# Credit Risk Analysis of Cash Flow CDOs by sdfwerte

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```									         ABN AMRO / Model & Tools / Credit Risk Scoring

Calibrating with Power
Marco van der Burgt
Uniform Counterparty Rating Team ABN AMRO
Eurobanking 2008

Agenda
1.   Introduction and background
2.   Modeling the power curve
3.   Description of the method
4.   Demonstration for artificial portfolios
5.   Demonstration for a portfolio of sovereign debtors
6.   Conclusion

1
1. Introduction and background

1. Credit rating models:
•   Classify counterparties by credit quality
•   Linked to a Probability of Default (PD)
•   Input for calculating EC, RC and loan pricing (RAROC)

2. Basel II: Banks are allowed to develop their own rating models (Advanced
Internal Ratings Based Approach)
3. Calibration: every rating is linked to a 1-year Probability of Default (PD R), derived
from observed 1-year default rates:             # defaults 
PD R                  R

# counterparties R
4. The default rate can not be properly estimated, when the number of
observations or number of defaults is low
5. New method: use all observations, not only those in rating class R, by fitting the
power curve to a closed form and derive the default rates from this closed form

2
2. Modeling the power curve

Construction
100%
•     Order counterparties from high risk to
Discrim inative m odel
low risk
Perfect m odel
•     Plot the cumulative % of defaults (Y)                               80%                            Random m odel

Cumulative % of defaults
versus the cumulative % of                                                                         Adverse selection
counterparties (X)
60%
•     Closed form (k = concavity):
1  e  kx
yx  
1  e k                                                  40%

Interpretation
1.     Perfect model: k = ∞                                               20%

2.     Discriminative: 0 < k < ∞
3.     Random model: k = 0                                                 0%
4.     Adverse selection: k < 0                                                  0%     20%      40%       60%       80%          100%
CCC     B     BB      BBB       A    AA      AAA

Cumulative % of debtors (high risk -> low risk)

3
3. Description of the new method

Derive the PD from the power curve:
1.    According to Falkenstein et al.: PDR   D dy
dx

1  e  kx
Using the closed-form y x  
k D
2.                                                for the power curve: PD R            exp  kxR 
1  e k                                  1  e k
with xR as the midpoint between cumulative % at R and cumulative % at R-1.

How to calculate the concavity k?
1  exp  kx 
2
   1   N
1.    Minimize Root-Mean-Square Error: RMSE            yi  1  exp  k i 
i 1    N               
2.    When A > 0.8, the following approximation can be used:
1
1  e  kx        1      1     1             1
A         k 
dx       k
  1      k
0 1 e            1 e     k     k            1 A

4
3. Description of the new method

Connection between AR (Gini index) and the PD curve:
Using the following equations:
1.0
A  Arandom
AR                      2A 1
Aperfect  Arandom                             0.9

Area under CAP (A)
0.8
1
k
1 A                                                                  0.7

k D
PD R   D                exp  kxR 
dy
                                                       0.6
dx 1  e  k
0.5
0   5   10       15        20   25   30

gives a relation between AR and the PD curve:                                               Concavity k

 2xR 
2 D exp         
PDR                1  AR 
           2 
1  exp
                 1  AR

        1  AR  

5
4. Demonstration for an artificial portfolio

•   Suppose: the PD is known and called PDreal.
•   Simulate defaults by drawing Bernouilli numbers B(PDreal ) for N counterparties
in each rating class
•   Apply the new method to calculate PD
•   Calculate PD as #defaults / #counterparties (traditional method)
•   Apply the method for homogeneous and inhomogeneous portfolios

6
4. Demonstration for an artificial portfolio

Homogeneous portfolio (100                                                                                 Cumulative Accuracy Profile

counterparties per rating):                                                  100%

•    average default rate = 2.53%,                                                80%

