Transactions of the Operations Research Society of Japan 2007 50 42-67 2 ( 2005 9 26 ; 2006 10 9 ) 2 :,,,, 1. 2 1 Gordy & Heitfield [11] 1 2 2 1 42
43 2 3 4 5 6 5 7 2. Merton [18] Structural Zhou [27] Merton 2 Gaussian Copula 2 A
2.1. 44 Lucas [15] Moody s 25 Nagpal & Bahar [20] Standard & Poor s 19 Servigny & Olivier [24] Gordy [10] 3 4 Gordy & Heitfield [11] Moody s Standard & Poor s 2.2. Gordy [10] 1 II 5 Dietsch & Petey [7] Coface Creditreform Gordy [10] 1 3 Dullmann & Scheule [6] Gordy & Heitfield [11] Hemerle et al. [13] Boegelein et al. [1] G7 3 Vasicek [26] CreditMetrics [12] 4 B 5 Duchemin [5] Schmit [22] Calem [2]
45 2.3 0.023 1 Nagpal & Bahar [20] 3. 3.1. 6 n 2 Frey and McNeil [9] V =(V 1,,V n ) n i i i θ 0,i = θ 1,i <θ 0,i <θ 1,i = S S i S i = j V i (θ j 1,i,θ j,i ], j {0, 1}, i {1,,n} (1) S i =0 S i =1 V i θ 0,i 7 V i G 0,i V i G 0,i p i p i = P [S i =1]=P [V i θ 0,i ]=G i (θ 0,i ) (2) i t V i,t G i 1 V i,t = ρ i X t + 1 ρ 2 i ε i,t (3) X t G 1 ε i,t G 2,i i ρ i 0 1 G 0,i G 1,G 2,i ρ i G 1,G 2,i i, j ρ i ρ j 6 Schönbucher [23] [14] 7 CreditMetrics 2 Gordy [10] Finger [8]
46 i g ρ i, θ 0,i g ρ g, θ g X g,t G 1 N(0, 1) ε i,t G 2,i V g,i,t G 0 i N(0, 1) V g,i,t = ρ g X g,t + 1 ρ 2 gε i,t (4) g ρ 2 g X g,t = x g,t p g,t p g (x g,t ) (1) (3) [ 1 ρ 2gε ] i,t θ g Xg,t = x g,t p g (x g,t ) = P ρ g X g,t + [ = P ] ( ) ε i,t θ g ρ g X g,t θ g ρ g x g,t X 1 ρ 2 g,t = x g,t = G 2 (5) g 1 ρ 2 g 3.2. ρ g Gordy & Heitfield [11] (4) 1 X g,t G g t (t =1,..., T ) N g,t 1 t D g,t X g,t t N(0, 1) X g,t t X g,t = x g,t D t p g (x g,t ) N g,t 1 2 2 b(d g,t ; N g,t 1,p(x g,t )) f(d g,t ; N g,t 1,p(x g,t )) (5) ( ) N g,t 1 f(d g,t ; N g,t 1,p(x g,t )) = p(x g,t ) Dg,t (1 p(x g,t )) N g,t 1 D g,t (6) D g,t X t =(X 1,t,X 2,t...X G,t ) G X g,t (6) X t ρ =(ρ 1,ρ 2...ρ G ) θ =(θ 0,1,θ 0,2...θ g ) G Σ G φ Σ (x) x =(x 1,x 2...x G ) { T } G l(ρ, θ) = ln (f (D g,t ; N g,t 1,p g (x g,t ))) φ Σ (x)dx (7) t=1 g=1 (7) ρ θ Σ Gordy & Heitfield [11]
