On Joint Marginal Expected Shortfall and Associated Contribution Risk Measures

TL;DR

This paper introduces the joint marginal expected shortfall (JMES) as a new systemic risk measure, along with contribution measures, and compares their properties and performance with existing measures through theoretical analysis and real stock market data.

Contribution

It proposes the JMES and related contribution measures, providing a novel approach to quantify systemic risk and its spillover effects, with theoretical properties and empirical validation.

Findings

JMES captures risk contagion more effectively than traditional measures.

The proposed measures have desirable properties like monotonicity and additivity.

Numerical examples and stock market analysis demonstrate the measures' practical usefulness.

Abstract

Systemic risk is the risk that a company- or industry-level risk could trigger a huge collapse of another or even the whole institution. Various systemic risk measures have been proposed in the literature to quantify the domino and (relative) spillover effects induced by systemic risks such as the well-known CoVaR, CoES, MES and CoD risk measures, and associated contribution measures. This paper proposes another new type of systemic risk measure, called the joint marginal expected shortfall (JMES), to measure whether the MES of one entity's risk-taking adds to another one or the overall risk conditioned on the event that the entity is already in some specified distress level. We further introduce two useful systemic risk contribution measures based on the difference function or relative ratio function of the JMES and the conventional ES, respectively. Some basic properties of these proposed measures are studied such as monotonicity, comonotonic additivity, non-identifiability and non-elicitability. For both risk measures and two different vectors of bivariate risks, we establish sufficient conditions imposed on copula structure, stress levels, and stochastic orders to compare these new measures. We further provide some numerical examples to illustrate our main findings. A real application in analyzing the risk contagion among several stock market indices is implemented to show the performances of our proposed measures compared with other commonly used measures including CoVaR, CoES, MES, and their associated contribution measures.