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**Distributionally robust return-risk optimization models and their applications.**
*(English)*
Zbl 1442.91120

Summary: Based on the risk control of conditional value-at-risk, distributionally robust return-risk optimization models with box constraints of random vector are proposed. They describe uncertainty in both the distribution form and moments (mean and covariance matrix of random vector). It is difficult to solve them directly. Using the conic duality theory and the minimax theorem, the models are reformulated as semidefinite programming problems, which can be solved by interior point algorithms in polynomial time. An important theoretical basis is therefore provided for applications of the models. Moreover, an application of the models to a practical example of portfolio selection is considered, and the example is evaluated using a historical data set of four stocks. Numerical results show that proposed methods are robust and the investment strategy is safe.

### MSC:

91G70 | Statistical methods; risk measures |

90C22 | Semidefinite programming |

90C46 | Optimality conditions and duality in mathematical programming |

91G10 | Portfolio theory |

### Software:

SDPT3
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\textit{L. Yang} et al., J. Appl. Math. 2014, Article ID 784715, 9 p. (2014; Zbl 1442.91120)

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### References:

[1] | Markowitz, H., Portfolio selection, The Journal of Finance, 7, 1, 77-91 (1952) |

[2] | Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 3, 203-228 (1999) · Zbl 0980.91042 |

[3] | Mausser, H.; Rosen, D., Beyond VaR: from measuring risk to managing risk, ALGO Research Quarterly, 1, 2, 5-20 (1999) |

[4] | Rockafellar, R. T.; Uryasev, S., Optimization of conditional value-at-risk, Journal of Risk, 2, 3, 21-41 (2000) |

[5] | Uryasev, S.; Rockafellar, R. T., Conditional value-at-risk: optimization approach, Stochastic Optimization: Algorithms and Applications. Stochastic Optimization: Algorithms and Applications, Applied Optimization, 54, 411-435 (2001), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0989.91052 |

[6] | Lin, X.; Gong, Q., Research on the efficient frontier of Mean-CVaR under normal distribution condition, Management Sciences in China, 17, 3, 52-55 (2004) |

[7] | Alexander, G. J.; Baptista, A. M., A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model, Management Science, 50, 9, 1261-1273 (2004) |

[8] | Andersson, F.; Mausser, H.; Rosen, D.; Uryasev, S., Credit risk optimization with conditional value-at-risk criterion, Mathematical Programming B, 89, 2, 273-291 (2001) · Zbl 0994.91028 |

[9] | Lim, C.; Sherali, H. D.; Uryasev, S., Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization, Computational Optimization and Applications, 46, 3, 391-415 (2010) · Zbl 1200.91283 |

[10] | Ogryczak, W.; Śliwiński, T., On solving the dual for portfolio selection by optimizing conditional value at risk, Computational Optimization and Applications, 50, 3, 591-595 (2011) · Zbl 1242.90102 |

[11] | Ben-Tal, A.; El Ghaoui, L.; Nemirovski, A., Robust Optimization (2009), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 1221.90001 |

[12] | Bertsimas, D.; Brown, D. B.; Caramanis, C., Theory and applications of robust optimization, SIAM Review, 53, 3, 464-501 (2011) · Zbl 1233.90259 |

[13] | Ogryczak, W., Tail mean and related robust solution concepts, International Journal of Systems Science, 45, 1, 29-38 (2014) · Zbl 1307.93384 |

[14] | Gao, J.; Bian, N., Portfolio selection based on the risk control of WCVaR, System Engineering: Theory & Practice, 29, 5, 69-75 (2009) |

[15] | Zhu, S.; Fukushima, M., Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57, 5, 1155-1168 (2009) · Zbl 1233.91254 |

[16] | Tong, X.; Liu, Q., WCVaR risk analysis under the box discrete distribution and application, System Engineering: Theory & Practice, 30, 2, 305-314 (2010) |

[17] | Bertsimas, D.; Doan, X. V.; Natarajan, K.; Teo, C. P., Models for minimax stochastic linear optimization problems with risk aversion, Mathematics of Operations Research, 35, 3, 580-602 (2010) · Zbl 1218.90215 |

[18] | Zymler, S.; Kuhn, D.; Rustem, B., Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming A, 137, 1-2, 167-198 (2013) · Zbl 1286.90103 |

[19] | Delage, E.; Ye, Y., Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58, 3, 595-612 (2010) · Zbl 1228.90064 |

[20] | Mansini, R.; Ogryczak, W.; Speranza, M. G., Twenty years of linear programming based portfolio optimization, European Journal of Operational Research, 234, 2, 518-535 (2014) · Zbl 1304.91202 |

[21] | Sharpe, W. F., The Sharpe ratio, Journal of Portfolio Management, 21, 1, 49-58 (1994) |

[22] | Shapiro, A., On duality theory of conic linear problems, Semi-Infinite Programming. Semi-Infinite Programming, Nonconvex Optimization and Its Applications, 57, 135-165 (2001), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 1055.90088 |

[23] | Deng, X., Decision on nonnegative investment proportional coefficient of portfolio for risk minimization, Pure and Applied Mathematics, 23, 4, 524-528 (2007) · Zbl 1150.91355 |

[24] | Tütüncü, R. H.; Toh, K. C.; Todd, M. J., Solving semidefinite-quadratic-linear programs using SDPT3, Mathematical Programming B, 95, 2, 189-217 (2003) · Zbl 1030.90082 |

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