QUANTITATIVE FINANCE RESEARCH CENTRE. Optimal Time Series Momentum QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE

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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 353 January 15 Opimal Time Series Momenum Xue-Zhong He, Kai Li and Youwei Li ISSN

2 OPTIMAL TIME SERIES MOMENTUM XUE-ZHONG HE, KAI LI AND YOUWEI LI *Finance Discipline Group, UTS Business School Universiy of Technology, Sydney 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 ime series momenum documened recenly. The underlying sochasic delay differenial sysem faciliaes he analysis of effecs of differen ime horizons used by momenum rading. By sudying an opimal asse allocaion problem, we find ha he performance of ime series momenum sraegy can be significanly improved by combining wih marke fundamenals and iming opporuniy wih respec o marke rend and volailiy. Furhermore, he resuls also hold for differen ime horizons, he ou-of-sample ess and wih shor-sale consrains. The ouperformance of he opimal sraegy is immune o marke saes, invesor senimen and marke volailiy. Key words: Momenum, reversal, porfolio choice, opimaliy, profiabiliy. JEL Classificaion: G1, G14, E3 Dae: January 1, 15. Acknowledgemen: Financial suppor from he Ausralian Research Council (ARC) under Discovery Gran (DP13131) is graefully acknowledged. We would like o hank Li Chen, Blake LeBaron, Michael Moore, Bern Øksendal, Lei Shi, Jun Tu and Zhen Wu for helpful commens and suggesions. The usual caveas apply. Corresponding auhor: Kai Li, Finance Discipline Group, UTS Business School, Universiy of Technology, Sydney, kai.li@us.edu.au. 1

3 HE, LI AND LI Opimal Time Series Momenum Absrac We develop a coninuous-ime asse price model o capure he ime series momenum documened recenly. The underlying sochasic delay differenial sysem faciliaes he analysis of effecs of differen ime horizons used by momenum rading. By sudying an opimal asse allocaion problem, we find ha he performance of ime series momenum sraegy can be significanly improved by combining wih marke fundamenals and iming opporuniy wih respec o marke rend and volailiy. Furhermore, he resuls also hold for differen ime horizons, he ouof-sample ess and wih shor-sale consrains. The ouperformance of he opimal sraegy is immune o marke saes, invesor senimen and marke volailiy. Key words: Momenum, reversal, porfolio choice, opimaliy, profiabiliy. JEL Classificaion: G1, G14, E3

4 OPTIMAL TIME SERIES MOMENTUM 3 1. Inroducion Recenly, Moskowiz, Ooi and Pedersen (1) empirically invesigae ime series momenum (TSM) ha characerizes srong posiive predicabiliy of a securiy s own pas reurns. This paper aims o heoreically examine how o opimally explore TSM in financial markes. We exend he sandard geomeric Brownian asse pricing model o incorporae a moving average momenum componen and a mean-revering fundamenal componen ino he drif. By sudying 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. To demonsrae he opimal performance of his sraegy, we esimae he model o monhly reurns of he S&P 5 index and use boh he uiliy of porfolio wealh and Sharpe raio o measure he performance of various rading sraegies. We show ha sraegies based on he pure momenum and pure mean-reversion models canno ouperform he marke bu he opimal sraegy of combining hem can. We compare he performance of he opimal sraegy wih he TSM sraegy used in Moskowiz e al. (1) and show ha he opimal sraegy ouperforms he TSM and passive holding sraegies in he empirical lieraure. 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, which is associaed wih marke volailiy. This paper conribues o he lieraure in hree main ways. Firs, we find ha he performance of TSM sraegy can be significanly improved by combining wih marke fundamenals. To demonsrae he ouperformance, we derive a subopimal porfolio purely based on he TSM effec. We find ha his porfolio is no able o ouperform he marke porfolio and he opimally combined porfolio. Wihou considering he fundamenals, he pure momenum porfolio is highly leveraged, and hence suffers from higher risks. For comparison, we also derive anoher subopimal porfolio, he pure mean-revering porfolio, by ignoring he TSM effec. We find ha his porfolio is based conservaively on fundamenal invesmens, leading 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 resuls, we furher invesigae he model-free performance of he TSM. We consruc he TSM porfolio following Moskowiz e al. (1) o verify he heoreical analysis. We find ha our opimal sraegy ouperforms he TSM sraegy wih respec o Sharpe raio and cumulaive excess reurn. Second, o he bes of our knowledge, his paper is he firs o heoreically examine he ime horizon effec on momenum rading. The ime horizon plays a crucial

5 4 HE, LI AND LI role in he performance of momenum sraegies, which have been invesigaed exensively in he empirical lieraure. 1 However, due o he echnical challenge, here are few heoreical resuls concerning he ime horizon. The asse pricing model developed in his paper akes he ime series momenum effec direcly ino accoun. Therefore hisorical prices underlying he ime series momenum componen affec asse prices, resuling in 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 which i is difficul o model he ime series momenum sraegy explicily. 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 using he dynamic programming mehod becomes more challenging because i involves infinie-dimensional parial differenial equaions. The only oher way o solve he problem is o apply a ype of Ponryagin maximum principle, which was 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 sraegies in closed form. This helps us o sudy horoughly he impac of hisorical informaion on he profiabiliy of differen sraegies based on differen ime horizons, in paricular of ime series momenum rading sraegies based on moving averages over differen ime horizons. Third, we show ha, in addiion o price rend, posiion size is anoher very imporan facor for momenum rading. The opimal posiion size derived in his paper is deermined by he level of rading signals and 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 porfolio is no able o ouperform he opimal porfolio for all ime horizons. The robusness of he performance of he opimal sraegy is also esed for differen sample periods, ou-of-sample predicions, shor-sale consrains, marke saes, invesor senimen and marke volailiy. This paper is closely relaed o he lieraure on reversal and momenum, wo of he mos prominen financial marke anomalies. 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), 1 See, for example, De Bond and Thaler (1985) and Jegadeesh and Timan (1993).

6 OPTIMAL TIME SERIES MOMENTUM 5 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 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) argue ha posiive auo-covariance is he main driving force for TSM and cross-secional momenum effecs, while he conribuion of serial crosscorrelaions 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 be profiable and ouperform pure momenum and pure mean-reversion sraegies. This paper esablishes a model o jusify his inuiion heoreically and empirically. 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 Exending he lieraure, his paper develops an asse price model by aking boh mean reversion and ime series momenum direcly 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). 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 6 HE, LI AND LI ino accoun and demonsraes he explanaory power of he model hrough he ouperformance of he opimal sraegy. 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. In his paper, we show heoreically ha a combined TSM and reversal sraegy is opimal. This paper is also inspired by Koijen, Rodríguez and Sbuelz (9), who propose a heoreical model in which sock reurns exhibi momenum and mean-reversion effecs. They sudy he dynamic asse allocaion problem wih consan relaive risk aversion (CRRA) uiliy. This paper differs from Koijen e al. (9) in wo respecs. Firs, our modelling of TSM is differen. In Koijen e al. (9), he momenum is calculaed from he enire se of hisorical reurns wih geomerically decaying weighs. This effecively reduces he pricing dynamics o a Markovian sysem, bu i is difficul o explicily sudy he impacs of differen look-back periods on momenum-relaed rading sraegies. In his paper, we measure momenum by he sandard moving average over a moving window wih a fixed look-back period, which is consisen wih he momenum lieraure. Second, our focuses are differen. Koijen e al. (9) focus on he performance of he hedging demand implied by he model. In our paper we focus on he performance of he opimal sraegy compared o 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. We hen invesigae he ime horizon effec in Secion 4. Secion 5 concludes. All he proofs and he 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, (7) presen an opion model o idenify observable firm-specific aribues ha drive momenum. Vayanos and Woolley (13) show ha slow-moving capial can also generae momenum. 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.

8 OPTIMAL TIME SERIES MOMENTUM 7 a riskless asse B saisfying db = rd (.1) B 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 Unied Saes are pas reurns, dividend yield, earnings/price raio, and erm srucure variables. Following his lieraure and Koijen e al. (9), we model he expeced reurn by a combinaion of a momenum erm m based on 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, 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 µ is defined by an Ornsein-Uhlenbeck process, dµ = α( µ µ )d+σ µ dz, 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 in his 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. This is differen from Koijen e al. (9), in which he momenum M a ime is defined by M = e w( u)ds u, S u which is a geomerically decaying weighed sum of he pas reurns over [,]. The advanage of M is ha he resuling price process is a Markovian process. However, i suffers from some drawbacks. Firs, M is no a weighed average since he weighs over [,] do no sum o 1. Second, M is no a moving average of pas reurns wih a fixed ime horizon as commonly used in he empirical lieraure on momenum.

