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A decomposition of general premium principles into risk and deviation

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  • Max Nendel
  • Frank Riedel
  • Maren Diane Schmeck

Abstract

We provide an axiomatic approach to general premium principles in a probability-free setting that allows for Knightian uncertainty. Every premium principle is the sum of a risk measure, as a generalization of the expected value, and a deviation measure, as a generalization of the variance. One can uniquely identify a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex premium principles, and study the consistency of premium principles with a financial market in which insurance contracts are traded.

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  • Max Nendel & Frank Riedel & Maren Diane Schmeck, 2020. "A decomposition of general premium principles into risk and deviation," Papers 2006.14272, arXiv.org, revised Dec 2020.
    • Handle: RePEc:arx:papers:2006.14272
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    Cited by:

    1. Santos, Samuel S. & Moresco, Marlon R. & Righi, Marcelo B. & Horta, Eduardo, 2024. "A note on the induction of comonotonic additive risk measures from acceptance sets," Statistics & Probability Letters, Elsevier, vol. 208(C).
    2. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2022. "Star-Shaped deviations," Papers 2207.08613, arXiv.org.
    3. Felix-Benedikt Liebrich & Cosimo Munari, 2022. "Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity," Mathematics and Financial Economics, Springer, volume 16, number 2, March.
    4. Max Nendel & Jan Streicher, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Papers 2303.08217, arXiv.org, revised Sep 2023.
    5. Nendel, Max & Streicher, Jan, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    6. Samuel Solgon Santos & Marlon Ruoso Moresco & Marcelo Brutti Righi & Eduardo de Oliveira Horta, 2023. "A note on the induction of comonotonic additive risk measures from acceptance sets," Papers 2307.04647, arXiv.org, revised Jul 2023.

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