24 [11] [4, 8, 10, 20, 21] CreditMetrics [8] CreditMetrics CreditMetrics CreditMetrics [13] [1, 12, 16, 17] [12] [1] Conditional Value-at-Risk C
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- かねろう いんそん
- 5 years ago
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1 Transactions of the Operations Research Society of Japan Vol. 54, 2011, pp c CVaR 1 FT ( ; ) Conditional Value-at-Risk : Conditional Value-at-Risk 1. BIS [3] structuralreduced form 2 [5] CreditMetrics [8] KMV [10] 1 23
2 24 [11] [4, 8, 10, 20, 21] CreditMetrics [8] CreditMetrics CreditMetrics CreditMetrics [13] [1, 12, 16, 17] [12] [1] Conditional Value-at-Risk CVaR [14, 15] [16] 1 CVaR CVaR [2] [14, 15] [7] CVaR CVaR N 2 N N CVaR [9] CVaR 15 5 [18] CVaR
3 CVaR 25 CVaR CVaR CVaR 10 7 LP 1/ LP 1/ CVaR , (G01) (G13) j g(j) r(j) g r n(g, r) 1 j R. Merton [13] j E j 1 Ẽj 1 ) (Ẽj m m log = γ δ g(j),k ε k + 1 γ 2 δg(j),k 2 ω j (2.1) E j k=1 k = 1,..., m ε k ω j j k=1
4 26 1: r g G G G G G G G G G G G G G ε k δ g(j),k γ ε 1, ε 2,..., ε m ω j log(ẽj/e j ) j r p r p r π r (2.1) j m m γ δ g(j),k ε k + 1 γ 2 δg(j),k 2 ω j < π r(j) (2.2) k=1 2 [19] 3 Ψ r k=1 2: r (%) p r (%) Ψ r γ II [3] γ γ γ γ δ g(j),k g t (t = 1, 2,..., 58) s g (t)
5 CVaR 27 S g = (s g (1),..., s g (58)) g = 1, 2,..., 13 S g S g = δ g,1 F δ g,5 F 5 + Ω g k = 1, 2,..., 5 F k = (f k (1),..., f k (58))Ω g = (e g (1),..., e g (58)) δ g,k 3 4 3: k : k g G G G G G G G G G G G G G = {G01, G03, G04, G05, G07, G10, G11, G13} (2.3) 2 = {G08, G09} (2.4) 3 = {G02, G06, G12} (2.5)
6 28 1: g r d g,r j d j := { 1 0 d j d g,r = d j j N(g,r) N(g, r) g r 1 n(g, r) N(g, r) T g r z g,r 1 φ g,r D D [ ] = dg,r L(z, D) := (g,r) G R z g,r T d g,r n(g, r) φ g,r (g,r) G R r 2 Ψ r f(z, D) : = L(z, D) z g,r T Ψ r = T (g,r) G R (g,r) G R z g,r ( dg,r n(g, r) φ g,r Ψ r ) (3.1)
7 z g,r CVaR 29 (g,r) G R z g,r = 1, z g,r 0 (g G, r R) (3.2) Ψ r z g,r (g,r) G R z g,r 0.2 for g G r R z g,r n(g, r) for g G, r R (3.3) 0.8% 1 20% % z = (z g,r ) (g,r) G R Z z = (z g,r ) (g,r) G R z = (z g,r ) (g,r) G R γ 4. CVaR Conditional Value-at-Risk CVaR β z Z Value-at-Risk (VaR) { α(z, β) := min α R Pr[f(z, D) } α] β CVaR 1 f(z, X)dP (X) 1 β f(z,x) α(z,β) CVaR P ( ) D 2 D ν G R i H i := ( ) h i g,r (g,r) G R (g, r) h i g,r g r D CVaR 1 ν minimize α + [f(z, H i ) α] + (P ) ν(1 β) i=1 subject to (α, z) R Z [14, 15] [ ] + [ a ] + := max{0, a} H i λ i [ ] + (P )
8 30 (P ) 1 ν minimize α + λ i ν(1 β) i=1 subject to (α, z) R Z λ i f(z, H i ) α for i = 1,..., ν λ i 0 for i = 1,..., ν f z 1 10 β 1 λ i f(z, H i ) α ν µ M (P (M)) 1 minimize α + λ i ν(1 β) i M (P (M)) subject to (α, z) R Z λ i f(z, H i ) α for i M λ i 0 for i M 2 µ(1 β) ν(1 β) 4.1 (P (M)) (α, z, (λ i ) i M ) i {1, 2,..., ν} \ M λ i = 0 (α, z, (λ i ) i=1,...,ν ) 4.1 (α, z, (λ i) i M ) (P (M)) i {1, 2,..., ν} \ M f(z, H i ) α 0 (4.1) (P ) (P ) i {1, 2,..., ν} \ M λ i f(z, H i ) α (P (M)) v (α, z, (λ i) i M ) (P (M)) λ i 1/ν(1 β) (P (M))(P ) v(p (M)) v(p ) v(p (M)) v(p ) (P ) 5.
