OPTIMALITY OF MOMENTUM AND REVERSAL

Transcription

1 OPTIMALITY OF MOMENTUM AND REVERSAL 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. This paper examines he opimaliy of a combined momenum and reversal sraegies. I is well documened ha financial markes exhibi shor-run momenum and long-run reversal. By proposing a linear combinaion of mean reversion and moving average as drif in a sandard geomeric Brownian moion of asse price model, we show ha he sraegies are opimal. We hen esimae he model o he S&P 5 and demonsrae ha, by aking he iming opporuniy wih respec o rend in reurn and marke volailiy, he sraegies ouperform no only pure momenum and pure mean reversion sraegies, bu also he marke. Furhermore he ouperforming is immune o marke saes, invesor senimen and marke volailiy. Key words: Momenum, reversal, porfolio choice, opimaliy, profiabiliy. JEL Classificaion: G1, G1, E3 Dae: April 5, 1. Acknowledgemen: We would like o hank Li Chen, Bern Øksendal and Zhen Wu for assisance in mahemaics. We are also graeful o Lei Shi 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

2 HE, LI AND LI 1. Inroducion This paper sudies he opimaliy of rading sraegies o reflec he shor-run momenum and long-run reversal in financial markes. We exend he sandard asse pricing model under geomeric Brownian moion o incorporae a weighed average of mean reversion and moving average ino he drif. When he preference is given by he log uiliy funcion, we obain he opimal porfolio, which includes Meron s opimal porfolio as a special case. We show ha a combined momenum and reversal sraegies are opimal. To demonsrae he opimaliy of he sraegies, we esimae he model o he S&P 5 index and show ha neiher pure momenum nor pure mean reversion sraegies can ouperform he marke; however, he opimal sraegies combining he momenum and mean reversion can ouperform he marke. In fac, differen from he momenum sraegies which are based on rend only, he opimal sraegies ake no only he rading signal based on momenum and reversal effecs bu also he marke volailiy ino accoun. Through regression analysis, we furher show ha he opimal sraegies are immune o marke saes, invesor senimen and marke volailiy. The asse pricing model developed in his paper akes he momenum and he reversal effecs ino accoun direcly. Therefore he hisorical prices underlying he momenum componen affec he 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 well documened in he lieraure (Meron 1969, 1971), which may rule ou he profiabiliy from momenum rading. In his case, he dynamic programming mehod hrough HJB equaion is mos frequenly used in solving he sochasic conrol problem. However, he dynamic programming mehod becomes very challenging o solve he opimal conrol problem for SDDEs because i involves infinie-dimensional PDEs. To overcome his challenge, we explore he laes developmen in he heory of maximum principle for conrol problem of SDDEs. By assuming he log uiliy preference, we derive he opimal sraegies in closed form. This helps us o sudy he impac of hisorical informaion on he profiabiliy of differen sraegies based on differen ime horizons. 1 This paper is closely relaed o he lieraure on reversal and momenum, wo of he mos prominen financial marke anomalies; in paricular, ime series momenum. Reversal, on he one hand, concerns predicabiliy of asses ha performed well (poorly) over a long period end o subsequenly underperform (ouperform). Momenum, on he oher hand, is he endency of asses wih good (bad) recen performance o coninue ouperforming (underperforming) in shor-run. Reversal and 1 The impac of he ime horizon on he profiabiliy has been exensively invesigaed in he empirical lieraure, see, for example, De Bond and Thaler (1985) and Jegadeesh and Timan (1993). However, due o he echnical challenge, here are few heoreical resuls on i.

3 OPTIMALITY OF MOMENTUM AND REVERSAL 3 momenum have been documened exensively for a wide variey of asses. Fama and French (1988) and Poerba and Summers (1988), among ohers, documen he reversal for holding periods more han one year, which induces negaive auocorrelaion in reurns. Jegadeesh (1991) finds ha he nex 1-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. Fama and French (199) documen he value effec, which 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)). Jegadeesh and Timan (1993) documen momenum for individual U.S. socks, predicing reurns over horizons of 3-1 monhs by 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)). More recenly, Moskowiz, Ooi and Pedersen (1) invesigae ime series momenum (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 TSM based on he pas 1 monh excess reurns persiss for 1 o 1 monhs ha parially reverses over longer horizons. They provide srong evidence on he 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 crosssecional 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 cross-correlaions and variaion in mean reurns is small. The size and apparen persisence of momenum and reversal profis have araced considerable aenion, and many sudies have ried o explain he phenomena. 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 sysemaic errors ha invesors make when hey use public informaion o form expecaions of fuure cash flows. Daniel, Hirshleifer and Subrahmanyam (1998) s model wih single represenaive agen and Hong and Sein (1999) s model 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. Vayanos and Woolley (13) show

4 HE, LI AND LI he slow-moving capial can also generae momenum. He and Li (1) find ha momenum sraegies are self-fulfilling. This paper is largely moivaed by he empirical lieraure esing rading signals wih combinaion of momenum and reversal. Balvers and Wu (6) and Serban (1) empirically show ha a combinaion of momenum and mean reversion sraegies can ouperform he pure momenum and pure mean reversion sraegies for equiy markes and foreign exchange markes respecively. Asness e al. (13) highligh ha sudying value and momenum joinly is more powerful han examining each in isolaion. Huang, Jiang, Tu and Zhou (13) find ha boh mean reversion and momenum can coexis in he S&P 5 index over ime. Koijen e al. (9) proposes a heoreical model in which sock reurns exhibi momenum and mean reversion effecs. They sudy he dynamic asse allocaion problem wih CRRA uiliy. However, he modelling of momenum in his paper is very differen from Koijen e al. (9). In Koijen e al. (9), he momenum is calculaed by all he hisorical reurns wih geomerically decaying weighs. This reduces he pricing dynamics o a Markovian sysem. In his paper, he momenum is measured by he sandard moving average over a moving window wih a fixed look-back period, which is consisen wih he momenum lieraure. Also, Koijen e al. (9) focuses on he performance of he hedging demand implied by he model, while he focus of his paper is on he performance of he opimal sraegies. By esimaing he model o he S&P 5, we are able o demonsrae ha he opimal sraegies based on he pure momenum and pure mean reversion models canno ouperform he marke bu a combinaion of hem can ouperform he marke. The opimal sraegies no only reflec he rading signal based on momenum and reversal effecs, bu also ake he volailiy ino accoun. The robusness of he opimaliy of he opimal sraegies are also esed in shor run and long run, wih differen esimaions, ou of sample predicions, marke saes, invesor senimen and marke volailiy. Finally, we compare he performance of he opimal sraegies wih he TSM sraegies used in Moskowiz e al. (1). Wih differen proxies, including he uiliy of wealh, Sharpe raio and average reurn, we show ha he profiabiliy paerns refleced by he average reurn in mos of he empirical lieraure underperform comparing o he opimal sraegies. The paper is organized as follows. We firs presen he model and derive he opimal asse allocaion in Secion. We hen esimae he model o he S&P 5 in Secion 3 and examine he performance of he opimal sraegies in Secion. They find ha separae facors for value and momenum bes explain he daa for eigh differen 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.

