Are Hedge Fund Capacity Constraints Binding? Evidence on Scale and Competition *

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1 Are Hedge Fund Capacity Constraints Binding? Evidence on Scale and Competition * Charles Cao Department of Finance Smeal College of Business Penn State University Raisa Velthuis Department of Finance Villanova School of Business Villanova University August 2017 We are grateful to Vikas Agarwal, Jeremiah Green, Matt Gustafson, Peter Iliev, Tim Simin, Ed Jenkins, Florian Weigert, Rich Evans, Charles Trczinka, and seminar participants at Penn State University, the FRB Board of Governors, EFA 2017, ECWFC 2017, and Villanova University for helpful comments and insights. Research grants from the Smeal College of Business and Villanova University are gratefully acknowledged. Correspondence: 338 Business Building, University Park, PA 16802, USA. Tel.: qxc2@psu.edu. Correspondence: Bartley Hall Rm 2085, 800 E. Lancaster Ave, Villanova, PA 19085, USA. Tel.: raisa.velthuis@villanova.edu.

2 Are Hedge Fund Capacity Constraints Binding? Evidence on Scale and Competition ABSTRACT An important question in hedge fund management is whether hedge funds experience decreasing returns to scale, as hedge fund managers often pursue arbitrage opportunities which are limited and short-lived. Extant literature has presented evidence of decreasing returns to scale at the hedge fund level based on OLS regressions. Employing a newly developed, unbiased estimation method based on recursive demeaning, we find no evidence of decreasing returns to scale at the hedge fund level. However, we do find evidence that hedge fund returns are decreasing in industry size. Further tests suggest that inter-hedge fund competition drives this result. Additionally, we examine the evolution of raw managerial skill of hedge funds over time and find that fund performance deteriorates as funds grow older, but this does not take away from the detrimental effects on performance due to the industry becoming more competitive. JEL classification: G23; G11 Keywords: Hedge funds; Capacity constraints; Returns to scale; Skill; Recursive demeaning

3 1. Introduction The hedge fund industry has experienced enormous growth over the last two decades. Hedge Fund Research, Inc. estimates that the total capital in the hedge fund industry stands at almost $3 trillion globally as of 2016, with over 10,000 active hedge funds. This compares to 1,200 hedge funds managing $200 billion in Yet, there is conflicting evidence on how this ten-fold increase in hedge fund activity affects hedge fund performance. We contribute to this literature by employing an unbiased methodology to empirically investigate how hedge fund performance responds to changes in fund size and industry size. Put differently, we provide new evidence on the extent to which hedge fund capacity constraints are binding at the fund-, strategy-, and industry-levels. This is relevant to academics, hedge fund managers, and investors alike, and provides insights into the efficiency and functioning of financial markets. There is an active and on-going debate on whether fund-level capacity constraints are binding. On the one hand, increasing fund size may reduce hedge funds profits by increasing the price impact of trading or increasing the costs of finding profitable investment opportunities. On the other hand, increasing fund size may increase profits if there are fixed costs to finding profitable investments or investments, such as large block purchases, that are only available to funds with large amounts of assets. Finally, since hedge funds have the option to close their funds to new investment, it is also conceivable that there are no detectable decreasing returns to scale effects if managers close their funds optimally. The existing literature has predominantly employed the OLS methodology and found evidence of decreasing returns to scale at the fund level. For example, Agarwal, Daniel, and Naik (2004), Fung, Hsieh, Naik, and Ramadorai (2008), Joenväärä, Kosowski, and Tolonen (2013), and Teo (2009) find evidence of diseconomies of scale in hedge funds. In this paper, we extend this literature in two important ways. First, we investigate the role of industry-level capacity constraints in determining hedge fund returns, while the existing hedge fund literature focuses on the effects of fund flows, strategy flows, and size on fund performance. Although hedge funds have the flexibility to close their fund and manage its size optimally, they do not have control over the size of the hedge fund industry. Increased competition may limit the ability of hedge funds to perform well, indicating that capacity constraints at the industry-level may be binding. For example, the continued availability of arbitrage opportunities given increased competition is not guaranteed. If there are still ample investment opportunities, and the current increased levels of competition 1

