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Statistical Arbitrage in the U.S. Equity Market
Marco Avellaneda∗† and Jeong-Hyun Lee∗
June 30, 2008
Abstract
We study model-driven statistical arbitrage strategies in U.S. equities.
Trading signals are generated in two ways: using Principal Component
Analysis and using sector ETFs. In both cases, we consider the residuals,
or idiosyncratic components of stock returns, and model them as a mean-
reverting process, which leads naturally to “contrarian” trading signals.
The main contribution of the paper is the back-testing and comparison
of market-neutral PCA- and ETF- based strategies over the broad universe
of U.S. equities. Back-testing shows that, after accounting for transaction
costs, PCA-based strategies have an average annual Sharpe ratio of 1.44
over the period 1997 to 2007, with a much stronger performances prior to
2003: during 2003-2007, the average Sharpe ratio of PCA-based strategies
was only 0.9. On the other hand, strategies based on ETFs achieved a
Sharpe ratio of 1.1 from 1997 to 2007, but experience a similar degradation
of performance after 2002. We introduce a method to take into account
daily trading volume information in the signals (using “trading time”
as opposed to calendar time), and observe significant improvements in
performance in the case of ETF-based signals. ETF strategies which use
volume information achieve a Sharpe ratio of 1.51 from 2003 to 2007.
The paper also relates the performance of mean-reversion statistical
arbitrage strategies with the stock market cycle. In particular, we study
in some detail the performance of the strategies during the liquidity cri-
sis of the summer of 2007. We obtain results which are consistent with
Khandani and Lo (2007) and validate their “unwinding” theory for the
quant fund drawndown of August 2007.
1 Introduction
The term statistical arbitrage encompasses a variety of strategies and investment
programs. Their common features are: (i) trading signals are systematic, or
∗Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012
USA
†Finance Concepts SARL, 49-51 Avenue Victor-Hugo, 75116 Paris, France.
1
rules-based, as opposed to driven by fundamentals, (ii) the trading book is
market-neutral, in the sense that it has zero beta with the market, and (iii) the
mechanism for generating excess returns is statistical. The idea is to make many
bets with positive expected returns, taking advantage of diversification across
stocks, to produce a low-volatility investment strategy which is uncorrelated
with the market. Holding periods range from a few seconds to days, weeks or
even longer.
Pairs-trading is widely assumed to be the “ancestor” of statistical arbitrage.
If stocks P and Q are in the same industry or have similar characteristics (e.g.
Exxon Mobile and Conoco Phillips), one expects the returns of the two stocks
to track each other after controlling for beta. Accordingly, if Ptand Qtdenote
the corresponding price time series, then we can model the system as
ln(Pt/Pt0) = α(t−t0) + βln(Qt/Qt0) + Xt(1)
or, in its differential version,
dPt
Pt
=αdt +βdQt
Qt
+dXt,(2)
where Xtis a stationary, or mean-reverting, process. This process will be re-
ferred to as the cointegration residual, or residual, for short, in the rest of the
paper. In many cases of interest, the drift αis small compared to the fluctua-
tions of Xtand can therefore be neglected. This means that, after controlling for
beta, the long-short portfolio oscillates near some statistical equilibrium. The
model suggests a contrarian investment strategy in which we go long 1 dollar of
stock P and short βdollars of stock Q if Xtis small and, conversely, go short P
and long Q if Xtis large. The portfolio is expected to produce a positive return
as valuations converge (see Pole (2007) for a comprehensive review on statistical
arbitrage and co-integration). The mean-reversion paradigm is typically asso-
ciated with market over-reaction: assets are temporarily under- or over-priced
with respect to one or several reference securities (Lo and MacKinley (1990)).
Another possibility is to consider scenarios in which one of the stocks is
expected to out-perform the other over a significant period of time. In this
case the co-integration residual should not be stationary. This paper will be
principally concerned with mean-reversion, so we don’t consider such scenarios.
“Generalized pairs-trading”, or trading groups of stocks against other groups
of stocks, is a natural extension of pairs-trading. To explain the idea, we con-
sider the sector of biotechnology stocks. We perform a regression/cointegration
analysis, following (1) or (2), for each stock in the sector with respect to a
benchmark sector index, e.g. the Biotechnology HOLDR (BBH). The role of
the stock Qwould be played by BBH and Pwould an arbitrary stock in the
biotechnology sector. The analysis of the residuals, based of the magnitude of
Xt, suggests typically that some stocks are cheap with respect to the sector,
others expensive and others fairly priced. A generalized pairs trading book, or
statistical arbitrage book, consists of a collection of “pair trades” of stocks rel-
ative to the ETF (or, more generally, factors that explain the systematic stock
2
returns). In some cases, an individual stock may be held long against a short
position in ETF, and in others we would short the stock and go long the ETF.
Due to netting of long and short positions, we expect that the net position in
ETFs will represent only to a small fraction of the total holdings. The trading
book will look therefore like a long/short portfolio of single stocks. This paper
is concerned with the design and performance-evaluation of such strategies.
The analysis of residuals will be our starting point. Signals will be based on
relative-value pricing within a sector or a group of peers, by decomposing stock
returns into systematic and idiosyncratic components and statistically modeling
the idiosyncratic part. The general decomposition may look like
dPt
Pt
=αdt +
n
X
j=1
βjF(j)
t+dXt,(3)
where the terms F(j)
t, j = 1, ..., n represent returns of risk-factors associated
with the market under consideration. This leads to the interesting question
of how to derive equation (3) in practice. The question also arises in classical
portfolio theory, but in a slightly different way: there we ask what constitutes
a “good” set of risk-factors from a risk-management point of view. Here, the
emphasis is instead on the residual what remains after the decomposition is
done. The main contribution of our paper will be to study how different sets of
risk-factors lead to different residuals and hence to different profit-loss (PNL)
for statistical arbitrage strategies.
