Is Factor Momentum More than Stock Momentum?
IIs Factor Momentum More than Stock Momentum? ∗ Antoine Falck † Adam Rej † David Thesmar †‡ September 11, 2020
Abstract
Yes, but only at short lags. In this paper we investigate the relationship between factormomentum and stock momentum. Using a sample of 72 factors documented in the literature,we first replicate earlier findings that factor momentum exists and works both directionallyand cross-sectionally. We then ask if factor momentum is spanned by stock momentum. Asimple spanning test reveals that after controlling for stock momentum and factor exposure,statistically significant Sharpe ratios only belong to implementations which include the lastmonth of returns. We conclude this study with a simple theoretical model that captures theseforces: (1) there is stock-level mean reversion at short lags and momentum at longer lags, (2)there is stock and factor momentum at all lags and (3) there is natural comovement betweenthe PNLs of stock and factor momentums at all horizons. ∗ We are especially grateful to Jean-Philippe Bouchaud for his feedback. We also thank Mark Potters, Philip Seager, as well as CFM seminarparticipants for useful comments. † Capital Fund Management International Inc., The Chrysler Building – 55th floor, 405 Lexington Avenue, NewYork, NY, 10174 ‡ MIT, NBER & CEPR a r X i v : . [ q -f i n . S T ] S e p Introduction
This paper analyzes factor momentum. In recent years, a body of literature in asset pricing hasanalyzed the properties of a large number of risk factors (Harvey and Zhu (2016), McLean andPontiff (2016), Hou and Zhang (2019), Giglio et al. (2020)). These factors are constructed bysorting stocks along various characteristics and are evidence that stock returns are predictable andco-move. As it turns out, recent research suggests that factor returns are themselves predictable(Cohen et al. (2003),Haddad et al. (2020)) and in particular exhibit momentum (Ehsani and Lin-nainmaa, 2019): A factor that was performing well tends to perform well in the future. At thesame time, it has been known for long that stocks also exhibit momentum (Jegadeesh and Titman(1993)). Factor and stock momentum are likely to be interconnected: Since stocks are exposedto factors, and factors have persistent returns, factor momentum may explain stock momentum.Conversely, since factors are portfolio of stocks, stock momentum may mechanically lead to factormomentum. This paper investigates the relation between these two momentums. We find that fac-tor momentum is difficult to distinguish from stock momentum: The only difference is at monthlytime scale, where stocks mean revert while factors exhibit strong momentum.We shall proceed in three steps. First, we use US data and build a dataset that contains 72 doc-umented signals published in the literature. We replicate the construction of factors as long-shortdollar-neutral portfolios and further hedge their residual market exposure dynamically. An equal-risk average of these factors generates a Sharpe ratio of 0.96, thereby confirming the soundnessof our replication exercise. Consistent with many investment strategies becoming crowded overtime, we find that average risk-adjusted performance tends to taper off in the late 2000s. We thendocument that these factors exhibit momentum. As Ehsani and Linnainmaa (2019), we exploreboth types of factor momentum: cross-sectional (buying the most successful factors and sellingthe least performing ones) and directional (buying factors with past positive returns and sellingfactors with past negative returns). We find that both strategies deliver significant and similar risk-adjusted performance, with a Sharpe ratio around 1. We then analyze “mechanical” drivers offactor momentum. Since factor momentum buys the best performing factors and sells the unper-forming ones, factor momentum contains a mechanical exposure to the mean and spread of factorreturns, a phenomenon shown in Lo and MacKinlay (1990). Controlling for mean factor return,we still find that factor momentum is present, in line with results shown by Ehsani and Linnainmaa(2019) using a different set of factors and a slightly different spanning approach. The bottom line2s that factor momentum is a resilient anomaly that begs for an explanation.In the second step of our analysis, we analyze the distinction between factor and stock momen-tum. We use the following approach. In the spirit of Jegadeesh and Titman (1993)’s analysis ofstock momentum, we analyze a full range of definitions of factor momentum using various lags(how many most recent months of data one discards when constructing the signal) and various“holding periods” (number of months worth of stock returns used to compute momentum). Wefirst find that factor momentum is present for a large range of lags and holding periods. We thencompute the excess performance of each one of these momentums after controlling for factor ex-posure. Consistently with our first step, we find that controlling for mean factor exposure removesabout 0.2 points of Sharpe from all factor momentums (i.e., irrespective of lag and holding period).Controlling for stock momentum further reduces the Sharpe of nearly all factor strategies, exceptfor the smallest possible lag (one month). Further, the excess return remains particularly strongwhen one focuses on one-month lag and one-month momentum time scale, i.e., the monthly meanreversion. Additional analysis confirms that factor momentum is “spanned” by stock momentumand factor exposure, except at one-month time scale. Put differently, factor returns are persistentat the monthly time scales, while stock returns mean revert. Otherwise, factor momentum is notdistinguishable from stock momentum.In our third step we develop a simple model designed to encapsulate these findings. The keyquestion is whether factor momentum can co-exist with stock mean reversion at short horizonsand stock momentum at longer horizons. In our model there are three types of investors: noisetraders, positive feedback traders who buy stocks exposed to an arbitrary factor when it performswell and