ABSTRACT KEYWORDS. Operational risk, Expected Shortfall, Lévy copula, regular variation. 1. INTRODUCTION

Transcription

1 ASYMPTOTICS FOR OPERATIONAL RISK QUANTIFIED WITH EXPECTED SHORTFALL BY FRANCESCA BIAGINI AND SASCHA ULMER ABSTRACT I his pper we esime operiol ris by usig he covex ris mesure Expeced Shorfll (ES) d provide pproximio s he cofidece level coverges o 00% i he uivrie cse. The we exed his pproch o he mulivrie cse, where we represe he depedece srucure by usig Lévy copul s i Böcer d Klüppelberg (2006) d Böcer d Klüppelberg, C. (2008). We compre our resuls o he oes obied i Böcer d Klüppelberg (2006) d (2008) for Operiol VR d discuss heir prcicl relevce. KEYWORDS Operiol ris, Expeced Shorfll, Lévy copul, regulr vriio.. INTRODUCTION Wihi he frmewor of Bsel II bs o oly hve o pu side equiy reserves for mre d credi ris bu lso for operiol ris. I 664 of Bsel Commiee of Big Supervisio (2004) he Bsel Commiee defies: Operiol ris is he ris of loss resulig from ideque or filed ierl processes, people d sysems or from exerl eves. The priculr difficuly i mesurig his ew ris ype rises from he fc h prilly he correspodig eves re exremely rre wih eormously high losses d he sme ime here re comprively few d. Bs hve o pply oe of hree mehods i order o clcule he cpil requireme: he Bsis Idicor Approch, he Sdrdized Approch or he Advced Mesureme Approch (AMA). Wihi he firs wo mehods, he cpil chrge is percege of he verge ul gross icome. Accordig o he AMA, b is llowed o develop ierl operiol ris model wih idividul disribuiol ssumpios d depedece srucures. Hece i is of gre ieres o develop suible mehods o esime he cpil reserve. The mos commo wy of esimig he mou of equiy reserve for operiol ris is by usig he ris mesure Vlue Ris (VR). I Böcer Asi Bullei 39(2), doi: 0.243/AST by Asi Bullei. All righs reserved.

2 736 F. BIAGINI AND S. ULMER d Klüppelberg (2005) he so-clled Operiol Vlue Ris (OpVR) level (0,) is defied s he -quile of he ggreged loss process. Operiol Vlue Ris hs bee exesively sudied boh i he uivrie d mulivrie cse respecively i Böcer (2006), Böcer d Klüppelberg (2005), (2006) d (2007) d (2008). A esseil disdvge of his ris mesure is h, i geerl, i is o cohere. I priculr, i c hppe h VR ribues more ris o loss porfolio h o he sum of he sigle loss posiios. Moreover, VR exclusively regrds he probbiliy of loss wheres is size remis ou of cosiderio. The mos populr lerive o VR is he Expeced Shorfll (ES), which is lso ow s Averge VR, Codiiol VR or Til VR. This ris mesure is cohere d idices he expeced size of loss provided h i exceeds he VR. I priculr, he ES seems o be he bes covex lerive o he VR, sice i is he smlles lw-ivri, covex ris mesure coiuous from below h domies VR (Theorem 4.6 of Föllmer d Schied (2004)). I ddiio, wihi he frmewor of Solvecy II d he Swiss Solvecy Tes, isurers hve o clcule heir rge cpil by usig he ES. The Federl Office of Prive Isurce jusifies his i chper 2.4. of Federl Office of Prive Isurce (2006) s follows: The ris mesure Expeced Shorfll is more coservive h he VR he sme cofidece level. Sice i c be ssumed h he cul loss profile exhibis severl exremely high losses wih very low probbiliy, he Expeced Shorfll is he more pproprie ris mesure, s, i cors o he VR, i regrds he size of his exreme losses. This rgumeio is lso suible for operiol ris, sice i is very similr o he quoed curil ris. I Chvez-Demouli d Embrechs (2004) d Moscdelli (2004) ES is he suggesed s lerive o VR for quifyig operiol ris. Hece, i his pper we evlue operiol ris by usig he Expeced Shorfll d derive sympoic resuls i uivrie d mulivrie models. The orgizio of he pper is he followig. Firs we cosider oedimesiol Loss Disribuio Approch (LDA) model. Sice i 667, Bsel Commiee of Big Supervisio (2004), he Bsel Commiee ses he cofidece level 99,9%, i is resoble o focus o he righ disribuio il ised of esimig he whole disribuio. Therefore we sudy he sympoic behvior of he righ disribuio il d, ssumig h he severiy disribuio hs regulrly vryig il wih idex >, we derive sympoic pproximio of he Operiol Expeced Shorfll: ES () - VR (), ", >. The we cosider mulivrie model, whose cells represe he differe operiol ris clsses, sice ccordig o he AMA, operiol ris shll be lloced o eigh busiess lies (Bsel Commiee of Big Supervisio

