Asymmetric Conditional Volatility Estimation of Stock ...

Asian Research Consortium Asian Journal of Research in Banking and Finance Vol. 8, No. 1, January 2018, pp. 47-56. ISSN 2249-7323 A Journal Indexed in Indian Citation Index DOI NUMBER: 10.5958/2249-7323.2018.00005.6 UGC APPROVED JOURNAL SJIF IMPACT FACTOR = 5.489 (2017)

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Asymmetric Conditional Volatility Estimation of Stock Prices in India Natchimuthu N*; Ashwati Jayakrishnan**; Bhuvana S*** *Assistant Professor, Department of Commerce, Christ University, Bengaluru, India. [email protected] **Post Graduate Scholar, Christ University, Bengaluru, India. [email protected] ***Post Graduate Scholar, Christ University, Bengaluru, India. [email protected]

Abstract This study was aimed at estimating the conditional variance of stock price returns in India using PGARCH model. Ten companies with highest market capitalization were chosen from National Stock Exchange (NSE). The data for a period from 2006 to 2016 were collected from PROWESS database. Volatility clustering feature was found in the estimated volatility of stock returns. The stock return volatility was also found with leverage or asymmetric effect. The residual diagnostic test (ARCH LM test) after conditional volatility estimation confirms the efficiency of PGARCH model.

Keywords: PGARCH, Conditional volatility, Unit Root test, ARCH LM test, Leverage effect, Indian Capital market.

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Natchimuthu et al. (2018). Asian Journal of Research in Banking and Finance, Vol. 8, No.1, pp. 47-56.

Introduction Volatility is the fluctuation of returns of a security or a market index, calculated over a period of time. A security with high volatility is usually considered as a high-risk security. It is an essential concept which has many financial and economic uses such as in risk management, portfolio optimization etc. (Ahmed & Suliman, 2011). It measures the variation in stock returns over a period of time. More precisely, it is the standard deviation of returns of the stock around the mean value (Kaur, 2004). Studies on volatility in the past had revealed some features of volatility. These features include time variance, clustering effect, fat tails, long term memory effect etc (Bhattacharya, Sarkar, & Mukhopadhyay, 2003). The time varying effect means that the volatility does not remain constant, it varies with time. This is the basic premise for the ARCH model which is widely used for forecasting volatility. The clustering effect of volatility implies that they are found in clusters, i.e. there will be periods of high volatility and periods of low volatility. Fat tail feature shows that when volatility is depicted in a graph the distribution is not closer to X axis at the ends, thereby creating a “fat tail”. The long-term memory effect of volatility indicates that when there is a news or shock in the market, the impact of the same has a long-lasting effect which fades away slowly. Another important feature of volatility is the asymmetric effect. This characteristic is based on the fact that the good news or positive innovation and the bad news or a negative innovation have different effects on volatility. Generally, it is believed or expected that the bad news will increase the conditional variance in the following period, that is, there will be an inverse relationship between the bad news or the returns of the security and the intensity of volatility. This is called the asymmetric effect(Zivot, 2008). Forecasting the volatility of stock returns can greatly help investors in risk analysis, portfolio selection etc. For this purpose, many models have been developed in the past. Initially research papers focused on to the use variance to measure the risk or volatility of stocks which was assumed to be constant. It was observed that the variance to be varying over a period of time. It was also found that variance had auto-regressive feature which means that the same patterns would repeat itself. Assuming both these features, auto regressive heteroscedasticity models were developed(Engle,1982). Heteroscedasticity is a condition where the stock returns have time varying volatility. Conditional heteroscedasticity is procedure where estimation of variance is done based on the information which is available till the previous period. Auto-regressive Conditional Heteroscedasticity (ARCH) model was developed assuming the conditions that volatility is time varying and auto regressive. These models cannot be used for all the financial assets. ARCH family of models may not be suitable for all the financial assets. If the residual from the mean estimation of asset returns has ARCH type of heteroskedasticity, then ARCH models can be applied for various financial assets.

