Data Analysis of Shenzhen and Shanghai Land Indices

Transcription

1 Poceedings of he 6h WSEAS Inenaional Confeence on Applied Compue Science, Hangzhou, China, Apil 15-17, Daa Analysis of Shenzhen and Shanghai Land Indices MEIFENG JI JUN WANG Depamen of Mahemaics, College of Science Beijing Jiaoong Univesiy Beijing Jiaoong Univesiy, Beijing , P.R.China CHINA Absac: - In his pape, he daa of Shenzhen and Shanghai land indices is analyzed, and he saisical popeies of Shenzhen and Shanghai land indices ae sudied. We selec he daa fo Shenzhen and Shanghai land indices duing he yea , and invesigae he saisical popeies, fa ails phenomena and he powe-law disibuions of euns fo hese wo land indices. Fuhe, we analyze he flucuaion of he elaive land pices, and discuss he coesponding pobabiliy disibuions of he elaive land pices. Key-Wods: - land index; euns; powe law disibuion; elaive pice 1 Inoducion In his pape, we analyze quoes daa fom Shenzhen Sock Exchange and Shanghai Sock Exchange duing he yea , especially we analyze he daa of he land indices fo hese wo Chinese sock makes. I is known ha he eal esae makes play a key ole in economy of a couny, i is one of he mos impoan asses in capialism. Shenzhen and Shanghai land indices ae wo synheical indices which consis of land socks of Shenzhen Sock Exchange and Shanghai Sock Exchange especively. They can eflec he aciviy and he end of Chinese land makes in lage degees, so his will be helpful fo us o undesand he saus of Chinese macoeconomic. Wih he efomaion and developmen of Chinese economic sysem, he Chinese eal esae and he Chinese eal esae makes develop apidly, now he movemen of land pices has a song influence on he economic behavio of individuals and fims, as a esul, i affecs he economic developmen of China diecly. In he pesen pape, ou sudies focus on he scaling behavio in land make, we analyze he saisical popeies of Shenzhen and Shanghai land indices, we discuss he powe law disibuions of euns fo he land indices, and sudy he pobabiliy disibuion of elaive pice, and hen we y o explain he end and flucuaions of Chinese land makes. The daabase which used in he pesen pape is fom he webses of Shenzhen Sock Exchange and Shanghai Sock Exchanges ( Consideing he hisoy of financial siuaion of Chinese sock makes, he daily pice limi (now 10%), he ading ules of he wo sock makes, and he financial policy of Chinese govenmen, we selec he daa of he daily closing pice (fo each ading day) fo each land index coveing he ecen 6-yea peiod duing he yea , he oal numbe of obseved daa fo one land index is abou The Saisical Popeies of Land Index In ecen yeas, he pobabiliy disibuion in financial make flucuaions has been sudied, he empiical eseach has shown he powe-law ails in pice flucuaions. Moe specifically, some eseach esuls show ha he disibuion of lage euns follow a powe law disibuion wih exponen 3, ha is P ( > x) x ξ, whee is he euns of he sock pices in a given ime ineval Δ, ξ 3, fo example see [1][6-7][9]. In his secion, accoding o he saisical mehod and daa analyzing mehod (see [2-3][5-7][9-10]), we will sudy he pobabiliy disibuions of daily euns and he powe-law chaace of daily euns fo Shenzhen and Shanghai land makes. Since he disibuion of euns is also chaaceized by skewness and kuosis, hee we also sudy he skewness and kuosis of euns fo Shenzhen and Shanghai land indices. 2.1 The Pobabiliy Disibuions of Reuns Le S () denoe he pice pocess a ime ( = 0,1,2, ), and le () be he eun of he pice which defined by (see [4][8][11])

2 Poceedings of he 6h WSEAS Inenaional Confeence on Applied Compue Science, Hangzhou, China, Apil 15-17, ( ) () = S ( + 1) S () S () = Δ S () S () whee Δ S () = S ( + 1) S (). Fo a sock pice S (), since he daily change of he pice Δ S () is elaive small wih espec o he pice S (), so we have ha he logaihm of eun R() = ln( S( + 1) S()) (). In his pape, fo = 1, 2,, le () be he daily eun pocess of -h ading day. Accoding o he daa obseved fom he wo land indices, we plo he daily euns figues as follows: [1-3][5-7][9]. The flucuaions of euns ae believed o follow a Gaussian disibuion fo long ime inevals bu o deviae fom i fo sho ime seps, especially he deviaion appeas a he ail pa of he disibuion, usually called he fa-ails phenomena. In he pesen pape, we y o sudy he pobabiliy disibuion of euns fo Shenzhen and Shanghai land indices duing he yea , and he daa comes fom Shenzhen Sock Exchange and Shanghai Sock Exchange. Compaing he daily euns of Shenzhen and Shanghai land indices wih he coesponding nomal disibuions, we have he following Figue 2. Figue1 (a) Figue2 (a) Figue1 (b) Fig.1. Daily euns plos of land indices fo Shenzhen (a) and Shanghai (b) duing he yea Recenly, a lo of eseach wok has been done o sudy he flucuaions of euns fo sock pices, see Figue2 (b) Fig.2. (a) The compaison of Shenzhen land index's daily euns (6-yea peiod) wih he coesponding nomal disibuion. (b) The compaison of Shanghai land index's daily euns (6-yea peiod) wih he coesponding nomal disibuion.

