Time Series Momentum Trading Strategy and Autocorrelation Amplification

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1 e Sere Moenu radng Sraegy and uocorreaon pfcaon K. J. Hong and S. Sache June 3 CWPE 3

2 e Sere Moenu radng Sraegy and uocorreaon pfcaon K. J. Hong a,* and S. Sache b Curren Veron: May 3, 3 a Unvery echnoogy of Sydney, Uo Rd, Hayarke NSW, uraa b rny Coege, Unvery of Cabrdge, ddre: rny Coege, Cabrdge, CB Q, U.K brac h arce aue genera aonary procee for prce and derve he auocorreaon funcon for a genera Movng verage M radng rue o nvegae hy h rue ued. he reu ho ha he M rue popuar becaue can denfy prce oenu and a pe ay of racng and expong prce auocorreaon rucure hou neceary knong prece rucure. We focu on anayzng he pac of prce oenu on he profaby of he M rue becaue he prce oenu effec end o be ronger and ore peren han he reurn oenu effec. * Correpondng auhor. e: E-a addre: [email protected]

3 Inroducon Whe fundaena anay udy a copany underyng ndcaor of prof uch a earnng, dvdend, ne produc and R&D, echnca anay focu any on prce and reurn bu, unngy or ohere, ao ake no accoun pychoogca apec n he deand for a copany ock. echncan epoy any echnque, one of hch he ue of char. Ung char, echnca anay eek o denfy prce paern and arke rend n fnanca arke and aep o expo hee paern. rader and porfoo anager connue o ue echnca anay ha foreca ock prce oveen ung horca prce o foruae buy and e decon. Moenu an acve reearch opc and recen ude ncude Grnba and Mokoz 3, Chora and Shvakuar 6, Sadka 6, Zhu and Zhou 9, Hou, Karoy and Kho, Marx, Faa and French, Mokoz, Oo and Pederen., MOP hereafer, Bagrocz and Sxae and Menkhoff, Sarno, Scheng and Schrpf. Soe earer ude ugge ha echnca anay bea he arke n rk neura er, hence popuary. One of he pe and o dey ued radng raege baed on echnca anay he Movng verage M rue. I an obecve rue-baed radng raegy n hch buy and e gna are deerned by he reave agnude of hor and ong er M. n obecve rue-baed radng ye ha e-defned and ndpuabe buy and e gna foong a decon rue baed on pa daa. here are a nuber of pubhed paper on M rue ncudng car and Sache 997, Charea, He and Hoe 6 and Menkhoff. he M rue ofen ead nveor o nve h or agan he rend, or oenu nce aue ha prce rend dreconay. I ake advanage of prce rend, hch coey nked h he raher nebuou dea of prce oenu. Prce Moenu defned by exng eraure a he produc of he reave prce over he hodng perod and a pa nner raegy o buy he op n-h of ock h h characerc n a porfoo. carefu readng of he oenu eraure fa o fnd carfcaon over heher oenu preen hen he prce of he ae foo a a rando ak b a rando ak h drf or c a ean reverng proce around a near rend. he ue furher copcaed by dfferen reaen for prce, reurn and earnng. One of he purpoe of h paper o ry and underand he e ere ba of a oenu raegy. Unke oher oenu eraure ha focu on he reave perforance of ecure n he cro-econ, e ephaze ha h paper nend o nvegae e ere

4 oenu, he oenu hn an ae prce proce. MOP ao ook a e ere oenu and argue ha e ere oenu drecy ache he predcon of any pronen behavora and raona ae prcng heore. Barber, Shefer and Vhny 998, Dane, Hrhefer and Subrahanya 998, and Hong and Sen 999 a focu on a nge rky ae, herefore havng drec pcaon for e ere, raher han croecona, predcaby. Lke-e, raona heore of oenu Berk, Green and Nak 999; Johnon, ; hn, Conrad and Dar, 3; Sag and Seahoe, 7; Lu and Zhang, 8 ao peran o a nge rky ae. e ere oenu focue purey on a ecury on pa proce. MOP nvegae 58 qud nruen of equy ndex, currency, coody, and bond fuure and fnd ex n a ae. We aue genera aonary procee for prce and derve he auocorreaon funcon for a genera M rue. We focu on anayng he pac of prce oenu, nead of reurn or earnng oenu, on he profaby of he M rue becaue o of, f no a, radng rue ung echnca anay are baed on pa prce, and he prce oenu effec end o be ronger and onger-ved han he reurn oenu effec. here ex anoher ype of prce rend bede he e ere and cro-econ oenu dcued, hoever. Equy prce end o ho a ong run arke-de rend. h can be due o nfaon or oher econoc condon. We defne h arke-de rend by he paraeer, μ. We reove he arke-de rend fro our prce proce n order o focu on prce oenu ony. h paper preen a pe and raghforard evauaon of he M rue h one nveor and one ae. We fnd ha ha here are o reaon hy he M rue popuar. Fr, he M rue can denfy he prce oenu rend, hch coforaby confr he reu of he prevou oenu eraure. Second, he M rue a pe ay of racng and expong he prce auocorreaon rucure hou acuay knong rucure. I very hard, f no pobe, o kno he rea rucure of he prce auocorreaon of an ae. he M rue provde a pe and cear ehodoogy ha can ake advanage of he prce auocorreaon rucure. he an conrbuon of h paper ha e ugge o reaon hy he echnca radng raegy, or Char, popuar and h ead o prce auocorreaon. Our fndng of he auocorreaon rucure of he radng rue ued n h udy drecy ache he fndng of any pronen behavora and raona ae prcng heore. For exape, f a conerva ba ex n a arke, he M rue oud be 3

