Volatility Modelling of the Johannesburg Stock Exchange All Share Index Using the Family GARCH Model

Abstract

In numerous domains of finance and economics, modelling and predicting stock market volatility is essential. Predicting stock market volatility is widely used in the management of portfolios, analysis of risk, and determination of option prices. This study is about volatility modelling of the daily Johannesburg Stock Exchange All Share Index (JSE ALSI) stock price data between 1 January 2014 and 29 December 2023. The modelling process incorporated daily log returns derived from the JSE ALSI. The following volatility models were presented for the period: sGARCH(1, 1) and fGARCH(1, 1). The models for volatility were fitted using five unique error distribution assumptions, including Student’s t, its skewed version, the generalized error and skewed generalized error distributions, and the generalized hyperbolic distribution. Based on information criteria such as Akaike, Bayesian, and Hannan–Quinn, the ARMA(0, 0)-fGARCH(1, 1) model with a skewed generalized error distribution emerged as the best fit. The chosen model revealed that the JSE ALSI prices are highly persistent with the leverage effect. JSE ALSI price volatility was notably influenced during the COVID-19 pandemic. The forecast over the next 10 days shows a rise in volatility. A comparative study was then carried out with the JSE Top 40 and the S&P500 indices. Comparison of the FTSE/JSE Top 40, S&P 500, and JSE ALLSI return indices over the COVID-19 pandemic indicated higher initial volatility in the FTSE/JSE Top 40 and S&P 500, with the JSE ALLSI following a similar trend later. The S&P 500 showed long-term reliability and high rolling returns in spite of short-run volatility, the FTSE/JSE Top 40 showed more pre-pandemic risk and volatility but reduced levels of rolling volatility after the pandemic, similar in magnitude for each index with low correlations among them. These results provide important insights for risk managers and investors navigating the South African equity market.

1. Introduction

1.1. Overview

It is common knowledge that financial markets are incredibly unpredictable and erratic. Historical evidence indicates that financial markets have a significant influence on economies across the globe. In many economies, return volatility is a gauge of the uncertainty around short-term monetary policy, which can either stimulate or restrain economic activity in the South African economy. Investors shifting their money in and out of the stock market or inside it but into other financial instruments causes the index’s fluctuation levels and, in turn, volatility [1]. The more the volatility of the market index, the greater the risk.

Furthermore, volatility is a basic topic in financial analysis since there is a vital demand for forecasting and modelling volatility in capital markets, specifically in investment strategies, techniques for managing risk, and assessing financial assets. The JSE ALSI is the primary market index in South Africa, assessing the performance of all listed companies. The JSE ALSI accounts for 98% of the total market capitalization value. Macroeconomic indicators like GDP growth, inflation, interest rates, commodity prices, market changes, liquidity, and sentiment influence it.

The leverage effect demonstrates how previous positive and negative values affect the prevailing stock market, with negative returns contributing to volatility more significantly than good returns [2]. Modelling time series with different variances and heteroskedasticity was consistently hidden. Ref. [3] attempted to address these issues using the family of models known as Autoregressive Conditional Heteroskedasticity (ARCH). Ref. [4] introduced the family of models known as Generalized Autoregressive Conditional Heteroskedasticity (GARCH), which is aimed at capturing leptokurtic returns and volatility clustering. Despite their advances, GARCH models have come under criticism for failing to accurately account for the leverage effect found in the financial markets [5]. There are various studies that focus on the stock market connectivity between countries. Ref. [6] examined the volatility levels in the stock markets of Singapore, Hong Kong, and India. According to this study, markets responded to information in a highly interconnected manner and had a significant GARCH impact, which affected both mean and volatility. Ref. [7] discovered an unbalanced association between South Korea and Japan’s stock price markets throughout the same period. Ref. [8] modelled the volatility of the JSE ALSI using Bayesian and frequentist methods. The empirical findings show that the conditional and unconditional volatility of the JSE ALSI is reflected in the Bayesian Autoregressive Moving Average-Generalized Autoregressive Conditional Heteroskedasticity (BARMA-GARCH-t) model.

This study aims to model and analyze the volatility of the JSE ALSI from 2014 to 2023 using the family GARCH model. The JSE ALSI serves as a key indicator of market performance in South Africa, but its volatility presents significant challenges for investors, risk managers, and policymakers. Traditional models often fail to capture complex volatility dynamics, necessitating more advanced techniques like the family GARCH model, which offers the potential for accurate volatility projections. Despite its ability to model volatility, applying GARCH models to the JSE ALSI remains limited, highlighting the need for this study. By investigating various GARCH models, this study seeks to enhance the understanding of volatility patterns in the South African equity markets, providing critical insights for informed decision making and effective risk management. The findings of this study offer valuable contributions to improving financial stability and efficiency, enabling investors and policymakers to navigate market volatility more effectively. This study’s scope includes collecting historical return data, preprocessing, and estimating multiple GARCH models to provide practical solutions for risk management and investment strategies.

1.2. Literature Review

Examining volatility in financial markets, especially by employing sophisticated econometric models like the GARCH model, is essential for effectively understanding and managing associated risks. The JSE ALSI, as a comprehensive indicator of the South African equity market, presents a compelling subject for volatility analysis due to its broad representation of diverse sectors within the economy. This review aims to examine how researchers have applied GARCH models to analyze and forecast volatility in the JSE ALSI. It outlines the models previously used to model the volatility of JSE ALSI. Over the past two decades, financial data analysis, particularly in volatility forecasting, has gained notable importance in the financial literature, particularly following the financial crisis. Numerous models have been introduced to predict volatility in financial modelling. Nonetheless, studies have yielded significantly varied outcomes regarding which models are most effective at forecasting volatility. One study laid the foundation for volatility modelling for financial time series data [3] with the introduction of Autoregressive Conditional Heteroskedasticity (ARCH) models, intended to account for the significant volatility observed in the log returns of financial data. Ref. [4] further developed these ARCH models into GARCH models to account for asymmetric effects and long memory in variance observed in financial time series. Several variations of the GARCH model have been introduced, such as the Integrated GARCH (IGARCH) model by [9], the Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) model by [10], the Asymmetric Power ARCH (APARCH) model developed by [11], the Exponential GARCH (EGARCH) model proposed by [12], and the family GARCH (fGARCH) model developed by [13].

After the recent national financial downturn, there has been an increased interest among academics and practitioners in analyzing financial data, particularly regarding the uncertainties of the stock market. Consequently, significant research has been directed towards modelling and forecasting stock volatility, especially in developed countries. Refs. [14,15] significantly contributed to the understanding that the uncertainty associated with stock prices, as indicated by variance, fluctuates over time. Ref. [15] also identified that volatility clustering and leptokurtosis frequently occur in financial time series. Additionally, reference [2] highlighted a phenomenon known as the leverage effect, where stock prices exhibit a negative correlation with changes in volatility. Ref. [16] further explored this leverage effect, suggesting that a decline in equity value results in an increased debt-to-equity ratio, which elevates the firm’s risk profile, thereby leading to heightened future volatility [17]. As a result of these observations of volatility clustering, the assumption of homoscedasticity is rendered less applicable, prompting researchers to focus on modelling techniques that account for time-varying variance.

