Pareto-Zipf’s Law in Variability of Financial Time Series
ROBERT KITT, JAAN KALDA
Department of Mechanics an Applied Mathematics
Institute of Cybernetics at Tallinn Technical University
Akadeemia tee 21, 12618, Tallinn
ESTONIA
Abstract: The goal of the paper is to reveal some new facts about financial time series. It is generally accepted
that financial time series are not following the Brownian random walk process, but rather (multi-)fractional
Brownian motion i.e. the fluctuations of financial time series tend to have a memory. While the (multi)fractal
description is adequate for the analysis of long-term dynamics, certain aspects of the variability of prices has
been left out of focus. In this paper the long-term correlations in short-term variability are studied. The trading
data are divided into two categories: “large variability” and “small variability”. These definitions are based on
the relative difference between the current price and the periods’ sliding average, using a (adjustable) threshold
value T. The sequential “low” volatility trading days make up a “silent” period; the “silent” periods are
“ranked” according to their length (measured in the number of trading days N). The rank R of a “silent” period
equals to the number of “silent” periods longer than N. The relationship between the rank R and N was studied
for various financial data. The analysis was conducted using different threshold values T ranging from 0.25% to
3.00%. The time series studied included daily closings of five equity and five currency series for a relatively
long time (each series containing at least 8000 data points). It appears that all the time series studied showed
relatively good Pareto-Zipf’s power-law: the amount of periods is decreasing according to power law. Besides,
the currency time series exhibited “super-universality”: the scaling exponent was similar in all currencies and
all definitions of “large changes” (i.e. T-values), with   -1.75. For equity time series, the scaling exponent
was sensitive with respect to the value of T; however, for a fixed T, different equities were described by similar
scaling exponents . Finally note that there is a simple implication of the very existence of such a power law:
the probability of terminating the “silence” tomorrow is inversely proportional to the length of the current
“silent” period.
Key Words: Pareto-Zipf’s law, scaling, econophysics, multifractality, power law
1. Introduction
Recent decade has opened a new era in financial
analysis. It is known that methods and assumptions
done half-century ago have been opposed and new
methods are introduced. Even a new term:
Econophysics is invented to describe the new
interdisciplinary field of physics. Finance has
gained from econophysics a lot: the econometrics
and time series analysis have got the new methods.
A lot of research is done, but even more is most
likely still ahead.
Econophysics is not only pioneering new field of
science. New methods of non-linear time-series
analysis, developed for econophysics, have made
significant contributions to many fields of physics.
Vice versa, the methods of intermittent time-series
analysis, developed in a different context, can be
successfully used to improve the understanding of
financial time-series.
Some stylized facts about financial time series can
be listed as follows. The time-series exhibit
multifractal structure [1-4]; the increments have
non-Gaussian distribution [5-6]; the autocorrelation
of returns drops quickly to zero [7-8]. This paper is
aimed to search for yet-unknown properties of
these time-series and is motivated by the recent
results about heart-rate variability indicating that
the low-variability periods of heart rate follow a
Zipf’s law [9].
2. Statement of the Problem
Pareto-Zipf’s law is a well-known phenomenon in
various fields of science. Italian economist Pareto
suggested [10] that in different countries and times,
the distribution of income and wealth amount
follows a logarithmic pattern: log N = log A + m
log x, where N is the number of income earners
who receive income higher than x and A and m are
constants.
Zipf found that similar relation is valid for word
distribution in language [11]: suppose every word
has assigned a rank, according to its “size” f,
defined as a relative number of occurrences in
some long text. Then, there is a power law
(actually, inverse proportionality law) between the
rank and size of a word. Such power laws have
been found in a vast variety of systems. A recent
example of Pareto-Zipf’s law in econophysics is
provided by Fujiwara, who found power law
distribution in companies’ bankruptcy events [12].
Also some other developments are recently done by
using Pareto-Zipf’s law [13-15].
The scaling exponent is calculated by using the
least square fit for various values of the threshold
parameter T.
2.2 Data and Analysis
We use daily closing data of different stock
exchange indices and also currencies. Table 1
describes the data used for our analysis.
Table 1 Description of the input data
2.1 The Method
For our analysis we are using the method
developed by Kalda et al [9].
