Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 23

 It is a widely held view that the validity of mean-variance analysis requires that investors have quadratic utility and that returns are normally distributed. This view is incorrect.

 For a given time horizon or assuming returns are expressed in continuous units, mean-variance analysis is precisely equivalent to expected utility maximization if returns are elliptically distributed, of which the normal distribution is a more restrictive special case, or (not “and”) if investors have quadratic utility.

 For practical purposes, mean-variance analysis is an excellent approx- imation to expected utility maximization if returns are approximately elliptically distributed or investor preferences can be well described by mean and variance.

 For intuitive insight into an elliptical distribution, consider a scatter plot of the returns of two asset classes. If the returns are evenly distributed along the boundaries of concentric ellipses that are centered on the average of the return pairs, the distribution is elliptical. This is usually true if the distribution is symmetric, kurtosis is relatively uniform across asset classes, and the correlation of returns is reasonably stable across subsamples.

 For a given elliptical distribution, the relative likelihood of any multivariate return can be determined using only mean and variance.

 Levy and Markowitz have shown using Taylor series approximations that power utility functions, which are always upward sloping, can be well approximated across a wide range of returns using just mean and variance.

 In rare circumstances, in which returns are not elliptical and investors have preferences that cannot be approximated by mean and variance, it may be preferable to employ full-scale optimization to identify the optimal portfolio.

 Full-scale optimization is a numerical process that evaluates a large number of portfolios to identify the optimal portfolio, given a utility function and return sample. For example, full-scale optimization can accommodate a kinked utility function to reflect an investor's strong aversion to losses that exceed a chosen threshold.