contrarian traders, who buy stocks whose price is high. The model captures the threefeatures present in the data: (1) there is stock-level mean reversion at short lags and momentum atlonger lags, (2) there is factor momentum at all lags and (3) there is natural comovement betweenthe PNLs of stock and factor momentums at all horizons (short- and long-term).Our paper contributes to the literature on factor timing. This literature looks for evidence thatrisk factor returns are themselves predictable. Most papers in this space focus on mean-reversionin factor returns. Cohen et al. (2003), and more recently Haddad et al. (2020) build on the ideathat factors are measures of risk premia, which tend to be mean-reverting (Campbell and Shiller,1987): A factor is expensive when stocks in its long (short) leg have a high (low) market to bookratio. Both papers find evidence consistent with the idea that expensive factors tend to have lowerfuture returns. Cohen et al. (2003) focuses on the value premium, while Haddad et al. (2020) focus3n the first PCs of a large group of factors. In the same spirit, Hanson and Sunderam (2014) findthat factors that short stocks that are otherwise heavily shorted tend to mean revert.Our paper focuses instead on shorter-term persistence in factor returns. We follow Ehsani andLinnainmaa (2019) and provide further evidence of factor momentum using a different set of fac-tors (we have 72 factors while they have 20 in their baseline study; moreover our factors are definedin US only) and a slightly different portfolio construction. Like these authors, we find that direc-tional factor momentum is marginally stronger than the cross-sectional implementation, with bothstrategies generating most of their returns from their long legs. But the main difference betweentheir study and ours is in the test differentiating stock and factor momentum. They construct factormomentum as a strategy based on the past 12 months of returns, while stock momentum is alsobased on the past 12 months of returns excluding the last month. They find that factor momentumis not spanned by stock momentum. Our strategy consists of looking at different definitions offactor momentum, where we vary both the lag and the holding period. This allows us to uncoverthe source of the discrepancy between the two momentums: At one-month time scales, stocksmean-revert while factors have persistent returns. If we exclude the last month of returns, factormomentum is subsumed by stock momentum. We provide a model consistent with our findings.The paper is structured as follows. Section 2 describes the data and the construction of ourfactor zoo. Section 3 provides evidence of factor momentum and tests for mechanical driversof momentum. Section 4 decomposes factor momentum into various lags and horizons. Thisdecomposition shows the key role of the lag in differentiating between both types of momentum.Finally, Section 5 proposes a model consistent with stylized facts that we uncovered. Section 6concludes.
We construct time series of returns for 72 risk factors documented in the literature (see list inAppendix A). These risk factors use data from a CRSP-COMPUSTAT merged sample, so theyonly use accounting and price information. This sample runs from January 1963 to April 2014. Inour merger of CRSP and COMPUSTAT, we make sure that there is no look ahead bias in financialstatement availability. We assume that financial statements for the last fiscal year become availablefour months later. This is conservative as earnings are typically anounced between 2 and 3 monthsafter the end of the fiscal year. But it ensures that there is little look ahead bias in our data.4hile our implementation of the 72 sorting variables is faithful to the papers we replicate, wepropose certain improvements to risk management in order to be closer to what is common practicein asset management. First, we restrict our analysis to the 1,000 most liquid stocks on CRSP. Thisis in order to ensure that the factors trade liquid stocks and have reasonable capacity. Second,we construct and risk manage each factor using the following methodology. For each factor inAppendix A, we first compute daily factor returns by implementing the characteristic proposed inthe original factor paper and using the portfolio construction method proposed by the authors. Weresample the PnLs monthly. Next, we remove any residual exposure to being long the market bybeta-hedging, i.e. we run a rolling 36 months regression of the factor PnL onto market returns todetermine the (past) beta and use this beta for the following month to debias PnLs. Finally, we riskmanage the resulting PnL by dividing it by the (rolling) standard deviation of returns computedover past 36 months and lagged one month. This ensures that each beta-hedged factor PnL has arelatively constant volatility and is thus closer to how factor investing would be implemented inpractice. We report the distribution of Sharpe ratios and t-stats of our 72 factors in Table 1. Asit becomes apparent, a typical factor Sharpe ratio is low (0.27), suggesting potential overfitting orarbitrage, an issue we explore in a companion paper.
Table 1: Summary Statistics on our Sample of 72 Factors
Mean Median p25 p75Annualized Sharpe 0.27 0.35 -0.02 0.55t stat 1.73 2.11 -1.25 3.64
Note: Annualised Sharpe and t stats, computed with monthly returns. All factor are beta-neutralised and risk-managed,computed on CRSP 1,000 most liquid stocks from 1963 to 2014.
We report in Figure 1 the cumulative return of factor risk-parity. This investment strategyconsists in being long an equal amount of risk of each of the 72 factors and is run at a constantoverall risk target. In what follows, we will refer to this strategy per “menagerie” (all of the“animals” in our “zoo” of factors). The Sharpe ratio of this strategy over the entire period is0.96. The cumulative return does however show signs of tapering off starting in the mid 2000s.5his is consistent with the idea, described in McLean and Pontiff (2016) among others, that afterpublication the strategies listed in these papers perform less well.