9 8 HE, LI AND LI Third, he reurn process and he momenum variable in he discreizaion of Koijen e al. s model have he same expression. 7 On he oher hand, he momenum m inroduced in (.4) overcomes hese drawbacks. I is a sandard moving average over amovingwindow[,]wihafixedlook-backperiodof(> )andheweighssum o 1. I is also consisen wih momenum sraegies used in he empirical lieraure ha explore price rends based on he average reurns over a fixed look-back period. The resuling asse price model (.) (.4) is characerized by a sochasic delay inegro-differenial sysem, which is non-markovian and lacks analyical racabiliy. We show in Appendix A ha he price process of (.) (.4) almos surely has a unique coninuously adaped pahwise soluion and he asse price says posiive for given posiive iniial values over [, ]. 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. The resul is presened in he following proposiion and he proof can be found in Appendix B. Proposiion.1. 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 σ S σ S + (z u) 3 σ S p 3 uσ S σ, (.7) S 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 7 As a resul, here is an idenificaion problem. To solve he problem, Koijen e al. (9) esimae he model using a resriced maximum likelihood mehod o ensure ha he model implied auocorrelaion fis he empirical auocorrelaion srucure of sock reurns.

10 and he opimal allocaion o socks is given by OPTIMAL TIME SERIES MOMENTUM 9 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 (see, for insance, Meron 1971). When γ = 1, he opimal sraegy (.8) characerizes he myopic behavior of he invesor wih log uiliy. This resul has a number of implicaions. Firs, 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). Second, 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 reurn m r only, wih = 1 monhs, he sraegy of long/shor when he rading signal is posiive/neaive 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 condiions and volailiy found in empirical sudies. In addiion, he opimal porfolio(.1) defines he opimal wealh fracion invesed in he risky asse. Hence he TSM sraegy of aking fixed posiions based on he rading signal may no be opimal in general.

11 1 HE, LI AND LI Third, 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 heoreical suppor for opimal sraegies ha combining of momenum and reversal in a simple asse price model. The simple model and closed-form opimal sraegy (.8) faciliae model esimaion and empirical analysis. In he res of he paper, we assume γ = 1. 8 We firs esimae he model o he S&P 5 and hen evaluae and demonsrae empirically he performance of he opimal sraegy comparing i o he marke and oher rading sraegies recorded in he lieraure. 3. Model Esimaion and Performance Analysis In his secion we firs esimae he model o he S&P 5. Based on hese esimaions, we hen use uiliy of porfolio wealh and he Sharpe raio o examine he performance of he opimal sraegy (.8), comparing i 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 and volailiy. 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 SdZ, dx = αx d+σ X dz, (3.1) 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 8 For γ 1, he opimal porfolio (.7) can be solved numerically, which leads o furher esimaion errors when esimaing he model. To reduce esimaion error and have a beer undersanding of he resuls, we ake he advanage of he closed-form soluion and consider γ = 1.

12 OPTIMAL TIME SERIES MOMENTUM 11 (5) and assume he Cholesky decomposiion on he volailiy marix Σ of he dividend yield and reurn, ( ) ( ) σ S σ S(1) Σ = =. σ X(1) σ X() 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 9 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 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 sudied in he nex secion. For comparison, we esimae he full model (FM) (3.) wih φ, 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 marke index can be explained by abou % of he momenum componen and 8% of he mean-revering componen. Oher parameer 9 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.

13 1 HE, LI AND LI 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) 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 o he pure momenum model (φ = 1) and pure mean-reversion model (φ = ). For he pure momenum model, he es saisic (131) is much greaer han 1.59, he criical value wih six degrees of freedom a he 5% significance level. For he pure meanreversion model, he es saisic (6) is much greaer han 3.841, he criical value wih one degree of freedom a 5% significance level. Therefore he full model is significanly beer han he pure momenum model and he pure mean-reversion model. This implies ha he (full) model capures shor-erm momenum and longermreversion inhemarke index andfis hedaabeer hanhepuremomenum and pure mean-revering models. 3.. Economic Value. Based on he previous esimaions, we examine he economic value of he opimal porfolio in erms of he uiliy of he porfolio wealh, comparing i o hose of he marke index and of he pure momenum and pure mean-reversion models. We evaluae he performance of a porfolio (or sraegy) in erms of he Sharpe raio in he nex secion. In he following performance analysis, we focus on he log uiliy preference for he invesor. 1 1 Since he model is non-markovian, he opimal invesmen sraegy for CRRA uiliy funcions in general involves a sysem of full-coupled forward-backward sochasic differenial equaions

14 OPTIMAL TIME SERIES MOMENTUM 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. 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 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 o he index from 11 January 1876 o December 1 is equal o For = 1, we calculae he moving average m of pas 1-monh reurns (FBSDEs), which are difficul o deal wih analyically and are usually analyzed numerically. For analyical racabiliy and o avoid simulaion errors, we consider log uiliy when examining he performance of he opimal sraegy. We show ha even he opimal sraegy associaed wih log uiliy can significanly ouperform he TSM sraegy and he marke index. Bu due o he above limiaion, he effec of hedging demand is no considered here and lef for fuure research. 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).

15 14 HE, LI AND LI 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 from January 1876 unil December 1 in Fig. 3.1 (a) and (b). Fig. 3.1 (c) repors 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 PM 3 1/1876 1/196 1/1976 1/1 (b) Uiliy of PM * 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. Nex, we compare he economic value of he pure momenum and pure meanrevering 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) illusraes 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

16 OPTIMAL TIME SERIES MOMENTUM 15 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 fromhigh risk and performworse 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) illusraes he ime series of he porfolio weigh and he uiliy of he wealh of he opimal porfolio for he pure mean-reversion model from January 1876 o December 1, showing ha he performance of he sraegy is abou he same as he sock 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 sraegies of he full model ln W * ln W /1945 1/1975 1/51/1 Figure 3.3. The ime series of he uiliy of opimal porfolio wealh and ha of he marke index porfolio from January 1945 unil December 1. There are wo ineresing observaions from Fig Firs, he opimal sraegies and index reurns are posiively correlaed in general. In fac, he correlaion is abou.335. Second, Fig. 3.1 (d) seems o indicae ha he improved uiliies of he opimal porfolio wealh were mainly driven by 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. To clarify his observaion, we also examine performance using daa from January 194 o December 1 o avoid he Grea Depression periods. We re-esimae he model, conduc he same analysis, and repor he erminal uiliies of he opimal porfolios in Fig. 3.3 over his ime period. I shows 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 condiions.

17 16 HE, LI AND LI 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 3.4. (a) Average uiliy and (b) one-sided -es saisics based on 1, simulaions for = 1. To provide furher evidence for he economic value 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. 3.4 (a), ogeher wih 95% confidence levels (he wo solid green lines ouside), comparing o he uiliy of he marke index (he doed blue line). I shows ha firs, he average uiliies of he opimal porfolios are beer han ha of he S&P 5. Second, 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 plo wo black dashed bounds for he 6% confidence level. I shows ha, a he 6% confidence level, he opimal porfolio significanly ouperforms he marke index. Fig. 3.4 (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 The Sharpe Raio. 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 on a porfolio and he sandard deviaion of he porfolio reurn. 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

18 OPTIMAL TIME SERIES MOMENTUM 17 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.1. For he opimal sraegy (.8), he reurn of he opimal porfolio wealh a ime is given by R = (W W 1)/W 1 = π 1R +(1 π 1)r. (3.3) Table 3. repors he Sharpe raios of he passive holding marke index porfolio and he opimal porfolios from January 1886 o December 1 for = 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 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). 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) In summary, we have used wo performance measures and 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. In he following secion, we conduc furher empirical ess on hese resuls. We firs conduc ou-of-sample ess on he predicaion power of he model and hen examine he performance of he opimal sraegy wih shor-sale consrains, marke saes, senimen and volailiy Ou-of-Sample Tess. We implemen a number of ou-of-sample ess for he opimal sraegies by spliing he whole daa se ino wo sub-sample 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,

19 18 HE, LI AND LI Fig. 3.5 (a) and (b) illusraes 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. 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 3.5. 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). 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. 3.5 (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 are robus.

