“Stress Testing Banking Book Positions Under Basel II”
Federal Reserve Bank of San Francisco
January 2009by
Paul Kupiec
Federal Deposit Insurance Corporation
The opinions expressed in this presentation represent those of the author. They are not the official views of
the FDIC.
Overview
• Basel II AIRB sets minimum capital using a modified version of the Vasicek credit loss model
• Capital covers 99.9% of all potential credit losses– Capital violations should happen only 1-in-1000 years
• Basel II requires supplemental stress tests• Question: What are appropriate stress scenarios if
AIRB capital is only breached 1-in-1000 years?
Stress tests: Why do we need them? How should we do them?
• Depends on….How well the AIRB model fits the data
– The AIRB has 3 basic parts where model fit may be an issue:
• The default rate model• The LGD assumption• The EAD assumption
– Other important issues not addressed in this talk» Asymptotic portfolio assumption» Maturity adjustment
–
Stress Testing Under Basel II
• Develop methods or techniques that enable an analyst to estimate the capital implications of relaxing inaccurate restrictive assumptions or modifying other unrealistic modeling features of the AIRB modeling framework.
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Portfolio Capital Requirement in %
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
99.9 percentile from the Vasicek portfolio default rate distribution model
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Portfolio exposure at default
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Portfolio exposure Loss Given Default
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Individual Credit Unconditional Probability of Default
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Default Correlation among portfolio credits
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Vasicek portfolio default rate
Regulatory correlation function “fine tuned” to reduce
procyclicality
Maturity adjustment factor specified to
mimic KMV estimates
These features are policy parameters and are not
derived from a formal credit risk model
Portfolio LGD and EAD
• AIRB does not specify EAD or LGD models– LGD and EAD are at the portfolio level
• AIRB measures them using a single parameter• No recognition or discussion that EAD and LGD have a
distribution at the portfolio level• Diversification issues are not modeled
– Basel provides broad regulatory guidance as to how these “parameters” should be estimated
• Minimum sample sizes for calibration• Minimum parameter values• LGD must be estimated in a way so that it reflects “downturn
conditions”
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
For many portfolios, EAD and LGD are more accurately
modeled as random variables with systematic risk
Stochastic LGD & EAD
• AIRB model can be generalized to account for stochastic LGD & EAD at individual exposure level– LGD and EAD realized values can be correlated in time– systematic time-variation in recovery and exposures – Leads to portfolio models for EAD and LGD
• Kupiec (2008) Journal of Derivatives
• Result: Once portfolio models for LGD and EAD are accounted for, portfolio credit loss rate distribution may have fatter tails relative to AIRB model– LGD and EAD correlation introduces additional systematic
risk & increases unexpected loss rates
Stochastic LGD and EAD
• Kupiec (2008) model provides a coherent framework for analyzing AIRB EAD and LGD parameters– A rigorous & consistent model for thinking about
stress or “downturn” LGD and EAD estimates• Fully accounts for diversification and systematic risk
– Given time constraints, I’ll skip this part of the paper and focus of the default rate model
AIRB Default Rate Model Fit
• Take a brief look at a large panel data set– Moody’s Corporate Bond Ratings and Default History,
1920-2006
• Fit the model to historical data• Evaluate default rate model fit• Upshot: AIRB model fit is poor
– Stress tests should account for AIRB default model risk
• What type of model generalizations may improve default rate performance?
