Operational Risk and Pareto L´ evy Copulas Claudia Kl¨ uppelberg Technische Universit¨ at M¨ unchen email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - B¨ ocker, K. and Kl¨ uppelberg, C. (2005) Operational VaR - a closed form approximation. Risk, December 2005. - B¨ ocker, K. and Kl¨ uppelberg, C. (2008) First order approximations to operational risk - dependence and consequences. To appear in: G.N. Gregoriou (ed.) Operational Risk Toward Basel III, Best Practices and Issues in Modeling, Management and Regulation. Wiley, New York.
Contents (1) Basel II (2) The Loss Distribution Approach (LDA) for the Single Cell (3) Modelling Dependence of L´ evy Processes (4) The Multivariate Subexponential Compound Poisson (SCP) Model (5) Estimating Total OpVar
(1) Basel II www.bis.org/bcbs/bcbscp3.htmcp3 Structure of Risk Management: • Pillar 1: minimal capital requirements • Pillar 2: supervisory review of capital adequacy • Pillar 3: public disclosure Definition of Operational Risk: The risk of losses resulting from inadequate or failed internal processes, people and systems, or external events.
Examples: 1995 Barings Bank: Nick Leeson (1.3b British Pounds) 2001 Enron (largest US bankruptcy ever) 2005 Brokerhouse Mizuho Securities: instead of selling 1 share for 610 000 Yen a trader wrote 610 000 shares for 1 Yen each (190 Mio Euro) 2008 Societ´ e G´ en´ erale: Jerome Kerviel (4.9b Euro) Basel II distinguishes 7 loss types and 8 business types
(2) The Loss Distribution Approach (LDA) for the Single Cell Subexponential compound Poisson (SCP) model (1) The severities ( X k ) k ∈ N are positive iid random variables with subexponential distribution function F . (2) The frequency process N ( t ) of loss events in the time interval [0 , t ] for t ≥ 0 constitutes a homogenous Poisson process with intensity λ > 0 . In particular, P ( N ( t ) = n ) = p t ( n ) = e − λt ( λt ) n n ! , n ∈ N 0 . (3) The severity process and the frequency process are independent . (4) The aggregate loss process is given by N ( t ) � S ( t ) = X k , t ≥ 0 . k =1
Let ( X k ) k ∈ N be iid random variables with distribution function F . Then F is said to be subexponential ( F ∈ S ) if P ( X 1 + · · · + X n > x ) lim P (max( X 1 , . . . , X n ) > x ) = 1 for some (all) n ≥ 2 . x →∞ Denote F ( x ) := 1 − F ( x ) = P ( X > x ) for x > 0 the distribution tail of F . If for some α ≥ 0 , F ( xt ) F ( t ) = x − α , lim x > 0 , t →∞ then F is called regularly varying with index − α , denoted by F ∈ R − α . The quantity α is also called the tail index of F or X . See Embrechts, Kl¨ uppelberg and Mikosch (1997) for details.
The aggregate loss distribution function: G t ( x ) = P ( S ( t ) ≤ x ) ∞ � = p t ( n ) P ( S ( t ) ≤ x | N ( t ) = n ) n =0 ∞ � p t ( n ) F n ∗ ( x ) , = x ≥ 0 , t ≥ 0 , n =0 where F is the distribution function of X k , and F n ∗ ( · ) = P ( � n i =1 X i ≤ · ) with F 1 ∗ = F and F 0 ∗ = I [0 , ∞ ) . Operational VaR up to time t at confidence level κ : VaR t ( κ ) = G ← t ( κ ) = inf { x ∈ R : G t ( x ) ≥ κ } , κ ∈ (0 , 1) , is the (left continuous) generalized inverse of G t . If G t is strictly increasing and continuous, then VaR t ( κ ) = G − 1 t ( κ ) .
Theorem [Embrechts, Kl¨ uppelberg and Mikosch, Theorem 1.3.9] Consider the standard LDA S ( t ) = � N ( t ) n =0 X i , t ≥ 0 . Assume that the severities X i have distribution function F ∈ S . Fix t > 0 and define the frequency distribution by P ( N ( t ) = n ) = p t ( n ) for n ∈ N 0 . Then, the aggregate loss distribution is given by ∞ � p t ( n ) F n ∗ ( x ) , G t ( x ) = x ≥ 0 , t ≥ 0 . n =0 Assume that for some ε > 0 , ∞ � (1 + ε ) n p t ( n ) < ∞ . n =0 Then for fixed t > 0 , G t ( x ) = EN ( t ) F ( x )(1 + o (1)) ∼ EN ( t ) F ( x ) , x → ∞ .