Previous studies on mean-reversion and contrarian strategies include Lehmann
(1990), Lo and MacKinlay (1990) and Poterba and Summers (1988). In a recent
paper, Khandani and Lo (2007) discuss the performance of the Lo-MacKinlay
contrarian strategies in the context of the liquidity crisis of 2007 (see also refer-
ences therein). The latter strategies have several common features with the ones
developed in this paper. Khandani and Lo (2007) market-neutrality is enforced
by ranking stock returns by quantiles and trading “winners-versus-losers”, in a
dollar-neutral fashion. Here, we use risk-factors to extract trading signals, i.e.
to detect over- and under-performers. Our trading frequency is variable whereas
Khandani-Lo trade at fixed time-intervals. On the parametric side, Poterba and
Summers (1988) study mean-reversion using auto-regressive models in the con-
text of international equity markets. The models of this paper differ from the
latter mostly in that we immunize stocks against market factors, i.e. we consider
mean-reversion of residuals (relative prices) and not of the prices themselves.
The paper is organized as follows: in Section 2, we study market-neutrality
using two different approaches. The first method consists in extracting risk-
factors using Principal Component Analysis (Jolliffe (2002)). The second method
uses industry-sector ETFs as proxies for risk factors. Following other authors,
we show that PCA of the correlation matrix for the broad equity market in the
U.S. gives rise to risk-factors that have economic significance because they can
be interpreted as long-short portfolios of industry sectors. Furthermore, the
stocks that contribute the most to a particular factor are not necessarily the
largest capitalization stocks in a given sector. This suggests that, unlike ETFs,
3
PCA-based risk factors are not biased towards large-capitalization stocks. We
also observe that the variance explained by a fixed number of PCA eigenvectors
varies significantly across time, leading us to conjecture that the number of ex-
planatory factors needed to describe stock returns is variable and that this vari-
ability is linked with the investment cycle, or the changes in the risk-premium
for investing in the equity market.1
In Section 3 and 4, we construct the trading signals. This involves the
statistical estimation of the process Xtfor each stock and the determination of
reasonable entry and exit points for trading. Based on these signals, and using
daily end-of-day (EOD) data, we perform a full estimation of stock signals on a
daily basis, going back to 1996 in some cases and to 2002 in others, across the
broad universe of stocks with market-capitalization of more than 1 billion USD
at the trade date.
The estimation and trading rules are kept simple to avoid data-mining. For
each stock in the universe, the the parameter estimation is done using a 60-day
trailing window, which corresponds roughly to an earnings cycle. The length
of the window is fixed once and for all in the simulations and is not changed
from one stock to another. We use the same fixed-length estimation window,
we choose as entry point for trading any residual that deviates by 1.25 standard
deviations from equilibrium, and we exit trades if the residual is less than 0.5
standard deviations from equilibrium, uniformly across all stocks.
In Section 5 we back-test different strategies which use different sets of fac-
tors to generate residuals, namely: synthetic ETFs based on capitalization-
weighted indices, actual ETFs, a fixed number of factors generated by PCA,
a variable number of factors generated by PCA. In all cases, we assume a
slippage/transaction cost of 0.05% or 5 basis points per trade (a round-trip
transaction cost of 10 basis points).
In Section 6, we consider a modification of the strategy in which signals
are estimated in “trading time” as opposed to calendar time. In the statistical
analysis, using trading time on EOD signals is effectively equivalent to multi-
plying daily returns by a factor which is inversely proportional to the trading
volume for the past day. This modification accentuates (i.e. tends to favor) con-
trarian price signals taking place on low volume and mitigates (i.e. tends not
to favor) contrarian price signals which take place on high volume. It is as if we
“believe more” a print that occurs on high volume and less ready to bet against
it. Back-testing the statistical arbitrage strategies using trading-time signals
leads to improvements in most strategies, suggesting that volume information
is valuable in the mean-reversion context, even at the EOD time-scale.
In Section 7, we discuss the performance of statistical arbitrage in 2007,
and particularly around the inception of the liquidity crisis of August 2007. We
compare the performances of the mean-reversion strategies with the ones studied
in the recent work of Khandani and Lo (2007). Conclusions are presented in
Section 8.
1See Scherer and Avellaneda (2002) for similar observations for Latin American debt se-
curities in the 1990’s.
4
2 A quantitative view of risk-factors and market-
neutrality
We divide the world schematically into “indexers’ and “market-neutral agents”.
Indexers seek exposure to the entire market or to specific industry sectors. Their
goal is generally to be long the market or sector with appropriate weightings in
each stock. Market-neutral agents seek returns which are uncorrelated with the
market.
Let us denote by {Ri}N
i=1 the returns of the different stocks in the trading
universe over an arbitrary one-day period (from close to close). Let Frepresent
the return of the “market portfolio” over the same period, (e.g. the return on
a capitalization-weighted index, such as the S&P 500). We can write, for each
stock in the universe,
Ri=βiF+˜
Ri,(4)
which is a simple regression model decomposing stock returns into a systematic
component βiFand an (uncorrelated) idiosyncratic component ˜
Ri. Alterna-
tively, we consider multi-factor models of the form
Ri=
m
X
j=1
βij Fj+˜
Ri.(5)
Here there are mfactors, which can be thought of as the returns of “benchmark”
portfolios representing systematic factors. A trading portfolio is said to be
market-neutral if the dollar amounts {Qi}N
i=1 invested in each of the stocks are
such that
βj=
N
X
i=1
βij Qi= 0, j = 1,2, ..., m. (6)
The coefficients βjcorrespond to the portfolio betas, or projections of the port-
folio returns on the different factors. A market-neutral portfolio has vanishing
portfolio betas; it is uncorrelated with the market portfolio or factors that drive
the market returns. It follows that the portfolio returns satisfy
N
X
i=1
QiRi=
N
X
i=1
Qi
m
X
j=1
βij Fj
+
N
X
i=1
Qi˜
Ri
=
m
X
j=1 "N
X
i=1
βij Qi#Fj+
N
X
i=1
Qi˜
Ri
=
N
X
i=1
Qi˜
Ri(7)
5
Thus, a market-neutral portfolio is affected only by idiosyncratic returns. We
shall see below that, in G8 economies, stock returns are explained by approxi-
mately m=15 factors (or between 10 and 20 factors), and that the the system-
atic component of stock returns explains approximately 50% of the variance (see
Plerou et al. (2002) and Laloux et al. (2000)). The question is how to define
“factors”.
2.1 The PCA approach: can you hear the shape of the
market?
