Pavel V. Shevchenko Quantitative Risk Management. CSIRO Mathematical & Information Sciences. Bridging to Finance

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1 Pavel V. Shevchenko Quanave Rsk Managemen CSIRO Mahemacal & Informaon Scences Brdgng o Fnance Conference Quanave Mehods n Invesmen and Rsk Managemen: sourcng new approaches from mahemacal heory and he real world Melbourne Cenre For Fnancal Sudes 0 h Sepember 007.

2 CSIRO Mahemacal & Informaon Scences Commonwealh Scenfc and Indusral Research Organzaon of Ausrala CSIRO Naonal research agency formed n 96. Approx 6500 saff Dvsons: Indusral Physcs Mnerals Mahemacal & Informaon Scences Marne and Amospherc Research ec. Dvson of Mahemacal and Informaon Scences CMIS over 00 researchers: Decson echnology Boechnology and Healh Informacs Envronmenal Informacs Quanave Rsk Managemen QRM group approx. 0 saff: fnancal rsk nfrasrucure envronmen rsk secury ar-ranspor. Acves/modes of engagemen: research consulng model developmen/valdaon sofware developmen. /QRM CSIRO Mahemacal & Informaon Scences

au Dvson of Mahemacal and Informaon Scences CMIS over 00 researchers: Decson echnology Boechnology and Healh Informacs Envronmenal Informacs Quanave Rsk

3 CSIRO Quanave Rsk Managemen Applcaon areas Fnancal Rsk Sraegc research Exremes sparse daa Exper elcaon Opmsaon Ar raffc Managemen Energy/commody modellng OAM Infrasrucure secury healh Real-me monorng Rsk assessmen CSIRO Mahemacal & Informaon Scences

Managemen Energy/commody modellng OAM Infrasrucure secury

4 CSIRO Mahemacal & Informaon Scences Soluons Developmen of mahemacal models and cusomsed sofware accordng o he clen mehodology and specfc needs. Asssng wh developmen of new models and her mplemenaon no sofware. Independen revew and advce on rsk models mehodology and sofware soluons. Independen valdaon of dervaves and rsk measuremen models. me-seres analyss of daa. Modes of engagemen here are many ways CSIRO can work wh you o undersand and quanfy fnancal rsks: Consulancy engagemen. Conrac research engagemen. Collaborave projecs. Sofware developmen. CSIRO Mahemacal & Informaon Scences

Independen valdaon of dervaves and rsk measuremen models. me-seres analyss of daa.

5 rack Records n Fnancal Rsk Dervave prcng: work on CSIRO opon prcng sofware Redus snce 999 consulng projecs n FX opon prcng: plug-ns for Fencs launched n users n overseas banks. Operaonal Rsk: valdaon projecs n 000 model developmen projecs n R&D/sofware projecs Marke Rsk: valdaon projecs n model developmen consulng projec n 007. Cred Rsk: valdaon projecs n ; valdaon and model developmen projecs 004-ongong. Underwrng rsk: consulng projecs n 999 curren proposals. Forecasng elecrcy/commodes: consulng projecs n curren proposals. Porfolo Managemen: model developmen projecs n 007 curren proposals. Waer/Carbon radng: curren proposals. Collaboraors: Monash Un Cambrdge Un EH Zurch UNSW US Macquare Un Sascal Research Assocaes NZ Indusry clens: CBA ANZ NAB S George Inegral Energy IAG Fencs FX Edgecap Moore Capal Eser Bank Cred Swss. CSIRO Mahemacal & Informaon Scences

Cred Rsk: valdaon projecs n 999 00; valdaon and model developmen projecs 004-ongong. Underwrng rsk: consulng projecs n 999 curren proposals.

6 Fnancal Rsk Lnks Managemen Oher Rsk Areas Marke Rsk Cred Rsk Operaonal Rsk Underwrng rsk Dervave prcng Ineres Raes radng sraeges Porfolo Managemen Commody/Energy Carbon/waer radng Model rsk Lqudy rsk Exreme Value modellng Dynamc conrol Exper Elcaon Bayesan mehods Dependence Modellng Model valdaon Compuaonal mehods PDE MCMC Mone Carlo mehods me seres analyss Hgh performance compung Ar ranspor Ecology/Envronmenal Infrasrucure Secury Weaher/Clmae Healh CSIRO Mahemacal & Informaon Scences

Elcaon Bayesan mehods Dependence Modellng Model valdaon Compuaonal mehods PDE MCMC Mone Carlo mehods me seres analyss Hgh

