=Paper=
{{Paper
|id=Vol-2713/paper41
|storemode=property
|title=Econophysics of sustainability indices
|pdfUrl=https://ceur-ws.org/Vol-2713/paper41.pdf
|volume=Vol-2713
|authors=Andriy Bielinskyi,Serhiy Semerikov,Oleksandr Serdiuk,Victoria Solovieva,Vladimir Soloviev,Lukáš Pichl
|dblpUrl=https://dblp.org/rec/conf/m3e2/BielinskyiSSSSP20
}}
==Econophysics of sustainability indices==
https://ceur-ws.org/Vol-2713/paper41.pdf
372
Econophysics of sustainability indices
Andrii Bielinskyi1[0000-0002-2821-2895], Serhiy Semerikov1,2,3[0000-0003-0789-0272],
Oleksandr Serdyuk4[0000-0002-1230-0305], Victoria Solovieva5[0000-0002-8090-9569],
Vladimir Soloviev1,4[0000-0002-4945-202X] and Lukáš Pichl6
1 Kryvyi Rih State Pedagogical University, 54 Gagarin Ave., Kryvyi Rih, 50086, Ukraine
2 Kryvyi Rih National University, 11 Vitalii Matusevych Str., Kryvyi Rih, 50027, Ukraine
3 Institute of Information Technologies and Learning Tools of NAES of Ukraine,
9 M. Berlynskoho Str., Kyiv, 04060, Ukraine
4 The Bohdan Khmelnytsky National University of Cherkasy,
81 Shevchenka Blvd., 18031, Cherkasy, Ukraine
5 State University of Economics and Technology,
16 Medychna Str., Kryvyi Rih, 50005, Ukraine
6 International Christian University, 3-10-2 Osawa, Mitaka-shi, Tokyo, 181-8585, Japan
krivogame@gmail.com, semerikov@gmail.com,
serdyuk@ukr.net, vikasolovieva2027@gmail.com,
vnsoloviev2016@gmail.com, lukas@icu.ac.jp
Abstract. In this paper, the possibility of using some econophysical methods for
quantitative assessment of complexity measures: entropy (Shannon,
Approximate and Permutation entropies), fractal (Multifractal detrended
fluctuation analysis – MF-DFA), and quantum (Heisenberg uncertainty principle)
is investigated. Comparing the capability of both entropies, it is obtained that
both measures are presented to be computationally efficient, robust, and useful.
Each of them detects patterns that are general for crisis states. The similar results
are for other measures. MF-DFA approach gives evidence that Dow Jones
Sustainability Index is multifractal, and the degree of it changes significantly at
different periods. Moreover, we demonstrate that the quantum apparatus of
econophysics has reliable models for the identification of instability periods. We
conclude that these measures make it possible to establish that the socially
responsive exhibits characteristic patterns of complexity, and the proposed
measures of complexity allow us to build indicators-precursors of critical and
crisis phenomena.
Keywords: Dow Jones Sustainability Index, measures of complexity,
precursors of stock market crashes.
1 Introduction
Current economic trends have convincingly demonstrated that green development is a
necessary condition for sustainable development, which is essential for a better life in
the future [40]. Economists have described climate change as a global market failure
___________________
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
373
estimating that without action, the rising overall costs of climate could result in losing
at least 5% of global GDP each year. A growing number of financial institutions are
joining in a constructive dialogue on the relationship between economic development,
environmental protection, and sustainable development. Financial institutions,
including banks, insurers, and investors, work with the United Nations Environment
Programme – Finance Initiative to better understand environmental, social, and
governance challenges, why they matter to finance, and how to take steps to address
them [30].
The availability of stock indexes based on sustainability screening makes
increasingly viable for institutional investors the transition to a portfolio based on a
Socially Responsible Investment (SRI) benchmark at a relatively low cost.
The 2008 subprime crisis and increased social awareness have led to a growing
interest in topics related to Socially Responsible Investment. SRI is a long-term
investment that integrates environmental, social, and corporate governance criteria
(ESG). According to the Global Sustainable Investment Alliance (GSIA), SRI reached
24 trillion euro’s in 2016, registering a growth of 25.2% between 2014 and 2016. So,
green and sustainable finance is more important nowadays than ever before [14].
This increased social interest coincides with international initiatives aimed at
developing environmental and social policies on sustainable finance issues, such as the
Action Plan on sustainable finance adopted by the European Commission in March
2018. This plan has three main objectives:
(i) to redirect capital flows towards sustainable investment to achieve sustainable
and inclusive growth,
(ii) to manage financial risks stemming from climate change, environmental
degradation, and social issues, and
(iii) to foster transparency and long-termism in financial and economic activity.
Therefore, the main purpose is to enhance the role of finance and to build an economy
that enables the goals of the Paris Agreement (2015) and the EU for sustainable
development to be reached [15].
The Dow Jones Sustainability Index (DJSI) comprises global sustainability leaders
as identified by SAM. It represents the top 10% of the largest 2,500 companies in the
S&P Global BMI based on long-term economic, environmental, and social criteria [59].
Founded in 1995, RobecoSAM is an investment specialist focused exclusively on
Sustainability Investing [38].
The S&P Global Broad Market Index (BMI) is the only global index suite with a
transparent, modular structure that has been fully float-adjusted since 1989. This
comprehensive, rules-based index series employs a transparent and consistent
methodology across all countries and includes more than 11,000 stocks from 25
developed and 25 emerging markets [39]. The SAM Corporate Sustainability
Assessment (CSA), established by RobecoSAM, is now issued by S&P Global.
RobecoSAM, an asset manager focused entirely on sustainable investing, established
the CSA in 1999. The CSA has become the basis for numerous S&P ESG Indices over
the last two decades attracting billions of USD in assets. Besides, S&P Global acquired
RobecoSAM’s ESG Ratings and Benchmarking businesses which operate out of S&P
Global Switzerland. SAM is a registered trademark of S&P Global. ESG is a generic
374
term used in capital markets and used by investors to evaluate corporate behavior and
to determine the future financial performance of companies. In the conditions of a wide
variety of sustainable development indices, investors need to have a comparative
characteristic of traditional indices with sustainable development indices obtained by
quantitative methods. At the same time, the set of tools of modern financial analysis
took shape in a separate rapidly growing applied science – fintech. Financial technology
(‘fintech’) is emerging as a core disruptor of every aspect of today’s financial system.
