The document discusses stress testing credit risk portfolios. It outlines the function of stress testing as investigating unexpected losses under extreme conditions not experienced in historical data. Stress testing is important for assessing capital adequacy and challenging risk models. The document describes regulatory requirements for stress testing, key credit risk parameters to consider in stress tests, and different types of stress test scenarios. It emphasizes that stress testing is a key risk management tool and supervisory expectation.

1. Stress Testing Credit Risk
Portfolios
Michael Jacobs, Ph.D., CFA
Senior Financial Economist
Credit Risk Analysis Division
U.S. Office of the Comptroller of the Currency
Risk / Incisive Media Training, March 2012
The views expressed herein are those of the author and do not necessarily represent the views of the Office of the
Comptroller of the Currency or the Department of the Treasury.

2. Outline
• Introduction
• The Function of Stress Testing
• Supervisory Requirements and Expectations
• The Credit Risk Parameters for Stress Testing
• Interpretation of Stress Test Results
• A Typology of Stress Tests
– Uniform Testing
– Risk Factor Sensitivities
– Scenario Analysis
• Historical Scenarios
• Statistical Scenarios
• Hypothetical Scenarios
• Procedures for Conducting Stress Tests
• A Simple Stress Testing Example

3. Introduction: Overview
• Modern credit risk modeling (e.g., Merton, 1974) increasingly
relies on advanced mathematical, statistical and numerical
techniques to measure and manage risk in credit portfolios
• This gives rise to model risk (OCC 2011-16) and the possibility
of understating inherent dangers stemming from very rare yet
plausible occurrencs perhaps not in our reference data-sets
• International supervisors have recognized the importance of
stress testing credit risk in the Basel framework (BCBS, 2009)
• It can and has been argued that the art and science of stress
testing has lagged in the domain of credit, vs. other types of risk
(e.g., market), and our objective is to help fill this vacuum
• We aim to present classifications & established techniques that
will help practitioners formulate robust credit risk stress tests

4. Introduction: Motivation in
the Financial Crisis* losses in
• Bank
Figure 3: Average Ratio of Total Charge-offs to Total Value of Loans for
Top 50 Banks as of 4Q09 the recent
0.035
(Call Report Data 1984-2009) financial crisis
exceed levels
0.03
observed in
0.025 recent history!
0.02 • This illustrates
0.015
the inherent
limitations of
0.01
backward
0.005 looking
0
models – we
must
84 1
85 1
86 0
87 0
87 1
88 1
89 0
90 0
90 1
91 1
92 0
93 0
93 1
94 1
95 0
96 0
96 1
97 1
98 0
99 0
99 1
00 1
01 0
02 0
02 1
03 1
04 0
05 0
05 1
06 1
07 0
08 0
08 1
09 1
30
19 033
19 123
19 093
19 063
19 033
19 123
19 093
19 063
19 033
19 123
19 093
19 063
19 033
19 123
19 093
19 063
19 033
19 123
19 093
19 063
19 033
20 123
20 093
20 063
20 033
20 123
20 093
20 063
20 033
20 123
20 093
20 063
20 033
20 123
09
84
19
anticipate risk
* Reproduced from: Inanoglu, H., Jacobs, Jr., M., and Robin Sickles, 2010 (July), Analyzing bank
efficiency: Are “too-big-to-fail” banks efficient?, forthcoming in the Journal of Efficiency

5. Introduction: Motivation in the
Imprecision of Value-at-Risk*
Gaussian Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR
• Sampling
variation in
6e-09
VaR inputs
leads to huge
5e-09
confidence
4e-09
bounds for
risk estimates
Density
3e-09
(coefficient of
variation
2e-09
=35.4%)
1e-09
• This is even
0e+00
assuming we
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
have the
99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)
correct model
VaR99.7%=7.64e+8, q2.5%=6.26e+8, q97.5%=8.94e+8, CV=35.37%
* Inanoglu, H., and Jacobs, Jr., M., 2009, Models for risk aggregation and sensitivity analysis: An application to bank economic
capital, The Journal of Risk and Financial Management 2, 118-189.

6. Conceptual Issues in Stress
Testing: Risk vs. Uncertainty
• Knight (1921): uncertainty is when a probability distribution is
unmeasurable or unknown, arguably a realistic scenario
• Rely upon empirical data to estimate loss distributions, but this
is complicated because of changing economic conditions
• Popper (1945): situations of uncertainty closely associated &
inherent to changes in knowledge & behavior (no historicism)
• Shackle (1990): predictions reliable only for immediate future,
as impact others’ choices after time has an appreciable effect
• This role of human behavior in economic theory was a key
impetus behind rational expectations & behavioral finance
• Implication is that risk managers must be aware of model
limitations & how an EC regime itself changes behavior
• Although we face uncertainty, valuable to estimate loss
distributions in that helps make explicit sources of uncertainty

