witzany creditriskman fg

8/18/2019 Witzany CreditRiskMan FG

1/152

European Social Fund Prague & EU: Supporting Your Future

Credit Risk Management andModeling

Doc. RNDr. Jiří Witzany, Ph.D.

[email protected] , NB 178Office hours: Tuesday 10-12

mailto:[email protected]:[email protected]


2/152


Literature

Requirement Title Author Year of Publication

Required Credit Risk Management and

Modeling

Witzany, J. 2010, Oeconomica,

pp. 214

Optional Managing Credit Risk – The

Great Challenge for Global

Financial Markets

Caouette J.B., Altman

E.I., Narayan O.,

Nimmo R.

2008, 2nd Edition,

Wiley Finance, pp.

627

Optional Credit Risk – Pricing,

Measurement, and

Management

Duffie D., Singleton

K.J.

2003, Princeton

University Press,

pp.396

Optional Consumer Credit Models:

Pricing, Profit, and Portfolios

Thomas L. C. 2009, Oxford

University Press, pp.

400

Optional Credit Derivatives Pricing

ModelsSchönbucher P.J. 2003, Wiley Finance

Series, pp. 375

2

Course Organization: project (scoring function development),small midterm test (26.3.), final test (14.5. ?)


3/152


Content

• Introduction

• Credit Risk Management – Credit Risk Organization

– Trading and Investment Banking

– Basel II

• Rating and Scoring Systems

– Rating Validation – Analytical Ratings

– Automated Rating Systems

3


4/152


Content - continued

– Expected Loss, LGD, and EAD

– Basel II Requirements

• Portfolio Credit Risk – CreditMetrics

– Credit Risk+

– Credit Portfolio View – KMV Portfolio Manager

– Basel II Capital requirements

4


5/152


Content - continued

• Credit Derivatives

– Overview of Basic Credit Derivatives

Products – Intensity of Default Stochastic Modeling

– Copula Correlation Models

– CDS and CDO Valuation

5


6/152


Introduction

• Classical commercial bank credit riskmanagement

– Corporate loans approval and pricing

– Retail loans approval and pricing

– Provisioning and workout

– Regulatory requirements – Basel II

– Loan portfolio reporting and management

– Financial markets (counterparty) credit risk

6


7/152European Social Fund Prague & EU: Supporting Your Future

Introduction

• Credit approval – can we get a crystalball?

• What is the first, the chicken or the egg,Basel II or credit rating?

• What is new in Basel III?

• Should we start the course with creditderivatives?

7



Credit Risk ManagementOrganization

8Separation of Powers – Risk Organization Independence!!!



Credit Risk Process

9

Importance of credit risk information management!!!



Counterparty and Settlement

Risk on Financial Markets

10

Settlement Risk only at the settlement dateCounterparty Risk over the transaction life



Counterparty Risk Equivalents

11

Exposure=max(market value,0) %·Nominal amount x

-

100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

Jun

98

Jun

99

Jun

00

Jun

01

Jun

02

Jun

03

Jun

04

Jun

05

Jun

06

Jun

07

Jun

08

Jun

09

B i l l i o n U S D

OTC derivatives

OTC derivatives

Growing global counterparty risk!!!Could be partially reduced through netting agreements



Trading Credit Risk

Organization

12

Importance of Back Office independence!!!



The risks must not be

underestimatedCraig Broderick, responsible for risk management in Goldman

Sachs in 2007 said with self-confidence: “Our risk culture hasalways been good in my view, but it is stronger than ever today.

We have evolved from a firm where you could take a little bit ofmarket risk and no credit risk, to a place that takes quite a bit of

market and credit risk in many of our activities. However, we do

so with the clear proviso that we will take only the risks that we

understand, can control and are being compensated for.” In spite

of that Goldman Sachs suffered huge losses at the end of 2008and had to be transformed (together with Morgan Stanley) froman investment bank to a bank holding company eligible for helpfrom the Federal Reserve.

13



Basel II - History• 1988: 1st Capital Accord – Basel I (BCBS,

BIS)

• 1996: Market Risk Ammendment

• 1999: Basel II 1st Consultative Paper

• 2004: The New Capital Accord - Basel II

• 2006: EU Capital Adequacy Directive

• 2007: CNB Provision on Basel II

• 2008: Basel II effective for banks• 2010: Basel III following the crisis (effective

2013-2019)14


15/152


Basel II - Principles

15


16/152


Basel II Structure

16


17/152


Basel II Qualitative Requirements• 441. Banks must have independent credi t r isk con trol uni ts that are

responsible for the design or selection, implementation and performance of

their internal rating systems. The unit(s) must be functionally independentfrom the personnel and management functions responsible for originating

exposures. Areas of responsibility must include:

• Testing and monitoring internal grades;

• Production and analysis of summary reports from the bank’s rating system,

to include historical default data sorted by rating at the time of default andone year prior to default, grade migration analyses, and monitoring of

trends in key rating criteria;

• Implementing procedures to verify that rating definitions are consistently

applied across departments and geographic areas;

• Reviewing and documenting any changes to the rating process, includingthe reasons for the changes; and

• Reviewing the rating criteria to evaluate if they remain predictive of risk.

Changes to the rating process, criteria or individual rating parameters must

be documented and retained for supervisors to review.

17


18/152


Basel II Qualitative Requirements

730. The bank’s board of directors has responsibility forsetting the bank’s tolerance for risks. It should also

ensure that management establishes a framework for

assessing the various risks, develops a system to relaterisk to the bank’s capital level, and establishes a method

for monitoring compliance with internal policies. It is

likewise important that the board of di rectors adopts

and supports stron g internal contro ls and wri t ten

pol ic ies and procedures and ensures that managementeffectively communicates these throughout the

organization….

