Chapter 9
Passive–Aggressive Mean Reversion
This chapter proposes a novel online portfolio selection (OLPS) strategy named
“passive–aggressive mean reversion” (PAMR) (Li et al. 2012). Unlike traditional
trend-following approaches, the proposed approach relies upon the mean reversion
relation of financial markets. We are the first to devise a loss function that reflects the
mean reversion principle. Further equipped with passive–aggressive online learning
(Crammer et al. 2006), the proposed strategy can effectively exploit mean reversion.
By analyzing PAMR’s update scheme, we find that it nicely trades portfolio return
with volatility risk and reflects the mean reversion principle. We conduct extensive
numerical experiments in Part IV to evaluate the proposed algorithms on various real
datasets. In most cases, the proposed PAMR strategy outperforms all benchmarks and
almost all state-of-the-art strategies under various performance metrics. In addition
to superior performance, the proposed PAMR runs extremely fast and thus is very
suitable for real-life online trading applications.
This chapter is organized as follows. Section 9.1 briefly reviews the ideas of
existing trend-following strategies and motivates the proposed strategy. Section 9.2
formulates the proposed PAMR strategy, and Section 9.3 derives the algorithms.
Section 9.4 further analyzes and discusses the algorithms. Finally, Section 9.5
summarizes this chapter and indicates future directions.
9.1 Preliminaries
9.1.1 Related Work
One popular trading idea in reality is trend following or momentum, which assumes
that historically outperforming stocks would still perform better than others in future.
Some existing algorithms, such as EG and ONS, approximate the expected loga-
rithmic daily return and logarithmic cumulative return, respectively, using historical
price relatives. Though this idea is easy to understand and makes fortunes for many
of the best traders and investors, trend following is hard to implement effectively. In
addition, in the short term, the stock price relatives may not follow previous
trends (Jegadeesh 1990; Lo and MacKinlay 1990).
59
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60 PASSIVE–AGGRESSIVE MEAN REVERSION
Besides trend following, another widely adopted approach is mean rever-
sion (Cover and Gluss 1986; Cover 1991; Borodin et al. 2004), which is also termed
as contrarian. This approach stems from the CRP strategy (Cover and Gluss 1986),
which rebalances to an initial portfolio every period. The idea behind this approach is
that if one stock performs worse than others, it tends to perform better in the following
periods.As a result, a contrarian strategy is characterized by the purchase of securities
that have performed poorly and the sale of securities that have performed well, or,
quite simply, “Sell the winner, buy the loser.”According to Lo and MacKinlay (1990),
the effectiveness of mean reversion is due to positive cross-autocovariances across
securities. Among existing algorithms, CRP, UP,
∗
and Anticor adopt this idea. How-
ever, CRP and UP passively revert to the mean, while empirical evidence from the
Anticor algorithm (Borodin et al. 2004) shows that active reversion to the mean may
better exploit the fluctuation of financial markets and is likely to obtain much higher
profit. On the other hand, althoughAnticor actively reverts to the mean, it is a heuristic
method based on statistical correlations. In other words, it may not effectively exploit
the mean reversion property.
Pattern matching–based nonparametric learning algorithms (B
K
,B
NN
, and
CORN, etc.) can identify many market conditions, including both mean reversion
and trend following. However, when searching similar price relatives, they may locate
both mean reversion and trend-following price relatives, whose patterns are essen-
tially opposite, thus weakening the following maximization of expected cumulative
wealth.
In summary, both trend following and mean reversion can generate profit in
the financial markets, if appropriately used. In the following, we will propose an
active mean reversion–based portfolio selection method. Though simple in update
rules, it empirically outperforms the existing strategies in most back-tests
†
with real
market data, indicating that it appropriately takes advantage of the mean reversion
trading idea.
9.1.2 Motivation
The proposed approach is motivated by the CRP (Cover and Gluss 1986), which
adopt the mean reversion trading idea. As shown in Chapter 5, the mean reversion
principle has not been widely investigated for OLPS.
Another motivation of the proposed algorithm is that, in financial crisis, all
stocks drop synchronously or certain stocks drop significantly. Under such situations,
actively rebalance may be inappropriate since it puts too much wealth on “mine”
stocks, such as Bear Stearns
‡
during the subprime crisis. To avoid potential risk
concerning such “mine” stocks, it is better to stick to a previous portfolio, which
∗
From the expert level, UP follows the winner. However, since its experts belong to CRP, it also follows
the loser in stock level. In the preceding survey, we classify it following the expert level.
†
Back-test refers to testing a trading strategy via historical market data.
‡
Bear Stearns was a US company whose stock price collapsed in September 2008.
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PRELIMINARIES 61
constitutes the CRP strategy. Here, the reason to choose a passive CRP strategy is that
these “mine” stocks are usually known only in hindsight, thus identifying them a priori
is almost impossible. Thus, to avoid suffering too much from such situations, the pro-
posed approach alternates between “aggressive” and “passive” reversion depending
on market conditions. The passive mean reversion avoids the high risk of aggressive
mean reversion, which would put most wealth on these “mine” stocks.
In the following, we propose a novel trading strategy named “passive–aggressive
mean reversion,” or PAMR for short. On the one hand, the underlying assumption
is that better-performing assets would perform worse than others in the next period.
On the other hand, if the market drops too much, we would stop actively rebalanc-
ing portfolios to avoid certain “mine” stocks and their associated risk. To exploit
these intuitions, we suggest adopting passive–aggressive (PA) online learning
(Crammer et al. 2006), which was originally proposed for classification. The basic
idea of PA is that it passively keeps the previous solution if the loss is zero, while it
aggressively updates the solution whenever the suffering loss is nonzero.
We now describe the proposed PAMR strategy in detail. Firstly, if the portfolio
period return is below a threshold, we will try to keep the previous portfolio such
that it passively reverts to the mean to avoid potential “mine” stocks. Secondly, if the
portfolio period return is above the threshold, we will actively rebalance the portfolio
to ensure that the expected portfolio daily return is below the threshold, in the belief
that the next price relatives will revert. This sounds a bit counterintuitive, but it is
indeed reasonable, because if the price relative reverts, keeping the expected port-
folio return below the threshold enables one to maintain a high portfolio return in the
next period. Here, the expected portfolio return is calculated with respect to historical
price relatives, for example, in our study, the last price relative (Helmbold et al. 1998).
To further illustrate that aggressive reversion to the mean can be more effective
than a passive one, let us continue the example that has a market going nowhere but
actively fluctuating. In such a market, the proposed strategy is much more powerful
than best constant rebalanced portfolio (BCRP), a passive mean reversion trading
strategy in hindsight, as shown in Table 9.1. As the motivating example shows,
BCRP grows to
5
4
n
for a n-trading period, while at the same time, PAMR grows
to
5
4
×
3
2
n−1
(the details of the calculation/algorithm will be presented in the next
section). We intuitively explain the success of PAMR below.
Assume the threshold for a PAMR update is set to 1, that is, if the portfolio period
return is below 1, we do nothing but keep the existing portfolio. Our strategy begins
with a portfolio
1
2
,
1
2
. For period 1, the return is
5
4
> 1. Then, at the beginning
of period 2, we rebalance the portfolio such that an approximate portfolio return
based on last price relatives is below the threshold of 1, and the resulting portfolio is
2
3
,
1
3
. As the mean reversion principle suggests, although we are building a portfolio
performing below the threshold in the current period, we are actually maximizing the
next portfolio return. As we can observe, the return for period 2 is
3
2
> 1. Then,
following the same rule, we will rebalance the portfolio to
1
3
,
2
3
. As a result, in such
a market, PAMR’s growth rate is
5
4
×
3
2
n−1
for a n-period, which is superior to
BCRP’s
5
4
n
.
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62 PASSIVE–AGGRESSIVE MEAN REVERSION
Table 9.1 Motivating example to compare BCRP and PAMR
BCRP PAMR
Period # Relatives Portfolio Return Portfolio Return Notes
1 (1/2, 2)(1/2, 1/2) 5/4 (1/2, 1/2) 5/4 Rebalance to
(2/3, 1/3)
2 (2, 1/2)(1/2, 1/2) 5/4 (2/3, 1/3) 3/2 Rebalance to
(1/3, 2/3)
3 (1/2, 2)(1/2, 1/2) 5/4 (1
/3, 2/3) 3/2 Rebalance to
(2/3, 1/3)
4 (2, 1/2)(1/2, 1/2) 5/4 (2/3, 1/3) 3/2 Rebalance to
(1/3, 2/3)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Remarks on Motivations:Although the motivating example in Table 9.1 demon-
strates the effectiveness of PAMR over BCRP, PAMR may not always outperform
BCRP. In general, PAMR is an online algorithm, whereas BCRP is an optimal offline
algorithm for i.i.d. markets (Cover and Thomas 1991, Theorem 15.3.1). Now, we
discuss some possible situations where PAMR may fail to outperform BCRP.
