modeling hedge fund returns

Modelling Hedge Fund Returns Beat Huggler Master of Advanced Studies in Finance Swiss Federal Institute of Technology of Zurich - University of Zurich December 28, 2004 Abstract The principal aim of this paper is the modelling of hedge fund portfolios using representative proxies. Two competitive models are investigated in this paper: one using manager prox- ies, a so called Manager-Up model, and the second based on strategy proxies, a so called Style-Up model. These proxies are constructed using publicly available hedge fund return series. The serial dependence observed in the proxies is described by a multivariate AR-GARCH model. There is strong empirical evi- dence that the marginal distributions of the model’s innovations are skewed and heavy tailed, and hence we propose the use of askewed-t distribution for their modelling. Under the assump- tion that no cross-lag correlation exists between the proxies, the cross-sectional dependence structure is only apparent in the inno- vations. Although there exists evidence of an asymmetric depen- dence structure between some of the proxies, a grouped-t copula is proposed to model the cross-sectional dependence structure of the innovations. Finally, the calibrated model is used to simu- late hedge fund portfolio return series, which then are compared with similar portfolios constructed from the data. Keywords: hedge funds, portfolio simulation, ARMA-GARCH process, skewed-t distribution, asymmetric dependence structure, grouped-t copula. Acknowledgments I would like to thank all those who in one way or another helped bring to completion my master thesis. I am highly appreciative of the interesting and helpful discussions with Anthony Ledford and Darren Upton from AHL Research, and also many thanks to the rest of the research team for their support. Furthermore, I would like to thank my supervisor, Dafydd Daniel, for his helpful input as well as my co-supervisor Prof. Paul Embrechts, whose help was essential for the development of the model presented in this paper. Also many thanks to Stefan Scholz, Andrew Davies, Yianna Tchopourian and Leonie Ryhnould-Troxler for their support. Moreover, I am very grateful to Man Investments for making it possible for me to spend these last few months in London, working with all these interesting people. Last, but not least, I would like to thank my girlfriend, Bojana Taraba, for her patience during this absorbing time and also for her emotional support. ii Contents 1 Introduction 1 2Data 4 2.1 Hedgefunduniverse 4 2.2 Datacleaning 4 2.3 Standardisingthedata 5 3Model 8 3.1 Thebasicstructureofthemodel 8 3.2 The skewed-t distribution 9 3.3 Normalmean-variancemixtures 10 3.3.1 The generalised hyperbolic distribution . . . . . . . . 11 3.4 Copulas 12 3.4.1 The grouped-t copula 12 4 Modelling 15 4.1 Methodology 15 4.2 Dataanalysis 16 4.3 Serialdependence 16 4.3.1 Analysesoftheserialdependence 16 4.3.2 Modelling the serial dependence 19 4.4 Modelling the innovations . 22 4.5 Modelling the cross-sectional dependence structure . . . . . . 25 5 Simulation 31 5.1 Styleportfoliosimulations 32 5.2 Diversifiedportfoliosimulations 33 5.3 Maximumdrawdownanalysis 36 6 Conclusions 38 7 Appendix 41 iii 1 INTRODUCTION 1 1 Introduction Over the last decade hedge funds have been the fastest growing asset class of the financial sector. From 1990 to 2003 the number of hedge fund managers increased four-fold and is now estimated to be over 8000. TASS Research estimated the total net assets managed by single hedge funds as approximately 890 billion USD at the end of October 2004. This rapid development is directly related to the regulatory exemptions that hedge funds enjoy as privately held investment vehicles. The fund managers have relatively greater flexibility when handling derivatives and employing leverage to achieve higher returns. Their trading strategies are very diverse in terms of trading approach and markets. For a detailed description of hedge funds, Ineichen [13] or Lhabitant [15] are recommended references. The performance of a hedge fund very often depends on the skills of the hedge fund manager and therefore rigorous due diligence and monitoring of the manager is crucial for a successful hedge fund investment. Further- more, many studies such as Lhabitant and Learned [17] and Bacmann and Huggler [3] have shown a diversification benefit from investing in more than one hedge fund. In addition, single hedge funds often require a minimum initial investment of more than a million USD. The previously mentioned characteristics have enhanced the popularity of fund of funds (FoF) in this industry. A FoF provider has the expertise to select the good managers and by pooling money from various investors is able to construct a well diversified portfolio. The assessment of the risk-return characteristics of a portfolio of hedge funds is essential for FoF providers to construct their products efficiently and manage their risk effectively. Consequently, modelling return distributions is an important research topic in the field. To this end, various industry specific problems need to be solved. For example, the heterogeneity of the hedge fund universe makes it difficult to classify adequately the funds into groups of similar strategies with comparable risk-return characteristics. Other problems are the lack of sufficiently long data history and poor data quality. Typically the hedge funds in a portfolio may have only a few years of monthly data. Though there exist a range of ways to model portfolio return distributions, it is almost impossible to calibrate a model to an individual