ABSTRACT JEL: G11, G20, G23. KEYWORDS: Hedge Fund Replication, Dynamic Portfolio Optimization, Martingale Method, Malliavin Calculus INTRODUCTION

Transcription

1 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 A NEW EDGE FUND REPLICAION MEOD WI E DYNAMIC OPIMAL PORFOLIO Akihiko akahashi, he Universiy of okyo Kyo Yamamoo, he Universiy of okyo ABSRAC his paper provides a new hedge fund replicaion mehod, which exends Ka and Palaro (5) and Papageorgiou, Remillard and ocquard (8) o muliple rading asses wih boh long and shor posiions. he mehod generaes a arge payoff disribuion by he cheapes dynamic porfolio. he work here exends he work of of Dybvig (988) o a coninuous-ime framework and dynamic porfolio opimizaion where he dynamic rading sraegy is derived analyically by applying Malliavin calculus. I is shown ha cos minimizaion is equivalen o maximizaion of a cerain class of von Neumann- Morgensern uiliy funcions. he mehod is applied o he replicaion of a CA/Managed Fuures Index in pracice. JEL: G, G, G3 KEYWORDS: edge Fund Replicaion, Dynamic Porfolio Opimizaion, Maringale Mehod, Malliavin Calculus INRODUCION edge fund replicaion producs appeared in he financial markes recenly. Invesmen banks and asse managemen companies launched he producs one afer anoher. Some of hese insiuions developed replicaion echniques collaboraing wih he pioneers in hedge fund research. herefore, hese producs have gained increased aenion. edge fund replicaion overcomes he following difficulies in hedge fund invesing: high cos, low ransparency, and low liquidiy. he imporance of ransparency and liquidiy was recognized afer he subprime and Lehman shocks. Exising mehods of hedge fund replicaion can be caegorized in hree ways: rule-based, facor-based, and disribuion replicaing approaches. Our mehod is caegorized as disribuion replicaing approach. he previous sudies are Ka and Palaro (5) and Papageorgiou e al. (8). Ka and Palaro (5) ried o replicae he reurn disribuion of he arge hedge fund and is dependence srucure on an invesor s exising porfolio hrough he dynamic rading of he porfolio (proxied by a porfolio of sock index and bond fuures) and anoher asse. Papageorgiou e al. (8) proposed an alernaive way o perform Ka-Palaro s replicaion mehodology by uilizing a hedging scheme of opions in an incomplee marke. his aricle exends hese previous works. he mehods developed by he wo earlier research papers use only one asse as a replicaion ool. Moreover, hey can ake only long posiions. herefore, he user of he mehods should pay aenion o asse choice. On he oher hand, our new mehod allows muliple asses wih boh long and shor posiions for creaing a replicaing porfolio. When he invesmen universe is exended o muliple asses, a crierion ha chooses a payoff should be inroduced because here are infiniely many payoffs ha have he same saisical properies as he arge hedge fund. his paper proposes o choose he cheapes one. By exending he work of Dybvig (988) o coninuous-ime framework, he cheapes payoff is obained. hen, he dynamic rading sraegy is derived analyically by applying Malliavin calculus. For replicaing he marginal disribuion of he arge hedge fund reurn, i is shown ha cos minimizaion is equivalen o maximizaion of a cerain class of von Neumann- Morgensern uiliy funcions. In he subprime and Lehman shocks, CA/Managed Fuures funds enjoyed high reurns by conrolling heir exposures efficienly. (See figure.) his paper replicaes CA/Managed Fuures Index. Oher 3

