Volatility Modeling for SENSEX using ARCH Family G. Arivalagan* Research scholar, Alagappa Institute of Management Alagappa University, Karaikudi-630003, India. E-mail: arivu760@gmail.com *Corresponding author Dr. S. Rajamohan Professor, Alagappa Institute of Management Alagappa University, Karaikudi-630003, India. E-mail: srajamohan1988@gmail.com Abstract The study aims to attain the volatility of the historical price in Sensex. The researcher has examined the historical volatility on the Sensex price for twenty years duration from 1997 to 2016. The daily closing price of Sensex for the study period was obtained from the Bombay Stock Exchange (BSE). In the current research three models were utilized to measure the volatility of Sensex closing price movements which are Auto Regressive Conditional Heteroscedasticity (ARCH), Generalized Auto Regressive Conditional Heteroscedasticity (GARCH) and Threshold Generalized Auto Regressive Conditional Heteroscedasticity (TGARCH). Based on the results obtained, the researcher identifies the model fit on the basis of Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) value. The empirical result of models has also been tested by serial correlation and the ARCH LM test. It was concluded that the Threshold Generalized Auto Regressive Conditional Heteroscedasticity (TGARCH) model is suitable method for the volatility modeling of Sensex closing price. Keywords: volatility; SENSEX; ARCH; GARCH; closing price 1. Introduction The study is undertaken utilizing the time series analysis perspective. In this study, the historical and implied methods are primary tools to construct the modeling of volatility. The historical volatility model has been employed to analyze the time series data. The model measures the changes in the value of an underlying security prices over the different period of time. The changes were calculated on the basis of daily settlement of BSE Index. The volatility increased by the various market determinants such as economic factors, political climate, special announcement, and Vol.2 No.2 July-December 2016 52
others into providing the opportunity for both sides of traders. The down market trend is an opportunity of the investor for investing in the market. BSE creates a potential platform for the market participants and also a leading exchange in India. In order to improve the economic wealth, most of the corporate companies get listed in the stock exchanges and improve their profitability. This will also enable them to gain increased credibility from public. However, BSE index sensitively fluctuates upon the domestic and international situations. The present study attempts to examine the volatility of BSE SENSEX. Further, the researcher applied the appropriate univariate analysis to observe the volatility of the SENSEX index. 2. Review of Literature 2.1 Studies on Asymmetric volatility Bekaert and Wu (2000) have studied the asymmetric volatility of equity market in Japan. They conclude from the empirical findings that negative shocks increase the conditional covariance substantially, whereas positive shocks have a mixed impact on conditional covariance. Karmakar (2005)analyzed the volatility modeling in Indian stock market. The researcher evaluated the forecasting model in out-of-sample technique and investigates the leverage effect in CNX Nifty and BSE Sensex. GARCH (1,1) model has been evaluated by estimating parameters initially over trading days of the in-sample period and then used the estimated parameters to later data, thus forming out-of-sample forecasts on two market indices. These out-of-sample volatility forecasts have been compared to true realized volatility. Further, Floros and Christos (2008) analyzed the application of asymmetric GARCH models for modeling volatility and explaining financial market risk. They used two major indices i.e., CMA General Index from Egypt and TASE- 100 index from Israel. Results of the study provided strong evidence that asymmetric GARCH models can better explain volatility in two countries stock markets. Results of the study also concluded that increased risk will not necessarily lead to increased return in the market. Mehta and Sharma (2011) have