Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 269
For the class of elliptical distributions, which includes the normal, Student’s-
t , and Laplace, Chamberlain (1983) has shown that mean – variance approxi-
mation of expected utility is exact for all utility functions. In real-world
applications, however, portfolios often contain derivatives and other exotic
products and so the elliptical assumption becomes less and less plausible and
questions surrounding the accuracy of mean – variance resurface. Indeed, casual
inspection of empirical return distributions suggests that even equity returns
themselves may not be elliptically distributed.
In a nonelliptically distributed world, mean – variance does not guarantee a
sufficiently good approximation to expected utility and therefore direct maxi-
mization may be preferred. To explore this idea in more detail, we randomly
select U 1 ~ Uniform [6, 18] months of data from U 2 ~ Uniform [15, 30] equi-
ties and then compare the 1-month out-of-sample performance of the TA
algorithm against Markowitz’s MV approach. The global MV weights are the
solution to the following quadratic programming problem:
wwMV w∗ arg min ′Σ
(11.17)^where w i [0, 1], i 1, ... , N , and ∑wi 1. This process is then repeated
N 250 times for different combinations of equities and time periods. A
range of out-of-sample performance metrics are reported in Table 11.2 for
four different levels of risk aversion, α { 1, 2, 5, 10 }. These include the
Sharpe ratio, R p / σ (^) p , the Sortino ratio, Rpp/σD, the Upside potential ratio,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
–6
–5.8
–5.6
–5.4
–5.2
–5
–4.8
–4.6
–4.4
–4.2
Disappointment aversion, A
Draw down (%)
Figure 11.7 Maximum monthly drawdown for various levels of disappointment
aversion.