Download Leverage Effects In The Mauritian`s Stock Market

2009 Oxford Business & Economics Conference Program
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TITLE PAGE:
TITLE:
LEVERAGE EFFECTS IN THE MAURITIAN’S STOCK MARKET
AUTHOR’S NAME:
Mr USHAD SUBADAR AGATHEE
PHONE NUMBER:
+230 4541041
AFFILIATION:
DEPARTMENT OF FINANCE AND ACCOUNTING, FACULTY LAW AND
MANAGEMENT, UNIVERSITY OF MAURITIUS.
EMAIL: [email protected]
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Leverage Effects in the Mauritian’s Stock Market
Abstract
This paper essentially aims at comparing the GARCH (1,1), T-GARCH and E-GARCH models in
the ability to describe volatility on the Stock Exchange of Mauritius (SEM). Daily observations from
the SEMDEX for the period July 1989 to December 2007 are used for the study. The results suggest
that the SEMDEX series exhibit some non-normal properties and fat tail characteristics. Using the
GARCH models, the results indicate that there is no leverage effect in contrast to most developed
and emerging markets. Also, the presence of a leverage effect cannot be found when splitting the
sample into a non-daily trading regime and a daily regime.
Keywords: Leverage effects; Stock markets; African Emerging Stock markets; Efficient market
hypothesis; SEMDEX
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1.0 Introduction
Since its inception in 1989, the Stock Exchange of Mauritius (SEM) have experience major
developments in terms of market size, trading volume, number of listed companies and contribution
to the Mauritian GDP. There have also been significant interests from foreign investors lately.
However, published research has been so far very limited. To this effect, it is important to know the
behaviour of stock market volatility in this emerging market.
There have been a number of models attempting to capture volatility. However, one of the wellknown and commonly used models is the so-called Generalised Autoregressive Conditional
Heteroskedasticity (GARCH) model suggested by Bollerslev (1986). Specific styles in financial time
series call upon the use of GARCH models as they can capture the volatility clustering effect
whereby large changes are likely to trail large changes and small changes tend to follow small
changes. However, there is usually an asymmetry in the stock return’s distribution. As such,
asymmetric GARCH models such as T-GARCH, proposed by Glosten et al. (1993), and E-GARCH
models, proposed by Nelson(1991), are more appropriate.
This study is an initial formal attempt to shed some lights on the presence of volatility patterns and
leverage effects. Indeed, the use of GARCH models fit in perfectly with the stylized characteristics
of financial time series. In this respect, this paper attempts to investigate the leverage effects on the
SEM using GARCH models.
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This paper is organised as follows; Section 2 reviews previous literature, Section 3 provides an
overview of the research methodology, Section 4 focuses on the analysis of results while section 5
concludes the paper.
2.0 Prior Research
Earlier models in explaining movements of asset prices focus on the methodology developed by Box
and Jenkins (1976). In particular, Box and Jenkins developed the Autoregressive Integrated Moving
Average (ARIMA) model to predict movement in equity prices. However, ARMIA models are
constrained by their assumptions of constant conditional variance over time. Essentially, the ARIMA
model cannot be used to capture volatility clustering effects present in financial time series. To this
effect, Engle (1982) suggested an ARCH model to explain volatility patterns. However, given some
limitations of ARCH models, Bollerslev (1986) extended the ARCH models by introducing a
GARCH model. The GARCH model itself is an infinite ARCH process. Another style feature of
financial time series is the leverage effect whereby there is an asymmetric reaction of volatility
changes in response to positive and negative shocks of the same magnitude. To this effect, Nelson
(1991) has developed and exponential GARCH model (E-GARCH) and Glosten et al. (1993) have
suggested a T-GARCH model.
