The Use of Analytical-Statistical Simulation Approach in Operational Risk Analysis

Transcription

1 he Use of Analytical-Statistical Siulation Approach in Operational Risk Analysis Rusta Islaov International Nuclear Safety Center Moscow, Russia Alexey Olkov he Agency for Housing Mortgage Lending Moscow, Russia Abstract Quantitative operational risk assessent is essentially based on stochastic scenario odeling of operational loss sequences. Given the lack of reliable historical data in ost cases, atheatical ethods should eet even stronger requireents in ters of results received and include specific features, naely: rare event analysis, uncertainty analysis and huan factor analysis, etc. In this paper, the analyticalstatistical siulation approach (ASSA) is considered as the ost flexible approach to stochastic scenario loss sequence odeling and copared with Marcov chain, fault/event tree, Monte Carlo in ters of scenario loss sequence odel adequacy and calculation cost. Probabilistic risk analysis software PRAISE and soe applications are given.

2 I. Introduction A series of corporate scandals and downfalls in the last two decades showed how operational risk could be a long-ter killer even for the ost respected copanies all over the globe. As noted in Böcker and Klüppelberg (2005) probably the ost draatic and well-docuented exaple of a bank collapse caused by operational risk losses was Britain s oldest erchant bank Barings after the rogue trader Nick Leeson had been hiding loss-aking positions in financial derivatives. Other wellknown lessons the financial industry was taught by are Société Générale loss of 4.9 billion due to trader fraud, foreign currency trader staff fraud at National Australia Bank ($A360 illion loss), currency trader fraud at Allfirst bank, then part of AIB Group (a loss near US $69 illion) and Bear Stearns near death since it was not able to price its ortgage portfolios. hese exaples obviously show the increased iportance of effective and reliable operational risk anageent, based on risk identification, onitoring and reporting, risk itigation, risk controlling and risk quantification. It is evident that ost of these catastrophic losses would have never been prevented just by estiation of a loss distribution on historical data or naïve LDA siulation techniques as they were due to flaws in internal control fraeworks and ineffective internal audit functions. he first step in risk itigation is an effective anageent and internal control processes. As for the relevance of operational risk odeling, the siplest arguent is regulatory requireents. he ost popular fraework of Basel II, being de-facto the standard of risk anageent in financial institutions, provides the arguent for quantification of operational risk for every financial institution. In this respect, the ain intention for the socalled advanced easureent approaches (AMA) is to calculate a capital charge as a buffer against potential operational risk losses. Another reason for building odels, besides to ake predictions is that odels can help us to gain a deeper understanding of a subject atter as that is noted in Böcker and Klüppelberg (2009). In Giacoelli and Pelizzon (2009) three classes of odels for operational risk assessent and quantification are discussed. he ost popular ethod, known as the loss distribution approach (LDA), is the paraetric estiation of a frequency and a severity distribution for individual loss types and their subsequent aggregation that ay incorporate dependencies. his general approach also includes the odeling of operational risk via extree value theory (EV), coon Poisson shock odels, also soe odels based on ruin theory. he second wide class of operational risk odels also eploys statistical techniques to quantify operational risk but uses ainly qualitative easures to calibrate the odel. Key features of these odels are qualitative-based scenario analyses and scorecards. Expert judgent, risk and control self assessents also fall within this class. he third class odels focuses on the functional odeling of operational loss sequences. Functional processes and dependencies of operational risk events are defined and odeled via interdependence of the individual processes. he proposed approach of quantitative scenario odeling of operational losses, based on causal event tree analysis, is tie- 2

