Do They Know What They Do? A Decomposition of Mutual Fund Performance using Transaction Data

Transcription

1 Do They Know What They Do? A Decomposition of Mutual Fund Performance using Transaction Data Ran Xing January 31, 2018 Abstract This paper decomposes the performance of active mutual funds based on the investment horizon of their holdings and trades, using both daily holdings and transaction data. Funds earn 0.73% per year from recent trades (in the last half year), and profit substantially from long-term holdings (1.25% per year). These results generally hold across all investment styles, with sales contributing as much as purchases. Trades made by growth, small-cap, momentum, or small funds contribute more in half a year and beyond one year but contribute negatively to performance from half a year to one year, indicating funds profit from their specializations. This work was partially done while the author was a visiting scholar at The Wharton School and at the Stockholm School of Economics. The data used in this paper belongs to The Wharton School. I thank Joost Driessen (supervisor), Jules van Binsbergen, Magnus Dahlquist, Mathijs van Dijk, Xi Dong, Roger Edelen, Miguel Ferreira, Adam Farago, Bruno Gerard, Mariassunta Giannetti, Sebastian Gryglewicz, Jungsuk Han, Erik Hjalmarsson, Frank de Jong, Gregory Kadlec, Lingtian Kong, Melissa Lin, Mancy Luo, Mike Mao, Massimo Massa, Stefan Obernberger, Hongxun Ruan, Jame Russell, Esad Smajlbegovic, Patrick Tuijp, Patrick Verwijmeren, George Wang, Bas Werker, Chengdong Yin, Marcin Zamojski, Xiaoyan Zhang, Xin Zhang, and seminar participants at Tilburg University, Erasmus University Rotterdam, Lancaster University and University of Gothenburg for helpful comments and discussions, and I also thank Yuehua Tang for sharing the methodology to identify mutual funds in ANcerno database. Xing is from the Finance Group of the Erasmus School of Economics in Erasmus University Rotterdam. xing@ese.eur.nl. 1

2 1 Introduction Mutual funds manage trillions of dollars on behave of retail investors. The majority of this money is actively managed 1. Therefore the question that whether mutual fund managers who actively trade stocks add value is of great importance to the welfare of our society. Academics have used fund returns to answer this question for half a century (since Jensen (1968)) and still have not reached a consensus. On the one hand, a thread of the literature (e.g. Daniel, Grinblatt, Titman and Wermers (1997) and Wermers (2000)) analyses funds quarterly holdings and finds positive results. Although quarterly holdings are useful, they are only infrequent snapshots of their portfolios and neglect the short-term effect of trades on fund performance. On the other hand, a growing body of literature uses transaction data to approach this question and finds evidence of trading skill (e.g. Puckett and Yan (2011) and Chakrabarty, Moulton and Trzcinka (2017)). Those analyses are mostly limited to the short-term performance of trades and to aggregate trades of institutional investors because of the anonymity of the data. Consequently, the literature using holding data and that using transaction data are disconnected from each other even though they focus on the same question. Moreover, the first thread of the literature uses actual holdings, whereas the second thread uses hypothetical holdings inferred from transactions and additional assumptions. The former is based on facts but less informative about funds trading motives while the latter is the opposite. In this paper, I design a unified framework which reconciles the inconsistencies in this literature. Transaction data, holding data, and daily stock data are used together in this framework to decompose fund performance. The specific research questions I ask in this paper are, do mutual funds trades contribute to (or harm) their performance at each investment horizon? If so how much? In addition, I explore how these differ across fund categories. I focus on the investment horizons of trades because different trades of one fund usually target profits at different horizons, but most existing analyses assume only 1 As of 2016, mutual funds worldwide have about $40.4 trillion of assets under management, $16.3 trillion are managed by U.S. funds. About 52% of U.S. mutual fund assets are held in equity funds, and 75.1% of the equity fundsâ total net assets are managed actively (Investment Company Institute, 2017) 2

3 one horizon for all the trades of each fund 2. My framework allows for different horizons for different trades. I use this framework to measure the contribution of holdings and trades to fund performance at each investment horizon. Doing so allows me to decompose funds performance based on their holdings, trades, and trading costs at a daily frequency. I also investigate how purchases versus sales affects performance through abnormal returns, factor exposures, and trading costs. This decomposition of fund performance based on holdings investment horizon at a daily frequency is only possible when daily holding data are available and can be combined with transaction data, and the analysis across fund categories is only achievable if funds identities are known. To the best of my knowledge, this work is the first to perform these analyses. Few databases provide transaction data on institutional investors and almost all of them are anonymous 3. This is one of the reasons why most empirical papers on mutual funds still use fund return data or quarterly holding data. By merging the transaction data provided by Abel Noser Solutions with the quarterly holding data in the Thomson Reuters database from 2001 to 2010, I identify 336 U.S. mutual funds for which I can construct their daily holdings. My methodology can be easily described. I start by writing down the mathematical relation between the contribution of sales, purchases, and holdings to performance by investment horizon and show that this can be implemented using transaction data and daily holding data. Specifically, to decompose funds performance on the basis of the investment 2 Papers using fund turnover and churn ratio as measures of the average investment horizon of a fund include Gaspar, Massa and Matos (2005), Carhart (1997), Bushee (1998, 2000, 2001), Yan and Zhang (2009) and Cella, Ellul and Giannetti (2013), and papers measuring the average investment horizon of a fund directly include Cremers and Pareek (2014, 2016) and Lan, Moneta and Wermer (2016). Most papers use quarterly holding data to construct their horizon measures. Di Mascio, Lines and Naik (2017) documents a large dispersion in the holding periods of trades from the same fund. 3 The most widely used transaction databases in academia are the ones provided by Abel Noser Solutions (formerly ANcerno Ltd. and the Abel/Noser Corporation), e.g. Irvine, Lipson and Puckett (2006), Puckett and Yan (2011), Goldstein, Irvine, Kandel and Wiener (2009), Chemmanur, He and Hu (2009), Hu (2009), Green and Jame (2011), Anand, Irvine, Puckett and Venkataraman (2011) (2013), Busse, Green and Jegadeesh (2012), Hu, McLean and Wang (2013), Brown, Wei and Wermers (2014), Green, Jame, Markov and Subasi (2014), Agarwal, Gay and Ling (2014), Chemmanur, Hu and Huang (2015), and Chakrabarty, Moulton and Trzcinka (2017); and the database from Plexus Group, e.g. Keim and Madhavan (1995), Da, Gao and Jagannathan (2010), and Busse, Green and Jegadeesh (2012). Neither of these two databases reveals the identities of these institutions. 3

4 horizon of their holdings, I express the daily fund performance as the daily performance of their holdings at the beginning of the day plus the profits/losses and costs of their trades on that day. I then decompose the current holding of each stock into the change in the holding caused by trades in the past n days and the holding n days ago. A positive change in the holding for that stock corresponds to a net purchase in the past n days, and a negative change to a net sale. By varying the number of days n (1 to 240 days in this paper) in this decomposition, the contribution of purchases and sales at each investment horizon to the fund s performance is calculated, as well as the contribution of holdings beyond 240 days. The contribution of net sales in this decomposition is the effect of the negative changes in holdings on fund performance relative to the hypothetical case in which those shares would not have been sold. My key findings are as follows. Mutual funds possess stock picking skills in both the short term and the long term, adding up to a gross alpha of 130 basis points (bps) per year before trading commissions 4. I measure stock and fund performance using the Fama- French-Carhart four-factor alpha (FFC alpha). Trades in the last 120 business days (half a year) contribute 73 bps per year to the gross alpha, and funds on average still earn a gross alpha of 123 bps per year if they stick to their holdings one year ago. These results generally hold across all fund categories covered in this paper, with high-turnover funds (growth/small-cap/momentum/small funds) mainly profit from their long-term holdings as well. The analysis by fund category shows that mutual funds have clear specializations. Value funds profit gradually from the information of fundamentals, and growth funds profit from short-term anomalies. The stocks bought by value funds outperform the benchmark and the profits decay gradually over time, whereas the stocks bought by growth funds outperform substantially within 20 days but underperform from 20 days to a year. Similarly, large-cap, non-momentum, and large funds profit from the information of fundamentals and small-cap, momentum, and small funds profit from short-term anomalies. 4 Because this paper focuses on the contribution of holdings and trades to fund performance before commissions, I define gross fund performance (or gross alpha) as the performance before both expenses and trading commissions. Usually, gross fund performance refers to performance before expenses but after commissions. I later compare the profits from trades with commissions. 4

5 One of the key insights of this paper is that sales are the hidden heroes of fund performance. It is largely neglected in the previous literature because the performance of stocks sold is not a reliable measure of the contribution of sales in all scenarios. The performance of stocks sold is a reasonable measure when funds sell stocks before they underperform, e.g. the sales of high-turnover funds. But it is not a proper measure when funds sell stocks to make room for better investment opportunities, e.g. the sales of lowturnover funds. Trades contribute to fund performance if and only if stocks purchased outperform stocks sold. The performance of stocks sold itself only reveals an incomplete aspect of the contribution of sales to fund performance. Even if I measure the contribution of sales as the underperformance of stocks sold, the sales in the past half a year contribute 39 bps per year to the gross alpha on average, which is about the same magnitude as the contribution of purchases (34 bps). Mutual funds mainly profit from their long-term holdings. Holdings longer than a year generate an annual alpha of 125 bps 5. Interestingly, high-turnover funds (growth/smallcap/mom/small funds) profit more from their holdings longer than one year than lowturnover funds do, even though long-term holdings represent a smaller fraction of their total holdings 6. Therefore high-turnover funds are actually long-term investors when measured by the contribution of long-term holdings to fund performance. Like angel investors, highturnover funds invest in a large number of stocks, mostly growth and small-cap stocks. Most stocks underperform and got sold from half a year to a year, and the small number of stocks left in their portfolios more than a year explain the majority of their gross alphas. The price dynamics of stocks after trades are informative about funds trading motives. However, net purchases and net sales used in the benchmark setting do not reflect round-trip trades within a given period, thus their contribution by investment horizon is not the same as the price dynamics of stocks after trades. Therefore, I also aggregate all purchases/sales of each stock to investigate the price dynamics after trades and calculate the contribution of round-trip trades within a year. I find that the prices of stocks bought by growth funds 5 On average, holdings longer than 240 days (a year) account for 69% of total holdings. The fraction of holdings at each investment horizon is reported in Table 8. 6 The fraction of holdings at each investment horizon is reported in Table 8. 5

6 substantially reverse 30 days after their purchases, in accordance with the conjecture that growth funds profit from short-term anomalies, and sales in their round-trip trades protect them to some extent from those reversals. In contrast, the prices of stocks bought by value funds increase gradually in the following year and do not reverse, which mimics the description of an investor who invests on the basis of information on fundamentals. Lastly, on average, mutual funds pay as much as 1.51% of their total net assets (TNA) per year for trading costs (76 bps for explicit costs such as trading commissions and 75 bps for implicit costs such as price impact costs) 7, which is approximately the same magnitude as the expense ratio and the gross alpha of those funds. Trading cost is one of the main costs of mutual funds daily operations, especially for small and high-turnover funds. This paper contributes to the discussion on whether active mutual funds are skillful 8 by using a unified framework to zoom in on their daily holdings and transactions. The analysis by investment horizon and by fund category draws a complete and detailed picture of their skills. Moreover, one reason that academics are still debating this question even though fund performance data are widely available is that funds realized returns (or alphas) are noisy (and perhaps biased) estimates of skill. It is not only because stock returns are volatile but also because their holdings and trading amounts vary over time 9. This additional variation in fund returns introduces more noise into the estimation of fund alphas and even causes bias in the estimates if their holdings or trading amounts are correlated with pricing factors over time 10. By separately estimating the alpha of each stock and aggregating them based on holdings and trading amounts, I exclude this additional noise and source of bias. 7 The average trading cost is 50 bps of fund TNA per year when weighted by fund TNAs. 8 e.g. Jensen (1968); Gruber (1996); Carhart (1997); Fama and French (2010) 9 Because the investment opportunities, sizes, and strategies of funds vary over time, their holdings and trading amounts vary over time as well. For example, Pastor, Stambaugh and Taylor (2017) document that mutual funds investment opportunities vary substantially over time, and their trading amounts and fund performance co-vary with them. Berk and Green (2004) and Berk and van Binsbergen (2016) argue that additional inflows to superior funds erode their performance. Carhart (1997) shows that mutual funds performance is not persistent over time, attributing it partially to frequent changes in their investment strategies. 10 As discussed in Jensen (1968), Dybvig and Ross (1985), Grinblatt and Titman (1989b) and Daniel, Grinblatt, Titman and Wermers (1997), depending on the structure of returns and the behavior of the portfolio manager, factor timing ability could bias the alpha estimated from fund returns downward. The reason is that, with positive timing ability, the estimated factor betas will be biased upwards. Similarly, if funds trade more and earn more through stock picking when market or other factors are high, the estimation using fund returns bias the betas upwards and the alpha downward. 6

