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The Relationship Between Implied Volatility and Stock Index Returns: Evidence from South Africa FINA700: Final Honours Dissertation Supervisor: Faeezah Peerbhai
No.
Name
Student Number
1
Damien Kunjal
215 057 603
2
Chad Mckenzie
214 567 190
3
Fabian Moodley
214 582 814
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Delane Naidu
214 549 400
5
Camiel Singh
214 505 909
Table of Contents Abstract ...................................................................................................................................... 3 Abbreviations ............................................................................................................................. 4 1. Introduction ............................................................................................................................ 5 1.1. Motivation for this Study ................................................................................................ 6 1.2. Research Objectives ........................................................................................................ 6 1.3. Research Questions ......................................................................................................... 7 2. Literature Review................................................................................................................... 7 2.1. Comparison of GARCH Models ..................................................................................... 7 2.2. The Relationship between Volatility and Stock Market Returns .................................. 10 2.2.1. Theoretical Considerations ..................................................................................... 10 2.2.2. Empirical Evidence................................................................................................. 12 Summary: ............................................................................................................................. 18 3. Methodology and Data ......................................................................................................... 19 3.1. Data ............................................................................................................................... 19 3.1.1. Preliminary Data Analysis ...................................................................................... 20 3.2. Methodology ................................................................................................................. 21 4. Findings................................................................................................................................ 27 4.1. Data Analysis ................................................................................................................ 27 4.1.1. Unit Root Tests ....................................................................................................... 29 4.1.2. Descriptive statistics ............................................................................................... 30 4.1.3. Testing for Heteroskedasticity/ARCH effects ........................................................ 31 4.2. Superior GARCH Model............................................................................................... 32 4.3. EGARCH Regression Results ....................................................................................... 33 5. Discussion ............................................................................................................................ 37 5.1. Summary of Findings .................................................................................................... 37 5.2. Limitations .................................................................................................................... 38 5.3. Recommendations for Future Research ........................................................................ 38 5.4. Conclusion..................................................................................................................... 39 References ................................................................................................................................ 41 Appendices ............................................................................................................................... 49 Appendix 1: Models with the SAVI as an exogenous variable............................................ 49 1
Appendix 1a: RALSH ...................................................................................................... 49 Appendix 1b: RTOP40 ..................................................................................................... 53 Appendix 1c: RSMCAP ................................................................................................... 57 Appendix 1d: RMDCAP .................................................................................................. 61 Appendix 2: Residual diagnostics test at 20 lags: ................................................................ 65 Appendix 2a: RALSH ...................................................................................................... 65 Appendix 2b: RTOP40 ..................................................................................................... 65 Appendix 2c: RSMCAP ................................................................................................... 65 Appendix 2d: RMDCAP .................................................................................................. 66 Appendix 3: Turn it in Report .............................................................................................. 67 Appendix 4: The Writing Place Proof .................................................................................. 69 Appendix 5: Supervisors Permission to Submit................................................................... 72
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Abstract The objective of this study is to expand on existing knowledge regarding the relationship between implied volatility and stock market returns. Whilst previous studies on the topic have all been from an international perspective, this paper provides evidence from South Africa by examining the impact of the South African Volatility Index(SAVI) on different Johannesburg Stock Exchange listed stock indices, namely; All Share, Top 40, Small Cap and, Medium Cap. The purpose of this study is twofold: to determine which GARCH model is the best for modelling volatility in South Africa and; to determine whether the SAVI displays any relationship with the returns on equity indices. The study finds that the EGARCH model, with the SAVI as an exogenous variable, is the most appropriate model for modelling volatility on the JSE. Thereafter, using an EGARCH model, it is found that the SAVI is significantly positively related to the returns of all the chosen indices and that a leverage effect exists between them. The results are consistent with the findings of studies from developing economies and are important for investors, risk managers, as well as policymakers.
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Abbreviations ALSI: All Share Index ARCH: Autoregressive Conditionally Heteroscedastic BRIC: Brazil, Russia, India and China CAPM: Capital Asset Pricing Model CBOE: Chicago Board of Options Exchange DAX: Deutscher Aktien-Index EFT: Exchange Traded Funds E-GARCH: Exponential Generalised Autoregressive Conditional Heteroskedasticity EWMA: Exponentially Weighted Moving Average FTSE: Financial Times Stock Exchange GARCH: Generalised Autoregressive Conditional Heteroskedasticity GJR-Garch: Glosten-Jagannathan-Runkle Generalised Autoregressive Conditional Heteroskedasticity JSE: Johannesburg Stock Exchange RMSE: Root Mean Square Error S&P 100: Standard and Poor 100 S&P 500: Standard and Poor 500 SAVI: South African Volatility Index T-GARCH: Threshold Generalised Autoregressive Conditional Heteroskedasticity VAR: Vector Autoregressive (VAR) VIX: Volatility Index VXN: Nasdaq Volatility Index
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1.
Introduction
One of the greatest difficulties facing investors is the ability to predict stock market returns. This is because the returns earned by investors are affected by stock market volatility. Volatility is a measure of the dispersion of the returns from an underlying asset, therefore, the more volatile or dispersed returns are, the greater their unpredictability. Volatility in financial markets cannot be ignored as all financial markets are exposed to volatility in some way. Volatility is an important variable in the pricing of options, since, according to the BlackScholes Model, the volatility of the underlying asset needs to be known. Volatility is also an important component that is considered when analyzing the performance of stocks, as the volatility of the returns of a stock can indicate the amount of risk tied to that stock. It is because of this, that companies allocate a large amount of resources to hedging strategies. Therefore, volatility also affects the strategies developed for risk management purposes. This being so, volatility is an important concept for investors, stock brokers, portfolio managers, risk managers, firms, and the economy itself. The volatility experienced in financial markets can be attributed to many causes. In the case of South Africa, however, a large amount of this volatility can be attributed to the economic and political instability within the country. One could also argue that volatility is caused by the transference of information across national borders. This was seen during the 2008 financial crisis when the international financial system experienced a great amount of volatility. The news that financial markets began failing travelled rapidly, causing investors to become panicstricken. It is this kind of uncertainty that creates volatility. A major problem in modern financial markets is finding an appropriate measure of volatility. In 1993 Robert Whaley attempted to solve this problem through the introduction of the Volatility Index (VIX hereafter), in the United States. The VIX is a measure of the volatility that investors anticipate and can therefore be used as a measure of implied volatility. It should be noted that implied volatility is seen to be superior to realised volatility since implied volatility is a measure of investors’ expectations of future volatility, whereas realised volatility is a measure of historical volatility that has already been observed in the market. In South Africa, the South African Volatility Index (SAVI hereafter) is used as a measure of implied volatility. 5
The aim of this study is to determine the relationship between implied volatility -represented by the SAVI Top 40- and the returns of different stock indices listed on the Johannesburg Stock Exchange (JSE hereafter). This study will investigate the effect of volatility on companies of different sizes by assessing the relationship between implied volatility and the returns of the Top 40 Index, the Medium Capitalization (Mid Cap hereafter) Index, and the Small Capitalization (Small Cap hereafter) Index. The relationship between implied volatility and the returns of the JSE All-Share Index will also be evaluated.
1.1. Motivation for this Study The volatility of stock markets is an important factor that is considered by investors when developing investment strategies. The volatility of markets is used to examine the outlook of investors in the market. Investors then formulate investment strategies that they expect to be profitable. It was found in a study carried out by Gokcan (2000) that minimal attention has been given to modelling volatility of stock market returns in emerging markets. Furthermore, there has not been any attempts, known to the authors of this paper, to investigate the relationship between the SAVI and the returns of the various indices listed on the JSE. Given the importance of volatility, this paper aims to contribute to existing knowledge on the relationship between implied volatility and stock index returns. However, this paper will be unique in that it provides evidence from South Africa, by using the SAVI Top 40 and stock index returns from the JSE. This study, therefore, aims to benefit academics, investors, portfolio managers, monetary policy makers, risk management strategists, and all other individuals concerned about the relationship between implied volatility and stock index returns.
1.2. Research Objectives The objectives of this study include the following: To determine which GARCH model is the superior model with regards to predicting volatility in the chosen indices. To determine whether the relationship between the SAVI Top 40 and the returns of the selected indices are positive or negative. To determine whether the relationship between the SAVI Top 40 and the returns of the chosen indices are asymmetrical.
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1.3. Research Questions This study aims to answer the following research questions: Which GARCH Model is the superior model with regards to predicting volatility in the chosen indices? Is the relationship between the SAVI Top 40 and the returns of the selected indices positive or negative? Is the relationship between the SAVI Top 40 and the returns of the chosen indices asymmetrical? This study will begin by reviewing existing literature on the relationship between implied volatility indices and the returns of stock market indices. The data used in this study will then be outlined, followed by a detailed discussion of the methodology used in this paper. The study will conclude by reviewing the limitations of this study.
