Random Walk of Security Prices: Empirical Evidence from KSE, LSE, and ISE

Transcription

1 andom Walk of Securiy Prices: Empirical Evidence from KSE, LSE, and ISE Yasir Kamal and Dr. Kashif-Ur-ehman * SZABIST Islamabad, Pakisan Absrac: Previously securiy marke research had been focused mainly on developed economies wih no aenion paid o he securiy markes of developing counries of Souh Eas Asia. In an aemp o fill his gap in he lieraure, his paper conducs an empirical invesigaion of he random walk of securiy prices in Pakisani sock markes. The Augmened Dickey fuller es, Ljung Box Q es, Variance raio es and a non parameric un es has been used for analysis of andom walk of securiy prices. esuls indicae he presence of some predicable elemens, which conradic wih previous sudies on Karachi sock marke. This is because of he difference in number of observaion used in previous sudies and his paricular sudy. To conclude, he Karachi sock exchange and Islamabad sock exchange does show a weak random walk of securiy prices, while Lahore sock exchange show srong random walk of securiy prices. Keywords: andom Walk of Securiy Prices, Efficien Marke hypohesis, ADF, un Tes, Variance raio es, Ljung Box Q-saisics.. INTODUCTION The behavior of securiy prices is one of he affluenly documened works in empirical finance; he mos enduring model used for his purpose is he random walk hypohesis. The andom Walk Hypohesis has an illusrious hisory, wih remarkable inellecual forbears such as Bachelier, Einsein, L'evy, Kolmogorov, and Wiener. eference [] gives firs conribuion o lieraure by using random walk hypohesis for financial markes. I was firmly believed amongs financial empiriciss ha sock marke price should reflec he inrinsic value of underlying asses. Over he pas wo decades ample research work has been underaken o es he efficien marke hypohesis he claim ha a marke in which prices fully reflecs available informaion is an efficien one. I was srongly argued ha here were no opporuniies for invesors o make abnormal profis by exploiing informaion conained in he hisory of fundamenal marke daa. eference [] caegorizes hree forms of marke efficiency: weak, semi-srong and srong. These hree forms differ in erms of he ypes of informaion which are used in developing invesmen sraegies. As [3] poins ou, a sufficien condiion for weak-form efficiency is ha he sock prices flucuae randomly. As a resul; a marke is efficien in he weak form if sock prices follow a random walk process. * Bahria Universiy, Islamabad Invesors in weak-form efficien markes canno expec o find any paerns in he hisorical seuence of sock prices ha would provide insigh ino fuure price movemens and allow hem o earn an abnormal rae of reurn. The securiy prices flucuae randomly if a marke is efficien, and he degree of randomness of securiy prices increases wih an increase in marke efficiency. The mos efficien marke of all is one in which price changes are compleely random and unpredicable. For hese reasons, he andom Walk Hypohesis and is close relaive, he Efficien Markes Hypohesis, have become icons of modern financial economics ha coninue o fuel he imaginaion of academics and invesmen professionals alike. Augmened Dickey Fuller Uni oo Tes and Serially Un- Correlaion Mehod have been used for he deecion of random walk in ime series daa, see, among ohers [4], [5], [6] and [7]. eference [8] shows ha common sock prices have properies analogous o he movemen of molecules. Osborne applies he mehods of saisical mechanics o he sock marke, wih a deailed analysis of sock price flucuaions from he poin of view of a physicis. Many research works on sock behavior sugges ha he expeced value of speculaive sraegies should be zero. In an efficien marke he sock prices would reflec all he available informaion and as a resul of differen favorable and unfavorable news he sock prices varies differenly. These differen variaions are ermed as random movemen and in economeric erminology i is called random walk of securiy prices. The purpose of his paper is o invesigae