7. The Function of Stress Testing
• A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary
realm of experience (e.g., extreme events not in our data-sets)
• Many reasons for conducting periodic ST are largely due to the
relationship between UL and economic capital (EC)
• EC is generally thought of as the difference between Value-at-
Risk (VaR), or extreme loss at some confidence level (e.g., a
high quantile of a loss distribution), and expected loss (EL)
• This purpose for ST hinges on our definition of UL – while it is
commonly thought that EC should cover this, in that UL may not
only be unexpected but not credible as it is a statistical concept
• Therefore some argue that results of an ST should be used for
EC vs. UL, but this is rare, as we usually do not have probability
distributions associated with stress events

8. Function of Stress Testing:
Expected vs. Unexpected Loss
Figure 1 Vasicek
80 distribution
(theta = 0.01,
rho = 0.06)
Expected Economic Capital
Losses
60
Probability
40 Unexpected Losses
20
EL “Tail of the VaR 99.95%
“Body of the
Distribution” Distribution”
Losses
0.01 0.02 0.03 0.04

9. The Function of Stress Testing
(continued)
• ST can and commonly have been used to challenge the
adequacy of regulatory (RC) or EC & derive a buffer for losses
exceeding the VaR, especially for new products or portfolios
• Another advantage to ST to determine capital is that it can
easily aggregate different risk types (e.g., credit, market &
operational), problematic under standard EC methodologies
– E.g., different horizons and confidence levels for market vs. credit risk
– Powerful dependencies between risk types in periods of stress
• Quantification of ST appear and can be deployed several
aspects of risk management with respect to extreme losses:
– Risk buffers determined or tested
– Risk capacity of a financial institution
– Setting sub-portfolio limits, especially if low-default situation
– Risk policy, tolerance and appetite

10. Function of Stress Testing: The
Risk Aggregation Problem
• Correlations
Pairwise Scattergraph & Pearson Correlations of 5 Risk Types
7
x 10 Top 200 Banks (Call Report Data 1984-2008)
4
Credit amongst different
2
0 x 10
7
risk types are in
4
Operat.
many cases large
2
corr(cr,ops)
= 0.6517
and cannot be
0 x 10
7
ignored
2
0 corr(cr,mkt) corr(ops,mkt)
Market
• As risks are
= 0.2241 = 0.1989
-2 x 10
8 modeled very
5
corr(mkt,liqu)
Liqu. different, it is
corr(cr,liqu) corr(ops,liqu)
0
8
= 0.5343 = 0.1533 = 0.1127 challenging to
-5 x 10
2
aggregate these
Int.Rt.
0
corr(cr,int)
= -0.1328
corr(ops,int)
= -0.1174
corr(mkt,int)
= 0.2478
corr(int,liqu)
= 0.1897
into an economic
-2 capital measure
0 2 4 0 2 4 -2 0 2 -5 0 5 -2 0 2
7 7 7 8 8
x 10 x 10 x 10 x 10 x 10
* Inanoglu, H., and Jacobs, Jr., M., 2009, Models for risk aggregation and sensitivity analysis: An application to bank economic
capital, The Journal of Risk and Financial Management 2, 118-189.

11. The Function of Stress Testing
(continued)
• Apart from risk measurement or quantification, ST can be a risk
management tool in analyzing portfolio composition and
resilience with respect to disturbances:
– Identify potential uncertainties and locate the portfolio vulnerabilities
– Analyze the effects of new complex structures and credit products
– Guide discussion on unfavorable developments like crises and abnormal
market conditions, which cannot be excluded
– Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the market
– Derive some need for action to reduce the risk of extreme losses and
hence economic capital, and mitigate the vulnerability to important risk
relevant effects
– Test the portfolio diversification by introducing (implicit) correlations
– Question the bank’s attitude towards risk

12. Supervisory Requirements and
Expectations
• ST appears in Basel II (BIS, 2006) framework under both Pillar I
(minimum capital requirements) and Pillar 2 (the supervisory
review process) with the aim of improving risk management
• Every IRB bank has to conduct sound, significant and
meaningful stress testing to assess the capital adequacy in a
reasonably conservative way.
– Major credit risk concentrations have to undergo periodic stress tests.
– ST should be integrated in the internal capital adequacy process (i.e.,
risk management strategies to respond to the outcome of ST)
• Banks shall ensure that they dispose of enough capital to meet
the regulatory capital requirements even in the case of stress
• Should identify possible future events / changes in economic
conditions with potentially adverse effects on credit exposures
& assess the ability of the bank to withstand such

13. Supervisory Requirements and
Expectations (continued)
• A quantification of the impact on the parameters probability of
default (PD), loss given default (LGD), exposure at default
(EAD) as well as rating migrations is required
• Special notes on how to implement these requirements include
the use of scenarios including things like:
– economic or industry downturn
– market-risk events
– liquidity shortage
• Consider recession scenarios (worst-case not required)
• Banks should use their own data for estimating rating
migrations & integrate the insight of such for external ratings
• Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment

14. Supervisory Requirements and
Expectations: Regulatory Capital
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Normal:PD=1%,LGD=40%,Rho=0.1
EL-norm=0.40% Stressed:PD=1.5%,LGD=60%,Rho=0.15
0.8
Regulatory Capital
EL-stress=0.90%
0.6
CVaR-norm=6.78%
Probability Density
CVaR-stress=15.79%
0.4
Stressed Capital
0.2
0.0
0.00 0.05 0.10 0.15
Credit Loss
• Shocking credit risk parameters can give us an idea of what
kind of buffer we may need to add to an EC estimate

15. Supervisory Requirements and
Expectations (continued)
• Though ST are mainly contained in Pillar 1, it is a fundamental
part of Pillar 2, an important way of assessing capital adequacy
• This explains the non-prescriptiveness for ST as Pillar 2
recognizes that banks are competent to assess and measure
their credit risk appropriately
• This also implies that ST should focus on EC as well as
regulatory capital, as these represent the supervisory and bank
internal views on portfolio credit risk
• ST has been addressed by regulators or central banks beyond
the Basel II framework, regarding the stability of the financial
system, in published supplements (including now Basel III)
• ST should consider extreme deviations from normal situations
& hence involve unrealistic yet still plausible scenarios (i.e.
situations with low probability of occurrence)

16. Supervisory Requirements and
Expectations (continued)
• ST should also consider joint events which are plausible but
which may not yet been observed in reference data-sets
• Financial institutions should also use ST to become aware of
their risk profile and to challenge their business plans, target
portfolios, risk politics, etc.
• ST should not only be addressed to check the capital
adequacy, but also used to determine & question credit limits
• ST should not be treated only as an amendment to the VaR
evaluations for credit portfolios, but as a complimentary
method, which contrasts the purely statistical approach of VaR-
methods by including causally determined considerations for
unexpected losses
– In particular, it can be used to specify extreme losses in a qualitative and
quantitative way

17. The Credit Risk Parameters for
Stress Testing (continued)
• A key aspect of ST mechanics in Basel II or EC is examining
the sensitivity to variation in risk parameters
• In the case of RC the risk parameters in the ST exercise are
given by the PD, LGD, EAD and Correlation
• PD has played a more prominent role since conditional upon
obligor default LGD & EAD tend to be adapted to malign
environments & the stress scenarios are more limited
• EAD may exhibit some sensitivity to certain exogenous factors
like FX rates, we would expect such to be in the usual estimate
• LGD ranges are largely dependent upon the quantification
technique (e.g., the discount rate used for post default cash
flows) which should be disentangled from the economic regime
– For most types of lending it is thought that collateral values should be
key & incorporate sufficient conservatism naturally, but that varies

18. The Credit Risk Parameters for
Stress Testing: LGD
• LGD: estimate of the amount a bank loses if a counterparty defaults
(expected PV of economic loss / EAD or 1 minus the recovery rate)
• Depends on claim seniority, collateral, legal jurisdiction, condition of
defaulted firm or capital structure, bank practice, type of exposure
• Measured LGDs depend on default definition: broader (distressed
exchange/reneg.) vs. narrow (bankruptcy,liquidation)->lower/higher
• Market vs. workout LGD: prices of defaulted debt shortly after
default vs. realized discounted ultimate recoveries up to resolution
• LGDs on individual instruments tends to be either very high (sub or
unsecured debt) or very low (secured bonds or loans) - “bimodal”
• Downturn LGD: intuition & evidence that should be elevated in
economic downturns – but mixed evidence & role of bank practice
• Note differences across different types of lending (e.g., enterprise
value & debt markets is particular large corporate)
Discounted Recoveries
LGD=1- EAD 1 RecoveryRate
Discounted Direct & Indirect Workout Costs

19. The Credit Risk Parameters for
Stress Testing: LGD (continued)
• Contractual features:
Employees, Trade
Creditors, Lawyers
more senior and secured
instruments do better.
Bank Loans Banks • Absolute Priority Rule:
S some violations (but
E usually small)
Senior Secured
N • More senior instruments
I tend to be better secured.
O Senior Unsecured
R Bondholders • Debt cushion as distinct
I
from position in the
Senior Subordinated
T
capital structure.
Y Junior Subordinated
• High LGD for senior debt
with little sub-debt?
Preferred Shares • Proportion of bank debt
Shareholders • The “Grim Reaper” story
Common Shares • Enterprise value
19

20. The Credit Risk Parameters for
Stress Testing: LGD (continued)
• Bankruptcies (65.2%) have higher LGDs than out-of-court settlements
(55.8%)
• Firms reorganized (emerged or acquired) have lower LGDs (43.9%) than
firms liquidated (68.9%)
*Diagram reproduced from: Jacobs, M., et al., 2011, Understanding and predicting the resolution of financial distress, Forthcoming
Journal of Portfolio Management (March,2012), page 31. 518 defaulted S&P/Moody’s rated firms 1985-2004.

