Pinelis, Iosif (2013): An optimal threeway stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality.

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Abstract
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.
Item Type:  MPRA Paper 

Original Title:  An optimal threeway stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality 
Language:  English 
Keywords:  probability inequalities, extremal problems, tail probabilities, quantiles, coherent measures of risk, measures of economic inequality, valueatrisk (VaR), condi tional valueatrisk (CVaR), expected shortfall (ES), average valueatrisk (AVaR), expected tail loss (ETL), meanrisk (MR), Gini's mean difference, stochastic dominance, stochastic orders. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C10  General C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C54  Quantitative Policy Modeling C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C58  Financial Econometrics C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65  Miscellaneous Mathematical Tools Z  Other Special Topics > Z1  Cultural Economics ; Economic Sociology ; Economic Anthropology Z  Other Special Topics > Z1  Cultural Economics ; Economic Sociology ; Economic Anthropology > Z13  Economic Sociology ; Economic Anthropology ; Social and Economic Stratification 
Item ID:  51361 
Depositing User:  Professor Iosif Pinelis 
Date Deposited:  14 Nov 2013 20:03 
Last Modified:  28 Sep 2019 04:54 
References:  [1] C. Acerbi. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance, 26:15051518, 2002. [2] C. Acerbi and D. Tasche. Expected shortfall: a natural coherent alternative to value at risk. Economic Notes, 31:379388, 2002. [3] P. Artzner, F. Delbaen, J.M. Eber, and D. Heath. Coherent measures of risk. Math. Finance, 9(3):203228, 1999. [4] A. B. Atkinson. On the measurement of inequality. J. Econom. Theory, 2:244263, 1970. [5] A. B. Atkinson. More on the measurement of inequality. J. Econ. Inequal., 6:277283, 2008. [6] F. Bassi, P. Embrechts, and M. Kafetzaki. Risk management and quantile estimation. In A practical guide to heavy tails (Santa Barbara, CA, 1995), pages 111130. Birkhäuser Boston, Boston, MA, 1998. [7] V. Bentkus. A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand. Liet. Mat. Rink., 42(3):332342, 2002. [8] V. Bentkus. On Hoeffding's inequalities. Ann. Probab., 32(2):16501673, 2004. [9] V. Bentkus, N. Kalosha, and M. van Zuijlen. On domination of tail probabilities of (super)martingales: explicit bounds. Liet. Mat. Rink., 46(1):354, 2006. [10] P. Billingsley. Convergence of probability measures. John Wiley & Sons Inc., New York, 1968. [11] A. Cillo and P. Delquie. Meanrisk analysis with enhanced behavioral content. Technical Report TR201111, Institute for Integrating Statistics in Decision Sciences, George Washington University, May 2011. [12] J. A. Clarkson. Uniformly convex spaces. Trans. Amer. Math. Soc., 40(3):396414, 1936. [13] E. De Giorgi. Rewardrisk portfolio selection and stochastic dominance. Journal of Banking and Finance, 29:895926, 2005. [14] P. Delqui and A. Cillo. Disappointment without prior expectation: A unifying perspective on decision under risk. Journal of Risk and Uncertainty, 33:197215, 2006. [15] J.M. Dufour and M. Hallin. Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications. J. Amer. Statist. Assoc., 88:10261033, 1993. [16] M. L. Eaton. A probability inequality for linear combinations of bounded random variables. Ann. Statist., 2:609613, 1974. [17] P. Embrechts, C. Kluppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. Springer, New York, 1997. [18] P. C. Fishburn. Continua of stochastic dominance relations for bounded probability distributions. J. Math. Econom., 3(3):295311, 1976. [19] P. C. Fishburn. Meanrisk analysis with risk associated with belowtarget returns. The American Economic Review, 67(2):116126, 1977. [20] P. C. Fishburn. Continua of stochastic dominance relations for unbounded probability distributions. J. Math. Econom., 7(3):271285, 1980. [21] M. Frittelli and E. Rosazza Gianin. Dynamic convex risk measures. In Risk Measures in the 21st century, pages 227248. Wiley, 2004. [42] I. Pinelis. Exact inequalities for sums of asymmetric random variables, with applications. Probab. Theory Related Fields, 139(34):605635, 2007. [43] I. Pinelis. On the BennettHoeffding inequality, a shorter version to appear in Annales de l'Institut Henri Poincaré. http://arxiv.org/abs/0902.4058, 2009. [44] I. Pinelis. Positivepart moments via the FourierLaplace transform. J. Theor. Probab., 24:409421, 2011. [45] I. Pinelis. A necessary and sufficient condition on the stability of the infimum of convex functions. http://arxiv.org/abs/1307.3806, 2013. [46] I. Pinelis. (Quasi)additivity properties of the LegendreFenchel transform and its inverse, with applications in probability. http://arxiv.org/abs/1305.1860, 2013. [47] S. T. Rachev, S. Stoyanov, and F. J. Fabozzi. Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures. John Wiley, 2007. [48] E. Rio. Local invariance principles and their application to density estimation. Probab. Theory Related Fields, 98(1):2145, 1994. [49] E. Rio. English translation of the monograph Théorie asymptotique des processus aléatoires faiblement dépendants (2000) by E. Rio. Work in progress, 2012. [50] E. Rio. Personal communication. 2013. [51] R. T. Rockafellar. Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original, Princeton Paperbacks. [52] R. T. Rockafellar and S. Uryasev. Optimization of conditional valueatrisk. Journal of Risk, 2:2141, 2000. [53] R. T. Rockafellar and S. Uryasev. Conditional valueatrisk for general loss distributions. Journal of Banking & Finance, 26:14431471, 2002. [54] R. T. Rockafellar, S. Uryasev, and M. Zabarankin. Generalized deviations in risk analysis. Finance Stoch., 10(1):5174, 2006. [55] A. D. Roy. Safety first and the holding of assets. Econometrica, 20:431449, 1952. [56] M. Shaked and J. G. Shanthikumar. Stochastic orders. Springer Series in Statistics. Springer, New York, 2007. [57] M. E. Yaari. The dual theory of choice under risk. Econometrica, 55(1):95115, 1987. [58] S. Yitzhaki. Stochastic dominance, mean variance, and Gini's mean difference. American Economic Review, 72:178185, 1982. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/51361 