Optimal Consumption and Investment with Bankruptcy

433 Pages·1997·9.543 MB·English

by  Suresh P. Sethi (auth.)

OPTIMAL CONSUMPTION
AND INVESTMENT
WITH BANKRUPTCY
OPTIMAL CONSUMPTION
AND INVESTMENT
WITH BANKRUPTCY
Suresh P. Sethi
Foreword by Harry M. Markowitz
~.
"
Springer-Science+Business Media, LLC
ISBN 978-1-4613-7871-6 ISBN 978-1-4615-6257-3 (eBook)
DOI 10.1007/978-1-4615-6257-3
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1997 Springer Science+Business Media New York
Origina11y published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover lst edition 1997
AII rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer-Science+Business Media, LLC.
Printed on acid-free paper.
This book is dedicated to my collaborators
Abel Cadenillas, Myron Gordon, Brian Ingham, Ioannis Karatzas,
John Lehoczky, Ernst Presman, Steven Shreve, and
Michael Taksar.
Contents
Foreword ........................................................ xiii
Preface .......................................................... xix
PART I. INTRODUCTION ......................................... 1
1. Consumption/Investment Problems ......................... 3
1. Introduction .............................................. 3
2. Importance of Bankruptcy ................................ 6
3. Organization of the Book ................................. 7
References ................................................... 14
PART II. MODELS WITH CONSTANT MARKET PARAMETERS
AND NONNEGATIVE CONSUMPTION .......................... 19
2. Explicit Solution of a General Consumption/Investment
Problem ......................................................... 21
(with I. Karatzas, J. P. Lehoczky and S. E. Shreve)
Abstract, Contents, and Key Words .......................... 23
1. Introduction ............................................. 24
2. Summary of Assumptions and Results .................... 25
3. Admissible Policies ....................................... 27
4. The Bellman Equation ................................... 29
5. Reduction of the Model: Mutual Fund Theorem .......... 30
6. Solving the Bellman Equation. Part I .................... 32
7. Candidate Optimal Policies .............................. 34
8. Performance Evaluation of Candidate Optimal Policies ... 35
9. Solving the Bellman Equation. Part II ................... 40
= =
10. Solution When U'(O) U(O) Is Finite, P U(O)//3 ... 41
00,
11. Solution When U(O) = 42
-00 ..............................
12. Solution in All Other Cases Where the Consumption
Constraint Is Inactive .................................. 46
13. Solution When the Consumption Constraint Is Active .... 46
14. Solutions for HARA Utility Functions .................... 52
15. Dependence of the Optimal Policy on the Prices .......... 53
16. The Model with Nonconstant /3, E and r ................. 54
17. Tabulated Results ........................................ 54
viii Contents
References ................................................... 56
3. A Note on Merton's "Optimum Consumption and Portfolio
Rules in a Continuous-time Model" ........................... 57
(with M. I. Taksar)
Abstract ..................................................... 59
1. Introduction .............................................. 59
2. Feasibility Violations ...................................... 60
3. Concluding Remarks ...................................... 64
References ................................................... 64
4. Infinite-Horizon Investment Consumption Model with a
Nonterminal Bankruptcy ...................................... 67
(with M. I. Taksar)
Abstract and Key Words ..................................... 68
1. Introduction .............................................. 69
2. Model .................................................... 70
3. Bellman Equation ......................................... 74
4. Reduction to the Problem with Terminal Bankruptcy ...... 76
5. Correspondence between P and /-L,
= =
Cases Where U(O) -00 or /-L 0 ..................... 78
6. Correspondence between P and /-L,
Cases Where U'(O) > -00, U'(O) = 00 .................. 78
7. Correspondence between P and /-L,
Cases Where U' (0) < 00. . •...........•..•.•............ 81
8. Tabulated Results ......................................... 82
References ................................................... 84
5. Risk-Aversion Behavior in Consumption/Investment Prob-
lems .............................................................. 85
(with E. 1. Presman)
Abstract and Key Words ..................................... 86
1. Introduction .............................................. 87
2. Statement of the Consumption/Investment Problem ....... 89
3. Solution of the Consumption/Investment Problem ......... 92
4. Results in the General Case ............................... 98
5. Special Case of HARA Utility Functions .................. 107
6. Summary of Results ...................................... 110
7. Discussion of Results ..................................... 113
Contents ix
References .................................................. 115
PART III: MODELS WITH CONSTANT MARKET PARAMETERS
AND POSITIVE SUBSISTENCE CONSUMPTION .............. 117
6. Explicit Solution of a General Consumption/Portfolio Prob-
lem with Subsistence Consumption and Bankruptcy ....... 119
(with M. I. Taksar and E. L. Presman)
Abstract .................................................... 120
1. Introduction ............................................. 121
2. Summary of Assumptions and Results .................... 124
3. The Bellman Equation and its Analysis .................. 126
4. Construction of the Value Function ....................... 131
5. Characterization of Optimal Policy ....................... 135
6. Tabular Summary of Results ............................. 141
References .................................................. 142
7. Distribution of Bankruptcy Time in a Consump-
tion/Portfolio Problem ........................................ 145
(with E. L. Presman)
Abstract and Key Words .................................... 146
1. Introduction ............................................. 147
2. The Consumption/Investment Problem ................... 147
3. Probability Distribution of Bankruptcy Time ............. 151
4. Concluding Remarks ..................................... 153
References .................................................. 154
