Lgd Risk Resolved Bog And Occ

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Please do not quote or distribute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 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
3

The 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 we'll show.
(It first appeared in Modest Means, Risk, January) 4

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 function
Using two assumptions, we can divide the
robust 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 a
specific relation between default and LGD,
because both depend on the same Z…
6

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

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

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.

9

The alternatives are more flexible
Figure 4: Default and LGD with Alternative A
Alternatives produce greater or 100%

less LGD risk than robust LGD.

e.g., Alternative A: cLGDa = 10%

⎡ N −1[PD ELGD 1− a ] + ρ z ⎤ a = 2: negative LGD risk
ELGD 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

The distribution of credit loss
The 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 enter

This is the first paper to derive this PDF…

11

PDF 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

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 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

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% 4
Ba3 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% 10
B1 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

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 - 1.92
Log 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

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 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.
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Practical contribution
Our 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 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.
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Lgd Risk Resolved Bog And Occ

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