ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応

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1 VaR VaR VaR VaR GARCH yoshitaka.andou@boj.or.jp

2 VaR VaR LTCM VaR VaR VaR VaR VaR VaR VaR

3 VaR t P(t) P(= P() P(t)) Pr[ P X] =, X t100 (1 )VaR VaR P100 P X X (1 ) VaR VaR

4 VaR VaR VaR VaR VaR VaR VaR VaR VaR VaR VaR Exponentially Weighted Moving Average Method t 2 t 2 t 1 r t 1 2 t = 2 t 1 + (1 )r 2 t 1 (0<<1) 2 t = (1 )Σ k =1 k 1 r 2 t k i, j 2 ij,t = 2 ij,t 1 + (1 )r i,t 1 r j,t 1

5 VaR VaR VaR t x t T x t T,, x t 2, x t 1 r t T+1,, r t 1, r t x t i +1 r t i +1 = 1, i = 1,,T. x t i P(x t ) P = P(x t +1 ) P(x t ) r t T+1,, r t 1, r t T{x (T) t +1,, x (2) t +1, x (1) t+1 } x (i) t +1 = x t (1+r t i +1 ), i = 1,,T, T{ P 1, P 2,, P T } P i = P(x (i) t +1 ) P(x t ), i = 1,,T. { P 1, P 2,, P T } P VaR P i { P (1 ), P (2),, P (T ) }, r t i +1 = ln(x t i +1 /x t i ) (i =1,,T )

6 VaR100(1 ) P ((T+1 )) VaR(T+ 1) (T+1) VaR VaRsample quantile t n{x 1,t, x 2,t,, x n, t }T {x 1,t T, x 2,t T,,x n, t T },,{x 1,t 2, x 2,t 2,,x n, t 2 },{x 1,t 1, x 2,t 1,, x n, t 1 }, {r 1,t T+1, r 2,t T +1,,r n, t T+1 },,{r 1,t 1, r 2,t 1,,r n, t 1 },{r 1,t, r 2,t,, r n, t }. P {x 1,t, x 2,t,, x n, t } P = P(x 1,t +1, x 2,t +1,, x n, t +1 ) P (x 1,t, x 2,t,, x n, t ) i = 1,2,, T t i + 1{r 1,t i+1, r 2,t i+1,, r n, t i+1 } T {x (i) 1,t +1, x(i) 2,t +1,, x (i) n, t +1 } x (i) 1,t +1= x 1,t (1+r 1,t i +1 ) x (i) 2,t +1= x 2,t (1+r 2,t i +1 ) x (i) n, t +1= x n, t (1+r n, t i +1 ), i = 1,,T. T P i = P(x (i) 1,t +1, x (i) 2,t +1,, x (i) n, t +1 ) P(x 1,t, x 2,t,, x n, t ), i = 1,,T, { P 1, P 2,, P T }VaR VaR{ P 1, P 2,, P T } P100

7 VaR Efron VaR { P (1 ), P (2),, P (T ) } T VaR VaRVaR VaR VaR Harrell and Davis { P (1 ), P (2),, P (T ) }100 P P (i) = T w i P, T ( i) i = 1 P, w T,i w T, i 1 = ( k, T k + 1) i / T y k 1 ( i 1) / T ( 1 y) T k dy, k = ( T+ 1), VaR Inui, Kijima and Kitano

8 HDHD HD = T = w T,i VaRi i = (T +1) w T,i i T == T == T == T == HD Sheather and Marron HD HD (a, b) (a, b) = 1 0 y a 1 (1 y ) b 1 dy (a, b >0)HD n i =1 w T,i = 1, w T,i > 0 X (i) L = n i =1 w i X (i), w i 0, n i =1 w i = 1LL HDL

9 {x 1, x 2,, x n } f (x) 1 f ( x) = nh n i. i= 1 x x K h h K(u) (2 ) e 2 /2 [, ] 3/4 (1 u 2 ) [ 1,1] 15/16 (1 u 2 2 ) [ 1,1] VaR VaRVaRHD VaR VaR VaRButler and Schachter VaR

10 VaR VaRVaR VaR HDVaR VaRHDVaR VaR HD VaR

11 VaR VaR VaR VaR Boudoukh, Richardson and WhitelawHull and WhiteBarone-Adesi, Giannopoulos and Vosper Boudoukh, Richardson and WhitelawVaR VaRBRW

