Transactions of the Operations Research Society of Japan Vol. 55, 2012, pp. 42 65 c ( 2011 10 25 ; 2012 3 1 ) ( PD) ( DR) 2007 DR PD 54 PD DR 0.72% PD :,,,,, 1. 1 ( PD: Probability of Default) 10 ( DR: Default Rate) PD () 54 1 [14] 42
43 2 PD DR Sommar and Shahnazarianka [19] Simons and Rolwes [17] GDP Figlewski, Frydman and Liang [7] GDP Bonfim [2] 3 Bhattacharjee et al. [1] 2002 [3] GDP [18] [16] () [20] (AR 3 ) [12] [15] PD 4 2005 2008 54 2 20 2010 3 1 602 102 [10] 3 AR (Accuracy Ratio) [21] 4 c
44 (1) AR (2) PD DR (TOPIX) (WTI) () PD DR 0.72% AR PD 2 3 PD 4 5 2. PD PD PD 2.1 2.2 2.3 2.4 2.1. CRD RDB 2 1 1 3 AR 5 10% AR AR AR c
45 PD 0 100 (1) i (g i ) k (i = 1,..., n; k = 1, 2, 3) J 1 f ij (i = 1,..., n; j = 1,..., J) α j (j = 1,..., J) β k (k = 1, 2, 3) 5 SAS/STAT R LOGISTIC p i n g i (61 61 ) z i ( ) 1 1 p i = 1 + e, z pi z i = ln = α 0 + i p i J α j f ij + j=1 3 β k (g i ) k (i = 1,..., n) (2.1) k=1 (2) zi i CS i ( ) z CS i = η 0 + (η 1 η 0 ) i Z(1%) Z(99%) Z(1%) Z(1%), Z(99%) zi 1 99 z i = Z(1%) η 0 zi = Z(99%) η 1 η 0 = 10, η 1 = 90 zi 2.2. PD 2007 1 RDB (2005 4 =100) 2007 2 60% DR 2005 2006 26 2007 2008 AR PD AR 1 2007 2008 4.5% PD DR (2.2) (Brier) N p i PD1 i 5 (2.1) ( ) 1 pi z i = ln = α 0 + α j f ij + 1 s i α j f ij + p i j V s S j V s 3 β k (g i ) k (i = 1,..., n) V V s s ( J )S s 1 s i i s 1 0 k=1 c
46 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 2005 2006 2007 2008 2009 2010 1: RDB 1: AR (IN2005 2006 OUT2007 2008 ) 2005 2006 2007 2008 46.10% 49.43% 47.76% 43.23% 1 0 0 = 1 N N (p i 1 i ) 2 (2.2) i=1 2 PD 2005 2008 0 2: (IN2005 2006 OUT2007 2008 ) 2005 2006 2007 2008 0.0053801 0.0090510 0.0142918 0.0171396 2 PD DR 6 2007 2008 DR PD 2008 0.96% PD CRD PD DR AR PD CRD 6 () 2005 DR 1% c
47 3.0% ± 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% ddr PD 2.21% 1.92% 1.38% 1.17% 1.21% 1.23% 1.25% 1.00% PDÊ 25% 75% 2005 2006 2007 2008 2: PD DR (IN2005 2006 OUT2007 2008 ) PD [4 6] 2.3. PD x 7 ε i (2.3) J 3 z i = α 0 + α j f ij + β k (g i ) k + ε i (2.3) j=1 z i i i p i (2.4) z i Q p i = Pr( z i Q) (2.4) k=1 α 0 = α 0 Q (2.4) (2.5) [ ( )] J 3 p i = Pr ε i α 0 + α j f ij + β k (g i ) k j=1 k=1 (2.5) ε i i ( ) J 3 Q i = α 0 + α j f ij + β k (g i ) k (2.6) j=1 k=1 7 [13] c
48 (2.5) ε i Q i i ε i X ξ i (2.7) 1 8 ε i = b X + 1 b 2 ξ i (2.7) X ξ i (2.5) (2.6), (2.7) ( p i = Pr b X + ) ( ) 1 b 2 ξ i Q i = Pr ξ i Q i b X (2.8) 1 b 2 1 b 2 X x p i (t X = x) (2.9) ( p i = Pr ξ i Q i 1 b 2 b 1 b 2 x ) = Φ ( Q i 1 b 2 ) b x 1 b 2 (2.9) Φ( ) 1 Gordy [8], Gordy and Heitfiled [9] BIS II 1 b b 2 (2.7) (2.3) z i (2.10) z i = α 0 J + α j f ij + 1 b 2 1 b 2 j=1 3 k=1 β k (g i) k b + 1 b 2 1 b 2 X + ξ i (2.10) (2.11) 1 b 2 = b 1 b 2 (2.11) 1 π 2 /3 3/π 1 b 2 8 X(t) i.i.d. X c
