new introduction to credit risk - higher school of economics. cech... · 2014. 3. 26. ·...

Introduction to Credit Risk

Guest lecture at the Higher School of Economics

Christian Cech, University of Applied Sciences bfi Vienna

April 2014

Christian Cech, „Introduction to Credit Risk“

2

FH bfi Vienna

• University of Applied Sciences bfi Vienna, Austria– 1,800 students

– 7 Bachelor‘s and6 Master‘s programmes

• ARIMAQuantitative Asset and Risk Management

– Student Exchange cooperationwith Higher School of Economics

• Christian Cech

– Deputy Director of Arima

– Researcher and lecturer at the UAS bfi Vienna

3

Introduction

• Introduction

– Definition of Credit Risk

– Probability of Default (PD)

– Loss Given Default (LGD)

– Exposure at Default (EAD)

– Expected Loss (EL)

– Value-at-Risk (VaR)

– Unexpected Loss (UL)


4

Introduction

• Credit Risk is the risk that a borrower of a loan is unable or unwilling to meet her contractual obligations.

• It is hence the risk of loss of principal and interest payments.

• If a borrower does not meet his contractual obligations this is called a default.

• Under European Banking regulations (“Basel II” and starting 2014 “Basel III”), the definition of a default is quite broad:Every loan that is 90 days past due is considered a default.

• The probability that an obligor defaults within a specified time horizon (Basel II/III: one year) is calledProbability of Default (PD).


5

Introduction

• Another important parameter in the context of Credit Risk is the Loss Given Default (LGD) .

• The LGD is the percentage of loss that is expected in the case of default of a borrower.

• Example: Borrower A owes us EUR 300,000. Because collaterals have been provided, we expect that in the case of default of borrower A, we will lose only EUR 120,000 of the EUR 300,000. Compute the LGD.

120,000300,000

40%

• Finally, the parameter Exposure at Default (EAD) is the expected exposure (amount owed by the customer) at the time of default.


6

Introduction

• The three parameters PD, LGD and EAD can be used to calculate the Expected Loss (EL):

∙ ∙

• Example: Compute the Expected Loss for borrower B with following parameters:

1%, 45%, 1,000,000!

0.01 ∙ 0.45 ∙ 1,000,000 4,500


7

Introduction

• In Risk Management we are often not interested in the Expected Loss but we want to examine more adverse scenarios.

• The Value-at-Risk (VaR) is the worst loss that is expected over a given time horizon (e.g. 1 year) for a given confidence level(e.g. 99.9%).

• Examples:

• VaR(1 year, 99.9%) = 500,000

• VaR(1 year, 99%) = 350,000

probability of 99.9%: The 1Y-loss isless than or equal to 500,000

probability of 0.1%: The 1Y-loss is greater than 500,000

probability of 99%: The 1Y-loss is less than or equal to 350,000

probability of 1%: The 1Y-loss isgreater than 350,000


8

Introduction

• The Unexpected Loss (UL) is defined as

Source:Cech (2004)

prob

abili

ty

Loss distributionloan portfolio A

Loss distributionloan portfolio B

loss


9

Probability of Default

• Probability of Default

– External versus Internal Ratings

– Models for Internal Ratings• Heuristic Models

• Statistical Models (Regression models)

• Causal Models

• Hybrid Forms

– Model Validation


10

External Rating Agencies

• Rating Agencies are companies that assign credit ratings to tocompanies and sovereigns (as well as to specific financial instruments)

• The rating code typically ranges from AAA (best rating) toD (defaulted)

• Important Rating Agencies

– Moody’s: introduced a rating for railway-bonds in 1909. In 1914 they started rating US cities and communities.

– Standard and Poor’s (S&P): established in 1941.

– Fitch: focusses on rating financial institutions.


