Characterization and Construction of Sequentially Consistent Risk Measures

In dynamic risk measurement the problem emerges of assessing the risk of a financial position at different times. Sufficient conditions are provided for conditional coherent risk measures, in order that the requirements of acceptance, rejection and sequential consistency are satisfied. It is shown that these conditions are often violated for standard methods of updating. A method is consequently proposed for constructing a sequentially consistent risk measure, which entails the modification of the set of probability measures used to obtain the risk assessment at an initial time. This is demonstrated for the coherent entropic risk measure and for the class of Choquet risk measures, which generalizes the well-known TVaR. Finally we consider the situation where the term of risk exposures is longer than the time horizon used in solvency assessment. Then, regulation such as Solvency II requires replacing the financial position itself with its fair value at the time horizon. We show that in this setting acceptance consistency can be preserved, though the same is not true about rejection consistency