Modelling Operational Risk in Financial Institutions using Hybrid Dynamic Bayesian Networks. Authors:

Transcription

1 Modellng Operaonal Rsk n Fnancal Insuons usng Hybrd Dynamc Bayesan Neworks Auhors: Professor Marn Nel Deparmen of Compuer Scence, Queen Mary Unversy of London, Mle nd Road, London, 1 4NS, Uned Kngdom Phone: +44 (0) mal: marn@dcs.qmul.ac.uk Agena Ld., Haon Garden, London C1N 8DL, Uned Kngdom -mal: marn@agena.co.uk Dr. Lasse B. Andersen Deparmen of Indusral conomcs, Rsk Managemen and Plannng, Unversy of Savanger, 4036 Savanger, Norway Phone: mal: lasse.b.andersen@us.no Davd Häger, PhD Suden Deparmen of Indusral conomcs, Rsk Managemen and Plannng, Unversy of Savanger, 4036 Savanger, Norway -mal: davd.hager@us.no Phone: , mob:

2 xecuve Summary Ths paper descrbes he use of Hybrd Dynamc Bayesan Neworks (HDBNs) o model operaonal rsk n an AMA conex. The approach focuses on causeeffec modellng ncludng neracons beween falure modes and conrols. Value a Rsk s calculaed by applyng a new sae-of-he-ar HDBN algorhm ha approxmaes connuous loss dsrbuons and aggregaes across loss ypes. In order o llusrae he naural mach beween he model and he underlyng process, ncludng he causal complexy underlyng known and possble severe operaonal rsk losses, we apply he generalsed model o a fnancal radng example rogue radng. We conclude ha he sascal properes of he model have he poenal o explan recen large scale loss evens and offer mproved means of loss predcon. Keywords: Operaonal Rsk Managemen, Bayesan Neworks, Causal Models, Dynamc Dscrezaon, Basel II, Advanced Measuremen Approach, Rogue Tradng. 2

3 Modellng Operaonal Rsk n Fnancal Insuons usng Hybrd Dynamc Bayesan Neworks Davd Häger, Marn Nel and Lasse B. Andersen Unversy of Savanger Bayes Rsk Managemen AS Deparmen of Compuer Scence, Queen Mary, Unversy of London Agena Ld Absrac Ths paper descrbes he use of Hybrd Dynamc Bayesan Neworks (HDBNs) o model he operaonal rsk faced by fnancal nsuons n erms of economc capal. I descrbes a mehodology for modellng fnancal losses resulng from nenonal or accdenal evens and characerses hese by her ably o evade conrols and ulmaely lead o ncreasngly severe fnancal consequences. The approach presened focuses on modellng he causes and effecs of loss evens usng a Dynamc Bayesan Nework model based on neracons beween falure modes and conrols. To calculae he Value a Rsk (VaR) for oal losses we apply a new sae-of-he-ar Hybrd Bayesan Nework algorhm, called dynamc dscrezaon. The algorhm approxmaes he connuous loss dsrbuon funcons requred for each loss even a each pon n me and s used o aggregae across loss ypes. In order o llusrae he naural mach beween he model and he underlyng process, ncludng he causal complexy underlyng known and possble severe operaonal rsk losses, we apply he generalsed model o a fnancal radng example rogue radng. We conclude ha he sascal properes of he model have he poenal o explan recen large scale loss evens and offer mproved means of loss predcon. 1. Inroducon The Basel Commee on Bankng Supervson, n reacon o a number of well-publcsed fnancal dsasers (e.g. Barngs bank 1995, Dawa bank 1995, and Alled Irsh Bank 2002) has mandaed a sysem of regulaon addressng he ssue of Operaonal Rsk (OpRsk) and s assessmen and managemen (Basel 2006). Key o he regulaory process s he need for fnancal nsuons o model her operaonal rsk, n erms of a varey of loss even ypes (ncludng unauhorzed radng) n order o effecvely manage rsk and esablsh an approprae regulaory capal charge. Basel II defnes hree approaches o esablshng regulaory capal for OpRsk: The Basc Indcaor Approach (BIS), he Sandardzed Approach (SA) and he Advanced Measuremen Approach (AMA). AMA s by far he mos demandng approach, bu also he mos rewardng n erms of poenally reducng he OpRsk regulaory capal charge by as much as 20-40% compared o BIS and SA. An AMA model requres approval by he local Fnancal Supervsory Auhory and mus a leas conan he followng mporan model arbues: quanfcaon of expeced/unexpeced losses a a gven 3

4 confdence level; a one year holdng perod; nclude scenaro analyses; reflec day o day OpRsk managemen pracce. Of course, operaonal rsk problems are no pecular o he fnancal secor and operaonal rsk s no a new opc. In hs book, James Reason argues ha operaonal rsk s faced by all organsaons and he uses examples from he Fnancal, Ral Transpor, Cvl Avaon and Nuclear power secors (Reason 1997) and he concludes ha accdens are no solely he resul of human fallbly bu are suppored by organsaonal feaures ha fal o defend agans all-oo-human msakes, slps and (n he case of fraud) malcous acs. Human error mgh be seen as an onse of a caasrophc even, bu whou laen weaknesses whn he organsaon he even would no reach caasrophc proporons. From hs we conclude ha operaonal rsk predcon s nexrcably enwned wh good managemen pracces and ha measuremen of operaonal rsk can only meanngfully be done f he effecveness of organsaonal specfc rsk managemen and conrol processes s regularly assessed and ncluded n he modellng. Ths conrass sharply wh a vew ha OpRsk modellng solely nvolves he nvesgaon of sascal phenomena ha characerses many of oday s daa drven AMA models. Moreover, because of hs we fnd ha managemen of OpRsk whn he banks busness processes are oo ofen deached from he models developed o quanfy regulaory capal. Leanng on more han half a cenury of experence wh OpRsk n safey crcal ndusres we fnd ha modellng for he analyss of operaonal rsk always should srve o suppor decsons by provdng answers o he followng quesons: Is he rsk hgh or low, s accepable? Whch causal facors are mos crcal? Wha are he dfferences n rsk consderng alernave soluons? Wha rsk reducng effec can be acheved consderng mplemenaon of alernave rsk reducng measures? I s consdered fundamenal n OpRsk managemen o be able o creae quanave rsk models ha consue a sound bass for fruful dscussons on organsaon specfc rsk. Thus, dealed causal modellng a busness process level s requred for he model o absorb organsaon specfc npu, hghlgh he crcaly of causal facors (rsk drvers), denfy he poenal lack of conrols/barrers, and quanfy he overall rsk level n erms of expeced and unexpeced losses. Such a model obvously creaes more value han a rsk model based solely on acuaral echnques. Varous modellng ools have been developed and appled for hs purpose n he safey crcal ndusres, he mos famous ones beng Faul Trees (Haasl 1965) and ven Trees (Nelsen 1971). ven hough hese ools have domnaed quanave rsk assessmen modellng for several decades we acknowledge varous serous shorcomngs n her use: common causes, me dynamcs, subjecve probably and large number of dependen parameers and mul-sae varables raher han bnary-sae varables. Ths makes hem unsuable for dealed cause-consequence modellng of large and complex phenomena. Thus, he las decade an ncreasng number of professonals n he safey crcal ndusry have adoped Bayesan Neworks (BNs) o handle such modellng challenges [Ale e al 2007, Nel e al 2003; Fenon e al 2004; Langseh 2002; Røed e al 2007]. Moreover, due o sgnfcanly ncreased oupu value, numerous researchers have suggesed causal modellng by means of BN raher han he radonal acuaral echnques [Alexander 2003, Aduse-Poku 2005, Cowell e al 2007, Nel e al 2005, Mnk and Sarobnskaya 2007]. I s argued ha a major benef of BNs s ha hey provde a srucured and scenfcally sound way of combnng sascal analyss wh oher sources of emprcal knowledge (e.g. knowledge on busness processes and/or near msses) usng Bayesan mehods (Nel e al, 2005). The ably of BNs o accoun for causal dependences enables rsk analyss o lnk he operaonal condons of he bank, ncludng conrol envronmen, drecly o he probably of losses occurrng, as well as he severy of he losses. In oher words quanavely assess he effec of rsk managemen decsons on rsk exposure. 4