7 Besides, stock return data are available for a longer period than for fund return data especially for newly developed small funds to estimate stock alphas. Therefore, using holdings and transaction data to construct mutual funds alphas with stock alphas gives us greater statistical power and less bias for investigating these funds skills. Consistently, I find highly significant results using the fund alphas constructed from stock alphas and holdings, whereas the results using fund alphas estimated from fund returns are mostly insignificant 11. This paper relates to several threads in the literature. A large thread of mutual fund literature uses quarterly holding data to explore which and how funds outperform/underperform the benchmark (e.g. Daniel, Grinblatt, Titman and Wermers (1997); Wermers (2000); Kasperczyk, Sialm and Zheng (2005); Cremers and Petajisto (2009); Da, Gao and Jagannathan (2010); Dong and Massa (2013); Cremers and Pareek (2015); Kacperczyk, van Nieuwerburgh and Veldkamp (2014) (2016); Dong, Feng and Sadka (2017)). Another thread uses the aggregate quarterly holdings (or changes in quarterly holdings) of a group of skillful funds to predict stock returns (e.g. Chen, Jegadeesh and Wermers (2000); Alexander, Cici and Gibson (2006); Wermers, Yao and Zhao (2012); Lan, Moneta and Wermer (2015)). The unified framework in this paper connects the holding literature with trading literature and the literature using fund returns, and the analysis at daily frequency complements the holding literature with trading profits and costs within a quarter, where the information density is the highest. Kasperczyk, Sialm and Zheng (2008) measure the effects of intra-quarter trades on performance by return gap (the gap between the reported fund return and the return implied by quarterly holdings). The daily holding data allows me to calculate the contribution of trades to fund performance at daily frequency directly. In addition, this paper is also closely related to the papers that analyse the relationship between the investment horizon of institutional investors and stock performance using quarterly holding data (e.g. Yan and Zhang (2009); Cella, Ellul and Giannetti (2013); Lan, Moneta and Wermer (2015)). 11 This is partially because the time span of the data used in this paper (11 years) is shorter than that of other papers (20 30 years). 7

8 The existing literature using transaction data 12 to answer the question of whether funds are skillful is also inconclusive. Firstly, only a fraction of funds holdings mostly short-term holdings are revealed by transaction data, whereas funds investment goal is to maximize the overall performance of both short-term holdings and long-term holdings. This paper complements this body of literature with funds long-term holdings and their contribution to fund performance. Secondly, the methodologies used in most trading studies are focused on the short-term performance of the stocks traded. Most commonly, these studies separately aggregate all purchases and sales and compare the performance of the stocks shortly thereafter. This paper presents a complete picture of the channels and mechanisms through which trades affect fund performance at each investment horizon using both transaction data and holding data in a unified framework. In addition, I analyse how the contribution of trades to fund performance differs across fund categories (such as growth funds vs. value funds). Such an analysis is not possible when the identities of the funds are unknown. In addition, the empirical evidence documented in this paper suggests from several aspects that the mutual fund industry is competitive and efficient 13. Firstly, I find evidence that each fund category focuses on and profits from its competitive advantage. Growth funds profit instantly after their purchases, and the prices of the stocks purchased subsequently reverse. In contrast, value funds profit more evenly from their short-term, mid-term, and long-term holdings, and the prices of those stocks do not reverse. This result is consistent with the common believe that growth funds profit from short-term anomalies, whereas value funds live on the information of stock fundamentals. In addition, although the profits from the holdings at each investment horizon differ significantly between growth funds and value funds, the differences in their gross and net alphas are relatively small this is expected from an efficient market of funds with heterogeneous investment skills. As Berk and van Binsbergen (2017) argue, The rational expectations equilibrium approximates the observed equilibrium in the money management space at least as well as it does in the 12 e.g. Puckett and Yan (2011), Chakrabarty, Moulton and Trzcinka (2017), Jame (2017), Busse, Tong, Tong and Zhang (2017). 13 Although such evidence is suggestive and needs to be tested formally in future research, it is already surprising that the evidence from different aspects point to the direction of an efficient mutual fund industry at the same time. 8

9 stock market. Secondly, funds in the sample used in this paper generate positive abnormal returns before fees but no abnormal return after fees, which is in accordance with the model in Berk and Green (2004) for which the capital provision of investors to mutual funds is competitive. Thirdly, transaction-level evidence also exists. I find that funds trades contribute to their performance in half a year before trading costs, but have almost no effect on fund performance after trading costs. As Pedersen (2015) argues in his book, Financial markets are inefficient enough that money managers can be compensated for their costs through the profits of their trading strategies and efficient enough that the profits after costs do not encourage additional active investing. This paper also relates to the trading cost literature, which estimates trading costs from transaction data (e.g. Chan and Lakonishok (1997), Keim and Madhaven (1996) (1997), Bikker, Spierdijk and van der Sluis (2007)). Busse, Chordia, Jiang and Tang (2017) estimate the magnitude of trading costs as a percentage of fund TNAs. Few other papers investigate the relationship between the liquidity provision of trades and fund performance (e.g. Keim (1999) and Jame (2017)). Section 2 introduces the data and presents the summary statistics of the sample of mutual funds in this paper. Section 3 explains the methodology used in the benchmark setting to decompose (reconstruct) fund performance and documents the contribution of holdings and trades within one year to fund performance. Section 4 extends the methodology to calculate the contribution of holdings longer than one year. In Section 5, I change the methodology from net purchases/sales to gross (all) purchases/sales to investigate the price dynamics of stocks after trades and the contribution of round-trip trades. In Sections 3, 4, and 5, I also conduct separate analysis for each fund category. Section 6 documents the contribution through factor exposures and Section 7 concludes. 9

10 2 Data 2.1 Sample of Mutual Funds To construct the daily holdings data of mutual funds, I merge three databases: transaction data provided by Abel Noser Solutions (formerly ANcerno Ltd. and Abel/Noser Corporation), quarterly holdings data in Thomson Reuters, and fund characteristics from the CRSP Mutual Fund Database. Mutual funds were identified by matching the changes in the stock holdings indicated by the transaction data in Abel Noser with the changes in the holdings reported in Thomson Reuters. Following Busse, Chordia, Jiang and Tang (2017), if the change in a stock holding for a fund in Abel Noser and a fund in Thomson Reuters matches exactly with each other, then I call this stock a matched stock between these two funds for that reporting period. I call a period a matched period between a fund in Abel Noser and a mutual fund in Thomson Reuters if the period meets three criteria: (i) there are at least five matched stocks, (ii) the number of matched stocks is at least 10% of the number of stocks with changes in holdings as reported in Thomson Reuters under this fund, and (iii) the ratio of the number of matched stocks to the number of stocks traded by this Abel Noser manager is at least 10%. I consider the funds in Abel Noser and Thomson Reuters a likely match if there is at least one matched period between the two. If there are likely matches between a fund in Abel Noser and multiple funds in Thomson Reuters, I choose the best match first by the number of matched periods, then by the ratio of matched stocks in Thomson Reuters, and then by the ratio in Able Noser if there is a tie. Following this procedure, matches between 803 Abel Noser managers and the corresponding Thomson Reuters funds were obtained. There are 564 unique Thomson Reuters funds in this list. Multiple Abel Noser managers may map to the same Thomson Reuters fund for different periods. I further match these Thomson Reuters funds to the CRSP mutual fund data through the MFLINK data provided by WRDS. I keep equity funds for this analysis by dropping all funds with less than an average of 70% of their holdings in equities, as reported in the CRSP mutual fund database. I then exclude funds with a 10

11 turnover of less than 70% of the turnover reported in the CRSP database to ensure that it is not only a fraction of the trades being reported to Abel Noser. Lastly, I manually verify the matches previously identified one by one using fund names from Thomson Reuters and CRSP Mutual Fund databases, and a manager name list disclosed by Abel Noser in equity funds are properly matched and have reasonable fund turnovers. After dropping six index funds, 336 funds remain for the analysis. Table 1 reports the summary statistics of these 336 mutual funds and, for comparison, the summary statistics of all equity mutual funds in the CRSP mutual fund database. The characteristics of the 336 funds merged are similar to the characteristics of an average fund in CRSP both in aggregate and separately for each year. The only difference is that they are on average larger and older than the average fund in CRSP. Abel Noser Solutions clients are more likely to be large and well-known funds than small, newly developed funds. This difference is also noticed by other studies using Abel Noser data, such as Jame (2017). [Insert Table 1 about here] I then construct the daily holdings of the 336 mutual funds by merging the transaction data with the quarterly holdings data for stocks that are ordinary common shares in the CRSP database. Firstly, I merge the Abel Noser transaction data with daily stock data in CRSP for all fund-stock-quarters with at least one transaction data record in the database. Secondly, I merge the daily CRSP data for the stocks with holding data in Thomson Reuters but no trading data in Abel Noser for the fund-quarters with Abel Noser trading data for other stocks. Thirdly, I merge the original holding data (both prior quarter and current quarter) for those fund-stock-quarters in the sample 15. Lastly, I first generate the daily holdings on the basis of the prior-quarter holding information and the current-quarter trading information, if they exist. If this information does not exist, I then use the current-quarter holding information as well. 14 The name list provided by Abel Noser Solutions only includes vague abbreviations for fund managers in the same fund family. Therefore, I cross-check those abbreviations with fund names provided in the CRSP and Thomson Reuters databases and only keep those matches that I am certain are correct. 15 Because stocks newly purchased in the quarter have no prior quarterly holding data, and stocks completely sold out have no current quarterly holding data, I need to use both the holding data reported at the end of the prior quarter and the data reported at the end of the current quarter to construct complete holdings data in the current quarter. 11

12 3 Contribution of Trades/Holdings within One Year In this section, I analyse the contribution of trades in the past 1 to 240 business days (one year) to fund performance 16. The contribution of holdings with an investment horizon of 1 to 240 days to fund performance is documented using the same framework. This contribution is mathematically equivalent to the contribution of net purchases in the past 1 to 240 days to fund performance. 3.1 Methodology Measures of the Contribution of Trades/Holdings to Performance The common practice of analysing investors trading performance is to separately aggregate all purchases and sales and then compare the performance of the stocks. Different from this common practice, this paper decomposes the daily performance of mutual funds using their transactions and daily holdings and estimates the contribution of trades to funds performance by investment horizon. Because complete information on funds holdings is available at a daily frequency, the contribution of holdings (net purchases) at each investment horizon to fund performance can be calculated directly. I design a methodology in the spirit of Daniel, Grinblatt, Titman and Wermers (1997) and the calendar-time portfolio approach 17 to measure the contribution of their trades and holdings to fund performance in a consistent framework, and to extend it to different investment horizons. Different from Daniel, Grinblatt, Titman and Wermers (1997), who measure the contribution at a monthly frequency, I measure it at a daily frequency and include the contribution during trade execution and within the trading day. Different from the calendar-time portfolio approach, I measure the contribution of trades to their funds overall performance, covering the performance of both short-term and long-term holdings instead of only the short-term performance of the stocks traded. 16 I choose one year as the maximum time horizon because the average time length of the transaction data for each fund in Abel Noser Solutions is approximately three years. Analysis longer than one year is based on less than two thirds of the data and tilts towards funds with longer time spans of transaction data. 17 Since Jaffe (1974) and Mandelker (1974), the calendar-time portfolio method has been used in many studies, such as Barber and Odean (2000 and 2001), Seasholes and Zhu (2010), and others 12

