Advances in Correlation Modeling for Credit Risk Jing Zhang Managing Director, Head of Research, Moody s KMV February 10 th, 2009
Outline How to incorporate correlations among multiple asset classes, from corporate to retail and commercial real estate? The dynamic relationship between average asset correlation and PD Why Basel II correlation function underestimates risk Modeling correlation for ABS and CDO Why senior tranches are riskier than people thought 2
Correlations Among Multi-Asset Classes 3
Historically, Default Risk of Consumer, Corporate, and CRE are Correlated 8% Mo ody 's K MV U S Pu blic Corpo rate D e fault R a te Am eric an C o un cil of Life Insure rs - Total C o m m erc ial D e linque nc y and Foreclos ur e R ate Fe de ral R e se rv e B oa rd D elin quen cy R ate - R e sid en tia l M ortga ge s Fe de ral R e se rv e B oa rd D elin quen cy R ate - C re dit C ards 7% 6% 5% 4% 3% 2% 1% 0% 1 988 19 89 199 0 1 99 1 19 92 199 3 1 99 4 19 95 1 99 6 199 7 19 98 1 99 9 20 00 2001 2 002 20 03 2004 2 005 20 06 200 7 2 008 Sources: ACLI, Fed, and Moody s KMV 4
Common Industry Practices Measure correlation within specific asset class Not integrated across asset classes Different approach for different asset class Very little empirical work for non corporate asset classes such as retail and CRE End results: Do not fully account for concentration risk and diversification effect Incoherent stress testing across asset classes 5
Moody s KMV Global Multi-Asset Class Correlation Model ri = βi φi + εi Obligor/Counterparty φ i Systematic Factors ε i Idiosyncratic Factor Geographic Factors Sector Factors Two obligors are correlated through their exposures to their systematic factors Such a framework captures both intra- and inter-asset classes correlations Explicitly covers all the major asset classes -- corporates (listed and unlisted), CRE, and retails 6
Corporate Exposures: Country and Industry Risk φ = r + r Country, Industry Country Industry Corporate Exposure Systematic Factor Idiosyncratic Factor 45 Country Factors 61 Industry Factors Estimated based on data of nearly 40,000 listed firms worldwide, updated annually Sovereigns and munis can be incorporated Equity can also be incorporated 7
Commercial Real Estate Exposure: Property-Type and Location Risk φ = r + r MSA, PropertyType MSA PropertyType CRE Exposure Risk Systematic Risk Idiosyncratic Risk For US: 73 MSA Factors For US: 5 Property Type Factors We provide out-of-self estimates for U.S. exposures For other countries, we work with clients to construct a customized model 8
Retail Exposures: Product-Type and Geography Risk φ = r + r state,product Type state Product Type Retail Exposure Risk Systematic Risk Idiosyncratic Risk For US: 50 State Retail Factors For US 6 Retail Product Type Factors We provide out-of-self estimates for U.S. exposures For other countries, we work with clients to construct a customized model 9
Bring Them All Together Two obligors are correlated through their common exposures to their systematic factors Such a framework captures both intra- and inter-asset classes correlations Explicitly covers all the major asset classes -- corporates (listed and unlisted), CRE, and retails Challenges: How do you estimate the model using various data sources? 10
Average Asset Correlation Within Each Asset Class 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% 11 US Retail: Student Loans US Retail: First Mortgage US Retail: Credit Cards US CRE: Office US CRE: Industrial US CRE: Hotels Global Financial Firms US Non-Financial Firms
Sample Portfolio Analysis: CRE, Retail, and Corporate Three portfolios were analyzed: Corporate Only Portfolio Consist of 800 exposures in various countries and industries CRE Only Portfolio Consists of 575 exposures in 5 property types and 27 MSAs Retail Only Portfolio Consists of 600 exposures in 6 product types and 50 U.S. States Corporate, CRE, and Retail Portfolio combined 12
Diversification Benefit Capital - With Respect to Expected Loss ($ millions) Risk Contribution Tail Risk Contribution Commitment Amount Stand Alone Capital Combined Capital Diversification Benefits Combined Capital Diversification Benefits Corporate Only 26890.29 1220.86 814.28-33.30% 874.53-28.37% CRE Only 18373.39 921.08 687.88-25.32% 684.50-25.68% Retail Only 10661.61 260.71 215.09-17.50% 158.22-39.31% Total 55925.29 2402.65 1717.25-28.53% 1717.25-28.53% 13
The Dynamic Relationship Between Average Asset Correlation and PD 14
Asset Correlation Parameter in Basel II IRB Framework A decreasing function of PD For all asset classes: Corporate, CRE and Retail 35% 30% Corporate & CRE (Low) CRE (High) Retail 25% Correlation 20% 15% 10% 5% 0% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% PD 15 15