Ref. [18] examined how news affects the volatility of the Nigerian stock market using ARCH class models. The findings suggested a leveraging effect, indicating that negative news affects volatility more than positive news. Ref. [19] investigated the influence of macroeconomic variables on the stock return volatility at the Nairobi Securities Exchange in Kenya. Their research examined how fluctuations in foreign exchange rates, interest rates, and inflation affect stock return volatility over a decade, utilizing monthly time series data from January 2001 to December 2010. The study employed EGARCH and TGARCH models for analysis. The findings revealed that the stock returns exhibited leptokurtic characteristics and deviated from a normal distribution. Additionally, the results confirmed that foreign exchange rates, interest rates, and inflation significantly impact the volatility of stock returns in Nairobi.

Ref. [20] examined the fluctuations in the NGN/USD exchange rates in Nigeria, employing several models, including EGARCH(1, 1), GJR-GARCH(1, 1), GARCH(1, 1), APARCH(1, 1), TS-GARCH(1, 1) and IGARCH(1, 1). This analysis utilized monthly time series data from January 1970 to December 2007. The findings indicated that the APARCH and TS-GARCH models provided the best fit for the observed data. Ref. [21] investigated the time-series characteristics of daily stock returns for four companies listed on the Nigerian Stock Market, focusing on data from 2 January 2002 to 31 December 2006. They applied three heteroscedastic models: GJR-GARCH(1, 1), EGARCH(1, 1) and GARCH(1, 1). The companies analyzed in the study included Unilever, Mobil, UBA and Guinness. The return series exhibited several common traits of financial time series, such as leptokurtosis, leverage effects, negative skewness and volatility clustering. The analysis revealed that the GJR-GARCH(1, 1) model yielded the best fit for the data examined.

Ref. [22] explored the efficacy of an asymmetric normal mixture generalized autoregressive conditional heteroskedasticity (NM-GARCH) model using standard benchmark GARCH models. The study results revealed that the NM-GARCH model effectively captures the variations in kurtosis and skewness over time. Furthermore, the study illustrated that implementing the NM-GARCH(1, 1) model with a skewed Student-t distribution enhances the predictive capabilities of volatility models. Ref. [23] analyzed volatility return and financial market risk on the JSE. The modelling process was executed in two phases. Initially, the mean returns were estimated using the ARMA(0, 1) model, followed by the application of several univariate GARCH models, including EGARCH(1, 1), GARCH-M(1, 1), TGARCH(1, 1), and GARCH(1, 1), to assess conditional volatility. The results indicated the presence of leverage and GARCH and ARCH effects in the JSE returns during the study period. Their evaluation of forecasts revealed that the ARMA(0, 1)-GARCH(1, 1) model yielded the most accurate predictions for out-of-sample returns over a three-month horizon. Ref. [24] used univariate and multivariate GARCH models to study the volatility of the JSE index. According to the study’s empirical findings, the leverage impact is clear in the log returns JSE index. Ref. [25] employed GARCH models to investigate the volatility of stock returns on the JSE. Their empirical findings revealed that leverage effects were absent and indicated that stock return volatility is persistent.

Ref. [26] illustrated extending the Beta-t-EGARCH model to a Skew-t model, demonstrating that this model provides a superior fit compared to the GJR-GARCH model, which served as a benchmark. Additionally, ref. [27] developed autoregressive models incorporating exogenous variables, power transformations, and threshold GARCH errors, referred to as ARX-PPTGARCH. The Bayesian method is used to estimate parameters. A model called ARX-GARCH is used to perform a comparative study. The study’s findings showed that the ARX-PPTGARCH model effectively represents key financial features, including heteroscedasticity and asymmetry. In the Australian equities index data study, ref. [28] concluded that the GJR-GARCH model, as established by [10], yields superior results in forecasting volatility. Ref. [29] found similar outcomes in their investigation of daily stock returns in Japan. Ref. [30] examined the impact of the financial crisis on the Malaysian stock market from 2007 to 2009. The outcomes were contrasted by the authors with the volatility that persisted following the Asian Crisis. The findings showed a modest reduction in volatility persistence, accompanied by a notable increase in overall volatility and a slight uptick in the leverage effect. This study utilized GARCH, EGARCH, and GJR-GARCH models for the analysis. Nonetheless, reference [31] examined volatility effects on the capital markets of the BRIC nations during the financial crisis of 2007 to 2009 using GJR-GARCH, GARCH and EGARCH models. The authors discovered that the market responds to volatility shocks more quickly, with asymmetry and less persistence. Ref. [32] analyzed South African stock market volatility surrounding the 2014 global oil crisis. They employed symmetric and asymmetric GARCH models to assess conditional volatility, concluding that the asymmetric GARCH model best captured the behavior of equity returns, including during a crisis period.

Ref. [1] applied asymmetric GARCH models, namely the Exponential-GARCH (EGARCH(1, 1)) and GJR-GARCH(1, 1), to investigate the behavior of returns on the Johannesburg Stock Exchange All Share Index, more particularly about market responses to the news. The findings indicated that the returns follow a fat-tailed and skewed distribution, which contradicts the characteristics of the normal distribution. Furthermore, the findings of their study indicate the GJR-GARCH(1, 1) model with the skewed Student-t distribution as an appropriate framework to model stock returns.

A study by [33] examined stock return–market volatility interaction in the South African and Chinese markets. The volatility of stock returns based on the GARCH model was examined with emphasis on the Johannesburg Stock Exchange FTSE/JSE All Bond Index and the Shanghai Stock Exchange Composite Index. With the data received from January 1998 to October 2014, the research identified high levels of volatility in both markets. Consistent with the prevailing academic literature on stock market return behavior, the findings ascertained that the trends in the two markets tend to experience movements in similar directions.

Ref. [34] examined the effects of economic recessions of South Africa on the financial markets, exacerbated by the COVID-19 pandemic. The results show that the hybrid exponential GARCH(1, 1) model with the SD distribution EGARCH(1, 1)-SD outperformed the GARCH-GPD model for FTSE/JSE ALSI returns at a 2.5% VaR level. The GARCH(1, 1)-SD model for FTSE/JSE Banks Index returns outperformed the GARCH(1, 1)-GPD at 95% and 97.5% VaR levels. Similarly, for FTSE/JSE Mining Index returns, the GARCH(1, 1)-SD model outperformed at the 5% and 97.5% VaR levels.

This subsection reviewed empirical studies on volatility modelling using GARCH models, focusing on the JSE ALSI and other financial markets. The evolution of GARCH models, including variants like EGARCH, GJR-GARCH, and fGARCH, highlights their adaptability in capturing key features such as volatility clustering, leverage effects, and asymmetric shock responses. Stock, commodity, and foreign exchange market applications emphasize the models’ efficacy under diverse market conditions, including crises. The fGARCH model is an appropriate choice for this study due to its ability to capture both volatility clustering and asymmetric responses to shocks, characteristic features observed in financial time series, including the JSE ALSI.