13 I express the daily fund performance measured by the Fama-French-Carhart four-factor alpha (FFC alpha) 18 as R t = Ni=1 Ni=1 H i,t 1 R i,t V i,t Ri,t e Ni=1 + H Ni=1, s.t. H i,t 1 0 (1) i,t 1 H i,t 1 where R t is the gross fund FFC alpha on day t. H i,t 1 is the fund s holding of stock i at the end of day t 1 in dollars. Because mutual funds do not have short positions, H i,t 1 is non-negative. R i,t is the abnormal return of stock i on day t. V i,t is the fund s trading amount of stock i in day t, which is positive for purchases and negative for sales, and R e i,t is the change in the stock price from the execution of the trade to the end of day t as a percentage of the closing price of stock i on that day 19. The first term on the right-hand side of equation (1) is the contribution of the holdings at the end of day t 1 to fund performance on day t, and the second term is the contribution of the trades on day t to the same-day fund performance. I distinguish the change in holdings caused by trades within the past n days (10/20/60/120/240 days), H s(n) i,t 1 = n s=1 V i,t s, from the holdings n + 1 days ago H p(n) i,t 1. Hs(n) i,t 1 can be either positive or negative and H p(n) i,t 1 is non-negative. H i,t 1 = H s(n) i,t 1 + Hp(n) i,t 1 (2) Using this holdings decomposition, I decompose the daily fund performance into the performance from trades on the same day, the changes in holdings in the past n days (10/20/60/120/240 days), and the holdings n days ago R t = ( Ni=1 V i,t Ri,t e Ni=1 H s(n) i,t 1 Ni=1 + R Ni=1 i,t H p(n) i,t 1 H Ni=1 ) + R i,t i,t 1 H Ni=1. (3) i,t 1 H i,t 1 18 The Fama-French-Carhart four-factor alpha of stocks is calculated using the one-year rolling regression, as on French s website. Because a large number of funds do not have enough historical data for the oneyear rolling approach, I estimate the Fama-French-Carhart four-factor alpha of funds using all available observations of each fund. 19 Because the exact time of trade execution within a day is not reported in the Abel Noser dataset, I use raw returns instead of risk adjusted returns for intra-day trading profits/losses. The effect of intra-day risk premiums to fund performance is negligible. As will be shown in Table 11, the contribution of trades to fund performance through factor exposures is approximately 2% per year. If we assume trades on average occur mid of the day, the contribution of trades through factor exposures within a day is less than (0.02/240/2 ) 0.5 bps per year. If weighted by fund TNA, it is less than 0.1 bps per year. 13

14 The first and second terms on the right-hand side of equation (3) together in the bracket measure the contribution of trades within n days to fund performance. I further distinguish between the purchases (B i,t = V i,t, if V i,t 0) and sales (S i,t = V i,t, if V i,t 0) on the same day, and between the positive changes in holdings (net purchases) in the past n days and the negative changes in holdings (net sales) in the past n days. I denote the net purchases in the past n days by B s(n) i,t 1 = 0, if H s(n) i,t 1 0 H s(n) i,t 1, and the net sales in the past n days by S s(n) i,t 1 = H s(n) i,t 1, if Hs(n) i,t 1 > 0, (4) if Hs(n) i,t 1 < 0 0, if H s(n) i,t 1 0. (5) Thus, H s(n) i,t 1 = Bs(n) i,t 1 + Ss(n) i,t 1. (6) A numerical example is included in the Appendix to illustrate the decomposition of the holdings. Then, equation (1) can be rewritten as R t = ( Ni=1 B i,t Ri,t e Ni=1 B s(n) i,t 1 Ni=1 + R Ni=1 i,t S i,t R e Ni=1 i,t S s(n) i,t 1 H Ni=1 )+( i,t 1 H Ni=1 + R Ni=1 i,t H p(n) i,t 1 i,t 1 H Ni=1 )+ R i,t i,t 1 H Ni=1. i,t 1 H i,t 1 The first and second term on the right-hand side of equation (7) together in the first bracket measure the contribution of net purchases within n days to fund performance. It is crucial to this methodology that the contribution of net purchases within n days to fund performance is equivalent to the contribution of the current holdings with an investment horizon shorter than n days to fund performance. The third and fourth terms on the righthand side of equation (7) together in the second bracket measure the contribution of net sales within n days to fund performance. (7) 14

15 3.1.2 Measures of the Contribution of Trading Costs to Performance I measure the contributions of both explicit and implicit trading costs to fund performance. Explicit trading costs include commissions, taxes, and fees. Implicit trading costs include the intra-day implicit costs related to the price impact of trades and the bid-ask spread, and the multi-day implicit costs related to the liquidity consumption/provision across days. Trades commissions, taxes, and fees are reported directly by Abel Noser Solutions in dollars. I calculate their contribution to daily fund performance as the total dollar amount of those costs each day divided by the TNA of the fund on the same day. I measure the intra-day implicit costs using the execution shortfalls of trades. The execution shortfall is the difference between the price at order placement and the actual execution price as a percentage of the price at order placement. The expression is Pi,t e ES i,t = D P i,t 0 i,t P 0 i,t, (8) where D i,t is 1 for buys and 1 for sells. Pi,t 0 is the stock price at order placement, and P e i,t is the order s actual execution price. This measure of execution shortfall captures several dimensions of institutional trading, including the bid-ask spread, price impact, and slippage costs attributable to delayed executions. Unlike the bid-ask spread, which is always positive, the execution shortfall can be positive or negative depending on market conditions and the extent to which an order demands or supplies liquidity. Funds split large trades into small trades to reduce the price impacts and use limit orders to reduce or even benefit from the bid-ask spreads. The implicit trading costs captured by the execution shortfall are paid during trade execution. I measure the contribution of the execution shortfalls of trades to daily fund performance as ES t = ni=1 ( D i,t V i,t ES i,t ) ni=1 H i,t 1, (9) 15

16 Although the execution shortfall captures a trade s liquidity consumption/provision within a day, it does not capture a trade s liquidity consumption/provision across days. A stock s expected return in the coming days may be positive or negative if the current stock price is at the bottom or the top of a liquidity shock. A trader may trade with or against multi-day liquidity signals. I measure the multi-day liquidity costs using the abnormal returns of the stocks implied from the short-term reversal strategy. Intuitively, a fund trading in accordance with a short-term reversal strategy benefits from providing liquidity to the market, whereas a fund trading in a direction opposite to the short-term reversal strategy pays the costs for consuming liquidity. I construct the short-term reversal signal on the basis of the past week s market-adjusted returns. Following de Groot, Huij and Zhou (2012), I skip one day before constructing the portfolio to control for the price impact of trades. Thus, the short-term reversal signal on day t is based on the returns from t 6 to t 1. I sort all stocks ever held by mutual funds in my sample periods into quintiles on the basis of their short-term reversal signals on each day t and use the average FFC alpha of all stocks in the same quintile as the expected return for all stocks in that quintile 20. The multi-day liquidity costs of each fund on day t caused by trades within the past n days is measured as Ni=1 SR s(n) H s(n) i,t 1 t = Rsr i,t Ni=1, (10) H i,t 1 where R sr i,t is the expected return of stock i on day t according to the short-term reversal quintile to which stock i belongs. Thus, SR s(n) t captures the contribution of the multi-day liquidity consumption/provision of the fund s trades in the past n days to its performance on day t. I use SR s(n) t liquidity cost 21. of the trades in the past 10 days as a measure of the multi-day It is worth noting that the abnormal return of the short-term reversal 20 Some may argue that the short-term reversal strategy is more prominent for small-cap stocks than largecap stocks; therefore, for my analysis, I also sort the short-term reversal quintiles within each size quintile and calculate the expected return of each quintile. The short-term reversal effect is only slightly larger for small-cap stocks than large-cap stocks. All current findings hold. Using deciles rather than quintiles also does not affect the results. 21 Trades more than 10 days ago have nearly no effect on this measure of multi-day liquidity costs. 16

17 strategy is an incomplete measure of multi-period liquidity costs because liquidity shocks can last for months (e.g. The monthly reversal of stock returns) or years (Asset Fire Sales: Coval and Stafford (2007)). 3.2 Empirical Results: All Funds I first present the average contribution of trades to daily fund performance within one year. The first hypothesis I am going to test is whether mutual funds profit from their trades. Hypothesis 1: Mutual funds profit from their trades at each investment horizon. H 0 : Contribution of trades to fund performance (gross alpha or raw return) is zero or negative at each investment horizon. H 1 : Contribution of trades to fund performance (gross alpha or raw return) is larger than zero at each investment horizon. Figure 1 plots the contribution of trades in the past 240 days (one year) to daily fund performance on the basis of equation (3). I measure the contribution using the FFC alpha, Fama-French three-factor alpha, CAPM alpha, and raw return. The alphas and returns are equally weighted across funds and the annualized return is reported. The dash line shows that trades gradually contribute 1.5% to the annual raw return in half a year (120 business days) and contribute 2.0% in a year. This contribution is about 1% (0.5%) per year when measured by CAPM gross alpha (Fama-French 3-factor alpha). When measured by FFC gross alpha, trades contribute positively, 73 bps, the in the first half of the year but negatively, -65 bps, in the second half of the year, adding up to 7 bps a year. Therefore the trades of mutual funds contribute positively to fund performance within a year at each investment horizon and under different measures, supporting Hypothesis 1 that mutual funds profit from their trades. Table 2 reports the average contribution (and costs) of trades to fund performance by investment horizon. Panel A decomposes funds gross alphas on the basis of equation (3) using daily holdings and trades in the same day and the past n (=10/20/60/120/240) days. The gross FFC alpha (Fama-French-Carhart four-factor alpha) is calculated as the FFC 17

18 alpha of daily holdings adjusted by intra-day trading profits/losses, as in equation (1). The contributions of buys/sells (stocks with positive/negative changes in holdings in the past n days) are calculated using equation (7). The contribution from the past days is the difference between the contributions in the past 120 and 240 days. Panel B reports the FFC alpha of fund net returns, the expense ratio, and fund turnover including both purchases and sales. Panel C reports the explicit and implicit trading costs per day. Explicit trading costs include commissions, taxes, and fees as reported in the Abel Noser database. Implicit trading costs include the execution shortfall of the trades and the multi-day liquidity costs implied by the short-term reversal phenomenon. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. Both equally-weighted and value-weighted averages across funds are reported. Robust standard errors are clustered per day because the holdings and trades across funds can be correlated 22. It is noteworthy that the contribution of net purchases within n days to fund performance is equivalent to the contribution of the holdings with an investment horizon shorter than n days. [Insert Table 2 about here] I find that mutual funds possess stock-picking skills in both the short term and the long term, adding up to a gross alpha of 130 basis points (bps) per year before trading commissions. Panel A of Table 2 shows that the contribution of their trades in the past 120 days (half a year) to fund performance is 73 bps per year of a fund s TNA, which represents a substantial fraction of their gross alpha. In addition, if funds do not trade in the past 240 days (maintain their holdings 240 days ago), they still earn a gross alpha of 123 bps per year, indicating that the majority of their value added are from long-term holdings. The total contribution of trades to fund performance is positive as long as the stocks purchased outperform the stocks sold. While the total contribution of purchases and sales (all trades) have been investigated in Hypothesis 1, investigating the contribution of purchases and sales separately reveals more information about funds trading strategies. So I also study a sufficient but not necessary condition of Hypothesis The results become even more significant when the standard errors are clustered at the fund level or when the Newey-West standard errors with lag order from three to 120 days are used to control for autocorrelations. 18

19 Hypothesis 2: Stocks purchased outperform the benchmark and stocks sold underperform the benchmark. H 0 : Contribution of purchases or sales to fund gross alpha is zero or negative. H 1 : Contribution of both purchases and sales to fund gross alpha is larger than zero. Figure 2 plots the contribution of purchases and sales to fund performance (the third and fourth columns of Panel A in Table 2) separately. It shows that both purchases and sales contribute positively to fund performance in half a year, indicating that funds are able to buy before stocks outperform and sell before stocks underperform on average. Among the 73 bps contribution of trades within half a year, 34 bps are from net purchases within 120 days and 39 bps from net sales. They are all statistically significant at the 1% confidence level. In contrast, trades between 120 days and 240 days ago (half a year to one year) cost funds 65 bps per year. The underperformance of their trades between 120 and 240 days ago does not necessarily indicate that funds are unskilful at trading 23. The contribution of purchases to fund performance decays over time. Purchases contribute 13 bps to a fund s performance on the same day, 29 bps within 10 days, 34 bps within 120 days, and only 5 bps within 240 days. The majority of the profits are realized within 10 days. This result is consistent with the results of previous studies (e.g. Jame (2017); Di Mascio, Lines and Naik (2017)) that show that the profits and information density of trades are the highest in the short term and decay over time. In contrast, sales cost the fund 5 bps per year on the same day and 15 bps within 10 days, and only start to contribute 20 days after a sale, with the majority of the contribution being realized from 20 to 120 days. Funds profit instantly after their purchases, but only days after the sale. This partially occurs because sales on average have larger price-impact costs than purchases 23 Firstly, funds may trade for short-term profits and reverse those positions within 120 days. In the sample of this paper, the average investment horizon of the round-trip trades is half a year (the average horizon of round-trip trades reported in Di Mascio, Lines and Naik (2017) is also half a year). Thus, the majority of the round-trip trades have been cancelled out by their counterparts within one year. The net purchases and sales in one year are on average approximately 19% of the fund s total turnover. In Section 5, I document that round-trip trades within one year reduce those losses. Secondly, funds may trade to increase their factor exposures and earn premiums from doing so. In Section 5.4, I show that the additional factor premiums introduced by those trades are large enough to cover the losses in FFC alpha. Thirdly, funds may trade for long-term profits and, thus, choose to bear mid-term losses, which is consistent with the significant contribution of long-term holdings to gross alpha, as documented in this paper. 19