3 ASYMPTOTICS FOR OPERATIONAL RISK 737 (2004), 654) d seve loss ypes (Bsel Commiee of Big Supervisio (2004), ppedix 7). I he lierure, he sigle ris clsses re prevlely modelled by compoud Poisso process, i.e. he loss i oe ris cegory i ime $ 0 is represeed by he rdom sum N i ( ) = S i () = X, where (X i ) is idepede d ideiclly disribued (iid) severiy process d (N i ()) $ 0 is Poisso process idepede of (X i ). The ol operiol ris is he sum S () =S () S d (). However i is o relisic o ssume h ris clsses re idepede. Hece i order o describe he depedecies bewee he S i (), # i # d, we follow he pproch of Böcer d Klüppelberg (2006) d use Lévy copul. This yields relively simple model wih comprively few prmeers s he depedecies bewee severiies d frequecies re modelled simuleously. I his seig, we derive sympoic coclusios for he OpES i vrious scerios. For furher deils, we lso refer o Ulmer (2007). Filly we exmie he prcicl relevce of our resuls. i 2. APPROXIMATION OF THE OPES IN A ONE-DIMENSIONAL MODEL We suppose h operiol ris follows LDA model. Defiiio 2.. Loss Disribuio Approch (LDA) model). The severiy process: The severiies re modelled by sequece of posiive iid rdom vribles (X ). Le F be he disribuio fucio (i shor, df) of he X. 2. The frequecy process: The rdom umber N() of losses i he ime iervl [0,] is couig process, i.e. for $ 0 N() := sup{ $ : T # } is geered by sequece of rdom pois i ime (T ), which sisfy 0 # T # T 2 #.s. 3. The severiy process d he frequecy process re ssumed o be idepede. N( ) 4. The ggreged loss process is defied s S() := = X. I order o mesure operiol ris, we iroduce he Operiol Vlue Ris (OpVR) d he Operiol Expeced Shorfll (OpES). I his pper we will he focus o he OpES.

4 738 F. BIAGINI AND S. ULMER Defiiio 2.2. (OpVR, OpES) Le G be he df of he ggreged loss process (S ) $ 0 of LDA model. The Operiol Vlue Ris uil ime level (0,) is he geerlized iverse G of G VR () :=G () = if{x : G (x) $ }. The Operiol Expeced Shorfll uil ime level [0,) is defied s ES () := VR u du. - # ]g I order o compue hese ris mesures, we eed o ow he df G of S(). Becuse of he idepedece ssumpios we ow G (x) = (S() # x) = 3 * F (x) (N() =), () = 0 where F * is he -h covoluio of F d F * = F d F 0* = [0,3). We sudy ow he sympoic behvior of G (x) = (S()>x) for x " 3 d derive sympoic resuls i uivrie d mulivrie models. We sy wo rel fucios F, G re sympoiclly equl for x " 3 (F(x) G(x), x " 3) if Fx ] g lim " 3 G ] x g x =. Remr 2.3. From he sympoic equliy of he summds we c ifer he sympoic equliy of he sum. The sme holds for he iegrd d he iegrl: ) Le F i, G i, i =,,d, be posiive rel fucios wih The F i (x) G i (x), x " 3. (2) F (x) F d (x) G (x) G d (x), x " 3. b) Le f, c :[0,]" [0,3) wih f() c(), ", d suppose here exiss [0,) such h # f ()d < 3 d # c ()d < 3. The # # f] d g c] d, g ". Furhermore, by 667, Bsel Commiee of Big Supervisio (2004), operiol ris usully preses hevy-iled disribuio. We e his io ccou by dmiig oly regulrly vryig disribuio ils.

