Tail variance allocation, Shapley value and the majorization problem

With focus to the risk contribution in a portfolio of dependent risks, Colini- Baldeschi, Scarsini and Vaccari (2018) have introduced the Shapley values for the variance and standard deviation games. In this note we extend their results introducing the tail variance as well as the tail standard deviation games. We derive closed-form expressions of 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 the reinsurance pricing for peer-to-peer insurance policies