fund’s historical track record due to the data limitation. Hence, making use of proxies is indispensable. To circumvent this problem, one can use a publicly available fund of hedge funds index. However, the strategy exposures vary among FoF resulting in different risk-return characteristics of the hedge fund portfolio. In order to account for the different strategy allocations, the modelling has to be done at either a strategy or a single manager level. A publicly available strategy index consists of a large number of funds which, however, still may not accurately replicate the characteristics of a more concentrated 1 INTRODUCTION 2 strategy exposure of a typical FoF. Furthermore, each data provider has its own style classification which might not correspond to that of the FoF provider. For some indices even the verification of their returns is not possible because their composition and the return series of their constituents are not published. Finally, the short index history of around ten years of monthly data limits analyses of the return characteristics to some degree. The construction of proxies suggests itself as an alternative for overcom- ing the limitations of the publicly available indices. Such proxies may be constructed using single manager data from a combined data base of TASS 1 , HFR 2 ,Stark 3 and RMF 4 . This project follows two different approaches; one uses style proxies leading to the (so called) “Style-Up” model, and the other uses single manager proxies leading to the (so called) “Manager-Up” model. The computed proxies in the first case reflect the risk-return characteristics of a “typical” strategy portfolio and, in the second case, that of a “typical” manager of a certain strategy. There are ample studies demonstrating that hedge fund return distribu- tions are skewed and leptokurtic. Good references on this topic are McFall Lamm [19] or Brooks and Kat [6]. Data analyses of single managers and strategy proxies also show some evidence of serial dependence in return and in volatility. Moreover, there is some support that the cross-sectional dependence lies beyond a simple correlation structure (see e.g. Czub [7]). One implication of these findings is that the commonly used multivariate normal distribution is not an appropriate model for hedge fund portfolio returns. Historical simulation 5 can be easily performed using the computed proxies as these incorporate the skewed and heavy tailed return distribution, as well as the complex serial and cross-sectional dependence structure. However, a major drawback of this method is the limitation to events observed in the past. In contrast, a parametric model allows events more extreme than the historical observations to be generated in simulations. The aim of this project is to specify a fully parametric model which is able to simulate accurately hedge fund portfolios. The serial dependence and heteroskedasticity observed in the data is modelled with an AR(2)- GARCH(1,1) process. Instead of using a Gaussian distribution to model the standardised innovations within the AR-GARCH process, a skewed-t distribution is proposed. Finally, the cross-sectional dependences are either described with a grouped-t copula or a t-copula. The remainder of this paper is organised as follows. In Section 2 the data 1 TASS is a public hedge fund data provider. 2 HFR is a public hedge fund data provider. 3 Stark is a public commodity trading advisory fund data provider. 4 RMF is an investment manager of Man Investments. 5 Historical simulation is a non-parametric simulation technique which uses the empirical distribution and dependence structure of historical data. 1 INTRODUCTION 3 is described and analysed in detail. Furthermore, the construction of stan- dardised manager proxies is explained in full detail. The parametric model to simulate hedge fund portfolio return series is introduced in Section 3. Section 4 is devoted to the model’s calibration and the model selection. Section 5 compares simulated hedge fund portfolio return series from the model with portfolio returns constructed from the data. Moreover, the difference between the simulation of the Style-Up model and the Manager- Up model are investigated. Finally, Section 6 concludes and lists ideas for further improvement of the model and for future research. 2DATA 4 2Data 2.1 Hedge fund universe Since hedge funds are only lightly regulated, performance reporting is voluntary. Hence, all available hedge fund databases suffer from various biases, such as selection bias 6 , instant history bias 7 and survivorship bias 8 . A good overview of the various biases in hedge fund databases is given in FungandHsieh[12],orinAminandKat[1]. With the growth of the hedge fund industry, the number of publicly available hedge fund databases has increased. This study is based on the monthly data from the TASS, HFR and Stark hedge fund return databases, which are considered to suffer the least from the biases mentioned above. In addition, hedge fund returns collected by RMF have also been included. The public databases used in this study all came into existence around 1994, and from this time on they started to keep dead funds in their database. Dead funds are funds which stopped reporting due to various reasons, e.g. a manager might stop reporting when he does not want to attract new capital or when he wants to hide poor performance, or when hedge funds are liquidated or merged with other funds. On the one hand pre 1994 data is heavily biased towards surviving funds but on the other hand the use of as long as possible data history is desired. Therefore this study is based on data between January 1993 and July 2004. 