## Measuring Volatility of Nairobi Stock Exchange Using Garch and Egarch Models

ASSESMENT OF CONSISTENCY OF THE NSE ALL SHARE INDEX AND NSE 20 SHARE INDEX IN MEASURING THE NAIROBI SECURITIES EXHANGE CHARACTERISTICS OSORE DAVID LUVEMBE A Research Proposal Submitted To The Department Of Business Administration In Partial Fulfillment Of The Requirement For The Award Of The Degree Of Bachelor Of Commerce Of Chuka University College CHUKA UNIVERSITY COLLEGE NOVEMBER 2011 DECLARATION AND RECOMMENDATION Declaration: I declare that this project is my own original work and has not been presented for award of any degree in this or any other university. Signed:…………………………… Date……………………………..

OSORE DAVID LUVEMBE. C12/60097/08 OR BB1/0096/08. Recommendation This research has been submitted for examination with my approval as University Supervisor. Signature………………………… Date……………………………… MR. WAGALA, A, Department of business administration. Chuka University College. ABSTRACT TABLEOF CONTENTS DECLARATION AND RECOMMENDATIONII ABSTRACTIII TABLEOF CONTENTSIV ABBREVIATIONS AND ACRONYMSVII CHAPTER ONE1 INTRODUCTION1 1. 2 Statement of the problem. 3 1. 3 Purpose of the study. 3 1. 4 Objectives. 3 1. 5 Hypothesis. 3 1. 6 Significance of the study3 1. 7 Scope of the study. 3 1. Limitation of the study4 CHAPTER TWO4 LITERATURE REVIEW4 2. 1 History of the NSE4 2. 2 The role of the Nairobi Securities Exchange6 2. 3 ARCH model7 2. 5 GARCH model. 8 2. 6 EGARCH model. 9 2. 7 Indices study for the Nairobi Securities Exchange and the New York Securities Exchange. 10 2. 7. 1 Index calculation methodology10 2. 7. 2 Index formula. 11 2. 7. 3 computational precision. 11 2. 7. 4 Data correction policy12 CHAPTER THREE13 RESEARCH METHODOLOGY13 3. 1 introduction13 3. 2 research design13 CHAPTER FOUR14 RESULTS AND DISCUSSION14 4. 1 introduction14 4. 2 Preliminary analysis. 14 Figure 4. 14 Figure 4. 215 Table 4. 316 Table 4. 416 4. 2 Summery statistics17 Table 1. 17 ABBREVIATIONS AND ACRONYMS ARCH – Autoregressive Conditional Heteroscedasticity. DASS – Delivery And Settlement System. GARCH – Generalized Autoregressive Conditional Heteroscedasticity IGARCH – Integrated Generalized Autoregressive Conditional Heteroscedasticity. NSE – Nairobi Securities Exchange. NASI – Nairobi All share Index. CHAPTER ONE INTRODUCTION 1. 1. Background of the Study Financial markets can move quite dramatically, and stock prices appear too volatile to be justified by changes in fundamentals.

Such observations have been under scrutiny over the years and are still being studied vigorously (LeRoy and Porter, 1981; Shiller, 1981; Zhong et al. , 2003). Volatility as a phenomenon as well as a concept remains central to modern financial markets and academic research. The link between volatility and risk has been to some extent elusive, but stock market volatility is not necessarily a bad thing. In fact, fundamentally justified volatility can form the basis for efficient price discovery, while volatility dependence implies predictability, which is welcomed by traders and medium- term investors.

The importance of volatility is widespread in the area of financial economics. Equilibrium prices, obtained from asset pricing model, are affected by changes in volatility, investment management lies upon the mean variance theory, while derivatives valuation hinges upon reliable volatility forecasts. Portfolio managers, risk arbitrageurs, and corporate treasures closely watch the volatility trends, as changes in prices could have a major impact on their investment and risk management decisions. Speculators are usually seen with some sort of resentment by the wider community. Form the early days, scholar ave either supported that speculators stabilize prices (Smith, 1776; Mill, Friedman, 1953) or argued that speculators make money at the expense of others. In emerging economies like Kenya, there have been a great focus on market volatility and its effect on speculation and forecasting. In Kenya the security market is usually known as the Nairobi Securities Exchange. Nairobi Securities Exchange (NSE) formerly referred to as Nairobi Stock Exchange deals in shares and was started by the British colony back in the 1920’s though it was not formal and trading took place on ‘gentleman’s agreement. It was registered under societies Act in 1954 and was thus constituted as a voluntary association of stock brokers. The NSE in itself is presently a very important aspect in the Kenyan economy and is usually used to gauge the performance of the country’s economic outlook. This is so because how the NSE reacts has historically been a reflection of how the economy was at that time. That being the case, there should be proper outlay of how the market is presented, since poor market indications will also tend to be reflected in the way the economy reacts.

