Seven Proofs for the Subadditivity of Expected Shortfall

被引：21

作者：

Embrechts, Paul ^{[2
]}

Wang, Ruodu ^{[1
]}

机构：

[1] Univ Waterloo, Dept Stat & Actuarial Sci, Waterloo, ON N2L 3G1, Canada

[2] Swiss Fed Inst Technol, Dept Math, RiskLab, CH-8092 Zurich, Switzerland

来源：

DEPENDENCE MODELING | 2015年 / 3卷 / 01期

基金：

加拿大自然科学与工程研究理事会;

关键词：

Expected Shortfall; TVaR; subadditivity; comonotonicity; Value-at-Risk; risk management; education;

D O I：

10.1515/demo-2015-0009

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof. With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.

引用

页码：126 / 140

页数：15

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