STYLE EXPOSURE AND LEVERAGE OF FUNDS OF HEDGE FUNDS WITH A VARIABLE FACTOR MODEL *)

STYLE EXPOSURE AND LEVERAGE OF FUNDS OF HEDGE FUNDS WITH A VARIABLE FACTOR MODEL ) Andreas Ruckstuhl (andreas.ruckstuhl@zhaw.ch) Institute of Data Analysis and Process Design Zurich University of Applied Sciences (ZHAW), Postfach 805, CH-840Winterthur, Switzerland Peter Meier (peter.meier@zhaw.ch) Centre for Alternative Investments Zurich University of Applied Sciences (ZHAW), Postfach 805, CH-840Winterthur, Switzerland Abstract An innovative statistical method is used to analyse the style exposures and the economic leverage of Funds of Hedge Funds (FoHF). FoHF returns are explained by style indices from well known index providers. A modified version of LASSO (least absolute shrinkage and selection operator) is developed and used to select relevant style factors and to estimate factor exposures (beta). The sum of betas is not restricted and is used as a measure of leverage. The method is applied to FoHF taken from hedgegate, a database which is specialised on FoHF and separates directional, non-directional and diversified FoHF. Interesting findings result: the factor exposures are generally consistent with the FoHF classification, the equally weighted index families HFR and EDHEC achieve better results than the capital-weighted CS/Tremont in terms of adjusted R-squares and the sum of betas seems to be a well working measures for overall leverage of FoHF with respect to the risk characteristics of relevant FoHF universes. ) This research was supported by CTI, the Swiss Confederation s innovation promotion agency as well as Complementa AG.

Introduction Hedge funds are difficult to analyse and to judge because they mostly pursue complex, diverse and dynamic investment strategies. Furthermore, a lot of strategies must protect themselves from imitators competing for the same opportunities. Hence they disclose their positions, intentions and strategies at best partially. It is even more challenging to reach transparency for funds of hedge funds (FoHF). FoHF are not only fixed baskets of mostly opaque hedge funds, some FoHF also follow dynamic allocations strategies in accordance with their views of the performances of different investment styles, and others use cash or leverage to steer their exposures. So, the return characteristics of FoHF are not only an aggregation of the returns of the underlying funds, but are also overlaid by the strategy and the tactic of the FoHF manager. It is important in most investor s views to create more transparency within hedge fund products in spite of the inherent barrier for managers to disclosure their investment details. This is why sophisticated statistical methods become important to measure performance and risk characteristics. Return patterns and their relations with benchmarks can reveal what is called the investment style of hedge funds. SHARPE s (992) method of style analysis for mutual funds is also applicable to hedge funds in advanced forms. To characterise the return pattern of an equity fund one can use established indices like one for small cap, blue chips or value stocks. Straightforward statistical methods tell the importance of the different indices for the fund s return. Hedge fund returns have more and additional return drivers compared to long-only equity fund returns. Since most hedge funds follow flexible and dynamic investment strategies, the relationships between driving factors and hedge fund returns may vary over time. To model a whole portfolio of hedge funds, like FoHF, many asset-based return factors from equity, bonds and commodities are necessary as well as, according to the nature of hedge fund strategies, nonlinear derivatives of these factors. FUNG and HSIEH (2004) use seven asset-based factors to explain FoHF-returns. This state-of-the-art asset-based multifactor model for FoHF explains up to 80% of the return variability of FoHF-indices. This paper follows another route. The aim is to give a comprehensive picture of the risk characteristics of distinct FoHF according to the more or less established hedge fund styles (GEHIN, 2006). If a FoHF invests in single hedge funds, and if different hedge fund styles are summarized by corresponding style indices, the FoHF return characteristics should be reflected by these style-indices. Such a style-index-based analysis has been suggested by LHABITANT (200, 2004). In this paper, we explore FoHF returns of the hedgegate database [] by a multifactor model with a dozen of hedge fund style indices as factors. To obtain reliable results, time periods must be identified within which the FoHF is invested in the same selection of hedge fund styles. Furthermore, not all of the style indices may contribute significantly in characterizing the FoHF. Hence, an appropriate style factor selection is needed. We show that promising results can be obtained once these fitting, factor selection and restriction problems are overcome. The risk patterns for the majority of the analysed FoHF can be reasonably well described by a limited number of style indices and their loadings. Furthermore, the method used allows to measure the total leverage of FoHF. Since these analyses are done for three index families, insights are also gained into characteristics, strength and weaknesses of these index families. Section 2 of the paper develops the method to model the FoHF returns within a style-index-based analysis framework. Section 3 describes the data bases from which the FoHF returns and the style 2