observed the time varying volatility of Indian stock market. They have considered S&P CNX Nifty index of NSE (National Stock Exchange) for a period of approximately a decade, i.e., March 2001 to October 2010. The findings of the study accredited that the Indian equity market has witnessed the prevalence of time varying volatility where the past volatility has more significant impact on the current volatility. The identification of persistence of conditional volatility can help the investors to forecast their returns from equity market under alternate market phenomenon. The study observed the market volatility of BSE 500 stock index during the financial crisis of 2008-09. The BSE500 returns series found to react to the good and bad news asymmetrically. The presence of the leverage effect would imply that the negative news has a greater impact on volatility than positive news. The researcher Vol.2 No.2 July-December 2016 53
concluded that the sign of the innovation has a significant influence on the volatility of returns and the arrival of bad news in the market would result in the volatility to increase more than good news. Therefore, the bad news generates volatility on Indian stock market more than good news, pointed out by (Goudarzi and C.S, 2011). In a study of volatility model in Karachi Stock Exchange using GARCH construction between 2004 and 2012 period of data. This implied that all estimates of risk in the period based on standard deviations must be flawed and would understate the actual risk This has serious implications because risk assessment plays a vital role in estimating cost of capital, firm valuations and capital budgeting. Based on their findings, that higher order moments of returns should be considered for prudent risk assessment, reported by (Nawazish and Mawal Sara, 2012). 2.2 Studies on Time Varying Volatility Abdalla Suliman and Zakaria (2012) have made an attempt to model volatility in Saudi stock market TASI index. They applied various asymmetric GARCH models such as EGARCH, TGARCH and PGARCH. They observed determination of conditional volatility and the results of his study were in favor of 'positive correlation hypothesis' which established positive relationship between volatility and expected stock return. In a similar study by Kalu O and Stephen Friday (2012) examined the volatility of Nigerian stock exchange(nse). They applied EGARCH (1, 1) and GJR-GARCH (1, 1) models to find the asymmetry volatility effects in the NSE stock returns. The researcher took daily stock returns of NSE from January 2 nd 1996 to December 30 th 2011. They found the positive asymmetry volatility effect from the EGARCH model. GJR-GARCH model show negative and significant asymmetric volatility coefficient, also supporting the existence of positive asymmetric volatility. Overall results from this study provide support for positive news creates higher volatility in the immediate trading day than negative news of the same magnitude in Nigeria. Som Sankar and Tanmay (2012) have made attempted to find the characteristic of conditional volatility and asymmetry leverage effect of BSE Sensex. They used GARCH (1, 1) model for to estimate the conditional volatility of the data. The GARCH (1, 1) results show model is good fit for the data and Sensex has the volatility over a long period of time. The TGARCH (1, 1) model applied to compute the leverage effect of the sample. The empirical results show that news asymmetry and leverage effect are present in the BSE market. In a study by Singhania and Anchalia (2013) reported the volatility analysis in Asian stock markets and global financial crisis. They used Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model. The researchers analyzed using time series data of daily returns for the period 2005-2011 of the major indices of these countries (Hang Seng, Nikkei 225, Shanghai Composite and Nifty for Hong Kong, Japan, China and India, respectively). The results revealed that the sub-prime crisis had a positive impact on the volatility of returns of Japan, China and India while it had no impact on Vol.2 No.2 July-December 2016 54
