This book provides a gentle introduction into the theory of probability
metrics and the problem of optimal portfolio selection, which is considered
in the general context of risk and rewardmeasures.We illustrate in numerous
examples the basic concepts and where more technical knowledge is needed,
an appendix is provided.
The book is organized in the following way. Chapters 1 and 2 con-
tain introductory material from the fields of probability and optimization
theory. Chapter 1 is necessary for understanding the general ideas behind
probability metrics covered in Chapter 3 and ideal probability metrics in
particular described in Chapter 4. The material in Chapter 2 is used when
discussing optimal portfolio selection problems in Chapters 8, 9, and 10.
We demonstrate how probability metrics can be applied to certain areas in
finance in the following chapters:
■ Chapter 5—stochastic dominance orders.
■ Chapter 6—the construction of risk and dispersion measures.
■ Chapter 7—problems involving average value-at-risk and spectral risk
measures in particular.
■ Chapter 8—reward-risk analysis generalizing mean-variance analysis.
■ Chapter 9—the problem of benchmark tracking.
■ Chapter 10—the construction of performance measures.
Chapters 5, 6, and 7 are also a prerequisite for the material in the last
three chapters. Chapter 5 describes expected utility theory and stochastic
dominance orders. The focus in Chapter 6 is on general dispersion measures
and risk measures. Finally, in Chapter 7 we discuss the average value-at-risk
and spectral risk measures, which are two particular families of coherent
risk measures considered in Chapter 6.
The classical mean-variance analysis and the more general mean-risk
analysis are explored in Chapter 8.We consider the structure of the efficient
portfolios when average value-at-risk is selected as a risk measure. Chapter
9 is focused on the benchmark tracking problem.We generalize significantly
the problem applying the methods of probability metrics. In Chapter 10,
we discuss performance measures in the general framework of reward-risk
analysis.We consider classes of performance measures that lead to practical
optimal portfolio problems.