21. The Credit Risk Parameters for
Stress Testing: LGD (continued)
• Distributions of
Distribution of Moody's Market LGD: All Seniorities (count=4400,mean=59.1%) Distribution of Moody's Market LGD: Senior Bank Loans (count=54,mean=16.7%)
2.5
1.5
*
2.0
Moody’s Defaulted
1.0
1.5
Density
Density
1.0
0.5
Bonds & Loan
0.5
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
LGD
Distribution of Moody's Market LGD: Senior Secured Bonds (count=1022,mean=46.7%)
LGD
Distribution of Moody's Market LGD: Senior Unsecured Bonds (count=2215,mean=60.0%) LGD (DRS
2.0
Database 1970-
1.5
1.5
1.0
Density
Density
2010)
1.0
0.5
0.5
• Lower the quality
0.0
0.0
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
LGD LGD
of collateral, the
Distribution of Moody's Market LGD: Senior Subordinated Bonds (count=600,mean=67.9%) Distribution of Moody's Market LGD: Junior Subordinated Bonds (count=509,mean=74.6%)
2.5
1.5
2.0
higher the LGD
1.5
1.0
Density
Density
1.0
0.5
• Lower ranking of
0.5
0.0
0.0
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
the creditor class,
LGD LGD
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
Reproduced with permission:
(Moody's Ultimate Recovery Database 1987-2010)2
Senior Senior Junior
the higher the LGD
Senior Secured Unsecured Subordinated Subordinated Subordinated
Moody’s, URD, Release 10-15-
10.
Bank Loans Bonds Bonds Bonds Bonds Bonds Total Instrument
• And higher
Collateral Type
Cash & Highly Liquid Collateral
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
32 -0.4% 7 8.7% 7 8.7% 1 0.0% 0 N/A 0 N/A 40 1.2%
seniority debt
tends to have
Major Collateral
Inventory & Accounts Receivable 173 3.6% 0 N/A 7 6.9% 0 N/A 0 N/A 0 N/A 180 3.8%
Category
All Assets, 1st Lien & Capital Stock 1199 18.8% 242 24.7% 242 24.7% 1 14.0% 2 30.8% 0 N/A 1444 19.8%
Plant, Property & Equipment
2nd Lien
Intangible or Illiquid Collateral
67
65
1
12.4%
41.2%
0.0%
245
75
5
49.6%
37.5%
72.2%
245
75
5
49.6%
37.5%
72.2%
2
4
0
39.6%
59.0%
N/A
0
5
0
N/A
50.6%
N/A
0
1
0
N/A
60.0%
N/A
314
150
6
41.6%
40.3%
60.2%
better collateral
Total Secured
Total Unsecured
1537
129
17.4%
43.1%
581
0
36.8%
N/A
0
1147
N/A
51.4%
8
451
41.2%
70.8%
7
358
44.9%
71.7%
1
64
60.0%
80.8%
2134
2149
22.9%
59.2%
* Reproduced with permision:
Total Collateral 1666 19.4% 581 36.8% 1147 51.4% 459 70.3% 365 71.2% 65 80.5% 4283 41.1% Moody’s Analytics.Default Rate
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par.
2 - 4283 defaulted and resolved instruments as of 8-9-10
Service Database, 10-15-10.

22. The Credit Risk Parameters for
Stress Testing: LGD (continued)
• Downturns: 1973-74, 1981-82, 1990-91, 2001-02, 2008-09
• As noted previously, commonly accepted that LGD is higher during
economic downturns when default rates are elevated
• Lower collateral values
• Greater supply of distressed debt
• The cycle is evident in time series, but note all the noise
* Reproduced with permission: Moody’s Analytics. Default Rate Service Database, Release Date 10-15-10.

