The Sound of Many Funds Rebalancing

Transcription

1 The Sound of Many Funds Rebalancing Alex Chinco and Vyacheslav Fos December 5, 2017 Abstract This paper proposes that long rebalancing cascades generate noise in financial markets. There are two components to our analysis. First, to show that long rebalancing cascades can generate noise, we analyze a random-networks model where it s possible to predict whether a particular stock s demand will be affected by a long rebalancing cascade but not how the stock s demand will be affected due to computational complexity. Then, to show that long rebalancing cascades actually do generate noise in real-world financial markets, we study the end-of-day holdings of exchange-traded funds (ETFs). We demonstrate the existence of long ETF rebalancing cascades, and we document evidence of market participants treating the resulting demand shocks as noise. Taken together, these two components suggest that noise can be an externality imposed by a collection of funds following simple rebalancing rules and not just the result of individual investors who each behave randomly. JEL Classification. G02, G12, G14 Keywords. Noise, Indexing, Thresholds We thank Kerry Back, Nick Barberis, Zahi Ben-David, James Choi, Tony Cookson, Xavier Gabaix, Itay Goldstein, Sam Hartzmark, Ralph Koijen, Pete Kyle, Chris Parsons, and Jeff Pontiff as well as seminar participants at CalTech, Colorado, Illinois, Maryland, Yale, the Young Scholars Finance Consortium, the FINRA Market-Structure Conference, the SFS Calvalcade, the Conference on the Econometrics of Financial Markets, and the Helsinki Behavioral Finance Conference for extremely helpful comments and suggestions. Current Version: University of Illinois at Urbana-Champaign, College of Business; alexchinco@gmail.com. Boston College, Carroll School of Management; fos@bc.edu. 1

2 1 Introduction Noise makes financial markets possible but also makes them imperfect (Black, 1985). Imagine you re a trader who s just discovered that stock Z is under-priced. In a market without noise, there s no way for you to take advantage of this discovery. The moment you try to buy a share, other traders will immediately realize that you must have uncovered some good news. And, you won t find anyone willing to sell at the old price (Milgrom and Stokey, 1982). Noise pulls the rug out from under this no-trade theorem. In a market with noise, someone is always trying to buy or sell stock Z for erratic non-fundamental reasons. So, when you try to buy a share, it won t raise too many eyebrows. Because your buy order could just be some more random noise, other traders won t immediately realize that you must have uncovered some good news. The existence of this plausible cover story allows you to both trade on and profit from your discovery. But, where exactly does this all-important noise come from? Who generates it? And, what are their erratic non-fundamental reasons for trading? The standard answer to these two questions is that i) noise comes from individual investors, and ii) their demand looks erratic and unrelated to fundamentals because individual investors are just plain bad traders. These are the standard answers for a reason. Not only do individual investors suffer from all sorts of behavioral biases when they trade (Barberis and Thaler, 2003) but they also trade far too often (Barber and Odean, 2000). So, individual investors clearly generate noise. But, are they the only source? It seems unlikely. The role of individual investors in financial markets has steadily declined over the past few decades. To give one representative statistic, individual investors held 47.9% of all U.S. equity in 1980, but that percentage was down to only 21.5% in 2007 (French, 2008). Yet, we haven t seen a corresponding drop in trading volume. Motivated by this logical gap, we propose another noise-generating mechanism: long rebalancing cascades (stock A s price jumps, which causes a fund to buy stock A and sell stock B, which causes a second fund to sell stock B and buy stock C, which causes... ). There are two parts to our analysis. First, we analyze a random-networks model to show how long rebalancing cascades can generate noise. Then, to show that long rebalancing cascades actually do generate noise in real-world financial markets, we study the end-of-day holdings of exchange-traded funds (ETFs). Theoretical Model. Our hypothesis that noise comes from long rebalancing cas- 2

3 cades was sparked by a second trend that s shaped financial markets over the past few decades, the rise of index-linked investing (Wurgler, 2010). At the same time as individual investors were shrinking in importance, passive investing indexing was becoming popular as an alternative to active investment management, while active managers... were becoming more index-like in their investing (Stambaugh, 2014). In fact, the number of indexes now exceeds the number of U.S. stocks. 1 Most of these new index-like funds were not created in Jack Bogle s image, however. Many include stocks based on custom criteria, such as having low volatility or high dividends, 1 and involve threshold-based portfolio rules. For example, the PowerShares S&P 500 Low-Volatility ETF [SPLV] tracks the quintile of S&P 500 stocks with the lowest volatility. We say that this fund uses a threshold-based rule because an arbitrarily small change in a stock s volatility can move it from 101st to 100th place on the low-volatility leaderboard. When this happens, the low-volatility ETF has to exit its position in one stock and build a new position in another. This rebalancing activity will affect both the stock being added to as well as the stock being deleted from the low-volatility index in equal-but-opposite ways. For example, due to the additional buying pressure, the price of the stock being added will rise while the price of the stock being deleted will fall. So, because there are so many of these index-like funds tracking so many different threshold-based benchmarks, a small change in stock A s price can cause some fund to buy stock A and sell stock B, which can then cause a second fund following a different index to sell stock B and buy stock C, which can then cause... We use a random-networks model to theoretically study how this sort of alternating buy-sell-buy-sell sequence might result in apparently random demand shocks (a.k.a., noise) even though the individual funds are following simple deterministic rebalancing rules. Each node in the network corresponds to a stock. And, two nodes are connected with an edge, not if the corresponding stocks both belong to the same benchmark index, but rather if a change in one of the stock s characteristics would cause it to swap places with the other stock in some fund s benchmark index. Thus, a node with many edges corresponds to a stock that s on the cusp of a rebalancing threshold for many different funds. The model explains how, even if it s possible to compute the likelihood that stock Z will be affected by a long rebalancing cascade that starts with an initial shock to stock A, it can still be computationally intractable to predict the direction, buy? or sell?, of the resulting demand shock. 1 Bloomberg. 5/12/2017. There Are Now More Indexes Than Stocks. 3

