severity modeling of extreme insurance claims...“modeling insurance claims via a mixture...

Severity Modeling of Extreme Insurance Claims

10. September 2018

Christian LaudagéJoint work with Sascha Desmettre and Jörg Wenzel

Department Financial MathematicsFraunhofer ITWMFraunhofer-Platz 167663 Kaiserslautern | Germany

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10. September 2018 2

Motivation

Expected claim severity: Generalized linear models based on gamma distributionProblem: Extreme claim sizes in dataConcentration on body of distribution leads to

bias in estimationmissing robustness in estimation

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Outline

Modeling frameworkThreshold severity modelEstimators below and above a given thresholdSimulation study

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Modeling framework

Probability space: (Ω,F ,P)Claim severity: X : (Ω,F)→ (R>0,B (R>0))Vector of tariff features: R = (R1, . . . ,Rd) with Ri : (Ω,F)→ (R>0,B (R>0))Information of tariff features: G := σ (R1, . . . ,Rd)

Lemma 1 (Expected claim severity)

Given X with E (|X |) 0 we have for P-almost every ω ∈ Ω

E (X |G) (ω) = EκX ,G(ω;·)(idR>0 1(0,u]

)+ EκX ,G(ω;·)

(u1(u,∞)

)+ EκX ,G(ω;·)

((idR>0 − u) 1(u,∞)

). (1)

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Conditional probabilities

Motivation: Calculation of EκX ,G(ω;·)(idR>0 1(0,u]

)and EκX ,G

((idR>0 − u) 1(u,∞)

)in the sense of a splicing model.

Definition 2

Let u ∈ R>0 and κX ,G be a regular conditional probability of X given G. We define for all A ∈ B (R>0) andω ∈ Ω

P≤u,ωκX ,G (A) :=∫A

1(0,u] (x)κX ,G (ω; (0, u])

κX ,G (ω; dx) , (2)

if κX ,G (ω; (0, u]) > 0 and P≤u,ωκX ,G (A) := κX ,G (ω; A) otherwise. Also, for all A ∈ B (R>0) and ω ∈ Ω set

P>u,ωκX ,G (A) :=∫A

1(u,∞) (x)κX ,G (ω; (u,∞))

κX ,G (ω; dx) , (3)

if κX ,G (ω; (u,∞)) > 0 and P>u,ωκX ,G (A) := κX ,G (ω; A) otherwise.

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Threshold severity model

Aim: Express EκX ,G(ω;·)(idR>0 1(0,u]

)and EκX ,G

((idR>0 − u) 1(u,∞)

)with help of P≤u,ωκX ,G and P

>u,ωκX ,G

.

Theorem 3 (Threshold severity model)

Given X with E (|X |) 0 there exists a regular conditional probability κX ,G of X given Gsuch that for P-almost every ω ∈ Ω it holds

E (X |G) (ω) = κX ,G (ω; (0, u]) EP≤u,ωκX ,G (idR>0) + κX ,G (ω; (u,∞))(u + EP>u,ωκX ,G (idR>0 − u)

). (4)

Tariff cell: Group of policyholders with same tariff featuresWhat is the premium amount for one tariff cell?

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Tariff cells

Proposition 4

Given X with E (|X |) 0)and ω ∈ {ω̃ ∈ Ω|P (R = R(ω̃)) > 0} the following holds:

κX ,G (ω,A) = P (X ∈ A|R) (ω) . (5)

Note: For every ω ∈ {R = r} with P (X ≤ u,R = r) > 0 it holds:

P≤u,ωκX ,G (A) = P (X ∈ A|X ≤ u,R = r) , ∀A ∈ B (R>0) .

Assumption 5

The distribution of R is discrete, i.e. there are countably many, pairwisedifferent values x1, x2, . . . such that ∑∞i=1 P (R = xi) = 1.

Example:

North SouthBrand A Cell 1 Cell 2Brand B Cell 3 Cell 4

x1 = (Brand A, North)x2 = (Brand A, South)x3 = (Brand B, North)x4 = (Brand B, South)

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Distribution function of a tariff cell

Tariff cell: {R = r}Insured sum given by r1 > uDistribution function of X given by

Fr (x) =

(1− q)Hr (x)Hr (u) , x ≤ u,(1− q) + q Gr (x) , x > u,

with Hr(u) > 0 and Gr(u) = 0.Note:

u independent of rq independent of r

Figure: Distribution function, threshold (blue) and insured sum (red).

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Generalized linear model (GLM)

Density of X is of the form

fX (x ; θ, ϕ) = expxθ − b(θ)

ϕ\ω+ c(x , ϕ, ω)

Gamma distribution:

fX (x ; θ, ϕ)exp(c(x , ϕ, ω)) = exp

xθ + log(−θ)ϕ\ω

With link function g we get:

θ b′−→ E (X |R = r) g−→

d∑i=1

ri αi

Logarithmic link function: E (X |R = r) = g−1(rβ) = erα

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Basic claim sizes: Truncated gamma GLM

GLM to model conditional distribution function:

Fr (X ≤ x |X ≤ u) =Hr (min(x , u))

Hr (u)Hr given by gamma distributionWeak convergence + gamma distribution continuous =⇒ uniform convergence:

limu→∞ sup0

Extreme claim sizes

Excess distribution:

Fr (X ≤ x |X > u) = Gr (x)

Real damage for policyholder: YDamage for insurer: X := min (Y , r1)Assumption: Real damages over threshold independent of tariff cell, i.e. {Y ∈ A ∩ (u,∞)}independent of {R = r} for all A ∈ B (R>0)Theorem of Pickands, Balkema and de Haan:

limu↑xF

sup0

Simulation study

Figure: Histogram of simulated claims (left) and simulated claims (right) with threshold of 1 million (red).

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Simulation study

Figure: Estimates of dispersion parameter (left) and average of the absolute relative differences over all tariff cells (right) for different thresholds.

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Simulation study

Figure: Variance (red) and squared bias (blue) for gamma GLM (left) and threshold severity model (right) based on 500 simulations.

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Outlook

Application to real dataFurther developments:

Estimation of dispersion parameter without approximationConsideration of further tariff features for excess distributionUsage of different thresholds

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Literature

C. Laudagé, S. Desmettre & J. Wenzel. “Severity Modeling of Extreme Insurance Claims forTariffication”. Available at SSRN: https://ssrn.com/abstract=3168441 (2018).

J. Garrido, C. Genest & J. Schulz. “Generalized linear models for dependent frequency and severity ofinsurance claims”. In: Insurance: Mathematics and Economics. 70 (2016) 205-215.

D. Lee, W.K. Li & T.S.T Wong. “Modeling insurance claims via a mixture exponential model combinedwith peaks-over-threshold approach”. In: Insurance: Mathematics and Economics. 51 (2012) 538-550.

T. Reynkens, R. Verbelen, J. Beirlant & K. Antonio. “Modelling censored losses using splicing: A globalfit strategy with mixed Erlang and extreme value distributions”. In: Insurance: Mathematics andEconomics. 77 (2017) 65-77.

P. Shi. “Fat-tailed regression models”. In: Predictive Modeling Applications in Actuarial Science. 1(2014) 236-259.

P. Shi, X. Feng & A. Ivantsova. “Dependent frequency–severity modeling of insurance claims”. In:Insurance: Mathematics and Economics. 64 (2015) 417–428.

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