Regression Models and Loss Reserving: Decision Points in Model Design

1. Regression Models andLoss Reserving:Decision Points in Model Design Casualty Loss Reserve Seminar September 10-12, 2001 Dave Clark American Re-Insurance

2. Preliminary Question Why create a regression model? • Smooth out development pattern • Impose “objectivity” on reserving process • Extrapolate a tail factor • Estimate variability around carried reserve • Razzle-Dazzle

3. Components of Reserve Model 1) Expected development pattern 2) Stochastic element (random variation from expected) 3) Measure of variability of estimators (“Parameter Variance” )

4. Decision Points 1) Expected Loss Development Pattern: Discrete points or parameterized curve 2) Dependence structure: Additive or Multiplicative 3) Distribution of the random element 4) Include Paid, Case Incurred, or both 5) Parameter Variance: Classical or Bayesian

5. #1: Expected Pattern Do we want to impose a pattern on the expected shape of the loss development?

6. #1: Expected Pattern Yes: Creates a smooth curve Fewer parameters to estimate Allows us to extrapolate a tail No: Hard to find a pattern that fits the whole curve Parameter estimation is harder

7. #1: Expected Pattern Recommendation: Assume that the form of the cumulative reporting pattern follows a known CDF. • Form of CDF only - we are not yet introducing a statistical model • We just want a curve that smoothly moves from 0% to 100%

8. #1: Expected Pattern • For Loglogistic CDF, Sherman’s “inverse power” curve results: • For immature ages, use infinitely decomposable theory to improve fit (Philbrick, Robbin & Homer)

9. #2: Dependency Structure Is the amount reported in one period a function of the earlier periods? Multiplicative Model: Chain Ladder (Hayne, Heyer, Mack, Murphy) Additive Model: Bornhuetter-Ferguson (Halliwell, England & Verrall)

10. #2: Dependency Structure Answer should be based on • analysis of residuals in the model (see Venter) • our understanding of the loss-generating phenomenon Remember… A triangle with 10 accident years has a total of 55 dollar “cells”, or 45 link ratios.

11. #2: Dependency Structure Recommendation: Use Additive Model • Requires more information: measure of exposure, such as on-level premium • Can also produce estimate of distribution of prospective loss ratio

12. #2: Dependency Structure Example of Additive Model: Incremental loss for accident year i, at time t: Ci,t = Premiumi * ELR * [ F(t+1|) – F(t|) ]

13. #3: Random Element How does the variance of a predicted loss amount relate to its expected value? For GLM Variance = constant Normal Variance = constant*E[loss] Poisson Variance = constant*E[loss]² Gamma (constant CV) (c.f., England & Verrall in 2001 PCAS)

14. #3: Random Element Recommendation: Assume that the ratio of variance to mean is constant for predicted points (“over-dispersed Poisson” model).

15. #3: Random Element Selection of a relationship between variance and mean does not fully determine the distributional form. If we assume that the random element has a gamma distribution, with constant “scale” parameter, then the total reserve will also have a gamma distribution.

16. #3: Random Element Another advantage of a Gamma distribution model:

17. #4: Paid or Incurred What data do we use in the model? Are we estimating total unpaid, or total “bulk”? Good proposal: Use BOTH simultaneously! (see Halliwell, 1997 Conjoint Prediction of Paid and Incurred Losses)

18. #4: Paid or Incurred Paid and Incurred losses should reach the same ultimate loss dollars: Paidi,t = Premiumi * ELR * [ F(t+1|P) – F(t|P) ] same Incdi,t = Premiumi * ELR * [ F(t+1|I) – F(t|I) ]

19. #5: Parameter Variance What do we mean by “Parameter Variance”? Classical model: Var(y - ŷ) = Var(y) + Var(ŷ) Total Variance Process Variance “Parameter Variance” “Parameter Variance” means the uncertainty in the estimate ofŷ, due to few number of observations in the historical data.

20. #5: Parameter Variance What do we mean by “Parameter Variance”? Bayesian model: Var(y) = E[Var(y| )] + Var(E[y| ]) Total Variance Process Variance Parameter Variance “Parameter Variance” means our level of uncertainty about . It can incorporate information other than the observed data points.

21. #5: Parameter Variance Some comments: • Under either Classical or Bayesian frameworks, Parameter Variance is very significant • see Kreps, 1997 PCAS • Bayesian theory is attractive • Allows use of information other than just the triangle • Classical models are more readily available

22. #5: Parameter Variance Recommendation: Start with Classical approach. Using “over-dispersed Poisson” model, with loglikelihood function: Expected loss  is a function of the ELR and the parameter vector .

23. #5: Parameter Variance The Covariance Matrix for the parameters is approximated by the inverse of the “Information Matrix” of second derivatives:

24. #5: Parameter Variance Important Notes: • Parameter Variance based on Information Matrix is the Rao-Cramer lower bound. For a small sample size, our true variance may be greater. • We are not including the Parameter Variance associated with the V/M of the random element. That is, s2 is treated as fixed.

25. #5: Parameter Variance A final thought on estimating variance… The main use of stochastic reserving methods is in the provision of estimates of reserve variability, not in the reserve estimates themselves. England & Verrall, 2001 ( Do you agree? )

26. Select Bibliography England & Verrall A Flexible Framework for Stochastic Claims Reserving, PCAS 2001 Kreps, Rodney Parameter Uncertainty in the (Log)Normal Distribution, PCAS 1997 Halliwell, Leigh Loss Prediction by Generalized Least Squares; PCAS 1996 Halliwell, Leigh Conjoint Prediction of Paid and Incurred Losses, CAS Forum Summer 1997

27. Select Bibliography Hayne, Roger An Estimate of Statistical Variation in Development Factor Models, PCAS 1985 Heyer, Daniel A Random Walk Model for Paid Loss Development, Discussion Paper 2001 McCullagh & Nelder Generalized Linear Models 2nd Edition, Chapman & Hall/CRC 1999 Murphy, Daniel Unbiased Loss Development Factors, PCAS 1994

28. Select Bibliography Philbrick, Stephen Reserve Review of a Reinsurance Company, Discussion Paper 1986 Robbin & Homer Analysis of Loss Development Patterns Using Infinitely Decomposable Percent of Ultimate Curves, Discussion Paper 1988 Sherman, Richard Extrapolating, Smoothing, and Interpolating Development Factors, Discussion Paper 1984 Venter, Gary Testing the Assumptions of Age-to-Age Factors, PCAS 1998