Theoretical Framework

Much research effort has been devoted to analyze the properties of portfolio performance measures, both risk and safety. The goal of this research is to understand which properties a measure should satisfy to be attractive to an investor. The theory developed is based fundamentally on one hand on the concept of stochastic dominance consistency and on the other hand on the concept of coherent measure. In this chapter, we introduce these two concepts and discuss their relevance for the portfolio optimization models. In particular, we review the LP portfolio optimization models with respect to their consistency with the stochastic dominance rules and other axiomatic theories. We show that for typical dispersion statistics as risk measures, like variance, the mean-risk approach consistency cannot be proved, whereas models based on a safety measure maximization are consistent. Finally, we show that the models based on the safety measure maximization meet also the coherent measurement rules. The mathematical side of the theory is kept as limited as possible.