4 have not yet resulted in binding capacity constraints, hedge funds might be able to maintain their outperformance. Intuitively, hedge fund industry size should become an increasingly important determinant of hedge fund returns as the industry approaches saturation. However, whether or not the ten-fold increase in industry size over the past two decades represents a material increase in industry competition is an empirical question. Secondly, we reexamine whether capacity constraints at the fund level are binding by estimating the effect of fund size on hedge fund performance using a newly developed and unbiased methodology. Specifically, this paper uses a new econometric technique, the recursive demeaning methodology, which is developed by Moon and Phillips (2000) and Hjalmarsson (2010) and augmented by Pástor, Stambaugh, and Taylor (2015), to reexamine the impact of fund size on performance. It is hard to separate the effects that size and skill have on fund performance, because skill is difficult to measure. Larger funds are more likely accompanied by skilled managers (Berk and Green (2004)), while at the same time the larger fund size makes it harder to outperform when there are decreasing returns to scale. Fund fixed effects allow researchers to control for fixed levels of skill per fund, but this method is biased in finite samples. The recursive demeaning methodology mitigates dependencies between persistent endogenous regressors (fund size) and residual performance (the error term) in panel data sets. The methodology is novel in hedge fund research, and an important feature of this approach is that it adequately accounts for the endogeneity of fund size. We begin our empirical analysis by performing a simulation exercise to evaluate the biases in the conventional OLS estimation methods compared to those generated using the recursive demeaning estimation method that we employ. As expected, this analysis shows that the OLS results are indeed biased toward finding returns to scale effects, whereas the recursive demeaning estimation method accurately retrieves the true returns to scale effect. It also confirms the accuracy statistical inferences of the recursive demeaning method. To evaluate the effect of hedge fund size on performance, we use the TASS hedge fund database. Conventional OLS regression results show statistically significant evidence of decreasing returns to scale. Yet, using the recursive demeaning methodology we do not find evidence that hedge funds experience decreasing returns to scale at the fund level. This finding indicates that the documented evidence at the fund level on hedge funds decreasing returns to scale disappears once finite sample biases are controlled for. 2

5 At the industry level, we find that hedge fund performance is negatively affected by increased competition. Our results suggest that a one standard deviation increase in the hedge fund industry size relative to the U.S. total stock market capitalization decreases hedge fund gross returns by 0.21% per month (or 2.5% per annum). Given that net-of-fee hedge fund returns are less volatile than gross returns, hedge fund investors experience a decrease in after-fee performance of 0.18% (or 2.2% per annum) for the same increase in hedge fund industry size. The impact that inter-hedge fund competition has on fund performance is sizeable relative to typical estimates of hedge fund alpha of around 3% to 5% (Ibbotson, Chen, and Zhu (2011); Kosowski, Naik, and Teo (2007)). Prior research focuses on the effect of capacity constraints over time. This paper goes further to examine the effect across hedge fund strategy categories, as capacity constraints can be more important for some hedge fund strategies than for others. For example, among equity-oriented hedge funds, long/short equity strategies are most common; 48% of equity-oriented hedge funds follow this strategy, which makes it one of the more crowded strategies from the fund s perspective. Besides the number of hedge funds pursuing similar strategies, the nature of the strategy itself also affects the presence of any capacity constraints. For example, event driven funds focus predominantly on a subset of firms that release news or undergo a major change, such as mergers, acquisitions, or restructurings. By nature, some strategies will be more limited in how much capital they can accommodate and will therefore be more competitive than others. Further analysis suggests that the industry-level decreasing returns to scale cannot be explained by a simple time trend, and thus the relation is more complicated than merely proxying for trends that developed during the last two decades. Moreover, the number of hedge funds that exist during a given month is found to negatively affect hedge fund performance while the concentration of hedge funds as measured by the Herfindahl index is positively related to fund performance, suggesting that competition among hedge funds is the main driver of the returns to scale effect at the industry level. However, the returns to scale relationship varies across hedge fund strategies. We also analyze how returns to scale effects are related to characteristics unique to hedge funds. Hedge funds that have a high watermark provision and onshore funds for which it is harder to attract capital are less subject to industry-level decreasing returns to scale. Hedge funds with these characteristics may have a better alignment between managerial incentives and the interests of the fund investors, causing the funds to attract more skilled managers that are able to mitigate the detrimental effects of 3

6 competition. Funds with larger restriction periods are also less affected by increased competition in the hedge fund industry, which may be the result of these funds being less affected by pressures from industry-wide capital withdrawals. We find that smaller hedge funds with less than $100 million in assets under management experience stronger decreasing returns to scale at the industry-level, while for large funds there is no relation. This adds texture to the claims made by hedge fund managers regarding minimum fund size requirements in order to be viable. Furthermore, levered hedge funds are shown to have a more negative performance-industry size relation. We do not find any differential returns to scale effect due to a fund purposefully closing to new investors to limit its growth in size, which is consistent with optimal judgement regarding the fund closure decision by these funds. Returning to the interaction of fund size and managerial skill, while the recursive demeaning methodology can account for the effects of static levels of hedge fund managerial skill that vary across funds, it is important to examine the changes in a fund s level of skill over time or the distribution of skill across funds. Given that hedge fund managers extract large rents, it is also important to know how skillful hedge fund managers truly are by controlling for the effects of fund and industry size on performance. Therefore, we measure the overall level of skill of hedge fund managers and its development over time as funds age. We find that in any given month, the median hedge fund generates an abnormal gross performance equal to 2.3% per month on the first dollar invested in the fund, that is, before returns to scale are taken into account. Furthermore, we find that fund performance deteriorates as funds grow older, but this does not take away from the finding of industry level capacity constraints. Finally, we find that the total dollar profits of the hedge fund industry in aggregate have increased over time. This paper contributes to the hedge fund literature in several important aspects. First, the use of a new methodology to reexamine the effect that hedge fund size has on fund performance overturns regression-based results in the existing literature that finds evidence of decreasing returns to scale. Our simulation analysis suggests that the OLS results are flawed. Second, we examine the capacity constraints at the fund-level and the industry-level, while previous literature focuses on the overall relation between performance and size. We also investigate the capacity constraints across hedge fund strategies and find that while such constraints appear to vary across hedge fund strategies, differences are not statistically significant, possibly due to lack of power. Finally, we provide new evidence regarding the raw skills of hedge fund managers, before returns to scale effects come into play. 4

