Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 53

For a given time horizon or assuming returns are expressed in continuous units, it is a remarkably robust portfolio formation process, assuming that at least one of two conditions prevails: either investor preferences toward return and risk can be well described by just mean and variance, or returns are approximately elliptically distributed.

The objective function for mean-variance analysis is a quadratic function, which many investors find problematic because it implies that at a particular level of wealth investors would prefer less wealth to more wealth. Of course, such a preference is not plausible, but as shown by Levy and Markowitz (1979),² mean and variance can be used to approximate a variety of plausible utility functions across a wide range of returns reasonably well. If this condition is satisfied, it does not matter how returns are distributed, because investors care only about mean and variance.

If this condition is not satisfied, however, mean-variance analysis requires returns to be approximately elliptically distributed. The normal distribution is a special case of an elliptical distribution, which is itself a special case of a symmetric distribution. A normal distribution has skewness equal to zero, and its tails conform to a kurtosis level of three.³ An elliptical distribution, in two dimensions (two asset classes), describes a scatter plot of returns in which the return pairs are evenly distributed along the boundaries of ellipses that are centered on the mean observation of the scatter plot.⁴ It therefore has skewness of zero just like a normal distribution, but it may have non-normal kurtosis. The same is true for symmetric distributions more generally, though they also allow for return pairs in a two-dimensional scatter plot to be unevenly distributed along the boundaries of ellipses that are centered on the mean observation of the scatter plot, as long as they are distributed symmetrically. A symmetric distribution that comprises subsamples with substantially different correlations would not be elliptical, for example. The practical meaning of these distinctions is that mean-variance analysis, irrespective of investor preferences, is well suited to return distributions that are not skewed, have correlations that are reasonably stable across subsamples, and have relatively uniform kurtosis across asset classes, but may include a higher number of extreme observations than a normal distribution.