in Economics and Finance
Final Thesis
Regime-switching betas for hedge funds
returns
Supervisor
Ch. Prof. Monica Billio Assistant supervisor Ch. Prof. Domenico Sartore Graduand
Stefano Vaccher
Matriculation Number 860769 Academic Year
From the lately 1990s hedge funds have been object of intense aca-demical research. After the financial crisis of 2008 and the disappear-ance of many funds, the interest has gradually faded. This work aims to review the current state of the hedge fund industry. In the introduc-tory section, different statistical and regression analysis are performed on hedge fund indices and show the persistency of biases (i.e. selection and survivorship). A seven-factors ABS model is then applied to give further evidence and to consider the exposures of hedge funds to risk factors. However, it is verified that betas are not constant in time. Therefore, the central part of this work is devoted to the implementa-tion of the regime-switching beta model proposed by Billio et al. [2010]. The model allows to study the dynamic risk exposures of hedge funds in different regimes of the market (up, down, and tranquil). Investors and regulators can gain insightful information on where hedge funds are putting their bets to react accordingly.
1 Introduction 8
2 Data Analysis 11
2.1 Data Presentation . . . 11
2.2 Preliminary Analysis of Hedge Fund Indices . . . 13
2.3 Non-Parametric Analysis of Hedge Fund Indices . . . . 22
3 Linear Risk Exposures in Hedge Funds 27 3.1 Main approaches . . . 28
3.2 The Seven-Factor ABS Model . . . 30
3.3 Conclusions on linear risk models . . . 35
4 Non-Linear Exposures in Hedge Funds 37 4.1 Time-varying Factors . . . 38
4.2 The Beta Regime-switching model . . . 42
4.2.1 Theoretical Framework . . . 43
4.2.2 Empirical Analysis . . . 48
4.3 Conclusions on non-linear models . . . 61
5 Conclusions 64
A Non-parametric Analysis of Hedge Fund Indices 73
B Time-varying Factors 75
Introduction
The attention to the hedge fund industry escalated in the 1990s. The number of hedge funds increased from few hundreds to more than 3000. Shortly after came the availability of data and the academic re-search. Publications addressed a wild range of topics. Fung and Hsieh [2000] and Liang [2000] analysed the presence of biases in the main hedge fund databases. Following Sharpe [1992], Fung and Hsieh [1997] and Mitchell and Pulvino [2001] considered hedge fund styles. Perfor-mance and risk exposure were extensively covered by Fung and Hsieh [2002], Agarwal and Naik [2004], Billio et al. [2010]. Liquidity and sys-temic risks were examined by Getmansky et al. [2004] and Chan et al. [2005]. The boom continued until 2008. The financial crisis wiped out the 22% of the operating funds in the following three years. The at-tention to the industry gradually faded. Together with the number of papers published on the topic that considerably decreased. The excep-tions regarded mainly the role of hedge funds in crises and contagion issues (Boyson et al. [2010], and Billio et al. [2012]). However, hedge funds had already recovered their pre-crisis dimensions in 2013 and are now touching new records in terms of asset under management.
The aim of this work is to analyse the current state of the hedge fund industry. At this scope, the main existing statistical tools and
models are borrowed to update former analysis. The intention is to have a well-rounded view on different aspects related to hedge funds.
In next chapter, hedge fund data are considered. As they constitute the ground of any analysis, reliable data are fundamental. It is well documented that hedge fund databases suffered from various biases, especially selection and survivorship. The collection of data on hedge fund performance started about 30 years ago. However, databases needed time to enlarge the universe of constituent hedge funds. Fur-thermore, they had to overcome problems related to the opaqueness and the few disclosure requirements imposed to hedge funds. Thus, the aim is to determine if such problems were partially, or even totally, solved. As in Billio et al. [2009], are performed a preliminary statistical and a non-parametric analysis on three global hedge fund indices and on single strategies indices. The persistence of such biases is verified by comparing simple statistics and estimated probability distributions of the returns of same indices belonging to different databases.
The remaining part of this dissertation is dedicated to the analysis of the exposures of hedge funds to risk factors. Chapter 3 investigates the three most recognized linear models in the study of hedge fund risk exposures. The approach of Sharpe [1992], although designed for mutual funds, establishes a suitable framework to build more tailored model for hedge funds. Mutual funds are constrained by rigid man-dates that set the minimum number of asset classes owned and are often not allowed to use leverage. Their performances are therefore more easily captured by the standard asset classes the model is based on. Hedge fund have instead much less limitations. They use dynamic strategies, often involving derivatives and high leverage. From these observations was conceived the model by Fung and Hsieh [1997]. It is the second model presented here and it is an extension of the former. The Authors divided the returns of the factors into quintiles to rep-resent the dynamic strategies changing in different market conditions.
The last model is the seven-factor ABS model by Fung and Hsieh [2004]. The model uses tailored factors that simulate the performance of different hedge fund strategies. It is the landmark in this area and for this reason was applied here to the global indices in post-crisis data. The results show interesting insights for investors that can monitor the risk exposures in their portfolios and for regulators that can supervise possible convergences of hedge fund to same sources of risk.
The last chapter opens verifying how betas vary over time. Factors indeed change in time drawing peculiar patterns that depend on market conditions. Linear models are not able to capture these dynamics. In the search of models that can resolve these issues, non-linear models are considered. The multi-factor model with exposure only to the S&P500 and the multi-factor model with exposures to all factors proposed by Billio et al. [2010] are presented. This approach offers a flexible solution where factors are allowed to vary depending on the state of the market. The regimes are estimated through a Markov-switching model and then betas are derived conditional to them. The two models were applied to the single strategies indices in the post-crisis period. Results found here are very different from the ones obtained by the Authors, however they allow to confirm and deepen some conclusions attained by the linear model.
Data Analysis
2.1
Data Presentation
The indices used in this work include the Credit Suisse Hedge Fund Index, the HFRI Asset Weighted Composite Index, and the CISDM Equal Weighted Hedge Fund Index, together with single strategy in-dices from each database. Data are collected with monthly frequency from January 1994 to May 2017.
Credit Suisse: The Credit Suisse Hedge Fund Index was the indus-try’s first asset-weighted hedge fund index. The index uses the Credit Suisse Hedge Fund Database, which tracks approximately 9,000 funds and consists only of funds with a minimum of US $50 million under management, a 12-month track record, and audited financial state-ments. The index is calculated and rebalanced monthly, and reflects performance net of all hedge fund component performance fees and expenses. The Index in all cases represents at least 85% of the AUM in each respective category of the index universe. The methodology analyses the percentage of assets invested in each subcategory and se-lects funds for the index based on those percentages, matching the shape of the index to the shape of the universe. Fund weight caps
may be applied to enhance diversification and limit concentration risk. The Index uses a rules-based construction methodology, identifies its constituent funds, and minimizes subjectivity in the index member se-lection process as a result of the rules-based method. It aims to achieve maximum representation of the index universe. To minimize survivor-ship bias, funds are not removed from the index until they are fully liquidated or fail to meet the financial reporting requirements. Credit Suisse Hedge Fund single strategies indices include: Convertible Bond Arbitrage, Dedicated Short Bias, Emerging Markets, Equity Market Neutral, Long/Short Equity, Distressed, Event Driven MS, Risk Arbi-trage, and Global Macro. The Credit Suisse Hedge Fund Indices are not investable indices.
HFR: The HFRI Asset Weighted Composite Index is a global, asset-weighted index comprised of over 2,000 single-manager funds that re-port to HFR Database. Constituent funds rere-port monthly net of all fees performance in US Dollar and have a minimum of $ 50 Million un-der management or a twelve-month track record of active performance. The Index does not include Funds of Hedge Funds. The constituent funds of the HFRI Asset Weighted Composite Index are weighted ac-cording to the AUM reported by each fund for the prior month. HFR single strategies indices include: Emerging Markets, Equity Market Neutral, Equity Long/Short, Distressed, Event Driven, Relative Value Arbitrage, and Global Macro. The HFRI Indices are investable via synthetic replication products.
