First Order Approximations to Operational Risk Dependence. and Consequences

Transcription

1 Frs Order Approxmaons o Operaonal Rsk Dependene and Consequenes Klaus Böker and Clauda Klüppelberg 2 May 20, 2008 Absra We nvesgae he problem of modellng and measurng muldmensonal operaonal rsk. Based on he very popular unvarae loss dsrbuon approah, we sugges an nvarane prnple whh should be sasfed by any muldmensonal operaonal rsk model, and whh s naurally fulflled by our modellng ehnque based on he new onep of Pareo Lévy opulas. Our approah allows for a fully dynam modellng of operaonal rsk a any fuure pon n me. We explo he fa ha operaonal loss daa are ypally heavy-aled, and, herefore, we nensvely dsuss he onep of mulvarae regular varaon, whh s onsdered as a very useful ool for varous mulvarae heavy-aled phenomena. Moreover, for mporan examples of he Pareo Lévy opulas and approprae severy dsrbuons we derve frs order approxmaons for mulvarae operaonal Value-a-Rsk. Rsk Inegraon, Reporng & Poles - Head of Rsk Analys and Mehods - UnCred Group, Munh Branh, /o: HypoVerensbank AG, Arabellasrasse 2, D Münhen, Germany, e-mal: klaus.boeker@unredgroup.de, 2 Cener for Mahemaal Senes, Munh Unversy of Tehnology, D Garhng, Germany, e-mal: klu@ma.um.de, hp://

2 . Inroduon: Three years ago, n Böker and Klüppelberg (2005), we argued ha operaonal rsk ould be a long-erm kller, and a hs me maybe he mos speaular example for a bank falure aused by operaonal rsk losses was Barngs Bank afer he rogue rader Nk Leeson had been hdng loss-makng posons n fnanal dervaves. In he meanwhle oher examples aheved doubful fame, mos reenly Soéé Générale's loss of EUR 4.9 bllon due o rader fraud and Bear Searns near-deah sne was no able o pre s morgage porfolos. Suh examples learly show he nreased mporane of a sound and relable operaonal rsk managemen, whh onsss of rsk denfaon, monorng and reporng, rsk mgaon, rsk onrollng, and las bu no leas rsk quanfaon. Needless o say, suh aasroph losses as menoned above would have never been prevened jus my measurng an operaonal Value-a-Rsk (OpVaR). Ofen rsk mgaon s prmarly a maer of hghly effeve managemen and onrol proesses. In he ase of Soéé Générale, for nsane, he queson s how Jerome Kervel was able o hde hs massve speulave posons o he Dow Jones Eurosoxx 50 jus by offseng hem wh fous rades no he bankng sysem. Havng sad hs, le us brefly no only for a movaon o read hs arle onsder he queson regardng he relevane of operaonal rsk modellng. Maybe he smples answer would be a referene o he regulaory requremens. Indeed, wh he new framework of Basel II, he quanfaon of operaonal rsk has beome a ondo sne qua non for every fnanal nsuon. In hs respe, he man nenon for he so-alled advaned measuremen approahes (AMA) s o alulae a apal harge as a buffer agans poenal operaonal rsk losses. Anoher reason for buldng models, besdes of makng predons, s ha models an help us o gan a deeper undersandng of a subje maer. Ths s one of our nenons n 2

3 wrng hs arle. We presen a relavely smple model wh only a few parameers, whh allows us o gan neresng nsgh no he general behavour of mulvarae operaonal rsk. Furhermore, our approah s appealng from a purely model-heore pon of vew beause, as we wll show n more deal below, essenally s a sraghforward generalzaon of he very popular loss dsrbuon approah (LDA) o any dmenson. The key feaure of he onedmensonal, sandard LDA model s splng up he oal loss amoun over a eran me perod no a frequeny omponen,.e. he number of losses, and a severy omponen,.e. he ndvdual loss amouns. In dong so, we assume he oal aggregae operaonal loss of a bank up o a me horzon 0 whh an be represened as o be represened by a ompound Posson proess ( ) 0 X, N = X = X, 0. (. ) Le us denoe he dsrbuon funon of X by ( ) ( = ) G P X. As rsk measure we use oal OpVaR up o me a onfdene level κ, whh s defned as he quanle { } ( κ ) = G ( κ ) = nf z G ( z ) κ κ, (.2 ) O pvar :, 0, for κ near, e.g for regulaory purposes or, n he onex of a bank's nernal eonom apal, even hgher suh as Now, s undoubed among expers and sasally jusfed by Mosadell (2004) ha operaonal loss daa are heavy-aled, and herefore we onenrae on Pareo-lke severy dsrbuons. In general, a severy dsrbuon funon F s sad o be regularly varyng wh ndex α for α> ( ) 0 α F R, f () F x lm = x α, x > 0. F 3

4 For suh heavy aled losses and aually also for he more general lass wh subexponenal dsrbuon funons s now well-known and a onsequene of Theorem.3.9 of Embrehs e. al (997) (see Böker and Klüppelberg (2005), Böker and Sprulla (2006)) ha OpVaR a hgh onfdene levels an be approxmaed by κ Op VaR ( κ ): = G ( κ) ~ F, κ λ (.3 ) wh λ = E N. Fgure abou here Fgure : Qualy of approxmaon for a ompound Posson model wh Pareo loss dsrbuon. Usually, however, oal operaonal rsk s no modelled by (. ) drely, nsead, operaonal rsk s lassfed n dfferen loss ypes and busness lnes. For nsane, Basel II dsngushes 7 loss ypes and 8 busness lnes, yeldng a marx of 56 operaonal rsk ells. Then, for eah ell =,, d operaonal loss s modelled by a ompound Posson proess model ( X ) 0, and he bank's oal aggregae operaonal loss s gven as he sum X X X X. 2 = d, 0 The ore problem of mulvarae operaonal rsk modellng here s, how o aoun for he dependene sruure beween he margnal proesses. Several proposals have been made, see e.g. Fraho e al. (2004), Powojowsk e al. (2002), or Chavez-Demouln e al. (2005) jus o 4

