Operational Risk and Pareto Lévy Copulas

Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR - a closed form approximation. Risk, December 25. - Böcker, K. and Klüppelberg, C. (28) 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 Lévy 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) 21 Enron (largest US bankruptcy ever) 25 Brokerhouse Mizuho Securities: instead of selling 1 share for 61 Yen a trader wrote 61 shares for 1 Yen each (19 Mio Euro) 28 Societé Générale: 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 [,t] for t constitutes a homogenous Poisson process with intensity λ >. In particular, λt (λt)n P(N(t) = n) = p t (n) = e, n N. n! (3) The severity process and the frequency process are independent. (4) The aggregate loss process is given by S(t) = N(t) k=1 X k, t.

Let (X k ) k N be iid random variables with distribution function F. Then F is said to be subexponential (F S) if lim x P(X 1 + + X n > x) P(max(X 1,...,X n ) > x) = 1 for some (all) n 2. Denote F(x) := 1 F(x) = P(X > x) for x > the distribution tail of F. If for some α, lim t F(xt) F(t) = x α, x >, 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, Klüppelberg and Mikosch (1997) for details.

Theorem [Analytical OpVaR] Consider the SCP model for fixed t > and a subexponential severity with distribution function F. (a) Assume that F R α for α (, ). Then as κ 1, ( ) ( ) ( ) ( ) 1 λt 1 λ VaR t (κ) t 1/α F 1 κ F 1 κ ( ) 1/α ( ) λt λt = t 1/α VaR 1 (κ) L. 1 κ 1 κ (b) If 1/F R α for α (, ] (α = means that F(t)/F(xt) for x > 1 and F(t)/F(xt) for x < 1), then VaR t (κ) ( 1 F ) ( ) λt, κ 1. 1 κ

Popular subexponential severity distributions Name Distribution function Parameters Lognormal ( ) lnx µ F(x) = Φ σ µ R, σ > Weibull F(x) = 1 e (x/θ)τ θ >, < τ < 1 Pareto ( F(x) = 1 1 + x α θ) α,θ >

Approximated VaR (dashed line) and simulated VaR (solid line) for the Pareto-Poisson LDA with θ = 1. 5 alpha = 1.5 4 alpha = 1.1 VaR 4 3 VaR 3 2 2.998.9985.999.9995 Confidence Level 1.998.9985.999.9995 Confidence Level

(3) Modelling Dependence of Lévy Processes Invoking the copula idea: A d-dimensional copula C is a distribution function on [,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 : [, 1] d [, 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 Lévy processes? Our LDA model is a compound Poisson process with positive jumps. Problems The law of a Lévy process X is completely determined by the distribution of X at time t for any t >. 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 Lévy copula [Cont & Tankov (24), Kallsen & Tankov (24), Barndorff-Nielsen & Lindner (24).

Problem Lévy measures may have a non-integrable singularity at. Define E := [, ] d \ {}. Let X be a spectrally positive Lévy process in R d with a Lévy measure Γ, which has standard 1-stable one-dimensional marginals (Γ i (x) = x 1 for x > ). Then we call Γ a Pareto Lévy measure and the associated tail measure Γ(x) = Γ([x 1, ) [x d, )) =: Ĉ(x 1,...,x d ), x E, is called Pareto Lévy copula Ĉ. Remark Extension to general Lévy processes by quadrantwise definition (singularity in!).

Lemma Let X be a spectrally positive Lévy process in R d with Lévy measure Π on E and continuous marginal tail measures Π 1,...,Π d, where Π i (x) := Π([x, )) for i = 1,...,d. Then ( Π(x) = Π([x 1, ] [x d, ]) = Ĉ 1 Π 1 (x 1 ),..., 1 ) Π d (x d ), x E, and Ĉ is a Pareto Lévy copula.