18


19/152


Content

Introduction

Credit Risk Management

Rating and Scoring Systems• Portfolio Credit Risk Modeling


19


20/152


20

Rating/PD Prediction -Can We Predict the Future?

Actual Client/

Exposure/ Application

Information

Present

time

PAST

Historical data

Experience

Know-how

FUTURE?

Probabilty of Default?

Expected Loss?

Loss distribution?

Time horizon?

Default/Loss Definition?


21/152


Rating and Scoring Systems

21

• Rating/scoring system could be in generala black box

• How do we measure quality of a ratingsystem?


22/152


Rating Validation – PerformanceMeasurement

• Validation – measurement of prediction power – does the rating meet our expectations?

• Exact definitions:

– Time horizon

– Debtor or facility?

– Current situation or conditional on a new loan?

– Definition of default? – Probability of default prediction or just

discrimination between better and worse?

22


23/152


Default History

23

Do we observe less defaults for better grades?

Source: Moody‘s


24/152


Rating System Continuous

Development Process

24

Developmentand performancemeasurement –

optimization onavailable data

Data collection –

ratings and defaults

Roll-outInto approval

process

Performancemeasure on new data –

validation

Redevelopment/calibration


25/152


Discriminative Power of Rating

Systems• Let us have a rating system and let usassume that we know the future (good andbad clients)

• Let us have a bad X and a good Y

• Correct signal: rating(X)


26/152


Discriminative Power of Rating

Systems• To measure prediction power let us

count/measure probabilities of the signals

• p1 = Pr[correct signal]• p2 = Pr[wrong signal]

• p3 = Pr[no signal]

• AR = Accuracy Ratio = p1- p2• AUC = Area Under Receiver Operating

Characteristic Curve = p1+ p3/2 = (AR+1)/2

26


27/152


Cumulative Accuracy Profile

27

AR R

P

a

a


28/152


Receiver OperatingCharacteristic Curve

28

AUC

Proportion of good debtors

P r o p o r t i o n

o f b a d

d e b t o r s

/ 2 KS

KS


29/152


AR, AUC, and KS

• The probabilistic and geometricdefinitions of AR and AUC areequivalent

• It follows immediately from theprobabilistic definitions that AR=2AUC-1

• Related Kolmogorov-Smirnov statistics

can be defined as the maximumdistance of the ROC curve from thediagonal multiplied by

29

2


30/152


Empirical Estimations of AR (and AUC)

• Empirical AR or AUC depend on thesample used

• Generally there is an error between the

theoretical AR and an empirical one

• This must be taken into accountcomparing the Accuracy ratios of two

ratings• The are however tests of statistical

significance30


31/152


In-sample or Out-sample

• A rating system is usually developed (trained)on a historical sample of defaults

• When the AR of the system is calculated on

the training sample (in sample) then wegenerally must expect better values than onan independent (out sample)

• Real validation should be done on a sampleof data collected after the development

31


32/152


S&P Rating Performance

32


33/152


Comparing two rating systems

on the same validation sample

33

Gini St.Error 95% Confidence interval

Rating1 69,00% 1,20% 66,65% 71,35%

Rating2 71,50% 1,30% 68,95% 74,05%

H0: Gini(Rating1)=Gini(Rating2)

2(1)= 9,86

Prob> 2(1)= 0,17%

We can conclude that the second rating has a betterperformace than the first on the 99% probability level


34/152


Measures of Correctness of

Categorical Predictions • In practice we use a cut-off score in

order to decide on acceptance/rejection

of new applications• The goal is to accept good applicants

and reject bad applicants

• Or in fact minimize losses caused badaccepted applicants and good rejectedapplicants (opportunity cost)

34


36/152



Categorical Predictions • Generally we need to minimize the

weighted cost of errors

36

( | ) ( | ) ( | ( |· ) )c c

c G c B G c c F s B q F s G q C F s B F WCE l s G

( ( | ) ( | ) ( | ) ( |) )· ·c

c c c c cw s C F s B F s G C C F s B F s G

B

G

l C

q


37/152


Cut-off Optimization• Example: Scorecard 0…100, proposed cut-off

40 or 50• Validation sample: 10 000 applications, 7 000

approved where we have information on

defaults• We estimate F(40|G)=12%, F(50|G)=14%,

F(40|B)=70%, F(50|B)=76%

• Based on scores assigned to all applications

we estimate B=10%.

• Calculate the total and weighted rate of errors ifl=60% and q=10%. Select the optimal cut-off.

37


38/152


Validation of PD estimates

• If the rating is connected with expectedPDs then we ask how close is ourexperienced rate of default on rating

grades to the expected PDs• Hosmer-Lemeshow Test – one

comperehensive statistics

asymptotically with number of ratinggrades – 2 degrees of freedom

38

22

1 1

2( ) ( )

(1 ) (1 )

N s s s s s

N s

N

s s s s s s s

n PD p n PD d S

PD PD n PD PD

2


39/152


Hosmer-Lemenshow Test

Example

39

H0 is not rejected on 90% probability levelLow p-value would be a problem

rating (s) PD_s p_s N_s

1 30% 35,0% 50 0,595238095

2 10% 8,0% 150 0,666666667

3 5% 7,0% 70 0,589473684

4 1% 1,5% 50 0,126262626

Hos.-Lem. 1,977641072

p-value 0,37201522


40/152


Binomial Test

• Only for one grade, one-sided• If then the probability of

observing or more defaults is at most

• Example: PD=1%, 1000 observations,

13 defaults, is it OK or not?• The issue of correlation can be solved

with a Monte Carlo simulation40

s P PD

, )( 1; ) ( s

s

N

j d

s N j j

s s s s N PD B D j

d N P PD


41/152


Analytical (Expert) Ratings

41

Standard & Poor’s Rating Process


42/152


Key Financial Ratios

42

Category Ratio

Operating performance Return on equity (ROE) = Net income/EquityReturn on assets (ROA) = Net income/Assets

Operating Margin = EBITDA/Sales

Net Profit Margin = Net income/Sales

Effective tax rate

Sales/Sales last year

Productivity = (Sales-Material costs)/Personal costs

Debt service coverage EBITDA/Interest

Capital expenditure/Interest

Financial leverage Leverage = Assets/Equity

Liabilities/Equity

Bank Debt/Assets

Liabilities/Liabilities last year

Liquidity Current ratio = Current assets/Current LiabilitiesQuick ratio = Quick Assets/Current Liabilities

Inventory/Sales

Receivables Aging of receivable (30,60,90,90+ past due)

Average collection period


43/152


External Agencies’ Rating Symbols

43

Rating Interpretation

Investment Grade Ratings

AAA/Aaa Highest quality; extremely strong; highly unlikely to be affected by

foreseeable events.