indices are drawn. Section 4 contains estimated equations for FoHF and interprets the results. Conclusions are drawn in the last section. 2 Methods 2. The Style-Based Multifactor Model If there is a set of hedge funds invested according to a common investment strategy, we expect that their returns follow a common return pattern representing the market opportunities of that investment style. These market opportunities are shown in the corresponding hedge fund style index. Deviations of single hedge funds from the common pattern are quite possible. But if they are too pronounced we doubt whether the management really follows the announced strategy. A fund of hedge funds is a portfolio of hedge funds and therefore its return should follow the weighted sum of the return patterns of the involved styles. This idea, which basically goes back to SHARPE (988, 992), has been suggested by LHABITANT (200, 2004) in his index-based style analysis which searches for the combination of hedge fund sub-indices that would most closely replicate the actual returns of the FoHF over a specified time period in the past. Technically, this approach is expressed by the multifactor model p R t = α + βkik, t + Et, () k = where R t denotes the return on the FoHF at time t, I k,t the return value of the style index (= factor) k at time t, and E t the residual at time t (also called error or idiosyncratic noise) which cannot be explained by the style indices. The parameter α is the intercept, β k is the factor loading that expresses the sensitivity of the FoHF fund returns to the style index k returns, and p the total number of used style indices. We assume that the p style indices I, I 2,, I p do represent the hedge fund market and its different underlying investment strategies. Hence, we can think about imposing constraints in the multifactor model () which allow a useful economic interpretation. The factor loadings must add-up to one so that they can be interpreted as portfolio weights within an asset allocation framework, p β = k. (2) k= However, if the FoHF may be leveraged, it is plausible that the factor loadings sum up to a value larger than one. The interpretation of the sum of the factor loadings is challenging. On one hand side, one must consider that the single hedge funds can be leveraged. Hence, an averaged leverage of the included hedge funds is captured in each of the style indices. If the hedge funds in which the FoHF is invested are not well represented by the style indices with respect to leveraging, the lack of representativeness will show up in higher or lower estimates of the factor loadings. On the other hand, one must consider that a fund of hedge fund can be leveraged itself. An example is the 3A Windrider Fund with a maximal leverage of 350%. But there are other effects which can influence the sum of the factor loadings on the level of the FoHF. For example, if the FoHF is not completely invested in hedge funds, or the style indices do not cover all hedge fund strategies, it is very plausible that the sum of the loadings β is smaller than one. Over all, the 3

interpretation of the sum of factor loadings has to integrate all the effects just mentioned. If a sum larger than is observed, it may be interpreted as a total leveraging with respect to the hedge funds included in the style indices. If the sum is smaller than one, there may be three different sources of the cause: (i) The hedge funds in which the FoHF is invested are less leveraged than those in the style indices, (ii) the FoHF is not completely invested in hedge funds or (iii) the style indices to not cover all hedge fund strategies used in the FoHF. Now, we turn to the other constraint. A negative factor loading β k may be interpreted as shortselling of the corresponding style index. However, FoHF only hold long positions of single hedge funds and therefore it is appropriate to postulate that all factor loadings must be non-negative, β 0 k =, Kp. (3) k Furthermore, the hedge fund strategy short-selling is covered by an own style index and may also capture net short equity exposures of hedge funds belonging to other style categories. The intercept α in model () may be interpreted as the excess return (a constant) of the FoHF and reflects the timing or fund selection skills of the FoHF manager after adjustment for systematic style dependent returns. Of course, with such an interpretation the intercept becomes a crucial parameter of model () since it allows identifying the manager s claim of adding value to their funds. Although this view may be of conceptual interest, it is empirically shaky. The estimated intercept α is affected unpredictably by any sort of shortcomings in the data gathering process or in the specification of the model. For example, if one of the style indices is systematically too high, e.g., because of the presence of survivorship bias, it will only affect the estimation of the intercept α but not the estimations of the factor loadings β k, k=,..,p. Such effects can even change the sign of the intercept. FUNG and HSIEH (2007) give a very detailed analysis of the effects of incorrectly specifying the factors on estimating α and the loadings β in multifactor models. However, the estimation of α is not an issue for assessing the strategies of FoHF. Concerning the loadings β, the effects depend very much on the completeness of the hedge fund style indices. Model () which is subject to either of the constraints (2) or (3) or to both constraints can easily be fitted to the data by computer programs resulting in estimates for the unknown intercept α and the unknown factor loadings β k, k=,,p. The usefulness of the multifactor model for the style analysis depends crucially on the set of style indices chosen for its implementation. This issue is further discussed in the next section. For the present section, we assume that basically a qualified set of style indices is available. But there still are practical hitches when fitting the model (). First of all, there is no guarantee that the FoHF is run under the same investment strategy over the considered period. Hence, a time period must be identified from present to the past over which the model is suitable. The following two subsections introduce procedures to cope with both of theses hitches. As a result, we propose a two-stage procedure to fit model () satisfying constraint (3) and making the most out of constraint (2), if the latter is imposed. 2.2 Selecting Suitable Time Periods Fitting model () without considering the constraints is usually done by ordinary least squares (OLS), which implies assuming that the errors E i are stochastically independent and normally distributed with mean 0 and standard deviation σ. In a first approximation, such a model is 4

