Operational Risk Modeling and Quantification

Transcription

1 herry Bud Workshop 5, Keo Unversy Operaonal Rsk Modelng and Quanfcaon Pavel V. Shevchenko SIRO Mahemacal and Informaon Scences, Sydney, Ausrala E-mal:

2 Agenda Loss Dsrbuon Approach for OR hallenges. Is possble o model/quanfy OR? OR Frequency and Severy OR Insurance and pon process Esmaon wh Truncaed Daa Dependence beween rsks Dsrbuon als: EVT, mxures

3 Advanced Measuremen Approach: boom-up Loss Dsrbuon Approach Inernal Loss Daa scalng, bas Exernal Loss Daa scalng, bas Insurance Dependence Facors onrol/rsk Indcaor Rsk Frequency and Severy dsrbuons Rsk annual loss dsrbuons Mone arlo smulaons Exper Opnons busness judgmens on #losses and $ranges Toal annual loss dsrbuon over all rsks Annual apal harge Expeced and Unexpeced losses apal Sensves, wha f scenaros

4 Annual apal harge, M smulaons unexpeced loss=var-expeced Loss; Pr [Loss<=VaR]= loss dsrbuon.3 Expeced Loss Unexpeced Loss $ loss

5 hallenges; Tools Defnon, denfcaon, measuremen, monorng, ndcaors/conrols Daa Truncaon: known hreshold, sochasc hreshold, unknown hreshold Lmed Daa: mxng nernal and exernal daa va credbly heory, Bayesan echnques Exernal daa relevance: scalng Exper opnons Daa suffcency: capal charge accuracy orrelaon beween rsks and s esmaon: copula, common process ompany ndcaors: regresson/facoral analyss OR nsurance: pon processes on-gaussan dsrbuons, Fa als: EVT, mxed dsrbuons, splces VaR pfalls: coheren rsk measures, expeced shorfall alculaon apal charge and s sensves: Mone arlo smulaon ompung me/ram, Fas Fourer Transform

6 Loss Dsrbuon Approach: key parameers Annual frequency of evens Loss severy of he even Insurance agans losses orrelaon beween rsks

7 loss of he even,.4.3. severy dsrbuon, pdf Sngle Rsk annual loss, Z = = annual number of evens,.4.3. aggregae dsrbuon, pdf.5 frequency dsrbuon Z

8 Severy Dsrbuons f: e.g. Logormal, Gamma, Webull Frequency dsrbuons P: e.g. Posson, egave Bnomal, Bnomal Annual loss Assumpons and and Z = = j =,..., are ndependen j Z are ndependen Dsrbuon of : sem-analyc, Mone arlo smulaons

9 Insurance for Operaonal Rsks Insurance: probably of coverage, nsurer defaul, cover lm, excess, regulaory cap Modellng of loss even mes s requred nsead of even frequency o address OR nsurance. pon process: < <... < < + <... e.g. homogeneous Posson process δ = + ~ Exponenal, ~ Posson non-homogeneous Posson: doubly sochasc Posson: ~ Gamma. => => egavebnomal. oday year me

10 Daa Truncaon models Known consan runcaon level > L, =,,... Known varable runcaon level > L, =,,... Unknown runcaon level Sochasc runcaon level L ~ g.

11 Known hreshold Known consan runcaed level,.e. loss daa Unruncaed severy dsrbuon Truncaed severy dsrbuon f > L f = ; Pr[ > L] L; Pr[ f, > L] = L f d L, =,,... Severy pdf f va e.g. Maxmum Lkelhood Mehod Ψ α,..., α = K f = Pr[ > L] rue Frequency adjusmen / Pr[ > L] obs

12 Unknown runcaon level L s an exra parameer n lkelhood funcon L s unknown: α L + β; L < γ obs µ L = mn α, β, γ α γ + β; L γ [ µ L µ L ] Example: ~Logormalmu=8, sgma=, L=4 esmaes: mu 8.5, sgma., L mu vs level, L sgma vs level, L 5 9 3

13 Sochasc runcaon level Severy dsrbuon of losses repored losses > L, =,..., K f, where L ~ g. condonal pdf f L = a = f ; Pr[ > a] Pr[ > a a > a = ; ] a f y dy margnal dsrbuon of repored losses ~ f = rue = g a da a] f L = a g a da = f Pr[ > obs / Pr[ > L]; Pr[ > L] = g L dl L f Ψ = d ~ f

