On the Haezendonck–Goovaerts risk measure for extreme risks

Abstract

In this paper, we are interested in the calculation of the Haezendonck–Goovaerts risk measure, which is defined via a convex Young function and a parameter q ∈ ( 0 , 1 ) representing the confidence level. We mainly focus on the case in which the risk variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonck–Goovaerts risk measure as q ↑ 1 . As a subsidiary, we also consider the case with an exponentially distributed risk variable and a general Young function, and we obtain an analytical expression for the Haezendonck–Goovaerts risk measure.