Operational risk quantification for loss frequency using fuzzy simulation

Transcription

1 COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) Lu Shuxa, M Haje Operatonal rs quantfcaton for loss frequency usng fuzzy smulaton Shuxa Lu,*, Haje M School of Busness Admnstraton, Hebe ormal Unversty of Scence & Technology, Qnhuangdao, Chna Chnese Uncom Shjazhuang branch, Shjazhuang, Chna Receved June 04, Abstract The estmaton of the frequency parameter of operatonal rs quantfcaton has receved ncreased attenton under the new Basel proposal. Ths paper proposes an advanced measurement approach usng fuzzy pont estmaton. In ths approach, pror membershp functon could be obtaned through fuzzy maxmum entropy rule. When operatonal rs loss data s gven, posteror membershp functon can be easly calculated by usng fuzzy pont theorem. After posteror mean s exploted as fuzzy pont estmate, loss frequency dstrbuton s gotten. Fnallyn emprcal analyss on ths model s conducted based hstorcal data obtaned from a Chnese commercal ban. The result shows that economcal can reduce the complexty and communcaton cost. Keywords: Operatonal Rs, Basel II Advanced Measurement Approach, Fuzzy Pont Estmaton, Loss Dstrbutonal Approach, Fuzzy Varable Introducton Operatonal Rs s an mportant quanttatve topc n the banng world. Under the Basel II requrements [5,6], many bans ntend to use the Advanced Measurement Approaches (AMA) for the quantfcaton of operatonal rs. Through the Advanced Measurement Approach, the bans are permtted sgnfcant flexblty over the approaches that may be used n the development of operatonal rs models. There are varous quanttatve operatonal rs models ncludng extreme value theory [, 6],Bayesan nference [4, 7, 4, 9, 3, 33], dynamc Bayesan networs [30], maxmum lelhood [] and EM algorthms [3], VAR technques [, 3, 4], other approaches [, 7]. Of the methods developed to model operatonal rs, the majorty follow the Loss Dstrbutonal Approach (LDA) [5]. The dea of LDA s to ft frequency dstrbutons over a predetermned tme horzon, typcally annual. he fnancal nsttutons use a wde varety of frequency and severty dstrbutons for ther operatonal rs data, ncludng exponental, webull, lognormal, generalzed Paretond g- and-h dstrbutons [8].There are potentally many deferent alternatves [6, 5] for the choce of severty and frequency dstrbutons. Several researchers [, 8, 8, 3] have expermented wth operatonal loss data by Basel II busness lne and event type over the past few years. Maryam Prouz[34]dscuss several statstcal methods for modelng truncated datand suggest the best one for modelng truncated loss data, the approach can be useful for ncreasng accuracy of estmatng operatonal rs captal charge n E-banng. Fengge Yao[35] used Condtonal value-at-rs (CVaR) models based on the pea value method of extreme value theory to measure operatonal rs. Younès, Moutassm[36] used separately a lognormal dstrbuton and a gamma dstrbuton n the mxture models for the zeros losses. an operatonal rs assessment model of dstrbuton networ equpment based on rough set and D-S evdence theory was bult[38]. Ahmed Baraat[39] nvestgates the drect and jont effects of ban governance, regulatonnd supervson on the qualty of rs reportng n the banng ndustry. Pjotrs Dorogovsa[40] dscussed new tendences of management and control of operatonal rs n fnancal nsttutons. Lu [] proposed credblty measure and credblty theorynd ntroduced random fuzzy varable as a measurable functon defned on a credblty space valued random varables. Chance measure was proposed by L and Lu [] to measure the chance of a