5 OPTIMALITY OF MOMENTUM AND REVERSAL 5 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 pricing model and sudy he opimal asse allocaion problem..1. The Model. We consider a financial marke wih wo radable securiies, a riskless asse B saisfying db B = rd, (.1) where he riskless ineres rae r is a consan, and a risky asse S. The uncerainy is represened by a filered probabiliy space (Ω, F, P, {F } ) on which a wodimensional Brownian moion Z is defined. Le S be he price of he risky asse or he level of a marke index a ime and he dividends are assumed o be reinvesed. Following Koijen e al. (9), we assume he insananeous reurn of he risky asse is arrived a via a momenum erm m and a long-run mean reversion erm µ and he dynamics of sock reurns is given by 3 ds S = [ φm + (1 φ)µ ] d + σ S dz, (.) where φ is a consan, measuring he weigh o he momenum componen, σ S is a wo-dimensional volailiy vecor, and Z is a wo-dimensional vecor of independen Brownian moions. The mean reversion process µ is a saionary variable, defined by an Ornsein-Uhlenbeck process, dµ = α( µ µ )d + σ µdz, α >, µ > (.3) where µ is he consan long-run expeced rae of reurn, α is he rae a which µ converges o µ, and σ µ a wo-dimensional vecor of insananeous volailiies. The momenum erm m is defined by a sardard moving average (MA) of pas reurns over [, ], m = 1 ds u. (.) S u The modelling of momenum in his paper is moivaed by he ime series momenum (TSM) sraegies documened in Moskowiz e al. (1) who demonsrae ha he average reurn over a pas period (say, 1 monhs) is a posiive predicor of is fuure 3 The mos powerful predicive variables of fuure sock reurns in he Unied Saes have been found o be pas reurns, he marke dividend yield, he marke earnings/price raio, and erm srucure variables. We refer o Fama (1991) for a summary of he empirical sudies on reurn predicabiliy.

6 6 HE, LI AND LI reurns, especially he reurn for he nex monh. This is differen from Koijen e al. (9). In Koijen e al. (9), he momenum a ime is defined by M = e w( u) ds u S u, which is based on he pas reurns over [, ] wih geomerically decaying weighs. The advanage of M is ha he price dynamics can be reaed as a Markovian sysem, bu i also makes he sysem unable o be calibraed by merely using pure maximum likelihood mehod. Noe ha he weighs over [, ] are no added up o one. Thus M canno be reaed as a sandard average of pas reurns. Also M is no rolling forward wih a fixed ime window. The momenum m inroduced in his paper is a sandard moving average over a moving window [, ] wih lookback period of > and he weighs over he period are added up o one. This definiion is consisen wih he empirical momenum lieraure, which explores he price rends based on he reurns over a fixed look-back period. The resuling asse price model (.)-(.) is characerized by a sochasic delay inegro-differenial sysem, which is non-markovian. In discree-ime, i is he wellknown Gaussian vecor auoregressive (VAR) model (e.g. Johansen 1991). We show in Appendix A ha he process has an almos surely coninuous adaped pahwise unique soluion and asse price is always posiive for given iniial values over [, ]... Opimal Asse Allocaion. Consider a ypical long-erm invesor who maximizes he expeced uiliy of he erminal wealh. Assume ha he preferences of he invesor can be represened by a log uiliy funcion. Le W be he wealh of he invesor a ime and π is he fracion of he wealh invesed in he sock. Then he change in wealh follows dw W = { π [φm + (1 φ)µ r] + r } d + π σ SdZ. (.5) The invesmen problem of he invesor is given by J(W, m, µ,, T ) = sup E [ln W T ], (.6) (π u) u [,T ] where T is he erminal ime of he invesmen, and J(W, m, µ,, T ) is he value funcion corresponding o he opimal invesmen sraegies. Then we show in Appendix B ha he opimal dynamic sraegic allocaion can be characerized by he following proposiion. Specifically, he reurn process and he momenum variable in he discreizaion of heir model have he same expression. I yields ha he log likelihood funcion has nohing o do wih volailiy of he reurn shock and hence pure maximum likelihood mehod canno esimae i. In heir paper, hey esimae he model using maximum likelihood resriced o ensure ha he model-implied auocorrelaion srucure fis he empirical auocorrelaion srucure of sock reurns.

7 OPTIMALITY OF MOMENTUM AND REVERSAL 7 Proposiion.1. For an invesor wih log uiliy, he opimal sraegic allocaion o socks is given by π = φm + (1 φ)µ r σ S σ. (.7) S The opimal porfolio (.7) reflecs he myopic behavior of he invesor wih log uiliy. Two special cases are ineresing. Firs, when φ =, he asse price does no depend on he momenum and follows a geomeric Brownian moion process wih drif µ. In his case, he opimal porfolio (.7) becomes π = µ r σ S σ, (.8) S which is he sandard opimal sraegy when he drif is mean-revering, see, for example, Campbell and Viceira (1999) and Wacher (). In paricular, when µ = µ is a consan, he opimal porfolio (.8) collapses o π = µ r, which is he σ S σ S opimal porfolio in Meron (1971). Secondly, when φ = 1, he asse price depends only on he momenum. Correspondingly, he opimal porfolio (.7) reduces o π = m r σ S σ. (.9) S If we consider he rading signal indicaed by he excess reurn m r only, wih = 1, he opimal porfolio (.9) is consisen wih he TSM sraegy in Moskowiz e al. (1) by consrucing porfolios based on he monhly excess reurns over he pas 1 monhs and holds i for 1 monh. Moskowiz e al. (1) demonsrae ha his sraegy performs he bes among all he momenum sraegies wih look-back period and holding period varying from 1 monh o 8 monhs. If we only ake he posiion and consruc simple buy-and-hold momenum sraegies over a large range of look-back period and holding period invesigaed in Moskowiz e al. (1), (.9) shows ha he TSM sraegies can be opimal when he mean-revering is no significan in markes. This explains he dependence of he momenum profiabiliy on marke condiions and volailiy. Noe ha he opimal porfolio (.9) defines he opimal wealh fracion invesed in he risky asse, depending on no only he excess reurn bu also he volailiy. In general, he opimal porfolio (.7) implies ha a weighed average of momenum and mean-revering sraegies is opimal. Inuiively, i akes ino accoun of boh he shor-run momenum and long-run reversal, which are well suppored empirically. I is he simple form of he opimal porfolio (.7) ha faciliaes he comparison of he performance wih oher rading sraegies and he marke. As demonsraed by he following analysis, is empirical implicaion can be very significan.

