An optimal threeway stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality
A certain spectrum, indexed by a\in[0,\infty], of upper bounds P_a(X;x) on the tail probability P(X\geq x), with P_0(X;x)=P(X\geq x) and P_\infty(X;x) being the best possible exponential upper bound on P(X\geq x), is shown to be stable and monotonic in a, x, and X, where x is a real number and X is a random variable. The bounds P_a(X;x) are optimal values in certain minimization problems. The corresponding spectrum, also indexed by a\in[0,\infty], of upper bounds Q_a(X;p) on the (1p)quantile of X is stable and monotonic in a, p, and X, with Q_0(X;p) equal the largest (1p)quantile of X. In certain sense, the quantile bounds Q_a(X;p) are usually close enough to the true quantiles Q_0(X;p). Moreover, Q_a(X;p) is subadditive in X if a\geq 1, as well as positivehomogeneous and translationinvariant, and thus is a socalled coherent measure of risk. A number of other useful properties of the bounds P_a(X;x) and Q_a(X;p) are established. In particular, quite similarly to the bounds P_a(X;x) on the tail probabilities, the quantile bounds Q_a(X;p) are the optimal values in certain minimization problems. This allows for a comparatively easy incorporation of the bounds P_a(X;x) and Q_a(X;p) into more specialized optimization problems. It is shown that the minimization problems for which P_a(X;x) and Q_a(X;p) are the optimal values are in a certain sense dual to each other; in the case a=\infty this corresponds to the bilinear LegendreFenchel duality. In finance, the (1p)quantile Q_0(X;p) is known as the valueatrisk (VaR), whereas the value of Q_1(X;p) is known as the conditional valueatrisk (CVaR) and also as the expected shortfall (ES), average valueatrisk (AVaR), and expected tail loss (ETL). It is shown that the quantile bounds Q_a(X;p) can be used as measures of economic inequality. The spectrum parameter, a, may be considered an index of sensitivity to risk/inequality.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Length:  
Date of creation:  22 Oct 2013 
Date of revision:  
Handle:  RePEc:pra:mprapa:51361 
Contact details of provider:  Postal: Ludwigstraße 33, D80539 Munich, Germany Phone: +49(0)8921802459 Fax: +49(0)892180992459 Web page: https://mpra.ub.unimuenchen.de More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
 Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244263, September.
 Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379388, 07.
 De Giorgi, Enrico, 2005.
"Rewardrisk portfolio selection and stochastic dominance,"
Journal of Banking & Finance,
Elsevier, vol. 29(4), pages 895926, April.
 Enrico De Giorgi, . "RewardRisk Portfolio Selection and Stochastic Dominance," IEW  Working Papers 121, Institute for Empirical Research in Economics  University of Zurich.
 Fishburn, Peter C., 1976. "Continua of stochastic dominance relations for bounded probability distributions," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 295311, December.
 Alessandra Cillo & Philippe Delquié, 2013.
"MeanRisk Analysis with Enhanced Behavioral Content,"
Working Papers
498, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
 Cillo, Alessandra & Delquié, Philippe, 2014. "Meanrisk analysis with enhanced behavioral content," European Journal of Operational Research, Elsevier, vol. 239(3), pages 764775.
 Yitzhaki, Shlomo, 1982. "Stochastic Dominance, Mean Variance, and Gini's Mean Difference," American Economic Review, American Economic Association, vol. 72(1), pages 17885, March.
 Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95115, January.
 Dufour, J.M. & Hallin, M., 1992.
"Improved Eaton Bounds for Linear Combinations of Bounded Random Variables with Statistical Applications,"
Cahiers de recherche
9224, Universite de Montreal, Departement de sciences economiques.
 Marc Hallin & JeanMarie Dufour, 1993. "Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications," ULB Institutional Repository 2013/2043, ULB  Universite Libre de Bruxelles.
 Dufour, JM. & Hallin, M., 1990. "Improved Eaton Bounds for Linear Combinations of Bounded Random Variables , with Statistical Applications," Papers 9104, Universite Libre de Bruxelles  C.E.M.E..
 Dufour, J.M. & Hallin, M., 1992. "Improved Eaton Bounds for Linear Combinations of Bounded Random Variables with Statistical Applications," Cahiers de recherche 9224, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
 R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 5174, 01.
 Fishburn, Peter C., 1980. "Continua of stochastic dominance relations for unbounded probability distributions," Journal of Mathematical Economics, Elsevier, vol. 7(3), pages 271285, December.
 A. Atkinson, 2008. "More on the measurement of inequality," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 6(3), pages 277283, September.
 Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 15051518, July.
 Fishburn, Peter C, 1977. "MeanRisk Analysis with Risk Associated with BelowTarget Returns," American Economic Review, American Economic Association, vol. 67(2), pages 11626, March.
 Philippe Artzner & Freddy Delbaen & JeanMarc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203228.
 Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional valueatrisk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 14431471, July.
 Philippe Delquié & Alessandra Cillo, 2006. "Disappointment without prior expectation: a unifying perspective on decision under risk," Journal of Risk and Uncertainty, Springer, vol. 33(3), pages 197215, December.
This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.
When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:51361. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Joachim Winter)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.