The market indicators should meet the demand of stock market performance. Such an indicator can be used to put in quantifiable terms the movement in stock market prices and also act as a standard in evaluating the returns on money invested in the stock market. NSE uses two kinds of indices that act as aggregate measures that can be used by potential investors since they usually give a picture of the country’s development and of the specified market segments. The two main index numbers that are used in the NSE market are the NSE 20 share index and the NSE ALL share index.

The NSE 20 share index is an index that is derived from the performance of 20 blue chip companies. These are companies with strong fundamentals and have returned positive financial results over the years. The companies included in the Index are; Mumias Sugar, Express Kenya, Rea Vipingo, Sasini Tea, CMC Holdings, Kenya Airways, Safaricom, Nation Media Group, Barclays Bank Kenya, Equity Bank, Kenya Commercial Bank, Standard Chartered Bank, Bamburi Cement, British American Tobacco, Kengen, Centum Investment Company, East African Breweries, EA Cables, Kenya Power Ltd. and Athi River Mining.

This index primarily focuses on price changes for these 20 companies The other share index is the NSE all share index (NASI). It shows the performance of all the companies that are listed in Nairobi Securities Exchange. It was introduced as an alternative index that would eventually replace the NSE 20 share index. In the NSE the volume share is the total number of shares traded on the Stock Exchange on a particular day, which together with the total value of all shares traded, (that is turnover) gives a measure of the amount of business activity on the Stock Exchange.

In their function as a basis of derivative instruments, stock market indices facilitate the application of certain portfolio strategies such as hedging and arbitrage (operative function). To perform these functions, a stock market index should fulfill statistical as well as economic requirements. Fisher (1922) and Diewart (1992) in general summarized the statistical requirements for indices. With this in mind, forecasting the volatility of the stock market is also a key aspect in the stock market.

To forecast the market successfully, the GARCH model (Generalized Autoregressive Conditional Heteroscedasticity) is usually one of the most advanced methods of doing it. This method was proposed by Engle (1982) and Bollerslev (1986) seems to be the most successful (see Bollerslev, Chou and Kroner, 1992, for a survey of GARCH applications). 1. 2 Statement of the problem. The volatility of the market says a lot about the way the economy of the country will be behaving. The Nairobi Stock market indices are usually used to measure the market behavior.

Not so much research has been done on the extent of studying the stock indices to try and assess the market characteristics. Not so much has also been done to assess the consistency of these indices to study how they affect forecasting and if they come up with a trend in time. GARCH and EGARCH model will try to look at the possibility of forecasting the Stock market. 1. 3 Purpose of the study. Based on the problem stated we will seek on ways in which we can come up with GARCH model to forecast the Nairobi Securities Exchange market and then see if we can compare the results.

We will thereafter try and compare the results of the two indices 1. 4 Objectives. 1. To estimate the volatility of the Nairobi Stock market using daily NSE all share index and the NSE 20 share index using the GARCH (Generalized Autoregressive Conditional Heteroscedasticity) model. 2. To determine the asymmetric properties of the NSE via EGARCH (exponential general autoregressive conditional heteroskedastic) model using the NSE all share index and the NSE 20 share index. 3. To compare the results obtained by modeling the NSE 20 share index and the NSE ALL share index so as to find out their consistencies. . 5 Hypothesis. Ho1 – the volatility in the stock market is not statistically significant. Ho2 – there is no significance in the asymmetric properties of the NSE via EGARCH model using the NSE indices. 1. 6 Significance of the study The study is 1. 7 Scope of the study. We will use the daily data of the NSE all share indices and NSE 20 share indices of the Nairobi Securities Exchange. 1. 8 Limitation of the study The Nairobi Securities Exchange is a relatively unstable market and the GARCH models are parametric specifications which operate mainly under stable market conditions.

The GARCH model also usually fails to capture the highly irregular fluctuations including the wild market fluctuations (including crashes and subsequent rebounding) CHAPTER TWO LITERATURE REVIEW 2. 1 History of the NSE In Kenya, dealing in shares and stocks started in the 1920’s when the country was still a British colony. However the market was not formal as there were no rules and regulations to govern stock broking activities. Trading took place on a ‘gentleman’s agreement. ’ Standard commissions were charged with clients being obligated to honor their contractual commitments of making good delivery, and settling relevant costs.

At that time, stock broking was a sideline business conducted by accountants, auctioneers, estate agents and lawyers who met to exchange prices over a cup of coffee. Because these firms were engaged in other areas of specialization, the need for association did not arise. In 1951, an Estate Agent by the name of Francis Drummond established the first professional stock broking firm. He also approached the then Finance Minister of Kenya, Sir Ernest Vasey and impressed upon him the idea of setting up a stock exchange in East Africa.

The two approached London Stock Exchange officials in July of 1953 and the London officials accepted to recognize the setting up of the Nairobi Securities Exchange as an overseas stock exchange. In 1954 the Nairobi Securities Exchange was then constituted as a voluntary association of stockbrokers registered under the Societies Act. Since Africans and Asians were not permitted to trade in securities, until after the attainment of independence in 1963, the business of dealing in shares was confined to the resident European community.

At the dawn of independence, stock market activity slumped, due to uncertainty about the future of independent Kenya. 1988 saw the first privatization through the NSE, of the successful sale of a 20% government stake in Kenya Commercial Bank. The sale left the Government of Kenya and affiliated institutions retaining 80% ownership of the bank. Notably, on February 18, 1994 the NSE 20-Share Index recorded an all-record high of 5030 points. The NSE was rated by the International Finance Corporation (IFC) as the best performing market in the world with a return of 179% in dollar terms.

The NSE also moved to more spacious premises at the Nation Centre in July 1994, setting up a computerized delivery and settlement system (DASS). For the first time since the formation of the Nairobi Securities Exchange, the number of stockbrokers increased with the licensing of 8 new brokers. In 1996, the largest share issue in the history of NSE, the privatization of Kenya Airways, came to the market. Having sold a 26% stake to KLM, the Government of Kenya proceeded to offer 235,423,896 shares (51% of the fully paid and issued shares of Kshs. 5. 00 each) to the public at Kshs. 11. 25 per share.

More than 110,000 shareholders acquired a stake in the airline and the Government of Kenya reduced its stake from 74% to 23%. The Kenya Airways Privatization team was awarded the World Bank Award for Excellence for 1996 for being a model success story in the divestiture of state-owned enterprises. On Monday 11 September 2006 live trading on the automated trading systems of the Nairobi Securities Exchange was implemented. (www. nse. co. ke) 2. 2 The role of the Nairobi Securities Exchange A stock market is an institution that deals in exchange of securities issued by publicly quoted companies and the government.

The stock market is part of the broader market referred to as financial market (Reilly, 1997; Fabbozi 1995). The major role that the stock markets have played, and continues to play in many economies is that they promote a culture of thrift, or saving. The very fact that institutions exist where savers can safely invest their money and in addition earn a return is an incentive to investors to consume less and save more. The growth of related financial services sector such as unit trusts investments clubs, pension and provident fund schemes have extensively contributed towards the deepening of the stock market.

It should be appreciated that in as much as an economy can have savings, there is usually lack of established mechanisms for channeling those savings into activities that create wealth. Therefore encouraging a culture of saving in less developed financial markets may first track economic growth (www. nse. co. ke). An efficient stock market sector will have the expertise, the institution and the means to prioritize access to capital by competing users so that an economy manages to realize maximum output at least cost.

This is what economist refers to as the optimum production level. If an economy does not have efficient financial markets there is always the risk that scarce capital could be channeled to non-productive investments as opposed to productive ones, leading to wastage of resources and economic decline (Lee,1998). The existence of stock markets promotes higher standards of accounting, resource management and transparency in the management of business. This is because financial markets encourage the separation of owners’ capital from managers of capital.

