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1 1 2, BIC,, 1, 2,, CRD 1

2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

3 1 BIS (Jarrow and Turnbull (1995), Duffie and Singleton (1999)) , ( Duffie (1996) ).. 2,

4 4, ( Anderson et al. (1996) ) Merton (1973) Vasicek (1977), Cox, Ingersoll and Ross (1985). Heath, Jarrow and Morton (1992) 6, Brace, Gatarek and Musiela (1997) 7 McCulloch (1971, 1975), McCulloch (Schaefer (1981), Vasicek and Fong (1982), Nelson and Siegel (1987), Steely (1991), (22)). LGD; Loss Given Default. Lane et al. (1986) (Merton (1974)) (Jarrow and Turnbull (1995), Duffie and Singleton (1999)) 8 (Duffie, Pan and Singleton (2), Kijima and Muromachi (2a, 2b), Kusuoka (1999), (21) (25)). 6 Amin and Morton (1994), (1999) HJM 7 (1999) BGM 8 Merton (1974), 2

5 Market Implied LGD 9 Workout LGD, Market LGD, Historical Implied LGD, Market Implied LGD Workout LGD Market LGD (Altman and Saunders (21), Altman et al. (22)) Workout LGD Market Implied LGD Market Implied LGD (Jarrow (21), (24)), (Unal et al. (23)), ( (22)) 2.2 C R T L L t =(t 1 ;:::;t L ) r(t) T L PV (Present Value) PV(r( ); t) = LX i=1 C exp ρ Z ti ff r(u)du + R exp ρ Z tl ff r(u)du n R o t d(t) = exp r(u)du PV, t, r(t) PV(r( ); t) 1 C R T L L t =(t 1 ;:::;t L ) 9 (24) 1, 3

6 fi P (t <fi» t + jt <fi) h(t) = lim t!. h(t) t t + t 11 ρ Z t ff P (fi >t) = exp h(u)du : ffi ( ffi ), h(t) ffi 12 PV(r( );h( );ffi; t) = LX i=1» Z ti C exp fr(u)+h(u)g du +R exp +R ffi» Z tl Z tl» fr(u)+h(u)g du h(t) exp ρ Z t fr(u)+h(u)g du (Jarrow andturnbull (1995), Duffie and Singleton (1999)) h(t), P (fi >t) 12 (Recovery of face value) t, (Recovery of market value). Duffie and Singleton (1999) t ( (2) ) 13 PV(r( );h( );ffi; t) 3 tl R ffi t = t L 1 =(t1;t1 + ], 2 =(t2;t2 + ],:::, K =(t K;t K + ] K ( t1 =,t L = t K +, t k = t k 1 + ) k, R ffi n R tk R ffi exp r(u)du k P (t k <fi» t k + ) R ffi t L t L KX P (t k <fi» t k + ) k=1 = R ffi = R ffi» R ffi exp o ρ Z t ff k r(u)du " KX P (t k <fi» t k + jt k <fi) P (t k <fi) exp k=1 " KX P (t k <fi» t k + jt k <fi) exp k=1 ρ Z t k ρ Z t k h(u)du ff r(u)du exp ff# ff dt ρ Z t ff # k r(u)du k! (K!1) PV(r( );h( );ffi; t) 3 (1) 4

7 d(t) d(t) =exp h R t fr(u)+h(u)g du i r(t) h(t) ffi p g i, pcp i a g i, acp i n. 1; :::; J J n 1,...,n J x =(x 1 ; :::; x p ). r(t) x h j (t; x), ffi j (x) (j =1; :::; J) PV g, PV cp 8 p g i + ag i =PVg i (r( ); t i)+" g i (i =1; :::; n ); >< >: ( 1) p cp i1 + acp i1 =PVcp i1 (r( );h 1( ; x i 1);ffi 1 (x i 1); t i 1)+" 1 i1 (i 1 =1; :::; n 1 );.. ( J) p cp i J + a cp i J =PV cp i J (r( );h J ( ; x ij );ffi J (x ij ); t ij )+" J i J (i J =1; :::; n J ): (2) " g i ;"1 i1 ; :::; "J i J ff, 2 ff1 2 ; :::; ff2 J t i t i, j i j t ij ( j) x ij, r(t) 5