20 during trade execution, as shown in Panel C. Sales are hidden heroes of fund performance. On average, they contribute as much as (if not more than) purchases do to fund performance. The delayed and gradual accumulation, the indirect nature, and the differences across fund styles make sales contributions invisible in the previous literature. As discussed in the previous paragraph, the contribution of sales is delayed and gradual and starts only from 20 days after the sale. Sales contribute indirectly because they contribute through their effects on the performance of long-term holdings. Panel A of Table 2 shows that if funds do not sell under-performing stocks and only stick to their holdings from 120 days prior, then holdings longer than 120 days contribute 39 bps less to a fund s performance. The mathematical relationship between sales within n days and holdings longer than n days is shown in equation (13). Moreover, the contribution of sales differs significantly across fund styles. For example, growth funds sell stocks to protect them from price reversals, whereas value funds sell old stocks that deliver a lower alpha than new stocks (as is shown in Section 5.3). Thus, the contribution of sales cannot simply be evaluated using the performance of the stocks sold for all fund styles. In contrast, the contribution of purchases is direct, instant, and similar across funds. Because net purchases within n days is mathematically equivalent to holdings shorter than n days, the column Buy in Panel A also indicates the contribution of their holdings to fund performance across investment horizons. Thus, holdings shorter than 120 days (half a year) contribute 34 bps per year to the gross alpha and holdings between 120 days and 240 days (half a year to one year) cost funds 29 bps per year. Moreover, mutual funds pay significant trading costs. On average, they pay 151 bps of their fund TNA per year in trading costs (76 bps explicit costs and 75 bps implicit costs), which is approximately the same magnitude of the expense ratio and the gross alpha of those funds. It still amounts to 50 bps when weighted by fund TNA. The trading costs documented are of the same magnitude as those in Busse, Chordia, Jiang and Tang (2017) and slightly larger when calculated as equally weighted. On average, funds pay more trading costs for sales than for purchases. In addition, I find that the multi-day liquidity cost (5 bps) is small relative to other costs. 20

21 Interestingly, the results in Table 2 are in accordance with the model proposed in Berk and Green (2004). When funds are skillful and investors capital provisions to mutual funds are competitive, their gross alphas are positive and their net alphas are close to zero. The commissions, taxes, and fees of their trades 34 bps of fund TNA per year for purchases and 42 bps for sales are approximately the same magnitude as the contribution of purchases and sales to fund performance in 120 days. Therefore, purchases and sales neither contribute nor harm the fund s performance in the short term after trading costs. This result is consistent with the efficient market hypothesis, which states that no arbitrage opportunity remains in the market and funds cannot benefit from their trades after costs. However, given the limits to arbitrage related to long-term mispricings 24, funds profit from their long-term holdings. Consistent with the equally-weighted results, the value-weighted results also show that mutual funds benefit from both their short-term (net purchases within 240 days) and longterm holdings (holdings from 240 days prior). However, large funds on average lose from their sales because either they need to sell relatively less attractive investments for more promising ones (as the data show, the profits from purchases outweigh the losses from sales) or the forced selling causes additional costs that dominate the benefits. Different from the equally-weighted results, the contribution of net purchases accumulates within 240 days and does not decline, which is consistent with the expectation that large funds are more likely to be value investors who invest on the information of fundamentals than short-term arbitrageurs. 3.3 Empirical Results: By Fund Categories Next, I study the contribution of trades to fund performance by fund categories. More specifically, I investigate the sources of their trading profits by documenting the contribution of their trades to fund performance at each investment horizon for each fund category separately. 24 Shleifer and Vishny (1997) argue that trading on long-term mispricings is generally more expensive and difficult because, for example, the fund manager may be fired before ex-post successful long-term bets pay off. 21

22 Hypothesis 3: Growth funds profit from short-term anomalies. H 0 : Stocks bought by growth funds do not outperform in the short term or experience no price reversal in the mid term. H 1 : Stocks bought by growth funds outperform in the short term and experience a price reversal in the mid term. Hypothesis 4: Value funds profit from information of fundamentals. H 0 : Stocks bought by value funds do not gradually outperform or experience a price reversal in the mid term. H 1 : Stocks bought by value funds gradually outperform over time with no price reversal in the mid term. Table 3 reports the results by funds book-to-market quintiles. I sort the funds every year into quintiles on the basis of the value-weighted average book-to-market values of their stock holdings at the end of last year. Figure 3 plots the contribution of purchases and sales to fund performance for growth funds (quintile 1) and value funds (quintile 5) separately, which are also reported in the second and sixth columns of Table 3. Completely in accordance with Hypothesis 3, stocks bought by growth funds contribute substantially to their fund performance in the first 10 to 20 days and experience a severe price reversal from 20 days to a year. Also in accordance with Hypothesis 4, stocks bought by value funds contribute gradually to their fund performance within a year, and the trading profits decay over time but with no price reversal. In addition, stocks sold by growth funds underperform in half a year, and the price reverses back in a year. The stocks sold by value funds slightly outperform the benchmark but still substantially underperform the stocks bought, consistent with the conjecture that value funds sell stocks to make room for better investment opportunities. Growth and value funds clearly profit from their specializations. Table 4 reports the results by funds stock-cap quintiles, which are sorted on the basis of the average market capitalization of their stock holdings at the end of last year; Table 5 reports by funds momentum quintiles based on the average past year performance of their stocks holdings; and Table 6 reports by the fund size quintiles sorted every quarter on the basis of funds TNAs at the end of last quarter. 22

23 [Insert Table 3, 4, 5 and 6 about here] Across different fund categories, I find that, firstly, almost all fund categories profit from their net purchases within 120 days. The value funds in BM quintiles 4 and 5 of Table 3 profit 48 bps and 85 bps per year 120 days from their purchases. The growth and blended funds in quintiles 1, 2, and 3 mainly profit from their purchases in the past 10 days, which amount to 40 bps, 15 bps, and 27 bps per year. The net purchases in the past 120 days contribute less, but still make a positive contribution to fund performance. This result is consistent with the common believe that growth funds profit from extremely shortterm investments (shorter than 10 days), whereas value funds profit from both short-term and mid-term investments. Table 4 shows that purchases of small-cap funds in 120 days contribute more (71 bps) than purchases of mid-cap and large-cap funds (ranging from 10 to 45 bps), probably because small-cap stocks experience more short-term mispricings than mid-cap and large-cap stocks. As expected, Table 5 shows that purchases of momentum funds contribute more (60 bps) than non-momentum funds (24 bps), and purchases of small funds contribute more in 120 days (70 bps) than large funds (37 bps). Moreover, almost all fund categories profit from their purchases on the same day and within 10 days of their trades but lose from their sales in the same period. As previously discussed, net purchases within 120 days are mathematically equivalent to holdings shorter than 120 days. Through the lens of their holdings each day, these results can be interpreted as growth funds mainly profiting from their extremely short-term holdings (holdings shorter than 10 days). In contrast, value funds profit from both their short-term and mid-term holdings, indicating that both growth funds and value funds are skillful at trading, and that their knowledge of investment opportunities differ. Interestingly, the contribution of net purchases within 240 days as reported in Panel A of Table 3 shows that holdings shorter than 240 days (a year) on average contribute to value funds (55 bps for quintile 4 and 96 bps for quintile 5), but harm the performance of growth and blend funds ( 28 bps for quintile 1, 56 bps for quintile 2, and 46 bps for quintile 3). On average, value funds profit from their holdings shorter than one year, but growth and blend funds lose from those holdings when measured by FFC alpha. However, the underperformance 23

24 of their holdings shorter than one year does not necessarily indicate that those funds are unskilful. Firstly, funds may hold those stocks for long-term profits (longer than one year), thus choose to bear their short-term losses, which is consistent with the large contribution of long-term holdings to gross alpha as subsequently documented in Section 4. Secondly, funds may hold those stocks to increase their factor exposures and earn premiums from them. In Section 5.4, I show that funds earn substantial factor premiums through their trades within one year. Secondly, I find that sales of growth, small-cap, momentum, and small funds contribute substantially to fund performance in 120 days, but not for value, large-cap, non-momentum, and large funds. Sales of growth funds and blend funds in BM quintiles 1, 2, and 3 contribute 51 bps, 21 bps, and 63 bps in 120 days, whereas sales of value funds in BM quintile 5 cost 31 bps. Sales of small-cap funds contribute 35 bps in 120 days, more than the 21 bps for large-cap funds; and sales of momentum and small funds contribute 62 bps and 98 bps, correspondingly, which are substantially higher than the 3 bps and 6 bps for nonmomentum and large funds. The fund categories profiting from their sales in 120 days are those that earn substantially from their short-term holdings (within 10 days or 120 days) but lose from their holdings within one year. They also have relatively higher turnover ratios and pay more trading costs than the other fund categories, likely because relying on sales to lock in short-term profits is an important part of high-turnover strategies. Thirdly, the gross alphas of all fund categories are significantly positive at the 5% level, whereas most net alphas are not significantly different from zero. In the sample of this paper, value funds outperform growth funds in term of both gross alphas and net alphas. Although the gross alphas differ across fund categories, the magnitudes of the differences are substantially smaller than the performance difference in their holdings within one year. For example, in Table 5, the contribution of holdings shorter than 240 days is 71 bps for momentum funds in quintile 5 and 22 bps for non-momentum funds in quintile 1. However, their gross alphas (125 bps and 100 bps per year) are similar. In Table 6, the contribution of holdings shorter than 240 days is 47 bps for large funds in quintile 5 and 81 bps for small funds in quintile 1. Yet, the difference in their gross alphas is only ( =) 43 bps. 24

25 Apparently, each fund category has a comparative advantage and their overall performance is not very different from one another s. This is consistent with what is expected from an efficient market of funds with heterogeneous investment skills and supports the argument in Berk and van Binsbergen (2017): The rational expectations equilibrium approximates the observed equilibrium in the money management space at least as well as it does in the stock market. In addition, the total contribution of purchases and sales to fund performance in 120 days is quite close to the commissions, taxes, and fees for most fund categories. In general, the contribution of their trades increases (or decreases) together with the commissions across fund categories. For example, the total contribution of purchases and sales in 120 days for small funds in Table 6 is = 168 bps, and the commission for small funds is 150 bps, whereas the total contribution of purchases and sales in 120 days for large funds is 37 + ( 6) = 31 bps, and the commissions is 24 bps. Similarly, the total contribution of trades in 120 days for momentum funds in Table 5 is = 122 bps, and the commission for momentum funds is 113 bps; the total contribution of trades for non-momentum funds is = 27 bps, and the commissions for non-momentum funds is 61 bps. This pattern approximately holds for most fund categories and indicates that the fund style that trades more actually profits more from its trades in the short term and pays more in trading commissions. In the short term, the net profits of their trades after commissions are mostly close to zero. These results are consistent with the conjecture that mutual funds are skillful at trading, and that the mutual fund industry is competitive and efficient. These results are also consistent with the efficient market hypothesis that states that the market is efficient enough to leave no free lunch in the short term to investors after commissions. Lastly, growth, small-cap, momentum, and small funds pay substantially more in trading costs, especially implicit trading costs as an execution shortfall, than do value, large-cap, non-momentum, and large funds. 25

26 4 Contribution of Long-Term Holdings to Fund Performance In this section, I investigate the contribution of holdings longer than half a year and one year to fund performance. 4.1 Methodology: Measure the Contribution of Long-Term Holdings Each day, I measure the holdings longer than n days by the current holdings already in the portfolio n days ago, and denote these holdings as H l(n) i,t 1. In other words, these holdings are from n days ago that have not been sold and are still in the current portfolio. H l(n) i,t 1 is non-negative. Then, the total holdings at the beginning of day t can be expressed as H i,t 1 = H l(n) i,t 1 + Bs(n) i,t 1, s.t. Hl(n) i,t 1 0 (11) which is the sum of holdings longer than n days and net purchases in the past n days. Intuitively, the holdings on day t can be regarded as the net purchases throughout the entire history of the fund before day t. Then, the contribution of the holdings longer than n days can be calculated as Ni=1 H l(n) i,t 1 R Ni=1 Ni=1 i,t H i,t 1 R i,t B s(n) i,t 1 Ni=1 = H Ni=1 R i,t i,t 1 H Ni=1. (12) i,t 1 H i,t 1 The H l(n) i,t 1 is affected by Ss(n) i,t 1, the net sales in the past n days. It is because the position will no longer be in the portfolio if it has already been sold. Substituting equation (2), (6) into equation (11) gives H l(n) i,t 1 = Hp(n) i,t 1 + Ss(n) i,t 1, (13) where holdings longer than n days can be expressed as holdings n days ago minus the positions sold in the past n days. To also account for the effect of trades on the same day to the contribution of long-term holdings, I deduct the profits/losses from the sales on that day from equation (12). 26

