ABS CDO ABS (CDO) ABS CDO 6 7 RMBS ABS ABS ABS CDO CDO 1 ABS CDO 2 3 4 ABS CDO (B)
1 2007 2009 RMBS CDO (ABS ABS CDO 2007 2008 IMF 2008 4 9,450 10 1.5 1 4,050 2009 2 2.2 IMF, Global Financial Stability Report 1) 2008 10 Alt-A 4,250 ABS ABS CDO 9,800 2/3 72.5% ABS CDO ABS 2 2008 10 2.8 ABS CDO ABS CDO 6 7 RMBS ABS RMBS RMBS CDO CDO 1) 2008 4 10 2
ABS CDO ABS CDO 2 CDO ABS CDO 3 4 5 6 2 CDO CDO 1 2.1 CDO CDO CDS 2000 RMBS ABS CDO CDO RMBS 2) ABS CDO CDO ABS RMBS 2004 06 RMBS ABS CDO 1 CDO ABS CDO ABS CDO 2007 ABS CDO 2) ABS CDO Gorton (2009) 3
(10 億ドル ) 400 300 ハイイールドローン投資適格債仕組債その他 200 100 0 2005 Q1-Q2 2005 Q3-Q4 2006 Q1-Q2 2006 Q3-Q4 2007 Q1-Q2 2007 Q3-Q4 2008 Q1-Q2 2008 Q3-Q4 1: CDO 51% RMBS CMBS CMO ABS CDO CDS SIFMA ABS CDO CDO RMBS RMBS CDO ABS CDO 2.2 RMBS t i L i (t) L i (t) A i i LGD i i τ i i L i (t) = A i LGD i ι {τi t} (1) ι t τ 1 0 3) 3) RMBS 4
N L L(t) = N L i (t) = i=1 N A i LGD i ι {τi t} (2) i=1 (2) A i LGD i τ i G(τ 1,, τ N ) G i (τ i ) τ i 4) 5) V i (i = 1,, N) τ 1,, τ N G i (τ i V ) G(τ 1,, τ N V ) = N G i (τ i V ) (3) G i (τ i V ) G(τ 1,, τ N V ) V G V i=1 4) CDO Laurent and Gregory (2005) Li (2008) 5) McNeil et al. (2005) 1 ρ 1 Heitfield (2008) 5
G(τ 1,, τ N ) = = i=1 G(τ 1,, τ N V = v)dg V (v) N G i (τ i V = v)dg V (v) (4) X i τ i τ i = G 1 ( i Φ(Xi ) ) Φ V X 1,, X N τ 1,, τ N X i (5) X i = ρ i V + 1 ρ i ε i (i = 1,, N) (5) V V ε i 6) 1 1 ρ i [0, 1] X i i ρ i Corr(X i, X j ) Corr(τ i, τ j ) 7) G i (τ i V ) G i (τ i V = v) = Prob{X i Φ 1( G i (τ i ) ) V = v} { = Prob ε i Φ 1( G i (τ i ) ) } ρ i v 1 ρi = Φ ( Φ 1 ( G i (τ i ) ) ) ρ i v 1 ρi (6) 8) 3 6) G i (τ i ) [0, 1] X i = Φ 1`G i (τ i ) 7) 2 Corr(X i, X j ) = ρ i ρ j ρ Corr(X i, X j ) = ρ ρ i = a 2 i X i = a i V + p 1 a 2 i ε i ρ i 0.1 a i 0.3 8) (6) (4) Li (2000) 6
LGD i (i = 1,, N) G i ( ) (i = 1,, N) ρ i (i = 1,, N) ρ i i 2.3 RMBS CDO RMBS RMBS RMBS RMBS 1 RMBS RMBS RMBS RMBS RMBS CDO RMBS RMBS CDO CDO RMBS CDO 1 3 RMBS CDO RMBS RMBS RMBS CDO 3.1 RMBS N RMBS M CDO RMBS 3 RMBS CDO CDO 7
RMBS N RMBS M CDO CDO RMBS 3 RMBS CDO 9) 1% 1 CDO 3.2 1,000 N = 1, 000 3% 5 LGD = 0.5 0.1 ρ = 0.1 RMBS CDO 5 10,000 Multiply-With-Carry 10) 3.3 9) 10) Ox 8
頻度 14% 12% 1 8% 6% 4% 2% 頻度累積頻度 累積頻度 10 8 6 4 2 2 4 6 8 10 損失率 2: 3.3.1 2 5 3% 7% 99% 99% 2 RMBS 8 1% 11) ρ 3 ρ 0.1 0.5 3.4 3.3.2 8 1% RMBS 8 1 1 4 5 11) Greenlaw et al. (2008) 2005 2006 8 A 9