Calibration Methodology
• If all credits in a rating grade have identical PD and correlation parameter and defaults are driven by a single common factor– to a close approximation….– the annual default rate of a credit grade with a
large number of credits should have an ASFM default rate distribution
New Panel Regression Approach
1
1Mti
it
ePD
graderatingscreditaforratedefaultlconditionaVasicek
New Panel Regression Approach
1
1Mti
it
ePD
graderatingscreditaforratedefaultlconditionaVasicek
Unconditional probability of default for credits in the portfolio
New Panel Regression Approach
1
1Mti
it
ePD
graderatingscreditaforratedefaultlconditionaVasicek
The default correlation parameter
New Panel Regression Approach
1
1Mti
it
ePD
graderatingscreditaforratedefaultlconditionaVasicek
Latent Gaussian “Macro factor” that drives individual credit default realizations
Panel Regression Approach
itMt
iit e
PD
~11
11
Fixed effect for credit rating category i
Year effect that is identical across all credit grades in the rating system for a given year
Random deviation from ASFM model
Transformation of annual default rate in year t for credit rating category i
Data
• Moody’s Corporate Bond Default History 1920-2006– Issuer rated annual default rates by credit grades
• Aa, A, Baa, Ba, B, Caa_C
– Number of issuers with a given Moody’s rating that default in a given year, divided by the total number of issuers with same Moody’s rating at the beginning of the year
• If Moody’s withdraws ratings the issuer is removed from numerator and the denominator
Model Fit
• Actual vs Predicted– For each Moody’s credit grade– 90% confidence intervals around predicted values
from bootstrapped sampling distribution
Baa Credit Grade Model Performance
0
0.25
0.5
0.75
1
1.25
1.5
1.75
21
92
0
19
24
19
28
19
32
19
36
19
40
19
44
19
48
19
52
19
56
19
60
19
64
19
68
19
72
19
76
19
80
19
84
19
88
19
92
19
96
20
00
20
04
de
fau
lt r
ate
(p
ct)
actual
Ba Credit ASFM Model Performance
0
2
4
6
8
10
1219
20
1925
1930
1935
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
def
ault
rat
e in
pct
actual
Model Performance Grade 'B' Credits
0
2
4
6
8
10
12
14
16
18
201
92
0
19
25
19
30
19
35
19
40
19
45
19
50
19
55
19
60
19
65
19
70
19
75
19
80
19
85
19
90
19
95
20
00
20
05
de
fau
lt r
ate
pc
t actual
Caa_C Grade Model Performance
0
10
20
30
40
50
60
70
80
90
100
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
de
fau
lt r
ate
pc
t
actual
Can the AIRB reproduce the default rate data? • Compare actual & predicted default rate distribution
using Kolmogorov-Smirnov Statistic– Statistic is based on the maximum distance between two
empirical CDFs
K-S Statistic
0
0.20.4
0.60.8
11.2
-10
-8.5
-7.1
-5.6
-4.2
-2.7
-1.3 0.1
1.6
3.0
4.5
5.9
7.4
8.8
K-S=.3235
CDF 1
CDF 2
Asymptotic K-S Statistics
Aa 1.9968 0.00069A 3.1838 0.000000003
Baa 3.9939 3E-14Ba 4.3997 0B 4.6633 0
CaaC 4.6368 0
Asymptotic K-S Statistic
Credit Grade
Probability that the two
distributions are identical
Default Rate Model Fit• AIRB model fits the Moody’s data poorly
– Portfolios perform “in the tails” relative to the model’s predictions
• Portfolios perform exceptionally well far too often• ….exceptionally poorly far too often
– Maybe a double stochastic boundary model or time-a time varying correlation will fit better
• Default boundary is stochastic• Correlation is a random variable
– What would models with these characteristics look like?
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Mounting academic evidence suggests default boundaries maybe stochastic perhaps with
systematic time variation
Why Stochastic Boundary?• Intuition: Market has Liquidity Cycles or
Cycles in Underwriting Standards– Firms must refinance maturing debt– When liquidity is plentiful, underwriting standards
are lax and it is easy for all firms to refinance– When liquidity is scarce, underwriting standards
tighten; all firms face higher refinance boundaries
Noncurrent Bank C&I Loan Rates and Bank Loan Underwriting Standards
-30-20-10
010203040506070
Jun-
90
Jun-
91
Jun-
92
Jun-
93
Jun-
94
Jun-
95
Jun-
96
Jun-
97
Jun-
98
Jun-
99
Jun-
00
Jun-
01
Jun-
02
Jun-
03
Jun-
04
Jun-
05
Jun-
06
Jun-
07
Jun-
08
FR
B L
oan
Off
icie
r S
urv
ey
0%
1%
2%
3%
4%
5%
6%
per
cen
t n
on
curr
ent
C&I underwriting standards for med and large firms (left scale)
C&I loan noncurrent rate
(right scale)
New Stochastic Default Boundary Asymptotic Portfolio Model
• The ex ante probability of default (the default boundary) is random 1,0~
~iP
iP~
is driven by a latent Gaussian factor, iQ~
,
.,0)~~()~~()~~(
),(~
)(~~
~1~~
jieeEeeEeeE
ee
ee
eeQ
jdiQjQMjQiQ
iQiY
MM
iQQMQi
Same common factor that drives Vi
Parameter determines the correlation among
accounts’ stochastic default boundaries
Stochastic Default Boundary
Account defaults when ii DV~~ .