CISDM: The CISDM Equal Weighted Hedge Fund Index reflects average performance of all hedge funds. It demonstrates the average return of all hedge funds, except funds of funds and CTAs. Only hedge funds that have reported net returns for the particular month are in-cluded in the index calculation. The calculation of the Index perfor-mance does not include outliers which are at least +/-3 standard devi-ation away from the average. In the calculdevi-ations of the CISDM Equal
Weighted Hedge Funds Index and of the individual CISDM Hedge Fund Strategy Indices duplicate funds have been eliminated. CISDM single strategies indices include: Distressed, Long/Short, Market Neu-tral, ED Multistrategy, Global Macro, and Merger Arbitrage. The CISDM Hedge Fund Indices are not investable indices.
2.2
Preliminary Analysis of Hedge Fund Indices
Table 2.1 shows simple statistics for all the hedge fund indices pre-sented in the previous chapter. Each sample has 281 monthly observa-tions collected from January 1994 to May 2017. In this first preliminary analysis, the moments of the first four orders of the distributions are employed. These statistics are not explicative of the whole distribu-tion, but they give a decent summary of the behaviour of the indices. Indices are created with the aim to summarize the performances of the entire hedge fund universe or to represent each hedge fund strategy. It is therefore reasonable to think that all indices with the same object should behave similarly and present comparable values. Thus, this analysis is carried out by comparing the simple statistics of each kind of index among the three different databases. The first two moments of a distribution of probability are the mean and the variance. Annual means and variance are exhibited in the table. The mean represents the average yearly returns obtained by each index, expressed in per-centage. The variance is a measure of the dispersion of the observations from their mean. It is informative when the underlying distribution is assumed to be symmetric, even not necessarily gaussian. Otherwise, it should be considered carefully, particularly in the case of strong asym-metry. Moreover, the comparison of two variances should be done only if they are associated to quite similar expected values. The symme-try of a distribution is given by the moment of third order, that is the skewness. When a distribution is symmetric, as in the case of a
normal distribution, the skewness is equal to 0. Finally, the kurtosis indicates how much probability is contained within the center and the tails of the distribution rather than the shoulders. Normal distribution has kurtosis equal to 3. The values in Table 2.1 refer to the excess of kurtosis, that is the value of the kurtosis in excess to the normal one.
Table 2.1: Summary Statistics hedge fund inidices for the period January 1994-December2016 N Beta Annual Mean Ret.(%) Annual SD (%) Min (%) Median (%) Max (%) Skewness Excess Kurtosis JB p-value Box-Pierce p-value Credit Suisse Index 281 0.27 7.50 6.87 -7.59 0.69 8.49 -0.13 0.17 0.00 0.00 Convertible Arbitrage 281 0.16 6.42 6.32 -12.60 0.73 5.81 -2.65 14.76 0.00 0.00 Distressed 281 0.26 9.05 6.13 -12.49 1.01 4.15 -2.05 8.20 0.00 0.00 Long/Short Equity 281 0.42 8.40 9.08 -11.47 0.75 12.97 0.01 0.89 0.00 0.00 Equity Market Neutral 281 0.18 3.95 9.32 -40.45 0.56 3.66 -12.48 183.19 0.00 0.27 Emerging Markets 281 0.50 6.68 13.42 -23.07 1.01 16.38 -0.79 3.39 0.00 0.00 Event Driven MS 281 0.26 7.44 6.56 -11.56 0.86 4.78 -1.61 3.77 0.00 0.00 Global Macro 281 0.15 9.62 8.84 -11.58 0.84 10.56 0.14 1.73 0.00 0.13 Risk Arbitrage 281 0.13 5.59 3.91 -6.19 0.51 3.77 -0.92 1.59 0.00 0.00 Dedicated Short Bias 281 -0.84 -6.04 16.11 -11.28 -0.83 22.67 0.72 -1.48 0.00 0.10 HFR Index 281 0.34 7.71 6.63 -8.74 0.77 7.61 -0.58 -0.10 0.00 0.00 Convertible Arbitrage 281 0.22 6.99 6.59 -16.02 0.75 9.74 -2.88 24.50 0.00 0.00 Distressed 281 0.25 8.33 6.11 -8.54 0.91 5.55 -1.37 1.80 0.00 0.00 Equity Long/Short 281 0.46 8.97 8.77 -9.47 0.87 10.84 -0.20 -0.75 0.00 0.00 Equity Market Neutral 281 0.06 4.99 2.98 -2.88 0.40 3.55 -0.24 -0.77 0.00 0.01 Emerging Markets 281 0.58 7.27 13.11 -21.06 1.19 14.76 -0.88 1.43 0.00 0.00 Event Driven 281 0.32 9.02 6.48 -8.94 0.99 5.09 -1.18 1.04 0.00 0.00 Global Macro 281 0.13 6.44 6.18 -6.43 0.40 6.78 0.31 -1.72 0.00 0.12 Relative Value Arbitrage 281 0.16 7.50 4.06 -8.04 0.71 3.93 -2.70 12.73 0.00 0.00 CISDM Index 281 0.36 9.02 7.01 -8.86 0.84 8.33 -0.49 0.48 0.00 0.00 Convertible Arbitrage 281 0.16 7.78 4.82 -11.50 0.76 4.71 -3.20 22.66 0.00 0.00 Distressed 281 0.24 8.38 5.63 -10.60 0.81 4.91 -2.14 9.17 0.00 0.00 Equity Long/Short 281 0.37 8.88 7.14 -9.46 0.81 9.36 -0.21 -0.03 0.00 0.00 Equity Market Neutral 281 0.06 6.78 2.16 -2.11 0.55 2.79 -0.20 -0.83 0.00 0.00 ED Multistrategy 281 0.28 8.74 5.57 -7.34 0.84 4.77 -1.48 2.40 0.00 0.00 Global Macro 281 0.11 5.37 4.27 -5.39 0.35 6.86 0.69 2.26 0.00 0.57 Merger Arbitrage 281 0.13 7.15 3.16 -5.65 0.57 2.80 -1.38 4.84 0.00 0.00
The three main indices have quite similar values for the mean and standard deviation, even though the CISDM index has both values slightly higher. However, these results have a relative meaning since all the indices are not normally distributed, as shown by the Jarque-Bera test. This test compares the sample skewness and sample kurtosis with the ones of a normal distribution, respectively 0 and 3. All the indices have negative skewness, meaning that a higher proportion of the distribution is concentrated in the right side of the probability
dis-tribution curve. The values are almost equal for HFR and CISDM, while Credit Suisse has a higher value of symmetry. The three indices present also positive values for the excess of kurtosis, their distribu-tions have fat tails. Hence, positive and negative returns occur with a higher probability than in a normal distribution. In this case, HFR has a quasi-normal kurtosis while Credit Suisse and CISDM show larger departures from normality. Moreover, the Box-Pierce test indicates the presence of serial autocorrelation at the first lag for the three indices.
From these simple statistics, the indices seem to behave very sim-ilarly even though they have different methods of construction and select candidate targets from different universes of hedge funds. It is important to underline that the Credit Suisse index and the HFR index are asset-weighted indices, while the CISDM is an equally-weighted in-dex. These two rebalancing rules are similar but can lead to different portfolio selections. Indeed, both rules imply a contrarian approach where better performing hedge funds are limited and under-performing hedge funds are enhanced in order to maintain the same weights. The difference lays in the fact that equal weighted indices aim to maintain the equal weighting among all the hedge funds that compose the in-dex, while asset weighted indices are partitioned in subcategories that must hold equal proportions in the portfolio. In this case there is no significant distance between the asset weighted indices Credit Suisse and HFR and the equally weighted index CISDM. This may be due to the broad level of diversification reached by these indices.