5 However, research on he applcaon of BN models for operaonal rsk has so far faled o address hree fundamenal modellng challenges. These are: Illusrae applcably o a complex bankng process. xamples so far have been smplfed and appled only o fragmens of a bankng processes hus no llusrang wde applcably for he full complexy of operaonal rsk n banks Accoun for me dynamcs n operaonal rsk loss evens. Operaonal losses evolve hrough seres of sequenal evens n me (e.g. sequenal checks performed by a se of conrols ha may, or may no be funconal) whch has no been capured by any BN model so far Implemen connuous varables o assess loss severes. Use of BN sofware ools (e.g. (Hugn 2008) or (Neca 2008)) usng sac dscrezaon (unform dscrezaon n fxed nervals) for connuous varables lm he applcaon of connuous varables and hus also he possbles o model loss severes accuraely If any large scale applcaon of BN mehodology s o be achevable n he conex of operaonal rsk hese hree challenges have o be overcome. Operaonal rsk praconers are of he vew ha BN models are suable as a ool for rsk managemen, bu no as a ool for esablshng economc capal for regulaory purposes (Aduse-Poku 2005). We are assumng ha he relucance o use BNs for quanfcaon of regulaory economc capal s he resul of he nably o effecvely and accuraely handle connuous varables. However, f regulaory capal s o reflec he ndvdual rsk exposure (as saed by Basel II as a qualfyng crera for he AMA) follows ha hs measure has o change n accordance wh effors made o nfluence rsk.e. rsk managemen effors and quanfcaon of regulaory capal mus be lnked. Thus we see no reason no o use BN for esablshng a regulaory capal charge once he above challenges have been properly addressed. In hs paper we sugges soluons o hese challenges o he applcaon of BNs n operaonal rsk modellng, usng new sae-of-he-ar algorhms. We am o use he fnancal radng process as an example o llusrae how BNs can be used and he background o fnancal radng s gven n Secon 2. Secon 3 nroduces BNs, how hey are defned and he algorhms needed o solve hem, ncludng approaches o deal wh dynamc behavour and hybrd mxures of varables. In Secon 4 we specfy a generalsed Hybrd Dynamc Bayesan Nework (HDBN) model o accoun for me dependences n he chan of evens affecng operaonal loss. Ths model has hree layers: a loss even model, a loss severy model and a loss aggregaon model. Secon 5 uses he generalsed HDBN alored for he radng process and uses causal loss facors derved from acual case daa, coupled wh fcous probably and loss severy esmaes, for wo dsnc scenaros, o produce an aggregaed loss dsrbuon for radng losses and accompanyng Value a Rsk (VaR) esmaes. Fnally, n Secon 6 we offer some conclusons. 2. Rogue Tradng The background nformaon for he modellng example presened n hs paper s gahered from case sudes of severe, radng loss evens whn he Tradng and Sales Busness Lne (BIS, 2006), more colloqually known as rogue radng. In parcular, we have consuled repors on he Barngs Bank collapse (Bank of ngland 1995, IMD Inernaonal 2002, Burke 2004), he Dawa bank losses ncurred by rader Toshhde Iguch (Sungard Bancware rsk 2001 wh references), he Alled Irsh Bank loss (Wachell e al 2002) and he more recen Socéé Générale scandal (Progress repor of he Specal Commee of he Board of Drecors of Socéé Générale, 2008). We fnd ha he wsdom of hndsgh provded by he nvesgaons of hese evens gve ample knowledge o consruc a model of a generalsed radng process n a bank, ncludng he mos mporan conrols and conrol falures. To accoun for learnng whn he fnancal ndusry we have also consuled bes pracce documens lke he Model Code The nernaonal Code of Conduc and Pracce for he Fnancal Markes (ACI The Fnancal Markes Assocaon, 2000) and Managemen of Operaonal Rsk n Foregn xchange (The Foregn xchange Commee, 2004). The radng process and he major losses ncurred as a resul of hs acvy sus our purpose of llusrang applcaon of BN models o complex bankng 5

