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1 (277) * R/S Hurst Hurst Hurst *

2 (278) Efficiet Market CAPM APT R/S Rescaled Rage Aalysis R/S Hurst 1951 Madelbrot 1972, 1982 Madelbrot ad Wallis 1969 Lo 1991 R/S R/S R/S Ceter Itelligece Agecy 2006 GDP 1 World Rakig Coutry U.S. Chia Japa South Korea Taiwa Australia 1 GDP Coutry GDP GDP World GDP billio U.S.$ GDP GDP Real Growth Rate $12, % 3.40% $10, % 10.50% $4, % 2.80% $1, % 4.80% $ % 4.40% $ % 2.80%

3 (279) World Rakig Coutry South Africa Jorda World Coutry GDP GDP World GDP billio U.S.$ GDP GDP Real Growth Rate $ % 4.50% $ % 6.00% $65, % 5.10% GDP GDP GDP GDP GDP GDP GDP 46.65% R/S Hurst 2. Hurst 3. Hurst Efficiet Market Hypothesis EMH EMH ( Pt + 1 ) = ( P t ) + ε t+ 1 P t+ 1 P t ε t+ 1

4 (280) iid 2 ε t+ 1 ~ N(0, σ ) 1. Fama 1965 Fielitz ad Rozelle 1983 Sterge Self-similar Fractal LeRoy ad Porter 1981Variacebouds TestsExcess Stock Price Volatility, Madelbrot 1977 A fractal is a shape made of parts similar to the whole i some way (1) (2) (3) Madelbrot

5 (281) 1994 Peters R/S Homogeeous

6 (282) Characteristics Market Character Equilibrium Status System Complexity Feedback Mechaism Retur Series Price Series Predictability The Order of Volatility Both Relatio 2 Efficiet Market Theory Fractal Market Theory Liear Idepedet System No-liear, ope, diverget system Equilibrium Allow No-equilibrium Simple System Fractal ad Chaos System No Feedback Positive Feedback White Noise, No Relatio Fractal White Noise, Log Memory Browia Motio Fractal Browia Motio No-predictable Provide a New Predictable Method No Order Has Order Efficiet Market is a Special Case of Fractal Market R/S Greee Fielitz 1977 R/S Hurst Hurst Hurst Peters 1989 R/S S&P Hurst , C t 16.8%

7 (283) 21.5% 24.5% Mckezie 2001 R/S Mckezie R/S Hurst Hurst Rescaled Rage Aalysis, R/S Hurst Hurst Expoet, H R/S R/S (1) X N N d = ( ) m=1 2 3 d X X = ( x1, x2, x3,..., x N ) = ( x1,1, x1,2, x1,3,..., x1,, x2,1, x2,2, x2,3,..., x2,,..., xd,1, xd,2, xd,3,... xd, ) (2) u m S m m=1 2 3 d ( x 1,1, x1,2, x1,3,..., x1, ) u 1 S 1

8 (284) u j= = 1 1 x 1, j S 1 = ( x1, j u1 ) j= 1 2 (3) Z m, i Z m, i = xm, i um i 1,2,3,..., = m = 1,2,3,..., d (4) Ym, i = Z m, j, i = 1,2,3,..., m = 1,2,3,..., d (5) Rm = max( Ym, i ) mi( Ym, i ) m = 1,2,3,..., d (6) R m m = 1,2,..., d S m (7) i j= 1 ( R / S) = d m= 1 ( R m d / S m ) N = 5,6,7,..., 2 (8) Hurst 1951 d H ( R / S) = c * c H Hurst l l( R / S) = l c + H *l( ) (9) H 0.5 T H : H 0. 5 H : H = Hˆ 0.5 T = S( Hˆ ) 1 Ĥ H S ( Ĥ ) Ĥ l( R / S) = l c + H * l( ) l R/S l H H Hurst d (1/ 2) [0,1] Hurst 1951 Feller 1951 ( R / S) = ( * π / 2) 2H 1 H=0.5 C( t) = 2 1 Fractioal Dimesio D=2-H C(t) D Hurst (1) H = 0. 5 C (t) =0 D=1.5 0

