Mixed Distributions for Loss Severity Modelling with zeros in the Operational Risk losses

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1 Iteratoal Joural of Appled Mathematcs & Statstcs, It. J. Appl. Math. Stat.; Vol. 21; Issue No. J11; Year 2011, ISSN (Prt), ISSN (Ole) Copyrght by IJAMAS, CESER Publcatos Mxed Dstrbutos for Loss Severty Modellg wth zeros the Operatoal Rsk losses S. El Adlou 1, E. Ezzahd 2 ad Y. Mouatassm 3, 1 Isttut Natoal de Statstque et d'ecoome Applquee (INSEA), Rabat, Morocco E-mal: el_adlou@yahoo.com 2 Mohammed V-Agdal Uversty, Rabat, Morocco 3 Casse Natoale de Securte Socale (CNSS), Casablaca, Morocco Abstract May evets have cotrbuted to the recogto of operatoal rsk as a major problem by baks ad ther regulators. Basel Commttee o Bakg Supervso (BCBS) adopted two ovatve features ts accord of Frst, t recogzed operatoal rsk as a dstct rsk besde credt ad market rsks. Secod, the methodologes proposed to calculate captal charges for the three types of rsk become more rsk-sestve. Ideed, the advaced measuremet approach ams to permt to baks to take to accout ther proper exposure to rsks. I developg ther proper models to quatfy ther exposure to operatoal rsk, baks should take to accout the fact that there are huge dffereces betwee the behavour of the cetral part ad the tal of the dstrbuto of losses. Ths s especally true the case of losses characterzed by the so-called low frequecy-hgh severty losses. I ths paper, mxture models wth a probablty cocetrato for the zeros losses are ftted to our dataset. We used separately a logormal dstrbuto ad a gamma dstrbuto the mxture models. Such models capture dfferetly tal behavour ad allow to take to accout the features that are mportat for operatoal rsk modellg; that s: the exstece of zeros, postve skewess ad heavy taledess of data. Both fts are doe wth a data o the damages of physcal assets curred by a Morocca bak durg two years. Results show sgfcat dffereces whe the logormal or the gamma mxture models are used to evaluate captal at rsk or equvaletly retur perod of a gve loss ad cofrm the dscusso o dstrbutos tals related to the mportace of model selecto. Key words: Operatoal rsk, Basel II Accord, Loss dstrbuto approach, Log-ormal dstrbuto, gamma dstrbuto. Mathematcs subject classfcatos: 46N30, 62G32, 62H10, 62P05 1. Itroducto Facal sttutos are at the core of the fuctog of the ecoomy. They use maly the moey of others to make loas to borrowg agets. Ay dsruptos or problems faced by these sttutos mpact adversely the stuato of other ecoomc agets ad, heceforth, the smooth fuctog of the whole ecoomy. For ths reaso, regulators

