Time-Consistent Decisions and Temporal Decomposition of Coherent Risk Functionals

In management and planning it is a daily reality that more and more information is available gradually over time. It is well known that most risk measures (risk functionals) are time inconsistent in this situation 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 almost surely. The extended conditional risk functionals introduced in this paper allows a temporal decomposition of the initial risk functional in a way, which is consistent with the past and the future. The central result is a decomposition theorem, which allows recomposing the initial coherent risk functional by compounding the conditional risk functionals without loosing information or preferences. It follows from our results that the revelation of partial information in time must change the decision maker’s preferences—for consistency reasons—among the remaining courses of action. Further, in many situations the extended conditional risk functional allows ranking different policies, even based on incomplete information. In addition we show by counterexamples that without change-of-measures the only time consistent risk functionals are the expectation and the essential supremum.