Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation

For the multiplicative background risk model, a distortion-type risk measure is used to measure the tail risk of the portfolio under a scenario probability measure with multivariate regular variation. In this paper, we investigate the tail asymptotics of the portfolio loss ∑i=1dRiS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^{d}R_iS$$\end{document}, where the stand-alone risk vector R=(R1,…,Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}=(R_1,\ldots ,R_d)$$\end{document} follows a multivariate regular variation and is independent of the background risk factor S. An explicit asymptotic formula is established for the tail distortion risk measure, and an example is given to illustrate our obtained results.