1.3. Research Highlights and Contributions

Based on the literature review in Section 1.2, the following are the contributions of this present study. The ARMA(0, 0)-fGARCH(1, 1) model with the skewed generalized error distribution was found to be the best model for describing JSE ALSI return volatility with evidence of high persistence in volatility and the leverage effect. Secondly, the research highlighted the significant impact of the COVID-19 pandemic on market volatility in determining the sensitivity of financial markets to outside influences and providing a comparative analysis of the FTSE/JSE Top 40, S&P500, and JSE ALLSI indices during the pandemic.

Study highlights of the research are as follows:

• The JSE ALSI returns possessed leptokurtic characteristics with slight leftward skewness, revealing fat tails and asymmetry of the return distribution.
• The study revealed strong persistence in volatility, which means that market shocks remain for a very long time and decline slowly with the passage of time.
• It was found that negative shocks have more effect on increasing volatility than positive shocks, suggesting the presence of a leverage effect in the market.
• S&P500 and FTSE/JSE Top 40 experienced higher early pandemic volatility, with the S&P500 capturing better long-term returns despite short-run volatility spikes. The Top 40 index was riskier before the pandemic but showed comparable rolling volatility after the pandemic.

The study is divided into four sections. Section 2 presents the research methodology. The results and discussion of the empirical results are presented in Section 3, while the study’s conclusion is presented in Section 4.

2. Methodology

2.1. Modelling Framework and Data

Figure 1 shows the flow chart of the modelling framework.

This study uses daily secondary data of the JSE ALSI closing stock prices given in RSA ZAR, which are freely available for use on https://www.wsj.com/market-data/quotes/index/ZA/XJSE/ALSH/historical-prices, (accessed on 7 October 2024). The period of the data ranges from 1 January 2014 to 29 December 2023. The data are collected over a five-day trading week. A comparative analysis is then performed with the JSE Top 40 index, which represents the liquid tradable focus of the South African equity market with a very liquid derivative market. The data for this index can be accessed on https://za.investing.com/indices/ftse-jse-top-40-historical-data (accessed on 15 January 2025). For comparative purposes, we compare our results with a study of the S&P500 index, which represents a developed market. The data can be accessed on https://www.wsj.com/market-data/quotes/index/SPX/historical-prices (accessed on 16 January 2025). For the analysis, daily log returns were utilized and were generated using Equation (1).

$$r_t=100\times \log \left({\displaystyle \frac{Z_t}{Z_{t-1}}}\right),$$

(1)

where $r_t$ is the log return at time t, $Z_t$ is the closing price at time t and $Z_{t-1}$ is the closing price of the previous day.

2.2. Models

This study employs ARCH, GARCH, and fGARCH models to capture the time-varying volatility of the JSE ALSI. The ARCH model captures volatility clustering using past squared residuals, while the GARCH model extends this by including lagged conditional variances. The fGARCH model generalizes these approaches, incorporating asymmetries to account for the leverage effect.

2.2.1. ARCH Model

The ARCH model, proposed by [3], was developed to capture the dynamic nature of return volatility, which varies over time rather than remaining constant. Consequently, rather than depending on standard deviations derived from short- or long-term samples, the ARCH model suggests using weighted averages of previous squared forecast errors, effectively serving as a form of weighted variance [35]. Models that fall within the ARCH classification employ sample standard deviations to derive the conditional variance of returns based on historical data, utilizing a maximum likelihood estimation approach [21].

The formulation of an ARCH(q) model is expressed as follows:

$$r_t={\mu }_t+{\epsilon }_t,$$

(2)

Here, $r_t$ represents the log-returns of the JSE ALSI at time t, while ${\mu }_t$ denotes the conditional mean of these returns, which is modelled using the ARMA($p,q)$ model defined as:

$${\mu }_t=\mu +\sum _{i=1}^p{\varphi }_i{r}_{t-1}+\sum _{i=1}^q{\theta }_i{\epsilon }_{t-i},$$

(3)

${\epsilon }_t\sim N(0,1)$ is the error term (returns shocks or innovations) defined as:

$${\epsilon }_t={\sigma }_t{Z}_t,$$

(4)

where $Z_t\sim iid(0,1)$ with mean zero and variance 1.

The equation representing conditional volatility can be expressed as follows:

$${\sigma }_t^2=\omega +\sum _{i=1}^q{\alpha }_i{\epsilon }_{t-i}^2,$$

(5)

where ${\sigma }_t^2$ is the conditional volatility at time t, $\omega$ is the constant term and ${\alpha }_i$ represents the impact of past squared returns, with constraints $\omega >0$ and ${\alpha }_i\ge 0$ (for all $i=1,\dots ,q$) to ensure that ${\alpha }_t^2$ is positive.

The key limitation of the ARCH model lies in its strict autoregressive structure for conditional variance, which often requires a large number of parameters (long lags) to capture volatility dynamics accurately. This violates the principle of parsimony, leading to overfitting and reduced model simplicity. Additionally, the ARCH(1) model, which relies solely on the previous period’s squared residuals to estimate current variance, fails to account for the varying persistence of shocks, particularly during crises. This shortcoming led to the development of more generalized models, such as GARCH, which better capture volatility persistence and reduce parameter complexity while improving model fit.

2.2.2. Standard GARCH Model

Ref. [4] expanded the ARCH model into GARCH, incorporating elements of an autoregressive moving average (ARMA) model. In GARCH, the conditional variance is influenced by past squared residuals and past variances, reducing overfitting. This makes GARCH the most widely used model for volatility modelling [36]. The GARCH($p,q$) model can typically be represented as follows:

$${\sigma }_t^2=\omega +\sum _{i=1}^q{\alpha }_i{\epsilon }_{t-i}^2+\sum _{j=1}^p{\beta }_j{\sigma }_{t-j}^2,$$

(6)

where p denotes the order of the GARCH component, while q represents the order of the ARCH component. Parameters are subject to specific constraints: $\omega >0$, $0\le {\alpha }_i<1$ for $i=1,\dots ,q$, and ${\beta }_j\ge 0$ for $j=1,\dots ,p$, ensuring the conditional variance ${\sigma }_t^2$ remains positive and ${\alpha }_i$ measures the influence of prior squared returns (ARCH effect), and ${\beta }_j$ captures the influence of past variances (GARCH effect) on ${\sigma }_t^2$. Stationarity of the conditional variance holds if ${\alpha }_i+{\beta }_j<1$. For a GARCH(1, 1) model, the conditional variance equation can be expressed as:

$${\sigma }_t^2=\omega +{\alpha }_1{\epsilon }_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2,$$

(7)

where $\omega >0$, ${\alpha }_1\ge 0$, ${\beta }_1\ge 0$, and ${\alpha }_1+{\beta }_1<1$. The sum ${\alpha }_1$ + ${\beta }_1$ reflects the persistence of volatility shocks, indicating how quickly volatility reverts to its average level. However, the standard GARCH model is limited in capturing asymmetries in volatility responses, such as differential impacts from positive versus negative news events.

