Volatility means unexpected changes in stock prices which have impact on its future returns. The main characteristic of any financial asset is its return. Return is typically considered to be a random variable. An asset’s volatility, which describes the spread of outcomes of this variable, plays an important role in numerous financial applications. Its primary usage is to estimate the value of market risk. Volatility is also used for risk management applications and in general portfolio management. It is crucial for financial institutions not only to know the current value of the volatility of the managed assets, but also to be able to estimate their future values. A large part of risk management is measuring the potential future losses of a portfolio of assets, and in order to measure these potential losses, estimates must be made of future volatilities and correlations.
Forecasting the future level of volatility is far from trivial and evaluating the forecasting performance presents even further challenge. Even if a model has been chosen and fitted to the data and the forecasts have been calculated, evaluating the performance of that forecast is troubling due to the latent nature of realized conditional volatility. A proxy for the realized volatility is therefore needed and moreover the choice of statistical measure is, as pointed out by Bollerslev, Engle and Nelson (1994), far from clear
Volatility is not the same as risk. When it is interpreted as uncertainty, it becomes a key input to many investment decisions and portfolio creations. Investors and portfolio managers have certain levels of risk which they can bear. A good forecast of the volatility of asset prices over the investment holding period is a good starting point for assessing investment risk. (Poon and Granger, 2003)
The following are Characteristics which are crucial to note for the purposes of modeling and forecasting:
Volatility Clustering:
Volatility is not constant over time. Moreover it exhibits certain patterns. This means that large movements in returns tend to be followed by further large movements. Thus the economy has cycles with high volatility and low volatility periods. High volatility periods usually refer to economic crises and recessions.
Leverage effect:
Price movements are negatively correlated with volatility. This means that the volatility of stock tends to increase when the prices drops. The term “leverage” refers to one possible economic interpretation of the phenomenon, as asset prices decline, companies become mechanically more leveraged since the relative value of their debt rises relative to that of their equity. As a result, it is natural to expect that their stock becomes riskier, hence more volatility. ( Black ,1976 and Christie, 1982)
Fat tails. Market returns have distributions with fatter tails than the normal distribution. This results to a higher kurtosis. The normal distribution has the fourth moment equal to three, however several studies, have shown that the distribution of market returns have sample fourth moments larger than three.
Risk premium:
If risk premium is positive which indicate that investors are compensated for assuming greater risk. But if risk premium is negative which indicate that investors react to factors other than the standard deviation of equities from their historical mean.
LITERATURE REVIEW:
This study is related to the comprehensive literature on volatility forecasting and application of GARCH models
Engle, R. F. (1982) introduced a new class of stochastic process called autoregressive conditional heteroscedasticity to generalize the implausible assumptions of the traditional econometric models by estimating the means and variances of inflation in the UK. The study found significant ARCH effect and substantial volatility increase during seventies.
Bollerslev (1986) introduced a new, more general class of processes, GARCH (Generalized Autoregressive Conditional Heteroskedastic allowing flexible lag structure. The extension of the ARCH process to the GARCH process bears much resemblance to the extension of the standard time series AR process to the general ARMA process and, permits a more parsimonious description in many situations
Akgiray, V. (1989) presented a new evidence about the time series behavior of stock price using 6,030 daily returns from Center for Research in Security Prices (CRSP) from January 1963 to December 1986. The findings observed the second order dependence of the daily stock returns which could not be modeled with linear white noise process. Therefore study concluded that the GARCH models are superior in forecasting volatility.
Nelson (1991) analyzed the daily returns of CRSP value weighted index from 1962 to 1987 to propose a new ARCH model to overcome the three major drawbacks of GARCH model. The findings contribute a new class of ARCH models that does not suffer from the drawbacks of GARCH model allowing the same degree of simplicity and flexibility in representing conditional variance as ARIMA and related models have allowed in representing conditional mean.
Engle, R. and Ng, V. K. (1993), attempted to estimate news impact on volatility using daily return from Japan stock market. The result suggested that the Glosten, Jagannathan and Runkle (GJR) is the best parametric model.