9 CVaR 31 1 ν µ 0 µ 0 M 0 M := M 0 µ := µ 0 2 (P (M)) (α, z, (λ i) i M ) 3 (α, z ) i {1, 2,..., ν} \ M f(z, H i ) α 0 (P ) 4 4 i {1, 2,..., ν} \ M f(z, H i ) α > 0 H i M M := M M µ := µ + M 2 µ ν (P (M)) (P ) (P ) ν µ (4.1) µ M H i d(h i ) d(h i ) = (g,r) G R hi g,r γ = (1) (2) d(h 1 ) d(h 2 ) d(h ν ) (3) 1,000 (α, z) H i f(z, H i ) α 3 15 (P ) β = 0.99 ˆλ i H i 2 3 ˆλ i ˆλ i % (3) (2) (1) (3) (2) (z, α ) f(z, H i ) α (z, α ) (z, α) (3) d(h i ) µ 0
10 : ˆλ i (3) (2) (1) 6. Dell Studio XPS 435MT CPU Intel Core i7 (2.66 GHz) 6GB OS Windows Vista LP FICO [6] FICO Xpress 7.0 (64bit) β = 0.99, φ g,r = 0.5 m = 5 γ = 0.45 µ ν (a) (b) (3.2) (c) (d) (3.3) (a) (c) M 0 (b) (d) M 0 LPopt total (P ) µ 0 µ 0 3 (a) µ 0 =4, (a) µ 0 =20, (a) (P ) /8 4 (a) (P ) / (a) (c) µ 0 5% γ
11 CVaR 33 µ 0 3 (a) (c) 4 (a) (c) ν (3.3) (a) (c)(b) (d) µ 0 (P ) LP 120 LP 100 opt 100 opt 80 total 80 total µ 0 µ (a) :(3.2) (b) :(3.2) LP 120 LP 100 opt 100 opt 80 total 80 total µ 0 µ (c) :(3.2)+(3.3) (d) :(3.2)+(3.3) 3: µ m γ g r z g,r n(g, r) 1 5
12 LP 1200 LP 1000 opt 1000 opt 800 total 800 total µ 0 µ (a) :(3.2) (b) :(3.2) LP 1200 LP 1000 opt 1000 opt 800 total 800 total µ 0 µ (c) :(3.2)+(3.3) (d) :(3.2)+(3.3) 4: µ 0 50 (2.2) m m γ δ g(j),k ε k + 1 γ 2 δg(j),k 2 ω j < π r(j) k=1 γ γ k=1 ω j < π r(j) j g(j) r(j) γ γ = (3.2) j 1 φ g(j),r(j) p r(j) j φ g(j),r(j) p r(j) > Ψ r(j)
13 CVaR 35 5: 1 50 (3.2) G (0.1) 1.4 (0.1) 1.2 (0.0) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.5 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.3 (0.1) 1.1 (0.1) 1.1 (0.0) 1.0 (0.1) 0.9 (0.0) 0.8 (0.0) 0.6 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.1 (0.0) 0.9 (0.1) 0.9 (0.1) 0.8 (0.1) 0.5 (0.1) 0.0 (0.0) G (0.1) 1.3 (0.1) 1.2 (0.1) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.6 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.9 (0.1) G (0.1) 1.0 (0.1) 1.1 (0.1) 0.9 (0.0) 0.8 (0.1) 0.6 (0.1) G (0.1) 1.1 (0.1) 1.1 (0.1) 1.0 (0.1) 0.9 (0.0) 0.8 (0.0) 0.5 (0.1) 0.0 (0.0) G (0.1) 1.4 (0.1) 1.2 (0.1) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.6 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.0) 1.1 (0.0) 1.1 (0.1) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.6 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.1 (0.2) 1.1 (0.1) 0.9 (0.1) 0.8 (0.1) 0.5 (0.1) 0.0 (0.0) G (0.2) 1.3 (0.0) 1.2 (0.1) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.6 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.3 (0.0) 1.2 (0.0) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.5 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.0 (0.1) 0.9 (0.1) 10 3 φ g(j),r(j) = Ψ r (3.3) 50 m γ (3.2) 7 8 γ 0.14 m m = 3 m = 5 m = 1 m = (2.5) 1 1 CVaR m = 5 m = m = 1 8 m = 0 γ γ = 0.45 m (3.2) (3.3)