2. Literature Review The following literature review seeks to explore the different studies conducted by academics with respect to volatility and several stock indices. These findings serve as a guide to determining whether international and local financial markets react in a complementary manner. The analysis herein is divided into two parts. Firstly, to enhance the relevant knowledge on research question 1, a review of different GARCH models will be undertaken. Secondly, the different theories and empirical evidence on the relationship between volatility and stock market returns will be analysed.
2.1. Comparison of GARCH Models According to Campbell, Lo and MacKinlay (1997), the risk-return tradeoff exhibits nonlinearity in some input variables. Furthermore, Brooks (2014) states that most financial data exhibit leptokurtosis, and volatility clustering or volatility pooling. These properties cannot be adequately captured by time series models, and thus, volatility models are the most appropriate alternative. A majority of the literature in forecasting volatility employs Bollerslev’s (1986) and Taylors (1986) Generalised Autoregressive Conditionally Heteroscedastic (GARCH) model. The 7
GARCH model became well established in 1986 after academics began seeking advancements to Engle’s (1982) Autoregressive Conditionally Heteroscedastic (ARCH) model. The ARCH model, had gained popularity within financial engineering during its early years, however, the GARCH model became a more widely used model due to it being more parsimonious and better fitting (Brooks, 2014). Brailsford and Faff (1996) conclude, in their study on Australian equity index data, that the GJR-GARCH model provides better results in forecasting volatility than any other model used. In addition, Peters (2001) examined the performance of four GARCH models (GARCH, EGARCH, GJR-GARCH and APARCH) as well as three distributions namely; normal, student-t and, skewed distributions. The study examined 3935 daily observations of the Financial Times Stock Exchange (FTSE) 100 Index and the Deutscher Aktien-Index (DAX) 30. The study found that the APARCH and GJR models outperformed the EGARCH model, and that non-normal distributions provided better in-sample and out-of-sample results. On the other hand, Li and Lin (2003) found that the Markov-switching ARCH (commonly referred to as the SWARCH model) framework is a better model than the ARCH and GARCH models when predicting the dispersion of the returns of Taiwan stock indices. In a local and more recent study, Kgosietsile (2014), who aimed to identify the most suitable tool to model and forecast volatility of stock returns on the JSE found the EGARCH model to be the superior model. This study examined three popular models in finance; GARCH (p, q), historical volatility and, the implied volatility index (SAVI). The JSE Top 40 index for the sample period beginning 1 February 2007 and ending 31 December 2013 was inspected in this study. Evidence of volatility clustering, leptokurtosis and leverage effects in the JSE Top 40 returns was found. The study concluded that the EGARCH (1,1) model proved superior in comparison to the GARCH (1, 1), GARCH-M (1, 1), GJR-GARCH (1, 1), APGARCH (1, 1), and EWMA models. A study conducted by Ezpeleta (2015) added exogenous variables to basic GARCH models in an attempt to determine whether the performance of these models can be improved, with the addition of certain exogenous variables. Six different GARCH models were used, such as the GARCH (1,1), the EGARCH (1,1), and these two models with the VIX or SIX VX as exogenous variables in the variance equations. The study period ranged from 5th January 2009 until 23rd June 2015, and was based on the daily closing prices of OM XS 30 index. The Mincer 8
Zaenowitz regression (MZ-regression) estimate and the Mean Squared Error (MSE) estimate were utilized to evaluate each model. The MZ-regression revealed the EGARCH model with the addition of the SIX VX produces a conditional variance that is unbiased, whilst the model with the lowest MSE was the combination of the EGARCH and VIX. This was due to the VIX index closing later than the SIX VX index, therefore capturing more relevant information to the forecast. Overall, both estimates choose the EGARCH model with the addition of an exogenous variable as the best model since it captured the characteristics of asset returns. Masinga (2015) examined different univariate GARCH models in the Johannesburg Stock Exchange (JSE) with the purpose of identifying the most appropriate model that can be used to model and forecast volatility on the JSE stock exchange. The data used in this study included daily returns of the JSE All Share Index, Industrial 25 Index, Resource 10 Index and Top 40 Index. The sample period under investigation ran from 1st October 2002 until 30th December 2014, translating into a total of 3067 trading days. Results based on information criteria such as AIC, SBIC, and HQIC, indicated that the JSE All Share Index and all other indices used in this study, are best modelled by the EGARCH. However, the likelihood function contradicts this result, by finding the GARCH (1,1) model to be the most efficient model. In addition, the study found that, when modelling out of sample JSE All Share Index data, GARCH (1,1) proved to be more accurately followed by the EGARCH and GRJ-GARCH. Contrastingly, Olberholzer and Venter (2015) studied 5 different indices on the JSE and used the GARCH (1,1), GJR-GARCH (1,1) and EGARCH (1,1) models to analyze and forecast their daily volatility changes. According to this study, the best fitting model was the GJR-GARCH model as it accurately assessed all the indices, except for the JSE top 40 index. A more recent study by Mokoena (2016), aimed to determine which GARCH model and distributions would be best suited to forecast domestic equity volatility on the JSE. The GARCH models were carried out on the daily returns of the JSE All-Share Index (ALSI) over a 10-year period ranging from 30 September 2003 to 14 August 2013. With regards to forecasting, the study found that the E-GARCH (1,1) model outperformed the various GARCH models after many different tests. In addition, research concluded that when forecasting, the most suitable distributions were found to be students t-distribution.
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2.2. The Relationship between Volatility and Stock Market Returns 2.2.1. Theoretical Considerations The following section provides a discussion of the different theories that seek to explain the underlying relationship between implied volatility and stock index returns. These theories include, but are not limited to; the leverage hypothesis, the intertemporal capital asset pricing model, behavioural finance, equilibrium models with divergence of opinion and, the positive risk-neutral relations. i. The Leverage Hypothesis The Leverage Hypothesis was proposed by Modigliani and Miller in 1970. According to Caselani and Junior (2005), the leverage hypothesis contains the view that, a given decrease in the firm’s value will bring about a complementary reduction in the firm’s equity. Therefore, indicating that the equity portion of the entire firm will become smaller as the value of the firm falls. According to this hypothesis, the equity of any firm bears the risk associated with the firm’s profits. Consequently, the volatility of equity will increase as the volatility of the firm's profitability increases. Therefore, the share price of the firm will decrease as the volatility of the equity increases, due to the systematic portion of risk that cannot be eliminated through diversification. A direct consequence of this decrease in the stock price is that the market returns of that stock decreases. This is known as the negative-return volatility relationship. ii. Intertemporal Capital Asset Pricing Model (ICAPM) Merton (1973) produced a theory associated with capital markets which is referred to as the intertemporal capital assets pricing model (ICAPM). The model assumes that there is a dependent relationship between an asset’s expected return and its covariance with the market portfolio. The utility of investors is depicted from the consumption flow over time, therefore investors do not maximize their utility in the middle of a time-period, but at the end of a whole period (Maio and Santa-Clara, 2012). The model illustrates that assets will demonstrate low average returns in equilibrium if the pricing of the asset is consistent with the long-term investor terms, as investment opportunities is not only affected by declining stock returns and increasing volatility of stock returns, but by earnings, bonuses and other sources of income (Campbell, Giglio, Polk and Turley, 2018).
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According to Merton (1973), the CAPM model ignores the fact that investors want to hedge state variable risks, so the author includes a risk premium in addition to the market portfolio to make it a multifactor model. The state variables are accompanied by additional betas which account for the demand by investors to hedge uncertainly about future investment opportunities and the risk associated with the investment decision (Merton, 1973). It is therefore known that the ICAPM model can be identified as a multi-beta version of CAPM, were all state variables need to demonstrate the various aspects associated with the investment decision (De Jong, 2011). Investors will therefore pay a high price for those assets that hedge against negative macroeconomic events as well as financial risk. iii. Behavioural Finance Kahneman and Tversky (1979) introduced the behavioural finance theory to corporate finance. The negative relationship between volatility and the returns from stock markets can also be attributed to rational behaviour of investors, which is influenced by their emotions. When conducting the necessary research, it was evident that the volatility index represents investors’ sentiment. Therefore, if investors foresee high volatility, it is rational for them to not participate in the given market, which results in a decrease in the market returns. The decrease in market returns can be attributed to the demand for a stock by investors. If investors predict an increase in the volatility of the stock, this will result in the demand for the stock decreasing, due to the belief that excess risk decreases portfolio value (Sewell, 2007) This in turn will cause the market returns to decrease, due to the decrease in demand of the stock. The reiteration of the validity of the explanation is substantiated by Tseng (2006), who indicates that rational behaviour is influenced by emotions and should be considered. iii. Equilibrium Models with Divergence of Opinion Campbell and Hentschel (1992) introduced the Equilibrium Models with Divergence of opinion. In a study conducted by Miller (1977), the equilibrium model with divergence of opinion was examined. The model shows that, investors hold on their investments because they are under the belief that the value of the stock will increase. However, investors incur losses when the price of the stock is the average opinion of the market. The disagreement between the fair price of the stock results in the market price of the stock and equilibrium value being different, and therefore, increasing the volatility of the listed stocks, which decreases the stock returns in the future.
