Volatility of Indian stock market: an emperical evidence.

Srivastava, Aman

Introduction

The study of volatility is always a serious concern for analysts and researchers because high degree of volatility can affect the smooth functioning of any stock market. It may also affect the economic growth and development of the economy through its effect on investor's confidence and risk taking ability. The researchers worldwide have attempted to identify the major factors affecting the level of volatility in the stock markets. The available theoretical and empirical literature suggests that the main source of volatility in any stock market is the arrival of new information or news. Many of them attempted to establish pragmatic relation between stock return volatility and trading volume, the number of dealings, the bid ask spread, or market liquidity, in general. As an effect, a whole new area of Finance, "market microstructure," has been developed by these theories and the models. The challenge of this field, though, is that the prerequisite of the method is still improvised. Stock return volatility is conventional and asymmetric in its retort to past negative price shocks compared to past positive price shocks, but what and even how many basic factors drive volatility over time is not clear. Stock market volatility also has many of adverse implications. Emerging economy like India is confronting the challenges of high volatility in abundant fronts together with volatility of its stock markets.

The review of existing literature suggests that stock market volatility may have an impact on economic growth and development (Levine and Zervos, 1996 and Arestis et al 2001) and business environment (Zuliu, 1995). An increase in stock market volatility can be interpreted as an increase in risk level of equity investment and therefore a transfer of funds from equity to less risky assets classes i.e. debt. This shift can result to an increase in cost of capital to firms and hence new firms might abide this effect as investors will turn to purchase of stock in superior, well known firms. Whereas there is a general agreement on what constitutes stock market volatility and, to a smaller extent, on how to quantify it, there is far less conventionality on the reasons of changes in stock market volatility. A number of researchers investigated the causes of volatility in the arrival of new, unexpected information that affect expected returns on a stock (Engle and Mcfadden, 1994). Thus, changes in market volatility would just reproduce changes in the domestic or global economic environment. Others maintain that volatility is caused largely by changes in trading volume, practices or tends, which in turn are resolute by factors such as changes in macroeconomic policies, shifts in investor's risk appetite and growing uncertainty.

Conditional Heteroscedasticity (ARCH) became a very popular method in the modeling of stock market volatility. As comparison to traditional time series models, ARCH models allowed the conditional variances to change during time as functions of precedent errors. First approach was to improve the univariate ARCH model with a different requirement of the variance function. One development was introduced by Bollerslev (1986) where the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) method was presented. Then after, the Integrated GARCH (IGARCH) Engle and Bollerslev (1994) and the exponential GARCH (EGARCH) Nelson (1991) were significant one wherever re-specification of variance equation was considered. Nevertheless, the extent of empirical research on stock return volatility in emerging markets like India was not plentiful. While Roy and Karmakar (1995) focused on the measurement of the average level of sample standard deviation to investigate whether volatility has gone up, Goyal (1995) used conditional volatility estimates, as recommended by Schwert (1989), to spot the trend in volatility. He also analyzed the impact of carry forward system on the intensity of volatility. ARCH/GARCH models have been worn by Pattanaik & Chatterjee (2000) to model the volatility in Indian financial market.

The purpose of this paper is to understand the daily return data volatility of stocks & to develop an asymmetric GARCH models can explain determination of shock and volatility. This study is an attempt to develop models to elucidate the volatility of the stock of the major indices of India. To this end, the study includes two main indices of Indian stock markets. The data consists of indices of Bombay Stock Exchange (SENSEX) and that of National stock exchange (NIFTY). This study uses the Autoregressive Conditional Heteroskedasticity (ARCH) models and its extension, the Generalized ARCH, EGARCH and TARCH models was used to find out the presence of the stock market volatility on Indian stock market. The objective is to model the phenomena of volatility clustering and persistence of shock using asymmetric GARCH family of models.