2 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 approaches and exising disribuion replicaing mehods do no work for his replicaion. Since mos CAs or managed fuures funds employ sysem rading sraegies based on some rading rules, rule-based replicaion ends up as a managed fuures fund iself. Facor-based replicaions for such funds are impossible or very ough o develop, because i is difficul o find facors ha drive he reurn of CAs or managed fuures funds (See, for example, akamada e al. (7)). As will be shown, he exising disribuion replicaing mehods fail o replicae high performance in he recen credi crisis, because hey can rade only one replicaing ool and canno ake shor posiions. Many CAs or managed fuures funds are seeking aracive invesmen opporuniies in financial markes around he world. hey employ dynamic rading sraegies including leverage and shor sales o exploi hem. Our mehodology is also able o reflec heir invesmen behaviors while he exising mehods are no. A hisorical ou-of-sample simulaion shows ha our scheme provides a much beer performance han he exising mehod especially in he recen credi crisis.his paper is oulined as follows. he following secion briefly discusses he relevan lieraure. Secion 3 describes he heory of our replicaion mehod is explained. Secion 4 describes daa and empirical procedures. Secion 5 provides he replicaion resul and compares i wih he replicaion resul by he exising mehod. Finally, secion 6 concludes he paper. Figure : Eurekahedge edge Fund Indices by Invesmen Sraegy Arbirage Ne Asse Value...9 CA / Managed Fuures Disressed Deb Even Driven Fixed Income Long / Shor Equiies Macro Muli- Sraegy.8 Dec-6 Jan-7 Feb-7 Mar-7 Apr-7 May- Jun-7 Jul-7 Aug-7 Sep-7 Oc-7 Nov-7 Dec-7 Jan-8 Feb-8 Mar-8 Apr-8 May- Jun-8 Jul-8 Aug-8 Sep-8 Oc-8 Nov-8 Dec-8 his figure shows he performance of Eurekahedge edge Fund Indices by invesmen sraegy from January 7 o December 8. he iniial ne asse values for all he sraegies are normalized o in 3 December, 6. he hedge fund indices is downloadable from he homepage of Eurekahedge or Bloomberg. his paper uses he daa downloaded from he homepage for he analysis. LIERAURE REVIEW he emergence and exising mehods of hedge fund replicaion are described in akahashi and Yamamoo (9a). As menioned in he previous secion, he mehodologies can be caegorized in he following hree approaches. he firs is a rule-based approach ha mimics rading sraegies he arge hedge fund employs. his is he mos primiive way. he mehod of Duare e al. (7) can also be regarded as rulebased fixed-income hedge fund clone echniques. he second approach is a facor-based clone ha ries o replicae risk exposure of he arge fund. If his 4

3 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 mehod succeeds, hen he reurn of he replica racks ha of he arge fund on a monh-o-monh basis. his is he ideal replicaion, bu his mehod requires finding radable marke facors ha drive he reurn of he arge hedge fund. Lo and asanhodzic (6) and Fung and sieh (7a, 7b) sudied a facorbased approach. he echniques of facor analysis for hedge funds ha have been developed from he lae 99s such as ha of Fung and sieh (997,, ) and Agarwal and Naik (4) are direcly applied o he replicaion. he hird approach is disribuion replicaion ha gives up monh-o-monh reurn replicaion, and aims o replicae he disribuion of hedge fund reurns. Amin and Ka (3) firs ried he replicaion. owever, an aracive characer of hedge fund reurns is low dependence wih reurns of radiional asse classes. Furher, Ka and Palaro (5) and Papageorgiou e al. (8) developed modified mehod o replicae he dependence srucure wih he invesor s exising porfolio, oo. his paper exends he mehods developed by Ka and Palaro (5) and Papageorgiou e al. (8) o muliple rading asses wih long and shor posiions. he dynamic replicaing porfolio is obained by applying Malliavin calculus. See, for example, Nualar (6) for he inroducion o Malliavin calculus. By exending heorem in Dybvig (988) o coninuous-ime framework, i is shown ha generaion of a arge marginal disribuion is equivalen o maximizaion of cerain class of von Neumann-Morgensern uiliy funcions. In a complee marke, maringale mehod is a powerful approach o obain he dynamic porfolio ha maximizes a von Neumann-Morgensern uiliy funcion. See, for example, Karazas and Shreve (998) for he basics of maringale mehod. o obain he concree expression of he opimal porfolio, Clark- Ocone formula in Malliavin calculus is useful. For example, Deemple e al. (8) surveyed he Mone Carlo mehods for he compuaion of he opimal porfolio policies. akahashi and Yoshida (4) applied an asympoic expansion