AA/Aa Very High quality; capacity for repayments is not significantly

vulnerable to foreseeable events.

A/A Strong payment capacity; more likely to be affected by changes in

economic circumstances.

BBB/Ba Adequate payment capacity; a negative change in environment may

affect capacity for repayment.

Below Investment Grade Ratings

BB/Ba Considered speculative with possibility of developing credit risks.

B/B Considered very speculative with significant credit risk.

CCC/Caa Considered highly speculative with substantial credit risk.

CC/Ca Maybe in default or wildly speculative.

C/C/D In bankruptcy or default.


44/152


Through the Cycle (TTC) or Pointin Time (PIT) Rating

• External agencies do not assign fixed PDs to theirratings

• Observed PDs for the rating classes depend on the

cycle• Should the rating express the actual expected PD

conditional on current economic conditions (PIT) orshould it be independent on the cycle (TTC)

• External ratings are hybrid - somewhere betweenTTC and PIT

44


45/152


The Risk of External Rating

Agencies• The rating agencies have become extremely

influential – cover $34 trillion of securities

• Ratings are paid by issuers – conflict of interest• Certain new products (CDO) have become

extremely complex to analyze

• Many investors started to rely almost completely

on the ratings• Systematic failure of rating agencies => global

financial crisis

45


46/152


Internal Analytical Ratings

• Asset based – asset valuation

• Unsecured general corporate lending orproject lending – ability to generatecash flow

• Depends on internal analytical expertise

• Retail, Small Business, and SME alsousually depend at least partially onexpert decisions

46


47/152


Automated Rating Systems

• Econometric models – discriminantanalysis, linear, probit, and logitregressions

• Shadow rating, try to mimic analytical(external) rating systems

• Neural networks

• Rule-based or expert systems• Structural models, e.g. KMV based on

equity prices47


48/152


Altman’s Z-score

• Altman (1968), maximization of thediscrimination power by on a datasetof 33 bankrupt + 33 non-bankrupt companies

48

1

i i

k

i

x z

Variable Ratio Bankrupt

group mean

Non-bankrupt

group mean

F-ratio

x1 Working capital/Total assets -6.1% 41.4% 32.60

x2 Retained earnings/Total assets -62.6% 35.5% 58.86

x3 EBIT/Total assets -31.8% 15.4% 25.56

x4 Market value of equity/Book

value of liabilities

40.1% 247.7% 33.26

x5 Sales/Total assets 1.5 1.9 2.84

1 2 3 4 51.2 1.4 2.3 0.6 0.999 Z x x x x x


49/152


Altman’s Z-score

• Maximization of the discrimination function – maximization between bad/good groupvariance and minimization within groupvariance

• Importance of selection of variables• In fact the approach is similar to the least

squares linear regression

49

1 2

2 2

1 1

2 2

1, 1

2 2

2

1

,

1

2

( ) ( )

( () ) N N

i

i

i

i

N z z N z z

z z z z

'·i i i

y uβ x


50/152


Logistic Regression

• Latent credit score• Default iff , i.e.

• If the distribution is assumed to benormal than we get the Probit model

• If the residuals follow the logisticdistribution than we get the Logit model

50

* '· i ii y uβ x

*0

i y

Pr[ 0]0 | ] r ( )P [i i i i i i

p u y F x β ·x β ·x

) 1

( )1 1

( x

x x

e F x

e e x


51/152


Logit or Probit?

51

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

-15 -10 -5 0 5 10 15

PDF

Probit

Logit

0

0,2

0,4

0,6

0,8

1

1,2

-15 -10 -5 0 5 10 15

CDF

Probit

Logit

• The models are similar, the tails are heavier forLogit

• Logit model is numerically more efficient• Its coefficients naturally interpreted – impact on

odds or log odds1

ii

i

pe

p

β ·xG/B

1ln ln i

i

i

p

pβ ·x


52/152


Maximum LikelihoodEstimation

• The coefficients could be theoreticallyestimated by splitting the sample intopools with similar characteristics and byOLS

• …or by maximizing the total likelihood

or log-likelihood

52

1( ) ( ) (1 ·('· )' )i i

y y

i i

i

L F F b xbxb

ln ( ) ( ) ln ( )'· 1 '( ) ln(1 ( )· )i i i

i

i y y L l F F b b b bx x


53/152


Estimators’ properties

• Hence

• The solution is found by the Newton’s

algorithm in a few iterations• The estimated parameters are

asymptotic normal and the s.e. are

reported by most statistical packagesallowing to test the null hypothesis andobtain confidence intervals

53

, for( ( )) 0 0,...,

i i i

i

j

j

l y x j k

b·' xb

b


54/152


Selection of Variables

• In-sample likelihood is maximized if allpossible variables are used

• But insignificant coefficients where we are not

sure even with the sign can cause systematicout-sample errors

• The goal is to have all coefficients significant

at least on the 90% probability level and atthe same time maximize the Gini coefficient

