1. Financial Modeling of Extreme Events Thomas Weidman CAS Spring Meeting May 19, 2003

2. Financial Modeling of Extreme Events  defining and modeling extreme events – insured vs. total financial impact  financial event modeling  correlated events: insured + financial  case study: capital management

3. Defining Extreme Events  Miami Hurricane  San Francisco EQ  September 11, 2001

4. Defining Extreme Events

5.  SARS  West Nile Virus  Spanish Flu

6. Defining Extreme Events  SARS virus – first outbreak, China Nov 2002  West Nile Virus – first cases in western world 1999  Influenza – first description from 412 B.C.

7. Defining Extreme Events  Asbestos  Tobacco  Shareholders’ Class Actions

8. Defining Extreme Events  Asbestos: $200 billion cost/$100 billion insured  Tobacco: $246 billion settlement with state governments  Tort Costs: $205 billion/$146 billion insured in 2001, a 14% increase over 2000 [source: US Tort Costs-2002 Update, Tillinghast]

9. Defining Extreme Events  Stock Market  Credit Markets

10. Defining Extreme Events Stock Market Returns:  (65)% in 1929-33  (37)% in 1973-4  (38)% in 2000-3 Bond Market Returns:  (8)% in 1999  (7)% in 1994  2 worst annual returns in past 100 years

11. Defining Extreme Events  Pricing Inadequacy  Reserving Inadequacy

12. Defining Extreme Events  Pricing Inadequacy AY loss ratios 10 points higher than CY loss ratios from 1997 through 2000  Reserving Inadequacy $48 to $92 billion at December 2001 excl asbestos and environmental (ISO)

13. Defining Extreme Events  Rogue Trader  Rogue Underwriter  Rogue Agent/Broker

14. Defining Extreme Events  Operational Risk: Risk of direct and indirect loss resulting from failed or inadequate process, systems, or people and from external events  Difficult to quantify, see Basel accords for treatment by banks

15. Summary of Risk Types and Models Risk Type:  Catastrophe  Non-catastrophe  Reserves  Market  Credit  Operational Risk Model:  AIR, RMS, EQE  Exposure x freq x sev  Reserve ranges  VaR models  Default models  ?? Basel II?

16. Quantifying Extreme Events  Historical data  Empirical distributions  Realistic Disaster Scenarios  Models  Fitted probability distributions  Extreme value theory

17. Extreme Value Theory  Based on work describing the extreme behavior of random processes  Extrapolate the tail of a distribution from underlying data  Distributions to fit tails: –Generalized Pareto Distribution (GPD) –Generalized Extreme Value (GEV) Extrapolate the tail of a distribution from underlying data  Provides a rigorous framework to make judgments on the possible tail

18. Extreme Value Theory  GEV family of distributions: M n = Max{x 1,x 2,x 3,….x n } for n sufficiently large  “What is the maximum loss to be expected in one year?”

19. Extreme Value Theory  Generalized Pareto Distribution (GPD) fits tails of distributions above a threshold Pr (Y>y+u|y>u) for large u  “What is the expected loss to an excess layer?”

20. Extreme Value Theory Resources: The Management of Losses Arising from Extreme Events, GIRO 2002 Kotz and Nadarajah, Extreme Value Distributions Coles, An Introduction to Statistical Modeling of Extreme Events Embrechts, etal., Modeling Extremal Events

21. Modeling Financial Events: VAR  VAR is a method of assessing market risk that uses standard statistical techniques routinely used in other technical fields.  VAR is the maximum loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger.  A bank might have a daily VAR of its trading portfolio of $35 million at the 99% confidence level.

22. Modeling Financial Events: Credit Risk Credit Risk Models  Default rates  Loss Given Default (LGD)  Migration matrices

23. Modeling Financial Events: Credit Risk Default Rates = Frequency of loss = Mortality Quantitative Models for Credit Assessment 1. Identify characteristics that differentiate defaulting firms (e.g., Altman 1968); credit scoring models 2. Use credit market prices to estimate default rates 3. Structural models – use equity option pricing techniques (both equity and debt are options on the value of a firm’s assets)

24. Modeling Financial Events: Credit Risk Loss Given Default = Severity  Many models assume a constant loss given default  Dependent on both exposure volatility and recovery rate volatility  Correlated with default rates?

25. Modeling Financial Events: Credit Risk Credit Migration Matrices  Historical changes in credit rating of obligors  ‘loss triangles’ for credit ratings  Use S&P or Moody’s data  Useful for portfolio risk assessment, pricing credit derivatives, capital requirements  Dependent on current and future economic conditions ( recession vs. expansion)

26. Summary of Risk Types and Models Risk Type:  Catastrophe  Non-catastrophe  Reserves  Market  Credit  Operational Risk Model:  AIR, RMS, EQE  Exposure x freq x sev  Reserve ranges  VaR models  Default models  ?? Basel II?

27. Capital Management  Market Share of Industry Loss  Probable Maximum Loss (PML)/Aggregate Exposure  Risk of Ruin Approach: Pr (insolvency) < p over time period t where p is small, e.g.,.01 or.001

28. Capital Management - Issues  Consistent definition across all risk types  Correlations across risks  Allocation/attribution of capital to product  Accounting framework: GAAP vs. Fair Value  Matching capital to management responsibilities, e.g., assets vs. liabilities

29. Correlated (Extreme) Events  Global warming – storms – viruses  Lawyers’ fees from tobacco/asbestos wins  Stock markets – D&O/E&O claims  Credit - Equity prices  Pricing – Reserving (e.g., B-F methods)  Catastrophes – Demand Surge – Reinsurance Recoverable

30. Correlated (Extreme) Events Exposure:catNon- cat reservesmarketcreditOps risk Property X X x x x ? Casualty x X X x X ? Surety x X x x X ? Inv Assets x x x X X ?

31. Correlated (Extreme) Events Cas cat Cas Non-cat Cas reserves Cas market Cas credit Cas ops Prop-cat Low ? Prop- non-cat LowMedLow ? Prop reserves Low MedLow ? Prop market Low HighMed ? Prop- credit Low MedHigh ?

32. Correlated (Extreme) Events Generally impossible to model joint distributions of risks (unless multivariate normal) Therefore:  Estimate distributions for each risk type  Combine distributions into a joint distribution using ‘copulas’

33. Correlated (Extreme) Events Copulas:  Multivariate functions that combine marginal distributions into a joint distribution  Using a normal copula leads to a simpler approach for Monte Carlo simulation of correlated variables  CAS papers by Wang (1998) and Meyers (1999)

34. Financial Modeling of Extreme Events  Past experience lacks credibility Current state of the art:  Sophisticated risk models across all types of risk  Integration/Correlation of risk models important to management, rating agencies and regulators  Major role for actuaries