Credit Risk Analysis of Cash Flow CDOs

Document Sample
Credit Risk Analysis of Cash Flow CDOs Powered By Docstoc
					         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%
                                                                                                                                                           Traditional method
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,
      negative for adverse selection)
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




                                                                                        12

				
DOCUMENT INFO