2 Iteratoal Joural of Appled Mathematcs ad Statstcs demad to baks ad other facal sttutos to behave prudetly. Oe drecto to push baks to adopt a prudet behavour s to force them to have a captal charge proportoal to the rsks assocated wth ther actvtes. Basel I Accord, ssued by the Bak of Iteratoal Settlemets (BIS) 1988, was fouded o ths approach. For ths reaso, a mmum rato called Cooke rato s defed betwee the captal of the bak ad ts assets weghted by ther respectve rsk exposure. Two crtcsms were addressed to Basel I Accord. Frst, debtors of the same category are treated as f they have the same rsk exposure. Secod, Basel I Accord dd ot take to accout operatoal rsk. For ths reaso, sce ts adopto by the Bak of Iteratoal Settlemets (BIS) 1988, professoals, regulators, ad academcas bega to propose mprovemets especally the sese of 1. the troducto of a dscrmatory treatmet betwee debtors of the same category accordgly to ther credt rsk profle, ad 2. the recogto of operatoal rsk as a major threat to the soudess ad eve to the very exstece of a bak. As a result, BCBS troduced ts accord of 2004 may ovatos the areas of captal adequacy ad the measuremet of the extet of baks exposure to rsks. I ths paper, we focus o losses due to oe type of operatoal rsk (.e. damages to physcal assets) recorded weekly durg two years by a Morocca Bak. We use a mxture model to determe the losses dstrbuto. The mportace of ths cotrbuto stems from the fact that there are o academc papers treatg operatoal rsk the Morocca cotext ad also because usg mxture models for operatoal rsk s more accurate to model operatoal losses. Our cotrbuto focuses especally o the case where there are frequet perods durg whch zero losses are recorded. The probablty model used to ft to the data s chose so as to take to accout the so called low frequecy-hgh severty losses. That s, the chose model must capture the pheomeo of fat/heavy tals the observed data. I ths regard, we should sgal that f theoretcal probablty models provde coverget vews about the cetral part of the data, that s the bulk or the body of the data, vews dverge about the behavour of extreme values or tal evets ( values correspodg to hgh probablty of o-exceedace). Ths paper s structured as follows. I the secod secto we dscuss the defto of OR ad the mportace of ts maagemet Basel II accord. The thrd secto s reserved to the presetato of the ma approaches proposed by BCBS for assessg requred captal agast the operatoal rsk exposure of a bak. The fourth secto summarzes some elemets about the stuato ad the preparedess of Morocca Baks to comply wth Basel II recommedatos especally the area of operatoal rsk ad the regulatory strategy ths area. I the ffth secto, we preset a mxture modellg approach to evaluate the VaR of a partcular type of operatoal rsk usg rescaled data related to a Morocca bak; after ths we preset ad dscuss the results. The last secto serves to coclude. 2. Operatoal rsk: Defto ad sources Operatoal rsk has ot bee completely gored by baks. Some dramatc evets forced t to the frot le of the preoccupatos of facal sttutos especally baks. Some dramatc evets llustrate the catastrophc ature of operatoal rsk. The case of the 97

3 Iteratoal Joural of Appled Mathematcs ad Statstcs oldest bak Europe (233 years), the Bargs, whch collapsed 1995 cosequetly to o authorzed actvtes that provoked 1,3 bllos of USD as operatoal losses. The terrorst attack o World Trade Ceter 2001 s a example of a exteral evet that may cause huge losses. Aother evet wth substatal losses s the wdespread electrcal falure expereced by over 50 mllo people the orth-easter Uted States ad Caada August I 2008, the losses of about 5 bllo of Euro curred by the Frech bak Socété Géérale due to the postos take by a youg trader are a more recet llustrato of the devastatg cosequeces of the materalzato of operatoal rsk. For a log tme, baks ad other facal sttutos have curred losses they could classfy ether to credt rsk or to market rsks. These losses were due to the teral fuctog of the bak or to the terplay betwee the bak ad ts exteral evromet ad are qualfed as operatoal losses. At preset, ther amout ad frequecy crease because of the globalzato of markets, the ever creasg complexty of trasactos, the looseg of credt polces, the wdespread use of techologes, ad the rapd chage of workg rules. The materalzato of these rsks leads to decreases the compettveess of facal orgazatos, degradato of ther facal equlbrum, ad sometmes to ther bakruptcy. Oe of the ma ovatos of Basel II accord s the recogto of operatoal rsk as a well defed specal category of rsk besdes market ad credt rsks. It s explctly recogzed as a major threat to the soudess ad durablty of a bak s actvty. Table 1. Comparso betwee Basel I ad Basel II accords Basel I Basel II Recogsed rsk Credt ad market rsk Credt, market, ad operatoal rsks Proposed methodology to calculate requred captal Oe sze fts all approach Rsk sestve methodologes There s o geerally accepted defto of ths category of rsk amog facal commuty. Its scope s vast ad cludes trasacto-processg errors, systems falure, teral or exteral fraud, ad loss or damage of assets. Amog dfferet rsks that affect facal orgazatos, operatoal rsks would be the most destructve ad most dffcult to forecast. For ths reaso, facal sttutos pay more ad more atteto to detfy, maage, cotrol, ad mtgate ths rsk. The prevalg defto of operatoal rsk comes from BCBS. It states that operatoal rsk s defed as the rsk of loss from adequate or faled teral processes, people, ad systems or from exteral evets. Ths defto cludes legal rsk, but excludes strategc ad reputato rsk (BIS-BCBS, 2006, p. 144). Jameso (1998) defed operatoal rsk as every rsk source that s outsde the areas covered by market ad credt rsks. Ths defto s cosdered too broad because t cludes ot oly operatoal rsk but busess, strategy, ad lqudty rsk as well (Ngel Da Costa, 2004, p 2). Brtsh Bakers Assocato (1997) provded a alteratve defto. It 98

