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 [

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1 Transactions of the Operations Research Society of Japan Vol. 55, 2012, pp c ( ; ) ( 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

2 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 [10] 3 AR (Accuracy Ratio) [21] 4 c

3 44 (1) AR (2) PD DR (TOPIX) (WTI) () PD DR 0.72% AR PD 2 3 PD PD PD PD CRD RDB AR 5 10% AR AR AR c

4 45 PD (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 RDB ( =100) % DR AR PD AR % 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

5 : RDB 1: AR (IN OUT ) % 49.43% 47.76% 43.23% = 1 N N (p i 1 i ) 2 (2.2) i=1 2 PD : (IN OUT ) PD DR DR PD % PD CRD PD DR AR PD CRD 6 () 2005 DR 1% c

6 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% : PD DR (IN OUT ) 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

7 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

8 GDP () (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 c

9 50 AR TOPIX WTI ()VXJ AR 3: AR IIP TOPIX WTI VXJ DR AR () (1.43) (1.00) (1.21) (0.66) (0.94) (0.57) (n =263,715) (p ) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) AR () (2.82) (1.67) (1.20) (1.33) (1.13) (1.83) (n =398,010) (p ) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) AR () (1.13) (0.00) (0.17) (1.85) (1.24) (2.73) (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 (n =263,715) 11.08% 8.54% 10.13% (n =398,010) 15.35% 15.23% 13.46% (n =541,423) 16.22% 17.50% 13.55% 5 5% p 11 A c

10 51 WTI (2.13) t p i t 1 ()h t 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: () , , ,423 AR 49.00% 50.90% 50.23% p p p (1 ) (2 ) (3 ) DR t 2 () t 1 () c

11 PD AR () AR AR PD 3.1. PD PD 3 2 PD DR 3() PD % % DR 0.37%0.34% (25% 75%) IN: , OUT: IN: , OUT: % 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% : PD DR () 2008 PD 2.21% 0.72% DR 6 AR AR AR PD c

12 53 6: AR % 47.8% 43.2% 49.0% 47.5% 42.4% % 43.6% 50.9% 42.9% AR PD PD 1 PD 4 PD ( 1 2 ) PD ( 1 2 ) % PD 4: PD () [14] α j j 1 PD PD c

13 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 PD PD DR 5 7 PD PD DR Ç È È «««s«d s Ç È È «««««d 5: PD DR ( 1: IN OUT2007 ) Ç È È «««s s«d Ç È È «««««d 6: PD DR ( 2: IN OUT2008 ) c

14 55 DR % 0.81% % 0.47% PD DR 7 PD Ç È È «««s«d s Ç È È «««««d 7: PD DR ( 3: IN OUT2008 ) PD PD PD PD DR DR ( ) DR % PD DR 0.3% c

15 56 PD Ø Ø j y µj y ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 8: PD DR ( 1: IN OUT2007 ) Ø Ø j y µj y ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 9: PD DR ( 2: IN OUT2008 ) PD % % 0.37% 0.4% 0.7% % %0.23% (3.1) c

16 57 j y µj y Ø ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ 10: PD DR ( 3: IN OUT2008 ) È Ç «««d s«««v t~ È Ç «««d s«««v t~ 11: PD DR ( 1: IN ) «««d s«««è Ç v t~ 12: PD DR ( 2: IN OUT2008 ) c

17 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 AR 50.71%2007 DR 7: p p (1 ) (2 ) (3 ) DR PD DR 13 12, 4. PD PD c

18 59 «««d s«««è Ç v t~ 13: PD DR (IN OUT2008 ) PD PD (EL: Expected Loss) ( PD ) () PD PD 0 (0 ) PD 1 1 (EAD) 1, 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

19 60 D = n i=1 1 i D PD DR 8 PD DR PD PD 6 8 8: (IN OUT2008 ) n PD DR n PD DR 1 16, % 0.82% 0.20% 4, % 0.82% 0.21% 2 16, % 1.03% 0.28% 7, % 0.80% 0.07% 3 17, % 1.21% 0.32% 10, % 0.77% 0.08% 4 17, % 1.48% 0.45% 12, % 1.11% 0.10% 5 18, % 1.74% 0.52% 16, % 1.35% 0.16% 6 18, % 2.32% 0.83% 20, % 1.57% 0.09% 7 18, % 3.16% 1.21% 27, % 2.11% 0.15% 8 20, % 5.24% 1.68% 44, % 3.93% 0.24% 143, % 2.21% 0.72% 143, % 2.21% 0.00% = PD DR % PD 5. PD 54 PD DR c

20 61 9: (IN OUT2008 ) ( : ) n I(A) n I(B) (B A) 1 16, , , , , , , , , , ,673 1,586 20, , ,997 2,377 27, , ,103 3,716 44, , ,413 10, , , % PD PD PD 2 (1) GDP (2) PD c

21 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). houkoku-new.pdf [5] CRD : CRD 3 4 (BS ) 3-a (2010). houkoku.pdf [6] CRD : CRD 3 3-a 4 (2011). 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 (2006). [8] M. Gordy: A comparative anatomy of credit risk models, Journal of Banking and Finance. 24-1&2 (2002), [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), [11], : (2010), [12], : :., 29-3 (2010), [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), [18] : (, 2003). c

22 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), [20],,,, : RDB (2007), [21], : AR (, 2011). A. [14] (pp ) 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

23 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= hibiki@ae.keio.ac.jp c

24 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 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

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