Backtesting Value at Risk and Expected Shortfall (eBook)

Simona Roccioletti obtained her Master of Arts degree in Quantitative Asset and Risk Management at the University of Applied Sciences (bfi) Vienna, Austria.

Acknowledgements 6
Contents 7
List of Figures 10
List of Tables 12
Abbreviations 13
Symbols 14
Abstract 15
Chapter 1 Introduction 16
Chapter 2 Risk Measures and their Properties 20
2.1 Definition of risk measure 20
2.2 Value at Risk 22
2.3 Expected Shortfall 24
2.4 Expectiles 26
2.5 Coherent risk measures 29
2.5.1 Coherence of VaR 31
2.5.2 Coherence of ES 33
2.5.3 Coherence of Expectiles 33
2.6 Risk Measures: a deeper view 34
2.6.1 Convexity 34
2.6.2 Comonotonic Additivity 36
2.6.3 Law Invariance 37
2.6.4 Robustness 38
Chapter 3 Elicitability 41
3.1 Evaluate Point Forecasts 41
3.1.1 Consistency 45
3.1.2 Back to Elicitability 47
3.2 Elicitability of VaR 50
3.3 Elicitability of ES 51
3.4 Elicitability of Expectiles 53
Chapter 4 Backtesting 56
4.1 The Backtesting Idea 57
4.2 Backtesting VaR 59
4.2.1 Unconditional Coverage Tests 61
4.2.2 Conditional Coverage Tests 63
4.2.3 Backtesting with Information Variables 67
4.2.4 Regulatory Framework 68
4.3 Backtesting ES 71
4.3.1 Test 1. 73
4.3.2 Test 2. 75
4.3.3 Test 3. 79
4.3.4 Power of the tests 81
Chapter 5 Empirical Analysis 83
5.1 Data 83
5.2 Models 84
5.2.1 Normal Distribution 85
5.2.2 Student’s t-distribution 86
5.2.3 Kernel Density Estimation 86
5.2.4 GARCH Models 89
5.3 Backtesting results 90
5.3.1 VaR results 92
5.3.2 ES results 99
Chapter 6 Conclusions 110
Appendix A MATLAB Code 112
A.1 MATLAB variables 112
A.2 ESTIMATION OF RISK MEASURES 115
A.2.1 Normal model 115
A.2.2 Student’s t model 116
A.2.3 Kernel model 117
A.2.4 Garch with normal innovations 118
A.2.5 Garch with Student’s t innovations 120
A.3 Value at Risk Tests 121
A.4 Expected Shortfall Tests 125
A.5 Monte Carlo p-values 127
Appendix B Figures 130
B.1 DAX 130
B.2 FTSE 100 135
B.3 NIKKEI 140
B.4 EURO STOXX 50 146
Bibliography 151