20 OPTIMAL TIME SERIES MOMENTUM 19 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 andrepor 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 oal reurn index of 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 he marke in mos ou-of-sample ess (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 in Fig. C.4 and he pure mean reversion in Fig. C.6 based on he -year rolling window esimaes in Fig. C.3 and Fig. C.5 respecively, in Appendix C. We also implemen he esimaions for differen window sizes of 5, 3 and 5 years (no repored here) and find ha he esimaed parameers are insensiive o he size of rolling window and he performance of sraegies is similar o he case of -year rolling window esimaion. 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 Consrains and Transacion Coss. Invesors ofen face shorsale consrains. To evaluae opimal sraegies under such consrains, we consider hem when shor selling and borrowing (a he risk-free rae) are no allowed. The porfolio weigh π in his case mus lie beween zero and 1. Since he value funcion is concave wih respec o π, he opimal sraegy becomes, if π <, Π = π, if π 1, 1, if π > 1. (3.4) 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 consrains, comparing wih he marke index porfolio. Uiliy Sharpe Raio Average weighs Sd of weighs Wih consrains Wihou consrains Marke index 5.76.

21 HE, LI AND LI Table 3.3 repors he erminal uiliies and he Sharpe raio of he opimal porfolio wih and wihou shor-sale consrains, compared o he passive holding marke index porfolio. I shows ha, based on he wo performance measures, he opimal porfolio wih shor-sale consrains ouperforms boh he marke and he opimal porfolios wihou shor-sale consrains. Therefore, consrains can improve performance. This observaion is 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. To suppor his argumen, we also examine he mean and sandard deviaion of he opimal porfolio weighs and repor he resuls in Table 3.3 wih and wihou shor-sale consrains. I shows ha, wih he consrains, he mean of he opimal porfolio weighs is high while he volailiy is low. On he oher hand, wihou consrains he mean of he opimal porfolio weighs is low bu he volailiy is high, which seems in line wih Marquering and Verbeek s argumen. In addiion, we examine he effec of ransacion coss. Wih a ransacion cos of.5% or 1%, we find lile change in he erminal uiliies and he Sharpe raios 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, Peng and Xiong (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 profiabiliy 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. Baker and Wurgler (6, 7) find ha invesor senimen affecs cross-secional sock reurns and he aggregae sock marke. Wang and Xu (1) 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. To invesigae he performance of opimal sraegies under differen marke saes, we follow Cooper e al. (4) and Hou e al. (9) and 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

22 OPTIMAL TIME SERIES MOMENTUM 1 he opimal sraegy and compare he average reurns beween up and down marke monhs. We see from Table D.1 in Appendix D 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 basis poins, which is no significanly differen from zero, based on a wo-sample -es (p-value of.87). We hen implemen he ime series analysis by regressing he excess porfolio reurns on a consan, an up-monh dummy and he excess marke reurn. We repor he resuls in Table D. in Appendix 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 consan erm α and he coefficien of he up-monh dummy κ are insignifican, indicaing ha he average reurn in down monhs and he incremenal average reurn in up monhs are boh insignifican for all oher sraegies; hese resuls are consisen wih hose in Table D.1. The resuls are robus when we replace he up-monh dummy wih he lagged marke reurn over he previous 36 monhs (no repored here). We also conrol for marke risk in up and down monhs, wih similar resuls. Simple predicive regression of excess porfolio reurns on he up-monh dummy indicaes ha he marke sae has no predicive power on porfolio reurns. The resuls are also robus for he CAPM-adjused reurn. Compared wih he findings in Hou e al. (9), where cross-secional momenum reurns are higher in up monhs, we do no find significan differences beween up and down monhs for he sraegies from our model and he TSM. Among hese only he TSM shows significan reurns in down monhs. In erms of he effecs of invesor senimen on porfolio performance, we regress he excess porfolio reurn on he previous monh s senimen index consruced by Baker and Wurgler (6). We find (see Table D.6 in Appendix D) ha invesor senimen has no predicive power on porfolio reurns. We also examine he predicabiliy of marke volailiy o he porfolio reurn. The ime series analysis of regressing he excess porfolio reurn on he pas monh s volailiy or on he pas monh s volailiy condiional on up and down marke sae, suggess ha marke volailiy has no predicive power on porfolio reurns (see Tables D.7 and D.8 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

23 HE, LI AND LI 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 sraegies when he posiion is deermined by he sign of he opimal porfolio sraegies (.8) wih differen combinaions of ime horizons and holding periods 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). Table E.1 in Appendix E 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. in Appendix E repors he average reurns (%) for he pure momenum model. 1 Noice ha Tables E.1 and E. indicae ha sraegy (9,1)performshebes. Thisisconsisen wihhefindinginmoskowizeal.(1) ha sraegy (9, 1) is he bes sraegy for equiy markes alhough he 1-monh horizon is he bes for mos asse classes. 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) Nex we use he Sharpe raio o examine he performance of he opimal sraegy π of (.8) andcomparei ohepassive index sraegyandwo TSM sraegies: one follows from Moskowiz e al. (1) and he oher is he TSM sraegy based on he 1 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).

24 OPTIMAL TIME SERIES MOMENTUM 3 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 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 size of he posiion is anoher very imporan facor for invesmen profiabiliy. Following Moskowiz e al. (1), we examine he cumulaive excess reurn. Tha is, he reurn a ime is defined by ˆR +1 = sign(π ).144 ˆσ S, R +1, (3.5) where.144 is he sample sandard deviaion of he oal reurn index and he ex-ane annualized variance ˆσ S, for he oal reurn index is calculaed as he exponenially weighed lagged squared monh reurns, ˆσ S, = 1 (1 δ)δ i (R 1 i R ), (3.6) i= here he consan 1 scales he variance o be annual, 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=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.

25 4 HE, LI AND LI 1 8 Opimal Sraegy Passive Long TSM Sraegy Growh of $1 (log scale) Figure 3.6. Log cumulaive excess reurn of he opimal sraegy and momenum sraegy wih = 1 and passive long sraegy from January 1876 o December 1. Wih a 1-monh ime horizon Fig. 3.6 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. 3.6, showing ha he TSM sraegy ouperforms he passive long sraegy. 13 In summary, we have shown ha he opimal sraegy ouperforms he TSM sraegy of Moskowiz e al. (1). By comparing he performance of wo TSM sraegies, we find ha he TSM sraegy based on momenum and reversal rading signal is more profiable han he pure TSM sraegy of Moskowiz e al. (1) Time Horizon Effecs The resuls in he previous secion are based on a ime horizon of = 1 monhs. In his secion, we examine he effec of a ime horizon varying from one o 6 monhs on he ouperformance of he opimal sraegies. The impac of ime horizon on invesmen profiabiliy has been exensively invesigaed in he empirical lieraure, for example, De Bond and Thaler (1985) and Jegadeesh and Timan 13 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. 14 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 counry equiy indexes, 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.

26 OPTIMAL TIME SERIES MOMENTUM 5 (1993). Because of he closed-form opimal sraegy (.8), we are able o explicily examine he dependence of he opimal resuls on differen ime horizons. α 1 x (a) Esimaes of α φ (b) Esimaes of φ 5 x µ ν (c) Esimaes of µ (d) Esimaes of ν σ S(1) (%) σ X(1) (%) σ X() (%) (e) Esimaes of σ S(1) (f) Esimaes of σ X(1) (g) Esimaes of σ X() Figure 4.1. The esimaes of (a) α; (b) φ; (c) µ; (d) ν; (e) σ S(1) ; (f) σ X(1) and (g) σ X() as funcions of Model Esimaions and Comparison. The esimaions are conduced separaely for given a ime horizon. Fig. 4.1 repors he esimaed parameers in monhly erms for ranging from one monh o five years, ogeher wih he 95% confidence bounds. Fig. 4.1 (b) shows ha he momenum effec parameer φ is

