Jacobs stress testing_aug13_8-15-13_v4

Stress Testing Credit Risk
Portfolios
Michael Jacobs, Ph.D., CFA
Senior Manager and Risk Advisor
Enterprise Risk Services / Government and Regulatory Services
Deloitte and Louche, LLP
AUgust 2013
The views expressed herein are those of the author and do not necessarily represent the views of Deloitte and Touche LLP

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
• Stress Testing Example: Ratings Based Approach
• Stress Testing Example: ARIMA / Time Series Based Approach

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 occurrences 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

Introduction: Motivation in the Financial
Crisis*
* Reproduced from: Inanoglu, H., Jacobs, Jr., M., and Robin
Sickles, 2013 (March), Analyzing bank efficiency: Are “too-big-to-
fail” banks efficient?, forthcoming Journal of Banking & Finance
• Bank losses in
the recent
financial crisis
exceed levels
observed in
recent history!
• This illustrates
the inherent
limitations of
backward
looking
models – we
must
anticipate risk
Figure 1: Average Ratio of Total Charge-offs to Total Value of Loans for
Top 50 Banks as of 4Q09
(Call Report Data 1984-2009)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035

Introduction: Motivation in the
Imprecision of Value-at-Risk*
Gaussian Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR
VaR99.7%=7.64e+8, q2.5%=6.26e+8, q97.5%=8.94e+8, CV=35.37%
99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)
Density
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+001e-092e-093e-094e-095e-096e-09
• Sampling
variation in
VaR inputs
leads to huge
confidence
bounds for risk
estimates
(coefficient of
variation
=35.4%)
• This is even
assuming we
have the
correct model!
*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.

An Evolving Regulatory Landscape: Continuum of New and Overlapping
Requirements
Regulatory supervisors have long advocated stress testing1 as an integral part of an overall risk management
framework; however recent proposed guidance on capital planning and annual stress testing requirements
formalizes stress testing as an integral component of determining capital adequacy
SCAPBasel II
CCAR / CapPR
(Capital Plan
Final Rule) &
Disclosures?
FASB Liquidity &
IRR Disclosures
FDIC/OCC ANNUAL
STRESS TESTING
2012-16, 0004
ICAAP
 The Supervisory Capital Assessment Program (“SCAP”) provided a granular view on supervisory expectations
 CCAR formalized regulatory expectations and provided fairly prescriptive guidance associated with the role of stress testing and
capital management, capital adequacy processes, and planning
 The fundamental principles of the Basel III Internal Capital Adequacy Assessment Process (“ICAAP”) framework still remain and
have been further supported by the recent capital planning and stress testing 7 capital principles/guidance
 Model validation and independent review extend to all models used for risk measurement, stress testing, ICAAP or any models
supporting the overall capital adequacy process models and should be supported by an overall model risk management
framework. These components together highlight the need to consider an end to end view of capital adequacy to help provide
clarity to internal and external stakeholders
 Strong governance and risk management processes are further emphasized as cornerstones to effective capital management
and planning
 Guidelines for Capital Planning published June 7, 2012 by the Treasury/OCC/FDIC suggests all risks should be estimated, plus
banks should calculate sensitivities, complete reverse stress tests and complete scenario analysis (idiosyncratic risk)
MODEL
VALIDATION
2011-7 or 12
DODD FRANK -
ENHANCED
PRUDENTIAL
SUPERVISION
(Proposed)
1 Enhanced Prudential Standards and Early Remediation Requirements for Covered Companies; Board of Governors of the Federal Reserve System (Board); & SR 2012-16 (6-7-2012)

SR 2012-07’s Stress Test
“Conceptual Process”
Materiality
Measures
Qualitative Factors
5 Principles,
Controls, Capital
& Liq. Policies,
and Procedures
Credit/PPNR
Loss Estimates &
Assumptions
Methodology
Documentation
Back Testing,
Validation
Balance
Projections,
Assets, Liabilities
& Income &
RWA Statements
Ongoing
Monitoring of
Transparent,
Repeatable &
Auditable
Process
Strengths and Weaknesses of Risk
Models
RiskAppetite
Governance
Stress Test
Results
Capital Buffers for
Uncertainty Risk
Quantitative Factors

20+ Major Steps to Governance: For $10B to $50 Billion Banks
Risk Identification & Measurement Ensure Integrity of Assessment
Set & Assess Internal Adequacy Goals
Related to Risk
P1: Risk
Identification &
Measurement
P2: Translate Risk
Into Loss Estimate
P3: Available Capital
Resources
P4: Impact of Loss
and Resource
Estimation on Capital
Adequacy
P5: Use Estimates to
Make Key Capital
Decisions
P6: Internal Controls
& Governance
P7: Effective Board &
Senior Management
Oversight*
1. Risk measurement
infrastructure
identifies and
assess all material
risk
2. Risk models meet
governance
expectations and
qualitative
processes are
transparent and
repeatable
3. Leverage
macroeconomic
assumptions for
capital planning
and stress testing
4. Leverage risk
measurement
infrastructure to
generate loss
forecast
5. Loss forecasting
sensitivity analysis
6. Clear definition of
available capital
composition and loss
absorption capability
7. Effective resource
forecasting process
using assumptions
consistent with loss
forecasting
8. PPNR/other models
meet governance
expectations and
qualitative processes
are transparent and
repeatable
9. Resource forecasting
sensitivity analysis
10. Consistent and
repeatable process
to aggregate loss
and resource
estimates
11. Establish buffer
for limitations and
uncertainty
12. Analyze
prospective capital
measures that
represent both
leverage and risk
13. Assess capital
adequacy vs.
stated goals for the
level and
composition of
capital
14. Capital policy
guides key
decisions:
• Establish capital
goals
• Determining
appropriate capital
levels
• Making decisions
about capital actions
• Maintaining capital
contingency plans
15. ICAAP governance
structure with
defined roles and
responsibilities
16. Robust internal
controls with
sufficient policy and
process
documentation
17. Sufficient model
documentation,
change control,
validation and
independent review
18. MIS to support
quantitative tools
with appropriate data
governance
19. Sufficient audit
testing
20. Appropriate reporting
on key risks, impact
loss/resource estimates
on capital v. goals, and
ICAAP weaknesses
and uncertainty
21. Senior
Management/Board
make informed capital
action
recommendations and
decisions
22. Documented approval
of planned capital
actions
23. ICAAP information
used to inform other
management and
decision making
processes
$10-$50 B, similar to CCAR/CapPR”
*Effective challenge and communication of limitations and uncertainty
ICAAP:

SR 2012-07 Stress Test “Governance
End State”

