Maximally Acceptable Portfolios

Portfolios are selected in non-Gaussian contexts to maximize a Cherny and Madan index of acceptability. Analytical gradients are developed for the purpose of optimizing portfolio searches on the unit sphere. It is shown that though an acceptability index is not a preference ordering, many utilities will concur with acceptability maximization. A stylized economy illustrates the advantages from the perspective of acceptability of nonlinear securities and options. In sample results for the year 2008 indicate that maximizing the acceptability index can lead to portfolios that second order stochastically dominate their Gaussian counterparts. Backtests over the period 1997 to 2008 re‡ect gains to maximizing acceptability over holding a maximal Sharpe ratio portfolio. Portfolio rebalancing is now possible and is being executed at much higher frequencies than has been possible in the past. Some algorithms trade every …ve to …fteen minutes a fairly large number of stocks ranging from a thousand stocks upward. It has been known for some time now that at such short horizons returns are extremely non-Gaussian displaying signi…cant levels of skewness and excess kurtosis. Additionally modern economies directly provide access to nonlinear cash ‡ows via the markets for options and variance swaps. Optimal portfolio selection in such non-Gaussian contexts is expected to diverge from the multivariate Gaussian model that essentially focuses on maximizing the Sharpe ratio. This is primarily due to the recognition that investors are not indi¤erent to other aspects of a return distribution and ceteris paribus they prefer positive skewness and peakedness and dislike tailweightedness. Kurtosis, as noted in Eberlein and Madan (2009), is a preferentially confused statistic as it combines both peakedness and tailweightedness. The choice of criterion on which to base portfolio selection is then a critical issue and many alternatives have been formulated in the literature. We refer to Biglova, Ortobelli, Rachev and Stoyanov (2004) for a survey and application of a