Cross-Sectional Variation of Intraday Liquidity, Cross-Impact, and their Effect on Portfolio Execution

Transcription

1 Cross-Sectional Variation o Intraday Liquity, Cross-Impact, and their Eect on Portolio Execution Seungki Min Costis Maglaras Ciamac C. Moallemi Initial Version: July 2017; December 2017 Current Revision: November 13, 2018 Abstract The composition o natural liquity has been changing over time. An analysis o intraday volumes or the S&P500 constituent stocks illustrates that i volume surprises, i.e., deviations rom their respective orecasts, are correlated across stocks, and ii this correlation increases during the last ew hours o the trading session. These observations could be attributed, in part, to the prevalence o portolio trading activity that is implicit in the growth o ETF, passive and systematic investment strategies; and, to the increased trading intensity o such strategies towards the end o the trading session, e.g., due to execution o mutual und inlows/outlows that are benchmarked to the closing price on each day. In this paper, we investigate the consequences o such portolio liquity on price impact and portolio execution. e derive a linear cross-asset market impact rom a stylized model that explicitly captures the act that a certain raction o natural liquity provers only trade portolios o stocks whenever they choose to execute. e ind that due to cross-impact and its intraday variation, it is optimal or a risk-neutral, cost minimizing liquator to execute a portolio o orders in a coupled manner, as opposed to a separable VAP-like execution that is oten assumed. The optimal schedule couples the execution o the various orders so as to be able to take advantage o increased portolio liquity towards the end o the day. A worst case analysis shows that the potential cost reduction rom this optimized execution schedule over the separable approach can be as high as 6% or plausible model parameters. Finally, we discuss how to estimate cross-sectional price impact i one had a dataset o realized portolio transaction records that exploits the low-rank structure o its coeicient matrix suggested by our analysis. 1. Introduction Throughout the past decade or so we have experienced a so-called movement o assets under management in the equities markets rom actively managed to passively and systematically managed strategies. This migration o assets has also been accompanied by the simultaneous growth o Exchange-Traded-Funds ETFs. In very broad strokes, such strategies tend to make investment and trade decisions based on systematic portolio-level procedures e.g., invest in all S&P500 constituents proportionally to their respective market capitalization weights; invest in low-volatility stocks; high-beta stocks; high divend stocks; etc. In contrast, active strategies, or example, may ocus on undamental analysis o indivual irms that may, in turn, result into discretionary investment decisions on the respective stocks. In the sequel, we will reer to passive strategies as index und strategies. Graduate School o Business, Columbia University; {smin20, c.maglaras, ciamac}@gsb.columbia.edu 1

2 This gradual shit in investment styles has aected the nature o trade order lows, which motivates our subsequent analysis. e make three speciic observations. First, passive and systematic strategies tend to generate portolio trade order lows, i.e., trades that simultaneously execute orders in multiple securities in a coordinated ashion; e.g., buying a $50 million slice o the S&P500 over the next 2 hrs that involves the simultaneous execution o buy orders along most or all o the index constituents. Second, passive strategies tend to concentrate their trading activity towards the end o the day; in part, so as to ocus around times with increased market liquity, and because mutual unds that implement such strategies have to settle buy and sell trade instructions rom their retail investors at the closing market price at the end o each day; ETF products exhibit similar behavior. Third, the shit in asset ownership over time and the changes in the regulatory environment, have, in turn, changed the composition and strategies under which natural liquity is proved in the market these are the counterparties that step in to either sell o buy stock against institutional investor interest so as to clear the market. In 2 we will prove some empirical evence that show that pairwise correlations amongst trading volumes across the S&P500 constituents are positive throughout the trading day, and increase by about a actor o two over the last 1-2 hrs o the trading day. That is, trading volumes exhibit common intraday variation away rom their deterministic orecast in a way that is consistent with our earlier observations. In this paper we study the eect o portolio liquity provision in the context o optimal trade execution. Speciically, we conser a stylized model o natural liquity provision that incorporates the behavior o single stock and portolio participants, and, in turn, leads to a market impact model that incorporates cross-security impact terms; these arise due to the participation o natural portolio liquity provers. e ormulate and solve a multi-period optimization problem to minimize the expected market impact cost incurred by a risk-neutral investor that seeks to liquate a portolio. e characterize the optimal policy, which is coupled i.e., the liquation schedules or the various orders in the portolio should be jointly determined so as to incorporate and exploit the cross-security impact phenomena. e contrast the optimal schedule against that o a separable execution approach, where the orders in the portolio are executed independently o each other; this is commonly adopted by risk-neutral investors. Separable execution is suboptimal, in general, and we derive a bound on it sub-optimality gap when compared to the optimal coupled solution, which can be interpreted as the execution cost reduction that an investor can achieve by optimizing around such cross-impact eects. In a bit more detail, the main contributions o the paper are the ollowing. Stylized model o cross-sectional price impact: Under the assumption that the magnitude o single stock and portolio liquity provision is linear in the change in short-term trading prices, we show that market impact is itsel linear in the trade quantity vector, and characterize the coeicient matrix that exhibits an intuitive structure: it is the inverse o a matrix that is a diagonal capturing the eect o single-stock liquity provers plus a non-diagonal low-rank matrix 2

