Optimal Stop-Loss Reinsurance Under the VaR and CTE Risk Measures: Variable Transformation Method

In this paper, we propose a variable transformation way and obtain the optimal stop-loss reinsurance under value at risk (VaR) and conditional tail expectation (CTE) criteria, respectively. Let X be the initial loss of an insurer with cumulative distribution function FX(x)=P(X≤x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_X(x)=P(X\le x)$$\end{document} and survival function SX(x)=1-FX(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_X(x)=1-F_X(x)$$\end{document}. Denote a transformation variable Y=-ln(SX(X))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y=-\,\ln (S_X(X))$$\end{document}. Firstly, we analyze properties of the variables X and Y. Then, under VaR- and CTE-optimization criteria, we provide the necessary and sufficient conditions for the optimal retention existence of Y, respectively. Further, the optimal retention of X is obtained. Some examples are given to illustrate these results.