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 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 a 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.