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

Artikel

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.

Object Details

• Language: Englisch

References and Relationships

• Bibliographic citation: Journal: Risks ; ISSN: 2227-9091 ; Volume: 2 ; Year: 2014 ; Issue: 3 ; Pages: 349-392 ; Basel: MDPI

Classification and Topics

• Classification: Wirtschaft
• Subject: 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

Contributors, Places and Time

• Event: Geistige Schöpfung
• (who): Pinelis, Iosif
• Event: Veröffentlichung
• (who): MDPI
• (where): Basel
• (when): 2014

Further information

• DOI: doi:10.3390/risks2030349
• Handle: http://hdl.handle.net/10419/167835
• Last update: 10.03.2025, 11:45 AM CET