Statistical Arbitrage in the U.S. Equities Market Marco Avellaneda∗†and Jeong-Hyun Lee∗ First draft: July 11, 2008 This version: June 15, 2009 Abstract Westudymodel-drivenstatisticalarbitrageinU.S.equities. Thetrading signals are generated in two ways: using Principal Component Analysis and using sector ETFs. In both cases, we consider the residuals, or idio- syncraticcomponentsofstockreturns,andmodelthemasmean-reverting processes. This leads naturally to “contrarian” trading signals. The main contribution of the paper is the construction, back-testing and comparison of market-neutral PCA- and ETF- based strategies ap- plied to the broad universe of U.S. stocks. Back-testing shows that, af- ter accounting for transaction costs, PCA-based strategies have an av- erage annual Sharpe ratio of 1.44 over the period 1997 to 2007, with much stronger performances prior to 2003. During 2003-2007, the aver- age Sharpe ratio of PCA-based strategies was only 0.9. Strategies based on ETFs achieved a Sharpe ratio of 1.1 from 1997 to 2007, experiencing a similar degradation after 2002. Wealsointroduceamethodtoaccountfordailytradingvolumeinfor- mationinthesignals(whichisakintousing“tradingtime”asopposedto calendartime),andobservesignificantimprovementinperformanceinthe caseofETF-basedsignals. ETFstrategieswhichusevolumeinformation 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 indetailtheperformanceofthestrategiesduringtheliquiditycrisisofthe summer of 2007. We obtain results which are consistent with Khandani and Lo (2007) and validate their “unwinding” theory for the quant fund drawdown of August 2007. ∗Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012 USA †FinanceConcepts,49-51AvenueVictor-Hugo,75116Paris,France. 1 1 Introduction Thetermstatisticalarbitrageencompassesavarietyofstrategiesandinvestment programs. Their common features are: (i) trading signals are systematic, or 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 2 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 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 is 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 3 the U.S. gives rise to risk-factors that have economic significance because they can be interpreted as long-short portfolios of industry sectors. However, the stocks that contribute the most to a particular factor are not necessarily the largest capitalization stocks in a given sector. This suggests that PCA-based risk factors may not be as biased towards large-capitalization stocks as ETFs, as the latter are generally capitalization-weighted. We also observe that the variance explained by a fixed number of PCA eigenvectors varies significantly acrosstime,whichleadsustoconjecturethatthenumberofexplanatoryfactors neededtodescribestockreturns(toseparatesystematicreturnsfromresiduals) is variable and that this variability is linked with the investment cycle, or the changes in the risk-premium for investing in the equity market.1 This might explain some of the differences that we found in performance between the PCA and ETF methods. InSection3and4,weconstructthetradingsignals. Thisinvolvesthestatis- ticalestimationoftheresidualprocessforeachstockatthecloseofeachtrading day, using 60 days of historical data prior to that date. Estimation is always done looking back 60 days from the trade date, thus simulating decisions which might take place in real trading. The trading signals correspond to significant deviations of the residual process from its estimated mean. Using daily end-of- day (EOD) data, we perform a calculation of daily trading signals, going back to1996inthecaseofPCAstrategiesandto2002inthecaseofETFstrategies, across the universe of stocks with market-capitalization of more than 1 billion USD at the trade date. The condition that the company must have a given capitalization at the trade date (as opposed to at the time when the paper was written), avoids survivorship bias. Estimationandtradingrulesarekeptsimpletoavoiddata-mining. Foreach stock, the estimation of the residual process is done using a 60-day trailing window because this corresponds roughly to one earnings cycle. The length of the window is not changed from one stock to another. We select as entry point for trading any residual that deviates by 1.25 standard deviations from equilibrium,andweexittradesiftheresidualislessthan0.5standarddeviations from equilibrium, uniformly across all stocks. In Section 5 we back-test several strategies which use different sets of fac- tors to generate residuals, namely: (i) synthetic ETFs based on capitalization- weighted indices2, (ii) actual ETFs, (iii) a fixed number of factors generated by PCA, (iv) a variable number of factors generated by PCA. Due to the mecha- nism described above used to generate trading signals, the simulation is always out-of-sample, in the sense that the estimation of the residual process at time t uses information available only for the 60 days prior to this time. In all trades, we assume a slippage/transaction cost of 0.05% or 5 basis points per trade (a round-trip transaction cost of 10 basis points). 1See Scherer and Avellaneda (2002) for similar observations for Latin American debt se- curitiesinthe1990’s. 2SyntheticETFsarecapitalization-weightedsectorindexesformedwiththestocksofeach industrythatarepresentinthetradinguniverseatthetimethesignalincalculated. Weused syntheticETFsbecausemostsectorETFswherelaunchedonlyafter2002. 4 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) contrarian 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 are 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,isvaluableinthecontextofmean-reversionstrategies,evenatthe EOD sampling frequency and not just only for intra-day trading. 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 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 5 portfoliobetas; itisuncorrelatedwiththemarketportfolioorfactorsthatdrive the market returns. It follows that the portfolio returns satisfy   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 systematic 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 AfirstapproachforextractingfactorsfromdataistousePrincipalComponents Analysis (Jolliffe (2002)). This approach uses historical share-price data on a cross-section of N stocks going back M days in history. For simplicity of expo- sition, 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 matrixthandatapoints. Inthispaper,wealwaysuseanestimationwindowfor the correlation matrix of 1-year (252 trading days) prior to the trading date. The commonly used solution to extract meaningful information from the data is to model the correlation matrix4. 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 fewer “detached” eigenvalues than industry sectors. Therefore, we expect that the boundary between “significant” and 4We refer the reader to Laloux et al. (2000), and Plerou et al. (2002) who studied the correlationmatrixofthetop500stocksintheUSinthiscontext. 7 Figure 1: Top 50 eigenvalues of the correlation matrix of market returns com- puted on May 1 2007 estimated using a 1-year window and a universe of 1417 stocks (see also Table 3). (Eigenvalues are measured as percentage of explained variance.) 8 Figure2: ThedensityofstatesforMay1-2007estimatedusinga1-yearwindow, correspondingtothesamedatausedtogenerateFigure1. Noticethatthereare some “detached eigenvalues”, and a “bulk spectrum”. The relevant eigenvalues includesthedetachedeigenvaluesaswellasaeigenvaluesintheedgeofthebulk spectrum. 9 “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 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 eigenportfolio 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 10