The Basel II IRB Approach and

Internal Credit Risk Models

Dr Michael Prinz

Kellogg College

University of Oxford

A thesis submitted in partial fulfilment of the requirements for the MSc in

Mathematical Finance

March 29, 2004

To Judith

Abstract

The following thesis is intended to build a link between the regulatory require-ments given by the Basel II capital accord and an internal Credit risk model.Based on model assumptions for the internal and external rating structure, theregulatory internal ratings based approach and the CreditMetricsTM method-ology are applied for a bond portfolio. The characteristic risk parameters ofthe internal model are extracted from the loss distribution which results fromMonte Carlo simulations of correlated scenarios. On the level of single positionsthe relationship between these results and the risk values following from theanalytic formulas of the capital requirement can be described with the use of apower law. This enables to connect both approaches and to account for portfolioeffects within the regulatory framework. Furthermore the introduced methodrepresents a framework of potential practical relevance that could be used forthe internal estimation of the probability of default.

Acknowledgements

I want to thank Dr William Shaw for supervision of the thesis.

Many thanks also to my colleague Dr Peter Gloßner for a lot of fruitful discus-sions. He gave me a much deeper insight in the field of credit risk. Some resultspresented in this work are due to his ideas.

Much gratitude is owed to my employer d-fine, especially the managing direc-tors, for giving me the opportunity to participate in this programme.

Finally, I want to thank my wife Judith for her patience, support and under-standing during the preparation of this work. Without her this project wouldhave taken much longer.

Contents

Introduction 1

1 The Internal Ratings-Based Approach 4

1.1 The New Basel Capital Accord . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Credit Risk – The Risk Weighted Assets . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Risk Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Risk Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Minimum Capital Requirements . . . . . . . . . . . . . . . . . . . . 7

2 The CreditMetricsTM Approach 9

2.1 Credit Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Exposure Values and Credit Spreads . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Obligor Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Monte-Carlo Simulation of Asset Returns . . . . . . . . . . . . . . . . . . . 17

2.5 The One-Factor CreditMetricsTM Approach . . . . . . . . . . . . . . . . . . 19

3 Obligor Ratings 22

3.1 Modeling of Agency Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Internal Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Relations between External and Internal Ratings . . . . . . . . . . . . . . . 30

4 Results on a Sample Portfolio 32

4.1 The Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 The Test Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Regulatory Capital Requirements . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 One-Factor CreditMetricsTM Approach . . . . . . . . . . . . . . . . . . . . . 36

i

4.5 Multifactor CreditMetricsTM Approach . . . . . . . . . . . . . . . . . . . . . 39

4.6 A Link between Internal Model and Basel II . . . . . . . . . . . . . . . . . . 44

Summary and Outlook 50

A Example of Evaluation 52

A.1 Instrument parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.2 Present Value and Future Exposures . . . . . . . . . . . . . . . . . . . . . . 52

A.3 Capital requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.4 One-Factor CreditMetricsTM Approach . . . . . . . . . . . . . . . . . . . . . 54

A.5 Multifactor CreditMetricsTM Approach . . . . . . . . . . . . . . . . . . . . . 54

B The Obligor Correlation Matrix 56

Bibliography 57

ii

List of Figures

1.1 Benchmark risk weight function of the IRB approach . . . . . . . . . . . . . 7

2.1 Asset return thresholds for a one-year risk horizon . . . . . . . . . . . . . . 12

2.2 Beta distributions for seniority classes . . . . . . . . . . . . . . . . . . . . . 14

2.3 Determination of the obligor correlation matrix . . . . . . . . . . . . . . . . 17

3.1 Mapping of scoring values to rating grades . . . . . . . . . . . . . . . . . . . 24

3.2 Analysis between risk perception and PD . . . . . . . . . . . . . . . . . . . . 26

3.3 Mapping of external to internal rating grades . . . . . . . . . . . . . . . . . 27

3.4 Analysis of internal risk perception . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Yield curve used for credit-risk valuation . . . . . . . . . . . . . . . . . . . . 33

4.2 Correlation dependence of the analytic one-factor Credit–VaR . . . . . . . 37

4.3 Loss distribution of the numerical one-factor Credit–VaR . . . . . . . . . . 38

4.4 Distribution of the total portfolio value at the risk horizon . . . . . . . . . . 40

4.5 Loss distribution of the test portfolio value at the risk horizon . . . . . . . . 40

4.6 Comparison of future portfolio value distributions . . . . . . . . . . . . . . . 41

4.7 Marginal Credit–VaR for the positions of the test portfolio . . . . . . . . . 43

4.8 Comparison of risk contributions on the level of single positions . . . . . . . 44

4.9 Rating dependent comparison of risk contributions . . . . . . . . . . . . . . 45

4.10 Difference between estimated PD and internal rating grade specific PD . . . 47

4.11 Regression analysis for the complete dataset . . . . . . . . . . . . . . . . . . 48

4.12 Difference between estimated PD and internal PD . . . . . . . . . . . . . . . 48

iii

Introduction

“Once a wallflower, Credit risk emerged in the late 1990s as the belle of the ball”, Michael B.Gordy underlines the importance of Credit risk with this statement in one of his contribu-tions within the Credit risk framework [14]. The original “Basel I” capital accord [2], issuedby the “Bank for International Settlement”, described detailed and differentiated methodsfor measuring market risk, up to internal models with which the banks could apply theirown measuring methodologies. However, this approach is less differentiated: banks have tounderpin the book value of a loan or a bond with 8% capital, irrespective of the inherentrisks. Such influences are taken into account in the new Basel capital accord (Basel II)which was developed since 1999 [3, 4]. In contrast to its first version where Credit risk as-sesment was expressed in a standardised way, the new accord provides several possibilitiesfor Credit risk assesment. Banks can choose among two essentially different approaches tomodel Credit risk: the standard approach which is close to the 1988 capital accord and theso-called internal ratings based (IRB) approaches.IRB approaches rely heavily upon a banks internal assessment of its counterparties andexposures and the Basel Committee believes that they can secure two key objectives con-sistent with those which support the wider review of the new Basel capital accord. Thefirst is additional risk sensitivity. A capital requirement based on internal ratings is moresensitive to the drivers of Credit risk and economic loss in a bank’s portfolio. The secondis incentive compatibility, in that an appropriately structured IRB approach can provide aframework which encourages banks to continue to improve their internal risk managementpractices.However, the IRB approach does not account for effects resulting from internal models.Capital requirements are calculated on the level of single positions, irrespective of effectsdue to the specific composition of a bank’s portfolio. Diversification can reduce the ex-pected and the unexpected loss characterising the portfolio’s Credit risk. In contrast, riskconcentration can dramatically increase this indicators. Internal portfolio models representtoolkits that allocate Credit risk contributions. For example, the CreditMetricsTM method-ology uses obligor correlations to determine the influence of single positions on the totalposition. It uses externally1 determined probability of default (PD) to obtain the Credit–VaR .Hence, regulatory approach and internal models start with different assumptions. And thisfact implies a lot of questions that are – at least for some of them – of importance for thedeeper understanding of the Credit risk framework as well as the capital accord.

1In this context, the term “external” means that the survey of historical data is done by rating agencies

that provide default probabilities.

1

• Are both approaches compatible?The IRB approach represents an additive method to determine Credit risk contribu-tions, whereas an internal model is a highly non-linear approach. At first glance, it isnot clear at all if a connection between both formulations exists.

• How can the approaches be transferred one into another?If one can find an analytic expression that transforms a characteristic value obtainedwith the CreditMetricsTM approach into a IRB capital requirement, this would be adirect proof of the correspondence between both approaches. In addition, this wouldenable the access to further studies, for example on the influence of portfolio effectson the IRB capital requirement.

• Can internal models be used for PD estimation?An analytic relationship between the two formalisms would enable to use the internalmodel to determine the risk contribution within the portfolio, to map it onto a IRBvalue and to extract PD that depends on a bank’s specific portfolio due to correlationeffects.

• Which role do correlations play?Internal models – the CreditMetricsTM approach in particular – are based on corre-lations between risk factors. It is interesting to find out if correlations influence theresults of other characteristic values within the approaches.

• Are the ideas relevant for practical use?One motivation for studies in this field is related to the practical relevance. Thereforeit is necessary to formulate ideas that can be easily implemented.

The aim of this work would certainly not be to give answers to all these questions. Butit will focus on the question of how internal portfolio models and the IRB approach arelinked. If there exists a systematic relationship between the two approaches, the answersto other questions would be not too far. In particular, it would be possible to account forportfolio effects in the IRB approach.

The first chapter of this work gives a brief description of the Basel II internal ratingsbased approach. The components, formulas and minimum requirements necessary to cal-culate the risk weighted asset capital requirement are introduced.One possible internal credit portfolio model, the CreditMetricsTM approach, is described inthe second chapter. Two versions of the portfolio model are considered. Following the pub-licly available Technical Document [15], the full CreditMetricsTM methodology is reviewedin detail. This part can also be used as an implementation guide. Special interest is focusedon the calculation of the future exposure values for the default case at the risk horizon. Incontrast to this multifactor approach, the one-factor CreditMetricsTM model is consideredon the other hand. This methodology is important because it hast a close connection tothe IRB approach.Nearly the entire work is based on model assumptions. Chapter 3 introduces a model usedto generate the external and the internal rating structure. The reason for modeling ratingtransitions and PD is that it is very difficult to have access to the entire agency rating processas well as to internal rating procedures. The main issue of the modeled rating processes isto obtain data sets that can be used for input in the Credit risk evaluation. In addition, it

2

will be shown that simulated results are of a similar quality as those provided from “true”rating processes and rating histories.The last chapter discusses the Credit risk results of a bond portfolio used for test pur-poses. The securities are evaluated with respect to the IRB approach as well as bothCreditMetricsTM methodologies in order to obtain approach dependent loss distributions.Finally, a link between the CreditMetricsTM results and the IRB values is established.A summary, together with an outlook concludes this work.

3

Chapter 1

The Internal Ratings-Based

Approach

This chapter is intended to give an overview of the new Basel capital requirement that willbe mandatory for banks with beginning of 2006. The main interest will be focused on theIRB approach that is used in the following chapters.

1.1 The New Basel Capital Accord

The main intension of the capital accord is to set the total minimum capital requirements formarket, credit and operational risk. Three main elements are related to these requirements:the definition of regulatory capital, risk weighted assets and the minimum capital ratio.The capital ratio is defined as the ratio between the regulatory capital and the total sumof risk weighted assets

capital ratio =regulatory capital

∑

risk weighted assets(1.1)

The denominator of Equation 1.1 is composed of three addends describing market, opera-tional and Credit risk. The first two addends are multiplied with a factor of 12.5, whereasthe third representing the the Credit risk contribution is described in the following sections.The capital ratio must be at least 8%. The minimum capital requirements for Credit riskis one of the three pillars of new Basel capital accord:

• Minimum capital requirement

• Supervisory review process

• Market discipline.

The relevant information used within the context of this work is documented in the firstpillar that describes a methodology on how risk weighted assets are treated for the deter-mination of the capital requirement.

4

1.2 Credit Risk – The Risk Weighted Assets

For Credit risk, the Basel capital accord proposes two methods: the standardised and theIRB approach. Hence, there are two ways to determine the risk weights serving as the inputfor Equation 1.1. Depending on the category of the claims that form the basis of Creditrisk, the calculation of risk weights is carried out differently. However, in this work onlyclaims on sovereigns, banks and corporates are taken into account.Based on credit ratings of external agencies, the standardised approach uses a mappingbetween rating classes and risk weights. The mapping rules and risk weights that are in arange between 0% and 150% are defined by the supervisory authorities for each category ofclaims. Financial institutions are very inflexible in their way to apply this method becausethe the risk weighted assets are calculated using these fixed weights.The IRB approach allows banks to be more flexible. Applying this method, the risk weightsare calculated using internal estimators for the risk factors. For the above mentioned cate-gories of claims, three key elements form the basis of the IRB approach: risk components,risk weight functions and minimum requirements.

1.2.1 Risk Components

The estimates of four risk factors used in the IRB approach are provided either by banks orby the supervising authorities, depending on whether the IRB foundation approach or theIRB advanced approach is applied. These risk components are the probability of default(PD), the loss given default (LGD), the exposure at default (EAD) and the effective maturity(M). They are defined as follows.

Probability of Default

PD is a measure of the probability that an obligor is not able to pay his debts within acertain time horizon. If it is not mentioned otherwise, this time horizon is one year. Forbanks and corporates, a minimum PDmin = 0.03% is required for obligors with the bestcredit quality. In the case of default, the boundary condition is PD = 100%.

Loss given Default

The loss given default describes the part of a credit that is not recovered in the caseof default. In the foundation approach, the value LGD is in the range between 45% and75% for unsecured claims. These values can be reduced if eligible financial collaterals arerecognised in the IRB sense. In this case, the effective loss given default is calculated fromthe exposure values after risk mitigation1

The advanced IRB approach allows the use of internally estimated LGD values.

Exposure at Default

1The treatment of collaterals is described in detail in the Basel capital accord [4] (Paragraphs 259 to

274).

5

The amount of the facility that is likely to be drawn if a default occurs is expressed asthe exposure at default for loan commitments. For on-balance sheet positions, a nettedexposure is considered. In the case of off-balance sheet positions a credit conversion factoris used to determine the EAD value. Depending on the maturity and the properties of theinstrument (e.g. ability of cancellation, type of commitment), this factor ranges from 0 to 1.