4 Iteratoal Joural of Appled Mathematcs ad Statstcs states that operatoal rsk s the rsk assocated wth huma error, adequate procedures ad cotrol, fraudulet ad crmal actvtes; the rsk caused by techologcal shortcomgs, system breakdows; all rsks whch are ot bakg ad arsg from busess decsos as compettve acto, prcg, etc.; legal rsk ad rsk to busess relatoshps, falure to meet regulatory requremets or a adverse mpact o the bak s reputato; exteral factors clude: atural dsasters, terrorst attacks ad fraudulet actvty, etc (p. 11). Besdes these deftos, may facal sttutos have ther ow defto, whch largely complemets that of BCBS. For stace, Crédt Susse defes operatoal rsk as the: potetal adverse mpact mproper or adequate busess coduct wll have o operatos (Chorafas., 2004., p. 34). I proposg a treatmet of operatoal rsk, BCBS uderled oly mmum stadards for facal sttutos. To tackle effcetly operatoal rsk, Basel Commttee o Bakg Supervso proposed to dstgush seve sources of occurrece of ths rsk.. Iteral fraud: a act teded to defraud, msapproprate property or avod regulatos, the law or compay polcy, excludg dversty/dscrmato evets, whch volve at least oe teral party,. Exteral fraud: a act of a type teded to defraud, msapproprate property or crcumvet the law, by a exteral party,. Employmet practces ad workplace safety: a act cosstet wth employmet, health or safety laws or agreemets, from paymet of persoal grevace clams, or from dversty/dscrmato evets, v. Clets, products, ad busess practces: a utetoal or eglget breakdow to meet a professoal oblgato to specfc clets (cludg fducary ad sutablty requremets), or from the ature or desg of a product, v. Damage to physcal assets: the loss or damage to physcal assets from atural dsaster or other evets, v. Busess dsrupto ad system falures: dsrupto of busess or system falures, ad v. Executo, delvery, ad process maagemet: faled trasacto processg or process maagemet, from relatos wth trade couterpartes ad vedors. Coscous of the devastatg cosequeces of operatoal losses o baks actvty ad ther facal equlbrum, BCBS recommeded for baks to develop relable models ad strateges to detfy, assess, cotrol, ad mtgate ths category of rsk. Takg to accout operatoal rsk faced by a bak meas that t wll be ecessary to quatfy the captal that a bak must take asde agast ths category of rsk. 3. Ma Basel approaches for calculatg baks exposure to operatoal rsk I the logc of BCBS, baks must have a certa captal asde to absorb losses due to the rsks they wll face (Fabozz, p. 754). Ideed, oe of the fuctos of bak captal s to provde a buffer to protect a bak s debt holders agast peak losses that exceed expected levels (BIS-BCBS, 2005, p. 2). To calculate the captal charge commesurate wth a bak s exposure to captal may methods ad approaches are avalable. Frst there are methods that are smple ad do ot requre sophstcated techques. The Loss Dstrbuto Approach s also possble but s more complex ad ecesstates more elaborated tools to be mplemeted. The scorecard approach s also avalable. Aother drecto to evaluate the operatoal rsk captal charge s fouded o the use of Bayesa methods whch tegrate avalable 99