The Rational and Implications There is a heuristic argument that the firm-specific risk increases as the defaulting company approaches default No empirical evidence provided The empirical work done on corporate data by Lopez (2002) examining the relationship between correlation and PD for corporate using MKMV data No empirical nor theoretic basis for the relationship provided for CRE and retail The relationship would suggest less punitive capital charge for: More risky borrowers (i.e. subprime vs prime) When borrowers default risk increases 16
The Relationship Between R-squared and PD for Corporate - You do observe a decreasing relationship 20% R-squared 15% 10% Industrial firms 5% 0% PD1 PD2 PD3 PD4 PD5 [0.06%] [0.15%] [0.42%] [1.55%] [12.5%] 25% 20% R-squared 15% 10% Financial firms 5% 0% PD1 [0.07%] PD2 [0.19%] PD3 [0.44%] PD4 [1.05%] PD5 [6.02%] 17
But after Controlling for Firm Size, there is No Strong Relationship Between R-squared and PD 30% 25% 20% Industrial firms R-squared 15% 10% 5% S4 S5 S3 0% PD1 PD2 PD3 PD4 PD5 S1 S2 18
Regression Result Further Confirms It Rsquared ε = α + β i PD i = α + β log( Size ) + ε i i i Coefficient Standard Error t Statistic R-squared Intercept 0.0749 0.0019 38.5918 0.4503 log(size) 0.0193 0.0003 57.9019 Regression result R-squareds on Size Coefficient Standard Error t Statistic R-squared Intercept -0.0014 0.0010-1.3188 0.0032 PD 0.0505 0.0139 3.6273 Regression result Residuals on PD 19
As Firms Approach Default The firm-specific risk increases, hence R-squared decreases However, one could argue that if there is a significant negative shock that triggers widespread defaults in the sector, then the R-squared for these default firms do not necessary decrease Next slide demonstrates this 20
R-squared Dynamics All Firms Defaulted in Telecom Industry, 2001 25% 20% Median P25 P75 R-squared 15% 10% 5% 0% 24m 18m 12m 6m 0m Month Prior to Default 21
For CRE, the Relationship is More Likely an Increasing One If we plot the default-implied asset correlation vs. default rate from the ACLI data for various property type, we see the following 50% 40% R-squared 30% 20% 10% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% P D Average Correlation vs PD 22
Similar Increasing Relationship 40% R-squared 30% 20% U.S. Retail Property 10% 0% 0.0% 0.5% 1.0% 1.5% 2.0% P D 50% 40% R-squared 30% 20% U.S. Office Property 10% 0% 0.0% 0.5% 1.0% 1.5% 2.0% P D 23
Now Let s Examine Retail Exposures Basel II correlation suggests that subprime borrowers would have lower asset correlation than prime borrowers But subprime borrowers are more sensitive to economic condition than prime borrowers: The ratios of financial obligations over disposable income tend to be higher for subprime borrowers Subprime customers spend higher percentage of their income on utility, driving, etc. Next slide illustrates the point 24
Low Income Borrowers are More Sensitive to Economic Shock 12 10 8 Source: Moody's Economy.com Fuel oil Electricity Natural gas Gas 6 4 2 Lowest 20% Second 20% Middle 20% Fourth 20% Top 20% Percentage of Consumer Spending 25
Delinquency Rates Implied Asset Correlation Median Retail Asset Correlations of 250 MSA Retail Markets Prime Sub Prime 14.00% 12.00% 10.00% Median R-Squared 8.00% 6.00% 4.00% 2.00% 0.00% AUTO BANKCARD CONSUMER FIRSTMORTGAGE HOMEEQUITY STUDENTLOAN Source: MKMV calculation using data from Creditforecast.com 26
Similar Results from an Academic Study D e fault C o rre latio n and A s s e t C o rre latio n Im p lie d b y D e fault R ate and 9 0 D ays D e linq ue nc y R ate Imp lie d Asse t C o rr (D e fa u lt) Imp lie d Asse t C o rr (9 0 D e lin q u e n t) D e fa u lt C o rr (D e fa u lt) D e fa u lt C o rr (9 0 D e lin q u e n t) Implied Asset Corr 2 0 % 1 8 % 1 6 % 1 4 % 1 2 % 1 0 % 8 % 6 % 4 % 2 % 7 % 6 % 5 % 4 % 3 % 2 % 1 % Default Corr 0 % A A A B C C C 0 % Default Correlation: An Empirical Investigation of Subprime Lender by Adrian M. Cowan and Charles D. Cowan 27
Capital Charge Under Basel II IRB ( 1 1 ) ( ) 1 (0.999) R 1 R capital = LGD N N PD N PD LGD 1 R + where R is the asset correlation, or Rsquare in a one-factor framework Capital is increasing function in PD. However, marginal increase diminishes since correlation decreases as PD increases Empirical evidence implies a positive relationship between PD and R-squared for CRE and retail exposures. We plot capital charge based on actual CRE estimates together with Basel s function 28