Effective Maturity

The effective maturity M is a measure for the remaining economic time to maturity of acredit. If the IRB foundation approach is applied, the value for M is 2.5 years in most ofthe cases. A value of 6 months is attributed to exceptions (Paragraph 288). The advancedIRB approach uses a cash-flow weighted formula in order to determine the value for theeffective maturity. In these cases the value for M ranges between 1 and 5 years. For shortterm credits this range is reduced.

For both, the foundation as well as the advanced IRB approach, PD is estimated bythe bank. The other risk components are provided by the regulatory authorities for thefoundation approach. If a bank uses the advanced IRB approach, it is required to estimateLGD, EAD and M in addition to PD.

1.2.2 Risk Weight Functions

The calculation of the risk weighted assets is carried out using Equations 1.2 to 1.7 that arethe central formulas of the IRB approach for claims on sovereigns, banks and corporates.The correlation ρ is derived from PD and is defined as

ρ = 0.121 − exp(−50 × PD)

1 − exp(−50)+ 0.24

[

1 − 1 − exp(−50 × PD)

1 − exp(−50)

]

. (1.2)

However, for the most practical applications this formula can be approximated by

ρ = 0.12 [1 + exp(−50 · PD)] . (1.3)

Using the definition of the maturity adjustment

b(PD) = (0.08451 − 0.05898 log(PD))2 (1.4)

yields the capital requirement K

K = LGD × BRW × (1 + (M − 2.5)b(PD)). (1.5)

In this last equation, BRW is the PD dependent benchmark risk weight function

BRW(PD) = Φ

[

1√1 − ρ

× Φ−1(PD) +

√

ρ

1 − ρ× Φ−1(0.999)

]

× 1

1 − 1.5 b(PD). (1.6)

The formula for the risk weighted asset used as the input in Equation 1.1 is then defined as

RWA = 12.5 × K × EAD (1.7)

6

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 0,05 0,1 0,15 0,2 0,25PD

BR

W

Figure 1.1: Benchmark risk weight function of the IRB approach. The central risk weight

formula (cf. Equation 1.6) is plotted against PD.

In these equations Φ and Φ−1 denote the cumulative distribution function for a standardnormal distributed random variable (i.e. the probability that a normal random variablewith mean zero and variance of one is less than or equal to x) as well as its inverse function,respectively.Figure 1.1 shows the dependence of the benchmark risk weight on PD as expressed in Equa-tion 1.6.

1.2.3 Minimum Capital Requirements

The minimum capital requirements describe the framework that must be applied in orderto estimate the risk components. The underlying data, processes and the methods used forthe determination of PD, LGD, EAD and M must fulfill the minimum requirements and aresubject to the regulatory supervision.The rating system must be composed of at least seven grades for non-defaulted borrowersand one additional grade for those who have defaulted. In the same way, the rating classi-fications of external rating agencies are composed. Two dimensions, the borrower’s defaultrisk and factors that are specific for the financial transaction have to be taken into accountin order to evaluate Credit risk and to associate a borrower with a rating grade and theprobability of default.The Basel capital accord explicitly allows the use of models in order to assign the bor-rower’s rating grade or the estimation of the risk components described in the previoussection. Such models also allow easy access to test the rating assessment processes. For

7

example, stress tests can be performed to simulate unfavorable effects on a bank’s creditexposures (economic or industry downturns, market risk events, . . .). However, informationwhich is outside the scope of the model should also be taken into account when determiningthe characteristic Credit risk values. This fact can result in adding a model-error dependentmargin of conservatism It should be clear that this work can only cover the model side ofthe estimation for PD.The minimum requirements specific to PD estimation allow the use of an appropriate tech-nique to estimate the average PD for each rating grade. Three techniques are explicitlyproposed: (i) the estimation of PD based on data of internal default experience, (ii) themapping of internal rating grades to the rating classes used by external agencies and attri-bution of the corresponding default rates, and (iii) statistical default models. In order toavoid potential sources of errors resulting – for example – from judgmental considerations,additional requirements like statistical relevance, compatibility to general market conditionsand comparability to external rating statistics and processes. A minimum five year timehorizon is required for historical data.Additional requirements are defined for data maintenance, use and validation of internalratings as well as for the disclosure of the IRB approach.

8

Chapter 2

The CreditMetricsTM Approach

Measuring and managing Credit risk, whether for bonds, loans or derivatives, has becomea key issue for financial institutions. Credit risk can be broadly defined as the possibilityof losses exceeding expectations in a portfolio of credit assets. A quantitative portfolio ap-proach is required providing estimates of the probability and magnitude of losses in orderto manage Credit risk. At the turn of the previous century, several models have been devel-oped and received a great deal of public attention, including J.P. Morgan’s CreditMetricsTM

[15], Credit Suisse Financial Products’ CreditRisk+ [8], McKinsey & Company’s CreditPort-folioView [23] and the KMV-Model [9].In 1997, the portfolio approach CreditMetricsTM was proposed by J.P. Morgan, together witha corresponding data service, for the estimation of Credit risk. The success of CreditMetricsTM

is in part due to the technical documentation making the methodology available to a broadaudience in a fully transparent manner. One motivation for this transparency certainly isthe creation of a benchmark for measuring Credit risk. It would become possible to measurerisks systematically across various credit instruments with the same yardstick if this aimis successful. A second aim could be derived from this motivation: the increase of marketliquidity for credit derivatives. Once credit derivatives are better understood they will be-come less frightening to investors, promoting liquidity. And it is obvious that promotingthe understanding of Credit risk increases the understanding of credit derivatives.CreditMetricsTM is a tool for assessing portfolio risk due to changes in debt value caused bychanges in obligor quality. Changes in value are not only caused by possible default eventsthat would represent a two state world (default and non-default), but also by migrationsof the credit quality. Using the CreditMetricsTM approach, the portfolio’s distribution offuture values at the risk horizon can be estimated taking effects of diversification acrossobligors into account. From this distribution the Credit–VaR can be determined to quan-tify the Credit risk.Apart from the determination of the Credit–VaR , this approach can be used to allocateportfolio risks and to manage the portfolio (eg. by hedging with credit derivatives). Finally,a quantitative portfolio model could contribute to meet the capital adequacy requirementsimposed on financial institutions in the future ’The Committee welcomes further develop-ments in risk management practices and modeling that may pave the way towards a tran-sition to portfolio Credit risk modeling in the future.’ [5]. Although internal Credit riskmodels are not yet allowed to meet supervisory requirements, it is only a matter of time

9

Initial Rating at Risk Horizon

rating AAA AA A BBB BB B CCC Default

AAA 93.27 6.16 0.45 0.09 0.03 0.00 0.00 0.00

AA 0.62 91.63 7.04 0.53 0.05 0.09 0.02 0.02

A 0.06 2.19 91.75 5.27 0.45 0.18 0.04 0.06

BBB 0.03 0.24 4.62 89.42 4.40 0.77 0.24 0.28

BB 0.02 0.07 0.45 6.28 82.97 7.70 1.19 1.32

B 0.00 0.09 0.30 0.39 5.33 82.87 4.36 6.66

CCC 0.13 0.00 0.25 0.77 1.66 10.23 58.70 28.26

Default 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00

Table 2.1: One-Year Transition Matrix (quotations in %). The rating classes are in accor-

dance with the Standard & Poors (8 classes) scheme.

before internal models will be the main Credit risk management tool [19].Credit risk is on the first view closely related to the probability of default. The holder of asecurity, for example a zero coupon bond, is interested in whether the nominal will be paidat maturity or not. Hence, Credit risk can be interpreted within a two-state world wherean obligor defaults or ”survives”. The CreditMetricsTM methodology uses an extended ap-proach and takes also into account changes in the obligor’s credit quality, expressed interms of a credit rating as designated by rating agencies as Standard & Poors, Moody’sInvestor Services or Fitch ratings. Table 2.1 gives an example of the transition probabilitiesbetween the different credit qualities (from AAA down to default) for a time horizon ofone year. It indicates that it is rather unprobable that a AAA rated obligor will defaultwithin the considered horizon, but will be downgraded to a AA rating with a probabilityof 6.16%. Another important feature of the CreditMetricsTM approach is the assessmentof risk within the full context of a portfolio. This is done by addressing the correlationof credit quality moves across obligors, allowing calculate diversification benefits or poten-tial over-concentrations across a portfolio. These effects occur due to correlations betweenobligors. They are calculated from correlations between industry groupings on which theobligors are mapped.The following subsections describe the different parts of the CreditMetricsTM methodol-

ogy. They can be used as a step by step implementation guide for the calculation of theCredit–VaR .

2.1 Credit Migration

Within the context of Table 2.1, the credit migration matrix was introduced in the previoussection. As in the case of the classical1 two state world, where only the default and non-default state are distinguished, a Merton ”value of the firm” model [18] is used to estimatethe credit rating at the risk horizon. In the two-state world, this approach is used to define

1The term classical is only used to distinguish between the two-state world and the possibilities of credit

migration

10

a default threshold for asset returns. Assuming that a firm’s assets have a value V andthat the returns on these assets are normally distributed, V follows a geometric Brownianmotion

dV

V= µdt + σdz, (2.1)

with constant drift µ and asset volatility σ. In the presence of perfect markets, V isindependent of the firm’s capital structure and simply given by the sum of the debt andequity values. Hence, the firm is considered to be simply financed only by equity and onezero-coupon bond, representing the firm’s liabilities, maturing at a time T with a principalvalue F . Assuming that all the assets of the firm could be converted into cash at time T , allof the firm’s debt will be paid off in full if the terminal value of the firm’s assets VT exceedsthe principal value of the liabilities F . Otherwise, the debtors receive the firm’s assets andthe firm defaults. In consequence, the value of the firm’s equity at time T is given by

ST = max[VT − F, 0], (2.2)

which corresponds to a call option on the firm’s assets with a strike price that equals thevalue of the liabilities of the firm. As the probability of default P [VT < F ] is known fromTable 2.1 and asset returns are normally distributed, one can calculate a default thresholdin terms of the normal normal distribution with mean µ and volatility σ. One can alsotranslate this normal distribution of the asset returns into a standard normal distributionwith mean zero and variance one. In this case the default probability only depends on theinitial rating of the firm and the default threshold ZDef is defined in terms of the assetreturn R as

PDef = P (R < ZDef) = Φ(ZDef). (2.3)

Hence, the threshold can be expressed using the inverse cumulative standard normal distri-bution Φ−1(x)

ZDef = Φ−1(PDef). (2.4)

If credit migration is considered, the Merton approach can be easily extended and thresholdsfor the rating changes are calculated in an analogue way as shown above. To give an example,the threshold for the migration into a BB rating are calculated. The migration probabilityin terms of standard normalized asset returns is given by

PBB = P (ZB ≤ R < ZBB) = Φ(ZBB) − Φ(ZB)

= Φ(ZBB) − PB − PCCC − PDef ,

and henceZBB = Φ−1(PBB + PB + PCCC + PDef). (2.5)

Rating at Risk Horizon

AAA AA A BBB BB B CCC Default

P (%) 0.03 0.24 4.62 89.42 4.40 0.77 0.24 0.28

Z - 3.43 2.78 1.66 −1.58 −2.23 −2.56 −2.77

Table 2.2: Asset return thresholds for initial BBB ratings. A risk horizon of one year is

assumed. The transition probabilities are the same as for Table 2.1.

11

0

0,1

0,2

0,3

0,4

-5 -4 -3 -2 -1 0 1 2 3 4 5

asset return

prob

abili

ty

BBB A AA AAABBDefault B

CCC

Figure 2.1: Asset return thresholds for a one-year risk horizon and an initial BBB rating.

A standard normal return distribution is assumed (see text). The vertical lines symbolise

the thresholds.

For an initial BBB rating, one obtains the thresholds summarized in Table 2.2. In additionthe situation is graphically sketched in Figure 2.1 for the same initial rating. In both cases,the migration probabilities shown in Table 2.1 for a one-year risk horizon are used for thisnumerical example.The calculation of the asset return thresholds from the migration probabilities is the basefor the Monte-Carlo simulations that will be described in Section 2.4.

2.2 Exposure Values and Credit Spreads

The second important step in estimating the Credit–VaR using the CreditMetricsTM method-ology is the calculation of the values of the securities at the risk horizon. These future valuesare called exposures and are subject to credit quality upgrade, downgrade or loss in the eventof default. There are different exposure types, e.g. bonds, loans and market-driven instru-ments (swaps, caps, floors, etc.). Since this work only treats zero and coupon bonds issuedby sovereigns, banks and corporates, this section only describes the calculation of exposurevalues for this class of financial securities.The future exposure values are calculated for each possible rating grade and do thereforenot depend on the migration probabilities of Table 2.1. With regard to the implementationof the internal portfolio model this fact is an advantage because one only needs to calculate8 values for each instrument. Of course, if one builds-up a model containing more or less

12

than 8 rating classes (e.g. 19 for the enlarged S & P scheme or 3 as in the following sectionused for this model approach), more or less than 8 exposure values must be calculated foreach instrument.In a general form, the future exposure value is defined as the today’s (t) forward value ofan instrument with maturity T at the risk horizon (tRH)

V (t, T ; tRH ,R) = PV (t, T ;R) · B−1(t, tRH ;R), (2.6)

where PV (t, T,R) denotes the today’s present value of the instrument and B(t, tRH ;R) isthe discount factor for the time period between today and the risk horizon. Both quantitiesare calculated with respect to the considered rating grade R. The yield curve used todetermine the corresponding discount rates consists of two contributions: the base curve(i.e. a government curve or the internal yield curve of a financial institute) and a ratingdependent spread2. This enables to calculate the exposure values of any instrument for theseven rating classes (AAA to CCC). However, in the case of default at the risk horizon,the situation is different. The future exposure value depends on which amount of theoutstandings will be recovered or lost, respectively. The estimation of recovery rates r (orloss given default3 LGD) is fraught with many practical problems. The market for suchrare events is quite illiquid so that it is difficult to obtain recovery values. An even if themarket would provide such values, the question about how and when to settle the value ofrecovered capital remains open. For bonds, the CreditMetricsTM approach uses studies ofseniority types to refine the estimates of recovery rates. For corporate bonds, Carty andLieberman [7] published recovery rates extracted from a large sample of defaulted bondsfor the time period between January 1970 and December 1995. Table 2.3 summarizes theresults of this investigation. The difference between secured versus unsecured senior is notstatistically relevant, In contrast, subordinated classes are appreciably different from oneanother in their recovery realisations.

