Skip to main content
SpringerLink
Log in

Multi-Period Risk Measures and Optimal Investment Policies

  • Chapter
  • First Online:
Optimal Financial Decision Making under Uncertainty

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 245))

Abstract

This chapter provides an in-depth overview of an extended set of multi-period risk measures, their mathematical and economic properties, primarily from the perspective of dynamic risk control and portfolio optimization. The analysis is structured in four parts: the first part reviews characterizing properties of multi-period risk measures, it examines their financial foundations, and clarifies cross-relationships. The second part is devoted to three classes of multi-period risk measures, namely: terminal, additive and recursive. Their financial and mathematical properties are considered, leading to the proposal of a unifying representation. Key to the discussion is the treatment of dynamic risk measures taking their relationship with evolving information flows and time evolution into account: after convexity and coherence, time consistency emerges as a key property required by risk measures to effectively control risk exposure within dynamic programs. In the third part, we consider the application of multi-period measures to optimal investment policy selection, clarifying how portfolio selection models adapt to different risk measurement paradigms. In the fourth part we summarize and point out desirable developments and future research directions. Throughout the chapter, attention is paid to the state-of-the-art and methodological and modeling implications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Artzner, F. Delbaen, J.M. Eber, D. Heath, Coherent measures of risk. Math. Financ. 9 (3), 203–228 (1999)

    Article  Google Scholar 

  2. P. Artzner, F. Delbaen, J.M. Eber, D. Heath, H. Ku, Coherent multi-period risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152, 5–22 (2007)

    Article  Google Scholar 

  3. S. Basak, G. Chabakauri, Dynamic mean-variance asset allocation. Rev. Financ. Stud. 23 (8), 2970–3016 (2010)

    Article  Google Scholar 

  4. S. Basak, A. Shapiro, Value-at-risk based risk management: optimal policies and asset prices. Rev. Financ. Stud. 14 (2), 371–405 (2001)

    Article  Google Scholar 

  5. B. Bäuerle, A. Mundt, Dynamic mean-risk optimization in a binomial model. Math. Meth. Oper. Res. 70, 219–239 (2009)

    Article  Google Scholar 

  6. A. Berkelaar, P. Cumperayot, R.R.P. Kouwenberg, The effect of VaR-based risk management on asset prices and volatility smile. Eur. Financ. Manag. 8, 65–78 (2002)

    Article  Google Scholar 

  7. M. Bertocchi, G. Consigli, M.A.H. Dempster, Handbook on Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Market Strategies. Fred Hillier International Series in Operations Research and Management Science (Springer, New York, 2011)

    Google Scholar 

  8. A. Ben-Tal, L. El Ghaoui, A. Nemirovski, Robust Optimization (Princeton University Press, Princeton, 2009)

    Book  Google Scholar 

  9. K. Boda, J.A. Filar, Time consistent dynamic risk measures. Math. Meth. Oper. Res. 63, 169–186 (2006)

    Article  Google Scholar 

  10. U. Çakmak, S. Özekici, Portfolio optimization in stochastic markets. Math. Meth. Oper. Res. 63, 151–168 (2006)

    Article  Google Scholar 

  11. E. Çanakoǧlu, S. Özekici, Portfolio selection in stochastic markets with exponential utility functions. Ann. Oper. Res. 166, 281–297 (2009)

    Article  Google Scholar 

  12. P. Carpentier, J.P. Chancelier, G. Cohen, M.D. Lara, P. Girardeau, Dynamic consistency for stochastic optimal control problems. Ann. Oper. Res. 200 (1), 247–263 (2012)

    Article  Google Scholar 

  13. U. Çelikyurt, S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach. Eur. J. Oper. Res. 179, 186–202 (2007)

    Article  Google Scholar 

  14. Z.P. Chen, J. Liu, Time consistent risk measure under two-level information structure and its application in dynamic portfolio selection. Optimization Online (2015). http://www.optimization-online.org/DB_FILE/2015/05/4906.pdf

  15. Z.P. Chen, G. Li, J.E. Guo, Optimal investment policy in the time consistent mean-variance formulation. Insur. Math. Econ. 52, 145–156 (2013)

    Article  Google Scholar 

  16. Z.P. Chen, J. Liu, G. Li, Z. Yan, Composite time consistent multi-period risk measure and its application in optimal portfolio selection. Working paper, Xi’an Jiaotong University (2015)

    Google Scholar 

  17. Z.P. Chen, J. Liu, G. Li, Time consistent policy of multi-period mean-variance problem in stochastic markets. J. Ind. Manag. Optim. 12 (1), 229–249 (2016)

    Article  Google Scholar 

  18. P. Cheridito, M. Kupper, Recursiveness of indifference prices and translation-invariant preferences. Math. Finan. Econ. 2, 173–188 (2009)