Consistent and transparent ICAAP,
Capital, & Governance process with
documented stress test models
At the loan and transaction level, any higher
risk assets can be isolated and the proper
economic capital allocated, Assets or
Geographic Regions with Risk Profiles
beyond Risk Appetite Limits can be sold.
Management (economic capital)
and regulatory stress test and risk
reports: Translate results into
appropriate dynamic and static
risk reports
Result: Integration of Stress Test Results,
Economic Capital, Concentration Management,
New Loan Pricing all integrated into Business
Line Processes and Results, including Capital
Usage, and Product Level Pricing. Full process
includes risk assessment and performance
measurements. Process evaluates shareholder
returns, rating agency ratings, and all regulatory
requirements.
Capital Policies level set expectations, roles
& responsibilities, capital buffers and trigger
levels, and required actions to preserve
capital
Types of scenarios:
•Expected Losses for all Risks
•~<1% Likely Unexpected Loss Views
•Idiosyncratic Scenarios
•Regulator-driven Scenarios
•Reverse Stress Test
• Risk Profile
• Risk Tolerance and Buffers
• Hightened Supervisory Review
Response Levels
• Concentrations: Uses of Capital by
Product
• Optimal Business Mix Profile
Stress Testing
Governance
Oversight
Capital
Policy
Scenario
Development
Business Mix,
Risk Appetite &
Concentration,
New Business
Profile
Expected &
Unexpected
Losses @
Transaction
Level
Stress
Results /
Annual Budget
Reporting
Concentration,
Uncertainty,
& De-Risking
Action Plans
Integrated
Capital & Liquidity,
Concentration &
Risk Appetite
Process
“Process End State”: Stress Test Results Integrated with Capital Planning, Economic
Capital, Concentration Mgmt & Business Line Risk/Return Results

Sample Credit Loss Modeling
Framework for Stress TestingPortfolio Segment Loan / Pool Data Modeling Approach Key DependenciesPortfolio Segment
C&I
- Major Industries
- Oil & Gas
- Agriculture
- etc.
CRE
- Construction
- Income-Producing
- Land
Retail
- Mortgage
- HELOC
- Credit Cards
- Small Business
- Other
Macroeconomic and External Data
- National
- State-level
- MSA-level
- Unemployment
- GDP Growth
- HPI
- T-Bill Rates
- etc.
- Property Prices
- Land Prices
- BBB Bond Spread
- Stock Price Volatility
Loan / Pool Data
Loan Level
- Ratings
- EAD / Balances
- Vintage
- NAIC Code
Loan Level
- Ratings / LTV / DSCR
- EAD / Balances
- Collateral Type (Retail,
Industrial, etc)
Portfolio Level
- Historical charge-offs
- Further segmentation
- Vintage/maturity
- Legacy acquisition
Modeling Approach
Time Series Analysis
- Predict quarterly changes in PD, LGD
using footprint-specific state-level macro-
factors (e.g. state-level Unemployment)
and prior-period levels, for each industry
segment
Time Series Analysis
- Predict quarterly changes in LTV and
DSCR using state-level macro-factors and
vended property price data
- Defaults trigger charge-offs
Time Series and Static Regression
- Predict Charge-offs as function of macro
factors, deposits, prior-period balances,
FICO, OLTV, Vintage, Status
- Choice of method depends on data quality
and history length
Footprint
States
- New York
- Connecticut
- New Jersey
- etc.
Key Dependencies
- Valid PDs, LGDs, or
charge-offs by Rating and
by risk factor or industry
segment
- Rating history or
reference data
− Accurate LTVs and
vintage
− Reference property price
data histories by region
and for CLTV
− Balance projections
− Charge-off reference data
across credit cycle by risk
factor
− Geography
− Loan Type

Key Success Factors for Stress-Test
Modeling Engagements
What’s
Appropriate
for the Bank
Alignment
with
Business
Knowing the
Bank’s Story
Using
Intuitive Key
Risk Drivers
Getting
Results
Together
Preparing
for
Challenges
Knowledge
and Tools
Transfer
•Do the proposed models fit yo
•What loss and risk data do
•Internal and
external
parties should
see a model
result and be
able to
understand
•Modeling can be
complex. Constant and
ongoing communication
•Driving ROI
into Process,
Concentration
Management
and Changing
Risk Profile
•Full
ownership by
the bank is the
goal, with
engagement in
the business
lines and
process going
forward
•Model
validation,
documentatio
n, model use,
and the bank
team, must be
•The Models
and the
narrative in
the Capital
Methodologie
s should be
consistent and
integrated

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

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

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

(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

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

(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

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

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

Expectations: Regulatory Capital
0.00 0.05 0.10 0.15
0.00.20.40.60.8
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
ProbabilityDensity
EL-norm=0.40%
EL-stress=0.90%
CVaR-norm=6.78%
CVaR-stress=15.79%
Normal:PD=1%,LGD=40%,Rho=0.1
Stressed:PD=1.5%,LGD=60%,Rho=0.15
Stressed Capital
Regulatory Capital
• Shocking credit risk parameters can give us an idea of what
kind of buffer we may need to add to an EC estimate

• 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)

• 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

The Credit Risk Parameters for Stress
Testing
• 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

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

Testing: LGD (continued)
• Contractual features: more
senior and secured
instruments do better.
• Absolute Priority Rule:
some violations (but
usually small)
• More senior instruments
tend to be better secured.
• Debt cushion as distinct
from position in the capital
structure.
• High LGD for senior debt
with little sub-debt?
• Proportion of bank debt
• The “Grim Reaper” story
• Enterprise value 26
S
E
N
I
O
R
I
T
Y
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees, Trade
Creditors, Lawyers
Banks
Bondholders
Shareholders

• 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.