Conversely to the general indices, single strategies exhibit peculiar features that need to be treated with attention. The following strate-gies, in common for the three databases, are considered:
• Convertible Arbitrage • Distressed Securities • Equity Long/Short
• Equity Market Neutral
• Event Driven Multi Strategy • Global Macro
First of all, all the single strategies are not normally distributed, so also in this case mean and standard deviation must be considered carefully. CISDM convertible arbitrage strategy has the highest mean and the lowest standard deviation compared to the other two. However, all the values are quite similar. All the probability distributions are left-skewed and have very high values of the excess of kurtosis equal to 14 for Credit Suisse and, 24 and 22 for HFR and CISDM. Comparable results hold for the distressed securities strategy. It is remarkable that HFR has a much lower excess of kurtosis (equal to 1.80), while Credit Suisse and CISDM have values higher than 8. Equity long/short strat-egy has negative skewness and positive excess of kurtosis with values closed to zero for the three databases. The same applies for event driven multi strategies indices. Credit Suisse global macro index has the mean and standard deviation that almost double the other two indices. For this strategy, there are some noteworthy discrepancies. Credit Suisse has negative almost-zero skewness and positive excess of kurtosis, HFR has positive skewness and negative excess of kur-tosis, and finally CISDM has both values positive. All three indices show the presence of serial autocorrelation. HFR and CISDM s equity market neutral indices have very similar values. Credit Suisse index shows in this case extreme results: the skewness is equal to -12 and the excess of kurtosis reaches an incredible value of 183. The rea-son behind these strange numbers is the capture of the famous fraud Bernard Madoff in December 2008 and the consequent disappearance of two of his fund-feeders Kingate Global and Fairfield Sentry Fund. These funds were largely included in the Credit Suisse/Tremont equity
market neutral hedge fund index. Indeed, the index ended up losing the 40% in the same month. This preliminary analysis shows that gen-eral indices seem to move together. The difference can be associated with the rules of construction and rebalancing, since it is reasonable to believe that biases, such as selection and survivorship, do not affect the results. Investors and academics can therefore choose the right index according to their needs, without concerns about their reliabil-ity. On the contrary, single strategy indices present large discrepancies among the three databases. It is very likely that the presence of such kind of biases affects the results. In general, selection bias arises when a sample is not representative of the entire population. In this con-text, hedge fund managers are not required to disclose their activities and their performances. They can freely decide whether to partake to a database and choose the one they prefer. Usually, hedge funds cannot publicly solicit new subscribers, so enter a database becomes a way of self-advertising. Thus, each database cannot have all the possible hedge funds belonging to the corresponding strategy. More-over, it is convenient for hedge fund managers to choose only their best performing funds to enter databases and to maintain private the less performing ones. Survivorship bias is related to hedge funds that have stopped reporting information or ceased operations. These hedge funds are often discarded from databases because they are no more of interest for investors. In this way, the bias favours the better forming hedge funds, since the discarded funds have usually bad per-formances. In the website of the Credit Suisse database it is stated that: ”A fund will be dropped from the index in the following circum-stances: The fund fails to report monthly performance and AUM for two consecutive months; the fund fails to comply with the rules relating to the provision of financial information; the fund ceases operation” . There is no such information for HFR and CISDM databases but it is reasonable to think that they also suffer from survivorship bias, as
for Credit Suisse. One may wonder why what just explained does not concern also general indices. However, it is a more serious problem for single strategy indices because they are composed by a much lower number of hedge funds, so the benefits of diversification are reduced. Hence, when working with single strategy indices, more attention is required for the selection of the right database. Because final results may be significantly different because of the database used.
The insights derived above refer to a long-time horizon that com-prises more than 22 years of monthly returns. The statistical signifi-cance of these results is out of discussion thanks to the large sample size and because it includes an entire boom-and-bust cycle. However, the long period could obscure the actual state of these indices, because the results could be largely affected by biases present in the obser-vations more distant in time. Indeed, the birth of the hedge fund industry dates back to the 1949 but only in the end of the 1980s it started to capture the public’s attention and a myriad of new hedge funds was established since then. The collection of hedge funds data came shortly after but had to deal with the opaqueness and the few disclosure requirements of hedge funds, as discussed above with regard to selection bias. Databases presumably needed time and an enlarging universe of available hedge funds to construct reliable indices. Hence, it is possible that these problems were partially, or even totally, fixed with time.
The idea of a comparison between indices from different databases applied above comes from Billio et al. [2009]. They analysed hedge fund indices belonging to the same three databases as in this work, with observations starting approximately in the same time1 and ending in June 2007. Since this timeframe corresponds to the first half of the larger sample of this work, it can be useful to confront the results of
1In Billio et al. [2009]: HFR indices go from January 1990 to June 2007, Credit Suisse indices go
the two analyses to check if there are remarkable differences. Then, the same analyses will be performed on the indices for the period that goes from July 2007 to May 2017. In this way, a picture of the current state of hedge fund indices will be obtained, together with their evolution and, eventually, the improvements they have made in order to eliminate every kind of possible distortion.
As pointed out by the three authors the general indices showed similar values of skewness and kurtosis, except for a slight positive skewness for the Credit Suisse index compared to negative symmetries for HFR and CISDM. Similarities are found also in this work and all three indices show values of the skewness and of the excess of kurtosis with the same signs. This could be a proof of improvement of the general indices. As far as the single strategies are concerned, convert-ible arbitrage indices show no big differences among databases in the short period as well as in the whole period. There is a curious large discrepancy between the two works regarding the values of the excess of kurtosis, where in the former the values are closed to 2 while in this analysis values largely exceed 14. The reason is convertible arbi-trage hedge funds suffered from big losses after the collapse of Lehman Brothers, event that is included only in this work. The big losses were due to a large withdraw in financing from prime brokers that caused a mismatch between the illiquidity of the convertible bond portfolios and the short-term financing positions that supported those positions. Dis-tressed securities have comparable results among the databases in both analysis, except that the excess kurtosis is particularly low in the HFR index in this analysis while the other analysis show that Credit Suisse has excess of kurtosis three times higher than the other indices. Credit Suisse event driven multi strategy index is notably high compared to HFR and CISDM in the shorter period, while all databases present very similar values in the larger period. Equity market neutral indices show in both analysis very similar values among databases. There
in only an interesting difference: here the skewness and the excess of kurtosis are all negative while in the former analysis are all positive. The right skewness is due to the favourable period that started during the crises of 2008, where equity market neutral hedge funds achieved outstanding performances and continued in the following years with results above the positive long-term mean. The change in the sign of the excess of kurtosis is caused by the crash related to Madoff s facts already explained.
There is little evidence of improvements for general indices that need to be confirmed by further analyses. For single strategies, there is no evident proof of improvement on the indices. As explained above, these indices had historically a lower number of hedge funds to choose from and therefore a higher risk of being exposed to biases. The birth of new hedge funds and the stabilization of the indices in time should have led to better results. However, the comparison of the two works does not show significant proofs of improvements in the longer period. Nevertheless, Madoff s fraud uncovered how this type of indices is still heavily sensitive to shocks and how difficult it is to cope with this kind of events.
