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Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Lévy copulas

Published online by Cambridge University Press:  11 December 2015

Benjamin Avanzi ,
Jamie Tao ,
Bernard Wong  and
Xinda Yang
Benjamin Avanzi
Affiliation:
School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia Département de Mathématiques et de Statistique, Université de Montréal, Canada, Montréal QC H3T 1J4
Jamie Tao
Affiliation:
School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia
Bernard Wong*
Affiliation:
School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia
Xinda Yang
Affiliation:
School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia
*
*Correspondence to: B. Wong, School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW, Sydney, NSW 2052, Australia. Tel: +61 2 9385 2837. Fax: +61 2 9385 1883. E-mail: bernard.wong@unsw.edu.au
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Abstract

The class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e.g. between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

Keywords

Lévy copulaExchangeabilityDependenceNested copulasInsurance claimsG22C10
Type
Papers
Information
Annals of Actuarial Science , Volume 10 , Issue 1 , March 2016 , pp. 87 - 117
Copyright
© Institute and Faculty of Actuaries 2015 

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