Modelling Dependence in Insurance Claims Processes with … · The author acknowledges support from an Ernst & Young Honours Scholarship Modelling Dependence in Insurance Claims Processes

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Modelling Dependence in Insurance ClaimsProcesses with Lévy Copulas

Benjamin Avanzi∗ Luke C. Cassar† Bernard Wong∗

Actuarial StudiesAustralian School of BusinessUniversity of New South Wales

ASTIN & AFIR ColloquiaMadrid, 19-22 June 2011

∗The authors acknowledge support from an Australian Actuarial Research Grant†

The author acknowledges support from an Ernst & Young Honours Scholarship

Modelling Dependence in Insurance Claims Processes with Lévy Copulas Avanzi, Cassar and Wong (2011)

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Outline

1 Motivation

2 Literature review

3 Lévy copulas for Compound Poisson processes

4 Presentation of some Lévy copulas

5 Fitting Lévy copulas to real data

6 Conclusion


Motivation 3/18

Outline

1 Motivation

2 Literature review




6 Conclusion


Motivation 3/18

Dependence between insurance claims processes

A single event may give rise to claims in multiple insurance riskprocesses due to dependence between types of claims or classesof business.

Such dependence has potential implications for pricing, reserving,solvency, and capital allocation of an insurance company.

Some standard approaches in the literature include (1) commonshock models, and (2) distributional copulas, applied to aggregateclaims/claim counts.

Lévy copulas straddles the advantages of these approaches. It isTime-consistent;Coherent modelling of dependence in frequency separate todependence in severity;Parsimonious;Make full use of the available data;Enables a "bottom-up" approach to model building.


Literature review 4/18

Outline

1 Motivation

2 Literature review




6 Conclusion


Literature review 4/18

Existing Literature

Theoretical results:original: Tankov (2003); Cont and Tankov (2004); Kallsen andTankov (2006)also: Barndorff-Nielsen and Lindner (2007); Bäuerle et al. (2008);Eder and Klüppelberg (2009)

Applications:Bregman and Klüppelberg (2005): ruin probabilities with twoprocesses (Clayton Lévy copula)Esmaeili and Klüppelberg (2010a,b): fitting of the Clayton LévycopulaBäuerle and Blatter (2011): optimal investment and reinsuranceproblemsOperational risk modelling: Böcker and Klüppelberg (2008);Biagini and Ulmer (2009); Böcker and Klüppelberg (2010)


Lévy copulas for Compound Poisson processes 5/18

Outline

1 Motivation

2 Literature review




6 Conclusion



Bivariate compound Poisson process

!

" # $%

constituted of unique (⊥)and common (‖) jumps:{

S1(t) = S⊥1 (t) + S‖1(t)

S2(t) = S⊥2 (t) + S‖2(t)

S‖1(t) and S‖2(t) with

intensity λ‖

Joint survival function ofcommon jumps F

‖(x1, x2)

(may be dependent)



Sklar’s Theorem for Lévy copulas

A Lévy copula C couples the marginal tail integrals and the joint tailintegral so that

C(U1(x1),U2(x2)) = U(x1, x2)

For the marginal compound Poisson processes Si(t) (i = 1,2)with Poisson parameter λi and jump size survival function F i(x),the tail integral Ui(x) is given by

Ui(x) = λiF i(x).

The joint tail integral measures jumps which occur simultaneouslyand is given by

U(x1, x2) = λ‖F‖(x1, x2).



Effect of the Lévy copula

On the common jumps:

λ‖ = C(λ1, λ2) (intensity of common jumps)

F‖(x1, x2) =

1λ‖

C(λ1F 1(x1), λ2F 2(x1)) (joint survival function)

F‖1(x) =

1λ‖

C(λ1F 1(x), λ2) (marginal survival function 1)

F‖2(x) =

1λ‖

C(λ1, λ2F 2(x)) (marginal survival function 2)

On the unique jumps:

λ⊥i = λi − λ‖ (intensity of unique jumps)

F⊥i (x) =

1λ⊥i

(λiF i(x)− λ‖F

‖i (x)