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iv. Positive Risk-Neutral Relations The Positive Risk-Neutral Relation theory was proposed by Banerjee, Doran and Peterson (2007) and it states that it is rational to consider implied volatility as a risk-neutral variable, as it is obtained under the condition of risk-neutral measures. The use of implied volatility in the variance equation can be utilised to identify a Positive Risk-Neutral Relation. Barr (2009) found implied volatility to be effective, and to be an upward-biased determinant of intended realised volatility. This demonstrates that above average returns in the market are positively related to current implied volatility, and therefore, any information set for modelling above average returns in the market should incorporate estimated volatility. The relation between risk and return is attributed to the variations in- and expectations of prospective returns. This being so, the information contained in implied volatility is a critical element of this study.
2.2.2. Empirical Evidence 2.2.2.1. The Volatility Index One of the greatest difficulties in contemporary financial markets is finding an appropriate measure of investor sentiment. However, in 1993 Robert Whaley attempted to overcome this problem through the introduction of the VIX in the United States. According to this study, the VIX is a measure of the volatility that investors anticipate and is therefore used as a measure of “implied volatility” (Whaley, 1993). The VIX is calculated using the weighted prices of all the closest at-the-money call and put options of the Standard and Poor 500 (S&P 500 hereafter) Index. Consequently, the VIX is a forward-looking measure that measures implied volatility and does not measure volatility that has already been realized. Arik (2011) who studied the VIX’s descriptive statistics for three different sample periods (namely, January 2001 to December 2003; during the housing bubble between January 2004 till December 2007; and after the housing bubble and 2008 financial meltdown), concluded that the VIX usually forms a mean-reverting trend in the short term, whilst forming cycles in the long term. He also suggests that a high VIX represents a large amount of investor fear. Therefore, as the VIX increases, the fear of the market increases, indicating that investors perceive higher risks; and as the VIX decreases, investors become more optimistic about the future of the market, indicating that investors perceive lower risks.
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Lonesco (2011) performed a study to determine whether implied volatility outperforms historical volatility. This study analyzed monthly data of the S&P 500, FTSE 100 and DAX equity and option markets from 2004-2010. Giving attention to the repercussions of the 2008 financial crisis, data was split into two samples of 2004-2007 and 2008-2010. The superior informational content of implied volatility from 2004 until 2007 was unequivocal. However, during the 2008-2010 period, the performance of implied volatility was impacted by presence of the financial crisis. Nevertheless, results during this period demonstrated that the superiority of implied volatility was robust, as implied volatility was as good as the other estimators during 2008-2010 period. A more recent study by Narwal, Sheera and Mithal (2016) examined the forecasting efficiency of the implied volatility index of India (IVIX). The study used daily volatility data for the CNX Nifty stock index ranging from 2 March 2009 until 30 June 2012. The GJR-GARCH model was used to compare the forecasting ability of the implied volatility index to the realized volatility index. This study also concluded that implied volatility measures provided a significant information content of future volatility in comparison to measures of realized volatility. Several studies that have been conducted have used the VIX, based on the S&P500 index, in their analysis of the relationship between volatility and local stock market returns. The VIX, based on the S&P500 index, was used as a proxy for international stock market volatility. On the other hand, some studies made use of the local volatility index when evaluating the relationship between volatility and local stock market returns. Hence, the following sections has been separated to provide a review of studies that have made use of the VIX based on the S&P500 index, and, thereafter, studies that have used the local volatility index.
2.2.2.2. The Effect of the VIX (based on the S&P 500 Index) on a Country's Stock Returns The following studies examine the relationship between the VIX, which is calculated based on the S&P 500 Index, and the returns of a country’s stock indices:
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Bali and Peng (2006) examined the presence of an intertemporal relationship between risk and return in the United States (US) stock market, by utilizing high-frequency data for a period of 40 years. The authors made use of daily logarithm returns of the Centre for Research in Security Prices (CRSP) value-weighted index, S&P 500 index futures and, S&P 500 cash index, to encapsulate the different stock market returns in the US. The determination of the existence and significance of a risk-return trade-off incorporated in the stock markets indices was conducted by implementing daily realized, Garch-in-mean, implied and range-based volatility estimators. The results of the study found a positive and highly statistically significant relationship between the conditional mean and conditional volatility of market returns. According to the study, the use of high-frequency data as an input in the GARCH models has consistently displayed a significant and positive risk-return relation. The authors suggest that daily volatility is closely related to illiquidity, which earns a positive premium, hence, daily data may provide stronger support for a positive risk-return relation. A study by Guo and Whitelaw (2006) used the Intertemporal Capital Asset Pricing Model to examine the relationship between S&P100 Index prices and the VIX. The monthly excess market return and realized variance used in this study was obtained from the daily excess market returns. A significant positive relationship between risk and stock market returns was found in this study. According to the authors of this study, these findings differ from those studies that found there to be a negative relationship between risk and return, due to the fact that other studies did not appropriately differentiate risk that is attributed to expected returns from the hedge component, which results in a bias. Similarly, Bollerslev, Tauchen and Zhou (2009) made use of various econometric “model-free” regression techniques to determine the risk-return relationship between S&P 500 index and the VIX from January 1990 until February 2002. The study found a positive risk and return relation, which is consistent with the idea that, when markets anticipate higher volatility, there is a discount in stock prices, which results in higher future returns. In a study conducted by Kanas (2012), the GARCH-M model was used to study the risk-return relationship for the S&P 500 Index. The study found that a significantly positive relationship exists between risk (measured by the VIX) and returns on the S&P500 Index, if the VIX squared is used as an additional variable in the conditional variance equation. According to Kanas (2012: pp.1313), “the relation holds even when we control for the Fama and French 14
(1993) factors in the conditional mean equation”. The study also revealed that the positive relationship holds for both daily and weekly observations. It was also found that the GARCH model combined with the volatility index has superior predictive power than the GARCH model without the volatility index, or the volatility index itself. Contrastingly, Sarwar (2012) found a strong negative relationship between the VIX and the returns on the S&P 100, S&P 500, and S&P 600. This study undertook an analysis in three subperiods: 1992-1997, 1998-2009, and 2004-2009, to negate the effects of structural shifts in VIX. This study made use of time series data while employing a cross correlation analysis and regression analysis. A negative relationship was discovered in all three subperiods and the strength of this relationship was found to grow in each consecutive subperiod. Furthermore, the relationship between the VIX and the 3 selected indices were found to be stronger when the VIX was higher and more volatile. Antonakakis, Chatziantoniou and Filis (2013) analysed the relationship between the returns in the U.S stock market, implied volatility, and the uncertainty of policy. This study used data from the policy uncertainty index series, introduced by Baker, Bloom and Davis (2012), returns from the S&P500, as well as changes in the VIX. This study utilised time series data, from January 1985 to January 2013, and employed a dynamic conditional correlation (DCC) model, as well as a GARCH model. The results of the study indicated that there was a dynamic relationship between policy uncertainty and the returns on the U.S stock market. It was also found that an increase in the stock market volatility (measured by VIX) and policy uncertainty, resulted in a reduction in stock market returns (measured by the returns of the S&P500). The results, therefore, concludes that there is a negative relationship between the VIX and stock market returns. A further study undertaken by Kang, Ki-Hong and Yoon (2014), investigated the relationship between the VIX and the Asian financial markets, including the equity and exchange rate markets. This study had a sample period of 2 January 2004 – 30 December 2014, and made use of daily data regarding the VIX, Japan-MSCI Index, Korea-MSCI Index, Japanese Yen (JPY) and Korean Won (KRW). This study employed a VAR model. VIX was found to have a negative relationship with both the Japanese and Korean equity indexes. In addition, VIX was found to have a negative relationship with JPY, but a positive relationship with the KRW. This
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indicates that during times of turbulent market conditions, investors demand greater portfolio insurance premiums. Wang, Tsai and Lu (2014) found that the VIX is negatively related to the returns in the Chinese stock markets. The GARCH, E-GARCH, and T-GARCH frameworks were used to study the monthly returns of the CSI 300 Index (an index that tracks the performance of 300 Chinese leading listed companies) and the monthly closing prices of the VIX. The Value-At-Risk and Capital Asset Pricing Models then proved the existence of a negative relationship between the VIX and CSI 300 Index. A contrary result was found in a study conducted by Mariničevaitė and Ražauskaitė (2015). Their analysis was based on the daily returns of different indices from 1 January 2009 to 30 September 2014. A multiple regression analysis was used to test the ability of the VIX (based on the S&P 500 Index) and the equity market’s localized VIX in determining equity market returns. The results from this study indicated weak correlations between the S&P 500 VIX and the local VIX, and therefore, contradicted the conventional belief that the S&P 500 VIX is a measure of global risk aversion. Based on these correlations, it was concluded that the S&P500 VIX is insufficient in determining equity market returns in emerging economies. Hence, local volatility indices should be considered when analyzing the relationship between stock market volatility and equity market returns. Therefore, the next section provides an evaluation of the relationship between local volatility indices and stock market returns.
