AnInvestigationofthe U ncond itional
Distrib utionofSouth Af
ric anStockInd ex
R eturns
O w enB eeld ers¤
Departm ent ofE c onom ic s
E m ory U niversity
Atlanta,G a 30 32 2 -2 2 4 0
e-m ail: ob eeld e@em ory.ed u
T el.No.: (4 0 4 ) 72 7-6650
J une 1,2 0 0 0
Ab strac t
W e investigate the d istrib utionoff
our b road stoc k ind exesand f
our
f
uturesind exesonthe J ohannesb urgStoc k E xc hange (J SE ).
W e ¯nd that
the b road ind exesare skew ed and highly leptokurtic . W hereasthe All
Share, Ind ustrialand F inancialInd exesare negatively skew ed , the G old
Ind exispositivelyskew ed .Inad d ition,the skew nessisnot onlypresent in
the tails,b ut also inthe c entralpart ofthe d istrib ution.None ofthese ind exesisc ovariance stationarity over the sam ple period ;thism ayb e d ue to
struc turalc hangesinthe m arket suc h asthe introd uc tionofanelec tronic
trad ing system in19 9 6 and the volatility introd uc ed b y the Asianc risis.
For the f
uturesind exes,w e ¯nd that only the G old Ind exisc harac teriz ed
b y (positive) skew ness.Allthe f
uturesind exeshave exc esskurtosisand
none ofthem isc ovariance stationary.T he f
uturesind exeshave lessserialc orrelationthanthe b road ind exesb ec ause they are c onstruc ted f
rom
large,highly liquid stoc ks.
¤Iw oul
d l
ike to thankM ic helle Francisofthe J SE f
or her generousand invaluab le assistance
and Xiw angG ao f
or her c apab le research assistanc e.Allrem ainingerrorsare m ine.
1
1
In
trod uc tion
T hedistributionofassetreturns plays animportantroleinderivativevaluation
and in the testing ofasset pricing theories. T he distribution determines the
expected pay-o®s and risks ofderivatives and statisticalinferencerelies on the
assumptions ofcorrectspeci¯cation and normality. T o date, research on the
distribution ofreturns has focused on the developed economies because ofthe
easy access to data. In particular, the U S asset returns have been studied
intensivelyduetotheavailabilityoftheCenterforR esearch in SecurityP rices
(CR SP )and Compustatdatabases.R ecently,therehas been agreaterinterest
in emerging markets and the emerging A sian markets have received similar
scrutiny.
T he key empiricalregularity orstylized factthathas emerged from these
studies is that the unconditional distribution of asset returns is non-normal
(P agan (1 996)).M ills (1 999)documents thefollowingcharacteristics:¯rst,the
distribution ofassetreturns has fattertails and is more peaked than the normaldistribution and thus is leptokurtic.Second,thedistribution is negatively
skewed and the skewness only appears in the tails.T hird, the centralpartof
thedistribution is symmetricand morelike astable process.Fourth,thetails
donothavethe shape ofastabledistribution,butareconsistentwith a¯nite
varianceprocess.Finally,returns havea¯nitevariance,buttheunconditional
varianceand covarianceis notconstant,i.
e.covariancestationarityis rejected.
T he objective ofthis paperis toanalyze the unconditionaldistribution of
stock returns on the Johannesburg Stock Exchange (JSE).In particular, we
focus on the returns ofthe A llShare, G old, Industrialand FinancialIndexes
thatre° ectthe broadersectors ofthe JSE and the fourfutures indexes that
mirrorthe broad indexes.T he outline ofthe paperis as follows.In section 2
we discuss the unique characteristics ofthe JSE thatmakes this topic worth
pursuing and section 3 covers the main institutional features of the market.
Section 4 contains the empiricalmethodology,application and results,and we
concludewith section 5.
2
W hy are South Af
ric anreturnsofin
terest?
T odatetherehas beenlittlepublishedresearchonSouthA fricanassetreturns.
In the ¯nance literature, most of the research has focused on the developed
economies forthesimplereasonthatdatais readilyavailable.Inthecaseofthe
U S,research has focused on theCR SP and Compustatdatabases,butthereis
always anaggingconcernthatdetectedanomalies arespurious andduetodata
mining (L akonishok and Smidt (1 988)). In the contextoftests of e±ciency,
researchers areconcernedaboutT ypeI errors,i.
e.incorrectlyrejectingthenull
ofmarkete±ciency,andtheseareboundtooccurwhenthesamedatasetis used
repeatedly.O ne solution tothis problem is to ¯nd newdatasets.D uringthe
1 990 's,theemergingmarkets providedatreasuretroveofnewdataanditis for
this reason thatweturn totheSouth A frican stockmarket,theJohannesburg
2
StockExchange(JSE).
T he JSE is alsoofinterestforotherreasons.First, itis a large and welldeveloped marketrelative to otheremerging markets. A tthe end of1 998, it
was ranked20 'thinterms ofmarketcapitalizationintheworld(SalomonSmith
B arneyG uidetoW orldEquityM arkets(1 999)).Second,theJSEhashadstrong
tieswithinternationalmarketsduetoitsendowmentofnaturalresources.South
A frica was the primary producerofgold duringthe twentieth century and its
goldmines weretheprimaryvehicleforinvestinginthisindustry.Infact,South
A frican mining companies such as D e B eers, ER P M and D urban D eep were
1
listed on theL ondon StockExchangein 1 950 .
T hird,politicalriskhasplayedaveryimportantroleinthehistoryoftheJSE
since1 960 .From January24,1 979 toFebruary7,1 983 and from September2,
1 985 toM arch1 0 ,1 995 adualexchangeratesystem was introducedtoinsulate
thetradebalancefrom politicalrisk.T hedualexchangeratesystem consisted
ofa commercialexchange rate - a managed exchange rate thatwas used for
2
trade ° ows - and a ¯nancialexchange rate that was used forcapital° ows.
T he discount of the ¯nancial rate relative to the commercial rate was used
as a measure ofpoliticalrisk because itprovided a directmeasure offoreign
con¯denceintheSouth A frican economy.
