Regular(ized) Hedge Fund Clones

Transcription

1 Regular(zed) Hedge Fund Clones Danel Gamourds a,1, Sandra Paterln b a Department of Accountng and Fnance, Athens Unversty of Economcs and Busness, Athens, Greece b Department of Economcs, Unversty of Modena and Reggo E., Modena, Italy Frst draft: October 28, 2008 Ths draft: January 6, 2008 Abstract Ths artcle addresses the problem of portfolo constructon n the context of effcent hedge fund nvestments replcaton. We propose a modfcaton to the standard à la Sharpe style analyss where we augment the objectve functon wth a penalty proportonal to the sum of the absolute values of the replcatng asset weghts,.e. the norm of the asset weghts vector. Ths penalty regularzes the optmzaton problem, wth sgnfcant mpacts on the stablty of the resultng asset mx and the rsk and return characterstcs of the replcatng portfolo. Our results suggest that the norm-constraned replcatng portfolos exhbt sgnfcant correlatons wth ther benchmarks, often hgher than 0.9, have a fracton,.e. about 1/2 to 2/3, of actve postons relatve to those determned through the standard method, and are obtaned wth turnover whch s n some nstances about 1/4 of that for the standard method. Moreover, the extreme rsk of the replcatng portfolos obtaned through the regularzaton method s always lower than that exhbted by currently avalable commercal hedge fund nvestment replcaton products. Danel Gamourds s at the Department of Accountng and Fnance n the Athens Unversty of Economcs and Busness, Athens, Greece. He s also Vstng Fellow at Cass Busness School, Cty Unversty, London, UK, and, Research Assocate at EDHEC Rsk and Asset Management Research Centre, EDHEC Busness School, Nce, France. Sandra Paterln s at the Department of Economcs, Unversty of Modena and Reggo E., Modena, Italy. She s also Senor Research Fellow at the Center for Quanttatve Rsk Analyss, Department of Statstcs, LMU Munch, Germany. We would lke to thank Alesso Matteuzz (Ct) for hs valuable comments on practcal ssues. The paper has benefted from the comments of Andrea Rest and one referee, both from Carefn-Boccon and Lonel Marteln. The paper has also benefted from the comments of partcpants at the followng meetngs: Unversty of Mnnesota School of Mathematcs, Fnancal Mathematcs Semnar Seres. Fnancal support from Carefn-Boccon Centre for Appled Research n Fnance s greatly apprecated. Any remanng errors are our own. 1 Correspondng author. Athens Unversty of Economcs and Busness, Department of Accountng and Fnance, 76 Patsson Street, GR Athens, Greece. Tel: ; fax: E-mal address: dgamour@aueb.gr (D.Gamourds) 1

2 Regular(zed) Hedge Fund Clones Abstract Ths artcle addresses the problem of portfolo constructon n the context of effcent hedge fund nvestments replcaton. We propose a modfcaton to the standard à la Sharpe style analyss where we augment the objectve functon wth a penalty proportonal to the sum of the absolute values of the replcatng asset weghts,.e. the norm of the asset weghts vector. Ths penalty regularzes the optmzaton problem, wth sgnfcant mpacts on the stablty of the resultng asset mx and the rsk and return characterstcs of the replcatng portfolo. Our results suggest that the norm-constraned replcatng portfolos exhbt sgnfcant correlatons wth ther benchmarks, often hgher than 0.9, have a fracton,.e. about 1/2 to 2/3, of actve postons relatve to those determned through the standard method, and are obtaned wth turnover whch s n some nstances about 1/4 of that for the standard method. Moreover, the extreme rsk of the replcatng portfolos obtaned through the regularzaton method s always lower than that exhbted by currently avalable commercal hedge fund nvestment replcaton products. Keywords: Hedge Funds; Replcaton; Portfolo Management; Index Trackng; Norm-constraned portfolos JEL Classfcaton: G11; G12; C00 2

3 1. Introducton The last couple of decades have wtnessed a rapdly growng nterest n hedge funds. Relatve to tradtonal nvestment portfolos, hedge funds exhbt some unque characterstcs: they are flexble wth respect to the types of securtes they hold and the type of postons they take; they are not subject to publc dsclosure of ther actvtes; and they are not evaluated aganst a passve benchmark. Ths set up encourages hedge fund managers to construct hghly dynamc, complex tradng strateges and, as a result, expose ther portfolos to a plethora of economc rsk factors. Not surprsngly the return on alternatve nvestments has shown lttle correlaton wth returns on tradtonal assets such as stocks and bonds. Moreover, n many cases returns on hedge fund nvestments have also been sgnfcantly hgh 2. Whether the latter characterstcs are due to skll or due to non-conventonal technques such as short sellng, leverage and the use of dervatves employed by hedge fund managers has been debated by many researchers. At the core of ths strand of lterature are studes such as Agarwal and Nak (2000) and Fung et al. (2006) whch conclude that the alpha 3 of the average hedge fund or fund of funds manager s very poor and not persstent. Khadan and Lo (2008) n a dfferent context hghlght that the fact that the entre class of long/short equty strateges moved together so tghtly durng August 2007 s consstent wth the hypothess of certan exstng common factors wthn that class. In other words the lterature suggests that hedge fund performance on average can be attrbuted prmarly to alternatve betas rather than skll. 2 Hedge Fund Research n ts Second Quarter 2007 Hedge Fund Industry Report whch covers the perod January 1997 through to the second quarter of 2007, reports a twelve-month correlaton between the overall hedge fund market and S&P 500 of 0.50 on average. The twelve-month correlaton between the overall hedge fund market and Lehman Government/Credt Index s at on average. Annual Sharpe ratos are 3.58 for the overall hedge fund market and 2.12 and 0.41 for the S&P 500 and Lehman Government/Credt Index respectvely. 3 The component of performance that s attrbuted to skll,.e. the ntercept n the regresson of the fund s excess return on the excess return of one or more passve benchmarks. 3