)(marginal survival function i)

(see Böcker and Klüppelberg, 2008)Modelling Dependence in Insurance Claims Processes with Lévy Copulas Avanzi, Cassar and Wong (2011)

Presentation of some Lévy copulas 8/18

Outline

1 Motivation

2 Literature review




6 Conclusion



Pure Common Shock Lévy copula

The Pure Common Shock (PCS) Lévy copula is given by

C(u1,u2) =δ(u1 ∧ λ1)(u2 ∧ λ2) + [u1 − δλ2(u1 ∧ λ1)]I{u2=∞}+ [u2 − δλ1(u2 ∧ λ2)]I{u1=∞},

for 0 ≤ δ ≤ min(

1λ1,

1λ2

).

where λ1 and λ2 are the Poisson parameters for the bivariatecompound Poisson process (when used in that context), and where δis the parameter which will determine the intensity of common jumps.

Leads to a bivariate compound Poisson process with dependencein frequency onlyUnique and common jump sizes are all independent andidentically distributed within the different processes



Archimedean models

Tankov (2003) shows that for some strictly decreasing and convexfunction φ,

C(u1,u2) = φ−1 (φ(u1) + φ(u2))

defines a bivariate Archimedean Lévy copula.(this can easily be extended to the multivariate case)

Example:The generator φ(u) = u−δ yields the Clayton Lévy copula

Cδ(u1,u2) =(

u−δ1 + u−δ2

)− 1δ.

The Clayton Lévy copula has unique properties that do not extend toother Lévy copulas in general.Bregman and Klüppelberg (2005); Böcker and Klüppelberg (2008); Esmaeili and Klüppelberg (2010a,b); Tankov (2003); Cont and Tankov (2004); Kallsen and Tankov (2006); Böcker and Klüppelberg (2010); Biagini and Ulmer (2009); Barndorff-Nielsen and

Lindner (2007); Bäuerle et al. (2008); Bäuerle and Blatter (2011); Eder and Klüppelberg (2009)



The Lévy copula density

is defined as

c(u1,u2) =∂2C(u1,u2)∂u1∂u2

, where ui = Ui(xi), i = 1,2.

It is useful to compare Lévy copulas:Its shape indicates the relative prevalence of common jumps atdifferent sizes in each component.It reflects the dependence in frequency, as the volume under thedensity on [0, λ1]× [0, λ2] is the expected number of commonjumps per unit time.

In what follows, consider a bivariate compound Poisson process withλ1 = 100, λ2 = 100 and a Lévy copula parameter such that λ‖ = 60.



Pure Common Shock and Clayton densities

Pure Common Shock Clayton



Archimedean model I & II densities

Archimedean model I Archimedean model II


Fitting Lévy copulas to real data 13/18

Outline

1 Motivation

2 Literature review




6 Conclusion



Description of the data

We use data provided by SUVA, a Swiss worker’s compensationcompany incorporated under public law.

The dataset consists of a random sample of 5% of claims from theconstruction sector for accidents incurred in 1999 (developed asat 2003).

It features two classes of claims: 2249 medical claims and 1099daily allowance claims.

1089 claims are common to both classes.

Compound Poisson processes S1(t) and S2(t), with dependencecharacterised by a Lévy copula, are fit to the SUVA dataset.

We use the Gumbel and Normal distribution for the log of claimsizes in the two classes.



Maximum likelihood estimation results

MLEs using new Lévy copulas are

Lévycopula

ln L δ̂ λ̂1 λ̂2 λ̂‖

AM1 8631.27 0.0025358 2239.42 1113.32 1093.74(0.0001110) (47.20) (32.75)

Clayton 8536.43 2.2632 2176.90 1066.27 984.16(0.0688161) (46.26) (31.68)

PCS 7845.03 0.0004406 2249.00 1099.00 1089.00(0.0000093) (47.42) (33.15)

Empirical: 2249.00 1099.00 1089.00



Comparing the fit of tail integrals

We can compare the theoretical tail integrals of unique and commonjumps for each class with the observed number of jumps above size x .

