268 Optimizing Optimization
11.7.2 The coefficient of disappointment aversion, A
Up to this point, we have always set the coefficient of disappointment aversion
equal to unity, i.e., A 1. This was partly for computational convenience but
mainly because we did not have much intuition about what value is reasonable
in the context of portfolio optimization. In order to remedy this problem, we
illustrate how a suitable value can be chosen by relating the idea of disappoint-
ment aversion to the concept of maximum drawdown (MDD). By doing so, a
preferred value of A can be chosen on the basis of the maximum drawdown
that one is willing to concede over a 1-month holding period.
We define a drawdown, DD N , as the price difference resulting from a con-
tinuous fall of a security over the next N days:
DDNtr
t
1τ
∑
(11.15)where τ N (inf { t 1 : r t 0 } 1) and DD N 0 if τ 0. Analogously,
we can define the N -day maximum drawdown, MDD N , as the maximum of
the j 1, ... , D N drawdowns occurring over the next N days:
MDDN DD
jDN Nj
max ,
(11.16)where D N [0, [( N 1)/2]] denotes the number of drawdowns in N days.^24
On the basis of these definitions, we calculate 1-month out-of-sample MDD
for various values of A [0, 1] using N 250 randomly generated asset
universes based on U 1 ~ Uniform [3, 18] months of data from U 2 ~ Uniform
[3, 10] equities. The mean MDD, described in Figure 11.7 for α 2, reveals
a monotonically decreasing relationship between disappointment aversion and
MDD, and can therefore be used as a guide for choosing A on the basis of a
given level of drawdowns. For instance, if a hypothetical investor is willing to
tolerate an MDD of 5% in the proceeding month, then the coefficient of disap-
pointment aversion should be set at approximately A 0.8.
11.7.3 The importance of non-Gaussianity
As we stated in the introduction, the primary purpose of this chapter is to advance
the use of non-Gaussian alternatives to traditional mean – variance analysis for
asset allocation purposes. In this section, we therefore provide a formal compari-
son between our Johnson-based TA algorithm and Markowitz’s mean – variance
approach to optimization. At the heart of this comparison is a fundamental ques-
tion about the accuracy and reliability of mean – variance as an approximation.
24 The maximum number of draw downs is bounded from above by [( N 1)/2], where [ z ] denotes
the integer part of z.