バリュー・アット・リスクのリスク指標としての妥当性について ― 理論的サーベイによる期待ショートフォールとの比較分析―

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2 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 2000Ulmer FISC VaR Rockafeller and Uryasev

3 VaR VaR VaR Artzner et alvar VaR VaR VaR VaR VaR VaR VaR

4 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

5 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

6 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 ) 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

7 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 σ + σ + σ σ σ X Y X Y XY X σ = σ + σ + 2 σ. X Y Y X Y VaREmbrechts et al.

8 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

9 VaR VaR VaR Artzner et al Artzner et al.1999 Pflug VaR Artzner et al. VaR Pflug

10 VaR VaR VaR VaR VaR VaR A B A AB Rootzén and Klüppelberg

11 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

12 VaR BIS VaR VaR VaR VaR Kp(K) i S i P i x u (W)u(W)=ln(W)E[u(W)] E.

13 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

14 VaR VaR VaR Ahn et al.var

15 VaR VaR VaR VaR VaRVaR 95%

17 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

18 E[ u( W)] = 100 n= ln X 100 n= ln X X X n X n ln X X X ( W 3 0 X1 X 2 X n= n= 1 n n n 100 n C 100 n 100 n C 100 n 100 n C 100 n X 100 n C 100 n n ( W0 X1 X X n ( W0 X1 X X ) n ln X X + X ( W0 X1 X ) 2 X 3 W W X A X B X ) ) 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

19 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

22 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

23 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

24 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,

25 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

26 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

27 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

28 VaR VaRVaR VaR VaR α α

29 VaR VaR VaRVaR VaR VaR VaR VaR VaR VaR a auryasev VaR VaR VaR

30 VaR VaRVaR VaR VaR VaR extreme value theory NeftciScaillet

31 VaR VaR VaR VaR VaR VaR VaR VaR VaR VaRVaR VaR VaRVaR

32 VaR VaR B VaR VaR VaR VaR VaR

33 VaR Credit Suisse Financial ProductsVaR quantified using scenario analysis and controlled with concentration limits VaRVaR VaR VaR VaRVaR VaR VaR

34 VaR VaRVaR VaR VaR VaR

35 B Ahn, D, J. Boudoukh, and M. Richardson, Optimal Risk Management Using Options, Journal of Finance, Vol. 54, No. 1, February 1999, pp Artzner, P., F. Delbaen, J. M. Eber, and D. Heath, Thinking Coherently, Risk, Vol. 10, No. 11, November 1997, pp , and, Coherent Measures of Risk, Mathematical Finance, Vol. 9, No. 3, July 1999, pp Basak, S., and A. Shapiro, Value-at-Risk Based Risk Management: Optimal Policies and Asset Prices, working paper, The Rodney White Center for Financial Research, Credit Suisse Financial Products, Credit Risk + : A Credit Risk Management Framework, Danielsson, J., The Emperor has no Clothes: Limits to Risk Modelling, Working Paper Series W00:04, Institute of Economic Studies, University of Iceland, June Embrechts, P., A. McNeil, and D. Straumann Correlation and Dependency in Risk Management: Properties and Pitfalls, Preprint, ETH Zürich, Fishburn, P. C., Mean-Risk Analysis with Risk Associated with Below-Target Returns, American Economics Review, Vol. 67, No. 2, March 1977, pp Greenspan, A., Remarks at the 36 th Annual Conference on Bank Structure and Competition, Federal Reserve Bank of Chicago, Kim, J., and J. Mina, RiskGrades Technical Document, RiskMetrics Group, May Klüppelberg, C. and R. Korn, Optimal Portfolios with Bounded Value-at-Risk, Working Paper, Munich University of Technology, Lotz, C. M. J., Optimal Shortfall Hedging of Credit Risk, Working Paper, Faculty of Economics, University of Bonn, Neftci, S. N., Value at Risk Calculations, Extreme Events, and Tail Estimation, Journal of Derivatives, Spring Pflug, G. C., Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk, Probabilistic Optimization: Methodology and Applications, Kluwer Academic Publishers, 2000, pp

36 Rockafeller R. T. and S. Uryasev, Optimization of Conditional Value-at-Risk, Journal of Risk, Vol. 2, No. 3, Spring 2000, pp Rootzén, H., and C. Klüppelberg, A Single Number Can t Hedge Against Economic Catastrophes, Working Paper, Munich University of Technology, Scaillet, O., Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall, Working Paper, IRES, Ulmer, A., Picture of Risk, RiskMetrics Group, 2000.

CVaR

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................................................