•    From fitting the power curve: R2 =
60%
99.7%, k = 12.23
40%

20%
Observed
Fit
0%
0%               20%               40%                      60%              80%              100%
Number of        Number of   Default rate   Default rate (new
Rating   PD       counterparties   defaults    (old method)   method)
CCC/C    22.92%   100              23          23.00%         21.59%
25%
B-       10.83%   100              8           8.00%          10.51%
B        7.97%    100              7           7.00%          5.12%                                                                                        PD real
B+       2.59%    100              4           4.00%          2.49%               20%                                                                      New method
BB-      1.64%    100              0           0.00%          1.21%
BB       0.90%    100              1           1.00%          0.59%
BB+      0.70%    100              0           0.00%          0.29%               15%

BBB-     0.28%    100              0           0.00%          0.14%
BBB      0.29%    100              0           0.00%          0.07%
10%
BBB+     0.20%    100              0           0.00%          0.03%
A-       0.04%    100              0           0.00%          0.02%
A        0.03%    100              0           0.00%          0.01%                5%
A+       0.04%    100              0           0.00%          0.00%
AA-      0.03%    100              0           0.00%          0.00%
AA       0.03%    100              0           0.00%          0.00%                0%
CCC/C

B-

B

B+

BB-

BB

BB+

BBB-

BBB

BBB+

A-

A

A+

AA-

AA

AA+

AAA
AA+      0.03%    100              0           0.00%          0.00%
AAA      0.03%    100              0           0.00%          0.00%

7
4. Demonstration for an artificial portfolio

Homogeneous portfolio (10 counterparties                                                                       Cumulative Accuracy Profile

per rating):                                                                 100%

•    average default rate = 4.12%,                                                80%

•    From fitting the power curve: R2 =
60%
98.9%, k = 9.8
40%

20%
Observed
Fit
0%
0%               20%                40%                     60%              80%              100%

Number of        Number of   Default rate   Default rate (new
Rating   PD       counterparties   defaults    (old method)   method)
CCC/C    22.92%   10               3           30.00%         30.24%              35%
B-       10.83%   10               2           20.00%         17.00%
B        7.97%    10               1           10.00%         9.55%               30%
PD real
B+       2.59%    10               0           0.00%          5.37%                                                                                        New method
BB-      1.64%    10               0           0.00%          3.02%               25%                                                                      Traditional method
BB       0.90%    10               1           10.00%         1.70%
BB+      0.70%    10               0           0.00%          0.95%               20%

BBB-     0.28%    10               0           0.00%          0.54%
15%
BBB      0.29%    10               0           0.00%          0.30%
BBB+     0.20%    10               0           0.00%          0.17%
10%
A-       0.04%    10               0           0.00%          0.10%
A        0.03%    10               0           0.00%          0.05%
5%
A+       0.04%    10               0           0.00%          0.03%
AA-      0.03%    10               0           0.00%          0.02%
0%
AA       0.03%    10               0           0.00%          0.01%
CCC/C

B-

B

B+

BB-

BB

BB+

BBB-

BBB

BBB+

A-

A

A+

AA-

AA

AA+

AAA
AA+      0.03%    10               0           0.00%          0.01%
AAA      0.03%    10               0           0.00%          0.00%

8
4. Demonstration for an artificial portfolio

Inhomogeneous portfolio:                                                                                                                                    Cumulative Accuracy Profile

•    average default rate = 1.15%,                                                                                                 100%

•    From fitting the power curve: R2 =                                                                                            80%

93.5%, k = 11.4
60%

600
500
40%
400
300
200

100                                                                                                                       20%                                                                              Observed
0

Fit
CCC/C
B-
B
B+
BB-
BB
BB+
BBB-
BBB
BBB+
A-
A
A+
AA-
AA
AA+
AAA
0%
0%               20%               40%                     60%            80%               100%

Number of                   Number of                  Default rate                Default rate (new
Rating   PD                 counterparties              defaults                   (old method)                method)
CCC/C    22.92%             50                          11                         22.00%                      12.23%              25%

