Do The Highly Improbable Events Exponential Or Power Law Distributed? Evidence From Pakistan

Hence, we conclude that KSE stocks daily returns follow the power law distribution in the tail part of distribution and the plausible model for the extreme events of KSE[r]

1

DO THE HIGHLY IMPROBABLE EVENTS EXPONENTIAL OR POWER LAW DISTRIBUTED? EVIDENCE FROM PAKISTAN

Saleem Khan

Assistant Director EOBI, Ministry of oversees Pakistanis & HRD Government of Pakistan Email: saleem.khan@eobi.gov.pk

Dr Zahid Asghar

Associate Professor and Chairperson of Department of Statistics, Quid-e-Azam University Islamabad

Email: g.zahid@gmail.com Wajid Ali

Project Associate, Sustainable Development Policy Institute Islamabad Email: wajid@sdpi.org

ABSTRACT

We study the distribution of fluctuations of daily both aggregated returns of 33 KSE stocks and individually stock and index for the period of june-2004 to Feb-2012 We present evidence that econometric techniques based on normality assumption cannot be trusted in true fat-tailed distribution The result is un-computability of role of tail events where one single observation explains 99% of total kurtosis properties It also classifies decision payoffs in two types: simple payoffs (true/false or binary) and complex (higher moments); and randomness into type-1 (thin tails) and type-2 (true fat tails) We find that both positive and negative tail follow power law and tail exponent (𝛼̂ >2) lie outside the levy stable regime, but not consistent with universal cubic law and shows asymmetry To test the robustness of the result we perform Goodness –of-fit test and comparing distribution, and find support for asymptotic power law against the exponential model which some classical study support for developing countries

2

Introduction

Most of the concepts in theoretical and empirical finance developed over the last 50 years were based upon the assumption that the price or returns distribution for financial assets follows a normal distribution Since 1960s studies that have probed the validity of this assumption fail to find support for the normal or Gaussian distribution with rare exception (Mandelbrot, 1963) The study rejected normality as a distributional model for asset returns and infers that financial returns are more properly explained by a non-normal stable distribution Moreover, his early examinations on asset returns are progressed by (Fama, 1965a, 1965b among others) and led to an establishment of the hypothesis that asset returns can be better explained as a stable Paretian distribution (Mandelbrot, 1997; Mittnik & Rachev, 1993a,b; Chobanov, Mateev , Mittnik & Rachev, 1996; Adler et al 1998) Moreover, there is reasonable empirical evidence that many, if not most, financial return series are fat tailed and possibly skewed (Mandelbrot, 1960; Fama, 1965; Mandelbrot, 1962, 1963a, b, 1967 among others

The main criticism on the stable Paretian distribution that although empirical findings are not consistent with Gaussian distribution, but it also fail to find support with Levy stable distribution For instance, it was noticed that variance returns distributions of financial asset is not infinite as predicted by Levy stable distribution Moreover, it also did not remain constant under time aggregation (Akgiray & Lamoureux, 1989; Akgiray & Booth, 1988)

The findings of empirical distributions for stock prices and returns reveal that the extreme values are more likely than would be predicted by the normal distribution Which infers that between periods where the market unveils relatively moderate changes in prices and returns, there will be periods where there are changes that, are much higher (i.e., crashes and booms) than predicted by the normal distribution This is associated to both financial theorist and practitioners while assessing the frequency of severe market down turns in the stock markets, upset by the “ compelling evidence that something is rotten in the foundation of the statistical edifice ” used, for example, to generate probability estimates for financial risk assessment (Hope, 1999) Heavy or fat tails can help to expound extreme fluctuations for stocks over short time periods that can’t be described by changes in fundamental economic variables (Shiller, 1981)

Stable Paretian hypothesis is severely criticized at theoretical level, because of infinite variance feature The basic principle of Markowitz (1952, 1959) is that the risk which is measured by variance of returns distribution of a portfolio (including stocks not perfectly positive correlated) would be less than the weighted average of the risk of the individual stocks including in portfolio Furthermore, the basic structure developed by Markowitz, usually stated as mean-variance analysis, illustrates the concept of how investors benefits from diversification

3

beyond the pure thin tailed Gaussian domain The tools designed in financial economics are based on Gaussian probabilities or normal distribution and have all finite moments mainly described by mean and variance (Duffie et Al, 2000; Gatheral, 2006) All of these models considered non-scalable fat tailed distribution which collapse to thin tail due to the finiteness of all moments Thanks to the portfolios which consist of large numbers of securities The reliance on these thin tailed or finite moment distributions, led to the development of technique based on normal assumption, such as correlation, variance and beta

Finite variance is not only the basic constituent for the modeling, but is also used as risk management tool in financial market The scaling of the distribution of stocks returns have higher fourth moments, which does not justify such reliance on the Euclidian metric So the question is: why we use variance? Although it may be very useful, as “summary statistic” which measures the dispersion of the random variable, it is often misleading in fat tailed domain outside Gaussian thin tailed environment which have finite higher moments But decision making under uncertainty becomes very difficult Taleb and Goldstein, (2007) illustrate that most professional operators and fund managers practice a mental measure of mean deviation as a substitute for variance, without understanding it: since the literature emphases solely on squaring metrics, such as “Sharpe ratio”, “portfolio deviations”, or sigma’s” Standard deviation is extremely unsteady related to mean deviation in fat tailed distributions

It has been long tradition to investigate financial variables by mean of techniques developed for complex physical systems (Montroll & & Roehner, 2002) In recent times, physicists have converged their attention on using the examination tools of complex and dynamic system utilizing the developed methods for the complex physical systems in modeling the financial and economic processes (Gopikrishnan, 1998, 1999) Latest literature has uncovered some remarkable results of the distribution of stock price movements Findings suggest that it obeys a power–law distribution with tail exponents consistent with an inverse cube at the tails (the exponent α≅ 3), which lies out of the Levy stable range (0>α<2) (Matia, Amoral, Goodwin & Stanley, 2002) This phenomena is observed as a universal one as asset in the United States (Gopikrishnan, 1999 & 2000) Furthermore, some market indices also have this feature, for instance as the index of S&P 500, Dow Jones, NIKKEI, Hang Seng, Milan, and DAX (Gopikrishnan, 1998, 1999 & Stanley, 2002) There is very little literature on the subject in Pakistani context As per our knowledge Hussain & Uppal (1998) is the only study which investigated Karachi Stock Exchange 36 companies’ data and found that stock returns in Pakistani market cannot be characterized by the normal distribution Moreover, the study found that the distributions of stock returns in the Pakistani equity market are leptokurtic, i.e non-normal and positively skewed Hence, theoretical models must be used with caution

4

events and its impact on decision making under uncertainty trying to find practical and robust techniques around the theoretical problems along with to summarize the effect of power laws and scalable distribution on practice

In order to check the universality of the inverse cubic law in Pakistani equity market, we probe the returns distribution KSE-100 stocks The adjusted closing prices from 33 stocks from Karachi Stock Exchange is accumulated for analysis, and also investigates each stock and index distribution separately After stress testing and simulation procedures, it is observed that the behavior of tail distribution of daily returns obey power–law distribution but both the negative and positive tails show both inconsistency of the exponent of an inverse cube and an asymmetry in individual stock analysis but show symmetry in accumulated analysis The main objective of this study is to investigate distribution characteristic of Pakistani equity market and outlined that Gaussian and related probabilities underestimate the role of extreme events in fat tail domain In section two the study takes some insights from the literature Section three presents methodology of estimating lower bound and tail exponent of power law distribution Section four provides empirical evidence to the problem under consideration The distributional analysis of stocks and index are described in section five, whereas section concludes the study and provide some policy recommendations

Methodology and Data

The extreme fluctuations in asset returns well described by power law distribution (Lora & Boda, 2011; Clauset et al, 2009; Gabaix, 2003; Stanley et al, 2008) adversely affect the derivatives hedging and portfolio management, especially in risk management Moreover, financial returns are often considered to be asymmetric in positive and negative tail of the distribution (Yana et al, 2005) This phenomenon is very crucial in risk analysis where extreme events have large impact on decision making Therefore, the investigation about the true tail of the return distribution is highly desirable, the employment of techniques which incorporate the extreme events/fluctuations and the estimation of the underlying true risk of assets return are highly desirable This study utilized the direct methodology proposed by (Clauset, Shalizi & Newman, 2009) to probe the behavior of the tail distribution of stock returns It narrows down the hypothesis that the KSE-100 stock returns either to be exponential or power law If x denotes log returns series of KSE-100 stocks whose concern is to analyze tail distribution, then the probability density p(x) and their corresponding cumulative distribution functions (CDF) of continuous power and exponential distributions can be written as follows:

𝑃(𝑥) = 𝛼−1

𝑋𝑚𝑖𝑛 (

𝑥

𝑋𝑚𝑖𝑛)

−𝛼

, 𝑥 ≥ 𝑋𝑚𝑖𝑛, 𝛼 > (1)

5

𝑃(𝑥) = Pr(𝑋 ≥ 𝑥) = ( 𝑥

𝑋𝑚𝑖𝑛)

−𝛼−1

, 𝑥 ≥ 𝑋𝑚𝑖𝑛, 𝛼 > (2)

A continuous exponential probability density P(x) 𝑃(𝑥) = 𝑒𝑥𝑝−𝜆(𝑥−𝑋min ), 𝑥 ≥ 𝑋

𝑚𝑖𝑛, 𝜆 > (3) The corresponding cumulative distribution function (CDF) is

𝑃(𝑥) = Pr(𝑋 ≥ 𝑥) = − 𝑒𝑥𝑝−𝜆(𝑥−𝑋min ), 𝑥 ≥ 𝑋

𝑚𝑖𝑛, 𝜆 > (4)

It is important to note that both distributional form have two parameter to estimate, λ or α and 𝑋𝑚𝑖𝑛

An appropriate fitting of power laws form to observe distribution is important in order to thoroughly investigate the tail part of the distribution of stock returns and provide reliable and robust parameters estimate to empirical distribution, which truly describe risk of underlying assets Fitting power laws to empirical distributions by using simple tools of histogram and executing an OLS regression on the log of the histogram to extracted slope as scaling parameter𝛼̂ will produce significant systematic errors and inconsistent results (Clauset et al, 2009) In this study we describe a usually precise technique of maximum likelihood for the estimation of power law parameters 3.2 Estimating the power law parameters

In order to investigate the form of the tail part of the return’s distribution We fitted empirical data to power law model by computing the two parameters: the lower bound of the tail (𝑋̂ ) and the 𝑚𝑖𝑛 scaling parameter (𝛼̂) by maximum likehood method

The lower bound 𝑋𝑚𝑖𝑛 specify the beginning of the power law tail The estimate of the scaling parameter 𝛼̂ is very sensitive to the selection of the lower bound𝑋𝑚𝑖𝑛 We use the procedure of (Clauset et al, 2009) for estimation of𝑋𝑚𝑖𝑛 The fundamental idea behind this process is simple: we pick and choose the value of 𝑋𝑚𝑖𝑛 that makes the probability distributions of the measured data and the best-fit power-law model as close as achievable above 𝑋𝑚𝑖𝑛

The Kolmogorov-Smirnov or KS statistic are commonly used statistic for computing the distance between two probability distributions and is fitted as follows

𝐷 = 𝑥≥𝑥𝑀𝑎𝑥𝑚𝑖𝑛|𝑆(𝑥) − 𝑃(𝑥)| (5)

Here S(x) is the CDF of the data for the observations with value at least 𝑋𝑚𝑖𝑛, and P(x) is the CDF for the power-law model that best fits the data in the region x ≥ 𝑋𝑚𝑖𝑛 Our estimate 𝑋𝑚𝑖𝑛 is then the value of 𝑋𝑚𝑖𝑛 that minimizes D

6

parameter estimates when the sample size is large (Barndorff-Nielsen & Cox, 1995; Wasserman, 2003)

The maximum likelihood estimators (MLEs) of the scaling parameter from power law probability density is algebraically written as follows

𝑃(𝑥) = 𝛼−1

𝑋𝑚𝑖𝑛 (

𝑥

𝑋𝑚𝑖𝑛)

−𝛼

(6) Where α is the scaling parameter and Xmin is the minimum value at which power-law behavior holds The probability that the data were drawn from the model is proportional to

𝑃(𝑥 𝛼⁄ ) = ∏ 𝑋𝛼−1

𝑚𝑖𝑛

𝑛

𝑖=1 (

𝑥

𝑋𝑚𝑖𝑛)

−𝛼

(7) This probability is called the likelihood of the data given the model

3.3 Testing power law hypothesis

The methods discussed in the previous sections only help us in fitting of power law distribution to empirical data and in their estimation of parameters 𝛼̂ and𝑿𝒎𝒊𝒏 They give no information regarding plausibility of the power law fitting to the data Hence, we need some tools to check the plausibility of power law model and tell us whether the fit is a good match to the data

Most of the previous work done on empirical distributions of power laws has not tested the power law hypothesis quantitatively Instead, they just rely on subjective methods to evaluation of the data, for example on graphical visualizations But these methods can be deceptive and misleading us to accept a power law model for the data which were actually not drawn from power law distribution at all (Clauset et al, 2009) Therefore, in order to test the power law hypothesis this study used two tests namely goodness of fit test and log likelihood ratio test

3.4 Results and discussions

The data we used in this study was obtained as follows We begin with the listed companies in the KSE-100 index over the years (Jun 11, 2004 to February 15, 2012) Out of the 200 hundred listed companies we select those companies that had a complete return history Firms were chosen on the basis of three criteria’s (1) Companies have continuous listing on exchange for the whole period of analysis; (2) virtually all the important sectors are covered in data, and (3) companies have more than 1600 observations For the period from 2004 to 2012, data information on dividends, right issues and bonus share book value of stocks are taken from KSE website and read board quotations issued by the KSE at the end of each day

7

Empirical evidence

The usual descriptive statistics to test the null hypothesis that returns distribution follow normal distribution are Skewness, Kurtosis and Jarque-Bara test which were computed for all stocks and index as presented in table 4.1

Under the assumption of normality, the coefficients of skewness and excess kurtosis are zero and JB exceeds the critical chi-square value at given level of significance Table 4.1 shows that return for all stocks and index, are negatively skewed, except for two stocks It can be noted that most of the returns distribution for stocks and index are highly negatively skewed This asymmetry shows the extreme events in left tail are highly consequential have large impact decision making under uncertainty and should be dealt with caution

Regarding the height of returns distribution of sample stocks, table 4.1 illustrates the excess kurtosis are not only significantly greater than zero in all case, but some have coefficients value larger than 100 ranges from to 961 for all stock and index This implies that the empirical distribution have higher peaks and fat tails and cannot be described by normal distribution at all These findings suggest that the distribution of stock returns in the Pakistani equity market are leptokurtic and have heavy mass in the tails, as evident in the other equity markets

8

Table: 4.1 Summary Statistics of Daily Stocks and Index Returns Stocks

&Index

Mean

Std.Dev

Skewness

excess

Kurtosis Jarque-Bera Probability Observations

DWAH -0.09 4.29 -18.88 588.72 24477025 0.000000 1688

DGKC -0.05 -0.67 5.22 2209.29 0.000000 1824

ENGRO 0.01 2.61 -1.56 21.94 37496.38 0.000000 1833

GLAXO -0.06 2.72 -3.18 71.91 392336.6 0.000000 1807

GTYR -0.06 2.77 -0.07 2.31 389.17 0.000000 1741

HCAR -0.12 4.77 -7.71 234.13 4147512 0.000000 1808

HINOON -0.01 2.64 -1.21 15.27 17511.74 0.000000 1759

HUBC 0.01 2.07 -1.01 9.24 6823.37 0.000000 1830

IBFL -0.02 2.38 -0.06 0.58 24.65 0.000004 1664

ICI 0.03 2.44 -1.04 14.07 15335.12 0.000000 1820

INDU 0.05 2.39 0.02 0.96 69.32 0.000000 1800

INIL -0.1 4.01 -5.44 96.84 703914 0.000000 1779

JDWS 0.02 2.97 -1.72 23 39625.21 0.000000 1759

JOVC -0.29 4.88 -1.58 36.64 103208 0.000000 1831

JSCL -0.13 7.62 -12.06 305.91 7046364 0.000000 1796

LUCK 0.04 2.61 -0.57 5.26 2197.87 0.000000 1823

MARI 3.92 -1.57 157.49 1854820 0.000000 1794

MCB 0.07 3.01 -1.63 21.32 35442.48 0.000000 1829

MEBL 0.01 -1.83 24.06 44849.93 0.000000 1817

MLCF -0.16 3.89 -0.07 17.97 24659.86 0.000000 1832

MTL 0.02 3.81 -8.53 178.99 2343789 0.000000 1740

NBP -0.02 3.15 -3.88 64.69 324734.6 0.000000 1836

NCL -0.06 4.1 -2.96 86.03 566400.9 0.000000 1828

NML -0.01 3.31 -1.6 38.75 115375.3 0.000000 1832

NRL 0.02 2.95 -5.8 119.55 1096482 0.000000 1824

PAKRI -0.07 9.46 -21.75 958.76 69696984 0.000000 1816

PICT 0.08 2.94 -0.71 16.08 19721.64 0.000000 1817

PKGS -0.05 2.92 81.12 482560.2 0.000000 1756

PNSC -0.08 2.81 0.14 1.37 139.13 0.000000 1718

POL 0.03 3.26 -7.71 218.47 3661332 0.000000 1832

PMSC -0.04 3.13 -6.78 210.37 3323804 0.000000 1795

PSO 2.31 -2.06 32.07 79629.5 0.000000 1828

PTC -0.07 2.47 -0.31 3.5 962.51 0.000000 1833

INDEXD 0.05 1.61 -0.22 5.04 5242.9 0.000000 4911

INDEXW 0.23 3.84 -0.63 2.94 449.67 0.000000 1055

INDEXM 0.91 9.53 -0.77 3.17 125.86 0.000000 243

9

Table 4.2: Fourth kurtosis at daily, 10-day, 66-day window for the stocks & index

Stocks &Index K(1) K(10) K(66) Max Quartic Observation

DWAH 588.02 39.44 2.73 0.99 1690

DGKC 5.22 3.95 2.31 0.93 1826

ENGRO 21.91 8.56 1.18 0.92 1835

GLAXO 71.82 4.08 2.33 0.9 1809

GTYR 2.31 6.44 3.18 0.89 1743

HCAR 233.87 12.48 1.07 0.87 1810

HINOON 15.25 2.24 0.85 0.82 1761

HUBC 9.23 9.26 3.10 0.82 1832

IBFL 0.58 6.30 4.14 0.79 1666

ICI 14.05 9.20 3.45 0.78 1822

INDU 0.96 6.01 2.61 0.76 1802

INIL 96.73 11.23 2.44 0.69 1781

JDWS 22.97 2.74 2.03 0.67 1782

JOVC 36.60 2.91 0.04 0.65 1833

JSCL 305.56 15.77 5.89 0.64 1798

LUCK 5.25 6.35 2.09 0.62 1825

MARI 157.31 10.31 2.49 0.6 1796

MCB 21.29 9.21 2.56 0.52 1831

MEBL 24.03 4.84 1.59 0.48 1819

MLCF 17.95 9.28 1.21 0.46 1834

MTL 178.78 9.95 0.78 0.44 1742

NBP 64.61 3.04 0.55 0.44 1838

NCL 85.93 3.97 0.78 0.39 1830

NML 38.70 5.87 1.29 0.39 1834

NRL 119.42 13.31 1.57 0.36 1826

PARI 957.70 79.86 8.25 0.35 1818

PICT 16.06 2.32 2.72 0.35 1819

PKGS 81.03 9.87 3.45 0.34 1864

PNSC 1.23 3.30 1.34 0.32 1783

POL 218.22 26.00 4.57 0.28 1834

PMSC 210.14 5.80 1.45 0.16 1797

PSO 32.03 21.79 7.09 0.12 1830

PTC 3.49 9.74 3.14 0.12 1835

INDEXD 5.04 3.77 1.37 0.12 4913

INDEXW 2.93 1.94 -0.44 0.12 1057

INDEXM 3.12 0.52 -1.24 0.07 245

10

Figure 4.1: Maximum contribution to the Fourth moment Kurtosis coming from the largest one single observation in ~2000 (8 years of daily

observations) for 33 stocks and one index (daily, weekly, monthly and quarterly) For the Gaussian the number is expected to be ~0.02 for n=2000

Fourth moment contribution =𝑴𝒂𝒙[𝒙𝟒 ]

∑ 𝒙𝟒

Figure 4.2: A selection of 12 most acute case among 36 stock and indexes variables, where one single observation explain 99 % of total kurtosis

properties

The above analysis of Table 4.1, 4.2 and Figures 4.1 and 4.2 shows that: 1) All the stocks and index of Pakistani equity market are fat tailed 2) Conventional techniques, not just those rest upon a Gaussian distribution, but those grounded on least-square methods, or using variance as a measure of dispersion, are according to the data, unable of tracing the kind of “fat-tails” The reason is that most of the kurtosis is massed in a few observations, making it almost unknowable using conventional techniques (see Figure 2) When one massive observation presented 99% properties of total kurtosis, display more instability This undermine OLS technique, linear regression, and related methods, containing risk management approaches like “Gaussian Copulas” that depend on product or correlations of random variables

3) Table 4.2 reveals the fat tailed feature of empirical data remains fat tailed even after the summation of the variables, and hence, rejects the hypothesis of convergence to normal

0 0.2 0.4 0.6 0.8 DG KC DWA H EN G RO G LA XO G TY R HCA R H IN O O N H

UBC IBFL ICI

IN DE XD IN DE XM IN DE XQ IN DE XW IN

DU INIL JDWS JO V C JS CL LUCK MA R I

MCB MEBL MLCF MT

L N BP N CL N M L N R L

PARI PICT PKG

S PMS C PN SC PO L

PSO PTC

11

distribution under aggregation The fourth moment excess kurtosis which is measure of degree of fat-tailedness of the process, significantly greater than zero even under time aggregation and massed in very small number of observations e.g one single observation explains 99% of the total properties of KSE stock for 2000 observations These findings suggest that Guassian framework fails us in fat tail domain

Distributional Analysis

Are the tails Power-law? Once we find that empirical distributions of stocks returns are leptokurtic and fat tailed than normal, the next step is to find an appropriate specification for the distribution’s tails of returns The behavioral study of the tail part of the distribution is very important in practice, for it provides an ample opportunity to pass on information about the risk i.e involved highly consequential extreme events and returns in a portfolio The present study analyzes the behavior of tail part of the distribution in Pakistani equity market

We have normalized the returns of each stock so that the normalized returns have a mean of and a standard deviation of The plot fig (a) shows the negative tail and fig 5(b) shows positive tail Empty circles are CDFs of normalized absolute aggregated data from 33 KSE stocks and the solid line which closely fits the data is MLE power-law model fitted line While MLE exponential fitted line is represented by the dashed line which deviates from the CDFs in extreme tail observation as shown in following figure

A very close fit between observed data and power law model is obvious for both positive and negative tail returns while exponential fit deviates from CDFs as we go deeper in the tail (see figure 5) Using maximum likelihood method we estimated scaling parameter 𝛼̂ for aggregated data

𝛼̂𝑎𝑔𝑡𝑒𝑑 = {

2.58 ± 0.007 (𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑡𝑎𝑖𝑙) 2.14 ± 0.007 (𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑡𝑎𝑖𝑙)

(a)

12

Figure – Empirical cumulative distribution of the normalized absolute value aggregated daily returns of the 33 largest stocks from the Karachi

Stock Exchange for the 8-year period 2004–2012 (59376 observations)

In order to compare the findings of individually analyzed stocks with aggregated data, we averaged the scaling parameters 𝛼̂ for both tails as follow

𝛼̂𝑎𝑣𝑔≡

33 ∑ 𝛼̂

33

𝑖=1 (13) And find

𝛼̂𝑎𝑣𝑔 = {

2.54 ± 0.05 (𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑡𝑎𝑖𝑙) 3.69 ± 0.05 (𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑡𝑎𝑖𝑙)

(b)

The scale parameter 𝛼̂ determines the size of the power tail The smaller the 𝛼̂ , the fatter is the tail The negative tail of the individually analyzed data is fatter with 𝛼̂𝑎𝑣𝑔= 2.54 than positive tail with 𝛼̂𝑎𝑣𝑔=3.69 This may imply that more extreme events occur in the negative tail than positive tail Matching with tail exponents of the aggregated data, shown in above equation (a)and (b), negative tail value is consistent with each other but positive tail exponents is remarkably different form each other which indicate asymmetry in the negative and positive tail This may be interpreted that under developed market usually affected more by unanticipated imitate market behavior and aggregation of data than developed one

13

tail ranges from 1.5 to 4.5 and 29 out of 33 stock tail exponents fall between and 3.5.While positive tail exponents ranges from to and seem to be uniformly distributed over the range

Tail exponent parameters 𝛼̂

𝑖 of (a) the negative tail and (b) the positive tail, where i = 1, 33 KSE stocks analyzed We employ maximum

likelihood power law fit to estimate the parameters 𝛼̂

𝑖 of each stock The lines show the average values defined in equation (13)

Figure 5.1: Histograms: (c) the negative-tail parameters αavg = 2.54 ± 0.05, (d) the positive-tail parameters αavg = 3.69 ± 0.05

Based on all these analysis we conclude that for most of the stocks and index of different time period (daily to quarterly) have exponent (α > 2) and it lies in the range from 1.5 to 4.5 for negative tail and from to for positive tail and nearer to for maximum the stocks data So we reject the Mandelbrot levy stable Paretian hypothesis that asset have infinite variance and tail index of levy stable lies in the range (0 <α> 2) We can make a statement that developing market like Karachi Stock Exchange is follow asymptotic power law as against the result found in the previous study that developing ( Indian stock market ) follow exponential law (Matia, Pal, Salunkay & Stanley, 2004)

14

like 1987 and 2007 financial crises are normal events not outlier so this model has the capabilities to incorporate these extreme events We can use power laws as risk-management tools; they let us to measure sensitivity to left- and right-tail measurement errors and rank situations grounded on the full effect of the unobserved We can acquire information about our exposure to the tail events by varying the power-law tail exponent, and seeing the impact on the higher moments or the downfall (anticipated losses in excess of some level) This is a fully designed stress testing, as the tail exponent declines, all likely states of the world are incorporated And skepticism about the tails can guide to action and allow ranking states built on the fragility of knowledge; as these mistakes are less consequential in some situation than others

15

Table 5.1: Estimation of tail exponent α for Stocks & Index data

Stocks &

Index combined tail (?) SE Lower tail (?) SE Upper tail (?) SE

DWAH 2.26 0.03 2.54 0.05 5.68 0.04

DGKC 3.39 0.06 3.14 0.08 2.68 0.08

ENGRO 4.01 0.07 2.11 0.10 5.80 0.10

GLAXO 2.39 0.03 2.43 0.05 2.38 0.04

GTYR 5.97 0.12 2.36 0.17 4.50 0.16

HCAR 3.88 0.07 2.14 0.10 3.99 0.10

HINOON 2.83 0.04 2.94 0.06 3.44 0.06

HUBC 5.65 0.11 2.69 0.15 2.43 0.16

IBFL 5.99 0.12 2.70 0.18 3.32 0.17

ICI 4.97 0.09 2.08 0.13 2.23 0.13

INDU 2.21 0.03 2.32 0.04 2.20 0.04

INIL 4.36 0.08 1.92 0.12 2.33 0.11

JDWS 2.62 0.04 2.53 0.05 3.52 0.06

JOVC 3.56 0.06 3.18 0.08 3.95 0.09

JSCL 3.15 0.05 2.82 0.07 3.40 0.07

LUCK 4.34 0.08 2.13 0.11 2.16 0.11

MARI 1.87 0.02 2.10 0.03 3.94 0.03

MCB 3.00 0.05 2.10 0.07 4.14 0.07

MEBL 3.92 0.07 3.03 0.09 4.77 0.10

MLCF 3.40 0.06 3.17 0.08 3.63 0.08

MTL 2.64 0.04 2.46 0.05 2.50 0.06

NBP 2.31 0.03 1.95 0.04 2.14 0.04

NCL 3.67 0.06 3.54 0.09 4.21 0.09

NML 2.09 0.03 2.03 0.04 3.98 0.04

NRL 2.31 0.03 2.42 0.04 4.73 0.04

PARI 1.84 0.02 1.92 0.03 4.90 0.03

PICT 2.89 0.04 2.40 0.06 3.71 0.07

PKGS 2.12 0.03 2.19 0.04 2.72 0.04

PNSC 4.75 0.09 4.29 0.13 5.10 0.13

POL 2.22 0.03 2.20 0.04 2.32 0.04

PSMC 2.40 0.03 2.53 0.05 2.11 0.05

PSO 4.39 0.08 3.19 0.11 6.89 0.11

PTC 4.17 0.07 2.14 0.11 5.89 0.10

INDEXD 5.02 0.06 5.44 0.08 4.74 0.08

INDEXW 3.85 0.09 5.67 0.11 6.64 0.14

INDEXM 4.28 0.21 4.41 0.31 3.21 0.29

16

5.1 Testing power law hypothesis:

Most prior empirical studies of likely power-law distributed data have not endeavored to test the power-law hypothesis quantitatively In its place, they usually trust on subjective methods to test plausibility hypothesized distribution, i.e based on graphical analysis as we have done in the previous section But these can be misleading and can lead to the conclusion power-law behavior that not sustain under rigorous analysis Statistical errors might lead to accept a power law hypothesis for an empirical data which were not actually drawn from power law distribution (Clauset et al, 2009)

The p-value is exactly equal to after 100 iteration, for both tails of daily returns of aggregated data This imply that power law model is an appropriate model to explain returns fluctuations of KSE stocks, and cannot be ruled out because the variation of empirical data from the model fit is just because of statistical fluctuations and not statistical significant P-value is illustrated in upper right corner for both tails in figure 5.2(a) and 5.2(b) below

17

Figure 5.2(b) Positive tail

Figure 5.2(a) is about negative tail while figure 5.2(b) is for positive tail, The negative tail illustrate

the mean estimate of parameters 𝑋𝑚𝑖𝑛 ,𝛼̂, and the p-value The positive tail shows the estimate of standard deviation for each parameter The dashed-lines give approximate 95% confidence intervals As we have seen that for both tails the p-value is plate line passed on For further confirmation we also compared to competing model exponential and power law and see which one is best through log likelihood test

5.2 Comparing distribution

“Are the tails better described by exponential rather than Power-law”? Even if we know the

true underlying process of given data set, empirical data can be well described by more than one distribution The goodness-of-test performed in the previous section only confirmed that power law is plausible model to explain returns fluctuations But did not tell us that is there any other competing distribution which also provide good fit to the data than power law

We found that the test statistics of both the tails is positive and highly significant, which reject the null hypothesis in favor of alternative that power law is better than exponential model The results of the likelihood test are given in table 5.2

Table 5.2: Log likelihood test statistics and P-values

Test Statistics P-value

Negative Tail 5.82 0.0000006

18

5.3 Parameters uncertainty

In order check the uncertainty and stability we use bootstrap method to simulate parameters and calculate their standard errors from an observed sample After 100 iterations, the standard error of 𝑿𝒎𝒊𝒏 and scaling parameter 𝛼̂ of the negative tail is 0.56 and 0.15 respectively And for the positive tail, the standard errors of the parameters estimated to be 0.50 and 033 respectively as shown in figure 5.3 and 5.8 These finding shows that parameters are stable and we can safely make inferences about parameters under uncertainty Further, histograms are also drawn for both tail to show the uncertainty in both parameters The 𝑿𝒎𝒊𝒏 which measure the extreme returns ranges between and and 70 out 100 lies in and 1.5 for negative tail and between 0.5 and 3.5 for positive tail as shown in fig 5.4 (a) and fig 5.6 (a) This imply that extreme losses tend to be bigger and wider in magnitude than extreme gains Fig 5.4 (b) and fig 5.6 (b) shows the uncertainty in tail exponents, it ranges between and for negative tail and and for positive These findings suggest that negative tail is fatter than positive tail, but inferences about parameters can be drawn safely because both the tail exponents and lower bound is stable after 100 iteration

Figure 5.3 (Negative tail): The top row shows the mean estimate of parameters𝑋

𝑚𝑖𝑛 and α The bottom row shows the estimate of standard

deviation for each parameter The dashed-lines give approximate 95% confidence intervals After 100 iterations, the standard deviation of 𝑋

𝑚𝑖𝑛

19

Figure 5.4 (Negative tail) : Characterizing uncertainty in parameter values (a) 𝑋

𝑚𝑖𝑛 uncertainty (Standard deviation 0.56) (b) α uncertainty

(std dev 0.15)

Figure 5.5 (Positive tail): The top row shows the mean estimate of parameter 𝑋

𝑚𝑖𝑛 and α The bottom row shows the estimate of standard

deviation for each parameter The dashed-lines give approximate 95% confidence intervals After 100 iterations, the standard deviation of 𝑋

20 Figure 5.6 (Positive tail): Characterizing uncertainty in parameter values (a) 𝑋

𝑚𝑖𝑛 Uncertainty (Standard deviation 0.50) (b) α uncertainty (std

dev 0.33)

Conclusion and Policy Outcomes

Using daily returns of 33 stocks and KSE-100 index (daily, weekly, quarterly and yearly), this study has shown that generally daily returns of Pakistani equity market are leptokurtic, negatively skewed and have fat tails This means that the errors would be dominated by larger deviations than estimated Further it was shown that fat tailed characteristic of the data did not disappear under aggregation, meaning that the sum of the returns remain fat- tailed, and rejected the hypothesis of convergence to normal distribution Kurtosis which measure the degree of fat-tailedness of the process massed in very small number of observation e.g one single observation explains 99% of the total properties of KSE stock for 2000 observations These findings suggest that Gaussian framework will fail in fat tail domain

Taking into account both large aggregated data set of ≈ × 104records and individually analyzed daily returns of 33 stocks various hypothesis regarding tail returns distribution of stock were tested The analysis reveals tail distribution of daily stock returns follow power law for daily returns of separately analyzed stock Moreover, using goodness-of-fit test and log likelihood ratio rejected exponential law which may be explained that we have experienced a severe financial crisis 2007-08 in our time period of the study and used recent data than previous studies Regarding the uncertainty the analysis shows that parameters remains stable after 100 iteration

Why we so care about power law? This is because most of the random variables behave much like power law in our economic and financial system- most people follow the trend of average and cluster, but also there is many outlier Most events cluster around a typical value, but also there is many unexpected events Extensive amount of research work has been done on averaging and clustering the behavior of random character until now Future research should therefore concentrate on the investigation of these outliers and unexpected events and their impact on stock market behavior

21

References

Akgiray, V., & Booth, G G (1988) The siable-law model of stock returns Journal of Business & Economic Statistics, 6(1), 51-57

Akgiray, V., & Lamoureux, C G (1989) Estimation of stable-law parameters: A comparative study Journal of Business & Economic Statistics, 7(1), 85-93

Bergomi, L (2015) Stochastic volatility modeling CRC Press

Bollerslev, T (1986) Generalized autoregressive conditional Heteroskedasticity Journal of econometrics, 31(3), 307-327

Blattberg, R C., & Gonedes, N J (2010) A comparison of the stable and student distributions as statistical models for stock prices In Perspectives On Promotion And Database Marketing: The Collected Works of Robert C Blattberg (pp 25-61)

Cox, D R., & Barndorff-Nielsen, O E (1994) Inference and asymptotic (Vol 52) CRC Press Clauset, A., Shalizi, C R., & Newman, M E (2009) Power-law distributions in empirical data SIAM review, 51(4), 661-703

Duffie, D., Pan, J., & Singleton, K (2000) Transform analysis and asset pricing for affine jump‐ diffusions Econometrica, 68(6), 1343-1376

DuMouchel, W H (1973) Stable distributions in statistical inference: Symmetric stable distributions compared to other symmetric long-tailed distributions Journal of the American Statistical Association, 68(342), 469-477

Edward, H., Chong-Ling, L (2018) Distribution in Taiwan Stock Market Fluctuations: Normal or Power-Law Driven?

Engle, R F (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation Econometrica: Journal of the Econometric Society, 987-1007 Fama, E F (1963) Mandelbrot and the stable Paretian hypothesis The journal of business, 36(4), 420-429

Fama, E F (1965) The behavior of stock-market prices The journal of Business, 38(1), 34-105 Gabaix, X (2009) Power laws in economics and finance Annual Review of Econ., 1(1), 255-294

Gabaix, X., Gopikrishnan, P., Plerou, V., & Stanley, H E (2003) Understanding the cubic and half-cubic laws of financial fluctuations Physica A: Statistical Mechanics and its Applications, 324(1-2), 1-5

Gabaix, X., Gopikrishnan, P., Plerou, V., & Stanley, H E (2003) A theory of power-law distributions in financial market fluctuations Nature, 423(6937), 267

Goldstein, D G., & Taleb, N N (2007) We don't quite know what we are talking about when we talk about volatility

Gopikrishnan, P., Plerou, V., Amaral, L A N., Meyer, M., & Stanley, H E (1999) Scaling of the distribution of fluctuations of financial market indices Physical Review E, 60(5), 5305

Gunner, S M., Brooks, L., & Storer, R G (2006) Asymmetry of returns in the Australian stock market International Journal of Modern Physics C, 17(01), 147-153

Huang, Z F (2000) The first 20 in the Hong Kong stock market Physica A: Statistical Mechanics and its Applications, 287(3-4), 405-411

22

Janicki, A., & Weron, A (1993) Simulation and chaotic behavior of alpha-stable stochastic processes (Vol 178) CRC Press

Lau, A H L., Lau, H S., & Wingender, J R (1990) The distribution of stock returns: New evidence against the stable model Journal of Business & Economic Statistics, 8(2), 217-223 LeBaron, B (2008) Robust properties of stock return tails International Business School, Brandeis University

Mandelbrot, B B (1997) The variation of certain speculative prices In Fractals and scaling in finance (pp 371-418) Springer, New York, NY

Matia, K., Pal, M., Salunkay, H., & Stanley, H E (2004) Scale-dependent price fluctuations for the Indian stock market EPL (Europhysics Letters), 66(6), 909

McNeil, A J (1997) Estimating the tails of loss severity distributions using extreme value theory ASTIN Bulletin: The Journal of the IAA, 27(1), 117-137

Montroll, E W., & Badger, W W (1974) Introduction to quantitative aspects of social phenomena Gordon and Breach

Newman, M E (2005) Power laws, Pareto distributions and Zipf's law Contemporary physics, 46(5), 323-351

Plerou, V., & Stanley, H E (2008) Stock return distributions: tests of scaling and universality from three distinct stock markets Physical Review E, 77(3), 037101

Rachev, S T., Stoyanov, S V., Biglova, A., & Fabozzi, F J (2005) An empirical examination of daily stock return distributions for US stocks In Data analysis and decision support (pp 269-281) Springer Berlin Heidelberg

Rachev, S T (Ed.) (2003) Handbook of heavy tailed distributions in finance: Handbooks in finance (Vol 1) Elsevier

Rachev, S., & Mittnik, S (2000) Stable Paretian models in finance John Wiley and sons, Chi Chester

Roehner, B.M (2002) Patterns of Speculation: A Study in Observational Econophysics, Cambridge University Press, Cambridge

Samoradnitsky, G (2017) Stable non-Gaussian random processes: stochastic models with infinite variance Routledge

Solomon, S., & Richmond, P (2001) Power laws of wealth, market order volumes and market returns Physica A: Statistical Mechanics and its Applications, 299(1-2), 188-197

Taleb, N N (2009) Errors, robustness, and the fourth quadrant International Journal of Forecasting, 25(4), 744-759

Taleb, N N (2007) The black swan: The impact of the highly improbable (Vol 2) Random house

Todorova, L., & Vogt, B (2011) Power law distribution in high frequency financial data? An econometric analysis Physica A: Statistical Mechanics and its Applications, 390(23-24), 4433-4444

Wang, B H., & Hui, P M (2001) The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market The European Physical Journal B-Condensed Matter and Complex Systems, 20(4), 573-579

Wasserman, L (2013) All of statistics: a concise course in statistical inference Springer Science & Business Media