3 dependent, thorough and realistic representation of real world business processes going on in organization. he proposed approach, which falls within the third class, is particularly useful when the underlying distributions exhibit rare events and coplex dependence structures that influence the tails, which produce significant challenges for Monte Carlo odeling. his paper is organized as follows. he second paragraph describes soe popular logical probabilistic techniques for scenario odeling used in industry and the role event tree analysis plays aong the. he third and forth paragraphs describes atheatics which is used to quantify the odels described in the second paragraph. he fifth paragraph illustrates advantages of analytical statistical siulation approach (ASSA) on a siple exaple. he sixth paragraph concludes and soe references are given in seventh. Appendix includes ters and definitions used in the paper. II. Approaches to scenario odeling Apart fro quantitative scenario odeling ost financial institutions eploy qualitative assessent of hazards by expert discussions and polling. Extensive literature is available on the use of expert aggregation techniques and questionnaires. Refer to Peters and Hübner (2009) for details. his paper is focused on foral logical probabilistic odeling of operational losses which are based on probabilistic risk assessents (PRA) originating fro coplex syste reliability engineering field, especially nuclear-power engineering. he proposed odel is based on reliability stochastic odels of coplex syste and operational loss sequences which are characterized by: logic and tie cause-consequence relationships; rando rare event are present in accident sequence. In nuclear industry the ost widely used ethod for syste reliability and safety analysis is logic-probabilistic (based on Boolean algebra) using cut set definitions and Marcov process. he discussion below is ainly based on Papushkin, Islaov, Volkov (999). Let s forulate the proble for analysis. Syste State is usually denoted by binary structural function: f(x, x 2,..., x n ) or n-diensional vector {x i }, i =,..., n, where x i - binary vector coponent corresponds to syste eleent state: x i = 0 is efficiency state, x i = is failure state, n denotes full nuber of eleents in considered structural schee, f = denotes efficiency syste state, f = 0 denotes failure syste state (operational risk realization). When rando vector x(ω) takes the value in the sapling space (Ω, A, P), where Ω is event space, A is event σ-algebra, P is probabilistic easure, then the range of f(x) fors the state set E x (phase space), where syste behavior rando process ξ(t) takes its value at the oent of tie t [0,]. 3

4 otal nuber of states can be equal to N(E x ) = 2 n. he process can be odeled as discrete Marcov process crossing fro one space E x state into other one. However, when the nuber of eleents n > 20, phase space increases N(E x ) > 0 6 and as a rule the Fault ree ethod (with cut set tool) is used. If the statistical siulation is used there is a proble of saple volue. Rando values, which define accident events, belong to so-called rare events. Satisfied nuber of saple is usually obtained as 0 2 np -. he nuber of siulation for one eleent is equal to 0 9 for probability estiation p = 0-7. Analytical ethods are siple to use and calculation-efficient but statistical siulation ones have no restrictions on used odels and have correct justification. In order to use advantages of both approaches the analytical - statistical siulation approach had been developed. III. Scenario odel essentials Once initial events have been identified and grouped, it is necessary to deterine the syste response on each initiating event group. he odeling of the syste response results in the generation of operational loss (accident) sequences due to operational risk. Accident sequences are odeled in the for of event trees that are the logic diagras reflecting success/failure of barrier functions required to prevent an accident in each initial event. here are two types of rando events: Rando State his state is described only by probability point estiation for corresponding event in considered accident sequence. In different accident sequences the probability frequency estiation of the sae barrier could be different. Rando Process his process consists of paraeter or paraeters, which are the rando values. A distribution function of this process and a tie interval where the process defined are considered to be known. In this case one can obtain ean value, variance, other oents and confidence interval. Data for each barrier analysis depend on the type of the event. One can obtain data for rando state or rando process by three ways: fro odeling fault trees for functional events, fro historical or general database or fro expert estiations. When the event frequency estiation or point probability estiation for the whole barrier in considered accident sequence is known fro historical evidence, the first way takes place. When estiations for the whole barrier in considered accident sequence is unknown fro statistics, and fault frequencies estiation of eleents which for the barrier are known the second way takes place. In this case the barrier is considered as a syste or a part of the syste. Given the data for eleents one can construct fault trees. he required event frequency estiation for the whole barrier is defined with the aid 4