3 capturing the eect o portolio liquity provers that are assumed to trade along a set o portolio weight vectors, such as the market and sector portolios. Cross-impact is the result o portolio liquity provision. The linear market impact model results in quadratic trading costs, which will allow or a tractable downstream analysis. The derivation o this model suggests that cross-impact eects will also arise in settings where liquity provision ollows more complex and possibly non-linear strategies, as long as a portion o that liquity is proved by portolio investors. Optimal trade scheduling or risk-neutral minimum cost liquation: e ormulate and solve an optimal multi-period optimization problem that selects the quantities to be traded in each security over time so as to liquate the target portolio over the span o a inite horizon a day in our case in a way that minimizes the cumulative expected market impact costs. Coupling is not the result o a risk penalty that captures the covariance o intraday price returns, as is typically the case, but the result o correlated liquity. The optimal trade schedule is coupled, and, speciically, incorporates and exploits the presence and intraday variation o cross-impact eects. e entiy the special cases where a separable execution approach would be optimal, namely: a i there was no portolio liquity provision, or b the intensity o portolio liquity provision was constant throughout the trading day. orst case analysis: e compare the optimal policy against a separable VAP-like execution approach, and characterize the worst case liquation portolios and the magnitude o the beneit that one derives rom the optimized solution. A straightorward estimation o the mixture o single-stock and portolio liquity provers that would be consistent with the intraday volume proile and the intraday proile o pairwise volume correlations, allows us to back into a numerical value or the aorementioned bound, which is 6%. The worst case analysis proves some intuition on the settings where this eects may be more pronounced. Eicient estimation o cross-impact: e suggest a practical scheme or estimating the time-varying coeicient matrix or price impact. A direct estimation procedure or all crossimpact coeicients between each pair o stocks seems intractable due to the low signal-to-noise ratio that oten characterizes market impact model estimation, and the increased sparsity o trade data when we study pairs o stocks. Exploiting the low rank structure o our stylized impact model derived above, we propose a procedure that only involves the estimation o a ew parameters, e.g., one parameter per sector. e do not calibrate the cross-impact model, as this typically requires access to proprietary trading inormation, but we speciy a tractable maximum likelihood procedure that one could make use Literature Survey One set o papers that is related with our work ocuses on optimal trade scheduling, where the investor consers a dynamic control problem or how to split the liquation o a large order over 3

4 a predetermined time horizon so as to optimize some perormance criterion. Bertsimas and Lo 1998 solved this problem in the context o minimizing the expected market impact cost, and Almgren and Chriss 2000 extended the analysis to the mean-variance criterion; see also Almgren 2003 and Huberman and Stanzl Bertsimas and Lo 1998 shows that the cost minimizing solution under a linear impact model schedules each order in proportion to the stock s orecasted volume proile. In these papers, multiple-security trading is shortly discussed as an extension o single-stock execution, and the similar setup can be ound in recent studies e.g., Brown et al. 2010, Haugh and ang A separate strand o work, which includes Obizhaeva and ang 2013, Rosu 2009 and Alonsi et al. 2010, treat the market as one limit order book and use an aggregated and stylized model o market impact to capture how the price moves as a unction o trading intensity. Tsoukalas et al builds on Obizhaeva and ang 2013 to conser a portolio liquation problem incorporating risk and cross-impact eects that shit the b/ask price levels across limit order books in a couple way. Finally, closest to our paper is the recent work Mastromatteo et al that looks at portolio execution with a linear cost model with crossimpact terms; their analysis predicates that the portolio impact matrix has the same eigenvectors as the return correlation matrix, and is stationary. The problem structure allows or their model to be estimable in a way similar to what we suggest in our paper, and the stationary model leads to a separable optimal trading schedule, which agrees with our results in that special case. Our stylized derivation o a price impact model, uses eas rom the market microstructure literature. Speciically, as in Kyle 1985, we assume that each investor s holding position on a particular security changes linearly with price, usually justiied under a CARA utility unction; the price is determined through a market clearing equilibrium condition among all participants. e don t explicitly speciy the trading volume generating process in this paper, the literature o sequential inormation arrival Copeland 1976, Jennings et al and Tauchen and Pitts 1983 proves an insightul connection between trading volume, return volatility and liquity. hile we will not estimate an impact model in this paper, we briely discuss in the last section how one would go about doing so given a set o proprietary portolio executions. e reer to Almgren et al or a procedure to estimate an impact model that allows or a linear permanent component and a possibly non-linear instantaneous component without cross-impact eects. Huberman and Stanzl 2004 showed using a no-arbitrage argument that the permanent price impact must be a linear unction o the quantity traded; see also Gatheral 2010 or an extension o that argument to a setting where market impact is transient with a speciic decay unction. Rashkovich and Verma 2012 make some interesting practical remarks in relation to this market impact model estimation procedure. The topic o cross-impact has recently started to be explored, speciically in Benzaquen et al and Schneer and Lillo The irst paper postulates and estimates a linear propagator impact model based on the trade sign imbalance vector in each period. The second paper explores the implications o the no arbitrage ea o Huberman and Stanzl 2004 to the structure and 4

