Chapter 15
If there’s one thing I learned in prison
it’s that money is not the prime commodity in our lives...
time is.
Wall Street 2: Money Never Sleeps
15.1 Conclusions
This book aims to advance the state of the art in online portfolio selection (OLPS).
Here, our objective is to achieve better performance on real markets. The main
principles we adopted are the principles of pattern matching and mean reversion.
For the principle of pattern analysis, we try to locate similar patterns from the
historical market and construct optimal portfolios based on these patterns. Observing
that existing pattern matching–based approaches often adopt Euclidean distance to
measure the similarity between two patterns, we find that Euclidean distance ignores
their linear similarity and whole-market movements. Thus, we proposed to measure
the similarity via a correlation coefficient, which considers both ignored aspects,
and designed the CORrelation-driven Nonparametric learning (CORN) approach for
OLPS. The proposed CORN performs much better than existing pattern matching–
based strategies, which validates its motivations.
For mean reversion, we directly output portfolios based on the principle, which
assumes that the price trends will revert to their previous trends. Firstly, we proposed
to exploit the principle via passive–aggressive learning, resulting in the passive–
aggressive mean reversion (PAMR). In particular, PAMR tries to obtain portfolios
that perform worse than a threshold on the last price relatives, and also close to
the last portfolio. PAMR’s formulation is clear to understand, and its closed-form
solutions reflect the mean reversion principle. PAMR can achieve the best empirical
performance on most datasets at the time.
Observing that most existing algorithms only exploit the first-order information
of a portfolio, we proposed to exploit the second-order information and the mean
reversion property via confidence-weighted learning, resulting in a new family of
strategies called confidence-weighted mean reversion (CWMR). It models the port-
folio as a Gaussian distribution and sequentially updates the distribution similar to
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PAMR, but exploits both first-order and second-order information. CWMR’s closed-
form updates effectively trade off between the first- and second-order information of
a portfolio. Empirically, it generally outperforms other existing strategies, including
CORN and PAMR, on most datasets.
Analyzing the existing algorithms via Kelly’s framework, we find that the above
two mean reversion algorithms follow the assumption of single-period mean rever-
sion, which leads to performance degradation on certain datasets. To handle the
degradation and to further exploit the market, we proposed two forms of multiperiod
mean reversion, both of which are based on a moving average, and the online mov-
ing average reversion (OLMAR), which is more robust than PAMR. Empirically,
OLMAR is currently the best strategy, beating all existing algorithms, including
Weconductedan extensive set of empiricalevaluations,in which the resultsclearly
validate the effectiveness of the proposed algorithms. In particular, the proposed
algorithms sequentially advance the state of the art in terms of cumulative return,
which is the main performance metric of our studies. Besides, they are fairly robust
to their parameter settings, and most of the algorithms are generally computationally
efficient and are thus suitable for real-life large-scale environments.
15.2 Future Directions
15.2.1 On Existing Work
We have presented a family of algorithms for OLPS that represent the state of the art
of OLPS in academia. However, there is still room to further improve the existing
First of all, we calculate CORN’s correlation coefficient using a univariate cor-
relation by concatenating each market window to a column vector, during which we
may lose useful structural information. Thus, it is possible to improve its performance
using a multivariate correlation coefficient on the matrices such that the information
is retained. Since information is crucial for a portfolio selection task, we can con-
struct more effective portfolios. Another potential improvement is to redesign the
CORN algorithm when transaction costs exist, since the cumulative wealth achieved
by CORN decreases exponentially with increasing transaction costs. One possible
solution is to add a regularization term to the portfolio maximization step, such that
we can maximize expected return and meanwhile constrain expected turnovers.
Moreover, currently we only consider the similarity between two market win-
dows with the same length and same interval; however, locating patterns with
varying timing is also attractive in the pattern matching–based approaches. Dynamic
time warping (DTW) (Rabiner and Levinson 1981; Sakoe and Chiba 1990; Keogh
2002) is a dynamic programming approach proposed to recognize humans’ spoken
words, which vary a lot in timing and pronunciation. Since similarity among time
series also varies in time, DTW has been successfully applied to find patterns in
time series (Berndt and Clifford 1994; Yi et al. 1998; Keogh and Pazzani 2000;
Rakthanmanon et al. 2012). In finding patterns in time series, DTW’s basic idea
is to stretch or compress the time axis, such that the distance between time series
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and a template is minimized. In this way, one template can match various time
series with varying time. Since the pattern matching–based approaches try to locate
similar patterns among the historical time series, DTW may be directly applied to
locate time-warping patterns. To the best of our knowledge, no existing algorithm
has ever considered the time compression or stretch of market windows. However,
one associated problem with DTW in practice is its expensive time cost (Wang et al.
Second, although the three mean reversion algorithms outperform the previous
algorithms on most datasets, they may fail in certain cases. For example, if the market
contains one stock that drops continuously and significantly over some periods, then
all the three algorithms may fail. Such scenarios may simulate a financial crisis when
all stocks continuously drop for a certain period, or the case that one stock is hit
by a series of bad news, and its price slashes continuously over a certain period. In
fact, for such cases, most current good-performing algorithms, including Anticor and
all pattern matching–based algorithms, will underperform the naive market strategy.