11


Source:http://en.wikipedia.org/wiki/Credit_rating,

13.09.2012

Investment grade

Speculative grade


12


S&P ratingJanuary 2013


13

Internal Ratings

• Since the implementation of Basel II in 2007, banks are allowed to use internal ratings to compute their minimum capital requirements

• The following models exist

– Heuristic Models: out-dated

– Statistical Models (e.g. regression models): now standard

– Causal Models: not applicable for all obligors

– Hybrid Forms


14

Heuristic Models

• The rating is based on the experience of credit experts and is based on

– subjective practical experience and observations

– conjectured business interrelationships

– business theories related to specific aspects

• The ratings do not undergo statistical validation

• Hence, these models are outdated and will not be accepted as an internal model in Basel II.


15

Statistical Models

• Based on historical data on customer attributes and on historical information on the default or non-default of the customers, these models try to predict the probability of default of existing and new customers.

• The following characteristics are frequently used fornew retail customers (“Application Scorecards”)

– Date of birth ( age)

– Education (school, university etc.)

– Marital status (single, married, divorced, widowed)

– Working since…, working in the current company since…

– Employment relationship (freelancer, employed, temporarily employed etc.)

– Duration of client relationship with the bank

– Current level of debt at bank

– Amount of the applied loan


16

Statistical Models

• The following characteristics are frequently used fornew company clients (“Application Scorecards”)

– Limited versus unlimited liability

– Business sector

– Turnover and profit

– Number of employees

– Foundation date of the company ( age)

• Additional characteristics for existing clients (“Behavioural Scorecards”)

– Overdue amount

– Overdue since

– Maximum overdue amount (historically)

– Number of times overdue, number of days overdue

– Date of the last default


17

Statistical Models: Regression Models

• The most widely used statistical models are Regression Models

• One widely-used model is the well-knownLinear Regression Model

– Here one uses the client’s characteristics (like age, income etc.) as regressors (explanatory variables) and the variable as regressand.

– The estimated probability of default is modelled as

where are the coefficients of the regression, , are the regressors and , are dummy variables (see next slide).

customersdefaultednonfor0

customersdefaultedfor1iY

iininiii DDxXYPiPD 2,11,2,21,101)(


18


• Linear Regression Models

• To incorporate categorical variables (e.g. gender: male or female) into the model, so-called dummy variables , that take on values of either 0 or 1 are used.If a categorical variable has m attributes (e.g. gender: 2 attributes), (m-1) dummy variables are introduced and one reference category has to be defined.Example gender: we choose “male” as reference category. If the customer is male, the dummy variable takes on a value of zero and the associated coefficient does not enter into the estimation of Yi . If the customer is female, the dummy variable takes on a value of 1 and the estimate of Yi changes by the value of the associated coefficient as compared to the reference category, i.e. a male customer with the same characteristics (apart from the gender).

• In practice, one often uses categorical variables, derived from interval variables. E.g. rather than using “age” as regressor, one constructs age-categories (e.g. 18-23 years, 24-27 years etc.).


19


• Developing Linear Regression Models

• When developing a Rating model, we employ only 70% of the data sets available. This sample is called the training sample. The remaining 30% constitute the testing sample that is (later) used for model validation. This ensures that the validation is an out-of-sample test.

• In a first step, univariate analyses are performed and it is examined whether the regressors are statistically significant(p-value ≤ 10%) and whether the results seem reasonable. The explanatory power of the single regressors can be assessed by examining goodness-of-fit measures such as .

• In a further step, multivariate models are constructed from the significant variables.(p-values, F-test and adjusted are examined)


O F U P N total

number of obs. 437 89 106 177 48 857

defaults 113 21 24 43 25 226

% non-defaults 51.35% 10.78% 13.00% 21.24% 3.65% 100%

% defaults 50.00% 9.29% 10.62% 19.03% 11.06% 100%

WE 0.0266 0.1482 0.2019 0.1099 -1.1102 0

20


• In practice measures other than are often used for ordinal ornominal variables (categorical variables).

• To assess the explanatory power of attributes of a variable theWeight of Evidence WE is often used:

• If the absolute value of WE is large, the attribute has a high explanatory power.

iattributeindefaultstotalof%

iattributeindefaults-nontotalof%lniWE

low high0.51350.5


21


• The Information Value IV measures the explanatory power of nominal or ordinal variables

– The IV is used to compare the explanatory power of different nominal or ordinal variables.

– The higher the IV, the higher is the explanatory power of the variable. Compute the IV.

