Loss Reserving Using Policy-Level Data

1. Loss Reserving Using Policy-Level Data James Guszcza, FCAS, MAAA Jan Lommele, FCAS, MAAA Frank Zizzamia CLRS Las Vegas September, 2004

2. Agenda • Motivations for Reserving at the Policy Level • Outline one possible modeling framework • Sample Results

3. Motivation Why do Reserving at the Policy Level?

4. 2 Basic Motivations • Better reserve estimates for a changing book of business • Do triangles “summarize away” important patterns? • Could adding predictive variables help? • More accurate estimate of reserve variability • Summarized data require more sophisticated models to “recover” heterogeneity. • Is a loss triangle a “sufficient statistic” for ultimate losses & variability?

5. (1) Better Reserve Estimates • Key idea: use predictive variables to supplement loss development patterns • Most reserving approaches analyze summarized loss/claim triangles. • Does not allow the use of covariatesto predict ultimate losses (other than time-indicators). • Actuaries use predictive variables to construct rating plans & underwriting models. • Why not loss reserving too?

6. Why Use Predictive Variables? • Suppose a company’s book of business has been deteriorating for the past few years. • This decline might not be reflected in a summarized loss development triangle. • However: The resulting change in distributions of certain predictive variables might allow us to refine our ultimate loss estimates.

7. Examples of Predictive Variables • Claim detail info • Type of claim • Time between claim and reporting • Policy’s historical loss experience • Information about agent who wrote the policy • Exposure • Premium, # vehicles, # buildings/employees… • Other specifics • Policy age, Business/policyholder age, credit…..

8. More Data Points • Typical reserving projects use claim data summarized to the year/quarter level. • Probably an artifact of the era of pencil-and-paper statistics. • In certain cases important patterns might be “summarized away”. • In the computer age, why restrict ourselves? • More data points  less chance of over-fitting the model.

9. Danger of over-fitting • One well known example • Overdispersed Poisson GLM fit to loss triangle. • Stochastic analog of the chain-ladder • 55 data points • 20 parameters estimated  parameters have high standard error. • How do we know the model will generalize well on future development? • Policy-level data: 1000’s of data points

10. Out-of-Sample Testing • Policy-level dataset has 1,000’s of data points • Rather than 55 data points. • Provides more flexibility for various out-of-sample testing strategies. • Use of holdout samples • Cross-validation • Uses: • Model selection • Model evaluation

11. (2) Reserve Variability • Variability Components • Process risk • Parameter risk • Model specification risk • Predictive error = process + parameter risk • Both quantifiable • What we will focus on • Reserve variability should also consider model risk. • Harder to quantify

12. Reserve Variability • Can policy-level data give us a more accurate view of reserve variability? • Process risk: we are not summarizing away variability in the data. • Parameter risk: more data points should lead to less estimation error. • Prediction variance: brute force “bootstrapping” easily combines Process & Parameter variance. • Leaves us more time to focus on model risk.

13. Disadvantages • Expensive to gather, prepare claim-detail information. • Still more expensive to combine this with policy-level covariates. • More open-ended universe of modeling options (both good and bad). • Requires more analyst time, computer power, and specialist software. • Less interactive than working in Excel.

14. Modeling Approach Sample Model Design

15. Philosophy • Provide an example of how reserving might be done at the policy level • To keep things simple: consider a generalization of the chain-ladder • Just one possible model • Analysis is suggestive rather than definitive • No consideration of superimposed inflation • No consideration of calendar year effects • Model risk not estimated • etc…

16. Notation • Lj = {L12, L24, …, L120} • Losses developed at 12, 24,… months • Developed from policy inception date • PYi = {PY1, PY2, …, PY10 } • Policy Years 1, 2, …, 10 • {Xk}= covariates used to predict losses • Assumption: covariates are measured at or before policy inception

17. Model Design • Build 9 successive GLM models • Regress L24 on L12; L36 on L24 … etc • Each GLM analogous to a link ratio. • The Lj  Lj+1 model is applied to either • Actual values @ j • Predicted values from the Lj-1Ljmodel • Predict Lj+1 using covariates along with Lj.

18. Model Design • Idea: model each policy’s loss development from period Lj to Lj+1 as a function of a linear combination of several covariates. • Policy-level generalization of the chain-ladder idea. • Consider case where there are no covariates

19. Model Design • Over-dispersed Poisson GLM: • Log link function • Variance of Lj+1 is proportional to mean • Treat log(Lj) as the offset term • Allows us to model rate of loss development

20. Using Policy-Level Data • Note: we are using policy-level data. • Therefore the data contains many zeros • Poisson assumption places a point mass at zero • How to handle IBNR • Include dummy variable indicating $0 loss @12 mo • Interact this indicator with other covariates.  The model will allocate a piece of the IBNR to each policy with $0 loss as of 12 months.