4 Economists have long had the intuition that apparent randomness might just be the result of computational complexity. For instance, way back in 1921 John Maynard Keynes pointed out that, although the population of France isn t random, whether or not this number happens to be an even or odd number at any given instant may as well be. Yet, in the past, this intuition has remained just that an intuition. The goal of analyzing this random-networks model is to describe precisely why it s computationally intractable to predict how a long rebalancing cascade will affect the demand for stock Z in as transparent a way as possible. The goal is to give a concrete foundation for this long-held intuition, making it possible to identify other situations where the same logic applies. Rebalancing Cascades. After describing how long rebalancing cascades can transform simple deterministic rules into apparently random demand shocks, we next give empirical evidence that this transformation actually takes place in the real-world. With this goal in mind, we first pick both i) a particular group of funds following threshold-based strategies and ii) a set of initial shocks. Then, we show that, in the wake of these initial shocks, it s possible to compute which stock Zs are most likely to be hit by a long rebalancing cascade involving this group of funds but not the direction of the resulting demand shock. For our particular group of funds, we study ETFs. This might seem like an odd choice at first. If you ask a random person on the street to name an ETF, you re probably going to get The SPDR S&P 500 ETF [SPY] for an answer. But, while this is the oldest and most well-known ETF, it is no longer representative of the industry as a whole. The vast majority of ETFs now track niche self-defined indexes. A recent article in the Financial Times described how Exxon Mobile was held by ETFs following numerous threshold-based indexes like active beta, momentum, dividend growth, deep value, quality, and total earnings. 2 And, more and more people are talking about how ETF rebalancing can influence trading in individual stocks. 3 We get data on each ETF s end-of-day portfolio holdings from XpressFeed. This data covers every trading day from January 2010 to December We restrict our sample to include only those ETFs that rebalance their positions daily think about the PowerShares S&P 500 Low-Volatility ETF rather than the SPDR S&P 500 ETF. To be sure, these ETFs are smaller than the broad market-index funds, but their rebalancing activity can still affect the underlying stocks because they tend 2 Financial Times. 10/7/2017. On The Perverse Economic Effects Created by ETFs. 3 Bloomberg. 4/10/2015. Tail Can Wag Dog When ETFs Influence Single Stocks, Goldman Says. 4

5 to do their trading during the final 20 or 30 minutes of the trading day. 4 It s also important to emphasize that we net out changes in ETF holdings due to creations and redemptions, which are executed as in-kind transfers for tax reasons (Madhavan, 2016) and so can t generate the distortions that underpin long rebalancing cascades. Then, for our set of initial shocks, we use M&A announcements, referring to the target of the announcement as stock A. This data comes from Thomson Financial. M&A deals are a natural choice for the initial shocks because, in the words of Andrade et al. (2001), a profusion of event studies has demonstrated that mergers seem to create shareholder value, with most of the gains accruing to the target company. While M&A targets are certainly not random, the exact date of the announcement (Tuesday, Wednesday, or Thursday?) effectively is. Here s how we structure our empirical tests. Following the announcement of each stock A as an M&A target, we collect the set of stock Zs that are unrelated to stock A. For a stock Z to be unrelated to a particular stock A, it has to be twice removed in the network of ETF holdings at the time of the M&A announcement. It can t have been recently held by any ETF that also recently held stock A. And, it can t have been held by any ETF that also held a stock that was held by another ETF that held stock A. The chain has to be A B C Z or longer. Note that, because there are smart-beta ETFs focusing on large-cap, value, and industry-specific benchmarks, this twice-removed criteria implies that the set of stock Zs doesn t share well-known characteristics such as size, book-to-market, or industry with the stock A. The random-networks model we study suggests that, all else equal, a stock that is on the cusp of more funds respective rebalancing thresholds will be more likely to be hit by a long rebalancing cascade. So, we split each set of stock Zs into two subsets: those with an above-median number of neighboring stocks, which are on the cusp of many different ETFs rebalancing thresholds, and those with a belowmedian number, which aren t. Consistent with our economic story, we find that ETF rebalancing volume grows by 169% more for the above-median group of stock Zs than for the below-median group in the 5 days immediately following an M&A announcement. But, we also find that this percentage increase in ETF order flow is no more likely to be made up of buy orders than of sell orders. In other words, we document that it s possible to predict which stock Zs are most likely to be affected by a long ETF rebalancing cascade but not the direction of the resulting demand shock. Demand Noise. Finally, after showing that long ETF rebalancing cascades are 4 Wall Street Journal. 5/27/2015. Stock-Market Traders Pile In at the Close. 5