5 menon a few. In general, however, all hese approahes lead o a oal aggregae loss proess( X ) 0 framework of, whh s no ompound Posson anymore and, hus, does no f no he.. More generally, s reasonable o demand ha ( ) 0 X does no depend on he desgn of he ell marx,.e wheher he bank s usng 56 or 20 ells whn s operaonal rsk model should n prnpal (.e. absrang from sasal esmaon and daa ssues) no affe he bank's oal OpVaR. In oher words, a naural requremen of a mulvarae operaonal rsk model s ha s nvaran under a re-desgn of he ell marx and, hus, also under possble busness re-organzaons. Hene, we demand ha every model should be losed wh respe o he ompound Posson propery,.e. every addve omposon of dfferen ells mus agan onsue a unvarae ompound Posson proess wh severy dsrbuon funon F j () and frequeny parameer λ j for j : X X j : = X j ompound Posson proesses. (.4 ) The nvarane prnple formulaed n (.4 ) holds rue, whenever he veor of all ell d proesses (,..., ) X X onsues a d -dmensonal ompound Posson proess. The 0 dependene sruure beween he margnal proesses s hen desrbed by a so-alled Lévy opula, or, as we wll do n hs arle, by means of a Pareo Lévy opula. 2. From Pareo opulas o Pareo Lévy opulas Margnal ransformaons have been ulsed n varous felds. Ceranly he mos promnen n he fnanal area s he vory marh of he opula, nvokng margnal ransformaons resulng n a mulvarae dsrbuon funon wh unform margnals. Whereas s eranly 5

6 onvenen o auomze eran proedures suh as he normalsaon o unform margnals, hs ransformaon s no always he bes possble hoe. So, was poned ou e.g. by Klüppelberg and Resnk (2008) ha, when asympo lm dsrbuons and heavy al behavour of daa s o be nvesgaed, a ransformaon o sandardsed Pareo dsrbued margnals s muh more naural han he ransformaon o unform margnals. The analog ehnque, however appled o he Lévy measure, wll prove o be useful also for our purpose, namely he examnaon of mulvarae operaonal rsk. Before we do hs n some deal, le us brefly reap some of he argumens gven n Klüppelberg and Resnk (2008). Le = (,..., ) Χ Χ Χ d be a random veor n d wh dsrbuon funon F and one- dmensonal margnal dsrbuon funons ( ) = ( ) F Ρ Χ and assume hroughou ha hey are onnuous. Defne for d wh ( ) = ( ) ( 2. ) F F. Noe ha P s sandard Pareo dsrbued;.e., for =,..., d holds ( P > ) = P x x, x. Defnon 2.. Suppose X has d.f. F wh onnuous margnals. Defne P as n (2.). Then we all he dsrbuon C of P a Pareo opula. Analogously o he sandard dsrbuonal opula, he Pareo opula an be used o desrbe he dependene sruure beween dfferen random varables. Here, we do no use dsrbuonal opulas drely o model he dependene sruure beween he ells' aggregae loss proesses ( X ) 0. One reason s ha, as desrbed n he 6

7 nroduon, we are lookng for a naural exenson of he sngle ell LDA model,.e. we requre ha also ( X ) 0 s ompound Posson. Ths an be aheved by explong he fa ha a ompound Posson proess s a spef Lévy proess, whh allows us o nvoke some Lévy sruure analyss o derve OpVaR resuls. Our approah s smlar o Böker and Klüppelberg (2006, 2008), where we used sandard Lévy opulas o derve analy approxmaons for OpVaR. In hs arle, however, we use a ransformaon smlar o ( 2. ), whh leads us o he onep of Pareo Lévy opulas. For a Lévy proess he jump behavour s governed by he so-alled Lévy measure Π, whh has a very nuve nerpreaon, n parular n he onex of operaonal rsk. The Lévy measure of a sngle operaonal rsk ell measures he expeed number of losses per un me wh a loss amoun n a pre-spefed nerval. Moreover, for our ompound Posson model, he Lévy measure Π of he ell proess X s ompleely deermned by he frequeny parameer λ > 0 and he dsrbuon funon of he ell's severy, namely ([ 0, x] ): P( X x) F ( x ) for [ ) Π =λ =λ x 0,. Sne here we are manly neresed n large operaonal losses, s onvenen o nrodue he onep of a al measure, somemes also referred o as al negral. A one-dmensonal al measure s smply he expeed number of losses per un me ha are above a gven hreshold, whh s n he ase of a ompound Posson model gven by: ( x) : [ x, ) P X x F x, x [ 0, ) ( 2.2 ) Π =Π =λ > =λ In parular, here s only a fne number of jumps per un me,.e. lm x 0 Π x =λ. Analogously, for a mulvarae Lévy proess he mulvarae Lévy measure onrols he jump behavour (per un me) of all unvarae omponens and onans all nformaon of 7