Study results Parameters

Seniority Average Std. Dev. a b

Senior Secured 53.80 26.86 2.91 2.50

Senior Unsecured 51.13 25.45 3.35 3.20

Senior Subordinated 38.52 23.81 3.79 6.05

Subordinated 32.74 20.18 5.08 10.44

Junior Subordinated 17.09 10.90 11.76 57.03

Table 2.3: Recovery rates (in %) for different seniority types. The data is extracted from

Moody’s Investor service [7]. The two left columns represent the parameters that are nec-

essary to generate a beta distribution function with the corresponding mean and variance

(see text).

An additional problem becomes obvious when analysing the data of Table 2.3. The standarddeviations of the recovery rates of all seniority classes have large values and indicate ahigh uncertainty of the results. Due to this fact the CreditMetricsTM approach proposes

2These datasets can be downloaded form the CreditMetricsTM website: www.riskmetrics.com

3The relation between recovery r rate and LGD is LGD = 1 − r.

13

0% 25% 50% 75% 100%Recovery rate

Fre

quen

cyJun. Subord.Subord.Sen. Subord.Sen. Unsec.Sen. Sec.

Figure 2.2: Beta distributions for seniority classes. With the information summarized in

Table 2.3, the Beta distribution density functions are plotted.

to characterize the recovery rate by a probability distribution. Two distributions fulfillthe requirements of the recovery rate (0% ≤ r ≤ 100%) and uncertainty. By default,the uniform distribution has a mean of µ = 0.5 and a standard deviation σ =

√

1/12 ≈0.29. These values correspond quite well to the senior secured and senior unsecured class.However, in order to cover the whole seniority range, a beta distribution seems to be moreappropriate. This distribution is described using the density function

f(x) =Γ(a + b)

Γ(a)Γ(b)xa−1 (x − 1)b−1 , 0 ≤ x ≤ 1 (2.7)

The parameters a and b determine the form of the density distribution. If a = b = 1, thedensity will be uniformly distributed. In addition, the density function includes the Gammafunction4 Γ(x). Mean and variance of the beta distribution depend on the parameters aand b.

µ =a

a + b(2.8)

σ2 =ab

(a + b)2 (a + b + 1)(2.9)

Accordingly, the density function parameters are defined using mean and variance

a =µ′

(1 + µ′)2 σ2− 1

1 − µ′(2.10)

b = µ′ a, (2.11)

4A description of the Gamma function is, for example, found in [6]

14

where

µ′ =1 − µ

µ.

Figure 2.2 shows the resulting distribution functions for the seniority classes summarizedin Table 2.3.For financial securities that are considered to be defaulted at the risk horizon, a simulatedrecovery rate rSim multiplied with the EAD of the instrument determines the future exposurevalue of the default state. The value for rSim is a beta distributed random number that canbe generated using the inverse cumulative beta distribution function applied to a uniformlydistributed random number. Such functions are available in many software applications(e.g. Excel, Mathematica, Matlab, Python). But at this point one is faced to anotherproblem. Which quantity represents the EAD in case of default? For low rated long terminstruments, it is possible to benefit from default arbitrage if the nominal amount is usedas a measure for EAD. In this case the today’s or one-year’s present value of the instrumentwould perhaps be lower than the product of recovery rate and nominal amount (or EAD).Hence, an investor would be interested in the default of this security. It is for this reasonthat the convention in this work will be to use the one-year’s present value to determineEAD.

EAD = PV (t, T )B−1(t, tRH ), (2.12)

where B(t, tRH) is the one-year discount factor obtained with the use of a spread–less yieldcurve.

2.3 Obligor Correlations

In order to account for portfolio effects in the estimation of Credit risk, correlations betweenthe issuers or counterparties of the different securities of the portfolio under investigationare included in the CreditMetricsTM approach. Therefore, it is assumed that the obligorscan be (at least partly) mapped onto industry indexes. This step is motivated by the factthat the obligors’ equity returns can be explained by some given percentage of index returns,given that these indexes represent the industrial sectors of the obligors’ business. As equityand asset values are linked via Equation 2.2, and as rating migration is driven by assetreturns, the mapping assumption seems to be reasonable and – from a practical point ofview – also applicable.Suppose that the obligor’s asset return rO can be sufficiently explained by N index returnsri with weight wi of the industry classifications to which the obligor belongs and a residualpart that can be explained solely by information that is unique and specific to the firm (rS

with weight wS). This obligor-specific or idiosyncratic information involves qualitative aswell as quantitative statements about the obligor that result in a statistical independencefrom the related industry movements. Therefore, the correlation between index returns andthe idiosyncratic returns is zero. Applying this method of index mapping, the asset returnof the obligor is

rO =

N∑

i=1

wi ri + wS rS , (2.13)

15

whereN

∑

i=1

wi + wS = 1 (2.14)

A correlation matrix whose components ρij describe the correlation between index i andindex j is used to quantify the dependence between industry indexes. Together with thevolatilities of the index returns σi, these correlations are the base for the calculation of theobligor correlations. As mentioned within the context of the calculation of return thresholds(cf. Equation 2.4), the obligor returns are standard normally distributed. Therefore thevariance of rO following from Equation 2.13 must be rescaled to a uniform value for thismapping method. In a first step, only the industry-specific weights are considered and ascaled volatility of the weighted indexes is defined by

σ =

√

∑

i

w2i σ2

i + 2∑

i,j

wi σi ρij wj σj . (2.15)

In the second step, the idiosyncratic weight is taken into account in order to calculate theindex weights with respect to their contribution to rO

wi = αwi σi

σ, (2.16)

where α = 1 − wS . Finally, the scaled idiosyncratic weight is adjusted to the constraint oftotal volatility to be one and thus

wS =√

1 − α2. (2.17)

The obligor correlations can now be obtained using matrix multiplication. The index cor-relations are known from time series analysis. This analysis is not described within thiscontext because this point is discussed in detail in many other works and could be a topic ofa stand-alone thesis. The CreditMetricsTM Technical Document [15] also gives explanationson how to calculate the index returns, volatility and the correlation between index returnson the base of sufficient historical data. The index correlations ρij form the upper left cornerof an enlarged correlation matrix C. Assuming m obligors that are mapped on N indexes,C is a (N +m)×(N +m) matrix that needs to be constructed as follows: in the first N rowsand N columns the index correlations are written. Further elements of the matrix have avalue of one if they are diagonal elements and zero otherwise because they correspond to theuncorrelated firm-specific risk of the m different obligors. The composition of this matrix isgraphically sketched in Figure 2.3. The scaled weights calculated in Equations 2.16 and 2.17form a (N + m) × m matrix W: in the first N rows the scaled index weights are written.In the lower part of the matrix (rows N + 1, . . . , N + m) the elements of the (N + i)th rowand ith column have the value of the scaled firm-specific weight concerning the ith obligor.All other elements of the lower part are zero. The (m × m) obligor correlation matrix CO

is then given by the matrix multiplication

CO = WT × C × W. (2.18)

This correlation matrix is used in the following section to generate correlated scenarios ofstandard normalized obligor asset returns.The advantage of the above described method to calculate obligor correlations is that it is

16

Figure 2.3: Graphical sketch of the composition of the enlarged correlation matrix C. E

denotes the unity matrix and 0 is the matrix with all elements zero.

a universal approach. In Equation 2.13 it is not explicitly stated that index returns have tobe used. Hence, one could also rely on other sources of returns, as for example the returnof the stock for the case that the issuer is a corporate quoted at a stock exchange. Eventhe returns on a basket of bonds issued by the evaluated obligor could be used to calculatecorrelations, provided that it is representative for the corporate. However, the reason forusing indexes is that they form a comparable measure and that they are liquid.

2.4 Monte-Carlo Simulation of Asset Returns

The estimation of the Credit–VaR is carried out using Monte-Carlo simulations of corre-lated asset returns. Assuming that the assets of the evaluated portfolio can be attributedto m different obligors, each scenario consists of a m-dimensional correlated random vector~x which is constructed by multiplying an m-dimensional vector of independent standardnormally distributed random variables zi (forming ~z) with the decomposed obligor correla-tion matrix. This decomposition is necessary in order to reproduce the obligor correlationsin the generated scenarios [10]. Based on Equation 2.18 the obligor correlation matrix CO

is is transformed into a set of two matrices

CO = AT × A. (2.19)

One simple method for matrix decomposition is the so called Cholesky-decomposition. Thismethod can be applied to positive semidefinite correlation matrices and results in a triangu-lar matrix A. The implementation of this method is quite easy [10, 20] and can be used inmost of the cases. However, sometimes the obligor correlation matrix could have negativeeigenvalues and is therefore not positive semidefinite. In such cases a more complicated de-composition method is applied. The main idea of this singular value decomposition (SVD)

17

is sketched in the following. With the use of orthonormal transformations Q it is possibleto extract the eigenvalues of the initial correlation matrix in the form of a diagonal matrixD

CO = Q × D× QT. (2.20)

If the diagonal matrix is be split into two parts, this would lead to the required decompo-sition

CO =(

Q ×√

D)

×(

Q ×√

D)T

. (2.21)

The decomposition of Equation 2.20 can be obtained within two steps. First, the obligorcorrelation matrix CO is transformed to a tridiagonal form using the Householder method.In a second step the QL-algorithm with implicit shifts is applied to transform the tridiagonalmatrix into a diagonal matrix D. The orthogonal transformations are accumulated in theover-all transformation matrix Q. A guideline for the application of these transformationsis given in [20]. For the case that some of the eigenvectors of the correlation matrix are neg-ative5 Jackel [16] proposes to apply a spectral decomposition for a workaround. Comparedto the Cholesky decomposition or the SVD for a matrix with only positive eigenvalues,this method leads to a slightly flawed decomposed matrix. However, it can also be used togenerate correlated scenarios by multiplying the decomposed correlation matrix with thevector of standard normally distributed random numbers

~x = A× ~z. (2.22)

For the case that SVD was used, the decomposed matrix consists of the matrix product

A = Q×√

D.

Each correlated scenario vector ~x is evaluated with respect to the thresholds calculatedin section 2.1 in order to determine the rating grade at the risk horizon. Once this is done,the future exposure value V (t, T ; tRH ,R) of each security in the portfolio can be extractedfrom the exposure matrix calculated in section 2.2. The sum of all future exposure valuesgives the expected value of the portfolio at the risk horizon for the corresponding scenario,accounting for credit migration based on the transition matrix. By repeating this procedurefor many scenarios, a distribution of portfolio values at the risk horizon is obtained.Characteristic values can be extracted from this distribution:

• Mean µThe arithmetic average of the future portfolio value over all scenarios. It is a measurefor the expected loss.

• Quantile qThe portfolio value V q

PF of the scenario for which q % of the simulated scenarios havea lower value VPF. It is a measure for the unexpected loss. Together with the meanit defines the Credit–VaR

VaR = V µPF − V q

PF. (2.23)

5In general, this unwanted effect only appears if the historical time series used for the calculation of

correlations has some lacks or if the calculated correlation matrix would be modified by hand.

18

• Loss distributionAssuming that all issuers contributing to the portfolio will remain in their initialrating class at the risk horizon, the expected future portfolio value can be determinedusing Equation 2.6. This value is used as the benchmark for the calculation of theloss distribution. Therefore all portfolio values of the simulated distribution that arebelow the benchmark will be interpreted as losses, whereas portfolio values that arelarger than benchmark are earnings.

• Marginal Credit Value–at–RiskThe marginal Credit–VaR is a measure of the contribution of a position Ik to thetotal Credit–VaR . It is defined as the difference between the Credit–VaR of theentire portfolio and the Credit–VaR of the portfolio without the position Ik

VaRm(Ik) = VaR (· · · , Ik−1, Ik, Ik+1, · · · , ) − VaR (· · · , Ik−1, Ik+1, · · ·) . (2.24)

Other authors define the marginal Credit–VaR as the impact on the VaR of increasingthe position by some small amount [1]. Regardless which definition is used, both servethe same purpose of measuring risk contribution to a portfolio, accounting for theeffects of diversification. Hence, the concept of marginal VaR enables to allocate riskconcentrations in the portfolio.

