Copyright by Cristian Ioan Tiu 2006 The Dissertation Committee for Cristian Ioan Tiu certifies that this is the approved version of the following dissertation: Systematic Risk in Hedge Funds Committee: Sheridan Titman, Supervisor Laura Starks Roberto Wessels Lorenzo Garlappi Rui de Figueiredo Systematic Risk in Hedge Funds by Cristian Ioan Tiu, B.A., Ph.D. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August 2006 To Mariana, who together with Andrei and Ioana (Ellie) supported me fully and unconditionally. Acknowledgments The author thanks Keith Brown, Lorenzo Garlappi, Ilan Guedj, Cathy Iberg, Andrea Reed, Laura Starks, Paul Tetlock, Sheridan Titman, Roberto Wessels and Uzi Yoeli for fruitful discussions, and Aleksey Bienneman and Tina Gatch for data support. Last but not least, I am thankful to UTIMCO for research support provided while this thesis has been written. Cristian Ioan Tiu The University of Texas at Austin August 2006 v Systematic Risk in Hedge Funds Publication No. Cristian Ioan Tiu, Ph.D. The University of Texas at Austin, 2006 Supervisor: Sheridan Titman We document that hedge funds with lower systematic risk exposures have higher Sharpe ratios. These funds are more successful: they are able to charge higher fees and manage more assets. When pressed with inflows, fund categories on average do not lose their ability to find investments bearing no systematic risk. By contrast, individual funds experiencing inflows above and beyond their category average increase their systematic risk exposures. The tradeoff between performance and the degree to which systematic risk exposures explain the returns of hedge funds has implications for the latter’s transparency: the risks taken by outperforming hedge funds cannot be understood solely from their monthly returns. vi Contents Acknowledgments v Abstract vi List of Tables x List of Figures xii Chapter 1 Introduction 1 Chapter 2 Literature Survey 5 Chapter 3 Framework and Hypotheses 9 3.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 4 Data 14 4.1 Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Risk factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 5 R-squares and Sharpe Ratios 5.1 20 Estimating R2 , Rsqr, T O1 and T O2 . . . . . . . . . . . . . . . . . . 20 5.1.1 20 Serial correlation in returns . . . . . . . . . . . . . . . . . . . vii 5.2 5.3 5.1.2 Results - R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.3 Results - Rsqr, T O1, T O2 . . . . . . . . . . . . . . . . . . . . 23 Testing Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2.1 Cross-sectional tests: overall R2 and Sharpe ratios . . . . . . 24 5.2.2 Time series tests: Rsqr, T O1, T O2 and Sharpe ratios . . . . 27 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3.2 Extensions: Sharpe ratios and R-squares, or appraisal ratios and R-squares? . . . . . . . . . . . . . . . 32 Extensions: Idiosyncratic Risk by Strategy . . . . . . . . . . 33 5.4 Economic Significance . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.5 R-square and inflows . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3.3 Chapter 6 Extensions: Other Measures of Success and R-squares 39 6.1 R-squares and fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 R-squares and assets under management . . . . . . . . . . . . . . . . 41 6.3 R-squares and age . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 R-squares and survival . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Chapter 7 Implications for Risk Management 45 Chapter 8 Conclusions 47 Appendices 50 Methodologies to fit the factor model . . . . . . . . . . . . . . . . . . . . . 50 Spurious estimation of systematic risk: Estimation of the systematic models 55 Tables and figures 56 Bibliography 88 viii Vita 94 ix List of Tables 8.1 Summary Statistics for the Hedge Funds Industry. 8.2 Heterogeneity across net returns and assets under management from the Altvest, HFR and TASS databases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8.3 Sample statistics for the risk factors used. 8.4 Summary of factor analysis applied to individual hedge funds. 8.5 R2 and talent proxies. 8.6 The relationship between the ability to explain a fund (high R2 or low T E) and fund characteristics. 8.7 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 The relationship between the ability to explain a fund (high R2 or low T E), and the fund’s performance, measured as Sharpe ratio adjusted for serial autocorrelation or as the probability of survival. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.8 Out-of-sample fit of stepwise regression vs. five principal components. . . . . . . . . 64 8.9 Fama-MacBeth regressions of Rsqr, T O1, T O2 on fund characteristics. . . . . . . . . 65 8.10 Performance and the ability to infer a fund’s risk exposures. . . . . . . . . . . . . 66 8.11 Economic significance of low Rsqr as talent indicator. . . . . . . . . . . . . . . . 67 8.12 Determinants of the out-of-sample Rsqr. . . . . . . . . . . . . . . . . . . . . 68 8.13 Determinants of the intra-factors timing. . . . . . . . . . . . . . . . . . . . . 69 8.14 Determinants of inter-factor timing T O2. . . . . . . . . . . . . . . . . . . . . 70 8.15 The link between R2 and fund fees. . . . . . . . . . . . . . . . . . . . . . . . 71 x 8.16 Stepwise regression vs. Principal components goodness of fit. 8.17 Average Sharpe ratios and R-squares for hedge fund strategies. 8.18 Robustness check: repeating out hypothesis tests by strategy. 8.19 The relationship between appraisal ratios and R-squares. xi . . . . . . . . . . . . 72 . . . . . . . . . . . 73 . . . . . . . . . . . . 74 . . . . . . . . . . . . . 76 List of Figures 8.1 Distribution of data across databases merged. 8.2 Estimated Total Assets Under Management in the hedge fund industry. 8.3 Estimated Quarterly Flows into the hedge fund industry. 8.4 Assets under management distributed across strategies, when the strategies are inferred using Sharpe regressions. 8.5 78 . . . . . . . . . . . . . 79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 81 Differences in the means of the assets under management between groups of Low - High . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Differences in the means of the management fees between groups of Low - High Rsqr, T O1, T O2. 8.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rsqr, T O1, T O2. 8.7 77 Differences in the means of the Sharpe ratios between groups of Low - High Rsqr, T O1, T O2. 8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Differences in the means of the incentive fees between groups of Low - High Rsqr, T O1, T O2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 . . . . . . . . . . . . . . . . . 85 . . . . . . . . . . . . . . . . . . . . . . 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.9 Stepwise regressions vs. principal components fit. 8.10 Length of backfilled histories in TASS. 8.11 Backfill bias correction. xii Chapter 1 Introduction The hedge fund industry has doubled almost every two years, contains more than 8,000 active funds, and manages more than $ 1.2 trillion. With management fees averaging 1.5% and 20% in incentive fees, hedge funds are also a very lucrative industry. Given this explosive growth, identifying talented managers has become an increasingly daunting task for the investor. The returns of any fund can be divided into two parts: systematic and idiosyncratic. Since the talent is unobservable, a funds wants its investors to have high confidence in its ability to generate returns. The investors’ t-statistic for ex√ cess returns is T (µ−rf )/σ , which justifies the motivation of the fund to maximize Sharpe ratios. This is also a justification of why the Sharpe ratios are often used as measures of performance by the hedge fund investors. In a simple setting in which funds maximize their Sharpe ratios , we show that while it is optimal for a manager to have some exposure to systematic risk, the proportion of the fund’s variance that is systematic is inversely related to the manager’s ability to generate performance. Simply put, if the manager is talented then the R-square of the regressions of the hedge fund returns on systematic factors should be small. This study tests the hypothesis that hedge funds which bear less systematic 1 risk outperform. Consistent with this hypothesis, we find that funds whose returns have lower R-squares with respect to systematic factors have higher Sharpe ratios. For example, comparing the funds in the highest R-square tercile with the funds in the lowest, we find that the latter has a Sharpe ratio that is 0.38 higher and that this difference is statistically significant. Furthermore, this relationship is economically significant: a portfolio consisting of the low past R-square decile of hedge funds has returns around 1% per year higher, and Sharpe ratios around 0.40 higher than the portfolio of the high decile of past R-square funds. This difference is robust to the size of the funds included in the analysis and to the frequency of rebalancing. Low R-squares are achieved not only when the fund does not bear systematic risk but also when the fund is dynamically changing risk exposures so when averaged, these exposures are small. We separate out the funds whose systematic exposures change through time and show that on average, their Sharpe ratios are lower. Therefore, we conclude that it is the ability of managers to make investments uncorrelated with the systematic factors, rather than their ability to time these factors that is positively related to the fund’s success. Strategies with no systematic risk exposures arguably exist in finite supply and may not accommodate all flows into hedge funds. Hence, the negative relationship between performance and systematic risk exposures may change when hedge funds receive inflows. This is similar to the mechanism in which the growth of a mutual fund renders the relationship between the manager’s talent and his performance insignificant, as in Berk and Green (2004). We show, however, that higher inflows into categories of funds do not impede managers’ ability to find proprietary strategies that are orthogonal to the space of systematic factors1 . Only in the case of a fund that receives inflows above and beyond its category average do we find evidence that manager’s ability to invest idiosyncratically is curtailed, causing the 1 This is not in contradiction with the fact that after receiving inflows, the performance of a strategy may decrease due to decreasing returns to scale. 2 manager to increase exposures to systematic risk and in turn to under perform. Do investors recognize the fact that low R-square funds are more talented? We document that these funds are able to charge higher fees: the funds in the lowest tercile of R-square charge, on average, 20 basis point more in management fees and 226 basis points more in incentive fees than the funds in the highest R-square tercile. Additionally, low R-square as an explanatory variable for fees dominates measures of realized performance such as the mean excess returns of the fund of the fund’s Sharpe ratio. Other measures of a fund’s success also favor the low R-square funds: the funds in the lowest R-square tercile manage $ 9 million dollars more on average, live about 10 months longer and are 5% less likely to exit the sample than the funds in the highest R-square tercile. Our paper extends the literature on understanding the sources of returns in hedge funds by focusing on the analysis of individual funds. In this respect our work differs from the literature directed at hedge fund risk adjusted performance, which studies only portfolios of funds. Instead, our research is close to earlier studies on risk factors that explain individual hedge fund returns such as Fung and Hsieh (1997), Liang (1998) or Dor and Jagannathan (2002). Additionally, it builds on more comprehensive and dynamic models, similar to Agarwal and Naik (2004), and accounts for more sophisticated statistical properties of the returns of the hedge funds a la Getmansky, Lo and Makarov (2004). Furthermore, this study performs a comprehensive out-of-sample analysis of the validity of risk models explaining individual hedge funds. As such, this analysis is also close to the literature that links mutual funds style stability to performance. From this point of view, we extend and apply to hedge funds what Chan, Chen and Lakonishok (2002) and Brown and Harlow (2005) did for mutual funds. Our conclusion is somewhat different from the conclusions of both of these studies: Chan, 3 Chen and Lakonishok (2002) asserts that managers stray from their “style course” following periods of bad performance; Brown and Harlow (2005) asserts that better performance is associated with “staying the course”. By contrast, our results on the relationship between performance and the extent to which funds are exposed to systematic factors show that hedge funds seem to be better off when they are orthogonal to any investment style, at any point in time. 4 Chapter 2 Literature Survey Our paper belongs mainly to the literature that examines whether hedge funds are able to generate excess returns and examines links between performance and fund characteristics. We draw from three relatively separate strands in the finance literature. The first strand, most related to this paper, addresses hedge funds performance. The fundamental question asked by this research is which hedge funds are most likely to outperform. In contrast with the literature that addresses performance of mutual funds (e.g. Carhart (1997)), the literature analyzing the performance of hedge funds seems to be almost unified in the belief that hedge funds are capable to produce positive risk adjusted excess returns (alpha). For example, Kazemi and Schneeweiss (2003) show that managers in the CIDR database add value. Liang (1998) argues that hedge funds offer better risk-return tradeoff than mutual funds. Novikov (2003) argues that the returns of hedge funds are persistent. Aggarwal and Naik (2004) find positive alphas on the hedge fund indices they study (except for the short-selling index). Malkiel and Saha (2006) take the opposite side on the hedge funds performance debate, arguing that various biases plague the performance studies and once they are taken into account hedge funds underperform 5 their benchmarks. Getmansky, Lo and Makarov (2004) report Sharpe ratios above 1 across hedge funds, way in excess of the modest Sharpe ratio of around 0.35 of the US market, or the 0.4 of the mutual funds. Authors such as Lo (2002), Getmansky, Makarov and Lo (2004) have argued that hedge funds exhibit smooth returns; after adjustment for serial autocorrelations the median Sharpe ratio of the hedge funds in the TASS database drops to a still impressive 0.69. Tied to this literature are the studies linking outperformance to various fund characteristics, and in that respect this study belongs to that literature, in the sense that we argue that there is a positive relationship between the degree to which a fund does not bear systematic risk and the performance of the fund. Other characteristics than systematic risk have been studied previously. One of these characteristics is the fund size. Gregoriou and Rouah (2003) and Koh, Koh and Teo (2003) find no relationship. De Souza and Gokcan (2003) and Amenc, Curtis and Martellini (2003) find a positive relationship between size and performance. Getmansky (2004) finds a inverse quadratic relationship between size and funds returns, suggesting that funds have an optimal size. We find that there is a positive albeit weak relationship between fund size and performance, consistent with the earlier studies. Another fund characteristic is the age of the fund. Amenc, Curtis and Martellini (2003) find that younger funds outperform the older ones, although this relationship varies across their models. Koh, Koh and Teo (2003) find that age is unrelated to the performance of Asian hedge funds. De Souza and Gokcan (2003) find that older funds outperform younger funds. We find a positive relationship between the lack of idiosyncratic risk and the hedge fund’s age, siding therefore with the latter studies in this area. A third fund characteristic is the compensation structure. Koh, Koh and Teo (2003) find a negative relationship between