5 profabe. conerva ba ndcae ha nveor are oo o oo conervave n updang her beef n repone o recen evdence. h ean ha hey gh nay underreac o ne abou a fr, eanng ha prce fuy refec ne nforaon ony graduay. Such a ba oud gve re o oenu prce auocorreaon n he ock arke. Ung R and M procee, e ake he dcuon aay fro hor er prce rend connuaon or ong er prce revera. In hor, no u abou buyng pa oer or nner bu ao abou prce rend and he auocorreaon rucure. h paper doe no nend o expan a he reaon for predcaby nor doe aep o expan hy arbrage fa o enae prcng. he conrbuon of he paper cear.. Unke prevou eraure ha any nvegae cro econa oenu aong varou eque, h paper nvegae e ere oenu. he reu ugge ha oenu radng raegy popuar no ony becaue ake advanage of rend hn he e ere daa bu ao becaue apfe he auocorreaon. h ho ha a e ere oenu radng raegy a pe ay of condonng on pa prce nforaon.. he paper furher nvegae he rucura change of auocorreaon o denfy he paern of he apfcaon. h poran becaue uch ncreae n auocorreaon no capured by he andard rk eaure uch a VaR. he paper provde a gudene of ho uch apfcaon occurrng. In h repec, he reu of he paper ha rk anageen pocy pcaon. 3. he paper ao provde heoreca ufcaon o de ue of onger perod M for prce oenu and horer perod M for reurn oenu. h paper coud be een a an exenon of MOP n ha nvegae anoher apec of e ere oenu, auocorreaon rucure. he curren udy herefore expand our knoedge of e ere oenu. he reu ha rk anageen and reguaory pocy pcaon. I repor un-capured poena ource of rk n one of he o popuar radng raege. 4

6 he re of he paper organzed a foo. Secon expan e ere oenu, he radng rue and ho reaed o prce oenu. Secon 3 preen he genera ode. Secon 4 ho he auocorreaon rucure of he radng raegy. Secon 5 preen eprca evdence and Secon 6 concude he paper. Background. e Sere Prce Moenu and echnca radng Rue Eprca ude of echnca anay, ncudng Brock, Lakonhok and LeBaron 99, Bue, Eaey and O Hara 994, Chan, Jegadeeh and Lakonhock 996, Bron, Goezann and Kuar 998, en and Karaanen 999, Chan, Haeed and ong, Lo, Maayky and Wang and Hou, Karoy and Kho focu on he reave perforance of ecure n he cro-econ. Many eraure fnd ha uch cro econa ode ork very e. Hoever, Leeen, Nage and Shanken argue ha he eprca ehodooge ued n cro econa ae prcng eraure are ofen hghy eadng, n he ene ha appareny rong expanaory poer hgh cro-econa R and a prcng error can provde que eak uppor for a ode. here anoher knd of oenu ha coud be nvegaed, e ere oenu. MOP argue, e ere oenu he oenu hn ae prce proce. I reaed o bu dfferen fro he phenoenon knon a oenu n he fnance eraure, hch prary croecona n naure. he oenu eraure focue on he reave perforance of ecure n he cro-econ, fndng ha ecure ha receny ouperfored her peer over he pa hree o onh connue o ouperfor her peer on average over he nex onh. Raher han focu on he reave reurn of ecure n he cro-econ, e ere oenu focue purey on a ecury on pa reurn. In h paper, e nvegae a echnca radng raegy baed on h e ere oenu h he focu on he auocorreaon rucure of he radng raegy. echnca radng rue ake advanage of e Sere Moenu. In uppor of MOP, e ao fnd ha our reu conen h na under-reacon and deayed over-reacon ore. We epoy and expand he defnon of he M ocaor by Brock, Lakonhock and LeBaron 99, hch ae: ccordng o he ovng average radng rue, buy and e gna are 5

7 generaed by o ovng average of he eve of he ndex a ong-perod average and a hor-perod average. In pe for h raegy expreed a buyng or eng hen he hor-perod ovng average re above or fa beo he ong-perod ovng average. he dea behnd copung ovng average o ooh ou an ohere voae ere. When he hor-perod average exceed he ong-perod ovng average, a rend condered o be naed. very popuar ovng average rue -, here he hor perod one day and he ong perod day. Whe nuerou varaon of h rue are ued n pracce, e aeped o eec evera of he o popuar one: -5, -5, 5-5, -, and -. Hoever, e can ao ake no conderaon he fp de of he raegy, here e buy hen he hor-perod M fa beo he ong-perod M and e e hen he oppoe happen. Le and for he e perod ha he M copued for a ong poon and for he e perod ha he M copued for a hor poon. Hence > and hey overap. Denoe SM a M copued over e perod and LM a M copued over e perod. For exape a e, f = 7 and = 3, hen SM copued over e 6 o and LM copued over e o. We cafy o oppoe M rue h SM and LM, Mbu, and Mbear,, and defne he Mbu, rue a akng a ong poon n he ae hen SM > LM and a akng no poon n he ae hen SM < LM and Mbear, rue doe he oppoe. Defnon : he Mbu, and Mbear, poon can be repreened a Mbu, ax, Mbear, ax, f LM SM here f LM SM and P SM and P LM he ogc behnd he Mbu rue e knon, hen a prce penerae he M fro beo, he bu rend beeved o be eabhed and a rader an o ake advanage of h 6