4 And ths trggers a natural queston: Can synthetc hedge funds be created at a lower cost to nvestors? The benefts from synthetc hedge funds are several, the most obvous relatng to cost, lqudty, transparency, and barrers to entry. A synthetc hedge fund product based on futures, total return swaps, ETFs, or other nstruments may nvolve much lower costs compared to the 2/20 fee structure,.e. 2% management fee and 20% success fee, whch s more or less the hedge fund ndustry s standard. Moreover, replcaton through exchange traded assets provdes sgnfcant lqudty. Hedge funds typcally requre at least one month notce before redempton and many of them have long lock-up perods. In addton, a synthetc structure s transparent as far as the underlyng holdngs are concerned. Fund nvestors on the other hand know very lttle on the allocaton of ther money,.e. whch assets/tradng strateges ther money s nvested on. Fnally, synthetc products may be open for nvestment to small nvestors or portfolos wshng to allocate small amounts n the hedge fund unverse. Many hedge funds requre a mnmum nvestment between $500,000 and $1,000,000 whle some funds are not open to new nvestors. These observatons have attracted the nterest of both academcs and practtoners and have stmulated a quest to develop effectve hedge fund replcaton strateges. Techncally, the development of synthetc hedge fund products can be related to the so-called ndex trackng problem. Index trackng s a form of passve management where the am s to reproduce as close as possble a stock market ndex or the performance of a mutual fund by usng a subset of assets or market ndexes (see, e.g. Corell and Marcellno, 2006). 4

5 In terms of hedge fund replcaton the current lterature focuses on two dfferent approaches 4 : moment matchng and factor based replcaton. Moment matchng has been ntroduced and thoroughly studed by Kat and Palaro (2005a, b) and used by Amenc et al. (2007), and Papageorgou et al. (2008). The core dea s developed n Amn and Kat (2003) n the context of hedge fund performance measurement. Central to ths approach s the conjecture that the return profle of an nvestment strategy can be replcated f ts rsk profle,.e. volatlty, skewness, kurtoss, correlaton to a specfed portfolo, are matched. The replcatng strategy nvolves holdngs on: a) a reserve portfolo - the return drver - whch s always a long poston, and b) the nvestor s portfolo, whch s the reference for the correlaton target wth a potental short poston. Whle favourable emprcal propertes of ths approach are acknowledged n the lterature several crtcsms are put forward (see, e.g. Amenc et al., 2007). The most mportant among them s that the probablty dstrbutons change and the replcatng portfolo may not be able to track those changes. It s also argued that that the frst moment may not match automatcally when matchng the rsk profle. Factor based replcaton has so far ganed sgnfcant consensus and s currently to our knowledge the most wdely appled methodology n the practce of hedge fund replcaton 5. Hasanhodzc and Lo (2007) provde a detaled overvew of ths approach as well as the results of a very comprehensve emprcal analyss. Bascally, factor based replcaton bulds upon Sharpe s 4 One thrd approach termed securty based replcaton s based on the mplementaton of a generc verson of a gven strategy. Fung and Hseh (2001) for example use lookback straddles to model trend followng strateges whch are typcal for CTAs. Schneewes and Kazem (2007) take long/short postons n commodty and fnancal futures usng movng average sgnals to replcate trend-followng CTAs. Mtchell and Pulvno (2001) study the rsk/return profle of the merger arbtrage strategy through long portfolos of target frms and short postons n acqurng frms. Convertble Arbtrage s replcated through long postons n convertble bonds and short postons n equtes n Agarwal et al. (2006) whle fxed ncome arbtrage s studed n Duarte et al. (2007). Securty based replcaton however may not be that dfferent from a typcal hedge fund. 5 Meryll Lynch s Equty Volatlty Arbtrage Index and Factor Index for example buld on the noton of factor based replcaton and so do Deutsche Bank s Absolute Return Beta Index, Goldman Sachs Absolute Return Tracker Index as well as Credt Susse s recent alternatve beta ntatve. 5

6 (1992) style analyss,.e. a constraned beta lnear regresson on a gven set of factors to buld the best mmckng portfolo. Hasanhodzc and Lo (2007) show that ths approach works well wthn the Equty Market Neutral, Global Macro, Long/Short Equty Hedge, Managed Futures, Mult- Strategy, and Fund of Funds space. Replcaton s not however successful for Event Drven and Emergng Markets funds. Jeager and Wagner (2005), Amenc et al. (2007), and Amenc et al. (2008) provde addtonal evdence for the factor based approach. Despte the attractveness of factor-based hedge fund replcaton, t has not yet to our knowledge consdered mportant emprcal conclusons from the hedge fund rsk lterature. Many studes (see, e.g. Agarwal and Nak, 2004; Fung and Hseh, 2000, 2001; Lang, 1999, Vrontos et al. 2008) argue that the true set of hedge fund rsk factors s vrtually unknown due to the lack of transparency and the large number of possble market and tradng strategy combnatons a manager can follow and use statstcal technques to dentfy mportant factors. Factor-based hedge fund replcaton has thus far consdered a fxed set of factors, typcally a small set, and reled on optmzaton to determne the amount by whch an asset contrbutes to the mmckng portfolo. As we show n our emprcal analyss ths approach tends to allocate captal to all assets n the orgnal unverse thus falng to dstngush n an effcent way between mportant and less mportant hedge fund return drvers. Ths s an extremely crtcal ssue when the possble number of factors s large. In addton, the constraned regresson seems to have been appled n a rather tradtonal way wthout accountng for the crtcsms and remedes related to such problems. As we show n our emprcal analyss standard constraned optmzaton of the weghts n the mmckng portfolo results n a very unstable mx over tme. Moreover many of the assets n the replcatng portfolo possess extreme weghts. Fnally, current studes have to our knowledge not looked at the 6

