Worst-case distortion risk measures of transformed losses with uncertain distributions lying in Wasserstein balls
Bibliographic record
Abstract
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) North American Actuarial Journal , 28 (3), 611–625), 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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".