Report

Sponsor: CAE Fall Meeting 2013 Dependencies in a Risky World Pedro Fonseca Head Risk Analytics & Reporting at SIX München, 27 September 2013 Agenda • Acknowledgements • The Risky World of SIX • Why Model Dependencies? • Modeling Dependencies • Compound Poisson Processes and Lévy-Copulas CAE Fall Meeting 2013 2 Acknowledgements I have benefited from discussion with: – Frank Cuypers (Prime Re Services) – Hansjörg Albrecher (University of Lausanne) – Andreas Troxler (Solen Versicherungen) – Marc Sarbach (Deloitte) – Fabian Qazimi (SIX) – Carlos Arocha (Arocha & Associates) – Christoph Hummel (Secquaero) CAE Fall Meeting 2013 München, 27 September 2013 3 The Risky World of SIX CHF 4 billion daily turnover Per second: 120 card transactions; CHF 5 million interbank payments Infrastructure for the Swiss financial center • User-owned Trading & Indices • AA- Rating Cash Transactions • > 3’000 people • 25 countries 1’500 corporate actions daily Handling of Securities CAE Fall Meeting 2013 Data on 7 million financial instruments • Predominantly Operational and Counterparty Risks • Many Dependencies Financial Data München, 27 September 2013 4 “The Swiss Fort-Knox” Olten • CHF 2,5 trillion assets • 800 tons of gold & silver = CHF 10b Seen from the outside Seen from the inside CAE Fall Meeting 2013 München, 27 September 2013 5 The Risky World of SIX Systemic Risk “Too big to fail” CAE Fall Meeting 2013 “Too important to fail” “Too big to fail” München, 27 September 2013 The Crash of 2:45 (aka Flash Crash) Systemic Risk Probability of INformed Trading* O’Hara et al, 1996 Incentive to develop internal risk models * Probability that informed traders adversely select uninformed traders. CAE Fall Meeting 2013 München, 27 September 2013 Questions • How to best model dependencies? • Causality behind copulas? • Causality behind Lévy-copulas? CAE Fall Meeting 2013 München, 27 September 2013 8 Why Model Dependencies? • Dependencies create riskier worlds • In particular, positive dependencies create: – Fatter-tails – Less diversification – Higher frequency of “rare” events – Increased Value-at-Risk – Increased Tail-Value-at-Risk CAE Fall Meeting 2013 München, 27 September 2013 9 Dependencies Create Riskier Worlds Example – Mergers/Consolidation Company 1 Company 2 Company 3 Company 4 Company 5 Merged Companies: New fat-tail risks due to new strong dependencies CAE Fall Meeting 2013 München, 27 September 2013 10 Why Model Dependencies? • Regulation: o Swiss Solvency Test and Solvency II (for Re- / Insurers) Dependencies modeled at the level of risk drivers Dependencies accounted by standard correlations between risk types (e.g. market, counterparty, life, health, nonlife) : Solvency Capital Requirement = , • Based on linear dependence • Might not reflect the specific situation of the insurer Incentive to develop risk internal models CAE Fall Meeting 2013 München, 27 September 2013 11 Why Model Dependencies? • Regulation: o Swiss Solvency Test and Solvency II (for Re- / Insurers) o Basel II/III with FINMA “Swiss finish” (for Banks) • Reserving / Risk Adjusted Capital • Pricing • Capital Allocation To Improve Strategy (Profitability, Survival, …) • Business Planning • Portfolio and Risk Management • For the fun of modeling • … CAE Fall Meeting 2013 München, 27 September 2013 12 Bottom Line • Modeling dependencies is important. • “Theorem”: To model independence there is only one choice. To model dependence there are infinitely many choices. • “Lemma”: Given three experts on modeling correlations, there is only one thing that two of them agree on… And that is, that the third one is doing something wrong. CAE Fall Meeting 2013 München, 27 September 2013 13 Modeling Dependencies Many Complex Levels Reality is made up of: Particles Atoms Molecules Proteins Cells Organs Organisms ... Emotions Companies Financial Markets Societies Models are made up of: Typically used in Processes & Controls “explicit” models Exposures Events Severities Fitted Distributions Networks/Hierarchies Stochastic Processes Typically used in Regressions, GLM “implicit” models Copulas Survival Copulas