Random Walk, Capital Market Efficiency and Predicting Stock Return: A Case Study of Karachi Stock Exchange

Capital market efficiency is an important issue in developing economy, as investors confidence depends on level of market efficiency [1]. Previous studies were focused on testing weak form market efficiency in Pakistani equity market, but they overlooked the stock return predictability. Further none of studies attempts to interpret their results based on country characteristics and foundation of developing economy. The literature concerning about capital market efficiency and Random walk is inconclusive and give mixed result in KSE. Karachi Stock Market is efficient in daily, weekly and monthly return that is random walk in stock prices is followed [2]. KSE is volatile, it goes up gradually but falls down sharply, and hence investors earn unusual return [3]. Patel et al. conducts a study on Asian Stock Market and repots weekend anomaly in most of Asian stock markets including KSE. Another study by Irfan et al. on KSE found non-randomness in stock prices and serial dependency among stock return and weekend effect.


Introduction
Capital market efficiency is an important issue in developing economy, as investors confidence depends on level of market efficiency [1]. Previous studies were focused on testing weak form market efficiency in Pakistani equity market, but they overlooked the stock return predictability. Further none of studies attempts to interpret their results based on country characteristics and foundation of developing economy. The literature concerning about capital market efficiency and Random walk is inconclusive and give mixed result in KSE. Karachi Stock Market is efficient in daily, weekly and monthly return that is random walk in stock prices is followed [2]. KSE is volatile, it goes up gradually but falls down sharply, and hence investors earn unusual return [3]. Patel  The mixed results in KSE create room for researcher not to only testify validity of Random Walk Model (RWM) but to check stock predictability in the KSE. In following we will present relevant work on it and proceed to literature gape and methodology of current Research work.

Background
The theory of EMH is widely tested in different markets and shows inconclusive result. For an equity market, it's an important concern. In behavioral finance different ways are discovered how to respond to coming forth information in financial market. There are few assumptions of EMH, one of them is a large number of investors trading in equity markets and their primary aim is to enhance profit for them. For profitable trading they evaluate each piece of information and then valued the security at market and shows bull or bear behavior. Information comes in market randomly and independently, information has a unique effect on security, so if the market is efficient in processing information, security prices must show random behavior. Return on one time must be independent of another time, which is the assumption of Random Walk Model. In evaluating bad or good news help the investor to determine the intrinsic value of a security. It means in an efficient market, prices uncorrelated and serially independent, which satisfies the Weak form of efficiency of markets. Conversely to this means market is inefficient, and there exists room for technical analyst to beat the market. As in developing market, there is informational inefficient, security prices do not follow a random walk. Pakistan is an emerging economy; weak form of market efficiency is primarily concerned for investors and academicians.
it an inefficient market. Abeysekera conducted a study on Colombo stock exchange, employing non-parametric test on daily stock return from 1991 to 1996 and revealed that Colombo market is not weak form of efficient. Random walk in stock prices is not followed in Karachi Stock Exchange, it is more volatile,it goes up gradually but falls down sharply, and hence investorsearn unusual return [4].

Literature on Monday Effect
Monday effect is a hot issue in literature on stock market efficiency. This is an anomaly that exists in share prices and a serious challenge for EMH. There is evidence that the day of week, that is Monday has a lower or negative return as compare to other day of week, and Friday has a higher return. This weekend effect is tested all over the world and reported for policy implication and restoring investors' confidence. Monday have significant effect in stock exchange and day of week effect exists in major stock exchange of world as reported [8][9][10][11]. Theories of finance give possible explanation of this weekend anomaly, but researchers thrust is still unfulfilled about it. One possible explanation they give the measurement of errors while taking the closing prices for calculating stock return is not considered as correct representation of intrinsic value of stock. Some argues that since Monday is the first day of trading and Friday is last trading day, there is a high trading on last day and low trading on first day, so Monday shows unusual return. Okpara studies the Ghana Stock Exchange and proved that GSE is inefficient market as it does not followed Random walk, and shows Serial correlation and Monday effect.
Gilmore and McManus investigated Indian stock Market and found that the Indian market is inefficient in processing information as Monday effect exists. In Pakistani equity market week of the day effect is significant and Monday have either negative or low return compare to other days of week [12][13][14]. However Ali and Akbar reports that there is no day of week affect in KSE and stock return follow a Random Walk. The Monday effect is vibrant for researchers of Behavioral finance and Stock market regulatory Authority.

