THE BEHAVIOUR OF STOCK MARKET RETURNS IN NIGERIA: A MARKOVIAN ANALYSIS APPROACH
ABSTRACT
This study presents a method of Markovian analysis of the behaviour of stock returns over time. It examines the Nigerian Stock market for the period December 2007 to December 2013. Given a time series of returns, a Markov chain is defined by letting one state represent a rise in stock returns, another state to represent a fall in stock returns and a third state to represent stability in stock returns. The assumption was that the transition probabilities of the Markov Chain were equal to one irrespective of prior years.
This definition of the set of states allows both the magnitude and the direction of change to be incorporated in the analysis. Standard statistical tests for homogeneity and order of the chain are applied. In addition, the hypothesis of stationarity and dependence in vector process Markov-Chain models is tested.
Empirical results for the individual process and vector process Markov Chain confirms heterogeneity in the chains. It also suggests that stock returns movements seem to be described by a first or higher order non-stationary Markov Chain. All these results indicate that Markov chains cannot be used to give a fairly conservative estimate of the future direction of stock market returns in the Nigerian bourse; and, also indicates that stock market returns are random. Because of the heterogeneity of the individual and collective vector processes, it is recommended that a three state of nature (rise, drop and stable) vector Markov Chain can be used to describe the dynamics of the daily stock returns behaviour in the Nigerian Stock market in the short run, most especially if it is aimed at describing the random nature of stock returns in the market only rather than for predictive purposes.
CHAPTER ONE
INTRODUCTION
1.1    BACKGROUND OF THE STUDY
Over the years, mainstream economists (both classical and neoclassical) have consistently maintained that an unregulated market return or price which is determined by the market forces of demand and supply is the best yardstick reflecting true scarcity or worth of a commodity; hence the developments of a unique body of research in the field of finance and economics, currently known as ‘Efficient Market Hypothesis’. Efficient market hypothesis (EMH) is based on the notion that stock prices are informationally efficient and reflect all available information about the value of an asset in the financial market at every moment. This therefore implies that in an efficient market, price discovery is easily achieved as stock prices behave in a manner that enables market participants to determine the true worth of the stock, defined as the discounted future cashflows (Fama, 1991, Eriki & Idolor, 2010; Idolor & Braimah, 2015).
Fama (1970) asserts that we can only test whether information is properly reflected in stock returns or prices in the context of a pricing model that defines the meaning of properly. As a result, when we find anomalous evidence on the behaviour of returns, the way it should be split between market inefficiency or a bad model of market equilibrium is ambiguous. Fama (1970) review also divides work on market efficiency into three categories: (1) Weak-form tests (How well do past returns predict future returns?), (2) semi-strong-form tests (how quickly do security prices reflect public information announcements?), and (3) strong-form tests (Do any investors have private information that is not fully reflected in market prices?). In the light of these classifications on market efficiency there still exist many disagreements in the extant literature on how best to capture the behavior of stock market return; especially in frontier capital markets like Nigeria (Eriki & Idolor, 2010; Idolor & Braimah, 2015).
Furthermore, numerous empirical studies, using different methodology have also appeared in recent years concerning the behaviour of stock market returns (see, for example Eriki & Idolor, 2010; Obodos, 2007; Mcqueen & Thorley, 1991; Hamilton, 1990; Cecchetti, Lam & Mark, 1990; Turner, Startz & Nelson, 1989; Samuelson, 1988; Gregory & Sampson, 1987; Ryan, 1973; Fielitz & Bhargava, 1972; Feilitz, 1969; Fielitz 1971 to cite only a few). While a few researchers believe the notion that certain price trends and behavioural patterns exists to enable the investor to make better predictions of the expected values of future stock market returns, the vast majority of these studies conclude that past prices or returns data alone cannot form the basis for predicting the expected values and behaviour of price movements in the stock market (Idolor & Okolie, 2014).     
To date, predicting the behaviour of stock returns still is a difficult and complex challenge. Though many methods may exist, there still remains many grey areas where controversies exist and for which new insights and methods will have to be developed to better capture the causes of price changes, and how best to predict its behaviour. Naturally, prediction of stock returns is usually characterized by a great degree of data intensity, uncertainty and hidden relationships. Many factors interact together in a disproportionate manner in influencing the general returns and prices of stocks which are traded on the floor of stock exchanges worldwide. These include, but are not limited to political events, general economic conditions and traders’ expectations. Therefore, determining the behavior and future prospects of securities has proved to be quite a difficult challenge; as increasingly academic investigations have tended to agree with the notion that stock returns movements are random and as such behaves in a dynamic and unpredictable manner (Obodos, 2005; Idolor & Okolie, 2014). Stock return prediction, for the short and long term, is basically an activity which is performed by many categories of capital market operators and participants who are very much interested in investing their funds in those stocks with good prospects and thus reaping capital gains in the future (Frear, 1988). It is also worthy of note that the techniques and methods adopted by numerous researchers over the years in ascertaining the behavior of stock market returns, whether random or not; have been very dynamic and in line with the ever changing financial environment. While many of these studies have provided empirical evidence bordering on the random nature of stock market returns and by implication the existence of some form of market efficiency; there seems to be a few dissenters (Okolie & Idolor, 2014; Obodos, 2005).
Theory has clearly made some progress in this subject area of research. However, very little is known about the empirical relevance of these theories to a third world frontier or developing financial market like Nigeria. Empirical work has unearthed some stylized facts on the random behavior of stock returns and market efficiency, but these evidence is largely based on foreign and highly advanced European and Asian economies; and it is not at all clear how these facts relates to different economic models of other developing countries. Without testing the robustness of these findings outside the environment in which they were uncovered, it is hard to determine whether these empirical regularities are merely spurious correlations; let alone whether they support one theory or another (Idolor, 2010; Rajan & Zingales, 1995). This study attempts to start filling some of this gaps in our knowledge.
Against this background, the purpose of this study is to reinvestigate in terms of a simple Markov Chain the question of dependency among the returns movements of equities traded in some regional West African bourses (stock exchanges). The procedure broadly employed is to consider the behaviour of changes in the natural logarithms of stock returns for the markets as a whole in terms of a vector Markov Chain, and for a single market in terms of its particular Markov Chain. Markov theory is seen to be likely relevant in the analysis of stock returns behaviour in two ways: firstly as a useful tool for making probabilistic statements about future stock returns level and secondly as an extension of the random walk hypothesis. In this role, it constitutes an alternative to the more traditional regression forecasting techniques to which it may be in some unique way superior in the analysis of stock returns behaviour. Markov theory is concerned with the transition of a system from one state to another. In the case of a sequence of observations on stock prices, the states of the system may be thought of as the set of all possible prices that might be observed for a given stock. Since the number of states so defined is virtually infinite, it is sometimes convenient to group prices into price ranges, or price classes. That security prices may be interpreted as a Markov process means certain theorems relating to the theory of Markov processes may be brought to bear, enabling us to answer certain questions concerning the future price level of a given stock (Ryan, 1973; Obodos, 2005; Eriki & Idolor, 2010).
For the study, the model considered is that of a first-order Markov Chain. Also, the particular Markov Chain studied here has an finite number of states and a finite number of points at which observations are made. In the analysis, use is made of standard methods, as developed by Anderson and Goodman (1957), which was adopted and applied by Bhargava (1962); Fielitz (1969 & 1971), Fielitz and Bhargava (1973), Obodos (2005), Eriki and Idolor (2010) and Idolor & Gani, (2015);  for drawing statistical inferences in time when Markov Chains are applied.
1.2    STATEMENT OF THE RESEARCH PROBLEM
The prediction of stock prices has been a major challenge to stakeholders in the literature. This has been as a result of a lack of appropriate data for statistical analysis, lack of technical skills to effectively utilize available statistical tools and a lack of in-depth research into that aspect of the capital market.
