Characteristics And Applications Of Multivariate Value At Risk

In economics,insurance and finance,value at risk(VaR)is a widely used measure of the risk of loss on a specific portfolio of financial assets.For a given portfolio,time horizon,and probability ?,the 100?% VaR is defined as a threshold loss value,such that the probability that the loss on the portfolio over the given time horizon exceeds this value is a.That is to say,it is a quantile of the distribution of the losses,which has both good analytic properties and easy interpretation as a risk measure.Based on the study of multivariate VaR by Raúl Torres et.al(2015),this paper proposes a multivariate VaR based on the high-dimensional median,that is MVaR_?~u(X),the related properties are discussed,the MVaR_?~u(X)-mean optimal investment portfolio problem is studied and the robustness of MVaR_?~u(X) is analyzed.First of all,this paper gives a definition of multivariate VaR,that is MVaR_?~u(X) based on the high-dimensional median,and then studies the good analytical properties.Considering that MVaR_?~u(X) is determined by multivariate quantile and high-dimensional median,the out-of-sample estimation of the multivariate quantile is obtained based on extreme value theory,and the algorithm of the high-dimensional median is given,which solves the computational problem.Then,for MVaR_?~u(X),similar to the case of univariate VaR-mean,the optimal portfolio problem with MVaR_?~u(X)-mean is proposed.Genetic algorithm is applied to the empirical analysis of MVaR_?~u(X)-mean model.The study generalizes the classic VaR-mean portfolio optimization theoretically,and the conclusion shows that the study has a good economic significance.Finally,the mean-based multivariate VaR.that is VaR_?~u(X),proposed by Raúl Torres et.al(2015) is used as a benchmark to analyze the robustness of MVaR_?~u(X).From the two aspects of outlier and risk level,the robustness of MVaR_?~u(X)is better than that of VaR_?~u(X).