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Modelling of Credit Risk Parameters: Probability of Default (PD) and Loss given Default (LGD)
Dr. Bodo Huckestein
XXV Heidelberg Physics Graduate Days,
October 6, 2010
Modelling of credit risk parameters: PD and LGD
Agenda
� Introducing credit risk
� Modelling Exposure at Default (very short)
� Modelling Probability of Default
� Score cards
� Calibration
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� Calibration
� Applications
� Modelling Loss Given Default
� Types of models
� Work-out LGD
� Example: mortgage loans
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
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Lends money
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
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Pays back loan+ interest
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
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Pays back loan+ interest
Or fails to do so: default!
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
But how likely is that going to happen?
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Pays back loan+ interest
Or fails to do so: default!
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
PD:
Estimated probability of
default
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Pays back loan+ interest
Or fails to do so: default!
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
And what am I going to lose if it happens?
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Pays back loan+ interest
Or fails to do so: default!
Modelling of credit risk parameters: PD and LGD
Credit risk
Bank Loan Borrower
LGD:
Estimated loss given
default
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Pays back loan+ interest
Or fails to do so: default!
Modelling of credit risk parameters: PD and LGD
Expected and Unexpected Loss
Credit defaults lead to
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Expected Loss (EL)(normal case)
Unexpected Loss (UL)(disaster scenario)
covered bycredit margin
covered by capital
Modelling of credit risk parameters: PD and LGD
Calculating Expected Loss
EL = EAD · PD · LGD
� Credit risk parameters� PD: Probability of Default
� EAD: Exposure at Default
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� EAD: Exposure at Default
� LGD: Loss Given Default
� Prerequisites for definition of credit risk parameters� consistent definition of default event
� consistent definition of loss amount
� From single credit to portfolio view� EL is additive � Summation over all credits
� EL does not depend on correlations within the portfolio
Modelling of credit risk parameters: PD and LGD
from credit portfolio model:
Calculating Unexpected Loss
,...),,,( ijiii EADLGDPDf ρ
pro
ba
bili
ty d
en
sity (
1 / th
ou
sa
nd
mio
. €
)
0.3
Loss/return distribution of risk type ECredit Risk Loss Distribution
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,...),,,( ijiii EADLGDPDf ρ
Example: 1-Factor-Model (Gordy)
� UL according to Basel II
loss (thousand mio. €)
pro
ba
bili
ty d
en
sity (
1 / th
ou
sa
nd
mio
. €
)
-8 -6 -4 -2 0 2
0.0
0.1
0.2
ELUL
Loss
99.9% quantile
0
Modelling of credit risk parameters: PD and LGD
Typical data environment in credit risk estimation
transaction data default data
data warehouseplatform for model
calibration and
samples from historic data
13
calculation kernel
PD, LGD, EAD, EL, UL, ...
data warehouse
with historic data
calibration and
validationmodel parameters
central bank reporting
regulatory capital requirement
according to Basel II
internal
reporting
integrated risk-return
management
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Modelling of credit risk parameters: PD and LGD
Statistical model and data collection
Calculation of EAD, PD and LGD based on historic data
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DataHistory
Statistical Model
effort
interdependencies
Modelling of credit risk parameters: PD and LGD
Statistical Model: Exposure at Default (EAD)
� Instalment credit
� Fixed payment schedule: exposure contains no stochastic component
� Collaterals may be considered (statistical methods for valuation where necessary)
� Deferrals have to be recognised
� Revolving credits (e.g. credit cards) and credit lines
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� Revolving credits (e.g. credit cards) and credit lines
� Exposure fluctuates stochastically
� Statistical method for EAD required: EAD = current availment (deterministic) + Add-On (stochastic)
� Off-balance sheet exposures and derivatives
� No deterministic component
� Purely statistical estimation
Modelling of credit risk parameters: PD and LGD
Statistical Model: Probability of Default (PD) (1/2)
� Find appropriate criteria with discriminatory power:
� Application criteria (income, domicile, etc.)
� Corporates: information from financial statements (sales, earnings, etc.)