47 1: ρ 2 0 X g,t αg 2 + βg 2 = ρ 2 g g α g α h = ρ g ρ h ρ 2 0 g h 4. 2 4.1. 2 t g i V g,i,t N(0, 1) 2 V g,i,t = α g Y t + β g Z g,t + 1 αg 2 βgε 2 i,t (8) Y t N(0, 1) Z g,t N(0, 1) ε i,t N(0, 1) i α g,β g 0 1 α 2 g + β 2 g 1 i t g (9) 0 1 ρ 0,ρ g α g = ρ g ρ 0,β g = ρ g 1 ρ 2 0 ) V g,i,t = ρ g (ρ 0 Y t + 1 ρ 20Z g,t + 1 ρ 2 gε i,t (9) (4) 1 X g,t = (ρ 0 Y t + 1 ρ 20Z ) g,t X g,t Σ 1 ρ 2 0... ρ 2 0 Σ= ρ 2 0 1......... ρ 2 0 ρ 2 0... ρ 2 0 1 X g,t ρ 2 0 g α 2 g + β 2 g = ρ 2 g g h α g α h = ρ g ρ h ρ 2 0 1 1 2 Y t = y t,z g,t = z g,t g p g (y t,z g,t ) (5) (10)
48 1: ρ 2 g ρ gρ h ρ 2 0 ( θ g ρ g ρ 0 Y t + ) 1 ρ 2 0 Z g,t p g (y t,z g,t ) = P ε i,t 1 ρ 2 g Y t = y t,z g,t = z g,t (11) ( = G 2 θ g ρ g ρ 0 y t + ) 1 ρ 2 0 z g,t 1 ρ 2 g 4.2. 2 X g,t 1 2 3 3 1 ρ 0 =0 2 ρ 0 =1 1 1 ρ 0 =0 g X g,t T G { } l(ρ, θ) = ln f (D g,t ; N g,t 1,p g (z g,t )) φ(z g,t )dz g,t (12) t=1 g=1 φ(z) X g,t (12) T { } l g (ρ g,θ g )= ln f (D g,t ; N g,t 1,p g (z g,t )) φ(z g,t )dz g,t t=1 2 1 ρ 0 =1 g X g,t X 1,t = X 2,t = X G,t { T } G l(ρ, θ) = ln (f (D g,t ; N g,t 1,p g (y t ))) φ(y t )dy t (13) t=1 g=1
49 3 2 0 ρ 0 1 g X g,t { T G { } } l(ρ, θ) = ln (f (D g,t ; N g,t 1,p g (y t,z g,t ))) φ(z g,t )dz g,t φ(y t )dy t (14) t=1 g=1 (9) 1 V g,i,t = ρ g Z g,t + 1 ρ 2 gε i,t 1 2 V g,i,t = ρ g Y t + 1 ρ 2 gε i,t 1 3 (9) 2 ρ 0 1 2 MATLAB C 4.3. Gordy & Heitfield [11] (7) 3 X g,t ρ 2 0 (10) (14) Demey et al. [4] (14) X g,t ρ 2 0 X g,t 9) α g = α h Gordy & Heitfield [11] (14) 1 2 3 ρ 0 =0(α g =0,β g = ρ g ) ρ 0 =1(α g = ρ g,β g =0) ρ 0 8 4.4. ρ 0 8 Demey et al. [4] (9)
50 5. 1 3 5.1. ρ θ X g,t ρ 2 0 (11) N g,t 1 D g,t T D 1 3 12 (13) (14) 9 ρ, θ,ρ 0 5.2. 3 1 2 3 ρ 1 = 0.15,ρ 2 =0.10,ρ 3 =0.05 θ g = 3.3 T =60 N g,t 1 = 2 16 1000 10. 2 1 2 3 ρ 0 =0, 1, 0.5 1000 X g,t ρ 2 0 0, 1, 0.5 ρ, θ,ρ 0 11 2 ρ 0 X g,t 9 1 Gordy & Heitfield [11] 1 10 Gordy & Heitfield [11] T 11 6 6 AIC
51 2: 1 3 1 2 3 ρ 0 = 0.5 1000 2 X g,t ρ 2 0 1 2 2 1 3 1 X g,t 0 2 X g,t ρ 0 1 ρ g 1 ρ 0 ρ g 1 ρ g 3 4 N g,t 1 =2 13 1000 12 1 12 Gordy & Heitfield [11] Demey et al.