Literature gap
Extensive research work has been done on testing weak form of market efficiency and reports mixed result. Few authors are in view that market is not efficient and other are opposite to the opinion. But no study reports the daily predictability in KSE. The current study aims to meet purposes, testing weak form and the extent of predictability of Stock return in Karachi Stock Exchange, being largest Market in Pakistan.

Statement of the problem
Academic motivation for the revise is that empirical evidence on weak form efficiency is dissimilar. Current research will use different tests to inspect the patterns of Stock return and predictability of future return in KSE based on past information.

Objectives of Study
• To empirically investigate the weak form of market efficiency in KSE.
• To test the validity of Random Walk Behavior at KSE.
• To empirically investigate the daily predictability in KSE.

Method of Study
In this study we used different statistical model to test the Random Walk Model. A data set of stock prices of Karachi Stock Exchange from Aug 1998 to Aug 2013 was obtained for analysis purposes and checks the presence of unit root and predictive properties in stock return.

The sample
Forty companies are selected, using proportionate sample from ten sectors. These ten sectors contribute more than 90% of total market capitalization. Their daily prices from August 1998 to August 2013 were obtained for analysis purposes.
There daily prices were obtained from business recorder and yahoo finance official websites. Then there Return is calculated by using formula in excel sheet, Yi t =In (Pit / In Pi t-1 ).
Where Yit is daily stock return, Pit is share price today and pit-1 is the share price of the previous day. And in is natural logarithmic of the fraction.

Sources of data
This research work based on secondary data. The stock prices have been taken from reliable sources. The KSE official websites in addition to yahoo finance and Business Recorder websites were used for the purpose.

Random Walk Model (RWM)
∆Y t =ѲY t-1 +ε t Where ∆Yt=Share price at time t, ѲYt-1=Share price at previous day and ε t is an unpredictable component. It is known as Random Walk Model (RWM), which states that the current price is equal to last day price plus an unpredictable term, thus if RWM follow on the stock market, prices follow Random Walk, otherwise prices are predictable and confirms weak form of market inefficiency.

Predictive model (OLS)
For stock return predictability in KSE we will use an OLS model, borrowed from Jarret. The model is Y it =β o +β 1 X 1 +β 2 X 2 +β 3 X 3 +β 4 X 4 +β 5 X 5 +ε t Where Y it is daily stock return and β o is intercept of model β 2 , β 3 , β 4 and β 5 is the slope of days of the week. X 1 is dummy variable for Monday, takes a value of 1, else 0, while X 2 , X 3 , X 4 and X 5 dummy variable for Tuesday, Wednesday, Thursday and Friday, takes value 1, if same day, else 0 and ε t is the error term of regression model (Table 1).

Descriptive statistics of daily return
The mean return is 0.000875 with a standard deviation of 0.01529. Similarly the Skewness is -0.290, and value of Kurtosis is 6.86, shows very little trouble in data for estimating our required result. We can proceed further with the said data set.

Run test
The Run test is also called as Geary test is a non-parametric statistical test that checks randomness hypothesis for a two-valued data sequence, more precisely it can be used to test that elements of the sequence are mutually independent.
A "run" of a sequence is a maximal non-empty segment of the sequence consisting of adjacent equal elements. For example, the sequence "++++−−−+++−−++++++−−−−" consists of six runs, three of which consist of +'s and the others of −'s. If +s and −s alternate randomly, the number of runs in the sequence N for which it is given that there are N + occurrences of + and N − occurrences of -If the expected number of Runs is different than the actual number of runs than the null hypothesis of randomness will be rejected. Table 2 the Z-statistics is -3.777, which is less than 1.96 critical values. The null hypothesis that data has a random behavior is rejected. The expected number of runs is less than the actual runs, which confirms the non-random behavior of stock return of KSE. So on the basis of Run Test, we are in no doubt that Karachi Stock Market does not follow RWM and the market is inefficient in processing past information, and create a room for investors to make unusual return. The result is consistent with Mustafa

Philip-Perron test
The Phillip-Perron (PP) test is widely used to test the RWM hypothesis. The null hypothesis of PP test claims that a unit root is present in series at a given level of confidence. The high negative t-statistics is proof of Non-Randomness in series.
In Table 3

Augmented Dickey-Fuller test
The Augmented Ducky fuller test result present here in the Table 4 above. ADF test is applied at level with intercept. The null hypothesis was that data contain a unit root, which is Random walk follow by stock prices in our case. The more is the negative test value, the stronger is the probability of rejecting null hypothesis. The value of t-statistics is -55.03 which is less than the 1% critical Value, hence we reject the null hypothesis of non-stationaoirty, and the data is stationary at 1% level. So it is concluded that there is non-randomness in data, the distribution follows non-random behaviors. Therefore on the Basis of ADF Result it is concluded that the daily Return of Karachi Stock Exchange do not follow Random walk, so we reject the null hypothesis that Karachi stock Exchange follow Random walk. The implication of ADF test result for investors is that the better forecast the stock return by analyzing past trend of securities in KSE. The result match with those studies done in emerging economy like Poshakwale, Mehdian and Perry, Aggarwal and Rivoli, Aly et al. and Guidi et al. [8][9][10][11][12].

Autocorrelation
Serial correlation measures the correlation between stock return in one period with stock return in the previous period. Positive or negative results indicate the possible existence of potentially profitable trading strategies. A zero correlation means that no trading strategy can yield potential.
A zero correlation is consistent with the random walk hypothesis. Durbin-Watson Coefficient is used to check presence of serial in residual.
In Table 5 the result of autocorrelation is presented; all the t-statistics are significant at the 1% level. The Q-Stat is ranging from 32.756 to 58.944, and Auto-correlations have values from -0.001 to 0.093. All these values are significant at 0.00 probabilities. Since Autocorrelation of 1ll 16 lags is not equal to zero, it shows the strong autocorrelation among prices and return of stocks at Karachi Stock Exchange. The today's return correlates with the last day return. Similarly the same result also revealed from Figure 1 Table 6 presents the Daily mean return on stocks at KSE. For this purpose retunes all days returns were sorted out in excel sheet and then calculate the descriptive statistics. The result clearly shows that Monday effect exists in KSE. The Monday has lowest return of 0.000249 and standard deviation of 0.007502. Since it is the lowest return in all days of the week. It means there is selling trend on Monday while Friday has a return of 0.000915, which clarifies the purchase trend. However Wednesday has the highest mean return among five trading days of the week. It means there is day effect in Karachi Stock Exchange; hence Random walk is not followed. Investors are involved in insider trading, because of Information asymmetry in the stock market. Our result confirms studies of Haroon and Nida (2013), Aggarwal and Rivoli [10], Aly et al. [11] and Guidi et al. [12]. However it is inconsistent with Ali and Akbar (2010).

OLS predictive model
For daily predictability we use OLS model with dummy variable for days of week as used by Jarret. The outputs of regression result present here in Table 7 for the dependent variable of daily return of ending prices. We reject the null hypothesis that β o =β 2 =β 3 =β 4 =β 5 =0, since each day has their own coefficients and t-value ranging from -1.23 to 6.22 and p-value from 0.001 to 0.07. The result of these time series estimation is good fit as our F-statistics has a value of 168.08 and highly significant as probability is 0.000. The adjusted coefficient of determination of the overall study time period is 0.244, it means the independent variable of the model is explaining 24% variation in our dependent variable, which is return on daily stock in KSE.
The coefficient of the days of the week not equal to zero for four trading days and all are significant up to 1% level. As we have five  *10% significant, **5% significant and ***1% significant    dummy variables in our model, so we will estimate only four dummy variables in order to avoid dummy trap. We take Monday as a base dummy as it is the first trading days in the stock market; rest of four dummy variables have been estimated and presented in Table 7. The intercept point of the model has a coefficient of 0.0362 with a standard error of 0.0056, the statistics is -1.23 and it is insignificant. While the coefficient of Tuesday is 0.0693 with a standard error of 0.0087 and it is significant at the 5% level. Similarly the coefficient of Wednesday, Thursday and Friday are 0.0119, 0.0181 and 0.0554 respectively, and all are significant up to 1% level.
There is clear evidence that there is day effect and short term predictability exits in the Pakistani equity market, the Random walk is in doubt. As we see from table that Tuesday have a coefficient of 0.0693, if we add coefficient of intercept, which is Monday have a coefficient of 0.0056, and the coefficient becomes 0.1055. It means that on Tuesday the return will be greater a unit of 0.1055 than Monday. Similarly for Wednesday it will be 0.048, Thursday 0.543, Friday will be increased by 0.0916 units from the starting day of the week. Hence we would conclude based on our results KSE is inefficient in its Weak, as it does not satisfy the assumptions of the EMH in Weak form. The results give clear evidence of getting an abnormal return based on past information as predictability in short run exits here.
The previous studies also prove the same result, that random walk does not follow most of the Asian Stock markets, which includes Bombay Stock Exchange Jethwani and Achuthan, Dhaka Stock Exchange, Sultana and Sharmin, and Colombo Stock market Almonte, Saudi Stock Market, Assaf and Hong Kong Stock Exchange and Clearing Limited, Jarrett. All these markets are inefficient in weak form and abnormal return can be made here based on past information. Our results of all tests revealed that Karachi Stock Exchange an emerging market shows non-random behavior. Our results are consistent with those done on developing markets across the Globe. Theses developing markets are inefficient in weak form and unable to absorb past information. The developing market is illiquid, inefficient, unstable political situation, hyperinflation and highly volatile. As in Pakistan goes under law and order situation, war against terrorism and natural disaster for the last of fifteen year, thus Stock Market do not perform well. Random Walk Model does not hold in our study time period, and shows a serious anomaly of weekend effect. This might occur because of high speculative pressure, weak regulator system and high rate of institutional investors. Another rational for such results might be the political shifts that occur between 1998 to 2013, dictatorships to democratic regime, and few mega corruptions Scandal like rental power project and Lawyers movement, due to which investors shows lack of confidence and hence market do not functioned well.
Regression result is subject to the limitation on sample element and study time period. Since our sample elements are forty and time period is fifteen years, it is lengthy enough period to minimize any short term effect on stock prices and return.

Political regime vise analysis
Descriptive Statistics of Stock Return in Political regime in Pakistan Table 8      *10% significant, **5% significant and ***1% significant. to 2009 and post financial crisis after 2010 to till and 1998-1999 may be due to post Asian crisis effect on the stock exchange. The Dictatorship time period 0f 2000 to 2006 was good as the stock market performs well in this era, as RWM follows in the time period while there was the highest mean return in that time period as shown in Table 8.

Conclusions
The behavior of stock return is a hot issue in behavioral finance and risk management of financial assets. The EMH theorem supports the Random Walk Model, which claims that the stock prices are random and unpredictable. In this research work we tried to empirically investigate the weak form of market efficiency and existence random walk in Karachi Stock Exchange, being largest stock market in Pakistan. For this purpose we took stock prices of KSE since 1998 to 2013. It is sufficient a large data set to absorb any short term fluctuation in the market. We apply Run test, ADF, PP and Autocorrelation to test Random Walk Model. These tests were used by different researchers across the world. The result of all tests of the random walk, model shows no evidence of Random behavior in the KSE. The return is serially correlated with previous return, which is violation of the Random Walk Model. The market is not able to absorb the past available information, which creates room for the investors to make unusual return. The result is consistent with those done in KSE and other stock market by developing market like BSE, DSE, and HKSE and alike. Furthermore to check the daily predictability in KSE, we apply the technique used by Jarret and Kyper, an OLS model with day's dummy variable. The model gives interesting results, as each day of the week has a separate co-efficient and significant up to 1% level. It means each day return is predictable, hence past information are useful in the KSE. Results of tests of the Random Walk Model and OLS model are consistent with each other. Further by dividing the whole set of data into four groups with respect to political setup, in a period of 2000-2006 KSE was functioning well and RWM is valid only in this period. This entire test supports the non-random behavior of KSE. We are in no doubt based on our result that KSE shows non-random behavior and there is predictive property of stock returns.

Policy implication
On the basis of our different test of Randomness of stock return it is concluded that Karachi Stock Market does not follow the Random Walk Model. The Karachi Stock Exchange is not efficient in its weak form. Investors can easily predict the future outcome of stocks by applying technical analysis. As in Karachi Stock Exchange the current prices do not reflect all available information, so one can massively beat the market. Furthermore, there is day of the week effect which evidences that the Market is not efficient in processing information coming out.
The policy maker need to review the pricing mechanism at KSE in order to achieve fair pricing of financial assets, at least to the extent of past prices. There is need a prudent policy gauge and regulatory structure to control trading activities at KSE.

Limitation of Study
We are only limited to, our result of the tool applied to data and sample period. Further study may cover diverse techniques of time series analysis and large sample period to measure the performance of equity markets.

Future Research Perspective
Further study may be conducted by taking Tuesday as a first day of the week, followed by a Monday holiday to testify daily predictability in the KSE. It would be a further confirmation of stock return predictability.