  The paper Zhao et al. (2015) shows that mean-CVaR-skewness portfolio
optimization problems based on asymetric Laplace (AL) distributions can be
transformed into quadratic optimization problems under which closed form
solutions can be found. In this note, we show that such result also holds for
mean-risk-skewness portfolio optimization problems when the underlying
distribution is 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 on 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
& Uryasev (2000) and in Landsman & Valdez (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 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 shortens the
computational time for portfolio optimization problems associated with VaR and
CVaR considerably.