evidence in suppor of random walk hypohesis in he Pakisani sock markes (LSE, ISE and KSE using hree echniues: he run es, Augmened Dickey fuller es for uni roo, Variance raio es and Ljung Box Q es. Of he four ess used, hree examine iner emporal srucure of sock reurns and one examines uni roo saionariy in he price revelaion process. As [9] poins ou, if he evidence fails o suppor weak-form ess here is no reason o examine semi-srong (and/or srong forms before declaring he marke is inefficien on he evidence. The significance of random walk hypohesis: The significance of he random walk heory can be explained boh pracically and heoreically. From pracical poin of view canvasser of he hypohesis deals wih he absence or presence of sysemaic elemens in heir empirical sudies, and he deecion of some nonrandom componens is no evidence agains he WH unless i can be shown ha Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006 7

2 sysemaic endencies in pas price behavior alone presen unexploied opporuniies o make above normal profis.. EVIEW OF LITEATUE Believer of he efficien-markes concep also end o espouse he concep of he random walk ha he marke behaves in discernible way. The advocaes of random walk holds ha i is impossible o predic he prices of a securiy from he pas performance because changes in economic condiion, securiies valuaions, corporae profis and marke as a whole all occur in a myriad of differen ways. In random walk process, successive sock reurns mus be idenically disribued and independen so ha he correlaion beween one period s reurn and he immediae following period is zero, [9], [0], and []. So he echnical analys would exploi any non-random flucuaion or speculaors would buy before an expeced rise in price or sell shor before an expeced fall in price. In random walk he flow of informaion is random, and if securiy prices adjused wih ha informaion so he new securiy prices would also be randomly auned, and hence each day securiies have differen prices depending on he flow of informaion. And no body can predic abou he fuure securiy prices. Wall sree journal has differen hinking on he subjec. I believes ha pas sock prices do show foreseeable rends and ha i is possible o forecas he marke based on pas performance. So by aking his issue several sudies have uncovered empirical evidence, which suggess ha sock reurns conained predicable componens (e.g. see [], [3] and [4]. Despie hese saggering findings heir resuls of failing o rejec he random walk applies only o he sock markes of indusrialized naions. I is of grea ineres o explore if he similar paerns can be idenified in he Asian sock markes. eference [5] repors ha sock reurns of emerging counries are highly predicable and have low correlaions wih sock reurns of developed counries. He suggesed ha emerging markes are less efficien han developed markes and ha higher reurn and lower risk can be obained by incorporaing emerging marke socks in invesor s porfolios. eference [6] used variance raio es o find ou random walk for Asian sock marke and rejeced random walk for all he observed Asian sock markes. eference [7] findings suggesed ha random walk hypohesis for London sock marke is rejeced bu he weak form of marke efficiency can no be rejeced. eference [8] findings sugges ha he monhly sock price for he Taiwan sock marke exhibis weakform efficiency. eference [9] used variance raio es and rejec he random walk hypohesis for Lain American sock markes, however run es indicae ha Lain American euiy markes are weak-form efficien. According o [0] Thus an accurae saemen of he narrow form of he random-walk hypohesis goes as follows: he hisory of sock price movemens conains no useful informaion ha will enable an invesor consisenly o ou perform a buy-and-hold sraegy in managing a porfolio. According o [], [5] and [] mos of he research has conduced on sock marke of developed counries and less aenion has been paid o emerging sock markes, so his sudy is a sep oward research on emerging sock markes. esearchers have consensus ha he power of variance