5 magnitude o cross-impact. An important motivation o our work is the gradual shit o assets under management rom active to passive and systematic strategies, and their implication to market behavior and the composition and timing o trading lows. This topic has been studied in the inancial econometrics literature and is summarized in Ben-Dav et al In particular, ocusing on the topic o liquity, which is our main concern, this literature has ound a causal relationship between ETF or mutual und ownership and the commonality in the liquity o the underlying constituents, e.g., see Karoli et al. 2012, Koch et al. 2016, Agarwal et al. 2018; the motivation o that cross-sectional dependency is attributed to the arbitrage mechanism o ETFs or the correlated trading o mutual unds Organization o Paper The remainder o the paper is organized as ollows. 2 proves some empirical evence o the cross-sectional variation o intraday liquity. 3 derives the unctional orm o a price impact model that incorporates cross-security impact terms that arise rom the presence o portolio index-und liquity provers. It subsequently characterizes the expected execution cost given a portolio trade schedule. 4 ormulates and solves an optimal portolio execution problem or a risk-neutral investor, 5 characterizes the perormance gains rom the optimal schedule over a separable VAPlike execution that is oten used in such a risk-neutral setting. e conclude in 6 with a brie overview o how to estimate price impact rom a set o proprietary record o portolio transactions, and discuss some additional practical conserations. 2. Preliminary Empirical Observations To motivate our downstream analysis, we prove some empirical evence regarding the crosssectional behavior o intraday trading volume, ocusing on the level and intraday variation o the pairwise correlations among trading volumes o the S&P 500 constituent stocks. e analyzed 459 stocks N = 459, indexed by i, that had been constituents o S&P500 throughout the calendar year o Our dataset contains 241 days D = 241 indexed by d, excluding days that are known to exhibit abnormal trading activity, namely: the FOMC/FED announcement days on 02/01, 03/15, 05/03, 06/14, 07/26, 09/20, 11/01, 12/13, and the hal trading days on 07/03, 11/24. e use a Trade-And-Quote TAQ database, and extract all trades, excluding those that a occur beore 09:35 or ater 16:00; b opening auction prints or closing auction prints COND ield contains O, Q, M, or 6 ; and c trades corrected later CORR ield is not 0, or COND ield contains G or Z. e dive a day into ive-minute intervals T = 77, 09:35-09:40, 15:55-16:00 indexed by t. e denote by DVol t the aggregate notional $ volume traded on stock i across all 1 The concentration o trading lows towards the end o the trading day has been a popular topic in the inancial press; see, e.g., Driebusch et al

6 transactions that took place in time interval t on day d. e deine DVol it to be the yearly average notional volume traded on stock i in time period t, and AvgVolAlloc t to be the cross-sectional average % o daily volume traded in period t daily volume in this deinition accounts or all trading activity between 9:35 and 16:00, excluding auction and corrected prints: DVol it 1 D DVol t, D d=1 VolAlloc it DVol it Ts=1 DVol is and AvgVolAlloc t 1 N N VolAlloc it. 1 i=1 For each pair o stocks i, j we denote by Correl ijt the pairwise correlation between the respective intraday notional traded volumes across days or each time period t. As a measure o crosssectional dependency, we subsequently calculate the average pairwise correlation over all pairs o stocks: Correl ijt AvgCorrel t Dd=1 DVol t DVol it DVol jdt DVol jt Dd=1 DVol t DVol it 2 D d=1 DVol jdt DVol jt 2, 2 1 NN 1 Correl ijt. 3 i j Figure 1: Cross-sectional average intraday traded volume proile let and cross-sectional average pairwise correlation right: S&P 500 constituent stocks in Figure 1 depicts the graphs o AvgVolAlloc t and AvgCorrel t. AvgVolAlloc t exhibits the commonly observed U-shaped behavior that shows that trading activity is concentrated in the morning and the end o the day. The graph o AvgCorrel t reveals that i trading volumes are positively correlated throughout a day, and ii that the cross-sectional average pairwise correlation increases signiicantly during the last ew hours o day. 2 2 Alternative calculations o the intraday volume and correlation patterns produce similar indings. For example, one could compute stock speciic average traded volume proiles, and or each day compute the stock-speciic normalized volume deviation proiles between the realized and orecasted volume proiles; these could be used or the pairwise correlation analysis. Similar indings are obtained when we study stocks clustered by their sector, e.g., 6

7 One possible explanation o the observed intraday volume correlation proile could be the nonstationary participation o portolio order low throughout the course o the trading session. Market participants that trade portolio order low cause correlated stochastic volume deviations across stocks that, in turn, could contribute to the observed pairwise correlation proile. Interpreting portolio order lows as the primary source o cross-sectional dependency in trading volume, AvgCorrel t indirectly relects the intensity o portolio order low within the total market order low. Our empirical observation indicates that i portolio order low contributes a certain raction o trading activity throughout the day, which ii is increasing towards the end o the day. In particular, with the increasing popularity o ETFs and passive unds in recent years, people now trade similar portolios which may incur stronger cross-sectional dependency; Karoli et al. 2012, Koch et al and Agarwal et al prove empirical evence showing that the commonality in trading volume indeed arises rom the trading activity in ETFs or passive unds. Similarly, transactions to buy or sell shares into mutual unds are settled to the closing prices, and mutual und companies tend to execute the net inlows or outlows near or at the end o the trading session. e will return to these indings about AvgVolAlloc t and AvgCorrel t in 5 to approximate the relative magnitude o the dierent type o natural liquity provers portolio vs. single stock investors, and characterize its eect o incorporating this phenomenon on the optimal execution schedule and execution costs. 3. Model e assume that there are two types o investors single-stock and index-und investors who prove natural liquity in the market. In this section, we derive the cross-sectional market impact model rom a stylized assumption on liquity provision mechanism o these investors. The term single stock here reers to discretionary or active investors that are willing to supply liquity on indivual securities Single-stock Investors and Index-und Investors Single-stock discretionary investors are assumed to trade and prove opportunistic liquity on indivual stocks by adjusting their holdings in response to changes in the undamental price o the stock. This change in single stock investor holdings in stock i is assumed to be linear in the change in the market price with a coeicient ψ,i. They will sell or buy ψ,i shares o stock i when its price p i rises or drops by one dollar. The assumption about a linear supply relationship between holdings and price is oten assumed in the market microstructure literature Tauchen and Pitts 1983, Kyle It is typically justiied under the assumption that risk averse investors choose their holdings to maximize their among inancial, energy, manuacturing, etc., stock sub-universes. 7