mehod o he opimal invesmen problems in he analyic approximaion mehods. Our mehod can be also considered as an alernaive dynamic porfolio opimizaion assuming no an agen s uiliy bu a arge payoff disribuion. E EORY Firs, we review he disribuion replicaion mehod of hedge fund reurns proposed by Ka and Palaro (5). Consider an invesor who has been invesing in radiional asses such as socks and bonds, and plans o inves in a hedge fund. I is assumed ha he is araced o he hedge fund because he disribuion of reurns and he dependence srucure in his exising porfolio of bonds and socks. Suppose ha S is a risk-free asse, S is an invesor s exising porfolio, and S is a risky asse (replicaing ool). Assume ha S is also radable. (I is proxied by a porfolio of sock index and bond fuures.) Le and be he sar and erminal daes of he invesmen, respecively. Ka-Palaro s approach ries o he reurn disribuion of he hedge fund and dependence srucure wih he invesor s exising porfolio by he dynamic rading sraegy of he invesor s porfolio S and anoher risky asse S. o implemen he replicaing sraegy, we need o proceed wih he following seps. Firs, he sochasic S processes { } S = = and { } are inferred. he iniial asse values are normalized so ha S = S. = Following ha, he join disribuion of R = log S and log R = S is also obained. Le be he random variable ha represens he log reurn of he arge hedge fund. Second, he join disribuion of he invesor s porfolio and hedge fund reurns ( R, ) is esimaed. his is he arge join disribuion R 5

4 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 o replicae. hird, he payoff funcion, which ransforms he join disribuion of ( R, R ) o ha of ( R, R ), is deermined. Finally, he payoff is priced is replicaed hrough he dynamic rading of S and S. In he firs sep ( inferenceof{ S } = and{ S } = ), Ka and Palaro (5) and Papageorgiou e al. (8) modeled by Gaussian and non-gaussian disribuions, and seleced he bes-fiing one. As hedge funds exhibi skewness and fa-ails, and are non-linearly relaed o radiional asse classes, in he second sep (esimaion of (R, R )), Ka and Palaro (5) proposed o model R and R separaely, and hen connec hem by a copula. For hedge fund reurns, i is desirable o use he disribuion class ha can capure is skewness and fa-ails. For example, Ka and Palaro (5) and Papageorgiou e al. (8) used Gaussian, Suden-, Gaussian mixure and Johnson disribuions. Some copulas can capure he asymmeric dependence srucure flexibly. Afer esimaing he parameers and selecing models of he asse reurns and price processes, a payoff funcion is deermined. o obain he join disribuion of he invesor s porfolio and hedge fund reurns hrough he dynamic rading of he invesor s porfolio and he replicaing ool, a funcion ha saisfies he following equaion is required. P( R ~ ( a, g R, R ) b) = P( R a, R b) for any a, b, () or equivalenly, ( ~ P g( R, R ) b R = a) = P( R b R = a) for any a, b. () hen, g ~ is given by g ~ ( a, b) = F R ( F ( b a R a )), g ~ ( a, b) = F R a ( F ( b)), R a (3) where F and R a F R a are he condiional disribuion funcions of R and R under R = a. In erms of he asse prices, he payoff funcion is represened as { ~ (log g ˆ( S, S ) = exp g S, log S )}. (4) If one obained he payoff funcion, he replicaing sraegy encouners he same problem wih pricing and hedging of derivaives. he dynamic replicaing sraegy is given by he dela-hedging sraegy of he payoff g ˆ( S, S ). 6

5 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 If he iniial cos for he rading sraegy is less (more) han, hen he arge payoff can be realized by a lower (higher) cos. he remaining (shorage of) money is invesed (funded) in he risk-free asse. he shape of he probabiliy densiy can be replicaed, bu he mean reurn is higher (lower) han he arge fund by he difference of he iniial cos. In his case, he replicaing ool does (does no) include greaer invesmen opporuniy han he arge hedge fund. Noe ha he payoff funcion g ˆ(, ) is an increasing funcion wih respec o he second argumen. hen, he dela-hedging sraegy never akes a shor posiion for S. In