8 8 HE, LI AND LI 3. Model Esimaion To demonsrae he opimaliy of he opimal porfolio (.7), we esimae he model o he S&P 5 in his secion. In line wih Campbell and Viceira (1999) and Koijen e al. (9), he mean-reversion variable is affine in he (log) dividend yield as follow, µ = µ + ν(d µ D ) = µ + νx, where ν is a consan, D indicaes he (log) dividend yield wih E(D ) = µ D, and X = D µ D denoes he de-meaned dividend yield. Thus he asse price model (.)-(.) becomes ds = [ φm + (1 φ)( µ + νx ) ] d + σ S SdZ, dx = αx d + σ XdZ, (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, he volailiy marix of he dividend yield and reurn is parameerized o be lower riangular, ( ) ( ) σ S σ S(1) Σ = =. σ X σ X(1) σ X() Tha is, Σ is he Cholesky decomposiion of he insananeous variance marix. Thus, he firs elemen of Z is he shock o he reurn and he second elemen of Z is he dividend yield shock ha is orhogonal o reurn shock. This seup follows Sangvinasos and Wacher (5) and Koijen e al. (9). By discreizing he coninuous-ime model a a monhly frequency o be consisen wih he momenum and reversal lieraure, sysem (3.1) resuls in a bivariae Gaussian vecor auoregressive (VAR) model for he simple reurn and dividend yield, which are observable, R +1 = φ (R + R R +1 ) + (1 φ)( µ + νx ) + σ S Z +1, (3.) X +1 = (1 α)x + σ X Z +1, where R = (S S 1 )/S 1 is he simple reurn of he sock a ime. 5 We esimae he model (3.) wih maximum likelihood mehod by employing monhly S&P 5 daa over he period January 1871 December 1 obained from he home page of Rober Shiller (hp://aida.wss.yale.edu/ shiller/). We se he insananeous shor rae o r = % annually. As in Campbell and Shiller (1988a, 5 Differen from Koijen e al. (9), we use he simple reurn o consruc m and also discreize he sock price process ino simple reurn raher han log reurn o be consisen wih he momenum and reversal lieraure.

9 OPTIMALITY OF MOMENTUM AND REVERSAL b), he dividend yield is defined as he log of he raio beween he las period dividend and he curren index. We consruc he oal reurn index using he price index series and he dividend series. 1 x α (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 3.1. 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. The esimaed parameers in monhly erms are illusraed in Fig. 3.1 for ranging from 1 monh o 5 years. As one of he key parameers of he model, Fig. 3.1 (b) shows ha he momenum effec φ is saisically differen from for 1, indicaing a significan momenum effec. Noe ha φ increases o 5% for

10 1 HE, LI AND LI [, 3] and hen decreases gradually when increases furher. Oher esimae resuls in erms of he level and significance in Fig. 3.1 are consisen wih Koijen e al. (9) AIC BIC HQ (a) AIC (b) BIC (c) HQ Figure 3.. (a) Akaike informaion crierion, (b) Bayesian informaion crierion and (c) Hannan Quinn informaion crierion for [1, 6]. Obviously, he esimaions depend on ime horizon. To explore he opimal value for, we compare differen informaion crieria for differen, including AIC, BIC and HQ from 1 o 6 monhs in Fig. 3.. AIC, BIC and HQ reach heir minima a = 3, 19 and respecively, implying ha he average reurns over a pas ime period of 1.5 years can predic fuure reurn bes. The increasing paern of he crieria for longer indicaes he reurn rend based on longer ime horizon window has less explanaory power. This resul is consisen wih he lieraure: here is no momenum effec for large ime horizon and he reurns exhibi mean revering in long run. Therefore, we should no expec he full model wih long ime horizons o well fi he daa σ S(1) (%) Figure 3.3. The esimaes of σ S(1) for he pure momenum model (φ = 1) for [1, 6].

11 OPTIMALITY OF MOMENTUM AND REVERSAL 11 To compare he performance of he opimal sraegies, comparing wih pure momenum or mean-revering sraegies, we also esimae he index o he model wih φ = 1 and φ = respecively. For he pure momenum model (φ = 1), he sysem (.)-(.) has only one parameer σ S(1) needed o be esimaed. Fig. 3.3 illusraes he esimaes of σ S(1) for [1, 6]. I shows ha he volailiy of he index decreases dramaically for small ime horizons and becomes sable for large ime horizons. I demonsraes ha he pas reurns over up o one year can explain par of he reurn volailiy bu longer hisorical reurns have less power in explaining reurn volailiy. We also compare differen informaion crieria for differen. AIC, BIC and HQ all reach heir minima a = 11 (no repored here). I implies ha he average reurns over he pas 11 monhs can predic fuure reurn bes for he pure momenum model. Parameers α µ ν σ S(1) σ X(1) σ X() Esimaes (%) Bounds (%) (.7, 1.3) (.31,.3) (-.6,.6) (3.97,.5) (-., -3.9) (1.3, 1.) Table 1. The esimaes of he parameers for he pure mean reversion model. Table 1 repors he esimaed parameers for he pure mean reversion model (φ = ). The resuls are comparable o hose for he full model illusraed in Fig Pure Momenum Pure Mean Reversion Likelihood raio es Figure 3.. The log-likelihood raio es for [1, 6]. For comparison, we repor he log-likelihood raio es in Fig. 3.. By comparing he full model o he pure momenum model (φ = 1) wih respec o each, he red solid line illusraes he es saisic which is much greaer han 1.59, he criical value wih 6 degrees of freedom a 95% confidence inerval. So he full model is significanly beer han he pure momenum model for all. The full model is

