The Influence of Investor Sentiment on the Formation of Golden-cross and Dead-cross

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1 From: AAAI Techical Report WS Compilatio copyright 2002, AAAI ( All rights reserved. The Ifluece of Ivestor Setimet o the Formatio of Golde-cross ad Dead-cross Kotaro Miwa Kazuhiro Ueda Depar~et of Systems Sciece, Uiversity of Tokyo 3-8-I Komaba, Meguro-ku, Tokyo, , Japa sti-miwa@mxq.mesh.e.jp ueda@gregorio.c.u-tokyo.ac.jp Abstract The so-ealled "golde-cross" ad "dead-cross" are said to be useful sigals to forecast market treds. I this paper, we focus o the Japaese stock market where gold-crosses ad dead-crosses are empirically cosidered as useful ivestmet sigals. First, we examied the usefuless of these sigals by usig historical Japaese stock price data. The results cofirmed that these crosses were useful as cofirmatory sigals for forecastig market treds. The results also showed that the miimum legth of period (days) of useful movig averages is shorter i the case of golde-cross tha i that of dead-cross. Secod, we tried to idetify the uderlyig reasos for the usefuless of the crosses. Because a model, which assumed all ivestors were ratioal fiacial experts, failed to explai the usefuless of the crosses, we were able to assume that the crosses reflected ivestors irratioality or behavioral bias: coservativeess ad represetativeess about treds (Tversky ad Kahema 1974). We the developed a model that icorporated this bias. Based o simulatios usig this model, we idetified the mechaism with which those crosses closely relate to ivestors irratioality or behavioral bias. The aalysis also revealed ivestors tedecy that they were coviced by a bull tred more easily ad quickly tha by a bear tred. This fidig is i lie with what is geerally observed as ivestors bull-bias i the Japaese stock market. 1. Itroductio 1.1 Defiitio of golde-cross ad dead-cross Ever sice Joseph Graville (1960) itroduced, the movig average lie has bee used as a tool for judgig coversio of a tred. The followig two patters, usig the movig average lies, are cosidered as "buy" or "sell" sigals: Whe a shorter (e.g., 5G-day) movig average lie crosses over a loger oe (e.g., 200-day), from below, while the both lies are risig, it is a major buy sigal called golde-cross, idicatig that the market is i a bull tred; The reverse is called as dead-cross, a sell sigal. Although Fama (1970) tried to dey the usefuless these crosses by usig a radom walk model, the golde-cross ad dead-cross are still supported by may experieced ivestors. 1.2 Purpose of the study The so-called golde-cross ad dead-cross are said to be useful sigals of market tred forecast i two aspects. Oe is that these crosses ca sigal (or spot) a tred s chage. Ad the other is that they ca cofirm the formatio of a ew sustaiable tred. There have bee researches that tried to show that the crosses were able to catch tred chages i several markets, such as US stock, US treasury bods, Japaese Ye forex, etc. However, there has bee o research o the Japaese stock market, particularly about the usefuless as a cofirmatory sigal of a ew sustaiable tred. First, we idetified, by applyig historical price data, the usefuless of those cross i the Japaese stock market with regard to the above metioed two aspects. Secod, we tried to explai, by ivestors behavioral biases, the reaso for the usefuless of the crosses, a approach similar to several previous studies explaiig abormal pheomea i fiacial market by the behavioral biases (Adrei S. 2000). I this paper, we first verify whether a radom walk model, which assumed all ivestor beig ratioal fiacial experts, ca explai the usefuless of the crosses. The we develop a model icorporatig ivestor s behavioral biases (hereafter we will call this model as a "ivestor setimet model"). Similarly, we verify whether a ivestor setimet model ca explai the usefuless of the crosses. Ira radom walk model fail to explai the usefuless of the crosses but ivestor setimet model ca explai it, we will show that the ivestors behavioral bias is the reaso that made the crosses effective as a buy or sell sigal. 