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International Conference on Stochastic Analysis and Applications
Measuring and analysing marginal systemic riskcontribution using copula
Brice Hakwa
1 Bergische Universitat Wuppertal
Join work with
Manfred Jaeger-Ambrozewicz2(2University of Applied Sciences HTW Berlin)
and
Barbara Rudiger1
International Conference on Stochastic Analysis and Applications,
Hammamet, Tunisia 14-19. October 2013Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Background
Financial system
Financial regulation
Value-at-Risk (as special case of regulatory risk measure)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Background
Financial system
Financial regulation
Value-at-Risk (as special case of regulatory risk measure)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Background
Financial system
Financial regulation
Value-at-Risk (as special case of regulatory risk measure)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Financial system
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Financial regulation
The aim of financial regulation is to ensure the stability ofthe entire financial system (and not that of financialinstitutions in isolation)
he principle of the financial regulation before the last crisis isto impose each bank to hold (or invested in risk-free assets) agiven amount (risk capital) in order to avoid its failure (toreduce the probability of failure).
The risk capital of a given financial institution is computingdepending on its loss L using risk measures (e.g.Value-at-Risk in Basell II)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Financial regulation
The aim of financial regulation is to ensure the stability ofthe entire financial system (and not that of financialinstitutions in isolation)
he principle of the financial regulation before the last crisis isto impose each bank to hold (or invested in risk-free assets) agiven amount (risk capital) in order to avoid its failure (toreduce the probability of failure).
The risk capital of a given financial institution is computingdepending on its loss L using risk measures (e.g.Value-at-Risk in Basell II)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Financial regulation
The aim of financial regulation is to ensure the stability ofthe entire financial system (and not that of financialinstitutions in isolation)
he principle of the financial regulation before the last crisis isto impose each bank to hold (or invested in risk-free assets) agiven amount (risk capital) in order to avoid its failure (toreduce the probability of failure).
The risk capital of a given financial institution is computingdepending on its loss L using risk measures (e.g.Value-at-Risk in Basell II)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Value-at-Risk
Definition: Value-at-Risk [McNeil and Frey and Embrechts(2005)]
Given some confidence level α ∈ (0, 1). The VaR of a portfolio at theconfidence level α is given by the smallest number l such that theprobability that the loss L exceeds l is no larger than (1− α). Formally
VaRα := inf {l ∈ R : Pr (L > l) ≤ 1− α}= inf {l ∈ R : Pr (L ≤ l) ≥ α} .
Recall that, for a distribution function F , the function defined by
qα (F ) := inf {x ∈ R : F (x) ≥ α} .
is called the α-quantile of F .Note that, if F is continuous and strictly increasing, we have .
qα (F ) = F−1 (α)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Assumption
Assumption
We consider here only random variables which have strictly positivedensity function, such that all distribution function F consideredhere are continuous and strictly increasing.
We have in this case
VaRα = F−1 (α) . (1)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Measuring systemic riskcontribution
Systemic risk
Regulation of systemic risk
CoVaR-Method
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Measuring systemic riskcontribution
Systemic risk
Regulation of systemic risk
CoVaR-Method
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Measuring systemic riskcontribution
Systemic risk
Regulation of systemic risk
CoVaR-Method
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Lessons from the Financial Crisis
With the last crisis it became clear that the failure of certainfinancial institutions (the so called system relevant financialinstitutions) can lead to failures by other financial institutionsthreatening in this way the stability of the financial system(systemic risk).A typical example is the failure of Lehman Brothers in Sept. 15th.2008
Figure: Bank Failures in the United States, from 2000 to 2010 (Source: FIDC)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Which factors promote systemic risk ?
