An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality

A certain spectrum, indexed by a∈[0,∞], of upper bounds Pa(X;x) on the tail probability P(X≥ x), with P0(X;x)=P(X≥ x) and P_∞(X;x) being the best possible exponential upper bound on P(X≥ 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 Pa(X;x) are optimal values in certain minimization problems. The corresponding spectrum, also indexed by a∈[0,∞], of upper bounds Qa(X;p) on the (1-p)-quantile of X is stable and monotonic in a, p, and X, with Q0(X;p) equal the largest (1-p)-quantile of X. In certain sense, the quantile bounds Qa(X;p) are usually close enough to the true quantiles Q0(X;p). Moreover, Qa(X;p) is subadditive in X if a≥ 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 Pa(X;x) and Qa(X;p) are established. In particular, quite similarly to the bounds Pa(X;x) on the tail probabilities, the quantile bounds Qa(X;p) are the optimal values in certain minimization problems. This allows for a comparatively easy incorporation of the bounds Pa(X;x) and Qa(X;p) into more specialized optimization problems. It is shown that the minimization problems for which Pa(X;x) and Qa(X;p) are the optimal values are in a certain sense dual to each other; in the case a=∞ this corresponds to the bilinear Legendre--Fenchel duality. In finance, the (1-p)-quantile Q0(X;p) is known as the value-at-risk (VaR), whereas the value of Q1(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 Qa(X;p) can be used as measures of economic inequality. The spectrum parameter, a, may be considered an index of sensitivity to risk/inequality.