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INVESTMENT IMPLICATIONS OF THE FRACTAL

MARKET HYPOTHESIS

ADAM KARP

*

and GARY VAN VUUREN

†

School of Economics

Department of Risk Management

North-West University

Potchefstroom Campus, South Africa

*

adam.karp@avivainvestors.com

†

vvgary@hotmail.com

Published

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 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

dimension.

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

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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 EMH’s 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. Hurst’s (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 former’s 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-

ket’s 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

time

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

(3–5 years)

New information rap-

idly incorporated

into asset prices

New information usually incorporated

rapidly into asset prices, with some

exceptions

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

“invariant”on 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

“value”outperform 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

returns

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

(1995)

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

periods

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: Author’s 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 H6¼ 0:5, 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 1992–December 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 markets’Hwas 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.

EMH 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

h0:5i

0.5 1.0

Auto-covariance <08lags ¼08lags >08lags

Behavior Anti-persistent Brownian Persistent

Statistical

interpretation

Decrements (incre-

ments) more likely

to be proceeded by

increments (decre-

ments)

Decrements/incre-

ments equally

likely

Increments (decre-

ments) more likely

to be proceeded by

increments (decre-

ments)

Character Reverts to the mean

more frequently

than a random one

Random motion Exhibit long-memory

and “trends”and

“cycles”of varying

length

Sources Barkoulas et al.

(2000); Kristoufek

(2010)

Osborne (1959) Mandelbrot and Van

Ness (1968)

(a) (b) (c)

Source: Author’s 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 1–July 07), Karangwa (2008) found

H0:5 for the JSE. Note that Karangwa’s (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 1995–August

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: Author’s calculations.

Figure 4. Average Hs measured on various JSE sectors over the period 2000–2010. 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 investors’behavior 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 (2003–2008), and considerable turbulence (1998–2000 (the Asian crisis

and the dotcom crash) and 2008–2011 (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 asset’s 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

p,

where "is a random number drawn from a standard normal distribution. The

solution of this differential equation is

St¼S0exp μt2

2tþWt

,ð2Þ

where S0is the initial asset price. In principle, Stdescribes the asset’s 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

Gaussianity”the 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 asset’s volatility and its returns (Chordia et al.,

2008),

(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)

becomes

dSt¼St(μdt þdZt),ð3Þ

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where dZt¼"ﬃﬃﬃﬃﬃﬃﬃ

t2H

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

2t2HþZt

:ð4Þ

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 Hausdorff–Besicovitch

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

“distance”the 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 series’standard deviation and ﬃﬃﬃ

n

p.

Hurst (1951) generalized this relationship to

R

n¼cnH,ð5Þ

where is the standard deviation (i.e. the realized volatility) of the stationary time

series’nobservations 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

c¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2H3

2H

1

2þH

(22H)

s,ð6Þ

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

n¼ln(c)þHln(n):

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

nPn

k¼1Nk,aas shown in Fig. 5.

The time series of cumulative departures from the mean, for each subperiod la,

are then

Xk,a¼X

k

i¼1

(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

la¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

nX

n

k¼1

(Nk,ae2

a)

v

u

u

t:

A rescaled range, Rla=lafor each subperiod, la, is then determined, the average of

which is

R

n¼1

AX

n

a¼1

Rla

la

:

The length nis then increased until there are only two subperiods (¼N

2).

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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: Author’s calculations.

Figure 6. Regression results, March 2006–March 2009. H¼0:509 and c¼exp

(0:009)¼1:009.

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 series’fractal 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 trader’s 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 asset’s price is Pion day i, its one-day log return, ri, on day iis

ri¼ln Pi

Pi1

:

The scaling factor, n, is used to determine the n-day log return, Ri,n, on day i:

Ri,n¼ln Pi

Pin

,

as well as the scaled return, Ni,n, on day i:

Ni,n¼Pi

inabs(ri)

abs Ri,n

n

¼Pi

inabs ln Pi

Pi1

abs

ln Pi

Pin

n

0

@1

A

,

and the scaled fractal dimension, Di,n, on day i:

Di,n¼ln(Ni,n)

ln(n)¼

ln Pi

in

abs ln Pi

Pi1

abs

ln Pi

Pin

ðÞ

n

2

6

6

43

7

7

5

ln(n):ð7Þ

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 economy’s 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),

(a)

(b)

Source: Author’s calculations.

Figure 7. Rolling (a) H(t)and (b) c(t)for the S&P 500 from January 1998–December

2017.

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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 “dotcom”crisis in 00 or the events of September 2001. These results

confirm and update those found by Karangwa (2008) and Chimanga and Mlambo

(2014).

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: Author’s calculations.

Figure 8. Rolling H(t)for the FTSE 100 from January 1998–December 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 “sectorized”or 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 D—i.e. the rate of change or

“speed”of the change of Dalso provides information about subsequent market

movements.

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: Author’s calculations.

Figure 9. Rolling H(t)for the JSE All Share from January 1998–December 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:

Dt0Dt5

Dt5

,

(a)

(b)

Source: Author’s 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

breaching.

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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

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: Author’s 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 “normal”market 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.

Acknowledgment

We are grateful to the anonymous reviewer for valuable comments and

suggestions.

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