A Gradual Non-Convexification Method for Minimizing VaR

Given a finite set of m scenarios, computing a portfolio with the minimium Value-at-Risk (VaR) is computationally difficult: the portfolio VaR function is non-convex, non-smooth, and has many local minima. Instead of formulating an n-asset optimal VaR portfolio problem as minimizing a loss quantile function to determine the asset holding vector R, we consider it as a minimization problem in an augmented space R, with a linear objective function under a probability constraint. We then propose a new gradual non-convexification penalty method, aiming to reach a global minimum of nonconvex minimization under the probability constraint. A continuously differentiable piecewise quadratic function is used to approximate step functions, the sum of which defines the probabilistic constraint. In an attempt to reach the global minimizer, we solve a sequence of minimization problems indexed by a parameter ρk > 0 where −ρk is the minimum curvature for the probability constraint approximation. As the indexing parameter increases, the approximation function for the probabilistic inequality constraint becomes more non-convex. Furthermore, the solution of the k-th optimization problem is used as the starting point of the (k + 1)-th problem. Our new method has three advantages. Firstly, it is structurally simple. Secondly, it is efficient since each function evaluation requires O(m) work. Thirdly, a gradual non-convexification process is designed to track the global minimum. Both historical and synthetic data are used to illustrate the efficacy of the proposed VaR minimization method. We compare our method with Gaivoronski and Pflug’s quantile-based smoothed VaR method in terms of VaR, CPU time, and efficient frontiers. We show that our gradual non-convexification penalty method yields better minimal VaR portfolio. We show that the proposed gradual non-convexification penalty method is computationally much more efficient, especially when the number of scenarios is large.