7 Exreme Value Analyss/Dependence Ar-ranspor devaons from ax cenerlne Weaher ranfall wnd speed Marke Rsk al of porfolo reurn dsrbuon dervaves Cred/Operaonal Rsk al of annual loss dsrbuon Elecrcy prcng prce spkes CSIRO Mahemacal & Informaon Scences

reurn dsrbuon dervaves Cred/Operaonal Rsk al of annual loss

8 Exreme Value Models M max X... Block maxma models for larges observaons n X X... are d e.g. daly daa grouped no quarerly blocks Lmng dsrbuon: Generalzed Exreme Value Dsrbuon GEV / exp[ ξ x / β H x exp[ exp x / β ] ξ ] ξ 0 ξ 0 X n hreshold exceedances models for large observaons exceedng some hgh level L e.g. operaonal losses exceedng mln Y X L Lmng dsrbuon: Generalzed Pareo Dsrbuon GPD / ξx / β H x exp[ x / β ]; ξ ; ξ 0 ξ 0 CSIRO Mahemacal & Informaon Scences

daly daa grouped no quarerly blocks Lmng dsrbuon: Generalzed Exreme Value Dsrbuon GEV / exp[ ξ x / β H x exp[ exp x

9 CSIRO Mahemacal & Informaon Scences Marke Rsk } 60 max{var Marke Rsk Regulaory Capal by s modelled ; For example : s saonary are d and... assume / ln ln q q q q VaR k C GPD Z Z CVaR CVaR Z VaR VaR X X X Z Z Z X S S S S S X σ µ σ µ βσ µ α α σ λ µ σ µ

10 Example: API reurns emprcal quanle emprcal quanle GARCH-Normal Asse resdual Q-Q plo heorecal quanle GARCH-Generalzed Pareo Asse resdual Q-Q plo heorecal quanle CSIRO Mahemacal & Informaon Scences emprcal quanle GARCH-Suden Asse resdual Q-Q plo heorecal quanle GARCH-volaly volaly 8/07/0 0//0 3/04/04 05/09/05 8/0/07

quanle CSIRO Mahemacal & Informaon Scences emprcal quanle GARCH-Suden Asse resdual Q-Q plo 5 4 3 0-5 -4-3 - - - - 0

11 Dervave Prcng Opon prcng : Q E[ Payoff S S ] Modellng volaly skew: e.g. local volaly models - ds / S r q d σ S dw sochasc volaly models jump dffuson Numercal Mehods: Fne Elemen Fne Dfference Mone Carlo mehods. CSIRO Mahemacal & Informaon Scences

12 Dependence modellng Dependence modellng va Copula mehod Consderrvs X... X wh F x Prob X x d F U ~ F X F U U ~ U0 F X U ~ U 0 F x x... x Prob[ X x X x... X d d x d ] s jon cdf F x x... x d margnal behavor F copula C u. u d Copula s mulvarae jon dsrbuon of unform random varables C u u... u d F F u F u... F u d d C F x F x... F d x d F x x... x d CSIRO Mahemacal & Informaon Scences

.. X d d x d ] s jon cdf F x x... x d margnal behavor F copula C u.

13 Dependence modellng va copula Y Gumble Copula X ~ Normal 0; Y corr X Y 0.7 Copula 4 3 ~ Normal 0 Gaussan Copula -3-4 X 4 3 Y X CSIRO Mahemacal & Informaon Scences 0 Y X 0

7 Copula 4 3 ~ Normal 0 Gaussan Copula -3-4 X 4 3 Y -4-3 -

14 Exper Elcaon/Bayesan mehods combnng nernal & exernal daa wh exper opnon Ecology esmaon of fsh densy from gllne caches Operaonal Rsk esmaon of loss frequency and severy Ar-ranspor ndvdual&collecve rsk of ar-collsons Insurance prcng of polcy premum Markov Chan Mone Carlo mehods sgnal processng Recen Publcaons: D. D. Lambrgger EH P.V. Shevchenko CSIRO and M. V. Wührch EH 007. he Quanfcaon of Operaonal Rsk usng Inernal Daa Relevan Exernal Daa and Exper Opnons. he Journal of Operaonal Rsk. Hans Bühlmann EH P.V. Shevchenko CSIRO and M. Wührch EH 006. A oy Model for Operaonal Rsk Quanfcaon usng Credbly heory. he Journal of Operaonal Rsk. P.V. Shevchenko CSIRO and M. Wührch EH 006. Srucural Modellng of Operaonal Rsk usng Bayesan Inference: combnng loss daa wh exper opnons. he Journal of Operaonal Rsk 3 pp.3-6. CSIRO Mahemacal & Informaon Scences

he Quanfcaon of Operaonal Rsk usng Inernal Daa Relevan Exernal Daa and Exper Opnons. he Journal of Operaonal Rsk. Hans Bühlmann EH P.V. Shevchenko CSIRO and M. Wührch EH 006.