Consider a special case where one stock crashes and the other explodes, for exam-
ple, a market sequence of two stocks as
1
2
, 2
,
1
2
, 2
,.... In this market, BCRP
increases at an exponential rate of 2
n
as it wholly invests in the second asset, while
PAMR keeps a fixed wealth of
5
4
over the trading period. Obviously, in such situation,
PAMR performs much worse than BCRP, that is, PAMR’s
5
4
versus BCRP’s 2
n
over
n periods. Though not shining in this example, PAMR still bounds its losses. More-
over, such a market, which violates the mean reversion assumption, is occasional,
at least from the viewpoint of our empirical studies.
9.2 Formulations
Now we shall formally devise the proposed PAMR strategy for the OLPS task.
PAMR is based on a loss function that exploits the mean reversion idea, which is
our innovation, and is equipped with the PA online learning technique (Crammer
et al. 2006).
∗
First of all, given a portfolio vector b and a price relative vector x
t
, we define an
-insensitive loss function for the t-th period as
(b;x
t
) =
0 b ·x
t
≤
b ·x
t
− otherwise
, (9.1)
∗
In fact, with the loss function, we can adopt any learning methods to exploit the mean reversion
property. We choose PA for its simplicity and effectiveness. Certainly, other learning techniques can be
adopted, if the new method can provide some new insights.
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FORMULATIONS 63
where ≥ 0 is a sensitivity parameter that controls the mean reversion threshold.
Since portfolio daily return fluctuates around 1,
∗
we empirically choose ≤ 1to
buy underperforming assets. The -insensitive loss is zero when return is less than
the threshold , and otherwise grows linearly with respect to portfolio return. For
conciseness, let us use
t
to denote
(b;x
t
). By defining this loss function, we can
distinguish the preceding two motivating cases.
Then, we will formulate the proposed strategy and will propose specific algorithms
to solve them. Recalling that b
t
denotes the portfolio vector for the period t, the first
proposed method for PAMR is formulated as a constrained optimization.
Optimization Problem 1: PAMR
b
t+1
= arg min
b∈
m
1
2
b −b
t
2
s. t.
(b;x
t
) = 0. (9.2)
The above formulation attempts to find an optimal portfolio by minimizing the
deviation from last portfolio b
t
if the constraint of zero loss is satisfied. On the
one hand, the above approach passively keeps the last portfolio, that is, b
t+1
= b
t
,
whenever the loss is zero, or the portfolio daily return is below the threshold .On
the other hand, whenever the loss is nonzero, it aggressively updates the solution
by forcing it to strictly satisfy the constraint, that is,
(b
t+1
;x
t
) = 0. Clearly, this
formulation is able to address the two motivations.
Although the above formulation is reasonable to address our concerns, it may have
some undesirable properties when noisy price relatives exist, which are common in
real-world financial markets.Forexample,a noisy price relative in a trendingsequence
may suddenly change the portfolio in a wrong direction due to the aggressive update.
To avoid such problems, we propose two variants of PAMR that are able to trade off
between aggressiveness and passiveness. The idea of the two variants is similar to soft
margin support vector machines by introducing some nonnegative slack variables into
optimization. Specifically, for the first variant, we modify the objective function by
introducing a term that scales linearly with respect to a slack variable ξ and formulate
the following optimization.
Optimization Problem 2: PAMR-1
b
t+1
= arg min
b∈
m
"
1
2
b −b
t
2
+Cξ
#
s.t.
(b;x
t
) ≤ ξ and ξ ≥ 0, (9.3)
where C is a positive parameter to control the influence of the slack variable on the
objective function. We refer to this parameter as an aggressiveness parameter similar
to PA learning (Crammer et al. 2006) and call this variant “PAMR-1.”
Instead of a linear slack variable, for the second variant, we modify the objec-
tive function by introducing a term that scales quadratically with respect to a slack
variable ξ, which results in the following optimization problem.
∗
Here we use simple gross return, as defined in Section 9.2. Financial literature often adopts simple net
return (Tsay 2002), which fluctuates around 0.
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