1. “ Stress Testing Banking Book Positions Under Basel II” Federal Reserve Bank of San Francisco January 2009 by Paul Kupiec Federal Deposit Insurance Corporation

2. The opinions expressed in this presentation represent those of the author. They are not the official views of the FDIC.

3. Overview Basel II AIRB sets minimum capital using a modified version of the Vasicek credit loss model Capital covers 99.9% of all potential credit losses –Capital violations should happen only 1-in-1000 years Basel II requires supplemental stress tests Question: What are appropriate stress scenarios if AIRB capital is only breached 1-in-1000 years?

4. Stress tests: Why do we need them? How should we do them? Depends on ….How well the AIRB model fits the data –The AIRB has 3 basic parts where model fit may be an issue: The default rate model The LGD assumption The EAD assumption –Other important issues not addressed in this talk »Asymptotic portfolio assumption »Maturity adjustment –

5. Stress Testing Under Basel II Develop methods or techniques that enable an analyst to estimate the capital implications of relaxing inaccurate restrictive assumptions or modifying other unrealistic modeling features of the AIRB modeling framework.

6. Basel II AIRB is a Modified Vasicek Model Portfolio Capital Requirement in %

7. Basel II AIRB is a Modified Vasicek Model 99.9 percentile from the Vasicek portfolio default rate distribution model

8. Basel II AIRB is a Modified Vasicek Model Portfolio exposure at default

9. Basel II AIRB is a Modified Vasicek Model Portfolio exposure Loss Given Default

10. Basel II AIRB is a Modified Vasicek Model Individual Credit Unconditional Probability of Default

11. Basel II AIRB is a Modified Vasicek Model Default Correlation among portfolio credits

12. Basel II AIRB is a Modified Vasicek Model Vasicek portfolio default rate Regulatory correlation function “fine tuned” to reduce procyclicality Maturity adjustment factor specified to mimic KMV estimates These features are policy parameters and are not derived from a formal credit risk model

13. Portfolio LGD and EAD AIRB does not specify EAD or LGD models –LGD and EAD are at the portfolio level AIRB measures them using a single parameter No recognition or discussion that EAD and LGD have a distribution at the portfolio level Diversification issues are not modeled –Basel provides broad regulatory guidance as to how these “parameters” should be estimated Minimum sample sizes for calibration Minimum parameter values LGD must be estimated in a way so that it reflects “downturn conditions”

14. Basel II AIRB is a Modified Vasicek Model For many portfolios, EAD and LGD are more accurately modeled as random variables with systematic risk

15. Stochastic LGD & EAD AIRB model can be generalized to account for stochastic LGD & EAD at individual exposure level –LGD and EAD realized values can be correlated in time –systematic time-variation in recovery and exposures –Leads to portfolio models for EAD and LGD Kupiec (2008) Journal of Derivatives Result: Once portfolio models for LGD and EAD are accounted for, portfolio credit loss rate distribution may have fatter tails relative to AIRB model –LGD and EAD correlation introduces additional systematic risk & increases unexpected loss rates

16. Stochastic LGD and EAD Kupiec (2008) model provides a coherent framework for analyzing AIRB EAD and LGD parameters –A rigorous & consistent model for thinking about stress or “downturn” LGD and EAD estimates Fully accounts for diversification and systematic risk –Given time constraints, I’ll skip this part of the paper and focus of the default rate model

17. AIRB Default Rate Model Fit Take a brief look at a large panel data set –Moody’s Corporate Bond Ratings and Default History, 1920-2006 Fit the model to historical data Evaluate default rate model fit Upshot: AIRB model fit is poor –Stress tests should account for AIRB default model risk What type of model generalizations may improve default rate performance?

18. Calibration Methodology If all credits in a rating grade have identical PD and correlation parameter and defaults are driven by a single common factor –to a close approximation…. –the annual default rate of a credit grade with a large number of credits should have an ASFM default rate distribution

19. New Panel Regression Approach

20. Unconditional probability of default for credits in the portfolio

21. New Panel Regression Approach The default correlation parameter

22. New Panel Regression Approach Latent Gaussian “Macro factor” that drives individual credit default realizations

23. Panel Regression Approach Fixed effect for credit rating category i Year effect that is identical across all credit grades in the rating system for a given year Random deviation from ASFM model Transformation of annual default rate in year t for credit rating category i

24. Data Moody’s Corporate Bond Default History 1920-2006 –Issuer rated annual default rates by credit grades Aa, A, Baa, Ba, B, Caa_C –Number of issuers with a given Moody’s rating that default in a given year, divided by the total number of issuers with same Moody’s rating at the beginning of the year If Moody’s withdraws ratings the issuer is removed from numerator and the denominator

25. Model Fit Actual vs Predicted –For each Moody’s credit grade –90% confidence intervals around predicted values from bootstrapped sampling distribution

26. actual

30. Can the AIRB reproduce the default rate data? Compare actual & predicted default rate distribution using Kolmogorov-Smirnov Statistic –Statistic is based on the maximum distance between two empirical CDFs CDF 1 CDF 2

31. Asymptotic K-S Statistics

32. Default Rate Model Fit AIRB model fits the Moody’s data poorly –Portfolios perform “in the tails” relative to the model’s predictions Portfolios perform exceptionally well far too often ….exceptionally poorly far too often –Maybe a double stochastic boundary model or time-a time varying correlation will fit better Default boundary is stochastic Correlation is a random variable –What would models with these characteristics look like?