This separation is important because people who have money may not have the best business ideas and people who have the best ideas may not have money to invest. The Stock Exchange thus becomes an important link. A private company in need of capital for expansion can therefore raise funds through the stock market. This arrangement benefits both those with excess funds and the company that raises funds because the manager of capital who is the entrepreneur, is able to access capital to turn his idea into a reality, while the owners of capita, who are the shareholders, receive a return on their investment (www. nse. o. ke). Improving access to finance by providing the flexibility for customization is an important role of the stock market. This is made possible as the financial sector allows the different users of capital to raise capital in ways that are suited to meeting their specific needs. Established companies for example can raise short term finance through commercial paper; small companies can raise long term capital through selling shares; the government and even municipal councils can raise funds by floating various types of bonds and other debt instrument as an alternative to borrowing from the external market (www. se. co. ke). Stock markets provide investors with an efficient mechanism to liquidate their investments. The very fact that investors are certain of the possibility of selling out what they hold as and when they want, is a major incentive for investment as it guarantees mobility of capital in the purchase of assets . The interactions of buyers and sellers in a stock market determine the price of traded assets ;or equivalently the required return that investors demand and is this feature of stock market that signals how funds in the economy should be allocated among financial assets (Fabozzi ,1995). . 3 ARCH model Recent developments in financial econometrics suggest the use of nonlinear time series structures to model the attitude of investors towards risk and expected return. Bera and Higgins (1993, P. 315) remarked that “a major contribution of ARCH literature is the finding that apparent changes in the volatility of economic time series may be predictable and result from specific type of nonlinear dependence rather than exogenous structural changes in variables” Campbell, Lo, and McKinley (1997, p. 81) argued that “it is both logically inconsistent and statistically inefficient to use volatility measures that are based on the assumption of constant volatility over some period when the resulting series moves through time. ” In the case of financial data, for example, large and small errors tend to occur in clusters, i. e. , large returns are followed by more large returns, and small returns by more small returns. This suggests that returns are serially correlated. When dealing with nonlinearities, Campbell, Lo, and MacKinlay (1997) make the distinction between nonlinear time series and non linear time series.

In linear time series, shocks are assumed to be uncorrelated but not necesarly identically distributed whereas in nonlinear time series shocks are assumed to be identically independent distributed, but there is a nonlinear function relating the observed time series {Xt}t=0? and the underlying shocks, {? t}t=0?. They suggest the following structure to describe a non linear process Xt=g? t-1,? t-2,…+ ? th(? t-1,? t-2, …) Where the function g(. ) corresponds to the conditional mean of Xt and the function h(. ) is the coefficient of probability between the innovation in Xt and the shock ? . The ARCH model though has its limitations that prompted the advancement of the model to more advance models that aimed to improve on it. 2. 5 GARCH model. Several studies indicate that the change in speculative prices and rates of return are approximately uncorrelated over time, but characterized by tranquil and volatile periods. To allow for such a dependence we shall here take the conditional mean. GARCH model was progressed by Bollerslev (1986) because ARCH model were hard to estimate and that they decay very slowly.

He proposed an extension of the conditional variance function which he termed too be a generalization of the ARCH model and suggested that the conditional variance be specified as, ht= ? 0+? 1 ? t-12+ ? 1ht-1+…+ ? pht-p 2. 5. 1 With the inequality conditions ? 0 >0, ? t ? 0 for i=1,…, q, ? t ? 0 for i=1…,p to ensure that the conditional; variance is strictly positive. A GARCH process with orders p and q is denoted as GARCH (p, q) and this essentially generalizes the purely autoregressive moving average model. The motivation for the GARCH process can be seen by expressing the formulae as, ht= ? + ? B? t2+? Bht 2. 5. 2 Where ? (B) = ? 1B+…+? qBq and <<>> are polynomials in the backshift operator B. now, if the roots of 1- ? (Z) lie outside the init circle, equation can be written as ht= ? 01-? (1)+? (B)1-? (B)? t2= ? 0*+t=1?? t? t-12 2. 5. 3 where ? 0*= ? 0[-? 1] and the co-efficient ? t is the co-efficient of Bi in the expression of ? B[1-? 1]-1. The slope parameter ? measures the combined marginal impacts of the lagged innovations while ? , on the other hand captures the marginal impacts of the lagged innovations in the conditional variance. When t=1p? t+ j=1q? <1, then the process is weakly stationery and (? t2 ) approaches the unconditional variance (? 2) as time goes to infinity i. e. E(? t+s2)>? 2 as s>?. However, when t=1p? t+ j=1q? j>1 then the process is non stationery. There exists some situations whereby parameters estimates in GARCH (p, q) models are close to the unit root but not less than unit, i. e. t=1p? t+ j=1q? j=1, for the GARCH process. Here the multi-step forecast of the conditional variance does not approach the unconditional variance. Engle and Bollerslev (1986) refer to these processes as the integrated GARCH or IGARCH.

The IGARCH process does not possess a finite variance but are stationary in the strong sense (Nelson, 1990). From 2. 5. 3, it is easy to see that a GARCH (p,q) process is an infinite order ARCH with a rational lag structure imposed on the co-efficient. The intention is that the GARCH process can parsimoniously represent a high-order ARCH-process (Bera and Higgins, 1993; Engle, 2004; Degiannakis and Xekalaki, 2004). The simplest GARCH (1,1) is often found to be the benchmark of financial time series modeling because such simplicity does not significantly affect the preciseness of the outcome.

