# Abstracts

**Consistency of risk measure estimates**

*Mark Davis (Imperial College London, UK)*

Thursday June 5, 11:30-12:00 | session 7.3 | Risk Measures | room EF

There has recently been renewed debate about the relative merits of the VaR and CVaR risk measures, particularly since the Basel Committee issued a recommendation that the former be replaced by the latter for bank regulation. The standard objections to VaR are that it is not coherent and takes no account of the magnitude of large losses. Recently, CVaR has in turn been criticised on the grounds of computational instability (Cont, Deguest and Scandolo, 2010) and for not being ‘elicitable’ (Gneiting, 2007, Ziegel 2013). We take a different approach to this question, addressing it from the point of view of probability forecasting and A.P. Dawid’s ‘prequential statistics’ (Dawid 1984). We introduce the concept of ‘consistency’ of a risk measure estimate and discuss its relation to elicitability. We study consistency for quantile estimates and expectation-based estimates like CVaR, and show that there are sharp differences between the two, quantile estimates being consistent under much more general conditions than any other risk measure. We use martingale limit theory to study the expectation-based case. Finally, we define a simple but remarkably effective algorithm for quantile prediction of financial data, designed to achieve the consistency criterion.

**On Robust Duality and Superhedging under Model Uncertainty**

*Michael Kupper (University of Konstanz, Germany)*

Thursday June 5, 12:00-12:30 | session 7.3 | Risk Measures | room EF

We focus on the robust representation of convex risk measures when there is no reference probability measure. Under a suitable tightness assumption it has been shown by Föllmer and Schied that convex risk measures on the space of bounded continuous functions have a robust representation in terms of probability measures. In this presentation we provide some representation results in the general case, when the reference and test probabilities are not tight. As an application we discuss the robust representation of the superhedging problem under model uncertainty.

**Equilibrium in Risk Sharing Games**

*Michail Anthropelos (University of Piraeus, Greece)*

Thursday June 5, 12:30-13:00 | session 7.3 | Risk Measures | room EF

We consider a market of several financial agents who aim to increase their expected utilities by optimally sharing their random endowments. In the classical optimal risk sharing scheme, agents with risky endowments design and price new financial contracts, the trading of which is mutually beneficial. The induced Arrow-Debreu equilibrium is based on the assumption that agents do not behave strategically and cannot apply any form of market power. The main goal of this paper is to relax this (unrealistic, in certain cases) assumption and examine how the risk sharing transaction changes when agents act strategically while negotiating the design of risk-sharing contracts.

We first focus on the actions of an individual agent who knows the sharing rules and the endowments that the other agents are willing to share. The agent's best response to this situation is to report as endowment the random quantity that maximises the agent's expected utility when the sharing transaction is applied (which is not necessarily equal to the true risk exposure that the agent has). Under exponential utility preferences, we provide an implicit formula of the so-called best endowment response, which implies (among other things) that reported endowments always differ for the agents' true risk exposures.

If all participating agents apply similar strategic behaviour, the market equilibrates at a Nash-type equilibrium, which can be considered as the outcome of negotiation among agents. Assuming that agents know only their own true risky endowments, we convert the problem of finding the Nash risk sharing equilibrium to a finite dimensional one, establishing a tractable characterisation of it. From this characterisation, it follows that (a) sharing contracts are different than the optimal ones in any non-trivial situation, (b) the equilibrium sharing contracts are endogenously bounded (even when the agents' endowments are not) and (c) the Nash equilibrium results in strictly positive risk sharing inefficiency. Furthermore, for the two-agent case, existence and the uniqueness of Nash equilibrium is proved and an implicit formula of the equilibrium sharing contracts is analysed and discussed.

Although the aggregate monetary utility is lower in the Nash risk sharing equilibrium when compared with the Arrow-Debreu one, there are cases where some agents achieve higher utility when a risk sharing game is played. In particular, it is shown that for agents with sufficiently high risk tolerance, the expected utility gain at Nash equilibrium is higher than the one induced by the optimal risk sharing transaction. Finally, the limiting behaviour of Nash equilibrium when the agent's risk tolerance approaches infinity (with the interpretation that agents become almost risk neutral) is analysed.