Economy & Finance

An Empirical Study of Exposure at Default

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1. An Empirical Study of Exposure at Default Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division Office of the Comptroller of the Currency…
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  • 1. An Empirical Study of Exposure at Default Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division Office of the Comptroller of the Currency December, 2008 The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.
  • 2. Outline • Background and Motivation • Introduction and Conclusions • Review of the Literature • Basel Requirements • Methodology • Measurement Issues • Empirical Results • Econometric Model & Out-of-Sample Validation • Summary and Future Directions
  • 3. Background and Motivation Why the special interest in understanding risk of committed revolving (unfunded) credit facilities? • Unique structural characteristics / complexities (optionality) and risk factors (adverse selection) • Represents a large exposure to the banking system and historically high risk / return tradeoff • Basel II requirements: Banks must empirically support assumptions on expected drawdowns given default • Relatively unstudied as compared with other aspects of credit risk (capital, PD, LGD, etc.) • Arises in many contexts / products (e.g., credit cards, market risk: trading CPC exposure, LCs) But focus here is on “standard”, “traditional” revolvers for U.S. large-corporates
  • 4. Formulation of the Research Problem: What Exactly is EAD? • Basel II definition: “A Bank’s best estimate of the amount drawn down upon on a revolving credit upon default in a year”? • Historical observation of a drawn (or fraction of previously undrawn) amount on a default in a reference data-set? • A random variable (or distribution) of future $ drawn (or % fraction of undrawn) amounts conditional upon default? • A feature of the EAD distribution (e.g., measure of central tendency or high quantile)? • The distributional properties of this feature (if we are modeling parameter uncertainty)? • A form of modeling framework (structural or reduced form) understanding or predicting EAD? We develop empirical methods potentially supporting EAD estimation in ALL of these senses
  • 5. Introduction and Conclusions • Empirical study of EAD for the large corporate defaulted (i.e., Chapter 11 & distress) universe (U.S., 1985-2007) • Builds upon previous practitioner literature and current practices in the industry • References issues in risk management and supervisory requirements (Basel II Advanced IRB) • Application of advanced statistical methods (beta-link GLM) • Highlights issues in measurement and data interpretation • Exploration of alternative measures of EAD risk • Confirms some previous findings: increased EAD risk with better rating, lower utilization or longer time-to-default • “New” findings: EAD risk found to increase (decrease) with company size, intangibility,% bank or secured debt (leverage, profitability, collateral quality, percent debt cushion), and • Counter-cyclicality: evidence that EAD risk is elevated during economic expansion periods
