ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL

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1 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL XUE-ZHONG HE, KAI LI AND YOUWEI LI *Universiy of Technology Sydney, Business School PO Box 13, Broadway, NSW 7, Ausralia **School of Managemen Queen s Universiy Belfas, Belfas, UK ony.he1@us.edu.au, kai.li@us.edu.au, y.li@qub.ac.uk Absrac. We develop a coninuous-ime asse price model o capure he well documened ime series momenum and reversal effecs. The opimal asse allocaion sraegy is derived heoreically and esed empirically. We show ha, by combining wih marke fundamenals and iming opporuniy wih respec o marke rend and volailiy, he opimal sraegy based on he ime series momenum and reversal ouperforms significanly, boh in-sample and ou-of-sample, he S&P5 and pure sraegies based only on eiher ime series momenum or reversal. The resuls are robus for differen ime horizons. Furhermore, he ouperformance is immune o shor-sale consrains, marke saes, invesor senimen and marke volailiy. Key words: Momenum, reversal, opimal asse allocaion, performance JEL Classificaion: G1, G14, E3 Dae: March 18, 16. Acknowledgemen: We would like o hank Li Chen, Blake LeBaron, Michael Moore, Bern Øksendal, Lei Shi, Jun Tu, Zhen Wu and Chao Zhou, and conference and seminar paricipans a he 14 Inernaional Conference on CEF (Olso), 15 MFA Annual Meeing (Chicago), Beijing Normal Universiy, Cenral Universiy of Finance and Economics, Chinese Universiy of Hong Kong, Naional Universiy of Singapore, Queen s Universiy Belfas, Renmin Universiy, Shanghai Universiy of Finance and Economics, Tianjin Universiy, Universiy of Technology Sydney, Sun Ya-sen Universiy, Universiy of Urbino and York Universiy for helpful commens and suggesions. Financial suppor from he Ausralian Research Council (ARC) under Discovery Gran (DP13131) is graefully acknowledged. The usual caveas apply. 1

2 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL Asse Allocaion wih Time Series Momenum and Reversal Absrac We develop a coninuous-ime asse price model o capure he well documened ime series momenum and reversal effecs. The opimal asse allocaion sraegy is derived heoreically and esed empirically. We show ha, by combining wih marke fundamenals and iming opporuniy wih respec o marke rend and volailiy, he opimal sraegy based on he ime series momenum and reversal ouperforms significanly, boh in-sample and ou-of-sample, he S&P5 and pure sraegies based only on eiher ime series momenum or reversal. The resuls are robus for differen ime horizons. Furhermore, he ouperformance is immune o shor-sale consrains, marke saes, invesor senimen and marke volailiy. Key words: Momenum, reversal, opimal asse allocaion, performance. JEL Classificaion: G1, G14, E3

3 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 3 1. Inroducion Equiy reurn momenum in he shor-run and reversal in he long-run are wo of he mos prominen financial marke anomalies. Though marke iming opporuniies under mean reversion in equiy reurn are well documened (see, for example, Campbell and Viceira (1999) and Wacher ()), ime series momenum (TSM) ha characerizes srong posiive predicabiliy of a securiy s own pas reurns has been explored recenly in Moskowiz, Ooi and Pedersen (1). If an invesor incorporaes boh reurn momenum and reversal ino a rading sraegy opimally, he invesor would expec o ouperform he sraegies based only on reurn momenum or reversal, and even he marke index. This paper examines heoreically and empirically how o opimally explore ime series momenum and reversal in financial markes. We firs inroduce an asse price model in financial marke o incorporae momenum and reversal componens. By solving a dynamic asse allocaion problem, we derive he opimal invesmen sraegy of combining momenum and mean reversion in closed form, which includes pure momenum and pure mean-revering sraegies as special cases. By esimaing he model o monhly reurns of he S&P 5 index, we show ha he opimal sraegy ouperforms, measured by he uiliy of porfolio wealh and Sharpe raio, no only he sraegies based on he pure momenum and pure mean-reversion models bu also he S&P 5 index. This paper makes hree conribuions o he lieraure. Firsly, we find ha he performance of TSM sraegy can be significanly improved by combining wih marke fundamenals, while he performance of mean-revering sraegy can be significanly improved by combining wih TSM. To demonsrae he ouperformance of he opimal sraegy over he pure TSM sraegy, we derive a subopimal sraegy purely based on he TSM effec, showing ha his sraegy is no able o ouperform he marke and he opimal sraegy. Comparing he performance of he opimal sraegy wih he TSM sraegy used in Moskowiz e al. (1), we show ha he opimal sraegy ouperforms he TSM and passive holding sraegies. Essenially, in conras o a TSM sraegy based on rend only, he opimal sraegy akes ino accoun no only he rading signal based on momenum and fundamenals bu also he size of posiion associaed wih marke volailiy. Wihou considering he fundamenals, he pure momenum porfolio is highly leveraged, and hence suffers from high risk. We also derive anoher subopimal sraegy, he pure mean-revering sraegy, by ignoring he TSM effec. We find ha his sraegy based conservaively on fundamenal invesmens leads o a sable growh rae of porfolio wealh, bu is no able o explore he price rend, especially during exreme marke periods, and hence underperforms he opimal porfolio. In addiion o he above model-based

