Preface to the 2nd edition xi

Preface to the 1st edition xiii

Some Hints for Troubleshooting xv

1 Estimating Credit Scores with Logit 1

Linking scores, default probabilities and observed default behavior 1

Estimating logit coefficients in Excel 4

Computing statistics after model estimation 8

Interpreting regression statistics 10

Prediction and scenario analysis 12

Treating outliers in input variables 16

Choosing the functional relationship between the score and explanatory variables 20

Concluding remarks 25

Appendix 25

Logit and probit 25

Marginal effects 25

Notes and literature 26

2 The Structural Approach to Default Prediction and Valuation 27

Default and valuation in a structural model 27

Implementing the Merton model with a one-year horizon 30

The iterative approach 30

A solution using equity values and equity volatilities 35

Implementing the Merton model with a T -year horizon 39

Credit spreads 43

CreditGrades 44

Appendix 50

Notes and literature 52

Assumptions 52

Literature 53

3 Transition Matrices 55

Cohort approach 56

Multi-period transitions 61

Hazard rate approach 63

Obtaining a generator matrix from a given transition matrix 69

Confidence intervals with the binomial distribution 71

Bootstrapped confidence intervals for the hazard approach 74

Notes and literature 78

Appendix 78

Matrix functions 78

4 Prediction of Default and Transition Rates 83

Candidate variables for prediction 83

Predicting investment-grade default rates with linear regression 85

Predicting investment-grade default rates with Poisson regression 88

Backtesting the prediction models 94

Predicting transition matrices 99

Adjusting transition matrices 100

Representing transition matrices with a single parameter 101

Shifting the transition matrix 103

Backtesting the transition forecasts 108

Scope of application 108

Notes and literature 110

Appendix 110

5 Prediction of Loss Given Default 115

Candidate variables for prediction 115

Instrument-related variables 116

Firm-specific variables 117

Macroeconomic variables 118

Industry variables 118

Creating a data set 119

Regression analysis of LGD 120

Backtesting predictions 123

Notes and literature 126

Appendix 126

6 Modeling and Estimating Default Correlations with the Asset Value Approach 131

Default correlation, joint default probabilities and the asset value approach 131

Calibrating the asset value approach to default experience: the method of moments 133

Estimating asset correlation with maximum likelihood 136

Exploring the reliability of estimators with a Monte Carlo study 144

Concluding remarks 147

Notes and literature 147

7 Measuring Credit Portfolio Risk with the Asset Value Approach 149

A default-mode model implemented in the spreadsheet 149

VBA implementation of a default-mode model 152

Importance sampling 156

Quasi Monte Carlo 160

Assessing Simulation Error 162

Exploiting portfolio structure in the VBA program 165

Dealing with parameter uncertainty 168

Extensions 170

First extension: Multi-factor model 170

Second extension: t-distributed asset values 171

Third extension: Random LGDs 173

Fourth extension: Other risk measures 175

Fifth extension: Multi-state modeling 177

Notes and literature 179

8 Validation of Rating Systems 181

Cumulative accuracy profile and accuracy ratios 182

Receiver operating characteristic (ROC) 185

Bootstrapping confidence intervals for the accuracy ratio 187

Interpreting caps and ROCs 190

Brier score 191

Testing the calibration of rating-specific default probabilities 192

Validation strategies 195

Testing for missing information 198

Notes and literature 201

9 Validation of Credit Portfolio Models 203

Testing distributions with the Berkowitz test 203

Example implementation of the Berkowitz test 206

Representing the loss distribution 207

Simulating the critical chi-square value 209

Testing modeling details: Berkowitz on subportfolios 211

Assessing power 214

Scope and limits of the test 216

Notes and literature 217

10 Credit Default Swaps and Risk-Neutral Default Probabilities 219

Describing the term structure of default: PDs cumulative, marginal and seen from today 220

From bond prices to risk-neutral default probabilities 221

Concepts and formulae 221

Implementation 225

Pricing a CDS 232

Refining the PD estimation 234

Market values for a CDS 237

Example 239

Estimating upfront CDS and the ‘Big Bang’ protocol 240

Pricing of a pro-rata basket 241

Forward CDS spreads 242

Example 243

Pricing of swaptions 243

Notes and literature 247

Appendix 247

Deriving the hazard rate for a CDS 247

11 Risk Analysis and Pricing of Structured Credit: CDOs and First-to-Default

Swaps 249

Estimating CDO risk with Monte Carlo simulation 249

The large homogeneous portfolio (LHP) approximation 253

Systemic risk of CDO tranches 256

Default times for first-to-default swaps 259

CDO pricing in the LHP framework 263

Simulation-based CDO pricing 272

Notes and literature 281

Appendix 282

Closed-form solution for the LHP model 282

Cholesky decomposition 283

Estimating PD structure from a CDS 284

12 Basel II and Internal Ratings 285

Calculating capital requirements in the Internal Ratings-Based (IRB) approach 285

Assessing a given grading structure 288

Towards an optimal grading structure 294

Notes and literature 297

Appendix A1 Visual Basics for Applications (VBA) 299

Appendix A2 Solver 307

Appendix A3 Maximum Likelihood Estimation and Newton’s Method 313

Appendix A4 Testing and Goodness of Fit 319

Appendix A5 User-defined Functions 325

Index 333