random fuzzy event. The condtonal chance measure was ntroduced by L and Lu [3] to measure the chance of a random fuzzy event after t has been learned that some other event has occurred. For the consdered ban, the unnown parameters (for example the Posson parameter or the Pareto tal ndex) of these dstrbutons have to be quantzed. Our approach to estmate the parameter of the loss dstrbuton s based on fuzzy pont nference. The dea s to use the bans collectve losses and expert opnons to mprove the estmates of the parameters of loss dstrbutons. We demonstrate how the parameter uncertanty can be taen nto account by ban nternal data and expert opnons and study the mpact on the captal charge. In any rs cell, we model the loss frequency and the loss severty by dstrbutons where the lognormal and Pareto dstrbutons are used for modellng severty dstrbutons and Posson dstrbutons for frequency dstrbutons, respectvely. The model mght be very useful at ths stage when the data are very lmted and t may also have educatonal mpact. Fnancally, we analyss the results of an emprcal study wth external operatonal loss data of some Chnese commercal bans. * Correspondng author s e-mal: xxgllushuxa@6.com 577

2 COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) Lu Shuxa, M Haje Fuzzy varables Defnton. [] A Fuzzy varable s defned as a functon from a credblty space (, ( ), Cr) to the set of real number. Defnton. [] Let be a fuzzy varable wth membershp functon. Then for any set B of real numbers, Cr{ }= ( sup ( x) sup ( x)). x x c Defnton 3. [] Let be a fuzzy varable on the credblty space (, ( ), Cr). The expected value E[ ] s defned as () 0 E[ ] 0 Cr{ r} dr Cr{ r} dr () Defnton 4. [] Suppose that s the contnuous fuzzy varable, then ts entropy s defned by H[ ] S( Cr{ x}) dx (3) where S( t) t ln t ( t) ln( t) Example. [] Let be a trapezodal fuzzy varable (a, b, c,d), then the expected value of s a b c d E[ ]. 4 Its entropy s defned by d a H[ ] ( ln 0.5) ( c d) Ch{ a a X ( a) x} Defnton 5. [34] Let X ( a), X ( a),, X ( a ) random fuzzy varables where a ( a m,) s fuzzy vector such that for each a ( a m, ) a ( ), X ( a), X ( a),, X ( a ) are d random varables wth functon (pdf ) or probablty mass functon(pmf) f ( x ). Then, gven the sample, the way to get the posteror membershp functon of pror membershp functon s called fuzzy pont estmaton. Theorem. [34] Let X ( a), X ( a),, X ( a ) be d cont- nuous random varables, where a ( a m, ) s fuzzy vector wth pror membershp functons a ( a),, m, such that for each a ( a m,) a ( ), X ( a), X ( a),, X ( a ) are d random varables wth pdf f ( x m ), let x be a sample. If f ( x m ) s contnuous wth respect to ( xa, ) then the posteror membershp functon of a can be deduced by a ( a X ( a) x) where sup Ch{ a ( a ) X ( a) x}, a m In ths secton, s vewed as a non-negatve fuzzy varable on the credblty space (, ( ), Cr) ) wth membershp functon, whch s called pror membershp functons. The parameters of pror membershp functons are called hyper-parameters (parameters for parameters). In a more general framewor the parameters of the pror membershp functon, are estmated by max- 0, f mn Cr{ a } 0 a m 0 n n f ( x ) f ( x, a) (5) n, f n 0.5 and mn Cr{ a a } 0 sup ( ) sup ( ) m m f x m f x, a a a y( where 0.5), otherwse 3 Fuzzy pont estmaton for loss frequency mum entropy rule and expert opnons. Expert opnons modfy ths characterstc accordng to the actual experence. Then P( ) can be consdered as random fuzzy Suppose that the frequences of operatonal rs losses s modeled by Posson dstrbuton P( ) wth a densty varable on the space (, ( ), Cr) (,, Pr). Let x denote the observatons sample x, x x, the sample can be observed and tae crsp values. Accordng to Equaton (4), the posteror membershp functon can be dedu- f ( ) e, 0,,... (6)! ced as ( X ( ) x) ( Ch{ }). (7) where (4) 578