6 equally likely to result in buy and sell orders for stock Z from a statistical perspective, we conclude our analysis by giving supporting evidence that market participants also treat the demand shocks coming from long ETF rebalancing cascades as noise. Specifically, we find that above-median subset of stock Zs tends to have higher liquidity than the below-median subset. This result is consistent with a model where the above-median stock Zs has higher demand-noise volatility. 1.1 Related Literature This paper connects to three main strands of literature. Noise. The problem we study is motivated by the central role that noise plays in asset-pricing theory. Noise plays a starring role in your favorite information-based asset pricing (Grossman and Stiglitz, 1980; Hellwig, 1980; Admati, 1985; Kyle, 1985) or limits-to-arbitrage model (Shleifer and Summers, 1990; Shleifer and Vishny, 1997; Gromb and Vayanos, 2010). This paper proposes an explanation for noise that does not rely on individual investors behaving randomly. Indexing. Our proposed explanation then connects our paper to the literature on index inclusion (Wurgler, 2010). These papers can be classified into two broad groups. The first group studies the predictable effects of stock A s inclusion in an index for stock A itself. For instance, Chang et al. (2014) shows that getting added to the Russell 2000 results in a price increase. For other examples involving ETFs, see Ben-David et al. (2016), Brown et al. (2016), and Israeli et al. (2017). The second group studies the predictable change in the correlation between the behavior of stocks A and B after stock A gets added to an index that stock B already belongs to. For instance, Barberis et al. (2005) shows that a stock s beta with the S&P 500 jumps sharply after it gets added to the index. For other well-known examples, see Greenwood and Thesmar (2011), Vayanos and Woolley (2013), and Anton and Polk (2014). In contrast to these papers, we focus on the unpredictable consequences of stock A s index inclusion for an unrelated stock Z as highlighted in Figure 1. Thresholds. And, this explanation implies that noise can be an externality imposed by funds following many different threshold-based rebalancing rules, which further connects our analysis to work studying simple financial decision rules. While we focus on ETFs, our central insight also applies to quantitative hedge funds following strategies like value and momentum (Khandani and Lo, 2007; Lou and Polk, 2013) and pension funds with strict portfolio mandates (Pennacchi and Rastad, 2011). Many 6

7 Amplification: Comovement: Demand Noise: (this paper) A buy A buy A buy A buy B buy B sell B sell C buy C buy Z sell Z buy Figure 1: How This Paper Is Different. There is an existing literature on index-linked investing. Papers in this literature fall into two broad groups. The first group studies how supplementary trading due to index inclusion can amplify initial shocks to stock A (Row 1). The second group of papers studies how stock A s returns suddenly move a lot more like stock B s returns whenever stock A and stock B happen to belong to the same index (Row 2). By contrast, this paper focuses on the unpredictable consequences of stock A s index inclusion, not for stock A or stock B, but for a seemingly unrelated stock Z (Rows 3 and 4). of these funds use strict portfolio rules with the form: Buy the top 30% and sell the bottom 30% of stocks when sorting on X. More generally, people use threshold-based rules to make all sorts of financial decisions (Gabaix, 2014). The existing literature measures the cost of using an overly simple rule in terms of losses in expected utility. Whereas, we look at how simple decision rules can affect demand volatility. 2 Theoretical Model This section introduces a random-networks model where stocks (the nodes) are connected to one another via fund rebalancing rules (the edges). In the model, an initial shock to a randomly selected stock A has the potential to trigger a long rebalancing cascade if there are many different funds following many different rebalancing rules. The model is designed to explain why predicting the effect of a long rebalancing cascade on the demand for any particular stock Z is computationally infeasible, making the resulting demand shock seem like noise. 2.1 Model Setup Here s how the random-networks model is set up. Nodes, Edges, and Neighbors. Consider a market with S 1 stocks indexed by s {1,..., S} def = S. Each of these stocks will correspond to a node in our network. In the analysis below, we will often look at the limiting case as the market gets infinitely 7

8 large, S. We will often refer to arbitrary stocks A, B, C,..., s, s,..., Z S when describing properties of the network. In addition to the S stocks, this market also contains a large collection of funds following a diverse collection of rebalancing rules equivalently, tracking a diverse collection of benchmark indexes. These rebalancing rules will define the edges in our network. Two nodes will be connected with an edge, not if the corresponding stocks are held by the same fund, but rather if a shock to one of the stocks would cause some fund to swap out its position in that stock for a new position in the other. If stock s and stock s share an edge, then we will call them neighbors. A node with many neighbors corresponds to a stock on the cusp of many funds rebalancing thresholds. def We use N s to denote stock s s neighbors and N s = N s to denote the size of this set. All neighbors are not created equal, though. To illustrate, consider the situation where stock s and stock s are the 2001st and 2000th largest stocks in the Russell universe. On one hand, a positive shock to stock A has the potential to move it from 2001st to 2000th position in the Russell ecosystem, which would force any fund tracking the Russell 2000 to swap its existing position in stock B for a new position in stock s. But, on the other hand, if stock s dropped from 2001st to 2002nd position, then these same funds wouldn t have to change a thing. With this asymmetry in mind, we use directed edges. If a negative shock to stock s would cause some funds to sell stock s and buy stock s, then we will draw an arrow from stock s to stock s. Whereas, if a positive shock to stock s would cause some fund to buy stock s and sell stock s, then we will draw an arrow from stock s to stock s. An arrow will always point towards the stock that realizes a positive shock. This distinction divides each stock s collection of neighbors into two different subsets. If s N + s, then there is an arrow pointing from stock s to stock s. This means that either a negative shock to stock s or a positive shock to stock s would cause some fund to dump its position in stock s and build a new position in stock s. We refer to this set as stock s s positive neighbors. Conversely, if s N s, then there is an arrow pointing from stock s to stock s. This means that either a positive shock to stock s or a negative shock to stock s would cause some fund to dump its position in stock s and build a new position in stock s. We refer to this set as stock s s negative neighbors. Clearly, if stock s is a positive neighbor to stock s, then stock s is a negative neighbor to stock s. Figure 2 gives some small-scale examples of this network structure. Degree Distribution. We want our random-networks model to capture the idea that 8