8 dependene beween he omponens. Hene, n hs framework, dependene modellng beween dfferen operaonal rsk ells s redued o hoosng approprae mulvarae Lévy measures. Sne jumps are reaed by posve loss severes, he Lévy measure Π s onenraed on he punured posve one n R d (he value 0 s aken ou sne Lévy measures an have a sngulary n 0 ) [ ] { } E : = 0, 0. Now, smlarly o he fa ha a mulvarae dsrbuon an be bul from margnal dsrbuons va a dsrbuonal (Pareo) opula, a mulvarae al measure (see also Böker and Klüppelberg (2006), Defnon 2.) ( x,..., xd) ([ x, )... [ xd, )) Π x = Π =Π, E x, ( 2.3 ) an be onsrued from he margnal al measures ( 2.2 ) by means of a Pareo Lévy opula. The margnal al measures are found from ( 2.3 ) as expeed by Π ( x) = Π( 0,..., x,...,0), [ 0, ) x. ) 0 Defnon 2.2. Le ( X be a Lévy proess wh Lévy measure Γ ha has sandard -sable one-dmensonal margnals. Then we all Γ a Pareo Lévy measure and he assoaed al measure ( x) ([, )... [ )) ˆ d, : (,..., d) Γ =Γ x x = C x x, x E, s referred o as Pareo Lévy opula C. We now an ransform he margnal Lévy measures of a Lévy proess analogously o ( 2. ), yeldng sandard -sable margnal Lévy proesses wh Lévy measures Γ = x x for x > 0. Noe ha he ransformed -sable Lévy proesses are NOT ompound Posson 8

9 anymore (even hough hey may have been before he ransformaon), nsead hey are of nfne varaon and have an nfne number of small jumps per un me expressed by lm Γ = x 0 x. For defnons and referenes of sable Lévy proesses see Con and Tankov (2004). Lemma 2.3. Le ( X ) 0 be a sperally posve Lévy proess (.e. a Lévy proess admng only posve jumps) wh Lévy measure Π on E and onnuous margnal al measures Π,..., Π. Then d Π ( x) = Π( [ x, ] [ x, ]) ˆ d = C,...,, x E, Π Π x d xd and Ĉ s a Pareo Lévy opula. Proof. Noe ha for all x E, ˆ C ( x,...,xd) =Π ( x),..., ( x d), Π Πd hs mples for he one-dmensonal margnal al measures ˆ C( 0,..., x,..., 0 ) =Π ( x) = Π x, [ 0, ) x The followng s Sklar's Theorem for sperally posve Lévy proesses n he onex of Lévy Pareo opulas. The proof s smlar o he one of Theorem 5.6 of Con and Tankov (2004). Theorem 2.4 (Sklar's Theorem for Pareo Lévy opulas) 9

10 Le Π be he al measure of a d -dmensonal sperally posve Lévy proess wh margnal al measures Π,..., Π d all x,..., x E d. Then here exss a Pareo Lévy opula ˆ [ 0, ] Π ( x,..., x ) ˆ d = C,..., Π Π x d xd If he margnal al measures are onnuous on[ 0, ], henĉ on Ran Ran Π Πd C:E suh ha for. ( 2.4 ) s unque. Oherwse, s unque. Conversely, f Ĉ s a Pareo Lévy-opula and Π,..., Π are d margnal al measures, hen Π defned n ( 2.4 ) s a jon al measure wh margnals Π,..., Π. d So-alled Lévy opulas, as nrodued n Con and Tankov (2004) and also used n Böker and Klüppelberg (2006, 2008), have one-dmensonal margnal Lebesgue measures. As a onsequene hereof, hey do no have an nerpreaon as he Lévy measure of a onedmensonal Lévy proess, beause a Lévy measure s, for nsane, fne on [, ). From he onsruon above s also lear ha, f (,, ) C x x s a Lévy opula, hen d he assoaed Pareo Lévy opula Ĉ an be onsrued by Cx ˆ(,, xd ) = C (/ x,,/ x d ). Hene, he followng examples follow mmedaely from hose gven n Böker and Klüppelberg (2006): Example 2.5. [Independene Pareo Lévy-opula]. Le ( d X =,, ) measures,, X X, 0, be a sperally posve Lévy proess wh margnal al Π Π. The omponens of d 0 X are ndependen f and only f 0

11 d = ( A) ( A ) A ( E) Π = Π { } B, where A = x : ( 0,...,0, x,0,...,0) A, x sands a he -h omponen, and B( E ) denoes he Borel ses of E. Ths mples for he al measure of ( X ) 0 Π ( x,, x ) = Π ( x ) I Π ( x ) I, d gvng a Pareo Lévy opula of { x =... = x = 0 } d d { x =... = x = } 2 d d- 0 ( x ) = {... 0 } x x d { x... x 0} Cˆ x I x I. 2= = d= = = d-= The resulng Lévy proess wh Pareo Lévy opula C s a sandard -sable proess wh ndependen omponens. Example 2.6. [Complee (posve) dependene Pareo Lévy opula]. Le ( d X =,..., ) X X, 0, be a sperally posve Lévy proess wh Lévy measure Π, whh s onenraed on an nreasng subse of E. Then The orrespondng Lévy opula s gven by ( ) ( x ) ( x ) ( x ) Π = mn Π,..., Π d d. ( x ) = ( d ) Cˆ mn x,..., x. Example 2.7. [Arhmedan Pareo Lévy opula]. Le φ :0, [ ] [ 0, ] be srly dereasng wh φ ( 0 ) = and φ ( ) = 0. Assume ha φ has dervaves up o order d wh ( ) k k d φ > 0 k d for k =,..., d. Then he followng s a Pareo Lévy opula ( x) = φ φ φ d Cˆ x... x.