1) The high interconnectedness and complexity of the modernfinancial system
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
2) The size of financial institutions
3) The non regulation of systemic risk (no systemic risk measure)
Remark
1), 2) and 3) ⇒ Too big to fail-, too interconnected to fail-theory.According to this , certain financial institutions are so large and sointerconnected that their failure will be disastrous to the wholefinancial system (Moral hazard, contagion)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
2) The size of financial institutions
3) The non regulation of systemic risk (no systemic risk measure)
Remark
1), 2) and 3) ⇒ Too big to fail-, too interconnected to fail-theory.According to this , certain financial institutions are so large and sointerconnected that their failure will be disastrous to the wholefinancial system (Moral hazard, contagion)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
2) The size of financial institutions
3) The non regulation of systemic risk (no systemic risk measure)
Remark
1), 2) and 3) ⇒ Too big to fail-, too interconnected to fail-theory.According to this , certain financial institutions are so large and sointerconnected that their failure will be disastrous to the wholefinancial system (Moral hazard, contagion)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Regulation of systemic risk
The main question here is:
How to quantify the systemic risk contribution of one singlefinancial institute to the whole financial system.
or how to quantify the adverse financial impact that thefailure of one given financial institution can cause to thewhole financial system.
Probabilistic framework
We denote by i and s the individual financial institution andthe the financial system respectively.
We assume that the losses of i and s are modeled by the r.v.Li and Ls respectively.
We assume that i and s are interconnected and thusnon-independent such the bivariate r.v.
(Li , Ls
)is assumed to
be statistically dependent
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Regulation of systemic risk
The main question here is:
How to quantify the systemic risk contribution of one singlefinancial institute to the whole financial system.
or how to quantify the adverse financial impact that thefailure of one given financial institution can cause to thewhole financial system.
Probabilistic framework
We denote by i and s the individual financial institution andthe the financial system respectively.
We assume that the losses of i and s are modeled by the r.v.Li and Ls respectively.
We assume that i and s are interconnected and thusnon-independent such the bivariate r.v.
(Li , Ls
)is assumed to
be statistically dependent
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
CoVaR-Method:CoVaRs|C(Li)α
As response to this question [Adrian and Brunnermeier(2011)] proposedthe so called CoVaR-method.
The CoVaR-method build on the term CoVaRs|C(Li)α .
Definition
CoVaRs|C(Li)α is defined as the Value-at-Risk at the level α of an financial
institution s (or a financial system) conditional on some event C(Li)
depending on the loss Li .
Pr
(Ls ≤ CoVaR
s|C(Li)α |C
(Li))
= α. (2)
CoVaRs|C(Li)α is thus as a conditional quantile of the loss distribution of
the system s.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
CoVaR-Method:CoVaRs|C(Li)α
As response to this question [Adrian and Brunnermeier(2011)] proposedthe so called CoVaR-method.
The CoVaR-method build on the term CoVaRs|C(Li)α .
Definition
CoVaRs|C(Li)α is defined as the Value-at-Risk at the level α of an financial
institution s (or a financial system) conditional on some event C(Li)
depending on the loss Li .
Pr
(Ls ≤ CoVaR
s|C(Li)α |C
(Li))
= α. (2)
CoVaRs|C(Li)α is thus as a conditional quantile of the loss distribution of
the system s.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
CoVaR-Method:CoVaRs|C(Li)α
As response to this question [Adrian and Brunnermeier(2011)] proposedthe so called CoVaR-method.
The CoVaR-method build on the term CoVaRs|C(Li)α .
Definition
CoVaRs|C(Li)α is defined as the Value-at-Risk at the level α of an financial
institution s (or a financial system) conditional on some event C(Li)
depending on the loss Li .
Pr
(Ls ≤ CoVaR
s|C(Li)α |C
(Li))
= α. (2)
CoVaRs|C(Li)α is thus as a conditional quantile of the loss distribution of
the system s.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
CoVaR-Method: ∆CoVaRs|iα
In the CoVaR-method the systemic risk contribution of a single
institution i is modeled (estimated) by the risk measure ∆CoVaRs|iα
Definition
∆CoVaRs|iα := CoVaR
s|Li=VaR iα
α − CoVaRs|Li=E(Li)α .
So, the main task here is the computation of the generelized
CoVaRs|Li=lα , l ∈ R
i.e.
Pr(Ls ≤ CoVaRs|Li=l
α |Li = l)
= α. (3)
Recall that Pr(Li = l
)= 0, for any l ∈ R. However for fixed a h a
conditional probability of the form Pr(Ls ≤ h|Li = l
)can be defined as
∀y ∈ R, Pr(Ls ≤ h|Li ≤ y
)=
∫ y
−∞Pr(Ls ≤ h|Li = l
)fi (l) dl (4)
(cf. e.g. [Breiman(1992)])Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Computation of CoVaRs|Li=lα
Method based on quantile regression.
