Application of Bayesian Inference to Operational Risk Management

Transcription

1 Applicatio of Bayesia Iferece to Operatioal Risk Maagemet Yuji Yasuda Doctoral Program i Quatitative Fiace ad Maagemet Advised by Professor Isao Shoji Submitted to the Graduate School of System ad Iformatio Egieerig i Partial Fulfillmet of the Requiremets for the Degree of Master of Fiace Uiversity of Tsukuba Jauary 3

2 Applicatio of Bayesia Iferece to Operatioal Risk Maagemet Yuji Yasuda Abstract Bayesia iferece that is able to combie statistical measuremet approach ad sceario aalysis is effective exceedigly for measurig operatioal risk. I choosig the prior distributio, takig idicators that may be predictive of the risk of future losses, exteral circumstace ad so forth ito cosideratio makes it possible to obtai more realistic risk amout, this process itself is a importat for ehacig operatioal risk maagemet. This paper proposes the examples of applicatio of Bayesia iferece to bakig practices. Keywords: Operatioal risk, Bayesia iferece, Prior distributio.

3 Ackowledgemet I am deeply grateful to my academic advisor, Professor Isao Shoji for his ecouragemet ad costructive commets. I also thak the member of Shoji Laboratory ad all my frieds for their support. Fially, I appreciate the Bak of Tokyo-Mitsubishi, Ltd. that gave me a good opportuity that I have studied at the Uiversity of Tsukuba.

4 Cotets Itroductio. Backgroud 3.. What is Operatioal Risk? 3.. Why is It Importat? 5.3. Why is Measurig It Necessary? 9. The Cocept of Measurig Operatioal Risk.. Top-Dow Method.. Bottom-Up Method 4 3. Applicatio of Bayesia Iferece What is Bayesia Iferece? Natural Cojugate Prior Distributio 3.3. Advatages of Bayesia Iferece 6 4. Examples of Applicatio to Bakig Practices Operatios Error Rate Number of Operatios Loss Evets Severity of Operatios Loss Evet Simulatio of Risk Amout Profitability Judgmet Discussios 36 Refereces 38

5 Itroductio Rapid ad extesive chages i the bakig eviromet make operatioal risk maagemet more importat. These chages result from ecoomic ad fiacial globalizatio ad cotiuig advaces i iformatio techology. The eed to maage operatioal risk through measurig methods becomes icreasigly urget with each passig year. The Basel Committee is curretly workig o ew BIS rules to iclude operatioal risk withi capital adequacy guidelies alog with market ad credit risk. Furthermore, it will become more ad more importat i the future to cotrol risk alog with cost reductio as low-cost operatio is ehaced through cotiuous ratioalizatio of bak maagemet. There are two mai methods i measurig operatioal risk: statistical measuremet approach ad sceario aalysis. Statistical measuremet approach is to be performed to the same stadard as for market risk ad credit risk, statistically measure the risk based o historical data o frequecy of loss occurrece, size of loss ad so forth. O the other had, uder sceario approach, as for evets with low frequecy ad high severity, losses would be estimated based o scearios, with referece to exteral data ad evets that occurred at other baks. However, if baks measure risk based oly o past evet data, they might ot capture those material potetial evets with low frequecy ad high severity ad, likewise, will ot capture the future impact of the chagig

6 eviromet both iterally ad exterally o future operatioal losses. Sceario aalysis teds to be less objective tha statistical measuremet approach. So, we propose we apply Bayesia iferece to measurig operatioal risk. Bayesia iferece combies statistical measuremet approach ad sceario aalysis. For measurig market risk ad isurace, the Bayesia aalysis of extreme value data has bee developed by Kotz ad Nadarajah, ad Smith. The idea of Bayesia etwork is applicable to oly settlemet risk. For operatioal risk maagemet, Cruz just metioed Bayesia techiques i his operatioal risk textbook briefly. So, we itroduce the examples of applicatio of Bayesia iferece to operatioal risk maagemet i view of bakig practices. By applyig Bayesia iferece, we ca use both data as likelihood, ad o-data iformatio as prior distributio. Predictive idicator ad qualitative data result ca be used for measurig. This paper aims to provide reader with possible solutios, or at least hits. This paper is orgaized as follows. We review the operatioal risk i sectio. Sectio summarizes the methods of measurig operatioal risk proposed util ow. The cocept of Bayesia iferece is described i Sectio 3. I sectio 4, the examples of applicatios for busiess practices are preseted ad discussios follow i sectio 5.