5 ASYMPTOTICS FOR OPERATIONAL RISK 739 Defiiio 2.4. A posiive mesurble fucio U o (0,3) is clled regulrly vryig i 3 wih idex r (U R r ) if Ux ] g lim x " 3 U ] x g r =, >0. A posiive mesurble fucio L o (0,3) is clled slowly vryig i 3 (L R 0 ) if Lx ] g lim x " 3 Lx ] g =, >0. From ow o we will cosider dfs wih regulrly vryig ils F R for $ 0. Noe h F becomes more hevy-iled for smller. Exmples for his id of dfs re he Preo d he Burr disribuio (see Exmples 2.6 d 2.7). Exmples for slowly vryig fucios re he logrihm d fucios h U coverge o posiive cos. For U R r, L(x) := ( x ) r R x 0. Thus, for every U R r here exiss L R 0 wih U(x) =x r L(x). By Theorem 2.3 of Böcer d Klüppelberg (2006) we obi h give LDA model for fixed ime > 0 wih severiy disribuio il F R, > 0, he followig sympoic equliy for he OpVR holds: VR () F - d -, 6 N ] g@ ", (3) if here exiss e > 0 such h 3 ] eg (N() =) < 3. (4) = 0 For furher deils bou (4), we refer o Theorem.3.9 of Embrechs, Klüppelberg d Miosch (997). Boh ecoomiclly relev frequecy processes, he Poisso process d he egive biomil process (see Embrechs, Klüppelberg d Miosch (997), Exmple.3.), sisfy codiio (4). To derive similr represeio of he OpES s i (3) we eed severl properies of regulrly vryig disribuio ils (see Appedix). We ow prove our mi resul. Theorem 2.5. (Alyic OpES) Cosider LDA model fixed ime >0, where he severiies hve disribuio fucio F such h he disribuio il F R for >. Assume h here exiss e >0such h 3 ] eg (N() =) < 3. = 0 The we hve he followig sympoic equliy for he OpES:

6 740 F. BIAGINI AND S. ULMER - ES] g F VR, - d - N - 6 ] g@ ] g ". (5) PROOF. Pu q := VR (). By Corollry 4.49 of Föllmer d Schied (2004) he Expeced Shorfll is give by 8S ] g S ( )> q ES] g = 6S ] g S ] g> VR] g@ = B ^S ] g > qh 3 = xdg x - # ] g q 3 = dq q x dx - G^ h # G] g q Sice codiio (4) is sisfied d he df F is subexpoeil due o Proposiio A. b), by Theorem.3.9 of Embrechs, Klüppelberg d Miosch (997) we hve h G (x) [N()] F(x), x " 3. Hece ( Rem. 23. b) 6N ] g@ 3 ES] g dq F q F x dx - ^ h # ] g q J 3 N F x dx N K # ] g 6 ] g@ O q = q F q - ^ h K O qf q K ^ h O L P ( Pr op. A. f ) 6N ] g@ q F q. - - ^ h (6) From Theorem 2.3 of Böcer d Klüppelberg (2006) we ow: ( 3) - q : = VR ] g F d -, 6 N ] g@ ". (7) Le X,, be posiive iid rdom vribles wih df F. The df F (or F) is clled subexpoeil, if F(x) = 0 for ll x, d if for ll $ 2: ^X X > xh,..., h > xi g lim x " 3 _ mx^x X =.

7 ASYMPTOTICS FOR OPERATIONAL RISK 74 Sice F R d by (7), from Proposiio A.c) wih c = we hve h - F^q h FdF d -, 6 N ] g@ ". (8) Moreover, sice F R, by Resic (987) pge 5 we hve h Puig everyhig ogeher we obi: F(F (x)) x, x > 0. (9) ES ] g ( 8) ( 9) ( 7) 6N ] g@ F - - F - - d - 6N ] g@ VR,, " - ] g - - d - F df d - 6N ] g@ 6N ] g@ h proves (5). Exmple 2.6 (Preo disribuio) If he severiies re Preo disribued, i.e. wih disribuio fucio - x Fx ] g = - c m,, q, x >0, q he F R. By (5) we obi ES ] g - F - d - 6N ] g@ 6N ] g@ q, ". - d - (0) Exmple 2.7 (Burr disribuio) Le J x N- Fx ] g = - K, q O,, q, x >0 L P be he Burr df. The F R sice F] x lim g q ] x - = lim f g p =, >0. " 3 F x x " 3 ] g q x x -