incentive fees and net returns; Amenc, Curtis and Martellini (2003) find a positive relationship between alphas and incentive fees. We find a positive link between the lack of idiosyncratic risk and both management and incentive fees 6 - suggesting that better funds are the ones charging higher fees. One novel characteristic related to the funds’ performance is manager tenure. Boyson (2003) argues that older managers are more risk averse and end up underperforming the younger ones. We add one more fund characteristic - the degree to which the fund takes systematic risk - that explains hedge funds performance. Another strand of the finance literature is concerned with the relationship between fund performance and a certain stability of the investment process. Studying mutual funds, Chan, Chen and Lakonishok (2002) find that following bad returns mutual funds change their style exposures. Brown and Harlow (2005) find that better performance is associated with more style stability. While styles are successful in explaining mutual funds returns (R squares in Brown and Harlow (2005) are higher than 80%), the similar literature proposing more sophisticated risk factors for hedge funds (Agarwal and Naik (2004)) has a more limited success. Our study could be seen as a generalization of the stability - performance studies from mutual funds to hedge funds. We find that style analysis has a relatively small success when used out-of-sample and applied to individual funds (rather than to indices of funds like Agarwal and Naik (2004)); and we find that not stability nor style instability is positively associated with performance, but rather that positive performance is associated with the failure of the style classification procedure altogether. In other words, maverick hedge funds who are not taking risks mapped and understood thrive. Lastly, we touch into a growing literature in investments that attempts to determine whether idiosyncratic risk is priced. Merton (1987) predicts that idiosyncratic risk should be priced when investors hold non-diversified portfolios, and that it should be positively related to returns. Miller (1977) argues that a negative relationship arises between idiosyncratic risk and returns in the presence of trading constraints. There is currently a vibrant debate in the empirical asset pricing liter- 7 ature siding with one side or another. For example, Ang, Hodrick and Zhang (2004) find evidence that idiosyncratic risk and returns are negatively related. Boehme, Danielsen and Sorescu (2006) argue that the two predicted effects are not complementary, and either one or the other may prevail. Fu (2006) estimates the idiosyncratic risk dynamically and finds a positive relationship between returns and idiosyncratic risk. Our findings offer a class of portfolios - the hedge funds - for which the relationship between performance and idiosyncratic risk is positive. 8 Chapter 3 Framework and Hypotheses 3.1 Framework Assume that a hedge fund manager i chooses between three investments1 : a risk-free asset, a publicly available index F and a proprietary strategy Ai for which E[F − rf ] = µ > 0; E[Ai − rf ] = αi > 0; Corr(Ai , F ) = 0 std[F ] =σ std[Ai ] = T Ei . We denote by wF,i and wA,i the weights allocated by the manager to his respective investment choices. The manager is able to borrow at the risk free rate (i.e., we do not require that wA,i + wB,i = 1). The excess returns of the manager are given by Ri − rf = wA,i (Ai − rf ) + wF,i (F − rf ), and his Sharpe ratio by SR(wF,i , wA,i ) = E[Ri − rf ] αi + βi µ =q , std[Ri ] T Ei2 + βi2 σ 2 1 (3.1) No generality is lost if we restrict the manager to two investments. A similar model would carry though with more than one systematic factor, only the notations would become more cumbersome. This model is a simple exposition of Treynor and Black (1973). 9 where βi = wF,i /wA,i . If the manager maximizes his Sharpe ratio2 , he solves max SR(wF,i , wA,i ). wF,i ,wA,i The solution to the above optimization problem is given by βi∗ = wF,i µ/σ 2 αi /T Ei2 . = βi∗ wA,i The Sharpe ratio of the optimal portfolio is sµ SRi∗ = αi T Ei ¶2 µ ¶2 + µ σ . An econometrician attempting to explain the systematic risk exposures of the fund by regressing Ri − rf on the systematic factor returns F would obtain an R2 of the OLS regression equal to (R2 )∗i = βi∗2 σ 2 = T Ei2 + βi∗2 σ 2 1+ 1 (αi /T Ei )2 (µ/σ)2 . The first observation from the equation above is that (R2 )i , like the Sharpe ratio, is independent of leverage; the second observation is that under the assumption that αi > 0, (R2 )i decreases with the information ratio αi /T Ei of the proprietary strategy, which leads to the following proposition: Proposition 1 If a hedge fund is maximizing its Sharpe ratio, then the R2 of the 2 Hedge funds are often evaluated by their Sharpe ratios. Also a way to convince an external observer that the fund generates statistically significant excess returns is for the fund to have a high t-statistics of the excess returns. The t-statistic of the excess returns is equal to the Sharpe ratio of the fund times square root of the length of the fund’s history. If we compare funds with histories of equal length then comparing the t-stats of their expected excess returns is equivalent to comparing their Sharpe ratios. 10 regression of the hedge fund’s excess returns on systematic factors is inversely related to the fund’s Sharpe ratio. 3.2 Hypotheses The main objective of this paper is to test the hypothesis that the fund’s performance - measured as the Sharpe ratio of its returns - is inversely related to the R-squares. To obtain the R-squares, we first explain the excess returns of individual funds Rt − rf , t ≥ t0 , from some initial time t0 until the time T , which is either the present period or the time the fund exited the sample, whichever is lower. Employing a factor model with K factors amounts to fitting every fund to Rt − rf = K X β T,k Ftk + (αT + ²Tt ), t = t0 , t0 + 1, ..., T. (3.2) k=1 R-square can be calculated from Equation (3.2), as R2 = 1 − V ar[²t ] V ar[Rt ] . We estimate the overall R-squares using the entire history of the fund. By doing so, only average exposures to systematic factors observed throughout the entire life of the fund are taken into account. We then test the following Hypothesis 1: Sharpe ratios are negatively related to R2 s. The model presented above is one period, hence it ignores timing - or dynamic portfolio allocation. In reality hedge funds attempt to time, with or without ability3 . Therefore, low R-square in the regressions of the funds on systematic factors may be due not only to predominant idiosyncratic investments but also to timing. To exemplify this case, imagine a fund who trades only in the S&P 500 and has a beta of -1 half of the time and a beta of +1 the rest. The fund’s R-square with respect to 3 Chen and Liang (2005) analyze the timing ability of 157 market timing funds from TASS, HFR and TUNA databases and show that timing is related to fund characteristics and more pronounced in bear markets. 11 the S&P 500 index is zero, however, the fund takes only systematic risks. Of course, the fund may exhibit talent and add value by timing, but just as well the fund may have no special information or timing ability. We therefore present tests which separate the low R-squares due to idiosyncratic investments from the low R-squares induced by timing. Note that a separation of the selectivity of idiosyncratic strategies from the timing of the systematic factors is impossible in the sense of Daniel, Grinblatt, Titman and Wermers (1997) as holdings data are not available for hedge funds. Thus, in order to separate them, we roll out-of-sample estimations of the extent to which past systematic exposures explain the returns of the fund; if past factor exposures explain the contemporaneous returns poorly, this may be due to the idiosyncrasy of the fund manager, or to its timing attempts. We employ measures of timing to separate away the latter case. The measure of goodness-of-fit out of sample is given by Definition 1 Given a hedge fund with rolling systematic risk exposures given by β T,k , k = 1 : K, T , the out-of-sample R2 is defined as V ar[Rt −rf −αT +β T,1 Ft1 +...+β T,K FtK ;t=T +1:T +12] RsqrT := 1 − . V ar[Rt −rf ;t=T +1:T +12] The Rsqr resembles an “out-of-sample” R-square or the statistic from a Chow (1960) test of a structural break in a linear model explaining the returns of a hedge fund. For example, suppose a fund has a beta of -0.5 with respect to the S&P 500 Index up to time T . Suppose that at time T the fund changes its exposure and its beta becomes +1. Then the out-of-sample RsqrT = 1 − V ar[(0.5RS&P 500 ) − (−0.5RS&P 500 )]/V ar[0.5RS&P 500 ] = −3 = −300%. The timing measures are defined below. The first measure describes changes in the exposures to the factors the fund continues to be exposed to. The cases in which a fund invests in a new systematic factor, or if the fund ceases to be exposed to a factor are not included in this measure. 12 Definition 2 Given a hedge fund with rolling systematic risk exposures given by β T,k , k = 1 : K, T , the intra-factor timing4 T O1T at time T is defined as T O1T := PK k=1 |β T −1,k − β T,k |1{β T −1,k 6=0} 1{β T,k 6=0} . The second measure is complementary to the first: it is the sum of the numbers of factors the fund ceased to be exposed to and the number of factors the fund started to be exposed to. Definition 3 Given a hedge fund with rolling systematic risk exposures given by β T,k , k = 1 : K, T , the inter-factor timing T O2T at time T is defined as T O2T := #[({k : β T −1,k 6= 0} ∪ {k : β T,k 6= 0}) − ({k : β T −1,k 6= 0} ∩ {k : β T,k 6= 0})]. A similar measure to T O2 may be constructed to capture not only changes in the number of the factors, but also the magnitude of these changes. Although this measure produces qualitatively the same empirical results exposed in this study, it is noisier that T O2 defined above5 , and hence we prefer to use T O2. We test that there is a negative relationship between Sharpe ratios and Rsquares, and that this is not due to timing: Hypothesis 2: Higher Sharpe ratios correspond to lower Rsqr, T O1, T O2. Simply put, managers who have zero exposures to systematic risk at each moment in time perform better. 4 “TO” is an acronym for turnover in the investment sense of the word, i.e., the change in weights in a portfolio. 5 Across funds with the same T O2T , the betas on the new factors may vary, thus making a measure designed to capture changes in betas noisier. 13 Chapter 4 Data 4.1 Hedge Funds We use a proprietary, comprehensive database consisting of the union of the Altvest, HFR and TASS databases. The proportion of funds coming from each database is described in Figure 8.1. For each of the databases, the graveyards, i.e. the lists of funds that dropped from the databases were obtained1 . Data of various complexity is collected for a total of 8,542 funds. While smaller subsets of the database contain more information, for 7,429 funds we have monthly returns net of fees, assets under management, whether the fund is a fund of funds, and the management and incentive fees the fund is charging. Missing records for the assets under management are filled assuming that between the dates at which information is provided the funds received uniform inflows (outflows). When a fund for which the data does not contain a name2 is added to the merged database, we follow a procedure which eliminates the 1 To the best of my knowledge this is the first paper that uses the Altvest graveyard. The graveyard data usually hide the names of the funds, due to non-disclosure agreements prohibiting making information on hedge funds that decided to stop reporting public. 2 Altvest and HFR cease to report the names of the funds once the funds stop reporting to the database, so their graveyards do not include names. 14 duplicates: when the correlation between the common net returns for two funds is greater than 99.9% and the correlation between the common assets under management(after filling the missing records as described above) is greater that 99% and the fees are identical the two funds are assumed duplicates and one is eliminated. This procedure eliminates only no-name database duplicates, but if one manager runs several funds, and the names of these funds are provided, then all the funds are kept (this may result in funds with the same series of returns but different sizes being kept in the database). The number of funds in the database is presented in Table 8.1, while the industry coverage of the data used is apparent in Figures 8.2 and 8.3. For December 2003, our database covers around $ 800 billion of assets under management. Several biases have been documented that plague hedge fund data; we list them below. • Survivorship bias. Databases drop from the sample the funds that stop reporting. This may be caused either by the fund going out of business or to the fund closing to new investments and not having incentives to report any longer. Fung and Hsieh (2000) estimate the difference in performance between the portfolio of all surviving funds and the portfolio of all the funds to be 3% annually. Similar estimations are found in Brown, Goetzmann, Ibbotson and Ross (1999). We included in the database the funds who ceased reporting, making thus the data free of survivorship bias. • Self selection bias. The reporting is voluntary, so a bad fund has no reason to report, and a fund that is too good closes quickly and does not have any reason to “advertise”3 ; Fung and Hsieh (1997b) claim that these effects even out. • Backfilling bias. The moment a fund decides to report to a database might not coincide with the date the fund started, and when the fund start re3 Reporting to a database may be an indirect form of advertising. 15 porting it is free to backfill its history. Typically, hedge funds smooth their back returns; in this study we use a procedure that removes the serial autocorrelation induced by smoothing, and hence correct for this bias. Potentially, this may bias the performance of hedge funds upward4 . As we do not have data on the exact time of each fund joining the database we have employed the following correction to the backfill bias. Some databases calculate indices of hedge funds using the fund currently existing in the database and these indices are not restated when a fund joins and brings its history of returns to the database. Unfortunately, as several funds join a database simultaneously, it cannot be directly inferred only from the returns of the funds and of the indices alone when each individual fund joined. However, we can discard a certain number of initial returns from the history of each fund, so that the index build after discarding some initial history matches closer the index reported by the database. Funds backfill great amounts of data (see a histogram for the funds in TASS constructed a la Novikov (2004) in Figure 8.10); as a consequence discarding initial histories may in fact reduce substantially the number of funds. If we repeat this procedure on the HFR database, which reports indices, we obtain that the distance between the HFR indices and the indices computed by us after discarding n months of history is minimized for n = 24 months. This is consistent with Novikov who finds n = 27 months. The correlation between the value weighted HFR index and the one calculated after discarding 24 months of each fund’s history is 0.988. The results of our analysis hold after correcting the history in this way. For a quick comparison, the median (mean) Sharpe ratio of a fund in our database, adjusted for serial autocorrelation of the returns a la Lo (2002) is 0.69 (0.80). To see the impact of the backfilling bias, we eliminated 24 months from the history of each fund. The new median (mean) Sharpe ratio (corrected for serial 4 See Posthuma and van der Sluis (2003). 