A first approach for extracting factors from data is to use Principal Components
Analysis (Jolliffe (2002)). This approach uses historical share-price data on a
cross-section of Nstocks going back, say, Mdays in history. For simplicity
of exposition, the cross-section is assumed to be identical to the investment
universe, although this need not be the case in practice.2Let us represent the
stocks return data, on any given date t0, going back M+ 1 days as a matrix
Rik =Si(t0−(k−1)∆t)−Si(t0−k∆t)
Si(t0−k∆t)
, k = 1, ..., M, i = 1, ..., N ,
where Sit is the price of stock iat time tadjusted for dividends and ∆t= 1/252.
Since some stocks are more volatile than others, it is convenient to work with
standardized returns,
Yik =Rik −Ri
σi
where
Ri=1
M
M
X
k=1
Rik
and
σ2
i=1
M−1
M
X
k=1
(Rik −Ri)2
The empirical correlation matrix of the data is defined by
ρij =1
M−1
M
X
k=1
YikYj k ,(8)
which is symmetric and non-negative definite. Notice that, for any index i, we
have
ρii =1
M−1
M
X
k=1
(Yik)2=1
M−1
M
P
k=1
(Rik −Ri)2
σ2
i
= 1.
2For instance, the analysis can be restricted to the members of the S&P500 index in the
US, the Eurostoxx 350 in Europe, etc.
6
The dimensions of ρare typically 500 by 500, or 1000 by 1000, but the data
is small relative to the number of parameters that need to be estimated. In
fact, if we consider daily returns, we are faced with the problem that very long
estimation windows MNdon’t make sense because they take into account
the distant past which is economically irrelevant. On the other hand, if we just
consider the behavior of the market over the past year, for example, then we are
faced with the fact that there are considerably more entries in the correlation
matrix than data points.
The commonly used solution to extract meaningful information from the
data is Principal Components Analysis.3We consider the eigenvectors and
eigenvalues of the empirical correlation matrix and rank the eigenvalues in de-
creasing order:
N≥λ1> λ2≥λ3≥... ≥λN≥0.
We denote the corresponding eigenvectors by
Figure 1: Eigenvalues of the correlation matrix of market returns computed
on May 1 2007 estimated using a 1-year window (measured as percentage of
explained variance)
v(j)=v(j)
1, ...., v(j)
N, j = 1, ..., N.
3We refer the reader to Laloux et al. (2000), and Plerou et al. (2002) who studied the
correlation matrix of the top 500 stocks in the US in this context.
7
A cursory analysis of the eigenvalues shows that the spectrum contains a few
large eigenvalues which are detached from the rest of the spectrum (see Figure
1). We can also look at the density of states
D(x, y) = {#of eigenvalues between xand y}
N
(see Figure 2). For intervals (x, y) near zero, the function D(x, y) corresponds
Figure 2: Density of States (DOS) for May 1-2007 estimated using a 1 year
window
to the “bulk spectrum” or “noise spectrum” of the correlation matrix. The
eigenvalues at the top of the spectrum which are isolated from the bulk spectrum
are obviously significant. The problem that is immediately evident by looking
at Figures 1 and 2 is that there are less “detached” eigenvalues than industry
sectors. Therefore, we expect that the boundary between “significant” and
“noise” eigenvalues to be somewhat blurred and to correspond to be at the
edge of the “bulk spectrum”. This leads to two possibilities: (a) we take into
account a fixed number of eigenvalues to extract the factors (assuming a number
close to the number of industry sectors) or (b) we take a variable number of
eigenvectors, depending on the estimation date, in such a way that a sum of the
retained eigenvalues exceeds a given percentage of the trace of the correlation
matrix. The latter condition is equivalent to saying that the truncation explains
a given percentage of the total variance of the system.
Let λ1, ..., λm, m < N be the significant eigenvalues in the above sense. For
8
each index j, we consider a the corresponding “eigenportfolio”, which is such
that the respective amounts invested in each of the stocks is defined as
Q(j)
i=v(j)
i
σi
.
The eigenportfolio returns are therefore
Fjk =
N
X
i=1
v(j)
i
σi
Rik j= 1,2, ..., m. (9)
It is easy for the reader to check that the eigenportfolio returns are uncorrelated
in the sense that the empirical correlation of Fjand Fj0vanishes for j6=j0. The
factors in the PCA approach are the eigenportofolio returns.
Figure 3: Comparative evolution of the principal eigenportfolio and the
capitalization-weighted portfolio from May 2006 to April 2007. Both portfo-
lios exhibit similar behavior.
Each stock return in the investment universe can be decomposed into its
projection on the mfactors and a residual, as in equation (4). Thus, the PCA
approach delivers a natural set of risk-factors that can be used to decompose our
returns. It is not difficult to verify that this approach corresponds to modeling
the correlation matrix of stock returns as a sum of a rank-mmatrix correspond-
ing to the significant spectrum and a diagonal matrix of full rank,
ρij =
m
X
k=0
λkv(k)
iv(k)
j+2
iiδij ,
where δij is the Kronecker delta and 2
ii is given by
9
2
ii = 1 −
m
X
k=0
λkv(k)
iv(k)
i
so that ρii = 1. This means that we keep only the significant eigenvalues/eigenvectors
of the correlation matrix and add a diagonal “noise” matrix for the purposes of
conserving the total variance of the system.
2.2 Interpretation of the eigenvectors/eigenportfolios
As pointed out by several authors (see for instance, Laloux et al.(2000)), the
dominant eigenvector is associated with the “market portfolio”, in the sense
that all the coefficients v(1)
i, i = 1,2.., N are positive. Thus, the eigenport-
folio has positive weights Q(1)
i=v(1)
i
σi. We notice that these weights are in-
versely proportional to the stock’s volatility. This weighting is consistent with
the capitalization-weighting, since larger capitalization companies tend to have
smaller volatilities. The two portfolios are not identical but are good proxies
for each other,4as shown in Figure 3.
