For portfolio construction, we need a new theory to replace mean variance.
We need a new theory, because we find Benoit Mandelbrot's paradigm shifting research into stock return distributions most convincing. These distributions are fat-tailed, most likely have infinite variances, and even, sometimes, infinite means. Figure 28.1, “Distribution of Stock Returns,” Figure 29.1, “Cash Economic Returns, Including Small Start-Ups,” and Chapter 29, “Producing Lower Fat-Tailed Risk with Higher Returns” summarizes the empirical evidence that we find so convincing.
So, any theory based on traditional statistics—mean, variance, standard deviation, correlation, least squares regression analysis, CAPM beta, and Sharpe ratios—do not describe the real world of investing. Thus, they all need to be revised, based on statistics that actually exist.
Not everyone agrees. Efficient market advocates, such as Eugene Fama and many others, tend to believe one or more assumptions.