Optimal Financial Decision Making under Uncertainty (eBook)

Giorgio Consigli is currently professor of applied mathematics in economics and finance at the University of Bergamo. Dr. Consigli is Coordinator of the Stochastic Programming technical section within the Italian OR society and Board Member of the European Working Groups of Stochastic Programming and Commodity and Financial Modelling within the European OR society. He is Research Fellow of the School of Mathematical Studies of the University of Cambridge (UK) and the UK Institute of Mathematics and Applications (FIMA). He holds an honours degree in Economics at the University La Sapienza in Rome, a Diploma in Financial intermediation in the same University and a PhD in mathematics at the University of Essex in the UK. Dr. Consigli has a substantial cooperation and R&D record with the insurance and financial industry in Italy and Internationally on the development of advanced tools for risk management and asset-liability management. Throughout the years he maintained an active cooperation with the academic and scientific communities specifically in the areas of stochastic optimization, financial modelling, risk modelling and static and dynamic portfolio selection. He is associate editor of the J of Management Mathematics (OUP), the J of Computational Management Science (Springer), the J of Financial Engineering and Risk Management (Inderscience), Quantitative Finance Letters (Taylor and Francis). Daniel Kuhn holds the Chair of Risk Analytics and Optimization at EPFL. Before joining EPFL, he was a faculty member at Imperial College London (2007-2013) and a postdoctoral researcher at Stanford University (2005-2006). He received a PhD in Economics from the University of St. Gallen in 2004 and an MSc in Theoretical Physics from ETH Zurich in 1999. His research interests revolve around robust optimization and stochastic programming. Paolo Brandimarte is full professor of quantitative methods at the Department of Mathematical Sciences of Politecnico di Torino, where he teaches Financial Engineering and Business Analytics. He is also adjunct professor at ESCP Europe. His primary research interests are in the application of optimization and statistical modelling to finance and supply chain management. He has written/edited more than ten books on these subjects.

Preface 6
References 12
Contents 14
1 Multi-Period Risk Measures and Optimal Investment Policies 19
1.1 Introduction 20
1.2 Dynamic Risk Control 20
1.2.1 Key Properties of Dynamic Measures 22
1.2.1.1 Extension of Risk Measures' Axioms 22
1.2.1.2 Dynamic Risk and Information Processes 24
1.2.2 Time Consistency 25
1.2.2.1 Time Consistency of Multi-Period Risk Measures 26
1.2.2.2 Time Consistency of Optimal Investment Policies 27
1.2.3 Discussion 28
1.3 Multi-Period Risk Measures 29
1.3.1 Statistical Estimates of Dynamic Risk Measures 31
1.3.1.1 Variance 31
1.3.1.2 Value-at-Risk 32
1.3.1.3 Conditional Value-at-Risk 33
1.3.2 Coherent and Time Consistent Risk Measures 34
1.4 Dynamic Risk Control and Risk Measures Selection 37
1.4.1 Mean-Variance Models 38
1.4.2 Time Inconsistent Mean-CVaR Models 40
1.4.3 Time Inconsistency and Time Consistent Revisions 41
1.4.4 Time Consistent Models 42
1.4.5 Practical Solution Methods for Optimal Dynamic Risk Control 45
1.5 Conclusions and Future Research 47
References 49
2 Asset Price Dynamics: Shocks and Regimes 53
2.1 Introduction 54
2.2 Risk Factors in Financial Markets 55
2.2.1 Regimes from Factor Thresholds 58
2.2.2 Regimes from Hidden States 60
2.2.3 Regime Fitting 61
2.3 Discrete Time Asset Pricing Model 63
2.3.1 Model with Jumps 64
2.3.2 Model with Regimes 65
2.4 Application: Exchange Traded Funds 66
2.4.1 Predicting Asset Returns 66
2.4.2 Portfolio Performance 67
2.5 Conclusion 70
References 71
3 Scenario Optimization Methods in Portfolio Analysis and Design 72
3.1 Introduction 73
3.1.1 Definitions and Preliminaries 74
3.2 Single-Period Analysis of Portfolio Shortfall Probability 75
3.2.1 The Shortfall Probability of the k-th Order Sample 75
3.2.2 The k-th Order Sample as an Approximator of the ?-Quantile 80
3.3 Single-Period Scenario Design 83
3.3.1 The Return Selection Rule 84
3.3.2 The Shortfall Probability 86
3.3.3 Shortfall Probability of the Optimal Data-Driven Portfolio 87
3.4 Multi-Period Scenario Design 90
3.4.1 Open-Loop Strategy 91
3.4.2 Closed-Loop Strategy with Affine Policies 94
3.4.3 Sliding-Horizon Implementation 96
3.5 Scenario Methods for Single-Period Robust Portfolio Design 96
3.5.1 Robust Portfolio Allocation Models 96
3.5.2 The Scenario Approach 98
3.6 A Practical Asset Allocation Example 100