8 742 F. BIAGINI AND S. ULMER Thus, he Burr disribuio sisfies he codiios of Theorem 2.5 if >, d we hve R J NV S N ES q K 6 ] g@ ] -, ". - d OW g K - S O () W T L PX For furher exmple, we lso refer o Secio 2 of Böcer (2006), where lyicl expressio for he ES of operiol ris hs bee compued for high-severiy losses followig geerlized Preo disribuio. Comprig our resul wih he oes of Böcer d Klüppelberg (2006), we hve ES] g lim >, " VR ] g d he closer is o, he higher is he differece bewee Expeced Shorfll d Vlue Ris. For isce if =, ES () VR (), ", =2 ES () 2 VR (), ". Hece usig OpVR d is sympoic esimio, we obi uderesimio of he cpil reserve h becomes bigger for smller. 3. TOTAL OPES IN THE MULTIVARIATE MODEL As meioed before, he bs usig AMA re required o divide heir operiol ris io severl ris clsses. Therefore, we ivesige ow higher dimesiol model, i which he sigle ris cells my be depede. Followig he pproch of Böcer d Klüppelberg (2006) we model he depedece srucure wih Lévy copul. From ow o we ssume h he frequecy process is Poisso process. Sice operiol riss re lwys losses, we cocere o Lévy processes dmiig oly posiive jumps i every compoe, herefer clled specrlly posiive Lévy processes. Beig ieresed i very high losses we iroduce he oio of il iegrl. Defiiio 3.. (il iegrl) Le L be specrlly posiive Lévy process o d wih Lévy mesure P. The il iegrl of L is he fucio P :[0,3] d " [0,3] wih he followig properies:. P(x) =P([x,3) [x d,3)), x [0,3) d, where P(0) = lim x. 0,, x d. 0 P([x,3) [x d,3)). 2. P(x) =0 if for y i {,, d} x i = 3.

9 ASYMPTOTICS FOR OPERATIONAL RISK P(0,,0,x i,0,,0)=p i (x i ), x i [0,3), i =,,d, where P i (x)=p i ([x,3)) is he il iegrl of he i-h compoe. For oe-dimesiol compoud Poisso process wih y jump size df F,we hve h P(x) = l F(x). We model he depedece srucure of he d compoes wih Lévy copul. By Defiiio 3. of Kllse d Tov (2004) we hve Defiiio 3.2. (Lévy copul) A d-dimesiol Lévy copul of specrlly posiive Lévy process is fucio C :[0,3] d " [0,3] such h. C(u,, u d ) 3 for (u,, u d ) (3,, 3), 2. C(u,, u d )=0 if u i =0 for les oe i {,, d}, 3. C is d-icresig, 4. he mrgis C i (u i ):= lim C(u,, u i,, u d )=u i for y i {,, d}. u j " 36, j i From ow o we cosider specil cse of he LDA model d ssume h he severiy disribuio sisfies ll he prerequisies of Theorem 2.5 such h i his model he sympoic pproximio (5) for he OpES holds. Defiiio 3.3. (RVCP model) A regulrly vryig compoud Poisso model cosiss of he followig elemes:. The severiy process: The severiies re modelled by sequece of posiive iid rdom vribles (X ). Le he disribuio il F of he X be regulrly vryig wih idex, >, d coiuous. 2. The frequecy process: The rdom umber N() of losses i he ime iervl [0,], $ 0, is Poisso process wih prmeer l >0. 3. The severiy process d he frequecy process re ssumed o be idepede. N( ) 4. The ggreged loss process is defied s S() :=, $ 0. X = The severiies X beig posiive, (S ) $ 0 is compoud Poisso process wih posiive jumps d il iegrl give by P(x) =lf(x), x $ 0. Accordig o he AMA operiol ris shll be divided io eigh busiess lies d seve loss ypes. We describe every sigle ris cell wih RVCP model i order o be ble o pproxime he OpES s i Theorem 2.5. As i Böcer d Klüppelberg (2006) we model he depedece srucure by Lévy copul d focus o mulivrie RVCP model.

