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

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 259

In the case of α 1 and A  1 for instance, the investor solves the following
maximization problem:

wwr

wr

t1

t1

wrt1

*arg max

w















U ≡

∝∫

EU

f

[( )]

()

(

()

1

1

1

1

′

′

′

α

Ω α

wwr′′t1)(dwrt1)

where r t (^)  1 is a vector of returns from N assets, w is the corresponding vector
of portfolio weights, and f ( w r t (^)  1 ) is the portfolio distribution that we estimate
using the Johnson density algorithm outlined in the previous section.
We also introduce two types of constraint on the portfolio weights. First, the
constraints Bw  b restrict the percentage allocations in each sector or indus-
try and define upper and lower bounds on the weights for each asset, whereas
the gross exposure constraint || w || 1  c prevents extreme positions. 16 In par-
ticular, c  1 corresponds to the case of no short selling, while c  means
that there are no constraint on short sales. In this study, we shall impose the
constraints w i  0 and Σ w i  1, which correspond to B   I N , b  0 , and
c  1 in the above notation.
16 || w || 1  || w 1 ||  ...  || w N || is the L 1 norm of the weight vector. This constraint can also be inter-
preted as the transaction cost incurred by implementing the portfolio.
–0.2 –0.15 –0.1 –0.05 –0 0.05 0.1 0.15 0.2
–1.6
–1.4
–1.2
–1
–0.8
–0.6
Gain/loss
Utility
A = 0.5
A = 1.0
Figure 11.2 Utility from gains and losses.