7 . Backgroud.. What is Operatioal Risk? Operatioal risk was a term that has a variety of meaigs withi the baks. Some baks defied operatioal risk as a o-measurable risk. A uiversal defiitio has ot yet bee established, but there has bee a high degree of covergece durig the past few years. I Jauary, the Basel Committee o Bakig Supervisio, which formulates broad supervisory stadards ad guidelies ad recommeds statemets of best practice, defied operatioal risk as: the risk of loss resultig from iadequate or failed iteral processes, people ad systems or from exteral evets. This defiitio was adopted from idustry as part of the Committee s work i developig a miimum regulatory capital charge for it. This defiitio icludes operatios risk, IT risk ad legal risk. Examples of operatioal risk evet iclude the followig: Executio, delivery ad process maagemet data etry errors, collateral maagemet failures, icomplete legal documetatio, uapproved access give to cliet accout, o-cliet couterpatry misperformace, vedor disputes, etc. System failure hardware ad software failures, telecommuicatio problems, utility outage, etc. 3

8 Iteral fraud itetioal misreportig of positios, employee theft, isider tradig o a employee s ow accout, etc. Exteral fraud robbery, forgery, cheque kitig, damage from computer hackig, etc Cliet, products ad busiess practices fiduciary breaches, misuse of cofidetial customer iformatio, improper tradig activities o the bak s accout, moey lauderig, sale of uauthorized products, etc. Damage to physical assets terrorism, vadalism, earthquakes, fires floods, etc. The Basel Committee s defiitio excludes strategic, reputatio ad systemic risk. They are ot suitable for capturig ad cotrol based o measurig risk at preset. This issue should always be revisited i the future i accordace with developmet of maagemet eviromet ad busiess structure. 4

9 .. Why is It Importat? Substatial Loss Evets The term operatioal risk was metioed for the first time after the ifamous Barigs bakruptcy evet i 995. Barigs lost $.9 billio through uauthorized tradig by their "star" trader, Nick Leeso. Breakdows i fudametal cotrols ca be attributed to these losses, as Leeso's activities wet uoticed util it was too late. Also, the Daiwa Bak s rogue trader, Toshihide Igushi, lost $. billio through tradig i US treasury bods. The trades took place uoticed over a log period of time, i fact years from 984 to 995. The trader covered up the tradig losses by falsifyig assets supposedly owed by the bak. It is clear that through these actios the trader was i effective cotrol of both frot ad back offices. This is a fudametal violatio of ay risk maagemet strategy. Ulike Barigs collapse, Daiwa survived although the icidet cost them oe seveth of their capital base. Ad US regulators prohibited them from cotiuig their operatios there, a uprecedeted move. Recetly, the terrorist attacks o the Uited States o September,, damaged may baks physical assets immesely. The computer systems problems ad operatioal cofusio, such as ATM services problems ad delays i automatic debit trasactios, i coectio with the lauchig of Mizuho Corporate Bak ad Mizuho Bak i April, is fresh i our memory. 5

10 Complexity of Bak s Operatios Over a few decades, baks have developed, ad have capitalized o, ew busiess opportuities give advaces made i IT, deregulatio, ad globalizatio. Also, sophisticatio of fiacial techology has bee growig. As the result of the faster pace of chage i the complexity of their operatios, the operatioal risks that baks face today have become more complex ad diverse tha ever before. Examples of these ew ad growig risks faced by baks iclude: Growth of e-commerce brigs with it potetial risks e.g., exteral fraud ad system security issues that are ot yet fully uderstood; Large-scale mergers, de-mergers ad cosolidatios test the viability of ew or ewly itegrated systems; If ot properly cotrolled, the use of more highly automated techology has the potetial to trasform risk from maual processig errors to system failure risk, as greater reliace is placed o globally itegrated systems; ad Baks may egage i risk mitigatio techiques e.g., collateral, credit derivatives, ettig arragemets ad asset securitizatios to optimize their exposure to market risk ad credit risk, but which i tur may produce other forms of risk. 6

11 3 The New Basel Capital Accord The Basel Committee o Bakig Supervisio of the Bak for Iteratioal Settlemets sets BIS guidelies that prescribe capital adequacy stadards for all iteratioally active baks. More tha a decade has passed sice the Basel Committee itroduced its 988 Capital Accord. The busiess of bakig, risk maagemet practices, supervisory approaches, ad fiacial markets each have udergoe sigificat trasformatio sice the. I Jauary, the Committee issued a proposal for a New Basel Accord. The proposal has three core elemets: required regulatory capital i lie with the risks at each fiacial istitutio; supervisory reviews by atioal bakig regulators, ad market disciplie through the disclosure of iformatio. The Committee believes that three pillars will collectively esure the stability ad soudess of fiacial systems. I respose to the growig eed for a system to cope with these operatioal risks, the Basel Committee is curretly workig o ew BIS rules to iclude operatioal risk withi capital adequacy guidelies alog with market ad credit risk. The 988 Accord set a capital requiremet simply i terms of credit risk the pricipal risk for baks, though the overall capital requiremet i.e., the 8% miimum ratio was iteded to cover other risks as well. I 996, market risk exposures were removed ad give separate capital charges. I its attempt to itroduce greater credit risk sesitivity, the Committee has bee workig with the idustry to develop a suitable capital 7

12 charge for operatioal risk. The ew regulatios themselves will reflect the ature of risks at baks more closely. To calculate the regulatory capital, the Committee has offered a meu of optios from which baks ca choose ot oly market risk, where the meu approach has bee implemeted sice 998, but also for credit risk ad operatioal risk i the proposed framework. Uder this framework, baks ca choose calculate their ow required regulatory capital based o their ow risk profiles ad risk maagemet methodology. Therefore, baks have started work to coform to the proposed regulatios. This icludes the selectio of a more advaced approach i the proposed meu i lie with their risk profiles. The Committee, i discussios with the idustry, is curretly fializig the proposal. The ew regulatios are expected to become effective i 6. 8

