In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level ...

We discuss a number of quantile-based risk measures (QBRMs) that have recently been developed in the financial risk and actuarial/insurance literatures. The measures considered include the Value-at-Risk (VaR), coherent risk measures, spectral risk measures, and distortion risk measures. We discuss and compare the properties of these different measures, and point out that the VaR is seriously fl...

We discuss a number of quantile-based risk measures (QBRMs) that have recently been developed in the financial risk and actuarial/insurance literatures. The measures considered include the Value-at-Risk (VaR), coherent risk measures, spectral risk measures, and distortion risk measures. We discuss and compare the properties of these different measures, and point out that the VaR is seriously fl...

We characterize time-consistent dynamic risk measures. In discrete time in context of uncertainty, we canonically associate a class of probability measures to any dynamic risk measure when the filtration comes from a process bounded at each time. Dynamic risk measures are conditional risk measures on a bigger space. In continuous time, we characterize time consistency, studying composition of c...

Significant changes in the insurance and financial markets are giving increasing attention to the need for developing a standard framework for risk measurement. Recently, there has been growing interest among insurance and investment experts to focus on the use of a tail conditional expectation because it shares properties that are considered desirable and applicable in a variety of situations....

The paper considers modeling of risk-averse preferences in stochastic programming problems using risk measures. We utilize the axiomatic foundation of coherent risk measures and deviation measures in order to develop simple representations that express risk measures via solutions of specially constructed stochastic programming problems. Using the developed representations, we introduce a new fa...

Significant changes in the insurance and financial markets are giving increasing attention to the need for developing a standard framework for risk measurement. Recently, there has been growing interest among insurance and investment experts to focus on the use of a tail conditional expectation because it shares properties that are considered desireable and applicable in a variety of situations...

This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The...

Mi Chen 1, Wenyuan Wang 2,4 and Ruixing Ming 3,* 1 School of Mathematics and Computer Science & FJKLMAA, Fujian Normal University, Fuzhou 350108, China; [email protected] 2 School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China; [email protected] 3 School of Statistics and Mathematics, ZheJiang GongShang University, Hangzhou 310018, China 4 School of Applied Mathemat...

Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analyses of the optimal behavior of a trader subject to VaR limits have produced a negative view of VaR as a risk-control tool. In particular, VaR limits have been found to induce increased risk exposure in some states and an increased probability o...