8 n he expecaon ha here be furher upard oveen n prce. he Mbear rue e u he oppoe ory, knon o praconer a hor er prce revera. When he prce fa enough o penerae he M fro above, h gh be due o arke over-reacon; hence a rader oud ake a ong poon, expecng ha he prce oud bounce edaey back. he reaonhp beeen he M rue and prce oenu e expaned n Fong and Yong 5. In he pe for, a M croover rue operae on he aupon ha buy gna are generaed hen he curren ock prce croe ovng average fro beo he e gna are generaed hen he ock prce croe ovng average fro above. he raonae for h nerpreaon ha a rend ad o have eerged hen he ock prce penerae he ovng average. Specfcay, an upard buh rend eerge hen he prce re above ovng average, he a donard bearh rend eerge hen he prce fa beo ovng average. herefore, he Mbu rue ake advanage of he buh rend he he Mbear rue ake advanage of he bearh rend. Mbu raegy can be hough a buyng a pa oer becaue ake a ong poon hen he ae prce reavey o and qudae he poon hen he ae prce reavey hgh.. Moenu Defnon and a Saca Mode Hong and Sache conder he proce dq q d dw here q : og P Wh he Ornen Uhenbeck proce above, hey nvegae he defnon of oenu propoed by che, oenu can be nerpreed a he envy of he expeced prce rao h repec o oday prce. If h envy pove hen here a buh oenu and f negave here a bearh oenu. Propoon of Hong and Sache ho he og condona popuaon oenu h repec o pervou perod og prce. hey noe ha for og OU, oenu abou auocorreaon. Equaon 3 ho ha hen he prce proce ean reverng, he eacy of oenu negave he pove f he prce expove. h exacy ha e oud ancpae. In oher ord, 7

9 gven he hgh pa prce, he fuure prce o f he prce proce rever around a ean bu he fuure prce hgh f he prce proce expove. If e ere o nerpre effcen arke a a og rando ak h drf, hen he eacy oud becoe zero. Fnay, he eacy ndependen of he rend..3 Carfcaon of Noaon We epoy a grea dea of noaon n h paper. Soe ony dffer n uper and ub crp, oe are conex pecfc. For exape, h or hou uper and ubcrp and for a covarance arx. and for he covarance arx of curren de-eaned prce proce, and for he covarance arx beeen he curren and -perod ag de-eaned prce proce and ndcae he covarance arx of he radng raegy crca vaue. We ncude a goary n ppendx o hep reader. 3 he Mode 3. Genera Mode Le he de-eaned og prce proce ha drbuon q ~, q og P. o denoe he auocorreaon beeen q and q h a h. Snce e are ung og prce, oud be enbe o ue geoerc prce M hch are equvaen o arhec og prce M. q SM exp and q LM exp here Under uch a fraeork, he reurn a e defned a q q og P og P 3 8

10 Defnon : he crca vaue of a M, rue defned a LM C, ; exp SM Q 4 here and Q q q q q he dervaon of defnon n ppendx. ake a ong poon f C,; becoe ore han and qudae he poon f C,; becoe e han. When he M rue apped baed on a pove rendng prce proce, he ex pon here he anager qudae a ong poon due o he curren prce penerang he M fro above aay be hgher han he enry pon hence he reurn aay pove. Hoever, he anager py pckng up he prce rend and he ae reurn can ao be ade h a buy and hod raegy. h e nereng. he porance of he auocorreaon hn a fnanca e ere ao noed n MOP, a hey ae We fnd ha pove auocovarance n our fuure conrac reurn drve o of he e ere and cro-econa oenu effec e fnd n he daa. he conrbuon of he oher o reurn coponen era cro-correaon and varaon n ean reurn a. In order o eparae he pac of oenu fro he arke-de rend, e ue he derended og prce, hch ao ha uefu aca propere for eaon. he derended M rue ao ha a pe prce nerpreaon a hon beo. Snce Mbear rue py he oppoe of he Mbu rue, e ony ook a he Mbu hereafer. Hoever he Mbear rue oud be uefu a one of benchark n evauang he profaby of Mbu rue. ung aonary n he econd oen, he bvarae drbuon of C and C- can be expreed a, Q ~ Q N 9

11 here he covarance arx of Q and C covarance arx beeen Q and Q-. For no, e do no aue anyhng abou he covarance arx bu py noe ha he auocovarance no zero. Lea : he Mbu, rue baed on a derended og prce equvaen o buyng a un of an ae hen P exp P 5 he dervaon of Lea n ppendx 3. Under our og Ornen Uhenbeck aupon, C,; ha a og nora drbuon. C, ; exp Q Q ~ N, hough e appy h ode o he prce proce ony, he appcaon of he ode doe no need o be rerced o he prce proce ony. I can ao be apped o reurn or earnng or o any proce ha oud have e ere oenu auocorreaon / rend. h pon be dcued agan aer n he paper. 3. he Profaby of he M Rue Before e dcu he profaby, e have o ephaze ha h paper doe no nend o nvegae he profaby of he M rue. he ony reaon ha e dcu he profaby becaue hep o nvegae he ranon vecor of he crca vaue, C,; drbuon. Hoever e do no h o ge no a fu engh dcuon of he profaby of he radng rue, hch no he an opc of nere n h paper. We pan o nvegae h opc n a eparae udy. h pon be ephazed agan hen e perfor eprca nvegaon. Defnon 3: Defne, ; a

12 f C, ; Q, ; f C, ; Q 6 hence E, ; probc, ; Reark : he uncondona and condona probaby of, ; akng he vaue of under noray n prce can be expreed a Uncondona: prob, ;. 5 Condona: prob, ; q SM 7 here : he fr order of he radng raegy auocorreaon and Q q SM. he proof of Reark n ppendx 4. Propoon : he h perod reurn fro he Mbu rue fro e k, r k h, k can be expreed a r, ; k h, k h, ; k k r k, k 8 here r, ; k h, k he h perod reurn fro he Mbu rue fro e k o e k+h, gven and. r k h, k he h perod ae reurn fro e k o e k+h. If r k h, k r k h, k, hen he M rue profabe. h ony pobe f ore key o ake a vaue of hen he ae reurn pove. h provde h u a condon ha oud ake he Mbu rue profabe. If and, he condona probaby of beng equa o one arger han.5 hen q q, hence he Mbu rue profabe.