7 propertes of the mmckng portfolo n detal. None of the studes we know are concerned wth the cost of mplementaton of the replcatng strategy or the extreme rsk of the mmckng portfolo for example. After all, settng up a hedge fund clone nvestment programme nvolves tactcal tradng of the underlyng factors as well as contnuous montorng and management of ts rsk. To fll the gaps outlned above, we propose a new methodology for the constructon of hedge fund clones. Our approach reles on mnmzng a penalzed verson of the trackng error volatlty between the hedge fund ndex and the clone. The penalty we consder s proportonal to the sum of the absolute values of the replcatng asset weghts,.e. the norm of the asset weghts vector. Ths penalty regularzes the optmzaton problem and jontly addresses model selecton and estmaton error. In the context of portfolo constructon a problem that s analogous to the constructon of mmckng portfolos norm-constraned portfolos have shown to possess superor propertes compared to those obtaned from conventonal approaches but also compared to those obtaned from a number of alternatve mproved strateges proposed n the lterature (see, e.g. Brode et al. 2007, DeMguel et al., 2007). It can be shown that our approach s a specal case of the standard à la Sharpe style analyss. In our sample the factor-based hedge fund clones are extremely stable n terms of the exposures to systematc rsks and possess mproved rsk and return characterstcs. In partcular, the norm-constraned replcatng portfolos exhbt sgnfcant correlatons wth ther benchmarks, often hgher than 0.9, have a fracton,.e. about 1/2 to 2/3, of actve postons relatve to those determned through the standard method, and are obtaned wth turnover whch s n some nstances about 1/4 of that for the standard method. Moreover, the extreme rsk of the replcatng portfolos obtaned through the regularzaton 7

8 method s always lower than that exhbted by currently avalable commercal hedge fund nvestment replcaton products. The contrbutons of ths artcle are several. Frst, we propose a method that has been successfully appled n portfolo constructon studes for the constructon of hedge fund clones. Second we study for the frst tme the propertes of the clones n a broader framework whch extends to the practcal mplementaton of the developed strategy. Thrd we study the out-of-sample propertes of the clone from the perspectve of a structured products manager who has not only to create the clone but also to manage ts rsks as well as the transacton costs of the replcatng strategy. Fourth we provde addtonal evdence for the use of regularzaton methods n emprcal fnance. The paper s organzed as follows. In Secton 2 we descrbe the proposed methodology that s used to determne the synthess of the replcatng portfolo. Secton 3 dscusses the data and the structure of our emprcal experments. In Secton 4 we present the results of the model selecton analyss. Secton 5 presents the results from the out-of-sample analyss of the propertes of the mmckng portfolos as well as the robustness checks for our results and Secton 6 concludes. 2. Methodologes Our approach falls n the class of factor based replcaton methods. As n Sharpe (1992) and Hasanhodzc and Lo (2007) among others, we use a decomposton of the return tme seres r of a hedge fund ndex nto several components factors and an dosyncratc component ε, accordng to: r= Fβ+ ε (1) In Equaton (1), r = (r t,...,r T ) s the (T-t+1)x1 vector of a hedge fund ndex return tme seres, F s the (T-t+1)xN matrx whose k th (k = t,,t) row s the 1xN vector f t of returns of the N factors at 8

9 tme k,.e. F = (f k ), β = (β 1,..β N ) T s the Nx1 vector of factor loadngs n the tme perod [t,t] and ε=(ε t,...,ε T ) s the (T-t+1)x1 vector of the dosyncratc component. To construct a mmckng portfolo for a hedge fund ndex wth ncreased trackng ablty t s necessary to dentfy the approprate replcatng factors frst. The standard approach nvolves a fxed set of factors whch are assumed a pror to be approprate for replcatng the returns of hedge fund ndex returns. Hasanhodzc and Lo (2007) for example undertake ther analyss wth fve factors,.e. proxes for the equty, bond, currency, credt, and commodty markets. Alternatvely, the hedge fund rsk lterature has suggested that typcal model selecton approaches such as stepwse regresson, selecton based on Akake s (1973) nformaton crteron or the Schwarz s (1978) Bayesan nformaton crteron (see, e.g. Agarwal and Nak, 2004, Vrontos et al. 2008) may be used to dentfy approprate factors. Wth the chosen set of factors, the composton of the replcatng portfolos can be determned through a constraned regresson,.e. beta coeffcents are estmated wth the constrant that they sum up to one, whch can be expressed as: ˆ Sharpe β = mnmze r Fβ s.t. 1 β = 1 N β 2 2 (2) where 2 r s the 2-norm of vector ( r Fβ ),.e. = ( ) 2 2 Fβ 2 2 T r Fβ r Fβ, the sum of squared resduals (SSR hereafter) or trackng error varance, and 1 N s a 1xN vector of ones. The elements of β ˆ Sharpe are then used as portfolo weghts for the chosen factors, hence the replcatng portfolo returns are equvalent to the ftted values r. t= 1 t Brode et al. (2007) hghlght that n the fnancal context t s lkely that a standard numercal procedure for the optmzaton n Equaton (2) may amplfy the effects of nose ansotropcally, 9

10 leadng to an unstable and unrelable estmate of the vector β. Moreover, wth lttle ex ante knowledge regardng the exact factors that drve hedge fund ndex returns t seems approprate that a broad set s ntally consdered. Ths renders the optmzaton problem more computatonally complex and ncreases the possblty that unstable estmates of the vector β are obtaned. A comprehensve set of replcatng factors may also nvolve collnear varables even just durng certan perods,.e. n dstress markets condtons whch ental addtonal obscurty for the numercal procedure (see, Brode et al., 2007). To tackle such ssues, recent works n the portfolo constructon lterature (see, e.g. Brode et al., 2007, De Mguel et al. 2007, Lobo et al. (n press), Welsch and Zhou, 2007) have proposed the use of regularzaton procedures such as rdge regresson (Hoerl and Kennard, 1970) or lasso regresson (Tbshran, 1996). The evdence so far suggests that such methods reduce the senstvty of the optmzaton to possble collneartes between assets, control transacton costs by promotng sparsty 6, and mprove the out-of-sample performance relatve to the classcal Markowtz mnmum varance portfolo. Furthermore, regularzaton methods have shown to possess nterestng Bayesan and moment-shrnkage nterpretatons (Tbshran, 1996, De Mguel et al., 2007). Despte these appealng propertes and the analogy of the portfolo constructon and ndex trackng problems, regularzaton methods have not to our knowledge yet been nvestgated n the context of hedge fund clonng or ndex trackng n general. Our nvestgaton s based on both rdge and lasso regresson. 6 A sparse model s a model where very few assets are selected. 10