T hedecadeofthe1 990 '
s has seentremendous politicalchangeand¯nancial
liberalization. T he exchange rate system was uni¯ed on M arch 1 0 , 1 995, the
¯rstfreeelections wereheld in A pril1 994,exchangecontrols on South A frican
residentswererelaxedon1 1 M arch1 998,afuturesexchangewasintroducedand
theJSE introducedanelectronictradingsystem onJune1 0 ,1 996.Inaddition,
an electronicsettlementsystem willbeintroduced in theyear20 0 0 .
Finally,theSouthA fricaneconomyhas experiencedaveryturbulentperiod
during the latter half ofthe past century. Following the oil price shocks in
the 70 's, itexperienced a boom in the early 80 's when the gold price soared
to over U S$80 0 per ounce. T he boom was short-lived and was followed by
foreigndisinvestmentaspoliticalproblemssurfaced.D uringthisturbulenttime,
South A fricaexperienced double digitin° ation and itwas only broughtunder
controlintheearly1 990 '
swhenthegovernoroftheSouthA fricanR eserveB ank
madeaconcerted e®orttorestrain moneysupplygrowth.In thedecadeofthe
1 990 's South A frican in° ation averaged 1 0 % and the economic environment
has stabilized. W ith this backdrop ofpoliticaland socioeconomic events, we
investigatethedistribution ofreturns ofthemajorindexes on JSE.
3 T he J SE
T he JSE is the oldest stock market in A frica and dates from N ovember 8,
1 887. Itis the mostwell-developed ofthe A frican Stock Exchanges and has
thelargestmarketcapitalization.A ttheendof1 998 themarketcapitalization
was U S$ 1 51 billionandwas rankedthe20 '
thlargeststockmarketintheworld.
1 Dan
ielSeatonofthe
2
Lond onStock E xc hange provid ed m e w ith thisinf
orm ation.
G arner (19 94 ) provid esanin-d epth stud y ofthe ¯nancialexchange rate.
3
R elativetootheremergingmarkets,itsmarketcapitalizationis9.
1 % ofthetotal
marketcapitalizationofallemergingmarkets.T heJSE isdividedintotheM ain
B oard, D evelopmentCapitalM arketand the V enture CapitalM arket.T here
are42 sectors on theJSE with thelargestbeingtheIndustrial,M iningH ouses
and Financialsectors with 1 7%, 1 4% and 8% ofthe marketcapitalization in
3
1 997,respectively.
A ttheendof1 998,therewere642 domesticand 26 foreign
companies listed on theJSE.
T he twobiggestdrawbacks ofthe JSE have been the lowlevelofliquidity
and the high level ofconcentration. In 1 998, the turnoverrate - de¯ned as
thetotalvaluetraded divided bytotalmarketcapitalization -was only30 .
2%.
A lthoughthisislowbyinternationalstandards,itisaconsiderableimprovement
overthe ¯rsthalfofthe 1 990 '
s.T he average turnoverrate from 1 990 to1 995
was 6.
4%,butithas increaseddramaticallysinceJune1 0 ,1 996 whenelectronic
trading was introduced. T urning to the high level of concentration, the 20
largest¯rms makeup 46% ofthetotalmarketcapitalizationandthetenlargest
companies make up 30 .
2% oftotalmarketcapitalization.M any ofthe largest
companies are characterized by pyramid structures thatare designed to keep
voting powerto a close-knitfamily group (B arr, G erson and Kantor(1 995)).
O ther companies were forced to become conglomerates within South A frica
because they were prevented from investing overseas afterthe dualexchange
ratesystem was introduced in 1 985.H owever,sincethechangein thepolitical
regimein 1 994,manyoftheconglomerates havebeen unbundled.
P riorto1 996,tradingwas carried outas an open outcryon atrading° oor
with ¯xed commissions. From M arch 1 , 1 996, the JSE phased in automated
trading and market sectors migrated from ° oortrading to electronic trading
overa 3-month period.O n June 7, 1 996 the ¯nalbellwas rung to end ° oor
tradingandM onday1 0 ,1 996 sawalltradingbeingconductedontheJohannesburgEquities T rading(JET )System.T he JET marketis a continuous orderdriven marketon a time priority basis with centralmarketprinciples.A dual
tradingcapacitywas alsointroduced,complementedby¯rms voluntarilyacting
as marketmakers.A specialistmanages an electronicorderbook forodd lots
againstwhich incomingorders automaticallytrade.M arketmakers voluntarily
quoteprices and abookfortradingorders with specialterms is provided.
T he removal of ¯xed brokerage commissions and the introduction of negotiated brokerage commissions4 coincided with the introduction ofelectronic
trading.Commissions are charged atan agreed ratein an agency transaction,
butmay notbe charged when the ¯rm is acting as principal. T rading hours
are from 9:30 am to4:0 0 pm M onday toFridays excludingpublicholidays (the
internationaltimezoneis G M T + 2).W iththeintroductionoftheJET system
a pre-opening period from 0 8:1 5am to 0 9:30 am and after-hours trading from
4:0 0 pm to 6:0 0 pm was introduced. O n September1 5, 1 999 the pre-opening
period was reduced to20 minutes and began at8:40 am.D uetoan increasein
3T he prim aryref
erencesf
or inf
orm ationinthissec tionare the U B S guid e to E m ergingm arkets,Clark(19 98),e-m ailc orrespond ance w ith M ic helle Francisat the J SE and c on
versations
w ith ind ivid ualsinthe South Af
ric ansec uritiesind ustry.
4 T he sc hed ul
e of¯xed b rokerage c om m isionsisavailab le f
rom G eorge (199 4 ,p.
2 62 ).
4
tradingvolume,theJSE announced on January 21 ,20 0 0 thatthepre-opening
period willbeextended and begin at8:25am.
A lltradingsettlements between members on theJSE areoperated through
the JSE clearinghouse in terms ofa¯xed settlementperiod,generally on the
following T uesday orthere-after. Settlementofprivate trades mustbe done
within seven tradingdays from thedayofthedeal.Settlementis physicaland
thereis nocentraldepository,however,theJSE intends tointroduceelectronic
settlementin theyear20 0 0 .