Simple statistics of the indices in the period that goes from July 2007 to May 2017, are presented in Table 2.2. Credit Suisse and CISDM s general indices behave very similarly regarding the mean and standard deviation, but also skewness and excess of kurtosis. HFR index differs in the value of the kurtosis that has opposite sign. Thus, in this last analysis appears a discordance among indices that did not show up in previous analysis. About single strategy indices, convertible arbitrage has all indices with similar values, and the same goes for long/short equity. Credit Suisse and HFR databases also show comparable results for distressed securities indices while CISDM has a very high value of the excess of kurtosis, probably due to the disappearance of one or more hedge funds included only in this database. In event driven
multi-strategy indices, Credit Suisse has the excess of kurtosis with opposite sign, but all other values are quite similar. Finally, equity market neutral indices in this period show the problem already explained for the whole sample suffered in this strategy.
Table 2.2: Summary Statistics hedge fund indices for the period June
2006-December2016 N Beta Annual Mean Ret.(%) Annual SD (%) Min (%) Median (%) Max (%) Skewness Excess Kurtosis JB p-value Box-Pierce p-value Credit Suisse Index 113 0.27 3.02 5.60 -6.60 0.43 4.10 -1.37 1.06 0.00 0.00 Convertible Arbitrage 113 0.29 3.63 8.20 -12.60 0.33 5.80 -2.52 10.59 0.00 0.00 Distressed 113 0.27 3.72 5.90 -5.70 0.57 4.10 -1.10 -0.97 0.00 0.00 Long/Short Equity 113 0.41 3.57 7.70 -7.80 0.46 5.20 -0.88 -1.24 0.00 0.02 Equity Market Neutral 113 0.31 -3.84 14.10 -40.50 0.16 3.70 -8.69 80.62 0.00 0.83 Emerging Markets 113 0.45 2.53 9.30 -13.60 0.45 7.00 -1.60 3.53 0.00 0.00 Event Driven MS 113 0.33 2.65 7.20 -6.20 0.53 4.80 -0.90 -1.81 0.00 0.00 Global Macro 113 0.10 4.27 5.40 -6.60 0.38 4.40 -0.93 0.67 0.00 0.09 Risk Arbitrage 113 0.13 2.95 3.50 -3.50 0.39 2.30 -0.97 -1.33 0.00 0.01 Dedicated Short Bias 113 -0.76 -11.48 14.90 -11.30 -1.35 9.70 0.39 -3.00 0.23 0.62 HFR Index 113 0.34 2.86 6.20 -6.80 0.41 5.10 -0.93 -0.41 0.00 0.00 Convertible Arbitrage 113 0.40 4.64 9.40 -16.00 0.61 9.70 -2.24 11.71 0.00 0.00 Distressed 113 0.33 3.82 7.10 -7.90 0.62 5.50 -1.06 -0.72 0.00 0.00 Equity Long/Short 113 0.50 2.52 8.80 -9.50 0.50 6.40 -0.88 -0.98 0.00 0.00 Equity Market Neutral 113 0.10 1.61 2.80 -2.90 0.26 1.80 -1.39 0.49 0.00 0.16 Emerging Markets 113 0.59 0.72 11.80 -14.50 0.31 9.60 -0.85 -0.11 0.00 0.00 Event Driven 113 0.35 3.97 6.60 -8.20 0.54 4.70 -1.23 0.31 0.00 0.00 Global Macro 113 0.05 1.84 4.60 -2.60 -0.04 4.20 0.44 -2.94 0.17 0.23 Relative Value Arbitrage 113 0.23 4.80 5.10 -8.00 0.58 3.90 -2.33 7.65 0.00 0.00 CISDM Index 113 0.36 3.90 6.80 -7.90 0.51 6.20 -0.94 0.33 0.00 0.00 Convertible Arbitrage 113 0.27 5.91 6.80 -11.50 0.60 4.70 -2.60 11.50 0.00 0.00 Distressed 113 0.27 4.85 6.10 -10.60 0.70 4.90 -2.31 9.60 0.00 0.00 Equity Long/Short 113 0.33 4.32 6.10 -5.40 0.64 4.30 -0.79 -2.08 0.00 0.02 Equity Market Neutral 113 0.06 4.35 2.30 -2.10 0.32 2.80 -0.17 -0.52 0.00 0.06 ED Multistrategy 113 0.32 4.23 6.20 -7.30 0.55 3.90 -1.76 2.19 0.00 0.00 Global Macro 113 0.04 2.86 2.90 -1.90 0.21 2.70 0.37 -2.48 0.14 0.12 Merger Arbitrage 113 0.10 4.78 2.40 -2.60 0.45 2.10 -1.40 1.17 0.00 0.48
This last analysis was performed with the aim to verify if indices have increased their ability to represent the hedge fund universe, fixing part of the problems they have being suffering from before. The com-parison between indices belonging to different databases demonstrates that discrepancies persist. Selection bias is still the main problem. In general, indices are still not enough representative of the whole hedge fund space and for this reason they often show considerably different results among different databases. This problem became clear with the crash caused by the Madoff s fraud, but it is also present among
the different economic phases where hedge funds behave somehow dif-ferently, and indices happen to fail to get the whole picture. Surely, databases can hardly control selection bias since they cannot freely choose all the hedge funds to insert in the indices; but a higher level of attention is required to prevent such kind of accidents from happening and to reduce the influence of the survivorship bias.
To conclude this first introductory section, the three analysis per-formed exhibit discrepancies in the results for the three databases and in the different sample periods considered. Caution is required when working with this kind of indices. The choice of the appropriate database to use should be clearly based on the rules of construction of each index. Particular attention should be paid to the number of hedge funds constituting the index and their weights, as a first check on the exposure of the index to selection bias and on how much the index is representative of the hedge fund universe. A difficult point regards the choice of the length of the sample. These analyses showed that there was not an improvement on the quality of the indices in time and that they still be very sensitive to shocks and to changes in the economic situation where some hedge funds can obtain odd results compared to their peers.
2.3
Non-Parametric Analysis of Hedge Fund
In-dices
Billio et al. [2010] completed their analysis with a non-parametric estimation of the indices. In this work, the same approach is imple-mented for all the indices in the entire sample period. Parametric approaches are very useful to summarize a large amount of informa-tion with few statistics. But this comes with a price. Indeed, the drawback of parametric distributions is that they rely on very strong
hypothesis. They assume a priori the data to belong to a certain distri-bution, usually normal. Table 1 showed that indices are not normally distributed and most of them are also serially autocorrelated. The use of mean, standard deviation, skewness, and kurtosis implies a para-metric approach since all these statistics work better when in presence of a normal, or quasi-normal, distribution. This is the reason why the previous analysis aimed to compare the indices solely looking for big differences and anomalous values. A deeper analysis is difficult to perform with this approach and can lead to aberrations. Instead, non-parametric approaches do not assume the data to belong to a cer-tain distribution and can give insightful information on the shape of the distribution. However, results are not always easy to obtain and very often they are not numerical. As in this case where a graphical analysis is used. The analysis employs the Kernel density estimation. It is a fundamental data smoothing problem where inferences about the population are derived from the kernel estimator :
ˆ f (xt∗) = 1 T b T X i=1 k xt − xt∗ b
where b is the bandwith and controls the resolution of the estimator, while k is the kernel function, a non-negative function that integrates to 1.
