On the Dynamics of Hedge Fund Strategies

Transcription

1 On the Dynamics of Hedge Fund Strategies Li Cai and Bing Liang Abstract Hedge fund managers are largely free to pursue dynamic trading strategies and standard static performance appraisal is no longer accurate for evaluating hedge funds. Accordingly, this paper presents some new ways of analyzing hedge fund strategies following a dynamic linear regression model. Statistical residual diagnostics are considered to assess the appropriate use of the model. We unveil dynamic alphas and betas for each investment style during the period of January 1994 to December We examine the in-sample goodness-of-fit and out-of-sample predictability on hedge fund performance. By simulating a hypothetical trading strategy, we demonstrate that the model-based predictability helps to implement a profitable fund selection process. Finally, timing skills can be directly examined with a dynamic model; we find significant evidence on market timing, volatility timing and liquidity timing, which is consistent with the timing literature in hedge funds. The First Version: August 2009 This Version: September 2010 Li Cai is an Assistant Professor of Finance at Stuart School of Business, Illinois Institute of Technology, (312) , lcai5@stuart.iit.edu; Bing Liang is a Professor of Finance at Isenberg School of Management, University of Massachusetts Amherst, MA 01003, (413) , bliang@isenberg.umass.edu. We would like to thank Hossein Kazemi, Tom Schneeweis, Mila Getmansky Sherman, Ben Branch, Matt Spiegel, an anonymous referee and seminar participants at the University of Massachusetts-Amherst for helpful comments. We are responsible for all remaining errors. Electronic copy available at:

2 With considerable freedom to trade various financial instruments, allocate among different asset classes and investment regions, and go long or short with financial leverage, hedge fund managers are enjoying unprecedented flexibility in pursuing their investment objectives. Hedge fund performance and risks has been studied intensively over the past decade; mostly remarkable work, among others, is a series of papers by Fund and Hsieh (2001, 2002a, 2002b, 2004). They identify seven asset based risk factors which help explain hedge fund return. Currently, appropriate risk measures for dynamic investment strategies are still lacking, and no convincing way exists to decompose hedge fund returns into the alpha, beta, and cost components. Lo (2001) illustrates the difficulties in measuring the risk of dynamic investment strategies by considering a hypothetical hedge fund. Novel variations to the conventional APT model are called for to accommodate hedge funds less restricted risk shifting behaviors since previous restrictions in modeling hedge fund return may root in model misspecification rather than inadequate risk factors. Patton and Ramadorai (2009) propose an innovative method to utilize high frequency instruments in order to capture hedge fund dynamics within a month. A very straightforward and easy-to-implement variation is the rolling-window OLS regression. More interesting variations also exist, including a regime-switching model introduced by Hamilton (1990) and applied in Billio, Getmansky, and Pelizzon (2006), and lately an optimal changepoint regression by Bollen and Whaley (2009). Both of these models allow for discrete and abrupt risk shifting behavior of hedge fund managers. While Bollen and Whaley (2009) show that an optimal changepoint regression model is 1 Electronic copy available at:

3 sufficient to reject the null hypothesis of constant risk exposures, we prefer models which allow but do not force the risk exposure to shift continuously for performing style analysis since deciding the number of break points is difficult but crucial if only discrete shifts are allowed. We consider mainly a dynamic linear regression model, in which alpha and betas are state variables and are governed by a state process. This model is applied in Mamaysky, Spiegel and Zhang (2008) for mutual fund timing, and in Roncalli and Weisang (2008) for hedge fund replication. We apply two models, i.e. rolling-window OLS and dynamic linear regression with three sets of risk factors for robustness: Carhart (1997) four equity factors plus a Pastor and Stambough (2003) liquidity factor; four asset factors covering equity, bond, commodity, and currency markets; the Fung and Hsieh (2004) asset based seven factors, a total number of six model-and-factor combinations on each equally weighted hedge fund style portfolio formed from the Lipper/TASS database. We plot the alpha time series for each hedge fund style with different factor sets. The alpha time series are significant and positive for most of the time covered although trending downward for many styles. We measure in-sample goodness-of-fit by likelihood value and out-of-sample predictability by simulating the return of prediction based strategy. Although a prediction model that relies on contemporaneous factors is out-ofsample statistically, it is not realistic since risk factors for the next time point are assumed to be available today. To make our prediction based investment strategy realistic, we fit prediction model on lagged risk factors. Besides looking at the time series trend of alphas and betas, studying the predictability, we also study the timing ability of each investment style including market timing, liquidity timing and volatility timing. Mamaysky, Spiegel, 2