2.2.3. The fGARCH Model

The family GARCH (fGARCH) model, developed by [13], is a flexible family of some important symmetric and asymmetric GARCH models as sub-models. The nesting includes the simple GARCH (sGARCH) model, the Absolute Value GARCH (AVGARCH) model, the GJR GARCH (GJRGARCH) model, the Threshold GARCH (TGARCH) model, the Nonlinear ARCH (NGARCH) model, the Nonlinear Asymmetric GARCH (NAGARCH) model, the Exponential GARCH (EGARCH) model, and the Asymmetric Power ARCH (apARCH) model. The sub-model apARCH is also a family model (but less general than the fGARCH model) that nests the sGARCH, AVGARCH, GJRGARCH, TGARCH, NGARCH models, and the Log ARCH model. The fGARCH$(u,v)$ model is stated as:

$${\sigma }_t^{\gamma }=\omega +\sum _{j=1}^v{\alpha }_j{\sigma }_{t-j}^{\gamma }(|z_{t-j}-{\lambda }_{2j}{|-{\lambda }_{1j}\{z_{t-j}-{\lambda }_{2j}\})}^{\delta }+\sum _{j=1}^u{\beta }_j{\sigma }_{t-j}^{\gamma }.$$

(8)

This robust fGARCH model allows different powers for ${\sigma }_t$ and $z_t$ to decompose the residuals in the conditional variance equation. Equation (8) is the conditional standard deviation’s Box–Cox transformation, where the transformation of the absolute value function is carried out by the parameter $\delta$, and $\gamma$ determines the shape. ${\lambda }_{2j}$ and ${\lambda }_{1j}$ control the shifts for asymmetric small shocks and rotations for large shocks, respectively. The fit of the full fGARCH model can be implemented with $\gamma$ = $\delta$.

2.3. Estimation of Parameters

This subsection covers how the best parameter estimations are estimated. Estimates can be obtained using many approaches, such as the moment method (MM), Maximum Likelihood Estimation (MLE), Bayesian Estimation, and Quasi-maximum Likelihood Estimation (QMLE). This study will employ the Maximum Likelihood Estimation (MLE) due to its consistency in numerous scenarios.

The parameters of the family of GARCH models are estimated using the maximum likelihood approach. The specific log-likelihood function is derived from the product of all conditional densities based on the assumption regarding prediction errors [1]. Ref. [12] explored maximum likelihood estimation under the premise that the errors follow generalized error distributions.

The rugarch R statistical package [37] is employed to obtain estimates for maximum likelihood.

2.4. Diagnostics and Evaluation

Model adequacy was evaluated using diagnostic tests. The Ljung–Box Q test was applied to detect residual autocorrelation, while the ARCH Lagrange Multiplier (LM) test assessed remaining heteroskedasticity. The Jarque–Bera test evaluated the normality of residuals. Model selection was based on information criteria, including the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Hannan–Quinn Information Criterion (HQIC), with lower values indicating better fit. These diagnostics confirmed that the selected model adequately captured the volatility dynamics of the JSE ALSI.

2.5. Risk Forecasting (i.e., Value-at-Risk (VaR) and Expected Shortfall (ES))

To further validate the model’s practical application in risk management, Value-at-Risk (VaR) and Expected Shortfall (ES) were computed based on the rolling volatility forecasts.

2.5.1. Value-at-Risk (VaR)

VaR estimates the maximum expected loss at a given confidence level. The 1% VaR is computed as:

$${\mathrm{VaR}}_{0.05}={\mathrm{\Phi }}^{-1}\left(0.95\right)\times {\sigma }_t,$$

(9)

where ${\mathrm{\Phi }}^{-1}\left(0.95\right)$ represents the inverse cumulative normal distribution function at the 95% confidence level.

2.5.2. Expected Shortfall (ES)

ES (also known as Conditional VaR) quantifies the expected loss beyond the VaR threshold:

$${\mathrm{ES}}_{0.05}={\displaystyle \frac{1}{0.05}}{\int }_{-\infty }^{{\mathrm{VaR}}_{0.05}}xf\left(x\right)dx,$$

(10)

This measure provides a more comprehensive assessment of potential losses in extreme market conditions.

2.6. Backtesting VaR Forecasts

The Kupiec likelihood ratio test was employed to assess the reliability of the VaR estimates. This test evaluates whether the proportion of VaR breaches (i.e., cases where actual losses exceed VaR) aligns with the expected failure rate.

The test statistic is defined as:

$$L{R}_{\mathrm{Kupiec}}=-2\ln \left[{\displaystyle \frac{{(1-p)}^{n-x}{p}^x}{{(1-x/n)}^{n-x}{(x/n)}^x}}\right],$$

(11)

where p is the expected failure probability (e.g., 5% for 95% VaR), x is the observed number of VaR breaches and n is the total number of observations. The test follows a chi-square distribution with one degree of freedom. A high test statistic suggests that the VaR model underestimates risk.

3. Empirical Results and Discussion

This section outlines and examines the exploratory data analysis, along with the model fitting process and diagnostics described in Section 2. The R statistical software (R version 4.4.0) is used for data analysis.

3.1. Exploratory Data Analysis

This section examines the structure of the data and its features, offering an overview of the dataset to be modelled. It outlines a clear direction for further data analysis.

3.1.1. Description of Data

Time Series Plots

Time series plots provide an overview of the sample data, illustrating its behaviour and variations across time. Figure 2 shows the JSE ALSI closing stock prices. It exhibits a clear upward trend over the long term, indicating that the JSE ALSI has generally appreciated. This upward trend reflects the overall growth in the South African equity market. There are periods of accelerated growth and intervals where the trend is less pronounced, suggesting varying economic conditions, investor sentiment, and market cycles. Significant volatility is evident in the time series, with noticeable spikes and drops. A marked decline in the index is observed in early 2020, aligning with the onset of the COVID-19 pandemic, which led to substantial economic disruptions and increased market volatility. Following these downturns, the index shows recovery periods, indicating market resilience. The series appears to be non-stationary as it shows a long-term trend and varying levels of volatility over time. Non-stationarity is often observed in financial time series, indicating that the mean and variance change across different periods. This characteristic necessitates differencing.

Figure 3 displays the results of applying Seasonal and Trend decomposition using Loess (STL) on the log-returns of JSE ALSI closing stock prices. The decomposition splits the time series into four components: the original data, the seasonal component, the trend component, and the remainder (residual) component. The top panel exhibits the original log-returns time series data. The data show a consistent fluctuation around zero, with a noticeable spike in volatility around 2020, likely corresponding to a significant market event such as the COVID-19 pandemic. The second panel shows the seasonal component. The seasonal component captures repetitive patterns or cyclical movements within each year. In this case, the seasonal pattern appears relatively weak, with minor fluctuations, indicating that seasonality is not a dominant feature in the log returns of JSE ALSI. This suggests that other factors largely drive the log returns beyond a regular seasonal pattern.

