Recent Developments in Tail Risk Measures

Dear Colleagues,

Tail risk measures are becoming an important attribute in the regulatory guidelines of banking, financial and insurance systems. Their popularity can be explained by their high sensitivity to low probability—extreme financial losses, such as bankruptcy, natural catastrophes, and so on. One of the most prominent of such measures, which is very popular in the economic, banking and insurance literature, is the quantile of a loss distribution corresponding to a high level of confidence, called Value-at-Risk (VaR). However, recently other risk measures quantifying the risk of the tail, being in some sense a projection of its expectation, have gained increasing popularity. These include, among others, the tail condition expectation (TCE), tail VaR (TVaR), expected shortfall (ES) and conditional VaR (CVaR). These risk measures, meet, in fact, the Euler capital allocation principles and provide the required capital allocation aimed at offsetting extreme losses. At the same time, the projection onto the tail serves the quantification of not only the loss but also the discrepancy of the loss from its expectation by means of covariance, skewness, kurtosis and higher moments.

Against this background, this Special Issue aims to assemble high-quality papers that offer a discussion of the state-of-the-art of the developments in tail risk measures, both from the theoretical and practical perspectives. We welcome papers related, but not limited to, the following topics—all associated with the tail of the loss distribution.

• Quantifying random risks and losses in the tails
• Projection onto the tails of dispersion, covariance and other dependence measures
• Translation invariance and positive homogeneous risk measures 
• Capital allocation of aggregate risks
• Optimal portfolio selection
• Robust tail risk measures
• Asymptotics with respect to a high level of confidence
• Multivariate tail risk measures

Prof. Dr. Zinoviy Landsman
Prof. Dr. Udi Makov
Dr. Tomer Shushi
Guest Editors

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• expected shortfall
• tail condition expectation
• tail variance
• portfolio selections
• translation invariant
• positive homogeneity
• allocation
• projection onto the tail