Theorem [Analytical OpVaR] Consider the SCP model for fixed t > 0 and a subexponential severity with distribution function F . Then as κ ↑ 1 , � 1 � ← � λ t � � � 1 − 1 − κ VaR t ( κ ) = F ← λ t (1 + o (1)) = 1 − κ (1 + o (1)) . F Proposition [Bingham, Goldie and Teugels (1987)] Let α ∈ (0 , ∞ ) . F ∈ R − α if and only if (1 /F ) ← ∈ R 1 /α . (1) G ( x ) ∼ cF ( x ) as x → ∞ iff (1 /F ) ← ( z ) ∼ (1 /G ) ← ( cz ) as z → ∞ . (2) F ( x ) = x − α L ( x ) for x ≥ 0 if and only if (1 /F ) ← ( z ) = z 1 /α � (3) L ( z ) for z ≥ 0 , where L and � L are slowly varying functions. �
Corollary Assume that F ∈ R − α for α ∈ (0 , ∞ ) . Then by (2), as κ ↑ 1 , � 1 � ← � λt � 1 � � ← � � λ ∼ t 1 /α VaR t ( κ ) ∼ 1 − κ 1 − κ F F � λ t � λ t � 1 /α � � t 1 /α VaR 1 ( κ ) ∼ = L . � 1 − κ 1 − κ Remark If 1 /F ∈ R α for α ∈ (0 , ∞ ] ( α = ∞ means that F ( t ) /F ( xt ) → ∞ for x > 1 and F ( t ) /F ( xt ) → 0 for x < 1 ), then � 1 � ← � λ t � VaR t ( κ ) ∼ , κ ↑ 1 . 1 − κ F
Popular subexponential severity distributions Name Distribution function Parameters � ln x − µ � Lognormal F ( x ) = Φ µ ∈ R , σ > 0 σ F ( x ) = 1 − e − ( x/θ ) τ Weibull θ > 0 , 0 < τ < 1 � � − α 1 + x Pareto F ( x ) = 1 − α, θ > 0 θ
Approximated VaR (dashed line) and simulated VaR (solid line) for the Pareto-Poisson LDA with θ = 1 . 40000 alpha = 1.5 alpha = 1.1 5000 30000 4000 VaR VaR 3000 20000 2000 10000 0.998 0.9985 0.999 0.9995 0.998 0.9985 0.999 0.9995 Confidence Level Confidence Level
(3) Modelling Dependence of L´ evy Processes Invoking the copula idea: A d -dimensional copula C is a distribution function on [0 , 1] d with standard uniform marginals. Theorem [Sklar’s Theorem] Let F be a joint distribution function with marginals F 1 , . . . , F d . Then there exists a copula C : [0 , 1] d → [0 , 1] such that for all x 1 , . . . , x d ∈ R = [ −∞ , ∞ ] F ( x 1 , . . . , x d ) = C ( F 1 ( x 1 ) , . . . , F d ( x d )) . (1) If the marginals are continuous, then C is unique. Otherwise it is unique on Ran F 1 × · · · × Ran F d . Conversely, if C is a copula and F 1 , . . . , F d are distribution functions, then the function F as defined in (1) is a joint distribution function with marginals F 1 , . . . , F d . �
Question Can we use copulas to model dependence of L´ evy processes? Our LDA model is a compound Poisson process with positive jumps. Problems The law of a L´ evy process X is completely determined by the distribution of X at time t for any t > 0 . The copula C t of ( X 1 ( t ) , . . . , X d ( t )) may depend on t . In general, C s cannot be calculated from C t , because C s depends also on the marginal distributions. For given infinitely divisible marginal distributions it is unclear, which copulas C t yield multivariate infinite divisible distributions. (Copulas are invariant under strictly increasing transformations, infinite divisibility is not!) Introduce a L´ evy copula [ Cont & Tankov (2004), Kallsen & Tankov (2004), Barndorff-Nielsen & Lindner (2004).
Problem L´ evy measures may have a non-integrable singularity at 0. Define E := [ 0 , ∞ ] d \ { 0 } . evy process in R d with a L´ Let X be a spectrally positive L´ evy measure Γ , which has standard 1-stable one-dimensional marginals. Then we call Γ a Pareto L´ evy measure and the associated tail measure Γ( x ) = Γ([ x 1 , ∞ ) × · · · × [ x d , ∞ )) =: � C ( x 1 , . . . , x d ) , x ∈ E , evy copula � is called Pareto L´ C . Remark Extension to general L´ evy processes by quadrantwise definition (singularity in 0!).
evy process in R d with L´ Lemma Let X be a spectrally positive L´ evy measure Π on E and continuous marginal tail measures Π 1 , . . . , Π d , where Π i ( x ) := Π([ x, ∞ )) for i = 1 , . . . , d . Then � � 1 1 Π( x ) = Π([ x 1 , ∞ ] × · · · × [ x d , ∞ ]) = � C Π 1 ( x 1 ) , . . . , , x ∈ E , Π d ( x d ) and � C is a Pareto L´ evy copula. �
Theorem [Sklar’s Theorem for Pareto L´ evy copulas] Let Π be the tail measure of a d -dimensional spectrally positive L´ evy process with marginal tail measures Π 1 , . . . , Π d . Then there exists a Pareto L´ evy copula � C : E → [0 , ∞ ] such that � � 1 1 Π( x ) = � C Π 1 ( x 1 ) , . . . , . (2) Π d ( x d ) If the marginal tail measures are continuous on [0 , ∞ ] , then � C is unique. � � � � 1 1 Otherwise, it is unique on Ran × · · · × Ran . Π 1 Π d Conversely, if � C is a Pareto L´ evy copula and Π 1 , . . . , Π d are marginal tail measures, then Π as defined in (2) is a joint tail measure with marginals Π 1 , . . . , Π d . �
Examples of Pareto L´ evy copulas Example Clayton Pareto L´ evy copula (special Archimedian Pareto L´ evy copula) � ϑ � − 1 /ϑ ϑ + · · · + x d � C ϑ ( x 1 , . . . , x d ) = x 1 Note: lim ϑ →∞ � C θ ( x 1 , . . . , x d ) = � C � ( x 1 , . . . , x d ) complete positive dependence , lim ϑ → 0 � C ϑ ( x 1 , . . . , x d ) = � C ⊥ ( x 1 , . . . , x d ) independence . evy copula � A Pareto L´ C is homogeneous (of order 1) , if for all t > 0 C ( x 1 , . . . , x d ) = t � � C ( tx 1 , . . . , tx d ) , ( x 1 , . . . , x d ) ∈ E .
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