The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or noncommercial research purposes only. Published by the (UCT) in terms of the non-exclusive license granted to UCT by the author.
11
N. m
1
1 2 3 1.1 1.2 1.3 1.4 2.1 3.1 3.2 nu'.i"k"'. movement. 16 16 PPI'OOC:nes to membersltup set.... 40 43 44 2
3.4.1 45 :3.5 NP<""I;l~l1tv measures 4 4.1 4.1.1 5 4.3 5.1 4.1.2.'1e<~esl)It" measures... states.... states. states. and 5.4.1 3
6 6.1 6.1.1 to,.,.....,,,,.,,,,,...,,"""... ""'u and ",...,,"'"'''' nun..,,,'.,,,,, 6.1.2 to 6.1.3 terms ma.xirnwn and "'...,...,a,u. 7 fj.5 T.l 7.2 1 markets mood.... u...,,,,,.,,,,... data the mood index 7.3.1 v 4
.2.:J indices values used.j
4-1 4-2 55 4-4 4-5 4-11 4-12 4-1-1 5-1 returns.. returns. returns.. returns... returns. returns. returns. returns. returns..... states: states: 55 59 5-3 states: 5--1 states: 6
5-5 states: 5-6 5-i 5-9 states: 6-1 - 1-1 6-5 6-11 V statistic: 6-19 1 2. 1 7
6-:U v 1 6-25 1 6-:!6 V 1 6-:!i 1 1 6-:J 1 1 1 6-45 mood of investors: 1 to 1 investors: 8
to 1. -1-2 -4-6 -11-16 -17 to to to to to to to to -21 9
to -26 mood to Index. mood investors to maximum and minimum: mood investors to UA<""""U~uu and -29 mood to... <""".u«... u and to to to 10
4.1 returns. 4.4 4.5 5.1 5.5 5.6 5.9 5. 5.11 1... measures measures states.. in UU'U.U.I"" state: UU'U.U.JL" state: states: states: state: state.. 61 5.12 5. 5.14 11
5.15 6.1 6.7 "',.,...'0<"'''... results:... 111... 1... 1... 1... 1 6.11 6. 6. 6.14 1
13
15
1 1 very to the economy a IY'I1I1n1r,rv are where Hij,CUj'...,l.:., securities are These among other economic The are into various sectors the economy. the an environment where r,,>rn,""'h'" a to movement ]s concern to ecl:>n()mls a The index is constructed movement. include company pertc)rn1al1ce shares. There are many,» 'TH"-" I't~SpOnSl to describe and n...-"<'''''' and every this has led to a wide range movement is to rl""=",m III investment u.~.v.~,.. on the
the ecc)llomv From the in uu.""..'_... markets is a cu.'"...".."... issue. the and other Iac:tOl~S is human movement of COJrlsl,del:ed to and natural t<""e>~.a.i!!i,<;; are and the 1 is to research one mathematical and "V'''Vti~V''''''''...,..., 19I1lorE~ and UU,Q""''''''' Vi""'''''U.. to vagueness are are not and nature its movement. states. of occurrence movement a market as measure movement index states 17
certain aslpel:::ts movement can be associated or states is not and... V.,l""'''''''.,... U an as to 1 and cornp,!l.r documented. structure movement. dence a or movement due to u."".u..w..'"'''~''',''' or movement due to index. index movement is not easy to used to de<~oi1npc)se
into illvestors exmovement. 1 to,.",nr'~'"'' movement It is to 1 is to: cases e.xpose a ~"~~,._ way. a create an 1. in movement
2. a to investors :1. a senstates: sets. measures to rlp'r.pr'itiitip IS three states is rel)resel movement <le]:>erl<ls to measure state. to in IS a
1 the movement are are memt;lojrle<l. to measure to is are IS ror.1rnr,nt measures to assess a process are to carry out are pre:sellted. is states are not measures are occurrence is ror.1rnr,,,1" is are: are It is states are or 5 not A COlnU()Sn
is the mood to illvestors in riii'+""e<>ylt and n,',f'!o:f'nt mood is a 22
literature review in this this is to a review to movement. share be (Peters 1999). vagueness in movement to the overall also be 1 economies of the world consi<ipl' stock rnarkpts wry ill that they act as econorillc stock market movement information underlying economic A summary of market movement IS measured a To create a stock the following are selection of included in the 1973, Fama to measure may be variallcp in tllt' other. In Sout}, bxchallge (.J S E) has
to measure the movement. the is cash resources into viable sectors the economy while enhancing. The overall index is the index of all the listed COl:np'anles on the Its main purpose is to moni tor the overall market Some kilown New York Stock EJ. the Dow the the etc. movement has been 1 Pratten A wide range have been movement. and dividend. Market movement is an index it to compare the in the market (benchmarking). IS among; stocks (Lorie and Harnilton Dist.rilmticllls of 1:larket illdices arc illlportallt and be considered ill t he next section. an have stated and of and movement is based on the ~u.,~.. ~U" that 1983). economical etc. of investment and (Lode and Hamilton to movement of funds and use the normal in the indices returns. use of the normal distribution 'u'.~n''v'' that knowledge the mean and variance of is to the movement the variance measures the ) of the illdt'x (Fam1i evidence bas that retul'lls Oil most stock the distribution. For Fama (1 the 110t more observations ill the le~'t tail than in the that the were fatter and the around the mean was tban that exlpe(~leu of the Ilormal distribution. A study 24
on the daily 500 and the Dow Jones the 1928 to that the were skewed, with around the mean ror.lry1yhh"qr! distribution (Turner and. The condition of fat around the mean that of the distribution is called" market indices 500 and Laibson (Sterge 1989). may not be ~"'Yn,,,,,,.. i deal with movement. has no with events standard deviations away the mean. The also assumes market movement in shown that market 'H'-U'-'(;<> 1991. 1 Literature of and memory to markets will be given in the later sections. This section will present some are to v11.o.his"" in the stock a stock is factors company actual pertc)rnlallce versus its ex"dec::t and the investors. actual ""oo"~'-'''''n news is vpry ai'(' III movement its movement is <JUI'vv""",,-, '-'H':'HI~O;:;<> in share as: economic vlillh"'"v, to t }I(' sharp OlJ the stock Infol'Inat ion if a futun-' Hi the share,""'0cl,ll to go down. If show an undervalued company, the share the rate at which certain will are U"'v~'Ul'C"U to determine and a company, mergers or overs can make the share to In a merger 25
believe and will illcn-'as(' and this pusll(~s up the share New product chal1ges in the board etc. also the share such as money and,,<::,~.uiiu':'" I.e. sales stock that cannot easily be "","".<,,,, 1 of in the usual manner, a direct on and Kochin Fisher et al. 1 Insider and... u.'~,-,vuu,-,.u estimates as well as corporate events (Biger and 2000) and location trade share subject of share price relationship with economic is wide economy have been -""',,",Hi;;"";' in money supply, (Lorie and Hamilton to when is low than when it is reason is that inflation is uo"v'~.lul"_"" with low level of industrial may induce a decrease in company for returns they accounts mon-~ attractive alld encollrage to sell off their shan"' and deposit their molley in low-risk fixed income are a share of slow \Vhen :nterest rates are rates affect the in turn the business Investors react to money been to pave way for "",:;o."!;"0 economic conditions. rate of the money on 1983). shows that,.",o,"!:;,,,,o the money supply some 26