49 2.4. GDP 9 54 1 2005 2008 () (2.1) h l,t 1 (l = 1,..., L) (2.12) h l,t 1 L J 3 L z i = α 0 + α j f ij + β k (g i ) k + γ l h l,t 1 (2.12) j=1 k=1 20 (TOPIX) (WTI) (IIP) (VXJ: Volatility Index Japan) 10 () 5 (IIP) (TOPIX) (WTI ) ( 10 ) (VXJ) ( DR) [11] (TOPIX) [15] () (WTI) [14] 1 DR 3 l=1 9 SAS/STAT R LOGISTIC 10 225 c
50 AR TOPIX WTI ()VXJ AR 3: AR IIP TOPIX WTI VXJ DR AR 48.43 49.86 49.43 49.64 49.09 49.37 49.00 2005 1 () (1.43) (1.00) (1.21) (0.66) (0.94) (0.57) 2006 12 0.139 0.119 0.130 0.102 0.089 0.086 (n =263,715) (p ) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) AR 49.07 51.89 50.74 50.27 50.39 50.20 50.90 2005 1 () (2.82) (1.67) (1.20) (1.33) (1.13) (1.83) 2007 12 0.206 0.166 0.128 0.136 0.119 0.154 (n =398,010) (p ) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) AR 47.51 48.64 47.51 47.68 49.36 48.75 50.23 2005 1 () (1.13) (0.00) (0.17) (1.85) (1.24) (2.73) 2008 12 0.135-0.042 0.137 0.098 0.178 (n =541,423) (p ) (0.00) - (0.00) (0.00) (0.00) (0.00) AR % AR 1 b 11 (2.11) b WTI WTI 4: b () DR WTI 2005 1 2006 12 (n =263,715) 11.08% 8.54% 10.13% 2005 1 2007 12 (n =398,010) 15.35% 15.23% 13.46% 2005 1 2008 12 (n =541,423) 16.22% 17.50% 13.55% 5 5% p 11 A c
51 WTI (2.13) t p i t 1 ()h t 1 1 12 γ ( ) 1 1 p i = 1 + e, z pi z i = ln = α 0 + i p i J α j f ij + j=1 3 β k (g i ) k + γh t 1 (2.13) k=1 5: () 2005 1 2005 1 2005 1 2006 12 2007 12 2008 12 263,715 398,010 541,423 AR 49.00% 50.90% 50.23% p p p 1 0.0735 0.0000 0.0796 0.0000 0.0869 0.0000 2 0.2206 0.0000 0.2173 0.0000 0.2027 0.0000 3 0.1498 0.0000 0.1770 0.0000 0.1883 0.0000 4 0.0902 0.0000 0.0828 0.0000 0.0885 0.0000 5 0.0718 0.0000 0.0416 0.0001 0.0264 0.0014 6 0.1266 0.0000 0.1322 0.0000 0.1260 0.0000 7 0.0716 0.0000 0.0779 0.0000 0.0780 0.0000 8 0.0699 0.0000 0.0734 0.0000 0.0667 0.0000 9 0.1522 0.0000 0.1477 0.0000 0.1244 0.0000 1 10 0.0424 0.0000 2 11 0.0790 0.0000 0.0743 0.0000 0.0734 0.0000 3 12 0.0362 0.0013 0.0305 0.0000 0.0252 0.0000 4 13 5 14 0.0264 0.0018 6 15 0.1080 0.0000 0.0745 0.0000 0.0599 0.0000 (1 ) 1.6970 0.0000 1.7396 0.0000 1.5903 0.0000 (2 ) 2.9035 0.0000 2.9748 0.0000 2.7020 0.0000 (3 ) 1.4741 0.0000 1.5106 0.0000 1.3849 0.0000 DR 0.0857 0.0000 0.1541 0.0000 0.1778 0.0000 2005 2006 0.0857 2007 2008 12 t 2 () t 1 () c
52 0.1541 0.1778 3. PD AR () AR AR PD 3.1. PD PD 3 2 PD DR 3() PD 2007 1.55%2008 1.87% DR 0.37%0.34% (25% 75%) IN: 2005 2006, OUT: 2007 2008 IN: 2005 2007, OUT: 2008 3.0% 3.0% ddr ddr ± 2.5% 2.0% 1.5% 1.0% 0.5% 1.12% 1.00% PD 2.21% 1.92% 1.38% 1.87% 1.55% 1.26% PDÊ 25% 75% ± ± 2.5% 2.0% 1.5% 1.0% 0.5% PD 2.21% 1.92% PD«s«2.21% 1.38% 1.76% 1.49% 1.36% 1.48% 1.18% 1.00% PDÊ 25% 75% 0.0% 0.0% 2005 2006 2007 2008 2005 2006 2007 2008 3: PD DR 2005 2007 2008 3() 2008 PD 2.21% 0.72% DR 6 AR AR AR PD c
53 6: AR 2005 2006 2007 2008 48.4% 47.8% 43.2% 49.0% 47.5% 42.4% 2005 2007 2008 49.1% 43.6% 50.9% 42.9% AR PD 3.2. 3 PD 1 PD 4 PD ( 1 2 ) PD ( 1 2 ) 2 3 50% PD 4: PD () [14] α j j 1 PD PD c
54 i j ij p i = ( ) 1 z i = ezi f ij z i 1 + e z i f ij (1 + e α z i ) 2 j = α j p i (1 p i ) PD α j PD ( PD ) 3.3. PD 3.3.1. PD PD DR 5 7 PD 8 1 8 PD DR Ç È È «««s«d s Ç È È «««««d 5: PD DR ( 1: IN2005 2006 OUT2007 ) Ç È È «««s s«d Ç È È «««««d 6: PD DR ( 2: IN2005 2006 OUT2008 ) 2005 2006 2007 2008 5 6 c
55 DR 6 1 1 8 5 1.92% 0.81% 6 2.26% 0.47% 2005 2007 2008 7 5 6 PD DR 7 PD Ç È È «««s«d s Ç È È «««««d 7: PD DR ( 3: IN2005 2007 OUT2008 ) 3.3.2. PD PD 8 10 10 PD PD DR DR ( ) 2005 2006 2007 2008 8 9 DR 8 9 0.6% 10 2005 2007 2008 PD DR 0.3% c
56 PD Ø Ø j y µj y ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 8: PD DR ( 1: IN2005 2006 OUT2007 ) Ø Ø j y µj y ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 9: PD DR ( 2: IN2005 2006 OUT2008 ) 3.3.3. PD 11 12 11 2005 2006 0.4% 12 2005 2007 0.10% 0.37% 0.4% 0.7% 11 0.87% 12 0.22%0.23% 3.4. 3.3.3 (3.1) c
57 j y µj y Ø ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 10: PD DR ( 3: IN2005 2007 OUT2008 ) È Ç «««d s«««v t~ È Ç «««d s«««v t~ 11: PD DR ( 1: IN2005 2006 ) «««d s«««è Ç v t~ 12: PD DR ( 2: IN2005 2007 OUT2008 ) c
58 h s t 1 t 1 s γ s z i = α 0 + J α j f ij + j=1 3 k=1 β k (g i ) k + s S 1 s i γ s h s t 1 (3.1) S s 1 s i i s 1 0 7 2005 2006 2005 2007 7 AR 50.71%2007 DR 7: p p 1 0.0800 0.0000 (1 ) 1.6970 0.0000 2 0.2122 0.0000 (2 ) 2.9035 0.0000 3 0.1794 0.0000 (3 ) 1.4741 0.0000 4 0.0827 0.0000 DR 5 0.0333 0.0023 0.3574 0.0000 6 0.1315 0.0000 0.4348 0.0000 7 0.0815 0.0000 0.5034 0.0000 8 0.0732 0.0000 0.3833 0.0000 9 0.1441 0.0159 0.4187 0.0000 1 10 0.0475 0.0000 2 11 0.0704 0.0000 3 12 0.0304 0.0000 4 13 5 14 6 15 0.0907 0.0000 PD DR 13 12, 4. PD PD c