9 (285) (2) 0.5 < H 1 0 < C( t) 1 1 D < H 0.5 (3) 0 H < C( t) < < D Mea Reversio V ( R/ S) = V Hurst 1951 R/S H ( R / S ) H (1/ 2) ( R / S) = c * V V V = = c * Hurst V (1) H=0.5 V c V l V (2) H 0.5 V (3) H 0.5 V V = d=

10 (286) R/S Hurst d 2 R/S = l R/S l Hurst V R/S V l V Global Fiacial Data Pt l Rt = l *100% l R P t t P t P t 1 t 1 t t-1 3 Coutry Japa U.S. Australia South Africa South Korea Taiwa Chia Jorda 3 Stock Price Idex No. of Sample Sample Period Nikkei ~2006 S&P ~2006 All Ordiaries ~2006 JSE All-Share Idex ~2006 Seoul Composite ~2006 Taiwa Weighted ~2006 Shaghai A Shares ~2006 AFM Geeral Idex ~2006

11 (287) Japa U.S. Australia South Africa South Korea Taiwa Chia Jorda 4 Mea Stadard Deviatio Skewess Kurtosis Kolmogorov -Smirov (Test) Sigificace Nikkei

12 (288) Kolmogorov-Smirov Kolmogorov-Smirov R/S Nikkei225 S&P500 All Ordiaries JSE All-Share Idex Seoul Composite Taiwa Weighted Shaghai A Shares AFM Geeral Idex R/S R/S l( R / S) = l c + H *l( ) Hurst Hurst () 2 H C t = 1 D=2-H R/S Itercept Hurst Hurst Idex *** *** Japa *** *** U.S *** *** Australia *** *** South Africa *** *** South Korea *** Taiwa *** *** Chia *** *** Jorda R R-square Correlatio Fractal R-square Adjusted Coefficiet Dimesio R-square *** t α =0.01

13 (289) 5 R/S Hurst Hurst R/S T 5 Hurst 0.5 Hurst 0.5 Hurst Hurst 0.5 Hurst 0.5 Hurst 0.6 Cluster Property Hurst Hurst Hurst L R/S l R-square R-square 95%~99% C() t D H=0.5 C() t D 1.5 Hurst 0.5 C() t D 1.5 C() t D l R/Sl V V l V l R/S l R/S V R/S V V V V V R/S

14 (290) 1(a) Nikkei 225 V 1(b) Nikkei 225 R/S 2(a) S&P500 V 2(b) S&P500 R/S 3(a) All-Ordiaries V 3(b) All-Ordiaries R/S

15 (291) 4(a) JSE All-Share Idex V 4(b) JSE All-Share Idex R/S 5(a) KOSPI-200 V 5(b) KOSPI-200 R/S 6(a) V 6(b) R/S

16 (292) 7(a) A V 7(b) A R/S 8(a) AFM Geeral Idex V 8(b) AFM Geeral Idex R/S 1(a) V l V 2(a) V l = V 2(b) R/S =138 廻 Hurst 138 Hurst Hurst V Hurst 3(a) V l = V 3(b) R/S =110 廻 Hurst 110 Hurst Hurst V Hurst 4(a) V l =152

17 (293) 152 V 4(b) R/S =152 廻 Hurst 110 Hurst Hurst V Hurst 5(a) V l V 6(a) V l = V 6(b) R/S =208 廻 Hurst 208 Hurst Hurst V Hurst 7(a) V l = V 7(b) R/S =202 廻 Hurst 202 Hurst Hurst V Hurst 8(a) V l V V South South Japa U.S. Australia Taiwa Chia Africa Korea Jorda Average Aperiodic Cycle

18 (294) Hurst 7 7 Hurst Developed Coutries Developig Coutries Coutry Hurst Hurst Idex Cycle Day Coutry Hurst Hurst Idex Cycle Day Japa South Korea U.S. Taiwa Australia Chia South Africa Jorda Average Average

