Severity Modeling of Extreme Insurance Claims for Tariffication Sascha Desmettre (joint work with C. Laudagé, J. Wenzel) OICA 2020 - Online International Conference in Actuarial Science, Data Science and Finance April 28-29, 2020 S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 1 / 15
Motivation Expected Claim Severity ◮ Usually modeled via generalized linear models (GLMs) based on gamma distribution (see e.g. [Ohlsson & Johansson (10), Wüthrich (17)]). Limitations ◮ Extreme claim sizes in data � The Gamma CDF is not heavy-tailed! Concentration on body of distribution may lead to ◮ bias predictions ◮ missing robustness in predictions � Extreme Value Theory might help! S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 2 / 15
Modeling Framework Claim severity : Positive iid random RVs X 1 , X 2 , · · · ∼ X Claim frequency : Positive discrete RV N , where N ind. of X Features like car brand, age of driver or power of car affects damage. Vector of tariff features : R = ( R 1 , . . . , R d ) with positive RVs R i Tariff cell : Concrete combination of tariff features, e.g. 60 kW 80 kW . . . 18 years Cell 11 Cell 12 . . . 19 years Cell 21 . . . Cell 22 . . . ... . . . . . . r = (19 years, 80 kW) What is the expected claim severity for a specific tariff cell r ? E ( X | R = r ) Total damage in the given time period: E ( S | R = r ) = E ( N | R = r ) · E ( X | R = r ) S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 3 / 15
Censoring by Insured Sum Primary insurers only pay for damages up to a specified amount. ◮ Considered as tariff feature R I . The actual damage Y may be larger than the insured sum. � Claim severity is then given by X := min( Y , R I ) . Insurer only observes realizations for X , i.e. right-censored data. � Determine the distribution of Y based on this censored data. S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 4 / 15
Threshold Severity Model (TSM) Split the distribution of Y at a certain threshold u > 0. � Body and tail of the claim size distribution can be modeled separately. Notation for a given tariff cell r : ◮ H r cdf for the body with parameter vector Θ H ◮ G r cdf for the tail with parameter vector Θ G ◮ q r prob. of exceeding the given threshold u with parameter vector Θ q Assumptions to obtain a contiuous distribution function: ◮ H r ( u ; Θ H ) > 0 ◮ G r ( u ; Θ G ) = 0 S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 5 / 15
Concrete Specification of the TSM Distribution function of Y with parameter vector Θ = (Θ H , Θ G , Θ q ): 0 , y ≤ 0 , (1 − q r (Θ q )) H r ( y ;Θ H ) F r ( y ; Θ) = , 0 < y ≤ u , H r ( u ;Θ H ) (1 − q r (Θ q )) + q r (Θ q ) G r ( y ; Θ G ) , y > u . Note: Threshold u independent of tariff cell r However, the exceeding probability depends on insured sum: 1 q r (Θ q ) = 1 + e − ( δ 0 + δ I r I ) with Θ q = δ. � ˆ � ˆ Θ H , ˆ Θ G , ˆ Θ = Θ q is estimated via maximizing the log-likelihood. � Obtain desired expectation for a tariff cell r by [ X = min( Y , R I )]: � r I � � � � �� y ; ˆ r I ; ˆ E ˆ Θ (min( Y , R I ) | R = r ) = yf r Θ dy + r I 1 − F r Θ . 0 S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 6 / 15
Recall: X := min( Y , R I ). S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 7 / 15
Estimators for Basic and Extreme Claim Sizes Use concrete distributions for the conditional distribution functions below and above the threshold for a tariff cell r . Claim severity below the given threshold: ◮ Use general regression methods, i.e., a generalized linear model (GLM). ◮ Assume a gamma distribution for H r . ◮ In particular, the conditional distribution function P ( Y ≤ y | Y ≤ u , R = r ) = H r ( y ; Θ H ) H r ( u ; Θ H ) , 0 < y ≤ u , describes a truncated gamma distribution. Claim severity above the given threshold: ◮ Apply the peaks-over-threshold approach from extreme value theory. ◮ I.e., the conditional distribution function P ( Y ≤ y | Y > u , R = r ) = G r ( y ; Θ G ) , y > u , is approximated by the generalized Pareto distribution (GPD). S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 8 / 15
Basic Claim Sizes: Truncated Gamma GLM We assume that for all covariates r ∈ R d ≥ 0 we have ( Y | Y ≤ u , R = r ) ∼ G ( φ, θ r , u ) with φ > 0 , θ r < 0 , i.e., they are truncated gamma distributed with dispersion φ , threshold u and scale θ r , depending on the tariff features r . GLM to model conditional distribution function of X = min( Y , R I ): P ( X ≤ x | X ≤ u , R = r ) = H r (min( x , u ); Θ H ) . H r ( u ; Θ H ) � d ′ θ ( b u ( ., ˆ φ ) ) g � − − − − − − → E ( X | X ≤ u , R = r ) − → α 0 + r i α i , i =1 with � 1 φ − 1 exp � � � − θ u θ u − u ′ := b ′ ( θ ) + φ φ ( b u ( θ, φ )) . � � 1 φ , − θ u γ φ S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 9 / 15
Extreme Claim Sizes We are looking at the excess distribution: F u ( y , r ) = P ( Y ≤ y | Y > u , R = r ) = G r ( y ; Θ G ) , y > u . Theorem of Pickands, Balkema and de Haan: � � lim sup � F u ( x ) − G ξ,β ( u ) ( x ) � = 0 . � � u ↑ x F 0 < x < x F − u Application to Y with Θ G = ( ξ, β ) provides approximation : G r ( y ; Θ G ) = G ξ,β ; u ( y ) = G ξ,β ( y − u ) , y > u . Conditional distribution function of X := min( Y , R I ): P ( X ≤ x | X > u , R = r ) = G ξ,β (min ( x , r I ) − u ) , x > u . S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 10 / 15
Simulation Study Goal: Show that the TSM outperforms the classical gamma GLM when fitting to simulated claim sizes from other regression models. � Use heavy-tailed regression models based on the log-normal and Burr Type XII distributions to generate claim sizes. Present and compare the predictions stemming from the gamma GLM and the TSM w.r.t. the different scenarios. Setting: ◮ Set the index of the insured sum to 1 and denote it by v (= r 1 = r I ). ◮ Insured sums: 5 million, 20 million, 50 million. ◮ Second tariff feature taking integer values from 1 to 10. [E.g. mileage or the car’s power; denoted by w (= r 2 )]. � 30 tariff cells in total. S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 11 / 15
Simulation Study: Log-Normal Regression 1 Simulate a normal random variable Z ∼ N ( µ, σ ) with mean µ = α 0 + α 1 v + α 1 w and standard deviation σ > 0. 2 Obtain the log-normal random variable by X = e Z . 3 In order to obtain a significant influence of the insured sum, we use the following parameters in this scenario: α 0 = 5 . 5 , α 1 = 4 × 10 − 8 , α 2 = 0 . 02 , σ = 2 . 75 . 4 Compare the classical gamma GLM with the TSM in this log-normal setting. S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 12 / 15
Simulation Study: Burr Regression 1 Simulate claim sizes from a Burr Type XII distribution, i.e, Y ∼ Burr ( β, λ, τ ) with density fucntion λβ λ τ y τ − 1 f B ( y ; β, λ, τ ) = ( β + y τ ) λ +1 , y > 0 , β, λ, τ > 0 . 2 To incorporate tariff cells, we use a regression for the parameter β , i.e., we obtain the conditional distribution ( Y | R = r ) ∼ Burr ( β ( r ) , λ, τ ) with β ( r ) := exp ( τ ( α 0 + α 1 v + α 1 w )) . 3 Parameter values in this scenario: α 0 = 8 , α 1 = 4 × 10 − 8 , α 2 = 0 . 02 , λ = 1 . 5 , τ = 0 . 7 ( ⇒ heavy tails) . 4 Compare the cl. gamma GLM with the TSM in this Burr-type setting. S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 13 / 15
Results - Observed Statistics Quantify the relative deviation between the true ( µ i ) and predictive mean ( ˆ µ i ) of a specific tariff cell. Calculate (weighted) averages of the relative differences for every scenario w.r.t. all tariff cells: 30 30 z 1 := 1 | ˆ µ i − µ i | m i | ˆ µ i − µ i | � � ¯ , ¯ z 2 := . 30 µ i m µ i i =1 i =1 Simulated Claims Model ¯ ¯ z 1 z 2 Log-Normal Gamma GLM 53.31% 14.58% Log-Normal TSM 21.67% 13.35% Burr Gamma GLM 74.82% 23.51% Burr TSM 17.78% 5.59% S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 14 / 15
Conclusion and Outlook TSM combines idea of GLMs with EVT for tariffication. Allows for simple interpretations. Robust against Log-Normal and Burr claim sizes. Outperforms the classical gamma-based GLM. Further tariff features for excess distribution. Usage of different thresholds. Transfer to risk management. S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 15 / 15
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