A Composite Risk Measure Framework for Decision Making Under Uncertainty

doi:10.1007/s40305-018-0211-9

Journal of the Operations Research Society of China ›› 2019, Vol. 7 ›› Issue (1): 43-68.doi: 10.1007/s40305-018-0211-9

所属专题： Stochastic optimization

收稿日期:2017-09-25 修回日期:2018-03-16 出版日期:2019-03-30 发布日期:2019-03-30

A Composite Risk Measure Framework for Decision Making Under Uncertainty

Peng-Yu Qian¹, Zi-Zhuo Wang², Zai-Wen Wen³

1 Graduate School of Business, Columbia University, New York, NY 10027, USA;
2 Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA;
3 Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China

Received:2017-09-25 Revised:2018-03-16 Online:2019-03-30 Published:2019-03-30
Contact: Peng-Yu Qian, Zi-Zhuo Wang, Zai-Wen Wen E-mail:pq2124@columbia.edu;zwang@umn.edu;wenzw@math.pku.edu.cn

摘要/Abstract

Abstract:

In this paper, we present a unified framework for decision making under uncertainty. Our framework is based on the composite of two risk measures, where the inner risk measure accounts for the risk of decision if the exact distribution of uncertain model parameters were given, and the outer risk measure quantifies the risk that occurs when estimating the parameters of distribution. We show that the model is tractable under mild conditions. The framework is a generalization of several existing models, including stochastic programming, robust optimization, distributionally robust optimization. Using this framework, we study a few new models which imply probabilistic guarantees for solutions and yield less conservative results compared to traditional models. Numerical experiments are performed on portfolio selection problems to demonstrate the strength of our models.

Key words: Risk management, Stochastic programming, Portfolio management

. [J]. Journal of the Operations Research Society of China, 2019, 7(1): 43-68.

Peng-Yu Qian, Zi-Zhuo Wang, Zai-Wen Wen. A Composite Risk Measure Framework for Decision Making Under Uncertainty[J]. Journal of the Operations Research Society of China, 2019, 7(1): 43-68.

参考文献

[1] Dantzig, G.B.:Linear programming under uncertainty. Manag. Sci. 1(3-4), 197-206(1955)
[2] Soyster, A.L.:Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21(5), 1154-1157(1973)
[3] Delage, E., Ye, Y.:Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595-612(2010)
[4] Dupačová, J.:The minimax approach to stochastic programming and an illustrative application. Stoch. Int. J. Probab. Stoch. Process. 20(1), 73-88(1987)
[5] Scarf, H., Arrow, K.J., Karlin, S.:A min-max solution of an inventory problem. Stud. Math. Theory Invent. Prod. 10, 201-209(1958)
[6] Gaivoronski, A.A., Pflug, G.:Value-at-risk in portfolio optimization:properties and computational approach. J. Risk 7(2), 1-31(2005)
[7] Rockafellar, R.T., Uryasev, S.:Optimization of conditional value-at-risk. J. Risk 2, 21-42(2000)
[8] El Ghaoui, L., Oks, M., Oustry, F.:Worst-case value-at-risk and robust portfolio optimization:a conic programming approach. Oper. Res. 51(4), 543-556(2003)
[9] Zhu, S., Fukushima, M.:Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155-1168(2009)
[10] Bertsimas, D., Brown, D.B.:Constructing uncertainty sets for robust linear optimization. Oper. Res. 57(6), 1483-1495(2009)
[11] Natarajan, K., Pachamanova, D., Sim, M.:Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129-1141(2009)
[12] Bertsimas, D., Gupta, V., Kallus, N.:Data-driven robust optimization. Math. Progr. 167(2), 235-292(2013)
[13] Wiesemann, W., Kuhn, D., Sim, M.:Distributionally robust convex optimization. Oper. Res. 62(6), 1358-1376(2014)
[14] Artzner, P., Delbaen, F., Eber, J.M., Heath, D.:Coherent measures of risk. Math. Finance 9(3), 203-228(1999)
[15] Shapiro, A., Dentcheva, D., Ruszczyński, A.:Lectures on Stochastic Programming:Modeling and Theory, vol. 9. SIAM, Philadelphia (2009)
[16] RiskMetrics.:Riskmetrics:Technical Document. Morgan Guaranty Trust Company of New York, New York (1996)
[17] Kast, R., Luciano, E., Peccati, L.:VaR and optimization. In:2nd International Workshop on Preferences and Decisions, Trento, vol. 1(1998)
[18] Lucas, A., Klaassen, P.:Extreme returns, downside risk, and optimal asset allocation. J. Portf. Manag. 25(1), 71-79(1998)
[19] Einhorn, D., Brown, A.:Private profits and socialized risk. Glob. Assoc. Risk Prof. 42, 10-26(2008)
[20] Birge, J.R., Louveaux, F.:Introduction to Stochastic Programming. Springer, Berlin (2011)
[21] Bertsimas, D., Brown, D.B., Caramanis, C.:Theory and applications of robust optimization. SIAM Rev. 53(3), 464-501(2011)
[22] Ben-Tal, A., El Ghaoui, L., Nemirovski, A.:Robust Optimization. Princeton University Press, Princeton (2009)
[23] Ben-Tal, A., Nemirovski, A.:Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1-13(1999)
[24] El Ghaoui, L., Lebret, H.:Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18(4), 1035-1064(1997)
[25] El Ghaoui, L., Oustry, F., Lebret, H.:Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9(1), 33-52(1998)
[26] Bertsimas, D., Pachamanova, D., Sim, M.:Robust linear optimization under general norms. Oper. Res. Lett. 32(6), 510-516(2004)
[27] Bertsimas, D., Sim, M.:The price of robustness. Oper. Res. 52(1), 35-53(2004)
[28] Chen, X., Sim, M., Sun, P.:A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058-1071(2007)
[29] Prékopa, A.:Stochastic Programming. Springer, Berlin (1995)
[30] Bertsimas, D., Popescu, I.:Optimal inequalities in probability theory:a convex optimization approach. SIAM J. Optim. 15(3), 780-804(2005)
[31] Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.:Robust solutions of optimization problems affected by uncertain probabilities. Manag. Sci. 59(2), 341-357(2013)
[32] Pardo, L.:Statistical Inference Based on Divergence Measures. CRC Press, Boca Raton (2005)
[33] Klabjan, D., Simchi-Levi, D., Song, M.:Robust stochastic lot-sizing by means of histograms. Prod. Oper. Manag. 22(3), 691-710(2013)
[34] Wang, Z., Glynn, P., Ye, Y.:Likelihood robust optimization for data-driven problems. CMS 13(2), 241-261(2016)
[35] Föllmer, H., Schied, A.:Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429-447(2002)
[36] Huber, P.J.:Robust Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1981)
[37] Luedtke, J., Ahmed, S.:A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674-699(2008)
[38] Nemirovski, A., Shapiro, A.:Scenario Approximations of Chance Constraints. Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3-47. Springer, Berlin (2006)
[39] Sarykalin, S., Serraino, G., Uryasev, S.:Value-at-risk vs. conditional value-at-risk in risk management and optimization. Tutorials in Operations Research. INFORMS, Hanover, MD (2008)
[40] Cui, X., Zhu, S., Sun, X., Li, D.:Nonlinear portfolio selection using approximate parametric valueat-risk. J. Bank. Finance 37(6), 2124-2139(2013)
[41] Wen, Z., Peng, X., Liu, X., Sun, X., Bai, X.:Asset allocation under the basel accord risk measures. arXiv:1308.1321(2013)
[42] Delbaen, F.:Coherent risk measures on general probability spaces. In:Sandmann, K., Schoenbucher, P.J. (eds.) Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002)
[43] Guigues, V., Römisch, W.:Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures. SIAM J. Optim. 22(2), 286-312(2012)
[44] Philpott, A.B., de Matos, V.L.:Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Eur. J. Oper. Res. 218(2), 470-483(2012)
[45] Shapiro, A.:On complexity of multistage stochastic programs. Oper. Res. Lett. 34(1), 1-8(2006)
[46] Wang, W., Ahmed, S.:Sample average approximation of expected value constrained stochastic programs. Oper. Res. Lett. 36(5), 515-519(2008)
[47] Alexander, S., Coleman, T.F., Li, Y.:Minimizing CVaR and VaR for a portfolio of derivatives. J. Bank. Finance 30(2), 583-605(2006)
[48] Xu, H., Zhang, D.:Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications. Math. Program. 119(2), 371-401(2009)
[49] Iyengar, G., Ma, A.K.C.:Fast gradient descent method for mean-CVaR optimization. Ann. Oper. Res. 205(1), 203-212(2013)
[50] Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.:Bayesian Data Analysis. CRC Press, Boca Raton (2013)

A Composite Risk Measure Framework for Decision Making Under Uncertainty

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