A Comprehensive System for Selecting and Evaluating DFA Model Parameters Chris Madsen, ASA, CFA, MAAA American Re-Insurance Company CAS DFA Forum, Chicago July 19th-20th, 1999 Chris Madsen, ASA, CFA, MAAA American Re-Insurance Company CAS DFA Forum, Chicago July 19th-20th, 1999

Discussion Overview Overview of a Integrated Risk Management System Focus on an Economic Model Calibration examples Optimization issues Conclusions Overview of a Integrated Risk Management System Focus on an Economic Model Calibration examples Optimization issues Conclusions

Model Structure

M2 Growth V2 Growth Inflation* GDP Growth* Interest Rates* (Forward, Spot, Yield) Equity Earnings Yield Equity Earnings Growth Asset Model * Currency Link (not currently modeled) Economic Model

SimulationDefining the r/i structure Modeling the portfolio Gross loss Net loss Ceded loss Retained premiums Ceded premiums Loss Simulation with DFA Loss data Premiums Customer requirements Limits Prices

What Makes A Good Scenario Generator? Logically defensible  Economic theory  Historical data Risk across time Logically defensible  Economic theory  Historical data Risk across time

Plausible Paths No negative interest rates Historical data does not necessarily equate expected value of statistics (trend sensitive) - rather, build distributions of statistic and ensure history is well-represented. No negative interest rates Historical data does not necessarily equate expected value of statistics (trend sensitive) - rather, build distributions of statistic and ensure history is well-represented.

Types of Models Strategic  Long-term planning  resource allocation (capital, business mix, asset mix, retro covers) Pricing  Risk-neutral (replication)  Does often generate unreasonable simulations (all returns = risk free rate) Strategic  Long-term planning  resource allocation (capital, business mix, asset mix, retro covers) Pricing  Risk-neutral (replication)  Does often generate unreasonable simulations (all returns = risk free rate)

Economic Model Long interest rates  dl t = a l ( l  - l t ) dt + l t  l dZ l Short interest rates  dr t = a r ( r  - r t ) dt + r t  r dZ r Long interest rates  dl t = a l ( l  - l t ) dt + l t  l dZ l Short interest rates  dr t = a r ( r  - r t ) dt + r t  r dZ r

Setting Targets Basic statistics (arithmetic mean, compound mean, st.dev., percentiles, min. & max., serial) Plausibility criteria (Becker - yield curve characteristics) Basic statistics (arithmetic mean, compound mean, st.dev., percentiles, min. & max., serial) Plausibility criteria (Becker - yield curve characteristics)

Target Example History Simulation

Calibration Example #1 Regressing on ‘74-’98, we get {A, B, C}={0.015, 1.3, } R 2 =58% 90% parameter confidence D=1.06 (ln(residual/mean)) Regressing on ‘74-’98, we get {A, B, C}={0.015, 1.3, } R 2 =58% 90% parameter confidence D=1.06 (ln(residual/mean))

Calibration Example #1 {A, B, C, D} = {0.75, 0.5, -0.04, 1.05}  The two are quite similar though at first glance…  Weight shifted from 30 Year Rate to Inflation  Mean reversion up  Volatility down slightly {A, B, C, D} = {0.75, 0.5, -0.04, 1.05}  The two are quite similar though at first glance…  Weight shifted from 30 Year Rate to Inflation  Mean reversion up  Volatility down slightly

Calibration Example #2

Conclusions Regression is a good starting point but may miss key statistics Key statistics may miss fundamental relationships Optimization is a valuable parameterization tool and enables us to monitor key statistics as well as fundamental relationships Regression is a good starting point but may miss key statistics Key statistics may miss fundamental relationships Optimization is a valuable parameterization tool and enables us to monitor key statistics as well as fundamental relationships