To interpret the other eigenvectors, we observe that (i) the remaining eigen-
vectors must have components that are negative, in order to be orthogonal to
v(i); (ii) given that there is no natural order in the stock universe, the “shape
analysis” that is used to interpret the PCA of interest-rate curves (Litterman
and Scheinkman (1991) or equity volatility surfaces (Cont and Da Fonseca
(2002)) does not apply here. The analysis that we use here is inspired by Scherer
and Avellaneda (2002), who analyzed the correlation of sovereign bond yields
across different Latin American issuers (see also Plerou et. al.(2002) who made
similar observations). We rank the coefficients of the eigenvectors in decreasing
order:
v(2)
n1≥v(2)
n2≥... ≥v(2)
nN,
the sequence nirepresenting a re-labeling of the companies. In this new order-
ing, we notice that the “neighbors” of a particular company are firms tend to be
in the same industry group. This property, which we call coherence, holds true
for v(2) and for other high-ranking eigenvectors. As we descend in the spectrum
towards the noise eigenvectors, the property that nearby coefficients correspond
to firms in the same industry is less true and coherence will not hold for eigen-
vectors of the noise spectrum (almost by definition!). The eigenportfolios can
the therefore be interpreted as “pairs-trading” or, more generally, long-short
positions, at the level of industries or sectors.
4The positivity of the coefficients of the first eigenvector of the correlation matrix in the
case when all assets have non-negative correlation follows from Krein’s Theorem. In practice,
the presence of commodity stocks and mining companies implies that there are always a few
negatively correlated stock pairs. In particular, this explains why there are a few negative
weights in the principal eigenportfolio in Figure 4.
10
Figure 4: First eigenvector sorted by coefficient size. The x-axis shows the ETF
corresponding to the industry sector of each stock.
Figure 5: Second eigenvector sorted by coefficient size. Labels as in Figure 4.
11
Figure 6: Third eigenvector sorted by coefficient size. Labels as in Figure 4.
Top 10 Stocks Bottom 10 Stocks
Energy, oil and gas Real estate, financials, airlines
Sunoco American Airlines
Quicksilver Res. United Airlines
XTO Energy Marshall & Isley
Unit Corp. First Third Bancorp
Range Resources BBT Corp.
Apache Corp. Continental Airlines
Schlumberger M & T Bank
Denbury Resources Inc. Colgate-Palmolive Company
Marathon Oil Corp. Target Corporation
Cabot Oil & Gas Corporation Alaska Air Group, Inc.
Table 1: The top 10 stocks and bottom 10 stocks in second eigenvector.
12
Top 10 Stocks Bottom 10 Stocks
Utility Semiconductor
Energy Corp. Arkansas Best Corp.
FPL Group, Inc. National Semiconductor Corp.
DTE Energy Company Lam Research Corp.
Pinnacle West Capital Corp. Cymer, Inc.
The Southern Company Intersil Corp.
Consolidated Edison, Inc. KLA-Tencor Corp.
Allegheny Energy, Inc. Fairchild Semiconductor International
Progress Energy, Inc. Broadcom Corp.
PG&E Corporation Cellcom Israel Ltd.
FirstEnergy Corp. Leggett & Platt, Inc.
Table 2: The top 10 stocks and bottom 10 stocks in third eigenvector.
2.3 The ETF approach: using the industries
Another method consists in using the returns of sector ETFs as factors. In
this approach, we select a sufficiently diverse set of ETFs and perform mul-
tiple regression analysis of stock returns on these factors. Unlike the case of
eigenportfolios, ETF returns are not uncorrelated, so there can be redundan-
cies: strongly correlated ETFs may lead to large factor loadings with opposing
signs for stocks that belong to or are strongly correlated to different ETFs. To
remedy this, we can perform a robust version of multiple regression analysis to
obtain the coefficients βij . For example, the matching pursuit algorithm (Davis,
Mallat & Avellaneda (1997)) which favors sparse representations is preferable to
a full multiple regression. Another class of regression methods known as ridge
regression achieves the similar goal of sparse representations (see, for instance
Jolliffe (2002)). Finally, a simple approach, which we use in our back-testing
strategies, consists in associating to each stock its sector ETF, following the
partition of the market in Figure 7, and performing a regression of the stock
returns on its returns.
Let I1, I2, ..., Imrepresent a class of ETFs that span the main sectors in the
economy, and let RIjdenote the corresponding returns. The ETF decomposition
takes the form
Ri=
m
X
j=1
βij RIj+˜
Ri.
The tradeoff between the ETF method and the PCA method is that in the
former we need to have some prior knowledge of the economy to know what
is a “good” set of ETFs to explain returns. The advantage is that the inter-
pretation of the factor loadings is more intuitive than for PCA. Nevertheless,
based on the notion of coherence alluded to in the previous section, it could
be argued that the ETF and PCA methods convey similar information. There
13
is an caveat, however: ETF holdings give more weight to large capitalization
companies, whereas PCA has no a priori capitalization bias. As we shall see,
these nuances are reflected in the performance of statistical arbitrage strategies
based on different risk-factors.
Figure 7 shows a sample of industry sectors number of stocks of companies
with capitalization of more than 1 billion USD at the beginning of January 2007,
classified by sectors. The table gives an idea of the dimensions of the trading
universe and the distribution of stocks corresponding to each industry sector.
We also include, for each industry, the ETF that can be used as a risk-factor
for the stocks in the sector for the simplified model (11).
3 A relative-value model for equity pricing
We propose a quantitative approach to stock pricing based on relative perfor-
mance within industry sectors or PCA factors. In the last section, we present
a modification of the signals which take into account the trading volume in
the stocks as well, within a similar framework. This model is purely based on
price data, although in principle it could be extended to include fundamental
factors, such changes in analysts’ recommendations, earnings momentum, and
other quantifiable factors.
We shall use continuous-time notation and denote stock prices by Si(t), ...., SN(t),
where tis time measured in years from some arbitrary starting date. Based on
the multi-factor models introduced in the previous section, we assume that stock
returns satisfy the system of stochastic differential equations
dSi(t)
Si(t)=αidt +
N
X
j=1
βij
dIj(t)
Ij(t)+dXi(t),(10)
where the term
N
X
j=1
βij
dIj(t)
Ij(t)
represents the systematic component of returns (driven by the returns of the
eigenportfolios or ETFs). To fix ideas, we place ourselves in the ETF framework.
In this context, Ij(t) represents the mid-market price of the jth ETF used to
span the market. The coefficients βij are the corresponding loadings.
