measuring and analysing marginal systemic risk...

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

Financial system


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)


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)


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)


Measuring systemic riskcontribution

Systemic risk

Regulation of systemic risk

CoVaR-Method


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)


Which factors promote systemic risk ?

1) The high interconnectedness and complexity of the modernfinancial system


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)


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


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.


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.


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.


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)


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.


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


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.


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


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.


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].


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)

).


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.


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.


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


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).


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)


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.


Examples: Bivariate gaussian copula

Assume that the copula of Li and Ls is the Gaussian copula, then


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.


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)


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


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:


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.


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


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


s

(g−1 (α,Fi (l))

). �


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)


Thank You


T. Adrian and M. K. Brunnermeier.

Covar.

Working Paper 17454, National Bureau of Economic Research,October 2011.

URL http://www.nber.org/papers/w17454.

L. Breiman.

Probability.

Society for Industrial and Applied Mathematics, Philadelphia, PA,USA, 1992.

ISBN 0-89871-296-3.

Hakwa and Jager-Ambrozewicz and Rudiger.

Measuring and analysing marginal systemic risk contribution usingcovar: A copula approach.

2011.

URL http://arxiv.org/abs/1210.4713.

M. Jager-Ambrozewicz.

Closed form solutions of measures of systemic risk.Brice Hakwa Measuring and analysing marginal systemic risk contribution using copula

http://www.nber.org/papers/w17454

http://arxiv.org/abs/1210.4713

SSRN eLibrary, 2010.

URL http:

//papers.ssrn.com/sol3/papers.cfm?abstract_id=1675435.

McNeil and Frey and Embrechts.

Quantitative Risk Management: Concepts, Techniques, and Tools.

Princeton Series in Finance. Princeton University Press, 2005.

R. B. Nelsen.

An Introduction to Copulas (Springer Series in Statistics).

Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006.

ISBN 0387286594.

T. Roncalli.

La Gestion des Risques Financiers.

Collection Gestion. Serie Politique generale, finance et marketing.Economica, second edition, 2009.

ISBN 9782717848915.


http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1675435

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1675435

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