4.5 Article

Contextual Areas Star-Shaped Risk Measures

Journal

OPERATIONS RESEARCH
Volume -, Issue -, Pages -

Publisher

INFORMS
DOI: 10.1287/opre.2022.2303

Keywords

convexity; capital charge; liquidity risk; competitive pricing; monotonicity along rays

Funding

  1. Italian Ministry of Education [PRIN-2017CY2NCA, PRIN-2017TA7TYC]
  2. Natural Sciences and Engineering Research Council of Canada [RGPIN-2018-03823, RGPAS-2018-522590]

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This paper characterizes and studies a type of monetary risk measures, called star-shaped risk measures, that possess positive superhomogeneity. These measures, which arise when the subadditivity property of coherent risk measures is disregarded and positive homogeneity is weakened, encompass all commonly used risk measures, including convex risk measures and value-at-risk. From a financial perspective, relaxing convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision-theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.
In this paper, monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties are studied. The measures in this class, which arise when the subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and value-at-risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.

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