Closed form formula under bivariate Gaussian distributionsetting.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Quantile regression approach
[Adrian and Brunnermeier(2011)] adopt a quantile regressionapproach as described in Koenker (1978)”Regression Quantiles”
1 Statical approach (i.e. no closed form or analytical formula)
2 Impose the bivariate normal distribution as a model for thesystem and the firm variables.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Quantile regression approach
[Adrian and Brunnermeier(2011)] adopt a quantile regressionapproach as described in Koenker (1978)”Regression Quantiles”
1 Statical approach (i.e. no closed form or analytical formula)
2 Impose the bivariate normal distribution as a model for thesystem and the firm variables.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Closed form formula when(Li , Ls
)is modeled by a
bivariate Gaussian r.v.
[Jager-Ambrozewicz(2010)] proposed a closed form formula for
CoVaRs|Li=lα by assuming that
(Li , Ls
)follows a bivariate Gaussian
distribution(LsLi
)∼ N2 (µ,Σ) . with µ =
(µs
µi
), Σ =
(σ2s ρσsσi
ρσsσi σ2i
).
Where ρ is the correlation between Li and Ls . In this case, it is well knowthat the conditional distributions of Ls given Li assume a value l isunivariate Gaussian distributed
Ls |Li = li ∼ N(µs + ρ
σsσi
(li − µs) , σ2s
(1− ρ2
))(cf. e.g. [McNeil and Frey and Embrechts(2005)]).
Hence CoVaRs|Li=lα = σs
√(1− ρ2)Φ−1 (α) + µs + ρ
σsσi
(li − µs) .
Where Φ is the distribution function of the standard Gaussian
distribution.Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Our approach: Computing CoVaRs|Li=lα through
Copula
Motivation
Copula
Our principal result (our formula)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Motivation
The computation method presented above have their relative advantagesand disadvantages but they share the common restriction that bothimpose bivariate Gaussian distribution as model for
(Li , Ls
). There is a
major reasons why bivariate Gaussian distribution can lead to difficulties.
So, we assert that the CoVaRs|Li=lα computed under the assumption that(
Li , Ls)
is bivariate Gaussian distributed is not consistent with thenotion of systemic risk
Recall that the aim of systemic measurement is to quantify the riskcontribution of distressed financial institution
In general institution defaults and systemic crisis can be consideredas extreme event. Indeed, the default which produces the contagioneffect corresponds generally to a shock (large loss) relative to anexpected loss.This can be characterized by an extreme value which appears inthe tail of the corresponding loss distributions.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Motivation
The computation method presented above have their relative advantagesand disadvantages but they share the common restriction that bothimpose bivariate Gaussian distribution as model for
(Li , Ls
). There is a
major reasons why bivariate Gaussian distribution can lead to difficulties.
So, we assert that the CoVaRs|Li=lα computed under the assumption that(
Li , Ls)
is bivariate Gaussian distributed is not consistent with thenotion of systemic risk
Recall that the aim of systemic measurement is to quantify the riskcontribution of distressed financial institution
In general institution defaults and systemic crisis can be consideredas extreme event. Indeed, the default which produces the contagioneffect corresponds generally to a shock (large loss) relative to anexpected loss.This can be characterized by an extreme value which appears inthe tail of the corresponding loss distributions.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Motivation
The computation method presented above have their relative advantagesand disadvantages but they share the common restriction that bothimpose bivariate Gaussian distribution as model for
(Li , Ls
). There is a
major reasons why bivariate Gaussian distribution can lead to difficulties.
So, we assert that the CoVaRs|Li=lα computed under the assumption that(
Li , Ls)
is bivariate Gaussian distributed is not consistent with thenotion of systemic risk
Recall that the aim of systemic measurement is to quantify the riskcontribution of distressed financial institution
In general institution defaults and systemic crisis can be consideredas extreme event. Indeed, the default which produces the contagioneffect corresponds generally to a shock (large loss) relative to anexpected loss.This can be characterized by an extreme value which appears inthe tail of the corresponding loss distributions.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Motivation
The computation method presented above have their relative advantagesand disadvantages but they share the common restriction that bothimpose bivariate Gaussian distribution as model for
(Li , Ls
). There is a
major reasons why bivariate Gaussian distribution can lead to difficulties.