13 .3. Why is Measurig It Necessary? Operatioal risk i the past used to cotrol based o qualitative risk maagemet practices ivolvig checklist ad operatios maual. But baks have foud limits to traditioal qualitative operatioal risk maagemet. Measurig risk is a effective tool for capturig ad cotrollig it. The mai reasos why baks try to measure operatioal risk are followig: Adequacy of Required Ecoomic Capital for Risk As market risk ad credit risk measuremet methods have bee developed, large baks have, i tur, established a capital allocatio system. By optimizig capital allocatio, baks aim to maximize retur after deducig cost of capital ad risk-adjusted performace measuremet, which assess their profitability ad efficiecy relative to risks. The capital allocatio system sets the amout of capital allowed to be placed at risk by each our busiess uits. The level of risk is the cotrolled ad maaged so as to remai withi that allocatio. Each busiess uits must take risks withi their capital adequacy. The capital allocated by this system seeks to cover all risks icludig operatioal risks. Thus, it is ievitable for baks to allocate their ecoomic capital to operatioal risk explicitly. Capturig risk o a itegrated basis by measurig each risk accordig to a commo stadard makes differet types of risk comparable with each other 9

14 ad thus leads to a effective ad efficiet use of maagemet resources. Performace Evaluatio It is importat to give employees the icetives to ehace risk maagemet through various methods such as performace idicators. It is commoly see i practice, however, that employees ted to focus o ways to icrease retur rather tha performace idicators such as retur o equity ROE. ROE could be based o measured risk, depedig o the balace betwee risk takig ad risk maagemet. Thus, baks seek to allocate ecoomic capital to operatioal risk based o risk measuremet ad results of risk assessmet, so employees have a icetive to improve risk maagemet. The improvemet, which turs out to be measured, reduces the allocated capital to their operatioal risk as their performace evaluatio improves. 3 Iteral Cotrol Framework Some baks seek to establish a basis for effective ad efficiet iteral cotrol measures. Subjective judgmets o iteral cotrol, however, ted to misguide the board of directors ad seior maagers with wrog priorities i ehacig operatioal risk maagemet. Operatioal risk measuremet eables baks to establish criteria of objectivity ad comparability i prioritizig risk cotrol amog differet busiess lies ad risk categories, i order to supplemet iteral cotrol i a more robust way. The result of operatioal risk measuremet ca be fed back to each busiess uit such as a

15 brach at sectio level ad serve as a icetive to improve iteral cotrol such as the revisio of operatig procedures. 4 Criterio for Use of Isurace It is worthwhile to keep abreast of effective risk trasfer methods such as isurace or ART Alterative Risk Trasfer. Isurace compaies start to supply ot oly traditioal BBB Bakers Blaket Bods but also more comprehesive isurace products which cover a wider rage of operatioal risks faced by baks. Measuremet of operatioal risk will serve as a importat criterio for determiig which is more advatageous i light of the cost of capital, maitaiig capital or buyig isurace.

16 . The Cocept of Measurig Operatioal Risk There are both Top-dow ad Bottom-up methods i measurig operatioal risk. At preset, several kids of measuremet methods are beig developed ad o idustry stadard has yet emerged. The details of these methods ca be see i Hiwatashi ad Ashida ad Marcelo G. Cruz... Top-Dow Method Top-dow method seeks to estimate operatioal risk o a macro basis without idetifyig evets or causes of losses. Table shows the examples of Top-dow method. I Top-dow method, the total amout or chage of profits or expese etc. derived from fiacial data i the balace sheet ad profit & loss statemet is coverted to risk amout. Although this method eables easy capturig of the overall risk, it is difficult i this way to determie icetives for risk mitigatio by idetifyig the areas eedig improvemet. It does ot lead to a appropriate capturig of risk accordig to the circumstaces or serves as adequate iformatio for market participats because it usually applies oe uiform set of multiplicatio factors regardless of the differeces betwee coutries i accoutig systems, employmet practice, ad the level of expectatio from customers for services provided by baks ad so forth.

17 The Basel Committee proposed a idicator approach as the most basic approach. Each bak calculates the capital for operatioal risk equal to the amout of a fixed percetage, α, multiplied by its idividual amout of gross icome. The approach is easy to implemet ad uiversally applicable across baks. However, its simplicity comes at the cost of oly limited resposiveess to firm-specific eeds ad characteristics. Therefore, the Committee expects iteratioally active baks ad baks with sigificat operatioal risk to use a more sophisticated approach. Table. Examples of Top-Dow Method Approaches Way to Measure Operatioal Risk Idicator approach It is assumed that, for example, gross icome or cost is a proxy, ad that a certai percetage is regarded as operatioal risk of baks. CAPM approach It is assumed that all the risks are measured based o Capital Asset Pricig Model CAPM; the, market risk ad credit risk, measured separately, are deducted from all risk measured by CAPM. Volatility approach Volatility of icome is regarded as a risk. For example, volatility of o-iterest icome, which is regarded as operatioal risk, is measured. Hiwatashi, Ashida 3

