Abstract
The particular subject of this paper, is to construct a general framework that can consider and analyse in the same time the upside and downside risks. This paper offers a comparative analysis of concept risk measures, we focus on quantile based risk measure (ES and VaR), spectral risk measure and distortion risk measure. After introducing each measure, we investigate their interest and limit. Knowing that quantile based risk measure cannot capture correctly the risk aversion of risk manager and spectral risk measure can be inconsistent to risk aversion, we propose and develop a new distortion risk measure extending the work of Wang (J Risk Insurance 67, 2000) and Sereda et al. (Handbook of Portfolio Construction 2012). Finally we provide a comprehensive analysis of the feasibility of this approach using the S&P500 data set from 01/01/1999 to 31/12/2011.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Regularly varying (heavy tailed distributions, fat tailed) non-degenerate tails with tail index \(\eta> 1\) for more detail see Danielson et al. (2005).
- 2.
- 3.
Artzner (2002) proposes a natural way to define a measure of risk as a mapping \(\rho: L^{\infty} \rightarrow \mathbb{R}\cup{\infty}\).
- 4.
\( VaR_{\alpha} (X) = q_{1-\alpha}= F_{X}^{-1}(\alpha)\)
- 5.
An extension can be found in Inui and Kijima (2005).
- 6.
In this last paper, the difference between ES and TCE is conceptual and is only related to the distributions. If the distribution is continuous then the expected shortfall is equivalent to the tail conditional expectation.
- 7.
If \(\rho_{i}\) is coherent risk measures for \(i = 1{\ldots} n\), then, any convex combination \(\rho=\sum _{1}^{n}\beta_{i} \rho_{i}\) is a coherent risk measure (Acerbi and Tasche 2002).
- 8.
The distortion risk measure is a special class of the so-called Choquet expected utility, i.e. the expected utility calculated under a modified probability measure.
- 9.
Both integrals in (1) are well defined and take a value in \([0,+\infty]\). Provided that at least one of the two integrals is finite, the distorted expectation \(\rho_{g}(X)\) is well defined and takes a value in \([-\infty,+\infty]\).
- 10.
This approach towards risk can be related to investor’s psychology as in Kahneman and Tversky (1979).
- 11.
This property involves that \(g'(S_{X}(x))\) becomes smaller for large values of the random variable X.
- 12.
\( 0.90 <0.99\) but \(0.031422>0.003081\).
References
Acerbi, C., and D. Tasche. 2002. On the coherence of expected shortfall. Journal of Banking and Finance 26 (7): 1487–1503.
Artzner, P., F. Delbaen, J.-M. Eber, and D Heath. 1999. Coherent risk measures. Mathematical Finance 9 (3): 203–228.
Bhansali, Vineer. 2012. Asset allocation and risk management in a bimodal world. Financial analysts seminar improving the investment decision-making process held in Chicago on 23–27 July 2012 in partnership with the CFA Society of Chicago.
Bookstaber, R., and R. Clarke. 1984. Option portfolio strategies: Measurement and evaluation. Journal of Business 57 (4): 469–492.
Barndorff-Nielson, O. E. 1977. Exponentially decreasing distribution for the logarithm of the particle size. Proceedings of the Royal society, Series A: Mathematical and physical sciences 350:401–419.
Chamberlain, G. 1983. A characterization of the distribution that imply mean-variance utility functions. Journal of Economic Theory 29:185–201.
Cotter, J., and K. Dowd. 2006. Extreme spectral risk measures: An application to futures clearinghouse margin requirements. Journal of Banking and finance 30:3469–3485.
Danielson, J., B. N. Jorgensen, M. Sarma, G. Samorodnitsky, and C. G. de Vries. 2005. Subadditivity re-examined: The case for value at risk. www.RiskResearch.org. (Unfinished working paper).
Delbaen, F. 2000. Coherent risk measures. Lecture notes of the Cattedra Galileiana at the Scuola Normale di Pisa, March 2000.