4 and Zhang (2008) documents that OLS timing model produce false positive timing results at a high rate while Kalman filter model produces them at a rate close to correct. In our results, positive and significant timing ability is presented by the corresponding timing coefficient in many cases, which is basically consistent with Chen and Liang (2007), and Cao, Chen, Liang and Lo (2009) that apply OLS timing models. Finally, we simulate the performance of an active alpha seeking strategy on individual hedge funds and find that alpha can be greatly enhanced. By simulating a baseline active strategy using OLS for comparison, we find that the improvement in alpha is partly due to active trading and partly due to the predictability carried by the dynamic linear model. The contribution of this paper to the literature is firstly to show that the dynamic linear regression methodology is more appropriate to model hedge fund performance, as shown by lower autocorrelation in residuals when used with the Fung and Hsieh seven factors. Residual diagnostics is a primary tool for model validation, which is very important and often overlooked. Secondly, we carefully examine the predictability on hedge funds performance. Compared to simply rolling static regression, a dynamic linear regression model is superior with the Fung and Hsieh seven factors to predict hedge fund return, as measured by the performance of a hypothetical fund of hedge funds. This superior predictability is more notable in crisis periods. Thirdly, with the use of this dynamic linear regression model, we can easily unveil dynamic alphas and betas for each hedge fund style. Lastly, this kind of analysis is very convenient and reliable for measuring the timing ability of fund managers. We extend Mamaysky, Spiegel, and Zhang (2008) and apply Kalman filter model for measuring volatility timing and liquidity timing. 3

5 The reminder of the paper is organized as follows. Section I describes the data. Section II discusses the models and statistical procedures for estimation/forecast. In Section III, we show four categorized analyses on style dynamics, goodness of fit and prediction, timing ability, and fund level studies. Finally, Section IV concludes. I.Data Description The hedge fund data used in our empirical analysis are from Lipper/TASS. We adopt eleven well recognized investment strategies from the TASS database, and construct equally weighted indexes for each of the eleven strategies, including Convertible Arbitrage, Emerging Market, Event Driven, Fixed Income Arbitrage, Multi- Strategy, Fund of Hedge Funds, Dedicated Short Bias, Equity Market Neutral, Global Macro, Long-Short Equity Hedge, and Managed Futures. Our overall sample period consists of 180 monthly observations from January 1994 to December To limit survivorship bias, we start with January 1994 when defunct fund information becomes available. We also require funds to have at least a 24 month return history. Other standard criteria to filter out noise funds include: minimum assets under management of $10 million, net return and asset information reported 1, monthly net of fee returns, and the US dollar denominated assets only. Finally, our 3,102 funds cover 1,590 live funds and 1,512 defunct funds. Details of the hedge fund data, including performance information and summery statistics for all eleven strategies can be found in EXHIBIT 1. Abbreviates for hedge fund styles can be found in EXHIBIT A1 of the Appendix. 1 Observation is deleted if net return or asset information is not reported for that observation. 4

6 We adopt factor models in modeling fund returns throughout this paper and consider three sets of risk factors for robustness. Set 1 has five equity factors, including Carhart (1997) four equity factors, excess market return, the size factor SMB, the value/growth factor HML, the momentum factor, and a Pastor and Stambaugh (2003) traded liquidity factor 2. Set 2 has four standard asset market index factors covering equity, bond, commodity, and currency markets. They are the excess return of the S&P 500 Index, the Barclays Aggregate Bond Index, the S&P GSCI Index, and the FRB broad dollar Index. Set 3 is the widely adopted Fund and Hsieh seven hedge fund factors, including three trend following factors on bond, currency, and commodity, the S&P 500 monthly return, the monthly return on Wilshire Small Cap 1750 minus Wilshire Large Cap 750, a bond market factor, and a credit spread factor. While these three sets of risk factors overlap mildly, they do, however, have distinguished focuses. Set 1 represents the well known equity factors. Set 3 is the wellaccepted hedge fund risk factors that are shown to have great power to capture the hedge fund monthly return variation, especially for well diversified hedge fund portfolio. Set 2 is a rather complete set of basic asset class factors. Although previous hedge fund researches generally conclude that these basic asset indices factors are incapable of explaining hedge fund returns. We intend to explore the power of these asset factors with dynamic models since the previously documented incapability might be driven by model misspecification rather than factor selections. Summery statistics regarding these three sets of risk factors are reported in EXHIBIT 2 and abbreviates for factors are included in EXHIBIT A2 of the Appendix. 2 Risk factor Set 1 is obtained from Ken French s website and WRDS. 5

7 II.Methodology A. Rolling Static Linear Regression Model and Dynamic Linear Regression Model In this paper, we firstly consider static the linear factor model with the ordinary least square (OLS) estimation, not on an in-sample fitting basis but rather on a month by month forward-rolling basis. The OLS approach is widely used in empirical finance studies, and has been conducted on hedge fund studies for a decade. However, we know that OLS assumes constant intercept and factor coefficients; OLS also assumes an IID normal distribution with innovations. Many hedge fund studies do not report residual check when OLS is performed while some studies report failed residual reports and come up with various alternative methods. Although many novel models have the advantage of allowing non-constant coefficients, no matter how flexible they are, they still require certain statistical assumptions, like normality and independency. To assess the appropriate use of statistical model, we test for autocorrelation and normality. Besides a static linear regression model with the rolling OLS estimation, we also consider a dynamic linear regression model. Dynamic linear models are presented as a special case of general state space models, being linear and Gaussian, for which, estimation and forecasting can be obtained recursively by the well known Kalman Filtering or Kalman Smoothing. State-space models consider a time series as the output of a dynamic system perturbed by random disturbances. The dynamic linear regression model that we apply can be displayed mathematically by the following two equations: 6