M ost shares are ordinary voting shares although there are also A and B
shares, and \ N " shares which have limited voting rights. T here are also a
limited numberofpreferenceshares and debentures.
Foreigninvestmentis encouragedandis notscreened.Infact,foreigners are
allowed 1 0 0 % ownership in South A frica. South A frican residents have been
constrained from investingoverseas byexchangecontrols,butthis was recently
relaxed on M arch 1 1 , 1 998. M any South A frican companies thatare seeking
greateraccess tocapitalhavesoughtlistingon theL SE orlisted A D R '
s in the
U S.W iththedemutualisationandlistingofO ldM utual-alargelifeassurance
companywith largeindustrialholdings - on theL ondon StockExchangethere
is concern thatthe JSE willbemarginalised as South A frican companies gain
access toglobalcapitalmarkets.
T herearenocapitalgains taxes,butthereis amarketablesecurities taxof
0.
5% thatispayableforeverypurchasethroughtheagencyoforfrom amember
oftheJSE forresidents ofSouthA frica,N amibia,L esothoandSwaziland.V A T
is payable on brokerage commissions. D ividends and interest payments are
exemptfrom normalorwithholdingtaxes.
T here is only one stock exchange in South A frica although the Stock ExchangeControlA ctof1 985 does allowfortheexistenceand operation ofmore
thanoneexchange.EachyeartheJSEmustapplytotheM inisterofFinancefor
an operation licence which vests externalcontroloftheexchange in theSouth
A frican FinancialServices B oard.
4
4.
1
E m piric alM ethod ol
ogy and R esul
ts
Data
W e focus on the broad marketindex and three broad sectorindexes thatrepresentthelargestpercentage ofmarketcapitalization,i.
e.theA llShareIndex
(A L SI),theIndustrialIndex(IN D I),theG oldIndex(G L D I)andtheFinancial
Index(FIN I).T hedailydatafortheA L SI,IN D I,G L D I andFIN I indexes were
obtained from D atastream forthe period January 1 988 toD ecember1 999.In
1 995 theJSE introduced indexes thatarebased on thelargerand morehighly
traded companies in each ofthese sectors forthe purpose ofbeingthe underlyingindex forfutures contracts on the South A frican Futures Exchange.W e
focus on the A L SI40 , G L D I1 0 , IN D I25 and FIN D I30 indexes thatmirrorthe
fourbroad indexes where the numberat the end ofeach index refers to the
5
numberofstocks thatis used in the construction ofthe index and FIN D I denotes theFinancialand IndustrialIndex.A lthough thefutures contractbased
on theG L D I1 0 indexwas discontinued in 1 7September1 998 andwas replaced
byafutures contractontheR esources Indexon 27February1 998,theindexis
stillcalculated by theJSE.T heFIN D I30 and FIN I1 5 indexes wereintroduced
on 2 O ctober1 995 and 27 February 1 998, respectively. W e use the FIN D I30
index as aproxyforthebroaderFinancialIndex becauseofthesamplesizeof
theFIN I1 5 indexis toosmall.M ichelleFrancis oftheJSE kindlyprovided the
data forthe indexes forthe period 1 5 June 1 995 to 31 D ecember1 999. T he
indexes are value-weighted and are adjusted formergers, de-listings, but not
5
dividends.
R eturns are computed as the naturallogofthe price relative,i.
e.
R t ´1 0 0 ¢
ln(P t=P t¡1 )whereP t is theindexattimet.
4.
2
Desc riptive Statistic s
T hepurposeofthissectionistocomparethedescriptivestatisticsoftheindexes,
to look for deviations from normality in the unconditional distribution and
to determine the amount of dependence in the ¯rst and second conditional
moments.T hedescriptivestatistics arereportedinT able1 andincludetests of
skewness and excess kurtosis based on theG M M estimatorand thetraditional
estimatorthatis derivedundertheassumption ofiid returns.W eincludetests
basedontheG M M estimatorbecausethetraditionalestimatorsu®ers from the
drawbackthatits varianceis underestimatedwhenreturns arenon-normaland
conditionallyheteroskedastic(P agan (1 996)).
T he mean returns ofallthe broad indexes are statistically di®erentfrom
zero.T here are a numberofpossible reasons forthis: ¯rst, we are analyzing
returns de¯ned as log price-relatives and not excess returns so the non-zero
mean may re° ectthe riskless rate ofreturn undera no-arbitrage equilibrium.
Second, during the sample period, the rate ofin° ation and the risk-free rate
werein excess of1 0 % perannum and this maybere° ected in therawreturns.
Finally,thestatisticalsigni¯cancemaybeastatisticalartifactas aresultofthe
largesamplesizeof2993 observations.
T urningtothe second moment,theG old indexis approximately¯vetimes
morevolatilethantheotherthreeindexes.T hedi®erenceinvolatilityisevident
inFigure1 whereweplotthetimeseriesofreturnsforthefourbroadindexes on
axeswiththesamescale.T heG oldindexalsodisplaysitsuniquenessinthethird
moment:in contrasttotheotherindexes,theG old indexis positivelyskewed.
T heimportanceoftestingforskewness and kurtosis usingtheG M M estimator
is apparentfrom thehugedi®erenceinthemagnitudeoftheteststatistics.For
example,theA llShareindexisnegativelyskewedandhasat-statisticof-28.
41 1
undertheassumptionofiidreturnswhereastheteststatisticisonly-1 .
997when
the assumption the iid assumption is relaxed.A llfourindexes display excess
kurtosis relativetothenormaldistributionthatis statisticallysigni¯cantatthe
1 % level.
5Detail
ed inf
orm ationab out the ind exc onstruc tionisavailab le at the w eb site ofthe J SE ,
http://w w w .
jse.
c o.
za/.