14 Dependence beween rsks Dversfcaon: Z=R+ +Rn<R+ +Rn VaR: VaR α Z Z = F α = mn{ z, F z α} ondonal VaR VaR: Dependence beween frequences Dependence beween even pon processes Dependence beween severes Dependence beween annual losses Z VaR Z = E[ Z Z VaR Z] α > α

15 oal annual loss : Z = K k= Z k = = + = K = K Basel ommee saemen: Rsk measures for dfferen operaonal rsk esmaes mus be added for purposes of calculang he regulaory mnmum capal requremen. However, he bank may be permed o use nernally deermned correlaons n operaonal rsk losses across ndvdual operaonal rsk esmaes, provded can demonsrae o a hgh degree of confdence and o he sasfacon of he naonal supervsor ha s sysems for deermnng correlaons are sound, mplemened wh negry, and ake no accoun he uncerany surroundng any such correlaon esmaes parcularly n perods of sress. The bank mus valdae s correlaon assumpons. Addng capals=>perfec dependence beween rsks oo conservave

16 Dependence beween frequences va copula e.g. Gaussan copula U, U,..., U K, U ~ Unform, = F [ U ],..., = F [ U ] K K K Ga ρ u corr, f ρ j ρ,..., ud = F F u,..., F uk Example corr,.5 vs ρ ~ Posson ~ Posson corr, ρ opula correlaon ρ =.5, = = 5, =

17 Dependence beween frequences=>dependence beween annual losses Example Z = FPosson U = = [ U], = FPosson[ ] Z = = Ga U, U = ρ U, U ~ Logormal,, ~ Logormal,, nd ρ S Z, Z vs ρ, ~ Posson = 5, = ~ Posson - opula correlaon ρ

18 Dependence va common Posson process Johnson, Koz and Balakrshnan /, ; ~ ; ~ ~ ; ~ ˆ ; ~ ˆ ˆ ~ ˆ ~ corr Posson Posson Posson Posson Posson + + = posve dependence; consan covarance + = p p prob wh ˆ prob wh ˆ exenson k k p p =, cov year s rsk evens nd rsk evens oday

19 Dependence beween ner-arrval mes < < e.g. Gaussan copula τ ~ Exp ; U, U,..., U K, U ~ Unform, K τ = F [ U ],..., τ = F [ U ] j = j... < + τ j Ga K τ,..., τ ~ F., =,..., K ρ uk ; d < ρ u,..., ud = F F ~ Posson ; corr, K Example: = 5, =, vs ρ corr + <... j u,..., F f ρ ;

20 Dependence beween severes va copula Gumbel opula 4 3 Y ~ ormal,; Y corr, Y =.7 opula 4 3 ~ ormal, -3-4 Gaussan opula 4 3 Y Y

21 Upper al dependence = lm Ρ[ Y α > F α Gaussan copula = > F α] = lm α + α, α α α Ga u,...,,,...,, u u d = F ρ F u F u F ρ u d -copula ν u,,...,,,..., u u d = F Σ u u u d Σ ν ν ν ν = β Gumble copula /... ln / β Gu u,..., u = exp lnu β + + u β, < β β d d

22 Severy Dsrbuon Tal Exreme Value Theory peaks over hresholds and splcng Generalzed Pareo Dsrbuon GPD / + ξx / β H x = exp[ x / β ]; ξ ; ξ ξ =. fx=qf +-q f.. f =Truncaed Logormal f =GPD a 4 8 f, < a f, a

23 Mxure of dsrbuons f = qf + q f f, < ; f, < Maxmum lkelhood Ψ = K = [ qf + q f ]. f. f

24 apal harge confdence nerval error n rsk dsrbuon parameers: Gaussan approx ML, boosrap uncerany n capal charge: Mone arlo Daa suffcency crera

25 References Johnson, Koz and Balakrshnan 997. Dscree Mulvarae Dsrbuons Arzner, Delbaen, Eber and Heah 999. oheren Measures of Rsk Klugman, Panjer and Wllmo 998. Loss Models Embrechs, Kaufmann, Samorodnsky. Run heory revsed: sochasc models for operaonal rsk Embrechs, Kluppelberg and Mkosch 997 Modellng Exremeal Evens for nsurance and fnance Embrechs, Mcel and Sraumann. orrelaon and dependence n rsk managemen: properes and pfalls ruz. Modelng, Measurng and Hedgng Operaonal Rsk, John Wley & Sons Kng. Operaonal Rsk, John Wley & Sons Joe 997 Mulvarae Models and Dependence onceps, hapman & Hall Fracho, Moudoulaud, Roncall 3 Loss Dsrbuon Approach n Pracce ommens and suggesons are welcomed