Capital Charge Based on Empirical Estimates and Basel s Function 30% BASEL CRE - Retail CRE - Office CRE - Multi-Family Housing CRE - Industrial CRE - Hotel 25% 20% Capital 15% 10% 5% 0% 0.1% 0.3% 0.5% 0.7% 0.9% 1.1% 1.3% 1.5% 1.7% 1.9% PD 29
On the Relationship Between Asset Correlation and Default Probability (Summary) After controlling for firm size, there is no strong negative relationship between asset correlation and default probability for corporate exposures Our estimates suggest an increasing relationship between asset correlation and default probability for CRE and Retail exposures The stylized decreasing relationship in Basel II doesn t have real theoretical or empirical support and may underestimate capital for: Risky exposures (such as subprime exposures) When exposure s default risk increases 30
Modeling Asset Correlations in ABS and CDO 31
Modeling ABS in a Portfolio Setting As a Structured Instrument Collateral pool data: default prob, LGDs, correlation structure ABS spreads/fees Waterfall structure As a Loan-Equivalent Default probability (the ABS Probability of Distress) Loss given default (the ABS Loss Given Distress) Correlation structure for an asset return process to integrate with the rest of the portfolio 32
Simulating the ABS to Parameterize a Loan- Equivalent Homogeneous ABS collateral For example: CA auto-loans with similar credit risk criteria Standard single-factor model Asset return process (r i ) for auto-loan i is modeled as: where R 2 is the RSQ value for the auto-loans, is the common factor, and ε i is the idiosyncratic noise. Pass-through waterfall r = R φ + R ε i 2 1 i. φ 33
ABS Probability of Distress Collateral R 2 = 10% Collateral LGD = 70% Prime Sub-prime 34
Correlation Between two ABS Simulate two similar ABS (same parameters, same systematic noise) together to obtain the ABS asset return correlation A using Normal copula: JPD( PD1, PD2, ρ 12) = JPD sim Collateral R 2 = 10% Collateral LGD = 70% 2 R max = 0.65 MKMV GCorr Corporate RSQ R = 0.201 2 mean Prime Sub-prime 35
ABS Pool Diversification Diversification level of ABS collateral pools varies RMBS examples: IndyMac MBS, Series 2007-AR21IP: CA (56-62%), NY (4-8%), FL (4-5%), CWABS Trust 2007-13: CA(29%), FL(12%), NY(6%), NJ(4%), WA(4%), TX(4%), 36
ABS Pool Diversification Diversified pool Underlying assets equally distributed between regions Asset return process for auto-loan i: r R R 2 i = φk ( i) + 1 i ε φ k ( i ) where is the factor that corresponds to loan i s region and ρ( φ, φ ) = 0.65 for k l. k l Collateral PD = 2% Collateral R 2 = 10% Collateral LGD = 70% 37
Loan-Equivalent Example: CDO of ABS ABS Data Subordination level = Mezzanine Collateral PD = 1% Collateral RSQ = 10% Tranche PD Collateral LGD = 70% 0.0350 0.0300 Loan-equivalent data 0.0250 0.0200 PD = 38 bp 0.0150 LGD = 29% 0.0100 0.0050 RSQ = 86% 0.0000 Junior Senior Subordination 38
CDO Correlations Why separate CDO Correlations from ABS correlations? Corporates have lower PDs and higher R 2 Overlapping names across CDOs Fitch s Synthetic CDO index (Ford - 56.52%, and GM - 52.40%) Overlapping names between a CDO and the single-name portion of the portfolio CDO collateral pools have a smaller number of instruments Making numerical calculations feasible 39
Correlation of CDOs with Overlapping names 1 Tranche 'Asset Return' Correlation Collateral PD = 2% Collateral R 2 = 25% Collateral LGD = 70% 0.9 Correlation 0.8 0.7 Mezzanine Tranches Junior Tranche 0.6 0% 5% 10% 20% 35% 65% 100% % Collateral Overlap Effect is greater for junior tranches 40
The Loan-Equivalent Approach (Summary) Estimating the loan-equivalent s parameters Loss Given Default and Default Probability (from simulation) Correlation structure (from simulation) Extremely high RSQ values for senior tranches Pool diversification effects Overlapping names effects Main Take-away: a portfolio of senior structured instruments may be under-diversified even if each collateral pool has a completely different focus (CA auto-loans, NY RMBS, corporate debt)! 41
References 42
References Jose A. Lopez, The Empirical Relationship between Average Asset Correlation, Firm Probability of Default, and Asset Size, working paper, 2002 Nihil Patel and Jing Zhang, "Modeling Asset Correlations for Commercial Real Estate Exposures in Credit Portfolios, Moody's KMV white paper, 2009. Joy Wang, Jing Zhang, and Amnon Levy, "Modeling Retail Correlations in Credit Portfolios, Moody's KM white paper, 2009. Jing Zhang, Fanlin Zhu, and Joseph Lee, Asset Correlation, Realized Default Correlation, and Portfolio Credit Risk, Moody's KMV white paper, 2008 Tomer Yahalom, Amnon Levy and Andrew Kaplin, Modeling Correlation of Structured Instruments in a Portfolio Setting, Moody's KMV white paper, 2008 43