3 Poceedings of he 6h WSEAS Inenaional Confeence on Applied Compue Science, Hangzhou, China, Apil 15-17, The above plos ae he nomal pobabiliy plos, which is a useful gaph fo assessing whehe daa comes fom a nomal disibuion. Fom Fig.2, we can see ha he euns pocesses of wo Chinese land indices ae nomal disibuions in he middle pa of Fig.2, since he `+ line (he obseved daa) coincides wih he dash line (he coesponding nomal disibuion) in he pa whee he pobabiliy is below he 75h and above 25h peceniles of he samples. Bu he pa (whee he pobabiliy is above he 75h o below 25h peceniles of he samples) deviaes fom he dash line. This implies ha he pobabiliy disibuion of he euns fo he wo land indices deviaes fom he coesponding nomal disibuions a he ail pas. This also ell us ha he fa-ail phenomenon of he euns may exis fo he wo land indices. 2.2 Skewness-Kuosis Tes of Reuns In his secion, we sudy he popeies of skewness and kuosis (see [10]) on he daa of euns fo Shenzhen and Shanghai land indices. Fis, we give he definiions of skewness and kuosis as following: n i= 1 n ( ) /( 1) i 3 3 Skewness = u n δ i= 1 ( ) /( 1) i 4 4 Kuosis = u n δ whee i denoes he eun of i -h ading day, u is he mean of, n is he oal numbe of ading daes, and δ is he coesponding sandad vaiance. I is known ha he skewness of sandad nomal disibuion is 0, and kuosis is 3. Fom he 6-yea daa of he wo land indices, we have he following Table 1. In Table 1(a)(b), he values of kuosis of euns fo he wo land makes ae boh bigge han 5, and he values of skewness ae close o This implies ha he saisical disibuion of he obseved daa deviaes fom he Gaussian disibuion in some pas (i may include he ail pas), and i also implies ha Table 1(a) The saisical popeies of euns Land Index of Shenzhen Sock Exchange skewness kuosis mean value e-004 vaiance e-004 maximum minimum Table 1(b) The saisical popeies of euns Land Index of Shanghai Sock Exchange skewness kuosis mean value e-004 vaiance e-004 maximum minimum he flucuaion of euns fo he wo land indices is geae han ha of he coesponding Gaussian disibuion. 2.3 The Powe-Law Behavio of Reuns In his secion, we sudy he saisical popeies of ail disibuions fo he wo Chinese land indices. Powe law scaling is he univesal popey ha chaaceizes collecive phenomena ha emege fom complex sysems composed of many ineacing unis. Powe law scaling has been obseved no only in physical sysems, bu also in economic and financial sysems. In his secion, we sudy he pobabiliy disibuion of euns fo he land indices, and y o show he powe law disibuion of euns. In ecen yeas, he empiical eseach has shown he powe-law ails in he eun flucuaions, ha is, P ( > x) x α, and some eseach esuls show ha he disibuion of lage euns follow a powe law disibuion wih exponen α = 3. Fis we plo he pobabiliy disibuion of euns in he yea 2004 in Figue 3, which is he log-log plo. Then we compae he exponens α fom he yea 2001 o 2006 in Table 2. Figue3 (a)