8 expected utility given their own belie on the uture price. ith a CARA utility unction and normally distributed belies, the optimal holding position is proportional to the gap between current price and their own reservation price, with a proportionality coeicient that incorporates their conence in their belie and their preerence regarding uncertainty. Our parameter ψ,i can be thought as a sum o the indivual investors sensitivity parameters. e conser a universe o N stocks, indexed by i = 1,..., N. Suppose that the change in the N-dimensional price vector is p R N. Let e i be the i th unit vector. Single-stock investors on stock i will experience the price change e i p and adjust their holding position by ψ,i e i p. In vector representation, the change in the holding vector o single-stock investors h R N can be written as N h p = e i ψ,i e i p = Ψ p R N, 4 i=1 where Ψ diag ψ,1,, ψ,n R N N. h p can be thought as signed -volume; i.e., it is positive when orders to buy are submitted in the market when the prices drop, and negative when orders to sell are submitted in the market when prices increase. In contrast to single-stock investors, index-und investors trade portolios based on some view on the entire market, a sector, or a particular group o securities such as high-beta stocks. This includes many institutional investors, but the indivual investors who hold ETFs or join index unds also belong to this group. e assume that there are K such unds, indexed by k = 1,..., K. Let w k = w k1,, w kn R N be the weight vector o index und k, expressed in # o shares: one unit o index und k contains w k1 shares o stock 1, w k2 shares o stock 2, and so on. Given a price change p R N, investors on index und k will experience the price change wk p. Analogous to single-stock investors, index-und investors adjust their holding position on index und k linearly to its price change wk p with a coeicient ψ,k. Since trading one unit o index und k is equivalent to trading a basket o indivual stocks with weight vector w k, we can state the change in index-und investors holding position vector h R N as a vector o changes in the constituents o that und: where h p = K w k ψ,k wk p = Ψ p R N, 5 k=1 w 1 w K RN K, Ψ diag ψ,1,, ψ,k R K K. 6 Throughout the paper, we assume that all ψ,i s and ψ,k s are strictly positive, and that w k s are linearly independent. 8

9 3.2. Cross-sectional Price Impact Conser a time period over which we wish to execute v R N shares. Each component can be positive or negative depending on whether we want to buy or sell. Our orders transact eventually against natural liquity provers proved by single-stock and index-und investors; market makers or high requency traders tend to maintain negligible inventories at the end o the day, so irrespective o who intermediates the market, we need to elicit v shares rom those two groups o investors. A price change o p R N will aect an inventory change o v shares i the ollowing market clearing condition is satisied: Based on equations 4 and 5, v + h p + h p = 0. 7 N K v = e i ψ,i e i + w k ψ,k wk p = i=1 k=1 Ψ + Ψ p. 8 The expression means that, out o v shares, Ψ p R N shares are obtained rom single-stock investors and Ψ p R N shares rom index-und investors. This linear relationship between v and p can be translated into the price impact summarized in the next proposition. Proposition 1 Cross-sectional price impact. hen executing v R N shares, the market clearing price change vector p R N is such that p = Gv and G Ψ + Ψ 1. 9 Note that the coeicient matrix G is an inverse o Ψ + Ψ which is composed o two symmetric and strictly positive-deinite matrices. Thereore, G is itsel a well-deined symmetric positive-deinite matrix, with the ollowing structure: a diagonal matrix plus a non-diagonal lowrank matrix. The ollowing matrix expansion derived rom an application o oodbury s entity will prove useul: G = 1 Ψ }{{} + Ψ }{{} diagonal rank K = Ψ 1 }{{} diagonal + Ψ 1 1 Ψ Ψ 1 Ψ 1 }{{} rank K Proposition 1 characterizes the structure o the cross price impact model. The cross-impact is captured by the non-diagonal entries in Ψ that result as a consequence o the natural liquity provision attributed to index-und portolio investors. e shall interpret the terms Ψ diag N i=1ψ,i and Ψ diag K k=1 ψ,k as liquity. ψ,i represents the amount o liquity proved by single stock investors in stock i and ψ,k represents the liquity supplied by o index und k investors. The sum Ψ + Ψ indicates the total 9

10 market liquity. As shown in 9, the price impact is inversely proportional to the liquity, which agrees with the conventional deinition o liquity as a measure o ease o trading. hen ψ,i or ψ,k is large, equivalently when the liquity is abundant, price impact is low. ψ,i and ψ,k were originally deined as the sensitivity o investors holdings to market price movements, and, as such, capture how many shares we can obtain rom these two types o investors when the price moves by a certain amount; a measure o price impact One-period Transaction Cost Conser a liquator that wishes to execute v R N shares over a short period o time, say 5 to 15 minutes. Let p 0 R N be the price at the beginning o this execution period. Assuming that v is traded continuously and at a constant rate over the duration o that time period, the liquator will realize an average transaction price given by p tr = p Gv + ɛtr, 11 where ɛ tr R N represents a random error term that captures unpredictable market price luctuations or the eect o trades executed in that period by other investors; this suggests that costs accumulate linearly over the duration o the period, and that the average execution price is halway the average impact plus a random contribution due to luctuations in the price due to exogenous actors. e will return to this assumption later on. e will assume that the error is independent o our execution v and zero mean: i.e., E [ ɛ tr v ] = 0. The single-period expected implementation shortall incurred by the liquator is given by C v E [v p ] tr p 0 = 1 2 v Gv. 12 Linear price impact induces quadratic implementation shortall costs; note that the resulting cost is always positive since G is positive deinite. The ollowing proposition briely explores how the mixture o natural liquity provers aects the expected execution cost. Proposition 2 Extreme cases. Conser a parametric scaling o the single-stock and index-und natural liquity, Ψ and Ψ, respectively, given by G = α Ψ + β Ψ 1, 13 or some scalars α 0, 1] and β 0, 1]. i I there are no index-und investors α 1 and β 0, the expected execution cost becomes separable across indivual assets: lim C v = 1 α 1,β 0 2 v Ψ 1 v = 1 2 N i=1 v 2 i ψ,i

