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Forecasting BET Index Volatility

MSc.: Răzvan GhelmeciSupervisor: Prof. Moisă Altăr

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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-at-Risk.

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.

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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.

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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.

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BET Index Return and Volatility We use BET index

historical data between January 3rd 2002 and May 31st 2007.

The daily return for day n is:

The daily volatility for day n is:-.16

-.12

-.08

-.04

.00

.04

.08

250 500 750 1000 1250

Return

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Data Series Statistics

0

40

80

120

160

200

240

280

-0.10 -0.05 0.00 0.050

200

400

600

800

1000

1200

1400

0.000 0.005 0.010

Property Daily Return Daily Volatility

Mean 0.001817 0.000201 Std. Deviation 0.014055 0.000575 Skewness -0.478088 12.85548 Kurtosis 9.660094 272.2391 Maximum 0.061451 0.014165 Minimum -0.119018 0.000000

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Data Series Statistics

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Models Definition

GARCH type models are defined as follows:

1. ARCH(1)

2. GARCH(1,1)

3. EGARCH(1,1)

4. TGARCH(1,1)

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Models Definition

The rest of the models are defined as follows:

1. EWMA

2. Moving Average

3. Linear Regression

4. Random Walk

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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

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Forecasting and testing Methodology Value-at-Risk Approach

The money-loss in a portfolio that is expected to occur over a pre- determined horizon and a pre-determined degree of confidence as a result of assets price changes.

The VaR computing equation is:

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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

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Empirical Results - Parameter Estimation Model ARCH(1) GARCH(1,1) EGARCH(1,1) TGARCH(1,1) Linear Regression

Coefficients statistics

µ 0.001388 0.001407 0.00128 0.001426

0 0 0 0

ω 0.000109 0.0000137 -1.092293 0.0000117 0.000152

0 0 0 0 0

α 0.494905 0.253566 0.424738 0.219874 0.243509

0 0 0 0 0

β 0.700499 0.91137 0.747767

0 0 0

γ 0.176084

0.0029

δ -0.195698

0.0005

R2 -0.000936 -0.000853 -0.001464 -0.000775 0.059303

AIC -5.872908 -5.932411 -5.928315 -5.935457 -12.14264

Residual statistics

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 -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

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Empirical Results – Parameter Estimation

GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*RESID(-1)^2*(RESID(-1)<0) + C(5)*GARCH(-1)

Coefficient Std. Error z-Statistic Prob.

C 0.001397 0.000326 4.286015 0.0000 Variance Equation

C 1.38E-05 1.93E-06 7.123177 0.0000 RESID(-1)^2 0.250256 0.032947 7.595741 0.0000

RESID(-1)^2*(RESID(-1)<0) 0.005947 0.033127 0.179512 0.8575 GARCH(-1) 0.700399 0.023058 30.37571 0.0000

R-squared -0.000895 Mean dependent var 0.001817

Adjusted R-squared -0.003901 S.D. dependent var 0.014055 S.E. of regression 0.014082 Akaike info criterion -5.930929 Sum squared resid 0.264144 Schwarz criterion -5.911489 Log likelihood 3969.826 Durbin-Watson stat 1.618621

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.

C 0.001276 0.000287 4.441619 0.0000 Variance Equation

C(2) -1.089850 0.117089 -9.307891 0.0000 C(3) 0.423750 0.031400 13.49539 0.0000 C(4) -0.003197 0.016512 -0.193587 0.8465 C(5) 0.911555 0.012029 75.77909 0.0000

R-squared -0.001487 Mean dependent var 0.001817

Adjusted R-squared -0.004494 S.D. dependent var 0.014055 S.E. of regression 0.014086 Akaike info criterion -5.926836 Sum squared resid 0.264300 Schwarz criterion -5.907396 Log likelihood 3967.090 Durbin-Watson stat 1.617665

Modified EGARCH(1,1)

Modified TGARCH(1,1)

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Empirical Results – Model Testing

ME RMSE MAE MAPE

Model Value Rank Value Rank Value 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

0.00000098 2 0.00036401 6 0.00019678 1 0.40326112 9

TGARCH(1,1) 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

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Empirical Results – Tests Based on Value-at-

Risk Approach VaR 1% TUFF Rank FT Rank

Linear Regression 7 2 0.024 1 EGARH(1,1) 7 2 0.026 2 Long Term MA 150 1 0.028 3 GARCH(1,1) 7 2 0.028 3 Long Term EWMA 7 2 0.03 5 TARCH(1,1) 7 2 0.032 6 ARCH(1) 7 2 0.036 7 Short Term EWMA 7 2 0.044 8 Short Term MA 7 2 0.052 9 Random Walk 4 10 0.288 10

VaR 5% TUFF Rank FT Rank

Long Term MA 7 1 0.088 2 Linear Regression 5 3 0.076 1 Long Term EWMA 7 1 0.094 3 EGARH(1,1) 5 3 0.096 4 GARCH(1,1) 5 3 0.096 4 ARCH(1) 5 3 0.1 6 TARCH(1,1) 5 3 0.104 7 Short Term EWMA 5 3 0.122 8 Short Term MA 5 3 0.124 9 Random Walk 4 10 0.354 10

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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.

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References

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 Techniques”, Journal of Banking and Finance, 20, 419-438.

Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, 31, 307-328.

Brooks, C. (1998), “Forecasting Stock Return Volatility: Does Volume Help?”, Journal of Forecasting, 17, 59-80.

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 Content of Stock Index Options”, Journal of Econometrics, 52, 267-287.

Engle, R.F. (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation”, Econometrica, 50, 987-1008.

Franses, P.H. and D. van Dijk (1996), “Forecasting Stock Market Volatility Using Non-Linear GARCH Models”, Journal of Forecasting, 15, 229-235.

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References

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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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Pagan, A.R. and G.W. Schwert (1990), “Alternative Models for Conditional Stock Volatilities”, Journal of Econometrics, 45, 267-290.

West, K.D. and D. Cho (1995),”The Predictive Ability of Several Models of Exchange Rate Volatility”, Journal of Econometrics, 69, 367-391.

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Thank you for your consideration!

Bucharest, July 2007