In practice, only ETFs that are in the same industry as the stock in question
will have significant loadings, so we could also work with the simplified model
βij =Cov(Ri, RIj)
V ar(RIj)if stock #iis in industry #j
= 0 otherwise (11)
where each stock is regressed to a single ETF representing its “peers”.
14
Figure 7: Trading universe on January 1, 2007: breakdown by sectors.
15
The idiosyncratic component of the return is given by
αidt +dXi(t).
Here, the αirepresents the drift of the idiosyncratic component, i.e. αidt is
the excess rate of return of the stock in relation to market or industry sector
over the relevant period. The term dXi(t) is assumed to be the increment of a
stationary stochastic process which models price fluctuations corresponding to
over-reactions or other idiosyncratic fluctuations in the stock price which are
not reflected the industry sector.
Our model assumes (i) a drift which measures systematic deviations from the
sector and (ii) a price fluctuation that is mean-reverting to the overall industry
level. Although this is very simplistic, the model can be tested on cross-sectional
data. Using statistical testing, we can accept or reject the model for each stock
in a given list and then construct a trading strategy for those stocks that appear
to follow the model and yet for which significant deviations from equilibrium
are observed.
Based on these considerations, we introduce a parametric model for Xi(t)
which can be estimated easily, namely, the Ornstein-Uhlembeck process:
dXi(t) = κi(mi−Xi(t)) dt +σidWi(t), κi>0.(12)
This process is stationary and auto-regressive with lag 1 (AR-1 model). In
particular, the increment dXi(t) has unconditional mean zero and conditional
mean equal to
E{dXi(t)|Xi(s), s ≤t}=κi(mi−Xi(t)) dt .
The conditional mean, or forecast of expected daily returns, is positive or neg-
ative according to the sign of mi−Xi(t).
The parameters of the stochastic differential equation, αi, κi, miand σi,are
specific to each stock. They are assumed to vary slowly in relation to the Brown-
ian motion increments dWi(t), in the time-window of interest. We estimate the
statistics for the residual process on a window of length 60 days, assuming that
the parameters are constant over the window. This hypothesis is tested for
each stock in the universe, by goodness-of-fit of the model and, in particular,
by analyzing the speed of mean-reversion.
If we assume momentarily that the parameters of the model are constant,
we can write
Xi(t0+ ∆t) = e−κi∆tXi(t0) + 1−e−κi∆tmi+σi
t0+∆t
Z
t0
e−κi(t0+∆t−s)dWi(s).
(13)
Letting ∆ttend to infinity, we see that equilibrium probability distribution for
the process Xi(t) is normal with
16
E{Xi(t)}=miand V ar {Xi(t)}=σ2
i
2κi
.(14)
According to Equation (10), investment in a market-neutral long-short portfolio
in which the agent is long $1 in the stock and short βij dollars in the jth ETF
has an expected 1-day return
αidt +κi(mi−Xi(t)) dt .
The second term corresponds to the model’s prediction for the return based on
the position of the stationary process Xi(t): it forecasts a negative return if
Xi(t) is sufficiently high and a positive return if Xi(t) is sufficiently low.
The parameter κiis called the speed of mean-reversion and
τi=1
κi
represents the characteristic time-scale for mean reversion. If κ1 the stock
reverts quickly to its mean and the effect of the drift is negligible. In our
strategies, and to be consistent with the estimation procedure that uses constant
parameters, we are interested in stocks with fast mean-reversion, i.e. such that
τiT1.
Figure 8: Empirical distribution of the characteristic time to mean-reversion τi
(in business days) for the year 2007, for the stock universe under consideration.
The descriptive statistics are given below.
17
Figure 9: Statistical averages for the estimated OU parameters corresponding
to all stocks over 2007.
18
Days
Maximum 30
75% 11
Median 7.5
25% 4.9
Minimum 0.5
Fast days 36 %
Table 3: Descriptive statistics on the mean-reversion time τ.
4 Signal generation
Based on this simple model, we defined several trading signals. We considered
an estimation window of 60 business days i.e. T1= 60/252. This estimation
window incorporates at least one earnings cycle for the company. Therefore, we
expect that it reflects so some extent fluctuations in the price which take place
along the cycle. We selected stocks with mean-reversion times less than 1/2
period (κ > 252/30 = 8.4). Typical descriptive statistics for signal estimation
are presented in Figure 9.
4.1 Pure mean-reversion
We focus only on the process Xi(t), neglecting the drift αi. We know that the
equilibrium variance is
σeq,i =σi
√2κi
=σirτi
2
Accordingly, we define the dimensionless variable
si=Xi(t)−mi
σeq,i
.(15)
We call this variable the s-score. See Figure 11 for a graph showing the evolution
of the s-score for residuals of JPM against the Financial SPDR, XLF. The s-
score measures the distance to equilibrium of the cointegrated residual in units
standard deviations, i.e. how far away a given stock is from the theoretical
equilibrium value associated with our model.
Our basic trading signal based on mean-reversion is
buy to open if si<−sbo
sell to open if si>+sso
close short position if si<+sbc
close long position si>−ssc
(16)
19
where the cutoff values are determined empirically. Entering a trade, e.g. buy to
open, means buying one dollar of the corresponding stock and selling βidollars
of its sector ETF or, in the case of using multiple factors, βi1dollars of ETF
#1, βi2dollars of ETF #2, ..., βim dollars of ETF #m. Similarly, closing a long
position means selling stock and buying ETFs.
Since we expressed all quantities in dimensionless variables, we expect the
cutoffs sbo, sbo , sbc, ssc to be valid across the different stocks. We selected the
cutoffs empirically, based on simulating strategies from 2000 to 2004 in the case
of ETF factors. Based on this analysis, we found that a good choice of cutoffs
is
sbo =sso = 1.25
sbc= 0.75 and ssc = 0.50
Thus, we enter trades when the s-score exceeds 1.25 in absolute value. Closing
short trades sooner (at 0.75) gives slightly better results than 0.50. For closing
long trades, we choose 0.50. (see Figure 10)
Figure 10: Schematic evolution of the s-score and the associated signal, or
trading rule.
