Abstract
Starting with a time-0 coherent risk measure defined for “value processes”, we also define risk measurement processes. Two other constructions of measurement processes are given in terms of sets of test probabilities. These latter constructions are identical and are related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity as in decision-making. We finally deduce risk measurements for the final value of locked-in positions and repeat a warning concerning Tail-Value-at-Risk.
Similar content being viewed by others
References
Artzner, P. (2002a). Conditional Value at Risk: Is it Good in the Multiperiod Case? IIR Conference on Volatility and Risk, London, Feb. 18–19.
Artzner, P. (2002b). Multiperiod Risk Measurement: Where are we? Quantitative Finance Seminar, Fields Institute, U. Toronto, Nov. 25, http://www.fields.utoronto.ca/audio/02-03/finance_seminar/artzner/.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1997). “Thinking Coherently,” Risk, 10, 68–71.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999a). “Coherent Risk Measures.” Mathematical Finance, 9, 203–228.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999b). Risk Management and Capital Allocation with Coherent Measures of Risk. http://symposium.wiwi.uni-karlsruhe.de/8thpapers/artzner.ps.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (2001). Coherent Measures of Multiperiod Risk. Presentation at the Workshop New Ideas in Risk Management, Carnegie Mellon University, Aug. 25–26.
Artzner, P., F. Delbaen, and P. Koch-Medina. (2005). “Risk Measures and Efficient Use of Capital,” Working Paper ETH, Zürich.
Bennet, O. (2001). “Reinventing RAROC.” Risk, 14, 112–113.
Cheridito, P., F. Delbaen, and M. Kupper. (2002). Convex Measures of Risk for Càdlàg Processes. Working Paper ETH, Zürich.
Cvitanić, J. and I. Karatzas. (1999). “On Dynamic Measures of Risk.” Finance and Stochastics, 3, 451–482.
Chow, Y.S., H. Robbins, and D. Siegmund. (1972). Great Expectations: The Theory of Optimal Stopping. Boston: Houghton Mifflin, reprinted by Dover, New York (1992).
Delbaen, F. (2000). Coherent Risk Measures on General Probability Spaces. Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann. New York: Springer.
Delbaen, F. (2001). “The Structure of m-Stable Sets and in Particular of the Set of Risk Neutral Measures”. Working Paper ETH, Zürich.
Delbaen, F. (2002). Coherent Risk Measures. Lectures given at the Cattedra Galileiana, March 2000. Pisa: Scuola Normale Superiore.
Dothan, M. (1990). Prices in Financial Markets. New York: Oxford University Press.
Embrechts, P. (1995). A Survival Kit to Quantile Estimation. Zürich: UBS Quant Workshop.
Epstein, L. and M. Schneider. (2003). “Recursive Multiple-Priors.” Journal of Economic Theory, 113, 1–31, earlier versions June 2001, April 2002.
Föllmer, H. and A. Schied. (2002). “Convex Measures of Risk and Trading Constraints.” Finance and Stochastics, 6, 429–447.
Föllmer, H. and A. Schied. (2004). Stochastic Finance, 2nd ed. Berlin: de Gruyter.
Gilboa, I. and D. Schmeidler. (1989). “Maxmin Expected Utility with Non-Unique Prior.” Journal of Mathematical Economics, 18, 141–153.
Heath, D. (1998) Coherent Measures of Risk. Documents from the 5th Annual Conference on Risk Management, International Center for Business Information, Geneva, December 8th.
Jaschke, S. and U. Küchler. (2001). “Coherent Risk Measures and Good-Deal Bounds.” Finance and Stochastics, 5, 181–200.
Nakano, Y. (2003). “Minimizing Coherent Risk Measures of Shortfall in Discrete-time Models with Cone Constraints.” Applied Mathematical Finance, 10, 163–181.
Neveu, J. (1972). Martingales à temps discret. Paris: Masson, English transl. Discrete-Parameter Martingales (1975), North-Holland.
Riedel, F. (2004). “Dynamic Coherent Risk Measures.” Stochastic Processes and Applications, 112, 185–200, earlier version November 2002.
Roorda, B., J. Engwerda, and H. Schumacher. (2002). Coherent Acceptability Measures in Multiperiod Models. Working Paper U. of Twente and Tilburg U., June, later version July 2003.
Wang, T. (1996). A Characterization of Dynamic Risk Measures. Working Paper Faculty of Commerce and Business Administration, U.B.C.
Wang, T. (1999). A Class of Dynamic Risk Measures. Working Paper Faculty of Commerce and Business Administration, U.B.C, later version 2002.
Wang, T. (2003). “Conditional Preferences and Updating,” Journal of Economic Theory, 108, 286–321, earlier version March 2002.
Wilkie, D., H. Waters, and S. Yang. (2003). “Reserving, Pricing and Hedging for Policies with Guaranteed Annuity Options.” Paper presented to the Faculty of Actuaries, Edinburgh, 20 January 2003. British Actuarial Journal, 9, 263–425.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Artzner, P., Delbaen, F., Eber, JM. et al. Coherent multiperiod risk adjusted values and Bellman’s principle. Ann Oper Res 152, 5–22 (2007). https://doi.org/10.1007/s10479-006-0132-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-006-0132-6