2.2.2.3. The Relationship between the Local Volatility Indices and Stock Returns The following studies examine the relationships between localised volatility indices and the returns of stock indices: Lee, Jiang and Indro (2002) conducted a study to examine stock market volatility, excess returns, and the role of investor sentiment. The study employed a GARCH framework to jointly test four behavioural effects outlined in the DSSW noise trading model. The sample used in the study ranged from January 5, 1973 to October 6, 1995. Three different market indices namely DJIA, S&P500 and NASDAQ are used to characterize the overall performance of the market. The study found higher excess returns to be associated with a decrease in conditional 16
volatility resulting from larger bullish shifts in sentiment for both small as well as large capitalization stocks. This relates to a positive relation between shifts in sentiment and excess returns across the three indices. In a study by Dowling and Muthuswamy (2005), the relationship between the Australian Market Volatility Index (AVIX) and the returns of the ASX 200 Index was analysed. The daily closing index levels from November 1999 to September 2007 were collected, a total of 2002 observations were examined in this study. A regression was used to test whether the contemporaneous relationship between stock returns and volatility in the Australian stock market is significant. The coefficient on the contemporaneous index returns was found to be negative and significant. The study concluded that there is a negative relationship between the AVIX and the returns from the ASX 200 returns. Furthermore, it was found that the AVIX displays no asymmetry in its response to positive and negative shocks in ASX stock returns. In addition, a study by Lunbald (2007), analysed the U.S and U.K stock market between years 1836-2003. The study made use of monthly time series data and utilised the returns for the NYSE, AMEX, and Nasdaq markets. The study made use of a Monte Carlo Analysis, GARCH, TGARCH, EGARCH, QGARCH and MGARCH Models. The results found there to be a significant positive relationship between risk and return on all stock markets. The finding of a positive relationship was attributed to the large study period. This was because small samples are negatively affected by the fact that conditional volatility has minimal explanatory power for realized returns, therefore resulting in the finding of a negative relationship. Guo and Neely (2008) investigated the risk-return relationship with Morgan Stanley Capital International (MSCI) data for 19 major international markets. The GARCH-M model was used to examine the daily stock market returns over the period from 7 January 1974 until 29 August 2003. The analysis found a positive relationship between risk and return in almost all markets and these relationships were often found to be statistically significant. On the other hand, Shaikh and Padhi (2014) examined the asymmetric inter-temporal relationship between India volatility index (NVIX) and stock market returns (Nifty S&P 50, 100, 200 and 500) in the Indian securities markets. It was based on the daily value of the volatility index and stock indices for the period from 2009 to 2015. Findings suggest a strong negative correlation between daily changes in the NVIX and stock returns. This relationship is 17
more prominent when NVIX is higher and more volatile. However, there is an asymmetry between India NVIX and the stock returns. In addition, the study also finds that a significant increase in the VIX (U.S.) index results into the momentous climb in the Nifty volatility index. Hence, Nifty volatility index is the best measure of the stock market turmoil on investors’ fear. Likewise, Chandra and Thenmozhi (2015) studied the daily closing prices of the Indian VIX and CNX Nifty Index for the period between March 2009 and ending November 2012. Volatility measures such the GARCH and EGARCH were then used to observe the linear relationships between the two indices. The regression estimations approach found that the CNX Nifty Index returns and the movements in the Indian VIX have an inverse relationship. However, the study found that in the case of high upward movements in the market, the movements of the Indian VIX and the returns on the CMX Nifty index tends to move independently. This was because the gains earned from the upward market movement were too small to sufficiently cover losses caused by volatility, leading to independent movements between the India VIX and the returns on the CMX Nifty Index. A more recent study by Emna and Myriam (2017) reaffirmed the presence of a relatively strong inverse relationship between the changes in implied volatility indices and the returns on equity markets. This was conducted through the examination of the daily closing prices of the volatility indices and equity market indices in France, Germany, Switzerland and the United Kingdom, from January 2010 until March 2015. The GARCH and GJR-GARCH models was then used to prove that future stock market volatilites can be forecasted using implied volatility indices. Thereafter, a multiple linear regression model confirmed the presence of a strong inverse and asymmetrical relationship between the changes in volatility indices and equity market returns.
Summary: The empirical evidence regarding the relationship between implied volatility is largely mixed, with some studies finding evidence of a negative relationship which is consistent with the leverage hypothesis and behavioural finance theories. However, other studies by Lee et al., (2002), Lunbald (2007), Bali and Peng (2006), Bollerslev, Tauchen and Zhou (2009), Kanas (2012) have all found a positive relationship which is consistent with with the positive riskneutral relation and intertemporal capital asset pricing model. There is currently no evidence on this subject in South Africa, which is the gap that this study aims to fill. 18
3. Methodology and Data 3.1. Data The aim of this study is to determine the relationship between implied volatility and the returns of different stock indices in South Africa. To examine the effect of implied volatility on the returns of companies of different sizes, this study will analyse the relationship between volatility and different South African indices, namely; Top 40, Mid Cap, and Small Cap. The JSE Top 40 Index is an index comprising of the largest 40 companies listed on the JSE. The Mid Cap Index is an index comprising of the largest 60 medium-sized companies. The companies in the Top 40 Index and the Mid Cap Index are ranked by market capitalisation. The Small Cap Index comprises of companies that form part of the JSE All-Share Index but are not large enough to form part of the Top 40 Index or the Mid Cap Index. This paper will also examine the relationship between implied volatility and the returns of the market as a whole. To measure the returns of the market, the JSE All-Share Index will be used. The JSE All-Share Index represents 99% of the full market capitalisation value, that is, before the application of any investable weightings, of all ordinary securities listed on the main board of the JSE, subject to minimum free-float and liquidity criteria. To measure implied volatility in South Africa, the South African Volatility Index (SAVI) Top 40 will be used. The SAVI Top 40 is a measure of investor sentiment in the South African equity market. Launched in 2007, the SAVI was introduced to measure investors’ expectation of the three-month market volatility (Kotze, Joseph and Oosthuizen, 2009). The SAVI was based on the JSE Top 40 Index and was calculated using the at-the-money volatilities. In 2009, however, the calculation of the SAVI was improved Kotze et al., (2009). Like the old SAVI, the new SAVI is determined by at-the-money volatilities and is grounded on the JSE Top 40 Index. However, the new SAVI includes a volatility skew. The volatility skew provides investors with a protection premium against market failure. This being so, the benefit of the new SAVI is that it considers all the dimensions of volatility, including the time dimension and the strike level dimension. The improved SAVI uses the weighted average prices of put and call options in its calculation. Therefore, it is a model-free volatility index. 19
The new SAVI is calculated as follows (Kotze et al., 2009):
𝑛𝑒𝑤 𝑆𝐴𝑉𝐼 =
∑
𝑤 𝑃 (𝐾 ) + ∑
𝑤 𝐶 (𝐾 )
(1)
In equation 1, 𝐹 represents the forward price of the JSE Top 40 Index. This forward price is determined using the dividend yield and risk-free interest rate. F marks the price boundary between the liquid put options 𝑃 (𝐾 ), and call options 𝐶 (𝐾 ), where 𝐾 represents the option’s exercise price. In equation 1, 𝑤 represents the respective weights of each option. Due to the improvement in the calculation of the SAVI in 2009, the sample period under investigation runs from 01 May 2009 through to 28 April 2017. This translates into 1999 trading days, and thus, observations in total. This study will use financial time series data that will be obtained from Bloomberg. The closing daily prices of the indices as well as the daily closing values of the SAVI will be examined in this study. In this study, daily data will be used due of the fact that equity markets typically assimilate information quickly. Furthermore, according to Harrison and Zhang (1999), the risk-return relationship is usually hidden in short-frequency data by the short-term noise caused by agents aiming to rebalance their portfolios. Therefore, by using daily data, this study will examine the ability of the SAVI to expose a risk-return relationship using data-frequencies which tend to hide it. It should be noted that, the daily price series is transformed via a logarithm into a return (or movements for the SAVI) series to make it stationary. The respective index daily price levels are transformed into returns by this simple operation: 𝑟 = 100 × ln(
)
(2)
where Pt and Pt−1 are current and previous stock prices for t = 1, 2, · · · , ∞.
3.1.1. Preliminary Data Analysis Before the GARCH models are estimated the data will be run through the following preliminary tests: Firstly, a test will be conducted to check the stationarity of the variables. The AugmentedDickey Fuller (ADF) test will be employed to verify that the data used in the study does not contain a unit root. The test will be conducted using a null hypothesis that states the series 20
contains a unit root and therefore is not stationary. The P-values for the ADF test will be examined, if we fail to reject the null hypothesis at level, the series will contain a unit root and the data will have to be run through the ADF test again at first differences until the null hypothesis can be rejected, in which case the time series data will be stationary (Kgosietsile, 2014). Once the level at which the time series data is stationary has been found, the descriptive statistics of the relevant series will be discussed. A Jarque-Bera test will be conducted to test departures from normality. The null hypothesis is of normality and will be rejected if the residuals from the model are either significantly skewed or leptokurtic. Following the descriptive statistics and the Jarque-Bera test for normality, the correlation coefficients between the indices and the SAVI will be tested to examine whether the indices and the SAVI are serially correlated or not. Lastly, an LM test will be conducted, to check whether the data presents any ARCH effects and heteroscedasticity. The ARCH-LM diagnostic test can be used to test the lack of fit of a time series model. The test is one of a joint null hypothesis that all q lags of the squared residuals have coefficient values that are not statistically different from 0 (Brooks, 2008). This study will use 5 lags of the squared residuals as a base, as this was suggested by Brooks (2008). If the squared residuals of the model exhibit autocorrelation, then ARCH effects are present. The estimated model will be tested for autocorrelation using the LM autocorrelation test. If there is a presence of autocorrelation in the estimated model, lagged values of the explained variable will be added to the right-hand side of the equation until serial correlation is eliminated (Brooks and Ragunathan, 2003).
3.2. Methodology According to Campbell, Lo and MacKinlay (1997), the risk-return trade off exhibits nonlinearity in some input variables. Furthermore, Brooks (2014) states that most financial data exhibit leptokurtosis, and volatility clustering or volatility pooling. These properties cannot be adequately captured by time series models, and thus, volatility models are the most appropriate alternative.
21
A majority of the literature in forecasting volatility employs Bollerslev’s (1986) and Taylors (1986) Generalised Autoregressive Conditionally Heteroscedastic (GARCH) model. This study will analyse different GARCH models to estimate which GARCH model most accurately predicts the relationship between implied volatility and the returns of Indices on the JSE. The GARCH model became well established in 1986 after academics began seeking advancements to Engle’s (1982) Autoregressive Conditionally Heteroscedastic (ARCH) model. The ARCH model, had gained popularity within financial engineering during its early years, however the GARCH model became a more widely used model due to it being more parsimonious and better fitting (Brooks, 2014). Following the development of the GARCH model, a variety of different GARCH models have been developed to forecast volatility more accurately. The Exponential GARCH (E-GARCH) model, proposed by Nelson (1991), models the logarithm of volatility, therefore there is no need to artificially impose non-negativity constraints on the model parameters and an additional variable to account for asymmetries has been added to the conditional variance equation (Brooks 2014). The GJR GARCH (T GARCH) model, proposed by Glosten, Jagannathan and Runkle (1993), is another extension of the GARCH model which includes a dummy variable to account for possible asymmetries in the market. Lastly the GARCH-M model will be analysed which adds the conditional variance term into the mean equation. The asymmetric GARCH models have the additional advantage of explaining the asymmetric nature of the market to past positive and negative shocks (Brooks, 2014). For the GARCH, EGARCH, GJR-GARCH and GARCH-M models, the mean equation used in the study will be based on the Fama-French (1993) Three Factor Model. The Fama-French (1993) Three Factor Model accounts for three risk factors, namely; market (MKT), size (SMB), and value (HML) factors. The market risk premium captures the excess returns of the market over (All Share Index) the risk-free rate of interest (3-month Treasury bill rate). The use of the 3-month T-Bill rate is supported by Strydom and Charteris (2013) who conducted a study on the South African risk-free rate anomaly and concluded that the relationship between the riskfree proxies, such as the 3-month T-bill rate, and the minimum required return does not differ in South Africa. Moreover, the size factor captures the excess returns of small stocks over large stocks, whilst the value factor captures excess returns of value stocks over growth stocks. These 22
three variables possess valuable information, regarding the risk present in financial markets, which in turn, influences the investment opportunity set. The use of the Fama and French (1993) 3 factor model is largely supported by previous literature. Kelly (2003) indicated that information about inflation and economic growth was included in the SMB and HML - information that is different from information represented by the market factor. Hahn and Lee (2006) found that GDP growth and business cycles were accounted for by SMB and HML. According to Chiah, Chai, Zhong and Li (2016, pp. 596), “The three-factor model has therefore become a benchmark model in the asset pricing literature”. Based on the aforementioned argument, the Fama-French (1993) three factor mean equation, for the GARCH-type models used in this study, will be consistent with Kanas (2012), and will be as follows: 𝑅 = µ + 𝛼 𝑀𝐾𝑇
+ 𝛼 𝑆𝑀𝐵
+ 𝛼 𝐻𝑀𝐿
(3)
+ 𝜀
However, the GARCH-M model will introduce the variance term into the mean equation, and therefore, the mean equation will be (Brooks, 2014): 𝑅 = µ + 𝛼 𝑀𝐾𝑇
+ 𝑎 𝑆𝑀𝐵
+ 𝛼 𝐻𝑀𝐿
+ 𝜓𝜎
+ 𝜀
(4)
The use of the lagged variables for the MKT, SMB, and HML factors are supported by Liew and Vassalou (2000). The study conducted by Liew and Vassalou (2000) found that the lagged variables are significant in predicting the future changes in the investment opportunity set and are accurate in predicting future economic growth. Moreover, a study conducted by Kanas (2012) found that the lagged variables are significant in capturing information about fundamental risks in the economy, which will affect the investment opportunity of the country. In equation 3 and 4 above, 𝑅 represents the returns of each stock index. μ is the constant term. 𝑀𝐾𝑇
represents the lagged variable of the market risk premium, which is calculated as the
difference between the expected return of the market and the risk-free rate of interest (3-month Treasury bill rate). 𝑆𝑀𝐵
represents the lagged variable of excess returns of small stocks
over large stocks, and is calculated by taking the difference between the Small Cap index and 23
the Top 40 index. 𝐻𝑀𝐿
represents the lagged variable of excess returns of value stocks
over growth stocks, and is calculated as the difference between the Value Style index (J330) and the Growth Style index (J331). 𝜀 is the white noise error term. To represent the lagged MKT variable, the excess returns of the market (JSE All Share Index) over the risk-free rate of interest (3-month Treasury bill rate) will be used. In equation 4, a variance term is introduced into the mean equation of the GARCH-M model. According to Brooks (2014), if the coefficient on the variance term (ψ) is positive and statistically significant, then increased risk leads to a rise in the mean return, thus ψ can be interpreted as a risk premium. With respect to the conditional variance equation for the GARCH-type models, the lagged value of the change in the SAVI will be added as an additional variable to each variance equation. The use of the SAVI as a lagged variable in the conditional variance equation is supported by previous literature. Ezpeleta (2015) conducted a study to test whether adding an exogenous variable -the implied volatility index- to the variance equation will increase the performance of the GARCH models. The study concluded that the volatility index contains relevant information when modelling financial volatility, as it can be seen as a general fear index in the stock market. Therefore, adding the implied volatility to the variance equation will increase the performance of the different GARCH models. The conditional variance equations for the various GARCH models are, therefore, specified in the table below:
Model
Conditional Variance Equation with SAVI
GARCH
𝜎 = 𝜔 + 𝛽𝑢
ln(𝜎 ) = 𝜔 + 𝛽 ln(𝜎
EGARCH
+ 𝛼
+ 𝛼𝜎
)+𝛾 |𝑢 𝜎
+ 𝛿𝑆𝐴𝑉𝐼
𝑢 𝜎 |
−
2 + 𝛿 𝑆𝐴𝑉𝐼 𝜋
24
GJRGARCH GARCHM
𝜎
= 𝜔 + 𝛽 𝑢
+ 𝛽𝜎 𝛿𝑆𝐴𝑉𝐼
𝜎 = 𝜔 + 𝛽𝑢
+ 𝛼𝜎
+ 𝛾𝑢
𝐼
+
+ 𝛿𝑆𝐴𝑉𝐼
The conditional variance equations presented in the table above are consistent with Ezpeleta (2015). These equations have the lagged SAVI as an exogenous variable. For the conditional variance equations in the table, ω is the constant term and 𝜎 represents the estimate of the conditional variance for period 𝑡. It is referred to as conditional variance as it is based on previous information that is assumed to be relevant. The parameter δ represents the effect of the exogenous variable SAVI on the conditional variance. The statistical significance of each coefficient will then be analysed using their relative p-values. For the EGARCH model the coefficient γ signifies the asymmetric effects of the shocks on volatility. These asymmetric effects can be tested by the hypothesis that γ=0. If the γ coefficient is zero, this would imply that positive and negative shocks of the same magnitude have the same effect on volatility of stock returns. If γ ≠ 0 the effect is asymmetric. According to Brooks (2014), the EGARCH model exhibits several advantages in comparison to the pure GARCH model. Firstly, since the log(𝜎) is modelled, even when parameters are negative, 𝜎 will be positive. Secondly, the EGARCH framework allows for asymmetries. Therefore, if the relationship between volatilities and returns are negative, 𝛾 will be negative. With regards to the GJR-GARCH model 𝐼 𝑢
< 0 𝑎𝑛𝑑 0 𝑖𝑓 𝑢
represents the dummy variable that equals 1 if
> 0. γ will depict the asymmetry coefficient. For leverage effects to
be inherent in the market a positive sign on the coefficient 𝛾 is expected as it relates to negative shocks having a greater impact on returns than positive shocks, the Dummy variable of 1 increases the variance by the leverage effect (Wang et al., 2014). Once all the GARCH models have been estimated, the Likelihood ratio test, Akaikes (1974) information criterion (AIC), Schwarz’s (1978) Bayesian information criterion (SBIC), and the 25
Hannan-Quin criterion (1979) (HQIC) will be employed to determine which GARCH model is the superior model, with respect to predicting volatility in the chosen indices (Brooks and Burke, 1998). The Likelihood ratio test follows a chi-squared distribution with m number of restrictions. The information criteria are designed to select the most relevant model for forecasting volatility and are able to compare models fit to the same data. The object of the information criteria is to choose the number of parameters which minimizes the value of the information criteria. SBIC is strongly consistent and inefficient and therefore works best for small samples as it will deliver the correct model order. AIC is not consistent but more efficient than SBIC and therefore will deliver on average too large a model. Due to the fact that this study has a larger data range, more emphasis will be put on SBIC as it is generally more superior to AIC with larger samples (Brooks, 2014). The best model for each index will then be interpreted. An ARCH residual diagnostic test will be conducted to show that the model being interpreted does not violate the requirement of homoscedasticity. The ARCH test will be run to check that the coefficients are significant at the conventional significant levels. As stated by Chandra and Thenmozhi (2015) the ARCH LM test on the residuals will check to see if the residuals are non-normally distributed; if the residuals are non-normally distributed no ARCH effects are present after estimating the model.
