Download - Simulating South African equity Index returns using GARCH models

Simulating South African equity Index returns using GARCH models

Niël Oberholzer* and Pierre J Venter

University of Johannesburg Department of Finance and Investment Management

*Corresponding author [email protected]

Abstract

The purpose of this paper is to simulate the GARCH processes obtained from fitting different

GARCH models to the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the logarithmic

returns of the US-Dollar South African Rand exchange rate. The results obtained for the

FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the USD/ZAR exchange rate indicate

that log returns are mean reverting with signs of volatility clustering. The datasets of the

FTSE/JSE All Share Index, FTSE/JSE Top 40 Index are slightly negatively skewed with signs

of leptokurtosis. The results obtain from the volatility simulation process, via the fitting of the

GARCH (1,1) and GARCH (2,1) models, to the log return datasets, indicate no similarity in

the results, even though the model assumption for both GARCH models regarding errors being

independent, was satisfied.

Keywords: GARCH (1,1), GARCH (1,2), All Share, Top 40, USD/ZAR, volatility, simulation

1. Introduction

Volatility is a key variable in financial applications and modelling. The estimation, calculation

and modelling of volatility is a critical topic of research for academics, policy makers and

market practitioners alike. Volatility is a measure of risk as it measures the size of errors,

disbursements, made in the modelling of asset returns. The greater the volatility in financial

markets the greater the impact on the stability of financial markets and the global economy

(Yu, 2002, Engle et al., 2007).

As a theoretical concept volatility is of importance in the pricing of derivatives instruments,

the estimations of market and financial risk, risk management, derivative pricing and hedging,

market making, market timing, portfolio selection and many other financial activities. In each

of these the computation, modelling and predictability of volatility is a prerequisite (Minkah,

2007, Engle et al., 2007).

In finance the common model for logarithmic returns of asset is the normal distribution.

However, volatility clustering and leptokurtosis or fat tails, are seen as common violations of

the assumption of normality of data distribution of financial time series (Mandelbrot, 1963,

Minkah, 2007). The non-normality of financial data regarding assets returns are strongly

rejected by Mandelbrot (1963) which in turn inference that financial return processes behave

like non-Gaussian stable processes.

Resulting from better dealing and information systems financial time series data are often

available as high frequency data. According to Harris and Sollis (2003) this high frequency

data display properties of long-memory in other words the data reflects the presence of

statistically significant correlations between observations that are not close together.

Financial time series data displays certain important features and characteristics when used in

the calculation of volatility, (1) the leverage effect as descript by Black (1976), and (2)

heteroskedasticity, variance or changes in volatility, of the data. The structures of variance

cannot be explain by simple linear models like to random walk (RW) or the ordinary least

squares (OLS). The random walk model postulates that each price made takes a random step

away from the previous price or asset value and the each step is independently and identically

distributed in size iid(N~(0,σ)). The random walk model normally uses t-1 value to estimate

volatility at time t0. (Gokbulut and Pekkaya, 2014).

The ordinary least squares method extend the principal of the random walk model, under the

assumption a stationary mean, implying that volatility forecasts depends on the long-term

average of past volatility observations. The heteroskedasticity in equity returns is explained by

the examination of the behaviour of the conditioning information variables in relationship to

the ARCH effects (Bollerslev, 1986, Engle and Ng, 1993, Gokbulut and Pekkaya, 2014).

The GARCH model is a dependent model as it does not have any independent random

variables. This implies that in GARCH models variance changes over time as a computed

function of past squared deviations from the mean and past variances (Johnston and Scott,

2000).

The purpose of this paper is to simulate the GARCH processes obtained from fitting different

GARCH models (Bollerslev, 1986) to the (1) FTSE/JSE All Share Index, (2) FTSE/JSE Top

40 Index and (3) the US-Dollar South African Rand exchange rate logarithmic returns. The

simulated process will then be visually compared to the actual returns.