reasonable. As the example in Figure shows, sometimes there are obvious irregularities in the error term E i. As the residuals are plotted against time, we observe that the residuals scatter much more at the beginning of the considered time period than later. These irregularities indicate a change in the factor model regime which may be triggered by market events as the Long-term Capital Management (LTCM) episode (September 998) or the internet bubble (March 2000), or changes in the investment strategy. In case of the FoHF in Figure, we know that the management of the FoHF has been replaced at the beginning of 2000 which may have resulted in a return pattern that could not be captured by the same model specification as the returns after that event. Although such irregularities are very interesting in themselves, we focus our attention on stable period in the near past in this paper. In what follows, we call a FoHF strategy persistent within a specified time period if the returns of the FoHF can be described by the same model specification, i.e., by model () with the same loadings. To use the result of fitting model () for assessing the current strategy of a FoHF, it is important to identify the most recent periods with a persistent investment strategy. Figure : Residuals against time of a robustly fitted model (). At the beginning of 2000, there might be a change in the investment strategy 8 6 4 Residual 2 0-2 -4-6 998 2000 2002 2004 2006 2008 Unfortunately, if we apply the OLS fitting method to returns from a FoHF containing irregularities, we will hardly find such structural breaks in the residuals because the OLS smears the irregularities over all residuals. Furthermore, the estimated loadings are influenced heavily by the presence of irregularities. Therefore FUNG and HSIEH (2004) devised a fitting method similar to running a Kalman filter with the time scale reversed. This approach allows to scan the whole period under investigation for structural breaks. However, since we are interested only in the most recent persistent investment period, we use so-called robust or resistant fitting methods which are hardly affected by structural breaks or other irregularities (cf. MARONNA, Martin, Yohai (2006)). Our preferred method is the MM-fitting technique [2] which is highly resistant against irregularities in the returns of the FoHF as well as in the returns of the style indices if needed. 5

The MM-fitting techniques allow us to check whether the investment strategy, as we have identified by the multifactor model (), is persistent over time by plotting the residuals against time (cf. Figure ). In presence of a persistent strategy, the same multifactor model should well describe the returns of the FoHF, and hence the residuals should scatter within a band of ±3σ [3], where the standard deviation σ is also estimated robustly within the robust MM-fitting technique. We consider the investment strategy as not persistent with respect to the index family when there will be time periods with many residuals clearly outside the band of ±3σ. For the example shown in Figure, we obtain an estimated standard deviation σ of 07. Such an order of magnitude of the estimated σ is very common in situations where the multifactor model fits well independent of the underlying family of hedge fund style indices. Figure indicates that the presented FoHF shows two periods with different investment strategies in their returns. In an early period (until April 2000), the multifactor model does fit poorly, whereas in a second period it predicts the returns very well. According to our approach of identifying persistent time periods, the presented FoHF may have changed its strategy or, at least, was not in line with the considered peer groups reflected by the style indices. We expect from the style analysis an identification of the investment strategy which may be applied in the near future. If it can be done based on past returns, the analysis must involve the most recent period with a persistent investment strategy. For the example in Figure, we thus select the period from May 2000 until December 2007. Since we do not expect any anomalies for this time period, we can fit the multifactor model by ordinary least squares. In few cases of the considered FoHF we identified structural breaks in the very near past and hence our method is not applicable. There are other rare cases where we observed single outliers, or two consecutive outliers of opposite sign. In the latter case, we suspect that a too optimistic NAV estimation was adjusted in the following period. In the former case, we suspect that the FoHF returns are affected by singular events like a collapse of FoHF position. In both situations, we remove the outliers and apply the second part of our method, described in the following section, only on the remaining data. 2.3 LASSO - a Method for Exploring the Constraints on the Factor Loadings When fitting the multifactor model () to the returns of a FoHF within a suitable time period, usually all loadings β k take on non-zero values. Hence, each FoHFs seem to be invested to all hedge fund styles. Such a result is neither practically relevant nor statistically significant. To select the statistically significant sensitivities of hedge fund styles, we may apply one of the statistical variable selection approaches as, i.e., a stepwise elimination of the least significant styles. By applying such a statistical selection procedure, we wish actually for selecting those styles which have a practically relevant influence on the returns of the FoHF as well. In practise however, we can just hope to achieve the latter intent by running a statistical selection procedure. As a result, we obtain portfolio weights just for the statistically important style indices in the multifactor model and hence, the actual returns are most closely replicated by a parsimonious selection of hedge fund style indices. In this paper, we propose to use the LASSO (least absolute shrinkage and selection operator) approach of TIBSHIRANI (996) for the selection step because it helps to explore the effects of including the constraints in the fitting process as well. The LASSO approach works as follow: Fit the multifactor model by least squares subject to the constraint that the absolute values of the factor loadings β sum up to a given bound b. If b is greater than or equal to the sum of the 6

