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- 1. LGD Risk Resolved Please do not quote or distribute Jon Frye Federal Reserve Bank of Chicago Mike Jacobs Office of the Comptroller of the Currency FR-BOG and OCC-RAD November 22-23, 2010 Any views expressed are the authors’ and do not necessarily represent the views of themanagement of the Federal Reserve Bank of Chicago, the Federal Reserve System, the Office of The Comptroller of the Currency or the U.S. Department of the Treasury. LGD Risk Resolved in a nutshellNobody cares about LGD by itself.This paper, despite the name, is about loss. Banks have credit loss models; we have Basel II-III.Our "robust" LGD function makes thesemodels fit credit loss data better than a model that assumes LGD is fixed.We test, using alternative LGD functions. None of them fits loss data better than robust LGD.Robust LGD is therefore the best we have. 2
- 2. LGD Risk Resolved—Topic List • Two loss functions • The robust LGD function • A quick comparison to historical data • Alternative LGD functions • The PDF of credit loss • Data, estimates, and test results • Incentives and downturn LGD • Science and practice 3The beginning: two loss functions This is the fixed-LGD loss function: ⎡ N −1[PD ] + R z ⎤ cLoss[ z; PD , ELGD, R ] = ELGD N ⎢ ⎥ ⎣ 1− R ⎦ This is the robust loss function: ⎡ N −1[PD ELGD ] + R z ⎤ cLoss[ z; PD , ELGD , R ] = N ⎢ ⎥ ⎣ 1− R ⎦ We call it "robust" because: It has only two parameters rather than three. It is powerful, as well show. (It first appeared in Modest Means, Risk, January) 4
- 3. Intuition behind robust LGD At the high percentiles, robust loss is greater than fixed-LGD loss. Perhaps the extra loss comes from the systematic variation of LGD. Following this intuition, we infer a behavior of LGD that can be tested against the evidence: robust LGD. 5 The robust LGD functionUsing two assumptions, we can divide therobust loss function by the default function: ⎡ N −1[PD ELGD ] + R z ⎤ N⎢ ⎥ ⎣ 1− R ⎦cLGD [ z; PD , R , ELGD ] = ⎡ N [PD ] + R z ⎤ −1 N⎢ ⎥ ⎣ 1− R ⎦This is the robust LGD function. It implies aspecific relation between default and LGD, because both depend on the same Z… 6
- 4. Default rates and LGD rates Figure 2. Conditional Default and LGD Rates Figure 2. Conditional Default and LGD Rates 100% 100% 80%Conditional LGD Rate Conditional LGD Rate 60% 10% 40% 20% 0% 1% 0% 20% 40% 60% 80% 100% 0% 1% 10% 100% Conditional Default Rate Conditional Default Rate N −1[EL ] − N −1[PD ] N −1[cLoss ] − N −1[cDR ] = 1− ρ 7 A comparison to historical data Figure 3. Data and Robust LGD 80% 70% 60% LGD Rate 50% 40% 30% Altman data cLGD [PD=4.59%, rho=10%, EL=2.99%] 20% MURD data cLGD [PD=4.54%, rho=10%, EL=1.94%] 10% 0% 2% 4% 6% 8% 10% 12% 14% Default Rate 8
- 5. Two things we needTo rigorously test robust LGD, we need:1. Alternatives to test against. We develop five LGD functions that can have greater or less LGD sensitivity than the robust LGD function.2. The PDF of loss in a finite portfolio. Modest Means assumes an asymptotic portfolio, but Moody’s data comes from finite portfolios. We maximize the likelihood, but only for alternatives. 9The alternatives are more flexible Figure 4: Default and LGD with Alternative AAlternatives produce greater or 100%less LGD risk than robust LGD. Conditional LGD Ratee.g., Alternative A: cLGDa = 10% ⎡ N −1[PD ELGD 1− a ] + ρ z ⎤ a = 2: negative LGD riskELGD a N ⎢ ⎥ a = 1: fixed LGD ⎣ 1− ρ ⎦ a = 0: robust LGD risk a = ‐1: high LGD risk ⎡ N −1[PD ] + ρ z ⎤ 1% N⎢ ⎥ 0.1% 1.0% 10.0% 100.0% 1− ρ Conditional Default Rate ⎣ ⎦ 10
- 6. The distribution of credit lossThe derivation of the PDF for a finite portfolio Asymptotic portfolios Finite portfolios Modest T. Default Default Default Default Means S. and and Loss (Loss) A. LGD LGD LGD Loss ("TSA" means "Two Strong Assumptions") This is where alternative LGD models enterThis is the first paper to derive this PDF… 11PDF of credit loss with robust LGD 25 For each PDF: PD = 10% 20 ELGD = 50% Asymptotic (EL = 5%) portfolio R = 15% 15 Portfolio containing 10 loans Prob [ Loss = 0 ] = 43% 10 5 0 0% 5% 10% 15% 20% 12 Credit loss rate