variatioll ill stock ret urns (Pesando 1 of The a the movement et ), IS various HlC'Tnle,", I-''-'A''''''''", on company news many or investors is sometimes news papers, Past work on of The mood of investors he In as season of year certain months l'ptpl','p,1 to as the have shown market movement such as those I are also to markets (cnn.colll 2UU1). IS HI 7. as volatility or rbk the 'H~'~v'''' to measure movement. stock will he these to
movement. The market movement is associated with many of the stock market as investment decisiolls, changes in money supply, corporate etc. returns may use marly studies. stationary au toregressi ve, 1) model was used to returns on te 12) wa.."l used to 1990). The ICAJJICI_C"'U stock returns and (French et al. 1 on stocks alld It been observed cannot be ear used in and ). 1997). The mean were ret urns rever:se to ones, and returns reduced risk U>-"~~"'~L case of av- erage (EWMA) models. has been used to as.uu'lhllal~,a.l WPl'e used to stock market alisation iwluced some in tlle returns of the Taiwall (Kwan in which the that ""_U'-''-''', the TlPl'fnl'n',Pr1 better 1 28
and ity is not constant. LH'V''UC;lO have a ----'1--~ to more recent r.h,,,,,,-,,,, is the most recent estimate of variance rate. and are Choll most recent these has Kroner (1992). variance to Tn'/vi""", estimates stock returns of excess returns LH~'U.v", such as the widento to HG.U\.Ll'" excess returns also measure and test realistic H.. J... L'" series for railroads and utilities discussed and 1 in series of the been used. news, and to in all series 1 "'.. Ju.'v<u where a process V<Oj.V~'C;U m and (1 LH""U<OlO are used
However some of these LUUU<:;W used to market A wide range of models been role. term structure' and portfolio time-models and markets. and as a result models years Lave seen great illterest stock of the methods models are surderivatives es- in the markets are not known the randomness of the models used to "~,~~r,~t \JuaVlvl\J a process is et al. the study of The of property is measured dimellsioll. dimension of Ii time serie:,; is to reveal its tor'<"" 'm rm the fractal dimension a time series can be computed as used HIJlalJl\';!':tl time series to ~. ~,y,,, the
of stock of fractal were GlJlJll'VU to the time series ~~'~''''U'''I''>'' index time series. The up to 15 the index was a has also U and German range was used to estimate computations H was to be 0.5 < < 1, H also range (RIS) was to examine fractal strllct un' in use 1991. market mit was market have tendencie,., consisteilt random walk (Ambrose et al. 1 chaos to stock are 1991 ). any theory is that it economy and HUUH'" canljot he halldled the llormal distrihution 1994). real So far the methods and reviewed deal with that account vagueness in markets have stock a increase could be movements the in lirlt' with the and due to randomness. dolj(~. reviewed in next section. review to set to markets. a long time was on
set was discovered not the of set the vagueness human as it is et al. and set theory ha..'l been et eds. 1995). last decade has seen been very technical. The motivation to ill the has been the 'v<nh~'u that market movement. movements because the worth and no consequences of ~'''~''I.,." investments, the use 1"''''''1'"'11'' in the as as vagueness in ov"no."r".ri returns (Mere~ and varying of the assets on also A of numerical values in the context of the economic by etc. are also to be for vagueness the (Tano to choose u.u"-'uu>,. n,,,u. to cause the the in the ellormous number ",,!!,..ub!', have 1''''a.",~.~ have been For stock "~l.~'-'vlvl1 results showed that fuzzy "<o~.uh1~1 overcome the the nonlinear the returns on the Data of the returns of five companies most all the were used. wen' also to data. 32
that a more "'...,(~o.",u'" stock returns. more than 2: 1 is is then used to neural Iletwork ha Vt' been used on the Stock index A return on the elllcl("nt use of data is ;:'H~Krnann et auu,"","" in A model that and where the r>01nf',>nt '"'-''"''''''' are... " n... '>l l1ipt1 ena in cases of is introduced (G rem. In used in the lul,u"'o" on models of to Massachusetts and 1 to vague the sharp IjV'UH'..LGl In '-"V'J'VU (Machado et over in the recogni tion Tn'~"''''T '-'."'"... 1".'-' rates information R. and that market.s informatioll art' (Tanaka 1 not 33
to vagueness thesis is ro"'n{',"t'r,,,,rl with "'AI'-'Ui"UJ'~ HA~"LU\O of 1 section will a summary the literature' l'e'view on stock review has shown that market movement is measured are summarised as company market indices. ill a market markets are IS measured by ""''''''-Lv''' used to measure and 1992). A to The models such as the used to measure volatility in mean and rescaled range most not 1 993). Models that take accoullt asymmetric news used (Nelsoll 1!NO) have been used <UH.l<::"";;' 2000). "'A.UH~H and LlGlU"UH to measure alld 1991, results have processes and thus UH'>'U"" ''''.JUG'''' to markets of of economy at the on ''''.U~A'''' are indices are chaos is made. Chaos awi thus posses charact('j'istics of about overall index to mernoriness. time series to will the Hurst and dimention 34
overall index to rla'~",,"'''"' nature of index. In movement markets. set IS to 1965). vagueness in market u.'-u'-',~u and Mesial'. Tano 1 and have been used to financial markets and some stock market indices et al. eds. also exi ts in selection and used to not (Tanaka and 1999a). in literature. fuzzy ~""UHJI4 to vagueness in movement 35
1 measures, be A is to nr''''''''nt",rl as a is vagueness in these."',",u.. U\.f that vagueness. in be to outcomes are "'.....,.. "'... are
a been Peters as now H process, can be -1 to L"AU'''''''' measure, = = o $ < a LUJ,.U'J.. A series. 0 $ < < 1.0 on are asa is an is al11tlp'erslst,ent ( IS as a COlDP'afl is an JH\.L"'~'''''''.L\OH. process.
nvn... "t"orf to also be is wi th variance Hare 1 =f the Tis is, randomness is "v,"""""t.,0/1 to be 3 to cornplilte Hurst exl)oilen,t, and Diamond and A summary req:re5islon nlodlels can be it mood many "'_... sensors and to to et 005. areas
1-'"."... "... " will continue to "'~"U'.' to and and aplprc)xllrna is ones and are are mentioned in UJ."~""J"" are most are a cornerstone any ""'''''&lulll'''' ~ A to take account context or is an extenlsloln ={~ the XEA x~a a set A a 1 or 0 to In set under elements set range and indicate set in a set set. denote a set. J.LA a set A is 39
{LA : X-I], 1] o to 1 1 pproa.ch~es to way to answers say yes to x x "is small". s mermann exc!.mldle, if U is u, s 0 u::;o 2 o::;u::;,8 0, 1')= 2,8::; u::; l' 1 u2;:"y U on an 0, l' are palranleters par~uneter,8 + is 0, 1') = 0.5. To vagueness, a...,,...,"'... cross over the source that
vagueness is a I".""'L" 1 a. an = o. a er",n,'"' ''' vagueness. A 1] IS,... "... """t to structure gener- are '-<"'.&AU"'''''' =exp +exp are IdE~nt:lned Le. three delmeid increas states two cannot of known to construct '..".UU..."'L.,U.i is based on the A set can L - R,.. "."".".. nt 41
