Risk Management in Stochastic Problems of Stock Investment

Abstract

We consider a game with nature as a decision-making model in stochastic problems of stock investment. The mathematical expectation of the investor’s gain is taken as an assessment of efficiency. The standard deviation is used as an assessment of risk. The emerging two-criterion problem is formalized using the method of translating one criterion into a constraint. The case of pure strategies (the choice of one investment option) and the case of mixed strategies (diversification of investment) are considered. For the case of mixed strategies, either the problem of maximizing a linear function with one linear and one quadratic constraint or the problem of minimizing a quadratic function with two linear constraints arises. Analytical solution methods based on the Karush – Kuhn – Tucker optimality conditions are obtained for both problems. Practical examples of the application of these methods using real statistical data are given.