Copula functions Advanced Methods of Risk Management Umberto Cherubini
Introducing Copula to Risk Management Presentation
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Transcript of Introducing Copula to Risk Management Presentation
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Introducing Copula to Risk Management
Introducing Copula to Risk Management
Wenting Li
SMU
April 2007
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Introducing Copula to Risk Management
An Overview
Copula
RiskApplicationSummary
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Introducing Copula to Risk Management
An Overview
CopulaRisk
ApplicationSummary
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Introducing Copula to Risk Management
An Overview
CopulaRiskApplication
Summary
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Introducing Copula to Risk Management
An Overview
CopulaRiskApplicationSummary
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Introducing Copula to Risk Management
Section I Concepts and Properties of Copula
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Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
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Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such that
C (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 9: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/9.jpg)
Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;
Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 10: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/10.jpg)
Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 11: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/11.jpg)
Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 12: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/12.jpg)
Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 13: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/13.jpg)
Introducing Copula to Risk Management
1.1 Concept of Copula
De�nition (Copula):
Multivariate distribution function C : IN ! I ,such thatC (u1, ..., uN ) is grounded and N-increasing;Marginals of Cn is uniformly distributed. i.e. Cn(u) = u for allu 2 [0, 1].
De�nition (Copula of F):Sklar�s Theorem:
F(x1, ..., xN ) = C(F1(x1), ...,FN (xN ))= C(u1, ...,uN )
The dependence is characterize by C and it is Unique.
The Copula of a distribution F can be considered to be the part ofF describing the dependence structure.
![Page 14: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/14.jpg)
Introducing Copula to Risk Management
1.2 Properties
Invariance of Copula:The copula of F is invariant under strictly increasingtransformations.If T1, ...,Td are strictly increasing, then (T1(X1), ...,Td (Xd ))t
has the same copula as (X1, ...,Xd )t .
Frechet Bounds
C� = max
(d
∑i=1ui + 1� d , 0
)� C (u) � min fu1, ..., udg = C+
Fr echet lower bound C� :: ;countermonotonicitycopula.(ρ = �1)Fr echet 2-dimensional upper bound C+(u1, u2) ::comonotonicity copula.(ρ = 1)
Independent Copula: i.e. Product Copula :
C?(u1, ..., ud ) =d
∏i=1ui .
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Introducing Copula to Risk Management
1.2 Properties
Invariance of Copula:The copula of F is invariant under strictly increasingtransformations.If T1, ...,Td are strictly increasing, then (T1(X1), ...,Td (Xd ))t
has the same copula as (X1, ...,Xd )t .Frechet Bounds
C� = max
(d
∑i=1ui + 1� d , 0
)� C (u) � min fu1, ..., udg = C+
Fr echet lower bound C� :: ;countermonotonicitycopula.(ρ = �1)Fr echet 2-dimensional upper bound C+(u1, u2) ::comonotonicity copula.(ρ = 1)
Independent Copula: i.e. Product Copula :
C?(u1, ..., ud ) =d
∏i=1ui .
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Introducing Copula to Risk Management
1.2 Properties
Invariance of Copula:The copula of F is invariant under strictly increasingtransformations.If T1, ...,Td are strictly increasing, then (T1(X1), ...,Td (Xd ))t
has the same copula as (X1, ...,Xd )t .Frechet Bounds
C� = max
(d
∑i=1ui + 1� d , 0
)� C (u) � min fu1, ..., udg = C+
Fr echet lower bound C� :: ;countermonotonicitycopula.(ρ = �1)
Fr echet 2-dimensional upper bound C+(u1, u2) ::comonotonicity copula.(ρ = 1)
Independent Copula: i.e. Product Copula :
C?(u1, ..., ud ) =d
∏i=1ui .
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Introducing Copula to Risk Management
1.2 Properties
Invariance of Copula:The copula of F is invariant under strictly increasingtransformations.If T1, ...,Td are strictly increasing, then (T1(X1), ...,Td (Xd ))t
has the same copula as (X1, ...,Xd )t .Frechet Bounds
C� = max
(d
∑i=1ui + 1� d , 0
)� C (u) � min fu1, ..., udg = C+
Fr echet lower bound C� :: ;countermonotonicitycopula.(ρ = �1)Fr echet 2-dimensional upper bound C+(u1, u2) ::comonotonicity copula.(ρ = 1)
Independent Copula: i.e. Product Copula :
C?(u1, ..., ud ) =d
∏i=1ui .