Denuit, M., J. Dhaene, and M. J. Goovaerts. 2006.Actuarial theory for dependent risks. New York: Wiley.
Dowd, K., G. Sorwar, and J. Cotter. 2008. Estimating power spectral risk measures. Mimeo. Nottingham University Business School.
Embrechts, P., R. Frey, and A. McNeil. 2005. Quantitative risk management: Concepts, techniques and tools. Princeton: Princeton University Press.
Föllmer, H., and A. Schied. 2004. Stochastic finance: An introduction in discrete time (2nd revised and extended edition). Berlin: Walter de Gruyter & Co. (de Gruyter Studies in Mathematics 27).
De Giorgi, E. 2005. Reward risk portfolio selection and stochastic dominance. Journal of Banking and Finance 29 (4): 895–926.
Hardy, M. R., and J. Wirch. 2001. Distortion risk measures: coherence and stochastic dominance. Working Paper, University of Navarra.
Huang, C.F., and R. H. Litzenberger. 1988. Foundations for financial economics. Englewood Cliffs: Prentice Hall.
Härlimann, W. 2004. Distortion risk measures and economic capital. North American Actuarial Journal 8 (1): 86–95.
Ingersoll, J. E. Jr. 1987.Theory of financial decision making. Totowa: Rowman and Littlefield.
Inui, K., and M. Kijima. 2005. On the significance of expected shortfall as a coherent risk measures. Journal of Banking and Finance 29:853–864.
Morgan, J. P. 1995.RiskMetrics technical document (4th ed.). NewYork: Morgan Guaranty Trust Company. http://www.riskmetrics.com/rm.html
Kahneman, D., & A. Tversky. 1979. Prospect theory: An analysis of decision under risk. Econometrica 47:313–327.
Markowitz, H. M. 1959. Portfolio selection: Efficient diversification of Investments. New York: Wiley (Oxford: Basil Blackwell, Yale University Press 1970, 1991).
Rockafellar, R. T., and S. Uryasev. 2000. Optimization of conditional value-at-risk. Journal of Risk 2 (3): 21–41.
Rockafellar, R. T., and S. Uryasev. 2002. Conditional value at risk for general loss distributions. Journal of Banking and Finance 26 (7): 1443–1471.
Rockafellar, R. T., S. Uryasev, and M. Zabarankin. 2006. Optimality conditions in portfolio analysis with generalized deviation measures. Mathematical Programming 108 (2–3): 515–540.
Samorodnitsky, G., and M. S. Taqqu. 1994. Stable non-Gaussian random processes: Stochastic models with infinite variance. New York: Chapman and Hall.
Sereda, E. N., E. M. Bronshtein, S. T. Rachev, F. J. Fabozzi, W. Sun, and S. V. Stoyanov. 2010. Handbook of portfolio construction, Volume 3 of Business and economics, Chapter Distortion risk measures in portfolio optimisation, 649–673. Springer US.
Tasche, D. 2002. Expected shortfall and beyond. Journal of Banking and Finance 26 (7): 1519–1533.
Tsanakas, A. 2004. Dynamic capital allocation with distortion risk measures. Insurance: Mathematics and Economics 35:223–243.
Wang, S. S., V. R. Young, and H. H. Panjer. 1997. Axiomatic characterisation of insurance prices. Insurance: Mathematics and Economics 21:173–183.
Wang, S. S. 2000. A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance 67 (1): 15–36.
Yaari, M.-E. 1987. The dual theory of choice under risk. Econometrica 55 (1): 95–115.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Guégan, D., Hassani, B. (2015). Distortion Risk Measure or the Transformation of Unimodal Distributions into Multimodal Functions. In: Bensoussan, A., Guegan, D., Tapiero, C. (eds) Future Perspectives in Risk Models and Finance. International Series in Operations Research & Management Science, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-07524-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-07524-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07523-5
Online ISBN: 978-3-319-07524-2
eBook Packages: Business and EconomicsBusiness and Management (R0)