5 Iteratoal Joural of Appled Mathematcs ad Statstcs quattatve data ad pror kowledge provded by experts. BCBS proposed 3 approaches for calculatg requred captal of operatoal rsk exposure. These methods are: 1. the Basc Idcator Approach (BIA), based o a percetage of gross come, 2. the Stadardzed Approach (SA), whch calculates a separate operatoal rsk charge for each le of busess, based, prmarly, o the sze of that busess, 3. the Advaced Measuremet Approach (AMA) whch allows sttutos to apply ther proper models order to evaluate ther exposure to operatoal rsk. The partcularty of the cotuum of methods s that they are, f we pass from the BIA, to the SA, ad the to the AMA, creasgly rsk-sestve ad adaptable to the stuatos of small baks, greater oes, ad teratoally operatg baks. Facal sttutos may select from ths meu, codtoal o the apprasal ad the approval of regulators, the approach whch s most sutable to ther stuato ad whch they ca apply most effectvely. The Basc Idcator Approach ad stadardsed approach are smlar that operatoal rsk requred captal s measured as a percetage of the gross come of the Bak. The major dfferece betwee the two methods s that the regulatory captal s calculated as a percetage of the overall Gross come of the Bak whlst the case of stadardsed approach the actvty of the bak s splted dow to 8 busess les ad bak s operatoal rsk requred captal s the summato of the captal charges calculated for the 8 busess les takg to accout the depedece betwee losses accrug the dfferet busess les. A bak plag to apply AMA should be able to develop ts proper Loss Dstrbuto models based o ts teral data ad, whe proved ecessary, the avalable exteral data. Baks plag to adopt AMA have to prove capabltes to costtute relable data sets, to measure ad maage ther operatoal rsk exposure ad hold commesurate captal (Medova ad Berg-Yue, p. 13). I ths case the actvty of the baks s broke dow to 8 busess les wth each oe of them the seve sources of operatoal rsks are tracked. The requred captal for operatoal rsk exposure s the the addto of captal charge of the 8 busess les. 4. Morocca baks: the road to the mplemetato of Basel II Accord Morocca bakg system s oe of the most lberalzed North Afrca. Sce the begg of the last decade of the tweteth cetury, broad reforms have bee troduced to deepe, to stregthe, ad to moderze the overall framework ad the fuctog of the Morocca facal sector (World Bak, 2001). Noetheless, t s hghly cocetrated, wth the sx largest baks accoutg for 85 percet of bakg sector assets (Achbae et Ezzahd, 2009). At the ed of 2008, there were 18 baks wth a balace sheet of 763 bllo of Morocca drham. The proftablty of Morocca baks s hgh compared to teratoal stadards (World Bak, 2001). I 2008, the retur o equty of Morocca baks was 15.5% ad the retur o assets was 1.1% (BAM-DSB, Exercse 2008, p. 7). The actual Morocca bakg law was eacted It ams to stregthe the supervsory power of the Morocca Cetral Bak (BAM) ad to mprove rsk maagemet 100

6 Iteratoal Joural of Appled Mathematcs ad Statstcs practces. I ths drecto, BAM was empowered to adapt prudetal measures to the rsk profle of each bak (FtchRatg, 2006). Morocca baks are largely complace wth the Basel I stadards ad are o-target to adopt Basel II as requred by the Morocca Cetral Bak. Complace of Morocca baks wth Basel II recommedatos about operatoal rsk s a hgh prorty sce For ths, a mpact study about the cosequeces of the troducto of the ew rules was realsed durg October A ma result of the study s that credt rsk cotrbutes by 83% to the rsks cofgurato of the Morocca Baks whlst the market rsk ad operatoal rsk cotrbute respectvely by 4.7% ad 12.3% (BAM-DSB, 2005, pp ). Durg 2005 ad 2006, a commsso of offcals, represetg the Morocca cetral bak, the Mstry of face, ad baks, had worked to prepare the full adopto of Basel II durg (BAM-DSB, 2006, p. 40). I 2007, BAM joed the Iteratoal Operatoal Rsk Workg Group. Furthermore, BAM s actve provdg assstace to Afrca cetral baks hadlg ad maagg operatoal rsk (Bak Al-Maghrb, 2008, p. 146). Meawhle Morocca baks have take may actos order to develop ther proper teral models order to gauge ther exposure to operatoal rsk. I fact, they mplemeted software to collect ecessary formato about losses ther dfferet ageces. I ther task to mplemet the Loss Dstrbuto Approach (LDA), whch use data collected to calculate captal allocated to operatoal rsk, Baks are geerally cofroted wth the structures of data. I fact, datasets are characterzed by the presece of Zero losses. Ths fact pose a problem the adjustmet of dscrete dstrbutos ad the the aggregato of dstrbutos. Uless ths problem s solved, the estmated captal at rsk would be over or uder estmated. 5. Methodology ad data The dstrbuto of losses due to the materalzato of operatoal rsk a bak may be characterzed by may features. Especally a bak may cur o losses a perod of tme. I the other extreme, there are evets wth low frequecy ad hgh severty that are mportat to cosder because ther occurrece may lead to the collapse of a bak. So, Value at Rsk wll be affected by these two features of ay loss dstrbuto. I the followg, we model the losses due to the damages of physcal assets a Morocca bak. The ftted model s a mxture of cotuous ad dscrete dstrbutos. I choosg the cotuous dstrbuto to be ftted we make the mplct hypothess that the true dstrbuto of operatoal losses the Morocca baks s oe from the class of heavy-taled dstrbutos as proposed by may papers (Shevcheko, 2009, Evas et al., 2007). Ths hypothess s geerally cosdered whe aalysg losses due to the materalzato of operatoal rsk (see for example: Uer, 2008) 101

