### Abstract

Original language | English |
---|---|

Pages (from-to) | 304-315 |

Journal | European Journal of Operational Research |

Volume | 266 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

### Fingerprint

### Keywords

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

### Cite this

*European Journal of Operational Research*,

*266*(1), 304-315. https://doi.org/10.1016/j.ejor.2017.09.009

}

*European Journal of Operational Research*, vol. 266, no. 1, pp. 304-315. https://doi.org/10.1016/j.ejor.2017.09.009

**Computing near-optimal Value-at-Risk portfolios using integer programming techniques.** / Babat, Onur; Vera, J. C.; Zuluaga, Luis F.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Computing near-optimal Value-at-Risk portfolios using integer programming techniques

AU - Babat, Onur

AU - Vera, J. C.

AU - Zuluaga, Luis F.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Risk analysis

KW - Value-at-Risk

KW - Integer programming relaxations

KW - Portfolio allocation

U2 - 10.1016/j.ejor.2017.09.009

DO - 10.1016/j.ejor.2017.09.009

M3 - Article

VL - 266

SP - 304

EP - 315

JO - European Journal of Operational Research

JF - European Journal of Operational Research

SN - 0377-2217

IS - 1

ER -