Extreme and Inference for Tail Gini Functionals With Applications in Tail Risk Measurement

Abstract

Tail risk analysis focuses on the problem of risk measurement on the tail regions of financial variables. As one crucial task in tail risk analysis for risk management, the measurement of tail risk variability is less addressed in the literature. Neither the theoretical results nor inference methods are fully developed, which results in the difficulty of modeling implementation. Practitioners are then short of measurement methods to understand and evaluate tail risks, even when they have large amounts of valuable data in hand. In this article, we consider the measurement of tail variability under the tail scenarios of a systemic variable by extending the Gini’s methodology. As we are very interested in the limit of the proposed measures as the risk level approaches to the extreme status, we showed, by using extreme value techniques, how the tail dependence structure and marginal risk severity have influences on the limit of the proposed tail variability measures. We construct a nonparametric estimator, and its asymptotic behavior is explored. Furthermore, to provide practitioners with more measures for tail risk, we construct three coefficients/measures for tail risks from different views toward tail risks and illustrate them in a real data analysis. Supplementary materials for this article are available online.

KEYWORDS:

• Extreme Value Theory
• Gini mean difference
• Measure of variability
• Tail copula
• Tail risk

Acknowledgments

We thank the associate editor and two anonymous referees for various constructive comments and suggestions.

Supplementary Materials

The supplementary materials contain some additional simulation studies, real data analyses, and all the proofs of the theorems in the article.

Additional information

Funding

Yanxi Hou’s research was partially supported by National Natural Science Foundation of China grants 71803026 and 71991471, and by Science and Technology Commission of Shanghai Municipality Project 19511120700.