Probability integral transform implies,
iii DQP~~
1~ 1
or,
ii QD~
1~ 11
.
So the credit defaults when,
ii QV~
1~ 11
Random default
boundary condition
Asymptotic Portfolio Default Rate Distribution
,~
RBX the portfolio default rate, is an implicit function of the
common latent factor, ,Me
,,,1
11~
~11
MMiQiQQ
MViQQMQRB eeee
eeepX
For any eM, the conditional default rate requires integrating out the idiosyncratic risk uncertainty in the default boundary……this
requires numerical procedures
Example: Stochastic Default Boundary
• Assume PD boundary is normally distributed with mean 1% and standard deviation of 0.2%
• Assume PD latent variables have 20% correlation • Assume firm latent default factors (firms value
proxies) have 20 percent correlation – Implies PD and Vi have a 20% correlations as well
Example: Asymptotic Portfolio Credit Loss Distribution when Default Boundaries are Stochastic
Influence of Default Boundary Correlation
Sensitivity of The Upper Tail of the Default Rate Density to Correlations Among Individual Credit Default Boundries
0 .08 0 .10 0 .12 0 .14 0 .16 0 .18 0 .20po rtfo l io defau l t rate0 .00
0 .01
0 .02
0 .03
0 .04p rob ab i l i ty
ii
V
PP~
,002.0,01.0~
20.
20.Q
50.Q
05.Q
p
iP~
As the default boundary correlation parameter increases, the 99.9% critical default rate increases
Influence of Default Boundary Standard Deviation
Sensitivity of The Upper Tail of the Default Rate Density to Correlations Among Individual Credit Default Boundries
0 .08 0 .10 0 .12 0 .14 0 .16 0 .18 0 .20po rtfo l io defau l t rate0 .00
0 .01
0 .02
0 .03
0 .04p rob ab i l i ty
ii
V
PP~
,002.0,01.0~
20.
20.Q
50.Q
05.Q
p
iP~
The critical value increases as the standard deviation of the default boundary distribution increases
Basel II AIRB is a Modified Vasicek Model
b
bMLGDPD
R
RPD
RLGDEADK
5.11
5.21999.
11
1 11 (7)
where,
50
50
50
50
1
1124.0
1
112.0
e
e
e
eR
PDPD
, 205478.11852.0 PDLnb .
Alternatively, default correlations may have
systematic time variation
Stochastic Correlation: Motivation
• Prior to 2006, rating agencies and investors calibrated sub-prime mortgage securitization models using a very low default correlation based on 1998-2005 data
• In 2006, these sub-prime mortgages began defaulting in large numbers– Default correlation had shifted from early data
Motivation II
• Popular credit loss models did not anticipate time-variation in default correlation
• Cause of shift? Housing prices– From 1998-2005 housing prices went up strongly
depressing sub-prime default correlations
– From 2006, housing prices started declining rapidly, increasing sub-prime default correlations
• AIRB model does not accommodate time-variation in default correlation parameter
• Lets see if we can fix this…….
Motivation III
• Stochastic correlation is a reduced-form model for contagion risk– Rarely, but with some positive probability, a
random factor causes a shift in default correlation patterns and very quickly, defaults become much more highly correlated………….
Model AssumptionsThe credit-specific correlation is represented by di~ which has
a cumulative distribution
1,1,~~ dididi .
Each credit (credit i ) has an associated latent unobserved factor, iT~
,
.,0)~~()~~(
),(~
)(~~
~~1~~~ 2
jieeEeeE
ee
ee
eeT
jdMjdid
idid
MM
iddiMdii
Model Assumptions
The credit-specific correlation is represented by di~ which has a cumulative distribution
1,1,~~ dididi .
Each credit (credit i ) has an associated latent unobserved factor, iT~
,
.,0)~~()~~(
),(~
)(~~
~~1~~~ 2
jieeEeeE
ee
ee
eeT
jdMjdid
idid
MM
iddiMdii
New notation for latent
factor proxy for firm value
Model Assumptions
The credit-specific correlation is represented by di~ which has a cumulative distribution
1,1,~~ dididi .