a IS if x S m,a > 0 x 2:: m, (3 > 0-0, L( ) is and m is mean of o and (3 are and and l. = = 2. = 1, 3. are are as p 2:: O. can as same as A~ its is x 2:: O. = 1- = = 1, see and pr :seilt~:1. set in is '-',",IIIUI,=-'" }.LA: -;. [0,
sets are is say, X= set A ~ X is as A =:...:..!.-'--...;:.;;.. + +... +:...:.!...:...;.;..:.. XI X2 Xn where + is meant in sense of mean variance set a set is ucuucu. ( as: = = ( mean a set as f.la IS to a = is = = serves as a nora set X2.. n = - f.la Pi i=l f.la IS of A. may be as a event 43
A of sates i = 1,2,..., k. enter IS on current state is jf where = {l. 2...., then IS i=i,2,...,k a It is transition... ~...,..."." i=i,2,...,k. k i=l transit to state. if» jf:: next state assume a = 1, E k«n.,i = 1,2,.,.,k, states its. "',Xn'..., Xn-l set n-l 1=1,i=I,2,...,k. = --,i = 1,2,...,k. n-l are U"'l,u"",U. as state is state 44
n-1 = ),i=1,2...,k. 1=1 transition f.j = a to state ),i=1.2...,k, )= 3 = terms next state = ), ),..., )) = to state )= )}, are about be use to the mood various
is two ways: as a measure as lmnrpr"i":,p are associated are associated VU~'U'U'" in much the same manner as is to ran,qolnnielss. A the tion IS a u a a set U is it as: u in U a in the 11, with to a a suppose we to U is its as the is ", into =
a is to to t.e., = (or IS 'itx and is to to Le.,.:. 11")( = If Let constraint is an JJ.U.~UI"'U. = =,..., n 1I"i = k=l = tl, is to to = as PI ~ P2 ~... ~ pn and min j = 1,2,...,n; A is, i = 1,...,n. F is constraint in = is is = are is u is must on ex-.~ is in and we to compare a measure measure 47
measure measure of a non A U is a set is ~ 7!'X 1) 1!" X (u) is the. If A is a U A is ~ /\ is a U If set is as ~ n, to an event A ~ n we ~n o.,""u... "o. measures CO]Cltlcierlce one may in occurrence event A account. If A is a sure event - 1, if A is an Iml)OSSl =0. u > ). UUll~JlUO measure IT u is for is '-''''...''"'-' as in = 1 means A is are two events a as supremum ) = 1 ntf~rn,ret;ed as two one at is
order to measure ",-,'n",-""",..", n"''',,,,,,,, is used. can the values u}, U2,...,Un with respei::t ITa IT with IS exl)ress ~ n It is as In order to measure In this 3 measures mood m(ilce~s events measures an event. measure CO]rlCEmt,s are and Prade measure aelrlot;ea is ~UU.lJ:U<;;'H\',"" measure n <
where 9 is '-'-"... "'... in 4.1. n = If means is N(A) = 1- I w ~ can a case a set A is as where P is a measure. in A over the amount _~... to the most is max If A = we event A ~ U is U = I j = 1,2,3,..., Pi =p PI ;::: P2 ;:::... ;::: Pn set n i=l Pi = 1 the extra amount )= i = 1,2,3,...,n j=1 =0 event events where convention Pn+l = O. If ne(:essl measure an event exits Le. zero, then pure occurrence the events is ""'Tn",,,,,,rl In other words there is the ALL'...,".. events to occur. the event is as of events not A
= 1 means A is sure measures =0 which two colrltraclllctc)i events necessary at same time. > 11) agrees an event becx>nrles bec::orrulllg necessary. used to > 0::::::> = 1 < 1::::::> = O. and ne(::es~nty elements of states are the nelce~;slt;y measures an of how and DrE~sente~d in common can... ",~:;"... ~n" '" set. will and ne(:essllt measures. are necessary to used to asses events in the movement of time states
an to states the ne(:ess:n measures 52
1 purpose is to """'''''.""""'",, be Drj~Seln.ted purpose is to be returns a the mean variance series are not constant. to is structure movement
.... ~ 1000 "1 1000 U 4000 2000 "22/114. 2112110.,5/15 1125/.1 l'hte +.1: to 1st r",,"1"i.,.,; out on as as Ut = can processes; is if S denotes an it can S={ te }. an In<leI>efldEmt increments sequence and t = 1,2,3,..., n S- te U = te a process motion is in
'" '" '" L-.--_--c-------'-,..,, ~, "".. "'.,,", '''. ~.,,,. Figure 4-2, Log ra.tio of the JSE Overall Ind~,,: 27th l>larch l!i85 \0 1.\ Xovemb~r 2001. It i. seen from the plot that moo;t Il, value. are near O. For such yalu s tlmt 5, and 5'_1 aloe no, very different_ It i" "Iso seen that.orne... ~,'alue. al"ij way below D. ill.uch cases 5'_1 is far much greater than S, and are asooci"te.ci with market crash. The insiances when u, is way above 0 aloe when S, i. much grea\er than 5' _1 and aloe aswciated with the rise in the index. The rnstogram of. u, is given in Fi~r e 4.3. u " '. ".. -....", ",, Figure 4-3: Histogram: JSE Over"Ulnd"". 27th March 198~ to 1st Xove::nber 2001. ". I 'the histogram shows a hii1j> peak at II, = D. This implies t~re w... U0t l1luch change in th., 5, villu,," fur most of \~ time the index was observed. The frequency ",f very SID",n "alues of u, "i~ify,he ti~s of rnal'ket cr(l.sh. The fre.quency of big values of ti,,;-!ow market ri..,. The estimated density of u, i r.:.wn in FiiUre 4..1.
:y----------------------------------------------,.. -O.1Q 0.00 I.eg feu. 0.05 to 1st mean not U""4O"''''''''.1 In is """"... Uv'u. to u = V.,]V"'V<J'V<.J'V =0 s= examination.. <>t<>..,~ to as is If is or 56
it may be PVi""'lJ'C to nh,:pr\,p and delscribe the movement of the I-"r,>VH"lll!': studies have do'culnelrlted persist1enc:e (IneInol:y pul)lislh.ed eviclellce of non rantdolm patt1ern the the Bradfield and th e DP.J'! oci!'; Polal<:o\\r(2000).,./\.<",U"CU the historical signitllcalrlt slcatistical memory. returns of the and Sh()WEtd The used sta,tlstlcleij. t(x;hlruq'ues to test memory the rescaled range will to imreslug,ate the Hurst phlenclmema of or for 27th is the last 4000 in the In Hurst Hurst 4.1. has the of the model time series that an the model is to time series. Thus to the two and ~tl'l.nctard error Y R squl'l.red Hurst exj)on.ent ~tl:mclard error the data are used. are i::)alnple size process can the and shown in the the The is in 4.5. 57
1.5 1.. H <II ~ 0 1.2 ~ 1,0 O.! 1.5 2.1 2.5 3.1 LogIn) is com- ~HI"llJIJ."'. ClW'''''''' test as a COlmJ:tar:lson is an ll(iej>er.lde:nt...,..r."'~'''' in deviates in an to occurs at n = < 1,
1.8 1.8 1.0 0.8 1.8 2.1 2.8 3.1 v may to at n= 1.3 1.2 n ::: 1.1 E ) returns. I~~----'---------~----------~--------~------~- 1.8 2.1 2.8 3.1 log(number of Oays) v returns. are
1. returns. is at n = $n$ the rella"el:;sictn
R error Y 5 3 5 3 ::; n::; a.re exlpollleilt IS 1 T = is
error Y R 1 9 9 7 7 nent 4.4: expo-- con- process once up, "... ",~.,.n or move the returns revert to some average mean
mean...,."o"""",,n 1 mean re\rer:.loln, 1 may two and 1U..."'= are prl~sentl~. 4.5 shows,,,... u..."" and 4.5: their pel:1o<is 2. are to as same as is in 63
.,.. 850 i 800 :; 550 500 815111', 2122199 119/00 tl25101 81UIOf is... un... "",.. in., i.. 500 400... loa.!! 200 100 9/13199 3131100 Oat. returns. 10/1 1/00 51510', tl2 I/O f returns. returns of the Dow Jones IS 64 '.~..
9113199 3131100 10117100 0...,. 515101 11121101 returns are ;0011 7000.. -...Ii '" i 8000 5000 4000 4.11. returns. 3000~--,---------,---------,---------,---------,---J- 9113119 3131100 10117100 Date 515101 4-11: returns. 11121101 returns are 4.
aoo 1111719a 5151G8 12122189 718100 Ollte 1125101 SIUIO, returns are 12122199 718100 O.. t. returns. 1125101 4. returns. 8113101 returns are in 4.14.