Stock market volatility has attracted great attention in the last decades and has as such been widely
discussed in several developed and developing capital markets. Among the pioneering studies, Black
(1975), using a sample of 30 industrial equities, considered the relationship between stock market
returns and volatility for the period 1962-1975. He found that changes in stock prices are negatively
related to changes in stock market volatility (leverage effects). Similarly, Christie (1982), using
quarterly data for the period 1962-1978 with a sample size of 379 firms, found that the average
regression coefficient between stock market prices and volatility to be negative. Additionally,
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Christie (1982) found that the debt to equity ratio could be a possible explanation for the negative
relationship between stock market returns and changes in volatility. Moreover, Koutmos and Saidi
(1995) and Henry (1998) support the claim of a leverage effect.
Other authors such as Haroutounian and Price (2001), Glimore and McManus (2001), Poshakwale
and Murinde (2001) have found significant high volatility persistence in Central and Eastern
European stock markets. There are also studies which consider volatility in periods around financial
crises. For instance, Schwert (1990) considered the stock market volatility before the October 1987
crash and after the crash. He found that stock market volatility was higher during the crash and after
that. Similarly, Kaminsky and Reinhart (2001) reported high volatility persistence in a post-crisis
period.
3.0 Research Methodology
According to Brooks (2004), a typical GARCH (1,1) is adequate for financial time series and it is
very uncommon to find advanced order of GARCH models in the academic finance literature. Daily
observations of the SEMDEXi are used to calculate returns for the months as from July 1989 to
December 2007. Daily stock returns are calculated as follows;
Rt= Ln(Pt)-Ln(Pt-1) where Pt is the index number at time t and Pt-1 is the index in the preceding
day.
Using GARCH models involve a number of advantages in that they assist in capturing the features of
financial time seriesii as they cater for volatility clustering and leverage effects. In particular, several
financial time series are subject to a period of successive strong volatility followed by a period of
low volatility. As a consequence, conditional variance is time-varying. However, the conditional
variance can exhibit a mixture of asymmetric behaviour. Thus, according to Engel (1982), Bollerslev
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(1986) and Bollerslev et al. (1992), autoregressive conditional heteroscedasticity models (GARCH)
are more suitable as they are more flexible in capturing dynamic structures of conditional variance.
Based on the empirical literature, the following regression models are used.
yt    yt 1   t
(1)
 t  1 ~ N (0, ht )
(2)
ht   0  1 2 t 1   2 ht 1
(3)
Where in equation (1), daily stock return, y, is regressed on a constant, μ and a time-lagged value of
return, y
t −1
; ε is an error term which is dependent on past information and ht is the conditional
variance. According to Zhang and Wirjanto (2006), “the purpose of using the AR(1) process is to
capture time dependence of the return series and to smooth the series of possible structural shifts
over the sample period”.
For the conditional variance, ht, to be nonnegative and positive, the following conditions must be
met:
 0  0; 1  0;  2  0and1   2  1
In general, the ARCH and GARCH terms,  1 and  2 indicate short run and long run shocks
persistence respectively.
Furthermore, based on the studies Black (1976) and Christie (1982), positive and negative shocks do
not have the effect on volatility. Essentially, shocks are asymmetric such that volatility is more
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sensitive to negative shocks. To this effect, the following two asymmetric GARCH models, namely
TGARCH and EGARCH, are employed
TGARCH:
ht   0  1 2 t 1   2 ht 1   3 2 t 1 I t 1
(4)
Where I t 1 =1 if  t 1  0 , or zero otherwise
EGARCH:
  t 1

2
ln( ht )   0   1 
    2 ln( ht 1 )   3 t 1
 
ht 1
 ht 1
(5)
For the TGARCH model, the leverage effect parameter,  3 , should be greater than zero. However,
restrictions are imposed on the parameters in that they must all be greater than zero for the
conditional variance to be non-negative. For the EGARCH model, there is no need for non-negativity
constraints on the parameters and the leverage effect is accounted for if the relationship between
volatility and returns is negative such that,  3 , will be negative. Finally, to assess the validity of the
model, the Ljung-Box Q statistics on the squared standardized residuals is used while the loglikelyhood value and the information criterion are used to assess which model is more appropriate.