26
4. Findings 4.1. Data Analysis Figure 1: Time series plot of the closing prices of indices. 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 2009
2010
2011
2012
2013
ALSH SmCap
2014
2015
2016
2017
TOP40 MdCap
In Figure 1 above is a plot of the closing prices of the All Share Index, the Top 40 Index, the Small Cap Index and, the Medium Cap Index for the period of study. It is evident that the closing prices of these indices have moved in an upward direction, indicating a growth in the closing prices for all 4 indices during the observed period.
However, from the visual
inspection, a substantial decrease in the closing prices of the indices in 2015 is exposed. This decrease was attributed to the decrease in performances of the different stocks that made up the respective indices in 2015. Nevertheless, the trend observed in the price series resembles a deterministic trend process. The price series do not contain a constant mean or constant variance over the period. This means that the price series is non-stationary.
27
Figure 2: Time series plot of the returns of the JSE indices. 8 6 4 2 0 -2 -4 -6 2009
2010
2011
2012
2013
RALSH RSMCAP
2014
2015
2016
2017
RTOP40 RMDCAP
Figure 2 is a plot of the transformed time series – the All Share Index return series, the Top 40 Index return series, the Small Cap Index return series, the Mid Cap Index return series. In Figure 2, each return series represents a white noise process as there are no visible trends and, each of the series’ frequently crosses its mean values. This may indicate that each of these series are now stationary. To confirm the stationarity of the return series, the Augmented Dickey-Fuller(ADF) test will be conducted later. In addition, in the plotted return series in Figure 2, the series displays a tendency for volatility to occur in clusters. Volatility clustering means that in the series we see periods where there are relatively higher and lower positive movements and periods of relatively lower positive and negative movements. This is a phenomenon commonly observed in financial time series.
28
Figure 3: Time series plot of the closing prices of the SAVI index SAVI 35
30
25
20
15
10 2009
2010
2011
2012
2013
2014
2015
2016
2017
Figure 3 is a plot of the closing prices of the SAVI. One major pattern that stands out in Figure 3 is a downward trend in the closing prices of the SAVI over time. This downward trend can be attributed to the decrease in volatility following the end of the 2008 global financial crisis. From the visual inspection it is therefore evident that the series may not be stationary as both the mean and variance are not constant over time. Therefore, in accordance with Wang et al., (2014), the first order logarithmic difference method will be applied to the closing prices series of the SAVI in order to obtain a return series of the SAVI.
4.1.1. Unit Root Tests To confirm the stationarity of the return series, the Augmented Dickey-Fuller(ADF) test will be conducted, as indicated above.
29
Using the Augmented Dickey-Fuller Test, we test the null hypothesis that each return series contains a unit root, in other words, the series is not stationary. Table 1 above depicts the Augmented Dickey-Fuller Test undertaken in level terms for each return series. In Table 1, the P-value of the ADF statistics for all of the series’ are 0.0000. This being so, for each of the indices, including the movements in the SAVI, the ADF test statistic is highly significant. Thus, the null hypothesis’ of non-stationarity in each of the series’ are rejected. Therefore, it is concluded that the return series for each of the 5 indices are stationary in level terms.
4.1.2. Descriptive statistics Table 2: Descriptive statistics of the index returns RALSH
RTOP40
RSMCAP
RMDCAP
RSAVI
a) Descriptive statistics of the returns of indices Mean
0.044
0.042
0.049
0.048
-0.037
Median
0.028
0.021
0.041
0.049
0.000
Maximum
4.233
4.678
6.248
4.247
19.447
Minimum
-3.693
-4.049
-3.868
-3.742
-21.784
Standard
0.974
1.065
0.544
0.747
2.820
Skewness
-0.186
-0.145
-0.037
-0.280
0.018
Kurtosis
4.345
4.354
14.647
5.201
10.474
Jarque-Bera
169.157
166.650
11781.34
448.297
4850.910
Probability
0.000
0.000
0.000
0.000
0.000
Observations
2084
2084
2084
2084
2084
-0.317
-0.397
1.000
Deviation
b) Correlation between the returns of indices RSAVI o
-0.568
-0.569
RALSH refers to the returns on the JSE All Share Index, RTOP40 refers to the returns on the JSE Top 40 Index, RSMCAP refers to the returns on the Small Cap Index, RMDCAP refers to the returns on the Medium Cap Index and RSAVI refers to the returns on the SAVI Top 40 Index.
The average return for the All Share Index, Top40 Index, Small Cap Index, and Medium Cap Index returns series were approximately 0.044%, 0.043%, 0.049%, and 0.048%. 30
In order for a distribution to be normal, it should have a skewness value of 0 and kurtosis of 3 (Brooks, 2008). The results in the table, along with Jarque-Bera tests statistics all prove nonnormality. Furthermore, we see that the returns on the SAVI Index is negatively correlated with returns on the on the JSE All Share Index, Top 40, Medium Cap and Small Cap Indices. The correlation is greater with the All Share and Top 40 indices. The above findings of non-normality, volatility clustering and leptokurtosis indicate that that the variance of the daily returns are not constant over time. This therefore results in the need to test for ARCH effects.
4.1.3. Testing for Heteroskedasticity/ARCH effects
Table 3: Test for ARCH effects F-statistic
Prob. F (5,2073) Obs*R-squared
Prob. Chi-Square (5)
RALSH
22.676
0.000
107.812
0.000
RTOP40
21.941
0.000
104.492
0.000
RSMCAP
68.551
0.000
294.957
0.000
RMDCAP
38.863
0.000
178.171
0.000
o RALSH refers to the returns on the JSE All Share Index, RTOP40 refers to the returns on the JSE Top 40 Index, RSMCAP refers to the returns on the Small Cap Index, RMDCAP refers to the returns on the Medium Cap Index.
The results of the ARCH tests are given in Table 3. Its shows that residuals contain ARCH effects, therefore the null hypothesis of the ARCH tests of all four indices can be confidently rejected. We reject the null hypothesis of linearity at the 1% level as all the p-values associated with the LM test (or F-test) are significantly zero for all indices. Therefore, this confirms that each index consists of ARCH effects and heteroskedasticity. Consequently, to model the relationship between the movements in the SAVI and the returns of the different indices, GARCH type models need to be used.
31
4.2. Superior GARCH Model Table 4: Models with SAVI as an exogenous variable (Please refer to Appendix 1) GARCH (1,1) LL AIC SBIC HQIC
-2741.702 2.640 2.661 2.648
LL AIC SBIC HQIC
-2927.337 2.818 2.840 2.826
LL AIC SBIC HQIC
-1448.841 1.398 1.420 1.406
LL AIC SBIC HQIC
-2188.481 2.108 2.130 2.116
o o o o
EGARCH (1,1) TGARCH (1,1) GARCH-M (1,1) RALSH -2723.546 -2728.829 -2741.233 2.623 2.628 2.640 2.648 2.653 2.665 2.632 2.637 2.649 RTOP40 -2908.960 -2914.798 -2926.934 2.801 2.807 2.818 2.826 2.831 2.843 2.810 2.816 2.827 RSMCAP -1449.836 -1448.809 -1448.479 1.400 1.399 1.399 1.425 1.424 1.423 1.409 1.408 1.408 RMDCAP -2179.735 -2181.673 -2187.918 2.101 2.103 2.109 2.125 2.127 2.133 2.110 2.112 2.118
LL represents Log-Likelihood AIC, SBIC and HQIC represent the Akaike, Schwartz and Hannan-Quinn Information Criterion. The selected model for each criteria is highlighted in bold. RALSH refers to the returns on the JSE All Share Index, RTOP40 refers to the returns on the JSE Top 40 Index, RSMCAP refers to the returns on the Small Cap Index, RMDCAP refers to the returns on the Medium Cap Index
The values of the log-likelihood statistic, AIC, SBIC and HQIC of the JSE All Share index and the JSE Top 40 index suggest that the EGARCH (1,1) model performs the best in modelling volatility. The Small Cap index does not concede EGARCH (1,1) to be the best model as the LL statistic, AIC and SBIC suggest that the GARCH (1,1) is the best model. Since our time series data of 2083 daily observations is considered large, SBIC will asymptotically deliver the correct model as it is strongly consistent, favouring large samples. This results in GARCH (1,1) being the most appropriate model to model volatility of the Small Cap index. These results indicate that modelling volatility is a tough task, as one model can be used for the JSE All Share Index, JSE Top 40 Index, and Mid Cap Index. However, this does not necessarily mean that other indices in the JSE, like the Small Cap Index, can be modelled by the same model, this is in line with studies of Poon and Granger (2003), and Masinga (2015). However, for this 32
study, the EGARCH model will be used to examine the relationship between the returns of the indices and the changes in the SAVI, as the EGARCH model is the most common superior model amongst the observed indices.
