1. Calibrating Stochastic Models for DFA John M. Mulvey - Princeton University François Morin - Tillinghast - Towers Perrin Bill Pauling - Towers Perrin

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

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 TreasuryYield Curve General PriceInflation Currencies Real Yields Expected Inflation Wage Inflation Fixed Income Returns Dividend Yields Stock Dividend Growth Rate Stock Returns Other Asset Classes

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

6. Overview:Stochastic differential equations • Stochastic differential equations generate time series for each variable: drt = f1(ru - rt)dt + f2(rt, pt,…)dt + f3(rt)dZ1 • 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 Random Element Variable Links

7. 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 Overview:Assumptions and Calibration

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. Bill Pauling: Step 1: Analyze Historical Data

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

19. Step 4: Review Model Output

20. 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 • Surplust = market value (assetst - liabilitiest) • 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 (ALEFsm)

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