11 ii I there are no single-stock investors α 0 and β 1, the liquator can only execute with inite expected execution costs portolio orders that can be expressed as a linear combination o the index-und weight vectors. Speciically: lim C v = α 0,β 1 { i v / span w 1,, w K 1 2 u Ψ 1 u i v = u. 15 The proo is proved in Appendix B.1. Thereore, separable security-by-security market impact cost models, oten assumed in practice, essentially predicate, per our analysis, that all natural liquity in the market is proved by opportunistic single-stock investors. And, in that case, 14 recovers the commonly used diagonal market impact cost model. The other extreme scenario assumes that all liquity is proved along the weight vectors o the index und investors, and the resulting cost then depends on how the target execution vector v can be expressed as a linear combination o the w 1,, w K. In practice, the latter case suggests that execution costs may increase in periods with relatively higher intensity o portolio liquity provision when the target portolio that is being liquated is not well aligned with the directions in which portolio liquity is supplied Time-varying Liquity and Multi-period Transaction Cost The stylized observations o Proposition 2 suggest that intraday trading costs may be aected by intraday variations in the mixture o natural liquity provers, and, in particular, i the relative contribution o index und investors increases signiicantly over time. e will conser the transaction cost o an intraday execution schedule v 1,, v T over T periods, in which v t R N shares are executed during the time interval t. e will make the ollowing assumptions on the intraday behavior o price impact, price dynamics, and realized execution costs. a e allow the mixture o liquity provision to luctuate over the course o the day. e denote the time-varying liquity by ψ,it and ψ,kt with an additional subscript t. e assume that the portolio weight vectors w k o index liquity provers are assumed to be ixed during a given day. Under this setting, the coeicient matrix o price impact can be represented as ollow: G t = Ψ,t + Ψ,t 1. b Let p t be the undamental price at the end o period t. The undamental price means the price on which the market agrees as a best guess o the uture price excluding the temporary deviation o the realized transaction price due to market impact. The undamental price process p 0, p 1,, p T is assumed to be a martingale independent rom the execution schedule: p t = p t 1 + ɛ t, or all t = 1,, T, 11

12 where the innovation term ɛ t satisies E [ɛ t F t 1 ] = 0 with all the past inormation F t 1. term ɛ t is commonly understood as the change in market participants belie perhaps due to the inormation revealed during the period t. e are implicitly assuming that our execution conveys no inormation about the uture price. c The realized transaction price in each period can deviate rom the undamental price temporarily, e.g., due to a short-term imbalance between buying order low and selling order low. In executing v t shares, the liquator is contributing to such an imbalance, which causes the temporary price impact according to the mechanism described earlier on. e assume that this impact is temporary, we particularly assume that the transaction price begins at the undamental price at each period regardless o the liquator s trading activity in prior periods. Given the coeicient matrix G t, when v t is executed smoothly, the average transaction price is where the error term ɛ tr t p tr t = p t G tv t + ɛ tr t, ] satisies E [ ɛ tr t v t = 0 as beore. Under these assumptions, the expected transaction cost o executing a series o portolio transactions v 1,, v T is separable over time and can be expressed as ollows: [ T C v 1,, v T E vt ] p tr t p 0 = 1 2 v t G t v t. This ormulation implicitly assumes that the intraday liquity captured through ψ,it s and ψ,kt s is deterministic and known in advance. Although intraday liquity evolves stochastically over the course o the day, its expected proile exhibits a airly pronounced shape that serves as a orecast that can be used as a basis or analysis as is done in practice; c.., the discussion in 6. The 4. Optimal Portolio Execution e will ormulate and solve the multi-period optimal portolio execution problem in 4.1, and subsequently explore the properties o the optimal solution as a unction o the intraday variation o the two sources o natural liquity provers in Optimal Trade Schedule Conser a risk neutral liquator interested in executing x 0 R N shares over an execution horizon T e.g., a day. e ormulate a discrete-time optimization problem to ind an optimal schedule 12

13 v 1,, v T that minimizes the expected total transaction cost: minimize 1 C v1,, v T = 2 v t G t v t 16 subject to v t = x Proposition 3 Coupled execution. The risk-neutral cost minimization problem has a unique optimal solution given by v t = G 1 t T G 1 s s=1 1 x 0 = Ψ,t + Ψ,t Ψ + Ψ 1 x0, 18 where the total daily liquity Ψ and Ψ are deined as ollows Ψ Ψ,t, Ψ Ψ,t. 19 e make the ollowing observations. First, the optimal solution is coupled across securities. Speciically, as long as the market impact is cross-sectional, the cost minimizing solution needs to conser all orders simultaneously in optimally scheduling how to liquate the constituent orders, as opposed to scheduling each order separately and attempting to minimize costs as i market impact is separable; such a separable execution approach is oten used in practice eectively assuming that there are no cross impact eects. The coupled execution recognizes that the blend o natural liquity changes intraday, and attempts to change the composition o the resual liquation portolio so as to take advantage o portolio liquity that may be available towards the end o the day, or example. e will explore this point urther in the remainder o this section. Second, it is typical to derive coupled optimal portolio trade schedules or investors that are risk-averse and conser the variance o the execution costs in the objective unction or as a constraint; in that case, the covariance structure o the portolio throughout its liquation horizon intuitively leads to coupled execution solution. In our problem ormulation, the coupling o the execution path is driven by the cross-sectional dependency o natural portolio liquity proved by index unds that leads to cross-impact, as opposed to the cross-sectional dependency o intraday returns. Third, we note that in the above ormulation we have not imposed se constraints that would enorce that the liquation path is monotone; we will return to this point later on. The structure o the optimal schedule in 18 takes an intuitive orm: the proportion o the trade that is liquated in period t is proportional to the available liquity in that period, as captured by the time-dependent numerator matrix Ψ,t + Ψ,t, normalized by the total liquity made available throughout the day, as captured by the time-independent denominator matrix Ψ + Ψ. An alternative interpretation also given by 18 is that the optimal schedule 13