absolute values of the ordinary least squares estimator, then that estimator is, of course, unchanged by the LASSO. For smaller values of b, the LASSO shrinks the estimated factor loadings β towards zero, typically by setting some of the loadings β k equal to zero. Thus, the LASSO combines characteristics of a shrinkage estimator like ridge regression and subset selection. The LARS algorithm described in detail in Efron, Hastie, Johnstone and Tibshirani (2004) simultaneously solves the entire set of lasso equations of the multifactor model () for all values of the bound b at the same computational cost as a least squares fit. [4] These solutions are shown in Figure 2 for one of the FoHF. At the right end of the graphic, there is the full least-squares (LS) solution (without constraints) of fitting the multifactor model (). On the right half (right of b=0.709) of the graphic, there are always some fitted portfolio weights which are of negative value. Figure 2: All LASSO solutions for the portfolio weights β. Fitted portfolio weights affiliated to the same style index are connected by lines and identified by numbers on the right hand side of the graphic. 0.3 5 Portfolio Weights 0. 8 4 0 2 3 2 7 9 Style Index -0. - 0.5 0.709.5 LS Bound b 6 When the bound b is set to and all the β k are positive, the LASSO fit is identical to a fit of the multifactor model () under the constraints (2) and (3). This shows the great similarity between the LASSO model and model () including both constraints.[5] On the other hand, if the constraint (3) is enforced and not inherent in the data, some of the factor loadings β k will be negative in the LASSO fit. Such negative values of factor loadings are statistically significant in the sense of factor selection. Hence, the LASSO selection approach results in a solution which is economically faulty according to the multifactor model () with the constraints (3) and irrespective of constraint (2); i.e. data and economic theory diverge. By a slight modification, the LASSO approach can however help to explore the impact of the constraints. Instead of fixing the bound to or not fixing at all, we choose the bound b as large as possible[6] under the constraint that all factor loadings β are non-negative. We may end up with a 7

solution where the β s do not sum up to but to a lower or larger value. A value of the bound b smaller than may indicate that the considered FoHF is not invested completely according the strategies suggested by the style indices. In section 4 will see that FoHF can have β s higher than because of leverage. Similar to the ordinary least-squares regression analysis, it is possible to calculate the coefficient of determination, also called R-square, for the LASSO fit. It is defined as the proportion of the variance explained by the LASSO fit of the multifactor model compared to the total FoHF s variance and is calculated as R 2 = var(e t )/var(r t ). To adjust for the numbers of involved 2 n 2 factors, the modified expression R adj = ( R ) is used, where q is the number of n q selected factors. Because the LASSO fit is an ordinary least-squares fit but considering constraints, the R-square of the LASSO fit is smaller than that of an ordinary least-squares fit, generally. As it can be observed in Section 4, the values of R-square are correlated with the optimal amount of the bound b. So far, our presentation of style analysis has given no information on the uncertainty surrounding the point estimations of the loadings which indicate the style exposures of a given FoHF. The presence of constraints in the fitting procedure considerably complicates the task of obtaining distributional information of the fitted loadings, however. Because the constraint in the LASSO approach forces some of the loading to be zero, we expect non-gaussian distributional characteristics of fitted loadings. Knight and Fu (2000) showed that bootstrapping, a resampling technique, can be used to obtain the (asymptotic) distribution of the fitted loadings. The R, I, L I is constructed from bootstrapping sample ( ) t, t, p, t t= L,, T p R t α ˆ + β ˆ ki k, t k= = + E t with E,L, E Tt sampled with replacement from the residuals and α ˆ,ˆβ, L, β ˆ p fitted to the original data by our modified LASSO approach. We then obtain a bootstrap estimation α ˆ,ˆβ, L, β ˆ p of α, β, Lβ p by applying our modified LASSO procedure to the bootstrap sample. These steps are repeated N times to obtain the N bootstrap estimations ( α ˆ,ˆβ, L,ˆβ p ). Knight and Fu (2000) showed that the empirical distribution of the j= L,, N bootstrap estimations approximates the true (asymptotic) distribution of the fitted factor loadings β ˆ, L, ˆ. β p 3 Data and Indices In this section we first discuss the potential families of style indices of hedge funds which will be at the heart of the multifactor model and then we describe the database of FoHF which are analysed by the modified LASSO approach. 8