1.3 Previous studies about Golde-cross ad Dead-cross Several people tried to aalyze the usefuless of goldead dead-crosses. Amog them, Stepha Tayler (1990) tried to aalyze it i the currecy futures market, by quatifyig the profitability of a ivestmet strategy, i.e., buyig a fixed amout whe a golde-cross appeared ad sellig off the same fixed amout whe a dead-cross appeared. Nauzer Balsara et al (1996) also applied similar ivestmet strategy to such markets as Comex gold, treasury bods, soybeas ad Japaese Ye forex. They examied the usefuless of the crosses ad of the optimal sets of dual movig lies. I Nauzer s paper, the short-term movig averages rage from 3 to 15 days ad the log-term oe rage from 9 to 45 days. 54

2 Both papers cocluded that, although the dual movig lie system was a effective techical tradig rule to some extet, there was foud o uiversally effective set of lies that did t deped o specific periods or markets. I this paper, we will examie the usefuless of these sigals ot oly as a idicator of a tred chage but also as a idicator of the cotiuity of a ew tred that has started. I additio, we focus o behavioral biases of ivestors. We idetify the uderlyig reasos for the usefuless of the golde-cross ad dead-cross sigals by usig a model that icorporates behavioral biases. 1.4 Costructio of this paper First, i Sectio 2, we quatify the usefuless of these sigals by calculatig various crosses performace by usig Japaese historical data. I Sectio 3, we verify whether or ot the pheomeo foud i Sectio 2 ca be simulated by the radom walk model. I Sectio 4, we itroduce a model that icorporates a behavioral bias. Fially, i Sectio 5, we try to explai the uderlyig reasos for the usefuless of the sigals. 2. Aalysis Usig Historical Data The purpose of the aalysis is to verify whether goldead dead-cross are useful as a buy or sell sigal ad what time-spas should be applied. 2.1 Data We used the historical daily closig prices of the followig stocks ad idices from August 27, 1991 to December 27, 2001 o the Tokyo Stock Exchage. Taisei, Shimizu, Kajima, Nisshi flour, Meiji Seika, Sow Brad, Kaebo, Teiji, Toray, Asahi Chem., Oji Paper, Mitsubishi Chem., Takeda Pharm., Daiichi Pharm., Nippo Steel, Kawasaki Steel, NKK, Hitachi, Toshiba, Soy, Rohm, Mitsubishi Heavy, Toyota, Nitedo, Mitsui & Co., Sumitomo Bak, Asahi Bak, Yokohama Bak, Daiwa See., Nikko Sec., Nomura See., Tokio Marie, Mitsui Marie, Nissay-Dowa, Tobu Railways, Seibu Railways, Sagami Railways, NTT, Tokyo Elec., Kasai Elec., Tokyo Gas, TOPIX, Nikkei 225 Average. 2.2 Procedure of the Aalysis The followig defiitios are used for this aalysis. (1) Moitor the crossig betwee a -day short-term movig average lie ad a 2-day log-term movig average lie. (2) Whe a short-term movig average lie crosses over log-term oe, from below, while the both lies are risig, this is defied as a golde-cross. (3) Whe a short-term movig average lie crosses over t Experieced ivestors ofte use approximately oe-to-two ratio i pickig up a short-term ad log-term movig averages, e.g., a pair of 13 ad 25 weeks lies or a pair of 90 ad 200 days oes. Accordigly, we decided to use a pair of -days ad 2-days lies i the aalysis. log-term oe, from above, while both lies are fallig, this is defied as a dead-cross. (4) While both short-term ad log-term movig average lies are risig but do ot cross, it is defied as a "quasi-golde-cross". (5) While both short-term ad log~term movig average lies are fallig but do ot cross, it is defied as a "quasi-dead-cross". (6) We measure the usefuless of a sigal for a tred chage, by quatifyig the profitability of a ivestmet strategy, i.e., buyig a frxed amout at the time of a golde-cross appears ad sellig offthe same fixed amout at the time of a dead-cross appears. These are compared with the average profitability durig the ivestigated spa. (7) To measure the usefuless of the crosses as sigals for forecastig a ew tred s cotiuity that has just bee formed, the retur o ivestmet is calculated as (2.1). rp=( ave= -1]/m (2.1) k Po )/ ave,. :the m-day average price after the evet (golde-cross, dead-cross, quasi-golde-cross or quasi-dead-cross) Po :the spot price o the day of the evet, This value (rp) is called as a "average price chage per day" (8) We set a sigificat level to be 5% ad do two side t-tests ad check whether there is a sigificat differece betwee the performace of a golde-cross ad quasi-golde-cross or betwee that of a dead-cross ad quasi-dead-cross. 