12 1 HE, LI AND LI also significan beer han he pure mean reversion model (φ = ) indicaed by he log-likelihood raio (he blue dash-doed line), which is much greaer han 3.81, he criical value wih 1 degree of freedom a 95% confidence inerval. In summary, we esimae he model o he index and show ha he opimal sraegies wih 1.5- year ime horizons fi he daa beer and also beer han pure momenum and mean-revering models. This observaion is furher suppored by performance analysis in he following secion.. Performance Based on he esimaion in he previous secion, we examine he opimaliy of he opimal porfolio in his secion. We firs examine he performance of he opimal rading sraegies (.7) by using wo proxies: he uiliy of porfolio wealh and he Sharpe raio. Then we sudy he sensiiviy of he performance of he opimal sraegies o he marke saes, invesor senimen and marke volailiy. Finally, following he momenum and reversal lieraure, we implemen some empirical analysis by focusing on he reurns implied by he opimal sraegies..1. Performance of he Opimal Sraegies. This subsecion provides empirical evidence on he opimaliy of he rading sraegies (.7). We use wo proxies o measure he performance of he opimal porfolio, he uiliy of he opimal porfolio wealh and he Sharpe raio The Full Model. To measure he performance of he opimal sraegies (.7), we compare he realized uiliy of he opimal porfolio wealh invesed in he S&P 5 index wih differen look-back period and 1-monh holding period o he uiliy of a passive holding invesmen in he S&P 5 index wih an iniial wealh of $1. We consider he look-back period from 1 monh o 6 monhs and inves monhly. For comparison, all he porfolios sar a he end of January 1876, (afer 6 monhs of January 1871 o calculae he rading signals). As he benchmark, he log uiliy of an invesmen of $1 o he index from January 1876 o December 1 is equal o For a fixed look-back period {1,,, 6}, say, = 1, we calculae he moving average m a any poin of ime (in monh) from January 1876 o December 1 using he index level over he ime period. Wih he iniial wealh of $1 a January 1876 and he esimaed parameers for = 1 in Fig. 3.1, we calculae he monhly invesmen of he opimal porfolio wealh W based on (.7). For = 1,,, 6, Fig..1 (a) illusraes he uiliy of he porfolio wealh from January 1876 unil December 1 for he opimal porfolio wih [1, 6] and he passive holding porfolio. For beer visibiliy, we also plo he uiliy of erminal wealh in Fig..1 (b). Boh of hem show ha he opimal sraegies ouperform he marke index in erms of he uiliy of wealh for [1, ]. As an

13 OPTIMALITY OF MOMENTUM AND REVERSAL * 8 ln W T (a) Series of uiliy (b) Terminal uiliy Figure.1. The uiliy of wealh from January 1876 unil December 1 for he opimal porfolio wih [1, 6] and he passive holding porfolio. alernaive benchmark, we also considered he sraegy holding 5% of wealh in he marke porfolio and 5% in he risk-free asse. Because he erminal uiliy of his sraegy is slighly high han he passive holding sraegy and we sill ge he same conclusions for he performance of he opimal sraegies, we do no repor i here. Moskowiz e al. (1) documen he TSM sraegy based on 1-monh horizon performs he bes. So we examine he performance for = 1 closely. Fig.. illusraes he ime series of he opimal porfolio and he uiliy of wealh from January 1876 unil December 1 for = 1. We also plo he corresponding index level and simple reurn of he oal reurn index of S&P 5 in Fig.. (a) and (b) over he same ime period. I shows ha he index reurn and π are posiively correlaed, wih a correlaion of.336. However, he correlaions become.398 and.139 for = 11 and = 7, a which he erminal uiliy has is maximum and minimum respecively as illusraed in Fig..1. From Fig.. (d), i seems ha he profis are mainly conribued by he Grea Depression in 193s. I is consisen wih Moskowiz e al. (1) who find ha he TSM sraegy delivers is highes profis during he mos exreme marke episodes. We also sudy he performance using he daa from January 19 o December 1 o avoid he Grea Depression periods. Based on he new esimaion (no repored here), we find ha he opimal sraegies sill ouperform he marke and he performances of he sraegies for he recen ime period become even beer for all ime horizons. This indicaes ha he opimal sraegies can ouperform he marke independen of marke condiion. To provide furher evidence, we conduc a Mone Carlo analysis. For = 1, wih he corresponding esimaed parameers in Fig. 3.1, we simulae he model (3.1). Fig..3 (a) illusraes he average porfolio uiliy based on 1, simulaions, ogeher wih 95% confidence levels (he wo green solid lines). I shows ha firs,

14 1 HE, LI AND LI P 8 6 1/1876 1/196 1/1976 1/ R /1876 1/196 1/1976 1/ π * /1876 1/196 1/1976 1/1 15 ln W * ln W /1876 1/196 1/1976 1/1 Figure.. The ime series of (a) he oal reurn 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 January 1876 unil December 1 for = 1.

15 OPTIMALITY OF MOMENTUM AND REVERSAL 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. (a) average uiliy and (b) one sided -es saisics based on 1 simulaions for = 1. he average performance of he opimal porfolio is beer han S&P 5 s. Secondly, he uiliy for he S&P 5 falls ino he area beween he wo bounds and hence he average performance of he opimal sraegy is saisically insignificanly differen from he marke index. We also plo wo black dashed bounds for he 6% confidence level. I shows ha, a 6% confidence level, he opimal sraegy significanly ouperforms he index. Fig..3 (b) illusraes he one sided -es saisics o es ln W SP 5 > ln W. The -saisics are above.8 in mos of he ime, which indicaes a criical value a 8% confidence level. Therefore, wih 8% confidence, he opimal sraegy significanly ouperforms he index * 1 ln W T Figure.. Average erminal uiliy based on 1 simulaions for [1, 6]. For [1, 6], Fig.. illusraes he average erminal uiliy based on he esimaed parameers in Fig. 3.1 and 1 differen random draws, which displays a differen erminal performance from Fig..1. In fac, he erminal uiliy in Fig..1 is based on only one specific rajecory (he real sock index), bu Fig.. provides average performance based on 1, rajecories. We can see ha

