大規模データベースを用いた信用リスク計測の問題点と対策(変数選択とデータ量の関係)

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1 ( ) CRD 450, , CRD,, CRD 1

2 ( )

3 AIC

4 1 ( ) ( ) (CRD ) CRD CRD 4

5 2 2.1 [Altman(1968)] (x 1, x 2,, x m ) (b 1, b 2,, b m ) ( ) z = b 1 x 1 + b 2 x b m x m (z = b 1 x 1 + b 2 x b m x m ) p(z) = exp(z) 1 + exp(z) [ (1996)] 1 3 BIS (ordered logit model) AAA = 1, AA = 2 [Kaplan,Urwitz(1979)] [ (1998)] [ (1999)] (2.1) 5

6 t h(t) [Lane, Looney,and Wansley(1986)] Cox Cox Cox h f(t) (x 1, x 2,, x m ) g(x 1, x 2,, x m ) h(t, x 1, x 2,, x m ) = f(t) g(t, x 1, x 2,, x m ) (2.2) 2.2 [Merton(1974)] ( ) ( ) [Duffie,Singleton(1999)] [Jarrow,Lando,and Turnbull(1997)] 2.1: 6

7 3 2 CRD n m (x i1, x i2,, x im ) z = b 1 x 1 + b 2 x b m x m (3.1) z p(z) = exp z 1 + exp z (3.2) ( 3.2 ) b = (b 1, b 2,, b m ) i p i L(b) L(b) = n i=1 p δ i i (1 p i) 1 δ i (3.3) δ i { 1 (i ) δ i = 0 (i ) 7

8 l(b), n l(b) = log L(b) = δ i log p i + (1 δ i ) log(1 p i ) (3.4) i=1 l(b) b b l(b) b j = n (p i δ i )x ij = 0, j = 0, 1, 2,, m (3.5) i=1 2 (p) 1 0 (Z) 3.1: 2 C [Press,Teukolsky,Vetterling, and Flannery(1993)] 8

9 (p) 1 0 (Z) 3.2: 3.2 CRD ( ) ) 0 1 9

10 Box-Cox , 2,, ( ) ,

11 3.1:

12 AIC AIC AIC = 2 ( ) + 2 ( ) (3.6) AIC AIC AIC 1 n N k k N k k l k m k b k = (b k1,, b kmk ) k x ik = (x ik1,, x ikmk ) 3 2 C 12

13 b seg x seg b seg = (b 11,, b 1m1, b 21,, b 2m2,, b n1,, b nmn ) ( x i11,, x i1m1 }{{} 1 (0,, 0, x i21,, x i2m2 x seg =, 0,, 0,, 0,, 0) ( 1 ) }{{} 2,, 0,, 0) ( 2 ). (0,, 0, 0,, 0,, x in1,, x inmn ) ( n ) }{{} n k b k x T ik = (b k1,, b kmk ) (x ik1, x ik2,, x ikmk ) T = (b 11,, b 1m1,, b n1,, b nmn ) (0,, 0, x ik1,, x ikmn, 0,, 0) T }{{} k = b seg x T seg l seg l seg = = = n l k k=1 ( n N k δ i log k=1 i=1 ( N i=1 exp(b k x T ik ) 1 + exp(b k x T ik )(1 δ i) log δ i log exp(b segx T seg) 1 + b seg x T seg ) exp(b k x T ik ) ) 1 + (1 δ i ) log 1 + b seg x T seg T AIC AIC seg n n AIC seg = 2 l k + 2 m k k=1 k=1 ) n = ( 2 l k + 2 m k k=1 AIC AIC AIC

14 t- AIC ( ) = ( ) ( ) + ( ) ( ) = ( ) + ( ) ( ) (3.7). x 1, x 2, x 3, x 4 z z = β 1 x 1 + ( ) (3.8) z = β 2 x 2 + β 3 x 3 + β 4 x 4 + ( ) (3.9) β 1, β 2, β 3, β 4 β 1 (3.7) (3.8) β 1 x 1 (3.9) β 2 x 2 + β 3 x 3 + β 4 x 4 14

15 AIC z (x 1, x 2,, x m ) z = β 1 x 1 + β 2 x β m x m z ( ) 2 z 2 [ (2001)] CAP(Cumulative Accuracy Profile) AR(Accuracy Ratio) N n CAP x x/n x n x n x /n % 60% CAP AR CAP 45 ( S 1 ) CAP 45 ( S 2 ) S 2 /S

16 AR AR 1 AR CAP AR "! #%$'&(*),+.-0/ : CAP ( ) 16

17 3.2: ( AIC: : ) t ) ( ) ( )

18 #!" $&%'(*),+.-0/ : CAP 3.3: CAP 10 ( ) 10 % % % % % % % % %

19 (1) (2) AIC AIC AR AR

20 ( ) = ( ) ( ) + ( ) = = 20

21 :

22 3.5: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

23 3.6: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

24 ! "$#&%&')(+*-,.0/ 21&3 3.5: CAP ( ) 3.7: CAP 10 ( ) 10 % % % % % % % % %

25 ! "$#&%&')(+*-,.0/ 21&3 3.6: CAP ( ) 3.8: CAP 10 ( ) 10 % % % % % % % % %

26 3.9: ( ) t AIC

27 3.10: ( ) t AIC

28 3.11: ( ) t AIC

29 4 4.1 AIC , 861 CRD 879, 430 ( 7126 ) ( ) (, ) AIC 29

30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4.1: ( ) ( ) ( ) ( ) AR 1 AR AR AR AIC AIC AR AIC