• If the WEs of different attributes do not differ strongly from each other, one should combine these attributes to a new attribute before continuing the calibration of the model.

ii

WEiiIV attributeindefaultstotalof%attributeindefaultsnontotalof%

O F U P N total

number of obs. 437 89 106 177 48 857

defaults 113 21 24 43 25 226

% non-defaults 51.35% 10.78% 13.00% 21.24% 3.65% 100%

% defaults 50.00% 9.29% 10.62% 19.03% 11.06% 100%

WE 0.0266 0.1482 0.2019 0.1099 -1.1102 0

0.0921


22


• Binary Choice Regression Models

• Are used as an alternative to linear regression models and have some advantages (in linear regression models it is, for example, [theoretically] possible that the PD takes values larger than one or less than zero).

• However, these models are more difficult to develop as the parameters are estimated with Maximum Likelihood Estimation (MLE) and hence an optimisation tool is required.Also, the coefficients are more difficult to interpret.

• Two model types are typically used in practice:

– Logit-Models

– Probit-Models

• Both models yield similar results (see next slide).Christian Cech, „Introduction to Credit Risk“

23


• Binary Choice Regression Models


Source: OeNB (2004)

24


• Binary Choice Regression Models: Logit Regression

• Logit Regressions model the PD as

• Example: We have calibrated a simple rating model and estimate the following parameters

What is the estimated PD of a customer aged 35 with annual available income of EUR 20,000?


)...(...

...

221221

221

1

1

1)1(

KKKK

KK

xxxx

xx

ii ee

eYPPD

b1: constant term -4b2: age in years -0.02b3: available income p.a. in EUR -0.00001

‐4

‐0,02 35

‐0,00001 20000

0,00739154

1

1 . ∙ . ∙ 0.74%

25


• Scorecards

• Customer consultants might have difficulties understanding therather complex computational algorithm for the PD.

• To prevent alienation, the regression models are often transformed into more simple scorecards.


Source: OeNB (2004), p. 34

26

Causal Models

• Causal Models: Option Price Models

• Option Price Models (e.g. „Merton‘s model“, Merton (1974)) model the probability of default on the basis of the level of debt(„default threshold“), the current stock price and the volatility ofthe stock returns. Hence, these models are only applicable forlisted stock companies.


Source: Cech (2004)

com

pany

valu

e

default threshold

default

time

27

Causal Models

• Causal Models: Cash Flow Models

• Are often used for specialised lending

• Definition of specialised lending: the source of repaying the debt is the project that is financed by the loan (e.g. office buildings, infrastructure projects).

• In these models the cash-flows that are generated by the financed project are simulated and compared to the interest and redemption payments to estimate the probability of default


28

Hybrid Models

• Hybrid Models

• In practice, the results of statistical and causal models are often examined by credit experts that have additional information (e.g. information on the quality of the management and business plans etc.).

• The experts may change the rating [to some predefined extent] based on the additional information.


29

Model Validation

• Model Validation

• Model validation is done both before and after a rating system is implemented.

• Model validation that is done before implementation is done with the testing sample (roughly 30% of the data sets) that is notused in the process of model calibration (model development).

• If the rating model is already in use it is regularly validated (at least annually) on the basis of new information concerning default/non-default of customers and comparing these to the estimated PD one year ago.


30

Model Validation

• Model Validation: PD per rating class

• One simple method of validation is to examine the (ex-post) PD per rating class. This PD should increase with the rating notch.

• In the above example this is the case.

• If too many rating classes exist, this often leads to a non-monotonic increase of the PD


Source: OeNB (2004), Chart 49

31

Model Validation

• Model Validation: ROC curve

• The ROC curve (Receiver Operator Curve) is one of the most widely used validation methods.

• A “good” rating model has acurve that increases quicklyfrom the origin[(0, 0)-coordinate].


source: Cech 2010

32

Model Validation


• Let us understand the ROC curve on the basis of a simple numerical example.

• We examine a loan portfolio with 15 customers.

• A rating model has estimated the PDs for each of the customers one year ago.

• (In this rating model we do not use rating classes but the model directly assigns PDs to each customer.)