頻度 5 4 ρ=0 ρ=0.1 ρ=0.5 3 2 1 1 2 3 4 損失率 3: 頻度 10 8 住宅ローンプール RMBSシニア RMBSメザニン RMBSエクイティ 6 4 2 2 4 6 8 10 超過損失率 4: RMBS 10 10 4 10 19% 0.2% 2 0.2% 97% 1 10
対数頻度 頻度 3 2.5 2 1.5 1 住宅ローンプール RMBS シニア RMBS メザニン RMBS エクイティ 10 1 1% 0.5 0 2 4 6 8 10 損失率 5: RMBS 4 5 12) 4 3.3.3 RMBS CDO RMBS RMBS 10 CDO CDO 10 M = 10 RMBS RMBS CDO 13) 6 CDO RMBS 99% 7 5 1% 3 12) 13) 11
頻度 8 7 6 5 4 3 2 1 頻度累積頻度 累積頻度 10 8 6 4 2 2 4 6 8 10 損失率 6: CDO RMBS 頻度 5 4 RMBS メザニンプール 住宅ローンプール 3 2 1 2 4 6 8 10 超過損失率 7: RMBS CDO 1% 3 1 6 7 RMBS CDO RMBS RMBS CDO RMBS 12
頻度 4 3 RMBSメザニンプール CDOシニア CDOメザニン CDOエクイティ 2 1 2 4 6 8 10 超過損失率 8: CDO 10 10 1: 99%VaR 99%ES 2 100. 7.1% 6% 17.8% 19.5% 0.3% RMBS 0.2% 0. 0. 0. 0. RMBS 19. 5.4% 77.5% 90. 10.2% RMBS 100. 65.2% 10 100. 100. 97. CDO 1.3% 0.7% 20.3% 63.3% 1. CDO 14.1% 5.8% 100. 100. 9.3% CDO 24.7% 17.4% 100. 100. 19.6% 14) 8 CDO 5 RMBS CDO RMBS 4 CDO 1 RMBS 0.2% 19. 100. CDO 1.3% 14.1% 24.7% 14) CDO CDO Whetten and Adelson (2005) 13
1 1 2 3 99% VaR 4 99% ES 99% 5 2 3 5 1 RMBS CDO RMBS 1 2 RMBS 5.4% 3 5 3 2 RMBS RMBS CDO 1% 99%VaR RMBS RMBS 99% CDO 99% 20.3% RMBS 3 CDO RMBS CDO CDO RMBS 99% 63.3% CDO 99%VaR 17.8% CDO 10 10 1 2 3.4 2 1.5 3% 4.5% 1 CDO % RMBS CDO RMBS 14
2: 99%VaR 99%ES 2 100. 10.3% 8% 23.4% 25.7% 3.4% (±0.) (+3.3%) (+2%) (+5.6%) (+6.2%) (+3.2%) RMBS 3.4% 0.1% 4.2% 7.2% 0. (+3.1%) (+0.1%) (±) (+4.2%) (+7.1%) (+0.) RMBS 47.1% 19. 100. 100. 31.8% (+28.1%) (+13.7%) (±) (+22.5%) (+10.) (+21.5%) RMBS 100. 83.3% 10 100. 100. 99.5% (0.) (+18.1%) (±) (±0.) (±0.) (+2.6%) CDO 9. 5.8% 100. 100. 7.5% (+7.7%) (+5.1%) (±) (+79.7%) (+36.7%) (+6.5%) CDO 39.1% 21.4% 100. 100. 30.3% (+25.) (+15.6%) (±) (±0.) (±0.) (+21.) CDO 55.3% 44.8% 100. 100. 48.4% (+30.7%) (+27.3%) (±) (±0.) (±0.) (+28.7%) 8 1 1 CDO 3 6 1 1.5 10.3% RMBS 1% ( RMBS CDO 9% RMBS CDO CDO 3 ρ 0.1 0.5 RMBS RMBS CDO CDO 99% 10 15
3: 99%VaR 99%ES 2 94.9% 7.1% 1% 39.4% 43.2% 10.7% (-5.2%) (+0.1%) (-5%) (+21.7%) (+23.7%) (+10.4%) RMBS 10.6% 1.2% 24.3% 29. 1.8% (+10.4%) (+1.2%) (±) (+24.3%) (+28.9%) (+1.8%) RMBS 24.9% 16.9% 100. 100. 20.8% (+5.9%) (+11.5%) (±) (+22.5%) (+10.) (+10.6%) RMBS 94.9% 45.1% 10 100. 100. 59.7% (-5.2%) (-20.) (±) (±0.) (±0.) (-37.3%) CDO 13.9% 12.1% 100. 100. 13.3% (+12.6%) (+11.4%) (±) (+79.7%) (+36.7%) (+12.3%) CDO 22.8% 17.9% 100. 100. 