7 The rest of the paper proceeds as follows. Section 2 develops testable hypotheses. Section 3 presents the recursive demeaning estimation methodology as well as the method used to derive gross fund returns from net returns. Section 4 provides simulation reuslts to verify the consistency and power of the recursive demeaning method. Section 5 describes the data used in the empirical analyses. Section 6 presents the empirical results on returns to scale effects at the hedge fund level and the industry level. Section 7 explores alternative explanations and the robustness of the results. Finally, Section 8 provides concluding remarks. 2. Development of hypotheses We study the relationship between fund size and performance in hedge funds, for which capacity constraints are likely much more severe than for mutual funds. Using standard regression frameworks, Agarwal et al. (2004), Joenväärä et al. (2013), and Teo (2009) find evidence of diseconomies of scale in hedge funds. This finding is consistent with the Berk and Green (2004) model. Teo (2009) finds that decreasing returns to scale effect is greater for funds engaged in illiquid strategies. Joenväärä et al. (2013) also find a decreasing returns to scale effect, and suggest that this may be partially due to managers seeking to increase their compensation by increasing the size of their funds beyond its optimal value. In contrast, Liang (1999) and Koh, Koh, and Teo (2003) find evidence of economies of scale in samples of U.S.-focused and Asian-focused hedge funds, respectively. Brown, Fraser, and Liang (2008) study funds-of-hedge-funds and find that larger funds-of-hedge-funds benefit from economies of scale as they are more effective at performing due diligence. In addition to this, a hedge fund manager has the flexibility and incentives to limit the size of the fund when optimal by returning outside investor capital. Thus, it remains an empirical questions as to whether hedge funds experience decreasing returns to scale, resulting in the following null hypothesis: H1: In the hedge fund industry there is a constant returns to scale effect at the fund-level, meaning that there is no relationship between current hedge fund size and future fund performance. Returns to scale may also occur at the industry or strategy level. Getmansky (2012) studies competition between hedge funds and finds that increased competition results in more funds getting liquidated, which points to strategy-level decreasing returns to scale. Naik, Ramadorai, and Stromqvist 5

8 (2007) directly examine capital inflows into hedge fund strategies, and find that for four out of eight strategies high capital inflows precede lower future abnormal returns. They conclude that investors likely choose between strategies before picking the hedge fund he invests in, making capacity constraints at the strategy level more likely. These arguments lead to the second hypothesis, stated as the null: H2: Hedge funds experience constant returns to scale at the industry-level. The literature has focused on the net returns generated by hedge funds to evaluate funds abnormal performance, which is important from an investor s standpoint. However, to evaluate managerial skill per se, it is equally important to look at funds gross outperformance while controlling for fund size. A fund s performance foremost depends on the manager s skill, but it is also contingent on the constraints that the fund faces. Gross returns are directly affected by the investment opportunities that are available to a manager, and are not influenced by the operational design of a hedge fund such as the fund fees. We expect that decreasing returns to scale effects are stronger from the managerial perspective. We also evaluate whether hedge funds make use of the option to close their fund to mitigate detrimental returns to scale effects. This is not necessarily the case as managers are at least partially incentivized to grow their funds in order to increase their compensation in form of management fees. However, if decreasing returns to scale are present, incentive fee compensation could be muted by managing a larger fund. Yin (2013) finds that for typical fund fee structures managers compensation is maximized when the fund s size is larger than optimal for fund performance. Liang and Schwarz (2011) find that hedge funds high pay-for-performance sensitivity are partially offset by investor outflow restrictions and are generally not strong enough to prevent the hoarding of assets, which hurts the fund s performance. Funds that close to new investors or funds that reopen are unable to generate the outperformance they achieved before they closed. In this paper, we rely on hedge fund size to evaluate capacity constraints, but there are other studies that rely on fund flows instead. For example, Agarwal et al. (2004) examine hedge fund flows and find that funds with larger inflows experience future performance deterioration. Fung et al. (2008) find evidence of decreasing returns to scale in fund-of-hedge-funds between 2000 and 2004, a period 6

9 when these funds received high capital inflows but subsequent performance declined. Zhong (2008) evaluates the distribution of hedge fund alphas over time and finds that fund flows are negatively related to the future performance of larger funds, while smaller funds have an opposite association. Although related, these studies don t necessarily measure the continuous or longer-term effects of capacity constraints that managers face. Furthermore, some of these studies can be viewed as trying to measure a smart money effect that focuses on whether investors are able to direct their money towards more profitable funds (Ozik and Sadka (2010)). Finally, a few recent studies explore how hedge fund firms deal with capacity constraints. For one, Chakravarty and Deb (2012) find that hedge fund management firms prefer to start new funds rather than grow their existing funds as existing funds approach their critical size, and that this relation is stronger for capacity constrained strategies. Furthermore, Fung, Hsieh, Naik, and Teo (2015) find that hedge fund firms with successful flagship funds tend to open new funds with higher fees and longer redemption and notification periods to mitigate capacity constraints and increase fee income. However, both new and existing funds performance suffers after the new funds are launched. This could cause an influx of low-quality hedge funds. Cao, Farnsworth, and Zhang (2014) scrutinize the environment in which a hedge fund is launched, and tease out a 4-5% differential in performance per year between new funds launched by skilled managers who may have identified new investment opportunities and other funds that are launched in response to investor demand. 3. Methodology 3.1. Biases when using OLS to measure returns to scale effects In the mutual fund and hedge fund literature, most researchers use ordinary least squares regressions to evaluate managerial skill and returns to scale (e.g.,chen, Hong, Huang, and Kubik (2004), Ferreira, Keswani, Miguel, and Ramos (2013), Goetzmann, Ingersoll, and Ross (2003)). However, ordinary least squares regressions that include persistent endogenous independent variables in the model have a tendency to generate biased coefficient estimates (Stambaugh (1999)). The reason is that size of a fund is affected by the fund s performance, in addition to fund fees and investor capital flows. Large funds are likely to be managed by skilled managers, and the lack of independence between fund size 7

10 and skill creates an endogeneity problem when examining the effect of size on performance. Fund size is also persistent as its value regularly remains close to its level from the previous month. Ordinary least squares (OLS) panel regressions without or with (equation (1) or (2)) fixed effects can be represented as follows: R aum ' z f, t OLS ft, 1 ft, 1 ft, (1) R aum ' z (2) f, t f OLSFE ft, 1 ft, 1 ft, where, denotes fund f s abnormal return in month t,, is fund f s assets under management in millions of dollars in month t-1, and, represents a vector of control variables, for example industry size or fund age. When correctly specified, measures the returns to scale effect. However, equation (1) suffers from an omitted variable bias, since managerial skill is not controlled for. Skilled managers likely attract more assets, resulting in an increase in the size of the funds they manage. Furthermore, skilled managers generate better performance. Thus, skill affects both size and performance, and leaving skill out of the model results in an omitted variable bias (Chen et al. (2004), Reuter and Zitzewitz (2015)). If we can assume that the relations between skill and size and skill and performance are both positive, the bias in turn causes the slope estimates to be more positive. Introducing fund fixed effects, as in equation (2), controls for a fixed level of skill per fund, and thereby alleviates the omitted-variable bias. This setup corresponds to the model by Berk and Green (2004). However, this model suffers from a bias due to the correlation between contemporaneous fund size and the residuals,,. Stambaugh (1999) shows that a predictive regression is downward biased in a finite sample when the error term (, ) is positively correlated with contemporaneous innovations in the regressor (, ). The positive correlation between the error term and changes in size is partly mechanical, as a high return during month t increases the fund s size in month t. In addition to this, a high return in month t also attracts new inflows at the end of month t, which in turn boosts the size of the fund. Pástor et al. (2015) deduce that this bias causes a spurious negative relation between changes in fund size and future fund performance in finite samples, as larger fund sizes due to unexpected large returns tend to be followed by months with lower returns in finite samples, and vice versa, due to the unpredictable nature of the error terms, which are on average zero. 8

11 Similarly, since introducing fund fixed effects is in practice the same as demeaning each variable using their fund-level average, Hjalmarsson (2010) highlights that the bias arises due to the forward-looking nature of the average fund size used to demean the variable, as it depends on the full sample. In particular, higher future fund size raises the time series average of the fund s size and in turn causes past demeaned size observations to be relatively smaller. This induces a negative correlation between the lagged size variable,, and the error term,. Industry-level returns to scale effects can be estimated in a panel regression using OLS with fixed fund effects, as this setup considers aggregated data and is therefore hardly affected by the contemporaneous dependence between individual hedge fund size and residual performance. To summarize, the empirical inference of the effect of hedge fund size on performance based on the OLS method are biased. Therefore, we rely on a new method to examine economies of scale in hedge funds Recursive demeaning methodology To mitigate the biases mentioned above, Hjalmarsson (2010) demonstrates an approach that can deal with persistent endogenous regressors in panel data sets. He builds on studies by Moon and Phillips (2000) and Sul, Phillips, and Choi (2005) that substitute all variables with their forward demeaned counterparts. Next, Pástor et al. (2015) improve upon the method by making it better suitable for the analysis of economies of scale. We follow Pástor et al. (2015) and Hjalmarsson (2010) by implementing their recursive demeaning method (RD) that corrects for omitted variable bias and correlation between independent variables and residuals. In order to measure the returns to scale effect, we estimate a two-stage leastsquares regression of recursively demeaned variables, using the following two-stage panel regression framework: 1 st stage: aum aum ' z f, t1 (3) f, t 1 f, t 1 f, t 2 nd stage: R aum z (4) ' f, t RD f, t1 RD f, t 1 f, t 9