3. Families of Style Indices for Hedge Funds Following Sharpe s recommendation, the factors, i.e. in our case the style indices, should be mutually exclusive, exhaustive with respect to the investment universe, and have differing returns. If some style indices are too closely correlated or not mutually exclusive, the multifactor model will yield unstable or oscillating results as a function of time. Likewise, if the set of indices does not span the investment universe, the methodology will fail in identifying a benchmark that consistently explains the fund s behaviour. Whether any existing hedge fund index family can satisfy such requirements remains to be shown. However, the return patterns of different style indices should show distinct patterns since otherwise it would not be economically sensible to define different styles. To explore this side of our approach further, the style analysis for each FoHF is completed with three different index families: Credit Suisse/Tremont, EDHEC and HFR [5]. The Credit Suisse/Tremont and the HFR Indices are internationally well recognised since the early nineties and serve as benchmarks for the hedge fund performance. The EDHEC indices were introduced in 2004 as indices of indices and are statistically constructed by taking the first component according to the principal component method of several existing indices. The choice of the EDHEC indices is evident, since it is a kind of super index which encompasses the main risk characteristic of all the underlying indices; it qualifies particularly well to explain the return pattern of FoHF. All other indices are prone of a selection bias because they contain only a small fraction of the hedge fund universe, and they are only partially representative for the hedge funds per se. The Credit Suisse/Tremont and the HFR indices also differ in their method of construction. While the former weights funds corresponding to their capital, the latter weights all funds equally in the construction of the indices. Table lists the three index families and their style indices (LHABITANT 2004, p). At least formally, the style indices correspond to each other as far as possible. There are only two style indices, Relative Value (RV), which has no correspondence with the Credit Suisse/Tremont index family, and Managed Futures (MF), which has no correspondence with the HFR index family. Effectively, the classification of the hedge funds is mostly done by self-declaration and the subindices of each family might comprise different funds so that the risk pattern of the style indices can substantially differ between index families. Furthermore, the HFR index family encompasses further hedge fund styles which have no correspondence with the Credit Suisse/Tremont indices and are therefore not considered by EDHEC and our own analyses. 9

Table : Index families and styles Style group Relative Value Event Driven Style Credit Suisse / Tremont EDHEC HFR Relative Value RV X X Convertible Arbitrage CA X X X Fixed Income Arbitrage FIA X X X Equity Market Neutral EMN X X X Event Driven Multi Strategy ED X X X Distressed Securities EDD X X X Merger Arbitrage EDRA X X X Long Short Equity LSE X X X Global Macro GM X X X Emerging Markets EM X X X Managed Futures MF X X Short Selling SS X X X 3.2 The hedgegate database The style analysis specified above was applied to FoHF of the hedgegate database (www.hedgegate.com [9]). Hedgegate is a unique database for FoHF containing nearly all products that are registered by Swiss legal authorities. Encompassing FoHF according to the Swiss mutual fund law, Swiss investment companies, special foundations for pension plans, hedgegate also includes offshore FoHF which are of interest for Swiss investors. Table 2 shows the total number of funds classified by currency and strategy as of February 2008. Many master funds are available in different currency-share-classes. The Swiss franc and Euro shares usually are nearly fully hedged against the original USD exposure. Table 2: Number of funds of hedge funds as of February 2008: Swiss registered / off shore domiciled. Source: www.hedgegate.com Diversified Focussed Focussed non directional directional Total CHF 54 / 6 2 / 2 4 / - 70 / 8 EUR 6 / 28 5 / 6 / - 82 / 39 GBP 4 / 5 / - / - 6 / 5 JPY 4 / / - - / - 5 / USD 72 / 50 6 / 8 7 / 4 95 / 72 Total 95 / 00 45 / 3 8 / 4 258 / 35 The FoHF are classified by the styles of their target funds (cf. Table 3 ) into diversified, focussed directional and focussed non-directional. Hence, this classification is done on the basis of the release of information by the FoHF manager and not on the return structures. The results of the style analysis will be presented in the order of this classification. So it can be seen, how far the style analysis is able to discriminate between the three FoHF strategies assuming that the a priori classification is correct. On the other hand, assuming that the style analysis is accurate, it allows checking if FoHF are properly classified. 0

