Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes

Abstract

We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time, and we show how this property manifests itself in the corresponding process of acceptance sets. For processes of coherent and convex monetary risk measures admitting a robust representation with sigma-additive linear functionals, we give necessary and sufficient conditions for time-consistency in terms of the representing functionals.

Tight Approximations of Dynamic Risk Measures

“…. , X T q into risk assessments at t. Following a large body of literature [Riedel, 2004, Artzner et al, 2007, Detlefsen and Scandolo, 2005, Roorda et al, 2005, Cheridito et al, 2006, Föllmer and Penner, 2006, Ruszczynski and Shapiro, 2006a, Ruszczyński, 2010, Cheridito and Kupper, 2011, we furthermore restrict the risk measurements at time t to only depend on the cumulative costs in the future, i.e., we take µ rt,T s : X T Ñ X t , and the risk of X rt,T s is µ rt,T s pX t`¨¨¨`XT q. While such measures have been criticized for ignoring the timing when future cashflows are received, they are consistent with the assumptions in many academic papers focusing on portfolio management under risk Chabakauri, 2010, Cuoco et al, 2008], as well as with current risk management practice [Jorion, 2006], and provide a natural, simpler first step in our analysis.…”

“…A central result in the literature [Riedel, 2004, Artzner et al, 2007, Detlefsen and Scandolo, 2005, Roorda et al, 2005, Cheridito et al, 2006, Roorda and Schumacher, 2007, Penner, 2007, Föllmer and Penner, 2006, Ruszczyński, 2010 is the following theorem, stating that any consistent measure has a compositional representation in terms of one-period risk mappings. Theorem 2.2.…”

“…. q˘, for a set of suitable mappings tµ τ u τ Ptt,...,T u (see, e.g., Epstein and Schneider [2003], Riedel [2004], Cheridito et al [2006], Artzner et al [2007], Roorda et al [2005], Föllmer and Penner [2006], Ruszczyński [2010]). Apart from yielding consistent preferences, this compositional form also allows a recursive estimation of the risk, and an application of the Bellman principle in optimization problems involving dynamic risk measures [Nilim and El Ghaoui, 2005, Iyengar, 2005, Ruszczynski and Shapiro, 2006a.…”

An axiomatic characterization of capital allocations of coherent risk measures

“…The extension to dynamic measures and allocations, which take into account the future evolution of losses, has been extensively studied in recent years. We refer to Cheridito et al (2006) and Cherny (2006) and the papers cited therein.…”

Robust Preferences and Robust Portfolio Choice

“…Indeed, dynamic consistency corresponds to the Bellman principle in dynamic programming and is the essential ingredient for the application of control methods. Recently, the dynamic consistency (8.9) of risk measures has been the subject of ongoing research; see, e.g., ARTZNER et al [2007], RIEDEL [2004], CHERIDITO et al [2004CHERIDITO et al [ , 2005CHERIDITO et al [ , 2006 Let A denote the set of all progressively measurable process π such that T 0 π 2 s ds < ∞ P-a.s. For π ∈ A we define…”