Testing: LGD (continued)*
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
Distribution of Moody's Market LGD: All Seniorities (count=4400,mean=59.1%)
LGD
Density
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.5
Distribution of Moody's Market LGD: Senior Bank Loans (count=54,mean=16.7%)
LGD
Density
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00.51.01.5
Distribution of Moody's Market LGD: Senior Secured Bonds (count=1022,mean=46.7%)
LGD
Density
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00.51.01.52.0
Distribution of Moody's Market LGD: Senior Unsecured Bonds (count=2215,mean=60.0%)
LGD
Density
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00.51.01.5
Distribution of Moody's Market LGD: Senior Subordinated Bonds (count=600,mean=67.9%)
LGD
Density
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00.51.01.52.02.5
Distribution of Moody's Market LGD: Junior Subordinated Bonds (count=509,mean=74.6%)
LGD
Density
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash & Highly Liquid Collateral 32 -0.4% 7 8.7% 7 8.7% 1 0.0% 0 N/A 0 N/A 40 1.2%
Inventory & Accounts Receivable 173 3.6% 0 N/A 7 6.9% 0 N/A 0 N/A 0 N/A 180 3.8%
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 67 12.4% 245 49.6% 245 49.6% 2 39.6% 0 N/A 0 N/A 314 41.6%
2nd Lien 65 41.2% 75 37.5% 75 37.5% 4 59.0% 5 50.6% 1 60.0% 150 40.3%
Intangible or Illiquid Collateral 1 0.0% 5 72.2% 5 72.2% 0 N/A 0 N/A 0 N/A 6 60.2%
1537 17.4% 581 36.8% 0 N/A 8 41.2% 7 44.9% 1 60.0% 2134 22.9%
129 43.1% 0 N/A 1147 51.4% 451 70.8% 358 71.7% 64 80.8% 2149 59.2%
1666 19.4% 581 36.8% 1147 51.4% 459 70.3% 365 71.2% 65 80.5% 4283 41.1%
Collateral Type
Junior
Subordinated
Bonds
Total Collateral
Total Secured
Total Unsecured
MajorCollateral
Category
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
Table 2 - Ultimate Loss-Given-Default1
by Seniority Ranks and Collateral Types
(Moody's Ultimate Recovery Database 1987-2010)2
Bank Loans
Senior Secured
Bonds
Senior
Unsecured
Bonds
Senior
Subordinated
Bonds
Subordinated
Bonds Total Instrument
• Distributions of
Moody’s Defaulted
Bonds & Loan LGD
(DRS Database
1970-2010)
• Lower the quality of
collateral, the higher
the LGD
• Lower ranking of the
creditor class, the
higher the LGD
• And higher seniority
debt tends to have
better collateral
* Reproduced with
permission: Moody’s
Analytics.Default Rate
Service Database, 10-
15-10.
*
* Reproduced with
permission: Moody’s,
URD, Release 10-15-10.

• 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.

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

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

Testing: EAD (continued)
• Typically banks estimate EAD by a loan equivalency quotient (LEQ): fraction
of unused drawn down in default over total current availability:
t tE ,t ,t,Tt
f t
,t,T t t t t t t
t t
O -O
EAD = O +LEQ × L -O O + | T × L -O
L -OXX X
• Where O: outstanding, L: limit, t: current time, τ: time of default, T:
horizon, X: vector of risk factors , Et (.) mathematical expectation
• 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)

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

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

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)

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

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

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

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

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:
– X: explanatory variables
– α,β: coefficient estimates
– Y: default indicator (=1,0 if default,survive)
– i,j,t,τ: indexes firm, variable, calendar time, time horizon
,
,
,
,
1
1
1|
1 exp
i t
j i t
i t
K
j
j
P Y
X
X
• “Leading” Jarrow-Chava model: 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.

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.
*

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
*Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A Guide,” (with.) in Ed.: Klaus Boecker, Rethinking
Risk Measurement and Reporting (Risk Books, London.)

Bayesian Model (cont.)
• 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%)
0.000 0.005 0.010 0.015 0.020 0.025 0.030
020406080
Smoothed Prior Density for Theta

Density
E(θ|R) σθ
95%
Credible
Interval E(ρ|R) σρ
95%
Credible
Interval E(τ|R) στ
95%
Credible
Interval
Acceptance
Rate
Stressed
Regulatory
Capital (θ)1
Minimum
Regulatory
Capital2
Stressed
Regulatory
Capital
Markup
1 Parameter
Model 0.00977 0.00174
(0.00662,
0.0134) 0.245 6.53% 5.29% 23.49%
2 Parameter
Model 0.0105 0.00175
(0.00732,
0.0140) 0.0770 0.0194
(0.0435,
0.119) 0.228 6.72% 5.55% 21.06%
3 Parameter
Model 0.0100 0.00176
(0.0069,
0.0139) 0.0812 0.0185
(0.043,
0.132) 0.162 0.0732
(-0.006,
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
Markov Chain Monte Carlo Estimation: 1 ,2 and 3 Parameter Models Default
(Moody's Ba Rated Default Rates 1999-2009)

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

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

Correlation Estimation for Credit Risk
Models – Empirical Example
• Jacobs et al
(2010)*: while not
directly related to
credit or default,
these show
important facts
about correlations
• The plot shows that
correlations are
time-varying and
can differ according
to time horizon
• The table shows
how correlations
amongst different
sectors’ indices can
vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the
30-yr T-Bond Yield vs. the S&P500
-0.82
-0.62
-0.42
-0.22
-0.02
0.18
0.38
0.58
0.78
19960102
19960625
19961217
19970612
19971204
19980602
19981123
19990520
19991111
20000508
20001030
20010426
20011019
20020417
20021009
20030404
20030929
20040324
20040917
20050314
20050906
20060302
20060824
Date (YYYY,MM,DD)
Correlation
30yr T-bond for
1mo rolling window
30yr T-bond for
3mo rolling window
30yr T-bond for
6mo rolling window
30yr T-bond for
1yr rolling window
30yr T-bond for
2yr rolling window
30yr T-bond for
3yr rolling window
S&P 500
Equity
Index
Goldman
Sachs
Commodity
Index
10 Year
Treasury
Yield
CRB
Precious
Metals
Index
CRB
Energy
Index
1 Year
Treasury
Yield
S&P 400
Equity
Index
NASDAQ
Equity
Index
Russel
2000
Equity
Index
S&P 600
Small
Cap
Equity
Index
PLX
Precious
Metals
Index
S&P 500 Equity Index - -0.0211 -0.1504 0.0056 -0.0602 -7.2E-04 0.8395 0.7852 0.7723 0.8071 0.0801
Golman Sachs Commodity Index 0.0456 - 0.0256 0.2520 0.8600 0.0257 0.0096 -0.0413 0.0188 0.0299 0.1849
10 Year Treasury Yield 3.39E-37 0.0382 - 0.0241 0.0632 0.5791 -0.0727 0.0302 -0.0509 0.1053 0.0881
CRB Precious Metals Index 0.6237 2.38E-112 0.0419 - 0.1528 -0.0414 0.0374 -0.0324 0.0649 0.0152 0.5978
CRB Energy Index 6.43E-06 0.00E+00 2.73E-06 1.12E-30 - 0.0145 -0.0255 -0.0467 -0.0356 0.0129 0.1538
1 Year Treasury Yield 0.9407 0.4185 0.00E+00 8.39E-05 0.2800 - 0.0785 0.1340 0.0757 0.1871 0.0086
S&P 400 Equity Index 0.00E+00 0.4478 1.27E-14 3.04E-03 0.0558 6.12E-10 - 0.8675 0.9224 0.9263 0.1232
NASDAQ Equity Index 0.00E+00 0.0025 0.0283 1.76E-02 6.43E-04 1.23E-22 0.00E+00 - 0.8701 0.8315 0.0512
Russsel 2000 Equity Index 0.00E+00 0.1211 1.27E-14 8.86E-08 7.63E-03 5.98E-10 0.00E+00 0.00E+00 - 0.9748 0.1353
S&P 600 Small Cap Equity Index 0.00E+00 0.1154 3.45E-08 0.4232 0.4972 4.93E-23 0.00E+00 0.00E+00 0.00E+00 - 0.1086
PLX Precious Metals Index 2.11E-09 4.26E-44 6.45E-11 0.00E+00 1.17E-30 0.5233 2.67E-20 1.73E-04 3.39E-24 9.66E-09 -
Table 3: Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estimates
P-Values
*Jacobs, Jr., M., and Karagozoglu, A, 2011 (June), Performance of time varying correlation
estimation methods, Forthcoming, Quantitative Finance (September, 2012).