18 .. Bottom-Up Method Bottom-up method measures operatioal risk based o idetified evets that explai the mechaism of how ad why it occurs. Table shows the examples of Bottom-up method. Table. Examples of Bottom-Up Method Approaches Statistical Measuremet Approach Sceario Aalysis Way to Measure Operatioal Risk The maximum amout of operatioal risk is measured based o idividual evets with frequecy ad severity usig Mote Carlo simulatio or a aalytical solutio. As for evets with low frequecy ad high severity, losses would be estimated based o scearios, with referece to exteral data ad evets that occurred at other baks. Factor Aalysis Approach Bayesia Network Model Factors related to losses such as trasactio volume ad error ratios are idetified ad are take ito accout with correlatio aalysis. Causes ad effects of operatioal risk are modeled. There are cases where this model is used i settlemet risk maagemet. Hiwatashi, Ashida 4

19 This method eables the aalysis of risk factors ad serves as a effective icetive for reductio operatioal cost ad mitigatio of operatioal risk icludig the review of operatioal work flows though it required complicated aalysis of risk by busiess lie. The followig two approaches are used widely i Bottom-up method: Statistical Measuremet Approach Uder the statistical measuremet approach, a bak, usig its iteral data, estimates two probability distributio fuctios for each busiess lie ad risk type; oe o sigle evet impact severity ad the other o evet frequecy for the ext oe year. Havig calculated separately both the severity ad frequecy processes, we eed to combie them ito oe aggregated loss distributio that allows us to predict figure for the operatioal losses with a degree of cofidece usig Mote Carlo simulatio or a aalytical solutio. The idea behid statistical measuremet approach is as follows: Measuremet of operatioal risk should be performed to the same stadard as for market risk ad credit risk. They should also be comparable with each other to esure cotrol o a itegrated basis. Market risk ad credit risk have bee statistically measured based o the aalysis of historical data o the market ad actual loss. As for the capital charge o operatioal risk, it will cofirm to the objective of risk maagemet, i.e. precise capturig risk, ad will lead to ehacemet of risk maagemet capabilities of baks to 5

20 statistically measure the risk i the bottom-up methods based o historical data o frequecy of loss occurrece, size of loss ad so forth. However, if baks measure risk based oly o past evet data, they might ot capture those material potetial evets with low frequecy ad high severity ad, likewise, will ot capture the future impact of the chagig eviromet both iterally ad exterally o future operatioal losses. I other words, the historical loss observatio may ot always fully capture a bak s true profile, especially whe the bak does ot experiece substatial loss evets durig the observatio period. To supplemet the limits of a iteral data with exteral data may very useful. However, i the course of implemetig them, they may face the challegig risk maagemet issue of mappig that exteral data ito a iteral database with differig trasactio volume. Sceario Aalysis Uder sceario aalysis, first, baks idetify ot oly the past evets of operatioal risk but also its potetial evets based o, for example, the past evets which happeed i other baks ad the impact of chages i eviromets o their operatios flows. Secod, baks estimate frequecy ad severity of these evets idetified by aalyzig causes of these evets ad factors of causig losses ad expadig loss amouts. I this process, a coordiated risk maagemet departmet i operatioal risk takes a iitiative i givig questioaires to busiess lies so that commo 6

21 uderstadig ad challegig issues ca be shared betwee the risk maagemet departmet ad busiess lies. Exteral data cotribute to the developmet of robust sceario aalyses. Sceario aalysis is flexible ad effective way of makig good use of iformatio obtaied by risk assessmet, risk mappig, key risk idicators, ad scorecards. I risk assessmet, bak s operatio ad activities are assessed agaist a meu of potetial operatioal risk vulerabilities. This process ofte icorporates checklists ad/or workshops to idetify the stregths ad weakesses of the operatioal risk eviromet. I risk mappig, various busiess uits, orgaizatioal fuctios or process flows are mapped by risk type. This exercise ca reveal areas of weakess ad help prioritize subsequet maagemet actio. Key risk idicators ca provide isight ito bak s risk positio. Scorecards provide a meas of traslatig qualitative assessmets ito quatitative metric that give a relative rakig of differet types of operatioal risk exposures. But sceario aalysis teds to be less objective tha statistical measuremet approach. Risk amout varies strikigly by which scearios are adopted. The opiio that it is just tool for supplemet statistical is isisted cosiderably. 7

22 3. Applicatio of Bayesia Iferece I short, it is ecessary to measure operatioal risk based ot oly o historical data, but also sceario data with forward lookig approaches, give the rapid chage i eviromet surroudig the bakig idustry. So, we propose to apply Bayesia iferece to measurig operatioal risk. Bayesia iferece combies statistical measuremet approach ad sceario aalysis. Cruz itroduced Bayesia techiques i his operatioal risk textbook briefly, too. But he oly itroduced quite simple cocept, we focus a poit of view from bakig practices. 3.. What is Bayesia Iferece? Bayes Theorem Suppose that z = x,..., x is a vector of observatios whose probability distributio pz depeds o the values of k parameters = The,,...,. Suppose also that itself has a probability distributio p. k p z p = p z, = p z p z Give the observed data z, the coditioal distributio of is Also, we ca write p z p p z = p z 8

