Value-at-Risk and Expected Shortfall for Quadratic portfolio of securities with mixture of elliptic Distributed Risk Factors

Generally, in the financial literature, the notion of quadratic VaR is implicitly confused with the Delta-Gamma VaR, because more authors dealt with portfolios that contained derivatives instruments. In this paper, we postpone to estimate both the expected shortfall and Value-at-Risk of a quadratic portfolio of securities (i.e equities) without the Delta and Gamma Greeks, when the joint log-returns changes with multivariate elliptic distribution. To illustrate our method, we give special attention to mixture of normal distribution, and mixture of t-student distribution.