Peng Shi. Actuarial Research Conference August 5-8, 2015

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1 Multivariate Longitudinal of Using Copulas joint work with Xiaoping Feng and Jean-Philippe Boucher University of Wisconsin - Madison Université du Québec à Montréal Actuarial Research Conference August 5-8, / 21

2 Outline / 21

3 modeling depends on the purposes of the models Model building often relies on the features of the data Some unique features in insurance claims data 1) Semi-continuous 2) Multivariate 3) Multilevel Frequency-severity (two-part) v.s. pure premium models More details in Predictive s in Actuarial Science, edited by Frees, Derrig, and Meyers 3 / 21

4 Some unique features in insurance claims data 1) Semi-continuous 2) Multivariate Automobile insurance: third party liability, collision, comprehensive... Homeowner multiperil: fire, lighting, hail, wind... Worker s compensation: medical care, income replacement... Health insurance: physician v.s. non-physician, outpatient v.s. hospital inpatient 3) Multilevel Automobile insurance: fleet/household, vehicle, multi-period Homeowner multiperil: neighborhood, houses, periods Worker s compensation: employer, occupation, individual Health insurance: employer/household, individual, periods 4 / 21

5 Automobile Personal automobile insurance in Canada Four types of coverage Accident benefit - no-fault insurance benefit to the injured insured Collision - damage to the policyholder s vehicle due to collision All risk - damage to the policyholder s vehicle due to risks other than collision Civil liability - bodily injury and property damage of third party We examine a longitudinal dataset for years Each policy could cover multiple vehicles Table : Distribution of number of vehicles per policy Total Number 77,352 10, ,670 Percentage / 21

6 Claim Costs 6 / 21

7 Predictors The data set contains basic rating variables policyholder characteristics, driving history, vehicle characteristics use binary predictors in the analysis Table : Summary of rating variables Variable Description Mean young =1 if age between 16 and seinor =1 if age more than marital =1 if married homeowner =1 if homeowner experience =1 if more than ten years of experience conviction =1 if positive number of convictions newcar =1 if new car leasecar =1 if lease car business =1 if business use highmilage =1 if drive more than 10,000 miles multidriver =1 if more than two drivers / 21

8 Multivariate Let Y ikjt denote the insurance cost for Household (Cluster) i (= 1,,M) Vehicle k (= 1,,K i ) Coverage type j (= 1,,J i ) Period t (= 1,,T i ) For example K 2 = 2, J i = 3, T i = 4, let Y i = (Y i111,,y i114,y i121,,y i124,,y i231,,y i234 ) Using Gaussian copula to build the joint distribution F i (y i ) = H(F(y i111 ),,F(Y i114 ),F(Y i121 ),,F(Y i124 ),,F(Y i231 ),,F(Y i234 );R) Need to specify F Need to specify the association matrix R in the gaussian copula 8 / 21

9 Multivariate Start from R = B (Σ P) where ( ) 1 δ B =,P = δ 1 and for example P = 1 ρ ρ 2 ρ 3 ρ 1 ρ ρ 2 ρ 2 ρ 1 ρ ρ 3 ρ 2 ρ 1 1 σ 12 σ 13 σ 21 1 σ 23 σ 31 σ 32 1 Fully separate 9 / 21

10 Multivariate Tweedie Define R = B P where Further ( 1 δ B = δ 1 ) and P = P 11 σ 12 P 12 σ 13 P 13 σ 21 P 21 P 22 σ 23 P 23 σ 31 P 31 σ 32 P 32 P 33 1 ρ j ρ 2 j ρ3 j P AR jj = ρ j 1 ρ j ρ 2 j ρj 2 ρ j 1 ρ j ρj 3 ρj 2 ρ j 1 Summarize dependence parameters Between vehicles δ Between coverage types σ 12, σ 13, σ 23 Temporal ρ 1, ρ 2, ρ 3 10 / 21

11 Composite Likelihood Use composite likelihood to estimate model parameters cl i (θ;y) = 1 (sum of bivariate loglik) m i 1 Use the inverse of the Godambe information matrix to get standard error G 1 N (θ) = H 1 N (θ)j N (θ)h 1 N (θ) 2 cl i (θ;y i ) θ θ and J N (θ) = 1 N N i=1 where H N (θ) = N 1 N i=1 Model Comparison CLAIC = 2cl(θ;y) + 2tr(J(θ)H(θ) 1 ) CLBIC = 2cl(θ;y) + log(dim(θ))tr(j(θ)h(θ) 1 ) cl i (θ;y i ) cl i (θ;y i ) θ θ 11 / 21

12 Bivariate Model Use Gaussian copula H(u1,u2) = Φ ρ (Φ 1 (u 1 ),Φ 2 (u 2 )) H(F 1 (0),F 2 (0)) if y 1 = 0 and y 2 = 0 f f (y 1,y 2 ) = 1 (y 1 )h 1 (F 1 (y 1 ),F 2 (0)) if y 1 > 0 and y 2 = 0 f 2 (y 2 )h 2 (F 1 (0),F 2 (y 2 )) if y 1 = 0 and y 2 > 0 f 1 (y 1 )f 2 (y 2 )h(f 1 (y 1 ),F 2 (y 2 ) if y 1 > 0 and y 2 > 0 Here h(u 1,u 2 ) = H(u 1,u 2 )/ u 1 u 2 h j (u 1,u 2 ) = H(u 1,u 2 )/ u j for j = 1,2 They have close-form expressions for Gaussian copula 12 / 21

13 Specification We set J = 2, K = 2, and T = 4, that is, a policy covers two vehicles and provides two types of coverage for each vehicle. We use Tweedie(µ j,p j,φ j ) with the following specification for coverage type j = 1 and 2: µ j = exp(β j0 + β j1 X 1 + β j2 X 2 ) φ j = exp(γ j0 + γ j1 X 1 + γ j2 X 2 ) X 1 Bernoulli(0.5) and X 2 Bernoulli(0.6) independently. 13 / 21

14 Specification We use the Gaussian copula with the correlation matrix specified as: ( 1 δ Σ = δ 1 ) σ 12 1 ρ 1 ρ1 2 ρ1 3 ρ 1 1 ρ 1 ρ1 2 ρ1 2 ρ 1 1 ρ 1 ρ1 3 ρ1 2 ρ ρ 1 ρ 2 1 ρ 3 1 ρ 2 1 ρ 1 ρ 2 1 ρ 2 2 ρ 2 1 ρ 1 ρ 3 2 ρ 2 2 ρ 2 1 σ 12 1 ρ 2 ρ2 2 ρ 3 ρ 1 1 ρ 2 ρ2 2 ρ1 2 ρ 1 1 ρ 2 ρ1 3 ρ1 2 ρ ρ 2 ρ 2 2 ρ 3 2 ρ 2 1 ρ 2 ρ 2 2 ρ 2 2 ρ 2 1 ρ 2 ρ 3 2 ρ 2 2 ρ 2 1 Use parameterization σ 12 = τ ρ1 2 1 ρ2 2/(1 ρ 1ρ 2 ). 14 / 21

15 Specification Table : results for sample size (number of policies) N = 200 and 500 Parameter Relative Bias SD SE MSE N = β 10 = β 11 = β 12 = β 20 = β 21 = β 22 = p 1 = p 2 = γ 10 = γ 11 = γ 12 = γ 20 = γ 21 = 0 NA NA γ 22 = ρ 1 = ρ 2 = τ = δ = / 21

16 Marginal Models Model dispersion as well as mean Different predictors for each type Table : Parameter estimates for accident benefit Mean Dispersion Est. S.E. Est. S.E. intercept intercept conviction homeowner homeowner experience experience multidriver young marrital senior senior highmilage p / 21

17 Dependence Strong association among coverage types Small serial correlation and cluster effect Table : Estimates of dependence parameters Est. S.E. ρ ρ ρ ρ τ τ τ τ τ τ δ / 21

18 Model Comparison Table below summarizes the goodness-of-fit statistics of alternative specifications Smaller statistics indicate better fit Table : Goodness-of-fit statistics Model Description CLAIC CLBIC M0 independence 958, ,142 M1 no temporal 957, ,375 M2 no cross-sectional 958, ,130 M3 no cluster 957, ,363 M4 no dispersion 958, ,120 M5 full model 957, , / 21

Individual Risk young senior marital homeowner experience conviction newcar leasecar business highmilage multidriver Excellent 0 1 1 1

19 Individual Risk young senior marital homeowner experience conviction newcar leasecar business highmilage multidriver Excellent Very Good Good Fire Poor / 21

20 Portfolio Risk Left: mean v.s. dispersion Right: independence v.s. copula 20 / 21

21 We focused on the multilevel structure of claims data Tweedie model was considered as an example Thank you for your kind attention. Learn more about my research at: 21 / 21

Ratemaking with a Copula-Based Multivariate Tweedie Model

Ratemaking with a Copula-Based Multivariate Tweedie Model with a Copula-Based Model joint work with Xiaoping Feng and Jean-Philippe Boucher University of Wisconsin - Madison CAS RPM Seminar March 10, 2015 1 / 18 Outline 1 2 3 4 5 6 2 / 18 Insurance Claims Two