PHYSICA ELSEVIER
Physica A 246 (1997) 454-459
Coherent and random sequences in financial fluctuations N. Vandewalle a,b,*, M. Ausloos a,b a SUPRAS, Institut de Physique B5, Sart Tilman, Universitk de Libe, B-4000 Lib#e, Bel#ium b ECOPHYNANCE, Av. Nouveau-Monde 327, B-7700 Mouscron, Belium Received 28 May 1997
Abstract
The detrended fluctuation analysis (DFA) is used to sort out temporal correlations in financial data. Its usefulness for the investigations of long-range power-law correlations in economic sequences is shown. Our findings of persistent and antipersistent sequences are surprisingly similar to those for DNA sequences which appeared as a mosaic of coding and non-coding patches.
Keywords: Econophysics; Brownian motion; Time series
1. Introduction
Recently, there have been several reports that financial fluctuations may display long-range power-law correlations [ 1-4] similar to turbulent processes [5] and selforganized critical systems [6], much beyond the first relevant considerations for economic organization or laws [7]. If the usual physical features of complex phenomena are found to be true also in some economic behaviors, this should allow physicists to develop economy or financial models [ 8-10] and consider predictability conditions. Different statistical techniques have been used up to now in order to sort out temporal correlations in economic fluctuations: L6vy statistics [3,4], wavelets [11], discrete scale invariance [12], auto-regressive conditional heteroskedasticity (ARCH) [13], rescaled range analysis (R/S) [14], a.s.o. We present here a method to sort out temporal correlations in financial data within the detrended fluctuation analysis (DFA) statistical method [15]. The latter has demonstrated its usefulness for the investigations of long-range power-law correlations in DNA sequences [16]. In such investigations, it * Correspondence address: SUPRAS, lnstitut de Physique B5, Sart Tilman, Universit6 de Liege, B-4000 Libge, Belgium. E-mail:
[email protected]. 0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 ( 9 7 ) 0 0 3 6 6 - X
N. Vandewalle, M. Ausloos/Physica A 246 (1997) 454~159
455
was asked where and whether DNA sequences, which appeared as a mosaic of coding and non-coding patches, contain the relevant genetic information through the correlated sequences or not [16]. Mutatis mutandis the same types of answer might be searched for the same type of question in economics. Our findings are surprisingly similar to those found for DNA sequences, i.e. persistent and antipersistent patches.
2. The D F A technique: a qualitative discussion The Detrended Fluctuation Analysis (DFA) technique consists in dividing a random variable sequence y(n) of length N into Nit nonoverlapping boxes, each containing t points. Then, the local trend z(n)=an + b in each box is defined to be the ordinate of a linear least-square fit of the data points in that box. The detrended fluctuation function F(t) is then calculated following: (k+l)t
F(tf=)-
Z
(Y(n)-z(n))2'
k = 0 , 1 , 2 .....
- 1 .
(1)
n=kt+l
Averaging F(t) over the N/t intervals gives the fluctuations (F(t)) as a function of t. If the y(n) data are random uncorrelated variables or short range correlated variables, the behavior is expected to be a power law (F) ~ t ~
(2)
with an exponent i [15]. An exponent ~ ¢ I in a certain range of t values implies the existence of long-range correlations in that time interval as, e.g. in the fractional Brownian motion [17]. An exponent ~ > ~I means persistent (smooth) excursion of the currency exchange ratio and an exponent ~ < 31 means antipersistentbehavior [ 18,19]. The main advantages of the DFA over other techniques like Fourier transform, or R/S methods are that (i) inherent trends are avoided at all scales t in Eq. (1), and (ii) local correlations can be easily probed as we will see below. In economic data like stock exchange and currency fluctuations, long or short scale trends are common evidence [14,20]. Indeed, the economic agents, i.e. traders, have different investment horizons: they are short-term, medium-term or long-term investors. The investment horizons range typically from one week to a few years in the economic markets [14] eventhough there are short scale (like daily) scale fluctuations [2]. The DFA method as introduced here for such a purpose allows one to avoid the trend effects. Moreover, we may expect that DFA will allow a better understanding of apparently complex economic signals.