  • 6. Review of the Literature Limited previous work, but some well-regarded benchmarks • The “classics”: Asarnow & Marker (1995 - ”The Citi Study”), Araten & Jacobs (2001 - “The Chase Study”) – Still the standard in methodology & concept • Multiple unpublished studies by financial institutions previously & in more recently preparation for Basel II – Much variation in degree to which differs from the above • Recent works in the academic & especially the supervisory / academic community (including this) – Moral* (2006): alternative frameworks for estimating EAD (optimal in regulatory sense, i.e. LEQ > 0, reg. capital not under-estimated) – Sufi (RFS, 2008): usage of credit lines in a corporate finance perspective (↑ historical profitability→more credit,revolvers=80% of all financing U.S.) – Jimenez et at (S.F. FRB, 2008): empirical EAD study for Spanish credit register data (defaulted firms -> higher usage up to 5 yrs. to default) – Loukoianova, Neftci & Sharma (J of Der., 2007): arbitrage-free valuation framework for contingent credit claims *In “The Basel II Risk Parameters: Estimation, Validation, and Stress Testing”
  • 7. Advanced IRB Requirements • Within the Basel II framework EAD is a bank’s expected gross dollar exposure to a facility upon the borrower’s default – EAD is meant to reflect the capital at risk • The general ledger balance is appropriate for fixed exposures, like bullet and term loans (see Paragraph 134) – But provides an allowance for allocated transfer risk reserve if the exposure is held available-for-sale • In the case of variable exposures, like revolving commitments and lines of credit exposures, this is not appropriate: banks must estimate the EAD for each exposure in the portfolio – But the guidance is not prescriptive about how to form this estimate – Ideally use internal historical experience relevant to the current portfolio • Note that there is no downward adjustment for amortization or expected prepayments – EAD is floored at current outstanding – At odds with empirical evidence (Banks seeing evidence ort paydowns) – Implications for properties of estimators (i.e., LEQ>0 or EAD>drawn)
  • 8. Methodology: The Loan Equivalency Factor (LEQ) • EAD: time t expected $ utilization (= availability) default time τ: ( ) ( ) EAD Xt ,t,T = E t UTIL Xτ ,τ | τ ≤ T, X t = E t AVAIL Xτ ,τ | τ ≤ T, X t • “Traditionally” estimated through an LEQ factor that is applied to the current unused: EAD Xt ,t,T = UTIL t + LEQ X ,t,T × ( AVAIL t − UTIL t ) f t ⎛ UTILτ - UTIL t ⎞ = Et ⎜ | τ ≤ T, X t ⎟ f LEQ X t ,t,T ⎝ AVAIL t - UTIL t ⎠ • The LEQ factor conditional on a vector of features X can be estimated by observations of changes in utilization over unused to default (typically averaging over “homogenous segments”): ⎛ UTIL X D ,TiD - UTIL Xti ,ti ⎞ Nx 1 ⎜ ⎟ ∑ ˆ LEQfX = Ti N X i=1 ⎜ AVAIL Xt ,ti - UTIL Xt ,ti ⎟ ⎝ ⎠ i i
  • 9. Methodology: The Credit Conversion Factor (CCF) • An alternative approach estimates a credit conversion factor (CCF) to be applied to the current outstanding (used amount): f EAD Xt ,t,T = UT IL t ×CCFXt ,t,T • The CCF is simply the expected gross percent change in the total outstanding: ⎛ AVAILτ ⎞ ⎛ UTILτ ⎞ | τ ≤ T, X t ⎟ = E t ⎜ | τ ≤ T, X t ⎟ f CCF = Et ⎜ X t ,t,T ⎝ UTIL t ⎝ UTIL t ⎠ ⎠ • CCF can be estimated by averaging the observed gross percent changes in outstandings: UTIL X NX ,TiD 1 ∑ ˆ TiD f CCF =X NX UTIL Xt ,ti i=1 i
