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Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach

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Abstract

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker’s risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme.

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Notes

  1. We use the phrase which may represent decision makers, regulators, engineers, customers etc depending on the nature of decision making problems.

  2. Similar inequalities also hold for the cases that \(q\nu =k/2\) and \(q\nu >k/2\), see [14, Theorem 2].

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Acknowledgements

We would like to thank the two anonymous referees for a number of valuable comments and suggestions which help us strengthen the presentation of this work. We would also like to thank the Associate Editor for effective handling of the review.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Shaoyan Guo

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

    Huifu Xu

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  1. Shaoyan Guo
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  2. Huifu Xu
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Correspondence to Shaoyan Guo.

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The research of the first author is supported by the National Natural Science Foundation of China (11801057). The research of the second author is supported by the GRC grant (14500620)

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Guo, S., Xu, H. Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach. Math. Program. 194, 305–340 (2022). https://doi.org/10.1007/s10107-021-01630-5

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