10 744 F. BIAGINI AND S. ULMER Defiiio 3.4. (Mulivrie RVCP model). Le every sigle ris cell be RVCP model wih ggreged loss process S i, severiy disribuio il F i R i, i =, d Poisso process N i wih prmeer l i,# i # d. 2. The depedece bewee cells is modelled by Lévy copul. More precisely, wih he il iegrl P i (x) =l i F i (x) of S i,# i # d, d Lévy copul C he il iegrl of (S,,S d ) is give by P(x,,x d )=C(P (x ),, P d (x d )), (x,,x d ) [0,3) d. 3. The ol ggreged loss process is defied s S () :=S () S d (), >0, wih il iegrl P (x) =P({(y,,y d ) [0,3) d d : y > x}), x $ 0. We deoe G he df of S (). Slr s Theorem (see Theorem 3.6 i Kllse d Tov (2004)) yields h (S,,S d ) is d-dimesiol specrlly posiive Lévy process. Defiiio 3.5. (ol OpES, ol OpVR) The ol Operiol Expeced Shorfll uil ime >0 level [0,) is defied s i = ES ] g : = VR u du, - # ] g where VR () :=if{x : G (x) $ } is he ol Operiol Vlue Ris uil ime level. I his seig we obi he followig resuls for he ol OpEs s cosequece of Theorem Oe domiig cell: Firs we cosider he cse where oe severiy disribuio is more hevy-iled h he oher severiy disribuios. Wihou loss of geerliy we ssume h i is he firs cell. By Theorem 2.5 d Theorem 3.4 of Böcer d Klüppelberg (2006) we obi he followig resul for he cse of OpES. Proposiio 3.6. Cosider mulivrie RVCP model wih < < i,2# i # d d jump size df F of he compoud Poisso process S. The F l (x) l F (x), x " 3, (2) d he ol OpES is sympoiclly equl o he OpES of he firs cell ES () ES (), ". i

11 ASYMPTOTICS FOR OPERATIONAL RISK 745 We see h i his cse he ol OpES is sympoiclly equl o he OpES of he firs cell idepedely of he geerl depedece srucure. Cosequely, huge operiol loss occurs very liely becuse of oe sigle loss i he firs cell ised of severl depede losses i differe ris cells. 2. Compleely depede cells: we ow ssume h i ll ris cells losses occur simuleously, i.e. h he compoud Poisso processes S,,S d lwys jump ogeher. Wih sligh buse of lguge, we sy h i his cse he Lévy processes S i (), # i # d, re compleely depede (see lso Böcer d Klüppelberg (2006) d Böcer d Klüppelberg (2007)). By Theorem 3.5 of Böcer d Klüppelberg (2006) he ol OpVR is sympoiclly equl o he sum of he OpVR of he cell processes d i = i VR ] g VR ] g, ". (3) We hve h he sme holds for he OpES by usig Theorem 2.5. Proposiio 3.7. Cosider mulivrie RVCP model fixed ime >0. We ssume h he ggreged loss processes S,,S d re compleely depede wih sricly icresig severiy dfs F,,F d. The he ol OpES sympoiclly equls he sum of he cell OpES d ES ] g ES ] g, ". (4) i = I 669 d) of Bsel Commiee of Big Supervisio (2004), he Bsel Commiee idices he sum over ll he ris cells s he sdrd procedure o quify he ol ris. Therefore, i seems h he Bsel Commiee cs o he ssumpio h he compleely depede cse is he wors cse h c hppe. If pplyig cohere, covex or subddiive ris mesure lie Expeced Shorfll, his ssumpio is rue, sice he ES of loss porfolio is lwys less or equl h he sum of he ES of he sigle losses, i spie of he previlig id of depedece. I fils, however, if VR is pplied. Noe h ES hs for > he sme properies of VR les sympoiclly. 3. Depede model wih b domiig cells: we ow ssume h he firs b {,, d } ris cells re more hevy-iled h he remiig ris cells. Also i his cse he ol OpES is sympoiclly equivle o he OpES of he domiig cells, s i lso hppes i he cse of he OpVR (see Proposiio 3.7 of Böcer d Klüppelberg (2006)). Proposiio 3.8. Cosider mulivrie RVCP model fixed ime >0. We ssume h he ggreged loss processes S,,S d re compleely depede wih sricly icresig severiy dfs F,,F d. Le b {,, d } d i