12 Example 2.8. [Clayon Pareo Lévy opula]. θ Take φ () = for θ > 0. Then he Arhmedan Pareo Lévy opula θ θ ( x) = ( ) / ˆ... Cθ x x d s alled Clayon Pareo Lévy opula. Noe ha Cˆ ( x) Cˆ ( x) = C lm ˆ ˆ θ 0 Cθ θ lm θ θ = x ;.e., hs model overs he whole range of dependene. x and 3. Undersandng he dependene sruure Reall our mulvarae operaonal rsk model, n whh oal aggregae loss s modelled by a ompound Posson proess wh represenaon., where ( ) 0 N s he Posson proess ounng he oal number of losses and,..., X X denoe all severes n he me nerval ( 0, ]. In hs model losses an our eher n one of he omponen proesses or resul from mulple smulaneous losses n dfferen omponens. In he laer suaon, of all losses, whh happen a he same me. N X s he sum Based on a deomposon of he margnal Lévy measures, one an show (see e.g. Böker and Klüppelberg (2008), Seon 3) ha he omponen proesses an be deomposed no ompound Posson proesses of sngle jumps and jon jumps. For d= 2 he ell's loss proesses have he represenaon (he me parameer s dropped for smply): X = X X = N N k l k= l= X X ( 3. ) 2 X = X X = 2 2 N N 2 2 X m X l m= l= 2

13 where X and X 2 desrbe he aggregae losses of ell and 2, respevely, ha are generaed by ommon shoks, and X and 2 X are ndependen loss proesses. Noe ha apar from X and 2 X all ompound Posson proesses on he rgh-hand sde of 3. are muually ndependen. If we ompare he lef-hand and rgh-hand represenaon we an denfy he parameers of he proesses on he rgh-hand sde. The parameers on he lef-hand sde are λλ, 2 > 0 for he frequenes of he Posson proesses, whh oun he number of losses, and he severy dsrbuon funons F,F; 2 for deals we refer o Böker and Klüppelberg (2008). Fgure 2 abou here Fgure 2: Deomposon of he doman of he al measure smulaneous loss par Π ( z )(grey) and ndependen pars Π Π 2 z (dashed blak lne). Π z for z = 6 no a z (sold blak lne) and The frequeny of smulaneous losses an be alulaed from he bvarae al measure Π ( x,x 2) as he lm of arbrarly small, smulaneous losses;.e. ˆ x x C ( x) ( x) lm Π, = λ, λ = lm Π = lm Π = λ 0, mn λ, λ x, x2 0 x 0 x 0 Consequenly, he frequeny of ndependen losses mus be 3

14 λ = lm Π x = λ -λ and λ 2= λ2 λ. x 0 By omparson of he Lévy measures we oban furher for he dsrbuon funons of he ndependen losses λ λ ( λ ) ˆ, F ( x ) = F ( x ) C λf ( x ) λ λ 2 λ ( ) F x C ˆ, F x. 2 F 2( x 2) = λ λ λ2 And, fnally, he jon dsrbuon funons of onden losses and her margnals are gven by ˆ, F ( x, x 2) = ( 2 P X > x, X > x 2) = ( ) λ ( ) λ2 2 2 λ C F x F x ( 3.2 ) F ( x ) = F ( x,x 2) = λ lm x2 0 ( λ ) 2 ˆ, λ C F x F 2( x 2) = F ( x,x 2) = λ λ lm x 0 ( ) ˆ, λ C F x. Summarsng our resuls, we an say ha he Pareo Lévy opula approah s equvalen o a spl of he ells' ompound Posson proesses no ompleely dependen and ndependen pars. All parameers of hese subproesses an be derved from he Pareo Lévy opula, whh we have shown here for he dsrbuon funons and frequenes of he dependen and ndependen pars. Moreover, we would lke o menon ha he smulaneous loss dsrbuons ( ) F and F () are no ndependen, nsead hey are lnked by a dsrbuonal opula, whh 2 an be derved from ( 3.2 ), see agan Böker and Klüppelberg (2008) for more deals. 4

15 4. Approxmang oal OpVaR In our model, OpVaR enompassng all rsk ells s gven by (.2 ), whh an asympoally be approxmaed by (.2 ). Needless o say, he parameers F and λ, whh desrbe he bank's oal OpVaR a aggregaed level, depend on he dependene sruure beween dfferen rsk ells and hus on he Pareo Lévy opula. We now nvesgae varous dependene senaros, for whh frs order approxmaons lke (.3 ) are avalable. Our resuls yeld valuable nsgh no he naure of mulvarae operaonal rsk. One domnang ell severy Alhough he frs senaro s raher smple by assumng ha hgh-severy losses manly our n one sngle rsk ell, s ye relevan n many praal suaons. Noe ha he assumpons n he followng resul are very weak, no proess sruure s needed here. Theorem 4. (Böker and Klüppelberg (2006), Theorem 3.4, Corollary 3.5) For fxed > 0 le X for =,..., d have ompound Posson dsrbuons. Assume ha F R α for α> 0. Le >α ρ and suppose ha regardless of he dependene sruure beween X,...,X d, P( X > x ) ( > ) VaR EN P X x, x, κ EN ( κ) F = OpVaR ( κ) ρ E X < for = 2,...,d. Then,, κ. Ths s a que mporan resul. I means ha oal operaonal rsk measured a hgh onfdene levels s domnaed by he sand-alone OpVaR of ha ell, where losses have 5