� Behavioural criteria (cumulative days past due, current arrears, etc.)
� Determine score function:
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� Determine score function:
score = weight1 · value(criterion1) + … + weightn · value(criterionn)
� Calibration: find the relationship between probability of default
and score
� Data basis: observed historic defaults
� Specify reference dataset
� Find appropriate counting method (e.g.: how to handle re-ageing of facilities?)
Modelling of credit risk parameters: PD and LGD
� With increasing „age“ of credit:
� Application criteria partly lose their discriminatory power
� Behavioural criteria gain importance
� Segregate exposures into different "age classes"
� Link between credit decision and PD estimation
Statistical Model: Probability of Default (PD) (2/2)
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� Link between credit decision and PD estimation
� Scoring methods (solely based on application criteria) for credit decision
(application score) have been established within banks for a very long
time, independent of Basel II
� Application score can be used directly within PD model
� Similar methods are well known in other fields like sociology or medical
science
� Score card
Modelling of credit risk parameters: PD and LGD
Score cards
� What is a score card?
� What kind of risk factors do we have?
� What is the discriminatory power of a score card?� histograms, ROC curve and CoC value
� How to determine the score function?
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� How to determine the score function?
� How to choose the risk factors?
� How can we transform scores into PDs?
Modelling of credit risk parameters: PD and LGD
What is a score card?
• abstractly: mapping from the risk factor space (age of obligor, profession, amount financed ...) to the real numbers (score)
score 2 factor risk
1 factor risk
function score →
M
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• example: assign points to the answers of a questionnaire and interpret sum
of these points as classification
• purpose: reduction of a multi-dimensional decision problem to a one-
dimensional scale
• goal: ex-ante separation between ex-post defaulted and non-defaulted obligors
N factor risk
M
Modelling of credit risk parameters: PD and LGD
What kind of risk factors do we have?
• metric risk factors
□ values are numbers
□ there is a ranking order
□ credit related examples: credit amount, instalment, time to maturity, age
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□ credit related examples: credit amount, instalment, time to maturity, age
of obligor ...
• categorical risk factors
□ values do not have to be numbers
□ typically no ranking order
□ credit related examples: marital status, gender, profession, purpose of financing, zip code …
Modelling of credit risk parameters: PD and LGD
Discriminatory power: histograms
• example: distributions of defaulted (red) and non-defaulted
(green) loans are separated
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risk factor 1 risk factor 2
frequency
Modelling of credit risk parameters: PD and LGD
Discriminatory power : ROC curve & CoC value
• ROC curve1. Order loans according to metric risk factor (if
necessary, one has to transform to metric)2. Start at the end where most non-performer are
expected (economic hypothesis)3. Looking at the first N loans, which percentage of
performers and which percentage of non-performers are captured?
80%
100%
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loan: 1 2 3 4 5
score: 10,5 8,35 8,3 7,3 -2,1
status: D D A D A
perform.: 0% 0% 50% 50% 100%
non-perf.: 33% 66% 66% 100% 100%
performers are captured?4. Use these percentages as x- and y-coordinates
to build up a curve
0%
20%
40%
60%
0% 50% 100%
perf.
no
n-p
erf
.
Modelling of credit risk parameters: PD and LGD
Discriminatory power : ROC curve & CoC value
• ROC curve1. Order loans according to metric risk factor (if
necessary, one has to transform to metric)2. Start at the end where most non-performer are
expected (economic hypothesis)3. Looking at the first N loans, which percentage of
performers and which percentage of non-performers are captured?
80%
100%
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performers are captured?4. Use these percentages as x- and y-coordinates
to build up a curve
0%
20%
40%
60%
0% 50% 100%
perf.
no
n-p
erf
.
� CoC value or Accuracy Ratio (AR)� Ratio of the area under the ROC curve and the maximum value ( = 1)� Lies between 50 % and 100 %� In the example: CoC = 83 % ( = 5/6)
Modelling of credit risk parameters: PD and LGD
score(D)>score(A)
Y
Y
Y
Example:
loan: 1 2 3 4 5
score: 10,5 8,35 8,3 7,3 -2,1
status: D D A D A
CoC value: what does it mean?