115191 12/22/111 719100 Date 1125101 1113/01 this the.. It is not account of vagueness returns. data have been are the index: a It a and is of the.jse methods the index. The next concerned with the not is
1 movement as accurate HUUU.J<;;A into time is an many etc. as an movement to "stable") or states:
"low" states. occurrence to 5 Drjese~nted the states movement ill order to assess the ~L'C;'<:;U'O::;U and occurrence will area movement. states is not to ever to assess the measures is measures to IS a hie,t"."..,un n = 1, 'V x E i = 1,2,..., n
is xin next section. set states: 1 u~ states: states: 1 are -.05 $ u $.05 u~
states were narrower COlmt)area to state. states 5 are states are pre~selntea state. TIl(' and I! states. _ :~ Ii I I all( I 4/1
each nh!,pt'vat tion states. states,",ujlhli\.-,"'<eu. as 1 1 1 1 1.9.3 =--+ +--+ + +--+--+-- 1.0 0.' 0,1 D.' ).2 0.0 =.1.3.7 1.8.5.1 + +-+0+-+-+-.2.5.9 1 1 1 =-+-+-+-+-+-. states is states: are '"'V."U"""'V"" as states 1 1 1.9.8.3 = --+--+-+-+--+-- -.11 -
.1.2.7 1.8.3.1 =--+--+--+-+-+-+- - 0 is... 1.0 0.1 e E" U! : 0.' o... 0.2 0.0 are "'....JU.!' U are as -4.05 0.00 0.05 Inde. 1.9.4 =--+--+-- states:.1.6 1.7.2 +--+0+-+-.3.8 1 1 =-+-+-+-. states
(I.'.... ~ o. :; : 0,4.. /.0004-0003 0002.0 001 0000 0001 Index states: are as state 1 1 1 1 1.9 =-+-+-+-+-+-+--+- -.11-1,0 : 0.1 " I! '" ~ 0.1 i....... 0.'.1.3 +-+.6 1.7.2.1 +-+-+-+o =-+-+-+-+-. ~ ~ 9 1 1 sates is 5-4. state -GoU5.c 100 -Go015 -GoOSO 410025 00000 00025 00010 If'H:fe. the states:
are {'''',-ntlnh>ri as state and the state the 1 1 + 1 1 1 +--+ + 1 1 + + + + 0.1 =--+--+ uu",".", IS.0.10.0.05 1 +-+ o 0.00 +-+ 0.1 1 1 1 +-+-. o.os states: the states the that the was was state in u = O. It was i5
u< - 31/ or u > "middle" state it was state. may states is movement. a sort measure were 2: 3: 4: to assess 5: states. state IS..""',.... 1">1"1 as vagueness states is in 76
.. 0. " 0.09 ~ 0.07... " 0.05 O.O~ t Middle 0,01 -'--r----..,-----,------,----.,.--l Tim.! I was state. were constant. to assess states were """'''&II<'U. states. COlnp:area to in state vagueness measures measures measure 1 1 5.i: measures states
any state measures or states. IS to In to measure events """n."ct.>n,'\1 were and occurrence 0.11 0.10 0.10..."'...,.." are prles ~nted in 0.10 occurrence events was in occurrence states are to movement measures measures time were COlmtlUt,ed. necesisu;y measures
5. states were zero. state were COlffi[)Ut,e<1 are in 5. 11. measures are COlillPU 1: NeCe!isit;y measures ne1oosisit,y measure react to...,.."",..., show a In "cu. In".'", are
states as: ( 0.10 ) ( O. 0.11 O. O..i3 ) Ull"LUJ.'I;;> state once it was "'nt"'''~>ri ( ) movement to are
( o.j-( O. O. O. 0.1~ ) = ( 01 = ) we ( o. ) -( ) = ( 0.14 0 o ) to ( oj-( o. ) o ) = ( 0 - we ( ) - (07.1I ) = ( o ) In movement in the movement movements most states were to compare the mean, have time
the means were ""cujl'c. the UL'PUoUJ.'"' state the 5 ona as iull,uwo. ( the at time t) is at t + 1, we the 0.08 ) =( ). shows in the state memor is "',",lnrr.v G Ut+1 = a E 1) I then Ut+l= =0
or a a: E ), Pl"!d.l(~t the the upper a: = the upper 1 the u.u.u."".u.~.j to 5.5. used the same -0.001-0.001-0.001 other cases the over ~~~'''''''''''"''' u the to the pr~~lctlon is the 5 section. the The purpose section is to results the """,,"!v... To this an processes been in
and ';~Jlln..Ul" process p is rlonnj'orl is '-."'"LA.""U. Ut = Q1 Ut-l +... + + tt, term f./ is "","lub",'v. to a O""HUH""U to as =0. u:;,.. UUO'" are =0. vv'iju'''l''... _! + exidet;~eu to tt = Utis... "'.u"""u... " carter were movement.
.. '".. 1000 2000 Time 5-7: mean variance the are as mean is The one <>o.1ui.ll<:; Test tails a at mean. ks=
p - value = 0 and that the the the a 0.7 1 0.7 0.1 +--+--+-+-+-+o an- as the the series states is m ~as'ur ~ terms a transition matrix. A the movement is states are,..nl'ylnnu>ri measures, ne<;es:sl 1 1 1 1 1 1 1 - --+ -0.11 + 1 1+--+ + +--+-- + + 0.1
to measures "... 1.0.. 0. ~ 0.' :! -., 0.4 o.. 0.2 0.0 0.15-0.10 0.05 0.00 0.05,,-.. jehu) state. states:... "AU""'... " -0.03 to -0.01 are measures state. are o is 1 0.99 1 1 0.96 o o.
state is more states. state is more ueigerrru:msl.!c to rar:lqomn.ess states are prone occurrence are a A If "'>AU ",,,,,,,IS at as: t is ) its to at are manner. t+ 1, states ( 0.8 )U 0.72 0.76 O. ) = ( 0.14 O. ). 88
at t + 1 is more to in state the the in,0.0 (t are t = Ut- so Ut= + tt, Ut = or 5.9. is = = + tt+l a o:::ui",.",u. state was 0 to v... v... u t+l thi is in
.001.00.00.8 0 7640.00.8 0.8 0.005 0 2 0.8 0.005 1 0.8 0 1 -.01.8 0 2 0, 01 0.004 0.8 009 0.003 0.8 009 0.011.01.8 0 0 0.012.8 0 3 0 0 1 0.8 0 0.8 0.8 0 0 0 -.003 0.8 -.009, 0 -.004 -.0006 0.8 -.OOll, 009 0 5.15: and a <J<,.'l.HU:U seemed to be zero in most cases. an
5 was states" measures were \.A.I.UlLJU',""-J. state. states were comtluted. states on states. to to states are not states are Dn~sentc~d. states to measure... nh..,~"'t... h~;'"''~''' vagueness ran- events were state. A nor events states. states. states. state
the was most in state than it was a states. state. state. states were to move movement movements "",,""r~'<>yi to measure of states were stable. he next movement over
lit 1 1 lit 1 to measure a lla'''j.i'''''" investors has wide HUVUlo.;"'L'''''l1''.u",... ". markets. The mood investors may act as an '"'.,,~''...''' is terms such as to investors. It is the purpose gap. lit The investors in are not easy to as: most way movement is and it may be to assume
that t"nnt"."nl"." are also to mood mood may some movement. in is to is movement. to to may compare...".""t,,,...,, may may causes low mood in nu,"""''''''''' may and 0..,""'''''''''-\ a demay occur movement or to the minimum and 94
terms to construct a "... rn'nn"" measure or or mum sections ions to investors may vui.. ""J'<J a- may measnre investors away -b may or
up away ratio or maximum -00 to +00. IS "","",u.u""... as amltoimed to b, = a= = b, 1 if..."..."'yu is K.t = if IS mood o if u.. It is ""'''''UL<O'''''',...,;;. IS O't = mood mood a= = =b or """.1..",_ manner mood is or measures or