Theorem [Sklar s Theorem for Pareto Lévy copulas] Let Π be the tail measure of a d-dimensional spectrally positive Lévy process with marginal tail measures Π 1,...,Π d. Then there exists a Pareto Lévy copula Ĉ : E [, ] such that ( Π(x) = Ĉ 1 Π 1 (x 1 ),..., 1 Π d (x d ) If the marginal tail measures are ( continuous on [, ( ], then 1 1 Otherwise, it is unique on Ran Ran. Π 1 ) Π d ) ). (2) Ĉ is unique. Conversely, if Ĉ is a Pareto Lévy 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 Lévy copulas Example Clayton Pareto Lévy copula (special Archimedian Pareto Lévy copula) Ĉ ϑ (x 1,...,x d ) = (x 1 ϑ + + x d ϑ ) 1/ϑ Note: lim ϑ Ĉ θ (x 1,...,x d ) = Ĉ (x 1,...,x d ) complete positive dependence, lim ϑ Ĉ ϑ (x 1,...,x d ) = Ĉ (x 1,...,x d ) independence. A Pareto Lévy copula Ĉ is homogeneous (of order 1), if for all t > Ĉ(x 1,...,x d ) = t Ĉ(tx 1,...,tx d ), (x 1,...,x d ) E.

θ =.3 1/2 stable severities, Clayton Levy copula: θ=.3 Jumps: 12.6 X 1.4.2 8 X, Y.1.2.3.4.5.6.7.8.9 1.6.7.8.9 1.6.7.8.9 1.6.7.8.9 1.6.7.8.9 1.6.7.8.9 1 t 6 8 4 Y 6 2 4 2.1.2.3.4.5.6.7.8.9 1.1.2.3.4.5 t t θ =2 1/2 stable severities, Clayton Levy copula: Jumps: θ=2 1.5 1 X.8.6.4 1.2 X, Y.1.2.3.4.5 t.2.5 Y.15.1.5.1.2.3.4.5.6.7.8.9 1.1.2.3.4.5 t t θ =1 1/2 stable severities, Clayton Levy copula: θ=1 Jumps:.5.15 X.45.4.5.35.3.1.2.3.4.5 t.25.2.15.15 Y X, Y.1.1.1.5.5.1.2.3.4.5 t.6.7.8.9 1.1.2.3.4.5 t

(4) The Multivariate SCP Model (1) All operational risk cells, indexed by i = 1,...,d, are described by an SCP model with aggregate loss process S i, continuous subexponential severity distribution function F i and Poisson parameter λ i >. (2) Dependence between different cells is modelled by a Pareto Lévy copula: Let Π i : [, ) [, ) be the tail measure to S i, i.e. Π i ( ) = λ i F i ( ), and let Ĉ : E [, ] be a Pareto Lévy copula. Then ( ) Π(x 1,...,x d ) = Ĉ 1,..., 1 λ 1 F 1 (x 1 ) λ d F d (x d ) defines the tail measure of the compound Poisson process S = (S 1,...,S d ). (3) The bank s total aggregate operational loss process is defined as with tail measure S + (t) = S 1 (t) + S 2 (t) + + S d (t), t, Π + (z) = Π({(x 1,...,x d ) [, ) d : d i=1 x i z}), z.

Proposition Consider the multivariate SCP model. Its total aggregate loss process S + is compound Poisson with intensity λ + = lim z Π + (z) and severity distribution F + (z) = 1 F + (z) = Π+ (z) λ +, z. Consider the multivariate SCP model with total aggregate loss S + (t) at time t > and denote G + t ( ) = P(S + (t) ). Total Operational VaR up to time t at confidence level κ is defined as VaR + t (κ) = G + t (κ), κ (,1), with G + t (κ) = inf{z R : G + t (z) κ} for < κ < 1.

(5) Estimating Total OpVar One cell dominant Theorem For fixed t > let S i (t) for i = 1,...,d have compound Poisson distributions. Assume that F 1 R α for α >. Let ρ > α and suppose that E[(X i ) ρ ] < for i = 2,...,d. Then regardless of the dependence structure between (S 1 (t),...,s d (t)), P(S 1 (t) + + S d (t) > x) EN 1 (t) P(X 1 > x), x, ( VaR + t (κ) F1 1 1 κ ) = VaR 1 EN 1 (t) t(κ), κ 1.

Multivariate SCP model with completely dependent cells - All cell processes jump together = λ := λ 1 = = λ d. - The mass of the Lévy measure is concentrated on {(x 1,...,x d ) (, ) d : Π 1 (x 1 ) = = Π d (x d )} = {(x 1,...,x d ) (, ) d : F 1 (x 1 ) = = F d (x d )}. Let F i be strictly increasing and continuous: F 1 i (q) exists for all q [, 1). Then Π + (z) = Π({(x 1,...,x d ) (, ) d : = Π 1 ({x 1 (, ) : x 1 + d i=2 d x i z}) i=1 F 1 i (F 1 (x 1 )) z}), z >.