33. Basel II AIRB is a Modified Vasicek Model Mounting academic evidence suggests default boundaries maybe stochastic perhaps with systematic time variation

34. Why Stochastic Boundary? Intuition: Market has Liquidity Cycles or Cycles in Underwriting Standards –Firms must refinance maturing debt –When liquidity is plentiful, underwriting standards are lax and it is easy for all firms to refinance –When liquidity is scarce, underwriting standards tighten; all firms face higher refinance boundaries

36. New Stochastic Default Boundary Asymptotic Portfolio Model The ex ante probability of default (the default boundary) is random Same common factor that drives V i Parameter determines the correlation among accounts’ stochastic default boundaries

37. Stochastic Default Boundary Random default boundary condition

38. Asymptotic Portfolio Default Rate Distribution For any e M, the conditional default rate requires integrating out the idiosyncratic risk uncertainty in the default boundary……this requires numerical procedures

39. Example: Stochastic Default Boundary Assume PD boundary is normally distributed with mean 1% and standard deviation of 0.2% Assume PD latent variables have 20% correlation Assume firm latent default factors (firms value proxies) have 20 percent correlation –Implies PD and V i have a 20% correlations as well

40. Example: Asymptotic Portfolio Credit Loss Distribution when Default Boundaries are Stochastic

41. Influence of Default Boundary Correlation As the default boundary correlation parameter increases, the 99.9% critical default rate increases

42. Influence of Default Boundary Standard Deviation The critical value increases as the standard deviation of the default boundary distribution increases

43. Basel II AIRB is a Modified Vasicek Model Alternatively, default correlations may have systematic time variation

44. Stochastic Correlation: Motivation Prior to 2006, rating agencies and investors calibrated sub-prime mortgage securitization models using a very low default correlation based on 1998-2005 data In 2006, these sub-prime mortgages began defaulting in large numbers –Default correlation had shifted from early data

45. Motivation II Popular credit loss models did not anticipate time- variation in default correlation Cause of shift? Housing prices –From 1998-2005 housing prices went up strongly depressing sub-prime default correlations –From 2006, housing prices started declining rapidly, increasing sub-prime default correlations AIRB model does not accommodate time-variation in default correlation parameter Lets see if we can fix this…….

46. Motivation III Stochastic correlation is a reduced-form model for contagion risk –Rarely, but with some positive probability, a random factor causes a shift in default correlation patterns and very quickly, defaults become much more highly correlated………….

47. Model Assumptions

48. New notation for latent factor proxy for firm value

49. Model Assumptions The default correlation parameter that multiplies the common Gaussian factor is random Correlation specification is slightly changed to simplify mathematical proofs Default correlation is now

50. Latent Correlation Factor

51. Latent correlation factor Common factor that drives correlation parameter

52. Latent correlation factor This model has two independent common factors --one drives firm values --one drives the correlation among firm value realizations Common factor that drives correlation parameter

53. Default Correlations

55. Unconditional default correlation distribution Default correlation distribution conditional on ek=-2 Correlations are independent →No time variability

56. Unconditional default correlation distribution Default correlation distribution conditional on ek=-2 Correlation among W i latent factor = 25% →Substantial shift in default correlation distribution

57. Unconditional default correlation distribution Default correlation distribution conditional on ek=-2 A shift in the default correlation distribution of “sub prime mortgage” proportions

58. Asymptotic Portfolio Default Rate Distribution when Correlation is Stochastic

59. Stochastic Correlation Example PD=1% Default correlation parameter is distributed uniformly over [.05,.35], average value=.20 –Average default rate correlation of 4% (.2^2) →default correlation distribution depends on the correlation parameter of, the Gaussian factor that drives default correlations

60. Stochastic Correlation Example pc parameter drives the distribution skew

61. Stochastic Correlation Example II Larger default correlation parameter→longer tail

62. Overall Conclusions Basel AIRB restrictive assumptions are likely to be violated –AIRB minimum capital requirements may be violated more frequently than the 1-in-1000 year nominal solvency standard Stress tests are a means for identifying capital needs on positions that are unlikely to adequately modeled under AIRB assumptions Stress tests can identify additional capital needs for scenarios that are more common than 1-in-1000 years Stress-testing based capital supplements should occur routinely for some AIRB banks if they are following a rigorous & well-designed stress testing regimen