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Introducing Copula to Risk Management
1.2 Properties
Invariance of Copula:The copula of F is invariant under strictly increasingtransformations.If T1, ...,Td are strictly increasing, then (T1(X1), ...,Td (Xd ))t
has the same copula as (X1, ...,Xd )t .Frechet Bounds
C� = max
(d
∑i=1ui + 1� d , 0
)� C (u) � min fu1, ..., udg = C+
Fr echet lower bound C� :: ;countermonotonicitycopula.(ρ = �1)Fr echet 2-dimensional upper bound C+(u1, u2) ::comonotonicity copula.(ρ = 1)
Independent Copula: i.e. Product Copula :
C?(u1, ..., ud ) =d
∏i=1ui .
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Introducing Copula to Risk Management
1.2 Properties
Figure: Distribution function plot of Frechet Bounds and Independentcopula.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Measure of Concordance
what is Concordance?Two points in R2, denoted by (x1, x2) and (x1, x2), are said tobe concordant if (x1 � x1)(x2 � x2) > 0and to be discordant if (x1 � x1)(x2 � x2) < 0 , e.g., if twopairs of observations from ra have same motion overtime thenit is said to be concordant and dependent to some extent.
Kendall�s tau
τ(X1,X2) = E (sign((X1 � X1)(X2 � X2)))
where (X1, X2) is an independent copy of (X1,X2).In bivariate case: τ = 4
R RI 2 C (u`, u2)dC (u1, u2)� 1 .
Gini index γ = 2R R
I 2(ju1 + u4 � 1j � ju1 � u2j)dC (u1, u2) isthe measure of inequality of distributions.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Measure of Concordance
what is Concordance?Two points in R2, denoted by (x1, x2) and (x1, x2), are said tobe concordant if (x1 � x1)(x2 � x2) > 0and to be discordant if (x1 � x1)(x2 � x2) < 0 , e.g., if twopairs of observations from ra have same motion overtime thenit is said to be concordant and dependent to some extent.
Kendall�s tau
τ(X1,X2) = E (sign((X1 � X1)(X2 � X2)))
where (X1, X2) is an independent copy of (X1,X2).In bivariate case: τ = 4
R RI 2 C (u`, u2)dC (u1, u2)� 1 .
Gini index γ = 2R R
I 2(ju1 + u4 � 1j � ju1 � u2j)dC (u1, u2) isthe measure of inequality of distributions.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Measure of Concordance
what is Concordance?Two points in R2, denoted by (x1, x2) and (x1, x2), are said tobe concordant if (x1 � x1)(x2 � x2) > 0and to be discordant if (x1 � x1)(x2 � x2) < 0 , e.g., if twopairs of observations from ra have same motion overtime thenit is said to be concordant and dependent to some extent.
Kendall�s tau
τ(X1,X2) = E (sign((X1 � X1)(X2 � X2)))
where (X1, X2) is an independent copy of (X1,X2).In bivariate case: τ = 4
R RI 2 C (u`, u2)dC (u1, u2)� 1 .
Gini index γ = 2R R
I 2(ju1 + u4 � 1j � ju1 � u2j)dC (u1, u2) isthe measure of inequality of distributions.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Correlation coe¢ cient
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Spearman�s rho:
ρS (X1,X2) = ρ(F1(X1),F2(X2))
or in bivariate case by : ρS = 12Z Z
I 2u1u2dC(u1, u2)� 3
is simply the linear correlation of probability transformed rv�s u.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Properties of Kendall�s tau and Spearman�s rho:
Both are measures of dependenceBoth τ and ρS 2 [�1, 1]Both τ and ρS = 0 if X1 and X2 are independent. butτ and ρS = 0; X1 and X2 are independent.τ and ρS = 1) X1 and X2 is comonotonic.τ and ρS = �1) X1 and X2 is countercomonotonic.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Properties of Kendall�s tau and Spearman�s rho:
Both are measures of dependenceBoth τ and ρS 2 [�1, 1]Both τ and ρS = 0 if X1 and X2 are independent. butτ and ρS = 0; X1 and X2 are independent.τ and ρS = 1) X1 and X2 is comonotonic.τ and ρS = �1) X1 and X2 is countercomonotonic.
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Upper tail dependence coe¢ cient λ
if a bivariate copula C is such that limu!1
C(u, u)1� u = λ exists
then it has upper tail dependence for λ 2 (0, 1] and no uppertail dependence for λ = 0.The measure λ is the probability that one variable is extremegiven that the other is extreme.