54


55/152



• Generally it is useful to performunivariate analysis to preselect thevariables and make appropriatetransformations

55Note that the charts use log B/G odds


56/152



• Correlated variables should beeliminated

• In the backward selection procedure

variables with low significance areeliminated

• In the forward selection procedure

variables are added maximizing thestatistic

56

likelihood of the restricted model

likelihood of the larger model2lnG

2

( )k l

C


57/152


Categorical Variables• Some categorical values need to be

merged

57

% bad

0,00%

2,00%

4,00%

6,00%

8,00%

10,00%

12,00%

Unmarried Married W idowed Divorced Separated

% bad

Marital Status of borrower

No. of good loans No. of bad loans Total % bad

Unmarried 1200 86 1286 6,69%

Married 900 40 940 4,26%

Widowed 100 6 106 5,66%

Divorced 800 100 900 11,11%

Separated 60 6 66 9,09%

Total 1860 152 2012 7,55%


58/152


Categorical Variables

• Discriminatory power can be measuredlooking on significance of the regressioncoefficients (of the dummy variables)

• Or using the Weight of Evidence (WoE)

• Interpretation follows from the Bayes

Theorem

58

ln Pr[ | ] ln Pr[ | ]WoE c Good c Bad

Pr[ | ] Pr[ | ] Pr[ ]

Pr[ | ] Pr[ | ] Pr[ ]·

Good c c Good Good

Bad c c Bad Bad


59/152


Weight of Evidence

• An overall discriminatory power of the

variable is measured by the informationvalue

59

Marital Status of borrower

Pr[c|Good] Pr[c|Bad] WoE

Unmarried 0,64516129 0,565789474 0,13127829

Married or widowed 0,537634409 0,302631579 0,57466264

Divorced or separated 0,462365591 0,697368421 -0,410958IV 0,24204342

1

· Pr[ | ] [) r | ]( PC

c

c Good c Ba IV Wo d E c


60/152


Weight of Evidence

• WoE can be used to build a naïve Bayesscore (based on the independence ofcategorical variables)

• In practice the variables are often correlated

• WoE can be used to replace categorical bycontinuous variables

1 2Pr[ | ] Pr[ | ] Pr[ | ]Good c Good c Good c

1 2Pr[ | ] Pr[ | ] Pr[ | ] Bad c Bad c Bad c

1( ( ))) (

nW WoE coE WoE cc

1( ) ( ( ))

Pop n s s WoE c WoE cc


61/152


Case Study – Scoring Function development

61

Scoring Dataset observations: 10,936

31 Jul 2009 14:56 variables: 21

variable name variable description # of categorical values

id_deal Account Number N/A

def Default (90 days, 100 CZK) 2

mesprij Monthly Income N/A

pocvyz Number of Dependents 7

ostprij Other Income N/A

k1pohlavi Sex 2

k2vek Age 15

k3stav Marital Status 6

k4vzdel Education 8

k5stabil Employment Stability 10

k6platce Employer Type 9

k7forby Type of Housing 6

k8forspl Type of Repayment 6

k27kk Credit Card 2

k28soczar Social Status 10

k29bydtel Home Phone Lines 3

k30zamtel Employment Phone Lines 4


62/152


Case Study – In Sample x Out Sample

62

Full Model GINI

Run In-sample Out-sample

1 71.4% 47.9%

2 69.2% 54.7%

3 71.1% 38.8%

4 70.9% 38.4%

5 69.2% 43.6%

Restricted Model GINI

Run In-sample Out-sample1 61.9% 57.7%

2 61.3% 58.7%

3 61.6% 58.0%

4 60.0% 62.7%

5 60.7% 59.5%

Full Model

Final Model – coarse classification, selection of variables


63/152


Combination of Qualitative

(Expert) and Quantitative Assesment

63


64/152


Rating and PD Calibration

64

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

3 0 J U N 2 0 0 5

3 0 S E P 2 0 0 5

3 1 D E C 2 0 0 5

3 1 M A R 2 0 0 6

3 0 J U N 2 0 0 6

3 0 S E P 2 0 0 6

3 1 D E C 2 0 0 6

3 1 M A R 2 0 0 7

3 0 J U N 2 0 0 7

3 0 S E P 2 0 0 7

3 1 D E C 2 0 0 7

3 1 M A R 2 0 0 8

3 0 J U N 2 0 0 8

3 0 S E P 2 0 0 8

3 1 D E C 2 0 0 8

3 1 M A R 2 0 0 9

3 0 J U N 2 0 0 9

3 0 S E P 2 0 0 9

Calibration date

Actual PD

Predicted PD no calibration

Predicted PD after calibration


65/152


Rating and PD Calibration

• Run a regression of ln(G/B) on thescore as the only explanatory variable

65

-2,0

-1,0

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

200 300 400 500 600 700 800 900

L n ( G B

O d d s )

NWU score

R2=99.7%


66/152


TTC and PIT Rating/PD

• Point in Time (PIT) rating/PD prediction isbased on all available information includingmacroeconomic situation – desirable from the

business point of view• Through the Cycle (TTC) rating/PD should be

on the other hand independent on the cycle – cannot use explanatory variables that are

correlated with the cycle – desirable from theregulatory point of view

66


67/152


Shadow Rating• Not enough defaults for logistic regression

but external ratings – e.g. Largecorporations, municipalities, etc.

