Structural Dependence and Stochastic Processes

1. Structural Dependence and Stochastic Processes Don Mango American Re-Insurance 2001 CAS DFA Seminar

2. Agenda • Just Say No to Correlation • Structural Dependence in Asset and Economic Modeling • Structural Dependence in Liability Modeling

3. Just Say No to Correlation

4. Just Say No to Correlation • Correlation has taken on something of a life of its own • It’s easy to measure • You can use Excel, or @Risk • People think they know what it means, and have an intuitive sense of ranges

5. Just Say No to Correlation • Paul Embrechts, Shaun Wang, and others tell us: • Correlation is simply one measure of Dependence, a more general concept • There are many other such measures • From a Stochastic modeling standpoint, simulating using Correlation surrenders too much control

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

7. 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

8. 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?

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

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

12. 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

13. 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 underwriting cycles? • What about the magnitude of year-over-year changes?

14. Example DFA Model - PROBLEMS • Bottom line: These scenarios (e.g., pricing cycle) could happen… • …but if they do, it’s “random” • …as in we don’t control in what manner and how often they happen, and in conjunction with what other events

15. 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.

16. Band Aid? • Problem: Resulting scenarios may not be internally consistent • Possible Improvement: a MEGA-CORRELATION matrix (Yield curves and Liabilities)... • …but that just treats the symptoms !! • Still have no guarantee of internal consistency

17. The Real Problem • No Overarching Structural Framework • “All Method, No Model” - LJH • We need a structural model of known relationships and dependencies… • …that has volatility and randomness, but we control how and where it enters… • … and the required internal consistency will be built-in (within constraints)

18. 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

19. Structural Dependence in Asset and Economic Modeling

20. 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, via randomly generated increments, calibrated to the starting point

21. Stochastic Difference Equations • Generate plausible future scenarios consisting of time series for each of many simulated variables • Preserve internal consistency within each scenario • Introduce volatility in a controlled manner

22. Stochastic Difference Equations • Begin with Driver Variables • “Independent”, Top of the food chain • Generate the simulated time series for these Drivers • Can either generate absolute level or incremental changes, but we need the increments (“D”) • Example: CPI and Medical CPI

23. Stochastic Difference Equations • The Next “level” of variables have defined functional relationships to the Drivers, plus error terms • “Volatility” or “Noise” • D GDP =f(DCPI, DMed CPI) + sdW • dW = “Wiener” term = Standard Normal • How we introduce volatility • s = scaling factor for that volatility

24. Stochastic Difference Equations • Each successive level of variables builds upon prior variables up the chain in a CASCADE…

25. Simple Economic Model Cascade CPI Medical CPI Real GDP Growth Unemployment Equity Index Yield Curve

26. Other Process Modeling Terms • Shocks = large incremental changes • Mean Reversion = process tends to correct back toward long term avg • Reversion strength = how quickly it reverts back • Calibration = tuning the parameters • See Madsen and Berger, 1999 DFA Call Paper

27. Structural Dependence in Liability Modeling

28. A Whole New Framework • Stochastic process modeling is about structure and control • Building in structural relationships we believe exist • Introducing volatility in the increments between periods • Controlling the resulting simulated values through parameters and calibration • Adds another dimension to simulation

29. Insurance Market Model • Following the hierarchical approach of capital markets models • Generate market time series for Product Costs and Price Levels by LOB • Not the same thing !! • Soft market: Costs > Price Levels (“under-pricing”)

30. Individual Company • Individual company product costs are partly a function of the Market Cost level and partly a function of their own book • Undiversifiable and Diversifiable • Individual company price levels behave similarly • Your price is some deviation above or below market • Like the tide

31. Insurance Market Model • What we are evaluating is participation in insurance markets • Market Cost shocks to product • Undiversifiable • Market prices will respond, but over how long? (Reversion strength) • How quickly does company price level respond to market price changes?

32. Market Cost Shock • Examples of a Market Cost shock • Asbestos • Pollution • Construction Defect • Benefit level change in WC • Hurricane Andrew

33. Insurance Market Model • Company-specific Cost shocks to product • Diversifiable • Market Prices will not respond • Company price level may respond, but will be out of step with market • Example: • North Carolina chicken factory that burned down with the doors locked

34. Insurance Market Model • Missing Links • Demand curves by LOB • Strength and nature of structural dependency relationships • This will require fundamental rewrites of our DFA models • Ultimately superior because it supports the scientific method • Requires hypothesis and testing

35. InsureMetricsTM • This is the development of InsureMetricsTM • The insurance kin to econometrics