Random walk and modelling stock return: Evidence from international stock markets

The debate on whether stock prices follow a random walk continues, becoming the foundation of modern portfolio theory. Supporters of the random walk theory still believe that modeling stock prices is a wasteful exercise. However, evidence from momentum pricing strategy and price reversals suggests otherwise. This study aimed to empirically explore the random walk theory in five international stock markets before and during the Covid-19 pandemic from June 30, 2017, to June 30, 2019, and January 1, 2020, to December 31, 2021. A multivariate runs test and a generalised distribution function was applied to investigate the theory. The results revealed no significant difference between the observed S-statistics runs and the expected ones, concluding that it is not common to consistently observe random walks in financial markets. Hence price changes in financial markets are significantly affected not only based on new information and investors’ expectations but also by irrationalities that exist among market participants. These irrationalities can be modeled using a generalised distribution function to produce price patterns. © 2023 by the authors. Licensee SSBFNET, Istanbul, Turkey. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY


Introduction
Random walk in asset pricing and stock markets is an ubiquitous application in finance literature and investment.According to the random walk theory, it is almost impossible to accurately forecast stock prices or the direction of the market (Fama, 1965).In essence, stock prices are perceived to follow an arbitrary mathematical process as it mimics a path that is lacking in direction and with an infinite pattern (Enow, 2021).Advocates of the random walk concept contends that; it is impossible to consistently produce superior performance above the market due to the unpredictable nature of stock prices.
The dynamic nature of security prices which changes all the time is considered as the main factor affecting the unpredictable pattern.Proponents of the random walk theory argue that, this dynamic nature is as a result of unbiased expectations made by investors and market participants about future expectations (Moosa, 2015).That is to say, stock prices are unbiased estimators of market value because market prices change with almost the same proportion as new information.Market participants and investors can therefore expect the value of a security to increase if new information entering the market is good.Under this theory, investors and market participants are not expected to model or perform relative valuation to identify under or value securities as it will have very little bearing in forecasting stock returns (Fama, 1965).Therefore, new information, rational investors and market expectations are the three building blocks of the random walk theory.This concept also has strong support from events studies that have examined the relationship between price changes over a very short period of time (MacKinlay, 1997).
In explaining the random walk, event studies contend that, there is little or no serial correlation between stock returns in the short run (Kalchev, 2009;Martin, 2021).In line with the findings of these event studies, market microstructures such as the lack of liquidity and bid-ask spread bias have been proposed as the main determinants of serial correlation rather than the absence of random walk (Le & Gregoriou, 2020).However, the random walk is not without criticism.The bone of contention from academics and chartists on the random walk theory is that market participants and investors are not always rational when they set their expectations (Al Mamun, Abu Syeedb & Yasmeen, 2015).These inconsistencies may lead to different expectations for different asset classes, thus new information into the market may positively impact one asset class and negatively impact the other.Also, price changes in themselves may provide useful trading information about financial markets because there are still many market participants who still use price charts and price patterns as a tool to predict future price movements.
Jegadeeh and Titman (1993) also presents price momentum evidence against random walk where stock prices that have gone up in the last six months have a higher propensity to continue in the same pattern in the next six months.The same can be said for the significant positive correlation between stock returns over longer periods (Hurly, 2022).Between 1945 and 2010, stocks in the Dow Jones that constitutes the upper 25 th percentile with respect to price performance had a 16.5% return higher than stocks in the lower 75 th percentile (Chaudhary, Bakhshi & Gupta, 2020).
Momentum stocks which are as a result of higher trading volumes are more sustainable than low volume trading stocks countering the proposition put forth by the random walk theory (Alhussayen, 2022).From the above mentioned, it is evident that there is still no consensus on the random walk theory.Several other authors (Enow, 2022;Rehman, Kashif, Chhapra, & Rehan, 2018;Ciftci, Ispir & Kok, 2019;Dias et al. 2022) have actively argued in favour and against this concept over the years.
In line with prior literature, this study extends the frontier of market efficiency and random walk by examining the following questions.Is there any evidence of random price movements during bullish periods in financial markets?Is there any evidence of random price movements during periods of financial distress?Can stock prices be modelled using scale and shape statistical properties before and during periods of financial distress?In answering these research questions, this study presents another significant contribution to the debate of random walk and market efficiency.This contribution is in twofold, firstly it examines the random walk theory for before and during periods of financial distress.Secondly, it uses a different approach from prior literature to investigate the theory hence, a noteworthy contribution.The next section highlights the theoretical perspective followed by the research methodology, the results and discussion and conclusion respectively.

Literature Review Theoretical and Conceptual Background
The random walk theory is based on the premise that stock prices have no memory and they are independent of their lag values (Woo et al. 2020).This hypothesis was postulated by Malkiel (1973) in his famous book titled A Random walk down wall street.Accordingly, there will be no meaningful correlation between stock returns as new information into the market will be quickly encapsulated in the market price.
Following the random walk, investment practitioners will add little or no value to the performance of a portfolio as technical and fundamental analysis have poor quality information (Enow, 2022).Also, investors are perceived to behave rationally with the demand and supply forces reflecting rational investment decision (Hirshleifer, 2001).The random walk theory also maintains that, it is impossible to outperform the market without taking a significant amount of risk.Investment practitioners that boast of superior performance usually don't disclose the risk involved.As such, risk averse investors will be better off investing in market portfolios.Therefore, the only consistent strategy of investing is a market buy and hold strategy.This theory is closely linked to the Efficient market hypothesis where asset prices are believed to always fully reflect all available information (Fama, 1965).
According to Fama (1965), investors will only profit if they assume the risk that is priced in the market index.This is because, financial markets are highly competitive.Buyers and sellers are competing to execute trades at the best possible price for their selfinterest.In a competitive market, we would expect buyers to buy securities that they feel are worth more than what they paid for based on their information and preference.On the other side, sellers believe that the cash they receive from selling those securities is worth more than what they sold.This competition between self-interest and opposite views is what makes stock prices to reflect the expectations of the market participants.Hence, price becomes a mechanism to aggregate disperse bits of information about fundamental value and expectations from all market participants leading to market efficiency.However, advocates of behavioural finance argue that there are many stock markets with large number of market participants and each trader spends different amounts of time on each market.Thus, savvy investors can outperform the market by strategically buying stocks when their prices are low and sell the securities when the price increase creating trends in the market within a short period of time.Also, investors do not always behave rationally even when all available information is presented to them as they tend to always favour their portfolios (Shiller, 2003).Table 1 below summarizes prior literature on random walk-in stock markets before and during the pandemic.

Source: Author
In summarizing prior literature, only studies from 2018 were deemed relevant in this study.from the tables above, random walk-in stock markets varies between stock markets depending on the methodology.However, Dias et al. (2022) study proposes that there is no evidence to support this theory.Despite the above relevance of these studies, none of them conducted a comparative analysis to test this concept before and during periods of financial distress.Also, none of the studies modelled the stock market returns.Hence this study attempts to fill in the gap in literature.

Research & Methodology
Multivariate runs test and generalised distribution function were used to test random walk in stock prices and model the stock returns for the Johannesburg Stock Exchange (JSE), the Nasdaq, the French Stock Market Index (CAC 40), the German blue chip companies (DAX) and the Tokyo Stock exchange (Nikkei 225) from June 30, 2017 to June 30, 2019 (before the pandemic) and January 1, 2020 to December 31, 2021 (during the pandemic).
A multivariate test separates the sample returns into two subsamples characterised by positive and negative returns above and below the mean (Paindaveine, 2009).The number of runs associated with each sub sample is then compared to a theoretical expected value that would have been realised if the returns followed a random walk (Sharp, 2007).These multivariate runs test provides an intuitive method of investigating random walk by determining whether the sequence in a time series have an observable pattern (Bramson, Baland & Iriki, 2019).Specifically, the multivariate runs test was used to determine whether the sequence of positive returns in the selected stock returns are followed and preceded by a series of negative returns taking into consideration their p-values.According to Barton and David (1957), the mathematical expression of the observed and expected statistic values of a multivariate runs test is given by; Where Fn is the sum of square for each subsample and N is the number of observations.A generalised distribution function was used to model the concentration of the returns around the mean and the pattern in the time series.Of particular interest was the scale and shape parameters of the returns.Where random walk was evident, the scale and shape parameters will exhibit values above and below one to mirror a stochastic process.In its simplest form, a generalised distribution function is given by Toulias (2015); Where  is Gamma,  the return of the asset and  −1 the lag values of the return.Only the daily share prices retrieved from yahoo finance for the selected stock markets were used.The results of the data analysis are presented below.

Results and analysis
The findings and the analysis are presented below in tables 3 and 4.

Before the pandemic
From table 2 above, the values of the observed S-stats are greater than the expected S-stats in the Nasdaq, the CAC-40 and mostly in the Nikkei 225 although the DAX had two observed stats values higher than and expected stats as well as two expected stats values higher than the observe stats.In essence, a positive return preceded another positive return and vice versa for negative returns.The switch from a positive run series to a negative run was less evident than expected.This finding suggests that, more runs irrespective of whether it was positive or negative were observed in the data than warranted by the random walk theory.This implies that the returns where switching between sub samples less than predicted by the random walk hypothesis.This finding also suggests that the returns for the sampled financial markets have less mean reverting properties where positive and negative returns tend to persist.The above finding could be observed for sample period 2, 3, 4 and 5.The observed S-stats values in the JSE were less than that of the expected for atleast three periods suggesting some form of efficiency.However, the p-values for all the financial markets under consideration where more than the 5% threshold confirming the absence of market efficiency.This finding contradicts the findings of Ciftci, Ispir & Kok, 2019;Amba & Camba, 2020 but is in accordance with the findings of Rehman, Kashif, Chhapra, & Rehan, 2018;Fada, 2019.

During the pandemic
During the Covid-19 pandemic, the JSE, CAC 40 and the DAX displayed high levels of market inefficiencies where their observed S-stats values were greater than the expected.In the Nasdaq, the opposite was true as the expected S-stat runs outweighs the observed S-stat runs and an equal weighting was seen in the Nikkei 225.In line with the findings before the pandemic, none of the p-values were significant at 5% except the DAX in period 4.This finding concurs with the findings of Enow (2022) who contends that, financial markets still exhibit irrationalities and market inefficiencies where returns can be predicted to a certain degree.However, the finding contradicts the Dias et al. ( 2022) results.The table below models the returns of the selected financial markets.3 where all the financial markets under consideration displayed heavy tail returns before and during the pandemic with the exception of CAC 40.As already highlighted in the literature, random walk hypothesis contends that stock market returns follow a stochastic process hence fat and long tails are expected (Enow, 2023).However, the shape of all the returns in table 4 displayed fat tails.This finding is strengthened by the results of the excess kurtosis where the values are more than 3 in all the cases.From these results, events in financial markets will always have a significant impact with predictable outcome to some extent.Furthermore, the scale parameters are much less than one inferring a slow switch between positive and negative runs (Stoyanov et al. 2011).The mean and standard deviation also present the stylised facts of the returns where Nasdaq had the highest risk before the pandemic and the DAX during the pandemic.The implications of this finding is that a buy and hold strategy is not an effective investment style despite its perceived risk tolerance threshold.Hence active market may significantly enhance the value of their portfolios by using active management strategies.

Conclusion
The random walk hypothesis is still considered as one of the most important concepts that is frequently used in asset price modelling.
According to this theory, investing based on technical and fundamental analysis is really a fruitless exercise as market participants will not be able to systematically outperform the market.This study set out to investigate the random walk theory empirically in five global stock markets before and after the Covid-19 outbreak.The findings showed no discernible difference between the observed and expected S-statistics runs, leading to the conclusion that random walk is not frequently observed consistently in financial markets.As a result, irrational behaviour among market participants has a considerable impact on price movements in financial markets in addition to new information and investor expectations.A generalized distribution function can be used to model these irrationalities and generate price patterns.In essence, active market participants can observe trends in security prices through charts and other market tools.Savvy investors can use factor investing to enhance the value of their portfolio.Hence market participants with superior skills can produce positive alpha over a period.In conclusion, the random walk hypothesis may not be a workable hypothesis in the current dispensation as market anomalies seemed to be prevailing.Future research should explore random walk before, during and after periods of financial distress.

Table 1 :
Prior literature on random walk before the Covid-19 pandemic

Table 2 :
Prior literature on random walk during the Covid-19 pandemic

Table 3 :
Multivariate runs test results

Table 4 :
Distribution function for returns