Severity Modeling of Extreme Insurance Claims for ... · Severity Modeling of Extreme Insurance Claims for Tariffication Author Sascha Desmettre (joint work with C. Laudagé, J. Wenzel)

Severity Modeling of Extreme Insurance Claimsfor Tariffication

Sascha Desmettre(joint work with C. Laudagé, J. Wenzel)

OICA 2020 - Online International Conferencein 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 SeverityI Usually modeled via generalized linear models (GLMs) based on gamma

distribution (see e.g. [Ohlsson & Johansson (10), Wüthrich (17)]).

LimitationsI Extreme claim sizes in data The Gamma CDF is not heavy-tailed!

Concentration on body of distribution may lead toI bias predictionsI missing robustness in predictions

y Extreme Value Theory might help!


Modeling FrameworkClaim severity: Positive iid random RVs X1,X2, · · · ∼ XClaim frequency: Positive discrete RV N, where N ind. of XFeatures like car brand, age of driver or power of car affects damage.Vector of tariff features: R = (R1, . . . ,Rd ) with positive RVs Ri

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)


Censoring by Insured Sum

Primary insurers only pay for damages up to a specified amount.I Considered as tariff feature RI .

The actual damage Y may be larger than the insured sum.

y Claim severity is then given by

X := min(Y ,RI).

Insurer only observes realizations for X , i.e. right-censored data.

y Determine the distribution of Y based on this censored data.


Threshold Severity Model (TSM)

Split the distribution of Y at a certain threshold u > 0.

y Body and tail of the claim size distribution can be modeled separately.

Notation for a given tariff cell r :I Hr cdf for the body with parameter vector ΘHI Gr cdf for the tail with parameter vector ΘGI qr prob. of exceeding the given threshold u with parameter vector Θq

Assumptions to obtain a contiuous distribution function:I Hr (u; ΘH) > 0I Gr (u; ΘG ) = 0


Concrete Specification of the TSMDistribution function of Y with parameter vector Θ = (ΘH ,ΘG ,Θq):

Fr (y ; Θ) =

0 , y ≤ 0,(1− qr (Θq)) Hr (y ;ΘH )

Hr (u;ΘH ) , 0 < y ≤ u,(1− qr (Θq)) + qr (Θq) Gr (y ; ΘG) , y > u.

Note: Threshold u independent of tariff cell rHowever, the exceeding probability depends on insured sum:

qr (Θq) = 11 + e−(δ0+δI rI )

with Θq = δ.

Θ̂ =(

Θ̂H , Θ̂G , Θ̂q)is estimated via maximizing the log-likelihood.

y Obtain desired expectation for a tariff cell r by [X = min(Y ,RI)]:

EΘ̂ (min(Y ,RI)|R = r) =∫ rI

0yfr(y ; Θ̂

)dy + rI

(1− Fr

(rI ; Θ̂

)).


Recall: X := min(Y ,RI).S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 7 / 15

Estimators for Basic and Extreme Claim SizesUse concrete distributions for the conditional distribution functionsbelow and above the threshold for a tariff cell r .

Claim severity below the given threshold:I Use general regression methods, i.e., a generalized linear model (GLM).I Assume a gamma distribution for Hr .I In particular, the conditional distribution function

P (Y ≤ y |Y ≤ u,R = r) = Hr (y ; ΘH)Hr (u; ΘH) , 0 < y ≤ u,

describes a truncated gamma distribution.

Claim severity above the given threshold:I Apply the peaks-over-threshold approach from extreme value theory.I I.e., the conditional distribution function

P (Y ≤ y |Y > u,R = r) = Gr (y ; ΘG ) , 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 GLMWe assume that for all covariates r ∈ Rd

≥0 we have

(Y |Y ≤ u,R = r) ∼ G (φ, θr , u) with φ > 0 , θr < 0 ,

i.e., they are truncated gamma distributed with dispersion φ,threshold u and scale θr , depending on the tariff features r .

GLM to model conditional distribution function of X = min(Y ,RI):

P (X ≤ x |X ≤ u,R = r) = Hr (min(x , u); ΘH)Hr (u; ΘH) .

y

θ(bu(.,φ̂))′−−−−−−→ E (X |X ≤ u,R = r) g−→ α0 +

d∑i=1

ri αi ,

with

(bu (θ, φ))′ := b′ (θ) +

−u(− θu

φ

) 1φ−1

exp(θuφ

)γ(

1φ ,−

θuφ

) .


Extreme Claim SizesWe are looking at the excess distribution:

Fu(y , r) = P (Y ≤ y |Y > u,R = r) = Gr (y ; ΘG) , y > u.

Theorem of Pickands, Balkema and de Haan:

limu↑xF

sup0<x<xF−u

∣∣∣Fu (x)− Gξ,β(u) (x)∣∣∣ = 0.

Application to Y with ΘG = (ξ, β) provides approximation :

Gr (y ; ΘG) = Gξ,β;u(y) = Gξ,β (y − u) , y > u.

Conditional distribution function of X := min(Y ,RI):

P (X ≤ x |X > u,R = r) = Gξ,β (min (x , rI)− u) , x > u.


Simulation Study

Goal: Show that the TSM outperforms the classical gamma GLMwhen fitting to simulated claim sizes from other regression models.

y Use heavy-tailed regression models based on the log-normal and BurrType XII distributions to generate claim sizes.

Present and compare the predictions stemming from the gamma GLMand the TSM w.r.t. the different scenarios.

Setting:I Set the index of the insured sum to 1 and denote it by v (= r1 = rI).I Insured sums: 5 million, 20 million, 50 million.I Second tariff feature taking integer values from 1 to 10.