    Article  Google Scholar 

  19. P. Cheridito, M. Kupper, Composition of time consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Financ. 14 (1), 137–162 (2011)

    Article  Google Scholar 

  20. P. Cheridito, M. Stadje, Time-inconsistency of VaR and time consistent alternatives. Finance Res. Lett. 6 (1), 40–46 (2009)

    Article  Google Scholar 

  21. P. Cheridito, F. Delbaen, M. Kupper, Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006)

    Article  Google Scholar 

  22. J. Claro, J.P. de Sousa, A multiobjective metaheuristic for a mean-risk multistage capacity investment problem with process flexibility. Comput. Oper. Res. 39 (4), 838–849 (2012)

    Article  Google Scholar 

  23. R. Collado, D. Papp, A. Ruszczyński, Scenario decomposition of risk-averse multistage stochastic programming problems. Ann. Oper. Res. 200, 147–170 (2012)

    Article  Google Scholar 

  24. G. Consigli, M.A.H. Dempster, Dynamic stochastic programming for asset-liability management. Ann. Oper. Res. 81, 131–161 (1998)

    Article  Google Scholar 

  25. G.M. Constantinides, A.G. Malliaris, Portfolio theory, in Handbooks on Operations Research and Management Science, vol. 9, ed. by R. Jarrow et al. (Elevier Science, Amsterdam, 1995)

    Google Scholar 

  26. X.Y. Cui, D. Li, S.Y. Wang, S.S. Zhu, Better than dynamic mean-variance: time inconsistency and free cash flow stream. Math. Financ. 22 (2), 346–378 (2012)

    Article  Google Scholar 

  27. X.Y. Cui, J.J. Gao, X. Li, D. Li, Optimal multi-period mean-variance policy under no-shorting constraint. Eur. J. Oper. Res. 234, 459–468 (2014)

    Article  Google Scholar 

  28. D. Cuoco, H. He, S. Isaenko, Optimal dynamic trading strategies with risk limits. Oper. Res. 56 (2), 358–368 (2008)

    Article  Google Scholar 

  29. C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time. Financ. Stoch. 17, 227–271 (2013)

    Article  Google Scholar 

  30. F. Delbaen, The structure of m-stable sets and in particular of the set of risk-neutral measures. Preprint, ETH Zürich (2003)

    Google Scholar 

  31. F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage (Springer, Berlin, 2006)

    Google Scholar 

  32. K. Detlefsen, G. Scandolo, Conditional and dynamic convex risk measures. Financ. Stoch. 9 (4), 539–561 (2005)

    Article  Google Scholar 

  33. D. Duffie, Security Markets: Stochastic Models (Academic Press, Boston, 1998)

    Google Scholar 

  34. J. Dupacŏvá, V. Kozmík, Structure of risk-averse multistage stochastic programs. OR Spectr. 37, 559–582 (2015)

    Article  Google Scholar 

  35. L.G. Epstein, S.E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57 (4), 937–969 (1989)

    Article  Google Scholar 

  36. C.I. Fábián, Handling CVaR objectives and constraints in two-stage stochastic models. Eur. J. Oper. Res. 191 (3), 888–911 (2008)

    Article  Google Scholar 

  37. H. Föllmer, I. Penner, Convex risk measures and the dynamics of their penalty functions. Stat. Decis. 24 (1), 61–96 (2006)

    Google Scholar 

  38. H. Föllmer, A. Schied, Convex risk measures, in Encyclopedia of Quantitative Finance, ed. by R. Cont (Wiley, New York, 2010), pp. 355–363

    Google Scholar 

  39. M. Frittelli, E. Rosazza Gianin, Dynamic convex risk measures, in Risk Measures for the 21st Century, ed. by G. Szegö (Wiley, Chichester, 2004), pp. 224–248

    Google Scholar 

  40. M. Frittelli, G. Scandolo, Risk measure and capital requirements for processes. Math. Financ. 18 (4), 589–612 (2006)

    Article  Google Scholar 

  41. H. Geman, S. Ohana, Time-consistency in managing a commodity portfolio: a dynamic risk measure approach. J. Bank. Financ. 32 (10), 1991–2005 (2008)

    Article  Google Scholar 

  42. N. Gülpinar, B. Rustem, Worst-case robust decisions for multi-period mean-variance portfolio optimization. Eur. J. Oper. Res. 183, 981–1000 (2007)

    Article  Google Scholar 

  43. N. Gülpinar, B. Rustem, R. Settergren, Multistage stochastic programming in computational finance, in Computational Methods in Decision-Making, Economics and Finance, ed. by E.J. Kontoghiorghes, B. Rustem, S. Siokos (Springer, New York, 2002), pp. 35–47