7 Iteratoal Joural of Appled Mathematcs ad Statstcs Let 5.1 Logormal ad Gamma mxed dstrbutos M0 F x; ; x0, be the model to be ftted to o-ull losses data. F s a cotuous loss Dstrbuto Fucto (DF). The vector of parameters to be estmated s ad s the space parameters. Geerally, the LDA approach the frequecy of ull losses s ftted separately to the severty of loss represeted here by the model M 0. To take to accout the presece of zeros the datasets, we propose the model M1 G x;, p ; x0,,0 p 1 whch s a mxture of the model M 0 of the oull observatos ad a sgular mass,1 p, at zero. Thus the modfed loss dstrbuto s gve by the followg DF: G x;, p 1 p f x0 1 p pfx; f x 0 The parameter estmato of ths type of models has receved cosderable atteto partcularly whe the dstrbuto F s expoetal (Atchso, 1955; Vama, 1995; Muraldhara, 1999, 2000). The expoetal dstrbuto has a very lght tal ad s ot recommeded for hghly skewed dataset caused by the presece of large extremes. May other dstrbutos ca ft adequately loss severty, whe less formato s avalable. Recetly, El Adlou et al. (2008) proposed a classfcato of dstrbutos that ca be cosdered the LDA. They dstgushed three ma classes: - Class C (regularly varyg dstrbutos): Fréchet (EV2), Halphe IB (HIB), Log- Pearso (LP3), Iverse Gamma (IG). - Class D (sub-expoetal dstrbutos): Halphe type A (HA), Halphe type B (B), Gumbel (EV1), Pearso type III (PIII), Gamma (Gam). - Class E (Expoetal dstrbutos). Fgure 1 presets expoetal (E), sub-expoetal (D) ad regularly varyg (C) classes ordered from lght taled (from the left) to heavy taled (to the rght) dstrbutos ad the lmtg cases (dow squares) whch are the dstrbutos the lmts of classes. The tal of the dstrbutos of the class C s heaver tha that of the dstrbutos belogg to class D. The tal of the late dstrbutos s heaver tha that of the dstrbutos belogg to the class E. Fgure 1: Dstrbutos ordered wth respect to the heavess of ther rght tals. 102