6 processes. We have however lmed he scope of he model only o address he rsk of unauhorsed propreary radng occurrng n he radng process. 2.1 The Tradng Process The process of radng can, more or less ndependen of he produc raded, be exhausvely descrbed by he followng seps; Trade reques Conduc Trade Regsraon of Trade Reconclaon check, Selemen and Neng nerng of Trade on Tradng Books. Any bank nvolved n radng also conducs connuous monorng of resuls and posons,.e. monorng of marke rsk exposure (every 1, 10 or 30 days dependng on he produc and he bank). Obvously here are ndvdual dfferences beween banks, he producs hey rade and sysems employed o carry ou he radng bu he basc general srucure of he process s essenally he same across banks. Recep of rade reques, conducng rade and regserng he rade n he radng sysem are done by he fron offce,.e. he raders. The back offce performs he confrmaon, reconclaon and selemen check whch nvolve checkng counerpary nformaon, auhorsaons, as well as check ha he rade s whn he erms of counerpary conrac (for non-propreary radng). The checks performed by he back offce have he poenal o uncover lm breaches by a rader bu are no desgned specfcally o serve hs purpose. Posons and resuls monorng s done by he mddle offce whch conans he rsk conrol funcons. Here he oal marke rsk exposure s monored and conrolled agans he rsk lms se by he bank. One of he major rsks denfed whn he radng process s ha of unauhorsed radng. Unauhorsed radng s essenally radng ousde lms specfed by he bank and he resul s, smply pu, an overexposure o marke rsk. However, radng ousde he lms may ncur caasrophc losses and nvolves hdng losses hrough fcve rades, fabrcang rades and manpulang marke posons monorng, as well as falsfcaon of documens, and n oher ways nfluencng he conrol srucure (see e.g. Wachell e al 2002). In oher words hese are very complex evens; however he complexy of he evens also provdes several opporunes o dscover he ransgresson assumng he bank has suffcen and funconal conrols n place. 2.2 The Conrols Process The poenal for hgh losses n radng acves, llusraed by evens lke Barngs, and more recenly confrmed by he Socéé Générale 3.7 bn loss, has resuled n banks mplemenng a complex conrol srucure o manage her radng. Conrols are mplemened o preven he raders (and/or oher employees) o conduc rades ha volae esablshed regulaons, lms and conracs. Should errors occur, here are also conrols desgned o uncover any rregulares and hus avod or lm he poenal loss. Conrols can be vewed as drec or ndrec; drec conrols beng specfc checks performed o ensure he rade s conduced n a correc manner (e.g. reconclaon check), and ndrec conrols beng operaonal elemens havng an ndrec mpac on he probably of ncorrec rades beng made and he performance of he drec conrols (e.g. organsaonal culure, rsk appee, ncenves srucure). The conrols perform checks a dscree mes n a sequenal order followng he rade progresson from begnnng o end n he process. Frsly here are conrols n he fron offce nended o monor and resrc he radng acvy so ha s kep whn he lms se by he bank. An example of such a conrol s nsrumened checks n he radng IT sysem, monorng he rades beng enered and provdng warnng should specfed consrans be volaed. Anoher example s a requremen for raders o ake a leas wo weeks vacaon and durng ha me hand over her radng porfolo o anoher rader o enable dscovery of unauhorsed radng (f any). Several oher conrols are mplemened on several levels and some bes pracces (no only regardng conrols bu also rounes) are provded by e.g. The nernaonal code of conduc and pracce for he fnancal marke (ACI, he Fnancal Marke Assocaon, 2000). Some of he conrols menoned are drec resuls of prevous unauhorsed radng evens (e.g. he vacaon rule). Secondly, once he rade has been processed and enered n he radng sysem by fron offce personnel, a number of consecuve checks are performed by he back offce. Inal checks performed n he back offce are done per rade and nclude checks such as reconclaon check/selemen check 6

7 and neng (see e.g. The Foregn xchange Commee, 2004). In addon he mddle offce performs connuous posons and resuls monorng ncludng ndependen prce checks and calculaon of marke VaR. Ths s done perodcally varyng from daly o every en days o every monh dependng on he produc and he bank. If unauhorsed posons are acqured by a rader hese wll show up n he marke VaR check provdng s no successfully hdden by he rader, hrough e.g. manpulaon of prce nformaon. Manpulaon of prce nformaon o change he value of posons may be uncovered hrough an ndependen prce check conduced n he mddle offce. Fnally here s also he conrol of perodcal or random auds of he radng operaon. The aud s dreced a he enre radng operaon ncludng processes and procedures, segregaon of dues, ec. and also ncludes checkng a sample of rades o check ha here are no false, unauhorsed or oherwse llegmae rades presen on he banks radng books. Based on he revewed case sudes we have arrved a he followng conrols as beng he mos mporan for prevenng and uncoverng unauhorsed radng here presened n sequenal order: Fron offce conrol envronmen (he conrol envronmen affecs he probably of unauhorsed radng) Back offce reconclaon checks (performed per rade) Marke posons and resuls monorng, VaR calculaon (perodcal) Aud checks (perodcal bu no as ofen as he marke checks) We are aware ha hese conrols are no exhausve wh regard o he radng process; we have for example chosen o exclude he specfc conrol of nsrumened checks n he fron offce, and only nclude he reconclaon check n he back offce. Ths s a conscous lmaon of he model n a necessary rade off beween complexy and reader frendlness. However, he modellng examples n secon 4 llusrae ha any number of conrols can easly be ncluded n he model. As we are focussng on propreary radng no clen s ncluded n he conrol srucure. For non-propreary rades we could vew he clen as a poenal conrol agans unauhorsed rades as s lkely ha he clen could dscover unauhorsed rades and make hs known o he bank. 2.3 Loss Severy The loss severy of unauhorsed rades, beng eher he resul of delberae acs or mshaps, s a complex phenomenon. Dependen on he me of dscovery and marke movemens a bank can suffer sgnfcan losses. The major unauhorsed radng evens are he resul of smple unhedged dreconal bes, very vulnerable o marke movemens. I s a combnaon of hese dreconal bes, volale markes and commmen o large posons ha gve rse o he al evens of he loss severy dsrbuon. The pon s ha he loss severy s he resul of a complex se of evens n self. We shall n hs paper only demonsrae ha BNs can handle connuous varables o an exen necessary o model complex loss severy dsrbuons; furher research s needed o esablsh a mehodology for loss severy modellng usng BNs o ensure ha he dsrbuons reflec he characerscs of operaonal rsk. 3. Bayesan Neworks A BN s a dreced acyclc graph, such as he one shown n Fgure 1, whose nodes represen he varables (wh each varable here s assocaed uncerany wh he sae of he varable) of neres and whose edges are he causal or nfluenal lnks beween he varables. Assocaed wh each node s a node probably able (NPT), a sascal dsrbuon or parameersed funcon. In he case of an NPT he relaonshp beween nodes s governed by a se of condonal probably values ha express he relaonshp beween he node and s parens ogeher wh any uncerany ha s presen n ha relaonshp. BNs enable reasonng under uncerany and combne he advanages of an nuve vsual represenaon wh a sound mahemacal bass n Bayesan probably. Wh BNs, s possble o 7

