Preference Robust Distortion Risk Measure and Its Application

63 Pages Posted: 18 Feb 2021 Last revised: 21 Oct 2021

See all articles by Wei Wang

Wei Wang

University of Southampton, Southampton Business School, Department of Decision Analytics and Risk; The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering and Engineering Management

Huifu Xu

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering & Engineering Management

Date Written: January 7, 2021

Abstract

Distortion risk measure (DRM) plays a crucial role in risk measuring and managing, especially in insurance pricing. Various DRMs have been introduced but little is discussed about which DRM at hand should be chosen to address a decision maker (DM)'s risk attitude. This paper aims to fill out the gap. Specifically we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk attitude is ambiguous. We introduce a preference robust distortion risk measure (PRDRM) which is based on the worst-case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well-known general principals such as concavity and inverse S-shapedness of distortion functions (over-weighting on events from impossible to possible or possible to certainty and under-weighting on those from possible to more possible) as well as new user-specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelop of a set of points to characterise the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst-case distortion function is a non-decreasing piece-wise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model.

Keywords: Distortion risk measure, Yaari's dual theory, pairwise comparison, certainty equivalent, risk capital allocation

Suggested Citation

Wang, Wei and Xu, Huifu, Preference Robust Distortion Risk Measure and Its Application (January 7, 2021). Available at SSRN: https://ssrn.com/abstract=3763632 or http://dx.doi.org/10.2139/ssrn.3763632

Wei Wang

University of Southampton, Southampton Business School, Department of Decision Analytics and Risk ( email )

Building 2, 12 University Rd
Highfield
Southampton, SO17 1BJ
United Kingdom

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering and Engineering Management ( email )

Hong Kong
China

Huifu Xu (Contact Author)

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering & Engineering Management ( email )

Shatin, New Territories
Hong Kong

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