Preference robust models in multivariate utility-based shortfall risk minimization

Abstract

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montréal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial information and consequently propose a robust formulation of SR based on the worst-case utility function. In this paper, we extend this new stream of research to a multi-attribute prospect space since multi-attribute decision-making problems are ubiquitous in practical applications. Specifically, we introduce a preference robust multivariate utility-based shortfall risk measure (PRMSR) and demonstrate that it is law invariant and convex. We then apply the PRMSR to an optimal decision-making problem where the objective is to minimize the PRMSR of a vector-valued cost function and propose some numerical scheme for solving the resulting optimization problem in the case when the underlying exogenous uncertainty is finitely distributed. Finally, we discuss statistical robustness of the PRMSR based optimization model by examining qualitative stability of the estimator of the optimal value obtained with potentially contaminated data. A case study is carried out to examine the performance of the proposed robust model and numerical scheme.

Keywords:

• Multivariate utility-based shortfall risk measure (MSR)
• worst-case multivariate loss function
• preference robust MSR
• tractable formulation
• statistical robustness

2010 Mathematics Subject Classifications:

• 90C15
• 90C31
• 90C29

Acknowledgements

The authors would like to thank the anonymous referee for insightful comments and constructive suggestions which help us significantly consolidate the paper and they would also like to thank the associate editor and the area editor Stefan Ulbrich for effective handling of the review. Finally the second author gratefully acknowledges the support of a CUHK startup Grant and RGC Grant (No. 14500620).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In literature, Young function is defined over ${\mathbb{R}}_{+}.$ Here we extend the notion to multivariate case.

Additional information

Funding

This work was supported by RGC Grant [grant number 14500620].

Notes on contributors

Yuan Zhang

Yuan Zhang is a PhD candidate in School of Mathematical Science, University of Southampton. His current research is on optimal decision making under uncertainty such as distributionally robust optimization and preference robust optimization which are associated with ambiguity about the distribution of exogenous uncertainty data and a decision maker's risk preference. His focus is on developing robust models and computational methods for these problems and applying them in finance and management sciences.

Huifu Xu

Huifu Xu is a Professor of the Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong. Prior to joining CUHK, he was a professor of Operational Research in the School of Mathematical Sciences, University of Southampton and the Director of the Centre of Operational Research, Management Science and Information Technology (2016-2018). Huifu Xu obtained a PhD degree from University of Ballarat (Federation University Australia) in 1999. His current research is on optimal decision making under uncertainty such as preference robust optimization and distributionally robust optimization which are associated with ambiguity in decision maker's utility preference or risk attitude and distribution of exogenous uncertainty data. He has published more than 70 papers in the international journals of operational research and optimization including Mathematical Programming, SIAM Journal on Optimization, Mathematics of Operations Research and Operations Research.

Wei Wang

Wei Wang is a PhD candidate funded by SCDTP, in the Business School, University of Southampton, and he is currently a research assistant in the Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong. His research includes optimal decision making under uncertainty such as preference robust optimization and distributionally robust optimization which are associated with ambiguity in decision maker's utility preference or risk attitude and distribution of exogenous uncertainty data; statistical robustness of risk measures and optimization models with applications in capital allocation.