20.4% (+8.6%) (+12.2%) (±) (±0.) (±0.) (+11.1%) CDO 27. 24.3% 100. 100. 25.2% (+2.4%) (+6.8%) (±) (±0.) (±0.) (+5.6%) 4: RMBS 99%VaR 99%ES 2 CDO 1.6% 0.9% 38.3% 75.6% 1.2% (+0.3%) (+0.2%) () (+17.9%) (+12.3%) (+0.2%) CDO 14.2% 6.1% 100. 100. 9.5% (+0.) (+0.3%) () (0.) (0.) (+0.2%) CDO 29.9% 17.8% 100. 100. 20.1% (+5.3%) (+0.3%) () (0.) (0.) (+0.5%) 3.5 RMBS RMBS RMBS RMBS CDO CDO RMBS 10 100 100 RMBS CDO 4 RMBS RMBS CDO 16
15) 4 ABS CDO 2 16) 4.1 9 1,000 RMBS 10 CDO RMBS 8 1 1 CDO 3 1 6 9 RMBS RMBS RMBS RMBS CDO 2 ABS CDO BIS (2008) Whetten and Adelson (2005) 15) RMBS 16) BIS (2008) ABS CDO ABS CDO ABS CDO RMBS BIS (2008) CDO RMBS 10 1, 000 = 10, 000 RMBS 10 17
損失率 ローンプール 損失率 RMBS メザニン 10 10 8 8 6 証券化 6 4 4 2 2 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 分散化 損失率 RMBS メザニンプール 損失率 CDO メザニン 10 10 8 8 6 6 4 4 2 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 再証券化 2 0.8 0.9 1.0 小 システマティック リスク 大 9: CDO RMBS CDO RMBS CDO V = v 1 Φ(v) 4.2 2 10 CDO 1 1 8 CDO 1 3 6 10 18
損失率 10 8 基準ケース トランシェの規模 : 大 信用補完 : 大 6 4 2 0.95 0.96 0.97 0.98 0.99 1.00 小 システマティック リスク 大 10: CDO 3 1 6 1 3 6 1 1 8 10 17) 4.3 3.4 17) L A B A l l = max{l A, 0} max{l B, 0} B A l dl/dl L < A A < L < B 1/(B A) B A 19
損失率 RMBS シニア 損失率 CDO シニア 10 8 6 4 基準ケースデフォルト確率 : 大デフォルト相関 : 大 10 8 6 4 2 2 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 損失率 RMBS メザニン 損失率 CDO メザニン 10 10 8 8 6 6 4 4 2 2 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 損失率 RMBS エクイティ 損失率 CDO エクイティ 10 10 8 8 6 6 4 4 2 2 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 0.0 0.2 0.4 0.6 0.8 1.0 小 システマティック リスク 大 11: 3% 0.1 4.5% 0.5 11 RMBS RMBS CDO 3.4 RMBS 8 1 1 CDO 3 6 1 RMBS CDO RMBS CDO 20
CDO RMBS RMBS CDO RMBS CDO 5 CDO 2007 RMBS (SEC) 2007 8 S&P 3 RMBS CDO 2008 7 8 2008 12 5.1 RMBS CDO 2002 RMBS CDO 21
ABS CDO CDO ABS RMBS RMBS CDO RMBS 5 CDO 3 RMBS 18) RMBS RMBS 5.2 BIS(2008) CDO CDO BIS (2008) 18) RMBS CDO CDO CDO CDO 22
II ABS CDO 2008 4 2009 1 6 CDO 1 ABS CDO 2 1 3 4 23
ABS ABS CDO 24
1 RMBS 1 i (= 1,, N) τ i τ i G i ρ G i (i = 1,, N) ρ τ i (i = 1,, N) 1. V ε i (i = 1,, N) V N(0, 1), ε i N(0, 1) (i = 1,, N) 2. X i = Φ 1( G i (τ i ) ) Φ X i = ρv + 1 ρε i 3. τ i τ i = G 1 ( i Φ(Xi ) ) t = 1,, T D(t) N(t) N D(t) = ι {τi =t} ι i=1 N(t) = N(t 1) D(t), N(0) = N RMBS j (= 1,, M) RMBS FSj R (t) F Mj R (t) FEj R (t) C R S CR M N j (t) 1 I(t) LGD 25