8 arculae dependences beween dfferen varables and o propagae conssenly he mpac of evdence (observaons) on he probables of unceran oucomes. The underlyng heory of BNs combnes Bayesan probably heory and uses condonal ndependence o represen dependences beween varables (Pearl 1986), (Spegelhaler and Cowell 1992). To dae BNs have proven useful n many areas of applcaon such as medcal exper sysems, dagnoss of falures, paern machng, speech recognon and, more relevanly for he operaonal rsk communy, rsk assessmen of complex sysems n hgh sakes envronmens, [Nel e al 2001, Ale e al 2007, Nel e al 2003; Fenon e al 2004; Langseh 2002; Røed e al 2007], ncludng fnancal nsuons [Alexander 2003; Nel e al 2005; Aduse- Poku 2005; Cowell e al 2007]. BNs were orgnally developed o model unceranes beween dscree varables wh fxed labels. Typcally, connuous varables are represened n a BN by a fxed se of dscree nervals and approxmaed by a pecewse consan funcon (.e. a hsogram) and hs dscresaon s hen fxed hroughou he nference process. 3.1 Defnon and Algorhm Formally, a BN s a probablsc graphcal model ha represens a se of dscree valued varables and her probablsc dependences. BNs are dreced acyclc graphs (DAGs), whose nodes represen varables n a probably dsrbuon, and whose arcs encode condonal dependences beween he varables. ach varable X wh parens pa( X ), has an assocaed probably able (also called poenal, condonal probably able or node probably able (NPT)), X pa( X )). A BN over a collecon of varables X = { X1, X 2, L, X n }, has a jon probably dsrbuon P( X ) whch s he produc of all condonal probables, as specfed by he gven BN: X ) = X pa( X )). A poenal s a real-valued able over doman of fne varables. Here φ X s used o denoe he poenal on he doman of a se of varables X. There are wo basc operaons on poenals: 1. Combnaon: If φ s a poenal over X, ψ s a poenal over Y, hen φ ψ s a poenal for X Y, where denoes combnaon. 2. Margnalzaon: If X1, X 2 are margnalzed ou of poenal φ over se of varables X = {,, L, }, hen: X1 X 2 X n X, L, X ) = X). 3 n Once a BN s desgned, can be execued usng an approprae propagaon algorhm, such as he Juncon Tree algorhm (Jensen 1996). Ths nvolves calculang he jon probably and all margnal probables for he model from he BN s condonal probably srucure n a compuaonally effcen manner. Ths s acheved by auomacally dervng from he BN an nermedae graph heorec represenaon of he BN, called he Juncon Tree (JT). The JT allows localsed, modular compuaons o be execued usng a message-passng algorhm. Ths s, n essence, an elaborae form of use of Bayes heorem. For full deals see (Jensen 1996), (Laurzen and Spegelhaler 1988), (Pearl 1986) or (Spegelhaler and Cowell 1992). Gven a BN over a se of varables X = { X X } X1, X 2, we can defne he JT, whch conans clusers and 1,..., n separaors. A cluser s a se whch conans one or more varables and s represened by a sngle node. The separaor s he nersecon of wo neghbourng clusers. A juncon ree over X s a ree of clusers of varables from X, such ha for each par of nodes, V, W, all nodes on he pah beween V and W conan he nerseconv W. The juncon ree assocaed wh he orgnal BN s consruced n he followng way: 8

9 1. Form a famly of nodes such ha for each varable X X, here s a leas one node V such ha{ X } pa( X ) V. 2. Organze he nodes as a ree wh each edge labelled wh separaors. ach node V and separaor S of he cluser ree s assocaed wh a facor or poenalφ V, ( φ S respecvely) over s varable se. 3. Gven all nodes V and separaors S assgn a able of ones o φv and φ S 4. For each varable combne φ V usng X pa( X )) X X choose exacly one node V conanng { X } pa( X ) V and Clearly he produc of all he poenals n he cluser ree s he produc of all condonal probably ables n he BN, so he poenal based represenaon of a BN s m X ) = φ φ L φ k = 1 Vk V1 V. m Propagaon nvolves desgnang a roo of he juncon ree frs, and hen, based on a message-passng scheme, every leaf cluser sends a message owards he roo. Once he roo has receved all he messages hen propagaes messages ouward o he leafs. Le V and W be neghbours n a juncon ree. Le S = V W be her separaor, andφ V,φ W and φ S be her poenals. A message pass from V o W occurs n wo major seps, called projecon and absorpon: V V W Projecon: φ compues he projecon of he poenal of V ono he separaor S,.e., he margnalsaon of he poenal φ V wh respec o S, sorng he message n he separaor s poenal: φ S = V W φ. V Absorpon: The message o cluser W s gven by he margnalsaon of he poenal φ V ono he doman S = V W. Updae he poenal of W wh he message from V, by sorng * he message: φw = φw, φv. We can hnk of absorpons as messages passed beween ( V W ) he nodes n he ree. Tha s, a node V sends a message o s neghbour W when W absorbs from V. The message self s he projecon of he poenal of V ono he separaor S : φ φ. S = V V W Afer all nodes n he JT have sen and receved messages from all neghbours he JT s sad o be globally conssen and margnal dsrbuons of neres can be relably obaned. The JT algorhm s enrely auomac and, n a ool lke AgenaRsk (AgenaRsk 2008) or Hugn, s hdden from he doman exper. When he BN s execued he effecs of daa enered no one or more nodes can be propagaed hroughou he BN, n any drecon, and he margnal dsrbuons of all nodes updaed. Ths makes deal for wha f? and scenaro analyss. Clearly, he key o he successful desgn of a BN model s he meanngful decomposon of a problem doman no a se of causal or condonal proposons abou he doman. Raher han ask an exper for he full jon probably dsrbuon of all he varables of neres, whch s obvously a very dffcul ask, we can apply a dvde and conquer approach and ask for paral specfcaons of he model ha are hemselves meanngful n he expers doman. In our case, for operaonal rsk he srucure s an obvous arefac dervable from he operaonal process, as wll become evden n laer dscusson. Nex, we need o model he condonal probably ables for each varable (node): hs can eher be done usng hsorcal daa (ncludng, for example, usng sandard Bayesan parameer learnng approaches or Mone Carlo smulaons), or by smply askng he exper o provde a seres of 9

