introducing copula to risk management presentation

Introducing Copula to Risk Management


Wenting Li

SMU

April 2007


An Overview

Copula

RiskApplicationSummary


An Overview

CopulaRisk

ApplicationSummary


An Overview

CopulaRiskApplication

Summary


An Overview

CopulaRiskApplicationSummary


Section I Concepts and Properties of Copula


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.




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].








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].








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].






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 .


1.2 Properties


has the same copula as (X1, ...,Xd )t .Frechet Bounds

C� = max

(d

∑i=1ui + 1� d , 0

)� C (u) � min fu1, ..., udg = C+



C?(u1, ..., ud ) =d

∏i=1ui .


1.2 Properties



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)


C?(u1, ..., ud ) =d

∏i=1ui .


1.2 Properties



C� = max

(d

∑i=1ui + 1� d , 0

)� C (u) � min fu1, ..., udg = C+



C?(u1, ..., ud ) =d

∏i=1ui .


1.2 Properties

Figure: Distribution function plot of Frechet Bounds and Independentcopula.


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.



Correlation coe¢ cient



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.



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.



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


1.4 Simulation of Copula and Meta Distribution



Gaussian copula, rho=0.3

t-copula, rho=0.3, v=4



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.



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)


Q-d measure for t-copula (v=5)


1.5 Copula Function and Copula Family

Gaussian Copula

t-CopulaGumbel CopulaFrank CopulaClayton Copula...et al



Gaussian Copulat-Copula

Gumbel CopulaFrank CopulaClayton Copula...et al



Gaussian Copulat-CopulaGumbel Copula

Frank CopulaClayton Copula...et al



Gaussian Copulat-CopulaGumbel CopulaFrank Copula

Clayton Copula...et al



Gaussian Copulat-CopulaGumbel CopulaFrank CopulaClayton Copula

...et al



Gaussian Copulat-CopulaGumbel CopulaFrank CopulaClayton Copula...et al



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.



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



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



Extreme-value Copula JOE [1997]

The EV copula C satis�es the following relationship:

C(ut1, ..., utd ) = Ct (u1, ..., uN ) 8t > 0


Section II Risk Management Basics


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


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.




ESα =1

1� α

Z 1

αqu(FL)du,


11�α


(α, 1)Estimation of VaR





ESα =1

1� α

Z 1

αqu(FL)du,


11�α



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.




ESα =1

1� α

Z 1

αqu(FL)du,


11�α



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.




ESα =1

1� α

Z 1

αqu(FL)du,


11�α





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.



Assumption:risk factor returns are always (jointly) normally distributed;

the change in portfolio value is linearly dependent on all riskfactor returns.

Weaknesses:





Assumption:risk factor returns are always (jointly) normally distributed;the change in portfolio value is linearly dependent on all riskfactor returns.

Weaknesses:






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.





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.



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.



strength: non-parametric;

weakness: su¢ ciency of data; the historical changes ineconomic policy limits the use of historical data.







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.


Section III Application of Copula in RiskManagement: Some Examples


Example 1:

Data: London Metal Exchange (LME)

Economic Capital Allocation


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.


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))


Example 2:


Section IV Open Field for Future Researchand Useful Resources


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


Open Fields




RM application

Market Risk of Investment

Credit Risk in commercial bankingOperational Risk of Insurance company


Open Fields




RM application

Market Risk of InvestmentCredit Risk in commercial banking

Operational Risk of Insurance company


Open Fields




RM application

Market Risk of InvestmentCredit Risk in commercial bankingOperational Risk of Insurance company


Useful Resources

http://www.rgemonitor.com/

http://ideas.repec.org/a/kap/geneva/

http://www.math.ethz.ch/~embrechts/RM/

http://www.gloriamundi.org


ThankYou!

introducing copula to risk management presentation

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