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In this paper, we present axiomatic characterizations of some simple risk-sharing (RS) rules, such as the uniform, the mean-proportional and the covariance-based linear RS rules. These characterizations make it easier to understand the underlying principles when applying these rules. Such principles typically include maintaining some degree of anonymity regarding participants' data and/or incident-specific data, adopting non-punitive processes and ensuring the equitability and fairness of risk sharing. By formalizing key concepts such as the reshuffling property, the source-anonymous contributions property and the strongly aggregate contributions property, along with their generalizations, we develop a comprehensive framework that expresses these principles clearly and defines the relevant rules. To illustrate, we demonstrate that the uniform RS rule, a simple mechanism in which risks are shared equally, is the only RS rule that satisfies both the reshuffling property and the source-anonymous contributions property. This straightforward axiomatic characterization of the uniform RS rule serves as a foundation for exploring similar principles in two broad classes of risk-sharing rules, which we baptize the $q$-proportional RS rules and the $(q_1,q_2)$-based linear RS rules, respectively. This framework also allows us to introduce novel particular RS rules, such as the scenario-based RS rules., Comment: 27

There is little disagreement among insurance actuaries and financial economists about the societal benefits of longevity-risk pooling in the form of life annuities, defined benefit pensions, self-annuitization funds, and even tontine schemes. Indeed, the discounted value or cost of providing an income for life is lower -- in other words, the amount of upfront capital required to generate a similar income stream with the same level of statistical safety is lower -- when participants pool their financial resources versus going it alone. Moreover, when participants' financial circumstances and lifespans are homogenous, there is consensus on how to share the "winnings" among survivors, namely by distributing them equally among survivors, a.k.a. a uniform rule. Alas, what is lesser-known and much more problematic is allocating the winnings in such a pool when participants differ in wealth (contributions) and health (longevity), especially when the pools are relatively small in size. The same problems arise when viewed from the dual perspective of decentralized risk sharing (DRS). The positive correlation between health and income and the fact that wealthier participants are likely to live longer is a growing concern among pension and retirement policymakers. With that motivation in mind, this paper offers a modelling framework for distributing longevity-risk pools' income and benefits (or tontine winnings) when participants are heterogeneous. Similar to the nascent literature on decentralized risk sharing, there are several equally plausible arrangements for sharing benefits (a.k.a. "skinning the cat") among survivors. Moreover, the selected rule depends on the extent of social cohesion within the longevity risk pool, ranging from solidarity and altruism to pure individualism. In sum, actuarial science cannot really offer or guarantee uniqueness, only a methodology.

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.

In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution ($\Delta$CoD) measures as measures of systemic risk and analyze their properties and representations. The classes include the well-known conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES as special cases. Sufficient conditions are presented for two random vectors to be ordered by the proposed CoD-risk measures and distortion risk contribution measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive, and excess wealth orders of the marginals and canonical positive/negative stochastic dependence notions. Numerical examples are provided to illustrate our theoretical findings. This paper is the second in a triplet of papers on systemic risk by the same authors. In \cite{DLZorder2018a}, we introduce and analyze some new stochastic orders related to systemic risk. In a third (forthcoming) paper, we attribute systemic risk to the different participants in a given risky environment., Comment: 41 pages, 7 figures