4.0 Analysis of Data and Results
The basic statistics are presented in Table 1 for return series on SEMDEX for the period July1989 to
December 2007.
[INSERT TABLE 1 ABOUT HERE]
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Table 1 shows the descriptive statistics for the whole sample period and on a year to year basis. At
first glance, most of the mean returns are positively skewed and have significant kurtosis. However,
from 1989 to 2007, there are six years where the returns have been negatively skewed and two years
where the kurtosis value has been lower than 3. On overall, for the whole sample, there is large
kurtosis value, suggesting that the series follow a fat tail distribution, and a positively skewed series.
With the exception of 3 years, the mean returns are found to be non-normal as the Jarque-Bera
statistics is significant at 1% level. In general, the series present some of the stylized facts of
financial series in that they are non-normal and exhibit fat tails, supporting the claim that GARCH
models appear to most appropriate.
Table 2 shows the results from the different GARCH models for the stock market returns for the
period 1989 to 2007.
[INSERT TABLE 2 AROUND HERE]
From table 2, the result shows that the E-GARCH model has the highest log-likelihood value as well
as the lowest AIC and SBIC values. Also, all coefficients on the E-GARCH model are statistically
significant. Also, based on the Ljung-Box statistics, there is no problem of autocorrelation for all the
three models. Finally, it is observed that all restrictions imposed on the GARCH models are met
while one non-negativity constraint on T-GARCH model is violated. Thus, in light of the above, the
E-GARCH is considered as the best model.
The ARCH and GARCH effects are significant in all three models. However, while the sum of
ARCH and GARCH coefficients are less than one for all the models, except for the E-GARCH
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model. Essentially, while there is shocks persistence for the E-GARCH model, the shocks to
volatility decay over few lags for the standard GARCH and T-GARCH models.
From the E-GARCH model, a significant asymmetry coefficient, B3 , is found. However, the leverage
effect is accounted if the coefficient is less than zero, where a negative surprises seem to increase
volatility more than a positive surprises. Contrary to the expectations, there seem to be no leverage
effects on the SEM as the coefficient is statistically positive. As such,
negative news on the SEM cause volatility to increase less than positive news of the same
magnitude.
However, the Stock Exchange of Mauritius has been trading at irregular intervals since its inception.
It only starts to trade on a daily basis for the full year in 1998. As such, structure of volatility could
have been different under the regime of non-daily trading relative to daily trading. To this effect, the
sample is segregated into two periods, namely, 1989-1997 and 1998-2007. The results are reported
below.
[INSERT TABLE 3 AROUND HERE]
[INSERT TABLE 4 AROUND HERE]
From Table 3, based on the Ljung-Box statistics, the T-GARCH and the standard GARCH models
seem to suffer from autocorrelation while the E-GARCH is valid model. As such, the E-GARCH
model is considered. It is observed that there is no leverage effect on the SEM for the period 19982007. As such, the results are in line with the earlier predictions. Considering Table 4, the EGARCH model is recommended for comparison purposes though results from T-GARCH and
GARCH models are reported. Considering the period 1989-1997, there seems to be no leverage
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effects on the SEM as the coefficient is positive though insignificant. In fact, a coefficient which is
statistically equal to zero will imply that the positive surprise will have the same effect on volatility
as the negative surprise of the same magnitude. As such, there are no leverage effects on the SEM
both under the regime of non-daily trading and daily trading.
5.0 Conclusion
This paper has investigated three GARCH models on the SEM. The descriptive statistics shows that
the mean returns exhibit some non-normal characteristics and excess kurtosis for most of the years as
well as for the whole sample period. With regards to the regression models, the results show that the
E-GARCH model is the most properly specified. The E-GARCH model suggests the absence of a
leverage effect on the SEM. Furthermore, the absence of a leverage effect is confirmed when
segregating the sample periods into two different regimes of daily and non-daily trading. As a
concluding note, it is suggested that negative news on the SEM cause volatility to increase less than
positive news of the same magnitude.
