Convex risk measures and the dynamics of their penalty functions

Abstract

We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

Representation results for law invariant time consistent functions

“…We here are mainly interested in law invariant risk measures which are studied for instance in Kusuoka [28], Frittelli and Rosazza Gianin [19], Jouini et al [26] and Cheridito and Li [10]. For dynamic risk measures their representations and related concepts such as time consistency, we refer to Artzner et al [3], Cheridito et al [8], Cheridito and Kupper [9], Föllmer and Penner [17] and the references therein.…”

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…Indeed, while necessary and sufficient conditions for temporal consistency of convex risk measures were established in [20], see also e.g. [6,28], such results are yet lacking for the quasiconvex case. 13 Finally, recall that the risk preferences of the investor are modelled via a standard continuous and concave utility function in (2.7).…”

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.…”