raio es is superfluous han any oher es used for random walk [0]. eference [3] derived he nonoverlapping V (NV saisic, which follows a Bea disribuion. As argued by [0], he OV es is expeced o have higher power han he NV es. The advanages of he V es are summarized by [4]. The applicaion of he Variance aio es o measure he ime series daa was observed by [5], [6] and [7]. For Karachi sock marke [8] observed he random walk of securiy prices raded in KSE 00 index. [9] ook he even of 9/ and check ou he presence of predicable elemen in sock reurn, his finding suggesed ha before 9/ he predicable elemen exiss in sock reurns (e.g. week day effec bu he securiy prices also followed a random walk. The sudy abou he marke efficiency of Karachi sock marke [30] suggesed ha afer 9/, securiy prices followed a random walk and no calendar relaed anomalies exis in KSE 00 index. 3. DATA & METHODOLOGY 3. Daa Daily sock index daa was aken for Karachi sock exchange, Lahore sock exchange, and Islamabad sock exchange from Business recorder, which is a well known and reliable source of business indicaors in Pakisan. The daily index for all hree markes consiss of five year daa and included almos 00 observaions for each sock marke. 3. Measure of Daily eurn The daily closing value of KSE, LSE and ISE were used for calculaing he daily reurns. The coninuously compounded annual rae of reurn is a well-acceped approach o measuring he daily reurns. The naural log of daily closing index value is, hus he measure of daily reurn used for his sudy. Following is he formula used o find he reurn: I = Ln I where: = reurn on day I = index closing value on day I - =index closing value on day - Ln= naural log. Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006 8

3 3.3 Mehod and Procedure The ADF uni roo, Variance raio es, Ljung Box Q saisics and a non parameric un es has been used for he analysis of he daa. The deails of mehod implemened are given below: 3.4 Augmened Dickey Fuller (ADF Uni roo es The Augmened Dickey Fuller (ADF uni roo es is given as: Y = β m + β + δy + α i Y + ε i= (A Where ε is a pure whie noise error erm and where Y = ( Y Y, Y = ( Y Y 3, ec. The number of lagged difference erms o include is ofen deermined empirically, he idea being o include enough erms so ha he error erm in euaion is serially uncorrelaed. Since E: A is a firs-order difference euaion, so he sabiliy condiion reuires ha P > 3.5 un Tes If a series of posiive and negaive residual occurred, hen he randomness of hese residual can be checked hrough runs es which is also known as Geary es, a nonparameric es. The run es deermined wheher he successive price changes are independen. Unlike o any parameric es he assumpion of normal disribuion is no applied on run es. If he reurn series exhibis greaer endency of change in one direcion, he average run will be longer and he number of runs fewer han ha generaed by a random process. Under he null hypohesis ha he successive oucomes are independen and assuming ha N >0 and N >0, he number of runs is (asympoically normally disribued wih NN Mean: E( = + N Variance: Where N=N +N NN (NN N = ( N ( N If he null hypohesis of randomness is susainable, following he properies of he normal disribuion, we should expec ha Prob [ E(.96 E( +.96 ] = Variance aio Tes Lo and MacKinlay variance raio es is derived on he assumpion ha if he naural logarihm of a ime series P is a pure random walk, he variance of is h difference grows proporionally wih he difference, ha is he variance of is h difference variable would be imes he variance of is firs difference. So if we obain n+ observaions P o, P, P,..,P n a eually spaced inervals, / of he variance of P, P - is expeced o be he same as he variance of P P -, for a ime series characerized by random walks. The variance raio, V (, is defined as V ( = or Var( P P Var( P P = Where is he unbiased esimaor of / of he variance of h difference of he logged securiy reurn (P -P - and is an unbiased esimaor of he variance of he logged reurn (P -P -. eference [0] demonsraes ha he esimaors and can be compued by: n = ( P P u m = n = ( P P u n = Where m = ( n + ( and u = ( P n P o n n Decision rule: he compued τ value should be more negaive han he criical τ value, and he large negaive τ value is generally an indicaion of saionariy. Decision rule: Do no rejec he null hypohesis of randomness wih 95% confidence if he number of runs, lies in he preceding confidence inerval; rejec he null hypohesis if he esimaed lies ouside hese limis. Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006 9