4 4 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL resuls, we furher invesigae he model-free performance of he opimal sraegy following Moskowiz e al. (1). We find ha our opimal sraegy ouperforms he TSM sraegy wih respec o he Sharpe raio and cumulaive excess reurn. Secondly, o he bes of our knowledge, his paper is he firs o heoreically examine he effec of he ime horizon of he TSM on he performance of he opimal porfolio. Empirically, TSM is measured by various moving averages over differen ime horizons wih fixed look-back periods, which play a very imporan role in he performance of momenum sraegies. This has been invesigaed exensively in he empirical lieraure. 1 However, dueoheechnical challenge, herearefewheoreical resuls concerning he effec of he ime horizon. The asse price model developed in his paper akes he ime horizon of TSM ino accoun direcly, which helps o examines he effec of ime horizon on he performance. As he resul, hisorical prices underlying he TSM componen affec asse prices, leading o a non-markov process characerized by sochasic delay differenial equaions (SDDEs). This is very differen from he Markov asse price process documened in he lieraure (Meron 1969, 1971). In he case of Markov processes, he sochasic conrol problem is mos frequenly solved using he dynamic programming mehod and HJB equaion. However, solving he opimal conrol problem for SDDEs becomes more challenging since i involves infinie-dimensional parial differenial equaions. One way o solve he problem is o apply a ype of Ponryagin maximum principle, which has been developed recenly by Chen and Wu (1) and Øksendal e al. (11) for he opimal conrol problem of SDDEs. By exploring hese laes advances in he heory of he maximum principle for conrol problems of SDDEs, we derive he opimal sraegy in closed form. This enables us o examine horoughly he impac of hisorical informaion on he performance of TSM rading sraegies based on moving averages over differen ime horizons. More ineresingly, we show ha he opimal sraegy based on he esimaed model performs he bes when he TSM is based on he moving averages of pas 9 o 1 monhs, which is consisen wih he empirical lieraure (see, for example, Moskowiz e al. (1)). Thirdly, we show ha, in addiion o price rend, posiion size is anoher very imporan facor for he opimal sraegy involving momenum rading. The opimal posiion size derived in his paper is deermined by rading signals and he level of marke volailiy. In he empirical lieraure, momenum rading only considers he rading signals of price rend and akes a consan posiion o rade. We show ha, if we only consider he sign of rading signals indicaed by he opimal sraegy and ake a uni posiion o rade, he sraegy is no able o ouperform he opimal sraegy (for all ime horizons). In addiion, he robusness of he performance of he opimal sraegy is also esed for differen sample periods, ou-of-sample 1 See, for example, De Bond and Thaler (1985) and Jegadeesh and Timan (1993).

5 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 5 predicions, shor-sale consrains, marke saes, invesor senimen and marke volailiy. This paper is closely relaed o he lieraure on reversal and momenum. Reversal is he empirical observaion ha asses performing well (poorly) over a long period end subsequenly o underperform (ouperform). Momenum is he endency of asses wih good (bad) recen performance o coninue ouperforming (underperforming) in he shor erm. Reversal and momenum have been documened exensively for a wide variey of asses. On he one hand, Fama and French (1988) and Poerba and Summers (1988), among many ohers, documen reversal for holding periods of more han one year, which induces negaive auocorrelaion in reurns. The value effec documened in Fama and French (199) is closely relaed o reversal, whereby he raio of an asse s price relaive o book value is negaively relaed o subsequen performance. Mean reversion in equiy reurns has been shown o induce significan marke iming opporuniies (Campbell and Viceira 1999, Wacher and Koijen, Rodríguez and Sbuelz 9). On he oher hand, he lieraure mosly sudies cross-secional momenum. 3 More recenly, Moskowiz e al. (1) invesigae TSM ha characerizes srong posiive predicabiliy of a securiy s own pas reurns. For a large se of fuures and forward conracs, Moskowiz e al. (1) find ha TSM based on excess reurns over he pas 1 monhs persiss for beween one and 1 monhs and hen parially reverses over longer horizons. They provide srong evidence for TSM based on he moving average of look-back reurns. This effec based purely on a securiy s own pas reurns is relaed o, bu differen from, he cross-secional momenum phenomenon sudied exensively in he lieraure. Through reurn decomposiion, Moskowiz e al. (1) show ha posiive auocovariance is he main driving force for TSM and cross-secional momenum effecs, while he conribuion of serial cross-correlaions and variaion in mean reurns is small. Inuiively, a sraegy aking ino accoun boh he shor-run momenum and long-run mean reversion in ime series should ouperform pure momenum and pure mean-reversion sraegies. In his paper, we provide a jusificaion o his inuiion heoreically and empirically. For insance, Jegadeesh (1991) finds ha he nex one-monh reurns can be negaively prediced by heir lagged muliyear reurns. Lewellen () shows ha he pas one-year reurns negaively predic fuure monhly reurns for up o 18 monhs. 3 Jegadeesh and Timan (1993) documen cross-secional momenum for individual U.S. socks, predicing reurns over horizons of 3 1 monhs using reurns over he pas 3 1 monhs. The evidence has been exended o socks in oher counries (Fama and French 1998), socks wihin indusries (Cohen and Lou 1), across indusries (Cohen and Frazzini 8), and he global marke wih differen asse classes (Asness, Moskowiz and Pedersen 13). See also Chuang and Ho (14) and Cremers and Pareek (15) on momenum rading.

6 6 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL The apparen persisen and sizeable profis of sraegies based on momenum and reversal have araced considerable aenion, and many sudies have ried o explain he phenomena. 4 This paper is largely moivaed by he empirical lieraure esing rading signals wih combinaions of momenum and reversal. 5 Asness, Moskowiz and Pedersen (13) highligh ha sudying value and momenum joinly is more powerful han examining each in isolaion. 6 Huang, Jiang, Tu and Zhou (13) find ha boh mean reversion and momenum can coexis in he S&P 5 index over ime. By aking boh mean reversion and ime series momenum direcly ino accoun, his paper develops an asse price model and demonsraes he explanaory power of he model hrough he ouperformance of he opimal sraegy. This paper is also largely inspired by Koijen, Rodríguez and Sbuelz (9), who propose a heoreical model in which sock reurns exhibi momenum and meanreversion effecs. This paper is however differen from Koijen e al. (9) in wo aspecs. Firsly, in Koijen e al. (9), he momenum is calculaed from he enire se of hisorical reurns wih geomerically decaying weighs, insead of a moving average wih a fixed look-back period. This effecively reduces he price dynamics o a Markovian sysem, and enables a horough analysis of he performance of he hedging demand implied by he model. In his paper, we follow he empirical lieraure and model TSM by he sandard moving average wih a fixed look-back period. Our model of momenum complemens in a unique way o he heoreical sudy of Koijen e al. (9) and many empirical sudies ha do no sudy sysemaically he role of moving averages wih differen look-back periods. We sudy explicily 4 Among which, he hree-facor model of Fama and French (1996) can explain long-run reversal bu no shor-run momenum. Barberis, Shleifer and Vishny (1998) argue ha hese phenomena are he resul of he sysemaic errors invesors make when hey use public informaion o form expecaions of fuure cash flows. Models Daniel, Hirshleifer and Subrahmanyam (1998), wih single represenaive agen, and Hong and Sein (1999), wih differen rader ypes, aribue he under reacion o overconfidence and overreacion o biased self-aribuion. Barberis and Shleifer (3) show ha syle invesing can explain momenum and value effecs. Sagi and Seasholes (7) presen an opion model o idenify observable firm-specific aribues ha drive momenum. Kelsey, Kozhan and Peng(1) explain momenum when invesors face Knighian uncerainy and reac differenly o pas winners and losers. Vayanos and Woolley (13) show ha slow-moving capial can also generae momenum. He and Li (15) find ha momenum sraegies can be self-fulfilling. 5 For example, Balvers and Wu (6) and Serban (1) show empirically ha a combinaion of momenum and mean-reversion sraegies can ouperform pure momenum and pure meanreversion sraegies for equiy markes and foreign exchange markes respecively. 6 Theyfindhaseparaefacorsforvalueandmomenumbesexplainhedaaforeighdifferen markes and asse classes. Furhermore, hey show ha momenum loads posiively and value loads negaively on liquidiy risk; however, an equal-weighed combinaion of value and momenum is immune o liquidiy risk and generaes subsanial abnormal reurns.