3. The D F A technique: an economic application We have considered the economic evolution of the USD/DEM currency exchange rate from 1 January 1980 till 15 October 1996. This represents N = 4383 data points.
N. Fandewalle, M. Ausloos/Physica A 246 (1997) 454-459
456 3.5
I
l
l
J
I
I
I
r
[
I
I
I
3.0
2.5
¢9
2.0
O
1.5
(a)
1.0
I
~ , l ~ , J , ~ , l ~
I
0.70
I
V
il
0.65
' , Wl ' ,
0.60
TZL
0.55 0.50
V
0.45 0.40 0.35
L
i
80
(b)
84
88
92
96
0.30
date
Fig. 1. (a) The jagged evolution of the USD/DEM currency exchange rate from 1 January 1980 to 15 October 1996 representing 4383 data points. (b) The evolution of the local value of ~ estimated with the DFA technique for boxes of 2 yr size. Typical error bars are indicated. The arrows indicate important economic events (notations in the text).
The week-ends and holidays are obviously not considered even though political or social events can occur during week-ends. The j a g g e d evolution o f the U S D / D E M currency is presented in Fig. la. At the time scale o f the figure, large trends are clearly observed as, e.g. the devaluation o f the USD between 1985 and 1988. In Fig. 2, a l o g - l o g plot o f the function (F(t)) is shown for the whole data o f Fig. 1. This function is very close to a power law with an exponent c~=0.56 + 0.01 holding over two decades in time, i.e. from about one week to about 1 yr. This resuits clearly support the existence o f long-range power-law correlations in currency fluctuations whatever the trend (see Fig. la). These power laws are the signature o f a propagation o f information across the economic system during very long times. For
N. Vandewalle, M. Ausloos/Physica A 246 (1997) 454-459
I 0°
'
'
'
'''"I
'
'
'
'''"I
'
'
' ' ' ' " I
457
,1,1I,
10"I
A V 10 ,2
I 0°
101
10 2
10 3
10 4
t
Fig. 2. The log-log plot of the (F(t)) function showing time scale invariance from one week to one year. The crossover is denoted by an arrow.
time scales above 1 yr, a crossover is however observed and is indicated by an arrow. This crossover suggests that correlated sequences have a characteristic duration of ca. 1 yr along the whole financial evolution at least for the case studied here over 16 yr. The range of validity for the power law (Eq. (2)) should be associated to the lower and larger investment horizons of traders. It should be underlined that the R/S analysis of currencies [14] has not been able to determine such a low and large scale investment horizons. Conversely, it is also of interest to check whether there are decorrelated sequences. In order to probe the existence of correlated and decorrelated financial-sequences, we first construct a so-called observation box (a probe) of "length" 2 yr placed at the beginning of the data, and we calculate 7 for the data contained in that box. Then, we move this box by 20 points (4 weeks) toward the right along the financial sequence and again calculate ~. Iterating this procedure for the 1980-1996 sequence, we obtain a "local measurement" of the degree of "long-range correlations". The results are shown in Fig. lb with error bars, i.e. at various typical points. Even within these error bars, one clearly observes that the c~ exponent value varies with the date. The ct exponent value is mostly above 1. However, both around spring 1983 and spring 1987, the exponent ~ has a sharp minimum below the ~ = ~1 value. The second event is the most dramatic one. This is similar to what is also observed along DNA sequences where the ~ exponent drops below ½ in so-called non-coding regions. By analogy, our findings suggest that for some yet unknown reason financial markets have lost control over the information propagation a such a time. It should be noted that both sequences observed around 1983 and 1987, respectively, were not directly visible from the data in Fig. la and would likely be missed by R/S and Fourier analysis.