3 COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) Lu Shuxa, M Haje Ch{ x} Ch{{ } { x}} Ch{{ } { x}}, f 0.5 Ch{ x} Ch{ x} y( 0.5), otherwse Then Pr{ x } 0.5, Cr{{ } { x}} 0.5, Ch{ x} sup( Cr{ } Pr{ x } 0.5, We can get Ch{{ } { x}} Cr{ } { x } (8) Ch{ x} sup( Cr{ } Pr{ x }) (9) Then posteror membershp functon s formulated as x n Cr{ } e t x! ( x) x, 0,,... (0) n sup( Cr{ } e ) t x! 4 Fuzzy pont estmaton of loss frequency parameter The sample of loss frequency n corporate fnance s x (480, 964), the experts gve the range of loss frequency of corporate fnance s The loss frequency wll be decreased by 00 tmes n order to prevent from the probablty of poson dstrbute n postvely nfnte, then wll be 5 35, x (5,0). Let be trapezodal fuzzy varable (5b,,35), then the pror membershp of s. ab40 max( ln 0.5)( b a) 5 M 3 ab, 4, (4) s. t.5 a b 35, where M s a suffcently larger number. By applyng the graphc method, then the value of a * * and b can be obtaned: a 7, b 7 s a trapezodal fuzzy varable (5, 7,7,35) then the pror membershp s 5, f 5 7, f 7 7 ( ) 35, f , otherwse (5) Accordng to equaton (), the posteror membershp of s ( x) x Cr{ } e t x! x sup( Cr{ } e ) t x! e 0 ), , otherwse (6) The fgure for the posteror membershp of can be depcted as FIGURE. 5, f 5 a a 5, f a b ( ) 35 f b 35 b 35 0, otherwse () Accordng by equaton () and (3), the expectaton of s ab40 E[ ], () 4 the entropy of s H[ ] ( ln 0.5)( b a) 5. (3) It follows from fuzzy maxmum entropy rule equaton max H[, ] M E[ ], where s the mean of those experts estmate. We can get FIGURE The posteror membershp functon of 5 Fuzzy smulaton for posteror mean E[ ] of the By the fuzzy smulatons technque we can calculate E[ ], E[ ], then posteror mean E[ ] of the s exploted as the fuzzy pont estmaton. For smplcty, A fuzzy smulaton wll be desgned to estmate E[ ] by the followng procedure. 579

4 COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) Lu Shuxa, M Haje ) Set e 0. ) Randomly generate ( ) from the -level set of and wrte ( x) for,,,, where the membershp of functon of. 3) Set a ( ) ( ), b ( ) ( ) 4) Unformly generate r from [a,b]. Set e e Cr{ r}, where Cr{ }= ( sup ( x) sup ( x)). x x c 5) Repeat the fourth step for tmes. 6) Compute E[ ] a 0 b 0 e ( b a) /, then output E[ ] E[ ]. By fuzzy smulaton technque, we can tae the posteror mean E[ ] of s fuzzy pont estmaton of s , to ampltude the result by 00 tmes fuzzy pont estmaton of s 806. Then densty functon of frequences n operatonal rs losses s m ( m) e, m 0,, (7) m! 6 Conclusons In ths artcle, by ntroducng novel estmaton approach we have substantally extended the range of models admssble for parameters of loss frequency under the LDA operatonal rs modelng framewor. We strongly advocate that fuzzy pont estmaton approaches to operatonal rs modelng should be consdered as a serous alternatve for practtoners n bans and fnancal nsttutonss t provdes a mathematcally rgorous paradgm n whch to combne observed data and expert opnon. We hope that the presented method provdes an attractve and feasble approach n whch to realze these models. The proposed measurng method allows usng the bans collectve losses data and expert opnons to mprove the correctness of estmates value. It s flexble and robust technque to adequately model the operatonal loss frequency and severty. Fnancally, we