Box-Copulas Lévy-Copulas Pareto-Copulas Pareto-Lévy-Copulas Semi-Martingale-Copulas C. Hummel (2009) arXiv:0906.4853 CAE Fall Meeting 2013 München, 27 September 2013 14 Modeling Dependencies Overview Models Explicit Models also known as: Causal Models, Common-Factors Models (or Common-Shocks) Implicit Models also known as: Copula Models, Covariance Models, “No Common-Factors” There is surely a copula (Sklar’s theorem) Causal interpretation? CAE Fall Meeting 2013 München, 27 September 2013 15 Modeling Dependencies Very Schematic Description… Common-Factors Like making your own pants Making sense out of the pieces CAE Fall Meeting 2013 “No Common-Factors” Like buying pants Making it fit München, 27 September 2013 16 Modeling Dependencies Schematic Description Common-Factors Stochastic Variable A “No Common-Factors” A Stochastic Variable A A’ Stochastic Common Factor Correlated a priori Stochastic Variable B B Correlating Algorithm Correlated a posteriori B’ CAE Fall Meeting 2013 Stochastic Variable B München, 27 September 2013 17 Common-Factors Example #1 Company Property in Germany Property in France Windstorm in France Fire in France Windstorm in Germany Fire in Germany Common Windstorm CAE Fall Meeting 2013 München, 27 September 2013 18 Common-Factors Example #2 Company Property in Germany Property in France Windstorm in France Fire in France Windstorm in Germany Fire in Germany Common Clients Common Event CAE Fall Meeting 2013 München, 27 September 2013 19 “No Common-Factors” Example Company Copula 3 Property in France Property in Germany Copula 1 Copula 2 Windstorm in France CAE Fall Meeting 2013 Fire in France Windstorm in Germany Fire in Germany München, 27 September 2013 20 Modeling Dependencies Common-Factors: • Intuitive • Potentially accurate Pros • Gives insight into business • Demanding in terms of input • Can lead to overly complicated models and a false sense of accuracy CAE Fall Meeting 2013 Cons München, 27 September 2013 21 Common-Factors Avoid overly complicated models Afghanistan war – social, political and economical risks: ‘ “When we understand that slide, we'll have won the war” Gen. Stanley McChrystal, US and NATO force commander CAE Fall Meeting 2013 München, 27 September 2013 22 Modeling Dependencies Common-Factors: • Intuitive • Potentially accurate “No Common-Factors”: Pros • Many types of dependencies • Explicit tail dependence • Gives insight into business • Demanding in terms of input • Can lead to overly complicated models and a false sense of accuracy CAE Fall Meeting 2013 • Calibration can be complicated Cons • Causal interpretation? München, 27 September 2013 23 Causal Interpretation for Copulas? Elliptical copulas: the Gaussian copula is about common-factors. Archimedean copulas: (1 , … , ) = ( −1 1 + ⋯ + −1 ) • Shared frailty models: Given cumulative hazard functions Λ , the Laplace transform of the frailty distribution (evaluated at the sum of the Λ ’s) is a survival copula. Suggested reading: 1) “Multivariate survival modelling (…) ”, P. Georges et al, 2001; 2) “Types of Dependence (…) in Single Parameter Copula Models”, Jaap Spreeuw, 2006. • Simplex distributions: Given equitable allocation of resources , the Williamson transform of is the generator of a survival copula. Suggested reading: “From Achimedean to Liouville Copulas”, A. McNeil and J. Nešlehová, 2009. Marshall-Olkin model: admits an elegant representation in terms of survival copulas. CAE Fall Meeting 2013 München, 27 September 2013 24 Causal Interpretation for Copulas? It seems to be all about survival! Life Medicine, Re/Insurance, … Solvency II, Basel III … Embrechts: “But why do we witness such an incredible growth in [copula] papers published starting the end of the nineties? Here I can give three reasons: finance, finance, finance.” CAE Fall Meeting 2013 München, 27 September 2013 25 Shared frailty models (also known as proportional frailty models) Consider the hazard functions (also known as failure rates) = − ln with cumulative hazard functions Λ . The survival functions are denoted by = 1 − , where are the cumulative distributions. • In the absence of dependence: = − . • In the shared multiplicative frailty model: = − , where is the shared frailty. The joint survival function is then = E − (Λ1+⋯+Λ) A Laplace transform! This is an Archimedean survival copula with generator = ℒ(), where is the distribution function of and Λ = −1 ( ). CAE Fall Meeting 2013 München, 27 September 2013 26 Shared frailty models Clayton Model (1978) • Suppose the shared frailty follows a gamma distribution with mean and variance both equal to 1/, that is: ~ Γ 1/, 1 • Its Laplace transform yields the generator () = (1 + )−1/ , and the joint survival function is the Clayton survival copula: = 1 − + 2 − −1 −1/ , with = −1 • Cross ratio function (association between two lifetimes): the ratio of one's failure risk at time 1 if the partner is known to have failed versus survived at time 2 , is constant (!): (1 2 = 2 ) (1 2 ≥ 2 ) CAE Fall Meeting 2013 =1+ München, 27 September 2013 27 Shared frailty models Hougaard Model (1986) • Suppose the shared frailty follows a stable distribution: ~ St 1/, 1, (cos ) , 0 2 • Its Laplace transform yields the generator = exp(− 1/ ), and the joint survival function is the Gumbel-Hougaard survival copula: (1 , 2 ) = exp − (−ln 1 ) +(−ln 2 ) 1/ , with = −1 • Cross ratio function - the 2 lives become less dependent as they age: (1 2 = 2 ) −1 =1− ln (1 , 2 ) (1 2 ≥ 2 ) CAE Fall Meeting 2013 München, 27 September 2013 28 Compound Processes • There are many ways to introduce dependence: 1 () = 1 () =0 1 and 2 () = 2 () =0 2 • Between final outcomes: o Copulas o Survival copulas o Regression o Autoregressive models o … CAE Fall Meeting 2013 München, 27 September 2013 29 Compound Processes • There are many ways to introduce dependence: 1 () = 1 () =0 1 and 2 () = = 2 () =0 2 , where is the intensity of common-jumps 1 2 • Between frequencies: o Copulas o Autoregressive models o Superposition & splitting/thinning for Poisson processes o Other common-factor models / causality o … CAE Fall Meeting 2013 München, 27 September 2013 30 Compound Processes • There are many ways to introduce dependence: 1 () = 1 () =0 1 and 2 () = 2 () =0 2 • Between severities: o Copulas o Survival copulas o Regression o Autoregressive models o Common-factors / causality o … CAE Fall Meeting 2013 München, 27 September 2013 31 Compound Processes • There are many ways to introduce dependence: 1 () = 1 () =0 1 and 2 () = 2 () =0 2 • Between frequencies and severities within a same business unit: o Copulas o Regression o Common-factors / causality o … CAE Fall Meeting 2013 München, 27 September 2013 32 Compound Processes • There are many ways to introduce dependence: 1 () = 1 () =0 1 and 2 () = 2 () =0 2 • Between frequencies and severities: o Common-factors / causality o Lévy copulas (for compound Poisson processes) o … Cont & Tankov, 2004 • They track losses originating from common events (sort of “common-factor” model) • They account for the dependence in both frequency and severity (“entanglement) Suggested reading: “Modelling and Measuring Multivariate Operational Risk with Lévy Copulas”, K. Böcker & C. Klüppelberg, 2008 (München!) CAE Fall Meeting 2013 München, 27 September 2013 33 Lévy Copulas • Intro: A copula parameterizes the joint cumulative distribution: 1 ≤ 1 ; 2 ≤ 2 = 1 1 , 2 2 • A survival copula parameterizes the joint survival function, with : 1 > 1 ; 2 > 2 = 1 1 , 2 2 They are related: 1 , 2 = 1 + 2 − 1 + 1 − 1 , 1 − 2 • A Lévy copula Lévy parameterizes the joint tail integral, the expected number of common losses above a certain value. For compound Poisson processes with intensities and common intensity , the copula for the tail integrals Π = λ is: 1 , 2 = 1 > 1 ; 2 > 2 = Lévy λ1 1 1 , λ2 2 2 for common losses CAE Fall Meeting 2013 München, 27 September 2013 34 Lévy Copulas • Examples of possible applications: o Motor insurance, where an accident can result in both injury and material claims. o Work-related accidents, which could give rise to both medical and allowance claims. = =0 , Common claims + , =0 , Unique claims • Dependent severities , • Independent , and , • Intensity: = Lévy 1 , 2 • Intensities: , = − • Survival functions: 1, 1 = Lévy 1 1 1 , 2 ; 2, 2 = Lévy 1 , 2 2 2 ; CAE Fall Meeting 2013 • Survival functions: , , = − , (1, , 2, ) = 1 Lévy München, 27 September 2013 1 1 , 2 2 35 Sampling Lévy Copulas 1) Compute the intensities and number of events: • Compute and draw • Compute 1, and 2, and draw 1, and 2, Common claims Unique claims • Dependent severities , • Independent , and , • Intensity: = Lévy 1 , 2 • Intensities: , = − • Survival functions: 1, 1 = Lévy 1 1 1 , 2 ; 2, 2 = Lévy 1 , 2 2 2 ; CAE Fall Meeting 2013 • Survival functions: , , = − , (1, , 2, ) = 1 Lévy München, 27 September 2013 1 1 , 2 2 36 Sampling Lévy Copulas 2) Compute the unique claims: • Draw independent 1 , 2 ~U 0,1 , as many times as dictated by 1, and 2, . • Find , = , −1 , with , = 1 − , Common claims Unique claims • Dependent severities , • Independent , and , • Intensity: = Lévy 1 , 2 • Intensities: , = − • Survival functions: 1, 1 = Lévy 1 1 1 , 2 ; 2, 2 = Lévy 1 , 2 2 2 ; CAE Fall Meeting 2013 • Survival functions: , , = − , (1, , 2, ) = 1 Lévy München, 27 September 2013 1 1 , 2 2 37 Sampling Lévy Copulas 3) Compute the common claims: • Draw independent 1 , 2 ~U 0,1 , as many times as dictated by . • Find 2 by solving 2 = (1 ,2 ) . 1 • Find , = , −1 , with , = 1 − , Common claims Unique claims • Dependent severities , • Independent , and , • Intensity: = Lévy 1 , 2 • Intensities: , = − • Survival functions: 1, 1 = Lévy 1 1 1 , 2 ; 2, 2 = Lévy 1 , 2 2 2 ; CAE Fall Meeting 2013 • Survival functions: , , = − , (1, , 2, ) = 1 Lévy München, 27 September 2013 1 1 , 2 2 38 A few Lévy Copulas Clayton: Lévy 1 , 2 = 1 − + 2 − Archimedean I: Lévy 1 , 2 −1/ 1 1 − −(1+2) = ln − 1 − 2 − 1 +2 + − 2 Archimedean II: Lévy 1 , 2 = ln 1 −1 − + 2 −1 − −1/ +1 • In all examples above > 0. • The Clayton and “Archimedean I” have upper-tail dependency. • The “Archimedean II” has both lower- / upper-tail dependencies and allows negative dependency between severities. CAE Fall Meeting 2013 München, 27 September 2013 39 Clayton-Lévy Copula Lévy 1 , 2 = 1 − + 2 − −1/ • Survival copula for common jumps: 1 , 2 = 1 − + 2 − − 1 • Time invariant: Lévy −1/ The Clayton copula! For the Clayton-Lévy copula 1 , 2 = Lévy 1 2 , = Lévy 1 , 2 • Intensity for common jumps: = 1 • Frequency correlation: = CAE Fall Meeting 2013 1 − + 2 − + 2 − −1/ − −1/ 1 2 München, 27 September 2013 40 Lévy Copula at Work Fit to claims arising from accidents in the construction sector. It features 2’249 medical claims and 1’099 daily allowance claims. Logarithm of claims Logarithm of claims CAE Fall Meeting 2013 Logarithm of claims Tail Integral Tail Integral Tail Integral Logarithm of claims Avanzi et al, 2011 Tail Integral Tail Integral Tail Integral • • Logarithm of claims Logarithm of claims München, 27 September 2013 41 Closing Remark #1 Solvency II on Internal Model Approval: 1. Senior management shall be able to demonstrate understanding of the internal model and how this fits with their business model. 2. Senior management shall be able to demonstrate understanding of the limitations of the internal model and that they account of it in their decisions. 3. The timely calculation of results is essential. (From “Use Test”, Section 3 of the CEIOPS Paper) CAE Fall Meeting 2013 München, 27 September 2013 42 Closing Remark #2 • Bad Governance influences model risk, among other things. • Example: the head of risk management tells you: “quantitative risk management is useless, because I can twist the knobs in such a way that the number being outputted is the number I want”. • A “dependence” between that person and the model parameters would be bad governance. CAE Fall Meeting 2013 München, 27 September 2013 43 Closing Remark #2 Standard safety rules for good governance should include these: Cut along dotted line Stochastic area Authorized Personnel only Thank you very much for your attention CAE Fall Meeting 2013 München, 27 September 2013 44 Casualty European Association – Fall 2013 Meeting Fell free to send me your questions as well as contact me for potential colaborations: Pedro Fonseca [email protected] Head Risk Analytics & Reporting