72 Optimizing Optimization
d. Mean – variance optimization: EU[]MV/,λ μ
λ
2 σλ^2
2, (^2)
e. Mean – variance optimization: EU[]MV/,λ μ
λ
10 σλ^2
2
, 10
It is interesting that the optimal GLO portfolio with l 4 is almost identi-
cal to the mean – variance portfolio with λ 10. They differ in weights by 1 – 4
basis points only. Increasing 1 to 20 changes the GLO portfolio substantially.
Another 38% is shifted to the asset class with the least negative skewness and
least kurtosis (resulting in a total weight of 90%). Also, a fourth asset class is
added, with even lower negative skewness and lower kurtosis.
3.5.9 Conclusions
We investigate the extent to which our GLO adds value to mean – variance opti-
mization. It became obvious that both optimization methods can lead to very
similar results. Differences are mainly caused by higher loss aversion and asym-
metric return distributions. In contrast to mean – variance optimization, the GLO
accounts for higher moments and, hence, captures asymmetries in return dis-
tributions. In summary, GLO seems to deliver results similar to mean – variance
optimization when it is applied to relatively normally distributed returns. Facing
asymmetries instead, the GLO is more preferable as it can distinguish by consid-
ering higher moments where mean – variance optimization has to assume normal
distributions.
To decide which optimization method is most adequate for an investor, he
or she has to consider his or her utility function and the distribution of returns
of his or her choice of asset classes. Variance is a measure of risk treating both
sides of the mean equally. As prospect theory and empirical evidence hypoth-
esize, there are investors who experience downside risk differently to upside
potential. Those would define their utility function rather through expected
gains and losses. If an investor cannot assume normal return distributions (and
Table 3.5 Summary of optimization results: IPD constrained to 10%Mean% StdDev% Skewness Kurtosis Gain% Loss% Omega Probability
(a) G/L l 1 7.74 4.80 0.20 3.61 8.00 5.06 1.581 0.60
(b) G/L^2 l 20 7.42 4.59 0.32 3.69 7.50 4.88 1.537 0.59
(c) G/L^2 l 4 7.60 4.70 0.31 3.70 7.75 4.96 1.563 0.61
(d) M/V λ 2 10.21 12.30 0.21 4.55 18.59 13.17 1.412 0.56
(e) M/V λ 10 8.12 5.30 0.00 3.40 9.07 5.75 1.577 0.58