16 Pareo als ha are heaver han losses of oher ells. Noe ha he assumpons of hs heorem are also sasfed f he loss severy dsrbuon s a mxure dsrbuon, n whh only he al s explly assumed o be Pareo-lke, whereas he body s modelled by any arbrary dsrbuon lass. We have elaboraed hs n more deal n Böker and Klüppelberg (2008), Seon 5. Mulvarae ompound Posson model wh ompleely dependen ells Complee dependene for Lévy proesses means ha all ell proesses jump ogeher,.e. losses always our a he same me, neessarly mplyng ha all frequenes mus be equal,.e. λ := λ = = λ d. I also mples ha he mass of he mulvarae Lévy measure Π s onenraed on Le {( x,...,xd) E: Π ( x ) = = Π d ( xd) } = ( x,...,xd) E : F( x ) = = Fd ( xd) F be srly nreasng and onnuous suh ha Π ( z ) = Π ({( x,, xd) E : x z}) Ths represenaon yelds he followng resul. { } d = = F q exss for all q [0,). Then d = 2 Π ({ x 0, :x F F x z }), z > 0. Theorem 4.2 (Böker and Klüppelberg (2006), Theorem 3.6) Consder a mulvarae ompound Posson proess X ( d ) = X,, X, 0, wh ompleely dependen ell proesses and srly nreasng and onnuous severy dsrbuons F. Then ( X ) 0 s ompound Posson proess wh parameers 6

17 λ = λ and = F z F H z, z > 0, where : = d H z z = 2F ( F( z )). If F R α for α (0, ), hen OpVaR d = ( κ) OpVaR ( κ) ~, κ, Where OpVaR ( ) denoes he sand alone OpVaR of ell. Corollary 4.3. Assume ha he ondons of Theorem 4.2 hold and ha F R α for α (0, ) and ( x) ( x) F lm = 0, x F [ ). Assume furher ha 0 for =,, b d and = 0 for = b,, d. Then OpVaR b α = / ( κ) ~ OpVaR ( κ), κ Noe how he resul of Theorem 4.2 resembles he proposals of he Basel Commee on Bankng Supervson (2006), n whh a bank's oal apal harge for operaonal rsk measured as OpVaR a onfdene level of 99.9 % s usually he sum of he rsk apal harges arbued o he dfferen rsk ype/busness lne ells. Hene, followng our model, regulaors mplly assume ha losses n dfferen aegores always our smulaneously. Mulvarae ompound Posson model wh ndependen ells The oher exreme ase we wan o onsder s ndependene beween dfferen ells. For a general Lévy proess ndependene means ha no wo ell proesses ever jump ogeher. Consequenly, he mass of he Lévy measure s onenraed on he axes, f. Example 2.5, so ha we have 7

18 Π ( z) =Π ( z ) Π ( z ) z 0. d d d Theorem 4.4. Assume ha X ( ) ( X ) 0 s a ompound Posson proess wh parameers = X,, X, 0, has ndependen ell proesses. Then = λ λ λ d and λ If F R α for α (0, ) and for all = 2,, d, F ( z) = F( z) λ d F d ( z) λ, z 0. F (x) lm = [0, ) x F (x), hen OpVaR ~ κ κ F ( λ 2λ2 dλd), κ. ( 4. ) If we ompare ( 4. ) o he formula for he sngle-ell OpVaR (.3 ), we see ha mulvarae OpVaR n he ase of ndependen ells an be expressed by he sand-alone OpVaR of he frs ell wh adjused frequeny parameer λ: = λ 2λ2 d λ d. Aually, we wll see n he nex seon ha hs s possble for muh more general dependene sruures, namely hose belongng o he lass of mulvarae regular varaon. Mulvarae regularly varyng Lévy measure Mulvarae regular varaon s an approprae mahemaal ool for dsussng heavy al phenomena as hey our for nsane n operaonal rsk. We begn wh regular varaon of random veors or, equvalenly, of mulvarae dsrbuon funons. 8

19 The dea s o have regular varaon no only n some (or all) margnals, bu along every ray sarng n 0 and gong hrough he posve one o nfny. Clearly, hs lms he possble dependene sruures beween he margnals, however, suh models are sll flexble enough o be broadly appled o varous felds suh as eleommunaon, nsurane, and las bu no leas VaR analyss n he bankng ndusry. Furhermore, many of he dependene models mplyng mulvarae regular varaon an sll be solved and analysed analyally. Le us onsder a posve random veor X wh dsrbuon funon F ha s as our Lévy measure Π onenraed on E. Moreover, we nrodue for ses (for any Borel se A E s omplemen n E s denoed by [, 0 x] = E [, 0 x] = { y E:max > }. d x y A ): x E he followng Assume here exss a Radon measure ν on E (.e. a Borel measure ha s fne on ompa ses) suh ha holds for all ( x) ( X [ 0, x] ) X [ ] F P lm = lm =ν F P 0, ([ 0, x] ) ν ( 4.2 ) x E, whh are onnuy pons of he funon [ 0, ]. One an show ha he above defnon ( 4.2 ) mples ha ν has a homogeney propery;.e. here exss some α> 0 suh ha α ([ sx] ) s [ x] ν 0, = ν 0,, > 0 s, ( 4.3 ) and we say ha F s mulvarae regularly varyng wh ndex α ( ) F R. Condon α ( 4.2) also says ha F( ) as a funon of s n R α. Defne now b( ) o sasfy ( ) ~ F b as. Then, replang by b( ) n 4.2 yelds 9

