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

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 265

returns.^22 Figure 11.4 illustrates how the posterior density reflects a compro-
mise between the prior and marginal densities for different values of the cor-
relation coefficient, φ  { 0.5, 0.9 } , and current forecast, Y  y  { 0, 0.3 }.
Although highly stylized, the figure illustrates the fundamental idea that the
the higher the correlation between returns and their forecasts, the closer the
posterior density resembles the forecast distribution and the more the location
is influenced by the current forecast, Y  y.

11.7 Empirical application

After outlining the baseline utility maximization algorithm and two extensions,
we now focus our attention on practical implementation and the choice of
“ tuning ” parameters. Wherever possible, we endeavor to describe data-driven

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Figure 11.4 Prior, marginal, and posterior density estimates: (A) φ  0.5 and y  0;
(B) φ  0.9 and y  0; (C) φ  0.5 and y  0.3; and (D) φ  0.9 and y  0.3.

22 This is likely to occur naturally in practical applications because of the inequality Var( X ) 
Var( E ( X Ω )), where Ω is the information set used to forecast X.