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 (<i>Q</i>_<i>α</i>(<i>X</i><i> </i>; <i>p) </i> _{<i>α</i>ε[0<i>,</i>∞]} on the (largest)  (1-p)-quantile <i>Q</i>(<i>X</i>;<i>p</i>)  of an arbitrary random variable  <i>X </i>is introduced and shown to be stable and monotonic in <i>α</i>, <i>p</i>, and <i>X </i>, with <i>Q</i>₀(<i>X</i><i> </i>;p) = <i>Q</i>(<i>X</i>;<i>p</i>).    If <i>p </i>is small enough and the distribution of <i>X </i>is regular enough, then Q_α(X ; p)  is rather close to <i>Q</i>(<i>X </i>; <i>p</i>).  Moreover, these quantile bounds are coherent measures of risk. Furthermore, <i>Q</i>_<i>α</i><i>(X</i><i> </i>; <i>p</i>) 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, <i>Q</i>₀(<i>X</i>;<i>p</i>) and <i>Q</i>₁(<i>X</i><i> </i>; <i>p</i>) are known  as the value at risk (VaR) and the conditional  value at risk (CVaR). The bounds <i>Q</i>_<i>α</i><i>(X</i><i> </i>; <i>p</i>) can also be used as measures of economic inequality. The spectrum parameter <i>α </i>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.