Correlation Estimation for Credit Risk
Models – Sensitivity Analysis
0.00 0.02 0.04 0.06 0.08 0.10
0.00.20.40.60.8
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions: Body & Tail of the Loss Distributions
PD=0.01, LGD=0.4,EAD=1
Credit Loss
ProbabilityDensity
EL=0.006 CVaR=0.0610 CVaR=0.0800 CVaR=0.0971
Rho=0.1
Rho=0.15
Rho=0.2
0.06 0.07 0.08 0.09 0.10 0.11
0.000.050.100.15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions: Tail of the Loss Distributions
PD=0.01, LGD=0.4,EAD=1
Credit Loss
ProbabilityDensity
CVaR=0.0610 CVaR=0.0800 CVaR=0.0971
Rho=0.1
Rho=0.15
Rho=0.2

Testing: Conclusion
• Some advanced models for estimating economic capital might
even require more information (e.g., economic conditions)
• Many portfolio models consider loan default and also value
changes using migration rates which can be stressed as well
• ST of risk parameters may be conducted for sub-portfolios &
the strength of the parameter modification might vary in these
• Such approaches are useful to model different sensitivities of
parts of the portfolio to risk relevant influences or to study the
vulnerability of certain (important) sub-portfolios
• They can be particularly interesting for investigations on
economic capital with the help of portfolio models
• Parameter changes for parts of the portfolio need not have a
smaller impact than analogous variations for the whole portfolio
due to effects of concentration risk or diversification

Interpretation of Stress Test Results
• As ST should be a component of the internal capital adequacy
assessment process (ICAAP), this requires comprehension of
how to utilize outputs to measure & manage portfolio credit risk
• The starting point for this should be the regulatory and EC as
outputs of the underlying ST & determining if the bank has
enough capital to absorb the stress requirements
• ST should be deployed in evaluating tools (limits, buffers and
policies) in place to guarantee solvency in such cases
• Since these might be applicable to different portfolio levels
(e.g., limits for sub-portfolios, countries, obligors), they should
be checked in detail
• The ST concept would be incomplete without knowing when
action has to be considered as a result of the outcome of tests

(continued)
• ST indicators & thresholds are typically introduced to:
– inform management about potential critical developments
– develop guidelines for new business to avoid extension of existing risk
– reduce risk for the portfolio through securitization and syndication
– readjust an existing limit management system & credit capital buffers
– to re-think the risk policy and risk tolerance
• Indicators for the “call to action” could be:
– an increase of EL, UL or ES over a threshold or by a specified factor
– the solvency ratio of capital and capital requirements under a threshold
– a low solvency level for meeting the EC requirements under stress
– quantile stress loss not within a specified quantile for the original portfolio
– stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfolio
– risk/return measured in UL lies above a specified threshold

(concluded)
• Interpretation of ST on EC outcomes can easily lead to inaction
if estimated on the basis of VaR having high confidence levels
• Motivation for latter approach is solvency avoidance by holding
enough capital except rare events simulated closely by ST
• Using large confidence levels for estimating EC offers the
possibility of comparing the capital requirements under different
conditions, but the resulting VaR should not question solvency
• In fact, it should be considered whether to use adapted
confidence levels for stress testing or to rethink the
appropriateness of high confidence levels
• One can see the probability of occurrence or the plausibility of a
ST as a related problem

A Typology of Stress Tests
• While supervisors require banks to perform ST on regulatory &
EC such differentiation is not essential but mainly technical as
inputs to these two forms of capital might be quite different
• A technical reason for this division of ST stems from different
regulatory capital calculations for performing vs. non-performing
– A performing loan gets downgraded but remains a performing loan: the
estimation of EC involves updated PD risk parameters
– A performing loan gets downgraded to non-performing: provisions have
to be estimated involving the net exposures calculated with the LGD
– A non-performing loan deteriorates – the provisions have to be increased
on the basis of an increased LGD
• ST can be performed by re-rating vs. adjusting PDs
– Former can accommodate transition of performing to nonperforming
– This can depend on economic states and are applied to the portfolio after
stressing the PDs

(continued)
• We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
• Market risk practice suggests ways to categorize ST, the most
important of which is methodology: statistically or model based
w.r.t. to conceptual design in sensitivity vs. scenario analysis
– While the latter is based upon hypothetical levels or changes in
economic variables, sensitivity analysis is statistically founded
• The common basis for all these specifications is the elementary
requirement for stress tests to perturb the risk parameters
– These can be the basic credit risk parameters (EAD, LGD, PD) as
mentioned previously with respect regulatory capital ST
– However, these can also be parameters in a portfolio model, like asset
value correlations or dependencies amongst systematic risk drivers
• The easiest way to perform ST is a direct modification of the
risk parameters and belongs to the class of sensitivity analysis

(continued)
• Uniform ST: risk parameters are increased simultaneously & we study
the impact on the portfolio values
– This depends on statistical analysis or expert opinion is not linked to any event
or context & for all loans without respect to individual properties
• Popular are flat ST for PDs, where the increase of the default rates is
derived from transition rates between the rating grades
– Advantage of these ability to perform simultaneously at different financial
institutions & aggregating results to check system’s financial stability
– Done by several central banks to checking the space & buffer for capital
requirements but it does not help for portfolio and risk management
• Model-based ST incorporate observable risk drivers
– Relies on the existence of a model, mainly econometric, that explains the
variations of the risk parameters by changes of such risk factors
– Can distinguish univariate vs. multivariate ST
– Can be seen as a refinement of those tests previously described

(continued)
• Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (e.g., up- or downgrades)
• Univariate ST can study specific & relevant impacts having the
benefit of isolating the influence of an important quantities
– Consequently can be used to identify weaknesses in portfolio structure &
are a kind of sensitivity analysis in terms of risk factors vs. parameters
– Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
• Multivariate ST avoids this problem at the potential price of
model risk in describing the correlation of the risk factors
• Scenario Analysis(SA):hypothetical, historical and statistically
determined scenarios determine stress values of risk factors
used to evaluate stress values for the risk parameters
– Distinguish bottom-up / BU vs. top-down / TD (portfolio vs. events)

