Abstract:In modern securities investment theories and methods, the optimal risk portfolio is one of the most important core concepts. This paper mainly studies the mathematical mechanism of the formation of the concept of the optimal risk portfolio in portfolio selection and makes mathematical derivation and solution. First of all, we use the mathematical analysis method to construct the corresponding relationship, that is, the risk, return of portfolio and the number pairs of plane coordinate system to establish one-to-one correspondence, and use the algebraic method to define an order relationship among these number pairs. Then, from the perspective of analytic geometry, using the relevant properties of conic in two-dimensional plane, through the analysis of the intersection coordinates of quadric clusters, the mathematical equations that meet the relevant conditions of portfolio theory are constructed, and then mathematical derivation and solution are carried out. Finally, by analyzing the mathematical model of the investment opportunity set of the binary portfolio, the mathematical expression of the optimal risk portfolio in the binary portfolio is determined, and this method is extended to the case of multiple portfolio.