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The paper Zhao et al. (2015) shows that mean-CVaR-skewness portfoliooptimization problems based on asymetric Laplace (AL) distributions can betransformed into quadratic optimization problems under which closed formsolutions can be found. In this note, we show that such result also holds formean-risk-skewness portfolio optimization problems when the underlyingdistribution is a larger class of normal mean-variance mixture (NMVM) modelsthan the class of AL distributions. We then study the value at risk (VaR) andconditional value at risk (CVaR) risk measures on portfolios of returns withNMVM distributions. They have closed form expressions for portfolios of normaland more generally elliptically distributed returns as discussed in Rockafellar& Uryasev (2000) and in Landsman & Valdez (2003). When the returns have generalNMVM distributions, these risk measures do not give closed form expressions. Inthis note, we give approximate closed form expressions for VaR and CVaR ofportfolios of returns with NMVM distributions. Numerical tests show that ourclosed form formulas give accurate values for VaR and CVaR and shortens thecomputational time for portfolio optimization problems associated with VaR andCVaR considerably.

Portfolio analysis with mean-CVaR and mean-CVaR-skewness criteria based on mean-variance mixture models