An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality

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

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 (1-p)-quantile of X is stable and monotonic in a, p, and X, with Q_0(X;p) equal the largest (1-p)-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 positive-homogeneous and translation-invariant, and thus is a so-called 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 Legendre--Fenchel duality. In finance, the (1-p)-quantile Q_0(X;p) is known as the value-at-risk (VaR), whereas the value of Q_1(X;p) is known as the conditional value-at-risk (CVaR) and also as the expected shortfall (ES), average value-at-risk (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
• Language: English
• Keywords: probability inequalities, extremal problems, tail probabilities, quantiles, coherent measures of risk, measures of economic inequality, value-at-risk (VaR), condi- tional value-at-risk (CVaR), expected shortfall (ES), average value-at-risk (AVaR), expected tail loss (ETL), mean-risk (M-R), 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

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