23 p z p d : cotiuous p z = E [ p z ] = c = 3 p z p : discrete where the sum or the itegral is take over the admissible rage of, ad where E[f ] is the mathematical expectatio of f with respect to the distributio p. Thus we may write alteratively as p z = cp z p. 4 The statemet of, or its equivalet 4, is usually referred to as Bayes theorem. I this expressio, p, which tells us what is kow about without kowledge of the data, is called prior distributio of, or the distributio of a priori. Correspodigly, p z, which tells us what is kow about give kowledge of data, is called the posterior distributio of give z, or the distributio of a posteriori. The quatity c is merely a ormalizig costat ecessary to esure that the posterior distributio p z itegrates or sums to oe. Likelihood Fuctio Now give the data z, pz i 4 may be regarded as a fuctio ot of z but of. It is called the likelihood fuctio of for give z ad ca be writte l z. We ca thus write Bayes formula as p z = l z p. 5 I other words, the, Bayes theorem tells us that the probability distributio for posterior to the data z is proportioal to the product of the distributio for prior to the data ad the likelihood for give z. That 9

24 is, posterior distributio likelihood prior distributio. Bayesia iferece is a method based o above geeral equatio. By Bayesia iferece, we ca take our earlier practical experiece ito accout explicitly. Also, we ofte obtai shorter cofidece itervals usig proper prior distributios tha we could obtai if we igored out practical experiece. 3 Poit Estimator The posterior distributio shows the distributio of parameter, ot the poit estimator. A poit estimator of parameter of a distributio, is ofte take to be the mea of the posterior distributio of. This is because the posterior mea miimizes the mea quadratic loss fuctio i a decisio-theoretic cotext. Other loss fuctio would imply other Bayesia estimator. A absolute error loss fuctio implies a posterior media estimator; for a uspecified loss fuctio we sometimes use the posterior mode.

25 3.. Natural Cojugate Prior Distributio Cojugacy is formally defied as follows. If F is a class of samplig distributios pz, ad P is a class of prior distributio for, the the class P is cojugate for F if p z P for all p F ad p P. 6 This defiitio is formally vague sice if we choose P as the class of all distributios, the P is always cojugate o matter what class of samplig distributios is used. So, we thik atural cojugate families, which arise by takig P to be the set of all desities havig the same fuctioal form as the likelihood. Cojugate prior distributios have the practical advatage of beig iterpretable as additioal data i additio to computatioal coveiece. Beroulli Trials Let there be idepedet trials of a experimet i which there are oly two possible outcomes o each trial, labeled success or failure, data xii=,,, each of which is either or. Let deote the probability of success o a sigle trial ad take z = x,..., x. The likelihood fuctio is biomial; xi = xi l z. 7 If were the radom variable, l z would look like a beta distributio. But we do t wat the beta prior family to deped upo sample data, so we use arbitrary parameters α, β, ad we orm the desity to make it proper, to

26 get beta prior desity, α β w = B α, β α >, β >. 8 Now we use out prior beliefs to assess the hyperparameters α ad β, which are the parameters of the prior distributio, that is, havig fixed the family of priors as the beta distributio, oly α, β remai ukow. We do ot have to assess the etire prior distributio. A additioal mathematical coveiece arises i computig the posterior. Sice the posterior desity is proportioal to the likelihood times the prior, i this case a beta prior, the posterior desity is give by w z [ xi xi ][ α β α xi β xi w z, 9 a beta posterior. I practice, we choose hyperparameters α, β usig these equatio that E = α/ α β V = αβ/ α β α β Mode = α-/ α β-. ] Poisso Distributio Suppose z = x,..., x is a radom sample from a Poisso distributio with mea. The likelihood fuctio of, xi l z = xi e e. i= xi! Suppose prior a parameter ~ Gamma α, β,

27 3 β e w a. The, the posterior distributio is the Gamma distributio, = = ', ' ' ' ' x a e w i a β β α β. I practice, we choose hyperparameters α, β usig these equatio that E = β/ α V = β/ α. 3 Normal Distributio Suppose,..., x x = z is a radom sample from a ormal distributio N,. The likelihood fuctio of, is,, ]. [ / x s x i e e l i = ϑ ν ϑ z 3 where = = = = = i i i i x x x x s.,., ν ν. i Estimatig the mea of a ormal distributio with kow Suppose prior a parameter follows a ormal distributio, that is ~N α, β, the posterior desity follows a ormal distributio,, ] [ ] /. / [ /. / µ µ µ e e e e p z l z p c x c c x c 4

28 4 where / /. µ µ = c c x, ο = c. ii Estimatig the variace of a ormal distributio with kow Suppose the prior distributio of is take to be the iverse- χ distributio with scale λ ad ν degrees of freedom, that is, ~ ν χ Iv. λ ν e p. 5 The result posterior distributio of is. λ ν λ ν s s e e e p z 6 Thus,, ~ s Iv λ ν χ z. iii Ifereces whe both mea ad variace are ukow We cosider first the coditioal posterior desity,, z p, ad the the margial posterior desity, z p. A coveiet parameterizatio is give by the followig specificatio:, ~ N p µ, ~ λ ν χ p, which correspods to the joit prior desity