Fintech covers everything from mobile payment platforms to high-frequency trading,
and from crowdfunding and virtual currencies to blockchain. In combination, such
forceful innovations will threaten the viability of today’s financial sector business
models, and indeed the effectiveness of current policies, regulations, and norms that
have shaped modern finance.
The use of financial technology innovations is of course not new – but a step change
is now expected with the novel application of several technologies in combination,
notably involving blockchain, the ‘Internet of things’, and artificial intelligence [6]. The
widespread introduction of fintech makes it possible to talk about green finance as a
strategy for the financial sector and broader sustainable development that is relevant
around the world [1; 7; 24; 33]. Green economy, green finance, and green development
are the peculiar coordinates of the phase space in which today it is generally accepted
to evaluate the sustainable development of world civilization.
Financial systems are complex systems and consist of a plurality of interacting
agents possessing the ability to generate new qualities at the level of macroscopic
collective behavior, the manifestation of which is the spontaneous formation of
noticeable temporal, spatial, or functional structures [54]. For many years financial
markets have been attracting the attention of many scientists like engineers,
mathematicians, physicists, and others for the last two decades. Such vast interest
transformed into a branch of statistical mechanics – econophysics [25]. Physics,
economics, finance, sociology, mathematics, engineering, and computer science are
fields of science which, as a result of cross-fertilization, have created the multi-, cross-,
and interdisciplinary areas of science and research such as econophysics and
sociophysics, thriving in the last two and a half decades. These mixed research fields
use knowledge, methodologies, methods, and tools of physics for modeling, explaining,
and forecasting economic and social phenomena and processes. Accordingly,
econophysics is an interdisciplinary research field, applying theories and methods
originally developed by physicists to solve problems in economics, usually those
including uncertainty or stochastic processes, nonlinear dynamics, and evolutionary
games. Obviously, quantitative econophysical methods for studying financial markets
are an interesting and promising area of fintech.
Our research structured as follows. Section 2 contains a brief description of socially
responsive indexes and an analysis of previous work on a comparative quantitative
analysis of this variety of indices. Section 3 describes algorithms for constructing
econophysical measures of complexity based on the informational, (multi-)fractal and
quantum physical properties of a time series. These measures are calculated based on
the DJSI index. Section 4 summarizes the results obtained and indicates the direction
of subsequent studies.
375
2 Review of previous research
In the last 20-25 years, a huge number of social responsibility or sustainability indices
have been created and their number continues to grow [13; 53]. Briefly consider the
most commonly used.
The Dow Jones Sustainability Indices are a family of best-in-class benchmarks for
investors who have recognized that sustainable business practices are critical to
generating long-term shareholder value and who wish to reflect their sustainability
convictions in their investment portfolios (http://www.sustainability-indices.com/).
The family was launched in 1999 as the first global sustainability benchmark and tracks
the stock performance of the world's leading companies in terms of economic,
environmental, and social criteria. Dow Jones Sustainability World Index, the most
important global stock market valuation index of corporate social responsibility.
FTSE4Good was created by the FTSE Group to facilitate investments in companies
that meet globally recognized corporate responsibility standards and constitutes an
important reference point for the establishment of benchmarks and ethical portfolios.
Companies in the FTSE4Good Index have met stringent environmental, social, and
governance criteria, and are therefore potentially better positioned to capitalize on the
benefits of responsible business practice (http://www.ftse.com/).
MSCI is a leading provider of investment decision support tools to investors
globally, including asset managers, banks, hedge funds, and pension funds. MSCI
Global Sustainability Indexes include companies with high ESG ratings relative to their
sector peers (http://www.msci.com/).
CDP (formerly the “Carbon Disclosure Project”) is one of the world’s leading not-
for-profit climate change organizations, assessing transparency in the disclosure of
information on climate change and greenhouse gas emissions, as well as in the
management of water resources (http://www.cdp.net/).
United Nations Global Compact 100 (“GC 100”), a global stock index developed
and released by the UN Global Compact in partnership with the research firm
Sustainalytics (https://www.unglobalcompact.org/). The index lists the 100 companies
which globally outstand for executive leadership commitment and consistent baseline
profitability, as well as their adherence to the Global Compact’s ten principles, on
human rights, labor, environment, and anti-corruption issues.
STOXX Global ESG Leaders Indices, a group of indices based on a fully transparent
selection process of the performance, in terms of sustainability, of 1,800 companies
worldwide (http://www.stoxx.com/). The ratings are calculated for three sub-areas –
environmental, social, and governance – and are then combined to form the overall
index. The indices are managed by STOXX, the owner of some of the most important
international stock indices, such as the STOXX50.
In our previous work [11], we performed a comparative analysis of the index DJSI
[62] with its classic and traditional counterpart – the index Dow Jones Industrial
Average (DJIA) [61].
In a comparative analysis of structural and dynamic properties of traditional stock
market indices and social responsibility indices, descriptive statistics methods are used
in most works [2; 31; 34; 43].
376
Descriptive statistics (mean, maximum, minimum, and standard deviation) of the
financial information required to apply the Ohlson [34] valuation model reviewed in
[31]. They were examining whether sustainability leadership – proxied by the
membership of the Dow Jones Sustainability Index Europe – is value relevant for
investors on the 10 major European stock markets over the 2001–2013 period. These
results reveal that there exist significant differences across markets.
The article [43] analyzes rate-of-return and risk related to investments in socially
responsible and conventional country indices. The socially responsible indices are the
DJSI Korea, DJSI US, and Respect Index, and the corresponding conventional country
indices are the Korea Stock Exchange Composite KOSPI, DJIA, and WIG20. Shown,
that conclude that investing in the analyzed SRI indices do not yield systematically
better results than investing in the respective conventional indices, both in terms of
neoclassical risk and return rate.
The authors [2] examined sustainable investment returns predictability based on the
US DJSI and a wide set of uncertainty and financial distress indicators for the period
January 2002 to December 2014. They employ a novel nonparametric causality-in-
quantile approach that captures nonlinearity in returns distribution. The authors
conclude that the aggregate Economic Policy Uncertainty indicator and some
components have predictive ability for real returns of the US sustainable investments
index. Paper [55] explores the relationship between sustainability performance and
financial performance by looking at the impact of sustainability index changes on the
market value of a company. The author has studied the price effects of changes in the
DJSI and FTSE4Good Index. He failed to observe statistically significant positive
abnormal returns for companies being added to a sustainability index. On the opposite,
he finds negative abnormal returns for companies being deleted from the FTSE,
however not in the case of the DJSI. This can be explained by studying the volume
effects and the behavior of investment managers.