- 7. Attention to the dataCell: Rating grade combined with seniority.Exposure: A rated non-defaulted firm has rated debt outstanding on January 1.Firm-default: Moodys records a default.Default: Moodys records post-default prices.LGD: 100 minus average post-default price.Loss rate: Total LGD / # of exposures.Default rate: # of LGDs / # of exposures. 13 Estimates using Moodys data PD : average annual default rate EL : average annual loss rate σ : average annual SD of LGD R : MLE using only default data We find the same results over a wide range of R. LGD parameters: MLE using loss data Only the alternatives see the loss data. 14
- 8. Testing cell-by-cell Senior Senior Senior Secured Loans Secured Bonds Unsecured Bonds EL D 0.2% 4 0.7% 3 0.4% 6 PD N 0.6% 616 2.1% 179 0.8% 703 ELGD D Years 42% 3 33% 3 49% 4Ba3 ρ N Years 7.6% 14 1.0% 26 27.5% 27 FirmPD FirmD 0.7% 5 2.1% 3 1.2% 9 a Δ LnL -9.00 0.37 1.45 0.01 2.07 0.17 e Δ LnL 20.4% 0.49 1.0% 0.00 11.9% 0.23 EL D 0.2% 9 0.2% 2 1.0% 13 PD N 0.8% 1332 0.6% 205 1.8% 757 ELGD D Years 28% 5 29% 2 53% 10B1 ρ N Years 14.4% 14 1.0% 27 1.0% 27 FirmPD FirmD 1.8% 25 0.8% 3 2.3% 17 a Δ LnL 0.82 0.04 -5.46 0.00 -14.28 0.29 e Δ LnL 12.3% 0.02 2.3% 0.00 3.4% 0.29 Nominal significance: ΔLnL > 1.92 15 Table 3. Testing cells in parallel Estimate Δ LnL Loans a 0.01 0.00 only b 0.19 0.31 σ = 23.3% c 0.11 0.19 ρ = 18.5% e 0.158 0.28 Estimate Δ LnL Bonds a -0.43 0.28 only b -0.03 0.03 σ = 19.7% c -0.03 0.06 ρ = 8.05% e 0.085 0.10 Estimate Δ LnL Loans and a -0.81 1.28 bonds b -0.10 0.41 σ = 20.3% c -0.09 0.55 ρ = 9.01% e 0.102 0.76 16
- 9. Correlation doesn’t matter All loans, Alternative A Figure 6. Log likelihoods for loans at assumed values of ρ 78 Max LnL Robust LnL 76 Max LnL - 1.92Log Likelihood 74 72 70 4.80% 45.4% 0% 10% 20% 30% 40% 50% Assumed value of rho 17 Incentives and downturn LGD Robust LGD produces a distribution of loss that is different from fixed LGD. Therefore, risk and incentives are different. If risk were controlled at the 99.9th percentile, robust LGD provides greater incentive to reduce PD and less incentive to reduce ELGD. 18
- 10. Robust LGD at the 99.9th percentile Figure 7. Downturn LGD less ELGD 20% PD = 10%, R = 12.1% PD = 3%, R = 14.7% 18% PD = 1%, R = 19.3% PD = 0.3%, R = 22.3% PD = 0.1%, R = 23.4% 16% PD = 0.03%, R = 23.8%Downturn LGD - ELGD 2006 Supervisory Mapping Function 14% 12% 10% 8% 6% 4% 2% 0% 0% 20% 40% 60% 80% 100% ELGD 19 Scientific contribution Robust LGD could be falsified by evidence, but it has not been falsified yet. This is the best that can be said in science. We dont think it can be falsified at present. In part, the robust LGD function isnt that bad. In part, there isnt enough data to show otherwise. But of course, we dont know. Anyone can try to show that we are just plain wrong. We hope someone tries and we expect that they fail. 20
- 11. Practical contributionOur LGD function uses PD, R, and ELGD. Banks have estimates of PD, R, and ELGD. We dont require any new estimates. A bank can adopt relatively easily.It is better than using fixed LGD.Nothing known is better than this. Unless you find something better than X, X is the best that you have. 21Summary of LGD Risk Resolved We present a robust LGD function. It attributes LGD risk to every exposure. LGD risk depends on PD, R, and ELGD. Banks estimate these parameters already. We find no evidence that the robust LGD function seriously misstates LGD risk. The robust LGD function can be used: to introduce LGD risk to existing models, to quantify downturn LGD, and as a null hypothesis in future research. 22
- 12. Questions? 23

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