Figure 1: Mean Factor ReturnWe report here the cumulative return of factor risk parity with 72 factors. The investment strategy is beta-hedged inorder to filter out exposure to the market and run at a constant risk target.
Let F t be a vector of monthly factor returns, with returns computed between months t − and t (and each factor constructed using information available at t − ). Each component of this vectorcorresponds to one of our risk-managed factors presented in Section 2, i.e. they are dynamically6edged with respect to market exposure and scaled by rolling volatility. N is the number of com-ponents. We first define the “menagerie” as investing $1 in each one of these factors (so the amountof risk in each factor is the same). Hence the PNL of the menagerie at the end of month t is: π met = e (cid:48) F t where e = (1 , .. is a vector of ones. We denote the transpose by (cid:48) .We then define cross-sectional and directional factor momentum as in Ehsani and Linnainmaa(2019). The key difference is that we sort factors based on past returns computed from t − m − n to t − m . Put differently, m is the most recent price used (the “lag”) and n is the number of monthsused (the “holding period”). To make things clear, we note: F t ( m, n ) = k = m + n − (cid:88) k = m F t − k the cumulative return between t − m − n and t − m . Note that as a special case F t (0 ,
1) = F t .The PNL of cross-sectional factor momentum is given by: π XSt ( m, n ) = rank ( F t ( m, n )) (cid:48) F t (1)where the rank operator transforms a vector with N components into a vector of ranks with N different, equally-spaced components in the range (cid:104)− , (cid:105) . By construction the cross-sectionalmomentum is always dollar neutral, i.e. the sum of weights is equal to zero.The PNL of directional momentum does not impose dollar neutrality. We define it as: π T St ( m, n ) = sgn ( F t ( m, n )) (cid:48) F t (2)For instance, if all factors have performed positively in the past period, the factor momentumportfolio will (temporarily) be the same as the menagerie. The net leverage of this strategy variesover time: it increases (decreases) when factors have positive (negative) returns.The above definitions (menagerie, XS and TS momentum) are that of “raw” portfolios, in thesense that they are not risk managed. The empirical results we present below are for the “risk-managed” versions of these PNLs, which are obtained by normalizing the returns with their rolling36 month volatility lagged one month. Remember that each one of the factors is itself already7isk-managed.Last, for clarity of exposition, we focus in the sections that follow on directional momentum,but our results with cross-sectional momentum are very similar. We report them in Appendix B. Here, we redo the analysis in Ehsani and Linnainmaa (2019) using our own set of factors andreport results in Table 2. The first line shows the performance of the menagerie strategy, which hasa Sharpe ratio of 0.96 (as we showed in Figure 1) and a t-stat of 6.86.
Table 2: Momentum in Our Set of Factors
Sharpe ratio t -valueMenagerie 0.96 6.86TS 1.16 8.22TS, winners 1.29 9.11TS, losers 0.43 3.03XS 1.01 7.12XS, winners 1.20 8.52XS, losers 0.46 3.24 Note: Annualised statistics computed with monthly returns. We compute factor momentum signals using n = 10 and m = 2 . Factors are beta-neutralised and risk-managed, computed on CRSP 1,000 most liquid stocks from 1963 to2014. The next lines of Table 2 show the performance of factor momentum. We calculate the directreturns of cross-sectional and directional momentum (using formulas (1) and (2)). We set n = 11 and m = 2 for now (we come back to varying m and n later). Time series and cross-sectionalmomentum both have high Sharpe ratios and the directional implementation outperforms the cross-sectional one by . in Sharpe. Ehsani and Linnainmaa (2019) also find that directional factormomentum outperforms, but they find a somewhat larger difference in Sharpe .26 (Their Table 2).There are three main differences between their methodology and ours. First, we are using a largerset of factors. Second, we are constructing our factors using the 1,000 largest U.S. stocks only. Ourconstruction thus excludes illiquid microcaps. Last, we risk-manage and beta-neutralize factors’PNLs. In spite of these significant differences, we manage to reproduce their headline results quiteclosely. 8nother salient feature of Table 2 is that most of the return on factor momentum is borne by thelong leg of the strategy (the “winners”). Ehsani and Linnainmaa (2019) find similar results. Onepossibility is that this whole effect is driven by somewhat unsophisticated, long-biased investorspiling into factors that performed well, as in Hanson and Sunderam (2014)). We rely on this ideain our model of Section 5. The key role of the long leg of factor momentum also suggests that part of the performance maysimply come from outright exposure to factors. This concern is particularly relevant for directionalmomentum. This intuition can easily be confirmed in a momentum decomposition à la Lo andMacKinlay (1990). Let us focus on one single factor f t (an element of F t ). Assume that this factorfollows an AR1 process f t = (1 − ρ ) µ + ρf t − + u t , where µ is the long-run excess return ofthe factor. Then, the conditional and unconditional PNLs of directional momentum on this factoralone are given