52 3: 2 ρ 0 = 0.5 3 ρ 3 13 2 3 3 ρ g (5) X g,t ρ g ρ g 1 3 3 ρ 0 ρ g 1 ρ g 3 ρ g 3 2 1 1 2 1 ρ g ρ h ρ 2 g ρ 2 g 3 [4] 13 10 5
53 4: 3 2 16 2 13 6. 14 6.1. D g,t 15 1,000 16 N g,t 1 17 18 1: 1 2: 5 1 3: 1 5 4: 500 1000 4 19 20 14 15 ( ) [25] http://www.tsr-net.co.jp/topics/zenkoku/ 16 17 http://www.nta.go.jp/category/toukei/tokei.htm 18 [3] [25] 19 20 500 1
54 1982 7 2002 7 241 21 2 3 120 1 T = 120 1 122 1 3 ρ, θ,ρ 0 ρ g ρ g ρ g =0.05, 0.10, 0.15 t 1 4 22 6.2. 4 5 6 1 2 3 ρ g 7 ρ 2 0 ρ 0 ρ 2 g 1 1 0 2 1 ρ g ρ h 3 2 ρ g ρ h ρ 2 0 5 0.01 0.03 ρ g 0.1 5 ρ g 23 ρ g ρ 0 X g,t 82 92 1 30 85 94 (92 02 ) 0.6 ρ 0 ρ g 80 (88 98 ) 30 85 94 1 4 ρ g 1 3 24 21 22 θ g θ g =Φ 1 ( D g,t / N g,t 1 ) ±0.5 23 5 24
55 2: 3:
56 4: ρ g 1 5: ρ g 2
57 6: ρ g 3 7: ρ 0 3
58 Dietsch & Petey [7] ρ g 3 1 2 3 1 2 X g,t 3 1 2 ρ 0 95 (90 00 ) 2 4 2 3 1 3 ρ 0 95 (90 00 ) 1 3 ρ g ρ 0 1 ρ 0 =0 1 3 1 2 3 4 1 3 ρ g 1 1 3 ρ g 5 8 9 10 1 2 3 θ g (1) 120 11 3 1 2 6.3. AIC 25 12 3 AIC 3 2 7 ρ 0 1 2 1 AIC ρ 0 1 1 2 AIC 1 2 3 ρ 0 =0 ρ 0 =1 ρ 0 1 2 3 3 (14) 2 ρ 2 g ρ 0 1 2 1 ρ 0 0 1 25 AIC (12) (13) (14) 1 2 8 3 9-2 ( - ) AIC [21]
59 8: θ g 1 9: θ g 2
60 10: θ g 3 11: θ g θ g =Φ 1 ( D g,t / N g,t 1 )
61 12: AIC=-2 ( - ) 3 AIC 2 2 3 1 X g,t ρ 0 ρ 2 g 1 1 3 1 3 1 ρ 0 3 10 ρ 0 3 7. 2 Gordy & Heitfield [11] 1
62 3 2 1 1 1 2 1 3 2 1 20 100 25 1 2 3 3 2 1 3
63 [1] L. Boegelein, A. Hamerle, R. Rauhmeier and H. Scheule: Modelling default rate dynamics in the CreditRisk+ framework. Risk, October (2002), 24 28. [2] P.S. Calem and J.R. Follain: The asset-correlation parameter in Basel II: for mortgages on single-family residences. Working Paper (Board of Governors of the Federal Reserve System, 2003). [3] : 2003 (, 2003). [4] P. Demey, J. Jouramin, C. Roget and T. Roncalli: Maximum likelihood estimation of default correlations. Risk, November (2004), 104 108. [5] S. Duchemin, M-P. Laurent and M. Schmit: Asset return correlation and Basel II: The case of automotive lease portfolios. Working Paper (CEB 03/007, 2003). [6] K. Dullmann, and H. Scheule: Determinants of the asset correlations of German corporations and implications for regulatory capital. Presentation paper (10th annual meeting of German Fianance Association, October 2003). [7] M. Dietsch and J. Perey: Should SME exposures be treated as retail or corporate exposures? A comparative analysis of default probabilities and asset correlations in French and German SMEs. Journal of Banking and Finance, 28 (2004), 773 788. [8] C.C. Finger: Conditional approaches for CreditMetrics portfolio distributions. Credit- Metrics Monitor (RiskMetrics Group, April 1999), 14 33. [9] R. Frey and A. McNeil: Dependent defaults in models of portfolio credit risk. Working Paper (University of Leipzig Department of Mathematics and ETHZ Department of Mathematics, 2003). [10] M. Gordy: A Comparative anatomy of credit risk models. Journal of Banking and