Abstract
In this paper, we consider both one-period and multi-period distributionally robust mean-CVaR portfolio selection problems. We adopt an uncertainty set which considers the uncertainties in terms of both the distribution and the first two order moments. We use the parametric method and the dynamic programming technique to come up with the closed-form optimal solutions for both the one-period and the multi-period robust portfolio selection problems. Finally, we show that our approaches are efficient when compared with both normal based portfolio selection models, and robust approaches based on known moments.
Similar content being viewed by others
References
Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Prog. 99(2), 351–376 (2004)
Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton, NJ (2009)
Chen, Z., Liu, J.: Multi-period robust risk measures and portfolio selection models with regime-switching. Working Paper. Xi’an Jiaotong University (2015)
Chen, L., He, S., Zhang, S.: Tight bounds for some risk measures, with applications to robust portfolio selection. Oper. Res. 59(4), 847–865 (2011)
Chen, Z., Consigli, G., Liu, J., Li, G., Fu, T., Hu, Q.: Multi-period risk measures and optimal investment policies. In: Consigli, G., Kuhn, D., Brandimarte, P. (eds.) Optimal Financial Decision Making Under Uncertainty, pp. 1–34. Springer, Berlin (2017)
Cheng, J., Delage, E., Lisser, A.: Distributionally robust stochastic knapsack problem. SIAM J. Optim. 24, 1485–1506 (2014)
Delage, E., Iancu, D.A.: Robust multistage decision making. In: INFORMS Tutorials in Operations Research, Operations Research Revolution, pp. 20–46 (2015)
Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58, 595–612 (2010)
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)
Ermoliev, Y., Gaivoronski, A., Nedeva, C.: Stochastic optimization problems with incomplete information on distribution functions. SIAM J. Control Optim. 23(5), 697–716 (1985)
Iancu, D.A., Sharma, M., Sviridenko, M.: Supermodularity and affine policies in dynamic robust optimization. Oper. Res. 61(4), 941–956 (2013)
Li, J.Y.: Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization. Working Paper. SSRN 2838518 (2016)
Li, D., Ng, W.L.: Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Math. Finance 10, 387–406 (2000)
Lobo, M.S., Boyd, S.: The worst-case risk of a portfolio. Working Paper. Stanford University (2000)
Lotfi, S., Zenios, S.: Equivalence of robust VaR and CVaR optimization. Working Paper. Wharton Financial Institutions Center WP, 16-03 (2016)
Mamani, H., Nassiri, S., Wagner, M.R.: Closed-form solutions for robust inventory management. Manag. Sci. (2016). doi:10.1287/mnsc.2015.2391
Mulvey, J.M., Shetty, B.: Financial planning via multi-stage stochastic optimization. Comput. Oper. Res. 31(1), 1–20 (2004)
Natarajan, K., Sim, M., Uichanco, J.: Tractable robust expected utility and risk models for portfolio optimization. Math. Finance 20(4), 695–731 (2010)
Paç, A., Pınar, M.: On robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity. TOP 22(3), 875–891 (2014)
Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Probabilistic Constrained Optimization, pp. 272–281. Springer, New York (2000)
Pflug, G.C., Analui, B.: On distributionally robust multiperiod stochastic optimization. Comput. Manag. Sci. 11, 197–220 (2014)
Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007)
Pflug, G.C., Pichler, A., Wozabal, D.: The \(1/N\) investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)
Rockafellar, R.T., Uryasev, S.P.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)
Rockafellar, R.T., Uryasev, S.P.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002)
Scarf, H.: A min-max solution of an inventory problem. In: Arrow, K.J., Karlin, S., Scarf, H.E. (eds.) Studies in the Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Stanford (1958)
Shapiro, A.: A dynamic programming approach to adjustable robust optimization. Oper. Res. Lett. 39, 83–87 (2011)
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)
Sharpe, W.F., Alexander, G.J., Bailey, J.: Investments. Prentice Hall, Englewood Cliffs (1995)
Steinbach, M.: Markowitz revisited: mean-variance models in financial portfolio analysis. SIAM Rev. 43(1), 31–85 (2001)
Wozabal, D.: Robustifying convex risk measures for linear portfolios: a nonparametric approach. Oper. Res. 62, 1302–1315 (2014)
Xin, L., Goldberg, D.A., Shapiro, A.: Time (in) consistency of multistage distributionally robust inventory models with moment constraints. arXiv preprint. arXiv:1304.3074 (2013)
Yu, Y., Li, Y., Schuurmans, D., Szepesvari, C.: A general projection property for distribution families. Adv. Neural Inf. Process. Syst. 22, 2232–2240 (2009)
Žácková, J.: On minimax solutions of stochastic linear programming problems. Časopis pro Pěstování Matematiky 91, 423–430 (1966)
Zhu, S.S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155–1168 (2009)
Acknowledgements
The authors are grateful to the editor and two anonymous referees for their insightful, constructive and detailed comments and suggestions, which have helped us to improve the paper significantly in both content and style. This research was supported by the National Natural Science Foundation of China under grant numbers 71371152 and 11571270, and Programme Cai Yuanpei under grant number 34593YE.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, J., Chen, Z., Lisser, A. et al. Closed-Form Optimal Portfolios of Distributionally Robust Mean-CVaR Problems with Unknown Mean and Variance. Appl Math Optim 79, 671–693 (2019). https://doi.org/10.1007/s00245-017-9452-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-017-9452-y