copulas are strict and the copula expression simplifies to

$C_{\theta}^{C}(u,v)=\left(u^{-\theta}+v^{-\theta}-1\right)^{-1/\theta}.$
(4.18)

Asymmetric tail dependence is prevalent if the probability of joint extreme
negative realizations differs from that of joint extreme positive
realizations. it can be seen that the Clayton copula assigns a higher
probability to joint extreme negative events than to joint extreme positive
events. The Clayton copula is said to display lower tail dependence
$\lambda_{L}=2^{-1/\theta},$ while it displays zero upper tail dependence
$\lambda_{U}=0,$ for $\theta\geq 0.$ The converse can be said about the Gumbel
copula (displaying upper but zero lower tail dependence). The margins become
independent as $\theta$ approaches to zero, while for
$\theta\rightarrow\infty,$ the Clayton copula arrives at the comonotonicity
copula. For $\theta=-1$ we obtain the Fréchet-Hoeffding lower bound and the
copula attains the Fréchet upper bound as $\theta$ approaches to infinity.

Clayton copula is the best suited for applications in which two outcomes are
likely to experience low values together, since the dependence is strong in
the lower tail and weak in the upper tail.

We take the same example as in the Subsection 4.1, we may now represents the
joint df’s $H_{i},$ $i=1,2,3,$ respectively by the Clayton copulas
$C_{\theta_{i}}^{C}$ given in (4.18) to have an idea about the effects of
lower tail dependence on our risk measurement.

The relationship between Kendall’s tau $\tau$ and the Clayton copula is given
by

$\tau=\theta_{i}/\left(\theta_{i}+2\right),$ (4.19)

we select a different dependents parameters corresponding to several levels of
positive dependency summarized in Table 4.4 for a weak, a moderate and a
strong positive association, to calculate and compare the CCTE’s of
$X_{i},i=1,2,3.$

$\lambda_{L}$ | $\theta_{i}$ | $\tau$
---|---|---
$0.250$ | $0.5$ | $0.200$
$0.707$ | $2$ | $0.500$
$0.943$ | $12$ | $0.857$
Table 4.4. Lower tail, Kendall’s tau and Clayton copula parameters used in
calculate of risk measures.

The CCTE of the rv’s $X_{i}$ with respect to the bivariate thresholds $(s,t)$
is given by

$\mathbb{CCTE}_{X_{i}}\left(s;t\right)=\frac{1}{\overline{C}_{\theta_{i}}^{C}\left(1-s,1-t\right)}\left(\frac{\alpha\left(1-s\right)^{-1/\alpha+1}}{\left(\alpha-1\right)}-{\displaystyle\int_{s}^{1}}\frac{\left(t^{-\theta_{i}}+u^{-\theta_{i}}-1\right)^{-1-1/\theta_{i}}}{\left(1-u\right)^{1/\alpha}u^{\theta_{i}+1}}du\right).$
(4.20)

The differences as reported in Table 4.5 and Figure 4.3 do not look very
significant, in the 6th column of Table 4.5, the relative difference between
$66.3802$ and $64.6330$ is only about $1.027\%.$ The differences is not found
also when $t$ is small compared to $s,$
$\mathbb{CCTE}_{X_{1}}\left(0.99;0.01\right)=64.6332$ and
$\mathbb{CCTE}_{X_{3}}\left(0.99;0.01\right)=64.6329$ the difference is about
$1\%.$