9 B C A N + A N A N A Figure 2: Nodes, Edges, and Neighbors. Nodes denote the same 3 stocks (A, B, and C) in various network configurations. Table reports number of neighbors for stock A in each configuration. There is an arrow from stock A to stock B if a negative shock to stock A would result in a positive shock to stock B or a positive shock to stock B would result in a negative shock to stock A. If there is an arrow from stock B to stock A, then stock B is a positive neighbors to stock A, B N + A. If there is an arrow from stock A to stock B, then stock B is a negative neighbor to stock A, B N A. long rebalancing cascades are possible if there are many funds following many different trading strategies. In this setting, nodes in the network will be densely connected to one another via a large number of edges. Ideally, there would be a single parameter that controls how dense these connections are i.e., a single parameter that controls how likely it is that two stocks are on the cusp of some fund s rebalancing rule. A natural way to do this is to study a family of random networks whose edges have been assigned according to some probabilistic rule. Then, a single parameter can control the likelihood that an edge connects any randomly selected ordered pair of stocks. We will assume that i) there is an arrow from stock s to stock s with probability κ/s for some parameter 0 κ < log(s) and that ii) the presence or absence of an arrow between any two stocks will be independent of the presence or absence of any other arrow. Given these two assumptions, the number of positive and negative neighbors for each stock s will obey a Poisson distribution as S : N ± s Poisson(κ) (1) See Appendix A for details. Again, this statistical rule is helpful because it means that market connectedness can be summarized using a single parameter: κ = E[N ± s ] (2) If κ 0, then the market is fragmented; the typical stock will tend to have very few neighbors. By contrast, if κ 0, then the market is densely connected. State Variables. We ve just seen how stocks in the model are connected to one 9

10 another via a network formed by various funds respective rebalancing rules. But, what we really want to know is: How do these connections propagate shocks through the market from one stock to the next? So, we need to introduce a set of state variables to keep track of how each stock has been affected by the network. X A s (t) { 1, 0, + 1} denote the current state of stock s following an initial shock to stock A at time t = 1. If X A s (t) = +1, then a rebalancing cascade has caused some fund to build a new position in stock s. Whereas, if X A s (t) = 1, then the opposite outcome has taken place, and a rebalancing cascade has caused some fund to exit its position in stock s. We will use the convention that all S stocks start out in their original state at time t = 0: Let X A s (0) = 0 for all s {1,..., S} (3) Then, at time t = 1, we assume that nature randomly selects one of the S stocks in the market and changes its state. We will refer to this randomly selected initial stock as stock A and assume that the initial shock is positive without loss of generality: X A A (1) = X A A (1) X A A (0) = +1 (4) Updating Rule. To see whether the state of stock s at time t will be affected by a rebalancing cascade that started with an initial shock to stock A at time t = 1, we first identify the subset of stock s s neighbors which realized shocks with the appropriate sign at time (t 1): U + def A s (t) = { s N s + X A s (t 1) < 0, s / U A s (t 1) } (5a) U def A s (t) = { s Ns X A s (t 1) > 0, s / U + A s (t 1) } (5b) Recall that a fund s rebalancing decision will have equal-buy-opposite effects on the stock being added to and the stock being removed from its portfolio. So, U + A s (t) represents the set of positive neighbors to stock s which realized negative shocks at time (t 1); whereas, U A s (t) represents the set of negative neighbors to stock s which realized positive shocks at time (t 1). The additional restrictions that s / U A s (t 1) and s / U + A s (t 1) in Equations (5a) and (5b) prevent a time (t 2) shock to stock s from affecting stock s at time (t 1) and then immediately rebounding back again to affect stock s at time t. Without this additional restriction, a rebalancing cascade would not have a well-defined direction. Finally, we use U A s (t) def = U + A s (t) U A s (t) to denote the combined set containing both kinds of recently updated neighbors for stock s at time t and assume U A s (1) = U A s (0) = 0 for all s S as initial conditions. 10

11 Then, following an initial shock to stock A at time t = 1, we compute the state of each stock s at times t 2 as follows: U A s (t) def = sign [ s U A s (t) X A s (t 1) ] (6a) X A s (t) = sign[x A s (t 1) U A s (t)] (6b) The state of stock s can change in 3 different ways at time t in response to an initial shock to stock A at time t = 1. First, stock s can realize a negative shock, X A s (t) 0, if most of its recently updated neighbors just realized positive shocks, U A s (t) = +1. This is what happens when several of stock s s neighbors get added to various benchmark indexes at stock s s expense. Second, stock s can realize a positive shock, X A s (t) 0, if most of its recently updated neighbors just realized negative shocks, U A s (t) = 1. This is what happens when stock s gets added to several funds benchmark indexes. And third, stock s might realize no shock at all, X A s (t) = 0, if the positive and negative shocks to its recently updated neighbors exactly cancel each other out, U A s (t) = 0. Color Coding. If stock s N s +, then we will draw an arrow from stock s to stock s in our diagrams, s s. But, there are two ways that this edge could transmit a shock. Stock s could be positively affected by a negative shock to stock s, or stock s could be negatively affected by a positive shock to stock s. We use colors to distinguish between these two possibilities, using the convention that edges in a cascade will always have a different color than the stock that initiated the rebalancing decision. We will represent cases where a negative shock to stock s causes some fund to sell stock s and buy stock s as s s. By contrast, we will represent cases where a positive shock to stock s causes the same fund to buy stock s and sell stock s as s s. Discussion. At this point, we re done describing the model s setup. But, before moving on to the model s analysis, it s worth pausing for a moment to talk about two important details. The first is that the model doesn t include a constrained optimization problem, which is unusual for theoretical models in economics. Economists usually apply network theory by adding a network structure to an existing assetpricing model and showing that this additional structure distorts how agents were previously solving their original constrained optimization problem (e.g., see Duffie et al., 2009; Ozsoylev and Walden, 2011; Atkeson et al., 2015). But, we re trying to make a different kind of point. We re trying to show that long rebalancing cascades can qualitatively change the 11

