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

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

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

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

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

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

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

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

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

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

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

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

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

Step 1: Analyze Historical Data Bill Pauling:

Step 2: Set Targets

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

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

Step 4: Review Model Output

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

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

Asset Liability Efficient Frontier (ALEF sm )

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