$s$ | ${\small 0.9000}$ | ${\small 0.9225}$ | ${\small 0.9450}$ | ${\small 0.9675}$ | ${\small 0.9900}$
---|---|---|---|---|---
${\small VaR}_{X_{i}}\left(s\right)$ | ${\small 4.641}$ | ${\small 5.501}$ | ${\small 6.914}$ | ${\small 9.819}$ | ${\small 21.544}$
$\mathbb{CTE}_{X_{i}}\left(s\right)$ | ${\small 13.924}$ | ${\small 16.503}$ | ${\small 20.743}$ | ${\small 29.457}$ | ${\small 64.633}$
$t$ | $\mathbb{CCTE}_{X_{1}}\left(s,t\right),\ \ {\small\theta=0.5}$
${\small 0.9000}$ | ${\small 14.088}$ | ${\small 14.092}$ | ${\small 14.096}$ | ${\small 14.101}$ | ${\small 14.105}$
${\small 0.9225}$ | ${\small 16.652}$ | ${\small 16.656}$ | ${\small 16.660}$ | ${\small 16.664}$ | ${\small 16.667}$
${\small 0.9450}$ | ${\small 20.874}$ | ${\small 20.878}$ | ${\small 20.881}$ | ${\small 20.884}$ | ${\small 20.888}$
${\small 0.9675}$ | ${\small 29.566}$ | ${\small 29.569}$ | ${\small 29.572}$ | ${\small 29.575}$ | ${\small 29.577}$
${\small 0.9900}$ | ${\small 64.706}$ | ${\small 64.707}$ | ${\small 64.709}$ | ${\small 64.711}$ | ${\small 64.713}$
$t$ | $\mathbb{CCTE}_{X_{2}}\left(s,t\right),\ \ {\small\theta=2}$
${\small 0.9000}$ | ${\small 14.500}$ | ${\small 14.536}$ | ${\small 14.572}$ | ${\small 14.610}$ | ${\small 14.648}$
${\small 0.9225}$ | ${\small 17.023}$ | ${\small 17.056}$ | ${\small 17.089}$ | ${\small 17.123}$ | ${\small 17.159}$
${\small 0.9450}$ | ${\small 21.199}$ | ${\small 21.227}$ | ${\small 21.257}$ | ${\small 21.288}$ | ${\small 21.319}$
${\small 0.9675}$ | ${\small 29.833}$ | ${\small 29.857}$ | ${\small 29.882}$ | ${\small 29.907}$ | ${\small 29.934}$
${\small 0.9900}$ | ${\small 64.882}$ | ${\small 64.898}$ | ${\small 64.915}$ | ${\small 64.932}$ | ${\small 64.950}$
$t$ | $\mathbb{CCTE}_{X_{3}}\left(s,t\right),\ \ {\small\theta=12}$
${\small 0.9000}$ | ${\small 15.605}$ | ${\small 16.118}$ | ${\small 16.743}$ | ${\small 17.494}$ | ${\small 18.383}$
${\small 0.9225}$ | ${\small 17.913}$ | ${\small 18.366}$ | ${\small 18.930}$ | ${\small 19.618}$ | ${\small 20.447}$
${\small 0.9450}$ | ${\small 21.888}$ | ${\small 22.274}$ | ${\small 22.762}$ | ${\small 23.371}$ | ${\small 24.119}$
${\small 0.9675}$ | ${\small 30.331}$ | ${\small 30.637}$ | ${\small 31.033}$ | ${\small 31.536}$ | ${\small 32.169}$
${\small 0.9900}$ | ${\small 65.169}$ | ${\small 65.363}$ | ${\small 65.619}$ | ${\small 65.951}$ | ${\small 66.380}$
Table 4.5. Risk measures of dependent pareto (1.5) rv’s with Clayton copula.
Figure 4.3. $\mathbb{CCTE}$, $\mathbb{CTE}$ and $VaR$ risks measures of PQD
pareto (1.5) rv’s with Clayton copula and $0.9\leq s=t\leq 0.99.$

##  5\. Application

The relationship between the parameter of an Archimedean copula and Kendall’s
tau has allowed us to calculate the value of this parameter assuming a well
precise Archimedean copula e.g., Gumbel copula. Once endowed with the
parameter value, we are able to compute any joint probability between the
stock indices.

For instance we analyzed $500$ observations from four European stock indices
return series calculated by $\log\left(X_{t+1}/X_{t}\right)$ for the period
July 1991 to June 1993 (see, Figure 5.4 ), available in ”QRM and datasets
packages” of R software, it contains the daily closing prices of major
European stock indices: Germany DAX (Ibis), Switzerland SMI, France CAC and UK
FTSE. The data are sampled in business time, i.e., weekends and holidays are
omitted. Table 5.6 summaries the Kendall’s tau between the four Market Index
returns.