27 6 HE, LI AND LI significanly differen from zero when ime horizon is more han half a year, indicaing a significan momenum effec for beyond six monhs. 15 Noe ha φ increases o abou 5% when increases from six monhs o hree years and hen decreases gradually when increases furher. This implies ha marke reurns can be explained by boh he momenum (based on differen ime horizons) and meanrevering componens. Oher resuls in erms of he level and significance repored in Fig. 4.1 are consisen wih Koijen e al. (9). Obviously, he esimaions depend on he specificaion of he ime horizon. To explore he opimal value for, we compare differen informaion crieria, including Akaike (AIC), Bayesian (BIC) and Hannan Quinn (HQ) informaion crieria for from one monh o 6 monhs in Fig. F.1 of Appendix F. The resuls imply ha he average reurns over he pas 18 monhs o wo years can bes predic fuure reurns and he explanaory power for he marke reurns is reduced for longer ime horizons. This is consisen wih sudies showing ha shor-erm (one o wo years), raher han long-erm, momenum beer explains marke reurns. Combining he resuls in Figs 4.1 (b) and F.1, we can conclude ha marke reurns are beer capured by shor-erm momenum and long-erm reversion. To compare he performance of he opimal sraegies wih he pure momenum sraegiesfordifferen, weesimaehemodel wihφ = 1. Forhepuremomenum model(φ = 1),hereisonlyoneparameerσ S(1) obeesimaed. Fig. F.(a)repors heesimaes of σ S(1) andhe 95%confidence bounds for [1,6]. I shows ha as increases, he volailiy of he index decreases dramaically for small ime horizons and is hen sabilized for large ime horizons. I implies high volailiy associaed wih momenum over shor ime horizons and low volailiy over long ime horizons. We also compare he informaion crieria for differen (no repored here) and find ha all he AIC, BIC and HQ reach heir minima a = 11. This implies ha he average reurns over he previous 11 monhs can predic fuure reurns bes for he pure momenum model. This is consisen wih he finding of Moskowiz e al. (1) ha momenum reurns over he previous 1 monhs beer predic he nex monh s reurn han oher ime horizons. In addiion, we conduc he log-likelihood raio es o compare he full model o he pure momenum model (φ = 1) and o he pure mean-reversion model for differen. We repor he log-likelihood raio es resuls in Fig. F. (b), which show ha he full model is significanly beer han he pure momenum model and pure mean-reversion model for all. 15 For from one o five monhs, φ is indifferen from zero saisically and economically. Correspondingly, for small look-back periods of up o half a year, he model is equivalen o a pure mean-reversion model. This observaion is helpful when explaining he resuls of he model for small look-back periods in he following discussion.

28 OPTIMAL TIME SERIES MOMENTUM * 8 ln W T * ln W T * 1 ln W T (a) (b) (c) Figure 4.. The erminal uiliy of he opimal porfolio wealh (a) from January 1876 o December 1, (b) from January 1945 o December 1, and (c) he average erminal uiliy of he opimal porfolios based on 1 simulaions from January 1876 o December 1, comparing wih he erminal uiliy of he marke index porfolio (he dash-doed line). 4.. The effec on performance. For = 1,,,6, based on he esimaions of he full model over he full sample, Fig. 4. (a) repors he uiliy of erminal wealh, compared o he uiliy of he marke porfolio a December 1. I shows ha he opimal sraegies consisenly ouperform he marke index for from five o monhs. 16 The corresponding uiliies are ploed in Fig. F.4. We observed from Fig. 3.1 (d) for = 1 ha he improved erminal uiliies of he opimal porfolio wealh migh be driven by he Grea Depression in he 193s. To clarify his observaion, we also examine performance using he daa from January 194 o December 1 o avoid he Grea Depression period. We re-esimae he model, conduc he same analysis, and repor he erminal uiliies of he opimal porfolios in Fig. 4. (b) over his ime period. I shows ha he opimal sraegies sill ouperform he marke and he performance of he opimal sraegies over he more recen ime period becomes even beer for all ime horizons. Consisen wih he resuls obained in he previous secion, he ouperformance of he opimal sraegy is no necessarily due o exreme marke episodes, such as he Grea Depression. 16 When is less han half a year, Fig. 4. (a) shows ha he opimal sraegies do no perform significanly beer han he marke. As we indicae in foonoe 15, he model wih a small lookback period of up o half a year performs similarly o he pure mean-reversion sraegy. Noe he significan ouperformance of he opimal sraegy wih a one-monh horizon in Fig. 4. (a). This is due o he fac ha he firs order auocorrelaion of he reurn of he S&P 5 is significanly posiive(ac(1) =.839) while he auocorrelaions wih higher orders are insignificanly differen from zero. This implies ha he las period reurn could well predic he nex period reurn.

29 8 HE, LI AND LI We also conduc furher Mone Carlo analysis on he performance of he opimal porfolios based on he esimaed parameers in Fig. 4.1 and 1, simulaions and repor he average erminal uiliies in Fig. 4. (c). The resul displays a differen erminal performance from ha in Fig. 4. (a). In fac, he erminal uiliy in Fig. 4. (a) is based on only one specific rajecory (he real marke index), while Fig. 4. (c) provides he average performance based on 1, rajecories. We find ha he opimal porfolios perform significanly beer han he marke index (he dash-doed consan level) for all ime horizons beyond half a year. In paricular, he average erminal uiliy reaches is peak a = 4, which is consisen wih he resul based on he informaion crieria in Fig. F.1, paricularly he AIC. Therefore, according o he uiliy of porfolio wealh, he opimal sraegies ouperform he marke index for mos of he ime horizons...15 Opimal Sraegy Passive Holding.1.1 Sharpe Raio.1.5 Sharpe Raio (a) (b) Figure 4.3. The Sharpe raio of he opimal porfolio (he solid blue line) wih corresponding 9% confidence inervals (a), he average Sharpe raio based on 1, simulaions (b) for [1,6], compared o he passive holding porfolio of marke index (he doed black line) from January 1881 o December 1. As he second performance measure, Fig. 4.3 (a) repors he Sharpe raio of he passive holding marke index porfolio from January 1881 o December 1 and he Sharpe raios of he opimal porfolios for from one monh o 6 monhs ogeher wih heir 9% confidence inervals (see Jobson and Korkie 1981). If we consider he opimal porfolio as a combinaion of he marke porfolio and a risk-free asse, hen he opimal porfolio should be locaed on he capial marke line and hence should have he same Sharpe raio as he marke. However Fig. 4.3 (a) shows ha, by aking he iming opporuniy (wih respec o he marke rend and marke volailiy), he opimal porfolios (he doed blue line) ouperform he marke (he solid black line) on average for ime horizons from six o monhs. The resuls are surprisingly consisen wih ha in Fig. 4. (a) under he uiliy measure. We also

30 OPTIMAL TIME SERIES MOMENTUM 9 conduc a Mone Carlo analysis based on 1, simulaions and repor he average Sharpe raios in Fig. 4.3 (b) for he opimal porfolios. I shows he ouperformance of he opimal porfolios over he marke index based on he Sharpe raio for he look-back periods of more han six monhs. The resuls are consisen wih ha in Fig. 4. (c) under he porfolio uiliy measure. In addiion, we show in Fig. F.3 ha he pure momenum sraegies underperform he marke in all ime horizons from one monh o 6 monhs. Therefore, we have demonsraed he consisen ouperformance of he opimal porfolios over he marke index and pure sraegies under he wo performance measures * ln W T (a) Terminal uiliy 3 1 ln W 1 6 1/8 4 1/1 (b) Series of uiliy Figure 4.4. The erminal uiliy of he wealh for he opimal porfolio, compared o he passive holding marke index porfolio (he doed line), wih ou-of-sample daa from January 8 o December 1 for [1,6] The effec on he ou-of-sample ess. For from one monh o 6 monhs, Fig. 4.4 repors he ou-of-sample uiliy of he opimal porfolio wealh from January 8 o December 1. I clearly shows ha he opimal sraegies sill ouperform he marke index for ime horizons up o wo years. We repor addiional ou-of-sample ess in Appendix F.4 and rolling window esimaes in Appendix F The effec on shor-sale consrains. For differen ime horizon, Fig. 4.5 (a) and (b) repor he erminal uiliies of he opimal porfolio wealh and he Sharpe raio for he opimal porfolio wih and wihou shor-sales consrains, respecively, comparing wih he passive holding marke index porfolio. We also examine he mean and sandard deviaion of he opimal porfolio weighs and repor he resuls in Fig. 4.5 (c) and (d) wih and wihou shor-sale consrains. The resuls for = 1 in he previous secion also hold. Tha is, wih he consrains, he opimal porfolio weighs increase in he mean while volailiy is low and sable. On he oher hand, wihou consrains, he volailiy of he opimal porfolio