7 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 7 he impacs of differen look-back periods on he performance of momenum-relaed rading sraegies. Secondly, insead of sudying he economic gains of hedging due o momenum in Koijen e al. (9), we focus more on he performance of he opimal sraegy comparing wih he marke, TSM, and mean-reversion rading sraegies. The paper is organized as follows. We firs presen he model and derive he opimal asse allocaion in Secion. In Secion 3, we esimae he model o he S&P 5 and conduc a performance analysis of he opimal porfolio and examine he impac of hedging demand. We hen invesigae he ime horizon effec in Secion 4. Secion 5 concludes. All he proofs and robusness analysis are included in he appendices.. The Model and Opimal Asse Allocaion In his secion, we inroduce an asse price model and sudy he opimal invesmen decision problem. We consider a financial marke wih wo radable securiies, a riskless asse B saisfying db B = rd (.1) wih a consan riskless rae r, and a risky asse. Le S be he price of he risky asse or he level of a marke index a ime where dividends are assumed o be reinvesed. Empirical sudies on reurn predicabiliy, see for example Fama (1991), have shown ha he mos powerful predicive variables of fuure sock reurns in he US are pas reurns, dividend yield, earnings/price raio, and erm srucure variables. More imporanly, reurns in financial markes display he shor-run momenum and long-run reversal, as we have discussed in he previous secion. Following his lieraure and moivaed by Koijen e al. (9), we model he expeced reurn by a combinaion of a momenum erm m based on moving average of he pas reurns and a long-run mean-reversion erm µ based on marke fundamenals such as dividend yield. Consequenly, we assume ha he sock price S follows ds S = [ φm +(1 φ)µ ] d+σ S dz, (.) where φ is a consan, 7 measuring he weigh of he momenum componen m, σ S is a wo-dimensional volailiy vecor (and σ S sands for he ranspose of σ S), and Z is a wo-dimensional vecor of independen Brownian moions. The uncerainy is represened by a filered probabiliy space (Ω,F,P,{F } ) on which he wodimensional Brownian moion Z is defined. As usual, he mean-reversion process 7 The dominance of marke fundamenals and TSM, measured by φ, can be ime-varying, depending on marke condiion. For simpliciy we ake φ as a consan parameer in his paper.

8 8 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL µ is defined by an Ornsein-Uhlenbeck process, dµ = α( µ µ )d+σ µdz, α >, µ >, (.3) where µ is he consan long-run expeced reurn, α measures he rae a which µ converges o µ, and σ µ is a wo-dimensional volailiy vecor. The momenum erm m is defined by a sandard moving average (MA) of pas reurns over [,], m = 1 ds u S u, (.4) where delay represens he ime horizon. The way we model he momenum inhis paper is moivaed by he TSM sraegy documened recenly in Moskowiz e al. (1), who demonsrae ha he average reurn over a pas period(say, 1 monhs) is a posiive predicor of fuure reurns, especially he reurn for he nex monh. The resuling asse price model (.) (.4) is characerized by a sochasic delay inegro-differenial sysem, which is non-markovian and has he following propery. Proposiion.1. The sysem (.)-(.4) has an almos surely coninuously adaped pahwise unique soluion (S,µ) for a given F -measurable iniial process ϕ : Ω C([,],R). Furhermore, if ϕ > for [,] almos surely, hen S > for all almos surely. We now consider a ypical long-erm invesor who maximizes he expeced uiliy of erminal wealh a ime T(> ). Le W be he wealh of he invesor a ime and π be he fracion of he wealh invesed in he sock. Then i follows from (.) ha he change in wealh saisfies dw W = [π [φm +(1 φ)µ r]+r]d+π σ S dz. (.5) Weassume hahepreferences ofheinvesor canberepresened byacrrauiliy index wih a consan coefficien of relaive risk aversion equal o γ. The invesmen problem of he invesor is hen given by [ W 1 γ T 1 ] J(W,m,µ,,T) = sup E, (.6) (π u) u [,T] 1 γ where J(W,m,µ,,T) is he value funcion corresponding o he opimal invesmen sraegy. We apply he maximum principle for opimal conrol of sochasic delay differenial equaions and derive he opimal invesmen sraegy in closed form. Proposiion.. For an invesor wih an invesmen horizon T and consan coefficien of relaive risk aversion γ, he opimal wealh fracion invesed in he risky asse is given by π u = φm u +(1 φ)µ u r γσ S σ S + (z u) 3 σ S γp 3 u σ S σ, (.7) S