Table 3: ZHAW-Classification of FoHF strategies Relative Value Event Driven Convertible Arbitrage Fixed Income Arbitrage Equity Market Neutral Merger Arbitrage Distressed Securities Long-Short Equity Global Macro Emerging Markets Managed Futures Focussed non-directional Focussed directional Diversified The style analyses in this paper are applied only to the FoHF in USD of the hedgegate database. This procedure was chosen for two reasons: First, most of the funds have a strong economic exposure in USD. The second reason is that an additional currency factor must be introduced in the multi- factor model for non-usd funds to capture the foreign currency-exposures. Such an extension of the multifactor model can be considered for future research. Some of the FoHF have closed-end structures and are listed investment companies at the Swiss stock exchange. The analysis for the closed-end funds is done with the returns based on their net asset values (NAV) and not on stock prices, which can substantially differ from the NAV. Since only NAV are used for this analysis, the closed-end puzzle is of no relevance. 4 Results The factor loadings of the analysed FoHF, estimated by our style analysis procedure, are summarized graphically by boxplots in Figure 3 5 according to the a priori classification focussed non-directional, focussed directional and diversified. All the FoHF are in USD and are either Swiss registered or off shore domiciled which should, however, not be relevant for the style analysis. Forty out of the 67 FoHF have been dropped from the analysis because they had less than 36 month of reported NAVs. Hence, 27 FoHF are available for the analysis. However, 5 FoHF have been dropped from the final analysis because the residual analysis indicated instability or missing style persistence over relevant estimation periods using any of the index families. In 2 cases of the remaining FoHF, no persistent investment style could be identified in the most recent period for a single index family. Otherwise, the considered analysis periods do vary according to the identification strategy for the persistency of the investment style introduced in Subsection 2.2. The periods always end as of December 2007 and have at least 36 months included. Applying our modified LASSO approach of subsection 2.3 to these FoHF, we identified b=0 in 36 cases. That is, it was not possible to fit the multifactor model without having at least one negative factor loading. Looking at the b=0 FoHF reveals that most of them follow very specific and focussed strategies as industry or geographically specific long-short equity, and evidently the broad style indices are not capable to explain the return behaviour of these specific investment objects.

In Table 4, the number of excluded FoHF and the number of finally analysed FoHF are summarized for each index family and each a priori classification. With respect to the analysed FoHF, the HFR index family is the most successful family, followed by EDHEC. The three index families are about equally successful in the number of identifications of the persistent investment period, but their success may differ with respect to a specific FoHF. There are only a few FoHF where the identification of the persistent investment period failed with more than one index family. The big differences in the number of finally analysed FoHF have their roots in the modified LASSO approach. Using the HFR index family, our approach could identify investments strategies in all of the FoHF except in 4 of them; i.e., b equalled 0 with 4 FoHF. Using the EDHEC index families, the number of failures doubled, namely b equalled 0 in 9 cases. Finally using the CS/T index family, the number of failures increased to 23 FoHF which is six times more than with HFR. Table 4: Overview of the number of excluded FoHF and of the number of FoHF finally summarized in the boxplots Index A priori FoFH 2 Excluded Analysed family classification Not persistent b=0 d 93 0 3 70 CS/T fd 25 7 8 0 fnd 9 0 2 7 d 93 4 6 73 EDHEC fd 25 7 2 6 fnd 9 0 8 d 93 2 80 HFR fd 25 8 6 fnd 9 0 8 Diversified (d), focussed non-directional (fnd), focussed directional (fd) 2 Number of FoHF in USD with at least 36 months of recorded NAVs 2

