Statistical Arbitrage in the U.S. Equities Market Marco Avellaneda∗†and Jeong-Hyun Lee∗ July 11, 2008 Abstract We study model-driven statistical arbitrage strategies in U.S. equities. Trading signals are generated in two ways: using Principal Component AnalysisandusingsectorETFs. Inbothcases,weconsidertheresiduals, oridiosyncraticcomponentsofstockreturns,andmodelthemasamean- reverting process, which leads naturally to “contrarian” trading signals. Themaincontributionofthepaperistheback-testingandcomparison ofmarket-neutralPCA-andETF-basedstrategiesoverthebroaduniverse ofU.S.equities. Back-testingshowsthat,afteraccountingfortransaction costs, PCA-based strategies have an average annual Sharpe ratio of 1.44 overtheperiod1997to2007,withamuchstrongerperformancespriorto 2003: during2003-2007,theaverageSharperatioofPCA-basedstrategies was only 0.9. On the other hand, strategies based on ETFs achieved a Sharperatioof1.1from1997to2007,butexperienceasimilardegradation 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 Thetermstatisticalarbitrageencompassesavarietyofstrategiesandinvestment 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 †FinanceConceptsSARL,49-51AvenueVictor-Hugo,75116Paris,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 mechanismforgeneratingexcessreturnsisstatistical. Theideaistomakemany 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-tradingiswidelyassumedtobethe“ancestor”ofstatisticalarbitrage. 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 P and Q denote t t the corresponding price time series, then we can model the system as ln(P /P )=α(t−t )+βln(Q /Q ) + X (1) t t0 0 t t0 t or, in its differential version, dP dQ t = αdt + β t +dX , (2) P Q t t t where X is a stationary, or mean-reverting, process. This process will be re- t ferred to as the cointegration residual, or residual, for short, in the rest of the paper. In many cases of interest, the drift α is small compared to the fluctua- tionsofX andcanthereforebeneglected. Thismeansthat,aftercontrollingfor t beta, the long-short portfolio oscillates near some statistical equilibrium. The modelsuggestsacontrarianinvestmentstrategyinwhichwegolong1dollarof stock P and short β dollars of stock Q if X is small and, conversely, go short P t andlongQifX islarge. Theportfolioisexpectedtoproduceapositivereturn t asvaluationsconverge(seePole(2007)foracomprehensivereviewonstatistical 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. “Generalizedpairs-trading”,ortradinggroupsofstocksagainstothergroups 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 Q would be played by BBH and P would an arbitrary stock in the biotechnology sector. The analysis of the residuals, based of the magnitude of X , suggests typically that some stocks are cheap with respect to the sector, t 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 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 returnsintosystematicandidiosyncraticcomponentsandstatisticallymodeling the idiosyncratic part. The general decomposition may look like n dPt = αdt + Xβ F(j) + dX , (3) P j t t t j=1 wherethetermsF(j),j =1,...,nrepresentreturnsofrisk-factorsassociatedwith t 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 that remains after the decomposition is done. The maincontributionofourpaperwillbetostudyhowdifferentsetsofrisk-factors leadtodifferentresiduals andhence todifferentprofit-loss (PNL)forstatistical arbitrage strategies. Previousstudiesonmean-reversionandcontrarianstrategiesincludeLehmann (1990),LoandMacKinlay(1990)andPoterbaandSummers(1988). Inarecent 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- encestherein). Thelatterstrategieshaveseveralcommonfeatureswiththeones 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. todetectover-andunder-performers. Ourtradingfrequencyisvariablewhereas Khandani-Lotradeatfixedtime-intervals. Ontheparametricside,Poterbaand 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 lattermostlyinthatweimmunizestocksagainstmarketfactors,i.e. weconsider 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- factorsusingPrincipalComponentAnalysis(Jolliffe(2002)). Thesecondmethod 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 alsoobservethatthevarianceexplainedbyafixednumberofPCAeigenvectors varies significantly across time, leading us to conjecture that the number of ex- planatoryfactorsneededtodescribestockreturnsisvariableandthatthisvari- 