  • 10. Methodology: The Exposure at Default Factor (EADF) • Alternatively, dollar EAD may be factored into the product of the current availability and an EAD factor: EAD Xt ,t,T = AVAIL t × EADfXt ,t,T • Where EADf is the expected gross change in the limit: ⎛ AVAILτ ⎞ | τ ≤ T, X t ⎟ f EAD = Et ⎜ X t ,t,T ⎝ AVAIL t ⎠ • May be estimated as the average of gross % limit changes: AVAIL X NX ,TiD 1 ∑ ˆ TiD f EAD = X NX AVAIL XX i=1 t i ,t i
  • 11. Methodology: Modeling of Dollar EAD • Most generally & least common, model dollar EAD as a function of used / unused & covariates (Levonian, 2007) • Restrictions upon parameter estimates could shed light upon the optimality of LEQ vs. CCF vs. EADF • We can set this up in a decision-theoretic framework as follows: { )} • ( EAD$ ( Yt ) = arg min E P ⎡ L EAD Yt − EAD$ ( Yt ) ⎤ ˆ ⎣ ⎦ EAD$ ( Yt ) • Where Y=(X,AVAIL,UTIL,T,t), L(.) is a loss metric, and EP is expectation with respect to physical (empirical) measure
  • 12. Methodology: A Quantile Regression Model for LEQ • Collect all the covariates into Yt with function g(.) (LEQ, CCF or EADF) & seek to minimize a loss function L(.) of the forecast error (Moral,2006): { } g * ( Yt ) = arg min EP ⎡ L ( EAD t,T − g ( Yt ) ) ⎤ ⎣ ⎦ g(Y ) t • Moral (2006) proposes the deviation in the quantile of a regulatory capital metric, which gives rise to an asymmetric loss function of the form: iff x ≥ 0 ⎧ax L ( x) = ⎨ b>a iff x < 0 ⎩ bx • Assuming that PD and LGD are independent & casting the problem in terms of LEQ estimation, we obtain the problem: { } LEQ* ( Yt ) = arg min EP ⎡ L ( EAD t,T − LEQ ( Yt ) × [ AVAILt − UTILt ]) ⎤ ⎣ ⎦ LEQ ( Y ) t • The solution to this is equivalent to a quantile regression estimator (Koenker and Bassett, 1978) of the dollar change in usage to default EADT,t-UTILt on the risk drivers Yt (the “QLEQ” estimator): 1 a LEQ* ( Yt ) = Q EAD t,T − UTILt , × a + b Yt AVAILt − UTILt P* • Key property: this estimator on raw data constrained such that 0<LEQ<1 is optimal also on censored data having this property (i.e., no collaring needed)
  • 13. Measurement Issues • The process is saturated with judgment & labor intensive (importance of documentation, automation & double checking work) • Data on outstandings and limits extracted from SEC filings: Lack of consistent reporting & timing issues (the Basel 1-Year horizon?) • Unit of observation: is it the same facility? – Amendments to loan agreements (“stringing together”) over time – Combining facilities for a given obligor • Need of a sampling scheme: generally at 1-year anniversaries, rating changes, amendments or “significant” changes in exposure – Avoid duplicative observations • Data cleansing: elimination of clearly erroneous data points vs. modifying estimates (capping / flooring, Winsorization) – When are extreme values deemed valid observations? – Treatment of outliers and “non-credible” observations • Repeat defaults of companies (“Chapter 22s”): look at spacing – Determine if it is really a distinct instance of default • Ratings: split between S&P & Moody’s? – Take to worst rating (conservativism)
