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Register a new userSystemic Risk: Conditional Distortion Risk Measures
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In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution ($Delta$CoD) measures as measures of systemic risk and analyze their properties and representations. The classes include the well-known conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES as special cases. Sufficient conditions are presented for two random vectors to be ordered by the proposed CoD-risk measures and distortion risk contribution measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive, and excess wealth orders of the marginals and canonical positive/negative stochastic dependence notions. Numerical examples are provided to illustrate our theoretical findings. This paper is the second in a triplet of papers on systemic risk by the same authors. In cite{DLZorder2018a}, we introduce and analyze some new stochastic orders related to systemic risk. In a third (forthcoming) paper, we attribute systemic risk to the different participants in a given risky environment.
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A growing body of studies on systemic risk in financial markets has emphasized the key importance of taking into consideration the complex interconnections among financial institutions. Much effort has been put in modeling the contagion dynamics of financial shocks, and to assess the resilience of specific financial markets - either using real network data, reconstruction techniques or simple toy networks. Here we address the more general problem of how shock propagation dynamics depends on the topological details of the underlying network. To this end we consider different realistic network topologies, all consistent with balance sheets information obtained from real data on financial institutions. In particular, we consider networks of varying density and with different block structures, and diversify as well in the details of the shock propagation dynamics. We confirm that the systemic risk properties of a financial network are extremely sensitive to its network features. Our results can aid in the design of regulatory policies to improve the robustness of financial markets.
Systemic risk arises as a multi-layer network phenomenon. Layers represent direct financial exposures of various types, including interbank liabilities, derivative- or foreign exchange exposures. Another network layer of systemic risk emerges through common asset holdings of financial institutions. Strongly overlapping portfolios lead to similar exposures that are caused by price movements of the underlying financial assets. Based on the knowledge of portfolio holdings of financial agents we quantify systemic risk of overlapping portfolios. We present an optimization procedure, where we minimize the systemic risk in a given financial market by optimally rearranging overlapping portfolio networks, under the constraints that the expected returns and risks of the individual portfolios are unchanged. We explicitly demonstrate the power of the method on the overlapping portfolio network of sovereign exposure between major European banks by using data from the European Banking Authority stress test of 2016. We show that systemic-risk-efficient allocations are accessible by the optimization. In the case of sovereign exposure, systemic risk can be reduced by more than a factor of two, with- out any detrimental effects for the individual banks. These results are confirmed by a simple simulation of fire sales in the government bond market. In particular we show that the contagion probability is reduced dramatically in the optimized network.
This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.
In our previous paper, A Unified Approach to Systemic Risk Measures via Acceptance Set (textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addition, the optimizer in the dual formulation provides a textit{risk allocation} which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.
In this note we consider a system of financial institutions and study systemic risk measures in the presence of a financial market and in a robust setting, namely, where no reference probability is assigned. We obtain a dual representation for convex robust systemic risk measures adjusted to the financial market and show its relation to some appropriate no-arbitrage conditions.
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