Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)

This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers. Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization......Page 6 Contents......Page 10 Preface......Page 16 Acknowledgments......Page 18 About the Authors......Page 20 1.1 INTRODUCTION......Page 22 1.3 DISCRETE PROBABILITY DISTRIBUTIONS......Page 23 1.4 CONTINUOUS PROBABILITY DISTRIBUTIONS......Page 26 1.5 STATISTICAL MOMENTS AND QUANTILES......Page 34 1.6 JOINT PROBABILITY DISTRIBUTIONS......Page 38 1.7 PROBABILISTIC INEQUALITIES......Page 51 1.8 SUMMARY......Page 53 BIBLIOGRAPHY......Page 54 2.1 INTRODUCTION......Page 56 2.2 UNCONSTRAINED OPTIMIZATION......Page 57 2.3 CONSTRAINED OPTIMIZATION......Page 69 BIBLIOGRAPHY......Page 79 3.1 INTRODUCTION......Page 82 3.2 MEASURING DISTANCES: THE DISCRETE CASE......Page 83 3.3 PRIMARY, SIMPLE, AND COMPOUND METRICS......Page 93 3.5 TECHNICAL APPENDIX......Page 111 BIBLIOGRAPHY......Page 123 4.1 INTRODUCTION......Page 124 4.2 THE CLASSICAL CENTRAL LIMIT THEOREM......Page 126 4.3 THE GENERALIZED CENTRAL LIMIT THEOREM......Page 141 4.4 CONSTRUCTION OF IDEAL PROBABILITY METRICS......Page 145 4.6 TECHNICAL APPENDIX......Page 152 BIBLIOGRAPHY......Page 157 5.1 INTRODUCTION......Page 160 5.2 EXPECTED UTILITY THEORY......Page 162 5.3 STOCHASTIC DOMINANCE......Page 168 5.4 PROBABILITY METRICS AND STOCHASTIC DOMINANCE......Page 178 5.6 TECHNICAL APPENDIX......Page 182 BIBLIOGRAPHY......Page 190 6.1 INTRODUCTION......Page 192 6.2 MEASURES OF DISPERSION......Page 195 6.3 PROBABILITY METRICS AND DISPERSION MEASURES......Page 201 6.4 MEASURES OF RISK......Page 202 6.5 RISK MEASURES AND DISPERSION MEASURES......Page 219 6.6 RISK MEASURES AND STOCHASTIC ORDERS......Page 220 6.7 SUMMARY......Page 221 6.8 TECHNICAL APPENDIX......Page 222 BIBLIOGRAPHY......Page 226 7.1 INTRODUCTION......Page 228 7.2 AVERAGE VALUE-AT-RISK......Page 229 7.3 AVaR ESTIMATION FROM A SAMPLE......Page 235 7.4 COMPUTING PORTFOLIO AVaR IN PRACTICE......Page 237 7.5 BACKTESTING OF AVaR......Page 241 7.6 SPECTRAL RISK MEASURES......Page 243 7.7 RISK MEASURES AND PROBABILITY METRICS......Page 245 7.9 TECHNICAL APPENDIX......Page 248 BIBLIOGRAPHY......Page 265 8.1 INTRODUCTION......Page 266 8.2 MEAN-VARIANCE ANALYSIS......Page 268 8.3 MEAN-RISK ANALYSIS......Page 279 8.5 TECHNICAL APPENDIX......Page 295 BIBLIOGRAPHY......Page 306 9.1 INTRODUCTION......Page 308 9.2 THE TRACKING ERROR PROBLEM......Page 309 9.3 RELATION TO PROBABILITY METRICS......Page 313 9.4 EXAMPLES OF r.d. METRICS......Page 317 9.5 NUMERICAL EXAMPLE......Page 321 9.7 TECHNICAL APPENDIX......Page 325 BIBLIOGRAPHY......Page 336 10.1 INTRODUCTION......Page 338 10.2 REWARD-TO-RISK RATIOS......Page 339 10.3 REWARD-TO-VARIABILITY RATIOS......Page 354 10.5 TECHNICAL APPENDIX......Page 364 BIBLIOGRAPHY......Page 380 Index......Page 382

S ince the 1990s, significant progress has been made in developing the concept of a risk measure from both a theoretical and a practical viewpoint. This notion has evolved into a materially different form from the original idea behind traditional mean-variance analysis. As a consequence, the distinction between risk and uncertainty, which translates into a distinction between a risk measure and a dispersion measure, offers a new way of looking at the problem of optimal portfolio selection.

In Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization, the authors assert that the ideas behind the concept of probability metrics can be borrowed and applied in the field of asset management in order to construct an ideal risk measure which would be ideal for a given optimal portfolio selection problem. They provide a basic introduction to the theory of probability metrics and the problem of optimal portfolio selection considered in the general context of risk and reward measures.

Generally, the theory of probability metrics studies the problem of measuring distances between random quantities. There are no limitations in the theory of probability metrics concerning the nature of the random quantities, which makes its methods fundamental and appealing. Actually, it is more appropriate to refer to the random quantities as random elements: they can be random variables, random vectors, random functions, or random elements of general spaces. In the context of financial applications, we can study the distance between two random stocks prices, or between vectors of financial variables building portfolios, or between entire yield curves that are much more complicated objects. The methods of the theory remain the same, no matter the nature of the random elements.

Using numerous illustrative examples, this book shows how probability metrics can be applied to a range of areas in finance, including: stochastic dominance orders, the construction of risk and dispersion measures, problems involving average value-at-risk and spectral risk measures in particular, reward-risk analysis, generalizing mean-variance analysis, benchmark tracking, and the construction of performance measures. For each chapter where more technical knowledge is necessary, an appendix is included.

"In Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization, the authors assert that the ideas behind the concept of probability metrics can be borrowed and applied in the field of asset management in order to construct an ideal risk measure which would be "ideal" for a given optimal portfolio selection problem. They provide a basic introduction to the theory of probability metrics and the problem of optimal portfolio selection considered in the general context of risk and reward measures." "Using numerous illustrative examples, this book shows how probability metrics can be applied to a range of areas in finance, including: stochastic dominance orders, the construction of risk and dispersion measures, problems involving average value-at-risk and spectral risk measures in particular, reward-risk analysis, generalizing mean-variance analysis, benchmark tracking, and the construction of performance measures. For each chapter where more technical knowledge is necessary, an appendix is included."--Jacket The finance industry is seeing increased interest in new risk measures and techniques for portfolio optimization when parameters of the model are uncertain. In this book, Fabozzi, Stoyanov, and Rachev intend to break new ground in tying together the theory of probability metrics to both risk measurement and portfolio optimization. Unlike current literature in this field, this book proposes applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance. An introduction to stochastic models for risk evaluation and portfolio selection enhanced by insights from the field of probability metrics and optimization theory. This book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into a single framework