4.3. EGARCH Regression Results
The constant term (µ) in the mean equation demonstrates positive intercepts for the JSE All Share, Top 40, Small Cap and Medium Cap indices, only the Small Cap and Medium Cap indices are statistically significant at a 1%, 5%, and 10% levels of significance. The positive and significant coefficients illustrate the presence of abnormal returns on the Small cap and Medium Cap indices. The market risk premium term (𝛼 ) is introduced in the EGARCH mean equation and the Fama And French three factor model to account for the likelihood that the indices returns may rely on its volatility (Kgosietsile, 2014). The coefficients associated with the JSE All Share, Top 40, Small Cap and Medium Cap returns are all found to be positive. However, the coefficient is statistically significant for the Small Cap and Medium Cap returns. The positive and significant parameters suggest a strong risk-return relation and it is robust to the inclusion of the FF factors.
33
The figure (𝛼 ) in the EGARCH mean equation illustrates the excess returns of small stocks over large stocks. The results in table 5 show that the coefficients of all four indices are positive, however, only the Small Cap returns are statistically significant. This indicates that companies listed on the Small Cap index are more sensitive to many risk factors due to their relatively undiversified nature and their reduced ability to absorb negative financial events. The evidence of a size effect in the Small Cap index can be attributed to the findings of Basiewicz and Auret (2010). The academics found that the persistence of the size effect on the JSE is not a direct consequence of irrational mispricing and that market microstructure effects are the main reason for the presence of the size effect. The High minus low ((HML) (𝛼 )) factor in the mean equation illustrate that only the parameter associated with JSE All Share and Top 40 is significant and negative. The negative and statistically significant parameters suggest that there is no evidence of a value effect being present in the South African environment.
Table 6: Coefficient Estimates of Conditional Variance for the estimated EGARCH model ⎡ ⎤ |𝒖𝒕−𝟏 | 𝟐 ⎥ + 𝜹 𝑺𝑨𝑽𝑰𝒕−𝟏 𝐥𝐧 = 𝝎 + 𝜷 𝐥𝐧 +𝜸 + 𝜶⎢ − 𝝅⎥ ⎢ 𝟐 𝟐 𝝈𝒕−𝟏 𝝈 ⎣ 𝒕−𝟏 ⎦ RALSH RTOP40 RSMCAP RMDCAP *** *** *** 𝝎 -0.046 -0.043 -0.152 -0.085*** (-4.536) (-4.247) (-9.237) (-6.291) 0.984*** 0.983*** 0.976*** 0.985*** 𝜷 (336.692) (327.764) (179.021) (277.824) 𝜸 -0.075*** -0.075*** -0.009 -0.037*** (-5.278) (-5.411) (-0.796) (-3.774) 𝜶 0.055*** 0.056*** 0.155*** 0.097*** (4.496) (4.385) (11.066) (6.164) 𝜹 0.020*** 0.022*** 0.019*** 0.016*** (4.377) (4.627) (3.928) (4.751) Notes: ***, **, * denotes significance at 1%, 5% and 10% levels respectively. Z statistics are represented in ( ). 𝝈𝟐
𝝈𝟐𝒕−𝟏
𝒖𝒕−𝟏
It is evident from the above table that the three coefficients depicted as constant (𝜔), GARCH term ( 𝛽 ) and ARCH term ( 𝛼 ) for the four indices (RALSH, RTOP40, RSMCAP and RMDCAP) are highly statistically significant. The interpretation of such with reference to the EGARCH model illustrates that news associated with the previous days volatility has 34
significant revelatory power on current volatility, due to the lagged conditional variance (GARCH term) and lagged squared disturbance (ARCH term) ability to impact conditional variance. The GARCH (beta 1) terms large coefficients demonstrate evidence to suggest that there is considerable volatility clustering in the four indices with JSE All Share Index depicting the highest. These findings are seen to be in line with the concluding remarks of Niyitegeka and Tewar (2013). The summation of the GARCH term (𝛽) and ARCH term (𝛼) of the chosen indices is seen to be greater than one, this clearly states that there is evidence of a considerable high degree of persistence in shocks to volatility, long memory in the conditional variance and a nonstationery variance. This means that shocks to volatility do not decay over time but merely suggest a very lengthy and unstable endurance in volatility (Frimpong and Oteng-Abayie, 2006). The constraints of the EGARCH model requires both the GARCH term and the ARCH term to be positive and the summation of these two terms to be less than one (Brooks, 2014). The results in table 6 show that only one constraint of the model is being met. The regressed EGARCH (1,1) model indicates that the JSE All Share, Top 40, Small Cap and Medium Cap indices asymmetry coefficient (𝛾) is negative, only the JSE All Share, Top 40 and Medium Cap indices are statistically significant. The result of the asymmetry coefficients being negative and statistically significant indicate that negative shocks affect volatility to a greater extent than positive shocks of the same magnitude. The EGARCH model demonstrates the existence of leverage effects in the three indices and therefore answers one of the research question imposed at the commencement of the research paper. The parameters associated with the SAVI ( 𝛿 ) are given to be positive and statistically significant for all four indices. This suggests a positive relationship between implied volatility and returns on the JSE All Share, Top 40, Small Cap and Medium Cap indices, with the JSE All Share and top 40 indices having the highest correlation. The positive relationship can be attributed to the risk-return trade-off, the principle states that potential returns rise with an increase in risk therefore, if volatility of the market increase there will be a complementary increase in returns of the index (Ghysels, Santa-Clara and Valkanov, 2005). Given a unit increase in the SAVI the market returns of all four indices are said to increase as volatility of the market increases, these findings are consistent with the concluding remarks of Sehgal and
35
Vijayakumar (2008) and Kanas (2012). The statistical significance of the coefficients illustrates that volatility measured by the SAVI does significantly contribute to the returns of each index being measured therefore, the SAVI and the various indices cannot be looked at in isolation as the contributing factors must be considered by market participants to ensure profound decisions are made. When comparing the SAVI with the JSE All Share, Top 40, Small Cap and Medium Cap indices under the descriptive statistics it is seen that a negative relationship is evident, however the proceeding table suggests a positive correlation with the respective indices. The contradiction in findings is a direct result in the different functions of descriptive statistics and GARCH modelling. Descriptive statistics takes into consideration the non-linear properties within the model and is used to identify and describe features of a specific data set by giving limited summaries about the sample and measures of data (Bartz, 1988), whereas GARCH modelling excludes non-linearity in the model and is used in finance as a result of its effectiveness in modelling asset returns and volatility (Alberg, Shalit and Yosef, 2008). The power of descriptive statistics in identifying a relationship between a volatility index and indices returns is very weak compared to GARCH models, therefore one would expect contradicting conclusions for both terms. Table 7: Diagnostics Results of Residuals for the Estimated EGARCH Model (Please refer to Appendix 2)
Q-stat ARCH-LM: Obs*R-squared
RALSH 25.573 [0.180] 19.426 [0.494]
RTOP40 26.626 [0.146] 20.553 [0.424]
RSMCAP 14.215 [0.819] 27.611 [0.119]
RMDCAP 10.451 [0.959] 18.001 [0.587]
Notes: ***, **, * denotes significance at 1%, 5% and 10% levels respectively. P values are represented in ( ).
The Box-Ljung (1978) and ARCH-LM diagnostic test can be used to test the lack of fit of a time series model. The test is applied to the residuals of the time series. If the autocorrelations are very small, we conclude that the model does not exhibit significant lack of fit however, if the autocorrelations are very big, we conclude that the model does exhibit significant lack of fit (Brooks, 2014). The results of Ljung-Box and ARCH-LM diagnostic tests are produced in 36
table 7. The null hypothesis that the residuals are not serially correlated for lag 20 cannot rejected by the Q-statistic and the ARCH-LM statistic in favour of the alternative hypothesis; that the residuals are serially correlated (these findings are supported by the high P- Values in Parenthesis). This informs us that the EGARCH model is successful at modelling any nonlinear dependence in the series, in the conditional mean equation and conditional variance.
