avellaneda2008statistical

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 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 Pt and Qt denote 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 + β dQt Qt + dXt, (2) where Xt is 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 α is small compared to the fluctua- tions ofXt and 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 β dollars of stock Q if Xt is small and, conversely, go short P and long Q if Xt is 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 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 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 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∑ j=1 βj F (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 that 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 Xt for each stock at the close of each trading 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 full calculation of daily trading signals, 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.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. In Section 5 we back-test different strategies which use different sets of factors 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. 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 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. 1See Scherer and Avellaneda (2002) for similar observations for Latin American debt se- curities in the 1990’s. 2The condition that the company must have a given capitalization at the trade date (as opposed to at the time this paper was written), avoids survivorship bias. 4 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. 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}Ni=1 the returns of the different stocks in the trading universe over an arbitrary one-day period (from close to close). Let F 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, Ri = βiF + R˜i, (4) which is a simple regression model decomposing stock returns into a systematic component βiF and an (uncorrelated) idiosyncratic component R˜i. Alterna- tively, we consider multi-factor models of the form Ri = m∑ j=1 βijFj + R˜i. (5) Here there arem factors, 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}Ni=1 invested in each of the stocks are such that βj = N∑ i=1 βijQi = 0, j = 1, 2, ...,m. (6) The coefficients βj correspond 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 5 N∑ i=1 QiRi = N∑ i=1 Qi  m∑ j=1 βijFj + N∑ i=1 QiR˜i = m∑ j=1 [ N∑ i=1 βijQi ] Fj + N∑ i=1 QiR˜i = N∑ i=1 QiR˜i (7) 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 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 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 i at time t adjusted 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∑ k=1 Rik and σ2i = 1 M − 1 M∑ k=1 (Rik −Ri)2 3For 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 empirical correlation matrix of the data is defined by ρij = 1 M − 1 M∑ k=1 YikYjk, (8) which is symmetric and non-negative definite. Notice that, for any index i, we have ρii = 1 M − 1 M∑ k=1 (Yik)2 = 1 M − 1 M∑ k=1 (Rik −Ri)2 σ2i = 1. 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 � N don’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.4 We 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 v(j) = ( v (j) 1 , ...., v (j) N ) , j = 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 D(x, y) = {#of eigenvalues between x and 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 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 4We 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 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 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 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∑ 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 Fj and Fj′ vanishes for j 6= j′. 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 m factors and a residual, as in equation (4). Thus, the PCA 9 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-m matrix correspond- ing to the significant spectrum and a diagonal matrix of full rank, ρij = m∑ k=0 λk v (k) i v (k) j + � 2 iiδij , where δij is the Kronecker delta and �2ii is given by �2ii = 1− m∑ k=0 λk v (k) i v (k) i so that ρii = 1. This means that we keep only the significant eigenvalues/eigenvectors of the correlation matrix and add a diagon