Introduction to Credit Risk Modeling, Second Edition / Edition 2

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Contains Nearly 100 Pages of New Material

The recent financial crisis has shown that credit risk in particular and finance in general remain important fields for the application of mathematical concepts to real-life situations. While continuing to focus on common mathematical approaches to model credit portfolios, Introduction to Credit Risk Modeling, Second Edition presents updates on model developments that have occurred since the publication of the best-selling first edition.

New to the Second Edition

  • An expanded section on techniques for the generation of loss distributions
  • Introductory sections on new topics, such as spectral risk measures, an axiomatic approach to capital allocation, and nonhomogeneous Markov chains
  • Updated sections on the probability of default, exposure-at-default, loss-given-default, and regulatory capital
  • A new section on multi-period models
  • Recent developments in structured credit

The financial crisis illustrated the importance of effectively communicating model outcomes and ensuring that the variation in results is clearly understood by decision makers. The crisis also showed that more modeling and more analysis are superior to only one model. This accessible, self-contained book recommends using a variety of models to shed light on different aspects of the true nature of a credit risk problem, thereby allowing the problem to be viewed from different angles.

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Editorial Reviews

From the Publisher
… this is a concise book for exploring the limitations of credit risk models and, to a lesser degree, asset valuation models. Read this book for a companionable journey through some of the limiting assumptions that make the models tractable. … it may be the first one [book] that wastes no time in getting to the point, and moving on.
Annals of Actuarial Science, Vol. 5, June 2011

Bluhm, Overbeck, and Wagner offer help to mathematicians and physicists leaving the academy to work as risk or portfolio managers. For this introduction, they focus on main themes rather than details, and on portfolio rather than single obligor risk. … this second [edition] takes account of problems in the banking industry [from] 2007-09.
SciTech Book News, February 2011

Having a valid and up-to-date credit risk model (or models) is one of the most important aspects in today’s risk management. The models require quite a bit of technical as well as practical know-how. Introduction to Credit Risk Modeling serves this purpose well. … it would best fit the practitioner’s needs. For students it can also be of great use, as an introductory course for credit risk models. A great first step into credit risk modeling. … The book provides a nice coherent overview of the methods used in capital allocation. … The book is written in a mixture of theorem-proof and applied styles. … I find this rather pleasing, as it gives the reader the edge of theoretical exposition, which is extremely important. … One really useful side of the book is that it provides step-by-step guide to methods presented. This should be really appreciated in industry and among students. …
MAA Reviews, January 2011

Praise for the First Edition
This is an outstanding book on the default models that are used internally by financial institutions. This practical book delves into the mathematics, the assumptions and the approximations that practitioners apply to make these models work.
—Glyn A. Holton, Contingency Analysis

There are so many financial tools available today and numbers are likely to grow in the future. If you work in this field of credit risk modelling it is worth looking at the theoretical background, and this book is a well-rounded introduction.
Journal of the Operational Research Society

As an introductory survey it does an admirable job. … this book is an important guide into the field of credit risk models. Mainly for the practitioner … It is well written, fairly easy to follow.
—Horst Behncke, Zentralblatt MATH

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Product Details

Meet the Author

Over the years, Christian Bluhm has worked for Deutsche Bank, McKinsey, HypoVereinsbank’s Group Credit Portfolio Management, and Credit Suisse. He earned a Ph.D. in mathematics from the University of Erlangen-Nürnberg.

Ludger Overbeck is a professor of probability theory and quantitative finance and risk management in the Institute of Mathematics at the University of Giessen. During his career, he worked for Deutsche Bundesbank, Deutsche Bank, HypoVereinsbank/UniCredit, DZBank, and Commerzbank. He earned a Ph.D. in mathematics from the University of Bonn.

Christoph Wagner has worked for Deutsche Bank, Allianz Group Center, UniCredit/HypoVereinsbank, and Allianz Risk Transfer. He earned a Ph.D. in statistical physics from the Technical University of Munich.

