Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 253
The corresponding cumulative distribution function (cdf) is derived by integrat-
ing Equation (11.2) over the relevant domain of integration:
Fx gX
()
Φγηε
λ⎛
⎝⎜⎜
⎜⎞
⎠⎟⎟
⎟⎛
⎝⎜⎜
⎜⎜⎞
⎠⎟⎟
⎟⎟
(11.3)^where Φ() ( )zw/ ( / )dw
z
22 π^122
∫∞
exp denotes the cdf of a standard
Gaussian distribution.
As we have already alluded to, the system of Johnson densities constitutes
a very flexible and comprehensive way of capturing the four-moment patterns
of financial time series. In fact, Johnson (1949) showed that the system can
capture all feasible combinations of skewness and kurtosis (see Figure 11.1 ).
In essence, the Johnson system divides the skewness – kurtosis plane into three
regions by defining two curves and the point where kurtosis 3 and skew-
ness 0 (corresponding to the normal distribution). The first region is the area
below the line defined by the relationship kurtosis skewness^2 1, which is
an impossibility for any probability distribution. The second region is between
this line and the curve that defines the lognormal distribution; this region is
covered by the S B distribution. The remaining region above the lognormal
curve is captured by the S U distribution. Thus, when viewed as a complete
system, the Johnson family is perfect for modeling the complex distributional
shapes exhibited by asset returns.
While the properties of the normal and lognormal distributions are reason-
ably well known, readers may not be familiar with the basic properties of the
0 0.5 1 1.5 2024681012SkewnessKurtosis
Normal BoundedImpossible regionUnboundedLog normalFigure 11.1 Skewness and kurtosis of the Johnson system of distributions.