,..., Xn. (Value-at-Risk) ( ) Jarque-Bera (Extreme Value Theory) ARCH/GARCH [1][2] Mcneil (Tail Index) [8][9] [10] ISE-100 ARCH/ GARCH , ; 2

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1 ,2, 60074; 2, Pareto ARCH/GARCH Pareto GARCH F224.0 A (Vale-at-Risk) ( ) Jarqe-Bera (Extreme Vale Theory) ARCH/GARCH [][2] Mcneil (Tail Index) [8][9] [0] ISE-00, ARCH/ GARCH 2 X, X 2,..., Xn F( x) a n > 0 b n R Gx ( ) lim P(max{ X, i n} a x+ b ) = G( x) n i n n

2 ( Gx ) ξ G ( ) exp( ( x) ), x 0, 0; ξ x = + ξ + ξ > ξ exp( e x ), x R, ξ = 0 ξ F( x) Gξ ( x) Gξ ( x) [2] G ( x) 0 F( x) Gξ ( x),( ξ > 0) F( x) = x ξ L( x) ( Lx) t- skewed-t Gξ ( x), ξ > 0 ξ ξ > 0 ξ ARCH/GARCH 3 ξ Hill ξ > 0 Pickands ξ R ξ R ξ > 0 ( Hmn, ) ξ 2 (threshold) GPD (General Pareto Distribtion GPD) F ( + y) F ( ) F ( y) = P( X y X > ) =, y > 0 F ( ) 0 > y ξ y ( + ξ ),+ ξ > 0, ξ 0; F ( ) ( ;, ) y G y σ σ σ ξ = y σ e, ξ = 0 Gyσ (,, ξ ) GPD N x F( x) = ( + ξ ) n σ ξ 2

3 N, ξ σ (sample mean exceedance) {( e, n( )), X, n < < Xn, n} ( en ) (sample mean excess fnction SME) en ( ) n + ( Xi ) i= + n ( ), :( ) n { X> } X X > X > e = X = I = 0 X 0 X I { Xi > } i= M ( ) = E( X X > ) F ( x),( x= y+ ) GPD( y, ξ, σ ) M ( ) = ( σ + ξ)/( ξ), σ + ξ> 0 Hill SME SME ξ ξ GPD (a) F ˆ ( x ) Gxξ (, ˆ, ˆ σ ) (b) N ˆ ( ) ( ˆ x ξ F x = + ξ ) Gxξ (, ˆ, ˆ σ ) n ˆ σ Gxξ (, ˆ, ˆ σ ) QQ ˆ ξ xi (c) : { log( + ), xi > } GPD ˆ ξ ˆ σ QQ 4 (normal market) { r t } α 3 r t t 3

4 4 VAR ( α) t Pr ( VAR( α) Ψ ) = α t t t -α GPD GPD N ˆ x ˆ ξ F( x) ( + ξ ) n ˆ σ ( F x) N n α ˆ σ nα ˆ ξ VAR( α) = + [( ) ] ˆ ξ N ARCH/GARCH J.P. GPD GARCH GPD 5 (volatile-ratio) T I{ ri> VARi( α )} i= n+ 0, if ri < VARi( α) v ratio=, I{ ri> VARi( α )} = T n, if ri VARi( α) α GARCH r = ε h, h = ω + α r + β h 2 t t t t t t ε t i.i.d. (GED) t GARCH VAR ( α) =Ψ ( α)* h, t = n+, n+ 2, L, T ( Ψ x) ε t t t GPD ˆ σ N ˆ VARm + ξ ( α) = + [( ) ], m n, n,, T ˆ ξ mα = + K N, m = n, n +, L, T ˆ σ ˆ ξ m 4

5 (percentage of log-retrn series) shr96t = 00 {log( Pt) log( Pt )} P t t ADF/PP (size and power) 6 shr96 ADF ADF ˆ ρ shr96t = ρshr96t + α shr96t + εt z t = ˆ σ 500 shr96 (α =0.0) 96 shr 5.2 ˆ ρ (0) I shr96 Hill shr ( 2 ) GPD shr96 2 GPD shr α = 0.00 GPD GPD GPD GARCH GARCH-normal GARCH-t GARCH- GED 000 GPD shr96 ˆ ξ = 0.267, ˆ σ =.0679 N / m,( m = 000,000 +, L,407) shr96 2 GPD GARCH-normal GARCH-GED GARCH-t GARCH GARCH-t GARCH-GED GARCH-normal 5

6 5-5-2 GARCH GPD shr96 GPD 2 ξ lower σ lower (0.0967).067(0.98) (0.0975).0692(0.98) (0.00).0697(0.32) α GARCH-normal GARCH-t GARCH-GED GPD (0.457).248(0.2092) GPD GARCH-normal GARCH-t GARCH-GED shr96 ADF 2 Hill SME 6

7 shr96 GPD () 3-, =.45, ˆ ) ξ = , σ =.0692, F ( x ) ; (2) 3-2 F( x) ; (3) 3-3 QQ ( α = 0.00, 0.95, ξ = , σ = ) GPD shr96 7