dist.ance the or the is the investors may more COlrlll.:1ellce or O't is ls Qe'pelLlQ,em on t1" n,."" is a 1 if 0.5 if o if is can be viewed and is in 1S is investors in terms or mum, the,",u""'u"'"' or or the hhr'~",..,t.. ",t"'l"'n( 'p t.ime the or to
last or A short Ul",u""'.lVC; or toward the last maximum or lowest same chc)sen in reveals COltlllcleI1Ce A low corlllo,en(;e may a A"<.~'."'''''U and = 1 if is in = 0.5 if market is = ::.t..~.;t-:~1n:.::2 1 o if is o stable o is aejllllea = = are low in terms '-"''''''1;0;:;'' in are... "',... view investors may view the construction view market movement terms in terms '-UlUl.. ""... ' is r1... nl orj
from,he mood of im'""tor, ", <T. and 0 oow(n>cled in,he previ""s sections. Ear.h lenn, 1<. <T. Bnd <p i" giwn equal weight in the construction d the oon1po:iite n100d index. Th IJS the fon",,),, for Lh~ composite mood index can he writ.ten as The mood index ~'. is e,>mpu(ed fel" the JS Ov~ralllndcx (3/27/85-11/1/01), JSE GOLD (7/13/99 _ 10/31/01), NYSE index (Q/13/99 _ 10/25/01), j',"asdaq (12/21/98 _ 10/25/01), DOW JOl\'ES (Q/lo/W - 10/25/01), DAX (9/28/99-10/26/01)' :--'lkkei500 (2/26/99 _ 10/26/0\), and the lbovespa (8/11/99-10/25/01) data, The computed ",<T, and Q for ~aeh market index are given in the appendix at the md of the study. The plot" and hi"togrbms d mood index"; for ~",-,h m"-tkel index are given in t he following- soction. 6.2.1 T h e mood indices plot~ Till. section provides the plots of th~ l1ieod index fc.- various matkets in th~ following figures. Th~ mood irukx for th~ JSE O"..,..allindex i. plotted in Figure 0.1. The hi,logrbm of the mood index is sho,'m in Figure 6.2. " i,, :; '.', " " ",'"C"---CC.C C.---.C"C"C. ---C.C.,,C,,---~ 0 Figu,.e 13---1: :>lood index: JSE Overalllnde~ 3/27/85-11/1/01
'"" '"" """ '" '" Figure 6-2: HistOl:ra.\ll; Mood index of th~ JSE O,-"r&lllnocx 3/2i/85-11/1/01 The mood in,j.,,, for the JSE Gold i. shown in Figure 6.3. Figure 5.4,hoWl! the hi.togram of the mood index for the JSE Gold. "" 1, "",, " "", "" "" ",.. "..,,"' FiguI'" 6-3: )Vlooci index: JSI:; Gold j:ldn 7/13/99" lo.,i.11/01 10i-, "
f " I Figure fi-~, Hi"ogram: Mood ind"" of the JSE gold ind~x 7/13/99 _ 10/31(01 Th... plol III Fi~r... 6.5 shem. the mood index for the NYSE. The his "'gram of,he "'(~J(t index is given in Fi~ 6.6 " j "..,. ".., " ''''.. ""'". m"..,,'''' Figure 6-5: "'lood index: NYSE i!rlex 9/13/99 - \O/2~!Ol,m
" "". ". " "",".. n..,'... Le - L",,"...,. Fi~lre 6-6: Histogram: mood index of the NYSE index 9/13/99 _ 10/25/01 The plo, of the mood indf'jc for the Nasd.&q is shown in figure 6.7. The histogram of the mood index foc the l\"asdaq i. soown in Fi~ 6.8...,, " :" " ;.",,, "''',. ",... " "". Figur~ (;.." ;Vi()(Hj 'll(icx: :\",,<1,,'1 inc,ex 12/21/98,.,,~, 102
"I... "..., >.".n.",.,.... ". "... ' " L",......,~,... Fi&'Ure 6-8: Histogram: ~Iood. index of the N~sdaq ina.::.: 12/21/98 The plot of th~ mood. index for the Dow Jon <! is shown in F'i~e 0.9, ~nd the histogr""m is given in Fi",re Ii 10., c_, ", " " '" ~" "".,, " " ~..", ''' ~, "''' ~, F:~llr~ f}-q, :>100d index: [)ow Jones index 9/16/99-10/25/01 103
'" " " ", --,C,'. '".".".",., '". " u...,.",." "-,,,......,,... Figure 6-10: Histogram: Mood index of l he Dow Jones ;r!dcx 9/16/99 - lo/2~/oi The mood index of the DAX is plotted in Figure 6.11. The histogram of the mood indr.< is given in Figure 6., 2.,,,,,,,,,, ri~lr~ 6_11: -"'1()()(~." "",.-,"" "",.., ".,~'",,"' ~, ill(iex: DAX ili28/i19 _ 10.-.-'26/01 10)
"!.. f, Figurr 0.12, Histogram:.\lood index of II... DAX 9/28/99 _ 10.,2,,/01 The plot and hi'~am of the mood ind"" for the Nikkei500 ar~ given in Figur ~ Figure 6.14 re,pectively.,,,,,.. ".. r ", ", " -,, 1 :::--;:;:0--,,-.-,-.-, --,,-,,-,-, --,-,,-,-,.",~,.",,~. "',,. Fi!';Ure 6-13: Mood index: Xikkei500 index 2/25/9<) _ 1O,i'W/OI... L G.:3 and 10"
., f r " Fi)\111"e ~-[.j, Hi'to~ram;!lj(xx! index of ~h~ :;ikkci~oo ind~x 2/28/99 _ lo/2<1!0l Thc ploi uf the "I(xxi mdcx for the lbovespa is,howl! in figur~ ~.15. the mood index for the lbovespa i, g;'en in Fi(l,ure 6.16., " - j 1 :" <. "! 1" ".~,..,,'n "~,,,,,.,.. "". "" ~, Figmc (,-15 :'Iood index, lbovesp.\ Sill/gO _ 10/15/01 c l r rhe hi,t,,~r"m of 11M}
--~--'-,, " Figtlre ii_i fi: Il;Ol~am: mood index of \he lbo\/espa 8/1 i/99-10/25/01 The plot. of t he moc>d indices for \'M;QUS rnarke(., show chac, (,h mood of il1vhst.ors for Ihe JSG had an upward -,[end from ahoul 5/19/87. The mood of investor. for the JSE &old index,howed a lipn",r.. l declil1e from around,/13/99 to about 1/'15/01 and then generally. tarted to ';&e, ThH mood of inv.,,;tas for the :"iyse, 000.>.' Jones and :-:asdaq had a general upward trend from about 9 ' i6/99 and then had a general dowilward t rend frqiil about.1/.11/00 for the SYSE and,]w Naodhq, whi]h lhh downward trhnd for thh Dow Jones was from about 5/5/0L T he mood of ;nve,tocs for Ihc Dow Jone, and t he ;'-as<.iaq was very low afvill,d the 8!1l/Ot. The mood of invhsc"''' for \hh DAX had.. general do'.viiward t",nd from about 3/31/00, The ""ikke500 had it, rr.ox.< i of investor generally rising: from 2/26/99 to ius, afler 12/'1'1/1<9. lhen a ", l1)n~d a,,;eneral c.:oi'.-nwanllreud, ThHmood of inv'"'tof' [or lhh IHOVESPA was characthrised by an upwa«; tthn(; b)[n about S/11/Y'J to about 11/22/99.md then had a general downward trhnd, Tlw mood wa, 'QWes ~ aftn 8/11/01. 6.2.2 Linear relationship between mood indices for various markets Correlations between ''''' mood indice, of various mark",", NYSE (NY), Na..Jaq C"-Q), Dow.Jones (OJ) DAX,DX). Xikk"i500 (XK). lbovespa (IB). JSE Overalllu'lex (JS) and Ihe JSG Gold!JG) ar computoo for the perio<i9/28/01 and lo/2/01 in onler ( 0 inve>;\igale their relationships. The res lit, a", sr.c,..,.'n in tbe following condation mmri~ given in Table 6.1- IO~
