Download Forecasting BET Index Volatility

Forecasting BET Index
Volatility
MSc.: Răzvan Ghelmeci
Supervisor: Prof. Moisă Altăr
1
Introduction

Into this paper we try to combine volatility forecasting and risk
management, analyzing if the intensely used predicting models
can be calibrated on data from Romanian stock exchange and if
the realized predictions can be employed as risk management
instruments using several test based on computing Value-atRisk.

The literature on forecasting volatility is significant and still
growing at a high rate.

The techniques of measuring and managing financial risk have
developed rapidly as a result of financial disasters and legal
requirements.
2
Literature Review

Akgiray (1989) showed that GARCH model is superior to ARCH
model, EWMA and models based on historical mean in predicting
monthly US stock index volatility.

A similar result was observed by West and Cho (1995) regarding
daily forecast of US Dollar exchange rate using root mean
square error test (RMSE). Nevertheless, for longer time horizons
GARCH model did not gave better results than Long Term Mean,
IGARCH or autoregressive models.

Franses and van Dijk (1996) compared three types of GARCH
models (standard GARCH, QGARCH and TGARCH) in
predicting the weekly volatility of different European stock
exchange indexes, nonlinear GARCH models bringing no better
results than the standard model.
3
Literature Review

Other papers tried to combine stock index volatility forecast
derived from traded options prices with those generated by
econometric models – Day and Lewis (1992).

Alexander and Leigh (1997) present an evaluation of relative
accuracy of some GARCH models, equally weighted and
exponentially weighted moving average, using statistic criterions.
GARCH model was considered better than exponentially
weighted moving average (EWMA) in terms of minimizing the
number of failures although the simple mean was superior to
both.

Jackson et al. (1998) assessed the empirical performance of
different VaR models using historical returns from the actual
portfolio of a large investment bank.
4
BET Index Return and Volatility

We use BET index
historical data between
January 3rd 2002 and
May 31st 2007.
.08
.04
.00
-.04

The daily return for day n is:
-.08
-.12

The daily volatility for day n is:
-.16
250
500
750
1000
1250
Return
5
Data Series Statistics
280
1400
240
1200
200
1000
160
800
120
600
80
400
40
200
0
0
-0.10
-0.05
Property
Mean
Std. Deviation
Skewness
Kurtosis
Maximum
Minimum
0.00
0.05
0.000
Daily Return
0.001817
0.014055
-0.478088
9.660094
0.061451
-0.119018
0.005
0.010
Daily Volatility
0.000201
0.000575
12.85548
272.2391
0.014165
0.000000
6
Data Series Statistics
7
Models Definition

GARCH type models are defined as follows:
1.
ARCH(1)
2.
GARCH(1,1)
3.
EGARCH(1,1)
4.
TGARCH(1,1)
8
Models Definition

The rest of the models are defined as follows:
1.
EWMA
2.
Moving Average
3.
Linear Regression
4.
Random Walk
9
Forecasting and testing Methodology

“Rolling Window” Method

Classical tests
1.
Mean Error
2.
Root Mean Square Error
3.
Mean Absolute Error
4.
Mean Absolute Percent Error
10
Forecasting and testing Methodology

Value-at-Risk Approach
The money-loss in a portfolio that is expected to occur over a predetermined horizon and a pre-determined degree of confidence as a
result of assets price changes.

The VaR computing equation is:
11
Forecasting and testing Methodology

Time Until First Failure
The first day in the testing period where the capital held is
insufficient to absorb the loss of that day.

Failure Rate
The percentage level of the times the computed value of VaR is
insufficient to cover the real losses during the testing period
12
Empirical Results - Parameter Estimation
Model
ARCH(1)
GARCH(1,1)
EGARCH(1,1)
TGARCH(1,1)
Linear Regression
Coefficients statistics
µ
0.001388
0
0
0
0
ω
0.000109
0.0000137
-1.092293
0.0000117
0.000152
0
0
0
0
0
0.494905
0.253566
0.424738
0.219874
0.243509
0
0
0
0
0
0.700499
0.91137
0.747767
0
0
0
α
β
0.001407
0.00128
0.001426
γ
0.176084
δ
-0.195698
0.0029
0.0005
R2
-0.000936
-0.000853
-0.001464
-0.000775
0.059303
AIC
-5.872908
-5.932411
-5.928315
-5.935457
-12.14264
Mean
0.013664
0.018883
0.028486
0.022724
3.65E-20
Maximum
4.845101
5.790628
5.328472
5.743252
0.013872
Minimum
-5.809138
-5.258532
-5.496103
-5.100418
-0.001504
Std. Dev.
1.000281
1.000309
1.000039
1.000266
0.000558
Skewness
Residual statistics
-0.278999
-0.028068
-0.068509
0.005634
13.45956
Kurtosis
5.746035
5.450277
5.544631
5.342954
297.5921
Jarque-Bera
437.4258
334.6405
361.7654
305.8143
4871342
13
Empirical Results – Parameter Estimation
GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*RESID(-1)^2*(RESID(-1)<0)
+ C(5)*GARCH(-1)
C
LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) +
C(4)*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))
Coefficient
Std. Error
z-Statistic
Prob.
0.001397
0.000326
4.286015
0.0000
C
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
1.38E-05
0.250256
0.005947
0.700399
-0.000895
-0.003901
0.014082
0.264144
3969.826
1.93E-06
0.032947
0.033127
0.023058
Std. Error
z-Statistic
Prob.
0.001276
0.000287
4.441619
0.0000
-9.307891
13.49539
-0.193587
75.77909
0.0000
0.0000
0.8465
0.0000
Variance Equation
Variance Equation
C
RESID(-1)^2
RESID(-1)^2*(RESID(-1)<0)
GARCH(-1)
Coefficient
7.123177
7.595741
0.179512
30.37571
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat

Modified EGARCH(1,1)

Modified TGARCH(1,1)
0.0000
0.0000
0.8575
0.0000
0.001817
0.014055
-5.930929
-5.911489
1.618621
C(2)
C(3)
C(4)
C(5)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
-1.089850
0.423750
-0.003197
0.911555
-0.001487
-0.004494
0.014086
0.264300
3967.090
0.117089
0.031400
0.016512
0.012029
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
0.001817
0.014055
-5.926836
-5.907396
1.617665
14
Empirical Results – Model Testing
ME
Model
Value
RMSE
Rank
Value
MAE
Rank
Value
MAPE
Rank
Value
Rank
Long Term EWMA
0.00000867
4
0.00035741
3
0.00019691
2 0.04543826
7
Linear Regression
0.00002593
8
0.00034979
1
0.00019857
6 0.01531826
2
EGARCH(1,1)
0.00002183
7
0.00035682
2
0.00019746
4 0.01809292
4
Short Term
EWMA
TGARCH(1,1)
0.00000098
2
0.00036401
6
0.00019678
1 0.40326112
9
0.00001365
5
0.00035940
5
0.00019700
3 0.02886367
6
GARCH(1,1)
0.00001956
6
0.00035901
4
0.00019756
5 0.02020098
5
Short Term MA
0.00000125
3
0.00036946
7
0.00020283
7 0.32339148
8
ARCH(1)
0.00002648
9
0.00036982
8
0.00020455
8 0.01544695
3
Long Term MA
0.00005273
10
0.00038389
9
0.00023225
10 0.00880840
1
Random Walk
0.00000036
1
0.00044743
10
0.00022390
9 1.25366463
10
15
Empirical Results – Tests Based on Value-at-Risk Approach
VaR 1%
Linear Regression
EGARH(1,1)
Long Term MA
GARCH(1,1)
Long Term EWMA
TARCH(1,1)
ARCH(1)
Short Term EWMA
Short Term MA
Random Walk
VaR 5%
Long Term MA
Linear Regression
Long Term EWMA
EGARH(1,1)
GARCH(1,1)
ARCH(1)
TARCH(1,1)
Short Term EWMA
Short Term MA
Random Walk
TUFF
Rank
FT
Rank
7
7
150
7
7
7
7
7
7
4
2
2
1
2
2
2
2
2
2
10
0.024
0.026
0.028
0.028
0.03
0.032
0.036
0.044
0.052
0.288
1
2
3
3
5
6
7
8
9
10
TUFF
Rank
FT
Rank
7
5
7
5
5
5
5
5
5
4
1
3
1
3
3
3
3
3
3
10
0.088
0.076
0.094
0.096
0.096
0.1
0.104
0.122
0.124
0.354
2
1
3
4
4
6
7
8
9
10
16
Conclusions

The results we obtained revealed that, although they can be easily
calibrated on Romanian stock exchange index, the models used for
predicting the volatility have a low performance, even unsatisfactory
compared with the results obtained using simpler methods.

The tests performed using a financial risk management framework
rejected all the models employed and showed that they could not be
successfully used in establishing a minimum capital requirement based
on the risk assumed by investing in portfolios that replicate the
Bucharest Stock Exchange index – BET.

The explanation for this failure is that the volatility on the Romanian
market is generally high, existing periods of accentuated turbulences
that make hard to use the classic econometric volatility forecasting
models.
17
References
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Akgiray, V. (1989), “Conditional Heteroskedasticity in Time Series of Stock
Returns: Evidence and Forecasts”, Journal of Business, 62, 55-80.
Brailsford, T.J. and R.W. Faff (1996), “An Evaluation of Volatility Forecasting
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Bollerslev,
T.
(1986),
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Autoregressive
Conditional
Heteroskedasticity”, Journal of Econometrics, 31, 307-328.
Brooks, C. (1998), “Forecasting Stock Return Volatility: Does Volume Help?”,
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Brooks, C. and G. Persand (2000), “Value at Risk and Market Crashes”, Journal
of Risk, 2, 5-26.
Cheung, Y.W. and L.K. Ng (1992), “Stock Price Dynamics and Firm Size: An
Empirical Investigation”, Journal of Finance, 47, 1985-1997.
Day, T.E. and C.M. Lewis (1992), “Stock Market Volatility and the Information
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Engle, R.F. (1982), “Autoregressive Conditional Heteroskedasticity with
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Franses, P.H. and D. van Dijk (1996), “Forecasting Stock Market Volatility Using
Non-Linear GARCH Models”, Journal of Forecasting, 15, 229-235.
18
References
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Granger, C.W.J. and S.H. Poon (2003), “Forecasting Volatility in Financial
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J.P. Morgan (1996), “RiskMetrics – Technical Document”, 4th Edition.
Jackson, P., D.J. Maude, and W. Perraudin (1998), “Testing Value at Risk
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Johansen, A. and D. Sornette (1999), “Critical Crashes”, Journal of Risk, 12, 9195.
Klaassen, F. (2002), “Improving GARCH Volatility Forecasts”, Empirical
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Pagan, A.R. and G.W. Schwert (1990), “Alternative Models for Conditional Stock
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West, K.D. and D. Cho (1995),”The Predictive Ability of Several Models of
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Memory”, Working Paper, Olsen Associates.
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Thank you for your consideration!
Bucharest, July 2007
20