4 And P o and P n are he firs and las observaions of he ime series and n is he sample size. The firs es saisic, Z ( is derived under he assumpion of homoscedasiciy, wih he asympoic variance of he V saisic φ ( defined as: ( ( φ ( = 3( n And he associaed sandard Z es saisic, Z ( as: V( Z( = N(0, φ( There is a growing consensus amongs finance empiricis ha volailiy is ime varying as documened by mos of he researchers ha variances of mos sock reurns are condiionally heeroscedasic wih respec o ime. As a resul, a linear relaion does no exis over he observaion inervals. To overcome his difficuly, Lo and McKinley derive he heeroscedasiciy-consisen variance esimaorφ * ( is given by: ( j φ * ( = ^ [ ] δ ( j j= Where δ ^ (J = = j+ ( P P [ n = u ( P P ( P j P ^ u ] j ^ u The V saisic can be sandardized asympoically o a sandard normal es-saisic, Z * ( which as repored by [0] Lo and McKinley (988, is compued as: V( Z * ( =... N(0, *( φ 3.7 Ljung Box-Q Saisics The firs approach we use here is o consider he auocorrelaion srucure of sock reurns and es he join significance of he auocorrelaion using he Ljung Box pormaneau saisic (Q. The Ljung Box Q-saisics are given by: 3 Q LB = T ( T + k j j= T r j Where r j is he jh auocorrelaion and T is he number of observaions. The Q saisic is ofen used as a es of wheher he series is whie noise. Q is asympoically disribued as a chi suare disribuion wih degree of freedom eual o he number of auocorrelaions. This es was developed by Ljung & Box, in 979. To compue auo correlaion and Ljung-Box Q saisics, we compue correlogram, which graphs he value of auo correlaion a successive lags agains he lengh of he lag and he las wo columns repored in correlogram are he Ljung-Box Q-saisics and heir P values DISCUSSION AND ANALYSIS 4. ADF uni roo es An Augmened Dickey-Fuller (ADF es is used for a uni roo in ime series sample; i is he advance version of he Dickey-Fuller es. The greaer he negaive value of he ADF es saisic he greaer he chance of he rejecion of he hypohesis of having a uni roo. From able, he ADF es resul shows ha for all he hree sock markes reurns, we can rejec he null hypohesis of uni roo. I was recommended by many financial researchers ha exisence of random walk componen does no mean ha fuure sock reurn is unpredicable. If sock reurns are characerized by a whie noise process, he correspondence price indices are said o follow he random walk. In ha case ha sock reurns are considered o be unpredicable. If he sock reurns do no follow whie noise or hey are inegraed of order one or I (, here exiss some predicable componens. The ADF is used o find ou only he sochasic rend componens while he purpose variance raio approach is o deec if he shorerm flucuaions dominae he sochasic rend componens or no. According o [30] he Karachi Sock marke follows a random walk for is daily reurns, he same resuls was also deermined by [8]. ADF Saisic KSE Table : ADF Tes esuls Tes - suared MacKinnon criical values for rejecion of hypohesis of a uni roo % Criical Value -.56 LSE % Criical Value -.93 ISE % Criical Value Decision ule: if Z ( and Z*( calculaed values is less han Z abulaed values hen we rejec H o: V ( =, Z abulaed =.96 for 5% significance level. 4 Decision rule: ejec null hypohesis under LB saisics if calculaed value of LB is greaer han abulaed value of chi suare disribuion able values wih m d.f. Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006 0