8 Y F v, v ~ (0, V ) t=1,...,n (1) t t t t t t G w, w ~ (0, W ) t=1,...,n (2) t t 1 t t (, ) t t t Equation (1) is called the observation equation; Equation (2) is state equation or system equation. If we have n observation and k risk factors, then Y t is a scalar and refers to the excess style portfolio return, F t is a known 1 by k matrix and represent factors in our context. At each time t t is an unknown k+1 by 1 state vector to be estimated and forecasted, and represent time-varying alpha and factor loadings in our context. G is k+1 by k+1 identity matrix and represents state transition matrix. V (Scalar) and W (k+1 by k+1 diagonal) are variance and covariance matrix of two independent white noise sequences, independent both between them and within each of them, with mean zero and unknown covariance matrices, which are also to be estimated. To make W non diagonal which means to make use of nonzero correlations may potentially further improve the model s predicting ability. But here, we consider a diagonal covariance matrix to keep things simple 3. B. Rolling-Window OLS and Kalman Filtering/Smoothing For forward rolling-window OLS, we use all data from January 1994 up to time t to do the OLS estimation, retain the estimates and use these estimates to forecast next time 3 It is interesting if future studies will treat this problem as seemingly unrelated time series equations so that the dependence structure among the state innovations will be specified. 7

9 period s observation. As a result, predictions produced are out-of-sample if the next period s factor realizations are known today. For our dynamic linear regression model Y F v, v ~ (0, V ) t=1,...,n (1) t t t t t t G w, w ~ (0, W ) t=1,...,n (2) t t 1 t t (, ) t t t Our statistical estimation and prediction procedures involve two major steps, Step one is to estimate the covariance matrices with both the observation innovation and state innovation ( and ) jointly by maximum likelihood; Step two is to do Kalman Filtering/Smoothing with the estimated parameters from Step one to obtain estimation and forecast on our state variables α t and, and further to do out-of-sample one-stepahead prediction on future observation Y. Still, predictions produced are totally out-of-sample if the next period s factor realizations are known today, which is not the case here. As mentioned in the introduction section, to make our prediction-based investment strategy more realistic, we use lagged risk factors to fit the model, do one-step-ahead prediction and simulate the performance of the hypothetical fund of hedge funds so that it is implementable. Another thing to mention is that to implement the MLE procedure in step one, we have to specify a vector of initial values (priors) of the unknown parameters for the optimization routine. For simplicity we assign zeros in all cases in this study. Conventionally, estimations produced at the beginning are not trustworthy since they 8

10 depend a lot on the precision of the initial values assigned 4. With the Kalman Filtering/Smoothing procedure the estimation precision improved dramatically with more and more observations involved. Considering all these, we exclude the very first 24 estimations (January 1994 to December 1995) from our original monthly time series (January 1994 to December 2008) in most analysis. We have to do the same for the results produced by the rolling-window OLS since OLS itself requires a fair number of observations for the initial estimation, and 24 is a reasonable choice for this purpose too. In the dynamic linear regression model set up, a static linear regression model corresponds to the case when is a zero matrix so that the state vector is constant over time. Thus, statistically speaking; this dynamic linear regression model relaxes assumptions of static linear regression model regarding state innovations. In this study, we use Kalman Filtering to do prediction and implement prediction based fund selections. We use Kalman Smoothing to do state variable estimation and the corresponding dynamic alpha, dynamic betas and timing analysis. We make such a choice since in the filtering distribution at time t one is conditioning on the observations from time 0 to time t only, while in the smoothing distribution the conditioning is with respect to all observations available. So that our one-step-ahead predictions, which are obtained from the filtering process, are out-of-sample, and can be compared directly with the rolling OLS out-of-sample prediction. With respect to state variable estimation and analysis, we utilize Kalman Smoothing since the resulting estimations become more precise with more data used. An easy way to understand the difference between Filtering 4 Subjectivity is a common critique of Bayesian statistics which arises through the necessity of specifying a prior and Bayesian statisticians sometimes analyze the sensitivity of their conclusions to the choice of prior. 9

11 and Smoothing is to remember that Filtering is a forward-recursive algorithm, while Smoothing is a backward-recursive algorithm for computing the conditional densities of state variables 5. III. Analysis and Results Our analysis and empirical results can be categorized into four sections, style dynamic analysis, goodness of fit and predictability, fund managers timing ability and fund level studies. A. Dynamic analysis of styles Firstly, questions arise regarding hedge fund dynamics: How does each hedge fund style s alpha change overtime? How does each hedge fund style portfolio adjust its exposure to different asset classes over time? We model the equally weighted style portfolio returns for each style on risk factor set 2 (the four asset index factors) with a dynamic linear regression model. At the same time, we estimate time-varying alpha and time-varying beta through applying Kalman Smoothing to the model. To answer the first question, we do a time series plot of alpha for each hedge fund style. To be robust, we also do the same with the other two risk factor sets, one has five equity market factors and the other is Fung and Hsieh seven hedge fund factors. These entire three alpha time series plots are presented in EXHIBIT 3. For the time horizon in 5 More details covering Kalman Filtering and Kalman Smoothing can be found easily in Bayesian or State Space Model related statistical text books. 10

