Dual representations of quasiconvex compositions with applications to systemic risk

Abstract

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.