8 h j (t; x) (j =1; :::; J) X m r(t; w ) = w k ffi k (t) =w ffi (t); k=1 " mj # ( X X p h j (t; x; w j ; fi h ) = w jk ffi jk (t) exp fi h`x` k=1 `=1 h i = w j ffi j(t) ) exp Φ fi h xψ ; (j =1; :::; J): (3) ffi j (t) = (ffi 1 (t); :::; ffi mj (t)) w j = (w j1 ; :::; w jmj ), fi h = (fi h1 ; :::; fi hp ) x (Cox (1972)), ffi j (x) (j =1; :::; J) x z ffi j (x) (j =1; :::; J) ffi 2 [; 1] ffi j (x;z;fi ffi ; fl;ff) = expφp m ffi k=1 fl kffi ffik (j)+ P Ψ p `=1 fi ffi`x` + ffz Ψ `=1 fi ffi`x` + ffz 1 + exp ΦP m ffi k=1 fl jkffi ffik (j)+ P p Φ = exp fl Φ ffi ffi (j)+fi ffi x + ffzψ 1 + exp fl ffi ffi (j)+fi ; (j =1; :::; J): (4) ffix + ffzψ ffi ffi (j) =(ffi ffi1 (j); :::; ffi ffimffi (j)) fi ffi = (fi ffi1 ; :::; fi ffip ), fl =(fl 1 ; :::; fl mffi ), ff McCulloch (1971) 2 (McCulloch (1971)), 3 (McCulloch (1975)), Bernstein (Schaefer (1981)) (Vasicek and Fong (1982)) ( (22)), B- (Steely (1991), Fisher et al. (1995)) 1 3 B- ffi j (t) t j 2 B- de Boor (1978) : ffi j (x; ) = ffi j (x; p) = ( 1; tj» x<t j+1 ; ; othejwise; x t j ffi j (x; p 1) + t j+p+1 x ffi j+1 (x; p 1): t j+p t j t j+p+1 t j+1 6

9 ffi j (x; p) p B- B- 2 B- ffi j (x;3) =ffi j (x) φ 1 (t)φ 2 (t)φ 3 (t)φ 4 (t)φ 5 (t)φ 6 (t)φ 7 (t) t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 1 t 11 1: 3 B-. (3), (4) T L L PV g j PV cp PV g (w ; t) = LX Φ Ψ Φ Ψ C exp w ψ (t i ) + R exp w ψ (t L ) (5) LX n o C exp w ψ (t i ) w j ψ j(t i ) exp(fi x) i=1 n o i=1 PV cp (x;z;w ; w j ; fi h ; fi ffi ; fl;ff;t) = +R exp w ψ (t L ) w j ψ j(t L ) exp(fi x) +R ffi j (x;z;fi ffi ; fl;ff) n w j μ ψ j (t L ; w ; w j ; fi) exp(fi x) t =(t 1 ;:::;t L ) ψ j (t) =(ψ j1 (t); :::; ψ jmj (t)), μ ψj (t; w ; w j ; fi) = ( μ ψ j1 (t; w ; w j ; fi); :::; μ ψ jmj (t; w ; w j ; fi)) o ψ jk (t) = μψ jk (t; w ; w j ; fi) = Z t Z t ffi jk (u)du; h ffi jk (u) exp n oi w ψ (u) w j ψ j(u) exp(fi x) du m j (2) (5) 7

10 8>< " Ψ 2 Φy f(yi g jt i; w ;ff)= 2 p 1 2ßff 2 exp i PVg i (w ;t i ) 2ff 2 # ; ( 1) f(yi1 cp jt i1 ; x i1 ;z i1; w ; w 1 ; fi h ; fi ffi ; fl;ff;ff1) 2 " = p 1 2ßff1 2 exp Φy cp i1 PVcp i1 (x i1 ;z i1 ;w ;w1;fi h ;fi ffi ;fl;ff;t i 1 )) Ψ 2 2ff 2 1 # ; (6) >:. ( J) f(y cp i J jt ij ; x ij ;z ij ; w ; w J ; fi h ; fi ffi ; fl;ff;ffj 2 ) = p 1 2ßff 2 J exp n y cp i J PV cp i J (x ij ;z ij ;w;w J ;fi h ;fi ffi ;fl;ff;t ij )) 2ff 2 J o : yi g = pg i + ag i, ycp i j = p cp i j + a cp i j (j = 1;:::;J) ( j) z ij x, p g i + ag i, p cp i j + a cp i j t, z, j (j =1; :::; R) =(w ; ff ; fi h ; fi ffi ; fl ;ff), w =(w ; :::; w J ), ff =(ff 2 ;ff2 1 ; :::; ff2 J ) (3), (4) x, B (6) `( ) = Xn i=1 log f(y g i jt i; w ;ff 2 )+ 2 JX X j=1 4 n j i j =1 log f(y cp i j jt ij ; x ij ;z ij ; w ; w j ; fi h ; fi ffi ; fl;ff;ffj 2 ) 5 : (7) 14 (3) (4) B-. 2 (McCulloch (1971)), 3 (McCulloch (1975)), Bernstein (Schaefer (1981)) (Vasicek and Fong (1982)) ( (22)) McCulloch (1971) (McCulloch (1975), (22)) 8 3