The rationale for opening trades only when the s-score siis far from equilib-
rium is to trade only when we think that we detected an anomalous excursion
of the co-integration residual. We then need to consider when we close trades.
Closing trades when the s-score is near zero also makes sense, since we expect
most stocks to be near equilibrium most of the time. Thus, our trading rule
detects stocks with large “excursions” and trades assuming these excursions will
revert to the mean in a period of the order of the mean-reversion time τi.
20
Figure 11: Evolution of the s-score of JPM ( vs. XLF ) from January 2006 to
December 2007.
4.2 Mean-reversion with drift
In the previous signal, the presence of the drift was ignored (implicity it was
assumed that the effect of the drift was irrelevant in comparison with mean-
reversion). We incorporate the drift by considering the conditional expectation
of the residual return over a period of time dt, namely,
αidt +κi(mi−Xi(t)) dt =κiαi
κi
+mi−Xi(t)dt
=κiαi
κi−σeq,i sidt.
This suggests that the dimensionless decision variable is the “modified s-score”
(see Figure 12 )
smod,i =si−αi
κiσeq,i
=si−αiτi
σeq,i
.(17)
To make contact with the analysis of the pure mean-reversion strategy, con-
sider for example the case of shorting stock. In the previous framework, we
short stock if the s-score is large enough. The modified s-score is larger if αiis
negative, and smaller if αiis positive. Therefore, it will be harder to generate
a short signal if we think that the residual has an upward drift and easier to
21
Figure 12: Including the drift in signal generation
short if we think that the residual has a downward drift. If the s-score is zero,
the signal reduces to buying when the drift is high enough and selling when the
drift is low. Since the drift can be interpreted as the slope of a 60-day mov-
ing average, we have therefore a “built-in” momentum strategy in this second
signal. A calibration exercise using the training period 2000-2004 showed that
the cutoffs defined in the previous strategy are also acceptable for this one. We
notice, however, that the drift parameter has values of the order of 15 basis
points and the average expected reversion time is 7 days, whereas the equilib-
rium volatility of residuals is on the order of 300 bps. The expected average
shift for the modified s-score is of the order of 0.15 ×7/300 ≈0.3. In practice,
the effect of incorporating a drift in these time-scales is minor.5
5 Back-testing results
The back-testing experiments consisted in running the signals through historical
data, with the estimation of parameters (betas, residuals), signal evaluations and
portfolio re-balancing performed daily. We assumed that all trades are done at
the closing price of that day. As mentioned previously, we assumed a round-trip
transaction cost per trade of 10 basis points, to incorporate an estimate of price
slippage and other costs as a single friction coefficient.
5Back-testing shows that this is indeed the case.We shall not present back-testing results
with the modified s-scores for the sake of brevity.
22
Let Etrepresent the portfolio equity at time t. The basic PNL equation for
the strategy has the following form:
Et+∆t=Et+Etr∆t+
N
X
i=1
Qit Rit + N
X
i=1
Qit!r∆t
+
N
X
i=1
Qit Dit/Sit −
N
X
i=1 |Qi(t+∆t)−Qit| ,
Qit =EtΛt,
where Qit is the dollar investment in stock iat time t,Rit is the stock return from
corresponding to the period (t, t + ∆t), rrepresents the interest rate (assuming,
for simplicity, no spread between long and short rates), ∆t= 1/252, Dit is the
dividend payable to holders of stock iover the period (t, t + ∆t)(when t=ex-
dividend date), Sit is the price of stock iat time t, and = 0.0005 is the
slippage term alluded to above. The last line in the equation states that the
money invested in stock iis proportional to the total equity in the portfolio. The
proportionality factor, Λt, is stock-independent and chosen so that the portfolio
has a desired level of leverage on average. For example, if we have 100 stocks
long and 100 short and we wish to have a ”2+2” leverage, then Λt= 2/100.
In practice this number is adjusted only for new positions, so as not to incur
transaction costs for stock which are already held in the portfolio. 6In other
words, Λtcontrols the maximum fraction of the equity that can be invested
in any stock, and we take this bound to be equal for all stocks. In practice,
especially when dealing with ETFs as risk-factors, we modulated the leverage
coefficient on a sector-by-sector basis.7
Given the discrete nature of the signals, the investment strategy that we
propose is “bang-bang”: there is no continuous trading. Instead, the full amount
is invested on the stock once the signal is active (buy-to-open, short-to-open)
and the position is unwound when the s-score indicates a closing signal. This
all-or-nothing strategy, which might seem inefficient at first glance, turns out to
outperform making continuous portfolio adjustments.
5.1 Synthetic ETFs as factors
The first set of experiments were done using 15 synthetic capitalization-weighted
industry-sector indices as risk-factors (see Figure 7). The reason for using syn-
thetic ETFs was to be able to back-test strategies going back to 1996, when
most ETFs did not exist. A series of daily returns for a synthetic index is calcu-
lated for each sector and recorded for the 60 days preceding the estimation date.
We performed a regression of stock returns on the corresponding sector index
6Hence, strictly speaking, the leverage factor is weakly dependent on the available signals.
7Other refinements that can be made have to do with using different leverage according to
the company’s market capitalization or choosing a sector-dependent leverage that is inversely
proportional to the average volatility of the sector.
23
and extracted the corresponding residual series. To ensure market-neutrality,
we added to the portfolio an S&P 500 index futures hedge which was adjusted
daily and kept the overall portfolio beta-neutral.
Since we expect that, in aggregate, stocks are correctly priced, we experi-
mented with adjusting the means of the OU processes so that the total mean
would be zero. In other words, we introduced the adjusted means for the resid-
uals
mi=mi−1
N
N
X
j=1
mj, i = 1,2, ..., N . (18)
This modification has the effect of removing “model bias” and is consistent with
market-neutrality. We obtained consistently better results in back-testing than
when using the estimated miand adopted it for all other strategies as well. The
results of back-testing with synthetic ETFs are shown in Figure 13 and Table
14.
Figure 13: Historical PNL for the strategy using synthetic ETFs as factors from
1996-2007
24
Figure 14: Sharpe ratios for the strategy using synthetic ETFs as factors : 1996-
2007. The Sharpe Ratio is defined as µ−r/σ, where µ, r, σ are the annualized
return, interest rate and standard deviation of the PNL.