So, we assert that the CoVaRs|Li=lα computed under the assumption that(
Li , Ls)
is bivariate Gaussian distributed is not consistent with thenotion of systemic risk
Recall that the aim of systemic measurement is to quantify the riskcontribution of distressed financial institution
In general institution defaults and systemic crisis can be consideredas extreme event. Indeed, the default which produces the contagioneffect corresponds generally to a shock (large loss) relative to anexpected loss.This can be characterized by an extreme value which appears inthe tail of the corresponding loss distributions.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Hence, in the context of the analysis and the measurement of thesystemic risk contribution of the financial institution i , thedistributions of Li and Ls have to be dependent in they tail. Thetail dependence of i and s have to be considered. This can beverify by a tail dependance measure e.g. the tail dependencecoefficient λ
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Tail dependence coefficient
Definition [cf. [McNeil and Frey and Embrechts(2005)]]
Let (X ,Y ) be a bivariate random variable with marginal distributionfunctions F and G , respectively. The upper tail dependence coefficient ofX and Y is the limit (if it exists) of the conditional probability that Y isgreater than the 100α− th percentile of G given that X is greater thanthe 100α− th percentile of F as α approaches 1, i.e.
λu := limα→1−
Pr(Y > G−1 (α) |X > F−1 (α)
). (5)
λu measures the probability that Y exceeds the threshold G−1 (α),conditional on that X exceeds the threshold F−1 (α). In other words, λumeasures the tendency for extreme events to occur simultaneously andthus in some sense also the contagion effect.
Hence, If(Li , Ls
)is modeled such that Li and Ls are asymptotically
independent in the upper tail (λu = 0) then extreme losses appear tooccur independently and they is in this case no contagion effect.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Tail dependence coefficient
Definition [cf. [McNeil and Frey and Embrechts(2005)]]
Let (X ,Y ) be a bivariate random variable with marginal distributionfunctions F and G , respectively. The upper tail dependence coefficient ofX and Y is the limit (if it exists) of the conditional probability that Y isgreater than the 100α− th percentile of G given that X is greater thanthe 100α− th percentile of F as α approaches 1, i.e.
λu := limα→1−
Pr(Y > G−1 (α) |X > F−1 (α)
). (5)
λu measures the probability that Y exceeds the threshold G−1 (α),conditional on that X exceeds the threshold F−1 (α). In other words, λumeasures the tendency for extreme events to occur simultaneously andthus in some sense also the contagion effect.
Hence, If(Li , Ls
)is modeled such that Li and Ls are asymptotically
independent in the upper tail (λu = 0) then extreme losses appear tooccur independently and they is in this case no contagion effect.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Examples
If(Li , Ls
)is modeled by bivariate Gaussian r.v. then
λu =
{0 if ρ < 11 if ρ = 1.
This means that the bivariate Gaussian distribution isunappropriated (unrealistic) for the analysis of systemic risk.
If(Li , Ls
)is modeled by bivariate t-student r.v. then
λu =
{> 0 if ρ > −1
0 if ρ = −1
The bivariate t-student distribution is thus a better alternativemodel for systemic risk analysis
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Copula
Definition (2-dimensional copula (cf. [Nelsen(2006)])
A 2-dimensional copula is a (distribution) functionC : [0, 1]2 → [0, 1] with the following satisfying:
Boundary conditions:
1) For every u ∈ [0, 1] : C (0, u) = C (u, 0) = 0.2) For every u ∈ [0, 1] : C (1, u) = u and C (u, 1) = u.
Monotonicity condition:
3) For every (u1, u2) , (v1, v2) ∈ [0, 1]× [0, 1] withu1 ≤ u2 and v1 ≤ v2
C (u2, v2)− C (u2, v1)− C (u1, v2) + C (u1, v1) ≥ 0.
Conditions 1) and 3) implies that the so defined 2-copula C is a bivariate
joint distribution function (cf. [Nelsen(2006)]) and Condition 2) implies
that the copula C has standard uniform margins.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Theorem [Nelsen(2006)] (Continuity)
For every u1, u2, v1, v2 ∈ [0, 1] with u1 < u2 and v1 < v2
|C (u2, v2)− C (u1, v1)| ≤ |u2 − u1|+ |v2 − v1| (6)
This means that copulas are lipschitz continuous with Lipschitz constant
equal to 1.
Theorem [Nelsen(2006)] (Differentiability)
Let C be a copula. For any v ∈ [0, 1], the partial derivative∂C (u, v) /∂u exists for almost all u, and for such v and u
0 ≤ ∂C (u, v)
∂u≤ 1.
Similarly, for any u ∈ [0, 1], the partial derivative ∂C (u, v) /∂vexists for almost all v , and for such u and v
0 ≤ ∂C (u, v)
∂v≤ 1.
Furthermore, the functions u 7→ ∂C (u, v) /∂v andv 7→ ∂C (u, v) /∂u are defined and non-decreasing everywhere on[0, 1].
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Sklar’s theorem
Sklar’s theorem, [Nelsen(2006)]]
Let H be a joint distribution function with marginal distribution functionsF and G , then there exists a copula C such that for allx , y ∈ R ∪ {−∞} ∪ {+∞}
H (x , y) = C [F (x) ,G (y)] . (7)
If F and G have density, then C is unique. Conversely, if C is a copulaand F and G are distribution functions, then the function H defined by(7) is a joint distribution function with margins F and G .
Corollary [Nelsen(2006)]
Let H denote a bivariate distribution function with margins F and Gsatisfying our assumption, then there exist a unique copula C such thatfor all (u, v) ∈ [0, 1]2 it holds:
C (u, v) = H(F−1 (u) ,G−1 (v)
).
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Interpretation of the Sklar’s theorem
1 The Copula - method allows the effectively separation of thedependencies from the margins
2 The Copula can be interpreted as the information missingfrom the individual margins to complete the joint distribution.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Copula’s Type
Depending on the way they are build. Copulas can be classify into twoclasses
1 Implicit copulas. i.e copulas which are derived from a multivariatedistribution . (e.g. elliptical copulas). Implicit bivariate copulas havein general the form
C (u, v) =
∫ u
−∞
∫ v
−∞f (s, t) dsdt,
where f (s, t) is the density of the corresponding bivariatedistribution.
2 Explicit copulas. i.e. copulas which are not derived from amultivariate distribution. (e.g. Archimedean copulas)
Remark
Implicit copula with corresponding margins corresponds to the underlyingmultivariate distribution.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Implicit copulas: examples of elliptical copulas
The bivariate Gaussian copula is defined as follows [Nelsen(2006)]
Cρ (u, v) = Φ2
(Φ (u)−1
,Φ (v)−1)
=
∫ u
−∞
∫ v
−∞
1
2π√
1− ρ2exp
(2ρst − s2 − t2
2 (1− ρ2)
)dsdt
where Φ2 isthe bivariate standard normal distribution with linearcorrelation coefficient ρ, and Φ the univariate standard normaldistribution.
The bivariate t copula with ν degrees of freedom is defined as
C tρ,ν (u, v) = tρ,ν
(t−1ν (u) , t−1
ν (v))
=
∫ t−1ν (u)
−∞
∫ t−1ν (v)
−∞
1
2π√
1− ρ2
(1 +
s2 + t2 − 2ρst
ν (1− ρ2)
)− ν+22
dsdt.
tν(x) is the univariate t-distribution with ν degrees of freedom.
These two copulas have in the central part the same behavior but showdifferent behaviors in the tail. This difference vanish with increasing ν
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Implicit copulas: examples of elliptical copulas
The bivariate Gaussian copula is defined as follows [Nelsen(2006)]
Cρ (u, v) = Φ2
(Φ (u)−1
,Φ (v)−1)
=
∫ u
−∞
∫ v
−∞
1
2π√
1− ρ2exp
(2ρst − s2 − t2
2 (1− ρ2)
)dsdt
where Φ2 isthe bivariate standard normal distribution with linearcorrelation coefficient ρ, and Φ the univariate standard normaldistribution.