the volatility of returns of Hong Kong. In addition, they stated that the period of Eurozone debt crisis has had a negative impact on the volatility of already highly volatile stock returns of countries such as India and China. Singhania and Prakash (2014) have conducted a study on cross-correlation of SARRC countries stock returns. The data consist of stock indices from India, Bangladesh, Sri Lanka and Pakistan, the daily closing price covered from 2000 to 2011. Results indicated the presence of serial autocorrelation in stock market returns, implying dependence of current stock prices on stock prices of previous times and leads to rejection of EMH. Correlation between stock indices of SAARC economies were found to be low which was in line with intra-regional trade being one of lowest as compared to other regional groups. The study results pointed out that there is a greater need for economic cooperation and integration between SAARC countries. Greater financial integration leads to the development of markets and institutions, effective price discovery, higher savings and greater economic progress. Further, in a study by Yao and Yao (2016) reported the impact of the future stock index on spot market volatility. The researcher used GARCH model with dummy variables. The data set comprised of CSI index from 2005 to 2015. The results indicated that after the launch of the CSI 300 index futures, the stock market volatility increased in the past five years. Policy measures such as improvement of both spot and futures market are necessary to contain the risks. 3. Research Methodology The current study demonstrates the appropriate method for the measurement of volatility of SENSEX for which daily data of closing price of Bombay Stock Exchange SENSEX for twenty years has been utilized from 1 st July 1997 to June 30 th 2016 covering 4,676 observations.. The main objective of the study is reached by employing the times series analytical tools ARCH, GARCH and TGARCH models utilizing eview software. 3.1 ADF The pioneering works were done by researchers in unit root time series. Early statisticians Dickey and Fuller (1979) developed the DF test, to test the null hypothesis about the presence of unit root in autoregressive model. Later, the augmented version of DF test was developed by Dickey and Fuller (1981) as ADF to test null hypothesis for measuring the presence of unit root in time series sample. Also, mentioned that if the null hypothesis is accepted, which implies non-stationarity with the variables in regression model, then it can be said that standard assumptions for asymptotic analysis is not valid. For a return series Rt, the Augmented Dickey- Fuller (ADF) test, which represents univariate time series, consists of a regression of the first difference of the series against the series lagged k times as follows: r t = α +δr t-1 + Σ β s r t-s + ε t.. (1) Where, α+δr t ; r t = r t - r t-1 ; r t = ln (R t ) Vol.2 No.2 July-December 2016 55
The study by Karmakar (2005) and Joshi (2012) also used ADF and corroborates this. 3.2 ARCH Engle (1982) proposed the idea of ARCH effect. Mentioned that the conditional variance h t can be modeled as a function of the lagged ε s, which is, the past news or history plays a vital role in predicting volatility. The detailed model developed by him is the qth order ARCH model, the ARCH (q): h t = ω +α 1 + ε 2 t-1 + α 2 ε 2 t-2 +..+ α q ε 2 t-q (2) Where, ω > 0, α 1, α 2,.., α q > 0 and ε t /Ψ t -1 ~ N(0,h t ). The effect of a return shock iperiods ago ( i q ) on current volatility is governed by the parameter α i. Normally, we would expect that α i <α j for i> j. That is, the older the news, the less effect it has on current volatility. In an ARCH (q) model, old news which arrived at the market more than q periods ago has no effect at all on current volatility. Alternatively, if a major market movement occurred yesterday, the day before or upto q days ago, the effect will be to increase today s conditional variance, which is also corroborated in the research of (Karmakar, 2005). 3.3 The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) Model In this model, the conditional variance is represented as a linear function of its own lags. This model is generalized by Bollerslev (1986) from ARCH (q) model to GARCH (p,q) model. The simplest model specification is the generic GARCH (1,1) model: h t = ω +α t ε 2 t-1+ β 1 h t-1...