smulate posteror mean of loss frequency of the operatonal rs.the method presented n ths paper performs better than the other mentoned methods when a few data are avalable. Acnowledgments Ths wor was supported by Doctoral Foundaton of Hebe ormal Unversty of Scence & Technology o.yb000, Hebe Provnce Department of Educaton of Youth Foundaton 0.SQ303, Qnhuangdao Admnstraton of Scence & Technology Project 0.SQ030a56.The natural scence foundaton of Hebe Provnce 0. G References [] Alderwereld, T., J. Garca and L. Leonard (006) A practcal operatonal rs scenaro analyss quantfcaton. Rs Magazne, [] Alexander J and Saladn T (003) Developng scenaros for future extreme losses usng the pot method. n extremes and ntegrated rs management, edted by Embrechts PME, publshed by RISK boos, London, [3] Bee M.( 006) Estmatng and smulatng loss dstrbutons wth ncomplete data. Oprs and Complance, 7, [4] BÄuhlmann H., Shevcheno P.V. andwäuthrch M. V.( 008) A toy model for operatonal rs quantfcatonusng credblty theory. the Journal of Operatonal Rs. [5] BIS. Basel II (005): Internatonal Convergence of Captal Measurement and Captal Standards: a revsed framewor. 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5 COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) Lu Shuxa, M Haje [9] Ramamurthy S., H. Arora and A. Ghosh (005). Operatonal rs and probablstc networs-an applcaton to corporate actons processng. Infosys Whte Paper. [30] Rosenberg J and Schuermann T (006). A general approach to ntegrated rs management wth sewed,fat-taled rss. Journal of Fnancal Economcs, 79, [3] [3] Shevcheno P and WÄuthrch M. The structural modellng of operatonal rs va bayesan nference: combnng loss data wth expert opnons. Journal of operatonal rs, 006,, 3-6. [3] Shevcheno P.V(008). Estmaton of operatonal rs captal charge under parameter uncertanty,journal of Operatonal Rs, 3(),5-63. [33] Tang W., Wang C., Zhao R., Fuzzy Parametrc Statstcal Inference, Informaton: An Internatonal Interdscplnary Journal, 0,4, 7-7. [34] Maryam Prouz, Mazar Salah (03) Modelng Truncated Loss Data of Operatonal Rs n E-banng, IJITCS, l5(), [35] Fengge Yao,Hongme Wen, Jaq Luan (03) CVaR measurement and operatonal rs management n commercal, bans accordng to the pea value method of extreme value theory, Mathematcal and Computer Modellng, 58, 5 7. [36] Younès, M.nd Adloun, S., et al. (0) Mxed Dstrbutons for Loss Severty Modellng wth Zeros n the Operatonal Rs Losses. Internatonal Journal of Appled Mathematcs & Statstcs,, -7. [37] Cunbn L.,Gefu Q. et al. (03) Operatonal Rs Assessment of Dstrbuton etwor Equpment Based on Rough Set and D-S Evdence Theory. Journal of Appled Mathematcs, Artcle ID 63905, 7 pages. [38] Ahmed B., Khaled H (03) Ban governance, regulaton, supervsonnd rs reportng: Evdence from peratonal rs dsclosures n European bans, Internatonal Revew of Fnancal Analyss, 30, [39] Pjotrs Dorogovsa, Irna Solovjovab, Andrejs Romanovsc(03) ew tendences of management and control of operatonal rs n fnancal nsttutons, Proceda-Socal and Behavoral Scences, 99, Authors ShuXa Lu, , Chna ShuXa Lu receved the Ph.D. degree n management from tanjn Unversty, Chna n 00. Currently, she s a researcher at, Hebe ormal Unversty of Scence & Technology of School of Busness Admnstraton, Chna. Hs research nterests nclude nformaton management andsystem,rs management. HaJe M, 983, Chna Haje M receved the master degree n management from Tanjn Unversty, Chna n 008. Her research nterests nclude rs management and statstcs Research, Mathematcal fnance fnancal engneerng. A few papers publshed n varous journals. 58