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The Efficient Market Hypothesis (EMH) has been repeatedly demonstrated to be an inferior — or at best incomplete — model of financial market behavior. The Fractal Market Hypothesis (FMH) has been installed as a viable alternative to the EMH. The FMH asserts that markets are stabilized by matching demand and supply of investors’ investment horizons while the EMH assumes that the market is at equilibrium. A quantity known as the Hurst exponent determines whether a fractal time series evolves by random walk, a persistent trend or mean reverts. The time dependence of this quantity is explored for two developed market indices and one emerging market index. Another quantity, the fractal dimension of a time series, provides an indicator for the onset of chaos when market participants behave in the same way and breach a given threshold. A relationship is found between these quantities: the larger the change in the fractal dimension before breaching, the larger the rally in the price index after the breach. In addition, breaches are found to occur principally during times when the market is trending.
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School of Economics
Department of Risk Management
North-West University
Potchefstroom Campus, South Africa
The Efficient Market Hypothesis (EMH) has been repeatedly demonstrated to be an
inferior or at best incomplete model of financial market behavior. The Fractal Market
Hypothesis (FMH) has been installed as a viable alternative to the EMH. The FMH asserts
that markets are stabilized by matching demand and supply of investorsinvestment hor-
izons while the EMH assumes that the market is at equilibrium. A quantity known as the
Hurst exponent determines whether a fractal time series evolves by random walk, a per-
sistent trend or mean reverts. The time dependence of this quantity is explored for two
developed market indices and one emerging market index. Another quantity, the fractal
dimension of a time series, provides an indicator for the onset of chaos when market
participants behave in the same way and breach a given threshold. A relationship is found
between these quantities: the larger the change in the fractal dimension before breaching,
the larger the rally in the price index after the breach. In addition, breaches are found to
occur principally during times when the market is trending.
Keywords: Efficient market hypothesis; fractal market hypothesis; hurst exponent; fractal
JEL Classifications: C52, G11
1. Introduction
A central tenet of modern portfolio theory (MPT) is the concept of diversification:
an assembly of several different assets can achieve a higher rate of return and a
lower risk level than any asset in isolation (Markowitz, 1952). MPT has enjoyed
remarkable success it is still in wide use today (2018) but it has also
Corresponding author.
Annals of Financial Economics
Vol. 14, No. 1 (March 2019) 1950001 (27 pages)
©World Scientific Publishing Company
DOI: 10.1142/S2010495219500015
January 11, 2019 12:08:43pm WSPC/276-AFE 1950001 ISSN: 2010-4952
attracted a large and growing critical literature (e.g. Michaud (1989), Elton and
Gruber (1997) and Mehdi and Hawley (2013) and references therein). An example
of these criticisms is that MPT relies on the statistical independence of underlying
asset price changes. This renders predictions of future market movements impos-
sible. Sources of instability and market risk are also assumed to be exogenous
under MPT. Were this true, the economic system would converge to a steady-state
path, entirely determined by fundamentals and with no associated opportunities for
consistent speculative profits in the absence of external price shocks. Empirical
evidence, however, shows that prices are not only governed by fundamentals,
but also by non-linear market forces and factor interactions which give rise to
endogenous fluctuations.
Asset returns are also assumed to be normally distributed, but this omits
(or assigns very low probabilities to) large return outliers. This is not an attribute of
financial markets: they are characterized by long periods of stasis, punctuated by
bursts of activity when volatility escalates often rapidly and without warning.
A consequence of the normal distribution assumption, then, is that these large
market changes occur too infrequently to be of concern. Classical financial models,
such as the efficient market hypothesis (EMH), embrace the precepts of MPT, so
these abrupt market events are omitted from their frameworks.
The EMH with its three varieties (weak, semi-strong and strong) evolved from
the MPT (Fama, 1965). Strong form efficiency is considered impossible in the real
world (Grossman and Stiglitz, 1980) so only the weak and semi-strong forms of the
EMH are empirically viable: both take for granted what Samuelson (1965) proved:
that future asset price movements are determined entirely by information not
contained in the price series; they must follow a random walk (Wilson and
Marashdeh, 2007). The literature is, however, replete with evidence that weak and
semi-strong forms of efficiency are inaccurate descriptions of financial markets
(for example, Jensen (1978), Schwert (2003) and Zunino et al. (2008)), so alter-
native descriptions must be sought.
Two alternatives to efficient markets have evolved: the Adaptive (AMH) and
Fractal (FMH) market hypotheses. The former offers a biological assessment of
financial markets specifically an evolutionary framework in which markets
(and market agents: assets and investors) adapt and evolve dynamically through
time. This evolution is fashioned by simple economic principles which, like natural
selection, punish the unfit (through extinction) and reward the fit (through survival)
as agents compete and adapt not always optimally (Farmer and Lo, 1999;
Farmer, 2002; Lo, 2002, 2004, 2005). Survival is paramount, even if that requires
temporarily abandoning profit and utility maximization. Unlike the EMH, the
AMH allows for an unstable, dynamic risk/reward relationship in which arbitrage
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opportunities arise and close depending on prevailing macro and microeconomic
conditions which in turn affect the success of investment strategies.
The FMH relaxes the EMHs random walk requirement of asset prices. Hurst
(1951, 1956) exploring the annual dependence of water levels on the river Nile
noted that these ebbs and flows were not random as expected but rather
displayed persistence and mean reversion. High levels one year tended to be fol-
lowed by high levels the next (and vice versa). In other periods, sharp reversions
toward the mean were recorded. Hursts (1956) observations led to the formulation
of the Hurst exponent, H, which effectively measures the degree of persistence
prevalent in a time series: higher values suggest directional similarity (persistence)
and lower values imply directional heterogeneity (reversion to the long-run mean:
the further away from the mean, the stronger the tendency to return to it).
The relationship between these competing hypotheses and some of the tests
used to determine their validity is summarized in Fig. 1.
The remainder of this paper proceeds as follows. The literature study in Sec. 2
provides a brief overview of salient features of the EMH. The EMH and the less-
explored FMH, which addresses some of the formers shortcomings, are also
discussed and compared here. Section 3 presents the data used to explore the FMH
approach. If market movements are indeed described by fractal geometry, the
implications for financial markets are profound. A diminishing fractal dimension,
for example, indicates herding behavior until critical values are breached, leading
to chaos. This section introduces the theoretical constructs of fractal geometry
prevalent in financial time series. The results of the investigation on some global
markets are presented in Sec. 4 as well as an empirical discussion on the impli-
cations of these results. Section 5 concludes.
