Zero Lower Bound Term Structure Modeling (eBook)

Leo Krippner is Senior Advisor to the Research Section of the Economics Department at the Reserve Bank of New Zealand.

Cover 1
Title 4
Copyright 5
Dedication 6
Contents 8
List of Figures 14
List of Tables 18
Preface 20
Acknowledgments 22
Selected List of Notation 24
Classification and Abbreviations for Term Structure Models 28
1 Introduction 30
1.1 Chapteroverview 31
1.2 Suggested reading 32
1.2.1 Group 1:Generalmonetary policy readers 33
1.2.2 Group 2:General financialmarket readers 33
1.2.3 Group 3:Termstructuremodelers 33
1.3 Data 34
1.4 Availability of results and code 35
1.5 References to the literature 35
1.6 Other preliminaries 36
2 ANew Framework for a New Environment 37
2.1 Monetary policy 37
2.1.1 Pre-GFC 37
2.1.2 Post-GFC 42
2.2 Termstructuremodeling 46
2.2.1 Pre-GFC 47
2.2.2 Post-GFC 51
2.3 Shadow/ZLBtermstructuremodels 57
2.3.1 ZLBmechanism 57
2.3.2 Options to hold physical currency 64
2.4 Monetary policy revisited 66
2.5 AlternativeZLBmodels 68
2.5.1 Square-root termstructuremodels 70
2.5.2 Log-normal termstructuremodels 71
2.5.3 QuadraticGaussian termstructuremodels 71
2.6 Summary 72
3 Gaussian Affine Term StructureModels 73
3.1 GATSMs 74
3.1.1 GATSMspecification 74
3.1.2 GATSMdynamics and related calculations 76
3.1.3 GATSMtermstructure 79
3.2 GATSMestimation 79
3.2.1 Kalman filter equations and related parameters 81
3.2.2 Partial estimation 83
3.2.3 Full estimation 85
3.3 Worked example:ANSM(2) 88
3.3.1 ANSM(2) Specification 89
3.3.2 ANSM(2)Termstructure 89
3.3.3 Partial ANSM(2) estimation with the Kalman filter 94
3.3.4 FullANSM(2) estimationwith theKalman filter 98
3.4 OtherGATSMs 103
3.4.1 ANSM(3) 104
3.4.2 Higher-orderANSMs 109
3.4.3 Non-arbitrage-freeANSMs 112
3.4.4 StationaryGATSMs 114
3.4.5 StationaryGATSMswith repeated eigenvalues 118
3.5 Empirical applications 119
3.5.1 Yield curve data set 119
3.5.2 ANSM(2) results 121
3.5.3 ANSM(3) results 125
3.6 Alternative estimationmethods 128
3.6.1 Estimation using forward rates 128
3.6.2 Estimation using bond prices 131
3.7 Summary 131
4 Krippner Framework for ZLB Term StructureModeling 133
4.1 K-AGMexposition 133
4.1.1 K-AGMIntuition 134
4.1.2 K-AGMoption effect 135
4.1.3 K-AGMforward rates 139
4.1.4 Observations on theK-AGMframework 140
4.1.5 Comparison with related results in the literature 142
4.1.6 K-AGMinterest rates and bond prices 143
4.2 K-AGMestimation 145
4.2.1 K-AGMs and nonlinearKalman filters 145
4.2.2 Partial estimation 146
4.2.3 Full estimation 153
4.3 Worked example:K-ANSM(2) 154
4.3.1 K-ANSM(2) specification 154
4.3.2 K-ANSM(2) shadowtermstructure 156
4.3.3 K-ANSM(2)ZLB termstructure 157
4.3.4 K-ANSM(2)Estimation 158
4.4 OtherK-AGMs 162
4.4.1 K-ANSM(3) 163
4.4.2 Higher-orderK-ANSMs 164
4.4.3 Non-arbitrage-freeK-ANSMs 164
4.4.4 StationaryK-AGMs 167
4.5 Empirical applications 169
4.5.1 K-ANSM(2) results 169
4.5.2 K-ANSM(3) results 173
4.6 Alternative estimationmethods 175
4.6.1 Estimation using forward rates 175
4.6.2 Estimation using bond prices 180
4.6.3 Iterative estimation usingGATSMs 183
4.7 Summary 185
5 Black Framework for ZLB Term StructureModeling 187
5.1 TheB-AGMframework principles 188
5.1.1 Initial comparison of the B-AGM and K-AGM frameworks 192
5.2 B-AGMimplementation 196
5.2.1 Customized calculationmethods 196
5.2.2 Finite differencemethods 197
5.2.3 Latticemethods 198
5.2.4 MonteCarlomethods 199
5.3 B-AGMMonteCarlo implementations 200
5.3.1 B-AGMMonteCarlo specification 201
5.3.2 Computing time for B-AGM Monte Carlo simulations 206
5.3.3 Antithetic sampling 208
5.3.4 Other potentially useful standard speed-up methods 209
5.3.5 Worked example:B-ANSM(2) 209
5.3.6 OtherB-AGMs 212
5.4 K-AGM as a control variate for B-AGM Monte Carlo simulations 213
5.4.1 Defining the control variate 214
5.4.2 B-AGM(1) illustration 216
5.4.3 Extensions tomore than one state variable 220
5.5 B-AGMestimation 226
5.5.1 Partial estimation 227
5.5.2 Full estimation 232
5.5.3 Alternative estimationmethods 234
5.6 Approximations toB-AGMs 234
5.6.1 K-AGMs as approximations toB-AGMs 234
5.6.2 B-AGMcumulant approximations 239
5.6.3 Non-arbitrage-free B-ANSMs 243
5.6.4 ApplyingB-AGMapproximations 244