29 5 ] [ /, µ λ ν = e p p p. 7 Multiplyig the prior desity 7 by the ormal likelihood yield the posterior desity,, ] ˆ [ / ]. [ / λ ν µ ν λ ν y s e e p z 8 where, y s = = =,. / /, µ ν λ λ ν ν,. ˆ y = µ The coditioal posterior desity of, give, is proportioal to the joit posterior desity 8 with held costat, /, ~, ^ N z p 9 The margial posterior desity of is scaled iverse- χ.., ~, λ ν χ λ ν = e d z p z p Itegratio of the joit posterior desity with respect to shows that the margial posterior desity for is., /, ˆ ~ ] ˆ [, ν ν λ λ ν t d z p z p =

30 3.3. Advatages of Bayesia Iferece Practical Use Both Data ad No-Data Iformatio By applyig Bayesia iferece, we ca use both data as likelihood, ad o-data iformatio as prior distributio. The problems of statistical measuremet approach are resolved i some extet. We ca take idicators that may be predictive of the risk of future losses ito cosideratio. Such idictors, ofte referred to as key risk idicators or early warig idicators, are forward-lookig ad could reflect potetial sources of operatioal risk such as rapid growth, the itroductio of ew products, employee turover, trasactio breaks, system dowtime, etc. Also, qualitative data, such as self-assessmet scorig result ca be used for measurig. This approach affords busiess lie maagers icetives to their risk maagemet through self-assessmet process. Cosider a situatio where huge operatioal losses occur i a busiess lie. The maximum amout of operatioal risk, or ecoomic capital charge, allotted to that lie becomes so large that lie maagers might have little icetive to improve their risk maagemet. I utilizig self-assessmet results through Bayesia iferece, icreasig levels of sophisticatio of risk maagemet should geerally be rewarded with a reductio i the operatioal risk capital requiremet. 6

31 Trasparecy of Measurig Process A lot of baks seem to use both statistical measuremet approach ad sceario aalysis to measure operatioal risk. Everyoe recogizes the limitatios of the statistical measuremet approach ad discout it heavily with a dose of judgmet. But it is ot clear how to adjust the result of statistical measuremet approach by the result of sceario aalysis. The amout of risk measured based o statistical methods teds to be modified ex post facto. Bayesia iferece accepts some degree of subjectivity i oly choosig the prior distributio. Auditig the process of choosig the prior distributio by iteral auditors ad supervisors warrats trasparecy. 3 Importace of the Process of Choosig Prior Distributio Choosig prior distributio meas that baks should idetify ad assess the operatioal risk iheret i all activities, processes ad systems. Therefore, this cosideratio itself is a process of great sigificace for operatioal risk maagemet as well as measurig. The approval for the prior distributio by the committee for example: Risk Maagemet Committee at which seior maagemet atted makes their recogitio of operatioal risk greatly improve, ad cotributes to ehace operatioal risk maagemet ad cotrol priciple. 7

32 4. Examples of Applicatio of Busiess Practices 4.. Operatios Error Rate The operatios error rate is the mai idicator ot oly to pla various policies of operatios maagemet but also to choose the prior distributio of the umber of operatios loss evets described i Sectio 4.. We assume that the operatios error rate follows Beroulli trials with parameter. The likelihood of Beroulli trials follows biomial distributio, so atural cojugate prior distributio is beta distributio. I order to kow the curret operatios error rate, it seems adequate to use the results of recet audit by iteral auditors. ad ot historical data. But iteral audit ca t cover all trasactios, has to draw samplig ispectio. For example, iteral auditors foud a error from samples, is the curret operatios error rate %? Is the rate % because auditors foud o error? We may felt that somethig was wrog with the results. Accordigly, Bayesia iferece appears. Based o the results of audits util ow ad statistics about operatios, with takig expected umber of trasactio ad clerks, procedures of risk maagemet, skills ad morals of clerks ito cosideratio, we choose the prior distributio for. I choosig, descriptive statistics of beta distributio such as mea, mode, media ad variace are used. Multiplied the prior distributio by likelihood obtaied from the latest result of audits, we 8

33 .5.45 Prior Be4,4 Postprior Be5, %.%.% 3.% 4.% 5.% 6.% 7.% 8.% 9.%.% Figure The prior ad posteror distributio of operatios error rate. ca get the posterior distributio. For example, the probability of the rate beig from % to 4% was judged about 5%, we might fit a beta 4,4 prior. If a error was foud from samples through the latest audit, the posterior distributio is beta 5,4. Figure shows the prior distributio ad the posterior distributio. The posterior mode of is.76% ad the posterior mea of is.8%. Reflectig the result of samplig audit makes it possible to estimate the rate i lie with the actual situatio. Moreover, other audit results improve the accuracy. 9

