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Robust portfolio asset allocation and risk measures

Abstract

Many financial optimization problems involve future values of security prices, interest rates and exchange rates which are not known in advance, but can only be forecast or estimated. Several methodologies have therefore been proposed to handle the uncertainty in financial optimization problems. One such methodology is Robust Statistics, which addresses the problem of making estimates of the uncertain parameters that are insensitive to small variations. A different way to achieve robustness is provided by Robust Optimization, which looks for solutions that will achieve good objective function values for the realization of the uncertain parameters in given uncertainty sets. Robust Optimization thus offers a vehicle to incorporate an estimation of uncertain parameters into the decision making process. This is true, for example, in portfolio asset allocation. Starting with the robust counterparts of the classical mean-variance and minimum-variance portfolio optimization problems, in this paper we review several mathematical models, and related algorithmic approaches, that have recently been proposed to address uncertainty in portfolio asset allocation, focusing on Robust Optimization methodology. We also give an overview of some of the computational results that have been obtained with the described approaches. In addition we analyze the relationship between the concepts of robustness and convex risk measures.

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Notes

  1. 1.

    In many risk-management applications, the multivariate normal distribution for asset returns does not reflect reality. In any case, the validity of such a claim may also depend on the investment horizon involved. Some evidence can be found in McNeil et al. (2005) where, by applying tests of normality to an arbitrary subgroup of 10 of the stocks in the Dow Jones index from 1993 to 2000, it can be seen that the daily, weekly and monthly return data fail the multivariate tests of normality, whereas return data over a quarter year are close to a normal distribution.

  2. 2.

    A way to define G is related to probabilistic guarantees in the likelihood that the actual realization of the uncertain coefficients will lie in the ellipsoidal uncertainty set S v . The definition of matrix G can be based on the data used to produce the estimates of the regression coefficients of the factor model (Fabozzi et al. 2007).

  3. 3.

    This kind of analysis has already been reviewed in Sect. 2.2.

  4. 4.

    In Artzner et al. (1999), a mapping ρ:Φ→ℝ is called a coherent measure of risk if it satisfies the four axioms of translation invariance, positive homogeneity, monotonicity and subadditivity, where subadditivity states that, ∀Y,ZΦ, ρ(Y+Z)≤ρ(Y)+ρ(Z); convexity is a consequence of these axioms.

  5. 5.

    A generalization of this theorem also exists in the case where Ω is an infinite set.

  6. 6.

    The goal is to form a portfolio in which the expected return is maximized, while some index of risk is minimized.

  7. 7.

    In presenting the soft robustness approach, we refer to D. Brown’s Ph.D. dissertation (2006), where emphasis is put on the concept of penalty functions.

References

  1. Alizadeh, F., & Goldfarb, D. (2003). Second-order cone programming. Mathematical Programming, 95(1), 3–51.

    Article  Google Scholar 

  2. Andersen, E. D., & Andersen, K. D. (2000). The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In H. Frenk, K. Roos, T. Terlaky, & S. Zhang (Eds.), High performance optimization (pp. 197–232). Dordrecht: Kluwer Academic. Code available from http://www.mosek.com/.

    Chapter  Google Scholar 

  3. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.

    Article  Google Scholar 

  4. Ben Tal, A., Bertsimas, D., & Brown, D. B. (2010). A soft robust model for optimization under ambiguity. Operations Research, 58(4), 1220–1234.

    Article  Google Scholar 

  5. Bienstock, D. (2007). Histogram models for robust portfolio optimization. Journal of Computational Finance, 11(1), 1–64.

    Google Scholar 

  6. Black, F., & Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28–43.

    Article  Google Scholar 

  7. Byrd, R., Hribar, M. E., & Nocedal, J. (1999). An interior point method for large scale nonlinear programming. SIAM Journal on Optimization, 9(4), 877–900.

    Article  Google Scholar 

  8. Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of Operations Research, 45, 21–58.

    Article  Google Scholar 

  9. Brown, D. B. (2006). Risk and robust optimization. Ph.D. Dissertation, Massachusetts Institute of Technology.

  10. Cavadini, F., Sbuelz, A., & Trojani, F. (2001). A simplified way of incorporating model risk, estimation risk and robustness in mean variance portfolio management (Working Paper).

  11. Chan, L. K. C., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5), 937–974.

    Article  Google Scholar 

  12. Chopra, V. K., & Ziemba, W. T. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choices. The Journal of Portfolio Management, 19(2), 6–11.

