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