Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/167835 
Authors: 
Year of Publication: 
2014
Citation: 
[Journal:] Risks [ISSN:] 2227-9091 [Volume:] 2 [Issue:] 3 [Publisher:] MDPI [Place:] Basel [Year:] 2014 [Pages:] 349-392
Publisher: 
MDPI, Basel
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 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.
Subjects: 
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
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