Paper: arXiv 2411.10139

Authors: Alfred Müller

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.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a solid theoretical exploration of infinite mean risks and non-diversification traps using formal stochastic dominance proofs and stable distribution theory. However, it is entirely mathematical/theoretical and lacks any empirical backtesting or real-world dataset analysis.

Research Flowchart

  flowchart TD
    A["Research Goal: Explain the nondiversification trap for infinite mean risks"] --> B["Input Models: Heavy-tailed distributions & catastrophic risks"]
    B --> C["Methodology: Compare distribution skewness against Cauchy distribution"]
    C --> D["Analysis: Model positive probability of infinite loss events"]
    D --> E["Key Outcome: Risk sharing fails for any distribution more skewed than Cauchy"]
    D --> F["Future Work: Formulate open problems & conjectures for actuarial science"]

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2 of 169 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.