Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 249
reduces to the much simpler task of minimizing the variance of portfolio returns
for a given mean return, or vice versa. That this approach still underpins many
of the asset allocation decisions carried out within the finance industry today
is a testament to the ease with which the optimal portfolio can be constructed.
Even in its many modern incarnations, the mean – variance optimal portfolio is
often no more than the solution to a simple quadratic programming problem.
Yet , even in the 1960s, researchers were already beginning to gather evidence
against the Gaussian hypothesis. 2 In this case, only under quadratic utility
does the mean – variance approach retain the property of directly maximizing
expected utility. 3 For all other utility functions, mean – variance optimization
can at best be interpreted as a second-order approximation of expected utility
maximization ( Rubinstein, 1973 ; Kraus & Litzenberger, 1976 ). Asset allocation
in this context is much more complicated and can quickly become intractable,
particularly in large dimensions.
Arguably , the most natural solution to this problem is to consider higher-
order approximations of expected utility. As Samuelson (1970) and Kraus and
Litzenberger (1976) demonstrate, we can obtain an expression for expected
utility that depends linearly on the higher moments of the portfolio return
by replacing the utility function with its infinite-order Taylor expansion.
Unfortunately however, for utility functions such as the power variety, the
series expansion only converges under very restriction conditions ( Lhabitant,
1998 ). And even for well-behaved functional forms, there is little consensus
over the appropriate point to truncate the expansion (cf. Bansal, Hsieh, &
Viswanathan, 1993 ). What’s more, this choice is made even more difficult
because of the fact that the inclusion of additional moments does not neces-
sarily improve the quality of the approximation ( Brockett & Garven, 1998 ;
Ber é nyi, 2001 ). Nevertheless, the inherent flexibility of this approach and the
obvious parallels with the original mean – variance framework mean that it is
still widely used in practice. Recent contributions include studies by Brandt,
Goyal, Santa-Clara, and Stroud (2005) , de Athayde and Flores (2004) , Harvey,
Liechty, Liechty, and M ü ller (2004) , and Jondeau and Rockinger (2006). All of
these studies use up to fourth-order expansions in order to capture preferences
over skewness and kurtosis.
The alternative to using the Taylor expansion is to continue maximizing
expected utility directly. In most applications however, this requires a cum-
bersome numerical integration that severely limits the number of assets that
can be included. For instance, in a recent study that is very much related to
our approach, Duxbury (2008) outlines a utility maximization problem for
Johnson-distributed asset returns and power utility. However, his choice of
2 There is strong empirical evidence that asset return distributions are characterized by leptokurto-
sis and negative skewness. See Mandelbrot (1963) and Fama (1965) for early evidence.
3 Quadratic utility has numerous unappealing properties. In particular, it implies increasing abso-
lute risk aversion and satiation, i.e., utility that is not everywhere monotonically increasing in
wealth.