25
5.2 Actual ETFs
Back-testing with actual ETFs was possible only from 2002 onward, due to the
fact that many ETFs did not exist previously. We simulated strategies going
back to 2002, using regression on a single ETF to generate residuals. The results
are displayed on Figure 15 and Table 16.
Figure 15: Historical PNL for the strategy using actual ETFs as factors, com-
pared with the one using synthetic ETFs : 2002-2007. Notice the strong out-
performance by the strategy which uses actual ETFs.
We observe that using actual ETFs improved performance considerably. An
argument that might explain this improvement is that ETFs are traded, whereas
the synthetic ETFs are not, therefore providing better price information.
5.3 PCA with 15 eigenportfolios
The back-testing results for signals generated with 15 PCA factors are shown
in Figures 17 and 18 and Table 19. We observe that the 15-PCA strategy
out-performs the actual ETF strategy after 2002.
26
Figure 16: Sharpe ratios for actual 15 ETFs as factors : 2002-2007. We observe,
for the purpose of comparison, that the average Sharpe ratio from 2003 to 2007
was 0.6.
27
Figure 17: PNL corresponding 15 PCA factors, compared with synthetic ETFs
from 1997-2007
5.4 Using a variable number of PCA factors
We also tested strategies based on a variable number of factors, chosen so as
to explain a given level of variance. In this approach, we retain a number of
eigen-portfolios (factors) such that the sum of the corresponding eigenvectors is
equal to a set percentage. The number of eigenvalues (or eigenvectors) which
are needed to explain 55% of the total variance of the correlation matrix varies
across time. This variability is displayed in Figure 20 and Figure 21. We also
looked at other cutoffs and report similar results in Figure 24. The periods over
which the number of eigenvectors needed to explain a given level of variance
is small, appear to be those when the risk-premium for equities is relatively
high. For instance, the latter parts of 2002 and 2007, which correspond respec-
tively the aftermath of the Internet bubble and the bursting of the subprime
bubble, are periods for which the variance is concentrated on a few top eigen-
vectors/eigenvalues. In contrast, 2004-2006 is a period where the variance is
distributed across a much larger set of modes.
Back-testing the strategy with 55% explained variance shows that it is com-
parable but slighly inferior to taking a fixed number of eigenvectors (see Figure
22 and Table 23). In the same vein, we studied the performance of other strate-
gies with a variable number of PCA eigenportfolios explaining different levels of
variance. In Table 27 and Figure 25, we display the performances of strategies
using 45%, 55% and 65% compared with the PCA strategies with 1 eigen port-
28
Figure 18: Comparison of strategies with 15 PCA factors and the using ac-
tual ETFs in the period 2002-2007. 15-PCA outperforms significantly the ETF
strategy.
29
Figure 19: Sharpe ratios for 15 PCA factors : 2002-2007
30
Figure 20: Number of significant eigenvectors needed to explain the variance
of the correlation matrix at the 55% level, from 2002 to February 2008. The
estimation window for the correlation matrix is 252 days. The boundary of the
shaded region represents the VIX CBOT Volatility Index (measured in percent-
age points).
Figure 21: Percentage of variance explained by the top 15 eigenvectors: 2002-
February 2008. Notice the increase in the Summer of 2007.
31
Figure 22: Comparison of the PNLs for the fixed explained variance (55%) of
PCA and the 15 PCA strategy: 2002-2007. The performance of the 15 PCA
strategy is slightly superior.
32
Figure 23: Sharpe ratios for the fixed explained variance (55%) of PCA : 2002-
2007
33
folio and with 15 eigenportfolios. The conclusion is that 55% PCA is the best
performing among the three strategies and is comparable, but slightly inferior,
to the 15 PCA strategy. We also observed that taking a high cutoff such as 75%
of explained variance leads to steady losses, probably due to the fact that trans-
action costs dominate the small residual noise that remains in the system after
‘defactoring’ (see Figure 26). Similarly, on the opposite side of the spectrum,
using just one eigen-portfolio, as in the Capital Asset Pricing Model, gives rise
low levels of mean-reversion, higher residual volatility and poor Sharpe ratios.
(See Figure (25) ).
Figure 24: Time-evolution of number of PCA factors for different levels of ex-
plained variance: 2002-2007
6 Taking trading volume into account
In this section, we add volume information to the mean-reversion signals. Let
Vtrepresent the cumulative share volume transacted until time tstarting from
an arbitrary reference time t0(say, the date at which the stock was first issued).
This is an increasing function which can be viewed as a sum of daily trading
volumes and approximated as an integral:
Vt=XδVk≈
t
Z
t0
˙
Vsds.
Historical prices can be viewed on a uniform “time grid” or on a uniform “volume
34
Figure 25: PNL for different variance truncation level:2002-2007
Figure 26: Truncation at 75 % of explained variance: 2007- Apr 2008
35
Figure 27: Sharpe ratios for variable PCA strategies: 2002- 2007
grid” (i.e. the price evolution each time one share is traded). If we denote the
latter prices by PV, we have
St+∆t−St=PV(t+∆t)−PV(t)
=PV(t+∆t)−PV(t)
V(t+ ∆t)−V(t)(V(t+ ∆t)−V(t)) .(19)
Thus, the average price change per unit share over the period of interest is
PV(t+∆t)−PV(t)
V(t+ ∆t)−V(t)=St+∆t−St
V(t+ ∆t)−V(t).
This suggests that, instead of the classical daily stock returns, we use the mod-
ified returns
Rt=St+∆t−St
St
hδV i
V(t+ ∆t)−V(t)=RthδV i
V(t+ ∆t)−V(t)(20)
where hδV iindicates the average, or typical, daily trading volume calculated
over a given trailing window. Measuring mean-reversion in trading time is
equivalent to using calendar time and ’weighting’ the stock returns as in (20).
The modified returns Rtare equal to the classical returns if the daily trading
volume is typical. Notice that if the trading volume is low, the the factor on
the right-hand side of the last equation is larger than unity and Rt> Rt.
Similarly, if volume is high then Rt< Rt. The concrete effect of the trading-
time modification is that mean-reversion strategies are sensitive to how much
trading was done immediately before the signal was triggered. If the stock rallies
on high volume, an open-to-short signal using classical returns may be triggered.