10 subjecve esmaes. The expers consuled have o suppor credbly n he assessmens made o movae ha he esmaes are based on experence and knowledge raher han blnd guesswork. 3.2 Hybrd Bayesan Neworks An exended noon of a BN s a hybrd BN (HBN) whch conans boh dscree and connuous varables and an nference algorhm wh specal operaons o deal wh connuous deermnsc and sascal funcons. In HBNs, local exac compuaons can be performed only under he assumpon of Condonal Gaussan (CG) dsrbuons (Laurzen and Jensen, 2001). The advanages and drawbacks of usng Condonal Gaussan dsrbuons are well known. They are useful o model mxures of Gaussan varables condoned on dscree varables and weghed combnaons of CG parens bu hey are much oo nflexble o suppor general-purpose nference over hybrd models conanng mxures of dscree labelled, neger and connuous ypes and non-gaussan dsrbuons. Mos real applcaons demand non-sandard hgh dmensonal sascal models wh nermxed connuous and dscree varables, where exac nference becomes compuaonally nracable. The presen generaon of BN sofware ools such as, [Hugn, 2008, Neca, 2008], aemp o model connuous nodes by numercal approxmaons usng sac dscrezaon. Alhough dscrezaon allows approxmae nference n a hybrd BN whou lmaons on relaonshps among connuous and dscree varables, curren sofware mplemenaons requre users o defne a unform dscrezaon of he saes of any numerc node (wheher s connuous or dscree) as a sequence of pre-defned nervals, whch reman sac hroughou all subsequen sages of Bayesan nference regardless of any new condonng evdence. The more nervals you defne, he more accuracy you can acheve, bu a a heavy cos of compuaonal complexy. Ths s made worse by he fac ha you do no necessarly know n advance where he poseror margnal dsrbuon wll le on he connuum for all nodes and whch ranges requre he fner nervals. I follows ha where a model conans numercal nodes havng a poenally large range, resuls are necessarly only crude approxmaons. We can embed connuous and dscree sascal dsrbuons whn he HBN model, as NPTs, and generae values for hese NPTs by approxmaon mehods, ncludng Mone Carlo smulaon. Unl very recenly, BN ools were unable o handle non-gaussan connuous varables, and so such varables had o be dscrezed manually, wh nevable loss of accuracy. However, a breakhrough dynamc dscrezaon algorhm presened n (Nel e al 2007) has now been mplemened n a sofware ool, (AgenaRsk 2007). Ths allows he approxmae soluon of classcal Bayesan sascal problems, nvolvng connuous varables, as well as hybrd problems nvolvng boh dscree and connuous varables. Ths algorhm overcomes mos of he problems nheren n he sac or unform dscrezaon approaches. In parcular, problems relaed o compuaonal neffcency caused by supporng oo many saes o represen he doman, hgh level of naccuracy n poseror esmaes for connuous varables, problems n nsanang evdence n areas of he doman ha are grossly under sampled ha lead o nconssency and error. Full deals of he dynamc dscrezaon algorhm are descrbed n (Nel e al 2007) bu here we provde an overvew of he algorhm. Le X be a connuous (or neger valued) random node n he BN. The range of X s denoed by Ω, and he probably densy funcon (PDF) of X, wh suppor Ω, s denoed by f. The dea X of dscrezaon s o approxmae f X as follows: 1. Paron X Ω no a se of nerval X { wj} Ψ =, and 2. Defne a locally consan funcon f % X on he paronng nervals. Dscrezaon operaes n much he same way when X akes neger values bu n hs paper we wll focus on he case where X s connuous. As n Kozlov and Koller (1997), we use an upper bound of he Kullback-Lebler (KL) merc beween wo densy funcons f and g : f ( x) D( f g) = f ( x)log dx g ( x ) S X X 10

11 as an esmae of he relave enropy error nduced by he dscrezed funcon. Under he KL merc, he opmal value for he dscrezed funcon f % s gven by he mean of he funcon f n each of he nervals of he dscrezed doman. The man ask reduces hen o fndng an opmal dscrezaon se Ψ = ω. X { j} Our approach o dynamc dscrezaon searches Ω X for he mos accurae specfcaon of he hghdensy regons gven he model and he evdence, calculang a sequence of dscrezaon nervals n Ω eravely. A each sage n he erave process, a canddae dscrezaon, Ψ, s esed o X deermne wheher he relave enropy error of he resulng dscrezed probably densy f % X s below a gven hreshold, defned accordng o he rade off beween he accepable degree of precson and compuaon me. By dynamcally dscrezng he model we acheve more accuracy n he regons ha maer and ncur less sorage space over sac dscrezaons. Moreover, we can adjus he dscrezaon any me n response o new evdence o acheve greaer accuracy. In oulne, dynamc dscrezaon follows hese seps: 1. Conver he BN o a juncon ree (JT) and choose an nal dscrezaon for all connuous varables. 2. Calculae he NPT of each node gven he curren dscrezaon 3. ner evdence and perform global propagaon on he juncon ree, usng sandard JT algorhms. 4. Query he BN o ge poseror margnals for each node, compue he approxmae relave enropy error, and check f sasfes he convergence crera. 5. If no, creae a new dscrezaon for he node by splng hose nervals wh hghes enropy error. 6. Repea he process by recalculang he NPTs and propagang he BN, and hen queryng o ge he margnals and hen spl nervals wh hghes enropy error. 7. Connue o erae unl he model converges o an accepable level of accuracy. 3.3 Hybrd Dynamc Bayesan Neworks Dynamc Bayesan Neworks (DBNs) allow us o model emporal dependences of complex sysems as a joned sequence of BNs, each represenng a parcular objec or he enre process a a parcular A = a,..., may change sae as some process, pon n me (called a me-slce). Some varable { 1 a n } O, progresses hrough dscree me nervals { 1,...,T } =. A each dscree me nerval, addonal nformaon s also receved abou he sae of A. The addonal nformaon may be observaons, specfc measuremens or asks performed a me affecng he sae of A. The sae of A a me s deermned based on he sae n he prevous me nerval,.e. 1. The sae of A mgh hen be deermned as A A 1, ). DBN models poenally overcome mos of he problems nheren n sae-space based models and n parcular, avods he sae-space exploson problem of he Markov chans based approaches. Hybrd DBNs are smply DBNs ha conan mxures of dscree and connuous varables ogeher. The curren sae-of-he-ar n approxmae nference echnques on HDBN models wh arbrary probably dsrbuons s prmarly based on sochasc algorhms and Markov Chan Mone Carlo mehods, (Murphy 2002), (e.g., mporance samplng, sequenal Mone Carlo, and Rao-Blackwellsed Parcle Flerng). These mehods rely on nensve samplng algorhms ha requre drawng ens of housands of dependen samples from, usually, hgh dmensonal probably dsrbuons. Ths presens wo man shorcomngs: makes smulaon echnques compuaonally neffcen and X 11