16 autocorrelation) becomes 0.66 (0.73). This difference could be explained not only by the backfill bias but also by the fact that we capture only the last months of returns of some defunct funds. The small difference between the backfilled, and non-backfilled Sharpe ratios may partly be attributed to the fact that correcting for smooth returns lowers the performance of the backfilling funds. • Smooth returns. Several authors (e.g. Asness, Krail and Liew (2002)) present evidence that the returns of hedge funds are serially autocorrelated, and so performance measures such as Sharpe ratios end up having difference statistical properties than being normally distributed. We adjust the fund returns for serial autocorrelations such as in Getmansky, Lo and Makarov (2004), the backfilled returns, which are generally smoother, decrease. • Late reporting bias. Databases usually wait for funds to report, and a fund can be as late as 8 months in reporting. This causes a fund to appear as “defunct” while in reality it still exists and it is still willing to report. I We correct for this bias using only data before December 2003, but collected in August 2004, when all the funds that are late5 came back to report again. This bias has not, to the best of my knowledge, been documented in the literature. The three databases merged do not contain homogeneous fund information; Table 8.2 documents the differences among them. To examine the extent to which data coming from different databases are structurally different, we perform a Kolmogorov-Smirnov test that net returns and assets under management from each database were drawn from the same distribution; results are presented in Table 8.2; the p-values of the statistic are high when differences between Altvest and HFR are the center of attention: we cannot reject the null that Altvest and HFR net returns and respectively assets under management were drawn from the same distribution. However, we can reject at better than 5 This problem seemed the most severe with TASS, where some funds were 8 months late. 17 95% confidence level the null hypothesis that HFR and TASS, respectively Altvest and TASS were drawn from the same distribution. The TASS database seems to pick smaller funds that also have lower performance, which indicates that performance tests ran on TASS tend to produce rather pessimistic results on hedge funds compared to studies based on different databases. This differences across databases may bias the results of studies based on a single source, and make generalization from one database to the whole industry of hedge funds problematic. 4.2 Risk factors Due to their dynamic nature, hedge fund portfolios are hard to explain by traditional buy-and-hold strategies, regardless of how sophisticated the latter might be. Failure of traditional risk factors to explain hedge fund returns6 lead to a search for nonlinear risk factors. Two main categories of nonlinear factors surfaced: the first category, pioneered by Glosten and Jagannathan (1994), is motivated by the observation that, unlike mutual funds7 , hedge funds employ derivative strategies, hence they may take asymmetrical positions relative to a factor’s performance. Risk factors resembling option payoffs are hence added to the factor models designed for hedge funds. The second category stems from an extension of the previous: if hedge funds employ derivatives, they are also cashing out or loosing on the premium that these derivatives command; such views were confirmed by detailed analysis of particular hedge fund strategies such as merger arbitrage (see Mitchell and Pulvino (2001)), whose returns resemble those of a strategy that sells “merger insurance”, or more 6 Fung and Hsieh (1997a, b) find “styles” that explain successfully returns of mutual funds more than 50% of the mutual funds they analyze have R2 greater than 78%. But these styles fail to explain hedge fund returns, nearly half of which have an R2 under 25%. The lack of success of this early analysis generated subsequent studies with identical methodologies but looking at different factors. Different asset classes or styles - as in Liang (1999) who uses 8 different “styles” or Schneeweis and Spurgin (1998) who look at 13 variables, out of which some are volatilities of asset classes, or Brealey and Kaplanis (2001) who use 31 factors - still kept the blur on the systematic hedge fund risk. 7 See Koski and Pontiff (1999) or Almazan et. al. (2001). 18 precisely, out of the money puts. The most important of these aspects (classical buy-and-hold risk factors and returns from option strategies have been combined by Agarwal and Naik (2004) and further refined by practitioners (See for example deFigueiredo and Meredith (2005)). A refined factor model considers, as explanatory risk factors, several market indices, both from US and abroad, as well as global; payoffs on a market index I of the form max(RI − k, 0) or min(RI − k, 0) (these serve as factors capturing market timing in the sense of Treynor and Mazuy (1966); market timing factors in the sense of Henriksson and Merton (1981), such as (Returns on the Russell 3000)×(Returns on the Russell 3000)8 ; market factors documented as successful in an equity factor model, such as the Fama-French two factors HML and SMB, and the Jegadeesh and Titman (1993) momentum factor MOM; returns on strategies involving selling short term at- or out-of-the-money options (calls and puts) on some market index I; fixed income factors, such as three “points” on the yield curve9 , indices of corporate, municipal and mortgage backed securities; inflation. The ideal factor model would include a minimal superset of the risk factors used in the previous research, but not to the extent to which we have factors that are collinear. Thus, this study uses to the 34 factors presented in Table 8.3. Although some of these factors are redundant, the procedure we use for fitting the model is insensitive to this problem. 8 In order to know that a fund is exposed to market timing factors dos not help an investor to replicate to fund, but merely to understand it. As Chen and Liang (2005) make the argument that some managers have the ability to time their focus market, we have included a term that model’s managers’ ability to time the S&P500. Note that these factors are not investable, but we can only assert that we “understand” the returns of fund or we can “proxy” them if we discover the fund is exposed to these factors. As a robustness check we eliminate the non-investable factors from our analysis, as shown in our section on robustness. 9 Litterman and Sheinkman (1991) showed that a three factor model explains 92% of the yield curve. 19 Chapter 5 R-squares and Sharpe Ratios In this section we shall argue that hedge funds performance, as expressed by the fund’s Sharpe ratio, is inversely related to the extent to which systematic risk exposures explain the fund’s returns. In the first subsection, we present the methodology used to estimate the R-squares. 5.1 Estimating R2 , Rsqr, T O1 and T O2 5.1.1 Serial correlation in returns The R-squares may be computed by estimating Equation (3.2) for the whole history of each individual hedge fund. However, Asness, Krail and Liew (2002) argue that hedge fund returns are autocorrelated; Getmansky, Lo and Makarov (2004) present some of the causes contributing to the serial correlation observed in hedge fund returns: 1. Hedge funds hold illiquid securities and use stale prices to compute returns; 2. Funds move their leverage ratios through time; 3. Managers smooth returns intentionally, for example by backfilling. 20 Getmansky, Lo and Makarov (2004) and Novikov (2004) estimate models in which the returns of the funds are autocorrelated. We follow practitioners (see deFigueiredo and Meredith (2005)) and assume serial correlation at the level of at most a lag. Following Getmansky, Lo and Makarov (2004), we estimate a model in which the observed returns of a fund follow: Rto = (1 − ρT )Rt + ρT Rt−1 , t = t1 + 1, ..., T. (5.1) If the true returns are given by (3.2), then the observed returns follow: 1 ) + ... + β T,K ((1 − ρT )F K + ρT F K ) + uT ; Rto − rf = αT + β T,1 ((1 − ρT )Ft1 + ρT Ft−1 t t t−1 uTt = (1 − ρT )²Tt + ρT ²Tt−1 , t = t1 + 1, t1 + 2, ..., T. (5.2) This is the factor model we estimate. In order to estimate the overall R2 used in the Hypotheses 1, we take t1 = (inception of the fund) and T = min{(Dec 2003), (exit time from the database)}. To estimate Rsqr, T O1, T O2 we estimate the model 5.2 starting with T being two years after the fund’s inception date and continuing to roll T until one year before the fund drops from the database or until December 2003, whichever comes sooner. Not every hedge fund is exposed to each of our 34 risk factors. To capture the factors relevant for each fund we shall use the stepwise regression to estimate the model (5.2). Other methodologies include principal components, or imposing a parsimonious model such as the CAPM or the Carhart (1997) 4-factors model. For arguments supportive of the virtues of using stepwise regressions to explain hedge fund returns, we send the reader to the appendix and we limit ourself to quote 21 Agarwal and Naik (2004), which assert that the “benefits of using the stepwise regression [to explain hedge fund returns] ... outweigh the costs [of potential data mining]”. 5.1.2 Results - R2 The summary of fitting model (5.2) for the whole life of individual hedge funds is presented in section. The distribution of the R2 estimates from the entire history of each fund using stepwise regressions is presented in Table 8.8, along with the distribution of the tracking errors and with the summary statistics of R2 if the estimation is applied instead to mutual funds. The quality of fit is about three times worse for hedge funds than for mutual funds - hedge fund explanatory regressions have a median adjusted R2 of 42.10%, contrasting the 66.86% for mutual funds - consistent with the findings of Fung and Hsieh (1997a) who show that hedge funds follow dynamic strategies, and with the findings of Griffin and Xu (2005), who show that the hedge fund turnover of the 13-F filed holdings alone is larger than that of mutual funds. Consistent with Getmansky, Lo and Makarov (2004), hedge funds seem to smooth returns. Consistent with Agarwal and Naik (2004), who show that hedge fund indices are exposed to systematic risk, we find that individual hedge funds bear systematic risk as well. The median adjusted R2 of 42.10% implies that hedge fund investors pay for quite a lot of “beta”. We note that is not timing related beta: when the non-investable factors related to the funds’ timing ability of the US equity market are taken out, the median adjusted R2 drops only 0.6% to 41.50%. On average, a fund is exposed to 5 different systematic factors from our set of 34; the minimum number of factors a fund is exposed to is 0, the maximum is 10. 22 5.1.3 Results - Rsqr, T O1, T O2 Rolling estimates of the model (5.2) are produced using stepwise regressions; at each time T , we estimate the model on the interval [T − 23, T ], and use the period [T + 1, T + 12] as the out of sample period to calculate Rsqr as in Definition 1. Out of sample tracking errors are also calculated, as well as in sample rolling R2 ’s and in sample rolling tracking errors. The measures T O1, T O2 are calculated as in Definitions 2 and 3. The summary statistics are reported in Table 8.8. The median of the out-of-sample Rsqr is negative, consistent with the fact that funds shift factor exposures. In sample R2 and tracking errors as comparable with those obtained estimating the whole history of the fund. For comparison reasons we report results produced using factor analysis (Panel B of Table 8.8) as well. We can paint the picture of the median hedge fund. The average fund is exposed to 5 different factors; from one month to another the average fund almost drops a factor in the favor of another (mean number of the factors switched out of and into is 1.67). However, the median out-of-sample Rsqr is around -3%, which indicates that the betas of the factor dropped and of the new factor as well are relatively small compared to the betas with respect to the factors kept. In order to compare the quality of the out-of-sample fit produced by the stepwise regression, we also reproduce the same analysis in the case principal components are used; the median out-of-sample Rsqr is -7.16%, hence, the 5 principal components model fares worse. However, a model with zero factors has an out-ofsample Rsqr of zero, thus seemingly exhibiting a better out-of-sample fit than both stepwise regressions and the principal components. Yet, the zero factors model does not exhibit any fit at all: the 80th percentile of the out-of-sample Rsqr estimated from the stepwise regressions is 18.23%, much better than the zero percent of the zero factors model ! 23 Are the Carhart (1997) four factors more, or less helpful in describing the hedge fund returns than the factors selected by the stepwise regression? In order to answer this question we check the former’s out-of-sample quality of fit. The median of the out-of-sample Rsqr is -19% - worse than the stepwise regression and much worse than the zero factors model. This suggests that out-of-sample model stability cannot be achieved simply by moving to more parsimonious models, and are better off including more factors with potential in explaining the hedge fund returns. When rolling the stepwise regression analysis, both in-sample and out-ofsample R-squares are serially autocorrelated across funds. The median autocorrelations are 0.6060 for in-sample R@ ’s and 0.5392 for the out of sample Rsqr (means are 0.5126 and 0.4528 respectively). 5.2 Testing Methodology In this subsection we present the methodology used to test Hypotheses 1 and 2. 5.2.1 Cross-sectional tests: overall R2 and Sharpe ratios As a preliminary test, funds are sorted in terciles according to their R2 , and the average size, fees charged, age, probability to remain in the sample and Sharpe ratios are computed for each of the R2 terciles. Under the null hypothesis of no relationship between the Sharpe ratios and R2 , the averages across R2 terciles should not differ. The results are in the second column of Table 8.5. The table also presents averages of other fund characteristics across different R2 terciles. Several fund characteristics may influence the R2 : the fund’s age, size, the degree to which the fund smooths its returns, the number of strategies in which the fund plays and the fees charged by the fund. Consequently, we first estimate the model: 24 Ri2 = f (sizei , size2i , mf eei , if eei , Agei , Alivei , rhoi , Complexityi , F OFi ) + ²i (5.3) for a linear specification f . We run a similar analysis using the tracking error as a depend variable. Although the tracking error is not formally related to our Hypotheses, we are interested if the idiosyncratic component of the funds’ returns is hedged (that is the tracking errors are small) or not. The test of Hypothesis 1 is performed by estimating the following model: c2 , size , Age , Alive , rho , Complexity , F OF , mf ee , if ee ) SRi = f (Ri2 − R i i i i i i i i i +²i SRi = f (T Ei − Td Ei , sizei , Agei , Alivei , rhoi , Complexityi , F OFi , mf eei , if eei ) + ²i (5.4) c2 are estimations from Equation for a linear specification f . Above, Td Ei and R i (5.3). The results are in the Panel A of Table 8.7. c2 )is now Note that R2 orthogonalized on fund characteristics (that is, R2 − R an exogenous variable. We have chosen to make this assumption because there is a large body of hedge fund literature studying the relationship between Sharpe ratios and fund characteristics1 and we had to control for the latter. Although the tracking error does not directly enter our tests, it is descriptive of whether the idiosyncratic 1 The relationship between size and performance has been studied by Getmansky (2004), by Gregoriou and Rouah (2002); by Koh, Koh and Teo (2003) among others. The relationship between performance and age of the fund has been studied by Howell (2001), Amenc, Curtis and Martellini (2003) and De Souza and Gokcan (2003) among others. The relationship between performance and fees has been studied by Kazemi, Martin and Schneeweis (2002), Koh, Koh and Teo (2003), De Souza and Gokcan (2003) and Amenc, Curtis and Martellini (2003). Other fund factors (e.g. manager tenure, redemption specifications, managerial investment in the fund) are also related to performance but we lack data on these variables. 