Each credit (credit i ) has an associated latent unobserved factor, iT~
,
.,0)~~()~~(
),(~
)(~~
~~1~~~ 2
jieeEeeE
ee
ee
eeT
jdMjdid
idid
MM
iddiMdii
The default correlation parameter that multiplies the common Gaussian factor is random
Correlation specification is slightly changed to simplify mathematical proofs
Default correlation is now djdiE ~~
Latent Correlation Factor
di~ are driven by a latent Gaussian factor iW~
through a probability integral transform.
iW~
has both common ( Ke~ ) and idiosyncratic sources of risk ( ice~ ),
.,0)~~()~~()~~()~~()~~(
),(~
)(~~
~1~~ 2
jieeEeeEeeEeeEeeE
ee
ee
eeW
MKjdKjdMjcicjcid
icic
KK
icciKcii
Latent correlation factor
di~ are driven by a latent Gaussian factor iW~
through a probability integral transform.
iW~
has both common ( Ke~ ) and idiosyncratic sources of risk ( ice~ ),
.,0)~~()~~()~~()~~()~~(
),(~
)(~~
~1~~ 2
jieeEeeEeeEeeEeeE
ee
ee
eeW
MKjdKjdMjcicjcid
icic
KK
icciKcii
Common factor that drives correlation parameter
Latent correlation factor
di~ are driven by a latent Gaussian factor iW~
through a probability integral transform.
iW~
has both common ( Ke~ ) and idiosyncratic sources of risk ( ice~ ),
.,0)~~()~~()~~()~~()~~(
),(~
)(~~
~1~~ 2
jieeEeeEeeEeeEeeE
ee
ee
eeW
MKjdKjdMjcicjcid
icic
KK
icciKcii
This model has two independent common factors
--one drives firm values
--one drives the correlation among firm value realizations
Common factor that drives correlation parameter
Default Correlations
jdid
Mjdid
Mjdid
ji
jiji
E
eEE
eETVarTVar
TTCovTTCorr
~~
~~~
~~~
)~
()~
(
)~
,~
(~,
~
2
2
jdidjdid EEE ~~~~
→The value of the correlation parameter ic
Changes the shape of the unconditional default correlation distribution Determines the default correlation between iT
~ and jT~ conditional on Ke .
→The unconditional correlation between iT~ and jT
~ is insensitive to the correlation
parameters ic
Default Correlations
jdid
Mjdid
Mjdid
ji
jiji
E
eEE
eETVarTVar
TTCovTTCorr
~~
~~~
~~~
)~
()~
(
)~
,~
(~,
~
2
2
jdidjdid EEE ~~~~
→The value of the correlation parameter ic
Changes the shape of the unconditional default correlation distribution Determines the default correlation between iT
~ and jT~ conditional on Ke .
→The unconditional correlation between iT~ and jT
~ is insensitive to the correlation
parameters ic
0icUnconditional
default correlation distribution
Default correlation distribution
conditional on ek=-2
Correlations are independent
→No time variability
5.0icUnconditional
default correlation distribution
Default correlation distribution
conditional on ek=-2
Correlation among Wi latent factor = 25%
→Substantial shift in default correlation distribution
95.0ic Unconditional default
correlation distribution
Default correlation distribution
conditional on ek=-2
A shift in the default correlation distribution of “sub prime mortgage”
proportions
Asymptotic Portfolio Default Rate Distribution when Correlation is Stochastic
As N , the conditional portfolio default rate converges to:
ic
ic
e
e
ic
iccKc
MiccKc
saKMRCN e
ee
eeePD
eeX
221
211
..
111
11
,|~
lim
This expression can be evaluated using numerical methods for given values of KM ee , .
Use Monte Carlo simulation to get unconditional default rate distribution.
Stochastic Correlation Example
• PD=1%• Default correlation parameter is distributed
uniformly over [.05,.35], average value=.20– Average default rate correlation of 4% (.2^2)
→default correlation distribution depends on the correlation parameter of , the Gaussian factor that drives default correlations
iW~
Stochastic Correlation Example
pc parameter drives the
distribution skew
Stochastic Correlation Example II
Larger default correlation
parameter→longer tail
Overall Conclusions• Basel AIRB restrictive assumptions are likely to be
violated – AIRB minimum capital requirements may be violated more
frequently than the 1-in-1000 year nominal solvency standard
• Stress tests are a means for identifying capital needs on positions that are unlikely to adequately modeled under AIRB assumptions
• Stress tests can identify additional capital needs for scenarios that are more common than 1-in-1000 years
• Stress-testing based capital supplements should occur routinely for some AIRB banks if they are following a rigorous & well-designed stress testing regimen