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 statisticalestimationoftheprocessX foreachstockatthecloseofeachtrading t day, using historical data prior to the close. Estimation is always done looking back at the historical record, thus simulating decisions which would take place in real automatic trading. Using daily end-of-day (EOD) data, we perform a fullcalculationofdailytradingsignals, goingbackto1996insomecasesandto 2002 in others, across the broad universe of stocks with market-capitalization of more than 1 billion USD at the trade date.2 The estimation and trading rules are kept simple to avoid data-mining. For each stock in the universe, the parameter estimation is done using a 60-day trailing estimation window, which corresponds roughly to one 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. InSection5weback-testdifferentstrategieswhichusedifferentsetsoffactors togenerateresiduals, namely: syntheticETFsbasedoncapitalization-weighted indices, actual ETFs, a fixed number of factors generated by PCA, a variable number of factors generated by PCA. Due to the mechanism described aboive used to generate trading systems, the simulation is out-of-sample, in the sense that the estimation of the residual process at time t uses information available only before this time. 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 volumeforthepastday. Thismodificationaccentuates(i.e. tendstofavor)con- trarian price signals taking place on low volume and mitigates (i.e. tends not tofavor)contrarianpricesignalswhichtakeplaceonhighvolume. Itisasifwe “believemore”aprintthatoccursonhighvolumeandlessreadytobetagainst 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. 1See Scherer and Avellaneda (2002) for similar observations for Latin American debt se- curitiesinthe1990’s. 2The condition that the company must have a given capitalization at the trade date (as opposedtoatthetimethispaperwaswritten),avoidssurvivorshipbias. 4 In Section 7, we discuss the performance of statistical arbitrage in 2007, andparticularlyaroundtheinceptionoftheliquiditycrisisofAugust2007. We comparetheperformancesofthemean-reversionstrategieswiththeonesstudied in the recent work of Khandani and Lo (2007). Conclusions are presented in Section 8. 2 A quantitative view of risk-factors and market- neutrality Wedividetheworldschematicallyinto“indexers’and“market-neutralagents”. Indexersseekexposuretotheentiremarketortospecificindustrysectors. Their goal is generally to be long the market or sector with appropriate weightings in eachstock. Market-neutralagentsseekreturnswhichareuncorrelatedwiththe market. Let us denote by {R }N the returns of the different stocks in the trading i i=1 universeoveranarbitraryone-dayperiod(fromclosetoclose). LetF represent 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, R = β F + R˜ , (4) i i i which is a simple regression model decomposing stock returns into a systematic component β F and an (uncorrelated) idiosyncratic component R˜ . Alterna- i i tively, we consider multi-factor models of the form m R = Xβ F + R˜ . (5) i ij j i j=1 Heretherearemfactors,whichcanbethoughtofasthereturnsof“benchmark” portfolios representing systematic factors. A trading portfolio is said to be market-neutral if the dollar amounts {Q }N invested in each of the stocks are i i=1 such that N X β = β Q =0, j =1,2,...,m. (6) j ij i i=1 Thecoefficientsβ correspondtotheportfoliobetas, orprojectionsoftheport- j folio returns on the different factors. A market-neutral portfolio has vanishing portfoliobetas; itisuncorrelatedwiththemarketportfolioorfactorsthatdrive the market returns. It follows that the portfolio returns satisfy 5   N N m N XQiRi = XQiXβijFj+XQiR˜i i=1 i=1 j=1 i=1 m " N # N = X Xβ Q F +XQ R˜ ij i j i i j=1 i=1 i=1 N = XQ R˜ (7) i i i=1 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- aticcomponentofstockreturnsexplainsapproximately50%ofthevariance(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? AfirstapproachforextractingfactorsfromdataistousePrincipalComponents Analysis (Jolliffe (2002)). This approach uses historical share-price data on a cross-section of N stocks going back, say, M days 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.3 Let us represent the stocks return data, on any given date t , going back M +1 days as a matrix 0 S −S R = i(t0−(k−1)∆t) i(t0−k∆t), k =1,...,M, i=1,...,N, ik S i(t0−k∆t) whereS isthepriceofstockiattimetadjustedfordividendsand∆t=1/252. it