• After one year we observe that 5 of the customers have defaulted while 10 have not defaulted.

• In the context of model validation, these are often called “bad” cases (default) and “good” cases (non-default).

• We rank the customers according to their estimated PD.


33

Model Validation


• In the adjacent table we find the 15 customers ranked according to their estimated PD (column PDi) and information whether they have defaulted or not (column default)

• Remark:– 1 default constitutes 20% of the total of 5 defaults

– 1 non-default constitutes 10% of the total of 10 non-defaults

• To draw the ROC curve we start at the origin at the bottom left corner (next slide).

• We process the customers sequentially. In the case of a default we go a 0.2-step upwards. In the case of a non-default we go a 0.1-step to the right.


# PDi

default (Yi)

1 80% 1

2 65% 1

3 55% 0

4 50% 1

5 45% 0

6 30% 0

7 25% 1

8 20% 0

9 15% 0

10 10% 0

11 5% 1

12 3% 0

13 2% 0

14 0.50% 0

15 0.03% 0

34

Model Validation



# PDi

default (Yi)

1 80% 1

2 65% 1

3 55% 0

4 50% 1

5 45% 0

6 30% 0

7 25% 1

8 20% 0

9 15% 0

10 10% 0

11 5% 1

12 3% 0

13 2% 0

14 0.50% 0

15 0.03% 00

0,2

0,4

0,6

0,8

1

0,2 0,4 0,6 10,1 0,5 0,70,3 0,9

0,8

35

Model Validation

• ROC curve: AUC

• The AUC (area under the curve) is a summarising measure.

• As the name suggests it is calculated as “the area under the curve”;in our example as

• AUC can take values between 1 (“perfect model” in the ROC context) and 0.5 (“random model”).

• The “perfect model” is defined as the model in which those customers with the highest estimated PD default.The ROC curve would go from the origin to the (0,1)-coordinate and then to the (1,1)-coordinate.


8.014.08.03.06.02.04.01.0 AUC

36

Model Validation

• ROC curve for rating classes

• The ROC curve can also be used for rating models that do not directly assign a PD to each customer but rather use rating classes.

• Here, the rating classes are processed sequentially, starting with the worst rating class.

• For each rating class, one goes the percentage of defaulted customers contained in that rating class upwards and the percentage of non-defaulted customers contained in that rating class to the right.

• One the next slide we see the ROC curve for the data-sample presented by OeNB (2004) that was shown some slides before.


37

Model Validation

• ROC curve for rating classes


source: OeNB (2004), Chart 55

38

Model Validation

• CAP curve

• The CAP curve (Cumulative Accuracy Profile; also: Power-Curve) is often used as an alternative to theROC curve.

• It contains the same informationalvalue as the ROC-curve.

• The difference is that the x-axis doesnot display the fraction ofnon-defaults (like in the ROC curve),but the fraction of total obligors.

• This means that for every obligor youalways go “one step to the right” andin the case of a default additionally“one step upwards”.


source: Cech (2010)

39

Model Validation

• CAP curve

• The AR (accuracy ratio, also called power statistic or Gini-coefficient) is a summarising measure for the CAP-curve.

Where A … the area between the examined rating model and the random modeland B … the area between the perfect model and the random model

• AR is 1 for the perfect model and 0 for the random model.

• Engelmann et al. (2003) show that the following relationships hold


B

AAR

12 AUCAR2

1

ARAUC

40

Model Validation

• Typical AUC and AR values

• In practice one can expect the following average AUC- and AR-values for different types of rating models (see OeNB(2004), Chart 60).


Model AUC ARUnivariate models(e.g. single accounting ratios) 0.7 0.4Qualitative systems 0.75 0.5Causal models 0.825 0.65Statistical models 0.825 0.65Hybrid models 0.875 0.75

41

Model Validation

• Non-concavity of ROC curves

• It is a bad sign if the ROC curve is not concave because this indicates that a rating class with a higher rating notch has a lower PD than the neighbouring rating class with a lower rating notch.



42

Model Validation

• Non-concavity of ROC curves

• While developing a model one should always examine the whole ROC curve and not just the summarising measure AUC.

