Risk Dependency Research: A Progress Report

1. Enterprise Risk Management Symposium Washington DC July 30, 2003 Risk Dependency Research:A Progress Report B. John Manistre FSA, FCIA, MAAA

2. Agenda • Nature of the project • Tool Development: • Risk Measures • Special Results for Normal Risks • Extreme Value Theory • Copulas • Formula Approximations • Toward Real Application • Literature Survey

3. Nature of the Project • Response to SoA’s Request for Proposal on “RBC Covariance” • Broad Mandate: “determine the covariance and correlation among various insurance and non-insurance risks generally, particularly in the tail”. • Phase 1: Theoretical Framework/Literature Search • Phase 2: Data Collection/Analysis - the practical element • Project organized at University of Waterloo • J Manistre (Aegon USA), H Panjer(U of W) & graduate students J Rodriguez, V Vecchione

4. Phase 1: Theoretical Framework • Tools: • Risk Measures • Extreme Value Theory • Copulas • Formula Approximations to Risk Measures • New results • Formula Approximations suggest measures of “tail covariance and correlation”

5. Phase 1: Risk Measures • Project focusing on risk measures defined by an increasing distortion function • For a random variable X risk measure is given by where • Capital is usually taken to be the excess of the risk measure over the mean

6. Phase 1: Risk Measures- Examples • Project does not take a position on which risk measure is best • Planning to work with the following: • Value at Risk • Wang Transform • Block Maximum • Conditional Tail Expectation

7. Phase 1: Risk Measures • For any Normal Risk X, • Risk measure is mean plus a multiple of the std deviation • Can use Kgas a tool to understand the risk measure

8. Phase 1: Risk Measures

9. Phase 1: Risk Measures - Aggregating Normal Risks • Suppose all risks normal and • Then • For any g conclude • This is “An exact solution to an approximate problem”.

10. Phase 1:Extreme Value Theory • EVT applies when distribution of scaled maxima converge to a member of the three parameter EVT family • Works for most ‘standard’ distributions e.g. normal, lognormal, gamma, pareto etc. • Key Result is the “Peaks Over Thresholds” approximation • When EVT applies excess losses over a suitably high threshold have an approximate generalized pareto distribution • Suggests that a generalized pareto distribution should be a reasonable model for the tail of a wide range of risks

11. Phase 1:Copulas • A tool for modeling the dependency structure for a set of risks with known marginal distributions • Technically a probability distribution on the unit n-cube • Large academic literature • Some sophisticated applications in P&C reinsurance • Project is concentrating on • t- copulas • Gumbel copulas • Clayton copulas

12. Phase 1:Copulas

13. Phase 1:Copulas

14. Phase 1:Copulas

15. Phase 1:Copulas

16. Phase 1: Formula Approximations • “Simple” Investment Problem. Let • Fix the joint distribution of the Ui and consider • Capital function is homogeneous of degree 1 in the exposure variables • Choose a target mix of risks • Put

17. Phase 1: Formula Approximations • Theoretical Result: The first two derivatives are given by • Some challenges in using these results to estimate derivatives. Second derivatives harder to estimate. • Some risk measures easier to work with than others. • Project team is working with a number of approaches.

18. Phase 1: Formula Approximations • Let ribe a vector such that then the homogeneous formula approximation agrees with the capital function and its first two derivatives at the target risk mix . • If riis a vector such that then a homogeneous formula approximation is

19. Phase 1: Formula Approximation #1 • When ri =0 • Suggests definition of “tail correlation”.

20. Phase 1: Formula Approximation #2 • Some simple choices • ri =0 • ri = Ci • ri = ci=Cg (Ui) • When ri =0 • Exact for Normal Risks

21. Phase 1: Formula Approximation #2 • When ri = Ci formula is essentially first order • “Factors “ Ci < ci already reflect diversification. • Suggests many existing capital formulas are as good (or bad) as first order Taylor Expansions.

22. Phase 1: Formula Approximation #3 • When ri = ci we get • Undiversified capital less an adjustment determined by “inverse correlation”

23. Phase 1: Formula Approximations • Practical work so far suggests • is a more robust approximation. In particular, when the risks are normal • Other homogeneous approximations are possible.

24. Phase 1: Numerical Example: Inputs • Three Pareto Variates combined with t-copula

25. Phase 1: Numerical Example: Results

26. Phase 2: Real Application • Phase 2 not yet begun • Will not be totally objective • Process: • Develop high level models for individual risks • e.g. model C-1 losses with a pareto dist’n. • Assume a copula consistent with “expert” opinion • Adopt a measure of “tail correlation” and calculate • Make subjective adjustments to final results as nec.

27. Literature Survey: Risk Measures • Artzner, P., Delbaen, F., “Thinking Coherently”, Eber, J-M., Heath, D., “Thinking Coherently”, RISK (10), November: 68-71. • Artzner, P, “Application of Coherent Risk Measures to Capital Requirements in Insurance”, North American Actuarial Journal (3), April 1999. • Wang,S.S., Young, V.R. , Panjer, H.H., “Axiomatic Characterization of Insurance Prices”, Insurance Mathematics and Economics (21) 171-183. • Acerbi, C., Tasche, D., “On the Coherence of Expected Shortfall”, Preprint, 2001.

28. Literature Survey:Measures and Models of Dependence (1) • Frees, E.W., Valdez,E.A., “Understanding Relationships Using Copulas”, North American Actuarial Journal (2) 1998, pp 1-25. • Embrechts, P., NcNeil, A., Straumann, D., “Correlation and Dependence in Risk Mangement: Properties and Pitfalls”, Preprint 1999 • Embrechts, P., Lindskog, F., McNeil, A., “Modelling Dependence with Copulas and Applications to Risk Management”, Preprint 2001. • McNeil, A., Rudiger, F., “Modelling Dependent Defaults”, Preprint 2001.

29. Literature Survey:Measures and Models of Dependence (2) • Lindskog, F., McNeil, A., “Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling”, Preprint 2001. • Joe, H, 1997 “Multivariate Models and Dependence”, Chapman-Hall, London • Coles, S., Heffernan, J., Tawn, J. “Dependence Measures for Extreme Value Analysis”, Extremes 2:4, 339-365, 1999. • Ebnoether, S., McNeil, A., Vanini, P., Antolinex-Fehr, P., “Modelling Operational Risk”, Preprint 2001.

30. Literature Survey:Extreme Value Theory • King, J.L., 2001 “Operational Risk”, John Wiley & Sons UK. • McNeil,A., “Extreme Value Theory for Risk Managers”, Preprint 1999. • Embrechts, P. Kluppelberg, C., Mikosch, T. “Modelling Extreme Events”, Springer – Verlag, Berlin, 1997. • McNeil, A., Saladin, S., “The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions”, XXVII’th International ASTIN Colloquim, pp 22-43. • McNeil, A., “On Extremes and Crashes”, RISK, January 1998, London: Risk Publications.

31. Literature Survey:Formula Approximation • Tasche, D.,”Risk Contributions and Performance Measurement”, Preprint 2000.