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Table of Contents

Preface to Second Edition ix

Preface xiii

About the Authors xv

List of Figures xvii

1 The Basics of Credit Risk Management 1

1.1 Expected Loss 2

1.1.1 Probability of Default (PD) 4

1.1.2 The Exposure at Default 15

1.1.3 The Loss Given Default 20

1.1.4 A Remark on the Relation between PD, EAD, LGD 21

1.2 Unexpected Loss 22

1.2.1 Economic Capital 27

1.2.2 The Loss Distribution 29

1.2.3 Modeling Correlations by Means of Factor Models 36

1.3 Regulatory Capital and the Basel Initiative 45

2 Modeling Correlated Defaults 51

2.1 The Bernoulli Model 53

2.1.1 A General Bernoulli Mixture Model 55

2.1.2 Uniform Default Probability and Uniform Correlation 56

2.2 The Poisson Model 58

2.2.1 A General Poisson Mixture Model 59

2.2.2 Uniform Default Intensity and Uniform Correlation 60

2.3 Bernoulli versus Poisson Mixture 62

2.4 An Overview of Common Model Concepts 63

2.4.1 Moody's KMV's and RiskMetrics' Model Approach 65

2.4.2 Model Approach of CreditRisk+ 68

2.4.3 CreditPortfolioView 71

2.4.4 Basic Remarks on Dynamic Intensity Models 78

2.5 One-Factor/Sector Models 80

2.5.1 One-Factor Models in the Asset Value Model Setup 80

2.5.2 The CreditRisk+ One-Sector Model 97

2.5.3 Comparison of One-Factor and One-Sector Models 98

2.6 Loss Dependence by Means of Copula Functions 99

2.6.1 Copulas: Variations of a Scheme 103

2.7 Working Example on Asset Correlations 111

2.8 Generating the Portfolio Loss Distribution 118

2.8.1 Some Prerequisites from Probability Theory 120

2.8.2 Conditional Independence 134

2.8.3 Technique I: Recursive Generation 136

2.8.4 Technique II: Fourier Transformation 140

2.8.5 Technique III: Saddle-Point Approximation 142

2.8.6 Technique IV: Importance Sampling 145

3 Asset Value Models 151

3.1 Introduction and a Brief Guide to the Literature 151

3.2 A Few Words about Calls and Puts 152

3.2.1 Geometric Brownian Motion 154

3.2.2 Put and Call Options 155

3.3 Merton's Asset Value Model 162

3.3.1 Capital Structure: Option-Theoretic Approach 162

3.3.2 Asset from Equity Values 167

3.4 Transforming Equity into Asset Values: A Working Approach 169

3.4.1 Itô's Formula "Light" 170

3.4.2 Black-Scholes Partial Differential Equation 171

4 The CreditRisk+ Model 179

4.1 The Modeling Framework of CreditRisk+ 180

4.2 Construction Step 1: Independent Obligors 183

4.3 Construction Step 2: Sector Model 184

4.3.1 Sector Default Distribution 186

4.3.2 Sector Compound Distribution 190

4.3.3 Sector Convolution 193

4.3.4 Calculating the Loss Distribution 193

5 Risk Measures and Capital Allocation 197

5.1 Coherent Risk Measures and Expected Shortfall 198

5.1.1 Expected Shortfall 202

5.1.2 Spectral Risk Measures 204

5.1.3 Density of a Risk Measure 206

5.2 Contributory Capital 208

5.2.1 Axiomatic Approach to Capital Allocation 209

5.2.2 Capital Allocation in Practice 213

5.2.3 Variance/Covariance Approach 215

5.2.4 Capital Allocation w.r.t. Value-at-Risk 217

5.2.5 Capital Allocations w.r.t. Expected Shortfall 218

5.2.6 A Simulation Study 220

6 Term Structure of Default Probability 225

6.1 Survival Function and Hazard Rate 225

6.2 Risk-Neutral vs. Actual Default Probabilities 228

6.3 Term Structure Based on Historical Default Information 230

6.3.1 Exponential Term Structure 230

6.3.2 Direct Calibration of Multi-Year Default Probabilities 231

6.3.3 Migration Technique and Q-Matrices 235

6.3.4 A Non-Homogeneous Markov Chain Approach 246

6.4 Term Structure Based on Market Spreads 248

7 Credit Derivatives 255

7.1 Total Return Swaps 256

7.2 Credit Default Products 258

7.3 Basket Credit Derivatives 262

7.4 Credit Spread Products 273

7.5 Credit-Linked Notes 276

8 Collateralized Debt Obligations 281

8.1 Introduction to Collateralized Debt Obligations 284

8.1.1 Typical Cash Flow CDO Structure 286

8.1.2 Typical Synthetic CLO Structure 296

8.2 Different Roles of Banks in the CDO Market 298

8.2.1 The Originator's Point of View 298

8.2.2 The Investor's Point of View 306

8.3 CDOs from the Modeling Point of View 309

8.4 Multi-Period Credit Models 314

8.4.1 Migration Model 314

8.4.2 Correlated Default Time Models 319

8.4.3 First-Passage-Time Models 320

8.4.4 Stochastic Default Intensity Models 325

8.4.5 Intertemporal Dependence and Autocorrelation 326

8.5 Former Rating Agency Model: Moody's BET 330

8.6 Developments, Model Issues and Further Reading 338

References 345

Index 359

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