Estimation of Stock Price Volatility – Conceptual Framework Robert Engle (1982) developed ARCH model for the estimation of conditional variance of financial assets which is an improvisation over the rolling standard deviation. This model required „n‟ number of lags to estimate the variance. Bollerslev (1986) modified ARCH into GARCH by introducing lagged forecasted variance as regressor. ARCH and GARCH models were symmetric 48


models are called symmetric models as these models do not estimate the effects of the signs of the innovations. These models assume that the effect of good and bad news on the volatility is the same. In order to assess the difference of impact between good and bad news or positive and negative innovation, Asymmetric conditional heteroskedasticity models were developed(Nelson, 1991). Symmetric conditional heteroskedasticity models had imposed restrictionson the estimation process. The coefficients are all made positive and the power transformation was assumed to be two. Threshold GARCH (hereafter TGARCH) model was developed with an asymmetric term along with ARCH and GARCH term to estimate the effect of negative innovation on the forecasted variance. Exponential GARCH (hereafter EGARCH) model was developed to remove the restriction of positive coefficients. In EGARCH model, the coefficients were allowed to be negative without making the conditional volatility estimates negative(Nelson, 1991). Long term memory and strong auto-correlation feature was observed in power transformed stock market returns (Ding, Granger, & Engle, 1993). Keeping this evidence, a power GARCH model was developed by Ding, Granger and Engle (1993). This model estimates the conditional variance as a function of innovations from the mean equation, lagged forecasted variance and an asymmetrical term. The unique feature of this model is that the power term which was assumed to be two in other models, is estimated as a coefficient.

Review of Literature Glosten et al (1993) found an evidence for an inverse relationship between conditional expected monthly return and conditional variance of monthly return. This was done using a modified GARCH-M model. A modified GARCH model was used because there was a difference between the results of a basic GARCH model and Campbell‟s Instrumental Variable model. In order to harmonize the difference between the two models, the modified GARCH model was used. The modifications were: 1) allowing volatility to show seasonal patterns, 2) allowing different impacts of positive and negative innovations to returns on conditional volatility and 3) conditional variance to be predicted by nominal interest rates. They also proved that their conclusion is not affected even when they use Nelson‟s EGARCH-M model. Engle R. (2001) conducted an in-depth analysis of the basic ARCH/GARCH models and discusses a few useful extensions of these models. This paper has tried to show, with example, the usefulness of studying the volatility or risk for taking various financial and economic decisions. They have proved the ability of GARCH models in analysing and forecasting the volatility. Engle & Patton (2001) attempted to find out a good volatility model by keeping in mind a few features of volatility models like its ability to forecast, the asymmetric impact of negative and positive innovations etc. Data from Dow Jones Industrial Index was used to demonstrate such features and the ability of various GARCH models to capture the same. It was found that despite being a useful model, GARCH model has its drawbacks. Most importantly, GARCH model is unable to specify correctly for data with different time scales. Also, portfolios may not exactly follow GARCH model like assets do. Batra (2004) examined the time varying volatility in stock returns and whether there is increase in volatility persistence in India with to reference to financial liberalization. Monthly data was used from 1979-2003 and asymmetric GARCH models are used in the study. This study reveals that Indian stock market was most volatile during the period of balance of payment crisis 49


and introduction of economic reforms. volatility in stock return seems to be influenced more by domestic, political and economic events rather than the global events. This study also finds that post reform period Indian stock markets have not intensified and there is generalized reduction in market instability. Syriopoulo et al (2015) examined the varying risk-return properties of BRICS capital markets and time varying correlation and the spill over effects of volatility in the US capital markets. VAR-GARCH framework was used in the study. This framework has enabled to give useful information about US-BRICS market interactions. This disaggregated approach gives importance to two important sectors i.e., industrial and financial sectors. The study reveals that the shocks and the transmission effects between US and BRICS markets and business sectors would impact efficient global portfolio diversification risk management strategies. Based on the evidence, this study also evaluates the various effective portfolio hedge ratios and to establish ideal portfolio weights for diversified asset allocation for US-BRICS MARKETS and business sectors.

Statement of the Problem It was found after the literature review that, there were a few limitations in the GARCH, EGARCH, TGARCH models. GARCH and TGARCH models assume the coefficients to be a positive number which may not necessarily true and all these three models assume a power transformation factor of two. This poses as a constraint for the development of better models. PGARCH model, on the other hand, removes this restriction and allows the coefficients to be negative without making the variance negative and also enables the estimation of power transformation coefficient. Due to these limitations of negative coefficients and the lack of the most fitting power term, analysts would not be able to forecast the volatility of stocks. These limitations do not prevail on the PGARCH model, that is why it was chosen as the tool for analysis in this study. PGARCH model was not applied to individual stocks in India. Hence in this study, an attempt was made to estimate the conditional volatility of stocks from National stock exchange (NSE India).

Objectives 1.

To test whether the PGARCH model is suitable to for the estimation of volatility of the Indian companies.

2.

To examine the presence of volatility clustering and asymmetric effect.

3.

To estimate the most appropriate power term.

Research Methodology This study was aimed at estimating the conditional volatility of stock returns in India. For this purpose, ten companies from (National Stock Exchange) Nifty 50 index were chosen based on their market capitalization. The companies chosen for study are presented in table 1. PGARACH model was chosen for the estimation of volatility of the stock returns, since it was considered superior over the other models (Engle & Patton, 2001).

