OPTIMIZATION THEORY
CONTENTS SECTION ONE PRACTITIONERS AND PRODUCTS 1 ROBUST PORTFOLIO OPTIMIZATION USING SECOND-ORDER CONE PROGRAMMING EXECUTIVE SUMMARY 1 1 INTRODUCTION 1 2 ALPHA UNCERTAINTY 1 3 CONSTRAINTS ON SYSTEMATIC AND SPECIFIC RISK 1 4 CONSTRAINTS ON RISK USING MORE THAN ONE MODEL 1 5 COMBINING DIFFERENT RISK MEASURES 1 6 FUND OF FUNDS 1 7 CONCLUSION REFERENCES 2 NOVEL APPROACHES TO PORTFOLIO CONSTRUCTION: MULTIPLE RISK MODELS AND MULTISOLUTION GENERATION 2 1 INTRODUCTION 2 2 PORTFOLIO CONSTRUCTION USING MULTIPLE RISK MODELS 2 2 1 OUT-OF-SAMPLE RESULTS 2 2 2 DISCUSSION AND CONCLUSIONS 2 3 MULTISOLUTION GENERATION 2 3 1 CONSTRAINT ELASTICITY 2 3 2 INTRACTABLE METRICS 2 4 CONCLUSIONS 3 OPTIMAL SOLUTIONS FOR OPTIMIZATION IN PRACTICE 3 1 INTRODUCTION 3 1 1 BITA STAR(™ ) 3 1 2 BITA MONITOR(™) 3 1 3 BITA CURVE(™) 3 1 4 BITA OPTIMIZER(™) 3 2 PORTFOLIO OPTIMIZATION 3 2 1 THE NEED FOR OPTIMIZATION 3 2 2 APPLICATIONS OF PORTFOLIO OPTIMIZATION 3 2 3 PROGRAM TRADING 3 2 4 LONG–SHORT PORTFOLIO CONSTRUCTION 3 2 5 ACTIVE QUANT MANAGEMENT 3 2 6 ASSET ALLOCATION 3 2 7 INDEX TRACKING 3 3 MEAN –VARIANCE OPTIMIZATION 3 3 1 A TECHNICAL OVERVIEW 3 3 2 THE BITA OPTIMIZER—FUNCTIONAL SUMMARY 3 4 ROBUST OPTIMIZATION 3 4 1 BACKGROUND 3 4 2 INTRODUCTION 3 4 3 REFORMULATION OF MEAN–VARIANCE OPTIMIZATION 3 4 4 BITA ROBUST APPLICATIONS TO CONTROLLING FE 3 4 5 FE CONSTRAINTS 3 4 6 PRELIMINARY RESULTS 3 4 7 MEAN FORECAST INTERVALS 3 4 8 EXPLICIT RISK BUDGETING 3 5 BITA GLO(™) GAIN /LOSS OPTIMIZATION 3 5 1 INTRODUCTION 3 5 2 OMEGA AND GLO 3 5 3 CHOICE OF INPUTS 3 5 4 ANALYSIS AND COMPARISON 3 5 5 MAXIMUM HOLDING 100% 3 5 6 ADDING 25% INVESTMENT CONSTRAINT 3 5 7 DOWN-TRIMMING OF EMERGING MARKET RETURNS 3 5 8 SQUARED LOSSES 3 5 9 CONCLUSIONS 3 6 COMBINED OPTIMIZATIONS 3 6 1 INTRODUCTION 3 6 2 DISCUSSION 3 6 3 THE MODEL 3 6 4 INCORPORATION OF ALPHA AND RISK MODEL INFORMATION 3 7 PRACTICAL APPLICATIONS: CHARITIES AND ENDOWMENTS 3 7 1 INTRODUCTION 3 7 2 WHY ENDOWMENTS MATTER 3 7 3 MANAGING ENDOWMENTS 3 7 4 THE SPECIFICATION 3 7 5 TRUSTEES’ ATTITUDE TO RISK 3 7 6 DECISION MAKING UNDER UNCERTAINTY 3 7 7 PRACTICAL IMPLICATIONS OF RISK AVERSION 3 8 BESPOKE OPTIMIZATION—PUTTING THEORY INTO PRACTICE 3 8 1 REQUEST: PRODUCE OPTIMAL PORTFOLIO WITH EXACTLY 50 LONG AND 50 SHORT HOLDINGS 3 8 2 REQUEST: HOW TO OPTIMIZE IN THE ABSENCE OF FORECAST RETURNS 3 9 CONCLUSIONS APPENDIX A: BITA ROBUST OPTIMIZATION APPENDIX B: BITA GLO 4 THE WINDHAM PORTFOLIO ADVISOR MARK KRITZMAN EXECUTIVE SUMMARY 4 1 INTRODUCTION 4 2 MULTIGOAL OPTIMIZATION 4 2 1 THE PROBLEM 4 2 2 THE WPA SOLUTION 4 2 3 SUMMARY 4 3 WITHIN -HORIZON RISK MEASUREMENT 4 3 1 THE PROBLEM 4 3 2 THE WPA SOLUTION 4 4 RISK REGIMES 4 4 1 THE PROBLEM 4 4 2 THE WPA SOLUTION 4 4 3 SUMMARY 4 5 FULL -SCALE OPTIMIZATION 4 5 1 THE PROBLEM 4 5 2 THE WPA SOLUTION 4 5 3 SUMMARY APPENDIX —WPA FEATURES SECTION TWO THEORY 5 MODELING, ESTIMATION, AND OPTIMIZATION OF EQUITY PORTFOLIOS WITH HEAVY-TAILED DISTRIBUTIONS ALMIRA BIGLOVA, SERGIO ORTOBELLI, SVETLOZAR RACHEV AND FRANK J FABOZZI EXECUTIVE SUMMARY 5 1 INTRODUCTION 5 2 