2.3 Result The differece i performace betwee golde-cross ad quasi-golde-cross as sigals for cotiuity of a ew tred is show i Fig.l. I Fig.1 (also Fig.2, Fig.3, Fig.4, Fig.5, Fig.6 ad Fig.7), the horizotal axis represets,i.e. the period of each short-term movig average lie, ad the vertical axis represets the correspodig average price chage per day. We chaged from 10 to 100 ad fixed m to 90. zthe differece betwee dead-cross ad quasi dead-cross is show i Fig.2. As show i Fig.l, a golde-cross is ot sigificatly useful for forecastig a ew tred s cotiuity whe relatively shorter movig average lies are applied, while it becomes sigificatly useful whe loger oes (>43 days) are applied. Likewise, as is show i Fig.2, a dead-cross is ot sigificatly useful for forecastig a ew tred s cotiuity whe relatively shorter movig average lies are applied, while it becomes sigificatly useful whe loger oes (>66 days) are applied. Moreover, the miimum period of movig average whe a golde-cross becomes useful is shorter tha the case of 2 We oly show the result where m was fixed 90 due to space limitatio. However, the similar results were also observed whe we fixed m to other values such as 30, 60 ad

3 dead-cross. The usefuless as a tred chage "spotter", whe chagig from 10 to 100 is show i Fig.3. I Fig.3, the horizotal axis represets ad the vertical axis represets profitability of the ivestmet strategy, described i Sectio 2.2, par day. Fig.3 shows that golde-crosses ad dead-crosses are useful as a idicatio of a tred chage to some extet. However, there seems o uiversal rule about the set of the lies for showig sigals of a tred chage. We ca coclude that those crosses are effective as sigals for idetifyig the cotiuity of a ew tred whe relatively loger-term movig average lies are applied (100O15 ~ ~ ~1 -o.00ol ~ ---- Without a s~dficat differece Wida a si~cat difl erese.~ A_ (Fig. 1) Result for golde-cross Without a sigificat.~.. differece (Fig.2) Result for dead-cross a $i~cat differece ~ k *~e a~ AI v - "~/ k/: I... o (Fig.3) Usefuless as a sigal for tred chage....i 3. Radom Walk Model We tried to verify if a radom walk model, which assumed all ivestor beig ratioal uder a weak form of a efficiet market hypothesis (EMH), explai the usefuless of the crosses as sigals that showed cotiuity of a ew tred. If a radom walk model ca ot explai the usefuless, we may coclude that the EMH caot explai the usefuless of the crosses. 3.1 The Model Used. We assume a world where ivestors are ratioal ad the price are ot effected by the past price data. Here, we assumed that a market satisfied the followig coditios. (These Coditios (1)-(4) will also apply to ivestor setimet model i Sectio 4) (1) I the market, oly oe asset X is traded at regular itervals. (I this paper, the umber of trasactio is fixed as 30,000) (2) Whe we deote X s price at time t (hereiafter "t" "today") as Pt, the price of the immediate future (hereiafter "t+l" or "Tomorrow") Pt+l satisfies (3.1). p,+~ = p, + y,(y, = +A,A : cost) (3.1) I short, the price moves oly at a costat otch. (3) There is o ews which affects the price of X. Every kows the theoretical price of X (which is deoted aspx)- (4) Ivestors decide the Tomorrow s price of X (or probability that the price will rise Tomorrow) based o the above metioed theoretical price ad o the series of prices i the past. We assume that ivestors predict future prices based o a radom walk model. I short, ivestors pay o attetio to previous prices. However, obviously o tedecy would appear i a pure radom walk model i which ivestors eve pay o attetio to the theoretical price. Because ivestors expect the probabilities that a price would go up or dow to be equal, the expectatio of 90-day average price after ay evet should be zero. Eve if prices go up or dow, ivestors predict a price by