Example:
D A
1 3
1 5
2 35/6
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Y
Y
N
Y
• Build all pairs of defaulted and non-defaulted loans
• Is score of defaulted loan higher than score of non-defaulted loan?
2 3
2 5
4 3
4 5
CoC value: probability that score of defaulted loan is higher than score of non-defaulted loan
5/6
Modelling of credit risk parameters: PD and LGD
ROC curves: limits
•Perfect discriminatory power : CoC = 100%
100
120
80%
100%
Distributions ROC curve
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0
20
40
60
80
1 3 5 7 9 11 13 15 17 19 21
frequency
0%
20%
40%
60%
0% 50% 100%
perf.
no
n-p
erf
.
risk factor
Modelling of credit risk parameters: PD and LGD
100
120
ROC curves: limits
•No discriminatory power: CoC = 50%
Distributions ROC curve
80%
100%
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0
20
40
60
80
1 2 3 4 5 6 7 8 9 10
frequency
risk factor
0%
20%
40%
60%
0% 50% 100%
perf.
no
n-p
erf
.
Modelling of credit risk parameters: PD and LGD
ROC curves: science
• Origin: signal detection theory
80%
100%
tru
e p
osit
ive r
ate
thresholdvalue
Actual value
p n
Prediction
pTrue
positive
False
positive
nFalse
negative
True
negative
Contingency table
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Examples:
� Radar
� Particle detectors
� Psychophysics
� Psychology
� Medicine
� Data mining
0%
20%
40%
60%
0% 20% 40% 60% 80% 100%
sensitivity
false positive rate
tru
e p
osit
ive r
ate
specificity: # true negatives / # all negatives
Modelling of credit risk parameters: PD and LGD
How to determine the score function?
distribution of the risk factors
X2
discriminatory direction distribution of the score
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X1
X2
X1
2211XXScore ⋅+⋅= ββ
Modelling of credit risk parameters: PD and LGD
How to choose the risk factors?
• Pros
□ Choice according to …
□ … univariate discriminatory power (CoC value)
□ … technical criteria• risk factors related to loan, obligor, behaviour, financed object ...
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• risk factors related to loan, obligor, behaviour, financed object ...
□ … contribution to increase in discriminatory power of score (CoC value)
• Cons
□ No substantial increase in discriminatory power of score
□ Mutually dependent risk factors (having high correlation)
Modelling of credit risk parameters: PD and LGD
How can we transform scores into PDs?
• Calibration
• Look at historic transaction data
• Calculate score for each transaction
• Determine whether or not transaction defaulted
• Assign probability of default to score
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• Assign probability of default to score
• Problem
• Default information is binary, PD is continuous
Modelling of credit risk parameters: PD and LGD
PD calibration
• Kernel density estimation
PD
1 ×××××××××× × Historic defaults
PD(score)
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Score
×××××××××××××××× Historic non-defaults
Kernel density
( )∑ −=idefaults
iscorescoreKernelscorePD )(
Modelling of credit risk parameters: PD and LGD
Why would a bank want to estimate the PD?
Portfolio management• Do we want to make (or keep)
the deal? (Limits)
• What should we charge for the
risk of default? (Pricing, standard
risk costs)
Risk management• How much do we expect to lose?
(Expected loss)
• How much should we set aside
for expected losses?
(Provisioning)
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risk costs) (Provisioning)
Capital requirements• How much capital do we need to
cover unexpected losses?
(Economic capital, regulatory
capital (Basel II))
Stress testing• How much could we lose under
adverse conditions?
• How would the PD change?
PDWhat PD?
Modelling of credit risk parameters: PD and LGD
How to define the probability of default?
The PD depends on a reference date and a time horizon:
PD = Probability to default between the reference date and the time horizon
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Dependence on time horizon and reference date:
Reference date Time horizon
PD
Time horizon
Fixed reference date
PD
Reference date
Fixed time horizon
Modelling of credit risk parameters: PD and LGD
How to estimate the probability of default?
Answer: Look at history and count defaults!