11 Rdge regresson nvolves the optmzaton of a penalzed verson of the SSR functon. The penalty term s proportonal to the 2-norm of the weghts. Ignorng optmzaton constrants for ease of exposton, we can express the rdge regresson method as: ˆ Rdge β = arg mn r Fβ + λ β β (3) We denote by x the sum p p T t= 1 x t p,.e. the p-norm of vector x. Addng the penalty n the objectve functon results n addng a postve constant to the dagonals of F T F such that the matrx F T F s non-sngular and computng the nverse does not affect the stablty of the SSR mnmzaton problem (see Appendx). Daubeches et al. (2004) have shown that any p-norm penalty (1 p 2) suffces to stablze the mnmzaton of Equaton (2) by regularzng the nverse problem. When p=2, the optmzaton problem s stll a contnuously dfferentable optmzaton problem and t can be easly tackled by conventonal technques, whle provdng more stable and accurate estmates than standard SSR mnmzaton. However, wth p=2 the coeffcents although shrunk thus stable cannot become 0. Hence, the resultng replcatng portfolo wll not be parsmonous. To accommodate ths possblty yet sustan useful propertes of regularzaton methods, Tbshran (1996) proposed to add a penalty proportonal to the 1-norm, an approach termed lasso regresson. Ignorng optmzaton constrants for ease of exposton, we can express the lasso regresson method as: ˆ Lasso β = arg mn r Fβ + λ β β (4a) 11

12 It can be easly shown (see Brode et al., 2007) that ths s equvalent to solvng the followng optmzaton problem: ˆ Lasso β = arg mn r Fβ s.t. β 1 1 t β 2 2 (4b) Due to the geometrcal nature of such constrants, the resultng procedure can gve coeffcent estmates that are exactly zero n the cases when the correspondng factors contan lttle nformaton about the hedge fund ndex (see Appendx). Thus, lasso regresson nvolves model selecton and estmaton n a sngle step. Brode et al. (2007) lst several addtonal useful consequences of the regularzaton wth the 1-norm penalty ncludng stablty, control of the short postons, and account of transacton costs. In partcular, under the budget constrant the λ penalty becomes a penalty on short postons. The larger the λ, the smaller the number of factors to be selected and the larger the number of no-short-postons n the replcatng portfolo. Transacton costs on the other hand can be controlled for through a modfed penalty term,.e. N k = 1 s β k k, where s k s the bd-ask spread for the k th securty. The choce of λ n rdge and lasso regresson s crtcal to control several aspects of the optmzaton problem. Although t s arbtrarly chosen n some nstances (see, e.g. Brode et al., 2007), more sophstcated cross-valdaton methodologes based on statstcal or economc crtera could be mplemented to determne ts value (see, e.g. De Mguel et al., 2007). 12

13 To construct a lnear clone for a hedge fund ndex we mpose the addtonal constrant that asset weghts vary between -1 and 1 7. So n the case of lasso regresson, the method we propose as most approprate for hedge fund ndex replcaton, the optmzaton problem becomes: ˆ Lasso β = arg mn r Fβ + λ β β s.t. (a) 1 β = 1 N (b) 1 β (5) When usng our approach, constrant (b) s not necessary to get realstc asset weghts snce ncreasng the value of λ, results n a decrease n the number of short postons and shrnks the portfolo weghts. However, we ntroduce ths constrant to have a more far comparson wth a standard quadratc programmng approach 8. Ths problem nests Sharpe s (1992) style analyss. In fact, strong style analyss 9 constrans the portfolo weghts to be postve and to add to one. Ths s equvalent to constranng the 1-norm of the factor loadngs to be equal to 1. Hence, t s not surprsng that mposng no short-sellng constrants on portfolo weghts as shown n Jagannathan and Ma (1993) has a shrnkng effect whch provdes better predcton accuracy. 7 β may n practce vary dependng on the manager s mandate. Gven that the replcatng portfolo s by constructon a low volatlty portfolo t should nvolve ether low beta exposures or hgh beta exposures that net each other out. Hgh values of beta however should be avoded n prncple for two man reasons. Frst, traders are averse to holdng hghly leveraged postons n sngle assets for rsk purposes. And second, mplementng strateges nvolvng hgh betas encompasses hgh transacton costs assocate wth the unwndng/creaton of large postons. From a methodologcal pont, allowng the weghts to vary n [-1, 1] allows enough space to nvestgate the stablty and assess the feasblty of the alternatve methodologes. Hence, our choce of β constrants whle by no means favour the proposed approach serve as reasonable choce from a practcal and methodologcal standpont. 8 Addng constrants (a) and (b) can affect the choce of λ n order to enforce sparsty. In fact, f no constrants are mposed on β, the 1-norm can vary between 0 and nfnty, whle f we mpose constrants (a) and (b) the 1-norm s bounded between 1 and N-1. Hence, larger λ values are requred when mposng such constrant to enforce sparsty nto the model. 9 Dependng on the constrants on the portfolo weghts, style analyss can be classfed as weak,.e. no constrants are mposed on the portfolo weghts, sem-strong,.e. only the budget constrant s consdered (the sum of portfolo weghts equals 1), and strong,.e. portfolo weghts are requred to be postve and sum up to 1 (see, e.g. Horst et al., 2004). 13