27 4.2 Empirical Results I first investigate whether mutual funds profit from their long-term holdings, and if so how much. Hypothesis 5: Mutual funds profit from their long-term holdings. H 0 : The contribution of long-term holdings to fund performance is zero or negative. H 1 : The contribution of long-term holdings to fund performance is larger than zero. Hypothesis 6: Most fund categories, including high-turnover funds, mainly profit from their long-term holdings. H 0 : The contribution of long-term holdings represents less than half of their gross alphas for most fund categories. H 1 : The contribution of long-term holdings represents more than half of their gross alphas for most fund categories. Table 7 reports the average contribution of long-term holdings to fund performance. Panel A reports the results for all funds and Panel B reports the results by fund category. The FFC alpha from holdings longer than n (=120/240) days is calculated as the difference between the alpha of the daily holdings and the contribution of the net purchases to the fund alpha within n (=120/240) days, as shown in equation (12). To account for trades on the same day, I deduct the profits/losses of the sales on that day. Gross alphas are reported for comparison. All numbers in this table are annualized and reported as a percentage of the fund s total dollar holdings. Both equally-weighted and value-weighted results are reported in Panel A, and the numbers reported in Panel B are equally weighted. Robust standard errors are clustered per day. [Insert Table 7 about here] The results show that the value added of mutual funds are mainly from their long-term holdings. As is reported in Panel A of Table 7, holdings longer than 120/240 days on average contribute 96 bps/126 bps per year (out of a total 130 bps) to fund gross alpha. This result is statistically significant at the 1% level. When weighted by funds TNAs, 27

28 holdings longer than 120/240 days on average contribute 112 bps/103 bps per year (out of a total 138 bps) to fund gross alpha. This finding is consistent with Cremers and Pareek (2016), who find that, among funds with large active shares, only those with long holding horizons outperform. Using daily holding data, I distinguish new holdings and old holdings every day for each horizon, which is more accurate than the investment horizons inferred from the quarterly holdings in previous papers. All fund categories profit from their holdings longer than 120 and 240 days, and the majority of funds gross alphas come from their long-term holdings for most fund categories. For example, for growth funds in quintile 1 of Table 3, 97 out of 98 bps are from holdings longer than 120 days. Holdings longer than 240 days contribute even more (126 bps) to fund performance, a result that holds for all other fund categories except for value funds in quintile 5 of Table 3. For these funds, 56 out of 142 bps of their fund gross outperformance are from their holdings longer than 120 days. The majority 85 bps are from their new holdings within 120 days. Therefore, despite their styles, most mutual funds in this industry mainly add value through their long-term holdings. Interestingly, the gross alphas of growth/small-cap/momentum/small funds from holdings longer than 240 days are larger than their overall gross alphas. The gross alphas for growth funds (quintile 1) from holdings longer than 240 days is 125 bps, which is larger than the overall gross alpha of 98 bps per year. For small-cap funds (quintile 1), the gross alpha from holdings longer than 240 days is 170 bps, which is larger than the overall gross alpha of 155 bps per year. The gross alphas from holdings longer than 240 days are 196 bps and 219 bps for momentum funds (quintile 5) and small funds (quintile 1), which are substantially larger than their overall gross alphas of 125 bps and 137 bps. These results are consistent with the findings in Panel A of Table 3, 4, 5, and 6, which show that their holdings shorter than 240 days harm their fund performance for these fund categories. Therefore, these funds not only benefit from their holdings longer than 240 days but also earn enough from their long-term holdings to cover their losses from holdings shorter than 240 days. Eventually, the overall gross alphas of growth/small-cap/momentum/small funds are not substantially different from those of other categories. 28

29 As I have shown in the previous section that high-turnover funds profit more from their short-term holdings than low-turnover funds do, it is easy to speculate that low-turnover funds profit more from their long-term holdings than high-turnover funds do. Is it indeed the case? Hypothesis 7: Low-turnover funds profit more from their long-term holdings than high-turnover funds do. H 0 : The long-term holdings of low-turnover funds contribute less (or the same) to their gross alpha than the long-term holdings of high-turnover funds do. H 1 : The long-term holdings of low-turnover funds contribute more to their gross alpha than the long-term holdings of high-turnover funds do. Table 8 reports the fraction of holdings at each investment horizon for each fund category. Mathematically, fund categories with higher turnover as reported in the tables in Section 3.3 also have shorter average investment horizons. Although growth/smallcap/momentum/small funds have shorter investment horizons and a smaller fraction of portfolios with long-term holdings, their gross alphas from holdings longer than 240 days are larger than value/large-cap/non-momentum/large funds, as shown in Panel B of Table 7. For example, the holdings longer than a year represent 76% of the total holdings of growth funds (in quintile 1), which is smaller than the 85% for value funds (in quintile 5). Whereas the holdings longer than a year contribute 1.25% per year to the gross FFC alpha of growth funds, which is substantially larger than the 0.45% for value funds. The same holds for small-cap funds vs. large-cap funds, momentum funds vs. non-momentum funds, and small funds vs. large funds. Figure 4 plots the fraction of long-term holdings (holdings longer than a year) in funds portfolios against the contribution of long-term holdings to funds annual gross FFC alpha for each fund category. The negative correlation between these two across all 4 fund categories shows clearly that high-turnover funds on average pick their long-term positions better than low-turnover funds do. This negative correlation is a strong evidence that mutual funds, especially high-turnover funds, possess stock picking skill in the long term. 29

30 The evidence from short-term and long-term holdings adds up and shows that mutual funds are skillful and add value to the market. Mutual funds in all categories profit from their purchases in the short term and holdings in the long term, and the fund categories (growth/small-cap/mom/small funds) that lose money from their mid-term holdings profit more from their short-term and long-term holdings than the other categories. One possible reason is that growth/small-cap/momentum stocks are mispriced more in both the short term and the long term. Greater opportunity means more work; therefore, these fund categories need to more frequently replace underperforming stocks with potentially outperforming stocks, leading to higher turnover. The underperformance of their mid-term holdings is the cost and time they need to pay to identify stocks with superior quality in the long term. This scenario is similar to the case in which angel investors need to invest in a large number of start-ups and sell most of them in a short period a necessary step for them to find the small number of high-quality firms that prosper and generate the most value for them in the long term. 5 Contribution of Gross Purchases and Sales Price Dynamics after Trades Until now, I have documented the contribution of net purchases and net sales to fund performance for each time horizon and fund category. However, this contribution is not informative about the motives of their trades. Firstly, it does not allow us to see the price dynamics of the stocks traded, which contain useful information on their trading motives. It is almost common knowledge in the asset pricing literature that price changes that do not reverse are caused by new information on stock (or market) fundamentals, and those that reverse are caused by short-term anomalies. Secondly, it does not tell us the degree to which their proper round-trip trades (selling at the right time) contribute to their performance because the round-trip trades were cancelled out in previous settings. In this section, I separately aggregate all purchases and sales of each stock at each time horizon and 30

31 investigate their contributions to fund performance 25. To distinguish from net purchases and net sales, I call them gross purchases and gross sales, enabling us to determine whether the prices of the stocks purchased subsequently reverse and, if so, how much those reversals would cost those funds if no sales protected them from such reversals. 5.1 Methodology: Contribution of Gross Purchases and Sales Price Dynamics after Trades In this section, I decompose funds performance using their trades every 10 days to investigate the effect of trades on fund performance at different time horizons irrespective of whether or not they have been cancelled out by opposite trades. R t = Ni=1 V i,t Ri,t e Ni=1 H s(10) i,t 1 Ni=1 + R i,t H Ni=1 + i,t 1 H i,t 1 Ni=1 H s(11 20) i,t 1 R Ni=1 i,t H s(21 30) i,t 1 R i,t Ni=1 + H Ni=1 i,t 1 H i,t 1 Ni=1 H s( ) i,t 1 R Ni=1 i,t H s( ) i,t 1 R i,t... + Ni=1 + H Ni=1 + i,t 1 H i,t 1 (14) Ni=1 H p(240) i,t 1 R i,t Ni=1 H i,t 1 Here, I define s(n) s(n) B i,t 1 as gross purchases of stock i within n days before day t and S i,t 1 as gross sales of stock i within n days before day t, and document the contribution of gross 25 If divided by fund turnover, the result of this analysis is equivalent to the average performance of purchases/sales in the following days/months, as reported in the previous literature. 31

32 purchases and gross sales every 10 days. R t = ( Ni=1 B i,t Ri,t e Ni=1 Bs(10) i,t 1 Ni=1 + R i,t H Ni=1 + i,t 1 H i,t 1 Ni=1 Bs(11 20) i,t 1 R i,t Ni=1 H i,t 1 + Ni=1 Bs(21 30) i,t 1 R i,t Ni=1 H i,t Ni=1 Bs( ) i,t 1 R i,t Ni=1 H i,t 1 + Ni=1 Bs( ) i,t 1 R i,t Ni=1 H i,t 1 )+ ( Ni=1 S i,t Ri,t e Ni=1 Ss(10) i,t 1 Ni=1 + R i,t H Ni=1 + i,t 1 H i,t 1 Ni=1 Ss(11 20) i,t 1 R i,t Ni=1 H i,t 1 + Ni=1 Ss(21 30) i,t 1 R i,t Ni=1 H i,t 1 (15)... + Ni=1 Ss( ) i,t 1 R i,t Ni=1 H i,t 1 + Ni=1 Ss( ) i,t 1 R i,t Ni=1 H i,t 1 ) + Ni=1 H p(240) i,t 1 R i,t Ni=1 H i,t 1 This decomposition is mathematically equivalent to the analysis of stock performance after purchases and sales. The only difference is that previous studies measure performance as a percentage of the trading amount, whereas I measure it as the contribution to fund performance as a percentage of fund TNAs to make it comparable to the contribution of net purchases and net sales to fund performance. By comparing the contribution of gross purchases with the contribution of net purchases at each time horizon, we can understand the degree to which sales of the round-trip trades within that time horizon reduce the gains/losses of those purchases. 5.2 Empirical Result: All Funds Figure 5 plots the total contribution of trades (both purchases and sales) on the same day and in the past 10 to 240 days (every 10 days) to the current daily fund performance, as described in equation (14). The grey bars represent the incremental contribution of the trades to fund performance as a percentage of fund TNA, and the black line plots the cumulative contribution. The exact numbers and their statistical significance are in the second and third columns of Table 9. The black line in Figure 5 shows that their trading 32

33 profits accumulate gradually within 120 days after the trades, which contribute a total of 73 bps per year to fund performance. In contrast, their trades hurt fund performance from 120 to 240 days after such trades, which cost them 65 bps per year. [Insert Figure 5 about here] Then, in Figure 6, I separately plot the contribution of gross purchases and sales to fund performance using equation (15). The exact numbers and their statistical significance are reported in columns 4 to 7 of Table 9. As is shown by the solid line in the left panel of Figure 6, funds profit instantly after their purchases but lose substantially in the subsequent price reversal. On average, the purchases contribute 41 bps per year to fund performance in the first 40 days, almost break even from 40 to 120 days, and gradually erode 52 bps from fund performance per year in the next 120 days. Most funds purchases profit from short-term overpricing that reverses in subsequent periods. Therefore, funds are more likely to profit from short-term anomalies than their information on stock/market fundamentals through their purchases. In stark contrast with purchases, sales (the solid line in the right panel) cost the fund 15 bps per year in the first 10 days. The contribution from sales only starts from 20 days after the trades and accumulates gradually to 40 bps per year from 20 to 160 days. This delayed and gradual accumulation of the contribution from sales to fund performance is one reason it is invisible in most existing literature. Moreover, it only reverses slightly from 40 to 22 bps from 160 to 240 days after the sales. Therefore, funds sales are more likely to be based on their information on fundamentals, although a small fraction of the sales are from anomalies. Next, I compare the contributions of gross and net purchases to understand the degree to which sales in round-trip trades reduce the losses of those purchases from the price reversals. The dotted line in the left panel of Figure 6 plots the contribution of net purchases within 240 days, as previously documented in Table 2. The only difference between the dotted line and the solid line is that the dotted line does not include the purchases that were cancelled out by the sales at each time horizon. This shows that including the purchases that were cancelled out by sales drives the contribution of the purchases in 240 days from 7 bps per year to 14 bps per year. In other works, sales in round-trip trades within one 33