6
T urningtothefourfutures indexes,themeans arenotstatisticallydi®erent
from zero and the standard deviation ofthe G L D I1 0 index is approximately
twiceas largeas thatoftheotherthreeindexes.A lthoughtheA L SI40 ,IN D I25
and FIN D I30 indexes have negative coe±cients ofskewness, none ofthem is
statistically signi¯cant.O n the otherhand, the returns ofthe G L D I1 0 index
are positively skewed and signi¯cantatthe 1 % level. A llfourfutures indexes
displayexcess kurtosis thatis signi¯cantatthe1 % level.
T he JSE is known forits low turnoverratio and thin trading and we expectthis to be evidentin the autocorrelations reported in T able 2.T he ¯rst
autocorrelation forthe fourbroad indexes is statistically signi¯cantalthough
this may be due toconditionalheteroskedasticity (D iebold (1 986)).T he ¯rst
autocorrelationoftheFinancialindexis thelargestofalltheindexes becauseit
contains stocks thatsu®ermostfrom thintrading.T heserialcorrelationinthe
FinancialIndexis notonlypresentatthe¯rstlag,butis alsosigni¯cantatthe
second and ¯fth lags.T heG old indexis leastlikelytosu®erfrom thin trading
because the gold stocks are highly traded by domestic and foreign investors.
T heL jung-B oxQ -statistics con¯rm thepresenceofserialcorrelation in allthe
indexes and are statistically signi¯cantatthe 1 % levelatboth 5 and 1 0 lags.
T he Q -statistics ofthe squared returns are also statistically signi¯cantatthe
1 % levelatboth 5 and 1 0 lags.T hepresenceofconditionalheteroskedasticity
is consistentwith theexcess kurtosis in returns.
T hefutures indexes displayasimilarpatternandhaveastatisticallysigni¯cantautocorrelationatthe¯rstlagalthoughitis theIN D I25 indexthathas the
lowestlevelofserialcorrelation.ExceptfortheG L D I1 0 index,themagnitude
ofthe autocorrelation coe±cients is smallerthan those ofthe broaderindexes
becausethe futures indexes areconstructed from asmallersubsetofmore liquid stocks.T he Q -statistics fortheA L SI40 ,G L D I1 0 and FIN D 30 indexes are
signi¯cantatthe 1 % levelatboth 5 and 1 0 lags, butthe Q -statistic forthe
IN D I25 index is onlysigni¯cantat5 lags atthe1 0 % levelofsigni¯cance.T he
Q -statisticsofthesquaredreturnsarealsohighlysigni¯cantandconsistentwith
thepresenceofconditionalheteroskedasticity.
B orrowingfrom P aganandKearns(1 993),welistthe¯velargestpositiveand
negativereturns forthe broad indexes in T able3.W hereas P agan and Kearns
were looking for changes in volatility across a 1 0 0 hundred year time span,
weare lookingforany correspondence between largechanges and thepolitical
events listed in A ppendix 1 .N oneofthelargechanges can beassociated with
apoliticalevent.In fact,mostofthechanges aretied totheemergingmarkets
crises that followed the A sian Crisis in 1 997 and the R ussian crisis in 1 998.
T heearliesteventinO ctober1 989 is related toaprecipitous declinein theU S
marketthatwas formented byportfolioinsurance(Jacobs (20 0 0 ),p.20 7-222).
O nce again the gold index displays its uniqueness and approximately 50 % of
thelargereturns correspondtomarketwideevents ontheJSE;theotherevents
particularlyin 1 993 areuniquetothegold market.
In conclusion, the gold sectoris di®erentto the othersectors in thatitis
morevolatileandpositivelyskewed.A lltheindexes displayexcesskurtosisthat
may be due to the presence ofconditionalheteroskedasticity orleptokurtosis
7
in the underlying distribution. T here is signi¯cant time dependence in the
returns of the broad and futures indexes especially at the ¯rst lag and the
serialcorrelationappears toberelatedtothintrading.T hereis alsosigni¯cant
serialcorrelationinthesquares ofthereturns thatis consistentwithconditional
heteroskedasticity.
4.
3 F inite or In¯nite Variance
T he discovery of leptokurtosis in the distributions of returns prompted the
searchforamorerealisticdescription ofthedistribution.A mongothers,M andelbrot(1 963a,b)proposedthestabledistribution,B lattbergandG onedes(1 974)
proposed the Student-tdistribution,and Kon (1 984)proposed adiscrete mixtureofnormals.T heclass ofsymmetricstabledistributions is ofmostinterest
becauseitnotonlynests thenormaldistribution,butalsoincludes in¯nitevariancedistributions.In fact,the stable distribution arises from ageneralization
ofthecentrallimittheorem,i.
e.ifthelimitingdistributionofasuitablyscaled
sum ofindependentandidenticallydistributedvariables exists,thenitmustbe
amemberofthestableclass,eveniftherandom variableshavein¯nitevariance.
T he stableclass has anotherdistinctiveadvantageovermany ofthesuggested
alternatives, it has a stability-under-addition property that is relevant in ¯nance:ifweeklyreturns arethesum ofdailyreturns,thenweeklyreturns have
6
theastabledistribution with thesametailindexas thedailyreturns.
T he symmetric stable distribution is characterized by two parameters - a
scale factorand acharacteristicexponentsometimes referred toas thetailindex.Itisthecharacteristicexponentthatdeterminesthefatnessofthetailsand
distinguishes itfrom thenormaldistribution.Forexample,whenthecharacteristicexponentequalstwo,thedistributionisnormalandwhenthecharacteristic
exponentequals one,thedistributionis Cauchy.W henthecharacteristicexponentis less than two,thevarianceofthedistribution is in¯nite.
L oretanandP hillips (1 994)de¯nethetailbehaviorofarandom variableX
as follows:
P (X > x) = C ³x¡³ (1 + ³R (x)); x> 0
P (X < ¡x) = C ³x¡³ (1 + ³L (x)); x> 0
where³i ! 0 fori = R (righttail);L (lefttail)as x! 1 ;³ is thecharacteristic
exponentand C is the parameterofscale dispersion. T he advantage ofthis
de¯nition ofthe tailbehavioris thatitcan be used to estimate C and ³ for
distributions otherthan thestabledist
¯ribut
¯ion.Fornon-stabledistributions,³
is themaximal¯niteexponent,i.
e.E ¯X k¯< 1 for0 ·k< ³.T his de¯nition
of tail behavior can also be used to distinguish between an in¯nite variance
(³ < 2)and a¯nitevarianceprocess (³ ¸2).IfX has astable distribution,³
is the characteristicexponentand ifX has a Student-tdistribution, ³ de¯nes
6T he stab l
e d istrib utionm ay al
so b e ab le to explainvol
atil
ity c lustering, b ut since this
isrelated to the c ond itionald istrib utionw e d o not investigate thisaspec t inthispaper (De
Vries(19 91)).
8
thedegrees offreedom.T hecharacteristicexponent,³;can beestimated using
orderstatistics.
Forthetimeseries ofreturns,fR tgTt=1 wede¯netheorderstatistics,R (1 )·
R (2) ·:::·R (T ).T heestimatorof³ fortheuppertailis de¯ned as
0
b
³ = @ s¡1
Xs
j=1
1 ¡1
lnR (T ¡j+ 1 )¡lnR (T ¡s)A
wheres= s(T )isselectedsuchthats(T )! 1 asT ! 1 .P hillipsandL oretan
suggestachoiceofsthatminimizestheM SEofb
³ bysettings(T )= ¸T 2=3 where
¸ is estimatedadaptivelyby
¯Ã
¯2=3
! µ ¶
¯ b
T ³b b ´¯
¯ ³1
¯
¸ = ¯ 1 =2
³1 ¡³2 ¯
¯ 2
¯
s2
and b
³1 and b
³2 are estimates from the truncations s1 = [T ¾ ]and s2 = [T ¿ ]
for¾ = 0 :6 and ¿ = 0 :9.T he estimate of³ forthe lowertailis obtained by
multiplyingtheorderstatistics by¡1 and repeatingthecalculations.
T heasymptoticdistribution ofb
³ is obtained from H all(1 982),
´
³
s1 =2 b
³ ¡³ »N (0 ;³2 )
andwecantestthehypothesis ofinterest,Ho :³ < 2 (in¯nitevariance)against
thealternativeH1 :³ ¸2 (¯nitevariance).
W e report the results of the hypothesis tests in T able 4A for the broad
indexes and T able 4B forthe futures indexes. W e can reject the nullof an
in¯nitevarianceforbothtails oftheA llShareandG oldIndexes,andtheupper
tailofthe IndustrialIndex, but we fail to reject the nullhypothesis forthe
FinancialIndex.T his resultis surprisingandmaybeduetothevolatilityafter
theA sian crisis thatonlya®ected thesectors thatareinvolved in international
trade. In T able 4b, there is overwhelming evidence in favor of the in¯nite
variancehypothesisintheFuturesindexes.W ecanonlyrejectthenullofin¯nite
variancefortheuppertailoftheA L SI40 andthelowertailoftheG L D I1 0 .T his
resultiscountertoourexpectationsbecausethefutures indexes areconstructed
from moreliquidstocks.T hreepossibleexplanations cometomind:¯rst,these
indexes onlybegin in1 995 and alargepartoftheirtailbehavioris determined
bytheA siancrisis.Second,thesamplesizemaysimplybetoosmalltoestimate
the tailindex precisely. T hird, the introduction ofelectronic trading in 1 996
and the marked increase in volume oftradingmay be su±cientevidence ofa
structuralchangein themarket.
4.
4
Covariance Stationarity
T he trade-o®between risk and return is fundamentalto¯nance and untilthe
developmentofportfoliotheory, the variance was regarded as the appropriate
9
measureofrisk.W iththeintroductionofportfoliotheoryandthecapitalasset
pricingmodel,moreinteresthasfocusedonboththevarianceandcovariancesof
returns.T hesimplifyingassumption thattheseparameters areconstantacross
timeis consistentwith thetimeseries conceptofcovariancestationarityand is
extremely convenientforempiricalanalysis, butis notimplied by the theory.
In factwe have noreason tobelieve thatthe second moments are constantif
information revelation reveals structuralchanges in risk-return relationships.
InthespiritofCU SU M test,P aganandSchwert(1 990 )proposethefollowing
teststatisticforcovariancestationarity,
à (r)=
where0 < r< 1 ,
¹
b2;T
v2
=
[T r]
¢
1 X ¡ 2
R j ¡b
¹ 2;T
Tb
v j=1
T
1 X 2
R ;
T j=1 j
= °0 + 2
¶
Xl µ
j
° ;j
1 ¡
l+ 1
j=1
°b0 ;:::;b
° larethecovariances oftheseries andv2 isakernelestimateofthe`longrun'varianceoftheseries.T hedistributionofthestatisticdependscriticallyon
thetailindexofthedistribution ofR t.For³ > 4 and T ! 1 ,Ã (r)converges
toaB rownianB ridge,butfor³ < 4,Ã (r)convergestoastandardizedtieddown
stable process.L oretan and P hillips (1 994)provide the criticalvalues forthe
supà (r)and infà (r)statistics.T he teststatistics forthe fourbroad indexes
andthefourfutures indexes arereportedinT able5 andweplotthetimeseries
ofthepartialsum process,Ã (r),foreach ofthebroad indexes in Figure2.W e
rejectthe nullofcovariance stationarity forallthe indexes atthe 1 % levelof
signi¯cance.From Figures 1 A to1 D itappears thatthe rejection ofthe null
is duetothehighlyvolatileperiod thatwas precipitated bytheA sian crisis in
1 997.
4.
5 Af
urther l
ook at Skew ness
Earlierwe found thatexceptforthereturns ofthe G old Index, the returns of
thebroadindexes areskewedtotheleft.Incontrast,thereturns ofthefutures
indexes aresymmetric.Inthis sectionwetakeacloserlookattheskewness and
whereitmanifests itselfinthedistributionofthebroadindexes.FollowingM ills
(1 999),ifthedistributionissymmetric,theorderstatistics,X (p)andX (T ¡p)for
p < [T =2]areequidistantfrom themedian,i.
e.aplotoftheupper-tailstatistics
againstthelower-tailstatistics shouldlieonalinewithslope-1 .InFigures 4A
to4D the positive skewness ofthe G old index versus thenegativeskewness of
the otherthree broad indexes is immediately evidentforp = 50 0 .
