Orlicz-Lorentz premia and distortion Haezendonck-Goovaerts risk measures
Aline Goulard, Karl Grosse-Erdmann
TL;DR
This work introduces distortion Haezendonck-Goovaerts risk measures, unifying distortion-based weighting of VaR with Haezendonck-Goovaerts Orlicz premia on an enlarged domain L_g^φ. It establishes that coherence holds when the distortion function g is concave and explores the rich structure of Orlicz-Lorentz premia, including existence of minimizers, Fatou-type properties, and connections to Lorentz spaces. The paper also analyzes domain properties, convex-cone conditions, and the α=0 boundary case, offering a comprehensive framework that generalizes both distortion and HG risk measures. The results provide new tools for coherent, law-invariant risk assessment in finance and actuarial science, with explicit conditions for domains, continuity, and comonotonic behavior. Overall, the DHG class broadens the toolkit for robust risk measurement by blending functional-analytic and probabilistic perspectives with practical coherence guarantees.
Abstract
In financial and actuarial research, distortion and Haezendonck-Goovaerts risk measures are attractive due to their strong properties. They have so far been treated separately. In this paper, following a suggestion by Goovaerts, Linders, Van Weert, and Tank, we introduce and study a new class of risk measure that encompasses the distortion and Haezendonck-Goovaerts risk measures, aptly called the distortion Haezendonck-Goovaerts risk measures. They will be defined on a larger space than the space of bounded risks. We provide situations where these new risk measures are coherent, and explore their risk theoretic properties.