Figure 5.4. Scatterplots of $500$ pseudo-observations drawn from a four European stock indices returns. Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small 1}$ | ${\small 0.4052}$ | ${\small 0.4374}$ | ${\small 0.3706}$
SMI | ${\small 0.4052}$ | ${\small 1}$ | ${\small 0.3791}$ | ${\small 0.3924}$
CAC | ${\small 0.4374}$ | ${\small 0.3791}$ | ${\small 1}$ | ${\small 0.4076}$
FTSE | ${\small 0.3706}$ | ${\small 0.3924}$ | ${\small 0.4076}$ | ${\small 1}$
Table 5.6. Kendall’s tau matrix estimates from four European stock indices
returns.

By assuming that Gumbel copula represents our four dependence structures, we
obtain the fitted dependence parameters of the six bivariate joint df’s,
presented in Table 5.7.

Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small\infty}$ | ${\small 1.6815}$ | ${\small 1.7777}$ | ${\small 1.5888}$
SMI | ${\small 1.6815}$ | ${\small\infty}$ | ${\small 1.6106}$ | ${\small 1.6459}$
CAC | ${\small 1.7777}$ | ${\small 1.6106}$ | ${\small\infty}$ | ${\small 1.6880}$
FTSE | ${\small 1.5888}$ | ${\small 1.6459}$ | ${\small 1.6880}$ | ${\small\infty}$
Table 5.7. Fitted copula parameter correspoding to Kendall’s tau, Gumbel
copula.

The $\alpha$-stable distribution offers a reasonable improvement to the
alternative distributions, each stable distribution
$S_{\alpha}(\sigma;\beta;\mu)$ has the stability index $\alpha$ that can be
treated as the main parameter, when we make an investment decision, skewness
parameter $\beta$, in the range $[-1,1]$, scale parameter $\sigma$ and shift
parameter $\mu.$ In models that use financial data, it is generally assumed
that $\alpha\in(1,2].$ By using the ”fBasics” package in R software, based on
the maximum likelihood estimators to fit the parameters of a df’s of the four
Market Index returns, the results are summarized in Table 5.8.

| DAX | SMI | CAC | FTSE
---|---|---|---|---
${\small\alpha}$ | ${\small 1.6420}$ | ${\small 1.8480}$ | ${\small 1.6930}$ | ${\small 1.8740}$
${\small\beta}$ | ${\small 0.1470}$ | ${\small 0.1100}$ | ${\small 0.0380}$ | ${\small 0.9500}$
${\small\sigma}$ | ${\small 0.0046}$ | ${\small 0.0045}$ | ${\small 0.0006}$ | ${\small 0.0053}$
${\small\mu}$ | ${\small-0.0001}$ | ${\small 0.0006}$ | ${\small-0.0001}$ | ${\small-0.0005}$
Table 5.8. Maximum likelihood fit of four-parameters stable distribution to
four European stock indices retuns data.

The $\alpha$-stable distribution has Pareto-type tails, it’s like a power
function, i.e., $F$ is regularly varying (at infinity) with index
$\left(-\alpha\right),$ meaning that
$\overline{F}\left(x\right)=x^{-\alpha}L\left(x\right)$ as $x$ becomes large,
where $L>0$ is a slowly varying function, which can be interpreted as slower
than any power function (see, Resnick; 1987 and Seneta; 1976 for a technical
treatment of regular variation). By using the Equations (4.16) for the Gumbel
copula fitting, we calculate for a fixed levels $s=t=0.95$ the CCTE’s risk
measures for the all cases, the results are summarized in Table 5.9.

Variable | DAX | SMI | CAC | FTSE
---|---|---|---|---
DAX | ${\small-}$ | ${\small 21.5009}$ | ${\small 21.0786}$ | ${\small 21.9731}$
SMI | ${\small 14.2812}$ | ${\small-}$ | ${\small 14.4703}$ | ${\small 14.3737}$
CAC | ${\small 18.8362}$ | ${\small 19.4915}$ | ${\small-}$ | ${\small 19.1671}$
FTSE | ${\small 13.9075}$ | ${\small 13.7593}$ | ${\small 13.6576}$ | ${\small-}$
Table 5.9. CCTE’s Risk measures for $s=0.99$ and $t=0.99$ with Gumbel copula
(left panel) and Clayton copula (right panel).

The smallest values in Table 5.9 gives the lowest risk. So, the less risky
couples $(X,Y)$ are: (DAX, CAC), (SMI, DAX), (CAC, DAX) and (FTSE, CAC), where
$X$ is the target risk and $Y$ is the associated risk.

##  6\. Conclusion notes

This paper discussed a new risk measure called copula conditional tail
expectation. This measure aid to understanding the relationships among
multivariate assets and to help us significantly about how best to position
our investments and improve our financial risk protection.

Tables 4.3 show that the copula conditional tail expectation measure become
smaller as the dependency increase. However, CTE and VaR are neither
increasing nor decreasing as the correlation increase. Therefore, the
dependency information helps us to minimize the risk.

Acknowledgements. The author is indebted to an anonymous referee for their
careful reading and suggestions for improvements.

##  7\. Appendix

###### Proof of Proposition 2.1.

By conditional probability is easily to obtain

$\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)=\frac{\mathbb{P}\left(x\geq
X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)}{\mathbb{P}\left(X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)}\vskip
6.0pt plus 2.0pt minus 2.0pt$

On the other hand, we have

$\displaystyle\mathbb{P}\left(x\geq
X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)$
$\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s\right)-\mathbb{P}\left(F_{2}\left(X_{2}\right)\leq
t\right)+\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right),$
$\displaystyle\mathbb{P}\left(X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)\
$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(F_{1}\left(X_{1}\right)\leq
s\right)-\mathbb{P}\left(F_{2}\left(X_{2}\right)\leq
t\right)+\mathbb{P}\left(F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right)$ $\displaystyle=$ $\displaystyle
1-s-t+C\left(s,t\right)$ $\displaystyle=$
$\displaystyle\overline{C}\left(1-s,1-t\right),$
$\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s\right)=s-VaR_{X_{1}}\left(x\right)$

and

$\mathbb{P}\left(F_{1}^{-1}\left(x\right)<F_{1}\left(X_{1}\right)\leq
s,F_{2}\left(X_{2}\right)\leq t\right)=C\left(s,t\right)-C(VaR_{X_{1}}(x),t).$

Then

$\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)=1+\frac{VaR_{X_{1}}\left(x\right)-C(VaR_{X_{1}}(x),t)}{\overline{C}\left(1-s,1-t\right)}$

Then the CCTE is given by

$\displaystyle\mathbb{CCTE}_{X_{1}}\left(s,t\right)$ $\displaystyle=$
$\displaystyle\int_{X_{1}>VaR_{X_{1}}\left(s\right)}xd\mathbb{P}\left(\left.X_{1}\leq
x\right|X_{1}>VaR_{X_{1}}\left(s\right),X_{2}>VaR_{X_{2}}\left(t\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{VaR_{X_{1}}\left(s\right)}^{\infty}xd\left(VaR_{X_{1}}\left(x\right)-C(VaR_{X_{1}}(x),t)\right).$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)d\left(u-C(u,t)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du-\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)dC(u,t)\right)$

This close the proof of Proposition 2.1.

###### Proof of Proposition 4.1.

Let’s denote by

$C_{u}\left(u,v\right):=\frac{\partial C\left(u,v\right)}{\partial u}$

then by (2.6), we have

$\mathbb{CCTE}_{X_{1}}\left(s;t\right)=\frac{1}{\overline{C}\left(1-s,1-t\right)}\left(\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du-\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)C_{u}(u,t)du\right).$

So, $C$ is Archimedean copula, then

$C_{u}\left(u,v\right)=\frac{\psi^{\prime}(u)}{\psi^{\prime}\left(C\left(u,v\right)\right)},$
(7.21)

Finely, we get (4.15) by substitution of

$\int_{s}^{1}F_{X_{1}}^{-1}\left(u\right)du=\left(1-s\right)\mathbb{CTE}_{X_{1}}\left(s\right)$

and (7.21) in (2.6).

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