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VaR VaRArtzner et al. VaR VaR VaR Artzner et al.var VaR VaR VaR ρ XY ρ (X+Y ) ρ(x) + ρ(y ) XY ρ VaRArtzner et al.1999basak and Shapiro1999Danielsson2000Rootzén and Klüppelberg VaR VaR VaRVaR Artzner et al.1999kim and Mina 2000Ulmer20002000FISC VaR Rockafeller and Uryasev
VaR VaR VaR Artzner et alvar VaR VaR VaR VaR VaR VaR VaR
α quantile X α α expected shortfall conditional VaRmean excess lossbeyond VaRtail VaR Artzner et al. VaR Artzner et a α VaR α ( X ) X VaR α ( X ) = inf {x P[X x ]> α } inf {x A} Axinf {x P[X x ]> α }α VaR α VaR Artzner et al Fishburnt γ γ =1 t = VaR VaR t F γ (t) = (t x) γ df( x) γ > 0 F (x )x t γ Fishburn
X α VaRVaR α (X) ES α (X ) ES α (X ) = E [ X X VaR α (X)]. X α α VaR αvar α E [x B]B x VaR X X
VaR VaR αvar VaR VaR VaR VaR VaR Artzner et al. VaRVaR VaR VaR 2 [ t E X I { X VaR ( X )}] VaR ( X ) α 1 α ( ) dt ESα X = E [ X X VaR ( X ) ] t e 2 α = = σ 2 X α ασx 2π q 2 VaR α ( X ) 2 2 2 α VaR α ( X ) t 2 q α σ X 1 σ = e X σ 2 σ e 2 X 2σ 2 2 X 2 X = e 2 σ X = e 2σ X = σ X ασ X 2π α 2π α 2π α 2π x I {A} A q α α VaR
VaRVaR VaR VaRVaR VaR VaR VaR VaRVaR VaR VaR VaR VaR Artzner et al Artzner et al. XY σ X σ Y XY σ XY σ XY σ X σ Y X+Y σ X+Y σ 2 2 2 2 σ + σ + σ σ σ X Y X Y XY X σ = + + 2 σ + σ + 2 σ. X Y Y X Y VaREmbrechts et al.
AuU B lllu LU AA BBA VaRU VaRA u BVaRB l VaR ABL UVaR VaRAB u l VaRAVaRB u lvar ABVaRAVaRB VaR ST L 0.8% u 1,000+l 1,000+u+l L ST U 98.4% u l u+l UST 0.8% 1,000+u l 1,000+u+l VaR u l 1,000 u l uvar
100 99 VaR VaR VaR Artzner et al. 1997 Artzner et al.1999 Pflug VaR Artzner et al. VaR Pflug
VaR VaR VaR VaR VaR VaR A B A AB Rootzén and Klüppelberg
Artzner et alvar VaR VaRVaR VaRVaR VaR VaR VaR VaRBasak and Shapiro1999Klüppelberg and Korn 1998Lotz VaR VaR VaR B In estimating necessary levels of risk capital, the primary concern should be to address those disturbances that occasionally do stress institutional insolvencythe negative tail of the loss distribution that is so central to modern risk management. Greenspan2000 Basak and Shapiro VaR
VaR BIS VaR VaR VaR VaR Kp(K) i S i P i x u (W)u(W)=ln(W)E[u(W)] E.
