# Optimal XL-insurance under Wasserstein-type ambiguity

*Corina Birghila* and
*Georg Ch. Pflug*

*Insurance: Mathematics and Economics*, 2019, vol. 88, issue C, 30-43

**Abstract:**
We study the problem of optimal insurance contract design for risk management under a budget constraint. The contract holder takes into consideration that the loss distribution is not entirely known and therefore faces an ambiguity problem. For a given set of models, we formulate a minimax optimization problem of finding an optimal insurance contract that minimizes the distortion risk functional of the retained loss with premium limitation. We demonstrate that under the average value-at-risk measure, the entrance-excess of loss contracts are optimal under ambiguity, and we solve the distributionally robust optimal contract-design problem. It is assumed that the insurance premium is calculated according to a given baseline loss distribution and that the ambiguity set of possible distributions forms a neighborhood of the baseline distribution. To this end, we introduce a contorted Wasserstein distance. This distance is finer in the tails of the distributions compared to the usual Wasserstein distance.

**Keywords:** Insurance contract; Model error; Minimax solution; Distributional robustness (search for similar items in EconPapers)

**JEL-codes:** G22 D81 (search for similar items in EconPapers)

**Date:** 2019

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:insuma:v:88:y:2019:i:c:p:30-43

**DOI:** 10.1016/j.insmatheco.2019.05.005

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