8. Risk-Aversion Behavior in Consumption/Investment Prob-
lems with Subsistence Consumption ......................... 155
(with E. 1. Presman)
Abstract and Key Words .................................... 156
1. Introduction ............................................. 157
2. Summary and Interpretation of Previous Results with
Subsistence Consumption and Bankruptcy ............ 159
3. Some Properties of Risk-Aversion Behavior in the
General Case ......................................... 163
4. Absolute Risk-Aversion Behavior in the HARA Case ..... 164
5. Relative Risk-Aversion Behavior in the HARA Case ...... 168
6. Proofs of Results ......................................... 176
x Contents
References .................................................. 183
9. Consumption Behavior in Investment/Consumption Prob-
lems ............................................................. 185
(with E. L. Presman)
Abstract and Key Words .................................... 186
1. Introduction ............................................. 187
2. Summary and Interpretation of Previous Results with
Subsistence Consumption and Bankruptcy ............ 188
3. Some Properties of Consumption Behavior in the
General Case ......................................... 194
4. Consumption Behavior in the HARA Case ............... 195
5. Proofs of Results ......................................... 199
References .................................................. 205
10. Equivalence of Objective Functionals in Infinite Horizon
and Random Horizon Problems ............................. 207
(with E. L. Presman)
Abstract and Key Words .................................... 208
1. Introduction .............................................. 209
2. Objective Functionals in the Infinite Horizon Case ......... 210
3. Objective Functionals in the Random Horizon Case ........ 211
4. Reduction in the General Case ............................ 212
5. Some Specializations and Discussion ....................... 213
References ................................................... 215
11. A Contribution to the Micro Foundation for Keynesian
Macroeconomic Models ....................................... 217
(with M. J. Gordon)
Abstract and Key Words .................................... 218
1. Introduction .............................................. 219
2. The Decision Model ....................................... 220
3. Discussion of Bankruptcy ................................. 224
4. 'f/ Equal to Zero ........................................... 227
5. 'f/ Greater Than Zero ...................................... 230
6. 'f/ Less Than Zero ......................................... 235
7. Macro Implications ....................................... 239
References ................................................... 242
Contents Xl
PART IV: MODELS WITH MORE GENERAL MARKETS AND POS-
ITIVE SUBSISTENCE CONSUMPTION ......................... 245
12. The Consumption-Investment Problem with Subsistence
Consumption, Bankruptcy, and Random Market
Coefficients ..................................................... 247
(with A. Cadenillas)
Abstract and Key Words .................................... 248
1. Introduction .............................................. 249
2. The Financial Market Model .............................. 251
3. Consumption and Investment ............................. 258
4. The Optimal Time of Bankruptcy ......................... 267
5. The Case of High Initial Wealth ........................... 271
6. The Optimal Portfolio and Consumption Processes:
The General Case ..................................... 273
7. Concluding Remarks ...................................... 276
References ................................................... 277
PART V: MODELS WITH CONSTANT MARKET PARAME
TERS, POSITIVE SUBSISTENCE CONSUMPTION AND BORROW-
INGjSHORTSELLING CONSTRAINTS ......................... 281
13. Optimal Dynamic Consumption and Portfolio Planning in
a Welfare State ................................................ 283
(with M. J. Gordon and B. Ingham)
Abstract .................................................... 285
1. The Model ................................................ 286
2. Solution of the Three Period Problem ..................... 288
3. Extension and Concluding Remarks ....................... 299
Appendix .................................................... 300
References ................................................... 302
14. Optimal Consumption and Investment Policies Allowing
Consumption Constraints, Bankruptcy and Welfare ....... 303
(with J. P. Lehoczky and S. E. Shreve)
Abstract and Key Words .................................... 304
1. Introduction .............................................. 305
2. Preliminary Statement of Results .......................... 310
3. Precise Model and Optimality Theorems .................. 319
4. Discount Rate Less Than Interest Rate (0 < (3 < r) ........ 353
xu Contents
5. Discount Rate Equal to Interest Rate (0 < {3 = r) ......... 364
6. Discount Rate Greater Than Interest Rate (0 < r < (3) .... 366
7. The HARA Utility Functions .............................. 372
References ................................................... 376
15. A Martingale Formulation for Optimal Consump-
tion/Investment Decision Making ............................ 379
(with J. P. Lehoczky and S. E. Shreve)
Abstract and Key Words .................................... 380
1. Introduction .............................................. 381
2. Summary of Wiener Process Results ....................... 384
3. Precise Model Formulation ................................ 388
4. Examples ................................................. 395
5. Appendix: Cumulative Utility of Consumption ............ 401
References ................................................... 405
PART VI: CONCLUSIONS ...................................... .407
16. Concluding Remarks and Open Research Problems ... 409
1. A Brief Summary of the Models Presented ................ .409
2. Important Extensions of the Single-Agent Models ......... .411
3. Where We Go From Here? ................................ 416
References ................................................... 419
Author Index .................................................. 423
Copyright Permissions ........................................ 427