Xl' :-<Q m DX :\K m.ls CG "y 0,139 0,375 0.232 U.I04 O,:j Jl 0008.0,329 :-'-Q o,ng O,W,/ O.:>ij,', OJi64 O,-I~Y -U,14' 0,il44 D.l O,lT,c, 0,057, O.OO,~ -0,167 0, J.l5 0.023-0,250 OX 0,~12 0.38~ 0,1)65, 0..100 0,309-0,002-0,061 :"K 0.10 1 0.664-0.167 0---100, c. \,)-1-0.16:3 0.093 m U;lJl 0, 169 0,1 1" 0.. '309 OA5,j -O,OO~ _0,162.lS -0,008 _1l.l47 0.023-0.002 -O,lG3 _0.006 0.0';5 CG -0,329 0,0 14-0.2,',(l -0.062 0.093-0.162 00.'>5, Table 6.1: Corrd"t;":l Ill""r;",,[ mood ;:Kii",. for var;""s Illarht ijj(jjc~' "IDOd of investo,", [or the DAX, l\"ikkei500 a:xi lbovespa ~ll nad " r~""o:l~bly hi(n,,-,rrelation with the -'iasdaq mood, Th~ DAX and lbovesp,'\' mood ~lso hftd >on", hi~h oorrd",;on with lh ~ XIKKEI500 Illood. Th~ mood o[ Ine.lSI': w~o negfttively oorrelatoo to the n~,od of njoo\ m"rke"" Tni; may be explabed by the fact thai whe:l em ~ rgilll> market. iih the JSE experience low mood. illv~.tors.llift to the de"doped mmkel, 6.3 Persistence and randomness of mood indices for various markets III,hi,,,,,,tim). tn~ :latllre of 'he computoo mxxi of i:wes='o fmm the IDJ<X! i::tdice, for variom mftrkets wiil be in\ sti\:ated, The R/S analy.;, will ~ app!i<.d \0 jllop"", wnelher th~ mood o[ inves\ors are i:xlepe::tdem procc",e. ur~, tne Hurst."p "ll ~ ll t memory I'he:lOme:la, P,er> (lw4) ha,.xplai:wd tn"t auloeorrelated da,,, ca::t influcnce the Hunt npo:l~nt AR(I) model,",'0.0 fit to the mood irkii"", [or tne "arim" rnark~t;. per["rrn.d OIl.he =iduftls,,[ th~ AB.(l) m(kl~b. The Al-t(l) n);)d<>l is given by Th", tne The R/S analy,i, was IDS
at t is ex'pre~sse~d as is at t, a is an 1) to at t - 1. summary un.u... ':;;" IS are CALlUJllCil~ IS is respec- are in the
I "j JSE ood!(! 15l '" ;;.:: 1.0... I 0.5 '-0 1.5 2.0 2.5 30 3.5 3.0 i: IS to is 2.5 JSE 2.0 1.5 1.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 log(nul'ft b., of Observations) V statistic: exhibits some IS 1
y R 15 15 _~u.a,j.u. error '" 1.5 ~ 1.0 o ~ 0.5 are is a Nasd is = IS 0.0 -r---,.----,----,----,----,----,----,--' 0.85 l.10 1.35 l.eo 1.85 2.10 2.35 2.60 LOQ(Num b.r of Ob,..lItion) 6-19: The is in 111
y is of result is His is 112
Tht' of and the for the liood are in 6.21. 1.4 Dow!!! 10 ~!: O.S E( 0.2 1.0 1.2 1.4 1.8 La 2.0 2.2 I.cg(Numbtr of Obl.Nation,) of the V statistic the Mood in V statistic is m(;re.aslng su~~ge:5ti[lg p,ersi.stence in the mood investors Dow 2.0 E O.'~--~--------------~--~r----r----~--------L,.0 12, 4 La los toi(numi ber of Otuu'I'V&ticuu) 6-22: V statistic: regression outj>ut is in 113
R error Y 5 5 1.2 ;;; 1.0 ;;: ;; o... o.a 0.8 0.4 E( is N / are is 0.15 1.10 1.35 I.SO 1.15 2.10 2.35 2.S0 IS 114
i 1.3 ~ I 1.1 Og N 0.1 0.5,... 0.15 1.10 1.35 1.50 l.n 2.10 2.35 LOGl(Hum b., oof Obl.rvations) R v y results: H 8 8 IS is is "";U,'U,..a,,~. the are in 115
1.4.. i!!! 1.0 «~ E / o.e 0.2~---r-----r----~----~----~----~----~---------- 1.0 1.2 1.8 1.1 2.0 2.2 log(null'llber ot Observations).. 1.1 u -= 1.4 M q; :> 1.0 O.8-L--~----~----~----~----~----~----~----------L 1.0 1.2 1.4 1.8 2.0 2.2 6-26: V is is 6.i. 1
y R.jOO 5 5.. 1.< 1.2 '!! 1.0 S.'! 0.8 is IS are 0.8 D is 0.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 in investors in 117
1.8 0 Mo 1.8 1.' E(R 1.2 10 0.8 1.0 1.2 1.4 1.8 1.1 2.0 2.2 Log(Num b.t of Ob vvations) R V y is is o. 5 5 is is in the are 118
Log(NuM ber 0' Obs.rvations) 3.0 2.5 2.0 1.5 1.0 0.5 E{R 1 is JSE 0 ood 1.0 1.2 1.4 1.5 1.6 2.0 2.2 LOI(NuM bar ot Observltion s) V of for the is in
y R ""ll... r.., 0.9994 1.4 0.5 is are is is O.2~--.-----~----r---~----~---- ---------------L 1.0 1.2 1.4 1.1 2.0 logtnumber of Observations} 1 : is in
R ""'''.. ',..., y 6. error 14.... 'VL1<:;.:> that investors is some on It also means that n",p"'e>nt in the "LU'L""'_" 121
to or most events in the the current true nature the where is the...,,,,,,<,,,t 6.11. """,b'u'u measure nv'"<;:~,r\''''' on the = -1, can exdfless;ed as a on is the the 1 1 are COlffitmt,ed at its space is in H the are 6. 122
1 1.13 1. 1.26 6. are IS in as a measure events. "...1".., us to compare as: events more may.08.45 1 1.57.08.1.2.3.4.5.6. i.8 1-0 +-+-+-+-+-+-+-+-+ +- "''''''''''~,IUH to its is in 6.33. 123
" "., /,. " \1 6-33: as the shown.64 1.41 =-+-+-+-+-+-+-+-+-+-+o.1.2.3.4.5.6.7.8.9 1..,.. " " -01. ~----~----~----~----~-----,----~, "... 1 +-+-+-+-+-+-+.1.2.3.4.5.6 as I 1'- +-+-+-.8.9 1 the
, Q o. I 0' -.,.,.,,.,., ~ \., 0' I I MoOt! " 1 = - + - + - + - + - + - + - + - + - + - + -. o.1.2.3.4.5.6.7.8.9 1 ",0..., 00 is IS as as 1.97.31 = - + - + - + - + - + - + - + - + - + - + -. o.1.2.3.4.5.6.7.8 1..,t,,,l',,..,,,... IS
0, 01 0, 0\ o I o. II is in,... o. 0'.84 1.1.2.3.4.5.7.8 +-+-+-+-+-+ +-+-+ "~-----'-------r------r------r------~------r..". as,.,.7,, is as.02..47 1 = - + - + - + - + - + - + - + - + - + - + -. o.1.2.3.4.5.6.7.8.9 1 The is in 6.39.
1,0...... 0.' r e.. 0.' 0.2 / 0.0 / o., 0.3 o 5 0.1 0.8 J$ E I"ruod 00' investors - 1 is. is.95.99 1 0 =-+-+-+-+-+-+-+-+-+-+. o.1.2.3.4.5.6.7.8.9 1 1,0.. 0.1! E 0.1 ~.. : 0.4 0,2 0,0 it is not easy to if their 1 in state.
states 1, a It can 1) rest have very...,,;"'u.a""'o>:> of the mood to is 6. n"'~':t'n.,.., can 0.74 to measure nu'><lt<.r", events is ~~""I-"u. In
... a... "''' to occur were In''.. ''''.r...,, events to to it is H"I->'-';:""
6.4 D COlIlpusition of s hare pnce of.1se Ang lo Gold The JSE m ~rall index is deriwd from all the listed companies and An~o Gold i. ooe 01 them. Anglo ",>ntribulps a signi ficanl port ion of 23% to the JSE. In this """lion the daily and monthly mood of investor. for.jse Anglo Gold shan" price ar~ comput~d. The mood index of the JSE Anglo G,>ld share price is computed from the monthly data. Th~ data used 10 cnrnpule Ih~ daily m()()(1 was l<>r the period 5/14/98 _ 11/22/01. The monthly data was fc>r the period.1/31/yo to 8/31/01. The computation. arc done as follow", 6.4.1 Daily mood Tn cnrnpute the d"ily mood of investor., the daily, egret ftlld daily COIlfid~nC>l These are ~ve "" f,>llow", Daily regret Th" riai.y r~l{l"~t ;, compmed and it' piot ;,!';Ivcn in Fi!{1"" 6. U. N -'"..- --,.,...,.------,.:..,..:..:.----.. "... arc compllted. F i ~e 6-41: Daily regret; JSE Anglo Gold share pri,.".\.'1-1/% - 11/22/ 01 The daily regret med t.o be ve, y 'piked. The,;ze of the, pike, ih~, e ~ ocd with '.; mc. Daily cnnfid" uce The dany confidetlcc is plotted in Fig"'lre R.12.
