Journal of Risk

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An examination of the tail contribution to distortion risk measures

Miguel Santolino, Jaume Belles-Sampera, José María Sarabia and Montserrat Guillen

  • The size of the risk measure estimate attributable to the tail of the distribution is analyzed.
  • The tail contribution to distortion risk measures is defined and their additive properties are investigated.
  • Decision makers are informed about the part of the regulatory reserve estimate attributable to the highest losses.
  • Decision makers will identify the part of the diversification benefits attributable to the right tail.

Extreme losses are specially relevant in the finance and insurance sectors. Here, we analyze the tail behavior of risks and its influence on risk measures. Specifically, we examine the part of the risk value of a distortion risk measure (DRM) that is attributable to extreme losses. We analyze the additive properties of tail contributions to risk values when several risks are aggregated. We show that the partial contributions are subadditive if the distortion function is concave in the tail. We examine the tail behavior for quantile-based DRMs, including value-at-risk and tail value-at-risk. We conclude that such an evaluation will allow decision makers to obtain relevant information about the contribution of extreme losses to risk values and about the fraction of the diversification benefit attributable to the tails. An example is used to illustrate our results.

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