16 16 HE, LI AND LI he average erminal uiliy reaches is peak a =, which is consisen wih he resul based on he informaion crieria in Fig. 3., especially he AIC. Therefore, we show ha he simulaed average erminal uiliy is a beer measure characerizing he uiliy of he porfolio wealh. According o his proxy, he opimal sraegies ouperform he marke for mos of he ime horizons.. Opimal Sraegy Passive Holding.15 Sharpe Raio Figure.5. The Sharpe raio for he opimal porfolio wih [1, 6] and he passive holding porfolio from January 1881 unil December 1. We also use Sharpe raio o es he performance, which is defined as he raio of he mean excess reurn on he (managed) porfolio and he sandard deviaion of he porfolio reurn. If a sraegy s Sharpe raio exceeds he marke Sharpe raio, he acive porfolio 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. The average monhly reurn on he oal reurn index of he S&P 5 over he period January 1871 December 1 is.% wih an esimaed (uncondiional) sandard deviaion of.11%. The Sharpe raio of he marke index is.1. Nex, we consider he opimal sraegies (.7). The reurn of he opimal porfolio wealh a ime is given by R = (W W 1)/W 1 = π 1R + (1 π 1)r. (.1) Fig..5 illusraes he Sharpe raio for he opimal porfolio wih [1, 6] and compares o he passive holding porfolio from January 1881 unil December 1. If we consider he opimal porfolio as a combinaion of he marke porfolio and a risk free asse, hen he opimal porfolio is on he capial marke line and hence i should have equally good performance as he marke according o he Sharpe crierion. However Fig..5 demonsraes ha he opimal porfolio (blue line) ouperforms he marke (black line) on average for small ime horizon by aking

17 OPTIMALITY OF MOMENTUM AND REVERSAL 17 he iming opporuniy. Ineresingly, he resuls are perfecly consisen wih he measure of erminal uiliy illusraed in Fig In conclusion, we have shown ha he opimal sraegies ouperform he marke index..1.. The Pure Momenum Model. We now examine he performance of he pure momenum sraegies and compare wih he marke index. 5 5 * ln W T Figure.6. The uiliy of erminal wealh for [1, 6]. Based on he esimaed parameers in Fig. 3.3, Fig..6 illusraes he uiliies of all he opimal porfolios for he pure momenum model (φ = 1) a December 1. I shows ha he pure momenum sraegies underperforms he index wih all he ime horizons from 1 o 6 monhs. Fig..7 illusraes he ime series of he opimal porfolio and he uiliy of wealh from January 1876 unil December 1 for = 1 for he pure momenum model. By comparing o he full model illusraed in Fig.., he leverage of he pure momenum sraegies is much higher indicaed by he higher level of π. The opimal sraegies for he pure momenum model suffer from high risk and perform worse han he opimal sraegies for he full model. Therefore, he pure momenum sraegies underperforms he marke and he opimal sraegies The Pure Mean Reversion Model. Similarly, we examine he performance of he pure mean reversion model and compare i wih he marke index. Based on he esimaes in Table 1, Fig..8 illusraes he ime series of he opimal porfolio and he uiliy of wealh from January 1876 unil December 1 for he pure mean reversion model. The performance of he sraegy is abou he 6 This resul is differen from Marquering and Verbeek () who argue ha Sharpe raio performance and he uiliy-based performance can be inconsisen because Sharpe raio does no appropriaely ake ino accoun ime-varying volailiy. However, if and when Sharpe raio is a good measure is no he focus of our paper.

18 18 HE, LI AND LI 5 3 π * /1876 1/196 1/1976 1/1 3 ln W * ln W /1876 1/196 1/1976 1/1 Figure.7. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from January 1876 unil December 1 for = 1 for he pure momenum model. same as he sock index, bu worse han he opimal sraegies (.7) for he full model illusraed in Fig Ou of Sample Tess. In his subsecion, we implemen some ou of sample ess o he opimal sraegies by spliing he whole daa se ino wo sub-sample periods. We use he firs sample period o esimae he model and hen apply he esimaed parameers o he second par of he daa o examine he performance of he sraegies. Many sudies (see, for example, Jegadeesh and Timan 11) show ha momenum sraegies perform poorly afer he subprime crisis. We now focus on he performance of he opimal sraegies afer he subprime crisis and use he las 5 years daa o es he performance. Fig..9 illusraes he uiliy of wealh for [1, 6] for he ou of sample ess based on he las 5 years. I clearly shows ha he opimal sraegies sill ouperform he marke for ime horizons up o years. To beer undersand he performance of he ou of sample ess, we fix he ime horizon = 1 and examine he ime series of he opimal porfolio and he uiliy of he porfolio wealh from January 8 unil December 1 in Fig..1. I is

19 OPTIMALITY OF MOMENTUM AND REVERSAL π * /1876 1/196 1/1976 1/1 6 5 ln W * ln W /1876 1/196 1/1976 1/1 Figure.8. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from January 1876 unil December 1 for he pure mean reversion model ln W 1 6 1/8 1/1 (a) Series of uiliy.6.. * ln W T (b) Terminal uiliy Figure.9. The uiliy of wealh from January 8 unil December 1 for he opimal porfolio wih [1, 6] and he passive holding porfolio wih ou of sample daa of he las 5 years. clear ha he opimal sraegy ouperforms he marke over he sub-sample period, in paricular, during he financial crisis period around 9.

20 HE, LI AND LI 3 1 * π ln W * ln W 6 1/8 1/9 1/1 1/11 1/1 1/1 1 1/8 1/9 1/1 1/11 1/1 1/1 Figure.1. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from January 8 unil December 1 for = 1 wih ou of sample daa of las 5 years * ln W T Figure.11. The uiliy of erminal wealh for [1, 6] based on he ou of sample period of he las 71 years. As anoher ou of sample es, we spli he whole daa se ino wo equal periods: January 1871-December 191 and January 19-December 1. We esimae he model o he firs sub-sample period and do he ou of sample es over he second sub-sample period. Noice he daa in he wo periods are quie differen. The marke index increases gradually in he firs period bu flucuaes widely in he second period as illusraed in Fig.. (a). For he ou of sample es, Fig..11 illusraes he uiliy of erminal wealh for [1, 6] using sample daa of he las 71 years. I becomes clear ha he opimal sraegies sill ouperform he marke for [1, 1]. Wih fixed = 1, Fig..1 illusraes he corresponding ime series of he opimal porfolio and he uiliy of he porfolio wealh by conducing he ou of sample es from January 19 unil December 1. I shows ha he uiliy of he opimal sraegy grows gradually and ouperforms he marke index. We also use he las 1 years and years daa as he ou of sample daa and find he resuls are robus.