5. Discussion 5.1. Summary of Findings Numerous observations from the review of the available literature in this study shows the EGARCH model to be the most suitable model to model volatility, conceding with this study. Results regarding the relationship between volatility indices and stock indices varied in literature as some authors found a negative relationship, whilst others found a positive relationship. The ADF test was conducted for confirmation of stationarity of the return series, results reveal that a level of stationarity is present in JSE All Share, Top 40, Small Cap, Medium Cap and SAVI indices. The descriptive statistics display non-normal distributions, and a presence volatility clustering and heteroskedasticity in the indices, resulting in the need for a test for ARCH effects. The test for ARCH effects demonstrates p-values of the L.M test to be zero, indicating that each index contains ARCH effects and heteroskedasticity, which is consistent with descriptive statistics. With the use of information criterion, the EGARCH (1,1) model is selected as the most appropriate model to forecast volatility in all indices, except the Small Cap index. After the regressed EGARCH (1,1) model. The conditional variance parameters were found to be statistically significant, this illustrated that current volatility of the indices is affected by news associated with past volatility. It was seen that the the JSE All Share, Top 40 and Medium Cap asymmetric coefficient depicted negative and statistically significant parameters which therefore demonstrated the presence of leverage effects. The summation of the GARCH and ARCH term illustrated high degree of persistence in shocks to volatility, long memory in the conditional variance and a non-stationery variance. The SAVI displayed positive and significant parameters for all indices, suggesting a positive relationship between implied volatility and the returns of the JSE All Share, Top 40, Small Cap and Medium Cap indices. Lastly, a diagnostic test on the Ljung-Box and ARCH-LM presents a serial correlation between 37
residuals, indicating the modelling success of the EGARCH (1,1) of any non-linear dependence in the series, in the conditional mean equation and conditional variance.
5.2. Limitations Limitations to any study serves as a recommendation to future scholars seeking to conduct research in the similar context, therefore allowing for more accurate findings. The evaluation of various literature has illustrated a few limitations in the current research study. The following has been observed: A major limitation in this study was utilization of daily closing prices. Research has demonstrated that the use of hourly closing prices could improve results and change conclusions of a study. However, hourly prices are only accommodated by Bloomberg for a given period of 120 days. The period of the study is given to be 10 years, as a result hourly prices were not accessible from various data sources. Due to the improvement of the SAVI in 2009, this has limited the current research period to 8 years. The limitation can be expected to have an influence on the results of the research, however this influence will not result in biased findings. The research period is given to be long-term, and therefore, any short or long-term effects will be factored into the data used. Due to the restriction on the length of this paper, this study did not attempt to measure volatility with the APGARCH (1,1) model, which could have possibly altered the model selection of each index, as a study by Brailsford and Faff (1996) found that the APGARCH model previously outperformed the EGARCH model.
5.3. Recommendations for Future Research This study showed us that the informational content of the individual models strengthened when they were jointly regressed with the SAVI, hence a recommendation for future research would be to combine several models into a single predictor to study the volatility on the JSE. Therefore, the shortfalls of individual models can be avoided by combining them into a single predictor. For future research we can also model volatility using GARCH models to assess the interrelation of volatility with other emerging stock markets like Nigeria, Botswana, Ghana, etc. 38
The findings from this paper bring about some recommendations for investors in South Africa and supervisory authorities. First, South African investors should be rational. We demonstrated how the JSE index is profoundly influenced by the SAVI and the presence of leverage is evident between the two. These results imply that South African investors possess inadequate judgment and their awareness to outside information. Accordingly, South African investors are encouraged to enhance their judgement, expand their learning on investing, develop a responsibility to invest individually and refrain from following what other investors do. Secondly, since different time zones lead to different exchange hours, there is an opportunity to utilize the VIX to forecast the South African stock markets. And, finally, South African financial services board can utilize the analyses of this study for a better understanding of the volatility present in the South African stock market and to facilitate South Africa’s financial regulatory agencies on the stock markets. In order to secure a more healthy and stable development of the South African market, South Africa’s regulatory agencies should take active part to improve its policies.
5.4. Conclusion Minimum attention has been given to the volatility stock markets in South Africa. When researching this topic, it became evident that majority of the literature is mainly aimed at the stock markets of the developed world. Since minimal literature exists regarding the South African context, the objective of this study was to examine the volatility in the South African stock markets. The purpose of this study was to determine the relationship between implied volatility and the returns of companies of varied sizes listed on the JSE. Different indices were used to represent companies of differences sizes namely; JSE All Share, Top 40, Mid Cap, and Small Cap Indices. The study had a sample period of 1 May 2009 through to 28 April 2017 and made use of the SAVI Top 40 Index as a measure of implied volatility. A further aim of this study was to analyse which GARCH Model best described the volatility in the South African stock markets. Many different GARCH Models were estimated namely GARCH, EGARCH, MGARCH AND TGARCH. The various GARCH Models were tested, and it was found that the EGARCH Model was the best suited model for the All Share, Top 40 39
and Medium Cap Indices. In contrast the GARCH Model was found to be the best suited model for the Small Cap Index. The use of the EGARCH model found there to be a positive significant relationship between the SAVI Top 40 Index and the JSE All Share, Top 40, Mid Cap, and Small Cap Indices. However, given the extensive effect of volatility, it is essential that more time and effort is directed at the study of volatility effects on the stock markets of the developing world.
40
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Appendices Appendix 1: Models with the SAVI as an exogenous variable Appendix 1a: RALSH GARCH (1,1) Dependent Variable: RALSH Method: ML ARCH - Normal distribution (BFGS / Marquardt steps) Date: 06/25/18 Time: 13:01 Sample (adjusted): 5/06/2009 4/28/2017 Included observations: 2083 after adjustments Convergence achieved after 46 iterations Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7) GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*GARCH(-1) + C(8)*RSAVI(-1) Variable
Coefficient Std. Error
z-Statistic
Prob.
C
0.038511 0.018624
2.067780
0.0387
MKT(-1)
0.001406 0.036351
0.038671
0.9692
SMB(-1)
0.021239 0.037650
0.564119
0.5727
HML(-1)
-0.048609 0.034722
-1.399936 0.1615
Variance Equation C
0.023745 0.003217
7.380145
0.0000
RESID(-1)^2
0.036246 0.007191
5.040176
0.0000
GARCH(-1)
0.936993 0.008255
113.5064
0.0000
RSAVI(-1)
0.028615 0.002974
9.622441
0.0000
R-squared
0.000890
Mean dependent var 0.044426
Adjusted R-squared -0.000552
S.D. dependent var
0.974493
S.E. of regression
0.974762
Akaike info criterion 2.640136
Sum squared resid
1975.386
Schwarz criterion
2.661803 49
Log likelihood
-2741.702
Hannan-Quinn criter. 2.648075
Durbin-Watson stat 2.011622
EGARCH (1,1) Dependent Variable: RALSH Method: ML ARCH - Normal distribution (BFGS / Marquardt steps) Date: 06/25/18 Time: 13:02 Sample (adjusted): 5/06/2009 4/28/2017 Included observations: 2083 after adjustments Failure to improve likelihood (non-zero gradients) after 52 iterations Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7) LOG(GARCH) = C(5) + C(6)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(7) *RESID(-1)/@SQRT(GARCH(-1)) + C(8)*LOG(GARCH(-1)) + C(9) *RSAVI(-1) Variable
Coefficient Std. Error
z-Statistic
Prob.
C
0.010418 0.018235
0.571288
0.5678
MKT(-1)
0.033791 0.037437
0.902613
0.3667
SMB(-1)
0.037716 0.039040
0.966098
0.3340
HML(-1)
-0.061624 0.034144
-1.804850 0.0711
Variance Equation C(5)
-0.045593 0.010051
-4.536114 0.0000
C(6)
0.055364 0.012314
4.496124
C(7)
-0.074614 0.014137
-5.278065 0.0000
C(8)
0.983942 0.002922
336.6921
0.0000
C(9)
0.020438 0.004669
4.377373
0.0000
R-squared
-0.000852
0.0000
Mean dependent var 0.044426
Adjusted R-squared -0.002296
S.D. dependent var
S.E. of regression
Akaike info criterion 2.623664
0.975612
0.974493
50
Sum squared resid
1978.829
Schwarz criterion
2.648040
Log likelihood
-2723.546
Hannan-Quinn criter. 2.632596
Durbin-Watson stat 2.048081
TGARCH (1,1) Dependent Variable: RALSH Method: ML ARCH - Normal distribution (BFGS / Marquardt steps) Date: 06/25/18 Time: 13:04 Sample (adjusted): 5/06/2009 4/28/2017 Included observations: 2083 after adjustments Convergence achieved after 48 iterations Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7) GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*RESID(-1)^2*(RESID(-1)