2.5 The One-Factor CreditMetricsTM Approach

Within the context of the above described CreditMetricsTM approach, Finger [12] proposeda simplified version of the methodology. This model is motivated by the Basel II capitalrequirement which is attributed to the class of risk-bucket capital rules. Risk-bucket capitalrules assign capital to an individual exposure based only on the characteristics of the expo-sure and not on information about the rest of the portfolio [13]. Taking into account two keyconditions, a portfolio model can also produce risk contributions that behave as risk-bucketcapital rules. First, only one systematic risk factor drives the performance of all obligors.The second key condition is that no exposure in the portfolio accounts individually for asignificant share of portfolio risk. In practice, this condition is satisfied if the idiosyncraticmovements of the obligors can be ignored and the portfolio is modeled as if its value werecompletely determined by the performance of the single risk factor.A model satisfying these two key conditions is presented as a modified version of theCreditMetricsTM model [11]. It describes the creditworthiness of obligor k within a portfo-lio of K credits using a characteristic variable Zk. This variable is modeled as a standardnormally distributed random variable that depends on one common market index Z anda term that is idiosyncratic to the obligor εk, both being stochastically independent fromeach other

Zk =√

ρk · Z +√

1 − ρk · εk, k = 1, . . . ,K (2.25)

The common market index for example represents the economic situation, whereas theidiosyncratic risk factor has only an influence on the credit standing of the obligor. Thecorrelation between Zk and the common risk factor Z is represented by ρk. The correlationbetween two creditworthiness variables Zi and Zj is

√ρi · ρj .

The former Basel II IRB approach [3] assumes a constant correlation ρi = ρj = ρ = 0.2 forall credits issued by corporates. The modifications of the IRB approach carried out in the

19

current version [4] assume a correlation that depends on the probability of default ρ(PD) asshown in Equations 1.2 or 1.3, respectively.The event of default can be modeled with equation 2.25 assuming that PD is determinedusing a rating process. In a two-sate world where obligors either default or not, but do notchange rating, the default event occurs if Zk < Φ−1(PDk). In other words, due to the factthat Zk is standard normally distributed,

P(

Zk ≤ Φ−1(PDk))

= PDk. (2.26)

Hence, given a realisation Z = z for the market index, PDk can be expressed as

P(

Zk ≤ Φ−1(PDk)|Z = z)

= P(√

ρk · Z +√

1 − ρk · εk ≤ Φ−1(PDk)|Z = z)

= P(√

ρk · z +√

1 − ρk · εk ≤ Φ−1(PDk)|Z = z)

= P(√

ρk · z +√

1 − ρk · εk ≤ Φ−1(PDk))

= P

(

εk ≤ Φ−1(PDk) −√

ρk · z√1 − ρk

)

= Φ

(

1√1 − ρk

· Φ−1(PDk) −√

ρk

1 − ρk

· z

)

. (2.27)

The third equality of this transformation is based on the stochastic independence betweenZ and εk. Equation 2.27 describes the conditional probability of default for the givenrealisation PDk(z). It is monotonously decreasing in z: for realisations with small z, theconditional default probability is high. If the quantity

Dk =

{

1, Zk ≤ Φ−1(PDk)0, Zk > Φ−1(PDk)

(2.28)

is a Bernoulli-distributed indicator for the default event of the k–th obligor and assumingthat bk is the contribution of the k–th credit to the portfolio, then LGD of the credit portfoliois

LGDPF =K

∑

k=1

LGDkbkDk, (2.29)

with the expectation value

E(LGDPF) =

K∑

k=1

LGDkbkPDk. (2.30)

The default probability in Equation 2.30 is the unconditional PD (cf. Equation 2.26). Theconditional expectation value for a given realisation Z = z is

E(LGDPF|Z = z) =

K∑

k=1

LGDkbkE(Dk|Z = z)

=

K∑

k=1

LGDkbkPDk(z)

=K

∑

k=1

LGDkEADkPDk(z). (2.31)

20

This result is very similar to the Basel II formula given in Equation 1.5 if z = Φ−1(0.001) =−Φ−1(0.999).

The specification of the market realization does not determine exactly what happensto the portfolio. It is also necessary to specify each εk in order to know the portfolio.However, as the realisation is conditioned on the common market factor, the individualobligor defaults are driven only by the idiosyncratic terms and are therefore independent.Thus, if K is large, the Law of Large Numbers (LLN) states that the fluctuation of the trueportfolio value around LGD(z) will be small. It is assumed that K is large enough to ignorethis fluctuation altogether and that given z, the portfolio loss is equal to its conditionalmean LGD(Z = z) [11]. The justification for utilizing the LLN is exactly the second keycondition described at the beginning of this section.This one-factor approach has the advantage that the portfolio VaR is easy to calculate. Ifone wants to extract the worst case loss at confidence level q, the value zq simply defines thethreshold P (Z > zq) = q, since the portfolio value is uniquely determined by the marketfactor. The worst case loss is then LGDPF(Zq). Due to Equation 2.31 the contribution ofan individual obligor k is just LGDk · PDk(zq). This relation satisfies the definition for arisk-bucket capital rule because it only depends on information about obligor k.Hence, the one-factor CreditMetricsTM approach offers a further methodology of calculatingthe Credit–VaR in an analytic as well as in an numerical way. It depends on the numberof securities that are within the portfolio and therefore on it’s granularity. On other words,the idiosyncratic risk must be be fully diversified. However, in first order approximation theanalytic solution can be used, even if it is only for the sake of comparing such a result withthe numerical one. The calculations performed and presented in Chapter 4 take advantageof this fact.

21

Chapter 3

Obligor Ratings

It is commonly accepted that the obligor’s PD is related to it’s rating grade. The betterthe grade, the smaller PD will be. Together with the rating grade migration probabilities,the PDs are one essential input for the calculation of the Credit–VaR within the frame-work of the CreditMetricsTM approach described in Chapter 2. Such migration matricesare generally provided by external rating agencies (Moody’s Investor Services, Standard &Poors or Fitch Ratings). However, the goal of a financial institution would be to use it’sinternal ratings based transition matrix to carry out the estimation of the Credit–VaR .But unfortunately many banks do not have a sufficient number of historical rating data setsto estimate their own rating migration probabilities with an acceptable statistic relevance.This is the reason for the use of publicly available information.Some questions arise in using the external data sets. Even if the Basel Capital accord re-quires the transparency of an agency’s rating process1, a bank will not have access to the fullnecessary information of a corporate in order to determine an internal rating in the sameway as external rating agencies do. In addition, it would be a difficult task for a financialinstitution to asses rating grades for all the obligors that contribute to Credit risk in theportfolio in the same way as rating agencies do. Finally, transition matrices are the resultof a rating grade time series analysis and reliable rating migration probabilities are relatedto a large pool of rated obligors. The situation becomes not easier if transition probabilitiesfor obligors in particular segments (e.g. industrial sectors or countries) are required.Hence, the question a bank will ask at this point is “How can we determine our internalrating for an obligor ?” Suppose that the financial institution has few, but reliable infor-mation about the obligors that is important – for example – from an investment decisionpoint of view or the prices of the instruments this obligor has emitted in the past. Thesefactors can be combined with the rating grade assigned by one or more agencies in order toobtain an evaluation of the obligor’s creditworthiness that takes into account the investorspecific classification.The following sections outline a methodology for an internal rating process that considersexternally assigned rating grades. As it is difficult to access the full agency information(i.e. input factors, method and output containing more than the grade), the agency ratingprocess is completely simulated assuming a simplified model. In order to obtain an artificialtransition matrix, the rating process is carried out twice (initially and at the risk horizon).

1See paragraph 61 of [4] for detailed requirements on external rating agencies.

22

Rating changes yield to the required migration probabilities, including PD. An additionalsimulation is used to obtain the internal rating as well as the internal transition matrix fora subset of the externally rated obligors.

3.1 Modeling of Agency Ratings

The processes of a rating agency are simulated using a simplified three-step approach basedon a credit scoring that divides the scoring interval into n sub-intervals in order to definethe n different rating grades. First, the initial score S0 of an obligor is obtained with theuse of N factors

S0 =N

∑

k=1

wk zk, (3.1)

where zk are independent identically distributed random variables and the weights aredefined as

N∑

k=1

wk = 100. (3.2)

The motivation for this approach is that agencies would also take their rating decisions onthe base of various factors with different weights. It is worth to mention at this point thatone can also take into account correlations between the different factors in the “real world”that are not regarded here. For the following discussion S0 is calculated using only twofactors (N = 2) with weights w1 = 60 and w2 = 40, respectively. This is sufficient becauseit sketches the main ideas of the method. This score value 0 ≤ S0 ≤ 100 is mapped ontoan initial rating grade in a second step. Three initial grades2 (n = 3) are used to describethe creditworthiness of the obligors: A (best rating for S0 > 70), B (intermediate gradefor 25 ≥ S0 ≤ 70) and C (near default for S0 < 25). The subdivision of the scoring rangeis also shown in Figure 3.1(a).The initial score was simulated for 50, 000 obligors and translated into a rating grade. Theresults are summarized in Table 3.1. Nearly 70 % of the obligors obtain an intermediaterating (B). This is due to the fact that two factors with different weights contribute to thescoring.

Rating A B C

Number 9, 238 31, 212 6, 596

Percentage 18.6% 68.2% 13.2%

Table 3.1: Distribution of initial rating grades. Equations 3.1 and 3.2 with parameters

N = 2, w1 = 60 and w2 = 40, respectively are used to extract the ratings for 50, 000

obligors.

In order to obtain the rating transition probabilities, the obligor’s rating grade at the one-year time horizon is determined in the third step. The scoring S1 is calculated using the

2As outlined in Chapter 1, the Basel Capital Accord requires at least seven different grades in addition to

default. The described rating process is only a simplified approach and therefore the choice of three rating

grades is sufficient for the purposes of this work.

23

Figure 3.1: Mapping of the initial (a) and one-year (b) scoring onto the rating grades A to

C and A to D, respectively.

time evaluation of the initial random factors zk. Two main assumptions build the boundaryconditions of this time evaluation: negative values for the rating score are not allowed(S1 ≥ 0) and on average the score should not change (S1/S0 ≈ 1). Hence, Equation 3.1becomes

S1 =N

∑

k=1

wk zk eNk(0,1), (3.3)

where the Nk(0, 1) are standard normally distributed random variables. Again, the weightssum up as for the initial score (see Equation 3.2). Equation 3.3 implies that the one-yearscoring can take values S1 > 100. In addition, the possible range of scoring values mustinclude default events. Therefore, compared to the initial rating, the mapping of S1 ontorating grades is slightly modified. The boundary of the best grade is shifted upwards anda new boundary for the defaulted grade (D) is introduced. Figure 3.1(b) represents themodified mapping that is used to assign rating grades at the time horizon. The resultingchanges of the rating grades yield to the transition matrix that is shown in Table 3.2.

Initial Rating at Risk Horizon

rating A B C D

A 62.32 33.93 3.58 0.17

B 35.22 46.79 15.69 2.30

C 6.17 30.02 37.54 26.27

Table 3.2: Simulated one-year transition matrix (quotations in %). Rating grade D repre-

sents the default state.

24

The features of this artificially obtained transition matrix are similar to those of theS & P migration matrix shown in Table 2.1. PD is reduced with increasing rating grade andthe probability to migrate from one rating grade to the next nearest neighbor (e.g. from Ato C) is lower than the transition probability to the nearest neighbor (A to B). In addition,the simulation results follow a behavioral model of rating assignment proposed by Keenanand Sobehart [17]. This model is based on a specific notion of the rating analyst’s perceptionof risk and an explicit description of the risk factors to which the analysts are assumed tobe responding. The authors argue that the risk assessment process is founded on a relativecomparison between perceived risk for pairs of risk exposures. Let E be the obligor’s riskexposure with the expected probability of default PD(E) and R the rating that expressesthe resultant risk perception. Taking also into account the saturation of PD ≤ 1 for largerisk exposures, a relative relationship between PD(E) and R can be obtained assuming thatthe perception of two risk exposures differs by a just noticeable amount when separated bya given increment. Then, when the risk exposures are increased, the increment must beproportionally increased for the difference in perception to remain just noticeable. In firstorder approximation, this can be described as

(1 − PD)∆R = a∆PD

PD, (3.4)

where PD = PD(E). In the limit ∆E → 0, the integration of Equation 3.4 yields to

log

(

PD

1 − PD

)

=1

a(T )(R − b(T )) . (3.5)

Here, a(T ) is a parameter describing the risk sensitivity with respect to the time horizon Tand b(T ) provides the reference risk rating, respectively.The simulation results of the external rating process summarized in Table 3.2 were analyzedwith respect to Equation 3.5. Therefore, the rating grades R are measured in numbersrepresenting the quantity R in the above outlined equations. The A rating is translatedinto R = 0, R = 1 represents the rating grade B and the C rating means R = 2, respectively.Figure 3.2 shows the relation between these numbers and the default rates in terms of theleft side of Equation 3.5. The most important feature of this plot is the linear behavior ofthe extracted data. This is an example for the Weber-Fechner law that is mainly observedin psychology and physiology. It states that human sensations tend to be measured in arelative sense yielding to logarithmic functions of the stimulus. One prominent example ofthis law is the physical feeling of sound [22]. The increase of the sound pressure level followsthe same law as Equation 3.5.In order to determine the specific parameters a and b, a linear fit was applied to thesimulated data (dashed line in Figure 3.2). For the one-year time horizon, a(1y) = 0.375and b(1y) = 2.394, respectively. As there are only three rating grades, these results cannotbe compared with the analysis of the S& P or Moodys data (17 grades in both cases) carriedout in the corresponding reference [17], where a ≈ 1.5 and b ≈ 18.5 for a one-year timehorizon. For the authors, these values are consistent with the notions that the agency ratingshave an accuracy of approximately one rating notch, and CCC(S& P) or Caa (Moodys)ratings show similar characteristics to defaulters.

