264 Optimizing Optimization
where B denotes the bivariate standard Gaussian distribution, φ is the associ-
ated Pearson correlation coefficient, g and k are the marginal densities of X and
Y , respectively, and Z X and Z Y are standard Gaussian transforms of ( X , Y ):
VZX
WZYX
Y
()
()where Z (.) is given by Equation (11.1). Applying these transforms to our sam-
ple data yields the joint sample of (transformed) realized returns and forecasts,
{ ( v , w ) } { ( Z X ( x ), Z Y ( y )) } , and an estimate of the correlation coefficient,
φˆCor v w(,).
Given this structure, we can straightforwardly derive the conditional distri-
bution of realized returns, X , given a point forecast, Y y. Since,
V
WN⎛
⎝⎜⎜
⎜⎜⎞
⎠⎟⎟
⎟⎟∼⎡⎣⎢
⎢⎤⎦⎥
⎥⎡⎣⎢
⎢⎤⎦⎥
⎥⎛
⎝⎜⎜
⎜⎜⎞
⎠⎟⎟
⎟⎟0
01
1,φ
φthen V | ( W w ) ~ N ( φ w , 1 φ 2 ), and so upon transforming the variates
into the original space, the distribution of X , conditional on Y y is given by:
HxyZxXYZy
()() ()
|
Φφ
1 φ^2⎛⎝⎜⎜
⎜⎜
⎜⎜⎞⎠⎟⎟
⎟⎟
⎟⎟
(11.11)where Φ (.) is the standard Gaussian cdf. Moreover, using Equation (11.10), the
corresponding density function is given by:
hxy
hx y
ky() ZxX Hxy
(,)
()|e xp|() ( ( ))
1
11
21(^22)
212
φ
⎡Φ
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢
⎢⎢
⎤
⎦
⎥
⎥
gx()
(11.12)
Thus, via the process of Bayesian updating, we have updated the original port-
folio return density, g ( x ), using the forecast Y y to yield a posterior density,
h ( x|y ), which can be used for the purposes of expected utility maximization as
described in Section 11.4.1.
To make these ideas concrete, and to illustrate the notion of Bayesian updat-
ing more generally, we conclude this section with a simple example. Suppose
that the prior return distribution, g ( x ), is described by an unbounded Johnson
density, S U , whereas the marginal forecast density, k ( y ), is of the bounded
variety, S B , and exhibits a much lower variance than the density of realized