A GARCH model can be applied with the assumption of normal, student t or general error distributions. Besides the empirical success, GARCH models have two major draw backs: First, they are unable to model asymmetry because in a GARCH model, positive and negative shocks of the same magnitude produce the same amount of volatility (i. e. only the magnitude and not the sign of the lagged residuals determines the conditional variance). However, volatility tends to rise in response to “bad” news and fall in response to “good” news (Nelson, 1991).

The second disadvantage of GARCH models is the non-negativity constraints imposed on the parameters which are often violated by estimated parameters (Curto, 2002). 2. 6 EGARCH model. Standard GARCH models usually assume that positive and negative error terms have a symmetric effect on the volatility. In other words, good or bad news have the same effect on the volatility in this model. In practice this assumption is usually violated, in particular by stock returns, in that the volatility increases more after bad news that after good news.

This so called ‘leverage Effect’ appears firstly in Black (1976), who noted that, “a drop in the value of the firm will cause a negative return on its stock, and will usually increase the leverage of the stock. […] That rise in the debt-equity ratio will surely mean a rise in the volatility of the stock. ” Therefore from an empirical point of view the volatility reacts asymmetrically to the sign of the shocks and therefore a number of parametized extensions of the standard GARCH which included the EGARH and the TGARCH model.

The general exponential; GARCH (EGARCH) model was given by Nelson (1991) where; log? t2= ? t+k=1?? kg(zt-k) 2. 6. 1 Where ? t, ? k are the deterministic coefficients and gzt= ? Zt+ ? ( It can be directly seen thatEgZt=0. The EGARCH model shows some differences from the standard GARCH model: Volatility of the EGARCH model, which is measured by the conditional variance? t2, is an explicit multiplicative function of lagged innovations. On the contrary, volatility of the standard GARCH model is an additive function of the lagged error terms ? , which causes a complicated functional dependency on the innovations. Volatility can react asymmetrically to the good and bad news. For the general distributions of Zt the parameter restrictions for strong and covariance-stationary coincide. 2. 7 Indices study for the Nairobi Securities Exchange and the New York Securities Exchange and calculation methodology. Real- time stock prices are provided by the NSE or the NYSE. The latest trading price is used for index calculation. The number of share is determined separately for each class of stock.

This information is obtains form the company itself Corporate actions are self sourced by the NSE index. The company itself may be used as an additional source. Data filters, audit and quality assurance tools are used to monitor and maintain the accuracy of the input data. Static data are verified against secondary sources and active data are monitored in real time. When adjusting closing prices of index components to reflect the effects of completed corporate actions, prices of securities involved in the transaction trading for regular-way settlement will be used whenever available.

If a company being spun or split off from a surviving company is trading only on a “when issued” basis, then the “when issued” price of the new company will be used to determine the adjust closing price or parent company. 2. 7. 1 Index formula. The NYSE sector indexes are calculates using Laspeyres formula. This formula is used for the calculation of the return index and the price index. The only difference is that the divisor Dt is different for the two indexes. The index is computed as follows Indext = i=1npit? qitCt? i=1npi0? i0 ? Base index value= MtBt? Base index value The above formula can be simplified as: indext = MtBt Where: Dt = Btbase index value=divisor at time (t) n= the number of stocks in the index. P10 = the closing price of stock I at the base date Pit = the price of stock at time (t) qit = the number of shares of company I at time (t) Ct = the adjustment factor for the base date market capitalization t= the time index is computes Mt = market capitalization of the index at time (t) Bt = adjusted base date market capitalization at time (t)

Dividends payments are not taken into account in the price index, whereas divided payments are reinvested in the index sample of the total return index. Any dividend larger than 10% of the equity price is considered a special cash dividend, which requires a divisor adjustment. The adjustment protects the index from the effects of changes in index composition and the impact of corporate actions. 2. 7. 2 Computational precision. Index values are rounded off to two decimal places and divisors are stored in a double precision floating point binary field.

Any values derived by the index calculation engine from corporate action used for the divisor adjustment and index computations are rounded to seven decimal places. 2. 7. 3 Data correction policy To maintain a high standard of data integrity, a series of procedures have been implemented to ensure accuracy, timeliness and consistency. Input prices are monitored using a variety of computerized range – check warning systems for both tickers – plant and real- time index systems. Redundant sources of market data and corporate action information are also used.

Various verification and audit tasks are performed to ensure the quality of the real-time data feeds and related market data. While every effort is taken to ensure the accuracy of the information used for index calculation, there is no guarantee that the index will be error- proof. An index may occur due to incorrect or missing data, including trading prices, exchange rates, and shares outstanding and corporate actions, due to operational errors or other reasons. CHAPTER THREE RESEARCH METHODOLOGY 3. 1 introduction

This chapter presents the research methodology techniques that were used to carry out the research; it contains the research design, target of the sample, sampling procedures and sample size, instrumentation, piloting, data collection procedures and data analysis. 3. 2 scope of the study This study was focused on the modeling of the daily NSE-share index and the NSE 20 share index using the GARCH and the EGARCH models. The two models are able to capture the volatility of the stock market. 3. 3 Data collection The data collected was secondary data from the Nairobi Securities Exchange through their official website (www. se. co. ke/resource-centre/information-products-and-services/historical-data. html) The NSE 20 share index and the NSE all share index were used so as to try and see if there is any comparison between the two indices in terms of volatility and there by seeing if both of them are viable. The NSE-20 share index and the NSE all share index that were used spanned a period of between 2nd January 2009 and 28th October 2010. These are in total 473 individual observation of the stock exchange. 3. 4 Data analysis Time plots for the data were obtained in order to check the empirical characteristics of the data.

MLE procedure assuming Gaussian was tested for each stock indices series when fitting the GARCH models while for the EGARCH maximum likelihood estimation assuming a normal distribution was utilized. The models were diagnosed using the log likelihood ratio test. Model adequacy was carried out for all the two series by examining the standardized residual and squared residual correlations through Ljung- pierce Q- Statistics. The MSE was used to check on the efficiency of various models in addition to the residual plots. The analysis was facilitated by use of computer software namely Time Series Modeling 4. 1(Davidson, 2008) CHAPTER FOUR RESULTS AND DISCUSSION 4. 1 introduction In this chapter we will look into the research methodology techniques that were used to carry out the research; it contains the research design, target population, sampling procedures and sample size, instrumentation, piloting, data collection procedures and data analysis. Introduction The study employed the use of two sets of data i. e. NSE 20 share index and NSE all share index. The NSE 20 share index is a weighted mean with 1966 as the base year at 100. It was originally based on 17 companies and was calculated on weekly basis.

In 1992, the numbers of companies were increased to 20 to represent nearly 90% of the NSE market capitalization and computation also changed to daily basis. The index is useful as it is used in calculating and measuring the general price in the listed shares in the stock market. The NSE all share index was introduced in April 2008 and they seek that it will replace the NSE 20 share index in the long term. It uses basically the same method of weighted mean with 2008 being the base at 100. 4. 2 Preliminary analysis. The preliminary analysis was done by looking at time plots for various series.

It is illustrated in figures 4. 1 and 4. 2 Figure 4. 1 Time plot for daily NSE 20 share index Figure 4. 2 Time plot for daily NSE all share index A visual inspection of the time plots clearly shows that the mean and the variance are not constant. This implies non stationarity of the data. The non constant mean and variance suggests the utilization of a nonlinear model and preferably a non-normal distribution for modeling the data. The series were transformed by taking the first differences of the natural logarithms of the values in each of the series.

The transformation was aimed at attaining stationarity in the first moment. The equation representing the transformation is given by the xt=Inpt- In(pt-1), where ptrepresents the daily average value for each series. The sequence plots for the returns are represented in figures 4. 3 and4. 4 Table 1. Daily statistics for daily stock prices. Asset| Mean| Median| Minimum| Maximum| Std deviation| Skweness| Kurtosis| Jb| NSE 20 share| 3730. 72| 3599. 93| 2840. 86| 4678. 81| 597. 818| 0. 172471| 1. 43463| 40. 0393| NSE all share| 80. 9933| 79. 17| 58. 99| 1002. 77| 12. 7661| 0. 75| 1. 59204| 32. 8561| Table 4. 3 Time plot for log differentiated NSE 20 share index Table 4. 4 Time plot for differentiated NSE all share index It is very apparent from the table 4. 3 and 4. 4 that the amplitude of the daily stock returns is changing in the Index futures market. The magnitude of this change is sometimes large and sometimes small. This is the effect that GARCH is designed to measure and that we call volatility clustering. There is another interesting feature in the above graphs that the volatility is higher when prices are falling than when prices are rising.

