Coupled Risk Measures and Their Empirical Estimation When Losses Follow Heavy-Tailed Distributions

Abstract

Considerable literature has been devoted to developing statistical inferential results for risk measures, especially for those that are of the form of L-functionals. However, practical and theoretical considerations have highlighted quite a number of risk measures that are of the form of ratios, or even more complex combinations, of two L-functionals. In the present paper we call such combinations ‘coupled risk measures’ and develop a statistical inferential theory for them when losses follow heavy-tailed distributions. Our theory implies – at a stroke – statistical inferential results for absolute and relative distortion risk measures, weighted premium calculation principles, as well as for many indices of economic inequality that have appeared in the econometric literature. Keywords : Risk measure, Heavy-tailed distribution, Distortion risk measure, Weighted risk measure, Proportional hazards transform, Conditional tail expectation, Premium calculation principle, Index of economic inequality, Statistical inference. Link http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1855623