34 year contribute (7 ( 14) =) 21 bps to fund performance per year. This contribution is only partial (out of a total 40 bps) because it can also prevent losses beyond 240 days after purchases. Interestingly, the cumulative contribution of both gross purchases and sales are mostly positive within one year (240 days), and both increase in the first half of the year and decline in the second half. Mutual funds on average seem to manage to ride the bubbles and avoid the valleys buy before the outperformance and sell before the underperformance. Otherwise, the cumulative contribution should be zero if these funds trade randomly or should be negative if they have been taken advantage by other investors. [Insert Figure 6 about here] [Insert Table 9 about here] 5.3 Empirical Result: Growth Funds vs. Value Funds Growth funds are mostly regarded as short-term investors who profit from short-term anomalies, value funds instead are regarded as value investors who profit from information on firm fundamentals. Therefore, in this section, I investigate the price dynamics of stocks purchased by those funds and the contribution of their sales to fund performance separately for the two fund categories. Figure 7 plots the contribution of gross purchases and sales to fund performance from 10 to 240 days separately for growth funds and value funds. The exact numbers are reported in Table 10. The solid line in the top-left panel plots the cumulative contribution of gross purchases to fund performance for growth funds (Holding BM quintile 1). This plot shows that growth funds profit instantly after their purchases but also that the price reverses starting from 30 days and continues to decline until at least 240 days after the purchases in accordance with the profile of an investor who profits from short-term anomalies. Also in accordance with the profile of a value investor who profits from information on fundamentals, the solid line in the bottom-left panel for value funds shows that they profit gradually after their purchases and the price does not reverse. Relative to the contribution 34

35 of net purchases the dotted line in each panel the top-left panel shows that sales of round-trip trades within one year save growth funds (( 28) ( 65) =) 37 bps per year from price reversals. Moreover, the bottom-left panel shows that sales of round-trip trades slightly reduce the profit from their purchases. Interestingly, the top-right and bottom-right panels show that the prices of stocks sold by growth funds reverse, whereas those sold by value funds do not. The negative contribution of sales of value funds to their performance indicates that stocks sold by such funds also outperform the market. However, the stocks that they purchase more significantly outperform the market relative to the stocks that they sell (as shown in the bottom-left panel). Those stocks seem to be sold for even better investment opportunities. Table 10 also reports the results for Holding BM quintiles 2, 3, and 4. Consistently, the prices of stocks bought by funds in quintiles 2 and 3 (moderate growth and blend funds) reverse, whereas the prices of stocks bought by funds in quintile 4 (moderate value funds) do not reverse. [Insert Figure 7 about here] [Insert Table 10 about here] 5.4 Contribution of Trades to Fund Performance by Factor Exposures In previous sections, I showed that trades of mutual funds cost 65 bps per year from 120 to 240 days after such trades when measured using FFC alpha and that funds on average lose 29 bps per year from their holdings from 120 to 240 days. Those trades and holdings may profit from their factor exposures in this intermediate period, and these profits are large enough to cover the losses measured by FFC alpha. In this section, I measure the contribution of the trades to fund performance through each factor exposure. Figure 1 plots the contribution of trades to daily fund performance using different measures. As discussed in previous sections, when measured by the FFC four-factor alpha, mutual funds profit from their trades in 120 days, but lose from their trades from 120 to 240 days. However, when measured using the CAPM alpha, they almost break even in 35

36 their trades from 120 to 240 days. When measured using the raw return, they even earn a profit. These results indicate that the premiums that they earn from momentum, value, and size anomalies are large enough to cover the losses of those trades from 120 to 240 days when measured by the FFC four-factor alpha, and the market premium compensates even further. This result suggests that mutual funds may use the CAPM alpha when making their investment decisions instead of the FFC four-factor alpha, which is consistent with Berk and van Binsbergen (2016), who show that investors of mutual funds mainly chase CAPM alphas. Table 11 reports the average contribution of trades to fund performance through different factor exposures. Consistent with Figure 1, Table 11 shows that funds trades contribute 61 bps per year from 120 to 240 days after such trades by increasing their exposures to momentum, HML, and SMB factors, which is close to the 65 bps per year losses in the same period when measured using FFC alpha. Among the 61 bps from factor exposures, 35 bps are from the exposure to momentum, 13 bps from HML, and 13 bps from SMB. The increase in market exposure contributes an additional 66 bps per year. The second column in Table 11 shows that trades in the past 40 days decrease the momentum exposure of funds, probably because of short-term reversal strategies applied, and trades in the past 40 to 240 days gradually increase the momentum exposure. On average, the momentum exposure of funds daily holdings contributes 76 bps per year to fund performance. The third column shows that trades in the past 120 days decrease the HML exposure and trades in the past 120 to 240 days increase the HML exposure. The average HML exposure of their daily holdings is negative but not statistically significant because the equally-weighted results tilt towards small growth funds that are more likely to take short-term investment opportunities through growth stocks in the short term and release them in the mid term. In sharp contrast, the value-weighted results in Table A1 report a premium of 98 bps from HML exposure of funds daily holdings because it tilts towards large value funds. Trades in the past 240 days gradually increase funds SMB and market exposure, and the SMB and market exposure of their daily holdings contribute 106 bps and 426 bps per year to fund performance, correspondingly. In contrast, the value- 36

37 weighted results in Table A1, which tilt towards large funds, report a slightly negative exposure to SMB ( 8 bps) and a much smaller premium from the exposure to the market risk (139 bps) of their daily holdings. [Insert Figure 1 about here] [Insert Table 11 about here] 6 Conclusion Understanding whether (and, if so, how) active fund managers add value is of significant importance. Existing mutual fund studies mostly rely on fund return data and have not reached a consensus to the answer to this question yet, partially because the time variation in funds investment strategies, characteristics, and investment opportunities reduces the statistical power of this test. The thread of the literature using quarterly holding data and the thread using transaction data to resolve this question are not conclusive as well. They each capture one incomplete aspect of a fund s decision, and the methodologies used in these two threads of the literature to measure their contribution to fund performance are inconsistent. In this paper, a unified framework is designed to decompose (reconstruct) funds performance on the basis of the investment horizon of their holdings and transactions. Daily holding and transaction data on 336 US mutual funds from 2001 to 2010 are used for this analysis. Building up from their daily holdings and transactions, I find that active mutual funds add value and possess stock picking skills in both the short and long terms. Funds generate a gross alpha of 130 bps per year before trading commissions. Holdings shorter than half a year contribute 34 bps per year to the gross alpha; holdings between half a year and one year cost funds 29 bps; and holdings longer than one year contribute 125 bps. All fund categories profit from their holdings shorter than half a year. Moreover, for all categories, most of their gross alphas are from their holdings longer than one year. Growth/smallcap/mom/small funds profit more (at least as much) from their holdings longer than one year than do other fund categories. 37

38 I also find clear evidence that funds are skillful at trading. Both purchases and sales of mutual funds contribute to the gross fund performance within half a year. Sales are the hidden heroes to fund performance and, on average, contribute as much as (if not more than) purchases do to fund performance by protecting funds from potential losses through the dumping of bad stocks. In addition, I find that the prices of stocks bought by growth funds reverse substantially from 30 days after their purchases, and sales in their round-trip trades to some extent protect them from those reversals. This finding suggests that growth funds profit from short-term anomalies. In contrast, the prices of stocks bought by value funds increase gradually in the following year and do not reverse, indicating that value funds profit from information on fundamentals. On average, mutual funds pay as much as 1.51% of the fund TNA per year for trading costs (76 bps for explicit costs and 75 bps for implicit costs), which is approximately the same magnitude as the expense ratio and gross alpha of those funds. This paper empirically links the trades and the performance of mutual funds and provides evidence that these funds are skillful at trading through the lens of their holding performance by investment horizon. Mutual funds know what they do. Because I only document the ex post performance of their trades, it would be interesting to further investigate the sources of their trading skills and the signals that they use. Given the limitation on the time length of the trading data, I only decomposed the contribution of trades and holdings to fund performance within one year, leaving the analysis of longer than one year to future research. Moreover, an analysis using a larger sample of mutual funds or other investment service providers for more recent years would be of significant assistance to assuring the generalization of the results in this paper. 38

39 7 Tables and Figures Table 1: Summary Statistics and Comparison with CRSP Sample This table reports the summary statistics of the 336 funds in the sample used in this paper and all equity mutual funds in the CRSP mutual fund database. Panel A is for the sample in this paper and Panel B is for the CRSP sample. Column Year represents the year of the records. Num. of Funds represents the number of funds in the sample. Fund TNA ($mn) represents the average TNA of the funds in millions of dollars. Stock Holding (%) represents equity holdings as a percentage of fund TNA, and Cash Holding (%) represents cash holdings as a percentage of fund TNA. Expense Ratio (%) represents the annual expense ratio and Management Fee (%) represents the management fee. Turnover (%) represents the annual turnover reported in CRSP, which is the minimum of aggregate purchases and aggregate sales during the calendar year divided by the average TNA of the fund. Fund Age represents the average age of the funds. All data reported are from the CRSP mutual fund database. Year Num. of Funds Fund TNA ($mn) Stock Holding (%) Cash Holding (%) Turnover (%) Expense Ratio (%) Manage -ment Fee (%) Fund Age Panel A: 336 funds merged , , , Panel B: All equity funds in CRSP mutual fund database ,486 1, , ,990 1, ,104 1,

40 Table 2: Contribution of Trades/Holdings to Performance within One Year This table reports the average contribution (and costs) of trades to fund performance by investment horizon. Panel A decomposes the fund s gross alpha using equation (3) and daily holdings and trades in the same day and within n (=10/20/60/120/240) days. The gross FFC alpha (Fama- French-Carhart four-factor alpha) is calculated as the FFC alpha of the daily holdings adjusted by intra-day trading profits/losses, as in equation (1). The contribution of buys/sells (stocks with positive/negative net changes in holdings in the past n days) are calculated using equation (7). The contribution of net purchases within n days is equivalent to the contribution of the holdings with an investment horizon shorter than n days. The contribution from the past days is the difference in the contributions in the past 120 days and 240 days. Panel B reports the average FFC alpha of the fund s daily net returns, the expense ratio, and the total turnover including all purchases and sales. Panel C reports the explicit and implicit trading costs, including commissions, taxes, and fees reported in the Abel Noser database, the execution shortfall of trades, and the multi-day liquidity costs implied by the short-term reversal phenomenon. All numbers in this table are annualized and represent a percentage of fund TNA. Both equally-weighted and value-weighted averages across funds are reported. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Equally-Weighted Value-Weighted Total Buy Sell Total Buy Sell Panel A: Decomposition of Fund Performance (FFC Alphas) from Daily Holdings 1.22*** 1.33*** from Trades in the same day 0.08*** 0.13*** -0.05*** 0.06*** 0.06*** Gross FFC Alpha 1.30*** 1.38*** from Trades (including the same day) within 10 days 0.14*** 0.29*** -0.15*** ** within 20 days 0.30*** 0.33*** * 0.10* within 60 days 0.52*** 0.30*** 0.22** 0.13** 0.28*** -0.15* within 120 days 0.73*** 0.34*** 0.39*** 0.09* 0.27** -0.19** within 240 days *** 0.36*** -0.11* from Holdings 240 days ago 1.23*** 1.14*** "from past days" -0.65*** -0.29*** 0.16** 0.09** Panel B: Fund Characteristics Fund FFC Alpha (net) Expense Ratio Turnover (buy + sell) Panel C: Trading Costs Explicit Costs - Commissions -0.68*** -0.34*** -0.34*** -0.18*** -0.09*** -0.09*** - Taxes and Fees -0.08*** 0.00** -0.08*** -0.02*** 0.00** -0.02*** Total (Explicit) -0.76*** -0.34*** -0.42*** -0.20*** -0.09*** -0.11*** Implicit Costs (deducted) - Execution Shortfall -0.70*** -0.34*** -0.37*** -0.28*** -0.13*** -0.15*** - Multi-Day SR Costs -0.05*** -0.02** ** -0.02*** -0.01* -0.01* Total (Implicit) -0.75*** -0.32*** -0.40*** -0.30*** -0.14*** -0.16*** Total Trading Costs -1.51*** -0.66*** -0.82*** -0.50*** -0.23*** -0.27***