referrig to the theoretical price based o the assumptio that they would ot deviate sigificatly. Therefore, the higher the curret spot price is, the lower the ivestors estimated probability that the future price will go up may become. Also, the lower the curret spot price is, the lower the ivestors estimated probability that the futtwe price will go dow may become. We will determie the tomorrow s price by (3.2). f(p,) = exp(-w-(p, - 2) px) /(P ) Co, -p. >- 0) Pr{p, İ -p, > 0} = 2 1 f(p ) (p,-p. <0) 2 p, :the price at the time t (3.2) w: parameter (the degree that the ivestors thik the theoretical price is importat.) 3.2 Result. We aalyzed the usefuless of golde- ad dead-cross as sigals by usig the price data that were obtaied through the simulatio of the radom walk model. We calculated average price chage per day i order for measurig the 56

4 performace of the crosses as the tred cofirmig sigals. No sigificat differece was foud betwee a golde-cross ad a quasi-golde-cross or betwee a dead-cross ad a quasi-dead-cross. This result idicates that the radom walk model caot explai the reaso why a golde-cross or a dead-cross is a useful as the tred cofu-mig sigals, whatever value was set to w. The result is show i Fig.4 ad Fig.5 (w was fixed to be 0.01). (Fig.4) Radom walk model simulatio for Golde-cross 00~15 OOG01 -~ 0OOOO5 ~ o --ooo015 Wlthout a sigllicat dew~rm, rt r~ (Fig.5) Radom walk model simulatio for Dead-cross 4. Ivestor Setimet Model The results of Sectio 3 idicates that we caot explai why a golde-cross or a dead-cross ca be a useful sigal as show i Sectio 2 if all ivestors are ratioal i a weak form efficiecy of the EMH. I this sectio, we preset a ew model to capture the result of Sectio 2 by icorporatig the ideas of the followig psychological study. This model will be called a ivestor setimet model. Now, we assume that a market also satisfies the coditios (1)-(4) i Sectio 3.1. However, we assume ivestors predict the future prices ot based o radom walk model hut based o upward ad dowward biased estimatio. 4.1 Behavioral Biases As a ature of the huma beig i the case of problem solutio, whe a certai ad uexpected pheomeo happes, people do ot believe it easily or they may ot otice the importace of that evet. However, if a certai pheomeo occurs cotiuously, they start to thik that the pheomeo will cotiue for a while without ay reasoig. The former (ot to believe certai pheomeo easily) is called as ma s "coservativeess" (Edward 1968) ad the latter (after a serious of pheomea, to start to believe the cotiuity of the pheomeo blidly) is called as "represetativeess" (Tversky ad Kahema 1974). Moreover, it is kow that coservativeess suddely (ot gradually) disappears after certai pheomeo occurs cotiuously, while represetativeess appears istead. The coservativeess ad represetativeess affect price tred i a market. Whe the price fluctuatio of a stock comes ito a ew tred without sigificat ews, people do o believe i the chage immediately. A pheomeo cotrary to the previous tred would ot immediately lead to the coversio of setimet. However if a ew pheomeo cotiues, a sudde setimet coversio would occur at a certai poit of time. Adrei S. (2000) focused o these pheomea ad costructed a ivestor setimet model to explai aomaly pheomea such as the retur reversal effects (De Bodt ad Thaler 1985). By takig his model ito accout, we costructed a model to explai why the golde-cross ad dead-cross ca be useful sigals. 4.2 Upward ad Dowward Biased Estimatio Our model, like Adrei s model, focuses o coservativeess ad the represetativeess. It is assumed that ivestors use the previous prices of a stock for predictig the future prices. Here, we assume that ivestors predict the future prices based o dual-state Marcov model: Upward tred or dowward tred. If a ivestor thiks that the market is i a upward tred, he or she thiks that the price ted to rise. If a ivestor thiks that the market is i a dowward tred, he or she thiks that the price will fall. I other words, if a ivestor thiks that the market is i a dowward tred (St=-l, St represets the market tred of time t) ad we igore the ifluece of the theoretical price (which is deoted as p~ = 0 ), Pr{y, >0[S, =-l,p, = 0} =re (4.1) : Estimated probability that the price will rise Pr{y, < 0IS, = -1,p, = 0} = I -x (4.2) : Estimated probability that the price will fall 0 < ~r < ½ (4.3) If a ivestor thik the market is i a upward tred (St=l), Pr{y~ > 0[ S~ = 1,p~ = 0} = 1-zr (4.4) Pr{y, <0IS ~ =l,p, = 0}=tr (4.5) The ( trasitio probability is fixed as Pr{S, = 11S,_, = 1} Pr{S, = 11S,_~ = -1} ] Pr{S,=-IIS,_,=I} Pr{S,=-IIS,_,=-I})=[I~, ~ 1_~3.21 (4.6) Now we set q, to be the probability that a ivestor thiks the state to be i a dowward tred at time t. 57