Figure 1. Relationship between efficient, FMH and AMH (Lo, 2012).
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2. Literature Survey
The phrase efficient market, introduced by Fama et al. (1965), originally defined
a market which received, processed and adapted to new information quickly.
A more contemporary definition, which considers rational processing of relevant
information, asserts that all available information is reflected in an efficient mar-
kets asset prices (Fama, 1991). If the relevant information was free, prices would
rise to their fundamental level, but financial incentives arise if procurement costs
are not zero. This is the strong form of the EMH (Grossman and Stiglitz, 1980).
The economically realistic, semi-strong version of the EMH, argues that prices
reflect information, but only to the point where the marginal costs of collecting the
information outweigh the marginal benefits of acting upon it (through expected
profits) (Jensen, 1978). The weak form of the EMH suggests that asset prices
reflect all past asset price data so technical analysis is of no help in forming
investment decisions.
The EMH generates several testable predictions regarding the behavior of asset
prices and returns, so much empirical research is devoted to gathering important
evidence about the informational efficiency of financial markets and establishing
the validity or otherwise of the EMH. Some of the more significant
assessments are summarized in Table 1.
MPT which arose from the tenets of EMH allows for the construction of
efficient portfolios (those which generate the highest return possible for a given
level of risk) while still maintaining the EMH assertion that outperforming the
market on a risk-adjusted basis is impossible (Elton and Gruber, 1997).
Far from an orderly system of rational, cooperating investors, financial markets
are instead characterized by nonlinear dynamic systems of interacting agents who
rapidly process new information. Investors with different investment horizons and
Table 1. EMH predictions and empirical evidence.
Prediction Empirical evidence Sources
Asset prices move as
random walks over
Approximately true. However, small
positive autocorrelation for short-
horizon (daily, weekly and monthly)
stock returns
Poterba and Summers (1988);
Fama and French (1992);
Campbell et al. (1997)
Fragile evidence of mean reversion
in stock prices at long horizons
(35 years)
New information rap-
idly incorporated
into asset prices
New information usually incorporated
rapidly into asset prices, with some
Chan et al. (1996); Fama and
French (1998)
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holding different market positions employ this information in different ways.
Considerable price fluctuations are observed, and these are indistinguishable or
invarianton different time scales, as illustrated in Fig. 2 which demonstrates this
phenomenon for crude oil prices using 70 daily, weekly, monthly and quarterly
prices. It is impossible to say which of these with the axes (deliberately, in this
case) is unlabeled.
This self-similarity implies market price persistence which would not be ob-
served if returns were indeed independently and identically distributed, as postu-
lated under the EMH. Further evidence of market persistence is shown by prices
which deviate from their fundamentals for prolonged periods, and by a greater
amount than allowed by the EMH (Carhart, 1997).
These empirical facts have created the need for a more realistic description of
market movements than that described by the EMH a need which was first
satisfied by Mandelbrot (1977) who argued that fractals (geometric shapes, parts of
Table 1. (Continued )
Prediction Empirical evidence Sources
Current information
cannot be used to
predict future ex-
cess returns
Short run, shares with high returns
continue to produce high returns
(momentum effects)
De Bondt and Thaler (1985);
Fama and French (1992);
Jegadeesh and Titman
(1993); Lakonishok et al.
(1994); Goodhart (1988)
Long run, shares with low price-earn-
ings ratios, high book-to-market-
value ratios, and other measures of
valueoutperform the market
(value effects)
FX market: current forward rate predicts
excess returns (it is a biased predic-
tor of future exchange rates)
Technical analysis
should provide no
useful information
Although technical analysis is in wide-
spread use in financial markets, there
is contradictory evidence about
whether it can generate excess
Levich and Thomas (1993);
Osler and Chang (1995);
Neely et al. (1997); Allen
and Karjalainen (1999)
Fund managers cannot
systematically out-
perform the market
Approximately true. Some evidence that
fund managers can systematically
underperform market
Lakonishok et al. (1992);
Brown and Goetzmann
(1995) Kahn and Rudd
Asset prices remain at
levels consistent
with economic fun-
damentals (i.e. they
are not misaligned)
At times, asset prices appear to be sig-
nificantly misaligned, for extended
Meese and Rogoff (1983); De
Long et al. (1990)
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which can be identified and isolated, each of which demonstrates a reduced-scale
version of the whole) provided such a realistic market risk framework. Prices
generated from simulated scenarios based on these fractal models reflect more
realistic market activity (Joshi, 2014a; Somalwar, 2016).
The quantification of self-similar structures is non-trivial: an analogy usually
invoked in the literature is that of the changing length of a coastline, depending on
the ruler used to measure it Feder (1988) and Cajueiro and Tabak (2004a). Dif-
ferences in estimation arise when line segments (as characterized by a ruler) are
used to measure lengths of nested, self-similar structures (Anderson and Noss,
2013). The fractal nature of financial markets has led to the formulation of the
FMH which replicates patterns evident in calm markets (predicted by MPT) as well
as highly turbulent trading conditions (not predicted by MPT). The FMH and
fractal price models may also be calibrated to replicate market price accelerations
and collapses, key features of heteroscedastic volatility.
The principal differences between the EMH and the FMH are summarized in
Table 2. Note that all the assumptions in the EMH column are false, whilst those in
the FMH column are true.
(a) (b)
(c) (d)
Source: Authors calculations.
Figure 2. (a) Daily, (b) weekly, (c) monthly and (d) quarterly crude oil prices measured
over 70 periods in each case. Without time-axis labels, these series trace a geometric
pattern which appears indistinguishable across different timescales.
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The FMH assumes that price changes evolve according to fractional Brownian
motion, a feature quantified by the Hurst exponent. Hurst (1956) explored long-
range time series component dependences and formulated the Hurst exponent, H,
which records both the level of autocorrelation of a series and estimates the rate at
which these autocorrelations diminish as the time delay between pairs of values
increases. The range of H2[0, 1]. The EMH is based upon standard Brownian
motion processes which assume that prices evolve by random walks, i.e. H¼0:5.
A natural consequence follows from this framework: forecasting future price
movements is impossible because price movements are independent and exhibit no
autocorrelation, thus technical analysis provides no assistance to investors.
Deviations from H¼0:5 indicate autocorrelation which violates a key principle of
the EMH. The finite nature of financial time series allows for H0:5, so this
possibility must be accounted for (Morales et al., 2012). Table 3 records the dif-
ferences in time series depending on subranges of H: Figure 3 shows different time
series for three sub-regions of H.
The literature exploring the Hurst exponent in finance and its relationship with
the EMH is rich. Using daily data from both emerging and developed market
indices spanning 10 years (January 1992December 2002), Cajueiro and Tabak
(2004a, b) calculated H(t), the time-varying H. For the emerging markets H>0:5,
but the long-term trend was towards H¼0:5, indicating increasing efficiency over
the observation period. Developed marketsHwas not statistically different from
0.5. The results for both markets were confirmed by Di Matteo (2007) who used 32
global market indices and Wang et al. (2010) who used daily data to explore the
efficiency of Shanghai stock market.
Grech and Mazur (2004) employed Hto forecast market crashes. Three such
crashes (1929 and 1987 in the US and 1998 in Hong Kong) were investigated using
two years of daily data prior to the relevant crash in each case. Before each crash, H
decreased sharply, an indication of vanishing trends and increasing volatility while
during each crash, H increased significantly, a sign of enhanced inefficiency. Using
Table 2. Summary of differences between the EMH and the FMH.
Return distribution is Normal (Gaussian) Return distribution is non-Normal (non-Gaussian)
Stationary process (distribution
mean does not change)
Non-stationary process (mean of distribution changes)
Returns have no memory (no trends) Returns have memory (trends)
No repeating patterns at any scale Many repeating patterns at all scales
Continuously stable at all scales Possible instabilities at any scale
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daily data from the Polish stock market, Grech and Pamuła (2008) reached the
same conclusions.
Alvarez-Ramirez et al. (2008) used daily data spanning 60 years from the S&P
500 and Dow Jones indices and found that Hdisplayed erratic dynamic time
Table 3. Characteristics of time series dependency on H.
Range H2[0, 0:5)H0:5H2(0:5, 1]
0.0 0.5
0.5 1.0
Auto-covariance <08lags ¼08lags >08lags
Behavior Anti-persistent Brownian Persistent
Decrements (incre-
ments) more likely
to be proceeded by
increments (decre-
ments equally
Increments (decre-
ments) more likely
to be proceeded by
increments (decre-
Character Reverts to the mean
more frequently
than a random one
Random motion Exhibit long-memory
and trendsand
cyclesof varying
Sources Barkoulas et al.
(2000); Kristoufek
Osborne (1959) Mandelbrot and Van
Ness (1968)
(a) (b) (c)
Source: Authors calculations.
Figure 3. S&P 500 price series for 18-month period in which (a) 0 <H<0:5
(mean reverting), (b) H0:5 (Brownian motion) and (c) 0:5<H<1:0 (trending).
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dependency. A time-varying evolution of market efficiency was observed with
alternating low and high persistent behavior, i.e. H>0:5 in both cases, with
different magnitudes.
The consequences for market efficiency of financial crises were explored by
Lim et al. (2008) who found that the 1997 Asian crisis dramatically reduced the
efficiency of global stock markets. Within three years, however, efficiency had
recovered to pre-crisis levels. The highest level of market efficiency was recorded
during post-crisis periods, followed by pre-crisis periods. During crises, markets
exhibit high inefficiency.
Using daily data for 19 months (January 1July 07), Karangwa (2008) found
H0:5 for the JSE. Note that Karangwas (2008) study concluded before the
onset of the 2008 credit crisis, so this event and its aftermath were not included in
the analysis. Using monthly data for a longer period (i.e. August 1995August
2007), Karangwa (2008) found H¼0:58. In a more recent study, Ostaszewicz
(2012) used two methods (Higuchi and absolute moments) to measure Husing JSE
price index data for both pre and post 2008 crisis periods and found H>0:5
predominantly in the pre-2008 crisis period and H<0:5 predominantly in the
post-2008 crisis period. Chimanga and Mlambo (2014) investigated the fractal
nature of the JSE and found H¼0:61 using daily data from 2000 to 2010. By
sector, the values for the JSE are shown in Fig. 4.
Sarpong et al. (2016) found H¼0:46 for the JSE using daily data from 1995 to
2015 (thereby embracing the full period investigated by Chimanga and Mlambo,
2014). In addition, Sarpong et al. (2016) used the BDS test (Brock et al., 1996) to
verify that JSE price index data exhibit non-random chaotic dynamics rather than
pure randomness. These results confirm those obtained by Smith (2008) who,
Source: Authors calculations.
Figure 4. Average Hs measured on various JSE sectors over the period 20002010. Error
bars indicate maximum and minimum values obtained from individual shares within the
relevant sector.
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using four joint variance ratio tests, rejected the random walk hypothesis
on the JSE.
Vamvakaris et al. (2017) examined the persistency of the S&P 500 index using
daily data from 1996 to 2010 and found that crises affect investorsbehavior only
temporarily (<six months). In addition, the index exhibited high anti-persistency
(an indication of investor nervousness,H<0:5) prior to periods of high market
instability. Considerable fluctuations of Hwere observed with a roughly annual
frequency and amplitude (from peak to trough) of 0.2 to 0.4. No prolonged trends
of Hwere recorded.
3. Data and Methodology
3.1. Data
The data used to calibrate the FMH (via the estimation of the Hurst exponent)
comprise 22.5 years (July 1995 to December 2017) of daily market index prices for
developed (S&P 500, FTSE 100) and emerging market stock exchanges (the JSE).
Three years (36 months) of daily index prices were used to determine H36. The
data sample was then rolled forward by one month and the next realization of
the Hurst exponent calculated, i.e. H37. This was repeated until the latest Hurst
exponent in the data sample was calculated, i.e. end of December 2017, using the
three years of data from January 2015 to December 2017.
This sample size was selected to include at least three full South African
business cycles. This has been shown to be seven years (Botha, 2004; Thomson
and van Vuuren, 2016). In addition, these data embrace a period of non-volatile
growth (20032008), and considerable turbulence (19982000 (the Asian crisis
and the dotcom crash) and 20082011 (the credit crisis)).
The same indices were used for the fractal dimension, D, analysis to establish
whether breaching of a given Dled to herding behavior (and a resulting collapse or
rally in price). The fractal dimensions of gold and oil prices were investigated over
the same period for calibration purposes and to confirm earlier work undertaken by
Joshi (2014a, b).
