“ Stress Testing Banking Book Positions Under Basel II” Federal Reserve Bank of San Francisco January 2009 by Paul Kupiec Federal Deposit Insurance Corporation

“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 %


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


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


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


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


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


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”


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


1

1Mti

it

ePD


Unconditional probability of default for credits in the portfolio


1

1Mti

it

ePD


The default correlation parameter


1

1Mti

it

ePD


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?


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


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 .


,

.,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 .


,

.,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



iW~


.,0)~~()~~()~~()~~()~~(

),(~

)(~~

~1~~ 2

jieeEeeEeeEeeEeeE

ee

ee

eeW

MKjdKjdMjcicjcid

icic

KK

icciKcii

Common factor that drives correlation parameter

Latent correlation factor



iW~


.,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



Correlation among Wi latent factor = 25%

→Substantial shift in default correlation distribution

95.0ic Unconditional default

correlation distribution



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

Documents

“ Stress Testing Banking Book Positions Under Basel II” Federal Reserve Bank of San Francisco January 2009 by Paul Kupiec Federal Deposit Insurance Corporation