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

Abstract

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 $$F_X(x)=P(X\le x)$$ and survival function $$S_X(x)=1-F_X(x)$$. Denote a transformation variable $$Y=-\,\ln (S_X(X))$$. 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.