9 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 9 where z u and p u are governed by a backward sochasic differenial sysem (B.6) in Appendix B.. Especially, when γ = 1, he preference is characerized by a log uiliy and he opimal allocaion o socks is given by π = φm +(1 φ)µ r σ S σ. (.8) S This proposiion saes ha he opimal fracion (.7) invesed in he sock consiss of wo componens. The firs characerizes he myopic demand for he sock and he second is he ineremporal hedging demand. When γ = 1, he opimal sraegy (.8) characerizes he myopic behavior of he invesor wih log uiliy. This resul has a number of implicaions. Firsly, when he asse price follows a geomeric Brownian moion process wih mean-reversion drif µ, namely φ =, he opimal invesmen sraegy (.8) becomes π = µ r σ S σ. (.9) S This is he opimal invesmen sraegy wih mean-revering reurns obained in he lieraure, say for example Campbell and Viceira (1999) and Wacher (). In paricular, when µ = µ is a consan, he opimal porfolio (.9) collapses o he opimal porfolio of Meron (1971). Secondly, when he asse reurn depends only on he momenum, namely φ = 1, he opimal porfolio (.8) reduces o π = m r σ S σ. (.1) S If we consider a rading sraegy based on he rading signal indicaed by he excess moving average reurn m r only, wih = 1 monhs, he sraegy of long/shor when he rading signal is posiive/negaive is consisen wih he TSM sraegy used in Moskowiz e al. (1). By consrucing porfolios based on monhly excess reurns over he pas 1 monhs and holding for one monh, Moskowiz e al. (1) show ha his sraegy performs he bes among all he momenum sraegies wih look-back and holding periods varying from one monh o 48 monhs. Therefore, if we only ake fixed long/shor posiions and consruc simple buy-and-hold momenum sraegies over a large range of look-back and holding periods, (.1) shows ha he TSM sraegy of Moskowiz e al. (1) can be opimal when mean reversion is no significan in financial markes. On he one hand, his provides a heoreical jusificaion for he TSM sraegy when marke volailiies are consan and reurns are no mean-revering. On he oher hand, noe ha he opimal porfolio (.1) also depends on volailiy. This explains he dependence of momenum profiabiliy on marke saes and volailiy documened in empirical sudies(hou, Peng and Xiong (9) and Wang and Xu (15)). In addiion, he opimal porfolio (.1) defines he opimal wealh fracion invesed in he risky asse. Hence

10 1 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL he TSM sraegy of aking fixed posiions based on he rading signal may no be opimal in general. Thirdly, he opimal sraegy (.8) implies ha a weighed average of momenum and mean-revering sraegies is opimal. Inuiively, i akes ino accoun he shorrun momenum and long-run reversal, boh well-suppored marke phenomena. I also akes ino accoun he iming opporuniy wih respec o marke rend and volailiy. In summary, for he firs ime, we have provided a heoreical suppor for opimal sraegies ha combining of momenum and reversal documened in he empirical lieraure (see, for example, Balvers and Wu (6) and Serban (1)). In he res of he paper, we firs esimae he model o he S&P 5 and hen evaluae empirically he performance of he opimal sraegy comparing i o he marke and oher rading sraegies recorded in he lieraure. In order o provide a beer undersanding of he performance, we firs consider γ = 1. The closed-form opimal sraegy (.8) faciliaes model esimaion and empirical analysis. We hen numerically solve he opimal porfolio (.7) and examine he values added by he hedging demand in Secion Model Esimaion and Performance Analysis In his secion we firs esimae he model o he S&P 5. Based on hese esimaions, we examine he performance of he opimal sraegy (.8) wih respec o he log uiliy (γ = 1) of porfolio wealh and he Sharpe raio, comparing o he performance of he marke index and he opimal sraegies based on pure momenum and pure mean-reversion models. To provide furher evidence, we conduc ou-of-sample ess on he performance of he opimal sraegy and examine he effec of shor sale consrains, marke saes, senimen, volailiy, and hedging (when γ 1). In addiion, we also compare he performance of he opimal sraegy o ha of he TSM sraegy Model Esimaion. In line wih Campbell and Viceira (1999) and Koijen e al. (9), he mean-reversion variable is affine in he (log) dividend yield, µ = µ+ν(d µ D ) = µ+νx, where ν is a consan, D is he (log) dividend yield wih E(D ) = µ D, and X = D µ D denoeshede-meaneddividendyield. Thusheassepricemodel(.)-(.4) becomes ds = [ φm +(1 φ)( µ+νx ) ] d+σ S S dz, dx = αx d+σ X dz, (3.1)

11 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 11 where σ X = σ µ /ν. The uncerainy in sysem (3.1) is driven by wo independen Brownian moions. Wihou loss of generaliy, we follow Sangvinasos and Wacher (5) and assume he Cholesky decomposiion on he volailiy marix Σ of he dividend yield and reurn, Σ = ( σ S σ X ) = ( σ S(1) σ X(1) σ X() Thus, hefirs elemen of Z is heshock o hereurn andhesecond is hedividend yield shock ha is orhogonal o he reurn shock. To be consisen wih he momenum and reversal lieraure, we discreize he coninuous-ime model (3.1) a a monhly frequency. This resuls in a bivariae Gaussian vecor auoregressive (VAR) model on he simple reurn 8 R and dividend yield X, R +1 = φ (R +R 1 + +R +1 )+(1 φ)( µ+νx )+σ S Z +1, (3.) X +1 = (1 α)x +σ X Z +1. Noe ha boh R and X are observable. We use monhly S&P 5 daa over he period January 1871 December 1 from he home page of Rober Shiller ( shiller/daa.hm) and esimae model (3.) using he maximum likelihood mehod. We se he insananeous shor rae r = 4% annually. As in Campbell and Shiller (1988a, 1988b), he dividend yield is defined as he log of he raio beween he las period dividend and he curren index. The oal reurn index is consruced by using he price index series and he dividend series. The esimaions are conduced separaely for given ime horizon varying from one o 6 monhs. Empirically, Moskowiz e al. (1) show ha he TSM sraegy based on a 1-monh horizon beer predics he nex monh s reurn han oher ime horizons. Therefore, in his secion, we follow Moskowiz e al. (1) and focus on he performance of he opimal sraegy wih a look-back period of = 1 monhs and a one-monh holding period. The effec of ime horizon varying from one o 6 monhs is examined in he nex secion. For comparison, we esimae he full model (FM) (3.) wih < φ < 1, he pure momenum model (MM) wih φ = 1, and he pure mean-reversion model (MRM) wih φ =. For = 1, Table 3.1 repors he esimaed parameers, ogeher wih he 95% confidence bounds. For he pure momenum model (φ = 1), here is only oneparameerσ S(1) obeesimaed. Forhefullmodel, asoneofhekeyparameers, i shows ha he momenum effec parameer φ., which is significanly differen from zero. This implies ha he marke index can be explained by abou % 8 To be consisen wih he momenum and reversal lieraure, we use simple reurn o consruc m and also discreize he sock price process ino simple reurn raher han log reurn. ).