8 Iteratoal Joural of Appled Mathematcs ad Statstcs The Logormal dstrbuto s a specal case. Ideed, ts tal behavour s a lmtg case betwee classes C ad D. Ths explas the flexblty of the LN dstrbuto ad ts use to ft extremes may appled statstc felds. I the preset study, the LN dstrbuto s cosdered to ft o-ull losses (dstrbuto F) ad wll be tegrated the mxture model (dstrbuto G). The proposed model s a Logormal mxed dstrbuto (LMD). The probablty desty fucto of the LN dstrbuto wth parameters ad, s gve by: 2 f x exp ;, 1 x 2 l x (1) To estmate the parameters of the mxture model, the maxmum lkelhood approach s used. For a gve dataset x 1, x, of loss severtes, we troduce the varable Z that dcates whether the observato x s ull or ot, we have: Z x 1 f x 0 0 f x 0 (2) The probablty desty fucto of the mxture model s gve by: 2 Z x 2 ;,, 1 ;, g x p p p f x 1Z x (3) Thus the lkelhood fucto of the LMD for a sample x, 1 x, s: 2 Z x 2 ;,, 1 ;, L x p p p f x 1 1Zx Z x Z x Zx p 1 p exp 1z x x 2 1 zx l x 2 Note that Z x s the umber of zeros the dataset. The maxmum lkelhood estmators 1 2 p,, of the vector of parameters p,, ˆ ˆ ˆ ˆ2 are solutos of the system of the dervatves of the logarthm of the lkelhood fucto wth respect to each of the three parameters. The solutos of ths system are: 103

9 Iteratoal Joural of Appled Mathematcs ad Statstcs pˆ ˆ x 0 x 0 2 ˆ l x 1 Z x Z x Z x l x ˆ 2 (4) Note that other dstrbutos ca be cosdered the mxture model ad thus t s mportat to compare dfferet models. I ths study we ll compare the LMD to the Gamma mxed dstrbuto (GMD). The GMD has smlar equatos as preseted for the LMD, where the o-ull severtes are represeted by Gamma dstrbuto. The probablty desty fucto (pdf) of a Gamma dstrbuted varable s gve by: 2 1 fg x;, x expx (5) where 0 ad 0 are respectvely, the scale ad shape parameters. The lkelhood fucto correspodg to a sample x x,, 1 x s gve by: 2 Z x ;,, 1 G ;, L x p p p f x 1 1Zx Zx Zx Z 1 x 1 1zx 1 1 p 1 p x exp 1zx x 1 1 (6) The maxmum lkelhood estmators (MLE) of the Gamma mxed model parameters are soluto to the system of equatos correspodg to the partal dervatve of the lkelhood fucto Lx; p,, wth respect to the parameters p, ad. The MLE of the GMD parameter are computed by umercal maxmsato of the lkelhood fucto (equato 6). For both model, the quatles correspodg to the probablty of o-exceedace q of the mxed dstrbuto G, ca be deduced from that of the F (Logormal or Gamma) dstrbuto by the followg expresso: 1 1q p1 Qq G q F p (7) 104

10 Iteratoal Joural of Appled Mathematcs ad Statstcs Note that the estmated losses ca be represeted as fucto of the probablty of oexceedace or retur perod. For example the loss x T whch s observed mea oe tme every T weeks, correspods to the quatle of the probablty of exceedace 1 T (The 1 probablty that these loss ca be exceeded s oe tme every T weeks : PX x T ). The T estmated quatles of both models are computed usg the mxture model (Equato 7) ad the ML estmated parameters Dataset, results ad dscussos The avalable dataset correspods to the weekly losses observed betwee March, ad February, The legth of the dataset s 98 ad the correspodg descrptve statstcs are summarzed Table 2. Results show that the dataset s hghly skewed ad have a large coeffcet of varato. Ths suggests the ecessty of the use of a model that takes accout of the exstece of zeros ad also ca model postve skewess ad large varablty (presece of extreme values). Table 2. Descrptve statstcs of observed loss severtes Basc statstcs Emprcal values Mmum 0.00 Maxmum 1.27E+006 Mea 1.55E+005 Stadard devato 2.10E+005 Meda 9.56E+004 Coeffcet of varato (Cv) 1.36 Skewess coeffcet (Cs) 2.95 Kurtoss coeffcet (Ck) 12.8 To model these data usg mxg model, some assumptos should be checked out such as depedece, statoarty ad homogeety of the observatos. Two tests, Wald- Wolfowtz ad Wlcoxo were respectvely cosdered to test the depedece ad homogeety. The thrd test s the modfed Kedall test for statoarty. All these tests lead to the acceptato of the ull hypothess, at 5% level of sgfcace, statg that the observatos are depedet ad detcally dstrbuted. The maxmum lkelhood estmators of the LMD parameters are gve by equato (4). 2 The correspodg estmatos are pˆ 0.14, ˆ ad ˆ 1.31 ad the GMD 5 parameter estmators are ˆ ad ˆ To llustrate the mportace of model 105