Since some stocks are more volatile than others, it is convenient to work with standardized returns, R −R Y = ik i ik σ i where M 1 X R = R i M ik k=1 and M 1 X σ2 = (R −R )2 i M −1 ik i k=1 3For instance, the analysis can be restricted to the members of the S&P500 index in the US,theEurostoxx350inEurope,etc. 6 The empirical correlation matrix of the data is defined by M 1 X ρ = Y Y , (8) ij M −1 ik jk k=1 which is symmetric and non-negative definite. Notice that, for any index i, we have M P(R −R )2 M ik i ρ = 1 X(Y )2 = 1 k=1 =1. ii M −1 ik M −1 σ2 k=1 i The dimensions of ρ are 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 M (cid:29) N don’t make sense because they take into account the distant past which is economically irrelevant. On the other hand, if we just considerthebehaviorofthemarketoverthepastyear,forexample,thenweare 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.4 We consider the eigenvectors and eigenvalues of the empirical correlation matrix and rank the eigenvalues in de- creasing order: N ≥λ > λ ≥ λ ≥...≥ λ ≥0. 1 2 3 N We denote the corresponding eigenvectors by (cid:16) (cid:17) v(j) = v(j),....,v(j) , j =1,...,N. 1 N 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 {#of eigenvalues between x and y} D(x,y) = N (see Figure 2). For intervals (x,y) near zero, the function D(x,y) corresponds to the “bulk spectrum” or “noise spectrum” of the correlation matrix. The eigenvaluesatthetopofthespectrumwhichareisolatedfromthebulkspectrum 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 4We refer the reader to Laloux et al. (2000), and Plerou et al. (2002) who studied the correlationmatrixofthetop500stocksintheUSinthiscontext. 7 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) Figure 2: The density of states for May 1-2007 estimated using a year window 8 edge of the “bulk spectrum”. This leads to two possibilities: (a) we take into accountafixednumberofeigenvaluestoextractthefactors(assuminganumber close to the number of industry sectors) or (b) we take a variable number of eigenvectors,dependingontheestimationdate,insuchawaythatasumofthe retained eigenvalues exceeds a given percentage of the trace of the correlation matrix. Thelatterconditionisequivalenttosayingthatthetruncationexplains a given percentage of the total variance of the system. Let λ ,...,λ , m<N be the significant eigenvalues in the above sense. For 1 m 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 v(j) Q(j) = i . i σ i The eigenportfolio returns are therefore XN v(j) F = i R j =1,2,...,m. (9) jk σ ik i i=1 Itiseasyforthereadertocheckthattheeigenportfolioreturnsareuncorrelated inthesensethattheempiricalcorrelationofF andF vanishesforj 6=j0. The j j0 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 m factors and a residual, as in equation (4). Thus, the PCA 9 approachdeliversanaturalsetofrisk-factorsthatcanbeusedtodecomposeour returns. It is not difficult to verify that this approach corresponds to modeling thecorrelationmatrixofstockreturnsasasumofarank-mmatrixcorrespond- ing to the significant spectrum and a diagonal matrix of full rank, m ρ = Xλ v(k)v(k)+(cid:15)2δ , ij k i j ii ij k=0 where δ is the Kronecker delta and (cid:15)2 is given by ij ii m (cid:15)2 = 1−Xλ v(k)v(k) ii k i i k=0 sothatρ =1. Thismeansthatwekeeponlythesignificanteigenvalues/eigenvectors ii 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 = 1,2..,N are positive. Thus, the eigenport- i folio has positive weights Q(1) = vi(1). We notice that these weights are in- i σi 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,5 as shown in Figure 3. Tointerprettheothereigenvectors, weobservethat(i)theremainingeigen- 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))doesnotapplyhere. TheanalysisthatweusehereisinspiredbyScherer 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) ≥ v(2) ≥ ... ≥ v(2), n1 n2 nN the sequence n representing a re-labeling of the companies. In this new order- i ing, we notice that the “neighbors” of a particular company tend to be in the 5The positivity of the coefficients of the first eigenvector of the correlation matrix in the casewhenallassetshavenon-negativecorrelationfollowsfromKrein’sTheorem. Inpractice, thepresenceofcommoditystocksandminingcompaniesimpliesthattherearealwaysafew negatively correlated stock pairs. In particular, this explains why there are a few negative weightsintheprincipaleigenportfolioinFigure4. 10