31 3 HE, LI AND LI Wih Consrains Wihou Consrains Passive Holding Wih Consrains Wihou Consrains Passive Holding * 8 ln W T 6 4 Sharpe Raio (a) Terminal uiliy (b) Sharpe raio Average Weighs Wih Consrains Wihou Consrains Sandard Deviaion of Weighs Wih Consrains Wihou Consrains (c) Mean of porfolio weighs (d) Sandard deviaion of porfolio weighs Figure 4.5. The erminal uiliy of wealh (a) and he Sharpe raio (b) for he opimal porfolio, he mean (c) and he sandard deviaion of he opimal porfolio weighs, wih and wihou shor-sale consrains, compared wih he marke index porfolio. weighs varies dramaically, which seems in line wih he argumen of Marquering and Verbeek (4) Comparison wih TSM wih differen ime horizons. As in he previous secion for = 1, we use he Sharpe raio o examine he performance of he opimal sraegy π in (.8) and compare wih he passive index sraegy and wo TSM sraegies for ime horizons from 1 monh o 6 monhs and one monh holding period. We repor he Sharpe raios of he porfolios for he four sraegies in Fig. 4.6 (a). 17 For comparison, we collec he Sharpe raio for he opimal porfolio and he passive holding porfolio repored in Fig. 4.3 and repor he Sharpe raios of he TSM sraegy using a solid green line and of momenum and mean-reversion sraegy using a doed red line ogeher in Fig. 4.6 (a) from January 1881 o December 1. We have hree observaions. Firs, he TSM sraegy ouperforms 17 The monhly Sharpe raio for he pure mean-reversion sraegy is.5, slighly higher han ha for he passive holding porfolio (.11).

32 OPTIMAL TIME SERIES MOMENTUM 31 he marke only for = 9,1 and he momenum and mean-reversion sraegy ouperform he marke for shor ime horizons 13. Second, by aking he meanreversion effec ino accoun, he momenum and mean-reversion sraegy performs beer han he TSM sraegy for all ime horizons. Finally, he opimal sraegy significanly ouperforms boh he momenum and mean-reversion sraegy (for all ime horizons beyond four monhs) and he TSM sraegy (for all ime horizons)...15 Opimal Sraegy Passive Holding Momenum and Mean Reversion TSM Sraegy Opimal Sraegy Passive Long TSM Sraegy.1 1 Sharpe Raio (a) Average Sharpe raio (b) Terminal log cumulaive excess reurn Figure 4.6. (a) The average Sharpe raio for he opimal porfolio, he momenum and mean-reversion porfolio and he TSM porfolio wih [1,6] and he passive holding porfolio from January 1881 unil December 1. (b) Terminal log cumulaive excess reurn of he opimal sraegies and TSM sraegies wih [1,6] and passive long sraegy from January 1876 o December 1. Fig. 4.6 (b) shows he erminal values of he log cumulaive excess reurns of he opimal sraegy (.8) and he TSM sraegy wih [1,6], ogeher wih he passive long sraegy, from January 1876 o December I shows ha he opimal sraegy ouperforms he TSM sraegy for all ime horizons (beyond four monhs), while he TSM sraegy ouperforms he marke for small ime horizons (from abou wo o 18 monhs). The erminal values of he log cumulaive excess reurn have similar paerns o he average Sharpe raio repored in Fig. 4.6 (a), especially for small ime horizons. 18 Noe ha he passive long sraegy inroduced in Moskowiz e al. (1) is differen from he passive holding sraegy sudied in he previous secions. Passive long means holding one share of he index each period; however, passive holding in our paper means invesing $1 in he index in he firs period and holding i unil he las period.

33 3 HE, LI AND LI 5. Conclusion To characerize he ime series momenum in financial markes, we propose a coninuous-ime model of asse price dynamics wih he drif as a weighed average of mean reversion and moving average componens. By applying he maximum principle for conrol problems of sochasic delay differenial equaions, we derive he opimal sraegies in closed form. By esimaing he model o he S&P 5, we show ha he opimal sraegy ouperforms he TSM sraegy and he marke index. The ouperformance holds for ou-of-sample ess and wih shor-sale consrains. The ouperformance is immune o he marke saes, invesor senimen and marke volailiy. The resuls show ha he profiabiliy paern refleced by he average reurn of commonly used sraegies in much of he empirical lieraure may no reflec he effec of porfolio wealh. The model proposed in his paper is simple and sylized. The weighs of he momenum and mean-reversion componens are consan. When marke condiions change, he weighs can be ime-varying. Hence i would be ineresing o model heir dependence on marke condiions. This can be modelled, for example, as a Markov swiching process or based on some raional learning process (Xia 1). The porfolio performance is examined under log uiliy in his paper. I would be ineresing o sudy he ineremporal effec under general power uiliy funcions. We could also consider incorporaing sochasic volailiies of he reurn process ino he model. Finally, an exension of he model o a muli-asse seing o sudy cross-secional opimal sraegies would be helpful o undersand cross-secional momenum and reversal.

34 OPTIMAL TIME SERIES MOMENTUM 33 Appendix A. Properies of he Soluions o he Sysem (.) (.4) Le C([,],R) be he space of all coninuous funcions ϕ : [,] R. For a given iniial condiion S = ϕ, [,] and µ = ˆµ, he following proposiion shows ha he sysem (.)-(.4) admis pahwise unique soluions such ha S > almos surely for all whenever ϕ > for [,] almos surely. Proposiion A.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. Proof. Basically, he soluion can be found by using forward inducion seps of lengh as in Arriojas, Hu, Mohammed and Pap (7). Le [,]. Then he sysem (.)-(.4) becomes ds = S dn, [,], dµ = α( µ µ )d+σ µ dz, [,], (A.1) S = ϕ for [,] almos surely and µ = ˆµ. where N = [ φ s s dϕ u ϕ u + (1 φ)µ s ] ds + σ S dz s is a semimaringale. Denoe by N,N = σ S σ Sds, [,], he quadraic variaion. Then sysem (A.1) has a unique soluion S = ϕ exp { N 1 N,N }, µ = µ+(ˆµ µ)exp{ α}+σ µexp{ α} exp{αu}dz u for [,]. This clearly implies ha S > for all [,] almos surely, when ϕ > for [,] almos surely. By a similar argumen, i follows ha S > for all [,]almossurely. ThereforeS > forall almossurely, byinducion. Noe ha he above argumen also gives exisence and pahwise-uniqueness of he soluion o he sysem (.)-(.4). Appendix B. Proof of Proposiion.1 To solve he sochasic conrol problems, here are wo approaches: he dynamic programming mehod (HJB equaion) and he maximum principle. Since he SDDE is no Markovian, we canno use he dynamic programming mehod. Recenly, Chen and Wu (1) inroduced a maximum principle for he opimal conrol problem of

35 34 HE, LI AND LI SDDE. This mehod is furher exended by Øksendal e al. (11) o consider a onedimensional sysem allowing boh delays of moving average ype and jumps. Because he opimal conrol problem of SDDE is relaively new o he field of economics and finance, we firs briefly inroduce he maximum principle of Chen and Wu (1) and refer readers o heir paper for deails. B.1. The Maximum Principle for an Opimal Conrol Problem of SDDE. Consider a pas-dependen sae X of a conrol sysem { dx = b(,x,x,v,v )d+σ(,x,x,v,v )dz, [,T], (B.1) X = ξ, v = η, [,], where Z is a d-dimensional Brownian moion on (Ω,F,P,{F } ), and b : [,T] R n R n R k R k R n and σ : [,T] R n R n R k R k R n d are given funcions. In addiion, v is a F ( )-measurable sochasic conrol wih values in U, where U R k is a nonempy convex se, > is a given finie ime delay, ξ C[,] is he iniial pah of X, and η, he iniial pah of v( ), is a given deerminisic coninuous funcion from [,] ino U such ha η sds < +. The problem is o find he opimal conrol u( ) A, such ha J(u( )) = sup{j(v( ));v( ) A}, (B.) where A denoes he se of all admissible conrols. The associaed performance funcion J is given by [ T ] J(v( )) = E L(,X,v,v )d+φ(x T ), where L : [,T] R n R k R k R and Φ : R n R are given funcions. Assume (H1): he funcions b, σ, L and Φ are coninuously differeniable wih respec o (X,X,v,v ) and heir derivaives are bounded. In order o derive he maximum principle, we inroduce he following adjoin equaion, dp = { (b u X ) p +(σx u ) z +E [(b u X + ) p + +(σx u + ) z + ] +L X (,X,u,u ) } d z dz, [,T], (B.3) p T = Φ X (X T ), p =, (T,T +], z =, [T,T +]. We refer readers o Theorems.1 and. in Chen and Wu (1) for he exisence and uniqueness of he soluions of he sysems (B.3) and (B.1) respecively.