14 3. Data and emprcal desgn We carry out our analyss wth monthly hedge fund ndex return data from the Hedge Fund Research (HFR hereafter) database. Pror studes that focus on hedge fund ndex replcaton nclude Amenc et al. (2007), Amenc et al. (2008). Other studes focus on the replcaton of ndvdual funds, see for example Hasanhodzc and Lo (2007), Jeager and Wagner (2005). Our dataset covers the perod February 1990 to July We consder the four prmary strategy ndces namely Equty Hedge, Event Drven, Macro, and Relatve Value. Equty Hedge comprses managers that mantan postons both long and short n prmarly equty and equty dervatve securtes. Even Drven classfed funds are those nvestng on companes that are lkely to be nvolved n corporate transactons of a wde varety. The funds consttutng the Macro ndex engage n a broad range of strateges n whch the nvestment process bulds upon predctons of movements n underlyng economc varables and the mpact these have on equty, fxed ncome, and currency and commodty markets. Relatve Value managers speculate on realzaton of a valuaton dscrepancy n the relatonshp between multple securtes. In addton to the prmary strategy ndces we consder an ndex wth a regonal focus, namely the Emergng Markets ndex. Ths ndex comprses managers ndependent of ther nvestment strategy. As a proxy for the aggregate market we consder the Fund Weghted Composte Index whch s an equal-weghted ndex made of over 2,000 ndvdual funds. Fnally, we consder a proxy for the overall Fund of Funds market, the Fund of Funds Composte Index whch comprses over 800 ndvdual Fund of Funds based on equal-weghtng. We nvestgate f these ndces can be explaned by common rsk factors representng almost all tradtonal asset classes,.e. equty, bond, commodty, currency, and real estate. Wthn each asset class we consder varous detaled nvestment styles to expand the cross-secton of rsk 14

15 exposures for the typcal hedge fund,.e. large cap stocks, volatlty, emergng markets, government bonds, credt, ol, gold among others. We nsure that each of the factor returns can be realzed through relatvely lqud nstruments, and therefore the returns of the mmckng portfolo may be achevable, by consderng ndces wth actve exchange traded or over-thecounter dervatves markets. Table 1 lsts the ndces that we consder as replcatng factors n our analyss. Our approach seeks to nvestgate the added value of model selecton technques n the context of hedge fund clonng hence the set of possble rsk factors s very comprehensve 10. Ths choce s motvated by the conclusons n several studes hghlghtng that model selecton may have mportant mplcatons for style analyss (see, e.g. Duplech et al., 2008), performance attrbuton (see, e.g. Agarwal and Nak, 2004, Vrontos et al. 2008), and hedge fund return predctablty (see, e.g. Amenc et al., 2003, Vrontos and Gamourds, 2008). Data for the replcatng factors are obtaned through Bloomberg for the perod February 1990 to July In the full sample the correlatons between factors range from to about 1. The average correlaton between the factors s 0.14 wth standard devaton Correlatons are dstrbuted wth postve skewness,.e. 0.62, whle the frst quartle s at and the thrd We choose not to remove factors that are hghly correlated to be able to nvestgate the mpact ths stuaton may have on the model selecton technques. Insert Table 1 somewhere here 10 Pror studes have concluded that hedge fund returns have non-lnear patterns and have proposed non-lnear rsk factors to explan ther returns. Agarwal and Nak (2004) for example show that hedge funds are exposed to a factor that captures the return characterstcs of an out-of-the-money put opton, Agarwal, et al. (2008) fnd that skewness and kurtoss factors obtaned through optons cross-sectons are mportant varables n hedge fund rsk modellng, whle Fung and Hseh (2001) show that trend followng strateges are explaned by lookback straddles. Whle these factors may be mportant for rsk management purposes, ther replcaton nvolves tradng nstruments that may n some nstances be beyond the mandate of the manager, e.g. over-the-counter lookback straddles. To mantan generalty and also snce our prmary purpose s to nvestgate the relatve performance of the proposed methodology we focus on a broad set of lnear factors motvated by current academc and practtoner work on hedge fund return replcaton. 15

16 The emprcal analyss s undertaken on a rollng-wndow bass. We ntally consder a rollngwndow of 120 months whch s then carred forward by one month as the wndow moves through the data set. Each month we seek the best model through fve dfferent model selecton strateges: (a) straght multple regresson (Straght hereafter),.e. estmaton of Equaton (1), where the best model s that contanng the sgnfcant factors, (b) factor selecton based on Akake s (1973) Informaton Crteron (AIC hereafter), (c) factor selecton based on Schwarz s (1978) Bayesan Informaton Crteron (BIC hereafter), (d) rdge regresson (Rdge hereafter) where the best model s the one determned by solvng the optmzaton problem n Equaton (3), and (e) lasso regresson (Lasso hereafter) where the best model s the one determned by solvng the optmzaton problem n Equaton (4a), wth two dfferent values for λ. For AIC and BIC we estmate all possble models wth respect to Equaton (1),.e. 2 K models arsng from all possble combnatons of the K factors. We dentfy the best model as that wth the lowest value of the respectve nformaton crteron, thus we jontly take nto account model ft and complexty. 4. Factor exposures To determne the explanatory power of the factors n our set, we perform a tme-seres analyss of each hedge fund ndex n our sample. In partcular, wthn a rollng-wndow set-up we apply the model selecton strateges (a) to (e) detaled n the Secton 3. We look nto varous aspects of the estmated models that are very mportant for hedge fund nvestment replcaton. Frst we dscuss the nature of the mportant factors. Next, we examne the n- and out-of-sample ft of the estmated models. We also report the sze of the selected model,.e. the number of mportant factors. And fnally we dscuss the maxmum and mnmum factor exposures. Table 2 reports the factors that are dentfed by the respectve model selecton strateges as valuable explanatory varables. The straght multple regresson and the rdge regresson ndcate 16