  • 14. Empirical Results: Data Description • Starting point: Moody’s Ultimate LGD Database™ (“MULGD”) • February 2008 release • Comprehensive database of defaults (bankruptcies and out-of- court settlements) • Broad definition of default (“quasi-Basel”) • Largely representative of the U.S. large corporate loss experience • Most obligors have rated instruments (S&P or Moody’s) at some point prior to default • Merged with various public sources of information • www.bankruptcydata.com, Edgar SEC filing, LEXIS/NEXIS, Bloomberg, Compustat and CRSP • 3,886 defaulted instruments from 1985-2007 for 683 borrowers • Revolving credits subset: 496 obligors, 530 defaults and 544 facilities
  • 15. Empirical Results: Data Description (continued) • MULGD has information on all classes of debt in the capital structure at the time of default, including revolvers – Exceptions: trade payables & other off-balance sheet obligations • Observations detailed by: – Instrument characteristics: debt type, seniority ranking, debt above / below, collateral type – Obligor / Capital Structure: Industry, proportion bank / secured debt – Defaults: amounts (EAD,AI), default type, coupon, dates / durations – Resolution types : emergence from bankruptcy, Chapter 7 liquidation, acquisition or out-of-court settlement • Recovery / LGD measures: prices of pre-petition (or received in settlement) instruments at emergence or restructuring – Sub-set 1: prices of traded debt or equity at default (30-45 day avg.) – Sub-set 2: revolving loans with limits in 10K and 10Q reports
  • 16. Empirical Results: Summary Statistics (EAD Risk Measures) • Various $ Table 1.1 - Summary Statistics on EAD Risk Measures S&P and Moodys Rated Defaulted Borrowers Revolving Lines of Credits 1985-2007 exposure Standard 25th 75th measures: EAD Cnt Average Deviation Minimum 5th Prcntl Prcntl Median Prcntl 95th Prcntl Maximum Skew Kurtosis & ∆ to default, Exposure at Default (EAD) 530 133,140 295,035 158 1,656 20,725 50,000 116,234 508,232 4,250,000 7.5099 82.1857 Dollar Change in Drawn to EAD (DCDE) drawn/ undrawn, 2118 48,972 279,972 (3,177,300) (3,177,300) (2,056) 7,514 36,617 275,400 4,250,000 6.8444 116.0538 LEQ (Raw) 1582 63.72% 2759.66% -21000.00% -21000.00% -12.75% 33.28% 87.64% 231.76% 106250.00% 35.7617 1391.0651 3 LEQ (Collared) 1582 42.21% 40.92% 0.00% 0.00% 0.00% 33.28% 87.64% 100.00% 100.00% 0.3054 -1.5700 limits, “race to LEQ (Winsorized) 1582 16.80% 210.38% -1165.74% -1165.74% -12.75% 33.28% 87.64% 231.76% 804.43% -1.9084 13.5038 CCF 1330 1061.8% 20032.7% 0.47% 0.47% 85.30% 111.11% 198.86% 860.29% 704054.38% 32.9416 1145.3158 default” CCF (Winsorized) 1330 190.4% 203.4% 26.29% 26.29% 85.30% 111.11% 198.86% 855.66% 860.29% 2.27 4.45 EAD Factor 1587 143.40% 2666.07% 0.37% 0.37% 42.46% 70.67% 95.96% 152.86% 106250.00% 39.80 1584.89 quantities, EAD Factor (Winsorized) 1587 70.76% 36.94% 11.24% 11.24% 42.46% 70.67% 95.96% 152.39% 152.86% 0.29 -0.39 Utilization 1621 45.85% 32.85% 0.00% 0.00% 14.00% 48.04% 74.27% 95.00% 100.00% -0.06 -1.35 Commitment 1621 184,027 383,442 217 217 40,000 80,000 176,400 570,000 4,250,000 6.24 48.28 • LEQ (CCF & Drawndown Rate 879 0.39% 7.00% -0.10% -0.10% -0.02% 0.01% 0.05% 0.41% 181.97% 23.17 561.82 Cutback Rate 1126 88.50% 2791.11% -96.07% -96.07% 0.00% 0.00% 0.00% 66.67% 93650.00% 33.54 1125.34 EADF) 2 (3 Drawn 1621 71,576 163,029 0 0 5,557 26,463 76,900 260,000 3,090,000 8.41 107.87 Undrawn 773 112,450 329,695 0 0 13,082 34,099 82,300 396,500 4,250,000 7.79 73.49 types) • This conveys a sense of the extreme values observed here – LEQ ranges in [-210,106], CCF (EADF) max at 704 (106) – Shows that you need to understand extremes & the entire distribution • Mean collared LEQ factor 42.2% in “ballpark” with benchmarks – Median 33.3% OK but mean 16.1% raw seems too low – Raw CCF, EADF better (natural flooring) but decide to Winsorize
