# 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.## Download Info

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## Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 51361.

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**Date of creation:**22 Oct 2013

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**Handle:**RePEc:pra:mprapa:51361

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## Related research

**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.;

**Find related papers by JEL classification:**

**C10**- Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General**C54**- Mathematical and Quantitative Methods - - Econometric Modeling - - - Quantitative Policy Modeling**C58**- Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics**C61**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis**C65**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools**Z1**- Other Special Topics - - Cultural Economics**Z13**- Other Special Topics - - Cultural Economics - - - Economic Sociology; Economic Anthropology; Social and Economic Stratification

**This paper has been announced in the following NEP Reports:**

- NEP-ALL-2013-11-22 (All new papers)
- NEP-RMG-2013-11-22 (Risk Management)

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