Dynamic quasi-concave performance measures

TL;DR

This paper introduces a broad class of dynamic performance measures based on conditional quasi-concavity, characterizes them via risk measures, and explores their time consistency and extension to dividend processes.

Contribution

It defines Conditional quasi-concave Performance Measures and Dynamic Performance Measures, providing their characterization, properties, and extension to dividend processes.

Findings

Characterization of CPMs via induced families of risk measures

Equivalence between time consistency and weak acceptance consistency

Extension of CPMs and DPMs to dividend processes

Abstract

We define Conditional quasi concave Performance Measures (CPMs), on random variables bounded from below, to accommodate for additional information. Our notion encompasses a wide variety of cases, from conditional expected utility and certainty equivalent to conditional acceptability indexes. We provide the characterization of a CPM in terms of an induced family of conditional convex risk measures. In the case of indexes these risk measures are coherent. Then, Dynamic Performance Measures (DPMs) are introduced and the problem of time consistency is addressed. The definition of time consistency chosen here ensures that the positions which are considered good tomorrow are already considered good today. We prove the equivalence between time consistency for a DPM and weak acceptance consistency for the induced families of risk measures. Finally, we extend CPMs and DPMs to dividend processes.