25 investments of a fund are hedged, and for comparison purposes we included in the results reported. The independent variables are described below. • size is the size of the fund. As the analysis of this section is purely cross-sectional, it is difficult to define the size of a fund (it varies through time). We transform the assets under management in December 2003 dollars, and as a proxy for size, we use the 25th percentile of the assets under management for the history of the fund. The relationship between this proxy for size and the extent to which a fund can be explained by systematic factors does not change if instead we use the 10th percentile of the 2003 assets under management or the median, but the significance decreases if we use the maximum assets under management or the mean. As habitually, we use the natural logarithm of size. Its square is included as well. • Age is the age of a fund, measured in months, from the time when the fund starts reporting until the minimum between December 2003 or the time the fund ceased reporting. • Alive is a dummy meant to separate funds that ceased to report from funds that were still in existence in December 2003. • rho is the degree to which returns are serially autocorrelated, precisely, it is the smoothing coefficient from model (5.2). • Complexity. There are structural differences in the fund managers’ talent, determined by the Complexity of the hedge funds they run; managers following certain investment strategies may have different Sharpe ratios, may grow to different sizes, or are more or less prone to smooth their returns. In order to address this problem we have to control for the investment strategy of the fund. Unfortunately most of the database used in this study does not contain information regarding the strategy of the hedge fund. To resolve this weakness, we estimate each fund’s 26 investment strategy as follows. A Sharpe (1992) style regression is performed, having the fund’s returns as dependent variable and the returns of 15 HFR strategy indices as independent variables. In this regression, some of the strategy indices have significant t-statistics2 . We retain the number of significant t-stats (at 5%) from each style regression, and define that number as the complexity of the fund. If no t-statistics were significant, then the complexity of the fund is taken to be equal to 16 (highest possible value). Funds of funds are also similar to funds we call complex; we do include a control for funds of funds. This rough classification results in a distribution of the assets under management across strategies as shown in Figure 8.4. • F OF is fund of funds dummy. • mf ee is the management fees charged by the fund in percents and if ee is the incentive fee. Note that the returns used throughout this study are net of fees. • SRi is the Sharpe ratio of the fund i, adjusted for serial correlation. With a few exceptions the variables above are relatively uncorrelated: the highest positive correlation is 30% (between Complexity and Age - we will therefore run robustness tests due to this potential problem as described in Section ??). The next positive correlation is 13% (between Alive and rho). The most negative correlation is -40% (between F OF and if ee - we address this issue in the robustness section as well). The next negative is -10% (between Lif e and mf ee). 5.2.2 Time series tests: Rsqr, T O1, T O2 and Sharpe ratios Low overall R-squares does not mean that a fund never bears systematic risk - it merely indicates that on average the systematic exposures are zero. In this section we present a methodology to test Hypothesis 2, which asserts that higher Sharpe 2 The t-statistics from the Sharpe (1992) style regression cannot be estimated as in an OLS regression, hence we use the methodology proposed by diBartolomeo and Lobosco (1997). 27 ratios correspond to low R-squares at any moment in time - precisely, high Sharpe ratios are associated with low Rsqr, but with a low degree of variability in systematic exposures - that is, low T O1 and T O2. Much in the spirit of the sorting performed in the previous section, every month T we sort portfolios based on Rsqr, T O1, T O2 and compute the difference between the average Sharpe ratio of the lowest, respectively the highest Rsqr, T O1, T O2 decile. The Sharpe ratios are from the period [T + 1, T + 12] and adjusted for serial autocorrelation à la Lo (1992). We also perform a difference in mean test between the Sharpe ratios in both deciles. The results are plotted in Figure 8.5. A star signifies that the difference in means between the low and the high Rsqr, T O1, T O2 deciles is significant at 5%. In the same spirit, we run Fama-MacBeth regressions to isolate the effect of fund characteristics on Rsqr, T O1, T O2: RsqrTi = c + b1 aumT,i + b2 Last1Y StdT,i + b3 AgeT,i + b4 P ressureT,i +b5 mktT + b6 F OFi + b7 mf eei + b8 if eei + b9 F lowF racT,i +b10 AliveT,i + ²T,i T O1iT = e + d1 aumT,i + d2 Last1Y StdT,i + d3 AgeT,i + d4 P ressureT,i +d5 mktT + d6 F OFi + d7 mf eei + d8 if eei + d9 F lowF racT,i +d10 AliveT,i + ²T,i T O2iT = +f1 aumT,i + f2 Last1Y StdT,i + f3 AgeT,i + f4 P ressureT,i +f5 mktT + f6 F OFi + f7 mf eei + f8 if eei + f9 F lowF racT,i +f10 AliveT,i + ²T,i The results are in Table 8.9. 28 (5.5) We then estimate the following Fama-MacBeth model: dT,i ) + d2 (T O1T,i − T O1 dT,i ) + d3 (T O2T,i − T O2 dT,i ) SharpeT,i = c + d1 (RsqrT,i − Rsqr +d4 aumT,i + d5 F lowT,i + d6 Last1Y StdT,i +d7 P ressureT,i + d8 F OFi + d9 mf eei + d10 if eei + ²T,i . (5.6) where Rsqr, T O1, T O2 are not simultaneously included in the equation, dues dT,i , T O1 dT,i , T O2 dT,i are estimations of to the fact that T O1, T O2 affect Rsqr. Rsqr RsqrT,i , T O1T,i , T O2T,i from Equations (5.5). Results are in Table 8.10. The variables in the right hand size are described below: • RsqrTi is the out of sample R2 as described in Definition 1 of Section 3. Note that the returns one year forward (the period [T + 1, T + 12] enter into the definition of Rsqr; • T O1iT and T OTi are the measures capturing changes in the systematic exposures from Definitions 2 and 3 in Section 3. • aumT,i is the natural logarithm of the assets under management of find i at time T ; • Last1Y StdT,i is the standard deviation of the monthly returns of the hedge fund i calculated over the period [T − 23, T ]. We include this variable as a control for the overall risk of the fund, as Kat and Menexe (2003) argued that the risk level of a fund is persistent (rather than the fund’s performance). Overall affects the out of sample Rsqr, and it also affects the extent to which funds may time systematic risk exposures. For example a fund who has a high leverage ratio is riskier than a fund with the same ratios of systematic exposures/idiosyncratic exposures which is unlevered. The inclusion of the overall risk in the model has the effect of equalizing two managers with the same proportion of idiosyncratic risk relative to the total risk, but differently levered. 29 • AgeT,i is the age of the fund i at time T . • P ressureT,i is a variable describing the flows into the category the fund belongs to. More inflows into the same category generate competition among the funds in that category. P ressureT,i is built as follows. First we employ the classification of the fund into strategies as in Section 5.2.1. The fund i is thus assigned weights i (single strategy funds will have all w’s but one into each of the strategies w1i , ..., w15 equal to zero). The inflows into each hedge fund (quarterly aggregation) are used to compute the aggregate flows into each of the 15 strategies. For each strategy j = 1, ..., 15, the flows fT,j into each strategy are computed as the natural logarithm of the (quarterly) dollar flow into the strategy divided by the assets under management for that respective strategy (f = log[(dollar f low)/(assets under management)]). The higher fT,j is, the more money flows into the strategy j at time T . The pressure on fund i is defined as P ressureT,i := 15 X wji fT,j . j=1 • M ktRetT represents alternatively the returns on the US market portfolio; mktcapT is the natural logarithm of the US market capitalization. Funds may be tempted to become more systematic during a boom to exploit the potential benefits of a bubble hence we included a control for this behavior. Also, when the market cap is larger, funds may have a higher potential to find idiosyncratic trades. • F OFi is a fund of funds dummy. • mf eei and if eei are the management and the incentive fees charged by the funds. • F lowF racT,i is the natural logarithm of the dollar flow into the fund i over the quarter [T − 3, T ] divided by the assets under management at time T . We include both the pressure on a fund and the flow into that fund as we are interested to capture the effect of inflows way and beyond the industry average on hedge fund 30 investment behavior (systematic/idiosyncratic/timing). • AliveT,i is a dummy indicating that the fund does not drop from the sample in the next two years. • SharpeT,i is the Sharpe ratio of the fund on the time interval [T + 1, T + 12], corrected for serial autocorrelation of the returns. 5.3 Test Results 5.3.1 Main Results Built based on the whole history of each individual fund, Table 8.5 shows that the average Sharpe ratio of the funds with low R2 is 0.38 higher than the average Sharpe ratios of high R2 funds, and that the difference is statistically significant. Results in Table 8.7 further show that a negative, significant relationship between Sharpe ratios and R2 ’s, hence we can reject the null of no relationship in favor of Hypothesis 1. On a rolling basis the statement is still valid: Figure 8.5 plots the differences between the average Sharpe ratios of low, vs. high Rsqr, T O1, T O2 groups of funds. In only two instances have the high Rsqr decile had an average Sharpe ratio higher than the one of the lower Rsqr decile. This time instances are September and October 1998 - the LTCM collapse3 . The lowest T O1, T O2 deciles also have the highest Sharpe ratios. This is evidence supporting Hypothesis 2. Table 8.10 further shows a negative, significant relationship between the Sharpe ratio and Rsqr, T O1, T O2, hence, the null of no relationship can be rejected in favor of Hypothesis 2. It is perhaps surprising that on average, funds attempting to time exposures to systematic factors under perform - after all, hedge fund managers may have the 3 Although LTCM did not report in any of the Altvest, HFR or TASS databases, its collapse meant a shock to the hedge fund industry, both literally and figuratively. 31 skill to time the markets. Despite evidence of timing abilities of certain managers4 , our results indicate that the average hedge fund is worse off by dynamically changing the systematic risk exposures. To some extent, this is less surprising if we consider recent evidence that simple, fixed allocation rules to asset classes dominate dynamic, sophisticated allocation strategies: DeMiguel, Garlappi and Uppal (2005) show that simple asset allocation rules such as the “1/N ” rule do not under perform sophisticated, dynamic asset allocation strategies. This result suggests that absent timing talent, a fund investing in publicly available factors is better off by not changing the allocation rules dynamically but by keeping them fixed. As the literature argues that mutual fund managers cannot out-perform simple benchmarks (a la Carhart (1997)), the same argument is consistent with the findings of Brown and Harlow (2004), who show that there is a positive relation between mutual funds’ performance and their style stability. 5.3.2 Extensions: Sharpe ratios and R-squares, or appraisal ratios and R-squares? Can the finding that low R-square is associated with higher Sharpe ratio be translated into a similar claim about the appraisal ratio? The short answer is yes, but there is a little caveat. The exception are a class of funds which define themselves as hedge funds and report to hedge funds databases, while in reality their nature is more that of an index fund. Precisely, there are 69 funds in the database whose tracking error is smaller than 3 bp per month (about two orders of magnitude less that the median of the sample). These funds have an average alpha of 1% per month (yet half of them have alpha less than or equal to zero), but the average information ratio are extremely large. We windsorize the data at 1% and carry out a analysis similar with what we ran for 4 Brunermeier and Nagel (2004) show that several large funds rode the technology bubble and Chen and Liang (2005) find managers may have timing skills. 32 Sharpe ratios. The results in Table 8.19. 5.3.3 Extensions: Idiosyncratic Risk by Strategy We can question whether our results regarding the negative relationship between R2 and performance is driven by a particular strategy who simultaneously have low R2 and outperforms. For example it is apparent from Table 8.17 that Convertible Arbitrage funds have smaller R-squares and higher Sharpe ratios compared to the rest of the funds, so it is legitimate to ask if our results are driven by this relationship. To check if this is the case, we have used our classification into strategies and added strategy dummies in the performance regressions, as well as in the regression studying the relationship between R-squares and fund characteristics. This was done by first dropping the Complexity variable from the regressions, and adding instead a strategy dummies. A strategy dummy can be computed in two ways: one is to assign a value of one to the predominant strategy of the fund and zero to the rest; another is to find all the strategies the fund is invested in, and assign to each of those dummies a 1 times the weight the fund has in that strategy. In both cases, c2 actually increased in magnitude when dummies were The t-statistics of the R2 − R added, so the documented relationship between R2 and performance is robust. A similar effect is obtained in the regressions linking R2 and fund characteristics. Furthermore, we reproduced our analysis within each strategy. The results are presented in Table 8.18. We observe that our claim, that lower R-squares correspond to higher Sharpe ratios, is robust to all categories, except for the Macro and Fixed Income Arbitrage categories. We note that the difference is negative (contrary to our claim) in only one category, and in that case it is statistically insignificant. The result in the Macro category may be due to excessive classification of the funds as ’Macro’, while in reality they may be multi-strategy funds. The claim that lower R-squares are associated with higher Management Fees 33 and Incentive Fees is robust to all categories. The claim that lower R-squares are associated with longer lives also holds for all categories but Event Driven and Emerging Markets. We note, again, that in the cases when the direction of the difference goes against our findings, it is statistically insignificant. Some of the variables used in the hypotheses exhibit what we may consider high correlations. Of concern is the correlation between Complexity and Age, at 30%. As point 6 below indicates, we have reproduced our results using strategy dummies and thus eliminating the variable Complexity from the regressions. The correlations between Age and these dummies are between less than 5%. Of concern is also the negative correlation of -40% between F OF and if ee. We have therefore projected if ee on F OF , and run the same analysis using F OF and if ee⊥ , and we obtained similar results. 