23. The Credit Risk Parameters for
Stress Testing: LGD (continued)

24. The Credit Risk Parameters for
Stress Testing: LGD (continued)
• Jacobs & Karagozoglu (2011)* study
Table 3 of Jacobs & Karagozoglu (2010):
Simultaneous Equation Modeling of Discounted Instrument & Oligor LGD: Full
Information Maximum Likelihood Estimation (Moody's URD 1985–2009)
ultimate LGD in Moody’s URD at the
Category
Instrument Obligor
Partial Partial
Variable Effect P-Value Effect P-Value
Debt to Equity Ratio (Market)
Book Value
-0.0903
-0.0814
2.55E-03
0.0174 loan & firm level simultaneously
Financial
Tobin's Q 0.0729 8.73E-03
Intangibles Ratio
Working Capital / Total Assets
Operating Cash Flow
0.0978
-0.1347
-8.31E-03
7.02E-03
4.54E-03
0.0193
• Empirically models notion that
recovery on a loan is akin to a collar
Industry
Profit Margin - Industry -0.0917 1.20E-03
Industry - Utility -0.1506 8.18E-03
option on the firm/enterprise level
Industry - Technology 0.0608 2.03E-03
Senior Secured 0.0432 0.0482
Senior Unsecured 0.0725 3.11E-03
Contractual
Senior Subordinated 0.2266 1.21E-03
Junior Subordinated
Collateral Rank
Percent Debt Above
0.1088
0.1504
0.1241
0.0303
4.26E-12
3.84E-03
recovery
Percent Debt Below -0.2930 7.65E-06
• Firm (loan) LGD depends on fin ratios,
Time
Time Between Defaults -0.1853 7.40E-04
Time-to-Maturity 0.0255 0.0084
capital structure, industry state,
Structure
Number of Creditor Classes 0.0975 1.20E-03
Capital
Percent Secured Debt -0.1403 7.56E-03
Percent Bank Debt
Investment Grade at Origination
-0.2382
-0.0720
7.45E-03
4.81E-03 macroeconomy, equity market / CARs
Credit Quality /
Principal at Default 8.99E-03 1.14E-03
(seniority, collateral quality, debt
Market
Cumulative Abnormal Returns -0.2753 1.76E-04
Ultimate LGD - Obligor 0.5643 7.82E-06
LGD at Default - Obligor 0.1906 4.05E-04
LGD at Default - Instrument 0.2146 1.18E-14
cushion)
Legal
Prepackaged Bankruptcy -0.0406 5.38E-03
Bankruptcy Filing 0.1429 5.00E-03
1989-1991 Recession 0.0678 0.0474
• Feedback from ultimate obligor LGD
Macro
2000-2002 Recession 0.1074 0.0103
Moody's Speculative Default Rate 0.0726 1.72E-04
S&P 500 Return
Number of Observations
In-Smpl Out-Smpl
568 114
-0.1392
In-Smpl
568
2.88E-04
Out-Smpl
114
to the facility level & at both level
Diagnostics
Log-Likelihood
Pseudo R-Squared
Hoshmer-Lemeshow
1.72E-10 9.60E-08
0.6997
0.4115
0.6119
0.3345
1.72E-10
0.5822
0.5204
9.60E-08
0.4744
0.3907
ultimate LGD depends upon market
Area under ROC Curve 0.8936 0.7653 0.8983 0.7860
Kolmogorov-Smirnov 1.12E-07 4.89E-06 1.42E-07 6.87E-06 *Jacobs, Jr., M., and Karagozoglu, A, 2011, Modeling ultimate loss given default on corporate
debt, The Journal of Fixed Income, 21:1 (Summer), 6-20.

25. The Credit Risk Parameters for
Stress Testing: EAD
• EAD: an estimate of the dollar amount of exposure on an instrument if
there is a counterparty / obligor default over some horizon
• Typically, a borrower going into default will try to draw down on credit
lines as liquidity or alternative funding dries up
• Correlation between EAD & PD for derivatives exposure: wrong way
exposure (WWE) problem: higher exposure & more default risk
• Derivative WWE examples
– A cross-FX swap with weaker a currency counterparty: more likely to
default just when currency weakens & banks are in the money
– A bank purchases credit protection through a CDS & the insurer is
deteriorating at the same time as the reference entity
• Although Basel II stipulates “margin of conservatism” for EAD, in the
case of loans greater monitoring->negative correlation with PD
• As either borrower deteriorates or in downturn conditions, EAD risk
may actually become lower as banks cut lines

26. The Credit Risk Parameters for
Stress Testing: EAD (continued)
• For traditional credit
products depends on loan
size, redemption schedule,
covenants, bank monitoring,
borrower distress, pricing
• Case of unfunded
commitments (e.g., revolvers):
EAD anywhere from 0% to
100% of line limit (term loans
typically just face value)
• Typically banks estimate EAD by a loan equivalency quotient (LEQ):
fraction of unused drawn down in default over total current availability:
O - Ot
EADXt ,t,T = Ot + LEQXf ,t,T × Lt - Ot Ot + E t | T , Xt × Lt - Ot
t
Lt - Ot
• Where O: outstanding, L: limit, t: current time, τ: time of default, T:
horizon, X: vector of risk factors , Et (.) mathematical expectation