• Regression is run on PDs or log odds

obtained from the ratings and withavailable explanatory variables

• The developed rating mimics the external

agency process• Useful if there is insufficient internal

expertise67

S i l A l i


68/152


Survival Analysis

68

So far we have implicitlyassumed that the probabilityof default does not depend onthe loan age – empirically itis not the case


69/152


Survival Analysis

• - time of exit

• - probability density andcumulative distribution functions

• - survival function

• - hazard function

• The survival function can be expressedin terms of the hazard function that ismodeled primarily

69

T

( ), ( ) f t F t

( ) 1 ( )S t F t ( )

( )( )

f t t

S t


70/152


Parametric Hazard Models

70uncensored all observationsobservations

) ln ( | ) nn l )l (( | L t S t θ θ θ

Parameters estimated by MLE


71/152


Nonparametric Hazard Models

• Cox regression

• Baseline hazard function and the risklevel function are estimated separately

71

0( , ) ( )exp( ' ),t t x x β

( , ) ))

( , ) )

exp( '(

exp( 'i i

j j

j

i i i

i

j

A j A

L t

t x

β

β

xxβ

x

( )

0 0

1

( )exp( ) exp ( )exp' ( )' ) ([ ] ( ),in

dN t

t i i i

i

L t t Y t x β x β

1

ln ln K

i

i L L

Cox Regression and an AFT Parametric


72/152


Cox Regression and an AFT ParametricModel Examples

72


73/152


Markov Chain Models

• Rating transitions driven by a migration matrixwith default as a persistent state

• Allows to estimate PDs for longer timehorizons

73


74/152


Expected Loss, PD, LGD, and

EAD• Loan interest rate = internal cost of

funds + administrative cost + cost of risk

+ profit margin• Credit risk cost = annualized expected

loss

• ELabs = PD*LGD*EAD

74

1

EL RP

PD

Approval Parametrization


75/152


75

Cut Off:

Changed

Increase of

Average

Default Rate

from 5% to 6%

Increase of

Marginal

Default Rate

from 12% to 14%

1

2

3

Accepted Applications

Approval Parametrization

YESNO

M i l Ri k


76/152


76

Marginal Default Rate According to Average Default Rate

0%

5%

10%

15%

20%

25%

30%

35%

1 %

2 %

3 %

4 %

5 %

6 %

7 %

8 %

9 %

1 0 %

Average Default Rate (in %)

M a r g i n a l D e f a u l t R a t e ( i n %

11.8%

13.7%

18.9%

Marginal Risk


77/152


Cut-off Optimization

77

Net Income According to Average Default Rate of Accepted Applicants

80

90

100

110

120

130

140

150

160

1 %

2 %

3 %

4 %

5 %

6 %

7 %

8 %

9 %

1 0 %

Average Default Rate (in %)

N e t I n c o m e

Income Line

Optimal Point

Net Income is Maximal

4.2%

Recovery Rates and LGD


78/152


Recovery Rates and LGDEstimation

• LGD = 1 – RR where RR is defined as themarket value of the bad loan immediatelyafter default divided by EAD or rather

• We must distinguish realized (expost) and

expected (ex ante) LGD• LGD is closely related to provisions and

write-offs78

1

1.

(1 )

i

i

t

t

n

i

CF RR

EAD r


79/152


Recovery Rate Distribution

79

LGD R i


80/152


LGD Regression

• Pool level estimations – basic approach• Advanced: account level regression

requires a sufficient number of

observations

• Link function should transform a normal

distribution to an appropriate LGDdistribution, e.g. Logit, beta or mixedbeta distributions

80

( ' )i i i

LGD f β x

RR P li lit


81/152


RR Pro-cyclicality

81

Defaulted bond and bank loan recovery index, U.S. obligors.Shaded regions indicate recession periods.(Source: Schuermann, 2002)

P i i d W it ff


82/152


Provisions and Write-offs• LGD is an expectation of loss on a non

defaulted account (regulatory and riskmanagement purpose)

• BEEL is an expectation of loss on a defaultedaccount

• Provisions reduce on balance sheet value ofimpaired receivables (IFRS) – created andreleased – immediate P/L impact

• Write-offs (with/without abandoning) – essentially terminal losses, but positiverecovery still possible

82

E pos re at Defa lt


83/152


Exposure at Default• What will be the exposure of a loan in case of

default within a time horizon?

• Relatively simple for ordinary loans butdifficult for lines of credits, credit cards,

overdrafts, etc.• Conversion factor – CAD mandatory

parameter

• Ex post calculation requires a reference pointobservation

83

Current Exposure ·Undrawn Limit EA F D C

( ) ( )( ,

( )) () d r

r

r r

Ex t Ex t CF CF a t

L t Ex t

Conversion Factor Dataset


84/152


Conversion Factor Dataset• The CF dataset may be constructed based on a

fixed horizon, cohort, or variable time horizonapproach

84

Conversion Factor Estimation


85/152


Conversion Factor Estimation• For very small undrawn amounts CF values

do not give too much sense, and so poolbased default weighted mean does notbehave very well

• A weighted or regression based approachrecommended

85

( )

1

( ) ( )| ( ) | o RDS l CF l CF o RDS l

( ( ) ( ))( )

( ( ) ( ))

EAD o Ex oCF l

L o Ex o( ) ( ) ( ( ) ( )) EAD o Ex o L o E o

2

( ( ) ( ))( )

( ( ) ( )

·( ( ) ( ))

)

L o EAD o Ex ol

L o Ex

Ex o

oCF


86/152


Basel II Requirements

86

Total CapitalCapital Ratio

Credit Risk Market Risk Operational Risk RWA


87/152


Stadardized Approach (STD)

87

Rating Sovereign RiskWeights

Bank RiskWeights

Rating Corporate RiskWeights

AAA to AA- 0% 20% AAA to AA- 20%

A+ to A- 20% 50% A+ to A- 50%

BBB+ to BBB- 50% 100% BBB+ to BBB- 100%

BB+ to B- 100% 100% BB+ to BB- 100%

Below B- 150% 150% Below BB- 150%

Unrated 100% 100% Unrated 100%

Table 1. Risk Weights for Sovereigns, Banks, and Corporates (Source: BCBC, 2006a)

I t l R ti B d A h (IRB)


88/152


Internal Rating Based Approach (IRB)

