Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 263

forecast whose realization y constitutes a point prediction of X , then the pur-
pose of this section is to derive the posterior distribution of X conditional on
the realized point prediction Y  y.
At the heart of this problem is the need to model the joint dependence
between realized returns, X , and their forecasts, Y , given that their marginal
distributions are members of the Johnson family. Fortunately, for our purposes,
there is now a burgeoning literature on the topic of constructing multivariate
distributions from specified marginals. 20 In theory, therefore, after appropri-
ately transforming the data, well-established results from copula theory can be
used to model the dependence between returns and their forecasts. The only
caveat is that the choice of copula needs to be computationally tractable.
Unlike traditional copula theory however, we choose to transform our
Johnson marginals into standard Gaussian distributions using Equation (11.1),
instead of uniform distributions, so that dependence can be modeled in the
computationally convenient bivariate Gaussian framework. This approach
leads to the well-known meta-Gaussian model of Kelly and Krzysztofowicz
(1995) , Kelly and Krzysztofowicz (1997).^21 In these papers, the bivariate meta-
Gaussian distribution, H , and density, h , of ( X , Y ) take the form:

Hxy() (PX xY, y) ((),()|),BZ x Z yXYφ (11.9)^

hx y

gx ky

() ZxXXYYZxZy Zy

() ()

()

, () () () ()

1 21

2

2 2

^2









exp

φ

φ

φ

()φφ^22

⎡

⎣

⎢

⎢⎢

⎤

⎦

⎥

⎥⎥

(11.10)

21 These results are easily extendable to the larger class of elliptical distributions. This leads to the
meta-elliptical model of Fang, Fang, and Kotz (2002).

20 See Sklar (1959) for an early reference, and Joe (1997) or Nelsen (2006) for a survey of the
recent literature.

0 20 40 60 80 100

0.009

0.01

0.011

0.012

0.013

0.014

0.015

Time

ρ = 0.90

ρ = 0.95

ρ = 1.00

0 20 40 60 80 100

0

0.02

0.04

0.06

0.08

0.1

0.12

Time

A Geometric B Hyperbolic

Weight

ρ = 0.90

ρ = 0.95

ρ = 1.00

Figure 11.3 Weight schemes: (a) geometric and (b) hyperbolic.