5 Therefore, the followig probabilities are give. Pr{y, >01p, =O}=(l-q,)(1-~r)+q,~r-g(t) (4.7) : Ivestor s estimated probability that the price will rise Pr{y, <0lp, =O}=l-g(t)=(1-q,)~+q,(l-~r) (4.8) : Ivestor s estimated probability that the price will fall qt.t is give as (4.9) ad (4.10). ((1-21 )q, + 22 (1 - q,))# qt+l = ((1 - I )q, +22(1 - q, )~r+ (2,q, + (122)(1- q,))(l- (if Y,-i > 0) (4.9) ((1 - I )q, + 22(1 - q, ))(1-7 q,+l = ((1-2, )q, + 22 (1 - q, )(1- #) + (~q, s )(1- q, ))Tr (if y,_~ < 0) (4.10) 4.3 Determiatio based o the Biases ad the Theoretical Price Ivestors predict the future prices based o the estimatio explaied i Sectio 4.2 ad the theoretical price explaied i Sectio 3.1. I this model, ivestors predictio follows fudametally a behavioral bias explaied i Sectio 4.2, but the model fulfills the property that a higher curret price leads to a lower probability, estimated by the ivestors, of further price icrease. Therefore we determie the tomorrow s price as (4.11). f(p,).g(t) (p,-p~>o) Pr{y, > 0} =[l-f(p,).(1-g(t)) (P,-Px <0) =,f exp(-w.~,,-p~) ).(q,.+(l-q,)o-.)) fp, -px >o) [l - exp(-w- (p, -~ p~)2 ). (q, (1 - :) - (1 Co, - p~ < O) (4.11) 4.4 Optimizatio This model ivolves may parameters (gr, w, ;11 &). We optimized the parameters 3 so that the simulatio results closely replicate that of the actual data (Fig. 1, Fig 2) i terms of the period (days) of movig average lies whe the crosses became useful for forecastig a ew tred s cotiuity. 4.5 Result of the Golde-cross ad Dead-cross Performace We measure the usefuless of golde- ad dead-cross by usig the price data obtaied from the simulatio of ivestor setimet model, i the same maer as it was doe i Sectio 2. By adjustig parameters (to ~ =0.39, w =0.01, 2j =0.001, 22 =0.008), we succeed i reproducig the results as show i Fig.6 ad Fig.74. Fig.6 shows that a golde-cross i a relatively shorter zt rages from 0 to 0.5.adjusted i icremet ofo.ol, w rages from 0 to 1, adjusted i icremet of O.O01.2~ rages from 0 to 0.5, adjusted i icremet of ,,l, 2 rages from 0 to 0.5, adjusted i icremet of I ease of the optimal solutio, the degree of coformity is sigificatly higher i predomiace compared to the other combiatio. movig average lie is ot sigificatly useful but the cross i a relatively loger movig average lie (>41) becomes sigificatly useful (i case of historical data aalysis result, the crosses i a relatively loger movig average lie (>43) become sigificatly useful). Likewise, Fig.7 shows that a dead-cross i a relatively shorter movig average lie is ot sigificatly useful but the cross i a relatively loger movig average lie (>67) becomes sigificatly useful (i case of historical data aalysis result, the crosses i a relatively loger movig average lie (>66) become sigificatly useful). 0~ c~.,~ (1 C015 m~ m " Without a sigificalat Vv~th a sigrdficat ~.. differece... ~=, e"... (Fig.6) Simulatio result for golde-cross O --Qo ~t ~lt~o ut a sigldfl d~ erece c a.rtt With a differece siga~ificam... ul (Fig.7) Simulatio result for dead-cross 4.6 Result of Aalysis o Ivestor Setimet We aalyzed how the ivestor setimet chages immediately before a cross emerged by calculatig the average qt before ad after the cross. Fig.8 is the result whe the spa of the shorter movig average is fixed to 80 days (=80). I Fig.8 the horizotal axis represets days before formatio of golde-cross ad the vertical axis represets qt. Fig.8 shows a sudde ad rapid chage of ivestor setimet from dowward bias to upward bias.! O.g days before 8olde cross erarse (Fig.8) Ivestor Setimet 58