12 where, represents the forward-demeaned (abnormal) return of fund f in month t,, is forward-demeaned size of fund f in month t-1,, is backward-demeaned fund size of fund f in month t-1, and, is a vector of other forward-demeaned variables, such as industry size or fund age, depending on the model specification. These forward- and backward-demeaned variables are defined as follows: f 1 R R R (5) f, t f, t f, Tf t 1 t T T f 1 1 aum aum aum (6) f, t1 f, t1 f, Tf t 1 t 1 t2 1 aum aum aum (7) ft, 1 f, t1 f, t 1 0 T f 1 1 z z z (8) f, t1 f, t1 f, Tf t 1 t 1 where is the length of fund f s history in months. If there are diminishing returns to scale, will be found to be negative. Note that the regression does not include an intercept in either the first or second stage. Equations (3) and (4) correct for the biases due to an omitted variable and correlation between regressors and residuals by recursively demeaning all variables included in the regression. The fixed effects model in equation (2) is equivalent to a setup where all variables are demeaned. However, demeaning a time series using their full sample mean uses at any point in time information about the variable from a later point in time. To avoid inducing the contemporaneous correlation between lagged fund size and current month s residual return, one can demean fund size recursively using a forward looking mean. Backward-demeaned size is used as an instrument for forward-demeaned size, to accommodate this regressor which does not have a zero overall time series mean and to avoid any contemporary dependencies between residual returns and fund size. As Pástor et al. (2015) and Moon and Phillips (2000) argue, the backward-demeaned variable is unlikely to be correlated with the forward 10

13 looking return information in the error term. The recursively demeaned estimator demeans all dependent and independent variables using their forwards-looking means, while mitigating the finite sample bias by using backward-demeaned size as an instrument. Backward-demeaned fund size,,, is a valid instrument if it satisfies the relevance and exclusion criteria (e.g., Roberts and Whited (2013)). Backward and forward demeaned size are generally positively related 1, which attests to the strength of the instrument. Some funds that show a strong upward or downward trend in their size are unfit for the recursive demeaning method, as the trend demands a non-zero intercept in the first-stage regression. Following Pástor et al. (2015), some of the funds that show a negative relation in the first stage regression are excluded from the sample to ensure the relevance of the instrument. Instrumental variable estimation with weak instruments may not be reliable. Stock, Wright, and Yogo (2002) advocate to use instrumental variables that produces an F- statistic of joint significance of all instruments that is above 10 in our case of a model with one endogenous variable. Among the set of funds that have a negative relation between forward demeaned size,,, and backward demeaned size,,, funds with large intercepts in a regression of forward demeaned size on backward demeaned size are iteratively removed as to ensure a first-stage Sanderson and Windmeijer (2013) F-statistic over When there are no controls, the returns to scale effect,, is found using the standard twostage least squares estimator: 1 RD ( aum aum' ) aum' R (9) where R,, and are 1 vectors, with F being the number of hedge funds and T the total number of months. 1 For the TASS hedge fund sample used in this paper, the average correlation between forward-demeaned and backward-demeaned fund size over all funds in the sample is Over 99.6% (0.4%) of the funds have a positive (negative) correlation between the two variables. 2 There are three different regressions at play here. The negative relation between, and, is estimated per fund using an OLS regression without an intercept. Other fund-level regressions of, on, and an intercept are performed to evaluate the appropriateness of the no intercept requirement. Finally, the relevance of the instrumental variable,,, is evaluated using the F-statistic of a panel regression without intercept of, on,, where standard errors are clustered by fund. In Panel A of Table 5, 6 funds (0.4% of our sample) are removed from the sample. 11

14 Since the recursive demeaning procedure can induce dependencies between the observations belonging to the same fund, the standard errors are clustered by fund. When there are no control variables, clustered standard errors of the returns to scale coefficient estimate,, are computed as follows (Cameron and Miller (2015)): 1 F F 1 se( RD) aum' aum aum f ' f f ' aum f aum' aum F 1 (10) f 1 where is the number of funds in the sample (and corresponds to the number of clusters), R f f RD aum and f are 1 vectors. f, subscripts f denote the subset of observations that belong to fund f, and R,, 3.3. Gross returns To examine the returns to scale effect, it is ideal to use gross returns in the analysis. Gross returns measure the investment return of the assets traded by the manager. However, returns reported by hedge funds to commercial databases are on a net-of-fee basis. If there are capacity constraints in the market, these will be imposed on the manager as they reduce his/her ability to generate abnormal performance before fees. In this section, we highlight the algorithm used to back out gross return from net returns, which is further described in the appendix. We expand on the procedure by Agarwal, Daniel, and Naik (2009) to extract gross returns from the reported net returns, which relies on the understanding provided by Goetzmann et al. (2003) to view investors assets as an option contract on the fund s assets. When a hedge fund has multiple investors that joined the fund at different times, the manager s incentive fee contract resembles a portfolio of call options, where each option represents a claim on an investor s assets in the fund and strike prices depend on the fund s high watermark and hurdle provisions as well as the time that the investor allocated capital to the fund. It is assumed that the fund starts with one investor, and additional investors are added to the fund s investor base during months when capital inflow is positive. Agarwal et al. (2009) s algorithm is based on annual performance data and fees are assessed in tandem. The value 12