5 of these fault trees. A syste description, syste boundaries, a syste schee, an eleent description, dependencies analysis and fault trees are initial data for the analysis. he third way takes place when data for the first and the second ways are unavailable. Initial data for this way are general databases, which are statistically analyzed or taken fro expert s estiations. Final states in event trees are grouped in categories. he first category is the safe state of the syste representing zero operational loss. Other categories can be defined by two ways. In the first case an operational loss interval fro noral value of loss to axiu value of loss divided into soe parts. Each resulted interval corresponds to the defined category. In PRA software results the total risk and the risk for each category is presented, so one can group final states such way that soe chosen ones concluded in a category. For each final state in each accident sequences the total operational loss is defined. It is odeled by a rando variable with one of any available probabilistic distributions. he data for a particular scenario loss distribution paraeters assessent is taken either fro historical based statistics or expert s estiations. IV. Scenario odel quantification For the syste first failure (sequence first operational accident) odeling the situation when syste or accident sequence (AS) reaches at least one failure (final) state on the denoted tie interval is considered. Let ξ(t) represent stochastic process on the probabilistic space (Ω, A, P), taking the value fro the finite set Z (possible states of the syste). he process ξ can be also represented by a sequence of pairs of rando values ζ n = (ξ n, τ n ), where ξ n denotes syste state at the oent of tie τ n when process changes its state, 0 < τ 0 < τ <... < τ n <... he set Z 0 Z designates all the states of interest and denotes final events (huan errors, equipent failures, etc.) Suppose that the failies of the conditional distributions F n (X, t ;Y 0, Y, Y 2,..,Y n- ) = P{ξ n =X, τ n t ζ 0 =Y 0, ζ = Y,.., ζ n- = Y n- }, () F 0 (X) = P{ξ 0 =X}, n =,2,..., where Y i =(X i,s i ), X,X i Z, 0 = s 0 <s <...s n <... are given. Consider tie interval [0,] and introduce the following event: A = { n : ξ n Z 0, τ n } (2) 5

6 It is usually a rare event, i.e. the probability of this event is very sall. he proble is to estiate P{A}. he ethod is based on odeling of rando oents of tie only inside the considered tie interval [0,]. Put A n = { ξ 0,..., ξ n- Z 0, ξ n Z, τ n }, (3) then P{ A} = P{ Z } + P{ A n } (4) ξ 0 0 n= Let us estiate P{A n }. We can write P{ An } = P{ An ξ0 = X 0} P{ ξ 0 = X 0} (5) X 0 Z0 Recurrently using the total probability forula we have for k=,..., n-2 P{ A ζ = Y,..., ζ = Y } = X n 0 0 k k Z k + 0 s k P{ A ζ = Y,..., ζ = Y } F ( X, ds ; Y,..., Y ) n 0 0 k + k + k + k + k + 0 k (6) And finally for k=n- we can write P{ An ζ 0 = Y0,..., ζ n = Yn } = Fn ( X n, dsn; Y0,..., Yn ) (7) X n Z0 sn Consequent application of forulas (5) - (7) yields function P{W} expressed by conditional distributions (). he P{A} estiation using analytical-statistical siulation is as follows. Let s transfor Equation (6) into a ore appropriate for P{ A ζ = Y,..., ζ = Y } = P{ A ζ = Y,..., ζ = Y } n 0 0 k k n 0 0 k + k + X k+ Z0 sk [ F ( X, ; Y,..., Y ) F ( X, s ; Y,..., Y )] k + k + 0 k k + k + k 0 k P{ ds ζ = Y,..., ζ = Y, ξ = X, s < s } k k k k + k + k k + (8) Each integral on the right-hand side of the Equation (8) is estiated using the Monte-Carlo ethod. At every step a rando value s k+ is siulated with the condition that ξ k+ = X k+, s k < s k+ and ζ 0 =Y 0,..., ζ k =Y k (9) he integrals on the last interval 6