This provides a global idea of the goodness-of-fit, as it considers bothfrequency (initial value) and severity (shape).

For the Pure Common Shock Lévy copula:Medical (Unique)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200

Logarithm of medical claims (unique)

Tail

inte

gra

l

Medical (Common)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200

Logarithm of medical claims (common)

Tail

inte

gra

l

Allowance (Common)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200

Logarithm of allowance claims (common)

Tail

inte

gra

l



For the Clayton Lévy copula:Medical (Unique)

0 2 4 6 8 10 12 14

02

00

40

06

00

80

01

00

01

20

0


Tail

inte

gra

l

Medical (Common)

0 2 4 6 8 10 12 14

02

00

40

06

00

80

01

00

01

20

0

Logarithm of medical claims (common)Ta

il in

teg

ral

Allowance (Common)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200


Tail

inte

gra

l

For the Archimedean Model I Lévy copula:Medical (Unique)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200


Tail

inte

gra

l

Medical (Common)

0 2 4 6 8 10 12 14

02

00

40

06

00

80

01

00

01

20

0

Logarithm of medical claims (common)

Tail

inte

gra

l

Allowance (Common)

0 2 4 6 8 10 12 14

0200

400

600

800

1000

1200


Tail

inte

gra

l


Conclusion 17/18

Outline

1 Motivation

2 Literature review




6 Conclusion


Conclusion 17/18

Summary

Introduction to Lévy copulas

Application to the modelling of insurance risk processes

Development of new Lévy copula models

Comparison of different Lévy copula models

Fitting to a new set of real data

Discussion of the goodness-of-fit of the candidate models

Also: Formal decomposition of the trivariate case in the paper


Main References 18/18

Main References

Barndorff-Nielsen, O. E. and Lindner, A. M. (2007). Lévy Copulas: Dynamics and Transforms of Upsilon Type. ScandinavianJournal of Statistics, 34:298–316.

Bäuerle, N. and Blatter, A. (2011). Optimal control and dependence modeling of insurance portfolios with Lévy dynamics.Insurance: Mathematics and Economics, 48(3):398–405.

Bäuerle, N., Blatter, A., and Müller, A. (2008). Dependence properties and comparison results for Lévy processes. MathematicalMethods of Operations Research, 67:161–186.

Biagini, F. and Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin,39(2):735–752.

Böcker, K. and Klüppelberg, C. (2008). Modelling and Measuring Multivariate Operational Risk with Lévy Copulas. Journal ofOperational Risk, 3:3–27.

Böcker, K. and Klüppelberg, C. (2010). Multivariate models for operational risk. Quantitative Finance, pages 1–15.

Bregman, Y. and Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure.Scandinavian Actuarial Journal, 2005(6):462–480.

Cont, R. and Tankov, P. (2004). Financial Modelling With Jump Processes. Chapman & Hall/CRC, London.

Eder, I. and Klüppelberg, C. (2009). The quintuple law for sums of dependent Lévy processes. The Annals of Applied Probability,19(6):2047–2079.

Esmaeili, H. and Klüppelberg, C. (2010a). Parameter estimation of a bivariate compound Poisson process. Insurance:Mathematics and Economics, 47(2):224–233.

Esmaeili, H. and Klüppelberg, C. (2010b). Two-step estimation of a multivariate lévy process. Available athttp://www-m4.ma.tum.de/Papers/.

Kallsen, J. and Tankov, P. (2006). Characterisation of dependence of multidimensional Lévy processes using Lévy copulas.Journal of Multivariate Analysis, 97(7):1551–1572.

Tankov, P. (2003). Dependence structure of spectrally positive multidimensional Lévy processes. Available athttp://www.math.jussieu.fr/ tankov/.


MotivationLiterature reviewLévy copulas for Compound Poisson processesPresentation of some Lévy copulasFitting Lévy copulas to real dataConclusion

Documents

Modelling Dependence in Insurance Claims Processes with … · The author acknowledges support from an Ernst & Young Honours Scholarship Modelling Dependence in Insurance Claims Processes