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# A Classification Problem of Credit Risk Rating Investigated and Solved by Optimization of the ROC Curve

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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 7.

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### A Classification Problem of Credit Risk Rating Investigated and Solved by Optimization of the ROC Curve

1. 1. 5th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 3-15, 2010 A Classification Problem of Credit Risk Rating Investigated and Solved by Optimization of the ROC Curve Gerhard-Wilhelm Weber * Kasırga Yıldırak and Efsun Kürüm Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey • Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia
2. 2. Outline • Main Problem from Credit Default • Logistic Regression and Performance Evaluation • Cut-Off Values and Thresholds • Classification and Optimization • Nonlinear Regression • Numerical Results • Outlook and Conclusion
3. 3. Main Problem from Credit Default  Whether a credit application should be consented or rejected. Solution  Learning about the default probability of the applicant.
4. 4. Main Problem from Credit Default  Whether a credit application should be consented or rejected. Solution  Learning about the default probability of the applicant.
5. 5. Logistic Regression P(Y 1 X xl ) log β0 β1 xl1 β2 xl 2 β p xlp P(Y 0X xl ) (l 1, 2,..., N )
6. 6. Goal Our study is based on one of the Basel II criteria which recommend that the bank should divide corporate firms by 8 rating degrees with one of them being the default class. We have two problems to solve here:  To distinguish the defaults from non-defaults.  To put non-default firms in an order based on their credit quality and classify them into (sub) classes.
7. 7. Data  Data have been collected by a bank from the firms operating in the manufacturing sector in Turkey.  They cover the period between 2001 and 2006.  There are 54 qualitative variables and 36 quantitative variables originally.  Data on quantitative variables are formed based on a balance sheet submitted by the firms’ accountants. Essentially, they are the well-known financial ratios.  The data set covers 3150 firms from which 92 are in the state of default. As the number of default is small, in order to overcome the possible statistical problems, we downsize the number to 551, keeping all the default cases in the set.
8. 8. We evaluate performance of the model non-default default cases cases cut-off value ROC curve test result value TPF, sensitivity FPF, 1-specificity
9. 9. Model outcome versus truth truth d n True Positive False Positive Fraction Fraction dı TPF FPF model outcome False Negative True Negative nı Fraction Fraction FNF TNF 1 1 total
10. 10. Definitions • sensitivity (TPF) := P( Dı | D) • specificity := P( NDı | ND ) • 1-specificity (FPF) := P( Dı | ND ) • points (TPF, FPF) constitute the ROC curve • c := cut-off value • c takes values between - and • TPF(c) := P( z>c | D ) • FPF(c) := P( z>c | ND )
11. 11. normal-deviate axes TPF Normal Deviate (TPF) FPF FPF(ci ) : Φ( ci ) TPF (ci ) : Φ(a b ci ) μn - μs σn a: b: σs σs Normal Deviate (FPF)
12. 12. normal-deviate axes TPF t Normal Deviate (TPF) FPF FPF(ci ) : Φ( ci ) TPF (ci ) : Φ(a b ci ) c μn - μs σn a: b: σs σs Normal Deviate (FPF)
13. 13. Classification Ex.: cut-off values actually non-default actually default cases cases c class I class II class III class IV class V To assess discriminative power of such a model, we calculate the Area Under (ROC) Curve: AUC : Φ(a b c) d Φ (c).
14. 14. relationship between thresholds and cut-off values Ex.: TPF FPF t0 t1 t2 t3 t4 t5 R=5 Φ(c) t c Φ 1(t )
15. 15. Optimization in Credit Default Problem: Simultaneously to obtain the thresholds and the parameters a and b that maximize AUC, while balancing the size of the classes (regularization) and guaranteeing a good accuracy.
16. 16. Optimization Problem 2 1 R 1 -1 i max α1 Φ( a b Φ (t )) dt α2 (ti 1 ti ) i 0 n a,b, 0 ti 1 subject to Φ(a b Φ 1(t ))d t δi (i 0,1,..., R 1) ti τ : (t1 , t2 ,..., tR -1 )T t0 0, tR 1
17. 17. Optimization Problem 2 1 R 1 -1 i max α1 Φ( a b Φ (t )) dt α2 (ti 1 ti ) i 0 n a,b, 0 ti 1 subject to Φ(a b Φ 1 (t ))d t δi 0 (i 0,1,..., R 1) ti ti 1 ti τ : (t1 , t2 ,..., tR -1 )T t0 0, tR 1
18. 18. Over the ROC Curve TPF 1-AUC AUC FPF t0 t1 t2 t3 t4 t5 1 AOC : (1 Φ(a b Φ 1 (t ))) dt 0
19. 19. New Version of the Optimization Problem 2 R 1 1 i 1 min α2 (ti 1 ti ) α 1 (1 Φ(a b (t ))) dt a, b, τ n i 0 0 subject to t j 1 1 (1 Φ(a b (t ))) dt tj 1 t j δj ( j 0,1, ..., R 1) t j
20. 20. Regression in Credit Default Optimization problem: Simultaneously to obtain the thresholds and the parameters a and b that maximize AUC, while balancing the size of the classes (regularization) and guaranteeing a good accuracy discretization of integral nonlinear regression problem
21. 21. Discretization of the Integral Riemann-Stieltjes integral AUC Φ(a b c) dΦ(c) Riemann integral 1 AUC Φ(a b Φ 1 (t )) dt 0 Discretization R AUC Φ(a b Φ 1(tk )) Δtk k 1
22. 22. Optimization Problem with Penalty Parameters In the case of violation of anyone of these constraints, we introduce penalty parameters. As some penalty becomes increased, the iterates are forced towards the feasible set of the optimization problem. 2 R 1 1 ΠΘ ( a,b, τ ) : i (ti 1 ti ) (1- Φ( a b -1 (t ))) dt 2 1 i 0 n 0 R-1 tj 1 1 3 θj δj Φ(a b (t ))) dt j 0 tj : j ( a , b, ) Θ : (θ1, θ2 ,..., θ R 1 )T θj 0 (j 0,1, ..., R 1)
23. 23. Optimization Problem further discretized 2 R 1 R ΠΘ (a,b, ) α2 i (ti 1 ti ) α1 ( (1- Φ(a b 1 (t j ))) Δt j )2 i 0 n j 1 2 nj δj Δην R-1 1( 3. j Φ(a b j )) j j 0 ν 0 tj 1 tj
24. 24. Optimization Problem further discretized 2 R 1 R ΠΘ (a,b, ) α2 i (ti 1 ti ) α1 ( (1- Φ(a b 1 (t j ))) Δt j )2 i 0 n j 1 2 nj δj Δην R-1 1( 3. j Φ( a b j )) j j 0 ν 0 tj 1 tj
25. 25. Nonlinear Regression N 2 min f dj g xj , j 1 N : f j2 j 1 T F( ) : f1 ( ),..., f N ( ) min f ( ) F T ( )F ( )
26. 26. Nonlinear Regression k 1 : k qk • Gauss-Newton method : T F( ) F ( )q F ( )F ( ) • Levenberg-Marquardt method : 0 T F( ) F( ) Ip q F ( )F ( )
27. 27. Nonlinear Regression alternative solution min t, t,q T subject to F( ) F( ) Ip q F ( )F ( ) t, t 0, 2 || Lq || 2 M conic quadratic programming
28. 28. Nonlinear Regression alternative solution min t, t,q T subject to F( ) F( ) Ip q F ( )F ( ) t, t 0, 2 || Lq || 2 M conic quadratic programming interior point methods
29. 29. Numerical Results Initial Parameters a b Threshold values (t) 1 0.95 0.0006 0.0015 0.0035 0.01 0.035 0.11 0.35 1.5 0.85 0.0006 0.0015 0.0035 0.01 0.035 0.11 0.35 0.80 0.95 0.0006 0.0015 0.0035 0.01 0.035 0.11 0.35 2 0.70 0.0006 0.0015 0.0035 0.01 0.035 0.11 0.35 Optimization Results a b Threshold values (t) AUC 0.9999 0.9501 0.0004 0.0020 0.0032 0.012 0.03537 0.09 0.3400 0.8447 1.4999 0.8501 0.0003 0.0017 0.0036 0.011 0.03537 0.10 0.3500 0.9167 0.7999 0.9501 0.0004 0.0018 0.0032 0.011 0.03400 0.10 0.3300 0.8138 2.0001 0.7001 0.0004 0.0020 0.0031 0.012 0.03343 0.11 0.3400 0.9671
30. 30. Numerical Results Accuracy Error in Each Class I II III IV V VI VII VIII 0.0000 0.0000 0.0000 0.0001 0.0001 0.0010 0.0010 0.0075 0.0000 0.0000 0.0000 0.0001 0.0001 0.0010 0.0018 0.0094 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0018 0.0059 0.0000 0.0000 0.0000 0.0001 0.0001 0.0006 0.0018 0.0075 Number of Firms in Each Class I II III IV V VI VII VIII 4 56 27 133 115 102 129 61 2 42 52 120 119 111 120 61 4 43 40 129 114 116 120 61 4 56 24 136 106 129 111 61 Number of firms in each class at the beginning: 10, 26, 58, 106, 134, 121, 111, 61
31. 31. Generalized Additive Models
32. 32. References Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004. Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004. Buja, A., Hastie, T., and Tibshirani, R., Linear smoothers and additive models, The Ann. Stat. 17, 2 (1989) 453-510. Fox, J., Nonparametric regression, Appendix to an R and S-Plus Companion to Applied Regression, Sage Publications, 2002. Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141. Friedman, J.H., and Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823. Hastie, T., and Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310. Hastie, T., and Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc. 82, 398 (1987) 371-386. Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer, 2001. Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990. Nash, G., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996. Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).
33. 33. References Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005). Nesterov, Y.E , and Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993. Önalan, Ö., Martingale measures for NIG Lévy processes with applications to mathematical finance, presentation in: Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006. Taylan, P., Weber, G.-W., and Yerlikaya, F., A new approach to multivariate adaptive regression spline by using Tikhonov regularization and continuous optimization, to appear in TOP, Selected Papers at the Occasion of 20th EURO Mini Conference (Neringa, Lithuania, May 20-23, 2008). Stone, C.J., Additive regression and other nonparametric models, Annals of Statistics 13, 2 (1985) 689-705. Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and data mining contributions dynamics and optimization of gene-environment networks, in the special issue Organization in Matter from Quarks to Proteins of Electronic Journal of Theoretical Physics. Weber, G.-W., Taylan, P., Yıldırak, K., and Görgülü, Z.K., Financial regression and organization, to appear in the Special Issue on Optimization in Finance, of DCDIS-B (Dynamics of Continuous, Discrete and Impulsive Systems (Series B)).