However, the first works appeared using more modern methods of analysis, using
the achievements of nonlinear dynamical systems and complexity theory [16; 26; 29;
32; 58]. The authors [58] constructed a sustainable regional green economy
development index system from five aspects - economic, social, technological,
resources, and environmental - using DPSIR (drivers, pressures, state, impact, response
model) and entropy-TOPSIS (a technique for order preference by similarity to an ideal
solution) coupling coordination to horizontally and vertically quantitatively analyze the
sustainable green economy development. The model was verified by the actual situation
of green economy development in Shandong Province from 2010 to 2016, which
confirmed the feasibility of the method.
A sustainable development capacity measure model for Sichuan Province was
established by applying the information entropy calculation principle and the
Brusselator principle [26]. Each subsystem and entropy change in a calendar year in
Sichuan Province were analyzed to evaluate Sichuan Province’s sustainable
development capacity. It was found that the established model could effectively show
actual changes in sustainable development levels through the entropy change reaction
system, at the same time this model could clearly demonstrate how those forty-six
indicators from the three subsystems impact on the regional sustainable development,
377
which could make up for the lack of sustainable development research.
A similar approach is implemented to measure the tourist attractiveness of the region
[16]. And in work [29] information and entropy theory used for the sustainability of
coupled human and natural systems.
Authors of [32] used R/S analysis to calculate the Hurst exponent as a measure of
persistence (efficiency of traditional stock market indices and social responsibility
stock market indices). The presence of persistence was evidence in favor of less
efficiency. According to empirical results, SRI has lower efficiency, in particular the
Dow Jones Sustainability Index. Lower efficiency was also observed in the emerging
markets with a responsible investment segment, compared to the traditional stock
market indices.
In paper [27] authors suggest three new indicators based on an engineering approach
of irreversibility. They allow evaluating both the technological level and the
environmental impact of the production processes and the socio-economic conditions
of the countries. Indeed, they are based on the energy analysis and on the irreversible
thermodynamic approach, in order to evaluate the inefficiency both of the process and
of the production systems, and the related consequences. Three applications are
summarized in order to highlight the possible interest from different scientists and
researchers in engineering, economy, etc. [19; 23], in order to develop sustainable
approaches and policies for decision-makers.
All mentioned measures can capture nonlinearity and complexity that peculiar even
for sustainability indices. Analysis of previous papers [47; 48; 49; 51; 52] shows that
indicators have theoretical perspectives and, in accordance with other studies, such
approaches are presented to be robust and computationally efficient. In some aspects,
the results of the multifractal analysis are presented to be better, but the computational
costs leave a lot to be desired. Therefore, due to computationally efficiency and ability
to monitor, and prevent crisis events in advance, the entropy measures present to be the
most attractive. However, empirical results of quantum and multifractal measures
present to be optimal that motivate further research work.
3 Econophysical measures of DJSI complexity and
precursors of crisis states
In a series of recent works [4; 44; 45; 50], we have demonstrated the possibility of using
the theory of complex systems and a set of developed analysis tools to calculate the
corresponding measures of system complexity. These complexity measures make it
possible to differentiate systems according to the degree of their functionality, to
identify and prevent critical and crisis phenomena.
Since the DJSI index is used as a calculation base, we will provide more detailed
information for it. DJSI measures the performance of companies selected for economic,
environmental, and social criteria that weighted by market capitalization using a best-
in-class approach. In assessing sustainability, the key factor in selecting components
for any DJSI index is the overall company sustainability rating (TSS). The first CSA
was undertaken in 1999 with the launch of the original DJSI family of indexes. The
378
annual CSA process begins in March of each year and is published with new estimates
in September. The index is calculated using the divisor methodology that is used for all
Dow Jones Index stock indices. Indices are calculated daily throughout the calendar
year. The exception is those days when all exchanges on which the index constituents
are quoted are officially closed or if the WM Reuters exchange rate services are not
published.
The table 1 and figure 1 provide information on the key companies in the index
basket and the weight of the respective economic sectors to which they belong.
Table 1. Top 10 components by index weight.
No Constituent Symbol Sector
1 Microsoft Corp MSFT Information
2 Technology Alphabet Inc C GOOG Communication Services
3 Nestle SA Reg NESN Consumer
4 Staples United health Group Inc UNH Health Care
5 Taiwan Semiconductor Manufacturing Co Ltd 2330 Information
6 Technology Roche Hldgs AG Ptg Genus ROG Health Care
7 Adobe Inc. ADBE Information
8 Technology Novartis AG Reg NOVN Health Care
9 Cisco Systems Inc CSCO Information
10 Technology Bank of America Corp BAC Financials
Fig. 1. Weights for each sector of the index, %.
For the daily DJSI time series { ( )| = 1, … , } we will carry out calculations of the
corresponding measures of complexity within the framework of the moving window
algorithm. For this purpose, the part of the time series (window), for which there were
379
calculated measures of complexity, was selected, then the window was displaced along
with the time series in a predefined value, and the procedure repeated until all the
studied series had exhausted. Further, comparing the dynamics of the actual time series
and the corresponding measures of complexity, we can judge the characteristic changes
in the dynamics of the behavior of complexity with changes in the cryptocurrency. If
this or that measure of complexity behaves in a definite way for all periods of crashes,
for example, decreases or increases during the pre-crashes or pre-critical period, then it
can serve as their indicator or precursor.
The returns over some time scale Δ is defined as the forward changes in the
logarithm of the corresponding time series: ( ) ≡ ( + Δ )/ ( ). We will
determine standardized returns ( ) ≡ [ ( ) − ⟨ ⟩]/ , where ≡ ⟨ ⟩ − ⟨ ⟩ is
the standard deviation of G, and ⟨… ⟩ denotes the average over the time period under
study.