(3) Where, ω>0, α 1 0, β 1 0. The stationary condition for GARCH (1, 1) is α 1 + β 1 < 1. Bollerslev, Chou, and Kroner (1992), stated that in the GARCH (1,1) model, the effect of a return shock on current volatility declines geometrically over time. As referred earlier, the GARCH (1,1) model is found to be an excellent model for a wide range of financial data. The sizes of the parameters α 1 and β 1 determine the short-run dynamics of the resulting volatility time series. Large GARCH lag coefficients β 1 indicate that shocks to conditional variance take a long time, so volatility is persistent. Large GARCH error coefficient α 1 means that volatility reacts quite intensely to market movements and so if α 1 is relatively high and β 1 is relatively low, then volatilities tend to be more spiky. In financial markets, it is common to estimate lag (or persistence ) coefficients based on daily observation in excess of 0.8 and error Vol.2 No.2 July-December 2016 56
(or reaction ) coefficients not more than 0.2, which is stated and corroborated by (Karmakar, 2005). 3.4 The Threshold GARCH (T-GARCH) Model Another volatility model generally used to handle the leverage effects is the threshold GARCH (or) TGARCH model (Zakoian 1994). In the TGARCH (1,1) version of the model (Tsay, 2001), the specification of the conditional variance is: ζ 2 t = ω + α t ε 2 t-1+ d t-1 ε 2 t-1 + β 1 ζ 2 t-1 (4) Where d t-1 is a dummy variable, which is as follows 1 if ε t-1 0 bad news 0 if ε t-1 0 good news The coefficient is known as the asymmetry or leverage term. When = 0, the model collapses to the standard GARCH forms. Otherwise, when the shock is positive (i.e., good news) the effect on volatility is 1, but when the news is negative (i.e., bad news) the effect on volatility is 1 + (Carter, Hill and William, 2007), which is cited and corroborated in the study by (Ahmed and Suliman, 2011) for measuring volatility. 4. Findings 4.1 Descriptive Statistics A summary of descriptive statistics for returns series of SENSEX of Bombay Stock Exchange for the period of 1 st July 1997 to June30 th 2016 is presentedintable-1. It includes various tests such as Mean, Maximum and Minimum value, Standard Deviation, Skewness and Kurtosis. Table 1 Descriptive Statistics of Sensex Returns Statistic Sensex Mean 0.052914 Maximum 17.33933 Minimum -11.13855 Std. Dev. 1.597175 Skewness 0.049826 Kurtosis 9.641458 Table-1describes the average daily return for SENSEX is found to be at 0.052 per cent. The Sensex has given maximum return of 17.33 percent and the risk for (-11.13) percent over the study period. The standard deviation 1.59 shows the risk involved in BSE s Sensex return. The coefficient of the Skewness is significant and positive for the Index returns. Similarly, the Coefficient of Kurtosis is found to be positive and is significantly higher than the bench mark level, it indicates highly Vol.2 No.2 July-December 2016 57
leptokurtic distribution when compared to the normal distribution of the returns. Kurtosis measures a fat-tail degree of distribution. 4.2 ADF Unit Root Test The hypothesis of the ADF is H 0 : There is no stationarity in the data series. Table 2 Augmented Dickey Fuller Unit Root Test Critical Value T statistic P Value Variable of Sensex 1% 5% 10% * @ intercept -3.43-2.86-2.56-48.414 0.0001 * @ intercept and trend -3.96-3.41-3.12-48.411 0 *1 st difference Table 2 of Augmented Dickey Fuller Unit Root Test shows the stability of the Sensex returns. The Sensex has not been unit rooted because the p value is less than 0.05level (intercept, intercept and trend). The null hypothesis is rejected which that implies there is no unit root test in the data of Sensex. 4.3 Clustering Volatility of the Data Figure 1.illustrates the clustering of the volatility of residuals. The graph shows clustering volatility data for the period of 2004, having low volatility trends till 2007, then in 2009 till 2016. Year 2009 is another start of low volatility that was prolonged till 2016. The high volatility occurred in the year 2008 is followed by the year 2009 for the small period. The low volatility period indicates that Sensex is at low risk and high volatility implies that it s at maximum risk. This residual checking confirms Sensex has the clustering of the volatility of data. 20 10 20 0 10-10 0-20 -10-20 98 00 02 04 06 08 10 12 14 16 Residual Actual Fitted Figure 1. Clustering the volatility of Residuals Vol.2 No.2 July-December 2016 58