(continued)
• BU tends to identify dependence on risk factors as starting
points, hence scenarios are chosen which involve risk factors
having the largest impact
• TD start with a chosen scenario (e.g., historical events) analyze
the impact of this on the portfolio, in order to identify those tests
which cause the most dramatic and relevant changes
– Extreme joint realizations of risk factors which were observed in the past
historical events / crises transferred to the current situation and portfolio
– A disadvantage of this is that transferred values may no longer be
realistic & generally not possible to specify the probability of the scenario
• Statistically determined scenarios might depend on historical
data based on the (joint) statistical distribution of risk factors &
scenarios might be specified by quantiles of such distributions
– While challenging to find suitable joint distributions, has the advantage
that if tells us the probability of a scenario occuring

(continued)
• The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for EC
• Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-
reliant on historic data can this be useful for risk management
• Finally, hypothetical scenarios of possible rare but never
observed events that might have a big impact on the portfolio
– Crucial point is the effect on the risk factors – may it is necessary to have
a macro-economic model of the dependence of the risk parameters
• If such a model is not part of the input for determining the stress
risk parameters, there are several steps required for macro ST
– Necessary to model the dependence of the risk parameters on factors
– Must choose values of risk factors representative for stress events
– Since intended to reproduce dependency structures between risk factors
and stress events, need intricate methods of estimation and validation

(concluded)
• In summary, a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not
in our reference data-sets
• However, a major advantage is forward-looking scenarios
based upon current conditions which do not necessarily reflect
historical events
• Thus, hypothetical scenarios present interesting supplements to
VaR-based analysis of portfolio credit risk and are a worthwhile
tool for portfolio management
• The use of risk factors as in the multivariate scenario analysis
has the additional advantage of allowing common ST for oher
risk types other than credit (e.g., market, liquidity or operational)
• Here, it is necessary to consider factors that influence several
forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress
Tests: Uniform ST
• One may analyze default rate (DR) data from either internal or
external ratings to assess deviations from expected PDs
– E.g., add a standard deviation of DR to the mean, or use a high quantile
from a posterior distribution of PD
• Develop stressed rating migration migrations (e.g., increase /
decrease downgrade / upgrade rates) and derive stressed
rating grades
• Investigate he effect of changing rating inputs (e.g., leverage
ratios) upon the final ratings
• LGDs may be stressed analogously to PDs, looking at historical
distributions, risk factors / regrading if there is a model, but we
would expect expert judgment to play a larger role
• EAD is much more problematic and if usually not done
• Correlations are likely shocked by purely by expert judgment

Tests: Risk Factor Sensitization
• Crucial to the task of identifying suitable risk factors & building
a robust macroeconomic model for risk parameter dependence
– Possible portfolio specific candidates: interest, inflation, FX rates; equity
indices, credit spreads, exchange rates, GDP, oil prices, credit losses
• Typically an econometric model links the risk parameters &
factors, with the challenge of determining restrictions on later
• Discovering which risk factors have the biggest impact on the
portfolio risk is a target and the benefit of sensitivity analysis
• Impact on risk parameters are calculated with the statistical
model & modified values used for evaluating capital
• Could also be used to verify uniform ST checking range of
parameter changes covered by the flat stress tests
• Pre-select scenarios: only those historical or hypothetical
involving risk factors showing large effects worth considering

Tests: Historical Scenarios
• Easy to implement: transfer the values or changes of risk
factors from historical event to the current situation
• Though risk management implications is a backward looking
approach, there are good reasons to use it
• Interesting historic scenarios which certainly would not have
been considered, as they happened by accident
– Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
• It can be assumed such events would rarely contribute to VaR
at the time of occurrence due to the extremely low probability
• Can be used to check the validity of the uniform ST and
sensitivity analysis & in designing hypothetical scenarios
• Offers unique possibility of learning about the joint occurrence
of major changes to risk factors & interaction several risk types

Tests: Statistical Scenarios
• A special role is played by the SA based on risk factor
distributions: not directly related to other types of SA
• While not be too difficult for isolated common risk to generate
such distributions on the basis of historic data, a situation
involving several factors can be far more intricate
• Nevertheless, distributions generated from historic data might
not be sufficient, so better to use such conditioned to the
situation applying at the time of ST
• If expected losses conditioned to a quantile are evaluated in
order to interpret them as unexpected losses and treat them as
economical capital requirement, then the risk factor distribution
should also be conditioned to the given (economic) situation

Tests: Hypothetical Scenarios
• Hypothetical SA is the most advanced means of ST in risk
management, combining experience in analyzing risk events,
expert opinion, economic conditions & statistical analysis
• Implementation of hypothetical SA is analogous to historical
except choice of values for the risk factors: can be based on
historical data or expert opinion might also be used
• The choice of scenarios should reflect the focus of the portfolio
for which the ST is conducted and should have the most
vulnerable parts of it as the target
• Hypothetical scenarios have the additional advantage that can
incorporate recent developments, events, news & prospects
• Note that scenarios involving market parameters like interest
rates are well suited for combinations with ST on market and
liquidity risk

Ratings Migration Model CreditMetrics
(RMM-CM) Stress Testing Example
• We present an illustration of one possible “bottoms-up”
approach to ST feasible in a typical credit portfolio
– This is has a bottoms-up flavor in that it accounts for loan ratings
• Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1/2/97 to 12/19/11, U.S. domiciled industrial
companies in 4 rating classes: Baa-A, Ba, B and C-CCC
• We calculate the risk of this portfolio in the CreditMetrics model,
which has the following inputs:
– A correlation matrix calculated from daily logarithmic returns
– A rating transition matrix amongst the rating classes from Moodys DRS
– Credit risk parameters LGD, EAD & a term structure of interest rates
• In order to compute stressed risk, we build regression models
for default rates (“DRs”) in the rating classes, and stress values
of the independent variables to compute stressed PDs
– The remainder of the correlation matrix is rescaled so that it is still valid

RMM-CM Stress Testing Example:
Default & Transition Rate Data
0
0.2
0.4
0.6
0.8
1
1.2
19791231
19800930
19810630
19820331
19821231
19830930
19840630
19850331
19851231
19860930
19870630
19880331
19881231
19890930
19900630
19910331
19911231
19920930
19930630
19940331
19941231
19950930
19960630
19970331
19971231
19980930
19990630
20000331
20001231
20010930
20020630
20030331
20031231
20040930
20050630
20060331
20061231
20070930
20080630
20090331
20091231
20100930
DefaultRate
U.S. Industrial Annual Default Rates (Moody's Default Rate Service
Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa

Default & Transition Rate Data
(cont’d.)
• Collapse the best ratings due to
paucity of defaults
• DR increase exponentially &
diagonals smaller as ratings worsen
• Correlations higher between adjacent
than more separated ratings
Rating Mean Median
Standard
Deviation
Coefficient
of
Variation Minimum Maximum
Baa-A 0.0930% 0.0000% 0.2140% 2.30 0.0000% 1.2926% 100.00% 15.88% 10.73% 13.90%
Ba 1.1341% 0.7246% 1.3265% 1.17 0.0000% 6.7460% 100.00% 70.88% 55.35%
B 6.1652% 5.2326% 5.7116% 0.93 0.0000% 33.0645% 100.00% 39.81%
Caa-C 31.1884% 20.0000% 29.1442% 0.93 0.0000% 100.0000% 100.00%
Through-the-Cycle Default Rates: U.S. Domiciled Industrial Obligors (Moody's DRS 1980-2011)
Correlations
Baa-A Ba B Caa-C Default
Baa-A 97.94% 1.62% 0.36% 0.04% 0.04%
Ba 1.29% 87.23% 9.52% 0.62% 1.34%
B 0.13% 5.11% 83.82% 5.25% 5.69%
Caa-C 0.23% 1.44% 8.10% 68.34% 21.89%
Through-the-Cycle Annual Transition
Matrix: U.S. Domiciled Industrial Obligors
(Moody's DRS 1980-2011)

Bond Index Return Data
-0.12
-0.07
-0.02
0.03
0.08
LogarithmicReturns
Bank of America-Merrill Lynch U.S. Industrial Bond Indices
(Source: Datastream)
Bond.US.Corp.Baa-A Bond.US.Corp.Ba
Bond.US.Corp.B Bond.US.Corp.C-Caa
Sector Rating Mean Median
Standard
Deviation
Coefficient
of
Variation Minimum Maximum
Aa-Aaa 0.0377% 0.0274% 0.7167% 18.99 -12.3977% 11.6545% 100.00% 36.07% 8.84% 8.26%
Baa-A 0.0433% 0.0331% 0.5247% 12.11 -11.5403% 7.4375% 100.00% 8.68% 16.46%
B-Ba 0.0372% 0.0418% 0.5308% 14.27 -6.0864% 10.8899% 100.00% 78.83%
C-Caa 0.0194% 0.0425% 0.4478% 23.12 -4.7283% 8.3753% 100.00%
Table 4: Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1/2/97 to
12/19/11 (Source: Datastream )
Correlations
Portfolio 1 -
Industrials
• Note the high variability relative
to the mean of these
• Higher ratings actually return &
vary more but CV is U-shaped
• Highest correlations between
adjacent ratings at the high &
low end
• Some of the correlations are
lower and some higher than
Basel II prescribed

Risk Factor Data
• A search through a large set of variables available on WRDS yielded this
set that are all significantly correlated to the Moody’s default rate
• VIX is a measure of volatility or fear in the equity markets
• The 4 Fama-French pricing indices (return on small & value stocks, broad
index and momentum) are found to be good predictors of DRs
• The year-over year changes in GDP, Oil Prices and Inflation are macro
factors found to be predictive
• The C&I charge-off rate is a credit cycle variable found to work well
VIX
Volatilit
y Index
Fama-
French
Size
Fama-
French
Value
Fama-
French
Market
Fama-
French
Risk-
Free
Rate
Fama-
French
Momen
tum
C&I
Chareg
off
Rates
GDP -
Level
GDP -
Annual
Change
CPI -
Annual
Change
Oil
Price -
Annual
Change
VIX Volatility Index 2.39% 2.18% 1.12% 46.73% 1.00% 6.19% 100.00% -1.12% 4.23% -14.52% 23.93% -10.78% 22.08% -26.08% -2.75% 34.05% -11.97%
Fama-French Size 0.00% 0.00% 0.08% 38.12 -0.20% 0.17% - 100.00% 15.47% -16.29% -9.67% 13.13% 11.84% 6.22% -10.40% 7.25% 6.42%
Fama-French Value 0.02% 0.01% 0.10% 6.30 -0.29% 0.35% - - 100.00% -37.86% 10.34% -16.99% 3.10% -5.09% 12.23% 6.41% -10.96%
Fama-French Market 0.03% 0.04% 0.13% 5.32 -0.37% 0.28% - - - 100.00% -5.18% -18.71% -3.40% -7.86% -13.48% 0.54% -8.92%
Fama-French Risk-Free Rate 0.02% 0.02% 0.01% 0.64 0.00% 0.06% - - - - 100.00% 15.54% -25.52% -79.15% 14.88% 77.91% 0.83%
Fama-French Momentum 0.03% 0.03% 0.12% 3.93 -0.58% 0.34% - - - - - 100.00% -9.98% -8.24% 11.39% 3.17% 8.67%
C&I Charegoff Rates 0.01% 0.91% 0.58% 62.84 0.10% 2.54% - - - - - - 100.00% -9.98% -27.66% 5.28% -11.74%
GDP - Annual Change 0.03% 3.00% 2.32% 88.46 -5.03% 8.48% - - - - - - - - 100.00% -23.86% 1.79%
CPI - Annual Change 0.04% 3.00% 2.65% 66.04 1.15% 12.96% - - - - - - - - - 100.00% -7.12%
Oil Price - Annual Change 0.10% 3.33% 34.71% 364.94 -56.14% 130.93% - - - - - - - - - - 100.00%
Variable
Correlations
U.S. Historical Macroeconomic Risk Factor Variables: Quarterly Data 1980-2010 (Source: Various)
MaximumMinimum
Coefficient
of
Variation
Standard
DeviationMedianMean

Risk Factor Data (continued)
-0.60%
-0.40%
-0.20%
0.00%
0.20%
0.40%
Date
19800630
19810331
19811231
19820930
19830630
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
20100630
%Return
Fama-French Equity Market Pricing Factor Returns
Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
Date
19800630
19810331
19811231
19820930
19830630
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
20100630
Annual%Change
Macroeconomic Indicators: GDP, CPI and Oil Prices Annual Changes
GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
Date
19800630
19810331
19811231
19820930
19830630
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
20100630
VIX Volatility Index and C&I Charge-off Rates
VIX Volatility Index C&I Charegoff Rates

