AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.

1. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Prediction of Credit Default
by Continuous Optimization
Gerhard-
Gerhard-Wilhelm Weber *
Efsun Kürüm, Kasırga Yıldırak
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

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. Main Problem from Credit Default
Whether a credit application should be consented or rejected.
Solution
Learning about the default probability of the applicant.

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. Logistic Regression
 P(Y = 1 X = xl ) 
log  = β0 + β1 ⋅ xl1 + β2 ⋅ xl 2 + K + β p ⋅ xlp
 P(Y = 0 X = x ) 
 l 
(l = 1, 2,..., N )

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. 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. 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. Model outcome versus truth
truth
d n
True Positive False Positive
Fraction Fraction
dı
TPF FPF
model outcome
False Negative True Negative
Fraction Fraction
nı
FNF TNF
1 1
total

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. 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. 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. 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. 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. 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. Optimization Problem
2
1
-1
α 1 ⋅ ∫ Φ( a + b ⋅ Φ (t )) dt − α 2 ⋅∑ 
 γi R −1 
max − (ti +1 − ti ) 
i =0  
n
a,b,τ 0
ti +1
subject to
∫ Φ(a + b ⋅ Φ −1 (t ))d t ≥ δi (i = 0,1,..., R − 1)
ti
τ := ( t1 , t 2 ,..., t R -1 ) T t0 = 0, tR = 1

17. Optimization Problem
2
1
-1
α 1 ⋅ ∫ Φ( a + b ⋅ Φ (t )) dt − α 2 ⋅∑ 
 γi R −1 
max − (ti +1 − ti ) 
i =0  
n
a,b,τ 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 , t 2 ,..., t R -1 ) T t0 = 0, tR = 1

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. New Version of the Optimization Problem
R −1 2
 γi  1
min α 2 ⋅ ∑  − (ti +1 − ti )  + α 1 ⋅ ∫ (1 − Φ(a + b ⋅ Φ −1 (t ))) dt
i =0  
a, b, τ n 0
subject to
t
j +1
∫ (1− Φ(a + b ⋅Φ−1(t ))) dt ≤ t j +1 − t j − δ j ( j = 0,1, ..., R −1)
t
j

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. 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 )) ⋅ ∆tk
k =1

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.
R −1 2
 γi  1
ΠΘ ( a,b, τ ) := α 2 ⋅ ∑  − (ti +1 − ti )  − α 1 ⋅ ∫ (1 - Φ( a + b ⋅ Φ -1 (t ))) dt +
i =0  
n 0
R -1   t j +1 
−1
α 3 ⋅ ∑ θ j ⋅  δ j −  ∫ Φ( a + b ⋅ Φ (t ))) dt  
  tj 
j =0  
1444442444444 4 
3
=: Ψ j ( a , b , τ )
Θ := ( θ1 , θ 2 , ..., θ R − 1 ) T θj ≥0 ( j = 0,1, ..., R − 1)

23. Optimization Problem further discretized
R−1 2
 γi
R

ΠΘ(a,b,τ ) = α2 ⋅ ∑  − (ti+1 −ti )  + α1 ⋅ ∑( (1-Φ(a + b⋅Φ−1(t j ))) ∆t j )2 +
i =0  
n j =1
 2
 nj    
R-1
α 3. ∑ θ j  ∑  −1(ην ) ) − δ j  ν 
 Φ(a + b ⋅Φ
 j 

∆η j
j =0 ν=0   t j +1 − t j   
   
 

24. Optimization Problem further discretized
R−1 2
 γi
R

ΠΘ(a,b,τ ) = α2 ⋅ ∑  − (ti+1 −ti )  + α1 ⋅ ∑( (1-Φ(a + b⋅Φ−1(t j ))) ∆t j )2 +
i =0  
n j =1
 2
R-1  nj  −1(ην ) ) − δ j  ∆ην  
α 3 . ∑ θ j  ∑   Φ(a + b ⋅Φ  
j  

ν=0  
j 
t j +1 − t j 
j =0  

 


25. Nonlinear Regression
2
∑ j ( j )
N
min f ( β ) =  d − g x ,β 

j =1 
N
=: ∑ f j2 ( β )
j =1
F ( β ) := ( f1 ( β ),..., f N ( β ) )
T
min f ( β ) = F T ( β ) F ( β )

26. Nonlinear Regression
β k +1 := β k + qk
• Gauss-Newton method :
∇F ( β )∇T F ( β )q = −∇F ( β ) F ( β )
• Levenberg-Marquardt method :
λ ≥0
( )
∇F ( β )∇T F (β ) + λ I p q = −∇F ( β ) F ( β )

27. Nonlinear Regression
alternative solution
min t,
t,q
subject to ( ∇F (β )∇ T
)
F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) )
2
≤ t , t ≥ 0,
|| Lq || 2 ≤ M
conic quadratic programming

28. Nonlinear Regression
alternative solution
min t,
t,q
subject to ( ∇F (β )∇ T
)
F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) )
2
≤ t , t ≥ 0,
|| Lq || 2 ≤ M
conic quadratic programming
interior point methods

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. 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. Generalized Additive Models
http://144.122.137.55/gweber/

32. Generalized Additive Models

33. References
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