Browsing by Subject "Risk measure"

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Open Access
Dual representations for systemic risk measures
(Springer, 2020) Ararat, Çağın; Rudloff, B.
The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg–Noe model as well as those of the resource allocation and network flow models.
Open Access
Dual representations of quasiconvex compositions with applications to systemic risk
(2021-07) Aygün, Mücahit
The importance of measuring risk in an interconnected ﬁnancial system has been appreciated recently, due in part to the global ﬁnancial crisis. In the literature, systemic risk measures are generally represented by the composition of a univariate risk measure and an aggrega-tion function, a function that encodes the structure of the ﬁnancial network. Having dual representations for systemic risk measures is helpful in providing economic interpretations and oﬀering duality-based computational methods. For a univariate risk measure, a key assumption is that diversiﬁcation should not increase risk. The mathematical translation of this assumption was considered as convexity earlier in the history of risk measures. Recently, quasiconvexity has been considered as a more accurate translation of diversiﬁcation. For a single quasiconvex risk measure, dual representations are available in the literature based on the so-called penalty functions. The use of a quasiconvex risk measure in composition with a concave aggregation function results in a quasiconvex systemic risk measure, a multivariate functional on a space of random vectors. Motivated by the problem of ﬁnding dual representations for quasiconvex systemic risk measures, we study quasiconvex compositions in an abstract inﬁnite-dimensional setting. We calculate an explicit formula for the penalty function of the composition in terms of the penalty functions of the ingredient functions. The proof makes use of a nonstandard minimax inequality (rather than equality as in the standard case) that is available in the literature. In the last part of the thesis, we apply our results in concrete probabilistic settings for systemic risk measures, in particular, in the context of the Eisenberg-Noe clearing model. We also provide novel economic interpretations of the dual representations in these settings.