19 (295) Asia Area Hurst Coutry Hurst Idex Cycle Day Japa South Korea Taiwa Chia Jorda Average Coutry U.S. Australia South Africa Average No-Asia Area Hurst Hurst Idex Cycle Day Hurst Hurst 30 Hurst Ma-Whitey H Hurst H Hurst x H 0 : H x = H y H1 : H x < H y U x ( x +1) U = Tx 2 x y Tx Hurst T Hurst y y

20 (296) U 0 p-value 0.05 H 0 Hurst Hurst Hurst Hurst Hurst Ma-Whitey S x S y H H 0 1 : S : S x x U U =0 p-value 0.05 H 0 = S < S Hurst Hurst Ma-Whitey Hurst H a Hurst H Hurst b H : H = H 0 1 a H : H < H a U =0 p-value 0.05 H 0 Hurst Ma-Whitey S a S b H 0 a H : S 1 : S a 0 p-value H = S < S b y y b b b

21 (297) R/S Hurst 0.6 Hurst Hurst Hurst % Hurst % Hurst % Hurst Hurst % = Hurst 0

22 (298) Cheol, S. Eu ad Jisoo Lee 2006 Mea-Variace Covergece Aroud the World AFA 2006 Bosto Meetigs Paper Hurst. C(t). cluster property Albert N.Shiryaev, Essetials of stochastic fiace facts, models, theory, World Scietific, p143~ 统 p13~ p59~66 1. Albert N. Shiryaev 1999, Essetials of Stochastic Fiace Facts, Models, theory, World Scietific. 2. Fama, E. 1965, The Behavior of Stock Market Prices, Joural of Busiess, Vol. 38, p Feller, W. 1951, The Asymptotic Distributio of the Rage of Sums of Idepedet Radom Variables, The Aals of Mathematical Statistics, Vol.22, p Fielitz, Bruce D., ad James P. Rozelle 1983, Stable Distributios ad the Mixtures of Distributios Hypotheses for Commo Stock Returs, Joural of the America Statistical Associatio, Vol. 78, p Greee, M. T. ad Fielitz, B. D Log-term Depedece i Commo Stock Returs, Joural of Fiacial Ecoomics 4: p Hurst, H. E. 1951, Log Term Storage Capacities of Reservoirs, Trasactios of the America Society of Civil Egieers 116, p LeRoy, Stephe F., ad Richard D. Porter 1981, The Preset-value Relatio: Tests Based o Implied Variace Bouds, Ecoometrica, Vol. 49, No. 3, p Lo, A. W. 1991, Log-term Memory i Stock Market Prices, Ecoometrica 59,

23 (299) 9. Madelbrot, Beoit B Statistical Methodology for No-Period Cycles: from the Covariace to R/S, Aals of Ecoomic ad Social Measuremet, 1, Madelbrot, Beoit B. ad M. S. Taqqu 1979, Robust R/S Aalysis of Log Ru Serial Correlatio, Biometrika, 68, Madelborot, Beoit B. 1977, Fractals, Form, Chace, ad Dimesio, W. H. Freema ad Compay. 12. Madelbrot, Beoit B. 1982, The Fractal Geometry of Nat. 13. Madelbrot, Beoit B. ad Wallis, J. 1969, Robustess of the Rescaled Rage R/S i the Measuremet of No-cyclic Log Ru Statistical Depedece, Water Resources Research 5, p Peters, Edgar E. 1989, Fractal Structure i the Capital Markets, Fiacial Aalysts Joural, Vol. 45, issue 4, pp32-37, July August. 15. Peters, Edgar E. 1991, Chaos ad Order i the Capital Market: a New View of Cycles, Prices, ad Market Volatility, New York: Joh Wiley & Sos Ic Peters, Edgar E. 1994, Fractal Market Aalysis: Applyig Chaos Theory to Ivestmet ad Ecoomics, New York: Joh Wiley & Sos Ic Sadique, S. ad P. Silvapulle 2001, Log Term Memory i Stock Market Returs: Iteratioal Evidece, Iteratioal Joural of Fiace ad Ecoomics, 6, Sterge J. A O The Distributio Of Fiacial Futures Price Chages, Fiacial Aalyst Joural, Vol. 45, No. 3: 74-78, W. H. Freema, New York.

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