21. Sample Results

22. Data • Policy Year 1991 – 1st quarter 2000 • Workers Comp • 1 record per policy, per year • Each record has multiple loss evaluations • @ 12, 24, …,120 months • “Losses @ j months” means: • j months from the policy inception date. • Losses coded as “missing” where appropriate • e.g., PY 1998 losses at 96 months

23. Covariates • Historical LR and claim frequency variables • $0 loss @ 12 month indicator • Credit Score • Log premium • Age of Business • New/renewal indicator • Selected policy year dummy variables • Using a PY dummy variable is analogous to leaving that PY out of a link ratio calculation • Use sparingly

24. Covariates • Interaction terms between covariates and the $0-indicator • Most covariates only used for the 1224 GLM • For other GLMs only use selected PY indicators • These GLMs give very similar results to the chain ladder

25. Results

26. Comments • Policy-Level model produces results very close to chain-ladder. •  is a proper generalization of the chain-ladder • The model covariates are all statistically significant, have parameters of correct sign. • In this case, the covariates seem to have little influence on the predictions. • Might play more of a role in a book where quality of business changes over time.

27. Model Evaluation Treat Recent Diagonals as Holdout 10-fold Cross-Validation

28. Test Model by Holding Out Most Recent 2 Calendar Years

29. Cross-Validation Methodology • Randomly break data into 10 pieces • Fit the 9 GLM models on pieces 1…9 • Apply it to Piece 10 • Therefore Piece 10 is treated as out-of-sample data • Now use pieces 1…8,10 to fit the nine models; apply to piece 9 • Cycle through 8 other cases

30. Cross-Validation Methodology • Fit 90 GLMs in all • 10 cross-validation iterations • Each involving 9 GLMs • Each of the 10 “predicted” pieces will be a 10x10 matrix consisting entirely of out-of-sample predicted values • Can compare actuals to predicteds on upper half of the matrix • Each cell of the triangle is treated as out-of-sample data

31. Cross-Validation Results

32. Reserve Variability Using the Bootstrap to estimate the probability distribution of one’s outstanding loss estimate

33. The Bootstrap • The Statistician Brad Efron proposed a very simple and clever idea for mechanically estimating confidence intervals: The Bootstrap. • The idea is to take multiple resamples of your original dataset. • Compute the statistic of interest on each resample • you thereby estimate the distribution of this statistic!

34. Motivating Example • Suppose we take 1000 draws from the normal(500,100) distribution  • Sample mean ≈ 500 • what we expect • a point estimate of the “true” mean • From theory we know that:

35. Sampling with Replacement • Draw a data point at random from the data set. • Then throw it back in • Draw a second data point. • Then throw it back in… • Keep going until we’ve got 1000 data points. • You might call this a “pseudo” data set. • This is not merely re-sorting the data. • Some of the original data points will appear more than once; others won’t appear at all.

36. Sampling with Replacement • In fact, there is a chance of (1-1/1000)1000≈ 1/e ≈ .368 that any one of the original data points won’t appear at all if we sample with replacement 1000 times.  any data point is included with Prob ≈ .632 • Intuitively, we treat the original sample as the “true population in the sky”. • Each resample simulates the process of taking a sample from the “true” distribution.

37. Resampling • Sample with replacement 1000 data points from the original dataset S • Call this S*1 • Now do this 399 more times! • S*1, S*2,…, S*400 • Compute X-bar on each of these 400 samples

38. The Result • The green bars are a histogram of the sample means of S*1,…, S*400 • The blue curve is a normal distribution with the sample mean and s.d. • The red curve is a kernel density estimate of the distribution underlying the histogram • Intuitively, a smoothed histogram

39. The Result • The result is an estimate of the distribution of X-bar. • Notice that it is normal with mean≈500 and s.d.≈3.2 • The purely mechanical bootstrapping procedure produces what theory tells us to expect. • Can we use resampling to estimate the distribution of outstanding liabilities?

40. Bootstrapping Reserves • S = our database of policies • Sample with replacement all policies in S • Call this S*1 • Same size as S • Now do this 199 more times! • S*1, S*2,…, S*200 • Estimate o/s reserves on each sample • Get a distribution of reserve estimates

41. Bootstrapping Reserves • Compute your favorite reserve estimate on each S*k • These 200 reserve estimates constitute an estimate of the distribution of outstanding losses • Notice that we did this by resampling our original dataset S of policies. • This differs from other analyses which bootstrap the residuals of a model. • Perhaps more theoretically intuitive. • But relies on assumption that your model is correct!

42. Bootstrapping Results • Standard Deviation ≈ 5% of total o/s losses • 95% confidence interval ≈ (-10%, +10%) • Tighter interval than typically seen in the literature. • Result of not summarizing away variability info?

43. Reserve Dist: All Years

44. Reserve Dist: 1992

45. Reserve Dist: 1993

46. Reserve Dist: 1994

47. Reserve Dist: 1995

48. Reserve Dist: 1996

49. Reserve Dist: 1997

50. Reserve Dist: 1998