Intuitively, we �nd from the meta-distribution of di¤erentcopulas with Gaussian margins, that copulas di¤er in their tailbehavior
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Introducing Copula to Risk Management
1.3 Measure of Dependence
Upper tail dependence coe¢ cient λ
if a bivariate copula C is such that limu!1
C(u, u)1� u = λ exists
then it has upper tail dependence for λ 2 (0, 1] and no uppertail dependence for λ = 0.The measure λ is the probability that one variable is extremegiven that the other is extreme.
Intuitively, we �nd from the meta-distribution of di¤erentcopulas with Gaussian margins, that copulas di¤er in their tailbehavior
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
Gaussian copula, rho=0.3
t-copula, rho=0.3, v=4
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
The converse statement of Sklar�s Theorem provides a verypowerful technique for constructing multivariate distributions witharbitrary margins and copulas.
Intuitively : F , C � C , F
In other words: if we start with a copula C and marginsF1, ...,Fd then F (x) = C (F1(x1), ...,Fd (xd )) de�nes amultivariate df with marginF1, ...,Fd .
e.g. consider building a distribution with the Gaussian copulabut arbitrary margins, such a model is known as ameta-Gaussian distribution.
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
The converse statement of Sklar�s Theorem provides a verypowerful technique for constructing multivariate distributions witharbitrary margins and copulas.
Intuitively : F , C � C , F
In other words: if we start with a copula C and marginsF1, ...,Fd then F (x) = C (F1(x1), ...,Fd (xd )) de�nes amultivariate df with marginF1, ...,Fd .
e.g. consider building a distribution with the Gaussian copulabut arbitrary margins, such a model is known as ameta-Gaussian distribution.
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
The converse statement of Sklar�s Theorem provides a verypowerful technique for constructing multivariate distributions witharbitrary margins and copulas.
Intuitively : F , C � C , F
In other words: if we start with a copula C and marginsF1, ...,Fd then F (x) = C (F1(x1), ...,Fd (xd )) de�nes amultivariate df with marginF1, ...,Fd .
e.g. consider building a distribution with the Gaussian copulabut arbitrary margins, such a model is known as ameta-Gaussian distribution.
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
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Introducing Copula to Risk Management
1.4 Simulation of Copula and Meta Distribution
Applying upper tail dependence measure: λQuantitatively, λ ! 0 as u ! 1, for all ρ 2 [�1, 1] in Gaussian copula(Fig 1), whereas λ > 0 for all ρ 2 [�1, 1] in t-copula.(Fig 2). but asthe degree of freedom increases in t-copula (Fig 3), the measure of λ isasymptotically zero which indicates tail independence as in the Gaussioncase.
Quantile-dependent measure for the Gaussian Copula Student t-copula (v=1)
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Introducing Copula to Risk Management
Q-d measure for t-copula (v=5)
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copula
t-CopulaGumbel CopulaFrank CopulaClayton Copula...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copulat-Copula
Gumbel CopulaFrank CopulaClayton Copula...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copulat-CopulaGumbel Copula
Frank CopulaClayton Copula...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copulat-CopulaGumbel CopulaFrank Copula
Clayton Copula...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copulat-CopulaGumbel CopulaFrank CopulaClayton Copula
...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Gaussian Copulat-CopulaGumbel CopulaFrank CopulaClayton Copula...et al
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Implicit copula:
Gaussian Copula:
CGaP (u) = P(Φ(X1) � u1, ...,Φ(Xd ) � ud )= ΦP (Φ�1(u1), ...,Φ�1(ud ))
Bivariate case: CGap (u1, u2) =R Φ�1(u1)�∞
R Φ�1(u2)�∞
12π(1�ρ2)1/2 exp
n�(s21�2ρs1s2+s22 )
2(1�ρ2)
ods1ds2
t-Copula:
C dv ,P (u) = tv ,P (t�1v (u1), ..., t�1v (ud ))
where P is the correlation matrix.