Srinivasan (2010) attempted to forecast the volatility (conditional variance) of the SENSEX Index returns using daily data, covering a period from 1st January 1996 to 29th January 2010. The result showed that the symmetric GARCH model do perform better in forecasting conditional variance of the SENSEX Index return rather than the asymmetric GARCH models.
Praveen (2011) investigated BSE SENSEX, BSE 100, BSE 200, BSE 500, CNX NIFTY, CNX 100, CNX 200 and CNX 500 by employing ARCH/GARCH time series models to examine the volatility in the Indian financial market during 2000-14. The study concluded that extreme volatility during the crisis period has affected the volatility in the Indian financial market for a long duration.
Dana AL-Najjar (2016) had examined the volatility characteristics on Jordan’s capital market that include; clustering volatility, leptokurtosis, and leverage effect,by applying ARCH, GARCH, and EGARCH to investigate the behavior of stock return volatility for Amman Stock Exchange (ASE) covering the period from Jan. 1 2005 through Dec.31 2014. The main findings suggest that the symmetric ARCH /GARCH models can capture characteristics of ASE, and provide more evidence for both volatility clustering and leptokurtic, whereas EGARCH output reveals no support for the existence of leverage effect in the stock returns at Amman Stock Exchange.
Naliniprava Tripathy and Ashish Garg (2013) This paper forecasts the stock market volatility of six emerging countries by using daily observations of indices over the period of January 1999 to May 2010 by using ARCH, GARCH, GARCH-M, EGARCH and TGARCH models. The study concludes that volatility increases disproportionately with negative shocks in stock returns. Hence investors are advised to use investment strategies by analyzing recent and historical news and forecast the future market movement while selecting portfolio for efficient management of financial risks to reap benefits in the stock markets.
M.Tamilselvan and Shaik Mastan Vali (2016) attempted to forecasts the stock market volatility of four actively trading indices from Muscat security market by using daily observations of indices over the period of January 2001 to November 2015 using GARCH(1,1), EGARCH (1,1) and TGARCH (1,1) models. The study discloses that the volatility is highly persistent and there is asymmetrical relationship between return shocks and volatility adjustments and the leverage effect is found across all flour indices.
Vasudevan and Vetrivel (2016) attempted to modelling and forecasting the volatility of the BSE-SENSEX Index returns of Indian stock market, using daily data covering a period from 1, July 1997 to 31, December 2015. Their result shows that the asymmetric GARCH models do perform better in forecasting conditional variance of the BSE-SENSEX returns rather than the symmetric GARCH model, confirming the presence of leverage effect.
Banerjee, A. and Sarkar, S. (2006), predicted the volatility using five-minute intervals daily return to model the volatility of a very popular stock market in India, called the National Stock Exchange. This result emphasized that the Indian stock market experiences volatility clustering and hence GARCH-type models predict the market volatility better than simple volatility models, like historical average, moving average etc. It is also observed that the asymmetric GARCH models provide better fit than the symmetric GARCH model, confirming the presence of leverage effect.
Yung-Shi Liau (2013) studied the stock index returns from seven Asian markets to test asymmetric volatility during Asian financial crisis. The empirical results showed that both volatility components have displayed an increasing sensitivity to bad news after the crisis, especially the transitory part.
Rakesh Gupta 2012 aimed to forecast the volatility of stock markets belonging to the five founder members of the Association of South-East Asian Nations, referred to as the ASEAN-5 by using Asymmetric-PARCH (APARCH) models with two different distributions (Student-t and GED). The result showed that APARCH models with t-distribution usually perform better
RESEARCH METHODOLOGY:
This study employ’s data including; 2289 daily closing observations of S&P BSE 500 index (Bombay stock exchange) for the period from sep. 17. 2007 till Dec. 30. 2016. The S&P BSE 500 index is designed to be a broad representation of the Indian market. Consisting of the top 500 companies listed at BSE Ltd., the index covers all 20 major industries in the Indian economy. S&P BSE 500 index represents nearly 93% of the total market capitalization on BSE.