15 and exercise price of each investor are tracked separately, which is consistent with the application of a share equalization method (Aragon and Qian (2010)). 3 We deviate from Agarwal et al. (2009) s algorithm to accommodate the use of monthly data which is preferable in this study. Our procedure of imputing gross returns from a given fund s net returns series is provided in Appendix A. In short, each month, inflows (if any) are attributed to the arrival of a new investor. The market value of each fund investor s assets is kept track of, along with each investor s exercise price that determines the payment of incentive fees. Throughout the year, yearto-date incentive fee accruals and net asset values are computed for each investor. Any incentive fees materialize at the end of each crystallization period, at which point taxes are taken out for onshore fund managers and the remainder of the manager s earnings are reinvested in the fund. Gross hedge fund returns are numerically solved for by equating the monthly percentage change in the hedge fund s after-fee market value to the net returns that were realized by the fund. Finally, end-of-month fund flows are (numerically) deduced by comparing the balance of each investor s account to the fund s reported asset value, while assuming that any outflows occur from the account of the oldest remaining investor first. This process is repeated every month. Simulation results are provided using both gross and net returns, to examine the magnitude of the effects of a number of fund characteristics on the intricacies of detecting decreasing returns to scale. The interpretation of the results differs depending on which returns are used in the analysis, but each analysis can provide new and interesting insights. In particular, capacity constraints with regard to the market conditions that the manager faces should be measured using gross returns. Net returns are used to determine whether funds or strategies are saturated from the perspective of the hedge fund investor. 3 Share equalization methods provide for a fairer assessment of incentive fees and prevent scenarios where some investors subsidize others, such as free rides where investors that join the fund at a later date pay less incentive fees in case of a decline in the fund s asset value, excessive burden of incentive fees by later investors in case of an increase in the fund s asset value, or unfair redistribution of reversals of incentive fees accrued by some investors. For onshore hedge funds that are structured as partnerships in the U.S., capital accounting replaces the need of share equalization. For more details, refer to section 2.2 in Lhabitant (2009). 13

16 4. Simulation study In this section, we conduct a simulation study to evaluate the accuracy and power of the OLS and the recursive demeaning methods in estimating returns to scale effects. We describe the process of simulating gross returns and fund sizes for a panel of hedge funds as well as the estimation of the returns to scale effect using different regression methods. The simulation exercise is inspired by Pástor et al. (2015), who study capacity constraints for mutual funds. We tailor to the hedge fund setting by taking into account the sensitivities of fund size and performance as empirically observed. We generate panel data such that funds have differential managerial skill and a correlation between size and returns over time is present. Abnormal fund returns are generated based on a timeinvariant, fund-specific level of managerial skill and a negative feedback effect through fund size. Fund size is updated each month based on its relation with fund returns as well as the overall growth trend in the size of the hedge fund industry. The process is represented by the following two equations: 4 R aum (2) f, t f f, t1 f, t aum aum f, t f, t1 c R (11) f, t f, t where, is the abnormal return of fund f in month t,, is the assets under management of fund f at the end of month t in millions of dollars, represents the skill of the hedge fund manager, is the constant monthly capital flow into the hedge fund industry in percent, and captures the comovement between fund size and returns over time. In this setup, measures the returns to scale effect, while captures heterogeneity in managerial skill that is not adequately captured by OLS setups, and induces the time series correlation between the contemporaneous size and return variables that is the cause of the bias. Other effects (e.g., correlations between funds) are abstracted away to focus the simulation to the issues at hand. The parameters are set based on empirical estimates obtained from the sample that is used in later sections. In a nutshell, the initial fund size of each fund,,, is set equal to $100 million, 4 The first equation is repeated from before for easier presentation. 14

