financial and econometric models for credit risk management

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financial and econometric models for credit risk management
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Contents
1 Introduction: The Context Of Credit Risk Management 11
1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Structural, Reduced-Form, And Hybrid Models . . . . . . . . . . . 15
1.3 The CreditMetrics Approach . . . . . . . . . . . . . . . . . . . . . . 16
1.4 The Normal Assumption . . . . . . . . . . . . . . . . . . . . . . . . 18

2 The Stable Distribution 19
2.1 Definition And Parameters . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Properties Of Stable RandomVariables . . . . . . . . . . . . . . . . 21
2.3 Dependence Among Stable Random Elements . . . . . . . . . . . . 24
2.4 Studies Of Stable Value at Risk (VaR) And The Normal Case . . . 25
2.5 Common Performance Measures Under Gaussian And Stable Assumption
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.1 The Capital Asset Pricing Model (CAPM) . . . . . . . . . . 27
2.5.2 The Stable CAPM . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.3 The Arbitrage Pricing Theory (APT) . . . . . . . . . . . . . 29
2.5.4 The Stable APT . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.5 The JensenMeasure . . . . . . . . . . . . . . . . . . . . . . 30
2.5.6 Regression Estimators . . . . . . . . . . . . . . . . . . . . . 31

3 Stable Modeling In Credit Risk 33
3.1 Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 A One-Factor Model For Stable Credit Returns . . . . . . . . . . . 34
3.3 A New Approach For The Returns . . . . . . . . . . . . . . . . . . 36
3.4 Excursus: Liquidity Risk . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Credit Risk Evaluation For Single Assets . . . . . . . . . . . . . . . 40
3.6 A Portfolio Model With Independent Credit Returns . . . . . . . . 41
3.7 A Stable Portfolio Model With Dependent Credit Returns . . . . . 43
3.8 Comparison Of Empirical Results . . . . . . . . . . . . . . . . . . . 45
3.8.1 The Observed Portfolio Data . . . . . . . . . . . . . . . . . 45
3.8.2 Generating Comparable Risk-Free Bonds From The Yield
Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.8.3 Fitting The Empirical Time Series For Ri, Yi, and ˆ Ui . . . . 46
3.8.4 VaR-Results For The Independence Assumption . . . . . . . 46
3.8.5 VaR-Results For The Dependence Assumption . . . . . . . . 48

4 The Economics of Financial Prices 53
4.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 The Application Of ARMA Models For Credit Returns . . . . . . . 55
4.3 Nonfractional ARIMAModels . . . . . . . . . . . . . . . . . . . . . 58
4.4 Modeling Credit RiskWith GARCH(p,q) . . . . . . . . . . . . . . . 59
4.5 Stable GARCHModels . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 ARMAModelsWith GARCH In Errors . . . . . . . . . . . . . . . 62
4.7 SubordinatedModels . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Long-Range Dependence In Financial Time Series 65
5.1 Self-Similar Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Fractional Processes And The Hurst Exponent . . . . . . . . . . . . 67
5.2.1 Stationary Increments . . . . . . . . . . . . . . . . . . . . . 68
5.2.2 Definition of Fractional Brownian Motion . . . . . . . . . . . 68
5.2.3 Definition of Fractional Gaussian Noise . . . . . . . . . . . . 69
5.2.4 Fractional Processes With Stable Innovations . . . . . . . . 70
5.3 Detecting andMeasuring LRD . . . . . . . . . . . . . . . . . . . . . 71
5.3.1 The Aggregated Variance Method . . . . . . . . . . . . . . . 71
5.3.2 Absolute Values Of The Aggregated Series . . . . . . . . . . 72
5.3.3 Classical R/S Analysis . . . . . . . . . . . . . . . . . . . . . 72
5.3.4 TheModified Approach By Lo . . . . . . . . . . . . . . . . . 74
5.3.5 The Mansfield, Rachev, And Samorodnitsky’s Statistic (MRS) 76
5.4 Empirical Results: LRD In Credit Returns . . . . . . . . . . . . . . 78
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 Modeling The Returns Of Different Credit Ratings 91
6.1 Empirical Evidence: New Phenomena In Credit Data . . . . . . . . 91
6.2 The Concept Of Cointegration . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 A CaseWith Two Variables . . . . . . . . . . . . . . . . . . 96
6.2.2 Error CorrectionModels . . . . . . . . . . . . . . . . . . . . 97
6.2.3 Testing For Cointegration . . . . . . . . . . . . . . . . . . . 98
6.2.4 Unit-Roots And Integrated Processes . . . . . . . . . . . . . 98
6.2.5 Unit Roots In The Stable Case . . . . . . . . . . . . . . . . 100
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Cointegrated VAR 103
7.1 Lag Order Of The VECM . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Estimating Cointegrating Relations When Cointegration Rank is
Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 Estimating Cointegrating Relations When Cointegration Rank Is
Unknown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4 Determining The Cointegration Rank Of A VAR . . . . . . . . . . . 108
7.5 The Trace Test And The Maximum Eigenvalue Test . . . . . . . . . 111
7.6 Determining Cointegration Rank With Model Selection Criteria . . 112
7.7 Cointegration In CreditModeling . . . . . . . . . . . . . . . . . . . 114
7.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 VAR Models For Credit Returns 117
8.1 The Data And Variables . . . . . . . . . . . . . . . . . . . . . . . . 118
8.2 Testing For Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3 Specification Of The VECM . . . . . . . . . . . . . . . . . . . . . . 122
8.4 Revised Cointegrating Relations . . . . . . . . . . . . . . . . . . . . 124
8.4.1 Checking Model Settings and Further Evaluation . . . . . . 127
8.4.2 Analysis Of The Residuals . . . . . . . . . . . . . . . . . . . 127
8.4.3 The Systematic Credit Risk Component . . . . . . . . . . . 128
8.4.4 Results for the Systematic Credit Risk Component . . . . . 131
8.5 The Behavior Of The Treasury Returns . . . . . . . . . . . . . . . . 133
8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