The S&P BSE 500 index stock return is calculated through:
Rt= ln(pt /pt-1)
Where
Rt is the return in the period t ,
pt is the daily closing price at a particular time t ;
pt-1 is the closing price for the preceding period and ln is the natural logarithm.
The graphs 1 and 2 are showing the prices and returns trend of the sample indices for the study period. Time series data are often assumed to be non-stationary. It is thus necessary to perform a pre-test to ensure that a stationary relationship is existed among the variables. This would avoid problems of spurious regressions. To test the presence of unit roots, the standard Augmented Dickey-Fuller (ADF) Test are employed in the study.
Augmented Dickey Fuller Test:
It is a test for unit root in a time series sample developed by Dickey and Fuller (1981). The Augmented Dickey-Fuller (ADF) statistics, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit roots at some level of 5 percent confidence. ADF test follows the below stated model:
ΔYt= α + βt + γYt-1 + βt ΔYt-1 + …+ δpΔYt-p + εt
Where α is a constant, β the coefficient on a time trend and p the lag order of the autoregressive process. Imposing the constraints α = 0 and β = 0 corresponds to modeling a random walk and using the constraint β = 0 corresponds to modeling a random walk with a drift. By including lags of the order p the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p has to be determined when applying the test. For the purpose of our analysis, the study has considered a lag of 1 variable.
GARCH (Generalized Autoregressive Conditional Hetroscedasticity) Model:
The Generalized Autoregressive Conditional Hetero scedasticity (GARCH) models were propounded by Bollerslev (1986). The distinctive feature of these models is that they recognize that volatilities and correlations are not constant: i.e. volatility clustering and excess kurtosis. The GARCH models are discrete-time models, attempting to track changes in the correlation and volatility over time. The GARCH model is used to estimate volatility for a variety of financial time series: stock returns, interest rates, and foreign exchange rates. GARCH models have been applied in various fields such as asset allocation, risk management, and portfolio management, and option pricing. In order to determine the nature of conditional volatility Garch model has been used. The GARCH(1,1) model can be specified as follows:
𝑅t = 𝑐 + 𝑅𝑡−1 + 𝜀𝑡
𝜀t /𝜀t−1 ~ N ( 0, ℎ𝑡 )
ℎ𝑡= ω + 
+ 
j ht-j
Where, 𝑅t in return equation is the stock market return in time period t and 𝜺t pure white noise error term. In variance equation ℎt is the conditional variance and ɷ, α1, α2 α q , β1..βp are parameters to be estimated. q is the number of squared error term lags in the model and p is the number of past volatility lags included in the model. The study has used the Garch (1,1) Model that assume ɷ > 0, α and β≥ 0. The stationary condition for Garch (1,1) is α +β< 1. If this condition is fulfilled, it means the conditional variance is finite. A straightforward interpretation of the estimated coefficient in above equation is that the constant ɷ is long – term average volatility where αi and βj represent how the volatility is affected by current news and past information regarding volatility, respectively.
EGARCH Model :
To ascertain the effect of unexpected shock on the mean return Exponential Garch or EGARCH(1,1) model has been used by the study as it is most popular among the asymmetric Garch models. The model is based on the log transformation of conditional variance, the conditional variance always remains positive. The model has been developed by Nelson (1991). The study used the following model specifications:
𝑅t = 𝑐 + 𝜌𝑅𝑡−1 + 𝜀𝑡
𝜀𝑡 /𝜀𝑡−1 ~ 𝑁 ( 0,ℎ𝑡 )
ℎ𝑡= 𝛼0 + 𝛼1 (|𝑍𝑡−1| − 𝐸|𝑍𝑡−1| + 𝛿𝑍𝑡−1) + 𝛽1 ln( ℎ𝑡−1 )
Here, 𝑍𝑡−1is the standard residual. The term (|𝑍𝑡−1| − 𝐸|𝑍𝑡−1|) measures the size effect of innovations in returns on volatility, while 𝛿𝑍𝑡−1 measures the sign effect. A negative value of 𝛿 is consistent with leverage effect, which explains that when the total value of a leveraged firm falls due to fall in price, the value of its equity becomes a smaller share of the total value. The total effect of a positive shock in return is equal to one standardized unit is (1+ 𝛿), that of a negative shock of one standardizes unit is (1- 𝛿). 𝛽1 is the coefficient of autoregressive term in variance equation. The value of 𝛽1 must be less than 1 for stationarity of the variance.