13 naycay anayzng equaon 8 oo copcaed hence uaon oud be uefu. We noed ha prce auocorreaon poran n deernng he condona probaby of, ;. hu an aupon regardng prce auocorreaon need o be ade and e aue ha he ae prce foo he R proce. We ue Godan Sach GS daa fro May 4, 999 o May 3, for eaon. here no peca reaon for eecng uch daa and e perod. he eaon ony o ocae oe enbe paraeer. he uaon proce uarzed n ppendx 5. h paper concerned h prce eve, hch end o have very hgh auocorreaon. Hence an nvegaon of hgh eve auocorreaon oud be ore eanngfu. For convenence, e exane he M, rue. abe : Suaed profaby of he M, rue hen ae prce foo an R proce for ρ =.99,.95,.995,.9975 and under noray nnua Prof Rho M bu 3.9% 3.45% 3.94% 5.9% M bear 5.54% 5.% 3.9%.73% Buy and Hod.79%.4% 3.55% 6.54% abe preen he uaed reu. he error er are uaed o have a nora drbuon. he uaed reu ho ha he condonng on prevou prce nforaon affec he profaby of he radng raegy hen here pove prce auocorreaon. he Mbu rue ouperfor he oher raege becaue here a pove rend n he uaed og prce proce hence prob, ;. 5 hen he ae reurn pove. h confr he reu of prevou eraure ncudng Fong and Yong 5 and ha e exacy ancpaed. he reu ao hghgh ha he degree of auocorreaon affec he perforance of he radng raegy a e. he porfoo poon deerned by he gn of C,;, herefore a furher nvegaon of C,; ranon h repec o e oud be eanngfu. o gan ngh no he propere of he M, rue, e cacuae a ranon arx for C,;. h cacuae dfferen vaue for he auocorreaon probaby of C,; ha ho change fro one gn o he oher. If he prce proce a rando and yerc, h oud be a quarer n a regon.

14 abe : C,; ranon Vecor of he Suaed Reu of he Mbu, rue R+, % 45.5% 49.6% 54.4% R+, 7.68% 8.89% 9.49% 7.37% R3,+ 7.58% 8.99% 9.6% 7.47% R4, 4.3% 36.87% 3.3% 3.% * R: C- > and C > R: C- > and C < R3: C- < and C > R4: C- < and C < he probabe of regon and 3 houd be yerc by he conrucon of he M rue. When C,; n Regon, he Mbu rue key o be profabe he hen n Regon 4, he Mbear rue key o be profabe. When n Regon 3, he Mbu rue ake qudae a poon n an ae and he Mbear rue qudae ake a poon n an ae. Wh a pove rend and auocorreaon n prce, e ee ha he ae ore key o be n Regon han n Regon 4, hch expan hy he Mbu rue ake he o prof. he probaby of Regon ncreae a prce auocorreaon ncreae, hch ao correpond o ha e have aed prevouy. he arger he agnude of he pove auocorreaon, he onger he pove rend rean, hence boh he M boh bu and bear rue houd be ore profabe. h ao hon n abe. 4 uocorreaon Fro he ode, ore pecfcay fro equaon 7, and he uaed reu e have een ha prce auocorreaon an poran facor n he profaby of he M rue. Hence h econ nvegae auocorreaon of he radng raegy. 4. Genera uocorreaon Srucure Le k be he ae prce auocorreaon n og prce beeen k perod. hen he covarance arx beeen Q and Q-,, can be expreed a 3

15 4 and k k nce e aue aonary n econd oen. Noe ha a covarance arx of Q, hch a vecor of q vaue =. a covarance arx beeen Q and Q-. nd he covarance arx beeen C and C-,, can be expreed a Propoon 3: he varance for C,; and he covarance beeen C,; and C,;- can be expreed a 9, ab ab b a here a and b Proof of Propoon 3 n ppendx 6. When he M rue praccay apped, anager ofen ake he horer e perod a =. When =, referred o a he M croover rue. h cae can be reaed a a peca cae nce ake he equaon gnfcany per by geng rd of he fr of boh doube uaon er.

16 5 Reark : he -h order auocorreaon beeen C = Q and C- = Q-, hen =,,,, can be expreed a,,,, ab b a ab b a he proof of Reark oed. Cacuaon check of Reark perfored h uaed reu and ncuded n ppendx 7. Choong a parcuar M, rue depend upon he auocorreaon propere of prce. When fxed, a conan. herefore for >, he envy of radng raegy auocorreaon h repec o ab b b a If, e have I ay e be reaonabe o degn opa propere. We do no purue h furher bu nead ook a he nk beeen he uocorreaon Funcon CF of he M rue and oe pe e ere ode. In he nex o econ, he auocorreaon rucure of he raegy nvegaed. Equaon 9 copued = 3 and ncreenng fro.5 o.95 h an nerva of.5.