by: E t − ( f t − f t ) = ρf t − + (1 − ρ ) µf t − E ( f t − f t ) = ρσ f + µ These two formulas illustrate why factor momentum is more likely to have a good PNL if theexpected return of the factor is positive µ > , independently of the strength of factor momentumitself ( ρ ). To fix ideas, set ρ = 0 . The first equation shows the conditional PNL of the factormomentum strategy, which is non-zero. Indeed when the factor has performed well ( f t − > ),the momentum trader will purchase it and will thus cash in the expected premium µ : expectedprofit will be µf t − > , even though factor returns have no real momentum. Another way tosee this is by looking at the unconditional PNL in the second equation. Even if ρ = 0 , andas long as µ (cid:54) = 0 , factor momentum always generates a positive PNL. On average, factors withpositive premia are more likely to experience two consecutive dates of positive returns. All thesame, factors with negative premia are more likely to underperform on two consecutive dates. Thisensures E ( f t − f t ) > even if there is no actual persistence in returns.To account for this mechanical relationship (and isolate the pure effect of factor momentum),we look at the residual of the regression of factor momentum PNL on the menagerie PNL. The9 igure 2: PNLs of Menagerie and Factor MomentumsCumulative sum of returns to two factor momentum strategies and the menagerie factor. All PnLs are rescaled (in acausal way) to target 1% monthly volatility. Factors are beta-neutralised and risk-managed, computed on CRSP 1,000most liquid stocks from 1963 to 2014. de facto exposure to the menagerie. We show the effect of this control in Figure 2where we report the momentum returns alongside with the corresponding residuals. We can seethat the cumulative PNLs of residuals are about 20% less than that of pure factor momentum. Thus,while the mechanical exposure of momentum to factor returns in non-negligible, factor momentumremains a strong force. We now turn to the analysis of the link between stock momentum and factor momentum. We startwith an analysis of the correlation between the two factors. We then break the analysis down intovarious lags and holding periods. In this section we use the directional implementation of factormomentum. The results for cross-sectional momentum are similar and are reported in AppendixB.
To investigate this, a natural object to look at is the correlation between stock and factor momen-tum. Let us first investigate this correlation analytically. To fix ideas, let us assume that the vectorof stock returns r t (between period t − and t ) follows a single factor structure: r t = βf t + e t where β is a vector of factor exposures of individual stocks and e t is the idiosyncratic shock,assumed to be stationary, i.i.d. and most importantly independent of f t for all lags. The factor canbe persistent but we assume it to be homoskedastic – its conditional variance does not vary withtime. We define here the PNLs of (directional) factor and stock momentum strategies as: π F t = f t (cid:48) f t π S t = r t (cid:48) r t m and holding period n . In this section we do not risk manage PNLs in order to obtain closed-form formulas (in theempirical analysis we will stick with the empirical definitions).In this setting one can show (See Appendix C.1) that the covariance between the two PNLs isgiven by: cov (cid:0) π Ft +1 , π St +1 (cid:1) = β (cid:48) β var (cid:16) f t f t +1 (cid:17) = β (cid:48) β var (cid:0) π Ft +1 (cid:1) > (3)which is generally positive, whether factor momentum has a positive PNL or not. The intuitionis simple: Buying winning stocks is equivalent to exposing oneself to the factor. So wheneverstock momentum performs well, it is likely that the underlying factor has performed well. In thisvery simple setup, all of stock momentum comes from factor momentum (as we assumed e t tobe i.i.d.). So even if factor momentum is not present, there will be instances when it works byaccident and consequently stock momentum will work too. The bottom line of this analysis is thatthe covariance between both strategies is mechanically positive, even when none of them has apositive Sharpe ratio. This due to the fact that β is stable.Let us now move to the empirical analysis and compute the correlation between the two typesof momentum. For factor momentum, we choose m = 1 and n = 12 . For stock momentum, weuse UMD, the stock momentum factor available from the Fama-French data library. UMD sets m = 2 and n = 11 in order to exclude short-term mean-reversion. We report this correlation andother statistics in Table 3. We use monthly returns. As expected from our algebra, the correlationof factor momentum with UMD is high, around 60%. This is much higher than the correlation be-tween UMD and the menagerie (0.04%). Note that this high correlation is thus unaffected whetherwe look at directional, cross-sectional or residual momentum. The analysis above has shown whypart of this correlation is likely to be mechanical.This very strong correlation between the two types of momentum begs for a “spanning test”. Todo this we regress the PNL of factor momentum on the PNL of the stock momentum strategy usingthe entire sample. We then compute the Sharpe ratio of the residual strategy. Using the abovedefinitions of stock and factor momentum we find a Sharpe ratio of 0.59, which is statisticallysignificant. We explore this in greater detail in Section 4.3. The fact that stock momentum does notentirely span factor momentum is consistent with Ehsani and Linnainmaa (2019) who implement12 able 3: Correlation with the momentum (UMD) factor. Menagerie Menagerie w/o UMD TS TS residual XS XS residualCorrelation 0.04 -0.08 0.57 0.59 0.58 0.58
