On the distortion of a copula and its margins

Abstract

This article examines the notion of distortion of copulas, a natural extension of distortion within the univariate framework. We study three approaches to this extension: (1) distortion of the margins alone while keeping the original copula structure; (2) distortion of the margins while simultaneously altering the copula structure; and (3) synchronized distortion of the copula and its margins. When applying distortion within the multivariate framework, it is important to preserve the properties of a copula function. For the first two approaches, this is a rather straightforward result; however, for the third approach, the proof has been exquisitely constructed in Morillas (2005). These three approaches unify the different types of multivariate distortion that have scarcely scattered in the literature. Our contribution in this paper is to further consider this unifying framework: we give numerous examples to illustrate and we examine their properties particularly with some aspects of ordering multivariate risks. The extension of multivariate distortion can be practically implemented in risk management where there is a need to perform aggregation and attribution of portfolios of correlated risks. Furthermore, ancillary to the results discussed in this article, we are able to generalize the formula developed by Genest & Rivest (2001) for computing the distribution of the probability integral transformation of a random vector and extend it to the case within the distortion framework. For purposes of illustration, we applied the distortion concept to value excess of loss reinsurance for an insurance policy where the loss amount could vary by type of loss.

Keywords:

• Multivariate distortion
• Ordering of risks
• Probability integral transformation
• Excess of loss reinsurance

Acknowledgements

The authors would like to thank Christian Genest, Roger Nelsen, and Shaun Wang for sharing and communicating their ideas about distortion of copulas during the conceptualization stages of developing this paper; and Michel Denuit for assisting us clarify some concepts on supermodular functions. Dr. Y. Xiao acknowledges the financial support of the National Philosophy and Social Science Foundation grant (No. 07BTJ002) and the National Nature Science Foundation grant (No. 10871201) of China. We are also thankful for the suggestions by an anonymous reviewer and the Editor who helped to improve the paper.