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Dependence Modeling

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Robustness regions for measures of risk aggregation

Silvana M. Pesenti / Pietro Millossovich / Andreas Tsanakas
Published Online: 2016-12-14 | DOI: https://doi.org/10.1515/demo-2016-0020


One of risk measures’ key purposes is to consistently rank and distinguish between different risk profiles. From a practical perspective, a risk measure should also be robust, that is, insensitive to small perturbations in input assumptions. It is known in the literature [14, 39], that strong assumptions on the risk measure’s ability to distinguish between risks may lead to a lack of robustness. We address the trade-off between robustness and consistent risk ranking by specifying the regions in the space of distribution functions, where law-invariant convex risk measures are indeed robust. Examples include the set of random variables with bounded second moment and those that are less volatile (in convex order) than random variables in a given uniformly integrable set. Typically, a risk measure is evaluated on the output of an aggregation function defined on a set of random input vectors. Extending the definition of robustness to this setting, we find that law-invariant convex risk measures are robust for any aggregation function that satisfies a linear growth condition in the tail, provided that the set of possible marginals is uniformly integrable. Thus, we obtain that all law-invariant convex risk measures possess the aggregation-robustness property introduced by [26] and further studied by [40]. This is in contrast to the widely-used, non-convex, risk measure Value-at-Risk, whose robustness in a risk aggregation context requires restricting the possible dependence structures of the input vectors.

Keywords: Convex risk measures; Aggregation; Value-at-Risk; Robustness; Continuity


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About the article

Received: 2016-09-30

Accepted: 2016-11-18

Published Online: 2016-12-14

Citation Information: Dependence Modeling, Volume 4, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2016-0020.

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© 2016 Silvana M. Pesenti et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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