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Risks 2014, 2(3), 349-392; https://doi.org/10.3390/risks2030349

An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality

Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA
Received: 15 June 2014 / Revised: 21 August 2014 / Accepted: 10 September 2014 / Published: 23 September 2014
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Abstract

A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained. View Full-Text
Keywords: quantile bounds; coherent measures of risk; sensitivity to risk; measures of economic inequality; value at risk (VaR); conditional value at risk (CVaR); stochastic dominance; stochastic orders quantile bounds; coherent measures of risk; sensitivity to risk; measures of economic inequality; value at risk (VaR); conditional value at risk (CVaR); stochastic dominance; stochastic orders
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Pinelis, I. An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality. Risks 2014, 2, 349-392.

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