Figure 2.1 shows the estimated conditional distributions of the gen-eral indices together with the normal distributions with the same stan-dard deviations as the corresponding indices. All three fitted distribu-tions have peculiar shapes and are clearly more kurtotic than normal distributions. Especially the Credit Suisse index seems to pinpoint the presence of, at least, a couple of crises. Indeed, two major events happened during the sample period causing large drops in the indices. In August 1998, all the three indices suffered losses between 7.5% and 8.8% because of the collapse of the renowned Long-Term Capital
Man-Figure 2.1: Non-Parametric Analysis of Global Indices
agement fund2. The fund, that used mainly arbitrage strategies (i.e. fixed income arbitrage) combined with high financial leverage, was very successful in the first three years, with outstanding annualized returns. However, the loss of $4.6 billion in less than four months after the Asiatic financial crisis in 1997 and the 1998 Russian finan-cial crisis, required the Federal Reserve to dismiss and liquidate the fund. The other large drop outlined occurred in September and Octo-ber 2008, concomitantly with the zenith of the financial crisis of 2008, the bankruptcy of Lehman Brothers. Hedge funds were affected by the liquidity shortage caused by the large amount of sales in the mar-ket. They ended up suffering big losses because of the impossibility to dismiss sizeable portions of their losing assets.
Figure 2.2 presents the estimated conditional distributions of the
2LTCM was founded in 1994 by John W. Merywether and included in its board of directors
financial auctoritas such as Myron Scholes and Robert Merton, who shared the Nobel Memorial Prize in Economic Sciences in 1997.
Figure 2.2: Non-Parametric Analysis of Equity Long/Short strategy
equity long/short strategy for the three databases. The estimated dis-tributions are not normally distributed and have peculiar behaviours. The non-parametric analysis seems to have uncovered different reac-tions of the indices to extreme events (both negative and positive). Indeed, the three plots show big difference especially in the tails. The estimated conditional distribution of the other strategies considered are enclosed in appendix. However, what just explained for the equity long/short strategy is valid also for the other strategies. All distribu-tions are far from normality and have unique shapes, in particular in the tails.
The non-parametric analysis confirms what found in the previous section through the preliminary analysis. The estimated distributions are almost all non-normal due to high values of the kurtosis. Moreover, this analysis adds some interesting information about the differences between the indices belonging to the three databases. Indices seem
to behave differently during extreme market conditions and this is reflected by the different shapes assumed by the curves in the tails. However, it is important to note that peculiar shapes are due also to a relative small number of observations. For the law of large numbers, increasing the sample size, the sample distribution tends to a normal distribution and therefore it becomes more bell-shaped.
Linear Risk Exposures in Hedge
Funds
The ultimate goal of the first chapter was to verify the unbiasedness of hedge fund indices. Simple statistics and non-parametric analysis, conducted on three among the most important hedge fund indices, revealed that biases are still present and that indices returns show significant distance among different databases.
Hereafter, it will be considered the exposition of hedge funds to risk factors. In this chapter the problem is tackled with a linear analy-sis, while afterwards risk exposures will be permitted to change dy-namically with time. The landmark in the field of risk factor models for hedge funds is the Seven-Factor ABS model from Fung and Hsieh [2002]. The Model will be applied to the post-crises period to study the current risk exposure of hedge funds. However, is important to firstly introduce how the Model works and what its strengths and weaknesses are. This way, it will be possible to draw up consciously partial conclu-sions that will be enriched with the analysis in the following chapter.
3.1
Main approaches
It is important to start this new section explaining what a factor model is. A factor model assumes that the rate of return of an asset is given by:
ri = a + b1F1 + ... + bkFk + e
where the Fi, i = 1, ..., k are random variables called factors, a and bi
are constants and, e is a zero-mean error term. The model assumes the uncorrelation between the factors and the error term (E(Fie) = 0),
while the factors are allowed to be correlated among themselves. Fol-lowing the equation, a linear regression is performed with the rate of return as the dependent variable and the factors as independent vari-ables. Factors can be statistical, macroeconomic, and/or fundamental and are chosen and combined together by the modeller with the aim to explain the greatest proportion of the rate of return of the asset.
There are in literature three main approaches for the analysis of hedge fund performances. The first successful attempt to analyse risk exposures goes back to Sharpe [1992]. He applied a factor model to study mutual fund performances, in order to help investors identify the sources of risk of their investments. Sharpe used twelve factors con-nected to different asset classes, to explain the performances of mutual funds through investable passive index funds. He achieved this result by performing what he called style analysis. The goal of style analy-sis is not to select the most performing combination of funds within the given factors. It is to choose the combination of styles (sets of asset class exposures) that minimizes the variance of the tracking er-ror. Where the tracking error can be defined, rearranging the previous formula, as:
ei = ri − [b1F1 + ... + bkFk]
The lower the variance of the tracking error, the closer the rate of return of the fund is mimicked by the weighted factors. Since each
factor can be linked to a source of risk, the model ultimately gives the exposures to risk of the fund.
Sharpe’s model had immediately great success among investors in mutual funds. It permitted to analyse asset allocation decisions and the ”style mix” of investor’s portfolios with ease and low-costly. However, mutual fund managers have usually rigid investment mandates. They are constrained to own a minimum number of asset classes in their portfolios and they are often not allowed to use leverage. Mutual fund performances are therefore highly correlated to standard asset classes and easily captured by the regression with the factors. Conversely, managers of alternative investment vehicles such as hedge funds have less limitations. They use dynamic strategies, often involving deriva-tives and high levels of leverage. Fung and Hsieh [1997] proposed an extension to Sharpe [1992] to attain an integrated framework to analyse a broader spectrum of investment funds. They observed that mutual funds were assimilable to buy-and-hold strategies. To explain their performances was sufficient to assess where mutual funds were investing but not how, as requested for hedge funds. Nevertheless, they concluded that finding other factors to proxy for the dynamic strategies was impossible in practice, since there was an infinite num-ber of strategies. Using principal component analysis1 on the sample of hedge fund returns, the two authors identified five components that explain a significant proportion of the hedge fund performance and that were linked to five common hedge fund strategies. These five categories (investment styles) were regressed with the eight factors previously exposed. The results were not satisfactory since they were too sensitive with extreme values, in line with what just said.
Ob-1Principal component analysis is a statistical procedure that applies an orthogonal
transforma-tion to convert a set of possibly correlated variables into a set of linearly uncorrelated variables, called principal components. Usually, the first principal components have the highest variance, (they account for as much variability in the data as possible). The understanding of the economic meaning of the principal components is left to the modeller.
serving this, they decided to divide the monthly returns of each asset class in five ”states of the world”, ranked by quintile and regressed for every state of each investment style. From the analysis, only one hedge fund category showed a buy-and-hold like behaviour, having all the coefficients aligned. While all the other categories use dynamic strategies in response to the states of the market. Hence, with this sort of non-parametric approach where returns are divided into quin-tiles, the authors gave an adequate starting point in the linear analysis of non-linear performances of alternative investment funds.
3.2
The Seven-Factor ABS Model
The last approach considered in the analysis of hedge fund returns, and the one that will be implemented in this chapter is the Seven-Factor ABS model. Fung and Hsieh [2004] understood that hedge funds returns were so different from traditional investment funds that needed a dedicated model. Sharpe [1992] is the right framework, but is not enough to simply modify the way standard asset classes are implied in order to correctly study hedge fund performances. Hedge funds need tailored factors that are able to catch their non-linear returns due to dynamic strategies and the use of derivatives and leverage. At this scope they used the so-called ”asset-based style” factors (ABS factors), factors that are constructed based on option-like payoffs. They proceed as follows. They started by extracting principal components from their sample of hedge fund returns. They called these principal components ”return-based” style factors. RBS factors were preferred to hedge fund returns to avoid data biases in hedge fund databases. The use of RBS factors also allowed to divide the hedge fund sample in style categories, without having to rely on the classifications of data vendors. They kept the first four principal components that they identified to be four common hedge fund styles: Trend-following funds, Merger arbitrage
funds, Fixed-income hedge funds and, Equity long/short hedge funds. The objective is to link these RBS factors to models involving only marketable risk factors to get the ABS factors. They ended up with the following asset-based style factors:
• Trend-following funds: in Fung and Hsieh [2001] they modelled this factor with three portfolios of lookback options on bonds, currencies, and commodities2. Trend followers bet on large move-ments, so they make money when the market is volatile, likewise option buyers.