5 4. Non-Parameric uns Tes A non-parameric runs es has been commonly used o examine he random walk hypoheses which furher implies he exisence of weak-form efficiency in sock markes reurn (e.g. see, [9] and [3]. A non-parameric runs es is applicable here as a es of randomness for he seuence of he reurn. In oher words, i ess wheher Pakisani sock markes reurns are predicable. If he successive change in he reurns is random, i is an indicaion of weak-form efficiency. A es for a run up and down reuires dividing he reurn series ino wo differen ypes, increasing and decreasing. A posiive sign (+ is assigned if he reurn in he seuence is higher han he preceding reurn, whereas a negaive sign (- is assigned if he reurn in he seuence is smaller han he preceding reurn. A zero (0 is assigned if here is no change in he reurns. For example, a series of would indicae five runs-wo posiive runs of lengh four, wo negaive runs of lengh five and one zero run of lengh five. The run-up and run-down es idenifies he exisence of persisence bu does no indicae he direcion and degree of persisence. ejecion of he null hypohesis of randomness indicaes he persisence in reurns does exis, which violaes he weak-form of marke efficiency. As suggesed in he able ha, he successive reurn for all he markes do no show independence and hence upon he above resuls we can rejec he null hypohesis of randomness which is inconsisen wih [30] and [8] sudy for KSE 00 index. 4.3 Variance raio es Variance raio es is developed by [0] and i is more powerful han any sandard auocorrelaions ools. Under he null hypohesis of variance raio es he sock reurn follows a random walk and he variance raio are expeced o be eual o one. Table no.3 repors he esimaed variance raios for lags =, 4, 8,, 6, and 0 for boh uns Tes Table KSE LSE ISE Tes Value Cases < Tes Value Cases >= Tes Value Toal Cases 5 4 Number of uns Z Asymp. Sig. (-ailed homoscedasic and heeroscedasic consisen Z ( and Z*( es saisics. The able no 3 repored ha excep LSE (under he assumpion of heeroscedasiciy no oher sock marke reurns follows a random walk hypohesis and hence he marke efficiency concep was void in his predicion of variance raio es. As concluded by [30] and [8] he presence of random walk in securiy price for he KSE 00 index has no more validaion under hese circumsances. Now o remove hese discrepancies in resuls furher heoreical suppors are needed. As for he [8] sudy is concerned he used he sandard ess for auocorrelaions, like Ljung Box Q saisics, ADF es for uni roo and correlogram mehod o find he random walk of securiy prices. So one reason of his discrepancy is ha [8] did no use a higher power es like Variance raio es, he oher reason for inconsisency is ha [8] used a weekly daa for his analysis while in his research daily observaion has is used. Now if we see Ljung Box Q saisics of our analysis we can easily conclude ha i was consisen wih our variance raio saisic resuls. The LB-Q saisics indicae auo correlaion for boh ISE and KSE bu no auocorrelaion for LSE a any observe lags. The validiy of Z ( depends o some exen on is asympoic disribuion owards normaliy and one can K=4 K=8 K= K=6 K=0 K=4 K=8 K=3 K=36 K=40 KSE (-3.9* (-0.4 * (-8.5 * (-7.47 * (-6.67 * (-6.09 * (-5.66 * (-5.3 * (-5 * (-4.75 * (-7.38* (-6.7 * (-5.49 * (-5 * (-4.58 * (-4.7 * (-4.05 * (-3.87 * (-3.7 * (-3.58 * LSE (-4.* (-0.37* (-8.56* (-7.45* (-6.* (-5.67* (-5.3* (-5.0* (-4.75* (-4.6* (-.6 (-.53 (-.5 (-.5 (-.48 (-.47 (-.45 (-.43 (-.4 (-.6 ISE (-3.9* (-0.36* (-8.48* (-7.44* (-6.66* (-6.07* (-5.65* (-5.9* (-4.98* (-4.73* (-4.7* (-3.99* (-3.79* (-3.67* (-3.54* (-3.44* (-3.36* (-3.8* (-3.* (-3.5* The bold numeric represens he variance raio while he values in parenhesis represens he Z ( and Z (* respecively and he value wih a (* represening he value is significan a 5%. Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006

6 also prefer a large sample such as in [30] sudy more han 000 daily observaions has been aken for analysis for KSE 00 index and only heeroscedasic consisen Z* ( is free of auocorrelaion and indicaing randomness. Since [30] concluded ha KSE 00 index is efficien in erm of informaion, so rejecion of random walk hypohesis do no imply ha marke is no efficien. In variance raio saisic when exceeds one leading o rejecion of null hypohesis, which is indicaion of posiive serial correlaion ha manifes iself in shor erm flucuaions. 4.4 Ljung Box Q es The LB-Q 5 saisic is ofen used as a es of whie noise for a ime series. LB-Q is asympoically disribued as a chi suare disribuion wih degree of freedom eual o he number of auocorrelaions. LB-Q saisics obained for Pakisani sock marke reurns (shown in Table 4, suggesed ha here exis auocorrelaion for KSE and ISE for few lags bu no shows any auocorrelaion a any lags for LSE. This resul was consisen wih no only o previous research work of [30], bu also consisen wih variance raio analysis of he same sock markes. Table 4: LB-Q saisics KSE LSE ISE Lags Q-Sa (P-value Q-Sa (P-value Q-Sa (P-value 0.8 ( ( ( ( ( ( ( CONCLUSION 0.54 ( (0.60. ( ( ( ( ( ( 0.00 In his paper we assessed he random behavior of he securiy prices for hree major Pakisani sock markes i.e. Karachi sock marke, Lahore Sock marke and Islamabad sock marke. To pu more validiy and accuracy o our analysis, we used a number of saisical models for random walk assessmens. These saisical ools sared from a non-parameric saisical es i.e. a un Tes, 5 The Ljung-Box Q saisic has an asympoic chi suare disribuion wih p degrees of freedom eualing he firs ρ lags. If he associaed probabiliies are less han 5%, he null hypohesis of no serial correlaion is rejeced a 95% level of confidence. followed by [0] Variance raio es and L-Jung Box Q saisics. For he assumpion of saionariy, he Augmened Dickey Fuller uni roo es is also applied. The analysis secion gives us conradicory resuls from differen saisical ools, since some of he ess are more saisically powerful han ohers ([0] and [33]. Tha is why we analyzed and inerpreed he resuls wih he appropriae assumpion and saisical power of he ool used. The Augmened Dickey fuller es of uni roo is used only for he exisence of uni roo in ime series daa, which sugges he randomness of he observed ime series for all he markes. The LB-Q saisic, un Tes and he variance raio es resuls were consisen wih each oher, bu LB-Q and V saisics conradic wih UN es in case of LSE. Our resuls of KSE also conradic wih he general assumpion of many researchers who suggesed ha KSE 00 index follows a andom walk hypohesis, for example [30] and [8]. The Pakisani sock markes follow a random walk hypohesis if he assumpion of variance raio es and LB- Q saisic is fulfilled, like large number of observaion, ec. One imporan finding of his resul is ha alhough his sudy conradics [30], bu i is consisen wih one par of his sudy where he argued ha in pre 9/ era KSE 00 index does no follow a random walk and Monday gives a highes posiive reurn which an indicaion of day of he week effec. EFEENCES [] Samuelson, Paul (965. "Proof Tha Properly Anicipaed Prices Flucuae andomly", Indusrial Managemen eview, 6, pp [] Fama, E.F. (970 Efficien capial markes: A review of heory and empirical work, Journal of Finance, 5, [3] McInish, H., and Pulisi, D.J. (98 The efficiency of he inernaional money marke, Journal of Business Finance and Accouning, 9(, [4] Barnes, P. (986, Thin Trading and Sock Marke Efficiency: The Case of he Kuala-Lumpur Sock Exchange, Journal of Business Finance & Accouning, pp. 609,7. [5] Cheung, Wong and Ho (993, The Pricing of isky Asses in wo Emerging Asian Markes, Korea and Taiwan, Applied Financial Economics, Vol. 3, No. 4, pp. 35, 4. [6] Dickinson, J.P. and Muragu, K. (994 Marke efficiency in developing counries: a case sudy of he Nairobi Sock Exchange, Journal of Business Finance and Accouning, (, [7] Ayadi, O. and C. Pyun (994, An Applicaion of Variance aio Tes o he Korean Securiies Marke, Journal of Banking and Finance, Vol. 8, pp. 643± 58. [8] Osborne, M. F. M. (96 Periodic srucure in he Brownian moion of he sock marke, journal of Operaions esearch, 0, Journal of Independen Sudies and esearch (JIS Volume 4, Number, January 006

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