Some remarks on the effect of risk sharing and diversification for infinite mean risks
Alfred Müller
TL;DR
The paper addresses risk-sharing in infinite-mean contexts, where diversification can fail or become harmful. It unifies this phenomenon with a simple principle: any distribution more skewed than a $Cauchy$ yields a nondiversification trap under increasing-convex risk-sharing rules, linking it to stable, Pareto, and Fréchet families and to deadly-risk scenarios. A core contribution is a general closure result for the $\mathcal{D}^-$ class under increasing-convex transformations, along with a detailed characterization of when stable laws belong to $\mathcal{D}^-$ and the demonstration that convex transforms of a $Cauchy$ distribution lie in this class; the paper also analyzes catastrophic tail risks with $P(X=\infty)>0$ and surveys related tail-risk frameworks and open problems. Overall, these results clarify when pooling can be detrimental for heavy-tailed or catastrophic risks and guide future work on tail-risk management and insurance design.
Abstract
The basic principle of any version of insurance is the paradigm that exchanging risk by sharing it in a pool is beneficial for the participants. In case of independent risks with a finite mean this is the case for risk averse decision makers. The situation may be very different in case of infinite mean models. In that case it is known that risk sharing may have a negative effect, which is sometimes called the nondiversification trap. This phenomenon is well known for infinite mean stable distributions. In a series of recent papers similar results for infinite mean Pareto and Fréchet distributions have been obtained. We further investigate this property by showing that many of these results can be obtained as special cases of a simple result demonstrating that this holds for any distribution that is more skewed than a Cauchy distribution. We also relate this to the situation of deadly catastrophic risks, where we assume a positive probability for an infinite value. That case gives a very simple intuition why this phenomenon can occur for such catastrophic risks. We also mention several open problems and conjectures in this context.