17 that all factors are mportant,.e. the coeffcents n Equatons (1) and (3) are not exactly equal to zero, therefore we report results only for AIC, BIC, and Lasso. Overall, the results ndcate that hedge fund portfolos nvolve holdngs n lterally all tradtonal asset classes. Equtes, bonds, commodtes, currences, and real estate assets appear as mportant drvers for the varous prmary strategy managers as well as for the emergng markets manager, the aggregate hedge fund manager, and the aggregate fund of funds manager. Among the most mportant varables are Russell 2000 Total Return Index and the MSCI Emergng Markets Index suggestng sgnfcant exposures to the US and Emergng Markets equty space. A sgnfcant exposure to the dollar ndex could be motvated by the substantal actvty n carry-tradng as well as crosscurrency nvestments. In terms of commodty nvestments managers seem to have favoured ol and gold over the study perod whle the rest of commodtes represented by the S&P GSCI Index seems to have also played an mportant role. Cash and government bonds are mportant factors for many strateges whle nvestment-grade bonds are related prmarly to event-drven strateges. Interestngly, Europe, Australasa, and Far East equty markets have not been among the top nvestment venues. Insert Table 2 somewhere here The ft of the estmated models s examned through the results presented n Table 3 where we tabulate the n- and out-of-sample Root Mean Squared Error (RMSE hereafter). The fgures n Table 3 are annualzed RMSE computed for the best model n each model selecton strategy over the entre perod. The mnmum n-sample RMSE,.e. 2.39%, s obtaned for Straght for the Fund Weghted Composte Index. The maxmum n-sample RMSE,.e. 7.41%, s observed for BIC and Lasso (hgh) for the Emergng Markets ndex. Out-of-sample RMSE s lowest for Lasso (low) for the Relatve Value managers and hghest for Straght appled to Emergng 17

18 Markets managers, 2.49% and 5.85% respectvely. These fgures are lower compared to those reported n Amenc et al. (2008) who consder lnear and the non-lnear models to replcate smlar ndces 11. Overall, we conclude that n terms of average ft both n- and out-of-sample nether of the methodologes seem to compare favourably to the others. Rdge regresson has a slghtly better out-of sample performance, whch seems to confrm the need to use penalzed least squared errors approaches when dealng wth fnancal factor models. A notceable property for Lasso however s that n most cases determnes models wth better out-of-sample ft relatve to ther nsample ft. Ths result provdes addtonal support to DeMguel et al. (2007) who argue that Lasso can be very successful out-of-sample n the presence of estmaton error. Insert Table 3 somewhere here Next, we turn our eye to examne the sze of the best model obtaned. Whle we have already hghlghted that Straght and Rdge result n non-zero factor exposures for all factors, AIC, BIC, and Lasso penalze large models. In Table 4 we report the average number of factors n the best models selected through AIC, BIC, and Lasso. We conclude that models obtaned through AIC and Lasso (low) are generally large. BIC and Lasso (hgh) determne models that are on average of smlar sze, although models obtaned through BIC are almost always smaller. Examnng these results jontly wth those dscussed above (see Table 3), we conclude that t s possble to acheve a sgnfcant degree of model ft by focusng on a subset of factors obtaned through AIC, BIC, and Lasso whch s on average half the sze of the full set. In partcular, BIC and Lasso (hgh) seem to offer a unque trade-off between parsmony and outof-sample ft. The best models selected through BIC and Lasso (hgh) are the smallest n sze 11 Ths s a rough comparson gven that Amenc et al. (2008) are studyng a dfferent perod,.e. January 1997 to December 2006, and a dfferent database,.e. TASS. Snce we do not test the effect of dfferences n the two sample we treat ths comparson as ndcatve. 18

19 whle at the same tme they provde comparable out-of-sample ft wth the other model selecton technques. Insert Table 4 somewhere here One fnal aspect we examne s the stablty of the estmated parameters, rather the tendency of the model selecton technques to result n models wth extreme factor exposures. In Table 5 we summarze the results of ths analyss. We show the mnmum and maxmum exposures on the factors obtaned through the alternatve model selecton technques over the entre perod. We observe ncredbly low exposures,.e. the mnmum exposure s , as well as unreasonably hgh exposures,.e. the maxmum exposure s 10.03, are obtaned wth Straght. Rdge, AIC and BIC compute exposures not as extreme as those n Straght, stll far smaller than and far bgger than 1.00 n many nstances. Interestngly, the pcture s very dfferent for Lasso. For both values low and hgh values of λ the mnmum factor exposures are close to zero, the mnmum of them beng The maxmum factor exposures are not extreme ether wth the maxmum of them beng Insert Table 5 somewhere here Overall, n ths secton we show that rgorous model selecton technques.e. AIC, BIC, and Lasso can be used to obtan reduced models, wth economcally relevant factors, wthout tradng off ft relatve to a model contanng a broader set of factors. Moreover we show that Lasso can be very successful out-of-sample. However AIC and Lasso (low) may result n models whose sze s sgnfcantly larger than the sze of models obtaned through BIC or Lasso (hgh). When we examne the magntude of the extreme factor exposures we conclude that AIC and BIC dentfy factor wth extremely hgh or extremely low exposures. 19

20 Collectvely, we fnd that Lasso mproves our ablty to select mportant economcally relevant factors wthout tradng off ft or parsmony. 5. Clones performance The analyss n Secton 4 reveals that Lasso posses some very attractve propertes that can make t a superor method n the context of hedge fund return replcaton. Whle we mpose no constrants on the weghts n the precedng analyss, we can draw mportant nferences for the core problem at hand the constructon of a hedge fund mmckng portfolo. We show that Lasso can acheve very good out-of-sample ft whch s crtcal for the assessment of the replcatng portfolo. We also show that the sze of the Lasso best model, and therefore the number of assets n the mmckng portfolo s reduced compared to the sze of the portfolo obtaned from conventonal technques. Ths s a very mportant aspect n the context of hedge fund replcaton as t s preferable to trade fewer assets n the replcatng nvestment programme. Fnally, we show that the extreme factor exposures are wthn reasonable bounds n Lasso relatve to all other approaches. As a result the weghts of the dfferent assets n the mmckng portfolo are more lkely to be stable over tme, consstent wth low transacton costs and ease of tradng. Ths Secton nvestgates whether these propertes can be exploted n practce. In partcular we consder the portfolo constructon problem descrbed n Equaton (5) whch nvolves some basc constrants that are typcally used n the constructon of hedge fund replcatng products. We also use a modfed framework of Sharpe s (1992) style analyss,.e. the optmzaton problem descrbed by Equaton (2) wth the addtonal constrant that 1 β 1 12, as a benchmark approach. We explore several propertes of the replcatng portfolo ncludng out-of-sample 12 See also footnote 7. 20