Default Rate Regression Model
Default
Rate
VIX
Volatility
Index
Fama-
French
Size
Fama-
French
Value
Fama-
French
Market
Fama-
French Risk-
Free Rate
Fama-
French
Momentum
C&I
Charegoff
Rates
GDP -
Annual
Change
CPI -
Annual
Change
Oil Price -
Annual
Change
R-Squared
Statistic
F Statistic
P-Value
Baa-A Coefficient Estimate 0.0665*** -0.118** -0.3047* -0.2055* 0.9276** -0.2872** 0.02354** -0.01956** -0.01936* 0.1654**
P-Value 2.98E-04 6.42E-03 1.53E-02 1.90E-02 7.74E-03 7.17E-03 5.26E-03 5.69E-03 2.29E-01 7.62E-03
Ba Coefficient Estimate 0.1973** -1.047** -1.055** -1.64** 0.8095*** -0.6578* 0.7042*** -0.2123*** -0.4336*** 0.1351**
B Coefficient Estimate 0.2129** -2.249** -5.488** -4.443* 1.706** -5.184* 1.5415*** -0.7663** -1.396** 0.1267*
Caa-C Coefficient Estimate 1.041*** -5.332** 3.242* -8.875** 3.208** -5.797** 8.58*** -3.246** -4.908** 0.1743**
***, **, * denotes statistical significance at the 0.1%, 1% and 5% confidence levels, respectively
4.47E-0437.19%
Regression Models for Through-the-Cycle Default Rates: U.S. Domiciled Industrial Obligors
(Moody's DRS 1980-2011)
1.19E-1244.78%
5.79E-0542.28%
2.59E-0838.80%
• Estimates are statistically significant across ratings (at least the 5% level)
• R-squareds indicate adequate fit (37-45%, better for lower grades)
• The 5 FF equity market factors indicate that default rates are lower if broad
market, small or value stocks are doing better & for higher momentum
• Higher market volatility, interest rates, chargeoffs or oil prices increase DRs
• DRs are lower if GDP growth or inflation rates are increasing
• Magnitude of coefficients varies across ratings, generally greater & more
precisely estimated for lower ratings

Results of Alternative Scenarios
• Uniform ST:
PD/LGD,
correlation & sys-
tematic factor
shocks has
greatest effect
• Generally,
economic has a
bigger stressed
capital than reg-EC
• The most severe
of the hypothetical
scenarios are
spike in market
volatility to geo-
political disaster &
stagflation redux
Expected
Loss -
Credit
Metrics
Economic
Credit
Capital -
Credit
Metrics
Regulatory
Credit
Capital -
Basel 2 IRB
Base Case 2.63% 7.17% 9.29%
Uniform 10% increase in LGD 3.16% 8.62% 11.23%
Uniform 50% increase in PD 4.05% 10.80% 11.35%
Uniform 10% & 10% increase in LGD & PD 6.33% 17.10% 13.72%
50% Decrease in CreditMetrics Systematic Factor 2.63% 13.21% 9.29%
Uniform Rating Downgrade by 1 Notch 3.05% 8.30% 10.54%
Uniform 20% Increase in Emprical Correlations 3.35% 15.21% 9.29%
Equity Market Crash: 50% decline across pricing factors 3.68% 10.29% 10.97%
Oil Price Spike: 50% increase in crude index 3.35% 8.94% 10.48%
Extreme Recession Scenario: 10% decline in GDP 3.92% 10.02% 11.27%
Geopolitical Disaster: 30% spike in VIX 4.46% 15.74% 11.90%
Credit crunch: doubling of C&I charegeoff rates 4.13% 10.86% 11.61%
1970s Stagflation Redux: 10% decline (increase) GDP (inflation) 5.03% 17.38% 15.27%
Stress Test Outcomes for Portfolio of U.S. Industrial Bond Indices: CreditMetrics vs.
Basel II IRB Models

Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario: Moody's Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 & DRS Database Annual Transitions 1980-2010)
Credit Losses
Probability
-0.10 -0.08 -0.06 -0.04 -0.02 0.00
050015002500
B2-cVar999=9.29%
CM-cVar999=7.17%
EL=2.63%
CreditMetrics Credit Loss Distribution under Stagflation Scenario: Moody's Stressed Rating Migration Matrix
Credit Losses
Probability
-0.15 -0.10 -0.05 0.00
050015002500
B2-cVar999=15.27%
CM-cVar999=17.38%
EL=5.03%

Autoregressive Integrated Moving
Average Time Series (ARIMA-TS)
Stress Testing Example
• We present an illustration of one possible “top-down” approach
to ST feasible in a typical credit portfolio
– This is tops-down in that it requires only portfolio level losses
• ARIMA econometric techniques utilized to project losses
• Macro-economic variables as specified by the Fed CCAR
• Charge-off loss rates from Fed Call / Y9 Reports

Theoretical ARIMA Construction – Functional Forms, Terms and Operators
Liquidityrisk
• Given a time series , where is an integer index and
are real numbers, then an model is given by:
• Where is the lag operator, are the
parameters of the autoregressive part of the model,
are the parameters of the moving average part
• The error terms are generally assumed to be
independent, identically distributed variables sampled from a
normal distribution with zero mean and constant variance.
• Assuming now that the polynomial has a unitary root
of multiplicity, it can be rewritten as:
tX t tX R ,ARMA p q
1 1
1 1
p q
i i
i t i t
i i
L X L
k
t t kL X X : 1,..,i i p AR p
: 1,..,i i q
2
~ 0,t NID
1
1
p
i
i
i
L d
1 1
1 1 1
p p d
di i
i i
i i
L L L
, ,ARIMA p d q
1 1
1 1 1
p q
di i
i t i t
i i
L L X L
,ARMA p d q
0d

Theoretical ARIMA Construction – Model Identification and Specification
• Identification and specification of appropriate factors in an
ARIMA model can be an important step in modeling as it can
allow a reduction in the overall number of parameters to be
estimated, while allowing the imposition on the model of types
of behavior that logic and experience suggest should be there.
• ARIMA models are used for observable non-stationary
processes that have some clearly identifiable trends:
• a constant trend (i.e. zero average) is modeled by
• a linear trend (i.e. linear growth behavior) is modeled by
• a quadratic trend (i.e. quadratic growth behavior) is
modeled by
• In these cases, the ARIMA model can be viewed as a "cascade"
of two models. The first is non-stationary:
0d
1d
2d
1
d
t tY L X
1 1
1 1 1
p q
di i
i t i t
i i
L L X L
0,1,0ARIMA
1t t tX X
, ,ARIMA p d q

ARIMA-SR ST: Macroeconomic
Variable Correlations

Aggregate Y9 Chargeoff Data
Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F
0 -19.2447 0.001 -3.51 0.0008
1 -9.9104 0.0237 -2.22 0.0272
2 -5.0654 0.1146 -1.5 0.1227
0 -19.249 0.0066 -3.47 0.0139 6.01 0.021
1 -9.9359 0.1122 -2.19 0.2117 2.43 0.4683
2 -5.0918 0.4032 -1.49 0.5296 1.11 0.7893
0 -19.3726 0.0428 -3.44 0.0603 5.92 0.0862
1 -9.8593 0.3927 -2.14 0.5083 2.35 0.7147
2 -4.9792 0.8053 -1.44 0.8333 1.11 0.9504
Zero
Mean
Augmented Dickey-Fuller Unit Root Tests
Trend
Single
Mean
• The log difference of the series shows some evidence of non-stationarity, but this
is the best that we can do
• Evidence of significance autocorrelation in the series implies time series is an
appropriate technique, but will the data support it?