59 «««d s«««è Ç v t~ 13: PD DR (IN2005 2007 OUT2008 ) PD PD (EL: Expected Loss) ( PD ) 2005 2007 2008 0 () PD PD 0 (0 ) PD 1 1 (EAD) 1,000 1 2 0 3 I (4.1) EAD i i r i i i PD n n I = EAD i r i (1 1 i ) EAD i 1 i (4.1) i=1 1 i i 1 0 EAD i i { n } I = EAD r i (1 1 i ) D (4.2) i=1 i=1 c
60 D = n i=1 1 i D 4 2005 2007 2008 PD DR 8 PD 8 2008 DR PD PD 6 8 8: (IN2005 2007 OUT2008 ) n PD DR n PD DR 1 16,564 0.62% 0.82% 0.20% 4,646 0.62% 0.82% 0.21% 2 16,409 0.75% 1.03% 0.28% 7,571 0.73% 0.80% 0.07% 3 17,118 0.88% 1.21% 0.32% 10,125 0.86% 0.77% 0.08% 4 17,466 1.03% 1.48% 0.45% 12,939 1.00% 1.11% 0.10% 5 18,083 1.22% 1.74% 0.52% 16,247 1.19% 1.35% 0.16% 6 18,673 1.49% 2.32% 0.83% 20,507 1.47% 1.57% 0.09% 7 18,997 1.95% 3.16% 1.21% 27,143 1.96% 2.11% 0.15% 8 20,103 3.57% 5.24% 1.68% 44,235 4.17% 3.93% 0.24% 143,413 1.49% 2.21% 0.72% 143,413 2.21% 2.21% 0.00% = PD DR 9 14 3 1 4 108 8 100 0.7% PD 5. PD 54 PD DR c
61 9: (IN2005 2007 OUT2008 ) ( : ) n I(A) n I(B) (B A) 1 16,564 337 4,646 97 240 2 16,409 458 7,571 52 406 3 17,118 562 10,125 83 644 4 17,466 794 12,939 138 655 5 18,083 967 16,247 264 703 6 18,673 1,586 20,507 216 1,370 7 18,997 2,377 27,143 467 1,909 8 20,103 3,716 44,235 353 4,069 143,413 10,796 143,413 800 9,996 0.72% PD PD PD 2 (1) GDP (2) PD c
62 [1] A. Bhattacharjee, C. Higson, S. Holly, and P. Kattuman: Business failure in UK and US quoted firms; Impact of macroeconomic instability and the role of legal institution. Cambridge Working Papers in Economics (2004). [2] D. Bonfim: Credit risk drivers: Evaluating the contribution of firm level information and of macroeconomic dynamics. Financial Stability Report 2006, Banco de Portugal (2006). [3] : 2002 (2002). [4] CRD : CRD 3 () (2009). http://www.crd-office.net/crd/img/model3-a houkoku-new.pdf [5] CRD : CRD 3 4 (BS ) 3-a (2010). http://www.crd-office.net/crd/img/model343-a houkoku.pdf [6] CRD : CRD 3 3-a 4 (2011). http://www.crd-office.net/crd/img/model33-a4 houkoku2011.pdf [7] S. Figlewski, H. Frydman, and W. Liang: Modeling the effect of macroeconomic factors on corporate default and credit rating transitions. Stern Finance Working Paper No. FIN-06-007 (2006). [8] M. Gordy: A comparative anatomy of credit risk models, Journal of Banking and Finance. 24-1&2 (2002), 119 149. [9] M. Gordy and E. Heitfield: Estimating default correlations from short panels of credit rating performance data. Working Paper, Federal Reserve Board (2002). [10],, :.,, (): (, 2010), 83 116. [11], :. 2010 (2010), 227 237. [12], : :., 29-3 (2010), 19 44. [13] G.S. ( ): (, 1996). [14] : (, 2009). [15], :. (2009). [16] :. No.28, (2003). [17] D. Simons and F. Rolwes: Macroeconomic default modeling and stress testing. International Journal of Central Banking, 5-3 (2009), 177 204. [18] : (, 2003). c
