Asset Allocation

CHAPTER 12Necessary Conditions for Mean-Variance Analysis

THE CHALLENGE

Mean-variance analysis is much more robust than commonly understood. For a given time horizon or assuming returns are expressed in continuous units, if returns are elliptically distributed and investor utility is upward sloping and concave, mean-variance analysis delivers the same result as maximizing expected utility. And even if returns are not elliptically distributed, mean-variance analysis yields the true utility-maximizing portfolio as long as utility is a quadratic function of wealth. (See Chapters 2 and 25 for more detail about elliptical distributions.)

But what if returns are not elliptical and utility is not quadratic? In this case, mean-variance analysis must be viewed as an approximation rather than an equality. But in almost all cases it is an exceptionally good approximation. Elliptical distributions are very good approximations of empirical return distributions. (For example, they allow for excess kurtosis.) Moreover, we can approximate many plausible utility functions as a quadratic function. Levy and Markowitz (1979), for example, used Taylor series to approximate a variety of power utility functions, thereby demonstrating the broad applicability of mean-variance analysis. Markowitz and Blay (2014) offer further discussion on the broad applicability of mean-variance analysis for a range of practical circumstances.

There are, however, plausible situations in which mean-variance analysis ...