    Article  Google Scholar 

  13. DeMiguel, V., & Nogales, F. J. (2009). Portfolio selection with robust estimation. Operations Research, 57(3), 560–577.

    Article  Google Scholar 

  14. Denneberg, D. (1994). Non-additive measure and integral. Boston: Kluwer Academic Publishers.

    Book  Google Scholar 

  15. Dhaene, J., Vanduffel, S., Tang, Q., Goovaerts, M. J., Kaas, R., & Vyncke, D. (2006). Risk measures and comonotonicity: a review. Stochastic Models, 22, 573–606.

    Article  Google Scholar 

  16. Dhaene, J., Laeven, R. J. A., Vanduffel, S., Darkiewicz, G., & Goovaerts, M. J. (2008). Can a coherent risk measure be too subadditive? The Journal of Risk and Insurance, 75(2), 365–386.

    Article  Google Scholar 

  17. El Ghaoui, L., Oks, M., & Oustry, F. (2003). Worst case value-at-risk and robust portfolio optimization: a conic programming approach. Operations Research, 51(4), 543–556.

    Article  Google Scholar 

  18. Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., & Focardi, S. M. (2007). Robust portfolio optimization and management. New York: Wiley.

    Google Scholar 

  19. Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: contributions from operations research and finance. Annals of Operations Research, 176, 191–220.

    Article  Google Scholar 

  20. Föllmer, H., & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447.

    Article  Google Scholar 

  21. Föllmer, H., & Schied, A. (2004). Stochastic finance: an introduction in discrete time. Berlin: de Gruyter.

    Book  Google Scholar 

  22. Gaivoronski, A., & Pflug, G. (2005). Value-at-risk in portfolio optimization: properties and computational approach. The Journal of Risk, 7(2), 1–31.

    Google Scholar 

  23. Goh, J., & Sim, M. ROME: robust optimization made easy. Version 1.0.4 (beta). Available at http://robustopt.com/.

  24. Goldfarb, D., & Iyengar, G. (2003). Robust portfolio selection. Mathematics of Operations Research, 28(1), 1–38.

    Article  Google Scholar 

  25. Halldórsson, B. V., & Tütüncü, R. H. (2003). An interior-point method for a class of saddle-point problems. Journal of Optimization Theory and Applications, 116(3), 559–590.

    Article  Google Scholar 

  26. Hellmich, M., & Kassberger, S. (2011). Efficient and robust portfolio optimization in the multivariate generalized hyperbolic framework. Quantitative Finance, 11(10), 1503–1516. Special issue: Themed issue on asset allocation.

    Article  Google Scholar 

  27. Huang, D. S., Zhu, S. S., Fabozzi, F. J., & Fukushima, M. (2008). Portfolio selection with uncertain exit time: a robust CVaR approach. Journal of Economic Dynamics & Control, 32, 594–623.

    Article  Google Scholar 

  28. Huber, P. J. (2004). Robust statistics. New York: Wiley.

    Google Scholar 

  29. Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: why imposing the wrong constraints help? The Journal of Finance, 58(4), 1651–1684.

    Article  Google Scholar 

  30. Lauprete, G. J., Samarov, A. M., & Welsch, R. E. (2002). Robust portfolio optimization. Metrica, 25, 139–149.

    Article  Google Scholar 

  31. Ling, A., & Xu, C. (2012). Robust portfolio selection involving options under a “marginal + joint” ellipsoidal uncertainty set. Journal of Computational and Applied Mathematics, 236, 3373–3393.

    Article  Google Scholar 

  32. Lobo, M. S., Vandenberghe, L., Boyd, S., & Lebret, H. (1998). Applications of second order cone programming. Linear Algebra and Its Applications, 284(1–3), 193–228.

    Article  Google Scholar 

  33. Lu, Z. (2011a). Robust portfolio selection based on a joint ellipsoidal uncertainty set. Optimization Methods & Software, 26(1), 89–104.

    Article  Google Scholar 

  34. Lu, Z. (2011b). A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set. Mathematical Programming Series A, 126, 193–201.

    Article  Google Scholar 

  35. Lutgens, F., Sturm, S., & Kolen, A. (2006). Robust one-period option hedging. Operations Research, 54(6), 1051–1062.

    Article  Google Scholar 

  36. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7, 77–91.