However, if the volume is sufficiently large, then the modified return is much
smaller so the residual will not necessarily indicate a shorting signal. Similarly,
36
buying stocks that drop on high volume is discouraged by the trading-time
approach.
We back-tested the previous strategies using the trading time approach and
found that this technique increases the PNL and the Sharpe ratios unequivocally
for stategies with ETF-generated signals (see Figure 28 and Table 29). For PCA-
based strategies, we found that the trading time framework does not lead to a
significant improvement. Finally, we find that the ETF strategy using trading
time is comparable in performance to the 15-PCA/55% PCA strategies (with or
without trading time adjustments) (see Figure 30 and Table 31 and also Figure
32).
Figure 28: Comparision of signals in trading time vs. actual time using actual
ETFs as factors : 2002-2007
7 A closer look at 2007
It has been widely reported in the media that 2007 was very challenging for
quantitative hedge funds; see Khandani and Lo (2007), Barr (2007), Associated
Press (2007), Rusli (2007). This was particularly true of statistical arbitrage
strategies, who experienced a large drawdown and subsequent partial recovery
in the second week of August 2007. Unfortunately for many managers, the size
of the drawdown was such that many had to de-leverage their portfolios and did
not recover to pre-August levels. Our backtesting results are consistent with
the real-world events of 2007 and show a strong drawdown in August 2007 (see
37
Figure 29: Sharpe ratios for signals in trading time using actual ETFs as factors
: 2002-2007
Figure 30: Comparision of signals in trading time vs. actual time using 15 PCAs
as factors : 2002-2007
38
Figure 31: Sharpe ratios for signals in trading time using 15 PCAs as factors :
2002-2007
39
Figure 32: Comparison of ETF and PCA strategies using “trading time”.
below). This drawdown was first reproduced in back-testing by Khandani and
Lo(2007) using contrarian strategies.
We analyzed the performance for our stategies in 2007 using ETFs with and
without trading time adjustment as well as the 15-PCA strategy (see Figure
33). First, we found that performance was flat or slightly negative in the first
part of the year. In early August, we found that mean-reversion strategies ex-
perienced a large, sudden drawdown followed by a recovery in about 10 days. In
certain cases, our strategies tracked almost identically the Khandani-Lo (2007)
simulation after adjusting for leverage (KL used 4+4 leverage and we used 2+2
in this paper). The PCA-based strategies showed more resilience during the
liquidity event, with a drawdown of 5% as opposed to 10% for the ETF-based
strategies (see Figure 33).
Khandani and Lo suggest that the events of 2007 could be due to a liquid-
ity shock caused by funds unwinding their positions. As we have seen, these
strategies result in levered portfolios with hundreds of long and short positions
in stocks. While each position is small and has probably small impact, the
aggregate effect of exiting simultaneously hundreds of positions may have pro-
duced the spike shown in Figure 34. A closer look at the PL for different sectors
shows, for example, that the Technology and Consumer Discretionary sectors
were strongly affected by the shock – and more so than Financials and Real
Estate; see Figure 36. This apparently paradoxical result – whereby sectors
40
that are uncorrelated with Financials experience large volatility – is consistent
with the unwinding theory of Khandani and Lo. A further breakdown of the
performance of the different sectors in August 2007 is given in Figure 35.
Figure 33: Zoom on 2007 of strategies of ETF factor with trading time, ETF
factor with actual time and 15 PCA.
8 Conclusions
We presented a systematic approach to statistical arbitrage and for constructing
market-neutral portfolio strategies based on mean-reversion. The approach is
based on decomposing stock returns into systematic and idyosincrating compo-
nents. This is done using different definitions of risk-factors: ETFs as proxies
for industry factors or a PCA-based approach where we extract factors, or eigen-
portfolios from the eigenvectors of the empirical correlation matrix of returns.
It is interesting to compare the ETF and PCA methods. In the ETF method,
we essentially use 15 ETFs to representing the ’market’ fluctuations. It is not
difficult to verify that, on average, the systematic component of returns in eq-
uity markets, explains between 40% and 60% of the variance of stock returns.
This suggests, on the PCA side, that the number of factors needed to explain
stock returns should be equal to the number of eigenvalues needed to explain
approximately 50% of the variance of the empirical correlation matrix. In prac-
tice, we found that this number to vary across time, somehwere between 10 and
30. More precisely, we find that the number varies inversely to the value of the
41
Figure 34: Comparison with Khandani & Lo:August 2007, 2+2 leverage on both
strategies
Figure 35: Sector view in Aug 2007
42
Figure 36: Technology & Consumer vs. Financials & Real Estate : Aug 2007
VIX Option volatilty index, suggesting more factors are needed to explain stock
returns when volatility is low, and less in times of crisis, or large cross-sectional
volatility.
On the performance side, we found that the best results across the entire
period were obtained using 15 ETFs or the 15-PCA strategy, or a variable
number of PCA factors explaining approximately 55% of the total variance.
Trading-time estimation of signals, which is equivalent to weighting returns
inversely to the traded volume, seems to benefit particularly the ETF strategy
and make it competitive with PCA.
We also note that the performance of mean-reversion strategies appear to
benefit from market conditions in which the number of explanatory factors is
relatively small. That is, mean-reversion statistical arbitrage works better when
we can explain 50% of the variance with a relatively small number of eigenval-
ues/eigenvectors. The reason for this is that if the “true” number of factors
is very large (>25) then using 15 factors will not be enough to ‘defactor the
returns’, so residuals ‘contain’ market information that the model is not able to
detect. If, on the other hand, we use a large number of factors, the correspond-
ing residuals have small variance, and thus the opportunity of making money,
especially in the presence of transaction costs, is diminished.
Finally, we have reproduced the results of Khandani and Lo (2007) and
thus place our strategies in the same broad universality class as the contrarian
strategies of their paper. Interestingly enough, an analysis of PNL at the sector
level shows that the spike of August 2007 was more pronounced in sectors such
as Technology and Consumer Discrectionary than in Financials and Real Estate,
lending plausibility to the “unwinding theory” of Khandani and Lo.
43
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