Despite the consistency of the simulated data it is worth to mention that the abovedescribed approach is only intended to get access to a market data set that provides in-formation beyond the rating grade. In addition, the complete external rating process is

25

-7

-6

-5

-4

-3

-2

-1

0

-1 0 1 2Rating notch

log(

PD

/(1-

PD

))

A

B

C

Figure 3.2: Analysis between risk perception and PD. The filled diamonds represent the

values extracted from Table 3.2, represented in a form of the left side of Equation 3.5, where

the rating grades are translated into numbers as discussed in the text. The dashed line is

a linear fit to the simulated data using a(1y) = 0.375 and b(1y) = 2.394.

accessible for simulation and Equations 3.1 to 3.3 can be extended in order to allow theconstruction of any scenario. However, there are disadvantages in the rating procedure.The subdivision into the scoring ranges shown in Figure 3.1 is carried out more or less ran-domly and is not based on any market observation. Compared to the three rating gradesused in the simulation experiment, the rating agencies provide a much more distinct ratingclassification (18 rating grades at the most). Finally, there is no plausible argument toobtain scoring values S1 > 100 at the risk horizon. This effect is due to Equation 3.3 and amodification seems to be required at this point. But the above described procedure is a verysimple way to obtain an artificial transition matrix for test purposes. The simulated resultis consistent with the main characteristics of the real transition matrix and a more detailedanalysis showed that the sensitivity of the rating assignment with respect to perceived riskfollows the same laws as the agencies results.

3.2 Internal Rating

The above gained information of the external rating process can now be combined withadditional information to obtain an internal rating. This procedure starts with the trans-formation of the initially assigned external rating grade to a fixed internal score. Themotivation for such a step is the following. Assuming that the bank intends to invest forthe first time in any security of an obligor, but has few information about and few expe-rience with the creditworthiness of the issuer (or counterparty) of this security. In sucha case the investment decision would probably depend on reliable rating information that

26

Figure 3.3: Mapping of the external rating grades (below the scoring range) to the internal

scoring (above the scoring). In addition, the scoring ranges for the internal rating grades

(a, b, c and d, respectively) are shown. A detailed description of the mapping procedure

and the assignment of the internal ratings is given in the text.

could be provided externally. Given that the bank has no additional information whichcontributes to the obligor’s rating, the external rating grade R is mapped on fixed internalscoring values Si

0 = M(R) (characterized by a superscript i) using the mapping rule

M(R) =

85 R = A45 R = B15 R = C

. (3.6)

M(R) is sketched in the lower part of Figure 3.3. The upper part of the same representationshows how the scoring, that again ranges from 0 to 100, is divided in the different scoringranges and internal rating grades necessary for the rating evaluation at the risk horizon. Thediscrete base values are not in the center of the corresponding ranges, but shifted towardsthe lower bounds. As in the external case, four rating grades describe the creditworthiness:a for the best, intermediate grade b, c for the worst and d describing the default case.Compared to the classification described in section 3.1, the division of the internal scoreinto the rating specific subranges is different but reasonable because internal rating criteriaare probably not the same as the external ones. It should also be pointed out that thismapping rule is applied only when the obligor “enters” the portfolio for the first timeand requires an initial internal rating grade. Obligors that already contribute to a bank’sportfolio are treated using the upper part of Figure 3.3. Therefore, a score at the riskhorizon T is derived from the initial grade. In this context “initial” corresponds to thetime of the last rating assignment and the score value Si

t is not restricted to the discretevalues obtained when using the mapping rule M(R). On the other hand the external ratinginformation at the risk horizon is also included in the internal score value via M(R).The time evaluation of the internal score takes into account the external rating information

27

at the risk horizon. Again a standard normally distributed random variable is used tosimulate the change of the internal score value that is always positive. Both, internal andexternal information contribute with equal weights to the scoring value at the risk horizon

Si1 =

Si0 eN(0,1) + M(R)

2, (3.7)

where M(R) is the same discrete mapping function that was used to obtain Si0 (see Fig-

ure 3.3). If more information is available, additional components with modified weightscan be included in the definition of Si. Obligors that are in the default state (externalD) contribute with M(D) = 0 to the internal one-year scoring. The upper part of thisrepresentation is again used to determine the internal rating grade of this scoring value.In the following it is assumed that 500 of the 50, 000 externally rated obligors actually have

or had any business relationship with the bank and therefore have also an internal rating.For this experiment, these obligors are arbitrarily chosen and the internal rating processwas applied as described above. In order to get a statistically relevant result, the ratingprocess for all obligors was carried out ten times. The resulting migration probabilities areshown in the left part of Table 3.3. It is noticeable that the probability for a transition ofan initially a rated obligor into the default state d is zero. This is not in accordance withthe capital accord described in Chapter 1. The IRB approach requires a minimum PD of0, 03% for a one-year risk horizon. Hence, the mirgation probabilities for initially a ratedobligors were adjusted in order to take into account for this fact. This adjustement is shownin the right part of Table 3.3. It has to be mentionned at this point that the supervisingauthorities would certainly have a problem if an IRB bank would apply such an adjustmentbecause the capital accord points out that the IRB approach requires a meaningful distri-bution across grades without excessive concetrations. Such concentrations would supportthe effect of PD = 0 in a speciffic rating class. The discussion at the end of the previoussection showed that the goal of this chapter is not to generate an exact copy of the completeexternal and internal rating process, but rather a schematic view of the context. This isalso the reason for the choice of only three (plus one) rating grades which is also not inaccordance with the IRB minimum requirements. The Basel II approach requires at leastseven non-default rating grades and one describing the borrower’s default case. However,for illustrative reasons the described approach provides all necessary information that isrequired to perform a Credit risk analysis. In analogy to the external rating simulation,Figure 3.4 shows the analysis of the risk sensitivity by applying Equation 3.5 to the PD

Internal Rating at Risk Horizon

Initial Simulated Adjusted

rating a b c d a b c d

a 47.14 50.48 2.38 0.00 47.13 50.46 2.38 0.03

b 20.82 62.78 15.85 0.54 20.82 62.78 15.85 0.54

c 2.19 27.66 47.03 23.13 2.19 27.66 47.03 23.13

Table 3.3: Simulated one-year internal transition matrix (quotations in %). The rating

grade d represents the default state. The right part of the table represents the transition

matrix after the IRB adjustment (see text).

28

-9

-8

-7

-6

-5

-4

-3

-2

-1

0 1 2Internal Rating notch

log(

p/(1

-p))

Figure 3.4: Analysis of internal risk perception. The left side of Equation 3.5 was applied

to the PD values of the adjusted internal transition matrix (Table 3.3, left part). Rating

notch 0 corresponds to the internal a rating, 1 to b and notch 2 represents c. The dashed

line is a linear fit to the simulated data using a(1y) = 0.289 and b(1y) = 2.402.

values summarized in Table 3.3. The linear fit to the simulated data points is shown asthe dashed line using ai(1y) = 0.289 and bi(1y) = 2.402. Compared to the results obtainedfrom the analysis of the external rating simulations, two noticeable effects are observed.First, the reference rating grade b(T ) remains nearly unchanged in both methods. This isdue to the fact that the external as well as the internal rating structure is composed ofthree plus one grades. Hence, the dimensionality for both problems is the same and thereference rating notch is in the range of 2.4 which is slightly “worse” than the external Cor the internal c grade.The second noticeable fact is the remarkably higher PD sensitivity of the external datacompared to the internally determined results. Table 3.4 summarizes the results of thesensitivity analysis as well as the absolute and relative differences of the parameters be-tween the internal and external rating processes for the same risk horizon. The relativedifference of approximately 30% in the sensitivity parameter indicates a difference in the

Parameter External Internal ∆abs ∆rel

a(T ) 0.375 0, 289 0.086 29.76%

b(T ) 2.394 2.402 0.008 0.33%

Table 3.4: Summary of the sensitivity analysis for the external and internal rating proce-

dures. The parameters a(T ) and b(T ) are obtained from a linear fit to the simulated data

using Equation 3.5.

29

accuracy of the rating grade. This accuracy is better for the internal ratings due to thelower value for a(T ). Reasons for this effect are the different choice of scoring ranges rep-resenting the rating grades (Figures 3.1 and 3.3) as well as the mapping rule (Equation 3.6).

3.3 Relations between External and Internal Ratings

The rating processes described in the previous sections provide the framework for the esti-mation of characteristic Credit risk values like the Credit–VaR , the total minimum capitaland other IRB values. External ratings reflect the evolvment of global economic circum-stances as well as firm specific developments for each obligor. It is therefore a view of theactual Credit risk market situation as well as on how this market evolves. The previoussection showed that the internal rating is assigned only for a subset of the externally ratedobligors and that the internal ratings differ slightly from the external ones. The constructionof the internal rating via the mapping rule (Equation 3.6) and the scoring value at the riskhorizon (Equation 3.7) implies that the internal rating is correlated to the market composedof external rating changes. But how can this correlation be measured? Correlations are theresult of the statistical analysis of two data sets. As for the internal rating assignment, abank can only use external information that is publicly available in order to improve theinternal ratings. There are different ways to get estimates for this correlation ρie.First, the rating grades can be used to obtain a value. But this approach is somewhat dif-ficult because both grades, external as well as internal must be transformed into numbers.Therefore it is reasonable to use the distribution of scoring values at the risk horizon as ameasure for the estimation of the correlation. Internal scoring values are calculated usingEquation 3.7, whereas the mapping rule3 defines the external scoring values. This methodyields to a correlation of ρ i e ≈ 0.38 for the subset of 500 obligors that are rated internallyand externally. It was outlined in section 3.1 that the scoring at the risk horizon could beabove the maximum value of the scoring interval [0, 100] used to identify the rating grades.This effect is also observed for the internal rating process (cf. Figure 3.3). If internal scoringvalues are adjusted to a maximum value Si

max = 100 for those scores that are outside theinterval, ρ i e ≈ 0.56. It is clear that the correlation is increased in this case because theexternal scoring values remain unchanged and hence the difference between internal andexternal scorings is reduced for the adjusted cases.Second, changes of scoring values for a given risk horizon could be used to calculate ρie. Butthis way implies more difficulties because it is not clear which deltas ∆S = S(T )−S(T = t0)should be used. The ideal approach would use internal as well as external scoring valuesat both time stamps. These are the true measures of the rating processes. Unfortunatelyfinancial institutes do not have access to the agencies’ scoring values and have to fall backon the mapping rule in order to generate artificial scoring values. Within the introducedmodel approach this ends up in the fact that the external rating for both times is translatedinto discrete values so that ∆S will also take discrete values. The internal rating processdescribed in section 3.2 also uses the mapping rule in order to obtain an initial scoring value.The internal scoring value Si

T is the only continuous value in this calculation that yields tocorrelation of ρ i e ≈ 0.21. If the external scorings were known, the ∆S can be calculated for

3As mentioned in the previous section, Equation 3.6 is extended to the default case (R = D), where

M(S) = 0.

30

internal as well as external scorings. Applying this method results in ρ i e ≈ 0.19. Hence,the methods that use the changes in the scoring values ∆S lead to similar results. Theestimation of the rating migration probabilities is also based on rating changes so that itseems reasonable to base the calculation of ρ i e on the scoring differences.The application of Equation 3.6 affects the result, but the observed perturbation of the cor-relation is within an acceptable statistical tolerance. Compared to the fluctuations observedwhen calculating ρ i e based on the scoring values at the risk horizon as described above,the differences of the correlation obtained by using ∆S are relatively small. This fact is anadditional indicator for the application of the ∆ based method.

31

Chapter 4

Results on a Sample Portfolio

The following chapter is intended to present the results of the various analytical and numer-ical methods that are used to determine the Credit–VaR as discussed within the previouschapters.Based on the artificial internal and external rating processes that are composed of three(or four, if the default case is also considered as a rating grade, respectively) rating classes,the CreditMetricsTM approach is fully applied. A distribution of portfolio values at the riskhorizon is generated and the values for expected (mean) loss, unexpected loss (quantile) aswell as characteristic values on the level of the different securities (marginal VaR) will bedetermined. It has to be mentioned here that the CreditMetricsTM methodology could alsobe performed using more that three rating grades (e.g. the Standard & Poors classification,cf. Table 2.1). However, this makes no difference with respect to the described approachbecause the simulation of rating transitions will be carried out on the same scale, but withmore than three thresholds that define the rating grade transition thresholds.In addition, the regulatory capital as defined in Equations 1.2 to 1.7 as well as the ana-lytic and the numerical VaR using the one-factor CreditMetricsTM model will be calculated.These approaches only depend on PD and the rating classification can be omitted. As inthe case of the multifactor CreditMetricsTM approach it is important to carry out all thecalculations on the same set of assumptions: rating structure, PD , evaluation parameters,transition matrix, portfolio.The chapter starts with the definition of the used yield curve and describes the portfoliothat is used for the calculations. The main part of the chapter is dedicated to the simu-lation of the future portfolio values within the CreditMetricsTM framework. The results ofthese calculations will be discussed with respect to its main features. In addition, it willbe analysed how the results of the multifactor simulations can be linked to the regulatoryrequirements or the one-factor approach, respectively.

4.1 The Yield Curve

Based on quoted market bond prices, a yield curve was defined. The result is shown inFigure 4.1. This yield curve is used for the valuation of the instruments as described inChapter 2.

32

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

5,5%

Jan-2004 Jul-2009 Jan-2015 Jun-2020

Date

Rat

e

Figure 4.1: Yield curve used for credit-risk valuation. The rates represent quoted market

values dated January 16, 2004.

Rates for times that do not correspond to the time buckets used for the generation of theyield curve are approximated using linear interpolation:

r(t) = r(ti < t) +r(ti+1) − r(ti)

ti+1 − ti(t − ti), (4.1)

where ti is the time bucket defining the rate ri before, and ti+1 is the time bucket definingthe rate ri+1 after t, respectively.In order to determine the rating grade dependent future exposure values of the differentsecurities with the use of Equation 2.6, a spread s(R) was added to the yield curve. Forthe sake of simplicity it is assumed that this spread is constant for all times so that eachrating grade uses the time dependence of the spread less yield curve

rR(t) = r(t) + s(R). (4.2)

Rating Grade Spread [bp]

A 15

B 150

C 460

Table 4.1: Spreads used for the shift of the yield curve for the different rating grades.

Table 4.1 summarizes the used spreads. These values are in good agreement with recent

33

spread values that are publicly available [24], assuming best (AAA), worst (CCC) and anintermediate(B) grades, regarding the S&P rating.

4.2 The Test Portfolio

The portfolio used to analyse the results of the applied methods is composed of 100 bonds(87 Coupon bonds and 13 Zero bonds), issued by 89 different obligors. The distribution ofrating grades for the contributing obligors and/or instruments is shown in Table 4.2. Asonly a subset of the obligors that are evaluated within the internal rating process is used,the condition for the choice of the ratings for the portfolio contributors was to be represen-tative for the internal pool of obligors. This requirement is fulfilled, even if the weight ofthe A-rated securities within the portfolio is much higher compared to the portion of A-rated counterparties that are subjected to the internal rating process. But this fact can beunderstood as an investment decision that prefers investments in securities of issuers withlower PD . Regarding the internal rating assignments, the mapping rule (Equation 3.6) isapplied because no historical information about the internal rating is available.

Portfolio Rated Obligors

Rating Grade Number Number Portion

A 29 84 16.80%

B 53 352 70.40%

C 12 64 12.80%

Sum 100 500 100%

Table 4.2: Composition of the portfolio with respect to the rating grades. The ratings of

the instruments represent a subset of the internally rated obligors.

Maturity Number

M ≤ 1y 9

1y < M ≤ 2y 8

2y < M ≤ 3y 13

3y < M ≤ 4y 6

4y < M ≤ 5y 13

5y < M ≤ 10y 24

10y < M ≤ 15y 17

15y < M ≤ 20y 10

Table 4.3: Maturities of the bonds contributing to the portfolio.

In addition, the nominal amount of the instruments ranges from 300, 000 EUR to 80Mio EUR.Maturities M range from one year to 20 years. The contribution to the different time buck-ets is listed in Table 4.3. With the use of the above introduced yield curve and Equation 2.6,

34

two characteristic portfolio values are calculated: the present value and the expected one-year’s exposure value provided that each obligor does not suffer any rating change. Thepresent value PV is the sum of all future cash flows that are discounted to the evalu-ation date using the official yield curve. The expected one-year’s future exposure valueV exp(t, T ; tRH ,R) assumes that the rating grade at the risk horizon R(tRH) remains thesame as today’s rating grade R(t). V exp is used to calibrate the loss distribution for themultifactor CreditMetricsTM simulations. Table 4.4 shows the results of these evaluations.It seems to be astonishing that today’s present value is larger than the expected future ex-posure value at the risk horizon, but this is a direct consequence of the evaluation methodthat uses the spread less yield curve to determine PV and includes spreads in order tocalculate V exp.An example for the valuation of a single position of the portfolio is given in Appendix A.

Quantity Value (Mio EUR)

Present Value (PV ) 1,079

Expected Value (V exp) 1,024

Table 4.4: Characteristic values of the test portfolio. The evaluation of these results is

carried out using the yield curve plotted in Figure 4.1 and Equation 2.6.

4.3 Regulatory Capital Requirements

In a first step, the positions of the portfolio are subjected to the Basel capital requirementformula using Equations 1.2 to 1.7. The amount of risk weighted assets RWA is calculatedwithin the framework of the IRB foundation approach. Therefore the following assumptionswere made for the determination of RWA:

• The effective maturity M is set to a value of 2.5 (Years). This value is required forthe use of the IRB foundation approach [4].

• The value for LGD depends on whether the seniority of the corresponding instru-ments are subordinated or not. For Senior Secured and Senior Unsecured instruments,LGD = 0, 45. Instruments with subordinated seniority classes (Senior Subordinated,Subordinated and Junior Subordinated) contribute with LGD = 0.75.

• Within the framework if the IRB approach, a bank has to use the internal estimatesfor PD. Hence, the adjusted internal PD following from the simulations described inChapter 3 and summarized in Table 3.3 are used.

• The estimation of EAD follows the concept presented in Section 2.2. The one-year’spresent value defined in Equation 2.12 is used in order to avoid default arbitrage.

As these assumptions are applied to all positions of the test portfolio, the risk weightedassets RWA are formed by the sum over all positions RWAP. The application of this for-malism leads to RWA = 1, 652.11 Mio EUR. The regulatory capital is just 8% of the RWA

35

and hence 132.17 Mio EUR, which is approximately in the order of 10 % of the portfoliovalue.

It was mentioned in Chapter 1 that banks can also apply the advanced IRB approach.In addition to the internal estimation of PD, this method also requires the estimate of LGD,EAD and M . Within this work, the CreditMetricsTM framework provides the estimates forLGD (via recovery rate) and EAD as described in Chapter 2. The estimation of the effectivematurity for instruments subject to a determined cash flow schedule is described in thecapital accord [4](paragraph 290).

M = min

∑

t

t · CFt

∑

t

CFt

, 5

, (4.3)

where CFt denotes the cash flows (principal, interest payments and fees) contractuallypayable by the borrower in period t measured in years. Even if this formula is applied, theeffective maturity must be at least one year (M ≥ 1 y).For the advanced IRB approach this leads to a value of RWA = 1, 728.91 Mio EUR andhence a regulatory capital of 138.31 Mio EUR. Compared to the IRB foundation approach,this value only differs by approximately 5%.Appendix A demonstrates the calculations of the regulatory capital requirement for a singleinstrument. This example gives an idea on how the IRB approach could be applied.

4.4 One-Factor CreditMetricsTM Approach

The one-factor approach can be used to determine the Credit–VaR in an analytical as wellas in a numerical way. The analytic VaR described in Equation 2.31 is linear with respect tothe single position Credit–VaR and it is therefore also possible to obtain the analytic valuefor each position. It makes use of the position’s risk components. LGD is the beta-simulatedvalue that was also used to obtain the capital requirement under the IRB advanced approachand EAD is the expected future exposure value V exp

k . The conditional default probabilityof each position PDk(z) is obtained using Equation 2.27. Here, z = 0.999 in order to beconsistent with the IRB approach (Equation 1.6).An important question that arises at this point is the use of the correlation that is linked tothe model variable of the one-factor approach. Equation 2.25 describes this characteristicvariable which depends on a common market factor, an idiosyncratic contribution andfinally on a market correlation ρk of each obligor. The IRB formula answers this questionby using a definition of the correlation, Equation 1.2. However, ρIRB = ρ(PD) is limited toa small range and it seems quite possible that correlations between obligors and the marketbehavior exceed 0.24 or are smaller than 0.12. The use of these correlations results in ananalytic Credit–VaR of 108.93Mio EUR for the test portfolio if the adjusted internal PD

are used. A value that is much smaller than the IRB capital requirement calculated in theprevious section. If the same calculation is carried out with the use of a common marketcorrelation ρ = 0.2 for all obligors, VaR = 122.21Mio EUR, a value that is much closer tothe IRB capital requirement.These results give some hints that the correlations of the Basel II formula do not represent

36

0

100

200

300

400

0 0,2 0,4 0,6 0,8 1

Correlation

Ana

lytic

VaR

[Mio

EU

R]

Figure 4.2: Correlation dependence of the analytic one-factor Credit–VaR . The correlation

is applied to all obligors contributing to the portfolio.

the “true” market correlation. But it is also difficult to find the exact definition of whatis meant by the market on one side and which indicator is measurable in the sense thatit can be used to determine correlations on the other side. One potential access to thisproblem was already mentioned in Section 3.3 within the context of the connection betweeninternal and external ratings. In that case the totality of all externally rated obligorswas defined as the market and the scoring and/or the rating changes are a measure thatcan be used to determine correlations. The applications of the described methods leadto common market correlations between 0.19 and 0.56. Even if the lower bound of thisrange corresponds to Basel II correlations, these results show that it is necessary to extendthe range of correlations defined within the IRB approach. The only problem in usingthe correlations revealing from the rating processes is that this method only provides thecorrelation between a portfolio of obligors and the market, but not between each obligorand the market as required by Equation 2.25. Such correlations would be the result of ananalysis of rating histories that is beyond the scope of this work.If a constant market correlation for all obligors is used for the calculation of the analyticCredit–VaR instead of Basel II correlations, it is possible to extract the dependence of theCredit–VaR on the correlation. Figure 4.2 shows this behavior that is approximately linearin ρ. This curve could be used to calibrate the analytic Credit–VaR with respect to theIRB capital requirement. For example, ρ ≈ 0.25 would lead to an analytic Credit–VaR of137.65Mio EUR that is in good agreement1 with the capital requirement resulting from theIRB advanced approach. The advantage of this calibration is its use on the level of singleobligors in the same way as for the entire portfolio.

1A more precise calibration can be obtained using a standard solver, e.g. the Excel solver.

37

0

5000

10000

15000

20000

25000

30000

0 50 100 150 200 250

Loss [Mio EUR]

Fre

quen

cyµ =29.95

q =143.85

Figure 4.3: Loss distribution of the numerical one-factor Credit–VaR . 650, 000 simulations

were performed on the test portfolio. The 99.9% quantile representing the Credit–VaR as

well as the mean µ of the loss distribution were plotted. In this example, a common market

correlation ρ = 0.2 for all obligors and adjusted internal PD as summarized in Table 3.3

were used. Note that about 13% of the paths leadt to a loss-free result. These realisations

are not plotted in this graph.

The numerical result of the one-factor CreditMetricsTM approach is obtained with theuse of Equation 2.25. Beside the common market factor Z, the idiosyncratic factor εk

for each obligor is also modeled so that the conditional default probability is not usedhere. The evaluation of each realisation is done with respect to the default event of eachobligor. This default event is characterised by a realisation that is below the obligor specificthreshold Zk < Φ−1(PDk). If an obligor defaults, the loss of the corresponding position isdetermined with the use of LGD and EAD, both determined as described above. In addition,the numerical implementation of this simplified internal Credit risk model allows to extracta loss distribution. The Credit–VaR is defined as the qth quantile of this loss distribution.The simulation of many realisations is necessary to approximate a stable value for theCredit–VaR. Figure 4.3 shows a loss distribution obtained for a simulation experiment with650, 000 paths. For small losses, the distribution shows a strong increase of the frequencywith increasing loss. In this range, the frequency of the loss contributions varies significantly.This is due to the fact that the test portfolio contains positions with a high PD. The defaultthreshold for those instruments is relatively low and as the experiment takes place in a two-state world (default or non-default) they contribute much more significantly to the totalportfolio loss. For high losses the frequency decreases monotonously indicating that highlosses in the range of 250Mio EUR are rare events. Two characteristic values are shown inthis loss distribution: the 99.9% quantile representing the Credit–VaR of the portfolio as

38

defined in Equation 2.31 and the distribution mean µ which is a measure for the portfolio’sexpected loss. Compared to the analytic Credit–VaR , the quantile value of the numericalimplementation of the one-factor CreditMetricsTM approach, q = 143.85Mio EUR, is about16% larger for the same model parameters (common market correlation, adjusted internalratings and PDs). This difference between the analytical and numerical result is due to theinsufficient diversification of the test portfolio. As discussed in Section 2.5, the fluctationsaround the expected Credit–VaR can be ignored for a portfolio consisting of a large numberof securities issued by a large number of obligors. This is not the case for the test portfolioand fluctuations caused by the idiosynchratic contributions to Equation 2.25 have an effecton the estimation of the Credit–VaR . Hence, it would be necessary to increase the number ofpositions in the portfolio in order to fulfil the law of large numbers. However, the numericalCredit–VaR determined in the described way is close to the advanced IRB regulatory capital.The relative difference between the results in this case is approximately 4%.

4.5 Multifactor CreditMetricsTM Approach

A third method for the estimation of the Credit–VaR is the application of the full or mul-tifactor CreditMetricsTM approach. The previous sections provided almost all the informa-tion necessary to perform the methodology: the portfolio information (securities, issuers,ratings and exposures); the transition matrices (internal and/or external), including PD.Only the obligor correlations are the missing key element for a successful realisation of theCreditMetricsTM Monte–Carlo simulations. Section 2.3 describes the procedure of mappingobligors onto industrial indexes and determining the obligor correlations on the base ofcorrelations between indexes. This formalism is not carried out here. The used correlationmatrix is based on the mapping of the obligors onto n indexes2 of a large index pool. Thetime series of the quoted index prices provides the information necessary to determine theindex correlations and to derive the obligor correlations. A part of the used 89× 89 obligorcorrelation matrix is shown in Appendix B.The distribution of portfolio values is obtained as described in Section 2.4. Figure 4.4shows the result at the risk horizon for 100, 000 correlated scenarios. The correspond-ing Monte-Carlo simulation uses the internally determined credit rating migration matrix.These adjusted transition probabilities are listed in Table 3.3. The pronounced asymmetricform of this histogram is characteristic for the calculation of the Credit–VaR with the useof the CreditMetricsTM approach [15]. The distribution has a maximum for portfolio valuesof the order to 1, 000Mio EUR that decreases sharply for higher portfolio values. The leftside of the histogram shows a fat-tail behavior. This asymmetric form is due to the factthat, compared to rating improvements, rating deteriorations have a much higher impacton the future exposure value at the risk horizon. In addition, the over-all probability thatan obligor sustains a downgrade is higher than the probability for an upgrade. This effectis also a reason for the asymmetric form shown in Figure 4.4. Moreover the correlationsbetween obligors are mainly positive so that a correlated scenario that results in a defaultevent for one obligor would probably imply the default of other obligors too. Such an effectis a further reason for the fat-tail behavior of the distribution of future portfolio values.However, the majority of the correlated scenarios result in an unchanged rating at the riskhorizon, leading to the described maximum.

2For each obligor issuing securities of the test portfolio n is between 1 and 5.

39

0

200

400

600

800

1000

1200

700 800 900 1000 1100

Value [Mio EUR]

Fre

quen

cy

Figure 4.4: Distribution of the total portfolio value at the risk horizon for 100, 000 correlated

scenarios.

0

200

400

600

800

1000

1200

-100 0 100 200 300 400

Loss [Mio EUR]

Fre

quen

cy

VaRq =203.51

µ =29.52

Figure 4.5: Loss distribution of the test portfolio value at the risk horizon. This plot follows

from Figure 4.4 assuming that the initial rating remains unchanged until the risk horizon.

40

0

200

400

600

800

1000

1200

700 800 900 1000 1100

Value [Mio EUR]

Fre

quen

cy

ExternalInternal

700 800 900

Figure 4.6: Comparison of future portfolio value distributions. Each curve bases on the

simulation of 100, 000 scenarios with the use of the adjusted internal as well as the external

rating transition matrix. The insert shows the behavior of the distributions’ fat tail on a

logarithmic scale for the frequency.

A loss distribution is extracted from Figure 4.4 by subtracting the future portfolio valuesfrom the expected value at the risk horizon V exp. This distribution is shown in Figure 4.5.It is just the mirror image of the portfolio value distribution that is shifted so that theresult for V exp is loss-free. The loss distribution also contains information about the meanµ and the 99.9% quantile q. The mean µ = 29.52Mio EUR, representing a measure for theexpected loss, is more or less exactly the same value that is obtained within the frameworkof the one-factor CreditMetricsTM approach. The reason for this effect is that the meanis essentially dominated by correlated scenarios that do not change the obligors’ ratings.The fat tail behavior of the loss distributions for both methodologies has a minor influenceon the expected loss. Obviously this is different when looking at the quantile value. Amuch higher result is obtained for the full CreditMetricsTM approach. And even the valueof the Credit–VaR defined in Equation 2.23 leads to a relatively high value compared to theone-factor result:

VaR = q − µ = 173.99Mio EUR,

where q represents the 100th lowest portfolio value of the 100, 000 simulations. This dif-ference in the Credit–VaR can be attributed to the use of correlated scenarios. In theone-factor approach independent random numbers are used to simulate the market realisa-tions. The multifactor CreditMetricsTM approach uses obligor correlations and hence themarket realisations depend on different random numbers. Therefore it is possible that abad asset return for one obligor could pull down other – highly correlated – obligor returnsyielding to a multiple default event. Such realisations form the fat tail which is much more

41

Approach Credit–VaR (Mio EUR)

IRB

Foundation 132.17

IRB advanced 138.31

One-Factor CreditMetricsTM

IRB correlations 108.93

ρ = 0.2 122.21

ρ = 0.25 137.65

ρ = 0.375 173.99

Multifactor CreditMetricsTM

Internal ratings 173.99

External ratings 227.01

Table 4.5: Summary of results of the Credit–VaR calculations using the various methods

described within this work.

pronounced in the multifactor case. It is interesting to note that the simulated Credit–VaRalso follows from the analytical result (cf. Equation 2.31 or Figure 4.2.) if a common marketcorrelation ρ = 0.375 is assumed. This corresponds approximately to the average obligorcorrelation ρ ≈ 0.35 following from the upper or lower triangle of the obligor correlationmatrix that was used for the CreditMetricsTM simulations.The same simulations can also be performed for the externally assigned ratings. One obtainsfuture portfolio value and loss distributions that have the same shape as the distributionsshown in Figures 4.4 and 4.5. A comparison of the resulting distributions of portfolio valuesis shown in Figure 4.6. The maximum of the external ratings based (ERB) portfolio distri-bution is shifted towards higher portfolio values. In addition, the fat tail behavior of the twodistributions is slightly different as shown on a logarithmic scale in the insert of Figure 4.6.The ERB fat tail is much more pronounced due to the increased probabilities for transitionsin deteriorated grades and PDs following from the external rating assignment. Both effectslead to an increased value for the Credit–VaR = 227.01Mio EUR (99.9% quantile) whenexternal rating transitions are used.In summary, the applied methods lead to somewhat different values for the Credit–VaRthat are resumed in Table 4.5. Pricipally, this summary shows that the order of magni-tude of the resulting values can be divided into two classes: IRB and ERB Credit–VaR. Aslong as one carries out the calculation of characteristic Credit risk values within the sameclass, it is simple to translate the IRB capital requirement into the results obtained withthe CreditMetricsTM (one or multi-factor) approaches. However, as the CreditMetricsTM

methodology is intended for the use of rating migrations probabilities and PDs providedby the rating agencies, the CreditMetricsTM Credit–VaR belongs to the ERB class. In thiscase the translation into an IRB Credit–VaR is not as simple as described above. A firststep for the definition of rules for the translation of ERB into IRB results is the analysisof the marginal Credit–VaR on the level of single positions. The marginal VaR was de-fined in Equation 2.24 as the difference between the Credit–VaR of the entire portfolio andthe Credit–VaR of the portfolio without the position. An interesting way to visualize themarginal Credit–VaR is to plot its relative contribution with respect to the present value,

42

0%

10%

20%

30%

40%

50%

0 20 40 60 80PV [Mio EUR]

mar

gina

l VaR

[%]

Figure 4.7: Marginal Credit–VaR for the positions of the test portfolio. The relative con-

tributions for each position are plotted against the corresponding present value. The solid

line represents a marginal VaR of 5Mio EUR.

VaRm(Ik)/PV (Ik) against the present value of each position. This is shown in Figure 4.7.Points in the upper left of the chart represent assets which are risky in percent terms, butwhose exposure sizes are small, while points in the lower right represent large exposureswhich have relatively small chances of undergoing credit losses. Note that the product ofthe two coordinates (that is, the relative marginal Credit–VaR multiplied by the presentvalue) gives the absolute marginal risk. The curve in Figure 4.7 represents points withthe same absolute risk. Points which fall above the curve have greater absolute risk, whilepoints which fall below have less.The marginal Credit–VaR indicates the effect of diversification if it is compared to a posi-tion’s stand–alone Credit–VaR (VaRs). This value is obtained if the future exposure valuesof the simulated scenarios are used to extract the value distribution at the risk horizon onthe level of single positions. In accordance with the CreditMetricsTM Technical Document ,it is observed that

VaRm ≤ VaRs. (4.4)

The diversification effect scales with the rating grade (for the higher rated securities, there isa greater reduction from the stand-alone to marginal risk than for the lower rated securities),with the correlation to other obligors (or instruments, respectively) as well as with thenumber of assets that are in the portfolio. A larger portfolio would be required to diversifythe effects of riskier credit instruments and hence to reduce VaRm. This observation in inline with the observations made in the context of the one-factor CreditMetricsTM approach(see Section 4.4).

43

4.6 A Link between Internal Model and Basel II

It was mentioned in the previous section that the original CreditMetricsTM approach usesexternal rating migration probabilities. In contrast, the basis of the IRB approaches isthe internal estimation of PD. Hence, it is remarkably useful to find a connection betweenthe regulatory capital requirements and the characteristic values extracted from an internalportfolio model.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10

CM-Value [Mio EUR]

IRB

-Val

ue [M

io E

UR

]

Figure 4.8: Comparison of risk contributions on the level of single positions on a linear

scale.

The IRB approach as well as the one-factor CreditMetricsTM approach are additive methodsso that the obtained results can be broken down to the level of single positions in ananalytic way. In contrast, the decomposition of the risk contributions for the multifactorCreditMetricsTM methodology is much more complex. The previous section revealed twopossible measures for such a decomposition: the marginal VaR (VaRm) and the stand-aloneVaR (VaRs). In Figure 4.8, the IRB results are plotted against the marginal VaRm. Mostof the points in this representation are concentrated in the lower left corner. In particular,the IRB results only depend weakly on VaRm, remaining below the value of ≈ 2Mio EURfor a majority of the positions. However, for high VaRm noticeable fluctuations in theIRB VaR are observed. In order to determine an functional relationship between the IRBand CreditMetricsTM results, the data of Figure 4.8 are plotted double-logarithmic scale.Figure 4.9 shows this representation, where the marginal VaR dependence of the IRB resultis additionally split into the three different external rating grades A, B and C. The linesin the three subfigures are least square fits [21] that are carried out for each data subset ofthe corresponding external rating grade R.

log(y) = mR · log(x) + bR (4.5)

44

-6-5-4-3-2-101

-5 -4 -3 -2 -1 0 1 2 3

-4

-3

-2

-1

0

1

2

-3 -2 -1 0 1 2

-1

0

1

2

3

-1 0 1 2 3

A

B

C

Figure 4.9: Rating dependent comparison of risk contributions on the level of single positions

on a double-logarithmic scale. The three subfigures represent the results of the different

external rating grades. Since the same data as in Figure 4.8 is shown, the axis labels are

omitted. Black lines are linear approximations to the corresponding data points.

45

Rating Symbol m b R2

A � 0.688 −1.809 0.675

B © 0.871 −0.901 0.663

C 4 0.870 0.675 0.754

Total 0.966 −0.986 0.644

Table 4.6: Results of the regression analysis for the comparison between IRB and portfolio

model Credit–VaR results. The results are split for the different rating grades R. The last

line represents the linear regression result independent from R.

The data points show significant fluctuations with respect to the estimated linear behav-ior, resulting in relatively low R2 values. Despite this fact, such a regression is one pos-sible approximation for the functional behavior between the Credit–VaR obtained withCreditMetricsTM and with the IRB approach. The results of this regression analysis aresummarized in Table 4.6. It is observed that, within these double logarithmic representa-tions, the slope of all trendlines is of the same range and that the difference of the ratingdependent functional behavior is mainly due to the offset bR. The rating grade dependencyof this offset indicates the risk of the rating grades in the low VaR limit. This standard riskis due to the initial PD that was attributed to each rating class. The lower bR, the lowerthe standard Credit risk. The linear assumption of Equation 4.5 for double logarithmicrepresentations can be translated into a power law

y = exp(bR) · xmR . (4.6)

This relationship is the approximation of an analytic expression to translate a Credit–VaRinto an IRB capital requirement. And once the value for the capital requirement is known,the internal PD can be deduced with the use of Equations 1.3 to 1.7. As it is a non-trivial taskto extract an analytic expression for PD(VaRIRB,R), this is done numerically by calculatingthe advanced IRB capital requirement with varying PD until the appropriate value is found.All other position specific input parameters that are necessary for the calculation of thecapital requirement remain constant for this method. Such an approach can be appliedbecause the capital requirement is a monotonous increasing function of PD.With this procedure it is possible to account for portfolio effects when estimating the internalPD. If, for example, an externally C-rated instrument with a low marginal VaR is addedto the portfolio, the estimated internal PD will be lower than the value that is generallyattributed to the internal rating grade c and vice versa. Hence, the PD values estimated withthe above mentioned method are adjusted to the risk structure of the underlying portfolio.The difference between the ERB based estimated internal PD and the internal PD displayedin Table 3.3,

∆PD = PDERB − PD

IRB, (4.7)

is a measure for the quality of the portfolio subjected to the transformation method. Fig-ure 4.10 shows the distribution of this difference for the analysis that is carried out foreach rating grade (upper parameter sets of Table 4.6). The shape of this distribution hasa pronounced peak at ∆ = 0 with extensions to positive as well as negative deviations. Inaddition, a tendency to ∆ > 0 is observed, indicating that the composition of the under-lying portfolio increases the Credit risk contribution of some of the securities. Of course,

46

-0,01 0,00 0,01 0,02Absolute Difference

Figure 4.10: Distribution of the difference between estimated PD and internal rating grade

specific PD. The y axis represents the frequency of the corresponding ∆ values.

the opposite effect is also observed because the portfolio includes risk reductions due to thecorrelations between obligors.In the same way the linear regression was done for Figure 4.9, one can apply the least squareprocedure to the entire dataset that is rating-independent. The linear approximation withinthe double logarithmic representation is shown in Figure 4.11. In contrast to the analysisabove, the estimation of PD (solid line) uses the same functional behavior for all securities.m = 0.966 and b = −0.986. But this method shows some artifacts. All C-rated securities(triangles) are above the linear fit. In addition, the slope parameter m is higher comparedto the rating dependent analysis. This is due to the summary of all rating grades. Theaccumulation of all data yields to an adjustment of the fitting parameters and hence a dis-tortion of the linear fit curve compared to the previous cases.Figure 4.12 shows the difference ∆ between the estimated and the internally used PD asdefined in Equation 4.7. The shape of the resulting distribution is broader than for therating-dependent analysis (Figure 4.10), indicating that the estimated PD deviate muchmore from those initially assumed for the calculation of the advanced IRB capital require-ment. This again is an artifact of using an general analytic expression for the mapping ofall data, independent of the rating grade.In summary, the above applied process represents an approximative method to assign in-ternal PD values based on a credit portfolio model. In this way, diversification effects aswell as risk concentrations of the underlying portfolio are reflected in the estimated PD .

