202 2 9 Vol. 9 yasuhisa.toyosawa@mizuho-cb.co.jp 3 3

Altman968 Z Kaplan and Urwitz 979 Merton974 Support Vector Machine SVM 20 20 2

SVM i s i x b si t = b x i i r i R * R r (R,R, L,R ), R < R < L < R i 2 K 2 K s i a k Pr(r i = Rk ) = + exp( (s i + a )) + exp( (s k i + a k )) N(*) Pr( r i = Rk) = N(a k s i) N(a k s ) i i rˆ i = arg maxpr(r i = x) x 200200220 Multi model 20Multi model 3

One model AIC SBCSchwarz One model EBITDA/ CF CF Debt/EBITDA EBITDA / / / 4

X: Log( ) Debt EBITDA EBITDA/ EBITDA = ( ) EBITDA EBITDA/ X 2 : X 3 : EBITDA EBITDA 2 X 4 : EBITDA X 5 : / X 6 : 3 5 NTT JR JT 2 Microsoft Excel Stdev 5

I * X7 : X8 : X9 : I I I {NTTJRJT} { } { } I * X0 : I { } AAAAA+BBB. 2. Accuracy RatioAR AR AR 6

Cumulative accuracy profilecap CAP X 30% Y 30% %,000 300 X 30% 300 300 Y 00% 2 CAP X 30%Y 30% 300 90 CAP X 30%Y 30% CAP Cumulative accuracy profiles 00 50 %% 0 0 50 00 AR A B 0 AR = B A + B Area Under CurveAUCAUC AR Engelmann and Tasche 2003 AR AUC 7

AR = 2AUC AR R k R k+ AR AR(R k,r k+ ) AR AAR K AAR = ( AR(R k= k,r k+ )) K AAR b AAR bˆ = arg max AAR( b) b 2009AR AUC Simulated annealing SA b 2 3 2 f(r k,r k + c ) c i q i HR HR N = I N { f(rˆ, q ) } i i i = HR a HR aˆ = arg maxhr( a) a 3 Moody s Investors Service 8

SA a 20 9 202 6 R&I 395 AAA AA+ AA AA- A+ A A- BBB+ BBB BBB- BB+ BB 0 2 4 3 2 4 7 2 4 0 2 4 23 3 2 2 2 8 3 3 5 0 3 6 2 43 2 3 3 2 2 2 5 2 2 2 2 2 0 2 2 6 2 5 9 2 3 2 3 7 9 5 2 30 5 6 0 0 7 3 2 47 2 2 5 3 4 8 28 2 2 2 8 2 3 2 2 3 8 2 2 7 2 3 5 4 5 4 25 2 4 2 6 2 3 4 2 4 3 5 3 4 4 20 2 9 4 6 3 3 2 2 2 5 5 0 0 2 4 36 83 80 55 47 5 5 2 395 3 9

AAA AAA BB+ BB 9 4,5,6 Log() / 2.0x 8 0.5x 4 0.0x AA+ AA A+ A A- BBB+ BBB BB* AA+ AA A+ A A- BBB+ BBB AA- BBB- AA- BBB- BB* 75 %th 25%th 2 BB*BB+ Pacific Data EBITDA/ EBITDA / 40% 0.30x 20% 0.5x 0% 0.00x AA+ AA A+ A A- BBB+ BBB BB* AA+ AA A+ A A- BBB+ BBB AA- BBB- AA- BBB- BB* 75 %th 25%th 2 BB*BB+ Pacific Data 0

/ / 0.8x 2x 0x 0.4x 2x 0.0x 4x AA+ AA A+ A A- BBB+ BBB BB* AA+ AA A+ A A- BBB+ BBB AA- BBB- AA- BBB- BB* 75 %th 25%th 2 BB*BB+ Pacific Data AAR SA AAR 7 95% 93.2 90% 85% 0,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 0,000 AAR % AR % AA AA- A+ A A- BBB+ BBB BBB- 93.2 97.9 94.9 94.3 93.2 90. 89.0 87.2 94.0 AAR 93%

8 AR 87% X Log() 7 8.3 X6 8 2.7 X2 / 4 3.2 X7 NTT,JR,JT 4.8 X3 EBITDA/ 2 0.7 X4 EBITDA/ 27.4 X5 / 3 0.3 X8, 2 3. X9 4 0.8 X0 0.5 9 s = 7X + 4X 4X 7 2 + 2X + 2X 8 3 + 4X 9 27X 4 0X 3X 0 5 + 8X 6 2 0 20 Multi model 88.9% 93.9% 00% 90% 93.9% 80% 70% 0,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 0,000 2

90% 2 5 80% 4 > % > % 0 0 0 -- 47 43 4 9 0 00 28 23 5 82 7 7 0 00 8 8 0 00 23 23 0 00 0 9 3 3 0 00 7 6 94 8 7 88 25 20 5 80 43 42 98 4 3 75 5 5 0 00 0 00 2 2 0 00 6 6 0 00 2 2 0 00 4 4 0 00 0 8 2 80 20 20 0 00 6 6 0 00 6 6 0 00 9 9 0 00 9 2 82 3 3 0 00 5 5 0 00 30 29 97 395 37 24 94 AA+ AA AA- A+ A A- BBB+ BBB BBB- BB* 8 9 39 34 78 77 53 45 7 37 > 2 2 2 2 5 3 2 2 4 0 24 0 2 4 36 83 80 55 47 5 7 395 % 80 90 95 94 94 96 96 96 73 00 94 BB*BB+ 2 AA+ BBB- 90% One model 4 20,000 5,000 5,000 76% 3

3,000,000,000,000,000 z XX4 0 = 7ΔX Δ 4 ΔX2 + 2 ΔX3 27 X4 X3EBITDA/ X4EBITDA / 0 = 7(Log(,000 +,000z) Log(,000)),000 +,000( z) 4( 3,000,000 ) 2,000 + 2ΔX 27Δ 3 X 4 4 7 7Log( + z) + z = 2ΔX + 27Δ 3 3 3 X 4 3 X4=+5% 5 5 EBITDA EBITDA / 4

X3= 5% 42%,000,420 % X4: EBITDA/ 22 +5.0 +2.5 0.0 2.5 5.0 X3 +5.0 27 20 4 8 2 EBITDA/ +2.5 3 24 8 5 0.0 34 28 2 5 9 2.5 38 3 25 8 2 5.0 42 35 28 22 6 0% 2%,000,20,000,790 3 One model 5

94% Altman, Edward L. 968, Financial Ratios, Discriminant Analysis and Prediction of Corporate Bankruptcy, The Journal of Finance, Vol.23, pp.589-609 Engelmann, Bernd and D. Tasche 2003, Testing rating accuracy, CREDIT RISK, RISK JANUARY 2003, pp.82-86 Gentle, James E. 2005, Random Number Generation and Monte Carlo Methods, Springer Hardle, Wolfgang, R. Moro and D. Schafer 2007, Estimating Probabilities of Default With Support Vector Machines, SFB 649 ECONOMIC RISK Kaplan, R. S. and G. Urwitz 979, Statistical model of bond ratings: A methodological inquiry, The Journal of Business, Vol.52, pp.23-26 200,,, 20,,, Liu, Jon S. 2008, Monte Carlo Strategies in Scientific Computing, Springer Merton, Merton C. 974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, The Journal of Business, Vol.29, pp.87-895 6

2009, AUC AR, FSA 5, pp.29-48 2002,, 50 2, yasuhisa.toyosawa@mizuho-cb.co.jp 7

9 202 No.6 24 2 9-3-3 Tel. (03) 5222-5075