72 CONFIDENCE-WEIGHTED MEAN REVERSION
Besides the virtual example in Section 9.1.2, we empirically analyze real market
data to show that mean reversion does exist.
Although measuring mean reversion in
a single stock is well studied (Poterba and Summers 1988; Chaudhuri and Wu 2003;
Hillebrand 2003),thestudy of meanreversionin a portfolio israre. Since, in ourformu-
lation, the portfolio is long-only,
we focus on whether we can obtain a higher return
than the market by investing on poor-performing assets.
With a threshold δ, let A
be the set of poor-performing stocks (x
< δ), B
be the set of mean reversion (MR)
< δ & x
> 1), C
be the set of non–mean reversion (non–MR) stocks
< δ & x
< 1), and D
be the setofremaining stocks (x
< δ & x
On period t, we calculate the percentage of a set U , which can be either A, B,
|, where |·|denotes the cardinality of a set, and the
gain of uniform investment in the set as G
|. For a total of n
periods, we further calculate their average values as
(U), respectively. In particular, we refer to the percentage of
mean reversion stocks as
P(B), and the gain of mean reversion stocks as
show whether buying poor-performing stocks is proﬁtable, we calculate the average
gain of uniform investment on poor-performing stocks, denoted as
G(A), and the
average gain of uniform investment in the whole market, denoted as
Table 10.1 gives the statistics on six real market daily datasets.
On the one hand,
except for the DJIA dataset (please refer to Chapter 12 for details), mean reversion
does exist (
and uniform investment on poor-performing stocks pro-
vides a greater proﬁt
than the market (
G(Market)). On the other hand, the
test failed on theDJIAdataset, and inthefollowingempirical evaluations, CWMR also
failed badly on the dataset, which motivates our next proposed method in Chapter 11.
Moreover, all state-of-the-art approaches only exploit ﬁrst-order information of a
portfolio vector, while higher order information may also beneﬁt the portfolio selec-
tion task (Harvey et al. 2010). Evidence (Chopra and Ziemba 1993) shows that in
portfolio selection, errors in variance have about 5% impact on the objective value
as errors in mean do. For simplicity, we exploit variance information while ignor-
ing covariance information, which has a much smaller impact on the ﬁnal objective
value. To take advantage of both ﬁrst- and second-order information, we adopt CW
online learning (Crammer et al. 2008; Dredze et al. 2008), which was originally pro-
posed for classiﬁcation. CW’s basic idea is to maintain a Gaussian distribution for a
The test program and datasets will be available at http://stevenhoi.org/olps
Long-only means if something is considered undervalued, managers would invest; if something is
considered overvalued, managers would avoid it.
If short is allowed, we can also show whether shorting good-performing stocks provides a higher
We list their details in Section 12.2. We empirically choose δ = 0.985 on all datasets. As we have
tested, other thresholds also release similar observations. For tests on other frequencies, please refer to
Li et al. (2013).
This indicates a higher probability of reversion, but we have no theoretical guarantee for the criteria.
The absolute return in the daily scale is relatively small. However, considering their net return, such a
strategy makes much higher proﬁt than the market does. Moreover, with compounding, such small absolute
differences will result in huge differences over time.
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