12 A B C A B C A B C A B C A B C X A A (t) X B A (t) X C A (t) t = 0 t = 1 t = 2 t = 3 t = Figure 3: Feedback Effects. An example rebalancing cascade on a network with 3 stocks. An arrow from stock B to stock A means that stock B is positive neighbor for stock A, B N + A. A node is colored blue if X A s(t) = +1; it s colored red if X A s (t) = 1; and, it s colored black if X s,a (t) = 0. At time t = 0, all 3 stocks start out with unchanged fundamentals, X A s (0) = 0 for all s {A, B, C}. At time t = 1, stock A realizes a positive shock, X A A (1) = +1, which is denotes by a blue star. This initial shock causes a fund to rebalance its position at time t = 2, buying stock A and selling stock B so that X A B (2) = 1. This secondary shock causes another fund to rebalance its position at time t = 3, selling stock B and buying stock C so that X A C (3) = +1. Finally, this tertiary shock causes one last fund to rebalance its position at time t = 4, buying stock C and selling stock A so that X A A (4) = 1. kind of constrained optimization problem that agents have to solve by introducing noise. More is different. And, the goal of the model is to show why. How is it that long cascades are able to turn simple deterministic rebalancing rules into apparently random demand shocks? This is a question about how stocks in the model are connected to one another not about any particular optimization problem agents are solving. So, we ve structured our model accordingly. The second important detail is that our model allows for feedback effects as illustrated in Figure 3. An initial positive shock to stock A might force a size-based fund to buy stock A and sell stock B, which might force a momentum-based fund to sell stock B and buy stock C, which might force a volatility-based fund to buy stock C and sell stock A, bringing the cascade full circle. Feedback effects are going to play a central role in the analysis below. But, it is crucial to point out that this loop stops once it returns to stock A. The updating rule in Equation (6a) implies that stock B s state will remain unchanged so long as X A A (t) = 0, which in turn implies that a rebalancing cascade can last no more than T def = (S + 1) periods. We ve made infinite feedback loops off limits to make our theoretical analysis more straightforward. But, rebalancing cascades would 12

13 clearly be more complex if they contained infinite feedback loops. So, if anything, this assumption that we ve made for analytical tractability works against us. 2.2 Likelihood We want to show that, while it s possible to compute which stocks are most likely to be hit by a long rebalancing cascade, it s computationally infeasible to predict the direction of this impact. So, in this subsection, we start by characterizing the probability that a particular stock will be hit by a long rebalancing cascade. Cascade Inclusion. Let I A s denote an indicator variable for whether the state of stock s S would ever be affected by a rebalancing cascade that starts with stock A: def ] I A s = max (7) t T [ 1{ XA s(t) 0} If I A s = 1, then we will say that stock s was included in the cascade started by an initial shock to stock A. Whereas, if I A s = 0, then the state of stock s would remain at X A s (t) = 0 for the entire duration of a rebalancing cascade starting with stock A. Note that stock A, which realized the initial shock, is always included in any resulting cascade, I A A = 1, since X A A (1) = +1 by construction. And, this fact remains true even if the state of stock A eventually returns to X A A (T ) = 0 due to some later feedback effect, just as we saw happen in Figure 3. We then use the variable L A [1, S] to denote the length of the rebalancing cascade that would be triggered by an initial shock to a randomly selected stock A at time t = 1: def L A = S s=1 I A s (8) The average rebalancing cascade has to have a length of at least E[L A ] = 1, and this can only occur if initial shocks never ever cause any funds to rebalance. Similarly, the average rebalancing cascade can have a length of no more than E[L A ] = S, and this can only occur if an initial shock to an stock always results in a rebalancing cascade that includes every stock in the market. Percolation Threshold. The length of the typical rebalancing cascade that would be set off by an initial shock to a randomly selected stock A changes dramatically as market connectivity crosses a critical threshold of κ = 1. This point is called a percolation threshold in the random-networks literature (Bollobás, 2001). Proposition 2.2 (Percolation Threshold). The expected cascade length in a large 13