17 4. Invegaon h an R Prce Proce h econ nvegae he nkage beeen echnca radng rue and he ochac proce. ue ha he de-eaned prce q foo an R proce. q q ~ N, hen he expeced vaue, varance, -perod auocovarance and auocorreaon of h proce e knon. E q, Varq Covq, q, Noe ha and for he varance of q and and for he -perod auocorreaon. he aued error er drbued h a varance of. he covarance arx beeen Q and Q-,, can be expreed a Ung h reu and Reark, e can derve he -h order auocorreaon beeen he crca vaue. Reark 3: he -h order auocorreaon beeen C = Q and C- = Q-, hen = and q foo Equaon,,,, can be expreed a 6

18 7,, ab b a ab b a here a and b he proof of Reark oed. h a drec reu of Reark. Coparng equaon h gve u exac agnude of auocorreaon apfcaon under he R prce proce fraeork. Propoon 4: he dfference beeen -h order M radng raegy = auocorreaon and -he order de-eaned ae prce auocorreaon can be expreed a 3 3,, ab b a ab b he proof of Reark n ppendx 8. Noe ha > 3 houd hod o ake equaon 3 eanngfu, hch a reaonabe aupon o ake. For >, he gn of equaon 3 depend on he er Fgure ho he M, rue auocorreaon rucure under R ae prce auocorreaon and ae prce auocorreaon rucure hen ae prce foo R proce h.5 fr order auocorreaon.

19 Fgure : radng Sraegy, M, Rue, uocorreaon Srucure v. e Prce uocorreaon Srucure hen he e Prce Foo an R Proce h.5 Fr Order uocorreaon Fr 3 Lag radng Sraegy e x ax: ag e perod, y ax: auocorreaon Fgure conen h behavora eraure n ha he CF of he M rue ho na under-reacon and deayed over-reacon or prce revera. We can ee ha he auocorreaon apfcaon of he radng raegy, copare o he underyng ae prce, occur hen prce revera happen. h eengy behavora reu drven hou aung any behavora apec n deernng ae prce proce. h gh ugge ha uch na under reacon and prce revera gh be a produc of he underyng ecury auocorreaon rucure and he radng rue adoped by arke parcpan. h oud be an nereng fuure reearch opc. 4.3 he M, v. Prce uocorreaon he benef of cacuang he auocorreaon funcon of he M, rue,,,, ha e can ee f he poon of he rue ead o an ncreae / decreae n auocorreaon n 8

20 he oney aoun aocaed h radng. o do h e need an overa eaure of auocorreaon and a naura one he ape ac anaogue o Lung-Box eaure. We ue u of quared auocorreaon and ca LB. Hence popuaon LB copued ung he popuaon auocorreaon, and ape LB ac copued ung he ape auocorreaon,. For a pe ode h can be cacuaed anaycay h popuaon vaue. For an R h correaon coeffcen ρ, P = β + ρp - + ν here P prce of an ae a e and ν he redua er fro he eaed regreon equaon, popuaon LB / ;. h econ preen reave LB baed on prce auocorreaon, hch aued o foo an R proce. By nvegang he ac of he Mbu, auocorreaon rucure e can gan a good dea of ho he auocorreaon rucure of he Mbu, rue apfe ha of he prce ef. MOP nvegae he correaon rucure of he oenu raegy o fnd e ere oenu raege are povey correaed hn an ae ca, bu e o han pave ong raege. fr gance, ook ke her reu conradcng our reu. Hoever, coe nvegaon of her radng raegy revea ha conen h our reu. her raegy aed a foo. We conder heher he exce reurn over he pa k onh pove or negave and go ong he conrac f pove and hor f negave, hodng he poon for h onh. h buyng he pa nner raegy. he Mbu rue nvegaed n our paper, a prevouy dcued, doe exacy he oppoe. hough e have no expcy hon h, exenon of our reu houd predc ha he pac of he radng raegy ued n MOP oud decreae he aoun of auocorreaon, and h ha hey fnd. Hence her reu can be een a a pobe eprca confraon of our heoreca reu. prevouy aed, ahough e appy h ode o he prce proce ony, he appcaon of he ode doe no need o be rerced o he prce proce ony. I can ao be apped o reurn or earnng or o any proce ha oud have e ere oenu. ypca reurn auocorreaon e beeen -. and. he ypca prce auocorreaon, f he proce aonary, hgher han.6. h ovae abe 3 and 4. abe 3 repor reave popuaon LB eaure for he fr order auocorreaon, ρ, fro -. o. and abe 4 repor for ρ arger han.6 n he ncreen of.5. ˆ 9

21 abe 3: Reave* Popuaon LB Meaure hen he e Prce Proce Foo R Proce h he paraeer ρ fro -. o., ρ ncreenng by.5 cuuaed up o Lag uocorreaon M,5 M, M, M,5 M, * Vaue are norazed reave o he underyng proce abe 4: Reave* Popuaon LB Meaure hen he e Prce Proce Foo R Proce h he paraeer ρ fro.6 o.95, ρ ncreenng by.5 cuuaed up o Lag uocorreaon M,5 M, M, M,5 M, * Vaue are norazed reave o he underyng proce ncreae, for =, he auocorreaon of he M rue apfe, ee abe 3 and 4. h reu hghgh he fac ha he M rue race he prce auocorreaon rucure and apfe, ahough he rue doe no expcy denfy he rucure. Lookng a abe 3 and abe 4, e ee ha for paube vaue, he raegy ha ed o ncreaed LB for hor rue 5 or and o auocorreaon reurn h ha ncreaed auocorreaon for ong rue 5 or and hgher auocorreaon prce. Wha hee reu ndcae ha hen e ere prce, earnng, reurn auocorreaon o an M rue h a a houd be ued h an M rue h a arge houd be ued h a hghy