27. EAD Example for Credit Models:
Jacobs (2010) Study
Table 6 - Generalized Linear Model Multiple • EAD risk increasing in time-to-
Regression Model for EAD Risk (LEQ Factor) - default; loan undrawn or limit
Moodys Rated Defaulted Revolvers (1985-2009)
Coeff. P-Value amount; firm size or intangibility; %
Utilization: Used Amount / Limit (%) -0.3508 2.53E-06
Total Commitment: Line Limit ($) 3.64E-05 0.0723
bank or secured debt
Undraw n: "Headroom" on line ($)
Time-to-Default (years)
3.27E-05
0.0516
7.42E-03
1.72E-05
• EAD risk decreasing in PD ( worse
Rating 1: BB (base = AAA-BBB) -0.1442 0.0426 obligor rating or aggregate default
Rating 2: B -0.0681 6.20E-03
Rating 3: CCC-CC -0.0735 1.03E-05
rate); firm leverage or profitability;
Rating 4: CCC -0.0502 2.08E-04 loan collateral quality or debt
Lev erage: L.T.Debt / M.V. Equity -0.0515 0.0714
Size: Book Value (logarithm) 0.1154 2.63E-03
cushion
I ntangibility: I ntangible / Total Assets 0.0600 0.0214
Table 5
1
Liquidity: Current Cssets / Current Liabilities -0.0366 0.0251 Estimated LEF by Rating and Time-to-Default
Profitabilty: Net I ncome / Net Sales -6.59E-04 0.0230 Moodys Rated Defaulted Borrowers Revolvers 1985-2009
Colllateral Rank: Higher -> Low er Quality 0.0306 3.07E-03 Risk Time-to-Default (yrs)
Debt Cushion: % Debt Below the Loan -0.2801 5.18E-06 Rating 1 2 3 4 5 >5 Total
Aggregate Speculativ e Grade Default Rate -0.9336 0.0635 AAA-BBB 64.56% 65.26% 84.93% 92.86% 84.58% 0.00% 69.06%
Percent Bank Debt in the Capital Structure 0.2854 5.61E-06 BB 38.90% 42.13% 45.91% 43.91% 42.35% 0.00% 40.79%
Percent Secured Debt in the Capital Structure 0.1115 2.65E-03 B 41.51% 43.92% 42.60% 52.77% 49.94% 14.00% 42.66%
Degrees of Freedom 455 CCC-CC 32.97% 47.38% 54.80% 55.05% 55.30% 0.00% 36.85%
Likelihood Ratio P-Value 7.48E-12
C 28.21% 9.71% 47.64% 25.67% 0.00% 0.00% 20.22%
Pseudo R-Squared 0.2040
Total 40.81% 44.89% 47.79% 54.00% 52.05% 14.00% 42.21%
Spearman Rank Correlation 0.4670
*Jacobs Jr., M., 2010, An empirical study of exposure at default,
MSE of Forecasted EAD 2.74E+15
The Journal of Advanced Studies in Finance, Volume 1, Number 1

28. The Credit Risk Parameters for
Stress Testing: PD
• In ST the PD risk parameter is the most common of the three
that risk managers prefer to shock
• PD varies for two principal reasons
– Obligors may be rated differently due to changes in risk factors that
determine the PD grade (e.g., increased leverage, decreased cash flow)
– Realized default rates upon which PD estimates with respect to a given
rating may change (e.g., economic downturn leads to more defaults)
• This gives rise to two design options for integration of PDs into
ST: altering either the assignment of rating or associated PDs
– Re-grading has the advantage that it admits the inclusion of transitions to
non-performing loans
– As varying PDs corresponds to a rating change, up-grades are possible
• Possibilities of variance & sensitivity of the input for the rating
process should be investigated to get a first estimate

29. The Credit Risk Parameters for
Stress Testing: PD (continued)
• ST should incorporate expert opinion on rating methodology in
addition analysis of hard reference data for transition & default
• Altering PDs associated with ratings could originate in the
variation of systematic risk drivers, an important theme in ST
• A common approach is as a 1st step to estimate the volatility of
PDs in ST of regulatory capital, with differential systematic &
idiosyncratic risk on PD deviations as 2nd step enhancement
• An analysis of the transition structure for rating grades might
also be used to determine PDs under stress conditions
• An advantage (disadvantage) of modifying PDs via rating
assignment is greater diversity change type (absence of a
modified assignment to performing & non-performing portfolio)

30. PD Estimation for Credit
Models: Rating Agency Data
• Credit rating agencies have a long
history in providing estimates of firms’
creditworthiness
• Information about firms’
creditworthiness has historically been
difficult to obtain
• In general, agency ratings rank order
firms’ likelihood of default over the
next five years
• However, it is common to take
average default rates by ratings as PD
estimates
• The figure shows that agency ratings
reflect market segmentations