11 , ` ( ) =`( ) JX j= n j 2 w j K jw j n fl 2 fl K fl fl n fi h 2 fi h fi h n fi ffi 2 fi ffi fi ffi : (8) n = P J j= n j j, K j m j m j (m j k) m j D jk = ( 1) kc ( 1) k kc k ( 1) kc ( 1) k kc k ( 1) kc ( 1) k kc k K j = D j2 D j2 k C i 2 B- (2 ) fl B- 2 Green and Silverman (1994), Eilers and Marks (1996) j 1 p(1) = P (fi» 1) j w j exp n o w j ψ j(1) =1 p(1): (9) p(1) w j w j 2 R m j r j (1999) ^ (6) / 15 1 C A 15 (1997), 9

12 (6) Bayes (Konishi, Ando and Imoto (24)), (Konishi and Kitagawa (1996)), (Stone (1974)), (Efron and Tibsirani (1993)) (6) Bayes ß( j ) m Z expf`( )gß( j )d Schwarz (1978) Bayes Schwarz (1978) Bayes Konishi, Ando and Imoto (24) (Tierney and Kadane (1986)) Bayes, (Konishi, Ando and Imoto (24)) (m n ρ ff 2)=2 ß( j ) = jk j 1=2 + exp n 2ß 2 w K w JY (mj r n j j 2)=2 ρ ff jk j j 1=2 + exp n j 2ß 2 w j K jw j j=1 (mffi 2)=2 n fl ρ ff jk fl j 1=2 + exp n fl 2ß 2 fl K ffi fl 16 w j 1

13 p=2 n fih ff expρ n p=2 fi n fifl ρ ff h 2ß 2 fi h fi h exp n fi fl 2ß 2 fi fl fi fl : jk j + K jk j j + (9) K j (Tierney and Kadane (1986)) Bayes (Konishi, Ando and Imoto (24)) BIC = 2`(^ )+n ^w K ^w + n +n fiffi ^fi ffi ^fi ffi JX j=1 JX j= JX j=1 j ^w j K j ^w j + n fl ^fl K fl ^fl + n fih ^fih ^fi h log jk j j + log jk fl j + +logjj(^ )j (m 2) log (1) (m j r j 2) log j (m ffi 2) log fl p log( fih fiffi ) + Const: Const J(^ ) = 1 (^ BIC J B- (Fisher et al. (1995), Eiler and Marxs (1996)) r(t) =:1t ( ). (J =1).», h(t;») =»t [; 1] (4) ffi( ) = exp( )=f1+exp( )g (1) LX ρ ff ρ ff PV cp (» +:1) (»; ; t) = C exp t 2 (» +:1) i + R exp t 2 L 2 2 i=1 +:5» R n 1 exp( t 2 L) 17 o ρ exp( ) 1 + exp( ) exp (» +:1) 2 ff : 11

14 f(y cp i jt i ;»; ; ff 2 )= " # 1 p exp fycp i PV cp (»; ; t i ))g 2 2ßff 2 2ff 2 : PV cp (»; ; t) n p cp 1 ; :::; pcp n L =2( ) [:2; 2] 2.5 R =1,» =:2, =1( ffi( ) =:73) ff =1 2 n», ffi( )» = =: n n (a):» (b): ffi( ) 2: n (a):», (b): ffi( ). 4, 12

15 Moody's (6) J J =5 1 Aaa, Aa1, Aa2, Aa3 (276 ) 2 A1, A2 (242 ), 3 A3, Baa1 (229 ), 4 Baa2, Baa3 (415 ) Ba1 5 Ba1 (199 ) (9) p(t) Moody's (2) 1983 ο21 Average Cumulative Default Rates by Alpha-Numeric Rating 19 (x 1 ), (x 2 ) 2 Moody's e-aurora. 4.2 r(t; w ) h j (t; x), ffi j (x; fi ffi ; fl;ff) m = 8, m j = 4 (j = 1; :::; 5), m ffi = 4 BIC BIC =:1, 1 =:1, 2 =:1, 3 =:1, 4 =:1, 5 =:1, fl =:1, fih =:1, fiffi =:1 3, 4 3, , 4 18 (Aaa, Aa1, Aa2, Aa3, A1,..., C) 19 Aa2, A1, A3, Baa2, Ba2 1 1 ο 5 2 (24) 21 ρ Z t ff d(t) = exp r(u; ^w)du» Z t Φ Ψ d j(t) = exp r(u; ^w)+hj(t; x; ^w j; ^fi h ) du ; (j =1;:::;J) 13