14 36 6: 1 10 (3.2) G (0.2) 1.4 (0.1) 1.1 (0.2) 1.1 (0.1) 0.9 (0.1) 0.9 (0.1) 0.8 (0.1) 0.6 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.3) 1.6 (0.2) 1.2 (0.2) 1.0 (0.0) 1.1 (0.1) 1.0 (0.1) 0.8 (0.1) 0.6 (0.0) 0.1 (0.1) 0.0 (0.0) G (0.3) 1.0 (0.3) 1.0 (0.3) 1.0 (0.2) 0.7 (0.1) 0.5 (0.3) 0.0 (0.0) G (0.1) 1.3 (0.1) 1.2 (0.0) 1.1 (0.0) 1.0 (0.0) 0.9 (0.1) 0.8 (0.1) 0.6 (0.1) 0.1 (0.0) 0.0 (0.0) G (0.5) 0.8 (0.5) G (0.2) 1.0 (0.2) 1.1 (0.3) 0.9 (0.1) 0.9 (0.1) 0.5 (0.1) G (0.4) 1.0 (0.3) 1.1 (0.2) 1.0 (0.2) 0.9 (0.1) 0.9 (0.1) 0.6 (0.2) 0.1 (0.1) G (0.3) 1.3 (0.1) 1.1 (0.0) 1.1 (0.1) 0.9 (0.1) 0.9 (0.1) 0.8 (0.0) 0.5 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.1 (0.1) 1.1 (0.1) 1.0 (0.0) 1.0 (0.0) 0.8 (0.1) 0.6 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.4) 1.0 (0.2) 0.9 (0.2) 0.9 (0.2) 0.4 (0.2) 0.7 (0.1) 0.0 (0.0) G (0.4) 1.3 (0.2) 1.1 (0.1) 1.0 (0.0) 1.0 (0.1) 0.9 (0.1) 0.8 (0.1) 0.5 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.3 (0.1) 1.2 (0.1) 1.1 (0.1) 1.0 (0.1) 0.9 (0.0) 0.8 (0.0) 0.6 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 1.0 (0.2) 1.0 (0.3) : 1 50 (m, γ) = (1, 0.14) (3.2) G (0.1) 1.1 (0.1) 1.1 (0.1) 0.9 (0.0) 0.8 (0.0) 0.7 (0.0) 0.6 (0.0) 0.2 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.9 (0.1) 1.7 (0.1) 1.6 (0.0) 1.5 (0.0) 1.4 (0.1) 1.2 (0.1) 0.9 (0.0) 0.2 (0.0) 0.0 (0.0) G (0.1) 0.9 (0.2) 0.8 (0.1) 0.5 (0.1) 0.5 (0.0) 0.2 (0.1) 0.0 (0.0) G (0.1) 1.2 (0.1) 0.9 (0.0) 0.8 (0.0) 0.7 (0.0) 0.6 (0.0) 0.5 (0.0) 0.2 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.4) 0.7 (0.1) G (0.2) 1.5 (0.2) 1.4 (0.1) 1.3 (0.1) 1.1 (0.1) 0.9 (0.1) G (0.2) 1.1 (0.1) 0.9 (0.1) 0.8 (0.1) 0.7 (0.0) 0.6 (0.1) 0.2 (0.1) 0.0 (0.0) G (0.1) 1.4 (0.1) 1.2 (0.1) 1.0 (0.0) 0.9 (0.0) 0.8 (0.1) 0.7 (0.0) 0.4 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.2 (0.1) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.8 (0.0) 0.5 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.9 (0.2) 0.7 (0.1) 0.6 (0.2) 0.4 (0.2) 0.1 (0.1) 0.0 (0.0) G (0.1) 1.2 (0.1) 1.0 (0.1) 0.9 (0.0) 0.8 (0.0) 0.7 (0.0) 0.6 (0.0) 0.2 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.7 (0.1) 1.5 (0.0) 1.4 (0.0) 1.3 (0.0) 1.2 (0.0) 1.0 (0.0) 0.7 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.7 (0.1) 0.8 (0.1) 10 3 γ = 0.3 m (3.2) m = 0 5 m = (2.5) 1 1 (2.3) 2 (2.4) 3
15 CVaR 37 8: 1 50 (m, γ) = (5, 0.14) (3.2) G (0.1) 1.4 (0.1) 1.3 (0.1) 1.1 (0.1) 1.0 (0.0) 0.8 (0.0) 0.7 (0.0) 0.3 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.4) 1.9 (0.1) 1.5 (0.1) 1.4 (0.0) 1.3 (0.0) 1.1 (0.1) 1.0 (0.1) 0.6 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.1 (0.2) 1.0 (0.1) 1.1 (0.2) 0.7 (0.1) 0.3 (0.1) 0.0 (0.0) G (0.2) 1.6 (0.1) 1.4 (0.1) 1.2 (0.0) 1.1 (0.0) 0.9 (0.0) 0.8 (0.0) 0.4 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 1.2 (0.3) G (0.2) 1.3 (0.2) 1.3 (0.1) 1.0 (0.1) 1.0 (0.1) 