page 7-8 of Ka and Palaro (5), he auhors claim ha users of his mehod should choose he replicaing ool S ha has he posiive expeced reurn facor uncorrelaed o he reurn of he invesor s porfolio. hen, he long posiion for S is jusified. In his case, he choice of a replicaing ool is crucial. As described here, his mehodology can replicae he shape of he probabiliy densiy, bu canno fi he mean. If you found a greaer invesmen opporuniy han he arge fund, he mean reurn would be superior and vice versa. herefore, he usage of only one risky asse is resricive. Papageorgiou e al. (8) synhesized muliple asses o creae a replicaing ool by equal-weighing, bu here would be inefficiencies in he ad hoc fixed weighed porfolio. he exension of he invesmen universe would bring in higher mean reurns. Le us exend he replicaion mehod o muliple asses. See akahashi and Yamamoo (9b) for a n rigorous argumen. Suppose ha S,, S i n processes of he financial asses { S } = = ) follow sochasic differenial equaions (SDEs) ds = r S d, (5) i i i i ij i j ds = µ S d + σ S dw ( i =,, n), (6) j= n,, W where W = ( W ) is n-dimensional sandard Brownian moion. r, µ, and σ saisfy appropriae measurabiliy and inegrabiliy condiions. All of he iniial asse values are normalized, so n ha S = = S =. he following noaions by n-dimensional vecors and a n n marix are inroduced: ( n n S = S,, S ), µ = ( µ,, µ ), = (,, ), and σ O σ =. (7) n nn σ σ Suppose ha is inverible almos surely. hen, here exiss he unique marke price of risk θ = σ ( µ r ). In oher words, he financial marke is complee. he financial marke is denoed by M= (r, μ, σ). i ij 7

6 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 In complee marke M, he unique sae price densiy process is given by exp = ru du. θ u du θudwu (8) he no-arbirage price of any measurable payoff X a is given by x = E[ X ]. By he sandard argumen of maringale mehod, X can be replicaed by a dynamic rading of he financial asses wih iniial cos x. For convenience, he minus logarihm sae price densiy process L is inroduced: L = +. + ru du θ u du θudwu (9) Le ξ be a posiive payoff a ime. Le F ξ s and F L s be he condiional disribuion funcions of ξ and L under condiion S = s. Assume ha F ξ s and F L s are coninuous and sricly increasing for any s >. If X is defined as follows, ( S, X ) has he same join disribuion as ( S, ξ ) : X = g( S, L ), () where g s, l) = F ξ ( F ( )). () ( s L s l he nex heorem assers ha X is he unique cheapes payoff among he random variables whose join disribuions wih S are same as ( S, ξ ). heorem in Dybvig (988) is he equally probable finie sae seing version of he heorem. heorem Assume ξ is a posiive payoff a ime, and F ξ s and F L s are coninuous and sricly increasing for any s >. In a complee marke M, he unique cheapes payoff X among he random variables whose join disribuions wih S are same as ( S, ξ ) by equaion (). See akahashi and Yamamoo (9b) for he proof. If condiion S = s is no aken accoun, he following claim can be proven. Le F ξ and F L be he disribuion funcions of ξ and L respecively. Assume ha F ξ and F L are coninuous and sricly increasing. If X is defined by X = F ξ ( F L ( L )), X is he unique cheapes payoff among he random variables ha has he same marginal disribuion wih ξ. he nex heorem assers ha he cos minimizaion for he marginal payoff disribuion is equivalen o an expeced uiliy maximizaion. heorem in Dybvig (988) is he equally probable finie sae seing version of he heorem. o sae he heorem in coninuous-ime framework, wo addiional condiions are required for he uiliy funcion. heorem Assume ξ is a posiive payoff a ime, and F ξ and F L are coninuous and sricly increasing. If X is he cheapes payoff ha has he same disribuion wih ξ, hen, in a complee marke M, here exiss a sricly increasing and sricly concave von Neumann-Morgensern uiliy funcion u ( ) such ha (a) lim u ( z) = +, z + (b) lim u ( z) =, z + and he dynamic rading sraegy ha aains payoff X 8