2.2 Data cleaning The voluntary reporting for hedge fund managers does not only result in a biased universe, but also in poor data quality. Furthermore, fund managers do sometimes report their performance to more than one database which results in double counting of funds. To overcome these and other anomalies, it is essential that some cleaning of the dataset is undertaken. Since the cleaning and classification procedure closely resembles a proprietary process developed by Man Investments, it cannot be published in full detail. However, the main steps are as follows: • Managers with less than 12 months data are discarded • Leading and finishing zero returns are removed 6 Selection bias is manifested in two ways: A data vendor only includes funds which fulfil certain criteria and a hedge fund manager generally only starts reporting if he can show a good track record. 7 The instant history bias is related to the selection bias and is caused by the fact that a hedge fund manager is able to report his pre-inclusion date performance. 8 If a data provider only keeps the surviving funds then this database is subject to survivorship bias. 2DATA 5 • Managers with consecutive repeated returns are excluded from the data set. Typically these arise through quarterly or semi-annual performance reporting • Where fund information is replicated only the fund with the longest track record is kept • A very small number of extraordinary returns are adjusted manually After the rigorous cleaning procedure the combined hedge fund universe contains 3491 funds. It is known that funds which follow a comparable trading strategy show similar risk-return characteristics. Hence, the funds are classified in nine distinct categories and are called risk-return or style buckets. A detailed description of each bucket can be found in the Appen- dix A. The number of managers in each style bucket is indicated in Table 1 below. Risk-Return Bucket No. Managers Convertible Bond Arbitrage 158 Event Driven 341 Fixed Income Arbitrage 99 Global Macro/Directional 343 Long/Short Equities 942 Market Neutral 161 Multi-Strategy Arbitrage 98 Non-Trend Following 971 Trend Following 378 Table 1: The total number of managers for each risk-return bucket. Figure 1 shows the evolution of the number of funds in each risk-return bucket. It is clear that the dataset is heavily biased towards the recent past. Given that the last classification has been performed early in 2004, no new funds have entered the universe since then. The decreasing number of funds over the most recent months is caused by the fact that some funds delay performance reporting and that each month a number of funds stop reporting. 2.3 Standardising the data The hedge fund industry is very young and most funds only have a track record of a few years. To make the best possible use of the available data, and to fit an accurate model, it is proposed to pool all the return data of a specific risk-return bucket. Thus, various artefacts of hedge fund return series have to be considered. Due to the underlying financing structures 2DATA 6 Convertible Bond Arbitrage 1990 1992 1994 1996 1998 2000 2002 2004 20 40 60 80 120 Event Driven 1990 1992 1994 1996 1998 2000 2002 2004 50 100 200 300 Fixed Income Arbitrage 1990 1992 1994 1996 1998 2000 2002 2004 0 204060 Global Macro/Directional 1990 1992 1994 1996 1998 2000 2002 2004 50 100 150 200 Long/Short Equities 1990 1992 1994 1996 1998 2000 2002 2004 0 200 400 600 800 Market Neutral 1990 1992 1994 1996 1998 2000 2002 2004 0 20 40 60 80 120 Multi-Strategy Arbitrage 1990 1992 1994 1996 1998 2000 2002 2004 0 20406080 Non-Trend Following 1990 1992 1994 1996 1998 2000 2002 2004 100 200 300 400 Trend Following 1990 1992 1994 1996 1998 2000 2002 2004 100 150 200 Number of funds in each risk-return bucket Figure 1: Evolution of the number of hedge funds in each risk-return bucket since January 1990. which hedge funds employ, the level of the returns is closely related to the level of the risk-free rate. Although the funds of a particular risk-return bucket follow a similar trading strategy, their level of returns and volatilities vary across managers. This is primarily due to the usage of different levels of leverage. Additionally, the risk-return characteristics of a specific strategy depend strongly on the market environment in which it is trading. Also, there exists evidence that the risk appetite of a hedge fund manager changes over the life-time of a fund and with the age of the manager 9 . A detailed discussion of this topic can be found in Boyson [5]. The main idea is to transform the observed hedge fund data onto the scale of a “typical” manager. To achieve this, all the temporal and manager specific patterns mentioned in the previous paragraph have to be removed from the data. The standardisation procedure of the returns series for each manager begins with computing the excess returns r ∗ t defined as r ∗ t = r t − r f t ,(1) where r t is the hedge fund return and r f t the risk-free rate 10 at time t.To 9 Of course, these two factors might well be correlated 10 The hedge fund returns are reported in different currencies and therefore the currency specific risk-free rate is subtracted. In the case of funds in US-Dollars the US 3-month Treasury Bill rates is used as risk-free rate and otherwise the particular 3-month LIBOR- rates. 