11 Iteratoal Joural of Appled Mathematcs ad Statstcs specfcato, the Logormal mxed model s compared to the Gamma mxed model followg the same steps detaled the last secto. The Gamma dstrbuto belogs to the class D of Subexpoetal dstrbuto (Fgure 1) ad has a lghter tal whe compared to the LN dstrbuto. Fgure 2 represets the probabltes of o-exceedace for the two models (gree curves for the Gamma ad blue for the LN mxed models). Fgure 2 shows that, for the probabltes betwee 0.1 (retur perod oe week) ad 0.9 (retur perod 10 weeks) the correspodg estmated losses gve by these models are practcally the same (the gree ad blue curves are dstgushable). However, for large retur perod (large tha 50 weeks) the LN estmated losses (Q2) are larger tha those estmated by the Gamma mxed model (Q1). Fgure 2: The o-exceedace probablty correspodg to the Logormal ad Gamma mxed models. Table 3 presets the estmated quatles for gve retur perod (colum 1) or correspodg o-exceedace probablty (q) estmated by the LN (Q1) ad Gamma (Q2) mxed models. The fourth colum gves the rato Q1/Q2 to show the dfferece betwee the two models. Table 3. Losses ad ther retur perods estmated by the Logormal-mxed ad Gammamxed dstrbutos T q 11/ T Q1 Q2 Q1/Q2 (weeks) Logormal Mxed Model Gamma Mxed Model E E E E E E E E E E E E E E E E E E

12 Iteratoal Joural of Appled Mathematcs ad Statstcs Results show table 2 cofrm the dscusso preseted o dstrbutos tals ad the mportace of model selecto. Ideed, the dfferece betwee the Logormal ad Gamma mxed models ca reach 2.83 tmes the estmated loss for hgh retur perods. For small probabltes of o-exceedace, the estmated quatles by the Logormal ad Gamma mxed models are close to each other. However, large dssmlartes are observed for hgh retur perods, large tha 50 weeks. The loss severty estmated by Logormal mxed model correspodg to ths perod s 50% larger tha that of the Gamma mxed model. Ths uder-estmato produced by to the Gamma mxed model ca lead to sgfcat dffereces the bak polces relato to operatoal rsk (Fgure 3). Recet developmets o extreme value theory allows to use graphcal approaches to select the correspodg class of dstrbutos that gve the most adequate ft to estmate large retur perod evets (El Adlou et al., 2008). We expect to use these methods future studes. Fgure 3: The probablty desty fuctos of the Logormal (LMD) ad Gamma (GMD) mxed models wth a zoom o extreme losses. 6. Cocluso Modellg operatoal rsk s at ts facy. There are problems operatoal rsk detfcato, assessmet, cotrol, ad mtgato. Destructve rare evets that may cause very hgh losses are oe of the features of operatoal rsk that must be well modelled ad take to accout. The models ftted to avalable data of losses due to the materalzato of operatoal rsk have to capture other features such as the heavess of the dstrbuto s rght tal. I ths paper, we have used a mxture model to estmate the extet of the exposure of a Morocca bak to operatoal rsk whe zero losses are recorded. Results show sgfcat 107