7 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 is he opimal invesmen sraegy for u ( ). Conversely, if a dynamic rading sraegy maximizes a sricly increasing and sricly concave von Neumann-Morgensern uiliy funcion u ( ) ha saisfies condiions (a) and (b), i aains he cheapes payoff for some disribuion. See akahashi and Yamamoo (9b) for he proof. his heorem suppors our mehod heoreically. Moreover, i ensures ha our mehod is applicable o no only hedge fund replicaion bu also dynamic porfolio opimizaion in invesmen managemen. i Le us see he dynamic porfolio ha replicaes he cheapes payoff. Le π (i =,, n) represen he i n money amoun invesed in asse i a ime. n-dimensional vecor π is defined by π = ( π,, π ), which denoes he porfolio of risky asses. Le x be he iniial cos required o realize he cheapes payoff X for some payoff disribuion. he iniial cos x is invesed in he financial asses by a dynamic selffinancing rading sraegy o generae payoff X. In oher words, he porfolio value a ime, X, saisfies = π + π () X for any. In a differenial form his is, dx = r X d + π µ r ) d + π σ dw. ( he dynamic porfolio can be obained for he case of Markovian coefficiens. (See akahashi and Yamamoo (9b).) his paper assumes ha are deerminisic funcions of. By applying Malliavin calculus, he dynamic porfolio generaing he cheapes payoff is obained. Proposiion Assume ha r, μ and σ are deerminisic funcions of. hen, in a complee marke M, he dynamic porfolio generaing payoff X = g( S, L ) is given by π = σ ( ) φ, (4) where = ( n φ,, ) φ φ is given by θ ( ) σ ( ) φ = E[ g ( S, L )] + E[ g( S, L ) S ], (5) i i ( ) φ = θ E[ g ( S, L )] for i =,, n, (6) where g represens he parial derivaive of g wih respec o argumen j. j he inerpreaions for he opimal porfolio consiuen facors are as follows. E[ g ( S, L )] is he presen value of he sensiiviy of he erminal payoff o he change of L. his quaniy corresponds o dela in he opion heory. he volume of he risky asse porfolio increases in his quaniy. his facor conribues o generaing he arge disribuion. In addiion, he replicaing sraegy allocaes wealh o radable asses according o he marke price of risk θ i (). hrough his operaion, he cheapes sraegy is realized. he second erm of φ is he presen value of he sensiiviy of he erminal payoff o he change of W. his erm conribues o he generaion of he dependence srucure on he invesor sexising porfolio. (3) 9

8 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 DAA AND MEODOLOGY Le us replicae Eurekahedge CA/Managed Fuures Index. he replicaion es is conduced on an ou-ofsample basis. his paper uses he following invesor s exising porfolio and risky asses. Assume ha he invesor s porfolio S is composed of 5% Japanese socks and 5% Japanese governmen bonds (JGB). Since hese asses are raded dynamically, OPIX fuures and long-erm JGB fuures were used as he proxies. Boh of hese securiies are lised on he okyo Sock Exchange. he S&P 5 fuures, NYMEX WI crude oil fuures, COMEX gold fuures, and JPY agains USD spo currency are used as replicaing ools. I is considered ha CAs and managed fuures funds inves in hese asses. All he daa are obained from Bloomberg. he log reurns on fuures are calculaed by rolling he fron conrac. he fron conrac is rolled on he las rading day of he mauriy monh. Our base currency is USD. Since OPIX and JGB fuures are denominaed in JPY, a currency hedge is applied. Accordingly, he log reurns of hese asses are adjused by he difference beween he ineres raes of USD and JPY. Libor raes are used for he ineres raes. Using pas daa, he following procedures were performed on a monhly basis o esimae and selec he bes-fied model of he arge reurn disribuion and is dependence srucure on he invesor s porfolio. Firs, he parameer esimaion and model selecion for he marginal disribuion of monhly log reurns of he CA/Managed Fuures Index are performed using he same mehod as wih Papageorgiou e al. (8). he bes-fied model is chosen from a Gaussian mixure wih m regimes (m =,, 3, 4, 5) and Johnson unbounded disribuion. Nex, he copula model beween he monhly log reurns of he replicaion arge and he invesor s porfolio is esimaed and seleced in he same manner as ha in