12 requres specalzed sascal knowledge. In parcular, specalsed knowledge s requred o ensure convergence of he dependen samples o a relable resul, and, even more dffcul, o denfy and deal wh he specal srucures whn he hdden varables of he HDBN requred o make samplng n hgh-dmensonal spaces a feasble ask, specfcally n Rao-Blackwellsed schemes. 4. The Generalsed HDBN Model for Operaonal Rsk Here we descrbe a generalsed HDBN model for operaonal rsk. The generalsed HDBN model for operaonal rsk comprses hree layers: Loss even model Loss Severy model Aggregaed loss model ach layer s represened by a dfferen BN, DBN or HDBN, as approprae wh nerface lnks beween hem comprsng common parameers. Use of Hybrd Bayesan Neworks have also been suggesed by Mnk and Sarobnskaya (2007) o model dependences beween operaonal rsk classes. We use he word generalsed o reflec he propery ha he layers are general enough o cope wh a wde varey of suaons, processes and conexs bu, n pracce, s necessary o nsanae hem by denfyng specfc varables local and meanngful n he process under sudy. 4.1 The Loss Model The frs layer s he loss even model whch models he poenal loss evens,, and how hese dynamcally evolve over me as hey are nfluenced by conrols, C, embedded whn he busness process. Ths dynamc me-based evoluon of an even gven he conrols, s modelled by 1, C ) wh me perods = 1,..., T. The performance of each conrol, C, s modelled as a funcon of a se of operaonal falure modes, O j. These falure modes are n urn nfluenced by a se of causal facors, F, whch n solaon or combned nae he operaonal falure. Dependence beween operaonal falure modes for dfferen conrols s modelled hrough dependency facors D k. The dependency facors D, are condoned on he occurrence of some operaonal falure mode specfc for a conrol, k C, and hey, n urn, nfluence he probably of occurrence of a falure mode n a secondary conrol, C +, where s = 1,..., T such ha: p + ( OC D ) ( ) s k p Dk OC s An operaonal falure mode may be modelled as a funcon of boh causal facors, F, and dependency facors, D k. Operaonal falures ha have no dependency relaonshp wh oher falure modes are smply modelled as a funcon of causal facors. The busness process s hen represened by a sequence of dscree me-dependen evens such ha we have a Dynamc Bayesan Nework as shown n Fgure 1 whch s a graph represenaon of he full jon dsrbuon: T m n C,O, F, D) = 1, C ) C O C ) O j FO, DO ) Dk Oq ) F ) C0 ), = 1 j= 1 = 1 k = 1 q= 1 o r j j 12

13 where j q, and, C, O, F, D are ses of loss evens, conrols, operaonal falure modes, causal facor varables and dependency facors such ha: {,..., }, C = { C, C,..., C }, O = { O, O,..., O }, F = { F, F,..., F }, { D, D,..., D } = D 0, m 1 2 n = 1 2 Furhermore, we assume ha he operaonal falure modes, O j, can nfluence any of he conrol varables whn any me perod,, and are hus me ndependen, usng ndex j nsead of. A smlar jusfcaon apples for he condonal relaonshp beween he causal facors and he operaonal falure modes. The use of ndex q for operaonal falure modes affecng dependency facors Dk denoes ha an operaonal falure mode canno have a dependency relaonshp wh self. By addng and C nodes and O j, F, and f applcable, D k varables we can easly ncrease he number of conrols, assumng a placemen on he dscree me connuum can be esablshed. l C 0 C 1 C 2 C 3 O 1 O 2 O 3 O 4 O 5 O 6 D 3 F 9 F 1 F 3 F 5 F 7 F 4 F 2 D 1 D 2 F 6 Fgure 1 DBN of Loss even model ach of he sae ransons 1, C ) s governed by a dscree node probably able ha models he ranson probably of he loss even from an undeeced (unsafe) o a deeced sae, dependen on he conrol sae. Should he conrol varable be operang correcly a me, he loss even a 1 would rans o a correc operang sae a usng he followng logcal condonal ranson probables: 13

14 = fal = fal = fal = fal = OK = OK = OK = OK = fal, C = fal, C = OK, C = OK, C = fal, C = fal, C = OK, C = OK, C = fal) = 1 = OK) = 0 = fal) = 0 = OK) = 0 = fal) = 0 = OK) = 1 = fal) = 1 = OK) = 1 I s easy o see how we could generalse hs o cope wh more loss even saes, ncludng hose showng a severy scale. So, f he nang loss even has no occurred, or has been deeced by a prevous conrol a a prevous me, he queson of wheher hs or laer conrols operae correcly become rrelevan. Thus he chance of deecon and ranson o a correc (safe) operang sae s deermned by he number of mplemened conrols as well as her ndvdual performance (relably). ach conrol varable, C, s dependen on a collecon of operaonal falure varables, O C, conanng a subse of he operaonal falure modes deemed o preven he conrol from performng s nended funcon. Thus he performance of each conrol s modelled by C O C ). Lkewse he operaonal falure modes, O j, are condonally dependen on a subse of he causal facors, F, and dependency facors D relevan o he occurrence of he ndvdual falure modes, and hs s modelled by p ( F, D ). O j O j O j k For smplcy we have chosen o represen all conrol, operaonal falure, causal facor and dependency facor varables usng Boolean nodes, for whch we can model C O C ) and p ( F, D ) usng Boolean logc o represen he ype and form of falure. Ths modellng choce O j O j O j allows use of Boolean operaors, such as AND, OR, XOR, NOT ec whn a dscree BN model. For example an OR operaor would be declared as: 1 f O1 O2 = fal C1 = fal O1, O2 ) = 0 oherwse The NPT for node C 1 wh parens O 1 and O 2 s hen generaed accordng o he above expresson. Alernavely, we can manually declare an NPT o represen he form of dependences ha bes maches he process, such as: C = fal O = fal) = C = OK O = fal) = C = fal O = OK) = C = OK O = OK) = Ths s equvalen o sayng we would expec he falure of conrol, C 1, o occur wh chance 80% when operaonal falure mode, O 1, has occurred. Ths conrass wh he use of he Boolean OR operaor whch assumes a 100% chance of he conrol falng. 14