31. PD Estimation: Rating Agency
Data – Migration & Default Rates
• Migration matrices Moody's Letter Rating Migration Rates (1970-2010)*
Panel 1: One-Year Average Rates
Default
summarize the average From/To:
AA
AA AA A BBB BB B CCC CC-C WR
87.395% 8.626% 0.602% 0.010% 0.027% 0.002% 0.002% 0.000% 3.336% 0.000%
Rates
rates of transition AA
A
BBB
0.971% 85.616% 7.966% 0.359% 0.045% 0.018% 0.008% 0.001% 4.996% 0.020%
0.062% 2.689% 86.763% 5.271% 0.488% 0.109% 0.032% 0.004% 4.528% 0.054%
0.043% 0.184% 4.525% 84.517% 4.112% 0.775% 0.173% 0.019% 5.475% 0.176%
between rating BB
B
0.008% 0.056% 0.370% 5.644% 75.759% 7.239% 0.533% 0.080% 9.208% 1.104%
0.010% 0.034% 0.126% 0.338% 4.762% 73.524% 5.767% 0.665% 10.544% 4.230%
CCC 0.000% 0.021% 0.021% 0.142% 0.463% 8.263% 60.088% 4.104% 12.176% 14.721%
categories CC-C 0.000% 0.000% 0.000% 0.000% 0.324% 2.374% 8.880% 36.270% 16.701% 35.451%
Panel 2: Five-Year Average Rates
• The default rates in the
Default
From/To: AA AA A BBB BB B CCC CC-C WR Rates
AA 54.130% 24.062% 5.209% 0.357% 0.253% 0.038% 0.038% 0.000% 15.832% 0.081%
final column are often AA
A
BBB
3.243% 50.038% 21.225% 3.220% 0.521% 0.150% 0.030% 0.012% 21.374% 0.186%
0.202% 8.545% 52.504% 14.337% 2.617% 0.831% 0.143% 0.023% 20.247% 0.551%
0.231% 1.132% 13.513% 46.508% 8.794% 2.827% 0.517% 0.083% 24.763% 1.631%
taken as PD estimates BB
B
0.043% 0.181% 2.325% 12.105% 26.621% 10.741% 1.286% 0.129% 38.668% 7.900%
0.038% 0.062% 0.295% 1.828% 6.931% 22.064% 4.665% 0.677% 43.918% 19.523%
CCC 0.000% 0.000% 0.028% 0.759% 2.065% 7.138% 8.234% 1.034% 44.365% 36.378%
for obligor rated CC-C 0.000% 0.000% 0.000% 0.000% 0.208% 2.033% 1.940% 2.633% 44.352% 48.833%
* Source: Moody's Investor Service, Default Report: Corporate Default and Recovery Rates (1920-2010), 17 Mar 2011
similarly to the agency
ratings
• Default rates are
increasing for worse
ratings & as the time
horizons increase

32. PD Estimation: Rating Agency
Data – Default Rates*
• Default rates tend to
Moody's Average Annual Issuer Weighted Corporate Default Rates by Moody's Average Annual Issuer Weighted Corporate Default Rates by
Year: Investment Grade Year: Speculative Grade
1.200 120.000
rise in downturns and 1.000
0.800
100.000
80.000
are higher for
Default Rate (%)
Default Rate (%)
Aaa Ba
Aa 60.000 B
0.600
A Caa-C
speculative than 0.400
Baa
All Inv. Grade
40.000
All Spec. Grade
investment grade 0.200
20.000
0.000
ratings in most years 0.000
• Investment grade
default rates are very All Inv. All
Aaa Aa A Baa Grade Spec.
0.15 Ba B Caa-C Grade
Mean 0.0000 0.0405 0.0493 0.2065 0.0928
6 Mean 1.2532 5.2809 24.0224 4.7098
Median 0.0000 0.0000 0.0000 0.0000 0.0000 Median 1.0020 4.5550 20.0000 3.5950
volatile and zero in St Dev 0.0000 0.1516 0.1089 0.3198 0.1420 St Dev 1.1982 3.8827 19.7715 2.9758
Probability Density
Probability Density
Min 0.0000 0.0000 0.0000 0.0000 0.0000 Min 0.0000 0.0000 0.0000 0.9590
Max 0.0000 0.6180 0.4560 1.0960 0.4610 Max 4.8920 15.4700 100.0000 13.1370
0.10
many years, with an
4
extremely skewed
0.05
2
distribution 0
0.0 0.1 0.2 0.3
Investment Grade Default Rates
0.4 0.5
0.00
0 4 8
Spec.Grade.Default.Rates
12 16
*Reproduced with permission from: Moody’s Investor Services / Credit Policy, Special Comment: Corporate Default an and Recovery
Rates 1970-2010, 2 -28-11.

33. PD Estimation: Rating Agency
Data – Performance of Ratings
• Issuers downgraded to the B1
level as early as five years
prior to default, B3 among
issuers that defaulted in 2010
• Cumulative accuracy profile
(CAP) curve for 2010 bows
towards the northwest corner
more than the one for the
1983-2010 period, which
suggests recent rating
performance better than the
historical average
• 1-year accuracy ratio (AR) is
positively correlated with the
credit cycle, less so at 5 years

34. PD Estimation for Credit Models:
Kamakura Public Firm Model*
• This vendor provides a suite of PD models (structural, reduced-form
& hybrid) all based upon logistic regression techniques
• Similar to credit scoring models in retail: directly estimate PD using
historical data on defaults and observable explanatory variables
• Kamakura Default Probability (KDP) estimate of PD:
1
– X: explanatory variables P Yi ,t 1| X i ,t
K
– α,β: coefficient estimates 1 exp X i j,t,
– Y: default indicator (=1,0 if default,survive) j 1
j
– i,j,t,τ: indexes firm, variable, calendar time, time horizon
• “Leading” Jarrow-Chava model: based on 1990-2010 actual defaults
all listed companies N. America (1,764,230 obs. & 2,064 defaults)
• Variables included in the final model:
• Accounting: net income, cash, total assets & liabilities, number of shares
• Macro: 1 mo. LIBOR, VIX, MIT CRE, 10 govt. bond yld, GDP, unemployment rate, oil price
• 3 stock price-related: firm & market indices, firm percentile rank
• 2 other variables: industry sector & month of the year
*Reproduced with permission from: Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document,
September, 2011.