21 OPTIMALITY OF MOMENTUM AND REVERSAL 1 3 π * /19 1/197 1/1 8 7 ln W * ln W /19 1/197 1/1 Figure.1. The ime series of (a) he opimal porfolio and (b) he uiliy of wealh from January 19 unil December 1 for = 1 for he ou of sample ess wih ou of sample daa of las 71 years. To avoid look ahead bias, we also esimae he models using rolling window daa and conduc he performance analysis based upon he rolling window esimaion. We repor he resuls in Appendix C. The resuls are consisen wih he main findings above. Overall, he analysis demonsraes he opimaliy of he opimal rading sraegies based on a large range of ime horizons, in paricular, we demonsrae he ouperformance based on 1 monh ime horizon... Marke Saes, Senimen and Volailiy. In addiion o he ime horizon, he cross-secional momenum lieraure has shown ha he momenum profiabiliy is also sensiive o marke saes, invesor senimen and marke volailiy. For example, Cooper, Guierrez and Hameed () find ha shor-run (6 monhs) momenum sraegies make profis in he up marke and lose in he 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 shor ime horizon (1 year) are no profiable in down marke, bu profiable in up marke. Similar resuls of profiabiliy are also repored in Chordia and Shivakumar () ha commonly using macroeconomic insrumens relaed o he business cycle can generae posiive

22 HE, LI AND LI reurns o momenum sraegies during expansionary periods and negaive reurns during recessions. Baker and Wurgler (6, 7) find ha invesmen senimen affecs he cross-secion sock reurns and he aggregae sock marke. Wang and Xu (1) find ha marke volailiy has significan power o forecas momenum profiabiliy. For he TSM, however, Moskowiz e al. (1) find ha here is no significan relaionship of he TSM profiabiliy wih eiher marke volailiy or invesor senimen. We are ineresed in he dependence of he performance of he opimal sraegies on marke saes, invesor senimen and marke volailiy. We regress he excess reurn of differen sraegies, including he opimal sraegies for he full model, he pure momenum model and he pure mean reversion model and he TSM sraegies, on differen proxies for he marke saes, invesor senimen and marke volailiy. The regression resuls are repored in Appendix D. Overall, we find ha he coefficiens for all he regressions are insignificanly differen from. In fac, he opimal sraegies have opimally 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 he marke saes, invesor senimen and marke volailiy..3. Comparison wih Moskowiz, Ooi and Pedersen (1). The momenum sraegies in he empirical sudies are based on rading signals. In his secion, we implemen analysis following he empirical momenum lieraure, especially Moskowiz e al. (1). Firs, o verify he profiabiliy of he TSM sraegies, we examine he excess reurn of buy-and-hold sraegies when he posiion is deermined by he sign of he opimal porfolio sraegies (.7) wih differen combinaions of ime horizons and holding periods (, h). Based on he index, for a given look-back period, we ake long/shor posiions based on he sign of he opimal porfolio (.7). Then for a given holding period h, we calculae he monhly excess reurn of he sraegy (, h). Table repors he average monhly excess reurn (in %) of he opimal sraegies by skipping one monh beween he porfolio formaion period and holding period o avoid he 1- monh reversal in sock reurns for differen look back period (in he firs column) and differen holding period (in he firs row). The average reurn is calculaed using he same mehod as in Moskowiz e al. (1). To calculae he momenum componen and o evaluae he profiabiliy of holding period, we calculae he excess reurn of he opimal sraegies over he period from January 1881 (1 years afer January 1871 wih 5 years for calculaing he rading signals and 5 years for holding periods) o December 1.

23 OPTIMALITY OF MOMENTUM AND REVERSAL 3 ( \ h) (1.8) (1.8) (3.9) (.83) (1.8) (.) (.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) (1.5) (.1) (-1.16) (-.17) (.) (-1.11) (1.85) (1.) (.8) (-.) (-.76) (-1.3) (-.1) (-.3) (-1.6) (-.3) (-.51) (-.79) (-.6) (-.61) (-.3) (.) (-.3) (-.15) (.35) (.59) (.5) (.3) (.) (.81) (.9) (.) (.6) (1.7) (1.3) (1.6) (.93) (.3) (-.1) (.) (.7) (.86) (-.5) (-.6) (-.81) (-1.) (-1.1) (-.9) (.55) (.69) (.9) Table. The average excess reurn (%) of he opimal sraegies for differen look back period (differen row) and differen holding period h (differen column). For comparison, Tables 3 repors he average reurn (%) for he opimal sraegies for he pure momenum model. 7 Noice ha Tables and 3 indicae ha he (9, 1) sraegy performs he bes. This is consisen wih he finding in Moskowiz e al. (1), which documens (9, 1) is he bes sraegy for equiy marke alhough 1- monh horizon is he bes for mos asse classes. Also noe ha he resuls are inconsisen wih he resuls in he previous subsecions using he processes of uiliy of wealh as a measure of performance. This is because he opimal sraegies no only explore he reurn signal bu also akes he volailiy ino accoun. 8 Therefore, we argue ha he profiabiliy paern characerized by he average reurns (or excess reurns) used by mos of he empirical momenum lieraure may no be refleced a porfolio wealh level. This conclusion is also confirmed by comparing wih Fig..5. In fac, he Sharpe raio in Fig..5 characerizes boh he reurn and risk. We 7 Noice he posiion is compleely deermined by he sign of he opimal sraegies. Therefore, he posiion used in Table 3 is he same as ha of he TSM sraegies in Moskowiz e al. (1). 8 In fac, Wang and Xu (1) find ha marke volailiy has significan power o forecas momenum profiabiliy.

24 HE, LI AND LI ( \ h) (-.1) (.89) (1.3) (1.5) (1.37) (-.1) (.) (.) (-.57) (1.61) (.16) (1.91) (.) (1.17) (-.69) (-.7) (-.5) (-1.38) (.78) (.79) (3.1) (.9) (1.3) (-.75) (-.6) (.) (-.77) (3.91) (3.78) (.6) (1.76) (.66) (-1.5) (-.38) (-.1) (-1.1) (.35) (1.67) (.9) (.13) (-.86) (-1.63) (-1.) (-.95) (-1.88) (.9) (-.) (-.81) (-1.3) (-1.6) (-.88) (-.13) (-.) (-.39) (-.1) (.19) (.3) (.) (.1) (.) (.18) (.37) (.33) (.7) (.73) (.) (.) (-.) (-.83) (-.1) (.61) (.55) (-.5) (-.8) (-1.1) (-1.77) (-.15) (-1.36) (-.6) (.) (.3) Table 3. 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. can see ha Fig..5 has a similar profiabiliy paern o Table, especially he firs column, bu differen from Fig.. for he average erminal uiliy of wealh...15 Opimal Sraegy Passive Holding Momenum & Mean Reversion Momenum Sraegy Sharpe Raio Figure.13. 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.