In our experiments, we choose blue chip stocks, by assuming that they would not
drop continuously for a long period. However, it is always possible that the “black
swan” (Taleb 2008) may exist in the financial market and hit your portfolio. Such a
hypothetical market sequence poses a serious challenge: How to achieve a reasonable
performance in such a market?
Third, in Section 9.4, we briefly connect PAMR’s update and the general form
of return-based contrarian strategies. Besides PAMR, we find that GP’s update in
Section 4.2 also coincides with the general form of return-based momentum strategies.
This finding is not abnormal as both adopt similar learning techniques but contrary
trading ideas, that is, contrarian for PAMR and momentum for GP. Based on such
a connection, we can anatomize a portfolio’s expected return and find its sources of
profit. Although pure trading strategies have been anatomized for a long time, OLPS
algorithms have yet to be anatomized. Such an analysis will help understand the
behaviors of OLPS algorithms, which is largely unknown in current research. The
resulting empirical observations can help validate the analysis, as is done in previous
studies (Lo and MacKinlay 1990; Conrad and Kaul 1998; Lo 2008).
Fourth, as empirically analyzed in Section 13.6 and discussed in Section 14,
differentassetpools are suitable for differentalgorithms.Also, as there exist thousands
of assets in the financial markets, it would be more computationally efficient to select
a subset of assets. This poses an open challenge, that is, how to prepare a proper asset
pool for specific OLPS algorithms. This challenge is similar to the task of feature
selection, and we plan to explore general methods to automatically select effective
subsets of assets.
Finally, although all the proposed algorithms significantly outperform the state of
the art, one important aspect missing in current study is their theoretical guarantees.
As discussed in Section 14.3, there exist several explanations and measures to handle
the issue. Nevertheless, the theoretical guarantees, either the universal property or
others, are still missing for the proposed algorithms. In future, we plan to present
some nontrivial theoretical guarantees for the proposed algorithms, such that we can
be more confident regarding the algorithms’ practical applicability.
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15.2.2 On Practical Issues
To improve the practicability of OLPS algorithms, one should also tackle the practical
issues in real markets, so as to relax the ideal assumptions of the market model and
provide possible applications.
A crucial practical issue is transaction cost (Kissell et al. 2003), which has been
addressed in some literatures (Blum and Kalai 1999; Iyengar 2005; Lobo et al. 2007;
Györfi and Vajda 2008; Kozat and Singer 2008, 2010). Among them, a research direc-
tion is to investigate a simple linear model with a proportional transaction cost (Blum
and Kalai 1999; Borodin et al. 2004; Györfi and Vajda 2008), which is easier to handle
than nonlinear models (Takano and Gotoh 2011). Different from existing solutions,
which mainly trade off between the expected return and expected transaction costs,
one can solve this by the regularization methods (Tibshirani 1996) to resolve ill-posed
problems and minimize portfolio turnover at a reasonable expected return.
In addition to addressing explicit transaction costs, another interesting future
direction is to ignore commission fees and taxes but consider them as part of the
bid–ask spread, which allows trading strategies to issue not only market but also limit
orders. This would allow for negative transaction costs,
which become profit of the
Another practical issue in the task is to incorporate useful side information, such
as experts’ opinions, firms’ fundamental information, and technical indicators. As
shown in existing studies (Cover and Ordentlich 1996; Fagiuoli et al. 2007), such
side information may facilitate the portfolio selection task. Moreover, other state-of-
the-art research (Zhang and Skiena 2008, 2010) focuses on learning real-time news
to facilitate a single stock trading, while portfolio trading is still new. Most existing
works manually evaluate the side information as integers, and one potential work is to
sequentially learn appropriate integers with the side information. The learned integers
can help us to handle the side information and thus improve the performance of the
proposed approaches.
15.2.3 Learning for Index Tracking
The literature (Meade and Salkin 1989, 1990) shows that most active management
funds focusing on absolute return usually do not outperform the corresponding mar-
ket index. Thus, index mutual funds, which try to track the market index, have
emerged in recent decades. Briefly speaking, index tracking is passive portfolio man-
agement aiming to construct a portfolio that tracks one real or virtual index using
stock shares of as few companies as possible. Currently, some computational intelli-
gence approaches (Gilli and Këllezi 2002; Beasley et al. 2003; Coleman et al. 2006;
Maringer 2008; Canakgoz and Beasley 2009) have been proposed to tackle this task.
We also notice the development of lasso techniques (Tibshirani 1996; Fu 1998; Zou
2006; Bach 2008; Yang et al. 2010; Zhou et al. 2010), which mainly control the spar-
sity of a decision variable. Thus, it is possible to handle the index-tracking problem
via the lasso techniques (McWilliams and Montana 2010). On the one hand, it can
By issuing limit orders, the algorithm would play the traditional role of a dealer.
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ensure that the required tracking error is minimized on the constraint that the number
of chosen companies is less than a predefined number. On the other hand, we can
stratify the companies into groups and achieve the sparsity among groups or within a
group. The main challenge to learn an index-tracking portfolio using lasso techniques
is the contradiction between the simplex constraint and the lasso constraint, as the
former deactivates the lasso constraint. Brodie et al. (2009) solved this challenge in
the traditional mean variance model, while how to tackle this challenge in the context
of online index tracking can be further investigated.
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