On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory

Transcription

1 On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory Danaïl Davidov under the supervision of Jean-Philippe Boucher Département de mathématiques Université du Québec à Montréal August 2009 D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

2 Table of Contents 1 Introduction 2 Tweedie GLM Model Definition Log-likelihood Overdispersion 3 Variance Modeling Variance in regressions Dispersion Models Double GLMs 4 Swiss Motor Industry Example Analysis of results 5 Conclusion D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

3 Introduction Definition A loss reserve is a provision for an insurer s liability for claims Notes D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

4 Introduction Definition Notes A loss reserve is a provision for an insurer s liability for claims A stochastic model uses random variables in a regression framework D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

5 Introduction Definition Notes A loss reserve is a provision for an insurer s liability for claims A stochastic model uses random variables in a regression framework Claims reserves models presented here use GLM theory as introduced in England, Verrall 2002 D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

6 Model Definition Notations C i,j Incremental payments D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

7 Model Definition Notations C i,j Incremental payments w i,j Exposure D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

8 Model Definition Notations C i,j Incremental payments w i,j Exposure r i,j Incremental number of payments D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

9 Model Definition Notations C i,j Incremental payments w i,j Exposure r i,j Incremental number of payments Y i,j Normalized incremental payments D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

10 Model Definition Notations C i,j Incremental payments w i,j Exposure r i,j Incremental number of payments Y i,j Normalized incremental payments Y i,j = C i,j w i,j D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

11 Model Definition Hypotheses C i,j is a compound Poisson-Gamma distribution D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

12 Model Definition Hypotheses C i,j is a compound Poisson-Gamma distribution Frequency Poisson with mean ϑ i,j w i,j and variance ϑ i,j w i,j D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

13 Model Definition Hypotheses C i,j is a compound Poisson-Gamma distribution Frequency Poisson with mean ϑ i,j w i,j and variance ϑ i,j w i,j Severity Gamma with mean τ i,j and variance ντ 2 i,j D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

14 Model Definition Hypotheses C i,j is a compound Poisson-Gamma distribution Frequency Poisson with mean ϑ i,j w i,j and variance ϑ i,j w i,j Severity Gamma with mean τ i,j and variance ντ 2 i,j Using the following parametrisation p = ν + 2 ν + 1 µ = ϑτ φ = ϑ1 p τ 2 p (2 p), p (1, 2) D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

15 Model Definition Tweedie Model Y i,j Tweedie(µ i,j, p, φ, w i,j ) D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

16 Model Definition Tweedie Model Y i,j Tweedie(µ i,j, p, φ, w i,j ) µ i,j = e Xβ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

17 Model Definition Tweedie Model Y i,j Tweedie(µ i,j, p, φ, w i,j ) µ i,j = e Xβ E[Y i,j ] = µ i,j, Var[Y i,j ] = φ w i,j µ p i,j D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

18 Log-likelihood function Tweedie Model l = i,j r i,j log ( ) (wi,j /φ) ν+1 yi,j ν (p 1) ν (2 p) log ( r i,j!γ ( r i,j ν ) y i,j ) + w i,j φ µ (y 1 p i,j i,j 1 p µ 2 p ) i,j 2 p D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

19 Parameters FIGURE: Parameter main influence D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

20 Parameter p Tweedie Model Can be estimated only when the number of payments is known D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

21 Parameter p Tweedie Model Can be estimated only when the number of payments is known Otherwise, it s supposed fixed and known D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

22 Parameter p Tweedie Model Can be estimated only when the number of payments is known Otherwise, it s supposed fixed and known p and φ need both to be estimated at the same time D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

23 Parameter φ Optimizing φ using the likelihood principle φ p = i,j w i,j (y i,j µ 1 p i,j 1 p µ2 p i,j 2 p (1 + ν) i,j r i,j ) D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

24 Parameter φ FIGURE: Optimizing p using the likelihood principle for φ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

25 Parameter φ Optimizing φ using the deviance principle φ p = i,j 2 y (y 1 p i,j µ 1 p i,j i,j N Q 1 p y 2 p i,j µ 2 p i,j 2 p ) D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

26 Parameter φ FIGURE: Optimizing p using the deviance principle for φ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

27 Parameter w i,j Note The exposure has been incorporated in the begining, within the initial hypothesis D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

28 Parameter w i,j Note The exposure has been incorporated in the begining, within the initial hypothesis Different ways of incorporating the exposure within the initial hypothesis would lead to different models D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

29 Frequency vs Severity Compound Poisson Model Y = N k=1 X i Case 1 Case 2 Case 3 E[N] Var[N] E[X] Var[X] E[Y] Var[Y] TABLE: Mean and variance of total costs for various situations D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

30 Frequency vs Severity Typical situation in a long-tail business Decreasing average frequency D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

31 Frequency vs Severity Typical situation in a long-tail business Decreasing average frequency Increasing average severity D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

32 Frequency vs Severity Typical situation in a long-tail business Decreasing average frequency Increasing average severity Increasing variance in the severity D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

33 Impact of the Distribution FIGURE: Fitted curve for Normal, Poisson and Gamma models D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

34 Dispersion Models Model Definition Y i,j Tweedie(µ i,j, p, φ i,j, w i,j ) D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

35 Dispersion Models Model Definition Y i,j Tweedie(µ i,j, p, φ i,j, w i,j ) µ i,j = e Xβ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

36 Dispersion Models Model Definition Y i,j Tweedie(µ i,j, p, φ i,j, w i,j ) µ i,j = e Xβ E[Y i,j ] = µ i,j, Var[Y i,j ] = φ i,j µ p i,j D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

37 Dispersion Models Model Definition Y i,j Tweedie(µ i,j, p, φ i,j, w i,j ) µ i,j = e Xβ E[Y i,j ] = µ i,j, Var[Y i,j ] = φ i,j µ p i,j φ i,j = e V γ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

38 Dispersion Models Log-likelihood l = ( ) (wi,j /φ i,j ) ν+1 yi,j ν r i,j log (p 1) ν log ( r i,j!γ ( r i,j ν ) ) y i,j (2 p) i,j + w i,j µ (y 1 p i,j i,j φ i,j 1 p µ 2 p ) i,j 2 p D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

39 Dispersion Models Notes The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

40 Dispersion Models Notes The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

41 Dispersion Models Notes The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter Due to lack of data in the last column, the model was build so that the last two columns have the same dispersion parameter D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

42 Dispersion Models Notes The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter Due to lack of data in the last column, the model was build so that the last two columns have the same dispersion parameter Possibility to incorporate trends in the dispersion parameter by using the Hoerl s curve parametrisation D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

43 Double GLMs Algorithm 1 Start with the initial exposure and find the normalized incremental payments D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

44 Double GLMs Algorithm 1 Start with the initial exposure and find the normalized incremental payments 2 Find the deviance between fitted and observed values D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

45 Double GLMs Algorithm 1 Start with the initial exposure and find the normalized incremental payments 2 Find the deviance between fitted and observed values 3 Model the deviance D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

46 Double GLMs Algorithm 1 Start with the initial exposure and find the normalized incremental payments 2 Find the deviance between fitted and observed values 3 Model the deviance 4 Establish the new exposure and start over D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

47 Double GLMs FIGURE: Inter-relationship between the two sub-models D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

48 Iterative Weighted Least Squares Algorithm additional specifications Analogous to Fisher s weighted scoring method for optimization D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

49 Iterative Weighted Least Squares Algorithm additional specifications Analogous to Fisher s weighted scoring method for optimization IWLS implicitly uses the deviance principle for estimating φ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

50 Restricted Maximum Likelihood Notes Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of observations D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

51 Restricted Maximum Likelihood Notes Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of observations REML produces estimators which are approximately and sometimes exactly unbiased D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

52 Restricted Maximum Likelihood Notes Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of observations REML produces estimators which are approximately and sometimes exactly unbiased Approximately maximizes the penalized log-likelihood l p (y; γ; p) = l(y; β γ ; γ; p) + 1 X 2 log T WX D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

53 Optimizing p Algorithm 1 Suppose p fixed and known D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

54 Optimizing p Algorithm 1 Suppose p fixed and known 2 Evaluate all the other parameters using the DGLM D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

55 Optimizing p Algorithm 1 Suppose p fixed and known 2 Evaluate all the other parameters using the DGLM 3 Evaluate the penalized log-likelihood D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

56 Optimizing p Algorithm 1 Suppose p fixed and known 2 Evaluate all the other parameters using the DGLM 3 Evaluate the penalized log-likelihood 4 Start over with different values for p and compare the penalized log-likelihood D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

57 Reserve Variability Mean Square Error of Prediction Dispersion Models : overdispersion included implicitly in the parameter covariance matrix D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

58 Reserve Variability Mean Square Error of Prediction Dispersion Models : overdispersion included implicitly in the parameter covariance matrix GLMs : overdispersion is included manually in the parameter covariance matrix D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

59 Data Analyzed D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

60 Data Analyzed D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

61 Data Analyzed D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

62 Optimizing p FIGURE: Restricted log-likelihood for various p in a DGLM D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

63 Analysis of Results D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

64 Analysis of Results D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

65 Analysis of Results FIGURE: Adjusting the most deviant observations has a bigger influence on the deviance principle D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

66 Analysis of Results FIGURE: Contribution of each cell to the dispersion. p = , w i,j 1 D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

67 Conclusion Further Discussion Lack of observations and abundance of parameters is a hostile environment for DGLMs D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

68 Conclusion Further Discussion Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

69 Conclusion Further Discussion Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

70 Conclusion Further Discussion Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters The algorithm does converge for p fixed for one parameter, but p cannot be optimized D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

71 Conclusion Further Discussion Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters The algorithm does converge for p fixed for one parameter, but p cannot be optimized The "bounds" of convergence need to be explored further D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

72 References England, PD and Verrall, RJ (2002). Stochastic Claims Reserving in General Insurance. Institute of Actuaries and Faculty of Actuaries Smyth, G. and Jorgensen, B. (2002). Fitting Tweedie s Compound Poisson Model to Insurance Claims Data : Dispersion Modelling. ASTIN Bulletin, 1, Wüthrich, MV (2003). Claims Reserving Using Tweedie s Compound Poisson Model. ASTIN Bulletin, D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37

73 The end D. Davidov (UQAM) Loss Reserves Variance Modeling August / 37