0.5 (0.1) G (0.2) 1.3 (0.1) 1.2 (0.1) 1.1 (0.1) 0.9 (0.1) 0.7 (0.1) 0.4 (0.1) 0.0 (0.0) G (0.1) 1.2 (0.1) 1.1 (0.1) 0.9 (0.0) 0.9 (0.0) 0.7 (0.0) 0.5 (0.0) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.1 (0.1) 0.9 (0.0) 0.9 (0.1) 0.7 (0.0) 0.5 (0.0) 0.2 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 1.1 (0.1) 1.0 (0.1) 0.8 (0.2) 0.6 (0.1) 0.3 (0.2) 0.0 (0.0) G (0.2) 1.6 (0.1) 1.4 (0.0) 1.2 (0.0) 1.1 (0.1) 1.0 (0.0) 0.8 (0.1) 0.4 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.6 (0.0) 1.3 (0.1) 1.1 (0.0) 1.0 (0.0) 0.9 (0.0) 0.7 (0.0) 0.3 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 1.0 (0.1) 0.8 (0.1) : 1 50 (m, γ) = (1, 0.45) (3.2) G (0.0) 0.3 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 3.5 (0.1) 3.3 (0.1) 3.0 (0.0) 2.9 (0.1) 2.7 (0.1) 2.4 (0.1) 1.9 (0.1) 0.8 (0.1) 0.0 (0.0) G (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 0.0 (0.0) G (0.1) 2.7 (0.3) 2.6 (0.1) 2.5 (0.1) 2.2 (0.1) 1.6 (0.1) G (0.2) 0.2 (0.1) 0.0 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.4 (0.1) 1.0 (0.1) 0.8 (0.1) 0.6 (0.1) 0.5 (0.1) 0.2 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 1.4 (0.1) 1.2 (0.1) 1.1 (0.0) 0.9 (0.1) 0.6 (0.0) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.7 (0.0) 0.3 (0.1) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 2.9 (0.1) 2.6 (0.0) 2.3 (0.0) 2.2 (0.0) 2.0 (0.0) 1.8 (0.0) 1.2 (0.1) 0.1 (0.0) 0.0 (0.0) G (0.0) 0.0 (0.0) 0.0 (0.0) 10 3 γ = 0.14 γ = 0.45 m = 1 9 m = 5 10 γ = m = 1 m = 5 γ = (3.3) 11 12
16 38 10: 1 50 (m, γ) = (5, 0.45) (3.2) G (0.7) 3.3 (0.4) 1.7 (0.5) 0.4 (0.2) 0.0 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.6) 5.8 (0.3) 4.6 (0.1) 3.3 (0.2) 2.6 (0.2) 1.9 (0.3) 0.4 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.6) 1.4 (0.6) 0.5 (0.4) 0.1 (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.4) 4.5 (0.2) 3.2 (0.1) 1.6 (0.1) 1.0 (0.1) 0.2 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 3.1 (0.7) G (0.5) 2.3 (0.4) 1.6 (0.5) 0.9 (0.4) 0.0 (0.1) 0.0 (0.0) G (1.2) 3.3 (0.6) 2.2 (0.3) 1.5 (0.4) 0.8 (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.4) 2.9 (0.4) 1.7 (0.3) 0.0 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.4) 1.4 (0.3) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.9) 0.6 (0.4) 0.4 (0.2) 0.0 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 4.7 (0.2) 3.7 (0.3) 2.0 (0.3) 1.4 (0.1) 0.9 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 3.9 (0.4) 2.5 (0.1) 1.0 (0.1) 0.5 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.6) 0.0 (0.0) 0.0 (0.0) : 1 50 (m, γ) = (1, 0.45) (3.3) G (0.1) 1.2 (0.3) 0.5 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 3.3 (0.1) 2.7 (0.2) 3.2 (0.1) 3.0 (0.0) 2.7 (0.1) 2.8 (0.1) 2.6 (0.1) 1.4 (0.1) 0.0 (0.0) G (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 0.8 (0.2) 0.2 