It means that the negative returns are more likely to be associated with greater volatility than positive returns. This is called asymmetric volatility effect. And, this is not captured by GARCH (1, 1) model. Hence, we will use Nelson’s Exponential GARCH (1, 1) model for stock return volatility estimation. In the EGARCH model, the mean and variance specifications are: 4. 2 Summery statistics for NSE daily prices Daily statistics for Nairobi stock exchange returns Asset| Mean| Median| Minimum| Maximum| Std deviation| Skweness| Kurtosis| Jb| NSE 20 share| 0. 131554| 0. 110451| -18. 4877| 10. 8003| 1. 774| -1. 518929| 1. 43463| 40. 0393| NSE all | 80. 9933| 79. 17| 58. 99| 1002. 77| 12. 7661| 0. 775| 1. 59204| 32. 8561| The mean returns are all positive and close to zero which is a characteristic common in the financial return series. All the four have very heavy tail showing a strong departure from the Gaussian assumption. The JB test also clearly rejects the null hypothesis of normality. Notable is the fact that all the four series exist 4. 2 Empirical results and discussions. In the study we try and look at the results that are portrayed by two models i. e. EGARCH model and GRACH model. 4. 2. GARCH model results and discussions. The GARCH model for different values of p and q were fitted to the data. and form the diagnosis, the GARCH (1,1) was found to be the best model. This is consistent with most empirical studies that have been done and do involve the application of GARCH models in financial time series. The Maximum Likelihood Estimation (MLE) method was used in the parameter estimates as illustrated below Table 1 Maximum likelihood estimates for the variance equation and parameter for the AR(p) model GARCH(1,1) for the NSE 20 share index Parameter| Estimate| Std Error| T Ratio| P-Value|

AR1| 1. 07097| 0. 11557| 9. 267| 0| AR2| 0. 05328| 0. 11549| 0. 461| 0. 645| AR3| -0. 13453| 0. 07897| -1. 704| 0. 089| AR4| 0. 00151| 0. 07576| 0. 02| 0. 984| AR5| 0. 00917| 0. 06214| 0. 148| 0. 883| GARCH intercept | 18. 9057| 2. 024| | | GARCH Alpha 1| 0. 67514| 0. 17815| 3. 79| 0| GARCH Beta 1 | 0. 16116| 0. 08351| 1. 93| 0. 054| Table 2 Maximum likelihood estimates for the variance equation and parameter for the AR(p) model GARCH(1,1) for the NSE all share index Parameter| Estimate| Std error| T value| p-value| AR 1| 1. 21515| 0. 06906| 17. 596| 0| AR 2| -0. 134| 0. 098| -1. 67| 0. 172| AR 3| -0. 06885| 0. 07614| -0. 904| 0. 366| AR 4| -0. 00929| 0. 06824| -0. 136| 0. 892| AR 5| -0. 00254| 0. 05208| -0. 049| 0. 961| GARCH intercept | 0. 37516| 0. 0379| | | GARCH Alpha 1| 0. 57866| 0. 1522| 3. 802| 0| GARCH Beta 1| 0. 19842| 0. 09869| 2. 011| 0. 045| [G1]NSE ALL RETURNS | 0. 02845| 0. 01541| 1. 846| 0. 066| Table 3 Diagnostics test on the standardized residuals for the GARCH models Series| Skweness | kurtosis| JB| Q (7)| Q2(12)| NSE all share| 0. 0854| 4. 0266| 16. 6087 {0}| 10. 3115 {0. 172}| 16. 6087 {0}| NSE 20 share| 0. 1888| 6. 178| 237. 54 {0}| 11. 3591 {0. 124}| | The GARCH parameters estimates for the variance equation was significant for all the series i. e the NSE 20 share index and the NSE all share index. In the GARCH model, the parameters usually must satisfy the their stationarity requirement for the two parameters ? and ? which requires that ? +? <1. This implies that the fitted GARCH is a strong stationary and the conditional variance (? t2) does approach the unconditional variance (? 2) and thus the series might have a finite unconditional variance. EGARCH models results and discussions.

Maximum likelihood estimates for the variance equation and parameter for the AR(p) model EGARCH(1,1) Table 4 Maximum likelihood estimates for the variance equation and parameter for the AR(p) model EGARCH(1,1) for the NSE all share index Parameter| Estimate| Std Error| T Ratio| P value| AR1| 0. 21474| 0. 0399| 5. 328| 0| AR2| 0. 07914| 0. 0396| 1. 999| 0. 046| AR3| 0. 00879| 0. 03424| 0. 257| 0. 798| AR4| -0. 00447| 0. 03448| -0. 13| 0. 897| AR5| 0. 02026| 0. 02144| 0. 945| 0. 345| EGARCH Intercept| 2. 47709| 0. 5515| | | EGARCH Asymmetry (nu)| -0. 09949| 0. 09993| -0. 996| 0. 2| EGARCH Alpha1| 0. 7076| 0. 11755| 6. 02| 0| EGARCH Beta1| 0. 70621| 0. 08308| 8. 5| 0| Despite the popularity and the apparent success of the GARCH models in practical applications, they cannot capture asymmetric response of volatility to news since the signs of returns play no role in the model specification. Statistically, the asymmetric effects occurs when an unexpected decrease in price resulting from bad news increases the volatility more than unexpected increase in the price of similar magnitude following good news. Therefore, Nelson (1991) fitted the EGARCH model.