    Google Scholar 

  37. McNeil, A. J., Frey, R., & Embrechts, P. (2005). Princeton series in finance. Quantitative risk management: concepts, techniques, tools. Princeton: Princeton University Press.

    Google Scholar 

  38. Mendes, B. V. M., & Leal, C. R. P. (2005). Robust multivariate modeling in finance. International Journal of Managerial Finance, 1(2), 95–106.

    Article  Google Scholar 

  39. Michaud, R. O. (1989). The Markowitz optimization enigma: is optimized optimal? Financial Analysts Journal, 45(1), 31–42.

    Article  Google Scholar 

  40. Natarajan, K., Pachamanova, D., & Sim, M. (2008). Incorporating asymmetric distributional information in robust value at risk optimization. Management Science, 54(3), 573–585.

    Article  Google Scholar 

  41. Nesterov, Y., & Nemirovski, A. (1993). Interior-point polynomial algorithms in convex programming. Philadelphia: SIAM.

    Google Scholar 

  42. Nesterov, Y., & Nemirovski, A. (1994). Interior point polynomial methods in convex programming: theory and applications. Philadelphia: SIAM.

    Book  Google Scholar 

  43. Perret-Gentil, C., & Victoria-Feser, M. P. (2004). Robust mean-variance portfolio selection (FAME Research Paper 140). International Center for Financial Asset Management and Engineering, Geneva.

  44. Quaranta, A. G., & Zaffaroni, A. (2008). Robust optimization of conditional value at risk and portfolio selection. Journal of Banking & Finance, 32(10), 2046–2056.

    Article  Google Scholar 

  45. Recchia, R., & Scutellà, M. G. (2012). Robust asset allocation strategies: relaxed versus classical robustness. IMA Journal of Management Mathematics. doi:10.1093/imaman/dps023.

    Google Scholar 

  46. Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press.

    Google Scholar 

  47. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.

    Google Scholar 

  48. Rockafellar, R. T., & Uryasev, S. (2002). Conditional value at risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.

    Article  Google Scholar 

  49. Saigal, R., Vandenberghe, L., & Wolkowicz, H. (2000). Handbook on semidefinite programming and applications. Dordrecht: Kluwer Academic.

    Google Scholar 

  50. Scutellà, M. G., & Recchia, R. (2010). Invited Survey: Robust portfolio asset allocation and risk measures. 4OR, 8(2), 113–139.

    Article  Google Scholar 

  51. Stürm, J. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods & Software, 11–12, 625–653.

    Article  Google Scholar 

  52. Todd, M. J. (2001). Semidefinite optimization. Acta Numerica, 10, 515–560.

    Article  Google Scholar 

  53. Tütüncü, R. H., & Koenig, M. (2004). Robust asset allocation. Annals of Operations Research, 132, 157–187.

    Article  Google Scholar 

  54. Vandenberghe, L., & Boyd, S. (1996). Semidefinite programming. SIAM Review, 38, 49–95.

    Article  Google Scholar 

  55. Waltz, R. A. (2004). KNITRO user’s manual. Version 4.0. Evanston: Ziena Optim.

    Google Scholar 

  56. Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.

    Article  Google Scholar 

  57. Welsch, R. E., & Zhou, X. (2007). Application of robust statistics to asset allocation models. REVSTAT Statistical Journal, 5(1), 97–114.

    Google Scholar 

  58. Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55, 95–115.

    Article  Google Scholar 

  59. Zhu, S. S., & Fukushima, M. (2009). Worst-case conditional value at risk with application to robust portfolio management. Operations Research, 57(5), 1155–1168.

    Article  Google Scholar 

  60. Zhu, S. S., Li, D., & Wang, S. (2009). Robust portfolio selection under downside risk measures. Quantitative Finance, 7, 869–885.

    Article  Google Scholar 

  61. Zymler, S., Rustem, B., & Kuhn, D. (2011). Robust portfolio optimization with derivative insurance guarantees. European Journal of Operational Research, 210(2), 410–424.

    Article  Google Scholar 

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Correspondence to Maria Grazia Scutellà.

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This is an updated version of the paper that appeared in 4OR, 8(2), 113–139 (2010).

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Scutellà, M.G., Recchia, R. Robust portfolio asset allocation and risk measures. Ann Oper Res 204, 145–169 (2013). https://doi.org/10.1007/s10479-012-1266-3

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Keywords

  • Portfolio asset allocation
  • Robustness
  • Risk measures
  • Mathematical models
  • Algorithmic approaches