15 The presened mehodology s by no means resrced o he use of Boolean varables. On he conrary one of he benefs of usng BNs s ha he modeller may declare cusomzed varables and sae spaces o f he problem doman. Choce of varable ypes and sae space s ulmaely lef o he modeller n he process of developng a model bes sued for he process beng analysed. Armed wh hs model specfcaon we can calculae he margnal probably of occurrence for each loss even a any dscree me sep n he DBN, hus: T m ) = 1, C ) C OC ) O j FO, DO ) Dk Oq ) F ) C0) C,O,F,D = 1 j= 1 = 1 k = 1 q= 1 n o r Inference here can be carred ou usng he JT algorhm descrbed n secon 3.1, and he model wll ypcally execue whn a few seconds wh dozens of nodes. The advanages of hs model over a Mone Carlo smulaon model are: Unlke n smulaon, all compuaons are exac so s emnenly suable for calculang ulra low probably evens. Can deal wh mulple, neracng, common causes. In addon o srong mahemacal foundaons he graph represenaon of he BN model maps nealy ono he underlyng process and clearly shows how losses propagae. We can generae a se of loss even probables for dfferen severes of evens a dfferen sages of propagaon hrough he process and use hs o calculae our severy and loss models. Noe ha some clarfcaon of wha we mean by causes and how we denfy hem s needed here snce here s much undersandable dspue and confuson of how hese erms mgh be producvely appled o operaonal rsk and oher problems. We recognse ha here are an nfne number of acual and possble neracons beween unque evens and ha s herefore mpossble o represen hese aomc level neracons n any model. Insead we seek o model broad classes of evens ha have characerscs, behavour and oucomes ha are smlar enough o be consdered homogenous and whch can reasonably be expeced o nerac n a smlar way. Gven hs any dsncon beween classes of causal evens s a praccal decson as much nformed by process undersandng, documenaon, daa collecon sandards and budge as any refned phlosophcal poson. For nsance n a praccal suaon we mgh class operaonal IT falure as a sngle causal class coverng secury lapses, sysems falures and performance degradaon smply because he consequenal effec on a conrol ha depends on he IT nfrasrucure would be smlar n all cases. 4.2 The Loss Severy Model Here we use he probables generaed by he loss even model o predc he oal losses by severy class, gven he severy dsrbuon and a measure of volume o scale he losses. We assume ha s a subse of he saes n ) we are neresed n here, specfcally hose ha ncur a loss or are L unsafe : = e ), and of course here may be more han one loss even for each dscree me sep,, hence he superscrp, L. For brevy we le L = e ) = p L. e We can model he oal losses usng a condonal dependency model, agan represened as a DBN, where he oal number of loss evens whn each me sep, N, s bnomally dsrbued, gven a volume measure, e.g. he oal number of rades conduced weekly, monhly or annually, V : L e j j 15

16 N L e ~ Bn( W L, p L ) e e, where L e 1 W = V N s he volume of ransacons mnus he cumulave number of loss evens = 0 L e predced for all prevous me seps,.e. -1 n he DBN. Graphcally hs model s shown n Fgure 2. Assocaed wh each L Fgure 2 Loss severy model e s a severy dsrbuon f ( S ), represenng he dsrbuon of fnancal losses, rework coss or oher penales. We assume ha he severy dsrbuon can ake any form and ha he condonally deermnsc varable, T, represenng he oal loss for a sngle even s: f ( T S, N ) = S N L L L L L e e e e e where S 0,.e. we only consder losses and no gans. L e L e Furhermore we assume ha he severy of evens ha are dscovered laer n me are more serous snce hey ypcally lead o larger expeced losses or have larger unceranes aached. arler, near mss losses wll end o be of a more predcable naure snce hey are lkely o be encounered more ofen. Also early dscovery by a conrol wll presumably provde greaer opporuny o lm he poenal loss of an unauhorsed rade. Therefore, one mgh expec o consran he model such ha: ( S ) < ( S ) <... < ( S ). L L L e1 e2 e T The margnal dsrbuon for he oal losses for each loss even s herefore: f ( T ) = f ( T S, N ) f ( N W, p ) f ( S ) L L L L L L L L e e e e e e e e S L, N L, W e e e L L e 4.3 The Aggregaed Loss Model L Now we have a se of oal loss varables, T L, for each loss even, e e and wsh o calculae he oal aggregaed losses, A, as he sum of oal losses assocaed wh each even n each me perod. Ths T aggregaed sum s smply he deermnsc funcon, A = T L. Ths can eher be solved by = 0 e 16

17 convoluon or samplng, however, n our BN framework we nsead use he dynamc dscrezaon algorhm descrbed n Secon 3.2. The oal aggregaed loss dsrbuon s obaned by margnalsng: = f ( A) f ( A T ) f ( T ) L L e e T e L, where T L s he collecon of oal losses for each loss even. e From f ( A ) we can now calculae he Value a Rsk (VaR); under he Basel regulaons he VaR sasc s he 99.9 h percenle of hs margnal dsrbuon: f ( A) 99.9% 5. An HDBN for he Rogue Tradng Process 5.1 The Loss ven Model α =. In he conex of he example model n hs paper unauhorsed radng s defned as a rader (nenonally or unnenonally) exceedng hs auhorsed radng lm. The man conrols n place o preven such an even, and deec should occur are: Fron offce conrol envronmen Reconclaon check carred ou by he back offce Poson and resuls monorng (marke rsk monorng) carred ou by he mddle offce Aud (perodcally or random) Any rade conduced s assumed o be n one of a fne se of muually exclusve saes correspondng o poenal loss evens,. ach of he mplemened conrols performs a check ha may change or confrm (.e. he sae remans as before) he sae of he rade gven s rue sae. We assgn he followng defnon o he loss even varables: 0 : Sae of rade when enered n he radng sysem by he rader, C 0 1: Loss even dscovered durng reconclaon check, C 1 2 : Loss even dscovered durng marke rsk VaR check, C 2 3 : Loss even dscovered by aud, C 3, or whch has escaped he conrol sysem The saes assgned o each varable are: = {Auhorsed, Accdenal, Illegal Fame, Illegal fraud} = {Dscovered, OK, Accdenal, Illegal Fame, Illegal fraud} = {Dscovered, OK, Accdenal, Illegal Fame, Illegal fraud} = {Dscovered, OK, Accdenal, Illegal Fame, Illegal fraud} The meanng of each of he saes are as follows: 17

