Time consistency conditions for acceptability measures, with an application to Tail Value at Risk

Abstract

An acceptability measure is a number that summarizes information on monetary outcomes of a given position in various scenarios, and that, depending on context, may be interpreted as a capital requirement or as a price. In a multiperiod setting, it is reasonable to require that an acceptability measure should satisfy certain conditions of time consistency. Various notions of time consistency may be considered. Within the framework of coherent risk measures as proposed by Artzner et al. [Artzner, Ph., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Math. Fin. 9, 203-228], we establish implication relations between a number of different notions, and we determine how each notion of time consistency is expressed through properties of a representing set of test measures. We propose modifications of the standard Tail-Value-at-Risk measure that have stronger consistency properties than the original.

Tight Approximations of Dynamic Risk Measures

“…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.…”

“…In particular, even if µ i corresponded to the same primitive risk measure µ, the overall compositional measure 5 µ r0,T s " µ˝µ˝¨¨¨˝µ would bear no immediate relation to µ. As an example, when µ i " AVaR ε , the overall µ r0,T s corresponds to the so-called "iterated CTE" [Hardy and Wirch, 2004, Brazauskas et al, 2008, Roorda and Schumacher, 2007, which does not lend itself to the same simple interpretation as a single AVaR. Furthermore, practitioners often feel that the overall risk measure µ r0,T s is overly conservative, since it composes what are already potentially conservative risk evaluations backwards in time -for instance, for the iterated TCE, one is taking tail conditional expectations of tail conditional expectations.…”

“…For instance, if µ I " AVaR -the case considered in [Roorda and Schumacher, 2007] -the compositional measure corresponds to the socalled "iterated tail-CTE", which takes tail conditional expectations of quantities that are already tail conditional expectations. We show by means of an example that this informal belief is actually not true, even in the case of AVaR.…”

Weakly time consistent concave valuations and their dual representations

“…This indicates that a conditional requirement in a future state, no matter how well chosen, does not provide sufficient information about the conditional position in that state if it comes to determining a reasonable capital requirement today. Under weak time consistency, the accumulation of conservatism can be avoided, as shown in [12].…”

“…It prescribes by definition a unique update that is obtained by checking the acceptability of a position restricted to all possible events at a future date. The notions of sequential and conditional consistency have been introduced in [12] in a simple setting for coherent risk measures on a finite outcome space. We refer to [13] for further motivation of these concepts in a more general setting that includes nonconvex risk measures.…”