4.4 Heteroscedasticity Test of ARCH A hypothesis is used to test the Heteroscedasticity ARCH effect on the data. Here the researcher has formulated the null hypothesis as there is no ARCH effect on the data. Table 3 Heteroscedasticity Test: ARCH F-statistic 141.2767 Prob. F(1,4572) 0 Obs*R-squared 137.102 Prob. Chi-Square(1) 0 Heteroscedasticity test is used to find the ARCH effect on the data. Here the probability chi-square value is zero. It is less than the five per cent significance level. Hence, the null hypothesis is rejected. It means that there is ARCH effect in the model. This model is fulfilling both conditions of clustering the volatility and ARCH test. Further, the research can use the ARCH family models such as GARCH and TGARCH. Table 4 ARCH, GARCH and TGARCH Model Comparison ARCH GARCH TGARCH 0.064105 0.099974 0.071959 α 0 (0.000) (0.000) (0.000) Α 0.366285 0.099471 0.051179 (0.000) (0.000) (0.000) Β 0.887299 0.875122 (0.000) (0.000) Ɣ 0.109259 (0.000) Mean value 0.052914 0.052914 0.052914 AIC 3.673356 3.496619 3.482157 SIC 3.677571 3.50224 3.489183 Table 4 shows the volatility modeling of Sensex. The ARCH parameter (α) and GARCH parameter (β) are significant in all the three models. It indicates the presence of ARCH and GARCH effect on the returns of the stock. The TGARCH model is also positive; it shows the leverage effect in the Bombay stock market. The estimated GARCH co-efficient is a positive sign and greater than 0.05. It means that the past news has effect on the subsequent trading day returns. The researcher used AIC and SIC criteria to choose the better model from these three models. The TGARCH model is got the lower value of AIC and SIC compared to other models. From this the TGARCH model is the best model to identify the historical volatility of Sensex. Vol.2 No.2 July-December 2016 59
4.5 Serial correlation and ARCH LM test Serial correlation and ARCH LM TEST are used for testing whether the model fit or not. In here, the null hypothesis is accepted because there is no serial correlation in the residual. Table 5 Q- statistics and ARCH LM test Lags Q- STATISTICS P VALUE 1 0.7427 0.389 2 0.7475 0.688 3 0.7961 0.85 4 0.8108 0.937 5 0.8315 0.975 6 1.3523 0.969 7 1.4156 0.985 8 2.6243 0.956 9 2.7279 0.974 10 4.503 0.922 11 6.738 0.82 12 7.1401 0.848 13 8.0123 0.843 14 8.1993 0.879 15 8.2364 0.914 16 8.7936 0.922 17 8.8824 0.944 18 10.1 0.929 19 10.468 0.94 20 10.806 0.951 21 10.97 0.963 22 10.971 0.975 23 11.448 0.978 24 12.253 0.977 25 14.228 0.958 26 14.247 0.97 27 14.617 0.974 28 15.093 0.977 29 18.225 0.94 30 18.722 0.946 31 18.749 0.959 32 18.752 0.97 33 19.149 0.974 34 21.817 0.947 35 29.683 0.722 36 30.642 0.721 ARCH LM test P value 0.389 Vol.2 No.2 July-December 2016 60
Table 5 explicates the serial correlation and ARCH LM test of the TGARCH model. The Q statistic corresponding probability values are greater than the 0.05 percent. It implies that the null hypothesis is rejected and similarly ARCH LM test is also rejected the null hypothesis. This TGARCH model is free from the serial correlation hence the model is fit. 5. Discussion The present study aims to examine the historical volatility of the Sensex for the two-decade period with the 4676 observations. The descriptive statistics analyses show; the Sensex has given maximum return of 17.33 percent. The standard deviation 1.59 percent show the lower market risk of the BSE- Sensex. The ADF test shows the stationarity of the data. The data get stationarity at the level. The figure 1 explicates the low and high volatility of the Sensex. Heteroscedasticity test is applied to analyses the ARCH effect on the data. The results indicate there is ARCH effect in the data. The result of ARCH family results reveals that, the ARCH (α) and GARCH (β) parameter are significant in all models. The TGARCH model show the leverage effect on BSE- Sensex, the model result concludes that the past news has effect on the subsequent trading returns. 