  • 17. Empirical Results: Distributions of EAD Risk Measures • Raw LEQ distribution: Figure 1.1: Raw LEQ Factor (S&P and Moody's Rated Defaults 1985-2007) akin to the return on 0.004 an option? • Collared LEQ: familiar 0.0 -200 0 200 400 600 800 1000 “barbell” shape (like EAD.Data.0$LEQ.Obs LGDs) Figure 1.2: W insorized LEQ Factor (S&P and Moody's Rated Defaults 1985-2007) 0.25 • Decide to go with collared measure 0.10 0.0 • Consistency with -10 -5 0 5 EAD.Data.0$LEQ.Obs.Wind common practice Figure 1.3: Collared LEQ Factor (S&P and Moody's Rated Defaults 1985-2007) • Numerical instability 4 of others -> 3 2 estimation problems 1 0 0.0 0.2 0.4 0.6 0.8 1.0 EAD.Data.0$LEQ.Obs.Coll
  • 18. Empirical Results: Distributions of EAD Risk Measures (continued) • More stable than Figure 2.1: Raw CCF Figure 2.2: Winsorized CCF LEQs 0.6 0.0015 • Natural floor at 0% 0.4 • Choose Winsorized 0.2 0.0005 measures 0.0 0.0 • As with LEQ, 0 2000 4000 6000 0 2 4 6 8 estimation issues EAD.Data.0$CCF.Obs EAD.Data.0$CCF.Obs.Wind S&P and Moody's Rated Defaults 1985-2007 S&P and Moody's Rated Defaults 1985-2007 with raw Figure 2.3: Raw EADF Figure 2.4: Winsorized EADF • Multi-modality 1.5 0.008 (especially EADF)? 1.0 0.004 0.5 0.0 0.0 0 200 400 600 800 1000 0.0 0.5 1.0 1.5 EAD.Data.0$EAD.Fact.Obs EAD.Data.0$EAD.Fact.Obs.Wind S&P and Moody's Rated Defaults 1985-2007 S&P and Moody's Rated Defaults 1985-2007
  • 19. Empirical Results: Estimation Regions of EAD Risk Measures Table 3.2 • About 1/3 LEQs Estimated Regions of LEQ, CCF and EAD Factors by Rating and Time-to-Default S&P and Moodys Rated Defaulted Borrowers Revolving Lines of Credits 1985-2007 <= 0% → LEQ Region Region Risk Years-to- paydowns <0 =0 .(0,1) =1 >1 <0 =0 .(0,1) =1 >1 Rating Defau;t AAA-BBB 7.27% 1.82% 45.45% 16.36% 29.09% 1 30.42% 5.51% 45.44% 8.37% 10.27% effectuated BB 32.00% 3.43% 52.00% 1.71% 10.86% 2 28.73% 0.81% 51.22% 5.15% 14.09% B 27.49% 4.04% 50.32% 4.67% 13.49% 3 26.98% 0.47% 49.30% 5.12% 18.14% • But 14% > 1 CCC-CC 33.89% 9.30% 36.54% 6.31% 13.95% 4 21.09% 0.78% 48.44% 4.69% 25.00% C 27.03% 18.92% 45.95% 2.70% 5.41% 5 16.67% 0.00% 52.56% 3.85% 26.92% → additional Total 28.63% 5.75% 45.26% 6.19% 14.16% Total 28.63% 5.75% 45.26% 6.19% 14.16% CCF drawdowns? Region Region Risk Years-to- <0 =0 .(0,1) =1 >1 <0 =0 .(0,1) =1 >1 Rating Defau;t AAA-BBB N/A N/A 11.43% 2.86% 85.71% 1 N/A N/A 33.76% 6.12% 57.17% • 34% CCFs < 1 → BB N/A N/A 38.36% 4.79% 56.85% 2 N/A N/A 35.45% 1.00% 61.87% B N/A N/A 33.69% 5.10% 61.21% 3 N/A N/A 34.94% 0.60% 62.65% balance shrinkage CCC-CC N/A N/A 41.53% 11.29% 47.18% 4 N/A N/A 29.03% 2.15% 66.67% C N/A N/A 30.30% 21.21% 48.48% 5 N/A N/A 31.71% 0.00% 65.85% • But 56% > 1 Total N/A N/A 34.14% 6.99% 56.32% Total N/A N/A 34.14% 6.99% 56.32% EADF → inflation? Region Region Risk Years-to- <0 =0 .(0,1) =1 >1 <0 =0 .(0,1) =1 >1 Rating Defau;t • 14% EADFs > 1 AAA-BBB N/A N/A 54.55% 16.36% 29.09% 1 N/A N/A 84.15% 6.04% 9.81% BB N/A N/A 86.93% 2.27% 10.80% 2 N/A N/A 81.40% 8.35% 10.25% B N/A N/A 81.74% 4.79% 13.48% 3 N/A N/A 80.81% 5.14% 14.05% • Larger limits? CCC-CC N/A N/A 79.93% 6.25% 13.82% 4 N/A N/A 76.74% 5.12% 18.14% C N/A N/A 91.89% 2.70% 5.41% 5 N/A N/A 69.77% 5.43% 24.81% Total N/A N/A 79.58% 6.30% 14.11% Total N/A N/A 79.58% 6.30% 14.11% • But this tendency to quirky values attenuated for worse rating and shorter time-to-default