ARIMA Model 1 for Aggregate Chargeoffs: 5
Factor
• The model produces good diagnostics
and all variables are significant /
intuitive sign, but no AR / MA terms -
> reduces to OLS
Approx
Pr > |t|
NUM1 0.14883 0.04664 3.19 0.0014 0 UNP_Rate 0
NUM2 0.01168 0.005665 2.06 0.0392 0 CPI_Rate 0
NUM3 -0.14767 0.0481 -3.07 0.0021 0 Tr_3Mo 0
NUM4 0.07882 0.04631 1.7 0.0887 0 Tr_10Yr 0
NUM5 0.0031685 0.001712 1.85 0.0642 0 Vol_VIX 0
Variance Estimate 0.010344
Std Error Estimate 0.101706
AIC -66.4038
SBC -57.8359
Number of Residuals 41
ShiftParameter Estimate
Standard
Error t Value Lag Variable
Maximum Likelihood Estimation

Factor (cont’d.)
• The model produces good diagnostics and all variables are significant / intuitive sign,
but no AR / MA terms -> reduces to OLS
To Lag
Chi-
Square DF
Pr >
ChiSq
6 4.09 6 0.665 0.162 0.144 0.138 -0.055 0.114 0.081
12 9.66 12 0.6455 0.202 -0.058 -0.188 0.032 -0.147 0.013
18 23.16 18 0.1844 -0.025 -0.272 -0.115 -0.239 -0.189 -0.119
24 30.8 24 0.1596 -0.083 0.017 -0.171 -0.04 -0.124 -0.165
Autocorrelation Check of Residuals
Autocorrelations

Factor
• The model produces good diagnostics and all variables are significant / intuitive sign,
with AR / MA terms , is more parsimonious & ARIMA, but is single factor
To Lag
Chi-
Square DF
Pr >
ChiSq
6 3.12 4 0.5387 -0.047 0 0.204 -0.072 0.049 0.119
12 7.56 10 0.6713 0.107 -0.155 -0.146 -0.029 -0.128 0.081
18 21.19 16 0.1713 0.022 -0.408 -0.071 -0.062 -0.166 0.03
24 23.45 22 0.3766 0.008 -0.024 -0.146 0.054 0.019 -0.011
Autocorrelation Check of Residuals
Autocorrelations
Appro
x
Pr > |t|
MA1,1 0.56717 0.286 1.98 0.0472 1 CO_All_Log 0
AR1,1 0.82343 0.204 4.05 <.0001 1 CO_All_Log 0
NUM1 0.127 0.066 1.94 0.053 0 UNP_Rate 0
Variance Estimate0.012238
Std Error Estimate0.110626
AIC -61.0311
SBC -55.8904
Number of Residuals41
Maximum Likelihood Estimation
Shift
Param
eter Estimate
Stand
ard
Error t Value Lag
Variab
le

Comparison of Multi- & Single Factor ARIMA
Models for Aggregate Chargeoffs
• Both models imply similar
behavior in the Fed severe
scenario – loses roughly
quadruple from the 2006 trough,
although the multi-for model is
about 25% more severe
• So which do we prefer, bearing
in mind that neither can match
the worst of the last financial
crisis?

References
• Araten, M. and M. Jacobs Jr., 2001, Loan equivalents for defaulted revolving credits
and advised lines, The Journal of the Risk Management Association, May, 34-39.
• Araten, M., Jacobs Jr., M., and P. Varshney, 2004, Measuring LGD on commercial
loans: An 18-year internal study, The Journal of the Risk Management Association,
May, 28-35.
• Artzner, P., Delbaen, F., Eber, J.M., and D. Heath, 1999, Coherent measures of risk,
Mathematical Finance, 9:3, 203-228.
• The Basel Committee for Banking Supervision, 2006, International convergence of
capital measurement and capital standards: A revised framework.
• The Basel Committee for Banking Supervision, 2009, Principles for sound stress
testing practices and supervision - consultative paper, May (No. 155).
• 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.
• 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

References (continued)
• Jacobs Jr., M., 2010, An empirical study of exposure at default, The Journal of
Advanced Studies in Finance, Volume 1, Number 1 (Summer.)
• Jacobs Jr., M., and A. Karagozoglu, 2010, Modeling ultimate loss-given-default on
bonds and loans, U.S. Office of the Comptroller of the Currency and Hofstra
University, Working paper.
• Jacobs Jr., M., and A. Karagozoglu, 2010, Modeling the time varying dynamics of
correlations: applications for forecasting and risk management, Working paper.
• Jacobs Jr., M., Karagozoglu, A., and C. Pelusso, 2010, Measuring Credit Risk: CDS
Spreads vs. Credit Ratings. Hofstra University & Goldman Sachs, Working paper.
• Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A
Guide,” (with.) in Ed.: Klaus Boecker, Rethinking Risk Measurement and Reporting
(Risk Books, London).
• Merton, R., 1974, On the pricing of corporate debt: The risk structure of interest
rates, Journal of Finance, 29, 4449-470.
• The U.S. Office of the Comptroller of the Currency (“OCC”) and the Board of
Governors of the Federal Reserve System (“BOG-FRB”), 2011, Supervisory
Guidance on Model Risk Management (OCC 2011-12), April 4, 2011.

Thanks and Please Reach Out
Michael Jacobs, Jr., Ph.D., CFA
Deloitte & Touche LLP
Audit & Enterprise Risk Services / Government, Risk
and Regulatory Services / Business Risk / Financial
Services
1633 Broadway, 36th Floor
New York, N.Y.. 10019
Office: (212) 436-2956
Home: (212) 369-0025
Cellular: (917) 324-2098
e-mail: mikjacobs@deloitte.com
Home email: mike.jacobs@yahoo.com
Personal Website: http://www.michaeljacobsjr.com
SSRN Author Page:
http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?p
er_id=97517
YouTube:
http://www.youtube.com/user/MikeJacobsJr/videos
LinkedIn:
http://www.linkedin.com/profile/view?id=17630774&tr
k=tab_pro

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