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 2009 1 / 37
Table of Contents Introduction 1 Tweedie GLM 2 Model Definition Log-likelihood Overdispersion Variance Modeling 3 Variance in regressions Dispersion Models Double GLMs Swiss Motor Industry Example 4 Analysis of results Conclusion 5 D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 2 / 37
Introduction Definition A loss reserve is a provision for an insurer’s liability for claims Notes D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 3 / 37
Introduction Definition A loss reserve is a provision for an insurer’s liability for claims A stochastic model uses random variables in a regression framework Notes D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 3 / 37
Introduction Definition A loss reserve is a provision for an insurer’s liability for claims A stochastic model uses random variables in a regression framework Notes Claims reserves models presented here use GLM theory as introduced in England, Verrall 2002 D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 3 / 37
Model Definition Notations C i , j Incremental payments D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 4 / 37
Model Definition Notations C i , j Incremental payments w i , j Exposure D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 4 / 37
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 2009 4 / 37
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 2009 4 / 37
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 2009 4 / 37
Model Definition Hypotheses C i , j is a compound Poisson-Gamma distribution D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 5 / 37
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 2009 5 / 37
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 2009 5 / 37
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 ν + 2 p = ν + 1 , p ∈ ( 1 , 2 ) µ = ϑτ ϑ 1 − p τ 2 − p φ = ( 2 − p ) D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 5 / 37
Model Definition Tweedie Model Y i , j ∼ Tweedie ( µ i , j , p , φ, w i , j ) D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 6 / 37
Model Definition Tweedie Model Y i , j ∼ Tweedie ( µ i , j , p , φ, w i , j ) µ i , j = e X β D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 6 / 37
Model Definition Tweedie Model Y i , j ∼ Tweedie ( µ i , j , p , φ, w i , j ) µ i , j = e X β w i , j µ p φ E [ Y i , j ] = µ i , j , Var [ Y i , j ] = i , j D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 6 / 37
Log-likelihood function Tweedie Model � � ( w i , j /φ ) ν + 1 y ν � i , j l = r i , j log ( p − 1 ) ν ( 2 − p ) i , j � � µ 1 − p µ 2 − p � � � � + w i , j i , j i , j − log r i , j !Γ r i , j ν y i , j y i , j 1 − p − φ 2 − p D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 7 / 37
Parameters F IGURE : Parameter main influence D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 8 / 37
Parameter p Tweedie Model Can be estimated only when the number of payments is known D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 9 / 37
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 2009 9 / 37
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 2009 9 / 37
Parameter φ Optimizing φ using the likelihood principle � � − � µ 1 − p µ 2 − p i , j i , j i , j w i , j y i , j 1 − p − 2 − p � φ p = ( 1 + ν ) � i , j r i , j D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 10 / 37
Parameter φ F IGURE : Optimizing p using the likelihood principle for φ D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 11 / 37
Parameter φ Optimizing φ using the deviance principle � � y 1 − p − µ 1 − p y 2 − p − µ 2 − p � 2 i , j i , j i , j i , j � φ p = y i , j − N − Q 1 − p 2 − p i , j D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 12 / 37
Parameter φ F IGURE : Optimizing p using the deviance principle for φ D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 13 / 37
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 2009 14 / 37
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 2009 14 / 37
Frequency vs Severity Compound Poisson Model N � Y = X i k = 1 Case 1 Case 2 Case 3 E[N] 10 20 10 Var[N] 10 20 10 E[X] 10 10 20 Var[X] 100 100 400 E[Y] 100 200 200 Var[Y] 2000 4000 8000 T ABLE : Mean and variance of total costs for various situations D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 15 / 37
Frequency vs Severity Typical situation in a long-tail business Decreasing average frequency D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 16 / 37
Frequency vs Severity Typical situation in a long-tail business Decreasing average frequency Increasing average severity D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 16 / 37
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 2009 16 / 37
Impact of the Distribution F IGURE : Fitted curve for Normal, Poisson and Gamma models D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 17 / 37
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 2009 18 / 37
Dispersion Models Model Definition Y i , j ∼ Tweedie ( µ i , j , p , φ i , j , w i , j ) µ i , j = e X β D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 18 / 37
Dispersion Models Model Definition Y i , j ∼ Tweedie ( µ i , j , p , φ i , j , w i , j ) µ i , j = e X β 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 2009 18 / 37
Dispersion Models Model Definition Y i , j ∼ Tweedie ( µ i , j , p , φ i , j , w i , j ) µ i , j = e X β 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 2009 18 / 37
Dispersion Models Log-likelihood � � ( w i , j /φ i , j ) ν + 1 y ν � � � � � i , j l = r i , j log − log r i , j !Γ r i , j ν y i , j ( p − 1 ) ν ( 2 − p ) i , j � � µ 1 − p µ 2 − p + w i , j i , j i , j y i , j 1 − p − φ i , j 2 − p D. Davidov (UQAM) Loss Reserves Variance Modeling August 2009 19 / 37
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 2009 20 / 37
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 2009 20 / 37
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