6.0 References
Black, F. (1976). Studies of Stock Price Volatility Changes. Proceeding of the meetings of the
American Statistics Association, Business and Economics Section, 177-181.
Bollerslev, T. (1986). Generalised Autoregressive Conditional Heteroscedasticity. Journal of
Econometrics, 31, 307-27.
Bollerslev, T. and Wooldridge, J. (1992). Quasi-maximum likelihood estimation and inference in
dynamic models with time varying covariances. Econometric Reviews,11, 143-72.
Box, G. E. P. and G. M. Jenkins, Time Series Analysis: Forecasting and Control, HoldenJune 24-26, 2009
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Day, 1976.
Brooks C. (2004). Introductory Econometrics for Finance. Cambridge University Press
Christie, A. (1982). The Stochastic Behaviour of Common Stock variances: Value, Leverage and
Interest Rate Effects. Journal of Financial Economics, 10, 407-432.
Engle, R.F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of Variables of
UK Inflation. Econometrica, 50, 987-1008.
Glimore, C.G. and McManus, G. M., (2001) “Random-Walk and Efficiency of Central European
Equity Markets”, Presentation at the 2001 European Financial Management Association, Annual
Conference, Lugano, Switzerland.
Glosten, L., R. Jagannathan, and D. Runkle (1993): “On the Relation between Expected
Return on Stocks,” Journal of Finance, 48, 1779-1801.
Haroutounian, M. and S. Price, (2001) "Volatility in transition market of Central Europe”, Applied
Financial Economics (11), pp 93-105
Henry, O., (1998) “Modelling the asymmetry of stock market volatility”, Applied Financial
Economics (8), pp 145-153
Kaminsky, G.L. and C.M. Reinhart, (2001) “Financial markets in times of stress”, NBER Working
paper 8569, www.nber.org/papers/w8569
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Koutmos, G. and R. Saidi, (1995) “The leverage effect in individual stocks and the debt to equity
ratio”, Journal of Business Finance and Accounting (22), pp 1063-1073
Nelson, D. (1991): “Conditional Heteroskedasticity in Asset Returns: A New Approach,”
Econometrica, 59, 349-370
Pagan, A.(1996). The Econometrics of financial markets. Journal of Empirical Finance, 3, 15-102.
Pashakwale, S. and Murinde, V., (2001) “Modelling the Volatility in East European Emerging Stock
Markets: Evidence on Hungary and Poland”, Applied Financial Economic (11), pp 445-456
Schwert, G.W., (1990) “Stock Volatility and the Crash of ‘87”, Review of Financial Studies (3), pp
77-102
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LIST OF TABLES:
Table 1: Descriptive statistics for SEMDEX returns
Period
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-
P-Value
Bera
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
0.006396
0.020392
0.054026
1.952512
1.155111
0.56127
0.00741
0.016196
-0.73256
3.903369
6.295627
0.04295
-0.0021
0.008095
-0.22425
2.495465
0.9494
0.62207
0.001742
0.006716
0.546948
4.105234
9.974867
0.00682
0.005176
0.00936
0.518875
3.941684
7.936589
0.01891
0.003104
0.00825
0.459189
4.080463
12.23246
0.00221
-0.00228
0.010135
-0.01359
7.552003
122.602
0.00000
0.000175
0.008792
-1.87144
19.55013
1775.482
0.00000
0.000633
0.005183
0.199473
3.610731
3.547673
0.16968
0.0007
0.006143
