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

Onur Babat, J.C. Vera, Luis F. Zuluaga

Research output: Contribution to journalArticleScientificpeer-review

11 Citations (Scopus)


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 upper 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.
Original languageEnglish
Pages (from-to)304-315
JournalEuropean Journal of Operational Research
Issue number1
Publication statusPublished - 1 Apr 2018


  • Risk analysis
  • Value-at-Risk
  • Integer programming relaxations
  • Portfolio allocation


Dive into the research topics of 'Computing near-optimal Value-at-Risk portfolios using integer programming techniques'. Together they form a unique fingerprint.

Cite this