Worst-case risk measures of stop-loss and limited loss random variables under distribution uncertainty with applications to robust reinsurance
Bibliographic record
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. (2023). 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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.023 | 0.058 |
| Meta-epidemiology (narrow) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".