Factors are beta-neutralised, risk-managed and computed on CRSP 1,000 most liquid stocks from 1963 to 2014. UMDsorts stocks based on the last 12 months of returns except for the most recent one ( m = 2 and n = 11 ). XS and TSfactor momentum sorts factors using the last 12 months of returns ( m = 1 and n = 12 ). similar spanning tests . This analysis is, however, incomplete as (1) it does not take into accountmechanical exposure to the menagerie (as shown in Section 3.3) and (2) the lags m are differentfor stock and factor momentum. Let us now investigate these two aspects. In this section we vary the holding period n and the lag m and control for their values whenanalyzing the correlation between the two types of momentum. It is a priori important to controlfor these parameters, as they greatly affect the performance of momentum, as it is well knownsince at least Jegadeesh and Titman (1993).In Figure 3, before looking at correlations, we first show the performance of both strategies fordifferent values of m and n . Panel A focuses on stock momentum and reproduces three classicalresults . First, there is short term reversal for ( m, n ) = (1 , as evidenced by the very dark cells(indicating negative Sharpe of momentum) in the upper left corner of the heat map. Second, theclassic momentum ( m = 2 , n = 10 ) has a high Sharpe ratio (0.8). Third, a strongly laggedmomentum ( m = 6 , n = 6 ) has an impressive Sharpe ratio, around 1 as previously noticed byNovy-Marx (2012). Put differently, there is a northeast-southwest diagonal of light colors on theright side of the heat map, which is an evidence of the strength of “lagged momentum”. In panel Bwe focus on factor momentum. Two salient patterns emerge. First, there is no short-term reversalin factor returns. Past factor returns always predict stronger future returns, even at short timescales. It is clear that having the most recent month of return increases the performance of thestrategy: A big fraction of the strength of factor momentum comes from the first lag, preciselywhere stock momentum has a negative performance . Another emerging pattern is that strongly13agged factor momentum, like for stocks, performs quite well. Overall, this analysis highlights theone key difference between stock and factor momentum: in the short-run, stocks exhibit reversalswhile factors momentum. Figure 3: Sharpe Ratios of Stock and Factor Momentumas a Function of m and n Top: Sharpe ratios of the cross-sectional stock momentum. Stock returns are risk-managed, the universe is CRSP1,000 most liquid stocks from 1963 to 2014.Bottom: Sharpe ratio of the directional factor momentum. Factors are beta-neutralised, risk-managed and computedon CRSP 1,000 most liquid stocks from 1963 to 2014.
We now move to computing the correlation between the two types of momentum. Results arereported in Figure 4. The heat map delivers a simple yet impressive result: the correlations areremarkably stable, around .5 across the board.The stability of the correlation across values of ( n, m ) could be a priori surprising since wehave just uncovered that for small values of m and n stocks and factor returns behave differently.So while it is natural to expect that the two strategies are correlated for m > , it is more surprisingto find that they are positively correlated when their Sharpe ratios have opposite signs. The analysis14 igure 4: Correlation between Stock Momentum and Factor MomentumFor different levels of m and n Stock returns are risk-managed, the universe is CRSP 1,000 most liquid stocks from 1963 to 2014. of Section 4.1 provides an explanation for this. As shown in Equation (3) persistent factor exposureof stocks mechanically drives part of the correlation between both strategies: when factors performwell two periods in a row, stocks with positive beta do the same, while stocks with negative betasconsistently sell off. Thus, this correlation is present, and does not depend on how well or badlythe two momentums perform unconditionally.
We now proceed to analyze the residual of factor momentum on stock momentum. We have seenin Table 3 that factor momentum is not spanned by UMD (the residual has a Sharpe of 0.6). Butnow we want to incorporate what we have learnt by varying m and n , i.e. that the Sharpe of thetwo momentums vary quite differently as functions of lag and holding period. In addition, we alsoknow from Section 3.3 that factor momentum is correlated with the menagerie factor, so we alsowant to include the latter in our spanning test.Thus, in Figure 5, we show the Sharpe ratio of the residual of factor momentum after con-trolling for stock momentum, the menagerie factor and the market. All regressions use the entire15ample (these are spanning tests, not hedging regressions). This residual is computed separatelyfor each value of the lag m and holding period n . Figure 5: Sharpe Ratio of Residual Factor MomentumAfter Taking out Stock Momentum, Menagerie and MarketStock returns are risk-managed, the universe is CRSP 1,000 most liquid stocks. Sample period from 1963 to 2014.