15 Combnng nernal daa ndusry daa and exper opnons observaons CSIRO Mahemacal & Informaon Scences Bayesan nference X X X... X n θ θ θ... θk h X θ h X θ π θ ˆ π θ X h X ˆ π θ X h X θ π θ π θ h X θ ϕ parameers pror dsrbuon s esmaed by exper/ndusry daa lkelhood of nernal observaons g X n θ X n X π θ X dθ Combnng nernal daa exernal daa ˆ π θ X υ h X θ h υ θ π θ π θ h X θ h υ θ ˆ predcve dsrbuon and exper opnons pror dsrbuon s esmaed by ndusry daa lkelhood of nernal observaons lkelhood of exper opnons

.. θk h X θ h X θ π θ ˆ π θ X h X ˆ π θ X h X θ π θ π θ h X θ ϕ parameers pror dsrbuon s esmaed by exper/ndusry daa lkelhood

16 Example: combnng exper opnon and nernal daa Annual couns N from Posson λ 0.6 exper opnons E[ ] 0.5 Pr[ ] / ˆ λ ˆ α ˆ β ~ λ k k k k k k N λ λ α β he Bayesan esmaor wh Gamma pror he Maxmum Lkelhood esmaor α 3.4 β 0.5 arrval rae j esmae of he arrval rae vs year Bayesan esmae maxmum lkelhood esmae year CSIRO Mahemacal & Informaon Scences

5 ˆ λ ˆ α ˆ β ~ λ k k k k k k N λ λ α β he Bayesan esmaor wh Gamma pror he Maxmum Lkelhood esmaor α 3.

17 Parameer Rsk uncerany of parameers ϕ Z Qˆ Qˆ θˆ N Z X B annual loss; Y g Z θ ˆ π θ Y dθ predcve dsrbuon ˆ π θ Y h Y θ π θ B Qˆ quanle of quanle of ϕ Z s pon esmaor e.g. maxmum lkelhood esmaor bas E[ Qˆ Normal approxmaon : ˆˆ π θ Y s ] Y X N pas g Z Y θˆ observaons Markov Chan Mone Carlo mehods Normal mean θˆcov I ln ˆ π θ Y ln ˆˆ π θ Y I j j θ ˆ θ θ j ˆ θ ; j I j ln ˆ π θ Y θ θ j θ ˆ θ P.V. Shevchenko 007. Esmaon of operaonal Rsk Capal Charge under Parameer Uncerany submed o he Rsk Magazne. CSIRO Mahemacal & Informaon Scences

maxmum lkelhood esmaor bas E[ Qˆ Normal approxmaon : ˆˆ π θ Y s ] Y X N pas g Z Y θˆ observaons Markov Chan Mone Carlo mehods Normal mean

18 Parameer Rsk uncerany of parameers quanle year MLE Bayesan RUE Comparson of he quanle esmaors. Parameer uncerany s gnored by MLE bu s aken no accoun by Bayesan. MLE - maxmum lkelhood esmaor Bayesan - quanle of predcve dsrbuon Losses were smulaed from Posson0 and LN. Non-nformave consan pror were used. % bas 50% 00% 50% 00% 50% Lognormal Pareo Relave bas n he quanle esmaor nduced by he parameer uncerany vs number of observaon years. Lognormal - losses were smulaed from Posson0 and LN. Pareo losses were smulaed from Posson0 and Pareo wh L. 0% Year CSIRO Mahemacal & Informaon Scences

MLE - maxmum lkelhood esmaor Bayesan - quanle of predcve dsrbuon Losses were smulaed from Posson0 and LN. Non-nformave consan pror were used.

19 Kalman/Parcle fler echnques sae-space models On-lne monorng of waer qualy Healh survellance Modellng commodes/neres raes CSIRO Mahemacal & Informaon Scences

20 CSIRO Mahemacal & Informaon Scences Sae-space models: Kalman Fler Measuremen Equaon ranson Equaon C S B A F S S E F d dw dw E dw d X d dw d S d F S δ δ ρ σ κ κ δ α δ σ σ δ µ δ ln ln ] [ ] [ ; ] [ ] [ ln fuures prces yeld convenence spo prce; Commody spo models:e.g. -facor convenence yeld Ineres rae spo models:e.g. Vascek model r B A P r B A P dw d r dr P r ln ln ] exp[ ] [ bond prce - erm rae; shor σ γ α e X B A F r r r r ˆ ε r r r r X M X ˆ

convenence spo prce; Commody spo models:e.g.

21 Emergng over-archng research opcs Mxng nernal & exernal daa wh exper opnons credbly heory Bayesan echnques Dependence beween rsks: copula mehods srucural models Compound pon processes Modellng dsrbuon al: EV mxed dsrbuons splces Effcen Markov Chan Mone Carlo Mone Carlo fne elemen/fne dfference mehods Sae-space models sequenal Mone Carlo Kalman fler me seres analyss va chaos heory mehods Modellng runcaed/censored daa Nonlnear opmzaon wh consrans CSIRO Mahemacal & Informaon Scences

22 CSIRO Mahemacal & Informaon Scences Pavel Shevchenko Phone: Emal: Pavel.Shevchenko@csro.au /Pavel.Shevchenko hank you

Fixed Income Attribution. Remco van Eeuwijk, Managing Director Wilshire Associates Incorporated 15 February 2006

Fixed Income Attribution. Remco van Eeuwijk, Managing Director Wilshire Associates Incorporated 15 February 2006 Fxed Incoe Arbuon eco van Eeuwk Managng Drecor Wlshre Assocaes Incorporaed 5 February 2006 Agenda Inroducon Goal of Perforance Arbuon Invesen Processes and Arbuon Mehodologes Facor-based Perforance Arbuon