31 ( X ) 1 X 1 X 2 2 (p) (Z) 4.1: AIC 31

32 (p) 1 0 (Z) 4.2: 2 ( ) (p) (Z) 4.3: 2 ( ) 32

33 : OF non-cut AIC cut AIC OF OF non-cut non-cut cut cut AIC AIC AIC AIC = 2 ( ) + 2 ( ) (4.1) 33

34 0 0 AIC (4.1) 2 2 AIC m l(b) AIC AIC all = 2 l(b) + 2 m (4.2) m 1, m 2,, m 5 l(b 1 ), l(b 2 ),, l(b 5 ) AIC 5 5 AIC seg = 2 l(b i ) + 2 m i (4.3) i=1 i=1 5 m m i i= OF non-cut, non-cut cut non-cut

35 (1 5 ) (5 15 ) OF cut non-cut OF AIC ( 4.4 ) (1 5 ) non-cut OF cut cut AIC ( 4.5 ) 4.3: (1 5 ) OF OF OF non-cut non-cut non-cut OF non-cut non-cut non-cut cut OF non-cut non-cut cut cut non-cut non-cut cut cut cut non-cut cut cut cut cut non-cut cut cut cut cut 35

36 4.4: (5 15 ) OF OF OF OF OF OF OF OF OF OF cut OF OF OF cut cut OF OF non-cut cut cut OF non-cut non-cut cut cut OF non-cut cut cut cut 4.5: (50 ) OF OF OF cut cut cut OF cut cut cut cut OF cut cut cut cut cut cut cut cut cut cut cut cut cut cut cut cut cut cut cut OF AIC cut 4.4: 36

37 OF AIC cut 4.5: [ (2002)] OF

38 OF non-cut cut cut AIC non-cut cut OF non-cut cut OF cut non-cut 4.6: (1 5 9 ) non-cut non-cut non-cut non-cut non-cut non-cut non-cut non-cut non-cut non-cut cut non-cut non-cut non-cut cut cut non-cut non-cut cut cut cut non-cut cut cut cut cut non-cut cut cut cut cut 4.7: ( ) OF OF OF OF OF OF OF OF OF OF cut OF non-cut non-cut cut cut OF non-cut non-cut cut cut OF non-cut non-cut cut cut OF non-cut cut cut cut 38

39 AIC OF non-cut, non-cut cut 2. AIC OF non-cut, non-cut cut 3. OF non-cut 4. cut non-cut AIC 2 3 CPU 2.0GHz 1.0Gbite

40 AIC Cp CAP ROC ( ) 40

41 ( ) ( ) LISREL LISREL 41

42 [Altman(1968)] Altman, E.I. Financial Ratios, Discriminant Analysis and The Prediction of Corporate Bankruptcy. Journal of Finance, 1968, 23(4), [ (1996)] 1996 (JAFEE) [Kaplan,Urwitz(1979)] Kaplan,R.S. and Urwitz,G. Statistical Models of Bond Rating: Methodological Inquiry. The Journal of Bisiness, 1979, 52, A [ (1998)] 1998 (JAFEE) [ (1999)] [Lane, Looney,and Wansley(1986)] Lane, W.R., Looney,S.W. and Wansley,J.W. An application of the Cox proportinal hazard model to bank failure. Journal of Banking and Finance [Merton(1974)] Merton,R.C. On The Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance, 1974, 29(2), [Duffie,Singleton(1999)] Duffie, D. and Singleton,K. Modeling term structures of defaultable bonds, Review of Financial Studies [Jarrow,Lando,and Turnbull(1997)] Jarrow, R.A., Lando,D. and Turnbull,S.M. A Markov model for the term structure of credit risk spread. Review of Financial Studies [ (2001)] WP01-5,2001 [ (2002)], 2002-J-6 [Press,Teukolsky,Vetterling, and Flannery(1993)] Press, W.H., Teukolsky,S.A., Vetterling, W.T. and Flannery,B.P. NUMERICAL RECIPES in C[ ],

43 [ (1999)], 1999 [ (1999)], 11, , 1999 [ (1992)],

44 The problems and solutions of credit risk measurement using large-scale data base (The relation between the variables selection and the amount of data) Abstruct This paper provides an advice for the credit risk measurement using a large-scale database. In previous researches, it was impossible to estimate credit risk using a large-scale database, because there were no sufficient accumulations of data. To solve this problem, CRD (Credit Risk Database Association) have been developing a large-scale database, which has 450,000 company records with 86 management indices. In this paper, we estimate binominal logit models using this database. However if we estimate the parameters of the model naively, it takes significant amount of time for the calculation. To cope with this problem, we improve the optimization algorithm to reduce the estimation time and then select the explanatory variables affecting the probabilities of default. This paper also shows the optimal number of segments depend on the amount of data. We divide the database into the several segments and estimated the probabilities of default. We evaluate the goodness of fit and the robustness of the estimation by AIC. It becomes clear that the estimation using divided data was often worse than the one with all data. Then, we discuss the optimal number of segments depending on the amount of data. As a result, we obtained a table that determines the optimal number of segments depending on the number of data, defaulted companies in it and candidate explanatory variables. YAMASHITA, Satoshi The Institute of Statistical Mathematics Assistant professor Credit Risk Database Association Adviser Financial Services Agency Special research fellow KAWAGUCHI, Sho Mathematical principle Science, Science and Engineering, Waseda Univercity Financial Services Agency Technical research fellow 44

IMES DISCUSSION PAPER SERIES Discussion Paper No. 99-J- 9 -J-19 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN

IMES DISCUSSION PAPER SERIES Discussion Paper No. 99-J- 9 -J-19 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN 100-8630 03 IMES Discussion Paper Series 99-J- 9 -J-19 1999 6 * * [1999] *(E-mail: