Ratemaking with a Copula-Based Multivariate Tweedie Model

Transcription

1 with a Copula-Based Model joint work with Xiaoping Feng and Jean-Philippe Boucher University of Wisconsin - Madison CAS RPM Seminar March 10, / 18

2 Outline / 18

3 Insurance Claims Two types of predictive models for insurance claims Frequency severity model: Two-stage framework Pure premium: Both belong to GLM and straightforward to implement More details in Predictive s in Actuarial Science, edited by Frees, Derrig, and Meyers 3 / 18

4 Multilevel Structure We often assume individual risks are independent which is not always the case We examine a specific case where dependence among risks is often introduced through the multilevel structure Many examples in property-casualty insurance Automobile insurance: household/fleet, multiple coverage Homeowner: neighborhood, multi-peril coverage Health: individual/group Worker s compensation. One can argue that dependence among risks is important for claim management Our goal is to incorporate the correlation into the claims modeling framework 4 / 18

5 Automobile Insurance 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 Percentage / 18

6 Claim Costs 6 / 18

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 / 18

8 A Poisson sum of gamma random variables Y = (X X N )/ω N Poisson(ωλ) Y j (j = 1,,N) gamma(α,γ) The belongs to the exponential familiy with the reparameterizations: λ = µ2 p φ(2 p), α = 2 p p 1, The density function is shown as [ ( ω f (y) = exp φ y µ2 p (p 1)µ p 1 2 p γ = φ(p 1)µp 1 ) ] + S(y;φ/ω) with E(Y) = µ and Var(Y) = φ ω µp Dispersion modeling? Tweed GLM: g µ (µ) = x β Dispersion model: g φ (φ) = z η 8 / 18

9 Bivariate 9 / 18 A copula is a multivariate distribution function with uniform marginals. Let U 1,...,U J be J uniform random variables on (0,1). Their distribution function H(u 1,...,u J ) = Pr(U 1 u 1,...,U J u J ) Consider two tweedie marginals Y 1 and Y 2 with cdf F 1 and F 2. Define the joint distribution using the copula H such that F(y 1,y 2 ) = H(F 1 (y 1 ),F 2 (y 2 )) 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

10 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 ) Each marginal follows distribution 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 )) Need to specify the association matrix R in the gaussian copula 10 / 18

11 Define R = B P where ( 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 Further 1 ρ j ρ 2 τ jj 1 ρj 2 1 ρ 2 j ρ3 j j σ jj = and P AR 1 ρ j ρ j 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 τ 12, τ 13, τ 23 Temporal ρ 1, ρ 2, ρ 3 11 / 18

12 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 12 / 18

13 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 / 18

14 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 / 18

15 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, , / 18

16 Individual Risk young senior marital homeowner experience conviction newcar leasecar business highmilage multidriver Excellent Very Good Good Fire Poor / 18

17 Portfolio Risk Left: mean v.s. dispersion Right: independence v.s. copula 17 / 18

18 We focused on the multilevel structure of claims data model was considered as an example Thank you for your kind attention. Learn more about my research at: 18 / 18