88

( ( ) · ·

RWA

)

· , ·12.5

LGD MA

EAD w

K UDR PD

w

P

K

D

0%

50%

100%

150%

200%

250%

0,00% 1,00% 2,00% 3,00% 4,00% 5,00% 6,00% 7,00% 8,00% 9,00% 10,00%

w

PD

Risk Weight as a Function of PD

w

Basel II IRB Risk weights for corporates (LGD = 45%, M = 3)

F d ti Ad d


89/152


Foundation versus Advanced

IRB• Foundation approach uses regulatory LGDand CF parameters (table give)

• Advanced approach – own estimates of LGD,CF

• For retail segments only STD or IRBA optionspossible

• In IRBA BEEL and overall EL must becompared with provisions – negativedifferences => additional capital requirement

89


90/152


Content

Introduction



Portfolio Credit Risk Modeling


90


91/152



Markowitz Portfolio Theory


92/152


Markowitz Portfolio Theory• Can we apply the theory to a loan

portfolio?

• Yes, but with economic capitalmeasuring the risk

92

Expected and Unexpected


93/152


Expected and Unexpected

Loss• Let X denote the loss on the portfolio ina fixed time horizon T

93

[ ] EL E X ( ) Pr[ ]

X x F x X

sup{ | ( ) } X

x F q x

[ ] X

UL q E X

·rel N UL VaR qNormality =>

Credit VaR


94/152


• Credit loss of a credit portfolio can be measured indifferent ways

– Market value based

– Accounting provisions

– Number of defaults x LGD

• Credit VaR at an appropriate probability level shouldbe compared with the available capital => EconomicCapital

• Economic Capital can be allocated to individualtransactions, implemented e.g. by Bankers Trust

• Many different Credit VaR models! (CreditMetrics,CreditRisk+, CreditPortfolio View, KMV portfolioManager, Vasicek Model – Basel II)

94

C ditM t i


95/152


CreditMetrics

• Well known methodology by JP Morgan• Monte Carlo simulation of rating migration

• Today/future bond/loan values determined byratings

• Historical rating migration probabilities

• Rating migration correlations modeled asasset correlations – structural approach

• Asset return decomposed into a systematic(macroeconomic) and idiosyncratic (specific)parts

95


96/152

R ti Mi ti


97/152


Rating Migrations

97

R R t


98/152


Recovery Rates

• CreditMetrics models the recovery rates asdeterministic or independent on the rate ofdefaults but a number of studies show anegative correlation

98


99/152

Credit Migration (Merton’s) Structural


100/152


Model

• In order to capture migrations parameterizecredit migrations by a standard normalvariable which can be interpreted as anormalized change of assets

• The thresholds calibrated to historicalprobabilities

100


101/152

Merton’s Structural Model


102/152


• Creditors‘ payoff at maturity = risk free debt –

European put option

• Shareholders’ payoff = European call option

• Geometric Brownian Motion

• The rating or its change is determined by the

distance to the default threshold or its change,which can be expressed in the standardizedform:

102

( ) max( ( ),0) D T D D A T

( ) max( ( ) ,0) E T A T D

( ) ( )) ( )( A t dt A t A t W t d d 2(ln ) ( / 2)d W A dt d

21 (1)ln ( / 2) (0,1)

(0)

Ar N

A

Rating Migration and Asset


103/152


Rating Migration and Asset

Correlation• Asset returns decomposed into anumber of systematic and anidiosyncratic factors

• The decomposition can be based onhistorical stock price data or on anexpert analysis

103

1

1

( )i j j

k

k i

j

r w r I w

Asset Correlation Example


104/152


Asset Correlation - Example

• 90% explained by one systematic factor

• 70% explained by one systematic factor

• Correlation:• More factors – index correlations must be

taken into account104

2

1 1 11 0.90.9 ( ) 0.9 ( ) 0.44r r I r I

2

12 20.7 ( ) 0.7 ( ) 01 0.7 .71r r I r I

1 2( ·0.7· ( ), ( ) 0.6, 0. 3) 9r r I r I r

Portfolio Simulation


105/152


Portfolio Simulation

105

0

5

10

15

20

25

30

135 140 145 150 155 160 165 170 175

P r o b a b i l i t y ( % )

Market Value

Market Value Probability Density Function

1% quant ile 5% quant i le

mean

Unexpected Loss

nominalvalue

exp. loss

median

0.1% percentile simulation

Sample systematic factors(Cholesky decomposition)and the specific factors – determine ratings, calculateimplied market values

Recovery Rates Distribution


106/152


Recovery Rates Distribution

106

Default Rate and Recovery


107/152


e au t ate a d eco e yRate Correlations

107

CreditRisk+


108/152


CreditRisk+

• Credit Suisse (1997)• Analytic (generating function) “actuarial”

approach

• Can be also implemented/described in asimulation framework:

1. Starting from an initial PD0 simulate anew portfolio PD

12. Simulate defaults as independent events

with probability PD1

108

Credit Risk+


109/152


The Full Model

• Exposures are adjusted by fixed LGDs• The portfolio needs to be divided into

size bands (discrete treatment)

• The full model allows a scale of ratingsand PDs – simulating overall meannumber of defaults with a Gamma

distribution• The portfolio can be divided into

independent sector portfolios109


110/152

Credit Risk+ Details


111/152



• For more rating grades just putas

• The Poisson distribution can beanalytically combined with a Gammadistribution generating the number of

defaults (i.e. overall PD)

111

1 2 1 21) 1) ( )( 1( ( ) z z z e e e

1

R

r r

1 1

0

where1

( ) , ( )( )

.