12 BRW BRW VaR 1,2,, T T { P 1, P 2,, P T } { P 1, P 2,, P T } (0 < < 1) {w 1, w 2,, w T } 1 w i = i 1. 1 T decay factor { P 1, P 2,, P T }{ P (1 ), P (2),, P (T ) }{w (1 ), w (2),, w (T ) } VaR100(1 ) VaR VaR w (1 ) P (1 ) VaR (a) (b) k k+ 1 w( i) < w( i) i = 1 i= 1 k k+1 VaR = {( w( i )) P( ( ) 1) ( )} k+ + w( i ) P k i= 1 i= 1 w ( k+ 1), w (1) VaR = P (1). T i =1 w i = T i =1 (1 ) i 1 /(1 T )=1 BRW P (i ) P j w (i ) = w j P (k) k i =1 w (i ) 100 { k 1 i =1 w (i ) + w (k) /2} 100 w (i ) = 1/n P (k) (k /n) 100 {(k 0.5)/n} 100 P (k) {k /(n+1)} 100nkk VaR BRWVaR VaR

13 BRWVaR VaR = = = = BRWVaR =0.99 N i =1 (1 )i 1 /(1 T )> 0.99N

14 VaR BRW VaR VaR VaR VaRVaR BRWVaR VaR Hull and White

15 HW HW r t r t = t t, t2 = 2 t 1 +(1 )r 2 t 1. t t t r t t +1 t +1 VaR r t +1 t +1 t +1 N(0,1) HW t i+1 (i = 1,,T ) t i+1 (i = 1,,T ) r t i+1 (i = 1,,T ) t i+1 (i = 1,,T ) t i+1 = r t i+1 / t i+1, i = 1,,T, t +1 r t +1 VaR {r t T+1,, r t 1,r t } { t T+1,, t 1, t } t+1 t +1 r r t i+ 1= t + 1 t + 1, i = 1, 2,, T, i t i+ 1 {r t T+1,, r t 1,r t } t +1 / t i +1

16 {r t T+1,, r t 1,r t } VaR VaR NYHWVaR = 0.94 VaR BRWVaR HW VaR HW t VaR

17 HWVaR VaR Barone-Adesi, Giannopoulos and Vosper GARCHHW FHS: filtering historical simulationfhs FHSHW t Barone-Adesi, Bourgoin and Giannopoulos

18 r t GARCH r t = t t, t2 = +r 2 t 1+ 2 t 1. t FHSGARCH HW GARCH VaR HW HW NYFHSVaR GARCHGARCH GARCH t t

19 HWVaR GARCH HWVaR BRWHWFHS HS VaR BRWHW HSBRWHWFHS BRW HSHWFHS VaRHSBRW HWFHS Boudoukh, Richardson and WhitelawHull and White

20 VCV EWMAHSBRWHWFHSVaR VaR VaR VaR VaR = =

21 Bloomberg VaR VaR

22 VaR LIBOR BPV VaR Hendricks

23 VaRT HSHDVaR HSHSSQHSHD VaRVaR VaRVaR VaR EWMABRWHW BRWEWMAHW EWMA HW

24 VaRVaR VaRVaR VaR VaR VaR X t 1 ( t Var) X t = 0 ( t Var). {X t } {X t } VaR VaR VaRVaRVaR VaR VaR VaR VaR VaR VaR HDVaR Hull and White

25 VaR VaR HWFHSHSBRW VCVEWMA BRW HS

26 HSVCV BRWHW

27 EWMABRWHW FHSVCVHS EWMABRWHW BRW BRW VaR VaRVaR VaRVaR VaR VaR VaRVaR

28 VCVHS VaR HW VaRVaR HWVaR HW VaR HW HWFHS VaR VaRHW

29 VaR VaRHW VaR HW VaR Inui, Kijima and Kitanot HSSQVaRVaR HSHDVaRHSSQ VaR Inui, Kijima and KitanoVaR VaRVaR HSHDHSSQ VaR Inui, Kijima and Kitano VaR t

30 VaR EWMABRW HSHW VaRVCV EWMAHS VaRBRWHW FHSVaR VaR

31 HSHWVaR HWFHS VaRVaR HWFHSVaR HW VaR VaR HSBRWHW FHS VaRVaR VaRVaR VaRVaR VaR VaRVaR VaR

32 VaR VaR VaR VaR VaR VaR VaR VaRHS BRWHS HWFHSHS HWFHSGARCH HS HWFHS VaR HS HSSQHSBRWBRWHW FHS VaRHS VaR Basel Committee on Banking Supervision VaR HSBRWHWFHS