12 consideration, January 1994 to December 2008, alpha is positive most of the time 6. Noticeable from the plot is that the alpha time series for some styles is trending downward and even negative for some for recent two years. This finding is consistent with documented attenuating hedge fund alpha in Fung, Hsieh, Naik, and Ramadorai (2007). Moreover, three valley bottoms show up in all three alpha time series plots, especially notable with Emerging Market (EM). The first valley point is year-end of 1994, which corresponds to the Mexico Peso Crisis; the second bottom point is in late 1998, which is probably from Russian Financial Crisis and LTCM fail while the third bottom is the most recent, which reads Subprime Mortgage Crisis. From all three alpha plots, styles like Emerging Market (EM) gives more volatile alpha time series while some styles, e.g. Managed Futures (MF), always show persistent alpha. Managed Futures (MF) for the whole time period under study show up steady and most positive alpha and it can be associated with Schneeweis and Georgiev (2002) that Managed Futures participate in a wide variety of new financial products and markets not available in traditional investor products. While consistent with EXHIBIT 3, EXHIBIT 4 gives more detailed summary statistics on the estimated alpha. EXHIBIT 4 tells that every style gives positive average alpha with pretty low standard deviation which helps to guarantee significance 7. Moreover, there exists variation in terms of the magnitude cross styles or cross models. Alpha estimated with the Fung and Hsieh seven factors in general carries more dynamics 6 In fact, alpha is significant and positive most of the time, Confidence Interval plot is omitted to better display the alpha time series itself. 7 To make sure that we do not produce spurious alpha, we try our model on CRSP value-weighted index return and the resulted alpha series is always within (-0.1%, 0.1%). We would like to thank Matt Spiegel for suggesting us to do this model check. 11

13 than the other two risk factor sets and zero standard deviation only exist in Global Macro and Managed Futures. Consistent with EXHIBIT 3, Managed Futures demonstrates greatest stable positive alpha and Emerging Markets carry most volatile alpha. To answer the second question, we plot beta time series in EXHIBIT 5. We can easily tell from the beta plots that EMN (Equity Market Neutral) portfolios have relatively neutral exposures to all four markets, equity, bond, commodity and currency, which can be connected with Patton (2009). Most other styles demonstrate more volatile and time-varying exposures. This fact implies that hedge funds dynamic exposures can t be fully captured by a model which only allow discrete shift since we do not observe any clear regimes or segments from the continuous dynamic plots. B. Goodness-of-Fit and Predictability Goodness of fit is assessed by residual diagnostics and likelihood value; the results are included in EXHIBIT 6. For residual diagnostics, we test for independency and normality. Independency is largely rejected when modeling with the first two factor sets, thus EXHIBIT 6 only includes the results when modeling with the Fung and Hsieh seven factors. While only the first order autocorrelation is reported, autocorrelation of higher order is rarely significant; we conduct a Ljung-Box test of up to lag 12 to test for the overall significance of serial autocorrelations. Suggested by both the first order autocorrelation and Ljung-Box test statistic, residuals of dynamic linear model carry much less significance in serial autocorrelation than static linear model. Normality is tested by Shapiro-Wilk W test statistics. Consistent with the fact that the Shapiro-Wilk test always reject, we observe significance in both models and all styles. Since normality 12

14 tests have little power 8, we do not include results of other normality tests. Log Likelihood value is included in EXHIBIT 6 to compare the overall in-sample goodness of fit. As expected, dynamic linear model consistently offer tighter fit than static linear model since the observation variance and state variance are estimated to maximize likelihood. According to Pastor and Stambaugh (2009), serial correlation within predictive regression residuals is an indication of imperfect predictors, which suggests that all the three risk factor sets are only correlated with expected hedge fund returns, but cannot deliver it perfectly since residuals of static model is always serially correlated. As mentioned, we construct a hypothetical fund of hedge funds which follows a simple investment strategy based on the one-step-ahead prediction and quantify the performance difference when different models are used. We use the Fung and Hsieh seven hedge fund factors (risk factor set 3) in this process since it proves to be more appropriate by residual diagnostics. This hypothetical fund of hedge funds simply invests all capital for one time period into one of the ten hedge fund styles which is predicted to have the best expected return based on the one-step-ahead prediction with the Fung and Hsieh seven factors. To make this investment strategy more practical in reality, we use lagged factors in both models as explained before. As previously, we exclude the very first 24 monthly returns in analyzing the performance of a hypothetical fund of hedge funds since we need a substantial number of observations in order to obtain reliable initial prediction. Thus, we retain the monthly returns of hypothetical fund of hedge fund following static linear 8 It is generally believed that the Shapiro-Wilk test always rejects and other normality tests, e.g. Jarque- Bera rarely reject. Although not included, a corresponding QQ plot seems to suggest the residuals of both models are almost normal. 13