14 splits the order inversely proportional to a normalized time-dependent market impact matrix. Corollary 1 No index und investors, Ψ,t = 0 or t = 1,..., T. hen there are no index und investors: i.e., ψ = 0, a separable VAP -like trade schedule is optimal: v it = ψ,it Ts=1 ψ,is x i0, or i = 1,, N. 20 Proo o Proposition 3 Note that since G t is symmetric, v t 1 2 v t G t v t = G t v t. The KKT conditions o the convex minimization problem in require that there exists a vector λ R N such that λ = 1 v t 2 v t G t v t = G t vt, or all t = 1,, T, vt=v t which together with the inventory constraint in 17 imply that It ollows that vt = G 1 t solution exists and is unique. λ = G 1 t x 0 = vt = G 1 t λ. Ts=1 1 G 1 s x0. Since all G t s are invertible, the optimal In a market where all natural liquity is proved by single stock, opportunistic investors, there are not cross-security impact eects, market impact is separable, and the minimum cost schedule or a risk-neutral liquator is also separable across securities the optimal solution simply needs to minimize expected impact costs separately or each order in the portolio. Each indivual order can be scheduled independently o the others, and the resulting schedule is VAP -like in that the execution quantity v it is proportional to the available liquity ψ,it at that moment. Indeed, treating the overall market trading volume proile as the observable proxy o the natural liquity proile, the solution spreads each order separately and in a way that is proportional to the percentage o the market volume that is orecasted or each time period; this is what a typical VAP execution algorithm does. Conversely, i some o the natural liquity is proved by index und investors that wish to trade portolios e.g., liquate some amount o an energy tracking portolio i the energy sector has had a signiicant, positive return intraday, we would expect that the separable VAP schedule does not minimize expected market impact costs, and it is not optimal or the motivating trade scheduling problem Optimal Trade Schedule under Parametric Liquity Proile To gain some insight on the structure o the optimal policy we explore a setting where the intensity o single stock and index und liquity provision varies parametrically as ollows: single-stock 14

15 investors liquity ψ,it varies over time t = 1, 2,, T according to a proile α t, and index-und investors liquity ψ,kt varies according to another proile β t. Ψ,t = α t Ψ, Ψ,t = β t Ψ, or t = 1,, T, 21 where T α t = T β t = 1. e will assume that all single stocks share the same time-varying proile α t, and likewise all index unds share the proile β t. The empirical indings o 2 indicate that pairwise correlations o trading volumes increase towards the end o the day. I a primary source o stochastic luctuations in intraday trading volumes is the stochastic arrivals o single stock and portolio trades, then one would expect that the proiles α t, β t vary intraday so as to generate the well known U-shaped volume proile, and to vary dierently rom each other so as as to generate the time-varying pairwise correlation relationship; this is supported by behavior o market participants towards the end-oday, discussed earlier. Indeed, i the two groups o natural liquity had the same trading activity proiles, i.e., α t = β t, then the average correlation in intraday trading volume would not vary intraday. e expect that towards the end o the day, the intensity o index und liquity provision β t increases relatively aster than the intensity o single stock liquity provision α t. Proposition 4 Optimal execution under structured variation. Under the parameterization o 21, the schedule v t is optimal or the risk-neutral cost minimization 16: vt = α t x 0 + β t α t Ψ x 0, 22 or, equivalently, K vt = α t x 0 + β t α t ŵk x 0 w k, 23 k=1 1, where Ŵ Ψ 1 Ψ 1 + and ŵk denotes the k th column o Ŵ. Beore oering an interpretation or 23 we state the ollowing corollary. Corollary 2 Optimal execution under common variation. I α t = β t or all t = 1,, T, a separable, VAP -like strategy is again optimal. v t = α t x The proo o Proposition 4 is given in the Appendix B.2. Corollary 2 states that when the intensity o natural liquity provision is the same or single stock and index und investors, α t = β t, the optimal schedule vt is again aligned with x 0 scaled by α t. As α t = β t represents the market activity at time t, the above policy can be interpreted as a VAP -like execution that spreads each indivual orders proportionally to the total volume available at each point in time; this is separable across orders. As noted earlier, the setting where α t = β t is, however, inconsistent with 15

16 the empirical indings regarding the intraday behavior o pairwise correlations o trading volumes. In contrast, 23 highlights that when the mixture o natural liquity varies intraday through the dierence between α t and β t, the optimal schedule tilts away rom the VAP -like execution encountered in 24 so as to take advantage o increased available index und liquity, e.g., oered along the direction o sector portolios. 5. Illustration o Optimal Execution and Perormance Bounds In this section we prove a brie illustration o the optimized execution path that incorporates the eect o index und portolio liquity. Risk-neutral investors oten adapt a separable execution style, i.e., trade each asset separately, most oten using a Volume-eighted-Average-Price VAP algorithm. As we show in 4, this separable strategy, under some assumptions, can be shown to minimize expected impact costs per order, but disregards the eect o portolio liquity and crossimpact costs when trading multiple orders se-by-se. For a stylized model o natural liquity o the orm introduced in 4.2 simpliied to the case o a single index und e.g., the market portolio, we establish a worst case bound on the sub-optimality gap o such a separable execution approach against the optimized portolio schedule derived earlier. Speciically, restricting attention to the parameterization introduced in 4.2 in a setting with a single index und K = 1: Ψ,t = α t Ψ and Ψ,t = β t Ψ with T α t = T β t = 1, and Proposition 4 states that the optimal execution v t is vt = α t x 0 + β t α t ŵ 1 x 0 w 1, or t = 1,, T, 25 where w 1 R N is the weight vector o the index und e.g., the market portolio, expressed in number o shares, and ŵ 1 ψ 1,1 + w Ψ w 1 Ψ 1 w 1. In contrast, the separable execution liquates each order in the portolio independently, allocating quantities to be traded in each v sep t period in a way that is proportional to the total traded volume that is orecasted to execute in that period: v sep it = VolAlloc it x 0i, or t = 1,, T, or each i = 1,, N, 26 where VolAlloc it is the percentage o the daily volume in security i that trades in period t, deined in and, in more detail, Appendix B.3 posits a stylized stochastic process generative model or single stock and portolio index und investor order low, that results in a simple parametric structure or the total traded volume proile VolAlloc it and the resulting pairwise correlation proile among traded volumes Correl ijt. The model s primitive parameters can be estimated so as to be consistent with the AvgVolAlloc t and AvgCorrel t depicted in proves analytic results on the optimality gap between the separable and the optimal execution schedules, in 26 and 25, 16