17 which is close to the median fund size in our sample. We perform an panel OLS regression of equation (11) using the TASS hedge fund data to obtain coefficient estimates of and, and obtain intercept and slope estimates equal to and 0.99, respectively. Therefore, we set equal to 0.206% as the baseline parameter in the simulation study and let range over (0.8, 0.9, 1.0, 1.1, 1.2). The standard deviation of the error term,,, is estimated to be 0.101, thus in the simulation, is drawn from a normal distribution with mean 0 and standard deviation 10.1%. and the standard deviation of, are obtained from an OLS panel regression of equation (2). Consequently, is drawn from a normal distribution with mean 1.00% and standard deviation 0.71% per month, and, is drawn from a normal distribution with mean 0 and standard deviation 3.10%. is set to equal to 0.5x10-5, 0, -0.5x10-5, -1x10-5, -5x10-5, and -10x10-5, which are reasonable estimates relative to the empirical findings presented in section To interpret the magnitude of the returns to scale effect, consider that when -5x10-5, a $100 million increase in assets under management leads to a decrease in performance of 0.5% ( ) per month. One thousand panels consisting of 150 hedge funds with a 75 month history are constructed during the simulation. 6,7 Moreover, net returns are derived from the gross returns generated in the simulation according to the procedure outlined in section The net returns depend on the same random draws that were used to generate the gross returns. Estimating the effect of fund size on returns using OLS is biased due to the positive (indirect) cross-sectional correlation between managerial skill,, and fund size,,. Estimations using OLS with fund fixed effects can accommodate different levels of skill across funds, but it does not adequately account for the time series correlation between returns and fund size,. Recursive demeaning is designed to alleviate both biases. 5 When is positive, fund size can grow exponentially if not offset by (the lack of) a fund s managerial skill or by the residual term. In this case, the generation of the time series of these funds returns and size observations is halted as soon as a fund s size becomes larger than 10 6 million dollars. As a result the generated panel data set will be unbalanced in this case. 6 Although the median number of funds per month is around 750, many funds have correlated return series. The number of independent observations is less, which is why a smaller number of funds is chosen in the simulation exercise. 7 The fund elimination process to ensure a first-stage F-statistic over 10 results in the deletion of the most fund when equals -5x10-5. In this case, 6 out of 150 funds are eliminated. 8 The simulation does not consider risk factors to avoid modeling a cross-sectional dependence between firms. In the empirical analysis, returns are adjusted for risk factors. In this case, net returns need to be derived before making any risk factor adjustments to the return series. 15

18 Table 1 shows the simulation results where the returns to scale effect is estimated using (1) an OLS regression of gross hedge fund returns on a constant and fund size (OLS), (2) an OLS regression of gross returns on fund size and fund fixed effects (OLS FE), and (3) the recursive demeaning method (RD), which consists of a two-stage least squares regression of forward-demeaned gross hedge fund returns on forward-demeaned fund size where forward-demeaned fund size is instrumented for by backward-demeaned fund size. In particular, estimates of the returns to scale coefficient are presented for different calibrations of the model, where variations in the and parameters are displayed across the rows and columns, respectively. Furthermore, the size and power of the three different regression methods are compared. Panel A shows the average returns to scale coefficient estimates over the one thousand samples, while Panel B presents the medians. The results show that OLS estimates tend to be biased upward in comparison to the true parameter specified in the model. For example, when the model is set up with constant returns to scale (i.e., 0), depending on the value of that was chosen, the OLS estimates range between 1.0x10-5 and 1.2x10-5, leading to the incorrect conclusion. Similarly, when is set equal to -5x10-5, the OLS regression estimates retrieve values that are closer to half in magnitude. Table 1 shows that, when including fund fixed effects in the OLS model, the regression estimates are often biased downward in comparison to the true returns to scale value that was used to generate the data. For example, when there are constant returns to scale, OLS FE estimates for are around -0.6x10-5. Estimates from regressions based on the recursive demeaning method are most sensible. They are consistently close to the true value of that was used to generate the panel dataset, ranaging from 0 to 0.07x10-5. There is also very little variation across the columns, which shows that different levels of dependencies imposed through do not affect the coefficient estimates much. The medians presented in Panel B are again accurate, when simulations are based on the RD method. For example, when 0, the estimated returns to scale effect is between 0 and 0.1x10-5 for the different levels of considered. The largest deviations occur when true is equal to -5x10-5, in which case the median estimates deviate at most 7% from the true value. Panel C compares the power of the different methods. The null hypothesis being tested is whether there is a constant returns to scale effect, i.e., 0. When is set equal to 0 in generating the panel data in the simulation, we expect to reject the null hypothesis of constant returns to scale 16