I. We define a local average, which in financial
community is often referred as moving average:
d 1
Pt ,d 
P
t k
k 0
.
d
(1)
Here: Pt denotes security’s price at time t, d denotes
the length of averaging period. In our analysis we
took d = 5.
II. We evaluate each trading day and label them as
follows:

lt   Pt Pt
1

1  T ,
Name
MSCI World
Nikkei 225
DAX
DJIA
S&P500
JPY/USD
FRF/USD
CAD/USD
GBP/USD
DEM/USD
Period
31/12/1971 – 25/11/2003
05/01/1970 – 26/11/2003
01/10/1959 – 26/11/2003
03/01/1900 – 26/11/2003
31/12/1940 – 26/11/2003
04/01/1971 – 26/11/2003
04/01/1971 – 28/11/2003
04/01/1971 – 28/11/2003
04/01/1971 – 28/11/2003
04/01/1971 – 26/11/2003
Frequency # Data points
Daily
8319
Daily
8393
Daily
11068
Daily
26104
Daily
15882
Daily
8377
Daily
8385
Daily
8385
Daily
8370
Daily
8382
The examples of rank-length curves R(n) are
provided in Figures 1.-3. In Figure 1, USDJPY
time series and T = 0.75% is used. The least-square
fit line corresponds to the scaling exponent
  1.6747 .
(2)
where θ(x) denotes the Heaviside function and T is
a threshold to describe a “large” price fluctuation
Fig.1 USDJPY R(n) – n plot in log-log scale for
T=0,75%
10000
III. Following steps I and II results in a new timeseries lt: a sequence of "0"-s and "1"-s. In this
sequence, "0" means that we have a "silent" day
and "1" means that we have a day with a large
fluctuation. As a step III we measure the lengths of
the periods of subsequent days with lt = 0. These
are the "silent" periods where price fluctuations
stayed below the threshold T.
1000
100
10
1
IV Further we define a function R(n), which gives
the number of such “silent” periods, the length of
which is at least n days. For example, R(1) is the
overall number of “silent” periods; R(5) is the
number of “silent” periods with fluctuations staying
below the threshold T for at least 5 days.
V Finally, we plot R(n) against n in log-log
coordinates, see Fig. 1.-3. The linear part of the
curve at the right-hand-side of the plot indicates the
presence of a power law
R  n  .
(3)
1
10
100
1000
0.1
In figure 2, S&P500 equity index is used; the plots
are given for T=0.5%, 1.0%. The scaling exponent
value increases with decreasing T. In figure 3,
USD/DEM exchange rate data are studied using the
same threshold parameter values. The scaling
exponent value is almost independent of T.
Fig.2 S&P 500 R(n) – n plot in log-log scale for
T=0.5% and T=1.0%
100000
10000
1000
100
fluctuations of financial time series are described
by Pareto-Zipf’s power-law.
The respective scaling exponent values are given in
Tables 2 and 3 (in order to get idea about the
“intrinsic” degree of fluctuations, this table
provides also the standard deviation of the input
data).
Table 2 Measured values of  for Equity time
series
10
Name
1
1
10
100
T=0.50%
1000
T=1.0%
Fig 3 USDDEM R(n) – n plot in log-log scale using
T = 0.5%, T = 1.0%
10000
MSCI World
Nikkei225
DAX
DJIA
SP500
Average
St.dev.
0.93%
1.31%
1.32%
1.09%
1.07%
Name
JPY
FRF
CAD
GBP
DEM
Average
10
1
1
10
100
1000
0.1
T=0.50%
T=1.0%
Now, the following natural questions arise:
1. How universal is such a power-law for
financial time-series?
2. How dispersed are the scaling exponent
values for different time-series, but for a
fixed threshold parameter value?
3. For a given time-series, how does the
scaling exponent depend on the threshold
parameter T?