Finance, 24 (2000), 119 149. [11] M. Gordy and E. Heitfield: Estimating default correlations from short panels of credit rating performance data. Working Paper (Federal Reserve Board, 2002). [12] G.M. Gupton, C.C. Finger, and M. Bhatia: CreditMetrics Technical Document (Risk- Metrics Group, 1997). [13] A. Hamerle, T. Liebig, and D. Rosch: Benchmarking asset correlations. Risk, November (2003), 77 81. [14] : t CDO. 2006 (, 2006), 83 117 [15] D. Lucas: Default correlation and credit analysis. Journal of Fixed Income, 5 (1995), 76 87. [16] A. Lucas, P. Klaassen, P. Spreij and S. Straetmans: An analytic approach to credit risk of large corporate bond and loan portfolios. Journal of Banking and Finance, 25 (2001), 1635 1664. [17] The MathWorks: Optimization Toolbox User s Guide (The MathWorks Inc, 2004). [18] R. Merton: On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29 (1974), 449 470.
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A. 65 ρ b ij = p ij p i p j pi (1 p i ) p j (1 p j ) p ij i, j p i,p j i, j V i,v j i, j ρ a ij = Cov(V i,v j ) Var(Vi ) Var(V j ) B. Gordy [10] 26 (5) p(x) X 1 1 2 ˆp, ˆn 1 p E[p(X)] = p = E[ˆp] 2 Var[p ( X)] 2 Φ 2 Φ p Var[p(X)] = Φ 2 (Φ 1 ( p), Φ 1 ( p), ρ 2 ) p 2 (15) ρ 2 2 Var[p(X)] Var[ˆp] ˆn (15) (16) ρ 2 Var[p(X)] = Var[ˆp] E[1/ˆn] p(1 p) 1 E[1/ˆn] (16) C. 2 27 28 1 26 Gordy [10] Dullmann & Scheule [6] 27 N g,t 1 p g (x g,t ) N g,t 1 (1 p g (x g,t )) 20 2 ( ) 1 f(d g,t ; N g,t 1,p g (x g,t )) Ng,t 1 p g (x g,t )(1 p(x g,t )) φ D g,t N g,t 1 p g (x g,t ) Ng,t 1 p g (x g,t )(1 p(x g,t )) 28 MATLAB Miranda and Fackler [19] qnwnorm 2 5 =32
66 0 1 29 MATLAB fmincon 30 D. (11) Y t Z g,t ε i,t 1. y t 2. G z g,t 3. g N g,t 1 ε i,t ɛ i,t 4. ɛ i,t θg ρg ρ 0 y t+ 1 ρ 2 0 zg,t 1 ρ 2 g i g D g,t 5. g 3 4 6. T t 1 5 29 10 5 ρ 0.5 30 fmincon 2 SQP MATLAB [17]
67 ABSTRACT ESTIMATION OF DEFAULT DEPENDENCY USING HISTORICAL DEFAULT DATA MAXIMUM LIKELIHOOD ESTIMATION OF ASSET CORRELATION WITH 2-FACTOR MODEL Toshiyuki Kitano Tokyo Institute of Technology In the thesis, I discuss the estimation of default dependency using historical default data. 2-factor model is used to represent category-separated variability of latent variables as proxy for obligor credit. For simplification, one numerical calculation for maximum likelihood estimation of factor loading and in turn asset correlation is employed. By Monte Carlo analysis of the estimator s robustness and by real estimation with historical default data, it is confirmed that the model requires less computing power than existing methods, and generates more robust estimator than other category-blinded models that deal with real data. This shows that it is important for the robust estimation of default dependency to use multiple-category default data and estimate inter-category correlation, even when only correlation within one category is required for estimation.