63 [19] P. A. Sommar and H. Shahnazarian: Interdependencies between expected default frequency and the macro economy. International Journal of Central Banking, 5-3 (2009), 83 110. [20],,,, : RDB. 2007 (2007), 249 263. [21], : AR (, 2011). A. [14] (pp.167 168) i Ỹi N(0, 1) X N(0, 1) b 2 i ξ i N(0, 1) 1 Ỹ i = b i X + 1 b 2 i ξ i (A.1) Ỹi Q i Pr(Ỹi < Q i ) P D i Pr(Ỹi < Q i ) = P D i (A.2) PD b 2 i P D i ( X) X X x (A.3) P D i (x) = Pr(Ỹi < Q i X = x) = Φ ( ) Q i b i x 1 b 2 i (A.3) n t DR(P D it ) t x t P D i ( X) DR Q i b 2 i P D i DR P D i = E[P D i ( X)] = 1 n (A.2) Q i = Φ 1 (P D i ) n P D it t=1 (A.4) (A.5) V[P D i ( X)] = E[P D i ( X) 2 ] P D 2 i = 1 n 1 n (P D it P D i ) 2 (A.6) t=1 c
64 (A.7) Φ ( Q i b i x 1 b 2 i ) 2 ϕ(x)dx P D 2 i = 1 n 1 n (P D it P D i ) 2 (A.7) ϕ( ) (A.7) b 2 i b 2 i t=1 223-8522 3 14 1 E-mail: hibiki@ae.keio.ac.jp c
65 ABSTRACT ESTIMATING THE PROBABILITY OF DEFAULT IN THE CREDIT SCORING MODEL WITH MACROECONOMIC VARIABLES Norio Hibiki Kenzo Ogi Masahiro Toshiro Keio University Japan Finance Corporation Probability of default (PD) of a small company is estimated by the credit scoring model which mainly includes financial indices. Default is affected by not only a specific factor but also common factors. It is desirable to include the macroeconomic factors as explanatory variables in order to improve the accuracy of the estimated PDs. However, we have a serious problem that there are not enough time series data of default to determine the macroeconomic indices by the regression model. Recently, we begin to recognize a strong need to model the credit scoring with macroeconomic variables because the actual default rates (DRs) are higher than the estimated PDs by the serious downturn in economy from about 2007. In this paper, we determine the macroeconomic indices by using about 540,000 of loan data in Micro Business and Individual Unit of Japan Finance Corporation, and compensating for the lack of the time series data of macroeconomic factors. As a result of the analysis, we find that the previous default rate in a month is significant. We improve the accuracy of the estimated PDs by using the modified credit scoring model with the previous default rate in a month. The difference between the estimated PDs and the actual DRs can be reduced at a maximum of 0.72%. c