New class of distortion risk measures and their tail asymptotics with emphasis on VaR

TL;DR

This paper introduces new classes of distortion risk measures using compositing, mixing, and copula-based methods, and investigates their tail subadditivity properties, especially focusing on VaR in bounded risk supports.

Contribution

It proposes three novel methods for constructing distortion risk measures and analyzes the tail subadditivity of VaR, highlighting conditions where VaR is subadditive in the tail.

Findings

VaR is tail subadditive for risks with bounded support.

New classes of distortion risk measures are constructed via compositing, mixing, and copula methods.

Examples illustrate the tail subadditivity properties of VaR.

Abstract

Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of copula. Subadditivity is an important property when aggregating risks in order to preserve the benefits of diversification. However, Value at risk (VaR), as the most well-known example of distortion risk measure is not always globally subadditive, except of elliptically distributed risks. In this paper, instead of study subadditivity we investigate the tail subadditivity for VaR and other distortion risk measures. In particular, we demonstrate that VaR is tail subadditive for the case where the support of risk is bounded. Various examples are also presented to illustrate the results.