Beyond tail median and conditional tail expectation: extreme risk estimation using tail Lp-optimisation

The Conditional Tail Expectation is an indicator of tail behaviour that, contrary to the quantile or Value-at-Risk, takes into account the frequency of a tail event together with the probabilistic behaviour of the variable of interest on this event. However, the asymptotic normality of the empirical Conditional Tail Expectation estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in the heavy-tailed models of interest in insurance and finance. One solution when this assumption fails could be to use the more robust Median Shortfall, but being a quantile, this quantity only gives information about the frequency of a tail event. We construct a synthetic class of tail Lp medians, encompassing the Median Shortfall and Conditional Tail Expectation. We show that, for 1 ? p ? 2, a tail Lp median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on a set of real fire insurance data showing evidence of a very heavy right tail