15 model rolling OLS and dynamic linear regression model from January 1996 to December Assuming a $1 initial investment in January 1996 and no more external capital invested afterwards, the ending value of this $1 initial investment is $16.77 in December 2008 if the dynamic linear regression model with Kalman Filtering is used to do the prediction and in contrast this is $11.40 if the rolling OLS is used. A whole path of this return cumulating process can be found in EXHIBIT 7. EXHIBIT 7 also plots the time series of style allocation by dynamic model. Style allocation plot tells that this hypothetical fund of hedge funds most heavily shift among four styles, Long/Short, Short Bias, Futures and Emerging Markets. While the underlying style allocation is adjusted frequently, this practice can hardly be implemented in reality considering share restrictions. More results regarding the performance of the hypothetical fund of hedge funds can be found in EXHIBIT 8. From the risk adjusted return as measured by Sharpe Ratio for each of 13 years from 1996 to 2008, this hypothetical fund of hedge funds does better when the dynamic linear regression model with Kalman Filtering is used to do the prediction in 8 out of the 13 years as compared to when the rolling static linear regression model is used. The outperformance of our dynamic linear model over a rolling OLS is more notable in down markets, e.g. when we have LTCM event in 1998, technology bubble in 2001, and the subprime mortgage crisis in 2007, Panel B of EXHIBIT 8 offers more details on performance evaluated on the whole period. Panel B tells that return given by dynamic linear model is more positively skewed, has fatter tails, and gives less number of negative months. Both models give exactly the same maximum 14

16 drawdown from August 1998 to April 2001, which relates to the technology bubble period. C. Timing skill We also try to capture hedge fund managers market timing ability to vary risk exposure with the corresponding market outlook. Very intuitively, the linear relationship between the time-varying beta and the underlying risk factor is able to capture the timing ability on this specific market. We apply our dynamic linear regression model on each of the eleven style portfolios and estimate time-varying betas through Kalman Smoothing. We use this timing coefficient introduced in Mammaysky, Spiegel, and Zhang (2008), the linear correlation between beta and the underlying factor as the market timing ability measure. Results are reported in EXHIBIT 9. While Mammaysky, Spiegel, and Zhang (2008) documents that OLS timing model produce false timing results at a high rate, our timing result from dynamic model is in general consistent with hedge fund timing literature of OLS timing models. Our result is in support of the significant market timing skill in the hedge fund industry 9. In EXHIBIT 9, we also extend the market timing measure of Mammaysky, Spiegel, and Zhang (2008) to liquidity timing and volatility timing. Statistically, style portfolios are modeled dynamically against risk factor set 2 (excess return of equity, bond, commodity, and currency market indexes). Then to capture market timing skill, the linear correlation between estimated factor loading (beta) and underlying factor realization is 9 Although not included, we check for spurious timing results by applying the model on CRSP equallyweighted index and we do not get any positive timing skill. 15

17 calculated and reported. Hedge funds are assumed to allocate assets across these four major markets and successful market timing is captured by significant and positive timing coefficient. A hedge fund with good market timing skill is supposed to transfer capital to the market with increasing return from the market with decreasing return. Here, many style portfolios demonstrate positive and significant equity market timing skills, including equity styles long/short equity hedge (LSEH) and equity market neutral (EMN), Global Macro (GM), Emerging Market (EM) and Managed Futures (MF) 10. Moreover, Convertible Arbitrage (CA) shows positive timing skill on bond market while Event Driven (ED) demonstrates positive timing skill on currency market. The idea of the market timing coefficient can easily be extended to study other timing skills. To capture liquidity timing/volatility timing, we calculate the correlation between equity market beta and equity market liquidity/volatility. We use 1) Pastor and Stambaugh non-traded market liquidity 2) CBOE volatility index VIX to represent equity market liquidity and volatility respectively. We do this to examine how hedge funds adjust their equity market exposure in response to market liquidity and market volatility 11. It is very possible although not known for sure that hedge fund predict these market conditions and use them as guiding instruments to adjust fund exposure to equity market. Our empirical results are able to show how the real time market exposure (realized beta on equity market) change with real time market conditions (equity market 10 We consider the fact that many of the styles with significant positive timing coefficient on equity market do high frequency trading and it raise the possibility of artificial timing from interim trading. But these styles show no timing in the other three markets, which tells that artificial timings do not dominate if any. 11 Some of the changes in asset allocations are active investment adjustments while other changes may be passive, due to margin call or funding liquidity shortage. Here, we do not disentangle active and passive adjustments considering the limited data availability on hedge funds. 16

18 liquidity and volatility). Intuitively, effective liquidity timing should increase equity market exposure when market liquidity is going to be higher while successful volatility timing should decrease equity market exposure when market volatility is going to be higher. In EXHIBIT 9, significant and positive liquidity timing is found in LSEH, GM, MF, FOF and EMN. At the same time, significant and negative correlations show up for DSB and CA, which also indicate positive timing skill since both of them are strategies that short equities. This documents positive liquidity timing ability for these equity hedge fund styles, consistent with Cao, Chen, Liang and Lo (2009). Similarly, good volatility timing ability can be read from the significant and negative correlation between equity market beta and VIX for many hedge fund styles including LSEH, EMN, GM, MF, MS, ED, and FOF. This volatility timing result is consistent with Chen and Liang (2007). D. Fund Level Studies To be robust, we conduct similar analysis on individual hedge funds by simulating the return of an active strategy investing on predicted alpha. Specifically, we extend the idea behind EXHIBIT 8, but in a more realistic setting and target on risk adjusted return. Here, we compare the performance of two strategies, one is a naïve index strategy, which is to invest equally in each hedge fund within the same style for the whole period under consideration; the other is an active alpha seeking strategy, which we assume can be implemented by a hypothetical style fund of hedge funds. This hypothetical fund of hedge funds does one-step-ahead prediction on alpha, and invests for four months in the single hedge fund that is predicted to have the highest expected alpha within a certain style. We allow this hypothetical style fund of hedge 17