21 correlaton wth the replcated ndex, the number of asset n the replcatng portfolo, performance measures such as the excess return, the trackng error volatlty, but also the turnover of the mmckng portfolo. In addton we compare the clones produced wth our analyss wth selected currently avalable hedge fund replcaton products. And fnally, we carry out a set of tests to check the robustness of our results. 5.1 Comparson wth the benchmark Frst we examne the dversty of the replcatng portfolo by lookng at the number of assets that are dentfed as mportant n replcatng the benchmark returns. We also examne the out-ofsample correlaton of the replcatng portfolo returns wth the respectve hedge fund ndces. We report the results of ths analyss n Table 6. These results suggest that the mx of assets obtaned wth Lasso s reduced n sze compared to the mx obtaned through the standard approach. Dependng on the consdered values of λ, Lasso can result to a reduced sze n the range of about 50% to 75% of the full set 13. On an absolute bass we observe that both methods are capable of generatng portfolo mxes that n most nstances have sgnfcant correlaton wth the benchmark ndces the correlaton for Equty Hedge, Emergng Markets, and the Fund Composte Index exceed The correlaton for Event Drven and Fund of Funds managers s about Macro and Relatve Value managers mmckng portfolos have lower correlaton 0.62 and 0.43 wth ther benchmarks respectvely. Insert Table 6 somewhere here Next we examne several metrcs related to the performance of the replcatng portfolo wth respect to ts benchmark. We compute the average excess return,.e. the dfference between the replcatng portfolo return and the return of the benchmark, the trackng error volatlty, and the 13 Wth Lasso, dependng on the choce of λ, we can determne models wth a number of factors varyng from 1 to N. 21

22 turnover that s requred by the rebalancng strategy. We report these results n Table 7. We observe that the constraned regresson method produces both postve and negatve excess returns whle the excess return of portfolos constructed through Lasso s always negatve on average. When we compare the excess returns of the portfolo obtaned wth constraned regresson and the returns of the Lasso portfolos, we cannot reject the hypothess that ther means are equal, at the 5% sgnfcance level. When we compare these results wth the results reported n Amenc et al. (2008) for ther best performng method,.e. Kalman flter approach, we conclude that Lasso mproves our ablty to construct hedge fund replcatng portfolos. For example for the Fund of Hedge Funds and the Event Drven ndces the average excess return reported n Amenc et al. (2008) s about -0.33% whch s worse than -0.13% and almost dentcal to -0.32% for the respectve ndces n our experments for Lasso (hgh). For Emergng Markets and Macro Amenc et al. (2008) report -0.79% and -0.67% respectvely whle our analyss yelds % and -0.23%. In terms of trackng error volatlty Lasso does as good and some tmes even better that the constraned regresson approach. One nterestng property of the replcatng portfolo rebalancng strategy s the turnover whch obvously relates to the transacton costs requred for obtanng the targeted return. We report the turnover of the portfolo strategy n the last columns of Table 7. The results are consstent wth the argument that Lasso allows mplctly to control for turnover. Compared wth the constraned regresson approach whch on average requres about 1/3 of the portfolo to rebalance at each pont of tme, Lasso portfolos drop that to about 1/15 for the hgh λ case. Transacton costs for the assets consdered n our mmckng portfolo may vary n the range of 5 to 30 bps 14. Gven that excess returns are not statstcally dfferent, the benefts n the return can be n the magntude of 1 to about 10 bps per month. The benefts can be even hgher when more frequent rebalancng s necessary,.e. when 14 Ths was an ndcaton provded by practtoners tradng such contracts. 22

23 structured products on the ndex should be managed. Ths ssue has not been dscussed n prevous works to our knowledge although we beleve s very crtcal for the success of the replcaton strategy. Insert Table 7 somewhere here Another aspect that has not been addressed n the current lterature relates to the concentraton of a large proporton of the replcatng portfolo captal on certan assets. In Table 8 we present evdence on the replcatng portfolo hgh weghts,.e. greater than 50%, ether long or short. In column headed no factors we present the number of factors whch had a hgh loadng durng the sample perod. In column headed average we present what proporton of the test perod on average have the loadngs on these factors been hgh. The column head max presents the max proporton of the test perod that a factor has exceeded the threshold. The results n Table 8 ndcate that wth conventonal portfolo optmzaton a large number of factors concentrate sgnfcantly hgh weghts whch sometmes are kept at hgh levels for the entre perod. When we consder the results reported for Lasso we observe that the number of factors that a large proporton of wealth s allocated to s sgnfcantly lower, for example 0 to 2 factors for Lasso (hgh) and shows very lttle persstence. Insert Table 8 somewhere here In summary, the results n ths secton show that norm-constraned hedge fund replcatng portfolos are relatvely small portfolos n terms of the number of assets they nvolve and have smlar correlaton propertes wth the benchmark ndces wth portfolos constructed wth conventonal methods. Whle ther excess return s slghtly lower - although not dfferent statstcally - the turnover s a fracton of that of a conventonally constructed mmckng 23