13 Iteratoal Joural of Appled Mathematcs ad Statstcs dffereces whe the logormal or the gamma mxture models are used to evaluate captal at rsk or equvaletly retur perod of a gve loss ad cofrm the dscusso o dstrbutos tals related to the mportace of model selecto. Operatoal rsk cotrol s ot just a problem of modellg rsks; t s also a problem of deep explorato to kow the mportat shortcomgs the whole framework of fulfllg the actvtes of a bak (Chapelle, 2005, p. 4). Oe of the by-products of the adopto of Basel II Accord wll be the mprovemet of baks effcecy ad the developmet of a more thorough ad robust kowledge of the sources of rsk they face (Raja ad Rzv, 2006, p. 254). Baks ad regulators have geerally opposte goals. For baks they are essetally terested to have small reasoable captal charges for rsks they face. Regulators have as a prmary goal the cotrol ad the lmtato of baks exposto to rsks. The use of data about the bakg dustry may be more robust provdg relable estmates for the true exposto of a typcal bak to losses due to operatoal rsk. Aother drecto for the developmet of relable models about the exposure of baks to OR s to create models wth experts kowledge as put to gauge the frequecy of losses of ay magtude. Refereces Achbae, M. et Ezzahd, E. Evoluto du système bacare maroca etre 1993 et 2004 : structures, comportemets et performaces, Repères et Perspectves, Atchso, J. O the dstrbuto of a postve radom varable havg a dscrete probablty Mass at the org, J. Amer. Stat. Ass., 1955, Vol. 50, p Bak Al-Maghrb. Rapport de la supervso bacare - exercce Bak Al-Maghrb. Aual Report, Bak of Iteratoal Settlemets- Basel Commttee o Bakg Supervso. Iteratoal covergece of captal measuremet ad captal stadards, A revsed framework, Comprehesve Verso, Jue Bak of Iteratoal Settlemets- Basel Commttee o Bakg Supervso. A explaatory ote o the Basel II IRB rsk weght fuctos, July Brtch Bakers Assocato. Operatoal rsk maagemet survey, Chapelle, A. The vrtues of operatoal rsk maagemet, Solvay Busess School, Cetre Emle Berhem, Dmtrs N. Chorafas. operatoal rsk cotrol wth Basel II, basc prcples ad Captal requremets., Elsever Butterworth-Heema., 2004, p. 34. Evas, J., Womersley, R., Wog, D. Operatoal rsks Baks: a aalyss of emprcal data from a Australa bak, Paper preseted to the Isttute of Actuares of Australa, Beual Coveto September Fay, S. The Collapse of Bargs, New York: W.W. Norto,

14 Iteratoal Joural of Appled Mathematcs ad Statstcs El Adlou, S., B. Bobée et T.B.M.J. Ouarda. O the tals of extreme evet dstrbutos. Joural of Hydrology, 2008, 355, Fabboz, F. J. ad Focard, S. M., The Mathematcs of Facal modelg ad vestmet maagemet. Joh Wley & Sos FtchRatg, Le système bacare maroca et ses règles prudetelles Herrg, R. J. The Basel 2 Aprroach to Bak Operatoal Rsk: regulato o the Wrog Track, Paper preseted to the 38th Aual coferece o Baks Structure ad Competto of the Federal Reserve Bak of Choocago o May 9, Jameso, R., 1998, Playg the ame game, Rsk 11(10): Medova, E. A. ad Berg, E. K. Bakg captal ad operatoal rsk : Comparatve aalyss of regulatory approaches for a bak, Joural of facal trasformato, 2009, 26, Muraldhara, K. Tests for the mxg proporto the mxture of a degeerate ad expoetal dstrbuto, J. Ida Stat. Ass., 1999, 37, ssue 2, p Muraldhara, K. The UMVUE ad Bayes estmate of relablty of mxed falure tme dstrbuto., Comm. Statst- Smulatos & Computatos, 2000, 29(2), p Ngel Da Costa L., (2004). Operatoal Rsk wth Excel ad VBA, Appled Statstcal Methods for Rsk Maagemet. 2004, p Raja, S. ad Rzv, H. S. Basel II ad rsk maagemet, The Chartered Accoutat, August Shevcheko, P. V. Implemetg loss dstrbuto approach for operatoal rsk Appled Stochastc Models Busess ad Idustry. 2009, I Press. Uer, S. G. Loss Dstrbuto approach for the operatoal rsk Ecoomc Captal, SAS global Forum, paper Vama K. O the dstrbuto of the estmated mea from the ostadard mxtures of dstrbuto. Comm Statst Theory Methods, 1995, 24(6): World Bak. Morocco s facal sector strategy ote

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