The remainder of the paper will be structured as follows, part 2 literature review, Part 2 the

date and part 3 for the methodology used in the research paper. Part 4 will be the description

of results and part 5 will be the conclusion of the study.

2. Data

This study uses daily data for the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the

US-Dollar Rand (USD/ZAR) exchange rate for the period 4 January 2010 until 31 December

2014, 1248 observations were included in this study. All the datasets were obtained from

Thomson Reuters Eikon. The Indices and the exchange rate were converted to their logarithmic

returns.

3. Methodology

By employing a similar method to Dukich et al.(2010), a GARCH (1,1) and a GARCH (2,1)

model was fitted to each of the returns series. Thereafter residual diagnostics were performed

in order to determine which of the two models fitted mostly reflects the empirical nature of the

returns series. Finally, simulated GARCH (1,1) and GARCH (2,1) processes with the

respective coefficients were simulated and visually compared to the actual logarithmic returns

for each variable included in the study.

With regards to model specification, according to McNeil et al. (2006), the GARCH (1,1) and

GARCH(2,1) models are specified as follows:

2 2 2

1 1 1t t tu

2 2 2 2

1 1 2 2 1t t t tu u

4. Results

The results of the study will be divided in to three section. Each time series results will be

discussed in full before considering the results obtained from the next time series. The sequence

of result to be discusses will be the FTSE/JSE All Share Index then the FTSE/JSE Top 40 Index

and lastly the US-dollar South African Rand exchange rate results.

4.1 FTSE/JSE All Share Index

Figure 1 below display the time series and histogram for the FTSE/JSE All Share Index. The

log returns are mean reverting and show signs of volatility clustering. In addition, the histogram

shows that the log returns are slightly negatively skewed and leptokurtic (fat tails).

Figure 1: Log returns for FTSE/JSE All Share Index

Source: Researchers’ analysis

The auto correlation function (ACF) indicates the correlations for a time series between the

series xt and it’s successive. Thus, the ACF is a set of correlations between the series and lags

of itself over time. The ACF indicates the possible structure of the time series data. The optimal

results for the ACF is that there is not and significant correlations for any of the lags.

The partial autocorrelation function (PACF) measures the amount of correlation between a

variable and a log of itself that is not explained by correlations at all lower-order-lags.

The ACF plot as indicate in Figure 2 below, indicates that there appears to be serial correlation

among the log returns for the FTSE/JSE All Share Index returns. The result obtained for the

PACF, for the FTSE/JSE All Share Index returns, indicates that there is a long-term

dependence among the observations, as can be seen in Figure 2 below.

Figure 2: ACF and PACF results for the FTSE/JSE All Share Index


In Figure 3 below, the ACF-plots of residuals and squared residuals for the two GARCH

models, GARCH (1,1) and GARCH (1,2), are shown. For the ACF plot of the residuals of

GARCH (1,1) and GARCH (1,2) only two observation at lag 2 and 18 are outside the two

standard deviation of the autocorrelation sample. As only two lag residuals are outside the two

standard deviations of the autocorrelation sample the supposition of independent errors are

accepted.

Figure 3: FTSE/JSE All Share Index ACF-plots of residuals and squared

residuals for the two GARCH models


Figure 4 below indicated the normal probability plots and histogram results for the normality

assumption of the errors. As shown for both GARCH models the errors reflect normality, which

implies that the residuals will follow an approximate normal distribution.

Figure 4: Normal probability plots and histograms for the fitted GARCH models

for the FTSE/JSE All Share Index


For further analysis, we examine the skewness and kurtosis of the residuals and finally perform

the Ljung-Box test in order to determine whether the residuals are independent. The histograms

and normal probability plots show that the residuals are approximately normal. The skewness

shows that the residuals are slightly negatively skewed with tails not too much heavier than the

normal distribution, as indicate in Table 1 below. Moreover, the Ljung-Box test indicates that

the residuals are independently distributed, as illustrated in Table 2 below.

