One of the key components of financial risk management is risk measurement. This typically requires modeling …

A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES) in financial applications … for the analysis is established with the quadrangle theory of risk functions. We derived relationships between elements of … CVaR quadrangle and mixed-quantile quadrangle for discrete distributions with equally probable atoms. The deviation in the …

standard sample sizes. In an application to international government bonds, we detect a high tail{risk and low return situation … during the last decade which can essentially be attributed to increased higher-order tail risk. We also illustrate the … empirical consequences from ignoring higher-dimensional tail risk. …