Abstract
Coherent risk measures play an important role in building and solving optimization models for decision problems under uncertainty. We consider an extension to multiple time periods, where a risk-adjusted value for a stochastic process is recursively defined over the time steps, which ensures time consistency. A prominent example of a single-period coherent risk measure that is widely used in applications is Conditional-Value-at-Risk (CVaR). We show that a recursive calculation of CVaR leads to stochastic linear programming formulations. For the special case of the risk-adjusted value of a random variable at the time horizon, a lower bound is given. The possible integration of the risk-adjusted value into multi-stage mean-risk optimization problems is outlined.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Acciaio, B., and Penner, I.: Dynamic Risk Measures, In: Di Nunno, G., Oksendal, B. (eds.) Adv. Math. Meth. for Finance, pp. 1-34. Springer, Berlin Heidelberg (2011)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance, 9, 203–228 (1999)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res., 152, 5–22 (2007)
Bertsimas, D., Lauprete, G.J., Samarov, A.: Shortfall as a risk measure: properties, optimization and applications. J. Econ. Dyn. Control, 28, 1353–1381 (2004)
Cheridito, P., Kupper, M.: Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theoretical Appl. Finance, 14, 137–162 (2011)
Cheridito, P., Stadje,M.: Time-inconsistency of var and time-consistent alternatives. Finance, 6, 40–46 (2009)
Densing, M.: Multi-Stage Stochastic Optimization of Hydro-Energy Plant with Time-Consistent Constraints on Risk. PhD, ETH Zurich (2007) doi: 10.3929/ethz-a-005464814
Eichhorn, A., Heitsch, H., Römisch,W.: Stochastic optimization of electricity portfolios: Scenario tree modeling and risk management. In: Pardalos, P.M. et al. (eds.) Handbook of Power Systems II, pp. 405-432. Springer, Berlin Heidelberg (2010)
Eichhorn, A., Römisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optimiz., 16, 69–95 (2005)
Föllmer, H., Schied, A.: Stochastic Finance. de Gruyter Studies in Mathematics (2002)
Geman, H., Ohana, S.: Time-consistency in managing a commodity portfolio: A dynamic risk measure approach. J. of Bank. Financ., 32, 1991–2005 (2008)
Kall, P., Mayer, J.: Stochastic Linear Programming: Models, Theory, and Computation. 2nd Edition, Intern. Series in Oper. Res. & Manag. Sci., 156. Springer, New York (2010)
K¨unzi-Bay, A., Mayer, J.: Computational aspects of minimizing conditional value-at-risk. Comput. Manag. Sci., 3, 3–27 (2006)
Kilianova, S., Pflug, G.C.: Optimal pension fund management under multi-period risk minimization. Ann. Oper. Res., 166, 261–270 (2009)
Kydland, F.E., Prescott, E.C.: Rules rather than discretion: The inconsistency of optimal plans. J. Polit. Econ., 85, 473–492 (1977)
Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surveys in Oper. Res. and Manag. Sci., 16, 49–66 (2011)
Pflug, G.C., Römisch, W.: Modeling, measuring, and managing risk. World Scientific (2007)
Ruszczy´nski, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res., 31, 544–561 (2006)
Roorda, B., Schumacher, J.M.: Time consistency conditions for acceptability measures, with an application to tail value at risk. Ins.: Mathematics Econ., 40, 209–230 (2007)
Szegö, G. (ed.): Risk Measures for the 21st Century. Wiley (2004)
Uryasev, S., Rockafellar, R.T.: Optimization of conditional value-at-risk. J. Risk, 2, 21–41 (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Densing, M., Mayer, J. (2012). Multiperiod Stochastic Optimization Problems with Time-Consistent Risk Constraints. In: Klatte, D., Lüthi, HJ., Schmedders, K. (eds) Operations Research Proceedings 2011. Operations Research Proceedings. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29210-1_83
Download citation
DOI: https://doi.org/10.1007/978-3-642-29210-1_83
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29209-5
Online ISBN: 978-3-642-29210-1
eBook Packages: Business and EconomicsBusiness and Management (R0)