LGD Risk ResolvedPlease do not quote or distribute

Jon FryeFederal Reserve Bank of Chicago

Mike JacobsOffice 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 the management 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 nutshell

Nobody 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 these models 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

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

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This is the fixed-LGD loss function:

This is the robust loss function:

We call it "robust" because:It has only two parameters rather than three.It is powerful, as we'll show.(It first appeared in Modest Means, Risk, January)

The beginning: two loss functions

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RzRELGDPDNNRELGDPDzcLoss

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1

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RzRPDNNELGDRELGDPDzcLoss

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1

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.

Intuition behind robust LGD

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Using two assumptions, we can divide the robust loss function by the default function:

This is the robust LGD function. It implies a specific relation between default and LGD,

because both depend on the same Z…

The robust LGD function

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RzRPDNN

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ELGDRPDzcLGD

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Default rates and LGD rates

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Figure 2. Conditional Default and LGD Rates

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0% 20% 40% 60% 80% 100%

Conditional Default Rate

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itional LGD Rate

Figure 2. Conditional Default and LGD Rates

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100%

0% 1% 10% 100%

Conditional Default RateCo

nditional LGD Rate

ρ−−

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1111 PDNELNcDRNcLossN

A comparison to historical data

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Figure 3. Data and Robust LGD

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0% 2% 4% 6% 8% 10% 12% 14%

Default Rate

LGD

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Altman data cLGD [PD=4.59%, rho=10%, EL=2.99%] MURD data cLGD [PD=4.54%, rho=10%, EL=1.94%]

Two things we need

To 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.

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The alternatives are more flexible

Alternatives produce greater or less LGD risk than robust LGD.

e.g., Alternative A: cLGDa =

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Figure 4: Default and LGD with Alternative A

1%

10%

100%

0.1% 1.0% 10.0% 100.0%Conditional Default Rate

Cond

itional LGD Rate

a = 2: negative LGD riska = 1: fixed LGDa = 0: robust LGD riska = ‐1: high LGD risk

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ρρ

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zPDNN

zELGDPDNNELGDa

a

The distribution of credit loss

The derivation of the PDF for a finite portfolio

This is the first paper to derive this PDF…

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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 enter

Finite portfoliosAsymptotic portfolios

PDF of credit loss with robust LGD

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0

5

10

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0% 5% 10% 15% 20%Credit loss rate

Asymptotic portfolio

Portfolio containing 10 loansProb [ Loss = 0 ] = 43%

For each PDF:PD = 10%

ELGD = 50%(EL = 5%)R = 15%

Attention to the data

Cell: Rating grade combined with seniority.Exposure: A rated non-defaulted firm has

rated debt outstanding on January 1.Firm-default: Moody's records a default.Default: Moody's 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 Moody's data

PD : average annual default rate

EL : average annual loss rate

σ : average annual SD of LGD

R : MLE using only default dataWe find the same results over a wide range of R.

LGD parameters: MLE using loss dataOnly the alternatives see the loss data.

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Nominal significance: ΔLnL > 1.92

Testing cell-by-cell

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EL D 0.2% 4 0.7% 3 0.4% 6PD 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% 9a Δ LnL -9.00 0.37 1.45 0.01 2.07 0.17e Δ LnL 20.4% 0.49 1.0% 0.00 11.9% 0.23

EL D 0.2% 9 0.2% 2 1.0% 13PD 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% 17a Δ LnL 0.82 0.04 -5.46 0.00 -14.28 0.29e Δ LnL 12.3% 0.02 2.3% 0.00 3.4% 0.29

Secured Loans Secured Bonds Unsecured BondsSenior Senior Senior

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Estimate Δ LnLLoans a 0.01 0.00only b 0.19 0.31

σ = 23.3% c 0.11 0.19ρ = 18.5% e 0.158 0.28

Estimate Δ LnLBonds a -0.43 0.28only b -0.03 0.03

σ = 19.7% c -0.03 0.06ρ = 8.05% e 0.085 0.10

Estimate Δ LnLLoans and a -0.81 1.28

bonds b -0.10 0.41σ = 20.3% c -0.09 0.55ρ = 9.01% e 0.102 0.76

Table 3. Testing cells in parallel

Correlation doesn’t matterAll loans, Alternative A

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Figure 6. Log likelihoods for loans at assumed values of ρ

45.4%4.80%70

72

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0% 10% 20% 30% 40% 50%Assumed value of rho

Log

Like

lihoo

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Max LnLRobust LnL Max LnL - 1.92

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 providesgreater incentive to reduce PD andless incentive to reduce ELGD.

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Robust LGD at the 99.9th percentile

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Figure 7. Downturn LGD less ELGD

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PD = 10%, R = 12.1% PD = 3%, R = 14.7% PD = 1%, R = 19.3% PD = 0.3%, R = 22.3% PD = 0.1%, R = 23.4% PD = 0.03%, R = 23.8% 2006 Supervisory Mapping Function

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 don't think it can be falsified at present.In part, the robust LGD function isn't that bad. In part, there isn't enough data to show otherwise.

But of course, we don't know.Anyone can try to show that we are just plain wrong. We hope someone tries and we expect that they fail.20

Practical contributionOur LGD function uses PD, R, and ELGD.

Banks have estimates of PD, R, and ELGD.We don't 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.

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Summary of LGD Risk ResolvedWe 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, andas a null hypothesis in future research.

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Questions?

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