17 respectively, which or the parameters estimated in 5.1 could be as high as 6.2% A Useul Parametrization o Intraday Liquity e will posit a simple generative model o single stock and portolio index und order low driven by two underlying Poisson processes. This mixture o order lows comprise the total volume or the day, and also generates a certain correlation structure in the traded volumes per period across securities. e will oer a brie overview in this section, and deer to the Appendix B.3 or additional detail on this model. Let θ i denote the raction o traded volume in a day or stock i that is generated by order low submitted by index und investors. Formally, θ i w 1i q q,i + w 1i q, 27 where q is the notional traded by portolio investors, w 1i is the weight o security i in this index und notionally weighted, and q,i is the notional traded by single stock investors in security i. For simplicity, urther assume that θ 1 = θ 2 = θ N = θ, i.e., all securities have the same composition o order low as contributed by single stock and portolio index und investors. In such a model, as explained in B.3 the intraday volume and pairwise correlation proiles are given by: AvgVolAlloc t = 1 N AvgCorrel t = N i=1 1 NN 1 E [DVol t ] Ts=1 E [DVol s ] = α t 1 θ + β t θ, 28 Correl ijt = i j β t θ 2 α t 1 θ 2 + β t θ e note that the assumption that θ i = θ or all securities i leads to the conclusion that all securities have the same intraday volume proile, and, perhaps, more importantly that the intraday volume correlation proile Correl ijt is the same across all pairs o stocks. The latter is arguably a airly strong restriction, and it is only imposed so as to allow or a more tractable closed orm perormance analysis. Given the empirically observed proiles or AvgVolAlloc t and AvgCorrel t, visualized in Figure 1, we can solve a set o coupled equations deined by where the respective let hand ses are given by the empirically estimated values, to entiy the values o θ, α 1,, α T and β 1,, β T. The results are summarized in Figures 2 and 3. θ was estimated to be 0.21, implying that 21% o total traded volume originate rom the index und. e can observe that, at the beginning o the day, the trading activity o index-und investors β t is smaller than that o single-stock investors α t, but β t ar exceeds α t in the last hour o the day, as expected. Such an intraday variation in the composition o order low is consistent with the increasing pairwise correlation in volumes towards the end o the trading day. 17

18 Figure 2: Let panel: the intensity o single stock investors, α t, and portolio index und investors, β t, calibrated to best match the empirical proiles o traded volume, AvgVolAlloc t and pairwise volume correlations, AvgCorrel t. Right panel: depicts the deviation o α t, β t orm the market proile AvgVolAlloc t. Figure 3: The intraday trading volume proiles α t 1 θ and β t θ let, and the proportion o index-und order lows β t θ α t 1 θ+β t θ right. Finally, Figure 4 proves a graphical illustration o the eect o these estimated to the optimal execution schedule in 25. The example depicts an investor wants to liquate a portolio x 0 with two orders, where the weights o the liquation portolio deviate signiicantly rom the weights o the index portolio w 1 as captured by the angle between x 0 and w 1. To exploit the increased end-o-day liquity in the direction o the index portolio, w 1, the optimal schedule trades more aggressively stock 2 in the morning session, as shown by vt, thus tilting away rom a separable VAP-like execution that would be aligned with x 0. As a consequence, the resual portolio executed towards the end o the day is better aligned with the index portolio in the aternoon, vt is closer to the index portolio w 1. 18

19 Figure 4: Illustration o the optimized schedule, which is shown to tilt away or toward the direction o index und depending on the dierence between single stock and index und liquity Implementation Shortall Comparison: Optimal vs. Separable Execution Schedules From 26 and 28 we get that the separable schedule v sep t is given by: v sep t = α t 1 θ + β t θ x 0, 30 and or vt and v sep t, the expected implementation shortall can be written as ollows: Cv t = 1 2 x 0 Ψ + Ψ 1 x0, 31 Cv sep t = 1 2 α t 1 θ + β t θ 2 x0 α t Ψ + β t Ψ 1 x0. 32 Note that under the assumption that there is only one index und, w 1, we could simpliy the above expressions and reduce Ψ to w 1 ψ,1 w1 sep. The expression or Cv t is obtained rom substituting 30 into 16. e deine as a relative perormance measure the ratio between the 19

