The Probabilistic Risk Measure VaR as Constraint in Portfolio Optimization Problem

The paper realizes inclusion of probabilistic measure for risk, VaR (Value at Risk), into a portfolio optimization problem. The formal analysis of the portfolio problem illustrates the evolution of the portfolio theory in sequentially inclusion of different market characteristics into the problem. They make modifications and complications of the portfolio problem by adding various constraints to consider requirements for taxes, boundaries for assets, cardinality constraints, and allocation of the investment resources. All these characteristics and parameters of the investment participate in the portfolio problem by analytical algebraic relations. The VaR definition of the portfolio risk is formalized in a probabilistic manner. The paper applies approximation of such probabilistic constraint in algebraic form. Geometrical interpretation is given for explaining the influence of the VaR constraint to the portfolio solution. Numerical simulation with data of the Bulgarian Stock Exchange illustrates the influence of the VaR constraint into the portfolio optimization problem.