CVaR

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2 CVaR

3 ,.,.,. Markowitz,., (Value-at-Risk, VaR) (Conditional Value-at-Risk, CVaR). VaR, CVaR VaR. CVaR, CVaR. CVaR,,.,.,,,.,,.

4 1 5 2 VaR CVaR VaR CVaR CVaR CVaR µ Σ µ

5 1,.,.,.,,., [11]. Markowitz [3],., 2.,, [7], (Value-at-Risk, VaR) [10]. VaR α β β = 0.95, 0.99 α.var, [4], (Conditional-Value-at-Risk, CVaR) [11]. CVaR (Expected Shortfall). CVaR VaR, VaR. CVaR,.

6 , CVaR.,,.,..,,,.. 2 VaR CVaR,VaR CVaR,. 3 CVaR. 4, CVaR. 5,. 2 VaR CVaR 2.1 VaR CVaR Rockafellar Uryasev [5],VaR CVaR. i = 1,, n, i x i, i y i., y i, x = (x 1,, x n ) T, y = (y 1,, y n ) T.. f(x, y), y p(y) ( f(x, y) = x T y )., α Ψ(x, α) = p(y)dy f(x,y) α

7 . x, Ψ(x, α) α,. Ψ(x, α) α. VaR α β α, VaR β (x) = min{α Ψ(x, α) β}. VaR β (x) α β (x). Ψ(x, α) α, α β (x) Ψ(x, α) = β α., CVaR VaR, f(x,y) α CVaR β (x) = β f(x, y)p(y)dy (x) f(x,y) α β (x) p(y)dy.ψ(x, α) α, f(x,y) α β p(y)dy = 1 β, CVaR (x) CVaR β (x) = 1 f(x, y)p(y)dy 1 β f(x,y) α β (x). β CVaR. F β (x, α). F β (x, α) = α β y R m [f(x, y) α] + p(y)dy (1), [t] + = max{t, 0}. CVaR β (x) ϕ β (x)., CVaR VaR, α β (x) ϕ β (x)., 2.1 x, F β (x, α) α, ϕ β (x) F β (x, α) α., ϕ β (x) = min α R F β(x, α).

8 Shapiro Wardi[8], x G(α) = y R m [f(x, y) α] + p(y)dy G G (α) = Ψ(x, α) 1. (1) F β (x, α) α, α F β(x, α) = (1 β) 1 [Ψ(x, α) β]., F β (x, α) α Ψ(x, α) = β α., min F β(x, α) = F β (x, α β (x)) = α β (x) + 1 [f(x, y) α β (x)] + p(y)dy α R 1 β y R m, (f(x, y) α β (x))p(y)dy f(x,y) α β (x) = f(x, y)p(y)dy α β(x) p(y)dy f(x,y) α β (x) f(x,y) α β (x). 1 (1 β)ϕ β (x), 2 α β (x)(1 β), min F β(x, α) = α β (x) + (1 β) 1 ((1 β)ϕ β (x) α β (x)(1 β)) = ϕ β (x) α R. 2.2 VaR X 1,X 2, ρ( ). Artzner [4]. (monotonicity): X 1 X 2 ρ(x 1 ) ρ(x 2 ) (subadditivity): ρ(x 1 + X 2 ) ρ(x 1 ) + ρ(x 2 ) (positive homogeneity): λ > 0 ρ(λx) = λρ(x) (translation invariance): c ρ(x + c) = ρ(x) + c (Coherent measure of risk). VaR,. VaR 1

