1. CAS 1999 Dynamic Financial Analysis Seminar Chicago, Illinois July 19, 1999 Calibrating Stochastic Models for DFA John M. Mulvey - Princeton University François Morin - Tillinghast - Towers Perrin Bill Pauling - Towers Perrin

2. Basic financial modeling architecture Economic Scenario Generation Optimization Objectives Vs. Risks Results Regulatory GAAP Tax Economic Cash Financial Model Asset Performance Liability Performance Global CAP:Link TAS: P/C Opt:Link

3. Towers Perrin’s Global CAP:Link Used by institutional investors around the world Calibrated for 10 currencies Widely recognized and published model Named honoree for Edelman Award

4. Global CAP:Link : General cascade structure Currencies Real Yields Stock Dividend Growth Rate Dividend Yields Fixed Income Returns Stock Returns Other Asset Classes General Price Inflation Treasury Yield Curve Expected Inflation Wage Inflation

5. Global CAP:Link : Multi-currency links Japan Europe U.S.A Other Countries Key links are: Currencies Dividend yields Bond yields

6. Overview: Stochastic differential equations Stochastic differential equations generate time series for each variable: dr t = f 1 (r u - r t )dt + f 2 (r t, p t,…)dt + f 3 (r t )dZ 1 Initial conditions set initial values for variables to current levels set ‘normative’ values to long-term expected values Normative conditions set initial and normative values equal to each other Mean Reversion Variable Links Random Element

7. Overview: Assumptions and Calibration Assumptions refer to the mean value of key economic variables: Bond yields Inflation Dividend yields Equity risk premium Well researched in a multi- period time frame We offer ‘Basic Expectations’ approach, but will implement other approaches Calibration refers to more subtle aspects of the models behavior Degree of mean reversion Probability of ‘extreme’ values Key linkages between variables (average correlations, etc.) The calibration comes as a part of the implementation of the model

8. Calibration Problem Stochastic differential equations (55 equations, 220 parameters) Non-convexity Many targets must be satisfied simultaneously

9. Calibration Independent parameter estimation (regression) produces inconsistent and unrealistic results Past is not a central estimate of the future Calibration follows a different approach Define behavioral characteristics of scenarios  emergent properties Set parameters in such a way as to produce required characteristics

10. Calibration Solution - Theory Generalized Method of Moments Simulated Moments Estimator Integrated Parameter Estimation

11. Calibration Solution - Integrated Parameter Estimation Extends simulated moment estimation Target vector can include a variety of descriptive statistics standard deviation correlation serial correlation distribution percentages other range estimates (inter-quartile ranges) frequency of inversion probability of extreme values Parameters are bounded

12. Calibration Solution - Calibration Tool Interface to Global CAP:Link Objective function target ranges target weights Non-convex optimizer

13. Example: Calibrating Scenario Generator with Both Assets and Liabilities Calibrate Global CAP:Link to produce liability growth as well as asset returns Determine target statistics & importance weights Use non-convex optimization model - based on Integrated Parameter Estimation approach Solve for best set of parameters

14. Step 1: Analyze Historical Data Bill Pauling:

15. Step 2: Set Targets

16. Step 3: Use Calibration Tool Enter target types, ranges and importance weights 100 scenarios per iteration Calibrate to normative conditions

17. Step 4: Review Model Output Use optimal parameters to generate 500 scenario run Critically examine full set of Global CAP:Link results Adjust targets and parameter ranges, if necessary

18. Step 4: Review Model Output

21. Example: Linking Assets and Liabilities for DFA Insurance line of automobile policies liabilities driven by both Medical CPI and Legal Services CPI Reward measure: ending surplus Risk measure: volatility of ending surplus Use OPT:Link to produce asset/liability efficient frontier

22. Surplus Optimization Framework Surplus t = market value (assets t - liabilities t ) Grow economic surplus over planning period t = {1, 2, …, T} maximize risk-adjusted profit for entire company analyze over representative set of scenarios Internal and external constraints on asset mix GAAP income other financial statement measures

23. Asset Liability Efficient Frontier (ALEF sm )

24. Conclusions Assets and liabilities should be calibrated together Integrated parameter estimation can be used with non- convex optimization techniques to calibrate the model