Research On The Properties Of Distortion Risk Measures

In the field of statistics,the subadditivity of risk measure is a strong condition.When dealing with fat tail losses(i.e.,low-frequency and large-loss events),risk managers are especially interested in the tail region.Fat right tails have been extensively studied in insurance and finance.To the best of our knowledge,however,previous studies of the sub-additivity of risk measures in the tail region are scarce.Belles-Sampera,Guillen,Santolino(2014)proposed a new family of distortion risk measures,GlueVaR,which popularized the traditional quantile-based approach to risk measurement and gave a completely new definition.The commonly used T VaR and VaR are included.Moreover,the GlueVaR risk measure satisfies tail sub-additivity propert.y under certain conditions.In the tail sub-additivity property all definitions and properties are studies of the right tail region of the α-quantile.In this paper we study the left part of theα-quantile and call it the accumulation region.The main results of this paper is to prove that when the GlueVaR,risk measure distortion function kβ,αh1,h2(u)is concave in [0,α],the GlueVaR risk moeasure also satisfies sub-additivity in the accumulation region and call it accumulation sub-additivity.And during the proof process,it was found that the accumulation sub-additivity is not just for distortion functions of GlueVaR kβ,αh1,h2(u),as long as it is in [0,α] is a coneave function in [0,α].So we get that as long as the distortion function g(u)is concave in [0,α],the corresponding distortion risk measure is acciumlation subadditivity.This is a further generalization of the theorem.At the same time,it is found that TVaR satisfies accumulation sub-additivity,and TVaR satisfies tail sub-additivity,so TVaR satisfies sub-additivity in the entire region.