It is clear that choice of the threshold is one of the
important issues. If T is too large, then we would
not see any movements outside the threshold, and
R(n)-curve would be equivalently zero. On the
other hand, if T is too small, the whole time-series
would be a single "large fluctuation" period. A
non-trivial scaling behavior of R(N)-curve appeared
to be provided by the values T {0.25%, 0.5%,
0.75%, 1.0%, 1.5%, 3.0%}. For all these values, the
scaling behavior was reasonably well described by
a power-law. Power law held extremely well in a
region 0.5%< T <1.5%. Therefore we conclude that
0.50%
2.5026
2.5465
2.7991
2.5511
2.6606
2.61
Deviation limit
0.75% 1.00%
2.0563 1.7625
2.0601 1.8555
2.2430 2.1432
2.3931 2.1557
2.1281 1.9800
2.18
1.98
1.50%
1.2623
1.4728
1.8859
1.9221
1.5623
1.62
3.00%
0.8493
1.2104
1.1479
1.2630
1.0544
1.11
Table 3 Measured values of  for Currency time
series
St.dev.
1000
100
0.25%
2.9398
3.1706
3.2746
3.0607
3.5657
3.20
0.72%
0.70%
0.31%
0.67%
0.72%
0.25%
1.6310
1.4511
1.6455
1.6336
1.8076
1.63
0.50%
1.7447
1.6377
1.4941
1.8648
1.7705
1.70
Deviation limit
0.75% 1.00%
1.6747 1.5800
1.5688 1.6172
1.2834 0.8372
1.8464 1.7674
1.7970 1.5784
1.63
1.48
1.50%
1.3486
1.4291
0.5388
1.3400
1.3512
1.20
3.00%
0.6322
0.7903
0.6622
0.7494
0.71
Regarding the question 2, the scaling exponent
values appeared to be universal within both classes
of data (equities and currencies), exhibiting very
limited fluctuations for a fixed threshold parameter
T. However, the scaling exponents’ values between
the two classes were clearly different. Therefore,
one can conclude that this scaling law has captured
certain universal feature of the underlying data.
As for questions 3, one could expect that  is a
decreasing function of T, because larger values of T
means that there are fewer long periods of
"silence", and therefore a less steep fall-off of the
R(n)-curve. While such a dependence was
observed, indeed, for equity indices, in the case of
currencies (except for CAD, which seemed to be a
special case, due to a strong coupling to USD), an
unexpected super-universality was observed: the
scaling exponent was almost constant,  ~ 1.74 for
the range 0.25%  T  1%.
3. Probability of “Silence breaking”
As we have shown, the length-distribution of the
“silent” periods in stock- and currency markets
follows a power law. The very presence of such a
power law has an interesting consequence for the
“silence-breaking” probability. Suppose today is
the n-th day of a “silent” period. What is the
probability p(n) that tomorrow will be a “nonsilent” day with lt = 1? This can be calculated as the
number of “silence-breaking” days at the end of
those “silent” periods, which are not shorter than n,
divided by the overall number M(n) of such “silent”
days, which follow at least n-days-long “silent”
period. Since each “silent” period is terminated by
exactly one “non-silent” day, the first number is
equal to R(n). The second number is calculated as
M (n)   R(m  1)  R(m)(m  n).
mn
Assuming n  1 , the sum can be substituted by
integral; according to the power law,
M
 (m  n)dR(m)  
mn
mn
R(m)
mn
dm   R(m)dm
m
mn
This integral converges for   1 , yielding
M  R(n)n . Thus, the “silence-breaking”
probability
p(n)  R(n) / M (n)  n 1 .
Therefore, we arrived at a super-universal law:
assuming the presence of a power law (3) with
  1, the “silence-breaking” probability is
inversely proportional to the observed “silence”
length.
4. Conclusion
We have shown that the length-distribution of the
"silent" periods in currency and equity markets
follows the Pareto-Zipf’s power law. The “silent”
periods are defined as sequences of such
subsequent days, for which the local index
variability stayed below a threshold level T. It was
established that within the two groups, equities and
currencies, the scaling exponent values were very
similar for a fixed threshold parameter T. For
currencies, a super-universality was observed: the
scaling exponent was also (almost) independent of
T, the values being scattered around  ~ 1.75.
Finally we have shown that the very existence of a
power law for the length-distribution of "silent"
periods implies that the “silence-breaking”
probability (the probability of terminating the
“silence” tomorrow) is inversely proportional to the
length of the current “silent” period.
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