,,,. ~ '''' - r "" i t ".." '''".,,,. ' ~' H. "' ~, Figure G-42: Daily oooodenc": JSE Anglo Gold "hare price 5/14/98. 11/22/01. The daily oonedence seemed to be more stable than daily regret. Sh;ap spike. are seen towards the end indicating high oonfidence in Ihe share price in this period. Daily mood The computed daily mood" plc,ttcd :n F:"1.lrc 6.13. Q " ".... " "..,,~. -~ ' -:I\'." ~. " 'R '~.,.'. :...,- ~..,:::...,.. ~' ',. ~'.. - ~';. ', l...,... ~...:.......,,...-..,......-.. ~..... ": "";... -...."..,.'..-.. ",:,. ~,.: :'''~.. ' ~;..:." :.:. :-:.::.-,. ~ :.:~.: "';.,....., _.. " -'~.,......,:.. _. '.-'.~: - "'..~.. :., :.,~...., ' '........,'.' -;;. ",., '. : ::"'. ' ><" '.. '... ~: -.:.;../ "...:..;~~:ti. ~.-.... "..,.",. "" ~, Om r Fii\ure 6-43, Daily mood: JS E An glo Cold,hM ~ price 5/14/98 11/22/01. The distribu.con of the daily mood j, represent.ed by the histogram in Fi~~ 6.44. 131
'",. '" ". " Figure 6-.. 4: Di"tributioo of daily nlooo 0[ inv",toro:.lsi::; Anila Gold,hare prjce,'5/1<1/98 _ 11/1'2/0 L Characteristir.s of the daily mood (Anglo Gold share pnc<o) The Hunt (Ii) exponent is computed to inve,tiga!e the nature of the illoi!y mood of investors for the Anglo Gold aha,,, price. The [e","ale<! ra!lgij analysis WILS U;;OO. The computed Hll1\';t expunent II = O.55~8 may imply that the daily mood for the A.nglo Gold j, perai,tent and thus exhibits the Hatst memory. The mood of the "hare price in the past affect s the present mood and that of the fut~, The log-lo~ plot of the analysis i" given in Figure 6..1~, Figure 6-1.~:, I ',~--- ".... H = 0.5578 \ RjS analysis, dail} 11100d of i11"''''';.''''.js A"~kl Gold,harte pric~,')/1,j,/98. 11,iZ':'/OI. Possibility dilstributioll The po"ibility dj'tribmion of th~ daily mood is computed in order to represent knowledge and informa lion ",bout the Anglo ~"Id ' MIe price "" ",dl to inve,tig"'l ~ "ub.iec t iv ~ belief of inv.stors 132
alxmt thl',hare pric<' Th~ p"",ib;li,,- &;[ ribl.l\ioll ;, co!)lp'~ted a" follows I,M,IS,SO _5_.60,~J,70,69.79,79 C:,-\NGLO"'-+-'- --'- -.-+-.-+-+-. T-, +----;::;-+-+-+o _.,2,3 "~I,v,6.,,8 _~ Accmdin~ "" kr ", w]"i)l~ and infornlar ion obl.incd fro:n d~ta,,he 1D',ves\ mood (nine 1) i~ Ilw most ~oss; hie, l',,,, hig;he"l mood i '-a]ue 1) io tile ll ~xt mom po~~ibl e daily mood of inve,\ms for t he Aug)o Guld,har~ price. Th~ low mood of inwsto.., i, the most possible for AnKlo Gold 6.1.2 IVlood of JSE Anglo Gold due to monthly share price changes Thi,,,,,,lion will cmnpur,e I,,,, mood ;[ode.'{ for tl.. An](lo Gold,hore pric<', Th~ index willlw based on ITl(}nt hly c..~an~" of,har ~ price_ Th ~ ",-,n\hl,. da)a for,h ~ p<>riod.1/~1/9{) to H/~l/Ol will be uski. To compute th~,,>xx:! index. ",ocod uf in\i",l.o,", rd"tiv ~ "", I""",:I maxi"",,n and rniuinnun (con, id ~ red 1"',-ioJ), nwnthly lllii..'{imnm and minimnm; and cumnlative maximum and minim\!!h ar~ C<)"'~" t,-j. The modd inde.." i, the composite of Ihe:;c thre~ mood" of inve' tors. To compllt ~ ~I>(;h m,x\j of inve,w",_ the regret and conndence arc computed. how"",,er these will not. b~ shown her~_ Th~ mood ui inye,lo'o dn~ to ",,,t>[ hiy m,."imll!)l. ",on\hly ",in I mum and t he mnnt h ly cl,>sing price,,-ill be us<'<j to d"<xlmp"'" the cl",;ng,hare price int-o ii, fund-.mcn'a18 and,entirnent, Thp d oo ing p,-ice and til~ df'<oomp:-,',-j (fundam",i"l,),har~ I)rice will b~ correlated wilh som ~ of the South African econc:mic indicators (fundamental, ) namely:!uon ~y supply. to in\ie,tigate an" rd auun,hip. The computed mood of in\i",lor,,-d ali,- ~ \ 0 record maxinnun. r!'coni ",iniruuru and the monthly closin~.,."luf>s ;s compmed and plotted in Figur~ 6,46,
,.... ",, " ", I I 'Ie, I ", ".,,",,..." "',,'..,. "".. ""., l'i~urc I}--Ui :tlood of in"",(or, rclati,' ~ '" l'e!:o, u ma_~imllm, rec;o... d lilhuj]~"n and monthly clc.,ing p'i~"', " I.', JSE Anglo Gok13,/:1l/00 -- 8,.-:31/01-8ctwe~n llj:lono ar.d Ji':1l/~3, I/:ll/% a".l 12/:11/%, :l.f:h/97 i!.ild ~/3l/!I'l the vl!.jue, of thi8 IDOiX\ of iil"cslot_' '.\'~,e low_ Between 4/:lO/9:l a1c,,j 12/JI/9-1, I/Jl/96 ar.d 2i2~/~" ~/JU/W and ~.'iju/o() The "~KJd the value, were hi~h_,i or illvesl,,!", due It} monthly m3xi",<lin, monthly minimum UI") til. monlhly d08iilg pric~ was (~)mput.ed and plml...:! in Fi~r~ I).':" Fi(l;urc 6-47:.,.. ",.. '.... ".,.",,.. "..,,,,,,. "'". M(x}d (}f investo,,-; due [.0 monthly maximwn, minimum and luor.thly dc",ing price- JSE Ar.glo,..,,~,, J Cold ;l/:l1/!lo -- S. 31/o.!- The ffi()(xl with rc'pect to I.he,,~"'thly challw" of dle,hare price WiI..' hi~h1y spiked I.hrough out the da\~ period. The momhty mood = med st..,ionary "bou~ the meail. The,LiscribUli(}ll of monthly mood,i given by I..he histo"rum in Figllr~ 6.18,
.. " ".., -'------, ~ "...,,' ",.., ", ' "... '",.". ~ ""... Fig,tll'e 6-18 DistributioL of moolhly mood of inve,lon ; JSE Anglo Gold 3/.31,..00-8/31/01. Til~ m,)nthly mc:... d of inncs\,)l's is charactcri&cd by low mood. The mood rd ative to cumulaliv" ma.ximum "lid minimlull Wag c,)mputcd.. nel p]ottt"d L!l l';gun 6..19., ",, -.,1',,.'1 1.1,I,,-,III i. iii I II I,,- i I ~ 1 'I,, - i u_ -" "" " I, 'i' II, ~ t,,- I " ', '.'.- '-".~., 'i- ",.,,,,., ",,",." "".. Figur~ (,..49: )"lood of investors relative to cumul.. tiv" maximmn. cumulative miuimum and m()nth,y clming ~, p:~,. : JSE Ang,o Gold 3.31/ro- 8/31/01. Th;,!ro od h"d bel""",n 3/.11/90 and ~/.11/!):l, 1/31/95 and 12,/:>1/95. ~rjl,/97 and 8/31/99 low value" [l elw""",, 4/3il.'9~ and 12.. /.11/9", 1/.3) '96 ami 2/28/97, 9.. /.10/~ and 9/:],0./00,he The mood index comput ed '" \he "v"u!. ~ e of th~ above moods of idve'tor, is computt'<l and the plot is given in Figure 6. )1).