7 F ( X, ds ; Y,..., Y ) (0) sn are estiated analytically. n n n 0 n he syste final state has the following probability estiation property. he statistical estiation for P{W} under -ties siulation is derived by selecting (P,...,P ). he value h Pi i ( W ) = = () is point estiation for P{W }. It has asyptotically the Gaussian distribution with paraeters Eh (W ) = P{W } Dh (W ) = P{W }(-P{W })/ (2) he upper and lower confidence boundaries, P u (h ) and P l (h ) respectively, can be found by solving the following equation S S P h ( W ) tα, < ( ) < ( ) + α, = α / 2 P W h W t / 2 (3) where, 2 Pi h ( W ) 2 i = S = ; t α,- - is the correspondent α-quantile for the Student distribution with - paraeters, α- is the confidence level. Estiation for odels (2)-(3) is siilar to (3). One of the iportant estiation properties is that variance of these estiations is not greater than in standard siulation procedures. he described atheatics of analytical-statistical siulation approach for event tree scenario odel quantification is effectively ipleented in Probabilistic Risk Assessent for Industrial Safety Evaluation software package (PRAISE code for short) developed by Xlerate echnologies LLC. his package handles event tree developent and quantification easily with user-friendly graphical user interface and cobines graphical siplicity with powerful nuerical procedures for iportance, sensitivity and uncertainty analyses. he ipleentation details of these ethods lies beyond the scope of this paper. Refer to both papers Islaov, 998 for details on ethods used. 7

8 V. Advantages of ASSA As it s entioned in the previous paragraph PRAISE/Eventree is focused on event tree representation of the Analytical-Statistical Siulation Approach. PRAISE/Eventree, according to its docuentation, has been developed to satisfy professional requireents in: analysis of tie dependant events; event rando tie siulation; event rando probability function siulation; event deterinistic tie process odeling (ostly for application in engineering); point and interval probability and risk estiate; siulation speed optiization. We ll illustrate the advantages of ASSA for quantitative operational risk analysis and odeling on a siplified scenario of two consequtive operational risk events. Consider the siplest event tree representing a logic odel of operational loss event sequence (accident sequence). he first considerable event, which is called Initial Event, ay cause several possible paths. Each of other events ay for the event tree branches by logic Yes/No Falsified ortgage credit purchased Borrower s default... Loss Yes No No Yes Final states Pic. branch is deterined by respective probability P of the event E: P{ E = Yes} + P{E = No} = In general case a final state probability is obtained as follows P f = P E i. For independent events (in probabilistic sense) a i final state probability can be given P = P{ }, where event E relates f E i i to an appropriate logic function (Yes/No). he path final states with respective probability P f and accident consequence C f characterize the risk R = P f C. f Consider a fragent fro a siple odel in the pic.. Shortcoings in underwriting process cause the initiating event, a erroneous purchase of a fraudulent ortgage credit on a rando tie t, then a consequent event, failure to satisfy the ters of a loan obligation or a 90+ days payent delinquency (ortgage credit default) at rando tie t 2 follows. So we have an accident. Other functional events include the possible reselling to the originator of the loan in case of default, securitization of the loan, f 8

9 possible foreclosure and so on. For illustrative purposes and siplicity we take only two events. So ties t, t 2 are independent in probabilstic sense and have distribution functions F, F 2 respectively. he ission tie is, which is the tie interval risk analysis is conducted on. Falsified ortgage credit purchased Borrower s default Yes No Pic. 2 t t 2 P f he accident final state probability is defined as P f = P{ t + t2 Let s copare standard Monte Carlo siulation approach with ASSA for a siple scenario of two consequentive events in the Pic. 2. Standard Monte Carlo siulation In standard siulation procedure we would have to:. siulate rando values 0 < r < for each distribution function F, F 2 ; 2. estiate rando ties t = F - (r) using inverse functions of F, F 2 ; 3. check the event {t +t 2 <}, whether it is successful or not; 4. repeat ites-3 N ties to estiate the probability P f = /N, where is a nuber of successful events. It is obvious, that the lower probability P f we estiate, the larger nuber of siulation N we have to carry out (so-called rare event proble): P f M ~ ~ Analytical-Statistical Siulation Approach(ASSA) Let s transfor the expression he integral P f = P t + t < } : < t { 2 = P{ t = P{ t2 < u t dp{ t < u t P{ t. 0 P{ t + t P = P{ t + t f 2 2 9

10 0 P{ t 0 2 < u} dp{ t F { u} dp{ t 2 < u t < u t P{ t P{ t = ay be considered as a atheatical expectation of the function F 2 ( - u), because 0 dp { t < u t =. In ASSA siulation procedure we should:. estiate F (); 2. siulate rando value 0 < r < ; 3. estiate rando tie F ( r F ( )) u = using inverse function of F ; 4. estiate F 2 (-u); 5. repeat ites 2-4 ties to estiate the probability P f = i= F2 ( ui ). hat is why ASSA has no rare event proble: P f ~ ~ VI. Conclusion An iportant quantitative operational risk analysis benefit fro the ability to estiate the precision of results received which cobines uncertainty analysis of the input data, the odel uncertainty, sensitivity and iportance analyses. Nowadays any of the are elaborated and well docuented steps described in probabilistic risk assessent procedure guides available. Given powerful estiation ethods, like ASSA one can fully use liited data available. his techniques are usually not used in siplier approaches, but non-realization of a rare event is useful inforation, which helps to build one-sided interval estiations. Stochastic odeling itself provides instruents taking into account the uncertainty in observed or just roughly estiated frequencies and probabilities. hat s why quantitative scenario odeling, linked with uncertainty analysis particularly helps fully use data avaiable on frequency and probability estiations of rare operational risk events. Ability to deal with analysis of tie dependant events (stochastic processes) akes the proposed odel suitable for coplex cases, including ulti event scenarios with coplex dependencies. Ability to deal with rare event proble, coputational efficiency and flexibility of statistical siulation apparatus akes ASSA well suitable 0

11 for non-trivial probles of reliability, engineering and quantitative operational risk analysis. VII. References Böcker, K., and C. Klüppelberg Operational VAR: A closed-for approxiation. RISK Magazine Deceber, Böcker, K. and Klüppelberg, C First Order Approxiations to Operational Risk Dependence and Consequences. In G.N. Gregoriou (ed.), Operational Risk oward Basel III, Best Practices and Issues in Modeling, Manageent and Regulation. Wiley, New York, Giacoelli, A. and Pelizzon, L Operational Risk Based on Copleentary Loss Evaluations. In G.N. Gregoriou (ed.), Operational Risk oward Basel III, Best Practices and Issues in Modeling, Manageent and Regulation. Wiley, New York, Peters, J-P. and Hübner, G Modeling Operational Risk Based on Multiple Expert s Opinions. In G.N. Gregoriou (ed.), Operational Risk oward Basel III, Best Practices and Issues in Modeling, Manageent and Regulation. Wiley, New York, 3 2. Papushkin, V., Islaov R., Volkov A Developent of Standard Probabilistic Risk Assessent (PRA) Procedure Guide: Syste odeling, NSI PREDRAF REPOR-999, Russian Acadey of Science, Nuclear Safety Institute. Islaov R., Uncertainty analysis, IBRAE RAN, Report for NRC, 998 Islaov R., Vulnerability Study, SCIENECH, Report for Sandia National Laboratory, 998 VIII. Appendix. ers and Definitions. hrough the paper soe specialized ters and definitions are used originating fro nuclear power safety engineering, operational risk and reliability analyses. Soe of these ters and definitions are given below. Initiating event an event that creates a disturbance in the business process functionality that directly leads to operational loss depending on the successful operation of the various itigating controls and systes in business process. Coponents equipent, devices, operations, and other eleents designated to perfor specific functions solely or as a part of the syste and considered as a design structural eleent when perforing reliability and safety analysis. Failure an event of disrupting operating conditions of a coponent, or whole syste.

12 Event tree a logical odel that expresses in graphical for the different ways of operational loss sequence developent for an initiating event group being considered that depends on accoplishing or failure of safety functions and successful or not successful personnel activities necessary for prevention of operational loss. Fault tree a logical odel presenting the various failure cobinations resulting to safety function non-perforance. Point Estiate (Paraeter Point Estiate) a single nuber that represents an estiate of a reliability paraeter, such as the ean, edian, or ode developed through axiu likelihood (in soe sense it gives the best estiate), oents ethod, Bayesian estiation ethods, or other ethods. he Point Estiate does not contain any inforation on possible discrepancy of the paraeter estiation. Syste a set of coponents designed for fulfilling specified functions. Uncertainty rando variability in given paraeters or easurable quantity or the iprecision in the knowledge of the paraeter or odel. 2