458
N. Vandewalle, M. Ausloos/Physica A 246 (1997) 454-459
4. Speculating monetary policies The observed features through DFA can be tentatively associated to economic events [21] following, on the one hand, political events and sometimes panic storm spreading over the financial markets as seen below. In fact, as Peters underlined [14]: "currency exchange rates are tied to relative interest-rate movements in the two countries. In addition, the markets themselves are manipulated by their respective governments for reasons that may not be considered as rational in an efficient market sense". The more so, economic events might be said to be emotionally motivated. Specific monetary events having profoundly marked their impact on economists and historians are indicated in Fig. la. These events were compiled from the reports published by the OCDE. The 1983-84 drop follows the Volker chairmanship (V) at the Federal Reserve Bank and subsequent world panic. The ~ turn over and rebound in 1985 indicates the "need for control" resulting in the so-called Plaza agreement (P) while the 1987 event follows the so-called "accord du Louvre" (L), and the Deutsche Bank anomalous interest rate increase (D). No need to say that the loss of correlations in 1991 follows the Gulf War (W) and the subsequent Recession thereafter (R). The i value more or less being stable since 1993 indicates the presently apparent calm and regularity of the exchange rates. A physicist (Alice or Robin Hood, i.e. a pedestrian walking in a rough economic landscape) would then propose an interpretation as follows. The c~ behavior seems to support quantitatively that after a policy move for better control and in order to avoid panic or to heal a crisis, the "market" nevertheless searches for the best subtle holes in the regulation in order to "avoid and respect" the most severe constraints, a so-called result rather than a hypothetic premise of "market efficiency". Then, the market makers, seen as "the market", slide off the policy main stream as proposed in "conspiracy theories". The market then goes out of control before new rules are decided upon.
5. Conclusion In summary, long-range power law correlations and anticorrelations have been shown to occur in economic systems. We have used the detrendedfluctuation analysis (DFA) statistical method [15], a method which allows to sort out correlations and decorrelations, here in financial data. Moreover, we have quantified that some sequences appear where the economic system loses some control over information propagation. It seems that these features can be associated with real economic events and policies.
Acknowledgements The comments of D. Stauffer and A. Pekalski are greatly appreciated. The "G6n~rale de Banque" of Belgium [21] provided the data of Fig. la.
N. Vandewalle, M. Ausloos/Physica A 246 (1997) 454~59
459
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
B.B. Mandelbrot, J. Business 36 (1963) 349. R.N. Mantegna, Physica A 179 (1991) 232. R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46. A. Ameodo, J.-P. Bouchaud, R. Cont, J.-F. Muzy, M. Potters, D. Somette, Nature, to be published. S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, Y. Dodge, Nature 381 (1996) 767. P. Bak, K. Chen, M. Creutz, Nature 342 (1989) 780. V. Pareto, Cours d'Economie Politique (Lausanne, 1897); see also E.W. Montroll, W.W. Badger, Introduction to Quantitative Aspects of Societal Phenomena, Gordon and Breach, New York, 1974. M. Levy, S. Solomon, Int. J. Mod. Phys. C 7 (1996) 595; S. Solomon, M. Levy, lnt. J. Mod. Phys. C 7 (1996) 745. P. Bak, M. Paczuski, M. Shubik, Physica A 246 (1997) 430. S.N. Durlauf, Statistical Mechanics Approaches to Socioeconomic Behavior, preprint, 1996. J.B. Ramsey, Z. Zhang, in: Predictability of Complex Dynamical Systems, Springer, Berlin, 1996, p. 189. D. Somette, A. Johansen, J.-P. Bouchaud, J. Phys. l (France) 6 (1996) 167. T. Bollerslev, R.Y. Chou, N. Jayaraman, K.F. Kroner, J. Econometrics 52 (1990) 5. E.E. Peters, Fractal Market Analysis, Wiley, New York, 1994. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1685. H.E. Stanley, S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, M. Simons, Physica A 200 (1996) 4. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York, 1982. J. Feder, Fractals, Plenum, New York, 1988. B.J. West, B. Deering, The Lure of Modem Science, World Scientific, Singapore, 1995. S.D. Howison, F.P. Kelly, P. Wilmott, in: Mathematical Models in Finance, The Royal Society, London, 1994, pp. 449-598. J. Pirard, P. Praet, private communication.
