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Modeling lies at the core of both the financial and the insurance industry for a wide variety of tasks. The rise and development of machine learning and deep learning models have created many opportunities to improve our modeling toolbox. Breakthroughs in these fields often come with the requirement of large amounts of data. Such large datasets are often not publicly available in finance and insurance, mainly due to privacy and ethics concerns. This lack of data is currently one of the main hurdles in developing better models. One possible option to alleviating this issue is generative modeling. Generative models are capable of simulating fake but realistic-looking data, also referred to as synthetic data, that can be shared more freely. Generative Adversarial Networks (GANs) is such a model that increases our capacity to fit very high-dimensional distributions of data. While research on GANs is an active topic in fields like computer vision, they have found limited adoption within the human sciences, like economics and insurance. Reason for this is that in these fields, most questions are inherently about identification of causal effects, while to this day neural networks, which are at the center of the GAN framework, focus mostly on high-dimensional correlations. In this paper we study the causal preservation capabilities of GANs and whether the produced synthetic data can reliably be used to answer causal questions. This is done by performing causal analyses on the synthetic data, produced by a GAN, with increasingly more lenient assumptions. We consider the cross-sectional case, the time series case and the case with a complete structural model. It is shown that in the simple cross-sectional scenario where correlation equals causation the GAN preserves causality, but that challenges arise for more advanced analyses.

Two by-now folkloric results in the theory of risk sharing are that (i) any feasible allocation is convex-order-dominated by a comonotonic allocation; and (ii) an allocation is Pareto optimal for the convex order if and only if it is comonotonic. Here, comonotonicity corresponds to the no-sabotage condition, which aligns the interests of all parties involved. Several proofs of these two results have been provided in the literature, mostly based on the comonotonic improvement algorithm of Landsberger and Meilijson (1994) and a limit argument based on the Martingale Convergence Theorem. However, no proof of (i) is explicit enough to allow for an easy algorithmic implementation in practice; and no proof of (ii) provides a closed-form characterization of Pareto optima. In this paper, we provide novel proofs of these foundational results. Our proof of (i) is based on the theory of majorization and an extension of a result of Lorentz and Shimogaki (1968), which allows us to provide an explicit algorithmic construction that can be easily implemented. In addition, our proof of (ii) leads to a crisp closed-form characterization of Pareto-optimal allocations in terms of alpha-quantiles (mixed quantiles). An application to collaborative insurance, or decentralized risk sharing, illustrates the relevance of these results.

This paper studies the quantile risk-sharing rule introduced in Denuit, Dhaene & Robert (2022). This rule is not actuarially fair, but instead satisfies another type of fairness, which is comparable with “solvency fairness” in classical centralized insurance. New properties are investigated and an axiomatic theory is developed for the quantile risk-sharing rule, which allows for a deeper understanding of its proper use. The axiomatic characterization of the quantile risk-sharing rule is based on aggregate and comonotonicity-related properties of risk-sharing rules.

This paper offers a systematic treatment of risk-sharing rules for insurance losses, based on a list of relevant properties. A number of candidate risk-sharing rules are considered, including the conditional mean risk-sharing rule proposed in Denuit and Dhaene (2012) and the newly introduced quantile risk-sharing rule. Their compliance with the proposed properties is established. Then, methods for building new risk-sharing rules are discussed. The results derived in this paper are shown to be helpful in the development of peer-to-peer insurance (or crowdsurance), as well as to manage contingent risk funds where a given budget is distributed among claimants.