    Chapter  Google Scholar 

  44. S. Gülten, A. Ruszczyński, Two-stage portfolio optimization with higher-order conditional measures of risk. Ann. Oper. Res. 229, 409–427 (2015)

    Article  Google Scholar 

  45. T. Homem-de-Mello, B.K. Pagnoncelli, Risk aversion in multistage stochastic programming: a modeling and algorithmic perspective. Eur. J. Oper. Res. (2015). doi:doi:10.1016/j.ejor.2015.05.048

    Google Scholar 

  46. L. Jobert, L.C.G. Rogers, Valuations and dynamic convex risk measures. Math. Financ. 18, 1–22 (2008)

    Article  Google Scholar 

  47. P.R. Kleindorfer, L. Li, Multi-period VaR-constrained portfolio optimization with applications to the electric power sector. Energy J. 26 (1), 1–26 (2005)

    Article  Google Scholar 

  48. S. Kloppel, M. Schweizer, Dynamic indifference valuation via convex risk measures. Math. Financ. 17 (4), 599–627 (2007)

    Article  Google Scholar 

  49. R. Kovacevic, G.C. Pflug, Time consistency and information monotonicity of multiperiod acceptability functionals. Radon Ser. Comput. Appl. Math. 8, 1–24 (2009)

    Google Scholar 

  50. V. Kozmík, D.P. Morton, Evaluating policies in risk-averse multi-stage stochastic programming. Math. Prog. 152, 275–300 (2015)

    Article  Google Scholar 

  51. M. Kupper, W. Schachermayer, Representation results for law invariant time consistent functions. Math. Financ. Econ. 2, 189–210 (2009)

    Article  Google Scholar 

  52. M. Leippold, F. Trojani, P. Vanini, Equilibrium impact of value-at-risk regulation. J. Econ. Dyn. Control 30, 1277–1313 (2006)

    Article  Google Scholar 

  53. D. Li, W.L. Ng, Optimal dynamic portfolio selection: multi-period mean-variance formulation. Math. Financ. 10, 387–406 (2000)

    Article  Google Scholar 

  54. D. Li, T.F. Chan, W.L. Ng, Safety-first dynamic portfolio selection. Dyn. Contin. Discrete Impuls. Syst. 4 (4), 585–600 (1998)

    Google Scholar 

  55. H. Markowitz, Portfolio selection. J. Financ. 7, 77–91 (1952)

    Google Scholar 

  56. Y. Matmoura, S. Penev, Multistage optimization of option portfolio using higher order coherent risk measures. Eur. J. Oper. Res. 227, 190–198 (2013)

    Article  Google Scholar 

  57. R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econ. Stat. 51, 247–257 (1969)

    Article  Google Scholar 

  58. A.F. Perold, Large-scale portfolio optiminization. Manag. Sci. 30, 1143–1160 (1984)

    Article  Google Scholar 

  59. G.C. Pflug, A. Pichler, Time-inconsistent multistage stochastic programs: martingale bounds. Eur. J. Oper. Res. (2015). doi:10.1016/j.ejor.2015.02.033

    Google Scholar 

  60. G.C. Pflug, W. Römisch, Modeling, in Measuring and Managing Risk (World Scientific, Singapore, 2007)

    Book  Google Scholar 

  61. A.B. Philpott, V.L. de Matos, Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Oper. Res. Lett. 218 (2), 470–483 (2012)

    Article  Google Scholar 

  62. A. Prékopa, Stochastic Programming (Akademiai Kiado, Budapest, 1995)

    Book  Google Scholar 

  63. F. Riedel, Dynamic coherent risk measures. Stoch. Proc. Appl. 112, 185–200 (2004)

    Article  Google Scholar 

  64. R.T. Rockafellar, Uryasev, S.P.: Conditional value-at-risk for general loss distributions. J. Bank. Financ. 26, 1443–1471 (2002)

    Google Scholar 

  65. B. Roorda, J.M. Schumacher, Time consistency conditions for acceptability measures, with an applications to Tail value at risk. Insur. Math. Econ. 40, 209–230 (2007)

    Article  Google Scholar 

  66. B. Roorda, J.M. Schumacher, J.C. Engwerda, Coherent acceptability measures in multiperiod models. Math. Financ. 15, 589–612 (2005)

    Article  Google Scholar 

  67. B. Rudloff, A. Street, D. Valladão, Time consistency and risk averse dynamic decision models: Definition, interpretation and practical consequences. Eur. J. Oper. Res. 234, 743–750 (2014)

    Article  Google Scholar 

  68. A. Ruszczyński, Risk-averse dynamic programming for Markov decision processes. Math. Prog. Ser. B 125, 235–261 (2010)

