Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation

Abstract

Consider a portfolio of n obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the n obligors jointly follow a multivariate regular variation structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Multivariate models involving Archimedean copulas and mixtures are revisited. As applications, we derive asymptotic estimates for the value at risk and conditional tail expectation of the loss given default and compare them with the traditional empirical estimates.

Acknowledgments

The authors would like to thank Johan Segers at Université catholique de Louvain for providing them with the proof of Lemma 5.3, thank Alexandru V. Asimit at City University London for his help on the numerical studies, and thank two anonymous referees for their useful comments. The research for this project was sponsored by The Actuarial Foundation (TAF) and the Casualty Actuarial Society (CAS) through the 2012 Individual Grants Competition.