Exploring the distribution of security index prices during periods of distress: Evidence from International stock markets

Stock return modeling is essential for active investors and market players. Value and risk analysis benefit from these simulations. Despite extensive study on stock price modeling, little is understood about how internal and external shocks affect stock market returns. Therefore, this study sought to fill this gap. A sample of five international financial markets from December 1, 2007, to June 30, 2009 and January 1, 2020 to December 31, 2021, the 2007-2008 financial crisis and most recent COVID-19 pandemic, were tested using a Cramer-von Mises and Watson test. Research showed that internal market volatility is more damaging than external shocks. During financial instability driven by external shocks, portfolio managers should expect two to three standard deviations of volatility. However, financial system shocks should cause a greater range of volatility. The authors believe this is the first study to predict stock market returns in reaction to regulatory announcements from market shocks. © 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) license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Investing in stock markets usually arouses interest in modelling and observing the pattern of stock prices.However, visualising the behaviour of future securing prices is one of the most difficult task any financial expert has had to do because of the concept of market efficiency.Market participants and active traders are concern about the distribution of stock market prices because they are one of the big unknowns in planning for the future.
The importance of modelling stock market prices and future returns especially during periods of financial distress has many expectations.The fundamental aspect of these expectations is to have a glimpse expected future price movements and analyse the degree of uncertainty.According to Nelson, Harald & Hélène (2022), equity returns for global portfolio was 5.2% in excess of inflation.
Shocks may rather be important sources of information that may help model the stock market price pattern (Sultonov, 2021).It may be rather prudent to build expectations about the future prices by investigating how financial markets reacts to changes in the environment.
Financial distress are remarkable episodes in the history of financial markets were record high and low are observed.For the purpose of this study, only the 2007-2008 financial crisis and the most recent covid-19 pandemic are used as sample periods of financial distress for internal and external market shocks respectively.One of the most frequently asked questions that arises among financial practitioners is the similarities and differences between the 2007-2008 financial crisis and covid-19.There are certainly some similarities which were centred around a dramatic rise in unemployment due to the hardships most business entities had faced.However, the covid-19 pandemic was different from the 2007-2008 financial crisis in many ways.During the 2007-2008 financial crisis, the most hit sectors where the financial and construction sectors in that most construction activities where halted resulting in significant bankruptcy in that sector (Nistorescu & Ploscaru, 2010).On the contrary, the most affected sectors during the covid-19 pandemic were the consumer sector, tourism and hospitality, leisure and retail dominated Small and Medium size (SMEs) firms (Rogerson, Lekgau, Mashapa & Rogerson, 2021).Another significant difference was the government response between these two periods.There were permanent shocks in the economy and government finances due to much reliance on credit growth during the 2007-2008 crisis (Fatás & Summers, 2018).This prompted most Federal governments to decrease their budgets and increase taxes to meet their expenditures.However, because the pandemic was temporal, Federal governments rather opted to borrow in other to support their operational activities and the economy (Reis, 2022).These borrowings were facilitated by central banks because the borrowing cost was set at the lowest level.Also, the Covid-19 pandemic was mostly influenced by an external factor outside the financial system while the 2007-2008 crisis came from within the system.The pandemic occurred when financial markets had fully recovered from the financial crisis and where portraying signs of elevated stock prices and compressed credit spreads.These was mostly in the area of corporate debt particularly high yield corporate bonds and leverage loan markets.Has the pandemic spread across stock markets, volatility followed suit which was also observed during the 2007-2008 financial crisis (Enow, 2023).
Hence the purpose of this study was to empirically model stock market prices during periods of financial distress in response to adverse circumstances.More specifically, this study investigates the following question; Are there observable price patterns during periods of financial distress as a result of market shocks?In providing answers to this question, this study is the first as per the author's knowledge to model stock market price distributions emanating from within and outside the financial system.Hence, a significant improvement is made in the frontier of modelling stock market prices during periods of financial distress.The next section highlights the literature review.

Literature Theoretical Foundation
Price distribution and patterns as opposed to their historical movements While the average performance of stock markets may be accurate, the pattern, shape and distribution of stock prices in the global equity is still not clear especially during periods of distress.Also considering that the typical investor in stock markets is myopic and is likely to evaluate the performance of a portfolio on a yearly basis, it may be relevant to model stock market prices to enhance the informative properties that affect their distributive parameters.Also, the dispersion of these informative properties are an integral part of their analysis as uncertainty is exuberated during periods of financial distress due to the notion that market prices are assumed to follow a Markov chain process (Enow, 2022).
Consider the Autoregressive one (AR1) process below; Modelling stock prices with some distributive properties over a period of time should also consider; The logarithmic returns of stock market prices is given by; Where  is the time period,  0    are the current and yesterday's stock prices respectively and  2 is the standard deviation of the security.From equations, the distributive properties of stock market price modelling using an AR1 process mimics that of a weak form efficiency.In essence, the expected price should be dependent on other factors rather than past price movement.
These factors are perceived to play a major role in determining the price distribution and patterns as opposed to their historical movements.These factors are perceived to play a major role in determining the price distribution and patterns as opposed to their historical movements.
Using the play book from the crisis, regulators and central banks provided various measures to support the financial system.However, in contrast to the 2007-2008 crisis, there were still huge bank deposits during the covid-19 pandemic.Physical cash was not short in supply but due to fear of the virus, cashless transaction through the banking system was actively promoted putting pressure on technological advancement.Also, the flight to equality that occurred during the crisis was larger and faster than that of the covid-19 pandemic which impaired the functioning of central banks as most of the reforms during the crisis were geared towards the banking system (Mosser, 2020).Credit facilities and other financial activities have been taken up by institutions which are not in banking sector post the covid-19 pandemic.Also, several other reforms have been proposed some of which are; growing investments in credit related products and retail investment in mutual funds that hold fewer liquid securities including corporate and emerging market debts as well as commercial real estate.Institutions that are not in the banking sector are much more active in areas of the economy that large commercial banks have found less profitable such as SMEs and fintech firms post the covid-19 era.There were some significant supply chain disruptions in the covid-19 pandemic than the financial crisis which suggest that the needs of financial intermediaries were not well meet.Also, it is possible that most central banks did not understand business models prior to the pandemic which led to the supply chain disruptions.This difference may have significant effects on the behaviour of price patterns.Kendall (1953) however proposes that stock prices evolved randomly and the likelihood of increasing and decreasing was equal regardless of their past performance.The theory of market efficiency which was first postulated in the early 60s by Fama (1965) likened the behaviour of stock prices to a Brownian motion pattern due to what he coined, highly unpredictable.This stochastic nature was due to the fact that stock prices quickly incorporate new information (Enow, 2023).Dimson and Mussavian (1998) concurred with the stochastic nature and also proposed that stock prices displayed a random pattern because new information entering the stock market is random.Dimson and Mussavian (1998) is also of the opinion that competition for profits causes stock prices to change quickly to incorporate any new information.Several studies that have been conducted on modelling stock prices in many exchanges.The table below highlights the most recent empirical studies.

Source: Author
From table 1 above provides the findings of prior studies that have attempted to model stock prices.Firstly, very few author have actually attempted to model stock prices which is evident in table 1.Secondly, the authors cited above mainly focused on explanatory power of their models.Irrespective if the relevance of their studies, none of the above examined price patterns as a result of market shocks.The aim of this study was to advance the frontier of modelling stock price during periods of financial distress by exploring the distribution of security index prices during periods of distress.The next section highlights the research methodology.

Research Method
A Cramer-von Mises test was used to explore the distribution fitting for the CAC 40 Index (the French stock market index), Frankfurt stock exchange index (DAX Index), the Johannesburg stock exchange (JSE Index), NASDAQ Index and Japanese stock index (Nikkei 225).The Cramer-von Mises was suitable for modelling because it calculates the minimum distance estimates between the theoretical and empirical distribution of the stock returns (Baringhaus & Henze, 2016).In so doing, the model enhances the goodness of fit modelling through cumulative distribution functions based on observed returns (Baringhaus, & Taherizadeh, 2013).The interpretation of the values is similar to that of kurtosis parameter analysis where a value of less than 3 refers to platykurtic distribution while values of more than 3 refers to leptokurtic distributions (Queiros, Crokidakis & Soares-Pinto, 2009).This method was very useful in modelling the effect of internal and external financial markets shocks on the distribution of stock prices as it also provides a visual shape of the time series returns in order to enhance the analysis.Cramer-von Mises test equation is given below; Where  is the Cramer-von mises test value, n is the number of observations, (  ) is the sample distribution function,  2 is the scaled random variable function.Also, a modified Cramer-von mises known as the Watson test was also conducted to estimate the generalised empirical observation to provide a robust analysis.The mathematical expression of Watson test is shown below.
The value of  2 is the Watson statistics value and  the observation intervals.The main variable used in this study was the daily closing stock prices retrieved from yahoo finance.The sample period was from December 1, 2007 to June 30, 2009 and January 01, 2020 to December 31, 2021 which were the financial crisis and Covid-19 pandemic respectively.The next section highlights the findings and discussion of the data analysis.

Results and discussion
The findings are presented below, these results are separated into two the financial crisis findings and the Covid-19 pandemic results.

Source:
Author Table 2's findings contain some intriguing observations.As expected, all of the selected financial markets under investigation had abnormal returns during the Covid-19 pandemic.That is, the dispersion of returns where either skewed to the left or to the right.This can be seen in the significant p-values for the Cramer-Von mises and Watson test results as well as the sigma parameters which are less than the 5% threshold.The coefficients of the Cramer von Mises and Watson tests, which are both less than three imply that the distribution of returns during the covid-19 pandemic is platykurtic.This platykurtic distribution contains outliers and deviates somewhat from the mean.These findings are consistent with the proposition put forth by Nada, Taher, Amine, Muhammad (2019) who contends that market participants prefer to copy the market portfolio during periods of financial distress due to fear and greed, and so the overall return of their portfolio should be similar to that of the market.The findings of the 2007-2008 financial crisis are presented below.However, the coefficients are much lower than that of the pandemic.This may signal some form of differences in distribution of the returns.A Kernel density plot was also generated to visualised these perceived difference as shown below.

Conclusions
Shocks to the market, whether they are internal or external, will always elicit adaptation processes, which are typically carried out through regulatory reactions.These shocks cause difficulties that have disastrous effects for those participating in the market as a result of fear.The purpose of this research was to investigate the price distribution of the stock market in response to both internal and external shocks.According to the findings, market shocks that originate from within the financial system, such as the financial crisis that occurred in 2007-2008, generate idiosyncratic risk that is difficult to handle.Because of this abnormally high and exaggerated risk, the contemporaneous valuation may result in mispricing, and portfolio management is likely to be exceedingly challenging as a result.As a result, the performance of a portfolio will be significantly improved if it is made up of assets that have a low correlation with one another.During times of financial difficulty that are caused by external shocks, however, portfolio managers should be prepared for their holdings to exhibit a level of volatility that ranges from about two to three standard deviations.
Modeling the distribution of the prices of security indices both before and during times of financial turmoil should be the subject of additional investigation.

Figure 1 :Figure 2 :
Figure 1: The distribution fitting of stock market returns during the Covid-19 pandemic

Table 1 :
Review of prior studies

Table 3 :
2007-2008 Financial crisis outputThe findings in table 3 is similar to that of table 2 where all the p-values are significant for the Cramer-von Mises and Watson test.