An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality

Risks

Iosif Pinelis¹ 

[1] Department of MathematicalSciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA;

关键词: 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;   

DOI  :  10.3390/risks2030349

来源: DOAJ

【 摘 要 】

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 andmonotonic 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.

【 授权许可】

Unknown