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iiiBanking Regulation with Value-at-Risk
Inauguraldissertation zur Erlangung des akademischen Grades
eines Doktors der Wirtschaftswissenschaften
der Universit at Mannheim
Raphael Paschke
vorgelegt im Herbstsemester 2009iv
Dekan: Prof. Dr. Hans H. Bauer
Referent: Prof. em. Dr. Dr. h.c. Wolfgang Buhler
Korreferent Prof. Dr. Hans-Jochen Bartels
Tag der mundlic hen Prufung: 02. Dezember 2009Acknowledgement
This project would not have been possible without the support and help of others.
First, I would like to express my gratitude to my advisor Prof. em. Dr. Dr. h.c.
Wolfgang Buhler for o ering me the possibility to work under his guidance and his
continuing support. I would also like to thank Prof. Dr. Hans-Jochen Bartels for
co-refereeing this thesis.
Many thanks go to my current and former colleagues, namely Dr. Jens Daum,
Dr. Christoph Engel, Dr. Sebastian Herzog, Dr. Christoph Heumann, Prof. Dr.
Christian Koziol, Dr. Marcel Prokopczuk, Christian Speck, Volker Sygusch, Prof.
Dr. Monika Trapp, and Volker Vonho , as well as our secretary Marion Baierlein.
Furthermore, I would like to express my gratitude to Julia for accompanying this
work and always having an open ear for my concerns.
Most of all, I thank my family { my parents Winfried and Isabella, and my brothers
Wilhelm, Christoph, and Tilmann. Their unfailing support and love throughout the
years made this thesis possible.
Mannheim, January 2010 Raphael Paschke
vviContents
Acknowledgement v
List of Tables xi
List of Figures xiv
List of Frequently Used Symbols xv
1 Introduction 1
1.1 Problem and Main Results . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Review of the Related Literature . . . . . . . . . . . . . . . . . . . . 5
2 Individual Decision and Valuation in Equilibrium 11
2.1 Valuation of Securities under No-Arbitrage . . . . . . . . . . . . . . . 13
2.2 Optimal Consumption and Optimal Terminal Wealth . . . . . . . . . 19
2.2.1 The Non-Regulated Investor . . . . . . . . . . . . . . . . . . . 19
2.2.2 The Regulated Investor . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Asset Prices in a Pure Exchange Equilibrium . . . . . . . . . . . . . . 37
2.3.1 De nition and Characterization of the Equilibrium . . . . . . 37
2.3.2 Equilibrium with VaR-constrained Agents . . . . . . . . . . . 45
3 Regulating the Banking Sector: The Banks’ Optimal Decision 53
3.1 The Banking Sector within the Economy . . . . . . . . . . . . . . . . 54
3.2 Characterization of the Optimal Solution . . . . . . . . . . . . . . . . 69
viiviii CONTENTS
3.2.1 Formulation of the Optimization Problems . . . . . . . . . . . 69
3.2.2 Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.3 Reaction of the Banking System to Regulation . . . . . . . . . 77
3.3 The Banking Sector in Partial Equilibrium . . . . . . . . . . . . . . . 85
3.3.1 Value of the Optimal Solution . . . . . . . . . . . . . . . . . . 86
3.3.2 Restrictiveness of the VaR Restriction . . . . . . . . . . . . . 89
3.3.3 VaR-induced Capital Provision . . . . . . . . . . . . . . . . . 93
3.3.4 The Impact of Regulation on Debt Markets . . . . . . . . . . 95
3.3.5 Implied Risk Aversion of the Banking System . . . . . . . . . 99
3.4 Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.4.1 Proof of the Optimal Solution . . . . . . . . . . . . . . . . . . 108
3.4.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . 115
3.4.3 Deduction of Dynamic Wealth . . . . . . . . . . . . . . . . . . 117
3.4.4 Proofs of the Results in Partial Equilibrium . . . . . . . . . . 122
3.4.5 Deduction of the Portfolio Decision . . . . . . . . . . . . . . . 125
4 Equilibrium Impact of a Regulated Banking Sector on Markets 127
4.1 Construction of the Pure Exchange Equilibrium . . . . . . . . . . . . 128
4.1.1 De nition and Existence . . . . . . . . . . . . . . . . . . . . . 128
4.1.2 The Investment Opportunity Set in Equilibrium . . . . . . . . 131
4.1.3 Asset Prices and Volatility in Equilibrium . . . . . . . . . . . 133
4.2 Comparing Partial and Pure Exchange Equilibria . . . . . . . . . . . 138
4.3 Beyond Full Rationality . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.1 Realized vs. Expected Shocks . . . . . . . . . . . . . . . . . . 143
4.3.2 Endogeneity of Volatility and VaR Estimations . . . . . . . . 146
4.3.3 Procyclicality and Credit Crunch . . . . . . . . . . . . . . . . 148
4.4 Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.4.1 Proofs for the General Equilibrium Section . . . . . . . . . . . 155
4.4.2 Comparison of Equilibria . . . . . . . . . . . . . . . . . . . . . 157CONTENTS ix
5 A Two-Sided Equilibrium Model 161
5.1 The Structure of the Two-Sided Equilibrium . . . . . . . . . . . . . . 163
5.2 Constant Leverage Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3 Controlling the Probability of Distress . . . . . . . . . . . . . . . . . 167
5.4 Appendix to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4.1 Constant Leverage . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4.2 Constant Probability of Distress . . . . . . . . . . . . . . . . . 174
6 Summary and Conclusion 177
6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 Robustness of the Results . . . . . . . . . . . . . . . . . . . . . . . . 180
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Bibliography 186x CONTENTS

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