The heat map in Figure 5 suggests that that there is more to factor momentum than stockmomentum, but only at lag m = 1 , i.e. when we include the last month of returns. As soon as m > , the Sharpe ratios of residuals become insignificantly different from zero. This is consistent16ith the observation made in the previous section. The main difference between factor and stockmomentum is that for m = n = 1 stocks mean revert in the cross-section while factor returnspersist. Our next section provides a framework that combines these effects.Before we proceed, however, we would like to know whether stock and factor momentum areidentical phenomena or whether factor momentum is just subsumed by stock momentum. In orderto do this, we implement a spanning test opposite to the one proposed above, i.e. we regress stockmomentum on factor momentum, menagerie and market for different values of m and n in orderto single out the effect of lagging the signals. We report the results in Figure 6. For m = 1 , dueto mean-reversion, stock momentum is either negative or completely explained by the independentvariables. For other values of m , however, residual stock momentum is still significant for manycells in the heat map. This suggest that, while factor momentum is just stock momentum when m > , stock momentum is really something else for all values of m . Figure 6: Sharpe Ratio of Residual Stock MomentumAfter Taking out Factor Momentum, Menagerie and MarketStock returns are risk-managed, the universe is CRSP 1,000 most liquid stocks, from 1963 to 2014. Framework
In this section, we will develop a heuristic model that encapsulates the three main stylized factsdiscussed in the previous Sections: (1) there is stock-level mean reversion at short lags and momen-tum at longer lags, (2) there is factor momentum at all lags and (3) there is a natural co-movementbetween the PNLs of stock and factor momentums at all horizons (short- and long-term).
We assume that there are N different stocks and that there exists a factor strategy that attractsinvestors’ interest. This factor strategy is captured by a set of (normalized) positions on each of the N stocks, which we assume fixed to simplify the exposition. We denote them by w . We postulatethat stock returns satisfy the equation below: r t = µ + (cid:15) t + ( price impact due to flows into/out of factor investing ) t which can be derived from an equilibrium asset pricing model where three types of investors arepresent: Noise traders (who generate the mean-reverting noise (cid:15) t ), factor investors (described be-low) and contrarian traders who have a limited ability to correct prices by leaning against thedemand of the two other types of investors.We assume the following behavior of factor investors. In line with the vast literature on theflow-performance relationship, we assume that these investors purchase additional amounts of thefactor when it has performed well in the previous period. Past factor performance is given bythe scalar w (cid:48) r t − and each stock is purchased in proportion w i , so that net factor purchases areproportional to the vector ww (cid:48) r t − . The price impact of these imbalances is then assumed to beproportional to trading so that ( price impact due to flows into/out of factor investing ) t = αww (cid:48) r t − This assumption is consistent with existing literature which documents that flow-induced tradinghas a predictable component (corresponding to pure scaling up or down of the fund’s portfolio inresponse to changes in AUMs) and that this component has a measurable price impact (Coval andStafford (2007),Greenwood and Thesmar (2011), Lou (2012)).We end up with the following equations of motion for stock returns:18 t = µ + (cid:15) t + α ww (cid:48) r t − which implies the following dynamics of factor returns F t = µ ( F ) + aF t − + (cid:15) ( F ) t where we introduced a = αw (cid:48) .w > which is a scalar. The law of motion of factor returnsimplies that there is autocorrelation of factor returns due to a (performance-flow induced trading). µ ( F ) = w (cid:48) µ is the unconditional expected return of the factor.Finally, we assume that the stochastic term, (cid:15) t , to be strongly mean-reverting. We assume itsdynamics is governed by an MA1 process (cid:15) t = e t − ρe t − where ρ is a scalar and e t ∼ N (0 , Σ) , where Σ is the covariance matrix. Strong mean-reversionat fast time scales is an empirical fact for stocks. It is thus a short-memory process, while returnsthemselves, through positive feedback trading, have a longer memory. Define A = αww (cid:48) . Stationarity implies that a < , which is an assumption we will be making.Then, returns can be shown to be described by the following equation: r t = 11 − A µ + e t + ( A − ρI ) e t − + ( a − ρ ) (cid:88) k ≥ a k − Ae t − k where the terms in A − ρI and ( a − ρ ) represent the opposing forces of flow-induced trading ( a )and noise trader demand ( ρ ).The autocovariance matrix of returns (of order k ) is given by: Ω = (1 − ρ ( a − ρ )) A Σ − ρ Σ + ( a − ρ ) A Σ A (cid:18) a − ρ ) a − a (cid:19) Ω k = ( a − ρ )(1 − ρa ) a k − A Σ + ( a − ρ ) a k − A Σ A (cid:18) a − ρ ) a − a (cid:19) if k ≥ Proposition 1.
Factor momentum PNL
Let V = w (cid:48) Σ w denote the expected factor variance. Then, the expected PNL of factor momentumis given by: E ( F t − k F t ) = w (cid:48) Ω k w + (cid:0) µ ( F ) (cid:1) = V (cid:18) ( a − ρ )(1 − ρa )1 − a (cid:19) a k − + (cid:0) µ ( F ) (cid:1) Thus, after controlling for mechanical exposure to factor investing ( (cid:0) µ ( F ) (cid:1) ), there is factormomentum at all lags if and only if a > ρ In the second term of the above expressions, we rediscover that the expected PNL of factormomentum is positive even if there is no flow-induced trading ( a = 0 ). This is a result we haveseen previously: There is a mechanical relation between factor momentum and outright factorexposure.But the interesting part of the above proposition is in the first term, which is the performanceof factor momentum in excess of mechanical exposure to factor investing . This first term has thesame sign at all horizons: If excess return of factor momentum is positive for k = 1 , it will bepositive at all horizons. This is consistent with what we find in the data (Table 3).Let us now turn to stock momentum. Stock momentum is defined thourgh the trace of theautocovariance matrices Proposition 2.