In our previous paper [11] we devoted to a comparative analysis complexity of
traditional stock market indices and social responsible indices in the example Dow
Jones Sustainability Indices and Dow Jones Industrial Average. As measures of
complexity, the entropies of various recurrence indicators are chosen – the entropy of
the diagonal lines of the recurrence diagram, recurrence probability density entropy and
recurrence entropy. It is shown that these measures make it possible to establish that
the socially responsive Dow Jones index is more complex. In this paper, we will
continue to use econophysical measures of complexity, considering other than recurrent
entropy measures, as well as fractal and quantum measures of complexity in relation to
the index DJSI.
4 Entropy complexity measures for an index DJSI
The most important quantity that allows us to parameterize complexity in deterministic
or random processes is entropy. Originally, it was introduced by Clausius [8], in the
context of classical thermodynamics, where according to his definition, entropy tends
to increase within an isolated system, forming the generalized second law of
thermodynamics. Then, the definition of entropy was extended by Boltzmann and
Gibbs [5; 18], linking it to molecular disorder and chaos to make it suitable for
statistical mechanics, where they combined the notion of entropy and probability.
After the fundamental paper of Shannon [42] in the context of information theory,
where entropy denoted the average amount of information contained in the message,
its notion was significantly redefined. After this, it has been evolved along with
different ways and successful enough used for the research of economic systems [57].
A huge amount of different methods, as an example, from the theory of complexity,
the purpose of which is to quantify the degree of complexity of systems obtained from
various sources of nature, can be applied in our study. Such applications have been
studied intensively for an economic behavior system.
The existence of patterns within the series is the core in the definition of randomness,
so it is appropriate to establish such methods that will be based on the different patterns
and their repetition [9]. In this regard, Pincus described the methodology Approximate
380
entropy (ApEn) [37] to gain more detail analysis of relatively short and noisy time
series, particularly, of clinical and psychological. Pincus and Kalman [36], considering
both empirical data and models, including composite indices, individual stock prices,
the random-walk hypothesis, Black-Sholes, and fractional Brownian motion models to
demonstrate the benefits of ApEn applied to the classical econometric modeling
apparatus. This research the usefulness of ApEn on the example of three major events
of the stock market crash in the US, Japan, and India. During the major crashes, there
is significant evidence of a decline of ApEn during and pre-crash periods. Based on the
presented results, their research concludes that ApEn can serve as a base for a good
trading system. Duan and Stanley [12] showed that it is possible to effectively
distinguish the real-world financial time series from random-walk processes by
examining changing patterns of volatility, ApEn, and the Hurst exponent. The empirical
results prove that financial time series are predictable to some extent and ApEn is a
good indicator to characterize the predictable degree of financial time series. Alfonso
Delgado-Bonal [10] gives evidence of the usefulness of ApEn. The researcher
quantifies the existence of patterns in evolving data series. In general, his results present
that degree of predictability increases in times of crisis.
Permutation entropy (PEn), according to the previous approach, is a complexity
measure that is related to the original Shannon entropy (ShEn) that applied to the
distribution of ordinal patterns in time series. Such a tool was proposed by Bandt and
Pompe [3], which is characterized by its simplicity, computational speed that does not
require some prior knowledge about the system, strongly describes nonlinear chaotic
regimes. As an example, Henry and Judge [20] applied PEn to the Dow Jones Industrial
Average to extract information from this complex economic system. The result
demonstrates the ability of the PEn method to detect the degree of disorder and
uncertainty for the specific time that is explored.
4.1 Approximate entropy
When ApEn is calculated, for a given time series { ( )| = 1, … , }, non-negative
embedding parameter , with ≤ , and a filter r we construct subsequences
⃗ ( )=[ ( ), ( + 1), … , ( + − 1)] and ⃗ ( )=[ ( ), ( + 1), … , ( + − 1)].
The relative neighborhoods in phase space are measures by ∞ norm between all pairs
of ⃗ ( ) and ⃗ ( ). Then, for each = 1, … , − + 1 we count the number of
= 1, … , − + 1 that lie within a suitable distance r and define it as ( ). For
further estimations, we need to define the probability of finding such patters of the
length that will be similar to the given pattern:
( )
( )= ,
( )
or it can be presented in an equivalent form
( )= ∑ ( − ⃗ ( ), ⃗ ( ) ),
where (⋅) is the Heaviside function which counts the instances where the distance d
381
is below the threshold r.
Next, we define the logarithmic average over all the vectors of the ( )
probability as
( )= ∑ ( ( ))
( )
and ApEn of a corresponding time series is defined as an increment of the absolute
entropy ( ) during the transition from a sequence of patterns of length to a
sequence of length + 1 according to the following formula:
( , )= ( )− ( ), (1)
i.e., equation (1) measures the logarithmic likelihood that sequences of patterns that are
close for observations will remain close after further comparisons. Therefore, if the
patterns in the sequence remain close to each other (high regularity), the ApEn becomes
small, and hence, the time series data has a lower degree of randomness. High values
of ApEn indicate randomness and unpredictability. But it should be considered that
ApEn results are not always consistent, thus it depends on the value of r and the length
of the data series. However, it remains insensitive to noise of magnitude if the values
of r and dE are sufficiently good, and it is robust to artefacts and outliers. Although
ApEn remains usable without any models, it also fits naturally into a classical
probability and statistics frameworks, and, generally, despite its shortcomings, it is still
the applicable indicator of system stability, which significantly increased values may
prognosticate the upcoming changes in the dynamics of the data.
The empirical results for the corresponding measure of entropy of DJSI are presented
in figure 2:
Fig. 2. ApEn dynamics of the entire time series of DJSI.
Long before the crisis, the value of this type of entropy begins to decrease, the
complexity of the system decreases. This measure, in our opinion, is one of the earliest
precursors of the crisis.
382
4.2 Permutation entropy
According to this method, we need to consider “ordinal patterns” that consider the order
among time series and relative amplitude of values instead of individual values. For
evaluating PEn, at first, we need to consider a time series { ( )| = 1, … , } which
relevant details can be “revealed” in -dimensional vector:
⃗ ( ) = [ ( ), ( + ), . . . , ( +( − 1) )],
where = 1, 2, … , − ( − 1) , and is an embedding delay of our time delayed
vector. By the ordinal pattern, related to the time i, we consider the permutation
( ) = ( , ,…, ) of [0, 1, … , − 1] where 1 ≤ ≤ !. Then each of the
subvectors is arranged in ascending order:
( + )≤ ( + )≤⋯≤ ( + ).
We will use ordinal pattern probability distribution as a basis for entropy estimation.
Further, the relative frequencies of permutations in the time series are defined as
ℎ ℎ ℎ
( )= ,
( )
where the ordinal pattern probability distribution is given by
= { ( )| = 1, … , !}. The Permutation entropy (denoted by [ ]) of the
corresponding time series presented in the following form:
!
[ ] = −∑ .
Then, to take more convenient values, we calculate Normalized permutation entropy as
[ ]
[ ]=
whose represents the maximum value of [ ] (a normalization constant),
and normalized entropy restricted between 0 and 1. Here, the maximal entropy value is
realized when all ! possible permutations are uniformly distributed (more noise and
random data). With the much lower entropy value, we get a more predictable and
regular sequence of the data. Therefore, the PEn gives a measure of the departure of the
time series from a complete noise and stochastic time series.
In figure 3 we can observe the empirical results for permutation entropy, where it
serves as indicator-precursor of the possible crashes and critical events.
Information measures of complexity due to their initial validity and transparency,
ease of implementation and interpretation of the results occupy a prominent place
among the tools for the quantitative analysis of complex systems.
383
Fig. 3. PEn dynamics of the entire time series of DJSI.
5 Fractal and multifractral measures of complexity
The economic phenomena that cannot be explained by the traditional efficient market
hypothesis can be explained by the fractal theory proposed by Mandelbrot [28]. Before,
fractal studies focus on the Rescaled Range (R/S) analysis were proposed by Hurst [21]
in the field of hydrology. Peng et al. [35] proposed a widely used Detrended Fluctuation
Analysis (DFA) that uses a long-range power-law to avoid significant long-range
autocorrelation false detection. As a multifractal extension (MF) of the DFA approach,
Kantelhardtet et al. [22] introduced the MF-DFA method that for a long time has been
successfully applied for a variety of financial markets. An especially interesting
application of multifractal analysis is measuring the degree of multifractality of time
series, which can be related to the degree of efficiency of financial markets [56].
Similarly to our article [17] where we applied the MF-DFA method to Ukrainian and
Russian stock markets, we use it here to explore the multifractal property of DJSI and
construct a reliable indicator for it.
As an extension to the original DFA, the multifractal approach estimates the Hurst
exponent of a time series at different scales. Based on a given time series
{ ( )| = 1, … , }, the MF-DFA is described as follows:
1. Construct the profile ( ) (accumulation) according to the equation below
()=∑ ( ( ) − ⟨ ⟩),
where ⟨ ⟩ denotes the average of returns.
2. Then we need to divide the profile { ( )} into ≡ ( / ) non-overlapping
segments of equal length s, and the local trend for each segment is calculated
by the least-square fit. Since time scale s is not always a multiple of the length of
the time series, a short period at the end of the profile, which is less than the window
size, may be removed. For taking into account the rejected part and, therefore, to use
384
all the elements of the sequence, the above procedure is repeated starting from the
end of the opposite side. Therefore, the total 2 segments are obtained together,
and the variance is computed as
( , )= ∑ [ (( − 1) + ) − ( )] , for = 1, … ,
and
( , )= ∑ [ ( −( − ) + ]− ( )] , for = + 1, … , 2 .
Various types of MF-DFA such as linear, quadratic, or higher order polynomials can
be used for eliminating local trend in segment .
3. Considering the variability of time series and the possible multiple scaling
properties, we obtain the q-th order fluctuation function by averaging over all
segments:
( )= ∑ [ ( , )] .
The index q can take any non-zero value. For q = 0, ( ) is divergent and can be
replaced by an exponential of a logarithmic sum
( )= ∑ ( ( , )) .
4. At least, we determine the scaling behavior of the fluctuation function by analyzing
( ) vs graphs for each value of q. Here, ( ) is expected to reveal
power-law scaling
ℎ( )
( )∼ .
The scaling exponent h(q) can be considered as generalized Hurst exponent. With q = 2
MF-DFA transforms into standard DFA, and h(2) is the well-known Hurst exponent.
5. Another way of characterizing multifractality of a time series is in terms of the
multifractal scaling exponent ( ) which is related to the generalized Hurst exponent
h(q) from the standard multifractal formalism and given by
( ) = ℎ( ) − 1. (2)
Equation (2) reflects temporal structure of the time series as a function of moments q
i.e., it represents the scaling dependence of small fluctuations for negative values of q
and large fluctuations for positives values. If (2) represents linear dependence of q, the
time series is said to be monofractal. Otherwise, if it has a non-linear dependence on q,
then the series is multifractal.
6. The different scalings are better described by the singularity spectrum ( ) which
can be defined as:
385
( ) ℎ( )
= = ℎ( ) + ,
( ) = [ − ℎ( )] + 1,
with is the Hölder exponent or singularity strength. Following the methods described
above, we present results that reflect multifractal behavior of the DJSI time series.
Fig. 4(a) presents ( ) in the log-log plot. The slope changes dependently on q,
which indicates the multifractal property of a time series. As it was pointed out,
multifractality emerges not only because of temporal correlation, but also because DJSI
returns distribution turns out to be broad (fat-tailed), and this distribution could
contribute to the multifractality of the time series. The same dependence can be
observed in the remaining plots. The scaling exponent ( ) remains nonlinear, as well
as generalized Hurst exponents that can serve as evidence that Bitcoin exhibit
multifractal property.
(a) (b)
(c) (d)
Fig. 4. The fluctuation function ( ) (a), multifractal scaling exponent ( ) (b), ℎ( ) versus q
(c), and singularity spectrum ( ) (d) of the DJSI return time series obtained from MF-DFA.
In the case of multifractals, the shape of the singularity spectrum typically resembles
an inverted parabola (see Fig. 4(d)); furthermore, the degree of complexity is
straightforwardly quantified by the width of ( ), simply defined as
386
Δ = − , where and correspond to the opposite ends of the
values as projected out by different q-moments.
In the figure below we present the width of the spectrum of multifractality that
changes over time accordingly to the sliding window approach. The whole figure
consists of both a three-dimensional plot (singularity spectrum) and two-dimensional
representation of its surface (fig. 5).
Fig. 5. Changes in the spectrum of multifractality in time.
If the series exhibited a simple monofractal scaling behavior, the value of singularity
spectrum ( ) would be a constant. As can be observed, here our series exhibits a
simple multifractal scaling behavior, as the value of singularity spectrum ( ) changes
dependently on , i.e., it exhibits different scalings at different scales. Moreover, with
the sliding window of the corresponding length, we understand that at different time
periods DJSI becomes more or less complex. The value of Δ gives a shred of
additional evidence on it (fig. 6).
As we can see from the presented results, the width of the singularity spectrum after
the crisis starts to increase, which tells us that more violent price fluctuations are usually
expected. With the decreasing width of the singularity spectrum, the series is expected
to hold the trend. As the rule, it reaches its minimum before the crash of DJSI value.
6 Heisenberg uncertainty principle and economic “mass”
as a quantum measure of complexity
In this section, we will demonstrate the possibilities of quantum econophysics on the
example of the application of the Heisenberg uncertainty principle [46]. In our paper
[41], we have suggested a new paradigm of complex systems modeling based on the
ideas of quantum as well as relativistic mechanics. It has been revealed that the use of
387
quantum-mechanical analogies (such as the uncertainty principle, the notion of the
operator, and quantum measurement interpretation) can be applied to describing socio-
economic processes. Methodological and philosophical analysis of fundamental
physical notions and constants, such as time, space, and spatial coordinates, mass,
Planck’s constant, light velocity from modern theoretical physics provides an
opportunity to search for adequate and useful analogs in socio-economic phenomena
and processes.
Fig. 6. The comparison of the DJSI time series with the width of the multifractality spectrum
measure.
To demonstrate it, let us use the known Heisenberg’s uncertainty ratio which is the
fundamental consequence of non-relativistic quantum mechanics axioms and appears
to be
ℏ
Δ ⋅Δ ≥ , (3)
where Δ and Δ are mean square deviations of x coordinate and velocity
corresponding to the particle with (rest) mass m0, ℏ – Planck’s constant. Considering
values Δ and Δ to be measurable when their product reaches their minimum,
according to equation (3) we derive:
ℏ
= ,
⋅Δ ⋅Δ
i.e., the mass of the particle is conveyed via uncertainties of its coordinate and velocity
– time derivative of the same coordinate.
Economic measurements are fundamentally relative, local in time, space and other
socio-economic coordinates, and can be carried out via consequent and/or parallel
comparisons “here and now,” “here and there,” “yesterday and today,” “a year ago and
now,” etc.
Due to these reasons constant monitoring, analysis, and time series prediction (time
series imply data derived from the dynamics of stock indices, exchange rates,
cryptocurrencies prices, spot prices, and other socio-economic indicators) become
388
relevant for the evaluation of the state, tendencies, and perspectives of global, regional,
and national economies.
Suppose there is a set of K time series, each of N samples, that correspond to the
single distance T, with an equally minimal time step Δ :
( ), =Δ , for = 0, 1, 2, . . . , − 1, for = 1, 2, . . . , .
To bring all series to the unified and non-dimensional representation, accurate to the
additive constant, we normalize them, have taken a natural logarithm of each term of
the series. Then, consider that every new series ( ) is a one-dimensional trajectory
of a certain fictitious or abstract particle numbered i, while its coordinate is registered
after every time span Δ , and evaluate mean square deviations of its coordinate and
speed in some time window Δ = Δ ⋅ Δ = Δ , 1<< ΔN< , − < > , ~ ,
( ) ( )
where mi – economic “mass” of an series, h – value which comes as an economic
Planck’s constant.
Since the analogy with physical particle trajectory is merely formal, h value, unlike
the physical Planck’s constant ℏ, can, generally speaking, depend on the historical
period, for which the series are taken, and the length of the averaging interval (e.g.,
economical processes are different in the time of crisis and recession), on the series
number i etc. Whether this analogy is correct or not depends on the particular series’
properties.
Fig. 7. Dynamics of measure m, and its dynamics with the window size of 250 days and step of
5 days.
389
Obviously, there is a dynamic characteristic values m depending on the internal
dynamics of the market. In times of crashes and critical events marked by arrows, mass
m is significantly reduced in the pre-crash and pre-critical periods (fig. 7). Obviously,
m remains a good indicator-precursor even in this case. Value m is considerably reduced
before a special market condition. The market becomes more volatile and prone to
changes.
7 Conclusions
In this paper, for the first time, econophysical measures of complexity based on the
analysis of entropy, multifractal, and quantum properties of time series are used for the
analysis of sustainable development indices. Using the DJSI index as an example, it is
shown that, firstly, all econophysical measures are complex measures and, secondly,
they respond to critical and crisis conditions of the stock market.
In the future, a similar study for a set of other indices would be of interest, as well
as a comparison with the results of using other quantitative measures of complexity.
References
1. Accelerating Green Finance: A report to Government by the Green Finance Taskforce.
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_d
ata/file/703816/green-finance-taskforce-accelerating-green-finance-report.pdf (2018).
Accessed 25 Oct 2018
2. Antonakakis, N., Babalos, V., Kyei, C.K.: Predictability of sustainable investments and the
role of uncertainty: Evidence from a non-parametric causality-in-quantiles test. Applied
Economics 48(48), 4655–4665 (2016). doi:10.1080/00036846.2016.1161724
3. Bandt, C., Pompe, B.: Permutation Entropy: A Natural Complexity Measure for Time
Series. Phys. Rev. Lett. 88, 174102 (2002). doi:10.1103/PhysRevLett.88.174102
4. Bielinskyi, A., Soloviev, V., Semerikov, S.: Detecting Stock Crashes Using Levy
Distribution. CEUR Workshop Proceedings 2422, 420–433 (2019)
5. Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen.
Sitzungsberichte Akademie der Wissenschaften 66, 275–370 (1872).
doi:10.1142/9781848161337_0015
6. Castilla-Rubio, J.C., Zadek, S., Robins, N.: Fintech and sustainable development: Assessing
the implications. International Environment House, Geneva. http://unepinquiry.org/wp-
content/uploads/2016/12/Fintech_and_Sustainable_Development_Assessing_the_Implicat
ions.pdf (2016). Accessed 21 Mar 2017
7. Cen, T., He, R.: Fintech, Green Finance and Sustainable Development. Advances in Social
Science, Education and Humanities Research 291, 222–225 (2018). doi:10.2991/meeah-
18.2018.40
8. Clausius, R., Hirst, T.A.: The Mechanical Theory of Heat: With Its Applications to the
Steam-engine and to the Physical Properties of Bodies. John van Voorst, London (1867)
9. Delgado-Bonal, A., Marshak, A.: Approximate Entropy and Sample Entropy: A
Comprehensive Tutorial. Entropy 21(6), 541 (2019). doi:10.3390/e21060541
10. Delgado-Bonal, A.: Quantifying the randomness of the stock markets. Scientific Reports 9,
12761 (2019). doi:10.1038/s41598-019-49320-9
390
11. Derbentsev, V., Semerikov, S., Serdyuk, O., Solovieva, V., Soloviev, V.: Recurrence based
entropies for sustainability indices. E3S Web of Conferences 166, 13031 (2020).
doi:10.1051/e3sconf/202016613031
12. Duan, W.-Q., Stanley, H.: Volatility, irregularity, and predictable degree of accumulative
return series. Phys. Rev. E 81, 066116 (2010). doi: 10.1103/PhysRevE.81.066116
13. Durand, R., Paugam, L., Stolowy, H.: Do investors actually value sustainability indices?
Replication, development, and new evidence on CSR visibility. Strategic Management
Journal 40(9), 1471–1490 (2019). doi:10.1002/smj.3035
14. Escrig-Olmedo, E., Fernández-Izquierdo, M.A., Ferrero-Ferrero, I., Rivera-Lirio, J.M.,
Muñoz-Torres, M.J.: Rating the Raters: Evaluating how ESG Rating Agencies Integrate
Sustainability Principles. Sustainability 11(3), 915 (2019). doi:10.3390/su11030915
15. Fabregat-Aibar, L., Barberà-Mariné, M.G., Terceño, A., Pié, L.: A Bibliometric and
Visualization Analysis of Socially Responsible Funds. Sustainability 11(9), 2526 (2019).
doi:10.3390/su11092526
16. Feng, H., Chen, X., Heck, P., Miao, H.: An Entropy-Perspective Study on the Sustainable
Development Potential of Tourism Destination Ecosystem in Dunhuang, China.
Sustainability 6(12), 8980–9006 (2014). doi:10.3390/su6128980
17. Ganchuk, A., Derbentsev, V., Soloviev, V. N. Multifractal Properties of the Ukraine Stock
Market. arXiv:physics/0608009v1 [physics.data-an] (2006). Accessed 17 Aug 2020
18. Gibbs, J.W.: Elementary Principles in Statistical Mechanics: Developed with Especial
Reference to the Rational Foundation of Thermodynamics. C. Scribner's Sons, New York
(1902). doi:10.5962/bhl.title.32624
19. Havrylenko, M., Shiyko, V., Horal, L., Khvostina, I., Yashcheritsyna, N.: Economic and
mathematical modeling of industrial enterprise business model financial efficiency
estimation. E3S Web of Conference 166, 13025 (2020).
doi:10.1051/e3sconf/202016613025
20. Henri, M., Judge, G.: Permutation Entropy and Information Recovery in Nonlinear
Dynamic Economic Time Series. Econometrics 7(1), 10 (2019).
doi:10.3390/econometrics7010010
21. Hurst, H.E.: A Suggested Statistical Model of some Time Series which occur in Nature.
Nature 180, 494 (1957). doi:10.1038/180494a0
22. Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley,
H.E.: Multifractal detrended fluctuation analysis of nonstationary time series. Physica A:
Statistical Mechanics and its Applications 316(1–4), 87–114 (2002). doi:10.1016/S0378-
4371(02)01383-3
23. Khvostina, I., Havadzyn, N., Horal, L., Yurchenko, N.: Emergent Properties Manifestation
in the Risk Assessment of Oil and Gas Companies. CEUR Workshop Proceedings 2422,
157–168 (2019)
24. Kung, O.Y.: "Green Finance for a Sustainable World" - Keynote Speech by Mr Ong Ye
Kung, Minister for Education, Singapore and Board Member, Monetary Authority of
Singapore, at SFF x SWITCH 2019 on 11 November 2019.
https://www.mas.gov.sg/news/speeches/2019/green-finance-for-a-sustainable-world
(2019). Accessed 28 Nov 2019
25. Kutner, R., Ausloos, M., Grech, D., Matteo, T.Di., Schinckus, C., Stanley, H.E.:
Econophysics and sociophysics: Their milestones & challenges. Physica A: Statistical
Mechanics and its Applications 516, 240–253 (2019). doi:10.1016/j.physa.2018.10.019
26. Liang, X., Si, D., Zhang, X.: Regional Sustainable Development Analysis Based on
Information Entropy-Sichuan Province as an Example. International Journal of
Environmental Research and Public Health 14(10), 1219 (2017).
391
doi:10.3390/ijerph14101219
27. Lucia, U., Grisolia, G: Exergy inefficiency: An indicator for sustainable development
analysis. Energy Reports 5, 62–69 (2019). doi:10.1016/j.egyr.2018.12.001
28. Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New
York (1982)
29. Mayer, A.L., Donovan, R.P., Pawlowski, C.W.: Information and entropy theory for the
sustainability of coupled human and natural systems. Ecology and Society 19(3), 11 (2014).
doi:10.5751/ES-06626-190311
30. Mills, M., Wardle, M.: The Global Green Finance Index 4 (2019).
doi:10.13140/RG.2.2.28337.33124
31. Miralles-Quiros, M.M., Miralles-Quiros, J.L., Arraiano, I.G.: Sustainable Development,
Sustainability Leadership and Firm Valuation: Differences across Europe. Business
Strategy and the Environment 26(7), 1014–1028 (2017). doi:10.1002/bse.1964
32. Mynhardt, H., Makarenko, I., Plastun, A.: Market efficiency of traditional stock market
indices and social responsible indices: the role of sustainability reporting. Investment
Management and Financial Innovations 14(2), 94-106 (2017).
doi:10.21511/imfi.14(2).2017.09
33. Nassiry, D.: The Role of Fintech in Unlocking Green Finance: Policy Insights for
Developing Countries. ADBI Working Paper Series 883.
https://www.adb.org/sites/default/files/publication/464821/adbi-wp883.pdf (2018).
Accessed 25 Oct 2019
34. Ohlson, J.A.: Earnings, Book Values, and Dividends in Equity Valuation: An Empirical
Perspective. Contemporary Accounting Research 18(1), 107–120 (2001).
doi:10.1506/7TPJ-RXQN-TQC7-FFAE
35. Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.:
Mosaic organization of DNA nucleotides. Physical Review E 49, 1685 (1993).
doi:10.1103/PhysRevE.49.1685
36. Pincus, S., Kalman, R.E.: Irregularity, volatility, risk, and financial market time series.
PNAS 101(38), 13709–13714 (2004). doi: 10.1073/pnas.0405168101
37. Pincus, S.M.: Approximate entropy as a measure of system complexity. PNAS 88(6), 2297–
2301 (1991). doi:10.1073/pnas.88.6.2297
38. RobecoSAM: About RobecoSAM. https://www.robecosam.com/en/about-us/about-
robecosam.html (2020). Accessed 17 Aug 2020
39. S&P Dow Jones Indices: S&P Global BMI.
https://www.spglobal.com/spdji/en/indices/equity/sp-global-bmi (2020). Accessed 17 Aug
2020
40. S&P Global Switzerland: The Sustainability Yearbook 2020.
https://www.spglobal.com/esg/csa/yearbook/ (2020). Accessed 17 Aug 2020
41. Saptsin, V., Soloviev, V.: Relativistic quantum econophysics – new paradigms in complex
systems modelling. arXiv:0907.1142v1 [physics.soc-ph] (2009). Accessed 17 Aug 2020
42. Shannon, C.E.: A mathematical theory of communication. The Bell System Technical
Journal 27(3), 379–423 (1948). doi:10.1002/j.1538-7305.1948.tb01338.x
43. Śliwiński, P., Łobza, M.: Financial Performance of Socially Responsible Indices.
International Journal of Management and Economics 53(1), 25–46 (2017)
44. Soloviev, V., Belinskij, A.: Methods of nonlinear dynamics and the construction of
cryptocurrency crisis phenomena precursors. CEUR Workshop Proceedings 2104, 116–127
(2018)
45. Soloviev, V., Bielinskyi, A., Solovieva, V.: Entropy Analysis of Crisis Phenomena for DJIA
Index. CEUR Workshop Proceedings 2393, 434–449 (2019)
392
46. Soloviev, V., Saptsin, V.: Heisenberg uncertainty principle and economic analogues of
basic physical quantities. arXiv:1111.5289v1 [physics.gen-ph] (2011). Accessed 17 Aug
2020
47. Soloviev, V., Semerikov, S., Solovieva, V.: Lempel-Ziv Complexity and Crises of
Cryptocurrency Market. Advances in Economics, Business and Management Research
129, 299–306 (2020). doi:10.2991/aebmr.k.200318.037
48. Soloviev, V., Serdiuk, O., Semerikov, S., Kohut-Ferens, O.: Recurrence entropy and
financial crashes. Advances in Economics, Business and Management Research 99, 385–
388 (2019). doi:10.2991/mdsmes-19.2019.73
49. Soloviev, V., Solovieva, V., Tuliakova, A., Ivanova, M.: Construction of crisis precursors
in multiplex networks. Advances in Economics, Business and Management Research 99,
361–366 (2019). doi:10.2991/mdsmes-19.2019.68
50. Soloviev, V.N., Belinskiy, A..: Complex Systems Theory and Crashes of Cryptocurrency
Market. In: Ermolayev V., Suárez-Figueroa M., Yakovyna V., Mayr H., Nikitchenko M.,
Spivakovsky A. (eds) Information and Communication Technologies in Education,
Research, and Industrial Applications. ICTERI 2018. Communications in Computer and
Information Science, vol 1007, pp. 276–297. Springer, Cham (2019)
51. Soloviev, V.N., Bielinskyi, A., Serdyuk, O., Solovieva, V., Semerikov, S.: Lyapunov
Exponents as Indicators of the Stock Market Crashes. CEUR Workshop Proceedings (2020,
in press)
52. Soloviev, V.N., Yevtushenko, S.P., Batareyev, V.V.: Comparative analysis of the
cryptocurrency and the stock markets using the Random Matrix Theory. CEUR Workshop
Proceedings 2546, 87–100 (2019)
53. Sustainalytics: Index Research Services. https://www.sustainalytics.com/index-research-
services/ (2020). Accessed 10 Apr 2020
54. Thurner, S., Hanel, R., Klimek, P.: Introduction to the Theory of Complex Systems. Oxford
University Press, Oxford (2018)
55. Tillmann, J.: The link between sustainability performance and financial performance – An
event study on the impact of sustainability index changes on the market value of a company.
Master Thesis, University of Tilburg (2012)
56. Tiwari, A.K., Albulescu, C.T., Yoon, S.-M.: A multifractal detrended fluctuation analysis
of financial market efficiency: Comparison using Dow Jones sector ETF indices. Physica
A: Statistical Mechanics and its Applications 483, 182–192 (2017)
57. Tsallis, C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex
World. Springer, New York (2009)
58. Wang, W., Zhao, X., Gong, Q., Ji, Z.: Measurement of Regional Green Economy
sustainable development ability based on Entropy Weight-Topsis-Coupling Coordination
Degree – A case study in Shandong Province, China. Sustainability 11(1), 280 (2019)
59. Wikipedia: Dow Jones Sustainability Indices.
https://en.wikipedia.org/w/index.php?title=Dow_Jones_Sustainability_Indices&oldid=97
4037263 (2020). Accessed 20 Aug 2020
60. Wikipedia: List of stock market crashes and bear markets.
https://en.wikipedia.org/w/index.php?title=List_of_stock_market_crashes_and_bear_mark
ets&oldid=984366080 (2020). Accessed 19 Oct 2020
61. Yahoo Finance: Dow Jones Industrial Average (^DJI).
https://finance.yahoo.com/quote/%5EDJI?p=^DJI (2020). Accessed 17 Aug 2020
62. Yahoo Finance: Dow Jones Sustainability World (^W1SGI) Charts, Data & News.
https://finance.yahoo.com/quote/%5EW1SGI/ (2020). Accessed 17 Aug 2020