B-       10.83%             75                          7                          9.33%                       10.28%                                                                                  PD real
B        7.97%              100                         10                         10.00%                      8.05%
20%                                                                 New method
B+       2.59%              150                         3                          2.00%                       5.68%
BB-      1.64%              225                         1                          0.44%                       3.36%                                                                                   Traditional method
BB       0.90%              300                         6                          2.00%                       1.62%               15%
BB+      0.70%              400                         1                          0.25%                       0.61%
BBB-     0.28%              500                         3                          0.60%                       0.17%
BBB      0.29%              550                         2                          0.36%                       0.04%               10%
BBB+     0.20%              500                         1                          0.20%                       0.01%
A-       0.04%              400                         0                          0.00%                       0.00%
5%
A        0.03%              250                         1                          0.40%                       0.00%
A+       0.04%              225                         1                          0.44%                       0.00%
AA-      0.03%              150                         0                          0.00%                       0.00%               0%
AA       0.03%              100                         0                          0.00%                       0.00%
CCC/C

B-

B

B+

BB-

BB

BB+

BBB-

BBB

BBB+

A-

A

A+

AA-

AA

AA+

AAA
AA+      0.03%              75                          0                          0.00%                       0.00%
AAA      0.03%              50                          0                          0.00%                       0.00%

9
5. Demonstration for portfolio of sovereign debtors

•     Portfolio of 82 sovereign debtors, rated at March 2004
•     2 defaults in the period March 2004 – March 2005: Grenada, Dominican
Republic
1  e  kx
•     Construction of the power curve and fitting to y  x              gives k = 8.03
with an R 2 = 75%                                        1  e k

Rating   Sovereigns   Defaults   X      Y      PD curve
0%     0%                                           100%
CC       1            1          1%     50%    17,83%
CCC+     1            0          2%     50%    16,24%
B-       5            0          8%     50%    12,27%                                80%

Cumulative % of defaults
B        6            0          15%    50%    7,34%
B+       3            0          19%    50%    4,82%
BB-      4            1          23%    100%   3,48%                                 60%
BB       8            0          33%    100%   1,99%
BB+      5            0          38%    100%   1,08%
BBB-     2            0          41%    100%   0,78%                                 40%
BBB      5            0          47%    100%   0,56%
BBB+     4            0          51%    100%   0,37%
A-       9            0          62%    100%   0,20%                                                                       Observed
20%
A        5            0          67%    100%   0,10%
A+       6            0          74%    100%   0,06%                                                                       Fit to closed
AA-      2            0          77%    100%   0,04%                                                                       form
AA       1            0          78%    100%   0,04%                                  0%
AA+      3            0          81%    100%   0,03%                                        0%      20%      40%       60%       80%       100%
AAA      16           0          100%   100%   0,01%                                         Cumulative % of debtors (high risk -> low risk)
Total    86           2

10
5. Demonstration for portfolio of sovereign debtors

What is the error in concavity k?
How sensitive is the concavity k to observations in rating classes?

Answer: Try different scenarios and investigate how k will change.

30%

Scenario                              Concavity                                                                                              PD curve
25%
PD curve at k=5.87
1 default in CC, 1 default in BB      6.15

Calibrated PD
PD curve at 11.70
1 default in CC, 1 default in BB-     8.03                        20%

1 default in CC, 1 default in B+      11.70
15%
1 default in CCC+, 1 default in BB    5.87
1 default in CCC+, 1 default in BB-   7.47                        10%
1 default in CCC+, 1 default in B+    10.10
Standard deviation                    2.28                        5%
95% Confidence level                  4.47
Maximum                               11.7                        0%

BBB+
BBB
BBB-
BB+
BB
BB-
B+
B
B-

A+
CCC+

AA+
A
CC

AA

AAA
A-

AA-
Minimum                               5.9
Average                               8.22                                                        S&P FC Sovereign Rating

11
6. Conclusions

1.    A new method for calibration of credit models, where the power curve is fit to
a functional form and the PD curve is derived from this form
2.    Key parameter is the concavity (0 for bad model, ∞ for perfect model,
3.    The new method adds value when data is sparse
4.    Credit Crisis 2007: due to securitization wave, risk is transferred from highly
regulated banks to non-regulated hedge funds and other FI: loss of
information and low amount of data

References:
Van der Burgt, M.J., Calibrating Low-Default Portfolios, using the Cumulative
Accuracy Profile, Journal of Risk Model Validation, February 2008
Falkenstein, E., Boral, A. and Carty, L., RiskCalc for private companies: Moody’s
default model: rating methodology, 2000, Moody’s Investor Service

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