36 OPTIMAL TIME SERIES MOMENTUM 35 Nex, define a Hamilonian funcion H from [,T] R n R n R k R k L F (,T+ ;R n ) L F (,T +;Rn d ) o R as follows, H(,X,X,v,v,p,z ) = b(,x,x,v,v ),p + σ(,x,x,v,v ),z +L(,X,v,v ). Assume (H): he funcions H(,,,,,p,z ) and Φ( ) are concave wih respec o he corresponding variables respecively for [,T] and given p and z. Then we have he following proposiion on he maximum principle of he sochasic conrol sysem wih delay by summarizing Theorem 3.1, Remark 3.4 and Theorem 3. in Chen and Wu (1). Proposiion B.1. (i) Le u( ) be an opimal conrol of he opimal sochasic conrol problem wih delay subjec o (B.1) and (B.), and X( ) be he corresponding opimal rajecory. Then we have max v U Hu v +E [H u v + ],v = H u v +E [H u v + ],u, a.e., a.s.; (B.4) (ii) Suppose u( ) A and le X( ) be he corresponding rajecory, p and z be he soluion of he adjoin equaion (B.3). If (H1), (H) and (B.4) hold for u( ), hen u( ) is an opimal conrol for he sochasic delayed opimal problem (B.1) and (B.). B.. Proof of Proposiion.1. We now apply Proposiion B.1 o our sochasic conrol problem. Le P u := lns u and V u := lnw u. Then he sochasic delayed opimal problem in Secion becomes o maximize E u [Φ(X T )] := E u [lnw T ] = E u [V T ], subjec o { dxu = b(u,x u,x u,π u )du+σ(u,x u,π u )dz u, u [,T], X u = ξ u, v u = η u, u [,], (B.5) where X u = P u µ u, σ = σ S σ µ, b = V u π u σ S φ (P u P u )+(1 φ)µ u (1 φ) σ S σ S α( µ µ u ) [ ]. φ +π u (P u P u )+ σ S σ S φ+(1 φ)µ u r +r π u σ S σ S

37 36 HE, LI AND LI Then we have he following adjoin equaion dp u = { (b π X ) p u +(σx π ) z u +E u [(b π X u+ ) p u+ +(σx π u+ ) z u+ ] } +L X du zu dz u, u [,T], where p T = Φ X (X T ), p u =, u (T,T +], z u =, p u = (p i u ) 3 1, (b π X u+ ) = φ u [T,T +], φ π u+ z u = (z ij u ) 3, (σ π X ) = (σ π X u+ ) = 3 3. (b π X ) =, Φ X (X T ) = φ φ π u (B.6), 1 φ α (1 φ)πu, L X =, e (1 γ)v T The Hamilonian funcion H is given by [ φ H = (P u P u )+(1 φ)µ u (1 φ) σ S σ ] S p 1 u +α( µ µ u)p u { + π u σ S σ S [φ +π u (P u P u )+ σ S σ S φ+(1 φ)µ u r ] } +r ( ) ( ) ( ) +σ S zu 11 +σ zu 1 zu 1 µ +π zu σ S zu 31, zu 3 so ha H π π = π u σ S σ S + φ (P u P u )+ σ S σ S φ+(1 φ)µ u r +σ S I can be verified ha E u [H π π u+ ] =. Therefore, Hπ π +E u[hπ π u+ ],π { [ =π u πu σ S σ S + φ (P u P u )+ σ S σ ] S φ+(1 φ)µ u r p 3 u +σ S Taking he derivaive wih respec o π u and leing i equal zero yields π u = φ (P u P u )+ σ S σ S φ+(1 φ)µ u r σ S σ S = φm u +(1 φ)µ u r σ S σ S + σ S(1)z 31 u +σ S()zu 3 p 3 uσ S σ S ( ( + σ S(1)zu 31 +σ S()zu 3 p 3 u σ S σ S, z 31 u z 3 u p 3 u z 31 u z 3 u ). ) }. (B.7) where z u and p u are governed by he backward sochasic differenial sysem (B.6). This gives he opimal invesmen sraegy.

38 OPTIMAL TIME SERIES MOMENTUM 37 Especially, if γ = 1, he uiliy reduces o a log one and Φ X (X T ) =. 1 Since he parameers and erminal values for dp 3 u are deerminisic in his case, we can asser ha zu 31 = z3 u = for u [,T], which leads o p3 u = 1 for u [,T]. Then he Hamilonian funcion H is given by [ φ H = (P u P u )+(1 φ)µ u (1 φ) σ S σ ] S p 1 u +α( µ µ u )p u { + π u σ S σ S [φ +π u (P u P u )+ σ S σ S φ+(1 φ)µ u r ] } +r p 3 u ( ) ( ) +σ S zu 11 +σ zu 1 zu 1 µ, zu and he opimal sraegy is given by which is myopic. πu = φm u +(1 φ)µ u r σ S σ, S Appendix C. Addiional ou-of-sample ess and Rolling Window Esimaions In his appendix, we provide some robusness analysis o ou-of-sample ess and rolling window esimaions. C.1. Rolling Window Esimaions. For fix = 1, we esimae parameers of (3.) a each monh by using he pas years daa o avoid look-ahead bias. Fig. C.1 illusraes he esimaed parameers. The big jump in esimaed σ S(1) during is consisen wih he high volailiy of marke reurn illusraed in Fig. C. (b). Fig. C.1 also illusraes he ineresing phenomenon: ha he esimaed φ is very close o zero for hree periods of ime, implying insignifican momenum bu significan mean-reversion effec. By comparing Fig. C.1 (b) and (e), we observe ha he insignifican φ is accompanied by high volailiy σ S(1). Fig. C. illusraes he ime series of (a) he index level and (b) he simple reurn of he oal reurn index of S&P 5; (c) he opimal porfolio and (d) he uiliy of he opimal porfolio wealh from December 189 o December 1 for = 1 wih -year rolling window esimaed parameers. The index reurn and π are posiively correlaed wih correlaion.6. In addiion, we find ha he profis are higher afer he 193s.

39 38 HE, LI AND LI α φ..1. 1/1891 1/1941 1/1991 1/1 (a) Esimaes of α.1 1/1891 1/1941 1/1991 1/1 (b) Esimaes of φ x µ ν /1891 1/1941 1/1991 1/1 (c) Esimaes of µ.5.1 1/1891 1/1941 1/1991 1/1 (d) Esimaes of ν σ S(1) (%) 4 3 1/1891 1/1941 1/1991 1/ σ X(1) (%) /1891 1/1941 1/1991 1/ σ X() (%) /1891 1/1941 1/1991 1/1 (e) Esimaes of σ S(1) (f) Esimaes of σ X(1) (g) Esimaes of σ X() Figure C.1. The esimaes of (a) α; (b) φ; (c) µ; (d) ν; (e) σ S(1) ; (f) σ X(1) and (g) σ X() for = 1 based on daa from he pas years. Fig. C.3 illusraes he esimaes of σ S(1) for he pure momenum model (φ = 1) based on daa from he pas years; he big jump in volailiy is due o he Grea Depression in he 193s. Fig. C.4 illusraes he ime series of (a) he opimal porfolio and (b) he uiliy of wealh from December 189 unil December 1 for = 1 for he pure momenum model wih he -year rolling window esimaed σ S(1). By comparing Fig. C.3 and Fig. C.4 (b), he opimal sraegy implied by he pure momenum model suffers huge losses during he high marke volailiy period.

