Inspired by the recent developments in risk sharing problems for the value at risk (VaR), the expected shortfall (ES), and the range value at risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the inf-convolution and Pareto-optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR, ES, and RVaR. Optimal allocations are obtained in the settings of elliptical models and model uncertainty. In particular, several results are established for tail risk measures in the presence of model uncertainty, which may be of independent interest outside the framework of risk sharing. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk sharing.

Measuring systemic risk plays an important role in financial risk management to control systemic risk. By means of a vine copula grouped-CoES method, this paper aims to measure the systemic risk of Chinese financial markets. The empirical study indicates that the banking industry has a low risk and a strong ability to resist risks, but also contributes the most of the systemic risk. On the other hand, insurance companies and securities have high ES but low $\Delta$CoES, indicating their low risk tolerance and small contribution to the systemic risk. Furthermore, this study employs a sliding window in Monte Carlo simulation to forecast systemic risk. The findings of this paper suggest that different types of financial industries should adopt different systemic risk measures.

Capital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

In this article, we introduce three new classes of multivariate risk statistics, which can be considered as data-based versions of multivariate risk measures. These new classes are multivariate convex risk statistics, multivariate comonotonic convex risk statistics and multivariate empirical-law-invariant convex risk statistics, respectively. Representation results are provided. The arguments of proofs are mainly developed by ourselves. It turns out that all the relevant existing results in the literature are special cases of those obtained in this article.