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Risk sharing under heterogeneous beliefs without convexity
TL;DR: In this paper, the authors consider the problem of finding Pareto-optimal allocations of risk among finitely many agents, where the associated individual risk measures are law invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures.
Abstract: We consider the problem of finding Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are law invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations. We do not assume their convexity though. A simple sufficient condition for the existence of Pareto optima is provided. Its proof combines local comonotone improvement with a Dieudonne-type argument, which also establishes a link of the optimal allocation problem to the realm of "collapse to the mean" results. Finally, we extend the results to capital requirements with multidimensional security markets.
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TL;DR: In this paper, the authors proposed the star-shaped acceptability indexes as generalizations of both the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013) in the same vein as Star-shaped risk measures generalize both the classes of coherent and convex risk measures.
Abstract: We propose the star-shaped acceptability indexes as generalizations of both the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013) in the same vein as star-shaped risk measures generalize both the classes of coherent and convex risk measures. We characterize acceptability indexes through star-shaped risk measures, star-shaped acceptance sets, and as the minimum of some family of quasi-concave acceptability indexes. Further, we introduce concrete examples under our approach linked to Value at Risk, risk-adjusted reward on capital, reward-based gain-loss ratio, monotone reward-deviation ratio, and robust acceptability indexes. We also expose an application regarding optimization of performance.
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Book•
01 Oct 2002
TL;DR: Convex analysis and set valued maps: A Review and a Review of Set Valued Maps and Set-Valued Maps: Existence and Stability in Optimization Problems, Minimum Monotone Maps and Variational Inequalities as discussed by the authors.
Abstract: Convex Analysis and Set Valued Maps: A Review.- Asymptotic Cones and Functions.- Existence and Stability in Optimization Problems.- Minimizing and Stationary Sequences.- Duality in Optimization Problems.- Maximal Monotone Maps and Variational Inequalities.
378 citations
TL;DR: A methodology for optimal design of financial instruments aimed to hedge some forms of risk that is not traded on financial markets and is reduced to a unique inf-convolution problem involving a transformation of the initial risk measures.
Abstract: We develop a methodology for optimal design of financial instruments aimed to hedge some forms of risk that is not traded on financial markets. The idea is to minimize the risk of the issuer under the constraint imposed by a buyer who enters the transaction if and only if her risk level remains below a given threshold. Both agents have also the opportunity to invest all their residual wealth on financial markets, but with different access to financial investments. The problem is reduced to a unique inf-convolution problem involving a transformation of the initial risk measures.
312 citations
01 Jan 2006
TL;DR: In this paper, a dual characterization of law invariant coherent risk measures, satisfying the Fatou property, was given, and it was shown that the hypothesis of Fatou properties may actually be dropped as it is automatically implied by the hypothesis for law invariance.
Abstract: S. Kusuoka [K01, Theorem 4] gave an interesting dual characterization of law invariant coherent risk measures, satisfying the Fatou property. The latter property was introduced by F. Delbaen [D 02]. In the present note we extend Kusuoka’s characterization in two directions, the first one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG 05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. Follmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.
253 citations
TL;DR: In this paper, the authors consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law invariant risk measures.
Abstract: We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents’ utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk. MSC 1991 subject classifications: Primary 91B06, 46A20; secondary 91B70.
231 citations
TL;DR: In this paper, a generalization of the concepts of convex and coherent risk measures to a multi-period setting, in which payoffs are spread over different dates, is proposed.
Abstract: In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.
199 citations