14 20 E[L s ] κ λ κ Figure 4: Percolation Threshold, Below. x-axis: expected number of positive/negative neighbors for each stock in the network, κ. y- axis: expected cascade length, E[L s ], as given by Equation (9). Dashed line: percolation threshold at κ = 1. : sample averages in 1000 simulations of a random network with 5000 nodes. Figure 5: Percolation Threshold, Above. x-axis: expected number of positive/negative neighbors for each stock, κ. y-axis: expected fraction of stocks included in a cascade, λ, as given by solution to Equation (10). Dashed line: κ = 1. : sample averages in 1000 simulations of a random network with 5000 nodes. market is given by lim E[L A] = S 1 1 κ if κ [0, 1) if κ > 1 (9) Let λ def = lim S E[L A /S] denote the fraction of stocks involved in an average cascade. If κ > 1, then λ satisfies the equation λ = 1 e κ λ (10) Figure 4 plots the expected cascade length, E[L A ], as a function of overall market connectivity, κ. It shows that increasing the number of neighbors per stock increases the expected cascade length. If the typical stock has less than 2 neighbors, one positive and one negative, then at some point this chain reaction will peter out i.e., cascades will have finite length. This is the region in Figure 4 to the left of κ = 1. If the expected cascade length is finite, then each individual rebalancing cascade will include an infinitesimal fraction of the countably infinite number of stocks in the market as S. This corresponds to the line at λ = 0 for all κ < 1 in Figure 5. But, as soon as the expected cascade length becomes infinite for κ > 1, the typical rebalancing cascade will then impact a finite fraction of this infinitely large market. And, as a result, Figure 5 shows that λ > 0 as soon as as κ > 1. Likelihood of Impact. We can use the result in Proposition 2.2 to make predictions about which stocks are most likely to be hit by a long rebalancing cascade. Bayes rule tells us that, conditional on an overall level of market connectivity, the probability 14

15 stock s with N s ± neighbors will be involved in a cascade is given by: Pr[I A s = 1 N ± s, κ] Pr[N ± s I A s = 1, κ] E[I A s κ] (11) This brings us to the following corollary, which states that stocks with more neighbors are more likely to be impacted by a long rebalancing cascade. Corollary 2.2 (Likelihood of Impact). The probability that a randomly selected stock s will be included in a rebalancing cascade that starts with an initial shock to a randomly selected stock A at time t = 1 is increasing in its number of positive/negative neighbors: N ± s E[I A s N ± s, κ] > 0 (12) Thus, to sum up, long rebalancing cascades occur in densely connected markets with κ > 1, and stocks with more neighbors are more likely to be affected by these cascades. 2.3 Direction We ve seen how to predict which stocks are more likely to be hit by long rebalancing cascades. So, now let s investigate why it s so hard to predict how these long rebalancing cascades will affect the demand for each stock. Direction of Effect. Let D A s denote the final state of stock s at the conclusion of a rebalancing cascade that started with stock A: def D A s = X A s (T ) (13) Recall that the final data T = S + 1. If D A s = +1, then the net effect of the rebalancing cascade on the demand for stock s was positive. If D A s = 1, then the net effect was negative. And, if D A s = 0, then either stock s wasn t involved in the cascade or there was a feedback loop that canceled out any initial effect that the cascade had, just as diagrammed in Figure 3. Whether vs. How. Before getting to the math, let s first build some intuition for what we want to prove. Take a look at Figure 6. Both panels display cascades unfolding on 4 different networks. The nodes and edges are just as described in Subsection 2.1. The thick edges denote the rebalancing decisions that were made in response to an initial positive shock to stock A, which is represented by the large blue star. The gray shaded edges denote the potential rebalancing decisions that were not triggered by this initial shock to shock A. We are interested in how each of the 4 cascades eventually affects the demand 15

16 for stock Z, which will be represented by the color of the large square. A large blue square will denote a positive impact on stock Z, D A Z = +1; whereas, a large red square will denote a negative impact on stock Z, D A Z = 1. Panel (a, left) does not report how each edge was involved in the rebalancing cascade or how stock Z was eventually affected. Panel (b, right) does. We ve omitted the arrows representing the direction of each edge to avoid visual clutter, so now s s denotes a negative shock to stock s causing some fund to sell stock s and buy stock s while s s denotes exact same transaction triggered by a positive shock to stock s. Looking at the left panel, you can easily tell whether stock Z will be involved in each of the 4 cascades. In all but the bottom-right network, there is a clear path from the large blue star denoting stock A to the large black square denoting stock Z. But, while it s easy to trace out this path, which immediately reveals whether stock Z will be hit by the rebalancing cascade, there s no easy way to predict how stock Z will be effected by looking at the left panel. There s nothing in the left panel that would make you think that the upper-left network should result in a positive shock to stock Z, D A Z = +1. The only way to figure this out is by pressing play and seeing how the cascade unfolds. This is what we want to show mathematically. Root of the Problem. The source of the computational complexity lies in the facts that i) rebalancing cascades have an alternating buy-sell-buy-sell structure and ii) they follow threshold-based rules. These two facts imply that evaluating how a long rebalancing cascade involving L A stocks will affect the demand for stock Z is equivalent to evaluating the output of a logical circuit with L gates. Tiny changes in the fine-grained structure of a logical circuit can flip the sign of its output. So, predicting how a long rebalancing cascade will impact stock Z means keeping track of every minute detail of how the cascade unfolds. There are no shortcuts. This is the root of the problem. To illustrate, imagine that stock A realizes a positive shock at time t = 1 and there s an alternating buy-sell-buy-sell-buy path from stock A to stock Z that would result in a positive shock to stock Z, ˆDA Z = +1, if it were the only branch of the cascade. But, suppose that stock A also has another neighbor that doesn t sit on this branch, which means that there might be another way for the initial shock to propagate through the network and affect stock Z. Depending on how this second branch of the cascade unfolds, it could either reinforce or interfere with the original path and alter your original guess of D A Z = +1. This is the situation outlined in the dashed box at the top of Figure 7. 16