NA
NA
-
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PD = ND
NA
Eg. PD = 2/5
NA -ND
Reference date Time horizon
Modelling of credit risk parameters: PD and LGD
Calibration: Point-in-time vs. Through-the-cycle
Long default history available:
NA
NA
-
ND
Reference date Time horizon
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2003 2004 2005 2006 2010 2010
Modelling of credit risk parameters: PD and LGD
Calibration: Point-in-time vs. Through-the-cycle
Long default history available:
NA
NA
-
ND
Reference date Time horizon
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2003 2004 2005 2006 2010 2010
Point-in-time (PIT) calibration
Modelling of credit risk parameters: PD and LGD
Calibration: Point-in-time vs. Through-the-cycle
Long default history available:
NA
NA
-
ND
Reference date Time horizon
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2003 2004 2005 2006 2010 2010
Through-the-cycle (TTC) calibration
Modelling of credit risk parameters: PD and LGD
Example: Automobile bank retail portfolio
3,50%
4,00%
4,50%
Point-In-
Through-the-cycle2003-2010:PD = 3.0%
ObservedDefault
Rate
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2,00%
2,50%
3,00%
3,50%
Jan 0
3
Apr
03
Jul 03
Okt
03
Jan 0
4
Apr
04
Jul 04
Okt
04
Jan 0
5
Apr
05
Jul 05
Okt
05
Jan 0
6
Apr
06
Jul 06
Okt
06
Jan 0
7
Apr
07
Jul 07
Okt
07
Jan 0
8
Apr
08
Point-in-
time2003:
PD = 3.8%
In-Time2010:
PD = 2.3%
Time
Modelling of credit risk parameters: PD and LGD
Economic interpretation of different calibrations
Through-the-cycle calibration
(average macroeconomic conditions)
PD
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Point-in-time calibration
(current macroeconomic conditions)
PD
Reference date
Idiosyncratic
Risk
Macroeconomic
Conditionsinfluences influencePD
Reference date
Modelling of credit risk parameters: PD and LGD
Why would a bank want to estimate the PD?
Portfolio management• Do we want to make (or keep)
the deal? (Limits)
• What should we charge for the
risk of default? (Pricing, standard
risk costs)
Risk management• How much do we expect to lose?
(Expected loss)
• How much should we set aside
for expected losses?
(Provisioning)
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risk costs) (Provisioning)
Capital requirements• How much capital do we need to
cover unexpected losses?
(Economic capital, regulatory
capital (Basel II))
Stress testing• How much could we lose under
adverse conditions?
• How would the PD change?
PDWhat PD?
Modelling of credit risk parameters: PD and LGD
What PD to use for portfolio management?
Deal initiation:
• PD can be used for risk adjusted pricing: risk over the life of the loan = standard risk costs
• PD can be used to limit risks (expected loss, unexpected loss)
41
PIT: adjusted to current risk, but pro-cyclical, decreases volume of business
TTC: less cyclical, but underestimates risk in bad times
Answer for portfolio management:
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Point-in-time PD, time horizon = maturity of loan
Modelling of credit risk parameters: PD and LGD
What PD to use for risk management?
Collective risk provisioning / impairment:
• Risk provisioning should compensate expected losses in the near future
• Time horizon = 1 year
• Losses are cyclical ⇒ risk provisioning should be cyclical
42
• Losses are cyclical ⇒ risk provisioning should be cyclical
• Pro-cyclical risk provisioning adds strain during economic downturn
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Modelling of credit risk parameters: PD and LGD
What PD to use for risk management?
Losses and provisions over time:
Realised losses, PIT provisions
Overestimated losses in good times compensatingunderestimated losses in bad times
43
Answer for risk management:
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Point-in-time/Through-the-cycle PD, one year time horizon
Time
TTC provisions
Modelling of credit risk parameters: PD and LGD
What PD to use for capital requirements?