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Table 1 List of companies S.No. 1 2 3 4 5 6 7 8 9 10

Company Name Ltd HDFC Bank HDFC Ltd Hindustan Unilever Ltd Infosys Ltd ITC Ltd Maruti Suzuki India Ltd. ONGC Ltd Reliance Industries Ltd. State Bank of India Ltd Tata Consultancy Services Ltd

Since this study was aimed at estimating conditional volatility of stock returns, daily frequency data was considered appropriate. The closing price for the chosen companies were collected from PROWESS database maintained by Centre for Monitoring Indian Economy private limited (CMIE). The closing stock price data from April 01, 2006 to March 31, 2016for all the chosen companies were downloaded from the database.

Analysis The returns from the daily closing stock price was computed by taking the logged differences as given in the equation (1) Rt = (log Pt)- (logPt-1)

--------(1)

Where Rt is the daily return for time t. Pt is the closing stock price for time t.The descriptive statistics for the stock returns is presented in table 2. Table 2 Descriptive Statistics of Stock Price Returns HUL INFOSYS ITC Ltd Maruti ONGC Reliance State Tata Suzuki Ltd Industries Bank of Consultancy India Services Mean 0.000780 0.000572 0.000458 0.000457 0.000481 0.000567 -9.33E-06 0.000373 0.000298 0.000663 Median 0.000138 -0.000249 0.000000 0.000438 0.000597 3.54E-05 -0.000184 0.000286 0.000584 0.000285 Maximum 0.151005 0.202720 0.160264 0.155149 0.105489 0.118628 0.152579 0.193672 0.182575 0.144092 Minimum -0.116554 -0.116757 -0.084669 -0.238999 -0.098481 -0.130989 -0.165355 -0.177619 -0.129768 -0.113063 Std. Dev. 0.020433 0.024032 0.018575 0.020448 0.018692 0.021639 0.022337 0.022341 0.024274 0.021191 Skewness 0.302093 0.368458 0.418079 -0.594949 0.006125 -0.089010 -0.030065 -0.109779 0.170156 0.187668 Kurtosis 7.113424 7.961343 7.565023 15.88751 5.784263 6.061533 7.419239 9.541987 6.165050 7.918218 Jarque-Bera 1785.428 2598.608 2224.758 17301.74 800.7451 971.4249 2017.626 4425.614 1046.691 2513.060 Probability 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Observations 2479 2479 2479 2479 2479 2479 2479 2479 2479 2479 Source: Authors‟ calculations HDFC Bank

HDFC ltd

It can be observed from table 2, that all the stock price returns were not normally distributed during the study period. Jarque-Bera test conducted on all the stock returns were significant at 5 percent level. All the stocks showed positive mean returns except ONGC ltd during the study period.

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The stock returns were also subjected to Augmented Dickey Fuller Unit root test to check for their stationarity. The stock returns were tested at level for stationarity and it was observed that all the stock returns were found stationary. The t statistics and the respective P value are presented in table 3. All the P values for the t-statistics were significant. The results suggest that the stock returns were stationary at level.

Table 3 ADF Unit Root Test Results Stock HDFC ltd HDFC Bank Ltd HUL Infosys ltd ITC ltd MSIL ONGC ltd Reliance industries Ltd State Bank of India Ltd Tata Consultancy Ltd

t - statistic -36.38675 -37.69927 -50.29653 -38.16347 -51.28276 -47.69061 -48.61293 -47.20074 -45.00014 -32.12745

P value 0.0000 0.0000 0.0001 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000

Source: Authors‟ Calculations

The volatility of the daily stock return was estimated through the PGARCH model. The basic form of PGARCH model is given by equation (2) ζδt = α0 +

p i=1 αi

( εt−i + γi εt−i )δ +

q δ j=1 βj ζt−j

--------(2)

where ζδt , is the estimated conditional variance for time t. αi and βj are coefficients. 𝛆𝐭−𝐢 is the residual from the previous period (t-i)and 𝛔δ𝐭−𝐣 is the lagged variance from the period t-j. The coefficient 𝛄𝐢 estimates the asymmetric feature of volatility. It shows how bad news or good news has impacted the conditional volatility. As mentioned earlier, PGARCH model estimates the most appropriate power term instead of keeping two as the power transformation factor. In equation (2), the value of the power term is denoted by δ.GARCH term (βj ) estimates the volatility clustering feature of the conditional volatility. If the GARCH term is significant, it indicates that the volatility is found in clusters (periods of high volatility and periods of low volatility was observed). GARCH family conditional volatility model estimate necessitates appropriate mean equation specification (Bhattacharya, Sarkar, & Mukhopadhyay, 2003). Hence, the mean equation was estimated in an autoregressive form as given in equation (3) taking into account the significant AR terms. R t = ω0 +

m k=1 θk R t−k

+ εt

-----------(3)

PGARCH model was applied to estimate the conditional volatility for all the ten companies chosen for the study. The coefficient estimates are presented in table 4. The respective P values are presented in parenthesis.

