Optimal Scenario-Dependent Multivariate Shortfall Risk Measure and its Application in Capital Allocation

40 Pages Posted: 1 Jun 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

Tiejun Ma

University of Southampton, Southampton Business School, Department of Decision Analytics and Risk

Date Written: May 19, 2021

Abstract

In this paper, we consider a multivariate shortfall risk measure with scenario-dependent allocation weights and examine its properties such as convexity and quasi-convexity. For fixed allocation weights, we show that the resulting risk measure is a convex systemic risk measure in which case the property of translation invariance is dependent on the allocation weights. However, if the allocation weights are chosen optimally on a feasible set, then the resulting risk measure is a quasi-convex systemic risk measure. To evaluate the systemic risk of a financial system with the proposed risk measure computationally, we reformulate it as a two-stage stochastic program which is a finite convex program when the underlying uncertainty is discretely distributed. In the case when the uncertainty is continuously distributed, we propose a polynomial decision rule to tackle the semi-infinite two-stage stochastic program which restricts the scenario-dependent allocation weights to be a class of polynomials of the underlying uncertainty parameters and subsequently reformulate the evaluation problem as a tractable optimization problem. Some convergence results are established for the approximation scheme. Moreover, we apply the proposed risk measure to capital allocation problem and introduce scenario-dependent allocation strategy and deterministic allocation strategy. Finally, we carry out some preliminary tests on the proposed computational schemes for a continuous system, a discrete system and a capital allocation example for a life insurance company.

Keywords: risk management, multivariate shortfall, scenario-dependent allocation weights, polynomial decision rule, scenario-dependent allocation strategy

Suggested Citation

Wang, Wei and Xu, Huifu and Ma, Tiejun, Optimal Scenario-Dependent Multivariate Shortfall Risk Measure and its Application in Capital Allocation (May 19, 2021). Available at SSRN: https://ssrn.com/abstract=3849125 or http://dx.doi.org/10.2139/ssrn.3849125

Wei Wang (Contact Author)

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

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

Shatin, New Territories
Hong Kong

Tiejun Ma

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

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

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