Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 275
values. This procedure is repeated 10,000 times in order to obtain an estimate
of the relative bias and RMSE for each parameter and length of data:
BiasRMSE()
1001000
0
0202EEˆˆεε
εεε
ε⎡⎣⎢
⎢⎤⎦⎥
⎥
⎡
⎣⎢⎤
⎦⎥where εˆ is estimator and ε 0 is the true parameter value. The results are pre-
sented in Table 11.3.^26
In the majority of cases, we find that a fixed value of z 0.7 and our opti-
mal z deliver the lowest levels of relative bias. However, the appeal of using
z when compared to a fixed value is highlighted by the lower values of the
relative RMSE. Allowing z to be chosen by the data leads to a lower variability
in the parameter estimates because of the enhanced ability to match the data-
generating process.
The quantile estimator under misspecification
In this section, we explore the limitations of the method of quantiles, described
in Section 11.3.2, when the true density is not one of the Johnson family. To
compare the performance of the estimation procedure, we choose three popu-
lar densities that have been used to model financial time series. These are the
Normal Inverse Gaussian ( Barndorff-Nielsen, 1997 ; Eriksson et al. , forthcom-
ing ), Pearson Type IV ( Heinrich, 2004 ), and Skew Student’s- t ( Jondeau &
Rockinger, 2003 ). In order to facilitate comparisons between the results, we
select their parameters such that the theoretical moments correspond to an annu-
alized mean return of 10%, a standard deviation of 20%, skewness of 0.5,
and kurtosis of 5. We then compute relative bias and RMSE estimates for the
first four moments of the estimated Johnson density as well as expected util-
ity. For completeness, we also report the mean absolute error, L 1 ( f ), the mean
squared error, L 2 ( f ), and the Kullback – Leibler distance, KL ( f ):
Lf 1 ()∫|fx fxdx() ()|ˆ
(11.18)Lf 2 fx fx dx2
()∫()() ˆ()
(11.19)KL f f xfx
f x() () dx
()
()
logˆ
∞∞
∫
(11.20)26 Note: We find almost identical results using the other members of the Johnson family.