The trend component in the third panel reflects the long-term movement in the log returns, with a noticeable drop around 2020 due to the COVID-19 pandemic, followed by recovery and stabilization. The residual component in the bottom panel shows high volatility during 2020, indicating significant outliers likely due to the pandemic. Beyond these spikes, the residuals likely remain stable, suggesting the STL decomposition effectively captured the trend and seasonal patterns. The JSE ALSI time series data for closing stock prices had no missing values.

Figure 4 depicts the trend component of the return series, which reflects the long-term movement in the log returns. A noticeable drop around 2020, attributed to the COVID-19 pandemic, is observed, followed by a period of recovery and stabilization.

Figure 5a indicates that the daily stock prices exhibit non-stationarity upon visual examination. In contrast, Figure 5b reveals that the logarithmic transformation of stock prices becomes stationary after applying the first differencing. Also, Figure 5b displays the return series, highlighting periods of high volatility and clustering effects. Figure 5c presents a density plot of the return series, which exhibits traits of a leptokurtic distribution. This observation aligns with the findings of the normal Q-Q plot, indicating that the returns do not follow a normal distribution. Finally, Figure 5d illustrates the tails of the quantile–quantile (Q-Q) plot for the daily log return series, which indicates a fat-tailed distribution.

Descriptive Statistics

Descriptive statistics provide insights into the dataset’s characteristics and overall structure. As illustrated in

Table 1. Results of normality and stationarity tests for returns.

Test	Results
KPSS	Test Statistic: 0.022348
	p-value: 0.1
	Decision: Stationary
ADF	Test Statistic: −13.955
	p-value: 0.01
	Decision: Stationary
Jarque-Bera	Test Statistic: 6565
	p-value: $2.2\times {10}^{-16}$
	Decision: Non-Normal

, the results from normality and stationarity tests align with the observations made in the previous plots in Figure 5 at a significance level of 5%.

Table 2. Descriptive statistics of the JSE ALSI closing stock prices.

Statistic	Estimate
Minimum	37,963
Maximum	80,791
1st Quartile	51,912
Median	55,840
Mean	58,725
3rd Quartile	66,351
Std. Deviation	9086.296
Kurtosis	−0.5821364
Skewness	0.7663588
n	2599

shows the summary statistics of JSE ALSI closing stock prices with a maximum value of 80,791 and a minimum value of 37,963 during the sample period. The mean (58,725) is higher than the median (55,840), indicating a right-skewed distribution, while the standard deviation (9086.296) reflects moderate variability around the mean. The skewness of 0.7663588 indicates a moderate rightward skew in the distribution. At the same time, the kurtosis of −0.5821364 suggests a slightly platykurtic distribution, characterized by lighter tails and a flatter peak compared to a normal distribution.

Table 3. Descriptive statistics of return series.

Statistic	Estimate
Minimum	−10.22682
Maximum	7.26147
1st Quartile	−0.51940
Median	0.00000
Mean	0.01956
3rd Quartile	0.61427
Std. Deviation	1.096424
Kurtosis	7.699379
Skewness	−0.5563153
n	2599

summarizes the return statistics, with the minimum and maximum values being −10.22682 and 7.26147, respectively. Although the median and mean are nearly zero, they differ slightly. The returns are centered around zero, displaying a standard deviation of 1.096424. The log-return series exhibits a heavy-tailed distribution, indicated by a kurtosis value greater than 3, as shown in Table 3. Additionally, the skewness coefficient of −0.5563153 reveals that return series data are skewed to the left, suggesting asymmetry. This non-normality in distribution is further supported by the Jarque–Bera test, which confirms deviation from normality in the return series data.

Autocorrelation and Partial Autocorrelation Functions

Figure 6 displays the ACF and PACF plots for the return series. The ACF and PACF indicate significant autocorrelation at lags 6, 7, 8, 18, and 27, suggesting that autocorrelation in the JSE index returns is significant. According to

Table 4. Summary of Box–Ljung Q test for detecting autocorrelation.

Lag	Results
	critical value: 18.30704
10	statistic: 34.347
	p-value: 0.0001613
	critical value: 24.99579
15	statistic: 40.713
	p-value: 0.0003537
	critical value: 31.41043
20	statistic: 59.853
	p-value: $7.505\times {10}^{-6}$
	critical Value: 50.99846
36	statistic: 78.612
	p-value: $5.22\times {10}^{-5}$

, all Box–Ljung statistics exceed their respective critical values, leading to the rejection of the null hypothesis at a significance level of 0.05. Consequently, we conclude that there is significant autocorrelation present in the returns.

Heteroscedasticity

To identify heteroscedasticity, we examine the autocorrelation and partial autocorrelation plots of JSE ALSI squared returns. The plots in Figure 7 reveal significant serial autocorrelation, indicating a strong relationship between past and future volatility. The slow decay of the ACF and the pronounced spikes in the PACF at lower lags suggest long-term dependencies in the volatility. Engle’s ARCH test was also applied to the returns, as shown in

Table 5. Summary of Engle’s ARCH LM test for detecting heteroscedasticity.

Lag	Results
	critical value: 18.30704
10	statistic: 778.48
	p-value: $2.2\times {10}^{-16}$
	critical value: 24.99579
15	statistic: 785.77
	p-value: $2.2\times {10}^{-16}$
	critical value: 31.41043
20	statistic: 792.26
	p-value: $2.2\times {10}^{-16}$
	critical value: 50.99846
36	statistic: 789.05
	p-value: $2.2\times {10}^{-16}$

. This table illustrates that all values from Engle’s Lagrange Multiplier (LM) ARCH test statistics exceed their critical values, leading to the rejection of the null hypothesis at a significance level of 0.05. Therefore, we conclude that heteroscedasticity is present in the returns.

3.2. ARMA(p,q) Model Determination

The mean function is characterized by an ARMA(p,q) model, as outlined in Equation (3). The auto.arima() function from the “rugarch” statistical package is used to select an appropriate mean function, which appears to be ARMA(0, 0). ARMA(0, 0) typically means that the data are assumed to have a constant mean, and no adjustment is made based on past values or shocks to the series.

3.3. Fitting of Volatility Models

3.3.1. Fitting the Standard GARCH(1, 1) Model

The sGARCH(1, 1) model is fitted using various distributions, including the Student’s t-distribution, the skewed Student’s t-distribution, the generalized error distribution (GED), the skewed generalized error distribution (SGED), and the generalized hyperbolic distribution.

Table 6. Information criteria for ARMA(0, 0)-sGARCH(1, 1) with various error distributions.

Information Criteria	M1	M2	M3	M4	M5
AIC	2.7682	2.7651	2.7639	2.7626	2.7633
BIC	2.7795	2.7787	2.7752	2.7761	2.7791
Hannan–Quinn	2.7723	2.7700	2.7680	2.7675	2.7690

indicates that M1 corresponds to the Student’s t-distribution, M2 to the skewed Student’s t-distribution, M3 to the generalized error distribution, M4 to the skewed generalized error distribution, and M5 to the generalized hyperbolic distribution, along with their respective information criteria values. The mean equation, ARMA (0, 0), determined in the previous section, is used when fitting the sGARCH(1, 1) model. The optimal standard GARCH(1, 1) model with a suitable conditional distribution is chosen based on the information criteria.

