Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 251
solution.^7 In fact, it is our belief that the choice of algorithm is secondary to
the method of choosing the “ tuning ” parameters that motivates our final deci-
sion to use the threshold acceptance algorithm. That is, we feel that the thresh-
old accepting (TA) algorithm has the highest level of parsimony among local
search algorithms, especially when we apply the data-driven method of tun-
ing the threshold sequence described by Gilli et al. (2006). However, before we
describe the specifics of this algorithm in greater detail, we first of all review
the properties of the Johnson family of distributions, which underlie our
expected utility calculations.
11.3 The Johnson family
The Johnson family is composed of four flexible distributions which include the
normal, lognormal, bounded, and unbounded ( Johnson, 1949 ). Johnson distri-
butions are uniquely determined by their first four moments which can be cho-
sen mutually independently, and so they are well suited to modeling the empirical
distribution of asset returns. This flexibility also means that Johnson distribu-
tions can capture the fourth-order moment patterns of many of the popular par-
ametric models used in financial econometrics such as the Skewed Student’s- t ,
Pearson Type IV, and Normal Inverse Gaussian distributions to name but a few.
Despite this flexibility, however, the use of Johnson distributions in finan-
cial applications has been severely limited. This is especially surprising given
the speed and computational ease with which the parameters can be estimated.
Notable exceptions include studies by Perez (2004) , Jobst (2005) , Yan (2005) ,
and Duxbury (2008). Specifically, Perez (2004) uses the Johnson system as a
tool to analyze and model the nonnormal behavior of hedge fund indices, while
Jobst (2005) uses the Johnson density for prewhitening before his GARCH
analysis of spread dynamics within European asset-backed securities markets.
Yan (2005) also uses Johnson distributions in a GARCH context, but as an
alternative to the more familiar Gaussian and Student’s- t error distributions.
And finally, as we have already discussed, Duxbury (2008) uses the system of
Johnson distributions for asset allocation purposes in the spirit of the approach
we adopt in this chapter.
11.3.1 Basic properties
In his seminal work, Johnson (1949) described a family of three probability
distributions based on various transformations to normality. The first, denoted
S L , is the familiar two- or three-parameter lognormal distribution; the second,
7 For instance, although Chang, Meade, Beasley, and Sharaiha (2000) found that simulated anneal-
ing and the genetic algorithm outperformed tabu search in a large-dimension asset allocation
problem, the general view among practitioners is that this result was due to an inappropriate
choice of tuning parameters rather than the poor performance of the algorithm itself.