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A Theory for Combinations of Risk Measures

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A Theory for Combinations of Risk Measures A theory for combinations of risk measures Marcelo Brutti Righi∗ Federal University of Rio Grande do Sul [email protected] Abstract We study combinations of risk measures under no restrictive assumption on the set of alternatives. We develop and discuss results regarding the preservation of properties and acceptance sets for the combinations of risk measures. One of the main results is the rep- resentation for resulting risk measures from the properties of both alternative functionals and combination functions. To that, we build on the development of a representation for arbitrary mixture of convex risk measures. In this case, we obtain a penalty that recalls the notion of inf-convolution under theoretical measure integration. As an application, we address the context of probability-based risk measurements for functionals on the set of distribution functions. We develop results related to this specific context. We also explore features of individual interest generated by our framework, such as the preservation of con- tinuity properties, the representation of worst-case risk measures, stochastic dominance and elicitability. We also address model uncertainty measurement under our framework and propose a new class of measures for this task. Keywords: risk measures, uncertainty, combination, representations, probability-based func- tionals. 1 Introduction The theory of risk measures in mathematical finance has become mainstream, especially since the landmark paper of Artzner et al. (1999). For a comprehensive review, see the books of Pflug and R¨omisch (2007), Delbaen (2012), R¨uschendorf (2013) and F¨ollmer and Schied (2016). arXiv:1807.01977v6 [q-fin.MF] 14 Aug 2020 Nonetheless, there is still no consensus about the best set of theoretical properties to possess, and even less regarding the best risk measure. See Emmer et al. (2015) for a comparison of risk measures. This phenomenon motivates the proposition of new approaches, such as in Righi and Ceretta (2016) and Righi et al. (2019) for instance. Under the lack of a universal choice of a best risk measure from a set of alternatives, one can think into considering the joint use of many candidates in the goal of benefit from distinct qualities. ∗We would like to thank the editor-in-chief, Professor Ulrich Horst, and the anonymous Associate Editor and Reviewer for their constructive comments, which have been useful to improve the technical quality of the manuscript. We would also like to thank Professors Ruodu Wang and Martin Schweizer for their comments, which have helped to improve the manuscript. We are grateful for the financial support of FAPERGS (Rio Grande do Sul State Research Council) project number 17/2551-0000862-6 and CNPq (Brazilian Research Council) projects number 302369/2018-0 and 407556/2018-4. 1 However, such choice can lead to multidimensional or even infinite dimensional problems that bring a level for complexity that may make the treatment of risk measurement impossible to handle. For example, in a portfolio optimization problem in order to take into account this variety of features from distinct risk measures the agent may end up with a very complex multi-objective function or even an exorbitant volume of constraints. Such situation would lead to elevated computational cost or even the impossibility for a feasible solution. Hence, the alternative is to consider a combination of all the candidates instead of all of them individually. In the context for portfolio optimization, we would have a single constraint or objective with a larger space of feasibility. The drawback for considering such a combination is that we may end up without the main characteristics that define risk measures. More precisely, the axiomatic theory of risk measures strongly relies on a set of financial and mathematical properties that are related to dual repre- sentations and acceptance sets. Thus, the understanding on how to preserve such properties in a general fashion is crucial to guarantee the usefulness of combinations. In this sense, the devel- opment of a theoretical body for combinations of risk measures is pivotal. A general approach must deal with arbitrary sets of candidates, in particular uncountable ones. Such situation may arise when the parameter that defines the candidates relies on subset of the real line, such as the significance level for Value at Risk or some probability of default in credit risk. The main challenge relies in the generality that is needed to perform this kind of task since we are not able to rely in methods for finite dimensional spaces that appear on the literature. For instance, when dealing with some uncountable set of candidates we are not able to even consider usual summation, crucial for averaging, having to replace it by integration. However, in this case we have measurability issues to take into account that may become complex. Also, set operations may not be preserving topological properties, such as uncountable union of closed sets does not have to be closed. Even the choice of suitable domains for combination functions in infinite dimensional spaces can be per se a source of complexity since there is no canonical functional space. Another source of difficult and complexity is that combinations may assume any functional form, such as averaging or supremum based worst cases. In fact, any function applied over the set of candidate risk measures can be considered. Thus, to study the impact of such combinations in properties, acceptance sets, dual representations among other features is not straightforward. Since specific combination functions can be more suitable to distinct contexts, to have a general treatment is very beneficial. In this sense, the existence of a theory to provide a solid and practical manner to preserve desired properties, obtain acceptance sets or dual represent a general combination of risk measures can help to improve other fields in mathematical finance as well. Under this background, in this paper we study risk measures of the form ρ = f(ρI), where i ρI = {ρ , i ∈ I} is a set of alternative risk measures and f is some combination function. We propose a framework whereby no assumption is made on the index set I, apart from non- emptiness. Typically, this kind of procedure uses a finite set of candidates, leading the domain of f to be some Euclidean space. In our case, the domain of f is taken by a subset of the random variables over a suitable measurable space created on I. From that, our main goal is to 2 establish general results on properties, develop dual representations and study acceptance sets for such composed risk measures based on the properties of both ρI and f in a general sense. For this purpose, we expose results for some featured special cases, which are also of particular interest, such as a worst case and mixtures of risk measures. There are studies regarding particular cases for f, I and ρI , such as the worst case in F¨ollmer and Schied (2002), the sum of monetary and deviation measures in Righi (2019), fi- nite convex combinations in Ang et al. (2018), scenario-based aggregation in Wang and Ziegel (2018), model risk-based weighting over a non-additive measure in Jokhadze and Schmidt (2020). Our main contribution is do not impose a restriction on the set of alternative risk measures and the generality of combination functions we consider. Furthermore, none of such papers does consider all the features we do take into account. It is worthy to mention that the well known concept of inf-convolution of risk measures as in Barrieu and El Karoui (2005) or Jouini et al. (2008) is not suitable to the approach in this paper since even in the most sim- ple case of two risk measures we cannot write it as a direct combination as f(ρ1(X), ρ2(X)) = infY {ρ1(X −Y )+ρ2(X)} since it is not only a function of X. More precisely, the inf-convolution depends of every allocation X1 +X2 = X. The work of Righi (2020) is focused to inf-convolution and optimal risk sharing for arbitrary sets of risk measures. The classic theory for risk measures is developed under the assumption of a given probability measure that represents beliefs of the world. However, we frequently do not know if there is a correct probability measure. We then apply our framework to the special situation where I is a subset of probability measures since we frequently do not know if there is a correct one, but we have instead a set of candidates. We consider the concept of probability-based risk measurement, which is a collection of risk measures from a functional on the set of distributions generated by probabilities in I. Because the choice of probabilities may affect the risk value for a large amount, we would like to have risk measures that are robust in the sense that functionals do not suffer relevant variations in their value from changes on the chosen probability measure. This is the case for combinations in our approach since they depend on the whole set of alternative probabilities. The probability based risk measurement is directly related to the notion of model un- certainty, or model risk, relative to uncertainty concerning the choice of a model, which is studied in detail for risk measures by Kerkhof et al. (2010), Bernard and Vanduffel (2015), Barrieu and Scandolo (2015), Danielsson et al. (2016), Kellner and Rosch (2016), Frittelli and Maggis (2018), Maggis et al. (2018) among others. From a practical point of view, the Turner Review raises the issue of model ambiguity, often called Knightian uncertainty. Despite there is a consensus that better methods for dealing with model uncertainty are pivotal to improving risk management, measures to quantify it are not consolidated at the same level as those for market risk.
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