Time-Consistent Decisions and Temporal Decomposition of Coherent Risk Functionals

In management and planning it is commonplace for additional information to become available gradually over time. It is well known that most risk measures (risk functionals) are time inconsistent in the following sense: it may happen that at a given time period, some loss distribution appears to be less risky than another one, 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 enable a temporal decomposition of the initial risk functional that can be used to ensure consistency between past and future preferences. The central result is a decomposition theorem, which allows recomposing the initial coherent risk functional by compounding the conditional risk functionals without losing 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 of different policies, even based on incomplete information. In addition, we use counterexamples to show that without change-of-measures, the only time-consistent risk functionals are the expectation and the essential supremum.