5.4 Economic Significance The previous section argues that across time, low Rsqr funds outperform; in this section we shall analyze the portfolio of the low Rsqr, respectively, the high Rsqr funds. For that, we form equally weighted portfolio of funds from the lowest past Rsqr decile, and rebalance this portfolio on a quarterly or annual basis. We also build portfolios drawing from all the funds, then from all the funds over $15 million, then over $ 100 and over $ 300 million. Note that in order to form portfolios based on past Rsqr, we need funds with 3 years of previous data, and in the case of annual rebalancing, with 1 more year of data after the formation period. We do the same for the highest past Rsqr decile. The timeline of portfolio formation is described in the diagram below: 34 T − 36 | {z T − 12 }|| construct benchmark {z calculate Rsqr T | }| {z } form low previous Rsqr portfolios; hold till rebalancing The summary statistics of these portfolios are reported in Table 8.11. The portfolio of low Rsqr exhibit higher returns (with only one exception) and higher Sharpe ratios (both on an adjusted, or not adjusted for serial correlation) that the portfolio of the high Rsqr funds. For example, if we look at the portfolio low Rsqr funds over $ 300 million of size, rebalanced annually, versus the portfolio of high Rsqr funds, they have close means but the standard deviation of the high Rsqr portfolio is almost three times higher than the standard deviation of the low Rsqr portfolio, and the Sharpe ratio is 0.45 higher. The portfolio consist of 69 funds, a reasonable number to describe a fund of funds portfolio. These differences prevail across portfolios drawn from funds of different sizes, rebalanced quarterly or annually. The means are, with one exception, higher in the case of the low Rsqr portfolio. Additionally, we can form random portfolio of funds, equally weighted and containing the same number of funds as the corresponding low Rsqr portfolio (for example we can form portfolio of funds over $ 300 million, choosing the funds randomly from all the funds over $ 300 million and weighting them equally). When the Sharpe ratio of the low Rsqr portfolio is compared to the Sharpe ratios of the random portfolios containing the same number of funds, the low Rsqr portfolio ends the contest in the top 5%. Note that a portfolio long the low Rsqr funds has a good Sharpe ratio (0.62 is the worse Sharpe ratio of such a portfolio in Table 8.11, for the portfolio of funds larger than $ 300 million and on an adjusted basis), while a portfolio long in the 35 low Rsqr funds and short in the high Rsqr funds5 has not. This is due to the fact that the short leg has a standard deviation high enough so the effect of adding it to the long leg would be a large drop in the Sharpe ratio. We conclude that the negative relation between Rsqr and fund performance is economically significant. 5.5 R-square and inflows We have argued that managers with low R-squares due to idiosyncratic investments are successful. Do managers keep their ability to find idiosyncratic investments when the hedge fund receives inflows? Where do managers place the inflows? There are four possibilities. One possibility is that managers invest more into their systematic component. In this case, Rsqr increases with the inflows, and the timing measure T O1 increases, capturing a raise in the betas of the systematic component os the fund. The second possibility is that the fund invests in new systematic factors. In this case, T O2 increases because it captures timing among the factors, other than ones the fund is already exposed to. The third possibility is that the fund de-levers (invest the inflows in cash). This would be consistent with a decrease in the riskiness of the fund following the inflows and with no changes in the Rsqr. The last possibility is that the fund invests idiosyncratically. The betas stay the same, so the timing decreases, and the Rsqr decreases because the model estimated before the inflows does even a worse predictive job than without the inflows. 5 It is also unrealistic to short hedge funds. There have been recent attempts to create hedge funds derivatives in the OTC markets, but shorting hedge funds has yet to be developed as a process. 36 In order to investigate the effect of inflows on the ability of funds to find investments uncorrelated with public indices, we estimate the following model: RsqrTi = ci + b1 aumT,i + b2 Last1Y StdT,i + b3 AgeT,i + b4 P ressureT,i +b5 mktT + b6 F OFi + b7 mf eei + b8 if eei + b9 F lowF racT,i +b10 AliveT,i + ²T,i T O1iT = ei + d1 aumT,i + d2 Last1Y StdT,i + d3 AgeT,i + d4 P ressureT,i +d5 mktT + d6 F OFi + d7 mf eei + d8 if eei + d9 F lowF racT,i (5.7) +d10 AliveT,i + ²T,i T O2iT = gi + f1 aumT,i + f2 Last1Y StdT,i + f3 AgeT,i + f4 P ressureT,i +f5 mktT + f6 F OFi + f7 mf eei + f8 if eei + f9 F lowF racT,i +f10 AliveT,i + ²T,i The results are presented in Tables 8.12, 8.13 and 8.14. To analyze the effect of the inflows on the extent to which managers can invest idiosyncratically, we look at the coefficient of P ressure, which measures the inflows into hedge fund strategies, from Tables 8.12, 8.13 and 8.14. We observe that as strategies of funds receive inflows, Rsqr decreases when funds of all sizes are included in the analysis. Both types of managers, of large and small funds, seems to be timing the factor exposures less after inflows are received, consistent with the hypothesis that managers invest idiosyncratically. The first three possibilities on placing inflows are therefore eliminated, the alternative remaining being that funds, on average, invest the new money idiosyncratically (so their talent materializes after money flows into hedge funds). However, analyzing the relationship between Rsqr, T O1, T O2 and inflows 37 not into strategies but into individual funds, controlling for the flows in strategies, we observe, from Tables 8.12,8.13 and 8.14, that what we just showed to hold for strategies pressed with inflows ceases to remain true at funds level. Precisely, if funds receive inflows (expressed by F lowF rac) above and beyond their strategy average inflows, we observe that Rsqr increases, while T O1 increases and T O2 is not significantly affected. This is evidence consistent with the fact that managers of funds receiving more money relative to their size than other funds in their industry fall into the trap of making easy fees and boost the allocation to their systematic component. As we have shown earlier, this behavior triggers under performance, which Agarwal, Daniel and Naik (2004) show it conducts to outflows. This seems to be a self regulatory mechanism of the size of hedge funds: managers receiving more money than their industry average invest more systematically, under-perform and subsequently experience outflows. 38 Chapter 6 Extensions: Other Measures of Success and R-squares The previous section argues that low R-squares are inversely related to one measure of a fund’s success: the Sharpe ratio. If low R-squares funds out-perform, do investors recognize this? If they do, we expect the low R-squares funds to be able to charge higher fees than the high R-square funds, have more assets under management, be more likely to remain in the sample (i.e., survive) and be older. In this section we investigate the relationship between R-squares and these other measures of a fund’s success: fees, assets under management, age and the likelihood to survive. 6.1 R-squares and fees The objective of this section is to test the following hypotheses: Hypothesis 3: Higher fees correspond to lower overall R2 . Hypotheses 4: Higher fees correspond to lower Rsqr, T O1, T O2. From Table 8.5, we observe that as we move from the low to high R2 terciles, 39 the average of both incentive and management fees decrease. There is a difference of 20.20 basis points between the average management fees charged by the funds with low R2 and the average management fees charged by the funds with high R2 . the corresponding difference is 226.39 basis points for the incentive fees and both difference are statistically significant. Hypothesis 3. is therefore strongly supported. Figures 8.7 and 8.8 both show that funds with lower Rsqr, T O1, T O2 charge higher fees. This is confirmed further by the Fama-MacBeth tests of Table 8.9. Hypothesis 4 is thereby strongly supported. Furthermore, Table 8.6 shows a negative and statistically significant relationship between R2 and the fees charged; we can therefore reject the null of no relationship in favor of Hypothesis 3. One result from Table 8.6 is that fees are not very strongly related to the tracking error. In fact only the management fee seems to be strongly related to the tracking error, and this is true only for the funds smaller than $ 15 million. If the structure of the fees is what incentivizes the manager to take more or less risk, then mutual funds, who charge only management fees, should have a different risk taking behavior from hedge funds, whose manager extract rents from investors through mostly through the incentive fees. Under this assumption, that fees drive the risk taking, the weak link we find between the incentive fees and the tracking errors is consistent with Brown, Goetzmann and Park (2001) who show that hedge funds are less likely to engage in tournament behavior than their mutual funds counterparts. Furthermore, the fact that management fees are positively related to tracking errors are then consistent with Brown, Harlow and Starks (1996), who show that mutual funds (who charge management fees) engage in tournaments behavior. An interesting question is what do investors pay fees for. Are the hedge funds compensated for performance, Sharpe ratios, or in fact investors recognize the importance of the idiosyncratic investment ideas in the hedge fund world and 40 compensate the managers with these ideas? In order to answer this question, we test whether fees explain the funds’ R2 s beyond Sharpe ratios or excess returns. The results are presented in Table 8.15. We observe that fees explain the R2 ’s of the funds above and beyond Sharpe ratios or raw returns and that this relationship is more stronger for incentive fees. If fees reflect talent, this evidence suggests that R2 is a better descriptor of the manager’s talent, as differentiated from luck or any other apparent manifestations of talent1 . The Fama-MacBeth tests in Table 8.9 reject the null of no relationship between the Rsqr, T O1, T O2 and the fees in favor of Hypotheses 3 and 4. The fixed fund effects tests continue to confirm the negative relationship between fees and Rsqr, T O1, T O2. The null of no relationship between Rsqr, T O1 and T O2 respectively and fees can further be rejected at more than 1% confidence given the results in Tables 8.12, 8.13 and 8.14, in favor of Hypothesis 4. 6.2 R-squares and assets under management The objective of this section is to test the following hypotheses: Hypothesis 5: Assets under management are inversely related to overall R2 . Hypothesis 6: Assets under management are inversely related to Rsqr, T O1, T O2. From Table 8.5, we observe that funds in the low R2 tercile are $ 9.21 million larger on average (also statistically significantly) than the funds in the high R2 tercile. To gauge the magnitude of the difference, recall that the median size of 1 For example, Berk and Green (2004) imply that the proposition “talented mangers outperform” is a myth (because talented managers get to manage larger funds and their talent cannot materialize in performance regardless of scale). Hence, measuring manager’s talent by performance may be wrong. Our results suggests that investors reward more a low R2 manager than a outperforming one. 41 the hedge funds is around $ 38 million. This is evidence in favor of Hypothesis 5. However the difference is not statistically significant (although economically large). There is a concave, statistically significant relationship between R2 and size of the fund (as captured by aum25, as apparent from Table 8.6. Thus we reject the null of no relationship between R2 and size of the funds in favor of Hypothesis 5: when we examine the concave relationship between R-squares and size, we observe that it becomes negative only past a certain fund size. The coefficient of aum25 is 0.0107 for all funds, while the coefficient of (aum25)2 is -0.0014. This means that the R2 decreases with fund size as the fund manages more that log(0.0101/0.0014) = $2.0338 million. On the panel data, Figure 8.6 indicates that low R-square funds manage more money; this is confirmed by the Fama-MacBeth regressions of Table 8.9. Note that the Fama-MacBeth regressions produce stronger results that a pure cross-sectional regression using the whole history of the funds; the relationship between Rsqr and size is linear, instead of quadratic. Although we can reject the null of no relationship between R-squares and size in favor of Hypothesis 6, the relationship between T O1, T O2 and size is inconclusive. Some of the larger funds may, therefore, time exposures as well as invest idiosyncratically. The panel regressions do not need a quadratic size term as they capture the dynamics of size through time. From Table 8.12, we observe that funds with larger assets under management (aum) have also lower Rsqr (the relationship is significant). Therefore the null of no relationship between Rsqr and aum is rejected in favor of the alternative, which is Hypothesis 6. Again, the relationship between growth ( as expresses by the coefficient on aum on the time series regressions) and timing is inconclusive. 42 6.3 R-squares and age The objective of this section is to test the following Hypotheses: Older funds have lower overall R2 . Older funds have smaller Rsqr, T O1, T O2. Table 8.5 shows that funds with a lower R2 have been reporting to our database 9 months more, on average, than the funds with a higher R2 . This difference is statistically significant and economically important as it represents 17% of the median age of a fund in our database (which is 53 months), and constitutes evidence in favor of Hypothesis 3a. The coefficient of Age in the R2 regressions from Table 8.6 is negative and statistically significant, so we can reject the null of no relationship between age and R2 . The same relationship carries to larger funds although the coefficient of Age is less significant. When the relationship between age and R-squares is studied inter-temporally, the sign of the coefficient on age in fact reverses: as the funds grow older, their Rsqr in fact increases (see Table 8.12). As the panel regressions capture time series effects, our result may be explained in the context of Boyson (2004), who finds that managers with longer tenure (which older funds are more likely to have) tend to become less risky, and take actions that are more moderate than managers who are younger. This causes the manager of an aging fund to invest more conventionally - or equivalently less idiosyncratically. Hence, as managers get older they are less likely to use their investment talent, they prefer to covary more with public indices and thus ensure that their potential failures are “conventional”. Although different in scope and performed on another database, this study also shows that the magnitude of the idiosyncratic risk borne by hedge funds is unrelated to management fees, while it is positively related to incentive fees. If the 43 pay structure is the driving force behind managers’ decisions on risk taking, then our findings suggest that managers paid predominantly in management fees take higher risk2 , while managers paid mostly in incentive fees do not3 . We confirm the results of Boyson (2004). She shows that as a manager’s tenure (interpreted here as the age of the fund) becomes longer, the manager’s career concerns increase, which induces him to take less risk and to herd more. Thus, with age, the conformity of a manager increases. This may be seen in our Table 8.12, which shows that as the fund ages, the Rsqr increases. 6.4 R-squares and survival The objective of this subsection is to test the following: Hypothesis 9: Funds with a lower probability to exit the sample have lower overall R2 . Hypothesis 10: Funds with a lower probability to exit the sample have lower Rsqr, T O1, T O2. From Table 8.5, funds with lower R2 have 5.19% more probability to remain in the sample than funds with high R2 , supporting Hypothesis 9. Table 8.6, as well as Panel B in Table 8.7 further shows a negative, statistically significant relationship between probability to remain in the sample and the R2 . We can reject the null of no relationship in favor of Hypothesis 9. When testing Hypothesis 10, however, the tests do not have enough power to be able to reject the null of no relationship. 2 Brown, Harlow and Starks (1996) show that mutual fund managers, who charge mostly management fees, engage in tournaments behavior. 3 Brown, Goetzmann and Park (2001) show that hedge funds, who profit mostly from charging incentive fees (and also have watermarks), do not engage in tournaments behavior. 