Unlike the GARCH (p, q) model, a negative shock can have a different impact on future volatility when it is compared to a positive shock f asymmetry parameter (nu) is not zero for the EGARCH model. It also does not need restrictions to be imposed on the parameters to ensure the non-negativity. In EGARCH model estimation, the MLE criterion was employed. Different orders for p and q in the variance were tested with the best results for p=q=1. The EGARCH model parameter estimates also reveal the persistence in volatility of the Nairobi equity market. The parameter is significant for NSE all share index.

This is so because the asymmetric parameter (nu) is negative and it means that the market’s negative shocks volatility are more than the positive shocks of equal magnitude. This proves therefore that the leverage effect (I. e. negative shocks increasing volatility more than the positive shocks ) is very much applicable in the study for the stock market. this concurs with the previous studies on the Nairobi Stock Exchange for instance Ogum et al. (2005, 2006) who found the asymmetric parameter to be positive when modeling the daily NSE 20 share index using EGARCH models.

We can also see that their has been some disparity from results when we try and look at the volatility of the EGARCH model. This is so because it does not fulfill the required ? +? <1 rule. CHAPTER FIVE SUMMERY, CONCLUSION, RECOMMENDATIONS AND SUGGESTIONS FOR FURTHER STUDIES 5. 1 Introduction The main purpose of this study was to look at the characteristic of the stock market characteristics and thereafter try and see the volatility and the relationship of the various stock market indices. This chapter provides a summery, conclusion, recommendations and suggestion for further studies on the area of stock market volatility. . 2 Summery of the major findings In line with the research objective and the analysis of the study, several findings played part in all: 1. The Nairobi Stock Exchange indices both give rise to the same kind of results for use of both GARCH an EGARCH models though the difference is only slight and does not as such give a different inference 2. That the stock market volatility is statistically significant and that that the NSE behaves in perfect sense as the rest of established stock market world wide. ITEMS DESCRIPTION| UNITS| COST PER UNITS| TOTAL COST(KES)| STATIONARY (a) Ball pens (b) Foolscaps (c) Photocopy papers. d) Flash disk| 52 reams2 reams1(2GB)| 10/=300/=350/=850/=| 50. 00600. 00750. 00850. 00| SUB TOTAL| | | | Purchase of data from NSE| 378| 30| 11,340| SUB TOTAL| -| -| -| SECRETARIAL/OTHER SERVICES (a) Typing proposal. (b) Typing project. (c) Printing final copy. (d) Photocopying. (e) Data processing. | 10| 30| 300| SUB TOTAL| | | | TOTAL| | | | WORK PLAN Time Activity| Week 1and2 Sept 2011| Week 3and4Sept 2011| Week 1 to 4Oct 2011| Week1 and 2 Nov2011| Week 3 and 4Nov 2011| Week 1 and 2 Dec 2011| Formulation of the problem| | | | | | | Writing proposal. | | | | | | | Presentation of Proposal| | | | | | |

Data collection| | | | | | | Data analysis| | | | | | | Editing of the proposal| | | | | | | Final presentation| | | | | | | References Bera, A. K. and Higgins, M. L. (1997) ARCH and bilinearity as competing models for nonlinear dependence. Journal of Business and Economic Statistics 15, 43–50. Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Bollerslev, T. (1988) On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Time Series Analysis 9, 121–131. Bollerslev, T. 1990) Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505. Bollerslev, T. (2008) Glossary to ARCH (GARCH). In T. Bollerslev, J. R. Russell and M. Watson (eds), Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle. Oxford: Oxford University Press. Bollerslev, T. , Chou, R. Y. and Kroner, K. F. (1992) ARCH modeling in finance: a review of the theory and empirical evidence. Journal of Econometrics 52, 5–59. Bollerslev, T. , Engle, R. F. and Nelson, D. B. (1994) ARCH models. In R. F Engle and D. L. McFadden (eds),