22 auocorreaed proce. e dcued earer, he appcaon of our curren ode no neceary rerced o he prce proce. Baed on our reu n h econ, hen he M rue epoyed h a prce M, here prce auocorreaon very hgh, an M rue h a arge oud be approprae. nd hen he M rue apped o a reurn proce, hch ha oer auocorreaon han he prce proce n genera, a aer oud be uabe. 5 Eprca Evdence 5. Derended Log Prce he prevou econ ha focued on anayng ho he M rue apfe prce auocorreaon rucure n a uaed eng here e kno he acua prce auocorreaon rucure. h aoed u o ee uaon here auocorreaon apfed. I oud ao be eanngfu o appy he M rue o rea horca daa. In h econ, e nvegae he M rue and apfcaon of he auocorreaon h 5-year day daa of aor nernaona ock ndce fro 998 o. abe 5 he nernaona equy ndce and abe 6 preen her decrpve ac. abe 5: L of Inernaona Equy Indce Counry US Canada UK France Gerany Iay Neherand Japan Korea Chna uraa Index S&P5 Index S&P/SX Index FSE Index CC 4 Index DX 3 Index FSE MIB Index EX Index NIKKEI 5 Index KOSPI Index Hang Seng Index S&P/SX Index abe 6: Decrpve Sac of Norazed reave o he fr oberved prce of 998 Inernaona Indce fro 998 o

23 Mean Varance Skene Kuro US Canada UK France Gerany Iay Neheran Japan Korea Chna uraa * Sape dae fro January, 998 o Deceber 3, Raher han reporng he rade reu over he enre ape perod ogeher, e repor each year radng eparaey o ha coud be eaer o nerpre. Snce e have approxaey 5 radng day each year, ee reaonabe o ue a day ook back perod n eang og prce rend µ and ong er ovng average SM. o day conen h our heoreca reu n abe 3 and 4 ha a onger perod M rue uabe o a e ere h hgh fr order auocorreaon. day ag choen o copue a ape LB a dcrezed n Secon 4.3. ag engh of he axu nuber of ag ha provde eanngfu auocorreaon. 4-day agged auocorreaon n 5 radng obervaon can be cacuaed bu oud key o be noy and very coe o. Fgure depc he near rend ne of S&P5 og prce fro January, 9 o May 3, a an exape. abe 5 repor he derended og prce fr order auocorreaon for nernaona ndce ued n h econ. Fgure : Lnear rend Lne of S&P5 Log Prce

24 * Sape dae fro January, 9 o May 3, In fgure, e can ee ha ahough he ndex eve ha oved up gnfcany fro 93.8 o beeen January, 9 and May 3,, he derended og prce ha acuay decreaed. If a anager ony ake advanage of upard arke oveen n genera o generae prof, he prof no due o he anager aby o acvey anage he porfoo. herefore h he reaon hy e ue derended prce proce nead. Of coure, he nuber of day can vary for dfferen paraeer. We unfory appy day o a he paraeer o keep he eprca nvegaon pe. We expec he an reu oud no change gnfcany n oher cae. ny acve or decon ade a e be condoned on he nforaon up o e - and hence e aue ha a anager ony ha nforaon up o e - a e. h ean µ be copued baed on he og prce fro - o -. hen µ ubraced fro he og prce a o eae q. he C- copued baed on q- and a M, a e -, hen he hodng poon a e decded ung C-. he profaby of he ovng average radng rue oud be anoher nereng opc for furher udy. h paper abou he auocorreaon rucure of he radng poon raher han he profaby of he radng rue. Devang fro he an focu and dcung 3

25 he eprca profaby coud keep he dcuon aay fro he an hee of he paper. We ry o avod h by focung ony on he eprca auocorreaon rucure of S&P5 derended og prce and he crca vaue. Hence n h eprca econ, e focu on he aggregae eve of he auocorreaon n he derended og prce proce, q, and he crca vaue, C. 5. Eprca uocorreaon Funcon Snce e nvegaed he ndex eve rend of nernaona ndce, an nvegaon of he auocorreaon rucure of he aor nernaona ndce houd foo. Fgure 6 preen he horca auocorreaon rucure. o be conen h he re of he eprca econ, e ue a ag engh of n Fgure 6. Fgure 3: Horca Day Maor Inernaona Indce Derended Log Prce Proce uocorreaon Srucure S&P5 FSE CC 4 DX 3 NIKKEI 5 KOSPI EX S&P/SX FSEMIB S&P/SX Hang Seng x ax: ag e perod, y ax: auocorreaon * Sape dae fro January, 998 o Deceber 3, Ju ke he uaed reu, he eprca reu n Fgure 3 ao conen h behavora eraure n ha hey ho na under-reacon and deayed over-reacon or prce revera. Sape LB ac are copued; hey are runcaed a he h auocorreaon and are preened n abe 8. 4

26 abe 8: Reave ape LB Sac for Indce Derended Log Prce and C S&P5 S&P/SX FSE CC 4 DX 3 FSE MIB EX NIKKEI 5 KOSPI Hang Seng S&P/SX N * Sape dae fro January, 998 o Deceber 3, abe 9 repor ean and he varance of he reave ape LB vaue and -ac abe 9: Mean, Varance and -ac of Reave Sape LB Sac Index S&P5 S&P/SX Mean Varance ac Index FSE CC 4 DX 3 FSE MIB EX Mean Varance ac Index NIKKEI 5 KOSPI Hang Seng S&P/SX Mean Varance ac We ee ha he expeced vaue of he reave ape LB ac for a he ndce are arger han. Our a o ee heher he ean of he reave ape LB acay arger han or no. H : ELB = and H : ELB > Baed on abe 8, for a ndce bu one, KOSPI, e reec he nu hypohe of he noe a 5% gnfcance eve. he degree of freedo are 4 and hence he crca vaue.45. For a ndce bu KOSPI, e canno reec he nu hypohe a 5% eve. % eve, e reec he nu for a ndce. Fro he above eprca nvegaon, e can concude ha 5