3.2. Methodology
Standard Brownian motion describes the trajectory of a financial asset price, St,
through time by integrating the differential equation (Areerak, 2014):
dSt¼St(μdt þdWt),ð1Þ
where Stis a financial asset price at time t,dStis the infinitesimal change in
the assets price over time dt,μis the expected rate of return that the asset will earn
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over dt and the expected volatility. dWtis a Weiner process described by "ffiffit
where "is a random number drawn from a standard normal distribution. The
solution of this differential equation is
St¼S0exp μt2
where S0is the initial asset price. In principle, Stdescribes the assets price tra-
jectory through time, but in practice, many features of financial assets are not
captured by this formulation. Cont (2001) assembled a group of stylized statistical
facts which describe several financial assets. While not exhaustive, the following
list includes the empirical evidence that financial asset returns are characterized by:
(1) insignificant linear autocorrelations (Cont, 2001),
(2) heavy tails and conditional heavy tails (even after adapting returns for
volatility clustering) of unconditional return distributions which can be
described by power laws or Pareto-like tails with finite tail indices (Horák
and Smid, 2009),
(3) asymmetric gains and losses larger drawdowns than upward movements
(Horak and Smid, 2009),
(4) different distributions at different timescales. Known as aggregational
Gaussianitythe return distribution approaches a normal distribution as t!
1(Cont, 2001),
(5) a high degree of return variability at all timescales (Di Matteo et al., 2005),
(6) homoscedasticity or volatility clustering: the clustering of high-volatility
events and low-volatility events in time (Cont, 2001),
(7) long-range dependence of return data, characterized by the slow decay (as a
function of time) of the autocorrelation of absolute returns, often as a power
law with exponent 0:2β0:4 (Cont, 2001),
(8) negative correlation of the assets volatility and its returns (Chordia et al.,
(9) higher-than-expected correlation between trading volume and volatility
(Blume et al., 1994) and
(10) time scale asymmetry: fine-scale volatility is better predicted than coarse-
grained measures rather than the other way around (Di Matteo et al., 2005).
These features are generally not captured by standard Brownian motion, which
has led to the development of fractional Brownian motion. In this formulation, (1)
dSt¼St(μdt þdZt),ð3Þ
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where dZt¼"ffiffiffiffiffiffi
pand H(0 H1) is the Hurst parameter. The respective
Wiener processes (dWtin (1) and dZtin (3)) have many features in common, but
also exhibit strikingly different properties. The Wiener process dZtis self-similar
in time, while dWtis self-affine (Mandelbrot, 1977; Feder, 1988). Fractional
Brownian Motion, for example, captures dependence among returns. A generalized
solution for (3) is
St¼S0exp μt2
If 0 H<0:5, changes in Stare negatively correlated and if 0:5H<1, they
are positively correlated. Correlation also increases with H(Shevchenko, 2014).
3.2.1. Hurst exponent, H
A variety of methods for estimating Hare discussed in the literature, each with
associated advantages and drawbacks. Approaches include rescaled-range analysis
(proposed by Hurst (1951) himself), wavelet transformations (Simonsen and
Hansen, 1998), neural networks (Qian and Rasheed, 2004) and the visibility-graph
approach (Lacasa et al., 2009). The most commonly used methodology is rescaled-
range analysis, and this will be adopted here as it is also the technique used to
determine the fractal dimension, D, also known as the HausdorffBesicovitch
dimension (Hausdorff, 1919; Manstavičius, 2007).
Hurst (1951) asserted that the variation of fractal time series is related to the
horizon over which the time series are assessed by a power law relationship.
Starting with a de-meaned time series (to ensure stationarity), define Ykas the sum
of kincrements of this series, extending to nincrements. The adjusted range (the
distancethe series travels over ntime increments) is defined as the difference
between the maximum and the minimum of the series:
Y1Y2,,Ynor Rn¼max (Yk)min (Yk),1<k<n:
If Yis a time series characterized by Gaussian increments (i.e. a random walk),
then this range increases with the product of the seriesstandard deviation and ffiffi
Hurst (1951) generalized this relationship to
where is the standard deviation (i.e. the realized volatility) of the stationary time
seriesnobservations and His the Hurst exponent. Rescaling the series by deter-
mining the quotient of the range and measures time series that do not exhibit finite
variance (or fractals). This method makes no assumption regarding the underlying
distribution of increments; only how they scale with time, as measured by H.
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The theoretical value of the positive constant, c,is
where ()is the Gamma function.
The Hexponent captures the degree of persistence in a time series, irrespective
of the time scale over which it is measured. For a time series with an observed
H>0:5 implies that a large value of the series in one period is likely to be
followed by a larger value in a later period (the reverse applies if H<0:5 so such a
series is mean reverting). Hmay be calculated using ordinary least squares re-
gression after taking the logarithm of (5):
ln R
Using many different increments, n, and regressing ln(R
)on ln(n)gives a straight
line with c¼exp (yintercept)see (6) and H¼regression line slope.
Peters (1991) provides the following process for determining H.
Using a time series of Nþ1 prices fPtg, calculate the time series of Nreturns,
{Xtgsuch that Xt¼ln(Pt=Pt1). Divide the return time series (length N) into A
contiguous subperiods, each of length n(so An¼N). Label each subperiod la
with a¼1, 2, 3,,A. Label each element in laas Nk, where k¼1, 2, 3,,n.For
each subperiod, calculate the mean: ea¼1
k¼1Nk,aas shown in Fig. 5.
The time series of cumulative departures from the mean, for each subperiod la,
are then
(Ni,aea)8k¼1, 2, 3,,n:
Define the range as the difference between the maximum and minimum value of
Xk,awithin each subperiod la:Rla¼max (Xk,a)min (Xk,a), where 1 <k<n.
The sample standard deviation, , for each subperiod lais
A rescaled range, Rla=lafor each subperiod, la, is then determined, the average of
which is
The length nis then increased until there are only two subperiods (¼N
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A least squares regression is performed, with ln(n)as the independent variable
and ln(R
)nas the dependent variable. The slope of the regression is Hand the
y-intercept, c, as shown in Fig. 6 for a single three-year period, as an example. In
the subsequent month, this process is followed again using three years of data prior
to that month, and the next Hand care calculated.
Source: Authors calculations.
Figure 6. Regression results, March 2006March 2009. H¼0:509 and c¼exp
Figure 5. Applying Peters (1991) procedure for measuring eas.
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Lo and MacKinlay (1988, 2001) developed a test statistic to determine the
statistical significance of H, i.e. whether the null-hypothesis (that H¼0:5) can be
rejected or not. Known as the variance ratio test, this tests whether the time series is
stationary (the variance of the series remains constant over time) or whether the
series is trending (non-stationary). In this latter case, the series variance increases
over time and has a unit root (Steffen et al., 2014). No statically significant evi-
dence for stationarity was found in any time series.
The evolution of Hwas examined using this technique over the two-decade
period spanning January 1998 to January 2018. This reveals the characteristic
nature of markets over this period: persistence, random walks or mean reversion.
The fractal dimension, D, discussed in the next section, and Hare related (8)
although a different technique (7) is used to measure Din this case as it provides
more granular (daily) estimates than (8). When Dapproaches and breaches a given
threshold, the market tends to become chaotic, and given that the market exhibits a
level of predictability after the onset of chaos (and the threshold breach), this
tendency may be exploited by investors.
3.2.2. Fractal dimension, D
Joshi (2014a, b) described the fractal structure of a financial market using the
definition of the fractal dimension, Dand the rescaled range. The estimation of the
time seriesfractal dimension rests on the assertion that stock markets are complex
adaptive systems and thus embedded within them is an endogenous tipping
point of instability (i.e. no explicit exogenous trigger is required).
Market stability rests on balancing supply and demand (liquidity) and the fractal
structure of financial markets optimizes this liquidity. When different investors
with many different investment horizons are all active in the market, the market is
characterized by a rich fractal structure. Investors with different investment
periods focus on different buy and sell signals: traders on technical data and
momentum (short horizons) and pension funds on structural fundamentals and
valuation (long horizons) for example. Sharp one day sell-offs will be interpreted
by traders as a sell signal while pension funds may interpret this event as a buying
opportunity. There is ample market liquidity: a large price move is not inevitable
(Joshi, 2014a).
If the traders horizon becomes dominant, however, and liquidity evaporates when
sell orders far outweigh the number of buy orders, the fractal structure of the market
collapses and violent price corrections become manifest. This is the endogenous
tipping point and by monitoring the fractal dimension, discussed below, such
thresholds may be monitored and employed as early indicators of market corrections.
The lower the fractal dimension, the more unstable the market it measures.
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Breaching a fractal dimension threshold of 1.25 triggers market corrections.
This empirical limit appears identical across asset classes, geographies and time
periods it is not theoretically derived. It is impossible, however, to ascertain the
magnitude of the subsequent adjustment or its direction, i.e. the ensuing correction
may be >0or<0 (Joshi, 2014b, 2017).
The measurement of D, the fractal dimension, is described by Joshi (2014a, b).
If an assets price is Pion day i, its one-day log return, ri, on day iis
ri¼ln Pi
The scaling factor, n, is used to determine the n-day log return, Ri,n, on day i:
Ri,n¼ln Pi
as well as the scaled return, Ni,n, on day i:
abs Ri,n
inabs ln Pi
ln Pi
and the scaled fractal dimension, Di,n, on day i:
ln Pi
abs ln Pi
ln Pi
The theoretical relationship between Hand Dis given by Schepers et al. (2002):
D¼H2, ð8Þ
but (7) provides a much more granular (daily) estimate of Dthan (8) since H(in 8)
is a monthly value, determined using (5).
4. Results and Discussion
4.1. Hurst exponent, H
How Hchanges over time is useful to market participants: economists to ascertain
the nature of the prevailing markets (persistent or mean-reverting), government
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strategists to establish the economys current position in the business cycle, long-
term investors to exploit market rallies and busts and short-term investors to exploit
mean reversion conditions.
The rolling Hwas explored for three market indices: two in developed markets
(US and UK) and one in an emerging market (South Africa). Figure 7(a) shows the
results for the S&P 500: Cajueiro and Tabak (2004a, b) found similar results for
developed markets (H0:5). Grech and Mazur (2004) found that Hdecreased
sharply before market crashes showing a rapid decrease in trend. This is clearly
shown for the September 2001 and September 2008 events particularly for the
latter. After this event, Hincreases steadily (over three years) from a market
dominated by mean reverting to one characterized by random walk prices.
The rolling Hfor the FTSE 100 is shown in Fig. 8 on the same vertical and
timescale as Fig. 7(a). Again, in line with the findings of Cajueiro and Tabak (2004a, b),
Source: Authors calculations.
Figure 7. Rolling (a) H(t)and (b) c(t)for the S&P 500 from January 1998December
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H0:5. Unlike the results obtained by Grech and Mazur (2004), no sharp decrease
of Hwas observed for the crisis which affected the S&P 500. The events of
September 2001 occurred on US soil and so were more damaging to the US economy
than the UK economy. The financial crisis of 2008, however, was global in impact
and of considerable severity, yet the UK market appears to have been unaffected.
The FTSE 100 exhibits slight persistence (H>0:5) between the time of the
onset of the 2008 crisis and early 2012 when the sovereign crisis (which affected
several European countries, including the UK albeit not as dramatically) began
(Gärtner et al., 2011) see Fig. 8. At this point, the market changes gradually to
become slightly mean reverting and has since followed a random walk since 2014.
From 2012, the behavior of Hfor the FTSE 100 closely resembles that of the S&P
500 over the same period. These developed market results reinforce results
obtained previously (e.g. Alvarez-Ramirez et al. (2008)).
The JSE All Share index displays behavior significantly different from that of
developed market indices (Fig. 9). Until 2006, the JSE trends are strongly unaf-
fected by the dotcomcrisis in 00 or the events of September 2001. These results
confirm and update those found by Karangwa (2008) and Chimanga and Mlambo
Between 2006 and the start of the 2008 financial crisis, market prices on the
JSE evolve by random walk, but changes to a trending market rapidly at the onset
of the crisis the opposite of what is observed in developed markets. This could
be because developing markets in particular South Africa largely escaped
the consequences of the crisis because it occurred in a period to sustained growth
for the country and strong fundamentals (Zini, 2008). South African financial
institutions were also relatively robust and did not issue credit as freely and loosely
Source: Authors calculations.
Figure 8. Rolling H(t)for the FTSE 100 from January 1998December 2017 (note the
same vertical scale as used for the S&P 500 for comparison).
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as their global counterparts (Mnyande, 2010). In a trend similar to global markets,
JSE prices have become slightly mean-reverting or become random walks since
2012. Smith (2008) also found statistically significant results that H<0:5, but
over a shorter horizon and using daily (rather than monthly) data. Sarpong et al.
(2016) using daily JSE index data spanning 20 years from 1995 to 2015 also found
H<0:5 prior to 2012 and H0 after 2012. The South African market was also
found to be more sectorizedor heterogenous with respect to H; different market
sectors are characterized by different values of Hand these values tend to persist
over time.
4.2. Fractal dimension, D
Analysis of Dfor the JSE All Share generated interesting results, previously un-
explored. The majority (95%) of threshold breaches occur when H>0:5. Only 5%
of breaches occur during periods when the market exhibits periods of random walk
or mean reversion behavior. This fact alone provides valuable information to
market participants, but the percentage change in Di.e. the rate of change or
speedof the change of Dalso provides information about subsequent market
A breach is classified as an event in which D!1:25 from above, i.e.
D>1:25. There is no theoretical explanation for why this threshold value is
significant. It does appear to be empirically consistent across markets, eras, ge-
ographies and asset types. When Dbreaches 1.25 from below(when preceding
fractal dimension is <1:25), this is not deemed to be a breach of interest. When
threshold breaches were first identified, these occurred primarily during times
when the South African market was trending, i.e. between 1998 and 2006 (the
same results were obtained for the two developed market indices). Four such
Source: Authors calculations.
Figure 9. Rolling H(t)for the JSE All Share from January 1998December 2017.
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prominent breaches are shown in Fig. 10(a). The behavior of the market index over
the same period is shown in Fig. 10(b), illustrating the impact of breaches. The
shaded area links the timescales on Figs. 10(a) and 10(b) during the four breaches
observed during this period.
Next, the rate of change of Dwas determined over one trading week (five days)
prior to the breach (over which time Ddecreases considerably and rapidly, but not
instantaneously). One day is too short for a time to capture this time and over two
weeks, Dhas often recovered to pre-breach levels, so one week appears to be an
appropriate time to capture a significant, persistent decrease:
Source: Authors calculations.
Figure 10. (a) Fractal dimension, Dover the three-year period between January 2001
and January 2004 showing several breaches (shaded), i.e. when D1:25 and (b) the
JSE All Share index over the same period showing the behavior of index prices post
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where t0is the time Dfirst breaches D¼1:25. After t0price changes tend to be
significant (generally >5%), sustained and positive. One trading month (22 days)
was selected over which to measure index price changes, i.e.
Ptþ22 Pt0
Of course, price changes could be measured over shorter or longer periods than one
month, and changes in Dcould be ascertained over shorter or longer periods than
one week, but this approach provides a convenient, simple framework to analyze
the effect of breaches on asset prices. The results are shown in Fig. 11.
Regression analysis indicates that the larger D=Dover the week prior to a
breach, the larger the positive change over a month of the index price.
R2¼0:85 indicates a statistically significant result. Similar results were obtained
for the developed market indices. All time series used in this analysis were found to
be stationary using the Augmented Dickey Fuller test.
The slope of the line is 1:2, so for a 1:0%five-day pre-breach drop in D=D,
ceterus paribus leads to a 1.2% increase in the post-breach, one-month price series.
These results could have significant consequences for investors, and could serve as
a complementary tool to support, rationalize and justify investment decisions.
5. Conclusions and Suggestions
This paper examined the fractal properties of developed and developing market
indices and examined the evolution of these fractal properties over a two-decade
period. The FMH, using empirical evidence, posits that financial time series are
Source: Authors calculations.
Figure 11. Simple regression of one-month index return post-breach (D1:25) against a
five-day pre-breach percentage change in fractal dimension (D=D). The period analyzed
was July 1995 to December 2017, i.e. the full data sample.
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self-similar, a feature which arises because of the interaction of investors with
different investment horizons and liquidity constraints. The FMH presents a
quantitative description of the way financial time series change; so after the testing
of observed, empirical properties of financial market prices, forecasts may be
formalized. Under the FMH paradigm, liquidity and the heterogeneity of invest-
ment horizons are key determinants of market stability, so the FMH embraces
potential explanations for the dynamic operation of financial markets, their inter-
action and inherent instability. During normalmarket conditions, different in-
vestor objectives ensure liquidity and orderly price movements, but under stressed
conditions, herding behavior dries up liquidity and destabilizes the market through
panic selling.
This work also established a relationship between the change in a time series
fractal dimension (before breaching a threshold) and both the magnitude and
direction of the subsequent change in the time series. This relationship was found
to be prevalent during times of strong price persistence a feature detectable by
elevated Hurst exponents. These results suggest potential investment strategies.
Additional extensions could include more detailed calibration perhaps by
OLS of the optimal pre-breach period for D=Dand optimal post-breach
period for P=P. A comprehensive application of these results to other market
indices and asset classes is also needed. Whether the relationship above holds for
all asset classes (and, if so, whether the requirement that H>0:5 is a necessary or
sufficient prerequisite) also needs to be ascertained.
We are grateful to the anonymous reviewer for valuable comments and
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... From a neopositivist perspective, econophysics emerges as a branch of market efficiency analysis, which is based on the fractal characteristics of the market and chaos theory (Peters, 1994). In this perspective of the socalled fractal markets hypothesis (FMH), the market has random movements in the short run due to new information; however, patterns of behavior could be identified in the long run (Karp & Van Vuuren, 2019). In this way, asset price movements would be based on moments of local mean holding or reversal over time. ...
... Among the research justifications, there is no consensus in the literature on the issue of market efficiency, and, according to Karp and Van Vuuren (2019), the identification of market behavior in different time windows is relevant for resource allocation, for investment strategies, and for understanding the dynamics of the comovements of emerging and developed countries. According to Mensi et al. (2016), both in market and economic terms, the BRICS requires specific studies, considering its role in the world economy and its peculiarities. ...
... The study by Karp and Van Vuuren (2019), in addition to providing a literature review of the evolution of FMH comparing it to EMH, analyzes the H and D of the American, British, and South African markets. Among the most important points of the paper, it highlights that the magnitudes of variations in the Ds impacted the magnitudes of price variations in the markets. ...
... Consequently, a key feature of the IVOL perspective is that the beta anomaly is a function of a mis-specification in the CAPM. One solution provided in the literature has been to improve the CAPM, as demonstrated by the introduction of the Fama-French 3, 5, and other factor models, which have been credited with severe reductions in the beta anomaly (Karp and Van Vuuren 2019). On the other hand, behavioral finance provides an equally plausible alternative through which to explain the beta anomaly in the literature. ...
... On the other hand, negative SML results are met with contention. Karp and Van Vuuren (2019) assert that the CAPM and other asset-pricing models perform poorly on the JSE due to poor proxies. Consequently, the beta anomaly may not be adequately tested for because of the inherent methodological limitations of applying the model in developing countries. ...
... That is, the returns are mostly negative irrespective of the risk level, which suggests that there is no discernible relationship between risk and return. Harvey (1995) has asserted that relatively low and insignificant betas are common for emerging markets; however, Karp and Van Vuuren (2019) have also suggested that bias and noise may lead to inconclusive results, particularity when the sample size and horizon are small. Abnormal events, such as COVID-19 and SA credit downgrading, have likely contributed to the noise in the period studied; therefore, the former explanation is likely. ...
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High-risk stocks tend to provide lower returns than low-risk stocks on a risk-adjusted basis. These results (referred to as the low-beta anomaly) run counter to theoretical expectations. This paper examines the beta anomaly in one of the largest emerging markets in Africa, the Johannesburg Stock Exchange (JSE). It employs both time-series and cross-sectional econometric techniques to analyze the risk–return relationship implied by the CAPM, using data that span over 5 years and 220 companies. To check for robustness, the analysis period was extended to 10 years, and we also applied the Fama–French three-factor model. The findings suggest the existence of the beta anomaly and a negatively sloped SML, indicating that beta is not the only determinant of risk in the South African stock market. We also found positive beta–idiosyncratic volatility (IVOL) correlations. However, after controlling for IVOL and the adverse effects of COVID-19 for an extended study period, the beta anomaly disappeared.
... From the mathematical finance perspective, there are two main categories. One is the approach based on the efficient market hypothesis (EMH) [5,6] and the other is based on the fractal market hypothesis (FMH) [7]. ...
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As one of the main areas of value investing, the stock market attracts the attention of many investors. Among investors, market index movements are a focus of attention. In this paper, combining the efficient market hypothesis and the fractal market hypothesis, a stock prediction model based on mixed fractional Brownian motion (MFBM) and an improved fractional-order particle swarm optimization algorithm is proposed. First, the MFBM model is constructed by adjusting the parameters to mix geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM). After that, an improved fractional-order particle swarm optimization algorithm is proposed. The position and velocity formulas of the fractional-order particle swarm optimization algorithm are improved using new fractional-order update formulas. The inertia weight in the update formula is set to be linearly decreasing. The improved fractional-order particle swarm optimization algorithm is used to optimize the coefficients of the MFBM model. Through experiments, the accuracy and validity of the prediction model are proven by combining the error analysis. The model with the improved fractional-order particle swarm optimization algorithm and MFBM is superior to GBM, GFBM, and MFBM models in stock price prediction.
... Karp ve Van Vuuren (2019)'un da belirttiği gibi etkin piyasa hipotezi ve hâkim finans anlayışında varlık getirilerinin normal dağılıma uyduğu ve rassal seyir izlediği varsayılmaktadır. FPH'ta ise varlık getirileri Hurst Üstelini takip etmektedir (Karp ve Van Vuuren, 2019). ...
... Fractal market hypothesis (FMH) provides a framework for modeling turbulence, feedback loops, jumps, discontinuity, irrational behaviour, butterfly effects, and non-periodicity that truly characterise real world financial markets (Mandelbrot, 1963;Peters, 1991;Peters, 1994;Liu and Song, 2012;Anderson and Noss, 2013;Karp and Van Vuuren, 2019). FMH provides the theoretical basis for modeling variables with wild randomness and mean reversion such as market crashes, bubbles, and natural disasters (Joshi, 2014). ...
Traditionally, financial risk management is examined with cartesian and interpretivist frameworks. However, the emergence of complexity science provides a different perspective. Using a structured questionnaire completed by 120 Risk Managers, this paper pioneers a comparative analysis of cartesian and complexity science theoretical frameworks adoption in sixteen Zimbabwean banks, in unique settings of a developing country. Data are analysed with descriptive statistics. The paper finds that overally banks in Zimbabwe are adopting cartesian and complexity science theories regardless of bank size, in the same direction and trajectory. However, adoption of cartesian modeling is more comprehensive and deeper than complexity science. Furthermore, due to information asymmetries, there is diverging modeling priorities between the regulator and supervisor. The regulator places strategic thrust on Knightian risks modeling whereas banks prioritise ontological, ambiguous and Knightian uncertainty measurement. Finally, it is found that complexity science and cartesianism intersect on market discipline. From these findings, it is concluded that complexity science provides an additional dimension to quantitative risk management, hence an integration of these two perspectives is beneficial. This paper makes three contributions to knowledge. First, it adds valuable insights to theoretical perspectives on Quantitative Risk Management. Second, it provides empirical evidence on adoption of two theories from developing country perspective. Third, it offers recommendations to improve Quantitative Risk Management policy formulation and practice.
... There are many techniques used to calculate the Hurst exponent, such as the rescaled range method (R/S), the whittle estimator method, and the absolute value method. Likewise, fractal theory is also important for analyzing fractional order characteristics [26,27]. It reflects the structural similarities across the whole time series, which further affects its stationarity. ...
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This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) is then proposed considering innovations with stable infinite variance. The existence and convergence of the model solutions are discussed in depth. Ensemble learning with an autoregressive moving average model (ARMA) is used to further improve upon accuracy and generalization. The proposed method is used to predict the energy consumption in a real cooling system, and superior prediction results are obtained.
... In this paper, we apply both the Fixed Effect and Random Effect Models and use panel data from the sustainable stock market returns of six Asian countries. Extension of our paper includes using the approaches employed by our paper to study agriculture (Aye and Odhiambo, 2021), social media (Kim, 2021), risk (Chow, et al., 2019), decision-making (Hasan-Zadeh, 2019), volatility (Demirer, et al., 2020), investment (Liew, et al., 2008;Mroua, et al., 2017;Nkeki, 2018;Karp and Van Vuuren, 2019;Yang, et al., 2019;Thanh, et al., 2021). There are many other applications, readers may refer to Hon, et al. (2021) and Wong (2020) for more information. ...
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Purpose: This research studies the impacts of the six Worldwide Governance Indicators (WGI) on sustainable investment returns in the Asian region. Design/methodology/approach: This research uses the WGI data as proxied of good governance and employs The Fixed Effect Model (FEM) and Random Effect Model (REM) on the panel data from the sustainable stock market returns of six Asian countries to examine the relationship among variables. Further, the Feasible Generalized Least Squares (FGLS) Regression panel regression was conducted to achieve robust findings. Findings: Our empirical analysis found that political stability and absence of violence (PSA) and regulatory quality (REQ) positively influence sustainable investment returns in the Asian region. While control of corruption (COC) exhibits a significant negative impact on sustainable investment returns. These findings imply that more excellent political stability and reasonable regulations contribute to higher stock market returns. Conversely, contradictory with the Control of Corruption leads to downward stock market returns as the growth of the COC index increases. Research limitations/implications: This research has several limitations, including the lack of comprehensive sustainable stock market data in the Asian region and a short transaction period. Consequently, future research could categorize the market as developed or emerging, classify the sample based on the efficient market category, expand the sample size and include data from outside the Asian region. Practical implications: This research has several crucial policy implications for sustainable investors concerning the country-level governance index to create profitable and sustainable portfolio strategies. Moreover, policymakers should strengthen the implementation of anticorruption to increase the sustainable investors in the Asian region. Originality/value: This research contributes to the recent literature presenting causal relations of quality country-level governance on sustainable investment returns in the Asian region. JEL Classifications: C33; E44; G15; O16 Keywords: Sustainable Investment; Market Returns; World Governance Indicators; Asian Market; FGLS
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In this paper we analyse the degree of market efficiency present in bank equities for selected emerging and frontier economies. Employing the Multifractal Detrended Fluctuation Analysis (MFDFA), we inspect bank equity across thirty-seven banks in these economies. The MFDFA methodology enables to characterize the strength of multifractal complexities and to identify the sources of multifractal behavior. Furthermore, the time evolution of the Hurst exponents are calculated for banks having long-term temporal correlation as a significant source of multifractality. The results confirm that the time variant Hurst exponent can serve as a leading indicator for guiding investment decisions.
A hedging strategy is designed to increase the likelihood of desired financial outcomes. Market speculators hedge investment positions if they are worth protecting against potential negative outcomes of turbulent market conditions and effective hedging implementation can reduce the impact severity on the underlying investment since these negative scenarios cannot be avoided. This paper provides a solution for investors to implement a trading strategy to effectively manage turbulent market conditions (such as the COVID pandemic) by implementing an investment trading approach. The investment strategy includes an index held by the investor (long position) and uses a fractal dimension indicator to warn when liquidity or sentiment changes are imminent within financial markets. When the threshold is breached at a predetermined level, the investor will take this observation as a change in liquidity in the market and a hedging position is undertaken. This sequence of events triggers the implementation of a hedging strategy by entering a buy put option position. The fractal indicator was found to be effective when applied to four of the six tested indices in terms of cumulative returns, but also in effect increased the risk taken by the investor for all six indices. The conclusion was made that where the outcome was similar for each economy type, both had a scenario where two out of the three economies outperformed the underlying index and had one index not outperforming the underlying index. This comparison was done to establish whether the hedging strategy had a more promising application to a developing or developed economy type. The fractal indicator was found to be effective when applied to four of the six tested indices in terms of cumulative returns, but also in effect increased the risk taken by the investor for all six indices.
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By modeling the volatility structure of banks, the characteristic structure of risks and uncertainties that concern the economy as well as banks are revealed. In this study, it is aimed to estimate the volatility of stock returns of banks in Turkey. The review period of the study is January 5, 2010 - December 31, 2020. The return volatility of banks' stocks was estimated with the nonlinear asymmetric conditional volatility analysis method (APGARCH) proposed by Ding, Granger, and Engle (1993). In the study, first of all, the stability of returns, ARCH effect, asymmetry structure, and linearity properties are tested. Then, with the APGARCH model, it was revealed that the shock in the return volatility of banks has high permanence, has an asymmetry effect and has a long-term memory feature. The findings support that the existence of Fractal Market Hypothesis rather than the Efficient Market Hypothesis in the stock return volatility of the banks in Turkey. Accordingly, dependency on stock prices has been determined. Therefore, it can be said that investors take into account the assumptions of technical analysis.
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This study investigates the existence of chaos on the Johannesburg Stock Exchange (JSE) and studies three indices namely the FTSE/JSE All Share, FTSE/JSE Top 40 and FTSE/JSE Small Cap. Building upon the Fractal Market Hypothesis to provide evidence on the behavior of returns time series of the above mentioned indices, the BDS test is applied to test for non-random chaotic dynamics and further applies the rescaled range analysis to ascertain randomness, persistence or mean reversion on the JSE. The BDS test shows that all the indices examined in this study do not exhibit randomness. The FTSE/JSE All Share Index and the FTSE/JSE Top 40 exhibit slight reversion to the mean whereas the FTSE/JSE Small Cap exhibits significant persistence and appears to be less risky relative to the FTSE/JSE All Share and FTSE/JSE Top 40contrary to the assertion that small cap indices are riskier than large cap indices.
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We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
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This study investigates the existence of chaos on the Johannesburg Stock Exchange (JSE) and studies three indices namely the FTSE/JSE All Share, FTSE/JSE Top 40 and FTSE/JSE Small Cap. Building upon the Fractal Market Hypothesis to provide evidence on the behavior of returns time series of the above mentioned indices, the BDS test is applied to test for non-random chaotic dynamics and further applies the rescaled range analysis to ascertain randomness, persistence or mean reversion on the JSE. The BDS test shows that all the indices examined in this study do not exhibit randomness. The FTSE/JSE All Share Index and the FTSE/JSE Top 40 exhibit slight reversion to the mean whereas the FTSE/JSE Small Cap exhibits significant persistence and appears to be less risky relative to the FTSE/JSE All Share and FTSE/JSE Top 40 contrary to the assertion that small cap indices are riskier than large cap indices.
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A Fourier transform analysis is proposed to determine the duration of the South African business cycle, measured using log changes in nominal gross domestic product (GDP). The most prominent cycle (two smaller, but significant, cycles are also present in the time series) is found to be 7.1 years, confirmed using Empirical Mode Decomposition. The three dominant cycles are used to estimate a 3.5 year forecast of log monthly nominal GDP and these forecasts compared to observed (historical) data. Promising forecast potential is found with this significantly-reduced number of cycle components than embedded in the original series. Fourier analysis is effective in estimating the length of the business cycle, as well as in determining the current position (phase) of the economy in the business cycle.
This article presents an overview of the assumptions and unintended consequences of the widespread adoption of modern portfolio theory (MPT) in the context of the growth of large institutional investors. We examine the many so-called risk management practices and financial products that have been built on MPT since its inception in the 1950s. We argue that the very success due to its initial insights had the unintended consequence, given its widespread adoption, of contributing to the undermining the foundation of the financial system in a variety of ways. This study has relevance for both the ongoing analyses of the recent financial crisis, as well as for various existing and proposed financial reforms.