• Expected losses correspond to rare events
• 99.9% one-year-quantile ≅ “once in a thousand years”
• Long term risk buffer: should not depend on economic cycle
• Point-in-time calibration would be pro-cyclical:• High capital requirements in bad times
44
• High capital requirements in bad times
• Low capital requirements in good times
• Bank regulators (Basel II) demand through-the-cycle calibration
• Time horizon: typically one year (Basel II)
Answer for capital requirements:
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Through-the-cycle PD, one year time horizon
Modelling of credit risk parameters: PD and LGD
What PD to use for stress testing?
• Stress testing analyses the effects of stressed economic conditions on the risk position of a bank
• Baseline for stress testing is the long-time average
• Example: • “mild recession” scenario might correspond to an increase of the PD by 50%
45
• “mild recession” scenario might correspond to an increase of the PD by 50%
relative to long-time averages
• Scenario looses meaning if applied to PIT-PDs in worst recession in decades
Answer for capital requirements:
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Through-the-cycle PD, one year time horizon
Modelling of credit risk parameters: PD and LGD
Summary
• There is not one correct way to calibrate the PD,
there are many!
• Which one to use depends on what you want to use the PD
for!
• Two important parameters to consider:
46
• Two important parameters to consider:
• Desired level of cyclicality
• Time horizon of PD estimation
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Modelling of credit risk parameters: PD and LGD
Types of LGD models
Workout LGDCash flow model
of LGDMarket LGD
Implied market LGD
EAD
Kredit-äquivalent
zum Ausfall-zeitpunkt
Erlöse aus
Verwertung
-Sicherheiten
-Sonstige Erlöse
Verlust
Buchwert
Zinsverlust
KostenWorkout
LGD
Nettoerlös
imInsolvenzfall latest
appraisaltime horizonfor collateral
value
discounting
1 2 3
default oflessee
lease rates
time off leasescheduled
time off leasedefault
scrapvalue
no of leasecontract
forecast
5% percentile
95% percentile
median
latest
appraisaltime horizonfor collateral
value
discounting
1 2 3
default oflessee
lease rates
time off leasescheduled
time off leasedefault
scrapvalue
no of leasecontract
forecast
5% percentile
95% percentile
median
default
LGD
47
Required Data:
� Account data
� Losses
� Collateral values
Required Data:
� Object data (e.g. area, postal code)
� Market data ( e.g. rental rate
value scheduled defaultvalue scheduled default default
Required Data:
� Market data (e.g. bond quotes, stock quotes)
Required Data:
� Market data (e.g. bond quotes, stock quotes)
Examples:
� Retail
� SME
� Large-sized corporates
Examples:
� CRE
� Project and Object financing
Examples:
� Large-sized corporates
� Sovereigns banks
Examples:
� Large-sized corporates
� Sovereigns banks
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Modelling of credit risk parameters: PD and LGD
Solutions for key challenges
Complexity
– scenarios– recovery rates– collateral attributes
Amount of data
– source systems– interfaces– input screens
48
� Increased cost efficiency by interleaved development processes
Calibration needs Maintenance costs
– collateral attributes– direct and indirect costs– pooling criteria– statistical uncertainties– yield curves for discounting
– input screens– new data fields– processes– retroactive changes– documentation
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Modelling of credit risk parameters: PD and LGD
Statistical Model: Loss Given Default (LGD) (2/4)
collectioncosts
other costs
proceeds from sale of securities
Workout-LGD for single exposure
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costs sale of securities
other recoveries
economicloss
EAD
LGD =Economic Loss
EAD
Modelling of credit risk parameters: PD and LGD
Project example: LGD for mortgage loans
• The client
• Mortgage loans
• Loss given default
• Scenario model
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• Recovery rate
• Transition matrix model
• Settlement rate
• Summary
Modelling of credit risk parameters: PD and LGD
The client
• German real estate bank
• Assets: 85.7 bn €
• Real estate lending: 23.5 bn €
• Residential real estate lending: 16.1 bn €