12 1 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL Table 3.1. Parameer esimaions of he full model (FM), pure momenum model (MM) wih = 1, and pure mean-reversion model (MRM). Parameer α φ µ ν FM (%) Bounds (%) (.3,.95) (8.7, 31.) (.6,.46) (-.6, 1.) MM (%) Bounds (%) MRM (%) Bounds (%) (.7, 1.3) (.31,.43) (-.46,.46) Parameers σ S(1) σ X(1) σ X() FM (%) Bounds (%) (3.95, 4.4) (-4.4, -3.93) (1.9, 1.39) MM (%) 4.3 Bounds (%) (4.9, 4.38) MRM (%) Bounds (%) (3.97, 4.5) (-4., -3.9) (1.3, 1.4) of he momenum componen and 8% of he mean-revering componen. 9 Oher parameer esimaes in erms of he level and significance in Table 3.1 are consisen wih hose in Koijen e al. (9). We also conduc a log-likelihood raio es o compare he full model ( < φ < 1) o he pure momenum model (φ = 1) and pure mean-reversion model (φ = ), showing ha he full model is significanly beer han he pure momenum model and he pure mean-reversion model. 1 This implies ha he (full) model capures shor-erm momenum and long-erm reversion in he marke index and fis he daa beer han he pure momenum and pure mean-revering models. 3.. Performance Analysis. Based on he previous esimaions, we examine he performance of he opimal porfolio (.8) based on log uiliy in erms of he uiliy of he porfolio wealh and Sharpe raio, comparing o hose of he marke index and of he pure momenum and pure mean-reversion models. We firs compare he realized uiliy of he opimal porfolio wealh invesed in he S&P 5 index based on he opimal sraegy (.8) wih a look-back period = 1 9 This is consisen wih Chu, He, Li and Tu (15) showing ha abou % of imes he marke index reurn can be explained by non-fundamenal variables, including various momenum variables, while abou 8% of imes can be explained by fundamenal variables. 1 The es saisic 131 (6) is much greaer han 1.59 (3.841), he criical value wih six (five) degrees of freedom a he 5% significance level, for he pure momenum (mean-reversion) model.

13 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL P /1876 1/196 1/1976 1/1 (a) Marke index R /1876 1/196 1/1976 1/1 (b) Marke reurn ln W * ln W π * /1876 1/196 1/1976 1/1 (c) Porfolio weigh 5 1/1876 1/196 1/1976 1/1 (d) Uiliy Figure 3.1. The ime series of marke index (a), he simple reurn of he S&P 5 (b); and he ime series of he opimal porfolio (c) and he uiliy (d) of he opimal porfolio wealh from January 1876 unil December 1 for = 1. In (d), he uiliies of he opimal porfolio wealh and he marke index are ploed in solid red and dash-doed blue lines respecively. monhs and one-monh holding period o he uiliy of a passive holding invesmen in he S&P 5 index wih an iniial wealh of $1. As a benchmark, he log uiliy of an invesmen of $1 a index from January grows o a December 1. For = 1, we calculae he moving average m of pas 1-monh reurns a any poin of ime based on he marke index from January 1876 o December 1. Wih an iniial wealh of $1 a January 1876 and he esimaed parameers in Table 3.1, we calculae he monhly invesmen of he opimal porfolio wealh W based on (.8) and record he realized uiliies of he opimal porfolio wealh from January 1876 o December 1. Based on he calculaion, we plo he index level and simple reurn of he S&P 5 index in Fig. 3.1 (a) and (b). Fig. 3.1 (c) repors 11 Considering he robusness analysis for varying from one o 6 monhs in he nex secion, all he porfolios sar a he end of January 1876 (6 monhs afer January 1871).

14 14 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL he opimal wealh fracions π of (.8) and Fig. 3.1 (d) repors he evoluion of he uiliies of he opimal porfolio wealh over he same ime period, showing ha he opimal porfolios ouperform he marke index measured by he uiliy of wealh * π ln W * ln W 5 1/1876 1/196 1/1976 1/1 (a) Porfolio weigh of MM 3 1/1876 1/196 1/1976 1/1 (b) Uiliy of MM * π ln W * ln W.144 1/1876 1/196 1/1976 1/1 (c) Porfolio weigh of MRM 1 1/1876 1/196 1/1976 1/1 (d) Uiliy of MRM Figure 3.. The ime series of he opimal porfolio weigh and he uiliy of he wealh for he pure momenum model wih = 1 (a) and (b) and he pure mean-reversion model (c) and (d) from January 1876 unil December 1. In (d), he uiliies of he opimal porfolio wealh and he marke index are ploed in solid red and dash-doed blue lines respecively. There are wo ineresing observaions from Fig Firsly, he reurns of opimal sraegies and index are posiively correlaed (wih a correlaion of.335). Secondly, Fig. 3.1 (d) indicaes big jumps in he uiliies of he opimal porfolio during he period of he Grea Depression in he 193s. This observaion is consisen wih Moskowiz e al. (1), who find ha he TSM sraegy delivers is highes profis during he mos exreme marke episodes. However, he performance of he opimal porfolio is no compleely driven by is performance during crisis periods. 1 1 To clarify his observaion, we also examine he performance using daa from January 1945 o December 1 o avoid he Grea Depression periods. We re-esimae he model, conduc he

15 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 15 Nex, we compare he performance of he pure momenum and pure mean-revering sraegies o ha of he marke index. For he pure momenum model, based on he esimaed parameers in Table 3.1, Fig. 3. (a) and (b) illusrae he ime series of he porfolio weighs and he uiliies of he opimal porfolio for he pure momenum model from January 1876 o December 1. Compared o he full model illusraed in Fig. 3.1, he leverage of he pure momenum sraegies is much higher, as indicaed by he higher level of π. The opimal sraegies for he pure momenum model suffer from high risk and perform worse han he marke and hence he opimal sraegies of he full model. Similarly, based on he esimaes in Table 3.1, Fig. 3. (c) and (d) show ha he performance of he pure mean-reversion sraegy is abou he same as he marke index bu worse han he opimal sraegies (.8). Noe ha in his case here is no much variaion in he porfolio weigh and he opimal porfolio does no capure he iming opporuniy of he marke rend and marke volailiy. Therefore, boh he pure momenum and pure mean-reversion sraegies underperform he marke and he opimal sraegy of he full model ln W * ln W One Side es Saisics /1876 1/196 1/1976 1/1 (a) Average uiliy 1/1876 1/196 1/1976 1/1 (b) One-sided saisics Figure 3.3. (a) Average uiliy ((he solid red line), he 95% confidence bounds (he solid green lines) and he 6% confidence bounds (he doed blue lines) and (b) one-sided -es saisics based on 1, simulaions for = 1. To provide furher evidence of he opimal sraegy, we conduc a Mone Carlo analysis. For = 1 and he esimaed parameers, we simulae model (3.1) and repor he average porfolio uiliies (he solid red line in he middle) based on 1, simulaions in Fig. 3.3 (a), ogeher wih 95% confidence levels (he wo solid green lines ouside), comparing o he uiliy of he marke index (he doed blue same analysis. Our resuls (no repored here) show ha he opimal sraegies sill ouperform he marke index over his ime period. This indicaes ha he ouperformance of he opimal sraegy is no necessarily due o exreme marke episodes, such as he Grea Depression. Laer in his secion, we show ha he ouperformance is in fac immune o marke saes.