41 Table 3: Contribution of Trades by Fund Categories: Value/Growth This table reports the average contribution of trades to daily fund performance by funds book-tomarket quintiles. I sort the funds every year into quintiles on the basis of the value-weighted average book-to-market values of their stock holdings at the end of the prior year. Panel A reports the contribution of trades to fund performance for purchases and sales separately using equation (7). Panel B reports fund characteristics. Panel C separately reports the trading costs for commissions, taxes, and fees, execution shortfalls, and multi-day liquidity costs. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Holding BM Quintile 1 Growth Value Panel A: Contribution of Trades to Fund Performance Gross FFC Alpha 0.98** 0.69* 1.09*** 2.28*** 1.41*** Net Purchases (Holdings by Investment Horizon) same day 0.25*** 0.19*** 0.10*** 0.10*** 0.03 within 10 days 0.40*** 0.15** 0.27*** 0.30*** 0.33*** within 120 days ** 0.10** 0.48*** 0.85*** within 240 days -0.28** -0.56** -0.46*** 0.55*** 0.96*** Net Sales same day *** -0.08*** -0.07* -0.05* within 10 days -0.03* -0.53*** * -0.08** within 120 days 0.51*** 0.21** 0.63*** * within 240 days * Panel B: Fund Characteristics Net FFC Alpha * Expense Ratio Turnover (buy + sell) TNA ($million) Panel C: Trading Costs - Commissions -0.79*** -0.72*** -0.56*** -0.64*** -0.66*** - Taxes and Fees -0.04*** -0.06*** -0.08*** -0.06*** -0.06*** - Execution Shortfall -1.07*** -1.05*** -0.64*** -0.50*** -0.14*** - Multi-Day SR Costs -0.13*** -0.09*** * 0.00 Total -2.02*** -1.92*** -1.28*** -1.20*** -0.85*** 41

42 Table 4: Contribution of Trades by Fund Categories: Small-Cap/Large-Cap This table reports the average contribution of trades to the daily fund performance by funds stockcap quintiles. I sort the funds every year into quintiles on the basis of the average market capitalization of their stock holdings at the end of the prior year. Panel A separately reports the contribution of trades to fund performance for purchases and sales using equation (7). Panel C separately reports the trading costs for commissions, taxes, and fees, execution shortfalls, and multi-day liquidity costs. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Stock Size Quintiles 1 Small-Cap Large-Cap Panel A: Contribution of Trades to Fund Performance Gross FFC Alpha 1.55*** 0.90** 1.35*** 1.13** 1.54*** Net Purchases (Holdings by Investment Horizon) same day 0.27*** 0.19*** 0.10*** 0.07*** 0.04** within 10 days 0.66*** 0.15** 0.30*** 0.24*** 0.09** within 120 days 0.71*** 0.10** 0.14*** 0.45*** 0.10** within 240 days -0.15** 0.28** -0.13* 0.35*** Net Sales same day -0.10*** -0.03* -0.04** -0.05*** -0.04** within 10 days -0.22*** -0.08** -0.11*** -0.12*** -0.14*** within 120 days 0.35*** 0.28*** 0.22** * within 240 days * Panel A: Fund Characteristics Net FFC Alpha * Expense Ratio Turnover (buy + sell) TNA ($million) Panel C: Trading Costs - Commissions -0.87*** -0.91*** -0.70*** -0.44*** -0.44*** - Taxes and Fees -0.06*** -0.06*** -0.05*** -0.05*** -0.05*** - Execution Shortfall -1.21*** -0.84*** -0.48*** -0.41*** -0.43*** - Multi-Day SR Costs -0.07*** -0.05*** 0.00* -0.03** -0.09*** Total -2.21*** -1.86*** -1.24*** -0.93*** -1.01*** 42

43 Table 5: Contribution of Trades by Fund Categories: Momentum This table reports the average contribution of trades to daily fund performance by funds momentum quintiles. I sort the funds every year into quintiles on the basis of the average past year performance of their stock holdings at the end of the prior year. Panel A separately reports the contribution of trades to fund performance for purchases and sales using equation (7). Panel B reports the fund characteristics. Panel C separately reports the trading costs for commissions, taxes, and fees, execution shortfalls, and multi-day liquidity costs. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Momentum Quintiles 1 non-mom Mom Panel A: Contribution of Trades to Fund Performance Gross FFC Alpha 1.00** 1.71** 1.33*** 1.20* 1.25** Net Purchases (Holdings by Investment Horizon) same day 0.09*** 0.03* 0.10*** 0.15** 0.31*** within 10 days 0.34*** 0.23** 0.36*** 0.20** 0.33** within 120 days 0.24** 0.38*** 0.54** *** within 240 days 0.22** 0.45** 0.57*** -0.44** -0.71*** Net Sales same day -0.05*** -0.06*** -0.05*** -0.02* -0.09*** within 10 days *** -0.07* -0.17*** -0.34*** within 120 days ** 0.12** 0.47*** 0.62*** within 240 days 0.09* -0.08* * 0.00 Panel B: Fund Characteristics Net FFC Alpha ** Expense Ratio Turnover (buy + sell) TNA ($million) Panel C: Trading Costs - Commissions -0.61*** -0.30*** -0.62*** -0.75*** -1.13*** - Taxes and Fees -0.05*** -0.05*** -0.04*** -0.04*** -0.08*** - Execution Shortfall -0.72*** -0.25*** -0.29*** -0.68*** -1.54*** - Multi-Day SR Costs -0.04*** 0.00** 0.00* -0.08* -0.14*** Total -1.41*** -0.60*** -0.95*** -1.54*** -2.90*** 43

44 Table 6: Contribution of Trades by Fund Size This table reports the average contribution of trades to daily fund performance by fund size quintiles. I sort the funds into quintiles every quarter using funds TNAs at the end of the prior quarter. Panel A separately reports the contribution of trades to fund performance for purchases and sales using equation (7). Panel B reports the fund characteristics. Panel C separately reports the trading costs for commissions, taxes and fees, execution shortfalls, and multi-day liquidity costs. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Fund Size Quintiles 1 Small Large Panel A: Contribution of Trades to Fund Performance Gross FFC Alpha 1.37** 2.09*** 0.74* 1.44*** 0.94** Net Purchases (Holdings by Investment Horizon) same day 0.16*** 0.16*** 0.14*** 0.13*** 0.09*** within 10 days 0.40*** 0.37*** 0.27*** 0.25*** 0.14*** within 120 days 0.70*** ** 0.25** 0.37*** within 240 days -0.81*** ** 0.47*** Net Sales same day -0.08*** -0.04*** -0.08*** * within 10 days -0.03* -0.36*** -0.18*** -0.11* -0.07** within 120 days 0.98*** ** -0.16* within 240 days * * Panel B: Fund Characteristics Net FFC Alpha * 0.14 Expense Ratio Turnover (buy + sell) TNA ($million) Panel C: Trading Costs - Commissions -1.50*** -0.55*** -0.51*** -0.57*** -0.24*** - Taxes and Fees -0.11*** -0.09*** -0.07*** -0.06*** -0.02*** - Execution Shortfall -0.90*** -0.82*** -0.68*** -0.53*** -0.48*** - Multi-Day SR Costs -0.02** -0.06*** -0.09*** -0.04*** -0.04** Total -2.54*** -1.52*** -1.35*** -1.20*** -0.77*** 44

45 Table 7: Contribution of Long-Term Holdings to Fund Performance This table reports the average contribution of long-term holdings to fund performance. Panel A reports the results for all funds. Panel B reports the results by fund category. The FFC alpha from holdings longer than n (=120/240) days is calculated as the difference between the alpha of the daily holdings and the contribution of net purchases to the fund alpha within n (=120/240) days, as shown in equation (12). To account for trades on the same day, I deduct the profits/losses of the sales on that day. Fund performance is measured using FFC alpha. Gross alphas are reported for comparison. All numbers in this table are annualized and reported as a percentage of the total dollar holdings of the fund. Both equally-weighted and value-weighted results are reported in Panel A, and the numbers reported in Panel B are equally weighted. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Panel A: All Funds Equally-Weighted Value-Weighted Gross FFC Alpha 1.30*** 1.38*** > 120 days 0.97*** 1.12*** > 240 days 1.26*** 1.03** Panel B: by Fund Categories Holding BM Quintile 1 Growth Value Gross FFC Alpha 0.98** 0.69* 1.09*** 2.28*** 1.41*** > 120 days 0.97** 0.54** 0.99*** 1.80*** 0.56** > 240 days 1.25*** 1.25** 1.55*** 1.73*** 0.45** Stock Size Quintiles 1 Small-Cap Large-Cap Gross FFC Alpha 1.55*** 0.90** 1.35*** 1.13** 1.54*** > 120 days 0.84** 0.81** 1.21*** 0.67** 1.44*** > 240 days 1.70*** 0.62* 1.48*** 0.77** 1.62*** Momentum Quintiles 1 non-mom Mom Gross FFC Alpha 1.00** 1.71** 1.33*** 1.20* 1.25** > 120 days 0.76* 1.33*** 0.80** 1.40*** 0.65** > 240 days 0.79** 1.26** 0.77** 1.64*** 1.96*** Fund Size Quintiles 1 Small Large Gross FFC Alpha 1.37** 2.09*** 0.74* 1.44*** 0.94** > 120 days 0.67** 2.17*** 0.56** 1.19*** 0.58** > 240 days 2.19*** 2.06*** 0.78** 1.20*** 0.47* 45

46 Table 8: Holdings by Investment Horizon (as a fraction of total holdings) This table reports holdings by investment horizon as a fraction of total holdings. Panel A reports the results for all funds. Panel B reports the results by fund category. Holdings shorter than (net purchases within) 20/60/120/240 days and holdings longer than 240 days are reported as a fraction of total holdings. Both equally-weighted and value-weighted results are reported in Panel A, and the numbers reported in Panel B are equally-weighted results. Panel A: All Funds Equally-Weighted Value-Weighted < 20 days < 60 days < 120 days < 240 days > 240 days Panel B: by Fund Categories Holding BM Quintile 1 Growth Value < 20 days < 60 days < 240 days > 240 days Stock Size Quintiles 1 Small-Cap Large-Cap < 20 days < 60 days < 240 days > 240 days Momentum Quintiles 1 non-mom Mom < 20 days < 60 days < 240 days > 240 days Fund Size Quintiles 1 Small Large < 20 days < 60 days < 240 days > 240 days

47 Table 9: Contribution of Gross Purchases and Sales Price Dynamics after Trades This table separately reports the contribution of gross (all) purchases and sales on the same day and in the past 10 to 240 days to fund performance using equation (15). The first row, Day 0, is for the same day of the trades. The contribution of Changes in Holdings to fund performance at each time horizon is calculated using equation (14), and the contribution of All Purchases and All Sales is based on equation (15). All numbers in this table are annualized and reported as a percentage of fund TNA. The equally-weighted average across funds is reported. Robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Changes in Holdings All Purchases All Sells Day Incremental Cumulative Incremental Cumulative Incremental Cumulative *** 0.08*** 0.13*** 0.13*** -0.05*** -0.05*** * 0.14*** 0.15*** 0.29*** -0.10** -0.15*** *** 0.30*** 0.07* 0.35*** 0.09** *** 0.42*** 0.04* 0.40*** 0.07** ** *** * 0.49*** ** 0.08** 0.11** *** *** ** *** *** 0.07** 0.19** ** 0.59*** 0.03* 0.37*** 0.03* 0.22** * 0.64*** ** 0.03* 0.25** * 0.67*** *** *** ** *** 0.04* 0.30*** * 0.73*** *** 0.05* 0.36*** *** 0.65*** -0.08*** 0.30*** ** * 0.63*** -0.04* 0.25** 0.03* 0.38*** *** -0.03* 0.22** 0.03* 0.40*** *** * *** *** 0.52** -0.07** 0.14* *** *** * *** * 0.50*** * -0.04* 0.35** *** 0.41** -0.11*** *** ** 0.37*** ** 0.31*** ** 0.29** -0.07** *** *** 0.18* -0.03* -0.05* -0.08*** 0.22** *** *** -0.14*** ** *** -0.52*** -0.14* 47