9 Dynamic Volatility 137
9.1 Dynamic Interdependence In A Multivariate Framework . . . . . . 138
9.2 The Multivariate GARCH Model With Constant Correlation Matrix 139
9.3 TheMultivariate EWMAModel . . . . . . . . . . . . . . . . . . . . 142
9.4 Stable Subordination For Multivariate Stable GARCH-Type Models 145
9.5 Performance Measures For Volatility And Covariance Models . . . . 146
9.5.1 Statistical Loss Function . . . . . . . . . . . . . . . . . . . . 147
9.5.2 Loss FunctionsWith Economic Inference . . . . . . . . . . . 148
9.5.3 Evaluation Of VaR Estimates For Unconditional Coverage . 149
9.6 Persistence Of Bond Market Volatility . . . . . . . . . . . . . . . . 150
9.7 Forecast Horizon Of Volatility . . . . . . . . . . . . . . . . . . . . . 152
9.8 Results For The Stable GARCH And Stable EWMA - Comparison 153
9.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10 Fractional Models For Credit Data 161
10.1 Fractionally Integrated Time Series . . . . . . . . . . . . . . . . . . 162
10.2 Motivation For LRD In Financial Data . . . . . . . . . . . . . . . . 165
8 Table Of Contents
10.3 Testing For LRD In The Data . . . . . . . . . . . . . . . . . . . . . 167
10.4 Models For Long Memory In Volatility . . . . . . . . . . . . . . . . 172
10.5 Multivariate LRDModels . . . . . . . . . . . . . . . . . . . . . . . 174
10.6 The Gaussian FARIMA . . . . . . . . . . . . . . . . . . . . . . . . 177
10.7 The Stable FARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10.8 TheMultivariate FARIMAModel . . . . . . . . . . . . . . . . . . . 181
10.9 Conditional Volatility And FARIMA . . . . . . . . . . . . . . . . . 181
10.10Developing A Multivariate LRD Process For The Innovations Of
The Credit ReturnModel . . . . . . . . . . . . . . . . . . . . . . . 182
10.11Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

11 Estimation Of The Long-Memory Parameter For The Credit Return
Model 187
11.1 Review Of Existing Estimators . . . . . . . . . . . . . . . . . . . . 187
11.2 The Conditional Sum Of Squares Estimator . . . . . . . . . . . . . 188
11.2.1 Properties Of The CSS Estimator . . . . . . . . . . . . . . . 188
11.2.2 Modification of The CSS Estimator . . . . . . . . . . . . . . 189
11.3 Checking Inference For FARIMA . . . . . . . . . . . . . . . . . . . 191
11.3.1 Robustness Of The Estimator . . . . . . . . . . . . . . . . . 191
11.3.2 Significance Of The Estimates . . . . . . . . . . . . . . . . . 192
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

12 Empirical Long-Memory Analysis 197
12.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.2 Analysis Of The SACF . . . . . . . . . . . . . . . . . . . . . . . . . 198
12.3 Estimation AndModel Testing . . . . . . . . . . . . . . . . . . . . 198
12.4 Results Of Robustness Check . . . . . . . . . . . . . . . . . . . . . 204
12.5 Results Of Moving-Block Bootstrapping . . . . . . . . . . . . . . . 204
12.6 Analyzing Dependence In ut . . . . . . . . . . . . . . . . . . . . . . 207
12.7 Copulas As Measure For Dependence . . . . . . . . . . . . . . . . . 212
12.7.1 General Idea of Copulas . . . . . . . . . . . . . . . . . . . . 213
12.7.2 Gaussian Copula with Stable Marginals . . . . . . . . . . . . 214
12.8 Value at Risk Estimation For The Long-Memory Model . . . . . . . 216
12.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

13 Outlook - Further Applications Of The Credit Return Model 225
13.1 Proposal For Bond-Portfolio VaR . . . . . . . . . . . . . . . . . . . 225
13.1.1 Simulation-Based Risk Measurement . . . . . . . . . . . . . 226
13.1.2 Modeling The Risk Of A Single Bond-Position . . . . . . . . 226
13.1.3 Measuring Bond Portfolio Risk . . . . . . . . . . . . . . . . 230
13.2 Scenario Optimization - An Overview . . . . . . . . . . . . . . . . . 232
Table Of Contents 9
13.2.1 The Risk Optimization Process . . . . . . . . . . . . . . . . 232
13.2.2 VaR Measures For Portfolio Optimization . . . . . . . . . . 234
13.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

14 Conclusion 237
14.1 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
14.2 Review Of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

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