16 16 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL line). I shows ha firsly, he average uiliies of he opimal porfolios are beer han ha of he S&P 5. Secondly, he uiliy for he S&P 5 falls ino he 95% confidence bounds and hence he average performance of he opimal sraegy is no saisically differen from he marke index a he 95% confidence level. We also plowo blackdashed bounds forhe6%confidence level. I shows ha, ahe6% confidence level, he opimal porfolio significanly ouperforms he marke index. Fig. 3.3 (b) repors he one-sided -es saisics o es lnw > lnw SP5. The -saisics are above.84 mos of he ime, which indicaes a criical value a 8% confidence level. Therefore, wih 8% confidence, he opimal porfolio significanly ouperforms he marke index. In summary, we have provided empirical evidence of he ouperformance of he opimal sraegy (.8) compared o he marke index, pure momenum and pure mean-reversion sraegies. Table 3.. The Sharpe raios of he opimal porfolio and he marke index wih corresponding 9% confidence inerval and he Sharpe raio of he opimal porfolio based on Mone Carlo simulaions. Opimal porfolio Marke index Mone Carlo Sharpe raio (%) Bounds (%) (1.86, 9.84) (-1.88, 6.1) (5.98, 6.7) We now use he Sharpe raio o examine he performance of he opimal sraegy. The Sharpe raio is defined as he raio of he mean excess reurn and he sandard deviaion of he porfolio. When he Sharpe raio of an acive sraegy exceeds he marke Sharpe raio, we say ha he acive porfolio ouperforms or dominaes he marke porfolio (in an uncondiional mean-variance sense). For empirical applicaions, he (ex-pos) Sharpe raio is usually esimaed as he raio of he sample mean of he excess reurn on he porfolio and he sample sandard deviaion of he porfolio reurn (Marquering and Verbeek 4). The average monhly reurn on he oal reurn index of he S&P 5 over he period January 1871 December 1 is.4% wih an esimaed (uncondiional) sandard deviaion of 4.11%. The Sharpe raio of he marke index is.11%. For he opimal sraegy (.8), he reurn of he opimal porfolio wealh a ime is given by R = (W W 1 )/W 1 = π 1 R +(1 π 1 )r. (3.3) Table 3. repors he Sharpe raios of he passive holding marke index porfolio and heopimal porfoliosfromjanuary 1886o December 1for = 1 ogeher wih heir 9% confidence inervals(see Jobson and Korkie 1981). I shows ha, by aking he iming opporuniy (wih respec o he marke rend and marke volailiy), he opimal porfolio ouperforms he marke. We also conduc a Mone Carlo analysis based on 1, simulaions and obain an average Sharpe raio of 6.1% for he

17 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 17 opimal porfolio. The resul is consisen wih he ouperformance of he opimal porfolio measured by porfolio uiliy(wih an average erminal uiliy of 8.71 for he opimal porfolio). In summary, using wo performance measures, we have provided empirical evidence of he ouperformance of he opimal sraegy (.8) compared o he marke index, pure momenum and pure mean-reversion sraegies. The resuls provide empirical suppor for he analyical resul on he opimal sraegy derived in Secion Ou-of-Sample Tess. We implemen a number of ou-of-sample ess on he performance of he opimal sraegies by spliing he whole daa se ino wo subsample periods and using he firs sample period o esimae he model. We hen apply he esimaed parameers o he second porion of he daa o examine he ou-of-sample performance of he opimal sraegies. In he firs es, we spli he whole daa se ino wo equal periods: January 1871 o December 1941 and January 194 o December 1. Noice he daa in he wo periods are quie differen; he marke index increases gradually in he firs period buflucuaeswidelyinhesecondperiodasillusraedinfig. 3.1(a). Wih = 1, Fig. 3.4 (a) and (b) illusrae he corresponding ime series of he opimal porfolio and he uiliy of he opimal porfolio wealh from January 194 o December 1, showing ha he uiliy of he opimal sraegy grows gradually and ouperforms he marke index. Many sudies (see, for example, Jegadeesh and Timan 11) show ha momenum sraegies perform poorly afer he subprime crisis in 8. In he second es, we use he subprime crisis o spli he whole sample period ino wo periods and focus on he performance of he opimal sraegies afer he subprime crisis. The resuls are repored in Fig. 3.4 (c) and (d). I is clear ha he opimal sraegy sill ouperforms he marke over he sub-sample period, in paricular, during he financial crisis period around 9 by aking large shor posiions in he opimal porfolios. We also use daa from he las 1 years and years as he ou-of-sample es and find he resuls (no repored here) are robus. As he hird es, we implemen he rolling window esimaion procedure o avoid look-ahead bias. For = 1, we esimae parameers a each monh by using he pas years daa and repor he resuls in Fig. C.1 in Appendix C. We hen repor he ime series of he index level (a), he simple reurn of he S&P 5 (b), he opimal porfolio (c), and he uiliy of he opimal porfolio wealh (d) in Fig. C. of Appendix C, showing a srong performance of he opimal porfolios over he marke. We also implemen he ou-of-sample ess for he pure momenum and pure mean-reversion models (no repored here) and find ha hey canno ouperform

18 18 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 4 3 * 1 π ln W * ln W 3 1/194 1/197 1/1 (a) The porfolio weigh: 1/194 1/1 1 1/194 1/197 1/1 (b) The uiliy: 1/194 1/ * π ln W * ln W 6 1/8 1/9 1/1 1/11 1/1 1/1 (c) The porfolio weigh: 1/8 1/1 1 1/8 1/9 1/1 1/11 1/1 1/1 (d) The uiliy: 1/8 1/1 Figure 3.4. The ime series of ou-of-sample opimal porfolio weighs and uiliy of he opimal porfolio wealh (he solid lines) from January 194 unil December 1 in (a) and (b) and from January 8 o December 1 in (c) and (d) wih = 1 compared o he uiliy of he marke index (he doed line). he marke in mos ou-of-sample ess (over he las 1, and 71 years), bu do ouperform he marke for ou-of-sample ess over he las five years. We also repor he resuls of ou-of-sample ess of he pure momenum and he pure mean reversion in Fig. C.3 based on he -year rolling window esimaes. Overall, he ou-of-sample ess demonsrae he robusness of he ouperformance of he opimal rading sraegies compared o he marke index, pure momenum and pure meanreversion sraegies Shor-sale Consrain. Invesors ofen face shor-sale consrain. To evaluae he performance of he opimal sraegies wihou shor selling and borrowing (a he risk-free rae), we resric he porfolio weigh π [.1]. Since he value 13 We also implemen he esimaions for differen window sizes of 5, 3 and 5 years and find ha he performance of sraegies is similar o he case of -year rolling window esimaion.