34 4.. Number of Operatios Loss Evets Operatios loss evet is resultig from maifestatio of operatios risk defied as the possibility of losses arisig from egliget admiistratio by employees, accidets, or uauthorized activities. It is atural that, eve if operatios loss rate is costat, as trasactio volume icreases, the umber of operatios loss evets icreases, because the umber of operatios loss evets = trasactio volume operatios loss evet rate. The umber of operatios loss evets per a year has oly oe or two digits, i cotrast that trasactio volume is at least te millios. Therefore, operatios loss evet rate is microscopic; to estimate the rate withi a acceptace error rage is very difficult. So, we fit a appropriate distributio to the umber of evets itself i order to estimatio. The applicatios of Bayesia iferece ca reflect the tred of trasactio volume. Because operatios loss evet rate is microscopic, we apply Poisso distributio. Natural cojugate prior distributio is gamma distributio. Based o historical data, after cosiderig the factors such as exteral circumstace, the situatio of other baks, the tred of operatios error rate metioed above, trasactio volume, their risk maagemet system, ad so o, we choose the prior distributio of the umber of operatios loss evet rate. The quality factors like professioal moral of employee ifluece so much that we should take ito accout. The, multiplyig the recet data of operatios loss evets as likelihood decides the posterior distributio. 3

35 .8.6 Prior Poisso8, Postprior Poisso, Figure The prior ad posteror distributio of the umber of operatios loss evets. We will give a example. By maximum likelihood method from historical data, we estimated that parameter is 7.5. But, takig ito revisio of operatio procedures immediately after revisio, loss evet teds to icrease temporarily, icreasig tred of operatios error, the prior distributio is determied gamma 8, i early i this year. For three moths sice this year starts, three loss evets have bee observed. The, we obtai the posterior distributio ~ gamma,.8. Assumig the quadratic loss fuctio, the umber of operatios loss evets is 8.8. Figure shows the prior distributio ad the posterior distributio. 3

36 4.3. Severity of Operatios Loss Evet The method for applicatio to severity of operatios loss evet is same. The distributios of parameters oly differ. Supposed the severity X follows logormal distributio. For example, by maximum likelihood method from historical 3 data, we estimated logx~n5.,.3. Takig a eormous loss evet occurred at other bak, the tred of icreasig amout per a trasactio, ad so o, we choose the prior distributio p ~N6, /3 p ~ χ - 3,48. About three loss evets occurred for three moths sice this year starts, log-mea is 5.67 ad log-variace is.6. Fially, we obtai the posterior distributio p z~t5.98,.4, 33 p z~ χ - 33, 5.3. Assumig the quadratic loss fuctio, the severity of operatios loss evets follows logx~n5.98,.68. Expectatio of X is,75,9. 3

37 4.4. Simulatio of Risk Amout Usig the examples metioed i Sectio 4. ad 4.3, we calculates operatios risk amout of a busiess uit. Each three types, i based o oly historical data, ii the prior distributio ad iii the posterior distributio, are calculated. First, we geerate Poisso radom umber frequecy with. I order to perform the severity we should geerate as may logormal radom variables as demaded by the frequecy. For example, if the frequecy states 3, we should geerate three logormal radom umbers. After processig this scheme, times, the results eed to be summed up. Afterwards, we just eed to order the results to get the aggregated distributio. After simulatig, we may put all loss amouts for oe year i order startig with low amout ad may take 9,9 th loss amout as maximum loss for 99 th percetile i the cofidetial iterval. We regard the average of the result five times simulatios as risk amout. I lie with other bakig risks, coceptually a capital charge for operatioal risk should cover uexpected losses =maximum losses expected losses due to operatioal risk. Provisios should cover expected losses. However, accoutig rules do ot allow a robust, comprehesive ad clear approach to settig provisios, especially for operatioal risk. So, a capital charge for operatioal risk should cover maximum losses. 33

38 Simulatio Results Number of Loss Evets Severitythousads ye Stadard Mea Error Risk Amout 99% thousads ye Historical 7.5 7,648, ,5 Prior 8. 8,8 35,,3,685 Posterior 8.8,76 4,55,83,58 The icreasig tred of both frequecy ad severity is captured well. The profit from operatios is ot so much that the chage i risk amout ifluece profitability judgmet. This result shows that based o historical data oly, allocate lack capital that covers risks, for risk maagemet i which coservatism is regarded as importat. If baks measure risk oly based o the past evet data, they might ot capture material potetial evets ad future importat impacts of chagig eviromet iterally ad exterally o future operatioal losses. Whe risk maagers report risk measuremet to the board of directors without explaiig limits of these assumptio o risk measuremet, it could be misleadig i the sese that operatioal risk would be very small ad that baks could be allowed to expose more operatioal risk compared with their ecoomic capital or buffer for maximum losses. 34

39 4.5. Profitability Judgmet This example is about etry or exit of subsidiaries. Suppose a bak established a subsidiary give operatios by other compaies icludig baks last year. From the first, the items o busiess pla were based o suppositio. Needless to say, the capital allocated by the paret compay, coverig the subsidiary s risk, was determied without historical data. I first year, oly a loss evet with small amout was occurred. Uder drawig up ext year s pla, ca the paret compay allocate the subsidiary capital based oly results last year? Is there o dager of uderestimatio? We should apply Bayesia Iferece. If the results like observed i first year cotiue for several years, the posterior distributio update by the excellet result, risk capital requiremet will decrease. Paret compay judge profitability of the subsidiary based o ot after-tax profit itself but the oe after deductig cost of capital. Eve if after-tax profit is i the black, whe the profit after deductig cost of capital goes ito red, the paret compay makes exactig demads to the subsidiary. I case of the bad future prospect, the paret compay may decide to dissolve the subsidiary. Risk amout iflueces seriously profit judgmet i the form of cost of capital. 35