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Explicit copula: (e.g. Archimedean Copula) :Gumbel
CGuθ (u1, u2) = expn�((� ln u1)θ + (� ln u2)θ)1/θ
o, 1 � θ < ∞
Frank
CFrθ (u1, u2) = �1θln(1+
(exp(�θu1)� 1) (exp(�θu2)� 1)exp(�θ)� 1 ), θ 2 R
Clayton
CClθ (u1, u2) = (u�θ1 + u�θ
2 � 1)�1/θ, 0 < θ < ∞
Generalized Clayton
CGCθ,δ (u1, u2) =n((u�θ
1 � 1)δ + (u�θ2 � 1)δ)1/δ + 1
o�1/θ, θ � 0, δ � 1
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Archimedean Copula:
Gumbel, Frank, Clayton, GC are all belong to theArchimedean Copula family which has the following form:
C (u1, ..., uN ) =�
ϕ�1(ϕ(u1), ..., ϕ(uN )), if ∑Nn=1 ϕ(un) � ϕ(0)
0 , otherwise
where ϕ(un) is so called generator of the copula
Arichmedean copula and related coe¢ cients are easy tocalculate.e.g. the Kendall�s tau is given by
τ = 1+ 4Z 1
0
ϕ(u)ϕ0(u)
du
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Introducing Copula to Risk Management
1.5 Copula Function and Copula Family
Extreme-value Copula JOE [1997]
The EV copula C satis�es the following relationship:
C(ut1, ..., utd ) = Ct (u1, ..., uN ) 8t > 0
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Introducing Copula to Risk Management
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Introducing Copula to Risk Management
Section II Risk Management Basics
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Introducing Copula to Risk Management
2.1 Value-at-Risk
Value-at-RIsk
A risk measure approach based on loss distribution;Given any con�dence level, e.g. 95%, and a holding period (1year), VaR is the 95 percentile of the loss distributionFL(l) = P(L � l);Or, the potential maximum loss under 95% chance.
De�nition (VaR)Given some con�dence level α 2 (0, 1).The value-at-risk ofour portfolio at the con�dence level is given by the smallestnumber l such that then probability that the loss L exceeds lis no larger than (1� α).
VaRα= inf fl 2 R : P(L > l) � 1� αg = inf fl 2 R : FL(l) � αg
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Introducing Copula to Risk Management
2.1 Value-at-Risk
Value-at-RIsk
A risk measure approach based on loss distribution;Given any con�dence level, e.g. 95%, and a holding period (1year), VaR is the 95 percentile of the loss distributionFL(l) = P(L � l);Or, the potential maximum loss under 95% chance.
De�nition (VaR)Given some con�dence level α 2 (0, 1).The value-at-risk ofour portfolio at the con�dence level is given by the smallestnumber l such that then probability that the loss L exceeds lis no larger than (1� α).
VaRα= inf fl 2 R : P(L > l) � 1� αg = inf fl 2 R : FL(l) � αg
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Introducing Copula to Risk Management
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Introducing Copula to Risk Management
2.2 Expected Shortfalls
Expected Shortfalls (ES) or Expected Tail Loss(ETL)De�nition (ES): For a loss L with E (jLj) < ∞ and df FL theexpected shortfall at con�dence level α 2 (0, 1), is de�ned as
ESα =1
1� α
Z 1
αqu(FL)du,
where qu(FL) = F�1L (u) is the quantile function of FL.OR ESα =
11�α
R 1α VaRudu, the weighted average of VaR over
(α, 1)
Estimation of VaR
Variance-covariance (VCV), assuming Normality distributionand Linear approximation.The Historical Simulation (HS), assuming that asset returnsin the future will have the same distribution as they had in thepast (historical market data),Monte Carlo simulation, where future asset returns arerandomly simulated.
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Introducing Copula to Risk Management
2.2 Expected Shortfalls
Expected Shortfalls (ES) or Expected Tail Loss(ETL)De�nition (ES): For a loss L with E (jLj) < ∞ and df FL theexpected shortfall at con�dence level α 2 (0, 1), is de�ned as
ESα =1
1� α
Z 1
αqu(FL)du,
where qu(FL) = F�1L (u) is the quantile function of FL.OR ESα =
11�α
R 1α VaRudu, the weighted average of VaR over
(α, 1)Estimation of VaR
Variance-covariance (VCV), assuming Normality distributionand Linear approximation.The Historical Simulation (HS), assuming that asset returnsin the future will have the same distribution as they had in thepast (historical market data),Monte Carlo simulation, where future asset returns arerandomly simulated.
![Page 55: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/55.jpg)
Introducing Copula to Risk Management
2.2 Expected Shortfalls
Expected Shortfalls (ES) or Expected Tail Loss(ETL)De�nition (ES): For a loss L with E (jLj) < ∞ and df FL theexpected shortfall at con�dence level α 2 (0, 1), is de�ned as
ESα =1
1� α
Z 1
αqu(FL)du,
where qu(FL) = F�1L (u) is the quantile function of FL.OR ESα =
11�α
R 1α VaRudu, the weighted average of VaR over
(α, 1)Estimation of VaR
Variance-covariance (VCV), assuming Normality distributionand Linear approximation.