• Below, we see two models with the same AUC

• The “black” model is better because the ROC curve increases more quickly at the beginning. (We ignore the slight non-concavity.)



43

Model Validation

• Brier Score

• One weakness of the ROC- and CAP-curves is that they only examine the ranking of the obligors and that they do not explicitly compare the estimated PD with the actual defaults.

• If we would increase the PD in the above presented example with 15 obligors by – let’s say – 20 percentage points for every single obligor, we would compute the same ROC curve as the ranking of the obligors would not change.

• The Brier score (Brier (1950)) examines explicitly the level of the estimated PD-values

where PDi …estimated PD for customer iYi … default/non-default of customer I

• The (theoretically) optimal value for the Brier score is 0.


n

YPDB

n

i ii

2

1

44

Model Validation

• Brier Score for our example


# PDi default (Yi) PDi-Yi (PDi-Yi)2

1 80% 1 -0.2 0.042 65% 1 -0.35 0.12253 55% 0 0.55 0.30254 50% 1 -0.5 0.255 45% 0 0.45 0.20256 30% 0 0.3 0.097 25% 1 -0.75 0.56258 20% 0 0.2 0.049 15% 0 0.15 0.0225

10 10% 0 0.1 0.0111 5% 1 -0.95 0.902512 3% 0 0.03 0.000913 2% 0 0.02 0.000414 0.50% 0 0.005 0.00002515 0.03% 0 0.0003 0.00000009

sum = 2.54632509B = sum / n = 0.169755006

1697755.0

2

1

n

YPDB

n

i ii

45

Loss Given Default

• Loss Given Default (LGD)

– Basel II/III regulations

– Present-Value models

– Advanced models


46

Loss Given Default

• Basel II/III regulations

• Under Basel II/III, banks can choose to apply the standardised approach (external ratings) or the IRB approach (Internal Ratings based approach).

• Within the IRB approach banks can choose to use the

– FIRB approach (Foundation IRB approach),where the LGD does not have to be estimated but is assumed to be 45% for senior claims and 75% for junior claims.The same LGDs are assumed in the standardised approach.

– AIRB approach (Advanced IRB approach),where the banks estimate the LGD (per customer and product class) themselves.

• Exception: For retail loans the LGD must be estimated in the FIRB approach as well.


47

Loss Given Default

• Present-Value models

• The bank collects all data concerning cash-flows after the default:

– Cash-Inflows (partial or full redemption payments by the customer, sale of collaterals etc.)

– Cost components (legal costs, costs of selling collaterals etc.)

• In a next step the present value for these cash-flows at the time of default is computed.

• The maturity of the interest rate is not the time difference between the default and the time of the payment, as at the time of default we do not know when the (future) payment will occur.

• By convention, one uses the 1 month Euribor interest rate.


48

Loss Given Default


• Calculation of the discount factor with the 1M Euribor:

– For the 1st month after the default:

– For the 2nd month after the default:

– etc.

where Di … discount factor in month iti … days in month iri … 1M Euribor at the beginning of month i

• The so computed discount factors are then used to compute the present values

• [remark: Euribor day count convention: act/360]]


3601

1

11

1 tr

D

3601

1

22

12 tr

DD

49

Loss Given Default


• The LGD for one specific default is then calculated as

• Example– An obligor defaults on June 15, 2010 with an outstanding balance of

EUR 10,000.

– On July 10 we pay lawyer fees of EUR 1,200.

– On September 18 the obligor pays in EUR 5,300 and the default case is closed.

– Relevant 1M Euribor interest rates:


EAD

components costPV -inflows-cashPV 1LGD

06/2010 07/2010 08/2010 09/2010

1M Euribor 0.45% 0.58% 0.64% 0.62%

50

Loss Given Default

• Present-Value models: Example (continued)

• Discount factor for lawyer fees(06/2010: 15 days, 07/2010: 10 days)

• Present value of lawyer fees: 1200·0.99965 = 1199.58

• Discount factor for redemption(06/2010: 15 days, 07/2010: 31 days, 08/2010: 31 days, 09/2010: 18 days)

• Present value of redemption: 5300·0.99845 = 5291.80

•


99965.0

36010

%58.01

1

36015

%45.01

1

99845.0

36018

%62.01

1

36031

%64.01

1

36031

%58.01

1

36015

%45.01

1

%08.59%92.40110000

58.119980.52911

LGD

51

Loss Given Default


• Finally the LGDs per obligor are averaged

– For unsecured loans: per product class (e.g. consumer loans, mortgage loans etc.)