27 he apfcaon of he overa correaon n og prce occur hen he M rue apped. We beeve h can expan hy he oenu radng raegy vad and dey ued n aured arke here here ha been no gnfcan upard rend n arke. 6 Concuon h paper fr preen a echnca defnon of oenu n order o nvegae ho echnca anay, epecay he M rue, becoe profabe and heorecay deonrae ha he radng rue profabe. he heoreca exanaon reveaed ha, n addon, prce auocorreaon gh be anoher fundaena reaon for profaby, and he furher nvegaon of he auocorreaon rucure of he M rue hoed ha he M rue refec he prce auocorreaon rucure, hou denfyng. h anay hed oe gh on hy anay ue echnca radng rue. here are hree an fndng n h paper:. he M rue denfe he preence of prce rend and ake advanage of.. he M rue a pe and raghforard ay of expong prce auocorreaon hou acuay knong rucure. 3. Eprcay e oberve ha n he cae of M, rue, here a e perod for he hor poon M, he apfcaon of he auocorreaon rucure ore pronen for a a hen he eve of auocorreaon LB o and for a arge, hen he eve of auocorreaon LB hgh he fr fndng aready e knon and dcued n he earer eraure. We confr by conrucng a pe ode h a nge ae. he econd fndng ha no been nvegaed n he prevou eraure due o he ack of a prece echnca defnon of oenu. We ho h h uaed reu and eprca daa fro 6 nernaona ndce fro deveopng and aure arke. he hrd fndng ugge ha engh of houd be apped n epoyng he M rue. In parcuar our reu ndcae he dfferen reaen for prce and reurn baed on he dfference n he eve of auocorreaon. hee fndng houd be condered poran n heeve nce hey denfy hy he M rue ued. Hoever, he pracca appcaon of h ao nereng. One of any naura queon ha can be aked ho e oud he M rue oud perfor f he fu prce auocorreaon rucure a knon. fer a, e a ha he M rue a 6

28 abou pfcaon of condonng on he pa prce nforaon. herefore, f e ue he enre pa prce nforaon, h houd ouperfor he M rue. Hoever, n pracce he radeoff beeen he neffcency and he co of knong he fu auocorreaon oud ufy he ue of he M rue. h eakne of auocorreaon rucure be exacerbaed by eaon error. Neverhee he exen of he effcency gan oud be a orhy area for fuure reearch. 7

29 Reference car, E., Sache, S., 997. heoreca anay of radng rue: an appcaon o he ovng average cae h Markovan reurn. pped Maheaca Fnance 4, 65-8 che, S.,. echnca nay fro o Z. nd Edon, McGra-H Copane hn, D. H., Conrad, J., Dar, R. F., 3. Rk aduen and radng raege. he Reve of Fnanca Sude 6, en, F., Karaanen R., 999. Earnng and prce oenu. Journa of Fnanca Econoc 5, Bagrocz, P., Sxae, O.,. Sze, vaue, and oenu n nernaona ock reurn. Journa of Fnanca Econoc 6, Barber, N., Shefer,., Vhny,R., 998. ode of nveor enen. Journa of Fnanca Econoc 49, Berk, J., Green, R.C., Nak, V., 999.Opa nveen, groh opon and ecury reurn. Journa of Fnance 54, Bue, L., Eaey, D., OHara, M. B., 994. Marke Sac and echnca nay: he Roe of Voue. Journa of Fnance 49, Brock, W., Lakonhok, J., LeBaron, B., 99. Spe echnca radng Rue and he Sochac Propere of Sock Reurn. he Journa of Fnance 47, Bron, S. J., Goezann W. N., Kuar., 998. he Do heory: Wa Peer Haon rack Record Recondered. Journa of Fnance 53, Chan, K., Haeed,., ong, W.,. Profaby of Moenu Sraege n he Inernaona Equy Marke. Journa of Fnanca and Quanave nay 35, Chan, K., Jegadeeh, N., Lakonhok, J., 996. Moenu Sraege. Journa of Fnance, 5, Charea, C., He, X. Z., Hoe, C., 6. dynac anay of ovng average rue. Journa of Econoc Dynac and Conro, 39-, Chora,., Shvakuar L., 6. Ung genec agorh o gnd echnca radng rue. Journa of Fnanca Econoc 8, Dane, K., Hrhefer, D., Subrahanya,., 998. heory of over-confdence, efarbuon, and ecury arke under- and over-reacon. Journa of Fnance 53, Faa, E.F., French, K.R.,. echnca radng reved: Fae dcovere, perence e, and ranacon co. Journa of Fnanca Econoc 5,

30 Fong, W. M., Yong, H. M., 5. Chang rend: recurve ovng average radng rue and nerne ock. Journa of Eprca Fnance, Grnba, M., Mokoz,. J., 4. Predcng ock prce oveen fro pa reurn: he roe of conency and ax-o eng. Journa of Fnanca Econoc 7, Hong, K., Sache, S.,. Defnng nge ae prce oenu n er of a ochac proce. heoreca Econoc Leer 3, Hong, H., Sen, J., 999. unfed heory of underreacon, oenu radng and overreacon n ae arke. Journa of Fnance 54, Hou, K., Karoy, G., Kho, B. C.,. Wha Facor Drve Goba Sock Reurn? he Reve of Fnanca Sude, 4 8, Jegadeeh N., an S.,. Profaby of oenu raege: an evauaon of aernave expanaon. Journa of Fnance 56, Leeen, J., Nage, S., Shanken, J.,. kepca appraa of ae prcng e. Journa of Fnanca Econoc 96, Lu, L., Zhang, L., 8. Moenu prof, facor prcng, and acro-econoc rk. Reve of Fnanca Sude, Lo,. W., Maayky H., Wang J.,. Foundaon of echnca nay: Copuaona gorh, Saca Inference, and Eprca Ipeenaon. Journa of Fnance 55, Marx, R.,. I oenu reay oenu?. Journa of Fnanca Econoc 3, Menkhoff, L.,. he ue of echnca anay by fund anager: Inernaona evdence. Journa of Bankng and Fnance, 34, Menkhoff, L., Sarno, L., Scheng, M., Schrpf,.,. Currency oenu raege. Journa of Fnanca Econoc 6, Mokoz., Oo, Y. H., Pederen L.H.,. e Sere Moenu. Journa of Fnanca Econoc 4, 8 5. Ng, S., Perron P.,. Lag engh eecon and he conrucon of un roo e h good ze and poer, Econoerca, 69, Rez, S., 6. On he predcve conen of echnca anay. he Norh ercan Journa of Econoc and Fnance, 7, -37. Rouenhor, K. G., 998. Inernaona Moenu Sraege, Journa of Fnance, Vo. 53, No., pp

31 Sadka, R., 6. Moenu and po-earnng-announceen drf anoae: he roe of qudy rk. Journa of Fnanca Econoc 8, Sag, J. S., Seahoe, M. S., 7. Fr-pecfc arbue and he cro-econ of oenu. Journa of Fnanca Econoc 84, Schager, J. D., 99. he Ne Marke Wzard: Converaon Wh erca op rader. John Wey and Son, pg. 4 Johnon,.C.,. Raona oenu effec. Journa of Fnance 57, Zhu, Y., Zhou G., 9. echnca anay: n ae aocaon perpecve on he ue of ovng average. Journa of Fnanca Econoc 93,

32 ppendx Goary arke de rend: a ong run arke-de rend ha can be due o nfaon or oher econoc condon. In h paper, noed a μ. de-eaned prce proce, q: prce proce hou e dependen arke de rend, copued a q og P, here P he ae prce proce. M,,, SM, LM: M ovng average; he e perod ha he M copued for ong poon; he e perod ha M copued for hor poon, hence > and hey overap. SM he M copued over he perod of, LM he M copued over he perod of. M, = Mbu, : M bu rue conruced h and M rend: rend, n h paper, refer o upard or donard oveen n de-eaned prce proce for ore han conecuve day. v, ; : varabe v a e ha dependen on ovng average e perod and. r, ;, : reurn copued over e and ha dependen on ovng average e perod and.,,, beeen Q and Q-; : he covarance arx of Q; he covarance arx arx beeen C,; and C,;- he covarance arx of C,;; he covarance and : he -perod auocorreaon of ae prce and he auocorreaon of radng raegy crca vaue. 3

33 3 popuaon LB and ape LB: LB he u of quared auocorreaon fro order o pre-pecfed order. 3 h n Secon 4 and h n Secon 5, popuaon LB copued h popuaon auocorreaon and defned a and ape LB copued h popuaon auocorreaon and defned a ˆ. Dervaon of Defnon exp exp exp exp exp Q q q q q q q C 3 Dervaon of Lea P P og og P P og og og og P P og og P P P P exp

34 P exp P herefore, P exp P 4 Proof of Reark X Snce EX Y EX Y EY and X Y X Y condona probaby drbuon of C can be ren a, he uncondona and he pdf prob C prob Q. 5 Q Q ~ N Q, q SM prob Q here : Q q SM Snce f C Q f C Q hence E probc Uncondonay, prob. 5 Condonay, prob q SM 33

35 5 Suaon Procedure. We have q ogp c, aue q q Fro GS daa May 4, May 3,, eae c, and c 4.8,.687,.8 and e Le % radng co = %, annua rk free rae: % day rk free rae:.8%. Suae N, ~ 3. q q, og P q c, C q / SM f C Q SM exp/ q, f C Q R og P og P 4. % Reurn fro he M rue: r C R R f 34

36 35 6 Proof of Propoon 3 Decopoe no four pece here,, and, ao decopoe here and Le a and b

37 36, a a and b b a b ab ab ab b a

38 8 Cacuaon Check of Reark In h econ, he prevou reu checked h a uaon reu aung one of he o popuar raege, = and = and ha he auocorreaon funcon,, foo he R proce. e of 5 obervaon of q are uaed here q q, ~ N, and. 95. LM and SM are copued fro he uaed q, hen ubraced o ge Q. he auocorreaon of Q copared h 4 h = and = and he percenage dfference are repored. abe : Suaon v. heoreca Reu: uocorreaon of Q hen = and = Suaed R heoreca R % Dfference O[] % O[] % O[3] % O[4] % O[5] % O[6] % O[7] % O[8] % O[9] % O[] % O[] % O[] % O[3].5.9.9% O[4] % O[5] % O[6] % O[7] % O[8] % O[9] % O[] % O[] % O[] % O[3] % O[4] % O[5].6.6.7% O[6] % O[7] % O[8] % O[9] % O[3] % abe 3 ho ha our copuaon correc 37