17 (a) Whether? (b) How? Figure 6: Whether vs. How. Both panels display the same 4 rebalancing cascades unfolding on the same collection of 4 random networks. Nodes and edges are as described in Section 2.1; however, we ve omitted the arrows representing the direction of each edge to avoid visual clutter. Thick edges denote rebalancing decisions made in response to an initial positive shock to stock A, which is represented by a large blue star. Gray shaded edges denote potential rebalancing decisions that were not involved in the rebalancing cascade. Stock Z in each network is represented by the large square. We are interested in how each rebalancing cascade will affect the demand for stock Z, and we represent this outcome with by the color of the large square in the right panel. Blue denotes a positive impact, D A Z = +1; whereas, red denotes a negative impact Z, D A Z = 1. Panel (a, left) does not report how each edge was involved in the rebalancing cascade. Panel (b, right) does. s s denotes a negative shock to stock s causing some fund to sell stock s and buy stock s; whereas, s s denotes the same event triggered by a positive shock to stock s. While it was easy to figure out whether stock Z is involved in each cascade by inspecting the left panel, there s simply no way to predict how stock Z will be effected i.e., is the large square in the right panel blue or red? without actually calculating how the entire cascade plays out. 17

18 ?? A Z Case 0: Case 1: Case 2: Case 3: Case 4: Figure 7: Root of the Problem. Dashed box shows situation where stock A, which has N + A = 2 positive neighbors, realizes a positive shock and one of those neighbors is involved in an alternating buy-sell-buy-sell-buy path from stock A to stock Z that would result in a positive shock to stock Z, ˆD A Z = +1, if it were the only branch of the cascade. Remaining panels, labeled Case 0,..., 4, show various ways that stock A s second positive neighbor might be involved in the rebalancing cascade, too. The rebalancing cascade s effect on stock Z will be positive unless this second path reconnects with the first path in exactly 2 steps. This example illustrates how, to predict how a long rebalancing cascade will affect the demand for stock Z, you have to keep track precise details about the cascade s global structure. 18

19 The remaining panels show various ways that this second branch might unfold. If the second branch of the cascade never reconnects as in Case 0, then your original guess of ˆD A Z = +1 will be correct. And, if this second branch reconnects in 1, 3, or 4 steps, then your original guess will also be correct. But, if the second branch happens to reconnect in exactly 2 steps, then your original guess will be wrong. This subtle change in the details of how the alternative branch of the cascade unfolded made the difference between the rebalancing cascade having a positive effect on stock Z and the rebalancing cascade have no effect on stock Z at all. Thus, predicting how a long rebalancing cascade will affect the demand for some stock Z is complex because you have to keep track of the cascade s global structure. To figure out how stock Z will be affected by a long rebalancing cascade, you have to check every detail of how the entire cascade unfolds. Computational Complexity. The proposition below encapsulates this basic insight. Proposition 2.3 (Computational Complexity). Suppose that there is alternating buysell-buy-sell path of rebalancing decisions from stock A to stock Z that would result in a demand shock of ˆDA Z to stock Z if it were the only branch in the cascade. If κ > 1 and at least 2 stocks on this path have 3 or more neighbors, then not only is determining whether D A Z = ˆD A Z an NP-complete problem but Pr[D A Z = d ˆD A Z ] = Pr[D A Z = d] for each d { 1, 0, 1} with high probability as κ. What this result is saying is that, in a market with many funds following many different rebalancing rules, figuring out how one particular branch of a long rebalancing cascade affects the demand for stock Z doesn t tell you anything about the net effect of the entire rebalancing cascade on the demand for stock Z. So, even if you know that stock Z will be affected by a long rebalancing cascade, you may as well just flip a coin when it comes to predicting how stock Z will be affected. Note that NP completeness is a statement about how fast the difficulty of a problem grows as its size increases. So, saying that determining whether D A Z = ˆD A Z is NP-complete problem means that this problem gets really hard really fast as the number of stocks grows large S. See Appendix A for details. Densely Connected. There are 3 important details about this result that are worth emphasizing. The first is that rebalancing cascades are only hard to predict in a densely connected market with κ > 1. If κ [0, 1), then the number of stocks in any secondary branch of the cascade that links stock A and stock Z will tend to be finite. So, the difficulty of figuring out whether this secondary chain will change 19

20 ˆD A Z will be capped as the market grows large, S. Corollary 2.3a (Densely Connected). Determining whether D A Z = ˆD A Z is not an NP-complete problem when κ [0, 1). Alternating Pattern. The second important detail is that rebalancing cascades are only hard to predict if they involve an alternating sequence of buy and sell orders. In a world where a positive shock to one stock can only ever result in a positive shock another stock, predicting how stock Z will be affected by a long rebalancing cascade starting with stock A is equivalent to predicting whether stock Z will be affected because all stocks will be affected in the exact same way. Thus, the alternating buy-sell-buy-sell nature of portfolio rebalancing rules plays a key role in the result. Corollary 2.3b (Alternating Pattern). Determining whether D A Z = ˆD A Z is not an NP-complete problem in a market with one-direction shocks. Threshold-Based Rules. The third and final important detail is that rebalancing cascades are only hard to predict if they involve threshold-based rules. To see why this is the case, just imagine changing the updating rule defined in Equation (6) to something that did not involve a threshold. For example, suppose that the update to stock s at time t is the average the shocks to its recently updated neighbors and this update altered the state of stock s by θ (0, 1): Ũ A s (t) def 1 = s U A s (t) U A s (t) X A s (t 1) (14a) X A s (t) = θ { X A s (t 1) ŨA s(t) } (14b) In this sort of market, longer chains will have smaller effects on the demand for stock Z in the same way that an AR(1) process s impulse-response function will be weaker at longer horizons. As a result, you could get a pretty good estimate of how a long rebalancing cascade starting with stock A would affect the demand for stock Z by checking the directions of the shortest paths from stock A to stock Z. Corollary 2.3c (Threshold-Based Rules). Determining whether D A Z = ˆD A Z is not an NP-complete problem in a market without threshold-based rebalancing rules. 3 Rebalancing Cascades The previous section explained how a long cascade might theoretically be able to transform a collection of simple deterministic rebalancing rules into random demand 20