Stock momentum PNL
Write V = w (cid:48) Σ w . The expected PNL of stock momentum at all horizons k is given by: E ( r t − r t ) = tr (Ω ) + µ (cid:48) .µ = αV (cid:18) a − ρ ) − a (cid:19) − ρ tr Σ + µ (cid:48) .µE ( r t − k r t ) = tr (Ω k ) + µ (cid:48) .µ = αV (cid:18) ( a − ρ )(1 − ρa )1 − a (cid:19) a k − + µ (cid:48) .µ if k ≥ Thus: • There is stock momentum at all horizons k ≥ as long as a > ρ • When − ρ tr Σ starts dominating, there is mean-reversion for k = 1 a > ρ ),there is also stock momentum for all k ≥ . In this sense factor momentum and stock momentumare indistinguishable: They occur for the same parameter values as soon as one excludes the mostrecent lag.Another inference is that when the stock universe becomes large (keeping ρ constant), − ρ tr Σ becomes dominating and stock momentum becomes mean reversion for k = 1 .One last outcome of this analysis is that, stock momentum PNL ( ∼ tr (Ω k ) ) is a natural correlateof factor momentum if the values of a or ρ change. The expression below α V ( a − ρ )(1 − ρa )1 − a a k − is present both in stock momentum (for k ≥ ) and in factor momentum. Variations in the afore-mentioned parameters tend to make the two strategies comove. This is because, in our model,stock momentum is subsumed in factor momentum (for k ≥ ). An interesting extension of thismodel would be to introduce “idiosyncratic” stock momentum, for instance through assuming that (cid:15) t follows an MA2 process with a positive loading on e t − . In this paper we have established that, for most implementations, factor momentum is fully ac-counted for by stock momentum and factor exposure. An important exception occurs at one monthlag (and various holding periods). Including the last month of stock returns in the definition ofstock momentum degrades its performance because stocks exhibit mean reversion at monthly timescale. This is not true for factor momentum, as factors returns are persistent at all time scales thatwe have studied. Consequently, “true” factor momentum only exists through the effect of the lastmonth of returns. If we exclude it, factor momentum is pure stock momentum.It is possible to reconcile these empirical observations with a simple model, which we provideand solve. This model predicts that both strategies (factor and stock momentum) are correlatedbetween each other to a constant level, independently of the lag and the holding period, consistentwith evidence that we find empirically. 21 eferences A BARBANELL , J. S.
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List of Factors
Table A.1: List of Factor Names and References
Factor Reference articlePrice earnings Basu (1977)Unexpected quarterly earnings Rendleman et al. (1982)Long term reversal De Bondt and Thaler (1985)Debt equity Bhandari (1988)Change in inventory-to-assets Ou and Penman (1989)Change in dividend per share Ou and Penman (1989)Change in capital expenditures-to-assets Ou and Penman (1989)Return on total assets Ou and Penman (1989)Debt repayment Ou and Penman (1989)Depreciation-to-PP&E Holthausen and Larcker (1992)Change in depreciation-to-PP&E Holthausen and Larcker (1992)Change in total assets Holthausen and Larcker (1992)Size Fama and French (1993)Book-to-market Fama and French (1993)Momentum Jegadeesh and Titman (1993)Sales growth Lakonishok et al. (1994)Cash flow-to-price Lakonishok et al. (1994)Working capital accruals Sloan (1996)Sales-to-price Barbee et al. (1996)Share turnover Haugen and Baker (1996)Cash flow-to-price variability Haugen and Baker (1996)Trading volume trend Haugen and Baker (1996) see next page able A.1 (Continued): List of Factor Names and References Factor Reference articleInventory Abarbanell and Bushee (1998)Gross margin Abarbanell and Bushee (1998)Capital expenditures Abarbanell and Bushee (1998)Industry momentum Moskowitz and Grinblatt (1999)F-score Piotroski (2000)Industry adjusted book-to-market Asness et al. (2000)Industry adjusted cash flow-to-price Asness et al. (2000)Industry adjusted size Asness et al. (2000)Industry adjusted momentum Asness et al. (2000)Industry adjusted long term reversal Asness et al. (2000)Industry adjusted short term reversal Asness et al. (2000)Dollar volume Chordia et al. (2001)Dollar volume coefficient of variation Chordia et al. (2001)Share turnover coefficient of variation Chordia et al. (2001)Change in inventory Thomas and Zhang (2002)Change in current assets Thomas and Zhang (2002)Depreciation Thomas and Zhang (2002)Change in accounts receivable Thomas and Zhang (2002)Other accruals Thomas and Zhang (2002)Illiquidity Amihud (2002)Liquidity