5.7 Summary 245
6 K-ANSMFoundations and EffectiveMonetary Stimulus 247
6.1 Overviewof the economicmodel and its development 248
6.1.1 Establishing theGCE 249
6.1.2 The nominal termstructure in theGCE 250
6.1.3 Amacroeconomic interpretation of theGCE 252
6.1.4 GCE generalizations 253
6.2 ANSMs as the reduced-formGCE termstructure 254
6.2.1 ANSMspecifications 255
6.2.2 GCE short rate andANSMshort rate 257
6.2.3 GCE and ANSM long-horizon short rate expectations 258
6.2.4 GCE andANSMshort rate expectations 259
6.2.5 Long-run GCE volatility effect and the ANSMLevel volatility effect 263
6.2.6 Other volatility effects 263
6.2.7 Parsimonious ANSMstate equations 268
6.2.8 ANSMmacroeconomic interpretation 273
6.3 UsingANSMs to represent the shadowtermstructure 275
6.4 Theoretical case for theK-AGMframework 276
6.4.1 Pricing in the B-AGM and K-AGM frameworks 277
6.4.2 The GCE market portfolio and discount rates without physical currency 280
6.4.3 The market portfolio and discount rates with physical currency 281
6.5 The EMSmeasure ofmonetary policy 282
6.5.1 EMS Principles 283
6.5.2 K-ANSMEMSmeasure 286
6.5.3 Worked example:K-ANSM(2)EMS 287
6.5.4 K-ANSM(3)EMS 291
6.6 Summary 298
7 Monetary Policy Applications 300
7.1 OverviewofK-ANSMs, estimation, and empirical results 301
7.2 Measures of the stance ofmonetary policy 305
7.2.1 Overviewof threemonetary policymeasures 306
7.2.2 Monetary policymeasures in theZLBenvironment 307
7.3 The ShadowShortRate (SSR) 313
7.3.1 Theoretical overview 313
7.3.2 Empirical overview 315
7.3.3 Empirical evidence 316
7.3.4 SSRsummary 321
7.4 The ExpectedTime toZero (ETZ) 322
7.4.1 Theoretical overview 323
7.4.2 Empirical overviewand evidence 324
7.5 The EffectiveMonetary Stimulus (EMS) 324
7.5.1 Theoretical overview 325
7.5.2 Empirical overviewand evidence 327
7.5.3 Linking the SSRand the EMS 332
7.5.4 EMS summary 332
7.6 K-ANSMmacrofinance relationships 333
7.6.1 K-ANSM Level and long-horizon macroeconomic surveys 334
7.6.2 K-ANSMEMS andmacroeconomic data 336
7.6.3 K-ANSM(3)EMS and currencies 338
7.7 Summary 343
8 FinancialMarket Applications 344
8.1 Fixed interest portfolio risk 345
8.1.1 Security and portfolio risk 345
8.1.2 Fixed interest security and portfolio risk in non-ZLB environments 346
8.1.3 Fixed interest security and portfolio risk in ZLB environments 349
8.2 Arisk framework based onANSMs 353
8.2.1 ANSMtermstructure shifts 354
8.2.2 ANSMfactor durations 354
8.2.3 ANSMfixed interest portfolio risk 356
8.3 Worked example:ANSM(2) 358
8.3.1 ANSM(2) yield curve shifts 358
8.3.2 ANSM(2)duration vector 361
8.3.3 Level duration approximation to Level shifts 362
8.3.4 Slope duration approximation to Slope shifts 363
8.3.5 ANSM(2)fixed interest portfolio risk 365
8.3.6 Extension toANSM(3) 367
8.4 Arisk framework based onK-ANSMs 367
8.4.1 K-ANSMyield curve shifts 368
8.4.2 K-ANSMfactor durations 369
8.4.3 K-ANSMfixed interest portfolio risk 371
8.5 Worked example:K-ANSM(2) 372
8.5.1 K-ANSM(2) termstructure specification 372
8.5.2 K-ANSM(2)duration vector 373
8.5.3 Non-ZLBenvironment 373
8.5.4 ZLB environment 379
8.5.5 A perspective on the K-ANSM versus ANSM factor duration results 386
8.6 Bond option pricing 387
8.6.1 ANSMoption pricing 388
8.6.2 K-AGMoption pricing 391
8.6.3 B-AGMoption pricing 393
8.7 Summary 395
9 Conclusion and Future Research Directions 396
9.1 Summary 396
9.2 The case forK-ANSMs 398
9.3 Future research directions 401
Appendix A: Matrix Notation 404
A.1 Scalars, vectors, andmatrices 404
A.2 Matrix transpose 405
A.3 Some special and usefulmatrices 405
A.4 Matrix addition and subtraction 407
A.5 Matrixmultiplication 407
A.5.1 Multiplying two vectors 407
A.5.2 Multiplying twomatrices 408
A.5.3 Need for matrix notation in term structure modeling 409
A.5.4 The identitymatrix revisited 409
A.5.5 Powers ofmatrices 410
A.5.6 Multiplying partitionedmatrices 410
A.5.7 Multiplying amatrix by a scalar 410
A.6 Matrix inverse 411
A.7 Matrix decompositions 412
A.7.1 Cholesky decomposition 412
A.7.2 Eigensystemdecomposition 412
A.7.3 Jordan decomposition 413
A.7.4 Using eigensystemand Jordan decompositions 415
A.8 Thematrix exponential 416
A.9 Matrix calculus 418
Notes 420
Bibliography 424
Index 430