Range Value-at-Risk: Multivariate and Extreme Values

TL;DR

This paper extends the univariate Range Value-at-Risk to multivariate settings, introducing robust measures for heavy-tailed distributions, with theoretical properties, estimators, and empirical validation.

Contribution

It develops multivariate Range Value-at-Risk definitions, robustness properties, and empirical estimators, addressing limitations of traditional risk measures for heavy tails.

Findings

Robust multivariate risk measures are theoretically established.

Empirical estimators are shown to be accurate through numerical examples.

Closed-form expressions are derived for special cases in extreme value theory.

Abstract

The concept of univariate Range Value-at-Risk, presented by Cont et al. (2010), is extended in the multidimensional setting. Traditional risk measures are not well suited when dealing with heavy-tail distributions and infinite tail expectations. The multivariate definitions of robust truncated tail expectations are provided to overcome this problem. Robustness and other properties as well as empirical estimators are derived. Closed-form expressions and special cases in the extreme value framework are also discussed. Numerical and graphical examples are provided to examine the accuracy of the empirical estimators.