It is well known that most risk measures (risk functionals) are time inconsistent in the following sense: It may happen that today some loss distribution appears to be less risky than another, but looking at the conditional distribution at a later time, the opposite relation holds. In this article we demonstrate that this time inconsistency disappears if the conditional functionals are defined in an extended manner, i.e. are evaluated under a specific change-ofmeasure. Information, which is revealed gradually in time, is an every day reality in management and planning. It follows from our results that – for consistency reasons – the revelation of partial information in time must dramatically change a decision maker’s preferences among the remaining courses of action ([Eck12]). The extended conditional risk functionals introduced here allow a temporal decomposition of the initial risk functional in a way, which is consistent with the past and the future. The central result of this paper is a decomposition theorem, which allows recomposing the initial positively homogeneous risk functional by compounding the conditional risk functionals without loosing information or preferences. Ín addition we show by counterexamples that without changeof-measures the only time consistent risk functionals are the expectation and the essential supremum.
Citation
University of Vienna, Department of Staistics and Operations Research