Time consistency conditions for acceptability measures, with an application to Tail Value at Risk

Abstract

An acceptability measure is a number that summarizes information on monetary outcomes of a given position in various scenarios, and that, depending on context, may be interpreted as a capital requirement or as a price. In a multiperiod setting, it is reasonable to require that an acceptability measure should satisfy certain conditions of time consistency. Various notions of time consistency may be considered. Within the framework of coherent risk measures as proposed by Artzner et al. [Artzner, Ph., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Math. Fin. 9, 203–228], we establish implication relations between a number of different notions, and we determine how each notion of time consistency is expressed through properties of a representing set of test measures. We propose modifications of the standard Tail-Value-at-Risk measure that have stronger consistency properties than the original.