18 Auhorsed/Ok rade. An auhorsed rade permed and approved by managemen beforehand. Accdenal unauhorsed rade. Trades ha are accdenal due o msakes by he rader,.e. rades ha were no nended o be unauhorsed and of course no nended o ncur losses for eher he bank nor s clens Illegal Fame rade. Unauhorsed rades wh he nen o furher he rader s career, saus whn he bank and/or sze of bonuses. For hs caegory of rades he rader nends o make money for he bank and subsequenly also provde benefs for hmself. The mporan aspec s ha he rade s no nended o drecly benef he rader bu ndrecly hrough provdng success for he bank Illegal Fraud rade. Unauhorsed rades wh he sole nen of benefng he perperaor, f he bank or anyone else suffers losses as a resul of he rade s rrelevan as long as he perperaor makes money off he rade Dscovered. A rade of any unauhorsed caegory s revealed by a conrol and s hus a dscovered unauhorsed rade. We could use only wo saes for he even varable, 0 ; Normal and Unauhorsed bu ha wll no properly reflec he possble severy of he even. Lkewse we dsngush beween llegal fame and fraud saes because we are assumng dfferen severes, and wsh o assgn dfferen severy dsrbuons o hem. Alernavely we could desgn hree ndvdual models usng he presened mehodology, each addressng one of he unsafe rade saes separaely (.e. accdenal, fame and fraud rades). Such an approach could poenally allow a greaer degree of alorng he model o he operaonal falure modes and nfluencng facors relevan specfcally o a parcular loss even. However such an approach would also resul n greaer model complexy and sze. The rade has a rue loss even sae when enered n he radng sysem by he rader. A = 0 he loss even sae s unknown o us and as he rade progresses hrough he conrols process and checks are performed he sae of he rade may change o dscovered f an unauhorsed rade s uncovered, oherwse wll reman n he OK sae as he rade s assumed auhorsed. There s also a possbly ha an unauhorsed rade wll no be dscovered and connue o be n an unknown unauhorsed sae and escape he conrol sysem and our model of he process. Such loss evens ypcally have hgher severy and her mpac fel much laer when subsequen frauds are dscovered and forensc analyss hrows up unauhorsed pas rades. We may also use escaped unauhorsed rades o assess he severy of loss where severy s assumed dependen on he occurrence of prevous unauhorsed rades, hs wll be addressed n more deal n furher research. The descrbed process can be modelled usng our HDBN approach and he correspondng loss even model s presened n Fgure 3. 18

19 Fgure 3 DBN Loss even model for rogue radng process In Fgure 3 each conrol s nfluenced by a seres of operaonal falure nodes, O C, where each operaonal falure nfluence he performance of one or more conrols. Here he operaonal falures are parcular o each conrol wh he excepon of he operaonal falure Acve Dsrupon (AD) whch s shared by all of he conrols, excep he fron offce. However snce he conrols normally are separaed from each oher, dfferen acons, and allances are needed by he rader o dsable a complee se of conrols. Thus we have ncluded an AD node, O AD for each conrol, C excep C 0. Also he probably of acve dsrupon n one conrol s dependen on wheher or no here has been acve dsrupon n he precedng conrol, assumng a rader ha has successfully dsruped one conrol s lkely o aemp o dsrup he nex. Hence gven acve dsrupon n he back offce here s an ncreased probably of acve dsrupon n he remanng conrols. Such dependency s modelled usng dependency facors, D k. Ths s a useful feaure and an mporan advanage of he BN approach n ha common causes of falure ha undermne he whole process can be accouned for. In he case of Acve dsrupon he dependency facors, D, nfluencng conrols falure nclude faled segregaon of dues (beween fron and back offce), fabrcaed npu daa (n he posons monorng), corruped rade samplng (n he aud process) and allances beween saff, whn and ou wh, he nsuon. The fac ha hs operaonal falure affecs all back offce, mddle offce and aud conrols means a rogue rader could negae or reduce he effecveness of all conrols across he board. Also, modellng he acve dsrupon of conrols as suggesed enables he analys o accoun for he complexy n he delberae acs o negae or reduce he effcency of conrols. In conras wh Acve Dsrupon each of he oher conrols are modelled usng causal facors, such as compeence falure, IT falure and poson/prcng msakes. To llusrae how our approach works we have assgned probably values o each of he causal facors as well as he fron offce conrol, for wo scenaros, as shown n Table 1. The frs scenaro O j 19

20 represens saus quo on a radng floor whls he second represens he case where he conrol envronmen n he fron offce has faled and here s acve dsrupon n he back offce. Table 1: Falure probables for causal facors n wo scenaros Varable (rue = faled ) Probables for scenaro 1 Probables for scenaro 2 p ( C0 = rue) F C 1 = rue) FIT = rue) FR = rue) FA 1 = rue) FI = rue) F = PC rue) F M = rue) F CM = rue) F C 2 = rue) For he dependency facors resulng n acve dsrupon n he mddle offce and he nernal aud, whch are also dependen on prevous acve dsrupon D AD O, he assgned probables are: D D D D D D D D A2 A2 FI FI A3 A3 CT CT = rue O = rue O = rue O = rue O = rue O = rue O = rue O = rue O AD2 AD2 AD2 AD2 AD3 AD3 AD3 AD3 = false) = = rue) = 0.7 = false) = = rue) = 0.8 = false) = = rue) = 0.6 = false) = = rue) = 0.6 Noe ha he probably of occurrence of he dependency facors s assumed o ncreases dramacally when here s acve dsrupon n he precedng conrol. For he nal even sae 0 C 0, he NPT s shown n Table 2. Noce ha, when fron offce conrols are workng he odds for llegal fame and fraud evens are n he regon of 1:1000 for llegal fame and 1: for llegal fraud. Ths rses o 1:100 and 1:10000 when conrols have faled. O AD 20

21 Table 2: NPT probables for 0 C 0) C0 = false C0 = rue 0 = Auhorsed = Accdenal = Illegal Fame = Illegal Fraud When we execue he DBN, nference s carred ou on all varables and margnal dsrbuons produced as shown n Fgure 4 for he frs scenaro. Fgure 4 DBN Loss even model for rogue radng process wh supermposed margnal probably dsrbuons From Fgure 4 we oban he loss even probables for scenaro 1, and hese are lsed n Table 3. These loss probables gve he probably ha a sngle rade wll belong o ha parcular loss class. From hese we can calculae ha he escape probably for a loss,.e. he probably he rade s unauhorsed and evades all of he conrols s n scenaro one and n scenaro 2. The probably of a rade beng unauhorsed and dscovered n scenaro one s and n scenaro 2. 21