EMPIRICAL EVIDENCE FROM THE DOW JONES INDUSTRIAL AVERAGE COMPONENTS 5 3 GENERATION OF SCENARIOS CONSISTENT WITH EMPIRICAL EVIDENCE 5 3 1 THE PORTFOLIO DIMENSIONALITY PROBLEM 5 3 2 GENERATION OF RETURN SCENARIOS 5 4 THE PORTFOLIO SELECTION PROBLEM 5 4 1 REVIEW OF PERFORMANCE RATIOS 5 4 2 AN EMPIRICAL COMPARISON AMONG PORTFOLIO STRATEGIES 5 5 CONCLUDING REMARKS 6 STAYING AHEAD ON DOWNSIDE RISK 6 1 INTRODUCTION 6 2 MEASURING DOWNSIDE RISK: VAR AND EVAR 6 2 1 DEFINITION AND PROPERTIES 6 2 2 MODELING EVAR DYNAMICALLY 6 3 THE ASSET ALLOCATION PROBLEM 6 4 EMPIRICAL ILLUSTRATION 6 5 CONCLUSION 7 OPTIMIZATION AND PORTFOLIO SELECTION 7 1 INTRODUCTION 7 2 PART 1: THE FORSEY–SORTINO OPTIMIZER 7 2 1 BASIC ASSUMPTIONS 7 2 2 OPTIMIZE OR MEASURE PERFORMANCE 7 3 PART 2: THE DTR OPTIMIZER APPENDIX: FORMAL DEFINITIONS AND PROCEDURES 8 COMPUTING OPTIMAL MEAN/DOWNSIDE RISK FRONTIERS: THE ROLE OF ELLIPTICITY 8 1 INTRODUCTION 8 2 MAIN PROPOSITION 8 3 THE CASE OF TWO ASSETS 8 4 CONIC RESULTS 8 5 SIMULATION METHODOLOGY 8 6 CONCLUSION 9 PORTFOLIO OPTIMIZATION WITH “ THRESHOLD ACCEPTING ” : A PRACTICAL GUIDE 9 1 INTRODUCTION 9 2 PORTFOLIO OPTIMIZATION PROBLEMS 9 2 1 RISK AND REWARD 9 2 2 THE PROBLEM SUMMARIZED 9 3 THRESHOLD ACCEPTING 9 3 1 THE ALGORITHM 9 3 2 IMPLEMENTATION 9 4 STOCHASTICS 9 5 DIAGNOSTICS 9 5 1 BENCHMARKING THE ALGORITHM 9 5 2 ARBITRAGE OPPORTUNITIES 9 5 3 DEGENERATE OBJECTIVE FUNCTIONS 9 5 4 THE NEIGHBORHOOD AND THE THRESHOLDS 9 6 CONCLUSION ACKNOWLEDGMENT 10 SOME PROPERTIES OF AVERAGING SIMULATED OPTIMIZATION METHODS 10 1 SECTION 10 2 SECTION 10 3 REMARK 10 4 SECTION 3: FINITE SAMPLE PROPERTIES OF ESTIMATORS OF ALPHA AND TRACKING ERROR 10 5 REMARK 10 6 REMARK 10 7 SECTION 10 8 SECTION 5: GENERAL LINEAR RESTRICTIONS 10 9 SECTION 10 10 SECTION 7: CONCLUSION 11 HEURISTIC PORTFOLIO OPTIMIZATION: BAYESIAN UPDATING WITH THE JOHNSON FAMILY OF DISTRIBUTIONS 11 1 INTRODUCTION 11 2 A BRIEF HISTORY OF PORTFOLIO OPTIMIZATION 11 3 THE JOHNSON FAMILY 11 3 1 BASIC PROPERTIES 11 3 2 DENSITY ESTIMATION 11 3 3 SIMULATING JOHNSON RANDOM VARIATES 11 4 THE PORTFOLIO OPTIMIZATION ALGORITHM 11 4 1 THE MAXIMIZATION PROBLEM 11 4 2 THE THRESHOLD ACCEPTANCE ALGORITHM 11 5 DATA REWEIGHTING 11 6 ALPHA INFORMATION 11 7 EMPIRICAL APPLICATION 11 7 1 THE DECAY FACTOR, ? 11 7 2 THE COEFFICIENT OF DISAPPOINTMENT AVERSION, A 11 7 3 THE IMPORTANCE OF NON-GAUSSIANITY 11 8 CONCLUSION 11 9 APPENDIX 12 MORE THAN YOU EVER WANTED TO KNOW ABOUT CONDITIONAL VALUE AT RISK OPTIMIZATION 12 1 INTRODUCTION : RISK MEASURES AND THEIR AXIOMATIC FOUNDATIONS 12 2 A SIMPLE ALGORITHM FOR CVAR OPTIMIZATION 12 3 DOWNSIDE RISK MEASURES 12 3 1 DO WE NEED DOWNSIDE RISK MEASURES? 12 3 2 HOW MUCH MOMENTUM INVESTING IS IN A DOWNSIDE RISK MEASURE? 12 3 3 WILL DOWNSIDE RISK MEASURES LEAD TO “ UNDER-DIVERSIFICATION ” ? 29 12 4 SCENARIO GENERATION I: THE IMPACT OF ESTIMATION AND APPROXIMATION ERROR 12 4 1 ESTIMATION ERROR 12 4 2 APPROXIMATION ERROR 12 5 SCENARIO GENERATION II: CONDITIONAL VERSUS UNCONDITIONAL RISK MEASURES 12 6 AXIOMATIC DIFFICULTIES: WHO HAS CVAR PREFERENCES ANYWAY? 12 7 CONCLUSION