Structural Dependence and Stochastic Processes

1. Structural Dependence and Stochastic Processes Don Mango American Re-Insurance 2000 CAS DFA Seminar

2. Question:What Does This Have To Do With “Assessing Balance Sheet Protection Using DFA” ?

3. Answer:Plenty !But you have to wait...

4. Agenda • The Enemy of a Balance Sheet: Dependence • Simulating with Correlation • Structural Dependence in Asset and Economic Modeling • Extreme Events

5. The Enemy of a Balance Sheet: Dependence

6. The Nature of (Re)Insurance • Leveraged contracts • Likely outcome: Small (expected) gain • Unlikely outcome: Big Loss • Hopefully accumulated risk loading absorbs “typical” variability • Diversification benefit • Expected slow growth of Capital

7. The Nature of (Re)Insurance • Leveraged portfolio • In-force Limits are multiples of Capital • Leveraged just like a bank • What kills banks? • Bank runs (don’t see those anymore thanks to FDIC) • Series of capital-destroying defaults • Poor underwriting decisions

8. The Nature of (Re)Insurance • What kills (re)insurers? • Too many negative contract outcomes in too short a time • Depletes Capital more quickly than it can be replenished by underwriting activities

9. The Nature of (Re)Insurance • But what is “too short a time” ? • “Companies don’t go out of business in one year” • The enemy is really Dependence between balance sheet elements

10. Modeling Dependence • We want to make sure we have all the dependencies modeled properly in our DFA models • The most common type of dependence is CORRELATION • Just one of many types of dependence • But it’s our favorite !!

11. Simulating with Correlation

12. Simulating with Correlation • We think we know how to induce correlation between variables in our simulation algorithms • Two major problems: • Correlation is not the same throughout the simulation space • Known dependency relationships may not be maintained

13. Correlation Not Always The Same... • Consider a well-known approach for generating correlated random variables • Using Normal Copulas • Similar to the Iman-Conover algorithm (in @Risk) which uses Normal Copulas to generate rank correlation

14. Normal Copulas • Generate sample from multi-variate Normal with covariance matrix S • Get the CDF value for each point [ these are U(0,1) ] • Invert the U(0,1) points to get target simulated RVs with correlation… • …but what correlation will the target variables have?

15. Problem • Correlation in the tails is near 0 - extreme values are nearly uncorrelated • Is this your intended result? • Example….

17. Known Dependencies Not Maintained • Simple example DFA Model for a company • Liabilities: • 4 LOB: Auto, GL, Property, WC • Assets: • Bonds

18. Example DFA Model • Liabilities: • 4 LOB: Auto, GL, Property, WC • Simulation: correlated uniform (0,1] matrix per time period used to generate the variables • Assets: • Bonds • Simulation: yield curve scenarios

19. Example DFA Model - PROBLEMS • Liabilities: • Getting dependence within a year, but what about serial dependence across years? • Could expand the correlation matrix to be [ # variables x # years ] • But what about cycles? • What about the magnitude of year-over-year changes?

20. Example DFA Model - PROBLEMS • Assets: • Including yield curve variation - good thing • What about linkages with liabilities? • Example: inflation will impact severities and yield curve • Naively-built yield curve simulation may actually reduce variability of overall answer !! • Independent asset values will dampen the variability of net income, surplus, etc.

21. Band Aid? • Problem: Resulting scenarios may not be internally consistent • Possible Improvement: a MEGA-CORRELATION matrix (Yield curves and Liabilities)... • …but still have no guarantee of internal consistency

22. The Real Problem • No Overarching Conceptual Framework • “All Method, No Model” • Need an explanatory, causative, structural model which builds in known relationships and dependencies • Still has volatility, randomness • But the required internal consistency is built in (within constraints)

23. The Real Problem • This represents a significant mindset shift in actuarial modeling for DFA • Moves you away from correlation matrices… • …and towards STOCHASTIC PROCESSES... • …prevalent in asset and economic modeling

24. Structural Dependence in Asset and Economic Modeling

25. Stochastic Difference Equations • Focus is on Processes, Increments, and Paths • Processes: Time series • Increments: changes from one time period to the next • Paths: simulated evolution of the time series, calculating interim values, calibrated to starting point

26. Stochastic Difference Equations • Begin with Driver Variables • Example: Change in Money Supply (M2) and Money Velocity (V2) • Next Level of variables have defined relationship to drivers, plus error terms • Change in GDP = fcn(M2, V2) + sdW • dW = “Wiener” term • N. Wiener of information theory fame • Often a standard normal

27. Stochastic Difference Equations • Each successive level of variables builds upon prior variables in a “cascade” • Initial conditions need to be calibrated to match current state • Crucial point: future values depend on current value • Conditional distribution

28. Stochastic Difference Equations • Advantage: internal consistency • Not relying on the dependence matrix • Example: Yield curves • “Stylized facts” (D. Becker) • Should be Arbitrage-free... • …and consistent with economic scenarios • Raises the complexity bar

29. Economic Model Cascade V2 Growth M2 Growth Inflation GDP Growth Interest Rates (Forward, Spot, Yield) Equity Earnings Yield Equity Earnings Growth Asset Model

30. American Re’s DFA Model

31. Extreme Events

32. Extreme Events • Method of linking impacts • Objects which are a level above the other simulated variables • Directly dictating extreme scenarios • When event “occurs”, override regular simulation of impacted variables • Can still have variability • E.g., event triggers conditional distribution

33. Extreme Events • Example: $50B EQ in Los Angeles • What should happen? • Property loss (Cat model distribution) • WC cat loss (based on conditional dist.) • Bond liquidation loss (P&C insurers trying to convert to cash at the same time - liquidity crunch) • Non-voluntary assessment (if still solvent !)

34. Extreme Events • Could simply pad out your cat distribution for these costs • But doesn’t that defeat the purpose of DFA to a large extent? • Loss of detail • Good functionality to build in regardless • Scenario testing is popular

35. Extreme Value Theory • Presentation here by Embrechts and Patrik • Book by Embrechts et al