Study of Some Problems of Multivariate Risk Measures

Abstract

In the past two decades, research on financial risk measures and risk management has become a hot research field in financial mathematics, among which the axiomatic method is one of the most important methods in financial risk measures research The study of multivariate risk measures refers to measuring the risk of multiple financial positions or measuring the financial risk of multiple institutions in the financial system. Building upon the uni-variate risk measures theories by Artzner et al., Föllmer and Schied, and Frittelli and Rosazza Gianin, multivariate risk measures have evolved in three key directions: scalar-valued, set-valued, and systemic risk measures. Burgert and Rüschendorf were among the first to investigate consistent (convex) scalar-valued multivariate risk measures, establishing axioms for portfolio risk measures and deriving representation theorems. Jouini et al. introduced axiomatic vector-valued consistent risk measures, while Hamel and Hamel and Heyde delved into convex set-valued measures. Chen et al. brought forth axiomatic methods for systemic risk measures, constructing an axiomatic framework for positively homogeneous systemic risk measures. Furthermore, Kromer et al. expanded Chen’s framework to cover convex systemic risk measures and general measurable spaces. This development marked the foundation of a well-structured axiomatic framework for multivariate risk measures. Scholars have conducted extensive research on multivariate consistent risk measures and multivariate convex risk measures from different perspectives.

This doctoral dissertation primarily focuses on systemic risk measures, examining several aspects of multivariate risk measures, including the construction of an axiomatically based framework for strong comonotonic additive systemic risk measures. It presents a systemic risk modeling approach and conducts empirical research on the impact of China’s secondary industry share on systemic risk. The study is organized as follows:

Chapter 1 reviews the historical development of risk measures for financial positions. It extracts axiomatic conditions from economic finance principles, highlighting the diverse manifestations of these conditions when extending the analysis from single positions to portfolios. This work centers on multivariate systemic risk measures, building upon previous important findings such as uni-variate coherent (convex) risk measures, uni-variate acceptable criteria, positively homogeneous systemic risk measures, and convex systemic risk measures –all established within the axiomatic framework.

Chapter 2 introduces a new class of systemic risk measures, which we term strong comonotonic additive systemic risk measures. It begins by introducing the concept through novel axioms, followed by the development of its structural decomposition. When the single company risk measures and aggregation functions are both convex, dual representations are also provided. Illustrative examples justify the proposed risk measures and compare them with existing ones.

Chapter 3 studies the systemic financial risks of the banking system and employs a structured analytical approach to categorize bank risks into inter-bank loans and non-inter-bank loans, considering the inter-liability networks for systemic risk modeling. The empirical results show significant differences in the contributions of these two categories to systemic financial risk, and interbank lending contributes more to the risk of the banking system than non interbank lending, providing valuable insights for regulatory and risk management practices.

Chapter 4 primarily studies corresponding Monte Carlo scenario simulation methods for strong comonotonic additive systemic risk measures. Secondly, we use fixed effect models to examine the relationship between China’s secondary industry share and systemic risk. Given the recent surge of real estate companies investing in financial institutions and forming or acquiring industry funds, this study incorporates the risk of the real estate sector, given its financial characteristics, into the systemic risk and comprehensively calculates the systemic financial risk measured by VaR, MES, CoVaR, and Δ CoVaR indicators, and explored the relationship between the proportion of China’s secondary industry and the systemic financial risk calculated by the above indicators. Through empirical analysis, the findings demonstrate that an increase in China’s secondary industry share corresponds to heightened systemic risk. This insight holds significant implications for a deeper understanding of China’s economic system, optimizing industrial structure, and boosting the level of systemic risk management.

Date of Award	6 Mar 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Xiang ZHOU (Supervisor) & Yijun HU (External Supervisor)