t = 0,, T 1 1. RMBS t+1 N j (t)i(t) RMBS FSj R (t)cr S + FMj R (t)cr M spr R j (t + 1) spr R j (t + 1) = N j (t)i(t) (F R Sj(t)C R S + F R Mj(t)C R M) 2. L R j (t + 1) L R j (t + 1) = (N j (t) N j (t + 1)) 1 LGD Loss R Ej (t + 1) Loss R Ej(t + 1) = min{f R Ej(t) + spr R j (t + 1), L R j (t + 1)} t + 1 F R Ej(t + 1) = F R Ej(t) + spr R j (t + 1) Loss R Ej(t + 1) 3. Loss R Mj (t + 1) { } Loss R Mj(t + 1) = min FMj(t), R max{0, L R j (t + 1) FEj(t) R spr R j (t + 1)} t + 1 F R Mj(t + 1) = F R Mj(t) Loss R Mj(t + 1) 4. Loss R Sj (t + 1) { } Loss R Sj(t + 1) = min FSj(t), R max{0, L R j (t + 1) FEj(t) R spr R j (t + 1) FMj(t)} R t + 1 F R Sj(t + 1) = F R Sj(t) Loss R Sj(t + 1) CDO CDO FS C(t) F M C (t) FE C (t) 26
C C S CC M t = 0,, T 1 1. CDO t + 1 RMBS M j=1 F R Mj (t)cr M CDO F C S (t)cc S +F C M (t)cc M spr C (t+1) spr C (t + 1) = M FMj(t)C R M R (FS C (t)cs C + FM(t)C C M) C j=1 2. CDO L C (t + 1) RMBS L C (t + 1) = M Loss R Mj(t + 1) j=1 Loss C E (t + 1) Loss C E(t + 1) = min{f C E (t) + spr C (t + 1), L C (t + 1)} t + 1 F C E (t + 1) = F C E (t) + spr C (t + 1) Loss C E(t + 1) 3. Loss C M (t + 1) { } Loss C M(t + 1) = min FM(t), C max{0, L C (t + 1) FE C (t) spr C (t + 1)} t + 1 F C M(t + 1) = F C M(t) Loss C M(t + 1) 4. Loss C S (t + 1) { } Loss C S (t + 1) = min FS C (t), max{0, L C (t + 1) FE C (t) spr C (t + 1) FM(t)} C t + 1 F C S (t + 1) = F C S (t) Loss C S (t + 1) 27
N I(t) G(t) ρ LGD RMBS CDO F S R, F M R, F E R, F S C, F M C, F E C, RMBS CDO CS R, CR M, CC S, CC M, RMBS CDO T 10,000 Multiply-With-Carry 2 ρ ρ τ h h 1 e ht (t 0) G(t) = Pr{τ t} = 0 (t < 0) G G 1 (s) = 1 log(1 s), (0 s < 1) h E[τ] = 1 h, Var(τ) = 1 h 2 τ i τ j h i h j Corr(τ i, τ j ) = E[τ iτ j ] E[τ i ]E[τ j ] Var(τi )Var(τ j ) = E[τ iτ j ] 1 h i h j 1 h i h j = h i h j E[τ i τ j ] 1 28
1 ρ X i X j τ i = G 1( Φ(X i ) ), τ j = G 1( Φ(X j ) ) E[τ i τ j ] = E[G 1( Φ(X i ) ) G 1( Φ(X j ) ) ] = 1 h i h j E[log ( 1 Φ(X i ) ) log ( 1 Φ(X j ) ) ] Corr(τ i, τ j ) = E[log ( 1 Φ(X i ) ) log ( 1 Φ(X j ) ) ] 1 = log ( 1 Φ(x) ) log ( 1 Φ(y) ) ϕ(x, y; ρ)dxdy 1 ϕ(x, y; ρ) ρ { } 1 ϕ(x, y; ρ) = 2π 1 ρ exp 1 2 2(1 ρ 2 ) (x2 2ρxy + y 2 ) 4 x 4 4 y 4 dx dy 0.01 12 ρ ρ = 0.5 ρ 2 1 数係関相の刻時トルォフデ 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 ρ ( 状態変数の相関係数 ) 12: ρ 45 29
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