A Comparison of Actuarial Financial Scenario Generators: CAS/SOA vs. AAA RBC C3 Kevin Ahlgrim, ASA, PhD, Illinois State University Steve D’Arcy, FCAS, PhD, University of Illinois Rick Gorvett, FCAS, ARM, FRM, PhD, University of Illinois 14th AFIR Colloquium Boston November 2004

Outline of Presentation Motivation for Financial Scenario Generators Motivation for Financial Scenario Generators Description of economic variables Description of economic variables Structure of each model Structure of each model Comparison of output Comparison of output Conclusion Conclusion

Motivation Provide public access model for use in Provide public access model for use in DFADFA RegulatoryRegulatory Rating agencyRating agency Internal management testsInternal management tests Conduct literature review Conduct literature review From finance, economics, and actuarial scienceFrom finance, economics, and actuarial science Develop financial scenario generator model Develop financial scenario generator model Generate scenarios over a 50 year time horizonGenerate scenarios over a 50 year time horizon Facilitate use of model Facilitate use of model

Relationships Among Modeled Economic Series InflationReal Interest Rates Real EstateUnemploymentNominal Interest Lg. Stock ReturnsSm. Stock ReturnsStock Dividends

Inflation (q) Modeled as an Ornstein-Uhlenbeck process Modeled as an Ornstein-Uhlenbeck process One-factor, mean-revertingOne-factor, mean-reverting dq t =  q ( q – q t ) dt +  q dB q

Real Interest Rates (r) Two-factor Vasicek term structure model Two-factor Vasicek term structure model Short-term rate (r) and long-term mean (l) are both stochastic variables Short-term rate (r) and long-term mean (l) are both stochastic variables dr t =  r (l t – r t ) dt +  r dB r dl t =  l ( l – l t ) dt +  l dB l

Nominal Interest Rates Combines inflation and real interest rates Combines inflation and real interest rates i = {(1+q) x (1+r)} - 1 where i = nominal interest rate q = inflation r = real interest rate Restriction against negative interest rates Restriction against negative interest rates

Motivation Provide guidance for setting Risk-Based Capital (RBC) requirements for variable products with guarantees Focus is on Interest rate risk Equity risk Recommend use of models Also provide 10,000 Pre-packaged scenarios Available at:

Relationships Among Modeled Economic Series 3-month U.S. Treasury yields 10-year U.S. Treasury yields U.S. Long Term Corporate Bonds Money Market 7-year U.S. Treasury yields Diversified U.S. Equity Diversified International Equity U.S. Intermediate Term Government Bonds Diversified Fixed Income Diversified Balanced Intermediate Risk Equity Aggressive or Specialized Equity Risk-free rate r = 5.5% (effective) for all markets, roughly equal to the average 6-month U.S. Treasury yield over the past 50 years

Nominal Interest Rates Three Processes For Three Time Scales Long term, 10-Year Treasury Yield Long term, 10-Year Treasury Yield Short term, 3-Month Treasury Yield Short term, 3-Month Treasury Yield Medium-term, 7 year Treasury Yield Medium-term, 7 year Treasury Yield

Properties of the Interest Rate Model Where Where Z1, Z2, Z3 are normal distributions with mean 0; Z1, Z2, Z3 are normal distributions with mean 0; α, Φ are mean-reversion strengths; α, Φ are mean-reversion strengths; λ, τ, ξt are regression parameters. λ, τ, ξt are regression parameters. Lognormal distribution at time t Lognormal distribution at time t Avoid negative nominal interest rates. Avoid negative nominal interest rates. Make the Kurtosis positive all the time Make the Kurtosis positive all the time Funnel of Doubt Graphs shift to the upper side. Funnel of Doubt Graphs shift to the upper side. Variance increases faster as t increases Variance increases faster as t increases

CAS/SOAAAA RBC C3 t i f (t, i) i t 00

Funnel of Doubt Graphs 3 Month Nominal Interest Rates (U. S. Treasury Bills)

Histogram of 3 Month Nominal Interest Rates Model Values and Actual Data (01/ )

Funnel of Doubt Graphs 10 Year Nominal Interest Rates (U. S. Treasury Bonds)

Histogram of 10 Year Nominal Interest Rates Model Values and Actual Data (04/53-05/04)

Equity Returns Both models use Regime Switching Lognormal Model with monthly data and 2 regimes (RSLN2) Both models use Regime Switching Lognormal Model with monthly data and 2 regimes (RSLN2) Model equity returns as an excess return (x t ) over the nominal interest rate Model equity returns as an excess return (x t ) over the nominal interest rate s t = i t + x t Two Regimes Two Regimes 1.High return, low volatility regime 2.Low return, high volatility regime Six parameters μ 1, σ 1; μ 2, σ 2; P 12, P 21 Six parameters μ 1, σ 1; μ 2, σ 2; P 12, P 21 Within Volatility Regimes μ 1, σ 1; μ 2, σ 2 Within Volatility Regimes μ 1, σ 1; μ 2, σ 2 Transition Matrix Transition Matrix

Parameter Differences Data Sources for Maximum likelihood estimates Parameter Differences (AAA Pre-packaged scenarios) Parameter Differences (AAA Pre-packaged scenarios)

Funnel of Doubt Graphs Large Stock Return (US Equity)

Histogram of Large Stock Return Model Values and Actual Data ( )

Funnel of Doubt Graphs Small Stock Return (Intermediate Risk Equity)

Histogram of Small Stock Return Model Values and Actual Data ( )

Conclusion Financial models are assuming greater importance for actuaries Financial models are assuming greater importance for actuaries Actuaries need to understand how to apply these models Actuaries need to understand how to apply these models CAS/SOA model generates greater variance CAS/SOA model generates greater variance AAA RBC C3 model provides returns on more types of investments AAA RBC C3 model provides returns on more types of investments Try out these models Try out these models Suggest additions or improvements Suggest additions or improvements Questions? Questions?