Various measures have been introduced in the existing literature to evaluate extreme risk exposure under the effect of an observable factor. Due to the nice properties of the higher-moment (HM) coherent risk measure, we propose a conditional version of the HM (CoHM) risk measure by incorporating the information of an observable factor. We conduct an asymptotic analysis of this measure in the presence of extreme risks under the weak contagion at a high confidence level, which is further applied to the special case of the conditional Haezendonck–Goovaerts risk measure (CoHG). Numerical illustrations are also provided to examine the accuracy of the asymptotic formulas and to analyze the sensitivity of the risk contribution of the CoHG. Based on the asymptotic result in the Fréchet case, we propose an estimator for the CoHM via an extrapolation, supported by a simulation study.