Many attempts have been made to predict stock prices in the past. Analysts have used fundamental and technical approaches and more tools are being evolved in the literature to deal with this aspect of the stock market. All the attempts are to see if an investor can beat the market and reap a windfall. The success of such analytical tool would lead to an  upward trend in the stock market and further lead to market vibrancy and economic growth and development.
    Some of these analytical tools have had some successes in terms of long-term prediction of prices. For example, the Markov Chain approach has attracted latter day analysts and has been adjudged a possible tool of the future in developed economies and developing economies. The use of Markov chains has received a new impetus and is at the front burners of stock price analysis in the literature.
    The Nigerian financial environment has changed greatly over the last decade. This is basically as a result of the various capital market reforms that have been implemented over the years. This has led to an increase in the amount of new issues as well as market capitalization. Furthermore, numerous investors (individual and institutional) are now indicating serious interests in the market as it grows. There will therefore be need for analytical tool leverage to sustain these various efforts at improving the growth of the Nigerian stock market. It is against this background that the study test the use of Markov Chains in the Nigerian stock exchange.
1.3    RESEARCH QUESTIONS
The following research questions emerged against the background of the statement of the research problem to guide the researcher in the study.
i.    Are the probability vectors in the Markov Chain model capable of giving a conservative future estimate of the direction of stock returns in the Nigerian and Ghanian bourse?
ii.    To what extent can Markov Chains be used to ascertain the behaviour of stock market returns in the Nigerian and Ghanaian equity Market?
iii.    To what extent can Markov Chains be used to ascertain if stock market returns follow the Random walk Hypothesis in the Nigerian and Ghanaian equity market?
1.4    OBJECTIVES OF THE STUDY
The main objective of the study is to ascertain the behavior of equity market returns in Nigeria. In specific terms, the research objectives are to:
i.    ascertain if the probability vectors in the Markov Chain model is capable of giving a conservative future estimate of the direction of stock returns in Nigeria;
ii.    determine the extent to which Markov Chains can be used to ascertain the behaviour of stock market returns in the Nigerian equity Market;
iii.    ascertain the extent to which Markov Chains can be used to ascertain if stock market returns follow the Random walk Hypothesis in Nigeria.
1.5    RESEARCH HYPOTHESES
The hypotheses to be tested will provide answers to the research questions, as well as, assist in dealing with issues raised in the research problems and objectives. The hypotheses in line with Markov chain analysis mathematical methodology are stated in the null form as follows:
HO1:    The probability vectors in the Markov Chain model is not capable of giving a conservative future estimate of the direction of stock returns in Nigeria.
HO2:    Markov Chains cannot be used to ascertain the behaviour of stock market returns in the Nigerian equity Market.
HO3: Markov Chains cannot be used to ascertain if stock market returns follow the Random walk Hypothesis in Nigeria.
1.6    SIGNIFICANCE OF THE STUDY
    The study will broadly be of invaluable benefit and usefulness to all categories of financial information users, such as managers, existing and potential shareholders, creditors and fund providers, stock brokers and government agencies. Besides, researchers and other students of finance and banking who want to know more about stock price behaviour and possible methods of predicting it; will also find the study beneficial.
Indeed, there are several compelling needs for the study as may be gleaned from the earlier sections of the study. It will help update existing body of knowledge by going a step forward to evaluate the level of stock market returns behavior in Nigeria. It is therefore expected that the findings of this study will be of immense benefit to policy makers in the Nigerian capital market, regulators in these capital markets, relevant government institutions, researchers and numerous other stakeholders. As was mentioned earlier, very few empirical studies have been conducted in Nigeria and indeed the whole of Africa in general to determine the extent of stock returns behaviour in the respective equity markets.