35. PD Estimation for Credit Models:
Kamakura Public Firm Model (cont.)
• Area Under the Receiver
*
Operating Curve (AUROC) :
measure rank ordering power of
models to distinguish default risk
at different horizon & models
decent but reduced form
dominates structural model
• Comparison of predicted PD vs.
actual default rate measures
accuracy of models: broadly
consistent with history & RFM
performs better than SFM
• Issues & supervisory concerns
with this: overfitting (“kitchen sink”
modeling) and concerns about out-
sample-performance
*Reproduced with permission from: Kamakura Corporation (Donald van Deventer),
Kamakura Pubic Firm Model: Technical Document, September, 2011.

36. PD Estimation for Credit
Models: Bayesian Model*
• Jacobs & Kiefer (2010): Bayesian 1 (Binomial – rating agencies), 2 (Basel II
ASRF) & 3-parameter extension (Generalized Linear Mixed Models) models
• Combines default rates for Moody’s Ba rated credits 1999-2009 in
conjunction with an expert elicited prior distribution for PD
• Coherent incorporation of expert information (formal elicitation & fitting of a
prior) with limited data & in line with supervisory validation expectations
• A secondary advantage is access to efficient computational methods such
as Markov Chain Monte Carlo (MCMC)
• Evidence that expert information can result in a reasonable posterior
distribution of the PD given limited data information
• Findings: Basel 2 asset value correlations may be mispecified (too high) &
systematic factor mildly (positively) autocorrelated
Markov Chain Monte Carlo Estimation: 1 ,2 and 3 Parameter Models Default
(Moody's Ba Rated Default Rates 1999-2009) Stressed
95% 95% 95% Stressed Minimum Regulatory
Credible Credible Credible Acceptance Regulatory Regulatory Capital
*Jacobs Jr., M., and N. M. Kiefer (2010) “The
E(θ|R) σθ Interval E(ρ|R) σρ Interval E(τ|R) στ Interval Rate Capital (θ)1 Capital2 Markup
Bayesian Approach to Default Risk: A
1 Parameter (0.00662,
Model 0.00977 0.00174 0.0134) 0.245 6.53% 5.29% 23.49% Guide,” (with.) in Ed.: Klaus Boecker,
2 Parameter (0.00732, (0.0435, Rethinking Risk Measurement and Reporting
Model 0.0105 0.00175 0.0140) 0.0770 0.0194 0.119) 0.228 6.72% 5.55% 21.06% (Risk Books, London)..
3 Parameter (0.0069, (0.043, (-0.006,
Model 0.0100 0.00176 0.0139) 0.0812 0.0185 0.132) 0.162 0.0732 0.293) 0.239 6.69% 5.38% 24.52%
1 - Using the 95th percentile of the posterior distribution of PD, an LGD of 40%, and asset value correlation of 20% and unit EAD in the supervisory formula
2 - The same as the above but using the mean of the posterior distribution of PD

37. PD Estimation for Credit
Models: Bayesian Model (cont.) Smoothed Prior Density for Theta
80
60
Density
40
20
0
0.000 0.005 0.010 0.015 0.020 0.025 0.030

• Ba default rate 0.9%, both prior &
posterior centered at 1%, 95%
credible interval = (0.7%, 1.4%)
• Prior on rho a diffuse beta
distribution centered at typical
Basel 2 value 20%, posterior
mean 8.2%, 95%CI = (4%,13%),
• Prior on tau uniform centered at
0%, posterior mean 16.2%, 95%
CI (-.01%, 29.2%)

38. The Credit Risk Parameters for
Stress Testing: Correlations
• Correlations of creditworthiness between counterparties critical
to credit models but hard to estimate & results sensitive to it
• The 1st source is the state of the economy, but extent & timing
of the rise in default rates varies by industry & geography
• Also depends upon degree to which firms are diversified across
activities (often proxied for by size: larger->less correlation)
• Contagion: apart from the broader economy, default itself
implies more defaults (interdependencies), which can worsen
the economy
• Time horizon over which correlations are measured matters –
shorter (longer) can imply see little (much) dependence
between sectors
• Some credit models have asset correlation decrease in PD
(Basel II), but weak evidence for this & not intuitive->need
economic source

39. The Credit Risk Parameters for
Stress Testing: Correlations (cont.)
• May use various types of data having sufficient history, but
beware of structural change & time variation (cyclicality-
increases in downturn)
• PD, LGD & EAD variations might not be sufficient in ST design:
we need parameters modeling portfolio effects (i.e.,correlations)
between the loans or the common dependence on risk drivers
• Analysis of historical credit risk crises reveal that correlations &
risk concentration exhibit huge deviations in these episodes
• Basis for widely used portfolio models (e.g., CreditMetrics) used
by banks for estimating the credit VaR are provided by factor
models to present systematic risk affecting the loans
• In such models it makes sense to stress strength of the factor
dependence & their variations in ST with portfolio models