21 shocks. This section gives empirical evidence that the transformation actually does take place in real-world financial markets. To do this, we focus on a particular group of funds following threshold-based strategies and study the rebalancing cascades that emerge following a set of initial shocks. Then, we show that, while it s possible to compute which stocks are most likely to be hit by each cascade, it s not possible to predict the direction of the resulting demand shock. 3.1 Group of Funds We choose exchange-traded funds (ETFs) as our particular group of funds following threshold-based strategies. Benchmark Variety. There are 3 reasons for this choice. The first is that we need a large group of funds following a very heterogeneous collection of trading strategies. Prior to January 2008, ETFs all looked like the SPDR S&P 500 ETF [SPY] in that they all tracked some sort of pre-existing market index, like the S&P 500. But, in early 2008, the SEC changed its guidelines so that an ETF could track its own selfdefined benchmark. After this change, Invesco PowerShares was free to create an ETF tracking the returns of the quintile of S&P 500 stocks with the lowest historical volatility even though there was no pre-existing low-volatility S&P 500 index. All Invesco had to do was promise to announce the identities and weights involved in the benchmark one day in advance. Now, there are more ETFs than stocks. 5 From ProShares we have CLIX (100% long internet retailers and 50% short bricks-and-mortar U.S. retailers) and EMTY (which just bets against bricks-and-mortar retailers)... meanwhile from EventShares, we have policy-factor ETFs... like... GOP (full of oil drillers, gun manufacturers, and so on that would benefit from Republican policies) and DEMS (with companies that should do well under Democrats, such as clean-energy companies). There is also an ETF called TAXR that invests in companies poised to benefit most from a successful attempt to pass a tax reform bill. 6 The sheer number and variety of these so-called smart-beta ETFs has become something of a hot-button issue of late. To be sure, niche funds tend to be smaller. But, even the rebalancing activity of small ETFs can affect a stock s fundamentals because ETFs do all of their rebalancing during the final 20 to 30 minutes of the trading day. Manager Discretion. The second reason for this choice is that ETF managers have 5 Bloomberg. 5/16/2017. Mutual Funds Ate the Stock Market. Now ETFs Are Doing It. 6 Financial Times. 11/21/2017. A ROSE by any other ticker symbol... 21

22 less ability to deviate from their stated benchmark than mutual-fund or hedge-fund managers. This is due to the underlying structure of the ETF market (Madhavan, 2016; Ben-David et al., 2017). The company running an ETF (a.k.a., its sponsor ) has an obligation to create or redeem shares of the ETF at the end-of-day market value of its stated benchmark. If an ETF s price is higher than the end-of-day market value of its benchmark, then an arbitrageur can sell shares of the ETF back to its sponsor and use the proceeds to buy shares of the underlying assets in the benchmark index. Conversely, if an ETF s price is lower than the end-of-day market value of its benchmark, then an arbitrageur can sell shares of the assets in the benchmark and use the proceeds to buy shares of the ETF from its sponsor. Thus, ETF managers will always hold a basket of securities that closely mirrors the end-of-day market value of their stated benchmark. Following this logic through to it s natural conclusion, if arbitrageurs are constantly asking an ETF sponsor to create or redeem lots of shares, then the sponsor must be losing lots of money. So, just like you d expect, creations and redemptions are only a small fraction of daily trading volume for ETFs, and these trades involve less than 0.5% percent of ETFs net assets (Investment Company Institute, 2015). Instead, ETF trading volume primarily comes from managers rebalancing activity just prior to market close. This end-of-day trading is how ETF sponsors make sure that there is very little difference between the market value of their end-of-day holdings and the market value of their stated benchmark. An ETF manager who does the bulk of his rebalancing right at market close will incur higher trading costs. But, the typical investor in a smart-beta ETF is not looking for a cheap way to buy and hold a broad market portfolio. ETF investors traded $20 trillion worth of shares last year even though ETFs only have $2.5 trillion in assets. That s 800% asset turnover, which is about 3-times more than stocks. 7 An investor interested in holding a smart-beta ETF is looking for quick access to a very targeted position. He d rather the ETF manager have slightly higher trading costs and be much more faithful to his stated benchmark. For a niche ETF, the additional trading costs incurred by the end-of-day trading are nothing compared to the costs associated with replicating the entire position from scratch. Data Availability. The third and final reason for choosing ETFs is data availability. Other papers in the ETF literature, such as Ben-David et al. (2016), impute daily portfolio positions from end-of-quarter financial statements. But, we are interested 7 Bloomberg. 3/3/ Ways Passive Investing Is Actually Quite Active. 22