.H owever,
10
contrary toM ills (1 999)results, we ¯nd thatthe tails and centralpartofthe
distribution are skewed because the lowertaillies uniformly above the upper
tailforthe G old Index and uniformly belowthe uppertailforthe A llShare,
Industrialand FinancialIndexes.
5 Concl
usion
W e conclude thatthe empiricalregularities found in many industrialized and
emergingmarkets arepresenton theJohannesburgStockExchange.In particular,thebroadJSEindexes areskewedandhighlyleptokurtic.W hereas theA ll
Share,IndustrialandFinancialIndexesarenegativelyskewed,theG oldIndexis
positivelyskewed.T heskewness does notonlyexistinthetails,butalsointhe
centralpartofthedistribution.Inaddition,noneoftheseindexes is covariance
stationarity overthe sample period;this may be due tostructuralchanges in
themarketsuchas theintroductionofanelectronictradingsystem in1 996 and
thevolatilityintroduced bytheA sian crisis.
Forthefutures indexes,we¯ndthatonlytheG L D I1 0 indexis characterized
byskewness,positiveskewness,and theotherdistributions aresymmetric.A ll
the indexes have excess kurtosis and none of them is covariance stationary.
Finally, the futures indexes haveless serialcorrelation than thebroad indexes
becausetheyareconstructed from highlyliquid largestocks.
G iven thepoliticalturmoilsurround South A fricaduringtheinitialpartof
sampleperiodandtheA siancrisis duringthelatterpartofthesampleperiodit
isnotsurprisingto¯ndthattheassumptionofcovariancestationarityisrejected
in theindexes.T heseresults alsosuggestavenues forfurtherresearch,namely,
themodellingoftheconditionaldistributionofreturns.Inparticular,itwillbe
interestingtodeterminewhatproportionofthevariationinreturns ontheJSE
was duetoinformation ° ows from theN Y SE orotheremergingmarkets.
11
R ef
erences
[1 ]B adrinath, S.
G .and Sangit Chatterjee (1 988)\ O n M easuring Skewness
and Elongation in Common Stock R eturn D istributions:T he Case ofthe
M arketIndex,"JournalofB usiness,61 ,4,451 -472.
[2]B arr, G raham, Jos G erson and B rian Kantor (1 995) \ Shareholders as
A gents andP rincipals:T heCaseforSouthA frica'
s CorporateG overnance
System,"JournalofA ppliedCorporate Finance,8,1 ,1 8-31 .
[3]B lattberg, R obert and N icholas G onedes (1 974)\ A Comparison of the
Stable and StudentD istribution as StatisticalM odels forStock P rices,"
JournalofB usiness,47,A pril,244-280 .
[4]B rooks, R obert D .
, Sinclair D avidson, and R obert W .Fa® (1 997)\ A n
Examination of the E®ects of M ajor P olitical Change on Stock M arket
V olatility:theSouthA fricanExperience,"JournalofInternationalFinancialM arkets,Institutions andM oney,7,255-275.
[5]Clark, R obertA .(1 998)A frica'
s Emerging Securities M arkets, Q uorum,
W estport,Conn.
[6]Cowitt,P hilipP.(1 996)1 990 -1 993 W orldCurrencyY earbook,International
CurrencyA nalysis,Inc.
,B rooklyn,N Y .
[7]D eV ries,C.
G .(1 991 )\ O n theR elationbetweenG A R CH and StableP rocesses,"JournalofEconometrics,48,31 3-324.
[8]D iebold,Francis X .(1 986),\ T estingforSerialCorrelation in theP resence
ofA R CH ,"P roceedings oftheA merican StatisticalA ssociation,B usiness
andEconomics Statistics Section,p.
323-328.
[9]T he D owJones G uide tothe W orld Stock M arket, (1 998)P rentice H all,
Englewood,N J.
[1 0 ]Fama,E.(1 965)\ T heB ehaviorofStockM arketP rices,"JournalofB usiness,38,January,34-1 0 5.
[1 1 ]G arner,Jonathan(1 994)\ A nA nalysis oftheFinancialR andM echanism,"
CenterforR esearch in Economics and Finance in South A frica, L ondon
SchoolofEconomics,R esearch P aperno.
9.
[1 2]G eorge,R obertL loyd (1 994)T he H andbookofEmergingM arkets,P robus
P ublishing,Cambridge,England.
[1 3]H all, P.(1 982)\ O n Some Simple Estimates of an Exponent ofR egular
V ariation,"Journalofthe R oyalStatisticalSociety,Series B ,44,37-42.
[1 4]Jacobs,B ruce(20 0 0 )CapitalIdeas andM arketR ealities :O ptionR eplication,InvestorB ehavior,and StockM arketCrashes,B lackwellP ublishers.
12
[1 5]Kantor,B rian(1 998)\ O wnership andControlinSouthA fricaunderB lack
R ule,"JournalofA ppliedCorporate Finance,1 0 ,4,69-78.
[1 6]Kon,StanleyJ.(1 984)\ M odelsofStockR eturns-A Comparison,"Journal
ofFinance,39,1 ,1 47-1 65.
[1 7]L akonishok, J.and S.Smidt (1 988)\ A re Seasonal A nomalies R eal? A
N inetyY earP erspective,"R eviewofFinancialStudies,1 ,40 3-425.
[1 8]L oretan,M .and P.
C.
B P hillips (1 994)\ T estingtheCovarianceStationarityofH eavey-T ailed T imeSeries:A n O verviewoftheT heorywith A pplications to SeveralFinancialD atasets," JournalofEmpiricalFinance, 1 ,
21 1 -248.
[1 9]M andelbrot, B enoit (1 963a) \ N ew M ethods in Statistical Economics,"
JournalofP oliticalEconomy,71 ,421 -440 .
[20 ]M andelbrot,B enoit(1 963a)\ T heV ariationofCertainSpeculativeP rices,"
JournalofB usiness,36,394-41 9.