• Merger arbitrage funds: Mitchell and Pulvino [2001] analysed more than 4500 previous mergers to characterize the risk and return of risk arbitrage funds. They observed that fund perfor-mances were positively correlated with market returns during se-vere declines, but uncorrelated during flat and appreciating mar-kets. They therefore created an ABS factor to replicate risk arbi-trage funds selling uncovered put options on the S&P500.
• Fixed-income hedge funds: Fung and Hsieh [2002a] found that these funds are typically exposed to interest rate spreads. The reason is that many fixed income hedge funds cover their positions in illiquid and/or low rated bonds by shorting treasuries that have higher credit rating and more liquidity. The interest rate spread is the difference between the yields of the two bonds. Moreover, this kind of funds are usually highly leveraged and, since the cost of financing positions depends also on the liquidity of the market, interest spread reflects also this aspect. Hence, the factor explains both the direction of the bets and their entities.
• Equity long/short hedge funds: these funds take long positions on stocks that are expected to appreciate and short positions on
stocks that are expected to depreciate. The objective is to mini-mize market exposure while profiting with the open positions in both directions. However, Fung and Hsieh (2003) showed that eq-uity long/short hedge funds are largely correlated with the stock market. Moreover, they found a significant positive exposure to the spread between the returns of the small cap and large cap stocks. This is consistent with the fact that equity long/short funds tend to be long on lower capitalized stocks and short on higher capitalized stocks.
Thus, from this analysis seven asset-based style factors were identi-fied:
Table 3.1: Asset-based style factors Equity ABS Factors S&P 500 (market)Small cap - Large cap spread
Fixed-Income ABS Factors ∆10-year treasury yields∆10-year treasury - Moody’s Baa bonds spread Trend-Following ABS Factors
Portfolio of loockback options on bonds Portfolio of loockback options on currencies Portfolio of loockback options on commodities
After testing the new ABS factors, the two authors reported that: ”these seven risk factors are found in 57% of the hedge funds in TASS3, and 37% in HFR”. Moreover, they applied the seven ABS factor model to the HRI equally-weighted composite index and the CSFB/Tremont asset-weighted composite index, for the period from January 1994 to December 2002. They found out that the model was very successful in explaining the returns of the HFRI, showing a very high R2 (84%), even though it was significantly exposed only to the equity ABS factors. The R2 reached with the CSFB/Tremont index is instead much lower, though still quite high (48%), but the index showed to be exposed to all
the factors except for the portfolios of lookback straddles on currency futures and on commodities futures.
The seven-factor ABS model is now applied to the three indices presented in the introductory section in the period between June 2006 and December 2016. The results are presented in Table 3:
Table 3.2: Seven-factor ABS Model results for Jun2006-Dec2016 period
Credit Suisse HFR CISDAM
Estimate t Estimate t Estimate t
Intercept 0.01 2.47 0.00 0.66 0.01 0.02 SPY 0.27 10.54 0.33 14.09 35.76 13.35 Small-Large -0.10 -2.62 -0.06 -1.81 -6.81 -1.72 ∆10yTreasury 0.01 0.88 0.02 1.83 2.12 1.92 CreditSpread -0.23 -2.02 -0.03 -0.27 6.46 0.54 PtfBD 0.00 -0.63 0.00 -0.53 -0.31 -0.38 PtfFX 0.00 0.53 0.00 0.49 0.27 0.45 PtfCOM 0.00 -0.60 -0.01 -0.96 -0.60 -0.88 R2 0.58 0.71 0.68
The HFR index and the CISDM index present comparable results. They are exposed to the same factors (equity ABS factors and the change in the 10-year Treasury), with the same signs but with very different coefficients. Moreover, the model has almost the same ex-plicative power for both indices, showing very similar values of the R2. Somehow different results are presented by the Credit Suisse Index that is exposed to the equity ABS factors, to the other fixed-income ABS factor, and also has significant positive alpha. This is quite un-expected. It was reasonable to await for similar results for the Credit Suisse and HFR indices, that are both asset-weighted indices. Con-versely to the CISDM that is a equally-weighted index. These are in fact the results observed also in Fung and Hsieh [2004]. There was therefore a remarkable change in time of the exposures of the consid-ered hedge fund indices with regard to the ABS factors.
Another interesting difference between the two works regards the intercept, or alpha. The authors pointed out in their analysis that all the indices showed significant alphas with high positive coefficients. A significant positive alpha is good news for investors for how the model is constructed. Indeed, the model respects the rationale of the APT. It is constructed so that the factors account for the return given by all the possible strategies used by hedge funds. Therefore the in-tercept gathers all the additional return variability that is left over. This is basically attributable to a combination of momentum, superior cash management, and the quid given by the skills of the hedge fund managers. An investor in alternative investments such as hedge funds should look at positive alphas, especially during bear markets, when bringing value is way harder. Unfortunately, Table 3.2 shows that only the Credit Suisse database has the intercept significant and positive. HFR and CISDM indices are instead not significant. The reasons for the vanishing alphas may be various. From one side, it may due to the structural changes in the regulation applied to hedge fund after the crises of 2008, (i.e., the Dodd-Frank Act, the Volker Rule and the Jobs Act). Stricter rules and capital constraints may have limited the scope of action of hedge funds that are now less free to place their bets. On the other side, many, as Lack [2012], blamed a mix of causes such as the high fees and the increasing dimensions in asset under management (AUM) of the hedge funds. The latter is the main problem for hedge fund managers since a larger AUM imposes to take larger positions to maintain satisfactory returns. But larger positions are not always vi-able without influencing the market. In other words, the scalability of hedge fund strategies depends on the liquidity of the markets. There-fore, a strategy that has always performed well in the past could stop to work properly if working with orders of higher dimension. With regard to the high fees, the traditional 20%-2% structure is not sus-tainable any more. Indeed, the higher level of competition brought
by the booming passive indices industry with almost-zero fees and the difficulty of hedge funds to bring home high returns as in the past, cannot justify such high fees.
3.3
Conclusions on linear risk models
Introducing the second approach in the previous section, was said that the model of Sharpe [1992] had great success. The model was able to explain a great proportion of the variability of mutual fund returns, maintaining a very easy implementability. The seven-factor model, as a tailored version of the former to analyse hedge fund returns, preserves these positive features. The two authors obtained very good results applying the model to the indices in the period between January 1994 and December 2002, as testified by the high R2s. The analysis imple-mented in this work verifies that the model has passed the test of time. The exposition of the indices considered to risk factors has changed, though with unexpected results as explained above. Nevertheless, the model still has good explanatory power. Investor can therefore rely on a consistent model in order to get a snapshot of the exposure to risk factors of their portfolios. Similarly, regulators can use the model to prevent possible convergences between different hedge fund strategies to same risk factors that may lead to difficult situations to hold.
As pointed out also by Fung and Hsieh, the selection of the right factors to use is not unique and other factors can be considered. In example, further analysis may be conducted on the trend-following ABS factors that proved to be non-significant also in this work, as it was in the original one. With regard to the other factors, all indices show positive exposure to the market with a high level of significance, as denoted by the high t-students. Indeed, except for few exceptions all hedge funds strategies followed the market, declining in the very first part of the sample and then recovering from 2009 onward. All
indices also show negative exposure to the Small-Large factor. The negative sign suggests that hedge funds returns are comparable to those achieved by going long large cap stocks and shorting small cap stocks. This may be a convenient opportunity to have an opposite directional exposure for the portfolio of the average investor, generally shifted to small cap stocks. Another possible interpretation for the negative sign is that this factor proxies for the liquidity risk. Small cap stocks are more sensitive to the illiquidity of the market than large cap stocks, so they have a higher liquidity risk. Hence, this factor may well identify the renown difficulties of hedge funds in dismissing their large positions without incurring in losses during distressed market conditions.