40 OPTIMAL TIME SERIES MOMENTUM P /1891 1/1941 1/1991 1/1 (a) R /1891 1/1941 1/1991 1/1 (b) * π ln W * ln W /1891 1/1941 1/1991 1/1 (c) 1/1891 1/1941 1/1991 1/1 (d) Figure C.. The ime series of (a) he index level and (b) he simple reurn of he oal reurn index of S&P 5; (c) he opimal porfolio and(d) he uiliy of wealh from December 189 unil December 1 for = 1 wih -year rolling window esimaed parameers σ S(1) (%) /1891 1/1941 1/1991 1/1 Figure C.3. Esimaes of σ S(1) for he pure momenum model (φ = 1) based on daa from he pas years. However, Fig C. illusraes ha he opimal sraegy implied by he full model makes big profis during he big marke volailiy period. Fig. C.5 illusraes he esimaed parameers for he pure mean-reversion model based on daa from he pas -years.

41 4 HE, LI AND LI 15 1 * π ln W * ln W /1891 1/1941 1/1991 1/1 (a) 1 1/1891 1/1941 1/1991 1/1 (b) Figure C.4. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from December 189 unil December 1 for = 1 for he pure momenum model wih -year rolling window esimaed parameers α µ 16 x ν (a) Esimaes of α (b) Esimaes of µ (c) Esimaes of ν σ S(1) (%) σ X(1) (%) σ X() (%) (d) Esimaes of σ S(1) (e) Esimaes of σ X(1) (f) Esimaes of σ X() Figure C.5. The esimaes of (a) α; (b) φ; (c) µ; (d) ν; (e) σ S(1) ; (f) σ X(1) and (g) σ X() for he pure mean-reversion model based on daa from he pas years. Fig. C.6 illusraes he ime series of he opimal porfolio and he uiliy of wealh from December 189 unil December 1 for he pure mean-reversion model wih

42 OPTIMAL TIME SERIES MOMENTUM * π ln W * ln W 6 1/1891 1/1941 1/1991 1/1 (a) 4 1/1891 1/1941 1/1991 1/1 (b) Figure C.6. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from December 189 unil December 1 for he pure mean-reversion model wih -year rolling window esimaed parameers. -year rolling window esimaed parameers. Afer eliminaing he look-ahead bias, he pure mean-reversion sraegy canno ouperform he sock index any longer. Appendix D. Regressions on he Marke Saes, Senimen and Volailiy D.1. Marke Saes. Firs, we follow Cooper e al. (4) and Hou e al. (9) and define marke sae using he cumulaive reurn of he sock index (including dividends) over he mos recen 36 monhs. 19 We label a monh as an up (down) marke monh if he marke s hree-year reurn is non-negaive (negaive). There are 1,165 up monhs and 478 down monhs from February 1876 o December 1. Table D.1. The average excess reurn of he opimal sraegy for = 1. Observaions (N) Average excess reurn Uncondiional reurn 1, (.37) Up marke 1, (4.9) Down marke (.87) 19 The resuls are similar if we use he alernaive 6-, 1- or 4- monh marke sae definiions, even hough hey are more sensiive o sudden changes in marke senimen. We exclude January 1876 in which here is no reurn o he opimal sraegies.

43 4 HE, LI AND LI We compue he average reurn of he opimal sraegy and compare he average reurns beween up and down marke monhs. Table D.1 presens he average uncondiional excess reurns and he average excess reurns for up and down marke monhs. The uncondiional average excess reurn is 87 basis poins per monh. In up marke monhs, he average excess reurn is 81 basis poins and i is saisically significan. In down marke monhs, he average excess reurn is 11 basis poins; his value is economically bu no saisically significan. The difference beween down and up monhs is basis poins, which is no significanly differen from zero based on a wo-sample -es (p-value of.87). 1 We use he following regression model o es for he difference in reurns: R r = α+κi (UP)+β(R r)+ǫ, (E.1) where R = (W W 1)/W 1 in(3.3) is hemonh reurnof he opimal sraegy, R r is he excess reurn of he sock index, and I (UP) is a dummy variable ha akes he value of 1 if monh is in an up monh, and zero oherwise. The regression inercep α measures he average reurn of he opimal sraegy in down marke monhs, and he coefficien κ capures he incremenal average reurn in up marke monhs relaive o down monhs. We also replace he marke sae dummy in (E.1) wih he lagged marke reurn over he previous 36 monhs (no repored here), and he resuls are robus. Table D.. The coefficiens for he regression (E.1). Full model Pure momenum Pure mean reversion TSM α (1.46) (1.34) (-.1) (3.3) κ (.6) (.1) (-.3) (-.63) β (-1.48) (-14.6) (.97) (-6.39) Table D. repors he regression coefficiens for he full model, he pure momenum model, pure mean-reversion model and he TSM sraegy in Moskowiz e al. (1) for = 1 respecively. Excep for he TSM, which earns significan posiive reurns in down marke, boh α and κ are insignifican for all oher sraegies; he resuls are consisen wih hose in Table D.1. 1 The p-values for he pure momenum sraegy, pure mean-reversion sraegy and TSM are.87,.87 and.67 respecively.

44 OPTIMAL TIME SERIES MOMENTUM 43 Table D.3. The coefficiens for he regression (E.). Full model Pure momenum Pure mean reversion TSM α (1.44) (1.3) (.18) (3.9) κ (-.11) (-.9) (-.14) (-.81) β (1.9) (1.7) (3.84) (4.3) β (-.16) (-5.31) (34.88) (-14.63) To furher conrol for marke risk in up and down marke monhs, we now run he following regression: R r = α+κi (UP)+β 1 (R r)i (UP)+β (R r)i (DOWN)+ǫ. (E.) The regression coefficiens are repored in Table D.3. These resuls are similar o hose in Table D.. Table D.4. The coefficiens for he regression (E.3). Full model Pure momenum Pure mean reversion TSM α (1.) (1.8) (.14) (3.3) κ (.7) (.8) (-.14) (-.53) We also run he following regression: R r = α+κi 1 (UP)+ǫ, (E.3) and Table D.4 repors he coefficiens. We find ha he dummy variable of up marke monh has no predicive power for reurns of all sraegies. Table D.5. The coefficiens for he regression (E.4). Full model Pure momenum Pure mean reversion TSM α (1.46) (1.34) (-.1) (3.3) κ (.6) (.1) (-.3) (-.63)

45 44 HE, LI AND LI We also sudy he bea-adjused reurns: R r = αcapm +β CAPM (R r)+ε, R r β CAPM (R r) = α+κi (UP)+ǫ, (E.4) and Table D.5 repors he coefficiens. Again, he difference of reurns beween up and down monhs is no significan for all he sraegies. Compared wih findings in Hou e al. (9), where cross-secional momenum reurnsarehigherinupmonhs, wedonofindsignificandifferencesbeweenupand down monhs for he sraegies from our model and he TSM. Among he sraegies from our model and he TSM, only he TSM has significan reurns in down marke monhs. Table D.6. The coefficiens for he regression (E.5). Full model Pure momenum Pure mean reversion TSM a (1.77) (1.74) (1.49) (.57) b (1.) (.87) (-1.1) (1.48) D.. Invesor Senimen. In his subsecion, we examine he relaionship beween he excess reurn of he opimal sraegies and invesor senimen by running he following regression: R r = a+bt 1 +ǫ, (E.5) where T is he senimen index measure used by Baker and Wurgler (6). The daa on he Baker-Wurger senimen index from 7/1965 o 1/1 is obained from he Jeffrey Wurglers websie (hp://people.sern.nyu.edu/jwurgler/). Table D.6 repors he coefficiens. The resuls sugges ha senimen index has no predicive power for reurns of opimal sraegies and of he TSM. We also examine monhly changes of he level of senimen by replacing T wih is monhly changes and heir orhogonalized indexes. The resuls are similar. Table D.7. The coefficiens for he regression (E.6). Full model Pure momenum Pure mean reversion TSM α (1.6) (1.11) (-1.) (.78) κ (.5) (.5) (1.1) (-1.48)

