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

The paper Zhao et al. (Ann Oper Res 226:727–739, 2015) shows that mean-CVaR-skewness portfolio optimization problems based on asymetric Laplace (AL) distributions can be transformed into quadratic optimization problems for which closed form solutions can be found. In this note, we show that such a result also holds for mean-risk-skewness portfolio optimization problems when the underlying distribution belongs to a larger class of normal mean–variance mixture (NMVM) models than the class of AL distributions.We then study the value at risk (VaR) and conditional value at risk (CVaR) risk measures of portfolios of returns with NMVM distributions.They have closed form expressions for portfolios of normal and more generally elliptically distributed returns, as discussed in Rockafellar and Uryasev (J Risk 2:21–42, 2000) and Landsman and Valdez (N Am Actuar J 7:55–71, 2003). When the returns have general NMVM distributions, these risk measures do not give closed form expressions. In this note, we give approximate closed form expressions for the VaR and CVaR of portfolios of returns with NMVM distributions.Numerical tests show that our closed form formulas give accurate values for VaR and CVaR and shorten the computational time for portfolio optimization problems associated with VaR and CVaR considerably.