Papageorgiou e al. (8), which is based on he ranking of he ime-series daa. he copula is seleced from he Gaussian, Suden, Clayon, Frank and Gumbel. In his example, i is assumed ha he coefficiens of he invesor s porfolio and he replicaing ool price processes are consan during a monh. he parameers for he invesor s porfolio and he replicaing ools are esimaed by maximum likelihood. he daily daa in he previous monh are used for he esimaion in order o reflec he rend-following invesmen sraegy. Finally, payoff funcion () is replicaed by dynamic replicaing porfolio (4). he porfolio is rebalanced on a daily basis. he condiional expecaions in equaions (5) and (6) can be numerically calculaed by Mone Carlo simulaions wih wo-dimensional Gaussian disribuions. herefore, he compuaional burden o obain he dynamic replicaing sraegy is no differen from ha of he one replicaing ool case. For he purpose of comparison, he replicaion resul of only one asse wih only long posiion (Ka- Palaro s mehod) is also shown. ere, he replicaing ool is he equally weighed porfolio of he four asses used by our mehod. he ransacion coss are assumed o be one basis poin for he sale and purchase of all asses. he incepion monh of Eurekahedge CA/Managed Fuures Index is January. he daa from he firs wo years since incepion are used o esimae he reurn disribuions while he monhly log reurns from January o Ocober 9 were replicaed by muliple asses (our mehod) and single asse (Ka and Palaro s mehod), which is composed by he equal-weighed porfolio of he four replicaing ools. RESULS Figure shows he growh of ne asse values of he arge and wo replicaion sraegies. he replicaion by muliple asses performed beer han he arge. owever, noe ha he performance of he replicaion arge is afer deducion of managemen and performance fees, while hose of replicaions are no. Alhough he replicaion by single asse wih only long posiions performed well before July 7, i incurred a large drawdown afer his period. his is because he replicaing ool subsanially declined bu he mehod could no ake he shor posiion for he asse. CAs or managed fuures funds arac invesors because hey enjoy high reurns during he financial marke crises. owever, he replicaion by single replicaing ool wih only long posiion failed o reproduce he characer. On he oher hand, he 3

9 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 replicaion by muliple asses aained near he reurn level o he arge even under he subprime and Lehman shocks. Figure : Performance of he arge and he Replicaed Sraegies during Jan -Oc Ne Asse Value.5.5 Dec- Mar- Jun- Sep- Dec- Mar-3 Jun-3 Sep-3 Dec-3 Mar-4 Jun-4 Sep-4 Dec-4 Mar-5 Jun-5 Sep-5 Dec-5 Mar-6 Jun-6 Sep-6 Dec-6 Mar-7 Jun-7 Sep-7 Dec-7 Mar-8 Jun-8 Sep-8 Dec-8 Mar-9 Jun-9 Sep-9 arge Muliple Asse Single Asse his paper shows he growh of he ne asse values of he arge and is wo replicas from January o Ocober 9. he arge is Eurekahedge CA/Managed Fuures Index. he replicas are creaed by muliple and single asse. he replicaion mehod by muliple asse is developed by his paper. he replicaion by single asse is based on Ka and Palaro (5) and Papagerogiou e al. (8) where replicaing ool is he equally weighed porfolio of asses used by our mehod. he iniial ne asse values for all he sraegies are normalized o in 3 December,. able shows he summary monhly saisics of he arge and replicaed reurns from January o Ocober 9. As seen in figure, he replicaion by muliple replicaing ools resuled in higher mean reurn han he arge, while ha by single replicaing ool wih only long posiion did no. Moreover, exension o he muliple asses succeeded in replicaing he posiive skew and negaive excess kurosis, while single asse did no. I also made he correlaion wih he invesor s exising porfolio closer o he arge. his confirms ha he exension of Ka and Palaro (5) and Papageorgiou e al. (8) o muliple rading asses wih boh long and shor posiions improves he performance of he replicaion. Le us see he replicaion performance by spliing he oal period in 8 and oher periods. As seen in figure, mos hedge fund sraegies