0.874326
11.20818
730.7312
0.00000
-0.00027
0.003926
-0.0196
4.252316
16.3524
0.00028
-0.00044
0.002275
-1.09035
8.100459
319.24
0.00000
-0.00055
0.004035
0.572336
15.6869
1663.243
0.00000
0.000637
0.004066
0.406741
4.729487
37.74641
0.00000
0.001268
0.005942
0.26814
11.31762
729.4384
0.00000
0.001013
0.003357
0.329357
6.794036
156.9362
0.00000
0.000495
0.004078
0.519162
10.16587
543.9384
0.00000
0.00161
0.007684
0.624211
11.29317
735.59
0.00000
0.001705
0.009451
0.737439
14.28066
1342.822
0.00000
0.000853
0.006922
0.506192
14.16385
17893.08
0.00000
All Sample
[89-07]
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Table 2: GARCH MODEL (1989-2007)
Coefficient


B0
B1
B2
B3
LBQ2(12)
Log-likelihood function
value
GARCH(1,1) P-value
TGARCH
P-value
EGARCH
0.000378
0.393481
0.0000
0.0000
0.000427
0.400311
0.0000
0.0000
0.000609
0.348677
0.0000
0.0000
5.37E-06
0.0000
5.21E-06
0.0000
-1.255971
0.0000
0.314019
0.0000
0.355528
0.0000
0.378340
0.0000
0.566946
0.0000
0.582654
0.0000
0.904904
0.0000
-0.120705
0.0000
0.026103
0.0020
1.5734
13120.02
1.000
2.5448
0.998
1.6673
1.000
13114.51
P-value
13134.70
-7.679856
-7.682494
-7.691095
AIC
-7.670870
-7.671710
-7.680312
SBIC
Note: AIC is Akaike Information Criterion, SBIC is the Schwarz criterion, LBQ2(12) is the Ljung-Box
statistics for serial correlation on squared standardized residuals at the 5% level of the order 12 lags
respectively
Table 3: GARCH MODEL (1998-2007)
Coefficient


B0
GARCH(1,1) P-value
TGARCH
P-value
EGARCH
0.000222
0.312496
0.0020
0.0000
0.000277
0.313831
0.0003
0.0000
0.000425
0.295464
0.0000
0.0000
7.89E-07
0.0000
8.54E-07
0.0000
-1.209301
0.0000
B1
B2
B3
LBQ2(12)
0.143323
0.0000
0.176021
0.0000
0.355157
0.0000
0.839631
0.0000
0.834078
0.0000
0.910862
0.0000
-0.063057
0.0000
0.045906
0.0000
32.902
0.001
13.990
0.301
Log-likelihood function
value
10040.75
34.713
0.001
10046.41
P-value
10024.80
-8.038250
-8.041982
-8.024668
AIC
-8.026590
-8.027990
-8.010676
SBIC
Note: AIC is Akaike Information Criterion, SBIC is the Schwarz criterion, LBQ2(12) is the Ljung-Box
statistics for serial correlation on squared standardized residuals at the 5% level of the order 12 lags
respectively
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Table 4: GARCH MODEL (1989-1997)
Coefficient


B0
GARCH(1,1) P-value
TGARCH
P-value
EGARCH
0.000399
0.560087
0.1213
0.0000
0.000553
0.560475
0.0367
0.0000
0.000927
0.532883
0.0006
0.0000
1.24E-05
0.0000
1.25E-05
0.0000
-2.367505
0.0000
B1
B2
B3
LBQ2(12)
0.260199
0.0000
0.349568
0.0000
0.393887
0.0000
0.565982
0.0000
0.565580
0.0000
0.786341
0.0000
-0.191066
0.0024
0.052265
0.1207
0.2941
1.000
0.1716
1.000
Log-likelihood function
value
3193.755
0.2980
1.000
3197.445
P-value
3188.352
-6.954755
-6.960622
-6.940790
AIC
-6.928468
-6.929077
-6.909246
SBIC
Note: AIC is Akaike Information Criterion, SBIC is the Schwarz criterion, LBQ2(12) is the Ljung-Box
statistics for serial correlation on squared standardized residuals at the 5% level of the order 12 lags
respectively
List of Footnotes:
i
The SEMDEX is the market index on the official market, comprising all stocks on the official market.
ii Pagan (1996)
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