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Research Methodology

The study spanned the period from April 2000 to March 2008. This period of study is selected because Indian stock market has witnessed a tremendous growth and development in the sampled period. There are two major stock exchanges in India: Bombay Stock Exchange (BSE) and National Stock Exchange (NSE). The sample population of the study consists of the daily returns of the two most prominent domestic indices, viz., SENSEX and NIFTY. The data was collected from official websites of respective stock exchanges. Daily closing prices of the two indices were considered for the period of study. These market indices were fairly representative of the various industry sectors. The daily stock prices were converted to daily returns. Logarithmic difference of prices of two successive periods was used to determine the rate of return. The study has calculated the natural log of the daily return [Y.sub.t] = ln ([index.sub.t]) - ln ([index.sub.t-1]), where t index is the close price in [t.sub.th] day. The econometric software package Eviews 5.0 has been used to do the estimations.

Arch and Garch Models

Conventional econometric models assume a constant one-period forecast variance. To simplify this implausible assumption, Robert Engle presented a set of methods called autoregressive conditional heteroscedasticity (ARCH). These are zero mean, serially uncorrelated methods with non constant variance conditional on the past. A practical generalization of this model is the GARCH parameterization introduced by Bollerslev (1986). This model is also a weighted average of past squared residuals, but it has waning weights that by no means go entirely to zero. The set of equation (1)--(3) represent the original GARCH model.

In the third equation ht= var ([[member of].sub.t]/[[psi].sub.t-1]), [[psi].sub.y.sub.t] it is the information prior to time t-1. Because GARCH (p,q) is an annex of ARCH model, it has all the properties of the original ARCH model. And because in GARCH model the conditional variance is not only the linear function of the square of the lagged residuals, it is

also a linear function of the lagged conditional variances, GARCH model is more precise than the original ARCH model and it is easier to compute. The most commonly used GARCH model is GARCH (1,1) model. The (1,1) in parentheses is a standard notation in which the first number refers to how many autoregressive lags, or ARCH terms, come into view in the equation, as the second number refers to how many moving average lags are specified, which here is frequently called the number of GARCH terms. Occasionally models with more than one lag are needed to find better variance forecasts. GARCH (1,1) is the most extensively used GARCH model because it is correctness and ease. Although GARCH model is very helpful in the predicting of volatility and asset pricing, there are still many problems GARCH model cannot clarify. The main difficulty is that standard GARCH models presume that positive and negative error conditions have a symmetric effect on the volatility. In other terms, good and bad news have the similar impact on the volatility in this model. In real life this hypothesis is often desecrated, in particular by stock returns, in that the volatility increases more often after a flow of bad news than after good news. According to the challenges in the standard GARCH model, a number of parameterized extensions of the standard GARCH model have been recommended in recent times.

E-Garch Model

Exponential GARCH (EGARCH) model was first developed by Nelson in 1991. The main purpose of EGARCH model is to explain the asymmetrical response of the market under the positive and negative shocks. EGARCH model in the study has been represented by equation (4)--(5).

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One can see in equation (4), Nelson has rewritten the conditional variance with the natural log of the conditional variance. When [phi] [not equal to] 0, the impact of information are asymmetry and when [phi] [not equal to] 0, there is an important leverage effect. If one compared the above equations with the premises of the conventional GARCH model, one can see that there are no constraints for the parameters. This is one of the biggest benefits of EGARCH model as compared to the standard GARCH model.

GARCH-M (GARCH-in-mean) Mode:

In GARCH-M (GARCH-in-mean) t h is added in the right hand side of equation (1) and hence is given by set of equation (6)--(7).

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TARCH Model

Threshold ARCH (TARCH) model was first developed by Zakoian in 1990. It has the conditional variance given by equation (8).

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Because t d is built-in, the rise (0 [epsilon]) and fall (0 [epsilon]) of stock prices will have different impact on conditional variance. When the stock prices increase, [[psi].sub.i] [[member of].sup.2] t-1 [d.sub.t-1] = 0 the impact can be explained by the parameter when the prices fall, the impact can be explained by the parameter [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] we conclude that the information has asymmetrical impact. If 0 [psi] we say that there is leverage effect.

Analysis and Discussion

Arch Test

A descriptive investigation of the plot of daily returns on SENSEX and Nifty (Figure 1 & Figure 2) reveal that returns incessantly fluctuated about the mean value that was close to zero. The return measures were both in positive and negative area. More fluctuations be tending to cluster together and were alienated by periods of relative calm. This was in agreement with Fama's (1965) observation of "volatility clustering". From the time series graph of the returns for both markets, it is analyzed that high volatilities are followed by high volatilities and low volatilities are followed by low volatilities. That means both time series have important time varying variances. Additionally, it is appropriate to put conditional variance into the function to clarify the impact of risk on the returns. Hence, GARCH class model is the excellent tool for the study.

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Descriptive statistics (Table 1) for both SENSEX and Nifty returns showed skewness statistic of daily returns different from zero which indicated that the return distribution was asymmetric. In addition, relatively large excess kurtosis recommended that the underlying data was leptokurtic (heavily tailed and sharp peaked). The Jarque--Bera statistic is calculated to test the null hypothesis of normality rejected the normality assumption. Both the indices appeared to have significant strong autocorrelations in one-day lag returns. In addition, the autocorrelation in the squared daily returns suggested incidence of clustering. The results ruled out the independence assumption for the time series of given data set. Stationary of the return series were tested by conducting both Dickey-Fuller and Phillip-Peron tests. The results of both the tests confirmed that the series is stationary at first difference (Table 2).

Before ARCH-GARCH is used in the study to approximation the model, the study is required to test whether the data has ARCH effect. The most commonly used method is Lagrange Multiplier test (LM). When the study used the LM test to the residuals of the returns for both indices, the study found that when q=12, the p-value of [chi square] is still less than 05. at [alpha] = 0. That means the residuals have high order ARCH effect. The present work used GARCH, GARCH-M, TARCH, TARCH-M, EGARCH and

EGARCH-M to estimate the data. Following is the table with the results estimated from different models. From this table, one can select the best model for the further forecasting of stock market volatility.From Table 3, one can see that for both markets EGARCH (1,1)-M has the lowest RSS and the relative high adjusted 2 R. That means, EGARCH (1,1)-M is superior to other models in the estimation. From the standard of AIC and SC, we can see that EGARCH (1,1) has the lowest value. That means EGARCH (1,1) is also a relative good model for the estimation. In addition, when the study use GARCH (1,1) to estimate the data, it is found that the 1 a and 1 e for both markets are 0.9733 and 0.9716. They are very close to 1. This demonstrates that there is high durability of the volatilities in both markets. That means if there is an expected shock in these markets, the sharp movements will not die out in the short run. That is a sign for high risk. At the same time, the study found that the summation of the parameters is less than 1, which indicates that the GARCH process for the stock return is wide-sense stationary.

When the study used TARCH (1,1) to estimate the model, it is found that the estimate of [psi]s are greater than 0 for both stock exchanges. When the study used EGARCH (1,1), it is found the estimates of [psi]s are less than 0 for both markets. Then one can conclude that there are leverage effects in both markets. That is to say the volatilities caused by negative shocks are greater than that caused by positive shocks. This is in consistent with most of the existing literature. The study also used the estimated EGARCH (1,1) to predict the volatilities for BSE and NSE. In figure 3 and 4, one can see that the model did a great job. Also one can see that these two exchanges are highly correlated and there is a considerable synchronization in their movements. This is not unexpected because these two stock exchanges are the principal stock exchanges of India and they are regulated by the government. Also the maximum volume (more on NSE as comparison to BSE) is traded on these stock exchanges only.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusion

The findings of this study suggest that both the Indian stock exchanges have significant ARCH effects and it is appropriate to use ARCH/GARCH models to estimate the process. The research found that both EGARCH (1,1) and EGARCH (1,1)-M did good jobs in fitting the process for both exchanges. Because 1 a and 1 e are 0.9733 and 0.9716, which are close to 1, one can conclude that both markets will fluctuate radically with new shocks and this is a sign of high risk in the markets. The study also demonstrated that there are leverage effects in the markets. That means the investors in those markets are not grown well and they will be heavily influenced by information (good or bad) very easily. This can easily be seen in current turmoil in Indian stock market. The study also found that the volatility of the Indian stock market exhibited features similar to those found earlier in many of the global stock markets, viz., autocorrelation and negative symmetry in daily returns. It was found that asymmetrical GARCH models do better than the ordinary least square (OLS) models and the Vanilla GARCH models. Perseverance of shock could be explained the time dependent risk premium. If it is found that the shock was of short term in nature, then the investor would be reluctant from making any modification in their discounting factor while calculating the present discounted value of the stock and therefore its price.

References

Bollerslev T, R F Engle and D B Nelson (1994), "ARCH Models in R.F. Engle and D. McFadden (eds.)", Handbook of Econometrics, Vol. 4, pp. 2959-3038.

Bollerslev T. (1986), "Generalized Autoregressive Conditional Heteroscedasticity", Journal of Econometrics, Vol. 31 (3), pp. 307-327.

Fama E. (1965), "The Behavior of Stock Prices", Journal of business, Vol. 38 (1), pp. 34-105.

Goyal R. (1995), "Volatility in Stock Market Returns", Reserve Bank of India Occasional papers, Vol. 16 (3), pp. 175-195.

Levine, R and S. Zervos (1996), "Stock Market Development and Long-Run Growth", World Bank Economic Review, Vol. 10 (1), pp. 323-339.

Nelson D B. (1991), "Conditional Heteroskedasticity in asset returns: A new Approach", Economertica, Vol. 59 (2), pp. 347-395.

Pattanaik S and B Chatterjee. (2000), "Stock Returns and Volatility in India: An Empirical Puzzle", Reserve Bank of India Occasional Papers, Vol. 21 (1), pp. 3760.

Roy M K and M Karmakar. (1995), "Stock Market Volatility: Roots & Results", Vikalpa, Vol. 20 (1), pp. 37-48.

Schwert W G. (1989) "Why Does Stock Market Volatility Changes Over Time?" Journal of Finance, Vol. 44 (5), pp. 1115-1151.

Zuliu, H (1995), "Stock market Volatility and Corporate Investment", IMF Working Paper, 95/102.

Aman Srivastava

Jaipuria Institute of Management, A/32a, Sector-62, Nokia--201301, U.P., India

E-mail: amansri@hotmail.com asrivastave@jimnoida.ac.in

Table 1: Descriptive statistics of NIFTY and SENSEX returns

 SENSEX Return NIFTY Return

Mean 0.000613 0.000561
Median 0.00139 0.00154
Maximum 0.079311 0.079691
Minimum -0.118092 -0.130539
Std. Dev. 0.015473 0.015388
Skewness -0.537631 -0.752137
Kurtosis 7.067238 8.50806
Jarque-Bera 1463.086 2695.063

 SENSEX Return NIFTY Return

Probability 0 0
Sum 1.215901 1.113573
Sum Sq. Dev. 0.474761 0.469532
Observations 1984 1984

Table 2: Unit root test results

Variables Augmented Dickey Fuller (ADF) Phillips
At Levels Intercept Intercept No Intercept
 Model A Model B Model C Model A

Sensex Return -1.526557 -1.202829 -2.173018 -1.53088
Nifty Return -1.265402 -1.626452 -1.553985 -1.247947
At 1st diff.
Sensex Return -21.41668 -21.41218 -21.42214 -41.06032
Nifty Return -22.74671 -22.88564 -22.9123 -423.6543

Variables Perron Test (PP)
At Levels Intercept No
 Model B Model C

Sensex Return -1.202829 -2.173018
Nifty Return -1.685434 -1.629474
At 1st diff.
Sensex Return -41.07341 -41.05463
Nifty Return -425.3925 -431.9216

Note: MacKinnon Critical values at level: for model A. -2.9851; model
B. -3.469; model C. -1.9439, and at 1st difference: for model A.
-2.8955; model B. -3.4626; model C. -1.9445

Table 3: Estimates using various models

Market Model RSS A-[R.sup.2] AIC SC

BSE GARCH-M -0.002259 -0.003777 -5.752968 -5.741692
 TARCH -0.00157 -0.002581 -5.740518 -5.732061
 E-GARCH -0.00157 -0.003087 -5.734045 -5.722769
 EGARCH-M -0.000265 -0.002286 -5.769098 -5.755003

NSE GARCH-M -0.002322 -0.003841 -5.764502 -5.753226
 TARCH -0.001331 -0.002342 -5.753229 -5.744772
 E-GARCH -0.001518 -0.003543 -5.784464 -5.770369
 EGARCH-M -0.000291 -0.002313 -5.786751 -5.772656