A D.C. Formulation of Value-at-Risk constrained Optimization

In this paper we present a representation of Value-at-Risk (V@R) as a difference of convex (D.C.) functions in the case where the distribution of the underlying random variable is discrete and has finitely many atoms. The D.C. representation is used to study a financial risk-return portfolio selection problemwith a V@R constraint. A Branch-and-Bound algorithm that numerically solves the problem exactly is given. Numerical experiments with historical asset returns from representative market indices are performed to apply the algorithm to real-world financial market data.