19 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 19 funcion is concave wih respec o π, he opimal sraegy becomes, if π <, Π = π, if π 1, (3.4) 1, if π > 1. Table 3.3. The erminal uiliy of he porfolio wealh, Sharpe raio, mean and sand deviaion of he porfolio weighs for he opimal porfolios wih and wihou shor-sale consrain, comparing wih he marke index porfolio. Uiliy Sharpe Raio Average weighs Sd of weighs Wih consrains Wihou consrains Marke index Table 3.3 repors he erminal uiliies and he Sharpe raios of he opimal porfolio wih and wihou shor-sale consrain, compared o he passive holding marke index porfolio. The resuls show ha he opimal porfolio wih shor-sale consrain ouperforms he marke, even he opimal porfolio wihou shor-sale consrain under he Sharpe raio. I seems ha he consrain improves porfolio performance. This less-inuiive observaion is acually consisen wih Marquering and Verbeek (4 p. 419) who argue ha While i may seem counerinuiive ha sraegies perform beer afer resricions are imposed, i should be sressed ha he unresriced sraegies are subsanially more affeced by esimaion error. Indeed, we see from Table 3.3 ha he esimaed opimal porfolio weigh wihou consrain has higher sandard error bu lower mean level han ha wih consrain Marke Saes, Senimen and Volailiy. The cross-secional momenum lieraure has shown ha momenum profiabiliy can be affeced by marke saes, invesor senimen and marke volailiy. For example, Cooper, Guierrez and Hameed (4) find ha shor-run (six monhs) momenum sraegies make profis in an up marke and lose in a down marke, bu he up-marke momenum profis reverse in he long run (13 6 monhs). Hou e al. (9) find momenum sraegies wih a shor ime horizon (one year) are no profiable in a down marke, bu are profiable in an up marke. Similar resuls are also repored in Chordia and Shivakumar (), specifically ha common macroeconomic variables relaed o he business cycle can explain posiive reurns o momenum sraegies during expansionary periods and negaive reurns during recessions.

20 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL To invesigae he performance of opimal sraegies under differen marke saes, wefollowcoopereal.(4), Houeal.(9) 14 andreporheresulsinappendix D. We see from Table D.1 ha he uncondiional average excess reurn is 87 basis poins per monh. In up monhs, he average excess reurn is 81 basis poins and i is saisically significan. In down monhs, he average excess reurn is 11 basis poins; his value is economically significan alhough i is no saisically significan. The difference beween down and up monhs is abou basis poins, which is no significanly differen from zero, based on a wo-sample -es (p-value of.87). Conrolling for marke risk, we use an up-monh dummy 15 o capure incremenal average reurn in up marke monhs relaive o down marke monhs. We repor he regression resuls in Table D. for he opimal sraegy, he pure momenum sraegy, pure mean-reversion sraegy and he TSM sraegy in Moskowiz e al. (1) for = 1 respecively. Excep for he TSM, which earns significan posiive reurns in down markes, boh down marke reurns α and he incremenal reurns in up marke κ are insignifican for all oher sraegies. These resuls are consisen wih hose in Table D.1. We also conrol for marke risk in up and down monhs separaely and obain similar resuls in Table D.. Oher way o see effecs of marke sae on porfolio reurns is o look a is predicive powers. Table D.3 repors predicive regression resuls of excess porfolio reurns on he up-monh dummy. We see ha up marke has no addiional predicive power o porfolio reurns over down marke (insignifican κ), down marke has significan predicive power o TSM reurns. Also down marke has insignifican predicive power o he full model, pure momenum, and pure mean reversion, bu among hem, he effec is relaively srong in he full model, and weak in he pure mean reversion. We obain similar resuls for he CAPM-adjused reurn. In erms of he effecs of invesor senimen and marke volailiy on porfolio performance, Baker and Wurgler (6, 7) find ha invesor senimen affecs cross-secional sock reurns and he aggregae sock marke. Wang and Xu (15) find ha marke volailiy has significan power o forecas momenum profiabiliy. For TSMs, however, Moskowiz e al. (1) find ha here is no significan relaionship of TSM profiabiliy o eiher marke volailiy or invesor senimen. We 14 We define marke sae using he cumulaive reurn of he sock index (including dividends) over he mos recen 36 monhs. We label a monh as an up (down) marke monh if he hree-year reurn of he marke is non-negaive (negaive). We compue he average reurn of he opimal sraegy, compare he average reurns beween up and down marke monhs. 15 The resuls are robus when we replace he up-monh dummy wih he lagged marke reurn over he previous 36 monhs (no repored here).