9 [4]. 1, A,B, β = 0.99, VaR 0.99 (A) = 30, VaR 0.99 (B) = 30, VaR 0.99 (A + B) = 120, VaR 0.99 (A + B) > VaR 0.99 (A) + VaR 0.99 (B).,VaR. A B A+B % % % % VaR 2.3 CVaR CVaR X Y, β (0, 1) β CVaR, ϕ β. ϕ β (X + Y ) ϕ β (X) + ϕ β (Y ) Z = X + Y. X β x β, Y β y β, Z β z β, ϕ β (X), ϕ β (Y ), ϕ β (Z) [4]. ϕ β (X) = 1 1 β E[X1 X x β ] ϕ β (Y ) = 1 1 β E[Y 1 Y y β ] ϕ β (Z) = 1 1 β E[Z1 Z z β ], 1 A A 1, 0., 1 X xβ 1 Z zβ 0 if X x β 1 X xβ 1 Z zβ 0 if X x β

10 , (1 X xβ 1 Z zβ )(X x β ) 0 (1 Y yβ 1 Z zβ )(Y y β ) 0., (1 β)(ϕ β (X) + ϕ β (Y ) ϕ β (Z)) = E[X1 X xβ + Y 1 Y yβ Z1 Z zβ ] = E[X(1 X xβ 1 Z zβ ) + Y (1 Y yβ 1 Z zβ )] x β E[1 X xβ 1 Z zβ ] + y β E[1 y yβ 1 Z zβ ] = x β {(1 β) (1 β)} + y β {(1 β) (1 β)} = 0. ϕ β (Z) ϕ β (X) + ϕ β (Y ). CVaR,,, CVaR. 2.4,, Ben-Tal Nemirovski [1]).,,., ( ),. CVaR,,, P,. 2.1 P, x X CVaR (Worst-case CVaR, WCVaR). WCVaR β (x) sup CVaR β (x) p( ) P

11 3 CVaR 3.1 y,., p( ) P,, p( ) P (2)., P, P p i ( ), i = 1,, l 1 P M., l l P M { λ i p i ( ) : λ i = 1, λ i 0, i = 1,, l} (3) i=1 i=1. P M, P M p i ( ), i = 1,, l., Λ Fβ i (x, α). Λ {λ = (λ 1,, λ l ) : l λ i = 1, λ i 0, i = 1,, l} i=1 Fβ(x, i α) α + 1 [f(x, y) α] + p i (y)dy, i = 1,, l 1 β y R n 1.1,. 3.1 x β, P M WCVaR β (x). WCVaR β (x) = min α R max i L F i β(x, α), L {1, 2,, l}. Fβ L (x, α) F L β (x, α) max i L F i β(x, α), 2.1.

12 3.1 β. min WCVaR β(x) = min F β L (x, α) x X (x,α) X R 3.2 WCVaR. 3.1 Fβ L (x, α), WCVaR. min (x,α,θ) X R R θ s.t. α β [f(x, y) α] + p i (y)dy θ, i = 1,, l y R m,. i = 1,, l, y i [k] k, Si, WCVaR. min (x,α,θ) X R R s.t. α + 1 S i (1 β) θ S i k=1 [f(x, yi [k] ) α]+ θ, i = 1,, l u = (u 1 ; ; u l ) R n, n = l i=1 S i, WCVaR. min s.t. θ x X α β 1 S i S i k=1 ui k θ, i = 1,, l u i k f(x, yi [k] ) α, k = 1,, Si, i = 1,, l u i k 0, k = 1,, Si, i = 1,, l, f(x, y) x, X,. 4 CVaR 4.1.,. =,, J [9, pp ],.,,.,

13 ., 1,.,.,.,, r, r e, r = log(1 + r e ), exp(r) > 0, r e 1.,,., 1 + r e.,.. p(x) = 1 (ln x µ)2 exp( 2πσx 2σ 2 ), µ σ,,.,., m. 1 p(y) = ( ) m 1 1 exp( 1 2π Σ y 1 y 2 y m 2 (log y µ)t Σ 1 (log y µ)), log y = (log y 1,, log y m ) T, µ R m Σ R m m. 4.2 µ Σ, µ,σ. µ,,., Σ,., Σ, µ C. 2.4, CVaR,