20 lm X,, () = ν P b [ 0 x] [ 0 x]. ( 4.4 ) In ( 4.4 ) he random varable X s normalsed by he funon b ( ). As explaned n Resnk (2007), Seon 6.5.6, normalsaon of all omponens by he same funon b ( ) mples ha he margnal als of X sasfy for, j {,, d } where, [ 0, ) j lm x F x = F x, j. Assumng > 0 we se w.l.o.g. =. Then we an also hoose b( suh ) ha for j F b b F ( ()) ~ () ~ () ( 4.5 ) and, by subsung n ( 4.4 ), we oban a lm on he lef-hand sde of ( 4.4 ) wh he same salng sruure as before. To formulae analogous defnons for Lévy measures noe frs ha we an rewre ( 4.2) by means of he dsrbuon of X as: and, smlarly, ( 4.4 ) as PX ( [ 0, x] ) lm =ν( [ 0, x] ) P [ 0,] X. [ ] ( ()[ 0, x] ) = X 0 () x =ν( 0 x ) ( 4.6 ) lmp b lm P, b, X Then, he analogue expresson o ( 4.2 ) for a Lévy measure Π s smply 20

21 for all ( [ 0, x] ) [ 0,] ({ y E : or or d d} ) Π { y E : y > or or y > } Π Π y > x y > x lm = lm =ν Π ( d ) ([ 0, x] ) x E, whh are onnuy pons of he funon ν [ 0, ]. Summarsng wha we ( 4.7 ) have so far yelds he followng defnon of mulvarae regular varaon for Lévy measures, now formulaed n analogy o ( 4.4 ) or ( 4.6 ), respevely: Defnon 4.5. [Mulvarae regular varaon for sperally posve Lévy proesses] Le Π be a Lévy measure of a sperally posve Lévy proess on E. Assume ha here exss a funon b :( 0, ) ( 0, ) sasfyng b( ) as and a Radon measure ν on E suh ha holds for all ( () 0, b x ) ([ 0 x ] ) ( 4.8 ) lm Π = ν, x E whh are onnuy pons of he funon ν [ 0, ]. Then ν sasfes he homogeney propery α ([ 0 sx] ) s [ 0 x] ν, = ν,, s> 0 for some α> 0 and he Lévy measure Π s alled mulvarae regularly varyng wh ndex α ( Π R α). As before n he ase of a regularly varyng random veor X, we assume ha n ( 4.8 ) we an hoose one sngle salng funon b ( ), whh apples o all margnal al measures. In analogy o ( 4.5 ) we an herefore se Π Π ( b ()) ~ b () ~ (),. ( 4.9 ) 2

22 As explaned above, normalsaon of all omponens by he same funon b () mples ha he margnal al measures sasfy for, j {,, d } ( x) ( x) Π lm =,, j 0, x Π j j [ ). ( 4.0 ) As we have already sad, mulvarae regular varaon s jus a speal way of desrbng dependene beween mulvarae heavy-aled measures. Therefore s naural o ask, under whh ondons a gven Pareo Lévy opula s n lne wh mulvarae regular varaon. Sarng wh a mulvarae al measure Π, we know from Lemma 2.3 ha we an derve s Pareo Lévy opula Ĉ by normalsng he margnal Lévy measures o sandard -sable margnal Lévy proesses,.e. Π x Γ x = Π ( x) =, x [0, ) Π x. ( 4. ) We now onsder he queson under whh ondons he resulng mulvarae al measure Γ of a sandardsed -sable Lévy proess orresponds o a mulvarae regularly varyng Lévy measure Γ (auomaally wh ndex ). Example 4.6. [Pareo Lévy opula and mulvarae regular varaon]. Le 2 X = (, ), 0 X X, be a sperally posve Lévy proess wh Lévy measure Π on [ ) 0, d. Transformng he margnal Lévy measure by ( / ) ( x ), we oban he Pareo Lévy opula Cˆ ( x, x2) =Γ( [ x, ) [ x2, ) ) of ( ) 0 X. Π 22

23 From ( 4. ) we know ha, f Γ s regularly varyng, hen wh ndex, and, herefore, we se b()=. Obvously, for b( ) = we are n he sandard ase and all margnals are sandard Pareo dsrbued wh α=. Then, beause n general we have Π,, 0 = Π Π Π, ( x x ) ( x ) ( x ) ( x x ), ( x, x2) E, ( 4.2 ) we mmedaely ge for he lef-hand sde of ( 4.8 ) for he ransformed Lévy measure Γ ( ˆ,(, 2) ) (, 2) Γ 0 x x = C x x x x. 2 For bvarae regular varaon wh ndex, he rgh-hand sde above mus onverge for o a Radon measure ν on E, more presely, wh Cˆ x, x ν,(, ) x x 2 ( 2) ( 0 x x2 ) 0 ( 2) 0 ( 2) ν, sx, sx = s ν, x, x, s > 0. Ths s learly he ase f he Lévy Pareo opula Ĉ s homogenous of order - and, more generally, f here s a Radon measure µ suh ha lm ˆ (, C x x2) =µ,(, 2), (, 2) 0 x x x x E. A more general resul s he followng, whh lnks mulvarae regular varaon o he dependene onep of a Pareo Lévy opula. Theorem 4.7 (Böker and Klüppelberg (2006), Theorem 3.6) 23