19 about 5% of the time at a 5% significance level. On the other hand, when the panel data is generated in the presence of an increasing or decreasing returns to scale effect, test statistics should reject the null hypothesis of constant returns to scale more frequently. Panel C shows that OLS and OLS FE methods consistently reject the null hypothesis of constant returns to scale, even when the null hypothesis is true. Regardless of the specification, regression statistics overwhelmingly reject the null hypothesis in favor of the biased regression estimate. The recursive demeaning method shows the most promise in avoiding Type I errors of rejecting the null hypothesis when it is true. In about 6-9% of the iterations the null hypothesis of constant returns to scale is rejected at a 5% significance level. Furthermore, in 24 out of 25 scenarios, the method is able to reject the null hypothesis of no returns to scale with increasing power as the true of the model is further away from zero. However, the power of the test does not increase much when moving from constant returns to scale to equal to -0.5x10-5. Next, we evaluate the effect of estimating the returns to scale effect based on net-of-fee returns. Doing so changes the analysis from a managerial perspective to the investor s perspective. How is after-fee fund performance affected by the size of the fund? Since incentive fees are non-linearly related to gross fund returns, it is not obvious how the returns to scale effect is affected when estimated using net returns. It is worthwhile to have a closer look at the impact of the crystallization frequency of fund fees on the derivation of net returns, as further elaborated on in the appendix. Figure A shows scatter plots of gross returns versus net returns for the hedge funds in the TASS sample. In Panel A, incentive fees are assessed at the end of each month. As a result, net returns tend to be lower than gross returns when gross returns are positive. Net returns are equal to gross returns when the high water mark is not met or when gross returns are negative. This is common, as 73% of the funds in the sample have a high water mark. There is a kinked relation between hedge funds gross and net returns, due to the nonlinear nature of the compensation fee structure, which likely changes the estimation results to some extent. In Panel B of Figure A, incentive fees are assumed to be realized once a year. In all other months, incentive fees are merely accrued. This means that incentive fee accruals may be reversed 17

20 during subsequent months with worse performance. As the figure shows, the reduction in accrued incentive fees can make net returns in negative return months less negative than their gross return counterparts. Consequently, net returns are in turn less volatile when incentive fees crystallize less frequently. In the simulation exercise, net returns are derived from gross returns under the assumption that all funds are onshore hedge funds that have a high water mark and charge incentive fees annually equal to 20% of profits, following the procedure outlined in section 3.3. It is assumed that managers of onshore hedge funds pay taxes equal to 35% on the incentive fees earned. Both gross and net returns are net-of-management fees, which are interpreted as a cost of doing business. The estimation results based on net returns are displayed in Table 2. Again, a clear upward bias is present among the OLS estimates, while estimates from OLS regressions with fund fixed effects are downward biased. The original returns to scale parameter as set in the simulation cannot be recovered. In sharp contrast, the recursive demeaning method is able to reasonably retrieve the true returns to scale effects. In conclusion, the results in Tables 1 and 2 show that OLS is upward biased. Furthermore, OLS with fixed fund effects is shown to be downward biased overall. Finally, the results in Panel C of Tables 1 and 2 show that the recursive demeaning estimation method is much more accurate and has the most power compared to both OLS methods. 5. Data 5.1. Hedge fund sample We estimate the level of skill and returns to scale in the hedge fund industry using performance and assets under management data from the 2014 Lipper Trading Advisor Selection System database (hereafter TASS), which is widely used in the hedge fund literature. Starting 1994, TASS keeps dead funds in its database. The hedge fund literature has documented a variety of biases, such as survivorship bias, backfill bias, and selection bias (see Fung and Hsieh (2000) or Lhabitant (2009) for an overview). Survivorship bias arises when a database only includes funds that are currently operating. A backfill bias occurs due to funds submitting a longer fund history when they start reporting to a database, which 18

21 often consists of better-than-average performance. Selection bias refers to the voluntary nature of reporting by hedge funds, which may result in the funds included in the database not being a good representation of the population. To mitigate survivorship bias, we start our sample in To alleviate the backfill bias, we only include data from the month in which a fund started reporting to TASS onward. The evidence in Edelman, Fung, and Hsieh (2013) and Agarwal, Fos, and Jiang (2014) indicate that various types of survivorship biases more or less cancel each other out. Funds that report their net-of-fee performance at a monthly frequency and have average inflation-adjusted assets under management (AUM) of at least $15 million dollars (in December 2014) are included in the sample. Assets of foreign domiciled funds are converted to USD using monthly exchange rates from the Federal Reserve Board s H.10 Report and Datastream. These sample criteria are common in the hedge fund literature. We require funds to follow equity-oriented strategies, because we evaluate the returns to scale effect at both the fund- and strategy-level. For hedge funds with equity-oriented strategies, the total value of the securities that they invest in can be accurately measured (e.g., the size of all firms listed on the NYSE, Amex, and NASDAQ). Therefore, we limit the sample to hedge funds that follow one of the following strategies: equity market neutral, event-driven, and long-short equity. Finally, funds are required to have at least 24 months of performance and assets under management data available over a 3-year period. The final sample contains 1,523 equity-oriented hedge funds over the sample period from 1994 to Of these, 498 hedge funds currently report to TASS and 1,025 funds have been liquidated or stopped reporting for other reasons. 179 funds are market neutral equity funds, 304 are eventdriven funds, and the remaining 1,040 funds are long-short equity hedge funds. Table 3 provides pooled summary statistics of the equity-oriented hedge funds in the sample. Assets under management and (abnormal) returns are winsorized at 1% and 99% in each cross-section. The average monthly net return across all fund-months is 0.63% (or 7.6% annualized), ranging from % per month at the first quartile to 2.17% at the 75 th percentile. Gross returns are slightly higher at 0.80% per month (or 9.6% per year). The returns variables are fat-tailed as their kurtosis are well above three. Abnormal returns are obtained for each fund as the intercept plus residuals from the Fung- 19