E [u (W )] = P i. ln {W0 + x. e r. P (K ) x. max [K Si,0]}. W W 0 VaRα% VaR VaR maxe [u (W )]. {x,k} VaR maxe [u (W )], {x,k} subject to VaR 5 maxe [u (W )], {x,k} subject to 7 VaR maxe [u (W )], {x,k} subject to VaR 5 maxe [u (W )], {x,k} subject to 7 VaR VaR
VaR VaR VaR Ahn et al.var
VaR VaR VaR VaR VaRVaR 95%
A B AB n n n 100 C n AB n n n 100 C n A Bn n n C n m C n m n
E[ u( W)] = 100 n= 1 0.96 0.995 0.05 0.95 ln 1.0475 X 100 n= 1 0.04 0.995 0.05 0.95 0.96 0.005 0.05 0.95 ln 1.0475 X + 1.0075 X + 1.0075 0.1 X 0.04 0.005 0.05 0.95 1 1 n 2 + 1.055 X 100 0.9n ln 1.0475 0.1X1 + 1.0075 X 2+ 1.055 X + 1.0025 ( W 3 0 X1 X 2 X 100 100 n= 1 100 n= 1 n n n 100 n C 100 n 100 n C 100 n 100 n C 100 n 2 3 + 1.055 X 100 n C 100 n 100 0.9n + 1.0025 ( W0 X1 X 100 3 X 100 0.9n + 1.0025 ( W0 X1 X 3 100 2 X ) 100 0.9n ln 1.0475 0.1 X 1.055 1.0025. 1+ 1.0075 0.1 X + X 2 3 + 100 ( W0 X1 X ) 2 X 3 W W X A X B X 2 3 3 ) ) VaR VaR VaR VaR VaR max E [u (W )]. {X 1,X 2,X 3 } VaR max E [u (W )], subject to {X 1,X 2,X 3 } VaR 3
max E [u (W )], {X 1,X 2,X 3 } subject to 3.5 VaR max E [u (W )], subject to {X 1,X 2,X 3 } VaR 3 max E [u (W )], subject to 3.5 {X 1,X 2,X 3 } VaR A VaR A VaRVaR AVaR A VaRA VaR VaR VaR A B BVaR VaR
VaR VaR B VaR VaR VaR VaR VaR Basak and ShapiroBasak and Shapiro Basak and Shapiro VaRVaR t =0W ( 0 ) t = T W (T ) W (T )u (W (T)) = lnw (T ) BS Basak and Shapiro
db (t) = B (t)rdt, ds(t) = S(t)[ µdt+σdw(t)], w ( t ) rµσ ξ ( t ) 2 1 µ r µ r ξ ( t) exp r + t w( t) 2 σ σ. E [ξ ( T ) W ( T )] W ( 0 ). ξ ( T ) max E [lnw (T )] W ( T ) subject to E [ξ ( T ) W ( T )] W ( 0 ). W ( 0 ) W ( T ) =. ξ ( T ) W ( T ) W ( 0 ) W ( 0 ) W ( T ) = = = A 1.W (0). S (T ) µ r σ, ξ ( T ) A. S( T ) µ r σ A>0 T T
W(T) T T W(T) = A 1 µ r W(0) S(T ) σ S(T) VaR VaR 1 α %VaRVaRα P(W (0) W ( T ) VaR (α)) 1 α. VaRVaR Capital VaR (α) Capital, W (0) CapitalW W T VaR (α) W (0) W,
P(W (T ) W ) 1 α, α% VaR VaR max E [u (W (T ))], W ( T ) subject to E [ξ ( T ) W ( T )] W ( 0 ), P (W ( T ) W ) 1 α. VaR W W W W(T) W(T) W W W(T) α α S α S S(T) T T
VaR η W E [W(0) W ( T ) W ( T ) W ] η. ε η W(0) +W E [ W W ( T ) W ( T ) W ] ε. W W W ( T ) ε Baaaa E [ξ ( T ) ( W W ( T ))1 {W ( T ) W }] ε. E [ξ ( T ) ( W W ( T ))1 {W ( T ) W }] = E [ξ ( T ) ( W W ( T )) W ( T ) W ] P(W (T) W ), W W ( T ) ξ ( T ) W ( T )W P(W (T) W ) Basak and ShapiroVaR VaR W = W(0) VaR(α)VaR(α) W(0) VaR(α) W A A
W(T) W (T) W(T ) W T T S T max E [u (W (T ))], W ( T ) subject to E [ξ ( T ) W ( T )] W ( 0 ), E [ξ ( T ) (W W ( T ))1 {W ( T ) W }] ε. VaR
VaR VaRVaR VaR VaR α α
VaR VaR VaRVaR VaR VaR VaR VaR VaR VaR a auryasev VaR VaR VaR
VaR VaRVaR VaR VaR VaR extreme value theory NeftciScaillet
VaR VaR VaR VaR VaR VaR VaR VaR VaR VaRVaR VaR VaRVaR
VaR VaR B VaR VaR VaR VaR VaR
VaR Credit Suisse Financial ProductsVaR quantified using scenario analysis and controlled with concentration limits VaRVaR VaR VaR VaRVaR VaR VaR
VaR VaRVaR VaR VaR VaR
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Rockafeller R. T. and S. Uryasev, Optimization of Conditional Value-at-Risk, Journal of Risk, Vol. 2, No. 3, Spring 2000, pp. 21-41. Rootzén, H., and C. Klüppelberg, A Single Number Can t Hedge Against Economic Catastrophes, Working Paper, Munich University of Technology, 1999. Scaillet, O., Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall, Working Paper, IRES, 2000. Ulmer, A., Picture of Risk, RiskMetrics Group, 2000.