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Table 4 PGARCH Coefficient Estimates Stock Returns

Leverage Term(𝛄𝐢 )

Reliance Industries Ltd State Bank of India Ltd

ARCH Term(𝛂𝐢 ) 0.044147 (0.000) 0.058553 (0.000) 0.272195 (0.000) 0.149149 (0.000) 0.066868 (0.000) 0.059764 (0.000) 0.099141 (0.000) 0.081543 (0.000) 0.081312 (0.000)

Tata Consultancy Services Ltd

0.0100158 (0.000)

HDFC Bank Ltd HDFC Ltd Hindustan Unilever Ltd Infosys Ltd ITC Ltd Maruti Suzuki India Ltd ONGC Ltd

0.445038 (0.000) 0.167626 (0.000) 0.070041 (0.0823) 0.166708 (0.0056)

GARCH Term(𝛃𝐣) 0.954316 (0.000) 0.924908 (0.000) 0.549654 (0.000) 0.761194 (0.000) 0.916070 (0.000) 0.935207 (0.000) 0.892600 (0.000) 0.915295 (0.000) 0.906730 (0.000)

POWER Term(𝛅) 1.787694 (0.000) 2.055562 (0.000) 1.569436 (0.000) 0.478744 (0.000) 0.947666 (0.000) 0.707661 (0.000) 1.451099 (0.000) 1.460168 (0.000) 0.964585 (0.000)

0.330348 (0.000)

0.895383 (0.000)

0.581011 (0.000)

0.264480(0.000) 0.259736 (0.000) 0.071726 (.0376) 0.120742 (0.000) 0.444231 (0.000)

Source: Authors‟ calculations

Discussion It can be observed that the ARCH term(αi ) was significant for all the ten companies studied. The asymmetrical term (γi )or the leverage term was significant and positive for all the companies with the exception of Reliance Industries Limited which indicates the presence of leverage effect in almost all the share price returns. The GARCH term was significant for all companies which denotes that the clustering feature was present in the volatility of the companies studied. The Power term estimates were ranging from 0.581 (Tata consultancy services) to 2.0555 (HDFC ltd).The residuals from the PGARCH estimation were tested for the presence of ARCH type of heteroskedasticity using ARCH LM test. The coefficients of the test were presented in table 5. ARCH LM test was applied on the residuals to check whether PGARCH model is efficient enough in explaining conditional heteroskedasticity of the stock returns.

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Table 5 ARCH LM Test Results Stock Returns HDFC Bank Ltd HDFC Ltd Hindustan Unilever Ltd Infosys Ltd ITC Ltd Maruti Suzuki India Limited ONGC Ltd Reliance Industries Ltd State Bank of India Ltd Tata Consultancy Services Ltd Source: Author‟s calculations

Obs*R-squared 0.328945 0.11292 0.368973 0.264025 7.129492 3.696530 1.557471 0.818325 1.470507 4.878800

P Value 0.5663 0.7375 0.5436 0.6074 0.0076 0.545 0.2120 0.3657 0.2253 0.0272

The ARCH LM test coefficients estimated for all the stock return was not significant at 5 percent level except for ITC Ltd. This result indicates that PGARCH model is efficient in estimating the conditional volatility of the stock returns.

Conclusion Studying the volatility is an essential process of forecasting the future returns of any financial asset. For this, it is imperative to study the features volatility. Various econometric models have been developed for the estimation of volatility, each more advanced than the previous one. Through this study, an attempt has been made to find out whether the PGARCH model is suitable to study the conditional volatility of stock returns in the Indian stock market. The top ten companies from the NSE index NIFTY 50 were selected for this study. The results suggest that ARCH family of models are suitable for the estimation of volatility of stock returns (Indicated by the significant ARCH (αi ) terms). The volatility estimates were also found to have the leverage effect (Indicated by the significant leverage (γi ) terms). Finally, it was inferred from the results that the stock return volatility has a clustering effect, i.e. they are found in clusters. Through this study, the most appropriate power coefficient for each stock was also estimated. The estimated power transformation factors ranged between 0.58 and 2.05.The diagnostic test (ARCH LM test) also confirms the efficiency of PGARCH model. It can be concluded that the PGARCH model is suitable to estimate the volatility of the returns in Indian stock market.

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