Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T14:26:59.524Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aas, K. & Berg, D. (2009). Models for construction of multivariate dependence: a comparison study. The European Journal of Finance, 15(7–8), 639–659.CrossRefGoogle Scholar
Avanzi, B., Cassar, L.C. & Wong, B. (2011). Modelling dependence in insurance claims processes with Lévy copulas. ASTIN Bulletin, 41(2), 575–609.Google Scholar
Barndorff-Nielsen, O.E. & Lindner, A.M. (2004). Some aspects of Lévy copulas, technical report no. 388, Sonderforschungsbereich 386, Munich.Google Scholar
Barndorff-Nielsen, O. & Lindner, A. (2007). Lévy copulas: dynamics and transforms of Upsilon type. Scandinavian Journal of Statistics, 34, 298–316.Google Scholar
Bäuerle, N. & Blatter, A. (2011). Optimal control and dependence modeling of insurance portfolios with Lévy dynamics. Insurance: Mathematics and Economics, 48, 398–405.Google Scholar
Bertoin, J. (1998). Lévy Processes. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK.Google Scholar
Biagini, F. & Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin, 39(2), 735–752.CrossRefGoogle Scholar
Böcker, K. & Klüppelberg, C. (2008). Modeling and measuring multivariate operational risk with Lévy copulas. The Journal of Operational Risk, 3(2), 3–27.Google Scholar
Böcker, K. & Klüppelberg, C. (2010). Multivariate models for operational risk. Quantitative Finance, 1, 1–15.Google Scholar
Bregman, Y. & Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scandinavian Actuarial Journal, 2005(6), 462–480.Google Scholar
Bücher, A. & Vetter, M. (2013). Nonparametric inference on Lévy measures and copulas. The Annals of Statistics, 41, 1485–1515.Google Scholar
Cassar, L.C. (2010). Dependence modelling in multivariate compound Poisson processes with Lévy copulas. Honours thesis. School of Risk and Actuarial Studies, UNSW Australia Business School.Google Scholar
Cont, R. & Tankov, P. (2004). Financial Modelling With Jump Processes. Chapman & Hall/CRC, London.Google Scholar
Eder, I. & Klüppelberg, C. (2009). The quintuple law for sums of dependent Lévy processes. The Annals of Applied Probability, 19(6), 2047–2079.Google Scholar
Esmaeili, H. & Klüppelberg, C. (2010). Parameter estimation of a bivariate compound Poisson process. Insurance: Mathematics and Economics, 47(2), 224–233.Google Scholar
Esmaeili, H. & Klüppelberg, C. (2011). Parametric estimation of a bivariate stable Lévy process. Journal of Multivariate Analysis, 102, 918–930.Google Scholar
Esmaeili, H. & Klüppelberg, C. (2013). Two-step estimation of a multi-variate Lévy process. Journal of Time Series Analysis, 34, 668–690.Google Scholar
Farkas, W., Reich, N. & Schwab, C. (2006). Anisotropic stable Lévy copula processes – analytical and numerical aspects, technical report no. 2006-08, Eidgenössische Technische Hochschule, Zurich.Google Scholar
Frees, E.W. & Valdez, E.A. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), 1–25.Google Scholar
Grothe, O. & Hofert, M. (2015). Construction and sampling of Archimedean and nested Archimedean Lévy copulas. Journal of Multivariate Analysis, 138, 182–198.Google Scholar
Grothe, O. & Nicklas, S. (2013). Vine constructions of Lévy copulas. Journal of Multivariate Analysis, 119, 1–15.Google Scholar
Hofert, M. (2008). Sampling Archimedean copulas. Computational Statistics and Data Analysis, 12, 5163–5174.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Kallsen, J. & Tankov, P. (2006). Characterisation of dependence of multidimensional Lévy processes using Lévy copulas. Journal of Multivariate Analysis, 97(7), 1551–1572.Google Scholar
Kurowicka, D. & Joe, H. (2011). Dependence Modeling Vine Copula Ha ndbook. World Scientific, Singapore.Google Scholar
McNeil, A.J. (2008). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78, 567–581.Google Scholar
McNeil, A.J., Frey, R. & Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, New Jersey.Google Scholar
McNeil, A.J. & Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l 1-norm symmetric distributions. Annals of Statistics, 37(5B), 3059–3097.CrossRefGoogle Scholar
Mikosch, T. (2006). Non-Life Insurance Mathematics: An Introduction with Stochastic Processes. Springer, Berlin Heidelberg.Google Scholar
Nadarajah, S. & Bakar, S. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014, 180–187.Google Scholar
Nelsen, R.B. (1999). An Introduction to Copulas. Springer, Springer-Verlag, New York.Google Scholar
Ross, S. (1983). Introduction to Stochastic Dynamic Programming. Academic Press, San Diego, CA.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.Google Scholar
Tankov, P. (2003). Dependence structure of spectrally positive multidimensional Lévy processes. Available online at the address http://www.proba.jussieu.fr/pageperso/tankov/.Google Scholar
Tao, J. (2011). Capturing non-exchangeable dependence in multivariate insurance claims processes with nested Lévy copulas, Honours thesis, School of Risk and Actuarial Studies, UNSW Australia Business School.Google Scholar