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Risk-Sensitive Capital Requirements
and Pro-Cyclicality in Lending
Inauguraldissertation zur Erlangung des akademischen Grades
eines Doktors der Wirtschaftswissenschaften
der Universit at Mannheim
vorgelegt an der Fakult at fur Betriebswirtschaftslehre
der Universit at Mannheim
Volker Sygusch
Mannheim, im Herbstsemester 2010ii
Dekan: Dr. Jurgen M. Schneider
Referent: Prof. em. Dr. Dr. h.c. Wolfgang Buhler
Korreferent: Prof. Dr. Peter Albrecht
Tag der mundlic hen Prufung: 24. Januar 2011Contents
List of Figures viii
List of Tables xii
List of Abbreviations xiii
List of Frequently Used Symbols xvi
Acknowledgements xix
I Introduction 1
1 The Issue of Pro-Cyclicality 3
1.1 They of Lending . . . . . . . . . . . . . . . . . . . . . 6
1.2 The Basel I Accord and Pro-Cyclicality . . . . . . . . . . . . . . . . . 8
1.2.1 The Introduction of Basel I and the 1990/91 Credit Crunch . 8
1.2.2 Theoretical Research on the Pro-Cyclicality of Basel I . . . . . 10
1.3 The Basel II Accord and Pro-Cyclicality . . . . . . . . . . . . . . . . 11
1.3.1 The Basel II Accord . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Pro-Cyclicality of Required Minimum Capital . . . . . . . . . 13
1.3.3y and Capital Bu ers . . . . . . . . . . . . . . . 16
1.3.4 Pro-Cyclicality and Loan-Rating Systems . . . . . . . . . . . . 20
1.3.5 Further Theoretical Research on the Pro-Cyclicality of Basel II 22
1.4 Further Regulations and Pro-Cyclicality . . . . . . . . . . . . . . . . 23
iiiiv CONTENTS
1.5 Risk-Taking, Systemic Risk, and Capital Requirements . . . . . . . . 24
1.5.1 Basel I and Risk-Taking . . . . . . . . . . . . . . . . . . . . . 24
1.5.1.1 Empirical Evidence and Criticism . . . . . . . . . . . 25
1.5.1.2 Theoretical Criticism . . . . . . . . . . . . . . . . . . 26
1.5.2 Basel II and Risk-Taking . . . . . . . . . . . . . . . . . . . . . 28
1.5.2.1 The Three Pillars of Basel II . . . . . . . . . . . . . 28
1.5.2.2 Systemic Risk . . . . . . . . . . . . . . . . . . . . . . 29
2 Framework of the Analysis 31
2.1 Exogenous Shocks and the Business Cycle . . . . . . . . . . . . . . . 31
2.1.1 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Expectation Shocks . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.3 Interference of Di erent Shocks . . . . . . . . . . . . . . . . . 32
2.2 Di erent Types of Cyclical Behavior . . . . . . . . . . . . . . . . . . 33
2.3 The Model’s Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 The Basic Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Fixed Bank’s Equity Capital and Uninsured Deposits . . . . . 36
2.3.3 The Bank’s Risk-Neutrality . . . . . . . . . . . . . . . . . . . 38
2.3.4 Market Power in Banking . . . . . . . . . . . . . . . . . . . . 38
II Bernoulli Distributed Loan Redemptions 43
3 Theoretical Analysis 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 The Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 The Repayment of Deposits . . . . . . . . . . . . . . . . . . . 49
3.2.3 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.4 The Bank’s Expected Pay-o . . . . . . . . . . . . . . . . . . 63CONTENTS v
3.3 Equilibrium and Sensitivities . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1 The Equilibrium without Regulation . . . . . . . . . . . . . . 65
3.3.1.1 General Characterization . . . . . . . . . . . . . . . 65
3.3.1.2 Characterization if Case 1 Prevails . . . . . . . . . . 67
3.3.1.3 if Case 2 with ‘ = 0 Prevails . . . 68
3.3.1.4 Characterization if Case 4 Prevails . . . . . . . . . . 72
3.3.1.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . 79
3.3.2 The Equilibrium with Regulation by Fixed Risk Weights . . . 80
3.3.2.1 General Characterization . . . . . . . . . . . . . . . 80
3.3.2.2 Characterization if Case 1 Prevails . . . . . . . . . . 84
S3.3.2.3 if Case 2 with ‘ = 0 Prevails . . . 87
3.3.2.4 Characterization if Case 4 Prevails . . . . . . . . . . 88
3.3.3 The Equilibrium with Regulation by a Value-at-Risk Approach 95
3.3.3.1 General Characterization . . . . . . . . . . . . . . . 95
V 23.3.3.2 Characterization of Case 1 if l = Prevails . . 101
+1 2
3.3.3.3 if Case 4 Prevails . . . . . . . . . . 106
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Numerical Analysis of Regulatory Impacts 117
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2.1 Total Loan Volumes and Pro-Cyclical E ects . . . . . . . . . . 119
4.2.2 Tighter Con dence Levels and Total Volumes . . . . . . . . . 123