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 0.1 (0.1) G (0.0) 5.0 (0.0) 5.0 (0.0) 5.0 (0.0) 5.0 (0.0) 4.8 (0.1) G (0.4) 1.3 (0.2) 0.5 (0.1) 0.3 (0.2) 0.0 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 3.3 (0.0) 2.6 (0.0) 2.0 (0.1) 1.7 (0.0) 1.4 (0.0) 0.8 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.2) 2.7 (0.1) 2.2 (0.0) 1.9 (0.1) 1.5 (0.1) 1.1 (0.0) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 2.0 (0.1) 1.3 (0.1) 0.7 (0.1) 0.5 (0.0) 0.1 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.3) 1.1 (0.1) 0.5 (0.1) 1.0 (0.1) 0.8 (0.1) 0.5 (0.0) 0.7 (0.1) 0.5 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.1) 0.0 (0.0) 0.0 (0.0) 10 3 m = 5 γ = 0.45 (3.2) 10 (3.3) m γ II γ (3.3)
17 CVaR 39 12: 1 50 (m, γ) = (5, 0.45) (3.3) G (0.2) 1.3 (0.4) 0.2 (0.2) 0.9 (0.2) 0.2 (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.0) 4.0 (0.3) 2.7 (0.2) 3.6 (0.4) 2.9 (0.1) 2.2 (0.3) 2.5 (0.2) 2.2 (0.2) 0.2 (0.2) 0.0 (0.0) G (0.8) 1.8 (0.6) 0.8 (0.4) 0.2 (0.2) 0.8 (0.2) 0.6 (0.5) 0.0 (0.0) G (0.0) 2.3 (0.4) 1.0 (0.3) 1.9 (0.2) 1.3 (0.2) 0.5 (0.1) 0.9 (0.1) 0.7 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.3) 3.4 (0.8) G (0.7) 2.9 (0.3) 2.2 (0.4) 1.4 (0.3) 1.6 (0.3) 1.4 (0.2) G (1.4) 1.4 (0.7) 2.7 (0.3) 2.0 (0.4) 1.2 (0.3) 1.4 (0.3) 1.1 (0.4) 0.0 (0.0) G (0.2) 1.4 (0.3) 0.2 (0.2) 0.6 (0.2) 0.2 (0.2) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.5) 0.1 (0.1) 0.6 (0.2) 0.2 (0.1) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) G (0.7) 1.1 (0.8) 0.8 (0.5) 0.3 (0.5) 0.2 (0.3) 0.1 (0.1) 0.0 (0.0) G (0.0) 2.8 (0.2) 1.7 (0.3) 2.5 (0.3) 1.8 (0.1) 1.2 (0.2) 1.4 (0.3) 1.3 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.0) 1.9 (0.3) 0.5 (0.1) 1.4 (0.1) 0.8 (0.1) 0.1 (0.1) 0.4 (0.1) 0.1 (0.1) 0.0 (0.0) 0.0 (0.0) G (0.7) 0.1 (0.2) 0.0 (0.0) 10 3 γ 5 (3.3) CVaR γ m = 3 m = 5 γ CVaR 1 CVaR 5: CVaR γ (3.3) 8. CVaR CVaR
18 40 CVaR FICO Xpress FICO 2 [1] F. Andersson, H. Mausser, D. Rosen, and S. Uryasev: Credit risk optimization with conditional value-at-risk criterion. Mathematical Programming, 89 (2001), [2] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath: Coherent Measures of Risk. Mathematical Finance, 9 (1999), [3] Basel Committee on Banking Supervision: Basel II: international convergence of capital measurement and capital standards: a revised framework comprehensive version. (2006). [4] Credit Suisse Financial Products: CreditRisk+: a credit risk management framework. (1997). [5] D. Duffie and K.J. Singleton: Credit Risk (Princeton University Press, 2003). [6] FICO, [7] J. Gotoh and A. Takeda: Portfolio learning via VaR/CVaR minimization. Department of Industrial and Systems Engineering Discussion Paper Series No.08-04, Chuo University (2008). [8] G.M. Gupton, C.C. Finger, and M. Bhatia: CreditMetrics: the