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The authors modeled the non-normal returns of multiple asset classes by using a multivariate truncated Lévy flight distribution and incorporating non-normal returns into the mean-conditional value at risk (M-CVaR) optimization framework. In a series of controlled optimizations, they found that both skewness and kurtosis affect the M-CVaR optimization and lead to substantially different allocations than do the traditional mean-variance optimizations. They also found that the M-CVaR optimization would have been beneficial during the 2008 financial crisis.lthough numerous alternatives to the mean-variance optimization (MVO) framework have appeared in the literature, no clear leader has emerged. The lack of an agreed-upon alternative to MVO has slowed the development of practitioner-oriented tools. Moreover, the difficulty in estimating the required inputs-returns, standard deviations, and correlations-for MVO is well known, a problem that can be substantially more difficult with more advanced techniques. The future is hard to predict accurately, especially in detail.Asset class return distributions are not normally distributed, but the typical Markowitz MVO framework that has dominated the asset allocation process for more than 50 years relies on only the first two moments of the return distribution. Equally important, considerable evidence shows that investor preferences go beyond mean and variance to higher moments: skewness and kurtosis. Investors are particularly concerned about significant losses-that is, downside risk, which is a function of skewness and kurtosis.Recent research suggests that higher moments are important considerations in asset allocation. Patton (2004) showed that knowledge of both skewness and asymmetric dependence (higher correlations in downside markets) leads to economically significant gains, in particular, with no shorting constraints. Harvey, Liechty, Liechty, and Müller (2010) proposed a method for optimal portfolio selection involving a Bayesian decision theoretical framework that addresses both higher moments and estimation error. They suggested that incorporating higher-order return distribution moments in portfolio selection is important.The financial crisis of 2008 has led many investors to search for tools that minimize downside risk. In our study, we explored one of the promising alternatives to MVO that incorporates non-normal return distributions: mean-conditional value at risk (M-CVaR) optimization. Modeling Non-Normal ReturnsEmpirically, almost all asset classes and portfolios have returns that are not normally distributed. Many assets' return distributions are asymmetrical. In other words, the distribution is skewed to the left (or occasionally the right) of the mean (expected) value. In addition, most asset return distributions are more leptokurtic, or fatter tailed, than are normal distributions. The normal distribution assigns what most people would characterize as meaninglessly small probabilities to extreme events that empirically seem to occur approximately 10 times...

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