Figure 3: Factor loadings of the focused non-directional FoHF Non-.2 CSFB/T RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared Non-.2 EDHEC RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared Non-.2 HFR RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared Figure 3 exhibits the focussed non-directional FoHF. Conceptually, the returns of these funds should be explained by those style indices which are defined as non-directional. Effectively, the sums of the loadings β predominantly come from the relevant non-directional style indices as it can be clearly observed in Figure 3. The results which most agree with the a priori classification are based on the HFR index family. The use of the other index families results in less clear patterns. The styles Long Short Equity (LSE) and, for EDHEC, Short Selling (SS) show some significant contributions for which many explanations can be plausible. Event driven strategies are products overlapping with long short equity or use equity short selling for hedging purposes. The style Managed Futures (MF), which is missing within the HFR index family, seems not to be relevant for focused non-directional FoHF as the results of using EDHEC or CS/T index family indicate. With respect to the predominant contribution of the non-directional styles, there is little agreement among the three index families. As the style Relative Value (RV) has some significant contribution within the EDHEC and the HFR style index families, we may trace back some of the differences between the CS/T index family and the other index families on the missing style within the CS/T index family. But EDHEC and HFR neither agree on the importance of merger arbitrage (EDRA) which shows that index families have to some degree different return patterns as a result of different constituencies and construction methods. The adjusted R-squared values (called R-Squared in the graphics) are between 5 and 0.90 which is of similar magnitude as those reported in FUNG and HSIEH (2004), where asset-based multifactor models are fitted to the HFR FoHF index for different time periods. All three index families indicate little leveraging. 3

Figure 4: Factor loadings of the focused directional FoHF Non-.4.2.5 0.5 CSFB/T RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared.4.2 Non-.5 0.5 EDHEC RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared.4.2 Non-.5 0.5 HFR RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared For the focused directional FoHF in Figure 4, we would expect that the directional style indices should show up covering most of the β exposure. This is almost true for the CS/T index family. The HFR and the EDHEC index families still show some significant contributions from nondirectional styles, particularly from Event Driven Merger Arbitrage (EDRA) and, with HFR, Relative Value (RV). One reason for this anomaly could be found in the fuzziness between event driven strategies and long-short equity strategies. Many long-short managers also exploit merger announcements or restructuring situations. With respect to the contributions of the different directional style indices, all three analyses agree in general. The missing Managed Future (MF) contribution is replaced by Global Macro and maybe some non-directional style contributions in the HFR index family. With median R-squared lying around 0.7 as the boxplots indicate, the multifactor models show the capability to explain the returns of focused directional FoHF as well as non-directional. With respect to the b value, we obtain different indications from the CS/T index family and the other two index families. The former indicates, that there is little leveraging whereas the latter two show some moderate leveraging. Three quarter of the FoHF have leveraging. In a few cases, even some single factor loadings are larger than one. Since leverage or style exposure can be achieved on the target single fund level and/or on the FoHF level an interpretation of these results is very difficult. 4

Figure 5: Factor loadings of the diversified FoHF Non-.5 0.5 4 3 2 CSFB/T 0 RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared.5 0.5 Non- 4 3 2 EDHEC 0 RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared.5 0.5 Non- 4 3 2 HFR 0 RV CA FIA EMN ED EDD EDRA LSE GM EM MF SS b R-Squared Figure 5 exhibits the diversified FoHF. Basically, the returns of these funds are explained by any combination of directional and non-directional style indices. The bounds b are rather large for some FoHF (up to 4) and larger than for most of the FoHF as the boxplots in Figure 5 reveal indicating that many FoHF are leveraged with respect to the style index factors. Again some single factor loadings are larger than one. From the hedgegate database we know that many diversified FoHF have maximal leverage ratios according to the prospect between 25% and 250%, but there is no information available about the effective leverages on the level of the FoHF. The R-squared is similar to the previous results except for a few FoHF. Table 5: Mean of sum of β (b), where b is positive CS/T EDHEC HFR Focussed non directional 0.79 0.92 0.94 Focussed directional 0.90.0.7 Diversified 7.27.27 β exposures for the same FoHF, but different index families can substantially differ, but at the same time they are closer for one and the same FoHF than across different FoHF. Since each index family is based on different hedge fund universes, index construction method, style classifications and procedures, differences must come up. Table 5 shows that the sums of β are highest across the FoHF for HFR and lowest for Credit Suisse/Tremont. The latter might be different from other hedge fund indices since it is capitalization weighted and not equally weighted as EDHEC and HFR indices. So by construction the Credit Suisse/Tremont index seems to be less suitable to explain exposures of FoHF than the EDHEC and the HFR indices 5