incurred drawdown in his year especially afer he Lehman shock, while our replicaion arge CA/Managed Fuures index performed very well. able and 3 exhibi he summary saisics for he year and he oal period excluding 8 respecively. Comparing mean log reurns in 8 o oher periods, CA/Managed Fuures index performed beer in he 8 s financial crisis han oher periods. he replicaion by muliple asses succeeded in replicaing his characer. owever, he replicaion by single asse wih only long posiions incurred loss in 8, because his sraegy canno ake shor posiion for he replicaing ool in he bear marke. Especially, he clone wih single asse recorded he minimum reurn in 8. Even spliing he period, he replicaing sraegy by 3

10 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 muliple asses succeeded in replicaing skew and excess kurosis, while he replicaion wih single asse did no. he improvemen of he replicaion performance in 8 indicaes ha our exension o muliple asses wih long and shor posiions is significan in pracice. able : Monhly Saisics of he arge and Replicaed Log Reurns in oal Period arge Muliple Asse Single Asse Mean.87%.8%.65% Sd. Dev..4%.64%.4% Mean/Sd. Dev Skew Excess Kurosis Max 6.77% 7.39% 7.69% Min -4.5% -5.45% -5.3% Correlaion wih Invesor's Porfolio his able shows he summary monhly saisics of he arge and replicaed log reurns from January o Ocober 9. he arge is Eurekahedge CA/Managed Fuures Index. he replicas are creaed by muliple and single asse. he replicaion mehod by muliple asse is developed by his paper. he replicaion by single asse is based on Ka and Palaro (5) and Papagerogiou e al. (8) where replicaing ool is he equally weighed porfolio of asses used by our mehod. Excess kurosis means kurosis minus 3. ha is, he excess kurosis of normal disribuion is. able : Monhly Saisics of he arge and Replicaed Log Reurns in 8 arge Muliple Asse Single Asse Mean.5%.64% -.65% Sd. Dev..49%.39%.4% Mean/Sd. Dev Skew Excess Kurosis Max 6.4% 6.47%.8% Min -3.8% -.49% -5.3% Correlaion wih Invesor's Porfolio his able shows he summary monhly saisics of he arge and replicaed log reurns in 8. he arge is Eurekahedge CA/Managed Fuures Index. he replicas are creaed by muliple and single asse. he replicaion mehod by muliple asse is developed by his paper. he replicaion by single asse is based on Ka and Palaro (5) and Papagerogiou e al. (8) where replicaing ool is he equally weighed porfolio of asses used by our mehod. Excess kurosis means kurosis minus 3. ha is, he excess kurosis of normal disribuion is. able 3: Saisics of he arge and Replicaed Log Reurns during he oal Period Excluding 8 arge Muliple Asse Single Asse Mean.83%.%.84% Sd. Dev..%.68%.36% Mean/Sd. Dev Skew Excess Kurosis Max 6.77% 7.39% 7.69% Min -4.5% -5.45% -4.68%.7..3 Correlaion wih Invesor's Porfolio his able shows he summary monhly saisics of he arge and replicaed log reurns from January o Ocober 9 excluding 8. he arge is Eurekahedge CA/Managed Fuures Index. he replicas are creaed by muliple and single asse. he replicaion mehod by muliple asse is developed by his paper. he replicaion by single asse is based on Ka and Palaro (5) and Papagerogiou e al. (8) where replicaing ool is he equally weighed porfolio of asses used by our mehod. Excess kurosis means kurosis minus 3. ha is, he excess kurosis of normal disribuion is. 3

11 GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 CONCLUDING COMMENS his aricle presened a new hedge fund replicaion mehod wih he dynamic opimal porfolio by exending Ka and Palaro (5) and Papageorgiou e al. (8) o muliple rading asses wih boh long and shor posiions. I generaes a arge payoff disribuion by he cheapes dynamic porfolio. By exending Dybvig (988) o coninuous-ime framework, i was shown ha cos minimizaion is equivalen o maximizaion of a cerain class of von Neumann-Morgensern uiliy funcions. he dynamic rading sraegy was derived analyically by applying Malliavin calculus. he mehod was applied o he replicaion of CA/Managed Fuures Index. he replicaion performance was examined on an ou-of-sample basis. Our