Chapter 10
Confidence-Weighted Mean Reversion
Empirical evidence (Borodin et al. 2004) shows that stock price relatives may follow
the mean reversion property, which has not been fully exploited by existing strategies.
Moreover, all existing online portfolio selection (OLPS) strategies only focus on the
first-order information of a portfolio vector, though second-order information may
also benefit a strategy. This chapter proposes a novel strategy named “confidence-
weighted mean reversion” (CWMR) (Li et al. 2011b, 2013). Inspired by the mean
reversion principle in finance and confidence-weighted (CW) online machine learning
technique (Crammer et al. 2008; Dredze et al. 2008), CWMR models the portfolio vec-
tor as a Gaussian distribution, and sequentially updates the distribution following the
mean reversion principle. Analysis of CWMR’s closed form updates clearly reflects
the mean reversion trading idea and the interaction of first-order and second-order
information. Extensive experiments, in Part IV, on various real markets show that
CWMR is able to effectively exploit the power of mean reversion and second-order
information, and is superior to the state-of-the-art techniques.
This chapter is organized as follows. Section 10.1 motivates the proposed CWMR
strategy. Section 10.2 formulates the strategy, and Section 10.3 derives the algorithms
based on the formulations. Section 10.4 further analyzes the algorithms. Finally,
Section 10.5 summarizes this chapter and indicates future directions.
10.1 Preliminaries
10.1.1 Motivation
The proposed method, similartopassive–aggressivemean reversion (PAMR), is based
on the meanreversiontradingidea, which, in the context of portfolio or multipleassets,
implies that good-performing assets tend to perform worse than others in subsequent
periods, and poor-performing assets are inclined to perform better. Thus, to maximize
the next portfolio return, we could minimize the expected return with respect to
today’s price relatives since next price relatives tend to revert. This seems somewhat
counterintuitive, but, according to Lo and MacKinlay (1990), the effectiveness of
mean reversion is due to the positive cross-autocovariances across assets.
71
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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
t
be the set of poor-performing stocks (x
t,i
< δ), B
t
be the set of mean reversion (MR)
stocks (x
t,i
< δ & x
t+1,i
> 1), C
t
be the set of non–mean reversion (non–MR) stocks
(x
t,i
< δ & x
t+1,i
< 1), and D
t
be the setofremaining stocks (x
t,i
< δ & x
t+1,i
= 1).
On period t, we calculate the percentage of a set U , which can be either A, B,
C,orD,asP
t
(U) =|U
t
|/|A
t
|, where |·|denotes the cardinality of a set, and the
gain of uniform investment in the set as G
t
(U) =
i∈U
t
x
t,i
/|U
t
|. For a total of n
periods, we further calculate their average values as
¯
P(U)=
1
n−1
n−1
t=1
P
t
(U) and
¯
G(U) =
1
n−1
n−1
t=1
G
t
(U), respectively. In particular, we refer to the percentage of
mean reversion stocks as
¯
P(B), and the gain of mean reversion stocks as
¯
G(B).To
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
¯
G(Market).
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 (
¯
P(B) >
¯
P(C)),
¶
and uniform investment on poor-performing stocks pro-
vides a greater profit
∗∗
than the market (
¯
G(A) >
¯
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
return.
§
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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FORMULATIONS 73
Table 10.1 Summary of mean reversion statistics on real markets
Dataset
¯
P(B)
¯
G(B)
¯
P(C)
¯
G(C)
¯
P(D)
¯
G(A)
¯
G(Market)
TSE 42.89% 1.022370 41.63% 0.978395 15.48% 1.000598 1.000405
MSCI 54.19% 1.015737 45.05% 0.984046 0.76% 1.001107 1.000053
NYSE (O) 43.43% 1.021599 39.86% 0.981949 16.71% 1.002523 1.000620
NYSE (N) 47.87% 1.019624 43.19% 0.982050 8.93% 1.001644 1.000610
DJIA 48.54% 1.018545 50.57% 0.980843 0.90% 0.999398 0.999719
SP500 50.20% 1.020692 47.96% 0.980502 1.84% 1.000881 1.000488
classifier, and sequentially update the distribution similar to passive–aggressive (PA)
learning (Crammer et al. 2006). Thus, CW learning can take advantage of both first-
and second-order information of the classifier.
To address the above two concerns, we present a novel OLPS method named
CWMR. To exploit the first- and second-order information of a portfolio vector,
we model the portfolio vector as a Gaussian distribution, which is probably the most
widely studied distribution and can satisfy our motivations. We do not consider higher
orders and other distributions for their complexities. Then, we sequentially update the
distribution following the mean reversion principle. On the one hand, we keep the
previous distribution if the portfolio is profitablebyusingmeanreversion.Onthe other
hand, we move the distribution to a new distribution such that the new distribution is
expected to make profit while keeping it close to the previous distribution. Different
from CRP and Anticor, CWMR actively exploits the mean reversion property of
financial markets with a powerful learning method. Moreover, compared with all
existing algorithms, including PAMR, which only consider the first-order information,
CWMR exploits both the first- and second-order information of a portfolio vector.
10.2 Formulations
We model b as a Gaussian distribution with mean μ ∈ R
m
and diagonal covariance
matrix ∈ R
m×m
with nonzerodiagonalelements and zero for off-diagonalelements.