x

x f x e e x x dx



112/152



• Gamma distribution parameters andcan be calibrated to and

112

-0,005

00,005

0,01

0,015

0,02

0,025

0,03

0,035

0,04

0,045

0 100 200 300 400 500

f ( x )

x

sigma=10

sigma=50

sigma=90

0 0· D N P 0 · PD N



113/152



• The distributions of defaults can beexpressed as

• And the corresponding generatingfunction

113

0

Pr[ ] Pr[ | ] ( ) x f x d X n X n x

( 1)

1

0

1( ) ( )

(1 )

x z G z e f x dx

z

Pr[ ] (1

whe11

) , re .n

n X q

nqn q



114/152



• To obtain the loss distribution weassume that the exposures aremultiples of L and

• Individual (RR adjusted) exposureshave size where hence

114

0

portfolio loss · ]( ) Pr[ · n

n

nG z L z

· j

m L 1,..., j k

( 1

0

)

( ) !

m j j j j

n

z n j

j

n

m

G z e e z n

(

1

1) ( ( ) 1

1

)( ) ( )

mm j j j j j

k k z z

j j

P z

jG z G z e e e

1

k

j

j

1

( ) jk

m j

j

P z z

where



115/152



• Finally the generating function needs tobe combined with the Gammadistribution

• Here implicitly adjusts to

• For more sector we need to assumeindependence (!?) to keep the solutionanalytical

115

( ( ) 1)

0

1

1( ) ( )

(1 ( )) x P z G z e f x dx

P z

0,

0

j

j

final

1

( ) ( )S

s

sG z G z

CreditPortfolio View


116/152



• Wilson (1997) and McKinsey• Ties rates of default to macroeconomic

factors

116



117/152


CreditPortfolio View• Macroeconomic model

• Simulate based oninformation given at

• Adjust rating transition matrix toand simulate rating migrations

117

, ,( )

j t j t P Y D

, , ,' j t j j t j t Y Xβ

, , , ,0 , ,0 , 1, , ,0 , 2, , , j t i j i j i j t i j i j t i j t i X X X e

, , 1, ,... j t j t PD PD

1t

, , 0/ j t j t M M PD PD


118/152

An Overview of the KMV


119/152


Model1. Based on equity data determine initial

asset values and volatilities

2. Estimate asset return correlations

3. Simulate future asset values for a timehorizon H and determine the portfoliovalue distribution

• Under certain assumptions there is ananalytical solution 119

,( ) ( ( ) ) A P H f A H


120/152

Distance to Default and EDF


121/152



• If just D is payable at T then the riskneutral probability of default would be

• But since the capital structure isgenerally complex we use it or infact , the distance to default in

one year horizon as a “score” setting

121

2Pr[ )(]T T Q V K d

/ 2 DPT STD LTD

2 DD d



122/152



122

PD callibration


123/152


PD callibration

• The distance to default is calibrated toPDs based on historical defaultobservations

123

EDF of a firm versus S&P


124/152


rating

124

KMV versus S&P rating transitionmatrix


125/152


matrix

125

Risk neutral probabilities


126/152


Risk neutral probabilities

• The calibrated PDs are historical, butwe need risk neutral in order to valuethe loans

126

1

1 1

discounted cash f [ low] cash f [ ]

(1 ) (1 )

lowi i

i i i i

nr T

Q Q i

i

r T r

i

T

i i

n

i

n

i

PV E e

e CF LGD e CF LGQ D

E

Time CFi Qi e^-rTi PV1 PV2 PV

1 5 000,00 3% 0,9704 1 940,89 2 824,00 4 764,892 5 000,00 6,50% 0,9418 1 883,53 2 641,65 4 525,18

3 105 000,00 9,90% 0,9139 38 385,11 51 877,48 90 262,59

Total 42 209,53 57 343,12 99 552,65

Example

Adjustment of the real world


127/152


EDFs

127

* * *dA d A d t A W

*Pr[ ( ) ]T T EDF A T K

dA r dt A AdW Pr[ ( ) ]T T Q A T K

02

2

2

ln / ( /( ),

2)T T

A K T EDF N d d

T

2*

2

0*

2)

ln / ( / 2),( T

T

A K d

T

r T Q N d

22

*( ) /T d r d 1 )(

T T Q N N EDF r T

r M r CAPM:

Vasicek’s Model and Basel II


128/152


Vasicek s Model and Basel II

• Relatively simple, analytic, asymptoticportfolio with constant asset returncorrelation

• Let be the time to default of anobligor j with the probability distributionthen corresponds to thestandardized asset return and iff

• Hence128

j

T

1( ( ))

j j j X N Q T

jQ

1 j

T

1( )

j X D N P

(1) j PD Q

Vasicek’s Formula


129/152


• Let us decompose the X to a systematic

and an idiosyncratic part and expressthe PD conditional on systematicvariable

129

1 j j X M Z

1

1

1 1

1( ) Pr 1| Pr ( ) |

Pr 1 ( ) |

( ) ( )Pr 1 1

j j

j

j

PD m T M m X N PD M m

M Z N PD M m

N PD m N PD m Z N

Vasicek’s Formula


130/152


Vasicek s Formula

• So the unexpected default rate on agiven probability level is

• If multiplied by a constant LGD andEAD we get a “simple” estimate of

unexpected loss attributable to a singlereceivable

130

1 1( ) ( )( )

1

N PD N UDR PD N

1

Basel II Capital Formula


131/152


Basel II Capital Formula

• Correlation is given by the regulation

and depends on the exposure class,e.g. for corporates

• Similarly the maturity adjustement

131

99.9% · ·· , ·

( )1 5

( )2.

LGD MA RWA EAD w w K UDR PD P

K D

50 50 50

50 50

10.12 0.24

1 1

PD PDe ee

e e

21 ( 2.5), (0.1182 0.05478 ln )

1·

1.5

M b MA b PD

b

Basel II Problems


132/152


Basel II Problems

• Vague modeling of unexpected LGD• The issue of PDxLGDxEAD correlation

• Sensitivity to the definition of default

• Procyclicality!!!

