A spectrum of upper bounds (Qα(X;p)) α∈[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.
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