20 expected transaction costs incurred by the two execution schedules. Υx 0 Cv sep t Cv t ; 33 this ratio is clearly greater or equal to 1, and captures the additional cost incurred by the separable VAP-like schedule over the optimized, coupled, execution schedule. Proposition 5 Exact cost ratio. For any x 0 R N, where Υx 0 = 1 + θ 2 T βt 2 1 α t + x 0 w1 1 Ψ x 0 Ψ 1 x η 1 1 ψ,1, 34 γ t β t α t, η 1 ψ,1 w 1 Ψ 1 w 1, and α t 1 θ 1 γ t 2 1 γ t 1 + η 1 γ t. 35 The proo is given in Appendix B.4.1. The parameter η 1 is the ratio o index und liquity 1/ ψ,1 over the liquity proved by single stock investors along the index und weights w1 1 Ψ w 1. Equivalently, it is the ratio o the price change o trading along the index und direction w 1 against only the portion o single stock investors in the market 3, versus the price change o trading along w 1 along only the index und investors, which is 1/ ψ,1. 4 The last expression in the perormance metric is a product two terms: the irst is associated with the intraday variation o liquity and trading volume, and the second is associated with the degree o alignment between the execution portolio x 0 and the index und weights w 1. orst case liquation portolios. First, we explore the structure o the portolios that would exhibit the largest optimality gap under a separable execution. Remark 1 Maximum/minimum cost ratio. Let Υ market and Υ orth be the cost ratio when, respectively, x 0 = w 1, and when x 0 = w 1 where w 1 is an arbitrary portolio such that w 1 Ψ 1 w 1 = 0 with 3 I we trade w 1 against single stock investors we will cause a change in prices given by p = Ψ 1 w1, which would imply a change in the price o the market portolio equal to w1 1 Ψ w1. 4 To gain some intuition as to the magnitude o that parameter, imagine wanting to buy a $100 million slice o the S&P500, where in one case this is acquired rom distinct liquity provers, each trading only one o constituent orders, while in the other case it is acquired rom the same portolio liquity prover. The mere dierence in the aggregate volatility held by the distinct liquity provers in the irst scenario versus the unique liquity prover trading the market portolio in the latter, would suggest a potentially signiicant dierence in trading costs, and thereore a high 1 value or η 1. 20

21 w 1 0. Υ market Υx 0 = w 1 = 1 + θ 2 Υ orth Υx 0 = w 1 = 1 + θ 2 T T βt 2 1 α t + η 1, 36 βt 2 1 α t. 37 Then, largest and smallest cost ratio are obtained at either x 0 = w 1 or x 0 = w 1 sign o. depending on the max {Υx 0} = x 0 R N min {Υx 0} = x 0 R N { Υmarket Υ orth i 0 i 0, 38 { Υorth Υ market i 0 i In particular, or ixed η 1, α 1,, α T, β 1,, β T, there exists θ [0, 1] such that 0 i θ θ and 0 i θ θ. 40 Similarly, or ixed θ, α 1,, α T, β 1,, β T, there exists η1 [ θ 1 θ, ] such that 0 i η 1 η 1 and 0 i η 1 η This remark entiies which portolios give rise to the largest and smallest cost ratios, respectively. It is straightorward that the cost ratio has extreme values at x 0 = w 1 and x 0 = w 1, i.e., when x 0 is most and least aligned with the market portolio w 1. From 30 and 23 we get that in these two extreme cases the separable and optimal schedules are given by: v sep t = α t 1 θ + β t θ x 0, and vt = α t 1 η 1 η 1+η 1 + β t 1 1+η 1 x 0 i x 0 = w 1, α t x 0 i x 0 = w1. hen x 0 = w 1, the sensitivity o the optimized schedule to the intensity o index-und liquity η provision, β t, is 1 1+η 1, whereas that o the separable execution is θ. I θ > η 1 1+η 1, the separable execution will trade more than is optimal to do in the morning, and trade less than optimal towards the end o the day; the opposite happens i θ < η 1 1+η 1. e can expect that the suboptimality o separable execution roughly scales with θ η η 1 A similar argument suggests that the suboptimality gap when x 0 = w1 roughly scales like θ 02. Comparing θ η η 1 and θ 2 as proxies or Υ market and Υ orth, respectively, leads to the indings o Remark 1. Perormance implications when trading the market portolio. Next we characterize Υ market as a unction o the parameter η 1. 21

22 Remark 2 Characterization o Υ market. For ixed θ, α 1,, α T, β 1,, β T, as a unction o η 1, Υ market η 1 decreases i η 1 For particular values o η 1, θ 1 θ, and Υ marketη 1 increases i η 1 θ 1 θ. 42 Υ market η 1 = 0 = 1 + θ 2 Υ market η 1 = θ 1 θ T lim Υ η 1 marketη 1 = θ 2 βt 2 1 α t, 43 = 1, 44 T αt β t It predicts that Υ market decreases irst and then increases as η 1 varies. This can be similarly understood as in Remark 1: separable execution correctly reacts to the liquity proved by indexund investors only when η 1 = θ 1 θ, and overreacts or underreacts when η 1 deviates rom θ 1 θ. Figure 5: Possible range o cost ratio Υ with respect to η 1 given the values o θ, α 1,, α T, β 1,, β T obtained in 5.1. The coupled execution could save up to 6.2 % when trading the market portolio. Our estimate or the raction o index und liquity, θ =.21, which suggests a threshold value or θ/1 θ.27. Even though the value o η 1 is unentiiable in our context, one would expect the value o η 1 to be moderately large c. Footnote 4, and that the realized beneits rom optimizing the coupled execution o the portolio over that o a separable execution to approach the upper bound in 45. That upper bound is equal to 6.2% or the parameters θ, α 1,, α T, β 1,, β T estimated in 5.1, and Figure 5 graphs Υ market and Υ orth as unctions o η 1.That is, under the assumptions o our stylized generative model o order low one could reduce execution costs over the separable VAP-like execution by as much as 6.2% by optimally coupling the execution schedules o the various orders that are being liquated so as to exploit the beneits due to portolio liquity 22