Abstract
The tail value at risk at level p, with p ∈ ( 0 , 1 ) , is a risk measure that captures the tail risk of losses and asset return distributions beyond the p quantile. Given two distributions, it can be used to decide which is riskier. When the tail values at risk of both distributions agree, whenever the probability level p ∈ ( 0 , 1 ) , about which of them is riskier, then the distributions are ordered in terms of the increasing convex order. The price to pay for such a unanimous agreement is that it is possible that two distributions cannot be compared despite our intuition that one is less risky than the other. In this paper, we introduce a family of stochastic orders, indexed by confidence levels p 0 ∈ ( 0 , 1 ) , that require agreement of tail values at risk only for levels p > p 0 . We study its main properties and compare it with other families of stochastic orders that have been proposed in the literature to compare tail risks. We illustrate the results with a real data example.
Highlights
In actuarial and financial sciences, risk managers and investors dealing with insurance losses and asset returns are often concerned with the right-tail risk of distributions, which is related to large deviations due to the right-tail losses or right-tail returns
The following result shows that two random variables, in which distribution functions cross a finite number of times, are ordered in the p0 -tail value at risk order for some p0 ∈ (0, 1)
We have introduced a family of stochastic orders indexed by confidence levels p0 ∈ (0, 1), which are useful when we are concerned with right-tail risks
Summary
In actuarial and financial sciences, risk managers and investors dealing with insurance losses and asset returns are often concerned with the right-tail risk of distributions, which is related to large deviations due to the right-tail losses or right-tail returns (see Wang [1]). The increasing convex order (which is formally defined below) requires agreement of tail values at risk for any confidence level p ∈ (0, 1). An advantage of this method is, obviously, its robustness toward changes in the confidence level, which can be interpreted in terms of a common agreement of different decision-maker’s attitudes. Given two random variables X and Y with the same mean, Cheung and Vanduffel [8] say that X is smaller than Y in the tail convex order with index x0 ∈ R (denoted by X ≤tcx( x0 ) Y) if. S− (h) denotes the number of sign changes of h on its support, where zero terms are discarded
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