which are weighted equally, and therefore closer reflect the portfolio construction of most of the FoHF [8]. 5 Conclusions Our index-based style analysis aims for identifying the characteristic hedge fund style exposures of complex FoHF-returns. As we are primarily interested in assessing the investment strategies of FoHF, we focus on the sum of the loadings of the style indices. A sum close to one indicates that the FoHF is invested in the hedge fund market as it is represented by the corresponding family of hedge fund style indices. Furthermore, the loadings of a FoHF have several practical implications. They reveal how focussed a FoHF is, when its return can be mainly explained by only few style indices, or how diversified it is, when many style factors contribute to the explanation. Sums of loadings greater than one indicate leverage, low sums indicate that a FoHF has no typical risk characteristics of hedge funds as indicated by the different hedge fund styleindices, or these characteristics do not show up in a stable manner during the considered period. For a few FoHF, our index-based style analysis is not reasonably applicable, because the residuals show too many irregularities. Hence, we may question the strategy descriptions by the management of the relevant FoHF. The index-based style analysis also widely confirms the ZHAW-classification of FoHF. The diversified FoHF are best explained by this method showing the highest sums of loadings. Although, focussed directional FoHF show up with the expected exposures, the overall factor loadings are low, because many of this focussed FoHF follow very specific strategies which are not covered adequately by the available style indices. Obviously, the analysis could be enlarged with additional and more specific style indices, once the FoHF industry becomes more differentiated, and this would improve the explanatory power for such focussed funds. Finally, our analysis showed that, from the tested index families, HFR and EDHEC index families are best to explain the returns of the investigated FoHF. Methodologically, the style index-based multifactor model is a way to learn about the investment style mix of specific FoHF by listening to the data without imposing strong a priori restrictions. The modified LASSO-method selects the important style indices and imposes sound parameter restrictions by shrinking the factor loadings. Another important step is the appropriate selection of the time period, to which the multifactor model is fitted. Although the hedge funds may run highly dynamic strategies and may change them at much higher frequency than monthly, we experienced that the style-based multifactor model is able to capture the essential elements of the strategies. But sometimes, the style-based multifactor model identifies strong irregularities which hint to shifts in the investment strategies or changes of the investment process. Such discontinuities must be scrutinised separately and the multifactor model will be fitted to periods free of such irregularities. A FoHF investor might also use the style indices loadings to avoid style risk concentration investing in several FoHF or he could also find particular style exposures. So our index-based style analysis is supposed to be a useful additional tool for the selection of FoHF. 6

FOOTNOTES [] hedgegate (database (www.hedgegate.com) is a unique database for FoHF containing nearly all products that are registered by Swiss legal authorities. [2] We used the open source software R, a language and environment for statistical computing (cf. R Development Core Team, 2006) and therein the function rlm() of the package MASS (cf. VENABLES and RIPLEY, 2002). [3] 99% of the normally distributed data are within the interval mean ± 2.5 standard deviation. Hence, we expect 0.9 data points out of 72 (i.e., six years of monthly data) outside the interval. [4] The LASSO solutions are computed with the open source software R (cf. R Development Core Team, 2006) and therein the function lars() of the package lars as it is described in EFRON et al. (2004). With the "lasso" option, it computes the complete lasso solution simultaneously for ALL values of the bound b in the same computational cost as a least squares fit. [5] Furthermore, it follows in the case of positive loadings that the least-squares fit of model () under both constraints is a shrinkage estimator. [6] If constraint (2) must be satisfied, the bound b is chosen as large as possible but not exceeding. [7] cf. http://www.hedgeindex.com for Credit Suisse/Tremont, http://www.edhecrisk.com/indexes for EDHEC, and http://www.hfr.com for HFR (Hedge Fund Research). [8] cf. also the analysis of Credit Suisse/Tremont indices in FUNG and HSIEH (2004), p 74. [9] www.hedgegate.com is a public website and needs only cost-free registration for the access to offshore FoHF. REFERENCES EFRON, B., T. HASTIE, I. JOHNSTONE and R. TIBSHIRANI (2004): Least Angle Regression, Annals of Statistics (with discussion), 32(2), pp 407-499. FUNG, W., and D.A. HSIEH (2004): Hedge Fund Benchmarks: A Risk-Based Approach, Financial Analyst Journal, 60 (5), pp 65-80. FUNG, W., and D.A. HSIEH (2007): Will Hedge Funds Regress Towards Index-Like Products? Journal of Investment Management, Vol. 5, No. 2 GEHIN, W. (2006), Hedge Fund Returns: An Overview of Return-Based and Asset-Based Style Factors, EDHEC, January (www.edhec.com) HASTIE, T, R. TIBSHIRANI and J. FRIEDMAN (200): Elements of Statistical Learning: Data Mining, Inference and Prediction, New York: Springer-Verlag. KNIGHT, K, W. FU (2000): Asymptotics for lasso-type estimators, The Annals of Statistics, 28 (5), pp 356-378. LHABITANT, F.S. (200): Assessing Market Risk for Hedge Funds and Hedge Funds Portfolios, Journal of Risk Finance, Springer, pp -7. LHABITANT, F.S. (2004): Hedge Funds: Quantitative Insights, Chichester: John Wiley & Sons Ltd. 7

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