Table 1: Skewness and Kurtosis

Skewness Kurtosis

JALSH -0.1108077 1.544576

GARCH(1,1) -0.2413688 0.6834049

GARCH(2,1) -0.2248144 0.7005966


Table 2: Ljung-Box Test

X-squared p-value

GARCH(1,1) 1.1134 0.2913

GARCH(2,1) 0 0.9987


Table 3 below lists the coefficients of the different fitted GARCH models. All three coefficients

of the GARCH (1,1) model are statistically significant at a five percent level of significance.

Furthermore, the sum of the alpha and beta coefficients is close to unity, this suggests that a

shock to the FTSE/JSE All Share Index will be persistent (Brooks 2014). Moreover, when the

GARCH (2,1) coefficients are considered, the intercept, alpha2 and beta coefficients are

statistically significant at a five percent level.

Table 3: GARCH coefficient(s) FTSE/JSE All Share

GARCH(1,1) Estimate t value

Omega 1.58E-06 2.815**

Alpha 0.0698 5.482***

Beta 0.9141 56.138***


Omega 1.80E-06 2.965**

Alpha1 0.0301 1.496

Alpha2 0.0454 2.046**

Beta 0.9059 53.061***

*(**) [***]: Statistically significant at a 10(5)[1] % level


Figure 5 below indicate, for purpose of evaluation, the similitude data results for the logged

returns for the FTSE/JSE All Share Index and the two fitted GARCH models. The model

assumption for both GARCH models regarding errors being independent is satisfied. However,

the simulated volatility for both the GARCH (1,1) and GARCH(1,2) does not fit the log returns

series for the FTSE/JSE All Share Index.

Figure 5: Simulated GARH and log return plots for the FTSE/JSE All Share Index


4.2 FTSE/JSE Top 40 Index

The log returns for the values of the FTSE/JSE Top 40 Index indicate large variability as

indicate in Figure 6 below. The large variability indicated the presence of volatility clustering

around the mean. The plot for the FTSE/JSE Top 40 Index is similar to the log return plot of

the FTSE/JSE All Share Index as can be seen in Figure 1 above. The histogram of the log

returns for the FTSE/JSE Top 40 Index is very slightly negatively skewed and leptokurtic (fat

tails).

Figure 6: Log returns for FTSE/JSE Top 40 Index


The ACF plot for the FTSE/JSE Top 40 Index as illustrate in Figure 7 indicate serial correlation

among the log returns. The PACF plot, indicates that there is a long-term dependence among

the observations.

Figure 7: ACF and PACF results for the FTSE/JSE Top 40 Share Index


The ACF-plot for the FTSE/JSE Top 40 Index residuals for the GARCH (1,1) and

GARCH(1,2) indicate that all values are within the two standard deviation autocorrelation

sample except for lag 2 and 18, as indicated in Figure 8 below. These result obtained is similar

to the residuals for the FTSE/JSE All Share Index as indicated in Figure 3 above. The ACF

squared for the GARCH (2,1) model indicate that lags 3, 7, 8, 16,18,21,27 and 28 are outside

of the two standard deviation autocorrelation sample. This implies that the squared residuals

show profound serial correlation.

Figure 8: FTSE/JSE Top 40 Index ACF-plots of residuals and squared residuals

for the two GARCH models


The normal probability plot and histogram results for the normality assumption of the errors of

the GARCH (1,1) reflect normality. However, the plot and histogram for the GARCH (1,2)

reflects insignificant negative skewness and slightly more peakedness (leptokurtic) than the

histogram for the GARCH (1,1) model. These observations are shown in Figure 9 below.