The seven-factor model is conceived to catch the intrinsic nature of hedge fund returns. It is a static model where the factors are ex-ogenously selected to reflect the dynamicity of the different hedge fund strategies. However, Fung and Hsieh [1997] revealed that the exposures to risk depend also on the state of the world, or market condition, in which hedge funds are operating. Factors should be free to vary over time in response to the changing market environments. One of the ABS factors of the model is the S&P500. The model therefore considers the linear relationship between hedge funds returns and the possible states of the market. The problem is that also the other factors are influenced by market conditions, and the exposure to them changes accordingly. The rigid structure of the seven-factor ABS model cannot overcome this aspect. In order to achieve this result another step is required, but this is left to the next chapter.
Non-Linear Exposures in Hedge
Funds
In Chapter 3 was outlined that one important drawback of the lin-ear models is that they cannot deal with the time-varying and state-dependent nature of the risk factors. Models such as the seven-factor ABS model successfully capture the dynamicity of the strategies used by hedge funds. However, identify the factors is just the first step in order to properly analyse the hedge fund returns. An additional step requires to study how these factors behave in different market conditions and change with time. To preserve the philosophy of this dissertation of checking every hedge fund-related fact without giving anything for granted, this new chapter opens verifying how exactly factors have behaved in the last ten years. The seven-factor model is applied to rolling windows of three years for every hedge fund strategy index. Collecting the coefficients of each factor from every window allows to verify how they have changed in time.
After this check, the very hearth of this work will be presented: the regime-switching model by Billio et al. [2010]. The model embodies the ideas and approaches of the three linear models presented in Chapter 3, but offering a more flexible framework and avoiding part of their weaknesses. It is designed to get over the problems just mentioned.
This approach follows the idea of Fung and Hsieh [1997] of considering the exposure to risk factors during different states of the world. How-ever, the use of quintiles implies the exogenous determination of the states. Here, the states of the world are determined step-by-step, to-gether with the market. The model also follows the spirit of Fung and Hsieh [2004], but with an inverse fashion. Instead of using rigid factors to capture non-linearities in the dynamic strategies of the hedge funds, factors are endogenously determined and are allowed to vary dynami-cally. Thus, on one hand, this model is more accurate in obtaining the same goal of the seven-factor ABS model to catch the dynamic nature of the hedge fund returns. On the other hand, it attains the flexibility of the style analysis of Sharpe [1992], by not using tailored ABS factors for the hedge fund strategies.
After explaining how the model is constructed and works, it will be applied to post-crisis hedge fund indices to have the final picture of the current situation of the hedge fund industry. Comments on the results obtained and on the usage of the model will close this chapter.
4.1
Time-varying Factors
The huge difference between the results obtained by the seven-factor ABS model in this work compared to Fung and Hsieh [2004] is a first proof of the fact that the exposure of hedge funds returns to risk factors changes over time. The focus of this section is on possible common pat-terns that factors may follow during time and in response to changes in the market. This way it will be verified if a non-linear model is actually needed. To accomplish this task, the seven-factor ABS model is ap-plied to the hedge fund indices, through rolling windows of 36 months. So, the indices are regressed to the seven factors in the first 36-month window starting in January 1994. Then, the regression is applied to the second time-window starting in February 1994. The process is
re-peated and the betas collected every time until the regression with the window containing the last observation in December 2016. This pro-cess was applied to the Credit Suisse Hedge Fund Index as a proxy for all global hedge fund indices. Then, also all the single strategy indices belonging to the Credit Suisse database are considered.
Figure 4.1: Time-variation of Credit Suisse index betas with 90% confidence interval and rolling windows of 36 months
Figure 4.1 shows the time-varying betas for the Credit Suisse Hedge Fund Index, computed with rolling windows of 3 years. The dashed lines represent non-significant betas at a 90% confidence interval. The
exposure to the market risk factor is almost always significant and varies between 0.17 and 0.6. Only during the financial crash of 2008 the S&P500 stopped to be significant. All the other betas are in-stead often non-significant and their values change frequently in time. However, it can be seen that there are at least three factors with sim-ilar behaviours. The market factor and the small-large factors moved almost together until 2008, when they started to assume opposite po-sitions. This may be due to the fact that in tranquil market conditions the small cap stocks have outperformed large cap stocks in line with the performances of the S&P500. After the crisis of 2008, small cap stocks heavily suffered from liquidity risk and needed more time than large cap stocks to recover. As was already explained in Chapter 2, the Small-Large factor can be considered a proxy for liquidity risk since small cap stocks are more sensitive to illiquidity in the market than large cap stocks. The Credit Spread factor shows a very similar be-haviour with the Small-Large factor all over the sample. Instead the other Fixed-Income ABS factor shows opposite movements. Indeed, until 2008 it moved symmetrically to the S&P500 factor, than started to behave accordingly. For the other factors is more difficult to cap-ture possible patterns in their changes over time since they are more stationary. However, looking closely, the Forex and Bond portfolios factors are very correlated and move partly with the market.
To deepen the analysis, the single strategy indices are now consid-ered. This way is it possible to verify if the movements of the betas of the global index may be influenced by one, or few, single strategy index that behaves anomaly. For sake of space will be presented here only the plot of the Long/Short Equity strategy index here, all the other can be found in the appendix.
Figure 4.2: Time-variation of Long/Short Equity strategy index betas with 90% confidence interval and rolling windows of 36 months
The analysis of this strategy leads to the same results found above for the global index. Betas are inconstant and change very steadily, but more importantly they behave very similarly to the ones of the global index. S&P500 and ∆10-year Treasury yields factors move al-most together, though the latter is rarely significant. The Small-Large factor follows them until 2008 and then move symmetrically, while the Credit Spread factor has the opposite behaviour. Also here the other factors are more stationary and it is difficult to capture possible common patterns, also because are rarely significant.
Emerging Markets, Distressed Securities, Event Driven MS, and Risk Arbitrage strategies present very similar results to the ones ob-tained for the global index and the Long/Short strategy. Dedicated Short Bias strategy shows also the same kind of patterns of the other strategies, but with symmetric values due to the nature of this strat-egy. Equity Market Neutral is the only one index with different results. Until the 2008 crisis, factors are all quite stationary and included be-tween -0.1 and 0.3. Then, they all explode and began varying bebe-tween -0.4 and 0.8.
The analysis performed on the Credit Suisse global index and on the single strategy indices show that betas change in time following precise patterns, mainly conditional on the state of market. For this reason, the use of non-linear models able to consider these behaviours is highly recommended.
4.2
The Beta Regime-switching model
One the aims of the previous sections was to demonstrate that hedge fund returns are so particular that traditional linear models cannot get the whole picture. In this new section the focus is on non-linear models. Firstly, it is presented a general Markov switching model. This lays the foundations to arrive finally to the core of this dissertation: the multi-factor model with non-linear exposures to all factors by Billio et al. [2010].
The linear models presented above are modifications belonging to the same framework:
Rt = α + βIt + t
where Rt is the rate of return (of the hedge fund index) at time t, It
is the risk factor, and t is an IID error. With this rigid structure,
group of factors. But, as underlined multiple times in this work, the exposures to the risk factors of the returns of alternative investments, i.e. hedge funds, are not stable in time and across different market conditions. Therefore, for such kind of data, a single linear model is not adequate to capture all the distinct behaviours.