33 BRWHWFHS VaR VaR HSBRWVaR VaRVaR HSBRW VaR HSBRWVaR HSBRW VaRHSBRW VaR

34 VaR VaR HSSQHSBRWHW FHS HS BRWHS HWFHS HWFHS VaR VaR BRWHWFHSVaR HWFHS VaR HWFHSVaR HSHDVCVEWMA

35 VaRVaRVaR VaRHSBRW VaR HSBRWVaR HS BRWHS VaR BRW HSHS BRWHWFHSVaRVCVEWMA VCV EWMA HSHS HS HS

36 VaR HS BRWHWFHS VaR VaRVaR HSBRW VaRVaR VaR VaR HS BRW VaR

37 F(x) f (x) {X 1, X 2,, X n } F n (x) =1/ n n i =1 1 {x X i } {X (1 ), X (2),, X (n) }k X (k) n n! i n i F ( x)(1 F( x, i = k ( n i)! i! P { X ( ) x} = )) k X (k) f k (x) 1 k 1 n k fk( x) = F ( x) {1 F ( x)} f ( x), ( k, n k + 1) X (k) E[ X ( k) 1 ] = ( k, n k + 1) 1 = ( k, n k + 1) 1 0 xf ( x) F 1 k 1 ( y) y {1 F ( x)} k 1 (1 y) n k n k df( x) dy, E[X (n+1 ) ] n 100F n (x) 100 HD 1 = (( n + 1), ( n + 1)(1 )) 1 0 F 1 n ( y) y ( n + 1) 1 (1 y) (n +) 1 (1 ) 1 dy, HD n HD = w n X. i = 1, i ( i) w n,i w n, i 1 = ( k, n k + 1) i / n ( i 1)/ n y k 1 n k ( 1 y) dy, k = ( n + 1). 1 {x Xi } x X i x < X i (, ) (k, n k + 1) = (k 1)! (n k)!/n! (n + 1)

38 (n + 1) = k HD {X (1 ), X (2),, X (n) } n k X (i) X (i) k X (i) n n! j ( i / n) {1 ( i / n)} ( n j )! j! j = k n j, X (i 1) k X (i 1) n ( j = k n n! j n j {( i 1) / n} [1 {( i 1) / n}], j )! j! kx (i ) w n,i w n, i = n n! j ( i / n) {1 ( i / n)} ( n j)! j! j = k n j = k n j n! j {( i 1) / n} [1 {( i 1)/ n}] ( n j)! j! n j, n i =1 w n,i X (i) X Beta (k, n k + 1) Y Bi (n, p) Pr ( X p) = Pr ( Y K) 1 ( k, n k + 1) p y k 1 0 n n k n! ( 1 y) dy = p j 1 p) ( n j )!j! j = k ( n j.

39 w n,i w n, i 1 = ( k, n k + 1) 1 ( k, n k + 1) i / n 0 y k 1 ( i 1)/ n 1 y k 0 n k ( 1 y) dy n k ( 1 y) dy 1 = ( k, n k + 1) i / n 1 y k ( i 1)/ n n k. ( 1 y) dy HD w n,i HD

40 {x 1, x 2,, x T }m x {x 1, x 2,, x T } T ( x x x x t k t )( t k ) = + 1 ( k) =, k = 1,2,, m, T 2 ( x x ) t = 1 t LB(m) 2 m (k) LB( m) = T( T + 2 ), k k = 1 T m m χ 2 (m) mχ 2 ( m ) LB(m)>χ 2 (m)m =15 LB(15)>χ 2 (15) =

41 VaR VaR Barone-Adesi, G., F. Bourgoin, and K. Giannopoulos, Don t Look Back, RISK, 11 (8), 1998, pp K. Giannopoulos, and L.Vosper, VaR without Correlations for Non-linear Portfolios, Journal of Futures Markets, 19, 1999, pp Basel Committee on Banking Supervision, Supervisory framework for the use of backtesting in conjunction with the internal models approach to market risk capital requirements, Basel Committee Publications, 22, January Bollerslev, T., and J. M. Wooldridge, Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances, Econometric Reviews, 11, 1992, pp Boudoukh, J., M. Richardson, and R. Whitelaw, The Best of Both Worlds, RISK, 11 (5), 1998, pp Butler, J. S., and B. Schachter, Estimating Value-at-Risk with a Precision Measure by Combining Kernel Estimation With Historical Simulation, Working Paper, Efron, B., Bootstrap Methods: Another Look at the Jackknife, The Annals of Statistics, 7, 1979, pp Harrell, F. E., and C. E. Davis, A new distribution-free quantile estimator, Biometrika, 69, 1982, pp Hendricks, D., Evaluation of Value at Risk Models Using Historical Data, Economic Policy Review, Federal Reserve Bank of New York, April 1996, pp Hull, J., and A. White, Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk, Journal of Risk, 1, 1998, pp Inui, K., M. Kijima, and A. Kitano, VaR is subject to a significant positive bias, Working Paper, 70, Graduate School of Economics Kyoto University, Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, Chicago, Sheather, S. J., and J. S. Marron, Kernel quantile estimators, Journal of the American Statistical Association, 85, 1990, pp

カルマンフィルターによるベータ推定（）

カルマンフィルターによるベータ推定（） β TOPIX 1 22 β β smoothness priors (the Capital Asset Pricing Model, CAPM) CAPM 1 β β β β smoothness priors :,,. E-mail: koiti@ism.ac.jp., 104 1 TOPIX β Z i = β i Z m + α i (1) Z i Z m α i α i β i (the