"j I~ II I!I I "! I I 'I I, ij II I, II, II,, " II' \! 'I I I.." c I.'. I, ~ I I ". " 1'111 I "l' Ii' I l " ',, ~~I ~_ \, 'r. " I \ ", '.. " l III, '! I,.,,\ I I I I " I,,,m,,,"~,.,,~,,,,,.... ~ "".,, ---~ fi;"hl~ 6-500: l,jood ind~x; JSE Anglo Gold monthly ~nj/oo - 8;":jl/01. T:~e movement 0;,he mood index was maracleri.ed by an upward ",ud downward movemellc, The ind"" show;; chal l:>et"=n -1/30/93 and12/.11/94. lj':ll/oo and 2/28/97 the mood for the An,:]o "old,hare pri"" was high. 6.4.3 Decomposition of share price using the monthly mood index Bo~h hndame[ltab and market 'entimeut. oontriblll~ to what CAns t ilutes a ShiH" pri",,_ In this """ioil.. n attempt i, ma& Co d~comp~ the JSE Anglo Gold shure pric~ into fundam~il I ~b and, entiment,. To decompose the,hare pri~e into fllndamentah and 8en\im~1Lt, it "'ill h~ a.,umed that price ch~r.:("" this Ca_,~ illv~'(ol- ' lj.jl(i dhj.llge ill the IT""'.! of investors a", relakd o<,ly to unant icipated evell\s, in may not have adeqnate ti me tn ""aly,e all (he necessary informatioll to m"ke rational decisions, decision'! a", influenced by sent iment, this rna)' can"" mood of investor. to,wing IO high or I""", Without the unant icipated event., prir~ movement are due to eco<,omic fundame"tal, "lld ~re,table, The mood of invest"'s is al", stable in thi,,ituation. ~l""ill!> T11 " mood of in"",\0<3 due to momhiy maximum. mont hlv minimum ami the m""t\oly a", 113ect ill th~ ctocomposi\ioll. Th~ monlhlr mood of in'-es,ors i, g'jven by c-m ~C.,. -,),!;,.. exp{- _~ ml.. ~, - 5,
,,~, wlinc 5;" is the mrh,l.hly rtlaxltll'<m, ~';" i, tl:e lllolllhl), mini')lum and Sf t"e momldy d",i::tg" pn"eo It i, ~,..<",,,~ Lhat Lh~ dooin~ priee.'j-r ca" be wrin~" 1«a "l(" "f r,'llda'n"::ttals and,~,,,irnent',uell lljl!.t w),, re lit is llj~ prie~due \" h"damental, a\ time 1 and (, lh~ eh~"ge i" p,ic~ due to,entime"t, at time t. The marke, Illf)l)(l is percei\'lx\10 '\I<bl~ wl,hi::t a ccrtain ran~e of \'ahe<. "I<Y (O_:J, U.7). If h:, = 0, 5 indicates,tahle,narke, 'll(~)(ll<,"l implie5 5;--- = 'h (i.e_ the,hare price wile" m"",,\ b.'table i.' d1re to fu"dametllai,) it he,bow" Ihat 5 H + c;", '" c, ".., ",- I + In 2 which is the (m,tri»1o(i"" p"rtjo" frr)m \h~ eco'",mic fundamentals. 'l'he market,hl<re pric~ dne r " fu"d"'n~ntah '7< i, comp1ltoo from the Debeers <h~r~ pric~ data for the period 'U/I\"l/~I; to 14/05/01. Tl:e pl"l of TI1- given ill Fi6'UTe 6,5], "'.. -, "'.. -, "..,,, ",,,, ',,',, ""'",---,,,..,,, ~ '"... ",.,,ca, ' Tlw plot j"wca\es thai. chl<nge i::t prie,," due co fundam~"tab ov~r the period w,,-, upward. Thi, wnld be explai::tcd by the changing oco::tomic f"udam~n'a:,,wer 'IIne_ Th~ dla::tge '" marke' m(neme"t dn~ II> rn~rket sentiments C for the period 24..-0.,)/8$ to
14/0'i/01 was compul... :1 a... 1,... -I,,,... ~,,.-.,... " '.frllll~'lj' \t,} 11!1 1~il'II'I'll lli' I I", I ~ il 1i l!'i' ~ I I~j lll ~ '" ~, I ~'",,",.. ".,"....-".,,'",.-",' Fi~\lr~ 6-.52: l'ri,,~ duc lo j"'"c8t-or o~mimcnl, JSf;, An;:>;o Gold monlhly pric~ :;/31/90 - d/:h/ol. Th~ ;;::r~ph shows that I'd"", cha,,:;c due co ""'nljm~nl. "'8.'1 8tacionHY ~b()ut lh~ m~a" with incr~asin~ variation. This imp/i"" that,'-ariations in,.,.ntim~nls ",.. s incrt'asing with tim~, Thi. c.." be cxplainci by incl..,asm awarenc,," bj' j"yc,tor, duc to r~adily available mark~l information bro\]~ht about by lochnology ",::h'anccrn~nt, jn",l'~a",,:1 a".. ilability of multiple analy.,," ~nd i"l~rprctalion of macht inim'malion induced incr~as,,:1 variation i, t".. der.' '~ntim~nt". Th~ [0"0' "I'ik~",,' Ih~ ~1"d of Ilw plot.u~""t thai'" lim'" mhk~t,~ntimcnl.' W~" quit~ big. The "~tio I "" t o compare market fundam~ntals and mmk('\ scnt im~nt. i. (lompul",!. The pin of tij, ls giv~n in Fi~'.l[c 6,53, '13
,,,-, -. I I '", 'i ' " rigur~ rh)j: RabJ of '"ntim~nt 0" fw1d~nwlltak I' f'l Ii' 11\1 (I'! I.,j, ~II ~I,''IIi,', ~ I'." ~, " : 'I I..1 '., I I', '''':1\ II, 'I'l\ I II' il '/ I I ~ i, I I I, "''''' ".,"."," "..." ',....,,'" JSf: Angh) Cold IIl0nthly ~/31/!lQ 8/.Jl/OL The rat,i.. " are mlall due \n " llali \'alu.. of ('0 IIl"",,'ever, the "~ik e, show,hat semiment Wa;; sometimes qllit~ infllj"n!;,,] in 'he share pri"" movement. Ti,.,pik~ in the direction may be ",,",,,,jated wilh,he daily market. n",hes. Al\hough Ij', i,,man due.. 0 sman value, of ('0 it has all imporian! eff"",\ nll 'he s11are pri"" m'nemenl in the markw. Correlation The correlati..,n, of the m fl'.1 ~)S of the monthly ocnrwilli~ indi~"i0[": Produ""r Inde.' (PROD), ),!,mey S,,,p!'!}' (MS), Manllfacturin~ l'r<x,ucti,~, (MP) and Com"oner Pr,,,,, [ll('ex (('0\'3), with log rmio of,he JSE Anglo Gt~d monthly cio,ing pric before d,,"oul~o,ition (A;'i"G) and aft r decomposition into fundamellt",l, (Al\"GF) are computoo and _,l''',,"ll heh)w, TIw correimion of the conomic indicators with A:\C ",n" A",Cl" (T",bl~ 0.1,,) ",re v ry,mall a"d showing an almost non nist nce of lill ar rdationship, Al\"G.INGF PROD MS.If P COt,'S A:\C,.OOC 0,97;] OJJ2.j 0.018 0.O~8 O,O~~ fuiigr' 0!173 Looo 0.1).1, 0.037 O.O!Kl o lot Table 6.15: CorrelatiolL' lundam lltals and ""lltimell: "I JSI:: Cold,har pric with tjc"nolili~ indi~ator'. Aft r 'he "ecompo,;i;on. lhe corr i",ti"n between 'he AngIe, Gold share pri"" due 10 lundamenlhl' wi[,h Ihe ec:r",om;c i)[(l;~at"'-s incr ased. Tin" removing ' elllimenl from the Anglo Gold