Figure 9: Normal probability plots and histograms for the fitted GARCH models

for the FTSE/JSE Top 40 Index


The histograms and normal probability plots show that the residuals are approximately normal

as indicated in Figure 9 above. Table 4 below indicates that the residuals are slightly negatively

skewed with kurtosis values that is close to normal. The results of the Ljung-Box test, Table

5, reflects that the residuals are independently distributed


Skewness Kurtosis

JTOPI -0.060748 1.51019

GARCH(1,1) -0.195276 0.7175

GARCH(2,1) -0.108237 0.85735



X-squared p-value

GARCH(1,1) 1.5348 0.2154

GARCH(2,1) 1.8488 0.1739


The GARCH coefficients of the different fitted GARCH models for the FTSE/JSE Top 40

Index is displayed in Table 6 below. The three coefficients for the GARCH (1,1) model are all

statistically significant at a five percent level of significance. The sum of the alpha and beta

coefficients is close to unity, this suggests that a shock to the FTSE/JSE All Share Index will

be persistent Brooks (2008).

Table 6: GARCH coefficient(s) FTSE/JSE Top 40 Index


Omega 1.87E-06 2.802***

Alpha 0.069 5.409***

Beta 0.915 56.224***


Omega 1.64E-05 4.952***

Alpha1 2.11E-09 0.0131

Alpha2 0.1770 5.394***

Beta 0.6740 13.501***



The results obtained for the GARCH (1,1) and GARCH (1,2) simulation for the FTSE/JSE Top

40 Index does not fit the log return series for the Index, as indicated in Figure 9.

Figure 9: Simulated GARH and log return plots for the FTSE/JSE All Share

Index FTSE/JSE Top 40 Index


4.3 USD/ZAR

Figure 10 display the log returns of the USD/ZAR exchange rate. The log returns are mean

reverting with signs of volatilty clustering. The frequncy histogramme is slightle positively

skewed.

Figure 10: Log returns for the USD/ZAR


In Figure 11 below the ACF-plot and PACF –plot results for the USD/ZAR exchange rate can

be seen. The plot results indicates that there appears to be serial correlation among the log

returns for the USD/ZAR exchange rate. The result obtained for the PACF, for the USD/ZAR

exchange rate, indicates that there is a long-term dependence among the observations.

Figure 11: ACF and PACF results for the USD/ZAR


The results for the ACF-plots residuals and squared residuals for the USD/ZAR indicate that

no autocorrelation values are within the two standard deviations autocorrelation sample. The

hypothesis of independent errors are accepted.

Figure 14: The ACF-plots of residuals and squared residuals for the two

GARCH models


Figure 12 below indicated the normal probability plots and histogram results for the normality

assumption of the errors for the USD/ZAR time series. The results obtained for both GARCH

models reflect normality, which implies that the residuals will follow an approximate normal

distribution

Figure 12: Normal probability plots and histograms for the fitted GARCH

models for the USD/ZAR


As shown in Figure 12 above the histograms and normal probability plots show that the

residuals are approximately normal. The normal distribution of the USD/ZAR time series is

slightly positive skewed as reflected in Table 12. The tails of the distribution is not much

heavier than those of a normal distribution. The result for the Ljung-Box test reflects that the

residuals are independently distributed, as illustrated in Table 8 below.


Skewness Kurtosis

ZAR 0.2026153 2.12348

GARCH(1,1) 0.2797318 0.7772856

GARCH(2,1) 0.2853443 0.8379899



X-squared p-value

GARCH(1,1) 1.1839 0.2766

GARCH(2,1) 0.0146 0.9039


The coefficients of the fitted GARCH models for the USD/ZAR time series are indicated in

Table 8 below. All three coefficients of the GARCH (1,1) model are statistically significant at

a five percent level of significance. The result obtained by the summation of alpha and beta

indicate unity, implying that that a shock to the USD/ZAR exchange rate will be persistent.

Furthermore, when the GARCH (2,1) coefficients are considered, the intercept, alpha2 and beta

coefficients are statistically significant at a five percent level.

Table 8: GARCH coefficient(s) USD/ZAR


Omega 1.42E-06 2.841***

Alpha 0.063 6.26***

Beta 0.919 69.681***


Omega 1.94E-06 2.862***

Alpha1 1.93E-02 0.0168

Alpha2 0.0571 2.502***

Beta 0.8990 48.63***



Figure 13 below reflect the simulation results for the logged returns for the USD/ZAR

exchange rate and the two fitted GARCH models. The simulated volatility for both the GARCH

(1,1) and GARCH(1,2) does not fit the log returns series for the USD/ZAR exchange rate, even

though, the model assumption for both GARCH models regarding errors being independent is

satisfied.

Figure 13: Simulated GARH and log return plots for the USD/ZAR


5. Conclusion

The purpose of this paper was to simulate the GARCH processes obtained from fitting different

GARCH models to the FTSE/JSE All Share Index, FTSE/JSE Top 40 Index and the US-Dollar

South African Rand exchange rate logarithmic returns. The results obtain for the FTSE/JSE

All Share Index, FTSE/JSE Top 40 Index and the USD/ZAR exchange rate indicate that log

returns are mean reverting with signs of volatility clustering. The datasets of the FTSE/JSE All

Share Index, FTSE/JSE Top 40 Index are slightly negatively skewed with signs of

leptokurtosis. This result is opposing the results of the USD/ZAR that is slightly positively

skewed.

The time series datasets for all the variables reflects serial correlation with a long-term

dependence amongst the variables. The ACF and PACF plots for the log returns for the

variables reflects the presents of auto correlation for certain of the lag periods. As certain of

the histogram values falls outside the two standard deviation parameter. The ACF plots of the

residuals for GARCH (1,1) and GARCH (2.1) for all the variables reflect normality under the

normality assumption of the errors.

The results obtain from the volatility simulation process, via the fitting of the GARCH (1,1)

and GARCH (2,1) models, to the log return datasets, indicate no similarity in the results. Even

though, the model assumption for both GARCH models regarding errors being independent

was satisfied.

References

BLACK, F. Studies of stock price volatility changes. Proceedings of the 1976 Business

Meeting of the Business and Economics Statistics Section. American Statistical

Association, 1976 Washington, DC., 177-181.

BOLLERSLEV, T. 1986. Generalised Autoregressive Conditional Hetroskedasticity. Journal

of Econometrics, 3, 307-327.

BROOKS, C. 2008. Introductory Econometrics for Finance, United Kingdom, University

Press, Cambridge.

DUKICH, J., KIM, K. Y. & LIN, H.-H. 2010. Modeling Exchange Rates using the GARCH

Model.

ENGLE, R., FOCARDI, S. M. & FABOZZI, F. J. 2007. ARCH/GARCH Models in Applied

Financial Econometrics.

ENGLE, R. F. & NG, V. K. 1993. Measuring and testing the impact of news on volatilty.

Journal of Finance, 20, 419 -438.

GOKBULUT, R. I. & PEKKAYA, M. 2014. Estimating and Forecasting Volatility of

Financial Markets Using Asymmetric GARCH Models: An Application on Turkish

Financial Markets. International Journal of Economics and Finance, 6.

HARRIS, R. & SOLLIS, R. 2003. Applied time series modelling and forecasting, England,

John Wiley and Sons Ltd.

JOHNSTON, K. & SCOTT, E. 2000. GARCH MODELS AND THE STOCHASTIC

PROCESS UNDERLYING EXCHANGE RATE PRICE CHANGES. Journal of

Financial and Strategic Decisions, 13, 13-24.

MANDELBROT, B. 1963. The variation of certain speculative prices. Journal of Business,

36, 394-419.

MCNEIL, A. J., FREY, R. & EMBRECHTS, P. 2006. Quantitative Risk Management

Concepts, Techniques and Tools

MINKAH, R. 2007. Forecasting volatility. Uppsala University.

YU, J. 2002. Forecasting volatility in the New Zealand stock market. Applied Financial

Economics, 12, 193-202.

Download - Simulating South African equity Index returns using GARCH models

Top Related