This article suggests a method to accomplish some problems considered in the model of securities portfolio optimization for the case of risk-free borrowing and lending. The calculating of the feasible set is carried out using the Lagrange function by introducing the linear equations of the agent utility function as crucial lines. Calculation has done including for coordinates returns — risk when the risk is taken as a standard deviation of return. The authors define the parameters of the tangent line to the efficient frontier, which is a ray which emanate from the risk-free rate of return point. To construct the efficient frontier and tangent to it the authors have developed an algorithm that allows you to build the efficient frontier and the tangent to it in the case where the approximating polynomial for the feasible set has an inflection point to the left of the point of tangency .