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Tail variance allocation, Shapley value, and the majorization problem

Part of: Game theory Linear inference, regression

Published online by Cambridge University Press:  06 June 2023

Marcello Galeotti  and
Giovanni Rabitti
Marcello Galeotti*
Affiliation:
University of Florence
Giovanni Rabitti*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Statistics, Informatics and Applications, University of Florence, Via delle Pandette 9, 50127 Florence, Italy. Email: marcello.galeotti@unifi.it
**Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K. Email: g.rabitti@hw.ac.uk
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Abstract

With a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.

Keywords

Covariance matrixsum of random variablesvector majorizationcooperative gamestail risk sharing

MSC classification

Primary: 91A12: Cooperative games
Secondary: 62J10: Analysis of variance and covariance

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 153 - 171
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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