2DATA 7 remove time and manager specific characteristics, each excess return series is standardised by the exponentially weighted moving average (EWMA) of its mean ˆµ t =(1− λ)r ∗ t−1 + λˆµ t−1 (2) and the square root of the EWMA of its variance ˆσ 2 t =(1− λ)r ∗2 t−1 + λˆσ 2 t−1 (3) where 0 <λ<1 can be interpreted as a persistence parameter. A 24-month EWMA was considered to be appropriate for both moving averages and thus the parameter λ was chosen to be 1 / 24 . It is assumed that a hedge fund will always have an expected positive absolute return over a long term horizon, hence ˆµ t is restricted to be non- negative. Additionally the volatility ˆσ t is restricted to be greater or equal to 1% in order to ensure a limit on the inflation of returns. Hence, the standardised hedge fund return series {z t } is defined as z t = r ∗ t − max(ˆµ t , 0) max(ˆσ t , 0.01) . (4) The aim is to have return series with a zero mean and unit volatility. Since this is not guaranteed by the standardisation procedure described in (4), in a second stage, each series is standardised by subtracting its mean and then dividing it by its standard deviation. [...]... Methodology The hedge fund universe has been divided into nine distinct style buckets where funds in each bucket share comparable risk-return characteristics Since the simulation engine must be able to simulate portfolios with different strategy exposures, either single hedge funds returns or strategy returns have to be modelled Given that the hedge fund industry is rather young and older funds are usually... mentioned in Section 2.1 and are used to calibrate a parametric model to generate hedge fund portfolio return series In contrast to most other studies in finance we model returns instead of log -returns Since the monthly hedge fund returns are rather small, the difference is negligible Manager-Up The idea is to simulate “typical” hedge fund return series for each risk-return bucket Thereafter, these series are... MODEL 3 8 Model 3.1 The basic structure of the model In McFall Lamm [19] or Brooks and Kat [6], it is well documented that hedge fund return distributions are skewed and heavy tailed Additionally, the serial dependence in returns and clustering in volatility are often present, and hedge funds within a certain risk-return bucket and across different buckets show cross-sectional dependence It is doubtful if... model is done using the standardised hedge fund return series as described in Section 2.3, where these series should reflect the risk-return characteristics of a “typical” hedge fund The main advantages of this approach are a high degree of flexibility in terms of the number of managers in each strategy as well as the weighting and volatility scaling of each single fund in the portfolio Among others Lhabitant... approach, this model assumes a return distribution shape which does not change with the number of funds in a risk-return bucket As most hedge fund diversification studies show, skewness and kurtosis do not change significantly for most style portfolios with 10 and more funds Since most larger FoFs consist of 50-100 funds, the Style-Up model should be able to accurately simulate their return series The model... 4.2 Data analysis To analyse the return distribution of a particular style, the returns of managers or style portfolios with that style are pooled together Table 2 suggests that hedge fund returns are skewed and heavy tailed and the estimates of the skewness and the kurtosis clearly differ among styles The manager proxy returns show very large excess kurtosis 17 for Convertible Bond Arbitrage and Fixed... model is only specified for a single time series and cannot be used to simulate a portfolio of hedge funds It is assumed that no crosslag correlation exists among the different series, and hence the extension to the multivariate case is made more straightforward 14 Cross-sectional dependence between hedge funds is caused by their similar reaction to shocks in financial markets and exposures to the same... 6950 6950 6600 5650 6950 6950 Table 2: Some statistics of the standardised manager returns The skewness (S), the excess kurtosis (K) and the number of observations (N) in each risk-return bucket 4.3 4.3.1 Serial dependence Analyses of the serial dependence In a next step, the serial dependence of the standardised hedge fund returns and the constructed portfolios are analysed Again, the idea is to model... previous section the proposed model was calibrated through various individual steps Although the quality of the model has been examined at each step, its ability to simulate hedge fund portfolios remains to be assessed Since data in the hedge fund industry is very limited, out of sample testing is almost impossible, and hence the simulations have to be compared with the data used for the calibration Consequently,... the innovations from the AR(2)-GARCH(1,1) style proxies versus the theoretical quantiles from the fitted s-t distribution Manager-Up In the hedge fund industry it is believed and empirically observed to some extent that during crises the strength of dependence between funds increases leading to an asymmetric dependence mechanism For a deeper understanding of the cross-sectional dependence structure the . either single hedge funds returns or strategy returns have to be modelled. Given that the hedge fund industry is rather young and older funds are usually closed to new investors, hedge funds in newly. a parametric model to generate hedge fund portfolio return series. In contrast to most other studies in finance we model returns instead of log -returns. Since the monthly hedge fund returns are rather small,. the various biases in hedge fund databases is given in FungandHsieh[12],orinAminandKat[1]. With the growth of the hedge fund industry, the number of publicly available hedge fund databases has increased.