Multi-period risk measures and optimal investment policies

Zhiping Chen, Giorgio Consigli, Jia Liu, Gang Li, Tianwen Fu, Qianhui Hu

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

8 Scopus citations


This chapter provides an in-depth overview of an extended set of multi-period risk measures, their mathematical and economic properties, primarily from the perspective of dynamic risk control and portfolio optimization. The analysis is structured in four parts: the first part reviews characterizing properties of multi-period risk measures, it examines their financial foundations, and clarifies cross-relationships. The second part is devoted to three classes of multiperiod risk measures, namely: terminal, additive and recursive. Their financial and mathematical properties are considered, leading to the proposal of a unifying representation. Key to the discussion is the treatment of dynamic risk measures taking their relationship with evolving information flows and time evolution into account: after convexity and coherence, time consistency emerges as a key property required by risk measures to effectively control risk exposure within dynamic programs. In the third part, we consider the application of multi-period measures to optimal investment policy selection, clarifying how portfolio selection models adapt to different risk measurement paradigms. In the fourth part we summarize and point out desirable developments and future research directions. Throughout the chapter, attention is paid to the state-of-the-art and methodological and modeling implications.

Original languageBritish English
Title of host publicationInternational Series in Operations Research and Management Science
Number of pages34
StatePublished - 2017

Publication series

NameInternational Series in Operations Research and Management Science
ISSN (Print)0884-8289


  • Bellman’s principle
  • Dynamic risk control
  • Information processes
  • Multi-period risk measures
  • Portfolio optimization
  • Recoursive risk measures
  • Time-consistency


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