CVaR

CVaR 20 4 24 3 24 1 31

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

1 5 2 VaR CVaR 6 2.1................................................ 6 2.2 VaR........................................... 8 2.3 CVaR...................................... 9 2.4.......................................... 10 3 CVaR 11 3.1.............................................. 11 3.2............................................... 12 4 CVaR 12 4.1.................................... 12 4.2 µ Σ................................... 13 4.3 µ................................... 14 4.4.......................................... 14 5 15 5.1.......................................... 17 6 18

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,.

, 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

. 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 + 1 1 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).

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

[4]. 1, A,B, 80 20. 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 1 98.0% 80 80 160 2 0.9% 20 100 120 3 0.2% 30 30 60 4 0.9% 100 20 120 1 VaR 2.3 CVaR CVaR 2.2. 2.2 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

, (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

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.

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 + 1 1 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 + 1 1 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.191-194],.,,.,

., 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,

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)

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 + 1 1 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.

C(r) µ i, = 1,, 7. µ i, i = 1,, 7, (µ i, Σ) 3 y[k] i, k = 1,, 4000.. min θ s.t. x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 1 α + 1 1 4000 1 0.99 4000 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 2006 2010,,. A, B, C. 2006 2010 A, B, C 2., 2 A 0.002588675 B 0.000477945 C 0.006046415 3. 4. 2 4 3 A B C A 0.002308732 0.001720457 0.001741415 B 0.001720457 0.003987745 0.00241946 C 0.001741415 0.00241946 0.004607697 S C(r), r, C(r),. 4.4

4 A B C 2006-0.005311089 0.001542907 0.000488414 2007 0.005230447-0.006659508 0.003530539 2008 0.014399919 0.017666839 0.018699048 2009-0.004727987-0.00749924 0.001646234 2010 0.003823477-0.001526867 0.006584478 S S 1 2.9785 0.7603 4.0348 S 1 = 1.0 10 5 0.7603 0.6183 1.5458 4.0348 1.5458 6.2775. r, r = 2.5, 2.0, 1.5, 1.0, 0.5. 5.1 5 x 1 x 2 x 3 r=2.5 0.8110 0.0000 0.1839 r=2.0 0.7026 0.0604 0.2370 r=1.5 0.7949 0.0642 0.1408 r=1.0 0.8202 0.0663 0.1135 r=0.5 0.7690 0.0345 0.1965 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

., S 1. 2.9785 4.0348 0.7603 S 1 = 1.0 10 5 4.0348 6.2775 1.5458 0.7603 1.5458 0.6183, 6 r = 2.5, 2.0, 1.5, r = 1.0, 0.5 B 6 2 x 1 x 2 x 3 r=2.5 0.8637 0.0989 0.0374 r=2.0 0.7172 0.1538 0.1290 r=1.5 0.7192 0.1880 0.0928 r=1.0 0.8152 0.0897 0.0951 r=0.5 0.7708 0.0550 0.1742 C.,,,,. 6 CVaR,., 2006 2010,.,, Σ, C(r).,.,.

[1] A. Ben-Tal and A. Nemirovski, Robust Optimization - Methodology and Applications, Mathematical. Programming, Vol.92, pp. 453-480, 2002. [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.1707-1711, 1994. [3] H. Markowitz, Portfolio Seletion, The Journal of Finance, Vol7 pp.77-91, 1952. [4] P. Artzner, F. Delbaen, J.M. Eber, and D. Heath, Coherent Measures of Risk, Mathematical Finance, Vol.9, pp.203-228, 1999. [5] R.T. Rockafellar and S. Uryasev, Optimization of Conditional Value-at-Risk, J. Risk, Vol.2, pp.21-41, 2000. [6] S. Zhu and M. Fukushima, Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management, Operations Research, Vol.57, pp.1155-1168, 2009. [7], 1,, 1995. [8],, 1,, 2003. [9],, J, ( ),, 2003. [10],,, 2001. [11],,, Journal of the Operations Research Society of Japan, Vol.45, pp.99-101, 2001.