Firstly, of the studies that examine regional equity markets behaviour in African equity markets within the Markov Chain framework, none in the literature, to the best knowledge of the researcher, examines it explicitly with Nigeria as the focal point, as intended in the study. Methodologically, most of the past research fail to jointly examine the relative importance of regional factors in stock returns behaviour of African equity markets by quantifying the magnitude and proportion of the behaviour driven by regional African factors and whether it has remained stable over time. In this respect, a notable novelty of our intended study is that in contrast to previous research, the study attempts to link in some measure the information content of African equity prices to time-variations in the regional African markets and their integration so as to establish, beyond speculative doubt, the factors that drive their returns processes both at the domestic and regional level. This is of fundamental importance for policy formulation. Also our focus on the equity market is not arbitrary. It is motivated by our view point that the stock market is the aspect of the financial market that is most likely to be readily affected by domestic and external shocks originating within and outside the African borders, in comparison to the money market that is a priori expected to be much more responsive to domestic monetary policy conditions.
Secondly, the exposure of emerging and frontier African equity markets like that of Nigeria to external financial risk emanating from other foreign emerging or mature equity markets in and outside Africa, is a matter of substantial concern for international investors. West African markets for instance, are often considered very risky. That however might not stem from the markets not being fundamentally sound; it might just be that investors do not know much about them. This study may provide international financiers with further information about potential investment and portfolio diversification opportunities in the focal regional African markets. Direct investment, project risk evaluation, cost of capital calculation, asset pricing and allocation, in addition to the development of risk hedging techniques can potentially benefit from this study as well.
Finally, exploration of potential inter-market linkages may also aid academics in shedding more light on the outcome of market liberalization on capital flows and mobility in Africa. Enhanced awareness of market returns behavior within the African clime may expand their understanding of the significance of the structure and potential effects of free cash flows as well as any subsequent restrictions. In addition, the study may assist academics in forecasting and evaluating the reaction of international financial markets to global and local shocks emanating from Africa, as well as in achieving deeper understanding of the shock transmission mechanisms. These can help in providing answers to the nagging question of how quickly, or how badly, the selected African equity markets were affected by any economic downturn within and outside Africa; and may even provide some insights about future financial and macroeconomic policy course of action. Besides, the findings of this study may lay the foundation for other academia and research students in African Tertiary institutions to carry-out further research related to the study, with particular emphasis on achieving a global research outlook rather than just laying emphasis on their indigenous local financial market or economy alone. The findings of this study, may also serve as very useful springboard for related studies in Nigeria and some other less developed countries (LDCs) or added experience for some others outside the African continent.
1.7    SCOPE OF THE STUDY
    Geographically, our scope is limited to two equity markets located in West Africa. The equity markets are located in Nigeria and Ghana. Temporarily, the scope is limited to a time frame that ranges from December 2007 to December 2013. Our subject matter scope is on the behavior of equity market returns in some selected regional African Equity markets, while the scope in terms of the population of interest covers all listed and statutorily recognised stock markets in Africa. For the study the stock market indexes for the selected stock markets proxy via the daily MSCI BARRA Index (Morgan Stanley Composite Index) will be  analysed from December 2007 to December 2013; a time frame of nine years. This period is considered long enough to assure the adequacy of data and reliability of results as it covers the period prior to and after the recent global financial crisis which could possibly have affected the behavior of the regional West African equity markets significantly.
1.8    LIMITATIONS OF THE STUDY
In general, the depth and thoroughness of this research may be greatly constrained by some factors which may be beyond the control of the researcher. For instance, the parameter values once incorporated within the Markov Chain model framework are assumed to be constant. This from a mathematical and econometric point of view implies that the estimated coefficients are time-specific in nature and tend to be silent on breaks that may occur in the system. However, in reality financial markets are dynamic and market conditions change continuously with time. The Markov Chain model framework may not completely capture these shifts in market conditions; and this therefore forms a part of the possible limitations of the study.
1.9    ORGANISATION OF THE STUDY
    The study is organised into five chapters as follows: Chapter one provides the background of the study, stating its objectives, hypotheses, scope of the study and the limitations of the study. A review of the related literature indicating the theoretical framework, methodological review and empirical review will be carried out in chapter two. The focus of chapter three is the research methodology with emphasis on model specification, estimation techniques, analytical tools, data collection and data requirements. Chapter four concentrates on data analysis as well as the various data presentation technique that will be used. The summary and conclusions from the study, recommendations offered and suggestions for further studies is covered in chapter five.

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