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Register a new userEquilibrium in risk-sharing games
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Added by
Constantinos Kardaras
Publication date
2014
fields
Financial
and research's language is
English
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The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents best response problems have unique solutions. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for arbitrary number of agents and be unique in the two-agent game. In equilibrium, agents declare beliefs on future random outcomes different than their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (among other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.
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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, individual risk assessments are assumed to be 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.
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.
The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite sets of risk measures. In this study, we extend the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite) set of alternatives. The intuitive principle of this approach a generalization of convex weights in the finite case. Subsequently, we extensively generalize known properties and results to this framework. Specifically, we investigate the preservation of properties, dual representations, optimal allocations, and self-convolution.
We propose a generalization of the classical notion of the $V@R_{lambda}$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{lambda}$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $mathcal{P}(% mathbb{R}).$
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