A distortion risk measure used in finance and insurance is defined as the expected value of potential loss under a scenario probability measure. In this paper, the tail distortion risk measure is introduced to assess tail risks of excess losses modeled by the right tails of loss distributions. The asymptotic linear relation between tail distortion and Value-at-Risk is derived for heavy tailed l...

In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (V aR), are studied from the perspective of the theory of coherent imprecise previsions. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalize...

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function. Taking advantage of this cocycle condition, we introduce a new methodology for the construction of time-consistent dynamic risk measures. Starting with BMOmartingales, ...

Starting with a time-0 coherent risk measure defined for “value processes”, we also define risk measurement processes. Two other constructions of measurement processes are given in terms of sets of test probabilities. These latter constructions are identical and are related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity as i...

Understanding the attitude to risk implicit within a risk measure sheds some light on the way in which decision makers perceive losses. In this paper, a two-stage strategy is developed to characterize the underlying risk attitude involved in a risk evaluation, when executed by the family of distortion risk measures. First, we show that aggregation indicators defined for discrete Choquet integra...

There are more to a risk-measure than being coherent. Both the popular VaR and the coherent Tail-VaR ignore useful information in a large part of the loss distribution; As a result they lack incentive for risk-management. I propose a new coherent risk-measure that utilizes information in the whole loss distribution and provides incentive for risk-management.

At present, all value at risk (VaR) implementations ± i.e., all risk measures of thè`maximum loss at a given level of con®dence'' type ± are based on the assumption that the portfolio mix will not change before the VaR horizon. This hypothesis may be unrealistic, especially when the VaR horizon is established by the regulators (BIS). At the opposite, we measure VaR dynamically, i.e., taking int...

We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem in which the decision maker uses a dynamic convex risk measure to evaluate future rewards.

We study on the minimal hedging risk for a bounded European contingent claim when we use a convex risk measure. We find the infimum of hedging risk by using a kind of min-max theorem, Also we show that this infimum is again regarded as a convex risk measure.