On tail-risk measures for non-integrable heavy-tailed random variables

The assessment of risk for heavy-tailed distributions is a crucial question in various fields of application. An important family of risk measures is provided by the class of distortion risk (DR) measures which encompasses the Value-at-Risk and the Tail-Value-at-Risk measures. The Tail-Value-at-Risk is a coherent risk measure (which is not the case for the Value-at-Risk) but it is defined only for integrable quantile functions that is to say for heavy-tailed distributions with a tail index smaller than 1. Moreover, it is a matter of fact that the performance of the empirical estimator is strongly deteriorated when the tail index becomes close to 1. The main contribution is the introduction and the estimation of a new risk measure which is defined for all heavy-tailed distributions and which is tail-equivalent to a coherent DR measure when the tail of the underlying distribution is not too heavy. Its finite sample performance is discussed on a fire claims dataset.