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Distributionally robust shortfall risk optimization model and its approximation

Distributionally robust shortfall risk optimization model and its approximation
Distributionally robust shortfall risk optimization model and its approximation
Utility-based shortfall risk measures (SR)have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk.In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions} is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure.We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through $\phi$-divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples,we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some preliminary numerical test results are reported for the proposed modeling and computational schemes.
0025-5610
1-26
Guo, Shaoyan
9058e0bb-fe6e-425a-ad0f-1b051279dc72
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Guo, Shaoyan
9058e0bb-fe6e-425a-ad0f-1b051279dc72
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5

Guo, Shaoyan and Xu, Huifu (2018) Distributionally robust shortfall risk optimization model and its approximation. Mathematical Programming, 1-26. (doi:10.1007/s10107-018-1307-z).

Record type: Article

Abstract

Utility-based shortfall risk measures (SR)have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk.In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions} is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure.We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through $\phi$-divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples,we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some preliminary numerical test results are reported for the proposed modeling and computational schemes.

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Accepted/In Press date: 29 May 2018
e-pub ahead of print date: 7 June 2018

Identifiers

Local EPrints ID: 421332
URI: http://eprints.soton.ac.uk/id/eprint/421332
ISSN: 0025-5610
PURE UUID: c7f62ffe-3053-422b-9299-39ead6ccd245
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

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Date deposited: 01 Jun 2018 16:30
Last modified: 17 Dec 2019 05:24

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