48 Table 10: Contribution of Gross Purchases and Sales: Growth Funds vs. Value Funds This table reports the contribution of trades in the past 20 to 240 days to fund performance using funds book-to-market quintiles. I sort the funds every year into quintiles on the basis of the valueweighted average book-to-market values of their stock holdings at the end of the prior year. Panel A indicates all purchases (not net purchases) in the past 20 to 240 days, and Panel B indicates all sales. All numbers in this table are annualized and reported as a percentage of fund TNA. The equally-weighted average across funds is reported. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 Holding BM Quintile 1 Growth Value Panel A: Purchases ** 0.09* 0.23** 0.39* 0.47** *** -0.13* 0.44*** 0.49*** 0.63** ** -0.18** 0.38*** 0.56* 0.77** ** -0.43* 0.34* 0.66*** 0.93** * -0.47*** 0.23* 0.68*** 1.14** ** -0.49*** 0.08* 0.65** 1.12*** ** -0.15* 0.58*** 1.11*** ** -1.09*** -0.33*** 0.74*** 1.22*** * -1.22** -0.39** 0.91** 1.18** *** -1.41* -0.59** 0.90** 1.21** ** -1.52*** -0.66*** 0.90*** 1.22* *** -1.79*** -1.01*** 0.85*** 1.27*** *** -1.29*** -1.08*** 0.20** 0.15* Panel B: Sales * -0.43*** 0.23** -0.04* -0.08* ** -0.25** 0.46*** -0.08** -0.24** * -0.08* 0.69* -0.09* -0.29* ** 0.12* 0.85** -0.02* -0.26** *** 0.12* 0.83*** *** *** 0.35** 1.02*** 0.09* -0.43*** ** 0.30*** 1.05*** ** ** 0.29*** 1.22*** 0.01* -0.43*** * 0.28** 1.34** 0.07* -0.43* ** 0.17* 1.46** 0.05** -0.34*** ** 1.51*** 0.09** -0.37** ** -0.06* 1.55*** 0.02* -0.39*** *** -0.41*** 0.54**

49 Table 11: Contribution of Trades by Factor Exposures This table reports the average contribution of trades to fund performance given different factor exposures. The contribution of trades from the momentum exposure is measured by the difference in the contributions as measured between the Fama-French three-factor alpha and the Fama-French- Carhart four-factor alpha. Similarly, the contribution from the value HML exposure is measured by the difference between the SMB and Market 2 factor alpha and the Fama-French three-factor alpha; the contribution from the size SMB exposure is measured by the difference between the CAPM alpha and the SMB and Market 2 factor alpha; and the contribution from exposure to the market factor is measured by the difference between the raw return and the CAPM alpha. All numbers in this table are annualized and reported as a percentage of fund TNA. The equally-weighted average across funds is reported. The robust standard errors are clustered per day. Sig. lvl: *** 0.01, ** 0.05, and * 0.1 from the Changes in Holdings in the past Days (Cumulative) Day Mom HML SMB Mkt Raw * 0.07* * 0.19** * *** * * *** * * ** * *** * 0.10** 0.16* 0.66*** * 0.24* 0.90*** * 0.17* 0.27** 1.01*** * ** 0.38*** 1.19*** * 0.37** 1.26** ** -0.10* 0.23* 0.49*** 1.43*** ** 0.61*** 1.53*** * ** 0.76** 1.72*** ** *** 0.90*** 1.88*** * * 0.96*** 1.92*** ** *** 0.88** 1.86** *** ** 0.77*** 1.80*** ** * 0.93*** 1.95*** *** *** 0.94*** 1.96*** *** ** 1.00*** 2.02*** ** *** 1.03** 1.98** *** ** 1.12*** 2.05*** *** *** 1.15*** 2.05*** days 0.35*** 0.13** 0.13** 0.66*** 0.62*** Daily Holding 0.76*** ** 4.26*** 7.26*** Holdings 240 days ago 0.34** ** 3.11*** 5.21*** 49

50 Figure 1: Contribution of Trades to Fund Performance by Different Measures This figure plots the contribution of trades in the past 240 days (one year) to daily fund performance. I measure the contribution using the Fama-French-Carhart four-factor alpha, Fama-French threefactor alpha, CAPM alpha, and raw return. The annualized return is reported and it is equally weighted across funds. 50

51 Figure 2: Contribution of Purchases and Sales to Fund Performance This figure plots the contribution of purchases and sales in the past 240 days (one year) to daily fund performance separately. The annualized FFC alpha is reported and it is equally weighted across funds. 51

52 Figure 3: Contribution of Growth Funds (Quintile 1) vs. Value Funds (Quintile 5) This figure plots the contribution of net purchases and sales on the same day and in the past 10 to 240 days to fund performance for growth funds and value funds separately. I sort the funds every year into quintiles using the value-weighted average book-to-market values of their stock holdings at the end of the prior year. The two panels at the top are for growth funds (quintile 1) and the two panels at the bottom are for value funds (quintile 5). The left panels are for purchases and the right panels are for sales. I measure fund and stock performance using the Fama-French-Carhart four-factor alpha (FFC Alpha). 52

53 Figure 4: Fraction and Contribution of Long-Term Holdings by Fund Category This figure plots the fraction of long-term holdings (holdings longer than a year) in funds portfolios against the contribution of long-term holdings to funds annual gross FFC alpha for each fund category. Top-left panel is for growth funds vs. value funds, top-right panel for small-cap funds vs. large-cap funds, bottom-left panel for momentum funds vs. non-momentum funds and bottom-right panel for small funds vs. large funds. 53

54 Figure 5: Contribution of Trades (Changes in Holdings) to Fund Performance This figure plots the contribution of all trades, including both purchases and sales, on the same day and in the past 10 to 240 days (every 10 days) to the current daily fund performance, as described in equation (14). The grey bars are incremental contributions of trades to fund performance as a percentage of the TNA of the fund, and the black line plots the cumulative contribution. I measure the fund and stock performance using the Fama-French-Carhart four-factor alpha (FFC Alpha). 54

55 Figure 6: Contribution of Gross Purchases and Sales Price Dynamics after Trades This figure plots the separate contributions of gross (all) purchases and sales on the same day and in the past 10 to 240 days to fund performance using equation (15). The left figure is for purchases and the right figure is for sales. The grey bars are incremental contributions of trades to fund performance as a percentage of the fund s TNA, and the black line plots the cumulative contribution. The dotted line plots the contribution of net purchases/sales reported in the benchmark setting (Section 3) for comparison. I measure fund and stock performance using the Fama-French-Carhart four-factor alpha (FFC Alpha). 55

56 Panel A: Growth Funds (Holding BM Quintile 1) Panel B: Value Funds (Holding BM Quintile 5) Figure 7: Contribution of Gross Purchases and Sales: Growth Funds vs. Value Funds This figure plots the contribution of gross (all) purchases and sales on the same day and in the past 10 to 240 days to fund performance for growth funds and value funds separately. I sort the funds every year into quintiles using the value-weighted average book-to-market values of their stock holdings at the end of the prior year. Panel A is for growth funds (quintile 1) and Panel B is for value funds (quintile 5). The left figure is for purchases and the right figure is for sales. The grey bars are incremental contributions of trades to fund performance as a percentage of the TNA of the fund, and the black line plots the cumulative contribution. The dotted line plots the contribution of net purchases/sales reported in the benchmark setting (Section 3.3) for comparison. I measure fund and stock performance using the Fama-French-Carhart four-factor alpha (FFC Alpha). 56

57 8 Appendix 8.1 A Numerical Example for the Decomposition of Holdings I use a simple numerical example to illustrate the decomposition of holdings. Assume a fund holds two stocks: stock 1 and stock 2. The price of both stocks at the beginning of day t is $1 per share. The fund holds 70 shares of stock 1 and 30 shares of stock 2 at the beginning of day t (or at the end of day t 1), as shown in Figure A1, H 1,t 1 = 70, H 2,t 1 = 30. (16) One Period: Past n Days Assume this fund sold 30 shares of stock 1 and bought 30 shares of stock 2 in the past n days, using equations (4) and (5), I have B s(n) 1,t 1 = 0, Ss(n) 1,t 1 = Hs(n) 1,t 1 = 30; Bs(n) 2,t 1 = Hs(n) 2,t 1 = 30, Ss(n) 2,t 1 = 0. (17) According to equation (2), the current holding (holding at the beginning of day t) of stock 1, H 1,t 1 = 70, can be decomposed into changes in holdings in the past n days, H s(n) 1,t 1 = 30, and the holding of stock 1 n days ago, H p(n) 1,t 1, which can be calculated as H p(n) 1,t 1 = H 1,t 1 H s(n) 1,t 1 = 70 ( 30) = 100, (18) and, similarly, the holding of stock 2 n days ago is H p(n) 2,t 1 = H 2,t 1 H s(n) 2,t 1 = = 0. (19) Then, the contribution of the trade of stock 1 in the past n days to fund performance is the product of the change in holding H s(n) 1,t 1 = 30 and the abnormal return of stock 1 on day t, R 1,t. Because not all shares of stock 1 held n days ago are currently held by the fund on day t, H p(n) 1,t 1 = 100 > H 1,t 1 = 70, I cannot directly use H p(n) 1,t 1 as a measure of the long-term holding of stock 1 in the current portfolio. Instead, I use the holding of stock 1 already in the portfolio n days ago to measure the long-term holding of stock 1 in the current portfolio on day t, H l(n) 1,t 1. This can be calculated as the current holding minus net purchases in the 57

58 past n days, as shown in equation (11), H l(n) 1,t 1 = H 1,t 1 B s(n) 1,t 1 = 70 0 = 70. (20) The same number can be derived using equation (13) as the holding n days ago plus net sales in the past n days, H l(n) 1,t 1 = Hp(n) 1,t 1 + Ss(n) 1,t 1 = ( 30) = 70. (21) Similarly, the long-term holding of stock 2 in the current portfolio (the holding of stock 2 already in the portfolio n days ago) is H l(n) 2,t 1 = H 2,t 1 B s(n) 2,t 1 = = 0. (22) Two Periods: Past n and 2n Days In addition, I assume that this fund bought 50 shares of stock 1 and sold 50 shares of stock 2 from 2n days ago to n days ago, B s(n 2n) 1,t 1 = H s(n 2n) 1,t 1 = 50, S s(n 2n) 1,t 1 = 0; B s(n 2n) 2,t 1 = 0, S s(n 2n) 2,t 1 = H s(n 2n) 2,t 1 = 50. (23) According to equation (2), the holding of stock 1 2n days ago is the holding n days ago minus the change in the holding from 2n days ago to n days ago, H p(2n) 1,t 1 = Hp(n) 1,t 1 Hs(n 2n) 1,t 1 = = 50, (24) and, similarly, the holding of stock 2 at 2n days ago is H p(2n) 2,t 1 = Hp(n) 2,t 1 Hs(n 2n) 2,t 1 = 0 ( 50) = 50. (25) The net change in the holding for stock 1 in the past 2n days is the sum of the change in the past n days and the change from 2n days ago to n days ago, H s(2n) 1,t 1 = Hs(n) 1,t 1 + Hs(n 2n) 1,t 1 = ( 30) + 50 = 20. (26) 58

59 The net change in the holding for stock 2 in the past 2n days is H s(2n) 2,t 1 = Hs(n) 2,t 1 + Hs(n 2n) 2,t 1 = 30 + ( 50) = 20. (27) Then, the net purchases and net sales of those two stocks in the past 2n days are defined using equation (4) and (5), as B s(2n) 1,t 1 = Hs(2n) 1,t 1 = 20, Ss(2n) 1,t 1 = 0; Bs(2n) 2,t 1 = 0, Ss(2n) 2,t 1 = Hs(2n) 2,t 1 = 20. (28) As the example shows, the net change in holdings (net purchase/sale as well), H s(2n) i,t 1 (Bs(2n) i,t 1 /Ss(2n) i,t 1 ), cancels out the round-trip trades within the past 2n days and directly measures the aggregate contribution of trades in the past 2n days to the current holding. Therefore, I use the net change in holdings (net purchases/sales) in my benchmark setting, Section 4 and Table 2, to calculate the contribution of trades to fund performance at different horizons. Different from H s(2n) i,t 1, Hs(n 2n) i,t 1 measures the change in holdings from 2n days ago to n days ago irrespective of whether or not it has been cancelled out by subsequent trades. Additionally, I denote the gross (all) purchases by with B s(2n) 1,t 1 = 50, Ss(2n) 1,t 1 = 30; B s(2n) i,t 1 Bs(2n) 2,t 1 = 30, and the gross sales by S s(2n) i,t 1, Ss(2n) 2,t 1 = 50. (29) Doing so allows us to study separately the contribution of trades at each intermediate horizon (as well as the purchase and sale of the round-trip trades) to fund performance. In Section 5, I use the change in holdings and gross purchases and sales every 10 days for this analysis. The long-term holding of stock 1 in the current portfolio based on the holding 2n days ago (holding of stock 1 already in the portfolio 2n days ago) can be calculated as the current holding minus the net purchases in the past 2n days, as shown in equation (11), H l(2n) 1,t 1 = H 1,t 1 B s(2n) 1,t 1 = = 50. (30) Similarly, the holding of stock 2 already in the portfolio 2n days ago is H l(2n) 2,t 1 = H 2,t 1 B s(2n) 2,t 1 = 30 0 = 30. (31) 59

60 Figure A1: A Numerical Example This figure plots the holdings of stock 1 and stock 2 from day t 2n to day t in the numerical example to illustrate the decomposition of the holdings. The solid lines are the trajectories of the holdings in number of shares (or in time-t dollars). 60