Computing optimal mean/downside risk frontiers: The role of ellipiticity

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Optimizing Optimization: The Next Generation of Optimization Applications and Theory, 2010, 1st, pp. 179 - 199
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The purpose of this chapter is to analyze and calculate optimal mean/downside risk frontiers for financial portfolios. Focusing on the twO important cases of mean/value at risk and mean/semivariance, we compute analytic expressions for the optimal frontier in the two asset case, where the returns follow an arbitrary (nonnormal) distribution. Our analysis highlights the role of the normality/ellipticity assumption in this area of research. Formulae for mean/variance, mean/expected loss, and meanlsemistandard deviation frontiers are presented under normality/ellipticity. Computational issues are discussed and two propositions that facilitate computation are provided. Finally, the methodology is extended to nonelliptical distributions where simulation procedures are introduced. These can be presented jointly with our analytical approach to give portfolio managers deeper insights into the properties of optimal portfolios.
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