12 746 F. BIAGINI AND S. ULMER < = = b =: < j, j = b,, d d le c i (0,3), i =2,,b such h Fi ] xg lim = ci. x " 3 F ] xg The wih c := d C := b c / i = i ES () C ES () - F - - lc, d ". (5) 4. Idepede cells: we ow ur o he cse where he ggreged loss processes S,,S d re idepede. This holds if d oly if hey lmos surely ever jump ogeher. By Theorem 2.5 i follows h ol OpES behves sympoiclly s i he oe-dimesiol cse, logously o he cse of he OpVR (see Theorem 3.0 of Böcer d Klüppelberg (2006)). Theorem 3.9. Cosider mulivrie RVCP model fixed ime >0 wih idepede ggreged loss processes S,,S d. ) The S is oe-dimesiol RVCP model wih Poisso prmeer d severiy disribuio il l = l l d F (x) = l (l F (x) l d F d (x)) R, := mi(,, d ). The ol OpES behves sympoiclly s i he oe-dimesiol cse, i.e. ES () - F d - -, ". (6) l b) Le < = = b =: < j, j = b,, dforb {,, d } d cosider for i =2,,bc i (0,3) wih F ] xg lim F xg i x " 3 ] = c. i The he ol OpES c be pproximed i he followig wy: ES () - F wih C l := l c 2 l 2 c b l b. We coclude he Secio wih some exmples. - - c C m VR, - ] g ", (7) l

13 Exmple 3.0. Le F i, i =,,d, he Preo disribuios wih prmeers i, q i >0 d suppose h for b {,, d}< = = b =: < j, j = b,, d holds. If he ggreged loss processes S,,S d re compleely depede, i.e. N i = N 6i =,,d, he for i =,,b i follows h x -i Fi x ] g ` q j ^q xh qi lim = lim = lim x F x x x - e o " 3 " 3 x " 3 i x q ] g ` ^q q j h q i q qi = d. q Hece, we ow h c i = i Proposiio 3.8. For he severiy disribuio il F of he compoud Poisso process S we hve F ASYMPTOTICS FOR OPERATIONAL RISK 747 b b - / qi x i F = q q i = i = ] xg e c o ] xg e o c m b b - qi q qi i = i = - = e o ^ xh e o x, x" 3. For he ol Operiol Expeced Shorfll we obi q l ES g C ES g o - - m b ( 0) i ] $ ] e q q c i = = e b i = l qi o - c - m, ". If he ggreged losses S,,S d re idepede, we ow from Theorem 3.9 wih b C l = i = c i l i h F C (x) l l F b - b qi x (x) = d l l q i c q m qi l i x, l i = if x " 3. I his cse he ol OpES c be pproximed: ES ( 6) ] g F d l J b N qi l i = i K O - K - O L P ( 0) e b i = i `ES ] gj o, ". i =

14 748 F. BIAGINI AND S. ULMER For ideicl frequecy prmeers l := l = = l b we obi ES i = l b ] g qi, ". - c - m e o Our resuls hold for >. A firs sigh his requireme my pper more resricive wih respec o he cse of OpVR, sice for he OpVR he prmeer c be chose from he iervl (0,3). The resricio o > i Theorem 2.5 ws resul of he Expeced Shorfll beig iegrl of he Vlue Ris. However, lso he OpVR co provide good ris mesure for he cse 0 < <, s show i he followig Exmple. Exmple 3.. Cosider ideicl frequecy prmeers l lso i he idepede cse d suppose h 0< = = b =: < j, j = b,, d for b {,, d } lie i Exmple 3.0. Deoe by VR ; he ol OpVR of he compleely depede Preo model d by VR = he OpVR of he idepede Preo model. The i Secio 3..2 of Böcer d Klüppelberg (2006) i is show h VR VR Z q <, > ] [ =, = ] >, <. \ b / = ] g i i = b ; ] g q i = i I he cse 0< <,he ol OpVR lloces more ris o he idepede model h o he depede model, VR = () >VR ; () ssumig close o. Hece, he Preo disribuio for (0,) is so hevy-iled h he OpVR is o subddiive or covex ymore. 4. PRACTICAL RELEVANCE We ow discuss he prcicl relevce of our resuls. Firs of ll url quesio is wheher regulrly vryig disribuios wih idex for > esime correcly rel loss size disribuios. Moscdelli exmied i Moscdelli (2004) over operiol losses of 89 bs for he yer 2002, cegorized ccordig o eigh busiess lies. Due o he scrciy of d, he represeio of he few high losses proves o be cosiderbly more compliced. Moscdelli herefore uses exreme vlue heory, i priculr he pes over hreshold mehod, d ssumes h he high loss sizes hve geerlized Preo disribuio, where he geerlized Preo disribuio (GPD z, b ) wih form prmeer z d scle prmeer b > 0 is defied s GPD z, b (x) := * x b - z - ` z j for z 0 - exp^- x / bh for z = 0,