25 OPTIMALITY OF MOMENTUM AND REVERSAL 5 We also sudy he Sharpe raio for he TSM sraegies (green line) documened empirically for differen ime horizons and he sraegies (momenum and mean reversion) (red doed line) which are similar o he TSM sraegies excep ha we use he sign of he opimal sraegies sign(π ) as he rading signal insead of he average excess reurn over a pas period for he TSM sraegies. For comparison, we also plo he Sharpe raio for he opimal porfolio and he passive holding porfolio which have been already repored in Fig..5. Fig..13 shows he average Sharpe raio for he above four sraegies wih [1, 6] from January 1881 unil December 1. There are hree observaions from Fig..13. Firs, he TSM sraegies ouperform he marke only for = 9, 1 and he momenum and mean reversion sraegies ouperform he marke for shor ime horizons 13. The second observaion is ha, by aking he mean reversion effec ino accoun, he momenum and mean reversion sraegies perform beer han he TSM sraegies for all ime horizons. Finally, he opimal sraegies significanly ouperform boh he momenum and mean reversion sraegies and he TSM sraegies. Noice he only difference beween he opimal sraegies and he momenum and mean reversion sraegies is ha he former considers he size of he porfolio posiion, which is he inverse of variance, while he laer only akes one uni posiion. This implies ha, in addiion o price rend, he posiion size (or volailiy) is anoher very imporan facor for exploring he iming opporuniy. Following Eq. (5) in Moskowiz e al. (1), we sudy he performance of he cumulaive excess reurn. Tha is, he reurn a ime is given by ˆR +1 = sign(π ).1 ˆσ S, R +1, (.) where.1 is he sample sandard deviaion of he oal reurn index. Following Moskowiz e al. (1), 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) i= where he scalar 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) = monhs. To avoid look ahead bias conaminaes he resuls, we use he volailiy esimaes a ime 1 applied o ime reurns hroughou he analysis.

26 6 HE, LI AND LI Opimal Sraegy Passive Long Momenum Sraegy Figure.1. Terminal log cumulaive excess reurn of he opimal sraegies (.7) and TSM sraegies wih [1, 6] and passive long sraegy from January 1876 unil December 1. Fig..1 illusraes he erminal values of he log cumulaive excess reurn of he opimal sraegies (.7) and TSM sraegies wih [1, 6] and he passive long sraegy from January 1876 unil December 1. 9 I illusraes ha he opimal sraegies ouperform he TSM sraegies. The TSM sraegies ouperform he marke for small ime horizons. The erminal values of he log cumulaive excess reurn have similar paerns as he average Sharpe raio in Fig Opimal Sraegy Passive Long Momenum Sraegy Growh of $1 (log scale) 6 1/1876 1/196 1/1976 1/1 Figure.15. Log cumulaive excess reurn of he opimal sraegy (.7) and momenum sraegy wih = 1 and passive long sraegy from January 1876 unil December 1. Wih a 1-monh ime horizon, Fig..15 illusraes he log cumulaive excess reurn of he opimal sraegy (.7) and momenum sraegy and he passive long sraegy from January 1876 unil December 1. I shows ha, firs, he opimal 9 Noice he passive long sraegy inroduced in Moskowiz e al. (1) is differen from he passive holding sraegy sudied in 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.

27 OPTIMALITY OF MOMENTUM AND REVERSAL 7 sraegy has he highes growh rae and he passive long sraegy has he lowes growh rae. Secondly, we can replicae he paern in Fig. 3 of Moskowiz e al. (1), which documens ha he TSM sraegy ouperforms he passive long sraegy. Noice he TSM sraegy documened in Fig..15 should have he same performance as he opimal sraegy for he pure momenum model if we only consider he sign of he average excess reurn over he pas period in (.). However, we have documened in he previous subsecions ha he opimal sraegies for he pure momenum model perform badly if we also ake he volailiy ino accoun. Therefore, we argue ha he momenum sraegies documened empirically may no have good performance in sock index when he performance is measured a porfolio wealh level. 1 To conclude his secion, by comparing he resuls for he opimal sraegies and he TSM sraegies in Moskowiz e al. (1), we find ha he rend is imporan. To explore i, he buy/sell signal (based on π ) is more imporan han he size of he porfolio. Also, his paper sudies he S&P 5 index over 1 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. 5. Conclusion To characerize he shor-run momenum and long-run reversal in financial markes, we propose a coninuous-ime model of asse price process wih he drif as a weighed average of mean reversion and moving average componens. By applying he maximum principle for conrol problem of SDDEs, we derive he opimal sraegies analyically. We show he opimaliy of he opimal sraegies comparing o pure momenum, pure mean reversion sraegies, and marke index. The opimaliy is immune o he marke saes, invesor senimen and marke volailiy. The profiabiliy paern refleced by he average reurn in mos empirical lieraure may no be refleced a porfolio wealh level. The model proposed in his paper is simple and sylized. The weighs o 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, based on 1 In fac, he profis of he diversified 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 sock index may have less significan profis han he diversified TSMOM porfolio.

28 8 HE, LI AND LI he replicaor dynamics inroduced by He and Li (1), or as a Markov swiching process, or based on some raional learning process. The opimizaion problem is solved under log uiliy in his paper. I would be also ineresing o sudy he ineremporal effec under general power uiliy funcions which is considered in Koijen e al. (9). Furhermore, we can consider sochasic volailiies of he index process. Finally, an exension o a muli-asse model o sudy he cross-secional opimal sraegies would be helpful o undersand he cross secional momenum.

29 OPTIMALITY OF MOMENTUM AND REVERSAL 9 Appendix A. Properies of he Soluions o he Sysem (.)-(.3) 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 (.)-(.) admis pahwise unique soluions such ha S > almos surely for all whenever ϕ > almos surely. Proposiion A.1. The sysem (.)-(.) has an almos surely coninuous adaped pahwise unique soluion (S, µ) for a given F -measurable iniial process ϕ : Ω C([, ], R). Furhermore, if ϕ > a.s., hen S > for all a.s.. 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 (.)-(.3) becomes ds = S dn, [, ], dµ = α( µ µ )d + σ µdz, [, ], (A.1) where N = [ φ s s S = ϕ and µ = ˆµ. dϕ u ϕ u + (1 φ)µ s ] ds + σ S dz s is a semimaringale. Denoe by [N, N] = σ S σ Sds, [, ], is quadraic variaion. Then he sysem (A.1) has a unique soluion S = ϕ exp { N 1 [N, N] }, µ = µ + (ˆµ µ) exp{ α} + σ µ exp{ α} exp{αu}dz u for [, ]. This clearly implies ha S > for all [, ] almos surely, when ϕ > a.s.. By a similar argumen, i follows ha S > for all [, ] a.s.. Therefore S > for all a.s., by inducion. Noe ha he above argumen also gives exisence and pahwise-uniqueness of he soluion o he sysem (.)-(.). 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. The SDDE is no Markovian so we canno use he dynamic programming mehod. Recenly, Chen and Wu (1) inroduce a maximum principle for he opimal conrol problem of SDDE, and his mehod is furher exended by Øksendal e al (11) o consider a one dimensional sysem allowing boh delays of moving average ype and jumps. Because he opimal conrol problem of SDDE is relaive new o he field of economics and