GARCH-M Model:
To determine the impact of conditional volatility on the return, Garch-M model has been used. The model has been developed by Engle, Lilien, and Robins (1987). The Garch-M specifies conditional mean return as a linear function of conditional variance. This study used the following Garch-M (1, 1) model which can be specified as follows:
𝑅t = 𝑐 + 𝜌𝑅𝑡−1+ 𝛾ℎ𝑡+ 𝜀𝑡
𝜀t /𝜀𝑡−1 ~ 𝑁 ( 0,ℎ𝑡 )
ℎt
= 𝛼0 + 𝛼1 
+ 𝛽1ℎ𝑡−1
Rt is the stock market return, ht is the conditional variance and εt stand error for a Gaussian innovation with zero mean. Where, in return equation c, ρ and γ are the parameters to be estimated. Among all the parameters γ is the most significant one as it describes the nature of relationship between stock market return and volatility. More precisely a positive and significant γ implies that increased risk given by an increase in conditional variance represented by ht leads a hike in the mean return or vice versa.
EMPIRICAL ANALYSIS:
In order to verify the relationship between return and volatility of S&P BSE 500 Index GARCH family models have been applied. Table1 reports the statistical description for daily observations of of S&P BSE 500 Index during the period of 2007-2016 that contains; mean, median, max, min, skewness, kurtosis, Arch-test and Shapiro-wilk results.
Table 1. Descriptive Statistics
|
Statistical Indicators |
Value |
Statistical Indicators |
Value |
|
Average: |
0.00025 |
Minimum: |
-3.4329 |
|
Standard Deviation: |
0.10 |
Maximum: |
3.445 |
|
Skew: |
0.16 |
Probability: |
0.00000 |
|
Kurtosis: |
1096 |
No. of observations: |
2286 |
|
Median: |
0.0008 |
Variance : |
1 |
|
Arch-Effect test: |
571.43 |
Shapiro-Wilk test: |
0.05 |
The average daily observations of S&P BSE 500 Index is 0.00025 which indicates that there were few extreme gains across the period of the study, also the standard deviation is 0.10 There is a substantial gap between the max (3.44) and min (-3.43) which gives support to the high variability of price changes. In a normally distributed series skewness must be 0 and kurtosis is around 3,regarding results the skewness is 0.16 positively skewed which implies that the distribution has a long right tail relative to the left tail and occurs frequent small losses and a few extreme gains .In addition, the returns are leptokurtic caused by large kurtosis statistics of 1096 that exceeds normal value of 3 indicating that the return is fat tailed. A leptokurtic distribution means that small changes happen less frequently because historical values have clustered by the mean. The fat tail means risk is coming from outlier events and extreme observations are much more likely to occur. Regarding Shapiro and wilk test for normality, it is consistent with the outcome provided by both statistics of kurtosis and skewness, since the Shapiro and wilk test is significant at 1% level, that means to reject null and accept the hypothesis which states that; returns are not normally distributed. Consequently, all the pre mentioned statistical analysis gives more support to the suitability of applying GARCH model for our data gathered from Bombay Stock Exchange, since the selected observations can be described as leptokurtic, fat tailed and not normally distributed.The return series are examined for heteroscedasticity. The Shapiro and wilk test indicates that the null hypothesis of normality is rejected and shows that all the series exhibit non-normality. Hence the study shows volatility clustering tendency. To determine the conditional volatility and volatility clustering, Arch-effect test is applied. Arch-effect test provides evidence for rejecting the null hypothesis indicating the presence of Arch effects in the residuals series of the mean equation. It is found from the table that variance of return series shows presence of volatility tendency.