Set H(x 1 ) := x 1 + d invertible. Thus i=2 F 1 i (F 1 (x 1 )) for x 1 (, ) and note that it is Π + ( (z) = Π 1 ({x 1 (, ) : x 1 H 1 (z)}) = Π 1 H 1 (z) ), z >. Theorem Consider a multivariate SCP model with completely dependent cell processes S 1,...,S d and strictly increasing and continuous severity distributions F i. Then, S + is compound Poisson with parameters λ + = λ and F + ( (z) = F 1 H 1 (z) ). If F + S R α for α (, ], then VaR + t (κ) d VaR i t(κ), κ 1, i=1 where VaR i t( ) denotes the stand alone OpVaR of cell i.

Corollary Assume that the conditions of the Theorem hold and that F 1 R α for α (, ) and F i (x) lim x F 1 (x) = c i [, ). Assume that c i for i = 1,...,b d and c i = for i = b + 1,...,d. Then VaR + t (κ) b i=1 c 1/α i VaR 1 t(κ), κ 1.

Multivariate SCP model with independent cells - Not two cell processes ever jump together. - The mass of the Lévy measure is concentrated on the axes. Π + (z) = Π 1 (z 1 ) + + Π d (z d ). Theorem Assume S 1,...,S d are independent. Then S + defines a one-dimensional SCP model with parameters λ + = λ 1 + + λ d and F + (z) = 1 λ + [ λ1 F 1 (z) + + λ d F d (z) ], z. If F 1 R α for α (, ) and for all i = 2,...,d, lim x F i (x) F 1 (x) = c i [, ), then, setting C λ = λ 1 + c 2 λ 2 + + c d λ d, VaR + t (κ) ( 1 F 1 ) ( ) Cλ t 1 κ = VaR 1 t ( ) Cλ t 1 κ, κ 1.

Multivariate SCP model with regularly varying Lévy measure (a) Let Π be a Lévy measure of a spectrally positive Lévy process in R d. Assume that there exists a Radon measure ν on E such that for x E lim u Π({y : y 1 > ux 1 or or y d > ux d } Π 1 (u) = ν({y : y 1 > x 1 or or y d > x d }) =: ν([,x] c ). Then we call Π multivariate regularly varying. (b) The measure ν has a scaling property: there exists some α > such that for every s > ν([, sx] c ) = s α ν([,x] c ), and Π is called multivariate regularly varying with index α.

Some result for multivariate regularly varying and stable processes Theorem Let the spectrally positive Lévy process X in R d have tail measure Π. Then Π is multivariate regularly varying with index α > if and only if (1) at least one marginal tail measure Π i R α for α > and all other marginal tail measures have the same order or are lighter, and (2) Γ is multivariate regularly varying with index 1. Theorem [Kallsen and Tankov (26)] For < α < 2 let X be a Lévy process in R d. The process X is α-stable if and only if it has α-stable one-dimensional marginals and it has a Pareto Lévy copula, which is homogeneous of order 1.

Theorem Consider an SCP model with multivariate regularly varying cell processes (S 1,...,S d ) with index α and limit measure ν. Assume further that the severity distributions F i for i = 1,...,d are strictly increasing and continuous. Then, S + is compound Poisson with parameters λ + F + (x) ν + (1, )λ 1 F 1 (x), x. where ν + (z, ] = ν{x : d i=1 x i > z} for z >. Furthermore, λ + F + ( ) R α and total OpVaR is asymptotically given by VaR t (κ) F 1 ( 1 ) 1 κ t λ 1 ν + (1, ], κ 1.

Example [Clayton Pareto Lévy copula] Assume that F 2 (x)/f 1 (x) c as x. Set c := (λ 2 /λ 1 )c. [( ν + (1, ] = 1 + c 1/α E c 1/α + Y 1/α ϑ ) α 1)], where Y ϑ is a positive random variable with density g(s) = (1 + s ϑ ) 1/ϑ 1. For α = 1 we have ν + (1, ] = 1 + c, independent of ϑ. Consequently, total OpVar is for all < ϑ < asymptotically equal to the independent OpVar. If αϑ = 1, then ν + (1, ] = c1+1/α 1 c 1/α 1

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