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

A spectrum of upper bounds ( Q α ( X ; p) α ε[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 Q 0 ( X ;p) = Q ( X ; p ). If p is small enough and the distribution of X is regular enough, then Q α (X ; p) is rather close to Q ( X ; p ). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Q α (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, Q 0 ( X ; p ) and Q 1 ( X ; p ) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Q α (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.