14 P,.,,, µ C(r) CVaR., r. 4.3 µ, [8, pp.32-36] µ.,, µ center. T, µ 1,, µ T., µ center = 1 T T t=1 µt, µ center T, µ center, µ center.,.,., S. (µ µ center ) T S 1 (µ µ center ) = r 2 (4) r > 0, r µ center., µ C(r). 4.4, WCVaR. min WCVaR β(x) = min x X x X sup p( ) P CVaR β (x), X, P (µ C, Σ)., µ C(r) 3.1., C(r), (2)

15 P,, µ i, i = 1,, l. µ i p i ( ) (3). m, m, l l = 2m + 1., p i (y), i = 1,, l (µ i, Σ) m, p i 1 (y) = ( ) m 1 1 exp( 1 2π Σ y 1 y 2 y m 2 (log y µ i) T Σ 1 (log y µ i )). i = 1,, l, p i ( ) y[k] i, k = 1,, Si, f(x, y[k] i ) = xt y[k] i, WCVaR. 5 min s.t. θ x X α β 1 S i S i k=1 ui k θ, i = 1,, l u i k xt y i [k] α, k = 1,, Si, i = 1,, l u i k 0, k = 1,, Si, i = 1,, l 4. Matlab optimiztion toolbox linprog. 4 β = 0.99 CVaR,. m = 3, x = (x 1, x 2, x 3 ) T R 3, X = {x R 3 x 1 + x 2 + x 3 = 1, x 1, x 2, x 3 0}. 4.5, m C(r) l l = 2m + 1, l = 7. S i = 4000, i = 1,, 7.. Σ (4) µ center S. r.

16 C(r) µ i, = 1,, 7. µ i, i = 1,, 7, (µ i, Σ) 3 y[k] i, k = 1,, min θ s.t. x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 1 α k=1 ui k θ, i = 1,, 7 u i k xt y[k] i α, k = 1,, 4000, i = 1,, 7 0, k = 1,, 4000, i = 1,, 7 u i k ,,. A, B, C A, B, C 2., 2 A B C A B C A B C S C(r), r, C(r),. 4.4

17 4 A B C S S S 1 = r, r = 2.5, 2.0, 1.5, 1.0, x 1 x 2 x 3 r= r= r= r= r= S 1 i j S 1 (i, j). 5 A r. 3,., B C C. µ., C(r) (4), S 1 (3, 3) S 1 (2, 2) 10, r C B. B C C

18 ., S S 1 = , 6 r = 2.5, 2.0, 1.5, r = 1.0, 0.5 B 6 2 x 1 x 2 x 3 r= r= r= r= r= C.,,,,. 6 CVaR,., ,.,, Σ, C(r).,.,.

19 [1] A. Ben-Tal and A. Nemirovski, Robust Optimization - Methodology and Applications, Mathematical. Programming, Vol.92, pp , [2] A. Shapiro and Y. Wardi, Nondifferentiability of The Steady-State Function in Discrete Event Dynamic Systems, IEEE Transactions on Automatic Control, Vol.39, pp , [3] H. Markowitz, Portfolio Seletion, The Journal of Finance, Vol7 pp.77-91, [4] P. Artzner, F. Delbaen, J.M. Eber, and D. Heath, Coherent Measures of Risk, Mathematical Finance, Vol.9, pp , [5] R.T. Rockafellar and S. Uryasev, Optimization of Conditional Value-at-Risk, J. Risk, Vol.2, pp.21-41, [6] S. Zhu and M. Fukushima, Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management, Operations Research, Vol.57, pp , [7], 1,, [8],, 1,, [9],, J, ( ),, [10],,, [11],,, Journal of the Operations Research Society of Japan, Vol.45, pp , 2001.

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01-._.. Journal of the Faculty of Management and Information Systems, Prefectural University of Hiroshima 2014 No.6 pp.43 56 43 The risk measure for resilience in the inventory control system Nobuyuki UENO, Yu