47

-6

-5

-4

-3

-2

-1

0

1

2

3

-5 -3 -1 1 3Log (CM Value)

Log

(IR

B V

alue

)ABC

Figure 4.11: Regression analysis for the complete dataset, independent of the external rating

grades (Although, a differentiation between the grades is made by different icons). The solid

line represents the linear behavior whose parameters are shown in the last line of Table 4.6.

-0,01 0,00 0,01 0,02Absolute Difference

Figure 4.12: Distribution of the difference between estimated PD and internal PD. Compared

to Figure 4.10, the analysis here is rating grade independent.

48

The quality of the estimator particularly depends on the quality of the calibration of themethod. In the described case, a power law was assumed for the description of the relation-ship between the marginal Credit–VaR and the advanced IRB capital requirement. Thequality of the linear regression showed that this assumption is a simple approximation toa rather complex problem that certainly needs some improvement. However, the methodworks whether a distinction between different rating grades for the power law assumptionis made or not.

49

Summary and Outlook

Credit risk evaluation of a bond portfolio with the use of the CreditMetricsTM approachforms the key element of this work. It is founded on artificial transition probabilities andPD resulting from simulated external and internal rating processes. It was shown that thecharacteristic Credit risk values of the different methods (foundation and advanced IRBapproach, one- and multifactor CreditMetricsTM methodology) can be transfered one intoanother. Apart from the analytic relationship between Credit–VaR and IRB result, it ismainly the correlation parameter that is responsible for such transformations.Under the model assumptions made in this work it is possible to find a connection betweenthe IRB approach and the CreditMetricsTM methodology on the level of single positions. Inother words, an answer to the first and central question initially asked is found, arriving atfull circle. In first approximation, the relationship between the marginal Credit–VaR andthe single position IRB capital requirement follows a power law.Internal portfolio models account for the issuer and/or position dependent risk contribu-tions within a specific portfolio. These portfolio effects can be directly mapped onto theIRB requirements an an analytic way and lead to the estimation of PD that also dependson the composition of the portfolio. As the CreditMetricsTM approach is based on externalrating grade transition probabilities, the described method even allows to assign an internalPD. This possibility is an interesting feature for initial assignments of internal ratings.

It is clear that the described calibration method requires the definition and the use ofan internal rating structure that is conform with the Basel capital accord. But even if themodeled rating structure differs from the commonly used scheme applied by rating agencies,it shows the same qualitative behavior with regard to the transition probabilities and PD.Hence, the model assumptions can easily be applied to extended rating structures that areof practical relevance. In addition, the implementation of an internal credit portfolio modelis not far from reality. However, it was shown that the power law assumption still needsimprovements. It is therefore necessary to implement the method within a more practicaland market related framework.Furthermore there are probably other ways to build the link between IRB and port-folio model approach. In this context it would be interesting to study the one-factorCreditMetricsTM approach if the correlation to the common market factor is defined oth-erwise. One can use the bank’s portfolio as a common (internal) market or index. Thecorrelations between this market and its securities can be determined using time seriesanalysis. In a second step, these correlations can be translated into default probabilities us-ing the IRB formulas. Values for PD obtained in this way should also reflect portfolio effects.

50

The described method and the obtained results can open new and interesting perspec-tives for Credit risk evaluation, estimation of PD and the incorporation of internal portfolioeffects within a regulatory framework. Up to now, the capital accord rejects full internalmodel approaches. But having a quick look at the developments of market risk evaluationshows that the regulatory acceptance of internal models was the result of an evolutionaryprocess. And – to end with a question – why should this not be the same for Credit risk?

51

Appendix A

Example of Evaluation

This appendix gives an example for the valuation of a Coupon bond. In the following, thepresent value PV as well as the future exposure value assuming an unchanged rating gradeV exp at the risk horizon are calculated.

A.1 Instrument parameters

One of the 100 bonds of the test portfolio has the following characteristics

Parameter Value

Instrument No. 20

Issuer EMIT 15

Nominal Amount 15M EUR

Maturity 16.01.2007

Coupon rate 4.00%

Obligor Rating B

Seniority Class Senior Secured

Recovery Rate 0.937

Table A.1: Instrument parameters for one Coupon bond of the portfolio. Instrument number

and issuer name are chosen arbitrarily. The indicated recovery rate is the result of a

simulation as describes in section 2.2.

A.2 Present Value and Future Exposures

This information is used to calculate the projected cash flows as well as the correspondingpresent values. The results are summarized in Table A.2. The yield rates are extractedform the yield curve (Figure 4.1) using Equation 4.1. The present values are calculated

52

using continuous compounding and the act/act accrual method.In order to determine the rating grade dependent exposure values, the projected cash flowsare discounted with the corresponding spreads (cf. Table 4.1) using Equation 2.6. Table A.3summarizes the results for instrument No. 20. The future exposure value for the B gradecorresponds to the expected exposure V exp. Referring to section 4.2, this example showsthe effect of a higher PV (today) compared to V exp (at the risk horizon).

Date Projected CF Yield Present Value

16/01/2005 600, 000.00 2.369% 585, 912.78

16/01/2006 600, 000.00 2.835% 566, 880.75

16/01/2007 600, 000.00 3.215% 544, 782.79

16/01/2007 15, 000, 000.00 3.215% 13, 619, 569.85

Sum 15, 317, 146.17

Table A.2: Cash flow table for instrument No. 20. Values are quoted in EUR currency.

Rating Exposure

A 15, 671, 403

B 15, 248, 841

C 14, 385, 119

Default 14, 695, 191

Table A.3: Future exposure values of instrument No. 20 at the risk horizon. Values are

quoted in EUR currency.

A.3 Capital requirement

Following the IRB foundation approach, the capital requirement can be calculated on thebase of the quantities listed in the upper part of Table A.4. Which quantities must beprovided with the use of an internal rating process depends on whether the IRB foundationor the IRB advanced approach is applied. PD and EAD are the same for both methods,whereas LGD follows from the regulatory instructions for the foundation approach (FA) andis estimated in the case of the advanced IRB approach (AA). In addition, Equation 4.3 isused to calculate the effective maturity based on the cash flow table A.2 in the latter case.The obtained values for the risk weighted assets differ with respect to the applied method(RWAFA = 11, 711, 055EUR and RWAFA = 1, 743, 699EUR, respectively). This is due tothe fact that the recovery rate (and therefore LGD) is simulated when applying the advancedIRB approach. If one would use the same LGD for both methods, the difference would be lessdistinct: RWAAA(LGD = 0.45) = 12, 419, 708EUR. For the determination of the regulatorycapital the RWA are multiplied with the factor 0.08. This results in a capital requirementof 936, 884 EUR when applying the foundation approach and 139, 495 EUR if the advancedapproach with the indicated estimation for LGD is used.

53

Quantity Value

PD 0.54%

EAD 15, 686, 231

b(PD) 0.1540

BRW(PD) 0.1327

IRB Foundation

LGD 0.450

K 0.05973

RWA 11, 711, 055

IRB Advanced

LGD 0.063

M 2.89 y

K 0.00889

RWA 1, 743, 699

Table A.4: Quantities necessary for the calculation of the Basel II capital requirement for

instrument No. 20. In this case, the IRB foundation approach is applied. Values for EAD

and RWA are quoted in EUR currency.

A.4 One-Factor CreditMetricsTM Approach

The estimates used to determine th capital requirement can also be used to calculate theanalytic Credit–VaR following from the one-factor CreditMetricsTM approach discussed inSection 2.5. The conditional default probability is calculated with the use of Equations 2.27and 2.31. This is possible because the one-factor approach is considered as a risk-bucketapproach and the expectation value for the portfolio loss is the sum of expectation valuesfor the single instruments. Therefore the estimates used for the calculation of the capitalrequirements within the IRB advanced approach can be applied here, leading to a Credit–VaR of 101, 146EUR. Here, the calculation is based on the adjusted internal PD, so thatρ(PD = 0.54%) = 0.211.

A.5 Multifactor CreditMetricsTM Approach

The application of the full CreditMetricsTM methodology requires the calculation of futureexposure values at the risk horizon for all possible rating grades. Table A.5 summarizesthe results following from the application of Equations 2.6 for the non-default as well asEquation 2.12 for the default case. The yield curve that was used to obtain the instrument’svalues at the risk horizon was introduced in Section 4.1 together with the rating gradedependent spreads.For the evaluation of the correlated scenarios it is necessary to know the thresholds for allpossible rating grades at the risk horizon. Table 2.2 showed this for an initially BBBrated obligor within the S & P scheme. As this work uses a simplified rating scheme and a

54

Rating a b c d

Exposure 15.67 15.25 14.39 14.70

Table A.5: Exposures for instrument No. 20 used for the CreditMetricsTM Monte Carlo

simulations. These results are valid for internal as well as external rating grades. The

future exposure values are quoted in Mio EUR currency.

transition matrix that is different to the one presented in Table 2.1, Table A.6 shows thethresholds revealing from the adjusted internal transition matrix (cf. Table 3.3).

Initial Rating at Risk horizon

Rating a b c d

a − 0.061 −2.057 −3.469

b − 0.810 −0.982 −2, 608

c − 2.017 0.529 −0.735

Table A.6: Rating migration thresholds of the internal rating grade migration matrix.

Regarding the results for instrument No. 20, the following table summarizes the differentvalues Credit–VaR (marginal and single VaR). These quantities are defined in Sections 2.4and 4.5. It is observed that the single VaR is higher than the corresponding marginal VaR.This is due to portfolio effects. For instrument No. 20 such effects lead to a risk reduction.

Quantity Value (EUR)

Present Value 15, 318, 933

single VaR 866, 555

marginal VaR 556, 483

Table A.7: Simulation results for instrument No. 20. The 99.9 % quantile was assumed

for the extraction of the values. The Monte Carlo simulations were carried out using the

external transition matrix.

55

Appendix B

The Obligor Correlation Matrix

The test portfolio is composed of 100 securities that are issued by 89 obligors. Hence, a89×89 matrix describes obligor correlations. The following table shows a part (10 obligors,E 12 to E 20) of this large correlation matrix.

E 12 E 13 E 14 E 15 E 16 E 17 E 18 E 19 E 20 E 21

E 12 1 0.723 0.057 0.117 0.674 0.677 0.023 0.062 0.672 0.109

E 13 0.723 1 0.057 0.117 0.674 0.677 0.023 0.062 0.672 0.109

E 14 0.057 0.057 1 0.330 0.026 0.034 0.100 0.324 0.016 0.360

E 15 0.117 0.117 0.330 1 0.054 0.069 0.132 0.546 0.035 0.646

E 16 0.674 0.674 0.026 0.054 1 0.639 0.010 0.007 0.639 0.031

E 17 0.677 0.677 0.034 0.069 0.639 1 0.014 0.023 0.638 0.053

E 18 0.023 0.023 0.100 0.132 0.010 0.014 1 0.130 0.007 0.144

E 19 0.062 0.062 0.324 0.546 0.007 0.027 0.130 1 −0.003 0.597

E 20 0.672 0.672 0.016 0.035 0.639 0.638 0.007 −0.003 1 0.009

E 21 0.109 0.109 0.360 0.646 0.031 0.053 0.144 0.597 0.009 1

Table B.1: Part of the 89 × 89 obligor correlation matrix.

In total, it was possible to map the obligors onto 88 different industrial indexes. The 15most frequently used indexes are:

DAX30, CDAX-BANKS, DAX100-BANKS, MSCIEURO-FINACIALS,

EUROSTOXX-50, DOW-JONES-STOXX, MIB30, EUROSTOXX-BANKS,

MSCIWORLD-REALEST, FTSE, IBEX, DOW-JONES-BANKS,

MSCIWORLD-BANKS, MIB-BANKING, NYSE-FINANCIAL.

56

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57

[15] Gupton, G. M., Finger, C. C. and Bhatia, M.: CreditMetricsTM – Technical Document.Morgan Guaranty Trust Co., 1997.Remark: Currently, the document can be downloaded from www.riskmetrics.com.

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[24] The RiskMetrics homepage www.riskmetrics.com provides some sets of spread curvesthat can be used for the purposes of this work by applying for a guest user account.

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