19 funds to rebalance every four months since average lock up is between three months and four months of all hedge funds under study. To do one-step-ahead prediction on alpha, we apply Kalman filtering and use dynamic linear regression model and the Fung and Hsieh seven factors since this pair violates statistical assumptions the least. To compare the risk adjusted performance of a naïve equally weighted index strategy and active alpha seeking strategy, we conduct OLS performance analysis with the Fung and Hsieh seven factors on the two return series generated. Performance evaluations are reported in EXHIBIT 10. EXHIBIT 10 suggests that alpha is greatly enhanced when we make use of the predictability and goes from a naïve index strategy to an active alpha seeking strategy within the same style category while two exceptions 12, e.g. Convertible Arbitrage (CA) and Managed Futures (MF) exist. One issue of implementing the active alpha seeking strategy is that a selected fund may stop reporting to the database for part the following four months 13. That happens since a selected fund is more likely to be successful fund and as a result may not want to take in more investors and choose to stop reporting. We deal with this problem by excluding the observation in performance study if return data are not available for the selected fund for the selected month. Another uniform observation for all styles in EXHIBIT 10 is that the adjusted R-Square drops a great deal 12 It is very possible that with strategies like Convertible Arbitrage (CA) and Managed Futures (MF), the active alpha seeking strategy works with risk factors other than the Fung and Hsieh seven factors. We do not search for other potential risk factors since it is not our ultimate goal to propose profitable trading strategy for each style category. 13 We will naturally underestimate the alpha of active alpha seeking strategy because of this data issue. From the available number of observations in the performance study, we can tell that this problem although exist is pretty minor since in the worst case (for all 10 styles) we lose only 9 observations out of 168 in the final performance study for this reason. 18

20 when we go from naïve index strategy to alpha seeking strategy. This is natural and further confirms our initial perception that the previous shortage in modeling hedge fund returns may be more due to model misspecification rather than inadequate risk factors since with slightly dynamic trading, we will have a low R Square using a static regression to do performance analysis. To assess how much of the improved alpha is due to dynamic model, we simulate a baseline active strategy that invest on historical estimated alpha cross all 10 styles 14. Under the same setting, the baseline active strategy generates an alpha of 1.41%, which measures the improvement due to active trading. Thus, the enhanced alpha can be partially due to active trading and partially due to the dynamic linear model. While the active strategy is designed to generate alpha, the return profile looks less attractive when evaluated on the unadjusted raw returns. Panel B of EXHIBIT 10 gives details on the raw returns. Panel B shows that in terms of moments, active strategy offers higher mean return, higher standard deviation, more positive skewness and fatter tails than a passive index strategy. Moreover, active strategy generates more negative months and greater maximum drawdown. For passive strategy, the drawdown is long term from December 1999 to September 2008; while active strategy is much more volatile, the drawdown is short term, from May 2007 to September 2008 for dynamic model and from May 2007 to November 2007 for OLS. Comparing the two active return series, the one generated by OLS seems to offer relatively better profile than the one generated by dynamic model, which enhances alpha better. 14 We also lose 9 observations for the baseline active strategy due to return data availability. However, these 9 observations are not all the same as the 9 observations lost in DLM active strategy. 19

21 IV. Summary and Conclusion Serially correlated residuals from predictive regression modeling on hedge fund style portfolios suggest imperfect predictors as indicated in Pastor and Stambaugh (2009), consistent with common thought on hedge fund studies that existing risk factors are only correlated with hedge fund returns but imperfectly. We suggest in this paper that hedge fund can potentially be more effectively modeled by these imperfect risk factors with dynamic linear regression model. This suggests that hedge fund managers dynamically adjust exposures even against those trading strategies. Equally speaking, existing risk factors especially the Fung and Hsieh seven factors are appropriate to capture hedge fund returns with the dynamic model. This is confirmed when autocorrelation in residuals becomes statistically insignificant. Thus, we are more confident to carry out further analysis with this dynamic linear model. Secondly, we adopt our dynamic liner model to analyze hedge fund performance and risk shifting behaviors. We plot how each style s alpha changes over time from January 1994 to December This result is consistent with previously documented downward trending alphas, which implies that diminishing returns to scale combined with the inflow of new capital leads to the erosion of superior performance over time. The alpha plot also suggests that although some style portfolios, e.g. Emerging Market (EM) portfolio s alpha drops significantly and even become negative during crisis years, in 1998 and 2008, some other hedge fund style portfolios are able to maintain a steady and positive alphas over time. From the plot of beta time series, we find that Equity 20