– For secured loans: per credit facility type(classes of collaterals)

• Of course, further segmentation is possible, e.g. mortgage loans in region A versus mortgage loans in region B.


52

Loss Given Default

• Advanced models

• There exist a number of advanced models that are briefly presented here.

• Explicit modelling of obligors’ payments on the one hand side and of the collateral realisation on the other. On the basis of these models one can construct option-based models that also allow for the estimation of confidence intervals for the LGD.

• Regression models(like Moody’s LossCalc)

• Models that do not use the expected LGDbut assume a LGD-distribution and usethis in simulation models.


53

Exposure at Default

• Exposure at Default (EAD)

– EAD and CCF

– Estimation of CCF (Hofmann et al. (2005))


54

Exposure at Default

• Exposure at Default (EAD) and Credit Conversion Factors

– For granted loans, the EAD is the exposure that we expect at the time of default, i.e. the original loan amount minus the capital repaid(= outstandanding balance)(Basel II/III: before write-offs in the standardised approach and after write-offs in the IRB-approach.)

– For non-utilised loan commitments: To calculate the EAD we use Credit Conversion Factors CCF.

EAD = CCF · non-utilised loan commitments

– Under Basel II/III only banks using the advanced IRB-approach (AIRB) estimate the CCFs themselves.


55

Exposure at Default

• CCF-estimation

– The following estimation procedure is presented in Hofmann et al. (2004).

– Step 1: CCF for every single historical default

where balancet … outstanding balance at defaultbalancet-1 … outstanding balance one year prior to defaultlimitt-1 … approved loan amount one year prior to default

– The CCF is calculated with reference to the non-utilised loan commitment one year prior to default, i.e. (limitt-1 – balancet-1).

– The CCF cannot take values less than zero.


0;balancelimit

balancebalancemax

11

1

tt

ttiCCF

56

Exposure at Default

• CCF-estimation

– Step 1: example

balancet = EUR 61,000balancet-1 = EUR 20,000limitt-1 = EUR 70,000

Compute the CCF.


limitt-1

limitt-1 balancet time

balance resp. limit

evolution limitlimit

balance

source: Hofmann et al. 2005

%82

82.0

0;2000070000

2000061000max

iCCF

57

Exposure at Default

• CCF-estimation

– Step 2:

For every year, calculate the average CCF per customer/product class (“segment”).

– Step 3:

Finally, calculate the average CCF over all years per customer/product class.

This is then the CCF that is applied for the customer/product class.


58

References

• Brier, G.W., 1950, „Verification of forecasts expressed in terms of probabilities“ , MonthlyWeather Review, Vol. 78 No. 1

• Cech, C., 2004, Basel II: Die IRB-Formel zur Berechnung der Mindesteigenmittel für Kreditrisiko, in: Wirtschaft und Management, No.1, pp. 53-71

• Cech, C., 2010, „Lecture Notes: Applied Econometrics and Quantitative Methods –Angewandte Ökonometrie“, University of Applied Sciences bfi Vienna

• Engelmann, B., Haydn, E., Tasche, D,, 2003, „Testing Rating Accuracy“, Risk, Vol. 16, No.1, S. 82-86.

• Hofmann, C., Lesko, M., Vorgrimler, S., 2005, „Eigene EAD-Schätzungen für Basel II“, Die Bank 06/2005, S. 48-52

• Merton, R., 1974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, in: The Journal of Finance, Vol. 29, pp. 449-470

• OeNB (Oesterreichische Nationalbank), 2004, Guidelines on Credit Risk Management: Rating Models and Validation, Vienna


59

Introduction to Credit Risk

new introduction to credit risk - higher school of economics. cech... · 2014. 3. 26. ·...

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