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with respect to ST. This includes a typology of ST (uniform testing, risk factor sensitivities, scenario analysis; and historical, statistical and hypothetical scenarios) and procedures for con-ducting ST. We conclude with two simple and practical stress testing examples, one a ratings migration based approach, and the other a top-down ARIMA modeling approach.

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- 1. 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
- 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 • Stress Testing Example: Ratings Based Approach • Stress Testing Example: ARIMA / Time Series Based Approach
- 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 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
- 4. 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
- 5. 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.
- 6. 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)
- 7. 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
- 8. 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:
- 9. SR 2012-07 Stress Test “Governance End State”
- 10. 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
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 15. 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
- 16. 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
- 17. 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.
- 18. 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
- 19. 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
- 20. 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
- 21. Supervisory Requirements and 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
- 22. 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)
- 23. 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
- 24. 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
- 25. The Credit Risk Parameters for Stress 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
- 26. The Credit Risk Parameters for Stress 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
- 27. 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.
- 28. The Credit Risk Parameters for Stress 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.
- 29. 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.
- 30. The Credit Risk Parameters for Stress Testing: LGD (continued)
- 31. 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.
- 32. 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
- 33. The Credit Risk Parameters for Stress 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)
- 34. 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
- 35. 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
- 36. 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)
- 37. 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
- 38. 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
- 39. PD Estimation: Rating Agency Data – 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.
- 40. 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
- 41. 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.
- 42. 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. *
- 43. 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 *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.)
- 44. PD Estimation for Credit Models: 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)
- 45. 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
- 46. 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
- 47. 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).
- 48. 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
- 49. The Credit Risk Parameters for Stress 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
- 50. 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
- 51. Interpretation of Stress Test Results (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
- 52. Interpretation of Stress Test Results (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
- 53. 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
- 54. A Typology of Stress Tests (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
- 55. A Typology of Stress Tests (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
- 56. A Typology of Stress Tests (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)
- 57. A Typology of Stress Tests (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
- 58. A Typology of Stress Tests (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
- 59. A Typology of Stress Tests (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
- 60. 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
- 61. Procedures for Conducting Stress 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
- 62. Procedures for Conducting Stress 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
- 63. Procedures for Conducting Stress 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
- 64. Procedures for Conducting Stress 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
- 65. 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
- 66. 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
- 67. RMM-CM Stress Testing Example: 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)
- 68. RMM-CM Stress Testing Example: 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
- 69. RMM-CM Stress Testing Example: 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
- 70. RMM-CM Stress Testing Example: 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
- 71. RMM-CM Stress Testing Example: Risk Factor Data (continued) -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
- 72. RMM-CM Stress Testing Example: Risk Factor Data (continued) 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
- 73. RMM-CM Stress Testing Example: 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** P-Value 5.00E-03 4.61E-03 3.75E-03 6.25E-03 1.78E-05 4.58E-02 9.25E-04 2.95E-04 4.14E-06 6.59E-03 B Coefficient Estimate 0.2129** -2.249** -5.488** -4.443* 1.706** -5.184* 1.5415*** -0.7663** -1.396** 0.1267* P-Value 6.48E-03 7.34E-03 3.23E-03 2.77E-02 4.53E-03 2.11E-02 1.83E-03 4.63E-03 1.14E-03 3.75E-02 Caa-C Coefficient Estimate 1.041*** -5.332** 3.242* -8.875** 3.208** -5.797** 8.58*** -3.246** -4.908** 0.1743** P-Value 7.27E-07 6.10E-03 1.72E-02 3.09E-03 3.75E-03 7.42E-03 6.18E-05 4.95E-03 6.95E-03 4.92E-03 ***, **, * 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
- 74. RMM-CM Stress Testing Example: 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
- 75. RMM-CM Stress Testing Example: 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%
- 76. 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
- 77. 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
- 78. 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
- 79. ARIMA-SR ST: Macroeconomic Variable Correlations
- 80. 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?
- 81. 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
- 82. ARIMA Model 1 for Aggregate Chargeoffs: 5 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
- 83. ARIMA Model 2 for Aggregate Chargeoffs: 1 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
- 84. 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?
- 85. 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
- 86. 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.
- 87. 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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