The Historical Simulation (HS), assuming that asset returnsin the future will have the same distribution as they had in thepast (historical market data),Monte Carlo simulation, where future asset returns arerandomly simulated.
![Page 56: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/56.jpg)
Introducing Copula to Risk Management
2.2 Expected Shortfalls
Expected Shortfalls (ES) or Expected Tail Loss(ETL)De�nition (ES): For a loss L with E (jLj) < ∞ and df FL theexpected shortfall at con�dence level α 2 (0, 1), is de�ned as
ESα =1
1� α
Z 1
αqu(FL)du,
where qu(FL) = F�1L (u) is the quantile function of FL.OR ESα =
11�α
R 1α VaRudu, the weighted average of VaR over
(α, 1)Estimation of VaR
Variance-covariance (VCV), assuming Normality distributionand Linear approximation.The Historical Simulation (HS), assuming that asset returnsin the future will have the same distribution as they had in thepast (historical market data),
Monte Carlo simulation, where future asset returns arerandomly simulated.
![Page 57: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/57.jpg)
Introducing Copula to Risk Management
2.2 Expected Shortfalls
Expected Shortfalls (ES) or Expected Tail Loss(ETL)De�nition (ES): For a loss L with E (jLj) < ∞ and df FL theexpected shortfall at con�dence level α 2 (0, 1), is de�ned as
ESα =1
1� α
Z 1
αqu(FL)du,
where qu(FL) = F�1L (u) is the quantile function of FL.OR ESα =
11�α
R 1α VaRudu, the weighted average of VaR over
(α, 1)Estimation of VaR
Variance-covariance (VCV), assuming Normality distributionand Linear approximation.The Historical Simulation (HS), assuming that asset returnsin the future will have the same distribution as they had in thepast (historical market data),Monte Carlo simulation, where future asset returns arerandomly simulated.
![Page 58: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/58.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:
risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:
Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
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Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;
the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:
Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 60: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/60.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:
Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 61: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/61.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:
Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 62: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/62.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)
The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 63: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/63.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 64: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/64.jpg)
Introducing Copula to Risk Management
Variance-covariance Method
Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.
Weaknesses:Linearization may not always o¤er a good approximation of therelationship between the true loss distribution as the risk factorchanges (esp. large)The assumption of normality is unlikely to be realistic forheavy-tailed distribution of the risk factor changes. It tends tounderestimate the tail of the loss distribution and thus thenumber of VaR and ES that based on this tail.
PS: Also, for conditional time series models like multivariateGARCH, though the risk factor changes for the next tmeperiod is conditional on information up to the present, it isnot multivariate Gaussian, but rather a distribution whosemargins have heavier tails.
![Page 65: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/65.jpg)
Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
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Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;
weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
![Page 67: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/67.jpg)
Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
![Page 68: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/68.jpg)
Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
![Page 69: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/69.jpg)
Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.
In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
![Page 70: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/70.jpg)
Introducing Copula to Risk Management
Historical Simulation (HS)
strength: non-parametric;weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.
Monte Carlo Simulation (MC)
weakness: It doesn�t solve the problem of �nding amultivariate model for Xt+1.In practice: GARCH structure + heavy-tailed multivariateconditional distribution like multivariate t is desirable.
![Page 71: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/71.jpg)
Introducing Copula to Risk Management
Section III Application of Copula in RiskManagement: Some Examples
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Introducing Copula to Risk Management
Example 1:
Data: London Metal Exchange (LME)
Economic Capital Allocation
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Introducing Copula to Risk Management
Example 1:
Methods:
the empirical correlation ρ
the canonical correlation ρCML
mapping data to empirical uniformstransformed with the Φ�1(X ),the correlation is thencomputed for the transformed data.
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Introducing Copula to Risk Management
Example 2:
Stress Testing
X (x1, x2) = (CAC40, Dow Jones)
The EV-copula is given byG (χ+1 , ...,χ
+N ) = C�(G (χ
+1 ), ...,G (χ
+N ))
where G is the Generalized EV univariate distributions ofmaxima and minima of CAC40 and DowJones respectively.