22 Market Neutral (EMN) is a style that has close to neutral exposures to all four asset markets; we can also observe that for other hedge fund styles, exposures to different markets are much more dynamic than what can be captured by a small number of regimes or segments. Thirdly, we study in-sample goodness-of-fit and predictability on hedge fund returns. With the Fung and Hsieh seven factors, the dynamic model predicts better than rolling static model, especially during crisis years. We illustrate this by simulating the performance of a hypothetical fund of hedge funds. Fourthly, this dynamic linear regression model is very convenient and robust for studying a fund manager s timing ability. We study and report each style s timing ability across four markets (equity, bond, commodity, and currency). While successful timing of equity market is observed for many styles, timing other markets are more difficult and less identified with a few exceptions. This kind of analysis can be easily extended to study other timing skills, e.g. liquidity timing and volatility timing. Finally, fund level results demonstrate the profitability behind predictability in a more realistic setting after taking into account hedge fund share restrictions. With an active alpha seeking strategy which invests on predicted alpha, the realized alpha can be greatly enhanced for different styles; the improvement in alpha is partially due to active trading and partially due to the predictability of dynamic model. 21

23 Reference Monica Billio, Mila Getmansky, and Loriana Pelizzon, 2006, Dynamic Risk Exposure of Hedge Funds: A Regime-Switching Approach, Working Paper, CISDM. Nicolas P.B. Bollen and Robert E. Whaley, 2009, Hedge Fund Risk Dynamics: Implications for Performance Appraisal, Journal of Finance 64, Carhart, M., 1997, "On Persistence in Mutual Fund Performance," Journal of Finance, 52, Charles Cao, Yong Chen, Bing Liang, and Andrew W. Lo, 2009, Can Hedge Funds Time Market Liquidity? Working Paper, University of Massachusetts Amherst Yong Chen, 2006, Timing Ability in the Focus Market of Hedge Funds, Working Paper, Boston College Yong Chen, and Bing Liang, 2007, Do Market Timing Hedge Funds Time the Market? Journal of Financial and Quantitative Analysis 42, William Fung, and David A. Hsieh, 1997, Empirical Characteristics of Dynamic Trading Strategies: The Case of Hedge Funds, Review of Financial Studies 10, William Fung, and David A. Hsieh, 2001, the Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers, Review of Financial Studies 14, William Fung, and David A. Hsieh, 2002a, Risk in fixed-income hedge fund styles, The Journal of Fixed Income 12, William Fung and David A. Hsieh, 2002b, Asset-Based Style Factors for Hedge Funds, Financial Analysts Journal 58, William Fung, and David A. Hsieh, 2004, Hedge fund benchmarks: A risk based approach, Financial Analyst Journal 60, William Fung, David A. Hsieh, Narayan Naik, and Tarun Ramadorai, 2008, Hedge funds: Performance, risk, and capital formation, Journal of Finance 63, Hamilton, James D, 1990, Analysis of Time Series Subject to Changes in Regime, Journal of Econometrics 45,

24 Andrew W. Lo, 2001, Risk Management for Hedge Funds: Introduction and Overview, Financial Analysts Journal 57, Harry Mamaysky, Mathew Spiegel, and Hong Zhang, 2008, Estimating the Dynamics of Mutual Fund Alphas and Betas, Review of Financial Studies 21(1), Lubos Pastor, and Robert F. Stambaugh, 2009, Predictive Systems: Living with Imperfect Predictors, Journal of Finance 64, AJ Patton, 2009, Are Market Neutral Hedge Funds Really Market Neutral? Review of Financial Studies 22 (7), Andrew J. Patton, and Tarun Ramadorai, 2009, On the Dynamics of Hedge Fund Risk Exposures, Working Paper, Duke University Thierry Roncalli and Guillaume Weisang,2008, Tracking Problems, Hedge Fund Replication and Alternative Beta, Working Paper. Thomas Schneeweis and Georgi Georgiev, 2002, the Benefits of Managed Futures, AIMA. 23

25 EXHIBIT 1 Summary Statistics of Hedge Fund Style Portfolios Monthly Returns: January 1994-December 2008 The hedge fund portfolios are equally weighted portfolios constructed by styles, which are identified by category in the TASS database. The eleven categories includes: Convertible Arbitrage (CA), Dedicated Short Bias (DSB), Emerging Market (EM), Equity Market Neutral (EMN), Event Driven (ED), Fixed Income Arbitrage (FIA), Fund of Funds (FOF), Global Macro (GM), Long/Short Equity Hedge (LSEH), Managed Futures (MF), and Multi Strategy (MS). To avoid survivorship bias, our sample contains monthly returns starting from January 1994, when information on defunct fund becomes available in the TASS database, to December We also apply other common filtering standards to filter out noise hedge funds. Time series means and standard deviations of portfolio returns are reported for each style portfolio, and for live fund portfolios, defunct fund portfolios, and the combined portfolios, respectively. Except for number of funds and Sharpe Ratios, all numbers are reported in percentages. Live Funds Defunct Funds Live and Defunct Funds # of Funds Mean (%) Std (%) Sharpe Ratio # of Funds Mean (%) Std (%) Sharpe Ratio # of Funds Mean (%) Std (%) Sharpe Ratio CA DSB EM EMN ED FIA FOF GM LSEH MF MS All