[21 ]M ills,T erenceC.(1 995)\ M odellingSkewness and Kurtosis in theL ondon
StockExchangeFT -SE IndexR eturn D istributions,"T he Statistician,44,
3,323-332.
[22]M ills, T erence C.(1 999), T he Econometric M odelling ofFinancialT ime
Series,CambridgeU niversityP ress,Cambridge.
[23]P agan,A drianR .(1 996)\ T heEconometricsofFinancialM arkets,"Journal
ofEmpiricalFinance,3,1 5-1 0 2.
[24]P agan,A drianR .and P.Kearns (1 993)\ A ustralianStockM arketV olatility:1 875-1 987,"T he EconomicR ecord,V olume69,20 5,1 63-1 78.
[25]P agan,A drian R .and G .W illiam Schwert(1 990 )\ T estingforCovariance
Stationarityin StockM arketD ata,"Economics L etters,33,1 65-1 70 .
[26]T heU B S G uidetoEmergingM arkets,B loomsbury,L ondon,p.
641 -657.
[27]T heSalomon Smith B arney G uide toW orld EquityM arkets 1 999 (1 999),
R esearchEditor:B rianM uggeridgeA nderson,Editors:JacquelineG rosch
L obo; R ob Irish, Euromoney Institutional Investor P L C and Salomon
Smith B arney,L ondon.
13
A ppendix1
T he U B S G uide to Emerging M arkets (1 997)provides a listofimportant
dates goingbackto circa 30 0 B CE.W e focus on the mostrecentevents that
relevantforthesampleperiod ofourdata.
²1 960 Sharpevillemassacre- A N C and P an-A fricanistcongress banned
²June1 6,1 976 -SowetoU prising
²January24,1 979 -thedualexchangeratewasintroducedandtherandwas
devalued by 1 7.
85% in terms ofgold.T he ¯xed o±cialR ate ofU S$1 .
15
was placed on a controlled, ° oating basis and renamed the Commercial
R and, applicable to foreign trade as wellas authorized capitaltransfers
and currentpayments includingthe remittance ofdividend and interest
payments.A lso, the Securities, R and used by nonresidents forthe purchaseofA fricansecurities withits exchangevaluedeterminedbydemand
and supply, was renamed the FinancialR and, applicable tovirtually all
¯nancialtransactions bynonresidents includingdirectforeigninvestment,
the repatriation ofcapitaland pro¯ts, outward capitaltransfers by residents and emigrants as wellas certain realestateincome.
²1 979 - B lackT radeU nions legalized
²1 983 - T ricamerialparliament- Coloreds and A sians given votingrights
²February7,1 983 thedualexchangerateregimewas abolished
²1 985- StateofEmergencydeclared.Internationalsanctions stepped up
²September2,1 985 thedualexchangeratesystem was reestablished
²1 990 -banonA N C,PA CandSA CommunistP artylifted.N elsonM andela
is freed and the¯rstformaltalks with theA N C arebegun.
²1 993 G overnmentand A N C agreetoshare powerfor¯ve years afterthe
¯rstall-raceelections.U N lifts mostsanctions
²A pril27,1 994 - interim constitution replaces thatofSeptember3,1 984
²M ay1 0 ,1 994 - N elson M andelaInaugurated
²M arch 1 2,1 995 - dualexchangerateregimeremoved.Exchangecontrols
toberelaxed
²M arch1 1 ,1 998 - ExchangeControls on residents R elaxed
²M ay 8, 1 996 - Constitutionalassembly votes 421 to 2 to pass the new
constitution;itwilltakee®ectover3 years
²A pril1 999 - N ewconstitution comes intoe®ect
14
T able1 :D escriptiveStatistics oftheJSE Indexes
B road Indexes
SampleSize
M ean
Standard D eviation
A llShare
G old
Industrial
Financial
2993
0.
0 224c
0.
21 1
2993
-0 .
0 0 6c
1.
0 52
2993
0.
0 267c
0.
1 78
2993
0.
0 326c
0.
226
Skewness
iid t-statistic
G mm t-statistic
-1 .
274
(-28.
441 )c
(-1 .
997)b
0.
562
(1 2.
543)c
(3.
871 )c
-1 .
667
(-37.
20 9)c
(-1 .
781 )a
-1 .
462
(-32.
655)c
(-1 .
90 6)a
Kurtosis
iid t-statistic
G M M t-statistic
1 3.
644
(1 52.
242)c
(2.
726)c
3.
20 1
(35.
70 0 )c
(4.
858)c
22.
345
(249.
347)c
(2.
831 )c
21 .
280
(237.
265)c
(4.
0 82)c
Futures Indexes
A L SI40
G L D I1 0
IN D I25
FIN D I30
n
mean
V ariance
1 1 72
0.
015
0.
362
1 1 72
0.
005
0.
924
Skewness
iid t-statistic
G M M t-statistic
-1 .
20 0
(-1 6.
754)c
(-1 .
243)
0.
597
(8.
339)c
(3.
746)c
Kurtosis
iid t-statistic
G M M t-statistic
1 4.
30 5
(99.
759)c
(2.
1 37)b
2.
61 9
(1 8.
236)c
(6.
21 5)c
1 1 72
0.
014
0.
429
1 0 99
0.
014
0.
475
-0 .
899
-0 .
873
(-1 2.
453)c (-1 1 .
778)c
(-1 .
0 58)
(-1 .
110)
1 2.
427
(86.
656)c
(2.
1 67)b
11.
51 7
(77.
61 8)c
(2.
325)b
Superscripta,b andcdenotestatisticalsigni¯canceatthe1 0 %,5% and1 %
levelofsigni¯cance,respectively.
15
T able2:A utocorrelations oftheJSE Indexes
lag
A llShare G old
Industrial Financial
0.
1 45c
0.
0 70 c
0.
019
-0 .
0 22
0.
001
0.
0 98 c
-0 .
019
0.
008
0.
0 21
0.
006
0.
1 68 c
0.
0 95c
0.
0 25
-0 .
007
0.
0 32a
0.
1 98 c
0.
1 1 6c
0.
0 58
-0 .
003
0.
0 46b
Q R (5)
Q R (1 0 )
80 .
469c
91 .
742c
31 .
663c
40 .
973c
1 1 6.
99c
1 29.
44c
1 67.
1 4c
1 85.
52c
Q R 2 (5)
Q R 2 (1 0 )
438.
75c
459.
64c
276.
45c
378.
35c
40 0 .
26c
434.
0 5c
735.
1 9c
863.
1 7c
1
2
3
4
5
lag
A L SII40
G L D I1 0
IN D I25
FIN D I30
1
2
3
4
5
0.
1 1 3c
0.
0 53c
-0 .
0 22
-0 .
0 73b
-0 .
015
0.
1 51 c
0.
016
0.
008
0.
012
-0 .
0 22
0.
0 67b
0.
011
-0 .
0 44
-0 .
0 39
0.
015
0.
1 33c
0.
0 42
-0 .
0 33
-0 .
0 53c
0.
009
Q R (5)
Q R (1 0 )
25.
346c
27.
644c
27.
926c
34.
40 9c
9.
655a
1 3.
220
25.
721 c
27.
969c
Q R 2 (5)
Q R 2 (1 0 )
30 4.
75c
321 .
67c
275.
42c
397.
96c
284.
66c
31 9.
44c
248.
91 c
277.
0 2c
T he L jung-B ox Q -statistics have a Chi-squared distribution with kdegrees of
freedom wherekis thenumberoflags included inthecalculation ofthestatistic.Fork= 5;thecriticalvalues are9.
24,1 1 .
0 7and 1 5.
0 9 atthe1 0 %,5% and
1 % levelofsigni¯cance, respectively.Fork= 1 0 ;the criticalvalues are 1 5.
99,
1 8.
31 and 23.
21 atthe 1 0 %,5% and 1 % levelofsigni¯cance,respectively.Superscripta,b andcdenotestatisticalsigni¯canceatthe1 0 %,5% and 1 % level
ofsigni¯cance,respectively.
16
T able3:T heFiveL argestand FiveSmallestR eturns oftheB road JSE Indexes
A llShare
R eturn
D ate
G old
R eturn
D ate
Industrial
R eturn
D ate
Financial
R eturn
D ate
SmallestR eturns
-5.
1 47 1 0 /28/97
-4.
863 1 0 /1 6/89
-2.
955 1 1 /0 2/98
-2.
779
8/26/98
-2.
532
8/27/98
-5.
1 69
-3.
838
-3.
688
-3.
372
-3.
353
1 /1 7/91
6/0 2/93
1 0 /1 6/89
9/22/98
3/26/90
-5.
378
-5.
102
-2.
930
-2.
669
-2.
460
1 0 /1 6/89
1 0 /28/97
1 /1 2/98
8/27/98
8/1 9/91
-5.
659 1 0 /28/97
-4.
887 1 0 /1 6/89
-4.
1 85
8/27/98
-3.
285
8/26/98
-3.
1 93
9/1 1 /98
L argestR eturns
2.
90 7
2.
543
2.
1 89
1.
998
1.
984
1 0 /29/97
1 0 /1 7/89
1 0 /1 6/98
1 0 /0 6/99
1 0 /31 /97
6.
652
6.
332
5.
70 2
5.
422
5.
41 5
9/28/99
9/25/98
9/27/99
5/1 3/93
5/1 9/93
3.
009
2.
630
2.
363
2.
0 71
1.
889
1 0 /29/97
1 0 /1 7/89
1 0 /1 6/98
9/22/98
1 /6/99
T able4 A :T heM aximalExponents oftheB road JSE Indexes
A llShare G old
Industrial Financial
U pperT ail
&
s
Z -statistic
p-value
3.
1 71
1 55
4.
599
0.
000
2.
773
21 3
4.
0 68
0.
000
2.
361
1 78
2.
0 40
0.
0 21
1.
963
20 6
-2.
655
0.
60 4
2.
0 81
1 60
0.
495
0.
31 0
1.
937
1 74
-0 .
427
0.
665
L owerT ail
&
s
Z -statistic
p-value
2.
51 4 3.
864
1 40
103
2.
422 4.
896
0.
007 0.
000
17
3.
31 5
3.
101
3.
0 40
2.
695
2.
41 8
1 0 /1 6/98
1 /6/99
1 0 /29/97
9/1 6/98
9/23/98
T able4 B :T hemaximalExponents fortheFutures Indexes
A L SI40
G L D I1 0
IN D I25
FIN D I30
U pperT ail
&
s
Z -statistic
p-value
2.
392
1 26
1.
842
0.
0 32
1.
882
1 76
-0 .
829
0.
796
1.
921
1 71
-0 .
539
0.
70 5
1.
939
1 37
-0 .
491
0.
688
L owerT ail
&
s
Z -statistic
p-value
1.
865
111
-0 .
762
0.
777
2.
688
68
2.
112
0.
017
1.
960
110
-0 .
21 3
0.
585
1.
935
100
-0 .
336
0.
632
B eforecomputingthecharacteristicexponent,thereturnswerepre-whitened
byanA R (1 )model.T hereportedp-valueis forthetestthat&·2,i.
e.returns
haveanin¯nitevariance.T hesamplesizeforcomputingthemaximalexponent
is denoted bys.
18
T able5: T heT estforCovarianceStationarity
B roadIndexes
supà (r)
infà (r)
Futures Indexes
sup à (r)
infà (r)
A llShare G old
1.
446
-4.
858 c
A L SI40
1.
697c
-7.
0 65c
0.
393
-7.
51 6c
G L D I1 0
5.
31 4c
-4.
532c
Industrial Financial
0.
20 9
-6.
790 c
IN D I25
1.
325c
-7.
846c
0.
0 64
-1 0 .
491 c
FIN D I30
1.
980 c
-7.
570 c
Superscripta, b and c denote statisticalsigni¯cance atthe 1 0 %, 5% and 1 %
levelofsigni¯cance,respectively,where thenullhypothesis is thatthe process
is covariancestationary.
19