Worst-case Distortion Risk Measures of Limited Stop-loss Transforms With Uncertain Distributions Lying in Wasserstein Balls

Abstract

The limited stop-loss transform, along with the stop-loss and limited loss transforms-which are special or limiting cases of the limited stop-loss transform-is one of the most important transforms used in insurance, and it also appears extensively in many other fields including finance, economics, and operations research. When the distribution of the underlying loss is uncertain, the worst-case risk measure for the limited stop-loss transform plays a key role in many quantitative risk management problems in insurance and finance. In this paper, we derive expressions for the worst-case distortion risk measure of the limited stop-loss transform, as well as for the stop-loss and limited loss transforms, when the distribution of the underlying loss is uncertain and lies in a general k-order Wasserstein ball that contains a reference distribution. We also identify the worst-case distributions under which the worst-case distortion risk measures are attained. Additionally, our results also recover the findings of Guan et al. (2023) regarding the worst-case stop-loss premium over a k-order Wasserstein ball. Furthermore, we use numerical examples to illustrate the worst-case distributions and the worst-case risk measures derived in this paper. We also examine the effects of the reference distribution, the radius of the Wasserstein ball, and the retention levels of limited stop-loss reinsurance on the premium for this type of reinsurance.

Keywords: Distortion risk measure, limited stop-loss transform, stop-loss transform, limited loss transform, distribution uncertainty, Wasserstein ball, tail value-at-risk, quantile function, Wang's premium