Agenda

2. Agenda • Acknowledgements • The Risky World of SIX • Why Model Dependencies? • Modeling Dependencies • Compound Poisson Processes and Lévy-Copulas 2

3. Acknowledgements I have benefited from discussion with: • Frank Cuypers (Prime Re Services) • Hansjörg Albrecher (University of Lausanne) • Andreas Troxler (SolenVersicherungen) • Marc Sarbach (Deloitte) • Fabian Qazimi (SIX) • Carlos Arocha (Arocha & Associates) • Christoph Hummel (Secquaero) 3

4. 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 • AA- Rating • > 3’000 people • 25 countries Cash Transactions Trading & Indices 1’500 corporate actions daily Data on 7 million financial instruments • Predominantly Operational and Counterparty Risks • Many Dependencies Handling of Securities Financial Data 4

5. “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 5

6. The Risky World of SIX Systemic Risk “Too big to fail” “Too important to fail” “Too big to fail”

7. The Crash of 2:45 (aka Flash Crash) Systemic Risk Probability of INformedTrading* O’Hara et al, 1996 Incentive to develop internal risk models * Probability that informed traders adversely select uninformed traders.

8. Questions • How to best model dependencies? • Causality behind copulas? • Causality behind Lévy-copulas? 8

9. 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 9

10. 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 10

11. Why Model Dependencies? • Regulation: • 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, non-life) : • Solvency Capital Requirement • Based on linear dependence • Might not reflect the specific situation of the insurer Incentive to develop risk internal models 11

12. Why Model Dependencies? • Regulation: • Swiss Solvency Test and Solvency II (for Re- / Insurers) • Basel II/III with FINMA “Swiss finish” (for Banks) • Reserving / Risk Adjusted Capital • Pricing • Capital Allocation • Business Planning • Portfolio and Risk Management • For the fun of modeling • … To Improve Strategy (Profitability, Survival, …) 12

13. 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. 13

14. Modeling Dependencies Many Complex Levels Models are made up of: Processes & Controls Exposures Events Severities Fitted Distributions Networks/Hierarchies Stochastic Processes Regressions, GLM Copulas Survival Copulas Box-Copulas Lévy-Copulas Pareto-Copulas Pareto-Lévy-Copulas Semi-Martingale-Copulas Reality is made up of: Particles Atoms Molecules Proteins Cells Organs Organisms ... Emotions Companies Financial Markets Societies Typically used in “explicit” models Typically used in “implicit” models C. Hummel (2009) arXiv:0906.4853 14

15. 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’stheorem) Causal interpretation? 15

16. Modeling Dependencies Very Schematic Description… Common-Factors Like making your own pants “No Common-Factors” Like buying pants Making sense out of the pieces Making it fit 16

17. Modeling Dependencies Schematic Description Common-Factors “No Common-Factors” Stochastic Variable A Stochastic Variable A A Stochastic Common Factor A’ Correlating Algorithm Correlated a priori Correlated a posteriori B’ Stochastic Variable B Stochastic Variable B B 17

18. Common-Factors Example #1 Company Property in Germany Property in France Windstorm in Germany Windstorm in France Fire in Germany Fire in France Common Windstorm 18

19. Common-Factors Example #2 Company Property in Germany Property in France Windstorm in Germany Windstorm in France Fire in Germany Fire in France Common Clients Common Event 19

20. “No Common-Factors” Example Company Copula 3 Property in Germany Property in France Copula 2 Copula 1 Windstorm in Germany Windstorm in France Fire in Germany Fire in France 20

21. Modeling Dependencies • Common-Factors: • Intuitive • Potentially accurate • Gives insight into business • Demanding in terms of input • Can lead to overly complicated models and a false sense of accuracy Pros Cons 21

22. Common-FactorsAvoid 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 22

23. Modeling Dependencies • Common-Factors: • Intuitive • Potentially accurate • Gives insight into business • Demanding in terms of input • Can lead to overly complicated models and a false sense of accuracy • “No Common-Factors”: • Many types of dependencies • Explicit tail dependence • Calibration can be complicated • Causal interpretation? Pros Cons 23

24. Causal Interpretation for Copulas? • Elliptical copulas: the Gaussian copula is about common-factors. • Archimedean copulas: • Shared frailty models: • Given cumulative hazard functions , the Laplace transform of the frailty distribution (evaluated at the sum of the ’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”, JaapSpreeuw, 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-Olkinmodel: admits an elegant representation in terms of survival copulas. 24

25. 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.” 25

26. Shared frailty models (also known as proportional frailty models) • Consider the hazard functions (also known as failure rates) • with cumulative hazard functions . The survival functions are denoted by , 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 • This is an Archimedean survival copula with generator , where is the distribution function of and . A Laplace transform! 26

27. Shared frailty models Clayton Model (1978) • Suppose the shared frailty follows a gamma distribution with mean and variance both equal to , that is: • Its Laplace transform yields the generator, and the joint survival function is the Clayton survival copula: • , with • Cross ratio function (association between two lifetimes): the ratio of one's failure risk at time if the partner is known to have failed versus survived at time , is constant (!): 27

28. Shared frailty models Hougaard Model (1986) • Suppose the shared frailty follows a stable distribution: • Its Laplace transform yields the generator , and the joint survival function is the Gumbel-Hougaard survival copula: • , with • Cross ratio function- the 2 lives become less dependent as they age: 28

29. Compound Processes • There are many ways to introduce dependence: • and • Between final outcomes: • Copulas • Survival copulas • Regression • Autoregressive models • … 29

30. Compound Processes • There are many ways to introduce dependence: • and • Between frequencies: • Copulas • Autoregressive models • Superposition &splitting/thinning for Poisson processes • Other common-factor models / causality • … , where is the intensity of common-jumps 30

31. Compound Processes • There are many ways to introduce dependence: • and • Between severities: • Copulas • Survival copulas • Regression • Autoregressive models • Common-factors / causality • … 31

32. Compound Processes • There are many ways to introduce dependence: • and • Between frequencies and severities within a same business unit: • Copulas • Regression • Common-factors / causality • … 32

33. Compound Processes • There are many ways to introduce dependence: • and • Between frequencies and severities: • Common-factors / causality • Lévy copulas (for compound Poisson processes) • … Cont& Tankov, 2004 • They track losses originating from common events • (sort of “common-factor” model) • They account for the dependence in bothfrequencyand severity (“entanglement) • Suggested reading: “Modelling and Measuring Multivariate Operational Risk with Lévy Copulas”, K. Böcker & C. Klüppelberg, 2008(München!) 33

34. LévyCopulas • Intro: A copula parameterizes the joint cumulative distribution: • A survivalcopula parameterizes the joint survival function, with : They are related: • A Lévycopula 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 is: for common losses 34

35. LévyCopulas • Examples of possible applications: • Motor insurance, where an accident can result in both injury and material claims. • Work-related accidents, which could give rise to both medical and allowance claims. Common claims Unique claims • Dependent severities • Intensity: • Survival functions: • ; • ; • Independent and • Intensities: • Survival functions: 35

36. Sampling Lévy Copulas 1) Compute the intensities and number of events: • Compute and draw • Compute and and draw and Common claims Unique claims • Dependent severities • Intensity: • Survival functions: • ; • ; • Independent and • Intensities: • Survival functions: 36

37. Sampling Lévy Copulas 2) Compute the unique claims: • Draw independent , as many times as dictated by and . • Find , with Common claims Unique claims • Dependent severities • Intensity: • Survival functions: • ; • ; • Independent and • Intensities: • Survival functions: 37

38. Sampling Lévy Copulas 3) Compute the common claims: • Draw independent , as many times as dictated by . • Find by solving . • Find , with Common claims Unique claims • Dependent severities • Intensity: • Survival functions: • ; • ; • Independent and • Intensities: • Survival functions: 38

39. A few Lévy Copulas Clayton: Archimedean I: Archimedean II: • In all examples above . • The Clayton and “Archimedean I” have upper-tail dependency. • The “Archimedean II” has both lower- / upper-tail dependencies and allows negative dependency between severities. 39

40. Clayton-Lévy Copula • Survival copula for common jumps: • Time invariant: • Intensity for common jumps: • Frequency correlation: The Clayton copula! For the Clayton-Lévycopula 40

41. 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. Avanziet al, 2011 Tail Integral Tail Integral Tail Integral Logarithm of claims Logarithm of claims Logarithm of claims Tail Integral Tail Integral Tail Integral Logarithm of claims Logarithm of claims Logarithm of claims 41

42. Closing Remark #1 Solvency II on Internal Model Approval: • Senior management shall be able to demonstrate understanding of the internal model and how this fits with their business model. • Senior management shall be able to demonstrate understanding of the limitations of the internal model and that they account of it in their decisions. • The timely calculation of results is essential. (From “Use Test”, Section 3 of the CEIOPS Paper) 42

43. Closing Remark #2 • Bad Governanceinfluences 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. 43

44. 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 44