26 EXHIBIT 2 Summary Statistics of Monthly Risk Factors All three sets of factors are obtained as monthly data from January1994 to December Time series means and standard deviations, plus correlation matrix for each factor set are reported. Panel A (Factor set 1) includes five equity factors: the excess market return (FF1), size factor SMB (FF2), value/growth factor HML (FF3), the momentum factor (FF4), the Pastor and Stambaugh (2003) traded liquidity factor (FF5). Panel B (Factor set 2) is for the four standard asset class factors; including the excess return of the S&P500 Total Return (AF1), of the Barclays Aggregate Bond index (AF2), of the S&P GSCI Total Return (AF3), and of the FRB broad dollar index (AF4), which together cover equity, bond, commodity, and currency markets. Panel C ( Factor set 3) is for seven Fung and Hsieh hedge fund risk factors, including three trend following factors, trend following factors of bond (FH1), of currency (FH2), and of commodity (FH3), an equity market factor (FH4), a size spread factor (FH5), a bond market factor (FH6), and the credit spread factor (FH7). ***, **, and * indicate significance levels at 1%, 5%, and 10%, respectively. Panel A: Risk Factor Set 1 Factors FF1 FF2 FF3 FF4 FF5 Mean STD Correlation Matrix FF2 FF3 FF4 FF5 FF *** *** *** *** FF *** *** * FF *** FF *** Panel B: Risk Factor Set 2 Factors AF1 AF2 AF3 AF4 Mean STD Correlation Matrix AF2 AF3 AF4 AF * *** AF *** ** AF *** ** 25

27 EXHIBIT 2 (Cont.) Panel C: Risk Factor Set 3 Factors FH1 FH2 FH3 FH4 FH5 FH6 FH7 Mean STD Correlation Matrix FH2 FH3 FH4 FH5 FH6 FH7 FH ** ** ** FH *** *** FH ** * FH *** ** *** FH *** *** FH *** *** 26

28 EXHIBIT 4 Summary Statistics of Estimated Alpha Series Alpha Series is estimated by Kalman Smoothing when each hedge fund style portfolio is modeled on a risk factor set; Panel A is for risk factor set 1 of five equity market factors, Panel B is for risk factor set 2 of four asset class factors and Panel C is for risk factor set 3 of seven Fung and Hsieh factors. While Figure 1 gives the time series plot of these estimated alpha, this EXHIBIT gives detailed summary statistics on the estimated alpha series, including mean (Mean), standard deviation (Std), Maximum (Max), Minimum (Min) and t-statistics. ***, **, and * indicate significance levels at 1%, 5%, and 10%, respectively. Panel A. Five Equity Factors-Risk Factor Set 1 Mean (%) Std (%) Min (%) Max (%) t-statistic CA >100 *** DSB *** EM *** EMN *** ED *** FIA >100 *** FOF >100 *** GM >100 *** LSEH >100 *** MF >100 *** MS >100 *** Panel B. Four Asset Class Factors-Risk Factor Set 2 Mean (%) Std (%) Min (%) Max (%) t-statistic CA >100 *** DSB >100 *** EM *** EMN >100 *** ED >100 *** FIA >100 *** FOF >100 *** GM >100 *** LSEH *** MF >100 *** MS >100 *** Panel C. Seven Fung and Hsieh Factors-Risk Factor Set 3 Mean (%) Std (%) Min (%) Max (%) t-statistic CA *** DSB *** EM *** EMN *** ED *** FIA *** FOF *** GM >100 *** LSEH *** MF >100 *** MS *** 27

29 EXHIBIT 6 Residual Diagnostics and Goodness of Fit Residuals are studied when each hedge fund style portfolio is modeled on risk factor 3, the Fung and Hsieh seven factors. Residual diagnostics test for independency and normality. 1 St order autocorrelation (AR1) and Ljung-Box test statistics (Ljung-Box) for up to lag 12 are included in the first two columns for testing autocorrelations; Shapiro-Wilk test of normality is conducted and resulted test statistic is included in the third column (Shapiro). ***, **, and * indicate significance levels at 1%, 5%, and 10%, respectively. Log likelihood value (LL) is computed for measuring goodness of fit and is included in the last column. Dynamic Linear Model Ordinary Least Squares AR1 Ljung-Box Shapiro LL AR1 Ljung-Box Shapiro LL CA 0.18** *** *** 22.40** 0.92*** 582 DSB * 0.95*** *** 359 EM ** 0.99* *** 32.61*** 0.98** 426 EMN *** 0.94*** *** 72.27*** 0.91*** 561 ED * ** 20.56* 0.98** 573 FIA * *** 24.82** 0.80*** 538 FOF 0.19** 20.99* 0.97*** ** ** 519 GM 0.14* *** *** 0.94*** 545 LSEH ** *** 27.87*** 0.98** 601 MF *** * 358 MS 0.17** *** *** 31.13*** 0.97***