16 7, Ba1 5 6 h j (t) = P m j k=1 ^w jkffi jk (t) 5 ( 1 ο 3) 5 t = 4 6 Moody's (2) 1983 ο21 1 ο 5 Aa2, A1, A3, Baa2, Ba ο 4 Moody's 5 Moody's 8 h j (t; x; w j ; fi h ) (3) fi h (3) ; ^fi h1 = :12527 (:4), ; ^fi h2 = :54 (:3) 9 1 ^ff 2 5 ^ff (4) ^fl ffi ffi (1) = 1:4, ^fl ffi ffi (2) = :99, ^fl ffi ffi (3) = :53, ^fl ffi ffi (4) = :41, ^fl ffi ffi (5) = :27 9 Hamilton et al. (21) 1981 ο2 d(t) r(t) r(t) = d (t)=d(t), d(t) o (t) (t) = log d(t)=t exp n^fi h x =1 22 (22) 14

17 :. fi ffi ^fi ffi1 =:386 (:58) ^fi ffi2 =:285 (:54) (4) z ^ff = :267 (:74) Forward rate (%) Time to maturity (years) 3:. 15

18 Zero coupon yield Time to maturity (years) 4:. Hazard func Time to maturity (years) 5: 16

19 Default prob Time to maturity (years) 6: Default prob Time to maturity (years) 7: Moody's (2) Aa2, A1, A3, Baa2, Ba2). 17

20 Default prob Time to maturity (years) 1 Default prob Time to maturity (years) 2 Default prob Time to maturity (years) 3 Default prob Time to maturity (years) 4 Default prob Time to maturity (years) 5 8: h j (t; x; w j ; fi h ) 18

21 : 1 Aaa, Aa1, Aa2, Aa3 2 A1, A2 3 A3, Baa1 4 Baa2, Baa3 5 Ba1 25% 75% 25% % 1.5 ) 5.,.,,,.,, 19

22 Altman, E. I. and Saunders, A. (21) An analysis and critique of the BIS proposal on capital adequacy and ratings. Journal of Banking and Finance, 25, Altman, E. I., Bharath, S. T. and Saunders, A. (22) Credit ratings and the BIS capital adequacy reform agenda. Journal of Banking and Finance, 26, Amin, I. K. and Morton, J. A. (1994) Implied Volatility Functions in Arbitrage-Free Term Structure Models. Journal of Financial Economics, 35, Anderson, N., Breedon, F., Deacon, M., Derry, A. and Murphy, G. (1996) Estimating and interpreting the yield curve. John Wiley and Sons, Chichester. Brace, A., Gatarek, D. and Musiela, M. (1997) The Market Model of Interest Rate Dynamics. Mathematical Finance, 7, Cox, D. R. (1972) Regression models and life-tables. Journal of the Royal Statistical Society B, 34, Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985) A Theory of the Term Structure of Interest Rates. Econometrica, 53, de Boor, C. (1978) Apractical guide to splines. Springer, Berlin. Duffie, D. (1996) Dynamic Asset Pricing Theory, Second Edition. Princeton University Press. Duffie, D., Pan, J. and Singleton, K. J. (2) Transform Analysis and Asset Pricing for Affine Jump Diffusions. Econometrica, 68, Duffie, D. and Singleton, K. J. (1999) Modeling Term Structures of Defaultable Bonds. Review of Financial Studies, 12, Efron, B. and Tibsirani, R. J. (1993) Anintroduction to bootstrap. Chapman & Hall. Eilers, P. H. C. and Marx, B. D. (1996) Flexible smoothing with B-splines and penalties (with discussion). Statistical Science, 11, Fisher, M. E., Nychka, D. and Zervos, D. (1995) Fitting the term structure of interest rates with smoothing splines. Federal Reserve Bank Finance and Economics Discussion Paper 95-1, January. Green, P. J. and Silverman, B. W. (1994) Nonparametric regression and generalized linear models. Chapman & Hall, London. Hamilton, D. T., Gupton, G. and Berthault, A. (21) Default and Recovery Rates of Corporate Bond Issuers: 2. Special Comment. Moody's Investor Service, Global Credit Research. Heath, D., Jarrow, R. and Morton, A. (1992) Bond Pricing and Term Structure of Interest Rates: ANew methodology for Contingent Claims Valuation. Econometrica, 6,