6 5. Discussio 6. Coclusio 5.1 Reasos for the Usefuless of the Crosses From this model, we idetified the mechaism with which those crosses closely relate to ivestors irratioality or behavioral bias. I additio, from Fig.8, we ca say that, i most of the case, golde-cross ad dead-cross are cosidered to be relevat to the chage of ivestors setimet from dowward bias to upward bias or vice-versa. Cosequetly, we coclude that the crosses ca be sigals idicatig cosesus that a ew tred has bee formed. If the period is too short, the crosses may ot become effective because the so-called "cheat" or "oise" are picked up. However, if the period is log eough, the crosses ca forecast a sustaiable tred. Therefore, after a golde-cross formatio, a upward tedecy cotiues for a while uder the ifluece of bull bias. Similarly, after a dead-cross formatio, a dowward tedecy cotiues for a while uder the ifluece of bear bias. However, there seem o obvious rules about a "right" period of the movig averages i which the crosses are particularly effective for idetifyig a tred chage. If you choose a relatively shorter period, the cross may pick up deceivig moves while, if you choose relatively loger period, the cross may pick up true treds too late. 5.2 What We Ca Say from the Parameter? From the model, the peculiarity of the Japaese stock market that has bee empirically kow was also idetified. After optimizatio, each parameter was calculated as 7/" =0.39, w =0.01, 3.~ =0.001, 3 2 = represets the probability that the ivestors thik the market tred from upward to dowward. 3, 2 represets the probability that the ivestors thik the market tred from dowward to upward. Therefore,3,t <3,2 represets that the ivestors are less resposive to dowward movemets tha upward movemets. This coclusio correspods to what is ofte observed i the Japaese stock market as oe of its peculiarities. The Japaese ivestor is said to be "bull-biased", i.e., to believe a bull tred more easily ad quickly tha a bear tred. A fact supportig the existece of this bull-bias is that, i a stock margi trasactio, log positios always exceed short positios (Nikkei News Article 2001). other words, more people prefer to "go log" tha to "go short". The simulatio result (21 < ~-2 ) supports the existece of this bull-bias. By usig the historical Japaese price data, it was cofirmed that golde- ad dead-cross are effective as sigal for cotiuity of a ew tred if appropriate days (relatively loger days) of movig averages are used, poit that was ot thoroughly discussed by the previous works. Moreover, a golde-cross becomes a effective sigal sooer (i shorter days) tha a dead-cross i terms of days of movig averages. From the simulatio, we succeeded to idetify the mechaism with which those crosses closely relate to ivestors irratioality or behavioral bias, which relatioship has ot bee well discussed. The aalysis also revealed ivestors tedecy that they are coviced by a bull tred more easily ad quickly tha by a bear tred. This fidig is i lie with what is empirically observed as ivestors bull-bias i the Japaese stock market. Refereces Adrei, S Iefficiet Markets: A Itroductio to Behavioral Fiace, Oxford Uiversity Press, De Bodt, W. ad Thaler, R Does the Stock Market Overreact?, The joural of Fiace, 40, Edwards, W Coservatism i Huma Iformatio Processig, Formal Represetatio of Huma Judgmet, Bejami Kleimutz (Ed.), New York: Wiley, Fama, E.F., 1970 Efficiet Capital Markets: A review of the Theory ad Empirical Evidece, The Joural of Fiace, 25 Graville, J Strategy of Daily Stock Market Timig, Pretice Hall; Eglewood Cliffs, NJ Balsala N., Carlso, K. ad Rao N. V Usystematic futures profits with techical tradig rules: A case for flexibility,joural of Fiacial ad Strategic Decisios, Vol. 9,1 Nikkei News Article, Nov. 21, Stock Margi Balace Shows Ivestors tur to Bearish, Nippo Keizai Shibu, 3 Taylor, S. J Profitable currecy futures tradig: a compariso of techical ad time-series tradig rules, i The Currecy Hedgig Debate, Lee R. Thomas III (Ed.) IFR Publishig, Lodo. Tversky, A. ad Kahema, D Judgmet uder ucertaily: heuristic ad biases, Sciece, 22,