Pastor and Stambaugh (2003)Idiosyncratic return volatility x book-to-market Ali et al. (2003)Price x book-to-market Ali et al. (2003) see next page able A.1 (Continued): List of Factor Names and References Factor Reference articleOperating cash flow-to-price Desai et al. (2004)Abnormal corporate investment Titman et al. (2004)Accrual quality Francis et al. (2004)Earnings persistence Francis et al. (2004)Smoothness Francis et al. (2004)Value relevance Francis et al. (2004)Timeliness Francis et al. (2004)Tax income-to-book income Lev and Nissim (2004)Price delay Hou and Moskowitz (2005)Firm age Jiang et al. (2005)Duration Jiang et al. (2005)Change in current operating assets Richardson et al. (2005)Change in non-current operating liabilities Richardson et al. (2005)Growth in capital expenditures Anderson and Garcia-Feijoo (2006)Growth in capital expenditures (alternative) Anderson and Garcia-Feijoo (2006)Low volatility Ang et al. (2006)Low beta ∆ VIX Ang et al. (2006)Zero trading days Liu (2006)Composite issuance Daniel and Titman (2006)Intangible return Daniel and Titman (2006)Earnings surprises x revenue surprises Jegadeesh and Livnat (2006)Industry concentration Hou and Robinson (2006)Change in shares outstanding Pontiff and Woodgate (2008)Seasonality Heston and Sadka (2008)Investment Lyandres et al. (2008)Investment growth Xing (2008)Change in asset turnover Soliman (2008)29
Replication of Our Main Results Using Cross-Sectional Mo-mentum
In this Appendix we replicate our main results from Figures 2, 3 and 4 using cross-sectional factormomentum instead of time-series factor momentum. The results are essentially identical. FigureB.1 shows the PNL of the cross-sectional factor momentum for various values of ( m, n ) . FigureB.2 reports the correlation between stock and (cross-sectional) factor momentum. Figure B.3shows the Sharpe ratio of the residual of factor momentum on stock momentum. Figure B.1: Sharpe ratio of the cross-sectional factor momentum. Factors are computed on CRSP 1,000 most liquidstocks (1963-2014) and are beta-neutralised and risk-managed. igure B.2: Monthly correlation of factor momentum (cross-sectional) and stock momentum.Stock returns are risk-managed and factors are beta-neutralised and risk-managed. The universe is CRSP 1,000 mostliquid stocks (1963-2014).Figure B.3: Sharpe ratio of the residual of factor momentum (cross-sectional) w.r.t. stock momentum. Stock returnsare risk-managed and factors are beta-neutralised and risk-managed. The universe is CRSP 1,000 most liquid stocks(1963 to 2014). Proofs
C.1 Covariance between Factor and Stock Momentum
Assume the vector of stock returns follows a single factor structure r t = βf t + e t where we further assume that e t and f t are independent at all lags.Then, the PNLs of (directional) factor momentum and stock momentum are given by the fol-lowing two equations π Ft +1 = f t f t +1 π St +1 = r (cid:48) t r t +1 where the lower bar stands for any combination of lag m and holding period n until date t .To calculate the covariance of these two PNLs, we start with the expectation of the product E ( π Ft +1 π St +1 ) = E (cid:104) E t (cid:16) f t f t +1 r (cid:48) t r t +1 (cid:17)(cid:105) = E (cid:104) f t r (cid:48) t E t ( f t +1 r t +1 ) (cid:105) = E (cid:104) f t r (cid:48) t E t ( f t +1 ( βf t +1 + e t +1 )) (cid:105) We note σ f and µ t the conditional covariance and conditional expectation of f t +1 . We assume f t homoskedastic, so that the conditional variance is constant, but we allow the conditional meanto vary over time (there can be factor momentum for instance). We use the fact that e t and f t areindependent so that E ( π Ft +1 π St +1 ) = E (cid:104) f t r (cid:48) t (cid:105) βσ f + E (cid:104) f t r (cid:48) t µ t (cid:105) β Finally, we exploit the fact that cumulative returns have the same factor structure as one period32eturns, i.e. that r t = βf t + e t . This leads to E ( π Ft +1 π St +1 ) = E (cid:104) f t (cid:105) β (cid:48) βσ f + E (cid:104) f t µ t (cid:105) β (cid:48) β = β (cid:48) β (cid:18) σ f Ef t + E (cid:16) f t µ t (cid:17) (cid:19) Now, we need to compute the product of expectations E ( π Ft +1 ) E ( π St +1 ) = (cid:16) Ef t µ t (cid:17) β (cid:48) β Combining the two expressions yields:cov (cid:0) π Ft +1 , π St +1 (cid:1) = β (cid:48) β (cid:18) σ f Ef t + E (cid:16) f t µ t (cid:17) − (cid:16) Ef t µ t (cid:17) (cid:19) Noting that µ t = − σ f + E t ( f t +1 ) we obtaincov (cid:0) π Ft +1 , π St +1 (cid:1) = β (cid:48) β var (cid:16) f t f t +1 (cid:17) = β (cid:48) β var (cid:0) π Ft +1 (cid:1)(cid:1)