46 OPTIMAL TIME SERIES MOMENTUM 45 D.3. Marke Volailiy. Finally, we examine he predicabiliy of marke volailiy for profiabiliy. Firs, we run he following regression: R r = α+κˆσ S, 1 +ǫ, (E.6) where he ex-ane annualized volailiy ˆσ S, is given by (3.6). Table D.7 repors he resuls. We see ha volailiy has no predicive power for reurns of opimal sraegies and of he TSM. Table D.8. The coefficiens for he regression (E.7). Full model Pure momenum Pure mean reversion TSM α (-.7) (-.36) (.8) (1.19) κ (1.34) (1.33) (-.8) (-.7) κ (1.8) (1.75) (-.84) (.53) Second, we run he regression by following Wang and Xu (1): R r = α+κ 1ˆσ + S, 1 +κ ˆσ S, 1 +ǫ, (E.7) where ˆσ + S, (ˆσ S, ) is equal o ˆσ S, if he marke sae is up (down) and oherwise equal o zero. Table D.8 repors he coefficiens. The resuls are similar o hose in Table D.7 even if we disinguish volailiy in up and down marke monhs. Appendix E. Comparison wih Moskowiz, Ooi and Pedersen (1)

47 46 HE, LI AND LI Table E.1. The average excess reurn (%) of he opimal sraegies for differen look-back period (differen row) and differen holding period h (differen column). ( \ h) (1.8) (1.84) (3.9) (.83) (1.84) (.4) (.63) (.9) (.66) (.93) (.93) (.93) (.93) (.93) (.93) (.93) (.93) (.93) (1.93) (.8) (.6) (1.75) (.88) (-.58) (.3) (.53) (.) (3.7) (3.1) (.8) (1.45) (.41) (-1.16) (-.17) (.) (-1.11) (1.85) (1.4) (.8) (-.4) (-.76) (-1.3) (-.41) (-.3) (-1.46) (-.3) (-.51) (-.79) (-.6) (-.61) (-.34) (.4) (-.3) (-.15) (.35) (.59) (.5) (.43) (.44) (.81) (.49) (.4) (.64) (1.74) (1.3) (1.6) (.93) (.43) (-.1) (.4) (.7) (.86) (-.5) (-.6) (-.81) (-1.) (-1.41) (-.49) (.55) (.69) (.9) Appendix F. The Effecs of Time Horizons F.1. Informaion Crieria. We presen differen informaion crieria, including Akaike (AIC), Bayesian (BIC) and Hannan Quinn (HQ) informaion crieria for from one monh o 6 monhs in Fig. F.1. We see ha he AIC, BIC and HQ reach heir minima a = 3,19 and respecively. Also, we observe a common increasing paern of he crieria level for longer.

48 OPTIMAL TIME SERIES MOMENTUM 47 Table E.. The average excess reurn (%) of he opimal sraegies for differen look-back period (differen row) and differen holding period h (differen column) for he pure momenum model. ( \ h) (-.14) (.89) (1.34) (1.5) (1.37) (-.1) (.) (.4) (-.57) (1.61) (.16) (1.91) (.) (1.17) (-.69) (-.7) (-.45) (-1.38) (.78) (.79) (3.1) (.9) (1.34) (-.75) (-.6) (.4) (-.77) (3.91) (3.78) (.6) (1.76) (.66) (-1.5) (-.38) (-.1) (-1.1) (.35) (1.67) (.94) (.13) (-.86) (-1.63) (-1.) (-.95) (-1.88) (.9) (-.4) (-.81) (-1.3) (-1.6) (-.88) (-.13) (-.4) (-.39) (-.1) (.19) (.3) (.) (.1) (.4) (.18) (.37) (.33) (.74) (.73) (.4) (.) (-.4) (-.83) (-.1) (.61) (.55) (-.54) (-.84) (-1.41) (-1.77) (-.15) (-1.36) (-.6) (.) (.34) AIC BIC HQ (a) AIC (b) BIC (c) HQ Figure F.1. (a) Akaike informaion crieria, (b) Bayesian informaion crieria, and (c) Hannan Quinn informaion crieria for [1,6]. F.. The Esimaion and Log-likelihood Raio Tes for he Pure Momenum Sraegy. We presen he esimaion of parameer σ S(1) for he pure momenum model in Fig. F.(a) for [1,6]. We also presen he resuls of he

49 48 HE, LI AND LI log-likelihood raio es o compare he full model o he esimaed pure momenum model (φ = 1) and pure mean-reversion model wih respec o differen in Fig. F. (b), where he solid red line shows he es saisic when compared o he pure momenum model. The saisic is much greaer han 1.59, he criical value wih six degrees of freedom a he 5% significance level. The dash-doed blue line illusraes he es saisic when comparing o he pure mean-reversion model. The es saisic is much greaer han 3.841, he criical value wih one degree of freedom a 5% significance level σ S(1) (%) Likelihood raio es Pure Momenum Pure Mean Reversion (a) The esimaes of σ S(1) (b) The log-likelihood raio es Figure F.. The esimaes of σ S(1) for he pure momenum model (a) and he log-likelihood raio es (b) for [1,6]. F.3. The Performance for Differen Time Horizons. To compare wih he performance of he pure momenum sraegies o he marke index, based on esimaed parameers in Fig. F., we repor he erminal uiliies of he porfolios of he pure momenum model a December 1 in Fig. F * ln W T Figure F.3. The uiliy of erminal wealh of he pure momenum model for [1,6]. For = 1,,,6, Fig. F.4 illusraes he evoluion of he uiliy of he opimal porfolio wealh (he dark and more volaile surface) and of he passive holding

50 OPTIMAL TIME SERIES MOMENTUM 49 index porfolio (he yellow and smooh surface) from January 1876 unil December 1. I indicaes ha he opimal sraegies ouperform he marke index for from five monhs o monhs consisenly. Figure F.4. The uiliy of he opimal porfolio wealh from January 1876 unil December 1 comparing wih he passive holding porfolio for [1,6]. F.4. The Ou-of-sample Tes for Differen Time Horizons. To see he effec of he ime horizon on he resuls of ou-of-sample ess, we spli he whole daa se ino wo equal periods: January 1871 December 1941 and January 194 December 1. For given, we esimae he model for he firs sub-sample period and do he ou-of-sample es over he second sub-sample period. We repor he uiliy of erminal wealh for [1,6] using sample daa of he las 71 years in Fig. F.5. Clearly he opimal sraegies sill ouperform he marke for [1, 14] * 4 ln W T Figure F.5. The uiliy of erminal wealh for [1,6] based on he ou-of-sample period January 194 o December 1. F.5. Rolling Window Esimaes for Differen Time Horizons. We also implemen rolling window esimaions for differen ime horizons. Fig. F.6 illusraes he correlaions of he esimaed σ S(1) wih (a) he esimaed φ and he reurn of he opimal sraegies for (b) he full model, (c) he pure momenum model and (d)

51 5 HE, LI AND LI Correlaions Correlaions (a) (b) Correlaions (c) Correlaions (d) Figure F.6. The correlaions of he esimaed σ S(1) wih (a) he esimaed φ and he reurn of he opimal sraegies for (b) he full model, (c) he pure momenum model and (d) he TSM reurn. he TSM reurn for [1, 6]. Ineresingly, higher volailiy is accompanied by a less significan momenum effec wih small ime horizons ( 13). Bu φ and σ S(1) are posiive correlaed when he ime horizon becomes large. One possible reason is ha a long ime horizon makes he rading signal less sensiive o changes in price and hence he rading signal is significan only when he marke price changes dramaically in a high volailiy period. Fig. F.6 (c) and (d) show ha he profiabiliy of he opimal sraegies for he pure momenum model and he TSM sraegies are sensiive o he esimaed marke volailiy. The reurn is posiively (negaively) relaed o marke volailiy for shor (long) ime horizons. Bu Fig. F.6 (b) shows ha he opimal sraegies for he full model perform well even in a highly volaile marke.

52 OPTIMAL TIME SERIES MOMENTUM Figure F.7. The fracion of φ significanly differen from zero for [1,6]. We also sudy oher ime horizons. We find ha he esimaes of σ S(1), σ X(1) and σ X() are insensiive o bu he esimaes of φ are sensiive o. Fig. F.7 illusraes he corresponding fracion of φ, which is significanly differen from zero for [1,6]. I shows ha he momenums wih 3 monh horizons occur mos frequenly during he period from December 189 unil December * ln W T Figure F.8. The uiliy of wealh from December 189 o December 1 for he opimal porfolio wih [1,6] and he passive holding porfolio wih -year rolling window esimaed parameers. Fig. F.8 (a) illusraes he uiliy of wealh from December 189 unil December 1 for he opimal porfolio wih [1,6] and he passive holding porfolio. Especially, he uiliy of erminal wealh illusraed in Fig. F.8 (b) shows ha he opimal sraegies work well for shor horizons and he erminal uiliy reaches is peak a = 1.