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Modelling of credit risk parameters: PD and LGD
• Financing of commercial/retail real estate
Mortgage loans
Bank CustomerMoney
Collateral Buys
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• Important concepts:
□ value of mortgage (“Grundschuld”): maximum value of collateral the
bank is entitled to
□ collateral value (“Beleihungswert”): conservative estimate of value of collateral
Real estate
Modelling of credit risk parameters: PD and LGD
• LGD = econ. loss / EAD = (EAD – discounted recoveries) / EAD
Loss given default
exposure
atdefault
(EAD)
discounted
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• Mortgage loans: one big recovery cash flow
timewrite-offdefault recovery cash flows
timewrite-off
repossession/
foreclosure
of mortgage
default
discountedexposure
at
default
(EAD)
Modelling of credit risk parameters: PD and LGD
Scenario model
• Alternative scenario to settlement of non-performing loan:
customer reperforms
• Two scenarios: “settlement” / “curing”
LGD = SR · LGDS + (1-SR) · LGDC
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LGD = SR · LGDS + (1-SR) · LGDC
• Risk drivers:
□ Settlement rate SR
□ Loss given settlement LGDS
settlement rate:
percentage of defaulted
loans going into settlement
LGD given
settlementcure rate LGD
in case of curing:
= 0
Modelling of credit risk parameters: PD and LGD
LGD model
• Recovery on mortgage depends on
□ collateral value (“Beleihungswert”) CV
□ recovery rate RR
• but is not more than mortgage value (“Grundschuld”) MV
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LGDS = max(EAD – min(RR · CV; MV) · DF · DTF; 0) / EAD
• for simplicity’s sake, direct and indirect costs are missing from
expression
recovery
up to mortgage value
discount factor
downturn factor
Modelling of credit risk parameters: PD and LGD
LGD pooling
• Recovery rates determined from foreclosure history (1863
foreclosures since 2004)
• Historic foreclosures pooled by
□ object type (single-family house, others)
□ usage (owner-occupied, rented out, mixed)
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□ usage (owner-occupied, rented out, mixed)
□ region (strong, average, below average economic strength)
• Aggregated to 10 distinct pools, e.g.
□ single-family house, owner-occupied, average region
□ other type, rented out, below average region
□ owner-occupied, strong region
• Objects of foreclosure history assigned to the 10 LGD pools
Modelling of credit risk parameters: PD and LGD
LGD pooling: average recovery rates per pool
Recovery rate
50%
60%
70%
80%
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0%
10%
20%
30%
40%
50%
1 2 3 4 5 6 7 8 9 10
Pool
Modelling of credit risk parameters: PD and LGD
LGD estimate for active loan
• Assign loan to one of the 10 LGD pools based on object
properties
• Calculate LGDS,i using position data (EAD, CV, MV) of loan
and recovery rate of each object in historic LGD pool
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LGDS,i = max(EAD – min(RRi · CV; MV) · DF · DTF; 0) / EAD
• Average LGDS,i over LGD pool
historic LGD pool objects
position data
Modelling of credit risk parameters: PD and LGD
Settlement rate
• Can in principle be determined from loan history:
active
default
settlement
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• Problem: don’t have data for long time!
• Solution: analyze short time, extrapolate to long time!
timedefault
settlement
maturity
long time!
Modelling of credit risk parameters: PD and LGD
Transition matrix model
• 6-state model: each contract is in one of 6 states
□ Active (A)
□ Defaulted, but not in settlement (D1)
□ Defaulted, in settlement, but no foreclosure sale, yet (D2)
□ Defaulted, after foreclosure sale, but not written-off (D3)
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□ Defaulted, after foreclosure sale, but not written-off (D3)
□ Written-off (F)
□ Settled, no economic loss (E)
• Assign a state to each contract each month
• Calculate monthly transition rates between states
• Extrapolate monthly transition matrix to reach long time!
Modelling of credit risk parameters: PD and LGD
Transition matrix based on exposure
• Recoveries reduce exposure: base transition matrix on
exposure, i.e. “Where is 1€ of exposure next month?”