Incorporated Gumbel copulaC�(u1, u2) = exp(�((� ln u1)δ + (� ln u2)δ)1/δ)with δ the dependence parameter
The Failure Area is then possible to construct given level ofprobability:Pr(χ+1 > χ1,χ
+2 > χ2) =
1� G1(χ1)� G2(χ2) + C�(G1(χ1),G2(χ2))
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Introducing Copula to Risk Management
Example 2:
Stress Testing
X (x1, x2) = (CAC40, Dow Jones)
The EV-copula is given byG (χ+1 , ...,χ
+N ) = C�(G (χ
+1 ), ...,G (χ
+N ))
where G is the Generalized EV univariate distributions ofmaxima and minima of CAC40 and DowJones respectively.
Incorporated Gumbel copulaC�(u1, u2) = exp(�((� ln u1)δ + (� ln u2)δ)1/δ)with δ the dependence parameter
The Failure Area is then possible to construct given level ofprobability:Pr(χ+1 > χ1,χ
+2 > χ2) =
1� G1(χ1)� G2(χ2) + C�(G1(χ1),G2(χ2))
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Introducing Copula to Risk Management
Example 2:
Stress Testing
X (x1, x2) = (CAC40, Dow Jones)
The EV-copula is given byG (χ+1 , ...,χ
+N ) = C�(G (χ
+1 ), ...,G (χ
+N ))
where G is the Generalized EV univariate distributions ofmaxima and minima of CAC40 and DowJones respectively.
Incorporated Gumbel copulaC�(u1, u2) = exp(�((� ln u1)δ + (� ln u2)δ)1/δ)with δ the dependence parameter
The Failure Area is then possible to construct given level ofprobability:Pr(χ+1 > χ1,χ
+2 > χ2) =
1� G1(χ1)� G2(χ2) + C�(G1(χ1),G2(χ2))
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Introducing Copula to Risk Management
Example 2:
Stress Testing
X (x1, x2) = (CAC40, Dow Jones)
The EV-copula is given byG (χ+1 , ...,χ
+N ) = C�(G (χ
+1 ), ...,G (χ
+N ))
where G is the Generalized EV univariate distributions ofmaxima and minima of CAC40 and DowJones respectively.
Incorporated Gumbel copulaC�(u1, u2) = exp(�((� ln u1)δ + (� ln u2)δ)1/δ)with δ the dependence parameter
The Failure Area is then possible to construct given level ofprobability:Pr(χ+1 > χ1,χ
+2 > χ2) =
1� G1(χ1)� G2(χ2) + C�(G1(χ1),G2(χ2))
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Introducing Copula to Risk Management
Example 2:
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Introducing Copula to Risk Management
Section IV Open Field for Future Researchand Useful Resources
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Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company
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Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company
![Page 82: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/82.jpg)
Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company
![Page 83: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/83.jpg)
Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company
![Page 84: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/84.jpg)
Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of Investment
Credit Risk in commercial bankingOperational Risk of Insurance company
![Page 85: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/85.jpg)
Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial banking
Operational Risk of Insurance company
![Page 86: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/86.jpg)
Introducing Copula to Risk Management
Open Fields
High dimensional copula modeling
Dynamic copula or time-varing copula
Back-testing copula and alternative approaches.
RM application
Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company
![Page 87: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/87.jpg)
Introducing Copula to Risk Management
Useful Resources
http://www.rgemonitor.com/
http://ideas.repec.org/a/kap/geneva/
http://www.math.ethz.ch/~embrechts/RM/
http://www.gloriamundi.org
![Page 88: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/88.jpg)
Introducing Copula to Risk Management
Useful Resources
http://www.rgemonitor.com/
http://ideas.repec.org/a/kap/geneva/
http://www.math.ethz.ch/~embrechts/RM/
http://www.gloriamundi.org
![Page 89: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/89.jpg)
Introducing Copula to Risk Management
Useful Resources
http://www.rgemonitor.com/
http://ideas.repec.org/a/kap/geneva/
http://www.math.ethz.ch/~embrechts/RM/
http://www.gloriamundi.org
![Page 90: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/90.jpg)
Introducing Copula to Risk Management
Useful Resources
http://www.rgemonitor.com/
http://ideas.repec.org/a/kap/geneva/
http://www.math.ethz.ch/~embrechts/RM/
http://www.gloriamundi.org
![Page 91: Introducing Copula to Risk Management Presentation](https://reader034.fdocuments.us/reader034/viewer/2022051211/55645c27d8b42a951e8b4b15/html5/thumbnails/91.jpg)
Introducing Copula to Risk Management
ThankYou!