6. Conclusion This study carried out the asymmetric and conditional volatility of the Bombay stock market returns for 20 years from July 1997 to June 2016. From the ARCH, GARCH and TGARCH analysis says that all the ARCH family employed to find the volatility of Sensex return are asymmetric. TGARCH model has the lowest value of AIC and SIC compared to the ARCH and GARCH models. Hence, among the three models used, it is TGARCH that is the best model to identify the volatility of Sensex. Overall results presents that the BSE Sensex returns have the volatility and previous day news affects the next day returns. The researcher covers only the BSE sensex index data and also an attempt to examine the univariate analysis. Future research would address sector wise index in BSE and compare with some macro level economic indicators for their studies. References Abdalla Suliman, & Zakaria, S. (2012). Modelling Stock Returns Volatility: Empirical Evidence from Saudi Stock Exchange. International Research Journal of Finance & Economics, 85, 166-179. Ahmed, A. E. M., & Suliman, S. Z. (2011). Modeling stock market volatility using GARCH models evidence from Sudan. International Journal of Business and Social Science, 2(23), 114-128. Vol.2 No.2 July-December 2016 61
Bekaert, G., & Wu, G. (2000). Asymmetric volatility and risk in equity markets. Review of Financial Studies, 13(1), 1-42. doi: 10.1093/rfs/13.1.1 Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3), 307-327. Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH modeling in finance. Journal of Econometrics, 52(1-2), 5-59. doi: 10.1016/0304-4076(92)90064-x Carter R, Hill E, William C. (2007). Principles of Econometrics, 3rd Edition, New York. John Wiley and Sons, Inc Dickey, D.A., Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-431. Dickey, D.A., Fuller, W.A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49, 1057-1072. Engle, R. F. (1982). Autoregressive conditional Heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007. doi: 10.2307/1912773 Floros, & Christos. (2008). Modelling volatility using GARCH models: evidence from Egypt and Israel. Middle Eastern Finance and Economics(2), 31-41. Goudarzi, H., & C.S, R. (2011). Modeling Asymmetric Volatility in the Indian Stock Market. International Journal of Business and Management, 06(03), 221-231. Joshi, P. (2012). Financial Crisis and Volatility Behaviour of Stock Markets of Asia. Quest-Journal of Management and Research, 2(2), 35-44. Kalu O, E., & Stephen Friday, A. (2012). Modeling Asymmetric Volatility in the Nigerian Stock Exchange. European Journal of Business and Management, 4(12), 52-59. Karmakar, M. (2005). Modeling Conditional Volatility of the Indian Stock Markets. Vikalpa,30(3), 21-37. Mehta, K., & Sharma, R. (2011). Measurement of time varying volatility of Indian stock market through GARCH model. Asia-Pacific Journal of Management Research and Innovation, 7(3), 34-46. doi: 10.1177/097324701100700304 Nawazish, M., & Mawal Sara, S. (2012). Time Varying Stock Market Volatility: The Case of an Emerging Market. Research Journal of Recent Sciences, 1(11), 41-46. Singhania, M., & Anchalia, J. (2013). Volatility in Asian stock markets and global financial crisis. Journal of Advances in Management Research, 10(3), 333-351. doi: 10.1108/jamr-01-2013-0010 Singhania, M., & Prakash, S. (2014). Volatility and cross correlations of stock markets in SAARC nations. South Asian Journal of Global Business Research, 3(2), 154-169. doi: 10.1108/sajgbr-04-2012-0056 Vol.2 No.2 July-December 2016 62
Som Sankar, S., & Tanmay, B. (2012). Characteristics of conditional volatility of BSE SENSEX: The impact of information asymmetry and leverage. South Asian Journal of Management, 19(4), 27-44. Tsay, R. S. (2001). Analysis of financial time series: Financial econometrics. New York: John Wiley & Sons. Yao, & Yao. (2016). The Impact of Stock Index Futures on Spot Market Volatility. Atlantis Press, 1244-1247. Zakoian J. (1994). Threshold Heteroscedastic Models. Journal of Economic Dynamics and Control, 18, 931-944. Vol.2 No.2 July-December 2016 63