40 5. Discussio Bayesia iferece is very powerful tool for measurig operatioal risk, but there are a umber of outstadig issues to be resolved. We are goig to examie the followig issues. Choosig Prior Distributio The prior distributios represet a descriptio of opiio ad kowledge about the parameters of a certai distributio. I the prior resides most of the criticism of Bayesia Iferece, ad we must be very reasoable i choosig oe prior over aother. I choosig the prior distributio, which factors must be take ito accout? We should research the methods for traslatig qualitative assessmets such as scorecards ito quatitative metric. It is importat to establish quatitative data such as self-assessmet scorig or results objectively. So called elicited prior is basically the subjective opiio of the value of parameters before ay data is available or if oly limited data is available. At the curret stage i Operatio Research, few measuremet software packages are available, but those usig Bayesia iferece use subjective opiio to evaluate the parameters. Suppose that several experts are asked to fill i their opiios about the quatiles of a certai kid of operatioal evet i a busiess uit. Give the results, we ca fit a prior distributio based o the elicited opiio from experts. 36

41 If the approach proposed i this paper spreads amog bak idustries, via supervisios, there is a fair possibility that the factors to be cosidered will be foud. Validatio Ayway, the first thig we have to do is to put high priorities o collectig robust loss database. Baks must begi to systematically track relevat operatioal risk data by busiess lie across the firm. The ability to moitor loss evets ad effectively gather loss data is a basic step for operatioal risk. Further work is eeded by both baks ad supervisors to develop a better uderstadig of the key assumptios of measurig techiques, the ecessary data requiremets, the robustess of estimatio techiques ad appropriate validatio methods e.g. goodess-of-fit tests of the distributio types ad iterval estimatio of the parameters that could be used by baks ad supervisors. 37

42 Refereces: Basel Committee o Bakig Supervisio 998 Framework for Iteral Cotrol Systems i Bakig Orgaizatios. Basel Committee o Bakig Supervisio 998 Operatioal Risk Maagemet. Basel Committee o Bakig Supervisio The New Basel Capital Accord, Secod Cosultative Package. Basel Committee o Bakig Supervisio Workig Paper o the Regulatory Treatmet of Operatioal Risk. Basel Committee o Bakig Supervisio Soud Practice for the Maagemet ad Supervisio of Operatioal Risk. Basel Committee o Bakig Supervisio Iteral Audit i Baks ad the Supervisor s Relatioship with Auditors. Basel Committee o Bakig Supervisio The Quatitative Impact Study for Operatioal Risk: Overview of Idividual Loss Data ad Lessos Leared. Cruz, M. G. Modelig, Measurig ad Hedgig Operatioal Risk, Lodo: Wiley. Frachot, A., George, P. ad Rocalli, T. Loss Distributio Approach for Operatioal Risk, Workig Paper. Gelma, A., Carli, J. B., Ster, H. S. ad Rubi, D. B. 995 Bayesia Data Aalysis, CHAPMAN & HALL. 38

43 Hiwatashi, J. ad Ashida H. Advacig Operatioal Risk Maagemet Usig Japaese Bakig Experieces, Bak of Japa, Audit Office Workig Paper No.-, February. Jorio, P. Value at Risk: The New Bechmark for Maagig Fiacial Risk, McGraw-Hill. Klugma, S. A., Pajer, H. H. ad Willmot, G. E. 998 Loss Models: from Data to Decisios, Joh Wiley & Sos. Kotz, S. ad Nadarajah, S. Extreme Value Distributios: Theory ad Applicatios, Imperial College Press, Lodo. Marshall, C. Measurig ad Maagig Operatioal Risks i Fiacial Istitutios: Tools, Techiques ad Other Resources, Joh Wiley & Sos. Mori, T., Hiwatashi, J., ad Ide, K. Measurig Operatioal Risk i Japaese Major baks, Bak of Japa, Fiacial ad Paymet System Office Workig Paper No.-, July. Mori, T., Hiwatashi, J., ad Ide, K. Challeges ad Possible Solutios i Ehacig Operatioal Risk Measuremet, Bak of Japa, Fiacial ad Paymet System Office Workig Paper No.-3, September. Mori, T., ad Harada, E. Iteral Measuremet Approach to Operatioal Risk Capital Charge, Bak of Japa, Fiacial ad Paymet System Office Workig Paper No.-, March. Matsubara, N. 979 Ishikettei o Kiso, Asakura Shote. Press, S. J. 989 Bayesia Statistics: Priciples, Models, ad Applicatios. New York: Wiley. 39

44 Smith, R. Bayesia Risk Aalysis, i Extremes ad Itegrated Risk Maagemet, P. Embrechts ed., Risk Publicatios, Lodo, Chapter 7, pp Shigemasu, T. 985 Bayes Tokei Nyuumo, Tokyo Daigaku Shuppa Suzuki, Y. 978 Toukei Kaiseki, Chikuma Shobou. Wikler, R. L. 97 Itroductio to Bayesia Iferece, Holt, Reiehalt & Wisto, Ic. Wataabe, H. 999 Bayes Tokeigaku Nuumo, Fukumura Shuppa. 4