Computing near-optimal Value-at-Risk portfolios using integer programming techniques

Abstract

Value-at-Risk (VaR) is one of the main regulatory tools used for risk management purposes. However, it is difficult to compute optimal VaR portfolios; that is, an optimal risk-reward portfolio allocation using VaR as the risk measure. This is due to VaR being non-convex and of combinatorial nature. In particular, it is well-known that the VaR portfolio problem can be formulated as a mixed-integer linear program (MILP) that is difficult to solve with current MILP solvers for medium to large-scale instances of the problem. Here, we present an algorithm to compute near-optimal VaR portfolios that takes advantage of this MILP formulation and provides a guarantee of the solution's near-optimality. As a byproduct, we obtain an algorithm to compute tight lower bounds on the VaR portfolio problem that outperform related algorithms proposed in the literature for this purpose. The near-optimality guarantee provided by the proposed algorithm is obtained thanks to the relation between minimum risk portfolios satisfying a reward benchmark and the corresponding maximum reward portfolios satisfying a risk benchmark. These alternate formulations of the portfolio allocation problem have been frequently studied in the case of convex risk measures and concave reward functions. Here, this relationship is considered for general risk measures and reward functions. To illustrate the efficiency of the presented algorithm, numerical results are presented using historical asset returns from the US financial market.

Model and efficient algorithm for the portfolio selection problem with real‐world constraints under value‐at‐risk measure

“…Our preliminary experiments and the results provided by Babat et al. (2018) indicate that the method proposed by Babat et al. (2018) is more efficient than other existing IBHAs in the literature.…”

“…(2018) indicate that the method proposed by Babat et al. (2018) is more efficient than other existing IBHAs in the literature. However, due to the need for shadow‐price information, it is just applicable to problems having no binary variables other than those used in the VaR definition, and hence it cannot be applied for PSPs with real‐world constraints.…”

Portfolio Optimization Problem: A Taxonomic Review of Solution Methodologies

“…Babat et al proposed two algorithms (Algorithm A and Algorithm B) that exploited a Mixed-Integer Linear Programming (MILP) formulation to solve a Value-at-Risk (VaR) portfolio model (US stock market data obtained from Kenneth R.French's Website) [23]. These two algorithms were developed to generate near-optimal solutions.…”

Robust optimization approaches for portfolio selection: a comparative analysis

“…Despite its widespread use, VaR has received criticism as being an incoherent risk measure (Artzner et al 1999). Moreover, the use of VaR in an optimization context is a difficult task, because it requires the solution of a non-convex problem (Babat et al 2018). As an alternative, Rockafellar and Uryasev (2000) introduced conditional VaR (CVaR), defined as the conditional expectation of the loss of a portfolio that is at least equal to VaR.…”