Abstract
In this paper, we find analytic expressions of the lower partialmoment and kappa index of linear portfolios when the returns are ellipticallydistributed. We also introduced the notion of Target Semi-Kurtosis of portfolioreturn and discuss the robust optimization Mean-LPM problem with non-gaussian risk factors. Special attention is given to the particular case of amixture of multivariate t-distributions with application for portfolio allocationof some ESG indices and the CAC 40 index.