15 ASYMPTOTICS FOR OPERATIONAL RISK 749 where x $ 0 for z $ 0 d 0 # x # b /z for z < 0. The GPD z, b is regulrly vryig wih prmeer =/z for z = 0. I Moscdelli (2004) he prmeers (z, b) re esimed for every busiess lie by mximum lielihood esimio. The resul of his iquiry is h i six ou of eigh busiess lies he prmeer is less h. If Moscdelli s lysis were ccure ccou of he cul operiol ris, he he codiios of Theorem 2.5 would be sisfied i 25% of he busiess lies, sice he GPD wih prmeer z > 0 hs decresig Lebesgue desiy. However, i Neslehov, Embrechs d Chvez-Demouli (2006) i is suggesed h he ggregio chose i Moscdelli (2004) is quesioble, sice he seve loss ypes re o of he sme id. Therefore he problem of esimig he prmeer is sill highly debed d eeds furher reserch. The secod problem o be discussed is which id of mesure is he mos suible for he esimio of cpil reserves for operiol ris. As soluio Moscdelli suggess i Moscdelli (2004) he ris mesure Medi Shorfll, which dds he medi of he exceedce disribuio o he hreshold u: MS(u) :=u F u 2 c m, u >0, wih Fx ] ug - Fu ] g F u (x) := (X u # x X > u) =, - Fu ] g 0 # x < x F u, (8) where x F # 3 is he righ ed poi of F. The dvge of he medi is h i miimizes he bsolue deviio. Reservig equiy i he mou of MS(u), b presumbly c py hlf of ll losses h exceed u. I order o iclude cofidece level io he ris mesure, we choose VR s he hreshold u = VR () =G (), d obi he followig represeio of he Medi Shorfll i our model: MS ] g = VR] g if ' y : ^S ] g -VR] g # y S ] g> VR] gh $ 2 ( 8) G` y G -G G ] gj = G ] g if * y : $ ] gj ` G `G ] gj If G is coiuous, we c simplify he secod summd

16 750 F. BIAGINI AND S. ULMER - if ' y : G `y G ] gj - > 2 = if ' x : G] xg $ - G 2 ] g = G c - G 2 m ] g d obi MS G VR 2 2 ] g = c m = c m. Hece, i he cse of coiuous ggreged loss df G, he Medi Shorfll cofidece level equls he Vlue Ris level, i.e. for =99.9% MS (0.999) = VR (0.9995). This direcly yields h Medi Shorfll is o cohere d hus is o idel cdide for mesurig operiol ris. To coclude we remr gi h he choice of VR is o compleely sisfcory, sice i is oo opimisic (see (5)) d o covex. Idicig oly he probbiliy of loss d o he size of i, i my uderesime he poeilly severe il loss eves (Bsel Commiee of Big Supervisio (2004), 667). I ddiio, for (0,) he mere summio of he OpVR of he sigle cells is o upper boud of he ol OpVR, s he Bsel Commiee ssumes i Bsel Commiee of Big Supervisio (2004), 669d). This is oly ccure if pplyig covex ris mesure lie he ES provided i exiss. 2 A. REGULARLY VARYING DISTRIBUTION TAILS The clss of regulrly vryig fucios hs severl properies, h we recll here for he reder s coveiece. For furher deils, see Bighm, Goldie d Teugels (987), Embrechs, Klüppelberg d Miosch (997) d Resic (987) (especilly Theorem.7.2 d Proposiio.5.0 of Bighm, Goldie d Teugels (987), Lemm.3. d Appedix A3 of Embrechs, Klüppelberg d Miosch (997), Proposiio 0.8 of Resic (987)). Proposiio A.. ) Le F R be he il of df d le X be disribued ccordig o F. The [X b ]<3 if b <. b) Every regulrly vryig disribuio il is subexpoeil. c) Le U R r wih r d f, g posiive fucios o (0,3) wih f(x) " 3, g(x) " 3, x " 3, d such h here exiss cos c (0,3) wih f(x) c g(x), x " 3.