[E.g. mileage or the car’s power; denoted by w (= r2)].y 30 tariff cells in total.


Simulation Study: Log-Normal Regression

1 Simulate a normal random variable Z ∼ N (µ, σ) with meanµ = α0 + α1 v + α1 w and standard deviation σ > 0.

2 Obtain the log-normal random variable by X = eZ .

3 In order to obtain a significant influence of the insured sum, we usethe following parameters in this scenario:

α0 = 5.5, α1 = 4× 10−8, α2 = 0.02, σ = 2.75.

4 Compare the classical gamma GLM with the TSMin this log-normal setting.


Simulation Study: Burr Regression1 Simulate claim sizes from a Burr Type XII distribution, i.e,

Y ∼ Burr (β, λ, τ) with density fucntion

fB (y ;β, λ, τ) = λβλτy τ−1

(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 (α0 + α1 v + α1 w)) .

3 Parameter values in this scenario:

α0 = 8, α1 = 4× 10−8, α2 = 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 (µi) and predictivemean (µ̂i) of a specific tariff cell.Calculate (weighted) averages of the relative differences for everyscenario w.r.t. all tariff cells:

z̄1 := 130

30∑i=1

|µ̂i − µi |µi

, z̄2 :=30∑

i=1

mim|µ̂i − µi |

µi.

Simulated Claims Model z̄1 z̄2Log-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%


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.


Literature

C. Laudagé, S. Desmettre & J. Wenzel. “Severity Modeling of ExtremeInsurance Claims for Tariffication”. Insurance: Mathematics and Economics.88 (2019) 77–92.

T. Reynkens, R. Verbelen, J. Beirlant & K. Antonio. “Modelling censoredlosses using splicing: A global fit strategy with mixed Erlang and extremevalue distributions”. In: Insurance: Mathematics and Economics. 77 (2017)65-77.

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

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


Literature

E. Ohlsson, B. Johansson. “Non-Life Insurance Pricing with GenerlaizedLinear Models”. Springer. (2010)

M. Wüthrich. “Non-Life Insurance: Mathematics & Statistics”. Lecture Notesavailable at SSRN. (2017).

J. Garrrido, C. Genest, J. Schulz. “Generalized linear models for dependentfrequency and severity of insurance claims”. In: Insurance: Mathematics andEconomics. 70 (2016) 205-215.

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


Definition Gamma and Truncated Gamma DistributionFor parameters α, β > 0 and the parametrization φ = 1/α > 0 andθ = −β/α < 0, we call a RV X ∼ G (φ, θ) with DF, resp. CDF

fG (x ;φ, θ) := βα xα−1 e−βx

Γ (α) , x ≥ 0,

FG (x ;φ, θ) := γ (α, βx)Γ (α) , x ≥ 0,

gamma distributed with dispersion parameter φ and scaleparameter θ.

For a given threshold u ∈ R>0, a RV X ∼ G (φ, θ, u) with DF

fTG (x ;φ, θ, u) := fG (x ;φ, θ)FG (u;φ, θ)1(0,u] (x) , x ≥ 0,

is said to be truncated gamma distributed.


Simulated Claims in the TSM

Histogram of simulated claims (left) and simulated claims (right).Used parameters:

u = 106, ξ = 0.4, β = 2400000, φ = 0.5,α0 = 10, α1 = 0, α2 = 1/5.


Short Repetition: Generalized Linear Models (GLMS)For now, let X denote the size of a claim.

Basic idea: Density of X belongs to the exponential dispersion family:

fX (x ; θ, φ) = exp(xθ − b (θ)

φ\ω+ c(x , φ, ω)

),

whereI φ is the dispersion parameterI θ is the scaling parameter,I b(θ) is the cumulant function,I ω is a weight for e.g. the duration of a contract,I c(x , φ, ω) is a normalization constant for fX .

Special case gamma distribution: b(θ) = − log(−θ), i.e.,

fX (x ; θ, φ)exp(c(x , φ, ω)) = exp

(xθ + log(−θ)φ\ω

)S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 20 / 15

Short Repetition: Functionality of GLMsGeneralization of linear regression that allows for response variablesthat have error distribution models other than a normal distribution.

With the derivative of the CF b′ and link function g it holds:

θb′−→ E (X |R = r) g−→ α0 +

d∑i=1

ri αi

y Distributional behavior of the claim size X , which is parametrized byθ, is described by the estimated regressors αi of the covariates ri .

Logarithmic link function [leeds to multiplicative structure of premia]:

I E (X |R = r) = g−1(α0 +d∑

i=1ri αi ) = exp(α0 +

d∑i=1

ri αi )

I θ = (b′)−1 (E (X |R = r)) = (b′)−1(

exp(α0 +d∑

i=1ri αi )

)S. Desmettre Modeling of Extreme Insurance Claims April 28-29, 2020 21 / 15

Definition Generalized Pareto Distribution

For shape parameter ξ ∈ R, threshold u ∈ R and scale parameterβ ∈ R>0 we define the distribution function Gξ,β;u by

Gξ,β;u (x) =

1−(1 + ξ x−u

β

)− 1ξ , ξ 6= 0,

1− e−x−u

β , ξ = 0,

where x ≥ u if ξ ≥ 0 and x ∈[u, u − β

ξ

]if ξ < 0.

Then Gξ,β;u is called a generalized Pareto distribution (GPD).

We denote the density of a GPD by gξ,β;u and set Gξ,β := Gξ,β;0.


Severity Modeling of Extreme Insurance Claims for ... · Severity Modeling of Extreme Insurance Claims for Tariffication Author Sascha Desmettre (joint work with C. Laudagé, J. Wenzel)

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