256 Optimizing Optimization
experience is that a value between z [0.65, 0.75] is preferred for financial
applications. This choice not only reflects the large amounts of financial data
available to practitioners, but also the greater emphasis on capturing the tail
properties of return distributions. That is, we are taking advantage of the fact
that when fitting the distribution, the Johnson approximating distribution will
fit the data exactly at the chosen quantiles.
To the extent that the quality of the parameter estimates are sensitive to the
choice of z , we also define an optimal value, z , which is based on a grid-search
procedure across a range of z values. For each z value in the grid, we obtain
parameter estimates using the quantile estimator and then apply Equation
(11.1) to yield an approximately Gaussian series ( Jobst, 2005 ). The Lilliefors
test for normality is then used to select the optimal value of z such that the
transformed data are the closest to Gaussian ( Lilliefors, 1967 ). A comparison
of our optimal z with various fixed values is provided in Appendix 11.9.3.
In the majority of cases, we find that a fixed value of z [0.65, 0.75] and our
optimal z consistently deliver the lowest levels of relative bias. However, the
appeal of using z is illustrated by the lower values of the relative root mean
square error (RMSE).
We conclude this section by investigating the finite sample properties of the
Slifker and Shapiro (1980) estimator and its performance under misspecifica-
tion. In the latter case, we estimate the parameters of the Johnson distribu-
tion using data from an alternative parametric density. If the quantile estimator
performs well, then our estimated distribution should capture important fea-
tures of the true underlying density and the true value of expected utility. Our
simulation results are reported in Appendix 11.9.4 for three alternative densi-
ties; they are the Normal Inverse Gaussian ( Barndorff-Nielsen, 1997 ; Eriksson,
Ghysels, & Wang, forthcoming ), Pearson Type IV ( Heinrich, 2004 ), and Skew
Student’s- t ( Jondeau & Rockinger, 2003 ). In accordance with our discussion
in Section 11.3.1, the results indicate that the method of quantiles can capture
the four-moment patterns of the three alternative densities with a high degree
of accuracy. For our purposes, however, the more important finding is that this
is also true for the estimates of expected utility. Thus, within a portfolio opti-
mization framework where the primary goal is to estimate expected utility, the
quantile estimator appears robust to potential misspecification.
11.3.3 Simulating Johnson random variates
To obtain draws from the Johnson system of distributions, we simply reverse
the operation described by Equation (11.1). That is, we generate a standard
normal variate Z ~ N (0, 1) and then apply the inverse translation:
Xg
Z
ελ
γ
η1 ⎛
⎝⎜⎜
⎜⎜⎞
⎠⎟⎟
⎟⎟
(11.4)