24 Le Π be a mulvarae al measure of a sperally posve Lévy proess on E.Assume ha he margnal al measures Π are regularly varyng wh ndex α for some α> 0. Then he followng holds. () The Lévy measure Π s mulvarae regularly varyng wh ndex α f and only f he sandardsed Lévy measure Γ s regularly varyng wh ndex. (2) If he Pareo Lévy opula Ĉ s homogeneous of order - and 0< α< 2, hen Ĉ s he Lévy measure of a mulvarae α -sable proess. Example 4.8 [Vsualsaon of he Clayon Pareo Lévy opula]. ˆ, θ θ Reall he Clayon Lévy opula C ( x x ) = ( x x ) / 2 2 θ for x, x 2 > 0. From Defnon 2.2 we know ha ( x, x2) ([ ) [ x2 )) ( x x2) Cˆ =Γ x,,,, E, and he orrespondng margnal proesses have been sandardsed o be -sable Lévy proesses. Sne Ĉ s homogeneous of order -, we know from Theorem 4.7 ha he bvarae Lévy proess s a -sable proess. We an also onlude ha, f he margnal Lévy al measures Π and Π 2 before sandardsng he margnals were regularly varyng wh some ndex α, hen he Lévy measure Π was bvarae regularly varyng al wh ndex α. Noe also ha θ θ ([ ] ) 2( 2) ˆ Γ 0, x =Γ x Γ x C( x, x2) =. x x2 x x2 / θ 24

25 The homogeney an be used as follows o allow for some vsualsaon of he dependene. We ransform o polar oordnaes by seng x = rosφ and x2 = rsn φ for r = x = x x and φ [ 0, π/ 2]. From he homogeney propery follows ([ 0, ] ) 0, Ths s deped n Fgure 3 where (, ) Γ x = r Γ os φ,sn φ = : Γ( r, φ ). Γ r φ s ploed for r = as a funon of φ, and hus he Clayon dependene sruure s ploed as a measure on he quaerrle. Fgure 3a and 3b abou here Fgure 3: Plo of he Pareo Lévy opula n polar oordnaes Cr ˆ(, φ) =Γ(, rφ) as a funon of he angle ( 0, π/ 2) φ for r = and dfferen values for he dependene parameer. Lef Plo: θ =,8 (doed lne), θ = 0,7 (dashed lne), θ = 0,3 (sold lne). Rgh Plo: θ = 2, 5 (sold lne), θ = 5 (dashed lne), θ = 0 (doed lne), and θ = posve dependene, long-dashed lne). (omplee I s worh menonng ha all we have sad so far abou mulvarae regular varaon of Lévy measures holds rue for general sperally posve Lévy proesses. We now urn bak agan o he problem of alulang oal OpVaR and onsder a mulvarae ompound Posson proess, whose Lévy measure Π s mulvarae regularly varyng aordng o ( 4.8 ). In parular, hs mples al equvalene of he margnal Lévy measures, and we an wre 4.0 wh some (0, ) as 25

26 ( x) ( x) ( x) ( x) Π λ F λ : = lm = = : x Π λ F λ ( 4.3 ).e. lm ( x) / F ( x) F. We avod, suaons, where for some we have = 0, x = orrespondng o ases n whh for x he al measure ( x ) deays faser han Π α x,.e. n ( 4.3 ) we only onsder he heaves al measures, all of al ndex α. Ths makes sense, beause we know from our dsusson a he begnnng of hs seon ha only he heaves-aled rsk ells onrbue o oal OpVaR, see Theorem 4.. When alulang oal aggregaed OpVaR, we are neresed n he sum of hese al equvalen, regularly varyng margnals,.e. we have o alulae he al measure Π =Π : d z x E x > z, z > 0. = Analogously o Resnk (2007), Proposon 7.3, p. 227, he al measure varyng wh ndex α, more presely we have d d α lm Π ( bz () ) =ν x : x > = z ν : > = : α z x x z ν (, ] = = Π s also regularly E E. ( 4.4 ) Now, le us hoose he salng funon b so ha ( z) ( z) ( b ) b () Π b. Then we have ~ Π Π lm = lm =ν, z Π Π Ths mples he followng resul for aggregaed OpVaR. ( ]. Theorem 4.9 (Böker and Klüppelberg (2006), Theorem 3.8). d Consder a mulvarae ompound Posson model X ( ) = X,...,X, 0, wh mulvarae regularly varyng Lévy measure Π wh ndex α and lm measure ν n ( 4.8 ). Assume 26

27 furher ha he severy dsrbuons F for =,, d are srly nreasng and onnuous. Then ( ) 0 X s a ompound Posson proess wh parameers sasfyng for z where ( ] λ F ( z) ~ ν (, ] λ F ( z) R, ( 4.5 ) ν, = ν : d x E x >. Furhermore, oal OpVaR s asympoally gven by = κ OpVaR ( κ) ~ F, κ ν (, ] λ α. ( 4.6 ) We noe ha for he wde lass of regularly varyng dsrbuons, oal OpVaR an effevely be wren n erms of he severy dsrbuon of he frs ell. Spefally, he rgh-hand sde of (4.6) an be undersood as he sand-alone, asympo OpVaR of he frs ell wh an adjused frequeny parameer, namely λ ν (, ]. Wha remans s o fnd examples, where ν (, ] he nfluene of eran dependene parameer. an be alulaed analyally or numerally o undersand beer Revsng he ase of ndependen operaonal rsk ells Before we presen some expl resuls for he Clayon Pareo Lévy opula below, le us onsder agan he parularly easy ase wh ndependen ells. Sne hen all mass s on he posve axes, we oban From (, ] (, ] d (, ]. ( 4.7 ) ν = ν ν Π b ~ as follows for he al measure of he frs ell For = 2,, d we oban by usng ( 4.3 ) ( b ( ) z α ) z ( z ]. ( 4.8 ) lm Π = = ν, 27