    Article  Google Scholar 

  69. A. Ruszczyński, A. Shapiro, Conditional risk mappings. Math. Oper. Res. 31, 544–561 (2006)

    Article  Google Scholar 

  70. A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming. Oper. Res. Lett. 37, 143–147 (2009)

    Article  Google Scholar 

  71. A. Shapiro, A dynamic programming approach to adjustable robust optimization. Oper. Res. Lett. 39, 83–87 (2011)

    Article  Google Scholar 

  72. A. Shapiro, Minimax and risk averse multistage stochastic programming. Eur. J. Oper. Res. 219, 719–726 (2012)

    Article  Google Scholar 

  73. A. Shapiro, D. Dentcheva, A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia, 2009)

    Book  Google Scholar 

  74. R.J. Shen, S.Z. Zhang, Robust portfolio selection based on a multi-stage scenario tree. Eur. J. Oper. Res. 191, 864–887 (2008)

    Article  Google Scholar 

  75. N.L. Stokey, R.E. Lucas, Recursive Methods in Economic Dynamics (Harvard University, Cambridge, 1989)

    Google Scholar 

  76. N. Topaloglou, H. Vladimirou, S.A. Zenios, A dynamic stochastic programming model for international portfolio management. Eur. J. Oper. Res. 185, 1501–1524 (2008)

    Article  Google Scholar 

  77. T. Wang, A class of dynamic risk measure. Working paper, University of British Columbia (1999). http://web.cenet.org.cn/upfile/57263.pdf

    Google Scholar 

  78. T. Wang, Conditional preferences and updating. J. Econ. Theory 108 (2), 286–321 (2003)

    Article  Google Scholar 

  79. J. Wang, P.A. Forsyth, Continuous time mean variance asset allocation: a time consistent strategy. Eur. J. Oper. Res. 209, 184–201 (2011)

    Article  Google Scholar 

  80. S. Weber, Distribution-invariant risk measures, information, and dynamic consistency. Math. Financ. 16, 419–441 (2006)

    Article  Google Scholar 

  81. S.Z. Wei, Z.X. Ye, Multi-period optimization portfolio with bankruptcy control in stochastic market. Appl. Math. Comput. 186, 414–425 (2007)

    Google Scholar 

  82. L.W. Xin, D.A. Goldberg, A. Shapiro, Distributionally robust multistage inventory models with moment constraints. Working paper, Georgia Institute of Technology (2013). http://arxiv.org/abs/1304.3074

    Google Scholar 

  83. L.W. Xin, A. Shapiro, Bounds for nested law invariant coherent risk measures. Oper. Res. Lett. 40 (6), 431–435 (2012)

    Article  Google Scholar 

  84. A. Yoshimoto, The mean-variance approach to portfolio optimization subject to transaction costs. J. Oper. Res. Soc. Jpn. 39 (1), 99–117 (1996)

    Google Scholar 

  85. S.S. Zhu, D. Li, S.Y.Wang, Risk control over bankruptcy in dynamic portfolio selection: a generalized mean-variance formulation. IEEE Automat. Control 49, 447–457 (2004)

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to two anonymous reviewers for their detailed and stimulating comments, which have helped us to improve the chapter significantly in both content and style. This research was supported by the National Natural Science Foundation of China (Grant Numbers 70971109, 71371152 and 11571270).

Author information

Authors and Affiliations

  1. Department of Computing Science, School of Mathematics and Statistics, Xi’an Jiaotong University, Xian Ning West Road 28, 710049, Xi’an, Shaanxi, People’s Republic of China

    Zhiping Chen, Jia Liu, Gang Li, Tianwen Fu & Qianhui Hu

  2. Department of Management, Economics and Quantitative Methods, University of Bergamo, Via dei Caniana 2, 24127, Bergamo, Italy

    Giorgio Consigli

Authors
  1. Zhiping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Consigli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jia Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tianwen Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Qianhui Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiping Chen .

Editor information

Editors and Affiliations

  1. University of Bergamo, Bergamo, Italy

    Giorgio Consigli

  2. EPFL, Lausanne, Cham, United Kingdom

    Daniel Kuhn

  3. Politecnico di Torino, Torino, Italy

    Paolo Brandimarte

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Chen, Z., Consigli, G., Liu, J., Li, G., Fu, T., Hu, Q. (2017). Multi-Period Risk Measures and Optimal Investment Policies. In: Consigli, G., Kuhn, D., Brandimarte, P. (eds) Optimal Financial Decision Making under Uncertainty. International Series in Operations Research & Management Science, vol 245. Springer, Cham. https://doi.org/10.1007/978-3-319-41613-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation