Communications on Stochastic Analysis
Volume 9 | Number 1
Article 8
3-1-2015
Analysing systemic risk contribution using a closed
formula for conditional value at risk through copula
Brice Hakwa
Manfred Jäger-Ambrożewicz
Barbara Rüdiger
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Recommended Citation
Hakwa, Brice; Jäger-Ambrożewicz, Manfred; and Rüdiger, Barbara (2015) "Analysing systemic risk contribution using a closed formula for conditional value at risk through copula,"Communications on Stochastic Analysis: Vol. 9 : No. 1 , Article 8.
DOI: 10.31390/cosa.9.1.08
ANALYSING SYSTEMIC RISK CONTRIBUTION USING A CLOSED FORMULA FOR CONDITIONAL VALUE AT RISK
THROUGH COPULA
BRICE HAKWA, MANFRED J ¨AGER-AMBRO ˙ZEWICZ, AND BARBARA R ¨UDIGER
Abstract. The main challenge by the analysis and the regulation of sys-temic riskis the measurement of the adverse financial effect that the bank-ruptcy of one single financial institution can cause to thefinancial system. One of the main tools that has been proposed for this purpose is the risk mea-sure ∆CoV aRof Adrian and Brunnermeier in [2]. The main contribution of this paper is to propose a general and flexible framework for the computation of ∆CoV aRin a more general stochastic setting compared to those provided so far. The formula that we propose here is based onCopula’s theory. It al-lows us to stay not only in the Gaussian but also in the non-Gaussian setting. We also discuss the properties of our formula and analyse many examples, involving in particularellipticalandArchimedeancopula, as well as con-vex combination of copulas. We also propose alternative models to those in [2].
1. Introduction
With the last crisis it became clear that the failure of certain financial institu-tions (the so called system relevant financial instituinstitu-tions) can produce an adverse impact on the whole financial system (systemic risk). The inability of standard risk-measurement tools like Value-at-Risk (V aR) to capture this kind of risk (since their focus is on an institution in isolation: micro risk management) poses a new risk-management challenge to the financial regulators and academics. This prob-lem can be summarized into two questions:
(1) How to identify System-relevant Financial Institutions ?
(2) How to quantify the marginal risk contribution of one single financial in-stitute to the system ?
As an academic response to the second question Adrian and Brunnermeier [2] proposedCoV aR method as a tool to analyse the adverse financial effect of the failure of a single financial institution on the financial system. They defined the term CoV aR as the Value-at-Risk (V aR) of the financial system conditional on the state of one given financial institution and quantified the risk contribution (i.e. how much an institution adds to the risk of the system) of a given financial
Received 2014-12-14; Communicated by P. Sundar.
2010Mathematics Subject Classification. Primary 62H20; 91B30; 91G40; 62P05; Secondary 91B82; 91G10.
Key words and phrases. Systemic risk, ∆CoV aR, copula, conditional probability, contagion effect.
131
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Communications on Stochastic Analysis Vol. 9, No. 1 (2015) 131-158
institution by the measure ∆CoV aR. This is defined as the difference between CoV aR conditional on the given financial institution being in distress and the CoV aRwhen it is not.
The implementation ofCoV aRinvolves variables characterizing a single finan-cial institution i (e.g. the loss incurred by the financial institution i denoted by Li) and the financial system s (e.g. the loss incurred by the financial system
s denoted by Ls) respectively and variables characterizing the interdependency
structure between single financial institutionsi and the financial system s. This macro-dimension ofCoV aRallows the integration of the dependence structure of i ands in risk-measurement contrary to the standard risk measures (”micro-risk measure” e.g. VaR) where only variables characterizing the financial institution
alone are considered. The CoV aR concept can be thus used by regulatory
in-stitutions as a macro-prudential tool (or as a basis for the development of other tools) for the regulation of systemic risk. Its computation represents an open problem, although some approaches have been proposed: Adrian and Brunner-meier [2] proposed an estimation method based on ”linear quantile regression”, J¨ager-Ambro˙zewicz, M. [12] developed a closed formula for the special case where the random vector(Li, Ls)is modelled by a bivariate normal distribution. In all these approaches there are some difficulties to flexibly model the stochastic be-haviours of financial institution’s specific variables and their dependence structure (interconnection), since only bivariate normal distribution is considered.
Our aim is thus to provide a more flexible framework, including also the
non-Gaussian case, for the implementation of the CoV aR concept which allows the
integration of stylized features of marginal losses as skewness, fat tails and in-terdependence properties like linear, non-linear and positive or negative tail
de-pendence. To do this, we first propose an improved definition of CoV aR which
makes it mathematically tractable (see Definition 1.5), and based on copula
the-ory we propose a general formula for CoV aR and hence also for ∆CoV aR (see
Theorem 3.2).
We first recall here the definition of the Value-at-Risk (V aR) in order to define theCoV aRas a conditionalV aR following [2].
Definition 1.1 (Value-at-Risk). Given some confidence levelα∈(0,1) theV aR of a portfolio at the confidence levelαis given by the smallest numberlsuch that the probability that the lossLexceedsl is no larger than (1−α). Formally
V aRα:= inf{l∈R:P r(L > l)≤1−α}
= inf{l∈R:P r(L≤l)≥α}.
We will employ the notation of quantile as provided in the following definition (cf. [15] Definition 2.12).
Definition 1.2 (Generalized inverse and quantile function).
a) Given an increasing function T : R→ R, thegeneralized inverse of T is defined by
b) Given a distribution function F, the generalized inverseF← is called the
quantile functionofF. Forα∈(0,1) we have
qα(F) =F←(α) := inf{x∈R:F(x)≥α}.
Note that, ifF is continuous and strictly increasing, we simply have
qα(F) =F−1(α), (1.1)
where F−1 is the (ordinary) inverse of F. Thus suppose that the distribution F
of the lossLis continuous and strictly increasing. It follows
V aRα=F−1(α). (1.2)
We note that typical values taken forαare 0.99 or 0.995.
Assumption1.3. Henceforth we consider only random variables which have strictly positive density function. Also in case we consider a bivariate joint distribution H(x, y) we assume that it has strictly positive density and its marginal distribu-tions have strictly positive densities.
Due to this assumption all considered distribution functionsF are continuous and strictly increasing.
Let us describe the loss incurred by the financial institutioniand that incurred by the financial systemsby two random variablesLiandLswith univariate dis-tribution functions Fi andFs respectively. At least since the financial crisis it is
clear that the dependency between the financial system and the financial institu-tionimust be analysed more seriously. A step towards such an analysis is done by assuming that the random variables Li andLs are stochastically dependent and
that their joint behaviour is determined by a bivariate joint distribution function. Following this, Adrian and Brunnermeier [2] define CoV aRs|C(L
i)
α as the
Value-at-Risk at the level αof an financial system s conditional on some eventC(Li)
depending on the lossLi incurred by financial institutioni. ThusCoV aRs|C(L
i) α
can be implicitly defined as the α-quantile of the conditional probability of the financial system’s loss.
P r ( Ls≤CoV aRs|C(L i) α |C ( Li))=α. (1.3)
In their work Adrian and Brunnermeier considered the case where the condition C(Li)refers to the lossLi incurred by the financial institutionibeing exactly at its Value-at-Risk and at its mean. We generalize this approach by allowingLi to
assumes any valuel∈R. We have in this case in the context of (1.3) the following expression P r ( Ls≤CoV aRsα|Li=l|Li=l ) =α. (1.4)
Remark 1.4. The so defined CoV aRsα|Li=l is consistent with respect to the
(sto-chastic) independence, in the sense CoV aRαs|Li=l = V aRsα when Li and Ls are
Following [5] (Definition 4.7) we can define in the context of Assumption 1.3, a conditional probability of the formP r(Ls≤h|Li=l)for fixedhas a function of l as follows
P r(Ls≤h|Li ≤y)=
∫ y
−∞
P r(Ls≤h|Li =l)fi(l)dl ∀y∈R. (1.5)
Consider the function Rl(h) :=P r
(
Ls≤h|Li=l).As Rl(h) is strictly
increas-ing, it follows that its is invertible. Based on this we provide an alternative defini-tion, forCoV aRsα|Li=lwhich is more tractable from a mathematical point of view
than that proposed in [2].
Definition 1.5. Assume thatLi andLssatisfy Assumption 1.3. then for a given
α∈(0,1) and for a fixedl,CoV aRsα|Li=l is defined as:
CoV aRsα|Li=l:= inf{h∈R: P r(Ls> h|Li=l)≤1−α} = inf{h∈R: P r(Ls≤h|Li=l)≥α} =R−l 1(α).
For a fixedαwe define the function
CoV aRαs|i(l) :=CoV aRαs|Li=l, ∀l∈ R. (1.6) ∆CoV aRsα|iis defined by Adrian and Brunnermeier [2] as the difference between
CoV aRs|C(L
i)
α condition on the institutioni being under distress (i.e. C
( Li)= { Li=V aRi α }
) and theCoV aRs|C(L
i)
α condition on the financial institution having
mean loss (i.e. C(Li)={Li=E[Li]}).
Definition 1.6.
∆CoV aRsα|i:=CoV aRs|Li=V aRiα
α −CoV aR
s|Li=E(Li)
α . (1.7)
Using (1.6) we can rewrite (1.7) as follows.
Definition 1.7. Forl1, l2∈Rsuch thatl1≥l2, we define
∆CoV aRαs|i(l1, l2) :=CoV aRαs|i(l1)−CoV aRsα|i(l2) (1.8)
2. A Brief Introduction to Copulas
In this section we introduce the notion of copula and give some basic definitions and important properties needed later. For detailed analysis of copulas, we refer the reader to e.g. [13], [15], [16] or [17] and the references therein.
In order to introduce the concept of copula, we recall some important remarks upon which it is built.
Remark 2.1 (cf. [15] Proposition 5.2). Assume F is a distribution function such that its inverse functionF−1 is well defined.
(1) Quantile transformation. Let U be a standard uniform distributed random variable (i.e. U ∼U(0, 1)), then P r(F−1(U)≤x)=F(x).
(2) Probability transformation. Let X be a random variable with dis-tribution functionF, thenF(X) has a uniform standard distribution i.e. F(X)∼U(0,1).
Definition 2.2(2-dimensional copula (cf. [16] Definition 2.2.2)). A2-dimensional copulais a (distribution) functionC: [0,1]2→[0,1] with the following satisfying:
• Boundary conditions:
1) For everyu∈[0,1] :C(0, u) =C(u,0) = 0. 2) For everyu∈[0,1] :C(1, u) =u and C(u,1) =u.
• Monotonicity condition:
3) For every (u1, u2),(v1, v2)∈ [0,1]×[0,1]with u1≤u2 and v1 ≤v2
we have
C(u2, v2)−C(u2, v1)−C(u1, v2) +C(u1, v1)≥0.
Conditions 1) and 3) imply that the so defined 2-copula C is a bivariate joint distribution function (cf. [16] Definition 2.3.2) and Condition 2) implies that the copula C has standard uniform margins. From this it follows that a bivariate copulas is increasing with respect to each of its arguments i.e. for every uand v ∈[0,1] the maps v7→C(u, v) andu7→C(u, v) are each increasing.
Theorem 2.3 (Sklar’s theorem, cf. [16] Theorem 2.3.3). Let H be a joint distri-bution function with marginal distridistri-bution functionsF and G, then there exists a copulaC such that for all x, y∈R∪ {−∞} ∪ {+∞}
H(x, y) =C[F(x), G(y)]. (2.1)
If F and G have density, thenC is unique. Conversely, if C is a copula and F
and Gare distribution functions, then the function H defined by (2.1)is a joint distribution function with marginsF andG.
This theorem is very important because it asserts that, using copula function, it is possible to represent each bivariate distribution function as a composition of two univariate distribution functions and a given copula function. Thus, we can
use the copula to extract the dependence structure among the componentsX and
Y of the vector (X, Y), independently of the marginal distributionF andG. This allows us to model the dependence structure and marginals separately.
Remark 2.4. Assume (X, Y) is a bivariate random variables with copula C and joint distributionH satisfying Assumption 1.3, with marginals distribution func-tionF andG, then the transformed random variablesU =F(X) andV =F(Y) have standard uniform distribution andC(U, V) is the joint distribution of (U, V). In fact
C(u, v) =C(P r(U ≤u), P r(V ≤v)).
Corollary 2.5(cf. e.g. [16] Corollary 2.3.7). LetHdenote a bivariate distribution function with margins F and G satisfying Assumption 1.3, then there exist an unique copulaC such that for all (u, v) ∈[0,1]2 it holds:
Remark 2.6. As long as Assumption 1.3 is satisfied any copulaC assumed here is associated to a joint distribution functions that have strictly positive density, i.e. there exist a unique strictly positive function c : [0,1]2 →[0,∞), called ”copula density”, such that
C(u, v) = ∫ v 0 ∫ u 0 c(s, t)dsdt ∀u, v∈[0,1]. (2.2)
3. Analysing Systemic Risk Contribution Using ∆CoV aR: A Copula Approach
The aim of this section is to improve the quality of systemic risk analysis by providing a general and flexible framework for the calculation and the theoretical analysis of the termCoV aRsα|Li=lfor a large class of stochastic setting. To do this,
we will use the relation between conditional probability and copula to rewrite the implicit definition of CoV aRsα|Li=l in terms of copula and obtain so a general
formula. Then using our formula we will highlight some important properties of CoV aRsα|Li=land ∆CoV aRs|i.
We provide in the following theorem a general formula for the computing of CoV aRsα|Li=l. We do this by assuming that the joint distributionH ofLi andLs
satisfy Assumption 1.3 and we denote byfi andfsthe density function ofLiand
Ls respectively.
LetC be the copula associated toH, i.e.
H(x, y) =C(Fi(x), Fs(y)). (3.1)
Due to Assumption 1.3 we have that the CopulasChave strictly positive density functionc. We define the function
g(v, u) :=∂C(u, v)
∂u .
Remark 3.1.Under Assumption 1.3 the functiong(v, u) is well defined and for each fixedu∈[0,1] invertible with respect to the parameterv. In fact, by differentiating (2.2) with respect touand applying the Fubini’s theorem we obtain
g(v, u) =∂C(u, v)
∂u =
∫ v
0
c(u, t)dt. (3.2)
Since the copula densityc is strictly positive, it follows that for a fixed u∈[0,1] the functiong(v, u) is strictly increasing and thus invertible with respect to v.
Theorem 3.2. Under Assumption 1.3 for all l ∈ R and a given α ∈ (0,1),
CoV aRsα|Li=l is given by
CoV aRαs|Li=l=Fs−1(g−1(α, Fi(l))
)
. (3.3)
Proof. Recall that the implicit definition ofCoV aRαs|Li=lis given by:
P r ( Ls≤CoV aRαs|Li=l|Li=l ) =α ⇔P r ( Fs(Ls)≤Fs ( CoV aRsα|Li=l ) |Fi ( Li)=Fi(l) ) =α.
We defineV :=Fs(Ls), U :=Fi ( Li), v:= F s ( CoV aRsα|Li=l ) and u:=Fi(l) such that P r ( Fs(Ls)≤Fs ( CoV aRsα|Li=l ) |Fi ( Li)=Fi(l) ) =P r(V ≤v|U =u).
Due to Assumption 1.3 it follows from Remark 2.1 that V and U are standard
uniform distributed and hence continuous.
The conditional probabilityP r(V ≤v|U =u) can be thus computed as follows (cf. e.g. ([5] Equation. (4.4)) and ([17] Page 263))
P r(V ≤v|U =u) = lim ∆u→0+ P r(V ≤v, u≤U ≤u+ ∆u) P r(u≤U ≤u+ ∆u) = lim ∆u→0+ C(v, u+ ∆u)−C(v, u) ∆u =∂C(v, u) ∂u =g(v, u). (3.4)
We have thus the following equivalence P r ( Fs(Ls)≤Fs ( CoV aRsα|Li=l ) |Fi ( Li)=Fi(l) ) =g ( Fs ( CoV aRsα|Li=l ) , Fi(l) ) .
Based on this equivalence and due to the fact that the functiong(v, u) is invertible with respect tov for any fixed u∈[0,1] (see Remark 3.1), we are able to derive the explicit expressions ofCoV aRs|L
i=l
α . We do this by expressingvas a function
ofαanduas follow v=g−1(α, u). By replacingv byFs ( CoV aRsα|Li=l ) andubyFi(l) we obtain Fs ( CoV aRαs|L=l ) =g−1(α, Fi(l)). Thus CoV aRsα|Li=l=Fs ( CoV aRsα|L=l ) =Fs−1(g−1(α, Fi(l)) ) . □
One important advantage of our formula is that, the expression ofCoV aRsα|Li=l
(see Equation (3.3)) can be separated into two distinct components.
(1) On the one hand the marginal distributions Fi and Fs, which represent
the purely univariate features of the single financial institution iand the financial systemsrespectively.
(2) On the other hand the function g−1, which represents the dependency
structure between the single financial institutioniand the system s. This separation in the spirit of Sklar’s theorem is very important for the analysis of systemic risk because it allows to investigate the effect of the marginal distributions Fi andFs and the assumed copulaCto the systemic risk contribution.
Remark 3.3. We can see from Equation (3.3) thatCoV aRsα|Li=l is nothing other
than a quantile of the loss distribution Fs of the financial system s. In fact we
have
CoV aRsα|Li=l=Fs−1( ˜α) with ˜α:=g−1(α, Fi(l)). (3.5)
So, asCoV aRsα|Li=l can be expressed as the quantile ofFswith respect to the
adjusted level ˜α, it follows that CoV aRsα|Li=l as a function of ˜α has the same
properties like a simply Value-at-Risk. For example, the following properties hold.
Property 3.4.
a) CoV aRsα|Li=lincreases when the marginal distribution of the system (Fs)
has leptokurtosis (heavy-tailed) and positive skewness. (cf. [3] IV.2.8.1). b) If the lossLsof the financial systemsis assumed to be normal distributed
with meanµsand standard deviationσs, then
CoV aRsα|Li=l=σsΦ−1( ˜α) +µs, (3.6)
Φ denotes the standard normal distribution function and ˜αis defined ac-cording to Equation (3.5). Moreover
∆CoV aRsα|i=σs
(
Φ−1( ˜αd)−Φ−1( ˜αm)
)
, (3.7)
where ˜αd and ˜αm are the adjusted levels when the financial institutioni
is under distress and has its mean loss respectively, i.e. ˜ αd=g−1 ( α, Fi ( V aRiα)) and ˜αm=g−1 ( α, Fi ( E[Li])). In general the following Corollary of Theorem 3.2 holds:
Corollary 3.5. Under Assumption 1.3, the risk measure∆CoV aRsα|i is computed using Definition 1.6 as follows:
∆CoV aRsα|i=CoV aRs|L i=V aRi α α −CoV aR s|Li=E[Li] α =Fs−1(g−1(α, Fi ( V aRiα)))−Fs−1(g−1(α, Fi ( E[Li]))) =Fs−1(g−1(α, α))−Fs−1(g−1(α, Fi(µi)) ) , whereµi=E [ Li].
Remark 3.6. If we assume a symmetric distribution forLi, then it holds
∆CoV aRsα|i=Fs−1(g−1(α, α))−Fs−1(g−1(α,0.5)). (3.8)
Remark 3.7. In practice the conditional level l for the financial institution i is implicitly defined through a given confidence levelβ ∈(0,1) by
l=Fi−1(β), (3.9)
The confidence level β is specified by the regulatory institution and represents the probability with which the financial institutioniremains solvent over a given period of time horizon.
Based on this information we can expressCoV aRαs|Li=las follow:
CoV aRαs|Li=l=Fs−1(g−1(α, β)). (3.10) We observe that for a given marginal distribution functionFs ,CoV aR
s|Li=l
α can
be expressed as a function ofαandβ. This motivates the following definitions.
Definition 3.8. CoV aRβα:=CoV aRs|L i=F−1 i (β) α ∆CoV aRβα:=CoV aR s|Li=F−1 i (β) α −CoV aR s|Li=E(Li) α :=CoV aRαβ−CoV aRs|L i=E(Li) α It follows ∆CoV aRβα=Fs−1(g−1(α, β))−Fs−1(g−1(α, Fi(µi)) )
The bivariate Gaussian copula is defined as follows (cf. [16] Equation 2.3.6 ): CρGau(u, v) = Φ2
(
Φ (u)−1,Φ (v)−1
)
,
where Φ2 denotes the bivariate standard normal distribution function with
lin-ear correlation coefficient ρ, and Φ the univariate standard normal distribution function. Hence, CρGau(u, v) = ∫ Φ−1(u) −∞ ∫ Φ−1(v) −∞ 1 2π√1−ρ2exp ( 2ρst−s2−t2 2 (1−ρ2) ) dsdt. According to Theorem 3.2 we have the following formula for CoV aRsα|Li=l when
the dependence betweenLi andLsis modelled by a Gaussian copula.
Proposition 3.9. Assume that the copula ofLi and Ls is the Gaussian copula, then CoV aRsα|Li=l=Fs−1 ( Φ ( ρΦ−1(Fi(l)) + √ 1−ρ2Φ−1(α))). (3.11)
where Fi and Fs represent the univariate distribution function of Li and Ls re-spectively.
Proof. We first note thatCρ(u, v) can be expressed as
CρGau(u, v) = ∫ u 0 Φ ( Φ−1(v)−ρΦ−1(t) √ 1−ρ2 ) dt. (3.12)
In fact letX = (U, V) be a standard Gaussian random vector with correlation ρ, then we have: Φ2(u, v) =P r(U ≤u, V ≤v) = ∫ u −∞ ∫ v −∞ 1 2π√1−ρ2exp ( 2ρst−s2−t2 2 (1−ρ2) ) dsdt. This implies ∂Φ2(u, v) ∂u =ϕ(u)·Φ ( v−uρ √ 1−ρ2 ) ,
whereϕ denotes the density function of the standard univariate normal distribu-tion. Therefore, we have
Φ2(u, v) = ∫ u −∞ ϕ(x)·Φ ( v−xρ √ 1−ρ2 ) dx. The expression of the bivariate Gaussian copula is thus
Cρ(u, v) = Φ2 ( Φ−1(u),Φ−1(v), ρ)= ∫ Φ−1(u) −∞ ϕ(x)·Φ ( Φ−1(v)−xρ √ 1−ρ2 ) dx. By making the substitutiont= Φ (x), we obtain
Cρ(u, v) = ∫ u 0 Φ ( Φ−1(v)−ρΦ−1(t) √ 1−ρ2 ) dt. According to Theorem 3.2 we have:
g(v, u) = ∂Cρ(u, v) ∂u = Φ ( Φ−1(v)−ρΦ−1(u) √ 1−ρ2 ) . (3.13)
The functiong(v, u) is strictly monotone with respect tovand its inverse is given by
g−1(α, u) = Φ
(
ρΦ−1(u) +√1−ρ2Φ−1(α)). (3.14)
Thus by Equation 3.3 it follows that CoV aRsα|Li=l=Fs−1 ( Φ ( ρΦ−1(Fi(l)) + √ 1−ρ2Φ−1(α))). □ Remark 3.10. If we set in Equation (3.14)ρ= 0 (i.e. we assume that the financial institutioniand the financial systemsare not correlated) we obtain
g−1(α, u) =α, u∈[0,1].
HenceCoV aRs|Li=V aRi =l for alll∈R. Consequently we have that ∆CoV aRsα|i
is equal to Zero (i.e. there is no systemic risk contribution fromitos) if theiand sare uncorrelated.
Remark 3.11. In the context of Remark 3.3, we have ˜ α= Φ ( ρΦ−1(Fi(l)) + √ 1−ρ2Φ−1(α)). (3.15) By Corollary 3.5 we have ∆CoV aRsα|i=Fs−1 ( Φ ( ρΦ−1(α) +√1−ρ2Φ−1(α))) −Fs−1 ( Φ ( ρΦ−1(Fi(µi)) + √ 1−ρ2Φ−1(α)))
and ifFi is symmetric, then ∆CoV aRsα|i=Fs−1 ( Φ ( ρΦ−1(α) +√1−ρ2Φ−1(α))) −Fs−1 ( Φ ( ρΦ−1(0.5) +√1−ρ2Φ−1(α))) =Fs−1 ( Φ ( ρΦ−1(α) +√1−ρ2Φ−1(α)))−F−1 s ( Φ(√1−ρ2Φ−1(α))).
It is important to remark that the distributions functionsFiandFscan be assumed
to be any type of univariate distribution function satisfying Assumption 1.3. Let us consider in the rest of this section the particular case whereLiandLsare both an univariate normal distributed with expected valuesµi, µs and standard
deviation σi, σs respectively. Let Ni and Ns be the distribution function of Li
andLsrespectively i.e. Ni :=N
( µi, σ2i ) and Ns:=N ( µs, σs2 )
. The formula for CoV aRsα|Li=lis given by the following corollary.
Corollary 3.12. CoV aRsα|Li=l=ρσs σi (l−µi) + √ 1−ρ2σ sΦ−1(α) +µs. (3.16) Proof. By Theorem 3.2 we have:
CoV aRsα|Li=l=Ns−1 ( Φ ( ρΦ−1(Ni(l)) + √ 1−ρ2Φ−1(α))) =Ns−1 ( Ns ( σsρΦ−1(Ni(l)) +σs √ 1−ρ2Φ−1(α) +µ s )) =ρσs σi (l−µi) + √ 1−ρ2σ sΦ−1(α) +µs. □ Remark 3.13. The last case considered above was a combination of a bivariate Gaussian copula with two univariate Gaussian distributed margins. This case was already analysed by J¨ager-Ambro˙zewicz in [12]. Differently from the method provided here, J¨ager-Ambro˙zewicz derived a closed formula forCoV aRs|Li=V aRi
α
by using the expression of the conditional probability for bivariate normal distri-bution (cf. e.g. [8] Equation 2.6). Equation (3.16) coincides with the formula provided in [12] showing thus that the formula proposed by J¨ager-Ambro˙zewicz is a particular case of the formula provided here in Theorem 3.2. We will not further consider this particular case here and remark that for this case also the expressions ofCoV aRsα|Li=land ∆CoV aRcan be derived from Property 3.4 b).
4. Tail Events and Systemic Crisis
Recall that the main idea of the measurement of systemic risk contribution throughCoV aRmethod is to capture the potential for the spreading of financial distress across financial institutions by estimating the increase in tail co-movement (cf. [2]). Hence, in the context of the analysis and the measurement of systemic risk the dependence between the financial institutioniand the financial systems have to be considered in the tail of their joint distribution. It is thus important to quantify the extreme (or tail) dependence of i and s when the systemic risk
contribution ofiis analysed. This can be done using the so called tail dependence coefficients.
Definition 4.1 (cf. [15] Definition 5.30). Let (X, Y) be a bivariate random vari-able with marginal distribution functions F and G, respectively. The upper tail dependence coefficient of X and Y is the limit (if it exists) of the conditional probability that Y is greater than the 100α−thpercentile of Ggiven thatX is greater than the 100α−thpercentile ofF as αapproaches 1, i.e.
λu:= lim
α→1−λu(α), withλu(α) :=P r
(
Y > G−1(α)|X > F−1(α)). (4.1) If λu ∈ (0, 1], then (X, Y) is said to show upper tail dependence or extremal
dependence in the upper tail; if λu = 0, they are asymptotically independent in
the upper tail. Similarly, the lower tail dependence coefficientλlis the limit (if it
exists) of the conditional probability thatY is less than or equal to the 100α−th percentile ofGgiven thatX is less than or equal to the 100α−thpercentile ofF asαapproaches 0, i.e.
λl:= lim
α→0+λl(α), with λl(α) := limα→0+P r
(
Y ≤G−1(α)|X ≤F−1(α)). (4.2) Note that λu measures the probability thatY exceeds the threshold G−1(α),
conditional on thatXexceeds the thresholdF−1(α). In other words,λ
umeasures
the tendency for extreme events to occur simultaneously. If(Li, Ls)does not show tail dependence (λ
u=λl= 0), the extreme events of
Li and Lsappear to occur independently in each margin. This means that there
is no systemic risk contribution betweeniands.
Proposition 4.2. cf. [6]Providing that they exist the upper and lower tail depen-dence coefficient can be expressed in term of copula as follows:
λu= lim u→1− 1−2u+C(u, u) 1−u (4.3) and λl= lim u→0+ C(u, u) u . (4.4)
The tail dependence coefficient of the Gaussian Copula is given by (cf.[6]) λu=2 lim α→1− [ 1−Φ ( Φ−1(α)−ρΦ−1(α) √ 1−ρ2 )] =2 lim α→1− [ 1−Φ ( Φ−1(α)√1−ρ √ 1 +ρ )] . It follows that λu= { 0 ifρ <1 1 ifρ= 1.
Therefore if we assume the bivariate Gaussian copula as the dependence model for (Li, Ls), then, regardless of how high a correlation we choose, if we go far enough into the tail, extreme events appear to occur independently inLi andLs.
This means that the Gaussian copula is related to the independence in the tail and hence does not capture tail co-movements. This presents a big gap since tail events especially tail co-movements are the main features of systemic financial crisis (cf. [1]). This is the reason why we connect theCoV aRconcept to copula’s theory in order to develop an analytical formula forCoV aRsα|Li=l allowing the analysis and
the computation of systemic risk contribution for a more general stochastic setting than only the bivariate Gaussian setting. Our formula in Theorem 3.2 allows to consider other dependence models, especially those which are appropriate for the modelling of the simultaneous tail behaviour of losses during a financial crisis. It is also more flexible in the sense that it allows each margin independently of other to take a large class of distributions functions (for example we can assume thatLi ist-distributed and thatLsis normal distributed).
5. Applications to Non-Gaussian Copulas
In this section we apply the formula provided in Theorem 3.2 to non-Gaussian copulas. Especially we consider the bivariate t- copula as special case of the class of bivariate elliptical copula. We also consider the case of Archimedean copula and the convex combination of copula.
Elliptical copulas are the most used copulas in modern finance and risk-management. They are derived from multivariate elliptical distributions function using the Sklar’s theorem (see Corollary 2.5). The two most important elliptical
copulas are the Gaussian and the t copulas (student’s copula). Both have in
their central part the same behaviour and properties as the multivariate normal distribution (This is one reason of their popularity), but show different behaviours in the tail.
The Student t copula can be considered as a generalization of the normal copula allowing the consideration of tail-dependence. It has in addition to the correlation coefficientρa second dependence parameter, the degree of freedomν, which controls the heaviness of the tails.
Definition 5.1. The distribution function of a bivariate t distributed random variablewith correlation coefficientρis given by:
tρ,ν(u, v) = ∫ u −∞ ∫ v −∞ 1 2π√1−ρ2 ( 1 + s 2+t2−2ρtst ν(1−ρ2) )−ν+2 2 dsdt, whereν denotes the number of degrees of freedom.
Forν <3 the variance does not exist, and forν <5 the fourth moment does not exist. The t copula and the Gaussian copula are close to each other in their central part, and become closer and closer in their tail only whenν increases. Especially both copulas are almost identical whenν→ ∞.
Definition 5.2. Thebivariate t copula,Ct
ρ,ν is defined as Cρ,νt (u, v) =tρ,ν ( t−ν1(u), t−ν1(v) ) = ∫ t−1 ν (u) −∞ ∫ t−1 ν (v) −∞ 1 2π√1−ρ2 ( 1 +s 2+t2−2ρtst ν(1−ρ2) )−ν+2 2 dsdt,
wheretνdenotes the distribution function of a standard t withνdegrees of freedom
univariate distributed random variable.
The tail dependence coefficients of the t CopulaCρ,νt is given by ([6])
λl=λu= 2−2tν+1 (( (ν+ 1) (1−ρ) 1 +ρ )1 2 ) . It follows that, λu= { >0 ifρ >−1 0 ifρ=−1 .
So, provided thatρ >1, the bivariatetcopula is able to capture the dependence of extreme values and is thus appropriate for the modelling and the analysis of systemic risk contribution.
ThetcopulaCt
ρ,ν(u, v) can be expressed as follows (cf. e.g. [17] Page 299) :
Cρ,νt (u, v) = ∫ u 0 tν+1 ( ν+ 1 ν+[t−ν1(u) ]2 )1/2 t−ν1(v)√−ρt−ν1(t) 1−ρ2 dt. (5.1)
Now based on Theorem 3.2 we compute the expression ofg(v, u). We obtain
g(v, u) = ∂C t ρ,ν(u, v) ∂u =tν+1 ( ν+ 1 ν+[t−ν1(u) ]2 )1/2 t−ν1(v)√−ρt−ν1(u) 1−ρ2 .
The function g is invertible and its inverse is obtained by solving the equation g(v, u) =αforv. This leads to,
v=g−1(α, u) =tν ρt−ν1(u) + v u u t(1−ρ2) ( ν+[t−ν1(u) ]2) ν+ 1 t −1 ν+1(α) .
From this, we obtain the following formula forCoV aRs|L
i=l
α andCoV aRβα Proposition 5.3. Let the t copula be the copula of(Li, Ls), then for everyl∈R
CoV aRsα|Li=l= Fs−1 tν ρt−ν1(Fi(l)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l)) ]2) ν+ 1 t −1 ν+1(α) and CoV aRβα=Fs−1 tν ρt−ν1(β) + v u u t(1−ρ2)(ν+[t−1 ν (β) ]2) ν+ 1 t −1 ν+1(α) ,
where Fi and Fs represent the univariate distribution function of Li and Ls re-spectively and β denotes the regulatory risk level of the financial institutioni. Corollary 5.4. Assume that Li and Ls each follow an univariate standard t distribution withν degrees of freedom, then for every l∈R
CoV aRsα|Li=l=ρl+ √ (1−ρ2) (ν+l2) ν+ 1 t −1 ν+1(α) ∆CoV aRsα|i(l1, l2) =ρV aRiα+ √ (1−ρ2) ν+ 1 t −1 ν+1(α) [ ν+V aRiα−ν ] =V aRiα [ ρ+ √ (1−ρ2) ν+ 1 t −1 ν+1(α) ] .
We use here the fact that the standardtdistribution with ν degree of freedom has a mean equal to zero (and a variance equal to ν
ν−2).
The standardt distribution can be extended through linear transformation of the form
X :=a+bZ, Z ∼tν.
The distribution ofX is called generalized t distribution (X ∼T(a, b2, ν)). The mean ofX is equal to a (E[X] =a) and its variance V[X] is given by
V[X] =b2V [Z] =b2 ν ν−2.
The corresponding density fT is obtained using the Transformation formula for
density (cf. e.g. [14], theorem. 1.101). Letftbe the density function of standard
tdistribution, then fT(x) =ft(g(z)) = ft(g−1(z)) |g′(g−1(z))|, withg(z) =a+bz such that fT(x) =ft ( x−a b ) 1 b , b̸= 0. We have that T(x) =P r(X ≤x) =P r(a+bZ≤x) =tν ( x−a b ) . T(x) =α ⇔x−a b =t −1 ν (α) ⇔x=bt−ν1(α) +a ⇔T−1(α) =bt−ν1(α) +a. (5.2)
(1) Let µs:=E(Ls). IfLs∼T ( µs, σs∗ 2 , ν), i.e. ( Ls−µ s σ∗s ) follows an univari-ate standard t distribution withν degrees, then
CoV aRsα|i(l) = Fs−1 tν ρt−ν1(Fi(l)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l)) ]2) ν+ 1 t −1 ν+1(α) = σs∗t−ν1 tν ρt−ν1(Fi(l)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l)) ]2) ν+ 1 t −1 ν+1(α) +µs= σs∗ ρt−ν1(Fi(l)) + v u u t(1−ρ2) ( ν+[t−ν1(Fi(l)) ]2) ν+ 1 t −1 ν+1(α) +µs (2) Let µi:=E ( Li)andµ s:=E(Ls). If Li∼T ( µi, σi∗ 2 , ν ) , Ls∼T ( µs, σs∗ 2 , ν ) , i.e. ( Li−µ i σ∗i ) and ( Ls−µ s σs∗ )
each follows an univariate standardt distribu-tion withν degrees of freedom, then
CoV aRsα|i(l) = σ∗s ρt−ν1(Fi(l)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l)) ]2) ν+ 1 t −1 ν+1(α) +µs= σ∗s ρt −1 ν ( tν ( l−µi σ∗i )) + v u u u t(1−ρ2) ( ν+ [ t−ν1 ( tν ( l−µi σ∗i ))]2) ν+ 1 t −1 ν+1(α) +µs=σs∗ ρ ( l−µi σi∗ ) + v u u u t(1−ρ2) ( ν+ ( l−µi σi∗ )2) ν+ 1 t −1 ν+1(α) +µs= σs∗ρ σi∗ (l−µi) +σ ∗ st− 1 ν+1(α) v u u u t(1−ρ2) ( ν+ ( l−µi σ∗i )2) ν+ 1 +µs.
By (1.8) we have that for givenl1, l2∈R ∆CoV aRsα|i(l1, l2) =CoV aRsα|i(l1)−CoV aRsα|i(l2) =Fs−1 tν ρt−ν1(Fi(l1)) + v u u t(1−ρ2) ( ν+[t−ν1(Fi(l1)) ]2) ν+ 1 t −1 ν+1(α) − Fs−1 tν ρt−ν1(Fi(l2)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l2)) ]2) ν+ 1 t −1 ν+1(α) .
Consider again the previous two cases (1) IfLs∼T(µ s, σs∗ 2, ν), then ∆CoV aRαs|i(l1, l2) = σs∗ ρt−ν1(Fi(l1)) + v u u t(1−ρ2)(ν+[t−1 ν (Fi(l1)) ]2) ν+ 1 t −1 ν+1(α) +µs − σs∗ ρt−ν1(Fi(l2)) + v u u t(1−ρ2) ( ν+[t−ν1(Fi(l2)) ]2) ν+ 1 t −1 ν+1(α) +µs =σs∗ρ(t−ν1(Fi(l1))−t−ν1(Fi(l2)) ) +σ ∗ st− 1 ν+1(α) √ 1−ρ2 √ ν+ 1 ([ t−ν1(Fi(l1)) ]2 −[t−ν1(Fi(l2)) ]2) =σs∗(t−ν1(Fi(l1))−t−ν1(Fi(l2)) ) × [ ρ+t −1 ν+1(α) √ 1−ρ2 √ ν+ 1 ( t−ν1(Fi(l1)) +t−ν1(Fi(l2)) )] .
(2) IfLi∼T(µi, σi∗ 2 , ν)andLs∼T(µs, σs∗ 2 , ν), then ∆CoV aRsα|i(l1, l2) = σ∗sρ σ∗i (l−µi) +σ ∗ st− 1 ν+1(α) v u u u t(1−ρ2) ( ν+ ( l−µi σ∗i )2) ν+ 1 +µs − σ∗sρ σ∗i (l−µi) +σ ∗ st− 1 ν+1(α) v u u u t(1−ρ2) ( ν+ ( l−µi σi∗ )2) ν+ 1 +µs = σ ∗ sρ σi∗ (l1−l2) + σ∗st− 1 ν+1(α) √ 1−ρ2 √ ν+ 1 (( l1−µi σ∗i )2 − ( l2−µi σ∗i )2) = σ ∗ sρ σi∗ (l1−l2) + σ∗st− 1 ν+1(α) √ 1−ρ2 √ ν+ 1 ( l1−l2 σi∗ ) ( l1+l2−2µi σ∗i ) = σ ∗ s σ∗i (l1−l2) [ ρ+t −1 ν+1(α) √ 1−ρ2 √ ν+ 1 ( l1+l2−2µi σ∗i )] . (5.3) Corollary 5.5. Let µi :=E ( Li) and µs := E(Ls). If Li ∼T ( µi, σ∗i 2 , ν) and Ls ∼T(µs, σ∗s 2 , ν), i.e. ( Li−µ i σ∗i ) and ( Ls−µ s σ∗s )
each follows an univariate stan-dard t distribution with ν degrees of freedom, then
∆CoV aRsα|Li=l=σ∗st−ν1(α) [ ρ+t−ν1(α)t −1 ν+1(α) √ 1−ρ2 √ ν+ 1 ] . (5.4)
Proof. By (5.3) we have that
∆CoV aRαs|Li=l= ∆CoV aRαs|i(V aRiα, µi ) = σ ∗ s σ∗i ( V aRiα−µi )[ ρ+t −1 ν+1(α) √ 1−ρ2 √ ν+ 1 ( V aRi α+µi−2µi σ∗i )] . Since the Value-at-Risk can expressed in term of a quantile (see (1.2)), It follows from (5.2) that V aRiα=µi+σi∗t− 1ν(α). ∆CoV aRαs|Li=l=σ ∗ s σ∗i ( µi+σi∗tν− 1(α)−µ i ) × [ ρ+t −1 ν+1(α) √ 1−ρ2 √ ν+ 1 ( µi+σ∗tν−1(α) +µi−2µi σi∗ )] =σs∗t− 1 ν (α) [ ρ+t−ν1(α) t−ν+11 (α)√1−ρ2 √ ν+ 1 ] . □
Note that the dependence in the Gaussian and t-copulas setting are essentially determined by the correlation coefficient ρ (elliptical copula). The correlation coefficient is often considered as being a poor tool for describing dependence when
the margins are non-normal (cf. [15]. This motivates the use ofArchimedean
copula.
Unlike as in the elliptical copulas, the dependence in a bivariate Archimedean copula is not controlled by a constant (the correlation parameter) but by a function φcalled generator. This gives to Archimedean copulas good analytical properties, and the ability to reproduce a large spectrum of dependence structures.
Theorem 5.6. [[16], Theorem 4.1.4] Let φ be a continuous, strictly decreasing function from [0, 1]to[0,∞]such that φ(1) = 0, and letφ[−1](t) be the
pseudo-inverse ofφdefined by φ[−1](t) = { φ−1(t) if 0≤t≤φ(0) 0 if φ(0)< t≤ ∞. ) ,
then the functionC from[0, 1]2 to[0, 1]given by
C(u, v) =φ[−1](φ(u) +φ(v)). (5.5)
is a copula if and only if φis convex.
Note that the composition of the pseudo-inverse with the generator gives the identity i.e.
φ[−1](φ(t)) =t. ∀t∈[0,∞].
Ifφ(0) =∞the generator is said to be strict and its pseudo-inverseφ[−1]coincide with the ordinary functional inverseφ−1 (cf. [16] Definition 4.1.1).
Definition 5.7. A function ϕsatisfying the conditions in Theorem 5.6 is called
generator of a copula. A copula constructed through a generator is called Archi-median copula.
The lower and upper tail dependence coefficient of an Archimedean copula can be computed using the following corollary.
Corollary 5.8 ([16] Corollary. 5.4.3). Let C be an Archimedean copula with a continuous, strictly, decreasing and convex generator φ, then
λu= 2− lim x→0+ 1−φ−1(2x) 1−φ−1(x) and λl= limx→∞ 1−φ−1(2x) 1−φ−1(x)
In the context of systemic risk analysis, we are interested by Archimedean cop-ulas showing positive (upper or lower) tail dependence (e.g. Gumbel and Clayton copula).
Remark 5.9. In the case we assume copula with positive upper (lower) tail depen-dence, the loss have to be defined as a positive (negative) number (cf. [15]).
Example 5.10(Gumbel Copula). The generator of the Gumbel copula is defined by
φθ(t) = (−ln(t)) θ
It holdsφθ(0) =∞, i.e. φθ is strict and its inverse isφ−θ1(t) = exp
(
−t1θ
)
. The Gumbel copula is then according to 5.5 given by:
CθGu(u, v) = exp ( −[(−ln(u))θ+ (−ln(v))θ ]1 θ ) , 1≤θ <∞,
where θrepresents the strength of dependence. By Corollary 5.8, the tail depen-dence coefficients of the Gumbel copula are given by:
λu= 2−2
1
θ and λl= 0.
The Gumbel copula is thus able to model contagion effect and is therefore a good alternative model for the analysis of systemic risk contribution.
According to Theorem 3.2 the corresponding functiongis given by: gGu(v, u) := ∂CθGu(u, v) ∂u = exp ( −((−ln(u))θ+ (−ln(v))θ )1 θ ) ×((−ln(u))θ+ (−ln(v))θ )−θ−1 θ · (−ln(u)) θ−1 u .
The functiong is for u∈ (0,1) and for all θ >1 strictly increasing with respect tov and therefore invertible.
So according to Theorem 3.2 we can computeCoV aRsα|Li=l by
CoV aRαs|Li=l=Fs−1(g−Gu1(α, Fi(l))
)
. (5.7)
By imposing some conditions to the generatorφof an Archimedean Copula, we
can derive, using Theorem 3.2, an explicit expression ofCoV aRsα|Li=lin terms of
φ.
Proposition 5.11. Assume that the copulaC associated to the joint distribution of (Li, Ls) is a bivariate Archimedean copula with generator φ. If φ is strict and its derivativeφ′ is invertible, then the explicit formula forCoV aRαs|Li=lfor a given level α,∈(0,1) is given by
CoV aRsα|Li=l=Fs−1 ( φ−1 ( φ ( φ′−1 ( φ′(Fi(l)) α )) −φ(Fi(l)) )) . (5.8)
Proof. In fact, letCbe an Archimedean copula with a strict generatorφsuch that C(u, v) =φ−1(φ(u) +φ(v)) and it holds φ(C(u, v)) =φ(u) +φ(v). (5.9) Hence, ∂[φ(C(u, v))] ∂u = ∂[φ(u) +φ(v)] ∂u i.e. ∂C(u, v) ∂u ·φ ′(C(u, v)) = ∂φ(u) ∂u =φ ′(u)
it follows that ∂C(u, v) ∂u = φ′(u) φ′(C(u, v))= φ′(u) φ′(φ−1[φ(u) +φ(v)]). We have thus g(v, u) = ∂C(u, v) ∂u = φ′(u) φ′(φ−1[φ(u) +φ(v)]).
Now, setg(v, u) =αand solve forv. Ifφ′ is invertible we obtain g−1(α, u) =φ−1 ( φ ( φ′−1 ( φ′(u) α )) −φ(u) ) , and by Theorem 3.2 we have
CoV aRαs|Li=l=Fs−1(g−1(α, Fi(l)) ) =Fs−1 ( φ−1 ( φ ( φ′−1 ( φ′(Fi(l)) α )) −φ(Fi(l)) )) . □ Corollary 5.12. CoV aRβα=Fs−1 ( φ−1 ( φ ( φ′−1 ( φ′(β) α )) −φ(β) )) .
Example 5.13(Clayton Copula). The generator of the the Clayton Copula
φ(t) = 1 θ
(
t−θ−1). θ∈[−1,∞)− {0}
and is strict forθ >0. The Clayton copula can be thus expressed in this case as follows CθCl(u, v) = ( u−θ+v−θ−1)− 1 θ, u, v∈(0,1). (5.10) and we have φ−θ1(s) = (1 +θs)−1θ, φ′ θ(t) =−t− θ−1, φ′−1 θ (z) =−z− 1 θ+1. And by Proposition 5.11, we have that
CoV aRsα|Li=l=Fs−1 ( φ−1 ( φ ( φ′−1 ( φ′(Fi(l)) α )) −φ(Fi(l)) )) =Fs−1 (( 1 +Fi(l)−θ ( α−θ+1θ −1 ))−1 θ ) .
The description of tail dependence structures arising from real financial data is very important for an effective estimation of systemic risk contribution. For this purpose convex combination of copulas are more appropriate than single
cop-ula (such as elliptical copcop-ula and Archimedean). Theconvex combination of
copulasprovides more flexibility by the description of tail dependence structures. In fact it is possible to describe a set of different tail dependence structures by combining two or more copulas.
As a bivariate copula can be seen as a specific bivariate distribution, it is clear that the convex linear combination of two copulas is again a copula (see e.g. [16],
Chapter 2). Formally, letC1andC2be two copulas. Then the functionCdefined
by
C(u, v) :=αC1(u, v) + (1−α)C2(u, v), u, v, α∈(0,1) is a copula.
The following remark specifies the effect of tail dependence of the underlying cop-ulas on that of their convex combination.
Remark 5.14. Let C be a convex combination of two bivariate copulas C1 and
C2. Denote by λ1u (λ 1 l), λ 2 u (λ 2
l) and λu (λl) the upper (lower) tail dependence
coefficients ofC1,C2,Crespectively, then
λu=αλ1u+ (1−α)λ
2
u and λl=αλ1l + (1−α)λ
2
l. (5.11)
In fact, as C is a copula, its tail dependence coefficients can be computed using Equation (4.3) and (4.4) respectively.