30 3 HE, LI AND LI finance, we firs inroduce he maximum principle in Chen and Wu (1) briefly and refer readers o he original for deails. B.1. Brief Inroducion o he 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 an 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 and he 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 he Theorem. and Theorem.1 in Chen and Wu (1) for he exisence and uniqueness of he soluions of he sysems (B.1) and (B.3) respecively. 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

31 OPTIMALITY OF MOMENTUM AND REVERSAL 31 have he following proposiion on he maximum principle of he sochasic conrol sysem wih delay by summarizing he Theorem 3.1, Remark 3. 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.) (ii) Suppose u( ) A and le X( ) be he corresponding rajecory, p and z be he soluion of he adjoin equaion (B.). If (H1), (H) and (B.) hold for u( ), hen u( ) is an opimal conrol for he sochasic delayed opimal problem (B.1) and (B.). B.. Proof of Proposiion.1. Nex, we apply he heory in Chen and Wu (1) summarized in Subsecion B.1 o our sochasic conrol problem. Le P u := ln S u and V u := ln W u. Then he sochasic delayed opimal problem in Secion becomes o maximize E u [Φ(X T )] := E u [ln W 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 Then we have he following adjoin equaion dp u = { (b π X ) p u + (σ π X ) z u + E u [(b π X u+ ) p u+ + (σ π X u+ ) z u+ ] + L X } du zu dz u, u [, T ], p T = Φ X (X T ), p u =, u (T, T + ], z u =, u [T, T + ],

32 3 HE, LI AND LI where p u = (p i u) 3 1, (b π X u+ ) = φ 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 1 φ α (1 φ)πu 1, L X =, Since he parameers and erminal values for dp 3 u are deerminisic, we can asser zu 31 = zu 3 = for u [, T ], which leads o p 3 u = 1 for u [, T ]. Then he Hamilonian funcion H is given by so ha H = [ φ (P u P u ) + (1 φ)µ u (1 φ) σ S σ S ] p 1 u + α( µ µ u )p u { + π uσ 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, zu H π π = π uσ Sσ S + φ (P u P u ) + σ S σ S φ + (1 φ)µ u r. I can be also obained ha E u [H π π u+ ] =. Therefore, [ Hπ π + E u [Hπ π u+ ], π = π u π u σ Sσ S + φ (P u P u ) + σ S σ S φ + (1 φ)µ u r ]. Taking he derivaive wih respec o π u and leing i equal zero yield φ πu = (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 p 3 u, Appendix C. Rolling Window Esimaions In his secion we implemen rolling window esimaion. We firs fix = 1 and esimae parameers of (3.) a each monh by using 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 big volailiy of marke reurn illusraed in Fig. C. (b). 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

33 OPTIMALITY OF MOMENTUM AND REVERSAL α φ..1. 1/1891 1/191 1/1991 1/1 (a) Esimaes of α.1 1/1891 1/191 1/1991 1/1 (b) Esimaes of φ x µ ν /1891 1/191 1/1991 1/1 (c) Esimaes of µ.5.1 1/1891 1/191 1/1991 1/1 (d) Esimaes of ν σ S(1) (%) 3 1/1891 1/191 1/1991 1/1 3 5 σ X(1) (%) /1891 1/191 1/1991 1/ σ X() (%) /1891 1/191 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 he daa of pas years. wealh from December 189 unil December 1 for = 1 wih years 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 193s. Fig. C.1 also illusraes an 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), he insignifican φ is accompanied by high volailiy σ S(1). Fig. C.3 illusraes he correlaions of he

34 3 HE, LI AND LI P 8 6 1/1891 1/191 1/1991 1/ R /1891 1/191 1/1991 1/ π * /1891 1/191 1/1991 1/1 1 1 ln W * ln W /1891 1/191 1/1991 1/1 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 years rolling window esimaed parameers.

35 OPTIMALITY OF MOMENTUM AND REVERSAL Correlaions Correlaions Correlaions Correlaions Figure C.3. 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.

36 36 HE, LI AND LI 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 he TSM reurn for [1, 6]. Ineresingly, bigger volailiy is accompanied by 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 big ime horizon makes he rading signal less sensiive o he changes in price and hence he rading signal is significan only when he marke price changes dramaically in high volailiy period. Fig. C.3 (c) and (d) shows ha he profiabiliy of he opimal sraegies for he pure momenum model and he TSM sraegies are sensiive ohe esimaed marke volailiy. The reurn is posiively (negaively) relaed o marke volailiy for shor (long) ime horizons. Bu Fig. C.3 (b) shows ha he opimal sraegies for he full model perform well even in high volailiy marke Figure C.. 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. C. illusraes he corresponding fracion of φ which is significanly differen from zero for [1, 6]. I shows ha he momenums wih -3 monhs horizons occur he mos frequenly during he period of December 189 unil December 1. Fig. C.5 (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. C.5 (b) shows ha he opimal sraegies work well for shor horizons and he erminal uiliy reaches is peak a = 1.

37 OPTIMALITY OF MOMENTUM AND REVERSAL * ln W T Figure C.5. The uiliy of wealh from December 189 unil December 1 for he opimal porfolio wih [1, 6] and he passive holding porfolio wih years rolling window esimaed parameers σ S(1) (%) 5 3 1/1891 1/191 1/1991 1/1 Figure C.6. Esimaes of σ S(1) for he pure momenum model (φ = 1) based on he daa of pas years. Fig. C.6 illusraes he esimaes of σ S(1) for he pure momenum model (φ = 1) based on he daa of pas years and he big jump in volailiy is due o he Grea Depression in 193s. Fig. C.7 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 years rolling window esimaed σ S(1). By comparing Fig. C.6 and Fig. C.7 (b), he opimal sraegy implied by he pure momenum model suffers huge losses during he big marke volailiy period. Bu Fig C. illusraes ha he opimal sraegy implied by he full model makes big profis during he big marke volailiy period. Fig. C.8 illusraes he esimaed parameers for he pure mean-reversion model based on he daa of pas years. Fig. C.9 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