Two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivity for random variables, and the insurance risk measure, which assumes additivity for comonotonic random variables. We propose a new, data based, risk measure, called natural risk statistic, that is characterized by a new set of axioms. The new axioms only require subadditivity for comonotonic ran...

Investors use different approaches to select optimal portfolio. so, Optimal investment choices according to return can be interpreted in different models. The traditional approach to allocate portfolio selection called a mean - variance explains. Another approach is Markov chain. Markov chain is a random process without memory. This means that the conditional probability distribution of the nex...

We compare capital requirements derived by tail conditional expectation (TCE) with those derived by tail conditional median (TCM) and find that there is no clear-cut relationship between these two measures in empirical data. Our results highlight the relevance of TCM as a robust alternative to TCE, especially for regulatory control.

The paper focuses on multi-period aspects of risk functionals. It discusses properties, provides dual representations and offers methods for constructing multiperiod risk functionals. On the way, existence results and representations for conditional risk mappings are derived. In particular, conditional, multi-period and nested versions of the average value-at-risk are given. Finally, the import...

This article develops a safetyanalysis method for stochastic systems that is sensitive to the possibility and severity of rare harmful outcomes. We define risk-sensitive safe sets as sublevel sets solution nonstandard optimal control problem, where random maximum cost assessed via Conditional V...

Risk measure forecast and model have been developed in order to not only provide better but also preserve its (empirical) property especially coherent property. Whilst the widely used risk of Value-at-Risk (VaR) has shown performance benefit many applications, it is fact a measure. Conditional VaR (CoVaR), defined as mean losses beyond VaR, one alternative measures that satisfies There several ...