44 Chapter 7 Implications for Risk Management This study contributes to the current debate on the hedge fund transparency needs of the hedge fund investors. Hedge funds do not disclose their positions and hedge fund investors may only use returns to infer the risks embedded in their hedge funds. In this section, we address an issue related to the risk management performed by investors in hedge funds, as opposite to that performed by the funds themselves. For the latter, see Lo (2001). To asses the risk associated with a hedge fund, each investor estimates a factor model from the history of the fund; this estimation is useful as long as it is valid out of sample. In the language of the measures employed in this study, inferring risk exposures from the monthly returns is useful only as long as Rsqr is positive (and as close to the maximum of 1 as possible). We argued that funds for which Rsqr is large are funds whose managers do not exhibit talent; in other words, we document a tradeoff between the quality of the fund and the extent to which monthly returns allow investors in hedge funds to extract the risks the fund is exposed to. In particular, the better the returns of 45 the fund can be “explained” by systematic factors, the worse the Sharpe ratio of the fund is. The top panel of Figure 8.5 plots the difference between the average Sharpe ratios of the funds that are least likely to be explained by public factors, compared to the average Sharpe ratios of the ones most likely to be explained. The tradeoff between performance, and the extent to which a fund may be explained by systematic factors exacerbates in 2000 (the difference in average Sharpe ratios is close to 3 !), and it seems to have levelled in the recent years (2003), when the volatility of the markets in general lowered. These findings suggest that if an investor could understand the risk exposures of her hedge funds from the past monthly returns of the fund, then the funds are under performers. Combined with the fact that Rsqr has a negative median (see Table 8.8), this seems to suggest that on the one hand, inferring risk exposures of hedge funds from the past time series of monthly returns does not have much success, and in the case it has, it is because the investor holds under performing hedge funds. This explains the thriving risk systems which, instead of using only the time series of returns of a hedge fund to infer the inherent risks of the funds, actually collect the fund holdings under mutual secrecy arguments and aggregate these holdings for the investor1 . The evidence presented in this study also suggests that more transparency is needed from the hedge funds to their investors. 1 So the investor does not get to observe the positions of the fund, only the aggregate risks. 46 Chapter 8 Conclusions It is almost a truism of the hedge fund universe that a talented manager is one with investment ideas that are “out of the box” and whose strategies are uncorrelated with publicly available indices. Intuitively, a less talented manager has to rely more on investing correlated to public indices in order to produce returns. In turn, this manager’s R-squares will be low, and a simple model shows that their performance will also be low. Consistent with the common wisdom confirmed by our simple model, we find evidence in support of this hypothesis. Our finding, that the lack of covariance with public indices is related to abnormal performance, is economically significant: we show that a portfolio of low R-square funds outperforms a portfolio of high R-square funds. In addition, we show that a portfolio of low R-square funds outperforms the average portfolio of hedge funds, and that this relationship is robust to the size of the funds considered or to the frequency of rebalancing the portfolio. Additionally, we find mixed evidence on the funds’ ability to invest inflows in idiosyncratic strategies. This study shows that inflows into fund strategies do not trigger an increase in the systematic exposures of the hedge funds, on average. Therefore, the relationship between R-squares and performance is robust to inflows 47 into hedge fund strategies. While strategies of hedge funds accept inflows if these inflows can be invested idiosyncratically, the same ceases to hold true for individual hedge funds who receive inflows above and beyond their category average. These funds succumb to taking some systematic risk - thus, potentially, making easy fees off beta exposures; eventually these funds under perform and thus experience outflows. Not only do we find that funds whose strategies are more idiosyncratic have higher Sharpe ratios, but also that these funds are recognized and compensated by investors. For example, these funds have more assets under management. Additionally, we show not only that low R-square funds are able to charge higher fees, but also that R-square dominates excess returns or Sharpe ratios as an explanatory variable of the fund fees. This relationship is stronger for incentive fees. Simply put, we show that investors pay for investment ideas with little systematic risk on top of what they pay for fund performance. To the best of our knowledge we are the first to document the tradeoff between the extent we can understand the risk exposures of a hedge fund, and its performance. We show that if a manager is good, then his risk exposures cannot be inferred only from the monthly returns of the fund. Thus, this study has implications for the risk management performed by investors holding hedge funds and attempting to understand the risks associated with them. If investors attempt to understand the systematic exposures of their funds using factor models, and succeed, then our results show that the fund manager either lacks talent, or that he is overinvested. Returns based risk management is therefore problematic if the hedge funds to be explained have talented managers and provide only monthly returns, and additional transparency may be required by the investors of these funds. 48 Appendices 49 Methodologies to fit the factor model In this section we argue that using stepwise regressions to estimate the model (5.2) is superior to methodologies such as principal component analysis. Equation (5.2) is fitted for each of the funds in our database; we use the whole history of the fund (in-sample estimations), and we also roll estimations across time (to study the out-of-sample goodness of fit). Hedge fund histories are generally not very long - the median life of a fund in our database is 53 months while 31% of the funds have less than 34 months of history - and we employ K = 34 factors. Thus, while evaluating (5.2) in or out-ofsample we cannot simply run regressions on the fund returns on the set of all the factors. We investigate two main methods used in the factor model literature to reduce the number of factors. Agarwal and Naik (2004) advocate the use of stepwise regressions to explain hedge fund returns, claiming that the “benefits of using the stepwise regression [to explain hedge fund returns] ... outweigh the costs [of potential data mining]”. The first method considered is therefore the stepwise regression; the stepwise regression is an OLS regression performed in stages, in which not all the independent variables are used. First only the independent variable the most correlated with the dependent variable is considered (along with its first order lag), then new independent variables are included (with their first order lags) in the regression if and only the introduction increases the R2 of the regression. Due to the presence of serial autocorrelations in the time series of the returns, this study adapts the stepwise regression so that the first order lags of the dependent variables are included along with their contemporary counterparts at each stage of the stepwise regression, a la Getmansky, Lo and Makarov (2004), Section 5.2. The alternative method is factor analysis, where the model consisting of 34 factors is reduced to a model including only n < 34 factors, usually a linear 50 combination of the set of 34 factors. We take the first n principal components as the factors. There are several criteria that help identifying the number n of principal components that ought to be used: the Kaiser criterion selects only the principal components whose corresponding eigenvalue is greater than one1 ; when used in our case, it recommends selecting the first 6 principal components. The Cattell criterion picks principal components until their explanatory power seems to “level out” when plotted; applied to out case this criterion recommends selecting 5 principal components. The Akaike and Schwartz criteria minimize an expression depending on the likelihood function of an explanatory regression of the factors selected. When minimized in our particular case, these two criteria recommend to pick 10, respectively 8 factors. Although we have experimented with two other methods of fitting, namely ridge regressions and the lasso method (see Hastie, Tibshirani and Friedman (2001)), the results were similar to those obtained from the factor analysis and hence we only present the latter. To compare the two methodologies, we have simulated 10,000 funds under the null that they hold systematic risk described by our set of 34 factors, and analyzed which method better captures these risk exposures. We first simulate entire histories (that is funds that invest in systematic risks on the whole time period January 1994 - December 2003), then we simulated funds whole life span spreads only on random periods 24 months in length. We made the assumption that each fund does not invest in more than 10 different factors2 . Each fund is simulated in the following way: 1. A random number of factors is selected from a uniform distribution on 1 This criterion includes a principal component if and only if it’s explanatory power is at least as high as of one individual factor. 2 When later fitting the model (3.2) to the hedge fund returns data, either using the whole history or on a rolling basis, we observed that in all but 1% of the fund-month cases in which we estimated the model, the funds were exposed to fewer than 9 different factors. 51 the interval [1, n]. We select n = 10 (the fund invests in maximum 10 factors), and also n = 3 and n = 5 to compare the quality of fit with the case when the number of factors decreases3 . On short, we choose k ∼ U [1..n], n = 3, 5, 10. 2. Once the number of factors, k, is selected, a number of k factors, j1 , ..., jk is randomly drawn from the set of 34 factors in our data set. We choose j1 ∼ U [1..34], then j2 ∼ U ([1..34] \ {j1 }), ..., jk ∼ U ([1..34] \ {j1 , ..., jk−1 }). 3. A beta for each of these factors is drawn from a normal distribution. That is, we choose (βj1 , βj2 , ..., βjk ) ∼ N (0k , Ik ). 4. Another number, to represent the monthly idiosyncratic volatility of the fund, is drawn from a normal distribution, scaled so that the monthly idiosyncratic volatility cannot be larger than 5% with a probability of 99.9% (the typical overall volatilities of funds in-sample have a median of around 1% per month in our sample, when model 5.2 is estimated on rolling two year windows). That is, we choose σ² ∼ |N (0, 1)|/50 (the median of this random variable is 1.00%). 5. If we are using rolling windows, we simulate a starting time t0 ∼ U [1..(120− 23)]. The ending time is t1 = t0 + 23. If the whole history of the fund is used, then t0 = 1, t1 = 120. 6. A series of normal random noise is generated with a length equal to the history of the fund (which may be January 1994 - December 2003 or random subperiods of 24 months). That is, we simulate ² ∼ N (0T , IT ), where T = 24 (rolling windows) or T = 120 (whole history). 7. The returns of the fund simulated are equal to the sum of the randomly selected factors F j1 , ..., F jk multiplied with their corresponding betas simulated at step 3, plus the noise simulated at step 6 multiplied with the idiosyncratic volatility simulated at step 4. That is, 3 The motivation to pick 5 factors comes from the fact that the median number of factors the fund is exposed to, as computed from fitting the model (3.2) to hedge funds data, is equal to 5. 52 Rt = 34 X βm Ftjm + σ² ²t , t = t0 , t0 + 1, ..., t1 , m=1 where βl = βj m , l = jm 0 , l 6∈ {j1 , ...mjk }. We then use stepwise regressions and factor analysis (we repeat the analysis for the cases when 5 to 16 factors are used, as various criteria advise on any number c1 , ...β d between 5 and 10) to infer the betas of these funds, (β 34 ), and we calculate the square distances between the true betas and the inferred betas, that is, 34 X 2 |βc m − βm | . m=1 The distribution of these distances is presented in Table 8.16 for the various cases considered. The median differences between the true and estimated betas is the smallest in the case we use the stepwise regression, in all cases analyzed. If the estimation of the models is made for the whole history, stepwise regression clearly dominates the other methods. We notice, however, that if the fund can be exposed to as many as 10 systematic factors and the estimation is done on periods of 24 months, the distribution of the differences from the stepwise regression is much flatter than the similar distributions obtained when using principal components, indicating that the stepwise regression has a potential for more serious spurious mismatches than the multivariate factor analysis (or principal components analysis). However, in the majority of cases the stepwise regression produces a better fit than the alternative methodologies. Figure 8.9 shows, in each of the cases considered4 , the percentage 4 The fund is exposed to 3, 5 respectively 10 factors and the estimation is done for the whole history of for rolling periods of 24 months. 53 of simulations in which the stepwise regression produces a better fit than using a number of principal components. These comparison results are consistent with the claim of Agarwal and Naik (2004) on the quality of the stepwise regression. Note, however, that their statement and our comparison results are contingent upon the time period used and the factors employed (specific for hedge funds), and it should not be considered as an absolute proof of virtues for the stepwise regression in general. 54 Spurious estimation of systematic risk: Estimation of the systematic models Although we show that stepwise regression is a better way of inferring funds’ systematic exposures, for comparison reasons, we perform our tests based on factor analysis as a methodology to infer funds’ exposures. The results are qualitatively similar, although the tests lose power. For example, the adjusted R2 of the panel regressions presented in Table 8.12 drops 28 times (!) when factor analysis is used instead of the stepwise regression. As low out-of-sample R-squares may be of some concern, we experimented with the CAPM and the Carhart (1997) four factor models as explanatory models for hedge fund returns. Both produce negative out-of-sample R-squares, with the Carhart (1997) model faring worse, at the median, than the model based on stepwise regressions. This result show that naive smoothing of the out-of-sample fit is not likely to be achieved. Because it is built out of sample, Rsqr is less likely to be the subject of spurious estimation and this is the reason we prefer to use it throughout this study. However, we may use pure rolling R2 (an in-sample measure). If we do so then the conclusions of our analysis do not change. 55 Tables and figures 56 Table 8.1: Summary Statistics for the Hedge Funds Industry. Databases started keeping track of the funds that ceased to report only after 1994. Second Panel presents cross-sectional summary statistics of the funds. Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Funds Born 83 159 214 278 404 508 506 683 722 666 667 611 803 877 767 25th percentile of size ($ mil) Age Still in-sample (Alive) FOF Management Fee (%) Incentive Fee (%) Mean Excess Return (annual) Sharpe Ratio (annual) Funds Died 2 1 0 1 0 61 186 182 316 497 458 381 316 174 516 Total Funds 363 520 733 1011 1414 1922 2367 2864 3404 3754 3924 4077 4499 5060 5653 Mean Median Std 53.3589 66.0582 72.84% 23.33% 1.3665 16.5776 7.32% 0.8018 11.9450 53.0000 N/A N/A 1.2500 20.0000 6.21% 0.6657 388.0830 50.2813 N/A N/A 0.7657 6.8582 14.69% 1.5797 57 Table 8.2: Heterogeneity across net returns and assets under management from the Altvest, HFR and TASS databases. Means (with standard deviations in parenthesis) and Medians of the monthly returns net of fees and Assets Under Management across Altvest, HFR and TASS, and the merged database. A Kolmogorov-Smirnov test to check whether the distributions of assets under management and net returns are similar among the databases. Low p-values for the Kolmogorov-Smirnov statistic means that we are more likely to reject the null that the two samples are drawn from the same distribution. TASS seems to be different than both Altvest and HFR, and seems to contain funds with lower assets under management and with lower net returns. Altvest Net AUM .0133 (.0370) 122.52 (351.78) HFR 2003 Mean (Std) .0129 (.0345) 140.40 (815.93) Entire database Mean (Std) Net AUM TASS .0131 (.0388) 99.05 (218.55) .0099 (.0626) 85.05 (443.95) Net AUM .0088 32 Net AUM Altvest-HFR .2922 .0351 2003 Median .0086 28.6 Kolmogorov-Smirnov p-values Altvest-TASS .0042 .0000 58 .0082 26.4 HFR-TASS .0400 .0000 59 Sample statistics for the risk factors used. FTSE 100 MSCI World ex US MSCI EM NAREIT Equity NASDAQ Russell 1000 RUSSELL 2000 Russell 3000 max(R3K, -.01) min(R3K, -.01) R3K*R3K Russell Midcap Value VIXret MSCI EAFE SMB HML MOM Mean 0.53 0.35 0.08 1.01 1.14 0.98 0.92 1.11 1.56 -1.45 0.12 0.89 1.88 0.24 0.27 0.14 0.89 Median 0.40 0.55 0.49 1.19 1.57 1.38 1.68 0.94 0.94 -1.00 0.03 1.41 0.17 0.75 -0.13 0.11 1.08 Std 5.93 4.39 6.86 3.43 8.31 4.59 5.70 3.25 2.62 1.17 0.19 4.34 18.10 4.36 3.82 4.65 5.69 DXY Lehman Agg Lehman Agg - MBS Lehman Munis 10Y SB Currency Hedged SB Wt Gvt 1 Yr SB US Treas 10 Yr SB US Treas 30 Yr SB US Treas 5 Yr DefSpread CPI AMEX Oil Index GSCI SPPo SPPa SPCo SPCa Mean -0.07 0.57 0.56 0.51 0.58 0.57 0.52 0.57 0.51 0.81 0.20 0.80 0.52 -8.99 -85.35 6.01 -9.04 Median -0.12 0.66 0.61 0.71 0.65 0.30 0.48 0.64 0.44 0.71 0.19 0.40 0.63 3.80 3.98 4.16 3.86 Std 2.21 1.15 0.90 1.34 0.93 1.91 2.14 3.16 1.34 0.23 0.23 5.25 5.54 124.22 1099.81 199.61 463.33 There are 17 equity indices (including timing terms), 11 fixed income indices, 2 commodities indices and 4 option strategies. All the option strategies are written on the S&P 500 index, and the first two symbols in the factor name, “SP”, symbolizes that. The third letter is the name of the option used, “P” for put and “C” for call. The last small letter indicates moneyness, with “a” indicating at the money and “o” indicating 1% out-the-money. The unit is percent-per-month. Three of the factors are not investable: max(R3K, -.01), min(R3K, -.01) and R3K*R3K. Table 8.3: Table 8.4: Summary of factor analysis applied to individual hedge funds. Summary statistics for R2 and tracking errors from fitting the model (5.2) to individual hedge funds and fitting the model (3.2) (no serial correlation) for mutual funds are presented. The standard errors for the hedge fund model are adjusted for serial correlation. Mean Std 25th percentile Median adjusted R2 (stepwise regression) adjusted R2 (investable factors) adjusted R2 (Carhart 4-factor) adjusted R2 (VW market returns) TE 42.50% 42.15% 15.85% 5.99% 3.80% 26.89% 26.98% 25.86% 9.59% 6.10% 21.77% 20.91% 1.46% 0.52% 1.64% 42.10% 41.52% 13.71% 2.43% 2.71% 75th percentile Hedge Funds 62.96% 62.22% 30.09% 7.16% 4.50% adjusted R2 (stepwise regression) TE 61.34% 1.82% 28.34% 1.90% 48.60% 0.53% 66.86% 1.26% Mutual Funds 83.45% 2.61% 60 61 R2 and talent proxies. 12.14%(10.02%) 41.66%(7.65%) 72.79%(12.60%) -0.6065 -6.0499 low R2 med R2 high R2 low-high mean low-high t-stat R2 0.3853 7.1255 1.0197(2.3064) 0.7564(0.9997) 0.6344(1.0644) SR 0.2029 8.7329 1.4763(0.8905) 1.3525(0.7485) 1.2734(0.6254) mfee 2.2639 11.0641 17.7291(6.3577) 16.5727(6.8616) 15.4652(7.1362) ifee 9.21 0.81 57.53(439.8348) 54.38(416.6248) 48.32(294.6952) aum25 9.2085 5.8381 69.2829(50.2678) 69.0009(45.6930) 60.0744(53.8956) Age 0.0519 3.8358 0.7473(0.4346) 0.7436(0.4367) 0.6954(0.4604) Pr(Alive) We sort the funds by the R2 terciles and compute the average Sharpe ratio, Management Fee, Incentive Fee, size represented by aum25, Life of the fund and probability of not exiting the database among each R2 tercile. The differences between the low R2 tercile and high R2 tercile and their t-statistics are presented. Table 8.5: 62 The relationship between the ability to explain a fund (high R2 or low T E) and fund characteristics. Adjusted R2 cons aum25 (aum25)2 Age Alive rho Complexity F OF mf ee if ee Variable Dependent Variable 0.0371 -0.0062 0.0003 0.0001 -0.0097 0.0344 0.0008 -0.0203 0.0030 0.0002 Coef 5.75% 4.6203 -7.2702 2.0468 2.1397 -5.6438 5.0552 0.4514 -8.3210 3.4015 0.9971 0.6126 0.0107 -0.0014 -0.0005 -0.0226 -0.1645 0.0004 0.0621 -0.0351 -0.0044 TE All funds (6525 obs.) t-stat Coef 6.62% 29.4416 2.8590 -1.9770 -4.0246 -2.9282 -10.0601 0.0868 6.6304 -5.3650 -7.6269 t-stat R2 0.0338 -0.0088 0.0007 0.0000 -0.0095 0.0254 0.0036 -0.0205 0.0016 -0.0002 Coef 8.36% 4.6101 -2.1628 1.4966 0.0539 -4.1099 3.6250 4.6738 -6.6159 1.6696 -0.8376 0.6197 0.0029 -0.0007 -0.0001 -0.0216 -0.1821 -0.0089 0.0868 -0.0287 -0.0034 7.60% 17.5849 0.1085 -0.2514 -0.3353 -1.7300 -6.3751 -1.3008 6.0016 -4.1779 -3.8812 TE R2 Funds over $ 15 mil (4528 obs.) t-stat Coef t-stat Determinants of the ability to explain the risks specific to individual funds. For every fund, a stepwise regression is performed; this procedure generates a benchmark for the fund. The adjusted R2 and the standard deviation of the error term (or the tracking error T E) are stored. We then run cross-sectional regressions R̄i2 = f (f undi characteristics) + ²i and T Ei = f (f undi characteristics) + ²i and the results of this second stage regressions are reported in the table. Table 8.6: Table 8.7: The relationship between the ability to explain a fund (high R2 or low T E), and the fund’s performance, measured as Sharpe ratio adjusted for serial autocorrelation or as the probability of survival. For every fund, a stepwise regression is performed and the R2 and T E are stored. Panel A reports c2 and T E − Tc the results of cross-sectional regression of Sharpe ratios on R2 − R E, controlling 2 2 c c for fund characteristics. R and T E are estimations of R and T E from model (5.3). Panel B presents the results of a probit regression where the dependent variable is a dummy for whether the fund remained or not in our sample. Panel A: OLS regressions explaining the Sharpe ratios. Dependent Variable: Coef t-stat Coef const 0.4400 3.9009 0.2416 c2 R2 − R c TE − TE aum25 Age Alive rho Complexity F OF mf ee if ee Adjusted R2 -0.3068 -4.5121 0.0007 -0.0001 0.4192 2.2341 -0.0982 -0.1330 -0.0085 0.0028 0.0633 -0.2694 10.2083 32.5069 -4.4663 -2.5702 -0.3634 0.8857 0.2593 0.0006 0.0000 0.4289 2.2753 -0.0985 -0.1464 0.0016 0.0041 18.53% Panel B: Probit analysis explaining the fund’s probability of survival. Dependent Variable: Coefficient p-value Coefficient d Rsqr − Rsqr -0.1830 0.0244 T E − Tc E -1.3075 aum25 0.1000 0.0000 0.0921 Age 0.0045 0.0000 0.0048 rho 0.4964 0.0000 0.5733 Complexity -0.1785 0.0000 -0.1794 F OF 0.4245 0.0000 0.3849 mf ee 0.1588 0.0000 0.1728 if ee 0.0112 0.0001 0.0122 Nobs=6525, McFadden R2 6.75% 63 SR t-stat 2.2910 0.8672 0.0504 0.0052 10.4109 33.1188 -4.4748 -2.8148 0.0709 1.3108 18.28% Alive p-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7.05% 64 Out-of-sample fit of stepwise regression vs. five principal components. 0.0000 41.66% are zeros; Panel B: 0.0854 0.0089 0.0103 -0.03 -0.8781 T O1: T O2: In sample T E Out-of-sample T E In sample adj. R2 Out-of-sample Rsqr T O1 (T O2 = 0) 0.28 -0.5608 0.0067 0.0106 In sample T E Out-of-sample T E In sample adj. R2 Out-of-sample Rsqr Panel A: 20th Percentile 0.11 -0.2064 0.18 -0.0716 0.0225 0.0261 0.3672 5 principal components 0.2442 0.0175 0.0200 0.0952 0.57 -0.0391 0.0093 0.0263 50th 0.0501 0.49 -0.1489 0.0083 0.0202 Stepwise regression model 40th 0.26 0.0277 0.0280 0.0334 0.5407 0.1595 mean=1.6708; 0.65 0.0012 0.0222 0.0334 60th 0.43 0.2611 0.0450 0.0577 1.2450 0.4022 std=1.9794 0.79 0.1823 0.0301 0.0567 80th Each month from January 1997 to December 2002, the model (5.2) is estimated using two years of data prior to the current month, then the out of sample Rsqr, T O1, T O2 and tracking errors are computed as described in Section 3. The fitting of the model (5.2) is first done by stepwise regression (Panel A), then for comparison purposes using the first five principal components of the factors as explanatory variables (Panel B). Table 8.8: 65 Fama-MacBeth regressions of Rsqr, T O1, T O2 on fund characteristics. -0.3372 -0.0434 0.7326 0.0016 45.5204 -0.1087 -0.1231 -0.0073 -0.0118 0.0006 3.96% const aum Last1Y Std Age P ressure F OF mf ee if ee F lowF rac alive R2 (t-stat) Coef 2.8124 -1.0801 -2.8089 3.0591 4.8491 1.6977 -1.7078 -1.7578 -6.0181 -1.3469 0.0131 Rsqr t-stat 2.70% 0.0569 0.0063 1.5360 -0.0001 -15.4336 0.0041 -0.0041 -0.0009 0.0003 -0.0062 Coef 10.0481 0.7282 1.2328 9.9399 -4.1735 -1.1248 0.6714 -2.0035 -2.3798 0.1626 -0.9509 T O1 t-stat 0.22% 0.5520 -0.0323 Coef 2.8729 17.9910 -5.9216 T O1 t-stat 0.89% 0.3808 0.0148 1.9101 0.0006 7.0870 0.2448 -0.0261 -0.0056 -0.0355 -0.0183 Coef 6.9527 0.429 1.4993 6.4909 3.0865 0.141 5.4464 -2.083 -2.6678 -2.0376 -0.7636 T O2 t-stat 0.15% 1.6754 0.0154 Coef 1.9052 23.0458 1.8434 Dep. Var. T O2 t-stat The Table shows Fama-MacBeth regressions of out-of-sample Rsqr, T O1, T O2 on fund characteristics. The average R2 of the cross-sectional regressions are reported along with their t-statistics. t-statistics are Newey-West corrected for serial autocorrelation and heteroscedasticity. Table 8.9: Table 8.10: Performance and the ability to infer a fund’s risk exposures. The Table shows Fama-MacBeth regressions of funds’ performance calculated as out of sample d T O1 − Td Sharpe ratio, on Rsqr − Rsqr, O1, T O2 − Td O2, the estimations are from cross-sectional regressions of Rsqr, T O1, T O2 on fund characteristics. The average R2 of the cross-sectional regressions are reported along with their t-statistics. t-statistics are Newey-West corrected for serial autocorrelation and heteroscedasticity. Similar results are obtained if instead of using Rsqr− d T O1 − Td Rsqr, O1, T O2 − Td O2, we use Rsqr, T O1, T O2. const T O1 − Td O1 T O2 − Td O2 d Rsqr − Rsqr aum F low Last1Y Std P ressure F OF mf ee if ee Average R2 Coefficient 2.7797 -0.1338 t-statistic 4.9571 -3.4956 Coefficient 2.7628 t-statistic 4.8282 -0.0237 -3.1949 Dep Var: SharpeRatio Coefficient 2.9070 t-statistic 5.0144 0.1338 0.0370 -29.2579 32.8599 0.2249 -0.0892 0.0296 4.7098 2.5797 -6.3776 0.4227 2.0833 -1.7663 8.8985 0.1342 3.7231 -29.3194 25.7952 0.2279 -0.0873 0.0298 3.1712 2.0421 -6.3447 0.2764 1.4895 -1.6846 9.4913 -0.8613 0.1112 0.0181 -29.0717 10.9429 0.1401 -0.1820 0.0234 -7.9673 3.7857 1.2003 -5.9801 0.1281 1.0547 -1.7683 8.0403 8.65% 12.0817 8.64% 9.4143 15.35% 8.1409 66 Table 8.11: Economic significance of low Rsqr as talent indicator. Hedge fund long portfolios formed based on out-of-sample goodness of fit. At the beginning of every formation period (every month, quarter or year, between Jan 1998 and Dec 2003) portfolios are formed containing all the funds in the smallest decile of out-of-sample Rsqr. These funds are held until the next rebalancing period, when the procedure repeats. The table shows summary statistics of these portfolios: mean, standard deviation, Sharpe ratios, Sharpe ratios adjusted for serial autocorrelation, average number of funds in the portfolio, and the turnover (in number of funds) at the time of the rebalancing. Similar statistics are reported for the portfolio of the high decile of Rsqr (the low talent funds). On a Sharpe ratio ranking (adjusted or not for serial autocorrelation) the portfolio of the low Rsqr funds is in top 5% of portfolios of hedge funds formed at random, on an equally weighted basis, containing the same number of funds as the low Rsqr portfolio. Panel A: quarterly rebalancing all funds funds over $ 15 mil funds over $ 100 mil funds over $ 300 mil low high low high low high low high Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Mean Std Sharpe Ratio Adj. Sharpe Ratio No. of funds 7.80% 6.78% 7.95% 6.76% 8.11% 7.32% 8.36% 7.19% 3.19% 14.20% 3.25% 14.17% 3.28% 13.73% 3.50% 13.88% 1.16 0.19 1.18 0.19 1.22 0.23 1.21 0.22 1.06 0.18 1.07 0.18 1.08 0.23 1.09 0.21 231 Turnover 89% 122 88% 100 87% 69 87% Panel A: annual rebalancing all funds funds over $ 15 mil funds over $ 100 mil funds over $ 300 mil low high low high low high low high Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Rsqr Mean Std Sharpe Ratio Adj. Sharpe Ratio No. of funds 8.19% 7.23% 8.03% 7.41% 8.58% 8.82% 8.11% 7.27% 4.55% 12.81% 4.59% 12.79% 5.13% 12.72% 5.83% 12.95% 0.90 0.24 0.85 0.25 0.87 0.37 0.69 0.24 0.80 0.23 0.75 0.24 0.77 0.34 0.62 0.23 231 67 Turnover 89% 122 88% 100 87% 69 87% 68 Determinants of the out-of-sample Rsqr. 1.2738 0.7881 0.1370 2.2281 0.8059 1.0133 -0.6308 -0.0484 -0.8295 0.9914 0.7165 17.4423 0.0025 57.5338 14.4138 2.0368 -0.7154 -0.0084 -0.0123 1.4388 aum Last1Y Std Age P ressure M ktRet mktCap F OF mf ee if ee F lowF rac alive 0.04% 64,366 t-statistic Coefficient Variable Adj. R2 Obs. 1.12% 116,167 -5.4897 -1.8663 -4.7817 2.8371 -0.0500 -0.1610 -0.1155 -0.0080 0.0001 -0.0031 Adj. R2 Obs. -5.9640 -0.6333 8.3820 -2.6255 5.2458 -0.0459 -0.2051 0.0018 -0.9785 0.4814 aum Last1Y Std Age P ressure M ktRet mktCap F OF mf ee if ee F lowF rac alive t-statistic Coefficient Variable -10.8935 -5.2042 -1.8877 -4.7273 3.2150 0.0441 -0.6806 -0.1514 -0.1165 -0.0078 0.0001 0.0027 2.7828 0.8242 -0.5766 -0.1061 -0.6923 0.9257 20.0465 1.5396 -0.6597 -0.0185 -0.0114 1.3340 0.09% 64,366 1.0670 0.1881 0.1523 0.1011 5 principal components model t-statistic Panel B: Coefficient All Funds 0.5594 4.0477 0.0027 2.0421 1.66% 116,167 -5.6648 0.7912 8.4150 -0.3351 Stepwise regression model t-statistic Panel A: Coefficient All Funds -0.0428 0.2744 0.0017 -0.1312 1.1464 -2.9739 0.0463 0.0057 -0.7477 -0.6373 -5.7762 0.0159 61.4839 -6.8190 Coefficient -0.1274 -0.1302 -0.0059 0.0001 -0.0409 -0.0604 0.2497 0.0018 -1.3409 0.3844 Coefficient 1.33% 56,320 t-statistic Coefficient Funds over $ 15 mil. -1.8110 -0.6561 -0.6444 -16.2182 3.2925 0.0158 4.5086 49.9335 -5.3073 10.7111 1.5005 0.9386 -1.2274 -2.9999 0.8303 0.0441 1.9633 0.0056 -0.3184 -0.8008 1.45% 82,739 t-statistic Coefficient Funds over $ 15 mil. -4.3821 -0.0577 0.6408 1.0073 7.7150 0.0018 -2.8110 -0.0911 5.0670 -0.8512 -3.8494 -0.1182 -1.6521 -0.1304 -3.4048 -0.0060 2.8978 0.0001 -0.4396 -0.0325 Dependent Variable: Rsqr 1.76% 56,320 6.0665 1.2652 -1.2376 0.7966 1.9681 -0.3427 -1.8676 -1.9815 3.2554 3.5157 t-statistic 2.44% 82,739 -10.7543 -3.6446 -1.6579 -3.4862 3.3808 -0.3509 -4.2543 2.5167 7.7581 -0.1777 t-statistic For every month from January 1996 until December 2002, and for every hedge fund in the sample that existed for at least two years before the current month, we use the previous years of data and a stepwise regression to determine the factors best explaining the fund returns. This procedure produces a i F i 2 benchmark Bτi = αit + β1t 1τ + ... + βkt Fkτ , τ = t − 23, ..., t. The out-of-sample R is the percentage of the variance of the fund returns not explained by the benchmark B i over the time interval [t, t + 12]. To find the determinants of the out-of-sample R2 we run a panel regression using the tracking error as dependent variable. Fixed fund effects are included. The t-statistics are Newey-West corrected for serial autocorrelation. The coefficients significant over 1% are bolded. Table 8.12: 69 Adj. R2 Obs. 0.0015 0.5590 0.0001 -0.1859 0.0511 aum Last1Y Std Age P ressure M ktRet mktCap F OF mf ee if ee F lowF rac alive 0.0513 -0.0033 -0.0016 0.0000 0.0145 Coefficient Variable 0.46% 116,167 t-statistic Coefficient All Funds 1.0380 0.0012 8.2644 0.5091 2.1656 0.0001 -1.4280 -0.3840 1.8104 0.0725 6.9990 0.0502 -0.8586 -0.0032 -3.9026 -0.0016 1.8669 0.0000 0.9582 0.0138 0.52% 116,167 6.1633 6.8909 -0.8336 -3.9770 1.7216 0.9023 0.8081 7.6045 2.1718 -2.9436 t-statistic 0.0585 0.0014 -0.0010 0.0000 -0.0129 0.0015 0.7348 0.0001 -0.0828 0.0788 Coefficient 0.52% 82,739 t-statistic Coefficient Funds over $ 15 mil. 0.6598 0.0012 8.5101 0.6733 1.4203 0.0001 -0.5377 -0.3047 2.5051 0.0717 6.9430 0.0577 0.3106 0.0014 -2.0426 -0.0010 1.9151 0.0000 -0.7637 -0.0138 Dependent Variable: Determinants of the intra-factors timing. The model (5.2) is estimated at time t − 1 then at time t. There are differences between the set of factors the funds has non-zero exposures to between time t − 1 and time t. The intra factor timing turnover measures the extent to which exposures to factors common in periods t − 1 and t changed. Table 8.13: 0.57% 82,739 5.2201 6.8617 0.3154 -2.0563 1.7513 -0.8132 0.5384 7.7269 1.4211 -1.9795 t-statistic TO1 70 Adj. R2 Obs. 0.0233 1.5063 0.0004 -0.0510 0.8557 aum Last1Y Std Age P ressure M ktRet mktCap F OF mf ee if ee F lowF rac alive 0.2531 -0.0277 -0.0059 0.0000 0.0189 Coefficient Variable 0.56% 116,167 t-statistic Coefficient All Funds 3.6362 0.0211 5.3183 1.2069 1.7606 0.0004 -0.0945 -1.8366 7.8133 0.4442 7.9401 0.2463 -2.1364 -0.0269 -3.2953 -0.0062 0.1307 0.0000 0.2810 0.0133 0.64% 116,167 8.3214 7.7944 -2.1159 -3.4449 0.0392 0.1969 3.3009 4.3291 1.7238 -3.4203 t-statistic 0.2771 -0.0173 -0.0048 0.0000 -0.0212 0.0334 2.0864 0.0003 -0.1499 1.0842 Coefficient 0.61% 82,739 t-statistic Coefficient Funds over $ 15 mil. 3.1060 0.0313 5.5722 1.7014 1.2531 0.0003 -0.2322 -2.1872 8.3552 0.4635 7.3189 0.2716 -1.2042 -0.0172 -2.2161 -0.0049 0.0862 0.0000 -0.2606 -0.0281 Dependent Variable: Determinants of inter-factor timing T O2. The model (5.2) is estimated at time t − 1 then at time t. The inter-factor timing measures the number of factor that dropped from the funds’ exposures plus the number of factors the fund started being exposed to from t − 1 to t. Table 8.14: 0.67% 82,739 7.3993 7.2291 -1.2214 -2.2724 -0.0264 -0.3444 2.9269 4.5282 1.2233 -3.3981 t-statistic TO2 71 The link between R2 and fund fees. Adjusted R2 of regression const Mean Excess Returns Sharpe Ratio Management Fee Incentive Fee Dependent 0.4291 -0.0556 0.07% 102.8040 -2.0421 -3.0605 -0.0186 1.17% 74.6021 0.4399 t-stat -0.0339 0.4713 Coef 0.91% -4.5487 44.4271 t-stat -0.0056 0.5183 Coef 2.04% -11.3786 58.9489 t-stat -0.0056 -0.0184 0.5326 Coef 3.19% -11.4321 -3.0507 54.1895 t-stat Coef -0.0186 -0.0346 -0.0056 0.5804 4.15% -3.0200 -5.1037 -11.4938 44.0571 t-stat Coef Coef t-stat R-square Variable: R2 ’s of the fund with respect to an exhaustive set of systematic factors care regressed on fees. Fees are related to R2 even after controlling for Sharpe ratios and raw returns. Table 8.15: 72 Stepwise regression vs. Principal components goodness of fit. P34 0.0006 0.0332 0.3982 1.9649 10.724 0.0010 0.0791 0.6453 2.6393 20.5660 0.0045 0.409 2.1402 6.6221 54.0681 5th 25th 50th 75th 95th 5th 25th 50th 75th 95th 5th 25th 50th 75th 95th Stepwise 0.0005 0.0167 0.2289 1.3934 8.2157 0.0001 0.0059 0.0705 0.6089 4.4015 5th 25th 50th 75th 95th 5th 25th 50th 75th 95th 0.0001 0.0045 0.0424 0.4252 3.8706 5th 25th 50th 75th 95th Stepwise 0.4128 2.1469 4.226 7.4305 12.6552 0.2326 1.1592 2.6098 4.7152 8.8259 0.1112 0.776 1.8253 3.7316 7.2368 0.3422 1.9728 4.2562 6.9316 12.129 0.2078 1.0772 2.6547 4.7633 8.6219 5 0.0963 0.8129 1.8966 3.5044 7.2898 0.395 2.0419 4.0945 7.2617 12.3001 0.2260 1.1154 2.5158 4.5037 8.6149 0.0923 0.7321 1.7651 3.5884 7.011 24 months history Panel B: 0.0926 0.6837 1.6644 3.3553 6.7443 holds 0.2177 1.0367 2.2479 4.1401 8.2041 holds 0.3836 1.8726 3.7838 6.6829 11.6437 Fund 0.2169 1.0962 2.3420 4.3273 8.3445 Fund 0.3889 1.9763 3.9075 6.9889 12.0878 holds 0.2871 1.7532 3.8386 6.2922 11.2975 0.0927 0.6989 1.69 3.417 6.9045 Fund 0.2889 1.8349 3.9366 6.3922 11.6054 holds holds 0.1644 0.9598 2.4003 4.3717 7.8541 Fund 0.1750 1.0117 2.5000 4.4586 8.0012 Fund 8 0.0832 0.6505 1.6843 3.0878 6.4751 holds 7 0.0854 0.6917 1.7504 3.2623 6.7747 Fund Entire history 0.3008 1.9392 4.0919 6.7303 12.042 0.1773 1.0305 2.5611 4.5881 8.2466 6 0.0901 0.7305 1.793 3.3824 7.0561 Panel A: 1–10 0.3731 1.7829 3.6521 6.4203 11.2178 1–5 0.2066 0.9995 2.1757 3.9903 7.9086 0.0974 0.6678 1.6248 3.2177 6.6679 Principal 1–3 0.2523 1.6429 3.7316 6.0814 10.8034 1–10 1–5 0.1559 0.9094 2.2972 4.2098 7.5084 9 0.0758 0.6243 1.6189 2.9971 6.3151 Principal 1–3 factors 0.3784 1.748 3.5436 6.1507 10.9514 factors 0.2105 0.9875 2.0890 3.8795 7.7715 0.1056 0.6464 1.5826 3.1192 6.4628 factors 0.2139 1.5898 3.509 5.9128 10.5786 factors factors 0.1477 0.8488 2.1730 3.9938 7.4333 10 0.0754 0.6074 1.5184 2.8948 6.2375 factors 0.3835 1.645 3.3791 5.8724 10.5735 0.2186 0.9261 2.0231 3.7976 7.4963 0.1064 0.6094 1.5144 2.9688 6.3216 Components 0.2078 1.5348 3.3976 5.5732 10.2806 0.1330 0.8281 2.0272 3.8137 7.3487 11 0.0762 0.5463 1.4573 2.7753 5.9294 Components 0.3739 1.6379 3.3098 5.6119 10.2542 0.2192 0.9047 1.9540 3.6162 7.2950 0.118 0.6099 1.4702 2.8734 6.2224 0.1997 1.4693 3.2243 5.4343 9.4435 0.1249 0.7824 1.9016 3.6730 7.0267 12 0.0708 0.5277 1.3766 2.6729 5.7499 0.3551 1.59 3.1792 5.4141 10.1234 0.2196 0.8861 1.9502 3.5261 7.0942 0.1248 0.6067 1.4628 2.8356 5.9364 0.181 1.412 3.0213 5.2021 9.0744 0.1205 0.7169 1.8104 3.5193 6.8754 13 0.063 0.4883 1.2816 2.5626 5.5738 0.3511 1.5817 3.1151 5.2542 9.7521 0.2108 0.8779 1.9013 3.4424 7.0530 0.1352 0.626 1.4188 2.7938 5.8114 0.1649 1.2458 2.8872 4.9472 8.7103 0.1100 0.6382 1.6740 3.3926 6.7625 14 0.0601 0.4333 1.209 2.4231 5.4058 0.364 1.5178 3.0059 5.1933 9.3876 0.2066 0.8898 1.8947 3.4434 7.0300 0.144 0.6468 1.4141 2.7719 5.6741 0.1612 1.1427 2.7179 4.757 8.3085 0.1060 0.6083 1.5948 3.2205 6.5826 15 0.0591 0.4086 1.1282 2.336 5.2101 0.3658 1.5095 2.9896 5.1849 9.1449 0.2055 0.9017 1.9063 3.4153 7.1437 0.1475 0.6734 1.4636 2.8199 5.6754 0.1457 1.0331 2.5862 4.4797 7.8198 0.0781 0.5456 1.5038 2.9971 6.4722 16 0.0471 0.3479 1.0427 2.2894 4.9446 bk using stepwise regression or principal components (5 to 16 components are used). For each method, the quantity k=1 |βbk − β k |2 , which of the betas, β represents the distance between the true betas and their estimation, is calculated. The distribution of this difference is presented in the Table. Panel A presents the case when the estimation is run over 120 months, while Panel B presents the case when the estimations are run over random periods of 24 months. There are 10,000 simulations in each case. Distributions of the differences between true betas and betas estimated by stepwise regression of by factor analysis, P under the null that funds bear a certain 34 number of randomly weighted systematic risks. For each simulated fund whose returns are given by Rtf und −rf = α+ k=1 β k Ftk +²t , we produce estimators Table 8.16: Table 8.17: Average Sharpe ratios and R-squares for hedge fund strategies. Funds are classified into strategies using Sharpe (1992) regressions and average Sharpe ratios and R-squares with respect to a comprehensive set of risk factors are calculated. Short Selling Relative Value Arbitrage Macro Fixed Income Arbitrage Event Driven Equity Market Neutral Emerging Markets Distressed Securities Convertible Arbitrage Mean R-square 0.4029 0.3605 0.3894 0.3666 0.5411 0.2954 0.4769 0.3983 0.2760 73 Mean Sharpe ratio 0.5426 1.0876 0.6137 0.9132 0.7598 1.0440 0.5881 1.0035 1.2968 Table 8.18: Robustness check: repeating out hypothesis tests by strategy. Short Selling (500 funds) R2 SR mfee ifee aum25 Life Alive low R2 med R2 high R2 0.0876 0.3601 0.7504 0.5780 0.7552 0.3020 1.5095 1.3936 1.3179 18.4434 20.0133 18.3843 45.8779 27.1642 25.6438 66.6667 65.3212 51.7118 0.6970 0.7515 0.7000 low-high mean low-high t-stat -0.6628 -58.0526 0.2760 1.5836 0.1916 2.5501 0.0591 0.0982 20.2341 1.5702 14.9549 2.6400 -0.0030 -0.0602 Relative Value Arbitrage (528 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.0869 0.3384 0.6465 1.2929 1.0415 0.9337 1.3882 1.2516 1.3981 16.4225 15.5948 14.5861 42.0909 40.7858 98.5649 72.0575 69.0632 48.2167 0.7816 0.8333 0.7556 low-high mean low-high t-stat -0.5595 -41.2381 0.3593 2.6767 -0.0098 -0.1537 1.8364 2.3368 -56.4740 -1.1439 23.8408 4.4663 0.0261 0.5795 R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.1295 0.3628 0.6673 0.6340 0.5779 0.6286 1.6984 1.6811 1.3124 18.1336 16.2985 14.5163 51.4024 132.1998 58.9728 85.8764 81.1517 61.8392 0.7612 0.7107 0.6649 low-high mean low-high t-stat -0.5378 -58.7237 0.0055 0.0847 0.3860 7.0427 3.6172 7.1709 -7.5704 -0.5027 24.0372 5.8824 0.0964 2.8746 Macro (1079 funds) Fixed Income Arbitrage (346 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.0637 0.2983 0.7252 0.9103 0.8431 0.9838 1.4252 1.7763 1.5312 18.9942 17.3202 15.6412 46.2808 38.0041 114.5614 63.7982 60.5000 26.8644 0.5965 0.7368 0.7288 low-high mean low-high t-stat -0.6616 -42.3199 -0.0735 -0.3080 -0.1061 -0.6072 3.3529 3.8013 -68.2805 -0.8482 36.9338 9.8339 -0.1323 -2.1448 Event Driven (1640 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.2730 0.5605 0.7822 1.0092 0.6714 0.6039 1.2539 1.1860 1.2080 17.2878 16.8510 15.7008 38.8002 28.4088 46.7416 73.6913 76.0351 77.4301 0.7616 0.7726 0.7366 low-high mean low-high t-stat -0.5093 -72.1377 0.4052 6.6809 0.0459 1.3337 1.5870 3.8345 -7.9414 -1.0038 -3.7388 -1.0262 0.0250 0.9547 continued on the next page . . . 74 Equity Hedge (444 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.0172 0.2406 0.6194 1.6161 0.7659 0.7558 1.3320 1.3281 1.2775 18.6723 17.2541 17.8576 170.6210 57.8628 57.7703 55.3605 65.8699 40.1722 0.7891 0.7260 0.6887 low-high mean low-high t-stat -0.6022 -39.2235 0.8603 1.5068 0.0545 0.9581 0.8147 1.3361 112.8507 0.8973 15.1884 3.4067 0.1004 1.9770 Emerging Markets (1050 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.2008 0.4872 0.7353 0.7692 0.5618 0.4375 1.3810 1.3575 1.3179 17.3804 15.7746 14.3870 42.0243 32.5751 28.5119 56.5447 68.0116 61.3585 0.6916 0.7139 0.7255 low-high mean low-high t-stat -0.5345 -57.6295 0.3318 3.6362 0.0630 1.8271 2.9934 5.7631 13.5123 1.1330 -4.8139 -1.4585 -0.0338 -0.9876 (cont.) Distressed Securities (462 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.1084 0.3699 0.7066 1.3228 0.9106 0.7849 1.3563 1.3651 1.2447 17.5164 15.2903 13.7480 61.1693 37.3040 38.7168 74.7829 67.2614 44.3248 0.7961 0.6667 0.4331 low-high mean low-high t-stat -0.5982 -41.6203 0.5380 3.1881 0.1116 2.1775 3.7685 4.7815 22.4525 1.0917 30.4581 5.1302 0.3629 7.0281 Convertible Arbitrage (476 funds) R2 SR mfee ifee aum25 Life Alive 2 low R med R2 high R2 0.0238 0.2139 0.5805 1.4052 1.3873 1.1042 1.5207 1.4022 1.2737 18.1985 16.8185 15.1687 59.5569 73.4153 41.8154 52.6752 66.2038 45.4444 0.8280 0.7771 0.7160 low-high mean low-high t-stat -0.5567 -32.6518 0.3010 2.2426 0.2470 3.2196 3.0298 4.2289 17.7415 1.1112 7.2307 1.5641 0.1120 2.3940 75 Table 8.19: The relationship between appraisal ratios and R-squares. We repeat our analysis, that low R-square is associated with good quality funds, using the appraisal ratio as a proxy for performance instead of the Sharpe ratio. R2 IR mfee ifee aum25 Life Alive low R2 med R2 high R2 0.1180 0.4056 0.6988 0.1745 0.1161 -0.1433 1.4753 1.3627 1.2598 17.7729 16.5946 15.5176 58.1031 55.2410 45.0337 69.1581 69.8308 64.2132 0.7492 0.7428 0.6987 low-high mean low-high t-stat -0.5809 -178.3719 0.3178 3.9397 0.2155 9.0955 2.2552 10.8607 13.0694 1.2086 4.9449 3.0972 0.0505 3.6717 76 Altvest HFR 13 % 18 % 29 % 15 % 7% 5% 13 % TASS Figure 8.1: Distribution of data across databases merged. 77 900 Total Assets Under Management (bil) 800 700 600 500 400 300 200 100 0 1988 Figure 8.2: 1990 1992 1994 1996 1998 2000 2002 Estimated Total Assets Under Management in the hedge fund industry. 78 Figure 8.3: Estimated Quarterly Flows into the hedge fund industry. 79 4% 5% 7% 1% 2% < 1% <<1% 1% 11% 24% < 1% Short Selling Relative Value Arbitrage Merger Arbitrage Macro Fixed Income Arbitrage Event Driven 25% Equity non Hedge Equity Market Neutral − Stat Arb Equity Market Neutral Equity Hedge Emerging markets Distressed Securities Convertible Arbitrage COnvertible Bonds Managed Futures 25% 18% < 1% Figure 8.4: Assets under management distributed across strategies, when the strategies are inferred using Sharpe regressions. 80 low Rsqr − high Rsqr 3 2 1 0 −1 1997 1998 1999 2000 2001 2002 2003 1998 1999 2000 2001 2002 2003 1998 1999 2000 Time 2001 2002 2003 low TO1 − high TO1 1 0.5 0 −0.5 −1 1997 low TO2 − high TO2 1 0.5 0 −0.5 −1 1997 Figure 8.5: Differences in the means of the Sharpe ratios between groups of Low - High Rsqr, T O1, T O2. Each month from December 1997 to December 2002 the hedge funds are sorted based on their Rsqr, T O1, T O2, and the average Sharpe ratios are computed across the lowest, respectively the highest Rsqr, T O1, T O2 deciles. These differences are plotted. A star on the x-axis signifies that the difference in means between the low and high Rsqr, T O1, T O2 deciles is significant at 5%. 81 low Rsqr − high Rsqr low TO1 − high TO1 300 200 100 0 −100 1997 1999 2000 2001 2002 2003 1998 1999 2000 2001 2002 2003 1998 1999 2000 Time 2001 2002 2003 100 0 −100 −200 −300 1997 low TO2 − high TO2 1998 100 50 0 −50 −100 1997 Figure 8.6: Differences in the means of the assets under management between groups of Low - High Rsqr, T O1, T O2. Each month from December 1997 to December 2002 the hedge funds are sorted based on their Rsqr, T O1, T O2, and the average assets under management are computed across the lowest, respectively the highest Rsqr, T O1, T O2 deciles. These differences are plotted. A star on the x-axis signifies that the difference in means between the low and high Rsqr, T O1, T O2 deciles is significant at 5%. 82 low Rsqr − high Rsqr low TO1 − high TO1 1 0.5 0 1997 1999 2000 2001 2002 2003 1998 1999 2000 2001 2002 2003 1998 1999 2000 Time 2001 2002 2003 0.2 0.1 0 −0.1 −0.2 1997 low TO2 − high TO2 1998 0.2 0.1 0 −0.1 −0.2 1997 Figure 8.7: Differences in the means of the management fees between groups of Low High Rsqr, T O1, T O2. Each month from December 1997 to December 2002 the hedge funds are sorted based on their Rsqr, T O1, T O2, and the average management fees are computed across the lowest, respectively the highest Rsqr, T O1, T O2 deciles. These differences are plotted. A star on the x-axis signifies that the difference in means between the low and high Rsqr, T O1, T O2 deciles is significant at 5%. 83 low Rsqr − high Rsqr low TO1 − high TO1 low TO2 − high TO2 6 4 2 0 −2 −4 1997 1998 1999 2000 2001 2002 2003 1998 1999 2000 2001 2002 2003 1998 1999 2000 Time 2001 2002 2003 4 2 0 −2 1997 2 1 0 −1 1997 Figure 8.8: Differences in the means of the incentive fees between groups of Low - High Rsqr, T O1, T O2. Each month from December 1997 to December 2002 the hedge funds are sorted based on their Rsqr, T O1, T O2, and the average incentive fees are computed across the lowest, respectively the highest Rsqr, T O1, T O2 deciles. These differences are plotted. A star on the x-axis signifies that the difference in means between the low and high Rsqr, T O1, T O2 deciles is significant at 5%. 84 % of cases when stepwise regression is better than princ. comp. 90 85 whole history, 3 factors whole history, 5 factors 80 whole history, 10 factors 2 years history, 3 factors 2 years history, 5 factors 75 2 years history, 10 factors 70 65 60 4 6 Figure 8.9: 8 10 12 Number of principal components used 14 16 Stepwise regressions vs. principal components fit. For each simulated fund whose returns are given by Rtf und −rf = α+ P34 k=1 β k Ftk +²t , we produce estimators of the betas, b using stepwise regression or principal components (5 to 16 components P34 bk − β k |2 , which represents the distance between |β are used). For each method, the quantity k=1 the true betas and their estimation, is calculated. 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Mazuy, 1966, ‘Can Mutual Funds Outguess the Market?’, Harvard Business Review, 44, July-August, 131-136 93 Vita Cristian Ioan Tiu was born in Giurgiu, Romania on 22 January 1975, the son of Ionel and Elena Tiu. He received a Bachelor of Science degree in Mathematics from the University of Bucharest in 1998 and a PhD in Mathematics from the University of Texas at Austin in 2002. He is married to Mariana Tiu and has two children, a son, Andrei and a daughter, Ioana (Ellie) . Permanent Address: 3573 E Lake Austin Blvd., Austin, TX 78703 This dissertation was typeset with LATEX 2ε 5 by the author. 5 A LT EX 2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademark of the American Mathematical Society. The macros used in formatting this dissertation were written by Dinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extended by Bert Kay, James A. Bednar, and Ayman El-Khashab. 94