Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 257
where for all real z we define the inverse translation function:
gzzS
zSN
L
1
1()()
[( (Normal Family:
exp Lognormal Family:
exp
zS
zz)) B
(() ())12Bounded Family:
exp exp / Unbounded Family: SU⎧⎨⎪⎪
⎪⎪
⎪⎪⎩⎪⎪
⎪⎪
⎪⎪Repeated application of this procedure will yield a vector of draws from the
appropriate Johnson density.
11.4 The portfolio optimization algorithm
In this section, we describe the investor’s asset allocation problem and how the
use of the threshold acceptance algorithm can be used to obtain an optimal set
of portfolio weights.
11.4.1 The maximization problem
Choosing the appropriate objective function that our hypothetical investor
wants to maximize is arguably the most important part of the entire portfolio
optimization problem ( Brandt, 2009 ). Although many different objective func-
tions have been suggested, the academic literature has mainly focused on the
class of hyperbolic absolute risk aversion (HARA) utility functions. 13 Within
the HARA class, power or constant relative risk aversion (CRRA) utility is by
far the most popular. The reason is that portfolio choices expressed as a per-
centage of wealth are independent of wealth, which facilitates normalizations
that improve the tractability and transparency of the optimization procedure.
Despite this analytical elegance, the HARA class of utility functions have the
unfortunate implication that downside losses and upside gains are treated sym-
metrically. This is contrary to a growing body of experimental evidence that
suggests that decision makers are distinctly more sensitive to downside losses
than upside gains ( Tversky & Kahneman, 1991 ). Even in Markowitz’s semi-
nal treatise on portfolio optimization, he advocated the use of downside semi-
variance instead of variance as the appropriate measure of risk. It was only the
intractability of semivariance that prevented him from pursuing this approach
further ( Granger, 2009 ).
In the academic literature, behavioral prospect theory has been the most
common method of incorporating loss aversion into utility maximization prob-
13 The utility function has also be replaced by numerous risk measures such as those described
by Artzner, Delbaen, Eber, and Heath (1999). In this case, the utility maximization problem
becomes a risk minimization problem.