21 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 1 find ha boh invesor senimen and marke volailiy have no predicive power on porfolio reurns (see Tables D.4 and D.5 in Appendix D). Overall, we find ha reurns of he opimal sraegies are no significanly differen inupanddownmarke saes. Wealso findha bohinvesor senimen andmarke volailiy have no predicive power for he reurns of he opimal sraegies. In fac, he opimal sraegies have aken hese facors ino accoun and hence he reurns of he opimal sraegies have no significan relaionship wih hese facors. Therefore, he opimal sraegies are immune o marke saes, invesor senimen and marke volailiy Comparison wih TSM. We now compare he performance of he opimal sraegy o he TSM sraegy of Moskowiz e al. (1). The momenum sraegies in he empirical sudies are based on rading signals only. We firs verify he profiabiliy of he TSM sraegies and hen examine he excess reurn of buy-andhold sraegy when he posiion is deermined by he sign of he opimal porfolio sraegies (.8) wih differen combinaions of ime horizon and holding period h. For a given look-back period, we ake long/shor posiions based on he sign of he opimal porfolio (.8). Then for a given holding period h, we calculae he monhly excess reurn of he sraegy (,h) and repor he resuls in Appendix E. Table E.1 repors he average monhly excess reurn (%) of he opimal sraegies, skipping one monh beween he porfolio formaion period and holding period o avoid he one-monh reversal in sock reurns, for differen look-back periods (in he firs column) and differen holding periods (in he firs row). The average reurn is calculaed in he same way as in Moskowiz e al. (1). We calculae he excess reurns of he opimal sraegies over he period from January 1881 (1 years afer January 1871 wih five years for calculaing he rading signals and five years for holding periods) o December 1. For comparison, Table E. repors he average reurns (%) for he pure momenum model. 16 Noice ha Tables E.1 and E. indicae ha sraegy (,h) = (9,1) performs he bes. This is consisen wih he finding in Moskowiz e al. (1) for equiy markes alhough he 1-monh horizon is he bes for mos asse classes. Nex we use he Sharpe raio o examine he performance of he opimal sraegy π of (.8) and compare i o he passive index sraegy and wo sraegies: one follows from TSM in Moskowiz e al. (1) and he oher is based on he sign of he opimal sraegies sign(π ) as he rading signal (insead of he average excess reurn over a pas period), which is called momenum and mean-reversion (MMR) sraegy for convenience. For a ime horizon of = 1 monhs, we repor he 16 Noice he posiion is compleely deermined by he sign of he opimal sraegies. Therefore, he posiion used in Table E. is he same as ha of he TSM sraegies in Moskowiz e al. (1).

22 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL Table 3.4. The Sharpe raio of he opimal porfolio, marke index, TSM and MMR for = 1 wih corresponding 9% confidence inerval. Opimal porfolio Marke index TSM MMR Sharpe raio (%) Bounds (%) (1.86, 9.84) (-1.88, 6.1) (-4.1, 3.96) (.18, 8.15) Sharpe raios of he porfolios for he four sraegies in Table 3.4 from January 1881 o December 1. I shows ha he TSM sraegy underperforms he marke while he MMR sraegy ouperforms i. The opimal sraegy also significanly ouperforms all he momenum, mean-reversion and TSM sraegies. Noe ha he only difference beween he opimal sraegy and he MMR sraegy is ha he former considers he size of he porfolio posiion, which is inversely proporional o he variance, while he laer always akes one uni of long/shor posiion. This implies ha, in addiion o rends, he posiion size is anoher very imporan facor for invesmen profiabiliy. 1 8 Opimal Sraegy Passive Long TSM Sraegy Growh of $1 (log scale) Figure 3.5. Log cumulaive excess reurn of he opimal sraegy and momenum sraegy wih = 1 and passive long sraegy from January 1876 o December 1. Following Moskowiz e al. (1), we also examine he cumulaive excess reurn defined by ˆR +1 = sign(π ).144 ˆσ S, R +1, ˆσ S, = 1 (1 δ)δ i (R 1 i R ), where.144 is he sample sandard deviaion of he oal reurn index, ˆσ S, is he ex-ane annualized variance for he oal reurn index calculaed as he exponenially weighed lagged squared monh reurns wih he consan 1 o scale he variance annually, and R is he exponenially weighed average reurn based on he weighs (1 δ)δ i. The parameer δ is chosen so ha he cener of mass of he weighs is i=

23 ASSET ALLOCATION WITH TIME SERIES MOMENTUM AND REVERSAL 3 i=1 (1 δ)δi = δ/(1 δ) = wo monhs. To avoid look-ahead bias conaminaing he resuls, we use he volailiy esimaes a ime for ime +1 reurns hroughou he analysis. Wih a 1-monh ime horizon Fig. 3.5 illusraes he log cumulaive excess reurn of he opimal sraegy (.8), he momenum sraegy and he passive long sraegy from January 1876 o December 1. I shows ha he opimal sraegy has he highes growh rae and he passive long sraegy has he lowes growh rae. The paern of Fig. 3 in Moskowiz e al. (1 p.39) is replicaed in Fig. 3.5, showing ha he TSM sraegy ouperforms he passive long sraegy. 17 In summary, we have shown ha he opimal sraegy ouperforms he TSM sraegy of Moskowiz e al. (1). By comparing he performance of TSM and MMR, we find ha he he fixed posiion sraegy based on momenum and reversal rading signal is more profiable han he pure TSM sraegy of Moskowiz e al. (1) Hedging Demand. Taking he advanage of he closed-form soluion, previous discussions concenrae on he case of γ = 1. For γ 1, he opimal porfolio weigh is he sum of myopic and hedging demands. In his case, he opimal sraegy (.7) is deermined by a coupled forward backward sochasic differenial equaions (FBSDEs) wih ime delay, o which here is no efficien way o find numerical soluion (Ma and Yong, 1999 and Delong, 13). Given he curren sae of he ar of FBSDEs wih ime delay, we are only able o conduc a limied exploraory analysis on he hedging demand. We follow he Picard ieraions scheme developed in Bender and Denk(7). Due o he non-markovian srucure of ime-delayed BSDEs, he condiional expecaion in (B.6) in Appendix B has o be aken wih respec o he whole informaion F. Therefore, we esimae he expeced values by approximaing he Brownian moion by a symmeric random walk as in Ma, Proer, Marin and Torres (). Specifically, we firs simulae he T rajecories of he forward processes S and µ forfrom1ot basedonheapproximaingbinomialrandomwalk. Theparameers are chosen based on Table 3.1 and he iniial values S = ϕ, [,] and µ = ˆµ are chosen as he corresponding iniial values of S&P 5 and he dividend yield. A unique soluion (p,z) o he backward par (B.6) is obained as he limi of he 17 In fac, he profis of he diversified ime series momenum (TSMOM) porfolio in Moskowiz e al. (1) are o some exen driven by he bonds when scaling for he volailiy in equaion (5) of heir paper, and hence applying he TSM sraegies o he sock index may have fewer significan profis han he diversified TSMOM porfolio. 18 This paper sudies he S&P 5 index over 14 years of daa, while Moskowiz e al. (1) focus on he fuures and forward conracs ha include equiy indices, currencies, commodiies, and sovereign bonds. Despie a large difference beween he daa invesigaed, we find similar paerns for he TSM in he sock index and replicae heir resuls wih respec o he sock index.