3.7 Conclusions 102
References 104
4 Robust Approaches to Pension Fund Asset Liability Management Under Uncertainty 105
4.1 Introduction 105
4.2 ALM Model for Pension Funds: Problem Statement 107
4.3 Scenario-Based ALM Model for Pension Funds 109
4.4 Robust Investment Decisions 112
4.5 Robust ALM Models for Pension Funds 115
4.5.1 Robust ALM Model Formulation with Symmetric Uncertainty Sets 117
4.5.2 Robust ALM Model Formulation with Asymmetric Uncertainty Sets 118
4.5.3 Selecting Inputs to the Robust Optimization Models 119
4.6 Computational Experiments 122
4.6.1 Design of Experiments and Data 123
4.6.2 Computational Results 126
4.7 Concluding Remarks 131
Appendix 131
References 133
5 Liability-Driven Investment in Longevity Risk Management 136
5.1 Introduction 136
5.2 The Asset-Liability Management Problem 138
5.3 Investment Strategies 139
5.3.1 Non-liability-Driven Investment Strategies 139
5.3.2 Liability-Driven Investment Strategies 140
5.4 Diversification Procedure 142
5.5 Numerical Results 144
5.6 Conclusions 146
5.7 Assets and Liabilities 147
References 150
6 Pricing Multiple Exercise American Options by Linear Programming 152
6.1 Introduction 152
6.2 The Stochastic Scenario Tree and AmericanContingent Claims 154
6.3 The Formulation 156
6.4 The Main Result 157
6.4.1 The Case of Non-zero Interest Rate 160
6.4.2 A Min–Max Representation 161
6.5 Conclusions 163
References 164
7 Optimizing a Portfolio of Liquid and Illiquid Assets 166
7.1 Introduction 166
7.2 All Bonds Strategy vs. All Alternative Strategy 168
7.3 Tracking Indexes for Alternative Asset Categories 172
7.4 A Portfolio of Tactics 181
7.4.1 Overlay Approach 181
7.4.2 Constructing Optimal Portfolios via Multi-Stage Stochastic Programming 183
7.5 Conclusions 188
References 189
8 Stabilizing Implementable Decisions in Dynamic Stochastic Programming 191
8.1 Introduction and Background 192
8.2 Review of Pioneer Guaranteed Return Funds 194
8.3 Evaluating Under-estimation of Portfolio Risk 195
8.3.1 Position Limits Based on a Volatility Constraint 197
8.3.2 Position Limits Based on Asset Returns and Volatility Proportional Constraints 202
8.3.3 Summary 202
8.4 Empirical Results 203
8.5 Conclusions and Future Directions 209
Appendix: Pioneer Guaranteed Return Fund Model Formulation Yong06 210
Objective 210
References 213
9 The Growth Optimal Investment Strategy Is Secure, Too 215
9.1 Introduction 215
9.2 Constantly Rebalanced Portfolio Selection 216
9.3 Time Varying Portfolio Selection 225
References 235
10 Heuristics for Portfolio Selection 238
10.1 Introduction 239
10.2 Of Problems, Models and Methods 241
10.2.1 A One-Period Investment Model 241
10.2.2 Reality to Model, and Back 242
10.2.2.1 Sources of Error 242
10.3 Heuristics 245
10.3.1 What Are Heuristics? 245
10.3.2 Principles 247
10.3.3 Constraints 249
10.3.3.1 Throw Away 249
10.3.3.2 Include Constraint in N 249
10.3.3.3 Transform x 249
10.3.3.4 Repair x 249
10.3.3.5 Penalise x 249
10.3.4 Random Solutions 250
10.3.4.1 Randomness 250
10.4 An Example: Threshold Accepting 251
10.4.1 The Algorithm 252
10.4.2 Implementation 253
10.4.2.1 The Objective Function 253
10.4.2.2 The Neighbourhood Function 254
10.4.2.3 The Threshold Sequence 254
10.4.2.4 Constraints 255
10.5 An Example: Portfolio Selection with ta 256
10.5.1 Data, Backtesting Scheme and Reporting of Results 256
10.5.2 `Genesis' of a Model 257
10.5.3 Step 1: Optimisation of Tracking Error and Excess Return 258
10.5.4 Step 2: Optimisation of Tracking Error, Excess Return and ?rP,rM 259
10.5.5 Step 3: Optimisation of Tracking Error, Excess Return, ?rP,rM and ?rP,rI 260
10.5.6 Step 4: Optimisation of Tracking Error, Excess Return, ?rP,rM, ?rP,rI and Dmax 261
10.6 Conclusion 263
References 264
11 Optimal Financial Decision Making Under Uncertainty 267
11.1 The Domain of Financial Optimization 268
11.2 A Changing Financial Landscape 269
11.3 The Elements of a Decision Model 271
11.3.1 Discrete-Time Stochastic Control 272
11.3.2 The Interplay Between Model Building and Solution Method 274
11.3.2.1 Scenario Tree, Non-anticipativity and Information 274
11.3.2.2 Stochastic, Robust and Distributionally Robust Optimization 277
11.3.2.3 Data-Driven Optimization 281
11.4 Asset-Liability Management 283
11.4.1 An Overview of Financial Planning Problems 284
11.4.2 A Simple ALM Model 286
11.5 Solution Methods and Decision Support 289
11.5.1 Stochastic Programming 290
11.5.2 Dynamic Optimization Via Decision Rules 292
11.6 Open Issues 294
11.6.1 Probability Distributions and Optimization 295
11.6.2 Dynamic Time Consistency 296
11.6.3 Practical Financial Optimization 297
11.7 Conclusions 298
References 299
Index 303