Abstract
The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its utility representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox.
Similar content being viewed by others
References
Acerbi C. (2007) Coherent measures of risk in everyday market practice. Quantitative Finance 7(4): 359–364
Acerbi C., Tasche D. (2002b) Expected shortfall: A natural coherent alternative to value at risk. Economic Notes 31(2): 379–388
Allais M. (1979) The foundations of a positive theory of choices involving risk and a criticism of the postulates and axioms of the American school. In: Allais M., Hagen O. (eds) Expected utility hypothesis and the Allais Paradox. Reidel, Dordrecht Holland
Allais, M. & Hagen, O. (Eds.) (1979). Expected utility hypothesis and the Allais Paradox. Dordrecht, Holland:D. Reidel.
Alexander G.J., Baptista A.M. (2004) A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model. Management Science 50(9): 1261–1273
Artzner P., Delbaen F., Eber J-M., Heath D. (1999) Coherent measures of risk. Mathematical Finance 3: 203–228
Baucells M., Sarin R.K. (2007) Evaluating time streams of income: Discounting what? Theory and Decision 63: 95–120
Basset G.W. Jr., Koenker R., Kordas G. (2004) Pessimistic portfolio allocation and Choquet Expected Utility. Journal of Financial Econometrics 24: 477–492
Ben-Tal A., Ben-Israel A. (1991) A recourse certainty equivalent for decisions under uncertainty. Annals of Operation Research 30: 3–44
Ben-Tal A., Teboulle M. (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance 17(3): 449–476
Birge J., Louveaux F. (1997) Introduction to stochastic programming. Springer, New York
Carrión M., Conejo A.J., Arroyo J.M. (2007) Forward contracting and selling price determination for a retailer. IEEE Transactions on Power Systems 22(4): 2105–2114
Ellsberg D. (1961) Risk ambiguity and savage axioms. Quarterly Journal of Economics 75: 643–669
Ferguson T.S. (1967) Mathematical statistics: A decision theoretic approach. Academic Press, New York
Fleten, S., Wallace, S., & Ziemba, W. (1997). Portfolio management in a deregulated hydropower based electricity market. Hydropower′97 Proceedings, Trondheim.
Higle J., Wallace S. (2002) Managing risk in the new power business: A sequel. IEEE Computer Applications in Power 15(2): 12–19
Kubrusly, C. S. (2001). Elements of operator theory. Birkhäuser, ISBN:0-8176-4174-2.
Machina M.J. (1982) Expected utility analysis without the independence axiom. Econometrica 50: 277–323
Machina M.J. (2009) Risk, ambiguity, and the rank-dependence axioms. American Economic Review 99(1): 385–392
Pagnoncelli, B. K., Ahmed, S., & Shapiro, A. (2009). Sample average approximation method for chance constrained programming: Theory and applications, accepted for publication in JOTA. Journal of Optimization Theory and Applications. doi:10.1007/s10957-009-9523-6.
Rockafellar R.T. (1970) Convex analysis. Princeton University Press, Princeton
Rockafellar R.T., Uryasev S.P. (2000) Optimization of Conditional Value-at-Risk. The Journal of Risk 2: 21–41
Rockafellar R.T., Uryasev S.P. (2002) Conditional Value-at-Risk for a general loss distribution. Journal of Banking and Finance 26: 1443–1471
Ruszczynski, A., & Shapiro, A. (2004) Optimization of measures, risk and insurance 0407002, EconWPA.
Shapiro, A., & Ruszczynski, A. (2007, November). Lectures on stochastic programming. Retrieved November 2007, from http://www2.isye.gatech.edu/~ashapiro/.
Street, A. (2008). Certainty equivalent and risk measures in electrical energy trade decisions. PhD Thesis, Electrical Engineering Department of the Pontifical Catholic University of Rio de Janeiro (PUC-Rio).
Street A., Barroso L.A., Chabar R., Mendes A., Pereira M.V. (2008) Pricing flexible natural gas supply contracts under uncertainty in hydrothermal markets. IEEE Transactions on Power Systems 23(3): 1009–1017
Street, A., Barroso, L. A., Flach, B., Pereira, M., & Granville, S. (2008). Risk constrained portfolio selection of renewable sources in hydrothermal electricity markets, Accepted for publication in IEEE Transactions on Power Systems. doi:10.1109/TPWRS.2009.2022981.
von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behaviour. Princeton Press, ISBN 0-691-00362-9.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Street, A. On the Conditional Value-at-Risk probability-dependent utility function. Theory Decis 68, 49–68 (2010). https://doi.org/10.1007/s11238-009-9154-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-009-9154-2