Remark 5.15. LetC1andC2 be two copulas satisfying Assumption 1.3. If we
as-sume that the copulaCassociated to the joint distribution of(Li, Ls)is a convex
combination ofC1andC2, then the functiong(v, u) :=αg1(v, u)+(1−α)g2(v, u)
(wheregi(v, u) := ∂Ci∂u(u,v), i∈ {1,2}) is invertible with respect to the parameter
v, and for alll∈Rand a givenα∈(0,1) CoV aRsα|L=l=F− 1 s ( g−1(α, Fi(l)) ) . (5.12)
In fact under Assumption 1.3g1 andg2 are each strictly increasing with respect
to v (see Remark 3.1). This implies, that g(v, u) is also strictly increasing with respect tov and thus invertible. (5.12) is then obtained by applying Theorem 3.2.
Example 5.16 (Convex Combination of Clayton and Gumbel Copula). For the Clayton copulaCθCl1 (u, v) =(u−θ1+v−θ1−1)−
1 θ1 we have: g1:= ∂CθCl 1 (u, v) ∂u =u −θ1−1(u−θ1+v−θ1−1)− θ1 +1 θ1 , λ u= 0 and λl= 2− 1 θ1.
Denote byC the convex combination of the Clayton CopulaCCl
θ1 and the Gumbel
copulaCθGu 2
C(u, v) :=αCθCl1 (u, v) + (1−α)CθGu2 (u, v), α∈(0,1).
Then by Lemma 5.14 the upper and the lower tail dependence coefficient ofC are
given by λu=α·0 + (1−α) ( 2−2θ12 ) = (1−α) ( 2−2θ12 ) .
The copula C has thus positive upper tail dependence coefficient and is hence
appropriate for the analysis of systemic risk contribution. We have that ∂C(u, v) ∂u =α ( u−θ1−1(u−θ1+v−θ1−1)− θ1 +1 θ1 ) + (1−α)× [ e ( −((−ln(u))θ2+(−ln(v))θ2) 1 θ2 )( (−ln(u))θ2+ (−ln(v))θ2 )−θ2−1 θ2 (−ln(u)) θ2−1 u ] =g(v, u). (5.13)
The functiong(v, u) is strictly increasing with respect tov and hence invertible. Based on this we derive the following Corollary of Theorem 3.2.
Corollary 5.17. If the copulaCof(Li, Ls)is a convex combination of the Clayton and the Gumbel Copula, namely
C(u, v) :=αCθCl1 (u, v) + (1−α)CθGu2 (u, v), α∈(0,1), θ1, θ2>0.
Then for a given l∈R
CoV aRsα|L=l=Fs−1( ˜α), (5.14)
whereα˜ is the solution of the equationg(α, F˜ i−1(l))=αandgis given by (5.13). 6. Alternative Model for Systemic Risk Contribution
In this section, we first argue that ∆CoV aR as defined in [2] by Adrian and Brunnermeier (in this article Definition 1.6) is not consistent with the notion of systemic risk contribution and hence non-adequate for the analysis of systemic risk. Then by changing the way how the condition C(Li) is defined, we define
alternative financial risk measures which are consistent with the notion of systemic risk contribution and hence more appropriate for the analysis of systemic risk. The reasonable first step towards this is to introduce the notion of ”distressed financial Institutions”.
Definition 6.1(cf. [9] Definition 4.1). LetLbe the class of all possible losses. A mappingR:L →Ris called amonetary measure of riskif it satisfies the following conditions for allL1, L2∈ L
(1) Monotonicity: IfL1≤L2, then R(L1)≤ R(L2)
(2) Cash invariance: IfL∈ Landm∈RthenR(L+m) =R(L)−m Monotonicity property means that high losses require high risk capitals. Cash invariance property is motivated by the interpretation of R(L) as a regulatory capital. It suggests that the regulatory capital associated to a loss L is reduced by the amountl >0 if this amount is add toL.
A lossLsuch thatR(L)≤0 is calledacceptable, in the sense that a financial institution whit loss L is not required by the regulator to keep any regulatory capital. The set of acceptable losses associated to a risk measureRis given by
AR={L∈ L | R(L)≤0}.
That is, a lossL isacceptablewith respect to a risk measureRifL∈ AR. LetL be a non-acceptable loss i.e. L /∈ AR. If we add to La cash amount of
R(L), that is, we define an adjusted loss ˜
L:=L+R(L),
then by the cash invariance property of monetary risk measure we have that
R( ˜L) =R(L+R(L)) =R(L)− R(L) = 0
so that ˜L∈ AR. Hence one can interpretR(L) as the minimum amount of capital that a financial institution with lossLshould keep as regulatory capital. Formally,
From a purely economic point of view, financial distress may be defined as a situation where a financial institution’s operating cash flows are not sufficient to satisfy current obligations (cf. e.g. [18], A7 3.1). From a quantitative risk management perspective we can characterize a distressed financial institution as follows.
Definition 6.2(Distressed Financial Institutions). LetLbe the loss incurred by one financial institution B. Let RC be the regulatory capital associated to the loss L. For a given time t we say that the financial institutionB is indistress if at this time the realizationlofLis greater than the associated regulatory capital RC i.e.
l > RC. (6.2)
If we assume that the regulatory capital RC is determined by the Value-at-Risk, then we say that the financial institutionB is in distress a the timetif
l >Value-at-Risk. (6.3)
The condition C(Li) = {Li=V aRiα
}
in Definition 1.6 does not fulfill the default condition 6.3. In fact a loss equal to the Value-at-Risk does not lead to a default. In fact, the financial institutioniis supposed to have a regulatory capital equal to its Value-at-Risk. So, any loss smaller or equal to its Value-at-Risk is absorbed. Such losses can therefore not lead to the default of i and hence to a systemic risk contribution. It is for this reason that we say that the initial definition of ∆CoV aRsα|Li=l is not consistent with the notion of systemic risk contribution.
We propose in the next alternative risk measures which are consistent with the notion of systemic risk contribution.
Definition 6.3. ECoV aRsα|i :=E [ CoV aRsα|i ( Li)|Li≥V aRi ] . (6.4) Proposition 6.4. ECoV aRαs|i= 1 1−Fi(V aRi) ∫ ∞ V aRi CoV aRsα|i(l)fi(l)dl. (6.5) Proof. From basic probability theories, (cf. e.g. [14] Def. 8.9) we have that
ECoV aRsα|i:=E ( CoV aRsα|i(Li)|Li≥V aRi ) = E ( CoV aRsα|i ( Li)1 {Li≥V aRi} ) P r(Li≥V aRi) = 1 1−Fi(V aRi) ∫ ∞ V aRi CoV aRsα|i(l)fi(l)dl. □ Remark 6.5. Assume that the confidence level for the calculation of V aRi is β,
then ECoV aRsα|i= 1 1−β ∫ ∞ V aRi β CoV aRαs|i(l)fi(l)dl. (6.6)