E
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• Transition of object from state X to state Y generates “flow”
from state X to state E
monthn-1 n
Y
E
XExposure
Exposure
Modelling of credit risk parameters: PD and LGD
Monthly transition rates
40,0%
50,0%
60,0%
70,0%
80,0%
90,0%
100,0%
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0,0%
10,0%
20,0%
30,0%
2005
0731
2005
0930
2005
1130
2006
0131
2006
0331
2006
0531
2006
0731
2006
0930
2006
1130
2007
0131
2007
0331
2007
0531
2007
0731
A->A A->D1 A->D2 A->D3 A->E A->F
D1->A D1->D1 D1->D2 D1->D3 D1->E D1->F
D2->A D2->D1 D2->D2 D2->D3 D2->E D2->F
D3->A D3->D1 D3->D2 D3->D3 D3->E D3->F
Modelling of credit risk parameters: PD and LGD
• Average monthly transition matrixA D1 D2 D3 E F
A 90,65% 7,21% 1,39% 0,00% 0,75% 0,00%
D1 15,29% 77,92% 5,59% 0,01% 1,19% 0,01%
D2 1,07% 0,17% 96,75% 0,96% 1,05% 0,00%
D3 0,00% 0,00% 0,00% 65,81% 1,21% 32,98%
E 0,00% 0,00% 0,00% 0,00% 100,00% 0,00%
Average and asymptotic transition matrices
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• Extrapolate behavior for long time!: asymptotic matrix
E 0,00% 0,00% 0,00% 0,00% 100,00% 0,00%
F 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%
A D1 D2 D3 E F
A 0,01% 0,00% 0,02% 0,00% 70,30% 29,66%
D1 0,01% 0,00% 0,02% 0,00% 69,22% 30,74%
D2 0,01% 0,00% 0,02% 0,00% 59,98% 39,99%
D3 0,00% 0,00% 0,00% 0,00% 3,53% 96,47%
E 0,00% 0,00% 0,00% 0,00% 100,00% 0,00%
F 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%
Modelling of credit risk parameters: PD and LGD
Settlement rate I
• Settlement in terms of 6-state model: entering state D3
• But D3 is only transient state: no asymptotic rate D1→D3
• Solution: adjust transition matrix (“make D3 absorbing”)
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• 31,7% of each Euro in D1 eventually pass through D3!
A D1 D2 D3 E F
A 90,65% 7,21% 1,39% 0,00% 0,75% 0,00%
D1 15,29% 77,92% 5,59% 0,01% 1,19% 0,01%
D2 1,07% 0,17% 96,75% 0,96% 1,05% 0,00%
D3 0,00% 0,00% 0,00% 65,81% 1,21% 32,98%
E 0,00% 0,00% 0,00% 0,00% 100,00% 0,00%
F 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%
A D1 D2 D3 E F
A 0,01% 0,00% 0,02% 30,63% 69,22% 0,12%
D1 0,01% 0,00% 0,02% 31,72% 68,10% 0,14%
D2 0,01% 0,00% 0,02% 41,41% 58,52% 0,05%
D3 0,00% 0,00% 0,00% 100,00% 0,00% 0,00%
E 0,00% 0,00% 0,00% 0,00% 100,00% 0,00%
F 0,00% 0,00% 0,00% 0,00% 0,00% 100,00%
D3 0,00% 0,00% 0,00% 100,00% 0,00% 0,00%
longtime!
Modelling of credit risk parameters: PD and LGD
Settlement rate II
• But the settlement rate is the relative amount that enters into settlement, not the residual claim after receiving the sales amount!
direct
salesamount
ED2
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• Compensate:
• Settlement rate SR = 66%
amount
residualclaim
monthn-1 n
D3
D3-
( ) ( ) ( )( )3D2DP
E2DP3D2DP3D1DPSR A
→
→+→⋅→=
Modelling of credit risk parameters: PD and LGD
Summary
• Major risk drivers:□ settlement rate: how much of the defaulted exposure ends up in settlement?
□ recovery rate: how much of the collateral value is recovered in settlement?
• Scenario model with scenarios “settlement” vs. “curing”
• Recovery rate: based on history of settled contracts
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• Settlement rate: direct determination based on history not feasible (long time!)
• Transition matrix model□ average monthly transition rates between default classes
□ asymptotic transition rates for long times!
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