23 Jarrow, R. (21) Default Parameter Estimation Using Market Prices. Financial Analysts Journal, 57, Jarrow, R. A. and Turnbull, S. M. (1995) Derivatives on Financial Securities subject to Credit Risk. The Journal of Finance, 5, Kijima, M. and Muromachi, Y. (2a) Credit events and the valuation of credit derivatives of basket type. Review of Derivatives Research, 4, Kijima, M. and Muromachi, Y. (2b) Evaluation of credit risk of a portfolio with stochastic interest rate and default processes. Journal of Risk, 3, Konishi, S., Ando, T. and Imoto, S. (24) Bayesian information criteria and smoothing parameter selection in radial basis function networks. Biometrika, 91, Konishi, S. and Kitagawa, G. (1996) Generalised information criteria in model selection. Biometrika, 83, Kusuoka, S. (1999) Aremark on default risk models. Advances in Mathematical Economics, 1, Lane, W. R., Looney, S. W. andwansley, J. W. (1986) An application of the Cox proportional hazards model to bank failure. Journal of Banking and Finance, 1, McCulloch, J. H. (1971) Measuring the term structure of interest rates. Journal of Finance, 26, McCulloch, J. H. (1975) The tax-adjusted yield curve. Journal of Finance, 3, Merton, R. C. (1973) Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4, Merton, R. C. (1974) On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, Moody's Investors Service (2) Historical Default Rates of Corporate Bond Issuers, Global Credit Research. Nelson, C. R. and Siegel, A. F. (1987) Parsimonious modeling of yield curves. Journal of Business, 6, Schaefer, S. M. (1981) Measuring a tax-specific term structure of interest rates in the market for British government securities. The Economic Journal, 91, Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics, 6, Steely, J. M. (1991) Estimating the gilt-edged term structure: basis splines and confidence intervals. Journal of Business, Finance and Accounting, 18, Stone, C. J. (1974) Cross-validatory choice and assessment of statistical predictions (with discussion). Journal of the Royal Statistical Society, Series B. 36, Tierney, L. and Kadane, J. B. (1986) Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, Unal, H., Madan, D. and Guntay, L. (23) Pricing the risk of recovery in default with absolute priority rule violation. Journal of Banking and Finance, 25, Vasicek, O. A. (1977) An Equilibrium Characterization of the Term Structure. Journal of 21

24 Financial Economics, 5, Vasicek, O. A. and Fong, H. G. (1982) Term structure modeling using exponential splines. Journal of Finance, 37, , (24)., Vol 15., (1999). IMES Discussion Paper Series, 99 J 19., (22)., , (1999). IMES Discussion Paper Series, 99 J 24.,, (21).. (1997). IMES Discussion Paper Series, 97 J 12. (22)., 5 2, , (2). IMES Discussion Paper Series, 2 J 3., (1999) BGM. IMES Discussion Paper Series, 99 J 39., (24) Reduced Form PD, LGD. The Institute of Statistical Mathematics, Research Memorandum, No 911., (25). 22

25 A reduced form approach for the simultaneous estimation of hazard term structure and LGD using rating and financial information Abstract After the recognition of importance of credit risk measurement, lenders realized the necessity of the evaluation of various types of credit risk factors such as the hazard term structure, loss given default (LGD) on the credit exposure, default correlations, and fluctuation of exposure. In this paper, we propose a statistical method for the simultaneous estimation of the hazard term structure and the LGD on credit exposure from a cross-section of coupon bonds. These two credit risk factors are inferred adding the information on bond issuer's financial ratios, bond rating and date of maturity in the context of reduced form approach. We propose the term structure model of hazard rate using a smoothing spline with cubic B-spline bases. Then, we estimate the model parameter based on a maximum penalized likelihood approach and evaluate the goodness of fit from a Bayesian point ofview. Monte Carlo experiments are conducted to show the merits of a penalized likelihood approach. We also applied the proposed method to Japanese bond market data. This study finds that the proposed method provides meaningful findings to calculate the term structure of interest rates, hazard rates and the loss given default. ANDO, Tomohiro Graduate School of Business Administration Keio University Assistant Professor YAMASHITA, Satoshi The Institute of Statistical Mathematics Assistant professor Credit Risk Database Association Adviser Financial Services Agency Special research fellow 23

本邦国債価格データを用いたゼロ・クーポン・イールド・カーブ推定手法の比較分析

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