benchmark for understanding credit risk. J.P. Morgan & Co. Incorporated (1997). [9] M. Kaut, H. Vladimirou, S.W. Wallace, and S.A. Zenios: Stability analysis of portfolio management with conditional value-at-risk. Quantitative Finance, 7 (2007), [10] S. Kealhofer and J.R. Bohn: Portfolio management of default risk. KMV Corporation (2001). [11] H.M. Markowitz: Portfolio selection. Journal of Finance, 7 (1952), [12] H. Mausser and D. Rosen: Applying scenario optimization to portfolio credit risk. ALGO Research Quarterly, 2 (1999), [13] R.C. Merton: On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29 (1974),
19 CVaR 41 [14] R.T. Rockafellar and S. Uryasev: Optimization of conditional value-at-risk. Journal of Risk, 2 (2000), [15] R.T. Rockafellar and S. Uryasev: Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26 (2002), [16] D. Saunders, C. Xiouros, and S.A. Zenios: Credit risk optimization using factor models. Annals of Operations Research, 152 (2007), [17] F. Schlottmann and D. Seese: A hybrid heuristic approach to discrete multi-objective optimization of credit portfolios. Computational Statistics & Data Analysis, 47 (2004), [18] A. Takeda and T. Kanamori: A robust approach based on conditional value-at-risk measure to statistical learning problems. European Journal of Operational Research, 198 (2009), [19], [20] T. Wilson: Portfolio credit risk I. Risk, 10-9 (1997), [21] T. Wilson: Portfolio credit risk II. Risk, (1997), W9-77
20 42 ABSTRACT CREDIT RISK OPTIMIZATION VIA CVAR AND ITS SOLUTION Jun-ya Gotoh Yuichi Takano Yoshitsugu Yamamoto Yasuno Wada Chuo University Tokyo Institute of Technology University of Tsukuba Mizuho-DL FT Credit risk is the risk of loss stemming from borrower s default. We consider the credit risk minimization problem and propose an optimization method for minimizing the risk measured by Conditional Value-at-Risk (CVaR) criterion. Default of firms is modeled by the corporate valuation model and the factor analysis of time series data of TOPIX Sector Indices, scenarios of defaults are generated, and then CVaR minimization problem is solved. By varying the number of factors incorporated in the model as well as the coefficient that determines the impact of factors peculiar to industry type, we observe how economic trend, industry type and rating of the firms influence the defaults and the credit risk. A large number of scenarios are required to obtain a reliable implication; however, the CVaR minimization problem becomes harder to solve. We propose a simple but effective pre-treatment of the scenarios and also a solution technique. We solved the problem with a hundred thousand scenarios in about 7 seconds and that with half a million scenarios in about 35 seconds.
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