Worst-case Risk Measures of Stop-Loss and Limited Loss Random Variables Under Distribution Uncertainty With Applications to Robust Reinsurance

41 Pages Posted: 25 Apr 2023 Last revised: 1 Mar 2024

See all articles by Jun Cai

Jun Cai

University of Waterloo - Department of Statistics and Actuarial Science

Fangda Liu

University of Waterloo - Department of Statistics and Actuarial Science

Mingren Yin

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: April 20, 2023

Abstract

Stop-loss and limited loss random variables are two important transforms of a loss random variable and appear in many modelling problems in insurance, finance, and other fields. Risk levels of a loss variable and its transforms are often measured by risk measures. When only partial information on a loss variable is available, risk measures of the loss variable and its transforms cannot be evaluated effectively. To deal with the situation of distribution uncertainty, the worst-case values of risk measures of a loss variable over an uncertainty set, describing all the possible distributions of the loss variable, have been extensively used in robust risk management for many fields. However, most of these existing results on the worst-case values of risk measures of a loss variable cannot be applied directly to the worst-case values of risk measures of its transforms. In this paper, we derive the expressions of the worst-case values of distortion risk measures of stop-loss and limited loss random variables over an uncertainty set introduced in Bernard et al (2022). This set represents a decision maker's belief in the distribution of a loss variable. We find the distributions under which the worst-case values are attainable. These results have potential applications in a variety of fields. To illustrate their applications, we discuss how to model optimal stop-loss reinsurance problems and how to determine optimal stop-loss retentions under distribution uncertainty. Explicit and closed-form expressions for the worst-case TVaRs of stop-loss and limited loss random variables and optimal stop-loss retentions are given under special forms of the uncertainty set. Numerical results are presented under more general forms of the uncertainty set.

Keywords: Stop-loss, limited loss, uncertainty set, Wasserstein distance, distortion risk measure, tail value-at-risk, quantile function, min-max problem, robust stop-loss reinsurance.

JEL Classification: C60, G22

Suggested Citation

Cai, Jun and Liu, Fangda and Yin, Mingren, Worst-case Risk Measures of Stop-Loss and Limited Loss Random Variables Under Distribution Uncertainty With Applications to Robust Reinsurance (April 20, 2023). Available at SSRN: https://ssrn.com/abstract=4424759 or http://dx.doi.org/10.2139/ssrn.4424759

Jun Cai

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

Fangda Liu (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Ave.
Waterloo, Ontario N2L 3G1 N2L3G1
Canada

Mingren Yin

University of Waterloo - Department of Statistics and Actuarial Science

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