17 The U( f (x)) c r U(g(x)), x " 3. d) Le F R, >0, disribuio il. The ` F j R /. e) Le F, G R, >0,F df, G decresig. If F(x) cg(x),x" 3, for some c =0, he / x c c m ] g c m ] xg, x " 3. (9) F G f) (Krm s Theorem) Le L be slowly vryig d r <. The 3 r - r # L] gd x L x, x " 3. r ] g (20) x ASYMPTOTICS FOR OPERATIONAL RISK 75 ACKNOWLEDGEMENT We h Sebsi Crses for ieresig discussios d remrs d wo oymous referees, whose commes hve coribued o improve he pper lo. REFERENCES BASEL COMMITTEE OF BANKING SUPERVISION (2004). Ieriol Covergece of Cpil Mesureme d Cpil Sdrds. Bsel. Avilble BINGHAM, N.H., GOLDIE, C.M. d TEUGELS, J.L. (987) Regulr Vriio. Cmbridge Uiversiy Press, Cmbridge. BÖCKER, K. (2006) Operiol Ris: lyicl resuls whe high-severiy losses follow geerlized Preo disribuio (GDP) oe. Jourl of Ris 8, -4. BÖCKER, K. d KLÜPPELBERG, C. (2005) Operiol VR: closed-form soluio. RISK Mgzie, December, BÖCKER, K. d KLÜPPELBERG, C. (2006) Mulivrie models for operiol ris. Acceped for publicio i Quiive Fice. BÖCKER, K. d KLÜPPELBERG, C. (2007) Mulivrie Operiol Ris: Depedece Modellig wih Lévy Copuls ERM Symposium Olie Moogrph, Sociey of Acuries, d Joi Ris Mgeme secio ewsleer of he Sociey of Acuries, Csuly of Acuries, d Cdi Isiue of Acuries Sociey of Acuries. BÖCKER, K. d KLÜPPELBERG, C. (2008) Modellig d Mesurig Mulivrie Operiol Ris wih Lévy Copuls. J. Operiol Ris 3(2), CHAVEZ-DEMOULIN, V. d EMBRECHTS, P. (2004) Advced Exreml Models for Operiol Ris. Tech. rep. ETH Zürich. Avilble EMBRECHTS,P.,KLÜPPELBERG, C. d MIKOSCH, T. (997) Modellig Exreml Eves for Isurce d Fice. Spriger, Berli. FEDERAL OFFICE OF PRIVATE INSURANCE (2006) Techisches Doume zum Swiss Solvecy Tes. Ber. Avilble FÖLLMER, H. d SCHIED, A. (2004) Sochsic Fice. degruyer, Berli. KALLSEN, J. d TANKOV, P. (2004) Chrcerizio of depedece of mulidimesiol Lévy processes usig Lévy copuls. Jourl of Mulivrie Alysis 97,

18 752 F. BIAGINI AND S. ULMER MCNEIL, A.J., FREY, R. d EMBRECHTS, P. (2005) Quiive Ris Mgeme. Priceo Uiversiy Press, Priceo d Oxford. MOSCADELLI, M. (2004) The modellig of operiol ris: experiece wih he lysis of he d colleced by he Bsel Commiee. Bc D Ili, Termii di discussioe No. 57. NESLEHOVÁ, J., EMBRECHTS, P. d CHAVEZ-DEMOULIN, V. (2006) Ifiie me models d he LDA for operiol ris. Jourl of Operiol Ris, (), RESNICK, S.I. (987) Exreme Vlues, Regulr Vriio, d Poi Processes. Spriger, New Yor. ULMER, S.I. (2007) Ei mehrdimesioles Modell für operioelles Risio quifizier durch de Expeced Shorfll, Diplomrbei, vilble FRANCESCA BIAGINI Deprme of Mhemics, LMU, Theresiesr. 39 D Muich, Germy Fx: E-Mil: bigii@mh.lmu.de SASCHA ULMER E-Mil: sschulmer@gmx.de.