VaR VaR VaR VaR GARCH E-mail : yoshitaka.andou@boj.or.jp
VaR VaR LTCM VaR VaR VaR VaR VaR VaR VaR
VaR t P(t) P(= P() P(t)) Pr[ P X] =, X t100 (1 )VaR VaR P100 P X X (1 ) VaR VaR
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
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 )
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
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
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
{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
VaR VaRVaR VaR HDVaR VaRHDVaR VaR HD VaR
VaR VaR VaR VaR Boudoukh, Richardson and WhitelawHull and WhiteBarone-Adesi, Giannopoulos and Vosper Boudoukh, Richardson and WhitelawVaR VaRBRW
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
BRWVaR VaR = = = = BRWVaR =0.99 N i =1 (1 )i 1 /(1 T )> 0.99N
VaR BRW VaR VaR VaR VaRVaR BRWVaR VaR Hull and White
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
{r t T+1,, r t 1,r t } VaR VaR NYHWVaR = 0.94 VaR BRWVaR HW VaR HW t VaR
HWVaR VaR Barone-Adesi, Giannopoulos and Vosper GARCHHW FHS: filtering historical simulationfhs FHSHW t Barone-Adesi, Bourgoin and Giannopoulos
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
HWVaR GARCH HWVaR BRWHWFHS HS VaR BRWHW HSBRWHWFHS BRW HSHWFHS VaRHSBRW HWFHS Boudoukh, Richardson and WhitelawHull and White
VCV EWMAHSBRWHWFHSVaR VaR VaR VaR VaR = =
Bloomberg VaR VaR
VaR LIBOR BPV VaR Hendricks
VaRT HSHDVaR HSHSSQHSHD VaRVaR VaRVaR VaR EWMABRWHW BRWEWMAHW EWMA HW
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
VaR VaR HWFHSHSBRW VCVEWMA BRW HS
HSVCV BRWHW
EWMABRWHW FHSVCVHS EWMABRWHW BRW BRW VaR VaRVaR VaRVaR VaR VaR VaRVaR
VCVHS VaR HW VaRVaR HWVaR HW VaR HW HWFHS VaR VaRHW
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
VaR EWMABRW HSHW VaRVCV EWMAHS VaRBRWHW FHSVaR VaR
HSHWVaR HWFHS VaRVaR HWFHSVaR HW VaR VaR HSBRWHW FHS VaRVaR VaRVaR VaRVaR VaR VaRVaR VaR
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
BRWHWFHS VaR VaR HSBRWVaR VaRVaR HSBRW VaR HSBRWVaR HSBRW VaRHSBRW VaR
VaR VaR HSSQHSBRWHW FHS HS BRWHS HWFHS HWFHS VaR VaR BRWHWFHSVaR HWFHS VaR HWFHSVaR HSHDVCVEWMA
VaRVaRVaR VaRHSBRW VaR HSBRWVaR HS BRWHS VaR BRW HSHS BRWHWFHSVaRVCVEWMA VCV EWMA HSHS HS HS
VaR HS BRWHWFHS VaR VaRVaR HSBRW VaRVaR VaR VaR HS BRW VaR
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)
(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.
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
{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) =
VaR VaR Barone-Adesi, G., F. Bourgoin, and K. Giannopoulos, Don t Look Back, RISK, 11 (8), 1998, pp. 100-104. K. Giannopoulos, and L.Vosper, VaR without Correlations for Non-linear Portfolios, Journal of Futures Markets, 19, 1999, pp. 583-602. 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 1996.http://www.bis.org/ http://www.boj.or.jp/ Bollerslev, T., and J. M. Wooldridge, Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances, Econometric Reviews, 11, 1992, pp. 143-172. Boudoukh, J., M. Richardson, and R. Whitelaw, The Best of Both Worlds, RISK, 11 (5), 1998, pp. 64-67. Butler, J. S., and B. Schachter, Estimating Value-at-Risk with a Precision Measure by Combining Kernel Estimation With Historical Simulation, Working Paper, 1997. Efron, B., Bootstrap Methods: Another Look at the Jackknife, The Annals of Statistics, 7, 1979, pp. 1-26. Harrell, F. E., and C. E. Davis, A new distribution-free quantile estimator, Biometrika, 69, 1982, pp. 635-640. Hendricks, D., Evaluation of Value at Risk Models Using Historical Data, Economic Policy Review, Federal Reserve Bank of New York, April 1996, pp. 36-69. Hull, J., and A. White, Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk, Journal of Risk, 1, 1998, pp. 5-19. Inui, K., M. Kijima, and A. Kitano, VaR is subject to a significant positive bias, Working Paper, 70, Graduate School of Economics Kyoto University, 2003. Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, Chicago, 2000. Sheather, S. J., and J. S. Marron, Kernel quantile estimators, Journal of the American Statistical Association, 85, 1990, pp. 410-416.