saisical procedure is he same as Papageorgiou e al. (8). he resul showed ha he performance of he replicaion was dramaically improved compared o he exising mehod (invesing in only one replicaing ool wih only long posiion). Mos noably, our replicaion mehod was able o obain high reurns afer he subprime and Lehman shocks as he replicaion arge while he replicaion based on he one replicaing ool wih only long posiion incurred a large loss during his period. Any change or exension of he invesmen universe would affec he performance of he replicaion. In our empirical replicaion, i is assumed ha he sochasic processes of rading asses have deerminisic coefficiens. he implemenaion for he Markovian coefficiens case including a sochasic volailiy model as well as a sochasic ineres rae model is also a challenging ask. Also, he applicaion of our mehod o creaing new aracive rading sraegies represens an avenue for fuure research. REFERENCES Agarwal, V. and N.Y. Naik (4) Risks and Porfolio Decisions Involving edge Funds, he Review of Financial Sudies, Vol. 7(), p Amin, G.S. and.m. Ka (3) edge Fund Performance 99-: Do he Money Machines Really Add Value? he Journal of Financial and Quaniaive Analysis, Vol. 38, p Deemple, J., R. Garcia, and M. Rindisbacher (8) Simulaion Mehods for Opimal Porfolios, andbooks in OR & MS, Vol. 5, p Duare, J., F. Longsaff, and F. Yu (7) Risk and Reurn in Fixed Income Arbirage: Nickels in fron of a Seamroller? he Review of Financial Sudies, Vol. (3), p Dybvig, P. (988) Disribuional Analysis of Porfolio Choice, Journal of Business, Vol. 6, p Fung, W. and D.A. sieh (997) Empirical Characerisics of Dynamic rading Sraegies: he Case of edge Funds, he Review of Financial Sudies, Vol. (), p Fung, W. and D.A. sieh () Performance Characerisics of edge Funds and Commodiy Funds: Naural vs. Spurious Biases he Journal of Financial and Quaniaive Analysis, Vol. 35(3), p Fung, W. and D.A. sieh () he Risk in edge Fund Sraegies: heory and Evidence from rend Followers, he Review of Financial Sudies, Vol. 4(), p Fung, W. and D.A. sieh (7a) Will edge Funds Regress owards Index-like Producs? Journal of Invesmen Managemen, Vol. 5(), p Fung, W. and D.A. sieh (7b) edge Fund Replicaion Sraegies: Implicaions for Invesors and Regulaors Banque de France, Financial Sabiliy Review, Special issue on hedge funds, Vol. April, p

12 A. akahashi, K. Yamamoo GJBR Vol. 4 No. 4 akamada,., A. akahashi, and K. Yamamoo (7) Selecion and Performance Analysis of Asia- Pacific edge Funds, he Journal of Alernaive Invesmens, Vol. (3), p. 7-9 Karazas, I. and S. Shreve (998) Mehods of Mahemaical Finance, Springer Ka,.M. and.p. Palaro (5) Who Needs edge Funds? A Copula-Based Approach o edge Fund Reurn Replicaion, Working Paper Lo, A. and J. asanhodzic (6) Can edge-fund Reurns Be Replicaed?: he Linear Case, Working Paper. Nualar, D. (6) he Malliavin Calculus and Relaed opics nd Ediion, Springer Papageorgiou, N., B. Remillard, and A. ocquard (8) Replicaing he Properies of edge Fund Reurns, he Journal of Alernaive Invesmens, Vol. (), p akahashi, A., and K. Yamamoo (9a) edge Fund Replicaion, he Recen rend of edge Fund Sraegies, Nova Science Publishers, forhcoming akahashi, A., and K. Yamamoo (9b). Generaing a arge Payoff Disribuion wih he Cheapes Dynamic Porfolio: An Applicaion o edge Fund Replicaion, Working Paper akahashi, A., and N. Yoshida (4) An Asympoic Expansion Scheme for Opimal Invesmen Problems, Saisical Inference for Sochasic Process, Vol. 7, pp ACKNOWLEDGEMEN his research was funded by he Global COE Program he Research and raining Cener for New Developmen in Mahemaics. BIOGRAPY Dr. Akihiko akahashi is a Professor of Economics a he Universiy of okyo. e can be conaced a: Graduae School of Economics, he Universiy of okyo 7-3-, ongo, Bunkyo-ku, okyo, Japan. Mr. Kyo Yamamoo is a Ph.D. suden a he Universiy of okyo and a research analys a GCI Asse Managemen. e can be conaced a: Graduae School of Economics, he Universiy of okyo 7-3-, ongo, Bunkyo-ku, okyo, Japan. ee773@mail.ecc.u-okyo.ac.jp, kyo.yamamoo@gci.jp 34