Bekaert and Harvey [1995] proposed a time-varying measure of capi-tal market integration, based on a regime-switching model. The model allowed them to describe expected returns of countries that showed market segmentation with respect to the rest of the world in some parts of the sample and market integration in others. Billio et al. [2010] followed this scheme to analyse the risk exposures of hedge fund indices with a factor model based on regime-switching volatility of the market risk factor, where the non-linear nature of the exposures is captured by factors that are state-dependent. The model identifies when the market is in tranquil, up-market, or down-market state and considers the loading of the factors accordingly.
4.2.1 Theoretical Framework
The model can be ideally divided into two steps: firstly the Markov switching model characterizes the market and then, conditional to this result, a factor model is applied to estimate the betas. After conducting various tests and in accordance with the main literature, Billio et al. [2010] decided to use three regimes (tranquil,up-market, down-market) to qualify the market. Moreover, they proposed three different factor models:
• one-factor model1
• multi-factor model with non-linear exposures only to the S&P500 • multi-factor model with non-linear exposures to all factors
The Markow switching model by Hamilton [1989] (also known as regime-switching model) is one of the most popular non-linear models. This model implies as many structures (equations) as the number of regimes considered. Each equation describes a different state. The model is able to catch complicated dynamics, such as discontinuous shifts in average return and volatility, by switching from a structure to another or, equivalently, from a regime to another. In formulas, the model can be represented as:
(
Rt = α + β(St)It + ωut
It = µ(St) + σ(St)t
(4.1)
where St is a Markov chain with n states and probability transition
matrix P. ut and t are independent and normally distributed errors
with zero mean and unit variance. A Markov chain is a stochastic event that experiences transitions from one state to another according to certain probabilistic rules. The probability of transitioning to any state is dependent solely on the current state and time elapsed. The regime-switching process is described by the following probability transition matrix P: P = p00 p01 p02 p10 p11 p12 p20 p21 p22
where p00, p11, and p22 are the probabilities to remain in the same state,
pij is the probability to shift from state i to state j.
Each state of the market I is described by its own mean and vari-ance. At each time t, the hedge fund return is given by a parameter α and a factor loading β on the conditional mean of the factor, related to the regime of the market. Analogously, the hedge fund volatility is also related to the regimes and specified by the factor loading β, on the conditional volatility of the factor plus the volatility of the idiosyn-cratic risk factor ω. In both cases β may assume different values with
regard to the related state of the market. Considering three regimes, the model can be written as:
Rt = α + β0It + ωut if St = 0 α + β1It + ωut if St = 1 α + β2It + ωut if St = 2
where β depends on the state variable S:
β(St) = β0 if St = 0 β1 if St = 1 β2 if St = 2
For all Markov states are assumed the hypothesis of normality of the errors and of homoskedasticity within regimes.
At any time, the Markov model determines stochastically the regime of the market, based on the previous state and on the transition prob-ability matrix. Once selected the regime, the associated structure is activated to determine the hedge fund return. Thanks to the Marko-vian property, a structure may prevail for a random period of time until a switching takes place and another structure replaces the for-mer. The switching is usually permitted in response to occasional and exogenous events. This feature makes the Markov switching model suitable to describe correlated data that exhibit dynamic patterns in time, such as hedge fund returns.
The state St is not directly observable. However, Hamilton [1989]
proposed a non-linear iterative filter and a smoother to draw proba-bilistic inference about the occurrence of shifts. The filter is important because allows the estimation of the population parameters through the method of maximum likelihood and provides the tools to forecast future values of the series. Since states are endogenously determined by the model, the estimated parameters permit the observer to eval-uate the economic meaning of the results. The filter starts from the
joint conditional probability:
P [St−1 = st−1, St−2 = st−2, ..., St−r = st−r|yt−1, yt−2, ..., y−r+1]
and gives back the following output:
P [St = st, St−1 = st−1, ..., St−r+1 = st−r+1|yt, yt−1, ..., y−r+1]
together with the conditional likelihood of yt:
f (yt|yt−1, yt−2, ..., y−r+1)
A modification of the basic filter where inference is based on the cur-rently available information is the smoother:
P [St = st|yt, yt−1, ..., y−r+1] = 1 X st−1=0 1 X st−2=0 ... 1 X st−r+1=0 P [St = st, St−1 = st−1, ..., St−r+1 = st−r+1|yt, yt−1, ..., y−r+1]
The usefulness of these formulas will be clear afterwards. The filter will be used to characterize the regimes for the factor loadings, while the smoother will be utilized to draw the three regimes to verify if they actually reflect observed past market events (i.e., during the end of 2008 the down-market regime should prevail).
Equation 4.1 can be extended in several ways. The modeller can freely choose the factors to use and to study their exposures linearly or non-linearly. The multi-factor model with non-linear exposures to all factors that will be applied to post-crisis data in the following section, is one of these possible extensions. Before to jump to explain this model that is quite complicated, the other two models presented by Billio et al. [2010] are introduced.
The first is a regime-switching model with non-linearity in the volatil-ity of residuals and in the intercept coefficient:
(
Rt = α(Zt) + β(St)It + ω(Zt)ut
It = µ(St) + σ(St)t
It is a one-factor model as Equation 4.1. As before, St is the Markov
chain that characterizes the three states of the S&P500. However, Zt
is another Markov chain attached to the intercept and to the residu-als. Zt is designed to catch non-linearities other than the non-linear
relationship between hedge funds and the market. It has two regimes so that residuals can be in tranquil or distressed state. The model has therefore a total of six regimes.
The second model proposed considers the non-linear exposure of the hedge funds to the market and the linear exposure to other risk factors:
(
Rt = α(Zt) + β(St)It+ PKk=1θkFkt + ω(Zt)ut
It = µ(St) + σ(St)t
(4.3)
where θ is the factor loading corresponding to the k-th risk factor Fkt.
It is a linear factor model inserted in a regime-switching framework. Zt should catch all the missing non-linear dynamics.
It is finally possible to present the multi-factor model with non-linear exposures to all factors:
( Rt = α(Zt) + β(St)It + PK k=1θk(St)Fkt+ ω(Zt)ut It = µ(St) + σ(St)t (4.4)
In the previous chapter was demonstrated that linear models cannot properly analyse hedge fund returns and the previous section showed how the betas change in time and in different market conditions. This model is appointed to take care to all of these issues. Indeed, it con-siders the exposures of the hedge funds to all factors, conditional on the state of the market. Also in this case Zt captures all the non-linear
4.2.2 Empirical Analysis
In this section will be finally applied two of the factor models ex-plained above: the multi-factor model with non-linear exposure only to the S&P500 and the multi-factor model with non-linear exposures to all factors. The models will be applied to the hedge fund strat-egy indices, in the period that goes from June 2006 to December 2016. Given the low number of observations and the computational complex-ity, the process is divided into two parts: firstly, are determined the regimes of the S&P500 and then the factor models are performed. All the computations are performed in Matlab.
This work follows the choice of Billio et al. [2010] to characterize the market with three regimes: tranquil, up-market, and down-market. The authors performed various tests (such as the model selection test from Krolzig [1997] and a simulated likelihood ratio test) on their dataset to finally confirm the main literature on the number of regimes for the market.
Regime 0 Regime 1 Regime 2
Estimate t-stat Estimate t-stat Estimate t-stat
Mean 1.25 4.15 9.15 21.43 -4.01 -3.17
Standard Deviation 2.74 12.40 0.91 2.90 5.25 6.66
Frequency 79% 4% 17%
Transition Probabilities Matrix
Regime 0 Regime 1 Regime 2
Regime 0 0.96 0.00 0.04
Regime 1 0.19 0.81 0.00
Regime 2 0.00 0.17 0.83
Table 4.1: Three Regime-switching model for S&P500 in the period
Jun2006-Dec2016.
Table 4.1 shows the estimated parameters for the three regimes, together with their frequency and the transition probabilities matrix.
