70 Optimizing Optimization
3.5.5 Maximum holding 100%
a. Gain/Loss optimization: E [ U G (^) / (^) L E [ gains ] (1 l ) E [ losses ], l 1
b. Mean – variance optimization: EU[]MV/,λ μ
λ
2 σλ^2
2
, 2
c. Mean – variance optimization: EU[]MV/,λ μ
λ
10 σλ^2
2
, 10
All three optimizations include emerging markets in their optimal portfolio.
However, GLO obviously considers that asset class ’ negative skewness and high
kurtosis and chooses a much lower weight (13% versus 100% and 25% for low
and high risk aversion mean – variance optimization, respectively). GLO and high
risk aversion choose the same three asset classes, while the weights differ depend-
ing on skewness and kurtosis, which are considered by GLO. Hence, the GLO
portfolio has least negative skewness and least kurtosis, suggesting that higher
moments are dealt with. Naturally, it also shows the largest Ω.
Broadly speaking, portfolios that control expected losses unsurprisingly
have lower expected gains, lower expected losses, and a higher probability of
achieving a target. Contrary to popular suspicion, they are not less diversified,
they have lower negative skewness, lower kurtosis, lower standard deviation,
and lower expected return compared to either high or low risk mean – variance
portfolios.
3.5.6 Adding 25% investment constraint
For investors who seek for more diversification, we repeat the analysis with the
additional constraint that only 25% may be invested in each asset class.
Again , the GLO seems to be more diversified than either mean – variance
solution. The extent to which the constraints bind can be assessed by the
number of holdings of 25%. For GLO, we have 3 with three other assets. For
high risk tolerant mean – variance, we have 4 and for low risk tolerant mean –
variance we have 2 with three other assets.
Again , the same asset classes are considered, as high risk aversion mean –
variance adds one class to the 4 chosen by low risk aversion mean – variance
and GLO adds an additional one to those 5. That class only picked by GLO,
symptomatically shows positive skewness and kurtosis 3.
Now , the improvements in terms of moment behavior are less, skewness is
ranked second for GLO, and kurtosis is highest. However, the standard devia-
tions are lowest, probability of achieving the target is highest, and the expected
loss is lowest.
3.5.7 Down-trimming of emerging market returns
It might be that the emerging market returns are epoch-dependent and we will
get a clearer picture if we trim them down. We do this by subtracting 1% per