4 Poceedings of he 6h WSEAS Inenaional Confeence on Applied Compue Science, Hangzhou, China, Apil 15-17, gadually deviaes fom he powe law as he absolue eun becomes small, his shows ha he ail disibuion of euns fo he land index of Shenzhen follows he powe-law disibuion. Similaly, fom Fig.3(b), we can discuss he pobabiliy disibuion and is ail disibuion of euns fo he land index of Shanghai in he yea In Table 2, hough analyzing he obseved daa, we show he values of exponens α fom he yea 2001 o 2006 fo he wo land makes, Shenzhen (a) and Shanghai (b), whee S.E. denoes sandad eos and R2 he coefficien of deeminaion, he ange of powe law exponens is fom o in Shenzhen land make, and o in Shanghai land make. Figue3 (b) Fig.3. The pobabiliy disibuions of euns of he land indices fo Shenzhen (a) and Shanghai (b) in he yea 2004, and α = (Shenzhen), α = (Shanghai). Table 2(a). Powe law exponens α of he pobabiliy disibuion fo Shenzhen land index Yea Esimaedα S.E. R Table 2(b). Powe law exponens α of he pobabiliy disibuion fo Shanghai land index Yea Esimaedα S.E. R The Pobabiliy Disibuion of Relaive Land Pice In his secion, we analyze he pobabiliy disibuions of he elaive pices fo Shenzhen and Shanghai land indices. Le = ( 0,1,2, ) denoe he -h ading day, he elaive pice is defined by P () = S ( + 1)/ S () whee S (), S+ ( 1) ae he closing pices of -h and ( + 1) -h ading days especively, i.e., he daily closing pices. Fom above definiion, he elaive pices expess he values which denoe he aio of S+ ( 1) o S (), and we can undesand he end (inceasing o deceasing) of land pices fom hese values. Fom he daa of land indices in Shenzhen and Shanghai sock makes duing he yea , we have he disibuions of he elaive pices, and plo he log-log pobabiliy densiy plos in following Figue 4. In Fig.3(a), he pobabiliy disibuion of euns follows a powe-law wih exponen 2 α = ± ( R = ), which is deemined by odinay leas-squaes egession in 2 log-log coodinaes, whee R denoes he coefficien of deeminaion. The disibuion follows he powe law behavio in he lage pice ange, bu Figue4 (a)

5 Poceedings of he 6h WSEAS Inenaional Confeence on Applied Compue Science, Hangzhou, China, Apil 15-17, Acknowledgemens The auhos would like o hank he suppo of Insiue of Financial Mahemaics and Financial Engineeing in Beijing Jiaoong Univesiy, and hank S.Z. Zheng and Z.Q. Zhang fo hei kind coopeaion on his eseach wok. The auhos ae suppoed in pa by Naional Naue Science Foundaion of China Gan No and Minisy of Educaion of China Gan No. 406 (2003). Figue4 (b) Fig.4. Pobabiliy disibuion of he elaive pices on a log-log scale fo land indices of Shenzhen(a) and Shanghai(b) fom 2001 o The hoizonal axis denoes he elaive pices, defined as he logaihm of he elaive pice P () = S ( + 1)/ S (), and he veical axis denoes pobabiliy densiy. Fom Fig.4, he numbe of dos in he lef side of 0 10 (he poin of hoizonal axis) and he numbe of 0 dos in he igh side of 10 implies he flucuaion and end of Chinese land makes. In Fig.4, hee ae some dos nea 10, his means ha he lage jump of land pices occued in wo Chinese land makes. 4 Conclusion In his pape, we sudied he saisical popeies of euns and elaive pices of land indices fo Chinese land makes. The empiical eseach shows ha he disibuions of euns have lage values of kuosis and negaive values of skewness fo he Chinese land indices, we also show ha he lage euns follow he powe law disibuion, a las we show he pobabiliy densiy of elaive pices. Thee ae wo sock exchange makes (Shenzhen and Shanghai) in he mainland of China, and he Chinese sock make is a developing make. Alhough i has a sho hisoy (less han 20 yeas), he oal value of Shenzhen and Shanghai sock makes is moe han 50% of GDP fo China in Januay 2007, so we can see ha he Chinese sock make plays an impoan ole in he developmen of Chinese economic. In he pesen pape, ou daabase is fom hese makes, so ou empiical eseach is helpful o undesand he flucuaions of Chinese land makes. Refeences: [1] X. Gabaix, P. Gopikishnan, V. Pleou, H.E. Sanley, A heoy of powe-law disibuions in financial make flucuaions, Naue, 423, 2003, pp [2] R. Gaylod and P. Wellin, Compue simulaions wih Mahemaica: exploaions in he physical, biological and social science, Spinge-Velag, New Yok, [3] K. Ilinski, Physics Of Finance: Gauge Modeling in Non-equilibium Picing, John Wiley, [4] D. Lambeon and B. Lapeye, Inoducion o Sochasic Calculus Applied o Finance, Chapman and Hall/CRC, [5] T.C. Mills, The Economeic Modeling of Financial Time Seies (Second Ediion), Cambidge Univesiy Pess, [6] A.R. Pagan, G.W. Schwe, Tesing fo covaiance saionaiy in sock make daa, Economics Lees, 33, 1990, pp [7] V. Pleou e. al., Scaling of he disibuion of flucuaions of individual companies, Phys. Rev. E, 60, 1999, pp [8] S.M. Ross, An Inoducion o Mahemaical Finance, Cambidge Univesiy Pess, [9] H. Tanaka, A Pecolaion Model of Sock Pice, Flucuaions, Mahemaical economics, 1264, 2002, pp [10] J. Wang, Pobabiliy Theoy and Mahemaical Saisics, New Wun Ching Developmenal Publishing, Taiwan, [11] J. Wang, Sochasic Pocess and Is Applicaion in Finance, Tsinghua Univesiy Pess and Beijing Jiaoong Univesiy Pess, Beijing, 2007.