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:  14. Nov 2013 21:05 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/51361 