Some new results on value ranges of risks for mean–variance portfolio models

Researchers in the past always laid more emphasis on constructing models, assessing model risks and developing algorithms to solve various models, paying little attention to the research on the value ranges of risks for portfolio models. This paper attempts to fill this gap by presenting some practical approaches to obtain new and accurate results on value ranges of risks for the traditional mean–variance portfolio models. The accurate upper and lower bounds are identified for the minimizing risk portfolio models with or without short selling. A rigorous mathematical proof is utilized to derive necessary and sufficient conditions of the equal weight portfolio model that is equivalent to the minimizing risk portfolio model. Two numerical examples are given to verify the effectiveness and correctness of the theorems and corollaries discussed in this paper.