Bivariate lower and upper orthant value-at-risk

Abstract

Value-at-risk (VaR) is an important risk measure widely used in actuarial science and quantitative risk management. Embrechts and Puccetti (J Multivar Anal 97(2):526–547, 2006a) have introduced the multivariate lower and upper orthant VaR. The practical applications of these risk measures is very promising, especially in actuarial science and quantitative risk management. Our objective is to study in details the multivariate lower and upper orthant VaR in the bivariate setting, their properties and their applications. In particular, new characterizations of the bivariate lower and upper orthant VaR and desirable properties are given, such as translation invariance, positive homogeneity and comonotonic additivity. Lower and upper confidence regions for random vectors are developed and used to provide new results on the convexity conditions and to suggest capital allocation techniques. We provide bounds on functions of random pairs and derive interesting relations with existing results. We motivate the use of the bivariate lower and upper ortant VaR for risk allocation, to represent bivariate ruin probabilities and for risk comparison. Practical illustrations and examples of the results are presented throughout the article.