  • 20. Empirical Results: Summary Statistics (Covariates) Table 1.2 - Summary Statistics: Borrower, Facility and Market Characteristics • Availability of S&P and Moodys Rated Defaulted Borrowers Revolving Lines of Credits 1985-20071 fin. ratios 25th 75th 95th Cnt Avg Std Dev Min 5th Prcntl Prcntl Median Prcntl Prcntl Max Skew Kurt limited vs. Time-to-Default 1616 1.7776 1.3167 -0.1644 -0.1644 0.7671 1.4986 2.7171 4.5671 6.4192 0.85 -0.07 Rating 622 2.9873 0.8672 1.0000 1.0000 3.0000 3.0000 3.0000 4.0000 5.0000 -0.45 0.51 instrument, cap Leverage 1 - LTD/ MV 537 0.7495 0.2188 0.0605 0.0605 0.6382 0.8190 0.9304 0.9878 1.0000 -1.06 0.26 Leverage 2 - TD / BV 722 0.9735 0.3760 0.1785 0.1785 0.7608 0.9155 1.0618 1.6661 4.1119 2.49 11.77 structure & Size - log(Book Value) 725 2.7746 0.5077 0.4396 0.4396 2.4236 2.7588 3.0826 3.5195 5.0167 0.48 2.30 Intangibility - Intangibles/Total Assets 474 0.3570 0.3669 0.0000 0.0000 0.0000 0.2593 0.6481 1.0834 1.3179 0.76 -0.53 macro Liquidity - Current Ratio 685 1.5296 0.9900 0.0606 0.0606 0.9230 1.3977 1.9879 3.2472 12.5570 2.88 23.36 Cash Flow - Free Cash Flow/ Total Aseets 672 -2.36 100.03 -434.16 -434.16 -0.16 0.02 3.58 28.49 1739.52 8.55 157.51 • Companies Profitabilty - Profit Margin 721 -20.23 354.98 -6735.49 -6735.49 -0.24 -0.05 0.00 0.04 0.81 -18.86 355.70 Discounted Ultimate LGD 707 7.76% 29.76% -90.12% -90.12% -5.73% 0.00% 6.24% 77.62% 100.00% 1.07 1.85 highly levered, Market Implied LGD at Default 175 31.16% 23.48% -3.72% -3.72% 10.25% 28.00% 49.63% 74.22% 90.00% 0.51 -0.68 Creditor Rank 1621 1.3967 0.7495 1.0000 1.0000 1.0000 1.0000 2.0000 3.0000 6.0000 2.38 6.80 unprofitable, Colllateral Rank 1621 3.2529 1.4428 1.0000 1.0000 3.0000 3.0000 3.0000 8.0000 8.0000 2.16 4.64 Debt Cushion 1621 25.70% 32.51% 0.00% 0.00% 0.00% 0.00% 52.00% 90.06% 99.48% 0.81 -0.84 intangible, Speculative Grade Default Rate 1621 5.67% 2.92% 0.00% 0.00% 3.15% 6.03% 7.05% 11.39% 13.26% 0.44 -0.50 Speculative Grade Default Rate - Industry 1621 5.90% 4.12% 0.00% 0.00% 2.96% 5.08% 7.95% 14.14% 20.00% 0.78 0.10 negative cash Risk-Free Return 1621 0.40% 0.14% 0.06% 0.06% 0.35% 0.43% 0.50% 0.61% 0.72% -0.78 0.18 Excess Equity Market Return 1621 0.52% 4.46% -10.76% -10.76% -0.46% 1.50% 3.41% 6.93% 8.00% -1.09 0.83 flow Equity Market Size Factor (Fama-French) 1621 0.26% 2.76% -5.74% -5.74% -1.64% 0.44% 1.52% 5.84% 8.43% 0.34 0.40 Equity Market Value Factor (Fama-French) 1621 2.08% 4.59% -5.68% -5.68% -0.74% 1.67% 4.23% 12.52% 13.80% 0.58 0.43 • Low LGDs (top Cumulative Abnormal Equity Return 525 -5.99% 66.63% -152.71% -152.71% -51.63% -6.96% 36.32% 117.66% 174.70% 0.31 -0.13 Number of Creditor Classes 1621 2.3307 0.8228 1.0000 1.0000 2.0000 2.0000 3.0000 4.0000 6.0000 0.91 1.51 of the capital Percent Secured Debt 1621 0.4776 0.3125 0.0000 0.0000 0.2354 0.4342 0.7004 1.0000 1.
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