8 5- (shr96) ( , ) 5-2 (shr96) GARCH(,)-GED GPD [] Embrechts, P., Klppelberg, C. & Mikosch, T. Modeling Extreme Events for insrance and finance [M]. Springer, New York, 997. [2] Reiss, R.D. & Thomas, M. Statistical analysis of extreme vales from insrance, finance, hydrology and other fields[m]. \ Springer, Berlin,997. [3] McNeil, A.J. Estimating the tails of loss severity distribtions sing extreme vale theory [J]. ASTIN Blletin, Vol. 27, 997: 7-37 [4] McNeil, A.J. Calclating qantile risk measres for financial time series sing extreme vale theory[r]. manscript, Department of Mathematics, ETH, Swiss Federal Technical University,998. [5] McNeil, A.J. Extreme vale theory for risk managers[c]. in Internal Modeling and CAD II, Risk Books, 999: [6] McNeil, A.J. & Frey, R. Estimation of tail-related risk measres for heteroscedastic financial time series: an Extreme vale approach[j]. Jornal of Empirical Finance, 2000(Vol. 7):

9 [7] McNeil, A.J. & Saladin, T. The peak over thresholds method for estimating high qantiles of loss distribtions[r]. discssion paper, Department of Mathematics, ETH, Swiss Federal Technical University, 997. [8] Silva, A. C. & B. V. Mendes. Vale-at-risk and Extreme Retrns in Asian Stock Markets[J]. International Jornal of Bsiness, Vol. 8, No., [9] Terence D. Agbeyegbe and Gene Leon. The tail behavior of stock index retrn on the Jamaican Stock Exchange[Z]. Pblications/ WP2002pp03_0.pdf. [0] R. Gençay, F. Selçk & A. Ulg lya gci. High volatility, thick tails and extreme vale theory in vale-at-risk estimation [J]. Insrance: Mathematics and Economics 33 (2003): [] Leadbetter, M. R., Lindgren, G., & Rootzen, H. Extremes and related properties of random seqences and processes[m]. Springer, New York, 983. [2] Resnick, S. I. Extreme vale, reglar variation, and point processes [M]. Springer, New York,987. [3] Hill, B. M. A sample general approach to inference abot the tail of a distribtion [J]. Ann Statist. Vol. 3, 975:63-74 [4] J. Pickands. Statistical inference sing order statistics [J]. Ann Statist., Vol. 3, 975:9-3. [5] Dekkers, A. L. M., Einmahl, J. H. J., & Haan, L. de. A moment estimator for the index of an extreme- vale distribtion[j]. Ann Statist., Vol. 7, 989: [6] Mills, T.C. The econometric modeling of financial time series (2 nd edition)[j]. Cambridge University Press, 999,Cambridge [7] Dekkers, A. L. M., Haan, L. On the estimation of the extreme vale index and large qantile estimation, Ann Statist, 7: [8] Loretan, M. and Phillips, P. C. B. Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets [J]. Jornal of Empirical Finance, Vol., 994:2-248 [9] Balkema, A. A., and Haan, L. de. Residal lifetime at great age [J]. Ann. Prob., Vol. 2, 974: [20] Gencay, R. & Selck, F. Extreme vale theory and vale-at-risk: Relative performance in emerging markets [J]. International Jornal of Forecasting Vol. 20, 2004: [2] Gencay, R., Selck, F. and Ulglyagci, A. EVIM: A software package for extreme vale analysis in MATLAB[J]. Stdies in Nonlinear Dynamics and Econometrics, Vol. 5 (3), 200: [22] Gencay, R. and Selck, F. Software Review: Matlab Neral Networks Toolbox[J]. International Jornal of Forecasting, Vol. 7, 200: [23]. [D].,,2003 7,. Fat-tails measrement and Vale-at-Risk estimation and prediction of Composite Index of Shanghai Stock Exchange Peng Zo-xiang,2, Li Shi, Pang Hao 9

10 ( School of Statistics, Sothwestern University of Finance and Economics, Chengd, 60074; 2 School of Mathematics and Finance, Sothwestern Normal University, Chongqing, 40075) Abstract: Based on extreme vale theory and General Pareto Distribtion (GPD), this paper analyzes and describes the performance of the thick-tail of the high freqency financial time series data with tail index which fitted by local fitness on tail distribtion of the data. Both process, one is procedres of estimating and testing of the tail index, another is estimating and forecasting methods of Vale-at-Risk, are given systematically. The one-step forward forecasting reslts of the Composite Index of Shanghai Stock Exchange by extreme vale theory and other well-known modeling techniqes, sch as ARCH/GARCH models, are empirically compared and contrasted. The empirical reslts arge that GPD method is sperior to GARCH models on estimating and forecasting of Vale-at-Risk. Key Words: Vale-at-Risk; Tail Index; Forecasting; Extreme Vale Theory; Empirical Analysis JB (967-) ( []) 2 ξ Hill H m, n = log X + log X m m ( ) ( n k, n) ( n m, n) k = X X... X, n 2, n n, n X, X2,..., Xn [7] H( m, n) m Hmn (, ) m= m( n) Hmn (, ) [8] m 0. n 3 α rt = 00*log( pt / pt ) { } 5 T n GARCH GPD sliding window 6 r t 0

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