28 ( ()) ( ) b () ( u) ( u) Π bz Π uz Π u lmπ b = lm = lm = z = ν z, α Π u Π Π d and, herefore, alogeher ν (, ] = 2. By reover he resul of Theorem 4.4. = ( ], ( 4.9 ) 4.5 ogeher wh λ = λ we fnally Two expl resuls for he Clayon Lévy opula Le us onsder a bvarae example where he margnal Lévy measures are no ndependen, and hus he lm measure ν ( z, ] s no jus he sum of he margnal lm measures as n ( 4.7 ). Insead, ν (, ] z has o be alulaed by akng also mass beween he posve axes no aoun, whh an be done by represenng ν ( z, ] as an negral over a densy. Frs, noe ha from ( 4.2 ) ogeher wh ( 4.8 ) and ( 4.9 ), follows n he ase of a Pareo Lévy opula ha for all ( x, x2) E whh afer dfferenang yelds Hene, we an alulae α α αθ θ αθ ( 0,( x, x2) ) x 2 x2 ( x 2 x2 ) / θ ν = αθ / θ 2 θ 2 α ( θ) αθ θ ( x x ) ( θ ) x x ( x x ) ν, = α / (, ] ((, ] [ 0, ]) ( x, x x ) dxdx ν = ν ν 0, 2 2 and, by subsuon u = x, we oban / θ = α x x dx 0 θ αθ α 2 28

29 ( ] /, θ αθ θ α ν = α u u du 2 0 θ / θ / α / α / α α = s ( s ) ds ( 2 ) = / α / α / α α 2 E Y θ ( 4.20 ) θ where Y θ s a posve random varable wh densy / parameers bu he Pareo opula parameer θ. g s = s θ, ndependen of all Example 4.0. (a) For α= we have (, ] 2 ν =, whh mples ha, regardless of he dependene parameer θ, oal OpVaR for all 0 < θ < s asympoally equal o he ndependen OpVaR. (b) If α θ =, hen wh 2 = ( 2 / ) 2 λ λ as defned n (, ] ν = 4.3. / α 2 / α 2, Fgure 4 abou here Fgure 4: Illusraon of he al measure (, ] ν as a funon of α for dfferen values of θ. We have hosen θ = 0.3(lgh dependene, sold lne), θ = (medum dependene, dashed lne), and θ = 0 (srong dependene, doed-dashed lne). Morover, he long-dashed lne orresponds o he ndependen ase. 29

30 Fgure 4 llusraes he al measure ν (, ] as gven n θ and α. Noe ha aordng o ( 4.6 ), OpVaR nreases wh (, ] 4.20 for dfferen values of ν. Hene, Fgure 4 shows ha n he ase of α >, a hgher dependene (.e. a hgher number of jon losses n he omponens) leads o a hgher OpVaR, whereas for α <, s he oher way around: a lower dependene (.e. a lower number of jon losses n he omponens) leads o a hgher OpVaR. Ths agan shows, how hngs may go awry for exremely heavy-aled dsrbuons. Due o he non-onvexy of he OpVaR for α < dversfaon s no only no presen, bu he oppose effe ours. Fnally, noe ha for θ, ndependene ours and (, ] 2 ν = s onsan as ndaed by he horzonal long-dashed lne n Fgure 4. Dslamer The opnons expressed n hs arle are hose of he auhors and do no neessarly refle he vews of UnCred Group. Moreover, presened measuremen oneps are no neessarly used by UnCred Group or any afflaes Referenes () Basel Commee on Bankng Supervson. Observed range of prae n key elemens of Advaned Measuremen Approahes (AMA), (2006). Basel. (2) Böker, K. Operaonal Rsk: Analyal resuls when hgh severy losses follow a generalzed Pareo dsrbuon (GPD) A Noe. The Journal of Rsk Vol. 8, No. 4, (2006), pp (3) Böker, K. and Klüppelberg, C. Operaonal VAR: a Closed-Form Approxmaon. RISK Magazne, Deember, (2005), pp

31 (4) Böker, K. and Klüppelberg, C. Mulvarae Models for Operaonal Rsk. Workng Paper, Tehnshe Unversä Münhen. (2006). Avalable a (5) Böker, K. and Klüppelberg, C. Modellng and Measurng Mulvarae Operaonal Rsk wh Lévy Copulas. Journal of Operaonal Rsk, Vol. 3, No. 2 (2008). (6) Chavez-Demouln, V., Embrehs, P. and Nešlehová, J. Quanave Models for Operaonal Rsk: Exremes, Dependene and Aggregaon. Journal of Bankng and Fnane, Vol 30, No. 0 (2005), pp (7) Con, R. and Tankov, P. Fnanal Modellng Wh Jump Proesses (2004). Chapman & Hall/CRC: Boa Raon. (8) Embrehs, P., Klüppelberg, C. and Mkosh, T. Modellng Exremal Evens for Insurane and Fnane (997). Sprnger: Berln. (9) Fraho, A., Ronall, T. and Salomon, E. Correlaon and Dversfaon Effes n Operaonal Rsk Modelng, Operaon Rsk: Praal Approahes o Implemenaon. (2005), eded by Ellen Davs, London: Rsk Books. (0) Klüppelberg, C. and Resnk, S.I. The Pareo Copula, Aggregaon of Rsks and he Emperor's Soks. Journal of Appled Probably. Vol. 45, No. (2008), pp () Mosadell, M. The Modellng of Operaonal Rsk: Experene wh he Analyss of he Daa Colleed by he Basel Commee. Preprn, Bana D'Iala, Termn d dsussone. No. 57 (2004). (2) Powojowsk, M.R., Reynolds, D. and Tuener, J.H. Dependen Evens and Operaonal Rsk. Algo Researh Quaerly. Vol. 5, No. 2 (2002), pp (3) Resnk, S.I. Heavy-Tal Phenomena (2007). Sprnger: New-York. 3