Table 7. Ranking of different conditional distributions fitted with ARMA(0, 0)-sGARCH(1, 1) from the lowest to the highest value based on their respective information criteria.

No.	AIC	BIC	Hannan–Quinn
1	M4	M3	M4
2	M5	M4	M3
3	M3	M2	M5
4	M2	M5	M2
5	M1	M1	M1

shows the arrangement of different conditional distributions fitted with ARMA(0, 0)-sGARCH(1, 1) from the lowest to the highest value based on their information criteria. This Table 7 is constructed so that we can properly visualize the dominant conditional distribution fitted with ARMA(0, 0)-sGARCH(1, 1) with the lowest values of information criteria. It is obvious that the skewed generalized error distribution (M4) is dominant, indicating that the ARMA(0, 0)-sGARCH(1, 1) model with a skewed generalized error distribution is the most suitable fit for the JSE ALSI returns based on the information criteria.

Estimation of the ARMA(0, 0)-sGARCH(1, 1) Model

Table 8. Estimates of parameters for the ARMA(0, 0)-sGARCH(1, 1) model.

Parameter	Estimate	Std. Error	t-Value	p-Value
$\mu$	0.023106	0.017586	1.3139	0.188869
$\omega$	0.022579	0.008160	2.7670	0.005657
${\alpha }_1$	0.077131	0.013233	5.8286	0.000000
${\beta }_1$	0.903484	0.017303	52.2147	0.000000
skew	0.937336	0.022573	41.5242	0.000000
shape	1.367764	0.053756	25.4440	0.000000

presents the estimated parameters derived from the maximum likelihood estimation method executed through the R software package (R version 4.4.0). The results indicate that the estimated mean is not statistically significant, as its p-value exceeds the significance level 0.05. In contrast, all other estimated parameters are statistically significant since their p-values fall below this threshold, and their respective standard errors are relatively small, suggesting a strong model fit. Additionally, the sum of the estimated parameters $\widehat{{\alpha }_1}+\widehat{{\beta }_1}$ equals $0.980615$, signifying that volatility shocks exhibit a high level of persistence.

Diagnostic Checking of the ARMA(0, 0)-sGARCH(1, 1) Model

Table 9. Results of the Weighted Ljung–Box test on standardized squared residuals.

Lag	Statistic	p-Value
1	0.1414	0.7069
5	1.7295	0.6836
9	3.6470	0.6491

presents the findings from the Ljung–Box test conducted on the standardized squared residuals of the ARMA(0, 0)-sGARCH(1, 1) model. The results indicate that the model’s squared residuals exhibit no remaining autocorrelation, as the p-values exceed the significance level of 0.05 for both tested lags.

The results of the ARCH LM test presented in

Table 10. Results of the Weighted ARCH LM test on standardized residuals.

Lag	Statistic	Shape	Scale	p-Value
3	0.7468	0.500	2.000	0.3875
5	3.2288	1.440	1.667	0.2583
7	3.8298	2.315	1.543	0.3718

indicate that the ARMA(0, 0)-sGARCH(1, 1) model appropriately captures the ARCH effects. Consequently, there are no remaining ARCH effects (heteroskedasticity) in the residuals of the ARMA(0, 0)-sGARCH(1, 1) model, as both p-values exceed the significance level of 0.05.

The plot of the estimated conditional standard deviations depicted in Figure 8 illustrates the presence of volatility clustering. However, when examining the time series of standardized squared residuals in Figure 9 (the second plot of the panel), it is evident that they tend to stabilize with minimal clustering. A Q-Q plot is presented in Figure 10 to evaluate the standardized residuals’ normality. This plot indicates a deviation from a straight line, suggesting that the standardized residuals do not follow a normal distribution. Additionally, the p-value obtained from the Jarque–Bera test is $2.2\times {10}^{-16}$, significantly lower than 0.05, leading to the rejection of the normality assumption for the standardized residuals. The ACF of the standardized squared residuals shown in Figure 9 (the third plot of the panel) reveals no signs of autocorrelation, except at lag 13. Similar findings were also reported by the Box–Ljung test conducted above.

3.3.2. fGARCH(1, 1) Model Fitting

The Student’s t-distribution, skewed Student’s t-distribution, generalized error distribution, skewed generalized error distribution, and generalized hyperbolic distribution are fitted with the ARMA(0, 0)-fGARCH(1, 1) model and

Table 11. Information criteria for ARMA(0, 0)-fGARCH(1, 1) with various error distributions.

Information Criteria	M1	M2	M3	M4	M5
AIC	2.7348	2.7288	2.7321	2.7274	2.7280
BIC	2.7528	2.7492	2.7501	2.7477	2.7506
Hannan–Quinn	2.7413	2.7362	2.7386	2.7348	2.7362

shows the information criteria obtained. Similarly, as we have discussed in the above subsection,

Table 12. Ranking of different conditional distributions fitted with ARMA(0, 0)-fGARCH(1, 1) from the lowest to the highest value based on their respective information criteria.

No.	AIC	BIC	Hannan–Quinn
1	M4	M4	M4
2	M5	M2	M5
3	M2	M3	M2
4	M3	M5	M3
5	M1	M1	M1

shows the ranking of different conditional distributions fitted with ARMA(0, 0)-fGARCH(1, 1) from the lowest to the highest value based on their respective information criteria in Table 11. Based on the outcomes presented in Table 12, the skewed generalized error distribution seems to be a dominant conditional distribution; thus, the ARMA(0, 0)-fGARCH(1, 1) with skewed generalized error distribution is the best-fitting fGARCH model. The fGARCH model was fitted with the ALLGARCH model as a submodel.

3.4. Overall Best Fitting Model

Table 13. Information criteria for ARMA(0, 0)-sGARCH(1, 1) and ARMA(0, 0)-fGARCH(1, 1) models using the skewed generalized error distribution.

Model	AIC	BIC	Hannan–Quinn
ARMA(0, 0)-sGARCH(1, 1) with SGED	2.7626	2.7761	2.7675
ARMA(0, 0)-fGARCH(1, 1) with SGED	2.7274	2.7477	2.7348

presents the ARMA(0, 0)-sGARCH(1, 1) and ARMA(0, 0)-fGARCH(1, 1) models, both fitted using the skewed generalized error distribution, along with their respective information criteria. The results indicate that the ARMA(0, 0)-fGARCH(1, 1) model exhibits the lowest information criteria when compared to the ARMA(0, 0)-sGARCH(1, 1) model. This suggests that the ARMA(0, 0)-fGARCH(1, 1) model with SGED is the most suitable for capturing the JSE ALSI returns’ volatility dynamics.

The other reason why the ARMA(0, 0)-fGARCH(1, 1) model outperforms the ARMA(0, 0)-sGARCH(1, 1) model is that the fGARCH model is more flexible in capturing asymmetries in the volatility process, such as leverage effects, where negative returns might lead to higher volatility than positive returns of the same magnitude. This flexibility allows it to better model the behavior of financial time series that exhibit such characteristics.