4.2.3 Single Loans and Pro-Cyclical E ects . . . . . . . . . . . . . . 124
4.2.4 Return Volatilities of Total Loans and Deposits . . . . . . . . 127
4.2.5 Deposit Interest Rates . . . . . . . . . . . . . . . . . . . . . . 131
4.3 Credit Risk Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.1 Total Volumes and Cyclical Dampening . . . . . . . . . . . . . 132
4.3.2 Single Loans and the Role of Constant Risk Weights . . . . . 141vi CONTENTS
4.3.3 Flexible Risk Weights . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.4 E ects under the Value-at-Risk Approach . . . . . . . . . . . 142
4.4 Productivity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.1 Total Volumes and Cyclical Dampening . . . . . . . . . . . . . 146
4.4.2 E ects under the Standardized Approach . . . . . . . . . . . . 148
4.4.3 E ects under the Value-at-Risk Approach . . . . . . . . . . . 151
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
III Normally Distributed Loan Redemptions 161
5 A Basis for the Normal Distribution 163
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.2 Firms and Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3 The m-Dependence Structure and Its Limit Distribution . . . . . . . 168
5.4 The Limit Distributions of the Aggregate Loans . . . . . . . . . . . . 170
6 The One-Period Model 177
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 The Bank’s Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4 Equilibrium without Regulation . . . . . . . . . . . . . . . . . . . . . 187
6.5 with Regulation by a Value-at-Risk Approach . . . . . . 191
6.6 Numerical Analysis of Regulatory Impacts . . . . . . . . . . . . . . . 194
6.6.1 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.6.2 Expected Return Shocks . . . . . . . . . . . . . . . . . . . . . 204
6.6.3 Return Volatility Shocks . . . . . . . . . . . . . . . . . . . . . 207
6.6.4 Correlation Shocks . . . . . . . . . . . . . . . . . . . . . . . . 213
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7 The Two-Period Model 221CONTENTS vii
7.1 Timeline and Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.2 Numerical Analysis of Regulatory Impacts . . . . . . . . . . . . . . . 230
7.2.1 Expected Return Shocks . . . . . . . . . . . . . . . . . . . . . 232
7.2.2 Return Volatility Shocks . . . . . . . . . . . . . . . . . . . . . 237
7.2.3 Return Correlation Shocks . . . . . . . . . . . . . . . . . . . . 239
7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
IV Final Remarks 243
Appendix 247
A Proofs to Chapter 3 249
A.1 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.1.1 Proof of Result 2 . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.1.2 Derivations concerning Result 3 . . . . . . . . . . . . . . . . . 251
A.1.3 Proof of Result 4 . . . . . . . . . . . . . . . . . . . . . . . . . 253
A.2 Results on Speci c Instances of the Equilibrium . . . . . . . . . . . . 254
A.2.1 The Equilibrium without Regulation . . . . . . . . . . . . . . 254
A.2.1.1 Proof of Result 5 . . . . . . . . . . . . . . . . . . . . 254
A.2.1.2 Proof of Result 6 . . . . . . . . . . . . . . . . . . . . 259
A.2.1.3 Proof of Result 7 . . . . . . . . . . . . . . . . . . . . 261
A.2.1.4 Proof of Result 8 . . . . . . . . . . . . . . . . . . . . 264
A.2.1.5 Proof of Result 9 . . . . . . . . . . . . . . . . . . . . 268
A.2.1.6 Proof of Result 10 . . . . . . . . . . . . . . . . . . . 271
A.2.1.7 Proof of Result 11 . . . . . . . . . . . . . . . . . . . 273
A.2.1.8 Proof of Result 12 . . . . . . . . . . . . . . . . . . . 275
A.2.2 The Equilibrium with Regulation by Fixed Risk Weights . . . 276
A.2.2.1 Proof of Result 14 . . . . . . . . . . . . . . . . . . . 276
A.2.2.2 Proof of Result 16 . . . . . . . . . . . . . . . . . . . 279viii CONTENTS
A.2.2.3 Proof of Result 17 . . . . . . . . . . . . . . . . . . . 282
A.2.2.4 Proof of Result 18 . . . . . . . . . . . . . . . . . . . 283
A.2.2.5 Proof of Result 19 . . . . . . . . . . . . . . . . . . . 284
A.2.2.6 Proof of Result 20 . . . . . . . . . . . . . . . . . . . 287
A.2.2.7 Proof of Result 21 . . . . . . . . . . . . . . . . . . . 288
A.2.3 The Equilibrium with Regulation by a Value-at-Risk Approach 290
A.2.3.1 The Regulatory Constraints . . . . . . . . . . . . . . 290
A.2.3.2 Proof of Result 24 . . . . . . . . . . . . . . . . . . . 291
A.2.3.3 Proof of Result 25 . . . . . . . . . . . . . . . . . . . 293
B Derivations to Chapter 5 297
B.1 Bernoulli Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . 297
B.2 Moments of Aggregate Loan Repayments . . . . . . . . . . . . . . . . 301
C Proofs to Chapter 6 303
C.1 Proof of Result 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.2 Proof of Result 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
C.3 Proof of Result 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
D Proofs to Chapter 7 315
D.1 Existence of the Deposit Supply Function in the First Period . . . . . 315
D.2 Proof of Result 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
Bibliography 331
Curriculum Vit 344List of Figures
3.1 The Case constraints given a xed deposit volume . . . . . . . . . . . 52
3.2 The Case constraints given the household’s deposit-supply function . 67
4.1 Total loan volume as a function of the bank’s initial equity: w/o
regulation, StdA with c =c = 1, VaR at 1:0%, and VaR at 0:5% . . 1201 2
4.2 Total loan volume as a function of the bank’s initial equity: w/o
regulation, StdA with c = 0:5 and c = 1, and VaR at 0:2% . . . . . 1211 2
4.3 Loan-allocation rate as a function of the bank’s initial equity: w/o
regulation, StdA with c =c = 1, VaR at 1%, and VaR at 0:5% . . . 1221 2
4.4 Loan-allocation rate as a function of the bank’s initial equity: w/o
regulation, StdA with c = 0:5 and c = 1, and VaR at 0:2% . . . . . 1231 2
4.5 Loan to Firm 1 as a function of the bank’s initial equity: w/o
regulation, StdA with c =c = 1, VaR at 1%, and VaR at 0:5% . . . 1251 2
4.6 Loan to Firm 2 as a function of the bank’s initial equity: w/o
regulation, StdA with c =c = 1, VaR at 1%, and VaR at 0:5% . . . 1261 2
4.7 Loan to Firm 2 as a function of the bank’s initial equity: w/o
regulation, StdA with c = 0:5 and c = 1, and VaR at 0:2% . . . . . 1271 2
4.8 Return volatilities as a function of the bank’s initial equity: w/o
regulation, StdA c =c = 1, and VaR at 1% . . . . . . . . . . . . . . 1281 2
4.9 Return volatilities as a function of the bank’s initial equity: w/o
regulation, StdA c = 0:5 and c = 1, and VaR at 0:2% . . . . . . . . 1291 2
4.10 Deposit interest rate as a function of the bank’s initial equity: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5 and c = 1,1 2 1 2
VaR at 1%, VaR at 0:5%, and VaR at 0:2% . . . . . . . . . . . . . . 132
ixx LIST OF FIGURES
4.11 Total loan volume as a function of shifts in success probabilities: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5 and c = 1,1 2 1 2
VaR at 0:5%, and VaR at 0:2% . . . . . . . . . . . . . . . . . . . . . 133
4.12 Total loan volume as a function of shifts in success probabilities: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5 and c = 1,1 2 1 2
VaR at 0:5%, and VaR at 1%. . . . . . . . . . . . . . . . . . . . . . . 134
4.13 Deposit interest rate as a function of the correlation . . . . . . . . . . 135
4.14 Total loan volume as a function of the correlation . . . . . . . . . . . 137
4.15 Return volatilities as a function of shifts in success probabilities: w/o
regulation, StdA with c =c = 1, and with c = 0:5 and c = 1. . . . 1391 2 1 2
4.16 Loan to Firm 1 as a function of the correlation: w/o regulation, StdA
with c =c = 1, StdA with c = 0:5 and c = 1, and VaR at 1% . . . 1401 2 1 2
4.17 Loan to Firm 1 as a function of the correlation: w/o regulation, StdA
with c =c = 1, StdA with c = 0:5 and c = 1, VaR at 0:5%, and1 2 1 2
VaR at 0:2% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.18 Return volatilities as a function of shifts in success probabilities . . . 145
4.19 Total loan volume as a function of shifts in productivity: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5;c = 1,1 2 1 2
VaR at 1%, and VaR at 0:5% . . . . . . . . . . . . . . . . . . . . . . 146
4.20 Total loan volume as a function of shifts in productivity: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5;c = 1,1 2 1 2
VaR at 0:5%, and VaR at 0:2%. . . . . . . . . . . . . . . . . . . . . . 148
4.21 Loan to Firm 1 as a function of its expected gross return: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5;c = 1,1 2 1 2
VaR at 1%, and VaR at 0:5%. . . . . . . . . . . . . . . . . . . . . . . 149
4.22 Loan to Firm 1 as a function of its expected gross return: w/o
regulation, StdA with c = c = 1, StdA with c = 0:5;c = 1,1 2 1 2
VaR at 0:5%, and VaR at 0:2%. . . . . . . . . . . . . . . . . . . . . . 150
4.23 Loan to Firm 2 as a function of shifts in productivity: w/o regulation,
StdA with c = c = 1, StdA with c = 0:5;c = 1, VaR at 1%, and1 2 1 2
VaR at 0:5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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