The i-th element of μ represents the proportion of the i-th element. The i-th diagonal
term of stands for the confidence on the i-th proportion. The smaller the diagonal
term, the higher the confidence we have in the corresponding μ.
At the beginning of period t, we figure out a b based on the distribution N(μ, ),
that is, b ∼ N(μ, ). Then, after x
t
is revealed, the wealth increases by a factor
of b
x
t
. It is straightforward that the return D = b
x
t
can be viewed as a random
variable of the following univariate Gaussian distribution:
D ∼ N
μ
x
t
, x
t
x
t
.
Its mean is the return of mean vector, and its variance is proportional to the projection
of x
t
on .
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74 CONFIDENCE-WEIGHTED MEAN REVERSION
According to the mean reversion idea, the probability of a profitable b with respect
to a predefined mean reversion threshold is defined as
Pr
b∼N(μ,)
[D ≤ ]=Pr
b∼N(μ,)
(
b
x
t
≤
)
.
For simplicity, we write Pr[b
x
t
≤ ] instead. Note that we are considering the mean
reversion profitability in a portfolio consisting of multiple stocks; thus, this definition
is equivalent to the motivating idea of buying poor-performing stocks or, equivalently,
selling good-performing stocks.
The algorithm adjusts the distribution to ensure that the probability of a mean
reversion profitable b is higher than a confidence-level parameter θ ∈[0, 1]:
Pr
(
b
x
t
≤
)
≥ θ.
This is somewhat counterintuitive but reasonable with respect to the mean reversion
idea. If it is highly probable that the portfolio return b
x
t
is less than a threshold, it
is also highly probable that its next return based on x
t+1
tends to be higher since x
t+1
will revert.
Then, following the intuition underlying PA algorithms (Crammer et al. 2006),
our algorithm chooses a distribution closest to the current distribution N(μ
t
,
t
) in
terms of Kullback–Leibler (KL) divergence (Kullback and Leibler 1951).As a result,
at the end of period t, the algorithm updates the distribution by solving the following
optimization problem.
The Raw Optimization Problem: CWMR
(μ
t+1
,
t+1
) = arg min D
KL
(N(μ, )N(μ
t
,
t
))
s.t. Pr[b
x
t
≤ ]≥θ
μ ∈
m
.
(10.1)
The optimization problem (10.1) clearly reflects our motivation. On the one hand,
if the current μ
t
is mean reversion profitable, that is, the first constraint is satisfied,
CWMR chooses the same distribution, resulting in a passive CRP strategy. On the
other hand, if μ
t
does not satisfy the mean reversion constraint, CWMR tries to
figure out a new distribution, which is expected to profit and not far from the current
distribution.
Let us reformulate the objective and constraints. For the objective part, the KL
divergence between two Gaussian distributions can be rewritten as
D
KL
(N(μ, )N(μ
t
,
t
))
=
1
2
log
det
t
det
+Tr(
−1
t
) +(μ
t
−μ)
−1
t
(μ
t
−μ) −d
.
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FORMULATIONS 75
For the constraint part, since b ∼ N(μ, ), b
x
t
has a univariate Gaussian
distribution with mean μ
D
= μ
x
t
and variance σ
2
D
= x
t
x
t
. Thus, the probability
of a return less than is
Pr[D ≤ ]=Pr
*
D −μ
D
σ
D
≤
−μ
D
σ
D
+
.
In the preceding equation,
D−μ
D
σ
D
is a normally distributed random variable; thus,
the probability equals
−μ
D
σ
D
, where is the cumulative distribution function of
Gaussian distribution. As a result, we can rewrite the constraint as
−μ
D
σ
D
≥
−1
(θ).
Substituting μ
D
and σ
D
by their definitions and rearranging the terms, we can obtain
−μ
x
t
≥ φ
,
x
t
x
t
,
where φ =
−1
(θ). Clearly, we require that the weighted summation of return and
standard deviation is less than a threshold. Till now, we can rewrite the preceding
optimization problem.
The Revised Optimization Problem: CWMR
(μ
t+1
,
t+1
) = arg min
1
2
log
det
t
det
+Tr(
−1
t
) +(μ
t
−μ)
−1
t
(μ
t
−μ)
s.t. −μ
x
t
≥ φ
,
x
t
x
t
μ
1 = 1, μ 0. (10.2)
For the optimization problem (10.2), the first constraint is not convex in , there-
fore we have two ways to handle it. The first way (Dredze et al. 2008) is to linearize it
by omitting the square root, that is, −μ
x
t
≥ φx
t
x
t
. As a result, we can finalize
the first optimization problem, named CWMR-Var.
The Final Optimization Problem 1: CWMR-Var
(μ
t+1
,
t+1
) = arg min
1
2
log
det
t
det
+Tr(
−1
t
) +(μ
t
−μ)
−1
t
(μ
t
−μ)
s.t. −μ
x
t
≥ φx
t
x
t
μ
1 = 1, μ 0. (10.3)
The second reformulation (Crammer et al. 2008) is to decompose the positive
semidefinite (PSD) , that is, = ϒ
2
with ϒ = Qdiag(λ
1/2
1
,...,λ
1/2
m
)Q
, where
Q is orthonormal and λ
1
,...,λ
m
are the eigenvalues of and thus ϒ is also PSD.This
reformulation yields the second final optimization problem, named CWMR-Stdev.
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