The fGARCH model can incorporate various forms of conditional heteroskedasticity, such as the EGARCH, ALLGARCH or TGARCH models. These can be more adept at capturing the actual volatility dynamics of the JSE ALSI, which might be influenced by factors like sudden market shifts or changes in investor sentiment, and the fGARCH model often provides better out-of-sample volatility forecasts than the sGARCH model.

3.5. Model Diagnostics for the Overall Best-Fitting Model

Figure 11 illustrates volatility clustering in the fitted standardized residuals. Additionally, the plot of standardized squared residuals in Figure 12 demonstrates minimal volatility clustering with stable behavior. The ACF plot for standardized squared residuals, shown in Figure 13, indicates no significant autocorrelation except at lag 13. The Box–Ljung test was conducted on the standardized squared residuals to confirm these results. As seen in

Table 14. Weighted Ljung–Box test on standardized squared residuals.

Lag	Statistic	p-Value
1	0.00383	0.9507
5	1.28506	0.7925
9	3.16065	0.7322

, p-values at all tested lags exceed the significance threshold of 0.05, suggesting the absence of residual autocorrelation.

Table 15. Weighted ARCH LM test on standardized residuals.

Lag	Statistic	Shape	Scale	p-Value
3	0.5065	0.500	2.000	0.4767
5	2.5569	1.440	1.667	0.3608
7	3.2562	2.315	1.543	0.4665

presents the ARCH LM test results for the ARMA(0, 0)-fGARCH(1, 1) model, where p-values at tested lags also exceed 0.05, indicating no remaining heteroskedasticity in the fitted model.

Table 16. Results of the sign bias test.

Test	Statistic	p-Value
Sign bias	0.5814	0.56101
Negative sign bias	0.2212	0.82498
Positive sign bias	1.8559	0.06358
Joint effect	3.8638	0.27656

displays the outcomes of the sign bias test, which assesses asymmetries in the volatility model to determine if positive and negative shocks have varying impacts on subsequent volatility. The p-values for the sign bias, negative sign bias, positive sign bias, and the joint effect exceed 0.05, indicating a meaningful response to positive and negative shocks. The ARMA(0, 0)-fGARCH(1, 1) model with SGED effectively reflects the leverage effect in the JSE ALSI returns and successfully captures volatility clustering.

The Nyblom stability test assesses the parameter’s stability over time. A stable model should have constant parameters throughout the sample period.

Table 17. Nyblom test for parameter stability.

Parameter	Estimate
$\mu$	0.4475
$\omega$	0.9950
${\alpha }_1$	0.5830
${\beta }_1$	0.6689
${\lambda }_{11}$	0.5457
${\lambda }_{21}$	0.5984
$\gamma$	0.6204
$\mathrm{skew}$	0.2675
$\mathrm{shape}$	0.2930

shows that the parameters are stable in the model, with both ARCH and GARCH effects being significant. The 1% significance level sets critical thresholds at 0.75 for individual parameters and 2.82 for the joint parameters. For this model, the joint statistic is 2.7179, below the critical value. Among individual parameters, the Nyblom stability test indicates that $\omega$ shows temporal variation while other parameters remain stable.

Table 18. Estimated parameters.

Parameter	Estimate	Std. Error	t-Value	p-Value
$\mu$	−0.008445	0.016695	−0.50585	0.612960
$\omega$	0.023173	0.002651	8.74128	0.000000
${\alpha }_1$	0.070950	0.019066	3.72123	0.000198
${\beta }_1$	0.865027	0.000470	1841.50524	0.000000
${\lambda }_{11}$	0.189341	0.082816	2.28629	0.022237
${\lambda }_{21}$	0.976316	0.109650	8.90396	0.000000
$\gamma$	1.351223	0.228573	5.91156	0.000000
skew	0.905463	0.022789	39.73301	0.000000
shape	1.469429	0.058833	24.97611	0.000000

lists the estimated parameters for the ARMA(0, 0)-fGARCH(1, 1) model with SGED, along with their respective p-values. The results in Table 18 reveal that all parameter estimates, except for the mean (with a p-value above 0.05), are statistically significant, with p-values below 0.05.

Figure 14 demonstrates that the empirical density better fits the skewed generalized error distribution (SGED) compared to the normal density. The Q-Q plot for SGED illustrates that most data points align closely with the fitted SGED line, though some deviation appears in the tails.

3.6. Impact on Volatility

The news impact curve presented in Figure 15 illustrates that negative news has a more pronounced effect on volatility than positive news, with volatility responding more sharply to negative shocks. The estimated impact for positive shocks is $\widehat{{\alpha }_1}=0.070950$, while for negative shocks, it is $\widehat{{\alpha }_1}+\hat{\gamma }=1.422173$. The positive value of $\hat{\gamma }=1.351223$ suggests that negative events influence volatility more than positive ones. Additionally, the high persistence in the JSE ALSI returns, with a value of 0.935977, indicates a prolonged period for volatility to subside following a shock.

The plot in Figure 16 compares the time-varying conditional variance (blue line) with the long-run unconditional variance (red dashed line) for the JSE ALSI. The conditional variance fluctuates significantly over time, indicating that market volatility is not constant but dynamic, responding to recent events. Notably, there are clear spikes in the conditional variance, with the most prominent peak occurring around 2020, likely corresponding to a significant market event (possibly the COVID-19 pandemic), which caused a sharp increase in volatility. This suggests that during periods of bad news, volatility rises sharply, reflecting the heightened uncertainty and risk in the market.

In contrast, the unconditional variance, represented by the red dashed line, remains relatively flat. This constant variance represents the long-term average level of volatility expected in the market. Considering that the unconditional variance is much lower than the conditional variance during the 2020 spike, this indicates that the market’s reaction to the shock was temporary, with volatility eventually expected to return to more normal levels.

Throughout the sample period, the conditional variance exhibits volatility clustering, where periods of high volatility tend to follow one another, as seen between 2016 and 2021. These fluctuations suggest that risk in the JSE ALSI varies significantly depending on market conditions, but over the long run, the market tends to revert toward the lower unconditional variance.

Overall, the plot demonstrates that while short-term volatility (conditional variance) can deviate significantly due to market shocks, the long-run volatility (unconditional variance) remains stable, suggesting that the market eventually returns to its historical volatility levels once the effects of shocks dissipate.

3.7. Out-of-Sample Analysis Using Ugarchroll

In this analysis, we utilized the ugarchroll function to fit the ARMA(0, 0)-fGARCH(1, 1) model to the dataset. The ugarchroll function performs a rolling forecast, where the model is re-estimated at each step using a moving window approach. This allows us to generate out-of-sample forecasts for volatility and returns, providing a dynamic risk assessment over time. These forecasts are essential for understanding how the model predicts future volatility in evolving market conditions. By utilizing rolling forecasts, we can observe the model’s predictive performance outside the in-sample data and evaluate its ability to forecast real-time volatility and risk measures.

3.7.1. Out-of-Sample Volatility Forecasts from ARMA(0, 0)-fGARCH(1, 1)

Table 19. Rolling forecast results.