24 portfolo. Fnally, although they comprse a sgnfcantly smaller number of assets, normconstraned hedge fund replcatng portfolos encompass more evenly weghted mxes of assets. 5.2 Comparson wth selected commercally avalable hedge fund clones Ths secton dscusses the propertes of replcatng portfolos constructed wth Lasso and conventonal methods wth the propertes of selected commercally avalable hedge fund clones that banks trade wth ther clents as plan vanlla notes or n the form of structured products. Our analyss nvolves Ct s HARP Index, Deutsche Bank s Absolute Return Beta Index, and Meryll Lynch s Factor Index. Ct s HARP Index and Deutsche Bank s Absolute Return Beta Index are benchmarked aganst the HFR Fund of Hedge Funds Index, whle Meryll Lynch s Factor Index benchmark s the HFR Fund Weghted Composte ndex. Data on these ndces are obtaned from Bloomberg. A word of cauton here s due for the fact that the return on the replcatng strateges s gross of transacton costs or any other fees. Table 9 tabulates the results of ths analyss. In partcular we report rsk and return characterstcs. We also report extreme rsk measures,.e. Value at Rsk and Condtonal Value at Rsk. When we compare the proposed portfolo constructon strategy benchmarked at the Fund Weghted Composte Index wth Meryll Lynch s Factor Index we observe several mportant mprovements. Not only the average return s hgher, closer to the benchmark, but also the return/rsk rato s hgher. Most mportantly Meryll Lynch s Factor Index seems to exhbt sgnfcant tal rsk gven the large value of VaR and CVaR at the 1% level,.e. 5.20% and 5.64% respectvely, compared to 3.17% and 3.27% for Lasso (low), and to 3.04% and 3.15% for Lasso (hgh). However we need to acknowledge that Meryll Lynch s Factor Index nvolves sx factors whereas Lasso about eleven to fourteen on average wth very low turnover however. For the products related to the Fund of Hedge Funds benchmark we observe that the average return of Ct s HARP Index 24

25 and Deutsche Bank s Absolute Return Beta Index are hgher than Lasso. Ther volatlty s such though that the return/rsk rato s hgher for Lasso. The results on the extreme rsk measures suggest that the tal rsk assocated wth Ct s HARP Index and Deutsche Bank s Absolute Return Beta Index s hgher than the extreme rsk of the Lasso portfolo especally for Deutsche Bank s Absolute Return Beta. Insert Table 9 somewhere here Overall the results n ths secton ndcate that Lasso provdes an attractve alternatve to the methodologes that appear to underle selected commercally avalable hedge fund clones, especally from a rsk management perspectve. 5.3 Robustness checks and addtonal results In a separate work (avalable n detal from the authors upon request) we have extended our analyss to ncorporate: (a) a varety of hedge fund ndces, (b) addtonal optmzaton constrants, (c) a range of values for λ, (d) dfferent estmaton wndows. We used a wde range of sub-strategy ndces. These ncluded: Equty Market Neutral, Quanttatve Drectonal, and Short Bas form the Equty Hedge unverse; Dstressed/Restructurng and Merger Arbtrage managers from the Event-Drven unverse; Systematc Dversfed from the Macro prmary strategy; Fxed Income-Convertble Arbtrage, Fxed Income-Corporate, and Mult-Strategy managers from the Relatve Value unverse; Asa ex-japan from Emergng Markets; and Conservatve, Dversfed, Market Defensve, and Strategc managers from the Fund of Funds unverse. The results we obtaned were qualtatvely smlar wth those we report n the prevous sectons. 25

26 We also tested a varety of optmzaton constrants. One of the man tests we carred out n ths context was to explore stuatons where leverage s allowed. We relaxed the assumpton that N β = 1 to allow for total wealth leveraged by a factor of 1.5,.e. = 1 N = 1 β = 1.5. Our approach for ncorporatng leverage ncluded allocatng the dfferental captal of our allocatons to the rsk free asset 15. The results of ths analyss showed that the performance of the replcatng portfolo does not mprove. Ths n fact s reasonable as requrng that leverage s 150% both n good and bad tmes may not be a wse choce n practce. An mprovement to ths may be to set a cap on leverage and/or lnk allowed leverage wth market condtons or momentum. We also vared the lower and upper bound constrants for β. Whle ths had a consderable mpact on the clones determned through constraned regresson t changed the behavour of the mmckng portfolos obtaned through Lasso only margnally. The results reported n the prevous Sectons are obtaned for arbtrarly values of λ. In partcular n Secton 5 (constraned optmzaton) values are such that for low (hgh) λ about 75% (60%) of the total number factors are beng pcked up. In Secton 4 (unconstraned optmzaton) a hgh λ that s four tmes the value of low λ dentfes models wth on average half the number of factors. We tested addtonal values for λ. We observed that f we keep λ too hgh we tend to overft the model n-sample (we do slghtly better than wth a lower penalty), but then out-of sample the performance becomes worse. Furthermore, we nvestgated standard statstcal cross-valdaton 15 In partcular we estmated the weghts n the replcatng portfolo through r = β F + ε and computed the return of the fnal portfolo through r K K = β F + rf 1 β = 1 = 1 one can estmate the weghts through the equaton r rf = β ( r rf ) + ε K = 1 K = 1, where r f s the rsk-free rate. Alternatvely. 26

27 methods. These, dd not n fact provde better results n term of accuracy, whle resulted n hgher turnover. Future research may focus on dfferent crtera for choosng λ. In terms of the estmaton wndow, our robustness analyss concluded that wth shorter estmaton perods,.e. a wndow of 60 months. The analyss showed that wth shorter estmaton wndows the weghts become less stable. In relatve terms however the mpact s consderably stronger for constraned regresson mmckng portfolos than t s for portfolos obtaned through Lasso. In summary, addtonal analyss concludes that the results we present n the prevous sectons are generally robust wth respect to varous parameter choces. 6. Concluson Motvated by recent advances n the portfolo constructon lterature, ths artcle proposes the use of norm-constraned portfolo optmzaton methodologes for effcent hedge fund clone constructon. Techncally the proposed method s a modfcaton to the standard à la Sharpe style analyss where the objectve functon s augmented wth a penalty proportonal to the sum of the absolute values of the replcatng asset weghts,.e. the norm of the asset weghts vector. We show that the propertes of the resultng portfolos are extremely attractve both relatve to ther benchmark hedge fund ndex but also relatve to selected commercally avalable hedge fund clone products that use propretary constructon technques. In partcular the mmckng portfolos obtaned through norm-constraned optmzaton nvolve a reasonable number of assets, about ten out of the ntal set of twenty, and are hghly correlated wth the benchmark ndces. Ther turnover s sgnfcantly lower than that mplct n conventonal technques. Fnally, they seem to provde an attractve alternatve to selected commercally avalable hedge fund clones, especally from a rsk management perspectve. 27