Table 2. Augmented Dickey-Fuller test
|
|
t-Statistic |
p-value |
|
Augmented Dickey-Fuller test statistic |
-23.4 |
0.0000 |
|
Test critical values: 1% level |
-2.6 |
|
|
5% level |
-1.9 |
|
|
10% level |
-1.6 |
|
Note: Null hypothesis is rejected at a level of 1 per cent significance
The present study employs the Augmented Dickey Fuller test to examine whether the time series properties are stationary or not. The results are presented in Table 2. Table 2 presents that all series are stationary at 1 percent, 5 percent and 10 percent level of significance. The main result based on this test is that; ADF test is statistically significant at 1% level. This indicates to reject null hypothesis and accept that the returns are stationery. That all confirms the non-existence of autocorrelation. Hence the null hypotheses of ADF test are rejected and concluded that the return series data are stationary at level.
GARCH(1,1):
The results of the Garch (1,1) model exhibits in Table 3. Table 3 presents that value of α1 and β1 is highly significant and sum of the both is less than 1. So it is interpreted that model is valid. α1 value shows that the recent news has a positive impact on the current market volatility. Historical volatility impact is represented by β1 which is also positive and equal to recent news impact. The α1+ β1 measures the degree of persistence of volatility shocks. It is also found from the analysis that the sum of Arch and Garch coefficients (α + β) is very close to one, indicating that volatility shocks are quite persistent and long memory in the conditional variance in all country’s stock market. Table 5 represents volatility forecasting for the period of 200 days. The GARCH(1,1) long-term volatility is 10.28%.When the current volatility is below the long-term volatility, it estimates an upward-sloping volatility term structure (graph.3). Therefore, GARCH (1, 1) model estimates a upward-sloping volatility term structure.
E-GARCH (1,1):
EGARCH
model helps to explain the volatility of spot market when some degree of
asymmetric and leverage effect is present in the price series. If the bad news
has a greater impact on volatility than good news, a leverage effect exists.
Table 4 presents the results of EGARCH model. In order to capture the
availability of asymmetric behavior and the existence of leverage effect in the
financial return of BSE index, the study applies EGARCH model in order to
detect the leverage effect (asymmetric). It is expected that the sign of gamma
(γ) in EGARCH model must be negative and significant. The leverage effect
refers to the observed tendency of an asset’s volatility to be negatively correlated with the asset’s returns. The coefficient
(
)is negative and highly significant which indicates a
strong presence of asymmetry effect in volatility, i.e., volatility increases
disproportionately with negative shocks in stock returns. So it is evident that the Indian stock market return
is affected with negative shocks. . Table 6 represents volatility forecasting
for the period of 200 days. The EGARCH(1,1) long-term volatility is 0.68%.When
the current volatility is above the long-term volatility, it estimates an
downward-sloping volatility term structure(graph.4). Therefore, EGARCH(1, 1)
model estimates a downward-sloping volatility term structure.
GARCH-M(1,1):
Mostly, the return of a security may depend on its volatility. In other for such a phenomenon to be modelled, there is the need to consider the GARCH-M model . It is an extension of the basic GARCH model which allows the conditional mean of a sequence to depend on its conditional variance or standard deviation. The risk premium (𝜆) coefficient is zero which implies there is a risk but no excess Changes in risk given by conditional variance ,leads to no rise or fall in the mean return. If 𝜆 is positive which indicate that investors are compensated for assuming greater risk. But 𝜆 is negative which indicate that investors react to factors other than the standard deviation of equities from their historical mean. So it is evident that the Indian stock market return has no risk premium. Table 7 represents volatility forecasting for the period of 200 days. The GARCH-M(1,1) long-term volatility is 10.28%.When the current volatility is below the long-term volatility, it estimates an upward-sloping volatility term structure(graph.5). Therefore, GARCH-M(1, 1) model estimates a upward-sloping volatility term structure.
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