The bivariate t copula with ν degrees of freedom is defined as
C tρ,ν (u, v) = tρ,ν
(t−1ν (u) , t−1
ν (v))
=
∫ t−1ν (u)
−∞
∫ t−1ν (v)
−∞
1
2π√
1− ρ2
(1 +
s2 + t2 − 2ρst
ν (1− ρ2)
)− ν+22
dsdt.
tν(x) is the univariate t-distribution with ν degrees of freedom.
These two copulas have in the central part the same behavior but showdifferent behaviors in the tail. This difference vanish with increasing ν
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Explicit copulas: Archimedean copulas
Definition [McNeil and Frey and Embrechts(2005)]
A copula of the form
C (u, v) = ϕ[−1] (ϕ(u) + ϕ(v)) . (8)
is an Archimedean copula If ϕ is a convex and strictly decreasingcontinuous function from [0, 1] to [0,∞] with ϕ (1) = 0. ϕ iscalled the generator of the corresponding Archimedean copula
If ϕ (0) =∞ the generator is said to be strict and it is equivalentto the ordinary functional inverse ϕ−1.
So, to construct a Archimedean copulas we need only to find agenerator function and than define the corresponding copulasthrough Equation (8).
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Our principal result
Theorem [Hakwa and Jager-Ambrozewicz and Rudiger(2011)]
(Generalized explicit formula for CoVaRs|Li=lα )
Let Li and Ls be two random variables representing the loss of thefinancial institution i and that of the financial system s respectively.Assume that the joint distribution of
(Li , Ls
)is defined by a bivariate
copula C with marginal distribution functions Fi and Fs respectively.
If Fi and Fs are continuous, strictly increasing and strictly positive andthe partial derivative
g (v , u) :=∂C (u, v)
∂u
is invertible with respect to the parameter v , then for all l ∈ R the
explicit formula for CoVaRs|Li=lα at level α, 0 < α < 1 is given by
CoVaRs|Li=lα = F−1
s
(g−1 (α,Fi (l))
). (9)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Remark
Like a Copula, CoVaRs|Li=lα can be separated into two distinct
components.
1 The margins Ls and Li , which represent the purely univariatefeatures of the system s and the single Firm i respectively
2 The function g−1, which represents the true interconnectionbetween i and s.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Examples: Bivariate gaussian copula
Assume that the copula of Li and Ls is the Gaussian copula, then
CoVaRs|Li=lα = F−1
s
(Φ(ρΦ−1 (Fi (l)) +
√1− ρ2Φ−1 (α)
)).
where Fi and Fs represent the univariate distribution function of Li andLs respectively.
In particular if Li and Ls are univariat Gaussian distributed then
CoVaRs|Li=lα = ρ
σsσi
(l − µi ) +√
1− ρ2σsΦ−1 (α) + µs .
This show that that the formula proposed by [Jager-Ambrozewicz(2010)]
is a special case of our generalized formula.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Examples: Bivariate gaussian copula
Assume that the copula of Li and Ls is the Gaussian copula, then
CoVaRs|Li=lα = F−1
s
(Φ(ρΦ−1 (Fi (l)) +
√1− ρ2Φ−1 (α)
)).
where Fi and Fs represent the univariate distribution function of Li andLs respectively.
In particular if Li and Ls are univariat Gaussian distributed then
CoVaRs|Li=lα = ρ
σsσi
(l − µi ) +√
1− ρ2σsΦ−1 (α) + µs .
This show that that the formula proposed by [Jager-Ambrozewicz(2010)]
is a special case of our generalized formula.
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Example: Bivariate t copula
If we assume the bivariate student t Copula as modeled for(Li , Ls
)then
In particular if If Li and Ls are univariate t distributed with degrees offreedom ν then
CoVaRs|Li=lα = (ρ · l) +
√(1− ρ2) (ν + l2)
ν + 1t−1ν+1 (α)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Example: Bivariate t copula
If we assume the bivariate student t Copula as modeled for(Li , Ls
)then
In particular if If Li and Ls are univariate t distributed with degrees offreedom ν then
CoVaRs|Li=lα = (ρ · l) +
√(1− ρ2) (ν + l2)
ν + 1t−1ν+1 (α)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Representation of CoVaRs|Li=lα for Archimedean copula
Proposition: [Hakwa and Jager-Ambrozewicz and Rudiger(2011)]
CoVaRs|Li=lα for Archimedean copula
Let Li and Ls be two random variables representing the loss of thefinancial institution i and that of the financial system s. Assume that thejoint distribution of
(Li , Ls
)is defined by a bivariate copula C with
marginal distribution functions Fi and Fs respectively.If C is an Archimedean copula with a continuous, strictly, decreasing and
convex generator ϕ, then the explicit formula for the CoVaRs|Li=lα at level
α, 0 < α < 1
CoVaRs|Li=lα = F−1
s
(g−1 (α,Fi (l))
).
= F−1s
(ϕ−1
(ϕ
(ϕ′−1
(ϕ′ (Fi (l))
α
))− ϕ (Fi (l))
))provided that the function
g (v , u) :=∂C (u, v)
∂u
is invertible with respect to the parameter v .Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Proof of the theoremRecall that the implicit definition of CoVaR
s|Li=lα is given by:
Pr(Ls ≤ CoVaRs|Li=l
α |Li = l)
= α
⇔Pr(Fs (Ls) ≤ Fs
(CoVaRs|Li=l
α
)|Fi
(Li)
= Fi (l))
= α.
Let V = Fs (Ls) , U = Fi
(Li), v = Fs
(CoVaR
s|Li=lα
)and u = Fi (l)
i.e.
Pr(Ls ≤ CoVaRs|Li=l
α |Li = l)
= Pr(Fs (Ls) ≤ Fs
(CoVaRs|Li=l
α
)|Fi
(Li)
= Fi (l))
= Pr (V ≤ v |U = u) .
Due to Assumption 1 it follows from Remark ?? that V and U arestandard uniform distributed. In this case we can refer to([Breiman(1992)]) and ([Roncalli(2009)]) and compute the conditional
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
probability Pr (V ≤ v |U = u), as follows:
Pr (V ≤ v |U = u) = lim∆u→0+
Pr (V ≤ v , u ≤ U ≤ u + ∆u)
Pr (u ≤ U ≤ u + ∆u)
= lim∆u→0+
Pr (U ≤ u + ∆u,V ≤ v)− Pr (U ≤ u,V ≤ v)
Pr (U ≤ u + ∆u)− Pr (U ≤ u)
= lim∆u→0+
C (u + ∆u, v)− C (u, v)
∆u
=∂C (u, v)
∂u=: g (u, v) .
assume that g is invertible with respect to the (non-conditioning)variable v , then
v = g−1 (α, u) .
Set v = Fs
(CoVaR
s|Li=lα
)and u = Fi (l). We obtain
Fs
(CoVaRs|Li=l
α
)= g−1 (α,Fi (l)) .
Thus
CoVaRs|Li=lα = F−1
s
(g−1 (α,Fi (l))
). �
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Summary:
Using copula theories we provide a general closed formula for
the computation of CoVaRs|Li=lα .
Our formula coincide with the formula proposed by[Jager-Ambrozewicz(2010)] when
(Li , Ls
)is modeled by a
Bivariate Gaussian copula with Gaussian margins. (i.e. it is aspecial case of our formula)
Our formula generates a characterization of CoVaRs|Li=lα in
term of a generator when the dependence between Li and Ls
is modeled by a Archimedeans copula
Our formula provides a framework for flexible modeling andanalysis of systemic risk contribution by allowing theintegration of stylized features of marginal losses as skewness,fat-tails and complex de dependence structure (e.g. linear-,non-linear-, tail-dependence)
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
Thank You
Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
T. Adrian and M. K. Brunnermeier.
Covar.
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URL http://www.nber.org/papers/w17454.
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Measuring and analysing marginal systemic risk contribution usingcovar: A copula approach.
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URL http://arxiv.org/abs/1210.4713.
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Closed form solutions of measures of systemic risk.Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula
SSRN eLibrary, 2010.
URL http:
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Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula