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

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 K g as 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 U i 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 r i be 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 r i is a vector such that then a homogeneous formula approximation is

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

20 Phase 1: Formula Approximation #2 Some simple choices – r i =0 – r i = C i – r i = c i =C g (U i ) When r i =0 Exact for Normal Risks

21 Phase 1: Formula Approximation #2 When r i = C i formula is essentially first order “Factors “ C i < c i 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 r i = c i 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:  Artzner, P, “Application of Coherent Risk Measures to Capital Requirements in Insurance”, North American Actuarial Journal (3), April  Wang,S.S., Young, V.R., Panjer, H.H., “Axiomatic Characterization of Insurance Prices”, Insurance Mathematics and Economics (21)  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  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  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  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, ,  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  Embrechts, P. Kluppelberg, C., Mikosch, T. “Modelling Extreme Events”, Springer – Verlag, Berlin,  McNeil, A., Saladin, S., “The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions”, XXVII’th International ASTIN Colloquim, pp  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.