Forecast Horizon	Sigma (Forecasted Volatility)	Realized (Actual Returns)
1	0.6746	0.2235
2	0.6527	−0.0151
3	0.6491	0.2188
4	0.6287	0.1618
5	0.6130	−0.0513
6	0.6139	0.0000
7	0.6107	1.7764
8	0.5653	−1.2568
9	0.6996	−1.8114
10	0.8961	−0.1621

shows the results of the rolling forecast for the forecasted volatility and realized returns over a 10-day horizon. The forecasted volatility follows a declining trend at first, but it increases sharply on the 9th and 10th days, capturing increased market uncertainty. However, the realized returns are extremely volatile with sudden increases and decreases, i.e., a huge increase on day 7 with a value of 1.7764 and a sharp decline on days 8 and 9 with values of $-1.2568$ and $-1.8114$, respectively. Figure 17 plots this relation, showing that while the model captures general volatility patterns, it does not predict extreme return movements. The noted deviations show that the fGARCH(1, 1) model successfully accounts for the persistence of volatility but fails to satisfactorily account for sudden market shocks.

3.7.2. Value at Risk (VaR) and Expected Shortfall (ES)

The fGARCH-based Value at Risk (VaR) values presented in

Table 20. fGARCH-based Value at Risk (VaR) and Expected Shortfall (ES) at 5% confidence level.

Observation Index	fGARCH VaR (5%)
1	−1.196
2	−1.158
3	−1.151
4	−1.116
5	−1.088
6	−1.090
7	−1.084
8	−1.005
9	−1.240
10	−1.584
Expected Shortfall (ES) at 5%	−2.178

reflect the 5% quantile of the forecasted return distribution over the sample period. The VaR series indicates the potential loss at the 5% confidence level for each observation, which means that, under normal market conditions, the loss is expected to exceed these values only 5% of the time. For instance, on the first observation, the VaR is $-1.196$, implying that there is a 5% probability that the return could be worse than this value.

The Expected Shortfall (ES), calculated as the mean of the losses exceeding the VaR, is $-2.178$. This suggests that when the return falls below the VaR threshold, the loss tends to be −2.178, representing the magnitude of risk exposure beyond the VaR. The combination of VaR and ES provides a comprehensive view of potential losses, with VaR highlighting the threshold of extreme risk and ES quantifying the average severity of those extreme losses. These measures are critical for risk management, helping to gauge potential losses in adverse market conditions.

Table 21. VaR Backtesting Results.

Test	Value
Failure Rate (VaR Exceedances)	0.0527
Kupiec Test Statistic	0.1221
Christoffersen Test Statistic (Simplified)	$3.16\times {10}^{-8}$

showed the failure rate of 5.27% is slightly above the expected 5% threshold for a 95% Value at Risk (VaR), indicating that the VaR model slightly underestimates risk. The Kupiec test statistic, which evaluates the accuracy of the exceedance frequency, is relatively small, suggesting no strong evidence of model miscalibration. The Christoffersen test statistic, which checks for clustering in exceedances, is extremely low, implying that exceedances occur independently over time, a desirable property in risk modelling.

3.8. Comparative Analysis with the JSE Top 40 and the S&P500 Indices

This section presents a comparative analysis of the FTSE/JSE ALLSI with the FTSE/JSE Top 40 index, representing the liquid tradable focus of the South African equity market. For completeness of the study, we also compare the study with the S&P500 index, which represents indices from developed markets. All the analysis in this section was performed using the R package ‘PerformanceAnalytics’ developed by [38].

A time series plot of the returns of the three indices is given in Figure 18, top panel. Both the FTSE/JSE Top 40 and the S&P500 indices showed increased volatility at the onset of the COVID-19 pandemic, and later, the JSE ALLSI showed similar increases in volatility. A summary of the distribution of the returns is given in Figure 18, bottom panel, which shows the box plots of the three indices. The Top 40 index shows a wider interquartile range, implying inconsistent performance of the index, while the S&P500 has the smallesr index. However, the S&P500 has longer upper and lower whiskers, suggesting the presence of high returns and low returns. However, it has a short box, which suggests a stable period.

Scatter plots of the pairs of indices are shown in Figure 19. Visual inspection of the three pairs of plots shows scattered distributions, suggesting a very weak correlation between the pairs of indices.

In Figure 20, the top panel shows how an investment would have grown over time based on the historical returns for each of the three indices. The S&P500 shows the steepest upward trend, followed by the Top 40 returns. This helps investors to see which index provides the best long-term returns. The bottom panel of Figure 20 shows the variability of the returns over a 30-day rolling period. The S&P500 shows a significant spike in volatility during the onset of the COVID-19 pandemic, which suggests increased risk, which was short-lived. However, the Top 40 appears to be riskier than the JSE ALLSI before the COVID-19 pandemic, but the rolling volatility appears to be the same after the pandemic.

4. Conclusions

This study modelled the volatility of JSE ALSI returns that exhibited leptokurtic and slightly leftward skewness, using sGARCH(1, 1) and fGARCH(1, 1) models with five assumptions of error distribution. The best-fitting mean model, identified as ARMA(0, 0) using the auto.arima() function, was combined with these volatility models. ARMA(0, 0)-fGARCH(1, 1) with skewed generalized error distribution provided the best fit based on information criteria. The primary findings indicated high persistence in volatility, i.e., shocks to the market take a long time to dissipate and the volatility decays very slowly over time. It also indicated high autocorrelation, heteroscedasticity, and a leverage effect, i.e., negative shocks had a greater impact on volatility than positive shocks. The COVID-19 pandemic had a large impact on volatility, indicating the responsiveness of the market to exogenous shocks for the three indices.

To test the model’s predictive capability, we used the ugarchroll function to perform a rolling forecast with the JSE ALLSI index in order to make out-of-sample volatility and return forecasts. This enabled dynamic risk assessment over time and provided a sense of how the model reacts to evolving market conditions. Rolling forecasts indicated that while the ARMA(0, 0)-fGARCH(1, 1) model is able to capture general volatility trends, it is not effective at forecasting extreme return movements, suggesting limitations in capturing abrupt market shocks.

A comparative analysis between the FTSE/JSE Top 40, S&P500 and JSE ALLSI return indices showed different trends in the COVID-19 pandemic period. Higher volatility in the FTSE/JSE Top 40 and S&P500 was observed at the start of the pandemic, followed by the same trend in the JSE ALLSI. The box plots showed that the Top 40 index had a broader interquartile range, suggesting greater variability in performance, but the S&P500 had more compact boxes with longer whiskers, indicating stability as well as the presence of extreme returns. Scatter plots showed low correlations between the indices, and comparing cumulative growth showed the S&P 500 delivered the best long-term returns despite its short-term volatility burst during the pandemic. The Top 40, on the other hand, was riskier before the pandemic; however, the rolling volatility appeared comparable in the post-pandemic environment. Future research can address these limitations by employing more advanced methods such as artificial neural networks, GAS models, or multivariate GARCH models to examine time-varying relations and sectoral dependencies.