132

Basel III ?!


133/152


• Recent BCBS proposals reacting to the crisis• Principles for Sound Liquidity Risk

Management and Supervision (9/2008)

• International framework for liquidity riskmeasurement, standards and monitoring -consultative document (12/2009)

• Consultative proposals to strengthen theresilience of the banking sector announcedby the Basel Committee (12/2009)

133

Basel III ?!


134/152


• Comments could be sent by 16/4/2010 (viz

www.bis.org)• Final decision approved 12/2010

• Key elements

– Stronger Tier 1 capital

– Higher capital requirements on counterparty risk

– Penalization for high leverage and complexmodels dependence

– Counter-cyclical capital requirement – Provisioning based on expected losses

– Minimum liquidity requirements

134

C t t

http://www.bis.org/http://www.bis.org/http://www.bis.org/


135/152


Content

Introduction




Credit Derivatives

135

Credit Derivatives


136/152


• Payoff depends on creditworthiness ofone or more subjects

• Single name or multi-name

• Credit Default Swaps, Total ReturnSwaps, Asset Backed Securities,Collateralized Debt Obligations

• Banks – typical buyers of creditprotection, insurance companies – sellers

136

Credit Default Swaps


137/152


p

137

-

10 000

20 000

30 000

40 000

50 000

60 000

70 000

B i l l i o n U

S D

Global CDS Outstanding Notional Amount

CDS Notional

Credit Default SwapsCDS d ll id i t l


138/152


• CDS spread usually paid in arrears quarterly

until default• Notional, maturity, definition of default

• Reference entity (single name)

• Physical settlement – protection buyer hasthe right to sell bonds (CTD)

• Cash settlement – calculation agent, or binary

• Can be used to hedge corresponding bonds

138

Valuation of CDS


139/152


• Requires risk neutral probabilities ofdefault for all relevant maturities

• Market value of a CDS position is then

based on the general formula

• Market equilibrium CDS spread is thespread that makes MV =0

139

1

discounted cash flow] cash flo[ w[ ]i in

rT

Q Q i

i

MV E e E

Estimating Default


140/152


Probabilities

140

Rating systems

Bond prices

Historical PDs

Risk neutral PDs

CDS Spreads

Credit Indices


141/152


• In principle averages of single name CDSspreads for a list of companies

• In practice traded multi-name CDS

• CDX NA IG – 125 investment grade

companies in N.America• iTraxx Europe - 125 investment grade

European companies

• Standardized payment dates, maturities(3,5,7,10), and even coupons – market valueinitial settlement

141

CDS Forwards and Options


142/152


p

• Defined similarly to forwards andoptions on other assets or contracts,e.g. IRS

• Valuation of forwards can be done justwith the term structure of risk neutralprobabilities

• But valuation of options requires astochastic modeling of probabilities ofdefault (or intensities – hazard rates)

142

Basket CDS and Total ReturnSwaps


143/152


Swaps• Many different types of basket CDS: add-up,

first-to-default, k-th to default … requirescredit correlation modeling

• The idea of Total Return Swap (compare to Asset Swaps) is to swap total return of a bond(or portfolio) including credit losses for Libor +spread on the principal (the spread covers

the receivers risk of default!)

143

Asset Backed Securities(CDO )


144/152


(CDO,..)

144

• Allow to create AAA bonds from aportfolio of poor assets

CDO marketGl b l CDO I


145/152


145

0

100

200

300

400

500

600

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010E

B i l l i o n U S D

Global CDO Issuance

Total

Year

High Yield

Bonds

High Yield

Loans

Investment

Grade Bonds

Mixed

Collateral Other

Other

Swaps

Structured

Finance Total

2000 11 321 22 715 29 892 2 090 932 1 038 67 988

2001 13 434 27 368 31 959 2 194 2 705 794 78 454

2002 2 401 30 388 21 453 1 915 9 418 17 499 83 074

2003 10 091 22 584 11 770 22 6 947 110 35 106 86 630

2004 8 019 32 192 11 606 1 095 14 873 6 775 83 262 157 821

2005 1 413 69 441 3 878 893 15 811 2 257 157 572 251 265 2006 941 171 906 24 865 20 14 447 762 307 705 520 645

2007 2 151 138 827 78 571 1 722 1 147 259 184 481 601

2008 27 489 15 955 18 442 61 887

2009 2 033 1 972 331 4 336

Cash versus Synthetic CDOs• Synthetic CDOs are created using CDS


146/152


• Synthetic CDOs are created using CDS

146

Single Tranche Trading


147/152


• Synthetic CDO tranches based on CDX

or iTraxx

147

John Hull, Option, Futures, and Other Derivatives, 7th Edition

Valuation of CDOs


148/152


Valuation of CDOs

• Sources of uncertainty: times of defaultof individual obligor and the recoveryrates (assumed deterministic in a

simplified approach)• Everything else depends on the

Waterfall rules (but in practice often

very complex to implement precisely)

148

Valuation of CDOs


149/152


• Monte Carlo simulation approach:

• In one run simulate the times to default ofindividual obligors in the portfolio using riskneutral probabilities and appropriate

correlation structure• Generate the overall cash flow (interest and

principal payments) and the cash flows toindividual tranches

• Calculate for each tranche the mean(expected value) of the discounted cash flows

149

Gaussian Copula of Time to


150/152


Default• Guassian Copula is the approach when

correlation is modeled on the standardnormal transformation of the time to

default

• The single factor approach can be used

to obtain an analytical valuation

• Generally used also in simulations150

1( ( )) j j j

X N Q T

1 j j X M Z

Implied Correlation


151/152


• Correlations implied by market quotesbased on the standard one factor model(similarly to implied volatility)

151John Hull, Option, Futures, and Other Derivatives, 7th Edition


152/152

witzany creditriskman fg

Documents